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 preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/directional-zero-one-finite-range.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-Fixed-Particle-Test-for-Computation-in-a-Forced-Viscous-Flow-September-27-2026/manuscript.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-group-without-fixed-price-October-5-2026/unequal-costs-rank-100-amalgam.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
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+preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/article.pdf filter=lfs diff=lfs merge=lfs -text
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+preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/quartic-query-separation.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
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+preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/nonspectrahedral-hyperbolicity-cone.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/voronoi-pivotal-amplitude.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
diff --git a/preprints/A-Fixed-Particle-Test-for-Computation-in-a-Forced-Viscous-Flow-September-27-2026/README.md b/preprints/A-Fixed-Particle-Test-for-Computation-in-a-Forced-Viscous-Flow-September-27-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..b23913d6afb47c3f9778ac78dd292237d6f7258e
--- /dev/null
+++ b/preprints/A-Fixed-Particle-Test-for-Computation-in-a-Forced-Viscous-Flow-September-27-2026/README.md
@@ -0,0 +1,17 @@
+# [A Fixed Particle Test for Computation in a Forced Viscous Flow](manuscript.pdf)
+
+**Author:** OpenAI  
+**Date:** September 27, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-Fixed-Particle-Test-for-Computation-in-a-Forced-Viscous-Flow-September-27-2026,
+  author = {{OpenAI}},
+  title = {{A Fixed Particle Test for Computation in a Forced Viscous Flow}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-Fixed-Particle-Test-for-Computation-in-a-Forced-Viscous-Flow-September-27-2026/manuscript.pdf}{OAI:A-Fixed-Particle-Test-for-Computation-in-a-Forced-Viscous-Flow-September-27-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-Fixed-Particle-Test-for-Computation-in-a-Forced-Viscous-Flow-September-27-2026/build/analytic-appendix.tex b/preprints/A-Fixed-Particle-Test-for-Computation-in-a-Forced-Viscous-Flow-September-27-2026/build/analytic-appendix.tex
new file mode 100644
index 0000000000000000000000000000000000000000..d1f5566880168d00e63810aba59d338a30d1e065
--- /dev/null
+++ b/preprints/A-Fixed-Particle-Test-for-Computation-in-a-Forced-Viscous-Flow-September-27-2026/build/analytic-appendix.tex
@@ -0,0 +1,129 @@
+\appendix
+\section{Comparison on noncompact domains and Fourier identification}
+The main particle theorem is already complete. We record two analytic
+details separately: a comparison theorem with precise noncompact
+pressure assumptions, useful for other prescribed flows, and the
+elementary Fourier uniqueness argument used in the periodic projection.
+
+\subsection{The precise comparison class}\label{app:comparison}
+Let $\Omega$ be $\T^3$, $\R^3$, or
+$\R^2\times\T$. A classical solution on $[0,T]$ has continuous
+$u$, $u_t$, all spatial derivatives of $u$ through order two, $p$, and
+$\nabla p$ on the closed time cylinder, and satisfies the equation
+pointwise. In the noncompact cases we additionally require
+\begin{equation}\label{eq:energy-class}
+ \begin{gathered}
+ u\in C([0,T];H^2(\Omega))\cap C^1([0,T];L^2(\Omega)),
+ \qquad p\in C([0,T];H^1(\Omega)),\\
+ \sup_{[0,T]\times\Omega}(|u|+|\nabla u|)<\infty.
+ \end{gathered}
+\end{equation}
+The Sobolev spaces are periodic in the torus coordinates. On $\T^3$
+the pressure has mean zero; on the noncompact domains its $H^1$
+membership fixes the representative, since a nonzero spatial constant
+is not in $L^2$.
+
+\begin{lemma}[Prescribed motion and uniqueness]\label{lem:realization}
+Fix $\nu>0$ and $\Omega\in\{\T^3,\R^3,\R^2\times\T\}$.
+Let $U$ be smooth and divergence free on $[0,\infty)\times\Omega$,
+with $U(0)=0$, and suppose $U$ and $\nabla U$ are bounded on every
+finite closed time cylinder. In the noncompact cases assume the
+velocity conditions in \eqref{eq:energy-class}. Then the force
+\[
+ \mathcal F_\nu[U]=U_t+(U\cdot\nabla)U-\nu\Delta U
+\]
+has the solution $(u,p)=(U,0)$, unique in the classical class just
+defined on each finite time interval. Its material trajectories exist
+uniquely for every finite time. On $\T^3$, mean-zero $U$ gives
+mean-zero $\mathcal F_\nu[U]$.
+
+More generally, if an independently constructed pair $(U,P)$ is a
+classical solution in the same class for a given force and initial
+velocity, it is the unique solution for those data in that class.
+In either noncompact uniqueness statement, the competitor's gradient
+bound can be omitted while keeping its bounded velocity and all other
+conditions. The reference velocity $U$ still has bounded gradient.
+For smooth $U$, each finite-time material map $\Phi_t$ is a smooth
+diffeomorphism and satisfies
+\[
+ U(t,\cdot)=(\partial_t\Phi_t)\circ\Phi_t^{-1},\qquad
+ \partial_{tt}\Phi_t(a)
+   =(f-\nabla P+\nu\Delta U)(t,\Phi_t(a)).
+\]
+\end{lemma}
+\begin{proof}
+Substitution proves the residual assertion. For either uniqueness
+statement, let $(v,p)$ be another solution for the same data, and set
+$w=v-U$ and $q=p-P$. Subtracting the equations gives
+\[
+ w_t+(v\cdot\nabla)w+(w\cdot\nabla)U
+   =-\nabla q+\nu\Delta w,\qquad \diver w=0.
+\]
+On the torus, multiplication by $w$ and periodic integration give
+\begin{equation}\label{eq:energy-comparison}
+ \frac12\frac{d}{dt}\|w\|_2^2+\nu\|\nabla w\|_2^2
+ \leq\|\nabla U(t)\|_{\infty,\mathrm{op}}\|w\|_2^2.
+\end{equation}
+The transport and pressure integrals vanish by incompressibility.
+
+Here are the cutoff details on the other two domains. Let $x'$ denote
+the noncompact coordinates, choose $0\leq\zeta\leq1$ smooth with
+compact support, equal to one on $|x'|\leq1$, and set
+$\zeta_R(x')=\zeta(x'/R)$. Multiply the difference equation by
+$\zeta_Rw$ and integrate in all coordinates. Integration by parts
+leaves, besides the term $-\int\zeta_Rw\cdot(w\cdot\nabla)U$,
+three boundary errors. Their absolute values are at most
+\[
+ \frac C R\|v\|_\infty\|w\|_2^2,\qquad
+ \frac C R\|q\|_2\|w\|_2,\qquad
+ \frac {C\nu}R\|\nabla w\|_2\|w\|_2,
+\]
+respectively for transport, pressure, and viscosity. For example,
+the pressure term is $\int q\nabla\zeta_R\cdot w$ because
+$\diver w=0$. Each displayed norm is uniformly bounded on $[0,T]$
+by \eqref{eq:energy-class}. Also $w\in C^1([0,T];L^2)$
+justifies differentiating $\int\zeta_R|w|^2$. Integrate the identity
+between any two times and then let $R\to\infty$. Dominated
+convergence applies to the quadratic terms and the error integrals
+tend to zero. This gives \eqref{eq:energy-comparison} in integrated
+form, equivalently almost everywhere in time. Thus no decay of the
+pressure beyond the stated $C_tH^1$ condition has been assumed.
+
+On every finite interval the coefficient in
+\eqref{eq:energy-comparison} is bounded. Its integrating factor and
+$w(0)=0$ imply $w=0$. The equation then gives $\nabla q=0$;
+the mean-zero or $H^1$ pressure convention gives $q=0$.
+The estimates use boundedness of $v$ but never a bound on $\nabla v$,
+which proves the stated competitor extension.
+
+Finally, bounded spatial gradient gives a uniformly Lipschitz vector
+field on each finite time interval, so the material ODE has unique
+local solutions. Bounded velocity prevents escape to infinity in
+finite time. Successive local solutions therefore cover each finite
+interval, and periodic coordinates descend to their quotients.
+Solving the same ODE backwards on $[0,t]$ gives the inverse map.
+Smooth dependence on the initial point proves the diffeomorphism
+assertion; differentiating the trajectory equation proves the two
+displayed identities.
+For the mean assertion, integrate the residual: the integrals of
+$\Delta U$ and $(U\cdot\nabla)U=\diver(U\otimes U)$ vanish on
+the torus.
+\end{proof}
+
+
+\subsection{Identifying continuous periodic functions}\label{app:fourier}
+A continuous function on $\T^3$ is determined by its Fourier
+coefficients. Here is the approximate-identity argument used in
+Lemma~\ref{lem:projection}.
+
+In one variable,
+\[
+ F_N(s)=\frac1N\left|\sum_{j=0}^{N-1}e^{2\pi i js}\right|^2
+\]
+is nonnegative, has integral one, and has Fourier coefficients
+$(1-|k|/N)_+$. Outside any fixed neighborhood of the integers,
+the geometric-sum formula bounds $F_N$ by $C/N$. Hence the product
+of three such kernels has mass tending to one in every neighborhood
+of zero. Uniform continuity shows that convolution with this product
+converges uniformly to any continuous function on $\T^3$.
+Vanishing Fourier coefficients therefore imply a vanishing function.
diff --git a/preprints/A-Fixed-Particle-Test-for-Computation-in-a-Forced-Viscous-Flow-September-27-2026/build/analytic-tools.tex b/preprints/A-Fixed-Particle-Test-for-Computation-in-a-Forced-Viscous-Flow-September-27-2026/build/analytic-tools.tex
new file mode 100644
index 0000000000000000000000000000000000000000..335f66644e3376e24fe38ef29dffed0ff60bafbf
--- /dev/null
+++ b/preprints/A-Fixed-Particle-Test-for-Computation-in-a-Forced-Viscous-Flow-September-27-2026/build/analytic-tools.tex
@@ -0,0 +1,109 @@
+\subsection{Uniqueness for the constructed torus flow}
+The residual identity gives the solution. We now show that its particle
+event is determined by the force and initial data. The proof is the
+classical difference-energy comparison \cite[Section~18]{Leray}.
+If $(v,p)$ is another solution in the classical class of
+Section~\ref{sec:statement}, set $w=v-U$. Subtraction gives
+\[
+ w_t+(v\cdot\nabla)w+(w\cdot\nabla)U
+  =-\nabla p+\nu\Delta w,\qquad \diver w=0.
+\]
+Multiplication by $w$ and periodic integration by parts yield
+\[
+ \frac12\frac{d}{dt}\|w\|_2^2+\nu\|\nabla w\|_2^2
+ \leq\|\nabla U\|_{\infty,\mathrm{op}}\|w\|_2^2.
+\]
+The stated continuity of $v_t$, the first two spatial derivatives and
+pressure gradient justifies all these operations. The transport and
+pressure terms vanish by incompressibility. Boundedness of $\nabla U$
+on each finite interval and $w(0)=0$ give $w=0$ by Gronwall's
+inequality. The equation then gives $\nabla p=0$, and mean-zero
+normalization gives $p=0$. Smoothness and bounded spatial gradient on
+the compact torus give a unique material flow through every finite
+time. Thus the event already proved for $U$ belongs to the unique
+fluid solution. Appendix~\ref{app:comparison} supplies the corresponding
+noncompact comparison theorem and the finite-time flow identities.
+
+\subsection{Solenoidal forcing with the same velocity}
+The Helmholtz--Leray projection removes the divergence of the force.
+Because it changes a gradient, it also changes the pressure. The
+following quantitative version is useful both for the present
+$L^2$ bounds and for other time profiles.
+
+\begin{lemma}[Effective periodic projection]\label{lem:projection}
+Let $g$ be a smooth mean-zero vector field on
+$[0,\infty)\times\T^3$, effectively evaluable together with all
+mixed derivatives and with effective bounds for those derivatives
+on every finite time interval. There is a unique smooth mean-zero
+$\phi$ satisfying $\Delta\phi=\diver g$. Both $\phi$ and
+$g_{\rm sol}=g-\nabla\phi$ are effective, and $g_{\rm sol}$ is
+divergence free and mean zero. For every $h\geq0$ and spatial
+multi-index $\mu$,
+\begin{equation}\label{eq:projection-estimate}
+ \|\partial_t^h\partial_X^\mu\nabla\phi(t)\|_\infty
+ \leq C_\mu\max_{1\leq i\leq3}
+ \|\partial_t^h\partial_{X_i}^{|\mu|+5}g(t)\|_\infty.
+\end{equation}
+Thus, when all mixed derivatives of $g$ are bounded, square-integrable
+in time in spatial supremum norm, or rapidly decaying, the same
+property holds for $g_{\rm sol}$. Here rapid decay means a finite
+supremum after multiplication by $(1+t)^A$, for every $A\geq0$ and
+every fixed derivative order. Time periodicity and stationarity
+after a given time are also preserved. If $(U,P)$ solves the equation
+with force $g$, then it solves it with force $g_{\rm sol}$ and
+pressure $P-\phi$.
+More precisely, any common pointwise time weight on the finitely many
+derivatives on the right of \eqref{eq:projection-estimate} passes to
+the derivative on its left. Thus a specified polynomial bound is
+preserved whenever those higher spatial derivatives have that bound.
+\end{lemma}
+\begin{proof}
+For the Fourier basis $e^{2\pi i k\cdot X}$ set
+\[
+ \widehat\phi(k)=-\frac{i k\cdot\widehat g(k)}{2\pi|k|^2}
+ \quad(k\ne0),\qquad \widehat\phi(0)=0.
+\]
+The multiplier for $\nabla\phi$ is $kk^{\mathsf T}/|k|^2$,
+whose operator norm is one. For a derivative of order $r$, integrate
+the coefficients of $\partial_t^hg$ by parts $r+5$ times in a
+coordinate with maximal $|k_i|$. A term of the differentiated series
+is bounded by a fixed multiple of $|k|^{-5}$ times the right-hand
+derivative norm in \eqref{eq:projection-estimate}. There are
+$O(n^2)$ points in the shell $\|k\|_\infty=n$. The series and its
+derivatives therefore converge absolutely, with an explicit tail
+bounded by $C\sum_{n>K}n^{-3}$. The series for $\phi$ has one
+further factor $|k|^{-1}$. Termwise differentiation proves the
+Poisson equation; the Fourier coefficients also prove uniqueness
+among mean-zero solutions and establish
+\eqref{eq:projection-estimate}.
+
+The Fourier identification used here is justified in
+Appendix~\ref{app:fourier}. Since $g$ is real,
+$\widehat\phi(-k)=\overline{\widehat\phi(k)}$, so $\phi$ is real.
+
+Riemann sums with a bound on one more spatial derivative compute
+each coefficient to any requested error. Combined with the tail
+estimate, this gives effective evaluation of every derivative.
+The estimate is pointwise in time and commutes with time
+derivatives, so it proves each stated norm and weighted-decay
+assertion. Linearity and uniqueness of the mean-zero Poisson
+solution preserve periodicity and eventual stationarity. Finally,
+\[
+ -\nabla(P-\phi)+(g-\nabla\phi)=-\nabla P+g,
+\]
+which proves the pressure formula.
+Conversely, a classical solution with force $g_{\rm sol}$ and pressure
+$p$ solves the equation with force $g$ and pressure $p+\phi$.
+This correspondence preserves the normalized classical class on the
+torus, since $\phi$ is smooth and mean zero. It therefore preserves
+uniqueness for fixed initial data, as well as the reference velocity.
+\end{proof}
+
+To apply Lemma~\ref{lem:projection} to the force in \eqref{eq:force},
+we still need to verify its effective-evaluation hypotheses. The next
+subsection supplies them and completes the optional solenoidal
+construction. This periodic projection does not assert preservation
+of compact support on a Euclidean domain, nor does it preserve
+pressure zero.
+
+\subsection{Effective evaluation without a simulated execution}
diff --git a/preprints/A-Fixed-Particle-Test-for-Computation-in-a-Forced-Viscous-Flow-September-27-2026/build/main.tex b/preprints/A-Fixed-Particle-Test-for-Computation-in-a-Forced-Viscous-Flow-September-27-2026/build/main.tex
new file mode 100644
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+++ b/preprints/A-Fixed-Particle-Test-for-Computation-in-a-Forced-Viscous-Flow-September-27-2026/build/main.tex
@@ -0,0 +1,663 @@
+\documentclass[11pt]{article}
+\ifdefined\pdftrailerid\pdftrailerid{}\fi
+\ifdefined\pdfinfoomitdate\pdfinfoomitdate=1\fi
+\usepackage[T1]{fontenc}
+\usepackage{lmodern,amsmath,amssymb,amsthm,mathtools,microtype,booktabs}
+\usepackage[margin=1in]{geometry}
+\usepackage{xcolor,tikz,needspace}
+\usepackage{xurl}
+\usetikzlibrary{arrows.meta,positioning}
+\usepackage[colorlinks=true,linkcolor=blue!50!black,citecolor=blue!50!black,urlcolor=blue!50!black]{hyperref}
+\hypersetup{pdftitle={A Fixed Particle Test for Computation in a Forced Viscous Flow},pdfauthor={OpenAI}}
+\newtheorem{theorem}{Theorem}
+\newtheorem{lemma}{Lemma}
+\newtheorem{corollary}{Corollary}
+\newcommand{\T}{\mathbb T}
+\newcommand{\R}{\mathbb R}
+\newcommand{\N}{\mathbb N}
+\newcommand{\Z}{\mathbb Z}
+\DeclareMathOperator{\diver}{div}
+\DeclareMathOperator{\supp}{supp}
+\title{A Fixed Particle Test for Computation\\ in a Forced Viscous Flow}
+\author{OpenAI}
+\date{September 27, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We construct smooth external forces for incompressible Navier--Stokes flow
+on a fixed flat three-torus at any fixed positive computable viscosity,
+from zero initial velocity, such that a fixed
+particle enters a fixed open set exactly when a prescribed Turing machine
+halts. Every mixed derivative of the force and velocity is bounded and
+square-integrable in time in spatial supremum norm. The construction uses
+the machine's ordinary instructions, records their history in a third
+coordinate, and compensates for finer spatial gates by longer time steps.
+\end{abstract}
+
+\section{Introduction}\label{sec:introduction}
+
+Fix a flat three-dimensional torus, a positive viscosity, zero initial
+fluid velocity, one particle label, and an open region that serves as a
+detector. Can a finite program for an external force make that particle
+enter the region exactly when a given Turing machine halts? The machine
+and its finite input may change the force; the geometry, initial velocity,
+particle label and detector remain fixed. We prove this for an explicit
+family of smooth forces. The resulting reachability problem is undecidable.
+
+The question connects computability with the dynamics of a continuous
+system. Turing proved that no machine can decide whether an arbitrary
+described machine ever prints a designated symbol \cite[Section~8]{Turing}.
+The halting formulation used here follows by a finite modification
+that halts on that event. A bounded
+continuous phase space can represent such a tape by an infinite positional
+expansion. Moore's generalized shifts made this connection explicit:
+local symbol changes and shifts of the tape become affine operations on
+real coordinates \cite{Moore1990,Moore}. Exact real coordinates supply the unbounded
+information capacity; this is not a bound on what a finite-precision
+measurement can recover.
+
+A further issue arises when such operations are realized by a smooth
+flow. A machine may overwrite a symbol or merge two states, whereas the
+flow between two finite times is invertible. Retaining intermediate
+information is a classical way to avoid this loss \cite[Section~3]{Landauer}.
+Bennett gave an explicit reversible simulation that records the index of
+each executed instruction on a history tape \cite[Eqs.~(9)--(11)]{Bennett}. Our construction
+uses the same information-retention principle in a different setting:
+a third real coordinate retains the selected branches, while two planar
+coordinates execute the original, possibly irreversible machine.
+
+Computational universality has also been realized in fluid trajectories.
+Cardona, Miranda, Peralta-Salas and Presas constructed stationary Euler
+flows on an adapted Riemannian three-sphere whose trajectories simulate
+Turing machines \cite{CMPP}. Their proof passes through area-preserving
+realizations of generalized shifts. Dyhr, Gonz\'alez-Prieto, Miranda and
+Peralta-Salas subsequently obtained stationary unforced Navier--Stokes universality
+using an adapted metric and Hodge viscosity \cite{DGMP}. Their velocity
+is harmonic for the chosen Hodge Laplacian, so the viscous term vanishes.
+
+Fixed-metric computation also has precedents. Cardona, Miranda and
+Peralta-Salas construct universal steady Euler flows on Euclidean $\R^3$
+with infinite energy, and robust tape-bounded simulations on the flat
+three-torus, observed before exit from a prescribed compact region
+\cite[arXiv version~3, Theorems~1 and~26]{CMPBeltrami}.
+Their unforced viscous version starts from nonzero Beltrami velocity
+and traverses a finite interval of its Euler trajectory, chosen to cover
+the bounded computation \cite[arXiv version~3, Remark~29]{CMPBeltrami}.
+
+Here the flat torus and viscosity are fixed in advance and the fluid
+starts at rest. The
+machine and input instead enter an external force prescription. These
+are different choices of data for the fluid equation, and the earlier
+results do not supply the construction below.
+
+There is an elementary way to force a chosen smooth incompressible
+velocity: subtract its viscous term from its material acceleration.
+The mathematical work lies in choosing that velocity from the finite
+instruction table. It must execute each instruction, retain the information
+needed for invertibility, and avoid the detector at every intermediate
+time of a nonhalting run. In particular, our force formula must prescribe
+all instruction branches; it must not first execute the input computation
+and then replay its trajectory.
+
+We implement an instruction in two pulses. The first records its branch
+number in the third coordinate. The second reads the newest record through
+a periodic gate and applies the corresponding planar motion. Earlier
+records remain present but do not affect that gate. As the history grows,
+the gates have finer spatial scales. We make successive steps longer so
+that the time-dependent velocity and force have square-integrable
+derivatives of every fixed order.
+
+The result concerns exact material trajectories for these explicitly
+constructed forces. It gives neither a uniform perturbation tolerance
+for arbitrarily long runs nor an efficient simulation-time bound. Its
+existence and uniqueness proof uses the prescribed smooth velocity;
+it is not a regularity theorem for general Navier--Stokes data.
+
+\section{The statement and the history invariant}\label{sec:statement}
+
+Write $\T=\R/\Z$ and fix a positive computable viscosity $\nu$.
+For velocity $u(t,X)$, pressure $p(t,X)$ and external force $f(t,X)$,
+the incompressible Navier--Stokes equations are
+\begin{equation}\label{eq:NS}
+ \partial_tu+(u\cdot\nabla)u=-\nabla p+\nu\Delta u+f,
+ \qquad \diver u=0,\qquad u(0)=0
+\end{equation}
+on the unit flat torus $\T^3$. Pressure is periodic and has mean zero.
+We write $X=(x,y,z)$ for a spatial point, use representatives in $[0,1)$
+when specifying a coordinate interval, and take $\|\cdot\|_\infty$ over
+space. The material flow of a smooth velocity $u$ is defined by
+\[
+ \frac{d}{dt}\Phi_u(t,a)=u(t,\Phi_u(t,a)),\qquad \Phi_u(0,a)=a.
+\]
+For a label $a_*\in\T^3$ and an open set $O\subset\T^3$, the
+particle event is $\Phi_u(t,a_*)\in O$ for some finite $t$.
+
+An effective force prescription is a finite program that evaluates the
+force and any requested mixed derivative at computably supplied space-time
+arguments, to any prescribed rational error. It also supplies the derivative
+bounds on compact sets used below. The program prescribes the transition
+mechanism on all its branches; it does not compute and replay the
+distinguished execution. No running-time bound is required.
+
+A classical competitor has continuous velocity, spatial derivatives through
+order two, time derivative, pressure and pressure gradient on every closed
+cylinder $[0,T]\times\T^3$, with the periodicity and pressure normalization
+above. The constructed solution is smooth, including all one-sided time
+derivatives at zero.
+
+\Needspace{23\baselineskip}
+\begin{theorem}\label{thm:main}
+There is an algorithm taking a deterministic single-tape machine $M$ and
+a finite word $w$ to an effective smooth mean-zero force $f_{M,w}$ for
+\eqref{eq:NS}, with the following properties.
+\begin{enumerate}
+\item The equation has a global smooth solution $u_{M,w}$, unique among
+classical solutions on every finite time interval.
+\item For every time order $h\geq0$ and spatial multi-index $\mu$, both
+\[
+ t\longmapsto\|\partial_t^h\partial_X^\mu u_{M,w}(t)\|_\infty,
+ \qquad
+ t\longmapsto\|\partial_t^h\partial_X^\mu f_{M,w}(t)\|_\infty
+\]
+belong to $L^\infty(0,\infty)\cap L^2(0,\infty)$.
+\item The particle initially at $a_*=(1/8,3/8,0)$ enters
+\[
+ O=\{(x,y,z)\in\T^3:1/2<x<1\}
+\]
+at a finite time if and only if $M$ halts on $w$.
+\end{enumerate}
+The force may additionally be chosen divergence free, with unchanged
+velocity and observation. Constants in the derivative estimates may
+depend on $M,w,\nu$ and the derivative orders. After a loading interval,
+the prescribed velocity mechanism depends on $M$ but not on $w$.
+\end{theorem}
+
+The proof has one particularly useful invariant. A branch specifies the
+current state and the symbols read by one local instruction; the precise
+list is given in Section~\ref{sec:planar}. Number these finitely many branches by $1,\ldots,J$ and put $\Lambda=J+1$.
+If the first $n$ selected branches are $j_0,\ldots,j_{n-1}$, we will arrange
+\begin{equation}\label{eq:history}
+ z_n=\sum_{i<n}j_i\Lambda^{-i-1}.
+\end{equation}
+Thus $\Lambda^n z_n$ is an integer. Translating $z$ by
+$j_n\Lambda^{-n-1}$ makes its new value satisfy
+\begin{equation}\label{eq:selector}
+ \Lambda^n z=\frac{j_n}{\Lambda}\pmod 1.
+\end{equation}
+A smooth periodic gate can now select instruction $j_n$. This is the
+continuous history record corresponding to the transition-index record
+in Bennett's reversible simulation \cite{Bennett}. The earlier
+digits have become an integer and do not affect this gate. They have not
+been erased: they remain in $z$, so two planar updates with overlapping
+images can still represent different pasts.
+
+We next construct the planar branches and then realize
+\eqref{eq:history} by a divergence-free field. The symbols $j_n$ describe
+the trajectory in the proof; they will not occur in the formula for the
+force.
+
+Here is the route through the proof. Section~\ref{sec:planar} converts each
+instruction into an area-preserving planar motion and proves that it cannot
+cause a false detection between nonhalting configurations.
+Section~\ref{sec:recording} combines the recorder and these motions into one
+smooth incompressible velocity, and proves the event equivalence.
+Section~\ref{sec:force} chooses the force, establishes the derivative bounds
+and uniqueness, and verifies effective evaluation without running the
+machine. The Fourier projection there gives the optional solenoidal force.
+
+A force that eventually stops changing in time poses a further question:
+its spatial mechanism must keep computing without progressively changing
+the gates. A separate companion gives that finite spatial mechanism
+and proves eventual stationarity \cite{Stationary}. The present
+proof uses the ordinary machine directly and requires no result
+from that companion.
+
+The optional comparison theorem in Appendix~\ref{app:comparison}
+also applies on $\R^3$ and $\R^2\times\T$, with explicit Sobolev
+and pressure assumptions for other constructed flows. The particle theorem above concerns $\T^3$ only.
+For a positive viscosity that is not assumed computable, the same formulas
+and estimates hold, with effective evaluation relative to the ability to
+approximate that fixed viscosity. Unqualified algorithmic assertions in
+this paper use the computable viscosity fixed above.
+
+\section{A machine instruction as a planar motion}\label{sec:planar}
+
+Our task in this section is to turn each machine instruction into a planar
+motion whose entire path respects the detector. First encode the tape,
+then interpolate the resulting affine update by a Hamiltonian field.
+
+First merge designated halting states into a single state $q_h$, or
+adjoin an unreachable $q_h$ if none is designated. Let $Q$ now have
+$N\geq1$ states, and let the tape alphabet $\Sigma$ have $m\geq1$
+letters, including a blank $\square$. Replace a missing instruction
+on symbol $a$ by $(q_h,a,S)$, and give $q_h$ stationary identity
+instructions. This finite normalization preserves the halting event,
+including an initially halted input. The tape is a sequence $(s_i)_{i\in\Z}$ in $\Sigma$, with finitely many
+nonblank cells initially. Place the input word starting at cell zero,
+put blanks elsewhere, and start the head at zero in the machine's initial
+state. Write
+$\delta(q,a)=(q',b,D)$, where $D$ is a left move, right move or stay.
+
+The positional coding is the generalized-shift representation of a tape
+\cite[Section~1, Lemma~0]{Moore}, with gaps inserted between symbol intervals.
+Give the letters the odd digits $d_a\in\{1,3,\ldots,2m-1\}$ in base
+$B=2m+1$, and set
+\[
+ v(a_0a_1\ldots)=\sum_{r\geq0}d_{a_r}B^{-r-1},
+ \qquad I_a=[d_a/B,(d_a+1)/B].
+\]
+The intervals $I_a$ are separated by gaps of length at least $1/B$.
+A code lies in the interval of its first letter, and
+$Bv(a\omega)-d_a=v(\omega)$ removes that letter. This proves unique
+decoding. A blank tail has value $d_{\square}/(B-1)$, so all initial
+codes are rational.
+
+If the head is at cell $k$, encode the two relative tape halves by
+\[
+ \ell=v(s_{k-1}s_{k-2}\ldots),\qquad r=v(s_ks_{k+1}\ldots).
+\]
+Put $\kappa=(16N)^{-1}$ and $y_0=3/8$. Enumerate nonhalting states as
+$q_0,\ldots,q_{N-2}$ and choose
+\[
+ c(q_i)=1/8+2i\kappa,\qquad c(q_h)=3/4.
+\]
+The planar code is $P=(c(q)+\kappa\ell,y_0+\kappa r)$. The state box
+$[c(q),c(q)+\kappa]\times[y_0,y_0+\kappa]$ lies in
+\[
+ K=[1/8,13/16]\times[3/8,7/16].
+\]
+Every nonhalting state box has $x\leq1/4$; the halting box has $x\geq3/4$.
+These inequalities leave room for a fixed detector between the two.
+
+Index the triples $j=(q,l,a)$ by $1,\ldots,J=Nm^2$. Here $a$ is the
+letter under the head and $l$ the letter immediately to its left. Their
+closed source rectangles are
+\[
+ R_j=(c(q)+\kappa I_l)\times(y_0+\kappa I_a).
+\]
+They are pairwise disjoint. If $\ell_*=B\ell-d_l$ and $r_*=Br-d_a$,
+the tape instruction is exactly
+\begin{equation}\label{eq:branches}
+\begin{array}{c|cc}
+D&\ell'&r'\\ \hline
+R&(d_b+\ell)/B&r_*\\
+L&\ell_*&[d_l+(d_b+r_*)/B]/B\\
+S&\ell&(d_b+r_*)/B.
+\end{array}
+\end{equation}
+For example, a right move prefixes the written letter $b$ to the left
+tape and removes $a$ from the right tape. A left move removes $l$ from
+the left tape and prefixes $l,b$ to the old right tail. These operations
+explain both entries in each row of the table.
+
+Let $F_j$ be the resulting affine map in the physical coordinates $P$.
+It maps the whole rectangle $R_j$ into the next state box, and its linear
+part is $\operatorname{diag}(\lambda_j,\lambda_j^{-1})$, with
+$\lambda_j=B^{-1},B,1$ respectively. Different image rectangles may
+overlap. We will distinguish them by history, not by assuming that an
+ordinary machine is reversible.
+
+We have obtained the correct endpoint update. We now realize it by a
+continuous motion, keeping every nonhalting-to-nonhalting path to the
+left of the detector. Local area-preserving realizations of bijective generalized
+shifts also occur in \cite[Proposition~5.1]{CMPP}; here a separate history
+coordinate will distinguish different branches with overlapping images.
+
+\Needspace{9\baselineskip}
+Let $C_j^0,C_j^1$ be the centers of $R_j,F_j(R_j)$ and
+$E_j=C_j^1-C_j^0$. For $0\leq s\leq1$ define
+\[
+ C_j(s)=C_j^0+sE_j,\quad g_j(s)=1-s+s\lambda_j,
+ \quad \gamma_j(s)=\frac{\lambda_j-1}{g_j(s)},
+\]
+\begin{equation}\label{eq:path}
+ \Psi_{j,s}(P)=C_j(s)+
+ \operatorname{diag}(g_j(s),g_j(s)^{-1})(P-C_j^0).
+\end{equation}
+This path starts at $P$ and ends at $F_j(P)$. Its first coordinate is
+the convex combination of the two endpoint coordinates. To control the
+second coordinate, let $a$ be the source rectangle's vertical half-width.
+Convexity of the reciprocal gives
+\[
+ \frac{a}{g_j(s)}\leq(1-s)a+s\frac{a}{\lambda_j}.
+\]
+Together with the linear motion of the center, this keeps the entire
+rectangle in $K$. In particular, an instruction with nonhalting source
+and target stays in $x\leq1/4$ throughout its motion.
+
+Choose a smooth cutoff $\chi$ equal to one on a neighborhood of $K$
+and supported in the open unit square. Periodize the compactly supported
+Hamiltonian
+\[
+ H_j(s,x,y)=\chi(x,y)\bigl[
+ E_{j,x}(y-C_{j,y}(s))-E_{j,y}(x-C_{j,x}(s))
+ +\gamma_j(s)(x-C_{j,x}(s))(y-C_{j,y}(s))\bigr].
+\]
+The planar field $V_j=(\partial_yH_j,-\partial_xH_j)$ is divergence
+free and has mean zero. On the moving rectangle its components are
+$E_{j,x}+\gamma_j(x-C_{j,x})$ and
+$E_{j,y}-\gamma_j(y-C_{j,y})$. Differentiating \eqref{eq:path} gives
+exactly these velocities. Uniqueness for the smooth ODE therefore shows
+that $V_j$ realizes the claimed path, including its intermediate-time
+bound. The cutoff introduces no error there because all its derivatives
+vanish on that path.
+
+\section{Recording and selecting the instruction}\label{sec:recording}
+
+Each instruction uses two pulses. The first reads the current source
+rectangle and adds its branch number to the history; the second uses that
+new digit to select the planar motion. Thus we can use the individual
+motions of Section~\ref{sec:planar} even when their planar images overlap.
+Figure~\ref{fig:pulses} shows one step; the formulas below implement it
+simultaneously for every possible branch.
+
+\begin{figure}[ht]
+\centering
+\begin{tikzpicture}[>=Latex,font=\small]
+\node[draw,rounded corners,align=center,minimum width=2.15cm,minimum height=1.15cm] (start) at (0,0)
+ {$P\in R_j$\\$z=z_n$};
+\node[draw,rounded corners,align=center,minimum width=2.8cm,minimum height=1.15cm] (record) at (4.8,0)
+ {$P\in R_j$\\$z=z_n+j\Lambda^{-n-1}$};
+\node[draw,rounded corners,align=center,minimum width=2.8cm,minimum height=1.15cm] (end) at (10.5,0)
+ {$F_j(P)$\\$z=z_n+j\Lambda^{-n-1}$};
+\draw[->,thick] (start) -- node[above,align=center] {record $j$} node[below] {$P$ fixed} (record);
+\draw[->,thick] (record) -- node[above] {apply $\Psi_{j,s}$} node[below] {$z$ fixed} (end);
+\node[align=center] at (4.8,-1.2)
+ {$\Lambda^n z=j/\Lambda\pmod 1$\\the new digit selects the next pulse};
+\end{tikzpicture}
+\caption{One simulated instruction, schematically. The first pulse leaves
+both tape coordinates fixed while recording the selected branch. The
+second leaves the history fixed while executing that branch's planar
+motion. The diagram describes the encoded particle; planar images of
+different branches need not be disjoint.}
+\label{fig:pulses}
+\end{figure}
+
+We specify the cutoffs to ensure that closed rectangle boundaries are
+covered. Let
+\[
+ \rho(s)=\begin{cases}e^{-1/s},&s>0,\\0,&s\leq0,\end{cases}
+ \qquad \sigma(s)=\frac{\rho(s)}{\rho(s)+\rho(1-s)}.
+\]
+For a closed interval $I=[a,b]$, set
+\[
+ \eta_I^\varepsilon(v)=
+ \sigma\!\left(\frac{2(v-a+\varepsilon)}{\varepsilon}\right)
+ \sigma\!\left(\frac{2(b+\varepsilon-v)}{\varepsilon}\right).
+\]
+The cutoff $\eta_I^\varepsilon$ equals one on a neighborhood of $I$
+and vanishes outside its $\varepsilon$ enlargement. Products define
+the analogous rectangle cutoffs. For the earlier Hamiltonians, choose
+$\chi=\eta_K^{1/32}$; its support is strictly inside the unit square.
+
+The source rectangles have gaps at least $\kappa/B$, so the supports in
+\[
+ G_0(x,y)=\sum_{j=1}^Jj\eta_{R_j}^{\kappa/(4B)}(x,y)
+\]
+are disjoint. Periodize this function from the unit-square chart and put
+\[
+ G(x,y)=G_0(x,y)-G_0(x,y-1/2).
+\]
+The original supports lie in $1/4<y<1/2$; their half-period translates
+miss every $R_j$. Consequently $G=j$ on $R_j$ and $G$ has zero mean.
+The correction permits a mean-zero vertical recorder without altering
+the tracked computation. Define the gates by periodizing the following
+bumps with period one:
+\[
+ b_j(z)=\eta_{\{j/\Lambda\}}^{1/(4\Lambda)}(z),
+ \qquad \Lambda=J+1.
+\]
+Their supports are disjoint and each equals one near its own center.
+
+Let $\theta(s)=\sigma(2s-1/2)$. It rises smoothly from zero to one
+inside $(0,1)$ and is constant near both endpoints. Set
+\[
+ S_n=2^{n^2},\quad t_0=1,\quad t_{n+1}=t_n+2S_n,
+ \qquad
+ \alpha_n(t)=\frac{t-t_n}{S_n},\quad
+ \beta_n(t)=\frac{t-t_n-S_n}{S_n}.
+\]
+Step $n$ has a recorder interval and an equally long planar interval.
+
+The initial planar code $P_{\rm in}$ is rational. Let $V_{\rm load}$
+be the Hamiltonian field of
+\[
+ \chi(x,y)[(P_{{\rm in},x}-1/8)y-(P_{{\rm in},y}-3/8)x].
+\]
+It translates $(1/8,3/8)$ along its straight segment to $P_{\rm in}$.
+Define the velocity on $[0,\infty)\times\T^3$ by
+\begin{equation}\label{eq:field}
+\begin{split}
+ U(t,x,y,z)={}&(\theta'(t)V_{\rm load}(x,y),0)\\
+ &+\sum_{n\geq0}\left(
+ \frac{\theta'(\beta_n(t))}{S_n}
+ \sum_{j=1}^J b_j(\Lambda^n z)
+ V_j(\theta(\beta_n(t)),x,y),\quad
+ \frac{\theta'(\alpha_n(t))}{S_n\Lambda^{n+1}}G(x,y)
+ \right).
+\end{split}
+\end{equation}
+The displayed sum is a formula for all possible instructions. It has no
+reference to which instruction the machine will actually select.
+
+At any time only one stage can be active. Since $t_n\geq1+2n$, the sum
+is locally finite; the zero collars make every join smooth. Multiplying
+a planar divergence-free field by a function of $z$ preserves its planar
+divergence, and the vertical component is independent of $z$. Thus
+$\diver U=0$. Each planar field has zero planar mean, while $G$ has
+zero planar mean, so $U$ has zero spatial mean. Also $U(0)=0$.
+
+After the loader, the particle has planar coordinate $P_{\rm in}$ and
+$z_0=0$. Suppose at $t_n$ it has the current planar configuration and
+history \eqref{eq:history}. It belongs to exactly one source rectangle,
+say $R_{j_n}$. During the recorder interval its planar coordinate stays
+fixed, while integration of the last component in \eqref{eq:field}
+adds $j_n\Lambda^{-n-1}$ to $z$. Equation \eqref{eq:selector} follows.
+On the planar interval only $b_{j_n}$ is nonzero at this particle, and
+its value is one. Its path is therefore
+$\Psi_{j_n,\theta(\beta_n(t))}(P(t_n))$, ending at the next machine code.
+The third coordinate stays fixed on this interval. This proves the
+induction and \eqref{eq:history} for every $n$.
+
+There is no wraparound ambiguity for the tracked history:
+$0\leq z_n\leq1-\Lambda^{-n}$. At a known step its finite radix
+expansion recovers every previous branch, hence every head displacement
+and the absolute head position. The planar coordinates recover the
+state and relative tape. This is an encoding of the computation, not
+merely a halting indicator.
+
+If the machine halts, a planar code with $x\geq3/4$ is reached after
+finitely many finite stages. If it is initially halted, the loader
+already reaches such a code. Conversely, for a nonhalting run the loader
+stays in $x\leq1/4$, a recorder interval does not change $x$, and
+\eqref{eq:path} stays in $x\leq1/4$ at every planar stage. This proves
+both directions of the event equivalence at every real time.
+
+\section{Slowing the gates and obtaining the force}\label{sec:force}
+
+The velocity now implements the machine and gives the correct event at
+every time. It remains to control its derivatives, realize it as the unique
+fluid velocity, and show that the force has a finite effective prescription.
+
+\subsection{Bounds for the prescribed velocity}
+
+Why choose the rapidly growing stage lengths $S_n$? Differentiating a
+gate $b_j(\Lambda^n z)$ $k$ times costs a factor $\Lambda^{nk}$.
+The factor $S_n^{-1}$ in the velocity compensates for this at every
+fixed derivative order. More precisely, for a time order $h$ and
+spatial multi-index $\mu=(\mu_x,\mu_y,\mu_z)$, the fixed smooth
+profiles above give, throughout step $n$,
+\begin{equation}\label{eq:bounds}
+ \|\partial_t^h\partial_X^\mu U(t)\|_\infty
+ \leq C_{h,\mu}S_n^{-1-h}\Lambda^{n\mu_z}.
+\end{equation}
+The profiles $V_j$ and their phase derivatives are uniformly bounded
+for $0\leq s\leq1$, because $g_j(s)>0$. The recorder has no $z$
+dependence and has an additional small factor $\Lambda^{-n-1}$, so it
+also satisfies this estimate. The constants depend on the finite table
+and derivative orders but not on $n$.
+
+The right side of \eqref{eq:bounds} is bounded, and integration over
+the two intervals of length $S_n$ gives
+\[
+ \int_1^\infty
+ \|\partial_t^h\partial_X^\mu U(t)\|_\infty^2\,dt
+ \leq 2C_{h,\mu}^2\sum_{n\geq0}
+ 2^{-(1+2h)n^2}\Lambda^{2n\mu_z}<\infty.
+\]
+The loader occupies a compact time interval and adds a finite amount.
+Now prescribe
+\begin{equation}\label{eq:force}
+ f=\partial_tU+(U\cdot\nabla)U-\nu\Delta U.
+\end{equation}
+Then $(u,p)=(U,0)$ solves \eqref{eq:NS}. Every mixed derivative of $f$
+is a finite sum of derivatives of $U$ and products of two such
+derivatives. Each factor is bounded, and each has the stated temporal
+$L^2$ norm; use one bounded factor and one $L^2$ factor in a product.
+All force estimates in the theorem follow. Integration of
+\eqref{eq:force} over the torus gives zero mean, because $U$ has zero
+mean and $(U\cdot\nabla)U=\diver(U\otimes U)$.
+
+\input{analytic-tools}
+
+It remains to check that the prescription really is effective. We
+record the elementary fact about smooth profiles separately so that
+its computational hypothesis is explicit.
+
+\begin{lemma}[Effective smooth profiles]\label{lem:profiles}
+The functions $\rho,\sigma$ defined in Section~\ref{sec:recording}
+and their finite products, sums, translations, and positive rescalings
+by computable parameters have effective evaluations of every derivative,
+with effective derivative bounds on compact sets. A reciprocal may be
+included when an effective positive lower bound for its denominator is
+given. A compactly supported profile periodized through zero collars
+has the same property if a rational box containing its support is
+supplied. An infinite sum of such profiles has this
+property on compact sets if a finite superset of all possibly active
+indices can be computed there.
+\end{lemma}
+\begin{proof}
+On $s>0$, derivatives of $\rho$ have the form $P(1/s)e^{-1/s}$,
+where $P$ is an effectively computable polynomial. For $v>0$,
+$v^m e^{-v}\leq(m+k)!v^{-k}$ bounds every monomial and gives an
+effective neighborhood of zero where each derivative is smaller than
+a requested error. Away from that neighborhood ordinary effective
+arithmetic and the exponential evaluate it. This avoids deciding
+whether the supplied real is exactly zero. The denominator of
+$\sigma$ is at least $e^{-2}$; quotient differentiation therefore
+gives effective bounds as well. The finite operations in the statement
+preserve these properties by their differentiation rules.
+
+For a profile $\psi$ with support in a supplied rational interval
+$[a,b]$, obtain a rational $q$ with $|s-q|\leq1$. A translate
+$\psi(s-k)$ can be nonzero only for an integer
+$k\in[q-b-1,q-a+1]$. Evaluating all integers in this rational interval
+therefore computes the periodization and each of its derivatives;
+no integer-part computation on the real $s$ is required. The same
+argument works coordinatewise for a supplied rational box.
+A supplied finite active-index
+bound reduces the last assertion to a finite sum. Local finiteness
+without that effective bound would not suffice for this argument.
+\end{proof}
+
+All tables, rational initial codes, affine maps and stage times are
+obtained from the finite input. Lemma~\ref{lem:profiles} applies to
+their cutoffs and phase functions; the denominators $g_j(s)$ have the
+effective positive lower bounds $\min\{1,\lambda_j\}$. The supports
+of $\chi$ and the rectangle cutoffs lie in the rational coordinatewise
+enlargements of $K$ by $1/32$ and of $R_j$ by $\kappa/(4B)$,
+respectively. The unperiodized gate defining $b_j$ is supported in
+$[(j-1/4)/\Lambda,(j+1/4)/\Lambda]$. Thus every periodization has
+the supplied support enclosure required by the lemma. For a time
+argument, obtain a rational
+upper bound $T>t$. The inequality $t_n\geq1+2n$ gives a finite superset
+of possible active stages in \eqref{eq:field}; evaluating all of them
+is sufficient. Periodizations are evaluated by the same finite-superset
+procedure in coordinate charts, without deciding a real argument's
+integer part. The Gaussian series above gives effective norm bounds:
+for example, if $n\geq4\Lambda\mu_z$, then
+$\Lambda^{2n\mu_z}\leq2^{n^2/2}$.
+
+We may now apply Lemma~\ref{lem:projection} to $f$ in
+\eqref{eq:force}. It gives an effective mean-zero solenoidal force
+$f-\nabla\phi$ with solution $(U,-\phi)$, the same derivative bounds,
+and the same particle event. The lemma's pressure correspondence
+also transfers uniqueness from the original force. Riemann sums
+with derivative error bounds compute the Fourier coefficients, and
+the summable tail controls truncation. Neither this procedure nor the
+raw-force evaluation uses the branch sequence $j_0,j_1,\ldots$.
+For $t\geq1$ the loader and all its derivatives vanish, so the raw
+force depends on $M$ and $\nu$ but not on $w$. Its spatial projection
+at that time has the same independence.
+This completes the proof of
+Theorem~\ref{thm:main}.
+
+\begin{corollary}
+No algorithm decides the fixed particle event for every force description
+produced by this construction.
+\end{corollary}
+\begin{proof}
+Compose a proposed decision procedure with the terminating compiler of
+Theorem~\ref{thm:main}. The event equivalence would decide whether an
+arbitrary machine halts on its finite input. This would also decide
+Turing's symbol-printing problem \cite[Section~8]{Turing}: modify the
+simulated machine to halt when the specified symbol is printed and
+to continue forever otherwise, including if the original machine stops
+without printing it. Turing proves that no such decision procedure exists.
+Equivalently, one may fix a universal machine and encode the machine
+and word in its finite input; the same event then tests its computation.
+\end{proof}
+
+\input{analytic-appendix}
+
+\begin{thebibliography}{99}
+\bibitem{Bennett} C.~H. Bennett, Logical reversibility of computation,
+\emph{IBM J. Res. Develop.} \textbf{17} (1973), 525--532.
+\href{https://doi.org/10.1147/rd.176.0525}{\nolinkurl{doi:10.1147/rd.176.0525}}.
+
+\bibitem{CMPBeltrami} R. Cardona, E. Miranda and D. Peralta-Salas,
+Computability and Beltrami fields in Euclidean space,
+\emph{J. Math. Pures Appl.} \textbf{169} (2023), 50--81.
+\href{https://doi.org/10.1016/j.matpur.2022.11.007}{\nolinkurl{doi:10.1016/j.matpur.2022.11.007}}.
+\href{https://arxiv.org/abs/2111.03559v3}{\nolinkurl{arXiv:2111.03559v3}}.
+
+\bibitem{CMPP} R. Cardona, E. Miranda, D. Peralta-Salas and F. Presas,
+Constructing Turing complete Euler flows in dimension 3,
+\emph{Proc. Natl. Acad. Sci. USA} \textbf{118} (2021), e2026818118.
+\href{https://doi.org/10.1073/pnas.2026818118}{\nolinkurl{doi:10.1073/pnas.2026818118}}.
+
+\bibitem{DGMP} S. Dyhr, \'{A}. Gonz\'alez-Prieto, E. Miranda and
+D. Peralta-Salas, Turing complete Navier--Stokes steady states via
+cosymplectic geometry, \emph{PNAS Nexus} \textbf{5} (2026), pgag131.
+\href{https://doi.org/10.1093/pnasnexus/pgag131}{\nolinkurl{doi:10.1093/pnasnexus/pgag131}}.
+
+\bibitem{Landauer} R. Landauer, Irreversibility and heat generation in
+the computing process, \emph{IBM J. Res. Develop.} \textbf{5} (1961),
+183--191.
+\href{https://doi.org/10.1147/rd.53.0183}{\nolinkurl{doi:10.1147/rd.53.0183}}.
+
+\bibitem{Leray} J. Leray, Sur le mouvement d'un liquide visqueux
+emplissant l'espace, \emph{Acta Math.} \textbf{63} (1934), 193--248.
+\href{https://doi.org/10.1007/BF02547354}{\nolinkurl{doi:10.1007/BF02547354}}.
+
+\bibitem{Moore1990} C. Moore, Unpredictability and undecidability in
+dynamical systems, \emph{Phys. Rev. Lett.} \textbf{64} (1990), 2354--2357.
+\href{https://doi.org/10.1103/PhysRevLett.64.2354}{\nolinkurl{doi:10.1103/PhysRevLett.64.2354}}.
+
+\bibitem{Moore} C. Moore, Generalized shifts: unpredictability and
+undecidability in dynamical systems, \emph{Nonlinearity} \textbf{4}
+(1991), 199--230.
+\href{https://doi.org/10.1088/0951-7715/4/2/002}{\nolinkurl{doi:10.1088/0951-7715/4/2/002}}.
+
+\bibitem{Stationary}
+OpenAI, Universal Computation with Eventually Stationary Navier--Stokes Forcing,
+OpenAI Math Release preprint
+\href{https://github.com/openai/math/blob/main/preprints/Universal-Computation-with-Eventually-Stationary-Navier-Stokes-Forcing-September-27-2026/manuscript.pdf}{OAI:Universal-Computation-with-Eventually-Stationary-Navier-Stokes-Forcing-September-27-2026},
+2026.
+
+\bibitem{Turing} A.~M. Turing, On computable numbers, with an application
+to the Entscheidungsproblem, \emph{Proc. London Math. Soc.} (2)
+\textbf{42} (1936--1937), 230--265.
+\href{https://doi.org/10.1112/plms/s2-42.1.230}{\nolinkurl{doi:10.1112/plms/s2-42.1.230}}.
+\end{thebibliography}
+\end{document}
diff --git a/preprints/A-Fixed-Particle-Test-for-Computation-in-a-Forced-Viscous-Flow-September-27-2026/manuscript.pdf b/preprints/A-Fixed-Particle-Test-for-Computation-in-a-Forced-Viscous-Flow-September-27-2026/manuscript.pdf
new file mode 100644
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diff --git a/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/.gitignore b/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/.gitignore
new file mode 100644
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+++ b/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/.gitignore
@@ -0,0 +1,4 @@
+/.build/
+/.paper.pdf.*
+/__pycache__/
+/verification/__pycache__/
diff --git a/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026.pdf b/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..5068f8c87bcc9ebd7970b348328c0237eac33c60
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+++ b/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
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diff --git a/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/README.md b/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..fc5991dffe7b77b711dc7e6362e5ab1bc5aceeab
--- /dev/null
+++ b/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/README.md
@@ -0,0 +1,17 @@
+# [A Fock-space inequality and the Laughlin spectral gap](A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 24, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026,
+  author = {{OpenAI}},
+  title = {{A Fock-space inequality and the Laughlin spectral gap}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026.pdf}{OAI:A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026}},
+  year = {2026}
+}
+```
diff --git a/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/figures/relative-transfer.tex b/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/figures/relative-transfer.tex
new file mode 100644
index 0000000000000000000000000000000000000000..9890e8dad48ecd250f71504282e24c55a466c0da
--- /dev/null
+++ b/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/figures/relative-transfer.tex
@@ -0,0 +1,13 @@
+\begin{tikzpicture}[
+  box/.style={draw=blue!45!black,rounded corners=2pt,align=center,
+              text width=0.265\linewidth,inner sep=6pt,font=\small},
+  >={Latex[length=2mm]}, node distance=6mm]
+\node[box] (limit) {\textbf{Finite limiting matrices}\\[2pt]
+  Exact positivity on the\\four-particle space};
+\node[box,right=of limit] (local) {\textbf{Fixed set of modes}\\[2pt]
+  Exact change of variables;\\error bounded by pair energy};
+\node[box,right=of local] (global) {\textbf{All particle sectors}\\[2pt]
+  Add spectator modes and\\average over rotations};
+\draw[->] (limit) -- (local);
+\draw[->] (local) -- (global);
+\end{tikzpicture}
diff --git a/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/main.tex b/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/main.tex
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+++ b/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/main.tex
@@ -0,0 +1,109 @@
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+\pdfglyphtounicode{hatwider}{005E}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage[utf8]{inputenc}
+\usepackage[margin=1in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{booktabs,microtype,xcolor,listings,needspace}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta,positioning}
+\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref}
+\hypersetup{pdftitle={A Fock-space inequality and the Laughlin spectral gap},pdfauthor={OpenAI},pdfsubject={The full fermionic V1 interaction on the round sphere},pdfkeywords={Laughlin state, spectral gap, fractional quantum Hall effect, pseudopotential, sphere}}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newcommand{\FF}{\mathcal F}
+\newcommand{\LL}{\mathcal L}
+\newcommand{\ran}{\operatorname{ran}}
+\newcommand{\tr}{\operatorname{tr}}
+\newcommand{\diag}{\operatorname{diag}}
+\newcommand{\eps}{\varepsilon}
+\newcommand{\ket}[1]{\lvert #1\rangle}
+\newcommand{\bra}[1]{\langle #1\rvert}
+\newcommand{\ip}[2]{\langle #1,#2\rangle}
+\newcommand{\norm}[1]{\lVert #1\rVert}
+\lstset{basicstyle=\ttfamily\scriptsize,columns=fullflexible,keepspaces=true,breaklines=true,frame=single}
+\title{A Fock-space inequality and the Laughlin spectral gap}
+\author{OpenAI}
+\date{September 24, 2026}
+\AddToHook{env/theorem/before}{\Needspace{4\baselineskip}}
+\AddToHook{env/proposition/before}{\Needspace{4\baselineskip}}
+\AddToHook{env/lemma/before}{\Needspace{4\baselineskip}}
+\AddToHook{env/corollary/before}{\Needspace{4\baselineskip}}
+\begin{document}
+\maketitle
+\begin{abstract}
+We prove the spherical fermionic Laughlin spectral-gap conjecture for the
+full \(V_1\) interaction. With coefficient one for each pair projector,
+the gap above the Laughlin state at filling \(1/3\) on the round sphere
+is at least \(1/25\) for all sufficiently large systems.
+More generally, \(H_Q^2\ge\gamma H_Q\) for some fixed \(\gamma>1/25\)
+on the entire lowest-Landau-level Fock space for all sufficiently large
+flux \(Q\), independently of particle number.
+\end{abstract}
+\input{sections/01-introduction}
+\input{sections/02-operators}
+\input{sections/03-spin}
+\input{sections/04-comparison}
+\input{sections/05-transfer}
+\input{sections/06-gap}
+\input{sections/07-plane}
+\appendix
+\input{sections/08-arithmetic}
+\bibliographystyle{plain}
+\bibliography{references}
+\end{document}
diff --git a/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/references.bib b/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/references.bib
new file mode 100644
index 0000000000000000000000000000000000000000..f95b4fdcc85812757014f0cf28925e52c9fe0833
--- /dev/null
+++ b/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/references.bib
@@ -0,0 +1,329 @@
+@article{Laughlin,
+  author  = {Laughlin, Robert B.},
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+  journal = {Physical Review Letters},
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+  number  = {18},
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+  year    = {1983},
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+  doi     = {10.1103/PhysRevLett.50.1395},
+  url     = {https://doi.org/10.1103/PhysRevLett.50.1395}
+}
+
+@article{Haldane,
+  author  = {Haldane, F. D. M.},
+  title   = {Fractional quantization of the {Hall} effect: a hierarchy of incompressible quantum fluid states},
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+  doi     = {10.1103/PhysRevLett.51.605},
+  url     = {https://doi.org/10.1103/PhysRevLett.51.605}
+}
+
+@article{Rougerie,
+  author  = {Rougerie, Nicolas},
+  title   = {On the {Laughlin} function and its perturbations},
+  journal = {S{\'e}minaire Laurent Schwartz --- EDP et applications},
+  year    = {2018--2019},
+  pages   = {1--17},
+  note    = {Exp.~II. \href{https://doi.org/10.5802/slsedp.131}{doi:10.5802/slsedp.131}},
+  doi     = {10.5802/slsedp.131},
+  url     = {https://doi.org/10.5802/slsedp.131},
+  eprint  = {1906.11656},
+  archivePrefix = {arXiv}
+}
+
+@article{NWY,
+  author  = {Nachtergaele, Bruno and Warzel, Simone and Young, Amanda},
+  title   = {Spectral gaps and incompressibility in a {$\nu=1/3$} fractional quantum {Hall} system},
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+  eprint  = {2004.04992},
+  archivePrefix = {arXiv}
+}
+
+@article{WY,
+  author  = {Warzel, Simone and Young, Amanda},
+  title   = {The spectral gap of a fractional quantum {Hall} system on a thin torus},
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+  doi     = {10.1063/5.0084677},
+  url     = {https://doi.org/10.1063/5.0084677},
+  eprint  = {2112.13764},
+  archivePrefix = {arXiv}
+}
+
+@article{Knabe,
+  author  = {Knabe, Stefan},
+  title   = {Energy gaps and elementary excitations for certain {VBS}-quantum antiferromagnets},
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+  url     = {https://doi.org/10.1007/BF01019721}
+}
+
+@article{Cruz,
+  author  = {Cruz, Esther and Baccari, Flavio and Tura, Jordi and Schuch, Norbert and Cirac, J. Ignacio},
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+  doi     = {10.1103/PhysRevResearch.4.023161},
+  url     = {https://doi.org/10.1103/PhysRevResearch.4.023161},
+  eprint  = {2105.06866},
+  archivePrefix = {arXiv}
+}
+
+@article{Rai,
+  author  = {Rai, Kshiti Sneh and Kull, Ilya and Emonts, Patrick and Tura, Jordi and Schuch, Norbert and Baccari, Flavio},
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+  doi     = {10.22331/q-2026-04-13-2065},
+  url     = {https://doi.org/10.22331/q-2026-04-13-2065},
+  eprint  = {2411.03680},
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+
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+
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+  journal = {Physical Review B}, volume = {91}, pages = {085115}, year = {2015},
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+  eprint = {1409.3577}, archivePrefix = {arXiv}, primaryClass = {cond-mat.str-el},
+  note = {\href{https://doi.org/10.1103/PhysRevB.91.085115}{doi:10.1103/PhysRevB.91.085115}}
+}
+
+@article{NakamuraWangBergholtz2012,
+  author = {Nakamura, Masaaki and Wang, Zheng-Yuan and Bergholtz, Emil J.},
+  title = {Exactly Solvable Fermion Chain Describing a {$\nu=1/3$} Fractional Quantum {Hall} State},
+  journal = {Physical Review Letters}, volume = {109}, number = {1},
+  pages = {016401}, year = {2012}, doi = {10.1103/PhysRevLett.109.016401},
+  eprint = {1110.5033}, archivePrefix = {arXiv},
+  url = {https://doi.org/10.1103/PhysRevLett.109.016401},
+  note = {\href{https://doi.org/10.1103/PhysRevLett.109.016401}{doi:10.1103/PhysRevLett.109.016401}}
+}
+
+@article{WangNakamura2013,
+  author = {Wang, Zheng-Yuan and Nakamura, Masaaki},
+  title = {One-dimensional lattice model with an exact matrix-product ground state describing the {Laughlin} wave function},
+  journal = {Physical Review B}, volume = {87},
+  pages = {245119}, year = {2013}, doi = {10.1103/PhysRevB.87.245119},
+  eprint = {1206.3071}, archivePrefix = {arXiv},
+  url = {https://doi.org/10.1103/PhysRevB.87.245119},
+  note = {\href{https://doi.org/10.1103/PhysRevB.87.245119}{doi:10.1103/PhysRevB.87.245119}}
+}
+
+@article{Jansen2012,
+  author = {Jansen, Sabine},
+  title = {Fermionic and bosonic {Laughlin} state on thick cylinders},
+  journal = {Journal of Mathematical Physics}, volume = {53},
+  pages = {123306}, year = {2012}, doi = {10.1063/1.4768250},
+  eprint = {1109.4022}, archivePrefix = {arXiv},
+  url = {https://doi.org/10.1063/1.4768250},
+  note = {\href{https://doi.org/10.1063/1.4768250}{doi:10.1063/1.4768250}}
+}
+
+@article{WarzelYoung2023BulkEdge,
+  author = {Warzel, Simone and Young, Amanda},
+  title = {A Bulk Spectral Gap in the Presence of Edge States for a Truncated Pseudopotential},
+  journal = {Annales Henri Poincar\'e}, volume = {24},
+  pages = {133--178}, year = {2023}, doi = {10.1007/s00023-022-01210-z},
+  eprint = {2108.10794}, archivePrefix = {arXiv},
+  url = {https://doi.org/10.1007/s00023-022-01210-z},
+  note = {\href{https://doi.org/10.1007/s00023-022-01210-z}{doi:10.1007/s00023-022-01210-z}}
+}
+
+@article{LemmNachtergaeleWarzelYoung2026Recursive,
+  author        = {Lemm, Marius and Nachtergaele, Bruno and Warzel, Simone and Young, Amanda},
+  title         = {Recursive spectral relations and the charge vs neutral gap in fractional quantum {Hall} systems},
+  journal       = {Journal of Mathematical Physics},
+  volume        = {67},
+  number        = {1},
+  pages         = {011901},
+  year          = {2026},
+  month         = jan,
+  doi           = {10.1063/5.0292975},
+  url           = {https://doi.org/10.1063/5.0292975},
+  eprint        = {2507.18814},
+  archivePrefix = {arXiv},
+  primaryClass  = {math-ph},
+  note = {\href{https://doi.org/10.1063/5.0292975}{doi:10.1063/5.0292975}}
+}
+
+@misc{SchravenWarzel2026IMPS,
+  author = {Schraven, Severin and Warzel, Simone},
+  title = {Fractional Quantum {Hall} States: Infinite Matrix Product Representation and its Implications},
+  year = {2026},
+  eprint = {2601.12165},
+  archivePrefix = {arXiv},
+  primaryClass = {math-ph},
+  note = {\href{https://arxiv.org/abs/2601.12165v1}{arXiv:2601.12165v1}, January 17, 2026},
+  doi = {10.48550/arXiv.2601.12165},
+  url = {https://arxiv.org/abs/2601.12165}
+}
+
+@article{OrtizNussinovDukelskySeidel,
+  author = {Ortiz, G. and Nussinov, Z. and Dukelsky, J. and Seidel, A.},
+  title = {Repulsive interactions in quantum {Hall} systems as a pairing problem},
+  journal = {Physical Review B},
+  volume = {88},
+  pages = {165303},
+  year = {2013},
+  doi = {10.1103/PhysRevB.88.165303},
+  url = {https://doi.org/10.1103/PhysRevB.88.165303},
+  eprint = {1306.3268},
+  archivePrefix = {arXiv},
+  note = {\href{https://doi.org/10.1103/PhysRevB.88.165303}{doi:10.1103/PhysRevB.88.165303}}
+}
+
+@article{Wick,
+  author = {Wick, G. C.},
+  title = {The Evaluation of the Collision Matrix},
+  journal = {Physical Review},
+  volume = {80},
+  number = {2},
+  pages = {268--272},
+  year = {1950},
+  doi = {10.1103/PhysRev.80.268},
+  url = {https://doi.org/10.1103/PhysRev.80.268},
+  note = {\href{https://doi.org/10.1103/PhysRev.80.268}{doi:10.1103/PhysRev.80.268}}
+}
+
+@article{BartlettRudolphSpekkens,
+  author = {Bartlett, Stephen D. and Rudolph, Terry and Spekkens, Robert W.},
+  title = {Reference frames, superselection rules, and quantum information},
+  journal = {Reviews of Modern Physics},
+  volume = {79},
+  pages = {555--609},
+  year = {2007},
+  doi = {10.1103/RevModPhys.79.555},
+  url = {https://doi.org/10.1103/RevModPhys.79.555},
+  note = {\href{https://doi.org/10.1103/RevModPhys.79.555}{doi:10.1103/RevModPhys.79.555}}
+}
+
+@article{PeyrlParrilo,
+  author = {Peyrl, Helfried and Parrilo, Pablo A.},
+  title = {Computing sum of squares decompositions with rational coefficients},
+  journal = {Theoretical Computer Science},
+  volume = {409},
+  number = {2},
+  pages = {269--281},
+  year = {2008},
+  doi = {10.1016/j.tcs.2008.09.025},
+  url = {https://doi.org/10.1016/j.tcs.2008.09.025},
+  note = {\href{https://doi.org/10.1016/j.tcs.2008.09.025}{doi:10.1016/j.tcs.2008.09.025}}
+}
+
+@inproceedings{HouConference,
+  author = {Hou, Shui-Hung},
+  title = {On the {Leverrier--Faddeev} Algorithm},
+  booktitle = {Electronic Proceedings of the Third Asian Technology Conference in Mathematics},
+  year = {1998},
+  url = {https://atcm.mathandtech.org/EP/1998/ATCMP002/paper.pdf},
+  note = {\href{https://atcm.mathandtech.org/EP/1998/ATCMP002/paper.pdf}{Full text}}
+}
+
+@article{GaoLanzettaLedwithWangKhalaf,
+  author = {Gao, Qiang and Lanzetta, Ryan A. and Ledwith, Patrick and Wang, Jie and Khalaf, Eslam},
+  title = {Bootstrapping the Quantum {Hall} Problem},
+  journal = {Physical Review X},
+  volume = {15},
+  pages = {031034},
+  year = {2025},
+  doi = {10.1103/csnn-vjhn},
+  url = {https://doi.org/10.1103/csnn-vjhn},
+  eprint = {2409.10619},
+  archivePrefix = {arXiv},
+  note = {\href{https://doi.org/10.1103/csnn-vjhn}{doi:10.1103/csnn-vjhn}}
+}
+
+@article{GirvinJach,
+  author = {Girvin, S. M. and Jach, Terrence},
+  title = {Formalism for the quantum {Hall} effect: {Hilbert} space of analytic functions},
+  journal = {Physical Review B},
+  volume = {29}, number = {10}, pages = {5617--5625}, year = {1984},
+  doi = {10.1103/PhysRevB.29.5617},
+  url = {https://doi.org/10.1103/PhysRevB.29.5617},
+  note = {\href{https://doi.org/10.1103/PhysRevB.29.5617}{doi:10.1103/PhysRevB.29.5617}}
+}
diff --git a/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/sections/01-introduction.tex b/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/sections/01-introduction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..ca4c9fb728940a87567fc0dc6954b8d7407e2eb6
--- /dev/null
+++ b/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/sections/01-introduction.tex
@@ -0,0 +1,200 @@
+\section{The gap and the model}
+\label{sec:intro}
+Laughlin's wave function gave an explicit description of the fractional
+quantum Hall state at filling $1/3$~\cite{Laughlin}. Haldane's spherical
+geometry and pseudopotentials identify a particularly simple parent
+Hamiltonian: every pair pays energy one in relative angular momentum one
+and zero in the other pair sectors~\cite{Haldane}. Its exact Laughlin
+zero mode is known, but a uniform lower bound on the positive spectrum
+requires control of all competing many-particle states. We prove this
+bound for the full interaction on the round sphere, resolving positively
+the spherical fermionic Laughlin spectral-gap conjecture.
+
+The gap question has accompanied this model since its early development.
+Haldane and Rezayi's finite-size study supplied numerical evidence and
+explicitly distinguished the expected hard-core gap from a formal
+proof~\cite{HaldaneRezayi1985FiniteSize}. The density-wave variational
+picture of Girvin, MacDonald, and Platzman explains important features of
+the excitation spectrum~\cite{GMP}. A variational excitation energy is
+an upper bound; the lower bound sought here must hold for every state
+orthogonal to the zero modes.
+
+\subsection{Normalization and main results}
+For an integer \(Q\ge1\), let \(U_Q\) be the spin-\(Q/2\) representation of
+\(SU(2)\), equipped with its invariant inner product. It is the one-particle
+lowest Landau level on the sphere with \(Q\) flux quanta. Equivalently, it is
+the space of homogeneous polynomials of degree \(Q\) in spinor coordinates
+\((u,v)\). For \(2\le N\le Q+1\), define
+\begin{equation}\label{eq:physicalH}
+ H_{N,Q}=\left.\sum_{1\le i<j\le N}P_{ij}^{(1)}
+          \right|_{\bigwedge^N U_Q},
+\end{equation}
+where \(P_{ij}^{(1)}\) is the orthogonal projector onto pair spin \(Q-1\)
+in tensor positions \(i,j\). This is relative angular momentum one. The sum
+preserves the antisymmetric subspace. Each unordered pair has coefficient
+one, with no rescaling depending on \(N\) or \(Q\).
+
+Set \(H_{0,Q}=H_{1,Q}=0\), and write
+\[
+ \FF_Q=\bigoplus_{N=0}^{Q+1}\bigwedge^N U_Q,\qquad
+ H_Q=\bigoplus_{N=0}^{Q+1}H_{N,Q},\qquad
+ \gamma_*=\frac{4616733319001}{10^{14}}> \frac1{25}.
+\]
+Our principal estimate concerns this entire Fock space.
+
+\begin{theorem}\label{thm:fock}
+For every \(0<\gamma<\gamma_*\), there is an integer \(Q_\gamma\) such that
+for every integer \(Q\ge Q_\gamma\),
+\begin{equation}\label{eq:architecture}
+ H_Q^2\ge\gamma H_Q\qquad\text{on }\FF_Q.
+\end{equation}
+The threshold \(Q_\gamma\) is independent of particle number.
+\end{theorem}
+
+Thus the spectrum is contained in \(\{0\}\cup[\gamma,\infty)\).
+In a particle sector without a zero mode, this statement bounds the energy
+away from zero; it does not assert a gap between that sector's two lowest
+eigenvalues. At Laughlin filling the zero mode is unique, and we obtain the
+usual ground-state spectral gap.
+
+\begin{corollary}\label{thm:main}
+There is an integer \(N_0\ge2\) such that, for every \(N\ge N_0\) and
+\(Q=3(N-1)\),
+\begin{equation}\label{eq:main}
+ H_{N,Q}\ge\frac1{25}\bigl(I-P_{\mathrm L,N}\bigr)
+ \qquad\text{on }\bigwedge^N U_Q,
+\end{equation}
+where \(P_{\mathrm L,N}\) projects onto the line spanned by
+\[
+ \Psi_{\mathrm L,N}=\prod_{i<j}(u_iv_j-u_jv_i)^3.
+\]
+\end{corollary}
+
+At fixed magnetic length, the sphere radius is proportional to \(\sqrt Q\),
+so this is the two-dimensional thermodynamic limit. The thresholds in
+Theorem~\ref{thm:fock} and Corollary~\ref{thm:main} are existential.
+The proof does not give an explicit flux threshold or the endpoint
+coefficient \(\gamma_*\) on finite spheres.
+
+\subsection{Prior work}
+The exact ground-state structure is an established part of the theory.
+Besides Haldane's spherical construction, Trugman and Kivelson obtained
+exact Laughlin ground states for repulsive interactions in a vanishing-range
+limit~\cite{TrugmanKivelson1985}. The second-quantized algebra of
+pseudopotentials was developed systematically by Ortiz, Nussinov, Dukelsky,
+and Seidel~\cite{OrtizNussinovDukelskySeidel}. Chen and Seidel, and
+Mazaheri, Ortiz, Nussinov, and Seidel, gave algebraic constructions and
+characterizations of their zero modes~\cite{ChenSeidel2015,MazaheriOrtizNussinovSeidel2015}.
+These results identify the zero modes that a spectral estimate must preserve. We include a short self-contained proof of the known spherical
+zero-mode uniqueness at the Laughlin flux in Lemma~\ref{lem:kernel}.
+
+For Laughlin states on a cylinder of any fixed radius, Jansen established
+thermodynamic correlation limits and axial periodicity~\cite{Jansen2012}.
+Exact ground states of related finite-range orbital chains, including
+squeezed and matrix-product states, were constructed by Nakamura, Wang, and
+Bergholtz~\cite{NakamuraWangBergholtz2012} and by Wang and
+Nakamura~\cite{WangNakamura2013}. Nachtergaele, Warzel, and Young proved
+uniform gaps for a truncated fermionic pseudopotential~\cite{NWY};
+Warzel and Young obtained thin-torus bounds that separate bulk behavior
+from open-chain edge modes~\cite{WY}. Their invariant-subspace method
+also treats a truncated bosonic pseudopotential~\cite{WarzelYoung2023BulkEdge}.
+The fermionic estimates already hold uniformly across particle sectors.
+The distinction in Theorem~\ref{thm:fock} is the full spherical pair
+interaction, with its fixed projector normalization and sphere radius
+growing in the thermodynamic limit. The finite support used in our
+auxiliary comparison operators does not truncate that interaction.
+
+Recent work gives further structural and spectral information.
+Lemm, Nachtergaele, Warzel, and Young derive recursions between particle
+sectors and compare charge and neutral gaps~\cite{LemmNachtergaeleWarzelYoung2026Recursive};
+a uniform bound for the full pseudopotential still requires control of
+the model-specific ground-space overlap matrices in those criteria.
+Schraven and Warzel prove entanglement-gap and clustering results for
+Laughlin states on sufficiently thin cylinders~\cite{SchravenWarzel2026IMPS}.
+The entanglement spectrum is distinct from the Hamiltonian spectrum
+considered here. Density rigidity within the family of fully correlated
+Laughlin states is another complementary form of incompressibility,
+established by Lieb, Rougerie, and Yngvason~\cite{LRYRigidity}.
+No density-screening assumption enters our argument.
+
+Rougerie's survey records the spectral-gap conjecture and its geometric
+formulations~\cite[Appendix~A]{Rougerie}. For $N$ fermions in the planar
+lowest Landau level, consider the sum over unordered pairs of the orthogonal
+projectors onto relative polynomial degree one, each with coefficient one
+and with center-of-mass degree unrestricted. Section~\ref{sec:plane} proves
+that its positive spectrum is bounded below by the endpoint $\gamma_*$,
+uniformly in particle number $N\ge2$ and total polynomial degree $L$.
+In particular, this proves the fermionic cubic case of the folklore
+spectral-gap conjecture in \cite[Conjecture~A.1]{Rougerie}, whose degree range
+is $L\le3N(N-1)/2$.
+
+\subsection{Proof strategy}
+Proving positivity of $H^2-\gamma H$ is a standard route to a gap,
+underlying finite-size criteria~\cite{Knabe} and methods based on positive
+local decompositions~\cite{Cruz,Rai}. Our argument uses rotations and
+angular-momentum decomposition to construct such a comparison.
+It starts from the normal-ordered expansion of \(H_Q^2\).
+It has two-, three-, and four-body terms. We compare them with the
+corresponding terms of a positive operator
+\[
+ \mathcal K_Q=(2Q-1)\int_{SU(2)}
+                  \sum_{\rho=1}^{7}F_\rho(g)^\dagger F_\rho(g)\,dg,
+\]
+where each \(F_\rho\) is a specified linear combination of a pair
+annihilator and operators that annihilate three particles and create one.
+Here \(F_\rho(g)\) denotes rotation by \(g\), and Haar measure has mass
+one. These auxiliary operators involve finitely many orbital modes;
+the Hamiltonian itself is never truncated.
+
+Rotational symmetry reduces the comparison to fixed spin blocks.
+The three-body blocks have an explicit spectrum and a summable tail.
+For four particles, coupling the interacting pair to the different spin
+sectors of a second pair can produce several copies of the same total spin.
+Their four-particle images under exterior multiplication can be linearly
+dependent.
+For the retained blocks, seven rows of rational data establish finite
+inequalities in their large-flux limits. The remaining issue is to transfer
+those inequalities to finite spheres without an error that grows with the
+number of particles.
+
+The key device is an exact diagonal change of variables for the pair
+coefficients. On a fixed set of orbital modes, it converts finite-flux
+annihilators into their limiting forms. Every perturbation is then bounded
+by the local pair energy. Rotation averaging turns this into
+\(o(1)H_Q\) on the full Fock space. This relative estimate remains valid
+when the Gram matrices of these limiting four-particle images have kernels.
+A second useful observation
+is to apply Bessel's inequality to all equivalent spin copies at once;
+this substantially reduces the four-body comparison error.
+
+The comparisons leave a positive margin:
+\begin{equation}\label{eq:budget}
+ H_Q^2\ge\mathcal K_Q+
+       \bigl(\gamma_*-o(1)\bigr)H_Q.
+\end{equation}
+All errors and convergence
+thresholds are independent of particle number.
+Figure~\ref{fig:transfer} isolates the passage responsible for this
+uniformity. A small norm error on a growing many-particle space would not
+suffice: it must vanish on the zero modes and be controlled by the pair
+energy. The fixed-mode comparison gives precisely that control before
+rotations distribute it over the sphere.
+
+\begin{figure}[t]
+\centering
+\input{figures/relative-transfer}
+\caption{The finite-sphere comparison preserves an error relative to the
+pair energy at every step. The limiting positivity is imposed only after
+wedging; no invertibility of its Gram matrices is required.}
+\label{fig:transfer}
+\end{figure}
+
+
+Section~\ref{sec:algebra} derives the normal-ordered square, and
+Section~\ref{sec:spin} establishes the spin formulas.
+Section~\ref{sec:squares} gives the finite comparisons.
+Section~\ref{sec:transfer} proves their uniform transfer, and
+Section~\ref{sec:finish} controls the tail and proves the spherical results.
+Section~\ref{sec:plane} derives the planar consequence.
+The exact row data, rational formulas, and complete arithmetic procedure
+for the positivity checks are in Appendix~\ref{app:certificate}.
diff --git a/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/sections/02-operators.tex b/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/sections/02-operators.tex
new file mode 100644
index 0000000000000000000000000000000000000000..b84b8e36b23bc1467fa20742fc8fb792662b442b
--- /dev/null
+++ b/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/sections/02-operators.tex
@@ -0,0 +1,60 @@
+\section{Exterior algebra and the square of the interaction}
+\label{sec:algebra}
+Our first task is to isolate the two-, three-, and four-body terms of $H_Q^2$. The lifting notation below keeps this calculation independent of particle number.
+
+Throughout the operator argument take $Q\ge2$; statements involving limits hold for all sufficiently large integer $Q$. We use the convention that wedges of distinct increasing orthonormal basis vectors have norm one. On
+\[
+ \FF_Q=\bigoplus_{n=0}^{Q+1}\bigwedge^n U_Q,
+\]
+let $B(v)$ be the adjoint of left wedge multiplication by $v$. Thus $B(v)$ is conjugate-linear in $v$, and
+\[
+ B(v\wedge w)=B(w)B(v).
+\]
+For an orthonormal basis $e_0,\ldots,e_Q$ of $U_Q$, write $c_j=B(e_j)$. These operators satisfy the canonical anticommutation relations, and $\norm{c_j}\le1$.
+
+For a matrix on $\bigwedge^k U_Q$, define its lift by linear extension of
+\begin{equation}\label{eq:lift}
+ \LL_k(\ket v\bra w)=B(v)^\dagger B(w).
+\end{equation}
+This map preserves positivity, intertwines rotations, acts as the original matrix on $k$ particles, and vanishes on fewer than $k$ particles. Positivity follows by decomposing a positive matrix into a sum of rank-one positive matrices.
+
+Put $a=Q-1$ and let $V_Q\subset\bigwedge^2 U_Q$ be the spin-$a$ summand. If $v_0,\ldots,v_{2Q-2}$ is an orthonormal basis of $V_Q$, set
+\begin{equation}\label{eq:fockH}
+ B_p=B(v_p),\qquad H_Q=\LL_2(\Pi_{V_Q})=\sum_{p=0}^{2Q-2}B_p^\dagger B_p.
+\end{equation}
+This pair-annihilator form is standard in the algebraic treatment of
+pseudopotentials~\cite[Sections~II.B--D]{OrtizNussinovDukelskySeidel};
+see also \cite[Equation~(2)]{ChenSeidel2015}. We verify its coefficient-one
+normalization here. Its restriction to $\bigwedge^N U_Q$ is \eqref{eq:physicalH}. Indeed contraction in two specified tensor positions corresponds to $B(v)/\sqrt{\binom N2}$ under the isometric identification of wedge and antisymmetric tensor spaces; summing over the $\binom N2$ unordered pairs gives \eqref{eq:fockH}.
+
+Define intertwining wedge maps
+\[
+ W_3:V_Q\otimes U_Q\longrightarrow\bigwedge^3U_Q,
+ \qquad W_4:V_Q\otimes V_Q\longrightarrow\bigwedge^4U_Q,
+\]
+by $W_3(v\otimes u)=v\wedge u$ and $W_4(v\otimes w)=v\wedge w$. We suppress the subscript $Q$ when no confusion can result.
+
+A finite normal-ordering calculation in the sense of Wick~\cite{Wick}
+now determines the square. Only contractions between the two middle
+pair operators are needed.
+
+\begin{lemma}[Normal-ordered square]\label{lem:square}
+On $\FF_Q$,
+\begin{equation}\label{eq:square}
+ H^2=H+\LL_3\!\left(W_3(W_3^\dagger W_3-I)W_3^\dagger\right)
+       +\LL_4(W_4W_4^\dagger).
+\end{equation}
+\end{lemma}
+\begin{proof}
+In normal ordering $B_p^\dagger(B_pB_r^\dagger)B_r$, only the two inner annihilators cross the two inner creators. Two, one, or zero contractions leave two-, three-, or four-body terms, respectively. On two particles, $H=\Pi_{V_Q}$ is a projection, so the two-body coefficient is $\Pi_{V_Q}$. The term with no contractions is
+\[
+ \sum_{p,r}B(v_p\wedge v_r)^\dagger B(v_p\wedge v_r)
+ =\LL_4(W_4W_4^\dagger).
+\]
+There is no extra factor of two: this sum, and the defining basis of $V_Q\otimes V_Q$, both use ordered pairs. On three particles, $H=W_3W_3^\dagger$, since $W_3$ restricted to $v_p\otimes U_Q$ is the map $B_p^\dagger$ from one to three particles. Subtracting its two-body contribution from $H^2$ therefore determines the remaining coefficient as
+\[
+ (W_3W_3^\dagger)^2-W_3W_3^\dagger
+ =W_3(W_3^\dagger W_3-I)W_3^\dagger.
+\]
+Normal-ordered coefficients are determined successively by these restrictions, proving the identity on Fock space.
+\end{proof}
diff --git a/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/sections/03-spin.tex b/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/sections/03-spin.tex
new file mode 100644
index 0000000000000000000000000000000000000000..003649eb5aa67a9eabc27ff8ee305973b8faa17b
--- /dev/null
+++ b/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/sections/03-spin.tex
@@ -0,0 +1,106 @@
+\section{Spin coefficients and their limits}
+\label{sec:spin}
+The three-body comparison will use the exact spin spectrum of the wedge map. The four-body comparison will use only finitely many coupling coefficients and their limits.
+
+The spherical pair coefficients can be expressed through angular-momentum
+coupling~\cite[Section~II.C and Appendix~A]{OrtizNussinovDukelskySeidel}.
+We derive the particular coefficients and fixed-deficit limits needed
+here to keep their phases and normalization explicit.
+
+We now fix $e_p$ and $v_p$ to be the normalized lowering bases of $U_Q$ and $V_Q$, respectively, with $v_0=-e_0\wedge e_1$. For spin $j$, lowering from label $p$ to $p+1$ has coefficient $\sqrt{(p+1)(2j-p)}$. Write $(s)_p=s(s-1)\cdots(s-p+1)$, with $(s)_0=1$.
+
+\begin{lemma}[Pair coefficients]\label{lem:pair}
+The coefficient of $e_i\wedge e_j$ in $v_p$, for $i<j$, vanishes unless $i+j=p+1$, and otherwise equals
+\begin{equation}\label{eq:pair}
+ v_p(i,j)=(i-j)\sqrt{\frac{(Q)_i(Q)_j\,p!}{Q(2Q-2)_p\,i!j!}}.
+\end{equation}
+For fixed $p,i,j$, its limit is
+\begin{equation}\label{eq:pairstar}
+ v_p^*(i,j)=(i-j)\sqrt{\frac{p!}{2^p i!j!}}\,\mathbf1_{i+j=p+1}.
+\end{equation}
+\end{lemma}
+\begin{proof}
+Apply simultaneous lowering $p$ times to $-e_0\wedge e_1$ and divide by $\sqrt{(2Q-2)_p p!}$. Expand the lowering operator binomially. Combining the two contributions to each increasing pair gives \eqref{eq:pair}. Taking the limit gives \eqref{eq:pairstar}.
+The normalization can also be checked directly: the square of the
+coefficient in \eqref{eq:pair} is
+\[
+ (i-j)^2\frac{\binom Qi\binom Qj}{Q\binom{2Q-2}{p}},
+\]
+and differentiating \((1+x)^Q\) gives
+\[
+ \sum_{i,j=0}^Q(i-j)^2\binom Qi\binom Qj x^{i+j}
+ =2Qx(1+x)^{2Q-2}.
+\]
+Take the coefficient of \(x^{p+1}\) and halve the sum to restrict to \(i<j\).
+\end{proof}
+
+The elementary tensor product rule decomposes spins $j_1,j_2$ into one copy of each spin $j_1+j_2-z$, for $0\le z\le\min(2j_1,2j_2)$. This rule can be seen directly in the
+polynomial model: at deficit $z$ the raising equation determines a highest
+vector, and normalized lowering generates its $2(j_1+j_2-z)+1$
+orthogonal weight vectors. Different highest spins give orthogonal
+invariant subspaces, and the sum of these dimensions is
+$(2j_1+1)(2j_2+1)$, exhausting the tensor product. Let $R_{3,z}$ be the corresponding projection in $V_Q\otimes U_Q$, and put
+\[
+ d_a=2Q-1,\qquad d_{3,z}=3Q-1-2z.
+\]
+For $Q\ge2$, the allowed indices are $0\le z\le Q$.
+
+\begin{lemma}[Three-body Gram eigenvalues]\label{lem:q}
+\begin{equation}\label{eq:q}
+ W_3^\dagger W_3-I=\sum_{z=0}^Qq_z(Q)R_{3,z},\qquad
+ q_z(Q)=(-1)^z\frac{(Q)_z}{(2Q-2)_z}
+          \left(3z-1-\frac{z(z+1)}Q\right).
+\end{equation}
+In particular $W_3R_{3,z}=0$ for $z=0,1,3$ whenever these indices are allowed. For fixed $z$,
+\begin{equation}\label{eq:qlimit}
+ q_z(Q)\longrightarrow (3z-1)(-1/2)^z.
+\end{equation}
+\end{lemma}
+\begin{proof}
+Identify $V_Q\otimes U_Q$ with tensors antisymmetric in their first two positions. Then $W_3$ is $\sqrt3$ times full antisymmetrization. Its Gram operator is the compression of $I-P_{13}-P_{23}$, where $P_{ij}$ swaps tensor positions. The two swap compressions coincide, since conjugation by $P_{12}$ interchanges them and $P_{12}=-I$ on the domain. By equivariance and multiplicity-freeness they act by the same scalar $b_z$ on each spin block.
+
+A highest vector of that block has coefficients $s_p$ on $v_p\otimes e_{z-p}$, $0\le p\le z$, satisfying
+\begin{equation}\label{eq:highest-ratio}
+ \frac{s_{p+1}}{s_p}=-\sqrt{\frac{(z-p)(Q-z+p+1)}{(p+1)(2Q-2-p)}}.
+\end{equation}
+This follows by applying the raising operator. Write
+$\xi_z=\sum_{p=0}^z s_pv_p\otimes e_{z-p}$. Since
+$v_0=(-e_0\otimes e_1+e_1\otimes e_0)/\sqrt2$, testing $P_{23}\xi_z$
+against $v_0\otimes e_z$ tests $\xi_z$ only on the ordered triples
+$(0,z,1)$ and $(1,z,0)$. Their pair labels are $z-1$ and $z$,
+respectively, with the first contribution absent at $z=0$.
+Substituting their pair coefficients from \eqref{eq:pair} gives
+\[
+ s_0b_z=\frac12\left[
+ s_{z-1}\sqrt{\frac{z(Q)_z}{Q(2Q-2)_{z-1}}}
+ -s_z(z-1)\sqrt{\frac{(Q)_z}{(2Q-2)_z}}\right].
+\]
+Here the ordered tensor coefficients of a normalized wedge include a factor $1/\sqrt2$. Iteration of \eqref{eq:highest-ratio} gives
+\[
+ \frac{s_z}{s_0}=(-1)^z\sqrt{\frac{(Q)_z}{(2Q-2)_z}},\qquad
+ \frac{s_{z-1}}{s_0}=(-1)^{z-1}
+ \sqrt{\frac{z(Q-1)_{z-1}}{(2Q-2)_{z-1}}}\quad(z\ge1).
+\]
+Substitution gives $q_z=-2b_z$ in the form \eqref{eq:q}. The three stated null sectors have $q_z=-1$, and \eqref{eq:qlimit} follows directly.
+\end{proof}
+
+We will repeatedly use a fixed-deficit limit of this same calculation. In a product of spins $j_1,j_2$, total lowering deficit means $T=j_1+j_2-m$, where $m$ is the total magnetic quantum number. Coefficients below are with respect to normalized monomials $X^pY^{T-p}/\sqrt{p!(T-p)!}$, and are zero outside the stated index ranges.
+
+\begin{lemma}[Fixed-deficit coupling limit]\label{lem:coupling}
+Suppose $j_1,j_2\to\infty$ and $j_2/(j_1+j_2)\to u^2$, where $0<u<1$. Put $v=\sqrt{1-u^2}$. For fixed integers $0\le z\le T$, the normalized coupled vector of spin $j_1+j_2-z$ at total lowering deficit $T$ can be chosen with real coefficients converging to the coefficients $s^{(u)}_{z,T,p}$ of
+\begin{equation}\label{eq:coupling}
+ \frac{(uX-vY)^z(vX+uY)^{T-z}}{\sqrt{z!(T-z)!}}.
+\end{equation}
+Descendants in every copy are obtained by normalized lowering, so the matching of bases between equal-spin copies is intertwining.
+\end{lemma}
+\begin{proof}
+At $T=z$, the raising equation gives the adjacent ratio
+\[
+ -\sqrt{\frac{(z-p)(2j_2-z+p+1)}{(p+1)(2j_1-p)}}.
+\]
+With sign $(-1)^z$ at $p=0$, normalization yields the coefficients of $(uX-vY)^z/\sqrt{z!}$ in the limit. The change of variables $(X,Y)\mapsto(uX-vY,vX+uY)$ is orthogonal, so these polynomials have norm one in the normalized-monomial inner product. Normalized lowering from $T$ to $T+1$ divides the sum of the two lowering operators by
+\[
+ \sqrt{(T-z+1)\{2(j_1+j_2-z)-(T-z)\}}.
+\]
+In the limit this acts by multiplication by $(vX+uY)/\sqrt{T-z+1}$. Induction proves the statement for fixed descendants.
+\end{proof}
diff --git a/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/sections/04-comparison.tex b/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/sections/04-comparison.tex
new file mode 100644
index 0000000000000000000000000000000000000000..c0095c2660834fe1eb073473db8ee474c73f4560
--- /dev/null
+++ b/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/sections/04-comparison.tex
@@ -0,0 +1,172 @@
+\section{Rotational averages of explicit squares}
+\label{sec:squares}
+We construct a positive rotational average and compare its three- and four-body terms with those of $H_Q^2$. This section specifies the finite inequalities needed; Appendix~\ref{app:certificate} proves them by exact rational arithmetic.
+
+Haar measure $dg$ on $SU(2)$ is normalized to have total mass one. For an operator $A$, write $A(g)=U(g)AU(g)^\dagger$, with the appropriate Fock representation. Schur averaging sends a rank-one operator on an irreducible spin-$j$ space to its trace times $I/(2j+1)$. For several equivalent copies, it acts in the same way on the spin
+factor and retains the matrix between copies. Explicitly, if the
+representation is $\bigoplus_j(V_j\otimes\mathbb C^{m_j})$ and $P_j$
+is its spin-$j$ projection, then
+\[
+ \int U(g)XU(g)^\dagger\,dg
+ =\bigoplus_j\frac{I_{V_j}}{2j+1}\otimes
+            \operatorname{Tr}_{V_j}(P_jXP_j).
+\]
+Here $\operatorname{Tr}_{V_j}$ is the partial trace over the spin factor.
+This follows from Schur's lemma applied to each pair of irreducible
+copies, with the scalar fixed by its trace; see
+\cite[Section~II.C]{BartlettRudolphSpekkens}. In particular, averaging
+preserves off-diagonal entries between equivalent spin copies.
+
+The positive-square construction belongs to the quadratic-gap approach
+described in Section~\ref{sec:intro}. Fermionic positivity constraints,
+normal ordering, and symmetry also appear in the quantum Hall bootstrap
+of Gao et al.~\cite{GaoLanzettaLedwithWangKhalaf}, which bounds ground-state
+energies. Here the explicit squares are compared with $H_Q^2-\gamma H_Q$
+to control the positive excitation spectrum.
+
+For each of the seven rows in Appendix~\ref{app:certificate}, the integer
+$t_\rho$ lies in $\{0,\ldots,7\}$. Set
+\[
+ \mathcal S_\rho=\{(p,j)\in\mathbb Z^2:0\le p\le7,\ 0\le j\le8,\
+                         0\le p+j-t_\rho\le8\}.
+\]
+The row specifies real coefficients $\lambda_\rho$ and
+$\alpha^\rho_{pj}$ for $(p,j)\in\mathcal S_\rho$; unlisted entries are zero.
+Every coefficient sum for row $\rho$ below is over $\mathcal S_\rho$.
+Our comparison operator is
+\begin{equation}\label{eq:F}
+ K=\sum_{\rho=1}^7F_\rho^\dagger F_\rho,\qquad
+ F_\rho=\lambda_\rho B_{t_\rho}
+       +\sum_{(p,j)\in\mathcal S_\rho}\alpha^{\rho}_{pj}\,
+                                  c_{p+j-t_\rho}^\dagger c_jB_p.
+\end{equation}
+Each summand of $F_\rho$ removes two particles and decreases the sum of
+occupied orbital labels by $t_\rho+1$: for the second term the decrease is
+$(p+1)+j-(p+j-t_\rho)=t_\rho+1$. This common weight allows the two parts
+to interfere in their square, adjusting the three-body comparison at the
+two-body cost $\lambda_\rho^2B_{t_\rho}^\dagger B_{t_\rho}$.
+Only modes $0,\ldots,8$ occur in these auxiliary squares.
+
+For a single row, omit $\rho$ and put $i=p+j-t$, $k=q+l-t$. Define
+\[
+ O_{p,j,k}=B(v_p\wedge e_j\wedge e_k)=c_kc_jB_p.
+\]
+Reordering $c_ic_k^\dagger=\delta_{ik}-c_k^\dagger c_i$ gives the two-, three-, and four-body terms of $F^\dagger F$:
+\begin{align}
+ &\lambda^2 B_t^\dagger B_t,\label{eq:normal2}\\
+ &\sum_{p,j}\lambda\alpha_{pj}\left[(c_iB_t)^\dagger c_jB_p
+                         +(c_jB_p)^\dagger c_iB_t\right]
+   +\sum_{p,j;q,l}\alpha_{pj}\alpha_{ql}\delta_{ik}
+                         (c_jB_p)^\dagger c_lB_q,\label{eq:normal3}\\
+ &-\sum_{p,j;q,l}\alpha_{pj}\alpha_{ql}
+                         O_{p,j,k}^\dagger O_{q,l,i}.\label{eq:normal4}
+\end{align}
+Let $A_2,A_3,A_4$ denote these terms, summed over rows and averaged with factor $d_a$. Thus
+\begin{equation}\label{eq:Adecomp}
+ \mathcal K_Q:=d_a\int K(g)\,dg=A_2+A_3+A_4,\qquad A_2=\eta H,
+ \quad \eta=\sum_\rho\lambda_\rho^2.
+\end{equation}
+
+\subsection{The three-body comparison}
+Before wedging, \eqref{eq:normal3} is a matrix on $V_Q\otimes U_Q$. Its nonzero entries preserve $T=p+j$, and $T\le15$. Averaging therefore gives
+\begin{equation}\label{eq:A3}
+ A_3=\LL_3\left(W_3\sum_{z=0}^{15}\frac{d_a}{d_{3,z}}e_z^Q R_{3,z}W_3^\dagger\right),
+\end{equation}
+for all sufficiently large $Q$, where $e_z^Q$ is the trace of the unaveraged matrix in that spin block. By Lemma~\ref{lem:coupling}, with $u=1/\sqrt3$, these traces converge to
+\begin{equation}\label{eq:ez}
+ e_z=\sum_\rho\sum_{T=\max(z,t_\rho)}^{15}
+ \left[2\lambda_\rho s_{z,T,t_\rho}
+                   \sum_{p+j=T}\alpha^{\rho}_{pj}s_{z,T,p}
+       +\left(\sum_{p+j=T}\alpha^{\rho}_{pj}s_{z,T,p}\right)^2\right],
+\end{equation}
+where $s=s^{(1/\sqrt3)}$ and empty inner sums are zero. The condition $i=k$ in \eqref{eq:normal3} is exactly the equality of the two values of $T$.
+
+The exact calculation in Appendix~\ref{app:certificate} proves
+\begin{equation}\label{eq:3certificate}
+ e_z<\frac32(3z-1)(-1/2)^z,
+ \qquad z\in\{2,4,5,\ldots,15\}.
+\end{equation}
+There are finitely many strict inequalities, $d_{3,z}/d_a\to3/2$, and the omitted blocks $z=0,1,3$ are annihilated by $W_3$. Consequently, for all sufficiently large $Q$,
+\begin{equation}\label{eq:A3bound}
+ A_3\le\LL_3\left(W_3\sum_{z=0}^{15}q_z(Q)R_{3,z}W_3^\dagger\right).
+\end{equation}
+
+\subsection{The four-body comparison before taking limits}
+Let
+\[
+ \widehat W:V_Q\otimes\bigwedge^2U_Q\longrightarrow\bigwedge^4U_Q
+\]
+be the wedge map. The second tensor factor decomposes into spins $Q-r$, with $r$ odd, $1\le r\le Q$. In its lowering bases use the sign of Lemma~\ref{lem:coupling}; at $r=1$ this is precisely the basis $v_p$.
+
+A copy of spin $2Q-1-D$ is obtained by coupling $a=Q-1$ with $Q-r$ at deficit $D-r$. Denote its vector at total deficit $T$ by $\ket{D,T;r}_Q$. Its physical orbital-index sum is $1+T$, and it is a highest vector when $T=D$. For fixed $D,T$ and sufficiently large $Q$, the allowed copy labels are
+\[
+ \mathcal R_D=\{1,3,5,\ldots\}\cap[1,D],\qquad D\le T.
+\]
+Define the real finite-flux coefficients by
+\[
+ \ket{D,T;r}_Q=\sum_{\substack{p+j+k=T\\j<k}}
+ S_r^Q(D,T;p,j,k)\,v_p\otimes(e_j\wedge e_k),
+\]
+and extend them antisymmetrically in $j,k$. By two applications of Lemma~\ref{lem:coupling}, they converge to
+\begin{equation}\label{eq:S}
+ S_r(D,T;p,j,k)=\sqrt2\,
+ s^{(1/\sqrt2)}_{r,j+k,j}\,
+ s^{(1/\sqrt2)}_{D-r,T-r,p},
+\end{equation}
+interpreted as zero if $r>j+k$. The formula holds for either ordering of $j,k$, by antisymmetry. It follows by applying Lemma~\ref{lem:coupling} first to the two single-particle spins and then to the two pair spins.
+
+Only $1\le D\le23$ contribute: every unaveraged term has
+\[
+ T=p+j+k=q+l+i\le23.
+\]
+Put $d_D=4Q-1-2D$. In the coupled bases $\ket{D,T;r}_Q$, the
+spin-$2Q-1-D$ block is identified with
+$\mathbb C^{d_D}\otimes\mathbb C^{\mathcal R_D}$: the first factor uses the
+$(T-D)$-th normalized spin descendant, and the second has basis vector
+indexed by $r$. Let $E_D^Q$ be the partial trace over the spin factor of
+the compression to this block of the matrix representing \eqref{eq:normal4}
+before wedging, summed over rows. Schur averaging then gives, for all sufficiently large $Q$,
+\begin{equation}\label{eq:A4}
+ A_4=\LL_4\!\left(\widehat W
+   \left[\bigoplus_{D=1}^{23}\frac{d_a}{d_D}
+       (I_{d_D}\otimes E_D^Q)\right]\widehat W^\dagger\right),
+\end{equation}
+where the direct sum acts as zero on all other spin blocks.
+For these finitely many $D$, entrywise convergence gives $E_D^Q\to E_D$, with
+\begin{equation}\label{eq:E}
+ (E_D)_{rs}=-\sum_\rho\sum_{\substack{p,j;q,l\\T\ge D}}
+ \alpha^{\rho}_{pj}\alpha^{\rho}_{ql}
+ S_r(D,T;p,j,k)S_s(D,T;q,l,i),
+\end{equation}
+where $i=p+j-t_\rho$, $k=q+l-t_\rho$, and $T=p+j+k$. The exact finite-flux matrix $E_D^Q$ is given by the same formula with each $S_r$ replaced by $S_r^Q$. Exchanging the two ordered entry pairs exchanges $r$ and $s$, so both matrices are real symmetric.
+
+The target term $\LL_4(W_4W_4^\dagger)$ is obtained from the $r=1$ summand, since that summand is $V_Q\otimes V_Q$. Thus its copy matrix, in the units just used, is $(d_D/d_a)\chi$, where
+\[
+ \chi_{rs}=\mathbf1_{r=s=1},\qquad d_D/d_a\longrightarrow2.
+\]
+For even $D$, the $r=1$ coupled vector is antisymmetric under exchange
+of its two spin-$a$ factors, since its coupling deficit is $D-1$.
+Wedging two degree-two vectors is symmetric, so this target column then
+vanishes after applying $\widehat W$. The formula with $\chi$ still
+represents it exactly; we never assume that the wedged copy vectors are
+independent.
+
+Target spin blocks outside $1\le D\le23$ contribute positively and may be discarded in a lower bound.
+
+Set
+\begin{align}
+ w_{Dr}^Q&=\widehat W\ket{D,D;r}_Q,\label{eq:wQ}\\
+ w_{Dr}^*&=\sum_{\substack{p+j+k=D\\j<k}}
+ S_r(D,D;p,j,k)\,v_p^*\wedge e_j\wedge e_k,\qquad
+ (G_D)_{rs}=\ip{w_{Dr}^*}{w_{Ds}^*}.\label{eq:wstar}
+\end{align}
+The star is a limit label, not an adjoint. Let $W_D^*$ be the map with columns
+$w_{Dr}^*$. With allowance $\eps=3\cdot10^{-6}$, the limiting comparison on
+these four-particle images asks that
+$W_D^*(2\chi-E_D+\eps I)(W_D^*)^\dagger$ be nonnegative.
+Lemma~\ref{lem:compression} identifies this with the exact certificate
+\begin{equation}\label{eq:4certificate}
+ G_D(2\chi-E_D+\eps I)G_D\ge0,
+ \qquad 1\le D\le23.
+\end{equation}
+Appendix~\ref{app:certificate} specifies and verifies it entirely by rational arithmetic.
diff --git a/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/sections/05-transfer.tex b/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/sections/05-transfer.tex
new file mode 100644
index 0000000000000000000000000000000000000000..2ee48bc039d212dffe573770a1d0c55e6142e788
--- /dev/null
+++ b/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/sections/05-transfer.tex
@@ -0,0 +1,232 @@
+\section{A relative error bound for finite spheres}
+\label{sec:transfer}
+The limiting Gram matrices have kernels, so their positivity cannot be
+transferred by assuming that their ranges vary continuously.  Instead, we
+compare the annihilators themselves.  An exact diagonal change of variables
+reduces every error to a fixed finite family of pair-generated annihilators.
+This gives an error relative to $H_Q$, uniformly on the full Fock space.
+
+Geometry-dependent diagonal changes of occupation-number coordinates
+already relate pseudopotential annihilators and their zero modes~\cite[Section~II.E]{OrtizNussinovDukelskySeidel}.
+For a gap estimate, the needed additional statement is quantitative:
+on a fixed set of modes the change of variables approaches the identity,
+and its error is controlled by pair energy. The proof below establishes
+that statement before adding spectator modes and averaging over rotations.
+
+For the local argument, let $\FF_{\mathrm{loc}}$ be the exterior algebra
+of $\mathbb C^{25}$ with orthonormal basis $e_0,\ldots,e_{24}$.
+For $Q\ge24$ we identify it with the Fock space of the first 25 orbital
+modes in $U_Q$. By \eqref{eq:pairstar}, each limiting pair vector $v_p^*$
+with $p\le23$ lies in $\bigwedge^2\mathbb C^{25}$.
+Contraction by these vectors is defined on $\FF_{\mathrm{loc}}$
+by the same exterior-algebra convention as before.
+
+\begin{lemma}[Compression and annihilator bounds]\label{lem:compression}
+Let $W$ be a linear map between finite-dimensional Hilbert spaces,
+$G=W^\dagger W$, and let $C$ be self-adjoint on the domain of $W$. Then
+\[
+ GCG\ge0\quad\Longleftrightarrow\quad WCW^\dagger\ge0.
+\]
+On $\FF_{\mathrm{loc}}$, for any fixed finite collection
+$A_b=c_kc_jB(v_p^*)$ with $0\le p\le23$ and $0\le j,k\le24$,
+\begin{equation}\label{eq:annihilatorbound}
+ \sum_b\norm{A_b\psi}^2\le C_0\ip\psi{h_*\psi},
+ \qquad h_*:=\sum_{p=0}^{23}B(v_p^*)^\dagger B(v_p^*),
+\end{equation}
+where $C_0$ depends only on the collection.
+\end{lemma}
+\begin{proof}
+The two positivity statements both assert that the quadratic form of $C$
+is nonnegative on $\ran G=\ran W^\dagger$.  For the second assertion, use
+$\norm{c_kc_jB(v_p^*)\psi}\le\norm{B(v_p^*)\psi}$ and sum; one may take
+$C_0$ to be the largest multiplicity of a pair label $p$.
+\end{proof}
+
+\Needspace{11\baselineskip}
+\begin{lemma}[Uniform transfer]\label{lem:transfer}
+For $Q\ge24$ and $1\le D\le23$, put
+\[
+ C_D^Q=(d_D/d_a)\chi-E_D^Q+\eps I,
+ \qquad h_Q=\sum_{p=0}^{23}B_p^\dagger B_p.
+\]
+Assuming \eqref{eq:4certificate}, there is a scalar $\rho_Q\ge0$,
+independent of $D$ and tending to zero as $Q\to\infty$, such that
+\begin{equation}\label{eq:relative}
+ \sum_{r,s\in\mathcal R_D}(C_D^Q)_{rs}
+ B(w_{Dr}^Q)^\dagger B(w_{Ds}^Q)
+ \ge-\rho_Qh_Q
+\end{equation}
+on the entire Fock space $\FF_Q$.
+\end{lemma}
+\begin{proof}
+\emph{An exact change of variables.}
+Take $Q\ge24$.  All annihilators in \eqref{eq:relative}, including those
+in $h_Q$, involve only the modes $0,\ldots,24$. First work on
+$\FF_{\mathrm{loc}}$. Define the positive diagonal operator
+\[
+ \Delta_Q\ket A=\left(\prod_{x\in A}d_Q(x)\right)\ket A,
+ \qquad d_Q(x)=\sqrt{(Q)_x/Q^x},
+\]
+on each basis wedge $\ket A$.  This local Fock space is fixed as $Q$
+varies.  Thus $\Delta_Q$ and $\Delta_Q^{-1}$ converge in norm to $I$;
+also $0<\Delta_Q\le I$.
+
+The pair coefficient formula \eqref{eq:pair} gives
+\[
+ v_p(x,y)=r_p(Q)d_Q(x)d_Q(y)v_p^*(x,y),
+ \qquad r_p(Q)=\sqrt{(2Q)^p/(2Q-2)_p}\ge1.
+\]
+Checking the factors attached to the removed orbital labels on a basis
+wedge gives the exact identities
+\begin{align}
+ B_p&=r_p(Q)\Delta_Q^{-1}B(v_p^*)\Delta_Q,
+       \label{eq:pairconjugation}\\
+ B(v_p\wedge e_j\wedge e_k)
+ &=\frac{r_p(Q)}{d_Q(j)d_Q(k)}\,
+   \Delta_Q^{-1}B(v_p^*\wedge e_j\wedge e_k)\Delta_Q.
+       \label{eq:fourconjugation}
+\end{align}
+These are identities on the local Fock space, with the inverse diagonal
+acting on the output particle sector.  Repeated removed labels make both
+sides zero.  In particular, solving \eqref{eq:pairconjugation} for
+$B(v_p^*)\Delta_Q$ and using $r_p(Q)\ge1$ and $\norm{\Delta_Q}\le1$
+gives
+\begin{equation}\label{eq:pairenergytransfer}
+ \Delta_Q h_*\Delta_Q\le h_Q.
+\end{equation}
+
+\emph{The error is controlled by pair energy.}
+Fix $D$.  Index a finite family by
+$b=(p,j,k)$ with $p+j+k=D$ and $j<k$, and put
+\[
+ A_b=B(v_p^*\wedge e_j\wedge e_k),\qquad
+ \beta_{rb}^Q=S_r^Q(D,D;p,j,k)
+                \frac{r_p(Q)}{d_Q(j)d_Q(k)}.
+\]
+Equations \eqref{eq:wQ} and \eqref{eq:fourconjugation} yield
+\[
+ B(w_{Dr}^Q)=\Delta_Q^{-1}
+             \left(\sum_b\beta_{rb}^QA_b\right)\Delta_Q.
+\]
+The coefficients are real, and Lemma~\ref{lem:coupling} gives
+$\beta_{rb}^Q\to\beta_{rb}^*:=S_r(D,D;p,j,k)$.  Hence the finite
+real symmetric matrices
+\[
+ M_Q=(\beta^Q)^TC_D^Q\beta^Q
+ \quad\hbox{converge to}\quad
+ M_*=(\beta^*)^T(2\chi-E_D+\eps I)\beta^*.
+\]
+The quadratic expression in \eqref{eq:relative} is consequently
+\[
+ \Delta_Q\left(\sum_{b,c}(M_Q)_{bc}
+                 A_b^\dagger\Delta_Q^{-2}A_c\right)\Delta_Q.
+\]
+Its limiting expression before the outer diagonals is
+\begin{equation}\label{eq:limitpositive}
+ \sum_{b,c}(M_*)_{bc}A_b^\dagger A_c
+ =\sum_{r,s}(2\chi-E_D+\eps I)_{rs}
+       B(w_{Dr}^*)^\dagger B(w_{Ds}^*)\ge0.
+\end{equation}
+Indeed, the certificate \eqref{eq:4certificate} and
+Lemma~\ref{lem:compression}, applied to the map with columns $w_{Dr}^*$,
+give positivity before lifting; the lift $\LL_4$ preserves positivity.
+
+For clarity, the perturbation estimate does not require $M_*\ge0$.
+Let $R_Q$ be the block matrix with entries
+\[
+ (R_Q)_{bc}=(M_Q)_{bc}\Delta_Q^{-2}-(M_*)_{bc}I.
+\]
+Regard $R_Q$ as an operator on the Hilbert direct sum of one copy of
+$\FF_{\mathrm{loc}}$ for each index $b$.  The diagonal $\Delta_Q^{-2}$
+acts on the output of $A_c$, hence on the sector with four fewer particles;
+using its direct sum over local sectors makes this a single operator
+identity. Its norm tends to zero, since both its index set and the local
+Fock space are fixed. For every vector $\phi$,
+\[
+ \left|\sum_{b,c}\ip{A_b\phi}{(R_Q)_{bc}A_c\phi}\right|
+ \le\norm{R_Q}\sum_b\norm{A_b\phi}^2
+ \le C_0\norm{R_Q}\ip\phi{h_*\phi}.
+\]
+Combine this estimate with \eqref{eq:limitpositive}, take
+$\phi=\Delta_Q\psi$, and apply \eqref{eq:pairenergytransfer}.  This
+proves \eqref{eq:relative} on the local Fock space with a coefficient
+tending to zero.  Taking a maximum over the fixed set $1\le D\le23$
+gives a common $\rho_Q$.
+
+\emph{Additional particles are spectators.}
+Order the 25 local modes before all other modes in the Fock
+factorization.  Under the isometry that sends a local wedge tensored with a spectator
+wedge to their ordered product, each $c_j$ with $j\le24$ is its local
+annihilator tensored with the identity. The fermionic parity factor belongs
+to annihilators in the later spectator modes, which do not occur here.
+Thus every operator in \eqref{eq:relative} acts on the local factor alone.
+Tensoring the local inequality with the identity proves
+it on $\FF_Q$, with the same $\rho_Q$, regardless of the number of
+occupied modes outside this factor.
+\end{proof}
+
+The allowance $\eps I$ in the certificate must also be paid for in pair
+energy.  Here it is useful to sum over the orthonormal highest vectors
+\emph{before} estimating their wedges.  Bessel's inequality then avoids
+an extra factor equal to the number of spin copies.
+
+\begin{proposition}[Four-body estimate]\label{prop:A4}
+For the coefficients in Appendix~\ref{app:certificate},
+\begin{equation}\label{eq:A4bound}
+ A_4\le\LL_4(W_4W_4^\dagger)+(1222\eps+o(1))H,
+ \qquad Q\to\infty,
+\end{equation}
+uniformly on $\FF_Q$.
+\end{proposition}
+\begin{proof}
+Haar averaging the matrix unit
+$\ket{D,D;r}_Q\bra{D,D;s}_Q$ gives the corresponding copy matrix unit
+tensored with the spin identity, divided by $d_D$.  Therefore, on the
+retained spin blocks, the target minus $A_4$, with the $\eps I$ allowance
+added, is
+\[
+ d_a\sum_{D=1}^{23}\int\sum_{r,s}(C_D^Q)_{rs}
+ B(w_{Dr}^Q(g))^\dagger B(w_{Ds}^Q(g))\,dg.
+\]
+Lemma~\ref{lem:transfer}, conjugated by each rotation, bounds this below
+by $-o(1)H$, because Schur averaging gives
+\[
+ d_a\int h_Q(g)\,dg=24H.
+\]
+The number of retained blocks is fixed, so this error remains uniform
+in particle number.
+
+To bound the allowance, fix $D$ and write
+\[
+ \ket{D,D;r}_Q
+   =\sum_{\substack{p+j+k=D\\j<k}}S_{rb}^Q\,
+                     v_p\otimes(e_j\wedge e_k),
+ \qquad b=(p,j,k).
+\]
+The uncoupled vectors on the right are orthonormal.  So are the highest
+vectors on the left: distinct $r$ belong to orthogonal spin summands of
+the second pair factor.  Thus $S^Q(S^Q)^T=I$.  Apply the contraction
+$S^Q\otimes I$ to the Hilbert-space-valued vector
+$(c_kc_jB_p\psi)_b$.  It follows that
+\[
+ \sum_{r\in\mathcal R_D}\norm{B(w_{Dr}^Q)\psi}^2
+ \le\sum_{\substack{p+j+k=D\\j<k}}\norm{c_kc_jB_p\psi}^2
+ \le\sum_{\substack{p+j+k=D\\j<k}}\norm{B_p\psi}^2.
+\]
+This uses orthogonality before wedging; the vectors $w_{Dr}^Q$ themselves
+need not be orthogonal.  The number of triples on the right is
+\[
+ n_D=\sum_{n=1}^D\left\lceil\frac n2\right\rceil
+     =\left\lfloor\frac{(D+1)^2}{4}\right\rfloor.
+\]
+Conjugate by rotations and integrate.  Each pair norm contributes
+$\ip\psi{H\psi}/d_a$, so the total allowance is bounded by
+\[
+ \eps\sum_{D=1}^{23}n_DH
+ =\eps\left(\sum_{m=1}^{12}m^2+
+                 \sum_{m=1}^{11}m(m+1)\right)H
+ =1222\eps H.
+\]
+The discarded target spin blocks are positive.  Removing the allowance
+therefore proves \eqref{eq:A4bound}.
+\end{proof}
diff --git a/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/sections/06-gap.tex b/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/sections/06-gap.tex
new file mode 100644
index 0000000000000000000000000000000000000000..cadd8d2f499095147179bfacd14d97a545406205
--- /dev/null
+++ b/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/sections/06-gap.tex
@@ -0,0 +1,154 @@
+\section{The tail estimate and the gap}
+\label{sec:finish}
+The finite comparisons are now in place. We first bound all remaining three-body blocks by a summable multiple of $H_Q$, then identify the zero mode at Laughlin flux.
+
+\begin{lemma}[Three-body tail]\label{lem:tail}
+For sufficiently large $Q$,
+\begin{equation}\label{eq:tailbound}
+ \LL_3\left(W_3\sum_{z=16}^Qq_z(Q)R_{3,z}W_3^\dagger\right)
+ \ge-\delta_QH,
+\end{equation}
+where
+\begin{equation}\label{eq:delta}
+ \delta_Q=\sum_{\substack{17\le z\le Q\\z\text{ odd}}}
+       \frac{d_{3,z}}{d_a}|q_z(Q)|(z+1),\qquad
+ \delta_Q\longrightarrow\frac{61}{4096}.
+\end{equation}
+\end{lemma}
+\begin{proof}
+For $16\le z\le Q$, the bracket in \eqref{eq:q} is positive, since it is at least $2z-2$. Thus only odd $z$ contribute negatively. A normalized highest vector of $R_{3,z}$ is $\sum_{p=0}^zs_pv_p\otimes e_{z-p}$. Its orbit resolves that spin projector with factor $d_{3,z}$. After wedging and lifting, its quadratic form is
+\[
+ \norm{\sum_{p=0}^zs_pc_{z-p}(g)B_p(g)\psi}^2
+ \le\sum_{p=0}^z\norm{B_p(g)\psi}^2.
+\]
+Averaging each summand proves \eqref{eq:tailbound} with \eqref{eq:delta}.
+
+For $Q\ge4$, the ratios in $(Q)_z/(2Q-2)_z$ are nonincreasing, so
+\[
+ \frac{(Q)_z}{(2Q-2)_z}\le\left(\frac Q{2Q-2}\right)^z
+ \le(2/3)^z,\qquad \frac{d_{3,z}}{d_a}\le2.
+\]
+The summands, extended by zero for $z>Q$, are therefore dominated by
+$6z(z+1)(2/3)^z$. Dominated convergence and \eqref{eq:qlimit} give
+\[
+ \lim_{Q\to\infty}\delta_Q
+ =\frac32\sum_{z=17,19,\ldots}(3z-1)(z+1)2^{-z}
+ =\frac{61}{4096}.
+\]
+For the last identity, write $z=17+2m$. Then
+$(3z-1)(z+1)=900+208m+12m^2$, while the sums of $4^{-m}$,
+$m4^{-m}$, and $m^24^{-m}$ are $4/3$, $4/9$, and $20/27$,
+respectively. Multiplication by $(3/2)2^{-17}$ gives $61/4096$.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{thm:fock}]
+Combine Lemma~\ref{lem:square}, \eqref{eq:Adecomp}, \eqref{eq:A3bound}, Proposition~\ref{prop:A4}, and Lemma~\ref{lem:tail}. They give
+\[
+ H^2\ge d_a\int K(g)\,dg+
+       (1-\eta-\delta_Q-1222\eps-o(1))H.
+\]
+The integral is positive. The certificate gives
+\[
+ \eta=\frac{93527408868499}{10^{14}},\qquad\eps=\frac3{10^6},
+\]
+so the coefficient tends to
+\begin{equation}\label{eq:finalmargin}
+ 1-\frac{93527408868499}{10^{14}}-\frac{61}{4096}
+   -1222\frac3{10^6}
+ =\gamma_*=\frac{4616733319001}{10^{14}}>\frac1{25}.
+\end{equation}
+The limit in \eqref{eq:finalmargin} is $\gamma_*$. For each fixed $0<\gamma<\gamma_*$, the coefficient therefore exceeds $\gamma$ for all sufficiently large $Q$. Every convergence threshold is independent of particle number, proving the assertion on $\FF_Q$.
+\end{proof}
+
+We now recover the known unique zero mode of Haldane's spherical
+parent Hamiltonian~\cite[p.~606, Equations~(6)--(7)]{Haldane}.
+The elementary divisibility proof also fixes the exact flux at which the
+kernel is a line.
+
+\begin{lemma}[Kernel at Laughlin flux]\label{lem:kernel}
+For $N\ge2$ and $Q=3(N-1)$, the kernel of $H_{N,Q}$ is the line spanned by $\Psi_{\mathrm L,N}$.
+\end{lemma}
+\begin{proof}
+Realize $U_Q$ as the homogeneous polynomials of degree $Q$ in spinor coordinates $(u,v)$. Write $b_{ij}=u_iv_j-u_jv_i$. The pair-spin $Q-r$ summand has highest polynomial
+\[
+ b_{ij}^{\,r}u_i^{Q-r}u_j^{Q-r}.
+\]
+Since $b_{ij}$ is invariant under simultaneous rotations of the pair, every vector in this summand is divisible by $b_{ij}^r$. Exchange of the pair has parity $(-1)^r$, so antisymmetric pair states have odd $r$.
+
+If $H_{N,Q}\Psi=0$, then, in the ambient tensor space,
+\[
+ 0=\ip\Psi{H_{N,Q}\Psi}=\sum_{i<j}\norm{P_{ij}^{(1)}\Psi}^2.
+\]
+Thus $P_{ij}^{(1)}\Psi=0$ for every pair. Expanding in the other particles' monomials, the remaining pair sectors all have odd $r\ge3$. Thus $b_{ij}^3$ divides $\Psi$ for every $i<j$. Each bracket $b_{ij}=v_ju_i-u_jv_i$ is primitive as a polynomial in $u_i$, since $v_j$ and $u_jv_i$ are coprime, and is linear over the fraction field of the other variables. Gauss's lemma therefore makes it irreducible in the polynomial ring. Brackets for distinct unordered pairs are nonassociate. They are consequently relatively prime. Unique factorization implies
+\[
+ \prod_{i<j}b_{ij}^3\mid\Psi.
+\]
+This product has degree $3(N-1)=Q$ in each particle's spinor variables, exhausting the multidegree of $\Psi$; the quotient is constant.
+
+Conversely, $\Psi_{\mathrm L,N}$ is antisymmetric. Its factor $b_{ij}^3$ leaves pair degrees $Q-3$ in each spinor, whose maximum total pair spin is $Q-3$. Multiplication by the invariant bracket cannot introduce spin $Q-1$. Hence every pair projector annihilates $\Psi_{\mathrm L,N}$.
+\end{proof}
+
+\begin{proof}[Proof of Corollary~\ref{thm:main}]
+Apply Theorem~\ref{thm:fock} with $\gamma=1/25$. On each particle sector,
+the spectral theorem gives
+\[
+ H_{N,Q}\ge\frac1{25}\bigl(I-P_{\ker H_{N,Q}}\bigr)
+\]
+once $Q$ exceeds the common threshold. Take $Q=3(N-1)$ and apply
+Lemma~\ref{lem:kernel}. Choosing $N_0$ so that $3(N_0-1)$ exceeds the
+threshold proves the claim.
+\end{proof}
+
+The following comparison specializes the recursive spectral relations of
+Lemm, Nachtergaele, Warzel, and Young
+\cite[arXiv v1, Theorem~2.1(I), Lemma~2.3, and (2.10)]{LemmNachtergaeleWarzelYoung2026Recursive}.
+
+\begin{corollary}[Charge versus neutral gap at fixed flux]\label{cor:charge-neutral}
+For \(N\ge2\), fix \(Q_N=3(N-1)\) and write
+\(E_0(n;Q_N)=\min\operatorname{spec}H_{n,Q_N}\) for the coefficient-one
+interaction \eqref{eq:physicalH}.  Let \(\operatorname{Gap}_N\) be the
+first positive eigenvalue of \(H_{N,Q_N}\), its neutral gap.
+The particle-number charge gap at this same flux,
+\[
+ \Delta_N=E_0(N+1;Q_N)+E_0(N-1;Q_N)-2E_0(N;Q_N),
+\]
+satisfies
+\[
+ \Delta_N=E_0(N+1;Q_N)
+       \ge\frac{N}{N-1}\operatorname{Gap}_N>0.
+\]
+In particular, \(\Delta_N\ge N/[25(N-1)]\) for \(N\ge N_0\), with
+\(N_0\) as in Corollary~\ref{thm:main}.
+\end{corollary}
+
+\begin{proof}
+All three sectors are nonempty since \(N+1\le\dim U_{Q_N}=3N-2\).
+Lemma~\ref{lem:kernel} gives \(E_0(N;Q_N)=0\) and a one-dimensional
+ground space.  The \(N\)-particle sector has dimension greater than one,
+so \(\operatorname{Gap}_N>0\).
+For a nonzero Laughlin vector \(\Psi\), each pair annihilator \(B_p\)
+commutes with every \(c_j\), while
+\(\sum_j\|c_j\Psi\|^2=N\|\Psi\|^2\).  Thus some \(c_j\Psi\) is a nonzero
+\((N-1)\)-particle zero mode, giving \(E_0(N-1;Q_N)=0\).
+The divisibility argument in Lemma~\ref{lem:kernel} excludes a zero mode
+in the \((N+1)\)-particle sector: the required product of cubed brackets
+would have degree \(3N>Q_N\) in each particle variable.
+
+Let \(P_n\) project onto the lowest-energy eigenspace of \(H_{n,Q_N}\),
+and define the overlap operator on the \((N+1)\)-particle sector by
+\[
+ \mathcal G_N=P_{N+1}\left(\sum_{j=0}^{Q_N}c_j^\dagger P_Nc_j\right)P_{N+1}.
+\]
+Here \(P_{N+1}\) projects onto a positive-energy ground eigenspace.
+For the pure two-body fermionic form \eqref{eq:fockH}, the cited
+Theorem~2.1(I), with \(m_0=m_1=2\), gives
+\[
+ E_0(N+1;Q_N)\ge\frac{N+1}{N-1}E_0(N;Q_N)
+   +\frac{N+1-\|\mathcal G_N\|}{N-1}\operatorname{Gap}_N.
+\]
+The cited Lemma~2.3 gives \(\|\mathcal G_N\|\le\operatorname{rank}P_N=1\).
+Since \(E_0(N;Q_N)=0\), this yields
+\(E_0(N+1;Q_N)\ge N\operatorname{Gap}_N/(N-1)\).
+The two zero energies give the asserted second difference, and
+Corollary~\ref{thm:main} supplies the final uniform bound.
+\end{proof}
diff --git a/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/sections/07-plane.tex b/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/sections/07-plane.tex
new file mode 100644
index 0000000000000000000000000000000000000000..101b6ed314171c2e74d515f4192692770b498f48
--- /dev/null
+++ b/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/sections/07-plane.tex
@@ -0,0 +1,86 @@
+\section{A planar corollary}
+\label{sec:plane}
+
+The full-Fock-space estimate also gives a gap on the plane. We use the
+analytic lowest-Landau-level representation described by Girvin and
+Jach~\cite[Section~II]{GirvinJach}, with Gaussian scale fixed below.
+Let
+$\mathcal B$ be the Bargmann space of entire functions with norm
+\[
+ \|f\|_{\mathcal B}^2=\frac1\pi\int_{\mathbb C}|f(z)|^2e^{-|z|^2}\,d^2z,
+ \qquad e_j(z)=\frac{z^j}{\sqrt{j!}}.
+\]
+Multiplication by $\pi^{-1/2}e^{-|z|^2/2}$ identifies this space unitarily
+with the planar lowest Landau level at magnetic field $2$; changing magnetic
+length is a unitary dilation.
+For a pair of coordinates set $w=(z_1-z_2)/\sqrt2$ and
+$Z=(z_1+z_2)/\sqrt2$.  Let $P^{(1)}$ be the orthogonal projector onto
+relative degree one, with no restriction on the degree in $Z$, and put
+\[
+ H_N^{\mathrm{pl}}=\left.\sum_{1\le i<j\le N}P_{ij}^{(1)}
+                         \right|_{\bigwedge^N\mathcal B}.
+\]
+Thus each pair projector has coefficient one, as on the sphere.
+
+\begin{corollary}[Planar spectral gap]\label{cor:plane}
+For every $N\ge2$, the untruncated planar Hamiltonian satisfies
+\[
+ (H_N^{\mathrm{pl}})^2\ge\gamma_* H_N^{\mathrm{pl}},
+\]
+where $\gamma_*$ is the constant in Theorem~\ref{thm:fock}.
+Equivalently, if $P_{0,N}^{\mathrm{pl}}$ projects onto its kernel, then
+$H_N^{\mathrm{pl}}\ge\gamma_*(I-P_{0,N}^{\mathrm{pl}})$.
+The constant is independent of both particle number and total angular
+momentum.  In particular, the case $\ell=3$ of the spectral gap conjecture
+in \cite[Conjecture~A.1]{Rougerie} holds, in its pair-projector normalization.
+\end{corollary}
+\begin{proof}
+The coordinate change $(z_1,z_2)\mapsto(w,Z)$ is unitary for the
+Bargmann inner product.  Consequently the vectors
+$wZ^p/\sqrt{p!}$, $p\ge0$, form an orthonormal basis of the relative-degree-one
+subspace.  Their coefficients in the normalized wedge basis are
+\[
+ v_p^{\mathrm{pl}}(i,j)
+   =(i-j)\sqrt{\frac{p!}{2^p i!j!}}\,\mathbf1_{i+j=p+1},
+ \qquad i<j.
+\]
+Indeed, expanding $(z_1-z_2)(z_1+z_2)^p$
+first gives the coefficient in the tensor basis; passage to the normalized
+antisymmetric wedge multiplies it by $\sqrt2$.  These are exactly the
+limits in \eqref{eq:pairstar}.  The second-quantized pair-projector identity
+therefore gives
+$H_N^{\mathrm{pl}}=\sum_{p\ge0}B(v_p^{\mathrm{pl}})^\dagger B(v_p^{\mathrm{pl}})$
+on antisymmetric polynomials.
+
+Fix $N$ and a total degree $L$.  The corresponding homogeneous subspace
+has the finite orthonormal basis
+\[
+ e_{i_1}\wedge\cdots\wedge e_{i_N},\qquad
+ 0\le i_1<\cdots<i_N,\qquad \sum_{k=1}^N i_k=L.
+\]
+For $Q\ge\max\{L,N-1,2\}$ this is also the entire spherical subspace of total
+lowering deficit $L$, under the identification of normalized one-particle
+bases.  Both Hamiltonians preserve this subspace: a pair is annihilated
+and created with the same sum of orbital labels.  Only $p+1\le L$
+can contribute.  The finitely many matrix entries of the spherical
+restriction therefore converge to those of the planar restriction by
+\eqref{eq:pair}--\eqref{eq:pairstar}; the convergence is in operator norm.
+
+Fix $0<\gamma<\gamma_*$.  Theorem~\ref{thm:fock} gives
+$H_Q^2\ge\gamma H_Q$ for all sufficiently large $Q$.
+Restrict to the invariant block just described and let $Q\to\infty$.
+Norm convergence passes through the square and gives
+$(H_N^{\mathrm{pl}})^2\ge\gamma H_N^{\mathrm{pl}}$ on that block.
+Now let $\gamma\uparrow\gamma_*$; the cone of positive matrices is closed,
+so the same inequality holds at $\gamma_*$.  The limit $Q\to\infty$ is taken
+separately for each fixed $\gamma$, allowing the flux threshold to depend
+on $\gamma$.  Here $N$ and $L$ were arbitrary, and the resulting constant is the
+same for all of them.
+
+Finally, $\bigwedge^N\mathcal B$ is the Hilbert direct sum of its homogeneous
+subspaces, and $0\le H_N^{\mathrm{pl}}\le\binom N2 I$.  The block inequalities
+thus extend by continuity to the whole space.  The spectral theorem gives
+the asserted gap.  Restriction to total degrees
+$L\le3N(N-1)/2$ gives exactly the sector in the cited conjecture for
+$\ell=3$.
+\end{proof}
diff --git a/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/sections/08-arithmetic.tex b/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/sections/08-arithmetic.tex
new file mode 100644
index 0000000000000000000000000000000000000000..34fd61b949f15491125fa2015454fd9bfc8d87f7
--- /dev/null
+++ b/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/sections/08-arithmetic.tex
@@ -0,0 +1,152 @@
+\section{The finite rational certificate}
+\label{app:certificate}
+This appendix specifies the seven rows and all arithmetic needed to verify
+\eqref{eq:3certificate} and \eqref{eq:4certificate}. Starting from the integer
+rows in Section~\ref{app:rows}, evaluate the rational three-body traces in
+Section~\ref{app:three-arithmetic} and the rational four-body matrices in
+Section~\ref{app:four-arithmetic}. The margin and coefficient-sign tables in
+those sections record the resulting checks. The formulas below determine
+every value using only integers and rational numbers.
+
+\subsection{Row data}
+\label{app:rows}
+Let $P=10^7$. The first two numbers of each row are $t$ and $\ell=P\lambda$. An entry $pj:a$ specifies the integer $a_{pj}$, with $p$ and $j$ parsed as separate single-digit labels; omitted entries are zero. Continuation lines belong to the preceding row. Define
+\begin{equation}\label{eq:alphadata}
+ \alpha_{pj}=\frac{a_{pj}}P
+       \sqrt{\frac{2^p\binom{p+j}{p}}{2^t\binom{p+j}{t}}}
+ =\frac{a_{pj}}P\sqrt{\frac{2^{p-t}t!(p+j-t)!}{p!j!}}.
+\end{equation}
+Every nonzero entry has $0\le p+j-t\le8$.
+\par\noindent\begin{minipage}{\linewidth}
+\lstinputlisting{../verification/rows.txt}
+\end{minipage}\par
+\smallskip
+\noindent In particular $P^2\eta=\sum_\rho\ell_\rho^2=93527408868499$.
+
+\subsection{Three-body arithmetic}
+\label{app:three-arithmetic}
+For $c\in\{1,2\}$, an integer $T\ge0$, and integers $z,p$, define
+\begin{equation}\label{eq:U}
+ \mathsf U(c,z,T,p)=
+ \sum_{h=\max(0,p+z-T)}^{\min(p,z)}
+ (-1)^{z-h}\binom zh\binom{T-z}{p-h}c^{p-h},
+\end{equation}
+and set it to zero if $z$ or $p$ lies outside $[0,T]$. Expanding \eqref{eq:coupling} gives
+\[
+ s^{(1/\sqrt3)}_{z,T,p}=\mathsf U(2,z,T,p)
+ \sqrt{\frac{2^z\binom Tz}{3^T2^p\binom Tp}}.
+\]
+Substitution in \eqref{eq:ez} and \eqref{eq:alphadata} cancels all radicals:
+\begin{equation}\label{eq:erational}
+ P^2e_z=\sum_\rho\sum_{T=\max(z,t_\rho)}^{15}
+ \frac{2^z\binom Tz}{3^T2^{t_\rho}\binom T{t_\rho}}
+ X_{\rho,T}\left(X_{\rho,T}+2\ell_\rho\mathsf U(2,z,T,t_\rho)\right),
+\end{equation}
+where $X_{\rho,T}=\sum_{p+j=T}a^\rho_{pj}\mathsf U(2,z,T,p)$, with value zero on empty levels. For each $z\in\{2,4,5,\ldots,15\}$, evaluate \eqref{eq:erational} and form the exact rational margin
+$m_z=\frac32(3z-1)(-1/2)^z-e_z$. Write $10^6m_z=n_z/d_z$ with integers $n_z,d_z$ and $d_z>0$. Integer division gives the floors $\lfloor n_z/d_z\rfloor$ listed below:
+\begin{center}
+\begin{tabular}{rrrrrrrrrrrrrr}
+\toprule
+$z$&2&4&5&6&7&8&9&10&11&12&13&14&15\\
+\midrule
+floor&1298313&211317&249&250&249&250&250&250&249&250&250&6470&250\\
+\bottomrule
+\end{tabular}
+\end{center}
+All are positive, proving \eqref{eq:3certificate}.
+
+\subsection{Four-body arithmetic}
+\label{app:four-arithmetic}
+Fix $1\le D\le23$; all copy indices below lie in $\mathcal R_D$. Put
+\[
+ u_r=\frac1{r!(D-r)!},\qquad
+ (E_D)_{rs}=\sqrt{u_ru_s}\,Y_{rs},\qquad
+ (G_D)_{rs}=\sqrt{u_ru_s}\,Z_{rs}.
+\]
+The matrices $Y,Z$ are rational. Define
+\[
+ \mathsf V_r(D,T;p,j,k)=
+ \mathsf U(1,r,j+k,j)\mathsf U(1,D-r,T-r,p),
+\]
+with value zero for $r>j+k$. Two applications of \eqref{eq:coupling} give
+\begin{equation}\label{eq:Srational}
+ S_r(D,T;p,j,k)=\mathsf V_r(D,T;p,j,k)
+ \sqrt{2^{1-(j+k)-T+r}\frac{j!k!p!\,u_r}{(T-D)!}}.
+\end{equation}
+Therefore
+\begin{equation}\label{eq:Yrational}
+ \begin{split}
+ P^2Y_{rs}=-\sum_\rho\sum_{\substack{p,j;q,l\\T\ge D}}
+ &a^\rho_{pj}a^\rho_{ql}\,
+ \mathsf V_r(D,T;p,j,k)\mathsf V_s(D,T;q,l,i)\\
+ &\times2^{1-2T+p+q-t_\rho+(r+s)/2}
+       \frac{i!k!t_\rho!}{(T-D)!},
+ \end{split}
+\end{equation}
+where $i=p+j-t_\rho$, $k=q+l-t_\rho$, and $T=p+j+k$. Both entry sums are ordered. To verify the cancellation, multiply the two factors in \eqref{eq:Srational} by the two factors \eqref{eq:alphadata}; the factorial product under the combined square root is
+$(t_\rho!i!k!)^2u_ru_s/((T-D)!)^2$.
+Using $j+k=T-p$ and $l+i=T-q$, its power of two has exponent
+$2+2p+2q-2t_\rho-4T+r+s$. Taking the positive square root and
+removing $\sqrt{u_ru_s}$ gives exactly the displayed factor.
+
+For an increasing quadruple $A$ of nonnegative indices with sum $D+1$, put $f(A)=\prod_{b\in A}b!$. If $(x,y,j,k)$ lists its elements, let $\sigma(x,y,j,k)$ be the sign of sorting this list into increasing order. In each summand below put $p=x+y-1$, and define
+\begin{equation}\label{eq:Lrational}
+ L_r(A)=2^{(1-2D+r)/2}
+ \sum_{\substack{x<y\text{ in }A\\\{j,k\}=A\setminus\{x,y\},\ j<k}}
+ \sigma(x,y,j,k)(x-y)\,p!j!k!\,
+ \mathsf V_r(D,D;p,j,k).
+\end{equation}
+Combining \eqref{eq:pairstar} and \eqref{eq:Srational}, the coefficient of $w_{Dr}^*$ on the Slater wedge $A$ is $\sqrt{u_r/f(A)}L_r(A)$. Thus
+\begin{equation}\label{eq:Zrational}
+ Z_{rs}=\sum_{\substack{A\text{ increasing quadruple}\\\sum_{b\in A}b=D+1}}
+                   \frac{L_r(A)L_s(A)}{f(A)}.
+\end{equation}
+All exponents of two in these formulas are integers, since $r,s$ are odd.
+
+Let $D_u=\diag(u_r)$ and $D_{\sqrt u}=\diag(\sqrt{u_r})$. The matrix in \eqref{eq:4certificate} equals $D_{\sqrt u}MD_{\sqrt u}$, where
+\begin{equation}\label{eq:M}
+ M=ZBZ,\qquad
+ B=\diag\bigl((2\mathbf1_{r=1}+\eps)u_r\bigr)-D_uYD_u.
+\end{equation}
+Thus it suffices to check $M\ge0$ using rational arithmetic. This
+congruence uses only the positive diagonal $D_{\sqrt u}$; it does not
+invert $G_D$ or presume that its columns are independent.
+
+To check positivity for each $D=1,\ldots,23$, form $Y$ and $Z$ from
+\eqref{eq:Yrational} and \eqref{eq:Zrational}, then form $B$ and $M$ from
+\eqref{eq:M}, using $\eps=3/10^6$. All sums range over the printed rows and
+the finite index sets specified above. Compute with exact fractions
+throughout. If $d=|\mathcal R_D|$, initialize $J=I_d$. For
+$b=1,\ldots,d$, compute in order
+\begin{equation}\label{eq:recurrence}
+ X=MJ,\qquad c_b=\frac{\tr X}{b},\qquad J=c_bI_d-X.
+\end{equation}
+This is the Leverrier--Faddeev recurrence applied to $-M$;
+see \cite{HouConference}. It gives
+$\det(xI+M)=x^d+c_1x^{d-1}+\cdots+c_d$. For example, this recurrence follows by multiplying the polynomial adjugate of $xI+M$ by $xI+M$ and comparing coefficients, together with the derivative-of-determinant trace identity. Exact evaluation of \eqref{eq:U}, \eqref{eq:Yrational}, \eqref{eq:Lrational}, \eqref{eq:Zrational}, and \eqref{eq:recurrence} gives:
+\begin{center}
+\begin{tabular}{rl@{\qquad}rl@{\qquad}rl}
+\toprule
+$D$&signs&$D$&signs&$D$&signs\\
+\midrule
+1&\texttt{0}&9&\texttt{++000}&17&\texttt{++++++000}\\
+2&\texttt{0}&10&\texttt{+0000}&18&\texttt{+++++0000}\\
+3&\texttt{00}&11&\texttt{+++000}&19&\texttt{+++++++000}\\
+4&\texttt{00}&12&\texttt{++0000}&20&\texttt{++++++0000}\\
+5&\texttt{+00}&13&\texttt{++++000}&21&\texttt{++++++++000}\\
+6&\texttt{000}&14&\texttt{+++0000}&22&\texttt{+++++++0000}\\
+7&\texttt{+000}&15&\texttt{+++++000}&23&\texttt{+++++++++000}\\
+8&\texttt{+000}&16&\texttt{++++0000}&&\\
+\bottomrule
+\end{tabular}
+\end{center}
+The signs list the coefficients after the leading $1$, in descending degree.
+For each computed coefficient, choose a positive denominator and inspect
+its integer numerator: \texttt{+} denotes a positive numerator and
+\texttt{0} denotes zero. We use the characteristic-coefficient positivity test for symmetric
+matrices~\cite[Theorem~5]{PeyrlParrilo}. Every coefficient is nonnegative,
+so $\det(xI+M)>0$ for $x>0$. Since $M$ is real symmetric, a negative eigenvalue would give a positive root of this polynomial. Therefore $M\ge0$, proving \eqref{eq:4certificate}.
+
+\begin{remark}[Nature of the certificate]
+The data above certify fixed spin-matrix inequalities. The positive comparison operator is a sum of squares by construction. No many-particle finite-size spectrum, numerical extrapolation, or assumed plasma property enters the argument. The exact equalities and inequalities in this appendix can be verified by a finite calculation using only integers and rational numbers.
+\end{remark}
diff --git a/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/verification/rows.txt b/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/verification/rows.txt
new file mode 100644
index 0000000000000000000000000000000000000000..eb01ff68d5554a223e77b6c30cc12f9794766d85
--- /dev/null
+++ b/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/verification/rows.txt
@@ -0,0 +1,20 @@
+0 7181518 05:-849651 07:-2958744 15:751659 16:-43084 17:-146216 32:720949 35:-127514
+  42:222437 43:-262822 53:-117700 61:-55024 71:-166049
+1 -112991 05:142772 06:-142342 07:527055 08:50742 16:197917 18:-18343 24:5764 27:-21183
+  32:75832 35:-6115 45:2660 53:-9455 61:-27091 63:-3494 71:479 72:-15331
+1 -4247311 05:-12207250 06:-14064191 07:-14063663 08:-15662679 16:483831 18:3987214
+  24:-814558 27:888887 32:966462 35:145978 45:-171722 53:34350 61:389319 63:97603
+  71:412447 72:-53786
+2 3438746 14:6689266 15:6620205 16:6625065 17:4969272 18:4878238 27:-1387380 36:-586188
+  37:344087 42:-1120634 45:-318530 46:559169 55:157455 61:1207103 63:211617 64:33240
+  71:577001 73:-142578
+5 -98645 05:-2719776 08:-8519913 14:-949523 17:-1952105 18:-1562606 28:-8919966 32:248826
+  35:-1017623 37:3656005 38:-2081871 42:-33163 43:-305905 46:35677 47:-1320749
+  48:-1040378 53:502094 55:1060633 58:-998003 61:-113617 63:184919 64:209239 65:-68198
+  67:-266297 72:-606696 73:-563620 74:437345 76:-432189
+5 -3160276 05:-1698129 08:10575875 14:-252576 17:-8002616 18:22592844 28:7370640
+  32:289294 35:1400647 37:1987740 38:4030699 42:-1731401 43:-1034062 46:-1106371
+  47:-1257051 48:1461735 53:1408149 55:1136847 58:1878501 61:-2838516 63:-734099
+  64:1062416 65:11745 67:611312 72:-480776 73:20459 74:316925 76:37092
+7 1441784 07:3441286 08:-6097155 37:-1002399 47:-251321 58:1143225 66:-17397 71:-1007464
+  73:-662286 76:-344621 78:-1115284
diff --git a/preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/README.md b/preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..69a465a830053b2f19d8664b1fc55d32e9e63503
--- /dev/null
+++ b/preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/README.md
@@ -0,0 +1,18 @@
+# [A Fully Polynomial Randomized Approximation Scheme for Perfect Matchings in General Graphs](main.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 23, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026,
+  author = {{OpenAI}},
+  title = {{A Fully Polynomial Randomized Approximation Scheme for Perfect Matchings in General Graphs}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/main.pdf}{OAI:A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/build/main.tex b/preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/build/main.tex
new file mode 100644
index 0000000000000000000000000000000000000000..4af2d19d7c5da08e9c30b953bef0c088b8945323
--- /dev/null
+++ b/preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/build/main.tex
@@ -0,0 +1,3471 @@
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+\usepackage[nameinlink,noabbrev,capitalise]{cleveref}
+
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
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+
+\AddToHook{env/theorem/begin}{\crefalias{theorem}{theorem}}
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+\hypersetup{pdftitle={A Fully Polynomial Randomized Approximation Scheme for Perfect Matchings in General Graphs},pdfauthor={OpenAI}}
+\title{A Fully Polynomial Randomized Approximation Scheme\\
+for Perfect Matchings in General Graphs}
+\author{OpenAI}
+\date{September 23, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We give a fully polynomial randomized approximation scheme (FPRAS) for
+counting perfect matchings in arbitrary finite simple undirected graphs,
+resolving the general-graph perfect-matching approximation problem.
+The algorithm returns zero with certainty when no perfect matching exists.
+Otherwise, it achieves relative error $\eps$ with failure probability at
+most $\delta$ in worst-case bit time polynomial in the input length,
+$\eps^{-1}$, and $\log\delta^{-1}$.
+\end{abstract}
+\tableofcontents
+\section{Introduction}
+\label{sec:introduction}
+
+Let $G=(V,E)$ be a finite simple undirected graph. A perfect matching is a
+set of edges incident to each vertex exactly once. Write
+\[
+ Z(G)=\#\{M\subseteq E:M\text{ is a perfect matching of }G\}.
+\]
+A fully polynomial randomized approximation scheme approximates $Z(G)$
+within a prescribed relative error, with a prescribed success probability,
+in time polynomial in the input length and the reciprocal error.
+The confidence parameter enters through its logarithm. We prove the
+following theorem, including a bound on the time of every execution.
+
+\begin{theorem}\label{thm:main}
+There is a uniform classical randomized algorithm which, given $G$ and
+rational parameters $0<\eps<1$ and $0<\delta<1/2$, returns a nonnegative
+rational number $\widehat Z$ such that
+\[
+ \Prb\bigl[(1-\eps)Z(G)\le \widehat Z\le(1+\eps)Z(G)\bigr]
+ \ge 1-\delta.
+\]
+If $Z(G)=0$, the output is zero with certainty. The worst-case bit running
+time is polynomial in the input encoding length, $\eps^{-1}$, and
+$\log\delta^{-1}$.
+\end{theorem}
+
+As consequences, \cref{sec:prescribed-degree} gives counting schemes for
+edge subsets with prescribed vertex degrees and for matchings of a
+specified size. On feasible inputs, its samplers always return a
+feasible object and approximate the corresponding uniform law in total
+variation.
+
+\subsection{History and significance}
+Perfect matchings illustrate the distinction between finding a
+combinatorial structure and counting its realizations. Edmonds gave a
+polynomial-time algorithm for finding a maximum matching in a general
+graph~\cite{Edmonds1965}. Valiant proved that exact counting is
+\#P-complete even for bipartite graphs, where it is the permanent of a
+zero--one matrix~\cite{Valiant1979}. Pfaffian methods give
+polynomial-time exact counting for planar graphs~\cite{Kasteleyn1963}.
+
+For approximation, three problems must be distinguished: counting all
+matchings, counting bipartite perfect matchings, and counting perfect
+matchings in general graphs. Jerrum and Sinclair gave an FPRAS for the
+total number of matchings in an arbitrary graph, summed over all
+cardinalities~\cite[Corollary~4.5]{JerrumSinclair1989}.
+Their perfect-matching algorithm applied when the ratio of near-perfect
+matchings, which leave exactly two vertices unmatched, to perfect
+matchings was bounded by a fixed polynomial~\cite[Theorem~5.3]{JerrumSinclair1989}.
+Without such a bound, perfect matchings can carry too little stationary
+mass for this sampling approach. Approximating the total number of
+matchings therefore does not isolate its perfect-matching contribution
+with relative accuracy.
+
+Jerrum, Sinclair, and Vigoda removed the ratio restriction in the
+bipartite setting and obtained an FPRAS for the permanent of every
+nonnegative matrix~\cite[Theorem~1.1]{JerrumSinclairVigoda2004}.
+Their method assigns weights to near-perfect states according to their
+two unmatched vertices and learns suitable weights while gradually
+changing edge activities. The general-graph question was explicitly
+raised by Jerrum and Sinclair~\cite[Section~7(i)]{JerrumSinclair1989}
+and remained a benchmark for approximate counting. Cai and Liu relate
+it to approximation for the eight-vertex model~\cite{CaiLiu2020};
+Fei, Goldberg, and Lu distinguish the spin-system parameter line
+equivalent to general perfect-matching approximation from the region
+covered by their FPRAS~\cite[Proposition~1.2 and Theorem~1.5]{FeiGoldbergLu2025}.
+\Cref{thm:main} answers the general-graph perfect-matching approximation
+problem affirmatively.
+
+A direct extension of the bipartite chain faces a precise obstruction.
+\v{S}tefankovi\v{c}, Vigoda, and Wilmes constructed graphs on which every
+JSV-type chain has either exponentially small stationary probability
+of perfect matchings or exponentially slow mixing, even with arbitrary
+weights depending on the hole pattern~\cite[Definition~2.1 and
+Theorem~2.2 in the full version]{StefankovicVigodaWilmes2018}.
+Here JSV type means the perfect-and-near-perfect state space with the
+specified Broder proposals and a Metropolis acceptance rule. The
+sampler below uses a different state space: a product of
+perfect-matching spaces on an enlarged colored graph, with exchanges
+of alternating cycles between coordinates. Its energy comparison is
+proved for these states and moves.
+
+Subsequent permanent algorithms sharpen the bipartite result.
+Bez\'{a}kov\'{a}, \v{S}tefankovi\v{c}, Vazirani, and Vigoda accelerated the
+cooling schedule and improved the analysis of the matching
+chain~\cite{BezakovaStefankovicVaziraniVigoda2008}.
+Chen, Vigoda, and Yang obtain a further improvement through restricted
+Poincar\'e inequalities and coupled flows~\cite{ChenVigodaYang2026};
+Chen, Guo, Vigoda, and Yang develop an energy-weighted
+multicommodity-flow comparison and improve the running time
+again~\cite{ChenGuoVigodaYang2026}. These results concern bipartite
+perfect matchings.
+
+For nonbipartite graphs, Yi gives deterministic relative approximation
+for hafnians, the weighted sums over perfect matchings, when the
+$n$-vertex support has minimum degree at least $(1/2+\gamma)n$ and
+nonzero weights lie in $[\theta,1]$~\cite[Theorem~1.1]{Yi2026}.
+Here $0<\gamma<1/2$ and $0<\theta\le1$ are fixed constants, and the
+polynomial running-time exponent may depend on them.
+\Cref{thm:main} has no density promise on its simple unweighted input.
+
+The proof below also builds on the methodological framework of weighted
+matching chains. Alternating overlays and weight-preserving switches
+underlie the encodings of Jerrum and Sinclair and of Jerrum, Sinclair,
+and Vigoda~\cite{JerrumSinclair1989,JerrumSinclairVigoda2004}.
+Their sampling-to-counting strategy learns local partition ratios while
+gradually changing activities.
+Here these ratios are exposed by marked edges in an enlarged graph,
+and vertex scaling maintains a row-wise balance condition. The
+capacity bound, subdivision and marked-edge identities needed for this
+construction are proved explicitly.
+
+Energy comparison through weighted demands has classical antecedents
+in multicommodity-flow bounds for mixing~\cite{Sinclair1992}.
+Our comparison differs from routing every demand along a path of chain
+transitions: long-arc conversions enter a signed identity and need not
+be legal single updates. The resulting estimate initially controls
+only additive functions of two matching coordinates. The separate
+product-space argument is essential to obtain a spectral gap for
+arbitrary functions. It uses the classical orthogonal decomposition
+into coordinate interactions~\cite[Section~2]{EfronStein1981}; the
+ordered assignment of residual terms and the replication estimate are
+proved below.
+
+\subsection{Proof architecture}
+
+The counting algorithm starts with the complete graph, whose perfect
+matchings are easy to count, and gradually suppresses the nonedges of
+$G$ by decreasing their positive activities. It estimates the ratios
+of successive partition functions and multiplies them. To keep the
+observations statistically useful, it updates vertex scales from
+estimates of the two-hole partition ratios: partition functions after
+deleting two vertices, divided by the current partition function.
+Accurate estimates keep each row maximum between two absolute
+constants. A vertex scale
+multiplies every complete matching weight by the same factor, preserving
+the probability law used to estimate the current counting ratio.
+
+The purpose of balance is to control the extra mass introduced by an
+auxiliary graph. We replace each logical edge by a weighted path, with
+a bijection between logical perfect matchings and matchings using only
+these real path edges. We then add weak clique edges in overlapping
+bags indexed by a tree. Under balance, the added edges increase the
+partition function by at most a factor of two, so a sample uses only
+real edges with probability at least one half. A specially marked
+auxiliary edge exposes the corresponding two-hole partition ratio.
+These are exact event identities; they provide both the counting
+estimate and the data for the next scale update.
+
+To prove the inflation bound, \cref{thm:logical-hole-bound} assigns a
+pairing capacity to each even hole set using the strongest bottleneck
+paths between its vertices. The capacity pays for the weights of
+possible added edges on those holes. Subdivision transfers this bound
+to the real graph while making incident effective strengths comparable.
+The resulting geometry supplies the tree of clique bags. Thus the
+hole estimates control the probability of the observable events,
+while the tree-bag structure supplies a sampler for their law.
+
+That sampler uses local switches in one matching and exchanges
+alternating cycles between two matching coordinates. Its energy
+comparison initially controls only additive functions of the two
+coordinates. A quadrangulation adapted to the bag labels gives an exact
+signed identity for a whole-cycle difference
+(\cref{lem:quadrangulation,thm:pair-energy}). The identity is not a
+route through chain transitions: long arc conversions may change many
+edges. Two-matching encodings instead compare its terms with local
+switches and cycle exchanges, with polynomially bounded multiplicity.
+
+A ladder of nearby weight distributions then places many independent
+coordinates at each tier. An ordered orthogonal decomposition ensures
+that each coordinate-interaction term is charged to at most one pair.
+Replication reduces the coefficient of their combined contribution
+enough to absorb it. This yields a spectral gap for
+arbitrary functions of the product state (\cref{lem:replica-gap}).
+The bottom tier has unit activities and admits exact counting and
+ideal sampling by a tree dynamic program. The sampler requires no
+balance assumption. It therefore remains defined, with the same time
+bound, after an inaccurate empirical estimate; balance is used only
+to prove that accurate estimates propagate through the counting stages.
+
+\Cref{tab:graph-laws} summarizes the changes of law. Finally,
+\cref{sec:counting,sec:complexity} account for all arithmetic and
+finite-bit random choices. Every polynomial has an absolute degree;
+the constants are chosen for transparent estimates rather than
+practical efficiency.
+
+\begin{table}[ht]
+\centering
+\small
+\begin{tabular}{@{}p{.18\textwidth}cp{.61\textwidth}@{}}
+\toprule
+Object & Vertices & Relation of matching laws \\
+\midrule
+Input $G$ & $n$ & The target is its unweighted count $Z(G)$. \\
+Complete graph & $n$ & Raw activities $w^{(j)}$ gradually suppress the nonedges of $G$. \\
+Vertex scaling & $n$ & Activities $\lambda$ preserve the normalized raw matching law. \\
+Real paths & $N$ & Subdivision gives a matching bijection with a common weight multiplier. \\
+Colored bags & $N$ & Their activities $W$ recover the real law when every sampled edge is revealed as real. \\
+Weight tiers & $N$ & Common-interval clamps of $W$ connect the unit-activity law to the bag law on one colored state space. \\
+\bottomrule
+\end{tabular}
+\caption{The successive graphs and probability laws. Here $n$ is the
+original vertex count and $N$ the enlarged count. Scaling, subdivision,
+conditioning, and clamping have distinct roles.}
+\label{tab:graph-laws}
+\end{table}
+
+The specialized combinatorial and sampling arguments are proved here.
+We use Edmonds's algorithm for the initial matching-existence test
+and Hoeffding's bounded-variable inequality for concentration and
+confidence amplification~\cite{Edmonds1965,Hoeffding1963}.
+\section{Preliminaries}
+\label{sec:preliminaries}
+
+Logarithms are natural unless a base is indicated.
+
+\subsection{Weighted matchings and holes}
+We allow parallel edges distinguished by colors. A matching specifies
+the actual colored edges it uses. For positive rational activities
+$a=(a_e)$, put
+\[
+ \wt_a(M)=\prod_{e\in M}a_e,\qquad
+ Z_a(-U)=\sum_{M\in\calM(H-U)}\wt_a(M),
+ \qquad Z_a=Z_a(-\varnothing),
+\]
+where $H$ is the underlying graph and $\calM(H-U)$ is its set of perfect
+matchings after deleting $U$. The empty graph has one perfect matching,
+of weight one. Whenever $Z_a>0$, define
+\[
+ g_a(U)=\frac{Z_a(-U)}{Z_a},\qquad
+ \pi_a(M)=\frac{\wt_a(M)}{Z_a}.
+\]
+These weighted colored graphs are intermediate spaces constructed from
+the simple input graph. Their representation is explicit: vertices,
+colored edges, and rational activities are stored individually. The
+main theorem retains the finite simple undirected input model; it does
+not assert a counting algorithm for compressed edge multiplicities or
+for singleton-covering loops.
+
+We call the vertices in $U$ \emph{holes}. For distinct vertices we abbreviate
+$g_a(\{i_1,\ldots,i_k\})$ to $g_a(i_1\cdots i_k)$.
+Collapsing parallel activities into their
+sum preserves every partition function. Conditional on a collapsed
+edge, its original type can be recovered with probabilities
+proportional to the summands.
+
+The union of two matchings, with common edges retained twice, has
+maximum degree two. Its nontrivial components are alternating paths
+and even alternating cycles. Swapping the layers along a component
+preserves the product of their weights. In the union of two perfect
+matchings, deleting common identical edges leaves disjoint alternating
+cycles; different colors on the same pair of vertices form a cycle
+of length two. We fix a vertex order and a color order once and for
+all. They resolve every combinatorial tie in the proof.
+
+\subsection{Bottleneck similarities}
+For a positive weighted connected graph define
+\begin{equation}\label{eq:bottleneck}
+ B_{ij}=\max\left\{1,\ \max_{P:i\leadsto j}\ \min_{e\in P}a_e\right\}
+ \quad (i\ne j).
+\end{equation}
+Simple paths suffice. This symmetric similarity satisfies
+$B_{ij}\ge\min\{B_{ik},B_{kj}\}$: concatenating two paths and erasing
+loops cannot lower their minimum activity. For $t>1$, its equivalence
+classes are the connected components of the graph retaining edges of
+activity at least $t$. Singletons are included. At $t=1$ we take one
+class containing every vertex, consistently with the floor at one.
+For an even set $U$ define its \emph{pairing capacity}
+\begin{equation}\label{eq:pairing-capacity}
+ \Phi_B(U)=\max_{\mathcal P}\prod_{ij\in\mathcal P}B_{ij},
+ \qquad \Phi_B(\varnothing)=1,
+\end{equation}
+where $\mathcal P$ runs over pairings of $U$. We use this quantity only
+in proofs; the counting algorithm never needs to enumerate pairings
+or compute $\Phi_B(U)$ for a general hole set.
+
+\subsection{Tree bags}
+\begin{definition}\label{def:tree-bags}
+A \emph{tree-bag graph} consists of a finite rooted tree of labels and
+a finite even vertex set. A label $l$ at level $k$ has height $h_l=2^k$;
+the root has level zero and a child has level one greater than its
+parent. Each vertex belongs to either one label or two adjacent
+labels. The bag at $l$ is the set of vertices belonging to $l$.
+For each unordered pair of distinct vertices and each common label
+$l$, there is one edge of color $l$. There are no other edges.
+For an integer parameter $D\ge1$, its reference activities obey
+\begin{equation}\label{eq:bag-weight-range}
+ \frac{h_l}{D}\le W_{xy,l}\le3h_l.
+\end{equation}
+We also consider activities obtained by clamping all $W_e$ to one
+common interval $[a,b]\subset(0,\infty)$.
+\end{definition}
+
+Every vertex pair has at most two edge colors. At any bag, the
+interfaces with its parent and its different children are disjoint:
+a vertex in two such interfaces would have at least three memberships.
+Empty bags and repeated underlying vertex sets are permitted.
+Clamping is monotone and does not increase the ratio of a larger
+activity to a smaller one. In particular, reference activities of
+nearby labels remain comparable after clamping.
+
+\subsection{Dirichlet energy and approximation}
+For a finite reversible Markov kernel $P$ with stationary law $\mu$,
+its Dirichlet energy is
+\begin{equation}\label{eq:dirichlet}
+ \calE_{\mu,P}(f)
+ =\frac12\sum_{x,y}\mu(x)P(x,y)(f(x)-f(y))^2.
+\end{equation}
+A Poincar\'e inequality $\Var_\mu f\le\gamma^{-1}\calE_{\mu,P}(f)$
+means that the spectral gap is at least $\gamma$. A symmetric proposal
+$Q$ has Metropolis acceptance probability~\cite{MetropolisEtAl1953,Hastings1970} $\min\{1,\mu(y)/\mu(x)\}$
+for a proposed move from $x$ to $y$. Its off-diagonal stationary capacity is
+\begin{equation}\label{eq:metropolis-capacity}
+ \mu(x)P(x,y)=Q(x,y)\min\{\mu(x),\mu(y)\}.
+\end{equation}
+We regard inapplicable proposals as holding moves. All state spaces
+used below contain at least one perfect matching; thus their positive
+activities define probability laws with full support on that space.
+
+Total variation is
+$\|\mu-\rho\|_{\TV}=\frac12\sum_x|\mu(x)-\rho(x)|$.
+A deterministic map, or a common randomized postprocessing kernel,
+does not increase total variation, by the triangle inequality.
+Two finite laws at total variation distance $\eta$ admit a coupling
+that disagrees with probability $\eta$: first match the common masses
+$\min(\mu(x),\rho(x))$, then couple the remainders. Applying this
+construction successively bounds the discrepancy probability of an
+adaptive simulation by the sum of its conditional one-step errors.
+\section{Controlling the cost of unmatched vertices}
+\label{sec:holes}
+
+This section proves the bound that will control the additional edges in
+our sampling graph.  It uses only alternating paths and differentiation
+of a finite partition function.
+
+Let $V$ have even cardinality $n\ge2$, and give every edge of the complete
+logical graph $K_V$ a positive activity $\lambda_{ij}$. Use the weighted
+matching notation of \cref{sec:preliminaries}, abbreviating
+$\calM(K_V-U)$ to $\mathcal M(-U)$. For any positive activity array $a$
+on this graph, define the row maxima
+\[
+ r_i(a)=\max_{j\ne i}g_a(ij).
+\]
+The activities $a$ are \emph{balanced} if $1/4\le r_i(a)\le1$ for
+every $i$. Let $B$ be the bottleneck similarity of $\lambda$ from
+\eqref{eq:bottleneck}, and let $\Phi_B$ be its pairing capacity from
+\eqref{eq:pairing-capacity}.
+
+The estimate we need controls every even hole set, although balance
+only bounds the two-hole ratios:
+
+\begin{theorem}[Logical hole bound]
+\label{thm:logical-hole-bound}
+Let $\lambda$ be balanced, let $B$ be its bottleneck similarity, and set
+\[
+ D_0=10^8(n+1)^4.
+\]
+For every even set $U\subseteq V$,
+\begin{equation}
+\label{eq:logical-hole-bound}
+ g_\lambda(U)\Phi_B(U)\le 2D_0^{|U|/2}.
+\end{equation}
+\end{theorem}
+
+To prove this, we will add a small activity proportional to $B_{ij}$
+on every pair. Bounds for two and four holes keep the resulting
+partition inflation below two. Any pairing of an arbitrary hole set
+can then be inserted using the added edges, giving the stated bound.
+
+We use two elementary alternating-path comparisons. Such weight-preserving
+overlays also underlie the weighted matching encodings in
+\cite{JerrumSinclair1989,JerrumSinclairVigoda2004}; the comparisons and
+their constants needed here are proved below. In an overlay of
+two matchings, edges retain their layer.  Common edges can be kept as
+two parallel copies.  A path starting at a vertex unmatched in exactly
+one layer is then uniquely determined, and switching its edges between
+the layers preserves the product of their weights.
+
+\begin{lemma}[Four-hole inequality]
+\label{lem:four-holes}
+For any positive activities $\mu$ and four distinct vertices $a,b,c,d$,
+\[
+ g_\mu(abcd)\le
+ g_\mu(ab)g_\mu(cd)+g_\mu(ac)g_\mu(bd)
+                         +g_\mu(ad)g_\mu(bc).
+\]
+Consequently, if every row maximum is at most $4$, then
+\begin{equation}
+\label{eq:small-hole-bounds}
+ g_\mu(\varnothing)=1,\qquad
+ g_\mu(U)\le4\quad(|U|=2),\qquad
+ g_\mu(U)\le48\quad(|U|=4).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Overlay $M\in\mathcal M(-\{a,b,c,d\})$ and
+$P\in\mathcal M(-\varnothing)$.  The alternating path from $a$ ends at
+one of $b,c,d$, say $b$.  Switch that path between the layers.  The
+output layers miss $\{c,d\}$ and $\{a,b\}$, respectively, and their
+product weight is unchanged.  This map is injective for the specified
+endpoint $b$: in the output overlay, follow the path from $a$ and switch
+it back.  The same argument applies to endpoints $c,d$.
+Summing the three injective comparisons and dividing by $Z_\mu^2$
+proves the inequality.  The remaining assertions follow immediately.
+\end{proof}
+
+The four-hole comparison controls small hole sets by row maxima. We
+next need an additional factor when two holes are joined by a strong
+path. Normalizing by the row maximum at the moving hole will make
+the successive relocation errors add, without multiplying along the
+path.
+
+The next lemma is stated with $\mu\ge\lambda$ so that it continues to
+apply while we add activities to the graph.  Its bottleneck similarity
+always remains the one defined from $\lambda$.
+
+\begin{lemma}[Relocating a hole along a strong path]
+\label{lem:hole-relocation}
+Let $B$ be defined by \eqref{eq:bottleneck}, and suppose that
+$\mu_e\ge\lambda_e$ for every edge and
+\[
+ \frac18\le r_i(\mu)\le4\qquad(i\in V).
+\]
+If $|U|\in\{2,4\}$ and $i,j\in U$ are distinct, then
+\begin{equation}
+\label{eq:hole-relocation-bound}
+ g_\mu(U)\le\frac{10^5n}{B_{ij}}.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Throughout this proof, omit the subscript $\mu$ from $g$ and $r$.
+Write $B=B_{ij}$.  For $B\le8$, the assertion follows from
+\eqref{eq:small-hole-bounds}.  Suppose therefore that $B>8$, and choose
+a simple path from $i$ to $j$ whose $\lambda$-activities, and hence its
+$\mu$-activities, are all at least $B$.
+
+We first give one step of the relocation.  Let $a$ be the hole currently
+on this path, let $S$ be the current hole set, and let $z$ be the next
+vertex.  Thus $a\in S$ and $\mu_{az}\ge B$.
+If $z\in S$, inserting $az$ in a matching missing $S$ is injective and
+increases its weight by $\mu_{az}$.  Hence
+\begin{equation}
+\label{eq:adjacent-hole-fill}
+ \frac{g(S)}{r_a}\le
+ \frac{g(S\setminus\{a,z\})}{B r_a}\le\frac{32}{B}.
+\end{equation}
+The last inequality uses $|S|\le4$, \eqref{eq:small-hole-bounds}, and
+$r_a\ge1/8$.
+
+Now suppose $z\notin S$.  Choose $p\ne z$ with $g(zp)=r_z$.
+Inserting $az$ gives $g(az)\le1/\mu_{az}\le1/B<1/8$, so $p\ne a$.
+Overlay $M\in\mathcal M(-S)$ and $P\in\mathcal M(-\{z,p\})$.
+The path from $a$ starts with an edge of $P$.  Its endpoint belongs to
+exactly one of the following three cases.
+\begin{enumerate}[label=\textup{(\roman*)},leftmargin=*]
+\item It ends at $z$.  Switching the path gives layers missing
+$S\setminus\{a\}\cup\{z\}$ and $\{a,p\}$.
+The product weight is unchanged.
+\item It ends at $p$, which requires $p\notin S$.
+Switching gives layers missing $S\setminus\{a\}\cup\{p\}$ and
+$\{a,z\}$.  Insert $az$ in the second layer, making it perfect.
+The product weight increases by $\mu_{az}\ge B$.
+\item It ends at $x\in S\setminus\{a,p\}$.
+Switching gives layers missing $S\setminus\{a,x\}$ and
+$\{z,p,a,x\}$.  Insert $az$ in the second layer; its remaining holes
+are $\{p,x\}$.  Again the product weight increases by at least $B$.
+\end{enumerate}
+These cases exhaust the possibilities.  Indeed, the degree-one
+vertices in the overlay are exactly the symmetric difference of the
+two hole sets; if $p\in S$, then $p$ is isolated and cannot be an
+endpoint of the path.
+
+The edge $az$ is absent from both input layers, since the first misses
+$a$ and the second misses $z$. Switching cannot create an edge absent
+from the overlay. Each map is injective when its case and endpoint are specified.
+For case (i), the inverse switches the output path from $a$.
+For cases (ii) and (iii), first remove the inserted edge $az$ from the
+second output layer, then switch the path from $a$ in the resulting
+overlay.  This reconstructs both input layers, including their common
+edges.  Summing over all inputs, using $g(ap)\le r_a$, and dividing by
+$Z_\mu^2r_z$ yields
+\begin{equation}
+\label{eq:relocation-step}
+\begin{split}
+ g(S)\le{}&\frac{r_a}{r_z}
+                g(S\setminus\{a\}\cup\{z\})\\
+ &+\frac{1}{B r_z}\left[
+   \mathbf 1_{\{p\notin S\}}g(S\setminus\{a\}\cup\{p\})
+   +\sum_{x\in S\setminus\{a,p\}}
+        g(S\setminus\{a,x\})g(px)\right].
+\end{split}
+\end{equation}
+An inapplicable first term inside the brackets is omitted entirely.
+The brackets are at most $96$: for $|S|=4$ their first term is at most
+$48$ and their at most three product terms are each at most $16$;
+for $|S|=2$ the corresponding bound is $8$.
+After division by $r_a$, therefore,
+\begin{equation}
+\label{eq:normalized-relocation}
+ \frac{g(S)}{r_a}\le
+ \frac{g(S\setminus\{a\}\cup\{z\})}{r_z}
+       +\frac{6144}{B}.
+\end{equation}
+
+Starting at $a=i$, apply this recurrence along the chosen path, keeping
+all holes in $U\setminus\{i\}$ fixed.  Stop just before reaching the
+first of these fixed holes; such a hole exists because the path ends
+at $j$.  The path is simple, so there are at most $n-2$ recurrence
+steps, followed by \eqref{eq:adjacent-hole-fill}.  The normalization by
+the current row maximum makes the leading terms telescope.  Since
+$r_i\le4$,
+\[
+ g(U)\le\frac{4\bigl(6144(n-2)+32\bigr)}{B}
+       \le\frac{10^5n}{B},
+\]
+as required.
+\end{proof}
+
+
+\begin{proof}[Proof of \cref{thm:logical-hole-bound}]
+Temporarily add a distinguished parallel edge of activity $tB_{ij}/D_0$
+to every pair $ij$, where $0\le t\le1$.  Collapsing parallel edges gives
+activities
+\[
+ \mu_{ij}(t)=\lambda_{ij}+tB_{ij}/D_0.
+\]
+Write $Z(t)=Z_{\mu(t)}$ and $g_t=g_{\mu(t)}$; $B$ remains fixed.
+All partition functions involved are polynomials in $t$, and $Z(t)>0$.
+Differentiation of their finite sums gives
+\begin{equation}
+\label{eq:partition-derivative}
+ \frac{d}{dt}\log Z(t)=\frac1{D_0}\sum_{ij}B_{ij}g_t(ij),
+\end{equation}
+where sums over edges are unordered, and for distinct $x,y$,
+\begin{equation}
+\label{eq:hole-derivative}
+ \frac{d}{dt}g_t(xy)
+ =\frac1{D_0}\sum_{ij:\{i,j\}\cap\{x,y\}=\varnothing}
+                  B_{ij}g_t(xyij)
+       -g_t(xy)\frac{d}{dt}\log Z(t).
+\end{equation}
+
+On any initial time interval during which all row maxima belong to
+$[1/8,4]$, Lemma~\ref{lem:hole-relocation} bounds every summand
+$B_{ij}g_t(ij)$ and every summand $B_{ij}g_t(xyij)$ by $10^5n$.
+Since
+\[
+ \frac{\binom n2\,10^5n}{D_0}<\eta,
+ \qquad \eta=10^{-3},
+\]
+we obtain, throughout that interval,
+\[
+ 0\le\frac{d}{dt}\log Z(t)\le\eta,
+ \qquad
+ -\eta g_t(xy)\le\frac{d}{dt}g_t(xy)\le\eta.
+\]
+Integrating gives
+\[
+ e^{-\eta t}g_0(xy)\le g_t(xy)\le g_0(xy)+\eta t.
+\]
+For the lower row bound, fix a neighbor maximizing the row at time zero;
+the moving row maximum is at least this one evolving entry.
+Consequently, balance at time zero implies
+\[
+ \frac14e^{-\eta}\le r_i(\mu(t))\le1+\eta.
+\]
+Both bounds lie strictly inside $[1/8,4]$.  By continuity, the row maxima
+cannot first leave this interval before time $1$: at a proposed first
+exit the displayed strict bounds still hold and persist on a
+neighborhood.  Thus all preceding estimates hold on $[0,1]$, and
+\begin{equation}
+\label{eq:logical-inflation}
+ \frac{Z(1)}{Z(0)}\le e^\eta<2.
+\end{equation}
+
+Choose a pairing $Q$ of $U$ attaining $\Phi_B(U)$.  At time $1$, keep the
+original and added parallel edges distinguished.  To each original
+matching in $\mathcal M(-U)$, add the edges of $Q$ using their
+\emph{added} copies, of activities $B_{ij}/D_0$.  This is an injective
+map into perfect matchings at time $1$, and it multiplies every weight
+by $\Phi_B(U)/D_0^{|U|/2}$.  Hence
+\[
+ Z_\lambda(-U)\frac{\Phi_B(U)}{D_0^{|U|/2}}\le Z(1).
+\]
+Divide by $Z(0)=Z_\lambda$ and apply
+\eqref{eq:logical-inflation}.
+\end{proof}
+\section{A subdivision with controlled holes}
+\label{sec:enlargement}
+
+The sampler will use additional clique edges. To control the mass these
+edges introduce, we first replace each logical edge by a path. The
+path profile has two roles: equal adjacent factors make the ratio of
+its two matching patterns telescope, and the gradual change from each
+endpoint to the middle makes incident effective strengths comparable.
+We first construct the enlarged graph and identify its marked laws,
+then bound the mass introduced by its additional edges.
+
+Throughout this section the logical graph is the complete graph on an
+even number $n\ge2$ of vertices, with positive rational activities
+$\lambda_{uv}\le4^K$, where $K\ge0$ is an integer. Write
+$B$ for its bottleneck similarity, and put
+\begin{equation}\label{eq:enlargement-parameters}
+ H_u=\max_{v\ne u}B_{uv},\qquad
+ p=2K+2,\qquad
+ N=n+4p\binom n2,\qquad D=100N^2D_0.
+\end{equation}
+Here $D_0=10^8(n+1)^4$ is the constant in
+\cref{thm:logical-hole-bound}. Thus $1\le B_{uv}\le H_u\le4^K$.
+
+\subsection{The path profile}
+
+Orient each logical edge $e=uv$ by the fixed vertex order, and replace
+it by a path
+\[
+ x_0=u,x_1,\ldots,x_{4p},x_{4p+1}=v.
+\]
+Internal vertices of different paths are distinct. Denote the activity
+of $x_{j-1}x_j$ by $t_j=t_j^e$. For $1\le r\le p$, set
+\begin{align}
+ t_{2r-1}=t_{2r}
+   &=\max\{B_{uv},H_u2^{-(r-1)}\},\notag\\
+ t_{4p+3-2r}=t_{4p+2-2r}
+   &=\max\{B_{uv},H_v2^{-(r-1)}\},\label{eq:path-profile}\\
+ t_{2p+1}&=\lambda_{uv}.\notag
+\end{align}
+We call these the \emph{real} edges and write $Z'$ and $g'$ for their
+partition function and hole ratios. Let $B'$ be their bottleneck
+similarity. The two innermost equal pairs have activity $B_{uv}$,
+because $H_u2^{-(p-1)},H_v2^{-(p-1)}\le1/2$.
+
+\begin{lemma}\label{lem:subdivision}
+The real graph has $N$ vertices. Its perfect matchings are in
+bijection with logical perfect matchings, and
+\begin{equation}\label{eq:real-partition}
+ Z'=C_0 Z_\lambda,
+ \qquad
+ C_0=\prod_e\prod_{\substack{1\le j\le4p+1\\j\text{ even}}}t_j^e.
+\end{equation}
+On a noncentral real edge, its effective strength $B'$ equals its
+activity. On the central edge of the path for $uv$, it equals
+$B_{uv}$. Incident effective strengths differ by a factor of at most
+two. For an internal vertex $x_j$, its maximum incident effective
+strength is
+\[
+ T_j=
+ \begin{cases}
+ t_j,&1\le j\le2p,\\
+ t_{j+1},&2p+1\le j\le4p.
+ \end{cases}
+\]
+Its \emph{home terminal} is $u$ in the first case and $v$ in the
+second. Every edge on its leg to its home has activity at least $T_j$.
+\end{lemma}
+
+\begin{proof}
+An intact path has exactly two ways to match its internal vertices:
+the even-indexed edges, which use neither terminal, or the
+odd-indexed edges, which use both. The ratio of the latter weight to
+the former is $\lambda_{uv}$, since all the noncentral factors cancel
+in equal adjacent pairs. A real perfect matching therefore chooses
+the odd tiling precisely on a logical perfect matching. This proves
+the bijection and \cref{eq:real-partition}.
+
+Every noncentral edge has an equal-activity twin sharing an internal
+vertex of degree two. A path connecting its endpoints that avoids
+that edge must use the twin. Its bottleneck therefore cannot exceed
+the edge activity, and the direct edge attains that activity.
+
+The minimum activity on the full path replacing $e$ is $\lambda_e$.
+Consequently, for every $t>1$, real edges of activity at least $t$
+connect two terminals exactly when logical edges of activity at least
+$t$ connect them: a simple path between terminals traverses complete
+subdivision paths. In particular $B'$ restricted to terminals is $B$.
+Let $a=x_{2p}$ and $b=x_{2p+1}$ be the ends of the central edge.
+The legs from $a,b$ to $u,v$ have minimum activity $B_{uv}$.
+If $B_{uv}>\max\{1,\lambda_{uv}\}$, a logical path attaining
+$B_{uv}$ avoids $uv$, and its subdivision together with these legs
+connects $a$ to $b$ with bottleneck $B_{uv}$. If
+$B_{uv}=\max\{1,\lambda_{uv}\}$, the central edge and the floor at
+one already give the lower bound. For the upper bound, any
+alternative path from $a$ to $b$ must first use an edge on a leg
+of activity $B_{uv}$. Thus $B'_{ab}=B_{uv}$ in all cases.
+
+Along each half of a path the displayed profile is nonincreasing
+toward the center and successive values differ by at most two.
+The effective strength at the center agrees with both innermost
+pairs. At a terminal $u$, every first edge has activity $H_u$.
+These facts give all remaining assertions.
+\end{proof}
+
+\subsection{The enlarged matching law}
+
+For each integer $0\le k\le\lfloor\log_2\max_uH_u\rfloor$, take
+one label for every component of the enlarged graph whose retained
+edges have effective strength at least $2^k$, including singleton
+components. A label at level $k>0$ has as parent its containing
+component at level $k-1$. At level zero there is one component;
+these labels therefore form a rooted tree. Components recurring at
+different levels are still different labels.
+
+Label each real edge $xy$ by the component containing it at level
+$\lfloor\log_2B'_{xy}\rfloor$. A vertex belongs precisely to the
+labels of its incident real edges. Equal-level incident labels are
+identical, and the factor-two bound in \cref{lem:subdivision} implies
+that there are at most two incident levels. If there are two, they
+are consecutive and their labels are parent and child. Thus these
+memberships satisfy \cref{def:tree-bags}. In particular, a bag is
+the set of vertices with that membership; it need not be the entire
+threshold component defining its label.
+
+Within every bag of label $l$ and height $h_l$, add a virtual edge
+of activity $h_l/D$ for each vertex pair. Collapse any real edge
+of color $l$ into the virtual edge with the same endpoints and
+color, adding its activity. This gives one colored edge per common
+label, of activity
+\[
+ W_{xy,l}=h_l/D+
+ \begin{cases}
+ t_{xy},&xy\text{ is a real edge with label }l,\\
+ 0,&\text{otherwise}.
+ \end{cases}
+\]
+Since a real edge assigned to $l$ has
+$t_{xy}\le B'_{xy}<2h_l$, these weights satisfy
+\cref{eq:bag-weight-range}. Also
+$1/D\le W_{xy,l}\le3\cdot4^K$.
+
+For each logical edge $e$, split off activity $1/D$ from the
+virtual activity on its central colored edge, and mark that portion
+as the \emph{probe for $e$}. This is possible because $h_l\ge1$.
+After sampling a colored matching, independently reveal the portion
+of each used edge in proportion to its real, probe, and remaining
+virtual activities; zero portions are omitted.
+
+\Cref{fig:subdivision-probe-example} shows the two real tilings and
+the probe on one illustrative path.
+\begin{figure}[ht]
+\centering
+\begin{tikzpicture}[x=.59cm,y=1cm,
+  reference/.style={draw=black!25,line width=.45pt},
+  selected/.style={draw=black,line width=1.5pt},
+  vertex/.style={circle,fill=black,inner sep=1.1pt}]
+  \foreach \row/\height in {even/3.6,odd/1.8,probe/0} {
+    \foreach \i in {0,...,17} {
+      \coordinate (\row\i) at (\i,\height);
+    }
+    \foreach \j [evaluate=\j as \i using int(\j-1)] in {1,...,17}
+      \draw[reference] (\row\i)--(\row\j);
+  }
+  \foreach \j [evaluate=\j as \i using int(\j-1)] in {2,4,...,16}
+    \draw[selected] (even\i)--(even\j);
+  \foreach \j [evaluate=\j as \i using int(\j-1)] in {1,3,...,17}
+    \draw[selected] (odd\i)--(odd\j);
+  \foreach \j [evaluate=\j as \i using int(\j-1)] in {1,3,5,7,11,13,15,17}
+    \draw[selected] (probe\i)--(probe\j);
+  \draw[draw=black,densely dotted,line width=1.8pt]
+    (probe8)--(probe9) node[midway,above=4pt] {$1/D$};
+  \foreach \row in {even,odd,probe} {
+    \foreach \i in {0,...,17}
+      \node[vertex] at (\row\i) {};
+    \node[below=3pt] at (\row0) {$u$};
+    \node[below=3pt] at (\row8) {$x_8$};
+    \node[below=3pt] at (\row9) {$x_9$};
+    \node[below=3pt] at (\row17) {$v$};
+  }
+  \foreach \i in {0,17}
+    \draw[fill=white,line width=.7pt] (even\i) circle[radius=1.7pt];
+  \foreach \j [evaluate=\j as \i using int(\j-1)] in {1,...,17} {
+    \ifnum\j<3
+      \node[font=\scriptsize,above=3pt] at ($(even\i)!.5!(even\j)$) {$4$};
+    \else\ifnum\j=9
+      \node[font=\scriptsize,above=3pt] at ($(even\i)!.5!(even\j)$) {$\tfrac12$};
+    \else
+      \node[font=\scriptsize,above=3pt] at ($(even\i)!.5!(even\j)$) {$2$};
+    \fi\fi
+  }
+  \node[anchor=east] at (-.9,3.6) {even tiling};
+  \node[anchor=east] at (-.9,1.8) {odd tiling};
+  \node[anchor=east] at (-.9,0) {sole probe};
+\end{tikzpicture}
+\caption{One subdivision path with $K=1$, $p=4$ and $4p+1=17$ edges.
+The top labels are the real activities: $H_u=4$,
+$H_v=B_{uv}=2$, and $\lambda_{uv}=1/2$; the central edge has
+effective strength $B'_{x_8x_9}=2$. Solid edges show the selected
+real tiling. The even tiling uses neither terminal, and the odd tiling
+uses both; the odd-to-even weight ratio is $\lambda_{uv}$.
+The last row uses the probe portion of the central colored bag edge
+and real portions elsewhere. Here the central label has height $h_l=2$,
+so its collapsed activity $1/2+2/D$ consists of real activity $1/2$,
+probe activity $1/D$, and remaining virtual activity $1/D$.
+The probe forces the real edges on the two outer legs to use both
+terminals. Their combined weight equals the even-tiling weight,
+so the remaining paths represent a logical matching with $u,v$ deleted.}
+\label{fig:subdivision-probe-example}
+\end{figure}
+
+Let $Z_{\mathrm{bag}}$ be the colored partition function of the enlarged
+graph, and put $I=Z_{\mathrm{bag}}/Z'$. This ratio measures the mass
+introduced by the auxiliary edges.
+
+\begin{proposition}[Marked matching laws]\label{prop:marked-laws}
+For arbitrary positive logical activities, sample from the bag matching
+law and reveal the edge portions as above. The all-real event has
+probability $1/I$;
+conditional on this event, the decoded logical matching has law
+$\pi_\lambda$. For each logical edge $uv$,
+\begin{equation}\label{eq:probe}
+ \Prb(\text{probe for }uv\text{ is used and all other edges are real})
+ =\frac{g_\lambda(uv)}{DI}.
+\end{equation}
+Conditional on this event, the residual logical matching has the
+$\lambda$-weighted perfect-matching law on the vertices other than
+$u,v$.
+\end{proposition}
+
+\begin{proof}
+Revealing portions recovers the weighted law before the real and virtual
+activities were collapsed. The all-real matchings have total weight
+$Z'=C_0Z_\lambda$, and their normalized law is transported to
+$\pi_\lambda$ by \cref{lem:subdivision}.
+
+Now require the probe on the path for $uv$ and real edges everywhere
+else. The probe matches $x_{2p}$ to $x_{2p+1}$. The remaining vertices on
+its path have unique real tilings: the left one uses $u$, and the right
+one uses $v$. Equal adjacent activities in \eqref{eq:path-profile}
+show that their combined weight is exactly the even-edge baseline
+weight of this path. Other paths incident to $u$ or $v$ must use their
+even tilings. The remaining paths retain their even and odd alternatives,
+with ratio $\lambda_e$. Consequently these marked
+matchings are in bijection with logical matchings missing $u,v$,
+and every weight is multiplied by $C_0/D$. Their total mass is
+$C_0 Z_\lambda(-uv)/D$. Dividing by $Z_{\mathrm{bag}}=IC_0Z_\lambda$
+proves \eqref{eq:probe}; the same constant weight multiplier proves
+the conditional-law assertion.
+\end{proof}
+
+The identities show what the enlarged graph provides to the counting
+algorithm. To use its all-real event with polynomially many samples,
+we also need a uniform lower bound on that event's probability.
+The next lower-bound estimate requires balance; the construction and
+the exact laws above do not.
+
+\begin{proposition}[Bounded inflation]\label{prop:bag-inflation}
+If the logical activities are balanced, then
+\begin{equation}\label{eq:inflation}
+ 1\le I\le2.
+\end{equation}
+\end{proposition}
+
+We prove this bound in the remainder of the section. A collection of
+virtual edges leaves a set $U$ of endpoints to be removed from a real
+matching. Its added weight must therefore be compared with $g'(U)$.
+The next two subsections transfer the logical hole estimate to these
+enlarged hole sets.
+
+\subsection{Exact accounting for deleted vertices}
+
+We next describe exactly what deleting internal vertices does to the
+logical graph. A path is \emph{broken} if at least one of its internal
+vertices is deleted. Let $E_{\mathrm{br}}(U)$ be the set of logical
+edges whose paths are broken by a hole set $U$.
+
+Consider the ordered internal holes on a broken path. If a real
+completion exists, successive hole indices alternate in parity,
+because the intervening internal run must have even order. A first
+hole of even index forces the prefix to match the terminal $u$;
+we say that this hole \emph{consumes} $u$ and map it to $u$.
+A last hole of odd index similarly consumes and maps to $v$.
+Remove all consumed holes (zero, one, or two) from the list. The remaining holes
+split, in their order, into pairs $(j,k)$ with $j$ odd and $k$ even;
+call these \emph{neutral pairs}. There is no ambiguity when the
+list has one hole, since it cannot both have even and odd index.
+
+Let $R$ be the set of explicitly deleted terminals together with all
+consumed terminals. Feasibility requires these terminals to be
+distinct. To each consumed hole $x_j$ assign the factor
+\[
+ f_j=
+ \begin{cases}
+ 1,&\text{if it maps to its home terminal},\\
+ \lambda_e/T_j,&\text{if it maps to the other terminal}.
+ \end{cases}
+\]
+For a neutral pair $(j,k)$ assign
+\begin{equation}\label{eq:neutral-factors}
+ f_{jk}=
+ \begin{cases}
+ T_j^{-1},&j,k\le2p,\\
+ T_k^{-1},&j,k\ge2p+1,\\
+ \lambda_e/(T_jT_k),&j\le2p<k.
+ \end{cases}
+\end{equation}
+Let $F=F(U)$ be the product of all these factors on all broken paths;
+an empty product is one. Both $R$ and $F$ depend only on $U$ and
+the fixed path profiles, not on a choice of completion.
+
+For a set $E_0$ of logical edges, write
+$Z_\lambda(-R;E_0\text{ forbidden})$ for the partition function
+after deleting $R$ and forbidding those edges.
+
+\begin{lemma}\label{lem:deletion-accounting}
+If an even hole set $U$ has a real completion, then $R$ is even,
+$|R|\le|U|$, and
+\begin{equation}\label{eq:deletion-identity}
+ Z'(-U)
+ =C_0 F(U)
+ Z_\lambda\bigl(-R;E_{\mathrm{br}}(U)\text{ forbidden}\bigr).
+\end{equation}
+In particular,
+$g'(U)\le F(U)g_\lambda(R)$.
+\end{lemma}
+
+\begin{proof}
+Fix a broken path. A prefix ending immediately before a first hole
+of index $j$ has $j-1$ internal vertices. If $j$ is odd it uses the
+even tiling and leaves $u$ unused; if $j$ is even it uses the odd
+tiling and consumes $u$. The suffix has the analogous alternatives.
+Every run between consecutive holes has a unique tiling, since its
+order is even. These parity conditions prove the stated description
+of consumed holes and neutral pairs. On each path the number of
+consumed holes has the same parity as the number of internal holes.
+Adding explicit terminal holes gives $|R|\equiv|U|\pmod2$.
+Distinctness and $|R|\le|U|$ follow from feasibility and the fact
+that each consumed terminal is charged to a different hole.
+
+For completeness, all factors can be obtained from one prefix
+ratio. For even $j$, define
+\[
+ P(j)=
+ \frac{\prod_{i\le j,\ i\text{ odd}}t_i}
+      {\prod_{i\le j,\ i\text{ even}}t_i},
+ \qquad P(0)=1.
+\]
+The profile gives
+\[
+ P(j)=
+ \begin{cases}
+ 1,&j\le2p,\\
+ \lambda_e/T_j,&j\ge2p+2.
+ \end{cases}
+\]
+This is the factor for a first even-indexed hole consuming $u$;
+reflection gives the factor for a last odd-indexed hole consuming
+$v$. For a neutral pair $(j,k)$, its shifted tiling relative to
+the even-edge baseline contributes
+\[
+ \frac{\prod_{i=j+2,j+4,\ldots,k-1}t_i}
+      {\prod_{i=j+1,j+3,\ldots,k}t_i}
+ =
+ \frac{P(k)}{P(j-1)t_j}.
+\]
+If $j$ is odd on the left, its denominator is $T_j$; if $j$ is odd
+on the right, its denominator is $\lambda_e$, also when
+$j=2p+1$. Substitution proves \cref{eq:neutral-factors}.
+Prefixes and suffixes that do not consume a terminal, and runs
+outside these shifted intervals, retain their baseline even edges.
+Thus the product $F$ accounts for all changes to the baseline
+on broken paths.
+
+Every completion now determines a logical matching on the intact
+paths, using exactly the terminals outside $R$. Conversely, take
+any logical perfect matching on the vertices outside $R$ that uses
+no broken edge. Give its selected intact paths their odd tilings,
+give other intact paths their even tilings, and fill all broken
+paths by the uniquely forced runs just described. The distinctness
+conditions ensure that no terminal is used twice or is both used
+and deleted. This reconstructs a real completion and is inverse
+to the first map. Its weight is $C_0F$ times the logical weight,
+which proves \cref{eq:deletion-identity}. Omitting the forbidden-edge
+condition and dividing by \cref{eq:real-partition} gives the inequality.
+\end{proof}
+
+\subsection{Charging the pairing capacity}
+
+The deletion identity gives a factor $F(U)$ and a logical hole set $R$.
+To transfer the logical bound, we must compare the pairing capacity of
+$U$ with that of $R$. The next lemma expresses both capacities through
+threshold classes. Throughout the comparison, these classes belong to
+the original undeleted graphs, so they are independent of the completion
+being counted.
+
+\begin{lemma}[Pairings in a hierarchy]
+\label{lem:ultrametric-pairing}
+Let $B\ge1$ be a symmetric similarity on a finite vertex set $X$,
+satisfying $B_{ij}\ge\min\{B_{ik},B_{kj}\}$. Let $\mathcal C_t$ be
+the equivalence classes of $B\ge t$, including singletons.
+For every even set $U\subseteq X$,
+\begin{equation}
+\label{eq:threshold-pairing}
+ \log\Phi_B(U)
+ =\int_1^\infty\sum_{C\in\mathcal C_t}
+       \left\lfloor\frac{|C\cap U|}{2}\right\rfloor\frac{dt}{t}.
+\end{equation}
+There is one pairing that attains the maximum number of internal pairs
+in every threshold class simultaneously.
+\end{lemma}
+
+\begin{proof}
+For any pairing $Q$,
+\[
+ \log\prod_{ij\in Q}B_{ij}
+ =\int_1^\infty
+       |\{ij\in Q:B_{ij}\ge t\}|\,\frac{dt}{t}.
+\]
+Inside a threshold class $C$, there are at most
+$\lfloor |C\cap U|/2\rfloor$ pairs of $Q$, proving the upper bound.
+
+The threshold classes form a laminar family: any two are disjoint or one
+contains the other.  Process its finitely many distinct classes from
+smaller to larger, starting with the singletons.  In each class, pair
+its currently unpaired elements of $U$ arbitrarily until at most one
+remains.  When a class $C$ has been processed, exactly
+$\lfloor|C\cap U|/2\rfloor$ pairs lie inside it.  Subsequent processing
+cannot create another pair inside $C$, because at most one element of
+$C\cap U$ remains unpaired.  The final class is $X$, and $|U|$ is even,
+so the procedure produces a pairing.  It attains every floor count,
+and therefore attains the integral upper bound.
+\end{proof}
+
+\begin{lemma}\label{lem:lifted-hole-bound}
+If the logical activities are balanced, then for every even set
+$U$ of enlarged vertices,
+\[
+ g'(U)\Phi_{B'}(U)\le2D_0^{|U|/2}.
+\]
+More precisely, for every $U$ with a real completion, the deterministic
+quantities above satisfy
+\begin{equation}\label{eq:threshold-charge}
+ F(U)\Phi_{B'}(U)\le\Phi_B(R).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+The assertion is immediate if $U$ has no completion, so suppose
+otherwise. For a threshold $t>1$ and any set $A$ of enlarged vertices,
+let
+\[
+ S'_t(A)=\sum_{C}\left\lfloor\frac{|A\cap C|}{2}\right\rfloor,
+\]
+where $C$ ranges over the threshold classes of $B'$. Define $S_t$ on
+logical vertices in the same way using $B$. These definitions make
+sense even when $A$ has odd cardinality.
+The threshold partitions remain those of the undeleted graphs
+throughout this proof; only the sets of holes are changed.
+
+By \cref{lem:subdivision}, an internal vertex $x_j$ is isolated in
+the threshold partition precisely when $T_j<t$; otherwise it is
+connected to its home terminal. Call it \emph{active} in the latter
+case. The threshold relation induced on terminals is exactly the
+logical relation. These statements can equivalently be checked using
+raw real edges of activity at least $t$, since taking the bottleneck
+closure leaves threshold components unchanged.
+
+Remove the neutral pairs one at a time from the current hole set.
+For a pair on the same side, its active vertices lie in one class.
+Removing it reduces $S'_t$ by at most
+\[
+ \one_{\{\max(T_j,T_k)\ge t\}}.
+\]
+Indeed, two active holes are removed from one class, one active hole
+reduces a class floor by at most one, and an inactive hole is in
+a singleton class and contributes zero. On the left
+$\max(T_j,T_k)=T_j$; on the right it is $T_k$. For a cross pair, the
+loss is at most
+\[
+ \one_{\{T_j\ge t\}}+\one_{\{T_k\ge t\}}
+ -\one_{\{\lambda_e\ge t\}}.
+\]
+To see the subtracted term, when $\lambda_e\ge t$ both vertices
+are active and are connected by the full path, so their removal
+reduces a single class floor by one. Otherwise the bound is the
+sum of the separate active-hole bounds. These arguments apply to
+the current set after any preceding removals.
+
+Next consider each consumed hole mapping away from home. If it is
+active and $\lambda_e<t$, discard it. The floor loss is at most
+\[
+ \one_{\{T_j\ge t\}}-\one_{\{\lambda_e\ge t\}}.
+\]
+The expression is nonnegative, since $T_j\ge B_e\ge\lambda_e$.
+Every remaining active consumed hole can be mapped to its consumed
+terminal within its threshold class: a home mapping always has this
+property, and an away mapping retained by the rule has
+$\lambda_e\ge t$. Together with explicitly deleted terminals, these
+maps are injective, by the distinctness in
+\cref{lem:deletion-accounting}. Inactive internal holes lie in
+singleton classes and contribute no floor. Hence the floor sum
+remaining after the removals and discards is at most $S_t(R)$.
+
+Integrate these bounds against $dt/t$ on $(1,\infty)$. By
+\cref{lem:ultrametric-pairing}, the integral of $S'_t(U)$ is
+$\log\Phi_{B'}(U)$, and that of $S_t(R)$ is $\log\Phi_B(R)$.
+For a same-side neutral pair the integrated charge is exactly
+$-\log f_{jk}$. For a cross pair it is
+\[
+ \log T_j+\log T_k-\log\max\{1,\lambda_e\}
+ \le-\log f_{jk}.
+\]
+For an away-mapped hole it is
+\[
+ \log T_j-\log\max\{1,\lambda_e\}\le-\log f_j.
+\]
+Home mappings cost zero. Summing proves
+$\log\Phi_{B'}(U)\le\log\Phi_B(R)-\log F$,
+which is \cref{eq:threshold-charge}. Finally,
+\cref{lem:deletion-accounting,thm:logical-hole-bound} give
+\[
+ g'(U)\Phi_{B'}(U)
+ \le g_\lambda(R)\Phi_B(R)
+ \le2D_0^{|R|/2}
+ \le2D_0^{|U|/2}.
+\]
+\end{proof}
+
+\begin{proof}[Proof of \cref{prop:bag-inflation}]
+Expand the partition function before collapsing real and virtual
+portions. A nonempty partial matching $Q$ of $a$ colored virtual
+edges, with endpoint set $U$, contributes relative to $Z'$ the mass
+\[
+ g'(U)\prod_{xy,l\in Q}\frac{h_l}{D}.
+\]
+Vertices sharing $l$ lie in the same threshold component at $h_l$,
+so $h_l\le B'_{xy}$. Their pairing therefore gives
+\[
+ g'(U)\prod_{xy,l\in Q}\frac{h_l}{D}
+ \le D^{-a}g'(U)\Phi_{B'}(U)
+ \le2(D_0/D)^a
+\]
+by \cref{lem:lifted-hole-bound}. A vertex pair has at most two
+common labels, so there are fewer than $N^2$ colored virtual edges
+and at most $N^{2a}$ choices for $Q$. The empty $Q$ contributes one.
+Consequently,
+\[
+ 1\le I\le1+2\sum_{a\ge1}(N^2D_0/D)^a
+ =1+\frac2{99}<2.
+\]
+
+\end{proof}
+
+\section{A quadrangulation adapted to the labels}
+\label{sec:cells}
+
+We now turn to the sampling branch of the proof. Its input is a
+tree-bag graph as in \cref{def:tree-bags}; no balance assumption is
+needed. To compare two alternating matchings on a cycle, we shall
+subdivide its polygon into four-corner cells. A cell side may stand
+for a long arc of the original cycle. We first describe the matching
+patterns on such an arc and the endpoint repairs the cells must
+permit, then construct a quadrangulation with those properties.
+
+Let $\mathcal C$ be a simple even cycle of length $m\ge4$ in the colored
+bag graph.  Each edge retains its color, which is a node of the label
+tree.  For a vertex $v$ of $\mathcal C$, define
+\[
+ S_v=\{\text{labels of the two edges of $\mathcal C$ incident to $v$}\}.
+\]
+Thus $S_v$ is a singleton or consists of two adjacent tree nodes.
+These sets refer to the \emph{original full cycle} throughout the
+construction; adding a diagonal never changes them.  In particular,
+the cyclic sequence of edge labels is a walk on the tree, with repetitions
+allowed.
+
+An \emph{arc} is one of the two paths in $\mathcal C$ between distinct
+vertices, specified together with its endpoints. Its length is its
+number of edges. We use only odd arcs. An arc is \emph{admissible} if
+its endpoints $u,v$ satisfy $S_u\cap S_v\ne\varnothing$. The clique
+property then supplies an edge joining them with any chosen label in
+this intersection.
+
+For an admissible odd arc $J=(v_0,\ldots,v_s)$, choose such a colored
+endpoint chord $e_J$ and put
+\[
+ T_J=\{v_0v_1,v_2v_3,\ldots,v_{s-1}v_s\},\qquad
+ N_J=\{v_1v_2,v_3v_4,\ldots,v_{s-2}v_{s-1}\},
+ \qquad C_J=N_J\cup\{e_J\}.
+\]
+All path edges retain their cycle colors. The \emph{through pattern}
+$T_J$ and the \emph{closed pattern} $C_J$ both perfectly match the
+vertices of $J$; the \emph{interior pattern} $N_J$ leaves only its
+two endpoints unmatched. The symbol $C_J$ denotes a matching, not the
+original cycle. Call $J$ \emph{short} if $s=1$ and \emph{long} otherwise.
+On a short arc choose its actual cycle edge as $e_J$, so $T_J=C_J$.
+When a diagonal later represents two complementary arcs, we use the
+same colored chord in both patterns. \Cref{fig:odd-arc-patterns} shows
+these patterns on a long arc.
+
+\begin{figure}[ht]
+\centering
+\begin{tikzpicture}[x=1.2cm,y=1cm,
+  reference/.style={draw=black!28,densely dashed,line width=.5pt},
+  selected/.style={draw=black,line width=1.4pt},
+  vertex/.style={circle,fill=black,inner sep=1.6pt}]
+  \foreach \i in {0,...,5} {
+    \coordinate (t\i) at (\i,1.7);
+    \coordinate (c\i) at (\i,0);
+  }
+  \foreach \i [evaluate=\i as \j using int(\i+1)] in {0,...,4} {
+    \draw[reference] (t\i)--(t\j);
+    \draw[reference] (c\i)--(c\j);
+  }
+  \foreach \i [evaluate=\i as \j using int(\i+1)] in {0,2,4}
+    \draw[selected] (t\i)--(t\j);
+  \foreach \i [evaluate=\i as \j using int(\i+1)] in {1,3}
+    \draw[selected] (c\i)--(c\j);
+  \draw[selected] (c0) .. controls +(1,1) and +(-1,1) .. (c5)
+    node[pos=.5,above=2pt] {$e_J$};
+  \foreach \i in {0,...,5} {
+    \node[vertex,label=below:{$v_{\i}$}] at (t\i) {};
+    \node[vertex,label=below:{$v_{\i}$}] at (c\i) {};
+  }
+  \node[anchor=east] at (-.5,1.7) {$T_J$};
+  \node[anchor=east] at (-.5,0) {$C_J$};
+  \node[anchor=west] at (5.5,1.7) {through};
+  \node[anchor=west] at (5.5,0) {$N_J\cup\{e_J\}$};
+\end{tikzpicture}
+\caption{The two patterns on a long odd arc, illustrated for
+$J=(v_0,\ldots,v_5)$. Solid edges form the indicated matching;
+faint dashed edges only locate the underlying path. Both $T_J$ and
+$C_J$ perfectly match the same vertices. The latter uses the interior
+pattern $N_J$ and the endpoint chord $e_J$; their union
+$T_J\cup C_J$ is the alternating cycle consisting of $J$ and its chord.
+Replacing $T_J$ by $C_J$ can change interior edges; this replacement
+need not be a single chain update.}
+\label{fig:odd-arc-patterns}
+\end{figure}
+
+Besides closing an arc with a chord, we shall sometimes need to free
+its endpoints while starting from its through pattern. Removing the
+first and last edges of a long arc from $T_J$ uncovers its endpoints
+and its two \emph{leaves} $v_1,v_{s-1}$. Call $J$ \emph{good} if it is
+short or if
+\[
+ S_{v_1}\cap S_{v_{s-1}}\ne\varnothing.
+\]
+For a good long arc, choose an edge between its leaves with a label in
+this intersection, called a \emph{leaf repair}. After the two boundary
+edges are removed, inserting this repair matches all internal vertices
+and leaves both endpoints free. For a short arc, deleting its sole
+edge does the same. This is the matching operation encoded by the
+good-arc condition. The \emph{boundary edges of an arc} mean only its
+first and last edges, which coincide for a short arc.
+
+Realize the vertices of $\mathcal C$ in their cyclic order as a convex
+polygon.  A quadrangulation is a subdivision of this polygon into
+quadrilaterals by noncrossing diagonals.  A side of a cell represents the
+arc of the original cycle from one corner to the next corner, containing
+no other corner of that cell.  Consequently the four sides of each cell
+represent four consecutive arcs partitioning the entire cycle.  An
+interior diagonal has two occurrences as a cell side, and their arcs
+are complementary in the original cycle.
+
+\begin{lemma}[Label-adapted quadrangulation]
+\label{lem:quadrangulation}
+The polygon of $\mathcal C$ has a quadrangulation such that every cell
+side represents an admissible odd arc and the following hold in every
+cell:
+\begin{enumerate}[label=\textup{(\roman*)}]
+ \item some opposite pair of sides consists of good arcs;
+ \item if both arcs in an opposite pair are long, both arcs in the
+ other opposite pair are good.
+\end{enumerate}
+There are $(m-2)/2$ cells and $(m-4)/2$ interior diagonals.  One can
+construct the quadrangulation in time polynomial in $m$ and the encoding
+length of the label tree.
+
+Each diagonal may be assigned any common label from the fixed sets at
+its endpoints, used consistently in its two occurrences.  For a short
+side, use the color of its actual cycle edge.  These choices need not
+coincide with the auxiliary labels used in the construction.
+\end{lemma}
+
+\begin{proof}
+We give a recursive splitting rule.  A subproblem is an admissible odd
+arc $J=(v_0,\ldots,v_s)$; its \emph{exterior} is its complementary arc
+in the full cycle.  Choose a label $P\in S_{v_0}\cap S_{v_s}$, and
+write $a_i$ for the label of $v_iv_{i+1}$, $0\le i<s$.
+Adjacent terms of this sequence are equal or adjacent in the tree.
+If $s=1$, no cell is needed.  For $s\ge3$, we find indices
+\begin{equation}
+\label{eq:cell-split-indices}
+ 0<p_1<p_2<s,\qquad p_1\text{ odd},\quad p_2\text{ even},
+\end{equation}
+such that the three subarcs with successive endpoint pairs
+\[
+ (v_0,v_{p_1}),\qquad (v_{p_1},v_{p_2}),\qquad
+ (v_{p_2},v_s)
+\]
+are admissible.  They and the exterior arc form the four sides of the
+new cell.  We call these the first, middle, third, and exterior sides,
+respectively.
+
+We repeatedly use the following elementary fact about a tree.
+If a contiguous part of the label walk avoids $P$ and its first and
+last labels are adjacent to $P$, these two labels coincide: the walk
+lies in a single component of the tree with $P$ removed, and that
+component has just one node adjacent to $P$.
+
+\paragraph{The exterior is good.}
+It suffices to make either the middle side short or both the first and
+third sides short.  Either alternative immediately implies
+\textup{(i)} and \textup{(ii)}, because the exterior is good.
+
+First ensure that some $a_i$ equals $P$.  If none does, $a_0$ and
+$a_{s-1}$ are adjacent to $P$, because the endpoint sets contain $P$.
+The elementary tree fact gives $a_0=a_{s-1}=Q$ for a neighbor $Q$ of
+$P$.  Replace $P$ by $Q$, which also lies in both endpoint sets.
+We now distinguish the exhaustive possibilities below.
+\begin{enumerate}[label=\textup{(\alph*)}]
+ \item Suppose $a_i=P$ for an odd $i$.
+ Set $p_1=i$ and $p_2=i+1$.  The middle side is short.  The first
+ side's two endpoints contain $P$, as do the third side's two
+ endpoints, so all three sides are admissible.
+
+ \item Otherwise all occurrences of $P$ have even indices.
+ If their first index is $i>0$, set $p_1=i-1$ and $p_2=i$.
+ The prefix $a_0,\ldots,a_{i-1}$ avoids $P$, and its first and last
+ labels are adjacent to $P$.  They therefore equal the same neighbor
+ $Q$.  The first side's endpoints both contain $Q$, and the third
+ side's endpoints both contain $P$.  The middle side is short.
+
+ \item Suppose the first occurrence is at $0$, but the last occurrence
+ is at $j<s-1$.  Set $p_1=j+1$ and $p_2=j+2$.
+ The suffix $a_{j+1},\ldots,a_{s-1}$ avoids $P$ and has its two end
+ labels adjacent to $P$, so they equal one neighbor $Q$.
+ The first side's endpoints contain $P$, the third side's endpoints
+ contain $Q$, and the middle side is short.
+
+ \item In the remaining case $a_0=a_{s-1}=P$.
+ Set $p_1=1$ and $p_2=s-1$.  Both outer sides are short, and the
+ middle side's endpoints both contain $P$.
+\end{enumerate}
+All indices satisfy \eqref{eq:cell-split-indices}.  For example, in
+\textup{(b)} the even index $i$ is at least $2$, and in \textup{(c)}
+the even index $j$ is at most $s-3$.
+
+\paragraph{The exterior is not good.}
+The exterior is then long.  At least one of its leaves lacks $P$ in
+its fixed label set, since otherwise its two leaves would share $P$.
+Reverse the indexing of $J$ if necessary so that this leaf is the
+exterior neighbor of $v_0$.  The exterior edge at $v_0$ cannot have
+label $P$.  Since $P\in S_{v_0}$, the first edge of $J$ must have
+label $a_0=P$.
+
+If $a_{s-1}=P$ as well, choose $p_1=1,p_2=s-1$.  Both outer sides
+are short, and the middle side is admissible.  Otherwise let $t-1$
+be the last index with $a_{t-1}=P$.  Thus $1\le t\le s-1$.
+The suffix $a_t,\ldots,a_{s-1}$ avoids $P$.  Its first label is
+adjacent to $P$, and its last label is adjacent to $P$ because
+$P\in S_{v_s}$.  Hence
+\[
+ a_t=a_{s-1}=Q
+\]
+for the same neighbor $Q$ of $P$.
+\begin{itemize}
+ \item If $t$ is even, set $p_1=1,p_2=t$.
+ The first side is short.  The middle side's endpoints both contain
+ $P$, and the third side's endpoints both contain $Q$.
+ The third side is good: if long, both its boundary edges have
+ label $Q$, so both its leaves contain $Q$.
+
+ \item If $t$ is odd, set $p_1=t,p_2=s-1$.
+ The third side is short.  The first side's endpoints both contain
+ $P$, and the middle side's endpoints both contain $Q$.
+ The first side is good because, if long, both its boundary edges
+ have label $P$.
+\end{itemize}
+The required strict inequalities follow from $t\ge2$ in the even
+case and $t\le s-2$ in the odd case.  In either case the first and
+third sides are good and at least one is short.  This proves
+\textup{(i)}.  The only opposite pair that can contain two long sides
+is the middle--exterior pair, and its other pair is good, proving
+\textup{(ii)}.
+
+\paragraph{Recursion and consistent diagonals.}
+Start with any arc of length $m-1$, whose exterior is the remaining
+single cycle edge.  Its endpoints share that edge's label.
+Apply the splitting rule and recurse on each of the three subarcs
+whose length exceeds one, always using the original sets $S_v$ and
+the complement in the original cycle as exterior.  The three
+subpolygons have disjoint interiors, so all diagonals are noncrossing.
+Each recursive child has odd length and admissible endpoints by the
+construction.  The splitting label $P$ is only an auxiliary witness
+to admissibility; it does not constrain the color of the closing
+diagonal.  Thus each diagonal can be colored once and the same color
+used on both sides.
+
+For an odd arc of length $s$, the resulting number of cells is
+$(s-1)/2$: this is zero for $s=1$, and a split into odd lengths
+$s_1+s_2+s_3=s$ gives
+\[
+ 1+\sum_{i=1}^{3}\frac{s_i-1}{2}=\frac{s-1}{2}.
+\]
+The root therefore gives $(m-2)/2$ cells.  Each nonroot recursive
+subproblem that creates a cell has one closing diagonal, giving one
+fewer interior diagonals than cells. A direct implementation scans
+at most $m$ labels in each of the $(m-2)/2$ nontrivial subproblems;
+the recursion also has $m-1$ short leaf arcs. Fixing total orders for
+any choices makes the construction deterministic, with $O(m^2)$
+label comparisons and polynomial bit cost.
+\end{proof}
+
+For each cell, choose its colored side chords as in
+\cref{lem:quadrangulation}; for each good long side, choose one leaf
+repair.  These are precisely the additional edges needed for the
+local matching constructions in the next section.
+
+\begin{lemma}[Comparability within a cell]
+\label{lem:cell-comparability}
+Consider the four colored side chords of a cell, the first and last
+cycle edges of each of its four arcs, and one leaf repair for each
+good long arc.  The labels of any two edges in this collection have
+tree distance at most $6$.  Consequently their activities lie within
+a factor
+\[
+ A_0=1000D
+\]
+of one another.  The same conclusion holds after clamping every edge
+activity to any common interval $[a,b]\subset(0,\infty)$.
+\end{lemma}
+
+\begin{proof}
+Two consecutive side chords have labels in the same fixed set $S_v$
+at their common cell corner, so their labels are equal or adjacent.
+Any two of the four side-chord labels therefore have distance at
+most $2$.  A boundary edge of an arc and its side chord have labels
+in the same $S_v$ at the relevant endpoint, giving distance at most
+$1$.  A leaf repair and the boundary edge at either of its leaves
+have labels in that leaf's fixed set, again giving distance at most
+$1$.  Thus each edge in the stated collection is at distance at
+most $2$ from its side chord, and the diameter is at most
+$2+2+2=6$.
+
+Adjacent labels have heights differing by a factor $2$.  If edges
+$e,f$ have labels $\ell,\ell'$ with distance at most $6$, the
+assumed activity bounds imply
+\[
+ \frac{W_e}{W_f}
+ \le \frac{3h_\ell}{h_{\ell'}/D}
+ \le 3D\,2^6=192D\le A_0.
+\]
+The same bound holds with $e,f$ exchanged.  Finally the common clamp
+$\psi(x)=\max\{a,\min\{b,x\}\}$ satisfies
+\[
+ 1\le\frac{\psi(x)}{\psi(y)}\le\frac{x}{y}
+ \qquad (x\ge y>0).
+\]
+For example this follows by observing that
+$\log\psi(e^u)$ is the clamp of $u$ to
+$[\log a,\log b]$, a nondecreasing $1$-Lipschitz function.
+Clamping therefore preserves the factor bound.
+\end{proof}
+\section{A two-coordinate energy inequality}
+\label{sec:energy}
+
+This section proves the sampling argument's main inequality. The four
+side chords of each cell give a two-edge switch whose changed edges
+have comparable activities. Long arc conversions are used in an exact
+signed identity and need not be individual chain moves. We control the
+resulting context errors by swapping an alternating cycle with an
+independent matching. Encodings by two matchings change only a bounded
+number of edges in their multiset union, and their preimage counts give
+polynomial load bounds.
+The variance-to-energy comparison is related to the flow method
+of~\cite{Sinclair1992}, but we do not invoke a path-congestion theorem:
+the signed identity and its context-error estimates are established
+for the present pair kernel.
+
+Fix a tree-bag graph on $N\ge2$ vertices, with at least one perfect
+matching, and let $w$ be either its reference activities or any common
+clamp of them.  Write $\pi(M)=\wt_w(M)/Z_w$ and $\calM$ for its colored
+perfect matchings.  Throughout this section put
+\begin{equation}
+\label{eq:energy-constants}
+ A_0=1000D,\qquad \calT=(10N)^{30},\qquad
+ L=(10^4ND)^{100}.
+\end{equation}
+
+\subsection{The pair kernel}
+
+On $(A,B)\in\calM^2$, choose each of the following proposal types with
+probability $1/3$.
+\begin{enumerate}[label=\textup{(\arabic*)}]
+ \item \emph{Two-edge switch.}  Choose an unordered pair of distinct
+ edges of $A$ uniformly, and choose either alternative pairing of
+ their four endpoints with probability $1/2$.  For each proposed
+ edge, choose index $1$ or $2$ with probability $1/2$ from the ordered
+ list of common labels of its endpoints.  If an index is absent,
+ stay put.  If $A$ has fewer than two edges, stay put.
+
+ \item \emph{Color change.}  Choose an edge of $A$ uniformly and
+ choose index $1$ or $2$ from the common-label list with probability
+ $1/2$.  An absent index gives a holding move.
+
+ \item \emph{Cycle swap.}  Choose uniformly one discrepancy cycle of
+ $A,B$ and interchange its two layers.  If there is no discrepancy
+ cycle, stay put.
+\end{enumerate}
+Accept each proposal by the Metropolis rule for $\pi\otimes\pi$.
+Denote the resulting kernel by $P$ and its energy by $\calE$.
+The proposal is symmetric.  For the first two types, the reverse
+choice has the same edge, pairing, and color-index probabilities;
+the possible holding choices do not change this fact.  For the third
+type, the discrepancy cycles and their number are unchanged by the
+swap.  In particular, a cycle swap always has acceptance probability
+one, since its product weight is unchanged.
+
+\begin{theorem}[Pair energy inequality]
+\label{thm:pair-energy}
+For every two real functions $f_1,f_2$ on $\calM$,
+\begin{equation}
+\label{eq:pair-energy}
+ \Var_\pi f_1\le L\,
+ \calE\bigl((A,B)\mapsto f_1(A)+f_2(B)\bigr).
+\end{equation}
+The same constant works for every common-interval clamp of the
+reference activities.
+\end{theorem}
+
+In the proof write $F(A,B)=f_1(A)+f_2(B)$ and $E=\calE(F)$.
+A two-edge switch is called \emph{safe} if its four removed or
+inserted colored edges have activities within a factor $A_0$ of one
+another.  Let $\mathcal S(A)$ be its possible safe destinations, and
+let $\mathcal K(A)$ be its possible destinations by changing one
+edge color.  Consecutive labels have comparable heights, so every
+color change has activity ratio in $[A_0^{-1},A_0]$.
+
+\begin{lemma}[Elementary energy bounds]
+\label{lem:elementary-pair-energy}
+With the notation above,
+\begin{align}
+ \sum_A\pi(A)\sum_{A'\in\mathcal S(A)}
+       (f_1(A)-f_1(A'))^2
+ &\le60N^2A_0^2 E,
+ \label{eq:safe-switch-energy}\\
+ \sum_A\pi(A)\sum_{A'\in\mathcal K(A)}
+       (f_1(A)-f_1(A'))^2
+ &\le12NA_0 E,
+ \label{eq:color-energy}\\
+ \sum_{A,B}\pi(A)\pi(B)\sum_{C\in\mathcal C(A,B)}
+       \bigl(F(A,B)-F((A,B)^C)\bigr)^2
+ &\le6NE.
+ \label{eq:cycle-energy}
+\end{align}
+Here $\mathcal C(A,B)$ denotes the discrepancy cycles, and $(A,B)^C$
+is the pair after swapping $C$.
+\end{lemma}
+
+\begin{proof}
+Every specified legal two-edge switch has proposal probability
+at least $1/(30N^2)$; its Metropolis acceptance is at least
+$A_0^{-2}$ when safe.  Every specified color change has proposal
+probability at least $1/(6N)$ and acceptance at least $A_0^{-1}$.
+These moves change only the $f_1$ summand of $F$, so summing the
+second-coordinate law and using the factor $1/2$ in the energy proves
+the first two bounds.  There are at most $N$ discrepancy cycles.
+Each specified swap therefore has proposal probability at least
+$1/(3N)$ and is accepted; this proves the third bound.
+\end{proof}
+
+\subsection{An exact identity on a cycle}
+
+Fix a pair of demand matchings $(I_1,I_2)$ and a simple discrepancy
+cycle $C$ of length at least four.  All orientations and choices below
+are deterministic, using the orders fixed in \cref{sec:preliminaries}.
+Let $R$ be the edges of $I_1$ outside $C$, and let $O_0=I_1$ and
+$O_1$ be the matching obtained by toggling $C$ in $I_1$.  Thus both
+orientations use $R$ outside $C$.  Put
+\[
+ D_C=f_1(O_0)-f_1(O_1).
+\]
+Apply \cref{lem:quadrangulation} and fix the colors of its diagonals
+consistently.  A cell's four sides always mean the four odd arcs of
+the \emph{original whole cycle}, not paths formed from earlier
+diagonals.
+
+Use the through, interior, and closed patterns $T_J,N_J,C_J$ from
+\cref{sec:cells}, with the chosen chord colors. A side $J$ is
+\emph{through} in $O_\varepsilon$ if $T_J\subseteq O_\varepsilon$.
+Exactly one orientation $O_\varepsilon$ contains $T_J$; define
+\begin{equation}
+\label{eq:canonical-gain}
+ H_J=(O_\varepsilon\setminus T_J)\cup C_J,
+ \qquad g_J=f_1(O_\varepsilon)-f_1(H_J).
+\end{equation}
+For a short side, the chord is its actual cycle edge, so $g_J=0$.
+
+The reason for using the original cycle on every cell side is already
+visible on a six-cycle $C=(v_0,\ldots,v_5,v_0)$. Take
+\[
+ O_0=R\cup\{v_0v_1,v_2v_3,v_4v_5\},\qquad
+ O_1=R\cup\{v_1v_2,v_3v_4,v_5v_0\},
+\]
+and suppose the diagonal $e=v_0v_3$ is admissible. In the two cells
+it represents the complementary arcs
+$J=(v_3,v_4,v_5,v_0)$ and $\bar J=(v_0,v_1,v_2,v_3)$, respectively.
+Both conversions yield the same matching
+\[
+ H_J=H_{\bar J}=H=R\cup\{v_1v_2,e,v_4v_5\}.
+\]
+Thus $g_{\bar J}-g_J=f_1(O_0)-f_1(O_1)=D_C$; the shared diagonal
+contributes one whole-cycle difference, not zero.
+\Cref{fig:complementary-arcs} displays this common converted matching.
+This example isolates the complementary-gain identity. Its cells have
+only one long side each, so the context errors introduced below vanish;
+cells with two opposite long sides require a further estimate.
+
+\begin{figure}[ht]
+\centering
+\begin{tikzpicture}[x=1cm,y=1cm,
+  vertex/.style={circle,fill=black,inner sep=1.3pt},
+  selected/.style={draw=black,line width=1.3pt}]
+  \foreach \prefix/\shift in {a/0,b/6.2} {
+    \coordinate (\prefix0) at (\shift,0);
+    \coordinate (\prefix1) at (\shift+.9,1.15);
+    \coordinate (\prefix2) at (\shift+2.6,1.15);
+    \coordinate (\prefix3) at (\shift+3.5,0);
+    \coordinate (\prefix4) at (\shift+2.6,-1.15);
+    \coordinate (\prefix5) at (\shift+.9,-1.15);
+  }
+  \fill[black!4] (a0)--(a1)--(a2)--(a3)--cycle;
+  \fill[black!8] (a0)--(a3)--(a4)--(a5)--cycle;
+  \draw (a0)--(a1)--(a2)--(a3)--(a4)--(a5)--cycle;
+  \draw[densely dashed,line width=.8pt] (a0)--(a3)
+    node[midway,fill=white,inner sep=2pt] {$e$};
+  \node at (1.75,1.75) {$\bar J$};
+  \node at (1.75,-1.75) {$J$};
+  \draw[black!25,densely dashed]
+    (b0)--(b1)--(b2)--(b3)--(b4)--(b5)--cycle;
+  \draw[selected] (b1)--(b2);
+  \draw[selected] (b0)--(b3) node[midway,above=2pt] {$e$};
+  \draw[selected] (b4)--(b5);
+  \foreach \prefix in {a,b} {
+    \node[vertex,label=left:{$v_0$}] at (\prefix0) {};
+    \node[vertex,label=above:{$v_1$}] at (\prefix1) {};
+    \node[vertex,label=above:{$v_2$}] at (\prefix2) {};
+    \node[vertex,label=right:{$v_3$}] at (\prefix3) {};
+    \node[vertex,label=below:{$v_4$}] at (\prefix4) {};
+    \node[vertex,label=below:{$v_5$}] at (\prefix5) {};
+  }
+  \node at (1.75,-2.35) {two cells, complementary arcs};
+  \node at (7.95,-2.35) {common converted matching $H$};
+\end{tikzpicture}
+\caption{An admissible diagonal $e=v_0v_3$ cuts a six-cycle into two
+cells. In the upper cell its side follows the lower boundary arc $J$;
+in the lower cell its side follows the upper boundary arc $\bar J$.
+The two occurrences use the same colored edge $e$. Converting the
+through pattern on either arc gives the restriction of $H$ shown on
+the right (the common outside matching $R$ is omitted),
+so their signed gains sum to $D_C$. The drawing concerns the signed
+identity, not a sequence of chain transitions.}
+\label{fig:complementary-arcs}
+\end{figure}
+
+
+In a cell, let $P$ be its opposite pair of sides through in $O_0$,
+and $Q$ its opposite pair through in $O_1$.  Let $A_P$ use $N_J$ on
+every side and the chords of the two sides of $P$, together with $R$.
+Define $A_Q$ analogously and put
+\[
+ \Delta=f_1(A_P)-f_1(A_Q).
+\]
+These are legal perfect matchings: the interior vertices are matched
+by the $N_J$'s, and the two chosen chords match the four corners.
+Their difference is a safe two-edge switch by
+\cref{lem:cell-comparability}.
+
+Order each opposite pair as $(Y,X)$.  For the orientation in which
+both are through, first convert $Y$ and then $X$.
+Write $A^{(0)}=O_\varepsilon$ and $A^{(1)}=H_Y$ for the two
+pre-conversion contexts of $X$.  For a matching $R'$ on the vertices
+outside $X$, define
+\begin{equation}
+\label{eq:arc-derivative}
+ \partial_X f(R')=f(T_X\cup R')-f(C_X\cup R').
+\end{equation}
+The context error of the ordered opposite pair is
+\begin{equation}
+\label{eq:context-error}
+ E_{Y,X}
+ =\partial_X f_1(A^{(1)}\setminus T_X)
+  -\partial_X f_1(A^{(0)}\setminus T_X).
+\end{equation}
+It is zero if either side is short.  Denote these errors by $E_P,E_Q$
+for the two ordered pairs of the cell.
+
+\begin{lemma}[Cell identity and cancellation]
+\label{lem:cell-identity}
+The quantities just defined satisfy
+\begin{equation}
+\label{eq:single-cell-identity}
+ D_C=\Delta+\sum_{J\in P}g_J-\sum_{J\in Q}g_J+E_P-E_Q
+\end{equation}
+for each cell.  Consequently,
+\begin{equation}
+\label{eq:global-cell-identity}
+ D_C=\sum_{\text{cells of }C}(\Delta+E_P-E_Q).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+The first conversion in an opposite pair has its canonical gain.
+The second has its canonical gain plus the error in
+\eqref{eq:context-error}.  Thus
+\[
+ f_1(O_0)-f_1(A_P)=\sum_{J\in P}g_J+E_P,
+ \qquad
+ f_1(O_1)-f_1(A_Q)=\sum_{J\in Q}g_J+E_Q.
+\]
+Subtracting proves \eqref{eq:single-cell-identity}.
+
+Consider an interior diagonal with complementary arcs $J,\bar J$.
+They are through in opposite whole-cycle orientations, and their
+converted matchings coincide:
+\[
+ H_J=H_{\bar J}=R\cup N_J\cup N_{\bar J}\cup\{e_J\}.
+\]
+Therefore the two signed canonical gains contributed by this
+diagonal sum to $f_1(O_0)-f_1(O_1)=D_C$.  The signs here are precisely
+those in \eqref{eq:single-cell-identity}: the arc through in $O_0$
+has positive sign and the other negative sign.  Boundary sides have
+zero gain.  Summing \eqref{eq:single-cell-identity} over the cells
+therefore gives one copy of $D_C$ per cell on the left and one per
+interior diagonal on the right, in addition to the displayed local
+terms.  The number of cells is one more than the number of diagonals,
+which proves \eqref{eq:global-cell-identity}.
+\end{proof}
+
+\subsection{Encoding the demands}
+
+A \emph{switch demand} specifies $(I_1,I_2)$, a simple discrepancy
+cycle $C$, and one cell of its fixed quadrangulation.  Its value is
+$\Delta$ above.  An \emph{error demand} specifies these data and one
+opposite pair with two long sides, in its fixed order $(Y,X)$.
+Its value is $E_{Y,X}$.  The weight of either demand is
+$\rho(d)=\pi(I_1)\pi(I_2)$.  Repeated demands from different cells or
+different cycles are included separately in all sums.
+
+\begin{lemma}[Encodings with a bounded number of preimages]
+\label{lem:encoding}
+Every switch demand has an encoding $(A_P,B)$ by two perfect
+matchings, and every error demand has encodings
+$(A^{(j)},B^{(j)})$ for $j=0,1$, with the following properties.
+\begin{enumerate}[label=\textup{(\roman*)}]
+ \item For each encoding $(A,B)$,
+ $\rho(d)\le A_0^4\pi(A)\pi(B)$.
+ \item For each encoding type and each fixed output pair, there are
+ at most $\calT$ demand preimages.  This bound includes the choices
+ of cycle, cell, opposite pair, and context.
+ \item For an error demand, both guide matchings satisfy
+ $C_X\subseteq B^{(j)}$.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+We first describe a repair operation.  Start with the union of the
+four through patterns $T_J$ in a cell.  Each corner is covered twice,
+once by each incident side.  Select a good opposite pair of sides.
+For each selected short side delete its sole edge.  For each selected
+long side delete its first and last edges, and insert its chosen leaf
+repair.  Each corner loses exactly one incident edge.  On a long
+selected side the only newly uncovered internal vertices are its
+two leaves, which the repair matches.  Thus the resulting edges form
+a perfect matching of all cycle vertices.  This verifies legality
+also for a side of length three: its two leaves are distinct and
+its repair is an ordinary colored edge between them.
+
+\paragraph{Switch demands.}
+Use $A_P$ as the first output.  In the second output retain the edges
+of $I_2$ outside $C$, start with all four through patterns, and apply
+the preceding repair operation to a good opposite pair, which exists
+by \cref{lem:quadrangulation}(i).  Call the result $B$.
+Before these modifications, the original multiset union contains
+exactly one $T_J$ and one $N_J$ on each side.  The output union adds
+the two $P$-chords and the leaf repairs and drops the selected boundary
+edges.  If $r\in\{0,1,2\}$ selected sides are long, the numbers added
+and dropped are both $2+r\le4$.
+
+\paragraph{Error demands, context zero.}
+The first output is $A^{(0)}=O_\varepsilon$, the canonical orientation
+in which $X,Y$ are through.  Start the second output with the
+complementary cycle orientation and the edges of $I_2$ outside $C$.
+It has the $N$ patterns on $X,Y$.  Insert their two chords.  The
+other opposite pair is good by \cref{lem:quadrangulation}(ii),
+because $X,Y$ are both long.  On that other pair delete boundary
+edges and insert leaf repairs as above.  The result is a perfect
+matching $B^{(0)}$, containing both $C_X$ and $C_Y$.  Again, the
+multiset union differs from the original by equally many added
+and dropped edges, at most four of each.
+
+\paragraph{Error demands, context one.}
+Interchange the patterns $T_Y$ and $C_Y$ between
+$A^{(0)},B^{(0)}$.  Each pattern perfectly matches the entire vertex
+set of $Y$, so the interchange is legal and gives
+$A^{(1)}=H_Y$ and a guide $B^{(1)}$.  The vertex sets of the opposite
+arcs $X,Y$ are disjoint; hence $B^{(1)}$ still contains $C_X$.
+This interchange does not change the multiset union or product weight.
+
+In every case all added and dropped edges are among the chords,
+boundary edges, and good-side leaf repairs of this one cell.
+They have pairwise activity ratio at most $A_0$ by
+\cref{lem:cell-comparability}.  Since the two lists have equal
+length at most four, the input product weight is at most $A_0^4$
+times the output product weight.  Division by $Z_w^2$ proves (i).
+
+For completeness we give an inverse, including the information it
+needs.  Attach a tag consisting of the four cell corners in cyclic
+order, lists of the added and dropped colored edges padded to length
+four each, one orientation bit, and fixed flags for the encoding type
+(switch, error context zero, or error context one), opposite pair, and
+order.  An edge in these lists is an occurrence in
+the multiset union; coincident colored edges retain their multiplicities.
+Given the output pair and tag, remove the added occurrences and
+reinsert the dropped occurrences in its multiset union.  This recovers
+the original colored union $I_1\uplus I_2$.  Its component containing
+the first tagged corner is the active simple cycle $C$.  The
+orientation bit specifies which of the two alternating classes at
+that corner belongs to $I_1$, so alternation determines both original
+layers on all of $C$.  Outside $C$, the output layers were unchanged;
+they recover $I_1,I_2$ there directly.  The fixed quadrangulation can
+now be reconstructed.  Its tagged corners identify the cell, and
+the flags identify the term and, if appropriate, its ordered pair.
+Thus this is an inverse on every tag produced by the construction.
+
+There are at most $2N^2$ possible colored edges.  Allowing a padding
+symbol, the tags have at most
+\[
+ 128N^4(1+2N^2)^8
+ \le128\cdot3^8N^{20}< (10N)^{30}=\calT
+\]
+possibilities for $N\ge2$.  Here the factor $128$ more than covers
+the orientation bit and the fixed flags; the lists themselves may
+be stored in the fixed colored-edge order.  This proves (ii), and
+the constructions already proved (iii).
+\end{proof}
+
+The encodings provide two kinds of control. Their bounded preimage
+count converts a sum over demands to a sum over pairs of matchings.
+For context errors, the guide also contains a prescribed closed arc;
+this additional condition must survive that summation. We use the
+bounded preimage count for switches below, then retain the closed-arc
+condition when treating context errors in the next subsection.
+
+We apply this lemma first to switches.  For a fixed encoded pair
+$(A,B)$ there are at most $\calT$ demands, and each has its associated
+destination $A_Q$ in $\mathcal S(A)$.  Therefore
+\begin{align}
+ \sum_{d\text{ switch}}\rho(d)\Delta_d^2
+ &\le A_0^4\calT\sum_{A,B}\pi(A)\pi(B)
+       \sum_{A'\in\mathcal S(A)}(f_1(A)-f_1(A'))^2 \notag\\
+ &\le60N^2A_0^6\calT E.
+ \label{eq:switch-demand-load}
+\end{align}
+
+\subsection{Controlling context errors by cycle swaps}
+
+Here an identity $X$ means a directed, colored, simple odd path of
+length at least three together with a specified colored edge joining
+its endpoints.  It determines $T_X,C_X$ and \eqref{eq:arc-derivative}.
+We may sum over all such identities: this is a finite proof device,
+and the algorithm does not enumerate them.  Put
+\[
+ p_X=\pi\{B:C_X\subseteq B\}.
+\]
+We omit identities with $p_X=0$.  Every identity arising from an error
+demand has $p_X>0$, by the guide constructed in \cref{lem:encoding}.
+For any first matching $A\supseteq T_X$ and either $j=0,1$,
+that lemma gives the following \emph{conditional demand-mass bound}:
+\begin{equation}
+\label{eq:conditional-demand-mass}
+ \sum_{\substack{d\text{ error}:X(d)=X\\ A^{(j)}(d)=A}}
+       \rho(d)
+ \le A_0^4\calT\,\pi(A)p_X.
+\end{equation}
+Indeed, for each guide $B\supseteq C_X$ there are at most $\calT$
+preimages of $(A,B)$; now sum its product weight over these guides.
+The factor $p_X$ is essential and will cancel the conditioning below.
+
+The next comparison averages over precisely those guides counted by
+$p_X$. That conditional law can be concentrated on a very rare event.
+We therefore retain $p_X$ in the demand bound instead of estimating it
+from below.
+
+For a fresh guide $B\sim\pi$, define
+\[
+ \mu_X=\E\bigl[\partial_X f_2(B\setminus C_X)
+                    \mid C_X\subseteq B\bigr].
+\]
+For an error demand, \eqref{eq:context-error} and
+$(a-b)^2\le2(a-c)^2+2(b-c)^2$ imply
+\[
+ E_d^2\le 2\sum_{j=0}^1
+  \bigl(\partial_X f_1(A^{(j)}\setminus T_X)-\mu_X\bigr)^2.
+\]
+Applying \eqref{eq:conditional-demand-mass}, and then Jensen's
+inequality to the conditional expectation defining $\mu_X$, gives
+\begin{align}
+ \sum_{d\text{ error}}\rho(d)E_d^2
+ &\le4A_0^4\calT
+    \sum_X\sum_{A\supseteq T_X}\pi(A)p_X
+    \bigl(\partial_X f_1(A\setminus T_X)-\mu_X\bigr)^2
+       \notag\\
+ &\le4A_0^4\calT\,\Sigma_{\rm swap},
+ \label{eq:error-demand-load}
+\end{align}
+where
+\begin{equation}
+\label{eq:swap-sum}
+ \Sigma_{\rm swap}=
+ \sum_X\sum_{\substack{A\supseteq T_X\\ B\supseteq C_X}}
+ \pi(A)\pi(B)
+ \bigl(\partial_X f_1(A\setminus T_X)
+       -\partial_X f_2(B\setminus C_X)\bigr)^2.
+\end{equation}
+In detail, for a fixed $X,A$ Jensen bounds the squared difference
+from $\mu_X$ by
+\[
+ \frac1{p_X}\sum_{B\supseteq C_X}\pi(B)
+ \bigl(\partial_X f_1(A\setminus T_X)
+       -\partial_X f_2(B\setminus C_X)\bigr)^2.
+\]
+Multiplication by the $p_X$ in \eqref{eq:conditional-demand-mass}
+removes this denominator exactly.  No lower bound on $p_X$ is used.
+
+\begin{lemma}[Swap sum]
+\label{lem:swap-sum}
+The sum in \eqref{eq:swap-sum} satisfies $\Sigma_{\rm swap}\le12N^2E$.
+\end{lemma}
+
+\begin{proof}
+For a pair in \eqref{eq:swap-sum}, the matchings $T_X$ and $C_X$
+cover exactly the same vertex set, and their union is one alternating
+cycle: the path $X$ together with its closing chord.  Their edges
+are disjoint since $X$ is long and simple.  Thus this is an entire
+discrepancy component of $A,B$, not a portion of a larger one.
+Swapping it changes $F$ by exactly
+\[
+ \partial_X f_1(A\setminus T_X)
+       -\partial_X f_2(B\setminus C_X).
+\]
+For a fixed fresh pair $(A,B)$ and one of its discrepancy cycles,
+an identity $X$ is determined by marking the closing chord in the
+second matching and choosing a direction around that cycle.  The
+remaining colored path is then read from the pair.  There are at
+most $2N$ such choices.  Consequently
+\[
+ \Sigma_{\rm swap}\le2N\sum_{A,B}\pi(A)\pi(B)
+     \sum_{C\in\mathcal C(A,B)}
+       \bigl(F(A,B)-F((A,B)^C)\bigr)^2
+ \le12N^2E
+\]
+by \eqref{eq:cycle-energy}.
+\end{proof}
+
+\subsection{Completion of the energy proof}
+
+\begin{proof}[Proof of \cref{thm:pair-energy}]
+Take independent $I_1,I_2$ with law $\pi$.  Toggle their discrepancy
+cycles in the order of their least vertices, converting $I_1$ to
+$I_2$.  Cauchy--Schwarz over the at most $N$ cycles gives
+\[
+ (f_1(I_1)-f_1(I_2))^2
+ \le N\sum_C
+   (\text{the $f_1$ difference at the step for $C$})^2.
+\]
+For each selected $C$, toggle all earlier cycles in \emph{both}
+demand matchings.  This preserves their product weight, their union,
+and the selected cycle.  It is its own inverse, since the union fixes
+the same cycle order.  The term at that step becomes $D_C^2$ for the
+new demand pair, with the definition used above.  Since
+$\Var_\pi f_1=\tfrac12\E(f_1(I_1)-f_1(I_2))^2$, we obtain the
+convenient slightly weaker bound
+\begin{equation}
+\label{eq:cycle-demand-variance}
+ \Var_\pi f_1\le
+ N\sum_{I_1,I_2}\pi(I_1)\pi(I_2)
+       \sum_{C\in\mathcal C(I_1,I_2)}D_C^2.
+\end{equation}
+
+A length-two discrepancy cycle is a color change in $I_1$.
+For a fixed $I_1$ and specified change, summing the mass of $I_2$
+that supplies the other color costs at most one.  Its total
+contribution to the sum in \eqref{eq:cycle-demand-variance} is
+therefore at most $12NA_0E$, by \eqref{eq:color-energy}.
+
+For a simple cycle, there are fewer than $N$ cells, so
+\eqref{eq:global-cell-identity} and Cauchy--Schwarz give
+\[
+ D_C^2\le3N\sum_{\text{cells}}
+                 (\Delta^2+E_P^2+E_Q^2).
+\]
+Sum over all such cycle demands.  The errors with a short side
+vanish; all others are precisely the error demands already bounded.
+Equations \eqref{eq:switch-demand-load},
+\eqref{eq:error-demand-load}, and \cref{lem:swap-sum} bound the
+simple-cycle contribution by
+\[
+ 3N\bigl(60N^2A_0^6\calT+48N^2A_0^4\calT\bigr)E.
+\]
+Combining the two cycle types in \eqref{eq:cycle-demand-variance}
+and using $A_0,\calT\ge1$ yields
+\[
+ \Var_\pi f_1
+ \le\bigl(12N^2A_0+324N^4A_0^6\calT\bigr)E
+ \le1000N^4A_0^6\calT E.
+\]
+Finally
+\[
+ 1000N^4A_0^6\calT
+ =10^{51}N^{34}D^6
+ \le(10^4ND)^{100}=L.
+\]
+All comparisons used only \cref{lem:cell-comparability} and the
+adjacent-label comparison for color changes.  Both hold under any
+common clamp, proving the final assertion as well.
+\end{proof}
+\section{A sampler from replicated adjacent tiers}
+\label{sec:ladder}
+
+The two-coordinate inequality controls an additive function of a pair;
+it is not a Poincar\'e inequality for all functions of that pair.
+We now turn it into a Poincar\'e inequality on a larger product space.
+The construction uses adjacent weight tiers, with many independent
+copies at each tier. Orthogonality of product projections controls the
+terms involving both members of a pair. The slot order ensures that
+no orthogonal interaction component occurs in two different pair
+residuals. Averaging over many guides then makes their total
+contribution small enough to absorb. Using several adjacent
+distributions in one product chain has antecedents in replica Monte
+Carlo~\cite{SwendsenWang1986,WangSwendsen2005}. Here the coordinates
+are colored perfect matchings, the pair moves exchange alternating
+cycles, and the effect of replication is quantified by the proof below.
+
+Throughout this section, the tree-bag graph has $N\ge2$ vertices,
+$b_{\rm tree}$ labels, maximum level at most $2K$, and the integer
+parameter $D$ of \cref{def:tree-bags}, where $K$ is a nonnegative integer.
+Its reference activities are
+positive rationals. Let $\calM$ denote its common, nonempty set of
+colored perfect matchings. By \cref{eq:bag-weight-range},
+\begin{equation}\label{eq:ladder-global-range}
+  D^{-1}\le W_e\le 3\cdot4^K.
+\end{equation}
+
+\subsection{An exact refresh at unit activities}
+
+\begin{lemma}\label{lem:base-refresh}
+The unit-activity partition function is computable by a dynamic program
+on the label tree. With exact rational categorical choices, its tables
+also produce an exact sample. The tables
+can be computed with $O((b_{\rm tree}+1)(N+1)^3)$ arithmetic
+operations on counts of $O(N\log(N+1))$ bits. Label and loop indices
+use $O(\log(b_{\rm tree}+1)+\log(N+1))$ additional bits. Once the tables are
+available, a sample uses polynomially many integer operations and at
+most $4(b_{\rm tree}+N+1)$ rational categorical choices, each with at
+most $N+1$ outcomes. A positive table also yields a matching by
+entirely deterministic backtracking.
+\end{lemma}
+
+\begin{proof}
+A colored matching assigns each vertex to the color of its matched
+edge. Conversely, assign each vertex to one of its one or two bags,
+and pair the vertices assigned to each bag. These two operations are
+inverse. Thus, if $a_l$ vertices are assigned to bag $l$, the number of
+matchings for that assignment is $\prod_l J(a_l)$, where
+\[
+ J(0)=1,\qquad J(a)=(a-1)!!\quad\text{for even }a>0,
+ \qquad J(a)=0\quad\text{for odd }a.
+\]
+The values of $J$ count pairings of a complete graph.
+
+For a nonroot bag $C$, write $S_C$ for its interface with its parent,
+and put $s_C=|S_C|$; for the root use $S_C=\varnothing$.
+Let $u_C$ be the number of vertices belonging to $C$ alone.
+All interfaces incident to a bag are disjoint.
+For a fixed subset of $k$ vertices of $S_C$ assigned to $C$, let
+$F_C(k)$ count the assignments and pairings within the subtree of $C$;
+the other vertices of $S_C$ are assigned to its parent.
+The answer depends only on $k$: vertices of $S_C$ belong to no bag
+other than $C$ and its parent, and they are interchangeable in the
+complete clique of color $C$.
+
+If $C_1,\ldots,C_d$ are the children of $C$, set $s_i=s_{C_i}$.
+Choosing $\ell_i$ vertices of the $i$th interface to be assigned
+downward gives the recurrence
+\begin{equation}\label{eq:base-dp}
+ F_C(k)=\sum_{\ell_1=0}^{s_1}\cdots\sum_{\ell_d=0}^{s_d}
+ J\!\left(u_C+k+\sum_{i=1}^d(s_i-\ell_i)\right)
+ \prod_{i=1}^d\binom{s_i}{\ell_i}F_{C_i}(\ell_i).
+\end{equation}
+The required partition function is $F_{\rm root}(0)$.
+There is no exponential enumeration in evaluating this recurrence.
+Form the polynomial
+\[
+ H_C(z)=\prod_{i=1}^d
+ \left(\sum_{\ell=0}^{s_i}
+       \binom{s_i}{\ell}F_{C_i}(\ell)z^{s_i-\ell}\right).
+\]
+Its degree is at most $N$, and
+\[
+ F_C(k)=\sum_{a=0}^N[z^a]H_C(z)\,J(u_C+k+a),
+\]
+with impossible indices contributing zero. Naive convolutions suffice
+for the asserted arithmetic bound. The intermediate counts enumerate
+assignments and partial pairings on at most $N$ vertices. There are
+at most $2^N$ assignments, and for any one assignment at most $N!$
+such pairings, so $2^N N!$ bounds these counts. This gives the bit bound.
+
+For sampling, begin at the root with $k=0$. At a visited bag, choose
+$a$ with probability proportional to $[z^a]H_C(z)J(u_C+k+a)$.
+Backtrack the stored convolutions to choose the individual $\ell_i$,
+then choose uniformly the corresponding interface subsets. Recurse
+on each child with its chosen subset, and independently choose a
+uniform pairing of the vertices assigned to the current bag. Every
+choice is proportional to its number of completions in
+\cref{eq:base-dp}; induction on the subtree therefore gives the
+uniform law on colored matchings.
+A uniform subset is generated by sequential inclusion choices, and a
+uniform pairing by repeatedly pairing the least remaining vertex
+with a uniformly chosen other vertex. The total number of interface
+vertices is at most $N$, since a shared vertex belongs to exactly one
+interface. There are at most $N/2$ partner choices, $b_{\rm tree}$ total-size
+choices, and $b_{\rm tree}-1$ convolution choices. This proves the stated draw bound. Choosing any
+positive-count branch instead yields a deterministic matching.
+\end{proof}
+
+\subsection{The product chain}
+
+Set
+\begin{equation}\label{eq:ladder-parameters}
+ c=1+N^{-2},\qquad
+ T=\min\{t\in\mathbb Z_{\ge0}:c^t\ge\max(D,3\cdot4^K)\},
+ \qquad q=32L,
+\end{equation}
+where $L=(10^4ND)^{100}$ is the integer in
+\cref{thm:pair-energy}. At tier $t\in\{0,\ldots,T\}$, clamp every
+reference activity to $[c^{-t},c^t]$ and denote the matching law by
+$\pi_t$. Thus $\pi_0$ is the uniform colored matching law and
+$\pi_T$ is the desired law. Moreover,
+\begin{equation}\label{eq:adjacent-density}
+ \frac12\le\frac{\pi_{t-1}(A)}{\pi_t(A)}\le2
+ \qquad(A\in\calM,\ 1\le t\le T).
+\end{equation}
+Indeed, an individual activity changes by a factor in $[c^{-1},c]$,
+so unnormalized matching weights and partition functions each change
+by factors in $[c^{-N/2},c^{N/2}]$. The probability ratio is therefore
+in $[c^{-N},c^N]$, and
+$c^N\le\exp(1/N)<2$.
+Also $\log(1+N^{-2})\ge(2N^2)^{-1}$, whence
+\begin{equation}\label{eq:ladder-tier-bound}
+ T\le1+2N^2\log\max(D,3\cdot4^K)
+   =O\bigl(N^2(K+\log D+1)\bigr).
+\end{equation}
+Here and below, unbased logarithms are natural.
+
+Use $q$ slots at each tier, with product stationary law
+\[
+ \nu=\bigotimes_{t=0}^T\pi_t^{\otimes q}.
+\]
+An allowed pair $(x,y)$ consists of a slot $x$ at tier $t\ge1$ and
+any slot $y$ at tier $t-1$. On this pair use exactly the symmetric
+proposal of \cref{thm:pair-energy}, with local proposals acting on
+$x$, and accept by Metropolis for $\pi_t\otimes\pi_{t-1}$.
+Across unequal tiers a cycle swap uses this Metropolis acceptance
+ratio as well; the acceptance-one observation for two identical laws
+in \cref{sec:energy} no longer applies. For each base slot there is
+instead the update that refreshes it from $\pi_0$, using
+\cref{lem:base-refresh}. There are
+\begin{equation}\label{eq:ladder-update-count}
+ M=Tq^2+q
+\end{equation}
+updates. The chain $P$ holds with probability $1/2$; otherwise it
+chooses one of these $M$ updates uniformly. Every update preserves
+$\nu$ and is reversible, so the same is true of $P$.
+
+Write $\calE_{xy}$ for the energy of one allowed pair update, with
+all other coordinates integrated against $\nu$, and $\calE_i$ for
+the energy of a refresh of base slot $i$. In this notation
+\begin{equation}\label{eq:ladder-scan-energy}
+ \calE_{\nu,P}(f)
+   =\frac1{2M}\left(\sum_{xy}\calE_{xy}(f)
+                         +\sum_{i\text{ base}}\calE_i(f)\right).
+\end{equation}
+
+For later use, the two-coordinate inequality also applies across
+adjacent tiers. The pair state space and symmetric proposal are
+identical for the targets $\pi_t\otimes\pi_t$ and
+$\pi_t\otimes\pi_{t-1}$. By \cref{eq:adjacent-density}, the latter
+target density is at least half the former at every pair state.
+The Metropolis capacity formula \cref{eq:metropolis-capacity} thus
+shows that the unequal-tier energy dominates half the equal-tier
+energy. Consequently, for any functions $a$ and $b$ of the two slots,
+\begin{equation}\label{eq:unequal-pair-energy}
+ \Var_{\pi_t}a
+ \le 2L\,\calE_{xy}\bigl(a(x)+b(y)\bigr).
+\end{equation}
+In this display the energy is on the two slots alone; it can also be
+applied conditionally on any fixed values of other slots.
+
+\begin{lemma}\label{lem:replica-gap}
+The lazy product chain has spectral gap at least $3/(8M)$, and hence
+at least $1/(20LM)$. The dependence on $L$ in the stronger bound
+is carried by $q=32L$, and hence by $M$.
+\end{lemma}
+
+\begin{proof}
+Order the slots from highest tier to lowest, resolving ties
+arbitrarily. Thus every allowed guide $y$ follows its upper slot $x$.
+For each slot $x$, let $U_x$ be the set of its predecessors and put
+\[
+ h_x=\E[f\mid U_x,x]-\E[f\mid U_x].
+\]
+These are the orthogonal martingale increments, so
+\begin{equation}\label{eq:ladder-martingale}
+ \Var_\nu f=\sum_x\|h_x\|_{L^2(\nu)}^2.
+\end{equation}
+Fix an allowed pair $(x,y)$, abbreviate $U=U_x$, and define
+\[
+ k_{xy}=\E[f\mid U,y]-\E[f\mid U],\qquad
+ g_{xy}=\E[f\mid U,x,y],
+\]
+\begin{equation}\label{eq:ladder-residual}
+ r_{xy}=g_{xy}-\E[f\mid U]-h_x-k_{xy}.
+\end{equation}
+For fixed $U$, $h_x$ is centered over $x$ and $k_{xy}$ over $y$.
+Applying \cref{eq:unequal-pair-energy} and then averaging over $U$
+gives
+\[
+ \|h_x\|_2^2\le2L\calE_{xy}(h_x+k_{xy}).
+\]
+Conditional expectation onto $U,x,y$ decreases $\calE_{xy}$: the
+pair kernel is independent of all the other coordinates, and Jensen's
+inequality applies to each difference across a transition.
+For any stationary Markov kernel, \cref{eq:dirichlet} and
+$(a-b)^2\le2a^2+2b^2$ give $\calE(v)\le2\|v\|_2^2$.
+Using \cref{eq:ladder-residual}, and observing that functions of $U$
+have zero pair energy, we conclude that
+\begin{align}
+ \|h_x\|_2^2
+ &\le 2L\bigl(2\calE_{xy}(g_{xy})+2\calE_{xy}(r_{xy})\bigr)
+ \nonumber\\
+ &\le4L\calE_{xy}(f)+8L\|r_{xy}\|_2^2.
+ \label{eq:ladder-one-pair}
+\end{align}
+
+We now have an estimate for one upper slot and one guide, but it
+contains a residual involving both slots. Replication helps only if
+these residual terms can be charged collectively. The ordering of
+slots makes that collective estimate possible.
+
+The key point is that the residuals in this bound are orthogonal
+\emph{over all allowed pairs}. To verify it explicitly, use the
+orthogonal product (ANOVA) decomposition
+of Efron and Stein~\cite[Section~2]{EfronStein1981}, written here as
+\[
+ f=\sum_{S}f_S,\qquad
+ f_S=\sum_{A\subseteq S}(-1)^{|S|-|A|}\E[f\mid A],
+\]
+where $S$ ranges over subsets of the slot set.
+Each $f_S$ depends only on the coordinates in $S$ and is centered in
+each coordinate of $S$. Therefore distinct $f_S$ are orthogonal, and
+$\E[f\mid A]=\sum_{S\subseteq A}f_S$.
+Substitution into \cref{eq:ladder-residual} gives
+\begin{equation}\label{eq:ladder-anova}
+ r_{xy}=\sum_{\substack{S\subseteq U_x\cup\{x,y\}\\x,y\in S}}f_S.
+\end{equation}
+Every set $S$ in this sum has $y$ last and $x$ penultimate in the
+fixed slot order: all its other coordinates lie in $U_x$. These two
+positions determine the allowed pair uniquely. Thus even pairs that
+share a slot use disjoint sets of orthogonal components. Since every
+component here is nonconstant,
+\begin{equation}\label{eq:ladder-residual-sum}
+ \sum_{xy}\|r_{xy}\|_2^2\le\Var_\nu f.
+\end{equation}
+This decomposition is only an identity in the proof; the algorithm
+does not compute it.
+
+For a base slot $i$, conditional expectation onto $U_i,i$ similarly
+decreases its refresh energy. The refresh energy of $h_i$ equals
+$\|h_i\|_2^2$, because $h_i$ is centered over $i$ given $U_i$.
+Hence $\|h_i\|_2^2\le\calE_i(f)$.
+Average \cref{eq:ladder-one-pair} over the $q$ guides of each nonbase
+slot and sum over slots.
+\cref{eq:ladder-martingale,eq:ladder-residual-sum} yield
+\[
+ \Var_\nu f
+ \le\frac{4L}{q}\sum_{xy}\calE_{xy}(f)
+       +\frac{8L}{q}\Var_\nu f
+       +\sum_{i\text{ base}}\calE_i(f).
+\]
+Since $q=32L$, absorption gives
+\[
+ \Var_\nu f
+ \le\frac16\sum_{xy}\calE_{xy}(f)
+       +\frac43\sum_{i\text{ base}}\calE_i(f)
+ \le\frac{8M}{3}\calE_{\nu,P}(f).
+\]
+This proves the first gap bound. The second follows from $L\ge1$.
+\end{proof}
+
+\subsection{A quantitative sampling guarantee}
+
+\begin{theorem}\label{thm:bag-sampler}
+Define $T,q,M$ by
+\cref{eq:ladder-parameters,eq:ladder-update-count}, and put
+\begin{align}
+ \Lambda&=10Nq(T+1)
+       \bigl(K+\lceil\log_2(ND+2)\rceil+1\bigr),
+       \label{eq:ladder-lambda}\\
+ s(\xi)&=\left\lceil20LM
+       \bigl(\Lambda+\lceil\log_2(1/\xi)\rceil+1\bigr)\right\rceil
+       \qquad(0<\xi<1).
+       \label{eq:ladder-steps}
+\end{align}
+Initialize all slots to any one colored perfect matching and run the
+exact chain for $s(\xi)$ steps. The resulting product law is at total
+variation distance at most $\xi$ from $\nu$. In particular, any
+top-tier slot has law within $\xi$ of the desired weighted matching
+law.
+
+All tables, state representations, and transition probabilities are
+computable by rational arithmetic. Their sizes, arithmetic operation
+counts, and rational bit lengths are polynomial in $N,D,K$,
+$b_{\rm tree}$, $\log(1/\xi)$, and the bit lengths of the reference
+activities. Exact rational choices here define an ideal process;
+the bounded-bit implementation in \cref{sec:complexity} adds at most
+$\xi$ to the total variation error.
+\end{theorem}
+
+\begin{proof}
+Every colored matching consists of $N/2$ vertex pairs, each with at
+most two color choices. In particular, $|\calM|\le(2N^2)^N$.
+At every tier, all activities remain in the interval
+\cref{eq:ladder-global-range}. For any $A\in\calM$ this gives
+\[
+ \pi_t(A)\ge
+ \frac1{(2N^2)^N(3D4^K)^{N/2}}.
+\]
+Taking the product over $q(T+1)$ slots and using $N\ge2,D\ge1$
+shows that
+\begin{equation}\label{eq:ladder-minimum-mass}
+ \log\frac1{\min\nu}\le\Lambda.
+\end{equation}
+For example, with $a=\lceil\log_2(ND+2)\rceil$, the logarithm of the
+denominator in the preceding display is at most
+$N(K+\tfrac52a+2)$, which is at most $10N(K+a+1)$.
+
+For completeness, the finite-state spectral argument gives a direct
+mixing bound. Reversibility makes $P$ self-adjoint on $L^2(\nu)$.
+Its laziness makes its eigenvalues nonnegative, and
+\cref{lem:replica-gap} bounds every eigenvalue on the centered subspace above by
+$1-\gamma$, where $\gamma=1/(20LM)$.
+Starting from a point $z$, the centered initial density has squared
+norm $1/\nu(z)-1$. By diagonalization, reversibility, and
+\cref{eq:ladder-minimum-mass},
+\[
+ \|P^s(z,\cdot)-\nu\|_{\TV}
+ \le\frac12(1-\gamma)^s\sqrt{\frac1{\nu(z)}-1}
+ \le\frac12\exp(-\gamma s+\Lambda/2).
+\]
+Substitution of \cref{eq:ladder-steps} makes this at most $\xi$.
+Projection onto a chosen top-tier coordinate cannot increase total
+variation.
+
+It remains to specify the computation represented by this chain.
+The unit-activity tables are supplied by \cref{lem:base-refresh}; a
+positive table supplies an initial matching if one has not already
+been constructed. There are at most $N^2$ colored edges and
+$q(T+1)$ matching coordinates. Precompute the clamped activities at
+every tier using rational powers of $c$. The tier bound
+\cref{eq:ladder-tier-bound} is polynomial in the stated parameters,
+and these powers have polynomial bit length.
+A scan choice can be made by an integer in $\{1,\ldots,M\}$,
+decoding either a base slot or an upper tier and two slot indices.
+A pair proposal is implemented by inspecting two matchings: the
+alternating components can be listed by following the union of their
+edges, and the local switch and color proposals require only finite
+lists of edges and at most two color choices per new edge.
+Metropolis acceptance uses a ratio of products of at most $2N$
+activities; the unknown partition functions cancel. A refresh uses
+the precomputed integer tables. Thus a step has polynomial arithmetic
+cost and rational probabilities of polynomial bit length.
+The bound $s(\xi)$ is itself polynomial in the displayed parameters,
+which proves the assertion about the ideal computation.
+
+For the finite-bit assertion in this generality, including arbitrarily
+many empty bags, use the draw and option bounds
+\[
+ R_{\rm draw}^{\rm gen}=100(s(\xi)+1)(b_{\rm tree}+N+1),\qquad
+ R_{\rm opt}=10(M+N^2+1).
+\]
+Indeed a refresh uses at most $4(b_{\rm tree}+N+1)$ categorical
+choices, and laziness and the scan use two more. A pair step uses at
+most eight choices. A final revelation of edge types, when present,
+uses at most $N$ further choices. These fit the displayed draw bound.
+Every choice has at most $R_{\rm opt}$ options. Take a fixed integer
+$k$ with $2^k\ge2R_{\rm draw}^{\rm gen}R_{\rm opt}/\xi$.
+\Cref{lem:finite-random-choice} and successive coupling then bound
+the cumulative simulation discrepancy by $\xi$. This precision and
+the resulting bit cost are polynomial in the stated parameters,
+including $b_{\rm tree}$. For the hierarchy constructed in
+\cref{sec:enlargement}, the smaller specialized bookkeeping in
+\cref{sec:complexity} follows from
+$b_{\rm tree}\le N(2K+1)$.
+\end{proof}
+\section{From samples to the number of perfect matchings}
+\label{sec:counting}
+
+We now give the counting algorithm. Its sampling calls use only the
+tree-bag sampler proved above. In particular, neither balancing nor
+estimating a partition ratio requires an additional oracle.
+Positive completion, gradual suppression of nonedges, and estimation
+of successive partition ratios follow the annealing framework used
+for the permanent~\cite{JerrumSinclairVigoda2004}. The marked observables
+and vertex-scale update below supply the general-graph implementation
+and its balance guarantees.
+
+\subsection{The schedule and vertex scales}
+If $n=|V|=0$, return one. If a binary vertex count is used and
+$n>2|E|$, return zero: the listed edges cannot cover all vertices.
+On the remaining nonempty input, perform a deterministic
+perfect-matching existence test and return zero if it fails. Thus the
+remaining case has even $n\ge2$ and $Z(G)\ge1$.
+Set
+\begin{equation}\label{eq:schedule}
+ b=1+\frac1n,\qquad
+ K=\min\left\{k\in\mathbb N:k\ge1,\quad
+                    n^{n/2}b^{-k}\le\frac\eps{32}\right\}.
+\end{equation}
+Since $\log(1+1/n)\ge1/(2n)$,
+\begin{equation}\label{eq:schedule-length}
+ K\le 1+2n\left(\frac n2\log n+\log(32/\eps)\right).
+\end{equation}
+At stage $j\in\{0,\ldots,K\}$ the complete logical graph has raw
+activities
+\[
+ w^{(j)}_{uv}=
+ \begin{cases}1,&uv\in E,\\ b^{-j},&uv\notin E.\end{cases}
+\]
+Write $Z_j$ for its perfect-matching partition function. Then
+\begin{equation}\label{eq:raw-telescoping}
+ Z_0=(n-1)!!,\qquad
+ Z_K=(n-1)!!\prod_{j=0}^{K-1}\frac{Z_{j+1}}{Z_j}.
+\end{equation}
+
+For sampling, we additionally maintain positive vertex scales $s_i$ and
+use activities $\lambda_{ij}=w^{(j)}_{ij}s_i s_j$. Every perfect matching
+has the same vertex factor $\prod_i s_i$. Consequently
+\begin{equation}\label{eq:scaling-identities}
+ Z_\lambda=Z_j\prod_i s_i,
+ \qquad
+ g_\lambda(ij)=\frac{g_{w^{(j)}}(ij)}{s_i s_j}.
+\end{equation}
+Initialize $s_i=1/\floor{\sqrt{n-1}}$. At stage zero,
+\[
+ g_\lambda(ij)=\frac{\floor{\sqrt{n-1}}^2}{n-1}\in[1/4,1],
+\]
+so the logical graph is balanced. The update defined below always
+multiplies each scale by a positive number at most two, whether or not
+the estimates succeed. Starting from scales at most one and making at
+most $K$ updates, we therefore have at every sampling call
+\begin{equation}\label{eq:unconditional-height-bound}
+ 0<\lambda_{ij}\le4^K.
+\end{equation}
+We use the same structural parameters throughout a trial:
+\begin{equation}\label{eq:counting-structural-parameters}
+ D_0=10^8(n+1)^4,\qquad p=2K+2,\qquad
+ N=n+4p\binom n2,\qquad D=100N^2D_0.
+\end{equation}
+For each current $\lambda$, construct the enlarged graph and marked
+bag edges of \cref{sec:enlargement}.
+
+\subsection{Observables from one marked sample}
+Let $\lambda^+$ denote the activities at the next raw stage, with the
+\emph{current} vertex scales retained. If $P$ is a logical perfect
+matching, or a logical matching missing one specified pair of vertices,
+put
+\[
+ u(P)=b^{-\#\{e\in P:e\notin E\}}.
+\]
+Thus $\wt_{\lambda^+}(P)=u(P)\wt_\lambda(P)$ and
+\begin{equation}\label{eq:bounded-reweighting}
+ \frac12<b^{-n/2}\le u(P)\le1.
+\end{equation}
+The strict left inequality follows from
+$(1+1/n)^{n/2}<\mathrm e^{1/2}<2$.
+
+Sample a colored perfect matching from the bag graph and reveal the
+real, virtual, and probe portions of its edges. Define the following
+observables, all in $[0,1]$:
+\begin{align*}
+ a&=\one_{\{\text{all edges real}\}},\\
+ d&=\one_{\{\text{all edges real}\}}u(P),\\
+ z_{ij}&=\one_{\{\text{probe }ij\text{ and all other edges real}\}}u(P).
+\end{align*}
+Here $P$ is decoded only on the indicated event; elsewhere the observable
+is zero. On the all-real event it is a full logical perfect matching.
+On the sole-probe event it is a logical perfect matching missing $i,j$.
+Write $I=Z_{\rm bag}/Z'$ for the inflation factor. Under the exact
+marked bag law, the means satisfy
+\begin{equation}\label{eq:observable-means}
+ \E a=\frac1I,\qquad
+ \E d=\frac{Z_{\lambda^+}}{I Z_\lambda},\qquad
+ \E z_{ij}=\frac{Z_{\lambda^+}(-ij)}{D I Z_\lambda}.
+\end{equation}
+The all-real probability in \cref{prop:marked-laws} gives the first
+equality. The other two follow by multiplying the weights in that event,
+or in the sole-probe
+identity \eqref{eq:probe}, by $u(P)$ and summing. In particular,
+\begin{equation}\label{eq:observable-ratios}
+ R:=\frac{Z_{j+1}}{Z_j}
+   =\frac{Z_{\lambda^+}}{Z_\lambda}
+   =\frac{\E d}{\E a},\qquad
+ g_{\lambda^+}(ij)=D\frac{\E z_{ij}}{\E d}.
+\end{equation}
+Thus one batch of marked samples will estimate both the next raw
+partition ratio and the two-hole ratios needed to choose the next
+vertex scales.
+
+\subsection{Estimating and restoring balance}
+Use
+\begin{equation}\label{eq:statistical-parameters}
+ \alpha=10^{-5},\qquad
+ \sigma=\min\left\{\frac\alpha{100D},\frac\eps{1000K}\right\},\qquad
+ S=\ceil{10\sigma^{-2}(n+K+1)},\qquad
+ \xi=\frac1{1000KS}.
+\end{equation}
+At each stage make $S$ independent fresh sampler runs, each with marked
+output law at total variation distance at most $3\xi$ from the exact
+marked bag law. \Cref{sec:complexity} implements these runs in bounded
+bit time. Let bars denote the empirical means of the observables.
+Set
+\begin{equation}\label{eq:ratio-estimate}
+ \widehat R=
+ \begin{cases}\bar d/\bar a,&\bar a>0,\\1,&\bar a=0.
+ \end{cases}
+\end{equation}
+If $\bar d>0$, form the symmetric array
+\begin{equation}\label{eq:upper-estimate}
+ G_{ij}=\max\left\{\alpha,
+             \min\left\{3,D\frac{\bar z_{ij}}{\bar d}+\alpha\right\}
+                  \right\}
+ \quad (i\ne j).
+\end{equation}
+If $\bar d=0$, set all $G_{ij}=1$. The following elementary balancing
+procedure will be useful.
+
+\begin{lemma}\label{lem:one-pass-scaling}
+Suppose $G$ is symmetric and $\alpha\le G_{ij}\le3$ for $i\ne j$.
+Initialize $v_i=2$ for every vertex and visit the vertices in their
+fixed order, setting
+\begin{equation}\label{eq:scale-update}
+ v_i\leftarrow\max_{j\ne i}\frac{G_{ij}}{v_j}.
+\end{equation}
+Every coordinate only decreases, all final coordinates lie in
+$[\alpha/2,2]$, and the final vector satisfies
+$v_i v_j\ge G_{ij}$ with an equality in every row.
+\end{lemma}
+\begin{proof}
+Initially all products equal four and hence dominate $G$. At an update,
+feasibility shows that the new value is at most the old value. The
+definition restores all constraints involving $i$ and leaves all other
+constraints unchanged. Since every other coordinate is at most two,
+the new value is at least $\alpha/2$. At least one constraint involving
+$i$ is tight. If its neighbor is visited later, that neighbor cannot
+decrease without violating this tight constraint. Thus the equality
+persists to the end.
+\end{proof}
+
+After estimating the current ratio, perform this procedure and replace
+$s_i$ by $s_i v_i$. These scales are used with the next raw stage.
+
+\paragraph{Algorithm outline.}
+After the zero test, one trial uses the parameters in
+\cref{eq:schedule,eq:counting-structural-parameters,eq:statistical-parameters}
+and the initial scales above. For $j=0,\ldots,K-1$:
+\begin{enumerate}[label=\textup{\arabic*.},leftmargin=*]
+\item Form $\lambda_{uv}=w^{(j)}_{uv}s_us_v$ and its marked bag graph.
+\item Obtain $S$ independent fresh marked samples using the bounded-bit
+implementation of \cref{thm:bag-sampler}. Compute the empirical
+means, then $\widehat R_j$ and $G$ by
+\cref{eq:ratio-estimate,eq:upper-estimate}, including their
+zero-denominator conventions.
+\item Run the one-pass update \eqref{eq:scale-update} on $G$, replace
+$s_i$ by $s_iv_i$, and advance to the next raw stage.
+\end{enumerate}
+The trial returns $Y=(n-1)!!\prod_{j=0}^{K-1}\widehat R_j$;
+the final algorithm takes the median of independent trials as specified
+below. The current scales are held fixed during each ratio estimate,
+so their common factors cancel. Every empirical history executes these
+same steps with the unconditional caps above. Balance is used only to
+prove the accuracy of histories whose estimates succeed.
+
+\begin{lemma}\label{lem:successful-stage}
+Suppose the current logical activities are balanced and every empirical
+mean is within $\sigma$ of its exact mean. Then the next scaled
+activities are balanced and
+\begin{equation}\label{eq:stage-relative-error}
+ \left|\frac{\widehat R}{R}-1\right|
+ \le10\sigma\le\frac\eps{100K}.
+\end{equation}
+\end{lemma}
+\begin{proof}
+By \cref{prop:bag-inflation} and \eqref{eq:bounded-reweighting},
+\[
+ \E a\ge\frac12,\qquad \E d\ge\frac14,
+ \qquad\frac12\le R\le1.
+\]
+Reweighting both a full matching and a matching with two holes by
+factors in $[1/2,1]$ gives
+\[
+ \frac12 g_\lambda(ij)\le g_{\lambda^+}(ij)
+                         \le2g_\lambda(ij).
+\]
+In particular the pre-update row maxima lie in $[1/8,2]$.
+Write $g^+_{ij}=g_{\lambda^+}(ij)$. Since $\bar d\ge\E d-\sigma>0$,
+\begin{align*}
+ \left|D\frac{\bar z_{ij}}{\bar d}-g^+_{ij}\right|
+ &\le\frac{D|\bar z_{ij}-\E z_{ij}|
+              +g^+_{ij}|\bar d-\E d|}{\E d-\sigma}\\
+ &\le\frac{(D+2)\sigma}{1/4-\sigma}<\alpha.
+\end{align*}
+The last inequality follows from $D\ge1$ and
+$\sigma\le\alpha/(100D)$. Since $0\le g^+_{ij}\le2$, both truncations
+in \eqref{eq:upper-estimate} preserve
+\begin{equation}\label{eq:upper-estimate-accuracy}
+ g^+_{ij}\le G_{ij}\le g^+_{ij}+2\alpha.
+\end{equation}
+
+During the scaling procedure every updated coordinate satisfies
+$v_i\ge\max_jG_{ij}/2\ge1/16$. At the end all entries of the newly
+scaled two-hole matrix are at most one, by feasibility and
+\eqref{eq:scaling-identities}. In each row choose a final tight neighbor.
+For that pair,
+\[
+ \frac{g^+_{ij}}{v_i v_j}
+ \ge1-\frac{2\alpha}{v_i v_j}
+ \ge1-512\alpha>\frac14.
+\]
+This proves balance.
+
+For the ratio estimate, using $R\ge1/2$ and $\bar a\ge1/2-\sigma$,
+\[
+ \left|\frac{\widehat R}{R}-1\right|
+ \le\frac{|\bar d-\E d|+R|\bar a-\E a|}
+            {R(\E a-\sigma)}
+ \le\frac{2\sigma}{(1/2)(1/2-\sigma)}
+ <10\sigma.
+\]
+The final bound in \eqref{eq:stage-relative-error} is the other choice
+in \eqref{eq:statistical-parameters}.
+\end{proof}
+
+\subsection{Success probability and output}
+The form of Hoeffding's inequality~\cite[Theorem~2]{Hoeffding1963}
+used here is:
+if $X_1,\ldots,X_S$ are independent, take values in $[0,1]$, and have
+a common mean $\mu$, then, for every $t>0$,
+\begin{equation}\label{eq:hoeffding}
+ \Prb\left[\left|S^{-1}\sum_{i=1}^S X_i-\mu\right|>t\right]
+ \le2\exp(-2St^2).
+\end{equation}
+The theorem applies separately to each observable; the observables
+within a single sample need not be independent.
+
+Condition on any history in which all preceding stages succeeded.
+By \cref{lem:successful-stage}, the current activities are balanced.
+For independent ideal samples, a union bound over the at most
+$n^2+2$ observables gives failure probability at most
+\[
+ 2(n^2+2)\exp(-2S\sigma^2)
+ \le2(n^2+2)\exp(-20(n+K+1))<\frac1{100K}.
+\]
+One elementary verification of the last strict inequality is
+$\log(200K(n^2+2))\le6+K+2n<20(n+K+1)$ for $n\ge2,K\ge1$.
+The actual fresh runs can be coupled to these ideal samples with
+disagreement probability at most $3S\xi=3/(1000K)$.
+Therefore the conditional probability of failure at the first
+unsuccessful stage is at most $13/(1000K)$. Summing over all $K$
+stages proves that a whole trial succeeds with probability at least
+$1-13/1000>0.9$.
+
+The trial output is the nonnegative rational
+\begin{equation}\label{eq:trial-output}
+ Y=(n-1)!!\prod_{j=0}^{K-1}\widehat R_j.
+\end{equation}
+On success, \cref{lem:successful-stage} and
+$\prod_i(1-t_i)\ge1-\sum_i t_i$ for $t_i\in[0,1]$ imply
+\[
+ (1-\eps/100)Z_K\le Y\le\exp(\eps/100)Z_K.
+\]
+Every matching using a nonedge has raw final weight at most $b^{-K}$.
+There are at most $(n-1)!!\le n^{n/2}$ perfect matchings on the
+complete graph, so
+\begin{equation}\label{eq:padding-bias}
+ Z(G)\le Z_K\le Z(G)+n^{n/2}b^{-K}
+             \le(1+\eps/32)Z(G).
+\end{equation}
+For $0<\eps<1$, the resulting lower bound exceeds
+$(1-\eps)Z(G)$, and the upper bound is at most $(1+\eps)Z(G)$.
+For example, $\exp(\eps/100)\le1+\eps/50$, which gives
+\[
+ \exp(\eps/100)(1+\eps/32)
+ \le1+\left(\frac1{50}+\frac1{32}+\frac1{1600}\right)\eps
+ <1+\eps.
+\]
+
+To amplify, let $\ell=\ceil{\log_2(1/\delta)}$, computed by rational
+comparisons with powers of two. Run
+$r=10(\ell+1)+1$ independent trials and return their median.
+This is an odd number of trials. Since each succeeds with probability
+at least $0.9$, Hoeffding's inequality for their success indicators
+bounds the probability of fewer than half succeeding by
+\[
+ \exp\bigl(-2(0.4)^2r\bigr)
+ \le\exp\bigl(-3.2(\ell+1)\bigr)
+ <2^{-\ell}\le\delta.
+\]
+Whenever a majority succeed, the median belongs to the desired
+interval. The deterministic zero branch was taken before any trials.
+This proves the correctness and probability assertions of
+\cref{thm:main}, subject only to the bit implementation established
+next.
+\section{A bounded-bit implementation}
+\label{sec:complexity}
+
+We finish by specifying a classical finite-bit implementation and proving
+a worst-case bound. The argument is uniform over all empirical histories,
+including histories on which balance is lost. Balance controls the
+statistical accuracy of a successful trial; the unconditional caps in
+the algorithm control its running time.
+
+\subsection{Sizes of the parameters and state spaces}
+Let $\mathsf b$ be the total binary input length. We use a standard
+explicit encoding of a finite graph and of the two rationals. If isolated
+vertices are specified by a binary vertex count, first return zero when
+$n>2|E|$. This test is valid for every nonempty graph admitting a perfect
+matching and ensures that the nonzero branch has $n\le2|E|=O(\mathsf b)$.
+The deterministic existence test~\cite{Edmonds1965} therefore has
+polynomial bit cost on the original graph. It is the only
+matching-existence oracle used.
+
+\Cref{eq:schedule-length} bounds $K$ polynomially in $\mathsf b$.
+It can be found without logarithms by iterating rational powers of $b$
+and comparing the two sides of \eqref{eq:schedule}. The parameters
+$p,N,D_0,D$ are integers of polynomial numerical size in $n,K$.
+The hierarchy has at most
+\begin{equation}\label{eq:hierarchy-size}
+ H_{\rm tree}=N(2K+1)
+\end{equation}
+nodes, since its levels are $0,\ldots,2K$ and every level has at most
+$N$ components. The number of colored edges is less than $N^2$.
+
+Use the ladder parameters $c,T,L,q,M,\Lambda$ from
+\cref{eq:energy-constants,eq:ladder-parameters,eq:ladder-update-count,eq:ladder-lambda}, and write
+\[
+ t_{\rm mix}=s(\xi),
+\]
+where $s(\xi)$ is defined in \eqref{eq:ladder-steps}. The structural
+parameters $N,D,K$ are fixed throughout a trial. Hence the same $T$
+makes the final tier unclamped on every empirical history, by
+\cref{eq:unconditional-height-bound,eq:ladder-global-range}.
+It can be found by iterating rational powers of $c$ until the defining
+inequality in \eqref{eq:ladder-parameters} holds. The bound
+\eqref{eq:ladder-tier-bound} is polynomial in $N,K,D$.
+Ceilings of binary logarithms are found by integer or rational
+comparisons with powers of two.
+These bounds and the product-state size
+$q(T+1)$ are polynomial in $\mathsf b$ and $\eps^{-1}$.
+Indeed,
+\[
+ \sigma^{-1}\le C(D+K\eps^{-1}),\qquad
+ S=O\bigl((D^2+K^2\eps^{-2})(n+K+1)\bigr),\qquad
+ \xi^{-1}=1000KS,
+\]
+for an absolute constant $C$. The exponent $100$ in $L$ is fixed.
+
+There is no initial-sampling problem at the upper tiers. Pair the logical
+vertices in their fixed order and expand this matching by the path
+tilings. All logical activities are positive, so this is an available
+real perfect matching with assigned colors. Initialize every slot to
+that matching. The mixing bound applies from this deterministic product
+state.
+
+\subsection{Rational bit lengths}
+We record explicitly why the adaptive scales do not create an
+arithmetic blowup. Put
+\begin{equation}\label{eq:empirical-bit-bound}
+ B_G=1+n\ceil{\log_2(n+1)}+\ceil{\log_2(S+1)}
+                      +\ceil{\log_2(D+1)}.
+\end{equation}
+Absolute multiplicative constants in the bit bounds below are
+independent of the input.
+
+Every nonzero sample contribution is $(n/(n+1))^a$ for some
+$0\le a\le n/2$. These fractions have the common denominator
+$Q=(n+1)^{n/2}$. Thus empirical means, their ratios, and the capped
+entries $G_{ij}$ have $O(B_G)$ numerator and denominator bits. This
+bound does not involve the current scales.
+
+An assignment in \eqref{eq:scale-update} chooses one maximizer $j$ and
+sets $v_i=G_{ij}/v_j$. At that moment $v_j$ is either the initial value
+two or an expression formed at an earlier update. Following such
+dependencies strictly decreases the update index, so it produces a
+chain of at most $n$ factors. Each $v_i$ consequently has $O(nB_G)$
+bits. Accumulating at most $K$ scale updates gives $O(KnB_G)$ bits per
+scale. The raw activities have $O(K\log(n+1))$ bits. These estimates
+apply even when all observations are unfavorable, because the clipping
+and branch conventions are unconditional.
+
+Bottleneck entries $B_{ij}$ and the maxima $H_i=\max_{j\ne i}B_{ij}$
+select from finitely many input activities and one; they do not involve
+sums over paths. Initialize $B_{ij}=\max\{1,\lambda_{ij}\}$ and,
+for each vertex $k$, update the entries with distinct $i,j\ne k$ by
+\[
+ B_{ij}\leftarrow\max\{B_{ij},\min\{B_{ik},B_{kj}\}\}.
+\]
+This is the max--min Floyd--Warshall recurrence: after processing $k$,
+the entries allow the processed vertices as internal path vertices.
+It uses $O(n^3)$ rational comparisons. The path profiles add only powers of two of
+bit length $O(K)$. Bag activities are sums of a real activity and a
+dyadic height divided by $D$. Finally $c^{\pm t}$ has
+$O(T\log(N+1))$ bits for $t\le T$.
+
+It follows that every single-edge activity used by the sampler has
+numerator and denominator lengths at most the integer bound
+\begin{equation}\label{eq:edge-bit-bound}
+ B_W=C_1\bigl(KnB_G+K+\ceil{\log_2(D+1)}
+                       +T\ceil{\log_2(N+1)}+1\bigr),
+\end{equation}
+where $C_1$ is a sufficiently large absolute positive integer.
+A product of at most $N$ such activities has $O(NB_W)$ bits. In a
+Metropolis acceptance ratio the unknown normalizing constants cancel.
+Thus even a cycle exchange at unequal tiers requires only rational
+products of this size. The real and probe revelation probabilities
+are likewise rational, respectively the real portion divided by the
+collapsed activity and $1/(DW)$ for the probe portion.
+
+By \cref{lem:base-refresh}, the bottom-tier dynamic program uses
+counts of $O(N\log(N+1))$ bits. Its categorical normalizers count
+the same partial completions and obey the same bound. Label and loop
+indices use an additional
+$O(\log(H_{\rm tree}+1)+\log(N+1))$ bits, since
+$b_{\rm tree}\le H_{\rm tree}$ in the constructed hierarchy.
+The table computation and sampling therefore use polynomially many
+integer operations with polynomial bit lengths.
+
+\subsection{Random choices with a fixed number of bits}
+\begin{lemma}\label{lem:finite-random-choice}
+Let $p_1,\ldots,p_r$ be nonnegative rational numbers summing to one.
+Choose a uniform integer $J\in\{0,\ldots,2^k-1\}$ using exactly $k$
+fair bits, and return the least index $i$ such that
+$J/2^k<\sum_{j\le i}p_j$. The output law is at total variation
+distance at most $2r\,2^{-k}$ from $(p_i)_{i=1}^r$.
+\end{lemma}
+\begin{proof}
+For any interval $[a,b)\subseteq[0,1)$, the number of grid points
+$j/2^k$ in the interval differs from $2^k(b-a)$ by at most two.
+Apply this to the successive cumulative-probability intervals and sum
+the absolute probability discrepancies. The factor one half in total
+variation only improves the stated bound. An outcome with $p_i=0$
+has an empty interval and is never returned.
+\end{proof}
+
+The exact categorical choices in the unit-tier dynamic program define
+an ideal transition law. The actual algorithm implements each such
+choice, along with every proposal and acceptance choice, by the fixed
+grid in \cref{lem:finite-random-choice}. Hence exactness of the dynamic
+program is a counting identity; sampling from its tables in bounded
+bit time incurs the error budget derived below.
+
+Every choice in the chain is either uniform on an explicitly indexed
+finite set, Bernoulli, or proportional to a list of nonnegative
+integers supplied by the dynamic program. For each used edge, final
+revelation is a rational categorical choice with at most three outcomes.
+Subsets and pairings are sampled sequentially, so no exponential list is needed. Uniform choices
+among the $M$ scan positions can be implemented by integer arithmetic
+on the index rather than by enumerating the scan positions.
+
+A bound on the number of random choices in one complete sampler run,
+including marking its output, is
+\begin{equation}\label{eq:draw-bound}
+ R_{\rm draw}=100(t_{\rm mix}+1)(N+1)^2(K+2).
+\end{equation}
+To see this, laziness and the scan take two choices per step, and a
+pair update takes at most eight further choices. By
+\cref{lem:base-refresh}, a refresh takes at most
+$4(H_{\rm tree}+N+1)$ choices. Allowing both bounds at every step
+and adding the at most $N$ choices for final marking gives
+\[
+ t_{\rm mix}\bigl(10+4(H_{\rm tree}+N+1)\bigr)+N.
+\]
+Since $H_{\rm tree}=N(2K+1)$, this is bounded by
+\eqref{eq:draw-bound}. The bound therefore includes every categorical
+choice made by the implemented sampler.
+The number of options in any one choice is at most
+\begin{equation}\label{eq:option-bound}
+ R_{\rm opt}=10(M+N^2+1).
+\end{equation}
+Choose the same bit budget $k$ for all these choices, with
+\begin{equation}\label{eq:coin-precision}
+ 2^k\ge\frac{2R_{\rm draw}R_{\rm opt}}\xi.
+\end{equation}
+This integer is found by doubling; $k$ is polynomially bounded.
+By \cref{lem:finite-random-choice}, the sum of all conditional
+simulation errors is at most $\xi$. Successive coupling, while the
+simulated and exact histories agree, bounds the total variation error
+of the whole run and its marking by $\xi$. The exact-kernel mixing
+error is at most $\xi$, so the implemented marked sample is within
+$2\xi$, and in particular $3\xi$, of the required law.
+The number of fair bits used has a deterministic bound
+$kR_{\rm draw}$. There is no rejection loop or random convergence test.
+
+\subsection{Total bit cost}
+The preceding bounds already give a polynomial-time implementation.
+To make the accounting explicit, take a common bit bound
+\[
+ B_* = C\bigl(NB_W+N\log(N+1)+\mathsf b
+                 +\log(M+1)+\log(t_{\rm mix}+1)
+                 +\log(\xi^{-1}+1)+1\bigr)
+\]
+with a sufficiently large absolute constant $C$ and $B_W$ from
+\eqref{eq:edge-bit-bound}. In particular $NB_W$ dominates $KB_G$,
+the bit cost of multiplying the $K$ empirical ratios.
+The bound $B_*$ covers all integers
+used in weight products, dynamic-program counts, cumulative choices,
+acceptance comparisons, and output products. All categorical laws just
+described admit common-denominator integer weights, so computing their
+cumulative sums respects the same bound. Schoolbook arithmetic and the
+Euclidean algorithm perform the needed operations within $O(B_*^3)$
+bit time per operation.
+
+A deliberately loose bound of
+$O((H_{\rm tree}+1)(N+1)^4)$ arithmetic operations covers graph
+construction, the bottom-tier tables, and any one transition or refresh;
+storing the product state additionally takes
+$O(q(T+1)N B_*)$ bits. Thus a trial, including all $K$ stages and $S$
+fresh runs at each stage, has bit cost bounded by a fixed polynomial
+in
+\[
+ K,\ S,\ t_{\rm mix},\ H_{\rm tree},\ N,\ q(T+1),\ B_*.
+\]
+For example, the sum of
+\[
+ CKS(t_{\rm mix}+1)(H_{\rm tree}+1)(N+1)^4B_*^3
+ \quad\text{and}\quad
+ CKSq(T+1)(N+1)B_*^3
+\]
+is a valid coarse envelope for arithmetic, repeated initialization,
+and storage operations. Each displayed parameter is polynomial in
+$\mathsf b$ and $\eps^{-1}$. The
+$r=10(\ceil{\log_2(1/\delta)}+1)+1$ repetitions and the rational
+comparisons used to select their median add a factor polynomial in
+$\log\delta^{-1}$ and the input length.
+
+All data structures, loop bounds, and rational comparisons have now been
+specified by finite procedures. Every auxiliary parameter is bounded
+by a polynomial of fixed degree, and the bound holds on every random
+history. Together with \cref{sec:counting}, this completes the proof
+of \cref{thm:main}.
+\section{Prescribed-degree subgraphs and fixed-cardinality matchings}
+\label{sec:prescribed-degree}
+
+We record two counting consequences of \cref{thm:main} and a corresponding
+output-sampling guarantee. Throughout this section, the input host
+$G=(V,E)$ is a finite simple undirected unweighted graph whose vertices and
+edges are explicitly listed, including any isolated vertices. Only edges
+in $E$ may be selected, so absent edges may be forbidden arbitrarily.
+The objects counted are labeled edge subsets, not isomorphism classes.
+For binary-encoded integer demands $f=(f_v)_{v\in V}$ and a binary-encoded
+integer $k$, define
+\[
+\begin{aligned}
+ \mathcal F(G,f)&=\{F\subseteq E:\deg_F(v)=f_v\text{ for every }v\in V\},\\
+ \mathcal M_k(G)&=\{M\subseteq E:M\text{ is a matching and }|M|=k\}.
+\end{aligned}
+\]
+
+\begin{corollary}\label{cor:prescribed-degree}
+For either family $\mathcal A=\mathcal F(G,f)$ or
+$\mathcal A=\mathcal M_k(G)$, there are uniform classical randomized
+algorithms with the following guarantees.
+\begin{enumerate}[label=(\roman*)]
+\item Given rational $0<\eps<1$ and $0<\delta<1/2$, the counting algorithm
+returns a nonnegative rational $\widehat N$ such that
+\[
+ \Prb[(1-\eps)|\mathcal A|\le \widehat N\le(1+\eps)|\mathcal A|]
+ \ge1-\delta.
+\]
+If $\mathcal A$ is empty, it returns zero with certainty. Its bit running
+time on every execution is polynomial in the full input encoding length,
+$\eps^{-1}$, and $\log\delta^{-1}$.
+\item Given rational $0<\tau<1$, the sampling algorithm reports infeasibility
+exactly when $\mathcal A$ is empty. Otherwise it returns a member $X$ of
+$\mathcal A$ on every execution, and
+\[
+ \TV(\mathcal L(X),U_{\mathcal A})\le\tau,
+\]
+where $U_{\mathcal A}$ is the uniform law on $\mathcal A$ and
+$\TV(\mu,\nu)=\frac12\sum_x|\mu(x)-\nu(x)|$. Its bit running time on every
+execution is polynomial in the full input encoding length and $\tau^{-1}$.
+\end{enumerate}
+\end{corollary}
+
+Only the stated total-variation and polynomial-in-$\tau^{-1}$ sampling
+guarantee is asserted. In particular, the statement does not assert exact
+or pointwise almost-uniform sampling. The explicit simple unweighted host
+model does not encode edge weights or compressed multiplicities.
+
+\subsection{Constant-fibre reductions}
+
+Write $n=|V|$, $m=|E|$, and $d_v=\deg_G(v)$.
+
+\paragraph{The prescribed-degree reduction.}
+We use Tutte's reduction of prescribed degrees to perfect
+matchings~\cite[pp.~348--349]{Tutte1954}, keeping the dummy vertices
+labeled so that its counting multiplicity is explicit.
+If some $f_v$ is outside $0\le f_v\le d_v$, the family is empty and we
+return zero or report infeasibility, as appropriate. For valid demands put
+$s_v=d_v-f_v$. Construct a simple unweighted graph $H_f$ with one tagged
+port $(v,e)$ for every incidence of an edge $e$ at $v$. For each original
+edge $e=\{u,v\}$, join $(u,e)$ to $(v,e)$ by a cross edge. At each $v$, add
+$s_v$ labeled private dummy vertices, each adjacent to all $d_v$ ports at
+$v$. There are no other edges. The tags distinguish ports and dummies, so
+simplicity of $G$ makes $H_f$ simple. Its sizes satisfy
+\[
+ |V(H_f)|=2m+\sum_v s_v=4m-\sum_v f_v\le4m,
+ \qquad
+ |E(H_f)|=m+\sum_v d_vs_v\le m+2m(n-1).
+\]
+Thus the gadget is explicit and polynomial in the host size; the initial
+demand check prevents expansion of arbitrarily large binary demands.
+
+Project a perfect matching of $H_f$ to the original edges whose cross edges
+it contains. Each dummy at $v$ uses a distinct $v$-port. The remaining
+$f_v$ ports must use cross edges, so the projection lies in $\mathcal F(G,f)$.
+Conversely, an $f$-factor fixes its cross edges. Its remaining $s_v$ ports at
+$v$ can be bijected to the labeled dummies in $s_v!$ ways, independently at
+each vertex. Hence every factor has the same number of preimages, and
+\begin{equation}\label{eq:prescribed-degree-fibre}
+ C_f=\prod_v(d_v-f_v)!,\qquad Z(H_f)=C_f|\mathcal F(G,f)|.
+\end{equation}
+The fibre may be factorial, but $\log C_f=O(m\log(n+1))$, so $C_f$ has
+polynomial bit length. For an edgeless host with all demands zero, both the
+unique factor and the gadget's unique perfect matching are empty, and
+$C_f=1$ by $0!=1$. Other infeasible valid inputs give gadgets
+without perfect matchings by the same correspondence.
+
+\paragraph{The fixed-cardinality reduction.}
+If $k<0$ or $2k>n$, the family is empty, so return zero or report
+infeasibility, as appropriate. Otherwise put $q=n-2k$. Form $H_k$
+by adding $q$ labeled dummy vertices to $G$, joining each dummy to every
+original vertex, and adding no edges between dummies. Keep all original
+edges. This is a simple unweighted graph with at most $2n$ vertices and
+$m+n^2$ edges. In every perfect matching, the dummies use $q$ distinct
+original vertices; the remaining $2k$ original vertices are paired by
+exactly $k$ host edges. Conversely, a $k$-matching leaves $q$ original
+vertices, which can be bijected to the dummies in $q!$ ways. Therefore
+\begin{equation}\label{eq:cardinality-fibre}
+ C_k=(n-2k)!,\qquad Z(H_k)=C_k|\mathcal M_k(G)|.
+\end{equation}
+Here $\log C_k=O(n\log(n+1))$. The same bijection covers $k=0$, $2k=n$,
+and $n=k=0$, using $0!=1$.
+
+\begin{proof}[Proof of \cref{cor:prescribed-degree}\textup{(i)}]
+Apply \cref{thm:main} to the appropriate gadget and divide its nonnegative
+rational output by the corresponding positive integer $C_f$ or $C_k$.
+Compute these factorial products exactly and perform exact rational
+division, without rounding to an integer. Their displayed bit lengths and
+the polynomial gadget sizes preserve the asserted relative error,
+confidence, certain-zero behavior, and every-execution bit bound. This
+proves the counting assertion. The correspondences also make feasibility
+equivalent to perfect-matching feasibility in the gadget.
+\end{proof}
+
+\subsection{A deletion sampler for perfect matchings}
+
+The conversion from counts to samples uses the self-reduction framework
+of Jerrum, Valiant, and Vazirani~\cite[Section~6]{JerrumValiantVazirani1986}.
+We give the deletion procedure explicitly to retain feasibility on every
+execution and to track its total-variation and bit-time bounds. It uses
+\cref{thm:main} only through its counting guarantee.
+
+\begin{lemma}\label{lem:deletion-sampler}
+Given a finite simple undirected unweighted graph $H_0$ whose vertices
+and edges are explicitly listed, and a rational $0<\tau<1$, there is a
+uniform classical randomized algorithm that reports infeasibility exactly
+when $H_0$ has no perfect matching. Otherwise it returns a perfect
+matching on every execution, and its output law is at total variation
+distance at most $\tau$ from the uniform perfect-matching law. Its bit
+running time on every execution is polynomial in the full input encoding
+length and $\tau^{-1}$.
+\end{lemma}
+
+\begin{proof}
+First decide perfect-matching feasibility by the deterministic
+polynomial-time matching algorithm of Edmonds~\cite{Edmonds1965}. Report
+infeasibility if appropriate, and return the empty matching when
+$V(H_0)=\varnothing$. Otherwise set $m_0=|E(H_0)|$ and
+$m_*=\max\{1,m_0\}$. Choose
+\[
+ \eta=\gamma=\frac{\tau}{8m_*},\qquad
+ t=\min\{j\in\mathbb N:2^j\ge 8m_*/\tau\}.
+\]
+The integer $t$ is computed by doubling and exact rational comparison.
+
+Maintain a feasible residual graph $H$ and the already selected edges.
+Choose the first residual edge $e=uv$ in a fixed deterministic ordering.
+Perfect matchings of $H$ excluding $e$ are counted by $Z(H-e)$; those
+including $e$ correspond bijectively, after deleting $e$, to perfect
+matchings of $H-u-v$. Test both child graphs for feasibility exactly.
+At least one is feasible. If only one is feasible, take it. If both are
+feasible, make two fresh calls to \cref{thm:main}, with relative error
+$\eta$ and failure probability $\gamma$, obtaining nonnegative rational
+estimates $A$ for $Z(H-u-v)$ and $B$ for $Z(H-e)$. If $A+B=0$, take the
+exclusion child. Otherwise put $\widehat p=A/(A+B)$, draw exactly $t$
+fresh fair bits as a uniform integer $J\in\{0,\ldots,2^t-1\}$, and take the
+inclusion child precisely when
+\[
+ J<\floor{2^t\widehat p}.
+\]
+Replace $H$ by the chosen child. Whenever the inclusion child is taken,
+including a forced choice, append $e$ to the selected edges. All ratio and
+floor operations are exact rational or integer operations.
+
+Every chosen child is feasible, regardless of the estimates. Each decision
+removes at least one residual edge, so there are at most $m_0$ decisions.
+When no residual edges remain, feasibility forces the residual vertex set
+to be empty. The selected edges then form a perfect matching of $H_0$ on
+every execution.
+
+For the error bound, compare this process to the ideal deletion sampler
+that takes the inclusion child with its exact count ratio. These exact
+ratios telescope, giving the uniform perfect-matching law. At a history
+where both children are feasible, write their positive counts as $z_+$ and
+$z_-$, with $p=z_+/(z_++z_-)$. When both estimates succeed, write
+$A=z_+(1+a)$ and $B=z_-(1+b)$, where $|a|,|b|\le\eta$. Then
+\[
+ |\widehat p-p|
+ =\frac{z_+z_-|a-b|}
+ {(z_++z_-)(z_+(1+a)+z_-(1+b))}
+ \le\frac{\eta}{2(1-\eta)}.
+\]
+The last step uses $z_+z_-\le(z_++z_-)^2/4$. Fresh call randomness bounds
+the conditional probability that either estimate fails by $2\gamma$,
+at every adaptive history. On failure the branch law can change by at most
+one in total variation, while the dyadic floor changes the successful
+branch probability by at most $2^{-t}$. The fallback $A+B=0$ occurs only
+on a failed estimate when both true counts are positive. Thus the
+conditional branch-law error is at most
+\[
+ \beta=\frac{\eta}{2(1-\eta)}+2\gamma+2^{-t}.
+\]
+Forced branches have zero error. Couple the two deletion processes until
+their prefixes differ. With at most $m_0$ decisions, the output
+total-variation error is at most $m_0\beta$. Since $\eta<1/8$ and
+$2^{-t}\le\eta$, we have $\beta\le4\eta$ and hence
+\[
+ m_0\beta\le4m_*\eta=\tau/2<\tau.
+\]
+
+There are at most $2m_0$ counting calls and $2m_0+1$ exact feasibility
+tests, all on simple graphs no larger than $H_0$. The inverse relative
+error is $8m_*/\tau$, and the logarithmic confidence cost is
+$\log(8m_*/\tau)$. The every-execution bit bound of \cref{thm:main} also
+bounds the bit lengths of its outputs on failed calls. Therefore the
+exact ratio and floor computations, the $t=O(\log(m_*/\tau))$ fresh bits per
+branch, and all graph updates have bit cost polynomial in the full input
+length and $\tau^{-1}$ on every execution. No rejection loop or
+random stopping criterion is used.
+
+\end{proof}
+
+\begin{proof}[Proof of \cref{cor:prescribed-degree}\textup{(ii)}]
+For sampling, apply \cref{lem:deletion-sampler} to $H_f$ or $H_k$ and
+decode by keeping the original edges represented by selected cross edges,
+or by discarding the dummy-incident edges, respectively. Exact feasibility
+testing of the
+gadget reports infeasibility precisely for an empty target family.
+Uniform perfect matchings project to the uniform target law because the
+fibres in \eqref{eq:prescribed-degree-fibre} and
+\eqref{eq:cardinality-fibre} are constant across target members.
+Deterministic decoding cannot increase total variation and takes polynomial
+time, without enumerating any fibre. Every decoded output is feasible, which
+proves the sampling assertion.
+\end{proof}
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+Robert~H. Swendsen and Jian-Sheng Wang.
+\newblock Replica Monte Carlo simulation of spin-glasses.
+\newblock \emph{Physical Review Letters}, 57\penalty0
+  (21):\penalty0 2607--2609, 1986.
+\newblock \doi{10.1103/PhysRevLett.57.2607}.
+
+\bibitem[Tutte(1954)]{Tutte1954}
+W.~T. Tutte.
+\newblock A short proof of the factor theorem for finite graphs.
+\newblock \emph{Canadian Journal of Mathematics}, 6:\penalty0 347--352, 1954.
+\newblock \doi{10.4153/CJM-1954-033-3}.
+
+\bibitem[Valiant(1979)]{Valiant1979}
+Leslie~G. Valiant.
+\newblock The complexity of computing the permanent.
+\newblock \emph{Theoretical Computer Science}, 8\penalty0 (2):\penalty0
+  189--201, 1979.
+\newblock \doi{10.1016/0304-3975(79)90044-6}.
+
+\bibitem[Wang and Swendsen(2005)]{WangSwendsen2005}
+Jian-Sheng Wang and Robert~H. Swendsen.
+\newblock Replica Monte Carlo simulation (revisited).
+\newblock \emph{Progress of Theoretical Physics Supplement}, 157:\penalty0
+  317--323, 2005.
+\newblock \doi{10.1143/PTPS.157.317}.
+\newblock Preprint version: \url{https://arxiv.org/abs/cond-mat/0407273v1}.
+
+\bibitem[Yi(2026)]{Yi2026}
+Zihong Yi.
+\newblock Diffuse Gaussian truncation for deterministic approximate counting.
+\newblock Preprint, arXiv:2609.04079v1, 2026.
+\newblock URL \url{https://arxiv.org/abs/2609.04079v1}.
+
+\end{thebibliography}
+\end{document}
diff --git a/preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/main.pdf b/preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/main.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..b33e939a2fe0465b728a5768fe55d0bd2c26c234
--- /dev/null
+++ b/preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026/main.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:1510b15725ee7e92a6c543484c6c02ce7f26347a32c1a11cd66c5890b94c6de1
+size 617073
diff --git a/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/README.md b/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..8637314ca233c4110bcd8c6df0cd1b2b7c54ffb6
--- /dev/null
+++ b/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/README.md
@@ -0,0 +1,16 @@
+# [A Gepner stability condition on every smooth quintic threefold](paper.pdf)
+
+OpenAI  
+September 24, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026,
+  author = {{OpenAI}},
+  title = {{A Gepner stability condition on every smooth quintic threefold}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/paper.pdf}{OAI:A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026}},
+  year = {2026}
+}
+```
diff --git a/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/build/figures/fifth-phase-grid.tex b/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/build/figures/fifth-phase-grid.tex
new file mode 100644
index 0000000000000000000000000000000000000000..363582f0eec1dd4ddbd384cd8980069a025aa21d
--- /dev/null
+++ b/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/build/figures/fifth-phase-grid.tex
@@ -0,0 +1,39 @@
+\begin{figure}[tbp]
+\centering
+\begin{tikzpicture}[x=1.45cm,y=1cm,>=Stealth,
+  every node/.style={font=\small}]
+  \foreach \k in {0,...,5} {
+    \node (v\k) at (\k,1.1) {$V_{\k}$};
+  }
+  \foreach \k in {0,...,4} {
+    \pgfmathtruncatemacro{\next}{\k+1}
+    \draw[->] (v\next.west) -- (v\k.east);
+  }
+  \draw[->] (v0.north) to[bend left=35]
+    node[above] {$\Phi$} (v2.north);
+  \draw[->] (v0.north) to[bend left=40]
+    node[above] {$[1]$} (v5.north);
+  \foreach \k in {0,...,4} {
+    \fill[blue!6] (\k,-0.05) rectangle (\k+1,0.4);
+    \node at (\k+0.5,0.175) {$\mathcal S_{\k}$};
+  }
+  \draw (0,-0.05) -- (5,-0.05);
+  \foreach \k in {0,...,5} {
+    \draw (\k,0.4) -- (\k,-0.12);
+  }
+  \node[below] at (0,-0.12) {$0$};
+  \foreach \k in {1,...,4} {
+    \node[below] at (\k,-0.12) {$\k/5$};
+  }
+  \node[below] at (5,-0.12) {$1$};
+  \node[below] at (2.5,-0.65) {charge phase};
+\end{tikzpicture}
+\caption{The fifth-phase grid. Inclusion arrows point toward the larger
+aisle. The autoequivalence $\Phi$ advances two indices, and the shift $[1]$
+advances five. A nonzero object of $\mathcal S_k$ has charge phase in the
+closed interval $[k/5,(k+1)/5]$, by Proposition~\ref{prop:block-support}.
+At a shared endpoint, generators from both adjacent blocks enter the
+definition of the slicing. The diagram shows the unperturbed grid;
+Proposition~\ref{prop:fine-grid} gives the independent-parameter nesting.}
+\label{fig:fifth-phase-grid}
+\end{figure}
diff --git a/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/build/paper.tex b/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/build/paper.tex
new file mode 100644
index 0000000000000000000000000000000000000000..47830d760c8519751a34bfded7405fc452b0f163
--- /dev/null
+++ b/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/build/paper.tex
@@ -0,0 +1,61 @@
+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage{lmodern}
+\usepackage[margin=1in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{enumitem}
+\usepackage{microtype}
+\usepackage{xcolor}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta}
+\usepackage[unicode,colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref}
+\numberwithin{equation}{section}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newcommand{\C}{\mathbb C}
+\newcommand{\R}{\mathbb R}
+\newcommand{\Z}{\mathbb Z}
+\newcommand{\Q}{\mathbb Q}
+\newcommand{\D}{\mathcal D}
+\newcommand{\OO}{\mathcal O}
+\newcommand{\Db}{D^{\mathrm b}}
+\newcommand{\Knum}{K_{\mathrm{num}}}
+\newcommand{\norm}[1]{\left\lVert#1\right\rVert}
+\DeclareMathOperator{\Coh}{Coh}
+\DeclareMathOperator{\ch}{ch}
+\DeclareMathOperator{\rk}{rk}
+\DeclareMathOperator{\Hom}{Hom}
+\DeclareMathOperator{\Ext}{Ext}
+\DeclareMathOperator{\im}{im}
+\renewcommand{\Re}{\operatorname{Re}}
+\renewcommand{\Im}{\operatorname{Im}}
+\setlist{itemsep=3pt,topsep=5pt}
+\setlength{\emergencystretch}{1em}
+\title{A Gepner stability condition on every smooth quintic threefold}
+\author{OpenAI}
+\date{September 24, 2026}
+\hypersetup{pdftitle={A Gepner stability condition on every smooth quintic threefold},pdfauthor={OpenAI}}
+\begin{document}
+\maketitle
+\begin{abstract}
+We prove Toda's normalized quintic Gepner conjecture. On every smooth complex
+quintic threefold, we construct a numerical Bridgeland stability condition for
+which tensoring by the hyperplane bundle, followed by the spherical twist at
+the structure sheaf, increases phase by $2/5$.
+\end{abstract}
+\input{sections/introduction}
+\input{sections/numerical}
+\input{sections/aisles}
+\input{sections/grid}
+\appendix
+\input{sections/periodicity}
+\bibliographystyle{plain}
+\bibliography{references}
+\end{document}
diff --git a/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/build/references.bib b/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/build/references.bib
new file mode 100644
index 0000000000000000000000000000000000000000..3a440b39c663815dec88c9005668e6f3468a880b
--- /dev/null
+++ b/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/build/references.bib
@@ -0,0 +1,212 @@
+@article{Bridgeland2007,
+  author = {Bridgeland, Tom},
+  title = {Stability conditions on triangulated categories},
+  journal = {Annals of Mathematics},
+  series = {2},
+  volume = {166},
+  number = {2},
+  year = {2007},
+  pages = {317--345},
+  doi = {10.4007/annals.2007.166.317},
+  note = {\url{https://doi.org/10.4007/annals.2007.166.317}},
+  url = {https://annals.math.princeton.edu/2007/166-2/p01}
+}
+
+@article{HappelReitenSmalo1996,
+  author = {Happel, Dieter and Reiten, Idun and Smal{\o}, Sverre O.},
+  title = {Tilting in abelian categories and quasitilted algebras},
+  journal = {Memoirs of the American Mathematical Society},
+  volume = {120},
+  number = {575},
+  pages = {viii+88},
+  year = {1996},
+  doi = {10.1090/memo/0575},
+  note = {\url{https://doi.org/10.1090/memo/0575}},
+  url = {https://doi.org/10.1090/memo/0575}
+}
+
+@article{SeidelThomas2001,
+  author = {Seidel, Paul and Thomas, Richard P.},
+  title = {Braid group actions on derived categories of coherent sheaves},
+  journal = {Duke Mathematical Journal},
+  volume = {108},
+  number = {1},
+  year = {2001},
+  pages = {37--108},
+  doi = {10.1215/S0012-7094-01-10812-0},
+  note = {\url{https://doi.org/10.1215/S0012-7094-01-10812-0}},
+  eprint = {math/0001043},
+  archivePrefix = {arXiv},
+  primaryClass = {math.AG},
+  url = {https://arxiv.org/abs/math/0001043}
+}
+
+@incollection{Orlov2009,
+  author = {Orlov, Dmitri},
+  title = {Derived categories of coherent sheaves and triangulated categories of singularities},
+  booktitle = {Algebra, Arithmetic, and Geometry: In Honor of {Yu. I. Manin}},
+  editor = {Tschinkel, Yuri and Zarhin, Yuri},
+  series = {Progress in Mathematics},
+  volume = {270},
+  publisher = {Birkh{\"a}user},
+  address = {Boston, MA},
+  year = {2009},
+  pages = {503--531},
+  doi = {10.1007/978-0-8176-4747-6_16},
+  note = {\url{https://doi.org/10.1007/978-0-8176-4747-6_16}},
+  eprint = {math/0503632},
+  archivePrefix = {arXiv},
+  primaryClass = {math.AG},
+  url = {https://arxiv.org/abs/math/0503632}
+}
+
+@article{BayerMacriToda2014,
+  author = {Bayer, Arend and Macr{\`i}, Emanuele and Toda, Yukinobu},
+  title = {{Bridgeland} stability conditions on threefolds {I}: {Bogomolov--Gieseker} type inequalities},
+  journal = {Journal of Algebraic Geometry},
+  volume = {23},
+  number = {1},
+  year = {2014},
+  pages = {117--163},
+  doi = {10.1090/S1056-3911-2013-00617-7},
+  note = {\url{https://doi.org/10.1090/S1056-3911-2013-00617-7}},
+  eprint = {1103.5010},
+  archivePrefix = {arXiv},
+  primaryClass = {math.AG},
+  url = {https://arxiv.org/abs/1103.5010v2}
+}
+
+@misc{Toda2013,
+  author = {Toda, Yukinobu},
+  title = {{Gepner} type stability conditions on graded matrix factorizations},
+  year = {2013},
+  eprint = {1302.6293v1},
+  archivePrefix = {arXiv},
+  primaryClass = {math.AG},
+  doi = {10.48550/arXiv.1302.6293},
+  url = {https://arxiv.org/abs/1302.6293v1},
+  note = {\href{https://arxiv.org/abs/1302.6293v1}{arXiv:1302.6293v1}. Version 1, 26 February 2013}
+}
+
+@misc{TodaQuintic2013,
+  author = {Toda, Yukinobu},
+  title = {{Gepner} point and strong {Bogomolov--Gieseker} inequality for quintic 3-folds},
+  year = {2013},
+  eprint = {1305.0345v3},
+  archivePrefix = {arXiv},
+  primaryClass = {math.AG},
+  doi = {10.48550/arXiv.1305.0345},
+  url = {https://arxiv.org/abs/1305.0345v3},
+  note = {\href{https://arxiv.org/abs/1305.0345v3}{arXiv:1305.0345v3}. Version 3, 7 May 2013; first posted 2 May 2013}
+}
+
+@article{Li2019,
+  author = {Li, Chunyi},
+  title = {On stability conditions for the quintic threefold},
+  journal = {Inventiones Mathematicae},
+  volume = {218},
+  number = {1},
+  year = {2019},
+  pages = {301--340},
+  doi = {10.1007/s00222-019-00888-z},
+  note = {\url{https://doi.org/10.1007/s00222-019-00888-z}},
+  eprint = {1810.03434},
+  archivePrefix = {arXiv},
+  primaryClass = {math.AG},
+  url = {https://arxiv.org/abs/1810.03434v2}
+}
+
+@misc{Xu2026,
+  author = {Xu, Chunkai},
+  title = {Stronger {Bogomolov--Gieseker} type inequality on quintic threefold},
+  year = {2026},
+  eprint = {2511.21288v2},
+  archivePrefix = {arXiv},
+  primaryClass = {math.AG},
+  doi = {10.48550/arXiv.2511.21288},
+  url = {https://arxiv.org/abs/2511.21288v2},
+  note = {\href{https://arxiv.org/abs/2511.21288v2}{arXiv:2511.21288v2}. Version 2, 4 January 2026; first posted 26 November 2025}
+}
+
+@article{Tattar2021,
+  author = {Tattar, Aran},
+  title = {Torsion pairs and quasi-abelian categories},
+  journal = {Algebras and Representation Theory},
+  volume = {24},
+  year = {2021},
+  pages = {1557--1581},
+  doi = {10.1007/s10468-020-10004-y},
+  note = {\url{https://doi.org/10.1007/s10468-020-10004-y}},
+  url = {https://link.springer.com/article/10.1007/s10468-020-10004-y}
+}
+
+@article{Douglas2001,
+  author = {Douglas, Michael R.},
+  title = {D-branes, Categories and {N=1} Supersymmetry},
+  journal = {Journal of Mathematical Physics},
+  volume = {42},
+  year = {2001},
+  pages = {2818--2843},
+  doi = {10.1063/1.1374448},
+  note = {\url{https://doi.org/10.1063/1.1374448}},
+  eprint = {hep-th/0011017},
+  archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/hep-th/0011017v4}
+}
+
+@misc{KontsevichSoibelman2008,
+  author = {Kontsevich, Maxim and Soibelman, Yan},
+  title = {Stability structures, motivic {Donaldson--Thomas} invariants and cluster transformations},
+  year = {2008},
+  note = {\href{https://arxiv.org/abs/0811.2435v1}{arXiv:0811.2435v1}},
+  eprint = {0811.2435},
+  archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/0811.2435v1}
+}
+
+@article{BallardFaveroKatzarkov2012,
+  author = {Ballard, Matthew and Favero, David and Katzarkov, Ludmil},
+  title = {{Orlov} spectra: bounds and gaps},
+  journal = {Inventiones Mathematicae},
+  volume = {189},
+  year = {2012},
+  pages = {359--430},
+  eprint = {1012.0864v3},
+  archivePrefix = {arXiv},
+  primaryClass = {math.AG},
+  url = {https://arxiv.org/abs/1012.0864v3},
+  note = {\href{https://arxiv.org/abs/1012.0864v3}{arXiv:1012.0864v3}. Version 3, 14 October 2013}
+}
+
+@misc{Aspinwall2004,
+  author = {Aspinwall, Paul S.},
+  title = {{D}-branes on {Calabi--Yau} manifolds},
+  year = {2004},
+  eprint = {hep-th/0403166v1},
+  archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/hep-th/0403166v1},
+  note = {\href{https://arxiv.org/abs/hep-th/0403166v1}{arXiv:hep-th/0403166v1}. Version 1, 16 March 2004}
+}
+
+@article{BorelSerre1958,
+  author = {Borel, Armand and Serre, Jean-Pierre},
+  title = {Le th{\'e}or{\`e}me de {Riemann--Roch}},
+  journal = {Bulletin de la Soci{\'e}t{\'e} Math{\'e}matique de France},
+  volume = {86},
+  year = {1958},
+  pages = {97--136},
+  note = {\url{https://www.numdam.org/item/BSMF_1958__86__97_0/}},
+  url = {https://www.numdam.org/item/BSMF_1958__86__97_0/}
+}
+
+@misc{Catanese2014,
+  author = {Catanese, Fabrizio},
+  title = {Topological methods in moduli theory},
+  year = {2014},
+  eprint = {1411.3235v1},
+  archivePrefix = {arXiv},
+  primaryClass = {math.AG},
+  url = {https://arxiv.org/abs/1411.3235v1},
+  note = {\href{https://arxiv.org/abs/1411.3235v1}{arXiv:1411.3235v1}. Version 1}
+}
+
diff --git a/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/build/sections/aisles.tex b/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/build/sections/aisles.tex
new file mode 100644
index 0000000000000000000000000000000000000000..58b9c114ef47b68b2403f4cdf86202588e8be174
--- /dev/null
+++ b/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/build/sections/aisles.tex
@@ -0,0 +1,492 @@
+\section{A uniform family of double-tilt aisles}
+\label{sec:aisles}
+
+Our goal is to construct bounded aisles $U_\eta$ for which
+$\Phi U_\eta\subset U_\theta$ holds uniformly with independent parameters.
+We begin with Toda's proposed two tilts for the quintic Gepner charge
+\cite[Sections~3.4--3.6]{TodaQuintic2013}, and vary the second cut in two
+numerical directions. The slope estimate below supplies the strict margins
+that make the comparison survive these variations.
+The two first-tilt slopes $-1/2$ and $1/2$ reflect the order of the factors
+in $\Phi$: tensoring by $\OO_X(1)$ moves the first cut from $-1/2$ to
+$1/2$, and the spherical twist will take the resulting aisle into one
+constructed at $-1/2$.
+
+We first isolate the geometric input.  Slope always means
+$\mu(F)=c(F)/r(F)$ for a torsion-free sheaf of positive rank.  We use
+ordinary slope Harder--Narasimhan filtrations, writing $\mu^+(F)$ and
+$\mu^-(F)$ for their largest and smallest slopes.  For an arbitrary
+sheaf, its torsion subsheaf is removed before these slopes are taken.
+
+\begin{lemma}[Slope input]\label{lem:slope-input}
+Put
+\[
+ f(x)=\max\left\{-\frac{3x}{25},\ \frac{x}{2}-\frac7{25},\
+                    \frac{28x}{25}-\frac{31}{50}\right\}
+                    \qquad(0\leq x\leq1).
+\]
+Every torsion-free slope-semistable sheaf $F$ on $X$ with
+$|\mu(F)|\leq1$ satisfies
+\begin{equation}\label{eq:slope-envelope}
+                 \frac{d(F)}{r(F)}\leq f(|\mu(F)|).
+\end{equation}
+The estimate extends under integral twists by
+$(x,y)\mapsto(x+n,y+nx+n^2/2)$.  In particular,
+$f(x)\leq0$ for $0\leq x\leq1/2$, and $f(1/2)=-3/100$.
+\end{lemma}
+
+\begin{proof}
+We use Xu's Theorem~1.3 \cite{Xu2026}.  Its hypotheses are precisely a
+smooth complex quintic threefold with its hyperplane polarization and a
+torsion-free slope-semistable sheaf of slope in $[-1,1]$; its coordinates
+$\mu_H$ and $\xi_H$ are our $c/r$ and $d/r$.  On $[0,1/2]$ its upper
+bound $g$ is
+\[
+g(x)=
+\begin{cases}
+-x/2,&0\leq x\leq1/4,\\
+x/2-1/4,&1/4\leq x\leq5/13,\\
+-3x/20,&5/13\leq x\leq6/13,\\
+x/2-3/10,&6/13\leq x\leq1/2.
+\end{cases}
+\]
+The first and third lines are at most $-3x/25$; the second is also at
+most $-3x/25$ because $5/13<25/62$; and the last is at most
+$x/2-7/25$.  Thus $g\leq f$ on this half interval.  The full upper
+bound $g$ in Xu's theorem and the maximum $f$ both satisfy
+\[
+ h(1-x)=h(x)+\frac12-x.
+\]
+For $f$, this reflection exchanges the first and third affine pieces
+and fixes the second.  This proves the comparison on $[0,1]$, and the
+theorem uses $|\mu|$ on the negative strip.
+
+For completeness, the symmetry is also compatible with the sheaf
+operations used here.  Twisting by $\OO_X(n)$ has the displayed effect
+on $(x,y)$ and preserves slope semistability.  If $F$ is not reflexive,
+passing to $F^{**}$ preserves slope semistability and $r,c$, and only
+increases $d$.  For a reflexive sheaf on a smooth threefold the higher
+sheaf Ext terms of its derived dual are supported in dimension zero,
+so its ordinary dual has characters $(r,-c,d)$ through degree two and
+is slope-semistable.  The operations of dualizing and twisting by
+$\OO_X(1)$ therefore give $(x,y)\mapsto(1-x,y+1/2-x)$ without changing
+the validity of an upper bound.  Finally the three defining lines are
+nonpositive on $[0,1/2]$, and their maximum at $1/2$ is $-3/100$.
+\end{proof}
+
+We use aisles in the convention $\D^{\leq0}$: they are closed under
+$[1]$, and the heart of an aisle $V$ is $V\cap(V[1])^\perp$, where
+$\perp$ denotes right Hom orthogonality.  Recall explicitly the tilt
+construction of Happel--Reiten--Smal\o\ \cite{HappelReitenSmalo1996}.
+A torsion pair $(\mathcal T,\mathcal F)$ in the heart of
+a bounded aisle $V$ gives the aisle
+\begin{equation}\label{eq:aisle-tilt}
+ V^\sharp=\{E\in V:H^0_V(E)\in\mathcal T\},\qquad
+ \mathcal H^\sharp=\langle\mathcal F[1],\mathcal T\rangle,
+ \qquad V[1]\subset V^\sharp\subset V.
+\end{equation}
+Here and below angle brackets denote extension closure.  To construct
+the new truncation triangle, first truncate at zero for $V$, then take
+the inverse image of the torsion subobject of its zeroth cohomology.
+The remaining triangle has terms from $\mathcal F$ and the old positive
+degrees.  The old orthogonality and
+$\Hom(\mathcal T,\mathcal F)=0$ give the required new orthogonality.
+Shift closure and the two inclusions prove that this is a bounded
+$t$-structure.  The same truncations show conversely that an aisle
+between $V[1]$ and $V$ gives a torsion pair in the heart of $V$.
+
+For $b\in\{-1/2,1/2\}$, let $\mathcal C_{\leq b}$ consist of zero
+and the torsion-free sheaves with $\mu^+\leq b$, and let
+$\mathcal C_{>b}$ consist of sheaves whose torsion-free quotient has
+$\mu^->b$, together with all torsion sheaves.  Ordinary slope
+filtrations give the torsion pair
+$(\mathcal C_{>b},\mathcal C_{\leq b})$ in $\Coh(X)$.  Define the
+first tilted heart and its aisle by
+\[
+ \mathcal B_b=\langle\mathcal C_{\leq b}[1],\mathcal C_{>b}\rangle,
+ \qquad W_b=\D^{\leq0}_{\mathcal B_b},
+ \qquad p_b=c-br.
+\]
+On $\mathcal B_b$, $p_b$ takes values in $\frac12\Z_{\geq0}$, by
+Lemma~\ref{lem:numerical}.  For $E\in\mathcal B_b$, the equality
+$p_b(E)=0$ holds exactly when $H^{-1}(E)$ is slope-semistable of slope
+$b$ (or zero), and $H^0(E)$ has dimension at most one.  This follows
+by applying $p_b$ to the slope factors of the two cohomology sheaves:
+each contribution is nonnegative, and a nonzero divisorial torsion
+sheaf has positive $c$.
+
+\begin{lemma}\label{lem:first-tilt-noetherian}
+The abelian categories $\mathcal B_{-1/2}$ and $\mathcal B_{1/2}$ are
+noetherian.
+\end{lemma}
+
+\begin{proof}
+We give the rational first-tilt argument; compare
+\cite[proof of Lemma~3.2.4]{BayerMacriToda2014}.
+Consider epimorphisms $E_0\twoheadrightarrow E_1\twoheadrightarrow
+E_2\twoheadrightarrow\cdots$ in $\mathcal B_b$, with kernels $K_i$
+for the individual arrows.  The nonnegative discrete numbers
+$p_b(E_i)$ eventually stabilize, so $p_b(K_i)=0$.  The ordinary
+cohomology sheaves $H^0(E_i)$ form a sequence of epimorphisms and
+eventually stabilize, since $\Coh(X)$ is noetherian.  The preceding
+description of $K_i$ gives $r(K_i)\leq0$, hence the integers $r(E_i)$
+are nondecreasing.  They are bounded above by the now fixed rank of
+$H^0(E_i)$, so eventually $r(K_i)=0$.  Each remaining $K_i$ is then an
+ordinary sheaf of dimension at most one.
+
+The cohomology sequence now reads
+\[
+ 0\longrightarrow H^{-1}(E_i)\longrightarrow H^{-1}(E_{i+1})
+   \longrightarrow K_i\longrightarrow0.
+\]
+These torsion-free sheaves agree in codimension one.  Their injections
+extend to isomorphisms of their reflexive hulls, so they form an
+increasing chain of coherent subsheaves of a fixed reflexive sheaf.
+That chain stabilizes, forcing $K_i=0$.  Thus every such sequence of
+epimorphisms stabilizes, as required.
+\end{proof}
+
+The first tilted hearts are now available. We define the second cut and choose
+constants for three uses: negativity at the slope boundary, exclusion of a
+small neighboring slope strip, and a strict comparison between the two cuts.
+At zero perturbation and $b=-1/2$, the cut is the imaginary part of the
+normalized Gepner charge, as in Toda's candidate heart
+\cite[Section~3.6]{TodaQuintic2013}. Fix
+\begin{equation}\label{eq:aisle-parameters}
+ \begin{gathered}
+ \delta=10^{-5},\qquad s_0=\frac{553}{1000},\qquad
+ B_\delta=\{\eta=(\eta_0,\eta_1)\in\R^2:|\eta_j|<\delta\},\\
+ N_b^\eta=d-bc+ur+\eta_1c+\eta_0r.
+ \end{gathered}
+\end{equation}
+The following exact margins will be useful:
+\begin{align}
+ m_b&=\frac7{25}-u-\frac32\delta>0,\label{eq:boundary-margin}\\
+ m_s&=\frac{13857}{50000}-u-(1+s_0)\delta>0,
+                                                    \label{eq:strip-margin}\\
+ m_c&=\frac12-\frac{u}{s_0}-3\delta>0.\label{eq:comparison-margin}
+\end{align}
+For example, $\sqrt5>223606/100000$ implies
+$u<138197/500000$, and gives respectively the positive lower bounds
+$3591/10^6$, $73047/10^8$, and $8941/55300000$ for these three
+expressions.  On a slope-semistable sheaf $F$ of slope $b$,
+Lemma~\ref{lem:slope-input} gives
+\begin{equation}\label{eq:boundary-negative}
+ \frac{N_b^\eta(F)}{r(F)}
+ \leq -\frac3{100}-\frac14+u+\frac32\delta=-m_b<0.
+\end{equation}
+Consequently $N_b^\eta\geq0$ on the $p_b=0$ objects of $\mathcal B_b$;
+equality holds precisely for zero-dimensional sheaves, including zero.
+Indeed their degree-minus-one part contributes strictly positively
+unless it vanishes, while a sheaf of dimension at most one has
+$N_b^\eta=d\geq0$, with equality exactly in dimension zero.
+
+Call a nonzero object of $\mathcal B_b$ \emph{high} if $p_b=0$ or
+$N_b^\eta>0$, and \emph{low} otherwise.  Thus low means $p_b>0$ and
+$N_b^\eta\leq0$.  Set
+\[
+\begin{split}
+ \mathcal T_b^\eta&=\{E:\text{every nonzero quotient of $E$ in
+                                  $\mathcal B_b$ is high}\},\\
+ \mathcal F_b^\eta&=\{E:\text{every nonzero subobject of $E$ in
+                                  $\mathcal B_b$ is low}\}.
+\end{split}
+\]
+
+\begin{lemma}[The second tilt]\label{lem:second-tilt}
+For every $b=\pm1/2$ and $\eta\in B_\delta$, the pair
+$(\mathcal T_b^\eta,\mathcal F_b^\eta)$ is a torsion pair.  Its tilt
+has bounded heart and aisle
+\[
+ \mathcal A_b^\eta=\langle\mathcal F_b^\eta[1],\mathcal T_b^\eta\rangle,
+ \qquad \mathcal U_b^\eta
+   =\{E\in W_b:H^0_{\mathcal B_b}(E)\in\mathcal T_b^\eta\}.
+\]
+On $\mathcal A_b^\eta$ one has $N_b^\eta\geq0$, and no nonzero object
+of this heart has zero numerical class.
+\end{lemma}
+
+\begin{proof}
+High objects are closed under extensions: if one end term has positive
+$N_b^\eta$, so does the extension, and otherwise both have $p_b=0$.
+It follows, by taking the induced filtration on a quotient, that
+$\mathcal T_b^\eta$ is closed under quotients and extensions.
+
+If an object has a high subobject, choose such a nonzero subobject $A$
+with $p_b(A)$ minimal.  When $p_b(A)=0$, all its quotients have
+$p_b=0$.  Otherwise $N_b^\eta(A)>0$, and a low quotient $Q$ would
+have $p_b(Q)>0$ and $N_b^\eta(Q)\leq0$.  Its kernel would then be a
+high subobject with strictly smaller $p_b$, a contradiction.  Thus
+$A\in\mathcal T_b^\eta$ in either case.
+
+By Lemma~\ref{lem:first-tilt-noetherian}, every object has a maximal
+$\mathcal T_b^\eta$ subobject; sums of such subobjects again belong to
+$\mathcal T_b^\eta$, so maximality has its usual torsion meaning.  The
+quotient has no high subobject, since such a subobject would contain
+a nonzero member of $\mathcal T_b^\eta$ by the preceding paragraph,
+and its inverse image would enlarge the maximal torsion subobject.
+The quotient therefore lies in $\mathcal F_b^\eta$.  Finally a
+nonzero image of a map from $\mathcal T_b^\eta$ to
+$\mathcal F_b^\eta$ would be both high and low.  This proves the
+torsion pair, and \eqref{eq:aisle-tilt} gives boundedness.
+
+Write an object of the new heart as an extension of $T$ by $F[1]$,
+where $T\in\mathcal T_b^\eta$ and $F\in\mathcal F_b^\eta$.  Then
+$N_b^\eta(T)\geq0$ and $N_b^\eta(F)\leq0$, proving nonnegativity.
+If $N_b^\eta(E)=0$, both terms have $N_b^\eta=0$.  Since $T$ itself
+is high when nonzero, this forces $p_b(T)=0$, hence $T$ is
+zero-dimensional.  If $F\ne0$, then $p_b(F)>0$ and
+$p_b(E)=-p_b(F)<0$.  Thus $[E]=0$ would force $F=0$; it would then
+force $T=0$, since $e(T)=\operatorname{length}(T)/5$.  This proves
+the last assertion.
+\end{proof}
+
+We next compare the two slope cuts with independent parameters.
+Subscripts $-$ and $+$ mean $b=-1/2$ and $b=1/2$, respectively.
+
+\begin{lemma}\label{lem:compare-slope-cuts}
+For every $\alpha,\theta\in B_\delta$,
+\[
+                       \mathcal U_+^\alpha\subset\mathcal U_-^\theta.
+\]
+\end{lemma}
+
+\begin{proof}
+Let $\mathcal M$ be the ordinary torsion-free sheaves all of whose
+slopes lie in $(-1/2,1/2]$, and let
+\[
+ \mathcal L=\langle\mathcal C_{\leq-1/2}[1],\mathcal C_{>1/2}\rangle
+             =\mathcal B_-\cap\mathcal B_+.
+\]
+To see the torsion pair $(\mathcal L,\mathcal M)$ in $\mathcal B_-$,
+take $E\in\mathcal B_-$ and let $M_E$ be the quotient of $H^0(E)$
+obtained by removing its torsion and its slope factors above $1/2$.
+Then $M_E\in\mathcal M$.  The kernel of $E\to M_E$ in
+$\mathcal B_-$ has degree-minus-one cohomology $H^{-1}(E)$ and
+degree-zero cohomology in $\mathcal C_{>1/2}$, so it lies in
+$\mathcal L$.  The slope vanishing and the ordinary cohomological
+degrees give $\Hom(\mathcal L,\mathcal M)=0$.
+Tilting this torsion pair gives
+$\mathcal B_+=\langle\mathcal M[1],\mathcal L\rangle$; in
+$\mathcal B_+$ the corresponding torsion pair is
+$(\mathcal M[1],\mathcal L)$.  In particular
+$W_-[1]\subset W_+\subset W_-$, and
+$W_+[1]\subset W_-[1]\subset\mathcal U_-^\theta$.
+It suffices to prove
+\begin{equation}\label{eq:comparison-hom}
+               \Hom(\mathcal T_+^\alpha,\mathcal F_-^\theta)=0.
+\end{equation}
+Indeed an object $E\in W_-$ has maps to a member of
+$\mathcal F_-^\theta$ exactly through $H^0_{\mathcal B_-}(E)$;
+vanishing of all such maps puts this cohomology object in
+$\mathcal T_-^\theta$.  Equation~\eqref{eq:comparison-hom} therefore
+puts $\mathcal T_+^\alpha$ in $\mathcal U_-^\theta$, and the preceding
+inclusion handles the negative $\mathcal B_+$ degrees.
+
+Suppose there is a nonzero map $E\to G$ with
+$E\in\mathcal T_+^\alpha$ and $G\in\mathcal F_-^\theta$.
+Its source decomposes in $\mathcal B_+$ as
+$0\to M_E[1]\to E\to E_0\to0$, with $E_0\in\mathcal L$.
+Because $\Hom(\mathcal M[1],\mathcal B_-)=0$, the map factors through
+$E_0$, which is still in $\mathcal T_+^\alpha$.  The
+$(\mathcal L,\mathcal M)$ decomposition of $G$ in $\mathcal B_-$
+then factors it through a subobject $G_0\in\mathcal L$ of $G$, since
+$\Hom(\mathcal L,\mathcal M)=0$.
+
+Take the kernel $K$ and the nonzero image $Q$ of $E_0\to G_0$ in
+$\mathcal B_-$.  Let $K_{\mathcal L}$ be the $\mathcal L$ torsion
+part of $K$, and put $U'=E_0/K_{\mathcal L}$ in $\mathcal B_-$.
+The quotient closure of $\mathcal L$ gives $Q,U'\in\mathcal L$.
+All three terms of
+$K_{\mathcal L}\to E_0\to U'$ belong to both hearts, so its triangle
+is short exact in $\mathcal B_+$ as well.  Thus $U'$ remains a
+quotient of $E$ in $\mathcal B_+$, and we have
+\begin{equation}\label{eq:common-heart-sequence}
+ 0\longrightarrow S\longrightarrow U'\longrightarrow Q\longrightarrow0
+ \quad\text{in }\mathcal B_-,\qquad
+ S=K/K_{\mathcal L}\in\mathcal M,
+ \quad U'\in\mathcal T_+^\alpha,
+ \quad 0\ne Q\in\mathcal F_-^\theta.
+\end{equation}
+We use the sequence~\eqref{eq:common-heart-sequence} only as a short exact
+sequence in $\mathcal B_-$; its term $S$
+need not belong to $\mathcal B_+$.
+
+We now use the slope envelope to exclude factors close to the two slope
+boundaries. The resulting rank bounds will contradict the opposite signs of
+the two cut forms on $U'$ and $Q$.
+
+Directly from the three affine functions defining $f$,
+\begin{equation}\label{eq:strip-test}
+ \max_{1/2\leq s\leq s_0}\bigl(f(s)-s/2+u\bigr)
+                 =u-\frac{13857}{50000}.
+\end{equation}
+The perturbation error on a slope $\pm s$ is at most
+$(1+s_0)\delta$, so \eqref{eq:strip-margin} makes the relevant
+sheaf value of $N_-$ on $[-s_0,-1/2]$, or of $N_+$ on
+$[1/2,s_0]$, strictly negative.
+
+Write $A=H^{-1}(Q)$.  If $A\ne0$, its top slope factor $A_1$
+is a saturated subsheaf and $A/A_1\in\mathcal C_{\leq-1/2}$.
+Consequently $A_1[1]$ is a subobject of $A[1]$, and then of $Q$, in
+$\mathcal B_-$.  If $\mu(A_1)\in[-s_0,-1/2]$, the preceding
+negative sheaf value makes $A_1[1]$ high, contradicting
+$Q\in\mathcal F_-^\theta$.  Thus $\mu^+(A)<-s_0$.
+
+For $U'$, put $A'=H^{-1}(U')$ and $D'=H^0(U')$.
+If $D'$ has positive rank, remove its ordinary torsion subsheaf and
+take its bottom positive-rank slope quotient $G'$.  The kernel of
+$D'\to G'$ consists of that torsion and the other slope factors,
+all of slope $>1/2$; this sequence is therefore exact in
+$\mathcal B_+$.  The ordinary cohomology sequence also makes $D'$
+a $\mathcal B_+$ quotient of $U'$, so $G'$ is such a quotient.
+A slope $\mu(G')\in(1/2,s_0]$ would give
+$p_+(G')>0$, $N_+^\alpha(G')<0$, contradicting
+$U'\in\mathcal T_+^\alpha$.  Every positive-rank slope of $D'$
+therefore exceeds $s_0$.
+
+For any $L\in\mathcal L$, its two ordinary cohomology sheaves give
+\begin{equation}\label{eq:overlap-rank-bound}
+ c(L)\geq\frac12\bigl(r(H^{-1}(L))+r(H^0(L))\bigr)
+                                     \geq\frac12|r(L)|.
+\end{equation}
+Divisorial torsion contributes nonnegative $c$, and torsion in lower
+dimension contributes zero.  The stronger slope conclusions just
+proved similarly give
+\begin{equation}\label{eq:two-slope-tests}
+          c(Q)\geq-s_0r(Q),\qquad c(U')\geq s_0r(U').
+\end{equation}
+More explicitly, with $D=H^0(Q)$ the estimates are
+\[
+ \begin{split}
+ c(Q)&\geq s_0r(A)+\tfrac12r(D)\geq-s_0r(Q),\\
+ c(U')&\geq\tfrac12r(A')+s_0r(D')\geq s_0r(U').
+ \end{split}
+\]
+They include torsion cohomology sheaves: their ranks are zero and
+their contributions to $c$ are nonnegative.  Also $d(S)\leq0$, by applying
+Lemma~\ref{lem:slope-input} to the ordinary slope factors of $S$.
+
+Set $C=c(Q)+c(U')$.  By \eqref{eq:overlap-rank-bound}, $C\geq0$.
+If $C=0$, then $c(Q)=r(Q)=0$, so $p_-(Q)=0$, impossible for a
+nonzero member of $\mathcal F_-^\theta$.  Hence $C>0$; and
+\eqref{eq:two-slope-tests} together with
+\eqref{eq:common-heart-sequence} gives $C\geq s_0r(S)$.
+By additivity,
+\begin{align*}
+ N_-^\theta(Q)-N_+^\alpha(U')
+ &=\frac12 C-u r(S)-d(S)
+       +\theta_1c(Q)+\theta_0r(Q)-\alpha_1c(U')-\alpha_0r(U')\\
+ &\geq \left(\frac12-\frac{u}{s_0}-3\delta\right)C-d(S)>0.
+\end{align*}
+Here the absolute value of the last four terms is at most $3\delta C$
+by \eqref{eq:overlap-rank-bound}; this applies even when a derived
+rank is negative.  The strict sign is \eqref{eq:comparison-margin}.
+But $N_-^\theta(Q)\leq0\leq N_+^\alpha(U')$, a contradiction.
+This proves \eqref{eq:comparison-hom} and the aisle inclusion.
+\end{proof}
+
+\begin{lemma}\label{lem:twist-aisle}
+For all $\alpha,\theta\in B_\delta$,
+\[
+ \OO_X\in\mathcal A_-^\theta,\qquad
+ \OO_X[2]\in\mathcal A_+^\alpha,\qquad
+ T_{\OO_X}(\mathcal U_+^\alpha)\subset\mathcal U_-^\theta.
+\]
+\end{lemma}
+
+\begin{proof}
+The object $\OO_X\in\mathcal B_-$ has the smallest positive value
+$p_-=1/2$ and $N_-^\theta=u+\theta_0>0$.
+It has no nonzero subobject of $p_-=0$: a $p_-=0$ object is an
+extension of a sheaf of dimension at most one by $F[1]$ of slope
+$-1/2$, and both have zero Hom to $\OO_X$.
+Any nonzero low quotient of $\OO_X$ would have $p_-=1/2$, leaving
+a kernel of $p_-=0$; that kernel must vanish, contradicting that
+$\OO_X$ is high.  Hence $\OO_X\in\mathcal T_-^\theta$.
+
+Similarly $\OO_X[1]\in\mathcal B_+$ has $p_+=1/2$ and
+$N_+^\alpha=-u-\alpha_0<0$.  A $p_+=0$ object has no map to it:
+for its slope-$1/2$ part this is
+$\Hom(F[1],\OO_X[1])=\Hom(F,\OO_X)=0$, and for its
+dimension-at-most-one part $T$ it follows from
+\[
+ \Ext^1(T,\OO_X)=H^2(X,T)^*=0.
+\]
+Here we use Serre duality and $K_X\simeq\OO_X$.  Every nonzero
+subobject $A$ of $\OO_X[1]$ therefore has $p_+(A)=1/2$, and its
+quotient $Q$ has $p_+(Q)=0$.  Hence
+$N_+^\alpha(A)=N_+^\alpha(\OO_X[1])-N_+^\alpha(Q)<0$.
+This proves $\OO_X[1]\in\mathcal F_+^\alpha$, and thus
+$\OO_X[2]\in\mathcal A_+^\alpha$.
+
+For $B\in\mathcal U_+^\alpha$ and $j\geq2$, Serre duality and
+the heart orthogonality give
+\[
+ \Ext^j(\OO_X,B)
+       =\Hom(B[j-1],\OO_X[2])^*=0,
+\]
+since $B[j-1]\in\mathcal U_+^\alpha[1]$.
+Lemma~\ref{lem:compare-slope-cuts} puts $B$ in
+$\mathcal U_-^\theta$.  In the twist triangle
+\[
+ B\longrightarrow T_{\OO_X}(B)\longrightarrow
+          \operatorname{RHom}(\OO_X,B)\otimes\OO_X[1]\longrightarrow B[1],
+\]
+the third term is a finite direct sum of copies of
+$\OO_X[1-j]$ with $j\leq1$, because bounded complexes of
+finite-dimensional vector spaces split into their cohomology.
+All these terms lie in $\mathcal U_-^\theta$.  Extension closure
+therefore gives the claimed twist inclusion.
+\end{proof}
+
+\begin{proposition}[Uniform independent-parameter inclusion]
+\label{prop:uniform-aisle}
+There is a fixed open square
+\begin{equation}\label{eq:fixed-omega}
+ \Omega=\{\eta=(\eta_0,\eta_1):|\eta_0|,|\eta_1|<5\cdot10^{-6}\}
+\end{equation}
+and bounded aisles $U_\eta=\mathcal U_-^\eta$, with hearts
+$\mathcal A_\eta=U_\eta\cap(U_\eta[1])^\perp$, such that
+\begin{equation}\label{eq:uniform-aisle}
+                    \Phi U_\eta\subset U_\theta
+                    \qquad(\eta,\theta\in\Omega).
+\end{equation}
+Moreover
+\begin{equation}\label{eq:perturbed-positive-form}
+ N_\eta=d+\frac12c+ur+\eta_1c+\eta_0r\geq0
+                    \quad\text{on }\mathcal A_\eta,
+ \qquad N_0=\frac{\Im Z}{5t_0},
+\end{equation}
+and no nonzero object of any $\mathcal A_\eta$ has zero numerical
+class.  The two parameters in \eqref{eq:uniform-aisle} vary
+independently throughout this same square.
+\end{proposition}
+
+\begin{proof}
+Tensoring by $\OO_X(1)$ takes $\mathcal B_-$ to $\mathcal B_+$
+and preserves the $p$ test.  Its effect on the other test is
+\[
+ N_+^\alpha(E(1))=N_-^\eta(E)
+ \quad\text{when}\quad
+             \alpha_1=\eta_1,\qquad\alpha_0=\eta_0-\eta_1.
+\]
+Thus it takes the second torsion pair and aisle to those with this
+parameter $\alpha$.  If $\eta\in\Omega$, then $\alpha\in B_\delta$;
+also $\theta\in\Omega$ implies $\theta\in B_\delta$.
+Lemma~\ref{lem:twist-aisle} now gives
+\[
+ \Phi U_\eta=T_{\OO_X}\bigl(\mathcal U_+^\alpha\bigr)
+                                      \subset U_\theta.
+\]
+The remaining assertions are Lemma~\ref{lem:second-tilt} and
+\eqref{eq:imaginary-charge}.
+
+The parameter shear was used once, in proving this inclusion for
+the original family.  In particular, applying any power $\Phi^i$
+to \eqref{eq:uniform-aisle} gives
+$\Phi^{i+1}U_\eta\subset\Phi^iU_\theta$ with the same
+$\eta,\theta\in\Omega$; it performs no further shear and requires
+no further shrinkage.  Negative powers are allowed by
+Lemma~\ref{lem:periodicity}.
+\end{proof}
diff --git a/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/build/sections/grid.tex b/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/build/sections/grid.tex
new file mode 100644
index 0000000000000000000000000000000000000000..f7333196e763ed551f77fb2fa9b650ae7e5aa9f0
--- /dev/null
+++ b/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/build/sections/grid.tex
@@ -0,0 +1,647 @@
+\section{From uniform aisle inclusions to a slicing}
+\label{sec:grid}
+
+We now use the bounded aisles $U_\eta$ and their hearts
+$\mathcal A_\eta$ constructed above.  The parameter $\eta=(\eta_0,\eta_1)$
+ranges over the fixed open square $\Omega$ of
+Proposition~\ref{prop:uniform-aisle}, and $0\in\Omega$.  The inputs to this
+section are
+\begin{equation}\label{eq:grid-inputs}
+ \begin{gathered}
+ \Phi U_\eta\subset U_\theta
+ \quad(\eta,\theta\in\Omega),\\
+ N_\eta=d+\tfrac12c+ur+\eta_1c+\eta_0r\ge0
+ \quad\text{on }\mathcal A_\eta,
+ \end{gathered}
+\end{equation}
+the absence of nonzero objects of zero numerical class in these hearts,
+and Lemma~\ref{lem:periodicity}, Lemma~\ref{lem:numerical}, and
+Proposition~\ref{prop:central-charge}.  In particular, throughout this
+section $\Phi$ is an autoequivalence, $\Phi^5\simeq[2]$, and
+$Z\Phi=e^{2\pi i/5}Z$.  All extension closures below mean finite extension
+closures, contain zero, and are strictly full subcategories of $\D$.
+For a subcategory $\mathcal C$, write
+$\mathcal C^\perp=\{E:\Hom(C,E)=0\text{ for all }C\in\mathcal C\}$;
+the left orthogonal ${}^\perp\mathcal C$ is defined similarly.
+
+The decisive use of the perturbations is twofold.  Simultaneous positivity
+in fixed middle hearts improves the coarse nesting to a grid with spacing
+$1/5$.  Positivity in every perturbed preceding heart then bounds the full
+numerical class of each grid block by its charge.
+
+\subsection{The finer grid}
+
+Put $D_i^\eta=\Phi^iU_\eta$ for $i\in\Z$.  The hypotheses give
+\begin{equation}\label{eq:coarse-grid}
+ D_{i+1}^\eta\subset D_i^\theta\quad
+ (i\in\Z,\ \eta,\theta\in\Omega),\qquad
+ D_{i+5}^\eta=D_i^\eta[2].
+\end{equation}
+All successive inclusions allow independent parameter choices.
+The charge rotation suggests how to insert a fifth-step between these
+coarse cuts: $\Phi^3[-1]$ multiplies $Z$ by $e^{i\pi/5}$.
+For its translates to give nested aisles, we need
+$\Phi^3U_\alpha[-1]\subset U_\beta$, or equivalently
+$D_3^\alpha\subset D_0^\beta[1]$.  The next proposition proves this
+inclusion before any slicing or semistable phase is defined.
+
+\begin{proposition}[Uniform refinement of the grid]\label{prop:fine-grid}
+There is an open square $\Omega_1$ about zero, contained in $\Omega$, such
+that
+\begin{equation}\label{eq:single-shift-inclusion}
+ D_3^\alpha\subset D_0^\beta[1]
+ \qquad(\alpha,\beta\in\Omega_1).
+\end{equation}
+For $k\in\Z$ and $\eta\in\Omega_1$, define
+\begin{equation}\label{eq:fifths-aisles}
+ V_k^\eta=\Phi^mU_\eta[j],\qquad k=2m+5j.
+\end{equation}
+This is independent of the integers $m,j$ chosen, and these bounded aisles
+satisfy
+\begin{equation}\label{eq:fifths-nesting}
+ \begin{gathered}
+ V_{k+1}^\alpha\subset V_k^\beta,\qquad
+ V_{k+5}^\eta=V_k^\eta[1],\qquad
+ \Phi V_k^\eta=V_{k+2}^\eta,\\
+ k\in\Z,\quad \alpha,\beta,\eta\in\Omega_1.
+ \end{gathered}
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+For $\alpha,\beta\in\Omega$, consider
+\[
+ \mathcal I_{\alpha,\beta}
+   =D_3^\alpha\cap(D_0^\beta[1])^\perp.
+\]
+Every object of this intersection belongs to the hearts of $D_0^\beta$
+and $D_3^\alpha$.  More strongly, for every $\gamma\in\Omega$ and
+$i\in\{1,2\}$, \eqref{eq:coarse-grid} gives
+\[
+ D_3^\alpha\subset D_i^\gamma\subset D_0^\beta,
+ \qquad D_i^\gamma[1]\subset D_0^\beta[1].
+\]
+The first inclusion gives aisle membership, and the second gives the
+orthogonal membership.  Thus the same object lies in the heart of
+$D_i^\gamma$ for every $\gamma$ in the original fixed square $\Omega$.
+This square is independent of the endpoints $\alpha,\beta$ and of the
+object.
+
+Fix a Euclidean norm on $\Knum(X)_\R$.  Suppose there were nonzero
+objects $E_n\in\mathcal I_{\alpha_n,\beta_n}$ with
+$\alpha_n,\beta_n\longrightarrow0$.  The endpoint heart has no nonzero
+zero-class object, so
+$v_n=[E_n]/\norm{[E_n]}$ is defined.  After a subsequence it tends to a
+unit vector $v$.  Heart positivity at the two endpoints, and at the
+unperturbed middle hearts, gives in the limit
+\begin{equation}\label{eq:four-halfplanes}
+ \Im\bigl(e^{-2\pi i q/5}Z(v)\bigr)\ge0
+ \qquad(q=0,1,2,3).
+\end{equation}
+These four inequalities force $Z(v)=0$.  Indeed, if $Z(v)\ne0$, choose its
+argument $\vartheta\in[0,2\pi)$.  The first three inequalities force
+$\vartheta\in[4\pi/5,\pi]$, whereas the fourth forces
+$\vartheta\in[0,\pi/5]\cup[6\pi/5,2\pi)$, which is impossible.
+
+Now fix any $\gamma\in\Omega$.  The middle-heart inequality at index one
+holds for every $n$, so the same convergent subsequence satisfies
+\[
+ N_\gamma(\Phi_*^{-1}v)\ge0.
+\]
+Since $\gamma$ was arbitrary, this holds simultaneously for all
+$\gamma\in\Omega$; there is no further subsequence or shrinking of the
+middle parameter set.  Write $w=\Phi_*^{-1}v$.  The eigencharge identity
+and \eqref{eq:imaginary-charge} give $Z(w)=N_0(w)=0$.  Varying
+$\gamma_0,\gamma_1$ with both signs in $\Omega$ now gives
+$r(w)=c(w)=0$.  The real-linear independence of the $d,e$ coefficients
+of $Z$ gives $d(w)=e(w)=0$.  This contradicts $\norm{v}=1$.
+
+Consequently some square $\Omega_1$ about zero has
+$\mathcal I_{\alpha,\beta}=0$ for every pair
+$\alpha,\beta\in\Omega_1$: otherwise choose counterexamples with both
+parameters within distance $1/n$ of zero.  To deduce
+\eqref{eq:single-shift-inclusion}, let $E\in D_3^\alpha$ and truncate in
+the bounded t-structure with aisle $D_0^\beta$:
+\[
+ A\longrightarrow E\longrightarrow H^0_{D_0^\beta}(E)
+   \longrightarrow A[1],\qquad A\in D_0^\beta[1].
+\]
+Here $E\in D_0^\beta$, and
+$A[1]\in D_0^\beta[2]=D_5^\beta\subset D_3^\alpha$.
+Extension closure therefore puts the zeroth cohomology in
+$D_3^\alpha\cap(D_0^\beta[1])^\perp=0$.  Hence $E\in D_0^\beta[1]$.
+
+Every integer is $2m+5j$.  Two such representations differ by
+$(m,j)\mapsto(m+5t,j-2t)$, so $\Phi^5\simeq[2]$ proves that
+\eqref{eq:fifths-aisles} is well defined.  Since
+$V_1^\alpha=\Phi^3U_\alpha[-1]\subset U_\beta=V_0^\beta$,
+transport by $\Phi^m[j]$ proves the first assertion of
+\eqref{eq:fifths-nesting} for all $k$.  Its other assertions follow
+directly from the definition.
+\end{proof}
+
+From now on $V_k=V_k^0$.  Define the heart and the block
+\begin{equation}\label{eq:block-definitions}
+ \mathcal H_k=V_k\cap V_{k+5}^\perp,
+ \qquad \mathcal S_k=V_k\cap V_{k+1}^\perp.
+\end{equation}
+The first is a bounded heart because $V_{k+5}=V_k[1]$.
+
+\begin{lemma}[Block filtrations]\label{lem:block-filtrations}
+The blocks are extension closed, and
+$\Hom(\mathcal S_j,\mathcal S_k)=0$ whenever $j>k$.
+Every object has a finite triangle filtration with nonzero factors in
+blocks of strictly decreasing indices.  In $\mathcal H_k$ there is a
+torsion pair
+\begin{equation}\label{eq:block-torsion-pair}
+ \begin{aligned}
+ \mathcal T_k=V_{k+1}\cap\mathcal H_k
+     &=\langle\mathcal S_{k+4},\mathcal S_{k+3},
+                \mathcal S_{k+2},\mathcal S_{k+1}\rangle,\\
+ \mathcal F_k&=\mathcal S_k.
+ \end{aligned}
+\end{equation}
+In particular, $\mathcal S_k$ is closed under subobjects in $\mathcal H_k$.
+\end{lemma}
+
+\begin{proof}
+Extension closure follows from \eqref{eq:block-definitions}.  If $j>k$,
+then $\mathcal S_j\subset V_j\subset V_{k+1}$ and
+$\mathcal S_k\subset V_{k+1}^\perp$, proving the Hom vanishing.
+Boundedness and $V_{k+5}=V_k[1]$ put any object $E$ in
+$V_l\cap V_h^\perp$ for some integers $l<h$.  Truncation at $V_{l+1}$
+gives
+\[
+ A\longrightarrow E\longrightarrow B\longrightarrow A[1],
+ \quad A\in V_{l+1},\ B\in V_{l+1}^\perp.
+\]
+Because $E,A[1]\in V_l$, we have $B\in V_l$, hence $B\in\mathcal S_l$.
+Also $B[-1]\in V_h^\perp$: for $F\in V_h$,
+$F[1]\in V_h\subset V_{l+1}$ gives $\Hom(F,B[-1])=0$.
+The rotated triangle thus shows $A\in V_h^\perp$.
+Repeat with $A\in V_{l+1}\cap V_h^\perp$.  At index $h$ the remaining
+object belongs to $V_h\cap V_h^\perp$ and is zero.  Reading the resulting
+triangles from the first subobject to the last quotient gives the stated
+decreasing order; zero factors are omitted.
+
+If $E\in\mathcal H_k$, apply the same construction with $l=k$ and
+$h=k+5$.  The first triangle has all its terms in $\mathcal H_k$, so it is
+a short exact sequence there, with $A\in\mathcal T_k$ and
+$B\in\mathcal S_k$.  The preceding Hom vanishing proves the torsion-pair
+axiom.  Repeating on $A$ proves the displayed four-block description.
+For completeness, a subobject $G$ of an object of $\mathcal S_k$ has
+$\Hom(\mathcal T_k,G)=0$, by composing with the inclusion, so its torsion
+part vanishes.  Thus $G\in\mathcal S_k$.
+\end{proof}
+
+Figure~\ref{fig:fifth-phase-grid} records the index conventions and the charge
+sectors that will be established in Proposition~\ref{prop:block-support}.
+\input{figures/fifth-phase-grid}
+
+\subsection{Full numerical support on the blocks}
+
+We first locate a block's charge using its two adjacent unperturbed hearts.
+To bound its numerical class, we then place the same object in every perturbed
+heart at the preceding grid position. This second step retains the two
+independent inequalities that control rank and first Chern character.
+
+\begin{proposition}\label{prop:block-support}
+For the fixed Euclidean norm on $\Knum(X)_\R$, there is a constant $C>0$
+such that, for every $k\in\Z$ and every nonzero $E\in\mathcal S_k$,
+\begin{equation}\label{eq:block-support}
+ \begin{gathered}
+ Z(E)\in\{t e^{i\pi\varphi}:t>0,\ k/5\le\varphi\le(k+1)/5\},\\
+ \norm{[E]}\le C|Z(E)|.
+ \end{gathered}
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+For $k=2m+5j$, the functor $\Phi^{-m}[-j]$ takes $\mathcal H_k$ to
+$\mathcal A_0$ and multiplies its charge by $e^{-i\pi k/5}$.  Positivity
+therefore gives $\Im(e^{-i\pi k/5}Z(E))\ge0$ on $\mathcal H_k$.
+An object $E\in\mathcal S_k$ belongs both to $\mathcal H_k$ and to
+\[
+ \mathcal H_{k+1}[-1]
+   =V_{k-4}\cap V_{k+1}^\perp.
+\]
+The two tests are consequently
+\[
+ \Im(e^{-i\pi k/5}Z(E))\ge0,\qquad
+ \Im(e^{-i\pi(k+1)/5}Z(E))\le0.
+\]
+Their intersection is the closed sector in \eqref{eq:block-support},
+together with the origin.
+
+Choose $\varepsilon>0$ such that the closed square
+$[-\varepsilon,\varepsilon]^2$ is contained in $\Omega_1$.
+For every $\eta$ in this square, independent-parameter nesting gives
+\[
+ V_k\subset V_{k-1}^\eta,
+ \qquad V_{k-1}^\eta[1]=V_{k+4}^\eta\subset V_{k+1}.
+\]
+Thus $E$ belongs to the heart of $V_{k-1}^\eta$ for every such $\eta$:
+the second inclusion supplies exactly its required right orthogonality.
+Write $k-1=2m+5j$, and set $w=[\Phi^{-m}E[-j]]$.  This is the class of an
+actual object of $\mathcal A_\eta$ for every parameter under consideration,
+including the shift in this formula.  Hence
+\[
+ N_0(w)+\eta_1c(w)+\eta_0r(w)\ge0
+ \quad\text{for all }|\eta_0|,|\eta_1|\le\varepsilon.
+\]
+Choosing the two signs independently proves
+\begin{equation}\label{eq:perturbed-support}
+ \varepsilon\bigl(|r(w)|+|c(w)|\bigr)
+ \le N_0(w)=\frac{\Im Z(w)}{5t_0}
+ \le\frac{|Z(E)|}{5t_0}.
+\end{equation}
+The invertible real coordinate map
+$w\mapsto(r(w),c(w),\Re Z(w),\Im Z(w))$, established in
+\eqref{eq:numerical-norm-control}, now bounds $\norm{w}$ by a constant
+times $|Z(E)|$.  Explicitly,
+$d(w)=\Im Z(w)/(5t_0)-c(w)/2-ur(w)$, and then the real part of
+\eqref{eq:central-charge} recovers $e(w)$.
+
+We can choose $m\in\{0,1,2,3,4\}$ according to $k-1$ modulo five.
+There are therefore only five norm-conversion operators $\Phi_*^m$;
+the arbitrary shift $j$ only changes the sign of a numerical class.
+Taking the maximum of their operator norms gives one $C$ for every $k$.
+If $Z(E)=0$, the estimate gives $w=0$.  The object
+$\Phi^{-m}E[-j]$ belongs to $\mathcal A_0$, so
+Lemma~\ref{lem:second-tilt} forces $\Phi^{-m}E[-j]=0$ and hence $E=0$.
+This excludes the origin for a nonzero block object and completes the proof.
+\end{proof}
+
+\subsection{Finite refinement inside a block}
+
+The support estimate is now in place. We use it with the discreteness of the
+full numerical lattice to obtain maximal phases and a bound on the length of
+refinement; no discreteness of the charge image is needed.
+
+Fix $k$, and put $a=k/5$, $b=(k+1)/5$.
+Each nonzero $E\in\mathcal S_k$ has a unique phase
+$\varphi_k(E)\in[a,b]$ with $Z(E)=|Z(E)|e^{i\pi\varphi_k(E)}$.
+A subobject $F\subset E$ in $\mathcal H_k$ is called \emph{saturated in
+the block} if $E/F\in\mathcal S_k$.  Its source already belongs to
+$\mathcal S_k$ by Lemma~\ref{lem:block-filtrations}.  We call $E$
+\emph{block-semistable} if
+\[
+ \varphi_k(F)\le\varphi_k(E)
+ \quad\text{for every nonzero subobject saturated in the block.}
+\]
+The whole object is allowed as such a subobject.
+
+\begin{lemma}[Finite block refinement]\label{lem:block-hn}
+Every nonzero object of $\mathcal S_k$ has a finite filtration by short
+exact sequences in $\mathcal H_k$, with all intermediate quotients in
+$\mathcal S_k$, whose nonzero factors are block-semistable with strictly
+decreasing phases.
+\end{lemma}
+
+\begin{proof}
+Let $\ell>0$ be a lower bound for the norms of nonzero elements of the
+full numerical lattice, supplied by Lemma~\ref{lem:numerical}, and define
+\[
+ q_k(z)=\Re\bigl(e^{-i\pi(a+b)/2}z\bigr),\qquad
+ d_* =\frac{\ell\cos(\pi/10)}{C}>0.
+\]
+For every nonzero block object, Proposition~\ref{prop:block-support}
+implies
+\begin{equation}\label{eq:block-mass}
+ q_k(Z(E))\ge\cos(\pi/10)|Z(E)|\ge d_*.
+\end{equation}
+This projection is additive in short exact sequences.  Hence every
+filtration of a fixed $E$ with nonzero block factors has length at most
+$q_k(Z(E))/d_*$.
+
+If $F\subset E$ is saturated in the block, then both $F$ and $E/F$ are
+block objects.  Their projections are nonnegative, so
+\[
+ \norm{[F]}\le C|Z(F)|
+ \le \frac{C}{\cos(\pi/10)}q_k(Z(F))
+ \le \frac{C}{\cos(\pi/10)}q_k(Z(E)).
+\]
+Only finitely many lattice classes, and therefore only finitely many
+charges and phases, can occur for such $F$.  Choose a nonzero saturated
+subobject $F\subset E$ of maximal phase, and among those of maximal
+$|Z(F)|$.  These maxima concern finite sets of numerical values; no
+finiteness of the set of subobjects is needed.
+
+This $F$ is block-semistable.  Indeed, a saturated subobject of $F$ is
+saturated in $E$, since the resulting quotient of $E$ is an extension of
+two objects of $\mathcal S_k$.  If $G\subset E/F$ is a nonzero saturated
+subobject, its preimage $\widetilde G\subset E$ is also saturated and is
+an extension of $G$ by $F$.  Charges in a sector of width less than $\pi$
+have the elementary strict see-saw property: the argument of their sum
+is strictly between their arguments when those arguments differ, and
+their magnitudes add when they agree.  Thus
+$\varphi_k(G)>\varphi_k(F)$ contradicts maximal phase of $F$ by using
+$\widetilde G$; equality contradicts maximal magnitude.  Every such $G$
+therefore has phase strictly less than $\varphi_k(F)$.
+
+Apply the same selection to the nonzero quotient $E/F$, if there is one,
+and continue.  Preimages give a filtration in the original heart with all
+factors and intermediate quotients in the block.  The phases strictly
+decrease by the preceding argument, and \eqref{eq:block-mass} bounds the
+number of nonzero factors, so the process terminates.
+\end{proof}
+
+\begin{lemma}[The unsaturated test and Hom vanishing]
+\label{lem:block-hom}
+If $E\in\mathcal S_k$ is block-semistable, then every nonzero subobject
+$G\subset E$ in $\mathcal H_k$ satisfies
+$\varphi_k(G)\le\varphi_k(E)$, without a saturation assumption.
+Consequently, for block-semistable $E,F\in\mathcal S_k$,
+\[
+ \varphi_k(E)>\varphi_k(F)\quad\Longrightarrow\quad\Hom(E,F)=0.
+\]
+\end{lemma}
+
+\begin{proof}
+Suppose $G\subset E$ violates the stated inequality.  Decompose $E/G$
+by \eqref{eq:block-torsion-pair} and take the preimage of its torsion
+part:
+\[
+ 0\longrightarrow T\longrightarrow E/G\longrightarrow F'\longrightarrow0,
+ \qquad
+ 0\longrightarrow G\longrightarrow\overline G\longrightarrow T
+      \longrightarrow0,
+ \quad T\in\mathcal T_k,\ F'\in\mathcal S_k.
+\]
+The object $\overline G\subset E$ lies in $\mathcal S_k$ by subobject
+closure, and is saturated because $E/\overline G=F'$.
+The four-block filtration of $T$ has charge rays with phases in
+$[b,a+1]$.  If $g=\varphi_k(G)$, then
+\[
+ a<g\le b,
+ \qquad 0\le\psi-g\le a+1-g<1
+ \quad\text{for every such phase }\psi.
+\]
+Thus $Z(G)$ and every added ray lie in the sector with phase interval
+$[g,a+1]$ of width strictly less than one.  Their sum is nonzero and has
+phase in that same interval.  Since $\overline G\in\mathcal S_k$, its
+phase in $[a,b]$ must therefore be at least $g$.  In particular,
+$\varphi_k(\overline G)>\varphi_k(E)$, a contradiction.
+The strict inequality $g>a$, supplied by the alleged destabilization,
+is what excludes an antipodal endpoint in this argument.  If $T=0$, the
+contradiction already comes from $G$ itself being saturated.
+
+For a nonzero map $E\to F$, take its image $I$ in $\mathcal H_k$.
+Since $I\subset F$, we have $I\in\mathcal S_k$.  The kernel in $E$ is
+therefore saturated.  If that kernel is zero, $\varphi_k(I)=\varphi_k(E)$;
+otherwise semistability and the see-saw property give
+$\varphi_k(I)\ge\varphi_k(E)$.  The unsaturated test applied to
+$I\subset F$ gives $\varphi_k(I)\le\varphi_k(F)$.  These inequalities
+exclude a nonzero map when the source phase is strictly larger.
+\end{proof}
+
+\subsection{The slicing and its phase cuts}
+
+For $\varphi\in\R$, let $\mathcal P(\varphi)$ be the extension closure
+of all block-semistable objects whose assigned phase equals $\varphi$.
+There is one possible block unless $5\varphi\in\Z$, in which case the
+two adjacent blocks are both included in this definition.
+
+\begin{proposition}\label{prop:slicing}
+The categories $\mathcal P(\varphi)$ form a slicing of $\D$ with finite
+Harder--Narasimhan filtrations.  They satisfy
+\begin{equation}\label{eq:slicing-covariance}
+ \mathcal P(\varphi+1)=\mathcal P(\varphi)[1],\qquad
+ \Phi\mathcal P(\varphi)=\mathcal P(\varphi+2/5)
+ \quad(\varphi\in\R),
+\end{equation}
+and every nonzero $E\in\mathcal P(\varphi)$ satisfies
+\begin{equation}\label{eq:semistable-support}
+ Z(E)\in\R_{>0}e^{i\pi\varphi},\qquad
+ \norm{[E]}\le C|Z(E)|.
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+First consider the generators.  If two are in different blocks and the
+source has strictly larger phase, its block index is larger: for $j<k$
+one has $(j+1)/5\le k/5$.  Thus their Hom group vanishes by
+Lemma~\ref{lem:block-filtrations}.  The case of a common block is
+Lemma~\ref{lem:block-hom}.  Induction along extensions in each variable
+now gives
+\[
+ \Hom(\mathcal P(\varphi),\mathcal P(\psi))=0
+ \qquad(\varphi>\psi).
+\]
+The categories are additive, since finite direct sums are split
+extensions.  The charge of an extension of phase-$\varphi$ generators
+is a sum of positive multiples of $e^{i\pi\varphi}$ and hence is again
+positive on that ray.  Likewise the triangle inequality gives
+\[
+ \norm{[E]}\le\sum_j\norm{[E_j]}
+ \le C\sum_j|Z(E_j)|=C|Z(E)|
+\]
+for any such extension filtration.  This proves
+\eqref{eq:semistable-support}.
+
+Refine the decreasing block filtration of any object by
+Lemma~\ref{lem:block-hn}, splicing the resulting triangles by the
+octahedral axiom.  The phases are weakly decreasing globally and strictly
+decreasing inside each block.  Adjacent equal phases can only occur at a
+shared endpoint; merging each run of equal-phase factors into its
+extension gives a finite filtration with strictly decreasing phases and
+factors in the corresponding $\mathcal P(\varphi)$.  This also explains
+why both blocks at a shared endpoint must enter the same phase category.
+
+By \eqref{eq:fifths-nesting}, the functors $[1]$ and $\Phi$ take
+$(\mathcal H_k,\mathcal S_k)$ to
+$(\mathcal H_{k+5},\mathcal S_{k+5})$ and
+$(\mathcal H_{k+2},\mathcal S_{k+2})$, respectively.  These equivalences
+take exact sequences in the hearts to exact sequences, preserve the
+condition of being saturated in a block, and change the assigned phase
+by $1$ and $2/5$.  They therefore preserve block-semistability with these
+phase increments.  Applying them and their inverses to the extension
+closures proves both equalities in \eqref{eq:slicing-covariance}.
+The phase-cut argument that follows proves closure under direct
+summands without presupposing a slicing heart.
+\end{proof}
+
+For any interval $I\subset\R$, write $\mathcal P(I)$ for the extension
+closure of the $\mathcal P(\varphi)$ with $\varphi\in I$.
+
+\begin{lemma}[Direct construction of the phase cuts]\label{lem:phase-cuts}
+For every real $c$, both pairs
+\begin{equation}\label{eq:phase-cuts}
+ \bigl(\mathcal P((c,\infty)),\mathcal P((-\infty,c])\bigr),
+ \qquad
+ \bigl(\mathcal P([c,\infty)),\mathcal P((-\infty,c))\bigr)
+\end{equation}
+are an aisle and its right orthogonal for a bounded t-structure.
+Their hearts are respectively
+$\mathcal P((c,c+1])$ and $\mathcal P([c,c+1))$.
+Every interval category is the intersection of its lower and upper cut
+categories, with the indicated endpoint conventions.  In particular,
+these categories, including each $\mathcal P(\varphi)$, are closed under
+direct summands.
+\end{lemma}
+
+\begin{proof}
+Denote either pair in \eqref{eq:phase-cuts} by $(\mathcal U,\mathcal L)$.
+Strict phase separation gives $\Hom(\mathcal U,\mathcal L)=0$ by
+extension induction.  Equation~\eqref{eq:slicing-covariance} gives
+$\mathcal U[1]\subset\mathcal U$ and
+$\mathcal L[-1]\subset\mathcal L$.  Cutting the finite decreasing
+filtration at $c$ and splicing triangles gives, for every $E$, a triangle
+\begin{equation}\label{eq:phase-truncation}
+ U_E\longrightarrow E\longrightarrow L_E\longrightarrow U_E[1],
+ \qquad U_E\in\mathcal U,\ L_E\in\mathcal L.
+\end{equation}
+All factors of phase exactly $c$ are assigned to the same side, according
+to the selected convention.  These are the t-structure axioms in the
+aisle/right-orthogonal convention, and we can verify the orthogonal
+characterizations directly before using any summand closure.  If
+$E\in{}^\perp\mathcal L$, applying $\Hom(-,L_E)$ to
+\eqref{eq:phase-truncation} shows that $\Hom(L_E,L_E)=0$, since its
+neighboring terms $\Hom(U_E[1],L_E)$ and $\Hom(E,L_E)$ vanish.
+Thus $L_E=0$ and $E\in\mathcal U$.  Similarly, if
+$E\in\mathcal U^\perp$, apply $\Hom(U_E,-)$ and use
+$\Hom(U_E,L_E[-1])=\Hom(U_E,E)=0$ to obtain $U_E=0$.
+Hence
+\[
+ \mathcal U={}^\perp\mathcal L,\qquad
+ \mathcal L=\mathcal U^\perp.
+\]
+They are consequently closed under direct summands, and
+\eqref{eq:phase-truncation} is a genuine truncation triangle.  Finite
+upper and lower bounds on the phases of each object's filtration, and
+the shift law, prove boundedness.
+
+We justify the interval description explicitly.  A nonempty decreasing
+filtration with nonzero phase factors cannot have zero total object.
+Indeed, the identity of its first factor maps nontrivially into the
+total object: at each subsequent triangle this map remains nonzero
+because $\Hom(F_1,F_j[-1])=0$.  Now group a chosen filtration of an
+object at the two endpoints of an interval.  If the object lies in the
+upper and lower cut categories, the preceding orthogonal
+characterizations force the exterior truncation objects to be zero.
+By the observation just made, their groups of factors are empty.  The
+remaining factors lie in the interval.  The reverse inclusion follows
+from extension closure.  This works with either choice at either
+endpoint, as well as for rays and singletons.
+Intersecting the aisle with the right orthogonal of its shift therefore
+gives precisely the two displayed hearts.  Finally, intersections of
+orthogonals are closed under summands, which proves the last assertion.
+\end{proof}
+
+\subsection{Quasi-abelian intervals and local finite length}
+
+It remains to verify local finiteness. We first identify the exact sequences
+in a short phase interval by realizing it inside a phase-cut heart. We then
+use a positive projection of the charge to bound the length of every strict
+filtration in that interval.
+
+\begin{lemma}[The exact structure on a short phase interval]
+\label{lem:interval-quasiabelian}
+For every interval $I$ of width less than one, with any choices of open
+or closed endpoints, $\mathcal P(I)$ is quasi-abelian.  Its strict exact
+sequences are exactly the short exact sequences of a suitable bounded
+heart whose three terms belong to $\mathcal P(I)$.
+\end{lemma}
+
+\begin{proof}
+The empty interval gives the zero category.  Otherwise let $a\le b$
+be its endpoints, with $b-a<1$.  If $a$ is excluded, use the heart
+$\mathcal H=\mathcal P((a,a+1])$; if it is included, use
+$\mathcal H=\mathcal P([a,a+1))$.  Inside this heart, the cut at $b$
+gives a torsion pair $(\mathcal T,\mathcal F)$ with
+$\mathcal F=\mathcal P(I)$.  Equality at $b$ is assigned to
+$\mathcal F$ exactly when $b\in I$.  For example, for $I=(a,b)$ it is
+\[
+ \mathcal T=\mathcal P([b,a+1]),\qquad
+ \mathcal F=\mathcal P((a,b))
+ \quad\text{in }\mathcal P((a,a+1]).
+\]
+For $I=(a,b]$ the corresponding torsion class is
+$\mathcal P((b,a+1])$.  The lower-closed variants use the other heart
+above and the same upper-endpoint rule.  Hom orthogonality follows from
+the phase inequality, and the cut triangles have all terms in
+$\mathcal H$, so they give the required short exact decompositions.
+
+We prove the quasi-abelian assertion directly for a torsion-free class
+$\mathcal F$ in an abelian heart $\mathcal H$; this standard fact is also
+recorded in \cite[Lemma~3.1]{Tattar2021}.
+It is closed under subobjects and extensions.  For a morphism
+$f:A\to B$ in $\mathcal F$, put
+$K=\ker_{\mathcal H}f$, $J=\im_{\mathcal H}f$, and
+$Q=\operatorname{coker}_{\mathcal H}f$.
+The kernel in $\mathcal F$ is $K$, and the cokernel is
+$Q/t(Q)$, where $t(Q)$ denotes the torsion part of $Q$.
+Indeed, maps from $Q$ to an object of $\mathcal F$ annihilate $t(Q)$;
+this proves the cokernel's universal property.  Both $K$ and $J$ lie in
+$\mathcal F$.  Consequently the internal coimage of $f$ is $J$, whereas
+its internal image is the preimage of $t(Q)$ under $B\to Q$.
+
+Recall that a morphism is strict if its canonical coimage-to-image map
+is an isomorphism.  The preceding formulas show that a strict
+monomorphism in $\mathcal F$ is exactly a monomorphism in $\mathcal H$
+whose cokernel belongs to $\mathcal F$.  A strict epimorphism in
+$\mathcal F$ is exactly an epimorphism in $\mathcal H$ between its
+objects: for an internal epimorphism the internal image is $B$, and
+strictness says precisely that $J=B$.  This statement is about strict
+epimorphisms, not all epimorphisms of the subcategory.
+
+The pushout in $\mathcal H$ of a strict monomorphism $A\hookrightarrow B$
+along $A\to A'$ fits into
+\[
+ 0\longrightarrow A'\longrightarrow B'\longrightarrow B/A
+     \longrightarrow0.
+\]
+Both outside terms are in $\mathcal F$, so $B'\in\mathcal F$ and the
+new monomorphism is strict.  The pullback in $\mathcal H$ of a strict
+epimorphism $A\twoheadrightarrow B$ along $B'\to B$ is a subobject of
+$A\oplus B'$, hence lies in $\mathcal F$, and its projection to $B'$ is
+an ambient epimorphism, hence strict.  These ambient universal
+properties remain universal in the full subcategory $\mathcal F$.
+Thus kernels and cokernels exist, pushouts preserve strict
+monomorphisms, and pullbacks preserve strict epimorphisms.  This proves
+quasi-abelianity and the stated description of strict exact sequences.
+\end{proof}
+
+\begin{proposition}[Local finiteness]\label{prop:local-finiteness}
+Every phase interval of width less than one generates a finite-length
+quasi-abelian category.  Hence the slicing in
+Proposition~\ref{prop:slicing} is locally finite.
+\end{proposition}
+
+\begin{proof}
+Let $I$ have endpoints $a\le b$ and width $w=b-a<1$, and put
+\[
+ q_I(z)=\Re\bigl(e^{-i\pi(a+b)/2}z\bigr),\qquad
+ d_I=\frac{\ell}{C}\cos(\pi w/2)>0,
+\]
+where $\ell$ is the full-lattice norm lower bound used in
+Lemma~\ref{lem:block-hn}.  A nonzero semistable $F$ of phase in $I$ has
+nonzero numerical class, since its charge is nonzero.  Therefore
+\eqref{eq:semistable-support} gives
+\[
+ q_I(Z(F))\ge\cos(\pi w/2)|Z(F)|\ge d_I.
+\]
+Every nonzero object of $\mathcal P(I)$ is a finite extension of nonzero
+semistable objects with phases in $I$.  Additivity thus gives the same
+lower bound $q_I(Z(E))\ge d_I$ for every such object.
+
+By Lemma~\ref{lem:interval-quasiabelian}, a strict exact sequence in
+$\mathcal P(I)$ is exact in its ambient heart, so the projection $q_I Z$
+is additive on it.  Every nonzero successive quotient in a strict
+filtration of a fixed $E\in\mathcal P(I)$ is again an object of
+$\mathcal P(I)$ and consumes at least $d_I$ of this projection.
+The number of such quotients is consequently at most
+$q_I(Z(E))/d_I$.  This bounds both increasing and decreasing strict
+subobject chains, and proves finite length in the quasi-abelian exact
+structure.  The argument uses discreteness of the full numerical
+lattice; it requires no discreteness of its image under $Z$.
+\end{proof}
+
+The charge is numerical and normalized on point sheaves by
+Proposition~\ref{prop:central-charge}; its eigencharge identity combines
+with \eqref{eq:slicing-covariance}.  Proposition~\ref{prop:slicing},
+Lemma~\ref{lem:phase-cuts}, and Proposition~\ref{prop:local-finiteness}
+give all slicing, Harder--Narasimhan, and local-finiteness requirements.
+Finally \eqref{eq:semistable-support} is the support property on the
+\emph{full} space $\Knum(X)_\R$.  This completes the proof of
+Theorem~\ref{thm:main}.
diff --git a/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/build/sections/introduction.tex b/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/build/sections/introduction.tex
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@@ -0,0 +1,147 @@
+\section{Introduction}
+
+Stability conditions organize the objects of a derived category by a complex
+central charge and a real phase, with finite filtrations into semistable
+objects. Their physical antecedent is Douglas's proposal of $\Pi$-stability
+for D-branes \cite{Douglas2001}; Bridgeland gave the mathematical framework
+used here \cite{Bridgeland2007}. At a Gepner point, one seeks a stability
+condition compatible with a distinguished autoequivalence: the functor should
+rotate the central charge and translate every semistable phase by a fixed
+amount. We construct such a condition for every smooth quintic threefold.
+
+Let $X\subset\mathbb P^4_{\C}$ be a smooth quintic hypersurface, and put
+$\D=\Db(\Coh(X))$ and $H=c_1(\OO_X(1))$. The structure sheaf is spherical:
+$\Ext^*(\OO_X,\OO_X)=\C\oplus\C[-3]$. Using the spherical twist of
+Seidel--Thomas \cite{SeidelThomas2001}, we consider the autoequivalence
+\[
+ \Phi=T_{\OO_X}\circ(-\otimes\OO_X(1)),\qquad
+ T_{\OO_X}(E)=\operatorname{Cone}\bigl(
+ R\Hom(\OO_X,E)\otimes\OO_X\longrightarrow E\bigr).
+\]
+We write $\Knum(X)$ for the Grothendieck group modulo the radical of its Euler
+pairing, and $\Knum(X)_{\R}=\Knum(X)\otimes_{\Z}\R$.
+A numerical central charge is a homomorphism $Z\colon\Knum(X)\to\C$.
+
+Following Bridgeland \cite{Bridgeland2007}, a slicing $\mathcal P$ is a family
+of full additive subcategories $\mathcal P(\varphi)\subset\D$, indexed by
+$\varphi\in\R$, with
+$\mathcal P(\varphi+1)=\mathcal P(\varphi)[1]$,
+vanishing $\Hom(\mathcal P(\varphi_1),\mathcal P(\varphi_2))$ for
+$\varphi_1>\varphi_2$, and finite Harder--Narasimhan filtrations by distinguished
+triangles with strictly decreasing phases. A nonzero object of
+$\mathcal P(\varphi)$ has charge in $\R_{>0}e^{i\pi\varphi}$. Local finiteness
+requires the extension categories of sufficiently short phase intervals to be
+quasi-abelian of finite length. We use the support property of
+Kontsevich--Soibelman \cite[Section~1.2]{KontsevichSoibelman2008}, expressed on
+the full numerical group: there exist a norm on $\Knum(X)_{\R}$ and a
+constant $C>0$ such that
+\[
+ \norm{[E]}\le C|Z(E)|
+ \quad\text{for every nonzero }E\in\mathcal P(\varphi).
+\]
+The pair $(Z,\mathcal P)$ is a numerical Bridgeland stability condition when
+these charge and slicing requirements hold, with local finiteness.
+We will also verify directly that each phase category is closed under direct
+summands, using the phase cuts constructed in Lemma~\ref{lem:phase-cuts}.
+
+Toda formulated Gepner-type stability for graded matrix factorizations
+\cite[Definition~2.3 and Conjecture~1.2]{Toda2013}, and gave the normalized
+quintic formulation in \cite[Conjecture~3.1]{TodaQuintic2013}.
+The following theorem positively resolves this quintic conjecture.
+
+\begin{theorem}\label{thm:main}
+Every smooth complex quintic threefold admits a numerical Bridgeland stability
+condition $\sigma=(Z,\mathcal P)$ with the support property on $\Knum(X)_{\R}$
+such that, for every $E\in\D$ and $\varphi\in\R$,
+\[
+ Z(\Phi E)=e^{2\pi i/5}Z(E),\qquad
+ \Phi\bigl(\mathcal P(\varphi)\bigr)=\mathcal P(\varphi+2/5),\qquad
+ Z(\OO_x)=-1
+\]
+for every closed point $x\in X$.
+\end{theorem}
+
+The normalization identifies our central charge with Toda's by the uniqueness
+of the normalized eigenform proved in Proposition~\ref{prop:central-charge}.
+
+\paragraph{Prior work and the remaining construction.}
+The functor relation $\Phi^5\simeq[2]$ and its eigencharge are established
+features of the quintic Gepner problem. Orlov's equivalence
+\cite[Theorem~2.5]{Orlov2009} and the compatibility with grade shift proved by
+Ballard--Favero--Katzarkov \cite[Proposition~5.8 and Remark~5.12]{BallardFaveroKatzarkov2012}
+identify the relevant categorical symmetry; Toda recalls this identification
+in \cite[Theorem~2.5]{TodaQuintic2013}. A direct computation with
+Fourier--Mukai kernels appears in Aspinwall \cite[Section~7.1.4]{Aspinwall2004}.
+Section~\ref{sec:numerical} fixes the numerical conventions used in our
+construction; Appendix~\ref{app:periodicity} supplies a relative-kernel proof
+of the functor relation.
+
+The double-tilt approach to threefold stability was developed by
+Bayer--Macr\`i--Toda \cite{BayerMacriToda2014}. For the Gepner charge, Toda
+already proposed the first slope tilt at $-1/2$ and the second tilt using
+the imaginary part of that charge
+\cite[Sections~3.4--3.6 and Conjecture~3.9]{TodaQuintic2013}.
+His Remark~3.10 identifies the existence of Harder--Narasimhan filtrations
+for this irrational charge as a remaining difficulty.
+The existence of numerical Bridgeland stability conditions on every smooth
+quintic was established by Li \cite[Theorem~1.3]{Li2019}.
+Theorem~\ref{thm:main} adds the specified Gepner symmetry.
+Our slope-sheaf input is a consequence
+of Xu's stronger Bogomolov--Gieseker inequality
+\cite[Theorem~1.3]{Xu2026}, whose hypotheses and precise comparison with the
+bound used here are recorded in Lemma~\ref{lem:slope-input}.
+
+\paragraph{Main idea and proof roadmap.}
+The central step is to retain two independent perturbations of Toda's second
+cut. They give inequalities controlling rank and first Chern character; the
+complex charge then controls the other two numerical coordinates. We obtain
+this support estimate before defining semistable objects. It makes finite
+refinement possible even though the image of the numerical lattice under the
+charge need not be discrete.
+
+The proof proceeds through four stages.
+\begin{enumerate}
+\item In Section~\ref{sec:numerical}, we record periodicity and compute the
+unique normalized eigencharge. We also identify the full numerical group
+with a rank-four lattice. The resulting coordinate estimate will turn
+control of rank, first Chern character, and charge into full support.
+
+\item In Section~\ref{sec:aisles}, we construct a family of bounded aisles
+$U_\eta$, where an aisle is the nonpositive part of a bounded
+$t$-structure and $\eta=(\eta_0,\eta_1)$ ranges over a fixed open square in
+$\R^2$. The construction tilts $\Coh(X)$ twice, beginning at slope
+$-1/2$, and varies the rank and first-Chern coefficients of the second cut.
+Xu's inequality supplies strict margins for the comparison. The crucial
+conclusion of Proposition~\ref{prop:uniform-aisle} is
+\[
+ \Phi U_{\eta}\subset U_{\theta}.
+\]
+Both parameters vary independently throughout the same square.
+
+\item In Section~\ref{sec:grid}, periodicity and positivity refine the
+translated aisles to a nested grid $V_k$, indexed by integers, with phase
+spacing $1/5$ (Proposition~\ref{prop:fine-grid}). The category
+$\mathcal S_k=V_k\cap V_{k+1}^\perp$ between consecutive aisles is called
+a block; here $\perp$ is right $\Hom$-orthogonality. Its charges lie in a
+closed sector of angle $\pi/5$. Each block object belongs to every member
+of a suitable perturbed family of hearts. Testing all those perturbations
+gives the full support estimate of Proposition~\ref{prop:block-support}.
+
+\item We then refine each block into semistable factors
+(Lemma~\ref{lem:block-hn}). Support and a positive projection of the charge
+bound the numerical classes of subobjects and the length of any refinement.
+The full lattice is discrete, so maximal phases exist and refinement stops.
+We merge shared endpoint phases to obtain the slicing, construct its phase
+cuts, and prove local finiteness through the exact structure of short phase
+intervals (Proposition~\ref{prop:local-finiteness}).
+\end{enumerate}
+
+\paragraph{Conventions and prerequisites.}
+Our aisle convention is degree $\le0$, so aisles are closed under $[1]$.
+For a subcategory $\mathcal C$, the notation
+$\mathcal C^\perp$ means its right $\Hom$-orthogonal. Extension closures always
+include the zero object and finite iterated extensions. We use standard
+coherent-sheaf cohomology, slope Harder--Narasimhan filtrations, derived
+truncations, Riemann--Roch, Serre duality, and the hyperplane theorem. The
+specific slope inequality is cited as above; the construction of the
+stability condition from it is given in full.
diff --git a/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/build/sections/numerical.tex b/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/build/sections/numerical.tex
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index 0000000000000000000000000000000000000000..212cf8959e9d153b546189fec3fcb8938cbfea7d
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+++ b/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/build/sections/numerical.tex
@@ -0,0 +1,207 @@
+\section{Periodicity and the numerical central charge}
+\label{sec:numerical}
+
+Throughout, $X\subset\mathbf P^4_{\C}$ is a smooth hypersurface of degree
+five, $i\colon X\hookrightarrow\mathbf P^4$ is its inclusion, and
+$\D=\Db(\Coh(X))$. Put $H=c_1(\OO_X(1))$, so that $\int_XH^3=5$.
+For an object $E\in\D$ we use the normalized characters
+\begin{equation}
+\label{eq:normalized-characters}
+ v(E)=(r,c,d,e)
+ =\left(\rk(E),\frac{\int_XH^2\ch_1(E)}5,
+                  \frac{\int_XH\ch_2(E)}5,
+                  \frac{\int_X\ch_3(E)}5\right).
+\end{equation}
+In particular, ordinary slope is normalized as $\mu_H(E)=c(E)/r(E)$
+for a sheaf of positive rank. We first record the functor relation that
+will make the subsequent family of aisles periodic, and then determine
+the central charge on the full numerical Grothendieck group.
+The periodicity is already visible in Aspinwall's direct kernel calculation
+\cite[Section~7.1.4]{Aspinwall2004}. It also follows from Orlov's equivalence
+\cite[Theorem~2.5]{Orlov2009} and its compatibility with grade shift
+\cite[Proposition~5.8 and Remark~5.12]{BallardFaveroKatzarkov2012},
+as recalled in \cite[Theorem~2.5]{TodaQuintic2013}.
+Appendix~\ref{app:periodicity} gives a direct proof, keeping track of the
+relative kernels and the natural transformations needed for a functor
+isomorphism.  Here we record the relation and compute its numerical action.
+
+\begin{lemma}\label{lem:periodicity}
+Let $\Phi=T_{\OO_X}\circ(-\otimes\OO_X(1))$. There is a natural
+isomorphism of exact functors
+\begin{equation}\label{eq:periodicity}
+ \Phi^5\simeq[2].
+\end{equation}
+Consequently $\Phi$ is an autoequivalence, with inverse $\Phi^4[-2]$.
+\end{lemma}
+
+To prove the relation, expand the ordered composition as
+\[
+ \Phi^5\simeq
+ T_{\OO_X}\circ T_{\OO_X(1)}\circ\cdots\circ T_{\OO_X(4)}
+ \circ(-\otimes\OO_X(5)).
+\]
+The rightmost functor acts first.  The relative-kernel argument in
+Appendix~\ref{app:periodicity} identifies this twist product with
+tensoring by $\OO_X(-5)$ followed by $[2]$.
+
+We next identify the entire numerical group and its integral discreteness.
+After computing the charge, these coordinates will also show that controlling
+rank, first Chern character, and charge suffices to control a numerical class.
+
+\begin{lemma}\label{lem:numerical}
+The map $v$ identifies $\Knum(X)_{\R}$ with $\R^4$. Under this
+identification, $\Knum(X)$ is a full discrete lattice contained in
+\begin{equation}\label{eq:numerical-lattice}
+ \Z\times\Z\times\tfrac1{10}\Z\times\tfrac1{30}\Z.
+\end{equation}
+For $v=(r,c,d,e)$ and $w=(r',c',d',e')$, the Euler form is
+\begin{equation}\label{eq:euler-form}
+ \chi(v,w)=5\left(re'-cd'+dc'-er'
+                   +\frac56(rc'-cr')\right).
+\end{equation}
+The matrix induced by $\Phi$ is
+\begin{equation}\label{eq:numerical-rotation}
+ M=
+ \begin{pmatrix}
+ -4&-20/3&-5&-5\\
+ 1&1&0&0\\
+ 1/2&1&1&0\\
+ 1/6&1/2&1&1
+ \end{pmatrix},
+ \qquad
+ \det(zI-M)=z^4+z^3+z^2+z+1.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+The Weak Lefschetz Theorem (in the integral form recalled in
+\cite[Theorem~3]{Catanese2014}) and Poincar\'e duality give
+$H^{2j}(X,\Q)=\Q H^j$ for $0\le j\le3$; the integral version also
+gives $H^2(X,\Z)=\Z H$. Thus the even Chern character is precisely
+\[
+ \ch(E)=r+cH+dH^2+eH^3.
+\]
+The tangent sequence gives
+$c(T_X)=(1+H)^5/(1+5H)$, so that $c_1(T_X)=0$,
+$c_2(T_X)=10H^2$, and
+\[
+ \operatorname{td}(X)=1+\frac56H^2.
+\]
+Hirzebruch--Riemann--Roch, obtained by taking the proper map to a point in
+\cite[Section~7]{BorelSerre1958}, now gives \eqref{eq:euler-form}, and in
+particular
+\begin{equation}\label{eq:euler-characteristic}
+ \chi(E)=5\left(e(E)+\frac56c(E)\right).
+\end{equation}
+The matrix of the bilinear form~\eqref{eq:euler-form} is
+\[
+ J=\begin{pmatrix}
+ 0&25/6&0&5\\
+ -25/6&0&-5&0\\
+ 0&5&0&0\\
+ -5&0&0&0
+ \end{pmatrix},
+ \qquad\det J=625.
+\]
+Moreover the four vectors
+\[
+ v(\OO_X(k))=(1,k,k^2/2,k^3/6),\qquad 0\le k\le3,
+\]
+span $\Q^4$: their row matrix has determinant one. It follows that
+the kernel of $v$ on $K_0(X)$ is exactly the radical of the Euler
+pairing. Indeed one implication follows from Riemann--Roch, and the
+other follows by testing against these four spanning vectors and using
+the nondegeneracy of $J$. Hence $v$ identifies the numerical group with
+its image, which spans $\Q^4$.
+
+For any class of $K_0(X)$, integrality of Chern classes and
+\[
+ \ch_2=\frac{c_1^2-2c_2}{2},\qquad
+ \ch_3=\frac{c_1^3-3c_1c_2+3c_3}{6}
+\]
+give $d\in\tfrac1{10}\Z$ and $e\in\tfrac1{30}\Z$; integral Weak
+Lefschetz gives $c\in\Z$. The image is therefore a subgroup of the
+discrete lattice in \eqref{eq:numerical-lattice}. Since it contains the
+four spanning line-bundle classes, it is itself a full lattice.
+In particular its real span is the full space $\R^4$.
+
+Tensoring by $\OO_X(1)$ sends
+\[
+ (r,c,d,e)\longmapsto
+ \left(r,c+r,d+c+\frac r2,e+d+\frac c2+\frac r6\right).
+\]
+The twist $T_{\OO_X}$ then subtracts $\chi(E(1))$ from the rank and leaves
+the other three coordinates unchanged. Formula~\eqref{eq:euler-characteristic}
+therefore gives $M$ as displayed. Expanding $\det(zI-M)$ gives
+\eqref{eq:numerical-rotation}; in particular its four eigenvalues are
+the nonidentity fifth roots of unity, each with multiplicity one.
+\end{proof}
+
+We denote by $\Phi_*$ the induced automorphism of $\Knum(X)$ and its
+real-linear extension to $\Knum(X)_{\R}$; both are represented by $M$ in
+these coordinates.
+
+Put
+\begin{equation}\label{eq:charge-constants}
+ \lambda=\exp(2\pi i/5),\qquad
+ t_0=\frac12\cot(\pi/5)>0,\qquad
+ u=\frac45\sin^2(\pi/5)=\frac{5-\sqrt5}{10}.
+\end{equation}
+
+\begin{proposition}\label{prop:central-charge}
+There is a unique numerical homomorphism $Z\colon\Knum(X)\to\C$
+such that $Z\circ\Phi=\lambda Z$ and $Z(\OO_x)=-1$ for closed points
+$x\in X$. It is
+\begin{equation}\label{eq:central-charge}
+ \frac{Z(r,c,d,e)}5
+  =\frac{\lambda-1}{5}r
+   +\left(t_0^2-\frac56+\frac{i t_0}{2}\right)c
+   +\left(-\frac12+i t_0\right)d-e.
+\end{equation}
+In particular,
+\begin{equation}\label{eq:imaginary-charge}
+ \frac{\Im Z(r,c,d,e)}{5t_0}=d+\frac12c+ur.
+\end{equation}
+For any fixed norm on $\Knum(X)_{\R}$, there is a constant $C_0>0$
+such that every real numerical class $v=(r,c,d,e)$ satisfies
+\begin{equation}\label{eq:numerical-norm-control}
+ \norm{v}\le C_0\bigl(|r|+|c|+|Z(v)|\bigr).
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+A closed point has $v(\OO_x)=(0,0,0,1/5)$, so the normalization fixes
+the coefficient of $e$ in $Z/5$ to be $-1$.
+Write $Z/5=Ar+Bc+Cd-e$. Reading the $e$, $d$, and $c$ columns of
+$ZM=\lambda Z$, in that order, gives
+\[
+ A=\frac{\lambda-1}{5},\qquad
+ C=-\frac{\lambda}{\lambda-1},\qquad
+ B=-\frac43+\frac{C-1/2}{\lambda-1}.
+\]
+The identity
+\[
+ \frac1{\lambda-1}=-\frac12-i t_0
+\]
+then gives $C=-1/2+i t_0$ and
+$B=t_0^2-5/6+i t_0/2$. Substitution in the remaining, rank-column
+equation reduces it to
+$1+\lambda+\lambda^2+\lambda^3+\lambda^4=0$.
+This proves both the eigenvalue identity and uniqueness with the stated
+normalization. Taking imaginary parts and using
+\[
+ \frac{\sin(2\pi/5)}{5t_0}
+       =\frac45\sin^2(\pi/5)=u
+\]
+proves \eqref{eq:imaginary-charge}.
+
+Finally, the real coefficient matrix of $(d,e)$ in
+$(\Re Z,\Im Z)$ is
+\[
+ \begin{pmatrix}-5/2&-5\\5t_0&0\end{pmatrix},
+ \qquad \det=25t_0\ne0.
+\]
+Thus $v\mapsto(r,c,\Re Z(v),\Im Z(v))$ is an isomorphism of real
+vector spaces. Boundedness of its inverse in finite dimension proves
+\eqref{eq:numerical-norm-control}.
+\end{proof}
diff --git a/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/build/sections/periodicity.tex b/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/build/sections/periodicity.tex
new file mode 100644
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+++ b/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/build/sections/periodicity.tex
@@ -0,0 +1,153 @@
+\section{A relative-kernel proof of periodicity}
+\label{app:periodicity}
+
+\begin{proof}[Proof of Lemma~\ref{lem:periodicity}]
+Write $T_j=T_{\OO_X(j)}$ and $L=(-\otimes\OO_X(1))$.
+We interpret the twist as the Fourier--Mukai transform whose kernel is
+the cone of the evaluation map
+\[
+ \OO_X(-j)\boxtimes\OO_X(j)\longrightarrow\OO_\Delta.
+\]
+We take cones in the standard enhancement by complexes of perfect
+kernels, so that the evaluation squares below induce maps of cones.
+Their transforms give natural transformations of functors. Tensoring the evaluation
+kernel gives $LT_jL^{-1}\simeq T_{j+1}$, and hence
+\begin{equation}\label{eq:twist-product}
+ \Phi^5\simeq T_0T_1T_2T_3T_4L^5.
+\end{equation}
+We will identify the kernel of $T_0T_1T_2T_3T_4$ with
+$\OO_\Delta(-5)[2]$, where the twist is in the second factor.
+
+Keep the first copy of $X$ as a source parameter. Set
+\[
+ f=1_X\times i\colon X\times X\longrightarrow X\times\mathbf P^4,
+ \qquad
+ p\colon X\times\mathbf P^4\longrightarrow X,
+ \quad q\colon X\times X\longrightarrow X.
+\]
+Here $p$ and $q$ are the first projections.
+The kernel convention is that $F\in\Db(\Coh(X\times Y))$ sends $E$ to
+$R\pi_{Y*}(\pi_X^*E\otimes^{\mathbf L}F)$.
+For such a kernel, let
+\[
+ a_j^Y(F)=R\pi_{X*}\bigl(F\otimes\pi_Y^*\OO_Y(-j)\bigr),
+ \qquad
+ \mathsf L_j^Y(F)=
+ \operatorname{Cone}\bigl(a_j^Y(F)\boxtimes\OO_Y(j)\longrightarrow F\bigr).
+\]
+The arrow is the adjunction evaluation. Projection formula and proper
+base change show that $\mathsf L_j^Y(F)$ represents postcomposition of
+the transform of $F$ with the evaluation cone for $\OO_Y(j)$.
+Indeed, on an input $E$ the first kernel in this cone gives
+\[
+ R\Gamma\bigl(X,E\otimes^{\mathbf L}a_j^Y(F)\bigr)\otimes\OO_Y(j)
+ \simeq
+ R\operatorname{Hom}\bigl(\OO_Y(j),\Phi_F(E)\bigr)\otimes\OO_Y(j).
+\]
+In particular the source coefficient here is $a_j^Y(F)$ itself.
+
+These coefficient functors let us compare the evaluation cones on $X$ with
+those on $\mathbf P^4$. We will show that the five projective-space
+evaluations remove all coefficients and leave the zero kernel, while the
+comparison preserves the cone of the divisor counit. Computing that cone
+will give the required shift.
+
+Start with
+\[
+ K_5=f_*\OO_\Delta,\qquad G_5=\OO_\Delta,\qquad
+ u_5\colon Lf^*K_5\longrightarrow G_5
+\]
+given by the counit. For $j=4,3,\ldots,0$, form successively
+\[
+ K_j=\mathsf L_j^{\mathbf P^4}(K_{j+1}),
+ \qquad
+ G_j=\mathsf L_j^X(G_{j+1}).
+\]
+We construct compatible maps $u_j\colon Lf^*K_j\to G_j$ and show
+that their cones are all isomorphic to $\operatorname{Cone}(u_5)$.
+
+For any map $u\colon Lf^*K\to G$, the unit followed by $f_*u$ gives
+$K\to f_*G$, hence comparison maps
+\[
+ \theta_l\colon a_l^{\mathbf P^4}(K)\longrightarrow a_l^X(G).
+\]
+For $(K_5,G_5,u_5)$ these are isomorphisms for every $l$: the composite
+$f_*\OO_\Delta\to f_*Lf^*f_*\OO_\Delta\to f_*\OO_\Delta$
+is the identity by the adjunction identity. Suppose that, before the
+step indexed by $j$, the maps $\theta_l$ are isomorphisms for
+$0\le l\le j$. The evaluation square
+\[
+\begin{array}{ccc}
+ q^*a_j^{\mathbf P^4}(K_{j+1})\otimes\OO_X(j)
+   &\longrightarrow& Lf^*K_{j+1}\\
+ \big\downarrow\scriptstyle{\simeq}&&\big\downarrow\scriptstyle{u_{j+1}}\\
+ q^*a_j^X(G_{j+1})\otimes\OO_X(j)
+   &\longrightarrow& G_{j+1}
+\end{array}
+\]
+commutes by adjunction. Taking its cones defines $u_j$; because its
+left vertical arrow is an isomorphism, the cone of $u_j$ is isomorphic
+to the cone of $u_{j+1}$.
+
+For $l<j$, apply the coefficient functor $a_l$ to the two evaluation
+triangles. The comparison on their first terms is
+\[
+ a_j^{\mathbf P^4}(K_{j+1})\otimes R\Gamma(\mathbf P^4,\OO(j-l))
+ \longrightarrow
+ a_j^X(G_{j+1})\otimes R\Gamma(X,\OO_X(j-l)).
+\]
+It is an isomorphism: $\theta_j$ is an isomorphism, and the divisor
+sequence
+\[
+ 0\longrightarrow\OO_{\mathbf P^4}(m-5)
+ \longrightarrow\OO_{\mathbf P^4}(m)
+ \longrightarrow i_*\OO_X(m)\longrightarrow0
+\]
+identifies the two cohomology complexes for $1\le m\le4$.
+The comparison on the middle terms is also an isomorphism by induction,
+so the remaining maps $\theta_l$ stay isomorphisms after the mutation.
+This proves the induction and the constancy of the cone.
+
+We next show that $K_0=0$. The step indexed by $j$ kills
+$a_j^{\mathbf P^4}$, since
+$R\Gamma(\mathbf P^4,\OO)=\C$. It preserves the already vanishing
+coefficients with indices $k>j$, since
+$R\Gamma(\mathbf P^4,\OO(j-k))=0$ for $1\le k-j\le4$.
+Thus $a_j^{\mathbf P^4}(K_0)=0$ for $0\le j\le4$.
+All these kernels are perfect, and proper base change implies that
+every derived source fiber $F$ of $K_0$ satisfies
+\[
+ R\operatorname{Hom}_{\mathbf P^4}(\OO(j),F)=0
+ \qquad(0\le j\le4).
+\]
+The exact Koszul complex of the five homogeneous coordinates propagates
+these five consecutive vanishings to
+$R\Gamma(\mathbf P^4,F(n))=0$ for every integer $n$.
+For sufficiently large $n$, Serre vanishing and global generation,
+applied to the finitely many cohomology sheaves of $F$, force all those
+sheaves to vanish. Therefore every derived source fiber is zero, and
+the derived Nakayama lemma gives $K_0=0$. It follows that
+\[
+ G_0\simeq\operatorname{Cone}(u_0)
+       \simeq\operatorname{Cone}(u_5).
+\]
+
+Finally, $f$ is a Cartier divisor embedding with normal bundle
+$\OO_X(5)$ in the target variable. Its two-term divisor resolution
+computes
+\[
+ \mathcal H^{-1}(Lf^*f_*\OO_\Delta)=\OO_\Delta(-5),
+ \qquad
+ \mathcal H^0(Lf^*f_*\OO_\Delta)=\OO_\Delta,
+\]
+with no other cohomology sheaves. The counit $u_5$ induces the identity
+in degree zero. Consequently its cone has the single cohomology sheaf
+$\OO_\Delta(-5)$ in degree $-2$, and
+\[
+ G_0\simeq\OO_\Delta(-5)[2].
+\]
+Since $G_0$ is the composition kernel for $T_0T_1T_2T_3T_4$, this is
+an isomorphism of kernels and therefore a natural functor isomorphism.
+Equation~\eqref{eq:twist-product} now gives $\Phi^5\simeq[2]$.
+The asserted inverse follows by composing this relation with $[-2]$.
+\end{proof}
diff --git a/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/paper.pdf b/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..87fa280062584a436700ea73a0818805034c4e97
--- /dev/null
+++ b/preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:f70b73596fffb0689874c768d3975cb8e16cbdede231c21baef0707ab1898f49
+size 482371
diff --git a/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/README.md b/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..c947149af9e2f7d328757044ff7829af4079590a
--- /dev/null
+++ b/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/README.md
@@ -0,0 +1,18 @@
+# [A Limiting Satisfiability Threshold for Every Fixed Clause Size](article.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 25, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026,
+  author = {{OpenAI}},
+  title = {{A Limiting Satisfiability Threshold for Every Fixed Clause Size}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/article.pdf}{OAI:A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/article.pdf b/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/article.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..aaf3a91cfd27beae946adebf8415fc0f72b9d2e1
--- /dev/null
+++ b/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/article.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:5318fe3d0e3db4e4c22f79dc8b342ae38fc1ddd60172ba78a00ee612c1576365
+size 213536
diff --git a/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/build/source/paper.tex b/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/build/source/paper.tex
new file mode 100644
index 0000000000000000000000000000000000000000..7bb94bb0ba96d8488ed745fab08d61f777c93df9
--- /dev/null
+++ b/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/build/source/paper.tex
@@ -0,0 +1,28 @@
+\documentclass[11pt]{article}
+\input{preamble}
+\title{A Limiting Satisfiability Threshold\\for Every Fixed Clause Size}
+\author{OpenAI}
+\date{September 25, 2026}
+
+\begin{document}
+\maketitle
+\begin{abstract}
+For every fixed integer \(k\ge3\), random \(k\)-SAT with independent
+uniformly signed clauses on distinct variables, sampled with replacement,
+has a finite positive limiting satisfiability threshold.
+We credit Gaia Carenini~\cite{Carenini2026Polynomial} with priority for
+resolving the satisfiability conjecture. This paper gives an alternative
+proof, using concentration of a capped last satisfiable index and a
+comparison between different system sizes.
+\end{abstract}
+
+\input{sections/introduction}
+\input{sections/forcing}
+\input{sections/concentration}
+\input{sections/anchors}
+\input{sections/convergence}
+\input{sections/three-clause}
+\input{sections/boundaries}
+
+\input{references}
+\end{document}
diff --git a/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/build/source/preamble.tex b/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/build/source/preamble.tex
new file mode 100644
index 0000000000000000000000000000000000000000..cbc538c786d14a1c80d5652e221680ad16a8cb6b
--- /dev/null
+++ b/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/build/source/preamble.tex
@@ -0,0 +1,43 @@
+\pdftrailerid{}
+\pdfinfoomitdate=1
+\pdfsuppressptexinfo=-1
+\usepackage[T1]{fontenc}
+\usepackage{mathpazo}
+\usepackage[margin=1.12in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{microtype}
+\usepackage[dvipsnames]{xcolor}
+\usepackage{needspace}
+\usepackage{hyperref}
+\hypersetup{
+  colorlinks=true,
+  linkcolor=MidnightBlue,
+  citecolor=MidnightBlue,
+  urlcolor=MidnightBlue,
+  pdfauthor={OpenAI},
+  pdftitle={A Limiting Satisfiability Threshold for Every Fixed Clause Size},
+  pdfsubject={Convergence of the random k-SAT satisfiability threshold for every fixed k at least three}
+}
+\urlstyle{same}
+\numberwithin{equation}{section}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newcommand{\PP}{\mathbb P}
+\newcommand{\EE}{\mathbb E}
+\newcommand{\Var}{\operatorname{Var}}
+\newcommand{\Pois}{\operatorname{Pois}}
+\newcommand{\Bin}{\operatorname{Bin}}
+\newcommand{\ind}{\mathbf 1}
+\newcommand{\sat}{\mathrm{SAT}}
+\newcommand{\fall}[2]{(#1)_{#2}}
+\newcommand{\eps}{\varepsilon}
+\newcommand{\F}{\mathcal F}
+\newcommand{\sd}{\operatorname{sd}}
+\setlength{\emergencystretch}{2em}
+\clubpenalty=10000
+\widowpenalty=10000
+\displaywidowpenalty=10000
diff --git a/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/build/source/references.tex b/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/build/source/references.tex
new file mode 100644
index 0000000000000000000000000000000000000000..34c8147d528b7060f64837a4b440ba7dc72abe26
--- /dev/null
+++ b/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/build/source/references.tex
@@ -0,0 +1,155 @@
+\begingroup\small
+\begin{thebibliography}{99}
+\setlength{\itemsep}{0pt}
+
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+\href{https://arxiv.org/abs/1006.3786v1}{arXiv:1006.3786v1}.
+
+\bibitem{AP}
+D.~Achlioptas and Y.~Peres,
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+{doi:10.1090/S0894-0347-04-00464-3}.
+
+\bibitem{BGT}
+M.~Bayati, D.~Gamarnik, and P.~Tetali,
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+
+\bibitem{BBLM}
+S.~Boucheron, O.~Bousquet, G.~Lugosi, and P.~Massart,
+\emph{Moment inequalities for functions of independent random variables},
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+{doi:10.1214/009117904000000856};
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+(electronic reprint).
+
+\bibitem{Carenini2026Polynomial}
+G.~Carenini,
+\emph{A polynomial scaling window for random \(k\)-SAT and a proof of
+the satisfiability conjecture},
+Electronic Colloquium on Computational Complexity,
+Report TR26-229 (2026), published 5 October 2026.
+\href{https://eccc.weizmann.ac.il/report/2026/229/}{ECCC Report TR26-229}.
+
+\bibitem{Carenini}
+G.~Carenini,
+\emph{The Scaling Window of Random \(k\)-SAT},
+Electronic Colloquium on Computational Complexity,
+Report TR26-145 (2026), preprint;
+\href{https://arxiv.org/abs/2609.26222v1}{arXiv:2609.26222v1}.
+\href{https://eccc.weizmann.ac.il/report/2026/145/}{ECCC Report TR26-145}.
+
+\bibitem{CR}
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+620--627.
+\href{https://doi.org/10.1109/SFCS.1992.267789}{doi:10.1109/SFCS.1992.267789}.
+\href{https://www.diochnos.com/teaching/CS5970/2020F/project-papers/ChvatalReed.pdf}{conference paper PDF}.
+
+\bibitem{COP}
+A.~Coja-Oghlan and K.~Panagiotou,
+\emph{The asymptotic \(k\)-SAT threshold},
+Adv. Math. \textbf{288} (2016), 985--1068.
+\href{https://doi.org/10.1016/j.aim.2015.11.007}
+{doi:10.1016/j.aim.2015.11.007};
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+
+\bibitem{DSS}
+J.~Ding, A.~Sly, and N.~Sun,
+\emph{Proof of the satisfiability conjecture for large \(k\)},
+Ann. of Math. (2) \textbf{196} (2022), no.~1, 1--388.
+\href{https://doi.org/10.4007/annals.2022.196.1.1}
+{doi:10.4007/annals.2022.196.1.1};
+\href{https://arxiv.org/abs/1411.0650v3}{arXiv:1411.0650v3}.
+
+\bibitem{EfronStein}
+B.~Efron and C.~Stein,
+\emph{The jackknife estimate of variance},
+Ann. Statist. \textbf{9} (1981), no.~3, 586--596.
+\href{https://doi.org/10.1214/aos/1176345462}{doi:10.1214/aos/1176345462}.
+
+\bibitem{FV}
+J.~Franco and A.~Van Gelder,
+\emph{A perspective on certain polynomial-time solvable classes of
+satisfiability},
+Discrete Appl. Math. \textbf{125} (2003), nos.~2--3, 177--214.
+\href{https://doi.org/10.1016/S0166-218X(01)00358-4}
+{doi:10.1016/S0166-218X(01)00358-4}.
+See also the \href{https://citeseerx.ist.psu.edu/document?doi=a79e4c87237fc557de1acc74371bc986f575a571&repid=rep1&type=pdf}{author manuscript dated 7 August 1998}, Section~8.
+
+\bibitem{FL}
+S.~Franz and M.~Leone,
+\emph{Replica bounds for optimization problems and diluted spin systems},
+J. Statist. Phys. \textbf{111} (2003), 535--564.
+\href{https://doi.org/10.1023/A:1022885828956}{doi:10.1023/A:1022885828956};
+\href{https://arxiv.org/abs/cond-mat/0208280v1}{arXiv:cond-mat/0208280v1}.
+
+\bibitem{FLT}
+S.~Franz, M.~Leone, and F.~L.~Toninelli,
+\emph{Replica bounds for diluted non-Poissonian spin systems},
+J. Phys. A \textbf{36} (2003), no.~43, 10967--10985.
+\href{https://doi.org/10.1088/0305-4470/36/43/021}
+{doi:10.1088/0305-4470/36/43/021};
+\href{https://arxiv.org/abs/cond-mat/0307367v2}{arXiv:cond-mat/0307367v2}.
+
+\bibitem{Friedgut}
+E.~Friedgut, with an appendix by J.~Bourgain,
+\emph{Sharp thresholds of graph properties, and the \(k\)-SAT problem},
+J. Amer. Math. Soc. \textbf{12} (1999), no.~4, 1017--1054.
+\href{https://doi.org/10.1090/S0894-0347-99-00305-7}
+{doi:10.1090/S0894-0347-99-00305-7}.
+
+\bibitem{Goerdt}
+A.~Goerdt,
+\emph{A threshold for unsatisfiability},
+J. Comput. System Sci. \textbf{53} (1996), no.~3, 469--486.
+\href{https://doi.org/10.1006/jcss.1996.0081}{doi:10.1006/jcss.1996.0081}.
+
+\bibitem{GT}
+F.~Guerra and F.~L.~Toninelli,
+\emph{The thermodynamic limit in mean field spin glass models},
+Comm. Math. Phys. \textbf{230} (2002), no.~1, 71--79.
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+{doi:10.1007/s00220-002-0699-y};
+\href{https://arxiv.org/abs/cond-mat/0204280v1}{arXiv:cond-mat/0204280v1}.
+
+\bibitem{Hall}
+P.~Hall,
+\emph{On representatives of subsets},
+J. London Math. Soc. \textbf{10} (1935), no.~1, 26--30.
+\href{https://doi.org/10.1112/jlms/s1-10.37.26}{doi:10.1112/jlms/s1-10.37.26}.
+
+\bibitem{MMZ}
+S.~Mertens, M.~M\'ezard, and R.~Zecchina,
+\emph{Threshold values of random \(K\)-SAT from the cavity method},
+Random Structures Algorithms \textbf{28} (2006), no.~3, 340--373.
+\href{https://doi.org/10.1002/rsa.20090}{doi:10.1002/rsa.20090};
+\href{https://arxiv.org/abs/cs/0309020v2}{arXiv:cs/0309020v2}.
+
+\bibitem{MPZ}
+M.~M\'ezard, G.~Parisi, and R.~Zecchina,
+\emph{Analytic and algorithmic solution of random satisfiability problems},
+Science \textbf{297} (2002), no.~5582, 812--815.
+\href{https://doi.org/10.1126/science.1073287}
+{doi:10.1126/science.1073287}.
+
+\bibitem{Steele}
+J.~M.~Steele,
+\emph{An Efron--Stein inequality for nonsymmetric statistics},
+Ann. Statist. \textbf{14} (1986), no.~2, 753--758.
+\href{https://doi.org/10.1214/aos/1176349952}{doi:10.1214/aos/1176349952}.
+
+\end{thebibliography}
+\endgroup
diff --git a/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/build/source/sections/anchors.tex b/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/build/source/sections/anchors.tex
new file mode 100644
index 0000000000000000000000000000000000000000..f8c68b7d83ea5d098f088ff18009267ae500819a
--- /dev/null
+++ b/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/build/source/sections/anchors.tex
@@ -0,0 +1,71 @@
+\section{Positive and finite center bounds}\label{sec:anchors}
+
+Before comparing different sizes, we place the centers in a fixed
+positive interval below the cap. At high density, an assignment first
+moment makes satisfiability unlikely. At a sufficiently small positive
+density, a matching from clauses to variables supplies a satisfying
+assignment regardless of the signs.
+
+The matching criterion is Hall's theorem~\cite{Hall}. Its application
+to satisfiability and the associated deficiency counting appear in
+Franco and Van Gelder~\cite[Section~8]{FV}. We include the
+alternating-path argument and the probability estimate.
+
+\begin{lemma}[Bounds on the centers]\label{lem:center-bounds}
+For each fixed \(k\ge3\), there is \(a_0>0\) such that, for all
+sufficiently large \(n\),
+\[
+ a_0\le\mu_n\le2^k+1.
+\]
+\end{lemma}
+
+\begin{proof}
+For any fixed assignment, each proper uniformly signed \(k\)-clause
+is satisfied with probability \(1-2^{-k}\). The clauses are independent,
+so at \(m=2^kn\) the expected number of satisfying assignments is
+\[
+ 2^n(1-2^{-k})^{2^kn}\le(2/e)^n=o(1).
+\]
+Consequently \(P_n(2^kn)=o(1)\). Since \(V_n\le L_kn\),
+\[
+ \mu_n\le2^k+(L_k-2^k)P_n(2^kn)=2^k+o(1).
+\]
+
+For the lower bound, fix \(c_0>0\) with \(e^2c_0<1\) and put
+\(m=\lfloor c_0n\rfloor<n\). A matching assigning each clause position
+a distinct variable occurring in that clause guarantees satisfiability:
+give each matched variable the value that satisfies its literal in the
+matched clause. These choices are compatible because the matched
+variables are distinct.
+
+If no such matching exists, some \(t\) clause positions use fewer than
+\(t\) variables. To see this directly, take a maximum matching and
+start an alternating-path search from an unmatched clause. Every
+reached variable must be matched, or an augmenting path would exist.
+Its matched clause is then reached as well. The starting unmatched
+clause gives more reached clauses than reached variables, which is
+the asserted obstruction.
+
+Enlarge a deficient variable set to a set of exactly \(t\) variables.
+A union bound gives
+\begin{align*}
+ \PP(\text{no matching})
+ &\le\sum_{t=1}^{m}\binom mt\binom nt(t/n)^{kt}\\
+ &\le\sum_{t=1}^{m}(e^2c_0\,t/n)^t=o(1).
+\end{align*}
+Indeed, the probability that a proper clause lies in a specified
+\(t\)-set is \(\fall tk/\fall nk\le(t/n)^k\), and it is zero for
+\(t<k\). Bounding the two binomial coefficients by \( (em/t)^t\) and
+\((en/t)^t\) first gives \([e^2c_0(t/n)^{k-2}]^t\).
+Since \(k\ge3\) and \(t<n\), this is at most the displayed summand.
+For the last limit, every fixed summand tends to zero, whereas the tail
+is bounded by the tail of the convergent geometric series
+\(\sum_{t\ge1}(e^2c_0)^t\).
+
+Thus \(P_n(m)=1-o(1)\). Because \(m\le M_n\), the survival identity
+\eqref{eq:survival} gives
+\[
+ \mu_n\ge\frac mn P_n(m)=c_0-o(1).
+\]
+An eventual lower bound \(a_0>0\) and the asserted upper bound follow.
+\end{proof}
diff --git a/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/build/source/sections/boundaries.tex b/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/build/source/sections/boundaries.tex
new file mode 100644
index 0000000000000000000000000000000000000000..e7ae5c2208bc37668073180574b71789443fb2c5
--- /dev/null
+++ b/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/build/source/sections/boundaries.tex
@@ -0,0 +1,46 @@
+\section{Lower arities and finite-size information}\label{sec:boundaries}
+
+The balanced comparison gives a one-sided quantitative bound on the
+finite-size centers. In \eqref{eq:balanced-comparison-bound}, take
+\(u_n=\mu_n\), \(A=4\), and \(\beta=\delta_k\), then let the larger
+size tend to infinity. For every sufficiently large integer \(s\),
+\begin{equation}\label{eq:one-sided-center-rate}
+ \mu_s\le\alpha_k+C_k s^{-\delta_k},\qquad
+ C_k=4\left(1+\frac{(3/2)^{\delta_k}}
+                         {1-(2/3)^{\delta_k}}\right).
+\end{equation}
+Thus the center cannot exceed its limit by more than the stated error.
+The proof gives no corresponding bound on \(\alpha_k-\mu_s\), and
+hence no effective two-sided rate for approximating \(\alpha_k\).
+Proposition~\ref{prop:concentration} controls fluctuations about
+\(\EE V_n\); it does not place a two-sided window of width
+\(n^{1-\delta_k}\) around \(\alpha_kn\). All constants concern a fixed
+\(k\), and are not uniform when \(k\) grows with \(n\).
+
+\begin{remark}[The two-clause boundary]\label{rem:two}
+Chv\'atal and Reed~\cite[Theorems~3--4]{CR} proved that independent
+uniform clauses on two distinct variables, sampled with replacement,
+are satisfiable with probability tending to one below density \(1\)
+and unsatisfiable with probability tending to one above density \(1\).
+Goerdt~\cite{Goerdt} obtained the threshold independently.
+Thus \(\alpha_2=1\) in the same proper-clause convention. This is a
+separate theorem: the bound \(O_k(n^{1+2/k})\) is only \(O(n^2)\) at
+\(k=2\) and gives no sublinear concentration window there.
+\end{remark}
+
+\begin{remark}[The unit-clause boundary]\label{rem:unit}
+For \(k=1\), the threshold on the linear-density scale is zero.
+Fix \(c>0\), choose \(0<c'<c\), and Poissonize the number of independent
+uniform unit clauses at mean \(c'n\). For each variable, the positive
+and negative occurrence counts are independent \(\Pois(c'/2)\)
+variables, independently across variables. The formula is satisfiable
+exactly when no variable receives both signs, an event of probability
+\[
+ \bigl(2e^{-c'/2}-e^{-c'}\bigr)^n\longrightarrow0.
+\]
+The Poisson clause count is at most \(\lfloor cn\rfloor\) with
+probability tending to one. Monotonicity in the clause count therefore
+implies that the fixed-count unit-clause formula is unsatisfiable with
+probability tending to one at every \(c>0\). At density zero it is
+empty and satisfiable.
+\end{remark}
diff --git a/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/build/source/sections/concentration.tex b/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/build/source/sections/concentration.tex
new file mode 100644
index 0000000000000000000000000000000000000000..4ed7c144de72eba5df23c4748bd35dd3e4a47e5a
--- /dev/null
+++ b/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/build/source/sections/concentration.tex
@@ -0,0 +1,178 @@
+\section{Clause omission and polynomial concentration}\label{sec:concentration}
+
+We now turn the integrated forcing estimate into a bound on the
+fluctuations of the last satisfiable prefix. Omitting one clause can
+delay the loss of satisfiability. The delay counts the prefixes where
+that clause is pivotal, and its square counts pairs of pivotal times.
+The key observation is that a future punctured prefix can survive only
+if every intervening clause avoids killing the present solution set.
+
+\subsection{Omitting a clause without changing time indices}
+
+Fix an integer \(M\ge1\) and the independent clauses \(C_1,\ldots,C_M\).
+For the next two lemmas, let
+\[
+ V=\max\{0\le m\le M:F_{n,m}\in\sat\}
+\]
+be the last satisfiable index capped at \(M\). For \(1\le i\le M\), set
+\[
+ V^{-i}=\max\left\{0\le m\le M:
+       \bigwedge_{\substack{1\le j\le m\\j\ne i}}C_j\in\sat\right\}.
+\]
+The time index \(i\) is skipped; the remaining positions are not
+renumbered. Thus \(V^{-i}\ge V\), and \(V^{-i}\) is independent of
+\(C_i\).
+
+The following coordinate-omission inequality belongs to the martingale
+variance method associated with Efron and Stein~\cite{EfronStein} and
+Steele's treatment of nonsymmetric statistics~\cite{Steele}.
+Boucheron, Bousquet, Lugosi, and Massart
+\cite[Section~3.2, equation~(3.6)]{BBLM} state the arbitrary-omission
+form. The short proof fixes the deletion convention and constant.
+
+\begin{lemma}[Deletion variance bound]\label{lem:variance-deletion}
+For the capped indices just defined,
+\[
+ \Var(V)\le\sum_{i=1}^{M}\EE(V^{-i}-V)^2.
+\]
+\end{lemma}
+
+\begin{proof}
+Let \(\F_i=\sigma(C_1,\ldots,C_i)\), with \(\F_0\) trivial, and put
+\[
+ D_i=\EE[V\mid\F_i]-\EE[V\mid\F_{i-1}].
+\]
+Independence of the clauses and the fact that \(V^{-i}\) does not use
+\(C_i\) give
+\[
+ \EE[V^{-i}\mid\F_i]=\EE[V^{-i}\mid\F_{i-1}].
+\]
+For \(Z_i=V-V^{-i}\), it follows that
+\[
+ D_i=\EE[Z_i\mid\F_i]-\EE[Z_i\mid\F_{i-1}].
+\]
+The tower property and conditional Jensen inequality yield
+\[
+ \EE D_i^2
+ =\EE\bigl(\EE[Z_i\mid\F_i]\bigr)^2
+  -\EE\bigl(\EE[Z_i\mid\F_{i-1}]\bigr)^2
+ \le\EE Z_i^2.
+\]
+The martingale differences are orthogonal and sum to \(V-\EE V\).
+Summing the displayed inequalities proves the lemma.
+\end{proof}
+
+\subsection{Pairs of pivotal times}
+
+Fix \(i\). For \(i\le m\le M\), the punctured prefix and its killing
+probability are
+\[
+ G_m=\bigwedge_{\substack{1\le j\le m\\j\ne i}}C_j,\qquad
+ q_m=q_k(G_m).
+\]
+The function \(q_k\) was defined in \eqref{eq:kill}; in particular,
+\(q_m=0\) when \(G_m\) is unsatisfiable.
+
+\begin{lemma}[Pivotal pairs]\label{lem:pivotal}
+For every \(1\le i\le M\),
+\begin{equation}\label{eq:pivotal}
+ \EE(V^{-i}-V)^2
+ \le2\sum_{m=i}^{M}\EE\min\{Mq_m,1\}.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Let
+\[
+ A_m=\{G_m\in\sat,\ G_m\wedge C_i\notin\sat\}.
+\]
+This event is exactly \(\{V<m\le V^{-i}\}\): prefix \(m\) is
+satisfiable after omission and unsatisfiable before it. Hence, including
+the endpoint \(m=M\),
+\[
+ V^{-i}-V=\sum_{m=i}^{M}\ind_{A_m}.
+\]
+For \(\ell\ge m\), on \(A_m\) the original prefix remains
+unsatisfiable, so
+\[
+ A_m\cap A_\ell=A_m\cap\{G_\ell\in\sat\}.
+\]
+
+Condition on
+\(\F_m^{(-i)}=\sigma(C_j:1\le j\le m,\ j\ne i)\).
+The clause \(C_i\) and the future clauses
+\(C_{m+1},\ldots,C_\ell\) are conditionally independent. The
+conditional probability of \(A_m\) is \(q_m\). If \(G_m\) is
+satisfiable, then \(G_\ell\) can be satisfiable only if none of these
+future clauses individually makes \(G_m\) unsatisfiable. Each does so
+with probability \(q_m\), independently. Thus
+\begin{equation}\label{eq:pivotal-pair}
+ \PP(A_m\cap A_\ell\mid\F_m^{(-i)})
+ =q_m\PP(G_\ell\in\sat\mid\F_m^{(-i)})
+ \le q_m(1-q_m)^{\ell-m}.
+\end{equation}
+If \(G_m\) is unsatisfiable, the relevant event and the right side are
+both zero.
+
+Expand the square of the indicator sum and bound its double sum by
+twice the sum over \(m\le\ell\). For \(q>0\),
+\[
+ \sum_{\ell=m}^{M}(1-q)^{\ell-m}\le\min\{M,1/q\},
+\]
+because \(m\ge1\). Multiplication by \(q\), followed by expectation
+and summation over \(m\), gives \eqref{eq:pivotal}. When \(q=0\), every
+term in the corresponding product is zero, so no division by zero is
+needed.
+\end{proof}
+
+\subsection{The variance bound}
+
+The pivotal-pair estimate has reduced the squared omission delay to a
+truncated killing probability. Taking a \(k\)-th root converts that
+quantity into the forced fraction controlled by
+Proposition~\ref{prop:forcing}.
+
+\begin{proposition}[Polynomial concentration]\label{prop:concentration}
+For every fixed integer \(k\ge3\), the statistic
+\(V_n=\min(H_n-1,L_kn)\) from \eqref{eq:statistic} satisfies
+\begin{equation}\label{eq:variance}
+ \Var(V_n)=O_k(n^{1+2/k}).
+\end{equation}
+In particular, with \(\mu_n=\EE V_n/n\),
+\begin{equation}\label{eq:concentration}
+ \PP\bigl(|V_n-n\mu_n|\ge n^{1-\delta_k}\bigr)
+ =O_k(n^{-\eta_k}),\qquad
+ \delta_k=\frac{k-2}{4k},\quad \eta_k=\frac{k-2}{2k}.
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+Set \(M=L_kn\), \(V=V_n\), and fix \(i\). With the notation of
+Lemma~\ref{lem:pivotal}, let \(\rho_m=b(G_m)/n\).
+The falling-factorial ratios in \eqref{eq:kill} give
+\(q_m\le\rho_m^k\). Since \(\min\{z,1\}\le z^{1/k}\) for \(z\ge0\),
+\[
+ \min\{Mq_m,1\}\le(Mq_m)^{1/k}\le M^{1/k}\rho_m.
+\]
+Unconditionally, \(G_m\) has the law of \(F_{n,m-1}\).
+Lemma~\ref{lem:pivotal} and Proposition~\ref{prop:forcing} therefore
+give, uniformly in \(1\le i\le M\),
+\[
+ \EE(V^{-i}-V)^2
+ \le2M^{1/k}\sum_{m=i}^{M}\EE\rho_m
+ \le2M^{1/k}\sum_{j=0}^{M-1}\EE\frac{b(F_{n,j})}{n}
+ =O_k(n^{2/k}).
+\]
+Summing over \(M=L_kn\) indices in
+Lemma~\ref{lem:variance-deletion} proves \eqref{eq:variance}.
+Chebyshev's inequality proves \eqref{eq:concentration}, because
+\[
+ 1+\frac2k-2(1-\delta_k)
+ =-\left(1-\frac2k-2\delta_k\right)=-\eta_k.
+\]
+\end{proof}
+
+The exponent \(\delta_k\) is positive for \(k\ge3\), so the window
+\(n^{1-\delta_k}\) is sublinear. The estimate is centered at the
+finite-size mean \(n\mu_n\); comparing those means across sizes is a
+separate task.
diff --git a/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/build/source/sections/convergence.tex b/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/build/source/sections/convergence.tex
new file mode 100644
index 0000000000000000000000000000000000000000..d174e828fd0d10f31896f27ef983ba8dd2ccdf16
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+++ b/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/build/source/sections/convergence.tex
@@ -0,0 +1,442 @@
+\section{From concentration to a limiting location}\label{sec:convergence}
+
+The preceding estimates control fluctuations at one size.  We now rule
+out drift of the centers as the number of variables grows.  The argument
+has two steps.  An auxiliary Poisson formula compares the satisfiability
+probabilities at a combined size and at two smaller sizes.  Concentration
+turns this probability comparison into inequalities between centers, whose
+errors can then be summed along a balanced tree.
+
+Recall that \(H_n=\inf\{m\ge1:F_{n,m}\notin\sat\}\) is the first
+unsatisfiable index in the proper-clause process.  To state the comparison
+argument independently of the cap used above, fix \(\lambda>0\) and define
+\begin{equation}\label{eq:general-capped-center}
+\begin{gathered}
+ M_n^{(\lambda)}=\lfloor\lambda n\rfloor,\qquad
+ V_n^{(\lambda)}=\min\{H_n-1,M_n^{(\lambda)}\},\\
+ x_n^{(\lambda)}=\frac{\EE V_n^{(\lambda)}}{n}.
+\end{gathered}
+\end{equation}
+Thus, for every integer \(0\le m\le M_n^{(\lambda)}\),
+\begin{equation}\label{eq:general-cap-survival}
+ P_n(m)=\PP\bigl(V_n^{(\lambda)}\ge m\bigr).
+\end{equation}
+This identity includes \(m=M_n^{(\lambda)}\).  The superscript keeps
+these statistics distinct from the particular \(V_n\) and \(\mu_n\)
+used in the preceding sections.
+
+\Needspace{24\baselineskip}
+\begin{samepage}
+\begin{theorem}[Concentration implies convergence]
+\label{thm:concentration-to-convergence}
+Fix an integer \(k\ge3\) and \(\lambda>0\).  Suppose that the proper
+independent uniformly signed \(k\)-clause process has constants
+\[
+ 0<a\le b<\lambda,\qquad 0<\delta<\frac12,\qquad
+ \eta>0,\qquad 0\le K<\infty
+\]
+such that, for all sufficiently large \(n\),
+\begin{equation}\label{eq:convergence-input}
+\begin{gathered}
+ a\le x_n^{(\lambda)}\le b,\\
+ \PP\left(\left|V_n^{(\lambda)}-nx_n^{(\lambda)}\right|
+             \ge n^{1-\delta}\right)\le K n^{-\eta}.
+\end{gathered}
+\end{equation}
+Then \(x_n^{(\lambda)}\) converges to some \(\alpha\in[a,b]\), and
+\[
+ \lim_{n\to\infty}P_n(\lfloor cn\rfloor)=
+ \begin{cases}
+  1,&0\le c<\alpha,\\
+  0,&c>\alpha.
+ \end{cases}
+\]
+Moreover, for all sufficiently large \(r,s\),
+\begin{align}
+ x_{r+s}^{(\lambda)}
+ &\ge\min\{x_r^{(\lambda)},x_s^{(\lambda)}\}
+       -4\min\{r,s\}^{-\delta},\label{eq:convergence-sum}\\
+ x_{s+1}^{(\lambda)}
+ &\ge x_s^{(\lambda)}-4s^{-\delta}.
+ \label{eq:convergence-neighbor}
+\end{align}
+The first comparison holds without a restriction on \(r/s\).
+\end{theorem}
+\end{samepage}
+
+The input is a tail estimate about a finite-size mean.  The conclusion
+gives convergence of that mean, while the quantitative comparison is
+one-sided.  We begin with the probability inequality that supplies it.
+
+\subsection{An auxiliary product inequality}
+
+For every positive integer \(n\) and \(c\ge0\), let \(s_n(c)\) be the
+satisfiability probability of the following auxiliary formula.  Its number
+of clauses is \(\Pois(cn)\).  Within each clause, the \(k\) variable indices
+are chosen independently and uniformly from \(\{1,\ldots,n\}\), and their
+signs are independent and fair.  The clauses are independent.  The
+resulting literals are disjoined in the usual Boolean sense, so repeated
+variables and tautologies are allowed.  This definition applies at every
+positive size, including \(n=1\).
+
+The following calculation is a Poisson form of the frozen-variable
+interpolation of Bayati, Gamarnik, and Tetali
+\cite[Section~4, Proposition~2]{BGT}.  Independence of the endpoint
+Poisson families gives the product that we need.
+
+\begin{lemma}[Auxiliary product inequality]\label{lem:auxiliary-product}
+For every pair of positive integers \(r,s\) and every \(c\ge0\),
+\begin{equation}\label{eq:auxiliary-product}
+ s_{r+s}(c)\ge s_r(c)s_s(c).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Let \(N=r+s\), and partition the variables into parts of sizes
+\(n_1=r\) and \(n_2=s\).  For \(0\le t\le c\), take three independent
+Poisson families: a mean \(Nt\) family of auxiliary clauses on all
+\(N\) variables and, for \(i=1,2\), a mean \(n_i(c-t)\) family of
+auxiliary clauses confined to part \(i\).  Write \(G_t\) for their
+conjunction and \(f(t)=\PP(G_t\in\sat)\).
+
+We use the elementary add-one formula for a marked Poisson family.
+If a bounded observable has conditional expectations \(a_j\) given
+exactly \(j\) independent marks, then its expectation at mean \(v\) is
+\(\sum_{j\ge0}e^{-v}v^ja_j/j!\), whose derivative is
+\(\sum_{j\ge0}e^{-v}v^j(a_{j+1}-a_j)/j!\).  Both series converge
+uniformly on compact intervals of \(v\), so the formula also applies
+after conditioning on the other two Poisson families.
+
+For a satisfiable background \(G_t\), let \(b_i\) be the number of
+variables in part \(i\) that are forced by the entire background: every
+satisfying assignment gives each such variable the same value.  Set
+\(b_i=0\) on unsatisfiable backgrounds.  For a satisfiable background,
+a new auxiliary clause destroys satisfiability exactly when each of its literals
+is false in every satisfying assignment.  Each literal must therefore
+oppose the value of a forced variable.  Conversely, that condition makes
+the whole new clause false in every solution.  The independent signed
+draws give killing probabilities
+\[
+ \left(\frac{b_1+b_2}{2N}\right)^k
+ \quad\text{and}\quad
+ \left(\frac{b_i}{2n_i}\right)^k
+\]
+for a global clause and a clause confined to part \(i\), respectively.
+This remains exact when an index is repeated: each independent signed
+draw must still be one of the literals false in every solution.
+
+An unsatisfiable background remains unsatisfiable after any clause is
+added.  The add-one formula, applied to the three means, therefore gives
+for \(0<t<c\)
+\begin{align*}
+ f'(t)
+ &=N\EE\left[\ind_{\{G_t\in\sat\}}
+   \left\{\sum_{i=1}^{2}\frac{n_i}{N}
+       \left(\frac{b_i}{2n_i}\right)^k
+       -\left(\frac{b_1+b_2}{2N}\right)^k\right\}\right]\\
+ &\ge0.
+\end{align*}
+The last inequality is convexity of \(z\mapsto z^k\) on
+\([0,\infty)\), since \((b_1+b_2)/(2N)\) is the weighted average of
+\(b_i/(2n_i)\) with weights \(n_i/N\).  At \(t=0\) the two restricted
+formulas are independent, so \(f(0)=s_r(c)s_s(c)\); at \(t=c\), only
+the global formula remains, so \(f(c)=s_N(c)\).  Continuity at the
+endpoints proves the assertion for \(c>0\), and \(c=0\) is immediate.
+\end{proof}
+
+\subsection{Transfer to proper clauses}
+
+For the rest of the proof of
+Theorem~\ref{thm:concentration-to-convergence}, assume its hypotheses
+and keep \(k,\lambda,a,b,\delta,\eta,K\) fixed.  Choose any density
+ceiling \(D>b\).  The elementary sandwich below does not use the
+concentration hypothesis; its tail consequences also use
+\eqref{eq:convergence-input}.  The ceiling is independent of the cap
+parameter \(\lambda\): the cap locates the survival events in
+\eqref{eq:general-cap-survival}, whereas \(D\) bounds the expected number
+of clauses with internal repetitions in the auxiliary formula.  Put
+\begin{equation}\label{eq:transfer-parameters}
+ R=D\binom{k}{2},\qquad
+ \gamma=\min\{\eta,1-2\delta\}>0,\qquad
+ \kappa=\frac{e^{-R}}2.
+\end{equation}
+
+\begin{samepage}
+\begin{lemma}[Proper-clause transfer]\label{lem:proper-transfer}
+For \(n\ge k\) set \(\theta_n=(n)_k/n^k\), where
+\((n)_k=n(n-1)\cdots(n-k+1)\).  If \(0\le c\le D\) and
+\(J\sim\Pois(cn\theta_n)\), then
+\begin{equation}\label{eq:proper-sandwich}
+\begin{gathered}
+ e^{-R}\EE P_n(J)\le s_n(c)\le\EE P_n(J),\\
+ cn-R\le\EE J\le cn,\qquad \Var(J)\le Dn.
+\end{gathered}
+\end{equation}
+If \eqref{eq:convergence-input} holds, then there are constants
+\(0\le K'<\infty\) and an integer \(n_1\ge k\) such that for every \(n\ge n_1\) and
+every \(c\in[0,D]\),
+\begin{align}
+ c\le x_n^{(\lambda)}-2n^{-\delta}
+ &\quad\Longrightarrow\quad s_n(c)\ge\kappa,
+ \label{eq:proper-transfer-low}\\
+ c\ge x_n^{(\lambda)}+2n^{-\delta}
+ &\quad\Longrightarrow\quad s_n(c)\le K'n^{-\gamma}.
+ \label{eq:proper-transfer-high}
+\end{align}
+The constants \(K'\) and \(n_1\) may depend on the fixed parameters,
+but not on \(c\) or \(n\).
+\end{lemma}
+\end{samepage}
+
+\begin{proof}
+An auxiliary clause has distinct indices with probability \(\theta_n\).
+Poisson thinning separates the clauses into a family of \(J\) clauses
+with distinct indices and a family of \(Q\) clauses with an internal
+repetition, where
+\[
+ J\sim\Pois(cn\theta_n),\qquad Q\sim\Pois(cn(1-\theta_n)).
+\]
+These counts are independent.  Indeed, their joint probability-generating
+function is
+\[
+ \EE[z^Jw^Q]
+ =\exp\bigl(cn[\theta_n z+(1-\theta_n)w-1]\bigr)
+ =e^{cn\theta_n(z-1)}e^{cn(1-\theta_n)(w-1)}.
+\]
+The same independent-mark construction shows that, conditional on these
+counts, the marks in the two families are independent with their
+respective conditional laws.  In the first family, every unordered set
+of \(k\) distinct variables has \(k!\) equally likely orders and the
+signs remain independent and fair.  Conditional on \(J\), this family
+therefore has exactly the law \(F_{n,J}\).
+
+A union bound over the \(\binom{k}{2}\) pairs of variable positions gives
+\[
+ cn(1-\theta_n)\le c\binom{k}{2}\le R.
+\]
+Consequently \(cn-R\le\EE J\le cn\) and \(\Var(J)=\EE J\le Dn\).
+Deleting the repeated-variable clauses can only help satisfiability.
+Requiring \(Q=0\), which is independent of the proper family and has
+probability \(e^{-cn(1-\theta_n)}\ge e^{-R}\), gives the lower bound.
+This proves \eqref{eq:proper-sandwich}.
+
+Write \(h=n^{1-\delta}\).  The concentration input and the survival
+identity give, for all sufficiently large \(n\) and integer \(j\),
+\begin{align*}
+ P_n(j)&\ge1-Kn^{-\eta}
+ &&\text{if }0\le j\le nx_n^{(\lambda)}-h,\\
+ P_n(j)&\le Kn^{-\eta}
+ &&\text{if }j\ge nx_n^{(\lambda)}+h.
+\end{align*}
+For the first estimate the indicated counts are below the cap because
+\(x_n^{(\lambda)}\le b<\lambda\).  For the second, first use
+\eqref{eq:general-cap-survival} up to the cap and then use monotonicity
+beyond it at \(j_0=\lceil nx_n^{(\lambda)}+h\rceil\).
+The inequality \(j_0\le M_n^{(\lambda)}\) holds for all sufficiently
+large \(n\): the gap \((\lambda-b)n\) dominates \(h+2\).  This also
+accounts for both integer roundings.
+
+If \(c\le x_n^{(\lambda)}-2n^{-\delta}\), then
+\(\EE J\le nx_n^{(\lambda)}-2h\).  Chebyshev's inequality yields
+\[
+ \PP\bigl(J>nx_n^{(\lambda)}-h\bigr)
+ \le\frac{Dn}{h^2}=D n^{-(1-2\delta)}.
+\]
+\begin{samepage}
+The lower bound in \eqref{eq:proper-sandwich} is thus at least
+\[
+ e^{-R}\bigl(1-Kn^{-\eta}-D n^{-(1-2\delta)}\bigr),
+\]
+which is at least \(\kappa\) for every sufficiently large \(n\).
+\end{samepage}
+
+If \(c\ge x_n^{(\lambda)}+2n^{-\delta}\), then
+\(\EE J\ge nx_n^{(\lambda)}+2h-R\).  For large \(n\), \(h>R\), and
+\[
+ \PP\bigl(J<nx_n^{(\lambda)}+h\bigr)
+ \le\frac{Dn}{(h-R)^2}=O\bigl(n^{-(1-2\delta)}\bigr).
+\]
+The upper bound in \eqref{eq:proper-sandwich} is at most this probability
+plus \(Kn^{-\eta}\), proving \eqref{eq:proper-transfer-high} with
+\(\gamma=\min\{\eta,1-2\delta\}\).  All estimates used the common
+bounds \(R\) and \(Dn\), so they are uniform on \([0,D]\).
+\end{proof}
+
+\subsection{Comparing centers at different sizes}
+
+The lower transfer bound \eqref{eq:proper-transfer-low} stays positive,
+whereas the upper bound \eqref{eq:proper-transfer-high} tends to zero
+uniformly.  We use this separation with the product inequality
+\eqref{eq:auxiliary-product} to compare the centers.
+
+Let \(u=\min\{r,s\}\), \(N=r+s\), and set
+\[
+ c=\min\{x_r^{(\lambda)},x_s^{(\lambda)}\}-2u^{-\delta}.
+\]
+For all sufficiently large \(u\), the center bounds in
+\eqref{eq:convergence-input} put \(c\) in \([0,D]\).  Since
+\(u^{-\delta}\ge r^{-\delta},s^{-\delta}\), the lower estimate in
+Lemma~\ref{lem:proper-transfer} applies at both sizes.  Hence
+Lemma~\ref{lem:auxiliary-product} gives \(s_N(c)\ge\kappa^2\).
+For all sufficiently large \(N\), the upper estimate in
+Lemma~\ref{lem:proper-transfer} is smaller than \(\kappa^2\); it rules
+out \(c\ge x_N^{(\lambda)}+2N^{-\delta}\).  Therefore
+\[
+ x_N^{(\lambda)}>c-2N^{-\delta}
+ \ge\min\{x_r^{(\lambda)},x_s^{(\lambda)}\}-4u^{-\delta}.
+\]
+This proves \eqref{eq:convergence-sum} for every pair of sufficiently
+large sizes.  No ratio between them was used.
+
+To reach every sufficiently large size from a fixed block size, we will
+use blocks of sizes \(s\) and \(s+1\).  We therefore also need a
+neighboring-size comparison.  Take
+\(c=x_s^{(\lambda)}-2s^{-\delta}\), which again lies in \([0,D]\)
+for large \(s\).  In the size-one auxiliary model, the empty-formula
+event gives \(s_1(c)\ge e^{-c}\ge e^{-D}\).  Consequently
+\[
+ s_{s+1}(c)\ge s_s(c)s_1(c)\ge\kappa e^{-D}.
+\]
+For large \(s\), the upper transfer estimate at size \(s+1\) is smaller
+than this fixed positive constant.  It follows that
+\[
+ x_{s+1}^{(\lambda)}>c-2(s+1)^{-\delta}
+ \ge x_s^{(\lambda)}-4s^{-\delta},
+\]
+which proves \eqref{eq:convergence-neighbor}.
+
+The tree argument below uses the sum comparison only when its child
+sizes are comparable.
+
+\subsection{Summing the errors along a balanced tree}
+
+The principle that errors summable over geometrically increasing sizes
+cannot sustain macroscopic drift parallels the approximate-superadditivity
+argument of Abbe and Montanari \cite[arXiv version, Lemma~8]{AM}.
+Here the operation is a minimum of normalized quantities; the following
+lemma proves the required form directly.
+
+\begin{samepage}
+\begin{lemma}[Balanced comparisons]\label{lem:balanced-comparison}
+Let \(n_0\ge1\) be an integer and let \((u_n)_{n\ge n_0}\) be a bounded
+real sequence.  Suppose that \(0\le A<\infty\), \(\beta>0\), and that,
+for all \(r,s\ge n_0\) with \(1/3\le r/s\le3\),
+\[
+ u_{r+s}\ge\min\{u_r,u_s\}-A\min\{r,s\}^{-\beta}.
+\]
+Suppose also that, for every \(s\ge n_0\),
+\[
+ u_{s+1}\ge u_s-As^{-\beta}.
+\]
+Then \(u_n\) converges.  More precisely, put
+\[
+ B_\beta=\frac{(3/2)^\beta}{1-(2/3)^\beta}.
+\]
+For every integer \(s\ge\max\{n_0,2\}\) and every integer \(n\ge s^2\),
+\begin{equation}\label{eq:balanced-comparison-bound}
+ u_n\ge u_s-A(1+B_\beta)s^{-\beta}.
+\end{equation}
+\end{lemma}
+\end{samepage}
+
+\begin{proof}
+Fix an integer \(s\ge\max\{n_0,2\}\).  For \(n\ge s^2\),
+write \(n=qs+t\), where \(q=\lfloor n/s\rfloor\ge s\) and \(0\le t<s\).
+Thus \(n\) is a sum of \(q-t\) blocks of size \(s\) and \(t\) blocks
+of size \(s+1\).  Both block counts are nonnegative, and the neighbor
+inequality puts the value of every block at least
+\(u_s-As^{-\beta}\).
+
+Arrange these \(q\) blocks in a binary tree by dividing each group of
+\(a\ge2\) blocks into groups of \(\lfloor a/2\rfloor\) and
+\(\lceil a/2\rceil\) blocks.  Each child contains at least \(a/3\)
+blocks, so its total size is at least \(sa/3\).  Moreover, the ratio of
+the larger child's total size to the smaller child's is at most
+\[
+ \frac{\lceil a/2\rceil(s+1)}{\lfloor a/2\rfloor s}
+ \le2(1+1/s)\le3.
+\]
+The sum inequality is therefore valid at every internal node, with
+error at most \(A(sa/3)^{-\beta}\).
+
+Iterating the minimum inequality from the leaves to the root gives a
+lower bound by the least leaf value minus the largest sum of errors on
+a leaf-to-root path.  If \(a_1,a_2,\ldots\) are the block counts at the
+successive internal nodes of such a path, then \(a_1\ge2\) and
+\(a_{j+1}\ge(3/2)a_j\), because a child contains at most two thirds of
+its parent's blocks.  Every path error is consequently at most
+\[
+ As^{-\beta}(3/2)^\beta
+   \sum_{j=0}^{\infty}(2/3)^{j\beta}
+ =AB_\beta s^{-\beta}.
+\]
+This proves \eqref{eq:balanced-comparison-bound}.  First let
+\(n\to\infty\) with \(s\) fixed to obtain
+\[
+ \liminf_{n\to\infty}u_n
+ \ge u_s-A(1+B_\beta)s^{-\beta}.
+\]
+Taking the limsup as \(s\to\infty\) gives
+\(\liminf_n u_n\ge\limsup_n u_n\).  Boundedness makes both quantities
+finite, and hence \(u_n\) converges.
+\end{proof}
+
+\begin{proof}[Completion of Theorem~\ref{thm:concentration-to-convergence}]
+Apply Lemma~\ref{lem:balanced-comparison} to
+\eqref{eq:convergence-sum}--\eqref{eq:convergence-neighbor}, with
+\(A=4\) and \(\beta=\delta\).  It gives convergence, and the center
+bounds place the limit \(\alpha\) in \([a,b]\).
+
+Fix \(0\le c<\alpha\).  Since \(x_n^{(\lambda)}\to\alpha\) and
+\(n^{-\delta}\to0\), for all sufficiently large \(n\),
+\(\lfloor cn\rfloor\le nx_n^{(\lambda)}-n^{1-\delta}\).  This count
+lies below the cap, and \eqref{eq:convergence-input} together with
+\eqref{eq:general-cap-survival} gives
+\(P_n(\lfloor cn\rfloor)\ge1-Kn^{-\eta}\to1\).
+
+Now fix \(c>\alpha\), and choose a real \(d\) with
+\(\alpha<d<\min\{c,\lambda\}\), which is possible because
+\(\alpha\le b<\lambda\).  For large \(n\), the integer
+\(\lfloor dn\rfloor\) lies below the cap and is at least
+\(nx_n^{(\lambda)}+n^{1-\delta}\).  The same two estimates give
+\(P_n(\lfloor dn\rfloor)\le Kn^{-\eta}\to0\).
+Monotonicity then yields
+\(P_n(\lfloor cn\rfloor)\le P_n(\lfloor dn\rfloor)\to0\).
+\end{proof}
+
+\subsection{Application to every fixed clause size}
+
+We finish by checking the inputs for the statistic used in this paper.
+This also identifies the repetition constant furnished by the general
+transfer lemma.
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+Fix \(k\ge3\).  Recall \(L_k=2^{k+1}\),
+\(V_n=\min\{H_n-1,L_kn\}\), and \(\mu_n=\EE V_n/n\).
+Since \(L_k\) is an integer, the definitions in
+\eqref{eq:general-capped-center} with \(\lambda=L_k\) give exactly
+\(V_n^{(L_k)}=V_n\) and \(x_n^{(L_k)}=\mu_n\).
+
+Proposition~\ref{prop:concentration} supplies the tail estimate in
+\eqref{eq:convergence-input} with
+\[
+ \delta=\delta_k=\frac{k-2}{4k},\qquad
+ \eta=\eta_k=\frac{k-2}{2k},
+\]
+and some \(K=K_k<\infty\).  These exponents satisfy
+\(0<\delta<1/2\) and \(\eta>0\).  Lemma~\ref{lem:center-bounds}
+supplies the center bounds with
+\(a=a_0\) and \(b=2^k+1<L_k\).
+
+For the transfer in this application choose \(D=L_k>b\).  Then
+\[
+ R=L_k\binom{k}{2},\qquad
+ 1-2\delta=\frac{k+2}{2k}>\eta,
+ \qquad \gamma=\eta,\qquad \kappa=\frac{e^{-R}}2.
+\]
+Thus all hypotheses of Theorem~\ref{thm:concentration-to-convergence}
+hold with the stated cap, center bounds, and exponents.  It follows that
+\(\mu_n\to\alpha_k\in[a_0,2^k+1]\) and that the proper-clause
+satisfiability probabilities tend to one for \(0\le c<\alpha_k\)
+and to zero for \(c>\alpha_k\), as claimed.
+\end{proof}
diff --git a/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/build/source/sections/forcing.tex b/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/build/source/sections/forcing.tex
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index 0000000000000000000000000000000000000000..9a1d4cf744871061bd56815cea657419a6f5d383
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+++ b/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/build/source/sections/forcing.tex
@@ -0,0 +1,213 @@
+\section{Replacement and integrated forcing}\label{sec:forcing}
+
+We begin the local concentration proof by controlling how many
+variables are forced along the clause process. The estimate is
+integrated over prefix lengths: erasing one variable turns forcing into
+a difference of satisfiability probabilities, and those differences
+can be summed by telescoping.
+
+\begin{samepage}
+For the fixed \(k\ge3\), write
+\[
+ H_n=\inf\{m\ge1:F_{n,m}\notin\sat\}.
+\]
+The value \(+\infty\) is allowed in this definition, although
+\(P_n(m)\le2^n(1-2^{-k})^m\) implies that \(H_n\) is finite almost
+surely. Put
+\begin{equation}\label{eq:statistic}
+ L_k=2^{k+1},\qquad M_n=L_kn,\qquad
+ V_n=\min\{H_n-1,M_n\},\qquad \mu_n=\frac{\EE V_n}{n}.
+\end{equation}
+Thus \(V_n\) is the last satisfiable index up to the cap \(M_n\), and
+\begin{equation}\label{eq:survival}
+ P_n(m)=\PP(V_n\ge m)\qquad(0\le m\le M_n).
+\end{equation}
+This identity includes \(m=M_n\). Later, when a clause is omitted, the
+same endpoint convention will count every additional satisfiable prefix.
+\end{samepage}
+
+\subsection{Killing a solution set with one clause}
+
+A variable of a satisfiable formula \(H\) is \emph{forced} if all
+satisfying assignments of \(H\) give it the same value. Let \(b(H)\) be
+the number of forced variables, and set \(b(H)=0\) if \(H\) is
+unsatisfiable. This definition uses all solutions of the entire
+formula, even when the formula contains clauses of different lengths.
+
+For a formula \(H\) on \(v\) variables and \(1\le r\le v\), define
+\begin{equation}\label{eq:kill}
+ q_r(H)=\frac{\fall{b(H)}r}{2^r\fall vr},
+ \qquad
+ \fall ur=u(u-1)\cdots(u-r+1),
+\end{equation}
+where \(\fall ur=0\) for nonnegative integers \(u<r\).
+Here a random \(r\)-clause uses \(r\) distinct uniform variables and
+independent fair signs.
+
+If \(H\) is satisfiable, \(q_r(H)\) is exactly the probability that
+adding one independent random \(r\)-clause makes it unsatisfiable.
+Indeed, every solution of \(H\) must falsify every literal in such a
+clause. Each chosen variable must therefore be forced, and its sign
+must oppose the forced value. Conversely those conditions make the
+clause false in every solution. For unsatisfiable \(H\), the convention
+\(q_r(H)=0\) records that there is no transition to count.
+
+\subsection{Replacing a shorter clause}
+
+The next estimate is uniform in the background formula. This
+uniformity will allow successive replacements after conditioning on all
+other clauses.
+
+\begin{samepage}
+\begin{lemma}[Replacing a \((k-1)\)-clause]\label{lem:replacement}
+Let \(k\ge3\) and \(n\ge k+1\) be integers, and put
+\[
+ v=n-1,\qquad \eps=n^{-(k-1)/k},\qquad
+ g=\lceil2kn^{1/k}\rceil.
+\]
+For any formula \(H\) on these \(v\) variables, let \(Q\) be an
+independent random \((k-1)\)-clause and let \(R_1,\ldots,R_g\) be
+independent random \(k\)-clauses, also independent of \(H,Q\). Then
+\[
+ \PP(H\wedge R_1\wedge\cdots\wedge R_g\notin\sat)
+ \ge \PP(H\wedge Q\notin\sat)-\eps.
+\]
+\end{lemma}
+\end{samepage}
+
+The power comparison below is the Boolean forced-variable instance of
+the codimension comparison in Carenini~\cite[Lemma~6.1]{Carenini}. The
+proof records the block size and error needed for the present summation.
+
+\begin{proof}
+If \(H\) is unsatisfiable, both unsatisfiability probabilities are one.
+Suppose it is satisfiable, and set \(x=q_{k-1}(H)\) and \(b=b(H)\).
+When \(x\le\eps\), the right side after subtracting \(\eps\) is
+nonpositive. Assume \(x>\eps\).
+
+First \(b\ge k\). Otherwise
+\[
+ x\le\frac{1}{2^{k-1}\binom{n-1}{k-1}}
+ \le\frac1n\le n^{-(k-1)/k}=\eps.
+\]
+The middle inequality follows from
+\(\binom{n-1}{k-1}\ge n-1\), valid for \(1\le k-1\le n-2\), and
+\(2^{k-1}(n-1)\ge n\). For \(b\ge k\), the exact killing probabilities
+give
+\[
+ q_k(H)
+ =x\frac{b-k+1}{2(v-k+1)}
+ \ge \frac{x}{k}\frac{b}{2v}
+ \ge \frac1k x^{k/(k-1)}.
+\]
+Here \(b-k+1\ge b/k\) and \(v-k+1\le v\). For the last inequality,
+each factor \((b-j)/(v-j)\) is at most \(b/v\), so
+\(x\le(b/(2v))^{k-1}\). Consequently,
+\[
+ gq_k(H)\ge2n^{1/k}x^{k/(k-1)}\ge2x,
+\]
+because \(x>n^{-(k-1)/k}\).
+
+The block makes \(H\) unsatisfiable whenever at least one of its
+clauses does so individually. These individual events are independent
+conditional on \(H\). Hence
+\[
+ \PP(H\wedge R_1\wedge\cdots\wedge R_g\notin\sat)
+ \ge1-(1-q_k(H))^g
+ \ge1-e^{-2x}
+ \ge x.
+\]
+For the last inequality, \(0\le x\le2^{-(k-1)}\le1/4\), and
+\(1-e^{-2x}-x\) vanishes at zero and has positive derivative on that
+interval.
+\end{proof}
+
+\subsection{Erasing one variable and summing over time}
+
+The following estimate is the input to the omission argument in
+Section~\ref{sec:concentration}. Unsatisfiable formulas contribute zero,
+and the sum includes the empty prefix and the cap.
+
+\begin{proposition}[Integrated forcing]\label{prop:forcing}
+For every fixed integer \(k\ge3\),
+\begin{equation}\label{eq:integrated}
+ \sum_{j=0}^{M_n}\EE\frac{b(F_{n,j})}{n}=O_k(n^{1/k}),
+ \qquad M_n=2^{k+1}n.
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+It is enough to consider \(n\ge k+1\). We first prove an explicit
+bound for every integer cap \(M\ge0\), and then take \(M=M_n\).
+The case \(M=0\) is immediate, so assume \(M\ge1\).
+By symmetry, the \(j\)-th
+summand is the probability that \(F_{n,j}\) is satisfiable and \(x_n\)
+is forced. Delete the clauses containing \(x_n\), and let \(d\) be
+their number. Then \(d\sim\Bin(j,k/n)\). Conditional on their positions,
+the untouched formula \(B\) is an independent proper \(k\)-SAT formula
+on \(v=n-1\) variables with \(j-d\) clauses. Erasing the literal on
+\(x_n\) from each deleted clause leaves independent uniform
+\((k-1)\)-clauses \(Q_1,\ldots,Q_d\) on those \(v\) variables,
+independent of \(B\).
+
+If \(F_{n,j}\) is satisfiable and \(x_n\) is forced, then
+\begin{equation}\label{eq:forced-reduction}
+ B\in\sat,\qquad B\wedge Q_1\wedge\cdots\wedge Q_d\notin\sat.
+\end{equation}
+Indeed, any solution of the latter formula would extend to a solution
+of \(F_{n,j}\) with either value of \(x_n\).
+
+Keep the incidence positions fixed. Replace each \(Q_\ell\) in turn by
+\(g\) fresh random \(k\)-clauses, with \(g,\eps\) as in
+Lemma~\ref{lem:replacement}. At a replacement step, condition on \(B\),
+all other remaining shortened clauses, and all previously inserted
+blocks. The clause being replaced is still uniform and independent of
+this background, so the lemma bounds the loss in unsatisfiability
+probability by \(\eps\). Averaging at each step yields
+\[
+ \PP(B\wedge Q_1\wedge\cdots\wedge Q_d\notin\sat
+       \mid\text{incidence positions})
+ \le1-P_v(j-d+dg)+d\eps.
+\]
+The formula after all replacements has \(j-d+dg\) independent proper
+\(k\)-clauses on \(v\) variables. Since \(B\notin\sat\) is contained in
+the event on the left and has conditional probability \(1-P_v(j-d)\),
+the conditional probability of \eqref{eq:forced-reduction} is at most
+\begin{equation}\label{eq:replacement-difference}
+ \Delta_{j,d}+d\eps,\qquad
+ \Delta_{j,d}=P_v(j-d)-P_v(j-d+dg)\ge0.
+\end{equation}
+
+To sum this estimate, let
+\(p_{j,d}=\PP(\Bin(j,k/n)=d)\). For \(0\le d\le j\le M\),
+\[
+ p_{j,d}\le\binom jd(k/n)^d\le\frac{(kM/n)^d}{d!}.
+\]
+For fixed \(d\), monotonicity of \(P_v\) and telescoping give
+\begin{equation}\label{eq:telescoping}
+ \sum_{j=d}^{M}\Delta_{j,d}
+ =\sum_{t=0}^{M-d}\bigl[P_v(t)-P_v(t+dg)\bigr]\le dg.
+\end{equation}
+For example, after shifting the second sum by \(dg\), only the first
+\(dg\) nonnegative terms can remain; if the shift exceeds the number
+of terms, that number is already at most \(dg\).
+
+Use the envelope \((kM/n)^d/d!\) only for the nonnegative
+\(\Delta_{j,d}\) term. The error \(d\eps\) is averaged with its original
+binomial weight, whose mean is \(kj/n\). Therefore
+\begin{align}
+ \sum_{j=0}^{M}\EE\frac{b(F_{n,j})}{n}
+ &\le
+ \sum_{d=0}^{M}\frac{(kM/n)^d}{d!}
+       \sum_{j=d}^{M}\Delta_{j,d}
+       +\eps\sum_{j=0}^{M}\frac{kj}{n}\notag\\
+ &\le
+ g\sum_{d=0}^{\infty}\frac{d(kM/n)^d}{d!}
+       +\frac{kM(M+1)}{2n}\eps\notag\\
+ &=\frac{kM}{n}e^{kM/n}g+\frac{kM(M+1)}{2n}\eps.
+ \label{eq:forcing-general-cap}
+\end{align}
+At \(M=M_n=L_kn\), the exponential factor depends only on \(k\).
+The last expression is therefore \(O_k(g+n\eps)=O_k(n^{1/k})\),
+because \(g=O_k(n^{1/k})\) and \(n\eps=n^{1/k}\).
+\end{proof}
diff --git a/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/build/source/sections/introduction.tex b/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/build/source/sections/introduction.tex
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+++ b/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/build/source/sections/introduction.tex
@@ -0,0 +1,181 @@
+\section{The limiting-threshold problem}\label{sec:intro}
+
+As independent clauses are added to a random Boolean formula,
+satisfiability changes from likely to unlikely. A narrow transition
+window answers only part of the threshold question: its location might
+still drift as the number of variables grows. We prove that the location
+converges for every fixed clause size at least three.
+
+A Boolean assignment gives each variable \(x_i\) a value in \(\{0,1\}\).
+A literal is a variable or its negation; a clause is a disjunction of
+literals, and a conjunctive normal form formula is a conjunction of
+clauses. A formula is satisfiable if some assignment makes every clause
+true. Random \(k\)-SAT asks how this event changes as independent clauses
+are added to a formula on \(n\) variables.
+
+Fix an integer \(k\ge3\). For \(n\ge k\), a \emph{proper \(k\)-clause}
+uses \(k\) distinct variables chosen uniformly from \(x_1,\ldots,x_n\),
+with independent fair signs. Thus each of the \(2^k\binom nk\) possible
+clauses has the same probability. Let \(C_1,C_2,\ldots\) be independent
+clauses with this law, and define
+\[
+ F_{n,m}=\bigwedge_{i=1}^{m}C_i,
+ \qquad P_n(m)=\PP(F_{n,m}\in\sat),\qquad m=0,1,2,\ldots.
+\]
+Here \(\sat\) denotes satisfiability, and the empty conjunction is true.
+Whole clauses are sampled with replacement: two positions may contain
+the same clause, although a variable cannot occur twice within one
+proper clause. The prefix coupling makes \(P_n\) nonincreasing and gives
+\(P_n(0)=1\). The ratio \(c=m/n\) is the clause density.
+
+\begin{theorem}\label{thm:main}
+For every fixed integer \(k\ge3\), there is a constant
+\(\alpha_k\in(0,\infty)\) such that, for independent proper uniformly
+signed \(k\)-clauses sampled with replacement,
+\[
+ \lim_{n\to\infty}P_n(\lfloor cn\rfloor)
+ =
+ \begin{cases}
+  1,&0\le c<\alpha_k,\\
+  0,&c>\alpha_k.
+ \end{cases}
+\]
+\end{theorem}
+
+The conclusion is a strict-side threshold statement; it makes no
+assertion at \(c=\alpha_k\). All estimates below may depend on the fixed
+integer \(k\). The proof does not require the centers of the transition
+windows to be known in advance. It constructs a finite-size center,
+controls the fluctuations about it, and then proves that these centers
+converge.
+
+\subsection{Prior work and the role of the present argument}
+
+Carenini~\cite[Theorem~1.1 and Corollary~1.2]{Carenini2026Polynomial}
+proves an \(O_{k,\eta}(n^{1/2+1/k})\) clause window between probability
+levels \(\eta\) and \(1-\eta\), for each fixed \(0<\eta<1/2\), and, using
+the Abbe--Montanari criterion, establishes the satisfiability conjecture for
+every fixed \(k\ge3\). We credit Carenini with priority for the resolution
+of the satisfiability conjecture. The present paper gives an alternative
+proof of the limiting-threshold conclusion.
+
+Chv\'atal and Reed~\cite{CR} explicitly formulated the conjecture that
+each fixed clause size greater than one has a limiting satisfiability
+density. Friedgut's sharp-threshold theorem, with Bourgain's appendix
+\cite[Theorem~1.3]{Friedgut}, established that random \(k\)-SAT has a
+sharp transition around a sequence of densities. It left open whether
+that sequence converges for each fixed \(k\). The classical two-clause
+problem had already been settled by Chv\'atal and Reed
+\cite[Theorems~3--4]{CR}, who also credit independent proofs by Goerdt
+and Fernandez de la Vega. Goerdt's journal treatment is~\cite{Goerdt}.
+The implication-graph mechanism is separate from the proof given here.
+
+For larger arities, Achlioptas and Peres~\cite{AP} obtained leading-order
+bounds \(2^k\log2-O(k)\), using weighted second moments. The
+statistical-physics work of M\'ezard, Parisi, and Zecchina~\cite{MPZ}
+and of Mertens, M\'ezard, and Zecchina~\cite{MMZ} developed threshold
+predictions through the cavity method and one-step replica symmetry
+breaking. In particular, the latter work predicted the large-\(k\)
+expansion
+\[
+ 2^k\log2-\frac{1+\log2}{2}+o_k(1),
+ \qquad o_k(1)\longrightarrow0\quad(k\longrightarrow\infty).
+\]
+Coja-Oghlan and Panagiotou~\cite[Theorem~1.1]{COP} proved that the
+liminf and limsup of the sharp-threshold sequence both lie within an
+additive error tending to zero as \(k\to\infty\) of
+\(2^k\log2-(1+\log2)/2\). These asymptotic-in-arity estimates are
+distinct from convergence in \(n\) at a fixed arity. Ding, Sly, and
+Sun~\cite{DSS} proved existence and identified the threshold with the
+one-step replica symmetry breaking prediction for all sufficiently
+large fixed \(k\). Theorem~\ref{thm:main} establishes existence for every
+fixed \(k\ge3\), without identifying the value of \(\alpha_k\).
+
+The first ingredient is a comparison between shorter clauses and
+blocks of ordinary clauses. Friedgut's half-cube replacement lemma
+\cite[Lemma~5.7]{Friedgut} gives a qualitative antecedent based on frozen
+coordinates and sequential replacement. Carenini
+\cite[Lemma~6.1 and Corollaries~6.2, 7.10--7.11]{Carenini} developed an
+averaged clause-replacement method. The cited version proves an
+\(O_k(n/\log n)\) window between any two fixed central probability
+levels, with constants also depending on those levels, and distinguishes
+window width from convergence of the location in Section~10.1.
+Here the replacement is applied after erasing one variable. Summing the
+resulting changes in satisfiability probabilities bounds the forced
+fraction along the clause process. A deletion argument then converts
+this integrated bound into polynomial concentration. We give the
+replacement, forcing, and deletion proofs in full.
+
+The second ingredient compares a whole system with independent
+subsystems. Guerra and Toninelli~\cite{GT} used interpolation to
+establish thermodynamic limits for mean-field spin glasses.
+Franz and Leone~\cite{FL}, and Franz, Leone, and Toninelli~\cite{FLT},
+developed interpolation for diluted systems. Their pressure and
+free-energy arguments, with the stated even-arity restrictions, do not
+supply the hard satisfiability-threshold conclusion used here.
+Bayati, Gamarnik, and Tetali~\cite[Section~4, Proposition~2]{BGT}
+established a frozen-variable satisfiability interpolation between
+clauses on a whole variable set and clauses confined to its parts.
+Our Poisson comparison is a continuous-parameter form of that method.
+Its sign follows from convexity of \(x\mapsto x^k\), applied to the
+proportions of globally forced variables in the parts. The comparison
+is exact for an auxiliary clause model that permits repeated variables;
+we prove the transfer back to proper clauses, including the constant
+loss caused by repetitions.
+
+Finally, errors that are summable over geometrically increasing sizes
+cannot produce macroscopic drift. This principle parallels the
+approximate-superadditivity analysis of Abbe and Montanari
+\cite[arXiv version, Section~4.3, Lemma~8 and Remark~3]{AM}, developed
+in their study of solution counts. Our comparison uses the minimum of
+two normalized centers. We prove the corresponding deterministic
+convergence lemma, in a form that requires the comparison only when
+the two sizes are comparable.
+
+\subsection{How the proof reaches the threshold}
+
+We cap the last satisfiable index at a fixed multiple of \(n\), and
+take its expectation divided by \(n\) as the finite-size center. The
+first task is to show that this capped index has sublinear fluctuations.
+In a satisfiable formula, a variable is \emph{forced} if every
+satisfying assignment gives it the same value. The clauses that avoid
+the variable form the untouched background. If the variable is forced,
+deleting its literal from each incident clause produces shorter clauses
+that eliminate every solution of this background. Replacing each
+shortened clause by a block of ordinary clauses turns this event into
+a decrease of the satisfiability probability. Summing over prefix
+lengths telescopes the decreases. Section~\ref{sec:forcing} thereby
+controls the total expected forced fraction along the process.
+
+Section~\ref{sec:concentration} omits a single clause while keeping the
+other time indices. The delay in losing satisfiability counts the
+prefixes where that clause is pivotal. Its square counts pairs of such
+prefixes. If the punctured formula survives from the earlier prefix to
+the later one, every intervening clause must avoid killing the earlier
+solution set. This bounds the pair probability in terms of the variables
+already forced at the earlier time. The integrated forcing estimate
+then yields the required variance bound. Section~\ref{sec:anchors}
+places the normalized means in a fixed positive bounded interval, using
+a clause-to-variable matching at low density and an assignment first
+moment at high density.
+
+Concentration alone still permits the centers to drift. In the auxiliary
+Poisson model, interpolation says that satisfiability at a combined size
+is at least the product of the probabilities in two independent parts.
+Below either finite-size center, transfer from the proper model gives a
+fixed positive probability; above the combined center, it gives a
+probability tending to zero. These facts prevent the combined center
+from lying much below both smaller centers. A comparison with one extra
+variable handles adjacent sizes. Section~\ref{sec:convergence} makes
+these steps precise and sums the errors along a balanced tree of nearly
+equal blocks. Its concentration-to-convergence theorem is then applied
+to the local variance and center bounds.
+
+Section~\ref{sec:three-clause} uses a first moment at density six to
+bound the means of the three-clause indices, and gives exact relations
+between their cap conventions. It then specializes the forcing estimate and deduces
+concentration at the smaller cap. The improved mean bound permits a
+smaller transfer interval for either cap, sharpening the numerical
+constants while keeping the roles of the cap and the interval distinct.
+Section~\ref{sec:boundaries} explains the lower-arity cases and the
+finite-size information supplied by the proof.
diff --git a/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/build/source/sections/three-clause.tex b/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/build/source/sections/three-clause.tex
new file mode 100644
index 0000000000000000000000000000000000000000..b9d0580e1c7ff3e0b5ad6eacb6d94d06e22f65bf
--- /dev/null
+++ b/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/build/source/sections/three-clause.tex
@@ -0,0 +1,202 @@
+\section{Three-clause refinements}\label{sec:three-clause}
+
+For three literals per clause, a first moment at density six improves
+the center bound enough to use the comparison interval \([0,8]\),
+including for the main cap \(16n\). We also compare that statistic with
+last-satisfiable and first-unsatisfiable indices capped at \(10n\).
+Changing the cap of the last-satisfiable index has an exponentially
+small effect on its normalized mean. Comparison with the first
+unsatisfiable index also includes a one-step shift.
+
+Throughout this section the clauses are independent uniformly signed
+proper triples, sampled with replacement, and \(n\ge3\). Recall the
+first unsatisfiable index \(H_n\), and write
+\begin{equation}\label{eq:three-statistics}
+ W_n=\min\{H_n-1,10n\},\qquad
+ T_n=\min\{H_n,10n\},\qquad
+ V_n=\min\{H_n-1,16n\}.
+\end{equation}
+Thus \(V_n\) is the statistic of the main proof at \(k=3\). Put
+\(\nu_n=\EE W_n/n\), while \(\mu_n=\EE V_n/n\) as before.
+For integer \(m\),
+\begin{equation}\label{eq:three-survival}
+\begin{aligned}
+ P_n(m)&=\PP(W_n\ge m)&&\quad(0\le m\le10n),\\
+ P_n(m)&=\PP(T_n>m)&&\quad(0\le m<10n).
+\end{aligned}
+\end{equation}
+Only the last-satisfiable identity includes its capped endpoint.
+
+\subsection{The density-six bound and changes of cap}
+
+Each fixed assignment satisfies a proper triple with probability \(7/8\),
+so \(P_n(m)\le2^n(7/8)^m\). Set
+\[
+ \sigma=2(7/8)^6<1,\qquad \rho=2(7/8)^{10}<1.
+\]
+The survival-sum identity for the positive integer \(H_n\) gives the
+cap-independent bound
+\begin{equation}\label{eq:three-h-mean}
+ \EE H_n=\sum_{m\ge0}P_n(m)
+ \le6n+\sum_{m\ge6n}2^n(7/8)^m
+ =6n+8\sigma^n.
+\end{equation}
+In particular, the normalized means of \(W_n,T_n,V_n\) are all at most
+seven for large \(n\). Splitting at \(6n\) also gives the direct estimate
+\begin{equation}\label{eq:three-center}
+ \nu_n\le6+4P_n(6n)\le6+4\sigma^n.
+\end{equation}
+The matching argument in Lemma~\ref{lem:center-bounds} gives
+\(P_n(\lfloor c_0n\rfloor)=1-o(1)\) whenever \(c_0>0\) and
+\(e^2c_0<1\). These clause counts lie below both caps, so there is
+\(a_0>0\) such that
+\begin{equation}\label{eq:three-anchors}
+ a_0\le\nu_n\le\mu_n\le7
+ \qquad\text{for all sufficiently large }n.
+\end{equation}
+The improved upper bound therefore applies to either cap.
+
+The exact relations explain the distinction between first and last
+indices at the capped endpoint. With the common cap \(10n\),
+\begin{equation}\label{eq:three-same-cap}
+ W_n=T_n-1+\ind_{\{H_n>10n\}}
+     =T_n-1+\ind_{\{F_{n,10n}\in\sat\}}.
+\end{equation}
+If \(H_n\le10n\), the indices differ by one; otherwise both equal
+\(10n\). To compare \(V_n\) with \(T_n-1\), count the extra satisfiable
+prefixes:
+\begin{equation}\label{eq:three-cross-cap}
+\begin{gathered}
+ V_n=T_n-1+E_n,\qquad
+ E_n=\sum_{m=10n}^{16n}\ind_{\{H_n>m\}},\\
+ 0\le E_n\le(6n+1)\ind_{\{H_n>10n\}}.
+\end{gathered}
+\end{equation}
+The first term is necessary: \(H_n=10n+1\) gives \(E_n=1\).
+Since \(P_n(10n)\le\rho^n\), these identities imply
+\begin{align}
+ 0\le\EE V_n-(\EE T_n-1)&\le(6n+1)\rho^n,
+       \label{eq:three-mean-bridge}\\
+ |\sd(V_n)-\sd(T_n)|&\le(6n+1)\rho^{n/2}.
+       \label{eq:three-sd-bridge}
+\end{align}
+For the second estimate, center the identity \(V_n=T_n-1+E_n\) and
+apply the reverse triangle inequality in \(L^2\), using
+\(\sd(E_n)\le(\EE E_n^2)^{1/2}\). Similarly,
+\begin{equation}\label{eq:three-last-cap-bridge}
+ 0\le V_n-W_n\le6n\ind_{\{H_n>10n\}},\qquad
+ 0\le\mu_n-\nu_n\le6\rho^n.
+\end{equation}
+Thus \(\nu_n\to\alpha_3\), since the main proof already gives
+\(\mu_n\to\alpha_3\).
+
+\subsection{Numerical forcing bound and concentration}
+
+The replacement parameters of Lemma~\ref{lem:replacement} become
+\(v=n-1\), \(\varepsilon=n^{-2/3}\), and
+\(g=\lceil6n^{1/3}\rceil\), for \(n\ge4\).
+For a satisfiable background \(B\), put \(b=b(B)\) and
+\(x=q_2(B)=b(b-1)/(4v(v-1))\). When \(x>\varepsilon\), the proof of
+that lemma gives \(b\ge3\), and the exact killing probabilities read
+\begin{equation}\label{eq:three-killing}
+ q_3(B)=x^{3/2}
+ \frac{b-2}{\sqrt{b(b-1)}}
+ \frac{\sqrt{v(v-1)}}{v-2}
+ \ge\frac13x^{3/2}.
+\end{equation}
+Here \(b-2\ge b/3\), \(\sqrt{b(b-1)}\le b\), and
+\(\sqrt{v(v-1)}\ge v-2\). Thus \(gq_3(B)>2x\), and the lemma's
+independent-block estimate is
+\(1-(1-q_3(B))^g\ge1-e^{-2x}\ge x\).
+Together with the case \(x\le\varepsilon\), this recovers the uniform
+replacement loss \(\varepsilon\).
+
+The numerical forcing bound follows directly from
+\eqref{eq:forcing-general-cap} with \(k=3\) and \(M=10n\):
+\begin{equation}\label{eq:three-forcing}
+ \sum_{j=0}^{10n}\EE\frac{b(F_{n,j})}{n}
+ \le30e^{30}\lceil6n^{1/3}\rceil
+       +(150n+15)n^{-2/3}
+ =O(n^{1/3}).
+\end{equation}
+The constant \(30\) is the maximum mean incidence count \(3M/n\).
+The general proof supplies the conditioning and telescoping needed
+for this estimate, including the prefix \(j=10n\).
+
+Concentration at the smaller cap follows even more directly from the
+main variance bound: \(W_n=\min\{V_n,10n\}\). If \(V_n'\) is an
+independent copy of \(V_n\) and \(W_n'=\min\{V_n',10n\}\), truncation
+is \(1\)-Lipschitz, so
+\[
+ \Var(W_n)=\tfrac12\EE(W_n-W_n')^2
+ \le\tfrac12\EE(V_n-V_n')^2=\Var(V_n).
+\]
+Proposition~\ref{prop:concentration} and Chebyshev's inequality give
+\begin{equation}\label{eq:three-concentration}
+ \Var(W_n)=O(n^{5/3}),\qquad
+ \PP\bigl(|W_n-n\nu_n|\ge n^{11/12}\bigr)=O(n^{-1/6}).
+\end{equation}
+The same powers hold for \(V_n\), with its own finite-size mean.
+
+\subsection{The comparison range is independent of the cap}
+
+In the auxiliary model, an ordered triple has distinct variable indices
+with probability
+\[
+ \vartheta_n=\frac{(n)_3}{n^3}=1-\frac3n+\frac2{n^2}.
+\]
+At density \(c\), Poisson thinning separates a proper-clause count
+\(J\sim\Pois(cn\vartheta_n)\) from an independent repeated-variable
+family with mean count \(c(3-2/n)\). Therefore
+Lemma~\ref{lem:proper-transfer} gives, for every fixed \(D\ge0\) and
+\(0\le c\le D\),
+\begin{equation}\label{eq:three-density-sandwich}
+ e^{-3D}\EE P_n(J)\le s_n(c)\le\EE P_n(J),\qquad
+ cn-3D\le\EE J\le cn,\qquad \Var(J)\le Dn.
+\end{equation}
+The parameter \(D\) controls the density at which probabilities are
+compared. The proof of Theorem~\ref{thm:concentration-to-convergence}
+requires \(D>b\); it need not equal the cap.
+
+By \eqref{eq:three-anchors} and \eqref{eq:three-concentration}, both
+\((\lambda,x_n^{(\lambda)})=(10,\nu_n)\) and
+\((\lambda,x_n^{(\lambda)})=(16,\mu_n)\) satisfy that theorem's inputs
+with
+\begin{equation}\label{eq:three-engine-parameters}
+ (a,b,\delta,\eta)=\left(a_0,7,\frac1{12},\frac16\right),\qquad
+ (D,R,\gamma,\kappa)=\left(8,24,\frac16,\frac{e^{-24}}2\right).
+\end{equation}
+Here \(b<\lambda\) for either cap, \(D>b\), and
+\(\gamma=\min\{\eta,1-2\delta\}=1/6\).
+On \([0,8]\), the repetition mean is at most \(24\), so the sandwich is
+\begin{equation}\label{eq:three-poisson-sandwich}
+ e^{-24}\EE P_n(J)\le s_n(c)\le\EE P_n(J),\qquad
+ cn-24\le\EE J\le cn,\qquad \Var(J)\le8n.
+\end{equation}
+For either choice \(z_n=\nu_n\) or \(z_n=\mu_n\), there is an absolute
+constant \(K_8<\infty\) such that, uniformly on \(0\le c\le8\) for all
+sufficiently large \(n\),
+\begin{align}
+ c\le z_n-2n^{-1/12}&\Longrightarrow s_n(c)\ge e^{-24}/2,
+       \label{eq:three-transfer-low}\\
+ c\ge z_n+2n^{-1/12}&\Longrightarrow s_n(c)\le K_8n^{-1/6}.
+       \label{eq:three-transfer-high}
+\end{align}
+The theorem also gives the corresponding size comparisons for either
+choice of center:
+\begin{align}
+ z_{r+s}&\ge\min\{z_r,z_s\}-4\min\{r,s\}^{-1/12},
+       \label{eq:three-center-sum}\\
+ z_{s+1}&\ge z_s-4s^{-1/12},
+       \label{eq:three-neighbor}
+\end{align}
+for all sufficiently large relevant sizes, without a restriction on
+\(r/s\) in the first inequality.
+
+If probabilities are instead compared over the wider interval
+\([0,16]\), the same calculation gives \(R=48\), lower sandwich factor
+\(e^{-48}\), and lower-transfer constant \(e^{-48}/2\). These are the
+constants obtained from the generic choices \(\lambda=16\), \(b=9\),
+and \(D=16\) in the main proof. The sharper mean bound permits \(D=8\)
+for that same cap, while the \(D=16\) calculation covers a larger range
+of densities.
diff --git a/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/README.md b/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..bde4800b6e6ca3884ace2102d5e401dd797124d2
--- /dev/null
+++ b/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/README.md
@@ -0,0 +1,18 @@
+# [A Linear Clock for Random Walk on Tree-Weighted Planar Maps](linear-clock-random-walk-tree-weighted-planar-maps.pdf)
+
+**Author:** OpenAI
+
+**Date:** October 5, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026,
+  author = {{OpenAI}},
+  title = {{A Linear Clock for Random Walk on Tree-Weighted Planar Maps}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/linear-clock-random-walk-tree-weighted-planar-maps.pdf}{OAI:A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/build/bibliography.tex b/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/build/bibliography.tex
new file mode 100644
index 0000000000000000000000000000000000000000..a6470bcf252d4b4978642b408e17995115cf6376
--- /dev/null
+++ b/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/build/bibliography.tex
@@ -0,0 +1,152 @@
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diff --git a/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/build/paper.tex b/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/build/paper.tex
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+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage[margin=1.1in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{microtype}
+\usepackage{enumitem}
+\usepackage{booktabs,array}
+\usepackage{xcolor}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta,calc,patterns,positioning}
+\usepackage[hyphens]{url}
+\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref}
+\hypersetup{pdftitle={A Linear Clock for Random Walk on Tree-Weighted Planar Maps},pdfauthor={OpenAI}}
+
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\numberwithin{equation}{section}
+
+\newcommand{\R}{\mathbb R}
+\newcommand{\C}{\mathbb C}
+\newcommand{\Z}{\mathbb Z}
+\newcommand{\N}{\mathbb N}
+\newcommand{\E}{\mathbb E}
+\renewcommand{\P}{\mathbb P}
+\newcommand{\dd}{\mathrm d}
+\newcommand{\1}{\mathbf 1}
+\newcommand{\en}{\mathsf e}
+\newcommand{\enstar}{\mathsf e^{*}}
+\newcommand{\cE}{\mathcal E}
+\newcommand{\cQ}{\mathcal Q}
+\DeclareMathOperator{\EL}{EL}
+\let\Cap\relax
+\DeclareMathOperator{\Cap}{Cap}
+\DeclareMathOperator{\supp}{supp}
+\DeclareMathOperator{\diam}{diam}
+\DeclareMathOperator{\dist}{dist}
+\DeclareMathOperator{\vol}{vol}
+\DeclareMathOperator{\osc}{osc}
+\DeclareMathOperator{\Law}{Law}
+\DeclareMathOperator{\Var}{Var}
+\DeclareMathOperator{\Area}{Area}
+\newcommand{\articleid}[1]{\begingroup\Urlmuskip=0mu plus 2mu\relax\nolinkurl{#1}\endgroup}
+\setlist{topsep=4pt,itemsep=3pt,parsep=0pt}
+\setlength{\emergencystretch}{2em}
+
+\title{A Linear Clock for Random Walk\\on Tree-Weighted Planar Maps}
+\author{OpenAI}
+\date{October 5, 2026}
+
+\begin{document}
+\maketitle
+\begin{abstract}
+We prove that stationary random walk on a planar map sampled with weight
+equal to its number of spanning trees converges to Liouville Brownian
+motion on the unit-area $\sqrt2$-Liouville quantum sphere. The convergence
+retains the conditional path law jointly with the measured metric space.
+The walk chooses uniformly among all incident half-edges and starts from
+the stationary degree measure. For total attempt rate one and the
+continuum Dirichlet form with factor $1/2$, the time acceleration is exactly
+the number of map edges. The result uses the companion contour and metric
+limits for this same ensemble.
+\end{abstract}
+
+\tableofcontents
+\input{sections/01-introduction}
+\input{sections/02-inputs}
+\input{sections/03-nondegeneration}
+\input{sections/04-energy}
+\input{sections/05-sphere}
+\input{sections/06-green}
+\input{sections/07-dynamics}
+\input{bibliography}
+\end{document}
diff --git a/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/build/sections/01-history.tex b/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/build/sections/01-history.tex
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+Mullin's enumeration of tree-rooted maps \cite{Mullin} gives the
+combinatorial starting point: the selected spanning tree and its
+complementary dual tree encode a map by a two-coordinate walk. The
+continuum counterpart is the mating-of-trees construction of
+Duplantier, Miller, and Sheffield \cite{DMS}. At $\gamma=\sqrt2$ its
+boundary-length coordinates are independent Brownian motions. Miller
+and Sheffield \cite{MSSphere} give the corresponding finite-area
+sphere description, and their imaginary-geometry construction
+\cite{IG4} supplies the space-filling $\mathrm{SLE}_8$ between the two
+continuum trees. Couplings of discrete and continuum encodings have
+also been used to compare graph distances across random-map models
+\cite{GHS}. The continuum metric itself is characterized in
+\cite{GMMetric}, with conformal covariance established in
+\cite{GMConformal}.
+
+Liouville Brownian motion was constructed by Garban, Rhodes, and Vargas
+\cite{GRV} and independently by Berestycki \cite{Berestycki} through a
+change of Brownian time determined by quantum area. Its Dirichlet-form
+description \cite{GRVForm} separates two structures: the energy is
+conformally invariant, while the speed measure is quantum area. This
+distinction explains why the clock is a separate part of a random-map
+scaling limit.
+
+Several works establish important parts of the discrete picture.
+Gwynne, Miller, and Sheffield prove convergence of the Tutte embedding
+of mated-CRT maps, including conditional walk-law convergence modulo
+time parameterization \cite{GMS}. Their estimates for harmonic
+functions compare discrete and continuum energies up to constants and control
+oscillation \cite{GMSHarmonic}; their invariance principle for ergodic
+scale-free environments develops a broader framework for embedded
+networks \cite{GMS2}. Gwynne and Miller obtain resistance,
+return-probability, and displacement estimates for several infinite
+random planar maps, including spanning-tree-weighted maps \cite{GM}.
+
+Berestycki and Gwynne prove parameterized, quenched convergence to
+Liouville Brownian motion for mated-CRT maps \cite{BG}. Their
+deterministic time scale is an annealed exit-time median, comparable
+to the inverse quantum area of a cell. Whether its ratio to that
+inverse area converges is left open in \cite[Remark~1.3]{BG}.
+Theorem~\ref{thm:main} concerns
+the finite combinatorial tree-weighted map and its stationary
+half-edge walk. It retains the conditional path law together with
+metric and area, and identifies the acceleration as exactly $n$
+for \eqref{eq:continuum-form}.
+
+Our geometric inputs are the companion results in \cite{MetricMaps}.
+The nondegeneration argument adapts the local-flow method in
+\cite[Section~``The canonical conformal structure'']{ConformalMaps}
+to a primal and a dual electrical network. The use of planar duality
+to determine an effective conductivity has classical precedents,
+including Dykhne's exact-conductivity argument \cite{Dykhne}. Here
+we must first construct the energy limit without regularity
+assumptions on the shapes of discrete faces. A separate Green
+estimate in the original finite-map law then controls sources of
+small stationary mass and supplies the missing clock information.
diff --git a/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/build/sections/01-introduction.tex b/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/build/sections/01-introduction.tex
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+\section{Introduction}\label{sec:introduction}
+
+A scaling limit for the metric and area of a random surface does not by
+itself determine the motion of a particle on that surface. The electrical
+conductances of the approximating graphs can change the limiting
+diffusion, even when their measured metric spaces converge. A further
+question is whether the time needed to see the continuum motion has a
+single deterministic normalization.
+
+We resolve this question for planar maps weighted by their numbers of
+spanning trees. The geometric input is the metric-measure limit and the
+contour reconstruction theory of \cite{MetricMaps}. We prove the additional
+convergence of the conditional walk laws, with an exact linear clock.
+The proof separates the identification of the electrical energy from the
+control of the speed measure. This separation is useful because neither
+part follows from metric-measure convergence alone.
+
+\subsection{The model and the continuum process}
+
+A rooted planar map is a finite connected graph embedded in the oriented
+sphere, up to orientation-preserving homeomorphism, with a distinguished
+oriented edge. Loops and multiple edges are allowed. For $n\ge1$, let
+$(M_n,T_n)$ be uniform among rooted maps with $n$ edges decorated by one
+of their spanning trees. Thus the marginal weight of $M_n$ is proportional
+to its number of spanning trees; conditional on $M_n$, the tree $T_n$ is
+uniform. Write $V_n$ for the vertex set and $d_n$ for graph distance using
+all map edges, each of length one. Set
+\begin{equation}\label{eq:degree-measure}
+ \mu_n=\sum_{v\in V_n}\frac{\deg(v)}{2n}\,\delta_v,
+\end{equation}
+where each loop contributes two to the degree.
+
+Conditional on the map, let $X^n$ start from $\mu_n$. At the times of an
+independent rate-one Poisson process it chooses an incident half-edge
+uniformly and moves to its other endpoint. Choosing a loop makes no
+displacement. Its generator is
+\begin{equation}\label{eq:walk-generator}
+ (L_nf)(v)=\frac1{\deg(v)}
+   \sum_{e\text{ a half-edge at }v}
+       \bigl(f(\operatorname{other}(e))-f(v)\bigr).
+\end{equation}
+The measure $\mu_n$ is stationary and reversible for this walk.
+
+Let $(S,h)$ be an ordinary unit-area $\sqrt2$-Liouville quantum sphere,
+with its distinguished points forgotten. We use the ordinary fixed-area
+sphere law of Duplantier, Miller, and Sheffield \cite{DMS}. Its quantum
+area measure $\mu_h$ has total mass one. Write $D_h$ for its Liouville
+quantum gravity metric in a fixed deterministic normalization. Identify
+the underlying conformal sphere with the round sphere. The continuum
+process $B^h$ is the conservative $\mu_h$-symmetric diffusion associated
+with the closure on $L^2(\mu_h)$ of
+\begin{equation}\label{eq:continuum-form}
+ \cE_h(f,g)=\frac12\int_{S^2}
+      \langle\nabla f,\nabla g\rangle_{\mathrm{round}}
+                    \,\dd\vol_{\mathrm{round}},
+ \qquad f,g\in C^\infty(S^2).
+\end{equation}
+The closed form uses quasi-continuous representatives. This is Liouville
+Brownian motion, started with conditional law $\mu_h$; see
+\cite{GRV,Berestycki} and the Dirichlet-form framework in \cite{FOT}.
+Conformal invariance of the two-dimensional Dirichlet integral makes
+\eqref{eq:continuum-form} independent of the conformal coordinate. Its
+factor $1/2$ fixes the continuum clock.
+
+For precision we fix the spatial normalization used throughout. For a
+finite real random variable $Y$, let $m(Y)$ be the unique real solution
+of $\E[\arctan(Y-m(Y))]=0$. Put
+\begin{equation}\label{eq:spatial-normalization}
+ \Delta_n=\max\{1,\diam(V_n,d_n)\},\qquad
+ \Delta=\diam(S,D_h),\qquad
+ a_n=\exp\{m(\log\Delta)-m(\log\Delta_n)\}.
+\end{equation}
+Both centers use unconditional laws, so $a_n$ is deterministic. We use
+the geometric limit of \cite{MetricMaps} in the form
+\begin{equation}\label{eq:metric-premise}
+ a_n\longrightarrow0,
+ \qquad (V_n,a_nd_n,\mu_n)
+       \xrightarrow{\ d\ }(S,D_h,\mu_h)
+ \quad\text{in Gromov--Hausdorff--Prokhorov topology}.
+\end{equation}
+The contour and topological results from that work needed in addition
+to \eqref{eq:metric-premise} are stated in Section~\ref{sec:inputs}.
+
+\subsection{Retaining the conditional path law}
+
+Fix $T>0$. If $(K,d,\mu)$ is a compact metric probability space, let
+$Q$ be a Borel probability measure on $D([0,T],K)$. We regard
+$(K,d,\mu,Q)$ up to isometries transporting both measures. For two such
+quadruples, define $\delta_T$ as the infimum over isometric embeddings
+$i:K\to Z$ and $j:K'\to Z$ into a common complete separable metric space
+of
+\begin{equation}\label{eq:delta}
+ \max\left\{
+ d_H^Z(iK,jK'),\ d_P^Z(i_*\mu,j_*\mu'),\
+ d_P^{D([0,T],Z)}(i_*Q,j_*Q')\right\}.
+\end{equation}
+Here $d_H$ and $d_P$ denote Hausdorff and Prokhorov distances. The last
+term uses the Skorokhod $J_1$ topology: time changes are increasing
+continuous bijections of $[0,T]$, and both their uniform time displacement
+and the uniform spatial discrepancy tend to zero. Each of the three
+distances may be truncated at one. An embedding acts on a path at every
+time.
+
+\begin{samepage}
+\begin{theorem}[Linear clock and conditional laws]\label{thm:main}
+For every finite $T>0$, define
+\[
+ Q_n^T=\Law\bigl((X^n_{nt})_{0\le t\le T}\mid M_n,T_n\bigr),
+ \qquad
+ Q_h^T=\Law\bigl((B^h_t)_{0\le t\le T}\mid h\bigr).
+\]
+With the spatial normalization \eqref{eq:spatial-normalization},
+\[
+ (V_n,a_nd_n,\mu_n,Q_n^T)
+       \xrightarrow{\ d\ }(S,D_h,\mu_h,Q_h^T)
+       \qquad\text{for }\delta_T,
+\]
+through all positive integers $n$. Thus the deterministic clock
+multiplier for \eqref{eq:walk-generator} and \eqref{eq:continuum-form}
+is $c=1$.
+\end{theorem}
+\end{samepage}
+
+The random probability law is part of the state in Theorem~\ref{thm:main}.
+For example, the same argument gives joint convergence of any fixed
+number of conditionally independent stationary walks on the same map:
+in a common embedding, the associated conditional joint laws are the
+finite products of $Q_n^T$. Their weak convergence follows from that of
+$Q_n^T$. This preserves information about the shared environment.
+
+\subsection{Context and related work}
+\input{sections/01-history}
+
+\subsection{Proof strategy and reusable estimates}
+
+For a finite graph embedded in the sphere, write
+\begin{equation}\label{eq:network-energy}
+ \en(u)=\sum_{ab\in E}(u(a)-u(b))^2,
+\end{equation}
+with one term for each unoriented edge, including multiplicities. Loops
+contribute zero. The same convention defines $\enstar$ on the dual graph.
+The two main tasks are to identify the limit of these unnormalized
+energies and to pass from that limit to motion in actual time.
+
+The contour description of the map gives local finite graph pieces in
+conformal coordinates. Section~\ref{sec:inputs} explains why the local
+reconstruction and conditional-kernel calculations of \cite{MetricMaps}
+also apply to extremal lengths and constrained energy minima on these
+pieces. Exact edge incidences matter here: approximation of graph
+distances alone would not identify an energy. The extension retains
+compact numerical outputs before extracting a subsequence, and uses a
+change of measure depending only on the contour inputs.
+
+Section~\ref{sec:nondegeneration} proves bounded-energy cutoff estimates
+without geometric control of individual faces. Its central argument is
+a planar obstruction to simultaneous degeneracy of the primal and dual
+networks. If annular extremal lengths could vanish, low-energy flows
+would have a nonzero limiting average length measure. Locality and
+triviality of field point germs produce transverse primal and dual
+crossing flows at a typical point. Every such pair crosses an edge and
+its dual, contradicting the vanishing traffic norms. The cutoffs then
+yield harmonic compactness.
+
+Section~\ref{sec:energy} identifies every local variational limit as
+$\lambda\int|\nabla u|^2$, with the same positive deterministic $\lambda$
+for the primal and dual. To determine $\lambda$, we compare a condenser
+on an annulus with the period of its rotated current. The comparison in
+one direction gives $\lambda\le1$; recovery of an angular cochain and
+the signed primal--dual pairing give $\lambda\ge1$. Thus the coefficient
+is exactly one in every subsequence. Section~\ref{sec:sphere} transfers
+the result to fixed-area spheres using interior excursion densities and
+removes the single omitted contour endpoint by zero Sobolev capacity.
+
+The remaining speed-measure issue is addressed by the following estimate,
+proved in Section~\ref{sec:green}. Let $H_n$ be the positive unweighted
+graph Laplacian. For $\mu_n(f)=0$, define $G_nf$ by
+$H_nG_nf=\mu_nf$ and $\mu_n(G_nf)=0$, where the product $\mu_nf$ on the
+right denotes its vector of vertex masses. For every $0<\rho<1/2$ there
+are map-measurable random variables $A_n$ with
+$\sup_n\E A_n<\infty$ such that, simultaneously for $\|f\|_\infty\le1$,
+\begin{equation}\label{eq:intro-green}
+ \|G_nf\|_\infty\le A_n\,\mu_n(|f|)^\rho.
+\end{equation}
+Its proof is entirely in the fixed-size map law. A spanning-forest
+identity expresses the centered inverse Laplacian through tree cuts;
+the contour bijection expresses each cut size as one plus a dual-tree
+distance. Dyck-excursion estimates then control the interval sums.
+This estimate can be used independently of the energy identification.
+
+Finally, Section~\ref{sec:dynamics} uses \eqref{eq:intro-green} to obtain
+uniform resolvent convergence and a stopping-time tightness estimate.
+The resolvents identify the conditional path laws by their integrated
+finite-dimensional distributions. The normalization is visible already
+at the discrete level:
+\[
+ -\langle f,nL_nf\rangle_{L^2(\mu_n)}=\tfrac12\en(f).
+\]
+Combining this identity with the coefficient $\lambda=1$ gives the clock
+in Theorem~\ref{thm:main}.
diff --git a/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/build/sections/02-inputs.tex b/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/build/sections/02-inputs.tex
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+\section{Sewing and local electrical observations}
+\label{sec:inputs}
+
+Electrical energy depends on every edge, including parallel edges.
+Consequently, a description that approximates graph distances does not
+by itself suffice for our purpose.  This section explains how the local
+sewing construction of \cite{MetricMaps} retains exact graph pieces and
+gives conditional laws for numerical observations on those pieces.
+The observations may be extremal lengths or constrained energy minima;
+we do not yet assert that they are bounded or identify their limits.
+The construction has three parts.  We recover complete finite graph
+pieces from contour cells, pass their numerical observations to
+conditional limiting laws, and then read those laws from the fields
+in the corresponding spatial regions.
+
+Throughout this section,
+\[
+ \gamma=\sqrt2,\qquad Q=\frac2\gamma+\frac\gamma2=\frac3{\sqrt2},
+ \qquad \chi=\frac1{\sqrt2}.
+\]
+All quantum surfaces use the field and boundary-length normalizations of
+\cite{MetricMaps}.  A compact containment $K\Subset U$ means that $K$ is
+compact and contained in the open coordinate domain $U$.  A buffer for
+$K$ is an open neighborhood whose closure is compactly contained in $U$.
+
+\subsection{The discrete graph and its continuum background}
+
+Let $Z^{\mathrm d}=(L^{\mathrm d},R^{\mathrm d})$ be a bilateral walk
+whose increments are independent and uniform on the four axial unit
+vectors.  For integer times $i\le j$, set
+\begin{equation}\label{inp:vertex-equivalence}
+ i\sim_L j
+ \quad\Longleftrightarrow\quad
+ L^{\mathrm d}_i=L^{\mathrm d}_j
+       =\min_{i\le k\le j}L^{\mathrm d}_k.
+\end{equation}
+The equivalence classes are the primal vertices.  Horizontal steps give
+tree edges between consecutive classes, with the two traversals of a
+tree edge counted as one edge.  A vertical up-step ending at $u$ and
+the vertical down-step ending at $v>u$ are matched when
+\begin{equation}\label{inp:step-match}
+ R^{\mathrm d}_{u-1}=R^{\mathrm d}_v
+      <\min_{u\le k<v}R^{\mathrm d}_k.
+\end{equation}
+Each matched pair contributes one further edge between $[u-1]_L$ and
+$[v-1]_L$.  Distinct matched pairs remain distinct edges when their
+endpoints agree.  In particular loops and parallel edges are retained.
+Interchanging $L^{\mathrm d}$ and $R^{\mathrm d}$ gives the dual graph
+and its complementary spanning tree.
+
+For a finite interval of indices, the \emph{internal graph} uses the
+same vertex relation, all horizontal steps in that interval, and the
+vertical pairs whose two occurrences lie in the interval.  Its vertices
+embed injectively into the full graph: the minimum test between two
+retained times is the same in both constructions.  It need not contain
+every full-graph edge between those vertices.
+
+\begin{proposition}[Sewing and continuum background]\label{inp:sewing}
+The following couplings and normalizations will be used.
+\begin{enumerate}
+\item A uniform rooted map--spanning-tree pair with $n$ edges is encoded
+by the axial walk conditioned to remain in $\mathbb Z_{\ge0}^2$ and
+return to zero after $2n$ steps.  The preceding rules recover every
+primal and dual edge with its incidences.
+\item Give the bilateral walk an independent Poisson clock of rate
+$m$ and multiply its heights by $m^{-1/2}$.  As $m\to\infty$, its
+interpolated contour converges locally uniformly in distribution to
+$Z=(L,R)$, whose independent Brownian coordinates have variance
+$1/2$ per unit time.  The limiting contour has the peanosphere coupling
+with a $\sqrt2$-quantum cone $h$ and an independent whole-plane
+space-filling $\mathrm{SLE}_8$, subsequently parameterized by $\mu_h$.
+Write this curve as $\eta$ and normalize $\eta(0)=0$.
+\item The imaginary field $\widehat h$ is independent of $h$ and is a
+whole-plane GFF modulo $2\pi\chi\mathbb Z$, with uniform phase in
+one period.  It determines the oriented curve before area
+parameterization.  The fields, curve, and contour are retained with
+their joint peanosphere law.
+\item A finite interval of contour increments determines its abstract
+decorated quantum surface, its restricted curve, and the quantum lengths
+of its incoming and outgoing frontiers, modulo conformal coordinates.
+The analogous statement holds for every symmetric interior interval
+of the Brownian-excursion encoding of the ordinary unit-area sphere.
+\end{enumerate}
+\end{proposition}
+
+The discrete assertion is Mullin's encoding \cite{Mullin}, in the form
+of Theorem ``Mullin encoding and its infinite version'' in
+\cite{MetricMaps}.  The continuum assertions are recorded in its
+Theorem ``Peanosphere and interval surfaces'' and its imaginary-field
+normalization.  Their principal continuum inputs are the mating of
+trees theorems of \cite[Theorems~1.9 and~1.11]{DMS} and the
+space-filling construction of \cite{IG4}.
+The use of a Poisson clock makes the two coordinate jump processes
+independent; index time divided by $m$ has the same limiting area time.
+
+We recall the geometric information accompanying this coupling.  Two
+continuum times have the same image under $\eta$ precisely when they
+are equivalent under the relation generated by
+\begin{equation}\label{inp:brownian-fibers}
+ L_s=L_t=\inf_{[s,t]}L
+ \quad\hbox{or}\quad
+ R_s=R_t=\inf_{[s,t]}R,\qquad s\le t.
+\end{equation}
+Except for the common endpoint in the sphere encoding, every class has
+at most three elements and belongs to one coordinate when it is
+nonsingleton.  In a class $t_1<t_2<t_3$, the middle time is a strict
+local minimum; distinct strict local minima have different heights.
+At parameter $8$, the interval cells are closed topological disks.
+The cone curve is proper, so inverse images of compact sets are bounded.
+These facts, including their simultaneous versions on a countable
+collection of intervals and independent Poisson cuts, are the
+topological input accompanying Theorem ``Peanosphere and interval surfaces'' of
+\cite{MetricMaps}; see also the identification description in
+\cite[Section~1.3.1 and Lemma~8.15]{DMS}.
+
+All subsequential arguments below can be placed in a coupling with
+locally uniform contour convergence.  We attach the full continuum
+background conditionally on the limiting contour, independently of
+the additional \emph{raw} discrete observations given that contour.
+Here raw means that the observation uses contour labels, without
+projecting them through $\eta$.  This convention is compatible with
+further extraction.  Indeed, for a bounded background test $F$, its
+conditional expectation $k_F(Z)$ can be approximated in $L^1$ of the
+fixed Brownian marginal by bounded continuous functions of $Z$.
+Multiplying by any bounded raw test leaves the approximation error
+uniformly bounded by the same $L^1$ error.  Joint weak convergence
+therefore preserves the conditional sampling identity.  Spatially
+projected observations will depend on both the raw data and the
+background; we only make this conditional sampling assertion before
+projection.
+
+\subsection{Projections and planar topology}
+
+Regard the map together with its dual as a cell decomposition of a
+surface, subdivided into triangles at its corners.  Assign a primal or
+dual vertex the point $\eta(t)$ corresponding to any of its visits.
+All choices at one vertex, and all corners of an incident triangle,
+have uniformly vanishing discrepancy.  Otherwise a subsequence of
+visit times would limit to a discrete equality or matching relation
+whose endpoints have distinct continuum images, contradicting
+\eqref{inp:brownian-fibers}.  Join nearby assigned points by short
+coordinate arcs and fill the triangles in small disks.  This gives
+continuous projections $\pi_m$ with vanishing projected triangle mesh.
+On the plane this construction is made in a bounded window containing
+the region under consideration, after preserving its necessary stars
+and capping the window.  This protection is obtained directly from
+lower contour visits, as in the proof below; it does not use a local
+instruction graph or any numerical-array limit.
+
+\begin{proposition}[Topological comparison]\label{inp:topology}
+In the contour coupling, the projections $\pi_m$ may be chosen so that
+the following conclusions hold on the sphere and on every buffered
+compact part of the plane.  Each conclusion is eventual for each
+fixed choice of margins.
+\begin{enumerate}
+\item If the projections of two primal vertices approach each other,
+they can be joined by primal paths with arbitrarily small projected
+diameter.  This is uniform over all lifts in a fixed compact region.
+The same statement holds on the dual graph and for surface paths.
+In particular fixed tubes with clearance contain paths following them
+on either graph.
+\item All lifts of a sufficiently small closed disk or other compact
+well can be included in a connected set over any fixed open
+enlargement.  A loop projecting into a small disk can be filled over
+a slightly larger disk, provided that both have the prescribed
+clearance.
+\item The projection has degree $+1$ or $-1$ in a compact spherical
+completion.  Robustly transverse primal and dual tube crossings
+intersect at an edge and its dual.  Projected circuits with nonzero
+winding difference about two separated wells separate all the
+corresponding lifted fibers.
+\item Integer winding differences about two wells agree upstairs and
+downstairs up to the common degree sign, when loops and wells have
+fixed separation.  These statements retain repeated edge and face
+incidences.
+\end{enumerate}
+\end{proposition}
+
+\begin{proof}
+The construction and the first three assertions are the construction
+preceding Lemma ``Uniform topological joining'', that lemma,
+Lemma ``Exterior connection and small-loop filling'', Proposition
+``Degree and robust planar topology'', and Lemma ``Separation of
+entire fibers'' in \cite{MetricMaps}.
+We explain their applicability to both graphs and to the plane.
+
+The hypotheses are uniform contour convergence, vanishing projected
+cell mesh, and approximation of the limiting fiber generators by
+discrete links with small projected connecting paths.  For a Brownian
+excursion above a level, use last and first crossings of nearby levels
+on the two sides of an interior time.  Their discrete analogues
+approximate the chord.  At a triple, apply this on the two excursions
+on either side of its middle minimum.  The fiber description above
+therefore supplies every generator required for topological joining.
+Interchanging the coordinates gives the same argument for the dual.
+Applying planar separation to the two embedded graph paths gives the
+mixed crossing assertion: a primal path and a dual path can meet only
+through a primal edge and its dual, with the usual corner subdivision
+making the intersection literal on the surface.
+
+For a bounded plane region, choose a window containing all its visits
+and its fixed additional star layers.  Both contour coordinates make
+strictly lower visits on both sides of a sufficiently large window.
+Lemma ``Protected capping of a bounded window'' in
+\cite{MetricMaps} then supplies a spherical completion preserving
+all these incidences.  Its continuum completion is absolutely
+continuous with respect to a Brownian excursion, which supplies the
+sphere quotient needed for the preceding topological results.
+
+For the integer assertion, connect all lifts of each well away from
+the tested loop, using the first two assertions.  A two-chain bounded
+by that loop has constant multiplicity along each such connected
+fiber.  Projection multiplies the difference of its multiplicities
+at the two fibers by $\deg\pi_m=\pm1$.  This is exactly the equality
+of winding differences.  Equivalently, one can use relative homology
+after a simplicial perturbation inside the fixed margins.
+\end{proof}
+
+\subsection{Finite instruction graphs and numerical tests}
+
+For a finite multigraph $G$, write
+\begin{equation}\label{inp:graph-energy}
+ \en_G(u)=\sum_{ab\in E(G)}(u(a)-u(b))^2.
+\end{equation}
+The sum has one term for each unoriented edge; loops contribute zero.
+An energy test is the infimum of \eqref{inp:graph-energy} subject to
+finitely specified value constraints at vertices of a specified
+subgraph.  We allow $+\infty$ for an infeasible test.  An extremal-length
+test for a family $\mathcal P$ of graph paths is
+\begin{equation}\label{inp:el-test}
+ \EL_G(\mathcal P)=
+ \sup_{w:E(G)\to[0,\infty)}
+ \frac{\bigl(\inf_{P\in\mathcal P}\sum_{e\in P}w(e)\bigr)^2}
+      {\sum_{e\in E(G)}w(e)^2},
+\end{equation}
+with the usual conventions for empty families and constant paths.
+Only nontrivial families will be used in the crossing arguments.
+
+The graph on which one computes these quantities must be specified
+before taking a limit.  We use the finite instruction graphs of
+Lemma ``Finite-band reconstruction'' in \cite{MetricMaps}, recalled next.
+Their inputs consist of finite strings and relative heights.  This
+description will also show why the conditional-law proof applies to
+our numerical tests.
+
+\begin{definition}[Finite packages]\label{inp:packages}
+A \emph{master block} consists of two increment strings on a time
+interval $[0,\ell_b]$, starting at zero, together with a finite list
+of internal cuts.  A finite package contains finitely many master
+blocks.  For each coordinate $a\in\{L,R\}$ choose a forest on these
+blocks and record its height differences
+$o^a_{bd}=s^a_d-s^a_b$.  The values $s^a_b$ are thereby determined
+within each forest component up to one additive constant.
+
+A test chooses finitely many \emph{active windows}, each between
+recorded cuts of a master block.  Each active window initially has its
+own copy of the contour vertices.  Its starting heights and increment
+strings are obtained by restriction of its master block.  Enable the
+ordinary sewing inside each active window.  The additional
+cross-window links are specified by the band rules below.  A package
+can retain several tests on the same master strings.
+\end{definition}
+
+For an active window $b$, write $z^a_b$ for its increment string,
+$s^a_b$ for its starting height, and $m^a_b=\min z^a_b$.
+Its incoming and outgoing record arcs are the times satisfying,
+respectively,
+\[
+ z^a_b(t)=\min_{0\le u\le t}z^a_b(u),
+ \qquad
+ z^a_b(t)=\min_{t\le u\le\ell_b}z^a_b(u).
+\]
+A band from window $b$ to window $d$ permits outgoing-to-incoming
+links with common absolute height $H$ satisfying
+\begin{equation}\label{inp:bands}
+ \alpha\le H-q\le\beta,
+ \qquad H\le s^a_w+\min_J z^a_w\quad(w,J\text{ in the cap list}).
+\end{equation}
+Here $\alpha<\beta$ are fixed rational parameters, $q$ is the height
+at a recorded anchor cut, and each $J$ is a recorded subwindow.
+The forest supplies every relative height appearing in the rule.
+Each selected minimum is evaluated from the discrete string itself.
+
+In the primal graph, an $L$-band identifies its permitted vertices.
+An $R$-band is a rule for pairs of oriented step occurrences: an
+outgoing up-step and an incoming down-step must carry the same unit
+height strip, and the \emph{whole strip} must satisfy the band and cap
+conditions.  Such a pair contributes an edge; it does not impose an
+$R$-vertex identification.  The dual uses the exchanged convention.
+If a tree edge is represented in several retained pieces it is counted
+once after the corresponding vertex identifications: its unordered
+endpoint pair identifies it, since a tree has no parallel edges.
+Each additional matched-step edge is counted once for that matched
+pair, even when another such pair has the same endpoints.
+
+The same active windows and band instructions continue to define an
+old test when cuts are added or master blocks are merged.  In the
+merged string, old starting heights are evaluated at the old cuts;
+they are no longer independent offset parameters.  A graph allowed
+to use the entire merged interval is a separate new test.  In
+particular an old test cannot be recovered by merely confining paths
+in that larger graph, since the larger graph may use newly enabled
+edges witnessed in a formerly omitted gap.
+
+For each instruction list we retain a countable family of numerical
+outputs.  A vertex is selected when one of its visits lies in a finite
+union of intervals of relative time.  Endpoints can be recorded cuts
+or rational fractions between cuts.  Such selectors specify the
+subgraph, its ports, and any confinement condition.  Rational value
+intervals at selected visits specify energy constraints; repeated
+visits impose simultaneous constraints on the same vertex.  The
+output library includes both graphs, finite joint lists of tests,
+and tests recomputed on union graphs.  Inapplicable tests receive a
+fixed default value.  Every nonnegative output is compactified in
+$[0,\infty]$, for example by $x\mapsto x/(1+x)$.
+We can retain outputs divided by any prescribed positive deterministic
+factor $b_m$, including factors tending to zero.
+
+This gives compact countable products of numerical observations.  The
+master strings, durations, cuts, and offsets remain separate input
+coordinates.  Along any sequence $m\to\infty$, one may therefore
+extract joint limits of all the raw tests and the locally convergent
+contours.  The issue is now to relate these formal tests to complete
+pieces of the actual map.
+
+\subsection{Exact cores and the selector sandwich}
+
+Add independent Poisson cuts in area time, using nested refinements.
+A cell is the image of a consecutive cut interval under $\eta$.
+Only cells compactly contained in the current testing domain will be
+used.  Their local data consist of their oriented traversals, quantum
+area parameterizations, and tagged incoming and outgoing frontier
+lengths, with repeated frontier occurrences retained.
+A \emph{run} is a finite union of consecutive cells, viewed with its
+concatenated traversal and tagged incoming and outgoing frontiers.
+For any finite selection of cells, its maximal runs are the maximal
+consecutive groups in that selection.
+
+Here is how these local data provide the relative heights used in a
+package.  Fix a run $b$ and one coordinate $a\in\{L,R\}$.  For an
+initial segment of the run, the incoming frontier length is the drop
+from its starting contour height to the minimum on that segment;
+the outgoing length is the rise from that minimum to its ending
+height.  Their difference is therefore the contour increment.
+Reading these two tagged lengths at every rational relative area
+time, and using continuity, recovers the whole increment path
+$z_b^a$ from the local traversal.  Put
+$m_b^a=\min z_b^a$ and $e_b^a=z_b^a(\ell_b)$.
+
+The incoming frontier of the whole run has relative-height range
+$[m_b^a,0]$, and the outgoing frontier has range $[m_b^a,e_b^a]$.
+For a tagged point $p$ on either frontier, define its
+\emph{relative height} $\vartheta_b^a(p)$ by starting with value
+$m_b^a$ at the minimum endpoint and increasing by quantum boundary
+length along that frontier toward its tip.  Thus the incoming tip
+has height $0$ and the outgoing tip has height $e_b^a$.
+This orientation is by increasing height, irrespective of the
+orientation in which the contour traverses the frontier.  Equivalently,
+if $p$ is represented by a record time $t$, then
+$\vartheta_b^a(p)=z_b^a(t)$.
+
+A \emph{directed shared seam} $b\to d$ is a common frontier subarc
+of positive quantum length, tagged as outgoing for $b$ and incoming
+for $d$, in the same coordinate.  At a paired point $p$ of this
+seam, the two absolute heights agree.  Hence
+\begin{equation}\label{inp:seam-offset}
+ s_b^a+\vartheta_b^a(p)=s_d^a+\vartheta_d^a(p),
+ \qquad
+ s_d^a-s_b^a=\vartheta_b^a(p)-\vartheta_d^a(p).
+\end{equation}
+The right side is constant along the seam and is determined by the
+two local tagged frontier parameterizations.  One may read it at an
+interior point, so the construction does not require selecting or
+assigning a height to a branch endpoint.  Taking a spanning forest
+of such seams supplies exactly the offsets in
+Definition~\ref{inp:packages}.  Their outgoing and incoming tags
+also specify which linked run precedes the other.  A chronological
+order between runs without a declared seam is unnecessary.
+
+There is one endpoint configuration requiring an exact comparison.
+Suppose three visits of one coordinate meet at height $H$, in three
+disjoint runs $b,w,d$ in that order, with the middle visit a strict
+local minimum.
+The outer runs $b,d$ share a positive interval of seam heights below
+$H$.  Above $H$, the two positive seam intervals are $b\to w$ and
+$w\to d$; see Figure~\ref{inp:triple-figure}.  Formula
+\eqref{inp:seam-offset} on these intervals recovers $s_w^a$ relative
+to the outer starts.  The tag directions identify $w$ as the middle
+run.  It lies between the two outer visits and contains the middle
+contact.  The contour stays at least $H$ throughout that interval
+by \eqref{inp:brownian-fibers}, and attains $H$ in $w$.  Hence
+\begin{equation}\label{inp:exact-cap}
+                       H=s_w^a+m_w^a.
+\end{equation}
+A band for the outer pair may extend above the continuum value
+$H$, but it must include the cap in \eqref{inp:bands} evaluated
+as $s_{w,m}^a+\min z_{w,m}^a$ at every discrete resolution.
+It is this exact minimum, rather than a rounded approximation of
+$H$, that excludes the outer-to-outer links above the branch.
+
+\begin{figure}[htbp]
+\centering
+\begin{tikzpicture}[x=0.85cm,y=0.9cm,>=Stealth,
+                    every node/.style={font=\small}]
+ \fill[blue!7] (-0.15,2.1) rectangle (6.15,2.55);
+ \fill[blue!7] (-0.15,0.4) rectangle (6.15,0.85);
+ \draw[->,gray] (-1.7,0) -- (-1.7,3.1) node[above] {height};
+ \draw[thick] (0,0.15) -- (0,2.95) node[above] {$b$};
+ \draw[thick] (3,1.5) -- (3,2.95) node[above] {$w$};
+ \draw[thick] (6,0.15) -- (6,2.95) node[above] {$d$};
+ \draw[->,blue!65!black,thick] (0.08,2.3) -- (2.92,2.3);
+ \draw[->,blue!65!black,thick] (3.08,2.3) -- (5.92,2.3);
+ \draw[->,blue!65!black,thick] (0.08,0.62) -- (5.92,0.62);
+ \node[above] at (1.5,2.3) {$b\to w$};
+ \node[above] at (4.5,2.3) {$w\to d$};
+ \node[below] at (3,0.62) {$b\to d$};
+ \draw[densely dashed] (-1.9,1.5) -- (6.4,1.5);
+ \node[left] at (-1.9,1.5) {$H$};
+ \fill (3,1.5) circle (2pt);
+ \node[anchor=west,fill=white,inner sep=2pt] at (3.25,1.73)
+            {$s_w^a+m_w^a$};
+ \draw[red!65!black,very thick] (-0.65,0.45) -- (-0.65,2.5);
+ \draw[red!65!black] (-0.85,0.45) -- (-0.45,0.45);
+ \draw[red!65!black] (-0.85,2.5) -- (-0.45,2.5);
+ \node[anchor=east,rotate=90,red!65!black] at (-0.93,2.45)
+            {uncapped band};
+\end{tikzpicture}
+\caption{A triple contact in height coordinates.  The shaded strips
+indicate positive intervals of shared quantum-boundary heights;
+they are not drawings of Euclidean seams.  The outer pair is linked
+below $H$, while above $H$ each outer run is linked to the middle
+run.  The middle run's exact discrete minimum caps the outer-pair
+band.}
+\label{inp:triple-figure}
+\end{figure}
+
+We use bands lying inside these shared height intervals with positive
+margins at their effective endpoints, with two exceptions prescribed
+by the record data.  At a natural lower record minimum the rational
+band starts below the minimum and is clipped by the exact record
+condition.  At an upper endpoint with the three-seam configuration,
+the band is clipped by the exact selected minimum
+\eqref{inp:exact-cap}.  These are the admissible band choices in
+the proof of Lemma ``Finite-band reconstruction'' of
+\cite{MetricMaps}.  All their tests use the local tagged frontiers,
+relative increments, and quantum lengths just described.
+
+\begin{samepage}
+\begin{lemma}[Complete graph pieces in a buffer]\label{inp:complete-core}
+Fix $K\Subset U$ in a coordinate chart.  Almost surely, there is a
+finite prescription, selectable from the local cells in $U$ and their
+frontier data, with the following property.  In a neighborhood of $K$,
+its instruction graph agrees with the actual discrete graph, including
+vertex equalities and all edge incidences, for all sufficiently large
+resolutions in the coupling.  The same prescription can retain both
+the primal and the dual graph.  One can require any fixed smaller
+collar of this neighborhood before restricting the tests to it.
+
+The prescriptions can be taken from the countable instruction scheme
+above and chosen measurably using the fields and spatial cut marks
+in $U$, with an arbitrarily small open buffer.  The assertion also
+holds in the interior of a fixed interval cell, using only its
+internal increment string.
+\end{lemma}
+\end{samepage}
+
+\begin{proof}
+We use the exact-incidence part of the proof of Lemma ``Finite-band
+reconstruction'' in \cite{MetricMaps}.  We give the argument because
+exactness, rather than an approximation of path costs, is essential
+for energy tests.
+
+Choose $r>0$ so that the closed $3r$-neighborhood of $K$ lies in $U$.
+Properness bounds all its visit times in a finite window.  Refine the
+Poisson cuts until every cell meeting this neighborhood has diameter
+less than $r$.  Select all cells meeting the closed $2r$-neighborhood
+of $K$ and merge consecutive selected cells into maximal runs.  These
+runs are disjoint intervals; all visits sufficiently near $K$ lie in
+them.  Their endpoints stay away from $K$, since otherwise the
+neighboring cell would also have been selected.  Local traversal
+data identify the selection: first inspect a slightly larger buffer
+and require the portions around every visit to the smaller compact
+to be flanked by marks while they remain in that buffer.  Properness,
+continuity, and refinement make this condition hold eventually.
+No global index for the retained runs is part of this inspection.
+
+Consider one coordinate and two selected runs $b<d$ in chronological
+order.  An outgoing point of $b$ and an incoming point of $d$ at
+height $H$ are linked exactly when
+\begin{equation}\label{inp:gap-test}
+ H\le\min_{\text{between the two runs}} Z^a.
+\end{equation}
+The same assertion holds discretely, with the full-strip convention
+for a matched step.  Split the intervening interval into selected
+runs and omitted gaps.  If a limiting link projects near $K$, any
+intervening time at its height represents that same continuum point.
+It must therefore belong to a selected cell.  Thus every omitted-gap
+minimum is strictly above the height of that link.  The relevant
+endpoint pairs form a compact set, and there are only finitely many
+gaps.  Their positive slack is consequently uniform: a sequence of
+slacks tending to zero would limit to a contact in an omitted gap,
+contrary to the selection.
+
+An obstruction without slack can only be a minimum of a selected run.
+At a triple contact, the three directed seams in
+Figure~\ref{inp:triple-figure} identify the middle run and recover
+its offset through \eqref{inp:seam-offset}.  Formula
+\eqref{inp:exact-cap}, evaluated on the discrete string, supplies
+the cap in \eqref{inp:bands}.  At a natural
+lower seam endpoint, choose the lower rational band endpoint below
+the record minimum; the exact incoming or outgoing record condition
+then clips the band at that minimum.  This retains links at the
+minimum as well as links at nearby lattice heights.
+
+For every pair of runs, cover the compact set of relevant link heights
+by finitely many such bands.  Choose each band to be admissible over
+its entire effective height interval: that interval lies in a genuine
+shared seam, its omitted-gap comparisons have positive slack, and
+every allowed non-strict cap is the exact selected minimum just
+described.  The natural record conditions provide the exact clipping
+at lower seam endpoints.  Include the positive seam intervals
+needed to read the relative heights, and retain a spanning forest
+of these comparisons in each coordinate.  All needed offsets are
+sums along that forest.  Seam directions specify the order of
+linked runs and distinguish each middle cap; the instructions
+do not need an order between unrelated runs.  Each band retains every discrete link
+satisfying its rule.  Uniform contour convergence, the positive
+omitted-gap slack, and the exact selected minima show two things:
+every actual link near $K$ belongs to one of these bands eventually,
+and every permitted link in those effective bands is an actual link
+eventually.  In particular the latter assertion applies to any
+permitted link with one endpoint over a smaller core collar: its
+two limiting endpoints lie on the same shared-seam point and hence
+remain inside the captured collar.  The
+assertions hold simultaneously for all links, regardless of their
+number.
+
+Vertex equality is the same first-coordinate minimum test, so it is
+captured as well.  Start a chain of enabled identifications at a
+core representative.  Its first link is an actual identification
+by the preceding band validity.  All visits to that actual vertex
+have uniformly close projections, independently of the instruction
+graph, and lie in the captured collar.  Inductively the next enabled
+link is therefore actual as well and remains at that vertex.
+Thus a peripheral chain cannot introduce an additional equality
+at the core.  Horizontal tree traversals with the same identified
+endpoint pair give one tree edge; no two distinct tree edges have
+that pair.  Added edges are instead identified by their matched
+step-occurrence pair, preserving distinct parallel edges.  Hence the
+result is equality of graphs with incidences, not merely equality
+of their path relations.  Applying the argument to the other
+coordinate gives the dual conclusion.
+
+All seam, cap, and positive-margin conditions are measurable in the
+oriented local cells and frontier lengths.  Rational bands, integer
+lower bounds on positive margins, and finite forest choices give a
+countable selection rule.  Taking a further collar before the final
+restriction gives the stated neighborhood of $K$.  Inside a fixed
+interval cell, every visit to a compact subset of its interior lies
+in that interval with a time margin; the same argument then uses
+only its internal string.
+\end{proof}
+
+An arbitrary union of overlapping formal window copies need not be
+an injective copy of a map subgraph.  The lemma asserts the existence
+of \emph{complete} prescriptions, obtained from disjoint maximal
+runs or from a freshly computed merged graph.  All comparisons to
+the actual map below use these complete prescriptions.
+
+\begin{lemma}[Spatial and value selectors]\label{inp:sandwich}
+In the setting of Lemma~\ref{inp:complete-core}, suppose
+$K\subset O$ with $K$ compact, $O$ open, and
+$\overline O\Subset U$.  There is an interval selector on a complete
+prescription which, eventually, includes every vertex projected into
+$K$ and includes only vertices projected into $O$.  Several such
+choices may be made together, for confinement and for each port set.
+
+Let $u$ be continuous on a neighborhood of $\overline O$.  For any
+two uniform value tolerances $0<\delta_-<\delta_+$, finite rational
+interval constraints on selected visit labels can be placed between
+the constraints $|v-u\circ\pi_m|\le\delta_-$ on a slightly larger
+compact set and $|v-u\circ\pi_m|\le\delta_+$ on a smaller one.
+All these comparisons hold simultaneously for every continuous
+target through the same retained test library.
+\end{lemma}
+
+\begin{proof}
+On the finitely many selected windows, the limiting parameterizations
+are continuous.  Cover the inverse image of $K$ by finitely many
+relative-time intervals with closures projecting inside $O$.
+Their endpoints may be chosen from the prescribed dense countable
+relative-time library.  Uniform projection accuracy and the complete
+collar of Lemma~\ref{inp:complete-core} give the desired inclusions.
+Select a vertex when any of its visits is in this union.  The same
+argument applies to each port, while vanishing edge mesh allows an
+induced-edge rule within the complete prescription.
+
+For value constraints, refine the intervals until the oscillation of
+$u\circ\eta$ on each is less than a fixed fraction of
+$\delta_+-\delta_-$.  Choose a rational value interval between the
+two allowed tolerances.  Impose it at every selected visit in that
+interval, retaining all simultaneous constraints when a vertex has
+several visits.  The strict tolerance gap and uniform projection
+error give the two implications.  The construction chooses among
+an already countable collection of tests; it does not add a new
+exceptional null set for each target $u$.  Compact exhaustion and
+rational positive tolerances therefore yield common measurable
+versions for all continuous targets.
+\end{proof}
+
+The selector lemma compares variational values with strict margins.
+It does not assert convergence for a prescribed sharp boundary.
+For a crossing event, for example, choose smaller port and confinement
+constraints and then relaxed constraints, both strictly inside the
+desired geometry.  An intermediate selector produces actual paths
+in the relaxed test, and any path family having the smaller clearance
+can be included by a sufficiently relaxed selector.  Thus existence
+of a strictly fitting test with finite limiting extremal length is
+independent of the particular complete prescription.  Exhausted
+energy values have the same property.  These conventions also make
+readouts with a varying spatial center jointly measurable: all
+membership and value comparisons use finite selectors with a strict
+gap, followed by countable exhaustion.
+
+\subsection{The conditional law of enlarged outputs}
+
+We now identify the conditional law of the finite numerical arrays.
+The proof is the finite-experiment argument of \cite{MetricMaps},
+with the output vector enlarged before taking a subsequence.
+The enlargement is harmless for a specific reason: every change
+of density in that argument is a function of the inputs alone.
+
+\begin{lemma}[Conditional transfer for numerical outputs]
+\label{inp:kernel-transfer}
+Consider finitely many packages as in Definition~\ref{inp:packages}.
+Different packages use disjoint master blocks and disjoint offset
+forests.  Sample their Poissonized strings independently; give their
+durations, cuts, and offset vectors smooth positive densities, with
+offsets discretized on the height lattice.  Retain any finite or
+countable vector of compactified numerical tests of their instruction
+graphs, including tests on both graphs.
+
+Along a common subsequence the reference laws have disintegrations
+\[
+ \nu_r(\dd x_r)K_r(x_r,\dd y_r),
+\]
+where $x_r$ denotes the durations, cuts, increment strings, and
+forest offsets, and $y_r$ denotes the numerical outputs.
+Insert these blocks in the actual bilateral walk with positive
+omitted gaps.  Conditional on the entire limiting contour and its
+cuts, their output law is
+\begin{equation}\label{inp:conditional-product}
+             \bigotimes_r K_r(x_r,\dd y_r).
+\end{equation}
+The kernels are invariant under changes of positive reference
+densities and under relabeling blocks or changing the forest basis.
+They are compatible with jointly retained refinements and merged
+tests.  These identities hold after selection from the countable
+instruction library using the limiting contour and cuts.
+\end{lemma}
+
+\begin{proof}
+This is the argument of Lemma ``Finite reference experiment and conditional
+transfer'' in \cite{MetricMaps}.  We include its measure calculation
+to specify its application to potentially discontinuous variational
+outputs.
+
+Write $\epsilon_m=m^{-1/2}$.  For one coordinate, order all selected
+blocks chronologically as $1,\ldots,k$.  Let $s_i$ be their starts,
+$e_i$ their net increments, and $g_i>0$ the intervening durations.
+Complete the union of the package forests to a tree $T$ on these
+blocks.  Give its extra offsets independent positive densities.
+If the density on an edge $e$ is $\rho_e$, use lattice mass
+\[
+ r_{m,e}(a)=\int_{a-\epsilon_m/2}^{a+\epsilon_m/2}
+                      \rho_e(u)\,\dd u,
+ \qquad a\in\epsilon_m\mathbb Z.
+\]
+Let $p_t(j)$ be the transition mass of one coordinate of the
+unscaled rate-one axial Poisson walk, so its variance rate is
+$\sigma^2=1/2$.  The exact density of the actual relative starts
+with respect to these reference offsets is
+\begin{equation}\label{inp:density}
+ W_m^a=
+ \frac{\displaystyle\prod_{i=1}^{k-1}
+       p_{mg_i}\bigl(\epsilon_m^{-1}(s_{i+1}-s_i-e_i)\bigr)}
+      {\displaystyle\prod_{e\in T}r_{m,e}(a_e)}.
+\end{equation}
+Indeed, tree differences and consecutive differences are related
+by an integer linear map with integer inverse, obtained by summing
+along paths in $T$.  Its determinant is $\pm1$, so there is no
+lattice multiplicity.  Multiply \eqref{inp:density} for the two
+coordinates and include the density ratio for the durations and
+cuts to obtain $W_m$.
+
+On compact input sets with all gaps bounded below by a positive
+constant, the local central limit theorem gives
+\[
+ \epsilon_m^{-1}p_{mt}(j)
+ \longrightarrow
+ (2\pi\sigma^2t)^{-1/2}
+       \exp\{-x^2/(2\sigma^2t)\},
+ \qquad x=\epsilon_m j,\quad \sigma^2=\tfrac12,
+\]
+uniformly in bounded $x$ and in such $t$.  The numerator and
+denominator of \eqref{inp:density} have the same number of lattice
+factors.  They cancel, leaving uniform convergence $W_m\to W$ to
+a continuous finite function on these compact sets.
+
+Conditional on the selected inputs, the omitted strings are
+independent bridges and are independent of the numerical outputs.
+Their rescaled laws converge uniformly on these input sets to
+Brownian bridge laws.  For completeness, their finite-dimensional
+densities are ratios of transition masses, so the preceding local
+limit theorem applies.  On the first half of a bridge the density
+relative to the free walk is bounded by
+\[
+ \frac{\sup_j p_{m(t-u)}(j)}{p_{mt}(j_0)}\le C,
+ \qquad u\le t/2,
+\]
+when $t$ and $\epsilon_mj_0$ range in the chosen compact sets.
+The numerator is at most $C\epsilon_m$ and the denominator at
+least $c\epsilon_m$.  Free-walk tightness and the reversed argument
+on the other half prove uniform bridge tightness.  The same applies
+to a finite list of gaps and to exterior contour portions on a
+bounded window.
+
+The full independent reference experiments converge jointly to the
+product of their limits, because their output spaces are compact
+and their input marginals converge.  Let $a$ denote the completion
+offsets and $z$ the omitted bridges.  Integrating bounded continuous
+bridge tests first, the preceding uniform convergence gives the
+limiting actual law
+\begin{equation}\label{inp:weighted-law}
+ W(x,a)\,\nu(\dd x,\dd a)\,
+       \mathcal B(x,a;\dd z)\prod_rK_r(x_r,\dd y_r).
+\end{equation}
+The weight and the bridge law depend only on contour inputs.
+The complete contour and cuts recover $(x,a,z)$, so disintegration
+of \eqref{inp:weighted-law} gives \eqref{inp:conditional-product}.
+
+To remove the compact restrictions, first keep all finitely many
+limiting gaps positive, then bound endpoints and offsets, and
+finally restrict the block paths to compact sets.  Under the actual
+law the excluded probability tends to zero by finite-window contour
+tightness.  If $P_m$ is the actual law and $Q_m$ the reference law,
+the exact identity
+\[
+             \E_{Q_m}[W_m\1_{A^c}]=P_m(A^c)
+\]
+controls the omitted density mass.  No global bound on $W_m$ is
+required.  Exhaustion, followed by a monotone-class argument, gives
+the assertion for bounded measurable tests and for the entire
+bilateral contour.
+
+A change of reference density adds another input-only density
+ratio, continuous in the limit on compact sets, and thus leaves
+$K_r$ unchanged.  Forest bases are related by unimodular integer
+maps.  The pushed-forward lattice cells and the directly discretized
+cells have the same volume and vanishing diameter.  Their mass
+ratios converge to one on compact sets; tightness makes the two
+lattice laws converge in total variation.  This comparison remains
+valid after applying a discontinuous graph-output map.  Absolute
+time labels and the order of unrelated blocks occur only in
+$W_m$, and disappear from the conditional output law.
+
+For merged tests one must also check the old output.  Begin with
+the old independent blocks, their offsets, and their old numerical
+outputs.  Insert the formerly omitted positive-duration gaps by
+the exact density and bridge construction.  The resulting string
+has the new master-block law.  Recompute the old graph from its
+unchanged active windows and bands, and compute the new union graph
+as a separate output on that same string.  Formula
+\eqref{inp:weighted-law}, applied before conditioning on the new
+master string, proves the old conditional marginal.  It never
+evaluates a previously chosen kernel on a forced-offset diagonal.
+Windows with a common endpoint were already within one maximal
+run, so no positive-density argument is applied to a zero-length
+gap.  Jointly retaining finite vectors proves consistency for all
+refinements and unions, and then for the countable library.
+
+Every step used only compactness of the output coordinates and
+their measurability from their own package.  Thus it applies to
+the energy and extremal-length tests, with any deterministic
+normalization.  It does not require continuity of those outputs
+or preservation of conditional independence under arbitrary weak
+limits.  Finally, after conditioning the identities hold for all
+countably retained templates on one event; contour- and
+cut-measurable selection from them is therefore permitted.
+\end{proof}
+
+\subsection{Spatial locality and changes of quantum units}
+
+The preceding kernels are indexed by contour blocks.  To use them
+in disjoint spatial regions, their inputs must be readable from
+the fields in those regions.  This is where oriented cells and
+relative frontier heights enter.
+
+\begin{proposition}[Local numerical-array kernels]\label{inp:local-kernels}
+On each open coordinate domain, take the limits of the numerical
+tests on complete buffered prescriptions, with relative labels
+localized to their cells.  Conditional on $(h,\widehat h)$, the
+resulting local arrays admit kernels depending only on the fields
+in an arbitrarily small open enlargement of the testing region.
+Arrays using disjoint such enlargements are conditionally
+independent given the complete fields.
+
+One may retain the local Poisson marks.  Their conditional laws are
+the corresponding Poisson processes with intensity proportional
+to $\mu_h$, and their restrictions to disjoint domains are
+independent.  The kernels are consistent under restriction,
+common refinements, and compact exhaustion.
+\end{proposition}
+
+\begin{proof}
+Proposition ``Background locality'' in \cite{MetricMaps}
+states that $\widehat h$ in a buffer determines the oriented curve
+segments through the testing region.  The restriction of $h$
+gives their area parameterizations and their directed frontier
+lengths.  These data determine the locally retained marked cells.
+For a segment from time $s$ to time $t$, its coordinate increment
+is the outgoing frontier length minus the incoming frontier
+length of that coordinate, with tagged multiplicities retained:
+\begin{equation}\label{inp:frontier-increments}
+ Z^q_t-Z^q_s
+   =\nu_h(\partial_q^{\mathrm{out}}\eta([s,t]))
+      -\nu_h(\partial_q^{\mathrm{in}}\eta([s,t])),
+ \qquad q\in\{L,R\}.
+\end{equation}
+Formula \eqref{inp:frontier-increments} reads
+the increments without an absolute contour height.  Positive
+shared seams read the forest offsets, and the three-seam
+configuration in Lemma~\ref{inp:complete-core} reads the offsets
+needed at a minimum cap.  Durations and relative cuts are local
+area data.  Thus all four types of input in
+Lemma~\ref{inp:kernel-transfer} are local.
+
+For finitely many prescriptions in disjoint spatial domains,
+merge within each domain.  Different domains use disjoint cells;
+two runs from different domains cannot share a cut, since its
+image would belong to both compactly contained cells.  Their
+intervening chronological gaps are therefore positive even if
+the runs interleave in time.  Lemma~\ref{inp:kernel-transfer}
+gives their product law conditional on the contour and cuts.
+Its invariance under labels and forest bases lets us enumerate
+the cells locally without retaining a global cut index.
+
+Adjoin the continuum background by the conditional sampling
+convention above.  The product formula remains valid upon
+conditioning additionally on the fields.  Its factors are now
+measurable functions of the local fields and spatial marks,
+by the preceding input reconstruction.  Conditional expectation
+therefore gives the same product formula given the complete
+fields and spatial marks.  This is exactly the argument of
+Proposition ``Spatial product kernels'' in \cite{MetricMaps}, applied to the
+enlarged output vector of Lemma~\ref{inp:kernel-transfer}.
+
+The time cuts are independent of the continuum background.
+Area parameterization and the Poisson
+mapping theorem give their spatial conditional intensity
+$\rho\mu_h$ for a layer of time intensity $\rho$.
+Write $\Pi_i$ for the marks in the $i$th domain and
+$K_i^{\mathrm{marks}}(h,\widehat h,\Pi_i;\cdot)$ for the
+conditional output kernel already obtained.  Given the fields,
+the marks have independent Poisson laws, and hence the output
+law after removing them is
+\[
+ \bigotimes_i\int K_i^{\mathrm{marks}}
+       (h,\widehat h,\pi_i;\cdot)\,
+             \operatorname{Pois}_{\rho\mu_h|_{U_i}}(\dd\pi_i).
+\]
+This proves the fields-only assertion.  It uses no independence
+between the marks and the numerical arrays, whose tests may
+depend on those marks.  Alternatively, retain each $\Pi_i$
+together with its array as the local output.
+Finite joint consistency and the countable prescription library
+then give the kernel on all tests.  Lemmas
+\ref{inp:complete-core} and~\ref{inp:sandwich} identify its
+buffered exhaustion with the corresponding observations from
+the actual graph.  No additional conditioning on exterior
+numerical outputs is used.
+\end{proof}
+
+\begin{proposition}[Coordinate and unit transport]\label{inp:transport}
+The kernels in Proposition~\ref{inp:local-kernels} can be retained
+jointly with the following transformations.
+\begin{enumerate}
+\item For a fixed conformal coordinate map $\phi$, pull back the
+fields by
+\[
+ g\longmapsto g\circ\phi+Q\log|\phi'|,
+ \qquad
+ \widehat g\longmapsto\widehat g\circ\phi-\chi\arg\phi'.
+\]
+The numerical-array law is transported by the corresponding
+spatial relabeling of its tests.
+\item Fix $v=2^k$, $k\in\mathbb Z$, and put
+$c_v=\gamma^{-1}\log v$.  Expressing the surface in the field
+$g-c_v$ replaces area time by time divided by $v$ and replaces
+rate $m$ by rate $mv$.  Electrical energies and extremal lengths
+are unchanged by this relabeling.  If a numerical output is
+divided by $b_m$, retain this same factor $b_m$ in the $mv$
+experiment of the joint extraction.
+\item Write $K_{(1)}$ for the original local law, including the
+mark kernel, and $K_{(v)}$ for the law of the corresponding
+transformed rate experiment.  Then
+\begin{equation}\label{inp:unit-kernel}
+ K_{(v)}(g,\widehat g)
+     =\operatorname{Relabel}_v K_{(1)}(g+c_v,\widehat g).
+\end{equation}
+All fixed dyadic indices, their transformed templates, and the
+required Poisson layers may be retained in one joint extraction.
+\end{enumerate}
+\end{proposition}
+
+\begin{proof}
+This is the finite-experiment identity underlying Proposition ``Intrinsic
+transport and fixed quantum units'' in \cite{MetricMaps}.
+A conformal change leaves quantum areas, frontier lengths,
+directed seam roles, and their relative heights unchanged.
+Hence it leaves the finite instruction graph unchanged and
+only changes the projection of its labels.
+
+Adding $c_v$ multiplies areas by $v$ and boundary lengths by
+$\sqrt v$.  Thus the block inputs transform as
+\begin{equation}\label{inp:block-units}
+ \ell\longmapsto v\ell,
+ \qquad z(t)\longmapsto\sqrt v\,z(t/v),
+ \qquad o\longmapsto\sqrt v\,o.
+\end{equation}
+Cuts transform with time; band endpoints, their anchors, and
+selected minima transform with height.  A rate-$m$ Poisson
+string of duration $v\ell$ is, after this time change, a
+rate-$mv$ string of duration $\ell$.  The height lattices agree
+because $\sqrt v\,(mv)^{-1/2}=m^{-1/2}$.  Matching strips and
+minimum comparisons are exactly the same integer tests.
+Therefore the numerical costs on the resulting graph are
+identical.  Smooth sampling densities may change, but
+Lemma~\ref{inp:kernel-transfer} leaves the conditional kernels
+unchanged.
+
+The cuts must be transformed too.  At field $g+c_v$, the layer
+corresponding to a fixed layer at field $g$ has time intensity
+divided by $v$.  Its spatial intensity is then unchanged.
+Choose a countable family of layers closed under these changes,
+with independent layers and nested refinements as required.
+Their joint input and output identities give
+\eqref{inp:unit-kernel}.  A fixed field translation is locally
+equivalent to the Gaussian reference law, as specified below;
+this makes the identity valid under that fixed reference
+marginal as well.  No estimate uniform in unbounded $v$ is
+asserted or needed.
+\end{proof}
+
+For exhaustion of the plane we retain the joint cone normalization,
+not only the deterministic-coordinate kernel identity.  In the
+circle-average embedding define
+\begin{equation}\label{inp:joint-zoom}
+\begin{split}
+ r_v&=\sup\{r>0:h_r(0)+Q\log r=c_v\},\\
+ h^{(v)}&=h(r_v\,\cdot)+Q\log r_v-c_v,
+ \qquad \widehat h^{(v)}=\widehat h(r_v\,\cdot),\\
+ \eta^{(v)}(t)&=r_v^{-1}\eta(vt),
+ \qquad Z^{(v)}(t)=v^{-1/2}Z(vt).
+\end{split}
+\end{equation}
+The complete marginal has the original cone peanosphere law, by
+the joint zoom in \cite{MetricMaps} and
+\cite[Proposition~4.13(i)]{DMS}.  The positive dilation preserves
+the independent stationary-phase imaginary-field law.  These
+transformations are applied to the contour, raw arrays, and full
+background together, before disintegration.  Thus field-dependent
+reembedding does not require substituting a random map into
+separately chosen versions of deterministic-coordinate kernels.
+Moreover $r_{2^k}\to\infty$ as $k\to\infty$: the level $c_{2^k}$
+diverges, whereas $h_r(0)+Q\log r$ is bounded above on compact
+sets of radii and tends to $-\infty$ as $r\downarrow0$.
+Ordinary patches inside these normalized disks therefore exhaust
+the plane away from the cone point.
+
+\begin{lemma}[Ordinary Gaussian reference patches]\label{inp:gaussian}
+On a buffered patch compactly inside the cone's normalization
+disk and away from its marked point, the actual real and
+imaginary fields are locally equivalent to Gaussian reference
+restrictions.  The real-field pinning circle can be fixed outside
+all testing buffers.  This comparison preserves the actual
+additive constant and can be made after any fixed affine change
+of coordinates.
+
+Adjoining any of the local kernels above preserves the same
+field-only Radon--Nikodym derivative.  Consequently one fixed
+density truncation transfers high-probability events uniformly
+over all kernels attached to that reference marginal.  Fixed
+additive real-field shifts and the fixed coordinate transforms
+used in Proposition~\ref{inp:transport} preserve the corresponding
+local null sets.
+\end{lemma}
+
+\begin{proof}
+This is Lemma ``Ordinary cone patches and exterior Gaussian pins''
+in \cite{MetricMaps}.  Inside the unit disk the circle-average
+cone field has the law of the circle-average-zero whole-plane
+GFF minus $\gamma\log|z|$.  On the patch, this deterministic
+term has a smooth extension supported away from the marked
+point and the pinning circle.  The Cameron--Martin theorem
+and Gaussian domain comparison give local equivalence, with
+the actual constant retained.  The same argument applies to
+a fixed additive shift on the patch.  The imaginary field is
+independent and is interpreted modulo its angular period;
+its local restrictions have the corresponding Gaussian
+equivalence and stationary-phase coordinate invariance.
+
+If $Z=\dd\nu/\dd\mu$ is a field density and $K_i$ any of the
+conditional kernels, then for every event $A$ and every $R>0$,
+\begin{equation}\label{inp:density-truncation}
+ (\nu\otimes K_i)(A)
+ \le R(\mu\otimes K_i)(A)
+      +\E_\mu[Z\1_{\{Z>R\}}].
+\end{equation}
+The last term does not depend on $i$.  This proves the uniform
+probability comparison; local equivalence gives both directions
+for probability-one events.  A test domain containing the full
+pinning circle is excluded by the stated buffer condition.
+\end{proof}
+
+\begin{lemma}[Primal--dual reflection]\label{inp:reflection}
+The joint local-array laws are invariant under exchanging the
+primal and dual graphs and reflecting their coordinate tests.
+They are also covariant under fixed rigid motions, with the
+natural transformations of the two fields.
+\end{lemma}
+
+\begin{proof}
+The discrete assertion follows by exchanging the two independent
+coordinate strings and their forests.  In the continuum,
+spatial reflection combined with negation of the reflected
+imaginary field exchanges the two angle trees and the two
+sides of the curve.  The space-filling ordering is preserved:
+it can be read either from the $+\pi/2$ tree with right merges
+or from the $-\pi/2$ tree with left merges, as in
+\cite[Section~1.2.3, Figure~1.16]{IG4}.  Both descriptions are
+exchanged by this operation, so no reversal of area time is
+involved.  The cone field and the independent stationary-phase
+imaginary field have the required reflected law.  Boundary
+lengths exchange $L$ and $R$, and conditional sampling of the
+background therefore gives the same operation on the joint
+raw-array limit.  The intrinsic kernels and the selector
+construction then transport the spatial tests.  Fixed rotations
+and translations follow from coordinate covariance and local
+field equivalence on buffered ordinary patches.  A countable
+group of such transformations can be retained simultaneously.
+\end{proof}
+
+\begin{proposition}[Local-array interface]\label{inp:array-interface}
+From every sequence of resolutions tending to infinity one can
+extract, jointly with the continuum peanosphere background,
+all compactified numerical tests described above, on both
+graphs and at every fixed dyadic area offset.  Prescribed
+deterministic normalizing factors may be retained even when
+they tend to zero.  In this extraction:
+\begin{enumerate}
+\item complete buffered graph pieces and the selector sandwich
+of Lemmas~\ref{inp:complete-core} and~\ref{inp:sandwich} identify
+local tests with actual graph observations;
+\item Proposition~\ref{inp:local-kernels} supplies buffer-local
+conditional kernels and product laws in disjoint domains;
+\item Proposition~\ref{inp:transport} and
+Lemmas~\ref{inp:gaussian}--\ref{inp:reflection} supply their
+coordinate, unit, Gaussian-reference, and primal--dual
+comparisons;
+\item these properties persist under further joint extraction
+and under countable exhaustion with strict margins.
+\end{enumerate}
+\end{proposition}
+
+\begin{proof}
+All numerical coordinates belong to compact spaces, and the
+input contours are tight on bounded windows.  Include the
+countable transformed instruction lists and mark layers before
+the diagonal extraction.  The preceding results apply to
+every finite collection.  Their exact density and bridge
+calculation passes to any further limit by compact truncation;
+the continuum background passes by the $L^1$ conditional
+sampling argument.  Countable intersection and the selector
+exhaustions give the stated simultaneous version.
+\end{proof}
+
+The interface provides laws and exact local observability.
+The bounds which prevent degeneration of the electrical tests,
+and the identification of their variational limits, are proved
+next.
diff --git a/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/build/sections/03-nondegeneration.tex b/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/build/sections/03-nondegeneration.tex
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@@ -0,0 +1,606 @@
+\section{Nondegeneration of the electrical networks}
+\label{sec:nondegeneration}
+
+We first prove that a fixed conformal annulus admits cutoffs of bounded
+electrical energy on both sewn graphs.  The bound will depend on the
+annulus aspect and the coordinate patch, but not on its radius or the
+discrete resolution.  This is the coercivity input for the variational
+argument in the next section.  Throughout this section we work with
+the bilateral sewing, at rates $m\to\infty$, and use the coupled
+extractions and projections $\pi_m$ of
+Proposition~\ref{inp:array-interface}.
+
+The proof combines two uses of extremal length.  A lower bound supplies
+densities that charge every crossing of an annulus.  If lower bounds
+failed, a suitable quantile would instead supply probability measures
+on crossing paths with very small squared traffic.  Annular densities
+make their projected limits rectifiable.  Locality then forces a
+transverse crossing witness on the other graph, contradicting the
+small traffic.  The annular-density, rectifiable-flow, and local-germ
+strategy is developed for flag extremal length in
+\cite[Section ``The canonical conformal structure'']{ConformalMaps}.
+We adapt it to edge energies on paired primal and dual graphs.
+
+\subsection{Extremal length and local germs}
+
+Let $\mathcal P$ be a nonempty family of paths in a finite graph, with
+no path of length zero.  Edge occurrences are counted with
+multiplicity.  Its edge extremal length is
+\begin{equation}\label{nd:el-definition}
+ \EL(\mathcal P)
+ =\sup_{a:E\to[0,\infty)}
+   \frac{\bigl(\inf_{P\in\mathcal P}\sum_{e\in P}a(e)\bigr)^2}
+        {\sum_{e\in E}a(e)^2},
+\end{equation}
+where the zero density is omitted.  For a probability measure $\nu$
+on paths, write
+$j_\nu(e)=\int \#\{\text{occurrences of }e\text{ in }P\}\,\nu(\dd P)$
+for its unsigned edge traffic.
+
+\begin{lemma}[Densities and traffic]\label{nd:el-traffic}
+For a finite family of simple paths,
+\begin{equation}\label{nd:traffic-identity}
+ \EL(\mathcal P)=\min_{\nu}\sum_e j_\nu(e)^2.
+\end{equation}
+If $\mathcal P$ consists of all paths between two disjoint wired sets
+in a finite subgraph, then $\EL(\mathcal P)$ is the reciprocal of
+the minimum energy $\en(u)$ subject to the values zero and one on
+the two sets.  In particular, $\EL(\mathcal P)\ge b>0$ supplies a
+density charging each path by one with squared norm at most $1/b$.
+\end{lemma}
+
+\begin{proof}
+The traffic vectors form the convex hull of the path-incidence
+vectors.  Let $j$ be its point nearest to the origin.  The supporting
+inequality $\langle j,j_P\rangle\ge\|j\|_2^2$ for every path $P$
+shows that the density $j$ gives a value at least $\|j\|_2^2$ in
+\eqref{nd:el-definition}.  Conversely, averaging path length under
+any $\nu$ and applying Cauchy--Schwarz gives
+$\EL(\mathcal P)\le\|j_\nu\|_2^2$.  This proves
+\eqref{nd:traffic-identity}.  In the wired-set case loops may be
+removed from paths.  Absolute potential increments give a charging
+density, while the distance from the first set in any charging
+density, clipped at one, gives an admissible potential whose energy
+is at most the squared density norm.  These two constructions prove
+the reciprocal identity.
+\end{proof}
+
+We will also use a consequence of the local conditional kernels from
+Section~\ref{sec:inputs}.  A \emph{point-germ readout} is a jointly
+measurable random function $Y(z)$ whose value at $z$ can be read from
+the fields and extracted arrays in every neighborhood of $z$.  The
+readout may use countably many tests with strictly positive domain
+margins.  In particular, an assertion that specified tests succeed
+at every sufficiently small scale is a point-germ readout.
+
+\begin{lemma}[Determinism of point germs]\label{nd:germs}
+In an ordinary field chart, every bounded real point-germ readout
+of the extracted local arrays is, almost surely, equal at
+Lebesgue-almost every point to a deterministic measurable function
+of the point.  The same conclusion holds for readouts with values
+in a separable metric space, using a countable family of bounded
+coordinates that separates its points.
+\end{lemma}
+
+\begin{proof}
+Let $\mathcal H$ denote the pair of fields.  By
+Proposition~\ref{inp:local-kernels},
+$\E[Y(z)\mid\mathcal H]$ is measurable with respect to the field
+restrictions to every neighborhood of $z$.  This field germ is
+trivial.  Indeed, for a zero-boundary Gaussian free field, the
+intersection of the Gaussian subspaces generated by restrictions
+to shrinking disks is orthogonal to the Gaussian variables
+associated with Dirichlet functions vanishing near $z$.  Such
+functions are dense in the Dirichlet space: logarithmic cutoffs
+express the zero capacity of a point.  The intersection is thus
+zero, and the Gaussian chaos decomposition gives triviality of
+the corresponding sigma-field.  Apply this to both independent
+fields.  Local equivalence in Lemma~\ref{inp:gaussian}
+transfers the conclusion to the ordinary chart, including the
+imaginary field modulo its period.  Consequently
+$\E[Y(z)\mid\mathcal H]=y(z)$ for a deterministic $y$.
+
+For distinct $z,w$, disjoint sufficiently small neighborhoods and
+the product-kernel assertion give conditional independence of
+$Y(z)$ and $Y(w)$ given $\mathcal H$.  Hence, for every rational
+disk $D$ compactly in the chart,
+\[
+ \E\left[\left(\int_D(Y(z)-y(z))\,\dd z\right)^2\right]=0.
+\]
+The diagonal in $D\times D$ has Lebesgue measure zero.  Countably
+many disks and differentiation of integrals prove the assertion.
+Applying the argument to bounded separating coordinates proves
+the final statement.
+\end{proof}
+
+\begin{remark}[Coordinate and dual symmetries]\label{nd:germ-symmetries}
+For a readout defined intrinsically from local tests, its deterministic
+values respect the coordinate transports of
+Proposition~\ref{inp:transport}.  More explicitly, let $\mathcal A$
+be the extracted local array and let $T\mathcal A$ denote its
+relabeling by a fixed translation or rotation $T$.  A covariant
+readout satisfies
+\[
+             Y_{T\mathcal A}(Tz)=T\cdot Y_{\mathcal A}(z),
+\]
+where $T\cdot$ is the natural action on its values or direction
+labels.  The transport proposition identifies the law of
+$T\mathcal A$ with the same extracted kernel attached to the
+transformed fields.  Local equivalence of these field laws and
+Lemma~\ref{nd:germs} therefore give
+$y(Tz)=T\cdot y(z)$ almost everywhere.  This uses a relabeling in
+the joint extraction, rather than a newly chosen subsequential
+kernel.  For translation-covariant scalar readouts it implies
+constancy: first use rational translations, then continuity of
+translations in local $L^1$.  Rotations act on direction or matrix
+readouts in the stated natural way, and a countable dense rotation
+group suffices.  Finally, Lemma~\ref{inp:reflection} exchanges the
+two graphs and reflects the tests.  These observations apply on
+overlapping ordinary charts and identify their deterministic values.
+The definition through strict margins is important here: a transported
+test is still an admissible test after an arbitrarily small reduction
+of its clearance, by Lemma~\ref{inp:sandwich}.
+\end{remark}
+
+\subsection{Annular barriers at absolute quantum scales}
+
+Choose once and for all an ordinary cone patch, strictly inside
+the normalization disk and away from the cone point.  After a
+fixed affine coordinate change, it contains the annulus
+$\{1/16<|z|<4\}$ and the buffers needed below.
+For each graph take paths between the vertex sets selected by visits
+in the bands
+\[
+ A_1=\{.9<|z|<1.1\},\qquad
+ A_2=\{1.9<|z|<2.1\},
+\]
+confined to the actual subgraph selected in
+$A_0=\{.7<|z|<2.3\}$.  Denote the extremal lengths of these primal
+and dual path families by $Z_m^{\mathrm p}$ and $Z_m^{\mathrm d}$.
+Here selection uses the continuum visit labels
+in the coupling, with a fixed step-endpoint convention.  This defines
+a comparison variable; it need not itself be a local instruction
+test.  Properness makes the selected labels finite.  All their
+incidences and adjoining star layers are protected in larger bounded
+contour windows.  The projection and tube properties of
+Proposition~\ref{inp:topology} therefore apply.  If the ports overlap
+or there is no crossing, give the comparison the value one.  The
+probability of these exceptions tends to zero.
+
+Let $Z_m=\min\{Z_m^{\mathrm p},Z_m^{\mathrm d}\}$.  Fix
+$\varepsilon>0$, to be chosen below, and choose deterministic
+$q_m>0$ such that
+\begin{equation}\label{nd:quantiles}
+ \P[Z_m\le 2q_m]\ge\varepsilon/2,
+ \qquad
+ \P[Z_m\ge q_m/2]\ge1-2\varepsilon.
+\end{equation}
+We use rates in a countable set closed under dyadic multiplication
+and division, for example the positive dyadic rationals.  The
+quantiles can be chosen bounded below on every bounded rate interval
+in $[1,\infty)$.  To see this, first bound the continuum preimage
+window with high probability, then bound the Poisson step count
+and the degrees of the finitely many encountered vertices.  Local
+finiteness of the bilateral sewing makes these bounds uniform at
+bounded rates.  A nonexceptional finite graph with at most $L$
+edges has positive crossing extremal length at least $1/L$.
+
+The comparison in \eqref{nd:quantiles} produces genuinely local
+barrier events.  For example, select ports near radii $.96$ and
+$2.04$, and confinement between $.94$ and $2.06$, with strictly
+positive slack.  Every path from $\{|z|<.6\}$ to
+$\{|z|>2.5\}$ contains such a crossing portion.  A complete local
+prescription covering this intermediate shell exists by
+Lemma~\ref{inp:complete-core}; Lemma~\ref{inp:sandwich} selects the
+ports and confinement.  Its path family is a subfamily of the
+comparison family, so its EL is no smaller.
+
+The selection can depend on the limiting fields without changing
+this probability comparison.  To see the quantifiers, let $j$
+enumerate pairs of primal and dual prescription templates.  Let
+$\mathsf A_j$ be the local event that both templates' seam, cap, and
+positive-margin conditions cover the required shell and implement
+the selector sandwich.
+These conditions are read from the local fields and marks;
+their proved consequence is eventual completeness on the actual
+graph.  They do not test completeness by inspecting exterior
+numerical arrays.  For any deterministic units $b_m>0$ retained
+in the extraction, write $\xi_j$ for the minimum of the two
+retained limits of $\EL/b_m$.  Thus success supplies barriers on
+both graphs.  If a comparison event asserts
+$Z_m/b_m\ge a>0$, retain its indicator in a further coupling.
+On the event $E$ where the limiting indicator is one, choose
+an applicable $j$ from the limiting local cell data, keeping it
+fixed as $m\to\infty$.  Its eventual domination of the comparison
+gives $\xi_j\ge a$.  Consequently
+\[
+ E\subseteq\bigcup_j\bigl(\mathsf A_j\cap\{\xi_j>a/4\}\bigr).
+\]
+The union on the right is a local event with at least the
+limiting comparison probability.  Simultaneous convergence of
+the countably retained tests justifies the random choice of
+$j$; no deterministic bound on that index is required.  This
+argument uses the comparison for its probability and the local
+union for its conditional kernel, without conditioning that
+kernel on $E$.
+
+We recall precisely the annulus input that spreads these barriers
+through a chart.  It is Lemma ``All-point absolute quantum-scale
+annuli'' in \cite[Section ``Annuli at absolute quantum scales'']{MetricMaps}.
+Write $v_i=2^{-i}$ for the backward quantum-area offsets.  By
+Proposition~\ref{inp:transport}, graph costs are unchanged at these
+offsets; any deterministic denominator $b_m$ is kept the same
+across the offset experiments in one extraction.
+
+\begin{lemma}[Aligned annular barriers]\label{nd:coverage}
+Fix an ordinary chart and a compact subset of its interior.  Suppose
+that, at every fixed backward index $i$, a local unit-annulus event
+has probability at least $1-\varepsilon_*$ in the reference law and
+supplies, on both graphs, a crossing density of charge one and
+squared norm at most $C/b_m$.  Here $b_m>0$ is deterministic and
+the events belong to the common extracted offset kernels.
+For a sufficiently small reference threshold $\varepsilon_*>0$,
+there are constants $0<a<B$, $c>0$ such that the following holds
+almost surely for all sufficiently large integers $N$.
+Every point in the compact set belongs to the central holes of at
+least $cN$ pairwise disjoint buffered annuli whose local barrier
+events hold.  Their radii belong to one fixed geometric sequence
+in $[e^{-BN},e^{-aN}]$, their centers lie on deterministic meshes
+of spacing comparable to their radii, and their assigned indices
+belong to $\{N,\ldots,2N\}$.
+For each fixed such finite configuration, its density conclusions
+hold on the discrete graphs for all sufficiently large $m$.
+\end{lemma}
+
+\begin{proof}
+Apply the cited lemma to the kernels of
+Proposition~\ref{inp:local-kernels}, with the intrinsic transport
+of Proposition~\ref{inp:transport}.  Its hypotheses are the local
+product rule, the actual-field coordinate rule, and a common
+fixed-index change of quantum unit; all are supplied by
+Proposition~\ref{inp:array-interface}.  The real-field parameter
+$Q=2/\gamma+\gamma/2$ is greater than two.  The cost-index factor
+in the stated annulus lemma is undone in our numerical tests;
+the proof of that lemma uses the factor only in the indexed
+observation rule.  Thus it applies equally to the present
+unchanged network costs.  Local Poisson marks are included in
+the kernels with their corresponding intensities.
+
+For completeness, the probability transfer needed here is a fixed
+one.  Work first with the zero-boundary Gaussian reference patch
+of that lemma.  Local equivalence, retaining the actual additive
+constant of the real field, gives a single absolute-continuity
+modulus from our chosen ordinary cone patch to this reference
+patch.  Choose $\varepsilon$ in \eqref{nd:quantiles} sufficiently
+small that a barrier event of probability $1-2\varepsilon$,
+with a strict threshold as above, has reference probability at
+least $1-\varepsilon_*$.  The annulus lemma gives an exponentially
+small failure probability at scale $N$.  Borel--Cantelli gives
+the eventual assertion.  A fixed local change of field law
+transfers this probability-one assertion to any ordinary chart.
+No likelihood bound uniform over those charts is needed.
+Finally, success of an annulus is the existence of a retained
+EL coordinate strictly above a fixed threshold.  No density or
+path is retained as an additional output.  All centers, radii,
+indices, and prescription templates belong to countable
+libraries.  On their simultaneous convergence event, the
+successful local prescription on each realized annulus gives
+an eventual discrete EL lower bound.  Lemma~\ref{nd:el-traffic}
+then constructs a density on that finite graph.  For fixed $N$
+only finitely many annuli are used, so their thresholds in $m$
+can be met simultaneously, even though the successful annuli
+and prescriptions were selected from the fields and arrays.
+Thus $N$ is fixed before $m$ is sent to infinity.
+\end{proof}
+
+The geometric content of these many annuli is an area bound for a
+density that charges macroscopic displacement.  For a set $U\subset\C$
+write $U^{+r}=\{z:\dist(z,U)<r\}$, and put $R_N=e^{-aN}$.
+
+\begin{lemma}[Densities controlling displacement]\label{nd:densities}
+Under the hypotheses of Lemma~\ref{nd:coverage}, let $U$ be a finite
+union of open boxes compactly in a fixed ordinary chart.  For all
+sufficiently large $N$, and then all sufficiently large $m$, there
+is on either graph a nonnegative density $g_{m,N,U}$, supported
+near $U$, such that
+\begin{equation}\label{nd:density-area}
+ b_m\sum_e g_{m,N,U}(e)^2
+       \le C\vol(U^{+CR_N}).
+\end{equation}
+Every path subinterval whose projection is contained in $U$ pays
+at least
+\begin{equation}\label{nd:density-charge}
+ c\diam(\pi_m(P))-CR_N-o_m(1).
+\end{equation}
+The constants depend only on the smaller chart and the fixed
+annular geometry.  The error and the threshold in $m$ may depend
+on $N$ and $U$.  Charges add over disjoint path subintervals.
+\end{lemma}
+
+\begin{proof}
+On a successful shell of radius $r$, multiply its barrier density
+by $r$.  Its charge is now $r$ and its squared norm is at most
+$Cr^2/b_m$.  Sum these densities over the shells whose central
+holes meet $U$, using one density for each geometric shell, and
+divide by $N$.  At any fixed radius the buffered shells have
+bounded overlap, because their centers lie on a mesh of comparable
+spacing.  There are at most $CN$ radii.  Cauchy--Schwarz across
+these radii gives \eqref{nd:density-area}; all the involved shells
+lie within distance $CR_N$ of $U$.
+
+For the charge estimate choose a coordinate in which the projected
+path has oscillation comparable to its diameter.  Consider levels
+between its extreme values, discarding $CR_N$ near either end.
+At a visit to each remaining level, the point lies in at least
+$cN$ central holes.  Since all these shells have radius at most
+$R_N$, the path crosses each of them before reaching a suitable
+extreme.  A shell of radius $r$ can account for at most $Cr$ of
+the levels.  Integrating over levels therefore bounds the total
+length in the summed densities from below by $cN$ times the
+remaining coordinate oscillation.  Divide by $N$.  At fixed $N$
+the vanishing projected mesh permits full-edge crossing portions;
+the discarded endpoint pieces contribute $o_m(1)$.  Distributing
+each edge's density uniformly along that edge permits fractional
+first and last edges and makes additivity over disjoint intervals
+explicit.
+\end{proof}
+
+\subsection{Small traffic would force transverse crossings}
+
+We now prove the quantitative fact needed for unnormalized energies.
+\begin{proposition}[Nondegeneration]\label{nd:nondegeneration}
+For the quantiles in \eqref{nd:quantiles}, with $\varepsilon$ chosen
+as in Lemma~\ref{nd:coverage},
+\begin{equation}\label{nd:positive-quantile}
+                    \liminf_{m\to\infty}q_m>0.
+\end{equation}
+The conclusion holds along all rates in the chosen countable set.
+\end{proposition}
+
+\begin{proof}
+Suppose otherwise.  Choose approximate record lows $m\to\infty$
+such that $b_m:=q_m\to0$ and
+\[
+ q_s\ge c b_m\qquad(1\le s\le m,\ s\text{ in the chosen rate set})
+\]
+for a fixed $c>0$.  The bounded-rate lower bounds ensure $m\to\infty$.
+For each fixed $i$, the quantile at $m2^{-i}$ therefore gives the
+high-probability barrier event in units $b_m$.  Extract jointly
+with all these backward offsets.  Lemmas~\ref{nd:coverage} and
+\ref{nd:densities} apply in the comparison region.
+
+The first inequality of \eqref{nd:quantiles} and
+Lemma~\ref{nd:el-traffic}, on the other hand, supply with positive
+probability one graph's probability flow $\nu_m$ between the two
+macroscopically separated ports, confined in a fixed compact set,
+with
+\begin{equation}\label{nd:small-traffic}
+                      \sum_e j_{\nu_m}(e)^2\le Cb_m.
+\end{equation}
+Retain the event indicator, choose a graph on a positive-probability
+subevent, and work pathwise in a further coupling.  We establish
+two properties of the projected flows before using locality.
+
+\emph{Compactness of the flows.}
+For a path $P$, let $J_h(P)$ be the maximum number of disjoint
+subintervals making a projected advance of at least $h$.  Choose
+$N$ large enough that $CR_N<ch/2$ in
+\eqref{nd:density-charge}, and then take $m$ large.  Use a fixed
+larger box union $U$ containing the compact confinement.  By
+\eqref{nd:density-area}, \eqref{nd:small-traffic}, and
+Cauchy--Schwarz,
+\begin{equation}\label{nd:crossing-count}
+ \int J_h(P)\,\nu_m(\dd P)\le C/h.
+\end{equation}
+Thus the paths are tight after reparameterization.  One explicit
+choice of parameters is useful.  At every dyadic $h=2^{-l}$ place
+clock atoms at successive completed moves of distance $h$, each
+with mass $2^{-l}/(1+J_h(P))$.  Add the original traversal time
+of total mass one, invert this clock with pauses at its atoms,
+and rescale its total duration to one.  On sets where every
+$J_{2^{-l}}$ has a fixed bound, these parameterized paths are
+equicontinuous: an oscillation of several times $2^{-l}$ must
+cross a whole atom interval of a fixed positive duration.
+The bounds \eqref{nd:crossing-count} and a countable union bound
+make such sets have arbitrarily large flow probability.  The
+finitely many initial resolutions omitted at an individual scale
+can be included by enlarging its bound.  A subsequence therefore
+converges to a probability law $\nu$ on continuous paths.  Their
+endpoints remain separated.
+
+\emph{An absolutely continuous measure of arclength.}
+For at most $J$ disjoint path intervals compactly confined in an
+open box union $U$, the same argument gives
+\begin{equation}\label{nd:variation-bound}
+ \int\sup\sum_{k=1}^{J}
+       |\pi_m(P(t_k))-\pi_m(P(s_k))|\,\nu_m(\dd P)
+ \le C\sqrt{\vol(U^{+CR_N})}+CJR_N+o_m(1).
+\end{equation}
+The supremum may be taken over rational intervals with closure
+in the prescribed open condition.  It is lower semicontinuous
+under uniform convergence.  Pass first to the path limit, next
+let $N\to\infty$, and finally let $J\to\infty$.  These interval
+sums exhaust variation on open sets.  With a larger $U$ containing
+all paths, they first show that $\nu$-almost every path is
+rectifiable with integrable length.  Its average arclength
+measure $\Lambda$ consequently satisfies
+\begin{equation}\label{nd:length-area}
+ 0<\Lambda(\C)<\infty,
+ \qquad \Lambda(U)\le C\sqrt{\vol(U)}.
+\end{equation}
+The positivity follows from the separated endpoints.  Open box
+covers now show $\Lambda\ll\vol$.
+
+We next convert a tangent of a limiting path into a local event.
+Choose a sufficiently fine finite grid $\mathcal W$ of unoriented
+directions, closed under quarter-turn and reflection in a coordinate
+axis.  For $z\in\C$, $w\in\mathcal W$, and $s>0$, consider the tube
+\begin{equation}\label{nd:tube}
+ T(z,w,s)=\{z+tw+lw^\perp: |t|<2s,\ |l|<s/16\}.
+\end{equation}
+A \emph{strict crossing test} in this tube has ports strictly
+beyond the two lines $t=-s$ and $t=s$, and has confinement and
+ports with positive clearance from the tube boundary.  It is
+computed on a complete local prescription.  Accept all countably
+many selectors with these properties, and require that the
+prescription be readable within a fixed-factor enlargement of
+the tube, with positive margins.  Lemma~\ref{inp:sandwich} makes
+existence of such a test independent of the particular selector
+library after arbitrarily small relaxation of the margins.
+
+At $\Lambda$-almost every point there is a direction $w\in\mathcal W$
+for which, at every sufficiently small dyadic scale $s$, one of
+these tests on the graph supplying $\nu_m$ has finite limiting
+$\EL/b_m$.  Indeed, arclength parameterization gives a unit tangent
+at almost every parameter of almost every rectifiable path.  The
+fine direction grid puts that tangent inside one of the tubes
+with strict slack at all sufficiently small scales.  Almost
+every path is in the support of $\nu$.  The existence of the
+strict crossing subpath is an open path event, so it has positive
+$\nu$-probability at each such fixed scale.  Restrict $\nu_m$ to
+this event, take the crossing portions, and normalize.  Its traffic
+still has squared norm $O(b_m)$, with a constant allowed to depend
+on the scale and the event probability.  The complete-selector
+sandwich gives the asserted finite limiting test value.  The
+statement can be imposed simultaneously at the countably many
+dyadic scales.
+
+Let $I^{\mathrm p}_w(z)$, respectively $I^{\mathrm d}_w(z)$, be the
+indicator of existence of these finite-$\EL/b_m$ tests at every
+sufficiently small dyadic scale, on the primal or dual graph.
+These are jointly measurable point-germ readouts: the tests and
+their fixed-factor buffers eventually lie in every neighborhood
+of $z$.  The preceding paragraph and \eqref{nd:length-area} give
+a positive-area set of witnesses on our positive-probability
+event.  Lemma~\ref{nd:germs} and
+Remark~\ref{nd:germ-symmetries} therefore imply that one of these
+indicators has deterministic value one almost everywhere in
+ordinary charts.  Reflection exchanges the graphs; rotations
+within the finite direction grid then give the other graph's
+indicator in the perpendicular direction.  The null sets can
+be discarded simultaneously, since only finitely many directions
+and two graphs are involved.
+
+Choose a point with both perpendicular indicators equal to one,
+and then a sufficiently small common dyadic scale.  We obtain
+primal and dual strict crossing families with
+$\EL=O(b_m)$ in perpendicular tubes, as illustrated in
+Figure~\ref{nd:transverse-figure}.  Every path of one family meets
+every path of the other upstairs by the transverse-tube rule in
+Proposition~\ref{inp:topology}.  Such an intersection uses an edge
+and its dual.  Choose traffic minimizers $j_m,j_m^*$ by
+Lemma~\ref{nd:el-traffic}.  Their unsigned pairing satisfies
+\[
+ 1\le\sum_e j_m(e)j_m^*(e^*)
+    \le\Bigl(\sum_e j_m(e)^2\Bigr)^{1/2}
+       \Bigl(\sum_{e^*}j_m^*(e^*)^2\Bigr)^{1/2}
+    =O(b_m),
+\]
+contradicting $b_m\to0$.  This proves
+\eqref{nd:positive-quantile}.
+\end{proof}
+
+\begin{figure}[t]
+\centering
+\begin{tikzpicture}[scale=1.05]
+ \fill[blue!7] (-3,-.42) rectangle (3,.42);
+ \fill[orange!12] (-.42,-2.25) rectangle (.42,2.25);
+ \draw[blue!55,dashed] (-3,-.42) rectangle (3,.42);
+ \draw[orange!70!black,dashed] (-.42,-2.25) rectangle (.42,2.25);
+ \draw[blue!75!black,thick]
+   (-2.7,.05) .. controls (-1.8,-.25) and (-1,.24) .. (-.25,.05)
+   .. controls (.55,-.15) and (1.6,.27) .. (2.7,-.03);
+ \draw[orange!80!black,thick]
+   (.08,-2) .. controls (-.26,-1.2) and (.23,-.6) .. (.02,.05)
+   .. controls (-.18,.7) and (.23,1.25) .. (-.02,2);
+ \draw[blue!75!black,thick] (-2.15,-.42)--(-2.15,.42);
+ \draw[blue!75!black,thick] (2.15,-.42)--(2.15,.42);
+ \draw[orange!80!black,thick] (-.42,-1.5)--(.42,-1.5);
+ \draw[orange!80!black,thick] (-.42,1.5)--(.42,1.5);
+ \node[blue!75!black,anchor=south] at (1.7,.49) {primal crossing};
+ \node[orange!80!black,anchor=west] at (.53,1.35) {dual crossing};
+ \fill (0,0) circle (1.3pt);
+ \node[anchor=north west] at (.1,-.14) {$z$};
+\end{tikzpicture}
+\caption{The two narrow tubes force transverse passages.  The drawing
+shows only projected path ranges; it asserts neither injectivity of
+the projection nor shape control of discrete edges.  The topological
+input lifts this configuration to an intersection of a primal edge
+with its paired dual edge.}
+\label{nd:transverse-figure}
+\end{figure}
+
+\subsection{Uniformly bounded ring cutoffs}
+
+We can now apply the annular construction in constant energy units,
+rather than the hypothetical vanishing units used in the contradiction.
+
+\begin{proposition}[Ring cutoffs]\label{nd:cutoffs}
+Fix a compact subpatch of an ordinary chart and an aspect ratio
+$\tau>1$.  Along a coupled further extraction from any rate sequence,
+there is a constant $C$, depending only on the chart and $\tau$,
+with the following property.  For every fixed disk
+$\overline{B(z,\tau r)}$ contained with a margin in that subpatch,
+both graphs eventually admit functions $\chi_m$ valued in $[0,1]$
+such that
+\[
+ \chi_m=1\quad\text{on every lift of }\overline{B(z,r)},
+ \qquad
+ \chi_m=0\quad\text{on every lift outside }B(z,\tau r),
+ \qquad
+ \en(\chi_m)\le C
+\]
+on the primal graph, with $\enstar(\chi_m)\le C$ on the dual.
+Their energy is supported in the indicated ring, up to vanishing
+projected mesh.  The constant is independent of $z,r,m$; the
+threshold in $m$ may depend on the fixed disk.  These assertions
+hold simultaneously for the countable buffered disk tests, and
+extend to arbitrary fixed disks by strict inner and outer margins.
+\end{proposition}
+
+\begin{proof}
+Proposition~\ref{nd:nondegeneration} and the second quantile bound
+in \eqref{nd:quantiles} give the barrier hypotheses with
+deterministic units equal to one, at every fixed backward offset.
+Choose an intermediate closed ring whose two radii lie strictly
+between $r$ and $\tau r$, with all three resulting radial widths
+comparable to $(\tau-1)r$.  Cover it by a finite union $U$ of open
+boxes with closure strictly inside the target ring and area
+$O_\tau(r^2)$.  Apply Lemma~\ref{nd:densities} to this $U$.
+First choose $N$ so large that $R_N$ is small compared with the
+intermediate width and both collars, and then take $m$ large.
+The density is supported in the target ring.  Every full ring
+crossing has a subcrossing through the intermediate ring,
+confined to $U$, with advance comparable to $(\tau-1)r$.
+The density therefore charges it by a fixed multiple of this
+width.  After division by the width, the squared density norm
+is bounded by $C(\tau)$ times the chart constant.
+The clipped-distance construction of
+Lemma~\ref{nd:el-traffic} gives the cutoff, with exact constants
+on all inner and outer lifts.  One may carry out the construction
+on a protected capped sphere and extend by constants beyond the
+chart.  Only the ring density changes the function.
+
+Countable buffered comparisons make all the fixed tests available
+in one extraction.  A disk outside the chosen countable library
+is handled by choosing a nearby library center, an inner library
+disk containing $\overline{B(z,r)}$, and a concentric outer library
+disk contained in $B(z,\tau r)$.  Their radii can be chosen
+comparable to $r$, with constants depending only on $\tau$, and
+their radial separation at least $c(\tau-1)r$.  The preceding
+density estimate bounds the cutoff energy by a chart constant
+times the squared ratio of outer radius to radial separation.
+It therefore gives a uniform $C(\tau)$ for these library tests,
+with the required constants on the original inner and outer
+sets.  This use does not require a resolution threshold uniform
+over all disk radii.
+\end{proof}
+
+All these conclusions are local in the actual conformal coordinate.
+They therefore also hold in each fixed forward quantum-area offset
+of Proposition~\ref{inp:transport}.  The joint cone zoom and
+ordinary-patch exhaustion of \cite[Lemma ``From indexed Gaussian charts
+to the whole cone'']{MetricMaps} let these patches exhaust the plane
+away from its marked cone point.  Only probability-one conclusions
+are transferred in this exhaustion.  The marked point will be
+handled by removability when the variational limits are assembled.
diff --git a/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/build/sections/04-energy.tex b/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/build/sections/04-energy.tex
new file mode 100644
index 0000000000000000000000000000000000000000..6e0031e25b6fea9ef17432b56cf7e5fee1994aa5
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+++ b/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/build/sections/04-energy.tex
@@ -0,0 +1,779 @@
+\section{The limiting electrical energy}\label{sec:energy}
+
+We now identify the unnormalized energies of the primal and dual graphs.
+The cutoffs of Proposition~\ref{nd:cutoffs} first give compactness for
+harmonic functions and coercivity for arbitrary continuous limits.  The
+locality of the extracted arrays then makes the limiting energy a
+deterministic scalar multiple of the conformal Dirichlet integral.
+An annular current--period argument will show that this scalar is exactly
+one.  Thus the normalization is obtained from the network itself.
+
+Throughout this section we work in an ordinary conformal chart, in a
+coupled extraction with the conclusions of
+Proposition~\ref{inp:array-interface} and Proposition~\ref{nd:cutoffs}.
+Write $G_m,G_m^*$ for the primal and dual graphs and $\pi_m$ for their
+surface projections.  All sets used below have fixed positive clearance
+inside the chart before $m$ tends to infinity.  The energy
+$\en_m(v;U)$ is the sum of $|v(y)-v(x)|^2$ over edges with both projected
+endpoints in $U$; we use the same convention for compact sets.  Replacing
+this convention by edge-location membership does not change an inner
+limit, because the projected mesh tends to zero.  Uniform convergence
+on a compact set always means convergence at every vertex over that set,
+including every lift of a point.
+
+\subsection{Circuits and harmonic compactness}
+
+A cutoff on one graph gives a short level circuit on the other, where
+``short'' is measured by the increments of the function under study.
+This elementary coarea observation is the bridge between the cutoffs
+and the analysis of functions.
+
+\begin{lemma}[A separating circuit]\label{en:circuit}
+Fix concentric disks with radii $0<r<R$ and with a neighborhood of the
+closed larger disk inside an ordinary chart.  Given a primal function
+$v_m$, there is a simple primal circuit $C_m$ in a slightly buffered
+annulus which surrounds all lifts of the smaller disk and whose inside
+contains no lift outside a slightly enlarged larger disk, such that
+\begin{equation}\label{en:circuit-bound}
+ \left(\sum_{e\in C_m}|\dd v_m(e)|\right)^2
+ \le C\,\en_m(v_m;A^+).
+\end{equation}
+Here $A^+$ is a fixed buffered annulus and $C$ depends only on its aspect
+and the chart constants.  The conclusion holds eventually for every
+fixed annulus.  It also holds with the primal and dual graphs exchanged.
+\end{lemma}
+
+\begin{proof}
+Take a dual cutoff $b_m$ which is one on the inner well and zero on the
+outer well, with energy at most $C$.  These wells include all relevant
+lifts, with fixed collars.  For $t\in(0,1)$ away from the finitely many
+vertex values, color the primal faces according to whether their dual
+vertices satisfy $b_m>t$.  The oriented boundary of the colored faces
+splits into simple circuits.  A face appearing on both sides of an edge
+contributes no boundary occurrence; other edge occurrences have their
+usual multiplicities.
+
+The boundary chain has winding difference one between an inner dual
+vertex and an outer dual vertex, because their face colors are one and
+zero.  After splitting the chain into simple circuits, at least one
+circuit separates these two vertices.  The inner dual well is connected
+through dual edges on which the cutoff is one; such a path cannot cross
+any boundary edge, including at a primal articulation vertex.  Hence
+every vertex of this inner well is on the same side of the chosen
+circuit.  The connected outer well has the analogous property.  The
+collars and vanishing overlay mesh put all primal inner lifts on the
+inner side as well, and all lifts beyond the larger well on the outer
+side.  This uses the all-lift joining statement of
+Proposition~\ref{inp:topology} and remains valid when the full boundary
+chain touches itself.  The selected circuit lies in the change annulus
+and has variation at most that of the entire boundary.
+The coarea identity and Cauchy--Schwarz give
+\[
+ \int_0^1\sum_{e\in\partial\{b_m>t\}}|\dd v_m(e)|\,\dd t
+ =\sum_e |\dd v_m(e)|\,|\dd b_m(e^*)|
+ \le \en_m(v_m;A^+)^{1/2}\en_m^*(b_m)^{1/2}.
+\]
+Choose a level at most this average.  This proves
+\eqref{en:circuit-bound}, with an arbitrarily small slack if necessary.
+The same argument applies after exchanging the graphs.
+\end{proof}
+
+\begin{lemma}[Harmonic compactness]\label{en:harmonic-compactness}
+Let $U$ be a connected open subset of an ordinary chart.  Suppose that
+$v_m$ is graph-harmonic at every vertex whose star projects into $U$,
+and that its energies are bounded on compactly contained neighborhoods.
+Then $v_m$ is macroscopically equicontinuous on every compact subset of
+$U$: for $K\Subset U$,
+\[
+ \lim_{\delta\downarrow0}\limsup_{m\to\infty}
+ \sup_{\substack{\pi_m x,\pi_m y\in K\\
+                  |\pi_m x-\pi_m y|<\delta}}
+                  |v_m(x)-v_m(y)|=0.
+\]
+If $x_m$ is a vertex sequence with $\pi_mx_m\to z_0\in U$ and
+$v_m(x_m)$ is bounded, the functions have locally uniformly convergent
+subsequences.  The assertions hold for both graphs and for local
+branches of potentials, with a normalization on each initial branch.
+\end{lemma}
+
+\begin{proof}
+Fix a disk with clearance inside $U$ and an energy bound $E$ on a larger
+disk.  Around its center place $J$ disjoint annuli of one fixed aspect,
+all inside a prescribed small neighborhood.  Choose all radii before
+taking $m$ large.  Their energy sum is at most $E$, so
+Lemma~\ref{en:circuit} supplies a surrounding circuit on which the
+oscillation is at most $(CE/J)^{1/2}$.  The discrete maximum principle
+bounds the oscillation everywhere inside by the same quantity.  Taking
+$J$ large, and then using finitely many such neighborhoods, proves
+equicontinuity.  No assertion is needed at a radius depending on $m$.
+
+The same construction at a fixed, nonvanishing scale bounds oscillation
+on compact balls.  Overlapping chains of balls propagate an additive
+normalization throughout a connected compact subset.  Vertex density
+and the small-image joining property identify the limits on overlaps.
+The usual finite-net proof of Arzel\`a--Ascoli now gives local uniform
+compactness.  A local potential branch is an ordinary harmonic function
+on each of its branch disks, so the proof also applies to branches.
+\end{proof}
+
+\subsection{The uniform relaxation and its energy measures}
+
+The extracted variational tests let us define a single limiting energy
+for every continuous target.  This definition does not assume a
+continuum Dirichlet form.
+
+For $u\in C(U)$, $K\Subset U$ compact, and $\delta>0$, let
+\[
+ M_m(u,K,\delta)=
+ \inf\left\{\en_m(v;K):
+  |v(x)-u(\pi_m x)|\le\delta
+  \text{ for every vertex }x\text{ over }K\right\}.
+\]
+Define the \emph{inner relaxed energy} by
+\begin{equation}\label{en:relaxation}
+ F(u,U)=\sup_{K\Subset U}\sup_{\delta>0}
+                  \liminf_{m\to\infty}M_m(u,K,\delta).
+\end{equation}
+One may use a countable exhaustion of compact finite unions of boxes
+and let $\delta$ decrease through positive rationals.  Define $F^*$ by
+the same formula on the dual graph.
+
+\begin{lemma}[Uniform variational limits]\label{en:envelope}
+The value in \eqref{en:relaxation} is unchanged if $\liminf$ is replaced
+by $\limsup$.  It is lower semicontinuous for locally uniform convergence
+and inner regular in $U$.  Moreover:
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item Every locally uniformly convergent sequence $v_m\to u$ has local
+energy lower bounds given by $F(u,\cdot)$, also along further
+subsequences.
+\item If $F(u,U)<\infty$, there are compact sets $K_m$ exhausting $U$
+and functions $v_m$ on their graph neighborhoods such that $v_m\to u$
+locally uniformly and
+$\en_m(v_m;K_m)\le F(u,U)+o(1)$.
+\end{enumerate}
+These statements hold simultaneously for all continuous targets.  The
+values $F(u,U)$ are locally readable from the arrays with arbitrarily
+small open buffers.
+\end{lemma}
+
+\begin{proof}
+Given $K\Subset K'\Subset U$ and $0<\delta'<\delta$, insert between the
+two tests a finite selector test from Lemma~\ref{inp:sandwich}.  Cover
+the relevant visit labels by finitely many selectors on which the target
+varies by less than a fixed fraction of $\delta-\delta'$, and use
+intermediate rational value constraints.  Completeness of the selected
+graph preserves every edge incidence.  Restriction of an admissible
+function for the larger test is admissible for the intermediate one;
+restriction from the intermediate test is admissible for the smaller
+one.  The intermediate infimum has a retained limit.  Consequently,
+\[
+ \limsup_m M_m(u,K,\delta)
+ \le \liminf_m M_m(u,K',\delta').
+\]
+Taking the two inner suprema proves the assertion about $\liminf$ and
+$\limsup$, including infinite values.
+
+The same comparison proves the lower bound in (i).  For (ii), choose
+almost minimizers on increasing compact sets with decreasing
+tolerances, and take a diagonal sequence using the $\limsup$ formula.
+Tolerance slack also proves lower semicontinuity, directly from
+\eqref{en:relaxation}.  Exhausting $U$ proves inner regularity.  The
+selector library is countable; approximation of a continuous target
+uses its modulus of continuity on each compact, rather than a new
+exceptional event.  This gives the simultaneous assertion.  The
+locality and measurability claims follow from the same selector
+sandwich, with its spatial buffers.
+\end{proof}
+
+We call the sequences in Lemma~\ref{en:envelope}(ii) recovery sequences.
+The next estimate explains why such sequences can be joined on overlaps.
+For an oriented edge $xy$, put
+$\overline v(xy)=(v(x)+v(y))/2$.
+
+\begin{lemma}[Gluing estimate]\label{en:fundamental}
+Let $v_m,w_m$ converge uniformly to the same continuous function on a
+fixed transition region.  Suppose $0\le b_m\le1$ has bounded energy,
+is supported where $v_m$ is defined, and equals one wherever $w_m$ is
+not used.  Then $z_m=b_mv_m+(1-b_m)w_m$ has, on the gluing region,
+\begin{align}\label{en:gluing-estimate}
+ \en_m(z_m)
+ &\le(1+\varepsilon)\sum_e
+ \left(\overline b_m(e)|\dd v_m(e)|^2
+ +(1-\overline b_m(e))|\dd w_m(e)|^2\right)\notag\\
+ &\quad +(1+\varepsilon^{-1})
+       \|v_m-w_m\|_{\infty,\mathrm{tr}}^2\en_m(b_m),
+ \qquad \varepsilon>0.
+\end{align}
+Finite partitions of unity have the corresponding estimate, with an
+error tending to zero when the local approximations agree uniformly on
+overlaps.  Bounded-energy gluing weights exist for every fixed finite
+open cover of a compact set.
+\end{lemma}
+
+\begin{proof}
+The exact difference identity is
+\[
+ \dd z_m=\overline b_m\,\dd v_m
+        +(1-\overline b_m)\,\dd w_m
+        +\overline{v_m-w_m}\,\dd b_m.
+\]
+Apply convexity of the square to the first two terms, followed by
+$|a+c|^2\le(1+\varepsilon)|a|^2+
+(1+\varepsilon^{-1})|c|^2$.  Only the transition region contributes to
+the last term.  For finitely many weights the same identity follows by
+summing and using $\sum_i\dd b_{i,m}=0$.
+
+To construct weights, choose finitely many disks whose smaller disks
+cover the compact set and whose larger disks stay in the prescribed
+members of the cover.  Use Proposition~\ref{nd:cutoffs}, sum the
+resulting cutoffs, and normalize where their sum is at least one.
+Clipping and division on this fixed range preserve bounded energy.
+Extra cutoffs with collars extend the construction to a neighborhood
+of the compact set.  All these choices precede the limit in $m$.
+\end{proof}
+
+\begin{proposition}[Quadratic energy measures]\label{en:measure}
+For each open $U$, the functional $F(\cdot,U)$ is quadratic on its
+finite-energy space and vanishes on constants.  If $u$ has locally
+finite relaxed energy, then
+\[
+                    U\longmapsto F(u,U)
+\]
+is the restriction to open sets of a Radon measure.  For smooth $u$,
+on a smaller ordinary chart,
+\begin{equation}\label{en:smooth-upper}
+                  F(u,U)\le C\int_U|\nabla u|^2\,\dd z.
+\end{equation}
+The same statements hold for $F^*$.
+\end{proposition}
+
+\begin{proof}
+Recovery sequences for $u$ and $v$, the discrete parallelogram identity,
+and the energy lower bound give
+\[
+ F(u+v,U)+F(u-v,U)\le2F(u,U)+2F(v,U).
+\]
+Applying this inequality to $u+v$ and $u-v$ gives the reverse inequality.
+Homogeneity follows directly from the tolerance definition.  Constant
+addition leaves every edge increment unchanged.
+
+For fixed $u$, restriction gives monotonicity and superadditivity on
+disjoint open sets.  To prove subadditivity on $U\cup V$, first fix a
+compact subset of this union.  Recover $u$ in $U$ and $V$, and join the
+recoveries with Lemma~\ref{en:fundamental}; the transition is compactly
+contained in $U\cap V$.  Let $m\to\infty$ and then
+$\varepsilon\downarrow0$ in \eqref{en:gluing-estimate}.  Exhausting the
+compact gives
+\[
+                   F(u,U\cup V)\le F(u,U)+F(u,V).
+\]
+Finite subcovers and inner regularity imply countable subadditivity.
+For completeness, these properties give the required measure criterion
+as follows.  Define
+$\nu(A)=\inf\{F(u,O):A\subset O,\ O\text{ open}\}$.
+This is an outer measure by countable subadditivity.  If two sets have
+positive distance, they have disjoint open neighborhoods; disjoint-open
+superadditivity shows that $\nu$ is additive on such sets.  Thus it is a
+metric outer measure and all Borel sets are measurable.  On an open set
+$O$, monotonicity gives $\nu(O)=F(u,O)$.  Local finiteness and inner
+regularity make this Borel measure Radon.
+
+It remains to prove \eqref{en:smooth-upper}; this will also establish
+local finiteness for smooth targets.  On a compact set inside $U$, use a
+square grid of spacing $r$ and a bounded-overlap family of disk cutoffs
+with radii comparable to $r$.  Normalize these cutoffs to weights
+$b_{z,m}$ which sum to one near the compact, and set
+\[
+                         v_m=\sum_z b_{z,m}u(z).
+\]
+For fixed $r$ this approximates $u$ within its modulus of continuity at
+scale $Cr$.  On each edge the weight increments sum to zero.  Subtracting
+one nearby node value and using bounded overlap bounds the energy by
+\[
+ C\sum_z r^2\sup_{B(z,C'r)}|\nabla u|^2.
+\]
+Indeed each unnormalized cutoff has bounded energy, and normalization
+has the same bound because the denominator is bounded below and only
+boundedly many supports meet an edge.  First take $m$ large for this
+finite collection, and then let $r\downarrow0$.  The displayed sum
+converges to at most a constant times the gradient integral over the
+chosen region.  Exhaustion inside $U$ proves
+\eqref{en:smooth-upper}.  The arguments are unchanged on the dual.
+\end{proof}
+
+\subsection{Integral representation and coercivity}
+
+The energy measures of the two coordinate functions determine the
+whole functional.  Coercivity is a separate step: it follows from
+intersecting level circuits and does not assume geometric regularity of
+the mesh.
+
+\begin{proposition}[Integral representation]\label{en:representation}
+In every ordinary chart there is a measurable symmetric matrix field
+$A$, locally bounded and uniformly positive definite on smaller charts,
+such that, for every open $U$ and every $u\in C(U)$,
+\begin{equation}\label{en:matrix-form}
+ F(u,U)=
+ \begin{cases}
+ \displaystyle\int_U\nabla u^{\mathsf T}A\nabla u\,\dd z,
+                &u\in W^{1,2}_{\mathrm{loc}}(U),\\[4pt]
+ +\infty,       &u\notin W^{1,2}_{\mathrm{loc}}(U).
+ \end{cases}
+\end{equation}
+The integral is allowed to be infinite.  There is an analogous field
+$A^*$ for the dual functional.
+\end{proposition}
+
+\begin{proof}
+We first identify the energy of smooth functions.  Polarize the Radon
+measures from Proposition~\ref{en:measure}.  The measures of the linear
+functions $\ell_1(z)=\Re z$, $\ell_2(z)=\Im z$ and their sums are
+absolutely continuous by \eqref{en:smooth-upper}.  Define the entries of
+$A$ by their densities:
+\[
+ A_{ii}\,\dd z=F(\ell_i,\dd z),\qquad
+ 2A_{12}\,\dd z=
+ F(\ell_1+\ell_2,\dd z)-F(\ell_1,\dd z)-F(\ell_2,\dd z).
+\]
+Quadraticity and the upper bound imply that $A$ is symmetric,
+nonnegative, and locally bounded.  At a Lebesgue point $z$ of these
+densities, compare a smooth $u$ with its affine first-order Taylor
+polynomial $p_z$.  The seminorm inequality and
+\eqref{en:smooth-upper} give
+\[
+ \left|F(u,B(z,r))^{1/2}-F(p_z,B(z,r))^{1/2}\right|
+ \le C^{1/2}|B(z,r)|^{1/2}
+          \sup_{B(z,r)}|\nabla u-\nabla u(z)|.
+\]
+Divide by $|B(z,r)|^{1/2}$ and let $r\downarrow0$.  Differentiation of
+Radon measures yields \eqref{en:matrix-form} for smooth $u$.
+
+We next prove the converse bound, including weak differentiability:
+\begin{equation}\label{en:coercivity}
+                 \int_U|\nabla u|^2\,\dd z\le C F(u,U).
+\end{equation}
+Suppose the right side is finite.  Put a square grid of fixed small
+spacing $r$ in a compactly contained region of $U$.  At every grid node
+$x$, apply Lemma~\ref{en:circuit} to a recovery sequence, using an
+annulus with inner and outer radii $0.7r$ and $0.8r$, and choose a value
+$a_{x,m}$ at a vertex on the resulting circuit.  Buffers are small fixed
+fractions of $r$.
+
+Circuits at neighboring grid nodes share a vertex.  Here is the
+necessary topological verification.  A point near the midpoint of the
+two centers is inside both inner disks.  Points on the two opposite
+sides belong to exactly one inner disk and lie beyond the other outer
+disk; a distant point lies outside both.  The all-lift separation rule
+therefore gives a common interior lift, a common exterior lift, and
+lifts with each of the two opposite memberships.  Two disjoint simple
+circuits on the sphere with these memberships could be neither nested
+nor disjoint in their interiors, a contradiction.  Since both circuits
+belong to the primal graph, their intersection contains a vertex.
+Consequently,
+\[
+ |a_{x,m}-a_{y,m}|^2
+ \le 2\left(\sum_{e\in C_{x,m}}|\dd v_m(e)|\right)^2
+      +2\left(\sum_{e\in C_{y,m}}|\dd v_m(e)|\right)^2
+\]
+for neighbors $x,y$.  Write $B_r$ for the union of the buffered grid
+annuli.  Bounded overlap and \eqref{en:circuit-bound} give
+\[
+ \sum_{x\sim y}|a_{x,m}-a_{y,m}|^2
+ \le C\en_m(v_m;B_r)
+ \le C\bigl(F(u,U)+o(1)\bigr).
+\]
+For the last inequality, this fixed finite union lies in the recovery
+set $K_m$ for all sufficiently large $m$.
+The piecewise affine interpolation of the $a_{x,m}$ on grid triangles
+has Dirichlet energy bounded by this sum times an absolute constant.
+Take $m$ large after fixing the grid, and then let $r\downarrow0$.
+Continuity of $u$ and uniform recovery imply that these interpolations
+converge locally uniformly to $u$.  Weak Sobolev compactness proves
+\eqref{en:coercivity} on the compact region; exhaustion proves it on
+$U$.  Applying it to smooth affine functions on arbitrary disks gives
+$A\ge cI$ locally.
+
+Finally, let $u\in C(U)\cap W^{1,2}_{\mathrm{loc}}(U)$.  On smaller
+regions, smoothing approximates $u$ both uniformly and in $W^{1,2}$.
+Lower semicontinuity and the smooth upper bound first extend that upper
+bound to such continuous Sobolev functions.  Apply the extended bound
+to the difference between $u$ and its smooth approximations.  Their
+$F$-seminorm difference tends to zero, so the smooth formula passes to
+$u$.  Exhaustion gives \eqref{en:matrix-form}, including infinite
+integrals.  If local Sobolev regularity fails, \eqref{en:coercivity}
+forces infinite energy.  All steps apply equally to $F^*$.
+\end{proof}
+
+\begin{lemma}[The deterministic scalar]\label{en:scalar}
+In each coupled extraction there is a deterministic
+$\lambda\in(0,\infty)$ such that
+\begin{equation}\label{en:scalar-form}
+                    A=A^*=\lambda I
+\end{equation}
+almost everywhere in all ordinary charts.  At this stage $\lambda$
+may depend on the extracted rate sequence.
+\end{lemma}
+
+\begin{proof}
+The derivatives of the energy measures of linear functions are point
+germs of the arrays: use the tolerance definition on successively
+smaller disks, with arbitrarily small open buffers.  Choose measurable
+$\limsup$ versions and a fixed default on the null set where
+Lebesgue differentiation fails.  Lemma~\ref{nd:germs} applies to these
+matrix-valued readouts (or to a bounded injective encoding of their
+entries), so $A$ and $A^*$ are deterministic almost everywhere.
+Translation covariance makes each field a constant matrix on overlapping
+ordinary charts.  Under a rotation, the matrix transforms by
+$A\mapsto RAR^{\mathsf T}$.  Rotation covariance, in particular a
+quarter-turn, therefore makes each symmetric matrix scalar.  These
+uses of covariance are precisely those in
+Remark~\ref{nd:germ-symmetries}, and compare local laws with their actual
+field transformations.  Primal--dual reflection
+(Lemma~\ref{inp:reflection}) equates the two scalar values.  The upper
+bound and coercivity give $0<\lambda<\infty$.
+\end{proof}
+
+\subsection{Annular reciprocity fixes the scalar}
+
+We have proved existence and isotropy of the limiting energy.  To
+identify its normalization, compare a condenser potential with a
+potential whose value increases by one on going around an annulus.
+The first comparison rotates a discrete current to a dual potential;
+the reverse comparison recovers an angular potential with its period
+\emph{exactly} preserved.  Both comparisons use only interior graph
+pieces with collars.  The current--potential correspondence is the
+standard planar-network duality; see \cite[Section~9.2]{LP}.
+Its use to determine a scalar effective conductivity has a classical
+precedent in Dykhne's planar self-duality argument \cite{Dykhne}.
+Here we give both inequalities, including the recovery needed on our
+irregular graph pieces.
+
+Let
+\[
+ A(r,R)=\{z:r<|z|<R\},\qquad
+ C_0=\frac{2\pi}{\log(R/r)},
+\]
+where the closed annulus and a neighborhood of it lie inside an ordinary
+chart.  Thus $C_0$ is the ordinary Dirichlet capacity between its two
+boundary circles.  The radial minimizer is
+$\log(|z|/r)/\log(R/r)$.  A locally Sobolev potential with additive
+period $P$ around the annulus has energy at least
+\begin{equation}\label{en:period-bound}
+             \int_{A(r,R)}|\nabla v|^2\,\dd z
+                    \ge\frac{P^2}{C_0}.
+\end{equation}
+Indeed, on almost every circle,
+$\int_0^{2\pi}\partial_\theta v\,\dd\theta=P$; apply
+Cauchy--Schwarz and integrate $\dd s/s$.  Equality holds for
+$v=P\theta/(2\pi)$, interpreted in local branches.
+
+\begin{lemma}[Condenser convergence]\label{en:condenser}
+Prescribe $u_m=0$ on every primal vertex over $|z|\le r$ and $u_m=1$
+on every vertex over $|z|\ge R$, using a protected cap outside the chart.
+Let $C_m$ be the minimum primal energy.  Then
+\begin{equation}\label{en:condenser-limit}
+                             C_m\longrightarrow\lambda C_0.
+\end{equation}
+The minimizers converge uniformly, including at the two plate
+boundaries, to the round radial condenser potential.
+\end{lemma}
+
+\begin{proof}
+All nonzero increments of the minimizer project into any fixed collar
+of the closed annulus eventually.  Indeed, both endpoints of an edge
+strictly on one plate have its prescribed value, and the overlay mesh
+vanishes.  Thus the cap contributes no uncontrolled energy term.
+
+For the upper bound, approximate the radial potential by smooth
+functions which are constant on slightly wider plate collars and whose
+energies tend to $C_0$.  Recover them uniformly on a neighborhood of the
+annulus.  Bounded-energy weights in the flat collars paste in the exact
+constants at a vanishing energy cost, by
+Lemma~\ref{en:fundamental}.  Lemma~\ref{en:scalar} gives
+$\limsup C_m\le\lambda C_0$.
+
+The minimizer lies in $[0,1]$ and is harmonic off the plates.  Its bounded
+energy and Lemma~\ref{en:harmonic-compactness} give compactness in the
+open annulus.  At a plate-boundary point, use the stacks of rings from
+that lemma, small enough to avoid the other plate.  Every surrounding
+circuit meets a path lying strictly on the plate side: choose a tube
+there from the inner well to outside the ring and lift it by
+Proposition~\ref{inp:topology}.  Thus the circuit contains a vertex
+with the prescribed value.  A circuit of small variation is uniformly
+close to that value.  Inside it all remaining vertices are harmonic or
+have that same prescribed value, so the maximum principle proves
+continuity uniformly up to the plate.  Fixed stacks followed by finite
+covers give the assertion simultaneously along both boundaries.
+
+Any uniform subsequential limit has the condenser boundary values.
+The variational lower bound and Lemma~\ref{en:scalar} give energy at
+least $\lambda C_0$.  This proves \eqref{en:condenser-limit}; equality
+in the round Dirichlet principle identifies the limit and gives the
+last assertion.
+\end{proof}
+
+\begin{figure}[tb]
+\centering
+\begin{tikzpicture}[scale=1.05,>=Stealth]
+ \fill[gray!10,even odd rule] (0,0) circle (1.45) (0,0) circle (.55);
+ \draw (0,0) circle (1.45);
+ \draw (0,0) circle (.55);
+ \node[font=\small] at (0,0) {$u_m=0$};
+ \node[anchor=east] at (-1.52,0) {$u_m=1$};
+ \foreach \a in {40,140,220,320}{
+   \draw[blue!65!black,->,thick]
+       (\a:.64) -- (\a:1.29);
+ }
+ \draw[red!65!black,->,thick] (18:1.04)
+       arc[start angle=18,end angle=153,radius=1.04];
+ \node[red!65!black,anchor=south] at (0,1.54)
+       {dual period $C_m$};
+ \node[blue!65!black,anchor=north] at (0,-1.6)
+       {primal current $\dd u_m$};
+ \draw[->,thick] (1.9,.6) -- (3.1,.6);
+ \node[align=center] at (2.5,.05) {unit-period\\recovery};
+ \begin{scope}[xshift=5cm]
+  \fill[gray!10,even odd rule] (0,0) circle (1.45) (0,0) circle (.55);
+  \draw (0,0) circle (1.45);
+  \draw (0,0) circle (.55);
+  \draw[red!65!black,->,thick] (18:1.04)
+       arc[start angle=18,end angle=330,radius=1.04];
+  \node at (0,0) {$\alpha_m$};
+  \node[red!65!black,anchor=south] at (0,1.54)
+       {exact period $1$};
+  \node[anchor=north] at (0,-1.6)
+       {$|\langle\dd u_m,\alpha_m\rangle|=1$};
+ \end{scope}
+\end{tikzpicture}
+\caption{The two annular comparisons.  Rotating the condenser current
+gives a dual potential with period $C_m$ and hence $\lambda\le1$.
+An exact unit-period dual recovery pairs with the unit condenser jump
+to give $\lambda\ge1$.  This is a schematic of the test annulus;
+the projected discrete graph need not resemble a regular mesh.}
+\label{en:fig-reciprocity}
+\end{figure}
+
+\begin{lemma}[Rotation of a current]\label{en:current-period}
+Rotate the primal current $\dd u_m$ from
+Lemma~\ref{en:condenser} to the dual edges, with one consistent crossing
+orientation.  In the annulus this is a closed dual cochain with harmonic
+local potential branches and period $\pm C_m$.  The branches have
+subsequential locally uniform limits with period $\pm\lambda C_0$ and
+with total dual relaxed energy at most $\lambda C_0$ over one turn.
+\end{lemma}
+
+\begin{proof}
+Closedness around a dual face is the primal harmonic equation.
+Harmonicity of a potential branch at a dual vertex is the telescoping
+sum of $\dd u_m$ around the corresponding primal face.  These identities
+count all incidences and are valid also for loops and repeated edges.
+The squared dual increments sum to at most $C_m$ on every interior
+annular piece.
+
+We justify the existence of branches without assuming regular discrete
+annular boundaries.  In a disk with clearance inside the annulus,
+arbitrary dual lifts in a smaller disk are connected within the larger
+one.  Loops used to integrate the cochain can be filled inside a further
+buffer, by Proposition~\ref{inp:topology}; every primal vertex there is
+harmonic.  Cellular closedness therefore makes the integral independent
+of the chosen path.  On connected overlaps the resulting potentials
+differ by constants.
+
+A dual loop with projected winding one can be built along fixed tubes
+and closed using small-image joining.  The integer-winding statement of
+Proposition~\ref{inp:topology} makes its winding upstairs one, up to the
+fixed orientation sign.  Its current integral is the total current
+entering one plate, since all intervening primal vertices are harmonic.
+Discrete summation by parts gives total current $C_m$: the potential
+difference is one and its energy is $C_m$.  Hence the period is
+$\pm C_m$.
+
+Normalize branches on a finite overlapping disk cover of any compact
+subannulus, propagating constants through chosen overlaps.
+In particular, fix a cyclic chain of disks around one positively
+oriented circle.  Continue a normalized branch along this chain,
+matching on each connected overlap; on returning to the first disk,
+the continued branch differs from the original by $\pm C_m$.
+Lemma~\ref{en:harmonic-compactness} gives local uniform subsequential
+limits.  Their transition constants converge, so their period is
+$\pm\lambda C_0$.  Exhausting the annulus yields branches everywhere.
+For the energy bound, use disjoint boxes lying in branch disks.  Their
+discrete edge energies do not overlap, and their sum is at most $C_m$.
+Apply the local lower bound and then exhaust the annulus up to sets of
+Lebesgue measure zero.  This gives the asserted bound over one turn.
+\end{proof}
+
+Combining Lemmas~\ref{en:scalar} and \ref{en:current-period} with
+\eqref{en:period-bound} gives
+\begin{equation}\label{en:lambda-upper}
+        \lambda C_0\ge
+        \lambda\frac{(\lambda C_0)^2}{C_0},
+        \qquad\text{hence }\lambda\le1.
+\end{equation}
+For the reverse inequality, angular recovery must retain its period
+before taking any limit.  We record first the measure observation that
+makes the gluing sharp.
+
+\begin{lemma}[Energy measures of recovery sequences]\label{en:recovery-measures}
+Let $V$ be a bounded patch and let $p$ be smooth on a neighborhood of its
+closure.  Choose a recovery sequence on compact sets exhausting $V$
+whose total energy tends to $F^*(p,V)$.  Count only edges with both
+endpoints in these recovery sets, and put their squared increments
+at any of the corresponding projected edge locations.  These measures
+converge vaguely in $V$ to $\lambda|\nabla p|^2\,\dd z$.  In particular,
+energies weighted by a compactly supported continuous function converge
+to the corresponding weighted integral.
+\end{lemma}
+
+\begin{proof}
+The total masses are bounded.  Let $\nu$ be a subsequential vague limit.
+To check its local lower bound, take open sets $W\Subset O\Subset V$
+and a compact set $K\subset O$ containing $W$ with a collar.  The energy
+tests inside $W$ are eventually counted by the edge measure $\nu_m(K)$.
+Lemma~\ref{en:envelope} and compact-set Portmanteau give
+\[
+ F^*(p,W)\le\liminf_m\nu_m(K)
+          \le\limsup_m\nu_m(K)\le\nu(K)\le\nu(O).
+\]
+Exhaust $O$ by such $W$.  Thus $\nu$ dominates
+$\lambda|\nabla p|^2\,\dd z$.  Its total mass is at most the limiting
+total recovery energy, which is the total mass of that same measure.
+Equality follows.  Every subsequential vague limit is consequently the
+stated measure, proving the claim.
+\end{proof}
+
+\begin{lemma}[Recovery with exact unit period]\label{en:angular-recovery}
+For every sufficiently small $\eta>0$, let
+$A_\eta=A(r-\eta,R+\eta)$.  On the dual edges in a neighborhood of the
+closed annulus there are closed cochains $\alpha_m$ with exact period
+one and
+\begin{equation}\label{en:angular-energy}
+ \limsup_{m\to\infty}\sum_{e^*\subset A_\eta}|\alpha_m(e^*)|^2
+ \le\lambda\left(\frac1{C_0}+o_\eta(1)\right),
+             \qquad o_\eta(1)\longrightarrow0.
+\end{equation}
+The cochains are closed at every cell whose star lies in this
+neighborhood; the region can be chosen to contain every edge where
+$\dd u_m\ne0$ for all large $m$.
+\end{lemma}
+
+\begin{proof}
+Take finitely many disks covering a slightly larger closed annulus,
+each avoiding the origin, with angle branches
+$p_i=\theta_i/(2\pi)$.  Recover $p_i$ uniformly on larger branch disks.
+Lemma~\ref{en:recovery-measures} applies to these recoveries, denoted
+$v_{i,m}$.  Take a smooth partition of unity $b_i$ with supports compactly
+inside the branch disks.  Recover its weights, clip them to be
+nonnegative, cut them off in fixed support collars, and normalize their
+sum.  This gives $b_{i,m}\to b_i$ uniformly, bounded energies, supports
+inside the branch disks, and $\sum_i b_{i,m}=1$ near the annulus.
+
+There is a useful canonical cochain even though its own energy need
+not be bounded.  On an oriented dual edge, let
+\[
+ a_m(e^*)=\frac1{2\pi}\int_{\pi_m(e^*)}\dd\arg z.
+\]
+Here the integral means the endpoint increment of a continuous argument
+lift along the projected edge.  Every sufficiently small edge image
+lies in a branch disk, so this is well defined.  Vanishing face mesh
+makes $a_m$ closed around every cell under consideration, and its
+period around a loop is that loop's projected integer winding.
+
+On the support of $b_{i,m}$ the error
+$v_{i,m}-p_i\circ\pi_m$ is a single-valued function tending uniformly
+to zero.  Define the globally single-valued error and cochain by
+\begin{equation}\label{en:angular-error}
+ q_m=\sum_i b_{i,m}(v_{i,m}-p_i\circ\pi_m),
+                 \qquad \alpha_m=a_m+\dd q_m.
+\end{equation}
+The support collars make this definition independent of how terms are
+extended by zero.  Since a gradient has zero periods, $\alpha_m$ is
+closed and has period exactly one, in the projected orientation.
+This conclusion is an identity at each finite resolution.
+
+To bound its energy, apply the discrete product identity on each edge.
+On a branch disk, $\dd(p_i\circ\pi_m)=a_m$.  Since the weights sum to
+one, the terms containing $a_m$ cancel to give
+\[
+ \alpha_m(e^*)=
+ \sum_i\overline b_{i,m}(e^*)\,\dd v_{i,m}(e^*)
+ +\sum_i
+ \overline{v_{i,m}-p_i\circ\pi_m}(e^*)\,\dd b_{i,m}(e^*).
+\]
+The second sum tends to zero in edge $\ell^2$, by uniform convergence
+of the errors and bounded energy of the finitely many weights.
+Convexity bounds the squared norm of the first sum by the weighted sum
+of local recovery energies.  Lemma~\ref{en:recovery-measures} and
+$b_{i,m}\to b_i$ turn its limit into
+\[
+ \lambda\sum_i\int_{A_\eta} b_i|\nabla p_i|^2\,\dd z
+ =\lambda\int_{A_\eta}\left|\nabla\frac{\theta}{2\pi}\right|^2\,\dd z
+ =\frac{\lambda}{2\pi}\log\frac{R+\eta}{r-\eta}.
+\]
+One may compute first on a slightly larger closed annulus inside the
+cover; its boundary circles have zero limiting energy measure.
+The edge averages $\overline b_{i,m}$ converge uniformly to $b_i$ at
+the chosen edge locations, by vanishing mesh.  This justifies both
+the weighted passage to the limit and restriction to the annulus.
+This proves \eqref{en:angular-energy}.  Finally, $u_m$ is constant on
+each plate, so its nonzero increments lie in $A_\eta$ eventually by
+vanishing edge mesh.  Choosing additional fixed collars gives the
+assertion about stars.
+\end{proof}
+
+\begin{proposition}[Exact conductance normalization]\label{en:reciprocity}
+The scalar in Lemma~\ref{en:scalar} equals one.
+\end{proposition}
+
+\begin{proof}
+Only $\lambda\ge1$ remains after \eqref{en:lambda-upper}.
+Pair $\dd u_m$ with the cochain $\alpha_m$ of
+Lemma~\ref{en:angular-recovery}, matching each primal edge with its
+crossing dual edge and using the consistent crossing sign.  Then
+\begin{equation}\label{en:pairing}
+                         \left|\sum_e\dd u_m(e)\alpha_m(e^*)\right|=1.
+\end{equation}
+To see this at the discrete level, replace $u_m$ by the indicator of the
+outer side of an intermediate circle.  Their difference is supported
+strictly inside the region of closedness of $\alpha_m$.  Cellular
+summation by parts therefore leaves the pairing unchanged.  The
+indicator's gradient pairs with $\alpha_m$ along the oriented dual
+boundary chain separating the two wells.  This chain has total winding
+one up to sign by Proposition~\ref{inp:topology}; the cochain has exact
+unit period by \eqref{en:angular-error}.  This proves
+\eqref{en:pairing}.  The argument uses the boundary chain with its
+multiplicities and does not require a simple discrete plate boundary.
+
+Cauchy--Schwarz, \eqref{en:condenser-limit}, and
+\eqref{en:angular-energy} now give
+\[
+ 1\le C_m\sum_{e^*\subset A_\eta}|\alpha_m(e^*)|^2
+ \quad\Longrightarrow\quad
+ 1\le\lambda C_0\,\lambda\left(C_0^{-1}+o_\eta(1)\right).
+\]
+Let $\eta\downarrow0$.  Thus $1\le\lambda^2$, and
+\eqref{en:lambda-upper} proves $\lambda=1$.
+\end{proof}
+
+\begin{theorem}[Uniform conformal energy limit]\label{en:variational}
+In every coupled extraction and every ordinary chart, both unnormalized
+network energies have the same inner relaxed limit:
+\begin{equation}\label{en:final-energy}
+                   F(u,U)=F^*(u,U)=\int_U|\nabla u|^2\,\dd z
+\end{equation}
+for every $u\in C(U)\cap W^{1,2}_{\mathrm{loc}}(U)$, with infinite value
+when the integral diverges or local Sobolev regularity fails.  The local
+uniform lower bounds and recovery conclusions of
+Lemma~\ref{en:envelope}, the gluing estimate, and harmonic compactness
+hold simultaneously.  In particular, the limiting energy has no
+subsequence-dependent multiplicative constant.
+\end{theorem}
+
+\begin{proof}
+Combine Proposition~\ref{en:representation}, Lemma~\ref{en:scalar}, and
+Proposition~\ref{en:reciprocity}.  The preceding construction used the
+same countable buffered tests throughout, so its simultaneous and
+locality assertions persist.  All arguments apply in every fixed
+quantum-unit offset of the extraction.  Their compatibility is the
+transport statement of Proposition~\ref{inp:array-interface}.
+\end{proof}
diff --git a/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/build/sections/05-sphere.tex b/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/build/sections/05-sphere.tex
new file mode 100644
index 0000000000000000000000000000000000000000..10840441ecd202474118c36eaaab25ce7d6a3899
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@@ -0,0 +1,420 @@
+\section{From local plane energies to the sphere}\label{sec:sphere}
+
+The energy identification of Section~\ref{sec:energy} was obtained in
+ordinary patches of a quantum cone.  We now transfer it to the fixed-area
+sphere and include the point at which the contour begins and ends.  The
+output is a global variational statement in the same coupling as the
+metric-measure convergence.  This common coupling will later allow us to
+compare the walks and their metric spaces simultaneously.
+
+Write $\dd A$ for round area on the conformal sphere, and set
+\[
+ \mathcal D(u)=\int_{S^2}|\nabla u|^2\,\dd A
+ \quad\text{for }u\in C(S^2)\cap W^{1,2}(S^2),
+\]
+with $\mathcal D(u)=+\infty$ for other continuous functions.  The same
+notation on an open coordinate domain means the integral over that
+domain, with local weak differentiability required.  Conformal
+invariance makes these definitions independent of the chosen chart.
+
+\subsection{Transfer on the interior of a contour interval}
+
+First extend the plane conclusion beyond a single ordinary patch.
+Retain all forward dyadic unit changes $v=2^k$ from
+Proposition~\ref{inp:transport}.  For each fixed $k$, the transformed
+rate still tends to infinity, so Theorem~\ref{en:variational} applies
+on the ordinary patches inside that cone's normalization disk.
+Under the joint zoom \eqref{inp:joint-zoom}, these disks exhaust the
+plane away from the cone point.  Pulling the countably many
+probability-one conclusions back to the original coordinate proves
+the local energy identity everywhere off that point.  Both the
+network costs and the two-dimensional Dirichlet integral are unchanged
+by these transformations.
+
+For $0<d<1/2$, let $I_d=[d,1-d]$.  The decorated interval surface
+associated with the contour on $I_d$ has an interior determined by those
+increments.  A compact subset of that interior, together with a small
+collar, has all its contour visits strictly inside $I_d$.  Thus its
+eventual discrete incidences are read from the internal word; no
+information from either omitted end of the tour is needed.  These are
+the interval-surface and complete-core assertions used in
+Lemma~\ref{inp:complete-core}.
+
+Here is the part of the excursion comparison that permits new
+variational observations.  It concerns the joint law of the arrays
+and the word, not independence after weak convergence.
+
+\begin{lemma}[Additional observations under excursion conditioning]
+\label{sph:block-density}
+Fix $0<a<b<1$.  Let $W_N$ be the diffusively rescaled free axial
+increment path on $[\lfloor aN\rfloor,\lfloor bN\rfloor]$, with
+linear interpolation, and let $Z_N$ be any measurable function of
+these increments with values in a fixed compact metrizable space.
+Suppose $(W_N,Z_N)\Rightarrow(W,Z)$ along a sequence of even $N$.
+There is a nonnegative continuous function $f_{a,b}$ of the limiting
+path, with expectation one, such that the corresponding pair under
+quadrant-excursion conditioning converges to the law
+\[
+ f_{a,b}(w)\,\Law(W,Z)(\dd w,\dd z).
+\]
+In particular, every probability-one property of $(W,Z)$ also
+holds for this conditioned limit.  A countable product of
+compact-valued observations is allowed for $Z_N$.
+\end{lemma}
+
+\begin{proof}
+The lemma \emph{Excursion densities} in the spanning-tree sphere
+section of \cite{MetricMaps} proves the following statements for
+the density $f_N$ of this increment block.  If $w_N\to w$
+uniformly, then $f_N(w_N)\to f_{a,b}(w)$; the limit depends
+continuously on the two coordinate minima and the net increment.
+Moreover, $\sup_N\E[f_N(W_N)^2]<\infty$ under the free law.
+These assertions follow there from the entrance-and-exit killed-walk
+density and its Gaussian bounds.
+
+Realize the assumed joint convergence almost surely.  For any bounded
+continuous test $\Phi$, first truncate the densities at height $K$.
+Sequential convergence and bounded convergence give convergence of
+the weighted expectations with truncated densities.  The error in
+removing truncation is at most
+$K^{-1}\|\Phi\|_\infty\sup_N\E[f_N(W_N)^2]$, and the analogous
+limit error has the same bound.  Hence
+\[
+ \E[f_N(W_N)\Phi(W_N,Z_N)]
+ \longrightarrow\E[f_{a,b}(W)\Phi(W,Z)].
+\]
+Taking $\Phi=1$ gives total mass one.  The finite-resolution
+change of density depends only on the increment block, so its
+weighted expectation is precisely that of the additional block
+data under excursion conditioning.  This proves the assertion.
+\end{proof}
+
+\begin{proposition}[Energy identification under the sphere conditioning]
+\label{sph:transfer}
+Consider any joint subsequential extraction of the sphere contour and
+the countable internal variational arrays on $I_d$, for every rational
+$d\in(0,1/2)$, using the extraction convention of
+Proposition~\ref{inp:array-interface}.  Almost surely, on every open
+domain compactly contained in the interior of one of these interval
+surfaces, both primal and dual inner relaxed energies equal
+$\mathcal D$.  This identity holds simultaneously for all continuous
+targets, with the value $+\infty$ when the target is not locally
+Sobolev or its Dirichlet integral diverges.
+\end{proposition}
+
+\begin{proof}
+At each resolution retain the energy minima with rational value
+constraints and finite unions of relative-time intervals as selectors.
+Only increments within $I_d$ enter these internal graphs and tests.
+Their compactified values are therefore permissible additional block
+data in Lemma~\ref{sph:block-density}.  The corresponding free
+experiment is the bilateral sewing at rate $m=2n$.  In that experiment
+adjoin these jump-indexed internal tests to the Poisson-instruction
+arrays before extraction.  Each added test is computed from its exact
+integer increment block and is unchanged when an independent Poisson
+clock is attached to the word.  The earlier local identity applies
+in this enlarged extraction.  To read it from the internal tests,
+use complete pieces and fixed spatial collars as described below.
+The vanishing discrepancy of the rescaled clocks only
+locates the selectors inside these collars; the numerical energy
+outputs themselves are retained unchanged, without a continuity
+claim for them under relabeling.
+
+Starting with the given conditioned extraction, take a further
+subsequence on which the corresponding free block and its internal
+array converge jointly.  This is possible by tightness of the free
+contour and compactness of the array space, simultaneously for the
+countably many intervals by diagonal extraction.  The weighted-limit
+lemma then identifies the already specified conditioned block limit
+with the density tilt of this free joint limit.
+
+Theorem~\ref{en:variational} identifies the inner relaxed energies in
+that bilateral experiment, away from its distinguished point.  The
+distinguished point does not belong to this interval surface:
+time zero is almost surely a singleton Brownian contour fiber.
+Indeed, each coordinate takes values strictly below its value at zero
+arbitrarily close to zero on both sides.  Thus no nontrivial
+identification link in either coordinate can have zero as an
+endpoint, and neither can the equivalence relation generated by
+those links identify zero with another time.  Since $I_d\subset(0,1)$,
+its image excludes the distinguished point.
+
+To express the local energy conclusion in terms of the block alone,
+reconstruct its abstract decorated interval surface $H_d$ from the
+limiting increments using the interval reconstruction in
+Proposition~\ref{inp:sewing}.  This reconstruction supplies the surface
+and its restricted curve modulo conformal coordinates.  Evaluate
+the local identity in a conformal representative, using a countable
+atlas and compact exhaustions of its interior; conformal invariance
+makes the event independent of that representative.  Both the
+restricted curve locating the labels and the coordinate domains for
+the targets belong to this reconstructed interval surface.
+For a compact $K$ in one such domain and an open enlargement
+$U$ with a collar, Lemma~\ref{inp:complete-core} supplies all incidences
+near $K$ from the internal word.  The selectors of
+Lemma~\ref{inp:sandwich} put every constraint on $K$ between
+constraints on internal graphs contained in $U$.  Rational tolerances
+and compact exhaustion therefore express the inner relaxed energy
+using only the limiting block and its retained arrays.  A change of
+the conformal representative preserves both the transported energy
+identity and the Dirichlet integral.  The resulting assertion is
+intrinsic to $H_d$; the surrounding cone or sphere embedding supplies
+no further data for its definition.
+
+The simultaneous quantifier on targets can also be retained in this
+transfer.  On a compact coordinate set use a countable dense library
+of continuous functions and rational tolerances.  Uniform closeness
+of two targets enlarges or shrinks a tolerance constraint by at most
+that distance.  The library therefore describes all the relaxed
+value tests, jointly measurably in the retained arrays and the target.
+This is a discretization of constraints, not an approximation of
+Dirichlet energy merely from values on a uniformly dense library.
+For varying charts use a countable atlas and compact exhaustions.
+The usual Dirichlet integral, with its infinite-value convention,
+is likewise Borel on continuous targets: weak derivatives and their
+energy can be tested against countably many smooth functions on the
+exhaustion.  Existence of a continuous target violating the identity
+is consequently an analytic event, obtained by projection from its
+Polish function space.  The event of equality for all targets is
+universally measurable.  Theorem~\ref{en:variational} gives precisely
+this simultaneous event under the free law, so absolute continuity
+applies to it without a target-dependent exceptional set.
+
+Absolute continuity of the joint block law now transfers the
+probability-one identity to the excursion experiment.  A countable
+intersection over $d$, charts, and exhaustions proves the result.
+The rest of the sphere contour may be retained in this coupling: the
+complement of the event already has zero probability in the internal
+marginal, hence also in the joint law.
+\end{proof}
+
+The interval interiors exhaust the sphere away from the tour endpoint,
+denoted by $p$.  In fact, the inverse image under the continuous contour
+of a compact set disjoint from $p$ is compact and avoids times $0$ and
+$1$; the same holds after a sufficiently small spatial enlargement.
+The interval-surface identification then places the set and its collar
+in an interior with $d>0$.  Consequently
+Proposition~\ref{sph:transfer} gives the local energy identity at every
+point other than $p$.
+
+\subsection{Removing the tour endpoint}
+
+There are two different issues at $p$.  A limit of bounded-energy
+functions must have no extra Sobolev singularity there, and recovery
+sequences must incur no extra discrete energy there.  The following
+elementary fact handles both.
+
+\begin{lemma}[Continuous Sobolev removability]\label{sph:removability}
+Let $p$ be a point of a smooth two-dimensional surface.
+If $u$ is continuous near $p$, is locally $W^{1,2}$ off $p$, and has
+finite Dirichlet integral near $p$, then $u$ is $W^{1,2}$ across $p$.
+Moreover, if $u$ is continuous and Sobolev on a compact surface, there
+are continuous Sobolev functions $u_r$, constant on a neighborhood of
+$p$ and equal to $u$ outside a shrinking disk, such that
+\[
+ \|u_r-u\|_\infty\longrightarrow0,
+ \qquad
+ \|u_r-u\|_{W^{1,2}}\longrightarrow0.
+\]
+\end{lemma}
+
+\begin{proof}
+Work in a coordinate disk centered at $p$.  A point has zero
+two-dimensional Sobolev capacity: radial logarithmic cutoffs equal to
+one on a shrinking inner disk and zero outside a fixed disk have
+Dirichlet energies tending to zero.  Testing the weak-derivative
+identity off $p$ with the complementary cutoffs proves the identity
+across $p$.  More explicitly, the additional term containing the
+cutoff gradient tends to zero by Cauchy--Schwarz, since $u$ is bounded
+on the support of a fixed smooth test function, while the term
+containing $\nabla u$ converges by its local square integrability.
+
+For the approximation choose a smooth cutoff $\chi_r$ equal to one
+on $B(p,r)$, zero outside $B(p,2r)$, taking values in $[0,1]$, and
+with $\int|\nabla\chi_r|^2\,\dd A\le C$ independently of $r$.
+Set
+\[
+ u_r=u(p)+(1-\chi_r)(u-u(p)).
+\]
+The uniform error is bounded by the oscillation of $u$ on $B(p,2r)$.
+For the gradient error,
+\[
+ \int|\nabla(u_r-u)|^2\,\dd A
+ \le 2\int_{B(p,2r)}|\nabla u|^2\,\dd A
+       +2C\sup_{B(p,2r)}|u-u(p)|^2,
+\]
+and both terms tend to zero.  The ordinary $L^2$ error follows from
+uniform convergence.
+\end{proof}
+
+In a coupled extraction, choose a location $\pi_n(v)\in S^2$ for each
+primal vertex from one of its contour visits; different choices have
+uniformly vanishing discrepancy.  Use $\pi_n^*$ on the dual.
+For functions on the varying vertex sets, uniform convergence to $u$
+means
+\[
+ \max_{v\in V_n}|u_n(v)-u(\pi_n(v))|\longrightarrow0.
+\]
+The small-image joining and mesh assertions in the input guarantee that
+this convention agrees with the local comparisons already used.
+Harmonicity on an open set $U$ means that the graph Laplacian vanishes
+at every vertex whose location is in $U$.  For a compact subset of
+$U$, all adjoining stars also lie in $U$ once $n$ is large; thus
+this convention permits the buffered uses of
+Lemma~\ref{en:harmonic-compactness}.
+
+\begin{theorem}[Global sphere energies]\label{sph:energy}
+From any subsequence of the tree-decorated sphere maps, one can take a
+further joint extraction and a coupling with the limiting ordinary
+quantum sphere such that the following statements hold almost surely.
+\begin{enumerate}
+\item If $u_n$ converges uniformly to $u\in C(S^2)$, then
+\[
+ \liminf_{n\to\infty}\en_n(u_n)\ge\mathcal D(u).
+\]
+\item For each $u\in C(S^2)\cap W^{1,2}(S^2)$ there are uniformly
+convergent approximations $u_n$ for which
+\[
+ \en_n(u_n)\longrightarrow\mathcal D(u).
+\]
+Both assertions hold on the dual with $\en_n^*$ and $\pi_n^*$.
+\item On every open set, graph-harmonic functions with bounded energy
+and bounded absolute value are locally equicontinuous under these
+locations and have locally uniformly convergent subsequences.  This
+holds also on the dual and at the tour endpoint.
+\end{enumerate}
+All these conclusions may be required in the same extraction as the
+metric and measure comparison below.
+\end{theorem}
+
+\begin{proof}
+Retain the internal tests of Proposition~\ref{sph:transfer} and also
+the whole-sphere tests defining uniform relaxed energy.  The latter
+are countable compactified arrays; they need no local conditional
+kernel.  The selector sandwich applies to them in the same way.
+
+Suppose first that $\liminf_n\en_n(u_n)<\infty$ and pass to a subsequence
+realizing this liminf.  On disjoint compact-interior patches away from
+$p$, Proposition~\ref{sph:transfer} bounds their limiting Dirichlet
+integrals by the discrete energies in those patches.  Exhausting the
+complement of $p$ gives
+\[
+ \int_{S^2\setminus\{p\}}|\nabla u|^2\,\dd A
+ \le\liminf_n\en_n(u_n).
+\]
+Lemma~\ref{sph:removability} supplies the weak derivative at $p$.
+This proves the first assertion, including its infinite-value case.
+
+For the upper bound, first take a continuous Sobolev target that is
+constant near $p$.  Cover the complement of a smaller constant disk
+by finitely many patches from Proposition~\ref{sph:transfer}.
+Subdivide it into finitely many smaller coordinate regions with
+piecewise smooth boundaries, each contained in one patch, and thicken
+these regions slightly.  The boundaries have zero round area, so
+their collars can be chosen with arbitrarily small total integral
+of $|\nabla u|^2$.  Local recovery on the thickened regions and
+successive applications of Lemma~\ref{en:fundamental} now give
+recovery on the compact complement: the main energy is bounded by
+the sum of the target energies on those regions, and only the
+chosen small collars are counted more than once.  The error terms
+from gluing tend to zero by uniform agreement on each fixed
+collar.  This proves recovery with arbitrarily small energy slack.
+Extend by the constant on the
+smaller disk.  To make the extension exact, use a cutoff whose
+transition lies in a fixed annulus where the target is constant.
+Apply the finite-cover recovery just proved to a slightly larger
+annular neighborhood and a smooth cutoff $b$ that is constant on
+its two collars.  If $b_n$ is its recovery and
+$\delta_n>\|b_n-b\|_\infty$ tends to zero, replace $b_n$ by
+\[
+ \left(\frac{b_n-\delta_n}{1-2\delta_n}\right)\vee0\wedge1.
+\]
+This is exactly zero and one on the respective flat collars, and
+its energy is at most $(1-2\delta_n)^{-2}\en_n(b_n)$.
+The functions being joined differ by $o(1)$
+on the transition, so the extra term in the gluing estimate is $o(1)$.
+The mesh tends to zero, hence every edge crossing either gluing
+boundary is contained in the fixed collar for all sufficiently large
+$n$.
+
+For a general continuous Sobolev $u$, apply the approximation
+$u_r$ of Lemma~\ref{sph:removability}.  Recover each fixed $u_r$ first,
+then use a diagonal choice of $r\downarrow0$.  Uniform convergence and
+convergence of $\mathcal D(u_r)$ give the desired upper bound; the
+first assertion makes it an equality.  The proof for the dual is
+identical.
+
+Away from $p$, harmonic compactness follows from
+Lemma~\ref{en:harmonic-compactness}.  To include $p$, take any fixed
+finite stack of disjoint annuli centered at $p$, each of the same
+aspect ratio, in a domain of harmonicity.  Every transition annulus
+avoids $p$.  By the preceding local recovery, its dual cutoff can
+be taken with energy bounded by a constant depending only on the
+aspect ratio, with exact constant values on its two sides.  Fill
+the inner side by the constant; this uses only a flat collar in the
+annulus.  Lemma~\ref{en:circuit} gives a primal circuit in each annulus
+whose oscillation squared is bounded by that constant times the
+primal energy in the annulus.  Among $k$ disjoint annuli, one
+therefore has oscillation at most $C k^{-1/2}$ when the total energy
+is bounded.  The maximum principle transfers this oscillation to
+all vertices inside its circuit.  First choose $k$, and then take
+large $n$; no annulus radius depends on $n$.  Finite covers give
+equicontinuity, and bounded values give compactness.  Exchanging
+primal and dual proves the remaining assertion.
+\end{proof}
+
+\subsection{Keeping the metric and the quantum area in the same coupling}
+
+The coordinate used for energy convergence must also describe the
+metric-measure limit.  The contour correspondence makes this precise.
+
+\begin{proposition}[Joint spatial comparison]\label{sph:comparison}
+The extraction in Theorem~\ref{sph:energy} can be chosen so that its
+limiting sphere is the one in the metric-measure limit and, almost
+surely,
+\begin{align}
+ \sup_{v,w\in V_n}
+ \big|a_n d_n(v,w)-D_h(\pi_n(v),\pi_n(w))\big|&\longrightarrow0,
+ \label{sph:distortion}\\
+ \sup_{x\in S^2}\dist_{D_h}(x,\pi_n(V_n))&\longrightarrow0,
+ \label{sph:dense}\\
+ (\pi_n)_*\mu_n&\Longrightarrow\mu_h.
+ \label{sph:measure}
+\end{align}
+The maximum round diameter of a projected edge tends to zero.
+The measure $\mu_h$ is nonatomic and has full support, and $D_h$
+induces the round topology.
+\end{proposition}
+
+\begin{proof}
+The sphere-identification proposition and the final normalization
+argument for spanning-tree maps in \cite{MetricMaps} give, jointly
+with the excursion, the contour correspondence
+\[
+ \bigl\{(v_{\lfloor2nt\rfloor},\eta(t)):0\le t\le1\bigr\}
+\]
+with distortion tending to zero.  Here $v_j$ is the primal vertex
+at the start of tour step $j$ and $\eta$ is the sphere contour.
+Those results identify the metric of this very peanosphere limit;
+they do not introduce an independent quantum sphere.  Their
+deterministic diameter normalization is the $a_n$ in
+Theorem~\ref{thm:main}.  Choosing one visit for each vertex changes
+the correspondence by a uniformly vanishing discrepancy, which
+gives \eqref{sph:distortion} and \eqref{sph:dense}.
+
+There are $2n$ tour step starts.  Their vertex occupation measure is
+$\mu_n$: horizontal tree traversals are counted at their starting
+vertices, and vertical steps count the non-tree half-edges at their
+incident vertices.  Every half-edge is counted once, including both
+half-edges of a loop.  The area parameterization of $\eta$ therefore
+gives \eqref{sph:measure}.  Mesh convergence is the sphere
+projection construction preceding Proposition~\ref{inp:topology}.
+Nonatomicity, full support, and agreement of the metric and conformal
+topologies are the quantum-area and metric properties in the same
+repository input.
+
+All these variables may be retained together with the compactified
+energy arrays before extraction.  The metric identification is
+unchanged upon a further extraction, so it holds in the coupling of
+Theorem~\ref{sph:energy}.
+\end{proof}
diff --git a/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/build/sections/06-green.tex b/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/build/sections/06-green.tex
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+\section{A Green estimate in the finite map law}
+\label{sec:green}
+
+Uniform approximation of energies concerns continuous functions.  To
+control the walk clock, we also need to show that bounded sources of
+small stationary mass have small potentials.  We prove this directly
+under the finite map law.  The argument combines a spanning-forest
+formula for the inverse Laplacian with the two tree contours in
+Mullin's encoding.
+
+Write $N=2n$.  Let $H_n$ be the positive graph Laplacian on $V_n$,
+using counting measure and unit conductances, so that
+\[
+ \langle u,H_nu\rangle=\en_n(u)
+       =\sum_{ab\in E(M_n)}(u(a)-u(b))^2.
+\]
+There is one term for each unoriented edge, and loops contribute zero.
+For a real function $f$ with $\mu_n(f)=0$, define $G_nf$ by
+\begin{equation}\label{gr:green-definition}
+ H_nG_nf=\mu_n f,
+ \qquad \mu_n(G_nf)=0.
+\end{equation}
+Here $\mu_nf$ on the right denotes the vector with entry
+$\mu_n(v)f(v)$ at $v$.  Connectedness gives existence and uniqueness.
+
+\begin{proposition}[Small-mass Green estimate]
+\label{gr:green-bound}
+For each fixed $\rho\in(0,1/2)$ there are nonnegative random variables
+$A_n$, measurable functions of $M_n$, such that
+\begin{equation}\label{gr:green-main}
+ \sup_{n\geq1}\E A_n<\infty,
+ \qquad
+ \|G_nf\|_\infty\leq A_n\,\mu_n(|f|)^\rho
+\end{equation}
+simultaneously for all real functions $f:V_n\to\R$ satisfying
+$\|f\|_\infty\leq1$ and $\mu_n(f)=0$.
+The expectation is under the map marginal of the uniform rooted
+map--spanning-tree pair law.  In particular, $(A_n)$ is tight.
+\end{proposition}
+
+We first identify the exact inverse formula.  We then estimate its
+terms by grouping tree cuts according to the amount of tour time on
+their smaller side.
+
+\subsection{The inverse Laplacian as a sum over cuts}
+
+The next identity is a consequence of the matrix-tree theorem and
+its forest expansions; see the all-minors theorem of \cite{Chaiken}
+and \cite[Chapter~4]{LP} for electrical networks and spanning trees.
+We give the determinant argument to specify the multiplicities and
+centering used here.
+
+\begin{lemma}[Forest representation]
+\label{gr:forest}
+Let $\mathcal G$ be a finite connected multigraph, with positive
+probability measure $\mu$ on its vertex set $V$, and let $H$ be its
+unit-conductance graph Laplacian.  Suppose $q:V\to\R$ satisfies
+$\sum_{v\in V}q(v)=0$, and let $u$ solve
+$Hu=q$, $\mu(u)=0$.  For a uniform spanning tree $T$ of $\mathcal G$
+and an edge $e\in T$, choose either component $S_e$ of $T\setminus e$,
+and let $b_e$ be the number of graph edges between $S_e$ and
+$V\setminus S_e$, with multiplicity.  Then
+\begin{equation}\label{gr:forest-identity}
+ u(x)=\E_T\sum_{e\in T}
+       \frac{q(S_e)\bigl(\1_{\{x\in S_e\}}-\mu(S_e)\bigr)}{b_e},
+ \qquad x\in V,
+\end{equation}
+where $q(S)=\sum_{v\in S}q(v)$.  The value does not depend on the
+choice of component for any $e$.
+\end{lemma}
+
+\begin{proof}
+The one-vertex case is immediate, so suppose $m:=|V|\geq2$.
+Orient the non-loop edges arbitrarily and delete the row of a fixed
+vertex $o$ from the incidence matrix.  Denote the resulting matrix by
+$B$, and write $\widehat q$ for $q$ with its $o$-coordinate deleted.
+The reduced Laplacian is $\widehat H=BB^{\mathsf T}$.  If $\tau$ is
+the number of spanning trees, the matrix-tree theorem and the
+determinant lemma give
+\begin{equation}\label{gr:rank-one}
+ \det(\widehat H+z\widehat q\widehat q^{\mathsf T})
+       =\tau\bigl(1+z\widehat q^{\mathsf T}
+                            \widehat H^{-1}\widehat q\bigr).
+\end{equation}
+For $z\geq0$, apply Cauchy--Binet to the matrix obtained by appending
+$\sqrt z\widehat q$ to $B$.  A term contributing to the coefficient
+of $z$ selects $m-2$ incidence columns and the source column.
+The incidence columns are independent exactly when their edges form
+a spanning forest with two components.  Indeed, a cycle gives a
+linear dependence, whereas an acyclic graph with $m-2$ edges has
+exactly two components, counting isolated vertices.
+
+For such a forest $F$, let $S_F$ be its component not containing $o$.
+Its corresponding determinant has absolute value $|q(S_F)|$.
+To see this, successively eliminate leaf rows and their incident
+forest columns.  Eliminating a leaf adds its source entry to its
+neighbor's entry.  The component containing $o$ disappears into the
+deleted row, and the last entry in the other component is
+$\sum_{v\in S_F}q(v)$.  Consequently, comparison of the coefficients
+of $z$ in \eqref{gr:rank-one} yields
+\begin{equation}\label{gr:two-forest-quadratic}
+ \widehat q^{\mathsf T}\widehat H^{-1}\widehat q
+       =\frac1\tau\sum_F q(S_F)^2.
+\end{equation}
+
+Let $H^+$ be the inverse of $H$ on counting-mean-zero vectors,
+extended by zero on constants.  Solving the equation with value zero
+at $o$ shows that the left side of
+\eqref{gr:two-forest-quadratic} equals $q^{\mathsf T}H^+q$.
+Polarization therefore gives, for any two vectors $q,r$ of total
+sum zero,
+\begin{equation}\label{gr:two-forest-bilinear}
+ q^{\mathsf T}H^+r=\frac1\tau\sum_F q(S_F)r(S_F).
+\end{equation}
+Replacing $S_F$ by its complement changes the sign of each factor,
+so the product is independent of this choice.
+
+Every two-component forest $F$ has exactly $b(F)$ extensions to a
+spanning tree: add any one of the graph edges across its cut.
+Thus, when all pairs $(T,e)$ are counted, $F=T\setminus e$ occurs
+exactly $b(F)$ times.  This explains the denominator in
+\eqref{gr:forest-identity}.  Apply \eqref{gr:two-forest-bilinear}
+with $r=\delta_x-\mu$.  Its left side is
+\[
+ (H^+q)(x)-\mu(H^+q)=u(x),
+\]
+and its right side, regrouped by $(T,e)$, is the asserted expression.
+Loops enter neither the reduced incidence matrix nor a cut; parallel
+edges are counted separately throughout.
+\end{proof}
+
+\subsection{What a tree cut records in the tour}
+
+We use the finite form of Mullin's correspondence \cite{Mullin}, with
+the incidence conventions of \cite[Section~``Independent-walk encoding
+and local quantum surfaces'']{MetricMaps}.  The $N$ steps form a
+quadrant excursion whose horizontal and vertical coordinates are
+denoted by $L$ and $R$.  Horizontal steps traverse the selected
+primal tree $T$; vertical steps traverse the complementary dual tree
+$T^*$.  In each coordinate an up-step is matched with the down-step
+that first returns to its initial height.  A matched horizontal pair
+represents a tree edge.  A matched vertical pair represents a
+non-tree map edge between the primal vertices at its two step starts.
+Let $v_i$ be the primal vertex at step start $i$, with all indices
+read cyclically modulo $N$.
+
+\begin{lemma}[Tour mass and cut size]
+\label{gr:tour-cut}
+For each $v\in V_n$,
+\begin{equation}\label{gr:tour-degree}
+ \#\{0\leq i<N:v_i=v\}=\deg_{M_n}(v).
+\end{equation}
+For $e\in T$, the step starts visiting either component of
+$T\setminus e$ form a cyclic interval.  Choose a component $S_e$
+whose interval $I_e$ has length $\ell_e\leq N/2$.  If the lengths
+agree, choose the component not containing $v_0$.  Then
+\begin{equation}\label{gr:tour-cut-identities}
+ \mu_n(S_e)=\frac{\ell_e}{N},
+ \qquad b_e=1+D_e^*,
+\end{equation}
+where $D_e^*$ is the distance in $T^*$ between the dual vertices at
+the two horizontal traversal steps of $e$.
+\end{lemma}
+
+\begin{proof}
+Associate each step start with the departing step.  The two
+horizontal traversals of a tree edge depart once from each endpoint,
+and hence count its two incidences.  The two steps of a vertical
+matched pair count the two incidences of its non-tree edge.  When
+that edge is a loop, both are assigned to the same vertex.
+Every step belongs to one such pair, proving
+\eqref{gr:tour-degree}; this is also the tour-measure lemma of
+\cite[Section~``From tree excursions to the measured quantum
+sphere'']{MetricMaps}.
+
+The primal vertex changes only along horizontal steps.  It passes
+between the two components of $T\setminus e$ exactly at the two
+traversals of $e$.  Thus each side occupies one cyclic interval, and
+\eqref{gr:tour-degree} proves the mass identity.
+
+There is exactly one tree edge across this cut, namely $e$.
+A non-tree edge crosses it precisely when exactly one step of its
+vertical matched pair belongs to $I_e$.  Ignore horizontal holds
+and follow the resulting dual-tree contour segment.  An edge of
+$T^*$ is traversed once on this segment exactly when its matched
+steps lie on opposite sides of $I_e$; every other dual edge is
+traversed zero or twice.  In a tree the edges traversed an odd number
+of times by a path are exactly the edges on the simple path between
+its endpoints.  These endpoints are the dual vertices at the two
+horizontal traversals of $e$, since horizontal steps do not change
+the dual vertex.  There are therefore $D_e^*$ non-tree edges across
+the cut, giving $b_e=1+D_e^*$.
+\end{proof}
+
+Hereafter expectations involving both $T$ and the map use the uniform
+pair law.  The sequence $(v_i)$ is determined by the horizontal
+contour and the locations of the horizontal slots.  Thus the cut
+intervals, their lengths, and the smaller-side convention in
+Lemma~\ref{gr:tour-cut} are determined without inspecting the shape
+of the vertical contour.  By \eqref{gr:tour-degree}, the same is true
+of the vertex masses.
+
+The conditional structure of the finite law is useful.  Let $J$
+be the number of horizontal steps.  Conditional on $J$, the horizontal
+slots form a uniform $J$-subset of $\{0,\ldots,N-1\}$.  Conditional
+on these slots, the two coordinate strings, with holds deleted, are
+independent uniform Dyck excursions of lengths $J$ and $N-J$.
+In fact, writing $C_k=(k+1)^{-1}\binom{2k}{k}$ for the Catalan
+numbers, the number of quadrant excursions with $J=2k$ is
+\begin{equation}\label{gr:schedule-count}
+ \binom{2n}{2k}C_kC_{n-k},
+ \qquad
+ \sum_{k=0}^n\binom{2n}{2k}C_kC_{n-k}=C_nC_{n+1}.
+\end{equation}
+The last equality is Vandermonde's identity after writing out the
+Catalan numbers.  These counts also prove the stated conditional
+independence.  Catalan estimates and a binomial tail bound give
+absolute $C,c>0$ such that
+\begin{equation}\label{gr:count-tail}
+ \P\{J\notin[N/4,3N/4]\}\leq C(N+1)^3e^{-cN}.
+\end{equation}
+The same schedule calculation is recorded in the excursion discussion
+of \cite[Section~``From tree excursions to the measured quantum
+sphere'']{MetricMaps}.  It is an exact calculation in the fixed-size
+law.
+
+\subsection{Distances in a uniform contour tree}
+
+The cut denominator in \eqref{gr:tour-cut-identities} is a dual-tree
+distance.  The next estimates quantify its typical size and also
+control how many primal cuts can surround a given tour time.
+
+\begin{lemma}[Contour distance estimates]
+\label{gr:dyck}
+Let $\mathcal T_s$ be a uniform rooted plane tree with $s/2$ edges,
+where $s\geq2$ is even, and write $w_i$, $i\in\Z/s\Z$, for its
+cyclic contour vertices.  If $i,j$ are deterministic with
+$0\leq j\leq s$, set
+\[
+ D=\dist_{\mathcal T_s}(w_i,w_{i+j}),
+ \qquad d=1+\min(j,s-j).
+\]
+There are absolute constants $C,c>0$ such that, for every integer
+$a\geq0$,
+\begin{equation}\label{gr:dyck-density}
+ \P(D=a)\leq C(a+1)^2d^{-3/2}e^{-ca^2/d}.
+\end{equation}
+Consequently, for every fixed $p>0$,
+\begin{equation}\label{gr:dyck-moments}
+ \|(1+D)^{-1}\|_{L^2}\leq C d^{-1/2},
+ \qquad \|D\|_{L^p}\leq C_p\sqrt d.
+\end{equation}
+For every deterministic cyclic segment $K$ from corner $i$ through
+corner $i+L$, where $1\leq L\leq s$,
+\begin{equation}\label{gr:contour-diameter}
+ \bigl\|\diam_{\mathcal T_s}\{w_{i+r}:0\leq r\leq L\}\bigr\|_{L^4}
+       \leq C\sqrt L.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Rerooting a plane tree at its $i$th contour corner is a bijection on
+rooted plane trees with $s/2$ edges.  Its inverse reroots by $s-i$
+corners.  Thus the rerooted tree is uniform, and $D$ has the law of
+the height at time $j$ of a uniform Dyck excursion of length $s$.
+
+Let $p_l(a)$ be the probability that a simple symmetric walk started
+at zero stays nonnegative for $l$ steps and finishes at $a$.
+Reflection and the binomial estimate give
+\begin{align}
+ p_l(a)
+ &=2^{-l}\left[
+       \binom{l}{(l+a)/2}-\binom{l}{(l+a+2)/2}\right],
+       \label{gr:killed-kernel}\\
+ p_l(a)&\leq C(a+1)(l+1)^{-3/2}
+                         e^{-ca^2/(l+1)},\nonumber
+\end{align}
+with inadmissible binomial coefficients interpreted as zero.
+For even $s$, $p_s(0)$ is comparable to $(s+1)^{-3/2}$.
+The Markov property and time reversal of the second segment give
+\[
+ \P(D=a)=\frac{p_j(a)p_{s-j}(a)}{p_s(0)}.
+\]
+Inserting \eqref{gr:killed-kernel} proves
+\eqref{gr:dyck-density}.  Summing it after multiplication by
+$(a+1)^{-2}$ or by $a^p$ proves
+\eqref{gr:dyck-moments}.  In the first case the remaining Gaussian
+sum is at most $C\sqrt d$, so the second moment is at most $C/d$.
+
+For completeness, the diameter estimate follows by dyadic chaining.
+Reroot at the deterministic first corner of $K$ and parametrize its
+steps by the integers in $[0,L]$.  At refinement level $l$, use the
+times $\lfloor kL/2^l\rfloor$, $0\leq k\leq2^l$, and stop at
+$l=\lceil\log_2L\rceil$.  These grids are nested and the last one
+contains every integer in $[0,L]$.  Include the initial increment
+from $0$ to $L$.  For $l\geq1$, there are $O(2^l)$ new-to-parent
+increments at level $l$, each spanning at most $C L2^{-l}$ steps.
+By the fourth-moment estimate and
+$\|\max_k Z_k\|_{L^4}^4\leq\sum_k\|Z_k\|_{L^4}^4$, the maximum
+tree distance across these increments has $L^4$ norm at most
+\[
+ C2^{l/4}\sqrt{L2^{-l}}=C\sqrt L\,2^{-l/4}.
+\]
+Every contour vertex can be joined to the first vertex through its
+ancestors in the dyadic grids.  The triangle inequality and the
+summability of $2^{-l/4}$ bound the maximum distance from that first
+vertex by $C\sqrt L$ in $L^4$.  The diameter is at most twice this
+maximum.  For $L=1$ the bound follows directly from the single edge.
+\end{proof}
+
+\subsection{The sum of reciprocal cut sizes at one scale}
+
+For a dyadic integer $M$ with $1\leq M\leq N/2$, let
+\[
+ \mathcal E_M=\{e\in T:M\leq\ell_e<2M\},
+ \qquad
+ U_{M,t}=\sum_{\substack{e\in\mathcal E_M\\t\in I_e}}\frac1{b_e},
+ \qquad t\in\Z/N\Z.
+\]
+The intervals in this sum come from the horizontal tree.  Their
+denominators are governed by the independent vertical tree once the
+schedule and the horizontal tree have been fixed.
+
+\begin{lemma}[One-scale estimate]
+\label{gr:one-scale}
+There is an absolute constant $C$ such that, for every $n$, every
+dyadic $M\leq N/2$, and every deterministic tour time $t$,
+\begin{equation}\label{gr:Ut}
+ \E U_{M,t}^2\leq C.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+First condition on a count $J\in[N/4,3N/4]$.  Consider all cyclic
+intervals containing $t$ whose lengths lie in $[M,2M)$ and are at
+most $N/2$.  There are at most $C(M+1)^2$ such intervals.  For each
+one, a uniform schedule puts at least $c_0M$ vertical steps in both
+the interval and its complement, except with probability
+$Ce^{-cM}$, for suitable absolute $c_0,c>0$.  Indeed, the vertical
+density is at least $1/4$, both sets have at least $M$ slots, and
+the hypergeometric lower-tail bound applies.  Taking a union bound,
+let $\mathcal B_{M,t}$ be the event that this property fails for
+some interval.  Uniformly over the balanced counts,
+\begin{equation}\label{gr:schedule-tail}
+ \P(\mathcal B_{M,t}\mid J)
+       \leq C(M+1)^2e^{-cM}.
+\end{equation}
+
+Fix a schedule outside $\mathcal B_{M,t}$ and then condition on the
+horizontal Dyck excursion.  This fixes all intervals in $U_{M,t}$.
+For each one, its dual endpoints are separated in the vertical
+contour by the number $j$ of vertical steps in that interval.
+Both $j$ and $(N-J)-j$ are at least $c_0M$.  Lemmas
+\ref{gr:tour-cut} and \ref{gr:dyck}, followed by the triangle
+inequality in $L^2$ of the remaining vertical excursion, imply
+\begin{equation}\label{gr:vertical-bound}
+ \|U_{M,t}\|_{L^2(R\mid\text{schedule},L)}
+       \leq\frac{C}{\sqrt M}\,K_{M,t},
+ \qquad
+ K_{M,t}:=\#\{e\in\mathcal E_M:t\in I_e\}.
+\end{equation}
+
+We next bound this number of horizontal cuts.  Any two components
+$S_e,S_{e'}$ containing $v_t$ are nested unless their union is all
+of $V_n$.  This elementary tree property follows by considering
+the components cut off from $v_t$: they are either nested or
+disjoint.  The union alternative cannot occur for the chosen
+smaller sides, since
+\[
+ \mu_n(S_e\cup S_{e'})
+ =\mu_n(S_e)+\mu_n(S_{e'})-\mu_n(S_e\cap S_{e'})<1.
+\]
+Here each of the first two terms is at most $1/2$, and the
+intersection contains $v_t$, whose mass is positive.
+Thus all components counted by $K_{M,t}$ are nested.
+
+If the family is nonempty, let $S$ be its largest component and
+choose a contour vertex immediately outside its interval.  It has
+a representative at cyclic distance at most $2M+2$ from $t$.
+The tree path from $v_t$ to that vertex crosses all $K_{M,t}$ cut
+edges.  Hence $K_{M,t}$ is at most the diameter of the horizontal
+contour image of the deterministic window of tour times within
+$2M+2$ of $t$.  After the schedule has been fixed and holds deleted,
+this is a deterministic contour segment with at most $4M+4$
+horizontal steps.  If the window covers the entire tour, the full
+horizontal contour instead has length $J\leq N\leq4M+4$.
+A segment with no horizontal steps has diameter zero; otherwise
+Lemma~\ref{gr:dyck} applies.  Uniformly in the schedule, we obtain
+\begin{equation}\label{gr:horizontal-count}
+ \E\bigl[K_{M,t}^2\mid\text{schedule}\bigr]\leq CM.
+\end{equation}
+Combining \eqref{gr:vertical-bound} and
+\eqref{gr:horizontal-count} proves the required bound on schedules
+outside $\mathcal B_{M,t}$ with balanced counts.
+
+For the exceptional schedules, use a deterministic bound.
+Every interval counted by $K_{M,t}$ has its two delimiting tree
+traversals within cyclic distance $2M+2$ of $t$.  Distinct tree
+edges have distinct traversal occurrences, so
+$K_{M,t}\leq C(M+1)$.  Since $b_e\geq1$, the same bound holds
+for $U_{M,t}$.  By \eqref{gr:schedule-tail}, the contribution of
+these schedules to its second moment is at most
+$C(M+1)^4e^{-cM}$, uniformly bounded in $M$.
+Finally, $U_{M,t}\leq N$ always.  The unbalanced counts contribute
+at most $CN^2(N+1)^3e^{-cN}$ by \eqref{gr:count-tail}, also
+uniformly bounded.  This proves \eqref{gr:Ut} under the full law.
+\end{proof}
+
+\subsection{Summing the scales}
+
+We now return to the inverse formula.  Lemma~\ref{gr:one-scale}
+controls the cut sum at a fixed time; a grid converts this into a
+uniform bound over vertices.
+
+\begin{proof}[Proof of Proposition~\ref{gr:green-bound}]
+The one-vertex case has $G_nf=0$, so set $A_n=0$ there.  Otherwise,
+for each dyadic $M\leq N/2$, choose a deterministic cyclic grid
+$\Gamma_M$.  For $M\geq8$, take its gaps at most $M/4$, its
+cardinality at most $CN/M$, and its distinct points separated by
+at least $cM$.  For $M<8$, use all step indices; the same
+cardinality and separation bounds hold with absolute constants.
+
+An interval of length in $[M,2M)$ containing a fixed time $t$
+contains a grid point within cyclic distance $2M$ of $t$.  There
+are only a bounded number of grid points in this neighborhood.
+Consequently,
+\begin{equation}\label{gr:grid-comparison}
+ \sup_t U_{M,t}\leq C\max_{t\in\Gamma_M}U_{M,t}.
+\end{equation}
+Every visit to a given vertex belongs to the same side of a tree
+cut.  Also, counting the visits in each interval gives
+\[
+ \sum_{e\in\mathcal E_M}\frac{\mu_n(S_e)}{b_e}
+       =\frac1N\sum_{t=0}^{N-1}U_{M,t}.
+\]
+It follows that
+\begin{align}
+ W_M&:=\sup_{x\in V_n}\sum_{e\in\mathcal E_M}
+        \frac{\1_{\{x\in S_e\}}+\mu_n(S_e)}{b_e}
+       \leq C\max_{t\in\Gamma_M}U_{M,t},
+          \label{gr:WM}\\
+ \E W_M&\leq C\left(\sum_{t\in\Gamma_M}
+                                    \E U_{M,t}^2\right)^{1/2}
+       \leq C\sqrt{N/M}.\label{gr:WM-mean}
+\end{align}
+No independence among the grid variables is needed here.
+
+Fix $f$ as in the proposition and write $m_f=\mu_n(|f|)$.
+For $e\in\mathcal E_M$, the source mass obeys
+\begin{equation}\label{gr:source-interpolation}
+ \left|\sum_{v\in S_e}\mu_n(v)f(v)\right|
+ \leq\min(2M/N,m_f)
+ \leq(2M/N)^{1-\rho}m_f^\rho.
+\end{equation}
+Define, for the realized pair $(M_n,T)$,
+\[
+ Y_n(M_n,T)=\sum_{\substack{M\leq N/2\\M\text{ dyadic}}}
+                   (2M/N)^{1-\rho}W_M.
+\]
+By \eqref{gr:WM-mean},
+\begin{equation}\label{gr:Y-mean}
+ \E Y_n\leq C_\rho
+       \sum_{\substack{M\leq N/2\\M\text{ dyadic}}}
+                        (M/N)^{1/2-\rho}
+       \leq C_\rho.
+\end{equation}
+The last bound is a geometric sum, using $\rho<1/2$.
+
+Conditional on $M_n$, the selected tree is uniform.  Define
+\begin{equation}\label{gr:An}
+ A_n(M_n)=\E\bigl[Y_n(M_n,T)\mid M_n\bigr].
+\end{equation}
+Apply Lemma~\ref{gr:forest} with $q=\mu_nf$.  Taking absolute
+values, grouping by $M$, and using
+\eqref{gr:source-interpolation} proves
+\[
+ \|G_nf\|_\infty\leq A_n(M_n)m_f^\rho.
+\]
+The variable in \eqref{gr:An} is independent of the choice of $f$,
+so this bound holds simultaneously for all admissible functions.
+The tower property and \eqref{gr:Y-mean} give
+$\sup_n\E A_n<\infty$, and Markov's inequality gives tightness.
+\end{proof}
+
+The estimate depends only on the finite map law and its spanning
+trees.  In particular, it controls the full potential of a small
+source, including any behavior on vertex sets that uniform
+approximation of continuous functions might fail to detect.
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+\section{Resolvents and conditional path laws}\label{sec:dynamics}
+
+Uniform convergence of energies concerns continuous functions, whereas
+the speed measure may be singular with respect to round area.  The
+Green estimate bridges this difference: it makes a bounded source on
+a set of small quantum mass produce a uniformly small potential.
+After removing that potential, a resolvent is locally harmonic, so
+the compactness theorem for harmonic functions applies.  We first
+give this argument for a deterministic sequence of graphs.  The last
+subsection applies it in the coupled random environments and proves
+Theorem~\ref{thm:main}.
+
+\subsection{A deterministic convergence criterion}
+
+Let $S^2$ carry its round distance $d_0$ and area $\dd A$.  Let $\mu$
+be a nonatomic probability measure with full support.  Suppose the
+quadratic form
+\begin{equation}\label{dyn:continuum-form}
+ \cE(u,u)=\frac12\int_{S^2}|\nabla u|^2\,\dd A,
+ \qquad u\in C^\infty(S^2),
+\end{equation}
+is closable in $L^2(\mu)$, and its closure $(\cE,\mathcal F)$ is a
+regular conservative Dirichlet form with an associated continuous-path
+diffusion $B$.  We always start $B$ with law $\mu$.  These hypotheses
+hold for the Liouville area measures used here; the time-change and
+Dirichlet-form constructions are treated in
+\cite{GRV,Berestycki}, with general form theory in \cite{FOT}.
+Our normalization is the explicit factor $1/2$ in
+\eqref{dyn:continuum-form}.
+
+For each $n$, let $\mathcal G_n$ be a finite connected graph, allowing loops and
+multiple edges, with vertex set $V_n$ and a strictly positive
+probability measure $\mu_n$ on $V_n$.  Write
+\[
+ (H_nu)(v)=\sum_{vw}\bigl(u(v)-u(w)\bigr),
+ \qquad
+ \en_n(u)=\sum_{\{v,w\}}(u(v)-u(w))^2.
+\]
+The first sum counts incidences at $v$, and the second counts
+unoriented edges with multiplicity.  Loop summands vanish.
+Let $Y^n$ be the chain with generator
+\begin{equation}\label{dyn:abstract-generator}
+ (\mathcal L_nu)(v)=-\frac{(H_nu)(v)}{2\mu_n(v)}.
+\end{equation}
+It is reversible with respect to $\mu_n$, and its form is
+$\cE_n(u,u)=\en_n(u)/2$.  Choose maps $\pi_n:V_n\to S^2$.
+Uniform comparison of functions always means
+\[
+ \|u_n-u\circ\pi_n\|_{\infty,V_n}\longrightarrow0.
+\]
+No injectivity of $\pi_n$ is required.
+
+\begin{theorem}[From energies and Green bounds to path laws]
+\label{dyn:conditional}
+In the deterministic setting just defined, assume the following.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item $(\pi_n)_*\mu_n\Rightarrow\mu$, and the maximum projected
+edge length tends to zero:
+\[
+ \max_{\{v,w\}\in E(\mathcal G_n)}d_0(\pi_n(v),\pi_n(w))\longrightarrow0.
+\]
+\item Every uniformly convergent sequence with continuous limit $u$
+satisfies
+\[
+ \liminf_n\en_n(u_n)\ge\int|\nabla u|^2\,\dd A,
+\]
+where the right side is $+\infty$ unless $u\in W^{1,2}(S^2)$.
+Every smooth $u$ has a uniformly convergent recovery sequence with
+energy tending to that integral.
+\item For every open $U\subset S^2$, sequences that are graph-harmonic
+at all vertices in $\pi_n^{-1}(U)$ and have uniformly bounded
+supremum and total energy are locally equicontinuous in $U$ under
+$\pi_n$, and locally uniformly precompact there.
+Here equicontinuity means that, for every compact $K\subset U$,
+\[
+ \lim_{\delta\downarrow0}\limsup_n
+ \sup_{\substack{\pi_n(v),\pi_n(w)\in K\\
+                  d_0(\pi_n(v),\pi_n(w))\le\delta}}
+       |u_n(v)-u_n(w)|=0,
+\]
+and precompactness means uniform comparison with a continuous
+function on each compact subset after a common subsequence.
+\item For the centered solution operator
+\[
+ H_nG_ng=\mu_n g,\qquad \mu_n(G_ng)=0,
+ \qquad \mu_n(g)=0,
+\]
+there are constants $C<\infty$ and $\rho>0$, independent of $n$, such
+that
+\begin{equation}\label{dyn:green-assumption}
+ \|G_ng\|_\infty\le C\,\mu_n(|g|)^\rho
+ \quad\text{if }\|g\|_\infty\le1\text{ and }\mu_n(g)=0.
+\end{equation}
+\end{enumerate}
+Then the laws of $(\pi_n(Y^n_t))_{0\le t\le T}$, with initial law
+$\mu_n$, converge in $D([0,T],S^2)$ to the law of
+$(B_t)_{0\le t\le T}$, for every finite $T$.  The limit has continuous
+paths.  In addition, for every $\alpha>0$ and $f\in C(S^2)$, the
+resolvents converge uniformly under $\pi_n$.
+\end{theorem}
+
+We prove the theorem through four lemmas.  Their hypotheses throughout
+are those of Theorem~\ref{dyn:conditional}.  Notice that full support
+and weak convergence of the measures already imply that $\pi_n(V_n)$
+becomes dense: otherwise a fixed open ball of positive $\mu$-mass
+would be missed along a subsequence.
+
+\begin{lemma}[Continuous Sobolev functions in the closed form]
+\label{dyn:smooth-core}
+If $u\in C(S^2)\cap W^{1,2}(S^2)$, then $u\in\mathcal F$ and its
+closed-form energy is the right side of
+\eqref{dyn:continuum-form}.  It has smooth approximations converging
+both uniformly and in the form norm.  Two continuous versions of
+the same element of $L^2(\mu)$ are equal everywhere.
+\end{lemma}
+
+\begin{proof}
+In coordinate disks, mollification of a continuous Sobolev function
+converges both uniformly on smaller disks and in Sobolev norm.
+A smooth partition of unity gives smooth functions $u_j$ on the
+sphere with these two convergences globally.  Since $\mu$ is finite,
+uniform convergence also gives $L^2(\mu)$ convergence.  Thus $(u_j)$
+is Cauchy for $\cE(\cdot,\cdot)+\|\cdot\|_{L^2(\mu)}^2$ and has
+limit $u$ in the stipulated closure.  The gradient convergence
+identifies its energy.  Polarization gives the same identity for
+the bilinear form on continuous Sobolev functions.  Finally a
+nonzero continuous difference is bounded away from zero on some
+open set, which has positive $\mu$-mass by full support.
+\end{proof}
+
+Let $R^n_\alpha=(\alpha-\mathcal L_n)^{-1}$, and let $R_\alpha$ be the
+$L^2(\mu)$ resolvent of $(\cE,\mathcal F)$.
+
+\begin{lemma}[Uniform resolvent convergence]\label{dyn:resolvent}
+For every $\alpha>0$ and $f\in C(S^2)$, $R_\alpha f$ has a unique
+continuous version, which belongs to round $W^{1,2}$, and
+\begin{equation}\label{dyn:resolvent-limit}
+ \bigl\|R^n_\alpha(f\circ\pi_n)
+              -(R_\alpha f)\circ\pi_n\bigr\|_\infty
+       \longrightarrow0.
+\end{equation}
+The same conclusion holds with right-hand sides $f_n$ converging
+uniformly under $\pi_n$ to $f$.
+\end{lemma}
+
+\begin{proof}
+Put $u_n=R^n_\alpha(f\circ\pi_n)$.  The maximum principle and the
+resolvent equation give
+\begin{equation}\label{dyn:resolvent-bounds}
+ \|u_n\|_\infty\le\frac{\|f\|_\infty}{\alpha},
+ \qquad
+ H_nu_n=\mu_n g_n,
+ \qquad
+ g_n=2(f\circ\pi_n-\alpha u_n).
+\end{equation}
+In particular $g_n$ is uniformly bounded, and multiplying the equation
+by $u_n$ shows that $\en_n(u_n)$ is uniformly bounded.
+
+We first prove equicontinuity.  Fix a point $x\in S^2$, and choose
+disjoint open neighborhoods $U$ of $x$ and $V$ of a different point.
+Take $\mu$-null boundaries, with $\mu(V)>0$.  The neighborhood $U$
+may be made arbitrarily small in $\mu$-mass because $\mu$ has no
+atoms.  Write $U_n=\pi_n^{-1}(U)$ and $V_n'=\pi_n^{-1}(V)$, and set
+\[
+ q_n=g_n\1_{U_n}
+       -\frac{\mu_n(g_n\1_{U_n})}{\mu_n(V_n')}\1_{V_n'}.
+\]
+For large $n$, the denominator is bounded below.  Hence $q_n$ has
+mean zero, bounded supremum, and
+$\mu_n(|q_n|)\le C_1\mu_n(U_n)$.  Scaling
+\eqref{dyn:green-assumption} yields
+\[
+ \|G_nq_n\|_\infty\le C_2\mu_n(U_n)^\rho.
+\]
+Also
+\[
+ \en_n(G_nq_n)=\mu_n((G_nq_n)q_n)
+ \le\|G_nq_n\|_\infty\mu_n(|q_n|).
+\]
+Thus $u_n-G_nq_n$ has bounded energy and supremum and is harmonic
+on $U_n$.  Hypothesis~(iii) makes this difference equicontinuous on
+compact subsets of $U$.  Given an oscillation tolerance, first
+choose $U$ so that $G_nq_n$ has smaller supremum, and then use
+harmonic equicontinuity inside $U$.  A finite cover of $S^2$ proves
+equicontinuity of the original sequence.  Boundedness and density of
+$\pi_n(V_n)$ now give uniformly convergent subsequences.
+
+Let $u$ be one such limit.  Hypothesis~(ii) and
+Lemma~\ref{dyn:smooth-core} show that $u\in\mathcal F\cap C(S^2)$.
+The discrete resolvent minimizes
+\[
+ J_n(v)=\frac12\en_n(v)+\alpha\mu_n(v^2)
+                       -2\mu_n((f\circ\pi_n)v).
+\]
+The energy lower bound, uniform convergence, and weak convergence of
+the measures imply
+\[
+ \cE(u,u)+\alpha\mu(u^2)-2\mu(fu)
+       \le\liminf_n J_n(u_n).
+\]
+For any smooth $v$, take its uniform recovery sequence $v_n$.
+Minimality gives $J_n(u_n)\le J_n(v_n)$, and the latter converges to
+$\cE(v,v)+\alpha\mu(v^2)-2\mu(fv)$.  Smooth functions are a core
+of the closed form by its definition, so $u$ minimizes this
+functional over all of $\mathcal F$.  Strict convexity for $\alpha>0$
+identifies $u=R_\alpha f$ in $L^2(\mu)$.  Full support makes its
+continuous version unique.  Every subsequence has the same limit,
+which proves \eqref{dyn:resolvent-limit}.
+
+Finally, $\|R^n_\alpha a-R^n_\alpha b\|_\infty
+\le\alpha^{-1}\|a-b\|_\infty$.  This contraction bound proves the
+claim for varying right-hand sides.
+\end{proof}
+
+\subsection{Controlling small time intervals}
+
+Resolvent convergence integrates the walk over time.  To recover
+the actual clock, we next construct functions that separate spatial
+sets and whose discrete generators are bounded.  The needed continuum
+approximation is uniform, rather than merely in $L^2(\mu)$.
+
+\begin{lemma}[Uniform resolvent approximation]\label{dyn:feller}
+For every $f\in C(S^2)$,
+\begin{equation}\label{dyn:approximate-identity}
+ \|\alpha R_\alpha f-f\|_\infty\longrightarrow0
+ \qquad\text{as }\alpha\longrightarrow\infty.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+First suppose $f$ is smooth, and put $w_\alpha=\alpha R_\alpha f$.
+The spectral theorem for the nonnegative self-adjoint form operator
+gives $w_\alpha\to f$ in the form norm.  Indeed the spectral
+multiplier $\alpha/(\alpha+\lambda)$ tends to one and is bounded by
+one, so dominated convergence applies with spectral weight
+$1+\lambda$.  By Lemma~\ref{dyn:smooth-core},
+\[
+ \|\nabla(w_\alpha-f)\|_{L^2(\dd A)}\longrightarrow0.
+\]
+The maximum principle, or its discrete version followed by
+Lemma~\ref{dyn:resolvent}, gives
+\begin{equation}\label{dyn:range}
+ \min f\le w_\alpha\le\max f.
+\end{equation}
+
+We also need convergence in round $L^2$, which does not follow by
+changing the measure in the form norm.  Let $c_\alpha$ be the round
+average of $w_\alpha-f$.  Round Poincare gives
+\[
+ \|w_\alpha-f-c_\alpha\|_{L^2(\dd A)}\longrightarrow0.
+\]
+The constants $c_\alpha$ are bounded by \eqref{dyn:range}.  If a
+subsequence has limit $c$, then $w_\alpha\to f+c$ in round $L^2$
+on that subsequence, and therefore along a further subsequence almost
+everywhere.  Consequently
+$\min f\le f+c\le\max f$ almost everywhere for round area.
+Continuity of $f$ makes its essential extrema equal to its extrema:
+every neighborhood of a minimizer or maximizer has positive round
+area.  The two inequalities force $c\ge0$ and $c\le0$.
+Thus $c=0$, and $w_\alpha-f\to0$ in round $W^{1,2}$.
+
+Fix a small coordinate disk about a point $x$, and a fixed annulus
+inside that disk surrounding a smaller neighborhood of $x$.
+Radial averaging of the round $W^{1,2}$ error supplies, for each
+large $\alpha$, a circle in this annulus on which the trace of
+$w_\alpha-f$ tends to zero in one-dimensional $H^1$.  The radii stay
+in a fixed compact subinterval of $(0,\infty)$, so the
+one-dimensional Sobolev inequality gives uniform convergence on
+these circles.  Almost every radius has the Sobolev trace of the
+continuous representative, which is the trace we choose.
+
+Let $D_\alpha$ be the disk inside the chosen circle.  The resolvent
+equation is
+\begin{equation}\label{dyn:weak-resolvent}
+ \cE(w_\alpha,v)+\alpha\int(w_\alpha-f)v\,\dd\mu=0,
+ \qquad v\in\mathcal F.
+\end{equation}
+Choose a constant $K$ strictly above the boundary values of
+$w_\alpha$ and at least $\sup_{D_\alpha} f$.
+The positive part $(w_\alpha-K)^+$ on $D_\alpha$, extended by zero,
+vanishes in a collar of the circle and is continuous Sobolev.
+Lemma~\ref{dyn:smooth-core} makes it an admissible test in
+\eqref{dyn:weak-resolvent}.  The two terms are nonnegative; the
+second is at least $\alpha\int ((w_\alpha-K)^+)^2\,\dd\mu$.
+Thus $w_\alpha\le K$ on the disk, first $\mu$-almost everywhere and
+then everywhere by continuity and full support.  The negative-part
+test gives the lower bound.  Letting the strict slack tend to zero
+shows that the values inside the circle are bounded by its boundary
+values and the extrema of $f$ in the disk.
+
+The oscillation of $f$ on sufficiently small disks is arbitrarily
+small.  The boundary convergence and this comparison therefore
+give uniform convergence on a neighborhood of $x$.  A finite cover
+proves \eqref{dyn:approximate-identity} for smooth $f$.
+For continuous $f$, approximate it uniformly by a smooth function
+and use the contraction $\|\alpha R_\alpha\|_{\infty\to\infty}\le1$.
+\end{proof}
+
+\begin{lemma}[An exit estimate uniform over starting vertices]
+\label{dyn:tightness}
+For $d>0$ and $v\in V_n$, let
+\[
+ \tau^n_d(v)=\inf\{t\ge0:
+   d_0(\pi_n(Y^n_t),\pi_n(v))\ge d\}.
+\]
+Then
+\begin{equation}\label{dyn:exit}
+ \lim_{s\downarrow0}\limsup_{n\to\infty}
+       \sup_{v\in V_n}\P_v[\tau^n_d(v)\le s]=0.
+\end{equation}
+The stationary projected path laws are tight in $D([0,T],S^2)$,
+and all their limits are supported on continuous paths.
+\end{lemma}
+
+\begin{proof}
+Cover the sphere by finitely many round disks of radius less than
+$d/8$.  For each choose a smooth function $f$ in $[0,1]$, zero on
+that disk, and one outside the concentric disk of radius $d/2$.
+If $\pi_n(v)$ is in the smaller disk and displacement from
+$\pi_n(v)$ reaches $d$, the value of $f$ at the new location is one.
+
+Fix $b>0$.  Lemmas~\ref{dyn:feller} and \ref{dyn:resolvent}, first
+with a sufficiently large fixed $\alpha$ and then with sufficiently
+large $n$, give for each of this finite family a function
+\[
+ v_n=\alpha R^n_\alpha(f\circ\pi_n),\qquad
+ \|v_n-f\circ\pi_n\|_\infty\le b.
+\]
+Its generator satisfies
+\[
+ \mathcal L_nv_n=\alpha(v_n-f\circ\pi_n),
+ \qquad \|\mathcal L_nv_n\|_\infty\le C_b,
+\]
+with a common $C_b$ for the finite family and all sufficiently large
+$n$.  Start the chain at $v$ in the corresponding small disk and
+stop at $\sigma=\tau^n_d(v)\wedge s$.  If
+$p=\P_v[\tau^n_d(v)\le s]$, then
+\[
+ \E_v[v_n(Y^n_\sigma)]\ge p-b,
+ \qquad
+ \E_v[v_n(Y^n_\sigma)]
+   =v_n(v)+\E_v\int_0^\sigma \mathcal L_nv_n(Y^n_t)\,\dd t
+   \le b+C_bs.
+\]
+Hence $p\le2b+C_bs$, uniformly over all starting vertices.
+First let $s\downarrow0$ and then $b\downarrow0$ to obtain
+\eqref{dyn:exit}.
+
+Apply the strong Markov property to the chain $Y^n$ after any
+stopping time.  Since \eqref{dyn:exit} is uniform over its current
+vertex, it gives the usual stopping-time increment condition for
+the projected paths.  The projection itself need not be Markov.
+Apply Aldous's stopping-time criterion \cite{Aldous} to the three
+coordinate functions of the round sphere in $\R^3$; their values
+are bounded.  This gives tightness of each coordinate path law.
+The largest coordinate jump is bounded by the maximum projected
+edge length, which tends to zero.  Under $J_1$ convergence any jump
+of a limit is approximated by jumps of the approximating paths,
+so every coordinate limit is continuous.  Coordinate tightness
+with continuous limits gives joint tightness with continuous
+limits in $\R^3$: on continuous limits, $J_1$ convergence agrees
+with uniform convergence, allowing a common time comparison for
+the three coordinates.  The limits remain in the closed set
+$S^2$, proving the assertion.
+\end{proof}
+
+\subsection{Identifying the entire conditional law}
+
+\begin{proof}[Completion of the proof of Theorem~\ref{dyn:conditional}]
+Take a subsequence of the stationary projected path laws with a weak
+limit.  By Lemma~\ref{dyn:tightness}, diagonal extraction on the
+horizons $1,2,\ldots$ gives a consistent continuous-path law on
+$[0,\infty)$, denoted by $Q$.  Restrictions are consistent because
+the limits are continuous at the restriction times.
+
+Let $f_0,\ldots,f_k\in C(S^2)$ and $\alpha_1,\ldots,\alpha_k>0$.
+Integrate the stationary finite-dimensional expectation at times
+$0,t_1,t_1+t_2,\ldots,t_1+\cdots+t_k$ against
+$\prod_{j=1}^k e^{-\alpha_jt_j}\,\dd t_j$.  The Markov property
+and the resolvent identity give exactly
+\begin{equation}\label{dyn:laplace}
+ \mu_n\!\left[
+  (f_0\circ\pi_n)R^n_{\alpha_1}
+  \left((f_1\circ\pi_n)R^n_{\alpha_2}
+    \left(\cdots(f_{k-1}\circ\pi_n)
+                R^n_{\alpha_k}(f_k\circ\pi_n)\cdots\right)\right)
+ \right].
+\end{equation}
+Successive applications of Lemma~\ref{dyn:resolvent}, including its
+varying-data assertion, and weak convergence of the measures show
+that \eqref{dyn:laplace} converges to
+\[
+ \mu\!\left[f_0R_{\alpha_1}
+       \left(f_1R_{\alpha_2}
+          \left(\cdots f_{k-1}R_{\alpha_k}f_k\cdots\right)\right)
+       \right].
+\]
+By the semigroup-resolvent relation for the closed form, this is the
+same integrated expectation for the stationary diffusion $B$.
+
+It is also the integrated expectation for $Q$.  On bounded time
+boxes this follows from weak path convergence, since evaluation at
+fixed times is continuous at continuous paths.  Dominated
+convergence then integrates over the gaps.  The contribution of
+the complement of a large time box is bounded uniformly by the
+tails of the product of exponentials, times
+$\prod_j\|f_j\|_\infty$.
+
+The product expectations under $Q$ and under $B$ are bounded
+continuous functions of $(t_1,\ldots,t_k)$, including zero gaps.
+Uniqueness of multivariate Laplace transforms makes these
+functions equal: one may apply one-dimensional uniqueness
+successively in each coordinate, and then use continuity to
+remove almost-everywhere qualifications.  Product tests determine
+probability measures on the compact product $(S^2)^{k+1}$.
+Thus $Q$ and the law of $B$ have the same finite-dimensional
+distributions, and hence the same law on continuous paths.
+Every subsequential limit is the same, proving the theorem.
+\end{proof}
+
+\subsection{The clock and the metric-space conclusion}
+
+We now return to the random maps.  The exact relation between the
+graph energy and the prescribed walk fixes the clock without any
+subsequence-dependent factor.
+
+\begin{lemma}[Exact energy of the accelerated walk]\label{dyn:energy-clock}
+For the half-edge walk of Theorem~\ref{thm:main},
+\[
+ -\langle u,nL_nu\rangle_{L^2(\mu_n)}=\frac12\en_n(u).
+\]
+Equivalently, $nL_n=-\tfrac12\mu_n^{-1}H_n$, where
+$\mu_n^{-1}$ denotes division by vertex mass.
+\end{lemma}
+
+\begin{proof}
+The generator convention gives $H_n=-\deg\,L_n$.  Since
+$\mu_n(v)=\deg(v)/(2n)$,
+\[
+ -\sum_v\mu_n(v)u(v)nL_nu(v)
+   =\frac12\sum_vu(v)H_nu(v)
+   =\frac12\sum_{\{v,w\}}(u(v)-u(w))^2.
+\]
+Loops contribute zero to both sides.  Their degree contributions
+are already included in the first identity, so no holding-time
+correction is needed.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]\label{dyn:main-proof}
+Start with any subsequence.  Retain the random variable $A_n$ of
+Proposition~\ref{gr:green-bound} together with all the contour,
+spatial, and local and whole-sphere array variables used in
+Theorem~\ref{sph:energy} and Proposition~\ref{sph:comparison}.
+Its uniform first-moment bound gives tightness, so these variables
+admit a joint further extraction.  Apply the sphere results to this
+enlarged extraction and use a Skorokhod representation.  On the
+resulting probability-one event, their conclusions hold and $A_n$
+converges to a finite random variable.  In particular
+$\sup_n A_n<\infty$ after discarding finitely many initial terms.
+
+Fix a realization in this probability-one event.  Its limiting
+quantum sphere is now a fixed measured conformal sphere.  The
+projections $\pi_n$, measures, edge mesh, uniform energy bounds and
+recovery, and harmonic compactness satisfy hypotheses~(i)--(iii)
+of Theorem~\ref{dyn:conditional}.  The realized bounded $A_n$ gives
+hypothesis~(iv).  The Liouville form is exactly
+\eqref{dyn:continuum-form} with $\mu=\mu_h$.  By
+Lemma~\ref{dyn:energy-clock}, the chain in that deterministic theorem
+is $Y^n_t=X^n_{nt}$.  Consequently its conditional projected path
+law converges to the conditional law of $B^h$, for every fixed
+finite horizon.  All probabilities in this invocation are taken
+on the individual fixed graphs.  The random object being compared
+is the conditional probability law itself.
+
+It remains to express this comparison in the metric $\delta_T$.
+The round topology and the $D_h$ topology agree on the compact
+limiting sphere, so the identity map between them is uniformly
+continuous in both directions.  The path-law convergence just
+proved therefore also holds in $D([0,T],(S^2,D_h))$.
+
+The correspondences in Proposition~\ref{sph:comparison} have
+distortions tending to zero.  The standard correspondence gluing
+constructs a common metric space containing isometric copies of
+all $(V_n,a_nd_n)$ and $(S^2,D_h)$, with
+\[
+ \sup_{v\in V_n}\dist_Z\bigl(v,\pi_n(v)\bigr)\longrightarrow0.
+\]
+For completeness, for each $n$ attach its copy to the sphere by
+cross-distances obtained by taking the infimum over corresponding
+pairs, adding numbers $\varepsilon_n\downarrow0$ larger than half
+the respective distortions;
+use the shortest-path metric on the union, identifying only the
+common sphere, and then take the completion.  This preserves the
+metrics of every copy.  The images become close in Hausdorff
+distance, and their measures become close in Prokhorov distance.
+
+Apply the embedding to each coordinate of a chain path.  Its
+distance in the common Skorokhod space from its $\pi_n$-projected
+path is bounded by the displayed supremum, using the identity time
+change.  The corresponding probability laws therefore have
+Prokhorov distance tending to zero.  Combining this with the
+projected law convergence shows that all three terms defining
+$\delta_T$ tend to zero in the coupled extraction.
+
+We have obtained almost-sure convergence in a further coupling from
+every original subsequence, to the prescribed quantum sphere and
+its conditional diffusion law.  The subsequence principle gives
+convergence in distribution through all positive integers $n$.
+Forgetting the root, the spanning tree, and the contour marks
+leaves precisely the states in Theorem~\ref{thm:main}.  The same
+choice $c=1$ works for every time horizon.
+\end{proof}
diff --git a/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/linear-clock-random-walk-tree-weighted-planar-maps.pdf b/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/linear-clock-random-walk-tree-weighted-planar-maps.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..e1f596e35dbaab3a5bda0d3e4c8a6db24fc93ca5
--- /dev/null
+++ b/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/linear-clock-random-walk-tree-weighted-planar-maps.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:0acb1c84bf79e9c86c03d6e1346b607ff7bcef8241321e49adf5ffb52f2aeec6
+size 628184
diff --git a/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/README.md b/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..eb5e46f81808355ed3c5a67ba0bfe91f922bceee
--- /dev/null
+++ b/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/README.md
@@ -0,0 +1,18 @@
+# [A logarithmic independence bound for clique-free graphs](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 25, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026,
+  author = {{OpenAI}},
+  title = {{A logarithmic independence bound for clique-free graphs}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/paper.pdf}{OAI:A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/build/figures/splitting.tex b/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/build/figures/splitting.tex
new file mode 100644
index 0000000000000000000000000000000000000000..f6522087da7f86fedd1eb9d9ec456c9646543608
--- /dev/null
+++ b/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/build/figures/splitting.tex
@@ -0,0 +1,69 @@
+\begin{figure}[htbp]
+  \centering
+  \begingroup
+  \definecolor{splitblue}{RGB}{34,92,140}
+  \definecolor{splitochre}{RGB}{166,104,24}
+  \begin{tikzpicture}[
+    x=1cm,y=1cm,
+    font=\small,
+    vertex/.style={circle,draw=black,fill=white,minimum size=5.5pt,
+      inner sep=0pt,line width=.6pt},
+    copy/.style={vertex,minimum size=7pt},
+    blueedge/.style={draw=splitblue,line width=.9pt},
+    ochreedge/.style={draw=splitochre,line width=.9pt},
+    every label/.style={font=\small,inner sep=2pt}
+  ]
+    \node at (1.15,2.22) {Before splitting};
+    \node at (7.35,2.22) {After splitting};
+
+    \draw[blueedge] (0,0) -- (2.55,1.55);
+    \draw[blueedge] (0,0) -- (2.55,.55);
+    \draw[ochreedge] (0,0) -- (2.55,-.55);
+    \draw[ochreedge] (0,0) -- (2.55,-1.55);
+    \draw[line width=.6pt] (2.55,.55) -- (2.55,1.55);
+    \node[vertex] (old-u) at (0,0) {};
+    \node[anchor=east,align=center,inner sep=0pt] at (-.17,0)
+      {$u$\\[-1pt]$w_u$};
+    \node[vertex,label=right:$v_1$] at (2.55,1.55) {};
+    \node[vertex,label=right:$v_2$] at (2.55,.55) {};
+    \node[vertex,label=right:$v_3$] at (2.55,-.55) {};
+    \node[vertex,label=right:$v_4$] at (2.55,-1.55) {};
+
+    \draw[-{Latex[length=2.2mm]},line width=.65pt]
+      (3.55,0) -- (4.8,0);
+    \node[align=center,font=\footnotesize] at (4.17,.48)
+      {split by\\label};
+
+    \draw[blueedge] (6.15,1.05) -- (8.7,1.55);
+    \draw[blueedge] (6.15,1.05) -- (8.7,.55);
+    \draw[ochreedge] (6.15,-1.05) -- (8.7,-.55);
+    \draw[ochreedge] (6.15,-1.05) -- (8.7,-1.55);
+    \draw[line width=.6pt] (8.7,.55) -- (8.7,1.55);
+    \node[copy,draw=splitblue] at (6.15,1.05) {};
+    \node[anchor=east,align=center,inner sep=0pt] at (5.96,1.05)
+      {$(u,y)$\\[-1pt]$w_u$};
+    \node[copy,draw=splitochre] at (6.15,-1.05) {};
+    \node[anchor=east,align=center,inner sep=0pt] at (5.96,-1.05)
+      {$(u,z)$\\[-1pt]$w_u$};
+    \node[vertex,label=right:$v'_1$] at (8.7,1.55) {};
+    \node[vertex,label=right:$v'_2$] at (8.7,.55) {};
+    \node[vertex,label=right:$v'_3$] at (8.7,-.55) {};
+    \node[vertex,label=right:$v'_4$] at (8.7,-1.55) {};
+
+    \draw[blueedge] (.45,-2.12) -- (.9,-2.12);
+    \node[anchor=west,font=\footnotesize] at (.99,-2.12)
+      {label $y$ at $u$};
+    \draw[ochreedge] (5.75,-2.12) -- (6.2,-2.12);
+    \node[anchor=west,font=\footnotesize] at (6.29,-2.12)
+      {label $z$ at $u$};
+  \end{tikzpicture}
+  \endgroup
+  \caption{Local view of splitting by admissible labels. The incidences
+    $uv_1,uv_2$ receive label $y$, and $uv_3,uv_4$ receive label $z$.
+    Each copy of $u$ has weight $w_u$. The remote endpoints are also split:
+    $v'_i$ denotes the copy selected by the incidence $v_i u$.
+    The displayed triangle survives because its two incidences agree at
+    each of its three corners, with all labels admissible. Each retained
+    edge is placed once, between its selected copies.}
+  \label{fig:label-splitting}
+\end{figure}
diff --git a/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/build/main.tex b/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/build/main.tex
new file mode 100644
index 0000000000000000000000000000000000000000..1f4decdea04c8b1799860c2e810c6489b4dde5e3
--- /dev/null
+++ b/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/build/main.tex
@@ -0,0 +1,29 @@
+\pdfinfoomitdate=1
+\pdftrailerid{}
+\pdfsuppressptexinfo=15
+\documentclass[11pt]{article}
+\input{preamble}
+\title{A logarithmic independence bound for clique-free graphs}
+\author{OpenAI}
+\date{September 25, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+For every fixed integer $r\ge4$, every $K_r$-free graph on $n$ vertices
+with average degree $d\ge2$ has an independent set of size at least
+$c_r n\log d/d$, where $c_r>0$ depends only on $r$.
+This proves the fixed-clique-size independence conjecture of Ajtai,
+Erd\H{o}s, Koml\'os and Szemer\'edi.
+\end{abstract}
+\setcounter{tocdepth}{1}
+\tableofcontents
+\clearpage
+\input{sections/introduction}
+\input{sections/optimizer}
+\input{sections/invariant}
+\input{sections/entropy}
+\input{sections/coupling}
+\input{sections/closure}
+\bibliographystyle{plain}
+\bibliography{references}
+\end{document}
diff --git a/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/build/preamble.tex b/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/build/preamble.tex
new file mode 100644
index 0000000000000000000000000000000000000000..4efe8c4cdee7a66ea794db9a650ce14fd0ecb323
--- /dev/null
+++ b/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/build/preamble.tex
@@ -0,0 +1,20 @@
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage[margin=1.05in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{microtype,enumitem,xcolor,tikz,flafter}
+\usetikzlibrary{arrows.meta,positioning,calc,fit}
+\usepackage[hyphens]{url}
+\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=green!35!black,urlcolor=blue!55!black]{hyperref}
+\hypersetup{pdftitle={A logarithmic independence bound for clique-free graphs},pdfauthor={OpenAI}}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\numberwithin{equation}{section}
+\setlist{nosep}
+\setlength{\emergencystretch}{1em}
diff --git a/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/build/references.bib b/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/build/references.bib
new file mode 100644
index 0000000000000000000000000000000000000000..00bdc614044cee1a13608baf321b16bbb1eb6f34
--- /dev/null
+++ b/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/build/references.bib
@@ -0,0 +1,166 @@
+@article{AKS1980,
+  author = {Ajtai, Mikl{\'o}s and Koml{\'o}s, J{\'a}nos and Szemer{\'e}di, Endre},
+  title = {{\href{https://doi.org/10.1016/0097-3165(80)90030-8}{A note on {Ramsey} numbers}}},
+  journal = {Journal of Combinatorial Theory, Series A},
+  volume = {29},
+  number = {3},
+  pages = {354--360},
+  year = {1980},
+  doi = {10.1016/0097-3165(80)90030-8}
+}
+
+@article{AEKS1981,
+  author = {Ajtai, Mikl{\'o}s and Erd{\H o}s, Paul and Koml{\'o}s, J{\'a}nos and Szemer{\'e}di, Endre},
+  title = {{\href{https://doi.org/10.1007/BF02579451}{On {Tur\'an}'s theorem for sparse graphs}}},
+  journal = {Combinatorica},
+  volume = {1},
+  number = {4},
+  pages = {313--317},
+  year = {1981},
+  doi = {10.1007/BF02579451},
+  url = {https://renyi.hu/~p_erdos/1981-19.pdf}
+}
+
+@article{Shearer1983,
+  author = {Shearer, James B.},
+  title = {{\href{https://doi.org/10.1016/0012-365X(83)90273-X}{A note on the independence number of triangle-free graphs}}},
+  journal = {Discrete Mathematics},
+  volume = {46},
+  number = {1},
+  pages = {83--87},
+  year = {1983},
+  doi = {10.1016/0012-365X(83)90273-X}
+}
+
+@article{Shearer1995,
+  author = {Shearer, James B.},
+  title = {{\href{https://doi.org/10.1002/rsa.3240070305}{On the independence number of sparse graphs}}},
+  journal = {Random Structures \& Algorithms},
+  volume = {7},
+  number = {3},
+  pages = {269--271},
+  year = {1995},
+  doi = {10.1002/rsa.3240070305},
+  url = {https://www.cs.toronto.edu/~molloy/2427-25/readings/Shearer1.pdf}
+}
+
+@article{Alon1996,
+  author = {Alon, Noga},
+  title = {{\href{https://doi.org/10.1002/(SICI)1098-2418(199610)9:3\%3C271::AID-RSA1\%3E3.0.CO;2-U}{Independence numbers of locally sparse graphs and a {Ramsey} type problem}}},
+  journal = {Random Structures \& Algorithms},
+  volume = {9},
+  number = {3},
+  pages = {271--278},
+  year = {1996},
+  doi = {10.1002/(SICI)1098-2418(199610)9:3<271::AID-RSA1>3.0.CO;2-U},
+  url = {https://web.math.princeton.edu/~nalon/PDFS/indrams1.pdf}
+}
+
+@misc{DJM2025,
+  author = {Dhawan, Abhishek and Janzer, Oliver and Methuku, Abhishek},
+  title = {{\href{https://arxiv.org/abs/2511.17191v2}{Independent sets and colorings of {$K_{t,t,t}$}-free graphs}}},
+  year = {2025},
+  eprint = {2511.17191},
+  archivePrefix = {arXiv},
+  primaryClass = {math.CO},
+  note = {arXiv:2511.17191v2},
+  url = {https://arxiv.org/abs/2511.17191v2}
+}
+
+@misc{Davies2026,
+  author = {Davies, Ewan},
+  title = {{\href{https://arxiv.org/abs/2609.04654v1}{Random independent sets and local sparsity}}},
+  year = {2026},
+  eprint = {2609.04654},
+  archivePrefix = {arXiv},
+  primaryClass = {math.CO},
+  note = {arXiv:2609.04654v1},
+  url = {https://arxiv.org/abs/2609.04654v1}
+}
+
+@article{KleinbergTardos2002,
+  author = {Kleinberg, Jon and Tardos, {\'E}va},
+  title = {{\href{https://doi.org/10.1145/585265.585268}{Approximation algorithms for classification problems with pairwise relationships: metric labeling and {Markov} random fields}}},
+  journal = {Journal of the ACM},
+  volume = {49},
+  number = {5},
+  pages = {616--639},
+  year = {2002},
+  doi = {10.1145/585265.585268},
+  url = {https://www.cs.cornell.edu/info/people/kleinber/focs99-mrf.pdf}
+}
+
+@misc{AngelSpinka2021,
+  author = {Angel, Omer and Spinka, Yinon},
+  title = {{\href{https://arxiv.org/abs/1903.00632v2}{Pairwise optimal coupling of multiple random variables}}},
+  year = {2021},
+  eprint = {1903.00632},
+  archivePrefix = {arXiv},
+  primaryClass = {math.PR},
+  note = {arXiv:1903.00632v2},
+  url = {https://arxiv.org/abs/1903.00632v2}
+}
+
+@article{DMS2012,
+  author = {Dutta, Kunal and Mubayi, Dhruv and Subramanian, C. R.},
+  title = {{\href{https://doi.org/10.1137/110839023}{New lower bounds for the independence number of sparse graphs and hypergraphs}}},
+  journal = {SIAM Journal on Discrete Mathematics},
+  volume = {26},
+  number = {3},
+  pages = {1134--1147},
+  year = {2012},
+  doi = {10.1137/110839023},
+  url = {https://homepages.math.uic.edu/~mubayi/papers/HypIndSetFinal3.pdf}
+}
+
+@misc{DKPS2020,
+  author = {Davies, Ewan and Kang, Ross J. and Pirot, Fran{\c{c}}ois and Sereni, Jean-S{\'e}bastien},
+  title = {{\href{https://arxiv.org/abs/2003.14361v1}{Graph structure via local occupancy}}},
+  year = {2020},
+  eprint = {2003.14361},
+  archivePrefix = {arXiv},
+  primaryClass = {math.CO},
+  note = {arXiv:2003.14361v1, 31 March 2020},
+  url = {https://arxiv.org/abs/2003.14361v1}
+}
+
+@article{DhawanLocallySparse2026,
+  author = {Dhawan, Abhishek},
+  title = {{\href{https://doi.org/10.1007/s00026-025-00778-7}{Bounds for the independence and chromatic numbers of locally sparse graphs}}},
+  journal = {Annals of Combinatorics},
+  volume = {30},
+  pages = {277--304},
+  year = {2026},
+  note = {Published online 5 August 2025},
+  doi = {10.1007/s00026-025-00778-7}
+}
+
+@article{BDKY2015,
+  author = {Benjamini, Itai and Duminil-Copin, Hugo and Kozma, Gady and Yadin, Ariel},
+  title = {{\href{https://doi.org/10.1214/14-AOP934}{Disorder, entropy and harmonic functions}}},
+  journal = {The Annals of Probability},
+  volume = {43},
+  number = {5},
+  pages = {2332--2373},
+  year = {2015},
+  doi = {10.1214/14-AOP934},
+  eprint = {1111.4853},
+  archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/1111.4853v2}
+}
+
+@misc{RamseyFive,
+  author = {{OpenAI}},
+  title = {{The sharp logarithmic exponent of $r(5,t)$}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/The-Sharp-Logarithmic-Exponent-of-r-5-t-September-24-2026/paper.pdf}{OAI:The-Sharp-Logarithmic-Exponent-of-r-5-t-September-24-2026}},
+  year = {2026}
+}
+
+@misc{RamseyFixed,
+  author = {{OpenAI}},
+  title = {{Sharp Logarithmic Exponents for Fixed Off-Diagonal Ramsey Numbers}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/Sharp-Logarithmic-Exponents-for-Fixed-Off-Diagonal-Ramsey-Numbers-September-24-2026/paper.pdf}{OAI:Sharp-Logarithmic-Exponents-for-Fixed-Off-Diagonal-Ramsey-Numbers-September-24-2026}},
+  year = {2026}
+}
diff --git a/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/build/sections/closure.tex b/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/build/sections/closure.tex
new file mode 100644
index 0000000000000000000000000000000000000000..98c3eaaa7d817a1cca601abf1b103ce69528a041
--- /dev/null
+++ b/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/build/sections/closure.tex
@@ -0,0 +1,132 @@
+\section{From triangle mass to independent sets}
+\label{sec:closure}
+
+We now use the triangle estimate to choose independent vertices at a
+controlled cost in $F_G^*$. For an $n$-vertex $K_r$-free graph of maximum
+degree at most $\Delta\ge3$,
+a constant test weight gives $F_G^*$ at least a constant multiple of
+$n(\log\Delta)^2/\Delta$. The triangle estimate will provide a vertex
+whose closed-neighborhood deletion costs at most a constant times
+$\log\Delta$. Induction compares these two bounds and gives the required
+independent set. We keep $\Delta$ fixed throughout the induction, then
+pass to average degree by deleting high-degree vertices.
+Let $B_r\ge0$ be the constant in~\eqref{eq:optimizer-triangle}, and set
+\begin{equation}
+ D_r=3+\frac32 B_r.
+ \label{eq:independence-constant}
+\end{equation}
+
+\begin{proposition}\label{prop:maximum-degree}
+For every integer $r\ge4$, every real number $\Delta\ge3$, and every finite $K_r$-free graph $G$
+with $n$ vertices and maximum degree at most $\Delta$,
+\[
+ \alpha(G)\ge \frac{F_G^*}{D_r\log\Delta}
+ \qquad\text{and}\qquad
+ \alpha(G)\ge \frac{n\log\Delta}{4D_r\Delta}.
+\]
+\end{proposition}
+
+\begin{proof}
+Fix $\Delta\ge3$.  We prove the first inequality by induction on $n$.
+For the empty graph both sides are zero.  Suppose $n>0$, let $w$ maximize
+$F_G$, and write $W=\sum_v w_v$.  By Lemma~\ref{lem:optimizer},
+$0<w_v\le1$.
+
+For a vertex $v$, put $A_v=\{v\}\cup N(v)$ and let
+$G-A_v$ denote the induced graph on the remaining vertices.
+Set $q=0$ on $A_v$ and $q=w$ elsewhere.  The variational identity
+for $F_G(w)-F_G(q)$ gives the exact deletion cost
+\begin{equation}
+ F_G^*-F_{G-A_v}(w|_{V(G)\setminus A_v})
+ =w(A_v)+M_{G[A_v]}(w).
+ \label{eq:closed-neighborhood-cost}
+\end{equation}
+Indeed, the vertex divergence is $w(A_v)$, and the quadratic term
+$M_G(q-w)$ is $M_{G[A_v]}(w)$.
+
+Average this cost over $v$ with probability $w_v/W$.
+An edge $ab$ lies in $G[A_v]$ precisely when $v=a$, $v=b$, or
+$v$ is a common neighbor of $a$ and $b$.  Hence
+\begin{align}
+ \sum_v\frac{w_v}{W}
+       \bigl(w(A_v)+M_{G[A_v]}(w)\bigr)
+ &=\frac{\sum_v w_v^2+2M_G(w)
+       +\sum_v w_v^2w(N(v))+3T_G(w)}{W} \notag\\
+ &\le 1+(4+3B_r)\frac{M_G(w)}{W}.
+ \label{eq:averaged-deletion-cost}
+\end{align}
+Here $w_v\le1$ gives $\sum_vw_v^2\le W$ and
+$\sum_vw_v^2w(N(v))\le2M_G(w)$; we then used the triangle
+bound~\eqref{eq:optimizer-triangle}.
+
+It remains to bound $2M_G(w)/W$ by $\log\Delta$. First, stationarity
+and Jensen's inequality for $-\log$ give
+\begin{equation}
+ \log\frac{n}{W}
+ \le \frac1n\sum_v\log\frac1{w_v}
+ =\frac1n\sum_v w(N(v))
+ =\frac1n\sum_v \deg_G(v)w_v
+ \le \frac{\Delta W}{n}.
+ \label{eq:optimizer-total-weight}
+\end{equation}
+If $W/n<1/\Delta$, then the left side exceeds $\log\Delta>1$
+and the right side is less than $1$, a contradiction. Therefore
+$W\ge n/\Delta$. Applying the concavity of $\log$ with probabilities
+$w_v/W$ now gives
+\begin{equation}
+ \frac{2M_G(w)}{W}
+ =\sum_v\frac{w_v}{W}\log\frac1{w_v}
+ \le \log\left(\sum_v\frac{w_v}{W}\frac1{w_v}\right)
+ =\log\frac{n}{W}
+ \le\log\Delta.
+ \label{eq:optimizer-edge-weight}
+\end{equation}
+Thus the average in \eqref{eq:averaged-deletion-cost} is at most
+\[
+ 1+\left(2+\frac32B_r\right)\log\Delta
+ \le D_r\log\Delta,
+\]
+where we used $\log\Delta\ge1$.
+
+Choose a vertex whose cost is at most the average.  Since the restriction
+of $w$ is a feasible weight vector for $G-A_v$,
+\[
+ F_{G-A_v}^*\ge F_G^*-D_r\log\Delta.
+\]
+The graph $G-A_v$ is $K_r$-free and still has maximum degree at most the
+same number $\Delta$.  The induction hypothesis, followed by adjoining
+$v$ to an independent set of $G-A_v$, yields
+\[
+ \alpha(G)\ge 1+\alpha(G-A_v)
+ \ge 1+\frac{F_{G-A_v}^*}{D_r\log\Delta}
+ \ge \frac{F_G^*}{D_r\log\Delta}.
+\]
+
+For the second inequality, test $F_G$ on the constant vector with entry
+$p=(\log\Delta)/\Delta$.  Since $|E(G)|\le n\Delta/2$,
+\[
+ F_G^*\ge np\left(1-\log p-\frac{\Delta p}{2}\right)
+ =np\left(1+\frac12\log\Delta-\log\log\Delta\right)
+ \ge \frac{np\log\Delta}{4}.
+\]
+The last inequality holds for every $\Delta>1$: the function
+$1+t/4-\log t$ on $t>0$ has minimum $2-\log4>0$ at $t=4$.
+Combining this bound with the first inequality proves the result.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+Let $G$ have $n$ vertices and average degree $d\ge2$.  Delete the vertices
+whose degrees in $G$ exceed $2d$, and let $H$ be the induced graph on
+the remaining vertices.  Fewer than $n/2$ vertices are deleted, since
+the sum of all degrees is $nd$.  Thus $|V(H)|\ge n/2$, and
+$H$ is $K_r$-free with maximum degree at most the real number $2d$.
+Proposition~\ref{prop:maximum-degree} gives
+\[
+ \alpha(G)\ge\alpha(H)
+ \ge\frac{|V(H)|\log(2d)}{8D_r d}
+ \ge\frac{n\log d}{16D_r d}.
+\]
+We may therefore take $c_r=1/(16D_r)$.
+All constants used in obtaining $B_r$, and consequently $D_r$ and $c_r$,
+depend only on $r$.
+\end{proof}
diff --git a/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/build/sections/coupling.tex b/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/build/sections/coupling.tex
new file mode 100644
index 0000000000000000000000000000000000000000..b2c757f80787d5618035ff4eeb11ad2629f49dc1
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+++ b/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/build/sections/coupling.tex
@@ -0,0 +1,318 @@
+\section{Splitting vertices and bounding triangle mass}
+\label{sec:splitting}
+
+The local walks give similar probability distributions to the two
+neighbors at a corner of a triangle, on average under the ordered
+triangle law. We now sample from these distributions jointly.
+Equal samples keep the two incident edges of a triangle at the same copy
+of a vertex. A condition on the reversed transition probability ensures
+that each copy has a small weighted neighborhood.
+
+\subsection{Sampling all rows together}
+
+We use the shared rejection construction of Kleinberg and
+Tardos~\cite[Section~3, Lemmas~3.1 and~3.2]{KleinbergTardos2002}.
+Angel and Spinka~\cite[Theorem~2 and Section~2.1]{AngelSpinka2021}
+give the sharper pairwise disagreement estimate stated below.
+The short proof supplies the joint law for all rows at once.
+
+\begin{lemma}[Shared rejection sampling]\label{lem:shared-rejection}
+Let $I$ and $J$ be finite nonempty sets, and let $(p_i)_{i\in I}$ be
+probability distributions on $J$. There are jointly distributed random
+variables $(Y_i)_{i\in I}$ such that $Y_i$ has law $p_i$ and, for every
+$i,i'\in I$,
+\[
+ \mathbb P(Y_i\ne Y_{i'})
+ \le \frac{2\|p_i-p_{i'}\|_{\mathrm{TV}}}
+ {1+\|p_i-p_{i'}\|_{\mathrm{TV}}}
+ \le 2\|p_i-p_{i'}\|_{\mathrm{TV}}.
+\]
+The resulting joint law has finite support.
+\end{lemma}
+
+\begin{proof}
+Write $m=|J|$. Take independent trials $(Z_t,U_t)_{t\ge1}$, with $Z_t$
+uniform on $J$ and $U_t$ uniform on $[0,1]$. For each $i$, let $Y_i=Z_t$
+at the first trial satisfying $U_t\le p_i(Z_t)$. A trial is accepted
+by row $i$ with probability $1/m$, and its accepted value has law $p_i$.
+More explicitly, for $y\in J$,
+\[
+ \mathbb P(Y_i=y)
+ =\sum_{t\ge1}(1-1/m)^{t-1}\frac{p_i(y)}m=p_i(y).
+\]
+Every row stops almost surely; since $I$ is finite, all rows stop after
+a finite number of trials almost surely.
+
+Fix two rows $p,q$, and put $d=\|p-q\|_{\mathrm{TV}}$. At the first
+trial accepted by either row, the probability that both accept is
+\[
+ \frac{\sum_{y\in J}\min(p(y),q(y))}
+ {\sum_{y\in J}\max(p(y),q(y))}
+ =\frac{1-d}{1+d}.
+\]
+On this event their labels agree. They may also agree otherwise, so
+the disagreement probability is at most $2d/(1+d)$. Finally, the label
+vector takes values in the finite set $J^I$, so its induced law has
+finite support. The infinite sequence of trials is only a convenient
+construction of that finite joint law.
+\end{proof}
+
+\subsection{Admissible labels and vertex splitting}
+
+Let $H$ be a finite graph with positive vertex weights. Assume that every
+edge $uv$ has common-neighbor weight $c_{uv}\ge1/x$, that every
+neighborhood has weight at most $e^x$, and that $H$ satisfies the
+cross-mass condition~\eqref{eq:cross-mass} with constant $C\ge1$. Here $x\ge32$. If $H$ has
+an edge, then it has a triangle; throughout this subsection suppose
+$T_H(w)>0$.
+
+Use the integer $j$ supplied by Lemma~\ref{lem:entropy-smoothing}, and
+for each non-isolated vertex $u$ write $K_u=K_u^j$ for the corresponding
+kernel on $N_H(u)$. Isolated vertices may be discarded, since they
+contribute neither edge nor triangle mass. The
+directed-edge probability measure is
+\[
+ \mu(u,v)=\frac{w_uw_vc_{uv}}{6T_H(w)}.
+\]
+Conditional on $u$, its second coordinate has distribution
+$\pi_u(v)=w_vc_{uv}/h_u$, and $K_u$ is reversible for $\pi_u$.
+We use the constants from that lemma:
+\[
+ A_C=10+42C^2+2C,
+ \qquad B_C=2\sqrt{A_C+1}.
+\]
+In particular, $K_u(v,y)\le xw_y$, the averaged row entropy is at
+most $A_C\log x$, and the average total variation distance between
+the rows indexed by the other two vertices of a weighted ordered
+triangle is at most $B_C\sqrt{\log x/x}$.
+
+For each non-isolated $u$, apply Lemma~\ref{lem:shared-rejection} to all rows of
+$K_u$. Denote the label assigned to $v\in N_H(u)$ by $Y_{u,v}$.
+These finite joint laws may be sampled independently for different
+vertices $u$.  We will group the neighbors of $u$ according to their
+labels, while keeping the total weight of each group at most $e^{\sqrt x}$.
+For this purpose, call a label $y$ at the incidence $(u,v)$
+\emph{admissible} if
+\begin{equation}\label{eq:admissible-label}
+ K_u(y,v)\ge w_v e^{-\sqrt x}.
+\end{equation}
+For a fixed outcome of the labels, define
+\[
+ A_{u,y}=\{v\in N_H(u):Y_{u,v}=y
+       \text{ and }K_u(y,v)\ge w_ve^{-\sqrt x}\}.
+\]
+The reversed transition in \eqref{eq:admissible-label} is chosen so that
+all vertices in one group can be compared with a single probability row:
+\begin{equation}\label{eq:label-group-weight}
+ w(A_{u,y})
+ \le e^{\sqrt x}\sum_{v\in A_{u,y}}K_u(y,v)
+ \le e^{\sqrt x}.
+\end{equation}
+This holds for every outcome of the labels.  The next lemma shows that
+the admissibility requirement rejects only a small fraction of incidences
+under the triangle distribution.
+
+\begin{lemma}\label{lem:reverse-admissibility}
+The probability that $Y_{u,v}$ is inadmissible, averaged over
+$(u,v)\sim\mu$, is at most
+\begin{equation}\label{eq:inadmissibility-bound}
+ \delta_x:=\frac{(A_C+1)\log x}{\sqrt x+\log x}.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Under the joint law $(u,v)\sim\mu$, $y\sim K_u(v,\cdot)$,
+reversibility gives
+\[
+ (u,v,y)\ \stackrel{\mathrm{law}}=\ (u,y,v).
+\]
+Both $K_u(v,y)$ and $K_u(y,v)$ are positive on this law's support:
+all $\pi_u$ are positive and
+$\pi_u(v)K_u(v,y)=\pi_u(y)K_u(y,v)$.
+Consequently
+\[
+ \ell:=\log\frac{w_v}{K_u(y,v)}
+ \quad\text{satisfies}\quad
+ \mathbb E\ell=H_j\le A_C\log x.
+\]
+The density bound applied to the reversed row gives $\ell\ge-\log x$.
+Inadmissibility is precisely the event $\ell>\sqrt x$. Applying Markov's
+inequality to the nonnegative random variable $\ell+\log x$ proves
+\eqref{eq:inadmissibility-bound}. No bound on an individual row's
+entropy is needed here.
+\end{proof}
+
+We now turn the groups into a graph, for each fixed outcome of the labels.
+Make one vertex $(u,y)$ of weight $w_u$ for each nonempty group.
+For each edge $uv$ of $H$ whose two incidence labels are admissible,
+put an edge between $(u,Y_{u,v})$ and $(v,Y_{v,u})$.
+Call the resulting graph $H'$. Figure~\ref{fig:label-splitting} shows
+the construction at one vertex.
+We take the maximum neighborhood weight of the empty graph to be zero.
+\input{figures/splitting}
+
+\begin{lemma}\label{lem:weighted-splitting}
+There is a choice of these labels for which the projection
+$(u,y)\mapsto u$ preserves weights, maps directed edges injectively
+to directed edges of $H$, and satisfies
+\begin{align}
+ \max_{a\in V(H')}w(N_{H'}(a))&\le e^{\sqrt x},
+ \label{eq:split-neighborhood}\\
+ T_{H'}(w)&\ge(1-a_C\log x/\sqrt x)T_H(w),
+ \label{eq:split-triangles}
+\end{align}
+where $a_C=6(B_C+A_C+1)$.
+\end{lemma}
+
+\begin{proof}
+Every old edge creates at most one new edge. The projection therefore
+maps directed edges injectively, and the new graph has neither loops
+nor multiple edges. At a fixed vertex $(u,y)$, different neighbors
+project to different vertices of $H$, because $H$ is simple and each
+edge $uv$ was used at most once.  By \eqref{eq:label-group-weight},
+\[
+ w(N_{H'}(u,y))
+ \le w(A_{u,y})
+ \le e^{\sqrt x}.
+\]
+In particular, duplicating a vertex does not duplicate any of its
+incident edges at one fixed copy.
+
+An old triangle $\{u,v,z\}$ survives whenever its two labels agree
+at each corner and all six incidence labels are admissible. These
+conditions are also necessary for its three edges to form a triangle
+in $H'$. A triangle of $H'$ projects to three distinct old vertices,
+and hence to an old triangle. Moreover, there is at most one triangle
+over each old triangle, since each old edge has at most one preimage.
+The weights of a surviving triangle are unchanged.
+
+Sample an ordered triangle $(u,v,z)$ with probability
+$w_uw_vw_z/(6T_H(w))$. Its ordered-edge marginal is $\mu$.
+At the corner $u$, the shared coupling and the row-distance estimate
+give
+\[
+ \mathbb E\,\mathbb P(Y_{u,v}\ne Y_{u,z})
+ \le 2B_C\sqrt{\frac{\log x}{x}}.
+\]
+Each of the other two corners has the same bound, by the permutation
+symmetry of the ordered-triangle law. Each of its six directed
+incidences has marginal $\mu$, so
+Lemma~\ref{lem:reverse-admissibility} bounds the average
+inadmissibility probability of each by $\delta_x$.
+A union bound, with no independence assumption between these events,
+therefore gives
+\[
+ \frac{\mathbb E(T_H(w)-T_{H'}(w))}{T_H(w)}
+ \le6B_C\sqrt{\frac{\log x}{x}}+6\delta_x
+ \le a_C\frac{\log x}{\sqrt x}.
+\]
+The last inequality uses $\log x\ge1$. All label vectors together
+have finite support. At least one of them has triangle mass at least
+its expectation, proving the claim.
+\end{proof}
+
+\subsection{Reduction and iteration}
+
+The preceding construction turns the walk estimates into a reduction
+of weighted neighborhoods. Choose $X=X(C)\ge32$ so large that
+\begin{equation}\label{eq:iteration-threshold}
+ 2a_C\frac{\log X}{\sqrt X}\le\frac12.
+\end{equation}
+Such a choice depends only on $C$.
+
+\begin{proposition}\label{prop:neighborhood-reduction}
+Let $G$ be a finite positive-vertex-weighted graph satisfying the
+cross-mass condition~\eqref{eq:cross-mass} with constant $C\ge1$, and suppose that
+$\max_vw(N_G(v))\le e^x$ for some $x>X(C)$. There is a finite
+positive-vertex-weighted graph $G'$ with a weight-preserving map to
+$G$ that is injective on directed edges, such that
+\begin{align}
+ \max_vw(N_{G'}(v))&\le e^{\sqrt x},\label{eq:step-neighborhood}\\
+ T_{G'}(w)&\ge
+ \left(1-a_C\frac{\log x}{\sqrt x}\right)T_G(w)
+       -\frac{M_G(w)}x.\label{eq:step-triangles}
+\end{align}
+It also satisfies the same cross-mass condition, and $M_{G'}(w)\le M_G(w)$.
+\end{proposition}
+
+\begin{proof}
+Peel edges whose current common-neighbor weight is less than $1/x$.
+If $H$ is the remaining graph, Lemma~\ref{lem:triangle-peeling} gives
+\[
+ T_H(w)\ge T_G(w)-M_G(w)/x.
+\]
+Edge deletion does not increase any weighted neighborhood, and it
+preserves the cross-mass condition. If $H$ has no edges, take $G'$ to
+be empty. Then $T_G(w)\le M_G(w)/x$, which implies
+\eqref{eq:step-triangles}.
+Otherwise, apply Lemma~\ref{lem:weighted-splitting} to $H$.
+Since $x\mapsto\log x/\sqrt x$ is decreasing for $x\ge e^2$,
+\eqref{eq:iteration-threshold} ensures
+$0\le1-a_C\log x/\sqrt x\le1$. Hence
+\begin{align*}
+ T_{G'}(w)
+ &\ge\left(1-a_C\frac{\log x}{\sqrt x}\right)
+           \left(T_G(w)-\frac{M_G(w)}x\right)\\
+ &\ge\left(1-a_C\frac{\log x}{\sqrt x}\right)T_G(w)
+           -\frac{M_G(w)}x.
+\end{align*}
+The projection through $H$ is still injective on directed edges.
+The edge-mass bound and inherited cross-mass condition follow from
+Lemma~\ref{lem:invariant-preservation}.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{thm:weighted-triangles}]
+If $G$ has no edges, both sides vanish. Otherwise choose $x_0\ge1$
+such that every neighborhood has weight at most $e^{x_0}$. While
+$x_i>X$, apply Proposition~\ref{prop:neighborhood-reduction} and
+set $x_{i+1}=\sqrt{x_i}$. Write $G_i$ for the successive graphs,
+$T_i=T_{G_i}(w)$ and $M_i=M_{G_i}(w)$; the weights at each stage
+are the inherited weights of its vertex copies. Stop after $k$ steps,
+when $x_k\le X$. The procedure is finite, and the $k=0$ case is
+allowed.
+
+We record bounds independent of $x_0$ and $k$. If $k>0$, put
+$y=x_{k-1}>X$. The used parameters, in reverse order, are
+$y,y^2,y^4,\ldots,y^{2^{k-1}}$. For
+$f(t)=\log t/\sqrt t$ and $t\ge X\ge32$,
+\[
+ \frac{f(t^2)}{f(t)}=\frac2{\sqrt t}\le\frac12.
+\]
+As $f$ is decreasing on $[X,\infty)$, it follows that
+\begin{equation}\label{eq:summable-losses}
+ \sum_{i<k}a_Cf(x_i)\le2a_Cf(X)\le\frac12,
+ \qquad
+ \sum_{i<k}\frac1{x_i}\le\frac{1/X}{1-1/X}\le1.
+\end{equation}
+Both inequalities also hold for empty sums when $k=0$.
+
+Let $\varepsilon_i=a_Cf(x_i)$. Iterating
+\eqref{eq:step-triangles}, using $M_i\le M_0$ and
+$0\le1-\varepsilon_i\le1$, gives
+\[
+ T_k\ge\left(\prod_{i<k}(1-\varepsilon_i)\right)T_0
+          -M_0\sum_{i<k}\frac1{x_i}
+ \ge\frac12T_0-M_0.
+\]
+Here $\prod(1-\varepsilon_i)\ge1-\sum\varepsilon_i$.
+Every neighborhood of $G_k$ has weight at most $e^X$, so
+\[
+ 3T_k
+ =\sum_{\{u,v\}\in E(G_k)}w_uw_vw(N_{G_k}(u)\cap N_{G_k}(v))
+ \le e^X M_k\le e^X M_0.
+\]
+Combining the last two displays proves the theorem with
+$C_\triangle(C)=2(1+e^{X(C)}/3)$. All
+thresholds and constants depend only on $C$; neither the number of
+vertices nor the initial neighborhood bound enters them.
+\end{proof}
+
+
+Applying Theorem~\ref{thm:weighted-triangles} to the cross-mass bound
+of Lemma~\ref{lem:cross-mass} gives the variational estimate needed
+for the independent-set argument: for every maximizer $w$ on a finite
+$K_r$-free graph,
+\begin{equation}\label{eq:optimizer-triangle}
+ T_G(w)\le B_r M_G(w),\qquad B_r=C_{\triangle}(16r^2).
+\end{equation}
+The zero-vertex case has both masses zero.
diff --git a/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/build/sections/entropy.tex b/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/build/sections/entropy.tex
new file mode 100644
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+++ b/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/build/sections/entropy.tex
@@ -0,0 +1,299 @@
+\section{Local walks and entropy}
+\label{sec:entropy}
+
+We construct probability distributions on each neighborhood so that rows
+indexed by the other two vertices of a triangle are close on average,
+with the average weighted by triangle mass.  We obtain the rows by running
+a weighted random walk.  The cross-mass estimate bounds the average entropy
+by showing that most starting rows, in this weighted average, put almost
+all their probability on sets of small weight inside their neighborhoods.
+The resulting estimate is independent of the number of vertices.
+The use of entropy increments to control averaged distances between
+walk distributions has a precedent in Benjamini, Duminil-Copin, Kozma
+and Yadin~\cite[Section~2]{BDKY2015}.
+We prove the finite weighted version needed here, using the cross-mass
+hypothesis to bound its entropy.
+
+Let $G$ be a finite simple graph with positive weights satisfying
+\eqref{eq:cross-mass} with constant $C\ge1$.
+Fix $x\ge32$ and suppose further that $G$ has an edge and
+\begin{equation}
+ s_u:=w(N(u))\le e^x\quad\hbox{for every $u$},\qquad
+ c_{uv}:=w(N(u)\cap N(v))\ge x^{-1}\quad(uv\in E(G)).
+ \label{eq:walk-hypotheses}
+\end{equation}
+Put
+\[
+ h_u=\sum_{v\in N(u)}w_vc_{uv}
+\]
+for every vertex $u$, with the empty sum equal to zero.  We use kernels
+only at non-isolated vertices.  At such a vertex $u$, set
+\[
+ \pi_u(v)=\frac{w_vc_{uv}}{h_u}\quad(v\in N(u)).
+\]
+These quantities are positive, and
+\begin{equation}
+ s_u\ge x^{-1},\qquad h_u\ge s_u/x.
+ \label{eq:walk-lower-bounds}
+\end{equation}
+On the finite set $N(u)$ define stochastic matrices
+\begin{equation}
+ P_u(v,z)=\frac{w_z\mathbf1_{\{vz\in E(G)\}}}{c_{uv}},\qquad
+ Q_u=\frac{I+P_u}{2},\qquad K_u^j=P_uQ_u^j\quad(j\ge0).
+ \label{eq:walk-kernels}
+\end{equation}
+Thus $P_u$ is the weighted random walk on the induced graph $G[N(u)]$,
+and $Q_u$ adds probability $1/2$ of staying at the current vertex.
+The row $K_u^j(v,\cdot)$ is the law after one $P_u$-step from $v$ followed
+by $j$ such lazy steps.  The initial step gives
+$P_u(v,y)\le xw_y$; we will show that subsequent steps preserve this
+density bound.  The holding probability will let us compare rows started
+at adjacent vertices through an entropy increment.
+The superscript in $K_u^j$ indexes this family, rather than denoting a
+power of a fixed kernel.  We shall choose one index $j$ for all neighborhoods.
+
+Here are the probability laws used in the argument.  Put
+\[
+ Z=\sum_u w_uh_u=6T_G(w)>0,
+ \qquad \mu(u,v)=\frac{w_uw_vc_{uv}}{Z}\quad(uv\in E(G)),
+\]
+where both orientations of each edge occur in the sum defining $Z$.
+The conditional distribution of $v$ given $u$ under $\mu$ is $\pi_u$.
+If, after sampling $(u,v)$ from $\mu$, we sample $z$ from $P_u(v,\cdot)$,
+then
+\begin{equation}
+ \mathbb P(u,v,z)=\frac{w_uw_vw_z}{Z}
+ \quad\hbox{for each ordered triangle $(u,v,z)$}.
+ \label{eq:ordered-triangle-law}
+\end{equation}
+We call this the ordered triangle law.
+
+For a probability vector $a$ supported on vertices with positive weights,
+write
+\[
+ \mathcal H_w(a)=\sum_{y:a(y)>0}a(y)\log\frac{w_y}{a(y)},\qquad
+ H_j=\mathbb E_{(u,v)\sim\mu}\mathcal H_w(K_u^j(v,\cdot)).
+\]
+Thus $H_j$ is an average of row entropies relative to the vertex weights;
+the weights themselves need not sum to one.  For probability vectors $a,b$
+on the same finite set, write
+\[
+ \|a-b\|_{\mathrm{TV}}=\frac12\sum_y|a(y)-b(y)|.
+\]
+
+The neighborhood bound $s_u\le e^x$ alone allows row entropy as large
+as $x$.  The next lemma improves the averaged upper bound to
+$O_C(\log x)$ over $\lceil x\rceil$ steps.  Since the entropy increments
+are nonnegative, some step has small averaged increment; this will give
+the required comparison under the ordered triangle law.
+
+\begin{lemma}[Smoothing the local walks]
+\label{lem:entropy-smoothing}
+Under \eqref{eq:cross-mass} and \eqref{eq:walk-hypotheses}, put
+\[
+ E_C=42C^2+2C,\qquad A_C=10+E_C,\qquad B_C=2\sqrt{A_C+1},
+ \qquad R=\lceil x\rceil.
+\]
+Each $K_u^j$ is reversible with stationary measure $\pi_u$, and
+\begin{equation}
+ K_u^j(v,y)\le xw_y\qquad(j\ge0).
+ \label{eq:walk-density}
+\end{equation}
+Moreover,
+\begin{equation}
+ -\log x\le H_0\le H_1\le\cdots\le H_R\le A_C\log x.
+ \label{eq:entropy-upper}
+\end{equation}
+There exists $j\in\{0,\ldots,R-1\}$ such that, under the ordered triangle
+law,
+\begin{equation}
+ \mathbb E\bigl\|K_u^j(v,\cdot)-K_u^j(z,\cdot)\bigr\|_{\mathrm{TV}}
+ \le B_C\sqrt{\frac{\log x}{x}}.
+ \label{eq:triangle-row-distance}
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+We first check the kernels and then bound the entropy they can acquire.
+Detailed balance follows from
+\[
+ \pi_u(v)P_u(v,z)
+ =\frac{w_vw_z\mathbf1_{\{vz\in E(G)\}}}{h_u}
+ =\pi_u(z)P_u(z,v).
+\]
+Thus $P_u$ is reversible and stationary for $\pi_u$.  Its powers, and hence
+$Q_u$ and $K_u^j$, have the same properties.  No irreducibility assumption
+is needed.  By $c_{uv}\ge x^{-1}$, every row of $P_u$ satisfies
+$P_u(v,y)\le xw_y$.  Since $P_u$ and $Q_u$ commute,
+\[
+ K_u^j(v,y)=\sum_zQ_u^j(v,z)P_u(z,y)\le xw_y,
+\]
+which proves \eqref{eq:walk-density}.  In particular, $H_j\ge-\log x$.
+
+\paragraph{Sets that contain most of the walk.}
+Write $L=e^x$.  For each fixed non-isolated starting vertex $v$, we build
+sets in the whole graph that contain almost all of the walk for every
+center $u\in N(v)$ simultaneously.  Using the same set for all these
+centers will allow a cross-mass estimate to control its intersections
+with their neighborhoods.  Define
+\[
+ S_0(v)=N(v),\qquad
+ S_{i+1}(v)=S_i(v)\cup
+ \{y:w(N(y)\cap S_i(v))\ge x^{-6}/L\}.
+\]
+We suppress the argument $v$ while it is fixed.  Double counting gives
+\[
+ \sum_yw_yw(N(y)\cap S_i)
+ =\sum_{z\in S_i}w_zs_z\le Lw(S_i).
+\]
+Consequently
+\[
+ w(S_{i+1})\le(1+L^2x^6)w(S_i),\qquad
+ w(S_R)\le e^x(1+e^{2x}x^6)^R\le e^{20x^2}.
+\]
+For the last inequality, $x\ge32$ implies $6\log x\le x$,
+$\log(1+e^{2x}x^6)\le4x$, and $R\le2x$.
+
+For every edge $uv$ and $0\le i\le R$, we claim that
+\begin{equation}
+ K_u^i(v,N(u)\setminus S_i(v))\le ix^{-4}.
+ \label{eq:walk-leakage}
+\end{equation}
+The claim holds at $i=0$, since $P_u(v,\cdot)$ is supported on $N(v)$.
+In the step from $i$ to $i+1$, mass already outside $S_i$ contributes at
+most $ix^{-4}$.  If $z\in S_i$ and $y\notin S_{i+1}$, then $y\ne z$,
+so $Q_u(z,y)\le xw_y\mathbf1_{\{zy\in E(G)\}}$.  By
+\eqref{eq:walk-density}, the additional mass reaching such a $y\in N(u)$
+is at most
+\[
+ \sum_{z\in N(u)\cap S_i\cap N(y)}xw_z\,xw_y
+ \le x^{-4}w_y/L.
+\]
+Summing over $y\in N(u)$ contributes at most $x^{-4}$, proving the claim.
+
+\paragraph{Most neighborhoods meet these sets in small weight.}
+The global set $S_R(v)$ has weight at most $e^{20x^2}$.  We now show
+that its intersection with $N(u)$ has weight at most $x^8$ for most
+pairs $(u,v)$ under $\mu$.
+For fixed $v$ let
+\[
+ B_v=\{u\in N(v):w(N(u)\cap S_R(v))>x^8\}.
+\]
+The first use of \eqref{eq:cross-mass}, with parameter $20x^2$,
+gives
+\[
+ x^8w(B_v)\le e_w(N(v),S_R(v))
+ \le Cs_v(20x^2+\log^+(1/s_v))
+ \le21Cx^2s_v,
+\]
+where $s_v\ge x^{-1}$ was used in the final inequality.  Thus
+$w(B_v)\le21Cx^{-6}s_v$.
+
+This weight estimate must be converted to the conditional law of $u$
+given $v$, which is not normalized vertex weight.  The symmetry of $\mu$
+gives the exact formula
+\[
+ \mathbb P_\mu(u\in B_v\mid v)
+ =\frac{\sum_{u\in B_v}w_uc_{uv}}{h_v}
+ =\frac{e_w(B_v,N(v))}{h_v}.
+\]
+The second use of \eqref{eq:cross-mass}, now with parameter $x$,
+applies because $B_v\subseteq N(v)$ and $s_v\le e^x$.  The function $g_x$
+is nondecreasing on $[0,\infty)$, and therefore, for $b\ge0$,
+\[
+ g_x(b)\le2x\max\{b,e^{-x}\}.
+\]
+Using $h_v\ge s_v/x$ and $s_v\ge x^{-1}$, we obtain
+\begin{align}
+ \mathbb P_\mu(u\in B_v\mid v)
+ &\le\frac{2Cx^2}{s_v}\bigl(w(B_v)+e^{-x}\bigr)\notag\\
+ &\le42C^2x^{-4}+2Cx^3e^{-x}
+ \le E_Cx^{-4}.
+ \label{eq:exceptional-pair-probability}
+\end{align}
+Here $e^{-x}\le x^{-7}$ for $x\ge32$.
+
+We can now bound $H_R$.  A probability vector $a$ supported on a set $D$
+of total weight $m>0$ satisfies
+\begin{equation}
+ \mathcal H_w(a)\le\log m.
+ \label{eq:weighted-entropy-support}
+\end{equation}
+Indeed, normalize the weights on $D$ and use nonnegativity of relative
+entropy.  If $u\notin B_v$, the set $D=N(u)\cap S_R(v)$ has weight at most
+$x^8$, and \eqref{eq:walk-leakage} shows that
+$K_u^R(v,D)=1-\delta$ with $0\le\delta\le Rx^{-4}<1$.  Condition the
+row on $D$ and its complement.  Formula \eqref{eq:weighted-entropy-support}
+and the binary entropy bound then give
+\[
+ \mathcal H_w(K_u^R(v,\cdot))
+ \le\log2+8\log x+\delta x.
+\]
+An empty complement, or zero probability on the complement, simply
+contributes zero.  On pairs with $u\in B_v$, use the unconditional bound
+$\mathcal H_w(K_u^R(v,\cdot))\le\log s_u\le x$.  Averaging and applying
+\eqref{eq:exceptional-pair-probability} yields
+\[
+ H_R\le\log2+8\log x+Rx^{-3}+E_Cx^{-3}
+ \le(10+E_C)\log x=A_C\log x.
+\]
+
+\paragraph{A small entropy increment gives close rows.}
+For probability vectors $a,b$, let
+\[
+ D(a\|b)=\sum_{y:a(y)>0}a(y)\log\frac{a(y)}{b(y)}
+\]
+denote relative entropy, interpreted as $+\infty$ if necessary.  Sample
+$(u,v)$ from $\mu$ and then $z$ from $Q_u(v,\cdot)$.  Since
+$Q_uK_u^j=K_u^{j+1}$, the mixture of $K_u^j(z,\cdot)$ over this conditional
+choice of $z$ is $K_u^{j+1}(v,\cdot)$.  Consequently, for fixed $u,v$,
+\[
+ \mathbb E_{z\sim Q_u(v,\cdot)}
+ D\bigl(K_u^j(z,\cdot)\|K_u^{j+1}(v,\cdot)\bigr)
+ =\mathcal H_w(K_u^{j+1}(v,\cdot))
+  -\mathbb E_{z\sim Q_u(v,\cdot)}\mathcal H_w(K_u^j(z,\cdot)).
+\]
+All these divergences are finite on events of positive probability,
+because the second argument is the mixture containing the first.
+Conditional on $u$, the law of $v$ is $\pi_u$, which is stationary for
+$Q_u$.  After averaging over $v$ and then $u$, the preceding identity
+becomes
+\begin{equation}
+ H_{j+1}-H_j
+ =\mathbb E_{\mu Q}
+ D\bigl(K_u^j(z,\cdot)\|K_u^{j+1}(v,\cdot)\bigr)\ge0.
+ \label{eq:entropy-increment}
+\end{equation}
+The stationarity step is an averaged identity; it asserts no equality
+between individual rows.  Telescoping the increments gives an index
+$0\le j<R$ for which
+\[
+ H_{j+1}-H_j\le\frac{(A_C+1)\log x}{R}
+ \le\frac{(A_C+1)\log x}{x}.
+\]
+
+For completeness, $\|a-b\|_{\mathrm{TV}}\le\sqrt{D(a\|b)}$ follows
+from
+\[
+ D(a\|b)\ge-2\log\sum_y\sqrt{a(y)b(y)}
+ \ge\sum_y(\sqrt{a(y)}-\sqrt{b(y)})^2
+\]
+and Cauchy--Schwarz.  Therefore \eqref{eq:entropy-increment} and Jensen's
+inequality imply
+\[
+ \mathbb E_{\mu Q}
+ \bigl\|K_u^j(z,\cdot)-K_u^{j+1}(v,\cdot)\bigr\|_{\mathrm{TV}}
+ \le\sqrt{H_{j+1}-H_j}.
+\]
+To pass to the ordered triangle law, put
+$a_v=K_u^j(v,\cdot)$ and $b_v=K_u^{j+1}(v,\cdot)$ within each fixed
+neighborhood.  The triangle inequality and $Q_u=(I+P_u)/2$ give exactly
+\begin{align*}
+ \mathbb E_{\mu P}\|a_v-a_z\|_{\mathrm{TV}}
+ &\le\mathbb E_\mu\|a_v-b_v\|_{\mathrm{TV}}
+     +\mathbb E_{\mu P}\|a_z-b_v\|_{\mathrm{TV}}\\
+ &=2\mathbb E_{\mu Q}\|a_z-b_v\|_{\mathrm{TV}}.
+\end{align*}
+Combining the last two bounds proves \eqref{eq:triangle-row-distance}
+with $B_C=2\sqrt{A_C+1}$.
+\end{proof}
diff --git a/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/build/sections/introduction.tex b/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/build/sections/introduction.tex
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+\section{Introduction}
+\label{sec:introduction}
+
+An independent set in a graph is a set of pairwise nonadjacent vertices.
+For a finite simple graph $G$, write $\alpha(G)$ for the largest size of
+an independent set. If $G$ has $n>0$ vertices, its average degree is
+$d(G)=2|E(G)|/n$. The graph is $K_r$-free if no $r$ vertices are pairwise
+adjacent; throughout, exclusion means exclusion as an ordinary subgraph.
+All logarithms are natural. For general graphs, the elementary bound
+$\alpha(G)\ge n/(d+1)$ is best possible, as disjoint unions of cliques
+show. Excluding a fixed clique forces larger independent sets. We prove
+that the improvement is a factor of order $\log d$.
+
+\begin{theorem}\label{thm:main}
+For every integer $r\ge4$, there is a constant $c_r>0$ such that every
+finite simple $K_r$-free graph $G$ with $n$ vertices and average degree $d\ge2$
+satisfies
+\[
+ \alpha(G)\ge c_r\frac{n\log d}{d}.
+\]
+\end{theorem}
+
+The constant is uniform in both the number of vertices and the average
+degree. The theorem gives a positive answer to the fixed-clique-size
+independence conjecture of Ajtai, Erd\H{o}s, Koml\'os and Szemer\'edi,
+also recorded as Erd\H{o}s Problem~802. The order $n\log d/d$ is best
+possible up to a constant factor even for triangle-free graphs, and
+hence for the larger class of $K_r$-free graphs
+\cite[pp.~314--315]{AEKS1981}. We do not optimize the constant $c_r$.
+
+\subsection{History and related results}
+
+Ajtai, Koml\'os and Szemer\'edi~\cite{AKS1980} proved the logarithmic
+improvement when triangles are excluded.
+Shearer~\cite{Shearer1983} subsequently obtained the asymptotic
+coefficient~$1$: triangle-free graphs of average degree $d\to\infty$
+have independent sets of size at least $(1-o(1))n\log d/d$.
+In 1981, Ajtai, Erd\H os, Koml\'os and Szemer\'edi conjectured the same
+order of growth for each fixed forbidden clique~\cite[p.~314, Equation~(3)]{AEKS1981}.
+They proved the lower bound $\Omega_r(n\log\log d/d)$ as $d\to\infty$,
+so excluding any fixed clique already improves on the general bound.
+Shearer~\cite[Corollary~2]{Shearer1995} improved this to
+$\Omega_r(n\log d/(d\log\log d))$, again in terms of the average
+degree. Theorem~\ref{thm:main} removes this remaining $\log\log d$
+denominator.
+
+Subsequent refinements describe more of the graph's local structure.
+Dutta, Mubayi and Subramanian~\cite{DMS2012} give bounds involving
+the full degree sequence. Davies, Kang, Pirot and
+Sereni~\cite[Theorem~30 and Corollary~31]{DKPS2020} develop a local-occupancy
+framework that improves constants and also gives coloring estimates.
+Dhawan~\cite[Theorems~1.3 and~1.4]{DhawanLocallySparse2026} allows bounded
+numbers of cliques, either globally or inside neighborhoods.
+For a fixed forbidden clique, these estimates retain a $\log\log$
+loss in the relevant degree parameter; they provide information beyond
+the uniform order sought in Theorem~\ref{thm:main}.
+
+Several stronger local assumptions yield logarithmic bounds.
+Alon~\cite[Theorem~1.1 and the remark following its proof]{Alon1996}
+treated graphs whose neighborhoods have
+bounded chromatic number, obtaining an average-degree bound of the
+required order. His argument also admits bounded fractional chromatic
+number in each neighborhood. Davies~\cite[Theorems~1 and~3]{Davies2026}
+establishes weighted local versions: he bounds individual vertex
+inclusion probabilities in a random independent set under bounded
+neighborhood maximum average degree or fractional chromatic number.
+For every fixed three-colorable forbidden graph, Dhawan, Janzer and
+Methuku~\cite[Theorems~1.4 and~1.6]{DJM2025} prove
+$(1-o(1))n\log d/d$ as $d\to\infty$, uniformly in $n$.
+This attains leading coefficient~$1$ in that restricted class.
+Neither bounded neighborhood fractional chromatic number nor exclusion of a fixed
+three-colorable graph covers all ordinary $K_r$-free graphs for $r\ge4$.
+For the latter restriction, complete tripartite graphs illustrate the
+difference: they are $K_4$-free and contain every fixed three-colorable
+graph once their parts are large enough.
+
+For comparison, let $r(s,t)$ be the least integer $N$ such that every
+$N$-vertex graph contains a $K_s$ or an independent set of size $t$.
+The fixed off-diagonal Ramsey estimates in
+\cite[Theorem~1.1]{RamseyFive} and \cite[Theorem~1.1]{RamseyFixed}
+determine logarithmic exponents for $r(5,t)$ and, respectively,
+for $r(s,t)$ with each fixed $s\ge6$.
+Together they state
+\[
+ r(s,t)=\frac{t^{s-1}}{(\log t)^{s-2+o(1)}}
+ \qquad\text{for each fixed }s\ge5.
+\]
+These are estimates for extremal graph orders as $t\to\infty$.
+Theorem~\ref{thm:main} controls every input graph using its own average
+degree, including graphs whose degree is small relative to their order.
+Neither companion is an input to the proof. In particular, the weighted
+triangle theorem used here is proved from its own cross-mass hypothesis.
+
+
+\subsection{The proof in outline}
+
+The proof uses positive vertex weights. For such a vector $w$, write
+$w(S)=\sum_{v\in S}w_v$, and define its edge and triangle masses by
+\[
+ M_G(w)=\sum_{\{u,v\}\in E(G)}w_uw_v,
+ \qquad
+ T_G(w)=\sum_{\{u,v,z\}\text{ a triangle}}w_uw_vw_z.
+\]
+Both sums are over unordered sets. The central step is to show that
+$T_G(w)$ is at most a constant depending only on $r$ times $M_G(w)$
+when $w$ maximizes, over nonnegative vertex weights,
+\[
+ F_G(w)=\sum_v w_v(1-\log w_v)-M_G(w).
+\]
+The zero-coordinate convention is $0\log0=0$.
+Three stages establish the theorem.
+
+\begin{enumerate}
+\item \emph{Extract a bound that survives changes to the graph.}
+Section~\ref{sec:optimizer} chooses a maximizing weight and uses random
+multipliers of mean one to control the directed edge mass between any
+two vertex sets. The multiplier construction is recursive on the excluded
+clique size. Section~\ref{sec:triangle} shows that this cross-mass
+bound survives edge deletion and splitting a vertex into copies carrying
+its original weight, provided each original edge is retained at most once.
+
+\item \emph{Reduce weighted neighborhoods without losing many triangles.}
+Sections~\ref{sec:entropy} and~\ref{sec:splitting} prove a triangle bound
+for every weighted graph with that cross-mass property. At each reduction
+step, we first delete edges whose common neighborhoods have small weight,
+charging the lost triangle mass to their edge mass. For each vertex
+$u$, walks inside $N(u)$ assign a probability distribution on $N(u)$ to
+each starting vertex $v\in N(u)$. The cross-mass bound limits the average
+entropy growth of these walks. At a suitable common time, the distributions
+started at $v$ and $z$ are close on average over ordered triangles $(u,v,z)$,
+weighted by $w_uw_vw_z$.
+
+Sampling these distributions together groups the incident edges at $u$;
+close distributions usually give the same group to the two edges of a
+triangle. A cutoff on reversed transition probabilities bounds each
+group's total neighbor weight; the entropy and density bounds, together
+with reversibility, control the average triangle mass discarded by the
+cutoff. Splitting vertices according to
+the resulting groups reduces the largest neighborhood weight from
+$e^x$ to $e^{\sqrt{x}}$. The losses are summable through all iterations.
+Once every neighborhood has bounded weight, triangle mass is directly
+bounded by edge mass.
+
+\item \emph{Choose independent vertices at controlled cost.}
+For a graph with maximum degree at most $\Delta\ge3$, evaluating $F_G$ at
+the constant weight $(\log\Delta)/\Delta$ gives an extremal value at
+least a constant multiple of $n(\log\Delta)^2/\Delta$.
+When we delete a closed neighborhood and retain the old maximizing weights
+elsewhere, edges between the chosen vertex's neighbors contribute to
+the cost. Averaging over the chosen vertex with probabilities proportional
+to its weight counts these edges as triangles.
+Section~\ref{sec:closure} uses the triangle bound to find a vertex whose
+closed-neighborhood deletion reduces this value by at most
+$D_r\log\Delta$. Selecting such a vertex and repeating on the remaining
+graph gives an independent set of size at least a constant multiple of
+$n\log\Delta/\Delta$. The same degree bound $\Delta$ is kept throughout
+this induction. Deleting the original graph's high-degree vertices then
+yields the average-degree bound in Theorem~\ref{thm:main}.
+\end{enumerate}
+
+The weighted triangle theorem is a separate statement about cross masses;
+it has no clique hypothesis. Preservation of the cross-mass bound under
+vertex splitting permits repeated reduction without requiring the new weights to
+remain maximizers. The entropy estimate controls only averages over a
+specified triangle distribution, and the splitting argument uses exactly
+those averages. Every threshold in these two steps depends only on the
+cross-mass constant. The proof gives an existence result with an
+$r$-dependent constant; it makes no claim of an efficient algorithm or
+an optimal leading constant.
diff --git a/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/build/sections/invariant.tex b/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/build/sections/invariant.tex
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+\section{From cross mass to triangle mass}
+\label{sec:triangle}
+
+The triangle argument will replace vertices by copies carrying their full
+original weights. Total vertex weight can therefore increase. The rule
+that makes this operation useful is that each original edge is retained
+at most once. We show that this rule preserves the cross-mass bound and
+controls edge and triangle masses, so the subsequent argument can proceed
+without maximizing weights anew.
+
+Retain the notation $M_G(w)$, $T_G(w)$, $e_w(A,B)$ and $g_x$
+from the preceding sections. We write $e_{G,w}$ when the graph must
+be specified. We say that $(G,w)$ satisfies
+the cross-mass bound with constant $C$ if, for every $x\ge1$ and every
+$A,B\subseteq V(G)$,
+\begin{equation}
+ \label{eq:cross-mass}
+ w(A),w(B)\le e^x
+ \quad\Longrightarrow\quad
+ e_{G,w}(A,B)\le Cg_x(w(A)).
+\end{equation}
+
+\begin{theorem}[Triangle mass from cross mass]
+\label{thm:weighted-triangles}
+For every $C\ge1$ there is a constant $C_{\triangle}(C)$ with the
+following property.  If $G$ is a finite simple graph with positive
+vertex weights $w$ satisfying \eqref{eq:cross-mass}, then
+\[
+ T_G(w)\le C_{\triangle}(C)M_G(w).
+\]
+\end{theorem}
+
+We will repeatedly split vertices so as to reduce neighborhood weights,
+while retaining a fixed proportion of triangle weight.  The cross-mass
+bound is the information that passes from one step to the next.  We first
+verify this preservation and record the effect of deleting edges whose
+common-neighbor weight is small.  The proof of
+Theorem~\ref{thm:weighted-triangles} concludes after the splitting
+construction and its iteration.
+
+\subsection{Preservation and triangle counting}
+
+\begin{lemma}[Preservation under splitting]
+\label{lem:invariant-preservation}
+Let $(G,w)$ satisfy \eqref{eq:cross-mass} with constant $C$.
+Let $H$ be a finite simple graph and let $\pi:V(H)\to V(G)$ satisfy
+the following conditions:
+\begin{enumerate}
+ \item every edge $\{a,b\}$ of $H$ maps to the edge
+       $\{\pi(a),\pi(b)\}$ of $G$;
+ \item the map $(a,b)\mapsto(\pi(a),\pi(b))$ is injective on
+       directed edges of $H$.
+\end{enumerate}
+Give each $a\in V(H)$ the weight $\widetilde w_a=w_{\pi(a)}$.
+Then $(H,\widetilde w)$ satisfies
+\eqref{eq:cross-mass} with the same constant $C$.  Moreover,
+\begin{gather*}
+ M_H(\widetilde w)\le M_G(w),\qquad
+ T_H(\widetilde w)\le T_G(w),\\
+ \widetilde w(N_H(a))\le w(N_G(\pi(a)))
+ \quad\text{for every }a\in V(H).
+\end{gather*}
+These conclusions are preserved under any finite composition of such maps.
+\end{lemma}
+
+\begin{proof}
+For $A,B\subseteq V(H)$, write $A_0=\pi(A)$ and $B_0=\pi(B)$.
+Duplicating vertices can increase their total weight, so
+\[
+ w(A_0)\le\widetilde w(A),\qquad
+ w(B_0)\le\widetilde w(B).
+\]
+The directed edges counted by $e_{H,\widetilde w}(A,B)$ map injectively
+to directed edges counted by $e_{G,w}(A_0,B_0)$, with their weights
+unchanged.  Consequently, if $\widetilde w(A),\widetilde w(B)\le e^x$,
+then
+\[
+ e_{H,\widetilde w}(A,B)
+ \le e_{G,w}(A_0,B_0)
+ \le Cg_x(w(A_0))
+ \le Cg_x(\widetilde w(A)).
+\]
+This also covers overlapping sets and the case of an empty set.
+
+The directed-edge condition implies injectivity on unordered edges:
+two edges with the same unordered image could be oriented to have the
+same directed image.  It therefore gives the assertion about $M$.
+For a fixed $a$, two different neighbors $b,b'$ cannot have the same
+image, since the directed edges $(a,b)$ and $(a,b')$ would then have the
+same image.  Thus $\pi$ maps $N_H(a)$ injectively into $N_G(\pi(a))$,
+giving the neighborhood bound.
+
+Every triangle of $H$ maps to a triangle of $G$, since a homomorphism
+to a simple graph is injective on the vertices of a clique.  Distinct
+triangles cannot have the same image: each of the three image edges
+has at most one preimage edge, and those three preimage edges determine
+the triangle.  Triangle weights are unchanged, proving the assertion
+about $T$.  Finally, compositions preserve vertex weights,
+adjacency and injectivity on directed edges.
+\end{proof}
+
+Edge deletion and vertex deletion are special cases of
+Lemma~\ref{lem:invariant-preservation}, using the inclusion map.
+The splitting operation used below is another special case: replace a
+vertex $u$ by copies indexed by groups of its incident edges, give every
+copy weight $w_u$, and retain each original edge at most once, between
+its assigned endpoint copies.  No assumption about optimization is
+required after these operations.
+
+\begin{lemma}[Triangle normalization and peeling]
+\label{lem:triangle-peeling}
+For a positive-weighted finite simple graph $G$, put
+\[
+ s_u=w(N_G(u)),\qquad
+ c_{uv}=w(N_G(u)\cap N_G(v)),\qquad
+ h_u=\sum_{v\in N_G(u)}w_vc_{uv}.
+\]
+Thus $h_u=2M_{G[N_G(u)]}(w)$ is twice the edge mass inside the
+neighborhood of $u$.
+Then
+\begin{equation}
+ \label{eq:triangle-normalizations}
+ \sum_{\{u,v\}\in E(G)}w_uw_vc_{uv}=3T_G(w),
+ \qquad
+ \sum_{u\in V(G)}w_uh_u=6T_G(w).
+\end{equation}
+In particular, if every neighborhood has weight at most $L$, then
+\begin{equation}
+ \label{eq:terminal-triangle-bound}
+ T_G(w)\le \frac{L}{3}M_G(w).
+\end{equation}
+
+For every $\tau>0$, there is a spanning subgraph $G'$ obtained by
+successive edge deletions such that every edge of $G'$ has common-neighbor
+weight at least $\tau$ and
+\begin{equation}
+ \label{eq:peeling-loss}
+ T_{G'}(w)\ge T_G(w)-\tau M_G(w).
+\end{equation}
+For each nonisolated vertex of $G'$, its neighborhood and common-neighbor
+weights, computed in $G'$, satisfy
+\[
+ s_u\ge\tau,\qquad h_u\ge\tau s_u.
+\]
+If $G$ satisfies \eqref{eq:cross-mass}, then so does $G'$,
+with the same constant.
+\end{lemma}
+
+\begin{proof}
+Each triangle contributes its weight once for each of its three edges
+to the first sum in \eqref{eq:triangle-normalizations}, and once for
+each of its six oriented edges to the second.  Since $c_{uv}\le L$,
+the first identity also proves \eqref{eq:terminal-triangle-bound}.
+
+Starting with $G$, delete any edge whose common-neighbor weight in the
+current graph is less than $\tau$, and continue until no such edge
+remains.  The procedure terminates because $G$ is finite.  If
+$\{u_i,v_i\}$ is the $i$th deleted edge and $c_i$ its common-neighbor
+weight immediately before deletion, the triangle weight lost at that
+step is exactly $w_{u_i}w_{v_i}c_i$.  Telescoping gives
+\[
+ T_G(w)-T_{G'}(w)
+ =\sum_iw_{u_i}w_{v_i}c_i
+ \le\tau\sum_iw_{u_i}w_{v_i}
+ \le\tau M_G(w).
+\]
+This counts each destroyed triangle at its first deleted edge.
+The defining stopping rule gives $c_{uv}\ge\tau$ on every remaining
+edge.  If $u$ is nonisolated, choose a remaining neighbor $v$ to obtain
+$s_u\ge c_{uv}\ge\tau$; summing $w_vc_{uv}\ge\tau w_v$ over the
+remaining neighbors gives $h_u\ge\tau s_u$.  The last assertion follows
+from Lemma~\ref{lem:invariant-preservation}.
+\end{proof}
diff --git a/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/build/sections/optimizer.tex b/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/build/sections/optimizer.tex
new file mode 100644
index 0000000000000000000000000000000000000000..dc64b895e6d6a7e2126b527bde21efefac3836a3
--- /dev/null
+++ b/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/build/sections/optimizer.tex
@@ -0,0 +1,253 @@
+\section{An extremal weight and its cross-mass bound}
+\label{sec:optimizer}
+
+We first choose vertex weights for which a variational inequality controls
+the mass of edges between two sets.  Clique exclusion enters through a
+random modification that keeps every weight unchanged in expectation while
+making the expected edge mass small.
+
+For a finite simple graph $G$ and a real vector $a=(a_v)_{v\in V(G)}$, put
+\[
+ M_G(a)=\sum_{\{u,v\}\in E(G)}a_ua_v.
+\]
+Thus this quadratic expression is defined even when some coordinates are
+negative.  For nonnegative weights $w$, write $w(S)=\sum_{v\in S}w_v$ and
+\[
+ e_w(S,D)=\sum_{u\in S}\sum_{v\in D\cap N_G(u)}w_uw_v.
+\]
+This is a directed cross mass: an edge with both ends in $S\cap D$ is
+counted twice.  All logarithms are natural, and $0\log 0=0$.
+
+\subsection{The extremal weight}
+
+The functional below is the unit-parameter specialization of the
+entropy-minus-edge potential used by Davies
+\cite[Section~3.3, proof of Theorem~17]{Davies2026}.
+We give its elementary maximizing properties here.
+
+Define
+\[
+ F_G(w)=\sum_{v\in V(G)}w_v(1-\log w_v)-M_G(w),
+ \qquad F_G^*=\max_{w\ge0}F_G(w).
+\]
+The maximum in this definition exists, as the following lemma shows.
+
+\begin{lemma}\label{lem:optimizer}
+For every nonempty finite simple graph $G$, the function $F_G$ attains its
+maximum.  Every maximizer $w$ has positive coordinates and satisfies
+\begin{equation}\label{eq:stationarity}
+ \log(1/w_v)=w(N_G(v)) \qquad(v\in V(G)).
+\end{equation}
+In particular, $w_v\le1$.  For every nonnegative vector $q$,
+\begin{equation}\label{eq:optimizer-divergence}
+ 0\le F_G(w)-F_G(q)
+ =\sum_v\left(q_v\log\frac{q_v}{w_v}-q_v+w_v\right)
+   +M_G(q-w).
+\end{equation}
+Here a term with $q_v=0$ in the sum equals $w_v$.
+\end{lemma}
+
+\begin{proof}
+The continuous function $f(t)=t(1-\log t)$ on $[0,\infty)$ is at most $1$
+and tends to $-\infty$ as $t\to\infty$.  Since $M_G(w)\ge0$ for $w\ge0$,
+\[
+ F_G(w)\le f(\|w\|_\infty)+|V(G)|-1.
+\]
+Consequently $F_G(w)\to-\infty$ as $\|w\|_\infty\to\infty$, so continuity
+gives a maximum.  If a maximizer had $w_v=0$, changing just this coordinate
+to $t>0$ would increase $F_G$ by
+\[
+ t\bigl(1-\log t-w(N_G(v))\bigr)>0
+\]
+for sufficiently small $t$.  All coordinates are therefore positive, and
+the first derivative in each coordinate gives \eqref{eq:stationarity}.
+To obtain \eqref{eq:optimizer-divergence}, expand the quadratic expression
+$M_G(q)$ around $w$.  Its linear term is
+$\sum_v(q_v-w_v)w(N_G(v))$, which cancels the corresponding term using
+\eqref{eq:stationarity}.  The inequality follows from maximality.
+\end{proof}
+
+For the empty graph on no vertices we set $F_G^*=0$; no positive-coordinate
+assertion is needed in that case.
+
+\subsection{Random multipliers with small expected edge mass}
+
+The recursive use of clique-free neighborhoods goes back to the
+sparse-subgraph argument of Ajtai, Erd\H{o}s, Koml\'os and
+Szemer\'edi~\cite[Section~3]{AEKS1981}. Here the recursion preserves
+each vertex weight in expectation. The next lemma applies to arbitrary
+positive weights and does not require the maximizing equations.
+
+The construction successively removes neighborhoods from the remaining
+induced graph, until every remaining vertex has small neighborhood weight.
+Each removed part excludes a smaller clique, and the remainder has small
+edge mass. We then retain one part at random and divide its weights by
+that part's selection probability.
+Every vertex keeps its original expected weight, while edges joining
+different parts disappear. Within a selected neighborhood, the same
+construction is applied recursively. The multiplier bound below measures
+the cost of these successive rescalings.
+
+\begin{lemma}\label{lem:sparse-multipliers}
+Let $r\ge2$ be an integer, let $G$ be a finite $K_r$-free graph with positive vertex
+weights $w$, and put $s=w(V(G))$.  For every $0<\varepsilon\le1/2$ there
+are jointly distributed random variables $(m_v)_{v\in V(G)}$ such that
+\begin{equation}\label{eq:sparse-multipliers}
+ 0\le m_v\le b_r(\varepsilon)
+ :=2^{r(r-2)}\varepsilon^{-(r-2)},\qquad
+ \mathbb E m_v=1,\qquad
+ \mathbb E M_G((w_vm_v)_v)\le\varepsilon s^2.
+\end{equation}
+Their joint distribution can be taken to have finite support.
+\end{lemma}
+
+\begin{proof}
+The empty graph on no vertices is immediate.  We induct on $r$, starting
+with $r=2$, when there are no edges and $m_v=1$ suffices.  Suppose $r\ge3$
+and set $\eta=\varepsilon/4$.
+
+Starting with all vertices, repeatedly choose a vertex whose open
+neighborhood in the remaining induced graph has weight greater than
+$\eta s$, and remove that neighborhood as a set $V_i$.  These disjoint
+sets have weights $s_i=w(V_i)>\eta s$, and each $G[V_i]$ is
+$K_{r-1}$-free.  Indeed, a $(r-1)$-clique in such a neighborhood together
+with its chosen vertex would form a $K_r$.  When the procedure stops,
+the remaining set $R$ satisfies
+\begin{equation}\label{eq:remainder-mass}
+ M_{G[R]}(w)
+ =\frac12\sum_{v\in R}w_vw(N_G(v)\cap R)
+ \le\frac12\eta s\,w(R)\le\frac12\eta s^2.
+\end{equation}
+
+If no sets $V_i$ were removed, take every $m_v=1$.  Otherwise put
+$b=\sum_i s_i>0$ and $p_i=s_i/(2b)$.  Choose $R$ with probability $1/2$
+and $V_i$ with probability $p_i$, putting all multipliers outside the
+chosen set equal to zero.  On choosing $R$, set $m_v=2$ for $v\in R$.
+On choosing $V_i$, use the induction hypothesis there with precision
+$\varepsilon/4$, and multiply its output by $1/p_i$.
+The option of choosing $R$ is retained even if $R$ is empty; in that
+event all multipliers are zero.
+
+Every vertex has mean multiplier one.  Since
+$p_i>\eta/2=\varepsilon/8$, each multiplier is at most
+\[
+ \max\left\{2,\frac8\varepsilon
+                    b_{r-1}(\varepsilon/4)\right\}
+ = b_r(\varepsilon).
+\]
+The equality follows directly from the displayed formula for $b_r$.
+Only edges within the chosen set contribute.  Therefore
+\begin{align*}
+ \mathbb E M_G((w_vm_v)_v)
+ &\le 2M_{G[R]}(w)
+       +\sum_i\frac{\varepsilon s_i^2}{4p_i}\\
+ &\le \eta s^2+\frac{\varepsilon}{2}b^2
+ \le\frac34\varepsilon s^2.
+\end{align*}
+The finite construction and the induction hypothesis also give finite
+support for the joint law.
+\end{proof}
+
+\subsection{The cross-mass estimate}
+
+For $x\ge1$, define
+\[
+ g_x(s)=
+ \begin{cases}
+ s\bigl(x+\log^+(1/s)\bigr),&s>0,\\
+ 0,&s=0,
+ \end{cases}
+ \qquad \log^+ a=\max\{0,\log a\}.
+\]
+The function $g_x$ is nondecreasing on $[0,\infty)$.  On $(0,1)$ its
+derivative is $x+\log(1/s)-1\ge0$, and on $(1,\infty)$ it is $x$;
+the formulas agree continuously at $s=1$ and $s=0$.
+
+\begin{lemma}\label{lem:cross-mass}
+Let $r\ge2$ be an integer and let $w$ maximize $F_G$ on a nonempty finite $K_r$-free
+graph $G$.  For every $x\ge1$ and all vertex sets $S,D$ with
+$w(S),w(D)\le e^x$,
+\begin{equation}\label{eq:optimizer-cross-mass}
+ e_w(S,D)\le C_r g_x(w(S)),
+ \qquad C_r=16r^2.
+\end{equation}
+The sets $S$ and $D$ need not be disjoint.
+\end{lemma}
+
+\begin{proof}
+First suppose $S\cap D=\varnothing$, and put $s=w(S)$ and $t=w(D)$.
+The case $s=0$ is immediate.  Initially assume $e^{-x}\le s\le e^x$.
+We will increase the weights on $S$ and set those on $D$ to zero.
+In the variational identity, this creates a negative quadratic term
+proportional to $e_w(S,D)$. The random multipliers keep the positive
+quadratic term within $S$ small, while their bounded range controls the
+vertex terms through a logarithm.
+Apply Lemma~\ref{lem:sparse-multipliers} to $G[S]$ with
+$\varepsilon=e^{-8x}$, and set $k=e^{4x}$.  Define a nonnegative random
+vector by
+\[
+ q_v=
+ \begin{cases}
+ (1+km_v)w_v,&v\in S,\\
+ 0,&v\in D,\\
+ w_v,&v\notin S\cup D.
+ \end{cases}
+\]
+The expected quadratic term in \eqref{eq:optimizer-divergence} is at most
+\begin{equation}\label{eq:cross-quadratic}
+ \mathbb E M_G(q-w)
+ \le k^2\varepsilon s^2+\frac12t^2-k e_w(S,D).
+\end{equation}
+Here the last term uses $\mathbb E m_v=1$; no independence between the
+multipliers is required. The choice $k^2\varepsilon=1$ keeps the first
+term controlled despite the increase in the weights on $S$.
+
+For $y\ge0$,
+$(1+y)\log(1+y)-y\le y\log(1+y)$.  Also
+\[
+ \log(1+kb_r(e^{-8x}))\le A_rx,
+ \qquad
+ A_r=4+8(r-2)+(1+r(r-2))\log2.
+\]
+It follows that the expected sum of vertex terms in
+\eqref{eq:optimizer-divergence} is at most $t+kA_rxs$.
+Combining this with \eqref{eq:cross-quadratic} and dividing by $k$ gives
+\begin{align*}
+ e_w(S,D)
+ &\le A_rxs+e^{-4x}s^2+e^{-4x}(t+t^2/2)\\
+ &\le (A_r+3)xs.
+\end{align*}
+For the last inequality, use $s,t\le e^x$ and $s\ge e^{-x}$: the three
+error terms divided by $s$ are at most $e^{-3x}$, $e^{-2x}$, and
+$e^{-x}/2$, respectively, and their sum is at most $3x$.
+
+If instead $0<s<e^{-x}$, apply the estimate just proved with
+$x'=\log(1/s)>x$.  Both set weights are at most $e^{x'}$ and
+$s=e^{-x'}$, so
+\[
+ e_w(S,D)\le(A_r+3)x's\le(A_r+3)g_x(s).
+\]
+Thus the bound with constant $A_r+3$ holds for all disjoint sets.
+
+For arbitrary $S,D$, assign each vertex independently to one of two
+classes with equal probabilities.  Let $S'$ be the part of $S$ in the
+first class and $D'$ the part of $D$ in the second.  These sets are
+disjoint, their weights satisfy the same upper bounds, and
+\[
+ \mathbb E e_w(S',D')=\frac14e_w(S,D).
+\]
+There are no loops, so each directed edge in this expectation has two
+distinct endpoints.  The disjoint-set bound and monotonicity give
+\[
+ \frac14e_w(S,D)
+ \le (A_r+3)\mathbb E g_x(w(S'))
+ \le (A_r+3)g_x(s).
+\]
+Finally, $A_r+3\le r^2+6r-8\le4r^2$ for $r\ge2$, which proves
+\eqref{eq:optimizer-cross-mass}.
+\end{proof}
+
+Lemma~\ref{lem:cross-mass} is the only consequence of extremality
+needed in the weighted triangle argument.  That argument can therefore
+be formulated for any positive weighted graph satisfying
+\eqref{eq:optimizer-cross-mass} with some constant $C$.
diff --git a/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/paper.pdf b/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..cf4f898d7bdc1b1dbe6e6fd91fbc7e38c4dc4e4e
--- /dev/null
+++ b/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:15717a7c82914df2f220fad05a3fd6de7fd15f2abe26a4d1454d2b005ef39606
+size 427255
diff --git a/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/README.md b/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..cd0664961ac150850c8c9ce3dca64a6a52bcb022
--- /dev/null
+++ b/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/README.md
@@ -0,0 +1,18 @@
+# [A Low-Space Algorithm for Worst-Case Subset Sum](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 26, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026,
+  author = {{OpenAI}},
+  title = {{A Low-Space Algorithm for Worst-Case Subset Sum}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/paper.pdf}{OAI:A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/build/bibliography.tex b/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/build/bibliography.tex
new file mode 100644
index 0000000000000000000000000000000000000000..9126e43795cb247e24bf56c42b5293db06335824
--- /dev/null
+++ b/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/build/bibliography.tex
@@ -0,0 +1,94 @@
+\begin{thebibliography}{99}
+\raggedright
+
+\bibitem{AKKM13}
+Per Austrin, Petteri Kaski, Mikko Koivisto, and Jussi M\"a\"att\"a,
+\emph{Space--Time Tradeoffs for Subset Sum: An Improved Worst Case Algorithm},
+Automata, Languages, and Programming (ICALP 2013), Part I,
+Lecture Notes in Computer Science \textbf{7965}, Springer, 2013, 45--56.
+\href{https://doi.org/10.1007/978-3-642-39206-1_5}
+{doi:10.1007/978-3-642-39206-1\_5}.
+
+\bibitem{AKKN16}
+Per Austrin, Petteri Kaski, Mikko Koivisto, and Jesper Nederlof,
+\emph{Dense Subset Sum May Be the Hardest},
+33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016),
+Leibniz International Proceedings in Informatics \textbf{47}
+(2016), 13:1--13:14.
+\href{https://doi.org/10.4230/LIPIcs.STACS.2016.13}
+{doi:10.4230/LIPIcs.STACS.2016.13}.
+
+\bibitem{BCKM24}
+Tatiana Belova, Nikolai Chukhin, Alexander S. Kulikov, and Ivan Mihajlin,
+\emph{Improved Space Bounds for Subset Sum},
+32nd Annual European Symposium on Algorithms (ESA 2024),
+Leibniz International Proceedings in Informatics \textbf{308}
+(2024), 21:1--21:17.
+\href{https://doi.org/10.4230/LIPIcs.ESA.2024.21}
+{doi:10.4230/LIPIcs.ESA.2024.21}.
+Full version, arXiv:2402.13170v3, August 1, 2024.
+\href{https://arxiv.org/abs/2402.13170v3}{arXiv:2402.13170v3}.
+
+\bibitem{CJRS23}
+Xi Chen, Yaonan Jin, Tim Randolph, and Rocco A. Servedio,
+\emph{Subset Sum in Time $2^{n/2}/\poly(n)$},
+Approximation, Randomization, and Combinatorial Optimization.
+Algorithms and Techniques (APPROX/RANDOM 2023),
+Leibniz International Proceedings in Informatics \textbf{275}
+(2023), 39:1--39:18.
+\href{https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.39}
+{doi:10.4230/LIPIcs.APPROX/RANDOM.2023.39}.
+Full version, arXiv:2301.07134v2, January 29, 2023.
+\href{https://arxiv.org/abs/2301.07134v2}{arXiv:2301.07134v2}.
+
+\bibitem{DDKS12}
+Itai Dinur, Orr Dunkelman, Nathan Keller, and Adi Shamir,
+\emph{Efficient Dissection of Composite Problems, with Applications to
+Cryptanalysis, Knapsacks, and Combinatorial Search Problems},
+Advances in Cryptology---CRYPTO 2012,
+Lecture Notes in Computer Science \textbf{7417}, Springer, 2012, 719--740.
+\href{https://doi.org/10.1007/978-3-642-32009-5_42}
+{doi:10.1007/978-3-642-32009-5\_42}.
+
+\bibitem{FLPS16}
+Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh,
+\emph{Efficient Computation of Representative Families with Applications
+in Parameterized and Exact Algorithms},
+Journal of the ACM \textbf{63} (2016), no.~4,
+Article 29, 29:1--29:60.
+\href{https://doi.org/10.1145/2886094}{doi:10.1145/2886094}.
+
+\bibitem{HJ10}
+Nick Howgrave-Graham and Antoine Joux,
+\emph{New Generic Algorithms for Hard Knapsacks},
+Advances in Cryptology---EUROCRYPT 2010,
+Lecture Notes in Computer Science \textbf{6110}, Springer, 2010, 235--256.
+\href{https://doi.org/10.1007/978-3-642-13190-5_12}
+{doi:10.1007/978-3-642-13190-5\_12}.
+
+\bibitem{HS74}
+Ellis Horowitz and Sartaj Sahni,
+\emph{Computing Partitions with Applications to the Knapsack Problem},
+Journal of the ACM \textbf{21} (1974), no.~2, 277--292.
+\href{https://doi.org/10.1145/321812.321823}
+{doi:10.1145/321812.321823}.
+
+\bibitem{NW21}
+Jesper Nederlof and Karol W\k{e}grzycki,
+\emph{Improving Schroeppel and Shamir's Algorithm for Subset Sum via
+Orthogonal Vectors},
+Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing
+(STOC 2021), 1670--1683.
+\href{https://doi.org/10.1145/3406325.3451024}
+{doi:10.1145/3406325.3451024}.
+Full version, arXiv:2010.08576v2, April 12, 2021.
+\href{https://arxiv.org/abs/2010.08576v2}{arXiv:2010.08576v2}.
+
+\bibitem{SS81}
+Richard Schroeppel and Adi Shamir,
+\emph{A $T=O(2^{n/2})$, $S=O(2^{n/4})$ Algorithm for Certain
+NP-Complete Problems},
+SIAM Journal on Computing \textbf{10} (1981), no.~3, 456--464.
+\href{https://doi.org/10.1137/0210033}{doi:10.1137/0210033}.
+
+\end{thebibliography}
diff --git a/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/build/compression.tex b/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/build/compression.tex
new file mode 100644
index 0000000000000000000000000000000000000000..109d46fc9c0179e4e64de093beac0ad421f3868c
--- /dev/null
+++ b/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/build/compression.tex
@@ -0,0 +1,268 @@
+\section{Compressing repeated sums}
+\label{sec:compression}
+
+The first trial applies when random blocks have many repeated subset
+sums. It retains only the distinct weights of seven disjoint blocks,
+uses those small lists to generate four larger groups, and joins the
+groups one residue class at a time. Because the domains are disjoint,
+we can forget which subset produced a retained weight. A hard
+capacity bounds memory; success requires only the residue class of
+one fixed solution to fit.
+
+This trial uses the classical stored-list and generated-stream
+viewpoint of Schroeppel and Shamir~\cite{SS81}. Its modular
+regeneration and overflow rule also belong to the dissection approach:
+Dinur, Dunkelman, Keller, and Shamir~\cite{DDKS12} developed the
+earlier framework under randomness assumptions, and Austrin, Kaski,
+Koivisto, and M\"a\"att\"a~\cite{AKKM13} adapted dissection to
+worst-case Subset Sum. The trial and its bounds here are proved
+directly, including the dependence on the input bit length.
+
+\subsection{Seven compacted blocks and four groups}
+
+Take the first $7s$ indices of a uniform random permutation and divide
+them into seven consecutive blocks $K_1,\ldots,K_7$ of size $s$.
+For each $i$, enumerate and sort all its subset sums, then compact
+repeated weights into a list $D_i$ of distinct actual weights.
+Continue only if every list has at most $2^a$ entries, where
+\begin{equation}
+\label{comp:block-cap}
+ a=\lfloor s-.0075n\rfloor.
+\end{equation}
+This preliminary enumeration stores at most $2^s$ weights at a time.
+The seven retained lists are stored separately.
+
+Of the remaining indices, give
+\[
+ r_R=\lfloor .05n\rfloor
+ \quad\text{to the right and}\quad
+ r_L=n-7s-r_R
+ \quad\text{to the left}.
+\]
+Choose these domains in their permutation order. Split the right raw
+domain into $V_{R1},V_{R2}$ of sizes $r_{R1},r_{R2}$ differing by
+at most one, and call the left raw domain $V_L$.
+All these sizes are nonnegative on sufficiently large main-branch
+inputs.
+
+For a domain $V$, write $\mathcal S(V)$ for its full list of indexed
+subset-sum occurrences, retaining repetitions. For two lists $A,D$
+write $A\oplus D$ for the list with one occurrence of weight $x+y$
+for each pair $(x,y)\in A\times D$. Table~\ref{tab:compression}
+specifies the four generated groups by their two stored lists.
+A group stream enumerates the pair sums of the two lists in its row.
+The eight stored lists use separate subdomains, and the four group
+domains partition the input indices.
+
+\begin{table}[htbp]
+\centering
+\begin{tabular}{@{}llll@{}}
+\toprule
+Group & Index domain & First stored list & Second stored list\\
+\midrule
+$\mathcal L_1$ & $K_5\cup K_6$ & $D_5$ & $D_6$\\
+$\mathcal L_2$ & $K_7\cup V_L$ & $D_7$ & $\mathcal S(V_L)$\\
+$\mathcal R_1$ & $K_1\cup K_2\cup V_{R1}$ & $D_1$
+ & $D_2\oplus\mathcal S(V_{R1})$\\
+$\mathcal R_2$ & $K_3\cup K_4\cup V_{R2}$ & $D_3$
+ & $D_4\oplus\mathcal S(V_{R2})$\\
+\bottomrule
+\end{tabular}
+\caption{The seven compacted block lists generate four groups. Each
+row is streamed from its two stored lists. The raw and product lists
+retain occurrences, while each $D_i$ retains only distinct weights.
+No two rows share an input index.}
+\label{tab:compression}
+\end{table}
+
+Sample $q$ uniformly from $\mathcal P_\ell$ with
+$\ell=\lfloor .249n\rfloor$. For each integer
+$z\in\{0,\ldots,q-1\}$, proceed as follows.
+\begin{enumerate}
+\item Generate $\mathcal L_1$ by its direct key modulo $q$ and
+$\mathcal L_2$ by its complementary key $z-\text{weight}$
+modulo $q$. Join equal keys using Lemma~\ref{lem:streams}, with
+$p=1$ and $P=q$. Insert each resulting actual total weight into
+a weight-only dictionary of capacity $C$.
+\item If this dictionary overflows, discard it and move to the next
+$z$. Otherwise retain its completed list of distinct weights.
+An empty completed dictionary may also be skipped.
+\item Generate and join $\mathcal R_1,\mathcal R_2$ in the same
+way, with target residue $t-z\pmod q$. For every resulting actual
+right weight $y$, search for $t-y$ in the left dictionary. A hit
+returns YES. Otherwise free the dictionary and continue the residue
+loop.
+\end{enumerate}
+A sampling failure also fails the trial. The bounded samplers and
+operation counter are those of Section~\ref{sec:implementation}.
+
+Every retained block weight has a subset witness in its block. The
+four group domains are disjoint, so any choices witnessing an exact
+left--right weight match combine into an indexed subset of weight
+$t$. Thus the trial has one-sided correctness. It need not retain
+witnesses or output them: its required output is the decision YES.
+
+\subsection{Work, space, and success}
+
+Let $\mu$ be the mean of $\defect(V)$ when $V$ is uniform among
+the size-$s$ domains of the given input. The algorithm never computes
+this mean. It runs this trial and the representation trial in the same
+round; the following hypothesis identifies the case in which this
+first trial supplies success.
+
+\begin{lemma}[Compressed-block trial]
+\label{lem:compressed}
+For sufficiently large main-branch inputs, the compressed-block trial
+uses at most $u^{15}2^{n/2}$ word operations on every random outcome,
+including bounded sampling, and uses
+$O(u(C+1)+B+2^s+2^{.14n})$ writable words. Every returned YES is
+correct. If the input is a YES instance and
+\begin{equation}
+\label{alg:high-deficiency}
+ \mu\ge .00765n,
+\end{equation}
+its success probability is bounded below by an absolute positive
+constant.
+\end{lemma}
+
+\begin{proof}
+First bound the stored and generated lists deterministically, assuming
+the seven size tests pass. To retain the effects of floors, put
+\[
+ \delta=.12n-s\in[0,4),\qquad
+ \epsilon=.05n-r_R\in[0,1).
+\]
+Then $r_L=.11n+7\delta+\epsilon$ and
+$a\le.1125n-\delta$. Each stored product list on the right
+has at most
+\[
+ 2^{a+\lceil r_R/2\rceil}\le 2^{.1375n+1/2}
+\]
+occurrences. The left raw list has at most $2^{.11n+29}$
+occurrences, and each $D_i$ has at most $2^{.1125n}$ entries.
+Hence every stored sublist in Table~\ref{tab:compression} has size
+at most $2^{.14n}$ for sufficiently large $n$.
+
+Let $N_{L1},N_{L2},N_{R1},N_{R2}$ denote the lengths of the four
+group streams, counting occurrences. Their direct product bounds are
+\[
+\begin{aligned}
+ N_{L1}&\le2^{2a}\le2^{.225n},&
+ N_{L2}&\le2^{a+r_L}\le2^{.2225n+25},\\
+ N_{Ri}&\le2^{2a+r_{Ri}}\le2^{.25n+1/2}
+ &&(i=1,2).
+\end{aligned}
+\]
+More importantly, the products that count complete combinations obey
+\begin{align}
+ N_{R1}N_{R2}
+ &\le2^{4a+r_R}
+ \le2^{.5n-4\delta-\epsilon}\le2^{n/2},
+ \label{comp:right-product}\\
+ N_{L1}N_{L2}
+ &\le2^{3a+r_L}
+ \le2^{.4475n+4\delta+\epsilon}
+ <2^{.4475n+17}.
+ \label{comp:left-product}
+\end{align}
+These count all combinations, not just distinct resulting weights.
+
+Each complete left combination has exactly one total residue modulo
+$q$, and each right combination has exactly one complementary
+residue $t-\text{weight}$. Across the whole residue loop, the
+number of potential Cartesian matches is consequently at most
+$N_{L1}N_{L2}+N_{R1}N_{R2}$. Overflow may stop a join partway, but
+Lemma~\ref{lem:streams} still charges it to this same potential
+count. Repeated weights therefore cause neither an uncounted replay
+cost nor a loss of a combination.
+
+Here is a concrete operation accounting. Generating an item, including
+its weight, uses at most $O(n^2)$ operations; comparison sorting and
+heap operations use $O(n)$ per item or advance; trie operations use
+$O(u)$ per key. The stream states for a group have $O(2^{.14n})$
+words and are therefore $O(B)$ for large $n$. Thus the match-dependent
+term in~\eqref{eq:tie-cost} has $O(n)$ overhead per potential match;
+its buffer and stream-length terms are charged separately below.
+We may even initialize $O(u(C+1))$ trie space afresh at each residue.
+If $A$ is the total number of stored sublist entries and
+$N=N_{L1}+N_{L2}+N_{R1}+N_{R2}$, all nonsampling work is bounded by
+\begin{equation}
+\label{comp:operation-ledger}
+ O\left(u^3\left[
+  2^s+A+q\bigl(N+B+u(C+1)+1\bigr)
+  +N_{L1}N_{L2}+N_{R1}N_{R2}\right]\right).
+\end{equation}
+The $2^s$ term pays the seven initial enumerations and compactions,
+the terms multiplied by $q$ pay regeneration and initialization, and
+the final two terms pay every potential match and dictionary action.
+The bound also covers a failed size test.
+
+Since $q<2^{.249n}$, the largest regenerated-stream exponent is
+$.249+.25=.499$. The buffer and dictionary initialization exponents
+are $.439$ and $.448$, respectively. Equations
+\eqref{comp:right-product}--\eqref{comp:left-product} bound the
+remaining products, so~\eqref{comp:operation-ledger} is
+$O(u^4 2^{n/2})$. A permutation uses at most $O(n^4)$ bounded
+range attempts and elementary operations, and the one prime search
+costs at most $\poly(n)2^{.129n}$ with the fixed elementary
+trial-division implementation of Lemma~\ref{lem:samplers}.
+In particular it is $O(n^4 2^{.129n})$. These smaller terms leave
+the total below $u^{15}2^{n/2}$ for sufficiently large $n$.
+This deterministic bound is below the trial cap; it does not assume
+any favorable bucket sizes or random events.
+
+For space, the initial enumerations use $O(2^s)$ words, the stored
+sublists and stream states use $O(2^{.14n})$, tie replay uses $O(B)$,
+and the one retained dictionary uses $O(u(C+1))$. Sorting needs only
+linear auxiliary arrays. The permutation and loop state are absorbed
+by these bounds. In particular, overflow enforces the space bound on
+every outcome.
+
+It remains to show that one solution bucket fits in the high-mean
+case. Each random $K_i$ is marginally a uniform size-$s$ domain;
+the seven blocks need not be independent. Lemma~\ref{lem:deficiency},
+Chebyshev's inequality, and a union bound give
+\[
+ \Pr\bigl[\defect(K_i)\ge .00755n
+             \text{ for all }i\bigr]=1-O(1/n)
+\]
+under~\eqref{alg:high-deficiency}. These inequalities imply
+$N(K_i)\le2^{s-.00755n}\le2^a$ for sufficiently large $n$,
+so all tests pass, including their floors.
+
+Fix an original solution before sampling the partition. Conditional
+on any passing partition, let $x$ be its left weight. There are fewer
+than $2^{.4475n+17}$ distinct left weights in all. For each different
+one $x'$, Lemma~\ref{lem:prime} and
+$\ell=\lfloor .249n\rfloor$ give
+\[
+ \Pr[x'\equiv x\pmod q]
+ \le 8u^2 2^{-\ell}\le u^3 2^{-.249n}
+\]
+for large $n$. By linearity of expectation, the number $Y$ of
+other distinct weights in $x$'s residue class satisfies
+\[
+ \mathbb E Y\le u^3 2^{.1985n+17}.
+\]
+Because $C\ge2^{.199n-1}$ and $\log u\le n/10^9$,
+\[
+ \frac{\mathbb E Y}{C}
+ \le 2^{(3/10^9-.0005)n+18}=o(1).
+\]
+Markov's inequality implies $Y+1\le C$ with probability tending
+to one. The complete dictionary in that residue then retains $x$,
+and the right solution weight produces a hit. Earlier residues can
+only produce an earlier correct YES or be skipped.
+
+This proves success with probability $1-o(1)$ in the ideal experiment.
+The permutation and prime draws have negligible total bounded-sampler
+coupling loss by Lemma~\ref{lem:samplers}; the deterministic work
+estimate prevents the operation counter from aborting a completed
+trial for sufficiently large $n$. Thus an absolute positive success
+probability remains for the implemented trial.
+\end{proof}
+
+Compression has now handled the case in which random blocks have few
+distinct sums. When $\mu<.00765n$, the next trial instead uses the
+many distinct sums to preserve a representation of a fixed solution
+through several modular filters.
diff --git a/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/build/implementation.tex b/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/build/implementation.tex
new file mode 100644
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@@ -0,0 +1,270 @@
+\section{A single uniform bounded algorithm}
+\label{sec:implementation}
+
+We now implement the ideal choices, bound every stored structure on
+every execution, and finish the proof of Theorem~\ref{thm:main}.
+The program runs both trials; the deficiency split selects a success
+proof, not a branch that the program must recognize. Its sampling
+limits and operation counters enforce finite running time even when
+the resource event of Section~\ref{sec:resources} fails.
+
+The full schedule is as follows. On either exceptional case in the
+guard, enumerate all indexed subsets exactly. Otherwise perform
+\(u^{120}\) rounds. In each round run the compressed-block trial and
+then the representation trial, sequentially, using fresh randomness
+and reset trial state. Abort each trial after
+\(u^{500}2^{\lceil n/2\rceil}\) underlying word operations; a sampling
+failure has the same effect as an abort. A YES from either trial stops
+the program. If no trial returns YES, return NO after the last round.
+
+\subsection{Bounded sampling and its exact law}
+
+An ideal draw is uniform on its specified finite set, using fresh
+randomness conditional on the preceding choices. The implementation
+uses a fixed attempt limit. We couple it to the ideal experiment,
+rather than condition the whole experiment on every sampler succeeding:
+that latter conditioning could bias earlier adaptive choices.
+
+\begin{lemma}[Bounded samplers]\label{lem:samplers}
+A uniform integer in \(\{0,\ldots,D-1\}\), with \(1\le D<2^w\),
+can be sampled using at most \(n^3\) rejection attempts, or the trial
+can report failure. Conditional on any preceding history and on this
+call accepting, its result is uniform in the specified range. It
+couples to an ideal draw with failure probability at most
+\(2^{-n^3}\).
+
+For \(4\le\ell\le n\), the same attempt limit gives a uniform prime
+in \(\mathcal P_\ell\), conditional on this call accepting, and
+coupling failure at most \(e^{-n^2/4}\). A full trial, including
+permutations, signs, residues and the four separating families, couples
+to the ideal experiment with total failure
+\(\exp(-\Omega(n^2))\). All constants are absolute. All permitted
+attempts are included in the deterministic work bounds.
+\end{lemma}
+
+\begin{proof}
+For \(D=1\), return zero without randomness. Otherwise take
+\(\lceil\log D\rceil\) low bits of a fresh random word and reject
+values at least \(D\). Each value in the range has the same acceptance
+probability, and an attempt succeeds with probability at least \(1/2\).
+The first accepted value is therefore uniform, including conditional
+on accepting within \(n^3\) attempts. Comparing the same random
+attempts with an ideal repeat-until-success sampler gives failure
+probability at most \(2^{-n^3}\). Computing the bit length and each
+attempt takes polynomially many charged word operations. Uniform
+permutations use successive range draws among the remaining positions;
+a uniform fixed-size subset takes the required initial part of such a
+permutation. Sign masks use fresh unbiased bits.
+
+For a prime, draw \(\ell\) low bits, reject integers below two or
+outside \(4\ell p\ge2^\ell\), and test primality by deterministic
+trial division. Every prime in \(\mathcal P_\ell\) has the same
+probability of being drawn. Lemma~\ref{lem:prime} gives acceptance
+probability at least \(1/(4\ell)\), so the attempt limit fails with
+probability at most
+\[
+ (1-1/(4\ell))^{n^3}\le e^{-n^3/(4\ell)}\le e^{-n^2/4}.
+\]
+The result is uniform conditional on this call accepting. In every
+application \(\ell\le.258n\), so trial division costs at most
+\(\poly(n)2^{.129n}\) operations per attempt, and its \(n^3\)
+attempts preserve that form of bound.
+
+The four families are the only exponentially numerous sampled objects.
+Each family has at most \(10\cdot2^g+1\) sets, each generated by at
+most \(g\) range calls. All other permutations, guesses and residues
+use polynomially many range calls, and there are constantly many prime
+calls. Thus each trial uses at most \(\poly(n)2^g\) range calls.
+The failure bounds hold conditional on every history, including
+adaptively chosen ranges. Summing them over the bounded number of
+calls gives
+\[
+ \poly(n)2^g2^{-n^3}+O(1)e^{-n^2/4}
+   =\exp(-\Omega(n^2)).
+\]
+This bounds the event that the bounded and ideal executions first
+diverge at an exhausted sampler. Retaining only current sampler state
+uses no uncounted random tape. The bounded cost, including all rejected
+attempts, is the cost charged by the trial analyses.
+\end{proof}
+
+On the main branch, \(\log u\le n/10^9\), so the coupling loss is
+smaller than any fixed multiple of \(u^{-110}\) for large \(n\).
+A watchdog interruption does not create an additional probabilistic
+assumption. On successful coupling, all sampled choices agree with the
+ideal choices, while the cost estimates already allow the worst
+permitted number of attempts. The favorable work events therefore
+complete before the watchdog; other executions may simply abort.
+
+\subsection{Exact word operations}
+
+The program scans the input and counts magnitude bits by shifts to
+obtain \(b\). Actual signed weights, the transformed target and all
+relevant differences have magnitude less than
+\[
+ 2^{b+\lceil\log(n+2)\rceil+4}.
+\]
+Even an invalid parent tuple uses an input index at most twice, so the
+same bound covers the quantities considered before disjointness is
+checked. Store a sign and a magnitude, with canonical zero. Signed
+addition, comparison and normalized remainders are implemented on
+these fields. When combining an actual weight with a residue, reduce
+the weight first. A fixed-width signed weight followed by the relevant
+mask bits supplies the trie key.
+
+Every mask has at most \(n\) bits, and every stored item has a constant
+number of word fields. Binomial coefficients on at most \(n\) elements
+can be computed multiplicatively; immediately before division the
+intermediate integer is at most \(n2^n\). Floor logarithms are bit
+lengths. Hence the background allocations, modulus parameters and
+rational floors are exact integer computations. The separator quantity
+\(3^m\) fits in a word as well. Feasibility tests precede unsigned
+subtractions, or those quantities are represented with a sign.
+No entropy or real-arithmetic oracle is used by the program.
+
+The largest product modulus satisfies \(P\le2^{s+.258n}\). Residue
+arithmetic therefore fits in a word, and trial division compares the
+candidate divisor with the corresponding integer quotient instead of
+forming a potentially unnecessary square. Masks for range sampling
+use fewer than \(w\) bits. The main-branch guard also gives
+\[
+ \log\bigl(u^{500}2^{\lceil n/2\rceil}\bigr)
+ \le (1/2+500/10^9)n+1<w,
+\]
+for large \(n\). Thus fuel, the \(u^{120}\) round count, intermediate
+powers, timestamps and counters have bounded word width. The compact
+memory addresses fit by the space calculation below. All arithmetic
+invoked as a unit-cost operation stays within the stated word model.
+
+Permutations, sign masks, moduli, residues, family masks and replay
+state remain in the counted workspace whenever reused. Once a random
+value is discarded, the program never reads it again.
+
+\subsection{Space on every random outcome}
+
+We specify the simultaneous storage terms, rather than infer space
+from the favorable resource event. Let \(L_{\max}=2^{.185n}\), an
+upper bound on every raw representation leaf list on the main branch.
+There are only eight such lists; an item, heap cell or stream-row
+state occupies constantly many words. Linear auxiliary arrays suffice
+for sorting. Hence the stored leaves, heaps, row states and a constant
+number of replay snapshots occupy \(O(L_{\max})\) words. The tie
+buffer uses \(O(B)\) words. Initial compressed-block lists and
+mixer-counting half lists have sizes \(2^s\le2^{.12n}\); the
+compressed trial's stored sublists have size at most \(2^{.14n}\).
+These fit the same bound, and the two trials run sequentially.
+
+A dictionary keeps at most \(C\) distinct records and stops when a
+\((C+1)\)st key is encountered. Its fixed-width keys have \(O(u)\)
+bits, so the binary trie, including transient nodes for that rejected
+key, has \(O(u(C+1))\) nodes. Each node has constantly many word
+fields. At most a constant number of dictionaries and retained parent
+arrays coexist. In the right prefix search, an overflowing task's
+workspace is discarded before its children are processed. An
+\(O(n)\)-depth stack stores prefixes and return information, not all
+previous task dictionaries. The completed left array is retained
+while the current right task or final check runs.
+
+For the final check, the four family-index trees have total active
+depth \(O(n)\); every depth keeps constantly many arrays of length at
+most \(C\). These are actual \(O(nC)\) retained record buffers,
+including linear sorting workspace, as in \eqref{sep:workspace}.
+Caller-retained parent arrays contribute another \(O(C)\), already
+covered by this bound. The incidence tables have eight lists per
+block mask across the four blocks, each of length at most
+\(10\cdot2^g+1\). Their total capacity is at most
+\[
+ 8\,2^g(10\cdot2^g+1)=O(2^{2g}).
+\]
+Each stored index uses at most \(g+4\) bits, and each family mask uses
+\(g\) bits, so both fit in one word. Directories and sampled families
+add only \(O(2^g)\) words. The incidence contribution has the smaller
+exponential factor
+\(2^{2g}\le2^{.12n}\).
+
+Combination generation, permutations, sampler state, integer
+parameters, prefix and interval control, counters, and a conservative
+binomial workspace use \(O(u^2)\) further words. In particular,
+none retains the hypothetical check list. Combining all terms, some
+absolute implementation constant \(K\) bounds simultaneous storage by
+\[
+ K\bigl[u(C+1)+(n+1)C+L_{\max}+B+2^{2g}+u^2\bigr].
+\]
+Now \(C\le2^{.199n}\), \(L_{\max}\le2^{.199n}\),
+\(B\le2^{.199n}\), \(2^{2g}\le2^{.199n}\), and \(n\le u\).
+Term by term this proves the required conservative bound
+\begin{equation}\label{eq:space-final}
+ O\bigl(u^{15}2^{.199n}\bigr)
+\end{equation}
+on \emph{every} random outcome, including an overflowing or aborted
+trial. No assumption about average bucket size or \(\mathcal E\) was
+used. Its address bit length is at most
+\(.199n+15\log u+O(1)<w\) on the main branch.
+Allocation and stack pointers are reset between tasks and trials;
+entries are initialized when reused. Even clearing the allocated
+storage linearly is below the trial time budget, so aborts accumulate
+neither stored data nor uncharged cleanup work.
+
+\subsection{A finite program, amplification and the theorem}
+
+There are finitely many sufficiently-large conditions in the argument,
+with absolute constants and fixed polynomial degrees. The main-branch
+guard makes each fixed exponential margin dominate the required
+polynomial factors uniformly in that branch. Choose one absolute
+integer \(n_0\) meeting all these conditions and fix it in the program,
+independently of \(c\).
+
+Small-word inputs need not hold \(n_0\) as a word. Place an exact
+subset-enumeration routine near the beginning of the finite program.
+Unroll the binary digits of \(n_0\), from least to most significant,
+using only literals zero and one. Maintain a shifted copy of \(n\)
+and a three-valued comparison state. At each digit, compare its low
+bit with the fixed threshold digit, replace the comparison state
+when they differ, and shift the copy right. A newly seen difference
+has higher significance. If the copy becomes zero before the
+threshold's leading digit has been processed, branch to the nearby
+exhaustive routine. After that leading digit, a nonzero remaining copy
+means \(n>n_0\); otherwise the comparison state decides.
+
+A short input executes only \(O(1+\log n)\) adjacent digit stages
+before exiting. Its instruction addresses and state therefore remain
+small; it never loads the full threshold or a distant main-program
+address. Once \(n\ge n_0\), enlarge this fixed threshold if necessary
+so that all other fixed literals and main-program addresses fit.
+The second guard, \eqref{eq:guard}, is then an integer comparison.
+Both exceptional cases use exact enumeration with an \(n\)-bit subset
+counter and \(O(n)\) words. This is one finite program with no advice
+and no input for the exponent \(c\).
+
+On a main-branch YES input, Lemma~\ref{lem:compressed} gives constant
+success probability when \(\mu\ge.00765n\), and
+Lemma~\ref{lem:resources} gives probability at least \(u^{-110}\)
+when \(\mu<.00765n\). Both trials run in each round with fresh words
+and reset state. Thus each round succeeds with probability at least
+\(u^{-110}\), also conditional on every previous round failing.
+After \(u^{120}\) rounds the probability of missing a solution is at
+most
+\[
+ (1-u^{-110})^{u^{120}}\le e^{-u^{10}}<1/3.
+\]
+Every YES is an exact valid match by Lemma~\ref{alg:validity}; the
+sign-toggle identity certifies an original solution. The exceptional
+branch is exact, so correctness holds on every individual input.
+
+Instrument each trial with a counter for underlying word operations.
+Its constant overhead, stopping and reset give per-trial time
+\(O(u^{500}2^{\lceil n/2\rceil})\) on every outcome. Input scanning
+and the initial guards take polynomial time in \(u\). All trial
+preprocessing, sorting, table construction and sampling are charged.
+The total main-branch time is consequently
+\[
+ O\bigl(u^{620}2^{\lceil n/2\rceil}+\poly(u)\bigr).
+\]
+For every fixed positive integer \(c\), all sufficiently large inputs
+with \(b\le n^c\) avoid the exceptional branch and have
+\(u=\poly_c(n)\). This time bound and \eqref{eq:space-final} hold
+simultaneously for every random outcome. Together with the correctness
+argument they prove Theorem~\ref{thm:main}. Finally, for each fixed
+\(\rho>.199\), the factor \(u^{15}=\poly_c(n)\) is absorbed by
+\(2^{(\rho-.199)n}\), proving Corollary~\ref{cor:target}; the
+polynomial factor remains at the endpoint \(.199\).
diff --git a/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/build/introduction.tex b/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/build/introduction.tex
new file mode 100644
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@@ -0,0 +1,227 @@
+\section{Introduction}
+\label{sec:introduction}
+
+Given positive integers $a_1,\ldots,a_n$ and a nonnegative target $t$,
+\emph{Subset Sum} asks whether there is a set of indices
+$I\subseteq\{1,\ldots,n\}$ such that $\sum_{i\in I}a_i=t$.
+Repeated values are allowed, and the empty set is a possible solution.
+The parameter $n\ge2$ counts the input integers, not their total
+encoding length. We study the memory needed to solve this problem
+while retaining running time polynomial times $2^{n/2}$.
+
+The classical meet-in-the-middle algorithm of Horowitz and
+Sahni~\cite{HS74} splits the indices in half, lists the subset sums
+of each half, and searches for two complementary weights. Its time
+and space are $O^*(2^{n/2})$, where $O^*$ suppresses factors polynomial
+in the input length. Schroeppel and Shamir~\cite{SS81} retained this
+time bound with space $O^*(2^{n/4})$: four smaller stored lists suffice
+to generate the two large lists in sorted order. Their construction
+makes the distinction between generated information and stored
+information central to the time--space question.
+
+Nederlof and W\k{e}grzycki~\cite{NW21} reduced the space bound to
+$O^*(2^{0.249999n})$ with a randomized worst-case algorithm combining
+multiple decompositions of a solution with searches for disjoint sets
+of complementary weight. Belova, Chukhin,
+Kulikov, and Mihajlin~\cite{BCKM24} obtained
+$O^*(2^{0.246n})$ space at the same time bound, for input integers
+of magnitude $2^{O(n)}$. Polynomial-factor improvements to the
+running time are also known in specific word-RAM models, through the work of
+Chen, Jin, Randolph, and Servedio~\cite{CJRS23}. The question here is
+how far the space can be reduced at the half-exponential time scale.
+
+\subsection{Model and main result}
+
+We give a uniform algorithm with ordinary $O(2^{n/5})$ working space
+under each fixed polynomial bound on the input-integer bit length.
+The distinction between ordinary and polynomial-suppressed space
+requires us to specify the model. Put
+\[
+\begin{gathered}
+ b=\max\left(1,\left\lceil
+ \log_2\bigl(1+\max(t,a_1,\ldots,a_n)\bigr)\right\rceil\right),
+ \qquad u=n+b+2,\\
+ w=\left\lceil4\bigl(n+b+\log_2(n+2)\bigr)\right\rceil.
+\end{gathered}
+\]
+The input occupies read-only random-access memory, one integer per
+word. Unit-cost operations are word reads and writes, indirect
+addressing, comparisons, addition, subtraction, multiplication,
+integer quotient and remainder with nonzero divisor, bitwise Boolean
+operations, and shifts. Each operation has a constant number of
+$w$-bit inputs and outputs; arithmetic exceeding these limits must
+be implemented and charged as multiple operations. A fresh independent
+uniform $w$-bit random word costs one operation.
+
+Every writable register, address, counter, stack, table, seed, and
+retained random value counts toward working space. Old randomness can
+be reused only if it remains in this counted storage. The only
+uncharged storage is the read-only input and the fixed finite program;
+there is no random oracle, advice, or externally supplied table
+depending on $n$ or the input. All preprocessing, input accesses,
+table construction, and random sampling count toward running time.
+Space is the maximum number of writable words simultaneously in use.
+
+Our polynomial-bit convention fixes a positive integer $c$ and
+considers inputs with $b\le n^c$. The algorithm is not given $c$.
+Its asymptotic resource constants and threshold may depend on $c$,
+whereas correctness is required on every individual input.
+
+\begin{theorem}
+\label{thm:main}
+There is a single uniform classical randomized algorithm in this model that
+never answers YES incorrectly and answers each YES instance correctly
+with probability at least $2/3$. For every fixed positive integer $c$,
+there are a constant $A_c>0$ and positive integers $k_c,N_c$ such that,
+whenever $n\ge N_c$ and $b\le n^c$, every outcome of the algorithm's
+randomness uses at most
+\[
+ A_c n^{k_c}2^{n/2}\quad\text{operations}
+ \qquad\text{and}\qquad
+ A_c(n+b+2)^{15}2^{0.199n}\quad\text{writable words}.
+\]
+The algorithm does not depend on $c$.
+\end{theorem}
+
+\begin{corollary}
+\label{cor:target}
+For every fixed $\rho>0.199$, the same algorithm has ordinary
+$O_{c,\rho}(2^{\rho n})$ word space under each fixed polynomial-bit
+convention, with the time and correctness guarantees of
+Theorem~\ref{thm:main}. In particular, it has
+$O_c(2^{n/5})$ word space.
+\end{corollary}
+
+For fixed $c$ and $\rho>0.199$, the factor $(n+b+2)^{15}$ is absorbed
+by $2^{(\rho-0.199)n}$. At the endpoint $0.199$ the polynomial
+factor remains. The resource bounds apply simultaneously on each run,
+including unsuccessful runs. Randomness affects the probability of
+finding a solution; it does not permit the algorithm to exceed its
+time or memory budget.
+
+\subsection{Representations and their predecessors}
+
+A representation method gives a solution many decompositions, then
+keeps only decompositions satisfying selected congruences. The
+multiplicity creates room to filter the search without losing every
+copy of the solution. Howgrave-Graham and Joux~\cite{HJ10} used this
+idea, including recursive representations and modular filtering, for
+random hard-knapsack instances. Dinur, Dunkelman, Keller, and
+Shamir~\cite{DDKS12} developed recursive dissection tradeoffs for
+knapsack under randomness assumptions. Austrin, Kaski, Koivisto, and
+M\"a\"att\"a~\cite{AKKM13} adapted that dissection framework to
+worst-case Subset Sum using randomized modular reductions and explicit
+resource controls.
+
+For the present argument, the number of \emph{distinct} partial sums
+matters as much as the number of decompositions. The residue-coverage
+argument of Austrin, Kaski, Koivisto, and
+Nederlof~\cite[Proposition~3.5]{AKKN16}, extended to product moduli by
+Nederlof and W\k{e}grzycki~\cite[full version, Lemma~3.2]{NW21},
+turns many distinct weights into many occupied residue classes.
+Nederlof and W\k{e}grzycki also combined multilevel representations
+with weighted disjointness tests. Their construction uses separating
+sets that contain one choice and avoid the other, building on the
+separating collections of Fomin, Lokshtanov, Panolan, and
+Saurabh~\cite[Section~4.2]{FLPS16}. We use the additional-prime
+filtering of Belova et al.~\cite[full version, Sections~1.2.3 and~4.1]{BCKM24},
+together with their separator cost estimate for the required range of
+mask sizes~\cite[full version, Section~4.7]{BCKM24}.
+The proofs below supply the modified enumeration, probability, and
+workspace arguments needed for our construction.
+
+\subsection{Two trials and the proof strategy}
+
+For a set $V$ of input indices, let $N(V)$ be the number of distinct
+subset sums on $V$, and define its \emph{deficiency} by
+\[
+ d(V)=|V|-\log_2N(V).
+\]
+A block of large deficiency has a short list of distinct weights;
+a block of small deficiency supplies many distinct choices for a
+representation. We analyze two trials according to the mean deficiency
+of a uniformly chosen block of a fixed size. The algorithm does not
+compute that mean: each round runs both trials with fresh randomness.
+
+\paragraph{Compress repeated sums.}
+The first trial chooses seven disjoint blocks and retains the distinct
+subset sums in each. In the high-mean case, concentration makes these
+lists short with high probability. Their disjoint domains allow us to
+forget which subset produced a retained weight. Sorted pair-sum streams
+combine the lists without storing their full Cartesian products. A
+random prime partitions the left combinations into residue classes;
+the trial stores one class at a time and skips it if its dictionary
+exceeds capacity. Success requires only the class containing a fixed
+solution's left weight to fit.
+
+\paragraph{Create and filter representations.}
+The second trial uses three disjoint blocks $M_0,M_L,M_R$. Its two
+partial solutions, called \emph{parents}, may both use indices of
+$M_0$. Each parent is itself a join of two \emph{children}, which
+may overlap on $M_L$ on the left or $M_R$ on the right. All remaining
+indices belong to disjoint background parts. Every join checks
+disjointness on its shared block. Each child is generated from two
+stored lists, giving eight \emph{leaf lists} altogether. Within each
+parent, every index of $M_0$ belongs to exactly one leaf domain.
+
+For a fixed solution to have many such decompositions, first choose a
+uniform set $F$ of indices, negate its weights, and replace the target
+by $t'=t-\sum_{i\in F}a_i$, using the original weights in the sum.
+An original solution $S$ becomes the
+uniform subset $S\mathbin{\triangle}F$ of the signed instance, while
+all deficiencies stay unchanged. In the low-mean case, suitable pieces
+of this signed solution have many distinct sums. Splitting those pieces
+among the parents and children gives representations that survive
+nested modular filters with inverse-polynomial probability.
+
+\paragraph{Separate generation work from stored records.}
+A parent record consists of its exact weight and its set
+$D\subseteq M_0$ of shared indices, encoded as a membership mask.
+Equal weights with different masks
+remain different records because they can have different compatibility
+with the opposite parent. Generation examines raw tuples of four leaf
+occurrences before rejecting local overlaps and deduplicating records.
+Its work therefore depends on tuple counts, not only on dictionary size.
+
+An additional independent prime subdivides the filtered parents into
+buckets. Different weights are unlikely to collide in the solution's
+bucket. Equal weights, however, always stay together. The prescribed
+left-mask size bounds their number below capacity, so a left bucket
+can simply be skipped if it overflows. On the right, overflow instead
+fixes successive bits of the $M_0$ mask and regenerates each part.
+Each restriction is imposed in the leaf lists before joining them.
+Consequently every raw tuple belongs to at most one task at each
+prefix depth, including tuples later rejected for overlap. Its repeated
+processing is therefore charged by depth. Leaf regeneration and
+child-stream scans are still paid per task; bounds on bucket and distinct
+record counts control those task starts and the repeated matching checks.
+
+\paragraph{Match disjoint masks within the budget.}
+The final check seeks records $(x,D)$ and $(y,E)$ with
+$x+y=t'$ and $D\cap E=\varnothing$. Partition $M_0$ into four
+blocks and sample separating sets on each block. Fixing, on every
+block, a sampled set containing the restriction of $D$ and avoiding the
+restriction of $E$ leaves only an exact two-sum test on weights.
+A depth-first traversal stores only
+small intermediate arrays; the expected number of appearances of each
+record controls its work.
+
+The success proof fixes a representation before exposing the extra
+prime or separating families. Because representation survival is rare,
+it bounds resource failure unconditionally and subtracts that bound
+before applying the later random choices. Dictionary capacities bound
+space on every run, and bounded sampling and an operation counter
+bound time. Independent repetitions then give Theorem~\ref{thm:main}.
+
+\paragraph{Reading path.}
+Section~\ref{sec:preliminaries} establishes the counting and sorted-stream
+tools. Section~\ref{sec:compression} gives the complete compressed-block
+trial. Section~\ref{sec:representation} specifies the representation layout,
+filters, and bounded dictionaries, then proves the deterministic accounting
+for prefix refinement. Section~\ref{sec:separator} supplies the
+weighted-disjointness procedure. Section~\ref{sec:survival} combines
+representation survival, joint resource estimates,
+the additional prime, and the separating families; it is here that
+the accounting bounds receive their numerical estimates.
+Finally, Section~\ref{sec:implementation} implements the sampling and arithmetic, enforces
+the budgets, and completes the amplification argument.
diff --git a/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/build/main.tex b/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/build/main.tex
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@@ -0,0 +1,60 @@
+\newcommand{\DoNotLoadEpstopdf}{}
+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage[margin=1in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{booktabs,array}
+\usepackage{microtype}
+\usepackage[hidelinks]{hyperref}
+\hypersetup{
+  pdftitle={A Low-Space Algorithm for Worst-Case Subset Sum},
+  pdfauthor={OpenAI},
+  pdfsubject={Uniform randomized Subset Sum with worst-outcome time and space bounds}
+}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newcommand{\poly}{\operatorname{poly}}
+\newcommand{\defect}{\mathrm{d}}
+\newcommand{\SSum}{\textsc{Subset Sum}}
+\newcommand{\E}{\mathbb E}
+\newcommand{\Prb}{\mathbb P}
+\newcommand{\Z}{\mathbb Z}
+\newcommand{\R}{\mathbb R}
+\newcommand{\1}{\mathbf 1}
+\allowdisplaybreaks[1]
+
+\title{A Low-Space Algorithm for Worst-Case Subset Sum}
+\author{OpenAI}
+\date{September 26, 2026}
+
+\begin{document}
+\maketitle
+
+\begin{abstract}
+We give a uniform classical randomized decision algorithm for worst-case
+Subset Sum. Under every fixed polynomial bound on input-integer bit length,
+it uses $\poly(n)2^{n/2}$ time and ordinary $O(2^{n/5})$ writable words
+of $O(n+b)$ bits, where $b$ is the largest input bit length. Both resource
+bounds hold on every execution. The error is one-sided: the algorithm always
+rejects unsolvable instances and accepts each solvable instance with
+probability at least $2/3$.
+\end{abstract}
+
+\input{introduction}
+\input{preliminaries}
+\input{streams}
+\input{compression}
+\input{representation}
+\input{separators}
+\input{survival}
+\input{resources}
+\input{implementation}
+\input{bibliography}
+\end{document}
diff --git a/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/build/preliminaries.tex b/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/build/preliminaries.tex
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+++ b/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/build/preliminaries.tex
@@ -0,0 +1,222 @@
+\section{Counting and enumeration tools}
+\label{sec:preliminaries}
+
+Both trials use the same three facts about an arbitrary fixed input.
+The number of distinct sums changes little when one index is replaced;
+prescribed-cardinality choices have accurate entropy estimates; and a
+random prime rarely identifies two fixed unequal integer weights.
+We prove these facts first. We then give the small-state streams that
+turn stored lists into much larger generated lists.
+
+All logarithms are base two unless written $\ln$. Every finite decimal
+constant is an exact rational. Polynomial factors below have absolute
+degrees independent of the input-bit promise. We use the parameters
+\begin{equation}
+\label{eq:parameters}
+\begin{gathered}
+ u=n+b+2,\qquad m=8\lfloor3n/100\rfloor,\qquad
+ s=m/2,\qquad g=m/4,\\
+ C=2^{\lfloor .199n\rfloor},\qquad
+ B=2^{\lfloor .19n\rfloor},\qquad h_*=h(1/4),
+\end{gathered}
+\end{equation}
+where $h(x)=-x\log x-(1-x)\log(1-x)$ has its continuous endpoint
+values. Blocks used for compression have size $s$; the representation
+trial will use blocks of size $m$ and their halves and quarters.
+The capacity $C$ counts distinct dictionary entries, whereas $B$
+counts buffered occurrences in a stream. An entry can have many
+occurrences, so these capacities have different roles.
+
+The constructions below concern the \emph{main branch}, defined by
+$n\ge n_0$ and
+\begin{equation}
+\label{eq:guard}
+ u\le 2^{\lfloor n/10^9\rfloor},
+\end{equation}
+for one sufficiently large absolute integer $n_0$ fixed in the
+program. Outside this branch the program uses exact subset
+enumeration. Section~\ref{sec:implementation} gives the finite
+implementation of this test, the repetition schedule, and the operation
+cap. Each eventual fixed polynomial-bit class enters the main branch.
+
+\subsection{How many distinct sums does a block have?}
+
+For a domain $V\subseteq\{1,\ldots,n\}$, let $N(V)$ count the
+\emph{distinct integer values} of its subset sums, and define
+\[
+ \defect(V)=|V|-\log N(V).
+\]
+We call this its deficiency. In particular, $N(\varnothing)=1$ and
+$\defect(\varnothing)=0$. Here an actual weight is an integer sum,
+as opposed to a residue; masks always encode sets of indices, even
+when some input values agree. The next lemma applies also to signed
+weights, which arise later after a sign transformation.
+
+\begin{lemma}[Deficiency]
+\label{lem:deficiency}
+For arbitrary fixed weights on the $n$ indices, deficiency is
+nonnegative, invariant under sign changes of individual weights,
+inclusion-monotone, and superadditive on disjoint domains. Replacing
+one index in a domain of fixed size changes its deficiency by at most
+one. If $V$ is uniform among the size-$v$ domains, $0\le v\le n$,
+then
+\begin{equation}
+\label{eq:deficiency-variance}
+ \operatorname{Var}(\defect(V))\le v.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+There are $2^{|V|}$ subsets, so $N(V)\le2^{|V|}$.
+Adding one index multiplies the number of sums by a factor between
+one and two. Its addition therefore increases deficiency by a quantity
+in $[0,1]$, proving inclusion monotonicity. For disjoint $V,W$,
+every sum on their union is a sum of one value from each domain, so
+$N(V\cup W)\le N(V)N(W)$. Taking logarithms gives
+$\defect(V\cup W)\ge\defect(V)+\defect(W)$.
+
+For sign invariance, let $F\subseteq V$ be the negated indices.
+Toggling $F$ in a chosen subset is a bijection on all subsets of $V$.
+The new weight of a subset is the old weight of its toggle minus the
+old weight of $F$, so the set of sum values is merely translated.
+Finally, two equal-sized domains differing in one index are extensions
+of their common part. Each extension increases deficiency by a
+quantity in $[0,1]$, which proves the replacement bound.
+
+For the variance, expose a uniform ordered size-$v$ sample without
+replacement. Let $M_r$ be the conditional expectation of the final
+deficiency after the first $r$ positions, $0\le r\le v$.
+Given the first $r-1$ positions, couple the remaining completions for
+two possible next indices by transposing those indices. The final
+domains then agree or differ in one index. Their deficiencies, and
+hence their conditional means, differ by at most one. Thus all
+possible values of $M_r$ given the past lie in an interval of length
+at most one, and $|M_r-M_{r-1}|\le1$. Orthogonality of martingale
+differences now gives
+\[
+ \operatorname{Var}(\defect(V))
+ =\sum_{r=1}^v\mathbb E(M_r-M_{r-1})^2\le v.
+\]
+The empty-sample case is immediate.
+\end{proof}
+
+Consequently, for any fixed $\epsilon>0$, a deviation of
+$\epsilon n$ from the size-specific mean has probability $O(1/n)$,
+by Chebyshev's inequality. This concentration concerns a random
+\emph{domain} in a fixed input; it assumes no distribution of the
+numerical input values.
+
+\subsection{Exact entropy estimates}
+
+Cardinality restrictions on a block reduce its subset count from
+$2^v$ to a binomial coefficient. We will need the exact exponential
+rate, including in a later estimate with no exponential slack.
+
+\begin{lemma}[Elementary entropy bounds]
+\label{lem:entropy}
+For integers $0\le k\le v$,
+\[
+ \frac{2^{v h(k/v)}}{v+1}\le \binom vk\le 2^{v h(k/v)}.
+\]
+For $v=k=0$, interpret $v h(k/v)$ as zero. Entropy is concave,
+symmetric about $1/2$, and increasing on $[0,1/2]$. Moreover,
+\begin{equation}
+\label{eq:entropy-numbers}
+ h_*<.8114,\qquad h(1/3)<14/15,\qquad h(1/6)>5/8.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+For $0<k<v$, consider the binomial distribution with success
+probability $k/v$. The probability of exactly $k$ successes is
+$\binom vk 2^{-v h(k/v)}$. It is at most one and at least
+$1/(v+1)$: adjacent probability ratios show that $k$ is a mode,
+and there are $v+1$ possible outcomes. This gives both estimates.
+The cases $k=0,v$ and $v=0$ are immediate. Differentiation gives
+the asserted monotonicity and concavity; symmetry follows from the
+formula for $h$.
+
+For the strict numerical inequalities, first
+$h_*=2-\tfrac34\log3$. We have
+$3^{12}/2^{19}>1+1/74$, whose fourth power exceeds
+$1+4/74=39/37>256/243$. Hence $3^{53}>2^{84}$ and
+$\log3>84/53>1.5848$, giving $h_*<.8114$.
+Next, $h(1/3)=\log3-2/3$ and $3^5<2^8$, so
+$h(1/3)<8/5-2/3=14/15$.
+Finally, $6^2>2^5$ and $(6/5)^4>2$ imply
+\[
+ h(1/6)=\tfrac16\log6+\tfrac56\log(6/5)
+ >\tfrac16\cdot\tfrac52+\tfrac56\cdot\tfrac14=\tfrac58.
+\]
+Thus none of the three bounds relies on floating-point optimization.
+\end{proof}
+
+\subsection{Random primes and the sampling interface}
+
+The modular filters must control collisions of actual weights, even
+when their bit length grows polynomially with $n$. We use the
+following elementary estimate with its full bit-length dependence.
+
+\begin{lemma}[Prime factors]
+\label{lem:prime}
+Let $4\le\ell\le n$ be an integer. The set
+\[
+ \mathcal P_\ell=
+ \{p\text{ prime}:p<2^\ell,\ 4\ell p\ge2^\ell\}
+\]
+has at least $2^\ell/(4\ell)$ elements. If $p$ is uniform on this
+set and $\Delta$ is a fixed nonzero integer with
+$|\Delta|<2^{2u}$, then
+\[
+ \Pr[p\mid\Delta]\le 8u^2 2^{-\ell}.
+\]
+In particular, this estimate applies to every nonzero actual-weight
+difference used by the two trials.
+\end{lemma}
+
+\begin{proof}
+Put $X=2^\ell$. The central binomial coefficient obeys
+$\binom{X}{X/2}\ge2^X/(X+1)$. In its factorial formula, the
+exponent of a prime $p$ is
+\[
+ \sum_{r\ge1}
+ \left(\left\lfloor\frac X{p^r}\right\rfloor
+      -2\left\lfloor\frac {X/2}{p^r}\right\rfloor\right).
+\]
+Each summand is zero or one, and at most
+$\lfloor\log_p X\rfloor$ of them are nonzero. Thus each prime
+contributes a factor at most $X$ to the binomial coefficient. If
+$\pi(X)$ is the number of primes at most $X$, taking logarithms gives
+\[
+ \ell\pi(X)\ge X-\log(X+1)\ge X/2
+ \qquad(X\ge16).
+\]
+Remove primes below $X/(4\ell)$. There are at most $X/(4\ell)$
+integers in that range, so at least $X/(4\ell)$ primes remain.
+The upper endpoint $X$ is composite, proving the cardinality bound.
+
+An integer $\Delta$ with $0<|\Delta|<2^{2u}$ has at most $2u$
+distinct prime divisors: the product of $r$ distinct primes is at
+least $2^r$. Dividing by the number of available primes and using
+$\ell\le n\le u$ gives the displayed collision bound.
+Actual weights and their relevant differences have magnitude
+$O(n2^b)$; the constructions sum each indexed weight at most twice,
+even before rejecting an overlapping tuple. They therefore meet
+$|\Delta|<2^{2u}$ for sufficiently large main-branch inputs.
+\end{proof}
+
+We analyze uniform independent draws, conditional on each specified
+finite range. The implementation uses at most $n^3$ attempts for each
+draw and fails the current trial if they all fail. The precise
+interface, proved in Lemma~\ref{lem:samplers}, is as follows. A draw
+from $\{0,\ldots,D-1\}$ has the uniform law on success and couples
+to an ideal draw with failure probability at most $2^{-n^3}$;
+$D=1$ needs no randomness. A draw from $\mathcal P_\ell$ has its
+uniform law on success and coupling failure at most $e^{-n^2/4}$.
+These conditional bounds remain valid for ranges determined by past
+choices. All requested prime parameters satisfy $\ell\le.258n$;
+bounded trial division therefore costs at most
+$\poly(n)2^{.129n}$ per prime search, including all its attempts.
+The final section proves the finite-word implementation and the
+aggregate coupling estimate. The trials below use no unbounded
+sampling operation in their resource guarantees.
diff --git a/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/build/representation.tex b/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/build/representation.tex
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@@ -0,0 +1,552 @@
+\section{Representations and bounded parent dictionaries}
+\label{sec:representation}
+
+The second trial uses shared index sets to give a solution many
+representations. It must generate those representations without storing
+all of them. We first specify the shared sets and the leaf lists,
+then the modular filters and a finite schedule of bounded dictionary
+tasks. The last part of this section proves deterministic bounds for
+that schedule. Lemma~\ref{lem:representation} will show that the filters
+retain a representation; Section~\ref{sec:resources} will bound the
+work needed to bring it to a completed check.
+
+The trial runs alongside the compression trial in every round. It does
+not test the mean deficiency $\mu$ or decide which trial should apply.
+The representation method and modular filtering have antecedents in
+Howgrave-Graham and Joux~\cite{HJ10}; the multilevel worst-case setting
+and its weighted disjointness step are developed by Nederlof and
+W\k{e}grzycki~\cite{NW21}. The construction below specifies the shared
+indices, exact integer parameters, and bounded recomputation needed
+for the present resource bounds.
+
+\subsection{Shared mixers and leaf lists}\label{alg:layout}
+
+This trial gives each parent two children and each child two leaves.
+The leaves are stored; children and parents are generated from them.
+We first describe their index domains and prescribed cardinalities,
+then balance the unconstrained background parts so that every leaf is
+small enough to store. The shared blocks, which we call \emph{mixers}, create multiple ways to
+represent a solution and require explicit disjointness tests.
+
+Select three disjoint size-$m$ sets uniformly from a random permutation.
+For each, compute its number of distinct actual subset sums exactly: store
+the subset-sum lists of two size-$s$ halves and count distinct weights in
+their actual-weight sorted pair stream.  Name a mixer with the largest
+such number $M_0$, and name the other two $M_L,M_R$ in a deterministic
+order, breaking ties deterministically.  Put
+\[
+ N_i=N(M_i),\qquad d_i=\defect(M_i),\qquad i\in\{0,L,R\}.
+\]
+Continue only if
+\begin{equation}\label{eq:spread}
+  \max_iN_i\le (\min_iN_i)2^{\lfloor .0005n\rfloor}.
+\end{equation}
+Thus $d_0\le d_L,d_R$ and the deficiency spread is at most $.0005n$.
+
+Choose a uniform subset $F$ of all input indices, negate its weights, and
+replace the target by
+\begin{equation}\label{alg:sign-flip}
+  t'=t-\sum_{f\in F}a_f,
+\end{equation}
+where the sum uses the original weights.  In the rest of this trial all
+weights are signed after this change.  The signed weight of any subset
+$I'$ is the original weight of $I'\mathbin{\triangle}F$ minus the
+original weight of $F$.  Thus the transformed instance is equivalent to
+the original, and Lemma~\ref{lem:deficiency} makes all $N_i,d_i$
+unchanged.
+
+Independently guess $k_i$ uniformly from
+$\{\lceil m/6\rceil,\ldots,m/4\}$, for $i=0,L,R$, and put
+$\lambda_i=k_i/m$.  At the top join, the left and right masks on $M_0$
+will have sizes $k_0$ and $s-k_0$.  Within parent $j\in\{L,R\}$, its
+two children will have masks on $M_j$ of sizes $k_j$ and $s-k_j$.
+Disjointness of these masks is checked at the corresponding join.  The
+left parent never uses $M_R$, and the right parent never uses $M_L$.
+
+Independently choose a uniform labeled partition
+\[
+ M_0=T_1\mathbin{\dot\cup}T_2\mathbin{\dot\cup}T_3\mathbin{\dot\cup}T_4,
+ \qquad |T_a|=g.
+\]
+Choose integers $c_1,\ldots,c_4$ as equally
+as possible with sum $k_0$, assigning the remainder in a fixed order.
+The left count in $T_a$ is $c_a$ and the right count is $g/2-c_a$.
+In each parent, child 1 uses $T_1,T_2$ and child 2 uses $T_3,T_4$,
+one block in each of its two leaves.  In particular, within a parent the
+top-mixer indices in its children are disjoint.
+
+Independently for $j=L,R$, partition $M_j$ uniformly into two labeled
+blocks $Q_{j1},Q_{j2}$ of size $s$.  Prescribe child-1 counts
+$a_{j1},a_{j2}$ as equally as possible with sum $k_j$.  The corresponding
+child-2 counts are $s/2-a_{j1},s/2-a_{j2}$.  Both children use the same
+two blocks of $M_j$, one in each leaf; their possible overlap is exactly
+what the join of the two children into their parent checks.
+Figure~\ref{tab:leaf-layout} shows both parents and all eight leaves.
+Each leaf also receives its own disjoint background part, defined next. A chosen subset of a top block is recorded
+in the $M_0$ mask, and a chosen subset of a local block is recorded in the
+$M_j$ mask.
+
+\begin{figure}[htbp]
+\centering
+\small
+\setlength{\tabcolsep}{5pt}
+\begin{tabular}{@{}cccc@{}}
+\multicolumn{4}{c}{Final join: complementary weights and disjoint masks on $M_0$}\\[2pt]
+\multicolumn{2}{c}{$\nearrow$}&\multicolumn{2}{c}{$\nwarrow$}\\[-2pt]
+\multicolumn{2}{c}{\fbox{Left parent}}&
+\multicolumn{2}{c}{\fbox{Right parent}}\\[3pt]
+\multicolumn{2}{c}{check disjointness on $M_L$}&
+\multicolumn{2}{c}{check disjointness on $M_R$}\\[1pt]
+$\nearrow$&$\nwarrow$&$\nearrow$&$\nwarrow$\\[-2pt]
+Left child 1&Left child 2&Right child 1&Right child 2\\[2pt]
+$\uparrow$&$\uparrow$&$\uparrow$&$\uparrow$\\[-2pt]
+\fbox{\begin{tabular}{@{}c@{}}$T_1,\ Q_{L1}$\\[-2pt]\scriptsize own background\end{tabular}}&
+\fbox{\begin{tabular}{@{}c@{}}$T_3,\ Q_{L1}$\\[-2pt]\scriptsize own background\end{tabular}}&
+\fbox{\begin{tabular}{@{}c@{}}$T_1,\ Q_{R1}$\\[-2pt]\scriptsize own background\end{tabular}}&
+\fbox{\begin{tabular}{@{}c@{}}$T_3,\ Q_{R1}$\\[-2pt]\scriptsize own background\end{tabular}}\\[2pt]
+$+$&$+$&$+$&$+$\\[2pt]
+\fbox{\begin{tabular}{@{}c@{}}$T_2,\ Q_{L2}$\\[-2pt]\scriptsize own background\end{tabular}}&
+\fbox{\begin{tabular}{@{}c@{}}$T_4,\ Q_{L2}$\\[-2pt]\scriptsize own background\end{tabular}}&
+\fbox{\begin{tabular}{@{}c@{}}$T_2,\ Q_{R2}$\\[-2pt]\scriptsize own background\end{tabular}}&
+\fbox{\begin{tabular}{@{}c@{}}$T_4,\ Q_{R2}$\\[-2pt]\scriptsize own background\end{tabular}}
+\end{tabular}
+\caption{The eight stored leaf lists and their joins, read from bottom
+ to top. The two boxed leaves in each column generate one child by
+ pair-sum streaming. Each $T_a$ occurs once on each parent side;
+ each $Q_{jr}$ occurs in both children of side $j$. The eight
+ background parts are pairwise disjoint and avoid all three mixers.
+ Local joins reject overlap on $M_L$ or $M_R$; the final join rejects
+ overlap on $M_0$. Boxes label allowed domains; repeated labels mark shared indices.}
+\label{tab:leaf-layout}
+\end{figure}
+
+\subsubsection*{Balancing the stored lists}
+
+There remain $r=n-3m$ background indices, whose selections have no
+cardinality constraints. The prescribed mixer counts may make one
+parent's lists larger than the other's, or one child's lists larger
+than its sibling's. We compensate by assigning more background indices
+to the smaller side, first between parents and then between children.
+The program makes these allocations with integer binomial logarithms:
+\[
+\begin{aligned}
+ b_{0L}&=\left\lfloor\log \binom{m}{k_0}\right\rfloor,&
+ b_{0R}&=\left\lfloor\log \binom{m}{s-k_0}\right\rfloor,\\
+ b_{j1}&=\left\lfloor\log \binom{m}{k_j}\right\rfloor,&
+ b_{j2}&=\left\lfloor\log \binom{m}{s-k_j}\right\rfloor
+ \quad(j=L,R).
+\end{aligned}
+\]
+Set
+\[
+\begin{aligned}
+ r_L&=\left\lfloor\frac{r+b_{0R}-b_{0L}}2\right\rfloor,
+ &r_R&=r-r_L,\\
+ r_{j1}&=\left\lfloor\frac{r_j+b_{j2}-b_{j1}}2\right\rfloor,
+ &r_{j2}&=r_j-r_{j1}.
+\end{aligned}
+\]
+Thus $r_L+r_R=r$ and $r_{j1}+r_{j2}=r_j$ exactly. Split each
+child's background count as equally as possible between its two leaves,
+using any fixed order on the background indices.
+
+To verify feasibility and list sizes, write
+\[
+\begin{aligned}
+ H_{0L}&=h(\lambda_0),& H_{0R}&=h(1/2-\lambda_0),&
+ \overline H_0&=(H_{0L}+H_{0R})/2,\\
+ H_{j1}&=h(\lambda_j),& H_{j2}&=h(1/2-\lambda_j),&
+ \overline H_j&=(H_{j1}+H_{j2})/2\quad(j=L,R).
+\end{aligned}
+\]
+Concavity gives $\overline H_0,\overline H_L,\overline H_R\le h_*$.
+Lemma~\ref{lem:entropy} translates the integer assignments into
+\begin{equation}\label{eq:allocation}
+\begin{split}
+ r_j&=r/2+(\overline H_0-H_{0j})m+O(\log n),\\
+ r_{ji}&=r_j/2+(\overline H_j-H_{ji})m+O(\log n).
+\end{split}
+\end{equation}
+These identities express the cancellation we need: the larger a
+prescribed mixer family is, the fewer unconstrained background indices
+its leaf receives.
+
+The assignments are feasible for sufficiently large $n$.  In every
+entropy pair the arguments lie in $[1/6,1/3]$, so its difference in
+absolute value is at most $1-h(1/6)<3/8$.  Consequently
+\[
+ r_j\ge .14n-\frac{3m}{16}-O(\log n)
+       \ge .095n-O(\log n),
+ \qquad
+ r_{ji}\ge \frac{.095n-.09n}{2}-O(\log n)>0.
+\]
+All prescribed mixer counts are feasible as well: they are rounded
+versions of fractions in $[1/6,1/3]$ of their block sizes.  The program
+may simply fail the trial if any integer feasibility test fails.
+
+A leaf list has one occurrence for each choice with its prescribed
+counts in its top and local mixer blocks and an arbitrary subset of its
+background part.  An occurrence retains its actual weight and mixer
+masks.  Enumerate the fixed-cardinality subsets directly, for example in
+lexicographic order on index combinations, rather than scanning all
+unconstrained masks.  This takes polynomial overhead per generated item.
+Raw counts retain occurrence multiplicity. A raw child pair chooses
+one occurrence from each of its two leaves, and a raw four-leaf tuple
+chooses one from each of a parent's four leaves. Its weight is the sum
+of the occurrence weights, even when local-mixer masks overlap.
+The balanced background allocation makes the four leaf sizes on each
+side comparable. Define their common exponential scale by
+\begin{equation}\label{eq:beta}
+ \beta_j=r/4+\overline H_0m/2+\overline H_jm
+ \le n/4+(-.75+1.5h_*)m.
+\end{equation}
+Every leaf on side $j$ and each two-leaf child satisfy
+\begin{equation}\label{alg:leaf-size}
+ \#\{\text{leaf items}\}\le u^{10}2^{\beta_j/2},
+ \qquad
+ \#\{\text{raw child pairs}\}\le u^{20}2^{\beta_j}.
+\end{equation}
+Indeed, before substituting~\eqref{eq:allocation}, the logarithm of the
+leaf size is at most
+\[
+ H_{0j}m/4+H_{ji}m/2+r_{ji}/2+O(1).
+\]
+The binomial logarithm errors in each background allocation are at most
+$\log(n+1)+1$, and a rounded prescribed block count changes its entropy
+upper exponent by $O(1)$: for large $n$ the relevant fractions lie in
+$[1/8,3/8]$, where $|h'|<3$.  More explicitly, the error in $r_j$ from
+its ideal value is at most $3\log u$, and that in $r_{ji}$ after both
+ideal substitutions is at most $5\log u$.  Substitution gives
+$\beta_j/2+O(\log n)$ with ample room in the factor $u^{10}$ for large
+$n$.  Finally $h_*<.8114$ and $m\le .24n$ imply
+\[
+ \beta_j/2\le .181052n.
+\]
+Since $\log u\le n/10^9$, every raw leaf list has fewer than
+$2^{.185n}$ items for sufficiently large main-branch inputs.
+
+\subsection{Nested modular filters}
+
+The first filtering level constrains each child's weight; the second
+constrains the sum of the two children. Their moduli are nested so that
+once a parent and its first child pass, the required congruence for its
+second child follows. An additional independent prime $q$ subdivides
+the parent residue classes into buckets, following the additional-prime
+filtering approach of Belova, Chukhin, Kulikov, and
+Mihajlin~\cite[full version, Sections~1.2.3 and~4.1]{BCKM24}.
+The low filters reduce raw
+generation; the extra prime controls the distinct entries in the
+solution's bucket.
+
+For analysis define the real quantities
+\begin{equation}\label{eq:filter-scales}
+\begin{gathered}
+ \tau=.0078n,\qquad f_i=\min(\tau,d_i/2),\qquad L_i=s-f_i,\\
+ f_{\max}=\max_i f_i,\qquad \xi=n/4+.032m-f_{\max}.
+\end{gathered}
+\end{equation}
+The program uses the integer parameters
+\[
+ \ell_i=\lfloor L_i\rfloor
+ =\max\left\{\lfloor s-\tau\rfloor,
+              \left\lfloor\frac{\log N_i}{2}\right\rfloor\right\},
+\]
+computed by bit lengths; flooring $\log N_i$ first gives the same second
+term.  Since $d_0$ is smallest, $L_0$ is largest.  Sort the three
+$\ell_i$ in nondecreasing order.  For each increment from the preceding
+value, taking the initial preceding value to be zero, independently
+sample a prime with that increment as parameter if the increment is at
+least four; otherwise use the factor one.  At each sorted index let
+$p_i$ be the cumulative product.  Tied parameters give the same product.
+In particular,
+\[
+ p_L\mid p_0,\qquad p_R\mid p_0.
+\]
+
+For every fixed nonzero relevant actual-weight difference $\Delta$,
+conditional on these log parameters, the construction satisfies
+\begin{equation}\label{eq:filter-bounds}
+ \frac{2^{L_i}}{u^5}\le p_i\le 2^{L_i},
+ \qquad
+ \Pr[p_i\mid\Delta]\le u^{10}2^{-L_i}
+ \quad (i=0,L,R).
+\end{equation}
+For the size estimate there are at most three sampled factors; the
+discarded increments total at most nine.  The prime lower ranges and
+the floor therefore lose a factor of at most $2^{10}(4n)^3$ relative to
+$2^{L_i}$, which is at most $u^5$ for large $n$.  For divisibility,
+$\Delta$ has at most $2u$ distinct prime divisors.  A factor with
+parameter $\ell$ divides it with probability at most
+$8u^2 2^{-\ell}$, by Lemma~\ref{lem:prime}.  All sampled factors are
+independent.  Divisibility by their product requires divisibility by
+each sampled factor, so multiplication of these bounds, with the same
+discarded-increment and floor losses, proves~\eqref{eq:filter-bounds}.
+This argument does not require the sampled primes to be distinct.
+
+Independently sample one more prime $q$, with parameter
+\[
+ \ell_q=\left\lfloor n/4+.032m-s+\min_i\ell_i\right\rfloor.
+\]
+Then $\xi-2<\ell_q\le \xi$ and $\xi\le .25768n$; in particular $\ell_q$ is
+positive and linear in $n$.  The same prime estimates give
+\begin{equation}\label{alg:q-bounds}
+ 2^\xi/u^2\le q\le 2^\xi,
+ \qquad \Pr[q\mid\Delta]\le u^3 2^{-\xi}
+\end{equation}
+for each fixed nonzero relevant difference.  Write $P=p_0q$.  Choose
+independent uniform residues $\rho_0\pmod {p_0}$ and
+$\rho_j\pmod {p_j}$ for $j=L,R$, independently also of $q$.  Their
+desired parent low bases are
+\[
+ w_L=\rho_0,\qquad w_R=t'-\rho_0.
+\]
+For side $j$, require child 1 to have weight $\rho_j\pmod {p_j}$ and
+child 2 to have weight $w_j-\rho_j\pmod {p_j}$.  The independent draw of
+$q$ may be deferred until after all these choices when analyzing them.
+
+\subsection{Parent dictionaries and prefix recomputation}
+\label{alg:parents}
+
+A parent dictionary stores records $(x,D)$: the exact signed integer
+weight $x$ and the indexed top mask $D\subseteq M_0$. Its key is both
+fields, so equal weights alone do not cause deduplication. A parent
+task consists of a side, a target residue modulo $P$, and a possibly
+empty prefix of the $M_0$ mask in a fixed index order. Left tasks use
+the empty prefix; right tasks may use a longer prefix. The task regenerates its four leaves, joins its two child
+streams, and either completes its dictionary or reports overflow.
+
+The final check, specified in Lemma~\ref{lem:separator}, tests whether two
+parent arrays contain disjoint top masks whose actual weights add to
+$t'$.  Its four random separating families are sampled once before the
+following loop, with the distribution specified in
+Section~\ref{sec:separator}. Conditional on the layout, they use fresh
+randomness independent of every parent-generation choice, including
+the moduli, low residues, and extra prime, and are reused for all checks.
+Their values do not affect which parent tasks are generated or how
+those tasks process entries; a check can only stop the trial with YES.
+
+For $z=0,\ldots,q-1$, set
+\[
+ v_L=\rho_0+p_0z,
+ \qquad v_R=(t'-v_L)\bmod P,
+\]
+with normalized residues.  On each side these range exactly once over
+the $P$-classes with the required parent low base.  Since $p_j\mid p_0$,
+the child low requirements do not change with $z$.
+
+First perform the left task.  Its two child streams are obtained from
+their leaf pairs by Lemma~\ref{lem:streams}, with modulus $P$ and low
+modulus $p_L$.  Stream the first child by its weight modulo $P$, and
+the second by $v_L$ minus its weight modulo $P$.  Join equal keys.
+For each matched pair, check that its two $M_L$ masks are disjoint.
+If so, retain just the actual total weight and the combined $M_0$ mask,
+deduplicating these pairs in a dictionary of capacity $C$.  If a new
+distinct pair would overflow the dictionary, abandon this $z$.  Also
+skip $z$ if the completed dictionary is empty.
+
+Otherwise retain the completed left array and perform the analogous
+right task with residue $v_R$, initially without an additional top-mask
+restriction.  If its completed dictionary has between one and $C$
+distinct entries, apply the final check to that right array and the
+stored left array.  If it is empty, do nothing.  If it overflows,
+discard the task's workspace and replace the task by two right tasks
+that fix the next bit of its $M_0$ mask to zero and one, respectively.
+Process these tasks depth first, in a fixed order of the top-mixer bits.
+An overflowing task whose prefix already specifies the entire mask is
+simply dropped.  Reuse or free the workspace after completing or
+discarding each task, and stop at the first YES.
+
+In every prefix task, remove incompatible items from each
+leaf array \emph{before} generating its streams.  Test in that leaf
+exactly the prescribed prefix bits belonging to its own top block.
+Regenerating and filtering its raw leaf list from scratch is permitted.
+As each top block occurs in exactly one leaf of a parent, no raw
+four-leaf tuple with an incompatible top mask is generated in that
+task.  This remains true even for tuples rejected later because their
+local-mixer masks overlap.  It is this early restriction that permits
+the later charging of repeated four-leaf tuple processing by depth
+rather than by number of tasks.
+
+\begin{lemma}\label{alg:validity}
+Every retained parent entry has a subset witness on its parent's
+indices.  Deduplicating entries by actual weight and top mask preserves
+the existence of every valid final match.  Every YES returned by either
+trial is correct.
+\end{lemma}
+
+\begin{proof}
+Within a parent the children share only $M_j$, and their disjointness
+there has been checked.  Their top blocks and background parts are
+disjoint.  Thus a retained entry has a subset witness.  The only
+indices shared by the left and right parents are in $M_0$.  Once their
+top masks are fixed, the choice among witnesses of a given actual
+weight has no effect on compatibility with the other parent.  Hence
+deduplication loses no valid match.  A final check returning YES has
+disjoint top masks and actual total weight $t'$, so its two witnesses
+give a signed solution, and~\eqref{alg:sign-flip} gives an original
+solution.  The compressed-block trial was already seen to have the
+same one-sided correctness property.
+\end{proof}
+
+\subsection{Three bounds for prefix recomputation}
+\label{sec:prefix-accounting}
+
+The schedule has two different sources of work. Starting a task
+regenerates leaves and initializes streams; processing its matches
+examines raw four-leaf occurrences. A small dictionary can have many
+raw representations, including invalid local overlaps. We therefore
+use distinct entries to count task starts and raw occurrences to count
+match processing. A third count measures how often retained arrays
+are passed to the final check.
+
+\paragraph{What the trial counts.}
+Fix the mixers, signs, cardinality guesses, layout, low moduli, and
+low residues, without fixing the extra prime $q$ or the separating
+families. The following three finite counts are already determined at this stage:
+\begin{itemize}
+\item $G$ is the sum, over the four children, of the numbers of raw
+leaf pairs passing that child's low filter, before any mask-prefix
+restriction.
+\item $\mathcal T_j$ contains all raw four-leaf occurrences on side
+$j$ passing the parent low filter modulo $p_0$ and the first-child
+filter modulo $p_j$. The second-child filter follows from
+$p_j\mid p_0$. Put $J_j=|\mathcal T_j|$, retaining every repetition
+and every local overlap.
+\item $\mathcal U_j$ contains all distinct valid
+$(\text{actual weight},\text{top mask})$ entries satisfying the
+parent low filter, ignoring child filters. Here validity means that
+the entry has a subset witness on side $j$'s indices with the
+prescribed top-block counts. This ambient set may contain entries
+not generated by the child lists. Put $U_j=|\mathcal U_j|$.
+\end{itemize}
+After $q$ is chosen, assign each occurrence and entry its unique
+bucket using the enumerated $P=p_0q$ classes. On the right the bucket
+label is the value of $z$ whose class is
+$(t'-\rho_0-p_0z)\bmod P$. These labels depend on $q$, but the three
+counts do not. Equal entry keys have equal actual weights and hence
+the same bucket. No coprimality of $p_0$ and $q$ is needed.
+
+The next lemma separates these deterministic charges from all
+probability estimates. The schedule is the one specified in
+Section~\ref{alg:parents}: left overflow skips a bucket, whereas right
+overflow refines a mask prefix. Only for counting, continue this
+schedule after any YES answer and ignore the operation cap. This
+finite hypothetical schedule need not be stored.
+
+\begin{lemma}[Prefix accounting]\label{lem:prefix}
+Fix positive integers $C,q$ and a nonnegative integer $m$. On each
+side $j\in\{L,R\}$, let $\mathcal T_j$ be a finite collection of
+raw tuple occurrences. Each occurrence has a bucket in
+$\{0,\ldots,q-1\}$ and a top mask in $\{0,1\}^m$. A valid occurrence
+produces an entry consisting of its exact weight and top mask.
+Assume that occurrences producing the same entry have the same
+bucket. Let $\mathcal U_j$ be a finite ambient set containing all
+distinct valid entries, with bucket and mask labels extending those
+of the generated entries. Write
+\[
+ J_j=|\mathcal T_j|,\qquad U_j=|\mathcal U_j|.
+\]
+A task for bucket $z$ and prefix $\sigma$ generates exactly the raw
+occurrences with these labels, unless it stops early, and reports
+overflow at the $(C+1)$st distinct valid entry. Suppose a task with
+$Z$ potential occurrences costs at most $H+KZ$, even if stopped
+partway, for fixed nonnegative $H,K$.
+
+Apply the left-skip and right-prefix schedule of
+Section~\ref{alg:parents} to these data, ignoring early YES answers and
+the operation cap. Then its number $T$ of parent-generation tasks,
+its total parent-generation work $W_{\mathrm{parent}}$, and the total
+input-array length $N_{\mathrm{check}}$ of its final checks satisfy
+\begin{align}
+ T&\le 2q+2(m+1)U_R/C,\label{eq:prefix-tasks}\\
+ W_{\mathrm{parent}}
+ &\le H\bigl(2q+2(m+1)U_R/C\bigr)
+       +K\bigl(J_L+(m+1)J_R\bigr),\label{eq:prefix-work}\\
+ N_{\mathrm{check}}&\le U_L+(2m+3)U_R.\label{eq:prefix-checks}
+\end{align}
+Every reuse of a left array counts separately in
+$N_{\mathrm{check}}$.
+\end{lemma}
+
+\begin{proof}
+\emph{Task starts.}
+At a fixed prefix depth, different right tasks have disjoint sets of
+possible entries: different buckets are disjoint, and different
+prefixes of the same length are disjoint within a bucket. Each
+overflow witnesses more than $C$ entries of $\mathcal U_R$.
+There are consequently at most $U_R/C$ overflow nodes at that depth,
+and at most $(m+1)U_R/C$ over all depths. There are at most $q$ left
+tasks and $q$ right root tasks. Every other task is one of at most two
+children of an overflow node. This proves~\eqref{eq:prefix-tasks},
+including the possibility that an overflowing full-mask node is
+dropped without children.
+
+\emph{Raw occurrences.}
+Each left occurrence has one bucket and can appear in at most one
+left task. A right occurrence has one bucket and one prefix at each
+depth, so it is a potential occurrence in at most $m+1$ right tasks.
+This counts occurrence identities, rather than their possibly equal
+weights or masks. It also counts invalid occurrences: their top masks
+are defined before the local overlap test, and incompatible prefix
+choices are excluded before the join. Thus the sum of all potential
+occurrence counts is at most $J_L+(m+1)J_R$. Summing the assumed
+stopped-task costs gives~\eqref{eq:prefix-work}. Only the overhead
+$H$ is multiplied by the number of tasks.
+
+\emph{Final-check inputs.}
+Completed right arrays contain disjoint sets of entries. Different
+buckets are disjoint. Within a bucket, completed prefixes are
+incomparable, since a completed task has no descendants, so their
+mask classes are disjoint as well. Right arrays therefore contribute
+at most $U_R$ to the total check-input length.
+
+Left arrays used at right-root checks have total length at most $U_L$,
+since each bucket supplies at most one such check. Every other check
+uses a left array of length at most $C$, and there are at most
+$2(m+1)U_R/C$ nonroot tasks. Their repeated left lengths total at most
+$2(m+1)U_R$. Adding these three contributions gives
+\eqref{eq:prefix-checks}.
+\end{proof}
+
+\paragraph{Applying the accounting bounds.}
+Every raw four-leaf tuple has a well-defined top mask because
+$T_1,\ldots,T_4$ partition $M_0$ and each occurs in exactly one leaf
+on each side. The union is defined even if local-mixer masks overlap.
+Testing a prefix in its responsible leaves therefore gives exactly
+the restricted occurrence collection required by Lemma~\ref{lem:prefix}.
+The dictionary removes duplicate keys only after the local
+disjointness test. This verifies the combinatorial premises of the
+lemma for the concrete schedule.
+
+\paragraph{The stopped-task cost.}
+For sufficiently large main-branch inputs, its cost premise holds with
+\begin{equation}\label{eq:prefix-task-cost}
+ H=u^8(2^{.2n}+G),\qquad K=u^8.
+\end{equation}
+Here is the implementation behind those constants. The four raw leaf
+arrays in a task have total length
+$A<4\cdot2^{.185n}$ by~\eqref{alg:leaf-size}; regenerating them,
+including their masks and prefix tests, takes $O(n^2A)$ operations.
+Sorting and stream initialization cost $O(nA)$, and the two child
+streams together have at most $G$ occurrences. Their arrays and
+mutable states occupy $O(A)$ words, which is at most $B$ for large
+$n$. The tie buffer has length $B$, and Lemma~\ref{lem:streams}
+charges all replay, including stopped replay, to stream advances and
+potential matching pairs. Dictionary initialization can be bounded by
+$O(u(C+1))$, and local mask checks and trie operations cost $O(u)$ per
+reported tuple. For $Z$ potential matching tuples, all this gives
+\[
+ O\bigl(n^2A+B+nG+u(C+1)+uZ\bigr).
+\]
+Since $C\le2^{.199n}$, $B\le2^{.19n}$, and
+$\log u\le n/10^9$, this is at most $H+KZ$ with
+\eqref{eq:prefix-task-cost}, after enlarging the absolute main-branch
+threshold. The raw term includes tuples later rejected for overlap
+and repeated occurrences. A small value of $U_j$ does not bound this
+term.
+
+Thus Lemma~\ref{lem:prefix} controls the complete schedule before any
+probability estimate is used. The numerical bounds for $G,J_j,U_j$
+and the proof that a solution reaches a completed right prefix belong
+to Section~\ref{sec:resources}. The remaining independent ingredient
+is the final check on two bounded arrays, which we now construct.
diff --git a/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/build/resources.tex b/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/build/resources.tex
new file mode 100644
index 0000000000000000000000000000000000000000..d39cfd9e1d67373975b2d7e1967fa5ae5fd57d3d
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+++ b/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/build/resources.tex
@@ -0,0 +1,272 @@
+\subsection{Unconditional resource bounds}\label{sec:resources}
+
+A surviving representation must reach a completed check before the
+operation cap. Recall the three ambient counts from the prefix
+analysis: \(G\) counts low-filtered raw child pairs, \(J_j\) counts
+low-filtered raw four-leaf tuples on side \(j\), and \(U_j\) counts
+distinct valid weight--top-mask entries passing only the parent low
+filter. None uses \(q\), a prefix restriction, or a separating family.
+The first two retain all occurrence multiplicities, including tuples
+with a local overlap. The last counts only valid entries and ignores
+the child filters. We shall bound these counts without conditioning on
+the rare survival event.
+
+For their first moments, condition on a passing mixer setup, signs,
+guesses, layout and allowed moduli \(p_i\), leaving the low residues
+\(\rho_0,\rho_L,\rho_R\) independent and uniform. Each child's required
+residue is marginally uniform modulo \(p_j\), including child~2 because
+\(\rho_j\) is independent of \(\rho_0\). Each fixed four-leaf tuple
+passes its parent filter and first-child filter with probability
+\(1/(p_0p_j)\); divisibility \(p_j\mid p_0\) then gives the second-child
+filter. Before its parent filter, the number of distinct valid entries
+on side \(j\) is at most
+\[
+ 2^{r_j}\,2^{H_{0j}m}\,2^{m-d_j}.
+\]
+Here the three factors count background choices, possible top masks,
+and distinct actual weights from disjoint use of \(M_j\). The last
+factor is \(N_j\), also for the signed weights. Each entry passes the
+parent filter with probability \(1/p_0\).
+
+Define the analysis scales
+\begin{equation}\label{res:scales}
+ \begin{aligned}
+ \Gamma&=\frac n4+\left(-\frac54+\frac32h_*\right)m+f_{\max},\\
+ d_*&=\min(d_L,d_R),\\
+ \Lambda&=\frac n2+(\overline H_0-1)m-d_*+f_0.
+ \end{aligned}
+\end{equation}
+By \eqref{eq:allocation}, \eqref{eq:beta}, and
+\eqref{eq:filter-bounds}, the conditional moments satisfy
+\begin{equation}\label{res:moments}
+ \mathbb EG\le u^{26}2^\Gamma,\qquad
+ \mathbb E(J_L+J_R)\le u^{51}2^{n/2},\qquad
+ \mathbb EU_j\le u^{10}2^\Lambda.
+\end{equation}
+For the first bound, use \(\beta_j-L_j\le\Gamma\), the
+\(u^{20}\) factor for raw pairs, and the \(u^5\) loss in the
+modulus lower bound. For the second, raw four-leaf tuple counts are at
+most \(u^{40}2^{2\beta_j}\), and their exponent after filtering obeys
+\begin{equation}\label{res:tuple-exponent}
+ \begin{aligned}
+ 2\beta_j-L_0-L_j
+ &\le \frac n2+\left(-\frac52+3h_*\right)m+2\tau\\
+ &\le \frac n2.
+ \end{aligned}
+\end{equation}
+The final inequality holds for large \(n\), since the excess is at
+most \(-.0658m+.0156n\), and \(m=.24n+O(1)\). For the third
+bound, the exponential part of the preceding universe count divided by
+the parent modulus is
+\[
+ \frac r2+\overline H_0m+m-d_j-L_0
+ =\frac n2+(\overline H_0-1)m-d_j+f_0
+ \le\Lambda;
+\]
+the allocation's logarithmic error and the modulus loss fit in the
+factor \(u^{10}\).
+
+Let \(\mathcal E\) be the resource event
+\begin{equation}\label{eq:resource-event}
+ G\le u^{200}2^\Gamma,\qquad
+ J_L+J_R\le u^{200}2^{n/2},\qquad
+ U_L,U_R\le u^{200}2^\Lambda.
+\end{equation}
+The scales in \eqref{res:scales} are fixed under the conditioning used
+in \eqref{res:moments}. Markov's inequality and a union bound, followed
+by averaging over that conditioning, give
+\begin{equation}\label{res:joint-failure}
+ \Pr[\text{spread test passes and }\mathcal E^c]
+ \le u^{-174}+u^{-149}+2u^{-190}
+ <\frac1{10}u^{-100}.
+\end{equation}
+This is a joint failure estimate, not an estimate conditioned on the
+rare event in Lemma~\ref{lem:representation}. Consequently that lemma's
+surviving-representation event and \(\mathcal E\) hold together with
+probability at least \(.9u^{-100}\) in the low-mean YES case.
+
+Three exponent relations control the subsequent costs:
+\begin{equation}\label{res:exponent-relations}
+ \xi+\Gamma\le\frac n2,\qquad
+ \Lambda+(1-h_*)m\le\frac n2,\qquad
+ \xi-(\Lambda-.199n)\ge .0015n.
+\end{equation}
+The first relation pays for scanning the low-filtered children over
+roughly \(2^\xi\) tasks. The second pays for the repeated separator
+incidences of distinct records. The third makes the expected number
+of different-weight records in a solution bucket smaller than its
+capacity \(C\). These relations hold for large \(n\) on every passing
+setup. The first follows
+from \(.032-1.25+1.5h_*<0\), with cancellation of \(f_{\max}\).
+For the second, use \(\overline H_0\le h_*\) and
+\(f_0\le d_*\), since \(M_0\) has least deficiency. For the
+third, direct subtraction gives the lower bound
+\[
+ -.051n+(1.032-h_*)m+d_*-f_0-f_{\max}.
+\]
+Here \(f_0\le d_*/2\), while the spread test implies
+\(f_{\max}\le d_*/2+.00025n\). Since
+\(1.032-h_*>.2206\) and \(m=.24n+O(1)\), this lower bound is
+at least \(.0015n\). The first two relations preserve the exact
+time exponent \(1/2\); no positive exponential slack is added there.
+
+\subsection{The extra prime and the solution bucket}
+
+Condition on a surviving representation and \(\mathcal E\), exposing
+all choices so far but still not \(q\). In its loop iteration, the
+expected number of entries in side \(j\)'s universe with weight
+different from \(x_j\) but congruent to it modulo \(P=p_0q\) is
+at most
+\begin{equation}\label{res:bucket-expectation}
+ U_j u^3 2^{-\xi}
+ \le u^{203}2^{.199n-.0015n}.
+\end{equation}
+The nonzero difference must be divisible by the independent prime
+\(q\), so the prime collision bound applies. No coprimality of
+\(q\) and \(p_0\) is required. Since
+\(C\ge 2^{.199n-1}\), the right side of
+\eqref{res:bucket-expectation} divided by \(C/2\) is at most
+\[
+ 2^{(203/10^9-.0015)n+2}
+\]
+on the main branch. Markov's inequality shows, with conditional
+probability at least \(.9\) for large \(n\), that both sides have
+at most \(C/2\) such different-weight entries.
+
+Equal-weight entries need a separate argument. On L, at most
+\(\binom m{k_0}\le 2^{h_*m}\le 2^{.194736n}<C/2\) distinct
+entries can have weight \(x_L\), since the smaller top mask has
+size at most \(m/4\). Thus the entire solution L bucket fits. On R,
+follow the prefix of the solution top mask. At a full prefix there is
+at most one entry of weight \(x_R\), by deduplication. Together
+with the different-weight bound, this full-prefix bucket fits, so the
+solution's prefix path terminates in a completed nonoverflow task.
+In every task compatible with the solution masks, its surviving leaf
+choices generate the corresponding parent entries. Hence the solution
+pair reaches a weighted disjointness check, unless an earlier YES or
+an operation or sampling abort has already occurred.
+
+\subsection{Applying the three prefix bounds}
+
+Consider the complete hypothetical parent schedule obtained by ignoring
+YES answers and the operation cap, while retaining the actual overflow
+rules. Its generation uses only the parent choices and \(q\). Thus it
+can be fixed without examining any separating family. The three
+bounds in Lemma~\ref{lem:prefix} now have their required numerical
+inputs; we apply each to the quantity it controls.
+
+First, the number of parent-generation tasks is at most
+\begin{equation}\label{res:task-count}
+ T\le 2q+2(m+1)U_R/C\le u^{203}2^\xi.
+\end{equation}
+Indeed, \(q\le2^\xi\), \(C\ge2^{.199n-1}\), and the last inequality in
+\eqref{res:exponent-relations} gives
+\(U_R/C\le2u^{200}2^{\xi-.0015n}\) on \(\mathcal E\).
+The main-branch guard absorbs the remaining polynomial factor.
+
+Second, the concrete stopped-stream implementation supplies
+\begin{equation}\label{res:nonmatch-cost}
+ H=u^8(2^{.2n}+G),\qquad K=u^8
+\end{equation}
+in the task-cost formula of Lemma~\ref{lem:prefix}. The term \(H\)
+pays for regenerating the leaves, prefix tests before streaming,
+sorting, stream initialization and traversal, the bounded tie buffer,
+and dictionary initialization. The term \(K\) pays for each potential
+matching tuple, including replay, invalid local overlaps and repeated
+entries. Consequently the full parent-generation work is at most
+\[
+ \begin{aligned}
+ H T+K\bigl(J_L+(m+1)J_R\bigr)
+ &\le u^{211}2^\xi(2^{.2n}+u^{200}2^\Gamma)
+       +u^8\bigl(J_L+(m+1)J_R\bigr)\\
+ &\le u^{420}2^{n/2}
+ \end{aligned}
+\]
+for sufficiently large main-branch inputs. Here \(\xi+.2n<n/2\) follows
+from \(\xi\le.25768n\), whereas \(\xi+\Gamma\le n/2\) is the first
+exact relation in \eqref{res:exponent-relations}. The raw-match term
+is charged by prefix depth; it is never multiplied by \(T\).
+
+Third, write \(N_{\mathrm{check}}\) for the total array length supplied
+to all final checks, counting every reuse of a left array. The final
+bound of Lemma~\ref{lem:prefix} gives
+\begin{equation}\label{res:check-inputs}
+ N_{\mathrm{check}}\le U_L+(2m+3)U_R.
+\end{equation}
+Completed right chunks pay for their own entries, and overflow entries
+pay for repeated left arrays. This is why the final checks use the
+distinct-entry counts \(U_j\), even though generation required the raw
+counts \(J_j\).
+
+\subsection{Independent families and bounded-trial success}
+
+Fix all choices through \(q\), on the joint survival, resource and
+bucket events. Choose deterministically the check on the completed
+solution-prefix path and the solution pair in it. The complete
+hypothetical check list, its layout parameters, this designated check
+and its pair are all functions of those nonfamily choices. Conditional
+on the layout, the program sampled the four indexed families from the
+product law specified in Section~\ref{sec:separator}, independently
+of every parent and filter choice. Therefore that same product law
+holds after conditioning jointly on this entire list, its parameters,
+and the designated pair. This fact, rather than delayed exposure by
+itself, verifies the hypothesis of Corollary~\ref{cor:separator-reuse}.
+
+On \(\mathcal E\), \eqref{res:check-inputs} and that corollary give
+conditional expected total check work at most
+\[
+ N_{\mathrm{check}}u^{22}2^{(1-h_*)m}
+ \le u^{230}2^{\Lambda+(1-h_*)m}
+ \le u^{230}2^{n/2}.
+\]
+The second exact relation in \eqref{res:exponent-relations} is used
+here. Markov's inequality bounds the chance of exceeding
+\(u^{240}2^{n/2}\) operations by \(u^{-10}\). The designated pair has
+coverage probability at least \(1-4e^{-10}\). A union bound gives
+both coverage and this work bound with conditional probability at
+least \(1-4e^{-10}-u^{-10}\); these events need not be independent.
+The hypothetical list is an analysis device and is never stored.
+
+\begin{lemma}\label{lem:resources}
+On a YES input with \(\mu<.00765n\), a representation trial with
+the bounded samplers and operation cap succeeds with probability at
+least \(u^{-110}\), for sufficiently large main-branch inputs.
+\end{lemma}
+
+\begin{proof}
+Lemma~\ref{lem:representation} and \eqref{res:joint-failure} give
+probability at least \(.9u^{-100}\) of simultaneous survival and
+resources. The independent extra prime gives the bucket event with
+conditional probability at least \(.9\). Given both stages, the
+family argument gives coverage and check work with conditional
+probability at least \(1-4e^{-10}-u^{-10}\). These multiply as
+conditional lower bounds, not as an assertion that the events are
+independent. On their intersection, parent generation costs at most
+\(u^{420}2^{n/2}\) and all checks cost at most \(u^{240}2^{n/2}\).
+
+The remaining setup work is smaller. Counting the three mixer
+sumsets by half-list streams takes \(\poly(n)2^m\) operations.
+There are constantly many prime calls, each using at most \(n^3\)
+attempts, with \(\poly(n)2^{.129n}\) work per attempt. Family
+sampling and incidence construction take \(\poly(n)2^{2g}\)
+operations, including their bounded range-draw costs. Direct
+combination generation and weight/mask formation cost at most
+\(O(n^2)\) per item; comparison sorting, heap operations and interval
+searches cost at most \(O(n)\) per item or advance; trie operations
+cost \(O(u)\) per key. These explicit operations fit the displayed
+loose polynomial powers. Including the worst permitted sampler costs,
+the joint favorable execution uses fewer than
+\(u^{500}2^{\lceil n/2\rceil}\) underlying word operations for large
+\(n\).
+
+Lemma~\ref{lem:samplers} couples the bounded draws to the ideal
+experiment with loss \(\exp(-\Omega(n^2))\) per trial. This is
+smaller than \(u^{-110}\) by any fixed factor on the main branch.
+Subtracting that loss from
+\[
+ .81u^{-100}(1-4e^{-10}-u^{-10})
+\]
+leaves at least \(u^{-110}\). On this intersection the trial completes
+before its cap and finds a solution. An earlier YES is already a
+success.
+\end{proof}
diff --git a/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/build/separators.tex b/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/build/separators.tex
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+\section{Weighted disjointness in bounded space}\label{sec:separator}
+
+The parent schedule produces two arrays of records, each consisting of
+an actual weight and a top-mixer mask. We want one record from each
+array whose weights sum to a given target and whose masks are disjoint. A separating set
+turns this into an ordinary weight search: retain the left records whose
+sets lie inside it, and the right records whose sets lie outside it.
+Every pair of retained records is then disjoint. Sorting their weights
+finds an exact complementary pair, if one exists.
+
+The simplest illustration uses a block of $g$ indices, with $g$ divisible
+by four. Suppose each side of a fixed disjoint pair occupies $g/4$
+indices. A uniformly sampled set of size $g/2$ separates this pair with
+probability
+\[
+ \frac{\binom{g/2}{g/4}}{\binom{g}{g/2}}.
+\]
+Indeed, the sampled set must include the left $g/4$ indices, exclude
+the right $g/4$ indices, and choose its remaining $g/4$ elements from
+the $g/2$ unused indices. Sampling a family whose size is a constant
+times the reciprocal of this probability gives a constant chance of
+covering the fixed pair. An individual record can occur in several
+members of this family; those occurrences determine the work.
+
+We will apply this idea on four blocks. The left and right cardinalities
+may differ, so the sampled set size is biased to control occurrences on
+both sides. We first specify the distribution. We then show how to
+search its useful index tuples without storing all tuples or testing
+all pairs of records.
+
+This construction follows the separating-set approach of Nederlof and
+W\k{e}grzycki~\cite[full version, Section~6]{NW21}, which builds on the
+separating collections of Fomin, Lokshtanov, Panolan, and
+Saurabh~\cite[Section~4.2, Definition~4.3 and Lemma~4.5]{FLPS16}.
+We use the affine bias from Nederlof and W\k{e}grzycki's
+construction~\cite[full version, Lemma~6.6]{NW21}, with the
+cardinality-range extension of its entropy estimate by Belova, Chukhin,
+Kulikov, and Mihajlin~\cite[full version, Section~4.7]{BCKM24}.
+The rounding, traversal, and exact estimate required here are proved below.
+
+\subsection{Inputs and sampled families}
+
+Recall the parameters $u,m,g,C$ from \eqref{eq:parameters}. The four
+blocks in the representation layout give the partition
+\[
+ M_0=T_1\sqcup T_2\sqcup T_3\sqcup T_4,\qquad |T_i|=g.
+\]
+Write $k=k_0/m\in[1/6,1/4]$, where $k_0$ is the prescribed left
+mask size. To recall the balanced block counts explicitly, write
+$k_0=4d+r$ with $0\le r<4$ and put $c_i=d+1$ for $i\le r$ and
+$c_i=d$ otherwise. Then $\sum_i c_i=k_0$ and $|c_i-kg|<1$.
+The left array $\mathcal L$ and right array
+$\mathcal R$ are nonempty and have lengths at most $C$. Their records
+are $(x,D)$ and $(y,E)$, respectively, with $D,E\subseteq M_0$ and
+\begin{equation}\label{sep:cardinalities}
+ |D\cap T_i|=c_i,\qquad |E\cap T_i|=g/2-c_i
+ \quad(1\le i\le4).
+\end{equation}
+These conditions do not assert that an arbitrary pair is disjoint.
+For the transformed target $t'$, the required output is YES only if some
+pair satisfies
+\begin{equation}\label{sep:target}
+ D\cap E=\varnothing,\qquad x+y=t'.
+\end{equation}
+Array lengths count occurrences; distinct records are not required.
+
+The actual weights and target have magnitude below
+$2^{b+\lceil\log(n+2)\rceil+4}$. The exact arithmetic, masks, and
+array indices used here fit in a constant number of the words specified
+in Section~\ref{sec:introduction}; all stored tables and retained
+random choices count as workspace. The space bound for this check
+will hold on every outcome of the randomness.
+
+Let
+\[
+ \alpha=\log 3,\qquad
+ \alpha_m=\frac{\lfloor\log(3^m)\rfloor}{m},\qquad
+ e=\left\lfloor g\left(\frac12+
+              \alpha_m\left(k-\frac14\right)\right)\right\rfloor.
+\]
+The logarithm floor in $\alpha_m$ is the bit length of the integer
+$3^m$ minus one; no real-number oracle is needed. For each block set
+\begin{equation}\label{sep:families}
+ \nu_i=\left\lceil
+  10\binom{g}{e}\bigg/\binom{g/2}{e-c_i}
+ \right\rceil,
+ \qquad \mathcal F_i=(S_{i,1},\ldots,S_{i,\nu_i}).
+\end{equation}
+All $S_{i,j}$ are independent, and $S_{i,j}$ is a uniform $e$-element
+subset of $T_i$. The families are indexed: equal sampled sets retain
+their separate indices. The arrays and target are fixed independently
+of all four families.
+
+The binomial arguments in \eqref{sep:families} are feasible for
+large $n$. To see this, put
+\[
+ \zeta=\frac12+\alpha\left(k-\frac14\right),\qquad
+ Y=\zeta-k=\frac14+(\alpha-1)\left(k-\frac14\right).
+\]
+Then $e=\zeta g+O(1)$ and $c_i=kg+O(1)$, uniformly in
+$k\in[1/6,1/4]$. Since $1<\alpha<2$, $Y$ stays strictly between
+$0$ and $1/2$ throughout this interval.
+
+As in Lemma~\ref{lem:entropy}, $h$ denotes binary entropy and
+$h_*=h(1/4)$.
+
+\begin{lemma}[Weighted disjointness]\label{lem:separator}
+For the input and independent families just specified, there is a
+weighted-disjointness check with the following guarantees, for all
+sufficiently large $n$.
+\begin{enumerate}
+\item It returns YES only for a pair satisfying
+\eqref{sep:target}. For every fixed pair satisfying that relation,
+the probability that the check finds a solution is at least
+$1-4e^{-10}$.
+\item Preprocessing the families takes at most
+$\poly(n)2^{2g}$ operations and words. Thereafter the expected check
+work is at most
+\[
+ (|\mathcal L|+|\mathcal R|)\,u^{22}2^{(1-h_*)m}.
+\]
+\item Workspace, including preprocessing, is deterministically at most
+$O(nC+n^2+2^{2g})$ words. Any execution stopped early has no
+larger work or space usage than the complete traversal used in these
+bounds.
+\end{enumerate}
+The probability statements concern the stated ideal uniform families.
+Generating them with at most $n^3$ attempts per ordinary rejection
+draw also has deterministic cost $\poly(n)2^{2g}$, or reports sampling
+failure. It can be coupled to the ideal families with failure
+probability $\exp(-\Omega(n^2))$.
+\end{lemma}
+
+\begin{proof}
+We first check coverage of one fixed pair. We then describe a traversal
+whose workspace is bounded for every family outcome. Its time will be
+charged to individual occurrences of input records, and an entropy
+estimate will bound their expectation.
+
+\paragraph{Coverage of one pair.}
+Fix a pair satisfying \eqref{sep:target}. On block $T_i$, a sampled
+set must contain its left mask and avoid its right mask. Their union
+has size $g/2$, so the separation probability is exactly
+\[
+ \theta_i=\binom{g/2}{e-c_i}\big/\binom{g}{e}.
+\]
+Equation~\eqref{sep:families} gives $\nu_i\theta_i\ge10$.
+The probability that no family member separates the pair on this
+block is at most $(1-\theta_i)^{\nu_i}\le e^{-10}$.
+A union bound over the four blocks shows that some tuple of family
+indices separates the whole pair with probability at least
+$1-4e^{-10}$. This is coverage of a fixed pair; no simultaneous
+coverage assertion for every pair is needed.
+
+\paragraph{Preprocessing and the traversal.}
+A left record is compatible with index $j$ on block $i$ when
+$D\cap T_i\subseteq S_{i,j}$; a right record is compatible when
+$(E\cap T_i)\cap S_{i,j}=\varnothing$.
+For every mask on each block, store both sorted lists of compatible
+indices, one for containment and one for avoidance. Since
+$\binom{g/2}{e-c_i}\ge1$,
+\[
+ \nu_i\le10\cdot2^g+1.
+\]
+Store a block mask and a family index in one word each. For each block
+and each of its $2^g$ masks, the two incidence lists together need at
+most $2\nu_i$ index words; their offsets and lengths use only a
+constant number of words. Across the four blocks, a flat array of
+capacity at most $8\cdot2^g(10\cdot2^g+1)$ suffices for their
+contents. Generating the families needs $O(2^g+n)$ words, including
+the current permutation. Local-index maps and other control data
+use at most $O(n^2)$ words. Thus the families and incidence tables
+occupy $O(2^{2g}+n^2)$ words. Filling the lists in increasing index
+order costs $\poly(n)2^{2g}$ operations.
+A binary search in the stored list decides whether a record has a compatible index in
+any given interval of family indices.
+
+Process the blocks in order. For the current block, begin with its
+whole index interval $[1,\nu_i]$. Filter both received arrays to
+records having a compatible index in this interval. If either filtered
+array is empty, finish this branch. Otherwise split a nonsingleton
+interval into two nearly equal intervals and visit both children. At
+a singleton, fix that family index and enter the next block with the
+filtered arrays. After fixing all four indices, sort the two arrays
+by actual weight and search for weights summing exactly to $t'$.
+
+Every pair at this last search is disjoint, because its left mask lies
+inside the chosen set and its right mask lies outside it on every
+block. Thus a reported YES is valid. A fixed pair separated by a tuple
+of indices remains in both arrays on the entire path to that tuple.
+It therefore reaches an exact-weight search, unless a previous search
+has already returned YES. This proves the coverage guarantee.
+
+Use depth-first traversal and retain the parent arrays until their
+children finish. Each depth has its own buffers, written only up to
+their current lengths. In particular, a small filtered array does
+not require clearing a length-$C$ buffer. The combined depth of the
+four interval trees is at most
+\[
+ 4+\sum_{i=1}^4\lceil\log\nu_i\rceil\le m+20=O(n).
+\]
+At each depth at most two arrays of length $C$ are retained, and each
+record occupies a constant number of words. The retained-buffer term
+is therefore $O(nC)$ words, including the input arrays. Sorting at a
+terminal branch uses only $O(C)$ additional words, and stack control
+uses $O(n)$ words. Together with the incidence tables this gives the
+explicit bound
+\begin{equation}\label{sep:workspace}
+ O(nC+n^2+2^{2g})\quad\text{writable words}.
+\end{equation}
+This bound applies to every choice of the families and every feasible
+input array. Any other arrays retained by the caller are additional.
+
+\paragraph{Charging the visited occurrences.}
+Fix one input occurrence, and let $t_i$ be its number of compatible
+indices in family $i$. For a compatible prefix of previously fixed
+indices, the occurrence is scanned at the next block's root. If
+$t_i=0$, that scan removes it. If $t_i>0$, it can pass the filter only
+on paths to its $t_i$ compatible leaves. The union of these paths has
+$O(t_i(1+\log\nu_i))$ nodes; scans into their children have the same
+order of cost. There are at most $\prod_{r<i}t_r$ compatible
+prefixes before block $i$. Its scans at that block are consequently
+bounded by a polynomial factor times
+\[
+ \left(\prod_{r<i}t_r\right)(1+t_i).
+\]
+The term $1$ is necessary even when $t_i=0$.
+
+An interval predicate costs $O(n)$ operations, including binary
+search and local-mask extraction, and each tree has depth $O(n)$.
+After the fourth block, sorting and searching cost $O(n)$ operations
+per surviving occurrence. These costs give an $O(n^2)$ factor in
+the preceding scan count.
+Node overhead can be charged to scans because a child or new block is
+entered only with nonempty arrays. For sufficiently large $n$, the
+complete charge to this input occurrence is at most
+\begin{equation}\label{sep:charge}
+ u^4\sum_{j=0}^4\prod_{i=1}^j t_i,
+\end{equation}
+where the empty product equals one. This charge bounds a traversal
+which completes all branches even after finding a solution. Stopping
+at a YES, an operation limit, or any earlier point can only reduce
+its work.
+
+\paragraph{Expected incidences on one block.}
+For a fixed left mask of size $c=c_i$, the probability of containment
+in a uniform $e$-set is
+$\binom{g-c}{e-c}/\binom{g}{e}$. Since
+$\nu_i\le11/\theta_i$,
+\begin{equation}\label{sep:incidence}
+ \E t_i\le
+ 11\binom{g-c}{e-c}\bigg/\binom{g/2}{e-c}.
+\end{equation}
+Apply the binomial bounds of Lemma~\ref{lem:entropy}.
+Set
+\[
+ X=1-k,\qquad
+ Y=\frac14+(\alpha-1)\left(k-\frac14\right),\qquad
+ E(k)=Xh(Y/X)-\frac12h(2Y).
+\]
+Those bounds show that the logarithm of the
+right-hand side of \eqref{sep:incidence} is at most
+\begin{equation}\label{sep:entropy-rate}
+ gE(k)+O(\log n).
+\end{equation}
+The bound is uniform in $k\in[1/6,1/3]$. Indeed, $Y$ lies between
+$k$ and $1/4$, and $Y/X$ and $2Y$ stay away from zero and one on
+this compact interval. Their entropy derivatives are bounded in a
+fixed neighborhood. The $O(1)$ rounding of $c$ and $e$ therefore
+changes each binomial entropy exponent by only $O(1)$. The lower
+binomial bound for the denominator contributes at most
+$\log(g/2+1)$ in addition to constants.
+
+For right records use the complements of the sampled sets. Write
+\[
+ k'=\frac12-k,\qquad c'=g/2-c,\qquad e'=g-e.
+\]
+The complemented sets are uniform $e'$-sets. Their ideal bias is
+$1-\zeta=1/2+\alpha(k'-1/4)$, and rounding again costs only a
+bounded amount. The family-size denominator stays the same because
+\[
+ e'-c'=g/2-(e-c),\qquad
+ \binom{g/2}{e'-c'}=\binom{g/2}{e-c}.
+\]
+Thus \eqref{sep:incidence}--\eqref{sep:entropy-rate} apply to
+right records with $k'\in[1/4,1/3]$. It remains to prove one sharp
+entropy bound covering both sides.
+
+\paragraph{The exact entropy bound.}
+We claim
+\begin{equation}\label{sep:sharp}
+ E(k)\le1-h_*\qquad(1/6\le k\le1/3).
+\end{equation}
+Let $F=(\ln2)E$, $Z=X-Y$, and $W=1/2-Y$. Expanding entropy gives
+\[
+ F=X\ln X-Z\ln Z-\frac12\ln\frac12+W\ln W.
+\]
+Consequently
+\[
+ F'=\ln\frac{Z}{X}+(\alpha-1)\ln\frac{Z}{W},
+ \qquad
+ F''=\frac1X-\frac{\alpha^2}{Z}
+                  +\frac{(\alpha-1)^2}{W}.
+\]
+All denominators are positive even on $[0,1/2]$: explicitly,
+\[
+ Z=\frac{2+\alpha}{4}-\alpha k,
+ \qquad W=\frac\alpha4-(\alpha-1)k,
+ \qquad 1<\alpha<2.
+\]
+The ratio $Z/W$ decreases on this interval, since
+\[
+ \left(\frac{Z}{W}\right)'=\frac{\alpha-2}{4W^2}<0.
+\]
+Hence $Z/W\le(2+\alpha)/\alpha$. Also $Z/X\le1$, since
+$Y>0$. Multiplying the second derivative by $Z$ now gives
+\[
+ ZF''=\frac{Z}{X}-\alpha^2+(\alpha-1)^2\frac{Z}{W}
+ \le1-\alpha^2+(\alpha-1)^2\frac{2+\alpha}{\alpha}
+ =-2+\frac2\alpha<0.
+\]
+Thus $E$ is strictly concave. At $k=1/4$ we have
+$X=3/4$, $Z=1/2$, and $W=1/4$, so
+$F'=\ln(2/3)+(\alpha-1)\ln2=0$. Its maximum is therefore
+\[
+ E(1/4)=\frac34h(1/3)-\frac12=1-h(1/4).
+\]
+The last identity follows from
+$h(1/3)=\log3-2/3$ and $h(1/4)=2-(3/4)\log3$.
+This proves \eqref{sep:sharp} without a positive error in its
+exponential rate.
+
+Combining \eqref{sep:incidence}--\eqref{sep:sharp}, for either
+side and every block,
+\[
+ \E t_i\le u^4 2^{(1-h_*)g}
+\]
+for sufficiently large $n$. The four block families are independent,
+so expectations of the products in \eqref{sep:charge} factor.
+Since $1-h_*>0$ and $m=4g$, each of the five terms in
+\eqref{sep:charge} has expectation at most
+$u^{20}2^{(1-h_*)m}$. Their sum is at most
+$u^{22}2^{(1-h_*)m}$ for large $n$. Summing over input occurrences proves the
+expected work bound. The preceding construction already proved the
+deterministic space bound.
+
+Finally, a uniform $e$-set is obtained by taking the first $e$
+positions of a uniform permutation of its block. Generate each
+permutation from fresh successive range draws, with the bounded
+implementation of Lemma~\ref{lem:samplers}. Across all four families
+there are at most
+\[
+ 4g(10\cdot2^g+1)
+\]
+such calls. Its conditional per-call bound gives a coupling failure
+probability at most $4g(10\cdot2^g+1)2^{-n^3}$, which is
+$\exp(-\Omega(n^2))$. The bound uses the ideal product law and
+subtracts the coupling failure; it does not condition that law on
+successful bounded sampling. Allowing all $n^3$ attempts per draw
+adds only a polynomial factor to the generation work, which fits
+$\poly(n)2^{2g}$ together with preprocessing.
+\end{proof}
+
+\subsection{Reusing the same families}
+
+The check can be called repeatedly with the same four families.
+Different calls are then correlated, but their work still adds by
+linearity of expectation. The necessary condition is that the full
+list of their input arrays be fixed independently of those families.
+
+\begin{corollary}[Reusing the families]\label{cor:separator-reuse}
+Fix a finite list of checks satisfying the input conditions of
+Lemma~\ref{lem:separator}, with common block parameters, and let
+\[
+ N_{\mathrm{check}}=\sum_{\text{checks}}(|\mathcal L|+|\mathcal R|).
+\]
+The list may itself be random, provided that, conditional on the list
+and its parameters, the four families have the independent
+distributions \eqref{sep:families}. In the ideal experiment, the
+expected total check work conditional on the list and its parameters
+is at most
+\[
+ W_0=N_{\mathrm{check}} u^{22}2^{(1-h_*)m}.
+\]
+Preprocessing is paid once. Processing checks one at a time uses the
+check workspace of Lemma~\ref{lem:separator}, in addition to any
+storage retained by the caller. For any $0<\delta<1$, total
+check work exceeds $W_0/\delta$ with probability at most $\delta$.
+Suppose a designated check and a solution pair in it are also chosen,
+and that conditional on the list, its parameters, and these choices,
+the families still have the independent distributions
+\eqref{sep:families}. Then with probability at least
+$1-4e^{-10}-\delta$ the pair is covered and the total check work is
+at most $W_0/\delta$.
+\end{corollary}
+
+\begin{proof}
+An empty list has $N_{\mathrm{check}}=0$ and zero check work. Otherwise
+condition on the complete list and its parameters, and apply Lemma~\ref{lem:separator}
+to each input occurrence and sum expectations. Markov's inequality
+gives the work bound. For the last assertion, condition additionally
+on the designated check and pair. The family law is unchanged, so
+the same expected-work bound holds and the pair has coverage
+probability at least $1-4e^{-10}$. A union bound intersects the
+coverage and work events; those events need not be independent.
+\end{proof}
+
+In the Subset Sum application, continue parent generation as if checks
+never returned YES and as if no operation cap were present, retaining
+all dictionary overflow rules. This defines the complete check list
+without consulting the separating families. Conditional on the layout,
+their sampled product law is independent of all parent-generation
+choices. It therefore remains unchanged after fixing that list and a
+designated check and witness selected from it using only those choices.
+Corollary~\ref{cor:separator-reuse} applies once
+Section~\ref{sec:resources} bounds the total input length of this list.
+The actual run visits only an initial portion of its work; the list
+itself is never stored, and coverage is required only for the selected
+pair.
diff --git a/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/build/streams.tex b/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/build/streams.tex
new file mode 100644
index 0000000000000000000000000000000000000000..e5b10fd3aebfb3260182f34eb91f93faac1db0f1
--- /dev/null
+++ b/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/build/streams.tex
@@ -0,0 +1,157 @@
+\subsection{Sorted modular pair streams}
+\label{sec:streams}
+
+A stream will enumerate a Cartesian product while retaining only its
+smaller input arrays. An item consists of its actual, possibly signed,
+weight and a constant number of index masks. Write $\omega(a)$ for
+the actual integer weight of an item $a$. Repeated occurrences
+remain separate even if all their fields agree. Each item occupies
+constantly many words; array indices have $O(n)$ bits. We assume that
+weights, moduli, keys, and indices fit in the stated constant-word
+representation. Section~\ref{sec:implementation} verifies these
+conditions for all uses below.
+
+This is the sorted pair-sum method associated with Schroeppel and
+Shamir~\cite{SS81}; see also the explicit four-list description in
+Nederlof and W\k{e}grzycki~\cite[full version, Lemmas~3.3 and~A.1]{NW21}.
+We give the modular ordering and stopped-execution argument locally,
+since those details determine the cost of regenerating a dictionary.
+A generator's mutable state identifies its next occurrence and can be
+copied while sharing its immutable input arrays.
+
+\begin{lemma}[Filtered streams and tie joins]
+\label{lem:streams}
+Let $A_1,A_2$ be stored constant-word item arrays whose indices have
+$O(n)$ bits. Let $p\mid P$ be positive integers, and let $c,v$ be
+specified integer residues. Assume that the data occupy constantly
+many words each and that arithmetic, comparisons, and indexed access
+on them take $O(1)$ word operations in the stated model. The pairs $(a,b)\in A_1\times A_2$
+whose actual weights sum to $c\pmod p$ can be streamed in
+nondecreasing order of either
+\[
+ (\omega(a)+\omega(b))\bmod P
+ \quad\text{or}\quad
+ (v-\omega(a)-\omega(b))\bmod P,
+\]
+where all residues are normalized to $\{0,\ldots,P-1\}$.
+Initialization costs $O(n(|A_1|+|A_2|+1))$ word operations, each
+subsequent advance costs $O(n)$, and the state together with the
+input arrays uses $O(|A_1|+|A_2|+1)$ words.
+
+Consider two such streams of lengths $N_1,N_2$, with combined states
+and arrays of at most $S$ words. If their equal-key Cartesian
+products contain $Z$ pairs in all, they can report all these pairs
+using $O(S+B)$ words and, apart from initialization and the work
+performed on reported pairs,
+\begin{equation}
+\label{eq:tie-cost}
+ O\left(B+n(N_1+N_2+Z)+\frac SB Z\right)
+\end{equation}
+operations. Here $B\ge1$ is an integer buffer capacity. In
+particular, if $S\le\poly(n)B$, the bound is
+$\poly(n)(B+N_1+N_2+Z)$. The same upper bounds hold for an execution
+stopped partway, with $Z$ still counting \emph{all potential} matches.
+No coprimality assumption is needed.
+\end{lemma}
+
+\begin{proof}
+First sort $A_2$ lexicographically by its residue modulo $p$ and
+then its residue modulo $P$. For a fixed $a\in A_1$, binary
+search finds the contiguous slice with residue
+$c-\omega(a)\pmod p$. This slice contains exactly the eligible
+partners of $a$, and is ordered by their $P$-residues, including
+repetitions. Omit the row if the slice is empty.
+
+For a direct key, put $r=\omega(a)\bmod P$. Start at the first
+entry of the slice whose residue is at least $P-r$; if there is
+none, start at the beginning. Traverse forward cyclically for exactly
+the slice length. The entries at or above the pivot give the wrapped
+values $r+(\omega(b)\bmod P)-P$ first, followed by the unwrapped
+values $r+(\omega(b)\bmod P)$.
+Both portions are nondecreasing and the former values are smaller
+than the latter. This includes $r=0$, when the pivot search simply
+returns the beginning.
+
+For a complementary key put $K=(v-\omega(a))\bmod P$.
+Start at the last entry whose residue is at most $K$; if there is
+none, start at the last entry of the slice. Traverse in reverse
+cyclic order for exactly one slice length. Residues at most $K$
+produce the values $K-(\omega(b)\bmod P)$ in nondecreasing
+order. The residues above $K$ follow and produce the larger values
+$K-(\omega(b)\bmod P)+P$, also in nondecreasing order.
+Choosing the first pivot in the direct case and the last pivot in
+the complementary case preserves every repeated residue.
+
+Merge the rows with a min-heap having at most one current pair per
+item of $A_1$. A row stores its slice endpoints, current position,
+and remaining length; a heap entry has constantly many words.
+Comparison sorting, binary searches, and heap operations have depth
+$O(n)$ because array indices have $O(n)$ bits. This proves the
+initialization, advance, and state bounds. The construction permits
+repeated prime factors and other common factors. Empty input arrays
+give an empty stream.
+
+To join two streams, advance past unequal keys until an equal key is
+found. For that key, read up to $B$ second-stream items into a reusable
+buffer. If this exhausts the second tie group, pair its buffered items
+with each consecutive first-stream item of the key. No generator state
+is copied in this case. Reuse the buffer by its current length, without
+clearing $B$ positions for a short group.
+
+If the second tie group is longer, save the mutable second-generator
+state immediately after the first $B$ items. The snapshot includes
+its current lookahead item, heap, row cursors, and control state, but
+shares its immutable arrays. For each first-stream item of the key,
+report its pairs with the buffer, restore the saved state when needed,
+and replay the remaining suffix. After the final replay the second
+stream is already beyond the tie group.
+
+A save or restore costs $O(S)$. It is made only for a first item
+having at least $B$ potential partners; at most a constant number of
+such operations per first item is required. Their total cost is
+therefore $O(SZ/B)$. Replayed advances are charged to matching pairs;
+other advances are charged to the original stream lengths. The
+$O(n)$ cost per advance and the optional $O(B)$ initial buffer
+allocation give~\eqref{eq:tie-cost}. If processing stops immediately
+after a snapshot or restore, the same charge remains valid because
+$Z$ counts potential pairs, including pairs not reached. One snapshot,
+the buffer, and the two generators give the space bound.
+\end{proof}
+
+An actual-weight stream needs no modular pivots. Sort the second array
+by actual weight, add each first-array weight to that sorted row, and
+heap-merge the rows. Reading this ordered stream while remembering
+only the last weight counts its distinct values without storing the
+Cartesian product. This will let the representation trial measure a
+block's deficiency exactly from two small half-block lists.
+
+\subsection{Bounded dictionaries}
+
+Streaming saves space only if we also control what is retained.
+For deterministic deduplication, use a binary trie on a fixed-width
+signed actual weight, followed by the relevant mask bits when a mask
+is part of the key. Give zero its canonical nonnegative sign. In our
+applications the complete key has $O(u)$ bits. Searching or inserting
+it takes $O(u)$ operations and introduces at most $O(u)$ nodes,
+with constantly many words per node.
+
+Keep a separate append-only array of the distinct entries if subsequent
+array access is needed. A duplicate changes neither the array length
+nor its mathematical content. On finding a new key that would increase
+the array beyond capacity $C$, report overflow. Even if nodes for this
+first rejected key have been allocated, the trie has at most
+$O(u(C+1))$ nodes. All nodes, addresses, and appended records count as
+workspace. A weight-only dictionary is appropriate for disjoint
+compression blocks; a weight--mask dictionary will be required when
+compatibility depends on shared indices.
+
+For the concrete implementations below, raw list generation can be
+done in $O(n^2)$ operations per item, sorting and heap work in $O(n)$
+per item or advance, and dictionary operations in $O(u)$ per key.
+Whenever the combined stream storage is $O(B)$, the term depending
+on $Z$ in~\eqref{eq:tie-cost} is $O(nZ)$, before the separately
+charged processing of those matches. The buffer and stream-length
+terms remain separate. These explicit
+degrees will be used when the task costs are bounded. The argument
+does not assign a cost merely from the number of distinct entries
+eventually retained.
diff --git a/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/build/survival.tex b/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/build/survival.tex
new file mode 100644
index 0000000000000000000000000000000000000000..bfc6dd2e3916e09b78aa4a886080b8980eecf4bc
--- /dev/null
+++ b/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/build/survival.tex
@@ -0,0 +1,203 @@
+\section{Survival within the resource budgets}\label{sec:survival}
+
+The preceding construction always respects its dictionary capacities,
+but it may discard a bucket containing a solution or stop at the
+operation cap. We now show that a fixed solution reaches a completed
+check within the cap with inverse-polynomial probability, enough for
+repetition to succeed. The order of the argument matters: first fix a
+surviving representation, subtract an unconditional resource-failure
+bound, and only then use the extra prime and separating families.
+
+\subsection{Fixing a witness before the last random choices}
+
+Throughout this section suppose that the input is a YES instance and
+\(\mu<.00765n\), where \(\mu\) is the mean deficiency of a uniform
+size-\(s\) subset of the input indices. We first use ideal exact sampling.
+The extra prime \(q\) and the separating families are not exposed in this
+subsection. Our objective is to select one representation before either
+of those sources of randomness is used. We first balance the fixed
+solution inside the mixers, retain many distinct choices with the
+prescribed leaf counts, and finally show that the low residues hit
+some of them. Every selection of a witness in this argument is an
+analysis choice; the program enumerates candidates without knowing it.
+
+\begin{lemma}\label{lem:representation}
+For a main-branch YES input with \(\mu<.00765n\), ideal sampling
+produces, with probability at least \(u^{-100}\), a passing spread test and a representation of a solution satisfying all prescribed leaf
+counts and all low modular filters. Its two parent weights
+\(x_L,x_R\), with \(x_L+x_R=t'\), and all of its masks can be fixed as
+functions of the nonfamily choices preceding \(q\). Conditional on
+these choices, \(q\) still has its prescribed independent prime law;
+the separating families also retain their prescribed product law,
+with the now fixed layout parameters.
+\end{lemma}
+
+\begin{proof}
+\emph{Balance the solution and its deficiencies.}
+Fix an original solution \(S\) for analysis. Under the random sign flip
+\(F\), its signed counterpart \(S'=S\mathbin{\triangle}F\) is a uniform
+subset of the input indices, independent of the mixers. Put
+\(A_i=S'\cap M_i\). By Lemma~\ref{lem:entropy}, the balance event
+\[
+ |A_0|=|A_L|=|A_R|=s
+\]
+has probability at least \((m+1)^{-3}\). This event has the same
+probability for every choice of the three mixers before ordering them.
+Conditional on it, their distribution is unchanged, and their selected
+halves are independent uniform halves given the mixers. Each selected
+half and its complement in a pre-order mixer is therefore marginally a
+uniform size-\(s\) set of input indices.
+
+By Lemma~\ref{lem:deficiency}, with conditional probability
+\(1-O(1/n)\), all six halves have deficiency at most
+\(\tau=.0078n\): a deviation of \(.0001n\) above their common mean
+already suffices. The same lemma, now for size-\(m\) sets, shows that
+each pre-order mixer's deficiency is within \(.0002n\) of the
+size-\(m\) mean, again with conditional failure probability \(O(1/n)\).
+Thus the spread test \eqref{eq:spread} passes jointly with these events
+for sufficiently large \(n\). These are statements about the original
+weights, equally valid after sign changes by deficiency invariance.
+For this analysis we may sample the signs even on choices on which the
+algorithm would have stopped at the spread test.
+
+If \(A,B\) partition a mixer \(M\) into halves, then
+\(N(M)\le N(A)N(B)\), so
+\[
+ \defect(A)+\defect(B)\le \defect(M).
+\]
+At least one half has deficiency at most \(\defect(M)/2\). Given the
+three unordered pairs of halves, each orientation as the selected half
+is uniform, independently across the mixers. Both preceding good events concern the original-weight deficiencies
+of the unordered halves and are invariant under these orientations,
+as is the ordering of the mixers. With further conditional probability at least \(1/8\), then,
+\begin{equation}\label{surv:small-deficiency}
+ \defect(A_i)\le f_i=\min(\tau,d_i/2)
+ \qquad(i=0,L,R).
+\end{equation}
+No conditional independence of the signs is asserted or needed here.
+Sign invariance and \eqref{surv:small-deficiency} give at least
+\(2^{s-f_i}=2^{L_i}\) distinct signed subset sums on \(A_i\).
+
+\emph{Retain representatives with the prescribed leaf counts.}
+Partition these sums according to subset size. Some size has at least
+\(2^{L_i}/(s+1)\) distinct sums. Choose a maximizing size at most
+\(s/2=m/4\), using complementation in \(A_i\), which preserves the
+number of distinct sums between complementary sizes. This size exceeds
+\(m/6\) for sufficiently large \(n\). Otherwise its number of subsets
+would be at most \(2^{s h(1/3)}\), whereas
+\[
+ \frac{s-\tau-\log(s+1)}{s}\longrightarrow .935
+ >\frac{14}{15}>h(1/3).
+\]
+For each mixer fix such a size and one subset for each distinct sum of
+that size. These are analysis choices made before the size guesses and
+subblock partitions. The probability that all three guesses \(k_i\)
+equal these sizes is at least \((m+1)^{-3}\).
+
+We next retain many of these representatives with the required finer
+counts. Fix one representative \(D\subset A_i\), and write \(l=4\)
+for the top mixer and \(l=2\) for either other mixer. In a uniform
+partition into \(l\) labeled equal subblocks, prescribe the counts of
+the three categories
+\[
+ D,\qquad A_i\setminus D,\qquad M_i\setminus A_i
+\]
+to be respectively the smaller-mask counts, their prescribed
+complementary counts, and an equal split of the outside category.
+These vectors fill every block exactly. In a top block their counts
+are \(c_a,g/2-c_a,g/2\); in a local block they are
+\(a_{jr},s/2-a_{jr},s/2\). The divisibility of \(m\) by eight makes
+all these complementary counts integral. Each category's counts are
+as equal as possible. For \(v\) objects,
+a most balanced multinomial coefficient is maximal, since transferring
+one object from a larger count to a count smaller by at least two
+increases the coefficient. There are at most \((v+1)^l\) possible
+count vectors, and their coefficients sum to \(l^v\). Each of our
+three favorable coefficients is therefore at least
+\(l^v/(v+1)^l\). Dividing their product by the multinomial coefficient
+for the whole partition, which is at most \(l^m\), shows that \(D\)
+survives with probability at least
+\[
+ (m+1)^{-3l}\ge u^{-3l}.
+\]
+This survival ensures the finer counts for both \(D\) and its
+complement in \(A_i\).
+
+Let \(Z\in[0,1]\) be the surviving fraction of a representative
+family. If \(\mathbb EZ\ge p\), then
+\(\mathbb EZ\le p/2+\Pr[Z\ge p/2]\); hence
+\(\Pr[Z\ge p/2]\ge p/2\). Apply this with \(p=u^{-3l}\).
+The three subblock partitions are independent, so with further
+probability at least \(u^{-24}/8\ge u^{-25}\), all three retain
+these fractions. Let \(X_i\) consist of the distinct signed weights of
+the surviving smaller masks, retaining their associated masks. For
+sufficiently large \(n\),
+\begin{equation}\label{surv:representative-count}
+ |X_i|\ge \frac{2^{L_i}}{u^{14}}.
+\end{equation}
+
+We have now retained many distinct weights whose masks already meet
+all cardinality prescriptions. It remains to ensure that the low
+modular filters keep compatible choices. The next argument first
+bounds collisions under the random moduli, and then exposes the
+uniform target residues in the order that fixes the top choice before
+its children's choices.
+
+\emph{Hit occupied low-residue classes.}
+This collision-to-occupancy argument is the residue-coverage method of
+Austrin, Kaski, Koivisto, and Nederlof~\cite[Proposition~3.5]{AKKN16}
+and its product-modulus version in Nederlof and
+W\k{e}grzycki~\cite[full version, Lemma~3.2]{NW21}; we give the bounds
+for the present nested moduli explicitly.
+Expose the prime factors forming \(p_i\). For fixed \(X_i\),
+the sum of squares of its residue-class sizes modulo \(p_i\) has
+expectation at most
+\[
+ |X_i|+|X_i|^2u^{10}2^{-L_i}
+ \le u^{15}|X_i|^2 2^{-L_i},
+\]
+by \eqref{eq:filter-bounds} and \eqref{surv:representative-count}.
+Markov's inequality and a union bound show that, with probability at
+least \(1-3/u\), all three sums of squares are at most
+\(u^{16}|X_i|^2 2^{-L_i}\). The moduli need not be independent of
+one another. Cauchy--Schwarz gives at least
+\begin{equation}\label{surv:occupied-residues}
+ \frac{2^{L_i}}{u^{16}}\ge \frac{p_i}{u^{16}}
+\end{equation}
+occupied residues for each \(X_i\).
+
+First expose \(\rho_0\). Assign a surviving smaller mask
+\(D_0\subset A_0\) to L and its complement in \(A_0\) to R. As
+\(D_0\) varies, the left solution-parent weight is its weight plus
+a fixed term: the selected left background weight and the weight of
+\(A_L\). Equation~\eqref{surv:occupied-residues} implies that a
+uniform \(\rho_0\bmod p_0\) hits one of these weights with
+probability at least \(u^{-16}\). On a hit, fix a representative
+deterministically, thereby fixing disjoint top masks and parent weights
+\(x_L,x_R=t'-x_L\).
+
+Splitting \(A_j\) between the two children cannot change parent \(j\)'s
+weight: their union still uses all of \(A_j\). For each side \(j\),
+assign a surviving smaller mask
+\(D_j\subset A_j\) to its first child and the complement to its
+second child. The first-child weight is the weight of \(D_j\) plus
+a now fixed term from its background and its portion of the top mask.
+Expose the independent residues \(\rho_L,\rho_R\). They both hit
+possible first-child residues with probability at least \(u^{-32}\).
+The second-child filters then follow automatically from
+\(p_j\mid p_0\) and the parent filters. All prescribed leaf counts
+hold by the surviving subblock choices; the selected solution has no
+background cardinality constraint to satisfy.
+
+The product of the conditional probability lower bounds, for large
+\(n\), is at least
+\[
+ u^{-3}\cdot\frac12\cdot\frac18\cdot u^{-3}
+ \cdot u^{-25}\cdot\frac12\cdot u^{-48}
+ \ge u^{-100}.
+\]
+All choices of representatives on hits may be deterministic analysis
+choices. They precede both \(q\) and the separating families. For any
+subsequent \(q\), one loop iteration has
+\(v_L=x_L\bmod P\) and \(v_R=x_R\bmod P\).
+\end{proof}
diff --git a/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/paper.pdf b/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..e10af466bf74c50a3c383fb610bd9b404e55ef90
--- /dev/null
+++ b/preprints/A-Low-Space-Algorithm-for-Worst-Case-Subset-Sum-September-26-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:88f951cb233cf765b963cc3881f761846175bf5932649404d6ef3f69756d78ce
+size 503282
diff --git a/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/README.md b/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..2659e8811baf8d34ab40825ff47d65373f62685d
--- /dev/null
+++ b/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/README.md
@@ -0,0 +1,18 @@
+# [A Modulus Proof of Cannon’s Conjecture](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 23, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026,
+  author = {{OpenAI}},
+  title = {{A Modulus Proof of Cannon's Conjecture}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/paper.pdf}{OAI:A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/build/bibliography.tex b/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/build/bibliography.tex
new file mode 100644
index 0000000000000000000000000000000000000000..fb3fda580f68b6f7948ebc21d3504d52fa4a56d7
--- /dev/null
+++ b/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/build/bibliography.tex
@@ -0,0 +1,141 @@
+\begin{thebibliography}{99}
+
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+
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+
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+\emph{Geom. Topol.} \textbf{9} (2005), 219--246.
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+
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+M.~Bourdon and B.~Kleiner,
+Combinatorial modulus, the combinatorial Loewner property, and Coxeter groups,
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+\href{https://doi.org/10.4171/GGD/177}{doi:10.4171/GGD/177}.
+
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+Grundlehren der mathematischen Wissenschaften 319, Springer, 1999.
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+
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+\emph{Acta Math.} \textbf{173} (1994), 155--234.
+\href{https://doi.org/10.1007/BF02398434}{doi:10.1007/BF02398434}.
+
+\bibitem{CFPSquaring}
+J.~W. Cannon, W.~J. Floyd, and W.~R. Parry,
+Squaring rectangles: the finite Riemann mapping theorem,
+in \emph{The Mathematical Legacy of Wilhelm Magnus: Groups, Geometry
+and Special Functions}, Contemporary Mathematics 169,
+American Mathematical Society, 1994, 133--212.
+\href{https://personal.math.vt.edu/floyd/research/papers/mageps.pdf}{Author text}.
+
+\bibitem{CFP}
+J.~W. Cannon, W.~J. Floyd, and W.~R. Parry,
+Sufficiently rich families of planar rings,
+\emph{Ann. Acad. Sci. Fenn. Math.} \textbf{24} (1999), 265--304.
+\href{https://www.acadsci.fi/mathematica/Vol24/cannon.pdf}{Journal text}.
+
+\bibitem{CFPFiniteSubdivision}
+J.~W. Cannon, W.~J. Floyd, and W.~R. Parry,
+Finite subdivision rules,
+\emph{Conform. Geom. Dyn.} \textbf{5} (2001), 153--196.
+\href{https://doi.org/10.1090/S1088-4173-01-00055-8}{doi:10.1090/S1088-4173-01-00055-8}.
+
+\bibitem{CannonSwenson}
+J.~W. Cannon and E.~L. Swenson,
+Recognizing constant curvature discrete groups in dimension 3,
+\emph{Trans. Amer. Math. Soc.} \textbf{350} (1998), 809--849.
+\href{https://doi.org/10.1090/S0002-9947-98-02107-2}{doi:10.1090/S0002-9947-98-02107-2}.
+
+\bibitem{Drilling}
+D.~Groves, P.~Ha\"issinsky, J.~F. Manning, D.~Osajda,
+A.~Sisto, and G.~S. Walsh, Drilling hyperbolic groups,
+preprint, 2026.
+\href{https://arxiv.org/abs/2406.14667v2}{arXiv:2406.14667v2}.
+
+\bibitem{Haissinsky}
+P.~Ha\"issinsky, Hyperbolic groups with planar boundaries,
+\emph{Invent. Math.} \textbf{201} (2015), 239--307.
+\href{https://doi.org/10.1007/s00222-014-0552-x}{doi:10.1007/s00222-014-0552-x}.
+
+\bibitem{HK}
+J.~Heinonen and P.~Koskela, Definitions of quasiconformality,
+\emph{Invent. Math.} \textbf{120} (1995), 61--79.
+\href{https://doi.org/10.1007/BF01241122}{doi:10.1007/BF01241122}.
+
+\bibitem{KahnMarkovic}
+J.~Kahn and V.~Markovi\'c,
+Immersing almost geodesic surfaces in a closed hyperbolic three manifold,
+\emph{Ann. of Math.} (2) \textbf{175} (2012), 1127--1190.
+\href{https://doi.org/10.4007/annals.2012.175.3.4}{doi:10.4007/annals.2012.175.3.4}.
+
+\bibitem{Markovic}
+V.~Markovi\'c, Criterion for Cannon's conjecture,
+\emph{Geom. Funct. Anal.} \textbf{23} (2013), 1035--1061.
+\href{https://doi.org/10.1007/s00039-013-0228-5}{doi:10.1007/s00039-013-0228-5}.
+
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+P.~Pansu, Dimension conforme et sph\`ere \`a l'infini des vari\'et\'es
+\`a courbure n\'egative,
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+\href{https://doi.org/10.5186/aasfm.1989.1424}{doi:10.5186/aasfm.1989.1424}.
+
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+K.~Rajala, Uniformization of two-dimensional metric surfaces,
+\emph{Invent. Math.} \textbf{207} (2017), 1301--1375.
+\href{https://doi.org/10.1007/s00222-016-0686-0}{doi:10.1007/s00222-016-0686-0}.
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+O.~Schramm, Square tilings with prescribed combinatorics,
+\emph{Israel J. Math.} \textbf{84} (1993), 97--118.
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+
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+D.~Sullivan, On the ergodic theory at infinity of an arbitrary discrete
+group of hyperbolic motions, in \emph{Riemann Surfaces and Related Topics:
+Proceedings of the 1978 Stony Brook Conference}, I.~Kra and B.~Maskit, eds.,
+Annals of Mathematics Studies 97, Princeton University Press, 1981, 465--496.
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+
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+
+\end{thebibliography}
diff --git a/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/build/figures/level-continuum.tex b/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/build/figures/level-continuum.tex
new file mode 100644
index 0000000000000000000000000000000000000000..f53e9c7ec1f27c6ef626204bc79449616136f875
--- /dev/null
+++ b/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/build/figures/level-continuum.tex
@@ -0,0 +1,50 @@
+\begin{figure}[htbp]
+\centering
+\begin{tikzpicture}[x=1cm,y=1cm,font=\small,
+  line cap=round,line join=round,>=Stealth]
+  \definecolor{levelblue}{RGB}{54,101,125}
+  \definecolor{levelochre}{RGB}{164,111,48}
+  \fill[black!3] (0,1.25) rectangle (7,1.72);
+  \fill[black!3] (0,-1.72) rectangle (7,-1.25);
+  \draw[black!45] (0,-1.72) rectangle (7,1.72);
+  \draw[black!45] (0,1.25)--(7,1.25) (0,-1.25)--(7,-1.25);
+  \node at (1.9,1.49) {top band: $H_N=0$};
+  \node at (1.9,-1.49) {bottom band: $H_N=1$};
+  \node[anchor=north west,text=black!65] at (.10,1.15) {$\mathcal Q$};
+  \draw[levelblue,line width=2pt]
+    (0,.13)--(.37,.03)--(.69,.19)--(1.04,.05)--(1.34,.20)
+    --(1.64,-.04)--(1.98,.08)--(2.28,-.09)--(2.60,.04)
+    --(2.91,-.13)--(3.19,.06)--(3.53,-.09)--(3.84,.09)
+    --(4.16,-.02)--(4.48,.15)--(4.81,.01)--(5.13,.18)
+    --(5.47,.04)--(5.80,.20)--(6.10,.05)--(6.41,.20)
+    --(6.70,.08)--(7,.20);
+  \draw[levelblue,line width=1.1pt]
+    (1.34,.20)--(1.42,.37)--(1.59,.32)
+    (2.28,-.09)--(2.18,-.28)--(2.38,-.24)
+    (4.81,.01)--(4.89,-.21)--(5.03,-.15)
+    (6.10,.05)--(6.04,-.14)--(6.22,-.19);
+  \node[levelblue,anchor=south] at (1.95,.40) {$C_{N,t}\subset\{H_N=t\}$};
+  \draw[levelochre,dashed,thick]
+    (4.9,1.72) .. controls (4.4,1.32) and (3.88,1.29) .. (3.72,.79);
+  \draw[levelochre,thick]
+    (3.72,.79) .. controls (3.50,.34) and (4.24,-.24) .. (4.10,-.80);
+  \draw[levelochre,dashed,thick]
+    (4.10,-.80) .. controls (4.00,-1.24) and (4.94,-1.29) .. (5.1,-1.72);
+  \fill[levelochre] (3.72,.79) circle (1.5pt) (4.10,-.80) circle (1.5pt);
+  \node[anchor=west] at (3.96,.74) {$H_N<t$};
+  \node[anchor=east] at (3.89,-.86) {$H_N>t$};
+  \node[levelochre,anchor=west] at (4.27,-.46) {$\eta$};
+  \node[anchor=west,text=black!85,align=left] at (7.32,.82)
+    {strict-side\\extension};
+  \draw[black!65,thin] (7.2,.86)--(5.10,1.18);
+  \node[anchor=west,text=black!85,align=left] at (7.32,-.85)
+    {strict-side\\extension};
+  \draw[black!65,thin] (7.2,-.82)--(5.05,-1.36);
+\end{tikzpicture}
+\caption{A separating level continuum and a straddling path. The dashed
+extensions stay in the strict sublevel and superlevel sets and complete
+$\eta$ to a top-to-bottom crossing. That crossing meets $C_{N,t}$, so
+$\eta$ itself meets $C_{N,t}$. The continuum is drawn schematically;
+neither smoothness nor path connectedness is assumed.}
+\label{fig:level-continuum}
+\end{figure}
diff --git a/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/build/figures/pair-cost.tex b/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/build/figures/pair-cost.tex
new file mode 100644
index 0000000000000000000000000000000000000000..a7cb57ad0e155388121ebfdb49040ad878298818
--- /dev/null
+++ b/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/build/figures/pair-cost.tex
@@ -0,0 +1,54 @@
+\begin{figure}[htbp]
+\centering
+\begin{tikzpicture}[x=1cm,y=1cm,font=\small,
+  line cap=round,line join=round,>=Stealth]
+  \definecolor{costblue}{RGB}{54,101,125}
+  \definecolor{costochre}{RGB}{164,111,48}
+  \node[font=\bfseries] at (2.75,4.95) {Source};
+  \node[font=\bfseries] at (11.40,4.95) {Target};
+
+  \fill[costblue!5] (3.25,2.80) circle (1.10);
+  \draw[costblue,thick] (3.25,2.80) circle (1.10);
+  \draw[costblue,dashed,thick] (3.25,2.80) circle (.52);
+  \node[costblue,anchor=south] at (3.25,4.00) {$\mathcal D_{ij}$};
+  \fill[black!70] (.55,2.80) circle (1.8pt);
+  \node[anchor=north] at (.55,2.54) {$x_i$};
+  \draw[black!40,thin] (3.25,2.73)--(3.25,1.47);
+  \fill[costblue] (3.25,2.80) circle (1.8pt);
+  \node[anchor=north] at (3.25,1.37) {$z_{ij}$};
+  \node[align=center] at (2.75,.57)
+    {$h_i^{-1}E_j\subseteq\overline B(z_{ij},s_{ij})$};
+  \node[align=center] at (2.75,.05)
+    {$s_{ij}\asymp R^2/d_{ij}^{\,2}$};
+
+  \draw[black!85,thick,->] (5.55,2.80)--(7.25,2.80)
+    node[midway,above=5pt] {$h_i$};
+
+  \fill[costochre!7] (9.00,2.80) circle (.50);
+  \fill[costochre!7] (13.20,2.80) circle (.50);
+  \draw[costochre,thick] (9.00,2.80) circle (.50);
+  \draw[costochre,thick] (13.20,2.80) circle (.50);
+  \draw[costblue,dashed,thick] (13.20,2.80) circle (1.10);
+  \node[costochre,anchor=south] at (9.00,3.57) {$E_i$};
+  \node[costochre,anchor=west] at (14.50,3.65) {$E_j$};
+  \draw[costochre!75,thin] (14.38,3.57)--(13.51,3.19);
+  \draw[black!40,thin] (9.00,2.73)--(9.00,1.47);
+  \draw[black!40,thin] (13.20,2.73)--(13.20,1.47);
+  \fill[costochre] (9.00,2.80) circle (1.8pt);
+  \fill[costochre] (13.20,2.80) circle (1.8pt);
+  \node[anchor=north] at (9.00,1.37) {$p_i$};
+  \node[anchor=north] at (13.20,1.37) {$p_j$};
+  \draw[black!85,<->] (9.00,.76)--(13.20,.76)
+    node[midway,below=3pt] {$d_{ij}\gg R$};
+  \node[costblue,align=center] at (11.10,-.30)
+    {$h_i\mathcal D_{ij}\subseteq\overline B(p_j,C_3R)$};
+\end{tikzpicture}
+\caption{The measure cost of a far pair. The dashed source ball contains
+$h_i^{-1}E_j$ and has radius $s_{ij}\asymp R^2/d_{ij}^2$.
+Its fixed enlargement $\mathcal D_{ij}$, drawn solid, still maps into
+the dashed target ball of radius $C_3R$. Here $p_i=h_i z_0$, $z_{ij}=h_i^{-1}p_j$, and
+$E_i,E_j$ are the solid puncture balls of radius $A_hR$.
+The circles are schematic; no connectedness or boundary regularity of
+metric balls is assumed.}
+\label{fig:pair-cost}
+\end{figure}
diff --git a/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/build/main.tex b/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/build/main.tex
new file mode 100644
index 0000000000000000000000000000000000000000..58bb81a34624fe3e386aa907e3852ba8cf471928
--- /dev/null
+++ b/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/build/main.tex
@@ -0,0 +1,321 @@
+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage[margin=1.05in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{microtype,enumitem,booktabs,aliascnt,flafter}
+\usepackage[dvipsnames]{xcolor}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta,calc,positioning}
+\usepackage{needspace}
+\usepackage{hyperref}
+\usepackage[nameinlink,noabbrev,capitalise]{cleveref}
+\hypersetup{colorlinks=true,linkcolor=MidnightBlue,citecolor=MidnightBlue,
+ urlcolor=MidnightBlue,pdftitle={A Modulus Proof of Cannon’s Conjecture},
+ pdfauthor={OpenAI},bookmarksdepth=2}
+\numberwithin{equation}{section}
+\newtheorem{theorem}{Theorem}[section]
+\newaliascnt{proposition}{theorem}
+\newtheorem{proposition}[proposition]{Proposition}
+\aliascntresetthe{proposition}
+\newaliascnt{lemma}{theorem}
+\newtheorem{lemma}[lemma]{Lemma}
+\aliascntresetthe{lemma}
+\newaliascnt{corollary}{theorem}
+\newtheorem{corollary}[corollary]{Corollary}
+\aliascntresetthe{corollary}
+\theoremstyle{definition}
+\newaliascnt{definition}{theorem}
+\newtheorem{definition}[definition]{Definition}
+\aliascntresetthe{definition}
+\theoremstyle{remark}
+\newaliascnt{remark}{theorem}
+\newtheorem{remark}[remark]{Remark}
+\aliascntresetthe{remark}
+\crefname{lemma}{Lemma}{Lemmas}
+\Crefname{lemma}{Lemma}{Lemmas}
+\crefname{proposition}{Proposition}{Propositions}
+\Crefname{proposition}{Proposition}{Propositions}
+\crefname{corollary}{Corollary}{Corollaries}
+\Crefname{corollary}{Corollary}{Corollaries}
+\crefname{definition}{Definition}{Definitions}
+\Crefname{definition}{Definition}{Definitions}
+\crefname{remark}{Remark}{Remarks}
+\Crefname{remark}{Remark}{Remarks}
+\newcommand{\RR}{\mathbb R}
+\newcommand{\NN}{\mathbb N}
+\newcommand{\ZZ}{\mathbb Z}
+\newcommand{\HH}{\mathbb H}
+\newcommand{\Sphere}{\mathbb S^2}
+\newcommand{\cF}{\mathcal F}
+\newcommand{\cQ}{\mathcal Q}
+\newcommand{\Qhat}{\widehat{\mathcal Q}}
+\DeclareMathOperator{\Mod}{Mod}
+\DeclareMathOperator{\osc}{osc}
+\DeclareMathOperator{\diam}{diam}
+\DeclareMathOperator{\dist}{dist}
+\DeclareMathOperator{\id}{id}
+\DeclareMathOperator{\Isom}{Isom}
+\DeclareMathOperator{\Stab}{Stab}
+\setlist{topsep=4pt,itemsep=2pt,parsep=0pt}
+\setlength{\emergencystretch}{1em}
+
+\title{A Modulus Proof of Cannon's Conjecture}
+\author{OpenAI}
+\date{September 23, 2026}
+
+\begin{document}
+\maketitle
+\begin{abstract}
+We prove that every hyperbolic group whose boundary is homeomorphic to
+the two-sphere admits a proper cocompact isometric action on hyperbolic
+three-space with finite kernel. This resolves Cannon's conjecture
+positively.
+\end{abstract}
+
+\setcounter{tocdepth}{1}
+\tableofcontents
+\newpage
+
+\section{Introduction}
+\label{sec:introduction}
+
+Cannon's conjecture asks whether a hyperbolic group whose boundary is a
+topological two-sphere acts geometrically on hyperbolic three-space.
+It links the topology at infinity of a group to its three-dimensional
+geometry. Here a group is \emph{hyperbolic} if a Cayley graph for a
+finite generating set has uniformly thin geodesic triangles; its boundary
+consists of asymptotic classes of geodesic rays, with the usual visual
+topology. An isometric action is \emph{geometric} if it is proper and
+cocompact.
+
+The following theorem resolves Cannon's conjecture positively.
+
+\begin{theorem}[Cannon's conjecture]
+\label{thm:main}
+Let $G$ be a hyperbolic group with $\partial G$ homeomorphic to $\Sphere$.
+There is a homomorphism $\rho:G\to\Isom(\HH^3)$ for which the action
+on $\HH^3$ is proper and cocompact. Its kernel is finite.
+\end{theorem}
+
+Properness means that $\{g\in G:\rho(g)K\cap K\ne\varnothing\}$ is
+finite for every compact $K\subset\HH^3$; cocompactness means that one
+compact set has translates covering $\HH^3$. A finite kernel is allowed, and the image may contain orientation-reversing
+isometries. In particular, the theorem does not require a choice of
+orientation preserved by $G$.
+
+For torsion-free $G$, the quotient is a closed hyperbolic
+three-manifold, possibly nonorientable. We prove this consequence and
+the resulting virtual-structure statements in \Cref{sec:consequences}.
+
+\paragraph{History and the classical reduction.}
+Cannon's 1994 combinatorial Riemann mapping theorem grew out of the
+problem of finding an analytic structure on a sphere supplied only
+with combinatorial boundary data. It recovers quasiconformal
+coordinates from discrete annular moduli \cite[pp.~155--160]{Cannon}.
+In their 1998 paper, Cannon--Swenson stated the group-boundary conjecture
+and related a geometric action to conformality of the canonical boundary coverings
+\cite[Theorem 5.1 and Conjecture 5.1]{CannonSwenson}.
+Cannon--Floyd--Parry showed in 1999 that it suffices to find, around
+each point in every prescribed neighborhood, an annulus whose discrete
+moduli for separating curves stay bounded away from zero
+\cite[Theorem 8.2]{CFP}.
+Their finite subdivision rules provide a related framework for recursive
+planar tilings \cite{CFPFiniteSubdivision}.
+These works turn the topological problem into quantitative control of
+crossings at small scales.
+
+Bonk--Kleiner's uniformization theory gives a metric formulation of
+this program \cite{BK}. We use the precise criterion in
+Bourdon--Kleiner \cite[Corollary 3.5]{BourdonKleiner}: on an
+approximately self-similar metric sphere, a uniform bound for the
+combinatorial $2$-modulus of all continuous curves of a fixed sufficiently
+small minimum diameter yields a quasi-M\"obius parametrization by the
+round sphere. The visual boundary has the required self-similarity.
+Thus the remaining analytic task is an upper modulus bound for its
+fixed-scale coverings. The criterion and its hypotheses are verified
+in \Cref{thm:uniformization}.
+
+A related route uses \emph{Ahlfors regular conformal dimension}: the
+infimum of the Hausdorff dimensions of Ahlfors regular metrics
+quasisymmetric to the visual metric. This is a version of the
+boundary invariant introduced by Pansu \cite{Pansu1989}.
+Bonk--Kleiner proved Cannon's conjecture when that infimum is attained
+\cite[Theorem 1.1]{BKConformalDimension}; their argument connects the
+minimizing metric to quantitative curve families through the Loewner
+property \cite[Theorem 1.3]{BKConformalDimension}. The existence of a
+minimizing metric is a hypothesis of that criterion.
+
+The algebraic approaches connect surface subgroups, cubulations, and
+Kleinian realizations. Kahn--Markovi\'c constructed almost geodesic
+surface subgroups in closed orientable hyperbolic three-manifold groups
+\cite[Theorems 1.1--1.2]{KahnMarkovic}. Bergeron--Wise used this
+abundance to cubulate those groups, and more generally proved that
+quasiconvex codimension-one subgroups separating every distinct boundary
+pair yield a proper cocompact cubical action
+\cite[Theorem 1.4 and Section 5]{BergeronWise}.
+Agol proved virtual specialness for hyperbolic groups with such an
+action \cite[Theorem 1.1]{Agol}; these results give the consequences
+recorded in \Cref{sec:consequences} once a hyperbolic three-manifold
+realization is known.
+
+In the recognition direction, Markovi\'c showed that enough
+quasiconvex surface subgroups separating every boundary pair give a
+Kleinian realization for groups whose sphere-boundary action is faithful
+and orientation-preserving \cite[Theorem 1.1]{Markovic}.
+Ha\"issinsky established a virtual Kleinian criterion for cubulated
+hyperbolic groups with planar boundary \cite[Theorem 1.10]{Haissinsky}.
+More recently, Groves, Ha\"issinsky, Manning, Osajda, Sisto, and Walsh
+constructed group-theoretic drillings and showed that the Toral Relative
+Cannon conjecture implies a virtual Kleinian realization for every
+residually finite hyperbolic group with sphere boundary
+\cite[Theorems A--B and Corollary C]{Drilling}.
+The present proof follows the analytic route, establishing the required
+modulus bound directly from hyperbolicity and the topology of the boundary.
+
+After boundary uniformization, the invariant-conformal-structure method
+of Sullivan and the detailed circumcenter construction of Tukia
+straighten the action \cite{Sullivan,Tukia}.
+We state the analytic inputs at their point of use and include the
+short argument with unoriented conformal structures, so that
+orientation-reversing elements are retained. Properness and
+cocompactness then follow from the boundary action on ordered triples
+(\Cref{prop:action}).
+
+\paragraph{The proof in outline.}
+Fix a visual metric on $Z=\partial G$. At scale $a^{-n}$, where $a>1$,
+cover $Z$ by balls centered on a separated net. Assign nonnegative
+weights to the balls so that every path of a fixed minimum diameter
+meets balls of total weight at least one. Let $M(n)$ be the least sum
+of squared weights. The classical criterion reduces the theorem to
+\[
+ \sup_n M(n)<\infty.
+\]
+Assume this fails. Record indices permit comparison of $M(N-l)$
+with $M(N)$, and lead to two growth cases.
+
+First, crossing duality in a coordinate rectangle produces weights
+with squared sum $O(M(N)^{-1})$. A planar regularization turns them
+into continuous functions $H_N$ on a slightly enlarged rectangle,
+with separating level continua of uniformly positive diameter.
+Annular estimates give a nonconstant locally uniform limit $u$ and
+a finite measure $\mu$ such that
+\[
+ \bigl(\osc_{\overline B(x,s)}u\bigr)^2
+ \le C\mu\bigl(\overline B(x,C_\mu s)\bigr).
+\]
+The approximants retain vanishing local squared oscillation sums
+after every \emph{fixed} group transformation. A finite-cell
+inequality compares two such functions by sweeping through the
+separating levels. This is the analytic construction in
+\Cref{sec:scalar}.
+
+The growth of the record moduli determines how we use this scalar
+limit. Exponential growth makes $u$ locally H\"older, which allows
+normalized magnifications of its variation to converge to a nonconstant
+function on a punctured sphere. The level comparison forces each transform with a different
+puncture to be constant on the variation set of the first. A second
+magnification contradicts this constraint (\Cref{sec:exponential}).
+
+For subexponential growth, we use the measure bound to distribute
+variation among many balls. A tree of separated balls carries nested
+value intervals and a probability flow. The transition probabilities
+obey linear and quadratic bounds in the ratio of child to parent
+interval lengths. These bounds force a definite total probability,
+at arbitrarily deep levels, onto nodes with two separated regions of variation
+(\Cref{sec:flow}). At a common depth, many expansions have the same
+two target regions but distinct punctures. Comparing them in pairs
+charges more mass to small balls than their overlap allows
+(\Cref{sec:pairs}). This excludes the remaining case.
+
+Discrete crossing duality is
+part of the extremal-length theory developed in
+\cite{Schramm,CFPSquaring}. The finite incidence-vector identity is
+also the probabilistic modulus formula of
+Albin--Poggi-Corradini \cite[Theorem 5.1]{AlbinPoggiCorradini}; we
+reprove it in the notation of our cell covers. Pansu's coarse conformal geometry uses
+image-diameter energies and condenser capacities \cite[\S6]{Pansu1989}.
+Scalar extremals and separating levels also enter Rajala's
+uniformization of metric surfaces \cite[Sections 4--6]{RajalaUniformization}.
+Here two path envelopes produce the required continua while preserving
+finite-cell oscillation bounds. The subsequent level integrations and
+pair counts are proved directly for the visual boundary. The two growth
+branches use this same scalar construction, and their combination
+finishes the theorem in \Cref{sec:completion}.
+
+\paragraph{Conventions.}
+A path is a continuous map of a compact interval, identified with its
+image when convenient. A continuum is a
+nonempty compact connected set; it need not be path connected.
+$B(x,r)$ is open and $\overline B(x,r)=\{y:d(x,y)\le r\}$ is closed.
+For real $f$, write $\osc_A f=\sup_A f-\inf_A f$.
+Constants in $\lesssim$ and $\asymp$ are independent of the scale
+indices being varied. Dependence on a compact region, a fixed group
+element, or a subdivision parameter is specified where needed.
+
+\input{sections/01_geometry}
+\input{sections/02_scalar}
+\input{sections/03_exponential}
+\input{sections/04_flow}
+\input{sections/05_pairs}
+
+\section{Completion of the proof}
+\label{sec:completion}
+
+\begin{proof}[Proof of \Cref{thm:main}]
+If $M(n)$ were unbounded, the record selection of \Cref{sec:scalar}
+would give either exponential records, excluded by
+\Cref{prop:no-exponential-records}, or subexponential growth,
+excluded by \Cref{prop:no-subexponential-records}. Thus $\sup_n M(n)<\infty$.
+\Cref{thm:uniformization} uniformizes the boundary, and
+\Cref{prop:action} supplies the proper cocompact action with finite kernel.
+\end{proof}
+
+\section{Consequences}
+\label{sec:consequences}
+
+\begin{corollary}
+\label{cor:torsion-free}
+If $G$ is a torsion-free hyperbolic group and $\partial G$ is homeomorphic to $\Sphere$,
+then $G$ is the fundamental group of a closed hyperbolic three-manifold.
+\end{corollary}
+\begin{proof}
+Point stabilizers in a proper action are finite, hence trivial when
+$G$ is torsion-free. Thus the action in \Cref{thm:main} is free, and
+$\HH^3/G$ is a closed hyperbolic three-manifold with fundamental group
+$G$. The manifold may be nonorientable.
+\end{proof}
+
+\begin{corollary}[Virtual structure in the torsion-free case]
+\label{cor:virtual-structure}
+Let $G$ be a torsion-free hyperbolic group with $\partial G$ homeomorphic
+to $\Sphere$. Some finite-index subgroup of $G$ is isomorphic to
+$\pi_1(\Sigma)\rtimes\ZZ$ for a closed connected orientable surface
+$\Sigma$. In particular, $G$ has positive virtual first Betti number.
+Moreover, $G$ is virtually compact special and residually finite.
+\end{corollary}
+\begin{proof}
+Let $M^+$ be the orientation cover of the closed hyperbolic manifold in
+\Cref{cor:torsion-free}, using the manifold itself when it is orientable,
+and let $G^+=\pi_1(M^+)$. Thus $G^+$ is normal of index at most two in
+$G$. Agol's virtual-fibering theorem
+\cite[Theorem 9.2]{Agol} gives a connected finite cover $N$ of $M^+$
+fibering over the circle with closed connected orientable fiber $\Sigma$.
+The fibration sequence gives
+$\pi_1(N)\cong\pi_1(\Sigma)\rtimes\ZZ$, with action induced by the
+monodromy, and its surjection onto $\ZZ$ gives $b_1(\pi_1(N))\ge1$.
+
+Bergeron--Wise's cubulation theorem \cite[Theorem 5.3]{BergeronWise}
+gives a proper cocompact action of $G^+$ on a CAT(0) cube complex, free
+because $G^+$ is torsion-free. Agol's virtual-special theorem
+\cite[Theorem 1.1]{Agol} then gives a finite-index subgroup of $G^+$
+with compact special quotient, so $G$ is virtually compact special.
+Agol's separability corollary \cite[Corollary 1.3]{Agol}, applied to the
+trivial subgroup, also makes $G^+$ residually finite. The finite quotient
+$G/G^+$ and intersections of the finitely many $G$-conjugates of
+finite-index normal subgroups of $G^+$ then separate every nontrivial
+element of $G$.
+\end{proof}
+
+\input{bibliography}
+\end{document}
diff --git a/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/build/sections/01_geometry.tex b/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/build/sections/01_geometry.tex
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+\section{Boundary geometry and the modulus criterion}
+\label{sec:geometry}
+
+We collect the geometric estimates used throughout the proof, then state
+the classical uniformization input. Let $\Gamma$ be the Cayley graph of
+$G$ for a finite symmetric generating set, with unit edge lengths. Write
+$e$ for its identity vertex and $|v|=d_\Gamma(e,v)$.
+The Gromov product is
+\[
+ (y|z)_o=\tfrac12\bigl(d_\Gamma(o,y)+d_\Gamma(o,z)-d_\Gamma(y,z)\bigr).
+\]
+Boundary products are defined by limits along rays, up to a uniformly
+bounded additive error. A visual metric $d$ on $Z=\partial G$ satisfies
+\begin{equation}
+ C_v^{-1}a^{-(y|z)_e}\le d(y,z)\le C_va^{-(y|z)_e},
+ \qquad y\ne z,
+ \label{eq:visual}
+\end{equation}
+for some $a>1$. Set $r_k=a^{-k}$ for integers $k\ge0$.
+We use the standard product inequality
+$(y|z)_e\ge\min\{(y|w)_e,(w|z)_e\}-O(1)$ and its geometric form:
+geodesics with product at least $j-C$ stay a bounded synchronous
+distance apart through depth $j$, provided both have that length.
+The bound depends on $C$ and the
+hyperbolicity constant. These facts, visual metrics, and thin ideal
+triangles are recalled in \cite[III.H.3]{BH}.
+
+\subsection{Expansions}
+
+Left multiplication by the inverse of a vertex far along a ray magnifies
+a small boundary ball. The following formula also controls the complement
+of that ball; this second feature will be essential.
+
+\begin{lemma}[Expansion formula]
+\label{lem:expansion}
+Let $v$ be the depth-$k$ vertex on a based ray to $x\in Z$, put
+$h=v^{-1}$ and $R=r_k$, and set $D_y=\max\{R,d(x,y)\}$.
+There are constants $c_E,C_E>0$, independent of these choices, such that
+\begin{equation}
+ c_E\frac{R\,d(y,z)}{D_yD_z}
+ \le d(hy,hz)\le C_E\frac{R\,d(y,z)}{D_yD_z}.
+ \label{eq:expansion}
+\end{equation}
+In particular, on $B(x,CR)$ the map $h$ is bi-Lipschitz with scale
+factor $R^{-1}$ and distortion depending only on $C$.
+If $d(x,z_0)\ge\varepsilon_*>0$, $R<\varepsilon_*/2$, and $p=hz_0$, then
+\begin{equation}
+ d(hy,p)\lesssim R/D_y\quad(y\in Z),\qquad
+ d(hy,p)\asymp R/D_y\quad(d(x,y)\le\varepsilon_*/2),
+ \label{eq:puncture}
+\end{equation}
+and
+\begin{equation}
+ (v^{-1}|p)_e\ge k-C(\varepsilon_*).
+ \label{eq:inverse-prefix}
+\end{equation}
+The constants in the last two estimates may depend on $\varepsilon_*$.
+\end{lemma}
+\begin{proof}
+Base change and thin triangles give
+\[
+ (y|z)_v=(y|z)_e+k-(v|y)_e-(v|z)_e+O(1),
+ \qquad (v|y)_e=\min\{k,(x|y)_e\}+O(1).
+\]
+Thus $a^{-(v|y)_e}\asymp D_y$. Exponentiating the first identity
+and using $(hy|hz)_e=(y|z)_v+O(1)$ proves
+\eqref{eq:expansion}. With $z=z_0$, the factor $D_{z_0}$ is bounded
+above and below, and $d(y,z_0)$ is bounded above. When
+$d(x,y)\le\varepsilon_*/2$, the latter is also bounded below.
+This proves \eqref{eq:puncture}. Finally,
+$(v^{-1}|v^{-1}z_0)_e=k-(v|z_0)_e+O(1)$, and
+$(v|z_0)_e$ is bounded in terms of $\varepsilon_*$.
+\end{proof}
+
+\begin{lemma}[Prefixes]
+\label{lem:prefix}
+Every vertex $v$ lies a uniformly bounded distance from a depth-$|v|$
+vertex on a based ray. With that ray's endpoint as $x$,
+\eqref{eq:expansion} remains valid for $h=v^{-1}$ and $R=r_{|v|}$,
+with uniform constants. Moreover:
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item If $d(\xi,\eta)\le Cr_j$, depth-$j$ vertices on any based
+rays to $\xi,\eta$ are a bounded distance apart, depending only on $C$.
+\item If $|v|=k$ and $(v|\xi)_e\ge k-C$, every geodesic from $e$
+to $v$ stays a bounded synchronous distance from a ray to $\xi$
+through depth $k$.
+\end{enumerate}
+\end{lemma}
+\begin{proof}
+Translate a bi-infinite geodesic so that it passes through $v$.
+The ideal triangle with third vertex $e$ puts $v$ uniformly near one
+of its based sides; the nearby parameter differs from $|v|$ by a
+bounded amount. A bounded change of basepoint changes products by a
+bounded amount, preserving \eqref{eq:expansion}. The two remaining
+assertions follow from \eqref{eq:visual} and the product estimate.
+\end{proof}
+
+Each fixed $g\in G$ acts bi-Lipschitzly on $Z$, with a constant
+depending on $g$. The whole action is uniformly quasi-M\"obius:
+base-change factors cancel in distance cross-ratios, giving a single
+$C\ge1$ such that
+\[
+ C^{-1}\frac{d(x,z)d(y,w)}{d(x,w)d(y,z)}
+ \le\frac{d(gx,gz)d(gy,gw)}{d(gx,gw)d(gy,gz)}
+ \le C\frac{d(x,z)d(y,w)}{d(x,w)d(y,z)}
+\]
+for every distinct quadruple and every $g$.
+
+\subsection{Packing and paths}
+
+\begin{lemma}[Packing]
+\label{lem:packing}
+There are $Q\ge2$ and $C_P<\infty$ such that an $s$-separated subset
+of a ball of radius $t$ has at most $C_P(t/s)^Q$ points whenever
+$0<s\le t\le\diam Z$. In particular, $Z$ is doubling.
+\end{lemma}
+\begin{proof}
+Choose $k$ with $r_k\asymp t$. Rays ending in the given ball have
+their depth-$k$ vertices in a uniformly bounded graph ball, by
+\Cref{lem:prefix}. At depth $k+j$ there are at most $CA^j$ possible
+continuations, where $A>1$ depends on the valence. Endpoints whose
+rays share such a vertex are at visual distance at most $C'r_{k+j}$.
+Choose the least $j\ge0$ with $C'r_{k+j}<s$. The desired bound
+follows with $Q\ge\max\{2,\log_a A\}$. Coarse scales are absorbed
+by increasing the constant.
+\end{proof}
+
+\Needspace{5\baselineskip}
+\begin{lemma}[Controlled paths]
+\label{lem:paths}
+There is $L\ge1$ such that any two points of $\overline B(x,s)$
+join by a path in $B(x,Ls)$. Any two points of a continuum can also
+be joined by a path in any prescribed open neighborhood of it.
+\end{lemma}
+\begin{proof}
+For small $s$, expand at $x$ with $R\asymp s$ and choose $z_0$ a
+uniformly positive distance from $x$; two fixed distinct points
+suffice for this choice. By \eqref{eq:puncture}, the two image
+points stay a uniform positive distance from $p=hz_0$.
+Fix a homeomorphism $\phi:Z\to\Sphere$. Uniform continuity of
+$\phi^{-1}$ places their spherical images a uniform distance from
+$\phi(p)$. Join them outside a smaller round ball about $\phi(p)$.
+Uniform continuity of $\phi$ makes the pullback path stay a uniform
+$d$-distance from $p$. The upper bound in \eqref{eq:puncture}
+puts its preimage under $h$ in $B(x,Ls)$. Increase $L$ for the
+remaining scales, using path connectedness of $Z$.
+
+For a continuum $C$ in an open set $V$, choose $\varepsilon>0$
+with its $L\varepsilon$-neighborhood in $V$. Connectedness supplies
+a finite $\varepsilon$-chain in $C$ between any two specified
+points: the set reachable by such chains is relatively clopen.
+Join consecutive points by the first assertion.
+\end{proof}
+
+\subsection{The classical reduction}
+
+Choose a maximal $r_n$-separated set $T_n\subset Z$ and its cells
+$B(q,2r_n)$, $q\in T_n$. These cover $Z$; fixed enlargements have
+bounded overlap. For a path family $\mathcal F$, define
+\[
+ \Mod_2(\mathcal F,T_n)=\min_{\omega\ge0}
+ \left\{\sum_{q\in T_n}\omega(q)^2:
+ \sum_{B(q,2r_n)\cap\pi\ne\varnothing}\omega(q)\ge1
+ \text{ for every }\pi\in\mathcal F\right\}.
+\]
+Only finitely many incidence vectors occur, so the minimum exists.
+Fix a sufficiently small $d_0>0$ and put
+\begin{equation}
+ \mathcal F_0=\{\pi:\diam\pi\ge d_0\},\qquad
+ M(n)=\Mod_2(\mathcal F_0,T_n).
+ \label{eq:macro-modulus}
+\end{equation}
+We require $d_0<\diam Z/10$ and choose it so that every path of
+diameter at least $R/2$ in $\overline B(x,3R)$ has image diameter
+at least $d_0$ under a scale-$R$ expansion. This is possible by
+\eqref{eq:expansion}. We also impose the smallness requirement in
+\Cref{thm:uniformization} before fixing $d_0$ and $M(n)$.
+The family is nonempty, and $0<M(n)<\infty$: all-one weights are
+admissible, while any one path gives the lower bound $1/\#T_n$.
+
+\begin{theorem}[Bonk--Kleiner; Bourdon--Kleiner]
+\label{thm:uniformization}
+If $\sup_n M(n)<\infty$, then $(Z,d)$ is quasi-M\"obius
+homeomorphic to the round sphere.
+\end{theorem}
+\begin{proof}[Verification of the cited criterion]
+Corollary 3.5 of \cite{BourdonKleiner}, derived from
+\cite[Theorem 10.4]{BK}, applies to an arcwise connected,
+approximately self-similar metric sphere. Approximate self-similarity
+means that each open ball, with distances divided by its radius,
+is uniformly bi-Lipschitz to an open subset of the space.
+\Cref{lem:expansion} supplies this property; intermediate and large
+radii change only the constant.
+
+The criterion assumes bounded $2$-moduli of paths of diameter at
+least a sufficiently small $d_0$ in successive graph approximations.
+For a dyadic scale $2^{-m}$ choose $n$ with
+$2^{-m}/a<r_n\le2^{-m}$. Our cells form a fixed $\kappa$-approximation
+for any $\kappa>2a$: the inner balls of radius $2^{-m}/\kappa$
+are disjoint, each is contained in its cell, and each cell is
+contained in the radius-$\kappa2^{-m}$ ball about its center.
+The modulus is exactly $M(n)$. Finitely many coarse scales have
+finite modulus. No Coxeter-group assumption occurs in this criterion.
+\end{proof}
+
+\begin{proposition}[From the boundary to hyperbolic space]
+\label{prop:action}
+If $(Z,d)$ is quasi-M\"obius homeomorphic to the round sphere,
+then $G$ admits a proper cocompact isometric action on $\HH^3$
+with finite kernel.
+\end{proposition}
+\begin{proof}
+Let $\psi:Z\to\Sphere$ be the uniformizing homeomorphism and write
+$\mathcal G=\{\psi g\psi^{-1}:g\in G\}$. The boundary cross-ratio estimate
+and the two fixed maps $\psi,\psi^{-1}$ give one quasi-M\"obius distortion
+function for every element of $\mathcal G$. Fix $f\in\mathcal G$ and $x\in\Sphere$,
+keep a fourth point away from $x$, and let two points at equal spherical
+distance from $x$ approach $x$. In the cross-ratio inequality the factors
+involving the fourth point tend to one, both before and after applying
+$f$. Thus the infinitesimal linear distortion has a bound independent
+of $f$ and $x$; the radius at which the estimate is accurate need not
+be uniform in $f$. The metric characterization of planar
+quasiconformality, applied in conformal charts, makes each $f$
+quasiconformal. At its almost-everywhere nonsingular differentiability
+points, the same infinitesimal bound controls the ratio of singular values
+of $Df$, and hence its analytic dilatation. This gives a common
+quasiconformal bound. For a reversing map apply the characterization
+after a conformal reflection; see \cite[Theorems 2.5.3--2.5.4]{AIM}
+and the metric characterization in \cite[Theorem 1.4 and Remark 1.8(b)]{HK}.
+
+We recall explicitly the invariant-structure argument, including
+orientation reversal. This is the invariant-conformal-structure method of
+Sullivan, with the circumcenter construction detailed by Tukia
+\cite[pp.~73--76]{Tukia}. Analytic quasiconformal maps have
+almost-everywhere defined nonsingular weak differentials, and they and
+their inverses preserve null sets \cite[Theorem 3.1.2]{AIM}.
+Their differentials satisfy the almost-everywhere composition rule
+\cite[Lemma 10 and Theorem 5]{AhlforsBers}.
+Since $\mathcal G$ is countable, remove the $\mathcal G$-orbit
+of the union of all exceptional sets for these assertions and for all
+pairs of maps. Its complement $X$ is invariant and conull. At $x\in X$,
+let $\mathcal C_x$ be the space of positive definite quadratic forms on
+$T_x\Sphere$ modulo positive scalars. This is a hyperbolic plane, and
+pullback by any invertible linear map, of either orientation, is an
+isometry between the corresponding fibers
+\cite[Section 2.3, equations (2.17)--(2.18) and Theorem 2.3.2]{AIM}. The orbit
+$\{f^*q_{\rm round}(x):f\in\mathcal G\}\subset\mathcal C_x$ is uniformly
+bounded, by the common quasiconformal bound, and contains the round
+form. Its unique hyperbolic circumcenter defines a uniformly bounded
+measurable conformal structure $q(x)$. Indeed, circumcenters of the
+increasing finite subsets from an enumeration are measurable and
+converge to that circumcenter: their radii increase, their centers stay
+in a fixed compact ball, and every subsequential limit is the unique
+center of the whole orbit. The composition rule permutes this orbit
+under pullback, so $f^*q=q$ on $X$ for every $f\in\mathcal G$. Extend $q$
+arbitrarily across the null complement of $X$.
+
+The measurable Riemann mapping theorem
+\cite[Theorem 6]{AhlforsBers} gives a quasiconformal
+homeomorphism $F:\Sphere\to\Sphere$ with $F^*q_{\rm round}=q$ almost
+everywhere. Invariance and the analytic composition rule show that
+every $FfF^{-1}$ is conformal or anticonformal, according to the
+orientation of $f$. Such sphere homeomorphisms are M\"obius or
+anti-M\"obius. Their unique extensions to hyperbolic isometries therefore
+define a homomorphism $\rho:G\to\Isom(\HH^3)$.
+
+For completeness, properness and cocompactness follow directly
+from the action on ordered distinct triples $\mathcal T(Z)$.
+On any compact set of triples all pairwise distances are bounded
+below. For $|v|\to\infty$, at least two points of such a triple
+stay away from the expansion center for $v^{-1}$; their image
+distance is $O(r_{|v|})$. Thus $v^{-1}K\cap K=\varnothing$
+eventually for every compact $K\subset\mathcal T(Z)$.
+Conversely, expand at one point of a closest pair with scale
+comparable to their distance. Formula \eqref{eq:expansion} makes
+all three image distances uniformly positive: if the third point
+is nearby, all denominator factors have that scale; otherwise
+its distances to the pair cancel its denominator factor.
+Triples already uniformly separated need no expansion.
+Hence the action on triples is proper and cocompact.
+
+These properties survive boundary conjugacy. Let $I=\Isom(\HH^3)$.
+The stabilizer $H$ of a standard ordered boundary triple has two
+elements: the identity and reflection in its hyperbolic plane.
+Choose $o\in\HH^3$ fixed by $H$, and let $K_o$ be its compact
+stabilizer in $I$. The equivariant map
+\[
+ \mathcal T(\Sphere)\cong I/H\longrightarrow I/K_o\cong\HH^3
+\]
+is continuous, onto, and proper: isometries sending $o$ into a
+compact set form a compact set. Properness and cocompactness
+therefore descend. The kernel fixes $o$, so is finite.
+\end{proof}
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+\section{A scalar limit from unbounded modulus}
+\label{sec:scalar}
+
+Assume that the macroscopic moduli $M(n)$ are unbounded. Crossing
+duality first gives transverse weights of energy $O(M(n)^{-1})$ along
+a subsequence. We turn those weights into continuous functions whose
+level continua meet every path straddling the corresponding value.
+An annular estimate then yields a nonconstant limit $u$ and a finite
+measure controlling its squared oscillation. The approximating functions
+retain vanishing local oscillation energy after each fixed group
+transformation; later comparisons will use this property as well as the
+limit itself. Constants are independent of the approximation index;
+thresholds may depend on the fixed compact subset of the domain under
+consideration.
+
+\subsection{Record scales and crossing duality}
+
+If $\limsup M(n)^{1/n}>1$, choose
+$1<\lambda<\limsup M(n)^{1/n}$; otherwise put $\lambda=1$.
+In both cases $M(n)/\lambda^n$ is unbounded. Retain increasing record
+indices $N$ whose record values tend to infinity. Thus $M(N)\to\infty$ and
+\begin{equation}
+ M(N-l)\leq\lambda^{-l}M(N)\qquad(0\leq l\leq N).
+ \label{eq:records}
+\end{equation}
+This inequality survives every further subsequence.
+
+\begin{lemma}[Localization to a rectangle]
+\label{lem:rectangle-localization}
+There are coordinate rectangles
+$\cQ=[0,1]^2\subset\Qhat=[0,1]\times[-1,2]\subset Z$ and a
+subsequence of the record indices such that the left-to-right crossing
+modulus of $\cQ$ is at least $cM(N)$, with $c>0$ independent of $N$.
+\end{lemma}
+\begin{proof}
+Cover $Z$ by finitely many coordinate boxes $[-1,1]^2$, each lying in
+a chart containing $[-8,8]^2$ of diameter less than $d_0$. A macroscopic
+path meeting an inner box must leave its larger box. Follow it to its
+first meeting with $\partial[-2,2]^2$. If it exits through $x=2$,
+its segment after the last preceding meeting with $x=1$ crosses
+$[1,2]\times[-2,2]$ from left to right. The other three exit sides
+give analogous strips. Each strip extends by its height in both
+transverse directions inside $[-8,8]^2$.
+
+Consequently there are finitely many crossing families
+$\mathcal F_1,\ldots,\mathcal F_J$ such that every macroscopic path
+contains a member of one of them. Summing admissible weights gives
+\[
+ M(N)\leq J\sum_{j=1}^J\Mod_2(\mathcal F_j,T_N)
+       \leq J^2\max_j\Mod_2(\mathcal F_j,T_N).
+\]
+One strip realizes this bound along a subsequence. Affine coordinates
+give the asserted rectangles.
+\end{proof}
+
+The edge $y=0$ is the top of $\cQ$. The rectangles
+$[0,1]\times[-1,0]$ and $[0,1]\times[1,2]$ are its top and bottom
+bands, respectively. Coordinates serve only interpolation and topology:
+all distances refer to $d$.
+
+\begin{lemma}[Crossings and blockers]
+\label{lem:planar-blocker}
+In a closed coordinate rectangle:
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item every continuum meeting both lateral edges meets every
+top-to-bottom path;
+\item every compact set meeting all top-to-bottom paths contains
+a continuum meeting both lateral edges.
+\end{enumerate}
+\end{lemma}
+\begin{proof}
+For (i), suppose a lateral continuum and a top-to-bottom path are
+disjoint. Enlarge the rectangle, extending the continuum horizontally
+and the path vertically to the corresponding new edges. The extensions
+remain disjoint: a new horizontal and vertical segment can meet only at
+an original corner, already a common point of the original sets.
+Approximate the extended path, within its positive clearance from the
+continuum, by a polygonal path with interior in the enlarged rectangle.
+After subdivision at intersections it contains a simple polygonal arc
+between the top and bottom edges. The Jordan curve theorem, applied
+after adjoining a boundary route, shows that this arc separates the
+lateral edges. The connected continuum cannot avoid it.
+
+For (ii), subdivide into finer square grids and mark the closed squares
+meeting the blocker $K$. No edge-adjacent chain of unmarked squares
+joins top to bottom. Label $A$ the unmarked squares reachable from the
+top through unmarked edge-adjacent squares, and append a top row of
+$A$ squares. Label all remaining squares and an appended bottom row $B$.
+Draw the interfaces between unlike labels. Interior vertices have even
+degree; each lateral boundary has an odd number of ends, since its
+labels begin with $A$ and end with $B$. There are no top or bottom ends.
+Some interface component therefore joins the lateral boundaries:
+otherwise every component contributes an even number of ends to the
+only lateral boundary it meets.
+
+Along a path in this component, the adjacent $B$ squares are original
+marked squares. An adjacent unmarked square would be reachable, while
+an interface with the appended bottom row would give an unmarked
+top-to-bottom chain. Successive $B$ squares meet at least at a corner.
+Their union is a connected compact lateral crossing. A Hausdorff limit
+as the mesh tends to zero is connected, meets both lateral edges, and
+lies in $K$.
+\end{proof}
+
+The next finite-dimensional argument belongs to discrete extremal
+length. Cannon--Floyd--Parry describe optimal weights through crossing
+vectors and their pairing with transverse cuts
+\cite[Sections 2.3 and 2.4.3]{CFPSquaring}; the probability formulation
+of Albin--Poggi-Corradini gives the same least-norm incidence-vector
+identity \cite[Theorem 5.1]{AlbinPoggiCorradini}. We give the argument
+for the cell families used here.
+
+\begin{lemma}[Crossing dual weights]
+\label{lem:dual-weights}
+For each retained $N$ there are nonnegative weights
+$\rho_q=\rho_q^{(N)}$, $q\in T_N$, admissible for top-to-bottom
+crossings of $\cQ$, such that
+\begin{equation}
+ \sum_{q\in T_N}\rho_q^2\leq \frac{C}{M(N)}.
+ \label{eq:dual-energy}
+\end{equation}
+\end{lemma}
+\begin{proof}
+Let $P$ minimize Euclidean norm in the convex hull of the finitely many
+cell-incidence vectors $v$ of left-to-right crossings. Its coordinates
+are nonnegative and sum to at least one, so $P\ne0$. Minimality gives
+$\langle P,v\rangle\geq\|P\|^2$, making $P/\|P\|^2$ admissible.
+Conversely any admissible $\omega$ satisfies
+$\langle\omega,P\rangle\geq1$. Thus the left-to-right modulus is
+$\|P\|^{-2}$. By \Cref{lem:planar-blocker}(i), each top-to-bottom
+incidence vector has inner product at least one with every $v$, hence
+with $P$. Set $\rho=P$ and apply \Cref{lem:rectangle-localization}.
+\end{proof}
+
+\subsection{Continuous functions with separating levels}
+
+For $\pi\subset\Qhat$ write
+$\ell_\rho(\pi)=\sum_{q:B(q,2r_N)\cap\pi\ne\varnothing}\rho_q$.
+For $x\in\cQ$, let $b_N(x)$ be the infimum of this cost over paths
+from the top edge to $x$ in $\cQ$. Define the raw function
+$\min\{1,b_N\}$, assigning exactly zero on the top edge, and extend
+it by zero and one to the top and bottom bands. Admissibility gives
+value one on the bottom edge.
+
+Along every path in $\Qhat$, the difference of raw endpoint values is
+at most its cost. Inside $\cQ$ this follows by concatenating with
+nearly minimizing paths and clipping at one; a path beginning on the
+top edge is itself a competitor. A path visiting both bands contains
+a crossing of cost at least one. If it visits only the top band, each
+endpoint value is bounded by the cost of its segment to that band.
+If it visits only the bottom band, each endpoint value is at least one
+minus that segment's cost, by concatenation with a top-to-endpoint
+competitor. In either case the endpoint difference is bounded by the
+cost of the entire path.
+
+Triangulate $\Qhat$ by a finite coordinate grid containing the band
+boundaries, with every triangle of $d$-diameter less than $r_N$.
+Interpolate the raw vertex values affinely to obtain a continuous
+$U_N^0:\Qhat\to[0,1]$. It is constant on the bands. Continuity alone
+does not ensure that one selected level continuum meets every
+straddling path. The following two path envelopes give the strict
+sublevel and superlevel sets the connections to the bands needed for
+that conclusion. Define
+\[
+ F_N(x)=\inf_{\pi:x\to\mathrm{top}}\max_\pi U_N^0,
+ \qquad
+ H_N(x)=\sup_{\pi:x\to\mathrm{bottom}}\inf_\pi F_N,
+\]
+where paths remain in $\Qhat$ and end in the indicated band.
+
+\begin{lemma}[Regularization and level continua]
+\label{lem:level-continua}
+The functions $F_N,H_N$ are continuous, with the prescribed band values,
+and for every path $K\subset\Qhat$,
+\begin{equation}
+ \osc_K H_N\leq\osc_K F_N\leq\osc_K U_N^0.
+ \label{eq:flood-osc}
+\end{equation}
+For every $0<t<1$, the set $\{H_N=t\}\cap\cQ$ contains a lateral
+crossing continuum $C_{N,t}$ of diameter at least a fixed $d_1>0$.
+Every path in $\Qhat$ whose endpoint values strictly straddle $t$
+meets $C_{N,t}$.
+\end{lemma}
+\begin{proof}
+We have $F_N\geq U_N^0$ and $H_N\leq F_N$; constant paths give
+the band values. Concatenation with nearly optimizing paths gives,
+for $x,y\in K$,
+\[
+ F_N(x)\leq\max\{\max_K U_N^0,F_N(y)\},\qquad
+ H_N(y)\geq\min\{\inf_K F_N,H_N(x)\}.
+\]
+If $F_N(x)>F_N(y)$, the first inequality places their values between
+$\min_K U_N^0$ and $\max_K U_N^0$. If $H_N(x)>H_N(y)$, the
+second places them between $\inf_K F_N$ and $\sup_K F_N$.
+This proves \eqref{eq:flood-osc}. Short coordinate segments and
+continuity of $U_N^0$ then prove continuity of both functions,
+including at the boundary.
+
+Every point of $\{H_N>t\}$ joins the bottom band within that set:
+choose a path with $\inf F_N>t$ and use its tails. Likewise every
+point of $\{F_N<t\}$ joins the top within $\{F_N<t\}$.
+To connect a point $x$ with $H_N(x)<t\leq F_N(x)$ to the top,
+choose $H_N(x)<s<t$ and follow any top-going path until its first
+point with $F_N=s$. On this prefix $F_N\geq s$, so concatenation
+with nearly maximizing paths from any prefix point $z$ gives
+\[
+ H_N(x)\geq\min\{s,H_N(z)\}.
+\]
+Hence $H_N(z)\leq H_N(x)<t$ throughout the prefix. Its endpoint
+joins the top through $F_N<t$, hence through $H_N<t$. Thus both
+strict sides of the level join their respective bands, and then the
+outer top and bottom edges, without meeting the level.
+
+The compact set $\{H_N=t\}$ blocks all top-to-bottom paths by
+continuity and lies in $\cQ$. By \Cref{lem:planar-blocker}(ii) it
+contains a lateral continuum $C_{N,t}$; take $d_1$ to be the distance
+between the lateral edges of $\cQ$. Complete any straddling path to
+a top-to-bottom path using the strict-side connections. The completed
+path meets $C_{N,t}$ by \Cref{lem:planar-blocker}(i), and the added
+pieces avoid the level, proving the last assertion.
+\end{proof}
+
+Figure~\ref{fig:level-continuum} isolates the last step: the strict-side
+extensions cannot account for the intersection with the selected
+continuum, so the original straddling path must meet it.
+
+\input{figures/level-continuum}
+
+\subsection{Oscillation and annular estimates}
+
+The level continua now let us estimate oscillation by crossing an annulus.
+Energies formed from image diameters and their relation to condenser
+capacity appear in Pansu's coarse conformal geometry \cite[\S6]{Pansu1989};
+the estimates below use the finite-cell energy of our regularized scalar
+functions. Put $U=\operatorname{int}_Z\Qhat$. For a continuous function $f$ on
+an open set containing a compact set $K$, and $J\geq2$, define
+\[
+ \mathcal E_n^{(J)}(f;K)=
+ \sum_{\substack{q\in T_n\\B(q,2r_n)\cap K\ne\varnothing}}
+       \bigl(\osc_{\overline B(q,Jr_n)}f\bigr)^2.
+\]
+The closed balls lie in the domain for all sufficiently large $n$.
+
+\begin{lemma}[Local oscillation]
+\label{lem:local-oscillation}
+For fixed compact $K\subset U$ and $J>0$, uniformly for $x\in K$
+and sufficiently large $N$,
+\begin{equation}
+ \osc_{\overline B(x,Jr_N)}H_N
+ \leq\sum_{\substack{q\in T_N\\d(q,x)\leq C(J)r_N}}\rho_q.
+ \label{eq:local-osc}
+\end{equation}
+Here $C(J)$ depends only on $J$ and the ball path constant.
+Consequently, for every fixed $g\in G$, compact $K'\subset gU$,
+and $J\geq2$,
+\[
+ \mathcal E_N^{(J)}(H_N\circ g^{-1};K')
+       \leq\frac{C(g,K',J)}{M(N)}\longrightarrow0.
+\]
+\end{lemma}
+\begin{proof}
+Join two points of $\overline B(x,Jr_N)$ inside $B(x,LJr_N)$
+by \Cref{lem:paths}. Vertices of triangles meeting this path lie
+in $B(x,(LJ+1)r_N)$; any two such vertices join inside
+$B(x,L(LJ+1)r_N)\subset U$ for large $N$. The raw endpoint
+inequality bounds every difference of their values by the right side
+of \eqref{eq:local-osc}, with $C(J)=L(LJ+1)+3$. It therefore bounds
+the interpolant's oscillation along the original path. Apply
+\eqref{eq:flood-osc}.
+
+Squaring and summing costs a bounded factor by \Cref{lem:packing}:
+both the number of nearby weights and the number of centers charging
+each weight are bounded. The same holds for $br_N$-separated centers,
+with constants depending on $b>0$. A fixed $g$ is bi-Lipschitz,
+so inverse images of target balls lie in source balls of radius
+$C_gJr_N$, their centers are $b_gr_N$-separated, and all these source
+balls lie in a fixed compact subset of $U$. The summation bound and
+\eqref{eq:dual-energy} finish the proof.
+\end{proof}
+
+The mesh-scale estimate preserves the small dual energy. We next
+compare oscillation across a ball of radius $R=r_l$ with the dual
+energy in a surrounding annulus. Expansion takes that annulus to a
+fixed-size region and takes the mesh $r_N$ to $r_{N-l}$; this is why
+the next bound contains $M(N-l)$.
+
+\begin{lemma}[Annular estimate]
+\label{lem:annular}
+For every compact $K\subset U$ there are $R_*>0$ and an integer
+$m_0$ such that, if $x\in K$, $R=r_l\leq R_*$, and $N-l\geq m_0$,
+then any two points joined by a path in $B(x,R/2)$ have
+$H_N$-value difference $\Delta$ satisfying
+\begin{equation}
+ \Delta^2\leq C M(N-l)
+       \sum_{\substack{q\in T_N\\R/4\leq d(q,x)\leq4R}}\rho_q^2.
+ \label{eq:annular}
+\end{equation}
+The constant is uniform; $N$ is sufficiently large for the fixed
+interior region.
+\end{lemma}
+\begin{proof}
+Choose $R_*$ with $4R_*<d_1$ and all closed $5R_*$-balls about
+$K$ contained in $U$. All following restrictions on $m_0$ concern
+only $r_N/R=a^{-(N-l)}$.
+
+Expand at $x,l$ by $h$, and choose macroscopic admissible weights
+$\omega$ at scale $N-l$ with squared mass $M(N-l)$. Put
+\[
+ \mathcal A=\{q\in T_N:R/2\leq d(q,x)\leq3R\},\qquad
+ \sigma_q=\sum_{\substack{\xi\in T_{N-l}\\
+               d(\xi,hq)\leq C_0r_{N-l}}}\omega_\xi,
+ \quad q\in\mathcal A,
+\]
+where $C_0>C_E+2$. Expansion on $B(x,3R)$ separates the $hq$ by
+a fixed multiple of $r_{N-l}$. Packing bounds both overlap
+multiplicities, giving
+\[
+ \sum_{q\in\mathcal A}\sigma_q^2\leq C M(N-l).
+\]
+
+Let $I$ be the open interval between the endpoint values. For $t\in I$,
+the continuum $C_{N,t}$ meets the given path and exits $B(x,2R)$,
+since its diameter exceeds $4R$. By \Cref{lem:paths}, join these
+two continuum points in its $r_N/4$-neighborhood and retain a radial
+subpath $\eta_t$ between radii $R$ and $2R$, contained in that closed
+annulus. Its diameter is at least $R$, so $h\eta_t$ is macroscopic
+by the choice of $d_0$ in \Cref{sec:geometry}.
+
+If a target cell $B(\xi,2r_{N-l})$ meets $h\eta_t$ at $hy$,
+choose $q\in T_N$ with $d(q,y)\leq r_N$. For large $m_0$, this
+center belongs to $\mathcal A$ and
+$d(hq,hy)\leq C_Er_N/R$, so $\omega_\xi$ occurs in $\sigma_q$.
+Some level point lies within $r_N/4$ of $y$, hence in
+$\overline B(q,2r_N)$. Thus $t$ lies in the closed range interval
+$I_q$ of $H_N$ on that ball. Accounting for all target cells gives
+\[
+ 1\leq\sum_{q\in\mathcal A}\sigma_q\mathbf1_{I_q}(t)
+ \quad(t\in I),
+ \qquad
+ \Delta\leq C M(N-l)^{1/2}
+       \left(\sum_{q\in\mathcal A}|I_q|^2\right)^{1/2}.
+\]
+The second inequality follows by integration and Cauchy--Schwarz.
+Apply \eqref{eq:local-osc} with $J=2$ and its bounded-overlap
+summation. Taking $C(2)r_N<R/4$ confines every charged center to
+$R/4\leq d(q,x)\leq4R$, proving \eqref{eq:annular}.
+\end{proof}
+
+\subsection{The limiting function and measure}
+
+The finite measures
+$\mu_N=M(N)\sum_{q\in T_N}\rho_q^2\delta_q$
+have uniformly bounded mass by \eqref{eq:dual-energy}.
+
+\Needspace{5\baselineskip}
+\begin{proposition}[Scalar limit]
+\label{prop:scalar-limit}
+Along a subsequence of the record indices,
+$H_N\to u$ locally uniformly on $U$ and $\mu_N\rightharpoonup\mu$,
+where $u:U\to[0,1]$ is continuous and nonconstant and $\mu$ is finite.
+The function $u$ is zero and one on the interiors of the respective
+bands, and is locally H\"older continuous if $\lambda>1$.
+For every compact $K\subset U$ and sufficiently small
+$s>0$, uniformly for $x\in K$,
+\begin{equation}
+ \bigl(\osc_{\overline B(x,s)}u\bigr)^2
+       \leq C\mu\bigl(\overline B(x,C_\mu s)\bigr).
+ \label{eq:measure}
+\end{equation}
+For each $0<t<1$ there is a continuum $C_t\subset\cQ$ of diameter
+at least $d_1$, with $u=t$ on $C_t\cap U$, meeting every compact
+path in $U$ whose endpoint values strictly straddle $t$.
+The fixed-transform energy bounds of \Cref{lem:local-oscillation}
+hold along the retained subsequence.
+\end{proposition}
+\begin{proof}
+Fix compact $K\subset U$, and let $x,y\in K$ with $d(x,y)\leq r_k$.
+They join inside $B(x,Lr_k)$. Choose $b$ with $a^b>2L$ and consider
+indices $l\leq k-b$ above a fixed interior threshold, in an arithmetic
+progression of step $b_1$ with $a^{b_1}>16$. The annuli
+$\{r_l/4\leq d(x,\cdot)\leq4r_l\}$ are disjoint. For $N$
+sufficiently large relative to $k$, \eqref{eq:annular} and
+\eqref{eq:records} give
+\[
+ |H_N(x)-H_N(y)|^2\leq C\lambda^{-l}
+ \mu_N\{r_l/4\leq d(x,\cdot)\leq4r_l\}.
+\]
+Summing over at least $ck$ indices yields
+$|H_N(x)-H_N(y)|\leq C_Kk^{-1/2}$. If $\lambda>1$, the single
+choice $l=k-b$ gives $C_K\lambda^{-k/2}$ instead.
+
+For each tolerance this controls a tail uniformly; the finitely many
+omitted functions are individually uniformly continuous. Thus the
+sequence is equicontinuous on $K$. Arzel\`a--Ascoli on a compact
+exhaustion of $U$, followed by weak compactness of the measures on
+$Z$, gives the asserted subsequence. The two constant band values
+prove nonconstancy. The stronger estimate gives local H\"older exponent
+$\log\lambda/(2\log a)$.
+
+For \eqref{eq:measure}, fix $A>2L$ and choose
+$As\leq R=r_l<aAs$. Any two points of $\overline B(x,s)$ join in
+$B(x,Ls)\subset B(x,R/2)$. The annular and record estimates give
+\[
+ \bigl(\osc_{\overline B(x,s)}H_N\bigr)^2
+       \leq C\mu_N\bigl(\overline B(x,4aAs)\bigr)
+\]
+for large $N$. Local uniform convergence passes the oscillation to
+the limit; the closed-set inequality for weak convergence bounds the
+right-hand limsup by $C\mu(\overline B(x,4aAs))$.
+Take $C_\mu=4aA$.
+
+For each fixed $t$, take a Hausdorff limit $C_t$ of $C_{N,t}$ in
+$\cQ$. It is connected and has diameter at least $d_1$. Local
+uniform convergence gives $u=t$ on $C_t\cap U$. A fixed compact
+straddling path also straddles $t$ for every sufficiently large
+$H_N$, hence meets those $C_{N,t}$ and their limit. The subsequence
+used here may depend on $t$; no simultaneous selection is needed.
+\end{proof}
+
+\subsection{Two comparison tools}
+
+The scalar construction will be used in three ways. The large continua
+$C_t$ provide levels for annular constructions, with $u=t$ on
+$C_t\cap U$; no value of $u$ is asserted outside $U$. The measure
+estimate \eqref{eq:measure} turns oscillation on a small ball into a
+lower bound for its enlarged ball's $\mu$-mass. To compare
+transformations, we retain $H_N$ and $C_{N,t}$: the identity $H_N=t$
+holds on the entire continuum, and the local oscillation energies
+tend to zero after each transformation has been fixed. Those
+comparisons will pass to subcontinua contained in compact subsets of
+the common open domains.
+
+The first tool integrates variation of a second function over a
+family of level continua. It is a finite-cell statement, independent
+of the preceding construction.
+
+\begin{lemma}[A finite-cell level comparison]
+\label{lem:cross-energy}
+Let $f,w$ be continuous on an open set $V\subset Z$, let $K\subset V$
+be compact, and let $I\subset\RR$ be an interval. Suppose that for
+every $t\in I$ there is a continuum $D_t\subset K$ with $f=t$ on
+$D_t$ and $\osc_{D_t}w\geq b_0>0$. For sufficiently large $n$,
+\begin{equation}
+ b_0|I|\leq
+ \sum_{\substack{q\in T_n\\B(q,2r_n)\cap K\ne\varnothing}}
+ \osc_{\overline B(q,2r_n)}f\,
+ \osc_{\overline B(q,2r_n)}w.
+ \label{eq:cross-energy}
+\end{equation}
+Consequently these hypotheses cannot hold for sequences with both
+local oscillation energies on $K$ tending to zero, if $b_0$ and
+$|I|$ stay bounded below by positive constants.
+\end{lemma}
+\begin{proof}
+Let $I_q(f)$ be the closed range interval of $f$ on the closed cell.
+For large $n$ these cells lie in $V$. The interval $w(D_t)$ is covered
+by the range intervals of $w$ on cells meeting $D_t$, and each such
+cell has $t\in I_q(f)$. Therefore
+\[
+ b_0\leq\sum_{q:B(q,2r_n)\cap K\ne\varnothing}
+ \osc_{\overline B(q,2r_n)}w\,\mathbf1_{I_q(f)}(t).
+\]
+Integrate over $I$ and use
+$|I\cap I_q(f)|\leq\osc_{\overline B(q,2r_n)}f$.
+Only finitely many interval indicators are integrated, so no
+measurable selection of $D_t$ is required. Cauchy--Schwarz gives
+the final assertion.
+\end{proof}
+
+\begin{lemma}[Stopping a continuum at a sphere]
+\label{lem:stop-continuum}
+Let $D$ be a continuum, $p\in Z$, and $r>0$. If
+$x\in D\setminus\overline B(p,r)$ and $D\cap B(p,r)\ne\varnothing$,
+then $D$ contains a subcontinuum through $x$ meeting the radius-$r$
+sphere and avoiding $B(p,r)$.
+\end{lemma}
+\begin{proof}
+In a compact metric space $X$, a point's component is the intersection
+$Q_x$ of its clopen neighborhoods. To verify connectedness of $Q_x$,
+suppose it is separated into two compact pieces in disjoint open sets
+$O_1,O_2$. Compactness of $X\setminus(O_1\cup O_2)$ gives a finite
+intersection $W$ of clopen neighborhoods of $x$ contained in their
+union. Its intersection with the $O_i$ containing $x$ is clopen in
+$X$ and misses the other piece of $Q_x$, a contradiction. The same
+compactness argument shows that every open neighborhood of a component
+contains a clopen neighborhood of it.
+
+Apply this to $X=D\setminus B(p,r)$ and the component $C$ of $x$.
+If $C$ missed the sphere, it would have a clopen neighborhood $W$ in
+$X$ missing the sphere. Compactness places $W$ a positive distance
+from $\overline B(p,r)$, so $W$ is also clopen in $D$. It is nonempty
+and misses $D\cap B(p,r)$, contradicting connectedness. Thus $C$ is
+the required subcontinuum.
+\end{proof}
diff --git a/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/build/sections/03_exponential.tex b/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/build/sections/03_exponential.tex
new file mode 100644
index 0000000000000000000000000000000000000000..d47792305d95265f0fd40fa20445ca626610f9ff
--- /dev/null
+++ b/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/build/sections/03_exponential.tex
@@ -0,0 +1,328 @@
+\section{Exponential records: comparing two magnifications}
+\label{sec:exponential}
+
+Suppose $\lambda>1$. The scalar limit $u$ is then locally H\"older.
+We magnify its variation to a nonconstant function on a punctured
+sphere. The approximating level continua force every transform with a different
+puncture to be constant wherever this function varies. A second
+magnification makes two such transforms vary on the same two disks,
+with disjoint value ranges, yielding a contradiction.
+Scalar approximation indices throughout belong to the subsequence
+of \Cref{prop:scalar-limit}.
+
+\subsection{Magnifying variation}
+
+\begin{lemma}[H\"older magnification]
+\label{lem:holder-magnification}
+Let $f$ be a real-valued, locally H\"older continuous function on an open set $\Omega\subset Z$.
+Suppose $O$ is open, $\overline O\subset\Omega$, and $O$ contains a
+compact path on which $f$ is nonconstant. Fix $z_0$ with
+$\dist(z_0,O)>0$. There are depths $k_i\to\infty$, centers $x_i\in O$,
+positive numbers $\sigma_i$, and expansions $h_i=v_i^{-1}$ at
+$(x_i,k_i)$ such that, after passage to a subsequence,
+\[
+ p_i=h_i z_0\longrightarrow b,
+ \qquad
+ \frac{\dist(x_i,Z\setminus O)}{r_{k_i}}\longrightarrow\infty.
+\]
+Every compact subset of $Z\setminus\{b\}$ eventually lies in
+$h_i\Omega$, and
+\begin{equation}
+ F_i(z)=\frac{f(h_i^{-1}z)-f(x_i)}{\sigma_i}
+ \label{eq:blowup}
+\end{equation}
+converges locally uniformly there to a nonconstant locally H\"older
+continuous function $F$.
+\end{lemma}
+
+\begin{proof}
+Write $\delta(x)=\dist(x,Z\setminus O)$ and fix $\eta>Q$.
+A finite cover of $\overline O$ gives a uniform H\"older bound,
+with some exponent $\alpha_0>0$, on a neighborhood of $\overline O$.
+For all large $k$ put
+\[
+ A_k=\sup_{x\in O}\delta(x)^\eta
+            \osc_{\overline B(x,r_k)}f.
+\]
+The balls lie in $\Omega$, and
+\begin{equation}
+ c a^{-Qk}\le A_k\le C a^{-\alpha_0 k}.
+ \label{eq:magnification-bounds}
+\end{equation}
+For the lower bound, cover a compact variation path $P\subset O$
+by at most $C r_k^{-Q}$ radius-$r_k$ balls centered on $P$, using
+\Cref{lem:packing}. Their range intervals cover the interval $f(P)$,
+so one ball has oscillation at least $c r_k^Q$; also
+$\min_P\delta>0$. The upper bound is the H\"older estimate.
+
+Fix $0<\alpha<\alpha_0$. Since $A_k a^{\alpha k}\to0$, there are
+arbitrarily late indices attaining their maximum over the remaining
+tail. At these indices,
+\begin{equation}
+ A_{k+j}\le a^{-\alpha j}A_k\qquad(j\ge0).
+ \label{eq:magnification-tail}
+\end{equation}
+Choose $x_k\in O$ attaining at least half the supremum, and set
+$\delta_k=\delta(x_k)$ and $\sigma_k=A_k/\delta_k^\eta$.
+Boundedness of $f$ and \eqref{eq:magnification-bounds} give
+$\delta_k^\eta\ge c r_k^Q$, hence
+$\delta_k/r_k\ge c r_k^{Q/\eta-1}\to\infty$.
+Expand at $(x_k,k)$ and take a subsequence with $p_k=h_kz_0\to b$.
+
+For a compact $K\subset Z\setminus\{b\}$, \eqref{eq:puncture} gives
+\begin{equation}
+ h_k^{-1}K\subset\overline B(x_k,C_Kr_k)\subset O
+ \label{eq:magnification-preimage}
+\end{equation}
+eventually. The expansion formula gives
+$d(h_k^{-1}z,h_k^{-1}w)\le C_Kr_kd(z,w)$ on $K$.
+On every fixed multiple of this source ball, $\delta$ is comparable
+to $\delta_k$. Thus \eqref{eq:magnification-tail} implies
+\begin{equation}
+ \frac{|f(y)-f(y')|}{\sigma_k}\le C a^{-\alpha j}
+ \quad\text{if }d(y,y')\le r_{k+j},\quad j\ge0.
+ \label{eq:magnification-local-holder}
+\end{equation}
+For sufficiently small $d(z,w)$, choosing $j\ge0$ with
+$a^{-j}\ge C_Kd(z,w)>a^{-j-1}$ proves a uniform local
+$\alpha$-H\"older estimate for $F_k$ on $K$.
+
+For boundedness, \Cref{lem:paths} joins $h_k^{-1}z$ to $x_k$
+inside a fixed multiple of $B(x_k,r_k)$. Packing covers this path
+by a bounded number of radius-$r_k$ balls centered on the path, each with oscillation
+at most $C\sigma_k$. Their range intervals cover the path's image,
+so $|F_k(z)|\le C_K$. Compact exhaustion and Arzel\`a--Ascoli now
+give a locally uniform, locally H\"older limit.
+Finally, $\osc_{h_k\overline B(x_k,r_k)}F_k\ge1/2$.
+The lower puncture bound places all these image balls in one compact
+set off $b$, so local uniform convergence makes the limit nonconstant.
+\end{proof}
+
+\subsection{A common mesh and escaping level continua}
+
+Apply \Cref{lem:holder-magnification} to $u$ on $U$, taking a small
+variation region $O$ and $z_0$ outside its closure. Call the limit
+$W$, its puncture $b$, and retain $x_i,k_i,h_i,\sigma_i$.
+
+\begin{lemma}[Diagonal approximants]
+\label{lem:exponential-diagonal}
+After thinning the expansions, there are increasing scalar indices
+$N_j$ such that
+\[
+ f_j(z)=\frac{H_{N_j}(h_j^{-1}z)-u(x_j)}{\sigma_j},
+ \qquad z\in h_j\Qhat,
+\]
+converges locally uniformly to $W$ on $Z\setminus\{b\}$.
+For every fixed $g\in G$, compact $K\subset Z\setminus\{gb\}$,
+and fixed $J\ge2$,
+\begin{equation}
+ \mathcal E_{N_j}^{(J)}(f_j\circ g^{-1};K)\longrightarrow0.
+ \label{eq:exponential-translated-energy}
+\end{equation}
+All indicated enlarged balls eventually lie in the functions' domains.
+
+If a compact path $P\subset Z\setminus\{b\}$ has endpoint values
+of $W$ strictly straddling a compact positive-length interval $I$,
+then, for all large $j$ and every $t\in I$, there is a continuum
+$D_{j,t}\subset h_j\Qhat$ with
+\[
+ f_j=t\text{ on }D_{j,t},\qquad D_{j,t}\cap P\ne\varnothing,
+ \qquad \sup_{t\in I}\dist(b,D_{j,t})\longrightarrow0.
+\]
+\end{lemma}
+
+\begin{proof}
+For $j\ge4$ put
+$K_j=\{d(z,b)\ge1/j\}$ and $L_j=\{d(z,b)\ge1/(2j)\}$.
+Choose the $j$th expansion so that $h_j^{-1}L_j\subset U$ and its
+normalized $u$-transform differs from $W$ by at most $1/j$ on $L_j$.
+\emph{Hold this expansion and its normalization fixed.} Then choose
+$N_j>N_{j-1}$ so large that
+\[
+ \sup_{L_j}\frac{|H_{N_j}(h_j^{-1}z)-u(h_j^{-1}z)|}{\sigma_j}
+ \le\frac1j,
+ \qquad \mathcal E_{N_j}^{(j)}(f_j;K_j)\le\frac1j,
+ \qquad (j+2)r_{N_j}<\frac1{2j}.
+\]
+The uniform approximation follows from \Cref{prop:scalar-limit}.
+With $h_j$ and $\sigma_j$ fixed, \Cref{lem:local-oscillation} bounds
+the displayed energy by $C(h_j,K_j,j)/(\sigma_j^2M(N_j))$, so the
+energy condition follows by increasing $N_j$. The last condition
+keeps all the enlarged balls in $L_j$. This proves local uniform
+convergence and vanishing energy at every fixed enlargement off $b$.
+
+For a fixed $g$, choose a compact neighborhood $K'$ of $g^{-1}K$
+avoiding $b$. Pulling back each target ball and choosing a nearest
+source net point gives, by bi-Lipschitz distortion and packing,
+\[
+ \mathcal E_n^{(J)}(F\circ g^{-1};K)
+ \le C_g\mathcal E_n^{(J')}(F;K')
+\]
+for some fixed $J'\ge2$ and all large $n$. Each source center receives
+only boundedly many target centers. Eventually $K'\subset K_j$ and
+$J'\le j$, proving \eqref{eq:exponential-translated-energy}.
+
+Uniform convergence makes the endpoint values of $f_j$ straddle $I$.
+The source level $\tau_j(t)=u(x_j)+\sigma_jt$ therefore lies in $(0,1)$,
+and \Cref{lem:level-continua} supplies
+$D_{j,t}=h_jC_{N_j,\tau_j(t)}$, meeting $P$.
+Its source continuum has diameter at least $d_1$, so contains a point
+at distance at least $d_1/2$ from $x_j$. By \eqref{eq:puncture},
+\[
+ \dist(b,D_{j,t})\le d(b,p_j)+Cr_{k_j}\longrightarrow0
+\]
+uniformly for $t\in I$.
+\end{proof}
+
+\subsection{Comparison across punctures}
+
+Let $\mathcal N(F)$ be the points of a continuous function's domain
+at which it is not locally constant.
+
+\Needspace{5\baselineskip}
+\begin{lemma}[Transform comparison]
+\label{lem:exponential-comparison}
+If $c=gb\ne b$ and $V=W\circ g^{-1}$, then
+\begin{equation}
+ V(x)=V(b)\qquad
+ \bigl(x\in\mathcal N(W)\setminus\{c\}\bigr).
+ \label{eq:transform-level}
+\end{equation}
+More generally, for $W_s=W\circ s^{-1}$ and $p_s=sb$, if $p_s\ne p_t$,
+\begin{equation}
+ W_t(x)=W_t(p_s)\qquad
+ \bigl(x\in\mathcal N(W_s)\setminus\{p_t\}\bigr).
+ \label{eq:two-transform-level}
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Suppose $x\in\mathcal N(W)\setminus\{c\}$ and $V(x)\ne V(b)$.
+Choose $\varepsilon>0$ and a coordinate disk about $x$, compactly
+contained off $b,c$, on which $|V-V(b)|\ge4\varepsilon$.
+It contains a compact variation path $P$ for $W$. Choose a compact
+positive-length interval $I$ strictly between its endpoint values,
+avoiding $W(c)$. Continuity gives $\epsilon_c>0$ such that
+$\overline B(c,\epsilon_c)$ avoids $b$ and its $W$-range has positive
+distance from $I$. Consequently, for all large $j$, the continua
+$D_{j,t}$ from \Cref{lem:exponential-diagonal} avoid this closed ball.
+
+Choose $\rho>0$ so that $\overline B(b,\rho)$ avoids $c,P$ and
+$|V-V(b)|\le\varepsilon$ there. For $w_j=f_j\circ g^{-1}$,
+local uniform convergence gives
+\[
+ |w_j-V(b)|\ge3\varepsilon\quad\text{on }P,
+ \qquad |w_j-V(b)|\le2\varepsilon\quad\text{on }\overline B(b,\rho)
+\]
+for all large $j$. Each $D_{j,t}$ meets $P$ and enters $B(b,\rho)$.
+By \Cref{lem:stop-continuum}, it has a subcontinuum joining a chosen
+point of $P$ to the radius-$\rho$ sphere and contained in
+\[
+ K=\{z:d(z,b)\ge\rho,\ d(z,c)\ge\epsilon_c\}.
+\]
+This subcontinuum lies in the level $f_j=t$ and has $w_j$-oscillation
+at least $\varepsilon$. A fixed compact neighborhood of $K$ avoiding
+both punctures eventually lies in $h_jU\cap gh_jU$ and contains every
+closed radius-$2r_{N_j}$ cell meeting $K$. Hence
+\Cref{lem:cross-energy} and Cauchy--Schwarz give
+\[
+ \varepsilon|I|\le
+ \bigl(\mathcal E_{N_j}^{(2)}(f_j;K)\bigr)^{1/2}
+ \bigl(\mathcal E_{N_j}^{(2)}(w_j;K)\bigr)^{1/2}\longrightarrow0,
+\]
+a contradiction. All sets and $g$ are fixed in this limit.
+Finally apply \eqref{eq:transform-level} at $s^{-1}x$ with group
+element $s^{-1}t$ to obtain \eqref{eq:two-transform-level}.
+\end{proof}
+
+\subsection{The second magnification}
+
+We first control repeated punctures. An accumulation direction of
+vertices going to infinity is a subsequential boundary limit.
+
+\begin{lemma}[Stabilizer directions]
+\label{lem:stabilizer}
+For every $b\in Z$, the vertices of $E=\Stab_G(b)$ have at most
+two accumulation directions.
+\end{lemma}
+
+\begin{proof}
+Let $\xi$ be a based ray to $b$. For $q\in E$, $|q|\le t$,
+thinness of the ideal triangle with sides $[e,q],\xi,q\xi$ puts
+$q\xi(n)$ uniformly near $\xi$ when $n$ is sufficiently larger
+than $t$. The nearby parameter lies in $[n-t-C,n+t+C]$.
+For one common integer $n$, these vertices are distinct as $q$ varies.
+Bounded valence therefore gives
+\begin{equation}
+ \#\{q\in E:|q|\le t\}\le C'(t+1).
+ \label{eq:stabilizer-growth}
+\end{equation}
+
+If another accumulation direction exists, take $q_n\to x\ne b$ and
+$q_n^{-1}\to y$ along a subsequence. Base change gives, uniformly
+on compact sets avoiding $y$,
+\[
+ (q_n|q_nz)_e=|q_n|-(q_n^{-1}|z)_e+O(1)\ge |q_n|-O(1).
+\]
+By \Cref{lem:prefix}, $q_nz\to x$ there, and $q_nb=b$ forces $y=b$.
+Choose disjoint small neighborhoods of $b,x$, with a point outside
+their union. For large $n$, $q=q_n$ sends the complement of the
+first into the second. Its positive iterates never return the outside
+point, so $q$ has infinite order. It is therefore loxodromic, with
+endpoints $b,x'$; see \cite[III.$\Gamma$.3.10]{BH}.
+
+If some $h\in E$ moved $x'$, the cyclic subgroups generated by $q$
+and $hqh^{-1}$ would intersect trivially: a nonzero common power
+would have the same pair of endpoints. Their $m^2$ products
+$q^i(hqh^{-1})^j$, $1\le i,j\le m$, would then be distinct and have
+length $O(m)$, contradicting \eqref{eq:stabilizer-growth}.
+Thus every element of $E$ fixes both endpoints. It maps a geodesic
+line between them to a line at uniformly bounded Hausdorff distance,
+so its orbit of $e$ stays a fixed distance from that line. Its only
+possible accumulation directions are the two endpoints.
+\end{proof}
+
+\begin{proposition}
+\label{prop:no-exponential-records}
+The contradiction setup cannot have $\lambda>1$.
+\end{proposition}
+
+\begin{proof}
+Let $E=\Stab(b)$ and let $\Lambda_E$ be its at-most-two accumulation
+directions. Choose a variation region $O$ for $W$ whose closure avoids
+$\{b\}\cup\Lambda_E$. Such a region exists: otherwise $W$ would
+be locally constant on the connected sphere minus this finite set,
+and continuity would make it constant throughout its domain.
+
+Apply \Cref{lem:holder-magnification} to $W$ on $Z\setminus\{b\}$,
+with this $O$ and $z_0=b$. Write the expansions as $s_i=v_i^{-1}$,
+with centers $y_i\in O$, depths $\ell_i\to\infty$, and punctures
+$p_i=s_i b\to b_*$. Positive affine normalizations of
+$W_i=W\circ s_i^{-1}$ converge locally uniformly off $b_*$ to a
+continuous nonconstant $W_*$.
+
+The punctures have a pairwise distinct subsequence. Indeed, if
+$v_i^{-1}b=v^{-1}b$ infinitely often, then $v_i v^{-1}\in E$.
+After a subsequence $y_i\to y\in\overline O$. Since $v_i$ lies at
+depth $\ell_i$ on a ray to $y_i$, \Cref{lem:prefix} gives $v_i\to y$
+in boundary direction. Bounded right multiplication preserves this
+direction, implying $y\in\Lambda_E$, a contradiction.
+
+Choose two coordinate disks $D_1,D_2$, with compact closures off $b_*$,
+having disjoint $W_*$-range intervals and each containing a compact
+variation path. To do this, follow a variation path to its first
+hitting points of two distinct intermediate values. Just before either
+first hitting time the values differ, so the function is not locally
+constant at either point. Continuity supplies disks with disjoint
+ranges, and non-local constancy supplies a variation path in each.
+
+For every sufficiently large $i$, the puncture $p_i$ is outside the
+closed disks. Uniform convergence of the normalized $W_i$ preserves
+both the disjoint range intervals and the positive variation on the
+two fixed paths. Positive affine normalization preserves these
+properties, so $\mathcal N(W_i)$ meets each disk.
+Fix two distinct such indices $i,j$. By
+\eqref{eq:two-transform-level}, $W_j$ has the single value $W_j(p_i)$
+at every point of $\mathcal N(W_i)$ in either disk. Taking one point
+in each contradicts the disjoint $W_j$-range intervals.
+\end{proof}
diff --git a/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/build/sections/04_flow.tex b/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/build/sections/04_flow.tex
new file mode 100644
index 0000000000000000000000000000000000000000..15a2c362851b167ee9b7534d26e8a0ecac017661
--- /dev/null
+++ b/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/build/sections/04_flow.tex
@@ -0,0 +1,463 @@
+\section{A probability flow in the subexponential case}
+\label{sec:flow}
+
+Assume now that the unbounded moduli satisfy
+\begin{equation}
+ \limsup_{j\to\infty}M(j)^{1/j}\leq1.
+ \label{eq:subexponential-growth}
+\end{equation}
+Let $u,\mu,C_t$ be given by \Cref{prop:scalar-limit}. We build a tree of
+small balls carrying nested intervals of values of $u$. At each node,
+either two separated variations survive expansion, or many children
+carry a substantial fraction of its interval. A probability flow will
+force the first alternative at arbitrarily deep levels.
+
+Choose $x_0\in U$ where $u$ is not locally constant, and choose
+$\epsilon_0>0$ with $\overline B(x_0,3\epsilon_0)\subset U$. Put
+\[
+ K=\overline B(x_0,\epsilon_0),\qquad
+ K^+=\overline B(x_0,2\epsilon_0),
+\]
+and, shrinking $\epsilon_0$ if necessary, choose $z_0$ a positive distance
+from $K^+$. All centers will lie in $K$ and all source neighborhoods in
+$K^+$. Use the constants of \eqref{eq:measure} for this fixed region.
+
+\subsection{Node intervals and transition probabilities}
+
+The tree will record both where variation occurs and which values it
+carries. A node has a source center $x$, a scale $R$, and a value interval
+$J$. Its children have scale $r=\theta R$, where $\theta=a^{-D}$ for
+one positive integer subdivision depth $D$ used throughout the tree. Their intervals
+are contained in $J$. We will choose probabilities on the children
+that control interval length linearly and quadratically; the two
+controls have different roles in the final comparison.
+
+We first define this data for geometric constants $c_0>2L+2$,
+$Lc_0+2<A<B_0-2$, and $H>0$. Here $c_0$ is the source-ball enlargement,
+$A,B_0$ locate an annulus of possible child centers, and $Hr$ is the
+separation required of retained children. The next subsection fixes
+these constants, before choosing $D$. We use only scales $R=r_k$
+small enough that all fixed enlargements below lie in $K^+$ and
+satisfy \eqref{eq:measure}, and that $(B_0+2)R<d_1/2$.
+
+A \emph{node} is a center $x\in K$, a scale $R=r_k$, and a nonempty
+open interval $J=(j_-,j_+)\subset(0,1)$ such that
+\begin{equation}
+ J\subset\left(\min_{\overline B(x,c_0R)}u,
+                   \max_{\overline B(x,c_0R)}u\right).
+ \label{eq:node-straddling}
+\end{equation}
+Write $\Delta=|J|$. For every $t\in J$, a path in $B(x,Lc_0R)$
+strictly straddles $t$ and hence meets $C_t$. Since
+$\diam C_t\geq d_1$, the continuum leaves
+$\overline B(x,(B_0+2)R)$ and meets every intervening radial sphere.
+
+Take a maximal $r$-separated set $Y$ in
+\[
+ \mathcal A_x(R)=\{y:(A-1)R\leq d(x,y)\leq(B_0+1)R\}.
+\]
+It is an $r$-net there, and packing gives
+\begin{equation}
+ \#Y\leq C_Ya^{QD},
+ \label{eq:candidate-count}
+\end{equation}
+with $C_Y$ independent of $D$. For $y\in Y$ let $I_y^{\mathrm{in}}$
+and $I_y^{\mathrm{out}}$ be the closed intervals between the extrema
+of $u$ on $\overline B(y,2r)$ and $\overline B(y,c_0r)$, respectively.
+These are range \emph{intervals}; the balls need not be connected. Put
+\[
+ J_y=J\cap\operatorname{int}I_y^{\mathrm{out}},\qquad
+ t_y=|J_y|/\Delta.
+\]
+A positive $t_y$ gives a nested open child interval satisfying
+\eqref{eq:node-straddling}, and, including singleton ranges,
+\begin{equation}
+ |J\cap I_y^{\mathrm{out}}|=t_y\Delta.
+ \label{eq:closed-range-length}
+\end{equation}
+
+For $0<c_*\leq1/(8\max\{C_Y,1\})$, put $\kappa=c_*a^{-QD}$.
+A node is \emph{good} if two candidates $y^-,y^+$ satisfy
+\begin{align}
+ |I_{y^-}^{\mathrm{in}}\cap(j_-,j_-+\Delta/4)|&\geq\kappa\Delta,
+ &|I_{y^+}^{\mathrm{in}}\cap(j_-+3\Delta/4,j_+)|&\geq\kappa\Delta,
+ \label{eq:good-inner}\\
+ \dist(I_{y^-}^{\mathrm{out}},I_{y^+}^{\mathrm{out}})&\geq\Delta/4.
+ \label{eq:good-outer}
+\end{align}
+Goodness uses the full candidate set, independently of the children
+later retained.
+
+The required probabilities are as follows. Fix the exponents
+\begin{equation}
+ b_* =\frac12,\qquad \tau=\frac1{20},\qquad
+ \beta=\frac{\tau}{4(Q+1)},\qquad \gamma=\frac\beta2.
+ \label{eq:flow-exponents}
+\end{equation}
+At every node we seek an $Hr$-separated set of children, with
+probabilities summing to one, such that
+\begin{align}
+ p_y&\leq a^{-\gamma D}t_y,
+ &p_y&\leq a^{-b_*D}t_y^2
+ &&\text{at nodes that are not good},\label{eq:transition-bad}\\
+ p_y&\leq a^{-\gamma D}t_y,
+ &p_y&\leq a^{\tau D}t_y^2
+ &&\text{at good nodes}.\label{eq:transition-general}
+\end{align}
+Multiplying the linear bounds along a branch will telescope the
+interval ratios and make the branch probability small. Multiplying
+the quadratic bounds will compare probability to squared interval
+length. The measure estimate bounds the sum of those squares at
+each separated generation. It will therefore bound the average
+probability of nodes with the stronger quadratic bound and force a
+substantial amount of good-node probability at arbitrarily deep levels.
+
+Two facts will supply these transitions. A node that is not good has
+many children with $t_y\ge1/8$, on which uniform probabilities suffice.
+At every node, a capacity estimate supplies enough total weight to
+satisfy the weaker pair of bounds \eqref{eq:transition-general}.
+After constructing the transitions, we show that goodness also gives
+two separated regions of variation in one of finitely many target
+patterns. That geometric property is the input to \Cref{sec:pairs}.
+
+\subsection{Geometric choices and many large children}
+
+We now fix the geometry used by the definitions above. Put
+$c_{\mathrm{in}}=2L$ and choose $c_0>c_{\mathrm{in}}+2$ so large that,
+with
+\[
+ A=4Lc_0+20,\qquad B_0=A+10,\qquad
+ c_{\mathrm{pat}}=\frac{2C_Ec_{\mathrm{in}}}{(A-2)^2},
+\]
+we have
+\begin{equation}
+ \frac{c_Ec_0}{(B_0+2)^2}>12c_{\mathrm{pat}}.
+ \label{eq:flow-constant-choice}
+\end{equation}
+The constant $c_{\mathrm{pat}}$ will be the target radius factor:
+paths joining extrema of the two inner child balls will expand into balls of radius
+$c_{\mathrm{pat}}\theta/2$, while the larger source balls will contain
+the preimages of radius-$10c_{\mathrm{pat}}\theta$ target balls.
+The ratio of the left side of \eqref{eq:flow-constant-choice} to the
+right side grows linearly in $c_0$, so this choice is possible.
+Next choose
+\[
+ H>\max\{4(B_0+1),4C_{\mu}c_0,2\}.
+\]
+The first inequality preserves separation between different branches;
+the second makes the measure enlargements at one generation disjoint.
+These choices fix $C_Y$ in \eqref{eq:candidate-count}, and we fix
+$c_*$ as specified above. Require $\theta<1/4$ and $c_0\theta<1$.
+All further lower bounds on $D$ will depend only on this fixed data.
+
+The uniform upper bound on node scales can be chosen now, before $D$:
+all source radius factors are fixed, and the subsidiary scales $r$
+and $s$ used below never exceed $R$. Thus the source neighborhoods
+and measure enlargements stay in $K^+$ for every node. We choose the
+root scale after $D$ has been fixed.
+
+\begin{lemma}[Many children at a node that is not good]
+\label{lem:bad-node-children}
+For sufficiently large $D$, every node that is not good has at least
+$c_Ha^D$ mutually $Hr$-separated candidates with $t_y\geq1/8$,
+where $c_H>0$ is independent of the node and $D$.
+\end{lemma}
+\begin{proof}
+Fix $b\in[AR,B_0R]$. Each $C_t$, $t\in J$, meets the radius-$b$
+sphere, so the inner intervals of candidates with
+$|d(x,y)-b|\leq r$ cover $J$. By \eqref{eq:candidate-count}, two of
+them meet the respective quarters in \eqref{eq:good-inner} by at least
+$\kappa\Delta$.
+
+If both have $t_y<1/8$, the lower-quarter outer interval ends below
+$j_-+3\Delta/8$, and the upper-quarter interval starts above
+$j_-+5\Delta/8$. Indeed each meets its quarter but overlaps $J$ in
+length less than $\Delta/8$; the assertion also holds for intervals
+extending beyond $J$. Their distance exceeds $\Delta/4$, making the
+node good. Otherwise one candidate has $t_y\geq1/8$. Apply this at
+radial values spaced by $(H+3)r$: there are at least $c_Ha^D$ such
+values, and the resulting centers are more than $Hr$ apart.
+\end{proof}
+
+At a node that is not good, put uniform probabilities on the
+candidates in \Cref{lem:bad-node-children}. For large $D$ they satisfy
+\eqref{eq:transition-bad}: indeed, $p_y\le c_H^{-1}a^{-D}$ and
+$t_y\ge1/8$, while $\gamma,b_*<1$.
+
+\subsection{The capacity estimate}
+
+It remains to construct transitions satisfying
+\eqref{eq:transition-general}. A single subdivision depth $D$ must
+work even when a node's interval $J$ is arbitrarily short. We therefore
+integrate over $J$ before choosing $D$; its length will cancel from
+the estimate.
+
+\begin{lemma}[Separated capacity]
+\label{lem:separated-capacity}
+For sufficiently large $D$, uniformly over all nodes, there is an
+$Hr$-separated subset $Y'\subset Y$ such that
+\begin{equation}
+ \sum_{y\in Y'}\min\{a^{-\gamma D}t_y,a^{\tau D}t_y^2\}\geq1.
+ \label{eq:capacity}
+\end{equation}
+\end{lemma}
+\begin{proof}
+Join two candidates in a graph when their distance is at most $Hr$.
+Packing bounds its degree, so graph coloring partitions $Y$ into at most
+$C_H$ separated classes, with $C_H$ independent of $D$. If the conclusion fails, then
+\begin{equation}
+ \sum_{y\in Y}\min\{a^{-\gamma D}t_y,a^{\tau D}t_y^2\}<C_H.
+ \label{eq:failed-capacity}
+\end{equation}
+Call $y$ \emph{large} when $t_y\geq a^{-(\tau+\gamma)D}$ and
+\emph{small} otherwise. On large candidates the minimum is linear, so
+\begin{equation}
+ \sum_{\text{large }y}t_y\leq C_Ha^{\gamma D}.
+ \label{eq:large-total-length}
+\end{equation}
+
+We will test levels at an intermediate radius $s$, larger than the
+child radius $r$ and smaller than the parent radius $R$. This provides
+more radial tests than the large ranges can occupy. At the same time,
+it lets us detect the remaining levels with a modulus cost at the
+relative scale $r/s$.
+
+\emph{Remove the large ranges.}
+Set $D'=\lceil\beta D\rceil$ and $s=a^{-D'}R$. For large $D$,
+$0<D'<D$ and $s<R/4$. Choose $s$-spaced radial bins $\mathcal B$
+in $[(A+1)R,(B_0-1)R]$, so $\#\mathcal B\asymp a^{D'}$.
+On $\mathcal B\times J$ use counting measure times Lebesgue measure.
+Mark $(b,t)$ if a large candidate satisfies
+\[
+ |d(x,y)-b|\leq5s,\qquad t\in I_y^{\mathrm{out}}.
+\]
+Each candidate meets at most 11 bins. By
+\eqref{eq:closed-range-length} and \eqref{eq:large-total-length}, the
+marked measure is at most $CC_Ha^{\gamma D}\Delta$. Since $\gamma<\beta$,
+the unmarked measure is at least
+\begin{equation}
+ ca^{D'}\Delta.
+ \label{eq:unmarked-mass}
+\end{equation}
+
+\emph{Detect the remaining levels by modulus.}
+Take a maximal $s$-separated set $\mathcal Z_s$ in $\mathcal A_x(R)$;
+its radius-$s$ balls cover that annulus, and it has at most $Ca^{QD'}$ points. Expand at
+each $z\in\mathcal Z_s$ at depth $k+D'$ by $h_z$. Put
+$q=r/s=a^{-(D-D')}$, and take macroscopic admissible weights
+$\omega_\xi$, $\xi\in T_{D-D'}$, of squared mass $M(D-D')$.
+For fixed $C_T>2+C_E$, define
+\[
+ \sigma_y^z=
+ \begin{cases}
+ \displaystyle\sum_{\xi:\ d(\xi,h_zy)\leq C_Tq}\omega_\xi,
+       &d(y,z)\leq3s,\\[3pt]
+ 0,&d(y,z)>3s,
+ \end{cases}
+ \qquad \sigma_y=\sum_{z\in\mathcal Z_s}\sigma_y^z.
+\]
+Within $\overline B(z,3s)$ the candidate images are separated by a
+fixed multiple of $q$. Thus every sum has boundedly many terms and
+every target weight is charged boundedly many times. Thus
+$\sum_y(\sigma_y^z)^2\leq CM(D-D')$ for each $z$. Also each $y$
+lies within $3s$ of boundedly many auxiliary centers. Packing and
+Cauchy--Schwarz give
+\begin{equation}
+ \sum_y\sigma_y^2\leq Ca^{QD'}M(D-D').
+ \label{eq:capacity-transfer-energy}
+\end{equation}
+
+Fix an unmarked $(b,t)$. Choose $w\in C_t$ on the radius-$b$ sphere
+and $z\in\mathcal Z_s$ with $d(w,z)\leq s$. The diameter bound gives
+a point of $C_t$ outside $\overline B(w,2s)$. Join these points in the
+$r/4$-neighborhood of $C_t$, by \Cref{lem:paths}, and stop at the first
+radius-$s$ hit about $w$. The resulting path $\pi$ has diameter at
+least $s$ and lies in $\overline B(w,s)\subset\overline B(z,2s)$.
+The scale-$s$ expansion $h_z$ therefore makes it macroscopic by the
+choice of $d_0$ in \Cref{sec:geometry}.
+
+The radial bin lies between $(A+1)R$ and $(B_0-1)R$, while $s<R/4$.
+Thus every point of $\pi$ lies between radii $(A+3/4)R$ and
+$(B_0-3/4)R$ about $x$. In particular it lies in the candidate
+annulus and has a candidate center within distance $r$. The same
+fixed source margin places every level point within $r/4$ of $\pi$
+in $K^+\subset U$. On these points the scalar-limit property gives
+$u=t$, even though it does not assert this identity on parts of
+$C_t$ outside $U$.
+
+For a target cell meeting $h_z\pi$, choose a preimage hit $v\in\pi$
+and $y\in Y$ with $d(y,v)\leq r$. Then $d(y,z)<3s$. If $\xi$ is the target cell's center,
+\eqref{eq:expansion} gives $d(\xi,h_zy)<(2+C_E)q<C_Tq$, so its
+weight occurs in $\sigma_y^z$. A point of $C_t$ is within $r/4$ of
+$v$, hence within $5r/4<c_0r$ of $y$. The preceding interior check
+gives $t\in I_y^{\mathrm{out}}$. Since
+$|d(x,y)-b|\leq s+r<5s$, unmarkedness makes $y$ small. Admissibility
+therefore yields
+\begin{equation}
+ \sum_{\substack{\text{small }y:\ |d(x,y)-b|\leq5s}}
+ \sigma_y\mathbf1_{I_y^{\mathrm{out}}}(t)\geq1
+ \quad\text{on every unmarked pair}.
+ \label{eq:unmarked-indicator}
+\end{equation}
+
+The unmarked levels now have an admissible detection weight supported
+on small candidates. We compare its squared mass with the surviving
+amount of level parameter.
+
+\emph{Compare the energies.}
+Integrating \eqref{eq:unmarked-indicator}, using bounded bin multiplicity,
+gives
+\[
+ ca^{D'}\Delta\leq C\Delta\sum_{\text{small }y}\sigma_y t_y.
+\]
+Only finitely many closed-interval indicators are integrated; no
+measurable selection of continua or paths is needed, and singleton
+ranges contribute zero. Cancel $\Delta>0$. Cauchy--Schwarz and
+\eqref{eq:capacity-transfer-energy} imply
+\begin{equation}
+ \sum_{\text{small }y}t_y^2
+ \geq\frac{ca^{-(Q-2)D'}}{M(D-D')}.
+ \label{eq:small-square-capacity}
+\end{equation}
+Subexponential growth gives $M(D-D')\leq a^{\tau D/4}$ for large $D$: first
+bound $M(j)\leq Ca^{\tau j/8}$ and then absorb $C$ by increasing $D$. Since $D'\leq\beta D+1$ and $(Q-2)\beta<\tau/4$,
+\[
+ \sum_{\text{small }y}a^{\tau D}t_y^2
+ \geq ca^{[3\tau/4-(Q-2)\beta]D}
+ \geq ca^{\tau D/2}.
+\]
+On small candidates this is exactly the capacity, contradicting
+\eqref{eq:failed-capacity}. Every threshold for $D$ is uniform over nodes.
+\end{proof}
+
+Fix such a $D$. At each good node, divide the positive capacities on a
+set $Y'$ from \Cref{lem:separated-capacity} by their sum. These
+probabilities satisfy \eqref{eq:transition-general}. At the remaining
+nodes use the uniform rule above. Every positive transition retains
+a nonempty open child interval $J_y\subset J$.
+
+\subsection{Common target patterns}
+
+The probabilities are now fixed. We verify the additional geometric
+information carried by a good node: its two source variations expand
+into a pair of balls chosen from a finite list, with separated range
+intervals. The list depends on the fixed $D$, but not on the node's
+generation or interval length.
+
+Fix a node's expansion $h=v^{-1}$ when constructing it. At a good node,
+join the extrema of each inner ball by a path in
+$B(y^\pm,c_{\mathrm{in}}r)$. Each path has endpoint variation at least
+$\kappa\Delta$. Its points lie at distance at least $(A-2)R$ from $x$,
+so \eqref{eq:expansion} puts its image within
+$c_{\mathrm{pat}}\theta/2$ of $hy^\pm$.
+Conversely, write $D_w=\max\{R,d(x,w)\}$. If $d(y,z)\geq c_0r$, then
+$D_z\leq D_y+d(y,z)$ and monotonicity of $s/(D_y+s)$ give
+\[
+ d(hy,hz)\geq\frac{c_ERc_0r}{D_y(D_y+c_0r)}
+ \geq\frac{c_Ec_0\theta}{(B_0+2)^2}>12c_{\mathrm{pat}}\theta.
+\]
+Thus
+\begin{equation}
+ \overline B(hy,10c_{\mathrm{pat}}\theta)
+       \subset h(B(y,c_0r)).
+ \label{eq:pattern-containment}
+\end{equation}
+Choose a fixed finite $c_{\mathrm{pat}}\theta$-net in $Z$. For each
+$hy^\pm$ choose a net point within $c_{\mathrm{pat}}\theta$ and take
+the closed radius-$3c_{\mathrm{pat}}\theta$ ball about it. The ordered
+pair $(\mathcal B^-,\mathcal B^+)$ is the node's \emph{pattern}.
+Each ball contains its path image and lies in the corresponding outer
+source-ball image. Consequently, for $u\circ h^{-1}$ the two whole-ball
+range intervals have distance at least $\Delta/4$, and each ball
+contains a path with endpoint variation at least $\kappa\Delta$.
+There are finitely many patterns, independently of depth.
+
+\subsection{The tree and its flow}
+
+Choose $k_0$ so large that $R_0=r_{k_0}$ satisfies all the source-scale
+conditions above and
+\[
+ 2(B_0+1)R_0<\epsilon_0/2,\qquad (B_0+2)R_0<d_1/2.
+\]
+Non-local-constancy at $x_0$ supplies a nonempty open interval
+$J_0\subset(0,1)$ strictly inside the range on
+$\overline B(x_0,c_0R_0)$. Write $\Delta_0=|J_0|$ and construct the
+tree recursively, storing each node's expansion and, at good nodes,
+its pattern.
+
+A child moves by at most $(B_0+1)R$ from a parent of scale $R$.
+Consequently all descendants stay within
+\begin{equation}
+ \frac{B_0+1}{1-\theta}R\leq2(B_0+1)R
+ \label{eq:descendant-displacement}
+\end{equation}
+of that parent, and all centers lie in $K$. Generation $n$ has scale
+$r_{k_0+Dn}$ and its centers are $H$ times that scale apart: this holds
+for siblings, while children of distinct parents of scale $R$ are
+separated by at least
+\[
+ [H-2(B_0+1)]R>HR/2>H\theta R.
+\]
+At any level of scale $R$, \eqref{eq:measure} bounds $\Delta_i^2$ by
+$C\mu(\overline B(x_i,C_\mu c_0R))$. These enlarged balls are disjoint
+because $H>4C_\mu c_0$, and hence
+\begin{equation}
+ \sum_i\Delta_i^2\leq E_0,\qquad E_0=C\mu(Z)<\infty.
+ \label{eq:level-energy}
+\end{equation}
+
+Give the root probability 1 and let $P_i$ be the product of transitions
+to node $i$. Every level has total probability 1. If $i$ has generation
+$n$, let $m_i$ count its ancestors at generations $0,\ldots,n-1$ that
+are not good. Multiplying the transition bounds and telescoping the
+interval ratios yields
+\begin{equation}
+ P_i\leq a^{-\gamma Dn}\frac{\Delta_i}{\Delta_0},\qquad
+ \Delta_i^2\geq\Delta_0^2P_i
+ a^{D((b_*+\tau)m_i-\tau n)}.
+ \label{eq:flow}
+\end{equation}
+Conditionally, if $i$ is $m$ generations below $j$, nesting gives
+\begin{equation}
+ \frac{P_i}{P_j}\leq a^{-\gamma Dm}\frac{\Delta_i}{\Delta_j}
+ \leq a^{-\gamma Dm}.
+ \label{eq:conditional-flow}
+\end{equation}
+
+\begin{lemma}[Levels with substantial good mass]
+\label{lem:good-levels}
+Infinitely many generations have total probability at least $1/2$
+on good nodes.
+\end{lemma}
+\begin{proof}
+By \eqref{eq:flow}, \eqref{eq:level-energy}, and Jensen's inequality,
+\[
+ \frac{E_0}{\Delta_0^2}
+ \geq\sum_iP_i a^{D((b_*+\tau)m_i-\tau n)}
+ \geq a^{D((b_*+\tau)\sum_iP_im_i-\tau n)}.
+\]
+Thus
+\[
+ \frac1n\sum_iP_im_i\leq\frac{\tau}{b_*+\tau}+O(1/n)
+ =\frac1{11}+O(1/n).
+\]
+Conservation of probability identifies the left side with the average
+bad mass over the first $n$ generations. If good mass were eventually
+less than $1/2$, that average would have lower limit at least $1/2$,
+a contradiction.
+\end{proof}
+
+\begin{proposition}[Properties of the flow]
+\label{prop:probability-flow}
+The construction gives a fixed $D$ and a tree with generation-$n$
+centers $Hr_{k_0+Dn}$-separated and descendant displacement
+\eqref{eq:descendant-displacement}. Its nonempty value intervals are
+nested, its probabilities satisfy \eqref{eq:flow} and
+\eqref{eq:conditional-flow}, and infinitely many levels have good mass
+at least $1/2$. Every node has a fixed expansion. Each good node has
+a pattern from a finite list of ordered target-ball pairs. For that expansion,
+each target ball contains a path with endpoint variation at least
+$\kappa\Delta_i$, their whole-ball range intervals have distance at
+least $\Delta_i/4$, and the balls lie in the images of the respective
+outer balls of radius $c_0\theta r_{k_0+Dn}$.
+\end{proposition}
diff --git a/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/build/sections/05_pairs.tex b/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/build/sections/05_pairs.tex
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@@ -0,0 +1,356 @@
+\section{Same-depth expansions and the measure comparison}
+\label{sec:pairs}
+
+We compare expansions at many nodes of a single generation. Almost
+every pair of punctures is sufficiently separated. A level-continuum
+argument then forces one of the two charts to spend a definite amount
+of oscillation near the other's puncture. Pulling those neighborhoods
+back gives more total measure than their overlap permits.
+
+\begin{proposition}
+\label{prop:no-subexponential-records}
+The moduli $M(n)$ cannot be unbounded while satisfying
+\[
+ \limsup_{n\to\infty}M(n)^{1/n}\leq1.
+\]
+\end{proposition}
+
+We prove this proposition through the pair comparison and counting
+estimates below. Fix the flow of \Cref{prop:probability-flow}, including $D$, and a
+compact neighborhood of its centers in $U$. Use the point $z_0$ chosen
+in the flow construction, a positive distance from that neighborhood.
+At generation $n$ write
+\[
+ k=k_0+Dn,\qquad R=a^{-k},\qquad p_i=h_i z_0,
+\]
+where $h_i=v_i^{-1}$ is the expansion already chosen for node $i$.
+The centers $x_i$ are $HR$-separated, and
+\begin{equation}
+ P_i\le a^{-\gamma Dn}\frac{\Delta_i}{\Delta_0},
+ \qquad \frac{P_j}{P_A}\le a^{-\gamma Dm}
+ \quad\text{if $j$ lies $m$ generations below $A$}.
+ \label{eq:pairs-flow-input}
+\end{equation}
+We use the arbitrarily large generations having good mass at least
+$1/2$, supplied by \Cref{lem:good-levels}.
+
+A good node has a pattern $(\mathcal B^-,\mathcal B^+)$ from a fixed
+finite list. Each target ball contains a compact path with endpoint
+$u\circ h_i^{-1}$-variation at least $\kappa\Delta_i$, and lies in
+the image of its outer source ball. The two target range intervals
+have distance at least $\Delta_i/4$. Throughout, a \emph{range interval}
+means the interval between the minimum and maximum; metric balls need
+not be connected.
+
+All geometric and counting constants below are independent of the
+generation, nodes, and approximating index $N$. Constants and thresholds
+in convergence statements may depend on a pair after it has been fixed.
+
+\subsection{A probability class and separated punctures}
+
+\begin{lemma}[Bounded puncture multiplicity]
+\label{lem:puncture-multiplicity}
+For every fixed $C>0$, the balls
+$\{\overline B(p_i,CR):i\text{ at generation }n\}$ have uniformly
+bounded multiplicity.
+\end{lemma}
+
+\begin{proof}
+Centers assigned to the same depth-$k$ vertex have mutual distance
+$O(R)$. Their $HR$-separation and \Cref{lem:packing} bound their number.
+If several puncture balls share a point, their punctures have mutual
+products at least $k-O_C(1)$. By \eqref{eq:inverse-prefix} and
+\Cref{lem:prefix}, all corresponding inverse vertices lie in one
+bounded Cayley-graph ball. There are boundedly many such vertices,
+each with the bounded multiplicity just proved.
+\end{proof}
+
+There are at most $Ca^{Qk}$ nodes. Discarding those with
+$P_i<a^{-2Qk}$ loses probability at most $Ca^{-Qk}$, so the remaining
+good nodes have mass at least $1/3$ for large $n$. Partitioning by
+pattern and into the $O(k)$ dyadic probability classes gives a class
+$\mathcal I$ with
+\begin{equation}
+ P_*\le P_i\le2P_*\quad(i\in\mathcal I),
+ \qquad \sum_{i\in\mathcal I}P_i\ge\frac c k.
+ \label{eq:pairs-class}
+\end{equation}
+Writing $\mathcal N=|\mathcal I|$, we have
+\begin{equation}
+ \mathcal NP_*\ge\frac{c}{2k}.
+ \label{eq:pairs-class-size}
+\end{equation}
+Also $P_i\le a^{-\gamma Dn}$ by \eqref{eq:pairs-flow-input}, so
+$\mathcal N\ge ck^{-1}a^{\gamma Dn}\to\infty$.
+We will use punctures separated at a scale $a^{-l}$ much larger than
+$R$. The final source-overlap estimate carries a factor $a^{Ql}$,
+so we arrange $Ql\leq\gamma Dn$. Put
+\[
+ \zeta=\gamma/Q,\qquad m=\lfloor\zeta n\rfloor,\qquad l=Dm.
+\]
+Both $l$ and $k-l$ tend to infinity.
+
+\begin{lemma}[Probability near a puncture]
+\label{lem:near-puncture-probability}
+For every generation-$n$ node $i$,
+\begin{equation}
+ \sum_{\substack{j\text{ at generation }n\\d(p_j,p_i)\le a^{-l}}}P_j
+ \le Ca^{-\gamma Dm}.
+ \label{eq:near-puncture-probability}
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Fix an ancestor $A$ at generation $n-m$, whose depth is
+$k-l=k_0+D(n-m)$. We claim that only boundedly
+many descendants $j$ have $d(p_j,p_i)\le a^{-l}$. Split a geodesic word
+\[
+ v_j^{-1}=w_ju'_j,\qquad |w_j|=l,\qquad |u'_j|=k-l.
+\]
+The puncture condition and \eqref{eq:inverse-prefix} give
+$(p_j|p_i)_e\ge l-C$ and $(v_j^{-1}|p_j)_e\ge k-C$.
+Thus $w_j$ lies a bounded distance from the depth-$l$ point on a fixed
+ray to $p_i$, leaving boundedly many possibilities.
+
+Reversing and inverting \emph{the same word} gives the geodesic
+$v_j=(u'_j)^{-1}w_j^{-1}$. Its depth-$(k-l)$ prefix is therefore
+$(u'_j)^{-1}$. Since
+\[
+ d(x_j,x_A)\le2(B_0+1)r_{k-l},
+\]
+thinness and \Cref{lem:prefix} put this prefix a bounded distance from
+the depth-$(k-l)$ point on a fixed ray to $x_A$. Hence $u'_j$ also has
+boundedly many possibilities. Their products determine boundedly
+many vertices $v_j$, and the first step of
+\Cref{lem:puncture-multiplicity} bounds the nodes assigned to each.
+We count group elements here, so nonuniqueness of geodesic spellings
+causes no additional multiplicity. The source and inverse prefixes
+together account for the full word.
+
+Each such descendant has $P_j\le P_Aa^{-\gamma Dm}$ by
+\eqref{eq:pairs-flow-input}. Sum first over these descendants and
+then over ancestors, whose probabilities sum to one.
+\end{proof}
+
+For each $i\in\mathcal I$, the fraction of its class with puncture
+distance at most $a^{-l}$ is bounded by
+\[
+ \frac{Ca^{-\gamma Dm}}{\mathcal NP_*}
+ \le Ck a^{-\gamma Dm}=o(1).
+\]
+This includes the diagonal. Consequently at least $c\mathcal N^2$
+unordered pairs satisfy
+\begin{equation}
+ d_{ij}:=d(p_i,p_j)>a^{-l}.
+ \label{eq:far-pair}
+\end{equation}
+Call them \emph{far pairs}. Uniformly for these pairs,
+$R/d_{ij}\le a^{-(k-l)}\to0$.
+
+\subsection{Comparison on a compact common domain}
+
+Let $K\subset U$ contain all centers, with its closed
+$\epsilon$-neighborhood still in $U$. The upper puncture bound gives
+\begin{equation}
+ d(h_i y,p_i)\ge A_hR/2
+ \quad\Longrightarrow\quad
+ d(x_i,y)\le D_y\le2C/A_h.
+ \label{eq:exterior-preimage}
+\end{equation}
+Choose $A_h$ so that $2C/A_h<\epsilon/2$ and $A_h/2>2C/d_1$, and set
+\[
+ E_i=\overline B(p_i,A_hR).
+\]
+Every $h_iC_{N,t}$ reaches $B(p_i,A_hR/2)$: the diameter bound supplies
+a source point at distance at least $d_1/2$ from $x_i$, whose image
+has puncture distance at most $2CR/d_1$.
+
+The target balls lie a uniform distance $\delta_*>0$ from $p_i$.
+Indeed, their preimages lie in the outer source balls, where
+$D_y\le(B_0+2)R$, so the lower puncture bound applies for large $n$.
+Because the pattern is shared, this holds for every $i\in\mathcal I$.
+For large $n$, uniformly for far pairs,
+\[
+ 2A_hR<a^{-l},\qquad A_hR<\delta_*/2.
+\]
+Thus $E_i,E_j$ are disjoint, each lies compactly in the other's
+transformed domain by \eqref{eq:exterior-preimage}, and the pattern
+balls avoid both. These thresholds are independent of $N$.
+
+\begin{lemma}[Pair alternative]
+\label{lem:pair-alternative}
+Every far pair at a sufficiently large generation has an endpoint
+$i$ satisfying
+\begin{equation}
+ \osc_{h_i^{-1}E_j}u\ge\frac{\kappa}{16}\Delta_i.
+ \label{eq:pair-alternative}
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Write $F_i=u\circ h_i^{-1}$ and $F_j=u\circ h_j^{-1}$ on their open
+domains. If $\osc_{E_i}F_j\ge\Delta_j/16$, endpoint $j$ qualifies.
+Here we use $\kappa\le1$.
+Otherwise, one pattern ball $\mathcal B^s$ has its $F_j$ range interval
+at distance at least $\Delta_j/16$ from that on $E_i$. Indeed, if
+neither did, the two pattern intervals would have distance less than
+$3\Delta_j/16$, contradicting their separation by $\Delta_j/4$.
+
+Node $i$ supplies a compact path $\eta\subset\mathcal B^s$ with
+endpoint $F_i$-gap at least $\kappa\Delta_i$. Suppose
+$\osc_{E_j}F_i<\kappa\Delta_i/4$. Removing this latter range interval
+from the endpoint gap leaves a positive-length closed interval
+$I\subset(0,1)$ strictly between the endpoint values and a positive
+distance from the range on $E_j$.
+To obtain positive margins, choose a smaller closed interval inside
+one of the remaining open intervals. Its length need not be uniform
+over pairs.
+
+\emph{Fix the pair and $I$ before letting $N\to\infty$.}
+Put $F_{i,N}=H_N\circ h_i^{-1}$ and $F_{j,N}=H_N\circ h_j^{-1}$.
+The fixed range margins and local uniform convergence imply, for
+every $t\in I$ and all sufficiently large $N$, that $h_iC_{N,t}$ hits
+$\eta$ and avoids $E_j$. It also reaches the inner $i$-puncture ball.
+For the first assertion, the endpoint values still strictly straddle
+all of $I$, so \Cref{lem:level-continua} applies to the preimage path.
+For the second, the range on $E_j$ remains disjoint from $I$.
+By \Cref{lem:stop-continuum}, it has a subcontinuum $K_{N,t}$ containing
+a hit on $\eta$, meeting the sphere $d(z,p_i)=A_hR$, and avoiding
+$B(p_i,A_hR)$. It remains in the level $F_{i,N}=t$ and avoids $E_j$.
+In particular,
+\[
+ K_{N,t}\subset K_{ij}:=
+ Z\setminus\bigl(B(p_i,A_hR)\cup B(p_j,A_hR)\bigr)
+ \Subset h_iU\cap h_jU.
+\]
+Compact containment follows from \eqref{eq:exterior-preimage} with
+half the radius. The sphere hit belongs to $E_i$; no identification
+of that sphere with the boundary of a metric ball is needed.
+
+Uniform convergence on $\eta\cup E_i$ preserves the $F_j$ range gap,
+giving $\osc_{K_{N,t}}F_{j,N}\ge\Delta_j/32$ uniformly for $t\in I$.
+The finite-cell comparison, \Cref{lem:cross-energy}, now gives
+\[
+ \frac{\Delta_j}{32}|I|\le
+ \sum_{\substack{q\in T_N\\B(q,2r_N)\cap K_{ij}\ne\varnothing}}
+ \osc_{\overline B(q,2r_N)}F_{i,N}\,
+ \osc_{\overline B(q,2r_N)}F_{j,N}.
+\]
+For large $N$ these closed cells lie in the common domain. Both squared
+oscillation sums tend to zero by \Cref{lem:local-oscillation}, so
+Cauchy--Schwarz contradicts the positive left side. Therefore
+$\osc_{E_j}F_i\ge\kappa\Delta_i/4$, proving the alternative.
+The argument applies separately to every pair above the geometric
+threshold; no convergence rate uniform over pairs is required.
+\end{proof}
+
+\subsection{Cost balls and total multiplicity}
+
+We now turn the pair alternative into a measure bound and control the
+overlap of the sets being charged. The two indices have distinct roles:
+with $i$ fixed, the target punctures control the number of partners $j$;
+at a source point, separation of the centers controls the possible
+indices $i$. Both estimates will apply to the enlarged balls required
+by \eqref{eq:measure}.
+
+For an ordered far pair put $z_{ij}=h_i^{-1}p_j$. The relevant source
+scales are
+\[
+ \begin{array}{c|c}
+ \text{puncture denominator }D_{z_{ij}}&\asymp R/d_{ij}\\
+ \text{radius of the charged neighborhood}&R^2/d_{ij}^2.
+ \end{array}
+\]
+The radius divided by the denominator is $O(R/d_{ij})\to0$. This separation of scales is what
+allows a fixed enlargement of the charged neighborhood without losing
+control of its image.
+
+If $y\in h_i^{-1}E_j$, then
+$d(h_i y,p_i)\ge d_{ij}-A_hR\ge d_{ij}/2$, whence
+$D_y\le CR/d_{ij}$. These points approach $x_i$ uniformly, so the
+two-sided puncture bound applies and gives
+\begin{equation}
+ D_{z_{ij}}\asymp R/d_{ij}.
+ \label{eq:cost-denominator}
+\end{equation}
+The lower expansion bound and $d(h_i y,p_j)\le A_hR$ then give
+$d(y,z_{ij})\le CR^2/d_{ij}^2$. Fixing a sufficiently large $C_2$, put
+$s_{ij}=C_2R^2/d_{ij}^2$; thus
+\begin{equation}
+ h_i^{-1}E_j\subseteq\overline B(z_{ij},s_{ij}).
+ \label{eq:cost-containment}
+\end{equation}
+Choose the enlargement constant $C_\mu$ in \eqref{eq:measure}
+uniformly on a compact neighborhood of the centers and define
+\[
+ \mathcal D_{ij}=\overline B(z_{ij},C_\mu s_{ij}).
+\]
+Their centers stay in this neighborhood and their radii tend uniformly
+to zero. If $i$ satisfies \Cref{lem:pair-alternative}, \eqref{eq:measure} gives
+\begin{equation}
+ \mu(\mathcal D_{ij})\ge c\kappa^2\Delta_i^2.
+ \label{eq:cost-lower-bound}
+\end{equation}
+
+The enlarged balls still have small images. Indeed, $D_y$ is
+$1$-Lipschitz and
+\[
+ \frac{C_\mu s_{ij}}{D_{z_{ij}}}
+ \le C\frac R{d_{ij}}\le Ca^{-(k-l)}\longrightarrow0.
+\]
+Thus $D_y\asymp D_{z_{ij}}$ on $\mathcal D_{ij}$, and the upper
+expansion bound yields
+\begin{equation}
+ h_i\mathcal D_{ij}\subseteq\overline B(p_j,C_3R).
+ \label{eq:cost-image}
+\end{equation}
+Figure~\ref{fig:pair-cost} shows the charged neighborhood and the image
+control that survives the fixed enlargement.
+
+\input{figures/pair-cost}
+For fixed $i$, \Cref{lem:puncture-multiplicity} therefore bounds the
+multiplicity of $\mathcal D_{ij}$ as $j$ varies. Also
+\begin{equation}
+ \mathcal D_{ij}\subseteq\overline B(x_i,C_4Ra^l),
+ \label{eq:cost-source}
+\end{equation}
+because $d(x_i,z_{ij})\le CR/d_{ij}$ and the cost radius is
+$o(R/d_{ij})$. The $HR$-separation of the $x_i$ and
+\Cref{lem:packing} bound the multiplicity of these larger balls by
+$Ca^{Ql}$.
+
+\begin{proof}[Conclusion of the proof of \Cref{prop:no-subexponential-records}]
+Assign each unordered far pair to a qualifying endpoint. At any point,
+\eqref{eq:cost-source} allows only $Ca^{Ql}$ paying indices $i$, and
+\eqref{eq:cost-image} allows boundedly many partners for each. Integrating
+this multiplicity bound gives
+\begin{equation}
+ \sum_{\{i,j\}\text{ far, assigned to }i}\mu(\mathcal D_{ij})
+ \le Ca^{Ql}\mu(Z).
+ \label{eq:cost-total-upper}
+\end{equation}
+This counts the enlarged balls directly and uses no invariance of $\mu$.
+The assignment may be arbitrary when both endpoints qualify: all
+lower bounds hold for either endpoint, and the multiplicity estimate
+already counts every possible ordered contribution.
+
+Conversely, every paying endpoint satisfies
+\[
+ \Delta_i^2\ge\Delta_0^2a^{2\gamma Dn}P_i^2
+ \ge\Delta_0^2a^{2\gamma Dn}P_*^2.
+\]
+There are at least $c\mathcal N^2$ far pairs, so
+\eqref{eq:cost-lower-bound} and \eqref{eq:pairs-class-size} give total cost
+\begin{equation}
+ c\mathcal N^2\kappa^2\Delta_0^2a^{2\gamma Dn}P_*^2
+ \ge c'k^{-2}a^{2\gamma Dn}.
+ \label{eq:cost-total-lower}
+\end{equation}
+Since $\kappa,\Delta_0>0$ are fixed and
+$Ql=QD\lfloor\gamma n/Q\rfloor\le\gamma Dn$, comparison with
+\eqref{eq:cost-total-upper} yields
+\[
+ a^{\gamma Dn}\le Ck^2.
+\]
+This contradicts $k=k_0+Dn$, excluding the unbounded subexponential case.
+\end{proof}
diff --git a/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/paper.pdf b/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..84396bb85a95551d68f92f112f3aa7aec7271040
--- /dev/null
+++ b/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:650697e8e95aa8cbf86c4f3898d399d75d3f1551d683f9db6e254b370aea080b
+size 519583
diff --git a/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/README.md b/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..cb29cb83f6ceb1c1abd2cbafd0eb915cc60130ae
--- /dev/null
+++ b/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/README.md
@@ -0,0 +1,18 @@
+# [A Nearly Quartic Separation Between Randomized and Quantum Query Complexity](quartic-query-separation.pdf)
+
+**Author:** OpenAI
+
+**Date:** October 5, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026,
+  author = {{OpenAI}},
+  title = {{A Nearly Quartic Separation Between Randomized and Quantum Query Complexity}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/quartic-query-separation.pdf}{OAI:A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/build/figures/outward-pointers.tex b/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/build/figures/outward-pointers.tex
new file mode 100644
index 0000000000000000000000000000000000000000..dee59367527f629f09cb6aa6786665308517c1d6
--- /dev/null
+++ b/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/build/figures/outward-pointers.tex
@@ -0,0 +1,26 @@
+\begin{figure}[t]
+\centering
+\begin{tikzpicture}[>=Latex,font=\small,
+ col/.style={draw,rounded corners=2pt,minimum width=2.7cm,minimum height=1.05cm,align=center},
+ cell/.style={draw,fill=blue!6,minimum width=3.7cm,minimum height=0.85cm,align=center}]
+\node[col] (left) at (-4,1.8) {$X_j$\\$\cdots\ 0\ \cdots$};
+\node[col,fill=green!7] (win) at (0,1.8) {$X_{i_\ast}$\\$1\quad1\quad\cdots\quad1$};
+\node[col] (right) at (4,1.8) {$X_{j'}$\\$\cdots\ 0\ \cdots$};
+\node[align=center,text width=3.7cm,font=\footnotesize] at (0,3.0) {no incoming pointer can find a zero};
+\node[cell] (cert) at (0,-0.2) {$B_{i_\ast,s_{i_\ast}}$\\certified address and pointers};
+\node[align=center] (addr) at (0,-1.75) {$A_{i_\ast,1},\ldots,A_{i_\ast,r}$\\$s_{i_\ast}=(P(A_{i_\ast,h}))_{h=1}^r$};
+\draw[->] (addr) -- (cert);
+\draw[->,thick,blue!65!black] (cert.west) to[out=170,in=-90] (left.south);
+\draw[->,thick,blue!65!black] (cert.east) to[out=10,in=-90] (right.south);
+\draw[dashed] (win.south) -- (cert.north);
+\node[align=center,text width=2.8cm,font=\footnotesize] at (-4,-0.85) {zero witness\\disqualifies $j$};
+\node[align=center,text width=2.8cm,font=\footnotesize] at (4,-0.85) {zero witness\\disqualifies $j'$};
+\end{tikzpicture}
+\caption{The relevant cell of a winning column points outward to a zero
+in every other column; two such pointers are shown. Only the indicated
+zeros are prescribed in the other columns. The cell's address is
+computed from the partial-function answers and certified by stored
+circuit transcripts. The winner's all-one string prevents every opponent
+from disqualifying it, even at an incorrectly decoded address.}
+\label{fig:outward}
+\end{figure}
diff --git a/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/build/main.tex b/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/build/main.tex
new file mode 100644
index 0000000000000000000000000000000000000000..f857c437fd0ac612ee260aa9cf3d457c3630df74
--- /dev/null
+++ b/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/build/main.tex
@@ -0,0 +1,27 @@
+\newcommand{\DoNotLoadEpstopdf}{}
+\documentclass[11pt]{article}
+\input{preamble}
+\title{A Nearly Quartic Separation Between\texorpdfstring{\\}{ }Randomized and Quantum Query Complexity}
+\author{OpenAI}
+\date{October 5, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We construct total Boolean functions with a nearly quartic separation
+between bounded-error randomized and quantum query complexity.
+Writing these complexities as $\R(f)$ and $\Q(f)$, the examples rule
+out every universal bound $\R(f)=O((1+\Q(f))^\alpha)$ with
+$\alpha<4$. Thus the known quartic upper bound has the optimal
+exponent, disproving the conjectured cubic bound. Both complexities
+count worst-case bit queries with error at most $1/3$ on every input.
+\end{abstract}
+\input{sections/01-introduction}
+\input{sections/02-construction}
+\input{sections/03-quantum}
+\input{sections/04-classical}
+\input{sections/05-exponent}
+\appendix
+\input{sections/06-upper-bound}
+\bibliographystyle{plain}
+\bibliography{references}
+\end{document}
diff --git a/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/build/preamble.tex b/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/build/preamble.tex
new file mode 100644
index 0000000000000000000000000000000000000000..73c3db62a6bfa0cff0c013931121f8fd0be587d9
--- /dev/null
+++ b/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/build/preamble.tex
@@ -0,0 +1,28 @@
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage[margin=1in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{microtype,enumitem,needspace,placeins,xcolor,tikz,booktabs}
+\usetikzlibrary{arrows.meta,positioning,calc}
+\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref}
+\hypersetup{pdftitle={A Nearly Quartic Separation Between Randomized and Quantum Query Complexity},pdfauthor={OpenAI}}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\newtheorem{remark}[theorem]{Remark}
+\newcommand{\R}{\mathrm R}
+\newcommand{\Q}{\mathrm Q}
+\newcommand{\D}{\mathrm D}
+\newcommand{\E}{\mathbb E}
+\newcommand{\bits}{\{0,1\}}
+\newcommand{\bs}{\operatorname{bs}}
+\newcommand{\poly}{\operatorname{poly}}
+\DeclarePairedDelimiter{\norm}{\lVert}{\rVert}
+\DeclarePairedDelimiter{\abs}{\lvert}{\rvert}
+\DeclarePairedDelimiter{\ip}{\langle}{\rangle}
+\setlist[enumerate]{label=\textup{(\roman*)},leftmargin=*}
+\setlength{\emergencystretch}{1.5em}
+\allowdisplaybreaks[1]
diff --git a/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/build/references.bib b/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/build/references.bib
new file mode 100644
index 0000000000000000000000000000000000000000..d5938de5ea9400f00079b1134928f92eab6a6cea
--- /dev/null
+++ b/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/build/references.bib
@@ -0,0 +1,329 @@
+
+@inproceedings{Grover96,
+  author = {Lov K. Grover},
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+  publisher = {ACM},
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+  eprint = {quant-ph/9605043},
+  archivePrefix = {arXiv}
+}
+
+@article{BBCMW01,
+  author = {Robert Beals and Harry Buhrman and Richard Cleve and Michele Mosca and Ronald de Wolf},
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+  archivePrefix = {arXiv}
+}
+
+@article{BBBV97,
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+  title = {Strengths and Weaknesses of Quantum Computing},
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+  archivePrefix = {arXiv}
+}
+
+@inproceedings{GPW15,
+  author = {Mika G{\"o}{\"o}s and Toniann Pitassi and Thomas Watson},
+  title = {Deterministic Communication vs. Partition Number},
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+  pages = {1077--1088},
+  year = {2015},
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+}
+
+@article{ABBLSS17,
+  author = {Andris Ambainis and Kaspars Balodis and Aleksandrs Belovs and Troy Lee and Miklos Santha and Juris Smotrovs},
+  title = {Separations in Query Complexity Based on Pointer Functions},
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+  eprint = {1506.04719},
+  archivePrefix = {arXiv}
+}
+
+@inproceedings{ABK16,
+  author = {Scott Aaronson and Shalev Ben-David and Robin Kothari},
+  title = {Separations in Query Complexity Using Cheat Sheets},
+  booktitle = {Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing},
+  pages = {863--876},
+  year = {2016},
+  publisher = {ACM},
+  doi = {10.1145/2897518.2897644},
+  eprint = {1511.01937},
+  archivePrefix = {arXiv}
+}
+
+@article{AA18,
+  author = {Scott Aaronson and Andris Ambainis},
+  title = {{Forrelation}: A Problem That Optimally Separates Quantum from Classical Computing},
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+  pages = {982--1038},
+  year = {2018},
+  doi = {10.1137/15M1050902},
+  eprint = {1411.5729},
+  archivePrefix = {arXiv}
+}
+
+@inproceedings{Tal20,
+  author = {Avishay Tal},
+  title = {Towards Optimal Separations between Quantum and Randomized Query Complexities},
+  booktitle = {Proceedings of the 61st Annual IEEE Symposium on Foundations of Computer Science},
+  pages = {228--239},
+  year = {2020},
+  publisher = {IEEE},
+  doi = {10.1109/FOCS46700.2020.00030},
+  eprint = {1912.12561},
+  archivePrefix = {arXiv}
+}
+
+@inproceedings{BS21,
+  author = {Nikhil Bansal and Makrand Sinha},
+  title = {{$k$-Forrelation} Optimally Separates Quantum and Classical Query Complexity},
+  booktitle = {Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing},
+  pages = {1303--1316},
+  year = {2021},
+  publisher = {ACM},
+  doi = {10.1145/3406325.3451040},
+  eprint = {2008.07003},
+  archivePrefix = {arXiv}
+}
+
+@article{SSW23,
+  author = {Alexander A. Sherstov and Andrey A. Storozhenko and Pei Wu},
+  title = {An Optimal Separation of Randomized and Quantum Query Complexity},
+  journal = {SIAM Journal on Computing},
+  volume = {52},
+  number = {2},
+  pages = {525--567},
+  year = {2023},
+  doi = {10.1137/22M1468943},
+  eprint = {2008.10223},
+  archivePrefix = {arXiv},
+  note = {Preliminary version in STOC 2021}
+}
+
+@inproceedings{ABKRT21,
+  author = {Scott Aaronson and Shalev Ben-David and Robin Kothari and Shravas Rao and Avishay Tal},
+  title = {Degree vs. Approximate Degree and Quantum Implications of {Huang}'s Sensitivity Theorem},
+  booktitle = {Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing},
+  pages = {1330--1342},
+  year = {2021},
+  publisher = {ACM},
+  doi = {10.1145/3406325.3451047},
+  eprint = {2010.12629},
+  archivePrefix = {arXiv}
+}
+
+@article{Huang19,
+  author = {Hao Huang},
+  title = {Induced Subgraphs of Hypercubes and a Proof of the {Sensitivity Conjecture}},
+  journal = {Annals of Mathematics},
+  volume = {190},
+  number = {3},
+  pages = {949--955},
+  year = {2019},
+  doi = {10.4007/annals.2019.190.3.6},
+  eprint = {1907.00847},
+  archivePrefix = {arXiv}
+}
+
+@misc{Midrijanis04,
+  author = {Gatis Midrijanis},
+  title = {Exact Quantum Query Complexity for Total {Boolean} Functions},
+  year = {2004},
+  eprint = {quant-ph/0403168},
+  archivePrefix = {arXiv},
+  note = {arXiv:quant-ph/0403168}
+}
+
+@article{Ambainis02,
+  author = {Andris Ambainis},
+  title = {Quantum Lower Bounds by Quantum Arguments},
+  journal = {Journal of Computer and System Sciences},
+  volume = {64},
+  number = {4},
+  pages = {750--767},
+  year = {2002},
+  doi = {10.1006/jcss.2002.1826},
+  eprint = {quant-ph/0002066},
+  archivePrefix = {arXiv}
+}
+
+@inproceedings{BSS03,
+  author = {Howard Barnum and Michael E. Saks and Mario Szegedy},
+  title = {Quantum Query Complexity and Semi-Definite Programming},
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+  pages = {179--193},
+  year = {2003},
+  publisher = {IEEE},
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+}
+
+@article{SpalekSzegedy06,
+  author = {Robert {\v S}palek and Mario Szegedy},
+  title = {All Quantum Adversary Methods Are Equivalent},
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+  year = {2006},
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+  eprint = {quant-ph/0409116},
+  archivePrefix = {arXiv}
+}
+
+@article{Nisan91,
+  author = {Noam Nisan},
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+}
+
+@article{NisanSzegedy94,
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+}
+
+@article{BBHT98,
+  author = {Michel Boyer and Gilles Brassard and Peter H{\o}yer and Alain Tapp},
+  title = {Tight Bounds on Quantum Searching},
+  journal = {Fortschritte der Physik},
+  volume = {46},
+  number = {4--5},
+  pages = {493--506},
+  year = {1998},
+  eprint = {quant-ph/9605034},
+  archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/quant-ph/9605034}
+}
+
+@incollection{BHMT02,
+  author = {Gilles Brassard and Peter H{\o}yer and Michele Mosca and Alain Tapp},
+  title = {Quantum Amplitude Amplification and Estimation},
+  booktitle = {Quantum Computation and Quantum Information},
+  editor = {Lomonaco, Jr., Samuel J.},
+  series = {Contemporary Mathematics},
+  volume = {305},
+  pages = {53--74},
+  year = {2002},
+  publisher = {American Mathematical Society},
+  doi = {10.1090/conm/305/05215},
+  eprint = {quant-ph/0005055},
+  archivePrefix = {arXiv}
+}
+
+@misc{DurrHoyer96,
+  author = {Christoph D{\"u}rr and Peter H{\o}yer},
+  title = {A Quantum Algorithm for Finding the Minimum},
+  year = {1996},
+  eprint = {quant-ph/9607014},
+  archivePrefix = {arXiv},
+  note = {arXiv:quant-ph/9607014}
+}
+
+@inproceedings{SYZ04,
+  author = {Xiaoming Sun and Andrew C. Yao and Shengyu Zhang},
+  title = {Graph Properties and Circular Functions: How Low Can Quantum Query Complexity Go?},
+  booktitle = {Proceedings of the 19th Annual IEEE Conference on Computational Complexity},
+  pages = {286--293},
+  year = {2004},
+  publisher = {IEEE},
+  doi = {10.1109/CCC.2004.1313851}
+}
+
+@inproceedings{MPS23,
+  author = {Nikhil S. Mande and Manaswi Paraashar and Nitin Saurabh},
+  title = {Randomized and Quantum Query Complexities of Finding a King in a Tournament},
+  booktitle = {43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science},
+  series = {Leibniz International Proceedings in Informatics},
+  volume = {284},
+  pages = {30:1--30:19},
+  year = {2023},
+  publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
+  doi = {10.4230/LIPIcs.FSTTCS.2023.30},
+  eprint = {2308.02472},
+  archivePrefix = {arXiv}
+}
+
+@misc{BK26,
+  author = {Shalev Ben-David and Robin Kothari},
+  title = {Randomized Query Complexity Can Beat Certificate Complexity},
+  year = {2026},
+  eprint = {2609.15063},
+  archivePrefix = {arXiv},
+  note = {arXiv:2609.15063}
+}
+
+@misc{AIK26,
+  author = {Andris Ambainis and J{\=a}nis Iraids and Martins Kokainis},
+  title = {Near-Optimal Separations of Certificate Complexity from Randomized and Quantum Query Complexity},
+  year = {2026},
+  eprint = {2609.11664},
+  archivePrefix = {arXiv},
+  note = {arXiv:2609.11664v2}
+}
+
+@article{Shaltiel03,
+  author = {Ronen Shaltiel},
+  title = {Towards Proving Strong Direct Product Theorems},
+  journal = {Computational Complexity},
+  volume = {12},
+  number = {1--2},
+  pages = {1--22},
+  year = {2003},
+  doi = {10.1007/s00037-003-0175-x}
+}
+
+@article{Drucker12,
+  author = {Andrew Drucker},
+  title = {Improved Direct Product Theorems for Randomized Query Complexity},
+  journal = {Computational Complexity},
+  volume = {21},
+  number = {2},
+  pages = {197--244},
+  year = {2012},
+  doi = {10.1007/s00037-012-0043-7},
+  eprint = {1005.0644},
+  archivePrefix = {arXiv}
+}
+
+@inproceedings{Yao77,
+  author = {Andrew Chi-Chih Yao},
+  title = {Probabilistic Computations: Toward a Unified Measure of Complexity},
+  booktitle = {Proceedings of the 18th Annual Symposium on Foundations of Computer Science},
+  pages = {222--227},
+  year = {1977},
+  publisher = {IEEE},
+  doi = {10.1109/SFCS.1977.24}
+}
diff --git a/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/build/sections/01-introduction.tex b/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/build/sections/01-introduction.tex
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index 0000000000000000000000000000000000000000..8a0b208fb30ff4a92d5c352e6caf5dfac02f9b0c
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+++ b/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/build/sections/01-introduction.tex
@@ -0,0 +1,157 @@
+\section{Introduction}\label{sec:introduction}
+
+How much can a quantum algorithm reduce the number of input bits needed
+to compute a total Boolean function? Let $f:\bits^n\to\bits$. We write
+$\R(f)$ and $\Q(f)$ for the minimum worst-case numbers of bit queries made
+by randomized and quantum algorithms that compute $f$ with error at most
+$1/3$ on every input. Quantum queries use the standard oracle
+\[
+ O_x\lvert i,b,z\rangle=\lvert i,b\oplus x_i,z\rangle.
+\]
+Computation between queries is unrestricted. In particular, all query
+bounds in this paper are worst-case bounds, including over internal
+randomness and measurement outcomes.
+
+Polynomial relations between these two measures express a basic
+restriction on quantum speedups for total functions. To ask for the sharp
+power without imposing a particular logarithmic correction, define
+\begin{equation}\label{eq:alpha}
+ \alpha_{\mathrm{tot}}=
+ \inf\left\{\alpha\ge0:
+ \begin{array}{l}
+ \text{there exists }C_\alpha<\infty\text{ such that, for every }n\ge1\nobreak\text{ and}\smallskip\\
+ f:\bits^n\to\bits,\quad
+ \R(f)\le C_\alpha(1+\Q(f))^\alpha
+ \end{array}\right\}.
+\end{equation}
+The additive $1$ includes constant functions. We prove the following.
+
+\begin{theorem}\label{thm:main}
+The optimal power is $\alpha_{\mathrm{tot}}=4$.
+More precisely, $\R(f)=O((1+\Q(f))^4)$ for every total Boolean function.
+For each fixed integer $k\ge2$, there are positive constants $c_k,C_k,b_k$
+and total Boolean functions $F_{k,m}$, for all sufficiently large powers
+of four $m$, such that
+\begin{equation}\label{eq:main-family}
+ \Q(F_{k,m})\le C_k\sqrt m\,(\log m)^{b_k},
+ \qquad
+ \R(F_{k,m})\ge c_k\frac{m^{2-1/k}}{(\log m)^2}.
+\end{equation}
+\end{theorem}
+
+The universal upper bound is known. The contribution is the family in
+\eqref{eq:main-family}, which rules out every smaller power, including
+the conjectured cubic relation.
+
+\subsection{Historical context}
+
+Grover's search algorithm~\cite{Grover96} gives the basic quadratic
+speedup for the OR function, with a matching quantum lower bound
+\cite{BBBV97}. Beals, Buhrman, Cleve, Mosca, and de Wolf~\cite{BBCMW01}
+proved the general polynomial relation $\D(f)=O(\Q(f)^6)$, where
+$\D(f)$ is exact deterministic query complexity. Their result builds
+on the combinatorial and polynomial measures studied by
+Nisan~\cite{Nisan91} and Nisan and Szegedy~\cite{NisanSzegedy94}.
+Thus totality rules out the exponential query speedups possible for
+partial functions, but it leaves a substantial range of possible powers.
+
+Pointer constructions introduced by G\"o\"os, Pitassi, and
+Watson~\cite{GPW15} and developed by Ambainis, Balodis, Belovs, Lee,
+Santha, and Smotrovs~\cite{ABBLSS17} gave a nearly quartic separation
+between deterministic and quantum queries. For bounded-error randomized
+queries, Aaronson, Ben-David, and Kothari~\cite{ABK16} introduced the
+cheat-sheet framework and obtained a power-$5/2$ separation, up to
+logarithmic factors. This framework places certificates in an array
+whose correct address is determined by answers to partial-function
+instances. Improvements in those partial-function separations led to
+power $8/3-o(1)$ by Tal~\cite{Tal20}, and then $3-o(1)$ by Bansal
+and Sinha~\cite{BS21} and independently Sherstov, Storozhenko, and
+Wu~\cite{SSW23}.
+
+On the upper-bound side, Aaronson, Ben-David, Kothari, Rao, and
+Tal~\cite{ABKRT21}, using Huang's sensitivity theorem~\cite{Huang19},
+proved $\D(f)=O(\Q(f)^4)$. They conjectured that randomized complexity
+satisfies the stronger relation $\R(f)=O(\Q(f)^3)$
+\cite[Conjecture~2]{ABKRT21}. Theorem~\ref{thm:main} disproves that
+conjecture and shows that the quartic upper bound already has the
+optimal power for bounded-error randomized complexity.
+
+There is also a direct connection to algorithms that compare candidate
+witnesses. Quantum minimum finding~\cite{DurrHoyer96}, sink finding
+\cite{SYZ04}, and king finding in tournaments~\cite{MPS23} provide
+related search procedures. Recent constructions of Ben-David and
+Kothari~\cite{BK26} and Ambainis, Iraids, and Kokainis~\cite{AIK26}
+use comparisons of candidate witnesses to separate certificate
+complexity from randomized and quantum query complexity. Certificate
+complexity measures the worst-case size of a smallest partial input
+assignment that fixes the function value. Their nearly quartic
+separations between certificate and quantum complexity concern a
+different classical measure, but their comparison-based viewpoint is
+particularly relevant here.
+
+\subsection{The construction and its two analyses}
+
+The input is organized into $m$ columns. Each column $i$ contains a string
+$X_i$ of $m$ bits and an array of certificate cells. The address of the
+relevant cell is the vector of answers to several disjoint instances
+of a partial function. Quantumly these answers are cheap to obtain on
+promised inputs; classically they are hard to guess. Each cell stores
+pointers to positions in all the other strings $X_j$, together with a
+circuit transcript certifying the address.
+
+A column wins when its own string is all ones, its address is defined,
+and its certificate is correct and points to a zero in every other
+column. The total function asks whether a winner exists. There can be
+at most one. An incorrect address or transcript simply fails verification;
+the input to the total function carries no promise.
+
+The direction of the pointers matters. A true winner can use its own
+certificate to disqualify every opponent. Conversely, no cell in any
+opponent can point to a zero in the winner's all-one string. Thus a cheap
+quantum comparison favors the winner with high probability even if the
+opponent's address instances violate their promises. Comparisons between
+two non-winners may have arbitrary probabilities. Section~\ref{sec:quantum}
+proves a tournament lemma that permits precisely this behavior. It
+produces a short list containing the winner in $\sqrt m$ times a
+polylogarithmic number of queries, after which quantum search checks the
+certificates.
+
+The classical proof starts with one independently hidden zero in every
+$X_i$ and blank certificate cells. We choose a uniformly random column,
+erase its zero, and fill its correctly addressed cell. These two changes
+produce a winner. A randomized algorithm can distinguish the two inputs
+only by finding the erased zero or querying the filled cell. The first
+event is unlikely for a uniformly chosen column; the second requires
+either many address queries in that column or a correct guess of an
+entire vector of hard answers. An adaptive direct-product argument makes
+this last assertion precise without conditioning on the algorithm's
+choice of which columns to investigate.
+
+The tournament analysis is useful independently of this construction.
+It uses the actual probabilities of comparisons as multiplicative
+weights; fresh coherent workspaces and amplitude amplification sample
+those weights exactly. Only comparisons involving a genuine winner
+need to be reliable. This extends the candidate-comparison approach
+to the arbitrary behavior caused here by off-promise address inputs.
+The classical analysis combines the planted input with a standard
+per-instance-capped direct-product argument
+\cite{Shaltiel03,Drucker12}; the separate cell arrays let this argument
+charge the cost of addressing many possible winners.
+
+Section~\ref{sec:construction} defines the function and its certificates.
+Sections~\ref{sec:quantum} and~\ref{sec:classical} prove the two bounds in
+\eqref{eq:main-family}; Section~\ref{sec:exponent} derives the optimal
+power. Appendix~\ref{sec:upper} gives a self-contained proof of the known
+quartic upper bound. The only external lower-bound input is the
+Forrelation theorem stated in Section~\ref{sec:construction}.
+
+\subsection{Conventions}
+
+We write $[m]=\{1,\ldots,m\}$ and use logarithms to base two, except in
+expressions where changing the base only changes an absolute constant.
+Constants in bounds for $F_{k,m}$ may depend on the fixed integer $k$.
+A partial Boolean function is specified on a subset of $\bits^N$;
+its algorithms must be correct on that subset. The quantum routines used
+here have a fixed query cap on every input, including inputs outside the
+promise. A circuit transcript lists proposed values of the non-input
+gates; the actual input bits are read directly when a gate check needs them.
diff --git a/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/build/sections/02-construction.tex b/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/build/sections/02-construction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..21d798a10a80cfb32f4972b0f8da0703d4bf28c4
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+++ b/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/build/sections/02-construction.tex
@@ -0,0 +1,221 @@
+\section{An addressed certificate with outward pointers}
+\label{sec:construction}
+
+We construct a total function whose accepting certificate belongs to one
+column.  The certificate must certify the answers to several Forrelation
+instances and point to a zero in every other column.  The latter requirement
+will let a genuine accepting column defeat any competitor, even when that
+competitor's Forrelation instances violate their promise.
+
+Throughout the construction, fix an integer $k\geq 2$ and let $m$ tend to
+infinity through powers of four.  Constants in $O_k(\cdot)$ may depend on
+$k$, which remains fixed as $m$ grows.  Write $[m]=\{1,\ldots,m\}$.
+
+\subsection{The partial function and its exact circuit}
+
+We use the $k$-fold Forrelation problem introduced by Aaronson and
+Ambainis~\cite{AA18}, with the thresholds analyzed by Bansal and
+Sinha~\cite{BS21}. Let $H$ be the normalized Walsh--Hadamard matrix of order $m$, and let
+$u=m^{-1/2}(1,\ldots,1)^{\mathsf T}$.  We identify a bit $a$ with the sign
+$(-1)^a$.  For a sign vector $z$, write $D_z$ for the diagonal matrix with
+diagonal $z$.  Given $k$ sign vectors $z_1,\ldots,z_k\in\{\pm1\}^m$, put
+\begin{equation}\label{eq:forrelation-form}
+ \Phi(z_1,\ldots,z_k)
+ =u^{\mathsf T}D_{z_1}H D_{z_2}H\cdots H D_{z_k}u,
+ \qquad \delta=2^{-5k}.
+\end{equation}
+Define the partial Boolean function $P=P_{k,m}$ on $km$ bits by
+\begin{equation}\label{eq:forrelation-promise}
+ P(z_1,\ldots,z_k)=
+ \begin{cases}
+  1,&\Phi(z_1,\ldots,z_k)\geq\delta,\\
+  0,&|\Phi(z_1,\ldots,z_k)|\leq\delta/2,
+ \end{cases}
+\end{equation}
+and leave it undefined on all other inputs.  The two displayed cases are
+disjoint.  Query complexity for this partial function requires correctness
+only on its domain.
+
+\begin{lemma}\label{lem:forrelation}
+For each fixed integer $k\geq2$, the functions $P_{k,m}$ have the following
+properties.
+\begin{enumerate}
+ \item For all sufficiently large powers of four $m$,
+ \begin{equation}\label{eq:L-def}
+  \R(P_{k,m})>L,
+  \qquad
+  L=\left\lfloor
+       \frac{m^{1-1/k}}{(\log_2 m)^2}
+     \right\rfloor.
+ \end{equation}
+ \item For every $0<\eta<1/2$, a quantum algorithm uses
+ $O_k(1+\log(1/\eta))$ bit queries and returns $P_{k,m}$ with error at most
+ $\eta$ on every input in its domain.  The algorithm can be implemented
+ coherently, retaining its workspace.  On every input, promised or not, it
+ produces a bit when its output register is measured.
+ \item There is a Boolean circuit $C=C_{k,m}$ of fan-in at most two and
+ size $S=S_{k,m}=O_k(m(\log m)^2)$ with two output bits $d,b$ such that
+ \[
+  d=1\quad\Longleftrightarrow\quad P_{k,m}\text{ is defined},
+  \qquad d=1\quad\Longrightarrow\quad b=P_{k,m}.
+ \]
+ The circuit computes these outputs exactly on every input.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+Theorem~1.3 and Corollary~1.4 of Bansal and Sinha~\cite{BS21} apply to
+exactly the normalized form and thresholds in
+\eqref{eq:forrelation-form}--\eqref{eq:forrelation-promise}.  They give,
+for fixed $k$ and error $1/3$,
+\[
+ \R(P_{k,m})=
+ \Omega_k\!\left(
+   \left(\frac{m}{\log(km)}\right)^{1-1/k}
+ \right).
+\]
+This exceeds the value of $L$ in \eqref{eq:L-def} for all sufficiently
+large $m$.
+
+For the quantum algorithm, prepare $u$, apply the unitary product in
+\eqref{eq:forrelation-form}, in its right-to-left order, and test the
+projection onto $u$.  Each $D_{z_h}$ uses one phase query, implemented by
+the standard bit oracle, while $H$ and the projection require no queries.
+The projection succeeds with probability $\Phi^2$.  This probability is
+at least $\delta^2$ on a 1-input and at most $\delta^2/4$ on a 0-input.
+Independent repetitions and a threshold between these two numbers give
+error $\eta$ with $O_k(1+\log(1/\eta))$ repetitions.  The measurement
+outcomes and the final threshold decision can instead be computed into
+registers and retained, giving the claimed coherent implementation.
+
+For the circuit, let $W=\sqrt m\,H$, whose entries are $\pm1$.  The scalar
+\[
+ N=z_1^{\mathsf T}W D_{z_2}W\cdots W D_{z_k}(1,\ldots,1)^{\mathsf T}
+\]
+is an integer, and
+\[
+ \Phi=\frac{N}{M_{k,m}},
+ \qquad M_{k,m}=m(\sqrt m)^{k-1}.
+\]
+The denominator is an integer because $m$ is a power of four.  Compute
+$N$ by $k-1$ unnormalized Walsh--Hadamard transforms, coordinatewise sign
+changes, and a final signed sum.  Each transform uses $O(m\log m)$
+additions and subtractions by the usual butterfly recursion.  All
+intermediate integers have absolute value at most $m^{O(k)}$, so
+$O_k(\log m)$ bits suffice.  Binary additions, subtractions, sign changes,
+and comparisons have fan-in-two Boolean circuits of size linear in this
+bit length.  Consequently the entire arithmetic circuit has Boolean size
+$O_k(m(\log m)^2)$.
+
+Finally, the two tests in \eqref{eq:forrelation-promise} are the exact
+integer comparisons
+\[
+ 2^{5k}N\geq M_{k,m},
+ \qquad 2^{5k+1}|N|\leq M_{k,m}.
+\]
+Set $d$ to the disjunction of these comparisons and $b$ to the first
+comparison.  This adds only $O_k(\log m)$ gates and proves the last
+assertion.
+\end{proof}
+
+The exact circuit serves a different purpose from the quantum algorithm.
+The algorithm will propose an address.  A transcript of the circuit will
+certify that this address consists of defined, correct answers, including
+on inputs where an arbitrary proposed address has no reason to be correct.
+
+\subsection{The total function}
+
+Set
+\begin{equation}\label{eq:r-def}
+ r=\lceil20\log_2m\rceil,
+ \qquad \ell=\log_2m.
+\end{equation}
+The input has $m$ columns.  Column $i\in[m]$ contains the following bits:
+\begin{enumerate}
+ \item A string $X_i\in\bits^m$.
+ \item Strings $A_{i,1},\ldots,A_{i,r}\in\bits^{km}$, with no promise
+ imposed on them.
+ \item A cell $B_{i,s}$ for each address $s\in\bits^r$.  This cell
+ contains a pointer $p_{i,s,j}\in[m]$ for each $j\ne i$, encoded by
+ $\ell$ bits using a fixed bijection, and an alleged evaluation transcript
+ of $C_{k,m}$ on each $A_{i,h}$, $h\in[r]$.
+\end{enumerate}
+An evaluation transcript consists of one bit for every non-input gate of
+$C_{k,m}$, including its output gates.  Circuit inputs are the actual bits
+of $A_{i,h}$; they need not be copied into the transcript.  We count these
+non-input gates in $S$.
+
+Call column $i$ a \emph{winner} if $X_i=1^m$, every $P(A_{i,h})$ is
+defined, and the cell at its true address
+\begin{equation}\label{eq:true-address}
+ s_i=(P(A_{i,1}),\ldots,P(A_{i,r}))
+\end{equation}
+contains correct evaluation transcripts and pointers satisfying
+\begin{equation}\label{eq:outward-pointers}
+ X_j[p_{i,s_i,j}]=0\qquad\text{for every }j\ne i.
+\end{equation}
+Define $F_{k,m}$ to be 1 if a winner exists, and 0 otherwise.  This is a
+total Boolean function: all the preceding conditions are predicates on
+the input bits, and no promise is made about the $A$-strings, the pointers,
+or the transcripts.
+
+The addressed transcripts have the certificate role used in cheat-sheet
+constructions~\cite{ABK16}, while pointers between columns also occur in
+earlier query-complexity separations~\cite{GPW15,ABBLSS17}. Here the pointers have
+a specific outward orientation: an accepting column supplies a direct
+zero witness against every competitor.  In particular, there can be at
+most one winner, since a winner's pointer to another column certifies that
+the latter is not all ones.  Figure~\ref{fig:outward} depicts this
+condition.
+
+\input{figures/outward-pointers}
+\FloatBarrier
+
+We next make certificate verification precise.  For a specified pair
+$(i,s)$, say that $i$ wins \emph{at address $s$} if $i$ is a winner and
+$s=s_i$.
+
+\Needspace{9\baselineskip}
+\begin{lemma}\label{lem:local-tests}
+For every input to $F_{k,m}$ and every pair $(i,s)\in[m]\times\bits^r$,
+the assertion that $i$ wins at address $s$ is a conjunction of
+\[
+ K_{k,m}=m+(m-1)+r(S_{k,m}+2)
+        =O_k(m(\log m)^3)
+\]
+Boolean tests.  Each test can be evaluated exactly with $O(\log m)$ bit
+queries.  The tests can also be evaluated coherently with the same query
+bound, with their indices supplied in superposition.  Their number and
+indexing convention are the same for every $(i,s)$.
+\end{lemma}
+
+\begin{proof}
+Use one test for each bit of $X_i$, requiring it to be 1, and one test
+for each $j\ne i$, reading the pointer $p_{i,s,j}$ and requiring the
+indicated bit of $X_j$ to be 0.  The latter uses $\ell+1$ queries.
+
+For each of the $r$ transcripts in $B_{i,s}$, use one test per gate,
+requiring its claimed value to be the prescribed Boolean function of its
+predecessors.  A predecessor value is read from the transcript for a
+non-input gate and from $A_{i,h}$ for an input wire.  Fan-in at most two
+therefore makes each such test use at most three queries.  Add two output
+tests, requiring $d=1$ and $b=s_h$.  By induction in a topological order
+of the circuit, all gate tests pass if and only if the transcript is the
+true evaluation.  The output tests then say exactly that $P(A_{i,h})$ is
+defined and equals $s_h$.  Together with the $X_i$ and pointer tests,
+these are precisely the winner conditions at $(i,s)$.
+
+The wiring of the circuit and the locations of all bits are fixed and
+require no queries to compute.  A test can load the required bits into
+ancillas, compute its result reversibly, and unload the bits by querying
+again.  The bit oracle is its own inverse.  A controlled use of that
+oracle, when needed, is likewise implemented by loading, controlled use
+of the loaded value, and unloading.  Thus coherent evaluation, including
+an input-dependent pointer lookup, changes the query count by at most a
+constant factor.
+\end{proof}
+
+Two consequences will be used repeatedly.  A winning column has one
+address at which every test passes; a nonwinning column has a failing
+test at every address.  Thus an address proposed on an unpromised input
+can always be checked against the total definition.
diff --git a/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/build/sections/03-quantum.tex b/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/build/sections/03-quantum.tex
new file mode 100644
index 0000000000000000000000000000000000000000..8f6be253fe44ea8b17c031272a90db280b36b5a6
--- /dev/null
+++ b/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/build/sections/03-quantum.tex
@@ -0,0 +1,298 @@
+\section{Quantum search with probabilistic comparisons}\label{sec:quantum}
+
+We first prove a general tournament statement. A comparison need not
+behave almost deterministically: its outcome probabilities may be
+arbitrary, provided one distinguished candidate beats each opponent
+with sufficiently high probability. This is the additional feature needed
+beyond comparison algorithms for a fixed tournament
+\cite{SYZ04,MPS23,BK26}. There is no promised gap between a nonwinner
+comparison probability and $1/2$, so ordinary error reduction does not
+provide uniformly cheap bounded-error access to a fixed tournament.
+After proving the statement, we
+construct the comparisons for $F_{k,m}$ and verify the candidates they
+produce.
+
+\subsection{Amplification without changing the sampled distribution}
+
+The tournament will sample a candidate conditioned on defeating every
+previously selected candidate. Amplification must preserve that
+conditional distribution while increasing the probability of obtaining
+a sample. This is a direct consequence of the
+two-dimensional rotation underlying amplitude amplification
+\cite{BBHT98,BHMT02}; we give the details because the distribution,
+as well as the success probability, is used in the tournament.
+
+\begin{lemma}[Bounded amplification]\label{lem:amplification}
+Let $A$ be a quantum state preparation with a designated success bit.
+Its success probability is $p$, and, if $p>0$, its normalized successful
+state is $\lvert g\rangle$. Let $0<\gamma\le1/2$ and $0<\eta<1/2$.
+There is a procedure using
+\[
+ O\bigl(\gamma^{-1/2}\log(1/\eta)\bigr)
+\]
+calls to $A$ and $A^{-1}$ that either reports failure or produces a
+successful outcome. Conditional on success, the output state is
+$\lvert g\rangle$ up to a global phase. If $p\ge\gamma$, its failure
+probability is at most $\eta$. If $p=0$, it always reports failure.
+The call bound holds for every $p$ and on every branch.
+\end{lemma}
+
+\begin{proof}
+For $0<p<1$, write
+\[
+ A\lvert0\rangle
+   =\sin\theta\lvert g\rangle+\cos\theta\lvert b\rangle,
+ \qquad 0<\theta<\pi/2,
+\]
+where the two normalized vectors lie in the successful and unsuccessful
+subspaces. Reflection about $A\lvert0\rangle$, followed in the appropriate
+order by a phase flip of the success bit, gives after $\ell$ Grover
+iterations the state
+\[
+ \sin((2\ell+1)\theta)\lvert g\rangle
+  +\cos((2\ell+1)\theta)\lvert b\rangle.
+\]
+Thus measuring success leaves exactly the same normalized successful
+state, independently of $\ell$. Each reflection about the prepared state
+uses $A$, a query-free reflection about $\lvert0\rangle$, and $A^{-1}$.
+
+Set $M=\lceil10/\sqrt\gamma\rceil$. A trial chooses, with equal
+probabilities, either $\ell=0$ or a uniform $\ell\in\{0,\ldots,M-1\}$,
+then measures the success bit. If $p\ge1/2$, the first choice alone
+gives success probability at least $1/4$. If $\gamma\le p\le1/2$,
+then
+\[
+ \sin(2\theta)=2\sqrt{p(1-p)}\ge\sqrt\gamma.
+\]
+Summing a geometric progression bounds the averaged success probability
+in the second choice by
+\begin{equation}\label{eq:grover-average}
+ \frac1M\sum_{\ell=0}^{M-1}\sin^2((2\ell+1)\theta)
+ \ge \frac12-\frac{1}{2M\abs{\sin(2\theta)}}
+ \ge\frac{9}{20}.
+\end{equation}
+Every trial therefore succeeds with probability at least $9/40$ when
+$p\ge\gamma$. Repeat a fixed
+$\lceil\log(1/\eta)/\log(40/31)\rceil$ number of trials with fresh
+registers, stopping at the first success. The probability that all fail
+is at most $\eta$. Each trial uses $O(M)$ calls, whether it succeeds or
+fails. Every successful trial has the same state $\lvert g\rangle$, so
+neither the random choice of length nor stopping at the first success
+changes that state. The cases $p=0$ and $p=1$ follow directly, with the
+successful component absent or equal to the entire prepared state.
+\end{proof}
+
+\subsection{The tournament}
+
+Fix an input to the bit oracle. For candidates in $[m]$, suppose a
+coherent comparison produces one victorious candidate for each unordered
+pair. It retains all workspace, so it is a unitary state preparation
+that can be reversed. Let $a_{ij}$ be the probability that $i$ defeats
+$j$ when the comparison result is measured. We always use the same
+circuit for the unordered pair, and set $a_{ii}=0$. Hence
+\begin{equation}\label{eq:complementarity}
+ a_{ij}+a_{ji}=1\quad(i\ne j),\qquad a_{ii}=0.
+\end{equation}
+The following result is pointwise in the fixed
+oracle input; its probabilities are over the algorithm's internal
+randomness and measurements.
+
+\begin{proposition}[A tournament for probabilistic comparisons]
+\label{prop:tournament}
+For $m\ge2$, put
+\[
+ T=\lceil\log(1000m)\rceil,
+ \qquad \epsilon=\frac{1}{1000T}.
+\]
+Suppose a coherent comparison circuit, controlled by two candidate
+indices and reversible with all workspace retained, uses at most
+$c\ge1$ bit queries. Let $a_{ij}$ be the probability that it reports
+$i$ defeating $j$, and assume \eqref{eq:complementarity}. There is
+a quantum algorithm making
+\[
+ O\bigl(c\sqrt m\,T^2\log T\bigr)
+\]
+queries on every branch and returning a list of at most $T$ candidates
+with the following guarantee. If $i_\ast\in[m]$ satisfies
+\begin{equation}\label{eq:dominant}
+ a_{i_\ast j}\ge1-\epsilon\qquad(j\ne i_\ast),
+\end{equation}
+then the list contains $i_\ast$ with probability at least
+$1-2m/2^T-T\epsilon$.
+\end{proposition}
+
+\begin{proof}
+We analyze an ideal sampling process and then implement each of its
+steps quantumly. Start with $w_i=1$ for all $i$, and write $W=\sum_iw_i$.
+When $W>0$, sample a pivot $p$ with probability $w_p/W$, append it to
+the list, and update
+\begin{equation}\label{eq:weights}
+ w_i\longleftarrow w_i a_{ip}\qquad(i\in[m]).
+\end{equation}
+When $W=0$, leave the list unchanged and keep all weights zero. The
+weights only define this ideal process; the algorithm will not compute
+them numerically.
+
+For $W>0$, complementarity gives the exact drift identity
+\begin{align}
+ \E[W_{\mathrm{new}}\mid w]
+ &=\frac{1}{W}\sum_{p,i}w_pw_i a_{ip}\notag\\
+ &=\frac{W^2-\sum_iw_i^2}{2W}\le\frac W2.
+ \label{eq:drift}
+\end{align}
+The same inequality holds when $W=0$, so after $T$ steps
+$\E W_T\le m/2^T$. If a candidate $i_\ast$ satisfying
+\eqref{eq:dominant} has never been selected, its final weight is at
+least
+\[
+ (1-\epsilon)^T\ge1-T\epsilon>\frac12.
+\]
+In particular, the process cannot reach zero total weight before
+selecting it. Markov's inequality implies
+\begin{equation}\label{eq:ideal-miss}
+ \Pr[i_\ast\text{ absent from the ideal list}]
+ \le\Pr[W_T\ge1/2]\le\frac{2m}{2^T}.
+\end{equation}
+
+Now fix an arbitrary list of previously selected pivots
+$p_1,\ldots,p_t$, where $t<T$. Prepare a uniform superposition of
+$i\in[m]$, and, for each pivot, compare $i$ with that pivot in a fresh
+workspace. Conditional on $i$, let $\lvert\phi(i,p_b)\rangle$ denote
+the complete comparison state, including its result bit. Before
+computing a success bit, the joint state is
+\begin{equation}\label{eq:product-state}
+ \frac1{\sqrt m}\sum_{i=1}^m
+  \lvert i\rangle\bigotimes_{b=1}^t
+  \lvert\phi(i,p_b)\rangle.
+\end{equation}
+Set the success bit precisely when every comparison says that $i$
+defeats its pivot. All garbage is retained. By the tensor product in
+\eqref{eq:product-state}, the conditional success probability for index
+$i$ is
+\[
+ \prod_{b=1}^t a_{i p_b}=w_i.
+\]
+The empty product is one. Distinct index registers are orthogonal,
+so the total success probability is $W/m$, and the conditional
+distribution of $i$ on success is exactly $w_i/W$.
+
+Apply Lemma~\ref{lem:amplification} with $\gamma=1/(2m)$ and
+$\eta=\epsilon$. A successful sample still has exactly this index
+distribution: amplification preserves the full successful state, not
+just its total probability. If sampling reports failure, stop and
+return the list already accumulated. Otherwise measure the index,
+append it, and continue, for at most $T$ stages.
+
+Until $i_\ast$ is first selected, its weight is at least $1/2$, so
+$W/m\ge1/(2m)$. At each such history the sampler has failure probability
+at most $\epsilon$ and, conditional on success, exactly the ideal pivot
+law. One can therefore couple that stage to an ideal pivot, with
+probability at most $\epsilon$ of terminating instead. The coupling
+is conditional on the entire previous pivot history; it does not
+require history-independent failure probabilities. Summing these
+failure probabilities over at most $T$ stages and using
+\eqref{eq:ideal-miss} gives the stated guarantee. Once $i_\ast$ has
+entered the list, later termination cannot remove it.
+
+At stage $t$, the preparation and its inverse use $O(c(t+1))$ queries.
+Lemma~\ref{lem:amplification} uses $O(\sqrt m\log(1/\epsilon))$
+such calls. Summing over $t<T$ gives
+$O(c\sqrt m\,T^2\log T)$ queries. The number of stages and all trial
+counts are fixed upper bounds, so this is a worst-case query bound
+even if there is no candidate satisfying \eqref{eq:dominant}.
+\end{proof}
+
+The proof uses no order or transitivity among the candidates. Its
+shrinking quantity is the sum of fractional weights in
+\eqref{eq:weights}, and complementarity alone yields
+\eqref{eq:drift}. This is why unpromised address computations will be
+permitted in the application.
+
+\subsection{Comparing columns and verifying the list}
+
+We now fix an arbitrary input to $F_{k,m}$, including possibly undefined
+Forrelation instances and incorrect circuit transcripts. Keep $T$ and
+$\epsilon$ as in Proposition~\ref{prop:tournament}. To decode column
+$i$, run the quantum algorithm for each of its $r$ partial-function
+instances, amplified to error at most $\epsilon/r$. Repetition and a
+majority computation take
+\[
+ O_k\bigl(r\log(r/\epsilon)\bigr)=(\log m)^{O_k(1)}
+\]
+queries. If all these instances are defined, the resulting address is
+correct with probability at least $1-\epsilon$. Otherwise the routine
+still returns some address. All repetitions and majority computations
+can be performed unitarily with their workspaces retained.
+
+For a pair of distinct columns, sort their indices and apply the
+following comparison in this fixed order. Decode the address of each
+column in a separate workspace. Read each column's pointer toward the
+other at its decoded address, and inspect the bit of the other's $X$
+string to which it points. A column \emph{disqualifies} its opponent if
+that bit is zero. If exactly one column disqualifies the other, declare
+it victorious; otherwise declare the smaller index victorious. For
+a self-comparison, return false without reading a pointer.
+
+These operations prepare a comparison state with
+$c=(\log m)^{O_k(1)}$ queries. The decoded addresses remain in their
+registers, and subsequent operations are controlled in the
+computational basis. Consequently its measurement probabilities agree
+with the description just given; different address outcomes are
+orthogonal and do not interfere. Reversing the complete preparation
+costs the same number of bit queries. Using the same sorted-pair
+circuit for either ordering ensures \eqref{eq:complementarity}.
+
+If $i_\ast$ is a winner, then $X_{i_\ast}$ is all ones. No opponent,
+at any decoded address, can point to a zero there. When the address
+of $i_\ast$ is correctly decoded, its own certificate points to a zero
+in every other column. Thus
+\[
+ a_{i_\ast j}\ge1-\epsilon\qquad(j\ne i_\ast),
+\]
+regardless of whether the opponent's address instances are defined.
+Proposition~\ref{prop:tournament} therefore returns a list containing
+the winner, when one exists, with probability at least
+$1-2m/2^T-T\epsilon$.
+
+For each listed column, decode and measure its address afresh. At that
+fixed address, the winner conditions are the exact local tests of
+Lemma~\ref{lem:local-tests}. Let their number be
+$K=m(\log m)^{O_k(1)}$. A uniform test index and an exact failing-test
+flag give a preparation whose success probability is zero when all
+tests pass, and at least $1/K$ otherwise. Lemma~\ref{lem:amplification},
+with $\gamma=1/K$ and $\eta=\epsilon$, searches for a failed test in
+\[
+ O\bigl(\sqrt K\log(1/\epsilon)\log m\bigr)
+   =\sqrt m(\log m)^{O_k(1)}
+\]
+queries. Here $K\ge m\ge4$. Reject this candidate if a failed test is
+found, and accept it otherwise. Exact testing cannot report a failure
+when all tests pass. If the column is not a winner, every address has
+a failed test, so it is accepted with probability at most $\epsilon$.
+A true winner is accepted whenever decoding is correct: in that case
+there is no failed test for the search to report. Its rejection
+probability is therefore at most $\epsilon$, solely from address decoding.
+
+Output one if any listed candidate is accepted. On a zero-input, a
+union bound gives error at most $T\epsilon$. On a one-input, the miss
+probability and the winner's verification error give total error at most
+\[
+ \frac{2m}{2^T}+T\epsilon+\epsilon<\frac13.
+\]
+The comparisons, tournament, and at most $T$ verifications prove
+\begin{proposition}\label{prop:quantum}
+For every fixed $k\ge2$, the total functions of
+Section~\ref{sec:construction} satisfy
+\[
+ \Q(F_{k,m})\le\sqrt m(\log m)^{O_k(1)}.
+\]
+\end{proposition}
+
+All superposed array accesses used above are ordinary bit queries.
+The global bit index is computed reversibly from the column, cell, and
+wire indices. A pointer is read one bit at a time in $O(\log m)$ queries,
+after which its target index is computed without queries. A controlled
+bit query can be implemented by loading the addressed input bit into
+an ancilla, using it under the required control, and unloading it with
+a second query. Thus neither coherent addressing nor retaining and
+reversing comparison garbage changes the query model.
diff --git a/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/build/sections/04-classical.tex b/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/build/sections/04-classical.tex
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+++ b/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/build/sections/04-classical.tex
@@ -0,0 +1,238 @@
+\section{The randomized lower bound}
+\label{sec:classical}
+
+We compare a distribution with one zero in every column to a coupled
+distribution in which one uniformly chosen column is made a winner.
+An algorithm can distinguish them only by finding the zero that was
+removed or by reaching the newly filled certificate cell.  The first
+event requires searching many columns; the second requires either many
+Forrelation queries in the chosen column or a successful guess of its
+entire address.
+
+\subsection{Independent answers under adaptive queries}
+
+We first record the precise amplification fact needed for address
+guessing.  We use the capped direct-product argument developed by
+Drucker~\cite[Section~1.3 and Lemmas~3.1--3.2]{Drucker12}, following
+earlier work of Shaltiel~\cite{Shaltiel03}.  Its query limit is imposed
+separately on each input, while queries to different inputs may be
+interleaved adaptively.
+
+\Needspace{12\baselineskip}
+\begin{lemma}\label{lem:adaptive-product}
+Let $P\colon\mathcal D\to\bits$, where $\mathcal D\subseteq\bits^n$, let
+$\mu$ be a distribution on $\mathcal D$, and let $L$ be a nonnegative
+integer.  Suppose every deterministic algorithm making at most $L$
+queries computes $P$ under $\mu$ with success probability at most $\beta$.
+If $U_1,\ldots,U_r$ are independent samples from $\mu$, every classical
+algorithm making at most $L$ queries to each $U_h$ has probability at most
+$\beta^r$ of outputting
+\[
+ (P(U_1),\ldots,P(U_r)).
+\]
+The algorithm may be randomized, choose its queries adaptively among the
+$r$ inputs, and receive arbitrary additional information independent of
+$(U_1,\ldots,U_r)$.
+\end{lemma}
+
+\begin{proof}
+Fix the additional information and all internal random coins, leaving a
+deterministic algorithm.  At any transcript $\tau$ of positive
+probability, the conditional law of $(U_1,\ldots,U_r)$ is a product of
+their separate conditional laws.  Indeed, the event of obtaining $\tau$
+requires exactly the observed bit values in each input.  Any tuple
+satisfying these requirements follows the same query schedule, since
+the next query is determined by the preceding answers.  Thus this event
+is a Cartesian product of constraints on the individual inputs.
+
+For each $h$, let $q_h$ be the number of queries already made to $U_h$,
+and let $V_h(\tau)$ be the largest success probability for computing
+$P(U_h)$ from its current conditional law using at most $L-q_h$ further
+queries.  This maximum is attained because the domain and the remaining
+query budget are finite.  If the next query addresses $U_h$, optimality
+gives
+\begin{equation}\label{eq:bellman-query}
+ \mathbb E[V_h(\tau')\mid\tau]\leq V_h(\tau),
+\end{equation}
+where $\tau'$ includes its answer: making that particular query and
+then using an optimal continuation is one allowed strategy in the
+definition of $V_h(\tau)$.  The conditional laws and budgets of the
+other inputs do not change.  Hence
+\[
+ \mathbb E\!\left[
+   \prod_{j=1}^r V_j(\tau')\,\middle|\,\tau
+ \right]
+ \leq \prod_{j=1}^r V_j(\tau).
+\]
+The product is therefore a supermartingale over the successive queries.
+There are at most $rL$ such queries; a run that stops earlier can be
+extended by steps leaving all values unchanged.
+
+At the final transcript, the conditional probability that the algorithm's
+chosen answer vector is correct is at most
+\[
+ \prod_{h=1}^r
+   \max_{b\in\bits}\Pr(P(U_h)=b\mid\tau)
+ \leq \prod_{h=1}^r V_h(\tau).
+\]
+The first inequality uses the product conditional law, and the second
+uses the option of making no further queries.  Initially every $V_h$
+is at most $\beta$.  Taking expectations proves the bound $\beta^r$.
+Averaging over the fixed information and coins proves the full statement.
+\end{proof}
+
+Return to $P=P_{k,m}$ and $L$ from \eqref{eq:L-def}, with $m$ large
+enough that $1\leq L<\R(P)$.  There are finitely many deterministic
+decision trees of depth at most $L$ on this finite domain.  A randomized
+algorithm of that query depth is a distribution on these trees. Finite
+minimax, in its query-complexity formulation~\cite{Yao77}, applied to the
+payoff that the answer is correct, gives
+\[
+ \begin{aligned}
+ &\max_{\text{distributions on depth-$L$ trees}}
+       \min_{x\in\operatorname{dom}P}\Pr(\text{correct on }x)\\
+ &\hspace{2em}=
+ \min_{\mu}\max_{\text{depth-$L$ trees }T}
+       \Pr_{x\sim\mu}(T(x)=P(x)).
+ \end{aligned}
+\]
+The left-hand side is less than $2/3$, since its maximum is attained
+and $L<\R(P)$.  Consequently there is a distribution $\mu$ on
+$\operatorname{dom}P$ for which every such deterministic tree has
+success probability at most $2/3$.  Lemma~\ref{lem:adaptive-product}
+therefore bounds the success probability of guessing $r$ independent
+answers by $(2/3)^r$.
+
+\subsection{Coupled zero and one inputs}
+
+\begin{proposition}\label{prop:classical}
+For each fixed integer $k\geq2$ and every sufficiently large power of
+four $m$, the total function of Section~\ref{sec:construction} satisfies
+\begin{equation}\label{eq:classical-lower}
+ \R(F_{k,m})>
+ G:=\left\lfloor\frac{mL}{100}\right\rfloor,
+ \qquad
+ L=\left\lfloor\frac{m^{1-1/k}}{(\log_2m)^2}\right\rfloor.
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+Generate a \emph{dummy input} $Y^{(0)}$ as follows.  Independently for
+each $i\in[m]$, choose $J_i$ uniformly from $[m]$ and put a single zero
+at position $J_i$ in $X_i$, setting its other bits to 1.  Draw all
+$A_{i,h}$ independently from $\mu$, independently also of the $J_i$.
+Set all cells $B_{i,s}$ to one fixed blank bit pattern.  Since each $X_i$
+has a zero, $F_{k,m}(Y^{(0)})=0$.
+
+Choose $I$ uniformly from $[m]$, independently of the entire dummy
+input.  Form a \emph{planted input} $Y^{(1)}$ by setting $X_I=1^m$ and
+filling its true-address cell $B_{I,s_I}$ with the correct circuit
+transcripts and the pointers $p_{I,s_I,j}=J_j$ for $j\ne I$.
+All $A$-strings are in the domain of $P$, so $s_I$ is well defined.
+Column $I$ is a winner, and therefore $F_{k,m}(Y^{(1)})=1$.
+
+Fix a deterministic algorithm making at most $G$ queries, and consider
+its run on $Y^{(0)}$.  The coupled inputs can differ only at the bit
+$X_I[J_I]$ and inside the cell $B_{I,s_I}$.  Until one of these locations
+is queried, the two executions have identical answers and hence
+identical subsequent queries.  It follows that the difference between
+their output-1 probabilities is at most
+\begin{equation}\label{eq:coupling-events}
+ \Pr(\text{the dummy run queries }X_I[J_I])
+ +\Pr(\text{the dummy run queries any bit of }B_{I,s_I}).
+\end{equation}
+Counting every visit to the indicated cell is an upper bound even if
+the queried bit happens to agree with its blank value.
+
+\paragraph{Finding the removed zero.}
+Let $Z$ be the number of columns whose zero is discovered in the dummy
+run.  Before the zero in a column is found, its location is uniform
+among the positions in that column not yet queried.  This remains true
+conditional on the full adaptive transcript: the other $X$-strings and
+all $A$-strings are independent of its zero location, and the blank
+cells supply no information about that location.
+
+Call a discovery early if fewer than $m/2$ distinct positions in its
+column were queried before it.  Conditional on the preceding transcript,
+each query has probability at most $2/m$ of an early discovery.  Thus
+the expected number of early discoveries is at most $2G/m$.  Each
+remaining discovery occurs in a column that receives at least $m/2$
+queries, and there are at most $2G/m$ such columns.  Hence
+\[
+ \mathbb E Z\leq\frac{4G}{m}.
+\]
+The dummy run is independent of $I$, so
+\begin{equation}\label{eq:zero-hit}
+ \Pr(\text{the dummy run queries }X_I[J_I])
+ =\frac{\mathbb E Z}{m}\leq\frac{4G}{m^2}.
+\end{equation}
+
+\paragraph{Reaching the planted cell.}
+Call a column heavy if, over the entire dummy run, the algorithm makes
+more than $L$ queries in total to its strings $A_{i,1},\ldots,A_{i,r}$.
+There are at most $G/L$ heavy columns.  Independence of $I$ gives
+\begin{equation}\label{eq:heavy-probability}
+ \Pr(I\text{ is heavy})\leq\frac{G}{mL}.
+\end{equation}
+For the other columns we use Lemma~\ref{lem:adaptive-product}.  The
+following simulation bounds a joint event and does not condition the
+input distribution on a column being nonheavy.
+
+Fix $i\in[m]$ and a time $t\in\{1,\ldots,G\}$.  Give a simulator all
+the dummy input outside $A_{i,1},\ldots,A_{i,r}$ for free.  This
+information is independent of those $r$ strings.  The simulator runs
+the deterministic algorithm up to its proposed $t$th query, but aborts
+before making an $(L+1)$st query to these $r$ strings in total.  If it
+reaches time $t$ and the proposed query is to a cell $B_{i,s}$, it outputs
+the address $s$ as its guess for the vector of $r$ answers.  Otherwise,
+including if the simulated run stops or aborts, it outputs any fixed
+vector.
+
+The simulator makes at most $L$ queries in total to the target strings,
+and in particular at most $L$ to each one.  By
+Lemma~\ref{lem:adaptive-product}, its probability of guessing $s_i$
+correctly is at most $(2/3)^r$.  Whenever the original dummy run queries
+$B_{i,s_i}$ at time $t$ and column $i$ is nonheavy, this simulator has
+not aborted and outputs the correct address.  Thus
+\begin{equation}\label{eq:capped-address}
+ \Pr\bigl(i\text{ is nonheavy and the $t$th query is in }B_{i,s_i}\bigr)
+ \leq (2/3)^r.
+\end{equation}
+Union bounding over $t$ and averaging over the independent uniform
+choice of $I$, then adding \eqref{eq:heavy-probability}, yields
+\begin{equation}\label{eq:cell-hit}
+ \Pr(\text{the dummy run queries }B_{I,s_I})
+ \leq\frac{G}{mL}+G(2/3)^r.
+\end{equation}
+
+Equations~\eqref{eq:coupling-events}, \eqref{eq:zero-hit}, and
+\eqref{eq:cell-hit} bound the difference of output probabilities by
+\begin{equation}\label{eq:classical-distinguishing}
+ \frac{4G}{m^2}+\frac{G}{mL}+G(2/3)^r.
+\end{equation}
+For all sufficiently large $m$, we have $1\leq L\leq m$ and
+$G\leq mL/100$.  The first two terms are therefore at most $4/100$
+and $1/100$, respectively.  Moreover, \eqref{eq:r-def} gives
+\[
+ G(2/3)^r
+ \leq\frac1{100}m^{\,2-20\log_2(3/2)}=o(1).
+\]
+In particular, \eqref{eq:classical-distinguishing} is strictly less
+than $1/3$ for large $m$.
+
+The same bound holds for a randomized algorithm of worst-case query
+cost $G$, by fixing its coins and then averaging.  Such an algorithm,
+if it had worst-case error at most $1/3$, would output 1 with probability
+at most $1/3$ on $Y^{(0)}$ and at least $2/3$ on $Y^{(1)}$.  Their
+output-1 probabilities would differ by at least $1/3$, contradicting
+the preceding estimate.  Hence $\R(F_{k,m})>G$.
+\end{proof}
+
+For fixed $k$, Proposition~\ref{prop:classical} implies
+\[
+ \R(F_{k,m})=
+ \Omega_k\!\left(\frac{m^{2-1/k}}{(\log m)^2}\right).
+\]
+The heavy-column estimate pays for learning an address, while the
+independent-answer estimate makes guessing an unlearned address
+negligible even over all $G$ adaptive queries.
diff --git a/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/build/sections/05-exponent.tex b/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/build/sections/05-exponent.tex
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index 0000000000000000000000000000000000000000..668200e19a40300031b8b57bca5b8c288c13c4f4
--- /dev/null
+++ b/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/build/sections/05-exponent.tex
@@ -0,0 +1,41 @@
+\section{The optimal power}\label{sec:exponent}
+
+We combine the quantitative bounds before recalling the universal upper
+bound in the appendix.
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+For each fixed $k\ge2$, Proposition~\ref{prop:quantum} gives
+$\Q(F_{k,m})\le C_k\sqrt m(\log m)^{b_k}$ for suitable constants.
+Proposition~\ref{prop:classical} gives
+\[
+ \R(F_{k,m})>
+ \left\lfloor\frac m{100}
+  \left\lfloor\frac{m^{1-1/k}}{(\log m)^2}\right\rfloor
+ \right\rfloor
+ \ge c_k\frac{m^{2-1/k}}{(\log m)^2}
+\]
+for sufficiently large $m$, after decreasing $c_k>0$.
+This proves \eqref{eq:main-family}.
+
+Now fix $0\le\alpha<4$ and choose an integer $k\ge2$ such that
+$2-1/k>\alpha/2$. The integer $k$ remains fixed as $m$ tends to infinity.
+The preceding bounds imply
+\[
+ \frac{\R(F_{k,m})}{(1+\Q(F_{k,m}))^\alpha}
+ \ge c_{k,\alpha}
+ \frac{m^{2-1/k-\alpha/2}}
+ {(\log m)^{2+\alpha b_k}}
+ \longrightarrow\infty.
+\]
+Hence no exponent smaller than four belongs to the set in
+\eqref{eq:alpha}. Conversely, Theorem~\ref{thm:quartic-upper} proves
+$\R(f)\le\D(f)=O((1+\Q(f))^4)$ for every total Boolean function.
+Therefore four does belong to that set, and its infimum is four.
+\end{proof}
+
+Separate address arrays force a classical algorithm to invest in many
+columns, while quantum comparisons locate a candidate before its
+certificate is checked. Quantum search then supplies the quadratic
+saving in checking the local tests. The direct-product estimate prevents
+a classical algorithm from bypassing the addressing cost by guessing
+certificate locations.
diff --git a/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/build/sections/06-upper-bound.tex b/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/build/sections/06-upper-bound.tex
new file mode 100644
index 0000000000000000000000000000000000000000..d25b3bb787f1a85fe9e4cbd2c0eef87b9d7aa4e3
--- /dev/null
+++ b/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/build/sections/06-upper-bound.tex
@@ -0,0 +1,202 @@
+\section{The universal quartic bound}\label{sec:upper}
+
+For completeness, we prove the upper bound used in
+Theorem~\ref{thm:main}. The argument is the one of Aaronson, Ben-David,
+Kothari, Rao, and Tal~\cite{ABKRT21}, combining Huang's signed-cube
+method~\cite{Huang19} with the adversary bound and the deterministic
+degree--block-sensitivity inequality. No part of the construction is
+used in this appendix.
+
+For a total Boolean function $f$, let $\D(f)$ be its exact deterministic
+query complexity and $\deg(f)$ its degree as a real multilinear polynomial.
+The degree is unchanged by replacing input or output bits by signs.
+A nonempty set $B\subseteq[n]$ is a \emph{sensitive block at $x$} if
+flipping every bit in $B$ changes $f(x)$. The block sensitivity $\bs(f)$
+is the maximum, over $x$, of the number of pairwise disjoint sensitive
+blocks at $x$.
+
+\begin{theorem}[\cite{ABKRT21}]\label{thm:quartic-upper}
+Every total Boolean function satisfies
+\[
+ \R(f)\le\D(f)=O((1+\Q(f))^4).
+\]
+\end{theorem}
+
+We establish the three ingredients separately. All matrix norms below
+are operator norms for the Euclidean norm.
+
+\begin{lemma}[Nonnegative spectral adversary bound]
+\label{lem:adversary}
+Let $f$ be a nonconstant total Boolean function. Let $\Gamma$ be a nonzero
+real symmetric matrix, indexed by inputs, with nonnegative entries and
+$\Gamma_{xy}=0$ whenever $f(x)=f(y)$. Let
+\[
+ (\Gamma_i)_{xy}=\Gamma_{xy}\,\mathbf1_{\{x_i\ne y_i\}}.
+\]
+Then, for an absolute constant $c>0$,
+\[
+ \Q(f)\ge c\frac{\norm{\Gamma}}{\max_i\norm{\Gamma_i}}.
+\]
+\end{lemma}
+
+\begin{proof}
+This is the spectral adversary argument of
+Barnum, Saks, and Szegedy~\cite{BSS03}; see also
+\v{S}palek and Szegedy~\cite{SpalekSzegedy06}.
+Purify all randomness and defer measurements. A bounded-query algorithm
+can be represented with a fixed number of queries, with at most a
+constant-factor overhead: branches that have finished can perform
+ineffective queries, and controlled queries can be simulated with two
+bit queries. Let $\lvert\psi_x^t\rangle$ be its normalized pure state
+on input $x$ after $t$ queries.
+
+By the Perron--Frobenius theorem, $\Gamma$ has a nonnegative unit
+eigenvector $v$ of eigenvalue $\norm{\Gamma}$. Track
+\[
+ W_t=\sum_{x,y}v_xv_y\Gamma_{xy}
+            \langle\psi_x^t\mid\psi_y^t\rangle.
+\]
+Initially the states are input-independent, so
+$W_0=v^\top\Gamma v=\norm{\Gamma}$. Input-independent unitaries
+preserve every overlap.
+
+Immediately before a query, write the normalized state as
+\[
+ \lvert\psi_x\rangle=\sum_i\lvert i\rangle\lvert\phi_{x,i}\rangle,
+ \qquad \sum_i\norm{\phi_{x,i}}^2=1.
+\]
+The bit oracle acts identically
+on inputs $x,y$ at that index unless $x_i\ne y_i$. When they differ,
+the overlap changes in magnitude by at most
+$2\norm{\phi_{x,i}}\norm{\phi_{y,i}}$, since the difference between
+the relevant unitary and the identity has norm at most two. Put
+$u_i(x)=v_x\norm{\phi_{x,i}}$. Nonnegativity of the weights gives
+\begin{align*}
+ \abs{W_{t+1}-W_t}
+ &\le2\sum_i u_i^\top\Gamma_i u_i\\
+ &\le2\max_i\norm{\Gamma_i}\sum_i\norm{u_i}^2
+  =2\max_i\norm{\Gamma_i}.
+\end{align*}
+The last equality follows from normalization of each query state and
+$\sum_xv_x^2=1$.
+
+At the end, measure the output bit. For opposite-output inputs, one
+output-one probability is at most $1/3$, and the other is at least
+$2/3$. Splitting the two states by that output and applying
+Cauchy--Schwarz bounds their overlap in absolute value by
+$2\sqrt2/3$. Hence the final value satisfies
+\[
+ \abs{W_{\mathrm{final}}}
+ \le\frac{2\sqrt2}{3}\sum_{x,y}v_xv_y\Gamma_{xy}
+ =\frac{2\sqrt2}{3}\norm{\Gamma}.
+\]
+The required total change in $W$ divided by the per-query bound proves
+the assertion.
+\end{proof}
+
+Let $S_f$ be the adjacency matrix of the \emph{sensitivity graph}: its
+vertices are $\bits^n$, and two vertices are adjacent if they differ
+in exactly one bit and have different function values. Write
+$\lambda(f)=\norm{S_f}$.
+
+\begin{lemma}\label{lem:bs-lambda}
+For every nonconstant total Boolean function,
+\[
+ \bs(f)=O(\Q(f)^2),\qquad
+ \lambda(f)=O(\Q(f)).
+\]
+\end{lemma}
+
+\begin{proof}
+Choose an input $x$ with $b=\bs(f)$ disjoint sensitive blocks. Take
+$\Gamma$ to be the adjacency matrix of the star whose center is $x$
+and whose leaves are the $b$ inputs obtained by flipping those blocks,
+with zero entries elsewhere. Its norm is $\sqrt b$. Each coordinate
+belongs to at most one block, so $\Gamma_i$ has at most one undirected
+edge and norm at most one. Lemma~\ref{lem:adversary} gives
+$\Q(f)=\Omega(\sqrt b)$. This recovers the usual block-sensitivity
+lower bound for quantum queries~\cite{BBCMW01}.
+
+For the second assertion, use $\Gamma=S_f$. The edges in any one
+coordinate form a partial matching, whose adjacency matrix has norm
+at most one. Another application of Lemma~\ref{lem:adversary} gives
+$\Q(f)=\Omega(\lambda(f))$, as in~\cite{ABKRT21}.
+\end{proof}
+
+\begin{lemma}[\cite{ABKRT21}, using \cite{Huang19}]
+\label{lem:degree-lambda}
+Every nonconstant total Boolean function satisfies
+$\deg(f)\le\lambda(f)^2$.
+\end{lemma}
+
+\begin{proof}
+Write the function with sign inputs and sign outputs, and set
+$d=\deg(f)\ge1$. Choose a monomial supported on a set of $d$ variables
+with nonzero coefficient, and fix all other variables to any signs.
+Its coefficient survives: any different monomial that could contribute
+to that coefficient would strictly contain its support and have degree
+larger than $d$. The restricted sign function $g$ on the $d$-cube
+therefore has nonzero correlation with parity. For some sign
+$\sigma\in\{-1,1\}$, the set
+\[
+ U=\{x\in\{-1,1\}^d:g(x)=\sigma x_1\cdots x_d\}
+\]
+has size greater than $2^{d-1}$. Every cube edge with both endpoints in
+$U$ changes $g$, since parity changes across each edge.
+
+Define a signed adjacency matrix of the cube recursively by
+\[
+ H'_1=\begin{pmatrix}0&1\\1&0\end{pmatrix},
+ \qquad
+ H'_{j+1}=\begin{pmatrix}H'_j&I\\I&-H'_j\end{pmatrix}.
+\]
+Its entrywise absolute value is the ordinary cube adjacency matrix.
+Induction gives $(H'_j)^2=jI$, and its trace is zero. Thus $H'_d$ has
+eigenvalues $\pm\sqrt d$, each with multiplicity $2^{d-1}$. The
+$\sqrt d$-eigenspace intersects the space of vectors supported on $U$
+nontrivially, because their dimensions sum to more than $2^d$.
+Choose a nonzero real vector $v$ in this intersection, and put
+$u=\abs v$ entrywise. If $A_U$ is the unsigned adjacency induced on
+$U$, then
+\[
+ u^\top A_Uu\ge v^\top H'_dv=\sqrt d\,\norm v^2
+             =\sqrt d\,\norm u^2.
+\]
+All edges of $A_U$ are sensitivity edges of the restriction, and hence
+of the original function. Extending $u$ by zero to the full input
+cube gives $\lambda(f)\ge\sqrt d$, as required.
+\end{proof}
+
+\begin{lemma}[\cite{Midrijanis04}]\label{lem:degree-bs}
+Every total Boolean function satisfies
+$\D(f)\le\bs(f)\deg(f)$.
+\end{lemma}
+
+\begin{proof}
+Consider the following deterministic algorithm. If the function
+restricted by the queries so far is nonconstant, select a monomial
+of maximum degree in that restricted polynomial and query all its
+variables. Continue until the restriction is constant.
+
+Fix the actual full input $x$. In a given round, let $S$ be the
+selected monomial's support. Fix all other still-unqueried variables
+to their values in $x$. The coefficient on $S$ remains nonzero, since
+no monomial in the current restriction strictly contains $S$.
+The resulting function on $S$ is nonconstant. Some assignment to $S$
+therefore differs in value from the actual assignment $x|_S$. The
+nonempty subset of coordinates on which these two assignments differ
+is a sensitive block at the original input $x$.
+
+Different rounds query disjoint sets, so these sensitive blocks are
+disjoint. There are at most $\bs(f)$ rounds. Restrictions do not
+increase degree, and each round queries at most $\deg(f)$ variables.
+The stated deterministic bound follows.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{thm:quartic-upper}]
+For nonconstant $f$, Lemmas~\ref{lem:bs-lambda} and
+\ref{lem:degree-lambda} give both $\bs(f)=O(\Q(f)^2)$ and
+$\deg(f)=O(\Q(f)^2)$. Lemma~\ref{lem:degree-bs} then gives
+$\D(f)=O(\Q(f)^4)$. Since a deterministic algorithm is also randomized,
+$\R(f)\le\D(f)$. Constant functions require no queries.
+\end{proof}
diff --git a/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/quartic-query-separation.pdf b/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/quartic-query-separation.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..6464e4cc1ccfbeeefc1a7b89486626af9eae6cdb
--- /dev/null
+++ b/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/quartic-query-separation.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:a99f1c38d086a33591b6204ae3957fe3f25c8f31d97c48087325a18fd66daf3e
+size 433829
diff --git a/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/README.md b/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..5d7fe33864c549fba29c4cf1b4a9b088da98f09c
--- /dev/null
+++ b/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/README.md
@@ -0,0 +1,17 @@
+# [A negative solution to the finite lattice representation problem](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 24, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026,
+  author = {{OpenAI}},
+  title = {{A negative solution to the finite lattice representation problem}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/paper.pdf}{OAI:A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026}},
+  year = {2026}
+}
+```
diff --git a/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/figures/boolean-faces.tex b/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/figures/boolean-faces.tex
new file mode 100644
index 0000000000000000000000000000000000000000..39f54abb93fbdc56426dda55d89275263ad076ef
--- /dev/null
+++ b/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/figures/boolean-faces.tex
@@ -0,0 +1,29 @@
+\begin{figure}[htbp]
+\centering
+\begin{tikzpicture}[x=1cm,y=1cm,
+  face/.style={fill=white,inner sep=3pt},
+  every node/.style={font=\small}]
+\node[face] (x) at (0,0) {$x=\{1\}$};
+\node[face] (u1) at (-3,1.45) {$u_1=\{1,2\}$};
+\node[face] (u2) at (0,1.45) {$u_2=\{1,3\}$};
+\node[face] (u3) at (3,1.45) {$u_3=\{1,4\}$};
+\node[face] (w12) at (-3,3) {$\{1,2,3\}$};
+\node[face] (w13) at (0,3) {$\{1,2,4\}$};
+\node[face] (w23) at (3,3) {$\{1,3,4\}$};
+\node[face,draw,rounded corners=2pt,align=center] (all) at (0,4.6)
+  {$\{1,2,3,4\}$\\full union};
+\draw (x)--(u1) (x)--(u2) (x)--(u3);
+\draw (u1)--(w12) (u1)--(w13);
+\draw (u2)--(w12) (u2)--(w23);
+\draw (u3)--(w13) (u3)--(w23);
+\draw[densely dashed] (w12)--(all) (w13)--(all) (w23)--(all);
+\end{tikzpicture}
+\caption{The original Boolean faces containing $x$. Each pair among
+$u_1,u_2,u_3$ has a proper union, but their total union is full.
+All displayed subset relations survive the private insertions.
+In the extension configuration of Proposition~\ref{ext:reduction},
+solid edges correspond to restrictions of compatible homomorphisms
+on actual nested domains. The dashed edges reach the full union,
+for which a common extension is forbidden.}
+\label{fig:boolean-faces}
+\end{figure}
diff --git a/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/figures/tested-chain.tex b/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/figures/tested-chain.tex
new file mode 100644
index 0000000000000000000000000000000000000000..e8458c6f33686e064f5425ef0a3be7aae3ea0ae5
--- /dev/null
+++ b/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/figures/tested-chain.tex
@@ -0,0 +1,24 @@
+\begin{figure}[htbp]
+\centering
+\begin{tikzpicture}[x=1cm,y=1cm,
+  vertex/.style={circle,fill=black,inner sep=1.8pt},
+  every node/.style={font=\small}]
+\node[vertex,label=left:{$x_i$}] (lo) at (0,0) {};
+\node[vertex,label=left:{$x_{i+1}$}] (mid) at (0,1.15) {};
+\node[vertex,label=left:{$x_j$}] (hi) at (0,3.2) {};
+\node[vertex,label=right:{$v_{ij}$}] (shortcut) at (2.5,1.6) {};
+\draw (lo)--(mid);
+\draw[densely dashed] (mid)--(hi);
+\draw[thick] (lo)--(shortcut)--(hi);
+\node[align=left,anchor=west] at (3.45,1.6)
+  {$x_i\prec v_{ij}\prec x_j$\\two covering steps};
+\node[align=center] at (0.5,-.55) {selected chain segment};
+\end{tikzpicture}
+\caption{One shortcut in a tested chain. The two thick edges are covers
+in the entire subgroup interval; the selected chain on the left can
+have arbitrarily many intermediate vertices and is not required to be
+saturated. Fences, coatom witnesses, and the $M_{16}$ tests are omitted.
+The proof compares the restrictions imposed by every short route with
+those imposed by traversing a long chain.}
+\label{fig:tested-chain}
+\end{figure}
diff --git a/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/main.tex b/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/main.tex
new file mode 100644
index 0000000000000000000000000000000000000000..a185fed327bdd8ff14808d8ccbd3f11d4c8d3120
--- /dev/null
+++ b/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/main.tex
@@ -0,0 +1,29 @@
+\documentclass[11pt]{article}
+\input{preamble}
+\title{A negative solution to the\\finite lattice representation problem}
+\author{OpenAI}
+\date{September 24, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We give a negative solution to the finite lattice representation problem.
+We prove that there is a finite nonempty lattice that is not the full
+congruence lattice of any finite nonempty algebra of any finite signature.
+\end{abstract}
+\tableofcontents
+
+\input{sections/introduction}
+\input{sections/framework}
+\input{sections/classification}
+\input{sections/growth}
+\input{sections/lifts}
+\input{sections/packets}
+\input{sections/rigidity}
+\input{sections/lattices}
+\input{sections/extensions}
+\input{sections/gluing}
+
+\bibliographystyle{plain}
+\bibliography{references}
+\end{document}
+
diff --git a/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/preamble.tex b/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/preamble.tex
new file mode 100644
index 0000000000000000000000000000000000000000..62c42623957201c239b211c3f4dd95dc5dcb84a6
--- /dev/null
+++ b/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/preamble.tex
@@ -0,0 +1,59 @@
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage{lmodern}
+\usepackage[margin=1in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{booktabs,longtable,array,enumitem}
+\usepackage{microtype}
+\usepackage{needspace,etoolbox}
+\usepackage{xcolor}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta,positioning}
+\usepackage{hyperref}
+\hypersetup{colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black,pdftitle={A negative solution to the finite lattice representation problem},pdfsubject={A group-theoretic obstruction to finite congruence-lattice representation},pdfauthor={OpenAI}}
+\usepackage[nameinlink,capitalize,noabbrev]{cleveref}
+\usepackage{url}
+\urlstyle{same}
+\setlength{\emergencystretch}{2em}
+\numberwithin{equation}{section}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\newtheorem{convention}[theorem]{Convention}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newtheorem{example}[theorem]{Example}
+\BeforeBeginEnvironment{theorem}{\Needspace{5\baselineskip}}
+\BeforeBeginEnvironment{lemma}{\Needspace{5\baselineskip}}
+\BeforeBeginEnvironment{proposition}{\Needspace{5\baselineskip}}
+\BeforeBeginEnvironment{corollary}{\Needspace{5\baselineskip}}
+\BeforeBeginEnvironment{definition}{\Needspace{5\baselineskip}}
+\BeforeBeginEnvironment{convention}{\Needspace{5\baselineskip}}
+\BeforeBeginEnvironment{remark}{\Needspace{5\baselineskip}}
+\BeforeBeginEnvironment{example}{\Needspace{5\baselineskip}}
+\pretocmd{\subsection}{\Needspace{8\baselineskip}}{}{\PackageError{article-layout}{Failed to patch subsection spacing}{}}
+\DeclareMathOperator{\Aut}{Aut}
+\DeclareMathOperator{\Inn}{Inn}
+\DeclareMathOperator{\Out}{Out}
+\DeclareMathOperator{\Soc}{soc}
+\DeclareMathOperator{\Core}{core}
+\DeclareMathOperator{\Rad}{rad}
+\DeclareMathOperator{\Con}{Con}
+\DeclareMathOperator{\rank}{rank}
+\DeclareMathOperator{\GL}{GL}
+\DeclareMathOperator{\SL}{SL}
+\DeclareMathOperator{\Sp}{Sp}
+\DeclareMathOperator{\SO}{SO}
+\DeclareMathOperator{\GU}{GU}
+\DeclareMathOperator{\SU}{SU}
+\DeclareMathOperator{\Hom}{Hom}
+\DeclareMathOperator{\End}{End}
+\newcommand{\F}{\mathbb F}
+\newcommand{\N}{\mathbb N}
+\newcommand{\Z}{\mathbb Z}
+\newcommand{\Q}{\mathbb Q}
+\newcommand{\C}{\mathbb C}
+\newcommand{\id}{\operatorname{id}}
diff --git a/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/references.bib b/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/references.bib
new file mode 100644
index 0000000000000000000000000000000000000000..fa30f431113886cfbda4bbaa5234796069a8cf12
--- /dev/null
+++ b/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/references.bib
@@ -0,0 +1,435 @@
+@article{GratzerSchmidt1963,
+  note = {\href{https://acta.bibl.u-szeged.hu/13983/1/math_024_fasc_001_002_034-037.pdf}{Introduction scan, pp.~34--37}},
+  author = {Gr{\"a}tzer, George and Schmidt, E. Tam{\'a}s},
+  title = {Characterizations of congruence lattices of abstract algebras},
+  journal = {Acta Scientiarum Mathematicarum (Szeged)},
+  volume = {24},
+  year = {1963},
+  pages = {34--59},
+  url = {https://acta.bibl.u-szeged.hu/13983/1/math_024_fasc_001_002_034-037.pdf}
+}
+
+@article{PalfyPudlak1980,
+  note = {\href{https://doi.org/10.1007/BF02483080}{DOI}; \href{https://math.hawaii.edu/~ralph/Classes/619/PalfyPudlak.pdf}{article scan}},
+  author = {P{\'a}lfy, P{\'e}ter P{\'a}l and Pudl{\'a}k, Pavel},
+  title = {Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups},
+  journal = {Algebra Universalis},
+  volume = {11},
+  year = {1980},
+  pages = {22--27},
+  doi = {10.1007/BF02483080},
+  url = {https://math.hawaii.edu/~ralph/Classes/619/PalfyPudlak.pdf}
+}
+
+@incollection{PalfyTwisted,
+  note = {\href{https://doi.org/10.1017/9781108692397.020}{DOI}; \href{https://real.mtak.hu/102571/1/twisted.pdf}{author version}},
+  author = {P{\'a}lfy, P{\'e}ter P.},
+  title = {Subgroups of twisted wreath products},
+  booktitle = {Groups St Andrews 2017 in Birmingham},
+  editor = {Campbell, C. M. and Parker, C. W. and Quick, M. R. and Robertson, E. F. and Roney-Dougal, C. M.},
+  series = {London Mathematical Society Lecture Note Series},
+  volume = {455},
+  publisher = {Cambridge University Press},
+  address = {Cambridge},
+  year = {2019},
+  pages = {455--468},
+  doi = {10.1017/9781108692397.020},
+  url = {https://real.mtak.hu/102571/1/twisted.pdf}
+}
+
+@article{Aschbacher1984,
+  note = {\href{https://doi.org/10.1007/BF01388470}{DOI}},
+  author = {Aschbacher, Michael},
+  title = {On the maximal subgroups of the finite classical groups},
+  journal = {Inventiones Mathematicae},
+  volume = {76},
+  number = {3},
+  year = {1984},
+  pages = {469--514},
+  doi = {10.1007/BF01388470}
+}
+
+@book{KleidmanLiebeck1990,
+  note = {\href{https://doi.org/10.1017/CBO9780511629235}{DOI}},
+  author = {Kleidman, Peter B. and Liebeck, Martin W.},
+  title = {The Subgroup Structure of the Finite Classical Groups},
+  series = {London Mathematical Society Lecture Note Series},
+  volume = {129},
+  publisher = {Cambridge University Press},
+  address = {Cambridge},
+  year = {1990},
+  doi = {10.1017/CBO9780511629235},
+  isbn = {9780521359498}
+}
+
+@article{BurnessLiebeckShalev2017,
+  author = {Burness, Timothy C. and Liebeck, Martin W. and Shalev, Aner},
+  title = {Generation of second maximal subgroups and the existence of special primes},
+  journal = {Forum of Mathematics, Sigma},
+  volume = {5},
+  year = {2017},
+  pages = {e25},
+  note = {41 pp. \href{https://doi.org/10.1017/fms.2017.21}{DOI}},
+  doi = {10.1017/fms.2017.21},
+  url = {https://www.cambridge.org/core/services/aop-cambridge-core/content/view/33DE48669A141E6D894F050AC4D4494E/S2050509417000214a.pdf/generation-of-second-maximal-subgroups-and-the-existence-of-special-primes.pdf}
+}
+
+@phdthesis{Basile2008,
+  author = {Basile, Alberto},
+  title = {Second Maximal Subgroups of the Finite Alternating and Symmetric Groups},
+  school = {Australian National University},
+  address = {Canberra},
+  year = {2001},
+  month = apr,
+  note = {Posted in 2008 as \href{https://arxiv.org/abs/0810.3721v1}{arXiv:0810.3721v1}},
+  url = {https://arxiv.org/abs/0810.3721v1}
+}
+
+@article{LarsenPink2011,
+  note = {\href{https://doi.org/10.1090/S0894-0347-2011-00695-4}{DOI}; \href{https://people.math.ethz.ch/~pink/ftp/LP5.pdf}{published offprint}},
+  author = {Larsen, Michael J. and Pink, Richard},
+  title = {Finite subgroups of algebraic groups},
+  journal = {Journal of the American Mathematical Society},
+  volume = {24},
+  number = {4},
+  year = {2011},
+  pages = {1105--1158},
+  doi = {10.1090/S0894-0347-2011-00695-4},
+  url = {https://people.math.ethz.ch/~pink/ftp/LP5.pdf}
+}
+
+@article{LandazuriSeitz1974,
+  note = {\href{https://doi.org/10.1016/0021-8693(74)90150-1}{DOI}},
+  author = {Landazuri, Vicente and Seitz, Gary M.},
+  title = {On the minimal degrees of projective representations of the finite {Chevalley} groups},
+  journal = {Journal of Algebra},
+  volume = {32},
+  number = {2},
+  year = {1974},
+  pages = {418--443},
+  doi = {10.1016/0021-8693(74)90150-1},
+  url = {https://hdl.handle.net/2027.42/22240}
+}
+
+@incollection{BurnessTesterman2018,
+  author = {Burness, Timothy C. and Testerman, Donna M.},
+  title = {Irreducible subgroups of simple algebraic groups---a survey},
+  booktitle = {Groups St Andrews 2017 in Birmingham},
+  editor = {Campbell, C. M. and Parker, C. W. and Quick, M. R. and Robertson, E. F. and Roney-Dougal, C. M.},
+  series = {London Mathematical Society Lecture Note Series},
+  volume = {455},
+  publisher = {Cambridge University Press},
+  address = {Cambridge},
+  year = {2019},
+  pages = {230--260},
+  doi = {10.1017/9781108692397.010},
+  url = {https://arxiv.org/abs/1803.01174v2},
+  note = {\href{https://doi.org/10.1017/9781108692397.010}{DOI}. Locators refer to the corrected \href{https://arxiv.org/abs/1803.01174v2}{arXiv:1803.01174v2} (2018)},
+}
+
+@article{CPSvdK1977,
+  note = {\href{https://doi.org/10.1007/BF01390106}{DOI}},
+  author = {Cline, Edward and Parshall, Brian and Scott, Leonard and van der Kallen, Wilberd},
+  title = {Rational and generic cohomology},
+  journal = {Inventiones Mathematicae},
+  volume = {39},
+  year = {1977},
+  pages = {143--163},
+  doi = {10.1007/BF01390106}
+}
+
+@article{BNP2004,
+  note = {\href{https://doi.org/10.1016/j.jalgebra.2003.04.003}{DOI}},
+  author = {Bendel, Christopher P. and Nakano, Daniel K. and Pillen, Cornelius},
+  title = {Extensions for {Frobenius} kernels},
+  journal = {Journal of Algebra},
+  volume = {272},
+  number = {2},
+  year = {2004},
+  pages = {476--511},
+  doi = {10.1016/j.jalgebra.2003.04.003}
+}
+
+@article{BNPPSS2015,
+  author = {Bendel, Christopher P. and Nakano, Daniel K. and Parshall, Brian J. and Pillen, Cornelius and Scott, Leonard L. and Stewart, David I.},
+  title = {Bounding cohomology for finite groups and {Frobenius} kernels},
+  journal = {Algebras and Representation Theory},
+  volume = {18},
+  number = {3},
+  year = {2015},
+  pages = {739--760},
+  doi = {10.1007/s10468-014-9514-7},
+  url = {https://arxiv.org/abs/1208.6333v1},
+  note = {\href{https://doi.org/10.1007/s10468-014-9514-7}{DOI}. Preprint locators refer to \href{https://arxiv.org/abs/1208.6333v1}{arXiv:1208.6333v1}, titled {\emph{Bounding extensions for finite groups and Frobenius kernels}}},
+}
+
+@article{GaribaldiNakano2016,
+  note = {\href{https://doi.org/10.4153/CJM-2015-042-5}{DOI}; \href{https://www.cambridge.org/core/services/aop-cambridge-core/content/view/8D44F6A8D2D19D18AC8A1F5CA17A554B/S0008414X00017132a.pdf/bilinear_and_quadratic_forms_on_rational_modules_of_split_reductive_groups.pdf}{published PDF}},
+  author = {Garibaldi, Skip and Nakano, Daniel K.},
+  title = {Bilinear and quadratic forms on rational modules of split reductive groups},
+  journal = {Canadian Journal of Mathematics},
+  volume = {68},
+  number = {2},
+  year = {2016},
+  pages = {395--421},
+  doi = {10.4153/CJM-2015-042-5},
+  url = {https://www.cambridge.org/core/services/aop-cambridge-core/content/view/8D44F6A8D2D19D18AC8A1F5CA17A554B/S0008414X00017132a.pdf/bilinear_and_quadratic_forms_on_rational_modules_of_split_reductive_groups.pdf}
+}
+
+@article{DowdSin1996,
+  note = {\href{https://doi.org/10.1080/00927879608542648}{DOI}; section and page locators use the \href{https://people.clas.ufl.edu/sin/files/representations-of-algebraic-groups-in-characteristic-2.pdf}{author version}},
+  author = {Dowd, Michael F. and Sin, Peter},
+  title = {On representations of algebraic groups in characteristic two},
+  journal = {Communications in Algebra},
+  volume = {24},
+  number = {8},
+  year = {1996},
+  pages = {2597--2686},
+  doi = {10.1080/00927879608542648},
+  url = {https://people.clas.ufl.edu/sin/files/representations-of-algebraic-groups-in-characteristic-2.pdf}
+}
+
+@article{Schaffer1999,
+  note = {\href{https://doi.org/10.1080/00927879908826749}{DOI}},
+  author = {Schaffer, Mark},
+  title = {Twisted tensor product subgroups of finite classical groups},
+  journal = {Communications in Algebra},
+  volume = {27},
+  number = {10},
+  year = {1999},
+  pages = {5097--5166},
+  doi = {10.1080/00927879908826749}
+}
+
+@article{Steinberg1963,
+  note = {\href{https://doi.org/10.1017/S0027763000011016}{DOI}; \href{https://www.cambridge.org/core/services/aop-cambridge-core/content/view/02AE5498AF62BCAB6B313C75016E6015/S0027763000011016a.pdf/representations_of_algebraic_groups.pdf}{published PDF}},
+  author = {Steinberg, Robert},
+  title = {Representations of algebraic groups},
+  journal = {Nagoya Mathematical Journal},
+  volume = {22},
+  year = {1963},
+  pages = {33--56},
+  doi = {10.1017/S0027763000011016},
+  url = {https://www.cambridge.org/core/services/aop-cambridge-core/content/view/02AE5498AF62BCAB6B313C75016E6015/S0027763000011016a.pdf/representations_of_algebraic_groups.pdf}
+}
+
+@book{SteinbergNotes,
+  author = {Steinberg, Robert},
+  title = {Lectures on {Chevalley} Groups},
+  publisher = {Yale University, Department of Mathematics},
+  address = {New Haven, CT},
+  year = {1967},
+  note = {Notes prepared by John Faulkner and Robert Wilson; \href{https://www.math.utah.edu/~ptrapa/math-library/steinberg/steinberg-yale-notes.pdf}{original Yale scan}, with acknowledgment dated June 1968. Locators use original pagination},
+  url = {https://www.math.utah.edu/~ptrapa/math-library/steinberg/steinberg-yale-notes.pdf}
+}
+
+@incollection{Conrad2014,
+  author = {Conrad, Brian},
+  title = {Reductive group schemes},
+  booktitle = {Autour des sch{\'e}mas en groupes, Vol.~I},
+  series = {Panoramas et Synth{\`e}ses},
+  volume = {42--43},
+  publisher = {Soci{\'e}t{\'e} Math{\'e}matique de France},
+  address = {Paris},
+  year = {2014},
+  pages = {93--444},
+  note = {Page locators refer to the \href{https://math.stanford.edu/~conrad/papers/luminysga3.pdf}{author version (390 pp.)}},
+  url = {https://math.stanford.edu/~conrad/papers/luminysga3.pdf}
+}
+
+@article{Hasa2014,
+  author = {H{\"a}s{\"a}, Jokke},
+  title = {Growth of cross-characteristic representations of finite quasisimple groups of {Lie} type},
+  journal = {Journal of Algebra},
+  volume = {407},
+  year = {2014},
+  pages = {275--306},
+  doi = {10.1016/j.jalgebra.2014.02.031},
+  eprint = {1112.3941v2},
+  archivePrefix = {arXiv},
+  primaryClass = {math.RT},
+  url = {https://arxiv.org/html/1112.3941v2},
+  note = {\href{https://doi.org/10.1016/j.jalgebra.2014.02.031}{DOI}; \href{https://arxiv.org/html/1112.3941v2}{arXiv:1112.3941v2}. Table locators use this version}
+}
+
+@article{Luebeck2001,
+  author = {L{\"u}beck, Frank},
+  title = {Small degree representations of finite {Chevalley} groups in defining characteristic},
+  journal = {LMS Journal of Computation and Mathematics},
+  volume = {4},
+  year = {2001},
+  pages = {135--169},
+  doi = {10.1112/S1461157000000838},
+  url = {https://www.cambridge.org/core/services/aop-cambridge-core/content/view/7E6B3F033AEC9B401CABF62A6B87792F/S1461157000000838a.pdf/small-degree-representations-of-finite-chevalley-groups-in-defining-characteristic.pdf},
+  note = {\href{https://doi.org/10.1112/S1461157000000838}{DOI}; \href{https://www.cambridge.org/core/services/aop-cambridge-core/content/view/7E6B3F033AEC9B401CABF62A6B87792F/S1461157000000838a.pdf/small-degree-representations-of-finite-chevalley-groups-in-defining-characteristic.pdf}{published PDF}}
+}
+
+@article{LiebeckSeitzClassical1998,
+  author = {Liebeck, Martin W. and Seitz, Gary M.},
+  title = {On the subgroup structure of classical groups},
+  journal = {Inventiones Mathematicae},
+  volume = {134},
+  number = {2},
+  year = {1998},
+  pages = {427--453},
+  doi = {10.1007/s002220050270},
+  url = {https://link.springer.com/article/10.1007/s002220050270},
+  note = {\href{https://doi.org/10.1007/s002220050270}{DOI}; \href{https://www.researchgate.net/publication/225672765_On_the_subgroup_structure_of_classical_groups}{author-uploaded text}}
+}
+
+@article{Steinberg1981,
+  author  = {Steinberg, Robert},
+  title   = {Generators, Relations and Coverings of Algebraic Groups, {II}},
+  journal = {Journal of Algebra},
+  volume  = {71},
+  number  = {2},
+  year    = {1981},
+  pages   = {527--543},
+  doi     = {10.1016/0021-8693(81)90193-9},
+  url     = {https://www-users.cse.umn.edu/~reiner/REU/REU2019notes/steinberg-covering-groups.pdf},
+  note    = {\href{https://doi.org/10.1016/0021-8693(81)90193-9}{DOI}; \href{https://www-users.cse.umn.edu/~reiner/REU/REU2019notes/steinberg-covering-groups.pdf}{original article PDF}}
+}
+
+@book{BrayHoltRoneyDougal2013,
+  author = {Bray, John N. and Holt, Derek F. and Roney-Dougal, Colva M.},
+  title = {The Maximal Subgroups of the Low-Dimensional Finite Classical Groups},
+  series = {London Mathematical Society Lecture Note Series},
+  volume = {407},
+  publisher = {Cambridge University Press},
+  address = {Cambridge},
+  year = {2013},
+  isbn = {9780521138604},
+  doi = {10.1017/CBO9781139192576},
+  url = {https://www.cambridge.org/core/books/maximal-subgroups-of-the-lowdimensional-finite-classical-groups/30A44E29EC099FA81BA309096F8A4DA0},
+  note = {\href{https://doi.org/10.1017/CBO9781139192576}{DOI}; \href{https://ndl.ethernet.edu.et/bitstream/123456789/33955/1/125.pdf.pdf}{book PDF}}
+}
+
+@article{BaddeleyLucchini1997,
+ author={Baddeley, Robert and Lucchini, Andrea},
+ title={On Representing Finite Lattices as Intervals in Subgroup Lattices of Finite Groups},
+ journal={Journal of Algebra}, volume={196}, number={1}, pages={1--100}, year={1997},
+ doi={10.1006/jabr.1997.7069},
+ note = {\href{https://doi.org/10.1006/jabr.1997.7069}{DOI}}}
+@article{Baddeley1998,
+ author={Baddeley, Robert W.},
+ title={A New Approach to the Finite Lattice Representation Problem},
+ journal={Periodica Mathematica Hungarica}, volume={36}, number={1}, pages={17--59}, year={1998},
+ doi={10.1023/A:1004607902533},
+ note = {\href{https://doi.org/10.1023/A:1004607902533}{DOI}}}
+@incollection{Borner1999,
+ author={B{\"o}rner, Ferdinand},
+ title={A remark on the finite lattice representation problem},
+ booktitle={Contributions to General Algebra}, volume={11},
+ publisher={Verlag Johannes Heyn}, address={Klagenfurt}, pages={5--38}, year={1999}}
+@article{Aschbacher2008,
+ author={Aschbacher, Michael},
+ title={On intervals in subgroup lattices of finite groups},
+ journal={Journal of the American Mathematical Society}, volume={21}, number={3}, pages={809--830}, year={2008},
+ doi={10.1090/S0894-0347-08-00602-4},
+ url={https://authors.library.caltech.edu/records/cm5e5-ptt87},
+ note = {\href{https://doi.org/10.1090/S0894-0347-08-00602-4}{DOI}}}
+@article{McNulty2015,
+ author={McNulty, George F.}, title={A juggler's dozen of easy problems},
+ journal={Algebra Universalis}, volume={74}, number={1-2}, pages={17--34}, year={2015},
+ doi={10.1007/s00012-015-0333-4},
+ note = {\href{https://doi.org/10.1007/s00012-015-0333-4}{DOI}}}
+@misc{DeMeoInterval2014,
+ author={DeMeo, William}, title={Interval enforceable properties of finite groups},
+ year={2014}, eprint={1205.1927v4}, archivePrefix={arXiv}, primaryClass={math.GR},
+ url={https://arxiv.org/abs/1205.1927v4}, note={\href{https://arxiv.org/abs/1205.1927v4}{arXiv:\allowbreak1205.1927v4}, April 7, 2014}}
+
+@book{Jantzen2003,
+  author = {Jantzen, Jens Carsten},
+  title = {Representations of Algebraic Groups},
+  edition = {Second},
+  series = {Mathematical Surveys and Monographs},
+  volume = {107},
+  publisher = {American Mathematical Society},
+  address = {Providence, RI},
+  year = {2003},
+  doi = {10.1090/surv/107},
+  note = {\href{https://doi.org/10.1090/surv/107}{DOI}}
+}
+
+@book{Taylor1992,
+  author = {Taylor, Donald E.},
+  title = {The Geometry of the Classical Groups},
+  series = {Sigma Series in Pure Mathematics},
+  volume = {9},
+  publisher = {Heldermann Verlag},
+  address = {Berlin},
+  year = {1992},
+  isbn = {3-88538-009-9},
+  url = {https://www.maths.usyd.edu.au/u/don/papers/gcg.pdf},
+  note = {Locators refer to the \href{https://www.maths.usyd.edu.au/u/don/papers/gcg.pdf}{author PDF corrected April 2021}}
+}
+
+@article{Chevalley1935,
+  author = {Chevalley, Claude},
+  title = {D{\'e}monstration d'une hypoth{\`e}se de {M. Artin}},
+  journal = {Abhandlungen aus dem Mathematischen Seminar der Universit{\"a}t Hamburg},
+  volume = {11},
+  year = {1935},
+  pages = {73--75},
+  doi = {10.1007/BF02940714},
+  note = {\href{https://doi.org/10.1007/BF02940714}{DOI}}
+}
+
+@article{Warning1935,
+  author = {Warning, Ewald},
+  title = {Bemerkung zur vorstehenden Arbeit},
+  journal = {Abhandlungen aus dem Mathematischen Seminar der Universit{\"a}t Hamburg},
+  volume = {11},
+  year = {1935},
+  pages = {76--83},
+  doi = {10.1007/BF02940715},
+  note = {\href{https://doi.org/10.1007/BF02940715}{DOI}}
+}
+
+@article{PudlakTuma1980,
+  author = {Pudl{\'a}k, Pavel and T{\r u}ma, Ji{\v r}{\'i}},
+  title = {Every finite lattice can be embedded in a finite partition lattice},
+  journal = {Algebra Universalis},
+  volume = {10},
+  year = {1980},
+  pages = {74--95},
+  doi = {10.1007/BF02482893},
+  note = {\href{https://doi.org/10.1007/BF02482893}{DOI}}
+}
+
+@article{RepnitskiiTuma2008,
+  author = {Repnitski{\u\i}, Vladimir and T{\r u}ma, Ji{\v r}{\'i}},
+  title = {Intervals in subgroup lattices of countable locally finite groups},
+  journal = {Algebra Universalis},
+  volume = {59},
+  number = {1--2},
+  year = {2008},
+  pages = {49--71},
+  doi = {10.1007/s00012-008-2045-5},
+  url = {https://ka.karlin.mff.cuni.cz/preprint/alg07_01.pdf},
+  note = {\href{https://doi.org/10.1007/s00012-008-2045-5}{DOI}; \href{https://ka.karlin.mff.cuni.cz/preprint/alg07_01.pdf}{author preprint}}
+}
+
+@article{Seitz1987,
+  author = {Seitz, Gary M.},
+  title = {The maximal subgroups of classical algebraic groups},
+  journal = {Memoirs of the American Mathematical Society},
+  volume = {67},
+  number = {365},
+  year = {1987},
+  pages = {iv+286},
+  doi = {10.1090/memo/0365},
+  note = {\href{https://doi.org/10.1090/memo/0365}{DOI}}
+}
+
+@article{CavallinTesterman2019,
+  author = {Cavallin, Mika{\"e}l and Testerman, Donna M.},
+  title = {A new family of irreducible subgroups of the orthogonal algebraic groups},
+  journal = {Transactions of the American Mathematical Society, Series B},
+  volume = {6},
+  year = {2019},
+  pages = {45--79},
+  doi = {10.1090/btran/28},
+  url = {https://infoscience.epfl.ch/server/api/core/bitstreams/702600e9-aef0-4ffb-8d9e-4964ee89d3b1/content},
+  note = {\href{https://doi.org/10.1090/btran/28}{DOI}; \href{https://infoscience.epfl.ch/server/api/core/bitstreams/702600e9-aef0-4ffb-8d9e-4964ee89d3b1/content}{published article PDF}}
+}
diff --git a/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/sections/classification.tex b/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/sections/classification.tex
new file mode 100644
index 0000000000000000000000000000000000000000..6044865a15f75edcfb9c2c2ce488912f36358569
--- /dev/null
+++ b/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/sections/classification.tex
@@ -0,0 +1,950 @@
+\section{Tested chains and the classification reduction}
+\label{sec:classification}
+\label{cls:section}
+
+All constants called absolute in this section are independent of the finite
+group, the lattice, the Lie rank, the characteristic, the field degree, and
+the length of the chain.  A bound depending on a fixed rank bound is stated
+as such.  We use the labels and factor transport of
+Lemma~\ref{frm:labels} and Proposition~\ref{frm:active-transport}.
+
+For $k\ge2$, write $M_k$ for the height-two lattice with $k$ atoms;
+thus $M_{16}$ has eighteen elements.
+
+\begin{definition}[A tested chain]\label{cls:tested-chain}
+Let $D<a<b$ belong to a finite group interval, and let
+\[
+ a=x_0<x_1<\cdots <x_N=b.
+\]
+A tested chain consists of these vertices and vertices $v_{ij}$ for
+$0\leq i<j\leq N$, with the following properties.
+\begin{enumerate}[label=\textup{(T\arabic*)},ref=T\arabic*]
+\item\label{cls:test-labels}
+Every $x_i$ and $v_{ij}$ is labelled: its interval above $D$ is fenced.
+Moreover, $x_i\prec v_{ij}\prec x_j$, where both relations are covers in
+the entire interval.
+\item\label{cls:test-fences}
+Every $[x_i,x_j]$, $i<j$, is fenced.
+\item\label{cls:test-saturation}
+Every $x_i$ with $i<N$, and every $v_{ij}$ with $j<N$, is an intersection
+of coatoms of $[a,b]$.  The $v_{iN}$ are coatoms by
+\ref{cls:test-labels}.
+\item\label{cls:test-alternating}
+Put
+\[
+ \mathcal U=\{x_j:1\leq j\leq N\}
+ \mathbin{\cup}\{v_{ij}:1\leq i<j\leq N\}.
+\]
+For each $u\in\mathcal U$, the interval $[a,u]$ is fenced, $a$ is an
+intersection of its coatoms, and there is $a<c_u<u$ such that the
+\emph{entire} interval $[c_u,u]$ is $M_{16}$.
+\item\label{cls:test-normal}
+For $i<j$ and $x_i\leq z\leq x_j$, either $[D,z]$ is fenced, or there is
+a labelled $z'$ with $x_i\leq z'<z$ which is comparable with every vertex
+of $[D,z]$.
+\end{enumerate}
+No additional requirement is imposed on $[a,v_{0j}]$.  In particular,
+\ref{cls:test-alternating} never subdivides the cover $a\prec v_{0j}$.
+\end{definition}
+
+Condition~\ref{cls:test-normal} ensures the following consequences at
+every overgroup between chain vertices: every soluble normal subgroup
+is contained in $D$, and, of two normal subgroups whose commutator
+lies in a normal subgroup contained in $D$, at least one is contained
+in $D$.  At a fenced vertex these follow from Lemma~\ref{frm:labels};
+at a labelled bottleneck they follow from
+Corollary~\ref{frm:bottleneck}.  The same consequences hold modulo a
+higher labelled core, which is normal in each intermediate vertex
+and contained in $D$.  No full supplementation dichotomy at an
+unlabelled bottleneck is asserted.
+The construction in Section~\ref{sec:lattices} will verify exactly
+Definition~\ref{cls:tested-chain}.
+Figure~\ref{fig:tested-chain} isolates the shortcut comparison; the
+remaining tests make this comparison effective uniformly over finite groups.
+\input{figures/tested-chain}
+
+\begin{theorem}[Chain rigidity]\label{cls:chain-rigidity}
+There is an absolute integer $B$ such that every tested chain of length
+$N\geq B$ satisfies $T_a\cong T_b$.
+\end{theorem}
+
+The proof proceeds by contradiction through arbitrarily long tested
+chains whose endpoint types differ.  This section gives the classification
+and uniformity reductions.  Proposition~\ref{grw:remaining-paths} removes
+the geometric and cross-characteristic alternatives; the proof of
+Theorem~\ref{cls:chain-rigidity} is completed in Section~\ref{sec:rigidity}.
+Passing to a subchain always retains the originally supplied shortcuts
+and tests.  It does not assert that the subchain, with a new initial
+endpoint, is a new tested chain.
+
+\subsection{Literature inputs and maximal-step factor actions}
+
+\begin{proposition}[Maximal-subgroup input]\label{cls:literature}
+The following results are used from the classification and representation
+theory of finite simple groups.
+\begin{enumerate}[label=\textup{(L\arabic*)}]
+\item A maximal subgroup of an almost simple classical group, not
+containing its socle, is a stabilizer in one of Aschbacher's eight
+geometric collections, or has an almost simple irreducible component.
+The geometric collections are subspace or flag stabilizers, direct-sum
+decompositions, extension-field structures, tensor products, subfield
+structures, symplectic-type groups, tensor-induced decompositions, and
+forms.  Graph automorphisms may interchange the relevant subspaces or
+flag members; triality and exceptional graph--field automorphisms occur
+only in bounded rank.  We use
+\cite{Aschbacher1984} and the almost simple formulation and descriptions
+in \cite[Table~1.2.A and Theorem~1.2.1, pp.~3--4]{KleidmanLiebeck1990},
+with the geometric descriptions in Chapters~3--5; the connected tensor
+structures and their finite fixed-point versions are also given in
+\cite[Theorem~2 and Section~4(5)]{LiebeckSeitzClassical1998}.
+\item The maximal subgroups of alternating and symmetric groups have
+the intransitive, imprimitive, affine, diagonal, product-action, or
+almost simple alternatives of the maximal-subgroup form of the
+O'Nan--Scott Theorem.  The relevant formulation and all the alternatives
+used here are recorded in the proof of
+\cite[Proposition~3.1]{BurnessLiebeckShalev2017}.
+\item For a fixed algebraic type, sufficiently general finite subgroups
+lie between the derived group and the inner-diagonal group of a finite
+fixed-point group.  The complementary alternative is containment in a
+proper algebraic subgroup of bounded complexity.  The bounded-envelope
+version permits passage to the reductive simple quotients of that
+subgroup; see \cite[Theorems~0.5 and~12.3]{LarsenPink2011}.
+\item A subgroup maximal in a maximal subgroup of an almost simple
+group has at most five nonabelian chief factors
+\cite[Proposition~8.1]{BurnessLiebeckShalev2017}.  The number refers to
+chief factors, not to their simple direct factors.
+\item If $[H,A]$ is a height-two interval and $A$ is alternating or
+symmetric of degree at least five, it has at most eleven interior
+vertices \cite[Theorem~D]{Basile2008}.  Here the entire interval is required to have height two: $H$ is
+maximal in every maximal overgroup. This is the hypothesis imposed
+whenever the result is used.
+\item Cross-characteristic irreducible projective degrees satisfy the
+Landazuri--Seitz lower bounds \cite{LandazuriSeitz1974}.  In particular,
+outside bounded rank, a source of natural dimension $n$ and field parameter $q$ requires
+degree at least $q^{c n}$ for an absolute $c>0$; the applicable
+classical bounds are recorded in \cite[Tables~3--6]{Hasa2014}.
+Above an absolute classical rank, every perfect central cover of a
+finite simple classical group is a central quotient of its usual
+simply connected fixed-point group, uniformly over all finite fields,
+including the standard graph twists
+\cite[Theorem~1.1 and p.~528]{Steinberg1981}.
+The central-cover deduction and preservation of representations
+are proved in Lemma~\ref{lift:finite-covers}.
+After passing to that cover, defining-characteristic irreducibles
+have the Steinberg highest-weight and tensor-product parametrization
+\cite[Theorems~1.1 and~1.3]{Steinberg1963}, including the twisted
+fixed-point groups \cite[Theorem~43]{SteinbergNotes}.
+\end{enumerate}
+\end{proposition}
+
+The clauses of Proposition~\ref{cls:literature} are literature inputs.
+We first control the actions on simple factors and the resulting chains
+of projected cores. Numerical field and dimension estimates will enter in
+Section~\ref{grw:section}, after the reduction obtained here. No classification of arbitrary subgroup
+chains is assumed.
+
+For a transitive action of a group $K$ on a factor set, with point
+stabilizer $K_\omega$, a tower
+$K_\omega=K_r\leq\cdots\leq K_0=K$ describes successive invariant
+block systems. We call the induced coset actions of $K_{j-1}$ on
+$K_{j-1}/K_j$ its \emph{action levels}. The action kernels remain in
+these stabilizer groups.
+
+\begin{proposition}[Factor actions at a maximal step]\label{cls:factor-actions}
+Consider a strict labelled cover and its one-coordinate maximal subgroup
+$M$ in an almost simple group, as supplied by
+Proposition~\ref{frm:projection}. Mark its active nonabelian chief factor.
+There are absolute positive integers $b_0,r_0$ with the following property.
+\begin{enumerate}[label=\textup{(C\arabic*)}]
+\item\label{cls:cat-actions}
+The action on the simple factors of the marked chief factor is obtained
+through at most $r_0$ transitive action levels.  Each level either has
+degree at most $b_0$, or is the natural action of $A_d$ or $S_d$.
+\end{enumerate}
+\end{proposition}
+
+\begin{proof}
+We spell out the factor actions.  A subspace or paired-flag
+stabilizer has at most a bounded number of Levi components.  A tensor
+product of two spaces has two component groups.  Extension-field,
+subfield, form, and almost simple cases have one, except for the
+bounded low-dimensional coincidences.  A direct-sum decomposition or
+a tensor-induced decomposition has the full symmetric action on its
+blocks, with a possible alternating restriction from the determinant
+or spinor condition.  A block contributes at most a bounded number of
+simple factors; for instance an orthogonal plus-type four-space
+contributes two.  Taking a point inside a block and then the block
+itself accounts for this extra bounded level.  In the alternating
+case, the intransitive and affine alternatives have bounded component
+actions, while the imprimitive, diagonal, and product-action
+alternatives have this same full block action.  Exceptional types have
+bounded rank and hence bounded numbers of components.  Central,
+determinant, and form quotients do not split a simple component into
+more factors.  This proves \ref{cls:cat-actions}; the explicit
+$\mathcal C_2$ and $\mathcal C_7$ component quotients can also be read
+in \cite[proofs of Lemmas~5.4 and~5.9]{BurnessLiebeckShalev2017}.
+\end{proof}
+
+Several structural consequences of the maximal-subgroup descriptions
+will be needed when we reduce to classical groups of large rank.
+For a classical target and a classical active source type of sufficiently
+large rank, a cover with more than one active simple factor belongs to the
+geometric block or tensor cases. A soluble classical component has absolutely
+bounded natural dimension: a linear group in growing dimension contains
+a growing special linear simple section, and the corresponding assertion
+for a form group follows from its natural hyperbolic subspaces.
+For a fixed nonabelian simple type, the possible natural classical
+realizations have boundedly many dimensions. Outside the bounded list
+of low-order isomorphisms, its field, family, and dimension are determined
+by its type; dual, form, and special-isogeny conventions add only
+boundedly many choices. Finally, a geometric stabilizer has boundedly
+many component slots apart from repeated blocks. Thus fixing the inactive
+simple labels, repetition degrees, and bounded structural choices leaves
+boundedly many possible omitted dimensions. These facts use the full
+geometric stabilizers in Proposition~\ref{cls:literature}(L1), not a bound
+on the orders of the components. Their numerical field and order data
+will be established in Section~\ref{grw:section}.
+
+\subsection{A uniform lemma for natural giant actions}
+
+The shortcuts will put the relevant factor stabilizers inside one fixed
+projected bottom group. We need to control their composition types and
+indices simultaneously. The next two lemmas do this for bounded
+stabilizer towers, including the overgroups that arise when equal-type
+strips are merged.
+
+For a finite group $H$, write $\operatorname{cf}_{\mathrm{na}}(H)$
+for its set of nonabelian simple composition-factor types.  Only these
+types will be bounded; the set of cyclic-prime factors need not be.
+
+\begin{lemma}[Bounded routes]\label{cls:routes}
+Fix positive integers $r,b$.  Suppose
+\[
+ A=H_r\leq H_{r-1}\leq\cdots\leq H_0=H
+\]
+and every induced action on $H_{j-1}/H_j$ is either of degree at most
+$b$ or is a natural alternating or symmetric action.  There are
+constants $h,c,k$, depending only on $r,b$, such that:
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item the interval $[A,H]$ has height at most $h$;
+\item the set $\{[H:U]:A\leq U\leq H\}$ has cardinality at most $k$;
+\item for every such $U$, its nonabelian composition types belong to
+\[
+ \operatorname{cf}_{\mathrm{na}}(H)
+ \cup\{A_{d-j}:A_d\in\operatorname{cf}_{\mathrm{na}}(H),\ 0\leq j\leq c\}
+ \cup\mathcal E_{r,b},
+\]
+where $\mathcal E_{r,b}$ is a fixed finite set of types.
+\end{enumerate}
+The conclusions also hold for stabilizers of all invariant quotient
+blocks of the action on $H/A$.
+\end{lemma}
+\begin{proof}
+We induct on $r$.  The assertion for one step follows from primitivity
+of a natural giant, or from the bounded index in the other case.
+For the induction put $N=\operatorname{core}_H(H_1)$.  For an
+arbitrary $U\in[A,H]$, the group $A(U\cap N)$ lies in $[A,H_1]$,
+and its intersection with $N$ is $U\cap N$.  The induction hypothesis
+therefore controls the intersection chain and its types.  A strict
+increase in $U$ must increase $U\cap N$ or $UN/N$, since equality of
+both gives equality of orders.  Furthermore
+\[
+ [H:U]=[H/N:UN/N]\,[N:U\cap N],
+\]
+and the possible second factors are a fixed multiple of the possible
+indices $[H_1:A(U\cap N)]$.  Thus it remains to control the images.
+
+If $H/N$ has bounded degree, its order is at most $b!$ and there is
+nothing more to prove.  Otherwise it is $A_d$ or $S_d$, and $H_1/N$
+is its natural point stabilizer.  Project the remaining route into
+$H_1/N$.  A natural-giant step projects either to the same action or
+to a trivial action, because it is primitive; a bounded-degree step
+retains bounded degree.  Starting with the natural $A_{d-1}$ in
+$H_1/N$, retain an alternating group normal in the current projected
+group and supported on all but at most $r$ points.  A natural-giant
+quotient either kills this alternating group, which is then retained
+in the next subgroup, or maps it onto the natural alternating socle
+of its image.  In the latter case equality of orders gives the same
+degree, and its point stabilizer removes one supported point.
+A step of bounded degree retains the alternating group once its
+order exceeds $b!$.  After a step, take the union of all conjugates
+of the retained support in the new subgroup.  The overlapping-support
+argument in the next paragraph supplies a normal alternating group
+on that union, without further loss of points.
+Apart from a possible sign on the large support, the residual
+quotient acts on the omitted points and has order at most $2r!$.
+For $d>2r+b+10$, each step therefore loses at most one supported
+point, and the terminal image contains the alternating group
+supported on at least $d-r$ points.  Here the identification
+of the next point stabilizer uses
+$\operatorname{Aut}(A_e)=S_e$ for $e\geq7$; the degree-six exception
+and all smaller first degrees are absorbed in the fixed finite
+universe.  Thus we may take $c=r$ in the large-degree argument.
+
+An overgroup of an alternating group supported on $d-c$ points has
+bounded height and boundedly many possible indices.  To see this
+directly, assume $d>2c+4$.  The supports of any two conjugates overlap
+in at least three points, so their alternating groups generate the
+alternating group on the union of their supports.  The union of
+all conjugate supports is an orbit of the overgroup; its conjugate
+alternating groups generate the alternating group on that orbit.
+The orbit omits at most $c$ points.  Modulo the
+alternating group on that orbit, the overgroup has order at most
+$2c!$.  Its large simple type is therefore $A_{d-j}$ with
+$0\leq j\leq c$, and its possible indices belong to the finite list
+of falling factorials $(d)_j$ divided or multiplied by integers
+bounded in terms of $c$.  The excluded small $d$ add only bounded
+groups.  Combining this with the intersection induction proves
+all three conclusions.  Invariant quotient blocks correspond
+exactly to the overgroups $U$ already considered.
+\end{proof}
+
+\begin{lemma}[A common index palette]\label{cls:index-recipes}
+Fix positive integers $R,b,K$.  Consider any collection of the
+routes in Lemma~\ref{cls:routes}, each of length at most $R$ and
+bounded degree at most $b$, such that all their natural giant
+degrees belong to one set $\mathcal D$ with $|\mathcal D|\leq K$.
+The union, over all groups and routes in the collection, of all
+indices $[H:U]$ with $A\leq U\leq H$ has cardinality bounded by a
+function of $R,b,K$.  The same holds through a bounded number of
+operations, each consisting of a permitted route followed by an
+arbitrary invariant quotient-block choice, with a fresh route
+inside the chosen block stabilizer permitted at the next operation.
+\end{lemma}
+\begin{proof}
+Put $d_0=2R+b+10$ and $M=d_0!$, and define the positive rational
+atoms
+\[
+ \Gamma(\mathcal D)=
+ \{u/v:1\leq u,v\leq M\}
+ \cup\{(d)_j:d\in\mathcal D,\ d>d_0,\ 0\leq j\leq R\}.
+\]
+We claim that every index on a route of length $r$ is a product of
+at most $L_r=r(r+3)/2$ atoms or their inverses.  Induct on $r$,
+putting $N=\operatorname{core}_H(H_1)$ and
+$B=A(U\cap N)$.  Since $U\cap N\trianglelefteq U$, we have
+$A\leq B\leq H_1$ and $B\cap N=U\cap N$.  Consequently
+\begin{equation}\label{cls:route-index-identity}
+ [H:U]=[H/N:UN/N]\,
+        \frac{[H_1:B]}{[H_1:AN]}.
+\end{equation}
+The image factor is bounded by $M$ when the first action is small.
+Otherwise the supported alternating-group calculation in
+Lemma~\ref{cls:routes} gives
+\[
+ [H/N:UN/N]=\frac{\varepsilon(d)_j}{t},\qquad
+ 0\leq j\leq r,\quad \varepsilon\in\{1,2\},\quad1\leq t\leq2j!.
+\]
+It therefore uses at most two atoms.  The numerator $[H_1:B]$
+uses at most $L_{r-1}$ atoms by induction.
+
+The denominator is controlled across the whole collection, not
+just held fixed within one route.  Project the suffix route into
+$H_1/N$.  Its degree is exactly
+\begin{equation}\label{cls:route-denominator}
+ [H_1:AN]=\prod_{j=2}^{r}
+ [H_{j-1}N/N:H_jN/N].
+\end{equation}
+Under any homomorphism with kernel $C$, a natural step $L_0>L_1$
+projects to the same full natural action or to a trivial action:
+maximality gives $L_1C=L_1$ or $L_0$.  A bounded step keeps bounded
+degree.  Thus every factor on the right of
+\eqref{cls:route-denominator} is an allowed natural degree, a
+bounded integer, or one, and is a single atom.  In particular no
+arbitrary scalar-kernel divisor enters the denominator.
+Equation~\eqref{cls:route-index-identity} gives
+$L_r\leq L_{r-1}+2+(r-1)$, proving the claim.
+
+There are at most $M^2+K(R+1)$ atoms, so the union of all index
+palettes has cardinality at most
+\[
+ \bigl(1+2(M^2+K(R+1))\bigr)^{L_R}.
+\]
+Only integral values of these rational recipes occur as actual
+indices.  Invariant quotient blocks correspond to the overgroups
+$U$ already allowed.  If $H'$ is any such block stabilizer, apply
+the argument anew to the next full route inside $H'$; it is not
+necessary that $H'$ itself admit a natural-step route from $H$.
+Indices multiply through a bounded number of these operations,
+giving the final assertion.
+\end{proof}
+
+\begin{lemma}[Uniform active actions]\label{cls:uniform-actions}
+Suppose a tested chain has $T_a\not\cong T_b$, and project at one
+coordinate of the socle of $b$.  Let $H_*$ be the image of the factor
+stabilizer in $a$.  Then:
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item $H_*$ has at most five nonabelian chief factors;
+\item there is one set of nonabelian types, of absolutely bounded
+cardinality, which contains the types of every whole projected
+stabilizer group in $H_*$ of an active factor, of an intermediate
+fiber, or of an invariant quotient block used along a two-edge
+shortcut, and the types of all their induced images;
+\item equal-type stretches of chain vertices have absolutely
+bounded length;
+\item the set of ratios $m_{x_i}/m_{x_j}$, $i<j$, has absolutely
+bounded cardinality after equal-type repetitions are removed.
+\end{enumerate}
+\end{lemma}
+\begin{proof}
+By Propositions~\ref{frm:strict-product} and~\ref{frm:projection}, the
+starting group is a nontrivial proper coordinate product, and all
+designated intersections of coatoms are saturated.  The shortcut
+$a\prec v_{0N}\prec b$ projects to a two-step maximal chain in an
+almost simple group.  Thus Proposition~\ref{cls:literature} gives
+the five-chief-factor bound for the actual projected bottom $H_*$.
+The image of the original $D$-stabilizer need not equal $H_*$ as a
+group: in general $H_*$ is that image multiplied by the starting
+coordinate product.  What is required is equality of the induced
+active permutation actions, as follows.
+
+For every label $z$, the equality $z=DN_z$ implies that $D$ and $z$
+have the same image on $I_z$, because $N_z$ fixes its individual
+simple factors.  If $I_z\to I_w$ is a product-step factor map and
+$g\in z$ fixes $i\in I_w$, write $g=dn$ with $d\in D$ and
+$n\in N_z$.  Then $d$ fixes $i$ and has the same action as $g$ on
+the fiber over $i$.  Thus the bottom stabilizer induces the
+\emph{full} local factor action at each level, not an arbitrary
+subgroup of the natural giant.  This property survives the initial
+projection by Proposition~\ref{frm:active-transport}.
+
+We now account for all the shortcuts, including middles below a
+proper endpoint.  An endpoint $x_j$ is reached from $b$ by the
+original shortcut $b>v_{jN}>x_j$; take an empty route for $j=N$.
+The first downward step is strict, because a proper subdirect group
+cannot contain the nontrivial full coordinate product at the bottom.
+Two strict covers give a concatenated route of length at most
+$2r_0$.  A strict cover followed by an equal-type merge instead
+makes the endpoint factor stabilizer an overgroup of the first
+route's terminal stabilizer.
+
+Starting at $x_j$, the middle $v_{ij}$ and the lower endpoint $x_i$
+are reached by at most two further original covers.  An equal-type
+operation takes an invariant quotient block.  A strict operation
+supplies a fresh full local route inside the actual current
+projected factor stabilizer $U\leq H_*$.  Indeed the image of the
+original $D$-stabilizer lies in $U$ and already has the full local
+permutation image, whereas $U$ acts through the labelled-node
+stabilizer because $a$ lies below that node.  The kernel of the
+original-coordinate projection centralizes that coordinate and acts
+trivially on its sections.  Thus these are equal local permutation
+images; no equality of the stabilizer groups themselves is needed.
+
+The factor maps and equal-type strips respect the original $b$
+coordinates.  Here is the intersection identity behind this assertion.
+For labels $x\leq z$ used in a route, work modulo $F_b$ and write
+\[
+ x\cap N_b=P_x^{I_b},\qquad
+ N_z\cap N_b=R_z^{I_b},\qquad F_z\cap N_b=Q_z^{I_b}.
+\]
+These product descriptions follow from saturation and
+Proposition~\ref{frm:active-transport}.  Since $F_z\leq D\leq x$ and
+$N_z=F_z(N_z\cap N_b)$, intersection and the quotient map give
+\[
+ \frac{x\cap N_z}{F_z}
+ \cong \frac{x\cap N_z\cap N_b}{F_z\cap N_b}
+ =\prod_{i\in I_b}\frac{P_x\cap R_z}{Q_z}.
+\]
+In an equal-type comparison this is the subdirect subgroup of $S_z$
+whose strips describe the merge.  Because it is a full product over
+$I_b$, no strip can link factors in different original coordinates:
+such a link would prohibit independent elements in the two factors.
+For a strict split followed by a merge inside $[x_i,x_j]$, repeat this
+identity with $x_j$ as upper vertex.  The fenced endpoint comparison
+makes $x_i$ saturated in those coordinates, and the strict shortcut
+cover makes its middle saturated there.  Thus the merge cannot cross
+those local starting fibers either.
+Altogether, every required factor, fiber, and quotient-block
+stabilizer uses at most four original covers, interpreted as full
+local routes or quotient operations.  Four successive operations,
+each with route length at most $2r_0$, give a conservative common
+bound, including all intermediate block stabilizers.
+
+We establish the type pool before the degree and index pools.  For
+fixed route parameters let
+\[
+ \Phi(\mathcal P)=\mathcal P
+\mathbin{\cup}\{A_{d-j}:A_d\in\mathcal P,\ 0\leq j\leq c,\ d-j\geq5\}
+\mathbin{\cup}\mathcal E,
+\]
+where $c$ and the finite exceptional set $\mathcal E$ are supplied
+by Lemma~\ref{cls:routes}.  If the whole current projected group has
+nonabelian composition types in $\mathcal P$, that lemma puts the
+types of every route prefix and every quotient-block overgroup in
+$\Phi(\mathcal P)$.  This includes any kernel of the corresponding
+permutation action.  Retain the original types at every iteration.
+Starting with $\operatorname{cf}_{\mathrm{na}}(H_*)$, of cardinality
+at most five, a bounded number of iterations gives one pool of
+absolute bounded cardinality for all these projected groups.
+Their induced images have no additional composition types.  This
+proves (ii); it makes no claim about entire unprojected stabilizers
+in the original group.
+
+For an equal-type stretch, fix its highest vertex.  Equal-type
+transport identifies all earlier vertices with strip subgroups
+of that socle, whose support partitions strictly change along the
+stretch.  Equal support partitions and subgroup inclusion would
+force equality of orders and hence equality of subgroups, including
+the diagonal twists.  The preceding coordinate compatibility
+restricts these to invariant partitions of a single original
+$b$-coordinate fiber.  Transitivity ensures that a strict change
+appears on that chosen fiber.  All their block stabilizers are
+overgroups of a single route terminal supplied by the shortcut
+from $b$ to the highest vertex.  The height assertion of
+Lemma~\ref{cls:routes} proves (iii).
+
+Each full giant of large degree $d$ contributes the composition
+type $A_d$ to its actual starting projected group.  Every such group is a route
+prefix already covered by the preceding pool.  Hence the set
+$\mathcal D$ of giant degree values occurring anywhere in these
+bounded operations has absolute bounded cardinality.  The values
+may be large.  Apply Lemma~\ref{cls:index-recipes} with this entire
+set $\mathcal D$, and multiply indices through the bounded number
+of operations.  This yields a common index palette across all the
+actions, including arbitrary quotient-block choices followed by
+fresh full routes.  In particular it is a union bound, not merely
+a separate cardinality bound for each action.
+
+More explicitly, put $q_i=m_{x_i}/m_b$.  The original shortcut from
+$b$ to $x_i$ realizes $q_i$ as the degree on its active-factor fiber:
+a strict/strict path gives a route degree, and a strict/merge path
+gives a quotient-block degree.  The common recipe palette bounds
+the cardinality of $\{q_i\}$.  Since
+$m_{x_i}/m_{x_j}=q_i/q_j$, its quotient palette also has bounded
+cardinality.  This proves (iv).
+\end{proof}
+
+\subsection{The core reduction}
+
+\begin{lemma}[Killed layers and core gaps]\label{cls:core-gaps}
+In the initial projection, consider chain vertices strictly above the
+original starting vertex $a$, and write
+\[
+ K_i=H_*P_i,\qquad C_i=\operatorname{core}_{K_i}(H_*),
+ \qquad Q_i=C_i\cap P_i,
+\]
+and let $R_i/Q_i$ be its active nonabelian chief factor.  For $i<j$,
+the inner comparison and its killed kernel are
+\[
+ J_{ij}=(P_i\cap R_j)/Q_j,
+ \qquad B_{ij}=(Q_i\cap R_j)/Q_j.
+\]
+There is an embedding $B_{ij}\hookrightarrow C_i/C_j$.
+After deleting at most five nonsoluble gaps, an arbitrarily long
+subchain remains on which all the $B_{ij}$, including the analogous
+shortcut kernels, are soluble.  If the chain multiplicities are
+equal, every endpoint inner comparison, projected to one factor of
+its upper active socle, has just its active simple factor as a
+nonabelian composition factor.
+\end{lemma}
+\begin{proof}
+The exact product/core identity in
+Proposition~\ref{frm:active-transport} gives, before projection,
+\[
+ (F_i\cap N_b)/F_b=Q_i^{m_b},\qquad
+ (N_i\cap N_b)/F_b=R_i^{m_b},\qquad
+ R_i/Q_i\cong T_i^{m_i/m_b}.
+\]
+Since $F_j\leq D\leq x_i$, we have $Q_j\leq P_i$.
+Intersecting with the lower group and then with its core gives the
+displayed formulas for $J_{ij}$ and $B_{ij}$.  Also $C_j\leq C_i$
+and $R_j\leq P_j$, whence
+\[
+ C_j\cap R_j=Q_j.
+\]
+Thus $uQ_j\mapsto uC_j$ embeds $B_{ij}$ in $C_i/C_j$.
+
+The $C_i$ form a decreasing chain of normal subgroups of the one
+group $H_*$.  Each nonsoluble successive quotient consumes at least
+one nonabelian chief factor of $H_*$.  There are at most five, by
+Lemma~\ref{cls:uniform-actions}.  Cutting at these transitions
+leaves at most six stretches; one is arbitrarily long.  The core
+of a shortcut middle lies between the cores of its two endpoints,
+so its killed layers are controlled by the same soluble quotient.
+No separate bound on a collection of unrelated shortcut cores is
+being assumed.
+
+For the final assertion work with the original labelled vertices
+$x_i<x_j$ and put
+\[
+ E=x_i\cap N_j,\qquad W=F_i\cap N_j,\qquad L=N_i\cap N_j.
+\]
+These subgroups are normal in $x_i$, with $F_j\leq W\leq L$.
+The kernel of the conjugation action of $E$ on
+$S_i=N_i/F_i$ is exactly $W$, because
+$C_{x_i/F_i}(S_i)=1$.  When $m_i=m_j$, the product support map
+$I_i\to I_j$ is a bijection.  Since $E\leq N_j$ fixes $I_j$,
+it fixes every factor of $S_i$.  Its socle preimage is $L$, and
+therefore
+\[
+ E/L\hookrightarrow\operatorname{Out}(T_i)^{m_j},
+\]
+which is soluble by Schreier's Theorem.  In the $x_j$ coordinates
+write
+\[
+ E/F_j=P_{ij}^{m_j},\qquad W/F_j=Q_{ij}^{m_j},\qquad
+ L/F_j=R_{ij}^{m_j}.
+\]
+Proposition~\ref{frm:active-transport} gives
+$R_{ij}/Q_{ij}\cong T_i$, and the preceding embedding makes
+$P_{ij}/R_{ij}$ soluble.
+
+The killed kernel is exactly the one controlled in the initial
+$b$ projection.  Indeed $N_j=F_j(N_j\cap N_b)$ and $F_j\leq F_i$
+give
+\begin{align}
+ F_i\cap N_j&=F_j(F_i\cap N_j\cap N_b),\label{cls:killed-intersection}\\
+ W/F_j&\cong
+ \bigl((Q_i\cap R_j)/Q_j\bigr)^{m_b}.\label{cls:killed-kernel}
+\end{align}
+It is soluble on the retained stretch, so $Q_{ij}$ is soluble as
+well.  Thus $P_{ij}$ has exactly its one active simple factor as a
+nonabelian composition factor.  The same intersection calculation
+identifies the killed kernels for either half of a shortcut.
+This argument takes place in the original labelled $x_i$; it uses
+neither projected quasiprimitivity nor a supplementation dichotomy
+at an unlabelled bottleneck.  A shortcut middle which
+still has several active factors can retain a nonabelian
+factor-permutation quotient; this is not asserted to be soluble.
+Its degrees are controlled by the common type pool, and it occurs
+only in the split--merge case considered separately below.
+\end{proof}
+
+\begin{lemma}[Exclusion of alternating upper types]\label{cls:alternating}
+For $u\in\mathcal U$ with $T_a\not\cong T_u$, the type $T_u$ is
+not alternating of degree at least seven.
+\end{lemma}
+\begin{proof}
+Project the strict comparison from $a$ to $u$ using
+\ref{cls:test-alternating}.  The bottom is a full product and an
+intersection of coatoms.  The coatoms and their intersection $c_u$
+are saturated, so Proposition~\ref{frm:projection} carries the entire
+interval $[c_u,u]$ to an $M_{16}$ interval ending at an almost simple
+group with socle $T_u$.  For an alternating socle of degree at least
+seven that group is alternating or symmetric, contradicting
+Proposition~\ref{cls:literature}.  The exceptional automorphisms in
+degree six concern only bounded-order types.
+\end{proof}
+
+\subsection{Why bounded rank cannot support a long strict chain}
+
+The bounded-rank argument needs a normalized subfield group, rather
+than just a numerical resemblance between two field sizes.  We give
+the details, including the automorphisms of the almost simple ambient
+group.
+
+\begin{lemma}[Bounded envelopes]\label{cls:envelopes}
+Fix a connected adjoint simple algebraic type $\Phi$.  There is a
+constant $c_\Phi$ such that a finite subgroup $P$ of an algebraic
+group $X$ of type $\Phi$ has one of the following descriptions.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item For a Frobenius map $F_0$,
+\[
+ E=(X^{F_0})'\leq P\leq X^{F_0},
+\]
+where $E$ is simple.
+\item Every nonabelian composition factor of $P$ of order greater
+than $c_\Phi$ is of Lie type in the characteristic of $X$, with
+algebraic dimension strictly less than $\dim X$.  The total number
+of nonabelian composition factors, counted with multiplicity, is at
+most $c_\Phi$.
+\end{enumerate}
+In particular, the ranks of nonabelian simple sections of finite
+groups of Lie type of rank at most $R$ are bounded by a function of
+$R$; simple sections of bounded order are included in this bound.
+\end{lemma}
+\begin{proof}
+Apply \cite[Theorem~0.5]{LarsenPink2011}.  Outside (i), $P$ lies in
+a proper algebraic subgroup $B<X$ from a fixed constructible family.
+Apply \cite[Theorem~12.3]{LarsenPink2011} to that family.  It supplies
+an envelope $H\leq B$, with $P\leq H$, whose connected reductive
+quotient has simple adjoint factors $X_1,\ldots,X_t$ such that the
+image $L_i$ of $P\cap H^\circ$ satisfies
+\[
+ E_i=(X_i^{F_i})'\leq L_i\leq X_i^{F_i},
+\]
+with $E_i$ simple.  Each $\dim X_i\leq\dim H^\circ<\dim X$.
+The kernel of the map from $H^\circ$ to the product of its simple
+adjoint quotients is soluble: it is an extension of the unipotent
+radical by the full center of the connected reductive quotient.
+That center can be disconnected, but is abelian.  If $L$ is the image of $P\cap H^\circ$
+in $\prod_i L_i$, then $L'$ is a subdirect subgroup of
+$\prod_iE_i$, because each $L_i/E_i$ is abelian.  The subdirect
+product Lemma makes $L'$ a product of full diagonals.  Its simple
+factors are therefore among the $E_i$, counted at most once per
+position.  There are at most $\dim X/3$ positions.
+
+The component bound in \cite[Proposition~1.4]{LarsenPink2011} bounds
+$|H/H^\circ|$ throughout this constructible family.  Its contribution
+has bounded order and bounded composition length.  This proves (ii).
+Applying the same argument to every subgroup of a finite group of
+rank at most $R$ proves the last assertion: the ambient algebraic
+dimensions range over a finite set, and the finitely many small
+orders and characteristics in the exceptional alternative are
+absorbed in a function of $R$.
+\end{proof}
+
+\begin{lemma}[Normalized subfield steps]\label{cls:subfields}
+Fix an adjoint simple type $\Phi$.  Let $T=(X^F)'$ be simple,
+$T\leq A\leq\operatorname{Aut}(T)$, and let $M<A$ be maximal with
+$T\not\leq M$.  Suppose that $P=M\cap T$ is in alternative (i) of
+Lemma~\ref{cls:envelopes}, and that the parameter of $F_0$ is large
+enough in terms of $\Phi$.  Then $F=F_0^r$ for a prime $r$.
+Every element of $M$ normalizes the derived fixed-point groups for
+all intermediate powers of $F_0$.
+\end{lemma}
+\begin{proof}
+First, the pointwise centralizer of $E=(X^{F_0})'$ in
+$\operatorname{Aut}_{\rm alg}(X)$ is trivial when the field parameter
+is sufficiently large.  Indeed the fixed groups $X^a$, for nonidentity
+algebraic automorphisms $a$, form a constructible family of proper
+closed subgroups: take the equalizer of the action and projection
+maps over the finite-type algebraic automorphism group, remove the
+identity section, and stratify.  Now use
+\cite[Proposition~3.5]{LarsenPink2011}.
+
+Here is the isogeny calculation needed to include field and exceptional
+graph operations.  Let $\mathcal I(X)$ be the group of bijections of
+$X(\overline{\mathbb F}_p)$ generated by self-isogenies and their
+inverses.  The isogeny Theorem
+\cite[Theorem~6.1.16(1) and Proposition~6.1.13(2)]{Conrad2014}
+gives a homomorphism
+\[
+ \nu:\mathcal I(X)\longrightarrow\mathbb Z,
+ \qquad\ker\nu=\operatorname{Aut}_{\rm alg}(X).
+\]
+For completeness, after fixing a Borel pair, a self-isogeny permutes
+the simple roots by $\pi$ and raises their parameters to powers
+$p^{e_i}$.  Use the row-coroot convention
+$C_{ij}=\langle\alpha_j,\alpha_i^\vee\rangle$ for the Cartan
+matrix.  The compatibility condition is
+\[
+ p^{e_j}C_{ij}=p^{e_i}C_{\pi(i)\pi(j)}.
+\]
+The labeled undirected Coxeter diagram is preserved.  Unless a
+multiple edge is reversed, connectedness forces all $e_i$ equal.
+An edge reversal is possible only for $B_2,F_4$ in characteristic
+two, or $G_2$ in characteristic three.  The exponents on the two root
+lengths then differ by one.  Thus every self-isogeny is $a\theta^m$,
+where $a$ is algebraic, $m\geq0$, and $\theta=F_p$ except in those
+three cases, when $\theta^2=F_p$; see also
+\cite[Remark~6.2.10]{Conrad2014}.  Root-group constants are absorbed
+by the algebraic automorphism, using the pinned uniqueness in the
+isogeny Theorem.  Conjugation by $\theta$ preserves algebraic
+automorphisms, so the same normal form holds with $m\in\mathbb Z$
+in $\mathcal I(X)$.  Inseparability degree proves uniqueness of $m$,
+and $\nu(a\theta^m)=m$ is the asserted homomorphism.  Positive
+$\nu$ gives a Frobenius map: Lang--Steinberg removes its inner
+factor, leaving a positive diagram--field map
+\cite[Proposition~2.7(i), p.~435]{LiebeckSeitzClassical1998}.
+Its field parameter
+is $p^{\nu}$, or $p^{\nu/2}$ in the three exceptional cases.
+
+Let $C_E\leq\mathcal I(X)$ fix $E$ pointwise.  The preceding
+centralizer observation makes $\nu|_{C_E}$ injective.  Its image is
+a nonzero subgroup of $\mathbb Z$, since it contains $\nu(F_0)$.
+Thus $C_E=\langle F_1\rangle$ for a Frobenius map $F_1$, and
+$F_0=F_1^b$ for some $b\geq1$.  Put $d=\dim X$ and let $q_0$ be
+the parameter of $F_0$.  The fixed-point order bounds and bounded
+diagonal index in \cite[Theorem~3.4(b),(d)]{LarsenPink2011} give
+\[
+ \frac{(q_0-1)^d}{z_\Phi}
+ \leq |E|\leq |X^{F_1}|
+ \leq c_\Phi q_0^{d/b}.
+\]
+Enlarging $c_\Phi$ includes the finitely many possibilities for the
+parameter of $F_1$ below four.  For sufficiently large $q_0$, this
+forces $b=1$.  Consequently $C_E=\langle F_0\rangle$.  Since $F$
+fixes $E$, it follows that $F=F_0^r$ with $r>0$.
+
+The quotient $P/E$ is abelian and $E$ is perfect, so $P'=E$.
+Every $m\in M$ therefore normalizes $E$.  The finite automorphism
+Theorem \cite[Theorems~30 and~36]{SteinbergNotes} lifts $m$ to an
+element $\widetilde m\in\mathcal I(X)$.  The commutator
+$\widetilde mF_0\widetilde m^{-1}F_0^{-1}$ has $\nu=0$ and fixes
+$E$ pointwise.  It is therefore the identity.  Thus $m$ normalizes
+$E_e=(X^{F_0^e})'$ for every divisor $e$ of $r$.
+
+We have $r>1$, since $r=1$ would give $T\leq M$.  If $r$ were
+composite, choose $1<e<r$ dividing $r$.  Fixed-point order estimates,
+after increasing the original field threshold, give
+$E_e\not\leq P$ and $1<E_e<T$.  Hence
+\[
+ M<ME_e\leq N_A(E_e)<A.
+\]
+The last inequality follows because $T$ is simple and cannot
+normalize a nontrivial proper subgroup of itself.  This contradicts
+maximality and proves that $r$ is prime.
+\end{proof}
+
+\begin{lemma}[Bounded-rank chains]\label{cls:bounded-rank}
+For every $R$ there is $b(R)$ such that at most $b(R)$ distinct Lie
+types of rank at most $R$ occur in a tested chain after the alternating
+exclusion.  In an arbitrarily long family of such chains with distinct
+types, one may therefore require the ranks of all retained endpoints
+and all their shortcut middles to exceed any prescribed constant.
+\end{lemma}
+\begin{proof}
+Consider a subsequence of distinct endpoint types of ranks at most
+$R$.  A lower simple type is a section of each later one by
+Proposition~\ref{frm:equal-transport}; in particular their orders increase strictly.
+Remove the boundedly many types below the order thresholds of
+Lemma~\ref{cls:envelopes}, for the finitely many relevant algebraic
+types.  Every two-edge shortcut has intermediate types of rank
+bounded in terms of $R$, by the last assertion of that lemma.
+Increase the threshold accordingly.  On each strict cover read
+downwards, the characteristic is then unchanged and the algebraic
+dimension does not increase.  The only case of equal algebraic
+dimension is the sufficiently general alternative.  An equal-type
+cover changes neither characteristic nor dimension.
+
+There are finitely many possible dimensions and algebraic families.
+If the subsequence were arbitrarily long, it would have an
+arbitrarily long further subsequence with one dimension and family.
+On every prescribed two-edge shortcut between its endpoints the
+dimension cannot drop.  Each strict cover is therefore a normalized
+prime subfield step by Lemma~\ref{cls:subfields}; a possible equal-type
+cover contributes no field change.  With one fixed family at the
+endpoints, the conversion between its usual exponent $f$ and the
+Frobenius parameter exponent cancels.  Thus, for every $i<j$,
+\[
+ f_j/f_i\in\mathbb Z_{>1},\qquad \Omega(f_j/f_i)\leq2,
+\]
+where $\Omega$ counts prime factors with multiplicity.  Four
+successive endpoints would give
+\[
+ \Omega(f_4/f_1)=\sum_{i=1}^3\Omega(f_{i+1}/f_i)\geq3,
+\]
+a contradiction.  All discarded sets and pigeonhole choices have
+bounds depending only on $R$, proving the first assertion.
+
+For any desired rank threshold, remove the boundedly many endpoints
+below a sufficiently larger threshold.  If a remaining shortcut
+middle had bounded rank, its lower endpoint, being a simple section
+of it, would have rank bounded by Lemma~\ref{cls:envelopes}.  The
+larger threshold was chosen to exclude this.  Arbitrarily long
+chains remain, proving the second assertion.
+\end{proof}
+
+\subsection{The configuration passed to the growth argument}
+
+\begin{proposition}[Reduction to classical one-factor comparisons]
+\label{cls:reduction}
+If Theorem~\ref{cls:chain-rigidity} is false, then for every $s$ and
+$R$ there is a retained chain of at least $s$ vertices, with all its
+original two-edge shortcuts and normal-group tests, such that:
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item all endpoint types are distinct classical types; their ranks,
+and the ranks of all shortcut middle types, exceed $R$;
+\item all endpoint multiplicities are equal.  Thus each comparison,
+projected to one upper active coordinate, has one active simple
+factor;
+\item the killed kernels of all endpoint and shortcut comparisons
+are soluble.  Every endpoint inner comparison has exactly its one
+active simple factor as a nonabelian composition factor;
+\item there is a pool of absolutely bounded cardinality containing
+all nonabelian simple types in every projected active-factor and fiber
+stabilizer in the fixed $H_*$, and in their induced images, used in
+these comparisons.  No assertion concerns the entire unprojected
+original stabilizers.  The splitting multiplicities
+and invariant-block indices have boundedly many possible values;
+\item each shortcut is either two strict steps, both with one
+active simple factor and all other inner composition factors
+soluble, or a strict split followed downwards by an equal-type
+merge.  The latter is geometric.  In geometric steps the inactive
+dimensions have boundedly many possible values once their simple
+labels and bounded structural choices are fixed; when all inactive
+components are soluble these dimensions are absolutely bounded.
+\end{enumerate}
+All bounds described as absolute are uniform over the original
+groups.  This proposition introduces no new lattice tests at the
+retained bottom.
+\end{proposition}
+\begin{proof}
+Take counterexamples of lengths tending to infinity, retaining the
+original endpoints initially.  A lower simple type is a section of
+an upper one, so equality of the endpoint types of a segment forces
+equality throughout that segment.  Lemma~\ref{cls:uniform-actions}
+bounds the lengths of these equal-type stretches.  Choose one
+vertex from each stretch and discard the original initial stretch.
+There are arbitrarily many remaining vertices, and none has the
+type of the original $a$.  Their shortcut middles are above a
+retained lower vertex, so also cannot have type $T_a$.
+Lemma~\ref{cls:alternating} now excludes all large alternating
+types at both endpoints and middles.  Sporadic groups and all other
+bounded-order coincidences form a finite list.  Bounded-rank Lie
+types can be removed using Lemma~\ref{cls:bounded-rank}.  This gives
+(i), with arbitrarily large minimum rank.
+
+All later coordinate projections still use the original tests.
+For a retained upper endpoint or middle $u$, condition
+\ref{cls:test-alternating} applies to $a<u$, with
+$T_a\not\cong T_u$.  Proposition~\ref{frm:projection}, with
+original lower vertex $a$, gives $H_a\cap T_u=P_a$.  A retained
+vertex $z$ saturated in these coordinates then has
+\[
+ K_z=H_aP_z,\qquad K_z\cap T_u=P_z,
+\]
+because $P_a\leq P_z$.  Saturation for a distinct-type endpoint
+comparison comes from its fence, and saturation at a strict
+shortcut cover comes from its coatom status.  Thus the required
+inner-intersection identity never relies on inheriting a local
+coatom-meet representation from the global one below $b$, or on a
+new test at a retained bottom.
+
+Use the common cardinality bound on ratios in
+Lemma~\ref{cls:uniform-actions} to color pairs by $m_i/m_j$.
+Finite Ramsey gives an arbitrarily long homogeneous subchain.
+On a homogeneous triple the common ratio $t$ satisfies $t^2=t$;
+since $t>0$, it is one.  This proves (ii).  Next apply
+Lemma~\ref{cls:core-gaps} in the original fixed projection: deleting
+the at most five nonsoluble transitions leaves an arbitrarily long
+stretch.  All shortcut cores between its endpoints lie in the
+corresponding soluble gap.  Passing to any later active coordinate
+takes subgroups and images of these soluble killed kernels, and
+hence preserves solubility.  Lemma~\ref{cls:core-gaps} and
+Proposition~\ref{frm:active-transport} give (iii).
+
+The common type pool and index bounds are precisely
+Lemmas~\ref{cls:uniform-actions} and~\ref{cls:index-recipes}; none of
+these bounds counts abelian factors.  They remain valid under all
+the preceding thinning, giving (iv).
+
+Read a shortcut downwards from its upper endpoint.  Its first
+cover cannot be a proper subdirect merge, because the lower endpoint
+is a nontrivial full product.  If both covers are strict, their
+positive integral splitting multiplicities multiply to the endpoint
+ratio one.  Both are therefore one.  Their killed kernels are
+soluble by (iii), and their tails are soluble by
+Proposition~\ref{frm:active-transport}.  If the second cover is an
+equal-type merge, the first must split; the only unbounded-rank
+maximal-subgroup cases with such an active split are the geometric
+block and tensor cases described after
+Proposition~\ref{cls:factor-actions}.  The
+merge stays inside each original upper coordinate.  Its possible
+indices are already covered by (iv).
+
+The inactive nonabelian labels in a shortcut belong to the pool in
+(iv). To see this, write its local inner group as $P\trianglelefteq K$,
+with soluble killed kernel $Q$ and active chief factor $R/Q\cong T^a$.
+The faithful conjugation argument in
+Proposition~\ref{frm:active-transport} gives
+$P/Q\hookrightarrow\operatorname{Aut}(T^a)$. The kernel of the
+factor-permutation map on $P/R$ lies in
+$\operatorname{Out}(T)^a$ and is soluble. Since $P$ is normal in $K$,
+its permutation image is normal in the full local permutation image.
+Lemma~\ref{cls:uniform-actions} identifies that full image with an
+image of one of the projected stabilizers counted in (iv). Its normal subgroup
+therefore has its nonabelian composition types in the same pool. Thus
+all nonabelian composition types outside the active chief factor
+$R/Q$ lie in this pool. The active type $T$ can still vary along the chain.
+
+Apply now the structural observations following
+Proposition~\ref{cls:factor-actions}. There are boundedly many choices
+of inactive simple labels by (iv), and the repetition indices also
+have boundedly many values. The resulting omitted dimensions
+therefore have boundedly many possible values. When all inactive
+components are soluble, their individual natural dimensions are
+absolutely bounded. This proves (v) and all the assertions passed
+to the next section.
+\end{proof}
diff --git a/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/sections/extensions.tex b/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/sections/extensions.tex
new file mode 100644
index 0000000000000000000000000000000000000000..2def0d2958e714ad2e023a651e24338169440b2c
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+++ b/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/sections/extensions.tex
@@ -0,0 +1,365 @@
+\section{From subgroup intervals to extensions}
+\label{ext:section}
+
+We now apply the decorated lattice of Proposition~\ref{lat:decorated-lattice}.
+The reductions in this section use its tests in both orientations.  The
+simple group attached to the top of the original group interval will be
+denoted by $T$.  The simple group attached to an interval of extension
+domains will later be denoted by $R$; these groups need not be isomorphic.
+
+\begin{lemma}\label{ext:almost-simple}
+Let $T$ be a finite nonabelian simple group and let
+$\operatorname{Inn}(T)\le H\le\operatorname{Aut}(T)$.
+Every nontrivial normal subgroup of $H$ contains $\operatorname{Inn}(T)$.
+In particular, $H$ has trivial soluble radical and has characteristic
+socle $\operatorname{Inn}(T)$.
+\end{lemma}
+\begin{proof}
+An automorphism $\sigma$ centralizing every inner automorphism satisfies
+$c_{\sigma(t)}=c_t$ for every $t\in T$.  Since $Z(T)=1$, this implies
+$\sigma(t)=t$ for every $t$, so
+$C_{\operatorname{Aut}(T)}(\operatorname{Inn}(T))=1$.
+If $K\trianglelefteq H$, then
+\[
+ [K,\operatorname{Inn}(T)]\le K\cap\operatorname{Inn}(T).
+\]
+Simplicity makes the intersection either trivial or all of
+$\operatorname{Inn}(T)$.  In the first case the centralizer calculation
+gives $K=1$.  This proves the assertions.
+\end{proof}
+
+\begin{lemma}\label{ext:soluble-interval}
+A nontrivial fenced interval cannot be an interval in a finite soluble
+group.  Consequently, if
+$\operatorname{Inn}(T)\le B\le H\le\operatorname{Aut}(T)$, the interval
+$[B,H]$ is not nontrivially fenced.
+\end{lemma}
+\begin{proof}
+Lemma~\ref{frm:labels} would give a nonabelian simple composition factor
+of the soluble top group.  For the second assertion, quotient by
+$\operatorname{Inn}(T)$, which is contained in the bottom $B$.
+The resulting interval lies in the soluble group
+$\operatorname{Out}(T)$, by Schreier's Theorem.
+\end{proof}
+
+\subsection{Minimality in both orientations}
+
+Let $\mathcal L$ be the lattice in
+Proposition~\ref{lat:decorated-lattice}.  Suppose that one of $\mathcal L$
+and $\mathcal L^{\mathrm{op}}$ is a finite group interval.  Among all
+representations of either orientation choose
+\[
+             [D,G]\cong\mathcal M,
+       \qquad \mathcal M\in\{\mathcal L,\mathcal L^{\mathrm{op}}\},
+\]
+with $|G|$ minimal.  Minimizing over both orientations is the same
+kind of reduction used in the subgroup-interval and signalizer setting
+of \cite[Theorem~3 and Section~6]{Aschbacher2008}; see also the comparison in
+\cite[Section~4]{PalfyTwisted}.  We give the needed complementary-socle
+argument in full.  Quotienting by $\operatorname{core}_G(D)$ preserves
+the interval; hence this core is trivial.  The global fence and
+Lemma~\ref{frm:labels} give
+\begin{equation}\label{ext:top-socle}
+ S=\operatorname{soc}(G)=\prod_{i\in I}T_i\cong T^m,
+ \qquad G=DS,
+\end{equation}
+where $T$ is nonabelian simple and $D$ acts transitively on the factor
+set $I$.  Proposition~\ref{lat:decorated-lattice} applies to the chosen
+orientation $\mathcal M$.
+
+\begin{proposition}\label{ext:trivial-intersection}
+In this minimal representation, $D\cap S=1$.
+\end{proposition}
+\begin{proof}
+Put $E=D\cap S$.  For every $Y\in[D,G]$, the identity $G=DS$ gives
+\[
+                  Y=D(Y\cap S).
+\]
+Thus $Y\mapsto Y\cap S$ is an isomorphism from $[D,G]$ to the interval
+of $D$-invariant subgroups between $E$ and $S$; its inverse is
+$K\mapsto DK$.
+
+First suppose that $E$ is subdirect in $S$.  By the subdirect-power
+description in Lemma~\ref{frm:subdirect}, its simple factors are
+full diagonal strips on a partition $\Pi$ of $I$.  Every subgroup
+between $E$ and $S$ is subdirect.  Its strips split the blocks of $\Pi$,
+and their diagonal identifications are forced by their containment of
+$E$.  Therefore the preceding interval is anti-isomorphic to the
+lattice of $D$-invariant equivalence relations refining $\Pi$.
+
+Fix $i_0\in I$, let $A=D_{i_0}$, and let $B$ be the stabilizer in $D$
+of the block of $\Pi$ containing $i_0$.  The block correspondence for a
+transitive action identifies these equivalence relations with $[A,B]$:
+the block containing $i_0$ associated with $U\in[A,B]$ is $U i_0$,
+and all its other blocks are its $D$-translates.  Conversely, the
+setwise stabilizer of the block containing $i_0$ recovers $U$.
+It follows that $\mathcal M^{\mathrm{op}}\cong[A,B]$.  But
+$|B|\le |D|<|G|$, contrary to the choice of $G$ over both orientations.
+
+The coordinate projections of $E$ have equal order, since $D$ is
+transitive on $I$.  If they are proper and nontrivial,
+Proposition~\ref{frm:strict-product}, applied with the possibly unlabelled
+lower endpoint $D$, and the global fence imply that $E$ is the product
+of those projections.  Choose a proper original Boolean vertex $Y$
+of the decorated lattice.  Its forward label equals the top label,
+so $Y\cap S$ is subdirect by Proposition~\ref{frm:equal-transport}.
+It is proper, because $Y<G$.
+
+A product of nontrivial coordinate subgroups cannot lie in a proper
+subdirect subgroup: the latter has a diagonal link between two
+coordinates, whereas the former contains an element that is nontrivial
+in one of those coordinates and trivial in the other.  This contradicts
+$E\le Y\cap S$.  The only remaining possibility is that all coordinate
+projections of $E$ are trivial, and then $E=1$.
+\end{proof}
+
+\subsection{The extension dictionary}
+
+We give the parametrization with its group actions and order relations
+explicit.  It is the induced-coordinate form of
+\cite[Proposition~2.2 and Lemma~2.3]{PalfyTwisted}.
+
+\begin{proposition}\label{ext:dictionary}
+Let a finite group $D$ act on $S=\prod_{i\in I}T_i$, where all $T_i$
+are isomorphic to a nonabelian simple group $T$, and suppose that its
+action on $I$ is transitive.  Fix $i_0\in I$, set $A=D_{i_0}$, identify
+$T_{i_0}$ with $T$, and let
+$\alpha:A\to\operatorname{Aut}(T)$ be the induced action.
+
+The $D$-invariant subdirect subgroups of $S$ are in bijection with pairs
+$(U,\beta)$, where
+\[
+ A\le U\le D,\qquad
+ \beta:U\longrightarrow\operatorname{Aut}(T),\qquad
+ \beta|_A=\alpha.
+\]
+Write $S(U,\beta)$ for the corresponding subgroup.  Then
+\begin{equation}\label{ext:dictionary-order}
+ S(U,\beta)\le S(V,\gamma)
+ \quad\Longleftrightarrow\quad
+ V\le U\ \hbox{ and }\ \beta|_V=\gamma.
+\end{equation}
+In particular, the interval of $D$-invariant subgroups from
+$S(U,\beta)$ to $S$ is anti-isomorphic to $[A,U]$, by restricting
+$\beta$.
+\end{proposition}
+\begin{proof}
+Use left conjugation for the given action and let $\pi_0:S\to T$ be
+the chosen coordinate projection.  The map
+\[
+ s\longmapsto f_s,\qquad f_s(d)=\pi_0(d\cdot s),
+\]
+identifies $S$ with
+\[
+ \mathcal F_\alpha
+ =\{f:D\to T:f(ad)=\alpha(a)(f(d))
+                       \text{ for }a\in A,d\in D\}.
+\]
+Indeed, values on representatives of $A\backslash D$ are independent
+coordinates of $S$.  Under this identification the action is
+$(g\cdot f)(d)=f(dg)$.
+
+An extension $\beta$ defines
+\[
+ \mathcal F_\beta
+ =\{f:D\to T:f(ud)=\beta(u)(f(d))
+                       \text{ for }u\in U,d\in D\}.
+\]
+This is a $D$-invariant subgroup of $\mathcal F_\alpha$, with independent
+values on the cosets in $U\backslash D$.  Consequently
+$\mathcal F_\beta\cong T^{[D:U]}$, and it projects onto every coordinate
+of $\mathcal F_\alpha$.  Set $S(U,\beta)=\mathcal F_\beta$ in these
+coordinates.
+
+Conversely, let $K\le\mathcal F_\alpha$ be a $D$-invariant subdirect
+subgroup.  Its diagonal-strip partition is invariant under the
+transitive action on $A\backslash D$.  The block containing $A$ has
+stabilizer $U$ for some $A\le U\le D$.  Projection at $1$ identifies
+the diagonal factor on that block with $T$.  The induced action of $U$
+on this factor gives a homomorphism
+$\beta:U\to\operatorname{Aut}(T)$ extending $\alpha$.
+For $f\in K$ and $u\in U$ one has $f(u)=\beta(u)(f(1))$.
+Applying this identity to $d\cdot f$ gives
+\[
+                     f(ud)=\beta(u)(f(d)).
+\]
+Thus $K\le\mathcal F_\beta$.  Both groups have one freely varying
+diagonal factor for each coset of $U$, so they are equal.
+
+Containment of two subdirect groups reverses refinement of their strip
+partitions.  Hence $S(U,\beta)\le S(V,\gamma)$ forces $V\le U$.
+For $v\in V$, varying $f(1)$ over $T$ in the two defining equations
+then gives $\beta(v)=\gamma(v)$.  The converse follows immediately
+from those equations.  Finally, every subgroup containing a subdirect
+subgroup is subdirect, so all members of the stated interval have
+already been included.
+\end{proof}
+
+Apply Proposition~\ref{ext:dictionary} to
+\eqref{ext:top-socle}.  By Proposition~\ref{ext:trivial-intersection},
+the full interval $[D,G]$ is the lattice of $D$-invariant subgroups of
+$S$, via $Y\mapsto Y\cap S$.  Whenever $Y\cap S$ is subdirect, write
+$(U_Y,\beta_Y)$ for its extension.  In particular,
+\begin{equation}\label{ext:actual-filter}
+                  [Y,G]^{\mathrm{op}}\cong[A,U_Y].
+\end{equation}
+For nested filters these are compatible actual group intervals: the
+larger domain's homomorphism restricts to the smaller domain's
+homomorphism.  No representation of the entire dual lattice is asserted
+by \eqref{ext:actual-filter}.
+
+\subsection{The detector and the common image}
+
+\begin{proposition}\label{ext:detector}
+In the minimal representation fixed above, the induced factor-stabilizer
+action $\alpha$ satisfies
+$\alpha(A)\ge\operatorname{Inn}(T)$.
+\end{proposition}
+\begin{proof}
+Let $X$ be the detector vertex of
+Proposition~\ref{lat:decorated-lattice}.  Its forward type equals $T$, so
+$E_X=X\cap S$ is subdirect in $S$.  The action of $D$ on the simple
+factors of $E_X$ is transitive.  Its factor-stabilizer action is
+$\beta_X:U_X\to\operatorname{Aut}(T)$ by the construction in
+Proposition~\ref{ext:dictionary}.
+
+Suppose that this action preserved a subgroup $1<P<T$.  Transporting
+$P$ among the factors of $E_X$ would give a $D$-invariant product $Q$
+of proper nontrivial coordinate subgroups of $E_X$.  Thus $DQ$ is
+strictly between $D$ and $X$.  Choose a coatom $Y$ of $[D,X]$ containing
+$DQ$.  The detector's coatom property gives a designated vertex
+$d\le Y$ whose intersection $d\cap S$ is subdirect in the global
+coordinates.  Hence $Y\cap S$ projects onto every simple factor of
+$E_X$: projection onto such a factor can be identified with projection
+onto any one global coordinate in its diagonal support.
+Thus $Y\cap S$ is a proper subdirect subgroup of $E_X$ containing the
+product $Q$, which is impossible by the diagonal-link argument in
+Proposition~\ref{ext:trivial-intersection}.
+
+The automorphism lemma recalled by P\'alfy
+\cite[Lemma~2.11]{PalfyTwisted} states that a subgroup of
+$\operatorname{Aut}(T)$ not containing $\operatorname{Inn}(T)$
+preserves some proper nontrivial subgroup of $T$.
+It follows that
+\[
+                 \beta_X(U_X)\ge\operatorname{Inn}(T).
+\]
+
+The reverse detector interval is the actual interval $[A,U_X]$ in
+\eqref{ext:actual-filter}, and it is fenced.  Set $K=\ker\beta_X$.
+Lemma~\ref{frm:normal-test} gives either $AK=U_X$ or $K\le A$.
+The first possibility gives
+$\beta_X(U_X)=\beta_X(A)=\alpha(A)$, as required.
+
+In the second possibility $K\le C_X=\operatorname{core}_{U_X}(A)$,
+and $[A,U_X]$ is represented as
+\[
+             [\alpha(A),\beta_X(U_X)].
+\]
+Write $H=\beta_X(U_X)$ and $B=\alpha(A)$.  If
+$\operatorname{core}_H(B)$ is nontrivial, it contains
+$\operatorname{Inn}(T)$ by Lemma~\ref{ext:almost-simple}.  Quotienting
+by that core leaves a soluble top, contradicting
+Lemma~\ref{ext:soluble-interval}.
+Otherwise $C_X=K$, and $U_X/C_X\cong H$ has socle $T$ with multiplicity
+one.  The reverse detector supplies a strictly smaller labelled domain
+in $[A,U_X]$ whose simple type is equal to that of $U_X/C_X$.
+The equality is supplied by the reverse tests in
+Proposition~\ref{lat:decorated-lattice}, applied through
+\eqref{ext:actual-filter}.  Proposition~\ref{frm:equal-transport} would make
+its positive socle multiplicity strictly smaller than one.  This final
+contradiction excludes $K\le A$.
+\end{proof}
+
+\begin{lemma}\label{ext:constant-image}
+With $\alpha$ as in Proposition~\ref{ext:detector}, let $A<U\le D$ and
+let $\beta:U\to\operatorname{Aut}(T)$ extend $\alpha$.
+Suppose that $[A,U]$ is fenced.  Then
+\[
+                  U=A\ker\beta,
+       \qquad \beta(U)=\alpha(A).
+\]
+\end{lemma}
+\begin{proof}
+The normal-subgroup test gives the asserted supplementation unless
+$\ker\beta\le A$.  In the latter case $[A,U]$ is isomorphic to
+$[\alpha(A),\beta(U)]$.  Proposition~\ref{ext:detector} puts the inner
+group in its bottom, contrary to Lemma~\ref{ext:soluble-interval}.
+Supplementation gives the equality of images.
+\end{proof}
+
+\begin{proposition}[The Boolean extension configuration]
+\label{ext:reduction}
+If either orientation of the decorated lattice has a finite-group
+interval representation, there are finite groups $A\le D$, finite
+nonabelian simple groups $T,R$, a homomorphism
+$\alpha:A\to\operatorname{Aut}(T)$, and extensions
+\[
+ \beta_J:U_J\to\operatorname{Aut}(T)
+       \qquad(\varnothing\ne J\subsetneq\{1,\ldots,4\})
+\]
+with the following properties.  Set $U_\varnothing=A$ and
+$\beta_\varnothing=\alpha$.
+\begin{enumerate}[label=\textup{(E\arabic*)}]
+\item\label{ext:e1}
+For nonempty proper $J$, $[A,U_J]$ is fenced.  If
+$C_J=\operatorname{core}_{U_J}(A)$, the group $U_J/C_J$ has unique
+minimal normal subgroup
+\[
+ N_J/C_J\cong R^{I_J},\qquad U_J=AN_J.
+\]
+\item\label{ext:e2}
+If $J\subsetneq K\subsetneq\{1,\ldots,4\}$, then $U_J<U_K$ and
+$\beta_K|_{U_J}=\beta_J$.  For nonempty $J$, the equal-type transport
+of Proposition~\ref{frm:equal-transport} applies: in particular
+\[
+ \frac{U_J\cap N_K}{C_K}\ \cong\ \frac{N_J}{C_J},
+ \qquad N_J=(U_J\cap N_K)C_J,
+\]
+and the canonical surjections $I_K\to I_J$ are $U_J$-equivariant and compose.
+Whenever a nonempty family of proper subsets has proper union $K$, its domains
+generate $U_K$.
+\item\label{ext:e3}
+All images $\beta_J(U_J)$ equal
+$H=\alpha(A)\ge\operatorname{Inn}(T)$.
+\item\label{ext:e4}
+If a family of proper nonempty subsets has union $\{1,\ldots,4\}$,
+its homomorphisms have no common extension to the subgroup of $D$
+generated by their domains.
+\end{enumerate}
+\end{proposition}
+\begin{proof}
+Use the minimal representation already chosen.  Index the original
+Boolean component in its reverse order by subsets $J$ of
+$\{1,\ldots,4\}$.  Let $Y_J$ be the corresponding forward group, so
+$Y_\varnothing=G$ and $Y_{\{1,\ldots,4\}}=D$.
+Every proper nontrivial $Y_J$ has the top forward type and hence a
+subdirect intersection with $S$.  Proposition~\ref{ext:dictionary}
+provides $(U_J,\beta_J)$.
+
+The reverse labels and tests in
+Proposition~\ref{lat:decorated-lattice} apply in the actual filters
+\eqref{ext:actual-filter}.  They give the fences and equal simple types
+on comparable proper Boolean nodes.  All the nonempty proper nodes
+therefore have one common simple type $R$: connect two atoms through
+their proper two-atom join, and connect any other node to an atom below
+it.  Lemma~\ref{frm:labels} and Proposition~\ref{frm:equal-transport} prove
+\ref{ext:e1} and the transport assertions in \ref{ext:e2}.
+The order relation and restrictions follow from
+\eqref{ext:dictionary-order}.
+
+For the assertion about a proper join, let $K$ be the union of a
+nonempty family of index sets, and put
+$V=\langle U_J:J\text{ in the family}\rangle$.
+Then $A\le V\le U_K$, and $\beta_K|_V$ is an extension of all the
+$\beta_J$.  Its subdirect group is contained in every $Y_J\cap S$
+and contains $Y_K\cap S$.  Original Boolean meets are preserved in the
+decorated lattice, so the intersection of those $Y_J$ is exactly
+$Y_K$.  The subdirect groups are equal; the dictionary gives $V=U_K$.
+
+Proposition~\ref{ext:detector} and Lemma~\ref{ext:constant-image} give
+\ref{ext:e3}.  Finally, a common extension for a family with full union
+would give a nontrivial subdirect subgroup of $S$ contained in all the
+$Y_J\cap S$.  Their intersection is $D\cap S=1$, because their reverse
+Boolean join is the top.  This proves \ref{ext:e4}.
+\end{proof}
diff --git a/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/sections/framework.tex b/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/sections/framework.tex
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@@ -0,0 +1,568 @@
+\section{Fences, socles, and coordinate transport}
+\label{sec:framework}
+
+All groups in this section are finite. A vertex of a subgroup interval is
+written as a capital letter and identified with the corresponding subgroup.
+The common bottom is denoted by $D$. We first record the normal-subgroup
+test that will give the vertices their simple-group labels. We then explain
+the two different kinds of factor transport: diagonal merging when the
+simple type is unchanged, and coordinate products when it changes.
+
+\subsection{The normal-subgroup test}
+
+\begin{definition}
+A bounded lattice interval $[a,b]$ is \emph{fenced} if it has an interior
+element and, for each $a<c<b$, there are $a<u<v<b$ such that
+\[
+ c\wedge u=c\wedge v=a,
+ \qquad c\vee u=c\vee v=b.
+\]
+Thus the two comparable elements $u,v$ are both complements of $c$ in
+the interval. The requirement of an interior element excludes vacuous
+fences on covering intervals. Length always counts edges.
+\end{definition}
+
+\begin{lemma}[Normal-subgroup test]
+\label{frm:normal-test}
+Suppose that $[D,X]$ is fenced. If $R\trianglelefteq X$, then either
+$R\le D$ or $DR=X$. In particular, $[D,X]$ is not modular.
+The same assertion applies to a normal subgroup of $X/F$ whenever
+$F\trianglelefteq X$ and $F\le D$, with bottom $D/F$.
+\end{lemma}
+
+\begin{proof}
+For $D\le U\le V\le X$, the normality of $R$ and the subgroup
+modular identity give
+\begin{equation}
+ \label{frm:normal-modular-identity}
+ (DR\vee U)\wedge V
+   =(RU)\cap V
+   =U(R\cap V)
+   =U\vee(DR\wedge V).
+\end{equation}
+If $D<DR<X$, apply the fence to $DR$ and its comparable complements
+$U<V$. Equation~\eqref{frm:normal-modular-identity} would give $V=U$.
+Thus $DR$ is an endpoint. In a modular lattice the same identity holds
+with any element in place of $DR$, so the existence of a fence also
+excludes modularity. Finally, taking inverse images identifies
+$[D/F,X/F]$ with $[D,X]$.
+\end{proof}
+
+We will repeatedly use the following explicit form of the subgroup
+correspondence. If $R\trianglelefteq X$ and $X=DR$, then
+\begin{equation}
+ \label{frm:supplement-correspondence}
+ [D,X]\longrightarrow
+ \{E\le R:D\cap R\le E,\ E\text{ is normalized by }D\},
+ \qquad U\longmapsto U\cap R
+\end{equation}
+is an order isomorphism, with inverse $E\mapsto DE$. Indeed
+$U=D(U\cap R)$ for $D\le U\le DR$, and $(DE)\cap R=E$ when
+$D\cap R\le E$. This correspondence includes every overgroup of $D$.
+
+\begin{lemma}[Labels]
+\label{frm:labels}
+Suppose that $[D,X]$ is fenced, and put
+\[
+ F_X=\operatorname{core}_X(D)
+     =\bigcap_{g\in X}D^g.
+\]
+Then $X/F_X$ has a unique minimal normal subgroup
+\begin{equation}
+ \label{frm:socle-label}
+ S_X=N_X/F_X\cong T_X^{m_X},
+\end{equation}
+where $T_X$ is nonabelian simple and $N_X$ is its full inverse image in
+$X$. Moreover,
+\[
+ X=DN_X.
+\]
+Every nontrivial normal subgroup of $X/F_X$ contains $S_X$, and
+\[
+ C_{X/F_X}(S_X)=1.
+\]
+In particular, $X/F_X$ has no nontrivial soluble normal subgroup.
+If $I_X$ is the set of simple direct factors of $S_X$, then $D$ and $X$
+induce the same transitive permutation group on $I_X$.
+\end{lemma}
+
+\begin{proof}
+Factor out $F_X$, so that the action of $X$ on the cosets of $D$ is
+faithful. Lemma~\ref{frm:normal-test} says that every nontrivial normal
+subgroup supplements $D$, or equivalently is transitive in this action.
+Let $M$ be a minimal normal subgroup. If $M$ were abelian,
+\eqref{frm:supplement-correspondence} would identify $[D,X]$ with an
+interval of $D$-invariant subgroups of an abelian group. The latter lattice
+is modular: intersection is its meet and subgroup product is its join.
+This contradicts Lemma~\ref{frm:normal-test}.
+
+If $M'$ were another minimal normal subgroup, then
+$[M,M']\le M\cap M'=1$. Every $D$-invariant subgroup of $M$ would
+therefore be normalized by $DM'=X$, and so would be either $1$ or $M$.
+The correspondence~\eqref{frm:supplement-correspondence} would leave
+at most two elements in $[D,X]$, again a contradiction.
+
+A finite minimal normal subgroup is characteristically simple, hence
+a direct product of isomorphic simple groups. Since $M$ is nonabelian,
+these simple groups are nonabelian. Conjugation by $X$ is transitive on
+the direct factors: otherwise the product of one orbit of factors would
+be a proper nontrivial normal subgroup of $X$ contained in $M$. This
+proves~\eqref{frm:socle-label}. Every nontrivial normal subgroup contains
+a minimal normal subgroup and therefore contains $M$. Its centralizer
+is normal and intersects $M$ in $Z(M)=1$, so that centralizer is trivial.
+The assertion about soluble normal subgroups follows as well.
+
+Finally $X=DM$. Inner automorphisms of $M$ fix each of its simple direct
+factors. Thus $D$ and $X$ have the same image on their factor set.
+Restoring $F_X$ proves the statements as written.
+\end{proof}
+
+We call an $X$ satisfying Lemma~\ref{frm:labels} a \emph{labelled
+vertex}, and call the isomorphism type of $T_X$ its \emph{simple type}.
+The preceding argument is also the strongly nonmodular normal-subgroup
+argument in \cite[Lemma~3.6]{PalfyTwisted}.
+
+\begin{corollary}[Bottlenecks]
+\label{frm:bottleneck}
+Let $D<B\le Z$, suppose that $[D,B]$ is fenced, and suppose that $B$
+is comparable with every member of $[D,Z]$. If $R\trianglelefteq Z$
+and $R\nleq D$, then
+\[
+ B=D(R\cap B).
+\]
+Consequently a soluble normal subgroup of $Z$ is contained in $D$.
+If $R,Q\trianglelefteq Z$ and
+\[
+ [R,Q]\le F\trianglelefteq Z,\qquad F\le D,
+\]
+then at least one of $R,Q$ is contained in $D$.
+These conclusions also hold with $B=Z$.
+\end{corollary}
+
+\begin{proof}
+The subgroup $R\cap B$ is normal in $B$. Suppose first that
+$R\cap B\le D$. If $B\le DR$, then
+\[
+ B=B\cap DR=D(B\cap R)=D,
+\]
+a contradiction. Comparability therefore gives $DR\le B$, whence
+$R=R\cap B\le D$. Its contrapositive, followed by
+Lemma~\ref{frm:normal-test} in $B$, gives the asserted supplementation.
+The image of $R\cap B$ modulo $F_B$ then contains the nonabelian
+socle $S_B$, by Lemma~\ref{frm:labels}. This is impossible if $R$ is
+soluble. For the final assertion, $F\le F_B$ because $F$ is normal in
+$B$ and contained in $D$. If both $R$ and $Q$ escaped $D$, their
+intersections with $B$ would both map over $S_B$, while their images
+would commute. This would make $S_B$ abelian.
+\end{proof}
+
+\subsection{Subdirect powers and unchanged simple types}
+
+For a direct product $T^I=\prod_{i\in I}T_i$, with each $T_i\cong T$,
+a subgroup is \emph{subdirect} if all coordinate projections are
+surjective. A \emph{full diagonal strip} on a nonempty subset $J\subseteq I$
+is a subgroup obtained by identifying all the $T_j$, $j\in J$, by
+isomorphisms, and putting identity elements outside $J$.
+We recall the standard diagonal-strip description
+\cite[Lemma~2.1]{PalfyTwisted}, including its behavior under inclusion.
+
+\begin{lemma}[Subdirect powers]
+\label{frm:subdirect}
+Let $T$ be nonabelian simple. Every subdirect subgroup $E\le T^I$
+is a direct product of full diagonal strips on the parts of a partition
+of $I$. Both the partition and the coordinate identifications within
+each part are determined by $E$. An inclusion of subdirect subgroups
+corresponds to a refinement of these parts, with the existing
+identifications restricted to the smaller parts.
+\end{lemma}
+
+\begin{proof}
+Induct on $|I|$, separating the last coordinate. The projection to the
+other coordinates is, by induction, a product $T^r$ of diagonal strips.
+We have a subdirect subgroup of $T^r\times T$. The kernels of its two
+projections are normal in the respective factors. The kernel on the last
+$T$ is either $T$, in which case this is the full product, or $1$, in
+which case the subgroup identifies a quotient of $T^r$ with $T$.
+Normal subgroups of $T^r$ are products of its simple direct factors, so
+that quotient is one of the factors. The last coordinate is therefore
+joined to exactly one existing strip.
+
+Two coordinates belong to the same strip precisely when the projection
+of $E$ to their product is the graph of an isomorphism rather than the
+full product. This characterizes the partition and its links, and also
+proves the assertion about inclusion.
+\end{proof}
+
+\begin{proposition}[Equal-type transport]
+\label{frm:equal-transport}
+Let $D<X<Y$ be labelled vertices. Then $F_Y\le F_X$, and $T_X$ is
+isomorphic to a section of $T_Y$. Put
+\[
+ E_X=(X\cap N_Y)/F_Y\le S_Y.
+\]
+The following conditions are equivalent:
+\begin{enumerate}[label=\textup{(\roman*)}]
+ \item $T_X\cong T_Y$;
+ \item $E_X$ is subdirect in $S_Y$.
+\end{enumerate}
+When they hold, the quotient map induces an isomorphism
+\begin{equation}
+ \label{frm:equal-socle-identification}
+ E_X\xrightarrow{\ \sim\ }S_X,
+ \qquad
+ N_X=F_X(X\cap N_Y),
+ \qquad F_X\cap N_Y=F_Y.
+\end{equation}
+Moreover $m_X<m_Y$, and the strip supports give a canonical
+$X$-equivariant surjection
+\begin{equation}
+ \label{frm:equal-support-map}
+ \sigma_{YX}:I_Y\longrightarrow I_X.
+\end{equation}
+For three comparable labelled vertices of the same simple type these
+maps compose:
+$\sigma_{ZX}=\sigma_{YX}\sigma_{ZY}$.
+\end{proposition}
+
+\begin{proof}
+The normal subgroup $F_Y$ of $X$ is contained in $D$, so $F_Y\le F_X$.
+Since $Y=DN_Y$ and $D\le X$,
+\[
+ X=D(X\cap N_Y).
+\]
+The image of $E_X$ in $X/F_X$ is a nontrivial normal subgroup and
+therefore contains $S_X$. Every simple composition factor of a subgroup
+of a direct product is a section of one of the factors: projection to
+one coordinate gives an exact sequence whose kernel lies in the product
+of the remaining coordinates, and induction applies. Thus $T_X$ is a
+section of $T_Y$.
+
+If $T_X\cong T_Y$, some coordinate projection of $E_X$ is all of $T_Y$;
+otherwise none of those smaller groups could have a composition factor
+of order $|T_Y|$. The transitive action of $D$ on $I_Y$ makes all the
+projections surjective. Conversely, suppose $E_X$ is subdirect.
+Lemma~\ref{frm:subdirect} expresses it as a product of diagonal copies
+of $T_Y$. The action of $D$ on those copies is transitive. Since
+$X/F_Y=(D/F_Y)E_X$, the subgroup $E_X$ is minimal normal in $X/F_Y$.
+Its intersection with $F_X/F_Y$ is either trivial or all of $E_X$.
+The latter would imply $X=D$, which is excluded. Its image is therefore
+the unique minimal normal subgroup $S_X$, proving the equivalence and
+\eqref{frm:equal-socle-identification}.
+
+There are $m_X$ strips on $m_Y$ coordinates. If $m_X=m_Y$, every strip
+is a singleton, so $E_X=S_Y$ and $X=Y$. Thus $m_X<m_Y$.
+The quotient identification with $S_X$ assigns to every strip its
+actual simple factor in $I_X$ and defines~\eqref{frm:equal-support-map}.
+The subgroup $E_X$ is normal in $X/F_Y$, and its quotient identification
+with $S_X$ commutes with conjugation by $X$; hence this support map is
+$X$-equivariant.
+
+For composition, factor out $F_Z$ and let $E_Y=Y\cap N_Z$.
+We have $N_Y=F_YE_Y$ and $F_Y\cap E_Y=1$. Since $F_Y\le X$,
+\[
+ X\cap N_Y=F_Y(X\cap E_Y)=F_Y(X\cap N_Z).
+\]
+Thus the subdirect subgroup used for $X<Y$ is the image of the one
+used for $X<Z$ under the already specified identification
+$E_Y\cong S_Y$. Refinement of its strip partition is exactly
+composition of the support maps.
+\end{proof}
+
+In particular, a change of simple type is strict in order:
+$T_X\not\cong T_Y$ implies $|T_X|<|T_Y|$. A section of a finite group
+having its full order must be the group itself.
+
+\subsection{Products and one-coordinate intervals}
+
+Fix a labelled upper vertex $Y$. Work modulo $F_Y$ when discussing its
+socle $S_Y=T^{I_Y}$. Conjugation by $D$ is transitive on $I_Y$.
+Choose a reference coordinate and transport identifications of the other
+coordinates by elements of $D$. Notation such as $P^{I_Y}$ always means
+the product of these transported copies. The subgroup in the reference
+coordinate must be invariant under the corresponding factor stabilizer;
+this is precisely the condition that makes the transported product
+invariant under the whole group in question.
+
+Call $D\le Z\le Y$ \emph{saturated relative to $Y$} if
+$(Z\cap N_Y)/F_Y$ is the product of its coordinate projections.
+The proof below uses the product--subdirect intersection arguments of
+\cite[Lemmas~2.12 and~2.13]{PalfyTwisted} to obtain the required saturation.
+
+\begin{proposition}[Strict products and saturation]
+\label{frm:strict-product}
+Let $D\le X<Y$, where $Y$ is labelled. Suppose that the coordinate
+projections of
+\[
+ B=(X\cap N_Y)/F_Y
+\]
+are proper and nontrivial. If $[X,Y]$ is fenced, or if $X$ is a coatom
+of $[D,Y]$, then $X$ is saturated relative to $Y$.
+
+Once $B=P^{I_Y}$ is such a product, no proper subdirect subgroup of
+$S_Y$ contains $B$. Every coatom of $[X,Y]$, and every intersection
+of these coatoms, is saturated relative to $Y$.
+These assertions do not require $X$ to be labelled. For labelled
+$X<Y$ of different simple types the projection hypothesis holds.
+\end{proposition}
+
+\begin{proof}
+Factor out $F_Y$, and let $C$ be the product of the coordinate
+projections of $B$. It is $X$-invariant. The correspondence
+\eqref{frm:supplement-correspondence}, with bottom $X$ and normal
+subgroup $S_Y$, identifies $[X,Y]$ with the $X$-invariant subgroups of
+$S_Y$ containing $B$.
+
+Suppose $B<C<S_Y$. If $U<V$ are comparable complements of $XC$,
+put $E_U=U\cap S_Y$ and $E_V=V\cap S_Y$. Their projections already
+contain the projections of $C$. Hence projecting
+$\langle E_U,C\rangle=S_Y$ shows that $E_U$ is subdirect; the same
+holds for $E_V$.
+
+Every link within a strip of $E_U$ carries the projection of $B$ in one
+coordinate onto its projection in the other, because $B\le E_U$.
+Consequently $E_U\cap C$ is a product of independent diagonal copies
+of those nontrivial projections, one for each strip of $E_U$.
+The proper inclusion $E_U<E_V$ splits at least one strip, by
+Lemma~\ref{frm:subdirect}. Intersecting with $C$ then permits an
+additional independent nonidentity coordinate value, and gives
+$E_U\cap C<E_V\cap C$. This contradicts the required equality of
+both intersections with $B$. A fence therefore forces $B=C$.
+If $X$ is a coatom, $B<C<S_Y$ directly contradicts maximality.
+
+A proper subdirect subgroup has a link between two distinct coordinates.
+The full product $P^{I_Y}$ contains an element which is identity in the
+first of these coordinates and nonidentity in the second, contradicting
+that link. Now let $Z$ be a coatom above $X$. If its projections were
+full, it would be such a proper subdirect subgroup. Otherwise its product
+closure is proper, and maximality forces it to equal that closure.
+Intersections of products are products.
+
+Finally, for different labelled simple types, the projections are
+nontrivial because their image contains $S_X$, and transitivity makes
+all of them nontrivial. If one were full, all would be full, contradicting
+Proposition~\ref{frm:equal-transport}.
+\end{proof}
+
+\begin{proposition}[One-coordinate projection]
+\label{frm:projection}
+Let $D\le X<Y$, with $Y$ labelled, and suppose that
+\[
+ (X\cap N_Y)/F_Y=P_X^{I_Y},\qquad 1<P_X<T_Y.
+\]
+Suppose also that $X$ is an intersection of coatoms of $[X,Y]$;
+a coatom itself is allowed. Fix $i\in I_Y$ and let
+\[
+ \rho:Y_i\longrightarrow\operatorname{Aut}(T_Y)
+\]
+be the action of its factor stabilizer on that coordinate. Identify
+$T_Y$ with its inner automorphism group, and put
+$H_* = \rho(X_i)$. Then
+\begin{equation}
+ \label{frm:inner-bottom}
+ H_*\cap T_Y=P_X.
+\end{equation}
+The saturated overgroups of $X$ form a sublattice of $[X,Y]$, and
+\begin{equation}
+ \label{frm:coordinate-interval}
+ Z\longmapsto K_Z=\rho(Z_i)=H_*P_Z,
+ \qquad (Z\cap N_Y)/F_Y=P_Z^{I_Y},
+\end{equation}
+is an isomorphism from that sublattice onto the entire ordinary subgroup
+interval $[H_*,H_*T_Y]$. Original covering relations between saturated
+vertices remain covering relations after projection. In particular, a
+strict change of simple type at a cover yields a maximal subgroup of an
+almost simple group, supplementing its simple socle.
+\end{proposition}
+
+\begin{proof}
+For a coatom $M$ above $X$, Proposition~\ref{frm:strict-product} writes
+its intersection with $S_Y$ as $P_M^{I_Y}$, with $1<P_M<T_Y$.
+Let $J=H_*\cap T_Y$. It normalizes $P_M$, so
+\[
+ P_MJ\le N_{T_Y}(P_M)<T_Y.
+\]
+The last inequality follows from simplicity and $1<P_M<T_Y$.
+Both $P_M$ and $J$ are $H_*$-invariant, and their product therefore
+transports to a saturated overgroup of $M$ which is still proper in $Y$.
+Maximality gives $J\le P_M$. Intersecting over the coatom representation
+of $X$ gives $J\le P_X$; the reverse inclusion is immediate. This proves
+\eqref{frm:inner-bottom}.
+
+Products above $P_X^{I_Y}$ correspond precisely to $H_*$-invariant
+subgroups $P_Z\le T_Y$ containing $P_X$. By
+\eqref{frm:inner-bottom}, these in turn correspond precisely to the
+subgroups $K_Z$ of $H_*T_Y$ containing $H_*$: the two maps are
+$P_Z\mapsto H_*P_Z$ and $K_Z\mapsto K_Z\cap T_Y$.
+Since $Z=X P_Z^{I_Y}$ and the product fixes the coordinate set,
+$Z_i=X_iP_Z^{I_Y}$, proving~\eqref{frm:coordinate-interval}.
+Intersections and generated subgroups of coordinate products are again
+coordinate products, proving the sublattice assertion. Any intermediate
+projected subgroup would lift to an original intermediate saturated
+subgroup, which proves the assertion about covers. At a strict cover
+Proposition~\ref{frm:strict-product} supplies the product hypothesis,
+and the bottom is itself a coatom. The ambient group $H_*T_Y$ is a
+subgroup of $\operatorname{Aut}(T_Y)$ containing $T_Y$, hence is almost
+simple.
+\end{proof}
+
+\subsection{The active chief factor and its core}
+
+We will need the action on a specified nonabelian chief factor in a
+projected group. It is not necessary to assert that every projected
+coset action is quasiprimitive. The following proposition keeps the
+original cores and the projected cores distinct.
+
+\begin{proposition}[Active-factor transport]
+\label{frm:active-transport}
+Let $D<Z\le Y$ be labelled vertices, and suppose that $Z$ is saturated
+relative to $Y$:
+\[
+ (Z\cap N_Y)/F_Y=P_Z^{I_Y}.
+\]
+Let $K_Z$ be the automorphism group induced by a factor stabilizer in
+$Z$ on a reference coordinate of $S_Y$. There are $K_Z$-normal
+subgroups $Q_Z\le R_Z\le P_Z$ such that
+\begin{align}
+ \label{frm:product-core}
+ (F_Z\cap N_Y)/F_Y&=Q_Z^{I_Y},\\
+ \label{frm:product-active-preimage}
+ (N_Z\cap N_Y)/F_Y&=R_Z^{I_Y},\\
+ \label{frm:coordinate-chief}
+ R_Z/Q_Z&\cong T_Z^{a_Z},\qquad m_Z=a_Zm_Y.
+\end{align}
+Here $a_Z$ is a positive integer. The section $R_Z/Q_Z$ is a chief factor for the action of $K_Z$ on
+$P_Z$: there is no $K_Z$-normal subgroup strictly between $Q_Z$ and
+$R_Z$. Its factors give a $Z$-equivariant surjection
+\begin{equation}
+ \label{frm:product-support}
+ \tau_{ZY}:I_Z\longrightarrow I_Y
+\end{equation}
+whose fibers have size $a_Z$. The stabilizers in $D$ and $Z$ of a
+coordinate in $I_Y$ induce the same transitive action on its fiber.
+
+In addition, suppose $X<Z$ and the hypotheses of
+Proposition~\ref{frm:projection} hold for $X<Y$. Set
+\[
+ H_* = \rho(X_i),\qquad K_Z=H_*P_Z,
+ \qquad C_Z=\operatorname{core}_{K_Z}(H_*).
+\]
+Then the subgroups in~\eqref{frm:product-core} satisfy
+\begin{equation}
+ \label{frm:projected-core}
+ Q_Z=C_Z\cap P_Z,
+ \qquad K_Z=H_*R_Z.
+\end{equation}
+Thus $R_ZC_Z/C_Z\cong R_Z/Q_Z$ is a transitive minimal normal
+subgroup in the coset action of $K_Z/C_Z$ on $H_*/C_Z$.
+The groups $H_*$ and $K_Z$ induce the same action on its simple factors.
+No assertion that this is the entire projected socle is required.
+
+If $a_Z=1$, the quotient $P_Z/R_Z$ is soluble.
+\end{proposition}
+
+\begin{proof}
+Factor out $F_Y$ throughout the proof. Put
+\[
+ E=Z\cap N_Y=P_Z^{I_Y},\qquad F=F_Z,
+ \qquad W=F\cap E.
+\]
+Let $Q_i$ be the projection of $W$ to the $i$th coordinate, and let
+$\widehat Q=\prod_iQ_i$. These projections are transported into one
+another by $D$, and each is invariant under its coordinate stabilizer.
+Thus $\widehat Q$ is normal in $Z$.
+
+We claim that $W=\widehat Q$. For $q,q'\in Q_i$, choose $w\in W$
+whose $i$th coordinate is $q$ and let $e_i(q')\in E$ be supported only
+on coordinate $i$. Since $W\trianglelefteq E$,
+\[
+ [w,e_i(q')]=e_i([q,q'])\in W.
+\]
+It follows that $[\widehat Q,\widehat Q]\le W$.
+The image of $\widehat Q$ in $Z/F$ is therefore the abelian normal
+subgroup $\widehat Q/W$. Lemma~\ref{frm:labels} forces that image to
+be trivial. Hence $W=\widehat Q$, proving~\eqref{frm:product-core}.
+This argument also covers $Z=X$ at a covering comparison; it does not
+use a projected core with bottom equal to the whole projected group.
+
+The normal subgroup $EF/F$ of $Z/F$ is nontrivial, since $Z=DE$ and
+$Z>D$. It contains $S_Z$. Therefore, if $L=N_Z\cap E$, then
+\[
+ N_Z=FL,\qquad L/W\cong S_Z.
+\]
+In particular $L/W$ is a perfect normal subgroup of
+$E/W=\prod_i(P_i/Q_i)$.
+
+Every perfect normal subgroup $V$ of a direct product $\prod_iB_i$ is
+the product of its intersections with the factors. To see this, for
+$v,w\in V$ the element $[v,e_i(w_i)]$ lies in $V\cap B_i$ and is
+the $i$th coordinate of $[v,w]$. Hence
+$V'\le\prod_i(V\cap B_i)\le V$, and $V=V'$ gives equality.
+Apply this observation to $L/W$. It splits as a product of its
+coordinate intersections, each a direct product of copies of $T_Z$.
+Taking full inverse images gives $L=R_Z^{I_Y}$ and
+\eqref{frm:product-active-preimage}--\eqref{frm:coordinate-chief}.
+Conjugation by $Z$ permutes these coordinate intersections and commutes
+with $L/W\cong S_Z$, so the resulting support map is $Z$-equivariant.
+Transitivity of $D$ on $I_Y$ makes the multiplicity $a_Z$ the same in
+every coordinate.
+
+Any $K_Z$-normal subgroup strictly between $Q_Z$ and $R_Z$ would
+transport to a $Z$-normal subgroup strictly between $W$ and $L$.
+Its image would be a proper nontrivial normal subgroup of $Z/F$
+inside $S_Z$. This proves the chief-factor assertion and also the
+transitivity of $K_Z$ on the $a_Z$ simple factors of $R_Z/Q_Z$.
+
+We have $Z=DN_Z=DL$. Since $L\le N_Y$ fixes $I_Y$, for a chosen
+coordinate $i$ this gives
+\[
+ Z_i=D_iL,\qquad K_Z=\rho(D_i)R_Z.
+\]
+The group $R_Z$ acts by inner automorphisms on $R_Z/Q_Z$ and fixes
+its simple factors. Thus $\rho(D_i)$ and $K_Z$ have the same image
+on that factor set. The kernel of $\rho$ centralizes the ambient
+coordinate and so acts trivially on this section. This identifies the
+action with the original fiber action and proves
+\eqref{frm:product-support} and its assertions about stabilizers.
+
+For the additional projected-core assertion, let
+$C_Z=\operatorname{core}_{K_Z}(H_*)$. The subgroup $Q_Z$ is normal
+in $K_Z$ and lies in $P_X$, because $F_Z\le D\le X$.
+By~\eqref{frm:inner-bottom} it lies in $H_*$, and therefore
+$Q_Z\le C_Z\cap P_Z$. Conversely, transport $C_Z\cap P_Z$ over all
+coordinates. Its product is normal in $Z$ and is contained in
+$P_X^{I_Y}\le X<Z$. Lemma~\ref{frm:normal-test} for the original
+interval $[D,Z]$ forces this product into $D$, hence into $F_Z$.
+Equation~\eqref{frm:product-core} now gives the reverse inclusion.
+
+Since $\rho(D_i)\le H_*\le K_Z$ and
+$K_Z=\rho(D_i)R_Z$, we obtain $K_Z=H_*R_Z$. Also
+$R_Z\cap C_Z=Q_Z$. The resulting image of $R_Z$ modulo $C_Z$ is
+minimal normal by the chief-factor assertion and is transitive by
+supplementation. Its factor action is the same for $H_*$ and $K_Z$,
+because it is already induced by $\rho(D_i)$.
+
+Finally suppose $a_Z=1$. The support map~\eqref{frm:product-support}
+is then a bijection. The subgroup $E$ fixes $I_Y$ and consequently fixes
+$I_Z$. Conjugation embeds $EF/F$ faithfully in
+$\operatorname{Aut}(S_Z)$, since $C_{Z/F}(S_Z)=1$ by
+Lemma~\ref{frm:labels}. Its image therefore lies in
+$\operatorname{Aut}(T_Z)^{I_Y}$ and contains the inner subgroup
+$S_Z$. The quotient $E/L\cong(P_Z/R_Z)^{I_Y}$ embeds in
+$\operatorname{Out}(T_Z)^{I_Y}$. Schreier's Theorem, a consequence
+of the classification of finite simple groups, says that
+$\operatorname{Out}(T_Z)$ is soluble; see also the explicit use in
+\cite[Section~1]{PalfyTwisted}. Hence $P_Z/R_Z$ is soluble.
+\end{proof}
+
+\begin{remark}
+The projected bottom in Proposition~\ref{frm:projection} is the image
+of $X_i$, not necessarily the image of $D_i$ alone: in general
+$H_* = \rho(D_i)P_X$. The smaller image already induces the full
+active factor actions, so the actual projected bottom does also.
+Likewise, the identity $Q_Z=C_Z\cap P_Z$ in
+Proposition~\ref{frm:active-transport} is asserted only for $X<Z$.
+The product-core and chief-factor statements preceding it apply also
+to the lower endpoint of a covering comparison. These distinctions
+allow us to use maximal-subgroup information without assuming more
+quasiprimitivity in the projected diagram than has been proved.
+\end{remark}
diff --git a/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/sections/gluing.tex b/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/sections/gluing.tex
new file mode 100644
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+++ b/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/sections/gluing.tex
@@ -0,0 +1,524 @@
+\section{Boolean gluing of extensions}
+\label{glue:section}
+
+We prove that the configuration in Proposition~\ref{ext:reduction}
+cannot exist.  Throughout this section, $T$ is the target simple group
+of the homomorphisms, whereas $R$ is the simple type of the domain
+socles.  We retain the notation $A,D,H,U_J,\beta_J,C_J,N_J,I_J$ of
+that proposition.  All extension domains lie in the finite group $D$.
+
+The extensions will glue if their domain cores have images containing
+$\operatorname{Inn}(T)$.  We first prove this criterion: a least normal
+subgroup of $A$ with that image is normalized by every domain, and
+conjugation on its simple quotient supplies the common extension.
+We then force the criterion on the Boolean faces.  For an atom $x$,
+the normal image $\alpha(C_x)$ is either trivial or contains the inner
+group.  In the trivial case, compatible lifts from the three rank-two
+faces through $x$ produce a different common extension, again forbidden
+by \ref{ext:e4}.  Hence every atom core retains the inner group;
+intersections pass this property to the larger proper faces.
+
+\subsection{A criterion for extending over generated domains}
+
+We begin with the elementary fact that makes least subgroups with a
+prescribed perfect image available.
+
+\begin{lemma}[Intersections retaining a perfect image]
+\label{glue:perfect-intersections}
+Let $f:M\to H_0$ be a homomorphism of finite groups, let $P\le H_0$
+be perfect, and let $B_1,\ldots,B_s\trianglelefteq M$, with $s\ge1$, satisfy
+$P\le f(B_i)$ for every $i$.  Then
+\[
+                  P\le f\left(\bigcap_{i=1}^s B_i\right).
+\]
+\end{lemma}
+\begin{proof}
+For two normal subgroups,
+\[
+ f(B_1\cap B_2)\ \supseteq\ f([B_1,B_2])
+   =[f(B_1),f(B_2)]\ \supseteq\ [P,P]=P.
+\]
+Induction proves the assertion for the finite family.
+\end{proof}
+
+The following criterion applies to any finite family of extension
+domains.  It will be used at the end with two rank-three Boolean faces
+whose union is full.
+
+\begin{lemma}[A common core and its conjugation extension]
+\label{glue:common-core}
+Let $A\le U_i\le D$ be a finite nonempty family of subgroups of a finite
+group.  Let $T$ be nonabelian simple and suppose that homomorphisms
+\[
+ \alpha:A\to\operatorname{Aut}(T),\qquad
+ \beta_i:U_i\to\operatorname{Aut}(T)
+\]
+satisfy
+\[
+ \beta_i|_A=\alpha,
+ \qquad
+ \alpha\bigl(\operatorname{core}_{U_i}(A)\bigr)
+                         \ge\operatorname{Inn}(T).
+\]
+Put $K=\langle U_i:i\rangle$.  Then
+$\alpha(\operatorname{core}_K(A))\ge\operatorname{Inn}(T)$, and there
+is a unique homomorphism $\beta:K\to\operatorname{Aut}(T)$ extending
+all the $\beta_i$.
+\end{lemma}
+\begin{proof}
+Write $J=\operatorname{Inn}(T)$ and let
+$\iota:T\to J$ be the isomorphism $t\mapsto c_t$.
+Consider
+\[
+ Q=\bigcap\{B\trianglelefteq A:\alpha(B)\ge J\}.
+\]
+The family is nonempty.  Since $J$ is perfect,
+Lemma~\ref{glue:perfect-intersections} shows that its finite
+intersection still maps over $J$.  The subgroup $\alpha^{-1}(J)$
+belongs to the family, so in fact
+\begin{equation}\label{glue:least-perfect-image}
+                         \alpha(Q)=J.
+\end{equation}
+Thus $Q$ is the least normal subgroup of $A$ with image containing $J$.
+
+Fix $i$ and put $C_i=\operatorname{core}_{U_i}(A)$.  The hypothesis
+gives $Q\le C_i$.  Make the analogous construction in $C_i$:
+\[
+ Q_i=\bigcap\{B\trianglelefteq C_i:\alpha(B)\ge J\}.
+\]
+The same intersection argument, using $C_i\cap\alpha^{-1}(J)$,
+gives $\alpha(Q_i)=J$.
+For $u\in U_i$ and $c\in C_i$,
+\[
+       \alpha(ucu^{-1})
+          =\beta_i(u)\alpha(c)\beta_i(u)^{-1}.
+\]
+Since $C_i\trianglelefteq U_i$ and
+$J\trianglelefteq\operatorname{Aut}(T)$, conjugation by $U_i$ permutes
+the defining family for $Q_i$.  Thus $Q_i\trianglelefteq U_i$, and in
+particular $Q_i\trianglelefteq A$.  Minimality of $Q$ gives $Q\le Q_i$.
+Conversely, $Q\le C_i$, $Q\trianglelefteq C_i$, and
+\eqref{glue:least-perfect-image} give $Q_i\le Q$.
+Consequently $Q_i=Q$ for every $i$.
+
+It follows that $Q\trianglelefteq K$ and
+$Q\le\operatorname{core}_K(A)$, proving the common-core assertion.
+Set $L=Q\cap\ker\alpha$.  Since
+$L=Q\cap\ker\beta_i$ for every $i$, the subgroup $L$ is normalized by
+each $U_i$ and therefore by $K$.
+The map
+\[
+ \vartheta:Q/L\longrightarrow T,
+        \qquad \vartheta(qL)=\iota^{-1}(\alpha(q))
+\]
+is an isomorphism.  Transporting the conjugation action of $K$ on
+$Q/L$ through $\vartheta$ defines a homomorphism
+$\beta:K\to\operatorname{Aut}(T)$.
+For $u\in U_i$ and $q\in Q$,
+\[
+ \alpha(uqu^{-1})
+ =\beta_i(u)\alpha(q)\beta_i(u)^{-1}
+ =\iota\bigl(\beta_i(u)(\vartheta(qL))\bigr).
+\]
+As $\vartheta$ is onto, this proves $\beta|_{U_i}=\beta_i$.
+These restrictions determine $\beta$ uniquely because the $U_i$
+generate $K$.
+\end{proof}
+
+\subsection{An atom whose core is killed}
+
+For $B\subseteq I_J$, let
+\begin{equation}\label{glue:support-preimage}
+ N_J(B)=q_J^{-1}\left(\prod_{i\in B}S_{J,i}\right),
+ \qquad
+ q_J:N_J\longrightarrow S_J=N_J/C_J,
+\end{equation}
+where $S_{J,i}$ are the simple direct factors of $S_J$.
+Thus $N_J(\varnothing)=C_J$.  For comparable nonempty index sets
+$J\subset K$, write $p_{KJ}:I_K\to I_J$ for the canonical factor map.
+
+\begin{lemma}\label{glue:factorization}
+If $x$ is an original Boolean atom and $\alpha(C_x)=1$, then
+$\beta_x$ factors through the permutation action of $U_x$ on $I_x$.
+\end{lemma}
+\begin{proof}
+By \ref{ext:e3}, $U_x=A\ker\beta_x$.  Since $A<U_x$ and $C_x\le A$,
+the kernel is not contained in $C_x$.  The assumption makes
+$\beta_x$ a homomorphism on $U_x/C_x$, so its nontrivial kernel there
+contains the unique minimal normal subgroup $N_x/C_x$.
+Consequently $\beta_x(N_x)=1$.
+
+Let $P$ be the kernel of the action of $U_x$ on $I_x$.
+Modulo $C_x$, the faithful conjugation action on $S_x$ embeds
+$P/N_x$ into $\operatorname{Out}(R)^{I_x}$.  To justify faithfulness,
+the centralizer of $S_x$ in $U_x/C_x$ is normal and disjoint from
+$S_x$; uniqueness of the nonabelian minimal normal subgroup makes it
+trivial.  Schreier's Theorem therefore makes $P/N_x$ soluble.
+Its image under $\beta_x$ is a soluble normal subgroup of
+$H=\alpha(A)$.  Lemma~\ref{ext:almost-simple} makes that image trivial,
+so $P\le\ker\beta_x$, as claimed.
+\end{proof}
+
+Fix an atom $x$.  To rule out $\alpha(C_x)=1$, we shall show that this assumption
+produces a subgroup $L$ normalized by $A$, generated by subgroups on
+which the local extensions are trivial.  The remaining
+condition for the formula $\theta(a\ell)=\alpha(a)$ to define an
+extension on $AL$ is $A\cap L\le\ker\alpha$.  By Lemma~\ref{glue:factorization}, it is
+enough to prove that every element of $A\cap L$ fixes the factor set
+$I_x$.  The three proper faces through $x$ provide the local generators;
+their proper pairwise joins will let us compare different factor labels.
+
+Set
+\[
+ \mathcal U_x=\{x\cup\{a\}:a\in\{1,\ldots,4\}\setminus x\}.
+\]
+This family has three members.  Distinct $u,v\in\mathcal U_x$ have a
+proper join $w=u\cup v$ of rank three.  For $j\in I_x$ and
+$y\supseteq x$, write
+\[
+                   F_{y,j}=p_{yx}^{-1}(j),
+       \qquad F_{x,j}=\{j\}.
+\]
+All factor maps in this notation are defined on proper nodes.
+Figure~\ref{fig:boolean-faces} displays these three faces and their
+proper pairwise joins.
+
+A lift of the fibre $F_{u,j}$ must lie over the corresponding support
+$F_{u\cup v,j}$ in each proper pair join $u\cup v$.  We therefore intersect its support
+preimages in all those joins.  The perfect-image intersection lemma
+ensures that these simultaneous requirements do not lose any factor
+of the desired fibre.
+
+\begin{lemma}[Compatible lifts of the fibres]
+\label{glue:lifts}
+For $u\in\mathcal U_x$ and $j\in I_x$, define
+\begin{equation}\label{glue:Luj}
+ L_{u,j}=N_u(F_{u,j})\cap
+       \bigcap_{\substack{v\in\mathcal U_x\\v\ne u}}
+       \bigl(U_u\cap N_{u\cup v}(F_{u\cup v,j})\bigr).
+\end{equation}
+These subgroups have the following properties.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item They are normal in $N_u$, and
+$q_u(L_{u,j})=\prod_{i\in F_{u,j}}S_{u,i}$.
+\item Conjugation by $U_x$ permutes the choices according to its action
+on $I_x$.  In particular, $C_x$ normalizes every $L_{u,j}$.
+\item $U_u=\langle A,L_{u,j}:j\in I_x\rangle$.
+\item For $j\ne k$, one has
+\[
+ [L_{u,j},L_{v,k}]\le
+ \begin{cases}
+ C_{u\cup v},&u\ne v,\\
+ C_u,&u=v.
+ \end{cases}
+\]
+Both right-hand sides are contained in $C_x$.
+\item If $\alpha(C_x)=1$, then $\beta_u(L_{u,j})=1$ for all $u,j$.
+\end{enumerate}
+\end{lemma}
+\begin{proof}
+Fix a proper pair join $w=u\cup v$.  Equal-type transport gives
+\begin{equation}\label{glue:Nu-transport}
+                  N_u=(U_u\cap N_w)C_u.
+\end{equation}
+The first factor on the right acts by inner automorphisms on
+$N_w/C_w$, hence preserves every coordinate subproduct there.
+The second factor satisfies $C_u\le C_x\le U_x$.  Since $C_x$ acts
+trivially on $I_x$, equivariance of $p_{wx}$ says that $C_u$ fixes the
+set $F_{w,j}$.  It too normalizes $N_w(F_{w,j})$.
+Thus $U_u\cap N_w(F_{w,j})$ is normal in $N_u$ and lies in $N_u$.
+
+In $N_w/C_w$, the subdirect group $(U_u\cap N_w)/C_w$ has its diagonal
+strips indexed by $I_u$.  Their supports are the fibres of $p_{wu}$.
+Because $p_{wx}=p_{ux}p_{wu}$, intersecting with the $F_{w,j}$-support
+product selects exactly the strips indexed by $F_{u,j}$.
+The isomorphism from this subdirect group to $N_u/C_u$ therefore shows
+that $U_u\cap N_w(F_{w,j})$ maps onto
+$\prod_{i\in F_{u,j}}S_{u,i}$ under $q_u$.
+The first group in \eqref{glue:Luj} is normal in $N_u$ and has the same
+image.  This image is perfect.  Lemma~\ref{glue:perfect-intersections}
+proves (i).
+
+Every domain occurring in \eqref{glue:Luj} contains $U_x$, and all the
+factor maps are $U_x$-equivariant.  Its conjugation therefore permutes
+the whole definition with $j$, proving (ii).  The images in (i)
+generate $N_u/C_u$, and $C_u\le A$; since $U_u=AN_u$, this proves (iii).
+
+For $u\ne v$, the two lifts in (iv) lie in disjoint coordinate support
+products of $N_{u\cup v}/C_{u\cup v}$.  They commute in that quotient.
+For $u=v$ the identical argument takes place in $N_u/C_u$.
+Core inclusion along $U_x\le U_u\le U_{u\cup v}$ gives the asserted
+containment in $C_x$.
+
+Finally suppose that $\alpha(C_x)=1$.  Then $\beta_u(C_u)=1$.
+The kernel of $\beta_u$ supplements $A$ and is not contained in $C_u$,
+so it contains $N_u$ by the unique-minimal-normal-subgroup argument in
+Lemma~\ref{glue:factorization}.  This proves (v).
+\end{proof}
+
+We next construct subgroups $B_k$ which distinguish the labels
+$k\in I_x$ and are normalized by every lift with a different label.
+Different labels commute only modulo $C_x$, and $C_x$ need not be
+normal in the group generated by all the lifts.  The following finite
+core argument keeps a full simple image without requiring that
+normality.
+
+\begin{lemma}[Taking cores without losing a simple image]
+\label{glue:retained-core}
+Let $R_0\le D$, let $C\trianglelefteq R_0$, and suppose that
+$R_0/C$ is nonabelian simple.  Let $L_1,\ldots,L_s\le D$ be normalized
+by $R_0$.  Suppose that, for every $t$, there is a subgroup
+$D_t\trianglelefteq R_0$ such that
+\[
+ D_tC=R_0,\qquad D_t^{L_t}=D_t,
+ \qquad [D_t,L_t]\le D_t\cap C.
+\]
+Then, for $W=\langle R_0,L_1,\ldots,L_s\rangle$, the subgroup
+$\operatorname{core}_W(R_0)$ maps onto $R_0/C$.
+No normalization of $C$ by $L_t$ is assumed.
+\end{lemma}
+\begin{proof}
+Let $\pi:R_0\to R_0/C$ be the quotient map.  Start with $B=R_0$.
+At every stage $B$ will be normal in $R_0$ and satisfy
+$\pi(B)=R_0/C$.  For a fixed $t$, replace $B$ by
+\begin{equation}\label{glue:core-operation}
+       F_t(B)=\bigcap_{\ell\in L_t}(B\cap D_t)^\ell,
+       \qquad K^\ell=\ell^{-1}K\ell.
+\end{equation}
+The subgroup $B\cap D_t$ still maps onto $R_0/C$ by
+Lemma~\ref{glue:perfect-intersections}.  Put $K=B\cap D_t$.  The subgroup $K$ is normal in $D_t$, because
+$B\trianglelefteq R_0$ and $D_t\leq R_0$.  Since $L_t$ normalizes
+$D_t$, every conjugate $K^\ell$ lies in $D_t$ and is normal in $D_t$.
+For $d\in D_t$ and $\ell\in L_t$, the assumption
+$[D_t,L_t]\leq D_t\cap C$ gives $\pi(d^\ell)=\pi(d)$.
+Consequently $\pi(K^\ell)=\pi(K)=R_0/C$ for every $\ell$.
+This applies the quotient map only to elements of $D_t\leq R_0$;
+it does not require $C$ to be normalized by $L_t$.  Applying
+Lemma~\ref{glue:perfect-intersections} inside $D_t$ shows that $F_t(B)$
+has full image.
+
+The new subgroup is normal in $R_0$.  Indeed, $R_0$ normalizes $B$, $D_t$, and $L_t$.  For $r\in R_0$
+one therefore has $(K^\ell)^r=(K^r)^{\ell^r}=K^{\ell^r}$, and
+$\ell^r\in L_t$.  Thus $R_0$ permutes precisely the conjugates
+intersected in \eqref{glue:core-operation}.  The operation is descending, since the
+identity conjugate occurs.  Apply $F_1,\ldots,F_s$ in repeated rounds.
+Finiteness forces stabilization, and at a stable round every
+intermediate inclusion is equality.  The resulting subgroup $B_*$
+is therefore normalized by each $L_t$ as well as by $R_0$.  Thus
+$B_*\trianglelefteq W$ and $B_*\le R_0$, while
+$\pi(B_*)=R_0/C$.  Since
+$B_*\le\operatorname{core}_W(R_0)$, the asserted core also has full
+image.
+\end{proof}
+
+\begin{lemma}[Subgroups detecting the factor labels]
+\label{glue:detectors}
+Put $L_j=\langle L_{u,j}:u\in\mathcal U_x\rangle$ and
+$R_k=N_x(\{k\})$.  There are subgroups $B_k\le R_k$, for $k\in I_x$,
+such that
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item $q_x(B_k)=S_{x,k}$;
+\item every $L_j$ with $j\ne k$ normalizes $B_k$;
+\item $C_x$ normalizes every $B_k$, and $A$ permutes the $B_k$ in the
+same way as it permutes $I_x$.
+\end{enumerate}
+In particular, the subgroups $B_k$ are pairwise distinct.
+\end{lemma}
+\begin{proof}
+For $u\in\mathcal U_x$ set
+\[
+                     D_{u,k}=U_x\cap N_u(F_{u,k}).
+\]
+The transported subdirect group $(U_x\cap N_u)/C_u$ maps isomorphically
+onto $N_x/C_x$.  Selecting its strips over $k$ shows that
+\begin{equation}\label{glue:Duk}
+ D_{u,k}\le R_k,\quad D_{u,k}C_x=R_k,
+ \quad D_{u,k}\cap C_x=C_u.
+\end{equation}
+It is normal in $R_k$: all elements of $R_k\le N_x$ fix the factor
+labels $I_x$, and the support-preimage construction is equivariant.
+
+For $j\ne k$, both $D_{u,k}$ and $L_{u,j}$ lie in $N_u$ and have
+disjoint supports modulo $C_u$.  Thus
+\[
+                    [D_{u,k},L_{u,j}]\le C_u.
+\]
+Since $C_u\le D_{u,k}$, this also proves that $L_{u,j}$ normalizes
+$D_{u,k}$.  Moreover $R_k$ normalizes $L_{u,j}$ by
+Lemma~\ref{glue:lifts}(ii), since $R_k$ fixes every label of $I_x$.
+We can therefore apply Lemma~\ref{glue:retained-core} to $R_k$, its
+normal subgroup $C_x$, and all the lifts $L_{u,j}$ with $j\ne k$, using
+the subgroup $D_{u,k}$ for the lift labelled by $(u,j)$.
+
+Define canonically
+\[
+ W_k=\langle R_k,L_j:j\ne k\rangle,
+ \qquad B_k=\operatorname{core}_{W_k}(R_k).
+\]
+That lemma proves (i); the definition proves (ii).
+Since $C_x\le R_k\le W_k$, it normalizes $B_k$.
+Conjugation by $A$ permutes the pairs $(W_k,R_k)$ and hence their
+cores, proving (iii).  Their images in $N_x/C_x$ are the distinct
+nontrivial coordinate factors, so the $B_k$ are distinct.
+\end{proof}
+
+To use the subgroups $B_k$, we must collect all letters with one
+chosen label at the end of a word in the lifts.  The errors lie in
+$C_x$, which normalizes each individual lift.  The following word
+calculation records exactly the form needed.
+
+\begin{lemma}[Collection with labelled factors]
+\label{glue:collection}
+Let $C\le D$ normalize subgroups $P_\lambda\le D$ indexed by a finite
+set.  Assign each index a label $c(\lambda)\in I$.  Suppose that
+\[
+ [P_\lambda,P_\mu]\le C
+                \quad\text{whenever }c(\lambda)\ne c(\mu).
+\]
+Put $L_j=\langle P_\lambda:c(\lambda)=j\rangle$ and
+$L=\langle L_j:j\in I\rangle$.  For every $k\in I$, every element
+$g\in L$ can be written
+\[
+        g=pqc,\qquad
+        p\in\langle L_j:j\ne k\rangle,\quad q\in L_k,\quad c\in C.
+\]
+\end{lemma}
+\begin{proof}
+Write $g$ as a word whose letters belong to the individual
+$P_\lambda$; inverses are allowed because these are subgroups.
+Interchange adjacent letters with different labels to move the
+$k$-labelled letters to the end.  Each interchange introduces a
+commutator in $C$.  Move that error to the right through the remaining
+word, using
+\[
+                    c\ell=(c\ell c^{-1})c.
+\]
+The conjugated letter remains in its original $P_\lambda$, so its
+label does not change.  Each interchange reduces the number of pairs
+consisting of a $k$-letter preceding a non-$k$-letter.  The process
+terminates with the asserted expression.  This is a word calculation;
+it does not form a quotient by $C$ or require $C\trianglelefteq L$.
+\end{proof}
+
+\begin{proposition}\label{glue:atom-cores}
+For every original Boolean atom $x$,
+$\alpha(C_x)\ge\operatorname{Inn}(T)$.
+\end{proposition}
+\begin{proof}
+The subgroup $\alpha(C_x)$ is normal in
+$\beta_x(U_x)=H$.  Lemma~\ref{ext:almost-simple} reduces the assertion
+to excluding $\alpha(C_x)=1$.  Assume that equality and use the lifts
+and detecting subgroups just constructed.  Put
+\[
+                         L=\langle L_j:j\in I_x\rangle.
+\]
+The group $A$ normalizes $L$.
+
+Let $a\in A\cap L$.  We show that $a$ fixes every member of $I_x$.
+For subgroup conjugation use the right-action notation
+$K^g=g^{-1}Kg$.  If $a$ sent $k$ to $h\ne k$, equivariance would give
+$B_k^a=B_h$.  Lemmas~\ref{glue:lifts} and~\ref{glue:collection}, with
+$C=C_x$ and the individual $L_{u,j}$ as labelled subgroups, give
+\[
+        a=pqc,\qquad
+        p\in\langle L_j:j\ne k\rangle,\quad q\in L_k,\quad c\in C_x.
+\]
+The element $p$ normalizes $B_k$, $q$ normalizes $B_h$, and $c$
+normalizes both.  It follows successively that
+\[
+                 B_k^{qc}=B_h,
+       \qquad B_k^q=B_h,
+       \qquad B_k=B_h,
+\]
+contradicting Lemma~\ref{glue:detectors}.  Thus $a$ fixes the factor
+set.  Lemma~\ref{glue:factorization} now gives $\alpha(a)=1$.
+
+Consequently $A\cap L\le\ker\alpha$.  Because $A$ normalizes $L$,
+the formula
+\[
+              \theta:AL\longrightarrow\operatorname{Aut}(T),
+                       \qquad \theta(a\ell)=\alpha(a)
+\]
+defines a homomorphism.  It is well defined by the intersection
+inclusion; multiplication is respected since $L\trianglelefteq AL$.
+For each $u\in\mathcal U_x$, Lemma~\ref{glue:lifts}(iii),(v) says that
+$U_u$ is generated by $A$ and lifts killed by both $\theta$ and
+$\beta_u$.  Hence $\theta|_{U_u}=\beta_u$.
+
+The three sets $u\in\mathcal U_x$ have full union
+$\{1,\ldots,4\}$.  Their generated domain is $AL$, so this common
+extension contradicts \ref{ext:e4}.  The assumption
+$\alpha(C_x)=1$ is impossible.
+\end{proof}
+
+\subsection{Compatible joins and the contradiction}
+
+\begin{lemma}[Cores under a compatible join]
+\label{glue:core-joins}
+For every nonempty proper original Boolean node $z$,
+\[
+             C_z=\bigcap_{x\subseteq z,\,|x|=1}C_x.
+\]
+Consequently $\alpha(C_z)\ge\operatorname{Inn}(T)$.
+\end{lemma}
+\begin{proof}
+The assertion is immediate for an atom, so suppose that $|z|\ge2$.
+Every $C_x$ in the intersection contains $C_z$.  Pass to $U_z/C_z$
+and use bars for the images of $A,U_x,C_x$.  Put
+\[
+ N=N_z/C_z,\qquad E_x=(U_x\cap N_z)/C_z.
+\]
+The transport isomorphism in \ref{ext:e2} gives
+$E_x\cap\overline C_x=1$.  Both are normal in $\overline U_x$, so
+$[E_x,\overline C_x]=1$.
+
+Since $U_z=AN_z$ and $A\le U_x$, one has
+$\overline U_x=\overline A E_x$.
+Let $M=\langle E_x:x\subseteq z,\ |x|=1\rangle$.
+Each $E_x$ is normalized by $\overline A$.  The compatible-join
+generation in \ref{ext:e2} therefore gives
+\[
+                       \overline U_z=\overline A M,
+\]
+so $M\trianglelefteq\overline U_z$.  It is a nontrivial subgroup of
+the minimal normal subgroup $N$, and hence $M=N$.
+Thus $\bigcap_x\overline C_x$ centralizes $N$.
+The centralizer is normal in $\overline U_z$ and intersects $N$ in
+$Z(N)=1$.  If it were nontrivial it would contain the unique minimal
+normal subgroup $N$, a contradiction.  The intersection is therefore
+trivial, which proves the equality of cores.
+
+Each $C_x$ is normal in $A$ and maps over $\operatorname{Inn}(T)$ by
+Proposition~\ref{glue:atom-cores}.  Apply
+Lemma~\ref{glue:perfect-intersections} with
+$P=\operatorname{Inn}(T)$ to obtain the image assertion.
+\end{proof}
+
+\begin{theorem}\label{glue:contradiction}
+The Boolean extension configuration of
+Proposition~\ref{ext:reduction} does not exist.
+\end{theorem}
+\begin{proof}
+Take the two proper Boolean nodes
+\[
+              z=\{1,2,3\},\qquad w=\{2,3,4\}.
+\]
+Lemma~\ref{glue:core-joins} gives
+$\alpha(C_z),\alpha(C_w)\ge\operatorname{Inn}(T)$.
+Lemma~\ref{glue:common-core} therefore extends $\beta_z$ and $\beta_w$
+to $\langle U_z,U_w\rangle$.  Since $z\cup w=\{1,\ldots,4\}$, this
+contradicts \ref{ext:e4}.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{main:group}]
+Choose the finite decorated lattice of
+Proposition~\ref{lat:decorated-lattice}.  If either it or its dual had a
+finite-group interval representation, Proposition~\ref{ext:reduction}
+would produce the configuration excluded by
+Theorem~\ref{glue:contradiction}.  Hence this lattice has no such
+representation, proving Theorem~\ref{main:group}.
+\end{proof}
+
+Theorem~\ref{main:congruences} now follows from
+Theorem~\ref{main:group} and the global P\'alfy--Pudl\'ak equivalence
+\cite[Theorem~2]{PalfyPudlak1980}.
diff --git a/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/sections/growth.tex b/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/sections/growth.tex
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+++ b/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/sections/growth.tex
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+\section{Fields, indices, and growth along shortcuts}
+\label{grw:section}
+
+We continue the reduction of Theorem~\ref{cls:chain-rigidity}, starting
+with the configurations supplied by Proposition~\ref{cls:reduction}.
+Their distinct classical endpoint types have arbitrarily large rank,
+their multiplicities agree, and their killed kernels are soluble.
+We now compare the numerical restrictions imposed by a two-edge shortcut
+with the growth obtained by traversing several successive comparisons.
+The three quantities are field exponents, natural dimensions, and inner
+order quotients. We first establish the maximal-step estimates needed
+for these comparisons. They will exclude wholly geometric paths and
+characteristic crossings, leaving common-field paths with quadratic
+dimension growth and controlled inner order.
+
+Write $z_0<\cdots<z_\ell$ for the retained original vertices, in their
+original order, and abbreviate $F_i=F_{z_i}$, $N_i=N_{z_i}$,
+$T_i=T_{z_i}$, $m_i=m_{z_i}$, and $I_i=I_{z_i}$. The common
+bottom $D$ and lower endpoint $a$ remain those of the original tested
+chain; every shortcut and normal-group test is inherited from it.
+Write
+\[
+ T_i=T_i(q_i),\qquad q_i=p_i^{f_i},\qquad n_i=\dim V_i,
+ \qquad a_i=|T_i|.
+\]
+Here $V_i$ is the natural simple module. In characteristic two the
+type $B_r$ parameter is $n_i=2r$, using its simple symplectic natural
+quotient. For a unitary group the parameter is $q_i=p_i^{f_i}$,
+although its natural matrix field is $\mathbb F_{q_i^2}$.
+Put $\epsilon_X=2$ for a unitary type and $\epsilon_X=1$ otherwise,
+so the natural matrix field has degree $\epsilon_Xf_X$ over its
+prime field. We retain this conversion for restriction of scalars;
+twisted-tensor steps will have the exact usual-field relation
+$f_s=b f_t$.
+
+For each $i<j$ choose the prescribed two-edge shortcut. We write its
+ascending steps as $B_{ij},A_{ij}$ and its intermediate simple label as
+$Y_{ij}$; a prime denotes an intermediate parameter. The letters $A_{ij}$
+and $B_{ij}$ denote steps, not subgroups. On an individual ascending
+step, subscripts $s,t$ denote source and target parameters.
+
+All uses of ``arbitrarily long'' below have the same quantifiers: for
+every integer $s$ there are such configurations with at least $s$ chain
+vertices, and their minimum rank can be required to exceed any prescribed
+constant. A finite coloring followed by finite Ramsey preserves this
+property. So does retaining every $d$th vertex for a fixed $d$. Constants
+in this section are independent of the configuration and of its length.
+An explicitly mentioned pool of parameter values may depend on the
+configuration, but its cardinality has a uniform bound.
+
+\subsection{Numerical structure of classical maximal steps}
+
+The factor actions needed for the whole-chain reduction were proved in
+Proposition~\ref{cls:factor-actions}. We now derive the field, dimension,
+and order estimates from the maximal-subgroup descriptions in
+Proposition~\ref{cls:literature}. The proofs retain the actual finite
+normalizers, since their field actions will enter both the growth
+argument and the later constituent calculation.
+
+\begin{proposition}[The maximal-step catalogue]\label{cls:catalogue}
+Consider a strict labelled cover and its one-coordinate maximal subgroup
+$M$ in an almost simple group with socle $T$, as supplied by
+Proposition~\ref{frm:projection}.  Mark its active nonabelian chief factor.
+The non-geometric irreducible category below means that its simple
+source acts absolutely irreducibly on the ambient natural module
+and that the maximal subgroup is outside all geometric collections,
+in particular the subfield collection.  All growing-rank assertions
+refer to growing \emph{source} rank.
+There are absolute constants $C,c>0$ with the following properties.
+\begin{enumerate}[label=\textup{(C\arabic*)},start=2]
+\item\label{cls:cat-geometric}
+For unbounded-rank classical active types, a geometric
+defining-characteristic step changes natural dimension by taking a
+subspace or quotient, a repeated natural module, a tensor product, or
+restriction of scalars.  Subfield steps leave natural dimension
+unchanged and multiply the field exponent by a prime.  An
+extension-field step, read upwards from the active factor, has
+\begin{equation}\label{grw:matrix-field-convention}
+ n_t=b n_s,\qquad
+ \epsilon_s f_s=b\epsilon_t f_t,
+ \qquad f_t=\frac{\epsilon_s}{b\epsilon_t}f_s,
+\end{equation}
+where $b$ is the prime degree between natural matrix fields.
+A twisted-tensor step has
+\begin{equation}\label{grw:twisted-field-convention}
+ n_t=n_s^b,\qquad f_s=b f_t,
+\end{equation}
+exactly; here $b$ is prime, after bounded source-rank cases are
+removed.  All other changes of field exponent
+belong to a fixed finite set of positive rational factors.
+\item\label{cls:cat-inactive}
+If there is one active simple component and the other geometric
+components are soluble, their natural dimensions are absolutely
+bounded.  The active dimension in a subspace case is comparable with
+the ambient dimension; a totally singular summand and its dual may
+represent the same active component twice.  A nontrivial parabolic
+radical in these cases has order between $q^{c n_t}$ and $q^{C n_t^2}$.
+For a fixed ambient group, once the active slot, other component
+dimensions, and the bounded flag/form choices are fixed, there are
+boundedly many possible orders of the full inner stabilizer divided by
+the order of its active simple factor, with multiplicity included.
+\item\label{cls:cat-symplectic}
+A symplectic-type step of unbounded active rank is
+cross-characteristic.  Its source is symplectic or orthogonal over a
+prime field of order $r$, of natural dimension $2d+O(1)$, and its
+target natural dimension is $n_t=r^d$.  If $P=M\cap T$, then
+\[
+ c n_t^2\leq |P|/|T_s|\leq C n_t^2.
+\]
+Its target field exponent is at most $|T_s|^C$.
+\item\label{cls:cat-nonlinear}
+After the twisted-tensor case in
+\ref{cls:cat-geometric} is separated off, a defining-characteristic almost simple
+irreducible step has a single nonzero restricted, nonnatural
+Frobenius digit, up to a common Frobenius twist.  This restricted
+module may have further tensor factors in a very special
+characteristic.  Its dimension is at least $c n_s^2$.  Its source and target field
+exponents differ by a rational factor from a fixed finite set.
+Natural, dual-natural, and natural-size special-isogeny representations
+are included among the geometric cases instead.
+\item\label{cls:cat-cross}
+For a cross-characteristic almost simple step,
+\[
+ q_s^{c n_s}\leq n_t\leq |T_s|^C,
+ \qquad f_t\leq |T_s|^C.
+\]
+The upper degree bound also holds in defining characteristic.
+For this and the preceding irreducible case, the homomorphism from the
+ambient automorphism normalizer of the source simple group to its
+automorphism group has kernel of absolute bounded order.  In
+particular, a subgroup of that normalizer has order at most
+$C$ times the order of its induced automorphism group of the source.
+\end{enumerate}
+\end{proposition}
+
+\begin{proof}
+For the dimension and field statements, take the full stabilizers of
+the indicated structures.  On a nondegenerate direct summand the
+active group has its natural module; on a totally singular summand
+the dual summand has the contragredient module.  A tensor stabilizer
+acts as the tensor product of the component natural modules.
+Restriction of scalars multiplies dimension by its degree $b$ and
+gives $\epsilon_s f_s=b\epsilon_t f_t$ for the natural matrix
+fields.  For example, the mixed $\operatorname{GU}_m(q)$ row in
+$\operatorname{Sp}_{2m}(q)$ has $b=2$ and equal usual exponents,
+since $\epsilon_s=2$ and $\epsilon_t=1$.  It is a restriction of
+scalars, with a direct sum after scalar extension, and is not a
+twisted-tensor row.  A subfield inclusion changes only the field.
+If an ordinary field degree were composite, its proper intermediate
+field structure would be preserved by the normalizer, contradicting
+maximality.  The classical stabilizer descriptions with prime degree
+are given explicitly in
+\cite[proofs of Lemmas~5.5 and~5.7]{BurnessLiebeckShalev2017}.
+Unitary conjugation and the diagram maps have bounded order, which
+accounts for the permitted bounded ratios in the remaining field
+operations.  The
+twisted-tensor assertion is proved separately in
+Lemma~\ref{cls:twisted-tensor} below.
+
+The only soluble classical component groups have bounded natural
+dimension: a linear group in growing dimension contains a growing
+special linear simple section, and the corresponding assertions for
+the form groups follow by their natural hyperbolic subspaces.
+Thus the omitted dimensions in a one-active-component subspace or
+tensor stabilizer are bounded.  The explicit block description of a
+parabolic gives a radical of order $q^e$, with $e\leq Cn_t^2$ and
+$e\geq c n_t$ whenever that radical is nontrivial in the retained
+large-rank cases.  For example, the linear radical attached to a
+$k$-space has exponent $k(n_t-k)$; the symplectic and orthogonal
+exponents count the off-diagonal blocks subject to one symmetric or
+alternating relation.  The smallest nonzero retained exponent is
+linear in $n_t$.  The other factors are the displayed classical
+groups and scalar groups.  Consequently their orders are determined
+by the dimensions, field, form signs, and the bounded choices of
+interchanged flags.  Intersecting the full stabilizer with the ambient
+socle adds only its prescribed determinant or spinor condition; it
+does not introduce a freely chosen subgroup of a scalar group.  This
+proves \ref{cls:cat-inactive}.
+
+For a symplectic-type normalizer put $N=n_t=r^d$, and let $W$ be
+the projective image of its normal $r$-group, of order $r^{2d}$.
+The full inner-normalizer rows, including their minimal-field
+conditions, are given in
+\cite[Section~2.2.6, pp.~70--71, Table~2.9]{BrayHoltRoneyDougal2013}.
+Their small exceptions disappear on retaining $d\geq4$.
+Ordinary linear conjugation fixes scalars pointwise, so it preserves
+the nondegenerate scalar commutator pairing
+$W\times W\to\mu_r$.  Moreover the projective linear centralizer
+of $W$ is $W$: a centralizing operator twists the representing
+matrices by a character of $W$, that character is implemented by
+an element of $W$ through the pairing, and the remaining centralizer
+is scalar by irreducibility.
+
+In the linear and unitary rows, the normal group is an exponent-$r$
+extraspecial group for odd $r$, and a central product
+$C_4\circ2^{1+2d}$ for $r=2$.  In both cases the ordinary projective
+normalizer has quotient $\operatorname{Sp}_{2d}(r)$ on $W$.
+Indeed every symplectic change of generators lifts to a center-fixing
+automorphism and is implemented by the irreducible representation
+with its specified central character.  For $r=2$, multiplying a
+generator by the central element of order four restores its required
+square; thus a square form attached to a pure extraspecial subgroup
+is not an invariant of this scalar-extended normalizer.
+In the unitary case an implementing operator preserves the Hermitian
+form up to a multiplier, which is removed by a scalar using the
+surjective finite-field norm.
+The standard tensor generators of $W$ have determinant one: for odd
+$r$ this holds for the cyclic shift and diagonal root-of-unity
+generators, and for $r=2$ their determinants are
+$(-1)^{2^{d-1}}=1$.  The map from this normalizer to the abelian
+quotient of the projective linear or unitary group by its simple
+socle therefore factors through $\operatorname{Sp}_{2d}(r)$.
+That group is perfect in the retained range, so the map is trivial.
+Hence the full ordinary projective normalizer is already the inner
+intersection $P$.  Quotienting by the entire ambient scalar center
+leaves kernel exactly $W$, and
+\[
+ P/W\cong\operatorname{Sp}_{2d}(r),\qquad
+ |P|/|T_s|=(2,r-1)N^2.
+\]
+This uses the quotient, without requiring the extension to split.
+
+The pure symplectic and orthogonal form rows have $r=2$.
+Here the extraspecial square form is invariant.  The cited inner
+rows give $P/W=O_{2d}^{\varepsilon}(2)$ or
+$\Omega_{2d}^{\varepsilon}(2)$, with active simple group
+$T_s=\Omega_{2d}^{\varepsilon}(2)$ and index at most two.
+These are the inner rows after the determinant and spinor conditions;
+the scalar kernel of the natural cover is removed in $W$.
+Consequently $N^2\leq |P|/|T_s|\leq2N^2$ in every retained row.
+There is no extra factor from $r-1$ or the ambient field exponent.
+
+The target-field bound uses the minimal-field condition of the
+maximal row.  For odd $r$, the degree
+$e=\operatorname{ord}_r(p)$ divides $r-1$; the usual target exponent
+is $e$ in the linear row and $e/2$ in the unitary row.
+The binary and pure form rows have usual exponent one.
+Thus $f_t\leq\max(1,r-1)\leq |T_s|$.
+The representation characteristic is different from $r$, and the
+active source is over the prime field of order $r$.
+This proves \ref{cls:cat-symplectic}.
+
+For an absolutely irreducible representation of a central cover
+$\widetilde T_s$, the dimension is at most
+$|\widetilde T_s|^{1/2}$: the square of each simple-module dimension
+is bounded by the dimension of the group algebra.  The field-of-definition bound can be obtained directly, including in
+modular characteristic.  For a finite group $K$ in characteristic $p$,
+there are at most $|K|$ absolute simple module types, by the semisimple
+quotient of its group algebra.  Frobenius permutes these types, so an
+absolute simple representation has a Frobenius orbit of length
+$a\leq |K|$.  An isomorphism with its $p^a$-Frobenius twist descends the
+representation to $\mathbb F_{p^a}$: choose a finite field containing
+its matrices and an intertwiner, scale the intertwiner to have norm
+one, and apply matrix Hilbert's Theorem~90.  The scalar norm adjustment
+is possible because finite-field norms are surjective.  Apply this
+with $K=\widetilde T_s$.  The Schur multipliers of the
+large classical groups have order at most a polynomial in $|T_s|$;
+the finitely many exceptional multipliers do not change the bound.
+Forms cost an extension of degree at most two.  Maximality outside
+the subfield collection makes these bounds applicable to the target
+field itself.  For the cross-characteristic lower bound, the
+classical tables in \cite[Section~3, Tables~3--6]{Hasa2014} give
+degree at least $q_s^r/3$ once the source rank $r\geq6$, including
+the smaller Weil degrees.  Their excluded families have bounded
+rank.  As $n_s\leq2r+1$, this is at least $q_s^{n_s/4}$ after
+increasing the rank threshold.  This proves the numerical assertions
+of \ref{cls:cat-cross}.
+
+For the defining-characteristic quadratic bound, use
+\cite[Theorem~5.1 and Table~2]{Luebeck2001} in rank $r>11$.
+Modules outside that table have dimension greater than $r^3/8$
+in type $A$, and greater than $r^3$ in types $B,C,D$.
+Every listed nontrivial, nonnatural, nondual degree is at least $r^2/2$.
+Consequently the retained restricted module has dimension at
+least $r^2/2\geq n_s^2/10$.  Type $B$ in characteristic two is
+read through its type-$C$ natural simple quotient, as in our
+parameter convention; the natural-size special-isogeny cases are
+already assigned to geometry.
+
+A single occupied ordinary Frobenius digit has full source period:
+any shift strictly between zero and $f_s$ moves it to a different
+position in the unique finite restricted label.  Returning through
+$f_s$ applies at most the bounded diagram operation.  An internal
+tensor decomposition of that digit changes neither its occupied
+position nor this uniqueness argument.  Thus the minimal scalar
+field degree is $f_s$ times one of the bounded diagram periods.
+The matrix degree is separately $\epsilon_t f_t$.  The
+form-compatible minimal-field argument in
+Lemma~\ref{cls:normalizer} bounds the ratio of this matrix degree
+to the minimal degree outside the geometric collections.
+This gives the fixed finite rational set of usual exponent ratios
+and proves \ref{cls:cat-nonlinear}.
+Lemma~\ref{cls:normalizer} below supplies
+the final normalizer assertion.
+\end{proof}
+
+\begin{lemma}[Irreducible normalizers]\label{cls:normalizer}
+In a non-geometric absolutely irreducible step of
+Proposition~\ref{cls:catalogue}, let $Y$ denote the simple source
+inside the ambient projective classical group $T$.  In particular,
+the maximal subgroup is not in a subfield collection.  Then
+\[
+ \bigl|\ker\bigl(N_{\operatorname{Aut}(T)}(Y)
+       \longrightarrow\operatorname{Aut}(Y)\bigr)\bigr|\leq C
+\]
+for an absolute $C$.  Moreover
+$|\operatorname{Out}(Y)|\leq C n_Y f_Y$ for a large-rank classical
+source with parameters $(n_Y,p^{f_Y})$.
+\end{lemma}
+\begin{proof}
+Lift the irreducible projective representation to the perfect central
+cover of $Y$.  A projective linear element centralizing its image
+commutes with that representation up to a linear character of the
+perfect cover, hence commutes exactly.  Schur's Lemma makes it scalar,
+so it disappears projectively.
+
+If a semilinear element in this kernel has scalar-field action $\sigma$,
+the representations $\rho^\sigma$ and $\rho$ are isomorphic.  If
+$\sigma$ has order $a$ on the scalar field $E$, its intertwiner $A$
+satisfies
+\[
+ A\sigma(A)\cdots\sigma^{a-1}(A)=cI,
+ \qquad c\in E^{\sigma,*}.
+\]
+Surjectivity of the scalar norm adjusts $A$ so that the product is
+one; matrix Hilbert's Theorem~90 then gives a model over $E^\sigma$.
+An invariant bilinear form is a line defined by linear equations
+over the minimal field, and invariant quadratic-form coefficients
+satisfy linear equations there as well.  In characteristic two,
+two invariant quadratic forms with the same polarization differ
+by the square of an invariant linear form.  Absolute irreducibility
+makes that difference zero, so the prescribed quadratic form itself
+descends up to scalar.  Matching finite-field form classes costs at
+most a quadratic extension.  For a unitary group the involution on
+the minimal field is either nontrivial, giving Hermitian descent
+and surjective norms, or trivial, giving a preserved bilinear
+structure and hence a geometric form case.  A larger field period would
+therefore provide a proper subfield structure normalized by the
+whole irreducible normalizer, contrary to this step being outside the
+subfield collection.
+
+To justify the assertion about the whole
+normalizer, put an absolutely irreducible model over its minimal
+field $E_0$.  A normalizing matrix intertwines this model with the
+model obtained by a source automorphism and a field automorphism;
+both have entries in $E_0$.  The intertwining equations therefore
+have coefficients in $E_0$.  Their solution space over the algebraic
+closure is one-dimensional by absolute irreducibility.  It consequently
+has a nonzero $E_0$-rational solution, necessarily invertible.  Up to a
+scalar, the normalizer thus preserves the projective $E_0$-structure.
+The same argument applies after dualizing in a graph component, and
+the uniqueness of the invariant form gives the already allowed
+quadratic extension.  A graph operation contributes at most another
+factor of two in unbounded rank.  This bounds the kernel independently
+of the ambient field degree.
+
+Finally the diagonal, field, and graph
+description of automorphisms gives respectively factors at most
+$n_Y$, $f_Y$, and an absolute constant in the outer group.
+\end{proof}
+
+Twisted tensor subgroups and their normalizers were studied in detail by
+Schaffer \cite{Schaffer1999}; the algebraic and finite tensor structures
+used here are those of \cite[Section~4(5)]{LiebeckSeitzClassical1998}.
+The next lemma proves the precise maximality consequence and field relation
+needed for these tested chains.
+
+\begin{lemma}[Twisted tensor maximality]\label{cls:twisted-tensor}
+An almost simple irreducible maximal subgroup with a nontrivial
+Steinberg tensor decomposition is, in growing source rank, a geometric
+twisted-tensor step with the exact dimension and field relations in
+\ref{cls:cat-geometric}.  Its tensor-orbit degree is prime.
+\end{lemma}
+\begin{proof}
+Let $S$ be the simple source, $M$ the almost simple maximal subgroup,
+and $T$ the ambient simple classical group.  Work over an algebraic
+closure on the cover supplied by Lemma~\ref{lift:finite-covers}, and
+write the irreducible module as
+\[
+ V=\bigotimes_{i\in I}V_i,
+ \qquad
+ \mathcal A_i=\operatorname{End}(V_i)\otimes1
+       \ \leq\operatorname{End}(V),
+\]
+where $V_i$ are the nonzero ordinary restricted Frobenius digits,
+with their indicated twists.  The dimensions of these nontrivial
+digits tend to infinity with the source rank.
+
+We first verify preservation of the actual tensor structure.  A
+source field automorphism translates ordinary digit positions,
+applying the diagram operation when a position passes the source
+period; inner, diagonal, and diagram automorphisms act on the
+restricted modules at those positions.  Ambient semilinear
+operations give the same shifts, and a graph operation dualizes
+the modules.  Uniqueness of the finite restricted highest-weight
+parametrization matches the resulting whole digit modules.  Tensor
+their individual intertwiners, with the matching permutation, to
+obtain an intertwiner of $V$.  The discrepancy between a projective
+and linear intertwining identity is a linear character of the
+perfect cover and is trivial.  Absolute irreducibility makes the
+space of whole-module intertwiners one-dimensional.  Every actual
+normalizing operator therefore differs from this tensor intertwiner
+by a scalar and permutes the embedded algebras $\mathcal A_i$.
+For a graph operation the corresponding map on endomorphisms is an
+anti-isomorphism, which has the same conclusion.  This argument
+preserves a whole ordinary digit even if that restricted module
+has an internal tensor decomposition in a very special
+characteristic.
+
+Replace each digit by its full connected special linear group, or
+by its full connected bilinear-form group when required.  Bilinear
+self-duality makes each ordinary digit self-dual separately.
+Unitary descent acts on the special linear factors, permitting
+duals.  In characteristic two every nontrivial irreducible self-dual
+digit has an alternating form: a nonalternating invariant form
+would give an invariant linear functional by taking square roots
+of its diagonal.  The tensor product of at least two nondegenerate
+alternating spaces has a quadratic form with the product
+polarization and value zero on pure tensors
+\cite[Proposition~2.2(iv)]{LiebeckSeitzClassical1998}.
+It agrees with the prescribed finite quadratic form, after a scalar
+normalization of the polarization: their difference is the square
+of an invariant linear form, which vanishes by nontrivial absolute
+irreducibility.  These constructions give a connected semisimple
+tensor subgroup $K$ in the ambient form group.
+
+The groups are intrinsic to the algebras $\mathcal A_i$ and their
+restricted adjoint involutions.  Hence the actual ambient Frobenius
+$F$ and the full finite normalizer preserve $K$.  In growing
+classical rank, finite ambient automorphisms have lifts commuting
+with $F$; diagonal lifts satisfy this projectively because
+$F(x)x^{-1}$ is central.  Graph and field lifts also commute, with
+only the bounded-rank triality and exceptional graph cases removed.
+This is the setup of
+\cite[Theorem~2 and Section~4]{LiebeckSeitzClassical1998}.
+Henceforth view $K$, and later $K'$, in the ambient adjoint group.  Thus $M$ normalizes the terminal derived subgroup
+$H=(K^F)^{(\infty)}$.  Since $H$ is perfect it lies in $T$, and
+$S\leq H$ because $S$ is a perfect subgroup of $K^F$.
+
+For each orbit of $F$ on the factors, fixed points are obtained by
+choosing one coordinate fixed by the corresponding power of $F$;
+see \cite[Proposition~2.8(ii)]{LiebeckSeitzClassical1998}.
+Consequently $H$ is a central product of classical components, one
+per orbit.  It is proper in $T$ also as a finite group.  To check
+this uniformly, put $q=p^{f_t}$, let the individual digit dimensions
+be $d_1,\ldots,d_k$, put $d_{\min}=\min_i d_i$, and write
+$n_t=\prod_i d_i$.  The earlier bound for nontrivial digits gives
+$d_{\min}\to\infty$ with the source rank.
+Elementary classical order bounds give
+\[
+ |H|\leq q^{2\sum_i d_i^2},\qquad
+ |T|\geq q^{n_t^2/8}.
+\]
+For $k\geq2$,
+$\sum_i d_i^2/n_t^2\leq k/d_{\min}^{2(k-1)}\leq2/d_{\min}^2$.
+Thus $H<T$ for all sufficiently large source ranks.  If $MH$
+were the entire almost simple ambient group, its normal subgroup
+$H\leq T$ would equal $T$.  Hence $MH$ is proper, and maximality
+gives $H\leq M$.  But the perfect group $H/S$ is a subgroup of the
+soluble group $M/S$, so $H=S$.  There is therefore one Frobenius
+orbit, and its full natural classical component is the source.
+If its length is $b$, the digit dimension is $n_s$ and
+$n_t=n_s^b$ exactly.
+
+The field relation is read from this actual natural-factor cycle.
+Put $t=f_t$.  For a linear target, $F^b$ on one factor is the
+standard field map with exponent $bt$, up to an inner conjugacy.
+For a unitary target, write $F=F_{p^t}\tau$ with $\tau$ the
+inverse-transpose operation.  On one factor the return map is
+$F_{p^{bt}}\tau^b$, again up to an inner conjugacy.  Its source is
+unitary when $b$ is odd and linear when $b$ is even, in both cases
+with usual exponent $bt$.  An orthogonal graph choice changes the
+form sign, not this exponent.  The characteristic-two quadratic
+construction likewise changes no field exponent.  Thus
+\begin{equation}\label{cls:tensor-field}
+ f_s=b f_t
+\end{equation}
+exactly, in the convention of this paper.  The natural factors have
+positions $0,t,\ldots,(b-1)t$ modulo $bt$, which are distinct.
+For the odd unitary convention using positions $2jt$, multiplication
+by two permutes the same residues modulo $b$; the prime two-cycle
+in a unitary target has a linear source.  These are precisely the
+usual-field alternatives in
+\cite[Section~4(5)]{LiebeckSeitzClassical1998}.
+
+Finally suppose $b=uv$, with $u,v>1$.  Number the factors by
+$\mathbb Z/b\mathbb Z$, and group the cosets of its unique subgroup
+of order $u$.  The normalizer's permutation normalizes the regular
+cyclic descent action and preserves this block system.  The
+algebra for a block $B$ is the algebra generated by its
+$\mathcal A_i$, namely $\operatorname{End}(\bigotimes_{i\in B}V_i)
+\otimes1$.  Replace each block by its full special linear or
+bilinear-form group.  For a characteristic-two quadratic target
+use the full symplectic group on each grouped alternating space.
+The resulting quadratic tensor form is unchanged: it has the
+same polarization and vanishes on every fully pure original
+tensor, and such tensors span $V$.  Hence this grouped subgroup
+$K'$ still preserves the prescribed ambient form and is invariant
+under $F$ and $M$.
+
+Its terminal derived fixed-point subgroup $H'$ has one classical
+component on dimension $n_s^u$ over the field with usual exponent
+$v f_t$, with the unitary parity or orthogonal sign prescribed by
+the return map.  For $n_s\geq16$, the elementary bounds
+$Q^{d^2/8}\leq|\operatorname{Cl}_d(Q)|\leq Q^{2d^2}$ give
+\[
+ |S|\leq q^{2uv n_s^2}
+   <q^{v n_s^{2u}/8}\leq|H'|,
+ \qquad
+ |H'|\leq q^{2v n_s^{2u}}
+   <q^{n_s^{2uv}/8}\leq|T|.
+\]
+The strict inequalities follow from
+$n_s^{2u-2}>16u$ and $n_s^{2u(v-1)}>16v$.
+As before $M$ normalizes $H'$ and $S\leq H'$.  Its properness in
+$T$ forces $H'\leq M$, and perfectness then forces $H'=S$,
+contradicting the first inequality.  Therefore $b$ is prime.
+\end{proof}
+
+\subsection{Estimates along a two-edge shortcut}
+
+We record how the catalogue applies to the reduced configurations.
+In the estimates below, a \emph{nonlinear} cover means a
+non-geometric defining-characteristic irreducible cover as in
+\ref{cls:cat-nonlinear}; a \emph{crossing} cover means an almost simple
+cross-characteristic cover or a symplectic-type cover.
+The field estimates constrain prime degrees, the dimension estimates
+can be iterated along the chain, and the inner order quotients will
+multiply through intermediate vertices.
+
+\begin{enumerate}[label=\textup{(G\arabic*)},ref=G\arabic*]
+\item\label{grw:input-fields}
+For a geometric or defining-characteristic almost simple cover,
+\ref{cls:cat-geometric} and \ref{cls:cat-nonlinear} express the field
+ratio as a fixed rational factor times the contribution of a subfield,
+extension-field, or twisted-tensor degree. An exceptional degree is
+prime or belongs to a fixed bounded set of positive integers.
+All bounded rational corrections, including two, are fixed before
+any prime coloring is made. The restriction-of-scalars and
+twisted-tensor formulas are \eqref{grw:matrix-field-convention}
+and \eqref{grw:twisted-field-convention}, respectively.
+The mixed unitary restriction of degree two is a bounded geometric
+case, with $\epsilon_s/\epsilon_t\in\{1/2,1,2\}$.
+A natural space and its dual give a separate dimension operation,
+not an additional restriction degree. Other form and special-isogeny
+rows retain their own fixed rational field factors; no telescoping
+of $\epsilon$ factors through those rows is used.
+Lemma~\ref{cls:twisted-tensor} supplies distinct tensor positions
+modulo $f_s$, including the unitary alternatives. Thus there is no
+matrix-field factor of two in \eqref{grw:twisted-field-convention},
+and the dimension formulas for a two-step subfield/extension or
+subfield/twisted-tensor path are exact.
+
+\item\label{grw:input-geometric}
+The geometric dimension operations in \ref{cls:cat-geometric},
+together with Proposition~\ref{cls:reduction}(v), give boundedly
+many omitted dimensions once the active slot, permutation data,
+and inactive simple labels are fixed. In paths with no replication
+and no inactive nonabelian factor, these dimensions are bounded
+integers; ordinary tensor factors other than the active one then
+have bounded dimensions. An unbounded replication degree belongs
+to the bounded-cardinality pool in Proposition~\ref{cls:reduction}.
+
+\item\label{grw:input-degrees}
+Clause~\ref{cls:cat-nonlinear} gives $n'\ge c n^2$ on a nonlinear
+cover, with nontrivial Steinberg tensor decompositions already
+assigned to geometry. Clauses~\ref{cls:cat-symplectic} and
+\ref{cls:cat-cross} give the common crossing estimate
+\begin{equation}\label{grw:cross-bounds}
+ n'\ge \exp(c n\log q),\qquad n',f'\le |T|^C.
+\end{equation}
+For an almost simple cover the upper degree bound also holds in
+defining characteristic.
+Every ascending same-characteristic geometric step decreases natural
+dimension by at most a fixed factor, by
+\ref{grw:input-fields}--\ref{grw:input-geometric}. It also decreases
+$n\log q$ by at most a fixed factor. For restriction of scalars,
+\eqref{grw:matrix-field-convention} gives
+\[
+ n_t\log q_t
+ =\frac{\epsilon_s}{\epsilon_t}n_s\log q_s
+ \ge\tfrac12 n_s\log q_s,
+ \qquad n_t=b n_s\ge n_s.
+\]
+For a twisted-tensor step,
+\eqref{grw:twisted-field-convention} gives
+$n_t\log q_t=(n_s^b/b)\log q_s\ge n_s\log q_s$ when $n_s,b\ge2$.
+The other geometric rows have fixed positive lower factors. A
+same-characteristic nonlinear step has both lower-bound properties
+as well, by its quadratic degree bound and fixed positive field ratio.
+
+\item\label{grw:input-multipliers}
+In a one-factor almost simple cover the inner order quotient is at
+most $|\operatorname{Out}(T)|$. Clause~\ref{cls:cat-symplectic}
+bounds the symplectic-type quotient between $cN^2$ and $CN^2$,
+where $N$ is the target natural dimension; the target dimension
+and field exponent are at most $|T|^C$ for source $T$.
+For a parabolic or unipotent subspace cover with one active factor,
+no inactive nonabelian factor, and bounded field ratio,
+\ref{cls:cat-inactive} makes the target dimension $n'$ comparable
+to the source dimension $n$ and places the inner quotient between
+$q^{c n'}$ and $q^{C n^2}$. For fixed target type and parameters
+that quotient has boundedly many possibilities. This is a bound
+on cardinality; the quotient itself is usually very large.
+\end{enumerate}
+
+A strict split followed by an equal-type merge is geometric.
+Every shortcut containing an almost simple irreducible or a
+symplectic-type cover has one active simple factor at both covers
+and no other nonabelian inner
+composition factor, by Proposition~\ref{cls:reduction}. For those
+shortcuts, the next lemma transfers the normalizer bound to the
+actual lower coordinate stabilizer.
+
+\begin{lemma}[Transport of a coordinate stabilizer]\label{grw:input-normalizers}
+Suppose that the last cover is an almost simple irreducible cover with
+source $Y$, and that all three shortcut multiplicities agree, as in
+every application below.  Denote the lower node's full upper-coordinate
+stabilizer image by $M_{ij}$, and let $P_B$ be the inner projection in
+the first cover.  Then
+\begin{equation}\label{grw:normalizer-input}
+ |M_{ij}|\le C\,|P_B|\,n'^{C}(1+f')^{C}.
+\end{equation}
+\end{lemma}
+\begin{proof}
+ Write the actual shortcut as
+$U=z_i\prec V\prec W=z_j$, and work modulo $F_W$.
+In one $W$-coordinate the last cover has
+$Y\trianglelefteq P_A\le\operatorname{Aut}(Y)$.  Its killed kernel
+from the middle is soluble by Proposition~\ref{cls:reduction} and
+normal in $P_A$, hence is trivial.  The active subgroup in that
+coordinate is therefore the actual simple group $Y$.
+Proposition~\ref{frm:active-transport} gives a $V$-equivariant bijection
+$\beta:I_V\longrightarrow I_W$ because the multiplicities agree.
+For $\alpha\in I_V$ and $t=\beta(\alpha)$, the actual stabilizers in
+$U$ coincide: $U_\alpha=U_t$.  Let $\rho_V,\rho_W$ be their middle
+and upper coordinate actions, and let
+$\phi:N_{\operatorname{Aut}(T_j)}(Y)\longrightarrow\operatorname{Aut}(Y)$
+be conjugation.  The quotient-induced actions satisfy
+\[
+ \phi\circ\rho_W|_{U_t}=\rho_V|_{U_\alpha},
+ \qquad \phi(M_{ij})=M_B,
+\]
+where $M_B$ is the actual first-cover coordinate stabilizer image.
+
+The inner-intersection assertions use the original lower endpoint $a$
+from Definition~\ref{cls:tested-chain}.  All retained comparisons
+start strictly above this $a$, so none uses an excluded middle
+$v_{0j}$.  Condition
+\ref{cls:test-alternating} makes $a$ an intersection of coatoms in
+$[a,V]$ and $[a,W]$ for every retained middle and upper endpoint.
+Proposition~\ref{frm:projection} in these original-$a$ projections
+identifies the inner part of a saturated image with its transported
+inner projection.  For $U\prec V$ the coatom case of that Proposition
+also applies directly.  In particular,
+\[
+ M_B\cap\operatorname{Inn}(Y)=P_B.
+\]
+There is no assumption that a meet of coatoms in the original global
+interval is a meet of coatoms in every smaller interval.
+The bounded kernel from Lemma~\ref{cls:normalizer} now gives
+\[
+ |M_{ij}|\le C|M_B|
+ \le C|P_B|\,|\operatorname{Out}(Y)|
+ \le C|P_B|n'f',
+\]
+which implies \eqref{grw:normalizer-input}.  This includes diagonal,
+field, and graph operations.  All higher-core losses are already
+kernels of the displayed actual coordinate actions; no uncontrolled
+factor from the field of the final target is inserted.
+\end{proof}
+
+\subsection{Finite colors for field degrees}
+
+For a positive rational $x$, let $v_\ell(x)$ be its valuation at a prime
+$\ell$.  If $S$ is a finite set of primes, put
+\[
+ \Omega_S^+(x)=\sum_{\ell\notin S}\max\{v_\ell(x),0\},
+ \qquad
+ \Omega_S^-(x)=\sum_{\ell\notin S}\max\{-v_\ell(x),0\}.
+\]
+Both sums have finite support.
+
+\begin{lemma}[Finite prime complexity]\label{grw:prime-colors}
+Let $x_1,\ldots,x_s$ be positive real numbers whose pair ratios are
+positive rationals.  Suppose that, for every $i<j$,
+\[
+ x_j/x_i=c_{ij}u_{ij}^{\varepsilon},\qquad
+ c_{ij}\in\mathcal C,\qquad \varepsilon\in\{1,-1\},
+\]
+where $\mathcal C$ is a fixed finite set of positive rationals,
+$u_{ij}$ is a positive integer with at most $h$ prime factors counted
+with multiplicity, and $\varepsilon$ is the same for all pairs.
+There is a number of colors depending only on $\mathcal C,h$ such that
+on every homogeneous triangle all three $x$'s are equal.
+Consequently arbitrarily long such arrays have arbitrarily long
+constant subarrays.
+\end{lemma}
+
+\begin{proof}
+Let $S$ contain the primes occurring in the numerators or denominators
+of members of $\mathcal C$.  For $\ell\in S$, the integer
+$v_\ell(x_j/x_i)$ lies in a fixed bounded interval: the contribution of
+$c_{ij}$ is bounded, and that of $u_{ij}$ has absolute value at most $h$.
+Outside $S$ all valuations have sign $\varepsilon$, and the sum of their
+absolute values is at most $h$.  Color $(i,j)$ by the valuation vector
+on $S$ and this last sum.  This is a finite coloring with the asserted
+uniformity.
+
+For $i<j<k$ we have
+\[
+ v_\ell(x_k/x_i)=v_\ell(x_j/x_i)+v_\ell(x_k/x_j).
+\]
+On a homogeneous triangle each coordinate of the common valuation
+vector therefore equals twice itself, so it is zero.  Outside $S$
+there is no cancellation because all valuations have the same sign.
+Their total absolute value likewise equals twice itself, hence is zero.
+Every valuation of every pair ratio is zero, so all ratios equal one.
+Finite Ramsey gives the last assertion.
+\end{proof}
+
+\begin{lemma}[Opposite field moves]\label{grw:opposite-fields}
+Arbitrarily long homogeneous shortcut chains cannot have, on every
+comparison, one proper subfield step and one proper extension step,
+including the twisted-tensor extension alternative.
+\end{lemma}
+
+\begin{proof}
+Color first by the order and kind of the two steps.  By
+\ref{grw:input-fields}, their combined dimension formula is either
+\[
+ n_j=b_{ij}n_i\quad\hbox{for every }i<j,
+ \qquad\hbox{or}\qquad
+ n_j=n_i^{b_{ij}}\quad\hbox{for every }i<j,
+\]
+where $b_{ij}>1$ is prime or belongs to a fixed bounded set of integers.
+In the first case put $x_i=n_i$, and in the second put $x_i=\log n_i$.
+In either case $x_j/x_i=b_{ij}$ has uniformly bounded prime complexity.
+Lemma~\ref{grw:prime-colors}, with $\mathcal C=\{1\}$, gives a
+homogeneous triangle with $b_{ij}=1$, a contradiction.  Notice that
+field ratios alone would not suffice: ratios $r_j/r_i$ of distinct
+primes are compatible on arbitrarily many vertices.  It is the exact
+dimension formula that excludes this possibility.
+\end{proof}
+
+\begin{proposition}[Purely geometric paths]\label{grw:no-geometric}
+An arbitrarily long reduced chain cannot have exclusively geometric
+two-edge shortcuts.  This includes strict split/equal-type merge
+shortcuts.
+\end{proposition}
+
+\begin{proof}
+Color by the two cover categories, the endpoint family, and whether a
+split and merge occur.  There are finitely many colors.  The endpoint
+characteristic is common because every step is in defining
+characteristic.  If both an increasing and a decreasing exceptional
+field move occur, Lemma~\ref{grw:opposite-fields} applies.  Otherwise
+all exceptional primes in a pair ratio have the same sign.  The product
+of the two bounded rational factors belongs to a fixed finite set, and
+at most two exceptional degrees occur.  A bounded exceptional degree
+has uniformly bounded prime complexity as well.  Apply
+Lemma~\ref{grw:prime-colors} to $x_i=f_i$.  We may therefore assume
+that all endpoint fields and families agree.
+
+The exceptional degrees at the individual covers are now bounded
+integers.  Indeed, with no opposite field moves their product, or its
+reciprocal, is a member of the fixed finite set of bounded twisting
+ratios, because $f_j/f_i=1$.  Each positive degree divides the numerator
+or denominator of one of these finitely many rationals.
+
+Fix a lower endpoint $i$.  We show that its upper natural dimension has
+only boundedly many possible values.  For block and parabolic shapes,
+the ambient dimension is the active dimension, or twice that dimension
+when its dual is present, plus the dimensions of the omitted blocks.
+The dimensions of nonabelian omitted blocks are determined, with bounded
+ambiguity, by the bounded-cardinality pool of inactive types.  Soluble
+omitted blocks have bounded dimensions.  A tensor shape multiplies the
+active dimension by a product of the omitted dimensions.  A replicated
+base multiplies it by a replication degree from its bounded-cardinality
+pool.  A form change has the same natural dimension, with the bounded
+natural-module exceptions included in the family color.  Field
+restriction multiplies dimension by one of the bounded degrees just
+obtained, and a twisted-tensor step raises it to one of those degrees.
+Thus one geometric cover gives a bounded list of possible outputs from
+a fixed input.  Two covers give a list whose size is at most the product
+of the two bounds.  For a split followed by a merge the second step has
+the same simple label, so gives no new natural dimension at all.
+
+All labels now have the same family and field.  Their strict growth in
+order makes their natural dimensions distinct.  There cannot be more
+upper endpoints than the bounded number of possible outputs from the
+fixed lower endpoint.  This contradicts arbitrary length.
+\end{proof}
+
+\subsection{Multiplicative inner quotients}
+
+From now on each comparison contains a nonlinear or crossing step.
+There is no splitting.  The multiplicity is the common integer $m$
+from Proposition~\ref{cls:reduction}, also at the two covers of every
+chosen shortcut.  Let $P_{ij}\le T_j$ be the saturated inner projection
+of the lower node at an upper coordinate, and let
+$M_{ij}\le\operatorname{Aut}(T_j)$ be its full induced coordinate
+stabilizer.  Applying Proposition~\ref{frm:projection} with the
+original $a$ and condition~\ref{cls:test-alternating}, as explained
+in Lemma~\ref{grw:input-normalizers}, gives
+$P_{ij}=M_{ij}\cap T_j$.
+
+\begin{lemma}[Index quotients]\label{grw:indices}
+The positive integers
+\[
+ e_{ij}=|P_{ij}|/|T_i|
+\]
+satisfy $e_{ik}=e_{ij}e_{jk}$ for every $i<j<k$.  They multiply in the
+same way through each selected shortcut.
+\end{lemma}
+
+\begin{proof}
+The active factor $T_i$ is a composition factor of $P_{ij}$, so its
+order divides $|P_{ij}|$.  Saturation and supplementation give
+\[
+ [z_j:z_i]=[T_j^m:P_{ij}^m]
+ =\left(\frac{|T_j|}{|P_{ij}|}\right)^m.
+\]
+Multiplying group indices through $z_j$ and taking the positive $m$th
+root gives
+\[
+ \frac{|T_k|}{|P_{ik}|}
+ =\frac{|T_k|}{|P_{jk}|}\frac{|T_j|}{|P_{ij}|}.
+\]
+Rearranging and dividing by $|T_i|$ proves the formula.  The same
+calculation applies at a shortcut because its three multiplicities
+agree.  No identification of its intermediate label with a chain label
+is required.
+\end{proof}
+
+We use the soluble-normal and commuting-normal conclusions of
+Lemma~\ref{frm:labels} and Corollary~\ref{frm:bottleneck}.
+Fix retained labelled endpoints $z_i<z_j$ and put
+$F_j=\operatorname{core}_{z_j}(D)$.  This original higher core is
+normal in $z_j$ and contained in $D$, hence normal in every
+intervening group.  Write
+\[
+ S_j=N_j/F_j=\prod_{t\in I_j}T_{j,t},\qquad
+ (z_i\cap N_j)/F_j=P_{ij}^{I_j}.
+\]
+For each $M_{ij}$-invariant subgroup $R\le T_j$, write
+$\widehat R\le S_j$ for its transported coordinate product.
+It is normalized by $z_i/F_j$.
+
+\begin{lemma}[Coordinate normal tests]
+\label{grw:coordinate-normal-tests}
+Let
+\[
+ z_i/F_j\le\overline U<z_j/F_j
+\]
+be an actual subgroup, and let $U$ be its full inverse image in
+$z_j$.  A transported product normal in $\overline U$ is called
+\emph{killed} if it is contained in $D/F_j$; it then lies in
+$\operatorname{core}_{\overline U}(D/F_j)$.
+\begin{enumerate}[label=\textup{(N\arabic*)},ref=N\arabic*]
+\item\label{grw:normal-soluble}
+If $\widehat R\trianglelefteq\overline U$ is soluble, then
+\begin{equation}\label{grw:normal-containment}
+ \widehat R\le D/F_j,\qquad R\le P_{ij}.
+\end{equation}
+\item\label{grw:normal-commuting}
+Let $R,Q\le T_j$ be $M_{ij}$-invariant, with
+$\widehat R,\widehat Q$ normal in the same $\overline U$.
+Suppose that
+\begin{equation}\label{grw:commuting-kernel}
+ [\widehat R,\widehat Q]\le\overline K
+       \trianglelefteq\overline U,\qquad
+ \overline K\le D/F_j.
+\end{equation}
+Then at least one of $\widehat R,\widehat Q$ is killed, and
+its coordinate subgroup is contained in $P_{ij}$.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+For $U>z_i$, the original test \ref{cls:test-normal} provides
+a fenced interval or a labelled bottleneck at $U$.  If $U=z_i$,
+the endpoint itself is labelled.  Since $F_j\trianglelefteq z_j$
+and $F_j\le D$, apply Lemma~\ref{frm:labels} or
+Corollary~\ref{frm:bottleneck} in the quotient by $F_j$, with
+bottom $D/F_j$.  The soluble-normal conclusion gives
+\eqref{grw:normal-containment}.  For
+\textup{(\ref{grw:normal-commuting})}, the full inverse image of
+$\overline K$ is normal in this same $U$ and contained in $D$,
+so the commuting-normal conclusion kills one product.
+Since $D\le z_i$, a killed product has its coordinate subgroup
+in $P_{ij}$.
+\end{proof}
+
+For \textup{(\ref{grw:normal-soluble})} we use the enlargement
+\begin{equation}\label{grw:normal-enlargement}
+ \overline U=(z_i/F_j)\widehat R;
+\end{equation}
+for \textup{(\ref{grw:normal-commuting})} we use
+$\overline U=\langle z_i/F_j,\widehat R,\widehat Q\rangle$.
+Properness and normality in this same enlargement are checked
+in each application.  If one tested product contains a transported
+active subgroup known to map onto the original lower labelled
+socle, it is not killed, so the other product is killed.
+Containment in $P_{ij}$ alone does not imply that a product
+is killed and cannot replace the normality and bottom-containment
+hypotheses on $\overline K$.  Solubility here is a property of
+the quotient product; its full inverse image need not be soluble.
+No normality of the lower core $F_i$ in the enlargement, or full
+supplementation dichotomy at an unlabelled bottleneck, is used.
+
+\Needspace{5\baselineskip}
+\subsection{A small projective group and a parabolic}
+
+\begin{lemma}[Small coordinate stabilizers]\label{grw:small-parabolic}
+There is an absolute constant $c_0>0$ with the following property.
+Let $T$ be a classical simple group of sufficiently large natural
+dimension $n$, over parameter $q=p^f$, and let
+$M\le\operatorname{Aut}(T)$ have order $m$.  If
+\[
+ n>2(m^2+m),
+\]
+then $M$ normalizes a proper parabolic subgroup of $T$ whose unipotent
+radical has order at least $q^{c_0 n}$.  Consequently the coordinate
+normal test excludes $M=M_{ij}$ whenever, in addition,
+$q_j^{c_0 n_j}>|M_{ij}|$.
+\end{lemma}
+
+\begin{proof}
+Use the natural-space field $E=\mathbb F_{p^e}$; thus $e=f$, or $e=2f$
+in the unitary case.  Apart from the bounded-rank graph exceptions
+already excluded, automorphisms in the form cases are represented by
+projective semilinear similarities.  In the linear case there may also
+be correlations, which interchange the natural space and its dual.
+The ordinary orthogonal graph operation is a similarity in this
+description; characteristic-two type $B$ uses its symplectic natural
+quotient.
+
+First suppose that $M$ acts on the natural-space side.  Choose one
+semilinear representative $s_g$ for each projective element, with
+$s_1=1$.  The products satisfy
+$s_gs_h=c(g,h)s_{gh}$ for nonzero scalars $c(g,h)\in E$.
+For an unknown vector $v\in E^n$ impose
+\[
+ B(s_gv,s_hv)=0\quad(g,h\in M)
+\]
+in a bilinear or Hermitian form case, and impose also
+$Q(s_gv)=0$ in a quadratic form case.  In a plain linear space no
+equations are necessary.  Field automorphisms and semilinear maps are
+$\mathbb F_p$-linear.  These are therefore at most $e(m^2+m)$ scalar
+polynomial equations of degree at most two in $en$ prime-field
+variables.  Their sum of degrees is less than $en$.
+
+For completeness, the elementary polynomial count giving a nonzero
+solution is as follows.  If $F_1,\ldots,F_t$ have total degree sum less
+than the number $N$ of prime-field variables, the number of their common
+zeros, reduced modulo $p$, is
+\[
+ \sum_{x\in\mathbb F_p^N}
+       \prod_{a=1}^t(1-F_a(x)^{p-1}).
+\]
+Every expanded monomial has degree less than $N(p-1)$, so some variable
+has exponent less than $p-1$.  The sum over that variable is zero in
+$\mathbb F_p$, including exponent zero.  The displayed sum is zero.
+All our equations vanish at zero, so their common zero set has another
+element.  This is the needed Chevalley--Warning argument
+\cite{Chevalley1935,Warning1935}.
+
+Choose such a nonzero $v$ and put
+\[
+ U=\operatorname{span}_E\{s_gv:g\in M\}.
+\]
+Then $1\le\dim U\le m$.  Scalar discrepancies between representatives
+do not change this span, so $U$ is $M$-stable.  The imposed equations
+make $U$ totally singular in the form cases.  In particular, in
+characteristic two we used the quadratic equations as well as the
+polar equations.  Thus the stabilizer of $U$ is a proper parabolic
+normalized by $M$.
+
+Now let $T$ be linear and suppose $M$ contains correlations.  Put
+$H=M\cap\operatorname{P\Gamma L}(V)$ and choose $d\in M\setminus H$.
+On subspaces the correlation has the form
+\[
+ d(S)=\operatorname{ann}(F(S))
+\]
+for an invertible semilinear map $F:V\longrightarrow V^*$.
+Choose representatives $s_h$, $h\in H$, and impose
+\[
+ F(s_hv)(s_kv)=0\qquad(h,k\in H).
+\]
+Again there are at most $e|H|^2$ quadratic scalar equations.  The same
+count gives $v\ne0$.  Let
+$U=\operatorname{span}_E\{s_hv:h\in H\}$ and $W=d(U)$.
+The equations say $U\le W$.  Since $H$ is normal in $M$, both subspaces
+are $H$-stable.  Moreover $d(W)=d^2(U)=U$, because $d^2\in H$.
+Thus the flag $U<W$ is preserved by all of $M$, as an unordered pair
+under correlations.  Its dimensions are $r,n-r$, where
+$1\le r\le |H|<n/2$, so it is a proper flag.  Its parabolic stabilizer
+is normalized by $M$.  No involution assumption on $d$ has been used.
+
+In these constructions the small subspace has dimension
+$1\le r\le m<n/4$ once $n$ is sufficiently large.  The upper triangular
+block matrices in the radical supply at least $n/4$ independent
+prime-parameter root coordinates, each of order at least $q$.
+For example the linear case has $r(n-r)$ coordinates, and the flag
+case has at least $r(n-2r)$; in the polar cases the block from the
+nondegenerate middle space to the singular $r$-space supplies at least
+$r(n-2r-2)$ coordinates.  These lower bounds are at least $n/4$ after
+enlarging the fixed dimension threshold.  We may take a smaller
+absolute $c_0$ to cover all the families.  Projectivization does not
+remove unipotent coordinates: the scalar center has order prime to $p$.
+
+Let $J$ be this proper parabolic and $R$ its radical.  The group $R$
+is soluble and normalized by $M$.  In the coordinate application,
+take the actual enlargement \eqref{grw:normal-enlargement}.  Its inner
+part is
+\[
+ \overline U\cap S_j=(P_{ij}R)^{I_j},\qquad
+ P_{ij}R\le N_{T_j}(J)<T_j.
+\]
+The last inequality holds because $J$ is a nontrivial proper subgroup
+of the simple group $T_j$.  Thus $U<z_j$: properness only in
+$\operatorname{Aut}(T_j)$ would not suffice.  The soluble coordinate
+normal test gives $\widehat R\le D/F_j$, hence
+$R\le P_{ij}\le M_{ij}$.  Its order contradicts
+$q_j^{c_0n_j}>|M_{ij}|$.
+\end{proof}
+
+\subsection{Eliminating every crossing path}
+
+Put $\mathcal E_0(x)=x$ and
+$\mathcal E_{h+1}(x)=\exp(\mathcal E_h(x))$.
+
+\begin{lemma}[Repeated crossing growth]\label{grw:towers}
+Suppose every comparison in a reduced homogeneous chain has a crossing.
+For every fixed $h,C$ and every sufficiently distant pair $i<j$,
+\[
+ n_j>\mathcal E_h(a_i^C).
+\]
+The required distance is a constant depending only on $h,C$ and the
+constants in the catalogue.  Also $n_j>a_i^C$ for every fixed $C$ after
+a fixed separation and a sufficiently large minimum rank.
+\end{lemma}
+
+\begin{proof}
+Every ascending same-characteristic geometric step preserves both
+natural dimension and $n\log q$ up to fixed positive lower factors,
+as established in \ref{grw:input-degrees}.  The same holds for a
+same-characteristic nonlinear step by its quadratic degree bound and
+fixed positive field ratio.  If the last cover crosses, a preceding
+same-characteristic cover leaves $n'\log q'$ at least a fixed multiple
+of $n_i\log q_i$.  A preceding crossing has the same consequence
+from $n'\ge\exp(c n_i\log q_i)$ and $\log q'\ge\log2$ at sufficiently
+large rank.  The last crossing then gives the endpoint lower bound.
+If only the first cover crosses, it gives
+$n'\ge\exp(c n_i\log q_i)$, and the last cover preserves natural
+dimension up to a fixed factor.  Decreasing $c$ and increasing the
+fixed minimum-rank threshold, we obtain in every case
+\[
+ n_{i+1}\ge\exp(c n_i\log q_i).
+\]
+Write
+$u=n_i\log q_i$.  Classical order formulas give
+\[
+ \log a_i\le C_1 n_i^2\log q_i\le C_2u^2.
+\]
+After the first step every additional comparison gives at least the
+map $x\mapsto F(x)=\exp(c\log(2)x)$.  For fixed $C_3$,
+\[
+ \frac{F(\exp(cu))}{\exp(C_3u^2)}\longrightarrow\infty
+       \qquad(u\longrightarrow\infty).
+\]
+If $L(u)/R(u)\longrightarrow\infty$ and $R(u)\longrightarrow\infty$,
+then $F(L(u))/\exp(R(u))\longrightarrow\infty$ as well.  Induction
+therefore shows that $h+2$ comparisons dominate
+$\mathcal E_h(\exp(C_3u^2))$, and hence
+$\mathcal E_h(a_i^C)$ when $C_3\ge CC_2$.  This also proves the last
+assertion.  The target prime characteristic need not be bounded in
+this argument: its only use in the iteration was $\log q\ge\log2$.
+\end{proof}
+
+\begin{proposition}[No characteristic crossings]\label{grw:no-crossing}
+After passage to a reduced homogeneous chain, neither cover of any
+comparison changes the defining characteristic.
+\end{proposition}
+
+\begin{proof}
+Suppose otherwise, and color by the two step categories.  Every pair in
+the resulting arbitrarily long chain has a crossing.  There are three
+possibilities for its last step $A$.
+
+\smallskip\noindent
+\emph{The last step is geometric in the same characteristic.}
+The first step $B$ crosses.  Hence $n',f'\le a_i^C$ by
+\eqref{grw:cross-bounds}.  An ordinary geometric last step changes
+dimension by bounded factors, except for an extension degree.  By
+\eqref{grw:matrix-field-convention} that degree is
+$b=\epsilon'f'/(\epsilon_jf_j)\le2f'$.  A last subfield degree can be
+arbitrarily large but leaves dimension unchanged.  A twisted-tensor
+last step has exponent $b=f'/f_j\le f'$ by
+\eqref{grw:twisted-field-convention}.  Therefore
+\[
+ n_j\le (C n')^{C(1+f')}+C(1+f')n'+C
+       \le\exp(a_i^{C'})
+\]
+for a fixed $C'$.  Lemma~\ref{grw:towers} contradicts this bound on a
+sufficiently distant pair.
+
+\smallskip\noindent
+\emph{The last step is symplectic type.}
+Its source $Y$ is a symplectic or orthogonal group over a prime field,
+say of order $r$, and its target dimension is $r^{d}$ with
+$2d+O(1)=n'$.
+First suppose $B$ is in the same characteristic.  Then $r=p_i$ and
+$f'=1$.  In a subfield first step the formula $1=\rho f_i b$, with
+$\rho$ in a fixed finite set of positive rationals, bounds both $f_i$
+and $b$; in the untwisted formula such a proper step is impossible.
+An ordinary extension degree in $B$ satisfies
+$b=\epsilon_i f_i\le2f_i$, because the target here has $f'=1$ and
+$\epsilon'=1$; a twisted-tensor exponent satisfies $b=f_i$.
+The ordinary geometric formulas give a fixed-height tower
+bound on $n'$ in terms of $a_i$.  If $B$ is nonlinear, its irreducible
+degree is at most $a_i^C$, giving the same conclusion.  Since
+$p_i\le a_i$, the formula $n_j=p_i^{d}$ gives
+$n_j\le\mathcal E_h(a_i^{C'})$ for fixed $h,C'$.  This contradicts
+Lemma~\ref{grw:towers}.
+
+It remains in this case that $B$ also crosses.  Let $b_{ij},s_{ij}$
+be the inner multipliers of $B,A$.  By
+\ref{grw:input-multipliers},
+\[
+ 1\le b_{ij}\le a_i^C,
+ \qquad c n_j^2\le s_{ij}\le C n_j^2,
+ \qquad e_{ij}=b_{ij}s_{ij}.
+\]
+The bound for $b_{ij}$ is the outer-order bound when $B$ is almost
+simple; in the symplectic-type case it follows from
+$b_{ij}\le C(n')^2\le a_i^{C'}$.
+For $i<j<k$, Lemma~\ref{grw:indices} now gives
+\[
+ c n_j^2\le e_{ij}
+ =\frac{b_{ik}s_{ik}}{b_{jk}s_{jk}}
+ \le\frac{C}{c}a_i^C.
+\]
+The factors involving the common upper dimension $n_k$ have canceled.
+This contradicts Lemma~\ref{grw:towers} on a sufficiently distant
+pair $i<j$.
+
+\smallskip\noindent
+\emph{The last step is almost simple irreducible.}
+Let its source be $Y$.  If $B$ crosses, then
+$n',f'\le a_i^C$ and $|P_B|\le a_i^C$: for an almost simple $B$ this
+uses $|P_B|\le a_i|\operatorname{Out}(T_i)|$, and for a symplectic-type
+$B$ it uses the preceding bound on its multiplier.  Equation
+\eqref{grw:normalizer-input} gives
+\begin{equation}\label{grw:small-M-first-cross}
+ |M_{ij}|\le a_i^{C'}.
+\end{equation}
+It is irrelevant here whether $A$ itself is a crossing.
+
+Suppose next that $B$ is in the same characteristic, so that $A$
+crosses.  If $B$ is not a subfield step, its geometric dimension
+formulas, or its irreducible degree bound when nonlinear, bound $n'$
+by a fixed-height tower in $a_i$.  Its field satisfies
+$\log q'\le C\log q_i$: restriction of scalars contributes the
+bounded factor $\epsilon_i/\epsilon'$, and the other non-subfield
+rows have their fixed field factors.  Thus $|Y|$ is bounded
+by a fixed-height tower in $a_i$, and
+$n_j\le |Y|^C$ contradicts Lemma~\ref{grw:towers}.
+
+Finally suppose $B$ is subfield.  Its rank equals the lower rank and
+its characteristic equals $p_i$.  Its inner projection has order at
+most $a_i^C$, but $f'$ can be unbounded.  The last crossing gives
+\[
+ \log n_j\ge c n' f'\log p_i,
+ \qquad f'\le C\log n_j.
+\]
+Equation~\eqref{grw:normalizer-input}, with $n'$ comparable to $n_i$,
+therefore gives
+\begin{equation}\label{grw:small-M-subfield}
+ |M_{ij}|\le a_i^{C'}(1+\log n_j)^{C'}.
+\end{equation}
+The same upper bound includes \eqref{grw:small-M-first-cross}.
+By Lemma~\ref{grw:towers}, after a fixed separation we have
+$a_i^{C'}\le n_j^{1/16}$; and, for sufficiently large rank,
+$(1+\log n_j)^{C'}\le n_j^{1/16}$.  Hence
+$|M_{ij}|\le n_j^{1/8}$.  This satisfies the dimension inequality of
+Lemma~\ref{grw:small-parabolic}, and
+$q_j^{c_0n_j}>n_j^{1/8}\ge |M_{ij}|$.  That Lemma gives the final
+contradiction.
+
+The three possibilities exhaust the last cover in a crossing path.
+\end{proof}
+
+\subsection{A common field and the exclusion of radicals}
+
+\begin{lemma}[Common defining field]\label{grw:common-field}
+On the remaining arbitrarily long chains, the endpoints can be taken
+to have one common family and one common field $q=p^f$.  The intermediate
+field ratios belong to a fixed finite set of positive rationals.  In
+particular, all extension and twisted-tensor degrees in these paths
+are bounded integers.  Every comparison satisfies
+\begin{equation}\label{grw:quadratic-growth}
+ n_j\ge c n_i^2.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Propositions~\ref{grw:no-geometric} and~\ref{grw:no-crossing} leave a
+defining-characteristic nonlinear cover on every comparison, with the
+common characteristic of the endpoints. Color by endpoint family
+and cover categories.  A nonlinear cover has a bounded rational field
+ratio; only the other cover can carry an exceptional prime degree.
+Lemma~\ref{grw:prime-colors} applied to $f_i$ therefore gives equal
+endpoint exponents.  The exceptional degree at the remaining cover
+must cancel a member of a fixed finite set of rational twisting ratios,
+so it is bounded.  The intermediate ratios consequently belong to a
+fixed finite set.
+
+The nonlinear cover increases natural dimension at least quadratically.
+All remaining geometric steps increase it by at least a fixed positive
+factor.  Thus its position as first or last cover makes no difference
+to \eqref{grw:quadratic-growth}, after decreasing $c$.
+\end{proof}
+
+\begin{lemma}[No parabolic cover]\label{grw:no-parabolic-cover}
+After a further homogeneous reduction, no selected shortcut contains
+a parabolic or subspace cover with a large unipotent radical.
+\end{lemma}
+
+\begin{proof}
+Use Lemma~\ref{grw:common-field}, and color by the position of the
+parabolic cover.  The other cover is nonlinear.
+
+If the parabolic cover is first, its active dimension is $n_i$ and its
+ambient dimension is comparable to $n_i$.  Its inner quotient is at
+most $q^{C n_i^2}$.  The multiplier of the last nonlinear cover is at
+most $|\operatorname{Out}(Y)|$, which is absorbed by this bound, since
+$n'$ is comparable to $n_i$ and $f'/f$ is bounded.  Hence
+\begin{equation}\label{grw:first-radical-upper}
+ e_{ij}\le q^{C_1 n_i^2}\qquad\hbox{for every }i<j.
+\end{equation}
+For $j<k$ the radical in its first cover also gives
+$e_{jk}\ge q^{c_1 n_j}$, with the bounded field ratio absorbed in $c_1$.
+By Lemma~\ref{grw:indices},
+\[
+ q^{c_1 n_j}\le e_{jk}\le e_{ik}\le q^{C_1 n_i^2}.
+\]
+Iterating \eqref{grw:quadratic-growth} through a fixed number of
+intermediate chain vertices makes $n_j/n_i^2$ tend to infinity as the
+minimum rank grows.  The displayed inequalities are incompatible.
+
+If the parabolic cover is last, write $e_{ij}=b_{ij}s_{ij}$, where
+$b_{ij}\le|\operatorname{Out}(T_i)|$ is the nonlinear multiplier and
+$s_{ij}$ is the parabolic multiplier.  For a fixed upper endpoint $k$,
+the values $s_{ik}$ have a uniformly bounded number of possibilities
+by \ref{grw:input-multipliers}.  List those values in increasing order
+and color $(i,k)$ by its position in that list.  This gives a uniform
+finite coloring even though the lists for different $k$ may be
+different.  On a homogeneous triple, $s_{ik}=s_{jk}$, and therefore
+\[
+ e_{ij}=\frac{e_{ik}}{e_{jk}}
+       =\frac{b_{ik}}{b_{jk}}
+       \le |\operatorname{Out}(T_i)|.
+\]
+But the parabolic radical on the pair $(i,j)$ gives
+$e_{ij}\ge q^{c_2n_j}$.  The classical outer-order bound is
+\[
+ |\operatorname{Out}(T_i)|\le C n_i f
+                         \le q^{C_2(1+\log n_i)},
+\]
+so even \eqref{grw:quadratic-growth} contradicts these inequalities
+at sufficiently large rank.
+\end{proof}
+
+\begin{lemma}[Inner order bound]\label{grw:inner-order}
+For the remaining shortcuts there is an absolute $C$ such that
+\begin{equation}\label{grw:inner-small}
+ |P_{ij}|\le q^{C n_i^2}.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+There is at least one nonlinear cover.  For a remaining geometric
+cover with active label $T$ and natural dimension $n$, the other
+classical factors have bounded natural dimension and are soluble.
+Their orders, scalar factors, and bounded component-permutation factors
+are bounded by
+\[
+ q^{C_0}(1+n)^{C_0}(1+f)^{C_0}.
+\]
+Field and twisted-tensor degrees are bounded by
+Lemma~\ref{grw:common-field}; their semilinear factors satisfy the same
+bound.  These are precisely the block, form, tensor, and field
+structures left after Lemma~\ref{grw:no-parabolic-cover}; no unipotent
+radical or extra nonabelian factor is being omitted from the estimate.
+
+If the geometric cover is first, its intermediate natural dimension
+is at most $C_1(1+n_i)^{b_1}$ for fixed constants.  Its multiplier has
+the displayed bound in the lower parameters, and the last nonlinear
+multiplier is at most $|\operatorname{Out}(Y)|\le Cn'f'$.
+Since $|T_i|\le q^{C n_i^2}$ and $f\le q$, their product satisfies
+\eqref{grw:inner-small}.
+
+If the nonlinear cover is first, its intermediate degree is at most
+$a_i^{C_2}$ and $f'/f$ is bounded.  The geometric multiplier is bounded
+by $q^{C_3}(1+n')^{C_3}(1+f')^{C_3}$, which is at most
+$q^{C_4 n_i^2}$.  Multiplying by $a_i$ and the first nonlinear
+outer multiplier proves the same bound.  With two nonlinear covers
+only the two outer multipliers occur, and the latter argument applies
+without the geometric factor.
+\end{proof}
+
+\begin{proposition}[The remaining paths]\label{grw:remaining-paths}
+If the reduced chains of Proposition~\ref{cls:reduction} have arbitrary
+length, then they still have arbitrary length subject to all of the
+following properties.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item The endpoints have a common classical family and field $q=p^f$,
+their natural dimensions tend to infinity, and their common factor
+multiplicity is $m$.  Every selected shortcut has one active factor
+at each cover and no other nonabelian factor in its inner projection.
+
+\item Every selected two-edge shortcut contains a defining-characteristic
+nonlinear cover.  Its representation is a single Frobenius digit after
+the twisted-tensor geometric alternative has been separated.  The
+other cover is another such nonlinear cover, or a block, form, tensor,
+extension-field, subfield, or twisted-tensor cover.  No crossing,
+symplectic-type cover, or parabolic cover with a large unipotent radical
+remains.  The characteristic-two nonsingular-line form inclusion is
+retained, with its natural simple quotient convention.
+
+\item There are fixed constants $b,C$ and a fixed finite set
+$\mathcal R\subset\mathbb Q_{>0}$ such that every intermediate exponent
+satisfies $f'/f\in\mathcal R$, every field-extension or twisted-tensor
+degree is at most $b$, and every inactive geometric natural dimension
+is at most $b$.  In particular each remaining geometric cover satisfies
+$n_{\mathrm{target}}\le C(1+n_{\mathrm{source}})^b$.
+
+\item There are fixed $c,C>0$ such that
+\[
+ n_j\ge c n_i^2,\qquad |P_{ij}|\le q^{C n_i^2}.
+\]
+For any prescribed exponent $d$, fixed-spacing thinning gives also
+$n_j\ge n_i^d$ for every retained pair, after raising the minimum rank.
+
+\item The coordinate normal tests of
+Lemma~\ref{grw:coordinate-normal-tests} remain available for every tested proper
+coordinate overgroup.  After one more fixed-spacing thinning,
+$M_{ij}$ normalizes no proper parabolic of $T_j$.
+\end{enumerate}
+\end{proposition}
+
+\begin{proof}
+Parts (i)--(iv) collect Propositions~\ref{grw:no-geometric} and
+\ref{grw:no-crossing} and Lemmas~\ref{grw:common-field},
+\ref{grw:no-parabolic-cover}, and \ref{grw:inner-order}.
+For the spacing assertion put $x_i=c n_i$.  Then
+\eqref{grw:quadratic-growth} says $x_j\ge x_i^2$ on every consecutive
+comparison.  After $s$ comparisons,
+$n_j\ge c^{-1}(c n_i)^{2^s}$.  Choose $2^s>d$ and then increase the
+minimum rank to obtain $n_j\ge n_i^d$.
+
+The coordinate normal test is inherited from the original tested
+intervals and is not altered by deleting chain vertices.  Every proper
+parabolic of a large-rank classical group has unipotent radical of
+order at least $q^{c_3 n_j}$ for an absolute $c_3>0$.
+One can see the uniform bound by choosing a maximal parabolic
+containing it: the omitted simple root has at least the relative rank
+many positive-root coordinates outside the Levi subsystem.  Their
+product contributes at least $q^{c_3 n_j}$; unitary root orbits can
+only increase the corresponding root-group orders.  The radical of
+the smaller parabolic contains that of the maximal one.
+Fixed-spacing thinning makes $c_3n_j>Cn_i^2$ uniformly.
+If $M_{ij}$ normalized such a parabolic $J$ with radical $R$, use
+\eqref{grw:normal-enlargement}.  Since
+$P_{ij}R\le N_{T_j}(J)<T_j$, its full inverse image is proper in
+$z_j$.  The soluble product $\widehat R\trianglelefteq\overline U$
+then satisfies \eqref{grw:normal-containment}.  In particular
+$R\le P_{ij}$,
+contrary to $|R|\ge q^{c_3 n_j}>q^{C n_i^2}\ge|P_{ij}|$.
+This proves (v).
+\end{proof}
+
+The Proposition makes no assertion that a unique nonabelian composition
+factor alone produces a quasisimple subgroup.  The actual components
+and their compatible rational maps will be obtained from the retained
+geometric structures in the next section.  In particular, the
+characteristic-two form case is still among those structures and
+requires its own form-preservation argument.
diff --git a/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/sections/introduction.tex b/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/sections/introduction.tex
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@@ -0,0 +1,243 @@
+\section{Introduction}\label{sec:introduction}
+
+The congruence lattice of an algebra records all of its quotients at once.
+The representation theorem of Gr\"atzer and Schmidt says that every
+algebraic lattice occurs in this way \cite[Theorem~I, p.~34]{GratzerSchmidt1963}.
+The finite version asks for a finite representing algebra whenever the
+given lattice is finite. The difference is substantial: a finite
+congruence lattice may be represented by an infinite algebra without
+providing any finite representation.
+
+Here a finite algebra is a finite nonempty set equipped with finitely many
+total operations, each of finite arity. The signature is allowed to
+depend on the lattice. Nullary operations and the empty signature are
+allowed. We write $\Con(A)$ for the \emph{entire} lattice of congruence
+relations on $A$, ordered by inclusion.
+Allowing infinitely many finitary operations would give no additional
+finite representations. Indeed, a finite carrier has only finitely many
+equivalence relations. For each relation that is not a congruence, retain
+one operation witnessing its failure of compatibility. The resulting
+finite signature has exactly the same congruence lattice.
+
+\begin{theorem}\label{main:congruences}
+There exists a finite nonempty lattice $L$ such that
+\[
+ L\not\cong\Con(A)
+\]
+for every finite nonempty algebra $A$ of every finite signature.
+\end{theorem}
+
+Thus the finite lattice representation problem has a negative answer.
+The theorem concerns unrestricted finite universal algebras; it is not
+a restriction to a variety, a bound on the carrier size, or an obstruction
+to a particular representation method.
+
+P\'alfy and Pudl\'ak proved that the following two universal assertions
+are equivalent \cite[Theorem~2]{PalfyPudlak1980}:
+every finite lattice is the congruence lattice of a finite algebra, and
+every finite lattice is an interval in the subgroup lattice of a finite
+group. We establish the failure of the second assertion.
+
+\begin{theorem}\label{main:group}
+There exists a finite lattice that is not isomorphic to $[D,G]$ for any
+subgroup $D$ of any finite group $G$.
+\end{theorem}
+
+Here $[D,G]=\{X:D\le X\le G\}$ carries subgroup inclusion. The equivalence
+is global: Theorem~\ref{main:group} implies
+Theorem~\ref{main:congruences}, without asserting that a prescribed
+individual lattice has the two representation properties simultaneously.
+The one-element lattice causes no exception; it is represented by a
+one-element algebra and by a trivial subgroup interval.
+
+The constructed subgroup-interval obstruction is self-dual.  It also
+cannot be the entire lattice of intermediate fields of a finite separable
+extension $E/F$.  Indeed, a
+finite normal closure $N/F$ gives an order anti-isomorphism with
+\[
+ [\operatorname{Gal}(N/E),\operatorname{Gal}(N/F)],
+\]
+by the Galois correspondence; self-duality would turn such a field
+representation into a forbidden subgroup-interval representation.
+This is the group--field correspondence discussed in
+\cite[Section~3]{PalfyTwisted}.
+
+\subsection{History and the subgroup-interval approach}
+
+The finite problem has several distinct formulations: existence of a
+representation, finding a representing algebra, and deciding from a finite
+order table whether a representation exists.  McNulty's account separates
+these questions and places them in their universal-algebraic setting
+\cite[pp.~29--30]{McNulty2015}.  We address the existence question by
+ruling out a finite subgroup interval.  The global transfer just stated
+is essential: the decorated lattice constructed below is a subgroup-interval
+obstruction, and we do not identify it pointwise with the lattice asserted
+by Theorem~\ref{main:congruences}.
+
+Two positive representation results clarify the finite requirement.
+Pudl\'ak and T\r{u}ma proved that every finite lattice embeds, preserving
+meets and joins, into the partition lattice of a finite set
+\cite{PudlakTuma1980}.  Such an embedding need not realize the given
+lattice as the entire congruence lattice of operations on that set.
+On the group side, Repnitski{\u{\i}} and T\r{u}ma represented every finite
+lattice as an interval in the subgroup lattice of a countable locally
+finite group \cite{RepnitskiiTuma2008}.  Local finiteness still permits
+the ambient group to be infinite.
+
+The subgroup-interval approach exposes normal subgroups and primitive
+sections to lattice constraints.  Baddeley and Lucchini developed the
+reduction of height-two interval questions to finite simple-group problems
+\cite{BaddeleyLucchini1997}; Baddeley extended the class of lattice
+configurations under consideration \cite{Baddeley1998}.
+Further interval reductions were developed by B\"orner and Aschbacher
+\cite{Borner1999,Aschbacher2008}.  These results have different hypotheses;
+P\'alfy's comparison explains their relationship and the role of almost
+simple groups and twisted wreath products \cite[Section~4]{PalfyTwisted}.
+In particular, the invariant-subdirect and homomorphism-extension viewpoint
+in \cite{PalfyTwisted} is the starting structure for our final reduction.
+Aschbacher's minimum-order and signalizer-lattice viewpoint is a related
+predecessor of the complementary-socle and extension-kernel arguments.
+
+Lattice shapes can enforce group-theoretic conditions; DeMeo develops
+this general perspective and the combination of such requirements
+\cite{DeMeoInterval2014}.  Here the particular shapes must do two jobs
+simultaneously.  They force uniform restrictions on every possible
+simple-group-labelled chain, and they retain the Boolean joins on which
+the final extensions must agree.  Uniformity is the obstacle: every
+lattice in the construction is finite, but a hypothetical representing
+group has no prescribed bound on its order, Lie rank, or field size.
+
+\subsection{The proof in two stages}
+
+Our construction has two stages. The first is a uniform rigidity result
+for finite configurations of subgroups. We attach a nonabelian simple
+type $T_X$ to selected vertices $X$.  If $D$ is the common bottom,
+the unique minimal normal subgroup of $X/\operatorname{core}_X(D)$
+is a direct power of $T_X$; here $\operatorname{core}_X(D)$ is the
+largest normal subgroup of $X$ contained in $D$.
+Special intervals, called fences, test normal subgroups.  Equal labels
+give subdirect powers, in which selected simple coordinates are linked
+diagonally.  At the comparisons used below, different labels give
+coordinate products that can be projected to subgroup intervals of
+almost simple groups.
+Long chains are supplied with two-edge shortcuts and further interval
+tests that make these projections available.  The crucial conclusion
+is an \emph{absolute} upper bound
+on the length of such a chain when its endpoint labels differ.
+The bound must be independent of the finite group in which the
+configuration is represented.
+
+The classification of finite simple groups enters this stage through
+maximal-subgroup structure and representation theory. A subgroup maximal
+in a maximal subgroup of an almost simple group has at most five nonabelian chief
+factors \cite[Proposition~8.1]{BurnessLiebeckShalev2017}.
+We combine that bound with the fact that coordinate stabilizers in the
+common bottom already induce the full local permutation actions on the
+simple factors carrying the lower labels.
+An elementary bounded-depth action argument then
+controls the union of nonabelian simple types in the relevant projected
+coordinate stabilizers, rather than just their number at one vertex. Decreasing
+bottom cores account for the nonabelian layers lost between socles. These are the
+uniformity statements needed before applying finite Ramsey theory.
+
+The remaining classical-group comparisons are controlled by field
+parameters, dimension growth, and irreducible restrictions.
+The structural starting points are Aschbacher's maximal-subgroup
+classification and its classical-group formulations
+\cite{Aschbacher1984,KleidmanLiebeck1990}, together with Larsen--Pink's
+fixed-type envelope theory \cite{LarsenPink2011}.
+Landazuri--Seitz degree bounds separate cross-characteristic growth
+\cite{LandazuriSeitz1974,Hasa2014}; Steinberg's representation theory and
+L\"ubeck's restricted small-degree classification handle the defining
+characteristic \cite{Steinberg1963,SteinbergNotes,Luebeck2001}.
+The paper derives the required bounds on whole chains from these inputs;
+none of these results alone bounds the lengths of the tested chains.
+For defining-characteristic representations, the relevant action on
+absolute constituent types is the difference between a source field
+shift and an ambient semilinear shift. Their synchronization bounds
+composition length independently of the field degree.
+After another Ramsey argument, natural modules at later endpoints
+restrict irreducibly to earlier ones.  The irreducible-triple input
+comes from Seitz's classification and the corrections of Cavallin and
+Testerman, in the formulation of Burness and Testerman
+\cite{Seitz1987,CavallinTesterman2019,BurnessTesterman2018}.
+We use it twice: first with a bound on the composition length of the
+ambient natural module on the smallest subgroup
+(Lemma~\ref{irr:rank-length}), and then with an intervening simple
+subgroup of strictly larger rank whose natural module is irreducible
+on that smallest subgroup (Lemma~\ref{irr:intervening}).
+These hypotheses control the twisted-diagonal spin families.
+The resulting rank bounds are incompatible with a sufficiently long
+chain and its two-edge shortcuts.
+
+In the second stage, we build one finite lattice from a Boolean lattice
+of rank four and a small additional configuration, called the detector.
+The chain tests are inserted in both required orders while preserving
+the original Boolean meets and joins.  Their length is chosen from the
+absolute bound, before any representing group is considered.
+
+Suppose the resulting lattice is represented by $[D,G]$, with $|G|$
+minimal among representations in either orientation.  The global fence
+gives a socle $S\cong T^I$ and $G=DS$; minimality gives $D\cap S=1$.
+Equality of the forward labels forces every proper nonzero original
+Boolean vertex $Y$ to meet $S$ in a subdirect subgroup, meaning one
+which projects onto every simple coordinate.  Fixing a coordinate
+gives its stabilizer $A\le D$ and an action
+\[
+                   \alpha:A\longrightarrow\Aut(T).
+\]
+A $D$-invariant subdirect subgroup corresponds to a subgroup $U$ with
+$A\le U\le D$ and a homomorphism $\beta:U\to\Aut(T)$ extending
+$\alpha$.  This correspondence reverses inclusion.  It realizes the
+reversed upper filter above $Y$ as the actual group interval $[A,U_Y]$.
+
+The reverse chain tests make the socles of
+$U_Y/\operatorname{core}_{U_Y}(A)$ powers of one simple group $R$,
+which need not equal $T$.
+The detector ensures $\alpha(A)\ge\Inn(T)$.  Read the original
+Boolean copy in reverse order.  Proper Boolean unions give extensions
+on generated domains that agree on their overlaps.  For a full union,
+the corresponding forward vertices meet at $D$, so their intersections
+with $S$ have trivial intersection.  A common extension would produce
+a nontrivial subdirect subgroup in this intersection and is therefore
+forbidden.
+
+The final argument forces that forbidden extension.  If the core of
+$A$ in an atom domain had trivial image, compatible lifts of socle
+factors on the three rank-two faces through that atom would already
+produce a common extension.  Hence every atom core has image containing
+$\Inn(T)$.  Intersections give the same
+property for larger proper faces.  A least normal subgroup of $A$
+with this inner image is then normal in all the relevant domains;
+conjugation on its quotient isomorphic to $T$ supplies their common
+extension.  This is the contradiction.
+
+\subsection{Structure and scope of the proof}
+
+Section~\ref{sec:framework} develops the lattice tests and factor
+transport. The central group-theoretic result is
+Theorem~\ref{cls:chain-rigidity}, stated in
+Section~\ref{sec:classification} and proved in
+Sections~\ref{sec:classification}--\ref{sec:rigidity}.
+These sections control, successively, factor actions and simple types,
+field and dimension growth, algebraic lifts along paths,
+natural-module composition length, and irreducible restrictions.
+Section~\ref{sec:lattices} constructs a finite lattice realizing all
+the tests. Sections~\ref{ext:section} and~\ref{glue:section} reduce a
+hypothetical representation to an extension configuration and rule
+it out by Boolean gluing, proving Theorem~\ref{main:group}.
+
+The finite lattice is specified by finitely many private-branch
+insertions with a sufficiently large finite chain parameter.
+An explicit numerical value of that parameter is unnecessary for the
+existence conclusion. Its existence follows from bounds that are
+uniform over all possible finite-group representations.
+
+All groups are finite unless called algebraic groups. Algebraic
+groups occur only in the representation-theoretic part of the proof
+and are taken over algebraically closed fields of positive
+characteristic, with central covers and the stated special isogenies
+when needed. We use the classification of finite simple groups and
+identify each additional classification or representation input at
+its point of application. The new uniformity, lifting, lattice
+bookkeeping and gluing deductions are proved explicitly.
diff --git a/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/sections/lattices.tex b/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/sections/lattices.tex
new file mode 100644
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+++ b/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/sections/lattices.tex
@@ -0,0 +1,562 @@
+\section{A finite lattice carrying all the tests}
+\label{lat:section}
+\label{sec:lattices}
+
+We now construct the lattice to which Theorem~\ref{cls:chain-rigidity}
+will be applied.  All choices in this section precede the choice of a
+representing group.  In particular, the length of a test chain will be
+chosen using the absolute bound in that Theorem.
+
+Throughout this section, an \emph{original} vertex means a vertex of one
+of the two small lattices specified below, before any insertion.  An
+original atom of a Boolean lattice need not remain an atom after
+insertions; its original meets and joins will, however, be preserved.
+
+\subsection{Two original components and their requested tests}
+
+There are two original components, each with endpoints $0,1$.
+For each of the requested pairs below, we shall insert a chain and the
+tests from Definition~\ref{cls:tested-chain}.  A forward pair is read
+in the displayed order, with bottom $0$; a reverse pair is read in
+the dual order, with bottom $1$.
+
+The \emph{Boolean component} is the Boolean lattice $B_4$ of subsets
+of a fixed four-element set.  Its forward requested pairs are
+\begin{equation}
+\label{lat:boolean-forward}
+   (v,1)\qquad(0<v<1\text{ original}).
+\end{equation}
+Its reverse requested pairs are all
+\begin{equation}
+\label{lat:boolean-reverse}
+   (u,v)\qquad(0<v<u<1\text{ original}),
+\end{equation}
+where a reverse pair is read in the reverse order: its test starts at
+$u$ and ends at $v$.  Thus every reverse test is contained in a
+forward interval with two proper original endpoints.
+
+The \emph{detector component} has a distinguished original vertex
+$X$.  We choose its lower half so that every coatom below $X$ contains
+one of two designated vertices $d_1,d_2$.  It is the horizontal sum of
+the two chains
+\begin{equation}
+\label{lat:detector-lower}
+   0<p_i<d_i<q_i<X\qquad(i=1,2),
+\end{equation}
+and its upper half is the horizontal sum of the two chains
+\begin{equation}
+\label{lat:detector-upper}
+   X<e_i<z_i<f_i<1\qquad(i=1,2).
+\end{equation}
+The two halves are joined at $X$, and we put $z=z_1$.
+Its forward requested pairs are
+\begin{equation}
+\label{lat:detector-forward}
+   (d_1,1),\quad(d_2,1),\quad(X,1),
+\end{equation}
+and its only reverse requested pair is $(z,X)$.
+Both $[0,X]$ and $[X,1]$ are already fenced, with two separate
+components containing comparable pairs.
+
+\subsection{The lattice properties required in the reduction}
+
+The forward requests will equate the indicated simple labels to the
+global top label.  The reverse requests will equate labels inside
+reversed upper filters.  These local representations are the ones
+that will arise in Section~\ref{ext:section}; a representation of the
+entire dual lattice is not needed.  The construction will prove the
+following statement while preserving the original Boolean operations.
+
+\begin{proposition}[The decorated lattice]
+\label{lat:decorated-lattice}
+There is a finite nonempty lattice $L$ with the following properties.
+The properties hold in either orientation of $L$, using the
+corresponding Boolean and detector components.
+\begin{enumerate}[label=\textnormal{(\roman*)}]
+\item The lattice contains a bounded copy of the original $B_4$.
+  Every proper nonzero original Boolean vertex $v$ has both $[0,v]$
+  and $[v,1]$ fenced.  The global interval $[0,1]$ is fenced.
+\item There are original detector vertices $d_1,d_2,X,z$ with
+  $0<d_i<X<z<1$.  The intervals $[0,d_i]$, $[0,X]$, $[X,1]$ and
+  $[z,1]$ are fenced.  Every coatom of $[0,X]$ is above a $d_i$.
+\item In any finite subgroup-interval representation $L\cong[D,G]$,
+  the simple labels of every proper nonzero original Boolean vertex,
+  of $d_1,d_2$, and of $X$ equal the simple label of $G$.
+\item Let $0<v<u<1$ be comparable original Boolean vertices.  In any
+  finite subgroup-interval representation
+  \[
+        [v,1]^{\mathrm{op}}\cong[A,U_v],
+  \]
+  the labels of the vertices corresponding to $u$ and $v$, taken
+  relative to the common bottom $A$, are defined and equal.
+  The same assertion holds for the detector pair $(z,X)$ in any
+  representation $[X,1]^{\mathrm{op}}\cong[A,U_X]$.
+\end{enumerate}
+In (iv), no representation of the entire dual lattice is assumed.
+If several such upper filters have compatible realizations with a
+common bottom $A$, the conclusions hold in that compatible system.
+\end{proposition}
+
+The remainder of this section constructs this lattice.  We first show
+how to add the chain tests while protecting their covering relations,
+then combine the forward and reverse tests and choose their length.
+
+\subsection{Private insertions and their persistence properties}
+
+Let $a<b$ in a finite lattice $P$, and let $Q$ be a finite bounded
+lattice whose bottom and top are identified with $a$ and $b$.
+A \emph{private insertion of $Q$ into $[a,b]$} retains the old orders
+and introduces no comparisons between an old vertex and a fresh
+interior vertex except those forced through $a$ and $b$.
+
+\begin{lemma}[Private insertion]
+\label{lat:insertion}
+A private insertion gives a finite lattice and preserves all old meets
+and joins.  If $w$ is a fresh interior vertex and $v$ is an old vertex,
+then
+\begin{equation}
+\label{lat:cross-order}
+ v\leq w\quad\Longleftrightarrow\quad v\leq a,
+ \qquad
+ w\leq v\quad\Longleftrightarrow\quad b\leq v.
+\end{equation}
+An old cover $c\prec d$ is destroyed by the insertion if and only if
+the insertion has endpoints $a=c$, $b=d$ and has a nonempty interior.
+\end{lemma}
+
+\begin{proof}
+The prescribed order is transitive: every route from an old vertex into
+the fresh interior passes through $a$, and every route back passes
+through $b$.  This proves \eqref{lat:cross-order} and antisymmetry.
+For an old vertex $v$ incomparable with a fresh interior vertex $w$,
+the cross operations are
+\[
+ v\wedge w=v\wedge a,
+ \qquad v\vee w=v\vee b,
+\]
+where the operations on the right are those of $P$.  In the comparable
+cases, the smaller and larger vertices give the meet and join.
+For two fresh vertices, the operations in $Q$ remain their operations:
+an old common lower bound lies below $a$, and an old common upper bound
+lies above $b$.  An analogous argument shows that an old meet or join
+cannot be changed by a new bound.  This proves the lattice assertion.
+
+If a fresh vertex $w$ lies strictly between the old cover $c\prec d$,
+then \eqref{lat:cross-order} gives $c\leq a<b\leq d$.
+The cover condition forces $a=c$ and $b=d$.  Conversely, an insertion
+with those endpoints and a fresh interior vertex destroys the cover.
+\end{proof}
+
+A \emph{chain insertion with $k$ interior vertices} means a private
+insertion of a chain with exactly $k$ fresh interior vertices.  A
+\emph{fencing insertion} means two separate chain insertions, each
+with exactly two fresh interior vertices, into the same interval.
+In particular, these are nontrivial fences in the sense used in
+Lemma~\ref{frm:normal-test}.
+
+\begin{lemma}[Persistent fences]
+\label{lat:persistent-fence}
+Suppose the proper part of an interval $[a,b]$ has two different
+components in its comparability graph, each containing two strictly
+comparable vertices.  Then $[a,b]$ is fenced.  This property persists
+under any finite sequence of private insertions into the surrounding
+lattice.  In particular, a fencing insertion makes an interval fenced
+permanently.
+\end{lemma}
+
+\begin{proof}
+Vertices in different components of the proper comparability graph
+have meet $a$ and join $b$.  Indeed, a common lower bound strictly above
+$a$, or a common upper bound strictly below $b$, would connect them.
+For a given interior vertex, choose the two comparable vertices in one
+of the two specified components different from its own.  They are its
+required complements.
+
+Consider an insertion into $[c,d]$.  A fresh vertex can belong to
+$[a,b]$ only if $a\leq c<d\leq b$, by
+Lemma~\ref{lat:insertion}.  If both $c,d$ are proper vertices of
+$[a,b]$, they lie in the same comparability component.  All comparisons
+from the new interior to the old proper part then enter that component.
+The same is true if just one of $c,d$ is proper.  If $c=a$ and $d=b$,
+the new interior has no comparisons with the old proper part.
+Thus no insertion merges two old components.  The two reserved
+comparable pairs survive, proving persistence.  A fencing insertion
+supplies two such separate components.
+\end{proof}
+
+We shall make several insertions in a \emph{batch}.  This means a
+finite list of private insertions all of whose endpoints were present
+before the batch.  Perform them in any fixed order.  All their fresh
+vertices are distinct, and Lemma~\ref{lat:insertion} applies at every
+step.  This convention makes each batch a completely specified finite
+operation; it does not call for repeating the batch on its new vertices.
+
+\subsection{The shape of one chain test}
+
+Fix an integer $N\geq2$ and two existing vertices $a<b$.  The following
+five batches constitute the \emph{shape construction} for a test on
+$(a,b)$.  All sets of vertices appearing in a later batch are the named
+sets from the earlier batches, not all vertices then present.
+
+\begin{enumerate}[label=\textnormal{(\arabic*)}]
+\item Insert a chain with $N-1$ fresh interior vertices and name its
+  vertices
+  \[
+    a=x_0<x_1<\cdots<x_N=b.
+  \]
+\item For each $0\leq i<j\leq N$, insert one fresh vertex $v_{ij}$
+  privately into $[x_i,x_j]$.  The designated shortcut is
+  $x_i\prec v_{ij}\prec x_j$.
+\item Make fencing insertions into every interval in the following
+  finite set:
+  \[
+    \bigl\{[x_i,x_j]:0\leq i<j\leq N\bigr\}
+    \ \cup\
+    \bigl\{[a,v_{ij}]:1\leq i<j\leq N\bigr\}.
+  \]
+\item Define the set of upper targets
+  \begin{equation}
+  \label{lat:targets}
+    \mathcal U=
+      \{x_j:1\leq j\leq N\}
+      \cup\{v_{ij}:1\leq i<j\leq N\}.
+  \end{equation}
+  For every $u\in\mathcal U$, insert privately into $[a,u]$ the
+  bounded lattice with fresh vertices $m_u,t_{u,1},\ldots,t_{u,16}$
+  and relations
+  \[
+    a<m_u<t_{u,k}<u\quad(1\leq k\leq16),
+  \]
+  with the $t_{u,k}$ pairwise incomparable.  Thus $a<m_u$ and
+  $[m_u,u]$ is exactly $M_{16}$ at this stage.
+\item For every pair $(a,u)$ with $u\in\mathcal U$, make two separate
+  chain insertions with one interior vertex.  Also do so for every
+  pair $(c,b)$ with
+  \begin{equation}
+  \label{lat:saturated-principals}
+    c\in\mathcal C=
+      \{x_i:0\leq i<N\}
+      \cup\{v_{ij}:0\leq i<j<N\}.
+  \end{equation}
+  Repeated pairs may be listed just once.  The two new vertices for
+  a pair $(c,d)$ are its reserved coatom witnesses: they are coatoms
+  below $d$, and their meet is $c$.
+\end{enumerate}
+
+The vertices $x_i$ and $v_{ij}$ are called the \emph{principal vertices}
+of the test.  The vertices $m_u,t_{u,k}$ and the reserved coatom
+witnesses are not principal vertices and are not used as endpoints of
+any subsequent shape insertion for this test.
+
+\begin{lemma}[A finite shape construction]
+\label{lat:shape}
+The five batches are finite and have the following properties.
+\begin{enumerate}[label=\textnormal{(\roman*)}]
+\item Every designated shortcut has both edges as covers.
+\item Every $[x_i,x_j]$ is fenced.  For every $u\in\mathcal U$,
+      $[a,u]$ is fenced, $a$ is a meet of coatoms below $u$, and
+      $a<m_u$ with $[m_u,u]\cong M_{16}$.
+\item Every vertex in $\mathcal C$ is a meet of coatoms below $b$.
+      Each $v_{iN}$ is itself a coatom below $b$.
+\end{enumerate}
+No interval $[a,v_{0j}]$ is included in the extra target operations.
+The construction introduces at most $27N^2$ vertices.
+\end{lemma}
+
+\begin{proof}
+Each shortcut edge is a cover immediately after its one-vertex
+insertion.  By the cover criterion in Lemma~\ref{lat:insertion}, a
+later shape insertion could destroy it only by using that exact edge
+as its endpoint pair.  The listed pairs never do so.  The only apparent
+possibilities would be $[a,v_{0j}]$, which is excluded from
+\eqref{lat:targets}, and $[v_{iN},b]$, which is excluded from
+\eqref{lat:saturated-principals}.  This proves (i).
+
+All pair fences are present in the third batch and persist by
+Lemma~\ref{lat:persistent-fence}.  The chain targets $x_j$ already have
+the fence on $[a,x_j]$ from that batch; the other upper targets are
+listed there expressly.  Each reserved coatom witness is fresh when
+inserted and is not an endpoint of a subsequent shape insertion.
+Its edge to the upper endpoint therefore survives.  Its meet with its
+companion remains the lower endpoint, by
+Lemma~\ref{lat:insertion}.
+
+When an $M_{16}$ branch is inserted, its interval $[m_u,u]$ has no
+other vertices: an old vertex above $m_u$ must be above $u$.
+A subsequent shape insertion could add a vertex inside this interval
+only if both its endpoints lay in $[m_u,u]$.  None of $m_u,t_{u,k}$
+is used subsequently as a shape endpoint.  The remaining possible
+endpoint is $u$, which cannot by itself form a nondegenerate interval
+inside $[m_u,u]$.  Thus this interval remains exactly $M_{16}$.
+These arguments give (ii) and (iii).
+
+For completeness, the numbers of new vertices in the first four
+batches are respectively
+\[
+ N-1,\qquad \frac{N(N+1)}2,\qquad 4N^2,
+ \qquad \frac{17N(N+1)}2.
+\]
+The fifth batch adds at most $2N(N+1)$ vertices.  Their sum is
+$15N^2+12N-1\leq27N^2$.
+\end{proof}
+
+The exclusions involving $v_{0j}$ and $v_{iN}$ are necessary.
+Inserting a fence into either of their designated cover edges would
+destroy the two-edge shortcut.  The target set
+\eqref{lat:targets} is exactly the one required by
+Theorem~\ref{cls:chain-rigidity}; no test at a target $v_{0j}$ is
+needed there.
+
+\subsection{Combining the forward and reverse tests}
+
+Here is the complete construction for either original component $C$.
+First perform the five shape batches for all its forward requested
+pairs, using fresh vertices for every pair.  One may perform a batch
+for all pairs before passing to the next batch.  Then apply the same
+five batches in the dual lattice for all reverse requested pairs,
+again with disjoint fresh vertices.  In each case a
+private insertion is made in the order currently being used; its dual
+is a private insertion in the forward lattice.
+
+Let $C^{\mathrm{sh}}_N$ be the finite lattice now obtained, and freeze
+its vertex set.  Define two subsets of this frozen set:
+\begin{align}
+\label{lat:label-sets}
+ \mathcal F_C
+   &=\bigcup_{(a,1)\text{ forward requested}}
+          [a,1]_{C^{\mathrm{sh}}_N},\\
+ \mathcal R_C
+   &=\bigcup_{(u,v)\text{ reverse requested}}
+          [v,u]_{C^{\mathrm{sh}}_N}.
+\end{align}
+The following is a \emph{single final batch}, with every endpoint
+chosen from this frozen set:
+\begin{itemize}
+\item make a fencing insertion into $[0,t]$ for every
+  $t\in\mathcal F_C\setminus\{1\}$;
+\item make a fencing insertion into $[t,1]$ for every
+  $t\in\mathcal R_C$.
+\end{itemize}
+In the detector omit the first insertion when $t=X$, since $[0,X]$
+is already fenced.  No insertion into $[0,X]$ is made anywhere in the
+construction.  Denote the resulting component by $C_N$.
+
+We emphasize that the new vertices of this final batch are not added
+to $\mathcal F_C$ or $\mathcal R_C$.  The next lemma proves that no
+such closure operation is needed.
+
+\begin{lemma}[Simultaneous tests and the final batch]
+\label{lat:simultaneous}
+For either component $C_N$, all the following statements hold.
+\begin{enumerate}[label=\textnormal{(\roman*)}]
+\item Every forward shape test retains all conclusions of
+  Lemma~\ref{lat:shape}.  Its principal vertices other than $1$ have
+  fenced intervals from $0$.
+\item Every reverse shape test retains all conclusions of
+  Lemma~\ref{lat:shape} in the reverse order.  Every vertex in each
+  of its comparison intervals has a fenced interval from the reverse
+  base $1$.
+\item For a forward chosen chain and any $i<j$, every vertex
+  $t\in[x_i,x_j]$, except possibly $t=1$, satisfies one of the
+  following alternatives:
+  \begin{enumerate}[label=\textnormal{(\alph*)}]
+  \item $[0,t]$ is fenced;
+  \item there is a vertex $s$ with $x_i\leq s<t$, with $[0,s]$
+    fenced, such that every vertex of $[0,t]$ is comparable to $s$.
+  \end{enumerate}
+\item In the detector the coatoms of $[0,X]$ are exactly $q_1,q_2$.
+  In particular, each contains one of $d_1,d_2$ below it.  The
+  intervals $[0,X]$ and $[X,1]$ remain fenced.
+\end{enumerate}
+The construction is finite.  If $C$ has $q_0$ vertices and altogether
+$h$ requested pairs in the two directions, then
+\[
+             |C_N|\leq9(q_0+27hN^2).
+\]
+\end{lemma}
+
+\begin{proof}
+We first record how the two shape constructions are separated.  Every
+fresh vertex of a forward shape construction lies below no proper
+original vertex.  This follows inductively from
+\eqref{lat:cross-order}: its upper insertion endpoint is either $1$
+or an earlier fresh forward vertex with the same property.  In
+contrast, every vertex of a reverse shape construction lies in a
+forward interval $[v,u]$ with $0<v<u<1$ original.  Thus every vertex
+of $\mathcal R_C$ lies below a proper original upper bound, and no
+fresh forward vertex belongs to $\mathcal R_C$.
+
+These observations also show that the shape constructions do not
+spoil each other's designated intervals.  Different forward tests
+have disjoint fresh interiors; their only shared shape endpoints are
+original vertices and $1$.  Hence neither uses a fresh endpoint of
+the other's designated cover.  No reverse shape insertion can use
+such a forward endpoint as its lower endpoint, since it lies below
+a proper original upper bound.  For reverse tests, two different
+tests again have disjoint fresh interiors and share only original
+endpoints.  Lemma~\ref{lat:insertion} therefore protects their covers.
+The same argument protects the exact $M_{16}$ intervals: a shape
+insertion could enter one only if its two endpoints lay inside it,
+whereas its fresh bottom and atoms are never shape endpoints.
+Fences persist by Lemma~\ref{lat:persistent-fence}.
+
+We next check the final batch.  A new vertex of $[0,t]$ belongs to no
+comparison interval $[a,b]$ with $a>0$, by
+\eqref{lat:cross-order}.  A new vertex of $[t,1]$ can belong to such
+a comparison only if $b=1$ and $a\leq t$.  The lower endpoints of
+the latter insertions are in $\mathcal R_C$, and consequently lie
+below proper original upper bounds.
+
+In particular, a reverse fence cannot subdivide a fresh forward
+coatom edge $c\prec1$: that would require its lower endpoint to be
+exactly $c$, which lies below no proper original vertex.  A forward
+cover with a proper upper endpoint cannot be subdivided by an
+insertion ending at $1$.  Thus all forward shortcut edges and all
+reserved coatom witnesses survive.  A forward $M_{16}$ interval with
+proper upper endpoint cannot be entered by a reverse fence.  If its
+upper endpoint is $1$, its fresh bottom lies below no proper original
+vertex; hence no point of $\mathcal R_C$ lies above that bottom.
+So this $M_{16}$ interval also remains exact.  Forward base insertions
+cannot enter any of these positive-bottom intervals.
+
+All reverse comparison intervals have two proper original bounds in
+the forward order.  Neither kind of final insertion can add a vertex
+to such an interval: an insertion beginning at $0$ cannot do so, and
+an insertion ending at $1$ cannot do so.  Their covers, witnesses and
+exact $M_{16}$ intervals are therefore unchanged.  The final reverse
+fences label every vertex in each reverse comparison interval by
+the definition of $\mathcal R_C$.  All required forward principal
+vertices belong to $\mathcal F_C$; the sole omitted proper target is
+the detector $X$, whose lower interval was fenced at the outset.
+This proves (i) and (ii).
+
+For (iii), an old vertex in a forward comparison belongs to
+$\mathcal F_C$, so its lower interval is fenced, with the stated
+exception for the global top.  The only possible new vertex is an
+interior vertex $w$ of a final reverse fencing insertion into $[s,1]$.
+If $x_i\leq w$, then $x_i\leq s$ by
+\eqref{lat:cross-order}.  Every vertex below $w$ either lies below
+$s$ or lies on the same private chain above $s$.  This assertion also
+includes vertices of other insertions in the final batch: a point
+from another branch can be below $w$ only by passing through the old
+endpoint $s$.  Thus $s$ is comparable to every vertex of $[0,w]$.
+Moreover $s\in\mathcal F_C$, since $x_i\leq s<1$, and its lower
+interval is fenced.  This is exactly alternative (b), including the
+requirement that the bottleneck lie above the lower chosen chain
+vertex.  It proves closure of the normal-group tests without any new
+round of insertions.
+
+For (iv), no shape insertion introduces a point strictly below $X$.
+Forward fresh points lie below no proper original vertex, and reverse
+shape insertions in the detector are contained in $[X,z]$.
+A final insertion ending strictly below $X$ introduces only points
+below that endpoint, so it cannot create a new coatom of $[0,X]$.
+A final insertion ending above $X$ introduces no new point below $X$.
+The only remaining possibility would be an insertion into $[0,X]$,
+which was omitted.  Hence the original covers $q_i\prec X$ are
+precisely the coatoms of the lower half.  Its two separate original
+branches, and those of the upper half, still give fences by
+Lemma~\ref{lat:persistent-fence}.
+
+Finally, Lemma~\ref{lat:shape} bounds the size before the final batch
+by $q=q_0+27hN^2$.  Each of $\mathcal F_C$ and $\mathcal R_C$ has
+at most $q$ members, and each fencing insertion adds four vertices.
+The final size is at most $q+4q+4q=9q$.  This also proves finiteness
+directly from the displayed batches.
+\end{proof}
+
+\subsection{Assembly and proof of the lattice properties}
+
+Let $B_N$ and $E_N$ denote respectively the decorated Boolean and
+detector components just constructed.  Define $L_N$ to be the
+horizontal sum
+\begin{equation}
+\label{lat:horizontal-sum}
+       L_N=B_N\boxplus B_N^{\mathrm{op}}
+                  \boxplus E_N\boxplus E_N^{\mathrm{op}},
+\end{equation}
+identifying only the common endpoints $0,1$.  Interiors of different
+components are incomparable, with cross meet $0$ and cross join $1$.
+The order reversal exchanging each component with its opposite is
+an order anti-isomorphism of $L_N$.  Thus $L_N$ is self-dual.
+
+Every component has two comparable proper original vertices.  The
+global interval $[0,1]$ is therefore fenced: for a vertex in one
+component, take a comparable pair in a different component.  A
+comparison with a proper starting point, or a principal interval with
+a proper upper endpoint, contains no interior vertex from a different
+component.  Thus horizontal summation preserves every local property
+proved above, and supplies the previously deferred label at the
+global top.
+
+\begin{proof}[Proof of Proposition~\ref{lat:decorated-lattice}]
+Let $B$ be the absolute chain threshold of
+Theorem~\ref{cls:chain-rigidity}, and choose the single integer
+$N=\max\{2,\lceil B\rceil\}$.  Construct $L=L_N$ by the finite operations above.
+This choice is independent of every possible representing group.
+Finiteness follows from Lemma~\ref{lat:simultaneous} and
+\eqref{lat:horizontal-sum}; no limiting lattice is used.
+
+All original lattice operations survive by
+Lemma~\ref{lat:insertion}.  Every proper original Boolean vertex is
+an endpoint of a forward requested pair and of a reverse requested
+pair, so its two principal intervals receive the required fences.
+The detector statements follow from
+Lemma~\ref{lat:simultaneous}.  This gives (i) and (ii).
+
+For (iii), take a finite representation $[D,G]$.  By
+Lemma~\ref{frm:labels}, the global fence and the required principal
+fences provide all forward labels.  In every forward requested test,
+the sequence $x_0<\cdots<x_N$ and its private shortcuts have the
+shape required by Theorem~\ref{cls:chain-rigidity}, including its
+nonconsecutive shortcuts.  The interval $[x_i,x_j]$ is fenced;
+every target in \eqref{lat:targets} has its bottom saturation,
+fence, and exact positive-bottom $M_{16}$ interval; and all the
+specified principal vertices are meets of coatoms below the largest
+endpoint, with $v_{iN}$ themselves coatoms.  Finally,
+Lemma~\ref{lat:simultaneous}(iii), together with the global fence,
+verifies condition~\ref{cls:test-normal} at every vertex of every comparison:
+either its lower interval is fenced or it has the specified labelled
+bottleneck above the lower chain vertex.  In the latter case,
+Corollary~\ref{frm:bottleneck} supplies the required soluble-normal
+and commuting-normal consequences.  All hypotheses of the
+chain Theorem hold, and $N\geq B$ forces equality of its endpoint
+types.  The forward requested pairs are precisely
+\eqref{lat:boolean-forward} and \eqref{lat:detector-forward}, giving
+the asserted equalities.
+
+For (iv), fix one of the indicated actual upper-filter
+representations.  Its least lattice vertex is the former $1$, and
+its group bottom is $A$.  The entire reverse test for $(u,v)$ lies
+inside this upper filter.  Its principal labels come from intervals
+$[w,1]^{\mathrm{op}}$, which are fenced by the final reverse batch.
+All vertices of its comparison intervals have those fences, by
+Lemma~\ref{lat:simultaneous}(ii).  The reverse shape construction
+supplies all the other hypotheses of the same chain Theorem.
+Consequently the two endpoint labels, now taken relative to $A$,
+are equal.  The proof for $(z,X)$ is identical.  If the actual
+intervals are compatible, their common subgroups and their labels
+are the same on overlaps; the argument makes no additional choice of
+a representation and thus applies simultaneously.
+
+Finally, the order reversal exchanging the paired components in
+\eqref{lat:horizontal-sum} supplies the same Boolean and detector
+data in the other orientation.  This proves the last assertion.
+\end{proof}
+
+We record explicitly the Boolean incidence used in the final gluing
+argument; Figure~\ref{fig:boolean-faces} shows its proper faces.  In the reverse original $B_4$, fix an original atom $x$ and
+let $a_1,\ldots,a_3$ be the other original atoms.  Put
+\[
+                    u_i=x\vee a_i\qquad(1\leq i\leq3),
+\]
+with operations in that reverse Boolean copy.  Each $u_i\vee u_j$
+for $i\ne j$ has rank three and is proper, while
+$\bigvee_{i=1}^3u_i$ is the reverse top.  Every proper original
+Boolean vertex is the join of the original atoms below it.  These
+equalities are still equalities in the full decorated lattice, since
+all insertions preserve old operations.  Thus the local proper joins
+and the incompatible total join required later are present in the
+single finite lattice of Proposition~\ref{lat:decorated-lattice}.
+
+\input{figures/boolean-faces}
diff --git a/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/sections/lifts.tex b/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/sections/lifts.tex
new file mode 100644
index 0000000000000000000000000000000000000000..57522a9d5387365da1865943b67c6e1b184b3c33
--- /dev/null
+++ b/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/sections/lifts.tex
@@ -0,0 +1,830 @@
+\section{Components, forms, and rational maps along a path}
+\label{lift:section}
+
+We now work in the remaining situation of Proposition~\ref{grw:remaining-paths}.
+In particular the characteristic is fixed, the classical ranks tend to infinity,
+and the geometric steps with a nontrivial unipotent radical have been excluded.
+All statements about classical groups below are made beyond the finitely many
+small parameters already discarded in that proposition.  We write
+$P^{(\infty)}$ for the terminal member of the derived series of a finite group
+$P$.  A group is \emph{quasisimple} if it is perfect and its quotient by its
+centre is nonabelian simple.  Soluble groups can occur beside an active
+component; their presence must be distinguished from a noncentral soluble
+extension underneath it.
+
+We need two kinds of structure from the remaining finite comparisons.
+The normal-subgroup tests will be applied to actual quasisimple components
+and their centralizers.  Later, the irreducible-triple classification will
+be applied to connected algebraic groups containing the finite images.
+Proposition~\ref{lift:edge-lifts} supplies both the components and rational
+lifts of their finite inclusions.  Lemma~\ref{lift:path-lifts} composes
+these lifts along one chosen path: irreducibility on its finite image
+then implies irreducibility on the containing algebraic image.
+Independently chosen rational paths need not agree, even when their
+finite inclusions do.
+
+The preparatory lemmas extract components from the retained subgroup
+structures, choose standard finite covers uniformly over the field,
+and lift the prescribed quadratic form in characteristic two.
+
+\subsection{Extracting actual finite components}
+
+\begin{lemma}[A two-step component lemma]\label{lift:component-preimage}
+Suppose $P_0\leq P_1$, $E_1\lhd P_1$ is quasisimple, and $P_1/E_1$ is
+soluble.  Let $\pi:P_1\longrightarrow\operatorname{Aut}(E_1)$ be conjugation.
+Suppose that $\overline P_0=\pi(P_0)$ has a normal quasisimple subgroup
+$\overline E_0$ with soluble quotient.  Then
+\[
+ E_0=P_0^{(\infty)}\leq E_1
+\]
+is quasisimple and $P_0/E_0$ is soluble.  In particular $E_0$ is normalized
+by every group normalizing $P_0$.
+\end{lemma}
+
+\begin{proof}
+Set $Q=P_0\cap E_1$ and $A=\pi(Q)$.  The group $\overline P_0/A$ is a
+quotient of $P_0/Q$, which embeds in $P_1/E_1$, so it is soluble.  A perfect
+group has trivial image in a soluble group.  Consequently
+$\overline E_0\leq A$.  Let $B$ be its full inverse image in $Q$.  Then
+$B\lhd P_0$ and
+\[
+ B/Z_0\cong\overline E_0,
+ \qquad Z_0=B\cap\ker\pi=B\cap Z(E_1)\leq Z(B).
+\]
+Because $B/Z_0$ is perfect, $B=B'Z_0$, and therefore
+$B'=[B'Z_0,B'Z_0]=B''$.  Thus $B'$ is perfect.  The inverse image in $B$
+of $Z(\overline E_0)$ has central commutators, and its intersection with
+$B'$ is central in $B'$: for an element of this inverse image, commutation
+with $B'$ is a homomorphism from the perfect group $B'$ to the abelian
+group $Z_0$.  It is trivial.  It follows that $B'$ is a perfect central
+extension of the nonabelian simple group
+$\overline E_0/Z(\overline E_0)$, hence is quasisimple.
+
+The quotients $P_0/Q$, $Q/B\cong A/\overline E_0$, and $B/B'$ are
+soluble.  Hence $P_0/B'$ is soluble.  Since $B'$ is perfect, it equals
+$P_0^{(\infty)}$.  Finally $B'\leq Q\leq E_1$, and the terminal derived
+subgroup is characteristic.
+\end{proof}
+
+\begin{remark}\label{lift:not-composition-count}
+The hypothesis about $\overline E_0$ concerns its structure as a normal
+subgroup.  The assertion that $\overline P_0$ has just one nonabelian
+composition factor would not suffice: a semidirect product of a nontrivial
+module by a simple group can have that property without having a component.
+The explicit structures in Proposition~\ref{cls:catalogue} supply the
+stronger hypothesis used here.
+\end{remark}
+
+\subsection{Standard finite covers at every field size}
+
+\begin{lemma}[Standard finite covers]\label{lift:finite-covers}
+There is an absolute rank bound, independent of the finite field,
+with the following property.  Let $\mathbf G$ be a simply connected
+classical algebraic group above this bound, let $\sigma$ be a standard
+split or graph-twisted Steinberg map, and put $U=\mathbf G^\sigma$.
+Then $U$ is perfect, $H_2(U,\mathbb Z)=0$, and its standard simple
+quotient map $u:U\twoheadrightarrow S$ is central.  For any quasisimple
+central extension $a:L\twoheadrightarrow S$, there is a unique
+homomorphism $\pi:U\to L$ satisfying $a\pi=u$.  It is surjective and
+has central kernel.  A specified projective representation of $L$
+also lifts uniquely to a linear representation of $U$ over an
+algebraically closed field.
+\end{lemma}
+
+\begin{proof}
+Steinberg's multiplier Theorem has only a finite list of exceptions
+for the split groups, and the accompanying twisted statement has
+another finite list
+\cite[Theorem~1.1 and the opening paragraph of p.~528]{Steinberg1981}.
+Their ranks are bounded, independently of the field size.  We supply
+perfectness separately: the root-presentation statement
+\cite[Lemma~$32'$, original pp.~51--53]{SteinbergNotes} applies to the
+universal split groups, and
+\cite[Theorem~34 and its proof, original pp.~187--188]{SteinbergNotes}
+gives the twisted case outside its finite list of exceptions.
+The root-generated groups used there are the full fixed-point groups
+for simply connected groups over finite fields: see
+\cite[Corollary~3 to Theorem~7, original p.~65, and the Corollary
+after Lemma~64, original p.~186]{SteinbergNotes}.
+Increase the discarded rank bound beyond these finite exception lists.
+Thus $H_1(U,\mathbb Z)=H_2(U,\mathbb Z)=0$ and $u$ is the usual
+central simple quotient.  This argument retains every field size,
+including $q=2,3,4$.
+
+Let $A=\ker a\leq Z(L)$.  Pull back $a$ along $u$ to obtain the
+central extension
+\[
+ 1\longrightarrow A\longrightarrow U\times_S L
+       \longrightarrow U\longrightarrow1.
+\]
+The action on $A$ is trivial.  The universal coefficient sequence
+\[
+ 0\longrightarrow
+ \operatorname{Ext}^1_{\mathbb Z}(H_1(U,\mathbb Z),A)
+ \longrightarrow H^2(U,A)
+ \longrightarrow\operatorname{Hom}(H_2(U,\mathbb Z),A)
+ \longrightarrow0
+\]
+gives $H^2(U,A)=0$.  The extension splits, and projection of a
+splitting to $L$ supplies $\pi$.  Two such maps differ by a
+homomorphism $U\to A$, which is zero because $U$ is perfect.
+If $H=\pi(U)$, then $L=HA$ and $A$ is central.  Therefore
+\[
+ L=[L,L]=[HA,HA]=[H,H]\leq H,
+\]
+so $\pi$ is onto.  Its kernel is contained in $\ker u\leq Z(U)$.
+
+For a projective representation, compose with $\pi$ and pull back
+$\operatorname{GL}(V)\to\operatorname{PGL}(V)$.  This is a central
+extension of $U$ by $k^\times$.  The same cohomology argument splits
+it; perfectness gives uniqueness of the resulting linear lift.
+\end{proof}
+
+\subsection{Quadratic forms in characteristic two}
+
+Here and below a Frobenius twist $M^{[a]}$ means pullback by the $p^a$-power
+Frobenius morphism.  In particular $[1]$ denotes the first Frobenius twist;
+we write $[f]$ when $q=p^f$.  For a dominant weight $\lambda$, the symbols $L(\lambda)$,
+$\nabla(\lambda)$, and $\Delta(\lambda)$ denote the simple,
+costandard, and Weyl rational modules of highest weight $\lambda$,
+respectively; we use the highest-weight, simple-head and simple-socle
+properties in \cite[Part~II, Chapter~2]{Jantzen2003}.  Write $X^+$ for the dominant weights and $w_0$ for
+the longest Weyl-group element, and put $\lambda^*=-w_0\lambda$.
+
+\begin{lemma}[Restriction of first cohomology]\label{lift:h1-restriction}
+Let $\mathbf G$ be a simply connected simple algebraic group over an
+algebraically closed field $k$ of characteristic $p$.  Let
+$\sigma=F_q\gamma$, where $q=p^f\geq2$ and $\gamma$ is a standard diagram
+automorphism commuting with $F_q$.  For every nonzero $q$-restricted
+dominant weight $\nu$, restriction is injective:
+\[
+ H^1(\mathbf G,L(\nu))\longrightarrow
+ H^1(\mathbf G^\sigma,L(\nu)).
+\]
+\end{lemma}
+
+\begin{proof}
+Put $I=\operatorname{ind}_{\mathbf G^\sigma}^{\mathbf G}k$.
+The induction filtration of
+\cite[arXiv version, Proposition~3.1.2]{BNPPSS2015}, obtained from the
+coordinate-ring filtration and the Lang map, gives $I$ a filtration with
+sections
+\[
+ \nabla(\lambda)\otimes\nabla(\lambda^*)^\sigma,
+ \qquad \lambda\in X^+,
+\]
+each once.  Write such a section as
+$\nabla(\lambda)\otimes\nabla(\mu)^{[f]}$, where $\mu$ is obtained
+from $\lambda^*$ by the diagram automorphism.  Choose the regular height
+\[
+ h(\xi)=\left\langle\xi,\sum_{\alpha>0}\alpha^\vee\right\rangle.
+\]
+It is invariant under diagram automorphisms and duality, is positive on
+every nonzero dominant weight and every positive root, and satisfies
+$h(\mu)=h(\lambda)$.
+
+We first justify removing the actual constants from this filtration.
+A nonconstant section cannot contain a trivial submodule.  Indeed a
+map from $k$ into it would transpose to a nonzero map
+$\Delta(\lambda^*)\to\nabla(\mu)^{[f]}$.  The image contains the
+simple socle $L(q\mu)$, so its highest weight must occur in the
+source.  This gives $qh(\lambda)\leq h(\lambda)$, impossible when
+$\lambda\neq0$.  Let $I_j$ be the first filtration stage containing
+the actual one-dimensional space of constants.  That space injects
+into $I_j/I_{j-1}$, so this section is the unique section for
+$\lambda=0$.  The constants split it and
+$I_j=I_{j-1}\oplus k$.  The images of the earlier stages in $I/k$,
+followed by $I_i/k$ for $i\geq j$, give a filtration of $I/k$;
+omitting the repeated stage removes exactly the constant section.
+Thus its sections are precisely the nonconstant ones.
+
+We prove that a nonzero $q$-restricted simple $L(\nu)$ cannot map to a
+nonconstant section.  Suppose otherwise.  Transposing the second tensor
+factor gives a nonzero homomorphism
+\[
+ L(\nu)\otimes\Delta(\mu^*)^{[f]}
+       \longrightarrow\nabla(\lambda).
+\]
+Its image contains the simple socle $L(\lambda)$.  Filter the Weyl module
+by simple composition factors and apply Steinberg's tensor product
+Theorem~\cite[Theorems~5.1 and~6.1]{Steinberg1963}.  Since $\nu$ is $q$-restricted, each simple
+composition factor of the source has highest weight $\nu+q\eta$ for
+some dominant $\eta$.  Consequently
+\begin{equation}\label{lift:height-lower}
+ \lambda=\nu+q\eta,\qquad h(\lambda)\geq h(\nu).
+\end{equation}
+Transposing the first tensor factor instead gives a nonzero homomorphism
+\begin{equation}\label{lift:transpose}
+ \Delta(\lambda^*)\otimes L(\nu)
+       \longrightarrow\nabla(\mu)^{[f]}.
+\end{equation}
+Its image contains $L(q\mu)$, so the weight $q\mu$ occurs in its source.
+The maximum height of a weight of that source is
+$h(\lambda^*)+h(\nu)=h(\lambda)+h(\nu)$.  Thus
+\begin{equation}\label{lift:height-upper}
+ qh(\lambda)\leq h(\lambda)+h(\nu).
+\end{equation}
+Equations~\eqref{lift:height-lower} and~\eqref{lift:height-upper} are
+inconsistent when $q>2$.
+
+When $q=2$, both inequalities are equalities.  Hence $\eta=0$ and
+$\lambda=\nu$.  The only weight of maximum height in the source of
+\eqref{lift:transpose} is $\lambda^*+\nu$, and its weight space is the
+one-dimensional span of the tensor $a\otimes b$ of highest-weight
+vectors.  If $2\mu\neq\lambda^*+\nu$, the required weight does not
+occur.  Otherwise the homomorphism must be nonzero on $a\otimes b$.
+Choose a simple root $\alpha$ with
+$\langle\nu,\alpha^\vee\rangle=1$, which is possible because $\nu$ is
+nonzero and $2$-restricted.  For the Chevalley raising and lowering
+operators,
+\[
+ e_\alpha(a\otimes f_\alpha b)=a\otimes b.
+\]
+The target of~\eqref{lift:transpose} is Frobenius twisted, so its Lie
+algebra action is zero.  Every module homomorphism to it kills the
+displayed vector, a contradiction.
+
+A homomorphism from a finite-dimensional simple module to $I/k$ has
+image in a finite stage of the filtration.  Taking the first stage with
+nonzero intersection with that simple image gives a nonzero map into
+one of its sections.  We have proved
+\[
+ \operatorname{Hom}_{\mathbf G}(L(\nu),I/k)=0.
+\]
+Apply this also to the dual of the coefficient module.  Tensoring
+$0\to k\to I\to I/k\to0$ with $L(\nu)$ shows that the map from
+$H^1(\mathbf G,L(\nu))$ to $H^1(\mathbf G,L(\nu)\otimes I)$ is
+injective.  The tensor identity for induction and derived Frobenius
+reciprocity identify this with the asserted restriction map.  For split
+groups this is also the special case $U=k$ of
+\cite[Theorem~7.4]{CPSvdK1977}; the argument above includes the diagram
+twist and the equality case $q=2$.
+\end{proof}
+
+\begin{proposition}[Lifting a prescribed quadratic form]
+\label{lift:quadratic}
+Let $k$ have characteristic two and let $\mathbf G$, $q$, and $\sigma$
+be as in Lemma~\ref{lift:h1-restriction}.  Suppose that $V=L(\nu)$ is
+nontrivial and $q$-restricted, and that its restriction to
+$\mathbf G^\sigma$ is irreducible and preserves a nonzero quadratic
+form $Q$.  If $V$ is not the natural module for a group isomorphic to
+$\operatorname{Sp}_{2n}$, $n\geq1$, then
+the rational action of $\mathbf G$ preserves the same quadratic form
+$Q$.
+\end{proposition}
+
+\begin{proof}
+The polar form $B$ of $Q$ is nonzero: otherwise $Q$ is the square of a
+linear form, and that linear form would be invariant under the
+nontrivial irreducible finite action.  Its radical is invariant, so $B$
+is nondegenerate.  Thus the finite representation is self-dual.  The
+injective parametrization by $q$-restricted highest weights in
+Steinberg's finite-group restriction Theorem~\cite[Theorem~43]{SteinbergNotes}
+implies that $V\cong V^*$ rationally.  Schur's Lemma, first for
+$\mathbf G$ and then for $\mathbf G^\sigma$, shows that their spaces of
+invariant bilinear forms are both one-dimensional and agree.
+In particular $B$ itself is $\mathbf G$-invariant.
+
+Polarization gives an exact sequence of rational modules
+\begin{equation}\label{lift:polarization}
+ 0\longrightarrow(V^*)^{[1]}
+ \longrightarrow\operatorname{Sym}^2(V^*)
+ \longrightarrow\bigwedge^2(V^*)\longrightarrow0.
+\end{equation}
+The first map sends a twisted linear form to its square.  The
+obstruction to refining $B$ by a rationally invariant quadratic form
+is its connecting class
+\[
+ c_B\in H^1(\mathbf G,(V^*)^{[1]}).
+\]
+This class restricts to zero on $\mathbf G^\sigma$, because $Q$ is a
+refinement there.  This description is also
+\cite[Proposition~3.6 and Corollary~3.9]{GaribaldiNakano2016}.
+
+Let $\mathbf G_1$ be the first Frobenius kernel.  Its action on
+$(V^*)^{[1]}$ is trivial.  The inflation--restriction sequence therefore
+begins
+\[
+ 0\longrightarrow H^1(\mathbf G/\mathbf G_1,(V^*)^{[1]})
+ \longrightarrow H^1(\mathbf G,(V^*)^{[1]})
+ \longrightarrow
+ \bigl(H^1(\mathbf G_1,k)\otimes(V^*)^{[1]}\bigr)
+       ^{\mathbf G/\mathbf G_1}.
+\]
+The required first-kernel calculation is
+\[
+ H^1(\mathbf G_1,k)^{[-1]}\cong
+ \begin{cases}
+ N,&\mathbf G\cong\operatorname{Sp}_{2n},\ n\geq1,
+       \quad N\text{ its natural module},\\
+ 0,&\text{otherwise}.
+ \end{cases}
+\]
+It is reproduced in the proof of
+\cite[Proposition~7.1, p.~411]{GaribaldiNakano2016}, citing
+\cite[Theorem~3.1(C)(a),(f)]{BNP2004}.  Here the symplectic case includes
+$A_1=C_1$ and $B_2=C_2$.  The exceptional module is $L(\omega_1)$ in
+$C_n$ numbering, including $n=1$, and $L(\omega_2)$ in $B_2$
+numbering.  It is simple and self-dual.  After untwisting, the last invariant space
+is therefore zero under our hypothesis.  Inflation, followed by
+untwisting the quotient by $\mathbf G_1$, gives an isomorphism
+\[
+ H^1(\mathbf G,V^*)\xrightarrow{\ \sim\ }
+ H^1(\mathbf G,(V^*)^{[1]}).
+\]
+
+Write $c_B=F^*c$, where $F$ is ordinary Frobenius.  Put
+$\Gamma=\mathbf G^\sigma$ and $M=V^*$.  Since $F$ commutes with
+$\sigma$, its restriction $F_\Gamma$ is an automorphism of $\Gamma$.
+Naturality gives the commutative square
+\[
+\begin{array}{ccc}
+ H^1(\mathbf G,M)&\xrightarrow{\ F^*\ }&H^1(\mathbf G,M^{[1]})\\
+ \operatorname{res}_M\downarrow&&\downarrow\operatorname{res}_{M^{[1]}}\\
+ H^1(\Gamma,M)&\xrightarrow{\ F_\Gamma^*\ }&
+ H^1(\Gamma,F_\Gamma^*M).
+\end{array}
+\]
+The lower arrow is an isomorphism, with inverse obtained by pulling
+back along $F_\Gamma^{-1}$ with the transported coefficients.
+This does not identify $M$ with its Frobenius twist.  Hence the
+vanishing of $\operatorname{res}(c_B)$ implies that of
+$\operatorname{res}(c)$.  This and
+Lemma~\ref{lift:h1-restriction}, applied to $V^*$, imply $c=0$ and
+$c_B=0$.  Sequence~\eqref{lift:polarization} supplies a
+$\mathbf G$-invariant quadratic refinement $Q_0$ of $B$.
+
+Finally $Q-Q_0$ has zero polarization, hence is the square of a linear
+form $\ell$.  Its finite invariance implies that $\ell$ is invariant
+under $\mathbf G^\sigma$.  Such a linear form is zero on this
+nontrivial irreducible module.  Therefore $Q_0=Q$.
+\end{proof}
+
+\begin{remark}\label{lift:natural-exception}
+The exclusion of the natural symplectic module, including the
+$A_1=C_1$ and $B_2=C_2$ conventions just stated, is required in the
+inflation argument.  The geometric natural modules and the special
+isogenies in characteristic two are constructed as classical algebraic
+maps below; Proposition~\ref{lift:quadratic} is used for the remaining
+defining-characteristic nonlinear steps.  Suzuki and Ree endomorphisms
+have bounded rank and do not occur here.
+\end{remark}
+
+\subsection{The retained edge maps}
+
+\begin{lemma}[A nonsingular-line stabilizer]\label{lift:nonsingular-line}
+Let $(W,Q)$ be a nonsingular quadratic space of even dimension $2r+2$
+over $\mathbb F_q$, where $r\geq3$ and $q$ is even, and let $v\in W$ satisfy
+$Q(v)\neq0$.  Write $B$ for the polar form and $\overline W=v^\perp/\langle
+v\rangle$.  The kernel of
+\[
+ O(W,Q)_{\langle v\rangle}\longrightarrow
+ \operatorname{Sp}(\overline W,B)
+\]
+is the central group generated by the symmetry
+\[
+ r_v(x)=x+\frac{B(x,v)}{Q(v)}v.
+\]
+The image is the full symplectic group.  The symmetry has nonzero
+Dickson invariant, and restriction gives
+\[
+ \Omega(W,Q)_{\langle v\rangle}
+       \cong\operatorname{Sp}_{2r}(q).
+\]
+In particular this stabilizer has no nontrivial finite unipotent
+kernel on $\overline W$.  Over an algebraic closure, the vector
+stabilizer $\operatorname{SO}(W,Q)_v$ is a smooth connected adjoint
+group of type $B_r$.  The ambient module $W$ restricted to this group has composition factors
+$k,N_{B_r},k$ along the flag
+$\langle v\rangle<v^\perp<W$.
+\end{lemma}
+
+\begin{proof}
+A line stabilizer fixes $v$, because a scalar preserving its nonzero
+quadratic value has square one, and the field has characteristic two.
+Choose a complement $U$ to $\langle v\rangle$ in $v^\perp$ and choose
+$t$ with $B(v,t)=1$ and $B(t,U)=0$.  The space $U$ is symplectic.  An
+isometry inducing the identity on $\overline W$ sends
+$u\in U$ to $u+\ell(u)v$.  Preservation of $Q$ forces $\ell(u)^2Q(v)=0$,
+so it fixes $U$ pointwise.  Preservation of $B$ then forces its image
+of $t$ to be $t+av$.  Preservation of $Q$ says
+$a^2Q(v)+a=0$, hence $a=0$ or $a=Q(v)^{-1}$.  These are exactly the
+identity and $r_v$.
+
+For completeness, let $s\in\operatorname{Sp}(U)$.  The difference
+$Q(u)+Q(su)$ has zero polarization, so, because the field is perfect,
+there is a unique linear form $\ell$ satisfying
+\[
+ \ell(u)^2Q(v)=Q(u)+Q(su).
+\]
+The map $v\mapsto v$, $u\mapsto su+\ell(u)v$ is an isometry of
+$v^\perp$.  Witt's extension theorem \cite[Theorem~7.4, pp.~57--58]{Taylor1992}
+extends this isometry to $W$: the ambient polar radical is zero, so its
+radical-intersection criterion is automatic even though $v^\perp$ has
+degenerate polar form.
+This proves surjectivity.  The formula for $r_v$ also shows that it
+commutes with every isometry fixing $v$.
+
+The Dickson invariant of an isometry in even characteristic is
+$\operatorname{rank}(g-1)$ modulo two; in particular it is one on
+$r_v$.  Its zero kernel is $\Omega(W,Q)$ over a finite field of even
+order \cite[Theorems~11.43 and~11.51, pp.~160, 164--165]{Taylor1992};
+the exceptional group $\Omega^+(4,2)$ is outside the dimension range here.  Each fibre of the displayed surjection has two elements
+differing by $r_v$, so exactly one lies in this kernel.  The last
+finite assertion follows.
+
+We also justify the algebraic assertion, including its scheme
+convention.  Put $W_0=v^\perp$.  Restriction defines
+\[
+ H=\operatorname{SO}(W,Q)_v\longrightarrow
+       \operatorname{SO}(W_0,Q|_{W_0}).
+\]
+The kernel of restriction from the full orthogonal vector stabilizer
+is the constant group of order two computed above: over an arbitrary
+$k$-algebra its remaining parameter satisfies $a+a^2Q(v)=0$, an
+\'{e}tale equation.  Its nonidentity element has Dickson invariant one,
+so the displayed map has trivial scheme-theoretic kernel.
+We give its inverse over an arbitrary $k$-algebra $A$.
+Put $c=Q(v)$ and retain the decomposition
+$W=kv\oplus U\oplus kt$ used above.  An element
+$h\in\operatorname{SO}(W_0)(A)$ has symplectic quotient action, of
+determinant one; the odd special orthogonal determinant condition
+therefore makes it fix $v$.  Write
+$h(u)=su+\ell(u)v$ for $u\in U_A$.  There is a unique $w\in U_A$
+such that $B(su,w)=\ell(u)$ for all $u$.  The extension fixing $v$
+and restricting to $h$ is obtained by setting
+\[
+ g(t)=t+w+av,\qquad ca^2+a+Q(w)=0.
+\]
+The bilinear condition is $B(su,w)+\ell(u)=0$, and the quadratic
+condition is exactly the displayed equation.  Its derivative is one
+and its leading coefficient is a unit, so adjoining a solution gives
+an \'{e}tale double cover.  The two extensions differ by $r_v$ and
+have opposite Dickson invariants.  The unique Dickson-zero extension
+on this cover agrees on overlaps and descends to $A$.  This is a
+functorial inverse, proving the asserted scheme isomorphism.
+The odd special orthogonal group is smooth, connected,
+and adjoint of type $B_r$ by
+\cite[author version, Theorem~C.2.11 and Proposition~C.3.10]{Conrad2014}.
+This restriction calculation is also the one in the proof of
+\cite[author version, Proposition~C.3.1]{Conrad2014}.
+Its action on $W_0/\langle v\rangle$ is the purely inseparable
+isogeny to the symplectic group described in
+\cite[author version, Remark~C.3.6]{Conrad2014}; consequently this
+quotient is the simple module $N_{B_r}$.  Both outer factors of the
+flag are trivial.  We used the vector stabilizer: the line-stabilizer
+scheme $\operatorname{SO}(W,Q)_{\langle v\rangle}$ is
+$H\times\mu_2$.  Indeed its scalar on $v$ has square one,
+and multiplying by the inverse central scalar makes it fix $v$;
+the two factors have trivial intersection.  Its reduced subgroup
+and its finite-field points are the ones just described.
+\end{proof}
+
+\begin{proposition}[Components and rational edge maps]
+\label{lift:edge-lifts}
+In the setting of Proposition~\ref{grw:remaining-paths}, each retained
+projected subgroup $P$ has a characteristic normal quasisimple subgroup
+$L=P^{(\infty)}$ and $P/L$ is soluble.  In a fixed higher coordinate,
+these components nest with the projected subgroups and are normalized
+by their common bottom action.  All intermediate components of a
+shortcut are normalized by its lower coordinate stabilizer.  The
+transported product of an active component maps onto the lower
+nonabelian socle and is not contained in the common bottom.
+
+For every retained geometric or defining-characteristic almost-simple
+edge, its actual inclusion on these components has a rational lift
+between their simple classical algebraic groups, after passage to
+simply connected central covers.  The lift agrees with the finite
+inclusion after the specified central quotients.
+
+On a geometric edge, every nontrivial absolute composition factor of
+the target natural module is a tensor product of Frobenius twists of
+the source natural simple module or its dual.  The number of tensor
+factors is uniformly bounded by the constants in
+Proposition~\ref{grw:remaining-paths}.  Trivial factors and repeated
+copies are allowed.  For type $B$ in characteristic two, ``natural
+simple module'' means the $2r$-dimensional quotient of the usual
+$(2r+1)$-dimensional module.
+\end{proposition}
+
+\begin{proof}
+\emph{Finite components and their normalization.}
+We first specify the components in the structures of
+Proposition~\ref{cls:catalogue}.  A nondegenerate block or form subgroup
+contains the derived classical group on each active block.  A tensor
+subgroup contains the central product of the derived groups on its
+tensor factors.  An extension-field or subfield subgroup contains the
+derived classical group over its indicated field.  In the retained
+twisted tensor construction, the single Frobenius orbit of the tensor
+factors has fixed-point group equal to the classical group over the
+corresponding extension field; its derived group is the active
+quasisimple group.  The almost-simple case contains its quasisimple
+socle preimage.
+
+The reductions leave one nonabelian active factor.  Hence every
+permitted normalizer preserves it.  All other base factors commute
+with it modulo scalars, and their surviving images are soluble.
+The retained diagonal, field, graph, and factor-permutation quotients
+are soluble as well; the exclusion of other nonabelian factors here
+includes factors in the permutation quotient, not just base factors.
+Thus the explicitly displayed active derived group is normal and the
+quotient is soluble.  Scalar preimages introduce only central kernels:
+if $C/Z$ is the displayed perfect active factor, with $Z$ central,
+then $C=C'Z$ and $C'=C''$, giving its quasisimple derived preimage.
+For the characteristic-two nonsingular-line case,
+Lemma~\ref{lift:nonsingular-line} gives the asserted component directly.
+For a two-edge projection, we verify the precise hypothesis of
+Lemma~\ref{lift:component-preimage}.  Use the common upper-coordinate
+notation $P_0\leq P_1$ and $Q_1\leq R_1\leq P_1$ from
+Lemma~\ref{cls:core-gaps}.  Here $Q_1$ is soluble, $R_1/Q_1$ is the
+middle simple active factor, and $P_1/R_1$ is soluble.  The upper
+retained edge supplies the quasisimple component
+$E_1=P_1^{(\infty)}$.  Perfectness gives $E_1\leq R_1$.
+Its image in $R_1/Q_1$ is nontrivial, since $Q_1$ is soluble, and is
+normal, hence is the full simple group.  Therefore
+\[
+ E_1Q_1=R_1,\qquad E_1\cap Q_1=Z(E_1).
+\]
+Both $E_1$ and $Q_1$ are normal in $P_1$.  Thus, for $x\in Q_1$,
+commutation with $x$ maps $E_1$ into $E_1\cap Q_1=Z(E_1)$.
+It is a homomorphism to an abelian group, so perfectness makes it
+trivial.  Consequently $Q_1$ centralizes $E_1$.  Conjugation on
+$E_1$ agrees with conjugation on
+$E_1/Z(E_1)\cong R_1/Q_1$: an automorphism of the perfect group
+$E_1$ acting identically modulo its centre differs from the identity
+by a homomorphism $E_1\to Z(E_1)$ and is therefore the identity.
+
+The original-core product identity in
+Proposition~\ref{frm:active-transport} gives $Q_1\leq P_0$.
+Indeed its transported product comes from the middle original core,
+which is contained in $D$ and hence in the lower labelled group.
+It follows that
+\[
+ J=(P_0\cap R_1)/Q_1
+\]
+is the actual first-edge inner projection of
+Lemma~\ref{cls:core-gaps}.  Its retained structure supplies the
+normal quasisimple component $J^{(\infty)}$ with soluble quotient.
+This is a structural assertion about this particular $J$; the
+excluded parabolic or module semidirect extensions have not been
+inferred to split.
+
+Let $\pi:P_1\to\operatorname{Aut}(E_1)$ be conjugation.  By the
+centralization and simple-quotient identification just proved,
+$\pi(P_0\cap R_1)$ is the inner image of $J$, isomorphic to $J$.
+It is normal in $\pi(P_0)$, and
+$\pi(P_0)/\pi(P_0\cap R_1)$ is soluble because it is a quotient of
+$P_0/(P_0\cap R_1)\leq P_1/R_1$.
+Hence the image of $J^{(\infty)}$ is normal quasisimple in
+$\pi(P_0)$, with soluble quotient.  These are exactly the hypotheses
+of Lemma~\ref{lift:component-preimage}, which produces the actual
+lower component inside $E_1$.
+
+Finally, if $P\leq P'$ and $L'$ is the
+component of $P'$, the image of the perfect group $L$ in the soluble
+group $P'/L'$ is trivial.  Hence $L\leq L'$.  Characteristicity gives
+normalization by the common bottom.  Each intermediate inner
+projection is normal in its own coordinate stabilizer, which contains
+the lower coordinate stabilizer; its characteristic component is
+therefore normalized by that lower stabilizer too.
+
+To check that the active component is not killed, use the notation
+$Q\leq R\leq P$ of Proposition~\ref{frm:active-transport}.  In our
+multiplicity-one coordinate, $R/Q$ is the nonabelian simple active
+factor and $P/R$ is soluble.  Hence $L\leq R$.  Since $P/L$ is
+soluble, the simple group $R/Q$ has trivial image modulo $LQ$, and
+$LQ=R$.  Thus $L$ maps onto $R/Q$.  Transporting $L$ over all upper
+coordinates gives a normal subgroup of the lower labelled group,
+mapping onto its socle modulo its original bottom core.  If this
+product were contained in the common bottom, normality would put it
+inside that core, a contradiction.
+
+For each actual finite component $L_i$, fix its standard cover
+$U_i=\mathbf G_i^{\sigma_i}$ and the surjection
+$\pi_i:U_i\twoheadrightarrow L_i$ of
+Lemma~\ref{lift:finite-covers}, over the specified simple quotient.
+
+\smallskip\noindent
+\emph{Rational maps for the geometric edges.}
+We have obtained the finite components, their nesting, and their survival
+under the original bottom-core quotient.  We now lift their inclusions.
+For the geometric cases we describe the maps on natural modules over $k$,
+which also proves the assertion about their composition factors.
+Write $N$ for the source natural simple module.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item A block embedding acts on its active block by $g$, on a dual
+block, when present, by $g^{-\mathsf T}$, and trivially on omitted
+spaces.  Its constituents are $N$, $N^*$, and trivial modules.
+A form embedding is the natural action with its prescribed invariant
+bilinear or quadratic form.  These are rational matrix maps.
+
+\item For a tensor factor with an omitted space $U$, the map is
+$g\mapsto g\otimes1_U$.  On the active factor its natural module is
+the direct sum of $\dim U$ copies of $N$, or of a prescribed dual or
+Frobenius twist of $N$.  The commuting omitted factor changes the
+multiplicity space, not this formula.
+
+\item Restriction of scalars from $\mathbb F_{p^{ab}}$ to
+$\mathbb F_{p^a}$, followed by extension of scalars to $k$, has the
+explicit decomposition
+\[
+ k\otimes_{\mathbb F_{p^a}}\mathbb F_{p^{ab}}
+       \cong\bigoplus_{j=0}^{b-1}k,
+ \qquad x\longmapsto(x,x^{p^a},\ldots,x^{p^{a(b-1)}}).
+\]
+Thus its map is the direct sum of the rational representations
+$N^{[aj]}$.  If a form pairs two summands, the corresponding summand
+is expressed as the required dual.  A subfield inclusion is the same
+natural matrix map, possibly precomposed with its specified field or
+diagram automorphism.  On natural modules in the ranks in use, a
+diagram automorphism gives $N$ or $N^*$; triality has already been
+discarded with bounded rank.
+
+\item In a twisted tensor orbit the map is
+\[
+ g\longmapsto
+       \rho_0(F^{a_0}g)\otimes\cdots\otimes\rho_{b-1}(F^{a_{b-1}}g),
+\]
+where each $\rho_j$ is natural or dual.  This is a rational
+representation and it is the defining matrix construction of the
+finite subgroup.  Its natural module is
+$\bigotimes_j N_j^{[a_j]}$, $N_j\in\{N,N^*\}$.
+The orbit lengths, field degrees, and omitted tensor dimensions are
+bounded by Proposition~\ref{grw:remaining-paths}.
+\end{enumerate}
+
+In characteristic two, two further classical maps must be interpreted
+literally.  The nonsingular-vector stabilizer in a group of type
+$D_{r+1}$ has connected reduced subgroup of type $B_r$; its action on
+the $2r+2$-dimensional natural space has, by
+Lemma~\ref{lift:nonsingular-line}, the invariant filtration
+\[
+ 0<\langle v\rangle<v^\perp<W
+\]
+with factors $k,N_{B_r},k$.  Its simply connected cover therefore
+supplies the required $B_r$-to-$D_{r+1}$ map.  The middle symplectic
+action is an isogeny from this \emph{adjoint} $B_r$ stabilizer to
+$\operatorname{Sp}_{2r}$.  It has no nontrivial
+kernel on $k$-points, consistently with
+Lemma~\ref{lift:nonsingular-line}; it is not a finite unipotent
+radical.  In the following display, instead,
+$\mathbf B_r=\operatorname{Spin}_{2r+1}$ and
+$\mathbf C_r=\operatorname{Sp}_{2r}$ denote the \emph{simply
+connected} groups.  The preceding vector stabilizer is the adjoint
+quotient of $\mathbf B_r$; the kernels of its symplectic action and
+of the simply connected maps are not being identified.  For the two
+special isogenies
+\[
+ s:\mathbf B_r\longrightarrow\mathbf C_r,
+ \qquad t:\mathbf C_r\longrightarrow\mathbf B_r,
+ \qquad st=F_p,\quad ts=F_p,
+\]
+one has
+\[
+ s^*N_C=N_B,\qquad t^*N_B=N_C^{[1]}
+\]
+by \cite[author preprint, \S I.3.2]{DowdSin1996}.
+If the inverse of $s$ on $\mathbb F_q$-points is needed, it is induced
+by the rational map $tF_p^{f-1}$, since $q=p^f$.  No inverse
+inseparable morphism is asserted.  These formulas preserve the
+claimed description of simple constituents, including the trivial
+factors of the full odd orthogonal module.
+
+\smallskip\noindent
+\emph{Rational maps for the nonlinear edges.}
+For a defining-characteristic almost-simple edge, inflate its actual
+finite irreducible representation along $\pi_i:U_i\twoheadrightarrow L_i$,
+using the unique linear lift of Lemma~\ref{lift:finite-covers}
+if the natural action was given
+projectively.  Surjectivity preserves every invariant subspace and
+every prescribed form.  A form similitude multiplier is a character
+of the perfect group $U_i$, hence is trivial.
+Only now apply Steinberg's restriction
+Theorem~\cite[Theorem~43]{SteinbergNotes}: the resulting $U_i$-module
+is the restriction of its $q_i$-restricted rational simple module,
+with the specified field and diagram twist.  Here
+$\sigma_i=F_{p^{f_i}}\gamma_i$ uses $f_i$ ordinary Frobenius digits,
+including in the unitary case; the unitary natural scalar field
+$\mathbb F_{p^{2f_i}}$ does not change this restriction parameter.
+This yields a rational map to
+$\operatorname{GL}(V)$ whose determinant is trivial: a connected
+semisimple group has no nontrivial rational character.  Thus it maps
+to a linear target, including the algebraic group underlying a
+finite unitary target.  In a symplectic target or an orthogonal target
+in odd characteristic, finite self-duality and the injective
+highest-weight parametrization identify the invariant rational
+bilinear form with the finite form, by Schur's Lemma.  Its symmetry
+or alternating property is consequently the prescribed one.
+For an orthogonal target in characteristic two the remaining
+nonlinear module satisfies Proposition~\ref{lift:quadratic}, which
+lifts its actual quadratic form.  The natural and special-isogeny
+cases are the geometric maps already constructed.  The image is
+connected and hence lies in the identity component of the isometry
+group.
+
+\smallskip\noindent
+\emph{Compatibility through the fixed central covers.}
+These constructions initially give maps into whichever central form
+of the target is specified by its natural action.  A homomorphism
+from a simply connected semisimple source lifts through a central
+isogeny of semisimple algebraic groups; see
+\cite[author version, Exercise~6.5.2(iii), p.~236]{Conrad2014}.  Thus they give maps
+between simply connected covers as claimed.  Let
+$\phi_i:\mathbf G_i\to\mathbf G_{i+1}$ be such a map and let
+$\iota_i:L_i\hookrightarrow L_{i+1}$ be the actual finite inclusion.
+By construction their induced maps to the target adjoint group
+agree on $U_i$.  For $u\in U_i$, the target adjoint image of
+$\phi_i(u)$ is fixed by $\sigma_{i+1}$, so
+\[
+ \delta(u)=\sigma_{i+1}(\phi_i(u))\phi_i(u)^{-1}
+       \in Z(\mathbf G_{i+1})(k).
+\]
+Centrality gives $\delta(uv)=\delta(u)\delta(v)$.
+Perfectness of $U_i$ forces $\delta=1$, so
+$\phi_i(U_i)\leq U_{i+1}$, including when the algebraic centre is
+nonreduced.  The two homomorphisms
+$\pi_{i+1}\phi_i|_{U_i}$ and $\iota_i\pi_i$ to $L_{i+1}$ agree
+modulo its centre.  Their discrepancy is consequently a
+homomorphism $U_i\to Z(L_{i+1})$, and again is trivial.  Thus we have
+the exact commutative diagram
+\begin{equation}\label{lift:finite-diagram}
+\begin{array}{ccc}
+ U_i&\xrightarrow{\ \phi_i|_{U_i}\ }&U_{i+1}\\
+ \pi_i\downarrow&&\downarrow\pi_{i+1}\\
+ L_i&\xrightarrow{\ \iota_i\ }&L_{i+1}.
+\end{array}
+\end{equation}
+This proves compatibility with the actual finite inclusions through
+the fixed central-cover maps.  It gives no uniqueness assertion for
+rational maps agreeing on finite points.
+\end{proof}
+
+\subsection{Composing a chosen path}
+
+\begin{lemma}[Pathwise lifting]\label{lift:path-lifts}
+Fix a path through retained edges with finite nested components
+$L_0\leq\cdots\leq L_s$.  Choose an edge lift
+$\phi_i:\mathbf G_i\to\mathbf G_{i+1}$ from
+Proposition~\ref{lift:edge-lifts} for each edge, with the finite
+central-cover identifications specified there.  Then
+\[
+ \Phi_{i,j}=\phi_{j-1}\cdots\phi_i\qquad(i<j)
+\]
+is a rational lift of the actual finite inclusion along that path.
+If a rational $\mathbf G_j$-module is irreducible on $L_i$ under this
+finite inclusion, it is irreducible on the algebraic image
+$\Phi_{i,j}(\mathbf G_i)$.  A nontrivial such image is a closed
+connected simple algebraic group; if its rank is strictly smaller
+than the rank of $\mathbf G_j$, it is proper.
+\end{lemma}
+
+\begin{proof}
+The finite compatibility statements for the edges compose, so the
+first assertion follows by induction on $j-i$.  Every subspace
+invariant under the algebraic image is invariant under the contained
+finite image.  Finite irreducibility therefore implies algebraic
+irreducibility, without a density assertion or a bound on the
+highest weight of the composite representation.  The kernel of a
+nontrivial map from a connected simple algebraic group is finite as
+a group scheme: its connected reduced normal subgroup cannot be
+the whole source.  Its image is closed and connected, and is simple
+up to the allowed central form.  Inseparability can interchange the
+types $B$ and $C$ in characteristic two, but does not change rank.
+The last assertion follows from the rank inequality.
+\end{proof}
+
+\begin{remark}\label{lift:frobenius-periods}
+The exponents in each chosen $\phi_i$ are retained when composing.
+For example, on $\operatorname{SL}_n(p^f)$ the representations
+$g\mapsto g\otimes F^a(g)$ and
+$g\mapsto g\otimes F^{a+f}(g)$ agree, although their rational
+representations need not agree.  Accordingly an independently
+chosen shortcut map is never identified with a composed path map.
+The only comparison needed later is their agreement on the actual
+finite component, to which the irreducibility implication in
+Lemma~\ref{lift:path-lifts} applies.  Also, a strict finite subfield
+inclusion can have full algebraic image; properness in the later
+application comes from rank growth.
+\end{remark}
+
+\begin{corollary}[Passing irreducibility backwards through geometry]
+\label{lift:geometric-irreducibility}
+Suppose that a retained geometric edge has source component $L$,
+and that the target natural module restricts absolutely irreducibly
+and nontrivially to a subgroup $H\leq L$.  Then the source natural
+simple module restricts absolutely irreducibly to $H$.
+\end{corollary}
+
+\begin{proof}
+Work over the algebraic closure.  Any proper submodule or nontrivial
+composition filtration for the $L$-action would be an $H$-invariant
+one.  Thus the target module is a single nontrivial composition
+factor for $L$.  Proposition~\ref{lift:edge-lifts} expresses it as a
+tensor product of twists of the source natural module or its dual.
+If one factor had a nonzero proper $H$-invariant subspace, tensoring
+that subspace with all the other factors would give a nonzero
+proper invariant subspace of the product.  Hence each factor is
+irreducible on $H$.  Taking a dual or applying a field Frobenius to
+all matrix entries preserves irreducibility, so the source natural
+module itself is irreducible on $H$.
+\end{proof}
diff --git a/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/sections/packets.tex b/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/sections/packets.tex
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+\section{A uniform bound on natural-module composition length}
+\label{pkt:section}
+
+We continue in the setting of Proposition~\ref{grw:remaining-paths}.
+The endpoint types have a common characteristic, field parameter
+$q=p^f$, and classical family.  Write $n_i$ for their natural
+dimensions, using the symplectic realization for a group of type
+$B$ in characteristic two.  The natural scalar field is
+$\mathbb F_q$, except in the unitary case, where it is
+$\mathbb F_{q^2}$.  All ranks in this section may be assumed to be
+at least nine.
+
+For a comparison $i<j$, let $P=P_{ij}$ be the inner projection
+in the upper simple group $T_j$, and let $M=M_{ij}$ be the
+corresponding lower coordinate stabilizer in
+$\operatorname{Aut}(T_j)$.  Proposition~\ref{lift:edge-lifts} supplies a
+quasisimple component $E=E_{ij}$, normal in both $P$ and $M$, with
+$P/E$ soluble.  In matrix representations we take the indicated
+central covers.  The same stabilizer normalizes the intermediate
+component on either edge of a shortcut.  These are structural
+conclusions about the remaining edges, not consequences of the
+composition-length bound to be proved here.
+
+We first pass to a semisimple natural section, losing at most two trivial
+factors, and then bound three sources of composition length.  A soluble
+centralizer bounds the multiplicity of each absolute simple type.
+Many distinct types could still occur, so we group them into orbits under
+descent, the normalizer action, and duality.  The commuting-normal test
+allows at most one large orbit of summands, with a bounded-dimensional
+remainder.  Finally, comparing the source and ambient field shifts along
+the same two-edge shortcut bounds the number of types in each orbit.
+This last step makes the length bound independent of $f$.
+
+We use Lemma~\ref{grw:coordinate-normal-tests}, with its tests
+\textup{(\ref{grw:normal-soluble})} and
+\textup{(\ref{grw:normal-commuting})}.
+Here $P=P_{ij}$, $M=M_{ij}$, and
+$F_j=\operatorname{core}_{z_j}(D)$ is the original higher core.
+For an $M$-invariant $H\le T_j$, write
+$\widehat H\le N_j/F_j$ for its transported product.
+A killed product lies in $D/F_j$, and hence $H\le P$;
+the converse is not asserted.
+For one product the enlargement is
+$(z_i/F_j)\widehat H$; for two it is
+$\langle z_i/F_j,\widehat H_1,\widehat H_2\rangle$.
+Each application checks properness of this enlargement and
+normality of every tested product in that same enlargement.
+The pairs used in this section commute exactly in $N_j/F_j$,
+so \textup{(\ref{grw:normal-commuting})} uses $\overline K=1$.
+Their full inverse images commute modulo $F_j$, which is
+normal in the actual enlargement and contained in $D$.
+The transported active component $\widehat E$ is not killed,
+by the component-survival conclusion of
+Proposition~\ref{lift:edge-lifts}.
+
+\subsection{Semisimplicity and the nonsingular-line exception}
+
+\begin{lemma}[Exclusion of invariant parabolics]
+\label{pkt:no-parabolic}
+After passing to a fixed-spacing subchain, no comparison stabilizer
+$M_{ij}$ normalizes a proper parabolic subgroup of $T_j$.
+\end{lemma}
+
+\begin{proof}
+This is Proposition~\ref{grw:remaining-paths}(v), with its
+fixed-spacing thinning already incorporated in the retained chain.
+\end{proof}
+
+\begin{lemma}[A semisimple natural section]
+\label{pkt:semisimple}
+Let $V$ be the upper natural module in a retained comparison.
+As a module for the lower quasisimple component, either $V$ is
+semisimple, or the characteristic is two, $V$ is an even-dimensional
+quadratic space, and there is an $M$-invariant nonsingular line
+$R$ such that
+\[
+                  V^{\circ}=R^\perp/R
+\]
+is a semisimple symplectic module.  In the latter case the
+composition length of $V$ is that of $V^{\circ}$ plus two.
+The lower component and its soluble-quotient and normal-subgroup
+properties pass to $V^{\circ}$.
+\end{lemma}
+
+\begin{proof}
+Work first over the finite natural scalar field.  Put
+\[
+ S=\operatorname{soc}_E(V),\quad
+ R_0=\operatorname{rad}_E(V),\quad
+ U=S\cap R_0,\quad W=S+R_0.
+\]
+If $V$ is not semisimple, $R_0\ne0$, and a simple submodule of
+$R_0$ shows that $U\ne0$.  Also $W<V$: the proper submodule $S$
+is contained in a maximal submodule, which necessarily contains
+$R_0$.  Socles and radicals are intrinsic, so every semilinear
+normalizer preserves these constructions.
+
+For a linear group without graph operations, $U$ gives an
+$M$-normalized parabolic, contrary to
+Lemma~\ref{pkt:no-parabolic}.  Suppose that $M$ also contains
+duality operations.  The side-preserving subgroup has index two.
+A duality acts on subspaces by a semilinear correlation, and its
+transport of a module to the dual exchanges the socle with the
+annihilator of the radical, and the radical with the annihilator
+of the socle.  Consequently it exchanges $U$ and $W$ as members
+of a flag.  Thus the flag $U\le W$ is preserved by all of $M$;
+if $U=W$ it is a one-member flag.  Both members are nonzero and
+proper.  This again contradicts Lemma~\ref{pkt:no-parabolic}.
+This argument uses correlations on subspaces and does not treat a
+graph automorphism as a linear operator on $V$.
+
+In a bilinear or Hermitian space, identify the module with its
+appropriate dual using the preserved form.  Duality of socles
+and radicals gives
+\[
+                       R_0^\perp=S,\qquad S^\perp=R_0.
+\]
+Hence $U$ is totally isotropic.  This supplies a forbidden
+parabolic in the symplectic, unitary, and odd-characteristic
+orthogonal cases.
+
+It remains to consider a quadratic form $Q_V$ in characteristic
+two.  On a totally polar-isotropic space $U$ its restriction has
+the form $\ell^2$, for a linear functional $\ell$, since the
+finite field is perfect.  Its zero space is an $M$-invariant
+singular subspace.  Unless $U$ is a nonsingular line, this zero
+space is nonzero and contradicts
+Lemma~\ref{pkt:no-parabolic}.  We are therefore left with the
+nonsingular line $R=U$.
+
+The space $R^\perp/R$ has a nondegenerate alternating form.
+If its $E$-module were not semisimple, the argument with socle
+and radical would give a nonzero $M$-invariant totally isotropic
+subspace $A$ of that quotient.  Its inverse image $B$ in
+$R^\perp$ is totally polar-isotropic and contains $R$.  On $B$
+the quadratic form is the square of a linear functional nonzero
+on $R$.  Its zero hyperplane is therefore a singular complement
+to $R$, projecting isomorphically onto $A$.  This complement is
+$M$-invariant and again gives a forbidden parabolic in the
+original quadratic space.  The quotient is semisimple.
+
+The flag $0<R<R^\perp<V$ adds precisely two one-dimensional
+composition factors to that quotient.  Both are trivial for $E$
+because $E$ is perfect.  By Lemma~\ref{lift:nonsingular-line},
+the inner stabilizer has the canonical finite-group isomorphism
+\[
+ \theta:\Omega(V,Q_V)_R\xrightarrow{\sim}
+                    \operatorname{Sp}(V^\circ).
+\]
+Every $m\in M$ induces a semilinear similarity $m^\circ$ on
+$V^\circ$.  Evaluation on quotient vectors gives
+\[
+ \theta(mhm^{-1})=
+       m^\circ\theta(h)(m^\circ)^{-1}
+       \qquad(h\in\Omega(V,Q_V)_R).
+\]
+Consequently inverse images under $\theta$ preserve normalization
+by the actual $M$, inclusions, and exact commutation of inner
+subgroups.  Since $P=M\cap T_j$ preserves $R$, it lies in this
+inner stabilizer.  Write $P^\circ=\theta(P)$ and
+$E^\circ=\theta(E)$; then $P^\circ/E^\circ\cong P/E$.
+Every candidate section subgroup is pulled back through
+$\theta$ before applying a normal test.  Every lifted enlargement
+lies in the proper original line stabilizer, even if its
+symplectic image is the whole symplectic group.  Transport over
+the original coordinates uses the same normal higher core
+$F_j$ in \textup{(\ref{grw:normal-soluble})} and \textup{(\ref{grw:normal-commuting})}.  There is no additional
+finite kernel in this inner-group isomorphism.
+
+Finally, extending the finite scalar field to its algebraic
+closure preserves semisimplicity here.  Indeed the semisimple
+image algebra of the finite group is a product of matrix
+algebras over finite fields; those fields are separable, and
+scalar extension preserves semisimplicity of that algebra.
+\end{proof}
+
+\subsection{Multiplicities and packets}
+
+In this subsection $V$ denotes the semisimple space supplied by
+Lemma~\ref{pkt:semisimple}; the two removed factors will be added
+back at the end.  All centralizers and groups on summands are
+taken in the indicated finite classical realization and then in
+the inner projective group.  Passing through the finite central
+covers does not change a solubility assertion.
+
+\begin{lemma}[Soluble centralizer]
+\label{pkt:centralizer}
+The centralizer of the lower quasisimple component in the inner
+classical group on $V$ is soluble.
+\end{lemma}
+
+\begin{proof}
+Write $C=C_{T_j}(E)$ in the original natural realization.
+The group $MC$ normalizes $E$ and cannot contain $T_j$: the
+proper nontrivial $E$ cannot be normal in the simple ambient
+group.  Hence the full preimage of
+$\overline U=\langle z_i/F_j,\widehat E,\widehat C\rangle$
+is proper in $z_j$.  The two products $\widehat E,\widehat C$
+are normal in $\overline U$ and commute exactly.  Their full
+preimages commute modulo $F_j\trianglelefteq U$, with
+$F_j\le D$.  The active product $\widehat E$ is not killed,
+so \textup{(\ref{grw:normal-commuting})} kills $\widehat C$ and forces $C\le P$.
+Since
+\[
+ C\cap E\le Z(E),\qquad C/(C\cap E)\hookrightarrow P/E,
+\]
+$C$ is soluble.
+
+For the symplectic section of Lemma~\ref{pkt:semisimple}, put
+$E^\circ=\theta(E)$ and take the inverse image under $\theta$
+of $C_{\operatorname{Sp}(V^\circ)}(E^\circ)$.
+The $M$-equivariance and injectivity of $\theta$ show that this
+inner inverse image and $E$ commute exactly and are normal in
+their enlargement with $M$.  Their transported products again
+commute in $N_j/F_j$, and their original preimages commute
+modulo the normal subgroup $F_j\le D$.  The enlargement lies
+in the proper original nonsingular-line stabilizer even if
+its symplectic image is the full symplectic group.
+Condition \textup{(\ref{grw:normal-commuting})} and the same quotient calculation prove
+that the symplectic centralizer is soluble.
+\end{proof}
+
+\begin{lemma}[An absolute multiplicity bound]
+\label{pkt:multiplicity}
+Every absolute irreducible constituent of $V$ has multiplicity at
+most $15$.
+\end{lemma}
+
+\begin{proof}
+We give the linear-algebra details needed for a bound independent
+of the fields and constituent dimensions.  Let $U$ be a simple
+module over the finite natural field $F$.  Its endomorphism
+division ring is a finite field $F'$.  On an isotypic summand
+$U^m$ the commuting endomorphism algebra is
+$\operatorname{Mat}_m(F')$.  After scalar extension, $U$ splits
+into distinct absolute simples, each once; thus $m$ is exactly
+their common absolute multiplicity.
+
+In the linear case the centralizer contains
+$\operatorname{GL}_m(F')$.  Its subgroup
+$\operatorname{SL}_m(F')$ has determinant one on the whole
+$F$-space: its determinant there is a power of the norm of its
+$F'$-determinant.  Thus its derived subgroup survives the
+ambient special-linear condition.
+
+For a form group, pair the simple isotypic summands by duality.
+A pair of distinct dual summands again supplies
+$\operatorname{GL}_m(F')$, acting on the second summand by the
+inverse adjoint.  On a self-paired summand the adjoint operation
+on $\operatorname{Mat}_m(F')$ is an involution.  If it acts
+nontrivially on $F'$, the commuting isometries are a unitary
+group on an $m$-dimensional multiplicity space.  If it fixes
+$F'$, they preserve a nonsingular symmetric or alternating
+bilinear form on that multiplicity space.  These descriptions
+follow by writing the invariant pairing as a matrix of pairings
+between the $m$ copies of $U$; Schur's Lemma makes each matrix
+entry an element of $F'$, and the symmetry of the original form
+is precisely the adjoint condition on that matrix.
+
+There is one useful even-characteristic qualification.  A
+nonsingular symmetric multiplicity form need not be alternating.
+Over a perfect finite field its diagonal function is the square
+of a linear functional.  The vector representing that functional
+allows one to split off a nonsingular subspace of dimension one
+or two; its perpendicular is alternating and nonsingular.
+Thus a multiplicity space of dimension $m$ still contains a
+nonsingular alternating subspace of dimension at least $m-2$.
+
+If $m\ge16$, every group just described contains a subgroup
+isomorphic to $\operatorname{SL}_3(F'')$ for a suitable finite
+field $F''$.  In a general linear group use a three-dimensional
+block.  In a symplectic, orthogonal, or unitary multiplicity
+space choose paired totally isotropic three-spaces and let
+$g\in\operatorname{SL}_3(F'')$ act as $g$ and its inverse
+adjoint, fixing their perpendicular complement.  Finite-field
+forms of dimension at least $16$ have such paired spaces;
+the anisotropic part has dimension at most two.  The preceding
+paragraph handles the nonalternating symmetric form in
+characteristic two in the same way.
+
+These embedded groups are perfect and have nonabelian simple
+projective quotients.  Being perfect, their determinants,
+similitude multipliers, spinor norms where applicable, and
+Dickson invariants in even-characteristic orthogonal groups
+are trivial.  Therefore
+they lie in the actual inner classical group, rather than
+merely its full similitude group.  A scalar quotient cannot
+make their images soluble.
+
+In an orthogonal target of characteristic two, preserving the
+polar form on a nontrivial isotypic summand also preserves the
+quadratic form.  Indeed two $E$-invariant quadratic refinements
+of that polar form differ by $\ell^2$ for an $E$-invariant linear
+functional $\ell$.  On a sum of nontrivial simple $E$-modules
+there is no such nonzero functional.  A commuting polar
+isometry consequently preserves the given refinement.  On the
+trivial isotypic space use its actual quadratic isometry group;
+the same paired-three-space construction applies.  The trivial
+space is nondegenerate, since it is orthogonal to every
+nontrivial isotypic summand.
+
+We have proved that $m\ge16$ would put a nonsoluble subgroup in
+the centralizer.  Lemma~\ref{pkt:centralizer} excludes this.
+\end{proof}
+
+Define a \emph{packet} to be an orbit of absolute simple types
+under finite-field descent, the action induced by $M$, and the
+duality pairing when a form or an ambient correlation is present,
+intersected with the set of types occurring in $V$.
+The sum belonging to a packet is defined over the finite natural
+field.  In a form group it is nondegenerate, because all dual
+partners belong to the same packet.  The packet decomposition
+and every union of its parts are normalized by $M$.
+For a linear graph automorphism this last statement refers to
+the groups on the parts: the induced adjoint anti-automorphism
+permutes the central isotypic idempotents and takes the full
+linear group on a part to the full linear group on its paired
+part.  It therefore normalizes the product groups used below,
+although its action on subspaces is a correlation.
+
+\begin{lemma}[At most one large packet]
+\label{pkt:one-packet}
+Two disjoint unions of packets on which $E$ acts nontrivially
+cannot both have dimension at least $16$.  Consequently either
+one packet accounts for all but at most $15$ dimensions, or
+the sum of all nontrivial packets has dimension at most $45$.
+\end{lemma}
+
+\begin{proof}
+Let $V_1,V_2$ be two such unions, and put all remaining parts
+in $V_3$.  Let $H_1,H_2$ be the derived groups of the full
+linear or isometry groups on $V_1,V_2$, acting trivially on the
+other parts.  They commute, are normalized by $M$, and are
+normal in the enlargement generated by them and $M$.
+Its inner part preserves the proper decomposition and cannot
+contain $T_j$.  Thus
+$\overline U=\langle z_i/F_j,\widehat H_1,\widehat H_2\rangle$
+has proper full preimage in $z_j$.  Both $H_1,H_2$ are
+nonsoluble: the explicit
+$\operatorname{SL}_3$ construction in the proof of
+Lemma~\ref{pkt:multiplicity} lies in each derived group and in
+the ambient inner group.
+
+The two transported products are normal in this $\overline U$
+and commute exactly; their original preimages commute modulo
+$F_j\trianglelefteq U$, with $F_j\le D$.  Condition
+\textup{(\ref{grw:normal-commuting})} kills one product and forces, say, $H_1\le P$.
+The kernel
+of the nontrivial action of $E$ on $V_2$ is central in $E$,
+so $H_1\cap E\le Z(E)$.  Hence
+\[
+                 H_1/(H_1\cap E)\hookrightarrow P/E
+\]
+would make $H_1$ soluble, a contradiction.  On the symplectic
+section, take inverse images under the finite $M$-equivariant
+isomorphism $\theta$ from Lemma~\ref{pkt:semisimple}.
+They commute exactly and their enlargement lies in the proper
+original line stabilizer, so the same application uses the
+original higher core $F_j$, as in Lemma~\ref{pkt:centralizer}.
+
+Suppose that a nontrivial packet has dimension at least $16$.
+If another nontrivial packet occurs, apply the first assertion
+to the large packet and its entire complement, including the
+trivial packet.  The complement has dimension at most $15$.
+If no other nontrivial packet occurs, the complement is the
+trivial isotypic space, whose dimension is at most $15$ by
+Lemma~\ref{pkt:multiplicity}.
+If no packet has dimension $16$, but their total dimension
+were at least $46$, accumulate packets until their dimension
+first reaches $16$.  The accumulated dimension is at most
+$30$, leaving at least $16$ dimensions in the other union.
+This contradicts what was just proved.  Thus in this case the
+total is at most $45$.
+\end{proof}
+
+\subsection{Field shifts and the size of a packet}
+
+The endpoint field agreement is useful only together with the
+normalizer action on both edges of a shortcut.  The quantity
+to control is the difference between the source field action
+and the scalar action on the upper natural space.
+
+\begin{lemma}[Field-shift synchronization]
+\label{pkt:field-shifts}
+Let the endpoint exponent be $f$, and let the intermediate
+exponent on a remaining shortcut be
+\[
+                f'=\frac uv f,\qquad (u,v)=1.
+\]
+Put $h=\gcd(f,f')=f/v$.  If an element of $M$ induces source
+field exponent $a$ and upper scalar exponent $s$, then
+\begin{equation}
+                   s-a\equiv0\pmod h,\qquad
+                   v(s-a)\equiv0\pmod f.
+\label{pkt:shift-congruence}
+\end{equation}
+Every packet of absolute
+constituent types has size at most $2v$.
+\end{lemma}
+
+\begin{proof}
+We work on simply connected covers and use Frobenius positions
+modulo the group parameter, rather than the degree of its
+natural scalar field.  Write a classical finite group with
+parameter $p^r$ as the fixed points of $\tau F_p^r$, where
+$\tau$ is a diagram operation.  Steinberg's Restriction and
+Tensor Product Theorems give the unique parametrization
+\[
+ \lambda=\lambda_0+p\lambda_1+\cdots+p^{r-1}\lambda_{r-1},
+ \qquad\lambda_i\text{ $p$-restricted},
+\]
+of its absolute simple modules; see
+\cite[Theorems~41 and~43]{SteinbergNotes} and
+\cite{Steinberg1963}.  For a nontrivial simple module define
+\[
+ \operatorname{supp}_r(\lambda)
+       =\{i\in\mathbb Z/r\mathbb Z:\lambda_i\ne0\}.
+\]
+For an arbitrary module take the union of these supports over
+its nontrivial composition factors.  A field twist by $F_p^b$
+translates support by $b$.  Diagram operations and
+contragredience preserve positions.  When a position wraps
+modulo $r$, its restricted weight is transformed by
+$\tau^{-1}$, which does not change the position.
+
+Consider one edge, with source and target exponents $r,t$, and
+let $d=\gcd(r,t)$.  Suppose that the nonempty union $\mathcal E$
+of source digit supports in the target natural module lies in
+one residue class modulo $d$.  If a normalizer has target
+scalar exponent $b$ and source field exponent $a_0$, its
+transport of a composition factor has type
+\[
+                  (L(\lambda)\circ\gamma^{-1})^{[b-a_0]},
+\]
+where $\gamma$ records the source diagram operation.  This
+formula follows directly from
+$x(mw)=m(\alpha_m^{-1}(x)w)$ and the $p^b$-semilinearity of
+$m$.  It applies to composition factors by transporting a
+composition series; semisimplicity on an individual edge is
+not required.  An ambient correlation adds contragredience.
+Inner-diagonal automorphisms do not affect the absolute module
+type, because they are algebraic inner conjugations on the
+cover.  Normalization therefore implies
+\[
+                  \mathcal E+(b-a_0)=\mathcal E.
+\]
+Since $\mathcal E$ is nonempty and contained in a single
+residue class modulo $d$, we obtain
+\begin{equation}
+                          b-a_0\equiv0\pmod d.
+\label{pkt:one-edge-shift}
+\end{equation}
+
+Assume for the moment that this support condition holds for both
+edges of the shortcut.  Their source and target exponents are $f,f'$
+in opposite orders, so their common divisor $d$ is the same integer $h$.
+For an element of $M$, denote its induced intermediate field
+exponent by $a_1$.  Its existence on the same intermediate
+component is the normalization conclusion of
+Proposition~\ref{lift:edge-lifts}.  The natural realization of that
+automorphism has scalar exponent $a_1$, with any graph part
+recorded separately.  Applying
+\eqref{pkt:one-edge-shift} to the two edges gives
+\[
+                 a_1-a\equiv0\pmod h,\qquad
+                 s-a_1\equiv0\pmod h.
+\]
+Their sum proves \eqref{pkt:shift-congruence}.
+
+To bound a packet, let $\Phi$ denote the operation of field twisting on
+absolute source types, and let $\Gamma$ be the diagram group.
+The rank assumption gives $|\Gamma|\le2$.  These operations
+commute, and $\Phi^f=\tau^{-1}$.  Every stabilizer action lies
+in $\langle\Phi^h,\Gamma\rangle$.  If $\tau=1$, that group has
+order at most $2f/h$.  If $\tau\ne1$, the diagram involution
+already belongs to $\langle\Phi^h\rangle$, whose order is at
+most $2f/h$.  Thus in either case the order is at most $2v$.
+Descent over the upper natural field has exponent $f$ or $2f$
+and belongs to the same group.  Duality is the opposition
+diagram operation, also already included.  The packet bound
+follows.
+
+It remains to verify the support hypothesis for every remaining edge of
+Proposition~\ref{grw:remaining-paths} and
+Proposition~\ref{lift:edge-lifts}.
+\begin{enumerate}[label=\textup{(\alph*)}]
+\item In a nonlinear edge the natural module is a Frobenius
+      twist of a nonzero restricted module.  Its support is
+      a singleton.  The starting Frobenius position may be
+      arbitrary; it cancels in
+      \eqref{pkt:one-edge-shift}.
+\item A block, form, or subfield edge has natural or dual-natural
+      nontrivial composition factors in one Frobenius position,
+      allowing repetitions and trivial factors.  A tensor edge
+      with one active component has the same property: its
+      other tensor factors are trivial for that perfect
+      component, so they supply repetitions.  The
+      characteristic-two nonsingular-line representation has
+      these composition factors as well, with the additional
+      trivial factors already described.
+\item For restriction of scalars, the source constituents are
+      a direct sum of the Galois conjugates of the natural
+      module.  To distinguish group parameters from natural
+      scalar fields, put $\epsilon_s,\epsilon_t=2$ for a
+      unitary source or target, respectively, and $1$
+      otherwise.  A scalar restriction of degree $b_{\rm sc}$
+      satisfies
+      \[
+          \epsilon_s r=b_{\rm sc}\epsilon_t t.
+      \]
+      Its constituent positions are
+      $t_0+j\epsilon_t t$, so they all lie in one residue
+      class modulo $d=\gcd(r,t)$.
+      For example, restricting the natural matrix field of a
+      unitary component with parameter $p^r$ to a symplectic
+      target with the same parameter has scalar degree two.
+      Over the algebraic closure its module is
+      \[
+                         N\oplus N^{[r]}
+                         \cong N\oplus N^*.
+      \]
+      Both positions are zero modulo $r$; this repetition
+      causes no tensor carry because the operation is a
+      direct sum.
+\item For a retained natural twisted-tensor edge, let
+      $b_{\rm cyc}$ be the prime tensor-cycle length.
+      Lemma~\ref{cls:twisted-tensor} gives the exact usual-field
+      relation
+      \[
+                           r=b_{\rm cyc}t.
+      \]
+      In the finite realization supplied by
+      Proposition~\ref{lift:edge-lifts}, the cyclic fixed-point
+      construction gives the following actual finite-source
+      modules, up to a common initial Frobenius twist
+      \cite[Proposition~2.8(ii) and Section~4(5), p.~449]
+      {LiebeckSeitzClassical1998}.
+      Here $W$ is the source natural module in each case.
+
+      \emph{Linear, symplectic, and orthogonal targets.}
+      The module is
+      \[
+                   \bigotimes_{j=0}^{b_{\rm cyc}-1}W_j^{[jt]},
+      \]
+      where each $W_j$ is $W$ or $W^*$ as prescribed by the
+      diagram operation.  Its positions are $jt$ for
+      $0\le j<b_{\rm cyc}$; diagram changes affect the
+      weights rather than these positions.
+
+      \emph{Unitary target, odd cycle.}
+      The source is $\operatorname{SU}_m(p^r)$ and the module is
+      \[
+                   \bigotimes_{j=0}^{b_{\rm cyc}-1}W^{[2jt]}.
+      \]
+      Indeed, the cycle initially gives $W^{[jt]}$ for even
+      $j$ and $(W^*)^{[jt]}$ for odd $j$.  On this finite source,
+      $W^*\cong W^{[r]}$.  The integers $j$ for even $j$ and
+      $j+b_{\rm cyc}$ for odd $j$ are precisely
+      $0,2,\ldots,2b_{\rm cyc}-2$, giving the displayed module.
+      Its positions $2jt$ are distinct modulo $r$ because
+      $b_{\rm cyc}$ is odd.
+
+      \emph{Unitary target, two-cycle.}
+      Since the cycle length is prime, this is the only even
+      case.  The source is $\operatorname{SL}_m(p^{2t})$ and
+      its module is
+      \[
+                             W\otimes(W^*)^{[t]},
+      \]
+      with distinct positions $0,t$ modulo $r=2t$.
+
+      Thus in every case the positions are distinct modulo
+      $r=b_{\rm cyc}t$ and lie in one residue class modulo
+      $d=\gcd(r,t)=t$.  A common initial twist adds the same
+      $t_0$ to every position and preserves both assertions.
+      Steinberg's Tensor Product Theorem therefore gives the
+      required support condition with no coincident-digit
+      carries.  This distinctness comes from the exact finite
+      cyclic construction.
+
+\end{enumerate}
+Changes between the type $B$ and type $C$ conventions in
+characteristic two do not alter the numerical field exponent.
+The special isogenies are defined over $\mathbb F_2$, commute
+with Frobenius, and have Frobenius as their composites
+\cite[Section~I.3.1]{DowdSin1996}.  Thus each edge can use the
+convention in which its displayed module has the indicated
+support, while the intermediate field action has the same
+exponent in either convention.
+\end{proof}
+
+\begin{theorem}[Uniform absolute composition length]
+\label{pkt:bounded-length}
+In the setting of Proposition~\ref{grw:remaining-paths}, let $B$ be
+an absolute bound for the denominators of the intermediate
+field ratios $f'/f$ on the remaining shortcuts.  After a
+further fixed-spacing thinning, every upper natural module,
+restricted to a lower active quasisimple component, has at most
+\[
+                              L=30B+62
+\]
+absolute composition factors, counted with multiplicity.
+The bound uses neither natural-module irreducibility between
+endpoints nor an irreducible-triples theorem.
+\end{theorem}
+
+\begin{proof}
+Apply Lemma~\ref{pkt:semisimple}.  On its semisimple natural
+section, Lemma~\ref{pkt:multiplicity} bounds each absolute
+multiplicity by $15$, and Lemma~\ref{pkt:field-shifts} bounds
+each packet by $2B$ absolute types.
+
+If there is a large nontrivial packet, it contributes at most
+$30B$ factors.  All other nontrivial packets have total
+dimension at most $15$ by Lemma~\ref{pkt:one-packet}.
+If no packet is large, their total dimension is at most $45$.
+In either case the number of nontrivial composition factors
+is at most $30B+45$.  The trivial isotypic space has dimension
+at most $15$, again by Lemma~\ref{pkt:multiplicity}.  Thus the
+semisimple section has length at most $30B+60$.  Adding the
+possible two factors from the nonsingular-line case gives the
+displayed bound.
+
+All constants came from the remaining-edge bounds, explicit
+finite-dimensional linear algebra, and the common cyclic
+congruence \eqref{pkt:shift-congruence}.  They are independent
+of field exponent, rank, and chain length.
+\end{proof}
diff --git a/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/sections/rigidity.tex b/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/sections/rigidity.tex
new file mode 100644
index 0000000000000000000000000000000000000000..1cd9e0fb5894429fdf6ace49ede59aafe6815337
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+++ b/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/build/sections/rigidity.tex
@@ -0,0 +1,532 @@
+\section{Irreducible restrictions and the chain bound}
+\label{sec:rigidity}
+
+We finish the proof of Theorem~\ref{cls:chain-rigidity}.  The essential
+representation-theoretic input is the corrected classification of irreducible
+triples in \cite[Theorem~4.1 and Table~3]{BurnessTesterman2018}.
+The composition-length bound of Section~\ref{pkt:section} is the starting
+point.  After stabilizing the numbers of trivial and nontrivial factors,
+a first rank bound forces the natural endpoint modules to restrict
+irreducibly on earlier endpoints.  A second rank bound uses this new
+irreducibility to limit the source ranks of the nonlinear edges on later
+shortcuts.  The long chain supplies more such ranks than the bound permits.
+
+We develop the two classification consequences together.  The first uses
+bounded composition length of a natural restriction; the second uses an
+intervening simple group of larger rank whose natural module remains
+irreducible.  These hypotheses control the twisted-diagonal spin families
+in different ways.  Simplicity of the smallest group alone gives no
+uniform bound on the ambient ranks.
+
+\subsection{The two rank bounds}
+
+Throughout this section, algebraic groups are defined over an algebraically
+closed field $k$ of characteristic $p>0$, and representations are rational.
+We pass to simply connected covers when defining highest weights.  For a
+classical simple group $J$, write $N_J$ for its natural simple module and
+$n(J)=\dim N_J$.  In characteristic two, for $J$ of type $B_r$, this means the
+$2r$-dimensional simple quotient of the orthogonal natural module of dimension
+$2r+1$.  Thus the finite $B_r$ and $C_r$ conventions agree.  Write
+$\ell_H(M)$ for the composition length of an $H$-module $M$.
+
+We call a nontrivial simple module \emph{of natural type} if it is a Frobenius
+twist of a natural simple module or its dual, allowing the central, diagram,
+and characteristic-two $B/C$ isogenies in the chosen realization.  At ranks
+at least nine, its highest weight on a fixed simply connected source is
+$p^a\omega_1$, or $p^a\omega_r$ for the dual in type $A_r$, for some $a\geq0$.
+The exclusions of natural type below therefore include the natural module,
+its dual, and the simple natural quotient in the characteristic-two odd
+orthogonal realization.
+
+We record explicitly the image convention in the irreducible-triple Theorem.
+If $\varphi:Y\longrightarrow\operatorname{SL}(V)$ is its representation, the
+table records $(\varphi(Y),\varphi(H),V)$.  These images have the same ranks as
+the original groups.  The possible $B/C$ change in characteristic two must
+be made using this fixed representation; see
+\cite[Lemma~2.1 and Remark~4.2(b)]{BurnessTesterman2018}.
+Let $\sigma:B_r\longrightarrow C_r$ and $\tau:C_r\longrightarrow B_r$ be the
+special isogenies.  With Bourbaki numbering, their pullbacks on fundamental
+weights are
+\begin{align}
+ \sigma^*(\omega_i^C)&=\omega_i^B &&(i<r), &
+ \sigma^*(\omega_r^C)&=2\omega_r^B,\notag\\
+ \tau^*(\omega_i^B)&=2\omega_i^C &&(i<r), &
+ \tau^*(\omega_r^B)&=\omega_r^C.
+ \label{irr:bc-weights}
+\end{align}
+Both composites are Frobenius; these are the root-isogeny formulas of
+\cite[Chapter~I, Section~3.2]{DowdSin1996}.  In particular,
+\begin{equation}
+ \sigma^*N_{C_r}=N_{B_r},\qquad
+ \tau^*N_{B_r}=N_{C_r}^{[1]}.
+ \label{irr:bc-natural}
+\end{equation}
+Here $[a]$ denotes the twist by $p^a$.
+Consequently the length and irreducibility of a natural restriction are
+preserved on passing to the image groups in the table.  Frobenius twists
+preserve these properties by transporting invariant subspaces by the
+entrywise field automorphism.  This argument does not require the subgroup
+embedding to be defined over the prime field.
+
+For reference, the complete part of Table~3 with a connected simple subgroup
+of rank $r\geq9$ is displayed in Table~\ref{irr:high-rank-table}.
+We omit restrictions on $p$ and on the weight of $V$ that only remove cases
+and hence cannot enlarge the set of ranks.  The two rows with an unbounded
+number of blocks retain their essential condition $p=2$.
+
+\begin{table}[htbp]
+\centering
+\small
+\begin{tabular}{lll}
+\toprule
+Rows of Table~3 & Subgroup image & Ambient image\\
+\midrule
+$\mathrm I_1,\mathrm I'_1$ & $C_r$ & $A_{2r-1}$\\
+$\mathrm I_2,\mathrm I_3$ & $B_r$ & $A_{2r}$\\
+$\mathrm I_4,\mathrm I_5$ & $D_r$ & $A_{2r-1}$\\
+$\mathrm I_6$ & $A_r$ & $A_{(r^2+r-2)/2}$\\
+$\mathrm I_7$ & $A_r$ & $A_{(r^2+3r)/2}$\\
+$\mathrm {IV}_1,\mathrm {IV}'_1$ & $B_r$ & $D_{r+1}$\\
+$\mathrm {IV}'_2$ & diagonal $B_r$ in $B_rB_r$ & $D_{2r+1}$\\
+$\mathrm S_6$ & diagonal $B_r$ in $B_r^t$ & $D_{tr+1}$, $t\geq2$, $p=2$\\
+$\mathrm {MR}_4$ & $D_r$ & $C_r$\\
+$\mathrm {MR}_5$ & diagonal $B_r$ in $B_r^t$ & $B_{tr}$, $t\geq2$, $p=2$\\
+\bottomrule
+\end{tabular}
+\caption{The high-rank simple-subgroup rows of the corrected irreducible-triple
+table.  Diagonal projections may involve different Frobenius morphisms.}
+\label{irr:high-rank-table}
+\end{table}
+
+Here and below ``diagonal'' allows the twisted projections explained in
+\cite[Remark~6.1(c)]{BurnessTesterman2018}.  A simple group projecting onto
+several simple blocks has an isogeny onto each block, so their ranks all equal
+$r$.  The exceptional projection of an $A_1$ into a $B_2$ block cannot occur
+when $r\geq9$.  Every row not displayed either has subgroup rank at most
+eight or specifies an actual nonsimple product.  This proves the exhaustion
+asserted in Table~\ref{irr:high-rank-table}; one cannot replace a product in
+an unprimed row by an arbitrary diagonal subgroup without again testing
+irreducibility.
+
+Put
+\begin{equation}
+ \mathcal R(r)=
+ \left\{r,r+1,2r-1,2r,2r+1,
+ \frac{r^2+r-2}{2},\frac{r^2+3r}{2}\right\}.
+ \label{irr:rank-set}
+\end{equation}
+All rows outside $\mathrm S_6$ and $\mathrm {MR}_5$ have ambient rank in this
+set, which has at most seven elements.
+
+\begin{lemma}[Rank bound with composition length]
+\label{irr:rank-length}
+Let $H<Y$ be connected classical simple algebraic groups with
+$r=\operatorname{rk}(H)\geq9$.  Suppose $V$ is a nontrivial irreducible,
+$p$-restricted, tensor-indecomposable $Y$-module which is not of natural type,
+and $V|_H$ is irreducible.  If $\ell_H(N_Y)\leq c$ for an integer $c\geq1$, then
+\[
+ \operatorname{rk}(Y)\in
+ \mathcal R(r)\cup\{tr,tr+1:2\leq t\leq c\}.
+\]
+In particular the number of possible ambient ranks is at most $2c+7$.
+The assertion is unchanged by central covers and the isogenies in
+\eqref{irr:bc-weights}.
+\end{lemma}
+
+\begin{proof}
+Apply the corrected irreducible-triple Theorem to $V$ and use
+Table~\ref{irr:high-rank-table}.  Only $\mathrm S_6$ and $\mathrm {MR}_5$
+require an additional argument.  Each of their $t$ blocks contributes a
+nontrivial constituent of the natural restriction, of dimension $2r$.
+In $\mathrm {MR}_5$, passing from the odd orthogonal natural module to its
+simple quotient removes one trivial constituent and leaves all $t$
+nontrivial constituents.  In $\mathrm S_6$, there are also two trivial
+constituents, which do not affect the lower bound $t$ on length.  Different
+Frobenius twists may change the isomorphism types of these constituents, but
+not their dimensions or their number counted with multiplicity.  Thus $t\leq c$.
+The image conventions and \eqref{irr:bc-natural} justify the same calculation
+on the original groups.  No complete-reducibility assertion is used.
+\end{proof}
+
+The length hypothesis cannot simply be dropped.  For example, in
+characteristic two fix $H=B_r$ and embed it in $B_r^t<B_{tr}$ by
+$g\mapsto(g,F(g),\ldots,F^{t-1}(g))$.  The spin module of $B_{tr}$ restricts
+to
+\[
+ \bigotimes_{a=0}^{t-1}L_H(\omega_r)^{[a]},
+\]
+which is irreducible by Steinberg's Tensor Product Theorem.  The ambient
+spin module is restricted, nonnatural, and tensor-indecomposable, whereas
+$t$ is arbitrary.  This is the simple diagonal case of $\mathrm {MR}_5$.
+
+\begin{lemma}[Rank bound with an intervening group]
+\label{irr:intervening}
+Let $H<X<Y$ be connected classical simple algebraic groups with
+$r=\operatorname{rk}(H)\geq9$ and $s=\operatorname{rk}(X)>r$.
+Suppose $N_X|_H$ is irreducible.  If $V$ is a nontrivial irreducible,
+$p$-restricted, tensor-indecomposable $Y$-module, not of natural type, and
+$V|_H$ is irreducible, then
+\[
+ \operatorname{rk}(Y)\in\mathcal R(r).
+\]
+The same conclusion holds when $X=Y$ and $N_Y|_H$ is irreducible.
+\end{lemma}
+
+\begin{proof}
+Use the same representation
+$\varphi:Y\longrightarrow\operatorname{SL}(V)$ throughout the proof.
+For $(Y,H,V)$, every row of Table~\ref{irr:high-rank-table} outside
+$\mathrm S_6$ and $\mathrm {MR}_5$ gives the conclusion.  Suppose one of
+these two rows occurs.  Then $p=2$, the image of $H$ is $B_r$, the image of
+$Y$ is $D_m$ or $B_m$, and $V$ is a half-spin or spin module, respectively.
+Let $W$ be the orthogonal natural module of this fixed ambient image, on
+a cover when necessary.  Every nontrivial composition factor of $W|_H$
+has dimension $2r$.
+
+Since $H<X$, irreducibility on $H$ implies irreducibility on $X$.
+Apply the irreducible-triple Theorem a second time to $(Y,X,V)$, keeping
+the ambient group and representation unchanged.  The theorem requires
+restrictedness and tensor-indecomposability on $Y$, so it makes no such
+assumption about the $X$-module $V|_X$.
+If the ambient image is $B_m$, the only row with a simple subgroup of rank
+at least ten is $\mathrm {MR}_5$.
+If it is $D_m$ and $V$ is half-spin, the only such rows are
+$\mathrm {IV}_1$, $\mathrm {IV}'_1$, $\mathrm {IV}'_2$, and $\mathrm S_6$.
+In all these cases the image of $X$ is $B_s$ and every nontrivial composition
+factor of $W|_X$ is a Frobenius twist of $N_X$, of dimension $2s$.
+
+By hypothesis and \eqref{irr:bc-natural}, each such constituent remains
+irreducible on $H$.  Restricting a composition series for $W|_X$ and refining
+it over $H$ therefore exhibits an $H$-composition factor of dimension $2s$.
+This contradicts the preceding dimension $2r$, since $s>r$.
+The argument works for nonsplit natural restrictions as well as for direct
+sums.
+
+There is no extra case from $\mathrm {MR}_4$: its ambient \emph{image} is
+$C_m$, whereas the image of the fixed representation here is $B_m$ or $D_m$.
+One cannot change that image between the two applications.
+Equations~\eqref{irr:bc-weights}--\eqref{irr:bc-natural} preserve the natural
+irreducibility hypothesis on passing to these image groups, including an
+inseparable isogeny.  They do not replace the fixed spin representation $V$
+by a natural representation.
+
+Finally, if $X=Y$, the irreducibility of $N_Y|_H$ directly excludes
+$\mathrm S_6$ and $\mathrm {MR}_5$, since their natural restrictions have at
+least two nontrivial composition factors.
+\end{proof}
+
+\subsection{From bounded length to natural irreducibility}
+
+We now use the finite components and algebraic lifts supplied by
+Proposition~\ref{lift:edge-lifts} and Lemma~\ref{lift:path-lifts}.
+Fix the last endpoint of a finite retained chain and one of its socle
+coordinates.  Proposition~\ref{lift:edge-lifts} realizes the earlier endpoint
+components in this coordinate as nested quasisimple subgroups: the terminal
+derived groups of the nested inner projections nest.  If $P_i\le P_j$ are
+two such projections and $Q_j\le R_j\le P_j$ are the core and active
+preimages of Lemma~\ref{cls:core-gaps}, their original pair comparison is
+$(P_i\cap R_j)/Q_j$.  The two-step component argument in
+Proposition~\ref{lift:edge-lifts} identifies its terminal derived subgroup
+with the conjugation image of the lower component on the higher one;
+the kernel is central.  Thus this common finite realization carries the
+pair restrictions bounded in Theorem~\ref{pkt:bounded-length}.
+
+Write $L_i$ for the finite component at a chain endpoint,
+$\mathbf G_i$ for its simply connected algebraic source, and $N_i$ for
+its natural simple module over $k$.
+In the notation $N_j|_{L_i}$, take the natural representation on the
+standard finite cover $U_j$ and pull it back along the finite map
+$U_i\to U_j$ from \eqref{lift:finite-diagram}, or a composite of those maps.
+Two such finite lifts inducing the same map to $L_j$ differ by a
+homomorphism from the perfect group $U_i$ to the central kernel of
+$\pi_j$, and hence are equal.  The finite pair maps therefore agree with
+their composites, so these restrictions compose.
+Algebraic restrictions are taken along one chosen
+composed path at a time.  We make no identification of independently chosen
+shortcut lifts.
+
+\begin{lemma}[Stabilized composition factors]
+\label{irr:constituent-count}
+Suppose that, for every $i<j$ in a chain of quasisimple components, the
+restriction $N_j|_{L_i}$ has exactly $a$ nontrivial and $b$ trivial
+composition factors.  Assume $a\geq1$ and that every nontrivial simple
+$L_j$-constituent restricts nontrivially to each earlier $L_i$.
+For every $i<j<k$, each nontrivial composition factor of $N_k|_{L_j}$
+restricts irreducibly and nontrivially to $L_i$.
+\end{lemma}
+
+\begin{proof}
+Let $U_1,\ldots,U_a$ be the nontrivial $L_j$-constituents, counted with
+multiplicity.  If $U_t|_{L_i}$ had only trivial composition factors, the
+image of the perfect group $L_i$ would be conjugate into an upper unitriangular
+group and hence would be both perfect and solvable.  It would be trivial,
+contrary to the nontriviality hypothesis.  Thus, writing $a_t,b_t$ for its
+nontrivial and trivial composition counts, we have $a_t\geq1$ and $b_t\geq0$.
+Additivity of composition multiplicities along restriction gives
+\[
+ a=\sum_{t=1}^a a_t,\qquad
+ b=b+\sum_{t=1}^a b_t.
+\]
+It follows that $a_t=1$ and $b_t=0$ for every $t$, as required.
+\end{proof}
+
+\begin{proposition}[Natural endpoint restrictions]
+\label{irr:natural-endpoints}
+In the remaining configurations of Proposition~\ref{grw:remaining-paths},
+arbitrarily long chains have arbitrarily long subchains on which every
+natural endpoint module restricts absolutely irreducibly to each earlier
+endpoint.
+\end{proposition}
+
+\begin{proof}
+By Theorem~\ref{pkt:bounded-length}, fix an absolute integer $c$ bounding
+$\ell_{L_i}(N_j)$ for every retained comparison $i<j$.
+The pair of nontrivial and trivial composition counts has at most
+$(c+1)^2$ possible values.  Finite Ramsey applied to these pair colors
+gives arbitrarily long subchains with constant counts $(a,b)$.
+These restrictions are nontrivial, and $a\geq1$ follows because a perfect
+group cannot act nontrivially with only trivial composition factors.
+Moreover, a nontrivial simple representation of a quasisimple group has
+central kernel.  Its restriction along the noncentral earlier component
+is therefore nontrivial.  The hypotheses of
+Lemma~\ref{irr:constituent-count} hold.
+
+For each $j<k$, realize every nontrivial constituent $U$ of $N_k|_{L_j}$
+as the restriction of the corresponding restricted finite-group highest
+weight module of $\mathbf G_j$.  Steinberg's Tensor Product Theorem
+decomposes that rational simple module into Frobenius twists of
+$p$-restricted simple modules; see \cite[Theorem~41]{SteinbergNotes}.
+If a restricted factor is tensor decomposable, continue factoring it
+until its nontrivial factors are tensor-indecomposable.
+This terminates since every factor has smaller positive dimension.
+Every resulting factor is $p$-restricted: its dominant highest weight
+is a summand of a $p$-restricted dominant weight.
+
+Color a triple $i<j<k$ red if some such $U$ has a nontrivial terminal
+factor $V$, not of natural type, and blue otherwise.  In a red triple,
+Lemma~\ref{irr:constituent-count} says that $U|_{L_i}$ is irreducible.
+Hence each tensor factor $V^{[d]}$ is irreducible on $L_i$.
+Use a path lift from $\mathbf G_i$ to $\mathbf G_j$ and replace its image
+by its $p^d$-Frobenius image.  The module $V$ is irreducible on this closed
+connected simple subgroup: any proper rational invariant subspace would
+also be invariant under the corresponding finite component.  The subgroup
+has rank $r_i$ and is proper because the endpoint ranks increase.
+
+The natural restriction on this subgroup has algebraic composition length
+at most $c$.  Indeed, restriction from an algebraic group to a finite
+subgroup is exact and cannot reduce composition length, and Frobenius
+transport preserves length.  The finite restriction here is a twist of
+$N_j|_{L_i}$.
+Lemma~\ref{irr:rank-length} consequently bounds the number of possible
+middle ranks $r_j$ by $2c+7$, for fixed $i$.
+
+A red homogeneous chain with more than $2c+9$ vertices is impossible:
+fix its first and last vertices $i,k$ and vary its middle vertex $j$.
+The distinct ranks $r_j$ would exceed that bound.
+A second application of finite Ramsey therefore gives arbitrarily long
+blue homogeneous subchains.
+
+Fix $i<j<k$ in a blue subchain and choose a nontrivial constituent $U$
+of $N_k|_{L_j}$.  There is at least one nontrivial terminal tensor factor,
+and every such factor is now of natural type.
+Since $U|_{L_i}$ is irreducible, a natural or dual natural Frobenius factor
+restricts irreducibly to $L_i$.
+Undoing its twist, dual, and the isogenies
+\eqref{irr:bc-weights} shows that $N_j|_{L_i}$ is irreducible.
+Delete the last vertex of the blue subchain, so that a suitable $k$ exists
+for every remaining pair $i<j$.
+All modules and restrictions were taken over $k$, so the resulting finite
+restrictions are absolutely irreducible.
+\end{proof}
+
+\subsection{Restricted natural factors and shortcut tails}
+
+The word ``restricted'' in the next lemma concerns one fixed algebraic
+source.  A representation with many different Steinberg digits need not
+have bounded tensor length.  The retained nonlinear edges of
+Proposition~\ref{grw:remaining-paths}, however, have one digit after a
+common Frobenius twist is removed.
+
+\begin{lemma}[A restricted product of natural factors]
+\label{irr:restricted-natural}
+Let $J$ be a classical simple algebraic group of rank at least nine.
+Let $M$ be a nontrivial $p$-restricted simple $J$-module.
+If every nontrivial tensor-indecomposable factor of $M$ is of natural type,
+then $M\cong N_J$ or $N_J^*$.
+In particular there is exactly one nontrivial terminal factor and
+$\dim M=n(J)$.
+\end{lemma}
+
+\begin{proof}
+A tensor factor of a simple module is simple, since any proper nonzero
+submodule of the factor would tensor to one of the whole module.
+Highest weights add in an irreducible tensor product.
+The highest weights of the natural-type factors have the form
+$p^a\omega_1$, or $p^a\omega_r$ in type $A_r$ for the dual.
+Equation~\eqref{irr:bc-weights} shows explicitly that the special isogenies
+add no further natural weights.
+Since their sum is $p$-restricted and all fundamental-weight coefficients
+are nonnegative, every exponent $a$ is zero.
+The terminal factors are therefore $N_J$ or $N_J^*$.
+
+Any two of these factors have reducible tensor product.
+For two identical factors, the kernel of the flip minus the identity is
+a nonzero proper invariant subspace: it contains $v\otimes v$, while
+$v\otimes w-w\otimes v\ne0$ for independent $v,w$.
+This also proves properness in characteristic two.
+For opposite factors, $N_J\otimes N_J^*\cong\operatorname{End}(N_J)$
+contains the proper invariant line of scalar endomorphisms.
+The same arguments apply to dual identical factors.
+Tensoring such a proper submodule with all remaining factors contradicts
+simplicity of $M$.  There is thus exactly one terminal factor.
+\end{proof}
+
+\begin{lemma}[Pulling irreducibility through a natural tail]
+\label{irr:natural-tail}
+Consider an edge between the finite components associated with algebraic
+sources $J$ and $K$.  Suppose every nontrivial absolute composition factor
+of the target natural module, restricted to the source, is a tensor product
+of natural-type $J$-modules.  If that target module is absolutely irreducible
+on an earlier noncentral quasisimple component $L$, then $N_J|_L$ is
+absolutely irreducible.
+This applies to each retained geometric edge and to every one-digit edge
+whose restricted terminal factors are all of natural type.
+\end{lemma}
+
+\begin{proof}
+Irreducibility on $L$ implies irreducibility on the source component, so
+the target restriction has a single absolute composition factor.
+It is nontrivial because the target natural dimension exceeds one.
+By hypothesis this factor is a tensor product of natural-type modules.
+Irreducibility of its further restriction forces each factor to restrict
+irreducibly.  Undoing a twist, a dual, and the natural isogenies yields the
+assertion for $N_J$.
+For retained geometric edges this also follows from
+Corollary~\ref{lift:geometric-irreducibility}.  For the stated one-digit edges it follows
+from Lemma~\ref{irr:restricted-natural}, after restoring the common twist.
+\end{proof}
+
+\subsection{The final contradiction}
+
+\begin{proof}[Completion of the proof of Theorem~\ref{cls:chain-rigidity}]
+Suppose tested chains with different endpoint labels have unbounded length.
+Apply Proposition~\ref{cls:reduction} and
+Proposition~\ref{grw:remaining-paths}.
+We obtain arbitrarily long configurations with strictly increasing endpoint
+ranks, with their minimum rank tending to infinity, and with two-edge
+shortcuts on all pairs.  There are absolute constants $c_0>0$, $C\geq1$,
+and $B\geq1$ such that the natural endpoint dimensions satisfy
+\begin{equation}
+ n_j\geq c_0 n_i^2\qquad(i<j),
+ \label{irr:quadratic-growth}
+\end{equation}
+and every retained geometric edge satisfies
+\begin{equation}
+ n_{\mathrm{target}}\leq C n_{\mathrm{source}}^B.
+ \label{irr:geometric-degree}
+\end{equation}
+Every other edge is a defining-characteristic irreducible representation
+with one restricted digit after removing a common twist.  At least one
+such nonlinear edge occurs in each shortcut.
+Proposition~\ref{lift:edge-lifts} and Lemma~\ref{lift:path-lifts} supply the actual
+component homomorphisms and their algebraic path lifts.
+By Proposition~\ref{irr:natural-endpoints}, pass to arbitrarily long
+subchains with all natural endpoint restrictions absolutely irreducible.
+
+Color a pair of endpoints red if its chosen shortcut has a one-digit edge
+with a nontrivial restricted tensor-indecomposable factor not of natural
+type.  Color the pair blue otherwise.
+Finite Ramsey leaves arbitrarily long homogeneous configurations of one
+of these two kinds.  We exclude both.
+
+First consider a blue configuration.  On every one-digit edge,
+Lemma~\ref{irr:restricted-natural} bounds the target natural dimension by
+the source natural dimension; restoring a Frobenius twist does not change
+dimension.  Thus every edge of every shortcut satisfies
+\eqref{irr:geometric-degree}, enlarging $C,B$ if necessary.
+Composing at most two edges gives the uniform bound
+\begin{equation}
+ n_j\leq A n_i^E\qquad(i<j),\qquad
+ A=C^{B+1},\quad E=B^2.
+ \label{irr:pair-degree}
+\end{equation}
+Choose a fixed integer $d$ with $2^d>E$.
+Along $d$ consecutive comparisons of a retained chain,
+\eqref{irr:quadratic-growth} gives
+\[
+ n_d\geq c_0^{\,2^d-1}n_0^{2^d},
+\]
+whereas the shortcut from its first to its last endpoint gives
+$n_d\leq A n_0^E$.
+These inequalities contradict each other as the minimum rank, and hence
+$n_0$, tends to infinity.  This proves the bounded-natural-factor case
+without imposing a bound on the number of Steinberg digits of an arbitrary
+composite endpoint representation.
+
+Now consider a red homogeneous configuration.  Fix an early endpoint $H$
+of rank $r\geq9$.  Choose eight later pair comparisons with disjoint ordered
+index intervals, leaving a gap between successive intervals.
+For each such comparison, let $X$ be its lower endpoint, and choose the
+last edge in its shortcut that has a restricted factor not of natural
+type.  Let $Y$ be the algebraic source of that edge and $Z$ its target.
+Every edge after it has a natural tail in the sense of
+Lemma~\ref{irr:natural-tail}: it is either geometric or a one-digit edge
+with only natural restricted terminal factors.
+The upper endpoint natural module is irreducible on the finite component
+of $H$.  Applying that lemma backwards along the tail shows that $N_Z$
+is irreducible on the same component.
+
+Remove the common Frobenius twist of the $Y$-representation on $N_Z$,
+and choose a nontrivial restricted tensor-indecomposable factor $V$ not
+of natural type.  Irreducibility of the whole restriction implies
+irreducibility of the corresponding twisted factor on the finite $H$.
+Compose the actual algebraic maps along the path from $H$ through $X$
+to $Y$, and apply the removed Frobenius morphism to both lower images.
+Then $V$ is irreducible on the resulting connected simple image of $H$.
+Its rank is $r$, and the resulting image of $X$ has its original rank
+$s>r$.  Its natural simple module remains irreducible on the image of
+$H$, by the endpoint conclusion and \eqref{irr:bc-natural}.
+Finite irreducibility implies each of these rational irreducibility
+statements, since a rational invariant subspace would also be invariant
+under the actual finite image.
+
+If the algebraic image of $X$ is proper in $Y$, apply
+Lemma~\ref{irr:intervening}.  If that image is all of $Y$, the map from
+$X$ to $Y$ is an isogeny.  This can occur even when the finite groups are
+strictly nested, for example on a subfield edge.  The pullback of $N_Y$
+is a natural simple module or its dual, up to Frobenius twist and the
+isogenies in \eqref{irr:bc-natural}, by the root-datum classification
+of isogenies
+\cite[Definition~6.1.8, Example~6.1.9 and Theorem~6.1.16(1)]{Conrad2014}.
+Indeed, for compatible simple roots an isogeny has a root bijection
+$\delta$ and powers $q_i$ of the characteristic.  Write the Cartan
+matrices in the row-coroot convention
+$A^X_{ij}=\langle\alpha_j,\alpha_i^\vee\rangle$ and
+$A^Y_{ab}=\langle\beta_b,\beta_a^\vee\rangle$.
+The compatibility equation is
+$q_j A^X_{ij}=q_i A^Y_{\delta(i)\delta(j)}$.
+Pairing the pullback of the natural fundamental weight with the source
+simple coroots gives $q_i$ at the node mapped to the natural node and
+zero elsewhere.  In the retained ranks this node is the natural node,
+or its type-$A$ dual; the only change of root lengths is the
+characteristic-two $B/C$ case in \eqref{irr:bc-natural}.
+Since $N_X$ is irreducible on the
+lower $H$, it follows that $N_Y$ is irreducible on the corresponding
+image of $H$.  The $X=Y$ case of Lemma~\ref{irr:intervening} therefore
+applies.  Thus every one of the eight chosen source ranks belongs to
+$\mathcal R(r)$.
+They are nevertheless distinct.  Along an algebraic edge the rank cannot
+decrease, because a maximal torus maps with finite kernel into a torus
+of the image.  Hence each chosen source rank lies between the ranks of
+the two endpoints of its comparison.  The gaps between the ordered
+comparison intervals, and strict increase of endpoint ranks, make these
+eight intervals disjoint in rank.
+This contradicts $|\mathcal R(r)|\leq7$.
+
+Both homogeneous alternatives are impossible.  All Ramsey bounds and
+all path lengths used here depend only on the absolute constants in the
+preceding lemmas.  In particular only fixed finite paths are needed
+before taking the large-rank limit.  The assumed unbounded family of
+tested chains cannot exist, proving the asserted absolute chain bound
+and completing Theorem~\ref{cls:chain-rigidity}.
+\end{proof}
diff --git a/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/paper.pdf b/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..2f540c3ca2261b0f60067fdc398964241097ca03
--- /dev/null
+++ b/preprints/A-Negative-Solution-to-the-Finite-Lattice-Representation-Problem-September-24-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:a102efd4a3009423f0e36d9e73a239a57f6507cb57d3e9e3fe6d8a22208af876
+size 850383
diff --git a/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/README.md b/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..1a6f7f622da50041da34a8680b8c7ec2e067111f
--- /dev/null
+++ b/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/README.md
@@ -0,0 +1,18 @@
+# [A nonspectrahedral hyperbolicity cone](nonspectrahedral-hyperbolicity-cone.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 24, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026,
+  author = {{OpenAI}},
+  title = {{A nonspectrahedral hyperbolicity cone}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/nonspectrahedral-hyperbolicity-cone.pdf}{OAI:A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/figures/tangent-variation.tex b/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/figures/tangent-variation.tex
new file mode 100644
index 0000000000000000000000000000000000000000..3673cfacd89bfb2b484cdfdbe24d8fef83645c4c
--- /dev/null
+++ b/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/figures/tangent-variation.tex
@@ -0,0 +1,48 @@
+\begin{figure}[htbp]
+\centering
+\begin{tikzpicture}[x=1cm,y=1cm,font=\small]
+  \definecolor{tangentblue}{RGB}{35,92,135}
+
+  \node[anchor=west,font=\footnotesize] at (0,3.02)
+    {Local sphere section};
+  \coordinate (O) at (1.65,1.45);
+  \coordinate (V) at (2.75,1.45);
+  \coordinate (Us) at (2.6469,1.9149); %
+  \draw[black!28,line width=0.6pt] (O) circle[radius=1.1];
+
+  \draw[tangentblue,line width=1.6pt]
+    (2.6962,1.1101) arc[start angle=-18,end angle=48,radius=1.1];
+  \draw[black!40,line width=0.5pt] (O) -- (V);
+  \draw[black!40,line width=0.5pt] (O) -- (Us);
+  \fill[black!55] (O) circle[radius=0.025];
+  \node[anchor=north,font=\scriptsize] at (1.65,1.36) {$0$};
+
+  \draw[->,line width=0.8pt] (V) -- (2.75,2.65);
+  \node[anchor=west,font=\footnotesize] at (2.86,2.49) {$h\perp v$};
+  \fill (V) circle[radius=0.035];
+  \node[anchor=north west] at (2.83,1.42) {$v$};
+  \fill[tangentblue] (Us) circle[radius=0.037];
+  \node[anchor=south east,text=tangentblue] at (2.54,1.96) {$u_s$};
+  \node[anchor=north,font=\scriptsize] at (1.65,0.22)
+    {$S^3\cap\operatorname{span}\{v,h\}$};
+
+  \node[anchor=west,font=\footnotesize] at (4.65,3.02)
+    {Only $R$ is differentiated};
+  \node[anchor=west] at (4.65,2.49)
+    {$\mathcal U\subset\Omega,\qquad \operatorname{rank}A(u)=\rho\quad(u\in\mathcal U).$};
+  \node[anchor=west] at (4.65,1.79)
+    {$\displaystyle u_s=\frac{v+sh}{\sqrt{1+s^2\|h\|^2}},
+      \qquad u_{s=0}=v,\quad \left.\frac{du_s}{ds}\right|_{s=0}=h.$};
+  \node[anchor=west] at (4.65,1.02)
+    {$P(v)\text{ stays fixed.}$};
+  \node[anchor=west,text=tangentblue] at (4.65,0.43)
+    {$R(u_s)=R(v)+s\,dR_v(h)+o(s)\qquad(s\to0).$};
+\end{tikzpicture}
+\caption{A schematic local tangent step, for a fixed nonzero $h\in v^\perp$.
+The highlighted arc lies in a chosen constant-rank neighborhood $\mathcal U$;
+the expansion of $R$ is used only for small $s$ with $u_s\in\mathcal U$.
+The pictured section belongs to the parameter sphere, whereas $P(v)$
+and $R(u_s)$ act on the separate block spaces $\mathbb R^a$ and
+$\mathbb R^c$.}
+\label{fig:local-tangent}
+\end{figure}
diff --git a/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/main.tex b/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/main.tex
new file mode 100644
index 0000000000000000000000000000000000000000..a39ddb8ae33077758448ad58d165ff7938feb7bf
--- /dev/null
+++ b/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/main.tex
@@ -0,0 +1,52 @@
+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage[margin=1.05in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{microtype}
+\usepackage{xcolor}
+\usepackage{tikz}
+\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref}
+\usepackage{bookmark}
+\usepackage[numbers,sort&compress]{natbib}
+\pdftrailerid{}
+\hypersetup{pdftitle={A nonspectrahedral hyperbolicity cone},pdfauthor={OpenAI}}
+\newcommand{\R}{\mathbb R}
+\newcommand{\Sym}{\mathbb S}
+\newcommand{\norm}[1]{\left\lVert#1\right\rVert}
+\newcommand{\op}{\mathrm{op}}
+\DeclareMathOperator{\adj}{adj}
+\DeclareMathOperator{\tr}{tr}
+\DeclareMathOperator{\proj}{proj}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\numberwithin{equation}{section}
+\setlength{\emergencystretch}{1.5em}
+\title{A nonspectrahedral hyperbolicity cone}
+\author{OpenAI}
+\date{September 24, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We construct a homogeneous polynomial of degree $16$ in $23$ real variables
+whose hyperbolicity cone has no representation by a finite homogeneous
+real symmetric linear matrix inequality. This disproves the geometric Generalized Lax
+conjecture.
+\end{abstract}
+\tableofcontents
+\input{sections/01-introduction}
+\input{sections/02-form}
+\input{sections/03-hyperbolic}
+\input{sections/04-pencil}
+\input{sections/05-norm}
+\input{sections/06-tangent}
+\input{sections/reduced-polynomial}
+\appendix
+\input{sections/07-comparison}
+\bibliographystyle{plainnat}
+\bibliography{references}
+\end{document}
diff --git a/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/references.bib b/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/references.bib
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index 0000000000000000000000000000000000000000..e933b3a37ee7f96f9a2c4cf681a2d7f3b08e8ebe
--- /dev/null
+++ b/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/references.bib
@@ -0,0 +1,213 @@
+@article{lax-1958,
+  author = {Lax, Peter D.},
+  title = {Differential equations, difference equations and matrix theory},
+  journal = {Communications on Pure and Applied Mathematics},
+  volume = {11},
+  number = {2},
+  pages = {175--194},
+  year = {1958},
+  doi = {10.1002/cpa.3160110203}
+}
+
+@article{helton-vinnikov-2007,
+  author = {Helton, J. William and Vinnikov, Victor},
+  title = {Linear matrix inequality representation of sets},
+  journal = {Communications on Pure and Applied Mathematics},
+  volume = {60},
+  number = {5},
+  pages = {654--674},
+  year = {2007},
+  doi = {10.1002/cpa.20155},
+  eprint = {math/0306180},
+  archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/math/0306180}
+}
+
+@article{lewis-parrilo-ramana-2005,
+  author = {Lewis, Adrian S. and Parrilo, Pablo A. and Ramana, Motakuri V.},
+  title = {The {Lax} conjecture is true},
+  journal = {Proceedings of the American Mathematical Society},
+  volume = {133},
+  number = {9},
+  pages = {2495--2499},
+  year = {2005},
+  doi = {10.1090/S0002-9939-05-07752-X},
+  eprint = {math/0304104},
+  archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/math/0304104}
+}
+
+@article{branden-2011,
+  author = {Br{\"a}nd{\'e}n, Petter},
+  title = {Obstructions to determinantal representability},
+  journal = {Advances in Mathematics},
+  volume = {226},
+  number = {2},
+  pages = {1202--1212},
+  year = {2011},
+  doi = {10.1016/j.aim.2010.08.003},
+  eprint = {1004.1382},
+  archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/1004.1382}
+}
+
+@article{netzer-sanyal-2015,
+  author = {Netzer, Tim and Sanyal, Raman},
+  title = {Smooth hyperbolicity cones are spectrahedral shadows},
+  journal = {Mathematical Programming},
+  volume = {153},
+  pages = {213--221},
+  year = {2015},
+  doi = {10.1007/s10107-014-0744-6},
+  eprint = {1208.0441},
+  archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/1208.0441}
+}
+
+@misc{netzer-2026,
+  author = {Netzer, Tim},
+  title = {{Clifford} realizations of hyperbolic cubics},
+  year = {2026},
+  eprint = {2609.12957},
+  archivePrefix = {arXiv},
+  primaryClass = {math.AG},
+  note = {Version 1, 11 September 2026},
+  url = {https://arxiv.org/abs/2609.12957v1}
+}
+
+@article{choi-1975,
+  author = {Choi, Man-Duen},
+  title = {Positive semidefinite biquadratic forms},
+  journal = {Linear Algebra and its Applications},
+  volume = {12},
+  number = {2},
+  pages = {95--100},
+  year = {1975},
+  doi = {10.1016/0024-3795(75)90058-0}
+}
+
+@article{choi-lam-1977,
+  author = {Choi, Man-Duen and Lam, Tsit-Yuen},
+  title = {Extremal positive semidefinite forms},
+  journal = {Mathematische Annalen},
+  volume = {231},
+  number = {1},
+  pages = {1--18},
+  year = {1977},
+  doi = {10.1007/BF01360024}
+}
+
+@article{ha-2013,
+  author = {Ha, Kil-Chan},
+  title = {Notes on extremality of the {Choi} map},
+  journal = {Linear Algebra and its Applications},
+  volume = {439},
+  number = {10},
+  pages = {3156--3165},
+  year = {2013},
+  doi = {10.1016/j.laa.2013.09.011},
+  eprint = {1306.0945},
+  archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/1306.0945v1}
+}
+
+@article{dey-gardoll-theobald-2021,
+  author = {Dey, Papri and Gardoll, Stephan and Theobald, Thorsten},
+  title = {Conic stability of polynomials and positive maps},
+  journal = {Journal of Pure and Applied Algebra},
+  volume = {225},
+  number = {7},
+  pages = {106610},
+  year = {2021},
+  doi = {10.1016/j.jpaa.2020.106610},
+  eprint = {1908.11124},
+  archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/1908.11124v2}
+}
+
+@misc{gonzalez-nevado-2026,
+  author = {Gonz{\'a}lez Nevado, Alejandro},
+  title = {The generalized {Lax} conjecture is true for topological reasons related to compactness, convexity and determinantal deformations of increasing products of pointwise approximating linear forms},
+  year = {2026},
+  eprint = {2601.12267},
+  archivePrefix = {arXiv},
+  note = {Version 1, 18 January 2026},
+  url = {https://arxiv.org/abs/2601.12267v1}
+}
+
+@inproceedings{raghavendra-ryder-srivastava-2018,
+  author = {Raghavendra, Prasad and Ryder, Nick and Srivastava, Nikhil and Weitz, Benjamin},
+  title = {Exponential lower bounds on spectrahedral representations of hyperbolicity cones},
+  booktitle = {Proceedings of the Thirtieth Annual {ACM--SIAM} Symposium on Discrete Algorithms},
+  editor = {Chan, Timothy M.},
+  publisher = {Society for Industrial and Applied Mathematics},
+  pages = {2322--2332},
+  year = {2019},
+  doi = {10.1137/1.9781611975482.141},
+  eprint = {1711.11497},
+  archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/1711.11497v2}
+}
+
+@article{garding-1959,
+  author = {G{\aa}rding, Lars},
+  title = {An inequality for hyperbolic polynomials},
+  journal = {Journal of Mathematics and Mechanics},
+  volume = {8},
+  number = {6},
+  pages = {957--965},
+  year = {1959},
+  doi = {10.1512/iumj.1959.8.58061},
+  url = {https://doi.org/10.1512/iumj.1959.8.58061}
+}
+
+@article{kummer-2016-vamos,
+  author = {Kummer, Mario},
+  title = {A note on the hyperbolicity cone of the specialized {V{\'a}mos} polynomial},
+  journal = {Acta Applicandae Mathematicae},
+  volume = {144},
+  number = {1},
+  pages = {11--15},
+  year = {2016},
+  doi = {10.1007/s10440-015-0036-z},
+  eprint = {1306.4483},
+  archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/1306.4483}
+}
+
+@misc{kummer-netzer-2026-strict,
+  author = {Kummer, Mario and Netzer, Tim},
+  title = {The Generalized {Lax} Conjecture for Strictly Hyperbolic Polynomials},
+  year = {2026},
+  eprint = {2609.24542},
+  archivePrefix = {arXiv},
+  primaryClass = {math.AG},
+  doi = {10.48550/arXiv.2609.24542},
+  note = {Version 1, 21 September 2026},
+  url = {https://arxiv.org/abs/2609.24542v1}
+}
+
+@misc{blekherman-raita-shankar-sinn-2022,
+  author = {Blekherman, Grigoriy and Rai{\c{t}}{\u{a}}, Bogdan and Shankar, Isabelle and Sinn, Rainer},
+  title = {Weak and Strong Extremal Biquadratics},
+  year = {2022},
+  eprint = {2204.10625},
+  archivePrefix = {arXiv},
+  primaryClass = {math.AG},
+  note = {Version 1},
+  url = {https://arxiv.org/abs/2204.10625v1},
+}
+
+@article{scheiderer-2018,
+  author = {Scheiderer, Claus},
+  title = {Spectrahedral Shadows},
+  journal = {SIAM Journal on Applied Algebra and Geometry},
+  volume = {2},
+  number = {1},
+  pages = {26--44},
+  year = {2018},
+  doi = {10.1137/17M1118981},
+  url = {https://epubs.siam.org/doi/pdf/10.1137/17M1118981},
+  eprint = {1612.07048},
+  archivePrefix = {arXiv},
+}
diff --git a/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/sections/01-history.tex b/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/sections/01-history.tex
new file mode 100644
index 0000000000000000000000000000000000000000..e5e7f60b13af814064a1c12ec9b9bbe2ee8d1f1e
--- /dev/null
+++ b/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/sections/01-history.tex
@@ -0,0 +1,67 @@
+G\aa rding's theory establishes the convexity of hyperbolicity
+cones~\cite{garding-1959}. The original Lax conjecture asks for a
+definite determinantal representation of homogeneous hyperbolic polynomials in three
+variables~\cite{lax-1958}. Helton and Vinnikov proved the corresponding
+definite determinantal representation theorem~\cite{helton-vinnikov-2007};
+Lewis, Parrilo, and Ramana established its equivalence with Lax's
+formulation~\cite{lewis-parrilo-ramana-2005}. In higher dimension, Br\"and\'en
+constructed a real-zero polynomial none of whose positive powers admits
+a definite determinantal representation~\cite[Theorem~3.3]{branden-2011}.
+That obstruction does not exclude a different determinant, with additional
+factors, from defining the same cone. The geometric conjecture concerns this remaining
+possibility. Kummer exhibited the distinction concretely: the cone of
+the specialized V\'amos polynomial has a $7\times7$ representation,
+although no positive power of that polynomial has a definite
+determinantal representation~\cite[Section~3]{kummer-2016-vamos}.
+Raghavendra, Ryder, Srivastava, and Weitz proved exponential
+lower bounds on the matrix size of spectrahedral representations for
+suitable hyperbolicity cones~\cite{raghavendra-ryder-srivastava-2018}.
+Recent positive results include Netzer's theorem for
+hyperbolic cubics in five variables~\cite[Theorem~3.9]{netzer-2026}.
+
+Kummer and Netzer prove spectrahedrality for cones of strictly
+hyperbolic polynomials~\cite[Theorem~1]{kummer-netzer-2026-strict}.
+Here \emph{strictly hyperbolic} means that $p(te-x)$ has distinct roots
+whenever $x\notin\R e$. The polynomial~\eqref{eq:polynomial} fails this
+hypothesis: at $w=(I_4,0,0)\notin\R e$ it gives
+\[
+ p(te-w)=(t-1)^{16}t^4.
+\]
+For the reduced polynomial of Corollary~\ref{cor:reduced}, the same line
+gives $q(te-w)=(t-1)^{12}t^4$. Thus neither defining polynomial satisfies
+the strict-case hypothesis.
+
+A \emph{spectrahedral shadow} is a linear image of a set defined by an
+affine real symmetric linear matrix inequality.
+Netzer and Sanyal proved that a hyperbolicity cone is a spectrahedral
+shadow when every nonzero boundary point is a smooth point of the
+defining polynomial~\cite[Theorem~1.1]{netzer-sanyal-2015}.
+Our result concerns a direct representation in the original variables;
+it makes no assertion about representations using auxiliary variables.
+
+The scalar ingredient is the coefficient-one Choi--Lam biquadratic
+form~\cite[Section~4]{choi-lam-1977}, which reduces the coefficient of
+the extra cyclic squared monomials from two in Choi's earlier
+nonnegative form~\cite{choi-1975} to one. Its inability to dominate a nonzero
+bilinear square follows from the extremality proved in
+\cite[Theorem~4.4]{choi-lam-1977}; we give a short proof of precisely this
+property. It is called \emph{weak extremality} by Blekherman, Rai\c{t}\u{a},
+Shankar, and Sinn~\cite[Definition~2.7]{blekherman-raita-shankar-sinn-2022},
+whose characterization uses Taylor expansions of dominated squares at
+zeros~\cite[Theorem~3.2]{blekherman-raita-shankar-sinn-2022}.
+The correspondence between nonnegative biquadratic forms and
+positive maps on real symmetric matrices is classical; see
+\cite[Section~3]{ha-2013}. Positive maps have also been used to obtain
+certificates of conic polynomial stability~\cite{dey-gardoll-theobald-2021}.
+Here the construction turns the scalar obstruction into an obstruction
+to every finite pencil describing one hyperbolicity cone. A related
+general strategy appears in Scheiderer's local sums-of-squares
+obstructions to spectrahedral shadows~\cite[Section~4]{scheiderer-2018}.
+Our proof extracts bilinear square minorants from a differentiated
+operator-norm identity; it does not invoke a criterion for shadows.
+
+Gonz\'alez Nevado states a positive solution of the geometric conjecture
+in~\cite[Theorem~53, pp.~16--17]{gonzalez-nevado-2026}.
+Appendix~\ref{app:opposition} records a counterexample to the
+path-containment assertion used in that argument. The construction
+and proof in the present paper are independent of that comparison.
diff --git a/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/sections/01-introduction.tex b/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/sections/01-introduction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..00003a8b2d24509cbe495182969f4ff9545c4d14
--- /dev/null
+++ b/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/sections/01-introduction.tex
@@ -0,0 +1,129 @@
+\section{Introduction}
+\label{sec:introduction}
+
+A homogeneous polynomial $p\in\R[x_1,\ldots,x_m]$ of degree $d\geq1$ is
+\emph{hyperbolic with respect to} $e\in\R^m$ if $p(e)\ne0$ and every root of
+$t\mapsto p(te-x)$ is real for every $x\in\R^m$.
+Write these roots, with multiplicity, as $\lambda_1(x),\ldots,\lambda_d(x)$.
+The closed hyperbolicity cone is
+\[
+ \Lambda_+(p,e)=\{x\in\R^m:\lambda_j(x)\geq0\text{ for }1\leq j\leq d\}.
+\]
+We write $\Sym^N$ for the real symmetric $N\times N$ matrices and
+$A\succeq0$ (respectively, $A\succ0$) for positive semidefiniteness
+(respectively, positive definiteness).
+A cone is \emph{spectrahedral} if it has the form
+\begin{equation}
+ K=\{x\in\R^m:L(x)\succeq0\},\qquad
+ L(x)=\sum_{i=1}^m x_iA_i,\quad A_i\in\Sym^N,
+ \label{eq:spectrahedral}
+\end{equation}
+for some finite $N\geq1$.
+The geometric Generalized Lax conjecture asserts that every hyperbolicity
+cone is spectrahedral. We give a counterexample.
+
+\subsection{The explicit polynomial}
+
+All indices in $\{1,2,3\}$ in the following construction are cyclic.
+For $z,y\in\R^3$, let
+\begin{equation}
+ b(z,y)=\sum_{i=1}^3 z_i^2y_i^2
+       -2\sum_{1\leq i<j\leq3}z_iy_iz_jy_j
+       +\sum_{i=1}^3z_i^2y_{i+1}^2.
+ \label{eq:form}
+\end{equation}
+This is the coefficient-one Choi--Lam biquadratic
+form~\cite[Section~4]{choi-lam-1977}.
+Define $Q(y)\in\Sym^3$ by
+\[
+ Q(y)_{ii}=y_i^2+y_{i+1}^2,\qquad
+ Q(y)_{ij}=-y_iy_j\quad(i\ne j),
+\]
+so that $b(z,y)=z^\top Q(y)z$.
+For $a\in\R$, $r\in\R^3$ and $T\in\Sym^3$, define the linear map
+$\Phi_y:\Sym^4\to\Sym^4$ by
+\begin{equation}
+ \Phi_y\!\begin{pmatrix}a&r^\top\\r&T\end{pmatrix}
+ =\begin{pmatrix}
+    \tr(Q(y)T)&-r^\top Q(y)\\
+    -Q(y)r&aQ(y)
+   \end{pmatrix}.
+ \label{eq:phi}
+\end{equation}
+The dependence on $y$ is homogeneous quadratic. If $\adj X$ denotes the
+adjugate of $X$, set
+\begin{equation}
+ p(X,Z,y)=\det\big((\det X)Z-\Phi_y(\adj X)\big),
+ \qquad (X,Z,y)\in\Sym^4\times\Sym^4\times\R^3.
+ \label{eq:polynomial}
+\end{equation}
+This formula is polynomial even when $X$ is singular. The ambient space
+has dimension $10+10+3=23$.
+
+\begin{theorem}\label{thm:main}
+The polynomial \eqref{eq:polynomial} is homogeneous of degree $20$ and
+hyperbolic with respect to $e=(I_4,I_4,0)$, with $p(e)=1$.
+Its closed hyperbolicity cone $K=\Lambda_+(p,e)$ is not spectrahedral:
+for every finite $N\geq1$ and every real linear map
+$L:\Sym^4\times\Sym^4\times\R^3\to\Sym^N$,
+\[
+ K\ne\{(X,Z,y):L(X,Z,y)\succeq0\}.
+\]
+\end{theorem}
+
+The conclusion concerns the cone itself, so it allows every possible
+defining pencil and every finite matrix size. The compact formula for $p$
+is convenient for the proof, but one determinant factor can be removed.
+
+\begin{corollary}\label{cor:reduced}
+The quotient $q=p/\det X$ extends to a homogeneous polynomial of degree
+$16$ in the same $23$ variables. It is hyperbolic with respect to $e$,
+with $q(e)=1$, and $\Lambda_+(q,e)=K$. In particular, its closed
+hyperbolicity cone is not spectrahedral.
+\end{corollary}
+
+We prove the degree reduction in Section~\ref{sec:reduced}, after the
+obstruction argument. We make no claim that the dimension or degree is
+minimal. The real symmetric convention also covers Hermitian pencils:
+for a Hermitian matrix $H=A+iB$, positivity is equivalent to positivity
+of the real symmetric matrix $\left(\begin{smallmatrix}A&-B\\B&A\end{smallmatrix}\right)$.
+
+\subsection{Historical context}
+
+\input{sections/01-history}
+
+\subsection{Proof strategy}
+
+The scalar obstruction is particularly rigid: $b$ is nonnegative and
+nonzero, but every bilinear form $\ell$ satisfying $\ell(z,y)^2\leq b(z,y)$
+for all $z,y$ is zero. Section~\ref{sec:form} gives a short proof using
+zeros and curves along which $b$ vanishes to fourth order.
+
+The map $\Phi_y$ translates this form into a matrix inequality. In
+Section~\ref{sec:hyperbolic} we prove hyperbolicity and identify the slice
+of $K$ with $X\succ0$ as
+\[
+ Z\succeq\Phi_y(X^{-1}).
+\]
+The remaining argument assumes an arbitrary pencil defining $K$ and
+extracts a bilinear square dominated by $b$.
+First, Section~\ref{sec:pencil} rescales the pencil along
+$(X,s^2Z,sy)$ and obtains an equivalent block matrix inequality.
+Section~\ref{sec:norm} then sends $X^{-1}$ and $Z^{-1}$ toward rank-one projections;
+the resulting comparison yields an exact operator-norm identity for
+kernel projections of the pencil blocks. These projections are
+parametrized by the unit sphere in $\R^4$.
+Finally, Section~\ref{sec:tangent} differentiates one kernel projection
+on a neighborhood of constant rank in that sphere. At a suitable base
+point, the three-dimensional tangent space can be identified with all
+the $z$-variables, so this
+local identity controls $b$ for every $z$ and $y$.
+Each entry of the resulting matrix
+is a bilinear form whose square is bounded by $b$, giving a contradiction.
+
+The mechanism connecting the biquadratic form to the geometry of the cone
+is the rescaling and subsequent first variation of these kernel
+projections. It applies to an arbitrary hypothetical pencil without
+comparing its determinant with $p$.
+All matrix norms below are Euclidean operator norms, denoted by
+$\norm{\,\cdot\,}_{\op}$; vector norms are Euclidean.
diff --git a/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/sections/02-form.tex b/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/sections/02-form.tex
new file mode 100644
index 0000000000000000000000000000000000000000..293f2b43fbe3ef3dfe18d99e39ec82184cc9d356
--- /dev/null
+++ b/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/sections/02-form.tex
@@ -0,0 +1,82 @@
+\section{A biquadratic form with no dominated square}
+\label{sec:form}
+
+The geometric argument will produce bilinear forms whose squares are
+bounded by the Choi--Lam form $b$ in~\eqref{eq:form}.
+We prove directly the particular consequence of its classical
+extremality~\cite[Theorem~4.4]{choi-lam-1977} that the geometric
+argument needs: every such bilinear form must vanish. This property is
+also called \emph{weak extremality}~\cite[Definition~2.7]{blekherman-raita-shankar-sinn-2022}.
+
+\begin{lemma}\label{lem:form}
+The form $b:\R^3\times\R^3\to\R$ defined in~\eqref{eq:form}
+is nonnegative and nonzero. If a real bilinear form
+$\ell:\R^3\times\R^3\to\R$ satisfies
+\[
+ \ell(z,y)^2\leq b(z,y)
+ \qquad\text{for every }z,y\in\R^3,
+\]
+then $\ell=0$.
+\end{lemma}
+
+\begin{proof}
+Put $a_i=z_i^2$, with indices read cyclically modulo $3$.
+The matrix of $b(z,y)$ as a quadratic form in $y$ is
+\[
+ M(z)=
+ \begin{pmatrix}
+ a_1+a_3&-z_1z_2&-z_1z_3\\
+ -z_1z_2&a_2+a_1&-z_2z_3\\
+ -z_1z_3&-z_2z_3&a_3+a_2
+ \end{pmatrix}.
+\]
+Its diagonal entries are nonnegative. Its three principal minors of
+order two are
+\[
+ \det M(z)[\{i,i+1\}]
+ =a_i^2+a_{i-1}(a_i+a_{i+1})\geq0,
+ \qquad i=1,2,3,
+\]
+where the brackets denote a principal submatrix. Finally,
+\[
+ \det M(z)
+ =a_1^2a_2+a_2^2a_3+a_3^2a_1-3a_1a_2a_3\geq0
+\]
+by the arithmetic--geometric mean inequality. All principal minors
+are therefore nonnegative, so $M(z)\succeq0$ and $b(z,y)\geq0$.
+For the standard coordinate vectors $e_1,e_2,e_3$ of $\R^3$,
+we have $b(e_1,e_1)=1$.
+
+Now write $\ell(z,y)=\sum_{i,j}c_{ij}z_i y_j$ and assume the stated
+bound. Zeros will eliminate the off-diagonal coefficients; fourth-order
+vanishing along curves will eliminate the diagonal coefficients.
+At every zero of $b$, the form $\ell$ vanishes.
+In particular,
+\[
+ b(e_i,e_{i-1})=0
+ \quad\Longrightarrow\quad c_{i,i-1}=0.
+\]
+For each sign vector $\sigma\in\{-1,1\}^3$, we also have
+$b(\sigma,\sigma)=3-6+3=0$, and hence
+\[
+ 0=\sum_i c_{ii}
+   +\sum_{i<j}(c_{ij}+c_{ji})\sigma_i\sigma_j.
+\]
+Multiplying by $\sigma_p\sigma_q$ and averaging over the eight sign
+vectors gives $c_{pq}+c_{qp}=0$ for every $p<q$.
+Together with $c_{i,i-1}=0$, these identities force all off-diagonal
+entries to vanish. Thus $\ell(z,y)=\sum_i c_i z_i y_i$ for some real $c_i$.
+
+For a cyclic index $i$ and $s\in\R$, set
+\[
+ z=e_i+s e_{i-1},\qquad y=e_{i-1}+s e_i.
+\]
+Direct substitution gives
+\[
+ b(z,y)=2s^2-2s^2+s^4=s^4,
+ \qquad \ell(z,y)=s(c_i+c_{i-1}).
+\]
+For $s\ne0$, the bound implies $(c_i+c_{i-1})^2\leq s^2$.
+Letting $s\to0$ yields $c_i+c_{i-1}=0$ for all three indices.
+These equations force $c_1=c_2=c_3=0$, proving the lemma.
+\end{proof}
diff --git a/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/sections/03-hyperbolic.tex b/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/sections/03-hyperbolic.tex
new file mode 100644
index 0000000000000000000000000000000000000000..073367a32035eeae6edefd5a0dbc7ef37eb0c64d
--- /dev/null
+++ b/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/sections/03-hyperbolic.tex
@@ -0,0 +1,148 @@
+\section{Hyperbolicity and two cone slices}
+\label{sec:hyperbolic}
+
+We first prove that the polynomial in \eqref{eq:polynomial} is
+hyperbolic.  We then identify the two slices of its closed cone that
+will constrain any representing pencil.
+
+For $u=(u_0,u')$ and $v=(v_0,v')$ in $\R^4$, the block formula
+\eqref{eq:phi} gives
+\begin{equation}
+ u^\top\Phi_y(vv^\top)u
+ = (v_0u'-u_0v')^\top Q(y)(v_0u'-u_0v')
+ = b(v_0u'-u_0v',y).
+ \label{eq:rank-one}
+\end{equation}
+Indeed, the three terms on expanding the middle expression are
+$v_0^2u'^\top Q(y)u'$, $-2u_0v_0u'^\top Q(y)v'$, and
+$u_0^2v'^\top Q(y)v'$, exactly the terms from the block formula.
+Lemma~\ref{lem:form} therefore implies that
+$\Phi_y(vv^\top)\succeq0$.  Every positive semidefinite real symmetric
+matrix is a nonnegative linear combination of such rank-one matrices.
+Thus $\Phi_y$ is a positive linear map: it preserves positive
+semidefiniteness, and hence the order $\preceq$.
+The same formula also gives
+\begin{equation}
+ \Phi_{s y}=s^2\Phi_y\qquad(s\in\R),
+ \label{eq:phi-scaling}
+\end{equation}
+so in particular $\Phi_0=0$ and $\Phi_{-y}=\Phi_y$.
+This rank-one verification is the real-symmetric positive-map
+correspondence for biquadratic forms~\cite[Section~3]{ha-2013}
+applied to the explicit map~\eqref{eq:phi}.
+
+\begin{proposition}\label{prop:hyperbolic}
+The polynomial $p$ in \eqref{eq:polynomial} is homogeneous of
+degree $20$ on $\Sym^4\times\Sym^4\times\R^3$, a real vector
+space of dimension $23$.  It satisfies $p(e)=1$ and is hyperbolic
+with respect to $e=(I_4,I_4,0)$.
+\end{proposition}
+
+\begin{proof}
+Each entry of $(\det X)Z-\Phi_y(\operatorname{adj}X)$ is homogeneous
+of degree $5$: the adjugate has degree $3$, and $\Phi_y$ is quadratic
+in $y$.  Taking its $4\times4$ determinant gives a homogeneous
+polynomial of degree $20$.  Since $\Phi_0=0$, evaluation at $e$
+gives $p(e)=1$, so this polynomial is nonzero.  The dimension is
+$10+10+3=23$.
+
+Extend $\Phi_y$ complex linearly to complex symmetric matrices.
+For invertible $X$, real or complex, linearity and
+$\operatorname{adj}X=(\det X)X^{-1}$ give
+\begin{equation}
+ p(X,Z,y)=(\det X)^4\det\bigl(Z-\Phi_y(X^{-1})\bigr).
+ \label{eq:inverse-formula}
+\end{equation}
+The adjugate formula remains the definition at singular $X$.
+
+Fix real symmetric $X,Z$ and real $y$, and let $t=a+i\eta$ with
+$\eta>0$.  Orthogonal diagonalization of $X$ gives
+\[
+ \operatorname{Im}(tI_4-X)^{-1}
+ =-\eta\bigl((aI_4-X)^2+\eta^2I_4\bigr)^{-1}\prec0.
+\]
+For a complex symmetric matrix, the entrywise imaginary part equals
+the Hermitian imaginary part $(A-A^*)/(2i)$ and is real symmetric.
+Complex linearity and positivity of $\Phi_{-y}$ consequently show
+that
+\[
+ H=tI_4-Z-\Phi_{-y}\bigl((tI_4-X)^{-1}\bigr)
+ \quad\text{satisfies}\quad
+ \operatorname{Im}H\succeq\eta I_4\succ0.
+\]
+Such an $H$ is invertible: if $Hw=0$ for a nonzero complex vector
+$w$, then $0=\operatorname{Im}(w^*Hw)=w^*(\operatorname{Im}H)w>0$.
+Also $tI_4-X$ is invertible.  Equation~\eqref{eq:inverse-formula}
+therefore implies $p(te-(X,Z,y))\ne0$.
+The coefficients of this polynomial in $t$ are real, so conjugation
+also excludes roots in the lower half-plane.  Finally, its leading
+coefficient is $p(e)=1$ by homogeneity; it has degree $20$ for every
+real input.  All its roots are therefore real, as required.
+\end{proof}
+
+\begin{proposition}\label{prop:slices}
+Let $K=\Lambda_+(p,e)$.  For real symmetric $X,Z$ and $y\in\R^3$,
+\begin{align}
+ X\succ0:\qquad
+ (X,Z,y)\in K
+ &\ \Longleftrightarrow\ 
+ Z\succeq\Phi_y(X^{-1}),
+ \label{eq:slice-positive}\\
+ (X,Z,0)\in K
+ &\ \Longleftrightarrow\ 
+ X\succeq0\ \text{and}\ Z\succeq0.
+ \label{eq:slice-zero}
+\end{align}
+Moreover, $e$ lies in the interior of $K$ in the full ambient space.
+\end{proposition}
+
+\begin{proof}
+If $\lambda_1(x),\ldots,\lambda_{20}(x)$ are the roots of
+$t\mapsto p(te-x)$, homogeneity gives
+\[
+ p(x+te)=\prod_{j=1}^{20}(t+\lambda_j(x)).
+\]
+Thus $x\in K$ if and only if $p(x+te)\ne0$ for every real $t>0$.
+The strict inequality on $t$ allows zero roots and hence includes
+the boundary of $K$.
+
+Suppose $X\succ0$ and set, for $t\ge0$,
+\[
+ F(t)=Z+tI_4-\Phi_y\bigl((X+tI_4)^{-1}\bigr).
+\]
+If $F(0)\succeq0$, order preservation yields
+\[
+ F(t)=F(0)+tI_4+
+ \Phi_y\bigl(X^{-1}-(X+tI_4)^{-1}\bigr)
+ \succeq tI_4\succ0\qquad(t>0).
+\]
+Equation~\eqref{eq:inverse-formula} then shows that there is no
+positive root, so $(X,Z,y)\in K$.
+
+Conversely, if $F(0)\not\succeq0$, its least eigenvalue is negative.
+For $t>0$, positivity gives
+\[
+ 0\preceq\Phi_y\bigl((X+tI_4)^{-1}\bigr)
+ \preceq t^{-1}\Phi_y(I_4).
+\]
+Consequently $F(t)\succ0$ for all sufficiently large $t$.
+Continuity of its least eigenvalue gives a $t>0$ at which $F(t)$
+is singular.  Since $X+tI_4\succ0$,
+Equation~\eqref{eq:inverse-formula} now gives a positive root of
+$p((X,Z,y)+te)$.  This proves \eqref{eq:slice-positive} in both
+directions, including equality in the matrix inequality.
+
+When $y=0$, the polynomial definition directly gives
+\[
+ p(te-(X,Z,0))=\det(tI_4-X)^4\det(tI_4-Z),
+\]
+also for singular $X$ and $Z$.  Its roots are the eigenvalues of $X$,
+each repeated four times, and the eigenvalues of $Z$.
+They are all nonnegative exactly when both matrices are positive
+semidefinite.  This proves \eqref{eq:slice-zero}.
+
+Finally, $X\succ0$ and $Z-\Phi_y(X^{-1})\succ0$ define an open
+subset of the full ambient space.  It contains $e$ and, by
+\eqref{eq:slice-positive}, is contained in $K$.  Thus $e$ is an
+interior point.
+\end{proof}
diff --git a/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/sections/04-pencil.tex b/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/sections/04-pencil.tex
new file mode 100644
index 0000000000000000000000000000000000000000..e190c9304333baa54cfbb9c6e6616b9edb536d8d
--- /dev/null
+++ b/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/sections/04-pencil.tex
@@ -0,0 +1,157 @@
+\section{What a semidefinite representation would imply}
+\label{sec:pencil}
+
+The two slices in Proposition~\ref{prop:slices} constrain every pencil
+representing $K$.  We now turn those constraints into an exact block
+matrix inequality whose off-diagonal block is linear in $y$.
+A positive linear map preserves positive semidefiniteness; it is
+called \emph{unital} if it sends the identity to the identity.
+
+\begin{proposition}\label{prop:pencil}
+Let $K=\Lambda_+(p,(I_4,I_4,0))$ for the polynomial
+in~\eqref{eq:polynomial}. Suppose that $K$ admits a representation by a finite homogeneous real
+symmetric linear pencil.  Then there are integers $a,c\geq1$, positive
+linear maps
+\[
+ D:\Sym^4\longrightarrow\Sym^a,
+ \qquad E:\Sym^4\longrightarrow\Sym^c,
+ \qquad D(I_4)=I_a,\quad E(I_4)=I_c,
+\]
+and a linear map $B:\R^3\longrightarrow\R^{a\times c}$ such that, for
+every $X\succ0$, $Z\in\Sym^4$ and $y\in\R^3$,
+\begin{equation}\label{eq:schur-equivalence}
+ Z\succeq\Phi_y(X^{-1})
+ \quad\Longleftrightarrow\quad
+ E(Z)\succeq B(y)^\top D(X)^{-1}B(y).
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+Write the hypothetical pencil as
+\[
+ L(X,Z,y)=L_1(X)+L_2(Z)+L_3(y).
+\]
+We first remove the pencil's common kernel and normalize
+two positive blocks.
+We then rescale the pencil and prove that its limit represents the
+same inequality.
+
+\emph{Compression and normalization.}
+We use the elementary fact that
+\[
+ A+H\succeq0,\quad A-H\succeq0
+ \quad\Longrightarrow\quad \ker A\subseteq\ker H.
+\]
+Indeed, if $w\in\ker A$, the two nonnegative quadratic forms on $w$
+sum to zero.  A positive semidefinite matrix annihilates every vector
+on which its quadratic form vanishes, so both $(A+H)w$ and $(A-H)w$
+are zero.
+
+Since $e$ is interior to $K$, for every ambient vector $x$ there is
+an $\varepsilon>0$ with $L(e)\pm\varepsilon L(x)\succeq0$.
+The preceding fact shows that $\ker L(e)$ is a common kernel of the
+whole pencil.  Compressing to its orthogonal complement preserves the
+represented cone and makes $L(e)\succ0$.  This complement is nonzero:
+otherwise the pencil would vanish identically, whereas
+$(-I_4,I_4,0)\notin K$ by~\eqref{eq:slice-zero}.
+
+The same slice shows that $L_1$ and $L_2$ are positive maps.  Set
+$U=\ker L_1(I_4)$.  Positivity at $I_4\pm\varepsilon X$, for small
+$\varepsilon>0$, and the same kernel argument show that every
+$L_1(X)$ annihilates $U$.  Thus, in $U^\perp\oplus U$,
+\[
+ L_1(X)=\begin{pmatrix}D(X)&0\\0&0\end{pmatrix}.
+\]
+Both summands are nonzero.  If $U^\perp=0$, then $L_1=0$ and the
+pencil accepts $(-I_4,I_4,0)$, contradicting~\eqref{eq:slice-zero}.
+If $U=0$, then $L_1(I_4)\succ0$, so
+$L_1(I_4)-\varepsilon L_2(I_4)\succ0$ for sufficiently small
+$\varepsilon>0$.  This would accept $(I_4,-\varepsilon I_4,0)$,
+again contradicting that slice.
+
+Let $a=\dim U^\perp$, $c=\dim U$, and let $E(Z)$ be the lower
+diagonal block of $L_2(Z)$.  Both $D$ and $E$ are positive maps.
+Moreover, $D(I_4)\succ0$ by the definition of $U$, and
+$E(I_4)\succ0$ because it is the compression of $L(e)\succ0$ to
+$U$.  Apply the fixed block diagonal congruence
+\[
+ \operatorname{diag}\bigl(D(I_4)^{-1/2},E(I_4)^{-1/2}\bigr)
+\]
+and retain the notation for the resulting blocks.  We now have
+$D(I_4)=I_a$ and $E(I_4)=I_c$.
+
+\emph{Rescaling the pencil.}
+Write the other blocks as
+\[
+ L_2(Z)=\begin{pmatrix}F(Z)&C(Z)\\C(Z)^\top&E(Z)\end{pmatrix},
+ \qquad
+ L_3(y)=\begin{pmatrix}A(y)&B(y)\\B(y)^\top&G(y)\end{pmatrix}.
+\]
+Fix $y$ and put $Z_0=\Phi_y(I_4)$.  By
+\eqref{eq:slice-positive} and $\Phi_{sy}=s^2\Phi_y$, the point
+$(I_4,s^2Z_0,sy)$ lies in $K$ for every real $s$.  Its lower pencil
+block therefore satisfies
+\[
+ s^2E(Z_0)+sG(y)\succeq0 \qquad(s\in\R).
+\]
+Dividing by $|s|$ and taking limits through positive and negative
+$s$ gives $G(y)\succeq0$ and $-G(y)\succeq0$.  Hence $G(y)=0$.
+
+For fixed $X\succ0$, $Z$ and $y$, and any real $s\ne0$, congruence
+of $L(X,s^2Z,sy)$ by $\operatorname{diag}(I_a,s^{-1}I_c)$ gives
+\begin{equation}\label{eq:scaled-pencil}
+ M_s(X,Z,y)=
+ \begin{pmatrix}
+ D(X)+sA(y)+s^2F(Z)&B(y)+sC(Z)\\
+ B(y)^\top+sC(Z)^\top&E(Z)
+ \end{pmatrix}.
+\end{equation}
+Congruence preserves positive semidefiniteness for either sign of
+$s$.  The exact slice~\eqref{eq:slice-positive} gives
+\begin{equation}\label{eq:scaled-test}
+ M_s(X,Z,y)\succeq0
+ \quad\Longleftrightarrow\quad Z\succeq\Phi_y(X^{-1}),
+ \qquad s\ne0.
+\end{equation}
+As $s\to0$, these matrices converge to
+\[
+ M(X,Z,y)=
+ \begin{pmatrix}D(X)&B(y)\\B(y)^\top&E(Z)\end{pmatrix}.
+\]
+It follows immediately that $Z\succeq\Phi_y(X^{-1})$ implies
+$M(X,Z,y)\succeq0$.
+
+\emph{Recovering the inequality from the limit.}
+For the converse, suppose that $M(X,Z,y)\succeq0$.  Positivity and
+unitality give
+\[
+ D(X)\succeq\lambda_{\min}(X)I_a\succ0.
+\]
+For every $\delta>0$,
+\[
+ M(X,Z+\delta I_4,y)
+ =M(X,Z,y)+\operatorname{diag}(0,\delta I_c)
+ \succ0.
+\]
+To see strict positivity, a vector with nonzero lower component has
+strictly positive contribution from the added term.  A nonzero
+vector with lower component zero has strictly positive quadratic
+form under $D(X)$.
+
+Consequently, for each fixed $\delta>0$, the matrices
+$M_s(X,Z+\delta I_4,y)$ are positive definite for sufficiently small
+nonzero $s$.  Equation~\eqref{eq:scaled-test} yields
+$Z+\delta I_4\succeq\Phi_y(X^{-1})$.  Letting $\delta\downarrow0$
+proves the converse.  We have therefore shown
+\[
+ Z\succeq\Phi_y(X^{-1})
+ \quad\Longleftrightarrow\quad M(X,Z,y)\succeq0.
+\]
+Taking the Schur complement of the positive definite block $D(X)$
+gives~\eqref{eq:schur-equivalence}.
+\end{proof}
+
+The maps and their finite dimensions are now fixed by the
+hypothetical pencil.  We next recover the scalar form $b$ from
+\eqref{eq:schur-equivalence}; this will force a nonzero bilinear square
+dominated by $b$ and contradict its defining obstruction.
diff --git a/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/sections/05-norm.tex b/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/sections/05-norm.tex
new file mode 100644
index 0000000000000000000000000000000000000000..571698a7d88268c53c9a4cf3faf23a8fa8e9ed34
--- /dev/null
+++ b/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/sections/05-norm.tex
@@ -0,0 +1,122 @@
+\section{Rank-one limits and a norm identity}
+\label{sec:norm}
+
+Let $D,E,B$ be the maps supplied by Proposition~\ref{prop:pencil}.
+Thus $D$ and $E$ are positive maps with output sizes $a,c\geq1$ and
+are \emph{unital}, meaning $D(I_4)=I_a$ and $E(I_4)=I_c$,
+and $B(y)$ is an $a\times c$ matrix.
+For unit vectors $u,v\in\R^4$, define the orthogonal projections
+\begin{equation}
+ P(v)=\operatorname{proj}_{\ker D(I-vv^\top)},
+ \qquad
+ R(u)=\operatorname{proj}_{\ker E(I-uu^\top)}.
+ \label{eq:projections}
+\end{equation}
+These act on $\R^a$ and $\R^c$, respectively.
+The next proposition recovers the scalar form~\eqref{eq:form} from the
+Schur threshold~\eqref{eq:schur-equivalence}.
+
+\begin{proposition}
+\label{prop:norm}
+Let $a,c\geq1$, let $D:\Sym^4\to\Sym^a$ and
+$E:\Sym^4\to\Sym^c$ be positive unital linear maps, and let
+$B:\R^3\to\R^{a\times c}$ be linear. Assume that
+\eqref{eq:schur-equivalence} holds for every $X\succ0$,
+$Z\in\Sym^4$ and $y\in\R^3$, with $\Phi_y$ defined
+by~\eqref{eq:phi}. Define $P(v)$ and $R(u)$ by~\eqref{eq:projections}.
+For every pair of unit vectors $u=(u_0,u'),v=(v_0,v')\in\R^4$
+and every $y\in\R^3$,
+\begin{equation}
+ b(v_0u'-u_0v',y)
+ =\norm{P(v)B(y)R(u)}_{\op}^{2}.
+ \label{eq:norm-identity}
+\end{equation}
+Here the operator norm uses the Euclidean structures on the two block spaces.
+\end{proposition}
+
+\begin{proof}
+Fix $u,v,y$. For $t\geq1$, set
+\[
+ X_t=vv^\top+t(I-vv^\top),
+ \qquad
+ Z_t=uu^\top+t(I-uu^\top).
+\]
+Both matrices are at least $I$. Positivity and unitality therefore give
+$D(X_t)\succeq I$ and $E(Z_t)\succeq I$, so all inverse square roots below
+exist. Put
+\[
+ A_t=Z_t^{-1/2}\Phi_y(X_t^{-1})Z_t^{-1/2},
+ \qquad
+ C_t=D(X_t)^{-1/2}B(y)E(Z_t)^{-1/2}.
+\]
+The matrix $A_t$ is positive semidefinite. For every real $r$, congruence
+by $Z_t^{-1/2}$ gives
+\[
+ rZ_t\succeq\Phi_y(X_t^{-1})
+ \quad\Longleftrightarrow\quad
+ r\geq\lambda_{\max}(A_t).
+\]
+Likewise, linearity of $E$ and congruence by $E(Z_t)^{-1/2}$ give
+\[
+ \begin{split}
+ E(rZ_t)\succeq B(y)^\top D(X_t)^{-1}B(y)
+ &\quad\Longleftrightarrow\quad rI\succeq C_t^\top C_t\\
+ &\quad\Longleftrightarrow\quad r\geq\norm{C_t}_{\op}^{2}.
+ \end{split}
+\]
+Equation~\eqref{eq:schur-equivalence} identifies these two closed rays in
+$r\in\R$, including their endpoints. Consequently,
+\begin{equation}
+ \lambda_{\max}(A_t)=\norm{C_t}_{\op}^{2}.
+ \label{eq:threshold-norm}
+\end{equation}
+This applies to rectangular $C_t$ and includes a zero threshold.
+
+The explicit formulas
+\[
+ X_t^{-1}=vv^\top+t^{-1}(I-vv^\top),
+ \qquad
+ Z_t^{-1/2}=uu^\top+t^{-1/2}(I-uu^\top)
+\]
+show that
+\[
+ A_t\longrightarrow
+ uu^\top\Phi_y(vv^\top)uu^\top
+ =\alpha uu^\top,
+ \qquad
+ \alpha=u^\top\Phi_y(vv^\top)u\geq0.
+\]
+The sign follows from positivity of $\Phi_y$. Thus the largest eigenvalue
+of this limit is $\alpha$, also when $\alpha=0$. Continuity of the largest
+eigenvalue gives $\lambda_{\max}(A_t)\to\alpha$.
+
+For the other side, unitality gives
+\[
+ D(X_t)=I+(t-1)D(I-vv^\top).
+\]
+If $H\succeq0$ is a fixed finite-dimensional matrix, diagonalizing $H$
+shows that
+\begin{equation}
+ [I+(t-1)H]^{-1/2}
+ \longrightarrow\operatorname{proj}_{\ker H}
+ \quad\text{in operator norm}.
+ \label{eq:kernel-limit}
+\end{equation}
+Indeed, the factor on an eigenvector of eigenvalue $\mu\geq0$ is
+$(1+(t-1)\mu)^{-1/2}$, which is one when $\mu=0$ and tends to zero
+otherwise. Applying~\eqref{eq:kernel-limit} to $D(I-vv^\top)$ and
+$E(I-uu^\top)$ yields
+\[
+ D(X_t)^{-1/2}\longrightarrow P(v),
+ \qquad
+ E(Z_t)^{-1/2}\longrightarrow R(u).
+\]
+Hence $C_t\to P(v)B(y)R(u)$ in operator norm. The dimensions $a,c$ are
+fixed throughout; the limits include zero and identity kernel projections.
+Taking limits in~\eqref{eq:threshold-norm} and using the rank-one
+identity~\eqref{eq:rank-one} proves~\eqref{eq:norm-identity}.
+\end{proof}
+
+At $u=v$, the left side of~\eqref{eq:norm-identity} vanishes, so
+$P(v)B(y)R(v)=0$ for every $y$.
+We next study the identity as $u$ varies near $v$.
diff --git a/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/sections/06-tangent.tex b/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/sections/06-tangent.tex
new file mode 100644
index 0000000000000000000000000000000000000000..a54eba6316a946afad1fc1588e90356a26c7d5e6
--- /dev/null
+++ b/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/sections/06-tangent.tex
@@ -0,0 +1,89 @@
+\section{The tangent obstruction}\label{sec:tangent}
+
+We finish the proof by taking a first-order limit in the norm identity
+at a point where the kernel projection varies smoothly. This produces
+a bilinear matrix whose entries must vanish by Lemma~\ref{lem:form}.
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+Proposition~\ref{prop:hyperbolic} establishes the polynomial's degree,
+normalization, and hyperbolicity. Suppose its closed hyperbolicity cone
+has a real symmetric pencil representation of some finite size.
+Fix the maps \(D,E,B\) supplied by Proposition~\ref{prop:pencil}, with
+their fixed finite dimensions \(a,c\). Proposition~\ref{prop:norm} gives
+\eqref{eq:norm-identity} for the projections in \eqref{eq:projections}.
+We shall show that this identity is impossible.
+
+Write \(S^3=\{u\in\R^4:\norm{u}=1\}\) and consider the symmetric matrix
+\[
+ A(u)=E(I-uu^\top),\qquad
+ u\in\Omega:=\{u\in S^3:u_0\ne0\}.
+\]
+Its ranks take only finitely many values, so their maximum \(\rho\) is
+attained at some \(v\in\Omega\). If \(\rho>0\), select \(\rho\) independent
+columns of \(A(v)\), and let \(W(u)\) contain the same columns of \(A(u)\).
+These columns remain independent on a neighborhood of \(v\) in
+\(\Omega\). By maximality of \(\rho\), they span the range of \(A(u)\)
+throughout that neighborhood. Symmetry of \(A(u)\) then gives
+\[
+ R(u)=I-W(u)\bigl(W(u)^\top W(u)\bigr)^{-1}W(u)^\top .
+\]
+This is a smooth matrix function. If \(\rho=0\), then \(R(u)=I\) on
+\(\Omega\), so it is smooth in this case as well.
+
+Keep \(v\) fixed. At \(u=v\), Equation~\eqref{eq:norm-identity} gives
+\[
+ P(v)B(y)R(v)=0\qquad(y\in\R^3),
+\]
+because \(b(0,y)=0\). For \(h=(h_0,h')\in v^\perp\), define
+\[
+ Jh=v_0h'-h_0v',
+ \qquad
+ u_s=\frac{v+sh}{\sqrt{1+s^2\norm{h}^2}}.
+\]
+For small \(s\), this path stays in the neighborhood where \(R\) is
+smooth, and its velocity at zero is \(h\). Moreover,
+\[
+ v_0u_s'-(u_s)_0v'
+   =\frac{s\,Jh}{\sqrt{1+s^2\norm{h}^2}}.
+\]
+Thus, for \(s\ne0\), homogeneity of \(b\) and
+\eqref{eq:norm-identity} give
+\[
+ \frac{b(Jh,y)}{1+s^2\norm{h}^2}
+ =
+ \norm{P(v)B(y)\frac{R(u_s)-R(v)}{s}}_{\op}^{2}.
+\]
+The matrix difference quotient converges to the differential
+\(dR_v(h)\). Passing to the limit by continuity of the operator norm
+yields
+\begin{equation}\label{eq:tangent-identity}
+ b(Jh,y)=\norm{P(v)B(y)dR_v(h)}_{\op}^{2}
+ \qquad(h\in v^\perp,\ y\in\R^3).
+\end{equation}
+Only \(R\) has been differentiated; the projection \(P(v)\) remains
+fixed. Figure~\ref{fig:local-tangent} illustrates this local variation.
+
+\input{figures/tangent-variation}
+
+The linear map \(J:v^\perp\to\R^3\) is an isomorphism. Indeed, if
+\(Jh=0\), then \(v_0\ne0\) gives
+\(h=(h_0/v_0)v\); since \(h\perp v\), this forces \(h=0\).
+Both spaces have dimension three.
+Consequently the matrix
+\[
+ F(z,y)=P(v)B(y)dR_v(J^{-1}z),\qquad z,y\in\R^3,
+\]
+has entries that are bilinear in \(z,y\), since \(B\) and the differential
+\(dR_v\) are linear. In fixed orthonormal bases,
+each entry \(\ell_{ij}(z,y)\) satisfies
+\[
+ \ell_{ij}(z,y)^2
+ \le \norm{F(z,y)}_{\op}^{2}
+ =b(z,y)
+ \qquad(z,y\in\R^3).
+\]
+Lemma~\ref{lem:form} forces every entry to vanish identically.
+Equation~\eqref{eq:tangent-identity} would therefore make \(b\)
+identically zero, contradicting \(b(e_1,e_1)=1\).
+This excludes the hypothetical finite pencil and completes the proof.
+\end{proof}
diff --git a/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/sections/07-comparison.tex b/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/sections/07-comparison.tex
new file mode 100644
index 0000000000000000000000000000000000000000..75ebf98af28f6399c5a4fe738bdd018cddff9f37
--- /dev/null
+++ b/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/sections/07-comparison.tex
@@ -0,0 +1,56 @@
+\section{A path-containment assertion}
+\label{app:opposition}
+
+Theorem~53 (pp.~16--17) of version~1 of Gonz\'alez Nevado's
+\cite{gonzalez-nevado-2026} states a positive solution of the geometric
+Generalized Lax conjecture. Its proof invokes Theorem~52 (p.~16), which asserts
+containment of a fixed rigidly convex set throughout a smooth real-zero
+deformation from a product of normalized tangent linear factors to a
+multiple of the defining polynomial. The following example violates
+that intermediate assertion under its stated hypotheses.
+
+A real polynomial $g$ with $g(0)=1$ is \emph{real-zero} if the roots of
+$s\mapsto g(sx)$ are real for every real vector $x$. Its closed
+rigidly convex set is the closure of the component of $\{g\ne0\}$
+containing the origin. Fix $a,b,\eta>0$ and, in one variable, put
+\[
+ g(x)=(1-x/a)(1+x/b),\qquad
+ c(t)=1+\eta\sin^2(\pi t),\qquad
+ H_t(x)=g(c(t)x),\quad 0\le t\le1.
+\]
+The zero set of $g$ is the compact smooth set $\{-b,a\}$.
+The normalized tangent linear factors at these two points are
+$1+x/b$ and $1-x/a$, whose product is $g$. Every $H_t$ has value $1$
+at the origin and two distinct real roots,
+\[
+ -\frac{b}{c(t)}\quad\hbox{and}\quad\frac{a}{c(t)}.
+\]
+Thus the path depends smoothly on $t$ and consists of real-zero
+polynomials with compact smooth zero sets. It even admits the smooth
+monic real symmetric determinantal representation
+\[
+ H_t(x)=\det\begin{pmatrix}
+ 1-c(t)x/a&0\\
+ 0&1+c(t)x/b
+ \end{pmatrix}.
+\]
+Both endpoints equal $g$, so the final polynomial is $g$ times the
+constant cofactor $1$, as allowed by the stated hypotheses.
+Nevertheless the rigidly convex set of $H_t$ is
+\[
+ \left[-\frac{b}{c(t)},\frac{a}{c(t)}\right],
+\]
+which is strictly smaller than $[-b,a]$ whenever $0<t<1$.
+Consequently it does not contain the rigidly convex set of $g$.
+
+The parameters $a,b>0$ describe the open family of normalized real
+quadratics with negative leading coefficient, so the failure is not
+confined to a symmetric or multiple-root example. The parameter $\eta$
+may be arbitrarily small. The cited Theorem~52 states no restriction
+to two or more variables, no lower bound on pencil size, and no
+requirement that the endpoint cofactor be nonconstant.
+
+This example refutes the all-intermediate-times conclusion of that
+theorem. Its endpoints coincide, so it does not refute an endpoint-only
+statement. The nonspectrahedrality result of this paper follows from
+the construction and obstruction proved above.
diff --git a/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/sections/reduced-polynomial.tex b/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/sections/reduced-polynomial.tex
new file mode 100644
index 0000000000000000000000000000000000000000..9086142f1ea38fb1f5c71e83145a3579f9555bed
--- /dev/null
+++ b/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/build/sections/reduced-polynomial.tex
@@ -0,0 +1,60 @@
+\section{Removing a determinant factor}\label{sec:reduced}
+
+The polynomial $p$ makes the positive-map construction explicit through
+a $4\times4$ determinant. We now remove a factor that is unnecessary for
+its hyperbolicity cone. A larger block determinant shows directly that
+the quotient remains polynomial at singular $X$.
+
+\begin{proof}[Proof of Corollary~\ref{cor:reduced}]
+For the standard coordinate vectors $e_i$ of $\R^3$, put
+\[
+ K_i=\begin{pmatrix}0&e_i^\top\\-e_i&0\end{pmatrix}\in\R^{4\times4},
+ \qquad C=\begin{pmatrix}K_1&K_2&K_3\end{pmatrix}\in\R^{4\times12}.
+\]
+Here $U\otimes V$ denotes the block matrix with blocks $U_{ij}V$.
+Direct block multiplication in~\eqref{eq:phi} gives, for $T\in\Sym^4$,
+\[
+ \Phi_y(T)=\sum_{i,j=1}^3 Q(y)_{ij}K_iTK_j^\top
+          =C\bigl(Q(y)\otimes T\bigr)C^\top.
+\]
+Define the polynomial
+\begin{equation}\label{eq:reduced-polynomial}
+ q(X,Z,y)=\det\begin{pmatrix}
+ I_3\otimes X&C^\top\\
+ C\bigl(Q(y)\otimes I_4\bigr)&Z
+ \end{pmatrix}.
+\end{equation}
+For invertible $X$, taking a Schur complement yields
+\[
+ q(X,Z,y)=(\det X)^3\det\bigl(Z-\Phi_y(X^{-1})\bigr).
+\]
+Comparison with~\eqref{eq:inverse-formula} proves the polynomial identity
+\begin{equation}\label{eq:reduced-factor}
+ p(X,Z,y)=(\det X)q(X,Z,y)
+\end{equation}
+first for invertible $X$ and then everywhere by continuity.
+Since $p$ and $\det X$ are homogeneous of degrees $20$ and $4$, this
+identity makes $q$ homogeneous of degree $16$; also $q(e)=1$.
+For each real $(X,Z,y)$, the polynomial $q(te-(X,Z,y))$ is a factor of
+the real-rooted polynomial $p(te-(X,Z,y))$. Hence $q$ is hyperbolic
+with respect to $e$.
+
+Write $K_q=\Lambda_+(q,e)$. The factorization~\eqref{eq:reduced-factor}
+gives
+\begin{equation}\label{eq:cone-intersection}
+ K=\{(X,Z,y):X\succeq0\}\cap K_q,
+\end{equation}
+because the roots contributed by $\det(tI_4-X)$ are the eigenvalues of
+$X$. It remains to show that $K_q$ already forces $X\succeq0$.
+The polynomial $q$ is even in $y$, so $(X,Z,y)\in K_q$ implies
+$(X,Z,-y)\in K_q$. By convexity of hyperbolicity
+cones~\cite{garding-1959}, their midpoint $(X,Z,0)$ also belongs to $K_q$.
+But~\eqref{eq:reduced-polynomial} gives
+\[
+ q(X,Z,0)=(\det X)^3\det Z,
+\]
+whose closed hyperbolicity cone in this slice is exactly
+$\{(X,Z,0):X\succeq0,\ Z\succeq0\}$.
+Thus $X\succeq0$ throughout $K_q$, and~\eqref{eq:cone-intersection}
+implies $K_q=K$. Theorem~\ref{thm:main} now supplies nonspectrahedrality.
+\end{proof}
diff --git a/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/nonspectrahedral-hyperbolicity-cone.pdf b/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/nonspectrahedral-hyperbolicity-cone.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..99bb0d5d8501b45930801d7c72c35536d9be58fc
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+++ b/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026/nonspectrahedral-hyperbolicity-cone.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:c0522a9b099318b2b79a90219749b18e1f917b4c0c0c5337e420640d1989dddf
+size 410499
diff --git a/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/README.md b/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..ec906f2f3169718478622f2debf4c92b60864a22
--- /dev/null
+++ b/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/README.md
@@ -0,0 +1,18 @@
+# [A PD4 group without an aspherical manifold model](paper.pdf)
+
+OpenAI · September 24, 2026
+
+## Citation
+
+To cite this manuscript:
+
+```bibtex
+@misc{OAI:A-PD4-group-without-an-aspherical-manifold-model-September-24-2026,
+  author = {{OpenAI}},
+  title = {{A PD4 group without an aspherical manifold model}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/paper.pdf}{OAI:A-PD4-group-without-an-aspherical-manifold-model-September-24-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/build/paper.tex b/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/build/paper.tex
new file mode 100644
index 0000000000000000000000000000000000000000..0e246e05387020fd1c0d767f0fc0402b0a819a73
--- /dev/null
+++ b/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/build/paper.tex
@@ -0,0 +1,242 @@
+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage[margin=1.05in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{microtype}
+\usepackage{enumitem}
+\usepackage{needspace}
+\usepackage{float}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta,positioning,calc}
+\usepackage[hidelinks]{hyperref}
+\usepackage{bookmark}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\newtheorem{convention}[theorem]{Convention}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newcommand{\Z}{\mathbb Z}
+\newcommand{\R}{\mathbb R}
+\newcommand{\Q}{\mathbb Q}
+\newcommand{\PD}{\mathrm{PD}}
+\newcommand{\CAT}{\operatorname{CAT}}
+\DeclareMathOperator{\Wh}{Wh}
+\DeclareMathOperator{\im}{im}
+\DeclareMathOperator{\coker}{coker}
+\DeclareMathOperator{\rank}{rank}
+\setlist[enumerate]{itemsep=3pt,topsep=5pt}
+\setlist[itemize]{itemsep=3pt,topsep=5pt}
+\emergencystretch=2em
+\clubpenalty=10000
+\widowpenalty=10000
+\hypersetup{pdftitle={A PD4 group without an aspherical manifold model},pdfauthor={OpenAI}}
+\title{A PD4 group without an aspherical manifold model}
+\author{OpenAI}
+\date{September 24, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We construct a finitely presented integral Poincar\'e duality group of
+dimension four that has a finite classifying space but is not the fundamental
+group of any closed aspherical topological four-manifold. This gives a negative
+answer to Wall's manifold-realization question in dimension four.
+\end{abstract}
+
+\section{Introduction}\label{sec:introduction}
+
+A closed aspherical manifold is determined up to homotopy by its
+fundamental group. Poincar\'e duality therefore supplies a necessary
+algebraic condition on that group. The manifold-realization question asks
+whether this condition, together with the appropriate finiteness
+assumptions, is sufficient. Here the category matters: a finite Poincar\'e
+complex, a homology manifold, and a smooth manifold are different possible
+realizations. We address closed \emph{topological} manifolds in the same
+dimension as the duality group.
+
+An oriented integral Poincar\'e duality group $G$ of dimension $n$, or
+\emph{oriented integral $\PD_n$ group}, has a finite resolution of the
+trivial $\Z G$-module $\Z$ by finitely generated projective modules;
+its cohomology with coefficients in $\Z G$ vanishes outside degree $n$
+and is $\Z$ in degree $n$, with trivial right $G$-action. A finite
+classifying space is a finite CW complex with fundamental group $G$
+and contractible universal cover. Such a space provides a finite free
+resolution, but its existence alone does not provide a manifold model.
+
+Wall asked whether every integral Poincar\'e duality group is the
+fundamental group of a closed aspherical manifold
+\cite[Problem~G2]{WallProblems}. The finiteness condition requires care.
+Davis used reflection groups to turn the finiteness examples of
+Bestvina and Brady into integral $\PD_n$ groups that are not finitely
+presented, for every $n\geq4$ \cite[Theorem~C]{DavisCohomology}.
+Since a closed manifold has finitely presented fundamental group,
+these examples answer the unqualified question. The remaining
+manifold-realization question asks the same thing for \emph{finitely
+presented} integral Poincar\'e duality groups; it is Question~3.4 in
+Davis's survey~\cite{DavisPD} and Conjecture~7.29 in the cited version
+of L\"uck's survey~\cite{LuckFJ}. Kielak's 2026 survey records that
+no counterexamples to this formulation were known
+\cite[Section~4.1]{Kielak}.
+
+The coefficient ring is equally important. Fowler constructed
+torsion-free finitely presented rational Poincar\'e duality groups that
+cannot be realized by closed aspherical ANR rational homology
+manifolds~\cite[Theorem~2.1]{Fowler}. His examples have no finite
+classifying space. The example below instead satisfies integral
+duality and already has a finite four-dimensional classifying space;
+its obstruction concerns the passage from that Poincar\'e complex to a
+closed topological manifold.
+
+\begin{theorem}\label{thm:main}
+There is a finitely presented group $G$ with a finite, oriented,
+four-dimensional aspherical Poincar\'e complex $Q$ such that
+\[
+ H^i(G;\Z G)=0\quad(i\ne4),\qquad H^4(G;\Z G)\cong\Z
+\]
+with trivial right $G$-action on the last group, but no closed aspherical
+topological four-manifold has fundamental group $G$.
+In particular the trivial left $\Z G$-module $\Z$ has a finite resolution
+by finitely generated free modules.
+\end{theorem}
+
+Theorem~\ref{thm:main} refutes the assertion quantified simultaneously
+over all dimensions $n\geq3$, without asserting a counterexample in every
+dimension. The orientation qualification causes no additional issue:
+the orientation character of any hypothetical aspherical manifold with
+group $G$ would equal the right action on its dualizing module and hence
+would be trivial.
+
+\subsection{The source of the obstruction}
+Our starting point is the marked geometric and tensor obstruction in
+\emph{A marked tensor obstruction to four-dimensional disk
+embedding}~\cite{TO}. Its geometric package records a finite family of
+based loop relations, including independently chosen conjugating paths,
+together with algebraic dual spheres. Its algebraic package turns a
+compatible cut configuration into potentials and exact identities over
+a ring with nilpotent parameters. A finite tensor theorem excludes those
+identities. Section~\ref{ch:inputs} recalls the complete interface,
+including the mixed nilpotent terms and the memberships in actual
+gradient ideals. The marked geometry, algebraicity, formal comparison,
+and finite tensor theorem are inputs from the companion; the transfer
+to arbitrary marked topological fillings is proved here.
+
+The argument has two principal geometric ingredients beyond that input.
+First, we construct a smooth chamber $D$ with boundary $S$ and a specified
+hyperbolic basis for its entire middle homology. Attaching three-cells along that basis
+gives a marked graph-type Poincar\'e pair $(P,S)$, but its marked boundary
+admits no topological graph-type filling.
+Here $H=\pi_1D$ is free and $BH$ denotes a finite graph with fundamental
+group $H$; the space $P$ is homotopy equivalent to $BH$.
+The marking is a specified map $S\to BH$. A
+\emph{marked graph-type filling} is a compact topological four-manifold
+with identified boundary $S$ and a homotopy equivalence to $BH$ extending
+that map up to homotopy. This requirement retains information about the
+boundary loops throughout the construction.
+The extension of the cut obstruction to topological fillings uses
+smoothing off isolated points and a differential-form complex that
+computes the cohomology \emph{with those points filled}. Second, we show
+that a hypothetical manifold model for a reflection of $P$ would produce
+exactly such a forbidden filling. This requires a stable product with a
+specified sphere marking, followed by a destabilization argument. An
+unmarked stable homeomorphism would not suffice.
+
+The chamber obstruction is a relative manifold-realization statement:
+the manifold must realize the homotopy type of the given Poincar\'e pair
+relative to its identified boundary. Davis and Hillman prove such relative realization for an
+aspherical Poincar\'e pair with nonempty closed manifold boundary in
+dimensions at least five, assuming the $K$- and $L$-theoretic
+Farrell--Jones conjectures, and in dimension four for elementary
+amenable fundamental groups~\cite[Theorem~D]{DavisHillman}. Our chamber
+has nonabelian free fundamental group, and its marked filling obstruction concerns
+the four-dimensional case outside that elementary amenable hypothesis.
+Reflection converts this relative obstruction into the closed
+manifold-realization question of Theorem~\ref{thm:main}.
+
+\subsection{Proof architecture}
+The reflection method developed by Davis~\cite[Sections~13 and~15]{Davis1983}
+provides the framework for closing the chamber. Our construction closes
+the pair to an aspherical Poincar\'e
+complex $Q$ and reflects $D$ to a closed manifold $N\to Q$. A
+support-tree construction gives the relevant right-angled Coxeter
+extension of the free chamber group a proper cocompact CAT(0) model.
+This auxiliary model supplies assembly information; the chamber
+development retains the four-dimensional topology and its sphere classes.
+The CAT(0) action places $G=\pi_1Q$ among the groups to which the Farrell--Jones
+theorems apply~\cite{BL,Wegner}. If $Q$ had a manifold model $M$, the
+four-dimensional surgery result of Kasprowski--Land~\cite{KL}, followed
+by a stable product theorem, would make all the prescribed hyperbolic
+pairs of a stabilization of $N$ geometrically standard.
+
+Lift these finite marked data to the reflection development. In a large
+finite union of chambers, cap every boundary summand except the central
+one. Remove standard pairs labelled by a large inner ball. The remaining
+sphere problems are far from the central chamber, where radial directions
+in the Davis complex make every bounded local construction small.
+Grope separation follows Freedman--Quinn's finite-cover strategy
+\cite[Section~2.10.2]{FQ} and groups the upper branches over finitely many
+contractible direction patches. The essential requirement is that an
+entire resulting loop system stays over one such patch, even when it
+passes through many intersections. This replaces precisely the goodness-dependent
+null-loop step of Powell--Ray--Teichner~\cite{PRT}, and not their theorem's
+group hypothesis by fiat. The subsequent disk construction is
+uncontrolled and may range through the whole compact manifold. Surgery
+then gives the forbidden filling of $S$.
+
+Section~\ref{ch:section} constructs the marked chamber and proves its
+filling obstruction. Section~\ref{rf:section} reflects it to obtain
+$Q$, verifies integral duality, and supplies the two geometric models
+used later. Section~\ref{st:setup} uses a hypothetical manifold model to
+realize every prescribed sphere pair after stabilization.
+Section~\ref{ct:section} caps a finite chamber union, removes the inner
+standard pairs, and uses direction control to remove the remaining
+middle homology. Its output is the marked filling excluded by
+Section~\ref{ch:section}.
+
+\begin{figure}[H]
+\centering
+\begin{tikzpicture}[>=Latex, node distance=14mm and 25mm,
+  every node/.style={align=center,font=\small}]
+\node (D) {$D\longrightarrow P\simeq BH$\\marked boundary $S$};
+\node[right=of D] (N) {$N\longrightarrow Q\simeq BG$\\reflection};
+\node[below=of N] (M) {hypothetical $M\simeq Q$\\marked stable product};
+\node[below=of D] (Z) {$Z$ with boundary $S$\\cap and remove inner pairs};
+\node[below=of Z] (W) {graph-type filling of $S$\\contradicts the cut obstruction};
+\draw[->] (D) -- (N);
+\draw[->] (N) -- (M);
+\draw[->] (M) -- (Z);
+\draw[->] (Z) -- node[right,text width=35mm] {direction control\\and sphere surgery} (W);
+\end{tikzpicture}
+\caption{The two uses of the chamber. Its middle spheres permit the
+reflection and stable-surgery construction; its marked boundary forbids
+the filling obtained at the end.}
+\label{fig:architecture}
+\end{figure}
+
+\subsection{Conventions}
+All manifolds in the surgery argument are oriented and all unqualified
+homology groups have integral coefficients. For a connected space with
+fundamental group $\Gamma$, regular coefficients $\Z\Gamma$ mean the
+homology of its universal cover. We fix lifts and paths to a basepoint
+when recording sphere classes. The involution on an oriented group ring
+is $\overline{\sum n_g g}=\sum n_g g^{-1}$.
+A hyperbolic plane over $\Z\Gamma$ has matrix
+$\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)$.
+For immersed sphere representatives, the phrase \emph{framed hyperbolic
+system} also includes vanishing Wall self-intersection obstructions.
+
+All stabilizations are interior connected sums with $S^2\times S^2$ and
+leave boundary markings unchanged. Specific references to
+Freedman--Quinn~\cite{FQ} use the January 2013 reformatted edition's
+numbering.
+
+\input{sections/chamber}
+\input{sections/reflection}
+\input{sections/stable}
+\input{sections/control}
+
+{\raggedright\input{references}}
+\end{document}
diff --git a/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/build/references.tex b/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/build/references.tex
new file mode 100644
index 0000000000000000000000000000000000000000..a789e502adef271f1d3adef7599f22d470fa4b02
--- /dev/null
+++ b/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/build/references.tex
@@ -0,0 +1,161 @@
+\begin{thebibliography}{99}
+
+\bibitem{BL}
+Arthur Bartels and Wolfgang L\"uck,
+\emph{The Borel Conjecture for hyperbolic and CAT(0)-groups},
+Annals of Mathematics (2) \textbf{175} (2012), no.~2, 631--689.
+\href{https://doi.org/10.4007/annals.2012.175.2.5}{doi:10.4007/annals.2012.175.2.5}.
+
+\bibitem{BH}
+Martin R. Bridson and Andr\'e Haefliger,
+\emph{Metric Spaces of Non-Positive Curvature},
+Grundlehren der mathematischen Wissenschaften, vol.~319,
+Springer, Berlin, 1999.
+
+\bibitem{DavisHillman}
+James F. Davis and J. A. Hillman,
+\emph{Aspherical manifolds with boundary},
+preprint, 2026,
+\href{https://arxiv.org/abs/2501.12509v2}{arXiv:2501.12509v2},
+version of June 29, 2026.
+
+\bibitem{Davis}
+Michael W. Davis,
+\emph{The Geometry and Topology of Coxeter Groups},
+London Mathematical Society Monographs Series, vol.~32,
+Princeton University Press, Princeton, NJ, 2008.
+\href{https://doi.org/10.1515/9781400845941}{doi:10.1515/9781400845941}.
+
+\bibitem{Davis1983}
+Michael W. Davis,
+\emph{Groups generated by reflections and aspherical manifolds not
+covered by Euclidean space},
+Annals of Mathematics (2) \textbf{117} (1983), no.~2, 293--324.
+\href{https://doi.org/10.2307/2007079}{doi:10.2307/2007079}.
+
+\bibitem{DavisCohomology}
+Michael W. Davis,
+\emph{The cohomology of a Coxeter group with group ring coefficients},
+Duke Mathematical Journal \textbf{91} (1998), no.~2, 297--314.
+\href{https://doi.org/10.1215/S0012-7094-98-09113-X}{doi:10.1215/S0012-7094-98-09113-X}.
+
+\bibitem{DavisPD}
+Michael W. Davis,
+\emph{Poincar\'e duality groups},
+in \emph{Surveys on Surgery Theory}, Vol.~1,
+Sylvain Cappell, Andrew Ranicki, and Jonathan Rosenberg, eds.,
+Annals of Mathematics Studies, vol.~145,
+Princeton University Press, Princeton, NJ, 2000, 167--193.
+\href{https://people.math.osu.edu/davis.12/pdgroup.pdf}{Author's version}.
+
+\bibitem{Fowler}
+Jim Fowler,
+\emph{Finiteness properties for some rational Poincar\'e duality groups},
+Illinois Journal of Mathematics \textbf{56} (2012), no.~2, 281--299.
+\href{https://doi.org/10.1215/ijm/1385129948}{doi:10.1215/ijm/1385129948}.
+The cited theorem locator refers to
+\href{https://arxiv.org/abs/1204.4667v1}{arXiv:1204.4667v1}.
+
+\bibitem{FQ}
+Michael H. Freedman and Frank Quinn,
+\emph{Topology of 4-manifolds},
+Princeton Mathematical Series, vol.~39,
+Princeton University Press, Princeton, NJ, 1990.
+\href{https://doi.org/10.1515/9781400861064}{doi:10.1515/9781400861064}.
+Section and lemma numbers refer to Frank Quinn's
+\href{https://archive.mpim-bonn.mpg.de/id/eprint/4789/2/FreedmanQuinn-TopologyOf4Manifolds-Reformatted2013.pdf}{January 2013 reformatted, unrevised version}.
+
+\bibitem{Hatcher}
+Allen Hatcher,
+\emph{Algebraic Topology},
+Cambridge University Press, 2002.
+\href{https://pi.math.cornell.edu/~hatcher/AT/ATpage.html}{Author's online edition}.
+
+\bibitem{KL}
+Daniel Kasprowski and Markus Land,
+\emph{Topological 4-manifolds with 4-dimensional fundamental group},
+Glasgow Mathematical Journal \textbf{64} (2022), no.~2, 454--461.
+\href{https://doi.org/10.1017/S0017089521000215}{doi:10.1017/S0017089521000215}.
+Cited theorem and lemma locators refer to
+\href{https://arxiv.org/abs/2007.03399v3}{arXiv:2007.03399v3}.
+
+\bibitem{KPR}
+Daniel Kasprowski, Mark Powell, and Arunima Ray,
+\emph{Counterexamples in 4-manifold topology},
+EMS Surveys in Mathematical Sciences \textbf{9} (2022), no.~1, 193--249.
+\href{https://doi.org/10.4171/EMSS/56}{doi:10.4171/EMSS/56}.
+
+\bibitem{Kielak}
+Dawid Kielak,
+\emph{Virtual Fibring of Manifolds and Groups},
+in \emph{Proceedings of the International Congress of Mathematicians 2026},
+Vol.~4, \emph{Invited Lectures (Sections 5--8)},
+Susan Friedlander and Yuri Tschinkel, eds.,
+Society for Industrial and Applied Mathematics, 2026, 343--362.
+\href{https://doi.org/10.1137/25M1804741}{doi:10.1137/25M1804741}.
+
+\bibitem{LuckFJ}
+Wolfgang L\"uck,
+\emph{Survey on the Farrell-Jones Conjecture},
+Bulletin of the American Mathematical Society (N.S.) \textbf{63}
+(2026), no.~1, 79--117.
+The cited conjecture locator refers to
+\href{https://arxiv.org/abs/2507.11337v1}{arXiv:2507.11337v1},
+version of July 15, 2025.
+
+\bibitem{MilnorCW}
+John Milnor,
+\emph{On spaces having the homotopy type of a CW-complex},
+Transactions of the American Mathematical Society \textbf{90} (1959),
+no.~2, 272--280.
+\href{https://webhomes.maths.ed.ac.uk/~v1ranick/papers/milnorcw.pdf}{Original article}.
+
+\bibitem{TO}
+OpenAI,
+\emph{A marked tensor obstruction to four-dimensional disk embedding},
+OpenAI Math Release preprint
+\href{https://github.com/openai/math/blob/main/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/paper.pdf}{OAI:A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026},
+2026.
+
+\bibitem{PRT}
+Mark Powell, Arunima Ray, and Peter Teichner,
+\emph{The 4-dimensional disc embedding theorem and dual spheres},
+Selecta Mathematica (N.S.) \textbf{31} (2025), article~80.
+\href{https://doi.org/10.1007/s00029-025-01069-y}{doi:10.1007/s00029-025-01069-y}.
+
+\bibitem{QuinnEnds}
+Frank Quinn,
+\emph{Ends of maps. III: Dimensions 4 and 5},
+Journal of Differential Geometry \textbf{17} (1982), no.~3, 503--521.
+\href{https://doi.org/10.4310/jdg/1214437139}{doi:10.4310/jdg/1214437139}.
+
+\bibitem{QuinnStable}
+Frank Quinn,
+\emph{The stable topology of 4-manifolds},
+Topology and its Applications \textbf{15} (1983), no.~1, 71--77.
+\href{https://doi.org/10.1016/0166-8641(83)90049-4}{doi:10.1016/0166-8641(83)90049-4}.
+
+\bibitem{Swiatkowski}
+Jacek \'Swi{\k a}tkowski,
+\emph{Hyperbolic Coxeter groups with Sierpi\'nski carpet boundary},
+Bulletin of the London Mathematical Society \textbf{48} (2016), 708--716.
+The cited Appendix Proposition~A.2 is in
+\href{https://arxiv.org/abs/1502.02044v1}{arXiv:1502.02044v1}.
+
+\bibitem{WallProblems}
+C. T. C. Wall,
+\emph{List of problems},
+in \emph{Homological Group Theory}, C. T. C. Wall, ed.,
+London Mathematical Society Lecture Note Series, vol.~36,
+Cambridge University Press, 1979, 369--394.
+
+\bibitem{Wegner}
+Christian Wegner,
+\emph{The $K$-theoretic Farrell-Jones conjecture for CAT(0)-groups},
+Proceedings of the American Mathematical Society \textbf{140} (2012),
+no.~3, 779--793.
+\href{https://doi.org/10.1090/S0002-9939-2011-11150-X}{doi:10.1090/S0002-9939-2011-11150-X}.
+Cited theorem locators refer to
+\href{https://arxiv.org/abs/1012.3349v2}{arXiv:1012.3349v2}.
+
+\end{thebibliography}
diff --git a/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/build/sections/chamber.tex b/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/build/sections/chamber.tex
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+\section{A chamber with no marked graph filling}
+\label{ch:section}
+
+The obstruction will first be placed in a compact chamber.  Its boundary
+marking is essential: the obstruction does not merely assert that some
+disk problem has no solution.
+
+\begin{proposition}[The marked chamber]
+\label{ch:chamber}
+There are a compact connected oriented smooth four-manifold $D$, its
+connected boundary $S$, a finitely generated free group $H$, and a map
+$c:D\to BH$ with the following properties.
+\begin{enumerate}
+\item The map $c$ induces an isomorphism on fundamental groups, and $D$
+has the homotopy type of a finite graph with $2k$ two-spheres attached
+at its basepoint.  Its module $\pi_2D$ has a specified hyperbolic basis
+over $\Z H$, represented by framed immersed spheres with vanishing
+quadratic self-intersection obstructions.
+\item Attach a three-cell to each basis sphere, and denote the resulting
+space by $P$.  Then $(P,S)$ is an oriented Poincar\'e pair of formal
+dimension four, and $P\simeq BH$.  The boundary marking is
+$c|_S:S\to BH$.
+\item There is no compact oriented topological four-manifold $D'$ with
+boundary identified with $S$ and a homotopy equivalence $c':D'\to BH$
+whose restriction to $S$ is homotopic to $c|_S$.
+\end{enumerate}
+\end{proposition}
+
+Here and below, sphere classes and intersections use fixed whiskers and
+deck translations.  A hyperbolic pair has zero squares, mutual product
+$1$, and zero framing and Wall self-intersection obstructions.  We prove
+the proposition without using an embedded surgery on $D$.
+
+\subsection{The geometric and tensor inputs}
+\label{ch:inputs}
+
+We use the marked geometric construction of \cite[Section~2]{TO}, rather
+than only its main disk-nonembedding theorem. The marked bodies and
+exterior duals are supplied by its Lemmas~2.3 and~2.4. Its
+Proposition~2.2 and Lemmas~2.5--2.6 give the cut consequences under
+hypothetical disk replacements; those conditional conclusions are
+distinct from the marked geometric construction itself.
+We recall the precise data needed here.
+Fix $m\geq5$ and $p\geq9$.  Start with a four-dimensional one-handlebody
+on $2m+p$ handles, attach cancelling two-handles to the first $2m$
+handles with product framings, and plumb these two-handles in $m$ pairs,
+interchanging the two coordinate directions at every plumbing.
+The resulting smooth oriented manifold $X$ has graph homotopy type.
+Its cover associated to the plumbing quotient $\pi_1X\to F_m$ is a
+tree of blocks, each a one-handlebody on $p$ extra generators; neighboring
+blocks meet in a product four-ball.  The exposed boundary of a block is
+\[
+ K_v=\#^p(S^1\times S^2)\setminus
+       \operatorname{int}\nu(L_{2m}),
+\]
+where $L_{2m}$ is an unlink in a ball.  Its port meridians
+$x_1,\ldots,x_{2m}$ and extra loops $l_1,\ldots,l_p$ form a free basis.
+At each join the meridian on one side is the longitude on the other.
+
+Write $\mathcal C=\{1,l_1,\ldots,l_p\}$.  The marked-body construction
+installs a disjoint two-stage grope body for every ordered triple of
+distinct ports and every independent choice
+\[
+ X_a=c_a x_{r_a}^{\delta_a}c_a^{-1},\qquad
+ c_a\in\mathcal C,\quad \delta_a\in\{1,-1\},\quad a=1,2,3.
+\]
+Its lower surface is a punctured torus: one branch has a single cap,
+and the other carries the upper punctured torus with its two caps.
+All tip tracks, conjugating paths, and product framings are specified.
+The initial boundary word $R$ satisfies
+\begin{equation}
+\label{ch:laws}
+ R|_{X_a=1}=1\quad(a=1,2,3),\qquad
+ R\equiv[X_1,[X_2,X_3]]\pmod{\Gamma_4F(X_1,X_2,X_3)}.
+\end{equation}
+Internal changes of the marked paths are allowed by these identities;
+one must retain the entire finite family, not replace it by one
+unconjugated commutator.
+
+Let $F_1,\ldots,F_k$ be the lower punctured tori, with symplectic bases
+$(a_j,b_j)$, where $a_j$ is the single-cap branch.  Delete their open
+product tubes, including the proper edge collars, to obtain $E$.
+Lemma~2.4 (Algebraic duals in the exterior) of \cite{TO} supplies
+the truncated single caps $f_j$ and framed spheres $g_j$ in $E$ such
+that
+\begin{equation}
+\label{ch:exterior-duals}
+ \lambda(f_i,g_j)=\delta_{ij},\qquad
+ \lambda(g_i,g_j)=0,\qquad \widetilde\mu(g_j)=0
+ \quad\text{over }\Z[\pi_1E].
+\end{equation}
+The framing on $f_j$ is the surface-compression framing.  The $g_j$
+are compressed normal-circle tori over the $b_j$ curves; their
+individual representatives have trivial normal bundle.  Equation
+\eqref{ch:exterior-duals} is a group-ring statement before any
+meridians are killed.
+
+We also recall exactly how the cut obstruction is used.  Suppose that a
+compact smooth oriented four-manifold with the marked graph type of
+the parity cover of $X$ has two connected vertex pieces $A,B$, with
+outside faces $K_A,K_B$, and $2m$ connected three-dimensional cuts
+$Y_i$, each with its prescribed coordinate torus as its entire boundary.
+Orient $Y_i$ as a face of $A$, so its orientation as a face of $B$
+is opposite.
+Suppose the differences of restrictions give an isomorphism in degree
+two and, in degree one, an isomorphism after the extra-loop evaluations
+are set to zero.  Suppose further that precisely $m$ joins are active:
+at an active join the surviving coordinate meridian is at $A$, whereas
+the other coordinate is the meridian at $B$.  Require every marked law
+\eqref{ch:laws} in its respective vertex group, and the free ambient
+background in which the port meridians are trivial and the named extra
+letters can be assigned independently.
+
+These data cannot exist.  Indeed, Proposition~6.1 (Vertex data) of
+\cite{TO} gives a characteristic-zero field $K$ (one may take
+$\mathbb C((\sigma))$), separate coordinate blocks $h_i$ of total
+dimension $N$, and algebraic power-series germs $F_i(h_i)$ and
+$b_i(h_i)$, with $b_i$ indexed by the active set $I$, $|I|=m$.
+The germs are normalized by $F_i(0)=b_i(0)=0$.  Set
+\begin{equation}
+\label{ch:tensor-ring}
+ T=K[t_i:i\in I]/(t_i^2:i\in I),\qquad
+ S_0=\sum_iF_i,\qquad S_t=S_0+\sum_{i\in I}t_i b_i.
+\end{equation}
+There are formal parametrizations
+\[
+ H_L(x,t)\in(x,t)T[[x]]^N,\qquad
+ H_R(y)\in(y)K[[y]]^N,
+\]
+where $x$ has $d$ coordinates and $y$ has $N-d$ coordinates.  Their
+central tangent maps $\partial_xH_L(0,0)$ and $\partial_yH_R(0)$
+are injective with complementary images in $K^N$; nilpotent constant
+displacements of $H_L$ are allowed.  They satisfy
+\begin{equation}
+\label{ch:tensor-vanishing}
+ S_t(H_L)=0,\qquad S_0(H_R)=0.
+\end{equation}
+For every three distinct active indices $i,j,k$, they satisfy
+\begin{align}
+ t_it_jt_k&\in
+ \bigl((\partial_{h_\nu}S_t)(H_L):\nu=1,\ldots,N\bigr)
+                  \subset T[[x]],\label{ch:tensor-left}\\
+ (b_ib_jb_k)(H_R)&\in
+ \bigl((\partial_{h_\nu}S_0)(H_R):\nu=1,\ldots,N\bigr)
+                  \subset K[[y]].\label{ch:tensor-right}
+\end{align}
+Theorem~3.1 (Finite tensor obstruction) of \cite{TO} rules out
+\eqref{ch:tensor-ring}--\eqref{ch:tensor-right} for $m\geq5$.
+In particular, the memberships are in the actual gradient ideals,
+not their radicals, and $T$ retains all mixed square-free monomials.
+The independent conjugators in the full law list are what supply all
+these triple identities. These two source results, together with the
+marked edge potentials, algebraicity and formal comparison of
+\cite[Proposition~4.5, Theorem~5.1 and Proposition~5.2]{TO}, are the
+tensor obstruction used here.
+
+\subsection{The exterior with its side marking}
+\label{ch:exterior}
+
+The imported geometry supplies the surfaces and their algebraic duals.
+We now use those data to construct $D$, keeping track of the actual
+tube-side maps. This marking will identify the boundary of a hypothetical
+filling with the loop data needed for the tensor contradiction.
+
+We make one placement refinement of the marked-body construction.
+Place its longitudinal tip annuli on disjoint parallels outside
+sufficiently thin attaching solid tori in the initial one-handlebody
+boundary, and put the finitely many connectors in their complement.
+This is possible by the marked handlebody model: the bodies are boundary
+joins of the separated annular collars.  The complement of the attaching
+solid tori survives in the boundary after the two-handles are attached;
+the plumbing ports are on the new belt-handle portions.  Thus the
+unpushed lower surfaces lie in the final exposed boundary of $X$.
+The independent ribbon twists matching the cap framings are made in
+the separated parallel tubes.  They do not pass a body across a
+plumbing join.  The returning tip collars alone are lengthened back
+to product parallels on the attaching regions and then into the cores.
+The construction and intersection calculation of
+\eqref{ch:exterior-duals} are unchanged.
+
+Consequently the $F_j$ are proper inward pushes of disjoint boundary
+surfaces, with returning collars at their initial boundary curves.
+Moreover $F_j\to X$ is null-homotopic.  The three tip loops bound the
+corresponding core caps in $X$; hence the bottom $a_j$ and $b_j$ loops
+die in $\pi_1X$.  Since $X$ has graph homotopy type and $F_j$ retracts
+to a graph, this proves the assertion about the whole surface map.
+
+\begin{lemma}[The marked exterior model]
+\label{ch:exterior-model}
+With the surface-product trivializations, the exterior and its tube-side
+maps have the homotopy-pushout model
+\[
+ E\simeq X\cup_{\coprod_jF_j}\coprod_j(F_j\times S^1),
+\]
+where each gluing map to $F_j\times S^1$ is the inward section.
+The displayed $F_j\times S^1$ maps represent the actual tube sides.
+\end{lemma}
+
+\begin{proof}
+It is enough to describe one surface; the constructions have disjoint
+collars.  Write $B=\partial X$ and $K=\partial F$.  Before rounding
+corners, put the pushed surface at a horizontal collar level, with
+vertical walls along $K$ returning to the boundary.  The inner part of
+the exterior has homotopy type $X$, the outer part has homotopy type
+$B\setminus K$, and their overlap has homotopy type $B\setminus F$.
+Product slices, thickened by a variable positive width off the deleted
+surface, give this homotopy-pushout decomposition.
+
+In $B\setminus K$, take a two-sided neighborhood of
+$F^\circ=F\setminus K$ whose width decreases to zero near $K$.
+Relative to $B\setminus F$, it is a bridge over $F^\circ$, joining
+the two sides of the cut.  Replace $F^\circ$ in this diagram by a
+slightly shrunken copy $F_0$; this is an objectwise homotopy equivalence.
+The actual deleted surface and $K$ are unchanged.  In the rectangular
+normal coordinates to its horizontal copy, split the tube circle over
+$F_0$ into its outer and inward semicircles.  The outer semicircle is
+the bridge.  The inward semicircle lies in the inner $X$ piece and
+gives a specified homotopy between the bridge's endpoint maps.
+
+Using that homotopy to identify the endpoints replaces the bridge by
+$F_0\times S^1$, attached along its inward section.  Its circle is
+precisely the outer bridge followed by the inward return, hence the
+actual normal circuit.  Finally $F_0\to F$ and
+$F_0\times S^1\to F\times S^1$ are homotopy equivalences.  This
+also accounts for the proper edge collars and proves the assertion
+about the side maps, not just the absolute homotopy type of $E$.
+\end{proof}
+
+For each $j$, regard a solid torus $V_j$ as a compression body from
+the punctured torus $F_j$ to a disk, compressing $a_j$.  Define
+\[
+ D=E\cup_{\coprod_j(F_j\times S^1)}
+                  \coprod_j(V_j\times S^1).
+\]
+All gluings use the just specified product framings; round corners.
+The remaining disk face of $\partial V_j$ contributes a solid torus
+to the new boundary.  Hence $S=\partial D$ is the surface-framed
+surgery of $\partial X$ on the initial grope-boundary link.  In
+particular it is connected.
+
+By Lemma~\ref{ch:exterior-model},
+\[
+ D\simeq X\cup_{\coprod_jF_j}\coprod_j(V_j\times S^1).
+\]
+Each $F_j\to X$ is null-homotopic.  The extra summand is therefore
+the homotopy cofiber of $F_j\to V_j\times S^1$.  Retract the latter
+space to the torus on its longitude $b_j$ and normal meridian $t_j$.
+The cofiber adds one two-cell killing $b_j$ and one on the null loop
+$a_j$.  Cancel the $b_j$ one/two-cell pair.  The torus two-cell then
+has null attaching map, so the result is
+\[
+ S^1_{t_j}\vee S^2\vee S^2.
+\]
+Thus $H=\pi_1D$ is the original free graph group with one new free
+meridian $t_j$ per surface, and $D$ has the graph-and-spheres homotopy
+type in Proposition~\ref{ch:chamber}.
+
+\subsection{The middle form and the Poincar\'e pair}
+\label{ch:duality}
+
+Close $f_j$ by a meridian disk of $V_j$ at its fixed circle coordinate,
+obtaining a sphere $u_j$.  The two compression framings agree, so its
+second Stiefel--Whitney evaluation is zero.  The $g_j$ remain in $E$;
+therefore \eqref{ch:exterior-duals} gives
+\[
+ \lambda(u_i,g_j)=\delta_{ij},\qquad \lambda(g_i,g_j)=0
+ \quad\text{over }\Z H.
+\]
+The hermitian matrix of the $u_i$ has even diagonal.  As $H$ has no
+element of order two, every such diagonal is $z+\bar z$: choose one
+representative from each nonidentity inverse pair and halve the even
+identity coefficient.  Add suitable combinations of the $g_j$ to the
+$u_i$, taking a triangular half off the diagonal and this half on it.
+The resulting spheres, denoted $a_i$, together with $b_i=g_i$, have
+hyperbolic intersection matrix.  The usual framed representative
+adjustments give zero Wall self-intersections: choose normal Euler
+number zero using the evenness, and then use
+$\lambda(a_i,a_i)=\mu(a_i)+\overline{\mu(a_i)}$ and the absence of
+order-two elements.  This is the ordinary immersed-sphere calculus
+of \cite{FQ}; no disk-embedding assertion enters it.
+
+These $2k$ classes form a basis.  Indeed, $\pi_2D$ is already known
+to be free of rank $2k$.  Its square matrix of the displayed classes
+has a left inverse obtained from their nonsingular pairing.  To see
+directly that it also has a right inverse, pass to each finite quotient
+of $H$.  The resulting matrices act on finite-dimensional complex
+vector spaces, where a left inverse is a right inverse.  Residual
+finiteness of the free group separates the finite support of any
+nonzero group-ring element, so the right-inverse identity holds over
+$\Z H$ itself.  We henceforth use this specified basis.
+
+Attach a three-cell on every $a_i$ and $b_i$ to form $P$.  A map from
+the graph with $2k$ two-spheres to $D$ realizing this basis is a
+homotopy equivalence: it induces an isomorphism on $\pi_1$ and on the
+homology of simply connected covers.  Thus $P\simeq BH$.
+
+We check Poincar\'e duality for the pair, since this step must not be
+confused with embedded surgery.  Set $\Lambda=\Z H$.  In the duality
+of $(D,S)$, the middle sphere module maps isomorphically onto a split
+summand of $H_2(D,S;\Lambda)$, by its nonsingular form.  Attaching the
+three-cells quotients both absolute and relative homology by these
+middle summands; the attaching map is injective on them, so no new
+third homology appears.  On cohomology the pullbacks identify the new
+groups with the annihilators of the sphere summands.  If
+$f:(D,S)\to(P,S)$ is the natural cell-attachment inclusion and
+$[P,S]=f_*[D,S]$, cap-product naturality reads
+\[
+ z\frown[P,S]=f_*\bigl(f^*z\frown[D,S]\bigr).
+\]
+Hence the old duality isomorphisms induce exactly the new ones on
+these annihilators and quotients.  The spaces have finite free chain
+complexes over $\Lambda$, so the regular-coefficient cap-product
+criterion gives Poincar\'e duality with arbitrary local coefficients.
+This proves parts (1) and (2) of Proposition~\ref{ch:chamber}.
+
+\begin{corollary}
+\label{ch:boundary-homology}
+The map $\pi_1S\to H$ is onto.  If $r=\operatorname{rank}H$, then
+$H_1(S;\Z)\to H_1(P;\Z)$ is an isomorphism and
+$H_2(S;\Z)\cong\Z^r$.
+\end{corollary}
+
+\begin{proof}
+Duality and the graph type of $P$ give
+$H_1(P,S;\Z H)=H_0(P,S;\Z H)=0$.  Therefore
+$H_0(S;\Z H)\to H_0(P;\Z H)$ is an isomorphism.
+The former is free abelian on the cosets of the image of $\pi_1S$,
+and the latter is $\Z$; there can be only one coset.
+With ordinary coefficients the pair sequence and duality give
+$H_2(P,S)=H_1(P,S)=0$ and
+$H_3(P,S)\cong H^1(P)\cong\Z^r$.  The assertions follow.
+\end{proof}
+
+\subsection{Topological fillings and the cut obstruction}
+\label{ch:forbidden-filling}
+
+The construction and duality calculation establish the first two parts
+of Proposition~\ref{ch:chamber}. For the third part we must analyze an
+arbitrary topological filling with the specified boundary map. The
+following cut construction supplies the hypotheses recalled in
+Section~\ref{ch:inputs}; it does not presume a smooth structure on that
+filling.
+
+Suppose a marked filling $D'$ as in part (3) existed.  Attach the
+dual two-handles that undo the surgery on the initial link.  Denote
+the result by $W$, so $\partial W=\partial X$.  Each attaching circle
+represents its prescribed new free generator $t_j$ under the marking
+$D'\simeq BH$.  The corresponding one/two-cell pair cancels in the
+homotopy type.  Thus $W$ has the graph type of $X$, with the original
+boundary-to-graph map: equality of the marked fundamental-group maps
+gives equality up to homotopy because the graph is aspherical.
+The two-handle cocores are disjoint proper locally flat disks bounded
+by the original initial link.  In particular all the original marked
+words are now represented by disk boundaries.
+
+Here is the cut construction with the hypotheses on $W$ made explicit.
+Use the marked quotient $\pi_1W\to F_m$ and its associated tree cover.
+Prescribe the equivariant map to be constant at the appropriate vertex
+on each outside face $K_v$ and each lifted disk neighborhood, and
+to cross the appropriate edge on each boundary torus collar.  These
+prescriptions agree where they meet.  A section of the associated
+bundle with contractible tree fiber extends them to the rest of $W$.
+The free boundary extra loops, with whiskers, generate the fundamental
+group of this tree cover, just as for the marked graph model of $X$.
+
+We must allow $W$ to be topological.  By Quinn's punctured smoothing
+theorem \cite[Corollary~2.2.3]{QuinnEnds}, delete an interior point
+away from the disks and extend the given boundary smoothing to the
+complement.  Prescribe the map to be constant near this point before
+extending the section.  Approximate it smoothly over the open middle
+intervals of the finitely many graph edges, away from the disks and
+puncture, and choose regular values.  The cuts are compact smooth
+three-manifolds, disjoint from the disks.  Their filled complementary
+pieces are topological four-manifolds, smooth away from the finite
+set of lifted punctures in each finite quotient.
+
+For completeness, connectedness is not automatic from a regular-level
+construction.  Form the component graph of the cuts in the tree cover.
+Its fundamental group is a quotient of that of the cover, by joining
+successive crossings through connected complementary pieces.  Every
+extra boundary generator lies in one piece, so maps trivially to this
+graph.  It follows that the component graph is a tree.  The outside
+faces and torus collars give an embedded principal copy of the original
+plumbing tree: different vertex labels lie in different components of
+the complement of the target midpoints.  Any remaining branch has a
+unique nearest principal vertex.  Its stabilizer is trivial, since it
+fixes that vertex.  The quotient has finitely many components; hence
+these branches have finite valence and uniformly bounded depth, and
+each is finite.  Ignore their closed cuts and merge them into their
+principal vertices.  This leaves connected vertex pieces and one
+connected cut with each prescribed torus as its entire boundary.
+The disk prescriptions have not changed.  This is precisely the
+argument of Lemma~2.5 (Connected tree cuts) of \cite{TO}, now
+applied to the marked $W$ rather than requiring a homeomorphism $W=X$.
+
+Pass to the parity quotient, sending every plumbing generator to the
+nonidentity element of $\Z/2$.  Its filled ambient space $W_2$ has
+the homotopy type of the graph with two vertices and $2m$ edges,
+and $p$ extra circles at each vertex.  Write $A,B$ for the filled
+vertex pieces and $Y_1,\ldots,Y_{2m}$ for their cuts.  Over any
+characteristic-zero field $K$, Mayer--Vietoris gives
+\begin{equation}
+\label{ch:restriction-two}
+ H^2(A;K)\oplus H^2(B;K)
+       \xrightarrow{\ \cong\ }\bigoplus_iH^2(Y_i;K),
+\end{equation}
+because $H^2(W_2;K)=H^3(W_2;K)=0$.  In degree one the difference
+of restrictions is onto.  Its kernel is the image of $H^1(W_2;K)$
+modulo the $2m-1$ graph classes, and evaluation on the $p$ named
+extra loops at each vertex identifies that kernel with $K^{2p}$.
+Thus the restriction map becomes an isomorphism on the subspace
+where these extra evaluations vanish.
+
+On a cut $Y_i$, half-lives/half-dies says that the image of
+$H_1(\partial Y_i;K)$ in $H_1(Y_i;K)$ is a line.  If both original
+coordinate loops had nonzero image, their images would be nonzero
+multiples.  A class from either vertex annihilates the longitude
+bounding in its outside face, and hence would annihilate both
+coordinates.  The degree-one surjectivity just proved would then
+force every class on $Y_i$ to do so, a contradiction.  Hence one
+original coordinate $\beta_i$ dies and the other $\alpha_i$ survives.
+The plus endpoint is the side at which $\beta_i$ is a longitude.
+The parity involution pairs the two lifts of each join and exchanges
+their plus endpoints, giving exactly $m$ active edges at $A$.
+
+The prescribed disks lie in their own vertex pieces.  They therefore
+give every law \eqref{ch:laws} at those vertices, with all the
+independently chosen conjugators and signs.  The marked ambient free
+group still has trivial port meridians and freely assignable named
+extra letters.  In particular it supplies the common flat background
+used in the Vertex data Proposition: assign eight extra letters the
+eight nonidentity matrices among
+$\exp(a\sigma e)\exp(b\sigma f)$, $a,b\in\{0,1,2\}$, for the
+standard nilpotent generators $e,f$ of $\mathfrak{sl}_2$.
+Together with the identity, their adjoint operators span all
+endomorphisms over $\mathbb C((\sigma))$, as in
+\cite[Proposition~6.1]{TO}.
+Thus the geometric and marked hypotheses of the tensor interface
+in Section~\ref{ch:inputs} have all been established.
+
+It remains to justify that its vertex calculation works for the
+punctured smooth structures just used.  We give this step rather
+than assuming that the filled vertices admit smooth structures.
+
+\begin{lemma}[Forms on a punctured vertex]
+\label{ch:puncture-forms}
+Let $V$ be one of the filled compact vertex pieces and let $Z$ be
+its finite set of punctures.  In degree zero take smooth functions
+on $V\setminus Z$ separately constant near each puncture, and in
+positive degrees take smooth forms vanishing near the punctures.
+This differential graded algebra computes $H^*(V;\mathbb C)$,
+naturally on the unpunctured faces.  It supports the vertex
+calculation in the Vertex data Proposition of \cite{TO}, with the
+filled cohomology groups and with the same Stokes identities.
+\end{lemma}
+
+\begin{proof}
+Regard these forms as a sheaf on $V$.  Its stalk at a puncture is
+$\mathbb C$ in degree zero and zero in higher degrees; elsewhere
+it is the ordinary de Rham resolution.  Partitions of unity can
+be chosen constant on smaller neighborhoods of the finitely many
+punctures.  The complex is therefore a fine resolution of the
+constant sheaf, so computes the filled cohomology.  A based condition
+in degree zero removes only $H^0$.  Linear splittings onto cohomology
+and contracting homotopies may consequently be chosen inside this
+complex, exactly as for the ordinary vertex de Rham complex.
+
+Wedges, Lie brackets, the curvature recursions, and differentiation
+in the formal parameters preserve this algebra.  Each fixed
+positive-degree coefficient vanishes near $Z$, so integration and
+Stokes take place on a compact smooth subdomain containing its
+support.  The additional end boundaries contribute zero.  An entire
+formal series need not have one common vanishing neighborhood:
+every calculation at a fixed coefficient uses only finitely many
+terms.  The source's formal lifts and divisions likewise use
+finite linear combinations coefficientwise, and extensions from
+face collars can be supported away from $Z$.
+
+Removing interior points in dimension four does not change the
+fundamental group.  At each finite nilpotent order the flat systems
+in question vanish near the punctures in their local trivializations;
+gauges between them can be separately constant at those ends.
+Thus the based holonomy and exact-gauge arguments use the same
+marked group as the filled vertex.  Allowing separate constants
+at distinct ends is important here.  These are the uses of vertex
+smoothness in the proof of Proposition~6.1 of \cite{TO}.
+Its edge-algebraicity theorem, \cite[Theorem~5.1]{TO},
+is still applied only to the unchanged compact smooth
+three-manifolds $Y_i$.
+\end{proof}
+
+Apply Lemma~\ref{ch:puncture-forms} to the two vertex pieces.
+Equation~\eqref{ch:restriction-two} and the degree-one calculations
+refer to their filled cohomology, which the lemma computes.  The
+Vertex data Proposition, \cite[Proposition~6.1]{TO}, now produces exactly
+\eqref{ch:tensor-ring}--\eqref{ch:tensor-right}.  The Finite tensor
+obstruction contradicts their existence.  Therefore $D'$ cannot
+exist, completing the proof of Proposition~\ref{ch:chamber}.
diff --git a/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/build/sections/control.tex b/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/build/sections/control.tex
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@@ -0,0 +1,683 @@
+\section{Controlled removal of the middle homology}\label{ct:section}
+
+Suppose, towards a contradiction, that the group constructed above is the
+fundamental group of a closed aspherical topological four-manifold.  We use
+the marked systems supplied by Proposition~\ref{st:periodic-data}.  Thus,
+after finitely many stabilizations on the compact quotient, the ordinary
+reflection development is a manifold
+\[
+                 Y=\bigcup_{v\in C}D_v.
+\]
+Here $D_v$ is a stabilized copy of the chamber, its boundary is $S$, and
+its group is the free group $H$.  Boundary identifications are understood
+with the signs induced by the orientation of $Y$; signs in hyperbolic
+planes are absorbed by changing one basis vector.  There are two
+labelled systems in $Y$:
+the chamber-wise immersed hyperbolic bases, denoted $\mathcal A$, and
+disjoint standard hyperbolic pairs, denoted $\mathcal B$.  Each pair of
+$\mathcal B$ has an embedded neighborhood homeomorphic to
+$(S^2\times S^2)\setminus\operatorname{int}D^4$.  Corresponding spheres in
+the two systems are based-homotopic.  All these data, including the
+homotopies, are lifts of finitely many compact data on the quotient.
+
+We will remove an expanding finite part of $\mathcal B$ and use the
+remaining part of $\mathcal A$ to perform surgery.  The latter operation
+requires geometric duals.  We obtain them by applying the proof of the
+disc embedding theorem to a specially constructed null-loop input;
+we do not assume that $H$ is a good group.
+
+There are three tasks. First we cap the noncentral boundary factors and
+remove the inner standard pairs, leaving a compact manifold with the
+original boundary marking. Next we prepare the remaining sphere
+systems and place their upper grope branches over contractible
+direction patches. This supplies the null-loop input. Finally the
+tower construction and sphere surgeries remove the remaining middle
+homology and produce a graph-type filling.
+
+\subsection{Complementary fillings and the peeling operation}
+
+Write $r=\operatorname{rank}H$.  A stabilization of $D$ has homotopy type
+\[
+        \bigvee^r S^1\vee\bigvee^{2k}S^2
+\]
+for some $k$, and its middle module has the specified hyperbolic basis.
+Neither the boundary nor its map to $BH$ changes under stabilization.
+
+\begin{lemma}[Complementary fillings]\label{ct:fillings}
+For every such stabilized chamber there is a simply connected compact
+oriented topological four-manifold $J$ with boundary $-S$ such that
+\[
+ H_2(S;\Z)\longrightarrow H_2(J;\Z)
+ \quad\text{is an isomorphism},\qquad H_3(J;\Z)=0.
+\]
+Moreover, $D\cup_S J$ is simply connected, and inclusion of $D$ identifies
+its second homology and intersection form with those of $D\cup_S J$.
+\end{lemma}
+
+\begin{proof}
+First recall the consequences of the graph-type Poincar\'e pair $(P,S)$
+recorded in Corollary~\ref{ch:boundary-homology}.  Duality and its
+ordinary homology sequence give
+\begin{equation}\label{ct:boundary-homology}
+ H_1(S;\Z)\cong H_1(P;\Z)\cong\Z^r,
+ \qquad H_2(S;\Z)\cong H^1(P;\Z)\cong\Z^r.
+\end{equation}
+The same corollary gives the surjectivity of $\pi_1(S)\to H$.
+
+Start with the oppositely oriented copy of $D$.  Represent a free basis
+of its fundamental group by disjoint embedded circles.  Their normal
+bundles are trivial, and general position makes the circles disjoint
+from the finite sphere system.  Interior surgery on these circles kills
+the free generators.  The old sphere classes retain their hyperbolic
+intersection and framing data after augmentation to $\Z$.  In the now
+simply connected manifold, the sphere embedding theorem with geometric
+duals realizes a Lagrangian by disjoint framed embedded spheres.  Surgery
+on these spheres preserves simple connectivity and removes the old
+hyperbolic planes.  This use of the disc theorem is in the simply
+connected case only; see \cite[Theorem~B]{PRT}.
+
+Let $J$ be the result.  Each circle surgery increases Euler characteristic
+by two, and each sphere surgery decreases it by two.  Hence
+\[
+ \chi(J)=(1-r+2k)+2r-2k=1+r.
+\]
+The boundary is connected and $J$ is simply connected.  Thus
+$H^1(J,S;\Z)=0$ by the cohomology sequence of the pair, and
+Poincar\'e--Lefschetz duality gives $H_3(J;\Z)=0$.  Since $J$ has nonempty
+boundary, $H_4(J;\Z)=0$, and therefore $b_2(J)=r$.
+
+There is no integral index ambiguity in this calculation.  The relevant
+part of the duality sequence is
+\begin{equation}\label{ct:cap-sequence}
+ 0\longrightarrow H_2(S;\Z)\longrightarrow H_2(J;\Z)
+ \longrightarrow H^2(J;\Z)\longrightarrow H_1(S;\Z)
+ \longrightarrow0.
+\end{equation}
+Here $H_3(J,S;\Z)=H^1(J;\Z)=0$ accounts for the initial zero.
+The universal coefficient theorem makes $H^2(J;\Z)$ free, because
+$H_1(J;\Z)=0$.  Every torsion element of $H_2(J;\Z)$ would consequently
+belong to the image of the injected free group $H_2(S;\Z)$, so there is
+no such element.  The first two nonzero groups in
+\eqref{ct:cap-sequence} now have the same rank $r$.  The middle map has
+rank zero and takes values in a free group, hence is zero.  Exactness
+proves that the first map is an integral isomorphism.
+
+The map $H_2(S;\Z)\to H_2(D;\Z)$ is zero: the next map
+$H_2(D;\Z)\to H_2(D,S;\Z)$ is the nonsingular hyperbolic intersection
+map.  Also $H_1(S;\Z)\to H_1(D;\Z)$ is the isomorphism in
+\eqref{ct:boundary-homology}.  Mayer--Vietoris, together with the
+isomorphism just proved for $J$, therefore identifies
+$H_2(D\cup_S J;\Z)$ with $H_2(D;\Z)$.  All products are unchanged,
+since their representatives lie in the interior of $D$.
+Finally, the surjectivity of $\pi_1(S)\to H$ and van Kampen show that
+$D\cup_S J$ is simply connected.
+\end{proof}
+
+Let $|v|$ denote Coxeter word length.  Lemma~\ref{rf:shelling} identifies
+the finite chamber union
+\[
+                  U_L=\bigcup_{|v|\leq L}D_v
+\]
+with an iterated boundary connected sum, with the natural markings on
+its boundary factors.  Fill every noncentral boundary factor by the
+corresponding $J_v$ of Lemma~\ref{ct:fillings}.  The resulting manifold
+$U_L^+$ has boundary $S$ and admits the description
+\begin{equation}\label{ct:filled-sum}
+ U_L^+\cong D_e\mathbin{\#}
+       \mathop{\#}_{0<|v|\leq L}(D_v\cup_S J_v).
+\end{equation}
+To see this description, retain the connecting three-balls in the
+boundary-sum construction.  Gluing a punctured filling to a noncentral
+factor closes that factor away from its connecting ball and turns the
+connecting boundary neck into an interior connected-sum neck.  Repeating
+this one factor at a time gives \eqref{ct:filled-sum}.
+
+It follows that $\pi_1(U_L^+)=H$, with the central boundary marking.
+On $\pi_1(U_L)$ the inclusion is the projection of the free product of
+the chamber groups onto its central factor.  The universal cover of
+$U_L^+$ has reduced homology only in degree two, freely based over
+$\Z H$ by all the chamber sphere systems, including stabilizations.
+Its intersection form is their orthogonal hyperbolic sum.  These facts
+also follow directly by lifting the connected-sum necks in
+\eqref{ct:filled-sum}; all the closed noncentral factors are simply
+connected and have precisely the middle classes computed above.
+
+For an integer $R$, replace every standard neighborhood in $\mathcal B$
+whose label belongs to a chamber $|v|\leq R$ by a four-ball.  Call this
+operation \emph{peeling}.  The common boundary is a standard $S^3$.
+There is a continuous pinch map from each old neighborhood to the new
+ball, fixed on that boundary, and these maps combine with the identity
+outside to a global pinch map.  Both pieces and the common boundary are
+simply connected, so peeling preserves the fundamental group.  The
+relative homology sequence deletes exactly the hyperbolic pair carried
+by that neighborhood and introduces no third homology.
+
+We first perform this operation in the unbounded $Y$, obtaining $Y_R$.
+For $L$ large enough to contain the selected neighborhoods and all their
+marking homotopies, it also defines a compact manifold
+\[
+                        Z=Z_{R,L}
+\]
+by peeling $U_L^+$.  The order of filling and peeling is immaterial
+because the standard neighborhoods are in the interior.  The included
+homotopies identify the removed summands with exactly the chamber pairs
+labelled by $|v|\leq R$.  We have therefore proved
+\begin{equation}\label{ct:Z-homology}
+ \partial Z=S,\qquad \pi_1 Z=H,\qquad
+ \widetilde H_i(\widetilde Z;\Z)=0\quad(i\ne2),
+\end{equation}
+and $H_2(\widetilde Z;\Z)$ is freely based by the projected chamber
+pairs with $R<|v|\leq L$, with their orthogonal hyperbolic form.
+
+\subsection{Uniform finite preparation}
+
+A compact object is \emph{carried within $b$ galleries of $v$} if its
+image is contained in chambers whose labels are within word distance
+$b$ of $v$.  Replacement balls are assigned the carriers of their old
+standard neighborhoods.  We use this terminology for sphere maps,
+paths, homotopies and grope systems, including all their caps.  A
+uniform propagation bound is one value of $b$ valid for the whole
+labelled family.  It is stronger than a bound after an unspecified
+permutation of the labels.
+
+\begin{lemma}[Finite preparation]\label{ct:preparation}
+Fix a positive integer $h$.  In $Y_R$, the projected pairs of
+$\mathcal A$ labelled by $|v|>R$ admit the following preparation, with
+propagation bounded independently of $R$:
+\begin{enumerate}
+\item The two halves are represented by framed immersed spheres
+      $a_i,b_i$ with $a_i\cap b_j$ consisting of the designated single
+      point for $i=j$ and empty for $i\ne j$.
+\item The remaining intersections among the $a_i$ are paired by
+      framed Whitney discs.  Each Whitney disc is replaced by a
+      properly immersed capped grope of height at least $h$ with the
+      same framed attaching region.
+\item These gropes have the Clifford dual capped surfaces used in
+      \cite[Definition~3.1 and Lemma~4.1]{PRT}.  In particular, their
+      caps miss the \emph{entire} dual capped surfaces, and the bodies
+      of the latter lie in a boundary collar of the complement of the
+      $a_i$.
+\end{enumerate}
+The construction uses no complementary filling $J_v$.  Beyond a fixed
+gallery distance from the peeling region, it can be carried out
+separately inside individual chambers.
+\end{lemma}
+
+\begin{proof}
+First project the outer immersed systems through the pinch and put them
+in general position.  The deleted and retained blocks were orthogonal,
+since their classes agree with the correspondingly labelled standard
+pairs.  The retained forms and quadratic self-intersection data are
+therefore still hyperbolic over $\Z\pi_1(Y_R)$.  Choose framed immersed
+representatives and cancel the algebraic intersection pairs.  The
+framing corrections and, where necessary, local cusp changes can be
+chosen from finitely many local drawings.  The group in question is a
+free product of free chamber groups, hence is torsion-free; there is no
+order-two ambiguity in these quadratic data.
+
+We justify carefully the uniformity of the Whitney discs required at
+this and subsequent finite stages.  The initial lifted data have
+finitely many types under a cocompact group of translations of $Y$.
+A gallery ball of fixed radius contains a uniformly bounded number of
+chambers and of pieces of those data.  There are thus finitely many
+local templates, even after specifying which of the standard
+neighborhoods meeting that ball have been peeled.  Paths on participating
+spheres are taken inside their bounded carriers.  Common long whiskers
+conjugate an intersection label but need not be included in the
+geometric Whitney circuit.
+
+If a circuit enters a replacement ball, move its portions there to
+chosen arcs on the boundary three-sphere.  This gives a circuit in the
+unpeeled space.  Nullity is unchanged, since both the deleted
+punctured $S^2\times S^2$ and the replacement ball are simply connected
+and have simply connected boundary.  For every null circuit among the
+finitely many resulting templates, choose one compact filling in $Y$.
+There is a finite maximum of their propagation bounds.  Translate the
+chosen filling to the location of the circuit and apply the actual
+global pinch map.  This constructs a filling in $Y_R$.
+
+A filling may encounter additional peeled neighborhoods.  No further
+choice of fillings is then needed: the global pinch is already defined
+on them, and each such neighborhood has the same bounded carrier.
+Consequently this procedure is uniform in $R$.  It is a finite-template
+argument for the specified circuits, not a bound for arbitrary null
+loops.  Repeating a fixed finite number of stages preserves the
+argument: enlarge the carrier radius and refine the finite template
+list at each stage.
+
+Now apply the geometric Casson lemma to the two halves of the sphere
+system, removing the paired mixed intersections by regular homotopies.
+There is no iterative repair cascade here.  Use the local construction
+of \cite[Lemma~3.3]{PRT} with the two surface arguments taken to be the
+entire possibly disconnected families.  Preselect all framed pairing
+Whitney discs with disjoint boundaries; their interiors miss the other
+Whitney boundary arcs by $2+1<4$ general position.  First push the first
+family off all these interiors by fingers along arcs in the Whitney
+discs.  Choose the arcs away from their finitely many intersections
+with other Whitney discs and the second family.  The finger supports
+then leave the other Whitney discs and attaching boundaries fixed.
+Next Whitney-move the second family using the same discs.  The new
+intersections in each phase are only within that phase's family;
+intersections between Whitney interiors cause only allowed
+second-family intersections.  Thus the whole batch stays in the
+original bounded Whitney neighborhoods.  No later disc is transported
+or chosen again through an expanding sequence of carriers.  Local
+finiteness permits these bounded batches on the unbounded $Y_R$.
+This produces the geometric duality in (1), while permitting
+self-intersections within either half.  The remaining double points
+of the first half admit framed Whitney discs.  Tube their interiors
+off the first-half spheres using the geometric duals.  All the
+Whitney circuits involved are the null circuits just discussed.
+
+For completeness, we verify the algebraic input to the grope preparation
+in the complement $M_0$ of small plumbed neighborhoods of the $a_i$.
+The punctured geometric dual $b_i$ bounds an $a_i$-meridian in $M_0$.
+Since these meridians normally generate the kernel, inclusion induces
+$\pi_1M_0\cong\pi_1Y_R$.  Thus the following calculation retains all
+intersection labels in the required complement group ring.
+
+At one point of every paired double point take its Clifford torus.
+Cap its two basis curves by standard meridional discs and tube each
+meridional disc off its incident $a_i$-sheet into a punctured parallel
+of $b_i$, with the product framing.  Denote these caps by $C_\ell$,
+and let $i(\ell)$ be the index of the dual used.  Their equivariant
+intersection calculation is the Clifford-cap calculation in the proof
+of \cite[Section~5]{PRT}.  The local meridional pieces have disjoint
+interiors.  Choose the thin tubing tracks away from the finitely many
+other designated dual points; local changes of tracks introduce only
+oppositely signed points with the same group label.  The remaining
+intersections are those between the corresponding based parallels of
+$b_{i(\ell)}$.  Hence their sums are obtained from the pairings of the
+$b_i$ by the basing units and orientation signs.  Likewise, a cap's
+self-intersection sum is that of its framed $b_i$ parallel, together
+with cancelling local terms.  Since the mixed Casson regular homotopies
+preserved $\lambda(b_i,b_j)=0$ and $\mu(b_i)=0$, this proves
+\[
+ \lambda(C_\ell,C_m)=0,\qquad \mu(C_\ell)=0
+ \quad\text{for every }\ell,m,
+\]
+including distinct parallel caps using the same dual sphere.
+
+The Clifford bodies lie in a boundary collar of $M_0$.  Following
+\cite[Section~5]{PRT}, contract temporary parallel copies and push the
+Whitney discs off those contractions before using them as geometric
+dual spheres.  At the remaining intersections with Whitney discs,
+tube the Clifford caps into parallel copies of these temporary duals.
+Lemma~4.1 of \cite{PRT}
+now applies with its full cap-pairing hypothesis.  Its two separation
+batches and height raising provide the prepared Clifford copies,
+whose entire capped surfaces are geometrically dual in the sense of
+\cite[Definition~3.1]{PRT}.  These prepared copies, including all their
+caps, are the protected surfaces used from this point onward.
+The preparation in \cite[Lemma~4.1]{PRT} produces height-two Whitney
+gropes, separates their caps from the protected Clifford caps, and
+raises the height to $h$. The collar levels and the full dual-cap
+disjointness are part of this preparation, not an assertion made after
+the caps have been changed.
+There are exactly two cap-family separation batches before height
+raising.  Each uses the same no-new-mixed-pairs invariant, with its
+protected bodies at fixed collar levels.
+
+Each of the operations before height raising involves only a fixed
+number of kinds of tubing, push-off, Whitney-disc choice and regular
+homotopy.  Their simultaneous use does not require traversing the
+whole list of sphere components.  At each fixed stage, bounded
+carriers and finite drawings bound the number of other carriers
+meeting a given one.  Divide the operations into finitely many
+classes with disjoint enlarged carriers and perform one class at a
+time.  This gives a number of local stages independent of $R$.
+New null circuits are handled by the finite-template procedure above.
+
+For height raising there is also an explicit size estimate:
+\cite[Section~2.8.3]{FQ}, in the January 2013 numbering, bounds the
+increase of component-image diameter by a factor of seven at each
+step.  A fixed requested height therefore has a fixed propagation
+bound.  One can apply this estimate to the gallery map into the
+Davis complex: extend that map over each replacement ball by coning
+its boundary image inside a metric ball of uniformly bounded radius.
+This is possible by CAT(0) convexity.  A point in a chamber is mapped
+within a fixed distance of its label, and properness of the Coxeter
+action turns a fixed metric bound into a fixed word-distance bound.
+The construction uses parallel transverse gropes and
+neighborhood operations; it preserves the protected Clifford capped
+surfaces.  All these operations can be made locally finite in the
+unbounded development.
+
+Finally, outside a fixed buffer of the peeling region the input is
+the original system in an individual chamber.  Its intersection
+cancellations hold already over that chamber's group ring.  The same
+preparation, including the null discs, can consequently be chosen in
+that chamber.  Only a bounded buffer of the switch between the
+peeled and unpeeled data requires the preceding translated templates.
+This proves the last assertion and permits later truncation without
+cutting off an unfinished preparation.
+\end{proof}
+
+\subsection{Directions at infinity}
+
+Let $\Sigma_C$ be the ordinary Davis complex and let $o$ be its vertex
+labelled by $e\in C$.  The gallery map of Lemma~\ref{rf:davis-map} sends
+the inner part of $D_v$ to $v$, interpolating across the dual collars.
+The link of $o$ is $S$.  A small metric sphere of radius $t$ about $o$
+is consequently identified with $S$ by radial scaling in its conical
+star.  Write $\rho$ for radial projection onto this sphere, defined
+outside the open ball of radius $t$.
+
+\begin{lemma}[Small directions and the central marking]\label{ct:directions}
+Fix a propagation bound $b$.  For objects carried within $b$ galleries
+of labels $v$ with $|v|\to\infty$, their direction images in $S$ have
+diameter tending uniformly to zero.  This assertion continues to hold
+after peeling for the whole region of bounded propagation around the
+outer systems, if $R$ is sufficiently large.
+
+Let $\beta:S\to BH$ be the central boundary marking.  On that region,
+the homomorphism induced by the direction map followed by $\beta$
+agrees, up to the common change of base path, with inclusion into
+$Z$ followed by its classifying map.  In particular, every loop in an
+image system whose entire direction image lies in a contractible
+patch of $S$ is nullhomotopic in $Z$.
+\end{lemma}
+
+\begin{proof}
+Here is the radial estimate that supplies the first assertion.  If
+$x,y\in\Sigma_C$ have distance at least $s>t$ from $o$, put them first
+on the same metric sphere, of radius the smaller of their two radii.
+The distance between the resulting points is at most $2d(x,y)$,
+by the triangle inequality and
+$|d(o,x)-d(o,y)|\leq d(x,y)$.  CAT(0) comparison on the two radial
+segments then gives
+\begin{equation}\label{ct:radial-estimate}
+                  d(\rho(x),\rho(y))
+                  \leq \frac{2t}{s}\,d(x,y).
+\end{equation}
+The gallery map takes each chamber into a set of uniformly bounded
+diameter.  Thus a bounded gallery carrier has bounded image diameter.
+Properness of the Davis complex, together with its cocompact Coxeter
+action, makes its distance from $o$ tend to infinity with the word
+length of its center.  Equation~\eqref{ct:radial-estimate} applies.
+
+We only need the direction map outside the central core.  Consider a
+peeled standard neighborhood that meets an outer object or any of
+its fixed enlarged carriers.  Both the object and the whole standard
+neighborhood have bounded propagation.  The label of this neighborhood
+therefore has length at least $R$ minus a fixed constant.  Its entire
+boundary three-sphere, not merely the arcs encountered by the object,
+has direction image of arbitrarily small diameter for large $R$.
+Choose a slightly larger coordinate ball of $S$ containing this image.
+Contracting that ball extends the boundary map across the replacement
+four-ball, with image still in the chosen ball.  These extensions
+agree with the unchanged direction map on all their boundaries.
+
+The near-central peeled neighborhoods need not admit such small
+extensions.  They are disjoint from all the carriers under consideration
+once $R$ is large, and no direction map on them is used.  This proves
+the asserted far-out extension and its uniform smallness.
+
+Before peeling, the composition with $\beta$ extends across the
+central chamber by its given map to $BH$.  Its restriction to the
+central boundary agrees with that map: this is the boundary-direction
+identification in Lemma~\ref{rf:davis-map}.  In a noncentral chamber
+the gallery map collapses its inner part to the noncentral vertex,
+so every loop of its chamber group has constant direction there.
+The shelling description makes the chamber groups the free factors
+of the fundamental group of a finite ball of chambers.  The induced
+homomorphism is thus exactly the projection retaining the central
+factor.  By \eqref{ct:filled-sum} this is also the homomorphism induced
+by passage to $U_L^+$.
+
+Replacing a simply connected piece by a simply connected ball does
+not alter this calculation.  Loops through the replacement can be
+moved to the boundary of the piece, and the direction extension
+agrees there with the old map.  The comparison therefore holds on
+the required region in $Z$.  If a whole image system maps into a
+contractible patch, every one of its loops, including loops passing
+through arbitrarily many intersections, has trivial image under
+$\beta$.  Since $BH$ is a classifying space and $\pi_1Z=H$, these
+loops are nullhomotopic in $Z$.
+\end{proof}
+
+\subsection{Upper grope separation}
+
+We state explicitly how the controlled data supply the input used
+after the good-group step of the disc embedding proof.  The units to
+be separated are the upper capped subgropes attached to individual
+tips of the retained stages.  Separating only whole Whitney gropes
+would not suffice to make the future cap discs mutually disjoint.
+
+\begin{lemma}[Null upper replacements]\label{ct:upper-replacement}
+Suppose a finite prepared Whitney-grope system as in
+Lemma~\ref{ct:preparation} has at least eight surface stages.  Retain
+its first five stages.  Suppose the upper capped subgropes, of height
+at least three, have uniformly small direction images in the region
+of Lemma~\ref{ct:directions}.  If their diameters are sufficiently
+small, depending only on a fixed finite coordinate cover of $S$, the
+upper subgropes can be replaced by mutually disjoint framed immersed
+cap discs.  Their self-intersection loops are nullhomotopic in $Z$.
+The retained stages and the entire protected Clifford capped
+surfaces remain unchanged.
+\end{lemma}
+
+\begin{proof}
+We first specify the relative setting. Remove a sufficiently small
+regular neighborhood of the five retained stages, truncating each upper
+attaching collar at its boundary. The upper components are now properly
+immersed disk-like capped gropes of height at least three. By the duality
+condition in Lemma~\ref{ct:preparation}, their entire images miss the
+protected Clifford capped surfaces. Choose a neighborhood of the upper
+system disjoint from those surfaces and from the retained stages except
+at the prescribed attaching regions. All the following modifications
+are neighborhood operations in this relative manifold. They therefore
+leave the lower stages, attaching framings, and whole Clifford systems
+unchanged, including the unused boundary-collar levels needed later.
+
+We use the following precise content of Freedman--Quinn's controlled
+separation lemma \cite[Lemma~2.10.1]{FQ}. For a map from a four-manifold
+to a locally compact metric space, a properly immersed union of
+disk-like capped gropes of height at least three, with sufficiently
+small component images, admits new caps for the \emph{same bodies}
+such that distinct components are disjoint and their images remain
+small. Its compact-case proof separates finitely many cover collections
+before separating the members of each collection
+\cite[Section~2.10.2]{FQ}. We retain this two-phase construction because
+we need a bound on gallery propagation only in its first phase.
+
+Choose compact sets $K_1,\ldots,K_q$ covering $S$, each contained in an
+open coordinate ball $V_j$. Choose $\epsilon>0$ so that the closed
+$\epsilon$-neighborhood of every $K_j$ lies in $V_j$. Assign each upper
+component to a $K_j$ met by its direction image. If every initial image
+has diameter less than $\delta$, that component lies within distance
+$\delta$ of its assigned $K_j$.
+
+For the first phase, regard all components assigned to $K_j$ as one
+collection. The construction in \cite[Section~2.10.2]{FQ} raises height
+by one, contracts the added stages of the first collection and pushes
+the later collections off, then continues through the collections in
+order. The original bodies are retained. Its component-diameter estimate is
+$3^{q-1}7^2\delta$. A point of each retained body stays within
+$\delta$ of $K_j$, so the entire resulting $j$th collection lies in
+the $3^{q+3}\delta$-neighborhood of $K_j$. Choose $\delta$ with
+$3^{q+3}\delta<\epsilon$, leaving room for a small regular-neighborhood
+thickening. The number of height-raising and contraction passes depends
+only on $q$. The same size estimates apply to the gallery map into
+$\Sigma_C$, extended over the replacement balls as in
+Lemma~\ref{ct:preparation}. Since each original body is retained, its
+image anchors the resulting component within a bounded metric distance
+of the original label. Properness of the Coxeter action converts this
+into a gallery bound independent of the number of components and of $R$.
+
+The collections now have pairwise disjoint images. Choose disjoint
+regular neighborhoods of their entire images, with the $j$th
+neighborhood still mapping into $V_j$. Inside each such neighborhood,
+apply the collection-separation procedure again, this time taking
+each individual upper component as a collection. There may be
+arbitrarily many of them. The height is raised once more inside that
+neighborhood before contracting the added stages, so this second
+phase again retains the original bodies. No estimate depending on
+their number is needed: all operations stay in the already chosen
+neighborhood. The resulting upper components have pairwise disjoint
+neighborhoods. Totally contract each in its own neighborhood to obtain
+the claimed mutually disjoint framed immersed cap discs.
+
+Every loop in a resulting cap image lies in its color neighborhood,
+whose entire direction image is contained in the contractible $V_j$.
+Lemma~\ref{ct:directions} therefore makes that loop nullhomotopic in
+$Z$. The second phase need not bound the gallery diameter of an
+individual cap. It preserves instead the containment of its whole
+image in a direction patch, which is exactly the condition needed
+for nullity.
+\end{proof}
+
+\subsection{Choice of parameters and the tower construction}
+
+Fix the coordinate cover in Lemma~\ref{ct:upper-replacement}. Request
+height eight in Lemma~\ref{ct:preparation}. Its uniform propagation
+bound, together with the bounds for the standard neighborhoods and
+marking homotopies, controls the entire preparation near the peeling
+boundary. The first phase of upper separation adds only the fixed
+height extension and $q$ collection passes described above. Let $b$
+bound the gallery propagation of all these operations, independently
+of $R$. Choose $R$ so large that the direction estimates of
+Lemma~\ref{ct:directions}, including the replacement-ball extensions,
+meet the chosen cover margins for every such carrier.
+
+We now truncate the prepared family, before carrying out the
+collection-separation operations. Choose $L$ large enough to contain
+every peeled standard neighborhood and its marking homotopies, and
+the whole $b$-buffer around the transition between peeled and
+unpeeled data. Beyond that buffer, Lemma~\ref{ct:preparation} constructs
+the sphere and Whitney-grope data entirely inside their own chambers,
+using their chamber group-ring pairings. Retain precisely the planes
+whose labels satisfy $R<|v|\le L$, together with their complete
+preparations. The preparations at the outer boundary lie in chamber
+interiors, so none has a partner or an attaching region outside
+$U_L$. This gives a finite family in the peeled copy of $U_L$, before
+any complementary filling is used.
+
+Cap its noncentral boundary factors to form $Z=Z_{R,L}$. Apply the
+first phase of
+Lemma~\ref{ct:upper-replacement} to this finite family. All of its
+operations can be made in arbitrarily small neighborhoods of the
+retained upper system inside that peeled region; the bound $b$ supplies
+the direction
+margins independently of the truncation. Apply the second phase inside
+the resulting disjoint color neighborhoods. Neither phase requires
+an enlargement of $L$ or uses the complementary fillings. The second
+phase supplies the immersed null-loop caps without a uniform gallery
+bound on their individual images.
+
+Write
+\[
+                  M'=Z\setminus\bigcup_i\operatorname{int}\nu(a_i)
+\]
+for the complement used in the Whitney construction.  The spheres
+$a_i$ may still be immersed; the notation denotes their standard
+plumbed neighborhoods.  The prepared geometric dual spheres imply
+\begin{equation}\label{ct:complement-group}
+                   \pi_1(M')\xrightarrow{\ \cong\ }\pi_1(Z).
+\end{equation}
+Indeed the kernel of the inclusion map is normally generated by
+meridians of the $a_i$, and the punctured dual spheres bound these
+meridians in the complement.  General position supplies surjectivity.
+This is the usual $\pi_1$-negligibility provided by geometric duality.
+
+The new caps therefore have null double-point loops in $M'$, not
+just in $Z$.  They are mutually disjoint, although each may have
+self-intersections. Direction control has now supplied the needed
+fundamental-group condition. The remaining constructions require that
+condition and the protected geometric duals, but no bound on the
+diameter of the null homotopies. Symmetrically contract the fifth retained
+surface stage.  The result is a height-four capped Whitney grope
+system, still with mutually disjoint caps.  Its cap
+self-intersections occur in algebraically cancelling pairs, and
+their loops are parallel copies of the previous null loops.  This
+is exactly the intermediate input in the proof of
+\cite[Lemma~4.2]{PRT}, after that proof has used goodness.  The
+retained Clifford capped surfaces are still geometrically dual in
+the stronger sense of \cite[Definition~3.1]{PRT}.
+
+We explain why the remainder of that proof applies without any
+assumption on $H$.  Cap the null double-point loops by immersed
+discs in $M'$.  The bodies of the Clifford capped surfaces lie at
+the protected boundary-collar levels.  The attaching loops of the
+new discs miss that collar, so the null homotopies can be chosen
+to miss the Clifford \emph{bodies} by pushing them out of the
+collar.  They may meet the Clifford caps and may meet the Whitney
+gropes arbitrarily.  Correct their boundary framings, push their
+intersections with the gropes down, and tube into geometrically
+dual spheres obtained by contracting suitable parallel copies of
+the Clifford systems.  These operations produce the tower caps.
+At the next collar level, contract the remaining Clifford capped
+surfaces and push the tower caps off those contractions.  The
+resulting spheres are geometric duals to the one-storey capped
+towers.  These are the successive operations in the remainder of
+the proof of \cite[Lemma~4.2]{PRT}; no subsequent step uses goodness.
+
+One-storey capped towers with four surface stages contain the
+embedded discs required for the Whitney moves, with the same
+framed attaching regions.  Applying the final steps of
+\cite[Section~5]{PRT}, first move the original sphere duals off the
+towers by tubing into the tower duals.  Then use the embedded
+Whitney discs to remove the intersections of the $a_i$.  We obtain
+disjoint framed embedded representatives of their original
+Lagrangian classes, together with geometric transverse spheres.
+The changes of upper caps do not change the retained Whitney
+attaching data or the corrected sphere classes.  In particular,
+the symmetric-contraction independence of cap choices in
+\cite[Lemma~3.4]{PRT} applies at the retained stages.
+
+There is deliberately no assertion that the null homotopies, the
+tower caps, or the final embedded Whitney discs have small gallery
+diameter.  From \eqref{ct:complement-group} onwards they may range
+throughout $Z$.  Only their existence, their relative attaching
+data, and the geometric duals are needed.
+
+\Needspace{7\baselineskip}
+\begin{proposition}[Removal of the remaining planes]\label{ct:removal}
+The compact manifold $Z$ admits interior surgeries producing a
+compact oriented topological four-manifold $V$ with boundary $S$,
+fundamental group $H$, and contractible universal cover.  The
+boundary map to $BH$ is the original central marking.
+\end{proposition}
+
+\begin{proof}
+Perform surgery on the disjoint framed Lagrangian spheres just
+constructed.  Their geometric duals kill the meridians in their
+complement, so the surgeries preserve the fundamental group.
+All changes are in the interior; the boundary and its marking are
+unchanged.
+
+For clarity, let there be $m$ remaining planes and let $W$ be the
+five-dimensional surgery trace.  With regular $\Z H$ coefficients,
+the relative chain complex of $(W,Z)$ is concentrated in degree
+three, with module $(\Z H)^m$.  Its boundary map to
+$H_2(Z;\Z H)$ sends the basis to the Lagrangian basis.  By
+\eqref{ct:Z-homology} this is a split injection.  Consequently
+\[
+ H_3(W;\Z H)=0,\qquad
+ H_2(W;\Z H)\cong(\Z H)^m,
+\]
+the latter module represented by the dual half, and the other
+positive-degree homology groups vanish.  Read the same trace from
+$V$.  Its relative chain complex is concentrated in degree two,
+again with module $(\Z H)^m$.  The map
+\[
+            H_2(W;\Z H)\longrightarrow H_2(W,V;\Z H)
+\]
+is the pairing of the surviving dual classes with the surgered
+Lagrangian, hence is an isomorphism.  The sequence of $(W,V)$
+therefore gives $H_i(V;\Z H)=0$ for every $i>0$.
+
+The universal cover of $V$ is simply connected and acyclic. A
+topological manifold has CW homotopy type~\cite[Corollary~1]{MilnorCW};
+in the boundary case one may first append an exterior open collar,
+which does not change homotopy type. The homological Whitehead theorem
+therefore makes this universal cover contractible
+\cite[Corollary~4.33]{Hatcher}. Thus $V$ is a $K(H,1)$ and is homotopy equivalent
+to the finite graph $BH$.  Under this equivalence its boundary
+map is the retained central marking, as asserted.
+\end{proof}
+
+The manifold $V$ is the marked graph-type filling excluded by
+Proposition~\ref{ch:chamber}.  This contradiction rules out the hypothetical
+closed aspherical topological four-manifold and completes the proof
+of Theorem~\ref{thm:main}.
diff --git a/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/build/sections/reflection.tex b/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/build/sections/reflection.tex
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+++ b/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/build/sections/reflection.tex
@@ -0,0 +1,427 @@
+\section{Reflection and the duality group}
+\label{rf:section}
+
+We now close the chamber without losing its labelled middle-dimensional
+classes, using the reflection framework of Davis
+\cite[Sections~13 and~15, especially Remark~15.9]{Davis1983}.
+A second construction will give the resulting group a proper
+cocompact action on a CAT(0) space.  These are different developments:
+the first retains the four-dimensional chambers, whereas the second is
+used only for the assembly theorems.
+
+The input from Proposition~\ref{ch:chamber} is a compact connected oriented
+smooth four-manifold $D$, with connected boundary $S$ and free fundamental
+group $H$, together with a map of pairs
+\[
+ (D,S)\longrightarrow(P,S).
+\]
+Here $D$ has the homotopy type of a finite graph with finitely many
+two-spheres, its $\Z H$-intersection form has a specified framed
+hyperbolic basis of $k$ planes, and $P$ is obtained by attaching
+three-cells along all $2k$ basis spheres.  The pair $(P,S)$ is an
+oriented Poincar\'e pair of dimension four and $P\simeq BH$.
+The map on fundamental groups is the given identification with $H$.
+We use a collar on $S$ in both spaces, inserting a mapping cylinder
+on the $P$ side if necessary.
+
+By Corollary~\ref{ch:boundary-homology}, the boundary map
+$\pi_1(S)\to H$ is surjective.  In particular its associated cover of
+$S$ is connected.
+
+\subsection{Panels and finite developments}
+\label{rf:panels}
+
+Choose a finite flag triangulation $\mathcal L$ of $S$, for example a
+barycentric subdivision of a smooth triangulation.  Write $V$ for its
+vertex set and $S_s$ for the closed dual block of $s\in V$ in the
+barycentric subdivision.  These blocks are three-dimensional panels.
+For nonempty $T\subseteq V$, the intersection $S_T=\bigcap_{s\in T}S_s$ is nonempty exactly when
+$T$ spans a simplex of $\mathcal L$; in that case it is a ball of
+dimension $4-|T|$.  Collar neighborhoods of these intersections have
+the usual orthant structure.
+
+Let
+\[
+ C=\left\langle s\in V\ \middle|\ s^2=1,\quad
+ st=ts\text{ whenever }\{s,t\}\in\mathcal L\right\rangle
+\]
+be the right-angled Coxeter group.  For a collared space $B$ with these
+panels in its boundary, its development is
+\[
+ \begin{gathered}
+ \mathcal U(C,B)=(C\times B)/\sim,
+ \qquad T(x)=\{s:x\in S_s\},\\
+ (c,x)\sim(c',x)\quad\Longleftrightarrow\quad
+ c^{-1}c'\in C_{T(x)}.
+ \end{gathered}
+\]
+Here $C_T$ is the subgroup generated by $T$ and $T(x)=\varnothing$
+off the boundary.  Each chamber embeds: the equivalence relation never
+identifies different values of $x$.  For nonempty $T(x)$ the subgroup
+$C_{T(x)}$ is $(\Z/2)^{|T(x)|}$, and its sectors fill the corresponding
+orthant neighborhood.
+
+We use the elementary Coxeter word facts that special subgroups embed,
+a finite special coset has a unique shortest element, and the right
+descent set
+\[
+ \operatorname{Des}(c)=\{s\in V:|cs|<|c|\}
+\]
+is a clique.  These follow from the right-angled reduced-word rule of
+commuting adjacent commuting letters and cancelling equal consecutive
+letters; see also \cite[Chapters~4--5]{Davis}.
+
+\begin{lemma}[Labelled finite developments]
+\label{rf:shelling}
+Put $Y=\mathcal U(C,D)$ and let $D_c$ be its chamber with label $c$.
+Order $C$ by nondecreasing word length, breaking ties arbitrarily.
+Every finite initial segment containing the identity is, after rounding
+its collar corners, an iterated boundary connected sum of its chambers.
+In particular
+\[
+ U_L=\bigcup_{|c|\leq L}D_c
+\]
+is a compact manifold with boundary the connected sum of the corresponding
+copies of $S$.  Its fundamental group is the free product of their
+chamber groups.  The same statements hold if any chambers have been
+stabilized in their interiors.
+\end{lemma}
+
+\begin{proof}
+The part of a new chamber $D_c$ already present is exactly
+\begin{equation}
+ \label{rf:incoming}
+ A_c=\bigcup_{s\in\operatorname{Des}(c)}S_s.
+\end{equation}
+To verify the possible equal-length overlaps, consider a point with
+panel set $T$.  Its incident chambers are the special coset $cC_T$.
+Write $a$ for the shortest element of that coset.  Its members have
+the form $a\prod_{s\in J}s$, with $J\subseteq T$, and length
+$|a|+|J|$.  If a distinct incident chamber occurs no later than $c$,
+then $c$ is not the shortest member, so it has a descent in $T$.
+This places the point in \eqref{rf:incoming}.  The converse follows
+from the shorter adjacent chamber across each descent panel.
+
+The set in \eqref{rf:incoming} is a three-ball.  More generally a union
+of dual panels indexed by a nonempty simplex is a ball.  One proves
+this by adding its panels one at a time.  A new panel is a ball, and
+its intersection with the preceding panels is the corresponding union
+of dual panels in its boundary link.  Induction on dimension makes
+that intersection a ball of one lower dimension.  The dual-block
+coordinates give the product collars needed to glue the two balls.
+This argument also applies to panels in any simplicial cover of $S$.
+
+It remains to check that $A_c$ is a ball in the \emph{boundary} of the
+preceding union.  Contractibility alone would not establish the claim.
+At a face incident to $q$ panels, label the $2^q$ local orthants by
+subsets of those panels, starting at the shortest incident chamber.
+The already present orthants form a Boolean downset $I$: every proper
+subset of a present subset has smaller word length.  This remains true
+with arbitrary choices within a length tie.
+
+When $I$ is nonempty and proper, its frontier is a PL hypersurface.
+For an explicit verification, write local coordinates as
+$x=u+t(1,\ldots,1)$ with $\sum u_i=0$, and label an orthant by its
+positive coordinates.  Its union is described by
+\[
+ t\leq f(u),\qquad
+ f(u)=\max_{A\text{ maximal in }I}\ 
+               \min_{i\notin A}(-u_i).
+\]
+The function $f$ is finite, continuous, and piecewise linear.  Hence
+the frontier is locally a graph.  Along the incoming faces some sectors
+are present and the new chamber sector is absent, so these faces lie
+on this frontier.  The panel identifications are PL, and the embedded
+incoming three-ball has locally flat boundary in this three-manifold
+frontier.  Gluing the new chamber along it is therefore a boundary
+connected sum, after rounding.  Induction proves the manifold and
+boundary assertions.  Van Kampen gives the labelled free product,
+and interior stabilizations do not affect any collar argument.
+\end{proof}
+
+\subsection{The universal development}
+\label{rf:universal}
+
+Let
+\[
+ \Delta=\ker\bigl(C\longrightarrow(\Z/2)^V\bigr),
+\]
+where each generator maps to its own coordinate vector, and define
+\[
+ N=\Delta\backslash\mathcal U(C,D),\qquad
+ Q=\Delta\backslash\mathcal U(C,P).
+\]
+The parity map is injective on every finite special subgroup.
+Thus $\Delta$ has trivial intersection with every point stabilizer
+in either development.  The quotient has finitely many chambers.
+For $D$ the orthant charts show that $N$ is a closed four-manifold.
+Give chamber $c$ the orientation $(-1)^{|c|}$ times the chosen chamber
+orientation.  Adjacent chamber orientations fit across a panel, and
+$\Delta$ preserves them.  Thus $N$ is oriented.  The same convention
+orients the reflected Poincar\'e pairs.
+
+Let $\widehat S\to S$ be the connected cover determined by
+$\pi_1(S)\to H$, with lifted triangulation $\widehat{\mathcal L}$.
+This is a flag triangulation.  Indeed, the vertices of a lifted clique
+project to distinct pairwise adjacent vertices downstairs.  They span
+a simplex there; lifts of its edges, and of its triangular relations,
+place all the vertices in one lift of that simplex.  Denote by
+$\widehat C$ the right-angled Coxeter group on
+$\widehat{\mathcal L}$.  The deck action of $H$ permutes its generators.
+Forgetting the lift gives an $H$-invariant homomorphism
+$q:\widehat C\to C$, and hence a surjection
+\begin{equation}
+ \label{rf:group}
+ \begin{gathered}
+ \rho:\widehat C\rtimes H\longrightarrow C,
+ \qquad \rho(\widehat c,h)=q(\widehat c),\\
+ G=\rho^{-1}(\Delta).
+ \end{gathered}
+\end{equation}
+
+Let $\widetilde P$ be the universal cover of $P$, with boundary
+preimage $\widehat S$ and its lifted panels.  On
+$\widehat Q=\mathcal U(\widehat C,\widetilde P)$ the action is
+\[
+ (\widehat c,h)[\widehat d,x]
+     =[\widehat c\,h(\widehat d),hx].
+\]
+The map $[\widehat d,x]\mapsto[q(\widehat d),\overline x]$ descends
+to the ordinary development.  On a chamber interior it is the usual
+covering map.  At a boundary point it is also a covering map: the
+lifted panel set maps bijectively to its downstairs panel set, and
+the corresponding finite special groups map isomorphically.  These
+local charts identify $\widehat Q/G$ with $Q$.  The action of $G$ is
+free: if an element fixes a point, its $H$-component fixes the chamber
+coordinate and is therefore trivial, after conjugating the chamber
+label; the remaining stabilizer is a lifted finite special group,
+on which $\rho$ is injective and whose image misses $\Delta$.
+
+The space $\widehat Q$ is contractible.  Filter its chambers by word
+length in $\widehat C$.  The first chamber $\widetilde P$ is
+contractible.  Every further chamber meets the shorter ones in the
+nonempty ball of its descent panels, by the argument of
+Lemma~\ref{rf:shelling}.  Its attachment is therefore a homotopy
+equivalence.  In an infinite word-length layer, new interiors are
+disjoint and their mutual intersections already lie in shorter layers.
+All these attachments may be performed simultaneously.  The CW
+direct limit is weakly contractible and hence contractible.  This
+proves, in particular, that $Q$ is a finite $K(G,1)$.
+
+\begin{proposition}[The reflected pair]
+\label{rf:development}
+The space $Q$ is a finite oriented Poincar\'e complex of dimension four.
+The natural map $f:N\to Q$ has degree one and induces an isomorphism
+on fundamental groups, identifying both with the group $G$ in
+\eqref{rf:group}.  If $b$ is the total number of chamber hyperbolic
+planes in $N$, then
+\[
+ \pi_2(N)\cong(\Z G)^{2b}
+\]
+has the specified orthogonal hyperbolic basis, and
+$\widetilde H_i(\widetilde N;\Z)=0$ for $i\ne2$.
+The group $G$ is finitely presented, is torsion-free, and satisfies
+\begin{equation}
+ \label{rf:pd}
+ H^i(G;\Z G)=0\quad(i\ne4),\qquad
+ H^4(G;\Z G)\cong\Z
+\end{equation}
+with trivial right $G$-action.
+\end{proposition}
+
+\begin{proof}
+Use the same universal development with $\widetilde D$ in place of
+$\widetilde P$.  The preceding covering argument is unchanged.
+The first chamber is simply connected; attaching subsequent simply
+connected chambers along balls shows that this development is the
+universal cover $\widetilde N$.  Its reduced homology is the direct
+sum of the chamber reduced homologies, each concentrated in degree two.
+There are finitely many chamber orbits in $N$.  For each one, induction
+from its chamber subgroup gives
+\[
+ \Z G\otimes_{\Z H}\pi_2(D)
+       \cong (\Z G)^{2k}.
+\]
+The spheres lie in chamber interiors, so different chambers are
+orthogonal.  Within a chamber the form is its given $\Z H$-form,
+extended to $\Z G$.  Reversing one vector in a plane absorbs any
+chamber orientation sign.  This gives the asserted basis and form.
+
+The construction of $P$ attaches exactly the indicated three-cells
+away from the boundary.  Consequently $Q$ is obtained from $N$ by
+attaching three-cells on this entire middle basis.  It follows at
+once that $f$ induces the displayed identification of fundamental
+groups.  These attachments split off a nonsingular middle summand
+in both absolute and relative duality: on homology they quotient
+by the sphere summand and on cohomology they restrict to its
+annihilator.  With $[Q]=f_*[N]$, naturality of cap product identifies
+the surviving duality maps with those of $Q$.  The finite free chain
+criterion therefore gives Poincar\'e duality for $Q$, just as for the
+chamber pair.  Equivalently one may glue the chamber duality maps
+across the manifold collar sectors, where the orientations cancel
+in pairs.  The top fundamental class is unchanged, so $f$ has degree
+one.
+
+For clarity, the regular-coefficient calculation is particularly
+short.  In the cohomology sequence for these three-cell attachments,
+the only potentially new map is evaluation
+\[
+ H^2(N;\Z G)\longrightarrow
+       \operatorname{Hom}_{\Z G}(\pi_2(N),\Z G).
+\]
+Under duality it is the adjoint of the nonsingular hyperbolic form,
+and hence an isomorphism.  The middle cohomology thus disappears,
+whereas $H^4(Q;\Z G)=H^4(N;\Z G)=\Z$; all other degrees vanish.
+The right action on this last group is trivial because the orientation
+chosen above is preserved by $G$.
+
+Finally, finiteness and asphericity of $Q$ give a finite presentation
+and a finite free resolution of $\Z$ from the chains of $\widehat Q$.
+They also give finite integral cohomological dimension, which excludes
+torsion: restriction to a nontrivial finite cyclic subgroup would give
+a finite projective resolution for that group, contrary to its
+nonvanishing cohomology in arbitrarily high degrees.  This proves
+all the assertions, including \eqref{rf:pd}.
+\end{proof}
+
+\subsection{A proper cocompact CAT(0) model}
+\label{rf:assembly}
+
+The lifted nerve has infinitely many vertices when $H$ is infinite.
+Its Davis complex alone need not give a cocompact action of the
+semidirect product.  The following support construction addresses this.
+
+\begin{lemma}
+\label{rf:cat0}
+The group $\widehat C\rtimes H$, and hence its finite-index subgroup
+$G$, acts properly and cocompactly by isometries on a proper CAT(0)
+cube complex of dimension at most five.
+\end{lemma}
+
+\begin{proof}
+Choose a locally finite tree $T$ on which the free group $H$ acts
+freely and cocompactly, without inversions.  For the trivial group
+one may take a single vertex.  Assign to each lifted-nerve vertex
+$s$ a point $a_s\in T^{(0)}$, equivariantly under $H$.  There are
+finitely many vertex and edge orbits in the lifted nerve.  Thus the
+distances $d_T(a_s,a_t)$ for adjacent $s,t$ have a common finite bound.
+Choose an integer $r$ large enough that the closed radius-$r$ subtrees
+$T_s$ about $a_s$ intersect for every such pair.
+
+For a vertex $v$ and an edge $e$ of $T$, put
+\[
+ A_v=\{s:v\in T_s\},\qquad
+ A_e=\{s:e\subset T_s\}.
+\]
+These sets have uniformly bounded finite size: there are finitely many
+generator orbits, and a fixed-radius ball in the free cocompact tree
+contains uniformly finitely many orbit points.  Let $W_v$ and $W_e$
+be the special subgroups of $\widehat C$ generated by $A_v$ and $A_e$.
+For $e=[v,w]$ we have $A_e=A_v\cap A_w$, so the edge maps are special
+subgroup inclusions.
+
+The graph of groups over the tree $T$ has colimit $\widehat C$.
+Indeed, occurrences of a generator $s$ are identified precisely over
+its connected support $T_s$.  Every defining commutation of
+$\widehat C$ occurs in some vertex group, because the two supports
+intersect at a vertex.  Conversely, each vertex group uses the
+\emph{induced} subgraph of the lifted nerve, and so introduces no
+additional relation.  The resulting presentation is exactly the
+right-angled presentation of $\widehat C$.  All the edge maps are
+injective, as are the maps of these special subgroups into
+$\widehat C$.
+
+Let $\Sigma_v$ and $\Sigma_e$ be the Davis cube complexes of these
+finite-generator right-angled groups.  They are CAT(0), and the special
+subcomplexes $\Sigma_e\subset\Sigma_v$ are convex in the standard
+piecewise Euclidean cube metric.  The CAT(0) assertion is
+\cite[Theorem~12.2.1(i)]{Davis}; special-subgroup metric convexity is
+\cite[Appendix, Proposition~A.2]{Swiatkowski}.
+Form a tree of spaces with vertex spaces
+$\widehat C\times_{W_v}\Sigma_v$ and edge cylinders
+$\widehat C\times_{W_e}(\Sigma_e\times[0,1])$.
+Its underlying tree is the Bass--Serre tree, whose vertices and edges
+are the cosets $\widehat cW_v$ and $\widehat cW_e$, with their
+indicated labels in $T$; it is not simply the original support tree.
+Glue each cylinder at its endpoints by the special-subcomplex maps.
+Convex gluing gives a CAT(0) space over every finite subtree, preserving
+the factors as convex isometric subspaces
+\cite[Theorem~II.11.1 and Remark~II.11.2]{BH}.
+One can apply this gluing first over finite subtrees and then pass
+to the union; the local finiteness verified next ensures completeness.
+
+There is no local-finiteness problem from potentially infinite
+Bass--Serre valence.  In a fixed vertex Davis complex, for any one
+incident support edge, translates of its special subcomplex have
+vertex sets equal to the cosets of $W_e$ in $W_v$.  These cosets
+partition the Cayley vertices.  A finite subcomplex therefore meets
+only finitely many of these translates.  Each support vertex has
+only finitely many incident edges.  Since each $\Sigma_v$ is locally
+finite, the whole tree of cube complexes is locally finite.  Its
+cubes have unit edge lengths and uniformly bounded dimension, so
+the induced length metric is proper and complete.
+
+Left multiplication gives the $\widehat C$-action.  The equivariance
+of the supports extends this to $\widehat C\rtimes H$.  An element
+stabilizing a vertex space projects to an element of $H$ fixing its
+support vertex, and that element is trivial.  Within the vertex
+space its point stabilizer is therefore a finite Coxeter stabilizer.
+The same reasoning applies to edge interiors.  The cellular action
+on this locally finite complex is proper.  There are finitely many
+$H$-orbits of support vertices and edges, and each corresponding
+Davis complex has compact quotient by its special group.  Hence
+the full action is cocompact.  Each clique of the lifted nerve has
+at most four vertices; the vertex spaces have dimension at most four
+and the edge cylinders dimension at most five.  Finally $G$ has finite
+index by \eqref{rf:group}, so its restricted action has all the same
+properties.
+\end{proof}
+
+It follows that $G$ satisfies the $K$- and $L$-theoretic Farrell--Jones
+conjectures with coefficients: use Bartels--L\"uck
+\cite[Theorem~B(ii)]{BL} and Wegner
+\cite[Theorems~1.1 and~3.4]{Wegner}.  In particular, because $G$ is
+torsion-free, the Whitehead group, reduced $K_0(\Z G)$, and negative
+$K$-groups vanish, and the simple $L$-theory assembly map is an
+isomorphism in every degree; see \cite[Proposition~0.3(i)]{BL}.
+These are the assembly consequences used below.  No four-dimensional
+rigidity or disk-embedding conclusion is being inferred from CAT(0)
+geometry.
+
+\subsection{The gallery map}
+
+We finish by recording the geometric map needed in the truncation
+argument. The preceding CAT(0) model supplies assembly; the map here
+instead records positions and directions in the ordinary chamber
+development. Let $\Sigma_C$ denote the ordinary Davis cube complex of
+$C$.  Its vertices are the chamber labels $C$, and the link of the
+identity vertex is $\mathcal L$, whose realization is $S$.
+
+\begin{lemma}
+\label{rf:davis-map}
+There is a $C$-equivariant continuous map
+\[
+ \eta:Y\longrightarrow\Sigma_C
+\]
+which sends the inner core of each chamber to its label, respects
+the panels, and has uniformly bounded image diameter on each chamber.
+On the boundary collar of the central chamber its radial directions
+at the identity vertex give the dual-block identification with $S$.
+After interior chamber stabilizations, the same construction gives a
+continuous map with the same panel, core, and diameter properties;
+it is equivariant under any subgroup preserving the stabilized pattern.
+\end{lemma}
+
+\begin{proof}
+The Davis chamber is the cone on the barycentric subdivision of
+$\mathcal L$, with its usual dual panels on the base.  Identify that
+base panelwise with $S$.  Map a boundary collar of $D$ to this cone
+by that identification on its outer boundary and by coning to the
+apex on its inner boundary; map the remainder of $D$ to the apex.
+The maps agree across reflected panels and hence define $\eta$.
+Every chamber image lies in the corresponding compact Davis chamber,
+whose diameter is independent of its label.  At the identity apex,
+radial projection of the cone base to a sufficiently small metric
+sphere gives exactly its link identification.  Stabilization changes
+only the collapsed inner core, so does not affect the construction.
+\end{proof}
diff --git a/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/build/sections/stable.tex b/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/build/sections/stable.tex
new file mode 100644
index 0000000000000000000000000000000000000000..9815eacfa80c93425c075bef4533bd660cd24299
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@@ -0,0 +1,397 @@
+\section{Marked stable realization}\label{st:setup}
+
+We now suppose, for a contradiction, that the group constructed above is
+the fundamental group of a closed aspherical topological four-manifold
+$M$.  The preceding construction gives a finite oriented Poincar\'e
+four-complex $Q=BG$ and a closed oriented manifold $N$.  Write $q$ for
+the map of Proposition~\ref{rf:development}:
+\[
+ q\colon N\longrightarrow Q
+\]
+which is an isomorphism on fundamental groups and has degree one.  The
+only reduced homology of the universal cover of $N$ is its second
+homology.  With $\Lambda=\Z G$, this is the free module
+\begin{equation}\label{st:module}
+ A=\pi_2(N)\cong\Lambda^{2b},
+ \qquad \lambda_N\cong\bigoplus_{a=1}^{b}
+ \begin{pmatrix}0&1\\1&0\end{pmatrix}.
+\end{equation}
+The displayed basis consists of the prescribed chamber sphere pairs.
+All identifications include whiskers and orientations.  Moreover, $G$
+is torsion-free, is an integral $\mathrm{PD}_4$ group, and satisfies the
+$K$- and $L$-theoretic Farrell--Jones conjectures by the CAT(0)
+construction.  These are conclusions of the construction, not
+additional assumptions on a putative realization.
+
+Both $Q$ and $M$ are $K(G,1)$ spaces.  Their equivalence identifies the
+orientation character with the action on the top dualizing module.
+Consequently $M$ is orientable.  Choose its orientation and its
+fundamental-group marking so that its classifying map
+$c_M\colon M\to Q$ has degree one, and put
+\[
+ L=M\mathbin{\#}\,b(S^2\times S^2).
+\]
+Collapsing the simply connected summands gives a degree-one classifying
+map $c_L\colon L\to Q$.  Identify $\pi_2(L)$ with $A$ by sending each
+specified chamber pair in \eqref{st:module} to a specified standard pair
+in $L$.  This is an isometry of equivariant forms.  We must retain this
+particular isometry throughout the argument.
+
+\subsection{An integral marked homotopy equivalence}
+
+We first realize the prescribed module isometry by a homotopy
+equivalence. The comparison takes place in a common second Postnikov
+stage: group duality splits that stage, and integral cup-product tests
+identify the two fundamental classes there. These are the inputs to
+the lifting argument below.
+
+\begin{lemma}\label{st:second-stage}
+The second Postnikov stages of $N$ and $L$, with the preceding markings,
+identify over $Q$ with the split stage
+\[
+ B=EG\times_G K(A,2).
+\]
+There are maps $u_N\colon N\to B$ and $u_L\colon L\to B$ inducing the
+specified identifications of fundamental and second homotopy groups.
+If $\Gamma(A)=H_4(K(A,2);\Z)$, then
+\begin{equation}\label{st:homology-extension}
+ 0\longrightarrow\Gamma(A)_G\longrightarrow H_4(B;\Z)
+ \longrightarrow H_4(Q;\Z)\longrightarrow0
+\end{equation}
+is a split exact sequence.
+\end{lemma}
+
+\begin{proof}
+Since $A$ is finite free over $\Lambda$, integral group duality gives
+\begin{equation}\label{st:cohomology-vanishing}
+ H^2(G;A)=H^3(G;A)=0.
+\end{equation}
+The second Postnikov invariant lies in the latter group, so it vanishes.
+The former removes the degree-two ambiguity in making the identification
+with the split stage for the prescribed module marking.  We use a
+functorial Eilenberg--Mac Lane model for $K(A,2)$, so the given action on
+$A$ defines the displayed homotopy quotient and its section.
+
+Separately, freeness gives $H_i(G;A)=0$ for $i>0$.  In total degree four,
+the homology spectral sequence of
+\[
+ K(A,2)\longrightarrow B\longrightarrow Q
+\]
+therefore has only $H_4(G;\Z)$ and $H_0(G;\Gamma(A))$ as possibly
+nonzero terms: the intervening term $H_2(G;A)$ is zero, and the fiber has
+no homology in degrees one and three.  The possible incoming
+differentials to $H_0(G;\Gamma(A))$ have sources $H_3(G;A)=0$ and
+$H_5(G;\Z)=0$.  The section preserves the base term.  This proves
+\eqref{st:homology-extension}, including its integral, rather than merely
+rational, exactness.
+\end{proof}
+
+The next elementary check is useful because the middle classes are
+specified over the group ring.  Ordinary intersection forms would lose
+the relative translates needed here.
+
+\begin{lemma}\label{st:quadratic-detection}
+With the identifications above,
+\begin{equation}\label{st:fundamental-equality}
+ (u_N)_*[N]=(u_L)_*[L]\quad\text{in }H_4(B;\Z).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Write $e_1,\ldots,e_{2b}$ for the chosen $\Lambda$-basis of $A$ and
+$\phi_i\colon A\to\Lambda$ for the coordinate functionals.  The
+low-degree cohomology sequence of the Postnikov fibration, with regular
+coefficients, contains
+\[
+ 0\longrightarrow H^2(G;\Lambda)
+ \longrightarrow H^2(B;\Lambda)
+ \longrightarrow\operatorname{Hom}_{\Lambda}(A,\Lambda)
+ \longrightarrow H^3(G;\Lambda).
+\]
+Both outside groups vanish.  Thus each $\phi_i$ is the restriction of
+a unique class $x_i\in H^2(B;\Lambda)$.
+
+Their cup products may be evaluated without multiplying away either
+coefficient factor:
+\begin{equation}\label{st:cup-tests}
+ z\longmapsto
+ \langle x_i\smile x_j,z\rangle
+ \in(\Lambda\otimes_{\Z}\Lambda)_G,
+ \qquad z\in H_4(B;\Z).
+\end{equation}
+The tensor product has the diagonal left action.  Its coinvariants
+identify, as an abelian group, with $\Lambda$ by
+$g\otimes h\mapsto g^{-1}h$.  For a closed oriented manifold with
+the indicated middle module, evaluations in \eqref{st:cup-tests} are
+the equivariant cup form on the coordinate functionals.  By Poincar\'e
+duality this is the dual form of its equivariant intersection pairing.
+The prescribed isometry therefore makes these evaluations equal on
+$(u_N)_*[N]$ and $(u_L)_*[L]$.
+
+For completeness, these tests detect the fiber term integrally.  The
+underlying abelian group of $A$ is free on the elements $g e_i$.
+The group $\Gamma(A)$ has a basis consisting of one square generator
+for each such element and one cross generator for each unordered pair
+of distinct such elements.  The action of $G$ permutes this basis.
+Hence its coinvariants are free abelian on the corresponding orbits.
+A nonidentity element stabilizing an unordered pair would exchange its
+two members and have order two; torsion-freeness rules this out.
+
+The square orbit represented by $e_i$ evaluates to $1$ in the $(i,i)$
+test.  For $i<j$, the cross orbit represented by
+$\{e_i,t e_j\}$ evaluates to $t$ in the $(i,j)$ test.  Finally, for
+$t\ne1$, the orbit represented by $\{e_i,t e_i\}$ evaluates to
+$t+t^{-1}$ in the $(i,i)$ test.  These last orbits are indexed by $t$
+modulo inversion.  Since $G$ has no elements of order two, their supports
+are disjoint two-element subsets of $G\setminus\{1\}$.  The displayed
+tests are therefore jointly injective on $\Gamma(A)_G$.  In particular,
+no integral square or hidden two-torsion is discarded.
+
+The two fundamental classes have the same image $[Q]$ in $H_4(Q;\Z)$.
+By Lemma~\ref{st:second-stage}, their difference lies in
+$\Gamma(A)_G$.  All its cup tests vanish, so the difference is zero.
+\end{proof}
+
+\begin{proposition}\label{st:marked-equivalence}
+There is an orientation-preserving homotopy equivalence
+\[
+ f\colon N\longrightarrow L
+\]
+inducing the specified isomorphism on $\pi_1$ and the prescribed
+isometry $A\to\pi_2(L)$.
+\end{proposition}
+
+\begin{proof}
+Replace $u_L\colon L\to B$ by a fibration.  Its fiber is
+two-connected, with third homotopy group $\pi_3(L)$ carrying the
+usual $G$-action.  The first obstruction to lifting $u_N$ is consequently
+the pullback of a universal class
+\[
+ \kappa\in H^4(B;\pi_3(L)).
+\]
+The pullback $u_L^*\kappa$ is zero, since $u_L$ lifts to its own
+fibration replacement.  Naturality of cap product and
+Lemma~\ref{st:quadratic-detection} give
+\begin{align*}
+ (u_N)_*\bigl((u_N^*\kappa)\mathbin{\cap}[N]\bigr)
+ &=\kappa\mathbin{\cap}(u_N)_*[N]\\
+ &=\kappa\mathbin{\cap}(u_L)_*[L]=0
+ \quad\text{in }H_0(B;\pi_3(L)).
+\end{align*}
+The map $u_N$ induces an isomorphism on fundamental groups, and hence
+on these degree-zero coefficient groups.  Poincar\'e duality identifies
+\[
+ H^4(N;\pi_3(L))\xrightarrow{\ \cap[N]\ }
+ H_0(N;\pi_3(L))\cong\pi_3(L)_G.
+\]
+Thus the lifting obstruction vanishes.
+
+The manifold $N$ has a finite four-dimensional CW structure from the
+smooth chamber construction and the panel collars.  There are no
+higher obstructions on this source.  No triangulation or smooth
+structure on the hypothetical $M$ is required.  The resulting lift
+$f$ has the stipulated maps on $\pi_1$ and $\pi_2$.  The universal
+covers of $N$ and $L$ have reduced homology only in degree two, where
+the lift is an isomorphism. These manifolds have CW homotopy type
+\cite[Corollary~1]{MilnorCW}. The homological Whitehead theorem on their
+simply connected universal covers, followed by Whitehead's theorem,
+shows that $f$ is a homotopy equivalence
+\cite[Corollary~4.33 and Theorem~4.5]{Hatcher}.
+Finally $c_Lf\simeq q$ and both classifying maps have degree one, so
+$f_*[N]=[L]$.
+\end{proof}
+
+\subsection{From the marked equivalence to a marked stable product}
+
+The marked homotopy equivalence is now available. The next two steps
+pass through an $s$-cobordism and then a stabilized product, retaining
+the same identification of every original sphere class. The product
+trace also specifies the new classes introduced by stabilization.
+
+\begin{lemma}\label{st:marked-bordism}
+There is an $s$-cobordism $(W;N,L)$ whose two end inclusions identify
+$\pi_1$ and $\pi_2$ by the marking of
+Proposition~\ref{st:marked-equivalence}.
+\end{lemma}
+
+\begin{proof}
+The universal intersection forms are even, so $N$ and $L$ are almost
+spin.  Indeed, evaluation of $w_2$ on every spherical class is its
+self-intersection modulo two, and on the simply connected cover these
+evaluations detect $w_2$.
+
+We apply the proof of Kasprowski--Land's Theorem~1.1, specifically
+Lemma~2.3, Lemma~3.3 and Theorem~3.1 of~\cite{KL}.  All hypotheses hold:
+$Q$ is an aspherical Poincar\'e four-complex; $G$ satisfies both
+Farrell--Jones conjectures; the two manifolds have the same orientation
+character and degree-one classifying maps; and they are almost spin.
+Their normal-invariant calculation, using assembly injectivity and the
+Atiyah--Hirzebruch spectral sequence, identifies the normal invariant of
+this particular $f$ with a class in
+\[
+ \ker\bigl(H_2(L;\Z/2)\longrightarrow H_2(Q;\Z/2)\bigr).
+\]
+The Serre spectral sequence for the universal-cover fibration shows
+that this kernel is represented by spherical classes. Since $L$ is
+almost spin, their $w_2$ evaluations vanish. Lemma~3.3 of~\cite{KL}
+therefore supplies a target self-equivalence $\phi$ with the same
+normal invariant as $f$, of the form
+\[
+ L\longrightarrow L\vee S^4
+ \xrightarrow{\operatorname{id}\vee\eta^2}L\vee S^2
+ \xrightarrow{\operatorname{id}+a}L.
+\]
+Here $a:S^2\to L$ represents the spherical class. This pinch acts
+identically on $\pi_1$ and $\pi_2$, because its extra part factors through
+the four-sphere. Thus $\phi^{-1}\circ f$ has trivial normal invariant
+and the same marking as $f$. Surgery of its normal bordism over $L$,
+as in Theorem~3.1 of~\cite{KL}, gives the required $s$-cobordism.
+Neither this conclusion nor
+the cited theorem requires $G$ to be a good group.  We have not asserted
+that $W$ itself is a product.
+\end{proof}
+
+We use the stable product theorem for this particular bordism.  Quinn's
+Theorem~1.1 in~\cite{QuinnStable}, in the form stated in
+\cite[Remark~3.7]{KPR}, says that a five-dimensional topological
+$s$-cobordism becomes a product relative to one end after finitely many
+connected sums along arcs with $(S^2\times S^2)\times[0,1]$.
+There is no restriction on its fundamental group.  The relative product
+statement is important: an arbitrary stable homeomorphism of the ends
+would not, by itself, retain the specified sphere classes.
+
+\begin{proposition}\label{st:stable-product}
+For some integer $k\ge0$ there is a homeomorphism
+\[
+ h\colon N\mathbin{\#}\,k(S^2\times S^2)
+ \longrightarrow
+ L\mathbin{\#}\,k(S^2\times S^2)
+\]
+which induces the prescribed original marking and matches every added
+standard pair with its specified added standard pair.  In particular,
+on the left, all the prescribed original hyperbolic pairs and all new
+standard pairs have simultaneous geometric representatives with
+pairwise disjoint punctured $S^2\times S^2$ neighborhoods.
+\end{proposition}
+
+\begin{proof}
+Apply the stable product theorem to $W$ from
+Lemma~\ref{st:marked-bordism}.  Denote its stabilized ends by $N^*$ and
+$L^*$ and the stabilized bordism by $W^*$.  We verify its marking,
+including the new summands.
+
+The operation removes disjoint normal $D^4\times[0,1]$ neighborhoods
+of arcs between the ends and glues
+\[
+ \bigl((S^2\times S^2)\setminus\operatorname{int}D^4\bigr)
+ \times[0,1]
+\]
+along the resulting $S^3\times[0,1]$ interfaces.  Removing these
+codimension-four arcs changes neither $\pi_1$ nor second homology of
+the universal cover.  Mayer--Vietoris across the lifted interfaces
+therefore gives a specified decomposition
+\begin{equation}\label{st:bordism-splitting}
+ \pi_2(W^*)\cong\pi_2(W)\oplus\Lambda^{2k}.
+\end{equation}
+The second summand is represented by the standard spheres in the added
+product pieces.  Choose their whiskers along the stabilization arcs.
+The two end inclusions $j_N\colon N^*\to W^*$ and
+$j_L\colon L^*\to W^*$ carry each new pair to this same pair, while
+their old summands map to $\pi_2(W)$ with exactly the original bordism
+identification.
+
+A product structure $F\colon N^*\times[0,1]\to W^*$ which is the
+identity on the incoming end restricts at the other end to a
+homeomorphism $h\colon N^*\to L^*$.  Use the product trace for the
+outgoing basepoint path.  Any discrepancy from the original bordism
+path can be corrected by pushing the outgoing basepoint around a loop
+in $L^*$, since $j_L$ is an isomorphism on fundamental groups.
+The resulting based homotopy given by $F$ implies
+\[
+ (j_L)_*h_*=(j_N)_*.
+\]
+Both inclusions are isomorphisms on $\pi_1$ and $\pi_2$.  Thus
+$h_*=(j_L)_*^{-1}(j_N)_*$, with the same base paths.  In the
+decomposition \eqref{st:bordism-splitting} it
+preserves the original marking and identifies the new standard basis
+element by element.  There is no arbitrary automorphism mixing the
+old and new summands.
+
+All $b+k$ standard pairs on
+$L^*=M\mathbin{\#}(b+k)(S^2\times S^2)$ have disjoint punctured
+$S^2\times S^2$ neighborhoods.  Give the two spheres of each pair
+the common whisker to their intersection used for its module marking,
+and pull the whole based neighborhood back by $h$.  A pair has
+its one prescribed transverse intersection; different pairs and their
+neighborhoods are disjoint.  The marking just proved identifies their
+based homotopy classes with the original chamber basis and the added
+standard basis, respectively.
+\end{proof}
+
+\subsection{Periodically lifted geometric data}
+
+Let $C$ be the right-angled Coxeter group used in the reflection
+construction, and let $\Delta\le C$ be the finite-index torsion-free
+normal subgroup used to obtain $N$ from its ordinary reflection
+development.  Place the finitely many stabilization balls on the
+$N$-side inside chamber interiors.  This may be done by moving their
+endpoints along paths before performing the arc connected sums.
+Track the new whiskers along those paths.  There is no requirement that
+different chambers receive equal numbers of stabilizations.
+
+Lift the resulting $N^*$ to the ordinary reflection development and
+call the lifted manifold $Y$.  Its chambers, indexed by $v\in C$, are
+stabilized copies $D_v$ of $D$.  The numbers of stabilizations are
+$\Delta$-periodic.  Gallery distance is the word metric on the chamber
+indices for the standard Coxeter generators.
+
+\begin{proposition}\label{st:periodic-data}
+The manifold $Y$ has two $\Delta$-periodic systems of sphere pairs,
+with the same chamber labels:
+\begin{enumerate}
+\item framed immersed hyperbolic pairs in the interiors of the
+chambers $D_v$, including the added standard pairs, with their
+equivariant products already correct over the individual chamber group;
+\item geometrically standard pairs with pairwise disjoint punctured
+$S^2\times S^2$ neighborhoods, based-homotopic sphere by sphere to
+the correspondingly labelled pairs in the first system.
+\end{enumerate}
+There is an integer $d_0$ such that the drawing of each pair, its
+standard neighborhood, and the homotopies relating the two systems
+are contained in the chambers at gallery distance at most $d_0$ from
+that pair's label.  Basings in this statement are local basings
+transported with the chamber; no bound is asserted for whiskers drawn
+from one global base point to all chambers.
+\end{proposition}
+
+\begin{proof}
+The original immersed representatives stay in their chamber interiors,
+and the stabilization spheres can be put in their interior connected
+summands.  Stabilization adds orthogonal standard planes and changes
+neither the chamber fundamental group nor the old pairings.
+
+Proposition~\ref{st:stable-product} supplies a finite collection of
+geometric pairs on the compact quotient $N^*$, representing the
+specified based classes.  Choose a based homotopy for each sphere
+between its two representatives.  Lift the neighborhoods, sphere
+drawings, and homotopies to $Y$, using the prescribed marking to choose
+the lift for each chamber label.  Covering maps preserve $\pi_2$ and
+lift these based homotopies, so they end at precisely the labelled
+geometric representatives.  Disjoint embedded neighborhoods lift to
+disjoint embedded neighborhoods.
+
+There are only finitely many labelled data on the quotient.  Each chosen
+lift of a drawing, closed standard neighborhood, local basing, or
+homotopy has compact image.  Local finiteness of the development implies
+that it meets only finitely many chambers.  Take $d_0$ to be the maximum
+gallery distance from its assigned label over this finite list of
+chosen lifts.  All remaining data are translates by $\Delta$, whose
+action preserves gallery distance.  The same $d_0$ therefore works
+everywhere.
+\end{proof}
+
+The last proposition is the stable information needed below.  It gives
+actual geometric summands with their original chamber labels and
+compactly supported homotopies to the chamber systems.  It makes no
+claim of controlled isotopy, and no claim that the unstabilized chamber
+obstruction has disappeared.
diff --git a/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/paper.pdf b/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..c770aec3a947731740805b7a01c43d592d9c3313
--- /dev/null
+++ b/preprints/A-PD4-group-without-an-aspherical-manifold-model-September-24-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:ccbbf7f15686b6f7ef6e36a633ae5549ec2d9d932560ff4be2b5b9367d2414f7
+size 435849
diff --git a/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/README.md b/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..fe1775d6c6ab4b45e1296a973e7837c3e9c6e25a
--- /dev/null
+++ b/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/README.md
@@ -0,0 +1,18 @@
+# [A Pivotal Amplitude for Voronoi Percolation from Cardy's Formula](voronoi-pivotal-amplitude.pdf)
+
+**Author:** OpenAI
+
+**Date:** October 5, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026,
+  author = {{OpenAI}},
+  title = {{A Pivotal Amplitude for Voronoi Percolation from Cardy's Formula}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/voronoi-pivotal-amplitude.pdf}{OAI:A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/build/figures/lobes.tex b/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/build/figures/lobes.tex
new file mode 100644
index 0000000000000000000000000000000000000000..39573ecff6a37f8028f715d13a6a4f297129eaaf
--- /dev/null
+++ b/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/build/figures/lobes.tex
@@ -0,0 +1,80 @@
+\begin{figure}[tb]
+\centering
+\begin{tikzpicture}[font=\footnotesize,>=Latex,line cap=round,
+  bank/.style={very thick},
+  barrier/.style={densely dashed,gray!80},
+  fluid/.style={fill=blue!8}]
+  \begin{scope}[xshift=0.3cm]
+    \fill[blue!8] (0,0) circle (2.65);
+    \fill[black!9]
+      (2,0) .. controls (1.25,.25) and (.25,1.25) .. (0,2)
+      .. controls (-.25,1.25) and (-1.25,.25) .. (-2,0)
+      .. controls (-1.25,-.25) and (-.25,-1.25) .. (0,-2)
+      .. controls (.25,-1.25) and (1.25,-.25) .. cycle;
+    \draw[barrier] (0,0) circle (2);
+    \draw[bank]
+      (2,0) .. controls (1.25,.25) and (.25,1.25) .. (0,2);
+    \draw[bank,gray!55]
+      (0,2) .. controls (-.25,1.25) and (-1.25,.25) .. (-2,0);
+    \draw[bank]
+      (-2,0) .. controls (-1.25,-.25) and (-.25,-1.25) .. (0,-2);
+    \draw[bank,gray!55]
+      (0,-2) .. controls (.25,-1.25) and (1.25,-.25) .. (2,0);
+    \foreach \p in {(2,0),(0,2),(-2,0),(0,-2)}
+      \fill \p circle (2pt);
+    \draw[gray!75] (0,0) circle (.58);
+    \node[align=center,font=\scriptsize] at (0,0) {starting\\side};
+    \node at (1.15,1.15) {$L_1$};
+    \node at (-1.15,1.15) {$L_2$};
+    \node at (-1.15,-1.15) {$L_3$};
+    \node at (1.15,-1.15) {$L_4$};
+    \node[align=center] at (0,2.97) {unqueried annular back};
+    \draw[->,thin] (.78,2.87)--(1.15,2.18);
+    \node[align=left,anchor=west] at (2.8,.95)
+      {four-tip\\frontier};
+    \draw[->,thin] (2.78,.77)--(1.43,.30);
+    \node[align=left,anchor=west] at (2.8,-.80)
+      {open attachment\\interval};
+    \draw[->,thin] (2.78,-.70)--(1.74,-.97);
+    \node[align=center] at (0,-3.05)
+      {Dashed circle: stopping barrier.\\
+       Its open intervals lie inside the glued domain.};
+  \end{scope}
+
+  \begin{scope}[xshift=8.85cm,yshift=.20cm]
+    \node[align=center] at (0,2.28) {Two distinct banks at a contact};
+    \fill[blue!8]
+      (-1.55,1.70)--(1.55,1.70)--(1.55,.85)
+      .. controls (.85,.85) and (.50,0) .. (0,0)
+      .. controls (-.50,0) and (-.85,.85) .. (-1.55,.85)--cycle;
+    \fill[blue!8]
+      (-1.55,-1.70)--(1.55,-1.70)--(1.55,-.85)
+      .. controls (.85,-.85) and (.50,0) .. (0,0)
+      .. controls (-.50,0) and (-.85,-.85) .. (-1.55,-.85)--cycle;
+    \draw[bank]
+      (-1.55,.85) .. controls (-.85,.85) and (-.50,0) .. (0,0)
+      .. controls (.50,0) and (.85,.85) .. (1.55,.85);
+    \draw[bank,gray!55]
+      (-1.55,-.85) .. controls (-.85,-.85) and (-.50,0) .. (0,0)
+      .. controls (.50,0) and (.85,-.85) .. (1.55,-.85);
+    \fill (0,0) circle (2.2pt);
+    \node at (0,1.30) {first access};
+    \node at (0,-1.30) {second access};
+    \draw[->,thin] (0,.96)--(0,.20);
+    \draw[->,thin] (0,-.96)--(0,-.20);
+    \node[align=center] at (0,-2.30)
+      {The contact is boundary,\\not a gluing interval.};
+  \end{scope}
+\end{tikzpicture}
+\caption{The decomposition at an inner four-tip frontier. The four
+monochromatic frontier arcs, drawn alternately black and gray, bound
+Jordan lobes $L_1,\ldots,L_4$ with the stopping barrier. Each lobe is
+attached to the untouched annular back along an open barrier interval;
+the common shading denotes the resulting unqueried domain. The starting
+side remains in the complement. At the other end the construction is
+reversed. The inset records two accesses at a possible contact between
+different banks: the contact is not added to the fluid and does not join
+the accesses. All curves are schematic. Actual banks may be rough and
+may touch each other, although each individual frontier arc is simple.}
+\label{fig:frontier-lobes}
+\end{figure}
diff --git a/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/build/paper.tex b/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/build/paper.tex
new file mode 100644
index 0000000000000000000000000000000000000000..0bb997cbdcd13f0fd4742c538b48a95dde3cc14b
--- /dev/null
+++ b/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/build/paper.tex
@@ -0,0 +1,62 @@
+\newcommand{\DoNotLoadEpstopdf}{}
+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage[margin=1.05in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{microtype}
+\usepackage{enumitem}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta,patterns,calc}
+\usepackage{booktabs}
+\usepackage{xurl}
+\usepackage[hidelinks]{hyperref}
+\hypersetup{pdftitle={A Pivotal Amplitude for Voronoi Percolation from Cardy's Formula},pdfauthor={OpenAI}}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\newtheorem{assumption}[theorem]{Assumption}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newcommand{\R}{\mathbb R}
+\newcommand{\C}{\mathbb C}
+\newcommand{\E}{\mathbb E}
+\newcommand{\Pp}{\mathbb P}
+\newcommand{\eps}{\varepsilon}
+\newcommand{\TV}{\mathrm{TV}}
+\DeclareMathOperator{\Var}{Var}
+\DeclareMathOperator{\dist}{dist}
+\DeclareMathOperator{\supp}{supp}
+\DeclareMathOperator{\rank}{rank}
+\numberwithin{equation}{section}
+\setlist[enumerate]{itemsep=3pt,topsep=5pt}
+\setlist[itemize]{itemsep=3pt,topsep=5pt}
+\emergencystretch=1em
+\title{A Pivotal Amplitude for Voronoi Percolation\newline from Cardy's Formula}
+\author{OpenAI}
+\date{October 5, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+Taking Cardy's conformal crossing formula for critical planar
+Poisson--Voronoi percolation in every bounded Jordan quadrilateral as an
+input, we prove that the expected number of color-pivotal cells for a
+unit-square crossing is asymptotic to a positive constant times
+$\eps^{-3/4}$, where the point intensity is $\eps^{-2}$.
+No rate of convergence in Cardy's formula is required.
+\end{abstract}
+\setcounter{tocdepth}{1}
+\tableofcontents
+\clearpage
+\input{sections/00-introduction}
+\input{sections/01-inputs}
+\input{sections/02-annuli}
+\input{sections/03-spectrum}
+\input{sections/04-shape}
+\input{sections/05-transfer}
+\input{sections/06-amplitude}
+\input{references}
+\end{document}
diff --git a/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/build/references.tex b/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/build/references.tex
new file mode 100644
index 0000000000000000000000000000000000000000..3e365b78c61ffc419aca88152d4e27591b11fe75
--- /dev/null
+++ b/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/build/references.tex
@@ -0,0 +1,113 @@
+\begin{thebibliography}{99}
+
+\bibitem{BL}
+H.~J. Brascamp and E.~H. Lieb,
+On extensions of the Brunn--Minkowski and Pr\'ekopa--Leindler theorems,
+including inequalities for log concave functions, and with an application
+to the diffusion equation,
+\emph{J. Funct. Anal.} \textbf{22} (1976), no.~4, 366--389.
+\href{https://doi.org/10.1016/0022-1236(76)90004-5}{doi:10.1016/0022-1236(76)90004-5}.
+
+\bibitem{CN}
+F.~Camia and C.~M. Newman,
+Two-dimensional critical percolation: the full scaling limit,
+\emph{Comm. Math. Phys.} \textbf{268} (2006), no.~1, 1--38.
+\href{https://doi.org/10.1007/s00220-006-0086-1}{doi:10.1007/s00220-006-0086-1}.
+
+\bibitem{Cardy}
+J.~L. Cardy,
+Critical percolation in finite geometries,
+\emph{J. Phys. A: Math. Gen.} \textbf{25} (1992), no.~4, L201--L206.
+\href{https://doi.org/10.1088/0305-4470/25/4/009}{doi:10.1088/0305-4470/25/4/009}.
+
+\bibitem{CardyCS}
+J.~Cardy,
+Calogero--Sutherland model and bulk-boundary correlations in conformal
+field theory,
+\emph{Phys. Lett. B} \textbf{582} (2004), nos.~1--2, 121--126.
+\href{https://doi.org/10.1016/j.physletb.2003.12.029}{doi:10.1016/j.physletb.2003.12.029}.
+
+\bibitem{DGLZ}
+H.~Du, Y.~Gao, X.~Li, and Z.~Zhuang,
+Sharp asymptotics for arm probabilities in critical planar percolation,
+\emph{Comm. Math. Phys.} \textbf{405} (2024), Article~182.
+\href{https://doi.org/10.1007/s00220-024-05028-0}{doi:10.1007/s00220-024-05028-0}.
+
+\bibitem{GPS}
+C.~Garban, G.~Pete, and O.~Schramm,
+Pivotal, cluster, and interface measures for critical planar percolation,
+\emph{J. Amer. Math. Soc.} \textbf{26} (2013), no.~4, 939--1024.
+\href{https://doi.org/10.1090/S0894-0347-2013-00772-9}{doi:10.1090/S0894-0347-2013-00772-9}.
+
+\bibitem{KM}
+A.~V. Kolesnikov and E.~Milman,
+Brascamp--Lieb-type inequalities on weighted Riemannian manifolds
+with boundary,
+\emph{J. Geom. Anal.} \textbf{27} (2017), 1680--1702.
+\href{https://doi.org/10.1007/s12220-016-9736-5}{doi:10.1007/s12220-016-9736-5}.
+
+\bibitem{LP}
+G.~Last and M.~Penrose,
+\emph{Lectures on the Poisson Process},
+Institute of Mathematical Statistics Textbooks, vol.~7,
+Cambridge University Press, Cambridge, 2017.
+
+\bibitem{LSW}
+G.~F. Lawler, O.~Schramm, and W.~Werner,
+Values of Brownian intersection exponents, II: Plane exponents,
+\emph{Acta Math.} \textbf{187} (2001), no.~2, 275--308.
+\href{https://doi.org/10.1007/BF02392619}{doi:10.1007/BF02392619}.
+
+\bibitem{V}
+OpenAI,
+\emph{From critical crossings to quenched near-critical universality in Voronoi percolation},
+OpenAI Math Release preprint
+\href{https://github.com/openai/math/blob/main/preprints/From-critical-crossings-to-quenched-near-critical-universality-in-Voronoi-percolation-October-5-2026/critical-crossings-quenched-near-critical-universality-voronoi-percolation.pdf}{OAI:From-critical-crossings-to-quenched-near-critical-universality-in-Voronoi-percolation-October-5-2026},
+2026.
+
+\bibitem{Schramm}
+O.~Schramm,
+Scaling limits of loop-erased random walks and uniform spanning trees,
+\emph{Israel J. Math.} \textbf{118} (2000), 221--288.
+\href{https://arxiv.org/abs/math/9904022}{arXiv:math/9904022}.
+
+\bibitem{SS}
+O.~Schramm and S.~Smirnov,
+On the scaling limits of planar percolation,
+\emph{Ann. Probab.} \textbf{39} (2011), no.~5, 1768--1814.
+\href{https://doi.org/10.1214/11-AOP659}{doi:10.1214/11-AOP659}.
+
+\bibitem{Smirnov}
+S.~Smirnov,
+Critical percolation in the plane: conformal invariance, Cardy's formula,
+scaling limits,
+\emph{C. R. Acad. Sci. Paris S\'er. I Math.} \textbf{333} (2001),
+no.~3, 239--244.
+\href{https://doi.org/10.1016/S0764-4442(01)01991-7}{doi:10.1016/S0764-4442(01)01991-7}.
+
+\bibitem{SW}
+S.~Smirnov and W.~Werner,
+Critical exponents for two-dimensional percolation,
+\emph{Math. Res. Lett.} \textbf{8} (2001), 729--744.
+\href{https://doi.org/10.4310/MRL.2001.v8.n6.a4}{doi:10.4310/MRL.2001.v8.n6.a4}.
+
+\bibitem{Tassion}
+V.~Tassion,
+Crossing probabilities for Voronoi percolation,
+\emph{Ann. Probab.} \textbf{44} (2016), no.~5, 3385--3398.
+\href{https://doi.org/10.1214/15-AOP1052}{doi:10.1214/15-AOP1052}.
+
+\bibitem{Vanneuville}
+H.~Vanneuville,
+Annealed scaling relations for Voronoi percolation,
+\emph{Electron. J. Probab.} \textbf{24} (2019), Paper~39, 71~pp.
+\href{https://doi.org/10.1214/19-EJP293}{doi:10.1214/19-EJP293}.
+
+\bibitem{Zhan}
+D.~Zhan,
+Stochastic Loewner evolution in doubly connected domains,
+\emph{Probab. Theory Related Fields} \textbf{129} (2004), no.~3,
+340--380.
+\href{https://doi.org/10.1007/s00440-004-0343-1}{doi:10.1007/s00440-004-0343-1}.
+
+\end{thebibliography}
diff --git a/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/build/sections/00-introduction.tex b/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/build/sections/00-introduction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..727d37fe4fee22635c207ec13741eeb42ceb424f
--- /dev/null
+++ b/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/build/sections/00-introduction.tex
@@ -0,0 +1,140 @@
+\section{Introduction}\label{sec:introduction}
+
+Let $\eta_\eps$ be a homogeneous Poisson process in $\R^2$ of intensity
+$\eps^{-2}$. Color its Voronoi cells independently black or white, each
+with probability $1/2$. For a bounded quadrilateral $Q$, a black crossing
+means a connected black-cell path inside the closed quadrilateral joining
+its two specified opposite sides. The cells are those of the full-plane
+tessellation: the process is not resampled at the boundary, and no
+boundary sites are wired together.
+
+A nucleus is \emph{color-pivotal} for this event if changing only its cell's
+color changes whether the crossing occurs. Write $N_\eps(Q)$ for the
+number of such nuclei, including nuclei outside $Q$ when their cells
+meet $Q$, and set $N_\eps=N_\eps([0,1]^2)$ for a left-to-right crossing.
+All expectations below average both the Poisson geometry and the colors.
+
+\begin{assumption}[Scalar Cardy limit]\label{ass:cardy}
+For every bounded Jordan quadrilateral $Q$ with four distinct marked
+boundary points, the critical crossing probability converges, as
+$\eps\downarrow0$, to Cardy's conformally invariant value.
+\end{assumption}
+
+Here a Jordan quadrilateral is a Jordan domain with four marked boundary
+points in cyclic order and a specified pair of opposite boundary arcs.
+In the upper half-plane normalization with marked points $0,x,1,\infty$,
+$0<x<1$, the crossing between $[0,x]$ and $[1,\infty]$ has Cardy value
+\[
+ \frac{\Gamma(2/3)}{\Gamma(1/3)\Gamma(4/3)}
+ x^{1/3}\,{}_2F_1\!\left(\frac13,\frac23;\frac43;x\right).
+\]
+Assumption~\ref{ass:cardy} concerns one crossing event at a time. It
+neither asserts a quantitative convergence rate nor assumes an
+exploration or multi-crossing scaling limit.
+
+\begin{theorem}\label{thm:main}
+Under Assumption~\ref{ass:cardy}, there exists a constant
+$c_V\in(0,\infty)$ such that
+\[
+ \lim_{\eps\downarrow0}\eps^{3/4}\E N_\eps=c_V.
+\]
+Consequently, the inverse expected pivotal count is asymptotic to
+$c_V^{-1}\eps^{3/4}$.
+\end{theorem}
+
+The constant refers to the specified square, crossing event, intensity
+convention, and color-pivotal count. The theorem is conditional on
+Assumption~\ref{ass:cardy}; it does not prove Cardy's formula for the
+Voronoi model.
+
+\subsection{History and significance}
+Cardy's formula~\cite{Cardy} predicts crossing probabilities at criticality
+from the conformal shape of a planar domain. Smirnov proved the formula
+and conformal invariance of crossing probabilities for critical site percolation on the triangular
+lattice~\cite{Smirnov}. Schramm's stochastic Loewner evolution,
+the exponent calculations of Lawler, Schramm, and Werner, and their
+percolation application by Smirnov and Werner~\cite{Schramm,LSW,SW}
+turned this conformal description into precise information about rare
+connection events. In particular, the alternating four-arm exponent is $5/4$.
+A bulk site is pivotal only when four alternating arms emerge from its
+neighborhood, so this exponent predicts a pivotal count of order
+$\eps^{-2}\eps^{5/4}=\eps^{-3/4}$.
+
+The distinction between an exponent and an amplitude is essential here.
+An estimate $\E N_\eps=\eps^{-3/4+o(1)}$ allows a slowly varying or
+oscillating multiplicative correction. Sharp arm asymptotics, with a
+limiting multiplicative constant, have been proved for critical
+triangular-lattice percolation by Du, Gao, Li, and Zhuang~\cite{DGLZ}.
+Theorem~\ref{thm:main} concerns a Poisson--Voronoi color-pivotal
+observable, under a qualitative crossing-limit hypothesis, rather than
+an unconditional triangular-lattice arm probability.
+
+For Voronoi percolation, box-crossing and arm estimates~\cite{Tassion,Vanneuville} provide the
+multiscale control needed to study critical and near-critical behavior;
+we recall the precise estimates used here in Section~\ref{sec:inputs}.
+Pivotal measures and their role in near-critical scaling limits were
+developed for triangular-lattice percolation by Garban, Pete, and
+Schramm~\cite{GPS}. The companion manuscript~\cite{V} derives a
+quenched near-critical theory for Voronoi percolation from the scalar
+Cardy hypothesis. In particular, it calibrates local pivotal
+observables against the inverse expected pivotal count. Its
+normalization remains defined through that count. Our task is to show
+that this normalization is a pure power of the Euclidean mesh, with a
+positive limiting coefficient.
+
+Several ingredients have established predecessors. The passage from
+local crossing data to joint crossing information uses the
+quad-crossing framework and gluing theory of Schramm and
+Smirnov~\cite{SS}. The annular exploration calculation uses Zhan's
+annulus SLE~\cite{Zhan}. We state the precise inputs from~\cite{V}
+before beginning the proof, and prove the additional annular and
+signed-transfer estimates needed for the amplitude.
+
+\subsection{The mechanism of the proof}
+Work temporarily at unit Poisson intensity and enlarge the observation
+window. Let
+\[
+ Q_0=[-1,1]\times[-1/2,1/2],\qquad
+ f(R)=\Pp\{\text{a site inserted at $0$ is color-pivotal in $RQ_0$}\}.
+\]
+The inputs from~\cite{V} imply $f(R)=R^{-5/4+o(1)}$ and allow the final
+answer to be recovered from the existence of
+$\lim_{R\to\infty}R^{5/4}f(R)\in(0,\infty)$.
+
+To improve the exponent to an amplitude, we compare \emph{differences}
+of connection states across an annulus. A connection state records
+which boundary color runs are joined through the interior, equivalently
+the noncrossing pairing of the interfaces between those runs. The
+transmitted difference vanishes unless at least four interfaces survive
+across the annulus. With exactly four surviving interfaces, the two
+possible interior states differ by a single scalar. This is the
+source of a rank-one approximation for long-annulus transmission.
+
+There is a second, stronger estimate. Adding one fair-colored Poisson
+point produces a signed change that is invariant under color reversal
+and rotations. In this symmetry class, the leading four-interface
+contribution cancels. An explicit spectral calculation shows that the
+remaining transmission decays faster than the inverse square of the
+annulus scale ratio. This threshold is important: the point to be
+inserted ranges over the two-dimensional plane, so the insertion
+effect must be integrable in its location. An unsigned four-arm bound,
+with exponent $5/4$, cannot supply that integrability.
+
+The proof has three stages. First, Sections~\ref{sec:annuli}--\ref{sec:shape}
+construct the annular transmission limits and establish their
+rank-one asymptotic and signed rotational cancellation. Next,
+Section~\ref{sec:transfer} makes these estimates uniform over arbitrary
+signed boundary data and iterates them in the discrete Poisson model.
+Finally, Section~\ref{sec:amplitude} differentiates $f$ with respect to
+Poisson intensity. The integrable insertion estimate controls distant
+insertions, while the rank-one estimate compares the normalized effect
+of nearby insertions at two outer scales. Together they show that the
+logarithmic derivative of $f$ has a summable error from $-5/4$.
+
+The quantitative estimates do not come from a rate in
+Assumption~\ref{ass:cardy}. We first choose a large but fixed annulus
+ratio, use qualitative convergence to obtain the required contraction
+beyond a sufficiently large starting scale, and only then iterate.
+The uniformity over arbitrary signed boundary data, and the separation
+between rank-one contraction and symmetry-enhanced decay, are the
+technical features that may be useful for other amplitude problems.
diff --git a/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/build/sections/01-inputs.tex b/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/build/sections/01-inputs.tex
new file mode 100644
index 0000000000000000000000000000000000000000..e97d3aca2c823f2619f602fa9c887c2a97a29cf9
--- /dev/null
+++ b/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/build/sections/01-inputs.tex
@@ -0,0 +1,239 @@
+\section{Critical inputs and normalization}\label{sec:inputs}
+
+This section specifies the consequences of Assumption~\ref{ass:cardy}
+that we import from the companion manuscript~\cite{V}. In particular,
+the exploration and normalization statements below are previously
+proved inputs, not additional hypotheses and not conclusions of scalar
+convergence alone without further argument. The remainder of this paper
+proves the annular estimates and the amplitude from these inputs.
+
+Throughout, $C,c$ denote positive constants whose values may change
+between occurrences. A comparison $u\asymp v$ means that $u/v$ lies
+between two fixed positive constants. Fixed-factor changes in arm
+radii and fixed buffer widths are absorbed in these constants. We use
+unit-intensity geometry except when explicitly introducing a mesh
+parameter. Let $b_*$ be a sufficiently large fixed microscopic radius.
+
+\subsection{Arms, localization, and local changes}
+An arm in an annulus is a monochromatic cell path joining its two
+boundary components. Alternating four arms have colors black, white,
+black, white in cyclic order. Let $A(r,s)$ be their probability in the
+square-annulus convention of~\cite{V}, for $b_*\le r<s$.
+
+\begin{proposition}[Critical arm and localization inputs]\label{in:arms}
+There are $g\in(1,2)$, $\alpha,\beta>0$, and constants $C,c>0$ such
+that, for $b_*\le r<s<t$,
+\begin{align}
+ c(r/s)^g&\le A(r,s)\le C(r/s)^{1+\alpha},\label{in:power}\\
+ C^{-1}A(r,s)A(s,t)&\le A(r,t)\le C A(r,s)A(s,t).
+ \label{in:quasimult}
+\end{align}
+The one-arm probability is at most $C(r/s)^\beta$. The probability of
+three alternating arms in a straight half-plane is at most
+$C(r/s)^2$.
+
+For a collar of diameter comparable to $s$, an event determined by the
+Poisson points in a slightly enlarged collar makes the graph and all
+vertex stars in the smaller collar local to those points, with mesh
+smaller than any prescribed fixed fraction of $s$. Its failure
+probability is at most $C\exp(-cs^2)$. The widths and constants can
+depend on that fraction. Such events may be chosen rotation equivariant
+about the center. Events with disjoint enlarged supports use independent
+Poisson variables.
+\end{proposition}
+
+These are Lemmas~2.1 and~2.3 of~\cite{V}. The quantitative arm theory
+builds on the box-crossing and annealed arm estimates for Voronoi
+percolation; see~\cite{Tassion,Vanneuville}. A localization event will
+be called a \emph{screen}. For example, requiring that every ball of
+a sufficiently small fixed relative radius in a collar contain a
+Poisson point is rotation equivariant and has the asserted exponential
+bound, by a finite covering and the Poisson empty-ball formula.
+Extra buffers give locality of both cells and vertex stars. Thus a
+point inserted beyond the enlarged support cannot send its cell through
+the screened collar.
+
+We will also use the necessary-arm conclusions of Lemma~2.2
+of~\cite{V}. A color change or a local insertion that changes a
+crossing requires four alternating arms out of its interior support.
+Near a straight side, these continue as three alternating half-plane
+arms; beyond the scale of a corner, at least one arm continues. These
+statements use the exact closed-cell crossing convention. They apply
+to a mixed difference of two local changes as well: if the mixed
+difference is nonzero, each local change affects the appropriate
+crossing in at least one of the other local states. Screens keep the
+other modification out of the arm probes used for this implication.
+
+For later independence arguments it is useful to allow failed screens
+inside an arm probe. Here is the convention and the estimate behind it.
+Place screens at geometrically spaced radii. If the largest failed
+radius is $v$, retain only the arm test starting at $C_0v$, with its
+screens, where $C_0$ provides a fixed unused gap. This retained test
+is independent of the failed screen. Quasi-multiplicativity and
+\eqref{in:power} give, when $C_0v<s$,
+\[
+ A(C_0v,s)\le C(v/r)^g A(r,s).
+\]
+Consequently, summing the failed-screen terms over geometric radii
+costs at most
+\[
+ C A(r,s)\sum_{v\ge r}(v/r)^g e^{-cv^2}
+ \le C A(r,s).
+\]
+If $C_0v\ge s$, the exponential bound and the polynomial lower bound
+give the same conclusion. The corresponding construction for one-arm
+and straight half-plane three-arm tests preserves their displayed
+power upper bounds. We call these \emph{screened arm probes}. Their
+supports can be separated by fixed-factor gaps, and they can be used
+in the presence of inserted sites outside their buffers.
+
+\subsection{Fixed-scale convergence and exploration domains}
+The following inputs are qualitative. Their order of limits will matter:
+all radii, buffer widths, and domain tests are fixed before the mesh
+tends to zero.
+
+\begin{proposition}[Scaling-limit inputs]\label{in:limits}
+Under Assumption~\ref{ass:cardy}, the following consequences hold.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item From every sequence of Poisson meshes tending to zero one can
+extract, after coupling, a further subsequence along which the graphs
+almost surely form a deterministic good sequence in the sense of
+\cite[Lemma~2.5]{V}. Along such a sequence, critical crossings of
+Jordan quadrilaterals converge, also for graph domains whose boundaries
+and marked points converge in curve order. Finite joint crossing tests
+and tests with finitely many separated prescribed squares have the
+common triangular-percolation limit.
+\item Chordal color explorations in converging Jordan graph domains
+converge to chordal $\mathrm{SLE}_6$. At a pre-query stop on a
+crosscut, the remaining target face, its boundary banks, and its
+conformal maps converge in the sense of the stopped-domain results
+of~\cite{V}. A bank is an individual boundary arc in prime-end order;
+distinct banks may have touching physical images. Each relevant bank
+is individually simple.
+\item For some $c_0>0$, in the mesh-first limit the probability of
+four bulk arms with any specified color order is at most
+$C(r/s)^{1+c_0}$, and that of six bulk arms with any specified color
+order is at most $C(r/s)^{2+c_0}$. Three free arms at a regular
+boundary, measured in a conformal chart, have an upper bound with
+exponent strictly larger than one.
+\end{enumerate}
+\end{proposition}
+
+For precise locators, part~(i) uses Proposition~2.4, Lemma~2.5, and
+Proposition~4.5 of~\cite{V}; part~(ii) uses Proposition~3.1 and
+Lemmas~3.4--3.7; part~(iii) uses Lemmas~2.7, 2.8, and~2.11.
+The free boundary-arm estimate in Lemma~2.11 has exponent $4/3$.
+``Free'' means that the arms use the independent fair interior sites,
+not the sites in the prescribed base arc.
+
+We emphasize two domain qualifications. First, a boundary chart must
+converge uniformly on its closure, have no constant boundary subarc,
+and project injectively on the compact bank interval where the
+estimate is used. Separate prime-end banks need not have disjoint
+physical projections. The local estimate applies in an annulus by
+choosing a small graph disk patch with one side on that bank and the
+remaining sides in the interior. These are Jordan approximations;
+reflection in the parameter coordinate gives bounded distortion on
+smaller patches. Second, product colors are conditional on the graph.
+After an exploration, its remaining colors are fresh, but a
+remaining-domain theorem can be used only after its geometric
+hypotheses have been established. We never infer uniform convergence
+after conditioning on arbitrary atypical explored colors. The uniform
+operator estimate in Section~\ref{sec:transfer} will instead use
+independent core variables and bounds on the signed weights.
+
+We use the site triangulation of the Voronoi tiling from~\cite{V}.
+Its faces are triangles, drawn through the tiling rather than required
+to have straight edges. An interface strand, or \emph{wall}, passes
+through a triangle between its two heterochromatic edges. Inside a
+screened collar, simple site cycles approximate circles in curve
+order. They may be constructed by following narrow strips and
+retaining simple cuts; same-boundary chords on the domain-facing side
+can be shortcut while retaining the specified inner region. Auxiliary
+uniform angular choices, if needed, preserve rotation invariance.
+
+All transfer cuts in this paper lie in the bulk. Their incident cells
+and stars stay away from the quad boundary, where site connectivity
+and cell connectivity could otherwise require additional care.
+At the actual quad boundary, we retain the original closed-cell event
+and use the necessary-arm estimates just stated. Null nongeneric
+geometries, and null sets of insertion locations in integrals, are
+discarded throughout.
+
+\subsection{Calibration and the preliminary exponent}
+Let $S_\eps$ be the probability that a site inserted at the center of
+$Q_0=[-1,1]\times[-1/2,1/2]$ is color-pivotal for its left-to-right
+crossing in intensity $\eps^{-2}$. Equivalently, $S_\eps$ is the
+expectation of the black-forced crossing indicator minus the
+white-forced crossing indicator for that site. Set
+\[
+ m_\eps=\eps^{-2},\qquad
+ q_\eps=\bigl(\E N_\eps(Q_0)\bigr)^{-1},\qquad
+ f(R)=S_{1/R}.
+\]
+The last equality agrees by Euclidean scaling with the unit-intensity
+definition in the introduction.
+
+\begin{proposition}[Calibration]\label{in:calibration}
+There are constants $c,B\in(0,\infty)$ such that
+\begin{equation}\label{in:calibration-eq}
+ m_\eps q_\eps S_\eps\longrightarrow c,
+ \qquad q_\eps\E N_\eps\longrightarrow B.
+\end{equation}
+Moreover,
+\begin{equation}\label{in:comparison}
+ f(R)\asymp A(b_*,R).
+\end{equation}
+\end{proposition}
+
+The limits are precisely the single-time Poisson specialization of
+Lemmas~7.1 and~7.2 of~\cite{V}, called ``Common calibration'' and
+``The prescribed normalization.'' The signed color weight at the
+inserted site is $\delta_1-\delta_0$; crossing monotonicity identifies
+its expectation with $S_\eps$. The comparison follows from the local
+pivotal bounds and the square-to-site calibration in Sections~2 and~6
+of~\cite{V}. These inputs do not assert that
+$R^{5/4}f(R)$ converges.
+
+\begin{lemma}[Exponent without an amplitude]\label{in:exponent}
+Under Assumption~\ref{ass:cardy},
+\begin{equation}\label{in:exponent-eq}
+ f(R)=R^{-5/4+o(1)}\qquad(R\to\infty).
+\end{equation}
+\end{lemma}
+\begin{proof}
+Force all cells meeting a central square of radius $h$ in $Q_0$
+first black and then white, and denote the difference of crossing
+probabilities by $D_\eps(h)$. Proposition~4.5 of~\cite{V} gives,
+for each fixed small $h>0$, a common limit of this quantity in the
+Voronoi and triangular models. The upper arm bound and
+\cite[Lemma~6.1]{V}, ``A square can transmit a pivotal change,''
+compare $D_\eps(h)$ with the four-arm probability from radius $h$
+to radius $1$, with constants uniform in small $h$, after taking
+the mesh limit. The corresponding triangular four-arm probabilities
+have exponent $\lambda=5/4$ by~\cite{SW}.
+
+Fix $\eta>0$. For an arbitrarily large but fixed ratio $L=1/h$,
+these comparisons and qualitative convergence imply, for all
+sufficiently large $r$ depending on $L$,
+\[
+ c_\eta L^{-\lambda-\eta/3}
+ \le A(r,Lr)\le C_\eta L^{-\lambda+\eta/3}.
+\]
+Choose $L$ so large that both these constants and the constants in
+\eqref{in:quasimult} are absorbed by an additional factor
+$L^{\eta/3}$. Only after fixing this $L$ choose a starting radius
+above the convergence threshold. Iterating over radii $r,Lr,L^2r,
+\ldots$ gives four-arm bounds with exponents between
+$\lambda-\eta$ and $\lambda+\eta$. The bounded initial range and
+the final incomplete ratio change only the multiplicative constants,
+by \eqref{in:power} and \eqref{in:quasimult}. Equation~\eqref{in:comparison}
+then yields
+$C_\eta^{-1}R^{-\lambda-\eta}\le f(R)\le C_\eta R^{-\lambda+\eta}$.
+Letting $\eta\downarrow0$ proves the lemma.
+\end{proof}
+
+We will henceforth write $\lambda=5/4$. The remaining task is to
+replace the $o(1)$ in this exponent by convergence of the multiplicative
+coefficient. The next two analytic sections identify which annular
+symmetries make that improvement possible.
diff --git a/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/build/sections/02-annuli.tex b/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/build/sections/02-annuli.tex
new file mode 100644
index 0000000000000000000000000000000000000000..7dcc91afc8d70850e0f17c346cdd13dd7188f500
--- /dev/null
+++ b/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/build/sections/02-annuli.tex
@@ -0,0 +1,487 @@
+\section{Four-interface transmission in an annulus}
+\label{sec:annuli}
+
+The continuum quantity used below is the probability that four interfaces
+cross an annulus from one boundary component to the other.  Its existence
+does not follow by applying the scalar Cardy hypothesis directly to a
+multiply connected domain.  We establish it from the Jordan-domain inputs
+of \cite{V}.  We also allow the boundary banks produced by stopped
+explorations: a bank is viewed with its access from the domain, so distinct
+banks may touch in the plane without being identified.
+
+\subsection{Connection states and two caps}
+
+Use the site triangulation drawn through the Voronoi tiling, with interface
+strands drawn across the heterochromatic edges of its triangular faces.
+These strands will be called \emph{walls}.  On a simple site cycle with
+known colors, a \emph{run} is a maximal consecutive set of sites of one
+color, and a \emph{switch} is the edge between consecutive runs.  Filling
+the disk bounded by the cycle determines a noncrossing pairing of its switches by
+walls.  The pairing, together with the run colors, determines connectivity
+between boundary sites.  Closed walls need not be recorded.  We call this
+pairing the connection state on the cycle.
+
+Suppose each boundary component of a graph annulus has four switches and
+four alternating prescribed runs.  All remaining sites have independent
+fair colors.  Let $\mathcal T$ be the event that all four outer wall ends
+reach the inner boundary.  There are two possible connection states in a
+disk filling the inner cycle: the cap joining its two black runs, denoted
+$\mathsf b$, and the cap joining its two white runs, denoted $\mathsf w$.
+These are formal planar caps; no random colors are sampled in them.  Write
+$H_{\mathsf b}$ and $H_{\mathsf w}$ for the indicators that the two outer
+black runs are connected after the corresponding cap is attached.
+Figure~\ref{fig:annular-caps} shows the two cap states when all four
+walls are transmitted.
+
+\begin{lemma}[The cap identity]\label{an:caps}
+For every coloring of the annulus,
+\begin{equation}\label{an:cap-equation}
+ H_{\mathsf b}-H_{\mathsf w}=\mathbf 1_{\mathcal T}.
+\end{equation}
+Consequently the signed difference between the two inner cap states is
+transferred to the same signed difference at the outer cycle, multiplied
+by the indicator of four-interface transmission.  The identity remains
+valid under composition across several cycles.
+\end{lemma}
+
+\begin{proof}
+Walls in a triangulation neither branch nor cross.  If fewer than four
+inner ends are connected to outer ends, the wall pairing visible from the
+outer cycle is independent of the inner cap: there is no choice of pairing
+on zero or two surviving ends.  Thus the left side of
+\eqref{an:cap-equation} is zero.  If all four are transmitted, their cyclic
+order is preserved, with any possible winding retained.  The intervening
+monochromatic bands carry black connectivity to black connectivity and
+white connectivity to white connectivity.  Hence $H_{\mathsf b}=1$ and
+$H_{\mathsf w}=0$.  This is also the usual Hex duality proof after each run
+is contracted to a vertex.  Composing the planar pairings proves the last
+assertion; any closed walls created in the composition have weight one.
+\end{proof}
+
+\begin{figure}[tb]
+\centering
+\begin{tikzpicture}[scale=0.87,line cap=round]
+\foreach \a/\txt in {0/{Black cap: $H_{\mathsf b}=1$},6/{White cap: $H_{\mathsf w}=0$}} {
+  \begin{scope}[xshift=\a cm]
+    \draw[gray] (0,0) circle (1.85);
+    \draw[gray] (0,0) circle (.76);
+    \draw[very thick] (45:1.85) arc (45:135:1.85);
+    \draw[very thick] (225:1.85) arc (225:315:1.85);
+    \draw[very thick] (45:.76) arc (45:135:.76);
+    \draw[very thick] (225:.76) arc (225:315:.76);
+    \foreach \t in {45,135,225,315} {
+      \draw[blue!65!black,thick] (\t:.76)--(\t:1.85);
+      \fill (\t:.76) circle (1.4pt);
+      \fill (\t:1.85) circle (1.4pt);
+    }
+    \node at (0,2.12) {$B$};
+    \node at (0,-2.12) {$B$};
+    \node at (-2.12,0) {$W$};
+    \node at (2.12,0) {$W$};
+    \node[align=center] at (0,-2.65) {\txt};
+  \end{scope}
+}
+\draw[blue!65!black,thick] (45:.76) .. controls (.20,.25) and (.20,-.25) .. (315:.76);
+\draw[blue!65!black,thick] (135:.76) .. controls (-.20,.25) and (-.20,-.25) .. (225:.76);
+\begin{scope}[xshift=6cm]
+\draw[blue!65!black,thick] (45:.76) .. controls (.25,.20) and (-.25,.20) .. (135:.76);
+\draw[blue!65!black,thick] (225:.76) .. controls (-.25,-.20) and (.25,-.20) .. (315:.76);
+\end{scope}
+\end{tikzpicture}
+\caption{The two cap pairings when all four walls are transmitted.
+Blue curves are walls; thick black boundary arcs are black runs, and the
+other runs are white.  The cap changes the outer black connection bit by
+one.  The straight, unwound walls are schematic: the identity permits
+winding and does not prescribe a pairing of numbered inner and outer
+switches.}\label{fig:annular-caps}
+\end{figure}
+
+\subsection{The boundary convergence used below}
+
+For $p>0$ let
+\[
+ \mathbb A_p=\{e^{-p}<|z|<1\}.
+\]
+Thus $p=\log(R/r)$ for a round annulus with radii $r<R$.  On both circles
+angles increase counterclockwise.  Write $x=(x_1,x_2,x_3,x_4)$ for the outer
+switches and $y=(y_1,y_2,y_3,y_4)$ for the inner switches, using increasing
+lifts with $x_4<x_1+2\pi$ and $y_4<y_1+2\pi$.  Common translation of either
+lift by $2\pi$ does not change the data.  The run following the first outer
+switch is black.  For the $+$ prescription the corresponding inner run is
+black; for the $-$ prescription it is white.
+
+All limits in this section are taken along deterministic good sequences
+of triangulations in the sense of \cite[Section 2, ``Deterministic
+sequences and Jordan tests'']{V}.  In particular the mesh tends to zero on
+compact sets, the joint Jordan-quad crossing limits hold, and the
+mesh-first arm estimates recalled above are available.  Probabilities are
+over product colors conditional on these triangulations.
+
+\begin{definition}\label{an:admissible}
+A sequence of marked graph annuli $D_n$ has \emph{admissible boundary
+convergence} if it lies in a fixed bounded set and satisfies the following
+conditions.
+\begin{enumerate}
+\item Each boundary is a simple site cycle.  With rotations normalized,
+its uniformization
+$\phi_n:\mathbb A_{p_n}\longrightarrow D_n$ extends continuously to the
+closed annulus, where $p_n\to p\in(0,\infty)$.  After linear identification
+in logarithmic radial coordinates, these maps converge uniformly on the
+closed annulus to $\phi$.  The map $\phi$ is conformal and one-to-one in
+$\mathbb A_p$, and its boundary traversal specifies the prime-end accesses
+of $D=\phi(\mathbb A_p)$.
+\item Each limiting boundary traversal is a concatenation of finitely
+many simple arcs, each understood with its own access.  Different arcs
+may meet in physical projection.  The two boundary components remain
+distinct as prime-end boundary components.
+\item The four switches on each component converge in these coordinates
+to distinct switches.  Between them the prescribed run colors alternate.
+The sites strictly between the two cycles have fresh independent fair colors.
+\end{enumerate}
+In graph constructions a boundary is taken without chords facing the
+domain when that is required for a local boundary chart.  Such cuts are
+obtained by the graph approximation and shortcut construction of
+\cite[Section 2]{V}; at an exploration frontier, edges between already
+revealed vertices are included before the frontier is taken.
+\end{definition}
+
+The definition permits a slit to have two different banks, even when
+their images coincide.  It does not identify these banks.  The next
+elementary convergence criterion explains how the domains arising from
+stopped explorations fit this definition.
+
+\begin{lemma}[A back with finitely many attached lobes]
+\label{an:back-lobes}
+Suppose graph annuli consist of an open annular back and finitely many
+Jordan lobes attached to it along open seam intervals.  Assume that the
+backs are Jordan annuli whose two complete boundary traversals converge
+in curve order to those of a fixed Jordan annulus, the seam endpoints
+converge in that order and the limiting seams remain open, and each lobe
+converges as a Jordan domain with its boundary traversal.  The seam
+incidences and cyclic order are fixed; lobes are glued only along their
+specified seams, and their interiors are disjoint from the back and from
+one another.  Suppose the resulting
+limiting domain is an annulus, its hole and a separating open part of the
+back persist, and the two boundary components stay physically apart.
+Different limiting lobes may touch in projection but have disjoint
+interiors.  Then the annular uniformizations converge uniformly on their
+closures, with the stated accesses and traversal order.  In particular,
+if the marked accesses converge and remain distinct, these domains have
+admissible boundary convergence.
+\end{lemma}
+
+\begin{proof}
+Every compact subset of a lobe persists by Jordan convergence.  The lobe
+joins the back across every compact subinterval of its open seam.
+Conversely, every point of the limiting boundary is approached by the
+approximating boundaries.  Contacts between different lobes do not open
+passages: the individual Jordan pieces are attached along their specified
+seams, not through a point of contact.  These facts give kernel convergence
+with the indicated accesses.  The persisting hole and separating back
+exclude degeneration of the modulus.  The conformal kernel theorem
+therefore gives convergence of normalized annular maps on interiors and
+convergence of their moduli.
+
+It remains to control the boundary parameters.  Fix a nonconstant open
+subarc of one of the finitely many Jordan boundary pieces and a point in
+the persisting back.  Jordan-map convergence in a lobe, or in a disk patch
+along an unchanged back boundary, gives a positive lower bound for the
+harmonic measure of a smaller subarc as seen from a point in that patch.
+A Harnack chain through persisting interiors gives the same kind of lower
+bound from the fixed back point.  In round annular coordinates the
+harmonic-measure densities from its preimage are bounded above uniformly:
+the moduli and the preimage stay in compact nondegenerate ranges.
+Consequently no fixed nonconstant subarc of the boundary traversal can be
+squeezed into an angular interval of length tending to zero.
+
+The Jordan convergence of the finitely many pieces gives convergence of
+the full boundary traversals in curve order.  Partition those traversals
+into finitely many small image arcs, and apply the preceding lower bound
+to intervening nonconstant subarcs.  A failure of equicontinuity in angular
+parameters would traverse one such fixed subarc in a vanishing angular
+interval, a contradiction.  This proves equicontinuity of the boundary
+maps.  Interior convergence then identifies every uniformly convergent
+subsequence, giving convergence on the closure.  Small crosscuts at the
+finitely many joins give the same conclusion there.  There is no constant
+boundary interval in the limiting conformal map, by uniqueness for a
+holomorphic function continuous on a boundary arc.  The boundary order
+and converging marked accesses thus also identify the limiting switches.
+\end{proof}
+
+\subsection{Boundary motion and annular crossing laws}
+
+We first remove the rough banks.  The arm bounds needed here are
+mesh-first statements: three free arms along one regular bank have an
+upper bound with exponent $1+\eta$ for some $\eta>0$, and a free arm from a
+shrinking neighborhood to a fixed scale has probability tending to zero.
+The word \emph{free} excludes prescribed sites and cap wires.  The local
+bank versions, including arms approaching the bank arbitrarily closely,
+are those of \cite[Section 2, ``Cardy tests in a boundary chart'' and
+``Free boundary-arm bounds'']{V}.
+
+\begin{lemma}[Continuity under boundary motion]\label{an:motion}
+For a sequence with admissible boundary convergence, move either cycle
+slightly into the annulus in round coordinates, and move its switches
+slightly with it.  Couple all unchanged free colors.  The probability
+that either capped connection test changes tends to zero as the size of
+the motion tends to zero, after taking the mesh limit.  The assertion also
+holds for small piecewise smooth motions in spare space around an inset
+annulus.  It is uniform along any convergent sequence of admissible data
+with distinct limiting switches.  More precisely, if $H_n$ and
+$H_{n,\delta}$ are either capped indicators before and after a motion of
+size at most $\delta$, preserving the run order and its labels away from
+the moving switches, then
+\begin{equation}\label{an:motion-limit}
+ \lim_{\delta\downarrow0}\limsup_{n\to\infty}
+ \Pp(H_n\ne H_{n,\delta})=0.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+First the mesh in annular coordinates tends to zero.  Otherwise a
+connected cell or triangle portion of vanishing physical diameter would
+have preimages with a subsequential nontrivial continuum limit on which
+$\phi$ is constant.  Such a continuum cannot meet the interior, where
+$\phi$ is one-to-one, and cannot lie on the boundary, whose finitely many
+simple arcs contain no constant interval.  Here graph portions, rather
+than whole cells crossing the domain boundary, are used.
+
+Remove small coordinate neighborhoods of the switches and of the
+finitely many joins of simple bank arcs.  On each remaining compact bank
+interval, use disk patches whose boundary consists of that bank and cuts
+strictly inside the annulus.  Their physical boundaries are Jordan and
+their graph approximations converge in curve order.  Their uniformizing
+coordinates agree, with fixed-scale margins, with the annular coordinates;
+reflection in the limiting round bank controls distortion on smaller
+patches.  Thus the local boundary-arm estimates of \cite{V} apply on these
+patches even when another prime-end bank has the same physical image.
+
+Cover these intervals by $O(s^{-1})$ coordinate disks of radius $s$.
+Implement a motion much smaller than $s$ by successively replacing bank
+intervals, with tapered ends.  At graph level choose short transverse
+joins in disjoint narrow strips from the old cycle to its replacement.
+Substitute the intervals in cyclic order, retaining the old cycle off the
+interval currently being changed and erasing the joins once the whole
+cycle is replaced.  The graph approximation in curve order ensures that
+these are simple cuts.  Each join has the same constant color as its run.
+Moves containing switches or bank joins are instead included in the
+exceptional neighborhoods.  Two nearby nonnested contours can be
+compared through a common slightly inset contour.
+
+Consider one ordinary move on a black bank.  If a capped connection test
+changes, choose a winning black connection in one test and a winning
+dual white connection in the other.  Contract the prescribed runs and
+use the same cap in both tests.  Purely wired connections do not change,
+so a free segment of a winning path must meet the support of the move.
+The test neighborhood avoids every other run and mark.  The white path
+has two free branches leaving that neighborhood.  At least one black
+branch must also leave freely: a black branch already reaching the same
+prescribed run outside the moved interval could be joined there without
+the local change.  Taking simple paths after contraction gives disjoint
+white branches, and the opposite-colored black branch is disjoint from
+them in the common fair region.  This gives three free arms in a slightly
+shifted half-patch, with fixed-factor margins.  The same argument applies
+with the colors interchanged.  Forced sites on intermediate cuts are not
+counted as free arms.
+
+For any fixed larger chart scale, the mesh-first bound for a single disk
+is $O(s^{1+\eta})$.  The union over $O(s^{-1})$ disks therefore costs
+$O(s^\eta)$.  The number of exceptional neighborhoods is fixed.  A change
+there requires a fair arm approaching from positive parameter distance:
+connections already provided by the same run or by the cap do not
+change.  This arm has positive physical diameter, uniformly after fixing
+its outer parameter scale, by the no-continuum-fiber observation above.
+The ordinary physical one-arm bound consequently makes the exceptional
+cost tend to zero, including at a contact between different accesses.
+One may move each exceptional neighborhood as a single piece, so this
+cost is not multiplied by the number of ordinary intervals.
+
+First take the mesh limit, then make the ordinary motions small compared
+with $s$, then let $s$ tend to zero, and finally shrink the exceptional
+neighborhoods.  This proves the assertion.  The same proof with moving
+marks, or with a moving round radius, proves the stated continuity.
+\end{proof}
+
+After applying Lemma~\ref{an:motion}, all prescribed banks can be placed
+on smooth contours strictly inside the parameter annulus, with spare
+space on both sides.  There are still two separate issues: joint ordinary
+crossings must have a conformally invariant annular law, and those
+crossings must determine the capped tests.  We address them in that order.
+
+\begin{lemma}[Ordinary crossings in an annular open set]
+\label{an:ordinary}
+In an annular open set compactly inside the parameter annulus, the joint
+limiting law of any countable collection of regular ordinary quad tests
+is conformally invariant.  The tests may be chosen from a countable dense
+family of polygon or piecewise smooth quads with positive margins.
+\end{lemma}
+
+\begin{proof}
+On every strict Jordan disk patch the joint law is the critical triangular
+percolation law by \cite[Sections 3--4]{V}.  This law is conformally
+invariant; see \cite[Theorems 1 and 7]{CN} and the quad-crossing
+formulation in \cite[Section 2.3]{GPS}.  Scalar Cardy sandwiches identify smooth images of polygon
+tests and nearby graph crossing conventions with the same limit.
+
+Choose a finite-length smooth cut opening the annulus.  Any finite
+collection of tests compactly contained off the cut fits in a strict
+Jordan disk, so its joint law agrees in the physical domain and in the
+conformal chart.  To recover tests meeting the cut, use the
+Schramm--Smirnov gluing theorem
+\cite[Theorem 1.19 and Proposition 4.1]{SS}, only in strict Jordan disk
+patches.  For the critical triangular crossing law, the crossing of a
+regular quad in such a patch is measurable, modulo null sets, with
+respect to countably many tests avoiding the finite smooth cut in that
+patch.  Its hypotheses hold here: independence in the discrete model,
+RSW, one-arm decay, and a four-arm exponent strictly greater than one.
+Regular continuity tests may be used throughout, by Cardy sandwiches.
+
+Each required quad meeting the annular cut has a slightly larger Jordan
+disk neighborhood inside the annulus.  The recovering measurable rule
+in that neighborhood is the same in the chart and in its image, since
+their whole disk laws already agree.  All recovering tests belong to the
+common countable off-cut family.  Hence the recovered crossings have the
+same joint law as well.  This proves the assertion.  White crossings are
+included by Hex duality and continuity.  No gluing assertion along a rough
+slit is used.
+\end{proof}
+
+\begin{proposition}[The annular transmission limit]\label{an:limit}
+For every sequence with admissible boundary convergence, the probability
+of $\mathcal T$ has a limit depending only on the round data $(p,x,y)$ and
+the choice of inner colors.  Denote these limits by
+\[
+ T^+(p,x,y),\qquad T^-(p,x,y).
+\]
+They are continuous for $p>0$ and distinct switches.  The same convergence
+holds for each capped connection probability.  In particular, it applies
+to random remaining domains whenever, conditional on their geometry and
+explored boundary, their interior colors are fresh and their marked
+domains have admissible convergence almost surely in a subsequential
+coupling.  In that case the difference between the conditional
+transmission probability and $T^\pm$ evaluated at the actual annular
+conformal data tends to zero in mean absolute value.
+\end{proposition}
+
+\begin{proof}
+By Lemma~\ref{an:motion}, it suffices first to consider smooth inset
+annuli with spare space.  We show that ordinary quad crossings determine
+the limiting value of each capped black connection test.  Sample fair
+colors throughout a slightly larger annular neighborhood in that spare
+space, overriding them on prescribed runs only when evaluating a capped
+test.  Ordinary tube tests use the fair colors.  Compare two
+candidate sequences, one of which may be in round coordinates.  Couple
+their countable ordinary crossing vectors through the common limiting
+law of Lemma~\ref{an:ordinary}.  Along further subsequences include the
+cap indicators in the coupling; the curve tightness proved below also
+allows the chosen winning paths to be included.  The two limiting
+collections are coupled to have the same ordinary crossing vector.
+
+Fix a small \emph{favorable} perturbation of the black test in the second
+sequence.  Move black banks into the annular fluid and extend their runs
+slightly past the old switches; retreat white banks from the old fluid.
+Use positive slacks and taper the motions within the old white
+intervals, putting each new switch on the retreated part.  Thus every old
+black attachment lies behind a new black bank, the new white banks are
+positively separated from the old fluid, and any removed fluid is in
+separate collars of the expanded black runs.  All these choices take
+place in the spare space.
+
+On a positive configuration in the first sequence, contract each
+prescribed run and attach the cap edge when it is black.  Take an
+outermost simple winning path between the two outer black terminals.
+For example attach an external terminal edge and take the appropriate
+boundary of the union of cycles containing it.  Contracting connected
+boundary runs and attaching the cap preserves the planar embedding.
+The resulting path has no bypass on its chosen exterior side.  Since it
+visits each contracted run at most once, it decomposes into a bounded
+number of free black segments joining different runs, together with
+wired or cap joins.  Crop their endpoints by a vanishing graph distance
+if they end adjacent to a wire.
+
+We recall why these free segments admit simple limiting curves.  Many
+oscillations across an annulus supply many disjoint free arms; BK and
+one-arm decay bound their probabilities.  Successive-excursion
+compactness therefore gives tightness in the curve metric.  A bulk
+self-contact with two positive excursions would give four black
+traversals.  In two gaps on the no-bypass side, intermediate transverse
+cuts and Hex duality give two additional white traversals: a black
+transverse crossing would instead be a bypass.  The six arms are
+disjoint after fixed-factor crops, also for backtracking pieces.
+The mesh-first six-arm exponent greater than two excludes such contacts
+by a covering argument.  At a smooth boundary bank, a repeated visit to
+the same point gives at least three free traversals, including when one
+of the visits is an endpoint; the boundary exponent greater than one
+excludes these repeated visits.  A single tangential bank contact need
+not be excluded: the favorable perturbation accommodates it.  At the finitely
+many switches one free arm suffices.  These are precisely the local
+no-bypass and free-segment arguments of \cite[Section 4, ``Free segments
+of a terminal path'']{V}.  Contracted runs do not change the bulk argument,
+whose working annulus avoids them.  Constant pauses may be removed.
+
+In a subsequential coupled representation, fix the favorable displacement
+and then crop each limiting free segment a small positive amount at its
+ends, keeping the new endpoints strictly behind the expanded black
+banks.  A cropped endpoint may still lie on an old bank; the ambient
+tube may straddle that bank within the spare space.  This positive crop is distinct from the vanishing
+graph crop used to remove incidences to a wire.  Surround the cropped
+simple segment by a thin Jordan tube, with end caps in those endpoint
+neighborhoods.  The approximating free graph segments supply ordinary
+fair-color crossings of this ambient tube.  The tube can be approximated with positive margins by a quad
+from the countable ordinary test family.  Its crossing occurs in the
+first sequence, and hence in the second for all sufficiently large
+indices in the coupling.  The witnesses may be chosen from the countable
+family after taking the subsequential limit; coordinatewise convergence
+still applies to each of the finitely many chosen witnesses.
+
+Clip these transferred crossings when they hit the new black banks.
+Excursions outside the new fluid can only lie in the collars now wired
+black.  They can be shorted through the corresponding run, and collars
+of different black banks are separated.  This operation also handles
+arbitrarily many backtracks: at graph level every passage between a
+removed collar and the fluid meets its boundary cycle.  The transferred
+tubes, together with the unchanged cap or wire joins, therefore give a
+winning black connection in the favorably perturbed second test.
+
+Apply this comparison to subsequences realizing the upper and lower
+limits of the unperturbed cap probabilities.  The upper limit of the
+first is bounded by the lower limit of the second plus its boundary-motion
+error.  Remove the favorable perturbations using Lemma~\ref{an:motion}
+and reverse the roles of the sequences.  Their upper and lower limits
+coincide.  This proves convergence and conformal invariance of each cap
+test on inset annuli.  Taking the inset size to zero proves the same
+assertions under Definition~\ref{an:admissible}.  The cap identity gives
+the transmission limit.  Allowing the marks and round radii to move in
+the same comparison proves continuity in $(p,x,y)$.
+
+Finally consider random remaining domains under the stated coupling.
+For almost every convergent domain realization, the deterministic
+statement applies to its fresh product-color law; continuity identifies
+the same limit for $T^\pm$ at the actual conformal data.  All probabilities
+are bounded by one, so bounded convergence gives convergence in mean
+absolute value.  This conclusion does not assert a uniform conditional
+estimate over arbitrary atypical explored boundaries.
+\end{proof}
+
+\begin{lemma}[Collisions and symmetries]\label{an:collision}
+The functions $T^\pm$ extend continuously by zero when switches collide
+on either boundary, uniformly for $p$ in compact subsets of $(0,\infty)$.
+They are invariant under simultaneous rotation of $x$ and $y$ and under
+global color reversal.  Cyclic relabeling by one switch in either tuple
+interchanges $T^+$ and $T^-$.
+\end{lemma}
+
+\begin{proof}
+\leavevmode\clubpenalty=10000
+If two consecutive switches coalesce, four-interface transmission
+requires a fair arm from their intervening shrinking run to the other
+boundary.  This arm cannot use another prescribed run: it lies between
+the two corresponding transmitted walls.  A mesh-first one-arm bound,
+with a fixed outer scale and then a shrinking inner scale, proves
+vanishing.  The bound does not depend on the positions of the other
+switches and therefore also covers simultaneous collisions.
+Conformal invariance gives simultaneous rotational invariance, and
+fair colors give global color symmetry.  Moving the first switch to the
+next switch reverses the alternating run labels on that component.
+By the defining convention, this interchanges the two inner prescriptions.
+\end{proof}
diff --git a/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/build/sections/03-spectrum.tex b/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/build/sections/03-spectrum.tex
new file mode 100644
index 0000000000000000000000000000000000000000..7e25eb320bb0afcd4d33bb22304c1a185ca40553
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@@ -0,0 +1,452 @@
+\section{The spectrum of four-interface transmission}
+\label{sp:section}
+\label{sec:spectrum}
+
+We now estimate the limiting transmission probabilities
+$T^\pm(p,x,y)$ of Section~\ref{sec:annuli} when the modulus $p$ is large.
+There are two different conclusions.  The leading term separates into an
+inner factor and an outer factor.  The signed difference $T^+-T^-$ has
+additional decay, and averaging the rotation of one end improves that
+decay beyond exponent $2$.  This last improvement will make the response
+to a fair Poisson insertion integrable over the plane.
+
+\subsection{The modulus equation}
+
+The space of four cyclically ordered, labeled angles is
+\[
+ \mathcal C=
+ \{w\in\R^4:w_1<w_2<w_3<w_4<w_1+2\pi\}
+       /(2\pi\mathbb Z\mathbf 1).
+\]
+Thus we quotient by simultaneous full turns, not by cyclic relabeling.
+We use the product chamber $\mathcal X=\mathcal C\times\mathcal C$ for
+$(x,y)$.  All statements about distances or derivatives on this chamber
+use local angle lifts.  Constants are uniform as angles approach a
+collision; moduli are bounded below by a fixed positive constant.
+
+\begin{proposition}[Modulus equation]\label{sp:annular-pde}
+The functions $T^\pm$ are smooth on $(0,\infty)\times\mathcal X$ and
+satisfy
+\begin{equation}\label{sp:pde}
+ \partial_pT=(L+B_p\cdot\nabla)T,
+ \qquad
+ L=\frac18\sum_{w=x,y}
+ \left(3\sum_{i=1}^4\partial_{w_iw_i}
+ +\sum_{i=1}^4\sum_{j\ne i}
+       \cot\frac{w_i-w_j}{2}\,\partial_{w_i}\right).
+\end{equation}
+For every $p_0>0$ and every fixed finite derivative order $k$,
+the vector field $B_p$ extends smoothly across angle collisions and
+\[
+ \|B_p\|_{C^k}\le C_{k,p_0}e^{-p}\qquad(p\ge p_0).
+\]
+It is equivariant under permutations within either tuple, common
+rotations, and cyclic relabelings.  In particular, its normal component
+at every collision face is zero.
+\end{proposition}
+
+\begin{proof}
+We first derive a local equation with just one switch active.  Write the
+annulus as the cylinder $0<\operatorname{Im}z<p$, with real period
+$2\pi$, and take the active switch on its lower boundary.  Parametrize
+its exploration by the decrease of the modulus.  In the annulus Loewner
+normalization with zero regular constant at the driving pole, the
+angular kernel is
+\[
+ H_p(u)=\lim_{N\to\infty}\sum_{k=-N}^N
+                     \cot\frac{u-2ikp}{2}.
+\]
+The pole has residue $2$, and the imaginary part on the other boundary
+is $-1$.  The locality theorem for annulus $\mathrm{SLE}_6$
+\cite{Zhan} identifies the local driving angle with Brownian motion
+of variance parameter $6$.  Only locality before leaving a small
+simply connected neighborhood of the active switch is used here.
+
+For completeness, the absence of an additional driving drift depends
+on the normalization.  In a chordal chart the pole part is
+$2/(z-U_t)$.  If $F_t$ is a changing local conformal coordinate, comparison
+of pole residues gives the new clock $F_t'(U_t)^2\,dt$.  Comparison of
+the regular terms, both normalized to have zero constant, gives
+$\partial_tF_t(U_t)=-3F_t''(U_t)$.  This cancels the It\^o correction
+$\kappa F_t''(U_t)/2$ precisely for $\kappa=6$.
+No condition on a remote target in the annulus enters this calculation.
+
+Other switches on the active boundary have velocity $H_p(u)$, at
+angular difference $u$ from the active one.  Switches on the opposite
+boundary have angular velocity $\operatorname{Re}H_p(u+ip)$.
+Pairing the positive and negative terms of the series gives, uniformly
+with any fixed number of derivatives,
+\begin{equation}\label{sp:kernel-errors}
+ H_p(u)-\cot(u/2)=O(e^{-2p}),
+ \qquad \operatorname{Re}H_p(u+ip)=O(e^{-p}).
+\end{equation}
+The remainders are smooth periodic functions of $u$, including at
+$u=0$.  Thus, when $x_i$ is the active switch, its equation is
+\begin{equation}\label{sp:active-equation}
+ \begin{aligned}
+ 0={}&-\partial_pT+3\partial_{x_ix_i}T
+       +\sum_{j\ne i}H_p(x_j-x_i)\,\partial_{x_j}T\\
+    &+\sum_{j=1}^4\operatorname{Re}H_p(y_j-x_i+ip)\,
+                                      \partial_{y_j}T.
+ \end{aligned}
+\end{equation}
+For an active switch $y_i$, the equation is obtained by exchanging
+$x$ and $y$.  Indeed the cylinder change $z\mapsto-z+ip$ exchanges
+the boundaries and reverses both angle coordinates; the angular
+kernels are odd, so transforming back preserves the displayed signs.
+Averaging these eight equations gives \eqref{sp:pde}.
+The series estimates give the asserted bounds on $B_p$.  The averaged
+expression treats the labels within each tuple symmetrically; therefore
+it is permutation equivariant.  At a collision its two corresponding
+components agree, proving tangency.  The other symmetries follow in
+the same way from the kernels.
+
+We justify the Markov identity underlying these equations before
+assuming differentiability.  On a round annulus choose an arbitrarily
+small crosscut around the active switch, separated from all other
+switches and from the other boundary component.  Approximate the smooth
+bank by chord-free graph boundaries and complete the neighborhood to a
+Jordan patch, putting its second color switch remotely.  Up to the
+pre-query stop at the crosscut, the exploration in this patch is exactly
+the exploration in the annulus: no color beyond the crosscut is read.
+The stopped-exploration results of \cite[Lemmas~3.6--3.7]{V} give convergence
+of the path together with its remaining domains.  Crosscuts may be
+chosen as the semicircle crosscuts in the stopped-target construction;
+the probability of an endpoint stop vanishes when endpoint
+neighborhoods shrink.
+
+The unaltered part, obtained by cutting off the fixed crosscut
+neighborhood, is a Jordan annular back with unchanged boundary
+traversal.  For the remaining face containing this back, the stopped-target-face
+and remaining-domain statements of \cite[Lemmas~3.6--3.7]{V}
+give finitely many Jordan side pieces attached to the back along
+open subintervals of that crosscut.  Components already disconnected
+from this face are discarded; no assertion of finiteness is being
+made about all components swallowed by the exploration.
+We check that its limit is still doubly connected.  In the closed
+crosscut neighborhood, remove the side-lobe interiors and their open
+attaching intervals.  The remaining closed set is connected to the
+original outer complement.  Indeed join any of its points to the old
+outer bank by a path in that neighborhood whose interior avoids the
+crosscut.  If this path enters a lobe, replace its portion from first
+entry to last exit by the corresponding subarc of the lobe boundary
+off its open attachment.  The entry and exit lie on this retained
+bank, not on the crosscut.  This bank meets the crosscut only at its
+own endpoints and enters no other lobe interior, since the Jordan
+interiors are disjoint.  Thus each replacement avoids all lobe
+interiors and open attachments; finitely many replacements give a
+path in the complement.  The unchanged inner hole is separated from
+this connected outer complement by the persisting annular back.
+These are precisely the two nondegenerate complementary components.
+Lemma~\ref{an:back-lobes} gives convergence of annular uniformizations
+on their closures, with the prime-end traversals and marks in order.
+The hole and a separating open bulk persist throughout this local
+exploration, so the moduli remain nondegenerate.  Thus the residual
+annuli satisfy the hypotheses of Proposition~\ref{an:limit}.  The stopping
+edge becomes the replacement switch, the other marks are unchanged,
+and the colors in the remaining face are fresh.  Transmission topology
+and bounded convergence therefore give the local mean-value identity
+for $T^\pm$ at each of these hull stops.
+
+This identity first implies the active equations in the viscosity
+sense.  Work modulo simultaneous rotation and let a smooth test
+function touch $T$ from above or below.  On a sufficiently small
+crosscut stop the modulus and image marks remain in the touching
+neighborhood.  Indeed the hull capacity tends to zero, and annular
+maps normalized at a remote mark tend to the identity on the other
+compact bank intervals, by reflection.  The image of the tip is
+squeezed between neighboring unaffected bank points.  Applying It\^o's
+formula to the touching function contradicts a strict inequality with
+the wrong sign in the active equation, because the stop takes positive
+time.  The same touching function can be used for all eight active
+equations, so their average also holds in viscosity form.  On every
+compact subset of the open chamber the averaged equation is a smooth,
+uniformly parabolic linear equation, also after the rotation quotient.
+Interior parabolic regularity then gives smoothness and the classical
+equation.  Simultaneous rotation invariance removes the quotient.
+\end{proof}
+
+\subsection{The ground-state transform}
+
+Define positive functions on the open chamber by
+\begin{equation}\label{sp:ground-state}
+ J(w)=\prod_{1\le i<j\le4}\sin\frac{w_j-w_i}{2},\qquad
+ h(x,y)=J(x)^{1/3}J(y)^{1/3},\qquad \lambda=\frac54.
+\end{equation}
+The product $J$ is unchanged by a cyclic relabeling.  For one tuple put
+$s_i=\sum_{j\ne i}\cot((w_i-w_j)/2)$.  The elementary three-angle
+cotangent identity gives
+\[
+ \sum_i s_i^2=\sum_{i\ne j}\csc^2\frac{w_i-w_j}{2}-20,
+ \qquad \partial_{w_i}\log J^{1/3}=\frac{s_i}{6}.
+\]
+Substitution in $L$ shows that each tuple contributes $-5/8$ to
+$h^{-1}Lh$.  Hence $Lh=-\lambda h$.
+
+This ground-state calculation is the cotangent-diffusion form of the
+SLE--Calogero--Sutherland structure studied in \cite{CardyCS}; we give
+the estimates needed here directly.  The transformed generator and
+its invariant density are
+\begin{equation}\label{sp:doob-generator}
+ D=\frac38\bigl(\Delta+\nabla\log m\cdot\nabla\bigr),
+ \qquad m(x,y)=J(x)^{4/3}J(y)^{4/3}.
+\end{equation}
+Let $\mu$ be the probability measure proportional to $m$ times Lebesgue
+angle measure.  For $T=T^+,T^-$, or a linear combination of them, set
+$W(p)=e^{\lambda p}T(p)/h$.  Equation~\eqref{sp:pde} becomes
+\begin{equation}\label{sp:transformed-pde}
+ \partial_pW=DW+B_p\cdot\nabla W+V_pW,
+ \qquad V_p=B_p\cdot\nabla\log h.
+\end{equation}
+The apparent singularities of $V_p$ at collisions are removable.
+Every cotangent is multiplied by a difference of two components of
+$B_p$, which vanishes on the corresponding collision face.
+Periodicity handles the face $w_4=w_1+2\pi$.  Thus $V_p$ has smooth
+bounds $O(e^{-p})$ up to the chamber boundary, as does $B_p$.
+
+\begin{lemma}[Smoothing on the chamber]\label{sp:smoothing}
+Fix $p_0>0$.  For either transmission function, or its sum or difference,
+the transformed solution satisfies
+\begin{equation}\label{sp:uniform-smoothing}
+ \|W(p)\|_\infty+\operatorname{Lip}(W(p))\le C_{p_0}
+ \qquad(p\ge p_0+2).
+\end{equation}
+The semigroup $P_t$ of $D$ is symmetric on $L^2(\mu)$ and, for every
+$q>1$, maps $L^q(\mu)$ to bounded functions at every fixed positive
+time.  At unit time the evolution in \eqref{sp:transformed-pde} maps
+bounded functions to Lipschitz functions with a constant uniform in
+the starting modulus $p\ge p_0$.
+\end{lemma}
+
+\begin{proof}
+We first justify using the diffusion all the way to the chamber
+boundary.  The drift in one tuple is
+$\frac14(\sum_{j\ne i}\cot((w_i-w_j)/2))_{i=1}^4$.
+On the lifted convex chamber, use $-\log m$, shifted to be nonnegative,
+as a Lyapunov function.  If $\alpha=4/3$ is the density exponent, the
+cotangent identity shows that its generator has a singular contribution
+proportional to $(\alpha-\alpha^2)\sum_{i<j}\csc^2((w_i-w_j)/2)$.
+This coefficient is negative, so the generator is bounded above.
+More explicitly, the one-tuple contribution is
+\[
+ D(-\log J^{4/3})=
+ \frac{10}{3}-\frac1{24}\sum_{i\ne j}
+                           \csc^2\frac{w_i-w_j}{2}.
+\]
+Stopping on inner exhaustions proves that no collision occurs in
+finite time.  Interior existence and pathwise uniqueness therefore
+give a global diffusion.  A bounded added drift, such as the
+time-dependent $B_p$, preserves noncollision by Girsanov's theorem on
+each bounded time interval.  The divergence form of $D$ gives symmetry
+and invariance of $\mu$, first on interior exhaustions and then by
+noncollision.
+
+The Feynman--Kac representation for
+\eqref{sp:transformed-pde} uses $D+B\cdot\nabla$ and potential $V$,
+with coefficients read at decreasing moduli from the evaluation
+modulus down to the initial modulus.  Its initial data at $p_0$ obey
+$|W(p_0)|\le C/h$.  To justify this representation despite the
+unbounded transformed data, first apply the killed representation to
+$T$ on inner chamber exhaustions.  Lemma~\ref{an:collision} says that
+the lateral values of $T$ tend uniformly to zero on each compact
+modulus interval.  Their contribution therefore disappears.
+The $h$-transform on the exhaustion gives the stated representation;
+pass to the limit using noncollision.  It suffices to do this for the
+nonnegative $T^\pm$ and then use linearity.
+
+Here are the uniform smoothing estimates.  The negative logarithm of
+$m$ is convex on the lifted chamber, so the drift of $D$ is monotone
+dissipative.  Couple two diffusions by the same Brownian motion and
+add to the second a drift toward the first of magnitude their initial
+distance divided by the allotted time, until they meet.  Dissipativity
+forces meeting by that time.  Girsanov's theorem removes the extra
+drift with all fixed moments bounded in terms of the time and the
+initial distance.  Localization inside the chamber justifies the
+coupling, and the preceding change-of-measure argument excludes
+collisions for the controlled diffusion.  H\"older's inequality gives
+the dimension-free form
+\[
+ |P_tF(z)|^q\le
+ C(q,t,\operatorname{dist}(z,z'))P_t(|F|^q)(z').
+\]
+One can always choose lifts of $z,z'$ at bounded distance.
+Integration in $z'$ against the invariant probability $\mu$ proves
+the asserted $L^q$-to-supremum estimate.
+
+A bounded drift perturbation and a bounded potential on a unit time
+interval preserve initial smoothing from $L^4(\mu)$.
+Indeed Cauchy--Schwarz applied to the Girsanov density bounds the
+perturbed expectation of $|f|$ by $C(P_1|f|^2)^{1/2}$;
+the unperturbed $L^2$-to-supremum bound for $|f|^2$ then gives
+$C\|f\|_{L^4(\mu)}$.  There is exactly this integrability here:
+$h^{-1}\in L^4(\mu)$, since $h^{-4}m=1$.
+After the first unit interval $W$ is bounded.  Its bound persists
+for all subsequent times because
+$\int_{p_0}^\infty\|V_p\|_\infty\,dp<\infty$.
+
+The same coupling proves a Lipschitz estimate for bounded data on a
+unit interval.  In the presence of $B$, a control of size at most a
+fixed constant times the initial distance forces meeting; before
+meeting the distance stays bounded by the same multiple of the
+initial distance.  The second moment estimate for the Girsanov
+exponential bounds its total-variation cost by that distance.
+The two Feynman--Kac weights differ by at most the same order, since
+$V$ has a uniform Lipschitz bound.  Thus the unit-time Lipschitz norm
+is bounded by a constant times the initial supremum norm.  Apply this
+on the last unit interval to obtain
+\eqref{sp:uniform-smoothing}.
+\end{proof}
+
+\subsection{The two symmetry gaps}
+
+The role of rotation averaging can now be seen explicitly.  For a
+tuple with real ordered lift $w$, write
+\[
+ \bar w=\frac14\sum_iw_i,\qquad
+ r=w-\bar w\mathbf1,\qquad \theta=\bar w\pmod{2\pi}.
+\]
+The relative coordinate lies in the bounded convex alcove
+\[
+ \mathcal A=\{r\in\R^4:\textstyle\sum_ir_i=0,
+                         \ r_1<r_2<r_3<r_4<r_1+2\pi\}.
+\]
+This identifies $\mathcal C$ with a product of a circle and
+$\mathcal A$.  Its invariant law is a product as well, since $J$
+depends only on the relative angles.  The center part of $D$ is
+$(3/32)\partial_\theta^2$ for each tuple.
+
+\begin{lemma}[Spectral gaps]\label{sp:gaps}
+The relative-angle diffusion for one tuple has Poincar\'e gap at least
+$1/2$.  Consider the subspace of $L^2(\mu)$ consisting of functions
+invariant under simultaneous rotation of the two tuples and changing
+sign under one cyclic relabeling of either tuple.  On this subspace
+$-D$ has gap at least $3/4$.  On its subspace with both center
+frequencies zero the gap is at least $1$.
+\end{lemma}
+
+\begin{proof}
+For a mean-zero vector $v\in\R^4$ the Hessian of the negative
+logarithm of the one-tuple density satisfies
+\[
+ \nabla^2(-\log J^{4/3})[v,v]
+ =\frac13\sum_{i<j}\csc^2\frac{r_i-r_j}{2}(v_i-v_j)^2
+ \ge\frac43|v|^2.
+\]
+The Brascamp--Lieb inequality \cite{BL,KM}, applied on the convex alcove,
+therefore gives
+\[
+ \operatorname{Var}(F)\le\frac34\int|\nabla_rF|^2\,d\mu_r.
+\]
+The Dirichlet form has factor $3/8$, yielding the gap $1/2$.
+One may first use inner convex exhaustions of the alcove and then
+pass to the limit.
+
+For the two tuples, simultaneous rotation invariance allows only
+center frequencies $(n,-n)$, whose energy cost is $3n^2/16$.
+A cyclic relabeling sends $\theta$ to $\theta+\pi/2$ and acts on
+$\mathcal A$ by a measure-preserving affine isometry.  If a function
+changes sign under this relabeling, its relative-angle mean must
+vanish unless $e^{in\pi/2}=-1$, that is, unless $n\equiv2\pmod4$.
+The same assertion holds separately for the other tuple.  Therefore
+frequencies not congruent to $2$ modulo $4$ pay at least $1/2+1/2=1$
+in relative energy.  The remaining frequencies have $|n|\ge2$ and
+pay at least $3/4$ in center energy.  At $(n,-n)=(0,0)$ the two
+relative means vanish, giving the gap $1$.
+
+These inequalities give $L^2$ semigroup decay through the symmetric
+Dirichlet form
+\[
+ \mathcal E(F,F)=\frac38\int|\nabla F|^2\,d\mu.
+\]
+There is no hidden absorbing boundary: the constant function is in
+the form domain.  Cutoffs approaching the collision faces have
+vanishing energy because the density vanishes with power $4/3>1$.
+\end{proof}
+
+Let $W_-=e^{\lambda p}(T^+-T^-)/h$, and define
+\[
+ (\Pi F)(x,y)=\frac1{2\pi}\int_0^{2\pi}
+                           F(x,y+\phi\mathbf1)\,d\phi
+\]
+on simultaneous-rotation-invariant functions.  Equivalently, $\Pi$
+is projection to center frequencies $(0,0)$.  A one-step cyclic
+relabeling switches $T^+$ and $T^-$, while $h$ is unchanged.
+Thus $W_-$ has exactly the symmetries of Lemma~\ref{sp:gaps}.
+
+\begin{proposition}[Uniform transmission asymptotics]
+\label{sp:transmission-asymptotics}
+There are constants $K\ge0$, $\sigma>0$, and $C<\infty$ such that,
+uniformly over the open angle chamber and all sufficiently large $p$,
+\begin{align}
+ \|W_-(p)\|_\infty&\le Ce^{-3p/4},
+ &\|\Pi W_-(p)\|_\infty&\le C(1+p)e^{-p},
+ \label{sp:odd-decay}\\
+ T^\pm(p,x,y)&=e^{-\lambda p}
+                   \bigl(Kh(x,y)+O(e^{-\sigma p})\bigr).
+ \label{sp:rank-one-round}
+\end{align}
+Moreover,
+\begin{equation}\label{sp:odd-regularity}
+ \operatorname{Lip}(W_-(p))\le Ce^{-3p/4},\qquad
+ \|W_-(p+\ell)-W_-(p)\|_\infty
+        \le Ce^{-3p/4}\sqrt\ell\quad(0\le\ell\le1).
+\end{equation}
+The function $h$ is uniformly $1/3$-H\"older on the closed chamber.
+\end{proposition}
+
+\begin{proof}
+For any transformed solution put
+$F_p=B_p\cdot\nabla W(p)+V_pW(p)$.
+Lemma~\ref{sp:smoothing} implies $\|F_p\|_\infty\le Ce^{-p}$
+once $p$ is sufficiently large.  Variation of constants gives
+\begin{equation}\label{sp:duhamel}
+ W(p)=P_{p-p_1}W(p_1)+\int_{p_1}^pP_{p-s}F_s\,ds.
+\end{equation}
+This identity also follows directly by It\^o's formula with $D$,
+first stopping on inner exhaustions: on the interval in question
+the solution and the forcing are bounded, and collisions are not hit.
+
+For $W_-$ the forcing respects simultaneous rotation invariance and
+the two cyclic sign changes.  Apply the $3/4$ gap to
+\eqref{sp:duhamel}, and then apply the unit-time $L^2$-to-supremum
+smoothing to all semigroup times at least one.  For shorter times
+use the supremum bound on $F_s$.  The convolution with $e^{-s}$ is
+$O(e^{-3p/4})$.  After projecting by $\Pi$, which commutes with $D$,
+the gap is $1$ and the convolution is $O((1+p)e^{-p})$.
+This proves \eqref{sp:odd-decay}.  It does not require $B_p$ to
+commute with independent rotations: the projected forcing is
+estimated after projection.
+
+For either $W^+$ or $W^-$ separately, $D$ has a positive gap off
+constants.  Indeed the center gap is $3/32$ and the relative gap is
+at least $1/2$.  The same argument shows exponential convergence of
+$W-\int W\,d\mu$ to zero.  Also
+$\partial_p\int W\,d\mu=\int F_p\,d\mu=O(e^{-p})$, so its mean
+converges exponentially.  The first estimate of
+\eqref{sp:odd-decay} makes the two limiting constants equal; call
+their common value $K$.  Multiplication by $e^{-\lambda p}h$
+proves \eqref{sp:rank-one-round}.
+
+The unit-interval Lipschitz estimate in Lemma~\ref{sp:smoothing},
+applied to the homogeneous perturbed evolution for $W_-$, improves
+its Lipschitz bound to $Ce^{-3p/4}$.  To obtain time regularity,
+the same diffusion has expected angle displacement at most
+$C\sqrt\ell$ in time $\ell\le1$, uniformly in its initial point.
+Here is a bound that remains valid near a collision.  Apply It\^o's
+formula to squared distance from the initial lift.  Each pair in
+the scalar product with the singular drift contributes a constant
+multiple of
+\[
+ (v-v_0)\cot(v/2),\qquad v,v_0\in(0,2\pi),
+\]
+which is bounded above uniformly: use $v-v_0\le v$ for $v\le\pi$
+and $v-v_0\ge v-2\pi$ for $v\ge\pi$.
+The bounded additional drift and the Brownian quadratic variation
+then give $\E|Z_\ell-Z_0|^2\le C\ell$, by localization and
+Gronwall's inequality.  The Feynman--Kac representation over time
+$\ell$, the spatial Lipschitz bound, and the bounded potential give
+the time estimate in \eqref{sp:odd-regularity}.
+Finally $J$ is a bounded Lipschitz nonnegative function on the closed
+chamber, and taking a cube root is $1/3$-H\"older.  The assertion
+for $h$ follows by taking products.
+\end{proof}
diff --git a/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/build/sections/04-shape.tex b/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/build/sections/04-shape.tex
new file mode 100644
index 0000000000000000000000000000000000000000..92fd5f539c23fa8c40b7351bbe78eb22b4c0ae8f
--- /dev/null
+++ b/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/build/sections/04-shape.tex
@@ -0,0 +1,230 @@
+\section{Separating the conformal data of the two ends}
+\label{sh:section}
+\label{sec:shape}
+
+The estimates of Section~\ref{sp:section} are expressed in the
+uniformizing coordinates of the entire annulus.  To use them in a
+transfer operator, their leading term must instead depend separately
+on the inner and outer explored regions.  We prove the requisite
+comparison uniformly over their shapes.  Only separation of the
+two ends is needed; no boundary smoothness bound is imposed.
+
+Let $K_i$ be a closed Jordan region containing $0$ in its interior,
+and let $U_o$ be a bounded Jordan domain containing $K_i$.
+The annulus under consideration is $U_o\setminus K_i$.
+Fix constants $c_*,C_*>0$ and assume that
+\begin{equation}\label{sh:shape-hypotheses}
+ K_i\subset\{|z|\le a\},\quad
+ \operatorname{diam}K_i\ge c_*a,\qquad
+ \{|z|<b\}\subset U_o,\quad
+ \min_{z\in\partial U_o}|z|\le C_*b.
+\end{equation}
+We take $b/a$ larger than a constant depending only on $c_*,C_*$.
+Each boundary carries four distinct marked prime ends and alternating
+color prescriptions.
+
+Normalize the two ends separately.  Let
+\[
+ w_i:\widehat\C\setminus K_i\longrightarrow\{|w|>1\},
+ \qquad w_i(z)=z/l_i+O(1)\quad(z\to\infty),
+\]
+with $l_i>0$, and let
+\[
+ w_o:U_o\longrightarrow\{|w|<1\},\qquad
+ w_o(0)=0,\quad w_o'(0)=1/l_o>0.
+\]
+Thus $l_i$ is the logarithmic capacity of $K_i$ and $l_o$ is the
+conformal radius of $U_o$ at $0$.  Write $y^0,x^0$ for the angles of
+the inner and outer marks in these maps.  Capacity-diameter bounds
+for continua and the Koebe estimates imply
+\begin{equation}\label{sh:end-scales}
+ l_i\asymp a,\qquad l_o\asymp b,
+\end{equation}
+with constants depending only on $c_*,C_*$.
+
+\begin{lemma}[Uniform end-map comparison]\label{sh:comparison}
+Let $(p,x,y)$ be the modulus and marked angles in a uniformization
+of $U_o\setminus K_i$, with angular direction the same on both
+boundary components.  Put $p'=\log(l_o/l_i)$.  The common rotational
+normalization of the annulus map can be chosen so that
+\begin{equation}\label{sh:comparison-estimate}
+ |p-p'|\le C\frac ab,\qquad
+ |x-x^0|+|y-y^0|
+       \le C\bigl(1+\log(b/a)\bigr)\frac ab.
+\end{equation}
+Distances in the second inequality use compatible cyclic lifts.
+The constant is uniform over all Jordan boundaries satisfying
+\eqref{sh:shape-hypotheses}, including all placements of the marks.
+\end{lemma}
+
+\begin{proof}
+Define the single-valued analytic corrections
+\[
+ g_i(z)=\log\frac{l_iw_i(z)}z,
+ \qquad g_o(z)=\log\frac{l_ow_o(z)}z,
+\]
+with branches vanishing at infinity and at zero, respectively.
+The quotient defining $g_o$ has no zeros in $U_o$ and the apparent
+singularity at zero is removable; the exterior quotient has zero
+winding and its logarithm is normalized at infinity.
+Univalent distortion and \eqref{sh:end-scales} give
+\begin{equation}\label{sh:correction-bounds}
+ |g_o(z)|\le Ca/b\quad (|z|\le C_1a),\qquad
+ |g_i(z)|\le Ca/b\quad (|z|\ge c_1b),
+\end{equation}
+for every fixed $C_1<\infty$ and $c_1>0$, once $a/b$ is small
+enough.  The relevant sets in these inequalities are contained in
+the respective map domains.  The same bounds hold in fixed round
+collars in the single-end coordinates.  If $\psi_i=w_i^{-1}$,
+the maximum principle applied to $\psi_i(w)/w$ gives
+$|\psi_i(w)|\le a|w|$ for $|w|>1$.
+For $\psi_o=w_o^{-1}$, Koebe's distortion bound gives
+$|\psi_o(w)|\ge l_o|w|/(1+|w|)^2$.
+Thus the inverse exterior map sends $1\le|w_i|\le2$ into
+$|z|\le2a$, and the inverse interior map sends, for example,
+$1/2\le|w_o|<1$ into $|z|\ge cb$.
+These observations are the reason arbitrary roughness of the physical
+boundaries does not change the constants.
+
+Consider the multivalued analytic function
+\[
+ H(z)=\log(z/l_i)+g_i(z)+g_o(z).
+\]
+It has period $2\pi i$, and its real part is within $Ca/b$ of $0$
+on the inner boundary and of $p'$ on the outer boundary.  Let
+$A(z)$ be an annulus logarithm, normalized by
+$0<\operatorname{Re}A<p$.  Its period is also $2\pi i$.
+The difference $H-A$ is therefore single-valued.  In the cylinder
+coordinate $A$, the circular average of the real part of any
+single-valued analytic function is constant across the cylinder.
+At the two boundary circles the averages of
+$\operatorname{Re}(H-A)$ are, respectively, $O(a/b)$ and
+$p'-p+O(a/b)$.  Equality of the averages proves
+$|p-p'|\le Ca/b$.  The maximum principle now yields
+\begin{equation}\label{sh:real-error}
+ |\operatorname{Re}(A-H)|\le Ca/b
+ \quad\hbox{throughout }U_o\setminus K_i.
+\end{equation}
+
+It remains to compare the angular parts; a bound on real parts
+alone would not control them on arbitrary boundaries.  Use instead
+the round collars in the single-end coordinates.  The relevant
+single-valued functions satisfy
+\[
+ A-\log w_i=(A-H)+g_o,\qquad
+ A-p-\log w_o=(A-H)+(p'-p)+g_i.
+\]
+Near the inner
+end, $A-\log w_i$ has real part bounded by $Ca/b$, by
+\eqref{sh:correction-bounds} and \eqref{sh:real-error}, and its
+real part is exactly zero on $|w_i|=1$.  Schwarz reflection across
+this circle, followed by interior gradient bounds on a smaller fixed
+collar, bounds the variation of its imaginary part along the entire
+end circle by $Ca/b$.  Near the outer end apply the same argument
+to $A-p-\log w_o$.  Its real part is zero on $|w_o|=1$ and is
+bounded by $Ca/b$ in a fixed inner collar.  This gives the analogous
+angular bound, initially with possibly different additive constants
+at the two ends.
+
+To match these constants, choose $C_1$ large and $c_1$ small, both
+fixed.  The round annulus $C_1a<|z|<c_1b$ is contained in the
+physical annulus.  Along a radial segment across it, use
+$O(1+\log(b/a))$ overlapping disks of fixed relative size.
+The gradient estimate for the analytic function $A-H$, whose real
+part obeys \eqref{sh:real-error}, bounds the variation of its
+imaginary part on each disk by $Ca/b$.
+At either end, distortion in the corresponding single-end chart
+connects this radial segment to the fixed collar with a bounded
+additional chart distance.  The same interior estimate and the
+collar reflection estimate apply there.
+Thus the two additive constants differ by at most
+$C(1+\log(b/a))a/b$.  Choose the common rotation to fix the inner
+constant.  The comparison of all marked angles, uniformly even
+when marks are very close, follows.  This proves
+\eqref{sh:comparison-estimate}.
+\end{proof}
+
+\begin{proposition}[Separated-end transmission]\label{sh:separated-ends}
+There are constants $\sigma_1,\sigma_2>0$ and $C<\infty$,
+depending only on the fixed constants in
+\eqref{sh:shape-hypotheses}, such that the transmission probabilities
+of every such marked annulus satisfy
+\begin{equation}\label{sh:rank-one}
+ \left|T^\pm-
+ K\,l_i^\lambda J(y^0)^{1/3}\,
+       l_o^{-\lambda}J(x^0)^{1/3}\right|
+ \le C(a/b)^{\lambda+\sigma_1}.
+\end{equation}
+Here $K$ is the common constant in
+Proposition~\ref{sp:transmission-asymptotics}.  If one rotates the
+inner end, with its marks and prescribed colors, through angle $\phi$
+about $0$ while keeping the outer end fixed, then
+\begin{equation}\label{sh:rotation-average}
+ \left|\frac1{2\pi}\int_0^{2\pi}
+                 \bigl(T^+_\phi-T^-_\phi\bigr)\,d\phi\right|
+       \le C(a/b)^{2+\sigma_2}.
+\end{equation}
+The same conclusion holds if the outer end is rotated instead.
+Both estimates remain uniform for Jordan approximants to the
+prime-end boundaries of Section~\ref{sec:annuli}, provided they
+satisfy \eqref{sh:shape-hypotheses} with the same fixed constants.
+\end{proposition}
+
+\begin{proof}
+Write $\rho=a/b$ and $q=\log(1/\rho)$.  By
+\eqref{sh:end-scales}, $p'=q+O(1)$.  Set
+$\eta=C(1+q)\rho$, the angular error in
+Lemma~\ref{sh:comparison}.  The $1/3$-H\"older bound for $h$
+and the first inequality of that lemma give
+\[
+ |e^{-\lambda p}h(x,y)-e^{-\lambda p'}h(x^0,y^0)|
+       \le C\rho^\lambda(\rho+\eta^{1/3}).
+\]
+Combining this with \eqref{sp:rank-one-round} proves
+\eqref{sh:rank-one}; for example any
+$0<\sigma_1<\min\{\sigma,1/3\}$ is sufficient after changing
+the constant.  Since
+$e^{-\lambda p'}=(l_i/l_o)^\lambda$, the leading term has exactly
+the displayed separated form.
+
+For the signed estimate put $U=T^+-T^-=e^{-\lambda p}hW_-$.
+Equations~\eqref{sp:odd-decay} and
+\eqref{sp:odd-regularity}, together with the H\"older bound for
+$h$, show that replacing the actual annular data by the two
+single-end data costs at most
+\begin{equation}\label{sh:signed-shape-error}
+ C\rho^2\bigl(\rho^{1/2}+\eta+\eta^{1/3}\bigr).
+\end{equation}
+This bound is uniform over the rotation of either physical end.
+In particular it is a power improvement on $\rho^2$.
+
+The capacity $l_i$ is unchanged by rotation of the inner end, and
+its normalized map is
+$w_{i,\phi}(z)=e^{i\phi}w_i(e^{-i\phi}z)$.
+Consequently its marked single-end angles are exactly
+$y^0+\phi\mathbf1$.  The factor $h$ is unchanged by rotating
+either tuple separately.  Thus the rotation average in the unperturbed
+data is
+\[
+ e^{-\lambda p'}h(x^0,y^0)
+                       (\Pi W_-)(p',x^0,y^0),
+\]
+whose absolute value is at most
+$C(1+q)\rho^{\lambda+1}=C(1+q)\rho^{9/4}$ by
+\eqref{sp:odd-decay}.  Add the uniform error
+\eqref{sh:signed-shape-error}.
+Any $0<\sigma_2<1/4$ is then admissible, after increasing the
+constant and restricting to sufficiently small $\rho$.
+Rotating the outer end gives the same projection, by simultaneous
+rotation invariance.
+
+Every estimate was obtained with constants depending only on the
+separation and scale conditions, not on boundary smoothness or mark
+separation.  It can therefore be applied directly to the continuum
+function $T$ evaluated on the conformal data of the Jordan cycles
+approximating an explored domain.  Under admissible convergence,
+Proposition~\ref{an:limit} separately identifies the limit of the
+discrete transmission probabilities with that continuum function; no
+reflection across a physically nonsimple limiting bank is being
+asserted.
+\end{proof}
diff --git a/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/build/sections/05-transfer.tex b/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/build/sections/05-transfer.tex
new file mode 100644
index 0000000000000000000000000000000000000000..9f0d8678c3af3c2a8c52d9181284ec503caac743
--- /dev/null
+++ b/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/build/sections/05-transfer.tex
@@ -0,0 +1,960 @@
+\section{Uniform transfer of signed connection states}
+\label{tr:section}
+\label{sec:transfer}
+
+The estimates of the preceding section concern annuli with four prescribed
+switches at each end.  A discrete crossing test, however, can present an
+arbitrarily complicated connection state to an annulus.  Moreover, the
+outside of that annulus need not have a scaling limit.  We now prove
+operator estimates that are uniform over these states and outside data.
+The important distinction is between the two annular ends, where arbitrary
+data are allowed, and an independent region between them, where the
+scaling-limit arguments take place.
+
+Throughout this section the Poisson intensity is one, the center of all
+circles is fixed, and $\lambda=5/4$.  We use the localization and mesh-first
+arm estimates recalled above from \cite[Sections~2--3]{V}.  The constants
+implicit in choices of collars and buffers are fixed once and for all.
+
+\subsection{Cuts, connection states, and transfer operators}
+
+Fix a sufficiently small constant $\eta>0$.  A \emph{seam at radius $r$}
+consists of the marked Poisson process in the collar
+\[
+             (1-\eta)r<|z|<(1+\eta)r,
+\]
+together with auxiliary randomness used to select a graph cycle.  A
+local occupancy test, denoted $G_r$, ensures that the graph and the vertex
+stars through a narrower collar are determined by these variables and
+that their diameters are smaller than a fixed small fraction of $\eta r$.
+We choose the test and the narrower collar with buffers at both ends, so
+that
+\begin{equation}
+                    \Pp(G_r^c)\le C e^{-cr^2}.
+\label{tr:screen}
+\end{equation}
+On $G_r$ choose a simple site cycle $\Gamma_r$ surrounding the center,
+inside the narrower collar.  The choice uses the geometry but not the
+colors.  The test and the choice are rotation equivariant; an independent
+uniform angular axis may be included among the seam variables to break
+ties.
+
+For completeness, such a screen can be obtained by requiring that every
+point in a closed intermediate collar be within a sufficiently small
+distance, proportional to $r$, of a nucleus in the larger collar.  This
+covering-distance condition is rotation invariant and has the
+exponential failure bound in \eqref{tr:screen}.  Following a circle in the
+site triangulation and deleting loops gives the required cycle.  Buffers
+and the same covering condition determine all its vertex stars.  A
+nucleus beyond either extremity of the collar cannot affect the graph
+on the other side: its cell would have to pass through the screened
+intermediate collar, where all cells are small.  Consequently the graph
+on each side of $\Gamma_r$, including its attachments to the cycle, can
+be read from the seam and the Poisson variables on that side.  Conditional
+on the seam, the variables used on the two sides are independent.
+
+The colors on $\Gamma_r$ determine an even number of switches.  An inside
+connection state is a noncrossing pairing of those switches by interface
+strands in the disk on the inside.  The pairing, together with the
+monochromatic runs on the cycle, determines all boundary connectivity.
+Closed interface loops carry no information and are not stored.  We
+allow every formal noncrossing pairing as a state, whether or not it is
+realized by a particular inside graph.  Such pairings respect the
+alternating color convention: the endpoints of each pair have opposite
+parity in cyclic order.  Gluing a pairing to the annular strands is
+therefore defined without a realizability restriction.
+
+Write $\Sigma_r$ for the seam variables and $\mathcal S_r(\Sigma_r)$ for
+this finite state set on $G_r$.  A \emph{message} is a measurable field
+of signed measures $v_{\Sigma_r}$ on $\mathcal S_r(\Sigma_r)$ such that
+\[
+ v_{\Sigma_r}(\mathcal S_r(\Sigma_r))=0,
+ \qquad v_{\Sigma_r}=0\ \hbox{on }G_r^c.
+\]
+Its norm is
+\begin{equation}
+  \|v\|_r
+    =\operatorname*{ess\,sup}_{\Sigma_r}
+                       \|v_{\Sigma_r}\|_{\TV}.
+\label{tr:message-norm}
+\end{equation}
+Here total variation is the sum of the absolute point masses, so that a
+difference of probability measures has norm at most two.  Finite state
+sets can be numbered measurably; the bounded fields of zero-mass measures,
+modulo seam-law null sets, form a Banach space $\mathcal B_r$.
+
+If the collars at radii $r<s$ are disjoint, gluing through the intervening
+annulus defines a linear map
+\[
+                  M_{r,s}:\mathcal B_r\longrightarrow\mathcal B_s.
+\]
+More explicitly, conditional on $\Sigma_s$, integrate $\Sigma_r$ and
+the intervening marked process, impose both seam successes, and push
+$v_{\Sigma_r}$ through the annular strands.  Closed loops have weight
+one.  For every fixed realization, this push is a map between finite
+state spaces and contracts total variation.  Its mass remains zero.
+The success restrictions are indicators in this integral, not a
+conditioning followed by renormalization.  Only the output seam is
+conditioned on in the definition of its message field.
+Intermediate successful seams therefore give exactly the composition of
+these operators, with the corresponding success indicators imposed.
+This construction does not assert independence for unrestricted Voronoi
+events in neighboring regions; independence is used only for the
+underlying marked Poisson variables separated by the specified buffers.
+
+Rotation and global color reversal act on seam variables and states.  A
+message is \emph{isotropic} if it is fixed by the rotation action and
+\emph{even} if it is fixed by color reversal.  Both symmetries are exact
+for $M_{r,s}$, since the screens and cycle choices have those symmetries.
+
+\begin{proposition}[Fixed-ratio transfer]
+\label{tr:block}
+There are $\delta>0$ and $C<\infty$ with the following property.  For every
+sufficiently large integer $n$, set $L=2^n$.  There is a threshold $r_0(L)$
+such that, for $r\ge r_0(L)$, the transfer $M=M_{r,Lr}$ satisfies
+\begin{align}
+ \|M\|&\le C L^{-\lambda}, \label{tr:block-size}\\
+ M&=M_1+E,\qquad \operatorname{rank}M_1\le1,
+       \qquad \|E\|\le C L^{-\lambda-\delta},
+       \label{tr:block-rank}\\
+ \|M|_{\mathcal B_r^{\mathrm{even,iso}}}\|
+       &\le C L^{-2-\delta}.
+       \label{tr:block-cancellation}
+\end{align}
+The constants $C,\delta$ do not depend on $L$ or $r$.  Every norm in the
+conclusion includes the essential supremum over successful output
+seams, including their colors.  The rank-one operator may depend on
+$r$ and $L$.
+\end{proposition}
+
+The reduction has a simple mechanism.  Six surviving interfaces already
+cost a power strictly greater than two, while four interfaces leave just
+one signed connection coordinate.  We isolate an independent region in
+which to obtain four-tip boundaries, verify their convergence, and then
+apply the separated-end estimates to that coordinate.  Summing over the
+first four-tip scales gives both the rank-one approximation and the
+color--rotation cancellation.
+
+\subsection{An independent region for the limiting argument}
+
+Fix $L$ for the moment and rescale by $r$.  Thus the mesh tends to zero
+as $r\to\infty$, whereas all radii used in the present argument lie in a
+fixed finite range.  Choose auxiliary starting cycles near radii $8r$
+and $Lr/8$.  Their geometry is determined in small collars separated
+from both original seams.  From either starting cycle we shall explore
+towards the middle of the annulus, stopping at one of finitely many
+\emph{barriers}: color-independent site cycles approximating specified
+concentric circles in curve order.  Their colors remain unread until
+the exploration proceeds past them.  A \emph{stopping circle} means the
+limiting circle of the current barrier.  All cycles may be
+made chord free on the side approached by the exploration.
+
+We impose auxiliary occupancy tests in this intervening region.  Unlike
+the actual seam tests, these may require a relative mesh tending to
+zero: use covering distances that tend slowly to zero in rescaled
+coordinates.  For the fixed finite collection of collars under
+consideration, their total failure probability is $o(1)$.  Their supports
+are separated from the input and output seams.  On success, geometry
+around the two starting cycles and throughout the region between them
+agrees with the geometry of a full-plane Poisson sample, obtained by
+adjoining independent variables outside that region.  In particular,
+all these $o(1)$ failure bounds hold uniformly conditional on either or
+both original seam variables.
+
+There is also a local version of this construction.  An end experiment
+stopped at a barrier uses only variables from a buffer just before its
+starting cycle through a small fixed-factor enlargement past that
+barrier, together with the variables between its starting cycle and its
+own seam.  Its localization tests stop at that enlargement.  If two
+barriers are separated by a sufficiently large fixed factor, the
+supports for the two end experiments are disjoint.  Extra tests across
+the middle can then be added when evaluating an actual annular
+transmission.  Deleting those extra tests from an integral with bounded
+weights changes it by $o(1)$, uniformly in the original seams.  It is
+only after this deletion that the two end integrals are factored.
+Figure~\ref{tr:figure} records the radial order and the gaps between
+the supports of the two end experiments.
+
+\begin{figure}[tb]
+\centering
+\begin{tikzpicture}[x=0.88cm,y=0.75cm,>=Latex,font=\small]
+  \fill[black!12] (0,-0.26) rectangle (0.42,0.26);
+  \fill[black!12] (11.58,-0.26) rectangle (12,0.26);
+  \fill[black!5] (1.55,-0.18) rectangle (4.05,0.18);
+  \fill[black!5] (7.95,-0.18) rectangle (10.45,0.18);
+  \draw[thick] (0,0)--(12,0);
+  \foreach \x in {0.2,2,3.7,8.3,10,11.8}
+    \draw (\x,-0.14)--(\x,0.14);
+  \node[align=center,below] at (0.2,-0.42) {input\\seam};
+  \node[align=center,below] at (2,-0.42) {inner start\\$8r$};
+  \node[align=center,below] at (3.7,-0.42) {barrier\\$a$};
+  \node[align=center,below] at (8.3,-0.42) {barrier\\$b$};
+  \node[align=center,below] at (10,-0.42) {outer start\\$Lr/8$};
+  \node[align=center,below] at (11.8,-0.42) {output\\seam};
+  \draw[->,thick] (2,0.65)--(3.7,0.65);
+  \draw[->,thick] (10,0.65)--(8.3,0.65);
+  \node[align=center] at (6,0.72) {unexplored annulus};
+  \draw[<->] (1.2,1.5)--(10.8,1.5);
+  \node[fill=white,inner sep=3pt] at (6,1.5) {independent Poisson region};
+\end{tikzpicture}
+\caption{The radial order of two end experiments with separated barriers,
+on a schematic logarithmic scale.  Explorations start inside the transfer
+block, independently of its two seams.  Each shaded end support extends
+a little beyond its last barrier; the gap between those supports permits
+their independent integration.  The original seam screens remain
+imposed throughout.}
+\label{tr:figure}
+\end{figure}
+
+The following elementary observation is the uniformity mechanism used
+below.  Its purpose is to avoid any assertion of a uniform scaling
+limit conditional on an atypical explored shape.
+
+\begin{lemma}[Domination before conditioning on end shapes]
+\label{tr:domination}
+Suppose a contribution to $M_{r,Lr}v$ is expressed, after exploring from
+the auxiliary starting cycles, as an integral of an inner scalar
+coefficient, a bounded middle test, and an outer signed state measure.
+If the middle test is replaced with absolute error $F\ge0$, where $F$
+depends only on the independent region just described, then the error
+in output norm is at most
+\begin{equation}
+              C\|v\|_r\,\E_{\mathrm{core}}F+o(1)\|v\|_r.
+\label{tr:domination-bound}
+\end{equation}
+The same bound applies to discarding an event there, by taking $F$ to
+be its indicator.  The expectation is under the unweighted full-plane
+comparison law; the bound is uniform over successful output seams.
+\end{lemma}
+
+\begin{proof}
+For a fixed realization, pushing $v_{\Sigma_r}$ to any intermediate
+state set contracts total variation.  When that set has two states, its
+zero-mass image has the form $\alpha(\delta_B-\delta_W)$, with
+$|\alpha|\le\|v\|_r$.  Pushing the two-state difference through the other
+end gives a signed measure $q$ with $\|q\|_{\TV}\le2$.  Hence the
+pointwise norm of the replacement error is bounded by
+$2\|v\|_rF$.  The same argument, summing absolute masses before a
+two-state reduction, handles discarded events.  No regularity of the
+dependence of $\alpha$ or $q$ on the seam data is needed.
+
+On the auxiliary localization successes, $F$ is a function of marked
+Poisson variables independent of both original seams and has the stated
+full-plane comparison law.  Integrate the pointwise bound conditional
+on the output seam.  Auxiliary failures contribute the final $o(1)$
+term, also uniformly, because their supports are off the original seams.
+The seam success indicators themselves are not deleted.
+\end{proof}
+
+\subsection{Exploration up to the first four-tip frontier}
+
+First push the input message from its original seam to the inner
+auxiliary starting cycle, using the marked variables in the intervening
+attachment region.  The subsequent exploration uses the colored
+starting cycle and the variables towards its barrier; it never consults
+an attached cap state.  Dependence on the input message stays in the
+pushed signed coefficients.  At the outer end the corresponding
+attachment operation pushes states from its auxiliary cycle towards
+the original output seam.
+
+At the inner auxiliary cycle, reveal its colors and trace every switch
+into the annulus.  A wall is followed until it pairs with another traced
+wall or until its next color query would read a vertex on the current
+barrier.  In the second case, stop before that query.  The stopped wall
+ends at a switch edge whose next triangle has its opposite vertex on
+the barrier.  No barrier colors have been read.  The analogous outer
+experiment is directed inwards.  Increasing the inner barrier radius
+by a factor two, or decreasing the outer barrier radius by a factor
+two, continues the same wall system.  Tracing all walls in a previously
+colored sample shows that the outcome does not depend on their tracing
+order.
+
+We call the unpaired stopped wall ends the \emph{surviving tips}.
+Their number is even and decreases as the barrier is advanced.  The
+first barrier with at most four surviving tips will be the end's
+reduction barrier.  A reduction to zero or two tips kills every
+zero-mass message.  A reduction to exactly four leaves precisely two
+cap states.  We next verify both this assertion and the geometric
+properties needed for the annular transmission limit.
+
+\begin{lemma}[Frontier and arm counts]
+\label{tr:frontier}
+For either end experiment, outside an event whose probability tends to
+zero as the permitted separation of tips tends to zero, mesh first,
+the tips at each of finitely many fixed-scale barriers have distinct
+limiting positions.  The boundary facing the unqueried barrier is a
+simple site cycle whose switches are exactly these tips.  In particular:
+\begin{enumerate}
+\item If there are zero or two tips, the induced map on zero-mass
+connection-state measures is zero.
+\item If there are at least six tips, six disjoint bulk arms cross the
+region from the starting cycle to the barrier, after fixed-factor
+crops at the two ends.
+\end{enumerate}
+All probabilistic assertions are made under the unweighted exploration
+law, before conditioning on a particular history.
+\end{lemma}
+
+\begin{proof}
+First consider two tips approaching the same point of the barrier.
+Use a graph disk patch with its base on the barrier and a half-annulus
+on the exploration side.  Extend each stopped wall through its last
+triangle to the base, only for this geometric argument; no barrier
+color is queried.  The two wall fingers give two proper traversals of
+the half-annulus, with fixed margins at its two semicircular ends.
+Following the sites on their banks gives three disjoint free arms in
+the sectors on the two exterior sides and between the fingers.  These
+sites are on the exploration side, not on the barrier.  Short spokes
+through successive wall triangles keep each bank in its sector.  The
+base through the tips prevents an exterior bank from escaping around
+a finger's end.  Cropping the last mesh steps gives the same count
+when two stopping triangles share a barrier vertex.
+
+This is the free-bank-shadow count for two traversals in the boundary
+argument of \cite[Section~3]{V}.  The local boundary three-arm exponent
+is strictly greater than one.  Covering the barrier by $O(s^{-1})$
+patches of radius $s$ therefore excludes colliding tips as $s\downarrow0$,
+mesh first.  The BK inequality applied to the free shadows also makes
+the number of fixed-scale traversals tight.
+
+For the frontier assertion, fill the starting side of the starting
+cycle with a disk, using the sphere for the outside experiment.
+Attach the queried triangles along the explored bands and add all
+edges between revealed vertices.  The resulting graph is
+two-connected: attaching a triangle along an existing edge preserves
+two-connectivity, as does adding an edge between existing vertices.
+Thus the boundary of its face containing the unqueried barrier is a
+simple cycle.  The barrier and its paths towards the unexplored middle
+remain in that face.  In particular each stopping triangle is incident
+with this same face on its unread side.
+
+There is a useful invariant: every revealed vertex has a revealed path
+of its own color to the starting cycle.  It holds initially, and a newly
+queried triangle vertex has the color of one endpoint of the incoming
+heterochromatic edge, extending such a path.  Thus an added
+heterochromatic edge between revealed vertices cannot belong to an
+unstarted wall.  Such a wall would either close in the annulus or return
+to the barrier without visiting the starting cycle; in either case it
+would separate one bank from that cycle, contradicting its revealed
+same-color path.  Every heterochromatic revealed-vertex edge therefore
+belongs to the traced wall system.
+
+To identify the colors along the frontier, extend the stopped walls
+to the barrier as above.  These proper arcs, together with the walls
+that paired back to the starting cycle, divide the explored collar
+into regions with their consistent bank colors.  Every revealed vertex
+is either a starting-cycle vertex or was queried in a wall triangle.
+Its color agrees with the relevant bank.  An added edge cannot cross
+one of these wall arcs, and the preceding invariant excludes any
+untraced switch there.  Previously crossed switch edges have triangles on
+both sides in the revealed bands.  The only switches exposed to the
+remaining face are consequently the surviving stopping edges.
+
+With zero or two switches there is just one noncrossing cap pairing.
+Every input state gives that same pairing, so a zero-mass difference
+vanishes.  With six or more surviving wall fingers, their bank shadows
+give six disjoint site arms in successive sectors between the starting
+cycle and the barrier.  Fixed-factor crops remove the two ends, where
+different shadows could otherwise meet.  The arbitrary-color six-arm
+bound is applicable, so no particular numbering or color order of the
+six arms is required.
+\end{proof}
+
+The next lemma verifies the non-Jordan boundary issue rather than
+presuming that scalar Cardy convergence applies to a random annulus.
+
+\begin{lemma}[Admissibility of four-tip frontiers]
+\label{tr:admissible}
+At a fixed barrier, restrict the unweighted exploration law to the event
+of exactly four surviving tips.  Along subsequences with good geometry
+as in \cite[Section~2]{V}, the four monochromatic frontier arcs are tight
+in curve order.  In every coupled subsequential limit, after constant
+pauses are erased, each arc is simple and meets the stopping circle
+only at its two prescribed endpoints.  The four endpoints are distinct.
+
+For two such experiments ending at well-separated barriers, their
+residual annulus satisfies the admissibility conditions of
+Proposition~\ref{an:limit}.  If $t_r$ is the discrete
+through-transmission probability conditional on the complete geometry
+and the two end histories, and $T_r$
+is the continuum transmission function evaluated on its actual
+conformal data, then, for fixed $L$ and the fixed finite list of barriers,
+\begin{equation}
+       \E_{\mathrm{core}}\!\left[
+          \boldsymbol 1_{\{\text{four tips at both ends}\}}
+                              |t_r-T_r|\right]=o(1).
+\label{tr:mean-limit}
+\end{equation}
+The expectation concerns unweighted histories, not a uniform conditional
+estimate for all possible histories.
+\end{lemma}
+
+\begin{proof}
+The BK and successive-excursion argument gives tightness of the four
+frontier arcs.  Indeed repeated annular traversals provide disjoint
+free arms; all sites used, including those on the auxiliary starting
+cycle, have their original fair colors.  This is applied first at
+fixed radii and then at successively smaller excursion scales.
+The contact arguments below are the face-boundary arguments in
+\cite[Section~3]{V}; we give the details relevant to the present
+frontier.
+
+Every frontier vertex off the starting cycle belongs to a revealed
+wall band.  On its revealed side it has an opposite-colored neighbor
+with a bank path back to the starting cycle.  Consequently a visit of
+a frontier arc to the stopping circle other than at its designated
+endpoint, with a positive excursion on both sides of the visit, gives
+three free arms in a half-patch: two along the arc and one along the
+opposite bank.  The same argument excludes an intermediate return to
+an endpoint.  The boundary three-arm exponent and a covering of the
+circle exclude all such visits.  A nonconstant interval contained in
+the circle is excluded by the one-arm bound at a countable dense set
+of circle points.  Constant pauses can be removed from the
+parametrization.  Distinctness of the four tips follows from
+Lemma~\ref{tr:frontier}.
+
+Consider next a self-return of one monochromatic frontier arc at a
+point away from the starting and stopping circles.  Separate the two
+visits by a fixed positive excursion scale.  In a smaller annulus
+around the proposed contact, the arc supplies four proper traversals.
+Its opposite bank paths lie on the revealed side of the frontier
+cycle.  They may all be joined through the starting cycle when making
+the planar separation argument.  This joining set need not be
+monochromatic at the starting cycle: the working annulus misses that
+cycle, and within the working annulus every part of the joining set
+is genuinely of the opposite color.
+
+At least two complementary sectors have a frontier traversal on their
+boundary whose revealed side faces the sector.  Each such sector
+contains an opposite-colored longitudinal crossing.  To verify this assertion,
+choose transverse cuts at intermediate scales with margins, leaving
+both ends of that frontier traversal beyond the cuts.  If the joined
+neighbor set did not cross the sector longitudinally, planar duality
+would provide a transverse crosscut avoiding it.  Start this crosscut
+from that frontier traversal and stop at its first subsequent hit of the
+frontier cycle.  In the disk on the revealed side, the two traversal ends
+and the two crosscut ends alternate along the frontier.  The crosscut
+therefore separates their neighbor spokes, contradicting the joined
+neighbor set, which lies on the revealed side and avoids the
+crosscut.
+
+The argument remains valid when the traversals backtrack.  Choose the
+transverse cuts along intermediate graph cycles, or apply Hex duality
+with such a cycle temporarily wired in the sector.  Two disjoint cuts
+bound a quadrilateral whose side arcs belong to the chosen traversals
+and whose interior avoids those whole proper traversals, not merely
+the selected side subarcs.  Thus the crosscut's first subsequent
+frontier hit lies outside the whole traversal from which it started,
+so the alternating boundary order used above persists under
+backtracking.  The crosscut is in the quadrilateral's interior, whereas the end spokes are beyond
+the cuts.  They thus avoid the crosscut even if other portions of the
+frontier enter the sector.  Cropping the margins leaves two disjoint
+opposite-colored arms in different sectors, in addition to the four
+arms along the monochromatic arc.  This gives six disjoint bulk arms.
+The bulk exponent strictly greater than two excludes these contacts
+by a covering argument.  At a self-contact on the starting circle,
+the four arc traversals alone suffice: the arbitrary-color four-arm
+exponent is strictly greater than one, and the possible contact
+locations lie on a fixed circle.
+
+All these exclusions are made for distinct visits with fixed positive
+excursion scales before those scales decrease.  Countably many scales
+and finite ball coverings make the assertions simultaneous.  They
+bound joint probabilities with the four-tip event; no division by the
+probability of a rare history is involved.
+
+We now identify the remaining domain; see
+Figure~\ref{fig:frontier-lobes}.  Cut along the barrier, using the
+four short incidences from the tip edges through their terminal
+triangles.  Between consecutive tips, the frontier arc and its
+corresponding barrier interval bound a lobe polygon.  The four lobes
+are attached over open intervals to the untouched region on the middle
+side of the barrier.  Each lobe converges in order to a Jordan lobe,
+because its frontier arc is simple and meets the barrier only at its
+endpoints.  Different lobe boundaries may touch in physical projection;
+their interiors remain disjoint, and their distinct accesses are
+retained.  With two well-separated barriers there is a fixed open
+annular region between the ends.
+
+We verify that adjoining the lobes still gives an annulus, even when
+their boundaries touch.  On either side of this open annular region,
+start with the closed disk bounded by the corresponding stopping circle,
+using the sphere for the outer side.  The complementary set on that
+side is obtained by removing the lobe interiors and their open
+attachment intervals.  Each lobe's free frontier arc remains in this
+complement: it cannot enter another lobe's interior, since their Jordan
+interiors are disjoint, and it meets the stopping circle only at its
+own endpoints, never on another open attachment.  Connect a point of
+the remaining closed set to the preserved starting-side region by a
+path in the closed disk, with its interior off the stopping circle.
+Whenever the path enters a lobe, replace the portion from its first
+entry to its last exit by the intervening subarc of that lobe's free
+frontier.  The entry and exit lie on this free arc.  The replacement
+avoids every lobe interior and every open attachment, so it cannot
+undo previous replacements.  There are only four lobes; after finitely
+many replacements the path is wholly in the complementary set.
+Thus each side has a connected complement.  The two such nondegenerate
+continua are separated by the open annular region and exhaust the
+complement of the residual domain.  The domain is therefore doubly
+connected, with its two boundary components physically apart.
+
+Lemma~\ref{an:back-lobes} now gives uniform convergence
+of the closed-annulus parametrizations with their prime-end order.
+The boundary traversal consists of finitely many individually simple
+arcs.  These are exactly the hypotheses required by
+Proposition~\ref{an:limit}.
+
+Finally extract, from any sequence of scales, a further sequence on
+which the localized full-plane geometries are good in the sense of
+\cite[Section~2]{V}.  Conditional on those geometries, couple convergent
+subsequences of the frontier histories just described.  All unqueried
+colors in the residual face remain independent and fair.  Apply
+Proposition~\ref{an:limit} path by path to the converging graph annuli.
+The continuum function on the actual approximating conformal data
+also converges, by its continuity.  Bounded convergence proves
+\eqref{tr:mean-limit} along the further sequence and hence along the
+original sequence.  Unused colors on the removed sides of the
+frontiers may be revealed or integrated in their own end variables;
+they do not affect this argument.
+\end{proof}
+
+\input{figures/lobes}
+
+The probabilistic conclusions of the last two lemmas all concern the
+independent region.  Lemma~\ref{tr:domination} therefore makes their
+discarded probabilities and the error in \eqref{tr:mean-limit} uniform
+for the operator norm, even when a message depends arbitrarily on the
+seam and on its attachment information.  This is the reason the
+explorations start at auxiliary cycles rather than at the original
+seams.
+
+\subsection{The first-reduction decomposition}
+
+Index the inner barriers by radii
+\[
+             a_j=8\cdot2^j r,
+\qquad j=1,2,\ldots,
+\]
+and the outer barriers by
+\[
+             b_k=2^{-k}Lr/8,
+\qquad k=1,2,\ldots.
+\]
+Each list is finite, ending before the opposite starting region.  Fix
+a large integer $d$ so that
+\begin{equation}
+                  j+k\le n-d
+\label{tr:used-pairs}
+\end{equation}
+ensures disjoint buffered end supports and sufficient separation for
+Proposition~\ref{sh:separated-ends}.  A \emph{used pair} is a pair
+satisfying \eqref{tr:used-pairs} at which the two experiments first
+reduce to exactly four tips.  A first reduction to fewer than four
+contributes zero by Lemma~\ref{tr:frontier}.
+
+Choose $\tau>0$ smaller than the excess of the mesh-first six-arm
+exponent over two.  Let $\rho_j^{\rm in}$ and $\rho_k^{\rm out}$ be the
+subprobability laws of the corresponding first four-tip histories,
+including local screens and the variables attaching each history to its
+original seam.  The inner law integrates the input seam.  All outer
+laws and outer integrals below are conditional on the fixed output seam
+$\Sigma_{Lr}$; in particular they define fields over that seam, not
+averages of it.  The dependence on seam variables in the
+attachments will be kept in the scalar and signed measure factors
+below, rather than in any assertion about typical shapes.  We first
+bound the masses of these unweighted history laws; pointwise bounded
+message factors will then preserve the bounds up to constants.
+
+\begin{lemma}[First-reduction tails]
+\label{tr:tails}
+For fixed $L$, the end-history subprobability masses satisfy, in the
+mesh-first limit,
+\begin{equation}
+ \|\rho_j^{\rm in}\|_{\TV}
+ \operatorname*{ess\,sup}_{\Sigma_{Lr}}
+         \|\rho_k^{\rm out}(\cdot\mid\Sigma_{Lr})\|_{\TV}
+      \le C 2^{-(2+\tau)j}2^{-(2+\tau)k}+o(1)
+\label{tr:history-weight}
+\end{equation}
+for each used pair $(j,k)$.  Consequently, after inserting arbitrary
+middle tests of modulus at most one, the same bound up to a constant holds
+for the corresponding operator contribution on inputs of norm at most
+one.  The part of the transfer not represented
+by used pairs has norm at most
+\begin{equation}
+             C(n+1)^2L^{-2-\tau}+o(1).
+\label{tr:unused}
+\end{equation}
+Both estimates are uniform for input messages of norm at most one and
+for successful output seams.
+\end{lemma}
+
+\begin{proof}
+Before the first reduction at level $j$, there are at least six tips
+at the preceding level.  Lemma~\ref{tr:frontier} then requires six arms
+from the starting region to that level.  The mesh-first six-arm bound
+gives $C2^{-(2+\tau)j}$, with a changed constant for the first few
+levels.  The same statement holds at the outer end, uniformly in its
+original seam: the necessary six-arm event is read in the independent
+region, while attachment variables and success indicators have total
+conditional mass at most one.  Multiplying the two separate mass
+bounds proves \eqref{tr:history-weight}.  For used pairs the physical
+end experiments also have disjoint buffered supports, so their joint
+law is the product used in the operator contribution.  Arbitrary input
+weights are bounded
+pointwise by their total variation, as in
+Lemma~\ref{tr:domination}; the number of switches on the input seam
+does not enter the estimate.
+
+To bound the remainder, run the inner exploration to its longest
+allowed level.  If it never reduces, it has paid a six-arm probability
+over $n-O(1)$ dyadic levels.  Otherwise let $j$ be its first reduction
+level.  If the reduction has fewer than four tips, its contribution
+is zero.  In the remaining case run the outer experiment as far as
+\eqref{tr:used-pairs} allows, possibly through no additional level.
+If it does not reduce in time, the two experiments together have
+paid six-arm probabilities over at least $n-O(1)$ levels.  Their
+buffered supports remain disjoint.  Summing the resulting bounds over
+the at most $O((n+1)^2)$ possible terminal levels proves
+\eqref{tr:unused}.  The finitely many auxiliary localization failures
+and near-coincidences of tips cost $o(1)$ for fixed $L$, uniformly by
+Lemma~\ref{tr:domination}.
+\end{proof}
+
+At a four-tip inner frontier let $B$ and $W$ denote respectively the
+black-join and white-join cap states.  The pushed input message is
+\begin{equation}
+                 \alpha_j(v;H_i)(\delta_B-\delta_W),
+                 \qquad |\alpha_j(v;H_i)|\le\|v\|_r,
+\label{tr:inner-coefficient}
+\end{equation}
+where $H_i$ includes the inner history and its attachment to the input
+seam.  The coefficient is linear in $v$.  Similarly, pushing
+$\delta_B-\delta_W$ from an outer four-tip frontier to the output seam
+gives a signed measure $q_k(H_o)$ satisfying
+\begin{equation}
+                         \|q_k(H_o)\|_{\TV}\le2.
+\label{tr:outer-coefficient}
+\end{equation}
+Lemma~\ref{an:caps} says that the residual annulus sends the first
+cap difference to the second multiplied by its through-transmission
+probability.  Thus the used-pair contribution has the form
+\begin{equation}
+ \int \alpha_j(v;H_i)\,
+       \E_{\rm mid}\!\left[
+          \boldsymbol1_{G_{\rm aux}}t_r\mid H_i,H_o\right]
+       q_k(H_o)\,d\rho_j^{\rm in}(H_i)d\rho_k^{\rm out}(H_o),
+\label{tr:integral}
+\end{equation}
+where $G_{\rm aux}$ denotes the additional middle screens and
+$\E_{\rm mid}$ integrates the middle Poisson geometry conditional on
+the two end histories.  The probability $t_r$ already integrates the
+fresh colors in the remaining face, conditional on all geometry.
+The end laws are a product because their buffered supports are
+disjoint.  The conditional factor in square brackets still couples
+the ends, and must not yet be factored.
+
+On the auxiliary localization successes, the residual frontiers and
+both $t_r$ and $T_r$ depend only on the auxiliary-cycle explorations and
+the intervening independent geometry.  Attachments to the original
+seams enter only the signed coefficients.  The error $|t_r-T_r|$ thus
+has exactly the independent-region measurability required by
+Lemma~\ref{tr:domination}.
+
+By Lemmas~\ref{tr:admissible} and~\ref{tr:domination}, replace $t_r$ by
+$T_r$ in \eqref{tr:integral} with $o(1)\|v\|_r$ error in output norm.
+For fixed $n$, this error may be summed over all used pairs.  The
+frontiers themselves are still the actual graph cycles.  Their
+single-end conformal maps and capacities can therefore be used
+directly in Proposition~\ref{sh:separated-ends}, without asserting
+smoothness at contacts of a limiting boundary.
+
+\subsection{Rank-one factorization and signed cancellation}
+
+We now complete Proposition~\ref{tr:block}.  An inner frontier at
+level $j$ surrounds the center, stays inside radius comparable to
+$a_j$, and reaches that radius at its tips.  Its logarithmic capacity
+$l_i$ is therefore comparable to $a_j$.  An outer frontier at level
+$k$ surrounds a disk of radius comparable to $b_k$ and has a point
+at distance comparable to $b_k$; its conformal radius $l_o$ is
+comparable to $b_k$.  Write $y^0$ and $x^0$ for their single-end
+switch angles.  By \eqref{sh:rank-one},
+\begin{equation}
+ T_r(H_i,H_o)=K\,U(H_i)V(H_o)
+                       +O\big((a_j/b_k)^{\lambda+\sigma_1}\big),
+\label{tr:separated-formula}
+\end{equation}
+where
+\[
+ U(H_i)=l_i^\lambda J(y^0)^{1/3},
+ \qquad
+ V(H_o)=l_o^{-\lambda}J(x^0)^{1/3}.
+\]
+These factors depend on their own end histories only, and
+\begin{equation}
+             |U(H_i)V(H_o)|
+                  \le C L^{-\lambda}2^{\lambda(j+k)}.
+\label{tr:factor-size}
+\end{equation}
+
+For the separated leading term, delete the extra middle screens.
+This changes the integral by $o(1)\|v\|_r$ for fixed $L$: the screens
+use independent interior variables and fail with probability $o(1)$,
+while all weights at these fixed scales are bounded.  Keep the original
+seam screens and the local end screens.  The remaining end experiments
+now use disjoint Poisson supports for every used pair.  Define the
+scalar functionals and output messages
+\[
+ u_j(v)=\int \alpha_j(v;H_i)U(H_i)\,d\rho_j^{\rm in}(H_i),
+ \qquad
+ v_k=\int V(H_o)q_k(H_o)\,d\rho_k^{\rm out}(H_o).
+\]
+The leading contribution of that pair is exactly $K u_j(v)v_k$.
+The same definitions make sense separately for every level in each
+finite list, whether or not a pair of levels has disjoint supports.
+For such additional pairs we use the product of the two separate
+integrals, not a claim about simultaneous experiments.
+
+Set
+\begin{equation}
+       M_1v=K\left(\sum_j u_j(v)\right)\left(\sum_k v_k\right).
+\label{tr:rank-one-definition}
+\end{equation}
+This operator has rank at most one.  By the separate six-arm bounds
+and \eqref{tr:factor-size}, its norm is at most
+\begin{equation}
+ C L^{-\lambda}
+       \sum_{j,k}2^{-(2+\tau-\lambda)(j+k)}
+                   \le C L^{-\lambda}.
+\label{tr:rank-one-size}
+\end{equation}
+The extra product terms introduced by allowing $j+k>n-d$ have norm
+at most
+\begin{equation}
+ C L^{-\lambda}
+      \sum_{j+k>n-d}2^{-(2+\tau-\lambda)(j+k)}
+                   \le C(n+1)^2L^{-2-\tau}.
+\label{tr:artificial-pairs}
+\end{equation}
+This numerical estimate requires no independence for overlapping
+actual experiments.
+
+Decrease $\sigma_1$ if necessary so that
+$\lambda+\sigma_1<2+\tau$.  The errors in
+\eqref{tr:separated-formula}, summed using
+Lemma~\ref{tr:tails}, have norm at most
+\begin{equation}
+ C L^{-\lambda-\sigma_1}
+      \sum_{j+k\le n-d}
+             2^{-(2+\tau-\lambda-\sigma_1)(j+k)}+o(1)
+       \le C L^{-\lambda-\sigma_1}+o(1).
+\label{tr:rank-error-sum}
+\end{equation}
+Together with \eqref{tr:unused} and
+\eqref{tr:artificial-pairs}, this proves the first two estimates of
+Proposition~\ref{tr:block}, with a smaller positive exponent saving,
+in the mesh-first limit.  All errors are measured in output total
+variation, not merely after testing a fixed continuum event.
+
+It remains to prove the stronger estimate for an even isotropic input
+$v$.  After the continuum replacement, delete the extra middle screens
+as above.  Only $T_r$ then depends on both ends; all other factors
+belong to their separate, independently integrated end laws.
+Rotate all inner end data, including the seam and the revealed
+history.  By equivariance of $v$, the coefficient
+$\alpha_j(v;H_i)$ in \eqref{tr:inner-coefficient} is unchanged.  Reverse
+all colors of the inner data instead.  The message is still unchanged
+as a geometrically transformed signed measure, but the reference
+vector $\delta_B-\delta_W$ changes sign.  Thus
+\begin{equation}
+ \alpha_j(v;\operatorname{rot}_\theta H_i)=\alpha_j(v;H_i),
+ \qquad
+ \alpha_j(v;\operatorname{flip} H_i)=-\alpha_j(v;H_i).
+\label{tr:coefficient-symmetry}
+\end{equation}
+The inner end law is rotation and color-reversal invariant.  The outer
+measure factor is unaffected by either transformation of the inner
+data.  Labels for the four tips may be chosen independently of colors
+using the auxiliary angular axis, and transported under rotations.
+For each fixed outer history $H_o$, color reversal gives the identity
+\begin{equation}
+\begin{aligned}
+ &\int \alpha_j(v;H_i)T_r(H_i,H_o)\,d\rho_j^{\rm in}(H_i)\\
+ &\quad=\frac12\int \alpha_j(v;H_i)
+    \big[T_r(H_i,H_o)-T_r(\operatorname{flip}H_i,H_o)\big]
+                                   \,d\rho_j^{\rm in}(H_i).
+\end{aligned}
+\label{tr:flip-integral}
+\end{equation}
+Now average rotations of the inner data in the right side.  Since
+$\alpha_j$ and the inner law are rotation invariant, its contribution
+is one half of the integral against the rotation average of
+$T^+-T^-$, with its appropriate inner prescriptions.  In particular
+one must retain the sign; replacing this difference by an unsigned
+four-arm event would lose the required exponent.
+
+Equation~\eqref{sh:rotation-average} bounds the averaged difference
+by $C(a_j/b_k)^{2+\sigma_2}$.  The local end experiments and screens
+can be chosen rotation and color-reversal invariant.  Decrease
+$\sigma_2$ so that $0<\sigma_2<\tau$.  The used-pair contributions
+are bounded in total variation by
+\begin{equation}
+ C L^{-2-\sigma_2}
+    \sum_{j+k\le n-d}2^{-(\tau-\sigma_2)(j+k)}+o(1)
+                  \le C L^{-2-\sigma_2}+o(1).
+\label{tr:signed-sum}
+\end{equation}
+Adding the unused contribution \eqref{tr:unused} proves the desired
+power strictly greater than two.
+
+Finally choose $\delta>0$ smaller than all the positive savings just
+obtained, decreasing it to absorb the polynomial factors in $n$.
+For each fixed sufficiently large $n$, choose $r_0(2^n)$ so that every
+$o(1)$ term is smaller than the required fixed power of $L$.  This
+proves \eqref{tr:block-size}--\eqref{tr:block-cancellation}.
+Notice the order: the ratio is chosen first and the scale threshold
+afterwards.  No rate of convergence in Cardy's formula, in the
+good-sequence extraction, or near a nonsimple limiting bank is used.
+
+\subsection{Iteration and failed screens}
+
+The block estimates have two consequences.  The first uses the signed
+symmetry estimate to control a Poisson insertion.  The second uses the
+rank-one approximation to compare two responses at different distant
+boundaries.
+
+\begin{corollary}[Fair-insertion message]
+\label{tr:insertion}
+There are $0<\nu<1$ and $C<\infty$ such that the following holds.  Insert
+a nucleus at the center of a cut, with an independent fair color.  At
+a seam of radius $s$, take the difference of the inside state laws
+``with this insertion'' minus ``without this insertion,'' conditional
+on the seam and with its screen success imposed.  Its message
+$I_s$ satisfies
+\begin{equation}
+                            \|I_s\|_s\le C s^{-2-\nu}
+\label{tr:insertion-bound}
+\end{equation}
+for all sufficiently large $s$.  The estimate remains valid in the
+presence of fixed data outside the screened collar that do not enter
+its inside.  The seam tests and the selected cycle are those of the
+base process, without the insertion.
+\end{corollary}
+
+\begin{proof}
+Choose a ratio $L$ from Proposition~\ref{tr:block} so large that the
+constant per block in \eqref{tr:block-cancellation} is absorbed by a
+smaller power saving.  Choose $s_*>r_0(L)$ sufficiently large.  For
+given $s$, use radii
+\[
+             r_j=L^j r_0,\qquad 0\le j\le N,
+             \qquad r_N=s,\quad s_*\le r_0<Ls_*.
+\]
+All screens use the base process.  At a successful seam, the inserted
+point cannot reach or alter its cycle, by localization.  The two
+inside state laws consequently have the same state space and each
+has mass one.  Their difference has zero mass, norm at most two,
+and is even and isotropic.
+
+If every seam is successful, iterating
+\eqref{tr:block-cancellation} bounds the message by
+$C(s/r_0)^{-2-\nu}$ for some $0<\nu<1$.  Since $r_0$ stays in a fixed
+compact range of positive scales, this gives
+\eqref{tr:insertion-bound} for that contribution.
+
+For the remaining contribution, group configurations by the largest
+failed radius $v=r_j<s$.  The exact partition, with the final success
+kept in every term, is
+\[
+ \boldsymbol1_{G_{r_N}}
+   =\prod_{i=0}^{N}\boldsymbol1_{G_{r_i}}
+    +\sum_{j=0}^{N-1}\boldsymbol1_{G_{r_j}^c}
+                      \prod_{i=j+1}^{N}\boldsymbol1_{G_{r_i}}.
+\]
+Form a fresh difference of inside state
+laws at the next cut $Lv=r_{j+1}$, multiplying both laws by the same
+base-process indicator $\boldsymbol1_{G_v^c}$.  Earlier screen
+outcomes are left unrestricted.  This weighted difference still has
+zero mass for each successful seam at $Lv$: for each common base
+configuration the two state laws have equal total mass.  It is also
+even and isotropic.  The collar at $v$ is disjoint from that at $Lv$,
+so conditional on the latter seam its norm is at most
+\begin{equation}
+                2\Pp(G_v^c\mid\Sigma_{Lv})
+                           \le C e^{-cv^2}.
+\label{tr:reset}
+\end{equation}
+Success at $Lv$ ensures that both configurations use the same cut
+there, independently of what happened at smaller radii.
+
+Transfer this reset message through the successful cuts above $Lv$.
+Those success restrictions ensure that $v$ is indeed the largest
+failed radius; no restriction on earlier failures was needed.
+The sum of these contributions is bounded by
+\[
+ C\sum_{v\in\{r_0,\ldots,r_{N-1}\}}
+             e^{-cv^2}\left(\frac{s}{Lv}\right)^{-2-\nu}
+       \le C s^{-2-\nu}.
+\]
+The final inequality follows by summing the exponentially decaying
+weights times $v^{2+\nu}$ over the geometric sequence.  This proves
+the claim.  Outside data beyond the final screened collar have no
+effect on the inside law or on the argument, which proves the stated
+stability.
+\end{proof}
+
+For a message space $\mathcal B_t$, the projective norm on its algebraic
+tensor square is
+\[
+ \|z\|_\pi
+   =\inf\left\{\sum_{i=1}^m\|u_i\|_t\|v_i\|_t:
+          z=\sum_{i=1}^m u_i\otimes v_i,\quad m<\infty\right\}.
+\]
+We may complete in this norm.  Bounded linear functionals $\ell,\ell'$
+on $\mathcal B_t$ satisfy
+\[
+         |(\ell\otimes\ell')(z)|
+                \le\|\ell\|\,\|\ell'\|\,\|z\|_\pi.
+\]
+In particular, evaluating $u\otimes v-v\otimes u$ gives the determinant
+$\ell(u)\ell'(v)-\ell(v)\ell'(u)$.
+
+\begin{corollary}[Contraction of antisymmetric products]
+\label{tr:wedge}
+There are $\nu'>0$, $s_0<\infty$, and a fixed ratio $L>1$ such that,
+for $r\ge s_0$ and $s/r=L^N$, the transfer
+\[
+          \mathcal M=M_{s/L,s}\cdots M_{r,Lr}
+\]
+with all intermediate seam successes imposed satisfies
+\begin{equation}
+ \|\mathcal M u\wedge\mathcal M v\|_\pi
+       \le C(s/r)^{-2\lambda-\nu'}\|u\|_r\|v\|_r
+\label{tr:wedge-bound}
+\end{equation}
+for any two zero-mass input messages $u,v\in\mathcal B_r$.  Here
+$u\wedge v=u\otimes v-v\otimes u$ and $\|\cdot\|_\pi$ is the
+projective tensor norm.  Fixed-factor unused intervals at the two
+ends change only the constant.  No assertion of this contraction
+through a failed intermediate cut is made.
+\end{corollary}
+
+\begin{proof}
+For one block write $M=M_1+E$ as in
+\eqref{tr:block-rank}.  Since $M_1$ has rank at most one,
+$M_1\otimes M_1$ vanishes on antisymmetric tensors.  On that subspace,
+\[
+ \|M\otimes M\|
+        \le 2\|M_1\|\,\|E\|+\|E\|^2
+        \le C L^{-2\lambda-\delta}.
+\]
+Choose $L$ large enough to absorb $C$ into a smaller exponent saving
+$\nu'>0$, simultaneously with its choice in
+Corollary~\ref{tr:insertion}.  Iterating the estimate and using
+$\|u\wedge v\|_\pi\le2\|u\|_r\|v\|_r$ gives
+\eqref{tr:wedge-bound}.  The argument holds first for finite tensor
+sums and then for their projective-norm completion.  Ordinary
+Markov pushes contract total variation, so fixed-factor omitted end
+intervals affect only the constant after rounding to the ratio-$L$
+grid.
+\end{proof}
diff --git a/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/build/sections/06-amplitude.tex b/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/build/sections/06-amplitude.tex
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@@ -0,0 +1,518 @@
+\section{The intensity derivative and the pivotal amplitude}
+\label{sec:amplitude}
+
+The preceding section supplies two estimates with different roles.
+Corollary~\ref{tr:insertion} makes the effect of a distant fair insertion
+integrable, even after division by the central pivotal probability.
+Corollary~\ref{tr:wedge} makes the normalized effect of a nearby insertion
+almost independent of the size of the rectangle. Together they give a
+summable error for the logarithmic derivative of $f$.
+Throughout this section the Poisson intensity is one, unless otherwise
+specified, and $\lambda=5/4$.
+
+\subsection{Differentiating the intensity}
+
+Let $X_R(\eta)$ be the indicator of a black left--right crossing of
+$RQ_0$ for a colored point configuration $\eta$, with the closed-cell
+convention of Theorem~\ref{thm:main}. If $0^{\rm b}$ and $0^{\rm w}$
+denote an inserted black and white point at the origin, respectively, put
+\[
+ H_R(\eta)
+ =X_R(\eta\cup\{0^{\rm b}\})-X_R(\eta\cup\{0^{\rm w}\}).
+\]
+Thus $H_R\in\{0,1\}$, and $f(R)=\E H_R(\eta)$ for the unit-intensity
+fairly colored process. For $y\ne0$, write $y^{\rm f}$ for an independently
+fairly colored inserted point and define
+\begin{equation}\label{cl:kernel}
+ K_R(y)=\E\bigl[H_R(\eta\cup\{y^{\rm f}\})-H_R(\eta)\bigr].
+\end{equation}
+The expectation includes the extra color. Values at $y=0$ and at null
+nongeneric insertion locations are immaterial to the integrals below.
+In particular, $K_R$ is an insertion difference of a \emph{color}-pivotal
+probability, not a probability of Poisson pivotality.
+
+\begin{lemma}[Intensity differentiation]\label{cl:derivative}
+The function $f$ is continuously differentiable on $(0,\infty)$. For
+every $R>0$, $K_R$ is absolutely integrable and
+\begin{equation}\label{cl:derivative-eq}
+ \frac R2 f'(R)=\int_{\R^2}K_R(y)\,dy.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Let $\eta_t$ have intensity $t$ and fair colors. Euclidean scaling gives
+\begin{equation}\label{cl:scaling-intensity}
+ G_R(t):=\E H_R(\eta_t)=f(\sqrt t\,R).
+\end{equation}
+We justify differentiation of the full-plane expectation before using
+this identity. All the estimates in this paragraph are uniform for
+$t$ in a compact subinterval of $(0,\infty)$.
+
+For $n\ge n_0(R)$, let $H_{R,n}$ be the same test computed from the
+points in $B(0,2^n)$, together with the central insertion. A locality
+screen in a collar well inside $B(0,2^n)$ determines every cell meeting
+$RQ_0$, including its intersections with the boundary. Consequently
+$H_{R,n}=H_R$ outside an event of probability at most
+$C_R\exp(-c_R4^n)$. The same screen gives
+$H_{R,n+1}=H_{R,n}$. Finite-volume Poisson differentiation follows by
+differentiating the absolutely convergent Poisson series; see also
+\cite[Theorem~19.1]{LP}. Equivalently,
+for a bounded function $F$ of the process in a bounded set $D$,
+\begin{equation}\label{cl:poisson-score}
+ \frac{d}{dt}\E F(\eta_t\cap D)
+ =\E\left[F(\eta_t\cap D)
+       \left(\frac{N_t(D)}t-|D|\right)\right].
+\end{equation}
+Here $N_t(D)$ is the number of Poisson points in $D$.
+Apply this to $F=H_{R,n+1}-H_{R,n}$ on $D=B(0,2^{n+1})$.
+The score in parentheses has second moment $|D|/t$. Cauchy--Schwarz
+therefore bounds the derivative by
+$C_R2^n\exp(-c_R4^n)$. The differences of the expectations and of
+their derivatives are uniformly summable. Hence $G_R$ is $C^1$ and
+its derivative is the limit of the finite-volume derivatives.
+
+The add-one version of finite-volume differentiation is
+\[
+ \frac{d}{dt}\E H_{R,n}(\eta_t)
+ =\int_{B(0,2^n)}
+  \E\bigl[H_{R,n}(\eta_t\cup\{y^{\rm f}\})
+                     -H_{R,n}(\eta_t)\bigr]\,dy.
+\]
+For fixed $y$ the integrand converges to its full-plane counterpart.
+For $|y|\ge C R+C$, a screen surrounding $RQ_0$ and the origin, at
+distance comparable to $|y|$ and strictly short of $y$, prevents the
+insertion from changing any cell used by the test. Its failure
+probability is at most $C_R e^{-c_R|y|^2}$. The screen can be taken
+inside $B(0,2^n)$ whenever $y\in B(0,2^n)$ and $n$ is sufficiently
+large; thus this is also a dominating bound for the finite-volume
+integrands. On a fixed bounded set the bound $2$ suffices.
+Dominated convergence proves absolute integrability and identifies
+$G_R'(1)$ with the integral in \eqref{cl:derivative-eq}.
+Equation~\eqref{cl:scaling-intensity} proves the asserted derivative
+formula and, locally at each $R$, the $C^1$ assertion for $f$.
+\end{proof}
+
+\subsection{An integrable tail after normalization}
+
+The exponential localization in Lemma~\ref{cl:derivative} is a
+finite-scale statement. We need a bound uniform as $R$ grows, with
+the small factor $f(R)$ retained. This is where the signed insertion
+estimate is essential.
+
+\begin{lemma}[Normalized insertion tail]\label{cl:tail}
+There are constants $\xi>0$, $m_0<\infty$, and $C<\infty$ such that
+for $m_0\le m\le R/100$,
+\begin{equation}\label{cl:tail-eq}
+ \int_{|y|>m}|K_R(y)|\,dy
+ \le C f(R)\bigl(m^{-\xi}+R^{-\xi}\bigr).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Use the exponent $0<\nu<1$ of Corollary~\ref{tr:insertion} and
+the one-arm exponent $\beta>0$ from the critical inputs. All screens
+and arm probes below have fixed-factor buffers. Independent probes
+mean probes determined by disjoint sets of Poisson coordinates,
+not unrestricted Voronoi events in disjoint planar regions.
+
+We first explain how arm requirements can be multiplied by a signed
+message estimate. The integrand in \eqref{cl:kernel} is a mixed
+difference: one difference changes the central color, and the other
+inserts the point at $y$. If this mixed difference is nonzero, changing
+the central color changes a crossing for at least one setting of the
+insertion, and changing the insertion changes a crossing for at least
+one setting of the central color. A screened probe separated from
+both changes reads the same base colors in all four configurations.
+Thus the necessary-arm implications for a local change apply to
+these common colors; see \cite[Section~2, ``Necessary arms for a
+local change'']{V}. They apply even when the two crossing changes
+occur in different configurations.
+
+The conditional version is equally important. Every formal noncrossing
+pairing can be drawn as disjoint wall arcs in a topological disk.
+Coloring its complementary regions alternately realizes the prescribed
+boundary runs and the pairing. Thus two formal cap states give two
+planar fillings agreeing on every exterior probe; the deterministic
+necessary-arm argument does not require Poisson geometry inside the
+replaced disk. Fix a successful
+cut about $y$, its seam, and all exterior variables. The exterior
+test is now a function of the inside connection state. If a
+necessary central or enclosing four-arm event does not occur,
+the central black-minus-white test is zero for every such state.
+If a necessary side or corner arm event does not occur, each
+crossing test with the central color fixed is constant across
+inside states. These are
+the same planar necessary-arm implications, applied to two
+possible cap states on the cut. The insertion message has zero
+mass, so it annihilates the latter constants. Its conditional
+answer is consequently bounded by its total variation norm times
+the necessary probe indicators. Those indicators are measurable
+outside the cut. Disjoint buffered supports then allow their
+probabilities to be multiplied.
+
+We use screened arm probes with failures allowed, whose probabilities
+are still bounded by a constant times their intended arm bounds;
+see the convention in Section~\ref{sec:inputs} and
+\cite[Section~2, ``Buffered localization'']{V}.
+Indeed, group failures by the largest failed radius $v$ and retain
+the arm test starting a fixed factor beyond $v$, independently of
+the failure. For a four-arm probe starting at $r$, the relevant
+estimate is
+\[
+ e^{-cv^2}A(C_0v,R)
+ \le C A(r,R)(v/r)^g e^{-cv^2}
+ \qquad(C_0v<R),
+\]
+where $g$ is the polynomial exponent in \eqref{in:power}.
+For a side or corner probe the corresponding endpoint losses
+are bounded by $C(v/d)^2$ or $C(v/u)^\beta$, respectively.
+If a failed screen crosses the junction between side and corner
+probes, the lost factors are therefore only polynomial. If the
+retained test has no remaining radial range, the same comparison follows from the
+polynomial lower bound, or directly from the displayed power
+factor for a side or corner probe. Summing over geometric $v$
+preserves the indicated arm cost as long as the retained probes
+and the failure event have disjoint buffered supports. A failure
+large enough to reach the central probe requires a separate
+argument below: that probe is then dropped, not treated as
+independent. An unweighted exponentially small error alone would
+not suffice when $R$ is arbitrarily large.
+
+\paragraph{Insertions near the center.}
+Put $t=|y|$ and suppose $m<t\le R/100$. First fix a small
+constant $c_1>0$, and then choose $a>0$ sufficiently small relative
+to $c_1$. Choose a cut about $y$ of radius $a t$,
+a central four-arm probe from $b_*$ to $a t$,
+and a four-arm probe surrounding both insertions from $a^{-1}t$
+to $aR$.
+Short or overlapping endpoint ranges are omitted at constant cost.
+The buffers of these three constructions can be chosen disjoint.
+On cut success, Corollary~\ref{tr:insertion} gives a bound
+$Ct^{-2-\nu}$ for the insertion message. The central color
+difference requires the first four-arm probe; transmission of a
+change from the region containing both insertions to the crossing
+requires the second. Thus the success contribution is at most
+$Ct^{-2-\nu}A(b_*,t)A(t,R)$.
+
+For failures, let $v$ be the largest failed radius about $y$, with
+the initial radius comparable to $t$. If $v<c_1t$, the
+central probe and an enclosing
+probe retain disjoint supports after the fixed buffer adjustments.
+The preceding failed-screen estimate bounds their total
+contribution by $Ce^{-ct^2}A(b_*,t)A(t,R)$.
+If $v\ge c_1t$, drop the central probe. Retain instead a screened
+four-arm probe centered at the origin, from radius $C_1(v+t)$
+to $aR$, enclosing both modifications and with its support
+separated from the failed screen. The same necessary-arm
+implication supplies this probe. When its radial range is nonempty,
+quasi-multiplicativity and \eqref{in:power} give
+\[
+ e^{-cv^2}A(C_1(v+t),aR)
+ \le C A(b_*,R)(v+t)^g e^{-cv^2}.
+\]
+Since $t\le c_1^{-1}v$, summation over these geometric $v$ is
+at most $C A(b_*,R)e^{-c't^2}$. If the retained radial range
+is empty, then $R\le C(v+t)\le C'v$; the absolute failure
+bound and $A(b_*,R)\ge cR^{-g}$ give the same conclusion.
+Combining success and both failure cases proves
+\begin{align}
+ |K_R(y)|
+ &\le Ct^{-2-\nu}A(b_*,t)A(t,R)
+             +C A(b_*,R)e^{-ct^2}\notag\\
+ &\le C f(R)t^{-2-\nu}.
+ \label{cl:bulk-kernel}
+\end{align}
+In the last line we used quasi-multiplicativity and
+\eqref{in:comparison}. In particular,
+\begin{equation}\label{cl:bulk-integral}
+ \int_{m<|y|\le R/100}|K_R(y)|\,dy
+ \le C f(R)\int_m^\infty t^{-1-\nu}\,dt
+ \le C f(R)m^{-\nu}.
+\end{equation}
+
+\paragraph{Insertions near a side or corner.}
+Consider $R/100<|y|\le10R$, including points outside the rectangle.
+Fix a sufficiently large constant $d_*$, and then a sufficiently
+small constant $\rho>0$. If $\mathcal C_R$ is the set of the four
+corners of $RQ_0$, set
+\[
+ d(y)=\min\{\rho R,\max\{d_*,\dist(y,\partial(RQ_0))\}\},
+ \qquad
+ u(y)=\min\{\rho R,\max\{d_*,\dist(y,\mathcal C_R)\}\}.
+\]
+We take $R$ large enough that $d_*<\rho R$. Always
+$d_*\le d(y)\le u(y)\le\rho R$; write $d=d(y)$ and $u=u(y)$.
+The constants in the estimates may depend on $d_*$ and $\rho$.
+There is a central four-arm probe up to radius $cR$, where $c$
+is fixed and much smaller than $1/100$, disjoint from the other
+regions used below. Its cost is at most $CA(b_*,R)$.
+
+At an interior point $y$, use the insertion estimate on a cut of
+radius a small fixed fraction of $d$. At the bottom cutoff a
+constant bound suffices, and can be bounded by $Cd^{-2-\nu}$
+because $d_*$ is fixed. At an exterior point with distance above
+the bottom cutoff, successful screening at this scale leaves the
+quad unchanged. Only screen failures contribute there, with a bound
+exponential in $d^2$, which is no larger than $Cd^{-2-\nu}$.
+
+If $u/d$ is larger than a sufficiently large constant, the closest
+part of the boundary is a straight side. A crossing change then
+requires three alternating free arms in a half-plane probe centered
+at the projection onto that side, from radius $Cd$ to radius $cu$.
+Its cost is at most $C(d/u)^2$. If the projection is at a corner,
+the two distances are comparable and this probe is unnecessary.
+When $u$ is uncapped and is a sufficiently small fraction of $R$,
+retain a one-arm probe centered at the nearest corner from radius
+$Cu$ to radius $cR$. Its cost is at most $C(u/R)^\beta$.
+For $u$ capped above, this factor is a fixed positive constant,
+so the corresponding probe can be omitted. These are precisely
+the local-change arm implications for the unwired, closed-cell
+crossing convention. They also cover a modified cell meeting a
+side or a corner.
+
+The insertion cut, the side probe, the corner probe, and the
+central probe can be given disjoint buffered supports. To arrange
+this, first fix the central radius $cR$, then choose the distance
+cap $\rho R$ much smaller. A corner probe is used only in a
+corner neighborhood, far from the center. Fixed multiples are
+lost between consecutive probes, which changes only constants.
+To include failures, choose a sufficiently small fixed $c_2>0$.
+For a largest failed radius $v<c_2R$ about $y$, the central
+probe remains disjoint from the failed-screen support. Retain
+the side and corner probes only beyond that support. If a
+junction has been passed, the center of the affected side or
+corner stage lies within $O(v)$ of $y$, so a fixed-factor
+enlargement provides the required separation. Omitting an entire
+side stage loses at most $C(1+v/d)^2$, and shifting or omitting
+a corner stage loses at most $C(1+v/u)^\beta$. Since the first
+possible failed radius is a fixed fraction of $d$,
+\[
+ \sum_{v\ge c_3d}e^{-cv^2}(1+v/d)^2(1+v/u)^\beta
+ \le Cd^{-2-\nu}.
+\]
+The retained central probe supplies $A(b_*,R)$ throughout this
+case. For $v\ge c_2R$, discard all arm probes and sum the
+absolute failure probabilities, obtaining $Ce^{-c'R^2}$.
+This is bounded by the right side of \eqref{cl:boundary-kernel}:
+$A(b_*,R)\ge cR^{-g}$ and
+$d_*\le d\le u\le\rho R$ make the reciprocal of that right
+side at most a fixed power of $R$. This case includes every
+failure whose support reaches the central probe, because
+$|y|\ge R/100$ and the central probe radius was chosen much
+smaller than $R/100$.
+Thus no overlapping supports have been treated as independent.
+The conditional total-variation argument on screen success,
+together with these failure bounds, proves uniformly that
+\begin{equation}\label{cl:boundary-kernel}
+ |K_R(y)|\le C A(b_*,R)\,
+     d^{-2-\nu}\left(\frac d u\right)^2
+                    \left(\frac u R\right)^\beta.
+\end{equation}
+
+For completeness, a dyadic class with distances comparable to
+$d,u$ has area at most $Cud$. Below the upper cap, it lies in
+corner disks of radius $Cu$ and boundary strips of width $Cd$.
+If $u$ is capped, the total strip length is $O(R)=O(u)$;
+if both distances are capped, the ambient region has area
+$O(R^2)=O(ud)$. These assertions apply equally outside the
+rectangle. The bottom classes have the same bounds with $d_*$
+in place of a smaller distance. Summing
+\eqref{cl:boundary-kernel} first over dyadic $d\le u$ gives
+\begin{align*}
+ \sum_{d\le u} Cud\,d^{-2-\nu}(d/u)^2(u/R)^\beta
+ &=\frac C u\left(\frac uR\right)^\beta
+                           \sum_{d\le u}d^{1-\nu}\\
+ &\le C u^{-\nu}(u/R)^\beta,
+\end{align*}
+where $\nu<1$ is used. The remaining dyadic sum is bounded by
+\[
+ C R^{-\beta}\sum_{d_*\le u\le\rho R}u^{\beta-\nu}
+ \le C R^{-\xi_0}
+ \qquad\text{for every }0<\xi_0<\min\{\beta,\nu\}.
+\]
+The strict inequality accommodates the logarithm when $\beta=\nu$.
+Together with \eqref{in:comparison}, this proves
+\begin{equation}\label{cl:boundary-integral}
+ \int_{R/100<|y|\le10R}|K_R(y)|\,dy
+ \le C f(R)R^{-\xi_0}.
+\end{equation}
+
+\paragraph{Far exterior insertions.}
+For $|y|>10R$ an agreement screen between the test rectangle and
+$y$ gives $|K_R(y)|\le Ce^{-c|y|^2}$. Its integral is at most
+$Ce^{-cR^2}$. The polynomial lower bound for $f$ following from
+\eqref{in:comparison} absorbs this into $Cf(R)R^{-\xi_0}$.
+Combining this with \eqref{cl:bulk-integral} and
+\eqref{cl:boundary-integral}, and decreasing the exponent, proves
+\eqref{cl:tail-eq}.
+\end{proof}
+
+\subsection{Comparing the normalized derivatives}
+
+The tail estimate permits us to discard insertions far from the
+origin. For the remaining insertions we use the same interior
+messages in two different rectangles. Rank-one approximation then
+controls a determinant, so that no unknown leading factor needs
+to be identified.
+
+\begin{lemma}[Two-size determinant]\label{cl:determinant}
+Let $\nu'>0$ be the exponent in Corollary~\ref{tr:wedge}.
+There are constants $C,c>0$ such that, for sufficiently large $m$,
+$100m\le R\le R'\le2R$, and almost every $0<|y|\le m$,
+\begin{equation}\label{cl:determinant-eq}
+ \bigl|K_R(y)f(R')-K_{R'}(y)f(R)\bigr|
+ \le C(m/R)^{2\lambda+\nu'}+Ce^{-cm^2}.
+\end{equation}
+The constants are independent of the distance from $y$ to the origin.
+\end{lemma}
+
+\begin{proof}
+Take concentric cuts about the origin, starting at radius $10m$
+and ending at a radius comparable with a small fixed fraction of
+$R$, with consecutive radii in ratio $L$. Round the final radius
+down to the last such scale. Its ratio to $R$ is bounded below
+by a positive constant depending only on the already fixed $L$.
+All cut collars lie strictly inside both rectangles and outside
+both inserted points. If necessary, enlarge the fixed lower
+bound on $R/m$; the remaining bounded ratios follow by increasing
+$C$, since all four scalar quantities in the determinant are bounded.
+
+First restrict every test to success of every cut screen, with
+the screens evaluated on the common base process. The probability
+discarded is bounded by
+\[
+ C\sum_{j\ge0}\exp\bigl(-cm^2L^{2j}\bigr)
+ \le Ce^{-c'm^2}.
+\]
+At the first cut let $a$ be the signed connection-state message
+obtained by changing the central color. Let $b_y$ be the
+difference of this message with and without the extra fair point
+at $y$. Both have zero mass conditional on each seam, and
+their total variation norms are bounded by $2$ and $4$,
+respectively. They depend on neither $R$ nor $R'$: on screen
+success their construction uses only the common interior
+Poisson variables and the first seam. In particular this
+description does not require disjoint neighborhoods of $0$ and $y$.
+
+Let $T$ be the composed transfer through the intervening cuts.
+The exterior crossing tests in $RQ_0$ and $R'Q_0$ define bounded
+linear functionals $\ell_R$ and $\ell_{R'}$ at the final cut.
+The four restricted expectations are
+\[
+ \ell_R(Ta),\quad \ell_{R'}(Ta),\quad
+ \ell_R(Tb_y),\quad \ell_{R'}(Tb_y).
+\]
+Their determinant is the evaluation of $T^{\otimes2}$ on
+$b_y\otimes a-a\otimes b_y$ by
+$\ell_R\otimes\ell_{R'}$. Corollary~\ref{tr:wedge} therefore
+bounds it by $C(m/R)^{2\lambda+\nu'}$.
+Restoring the discarded events changes the determinant by at
+most $Ce^{-cm^2}$, because each scalar factor is bounded.
+This proves \eqref{cl:determinant-eq}.
+\end{proof}
+
+\begin{proposition}[Summable logarithmic error]\label{cl:log-derivative}
+There are constants $\alpha>0$ and $C<\infty$ such that
+\begin{equation}\label{cl:log-derivative-eq}
+ \left|\frac{Rf'(R)}{f(R)}+\lambda\right|
+ \le CR^{-\alpha}
+\end{equation}
+for all sufficiently large $R$.
+\end{proposition}
+
+\begin{proof}
+Set $g(R)=Rf'(R)/f(R)$, which is defined and continuous for
+large $R$ by Lemma~\ref{cl:derivative} and
+\eqref{in:comparison}. We first prove a uniform estimate on
+each interval $[R,2R]$. Fix
+\[
+ \zeta=\frac{\nu'}{4(2+2\lambda+\nu')},\qquad
+ \eta=\frac{\nu'}8,\qquad m=R^\zeta.
+\]
+The exponent estimate \eqref{in:exponent-eq} implies
+$f(s)\ge s^{-\lambda-\eta}$ for all sufficiently large $s$.
+For $R\le R'\le2R$, divide \eqref{cl:determinant-eq} by
+$f(R)f(R')$ and integrate over $|y|\le m$. The result is
+\begin{align}
+ \left|\int_{|y|\le m}\frac{K_R(y)}{f(R)}\,dy
+       -\int_{|y|\le m}\frac{K_{R'}(y)}{f(R')}\,dy\right|
+ &\le C R^{2\lambda+2\eta}m^2
+          \bigl[(m/R)^{2\lambda+\nu'}+e^{-cm^2}\bigr]\notag\\
+ &\le C R^{-\nu'/2}.
+ \label{cl:normalized-near}
+\end{align}
+Indeed the power in the first term is exactly
+\[
+ -\nu'+2\eta+\zeta(2+2\lambda+\nu')=-\nu'/2,
+\]
+and the exponential term is smaller. For large $R$, the chosen
+$m$ satisfies all the scale restrictions of
+Lemmas~\ref{cl:tail} and~\ref{cl:determinant}.
+Lemma~\ref{cl:tail} bounds each complementary normalized integral
+by $C R^{-\zeta\xi}$. Since
+$g(R)=2\int K_R(y)/f(R)\,dy$ by
+\eqref{cl:derivative-eq}, we obtain
+\begin{equation}\label{cl:dyadic-oscillation}
+ \sup_{R\le R'\le2R}|g(R')-g(R)|\le CR^{-\alpha},
+ \qquad
+ \alpha=\min\{\nu'/2,\zeta\xi\}>0.
+\end{equation}
+
+The values $g(2^nR_0)$ form a Cauchy sequence, because the
+successive errors in \eqref{cl:dyadic-oscillation} are summable.
+The same estimate between dyadic points shows that $g(R)$ has
+a finite limit $\gamma$ as $R\to\infty$, and summing along
+$R,2R,4R,\ldots$ gives
+\[
+ |g(R)-\gamma|\le C\sum_{j\ge0}(2^jR)^{-\alpha}
+ \le C'R^{-\alpha}.
+\]
+It remains to identify this limit. Integration yields
+\[
+ \log f(R)-\log f(R_0)=\int_{R_0}^R g(s)\,\frac{ds}s.
+\]
+Dividing by $\log R$ shows that
+$\log f(R)/\log R\to\gamma$. By \eqref{in:exponent-eq} the
+same limit is $-\lambda$. Thus $\gamma=-\lambda$, proving
+\eqref{cl:log-derivative-eq}.
+\end{proof}
+
+\subsection{Completion of the normalization}
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+By Proposition~\ref{cl:log-derivative},
+\[
+ \frac{d}{dR}\log\bigl(R^\lambda f(R)\bigr)=O(R^{-1-\alpha}).
+\]
+The right side is integrable at infinity. Since $f(R)>0$ for
+large $R$, the logarithm has a finite real limit. Consequently
+\begin{equation}\label{cl:amplitude}
+ R^{5/4}f(R)\longrightarrow a\in(0,\infty).
+\end{equation}
+Positivity and finiteness follow from convergence of the
+\emph{logarithm}, rather than merely boundedness of the normalized
+probabilities. In fact the argument also gives
+$R^{5/4}f(R)=a(1+O(R^{-\alpha}))$.
+
+Recall that $S_\eps=f(1/\eps)$,
+$m_\eps=\eps^{-2}$, and
+$q_\eps=(\E N_\eps(Q_0))^{-1}$.
+The calibration limits \eqref{in:calibration-eq} are
+\[
+ m_\eps q_\eps S_\eps\longrightarrow c\in(0,\infty),
+ \qquad
+ q_\eps\E N_\eps\longrightarrow B\in(0,\infty).
+\]
+Equation~\eqref{cl:amplitude} gives
+$S_\eps\sim a\eps^{5/4}$, hence
+$q_\eps\sim(c/a)\eps^{3/4}$. It follows that
+\[
+ \eps^{3/4}\E N_\eps\longrightarrow \frac{Ba}{c}
+ =:c_V\in(0,\infty).
+\]
+Here $N_\eps$ counts color-pivotal sites for the original unwired
+unit-square event. The Poisson insertion was used only to
+differentiate $f$; it has not changed the pivotal count in this
+last formula. Taking reciprocals proves the equivalent assertion
+for the near-critical scale.
+\end{proof}
diff --git a/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/voronoi-pivotal-amplitude.pdf b/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/voronoi-pivotal-amplitude.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..1ed24bf23ef934ecc399aaf1af42098f4cf2cf42
--- /dev/null
+++ b/preprints/A-Pivotal-Amplitude-for-Voronoi-Percolation-from-Cardys-Formula-October-5-2026/voronoi-pivotal-amplitude.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:50c54704fe210d89ed392432d8b3e61c9d950c6a1d6dd171169362a41ce5d305
+size 601111
diff --git a/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/README.md b/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..8ee844322429081628ba1d5d4ec906636df46478
--- /dev/null
+++ b/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/README.md
@@ -0,0 +1,18 @@
+# [A Polynomial-Time 2-Approximation for Shortest Common Superstring](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 24, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026,
+  author = {{OpenAI}},
+  title = {{A Polynomial-Time 2-Approximation for Shortest Common Superstring}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/paper.pdf}{OAI:A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/figures/collective.tex b/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/figures/collective.tex
new file mode 100644
index 0000000000000000000000000000000000000000..856f588524a9c7f54450e50faf4c17ed3802fe2b
--- /dev/null
+++ b/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/figures/collective.tex
@@ -0,0 +1,44 @@
+\begin{tikzpicture}[font=\small,>=latex,
+  layer/.style={line width=.65pt,draw=black!55},
+  connection/.style={line width=1.35pt,draw=blue!45!black},
+  guide/.style={draw=black!25,densely dotted,line width=.45pt}]
+
+  \begin{scope}[x=.64cm,y=.25cm]
+    \node[anchor=west,font=\small\bfseries] at (-.75,24.2)
+      {A collective link through ordered layers};
+    \draw[->,black!65] (-.75,3) -- (10.05,3) node[right] {start};
+    \draw[->,black!65] (-.75,3) -- (-.75,21.7) node[above] {end};
+    \draw[guide] (0,3) -- (0,12);
+    \draw[guide] (9,3) -- (9,12);
+    \draw[guide] (-.75,12) -- (0,12);
+    \node[left,inner sep=3pt] at (-.75,12) {$Y=b$};
+    \node[below,inner sep=3pt] at (0,3) {$a$};
+    \node[below,inner sep=3pt] at (9,3) {$a+K$};
+
+    \foreach \c/\shade in {11/75,8/55,5/40} {
+      \draw[layer,draw=black!\shade] (-.55,\c-1) -- (0,\c-1);
+      \foreach \j in {0,...,8} {
+        \draw[layer,draw=black!\shade]
+          (\j,\j+\c-1) -- (\j,\j+\c) -- (\j+1,\j+\c);
+      }
+      \draw[layer,draw=black!\shade]
+        (9,9+\c-1) -- (9,9+\c) -- (9.55,9+\c);
+    }
+
+    \draw[connection,->] (0,12) -- (9,12);
+    \draw[connection,->] (9,12) -- (9,21);
+    \foreach \j/\name in {1/top,4/middle,7/bottom} {
+      \fill[blue!45!black] (\j,12) circle (2.1pt);
+      \node[above,inner sep=4pt,font=\scriptsize] at (\j,12) {\name};
+    }
+    \filldraw[connection,fill=white,line width=.9pt] (0,12) circle (2.4pt);
+    \node[above left,inner sep=4pt] at (0,12) {$R$};
+    \filldraw[connection,fill=white,line width=.9pt] (9,21) circle (2.4pt);
+    \node[above left,inner sep=4pt] at (9,21) {$R+K$};
+    \draw[<->,black!65] (10.1,12) -- (10.1,21)
+      node[midway,right,inner sep=4pt,align=left,font=\scriptsize]
+      {$K$ up steps};
+    \draw[<->,black!65] (0,.55) -- (9,.55)
+      node[midway,above,inner sep=3pt,font=\scriptsize] {$K=kp$};
+  \end{scope}
+\end{tikzpicture}
diff --git a/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/figures/cycles.tex b/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/figures/cycles.tex
new file mode 100644
index 0000000000000000000000000000000000000000..b0d6054c14796f5fa54be4ccf9562ac6d19a98ff
--- /dev/null
+++ b/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/figures/cycles.tex
@@ -0,0 +1,44 @@
+\begin{tikzpicture}[font=\small,>=latex,
+  block/.style={draw=black!60,rounded corners=2pt,fill=black!3,
+    minimum width=3.05cm,minimum height=1.15cm,align=center,inner sep=5pt},
+  link/.style={->,draw=black!65,line width=.8pt},
+  selected/.style={->,draw=blue!45!black,line width=1.4pt}]
+  \node[block] (source) at (0,2.8)
+    {Selected source\\$k$ layers, period $p$};
+  \node[block] (target) at (7.8,2.8)
+    {Target block\\$\ell$ layers, period $q$};
+  \node[block] (next) at (7.8,0)
+    {Next block};
+  \node[block] (last) at (0,0)
+    {Another block};
+
+  \draw[selected] (source.east) -- (target.west)
+    node[midway,below=7pt,align=center,font=\scriptsize]
+      {retain selected link\\cost at most $kp$};
+  \draw[link] (next.west) -- (last.east);
+  \draw[link] (last.north) -- (source.south);
+
+  \draw[->,draw=black!35,densely dashed,line width=.8pt]
+    (target.south) -- (next.north);
+  \draw[black!70,line width=1.1pt] (7.59,1.21) -- (8.01,1.63);
+  \draw[black!70,line width=1.1pt] (7.59,1.63) -- (8.01,1.21);
+  \node[anchor=west,align=left,font=\scriptsize] at (8.17,1.42)
+    {omit target's\\outgoing link};
+
+  \coordinate (contact) at (3.9,2.8);
+  \fill[blue!45!black] (contact) circle (2.4pt);
+  \node[anchor=south east,inner sep=5pt] at (contact) {$s$, $|s|\le q$};
+  \node[circle,draw=blue!45!black,fill=white,line width=.9pt,
+    minimum size=.48cm,inner sep=2pt] (root) at (3.9,4.65) {$\varepsilon$};
+  \node[above=3pt,font=\scriptsize] at (root.north) {empty word};
+  \draw[<->,draw=blue!45!black,line width=1.05pt]
+    (contact) -- (root.south)
+    node[midway,right=5pt,align=left,font=\scriptsize]
+      {round trip\\cost at most $q$};
+
+  \node[above=6pt,align=center,font=\scriptsize] at (source.north)
+    {greatest distinguished\\rotation on the cycle};
+  \node[anchor=north,align=center] at (3.9,-.93)
+    {Target budget $\ell q$: internal joining costs at most $(\ell-1)q$;\\
+     at least $q$ remains to root the short contact.};
+\end{tikzpicture}
diff --git a/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/figures/layers.tex b/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/figures/layers.tex
new file mode 100644
index 0000000000000000000000000000000000000000..c815e0fa872ce8f640d82cbe39e7db2a11b47726
--- /dev/null
+++ b/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/figures/layers.tex
@@ -0,0 +1,28 @@
+\begin{tikzpicture}[font=\small,>=latex,
+  layer/.style={line width=.65pt,draw=black!55},
+  connection/.style={line width=1.35pt,draw=blue!45!black},
+  guide/.style={draw=black!25,densely dotted,line width=.45pt}]
+
+  \begin{scope}[shift={(0,.75)}]
+    \node[anchor=west,font=\small\bfseries] at (0,5.3) {A window layer};
+    \draw[->,black!65] (0,0) -- (3.9,0) node[below] {start $x$};
+    \draw[->,black!65] (0,0) -- (0,4.5) node[above] {end $e$};
+    \draw[layer] (.55,1.25) -- (1.6,1.25) -- (1.6,3.05)
+      -- (3.1,3.05) -- (3.1,4.05) -- (3.75,4.05);
+    \fill[black!65] (1.6,1.25) circle (1.6pt);
+    \fill[black!65] (1.6,3.05) circle (1.6pt);
+    \node[anchor=north east,inner sep=3pt] at (1.6,1.25) {$f_i(x)$};
+    \node[anchor=south east,inner sep=3pt] at (1.6,3.05) {$l_i(x)$};
+    \draw[connection,->] (1.25,1.8) -- (1.25,2.55)
+      node[midway,left,inner sep=3pt] {up};
+    \draw[connection,->] (1.95,3.4) -- (2.8,3.4)
+      node[midway,above,inner sep=2pt] {down};
+    \fill[blue!45!black] (1.6,2.1) circle (2pt);
+    \node[anchor=west,inner sep=5pt] at (1.6,2.1) {$[x,e)$};
+    \draw[guide] (1.6,0) -- (1.6,1.17);
+    \node[below,inner sep=3pt] at (1.6,0) {$x$};
+    \node[anchor=north,align=center,font=\scriptsize] at (1.8,-.55)
+      {$f_i(x)=z_i(x-1)$,\quad $l_i(x)=z_i(x)$};
+  \end{scope}
+
+\end{tikzpicture}
diff --git a/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/paper.tex b/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/paper.tex
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+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage[margin=1in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{microtype}
+\usepackage{enumitem}
+\usepackage{booktabs}
+\usepackage{xcolor}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta,positioning,calc,decorations.pathreplacing}
+\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref}
+\usepackage[nameinlink,noabbrev,capitalise]{cleveref}
+\pdfinfoomitdate=1
+\pdftrailerid{}
+\pdfsuppressptexinfo=15
+\numberwithin{equation}{section}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\newtheorem{remark}[theorem]{Remark}
+\newcommand{\OPT}{\operatorname{OPT}}
+\newcommand{\pref}{\operatorname{pref}}
+\newcommand{\suf}{\operatorname{suf}}
+\newcommand{\len}[1]{\lvert #1\rvert}
+\newcommand{\eps}{\varepsilon}
+\newcommand{\ZZ}{\mathbb Z}
+\newcommand{\NN}{\mathbb Z_{\ge0}}
+\newcommand{\cost}{\operatorname{cost}}
+\setlist{itemsep=3pt,topsep=5pt}
+\emergencystretch=1em
+\title{A Polynomial-Time 2-Approximation for\protect\\ Shortest Common Superstring}
+\author{OpenAI}
+\date{September 24, 2026}
+\hypersetup{pdftitle={A Polynomial-Time 2-Approximation for Shortest Common Superstring},pdfauthor={OpenAI}}
+\begin{document}
+\maketitle
+\begin{abstract}
+We give a deterministic algorithm that, for every finite family of explicitly
+represented ordinary strings, outputs a common superstring of length at most
+twice the optimum in time polynomial in the total encoded input size, including
+symbol labels. The guarantee applies to the algorithm constructed here, not the
+classical maximum-overlap Greedy procedure.
+\end{abstract}
+\input{sections/01-introduction}
+\input{sections/02-counts}
+\input{sections/03-layers}
+\input{sections/04-connections}
+\input{sections/05-processing}
+\input{sections/06-cycles}
+\input{sections/07-algorithm}
+\input{references}
+\end{document}
diff --git a/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/references.tex b/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/references.tex
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+++ b/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/references.tex
@@ -0,0 +1,76 @@
+\begingroup\small
+\begin{thebibliography}{99}
+
+\bibitem{blum1994}
+Avrim Blum, Tao Jiang, Ming Li, John Tromp, and Mihalis Yannakakis.
+\newblock Linear approximation of shortest superstrings.
+\newblock \emph{Journal of the ACM}, 41(4):630--647, 1994.
+\newblock \href{https://doi.org/10.1145/179812.179818}{doi:10.1145/179812.179818}.
+
+\bibitem{breslauer1997}
+Dany Breslauer, Tao Jiang, and Zhigen Jiang.
+\newblock Rotations of periodic strings and short superstrings.
+\newblock \emph{Journal of Algorithms}, 24(2):340--353, 1997.
+\newblock \href{https://doi.org/10.1006/jagm.1997.0861}{doi:10.1006/jagm.1997.0861}.
+
+\bibitem{cycle2026}
+Nikolai Chukhin, Alexander S. Kulikov, Ivan Mihajlin, and Alexander Smal.
+\newblock A tight cycle-cover inequality for shortest common superstring.
+\newblock \emph{Electronic Colloquium on Computational Complexity},
+Report TR26-157, August 2026. Preprint.
+\newblock \url{https://eccc.weizmann.ac.il/report/2026/157/}.
+
+\bibitem{emv2023}
+Matthias Englert, Nicolaos Matsakis, and Pavel Vesel\'{y}.
+\newblock Approximation guarantees for shortest superstrings: Simpler and better.
+\newblock In \emph{34th International Symposium on Algorithms and
+Computation (ISAAC 2023)}, volume 283 of \emph{LIPIcs}, pages 29:1--29:17, 2023.
+\newblock \href{https://doi.org/10.4230/LIPIcs.ISAAC.2023.29}{doi:10.4230/LIPIcs.ISAAC.2023.29}.
+
+\bibitem{finewilf}
+N. J. Fine and H. S. Wilf.
+\newblock Uniqueness theorems for periodic functions.
+\newblock \emph{Proceedings of the American Mathematical Society}, 16:109--114, 1965.
+\newblock \href{https://doi.org/10.1090/S0002-9939-1965-0174934-9}{doi:10.1090/S0002-9939-1965-0174934-9}.
+
+\bibitem{collapsing2020}
+Alexander Golovnev, Alexander S. Kulikov, Alexander Logunov,
+Ivan Mihajlin, and Maksim Nikolaev.
+\newblock Collapsing superstring conjecture.
+\newblock In \emph{APPROX/RANDOM 2019}, volume 145 of \emph{LIPIcs},
+article 26, 2019.
+\newblock \href{https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.26}{doi:10.4230/LIPIcs.APPROX-RANDOM.2019.26};
+\href{https://arxiv.org/abs/1809.08669v3}{revised arXiv version, 4 June 2020}.
+
+\bibitem{golovnev2014}
+Alexander Golovnev, Alexander S. Kulikov, and Ivan Mihajlin.
+\newblock Solving {SCS} for bounded length strings in fewer than $2^n$ steps.
+\newblock \emph{Information Processing Letters}, 114(8):421--425, 2014.
+\newblock \href{https://doi.org/10.1016/j.ipl.2014.03.004}{doi:10.1016/j.ipl.2014.03.004}.
+
+\bibitem{mucha2013}
+Marcin Mucha.
+\newblock Lyndon words and short superstrings.
+\newblock In \emph{Proceedings of SODA 2013}, pages 958--972, 2013.
+\newblock \href{https://doi.org/10.1137/1.9781611973105.69}{doi:10.1137/1.9781611973105.69}.
+
+\bibitem{greedy2026}
+Hiroki Shibata.
+\newblock Disproving the greedy superstring conjecture.
+\newblock \href{https://arxiv.org/abs/2609.01365v1}{arXiv:2609.01365v1},
+1 September 2026. Preprint.
+
+\bibitem{sweedyk1999}
+Z. Sweedyk.
+\newblock A 2.5-approximation algorithm for shortest superstring.
+\newblock \emph{SIAM Journal on Computing}, 29(3):954--986, 1999.
+\newblock \href{https://doi.org/10.1137/S0097539796324661}{doi:10.1137/S0097539796324661}.
+
+\bibitem{tarhio1988}
+Jorma Tarhio and Esko Ukkonen.
+\newblock A greedy approximation algorithm for constructing shortest common superstrings.
+\newblock \emph{Theoretical Computer Science}, 57(1):131--145, 1988.
+\newblock \href{https://doi.org/10.1016/0304-3975(88)90167-3}{doi:10.1016/0304-3975(88)90167-3}.
+
+\end{thebibliography}
+\endgroup
diff --git a/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/sections/01-introduction.tex b/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/sections/01-introduction.tex
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+++ b/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/sections/01-introduction.tex
@@ -0,0 +1,31 @@
+\section{Introduction}\label{sec:intro}
+
+In the \emph{shortest common superstring problem}, the input is a finite collection $\mathcal S$ of finite strings, and the task is to find a shortest string containing every member of $\mathcal S$ as a contiguous substring. Write $\OPT(\mathcal S)$ for its minimum possible length. The strings are represented explicitly, with encoded symbol labels and string boundaries. Let $N$ be the total bit length of this representation and $L$ the sum of the string lengths measured in symbols. Each symbol and boundary occupies at least one bit, so $L,|\mathcal S|\le N$; operations on symbol labels are charged for their encoded lengths.
+
+\begin{theorem}\label{thm:main}
+There is a deterministic algorithm, polynomial in the total encoded input size, that on every finite collection $\mathcal S$ of explicitly represented ordinary strings outputs a common superstring $T$ satisfying
+\[
+\len{T}\le 2\OPT(\mathcal S).
+\]
+\end{theorem}
+
+\paragraph{History and significance.}
+Tarhio and Ukkonen proposed the factor-$2$ conjecture for the maximum-overlap Greedy procedure~\cite{tarhio1988}. Blum, Jiang, Li, Tromp, and Yannakakis established a polynomial-time $3$-approximation~\cite{blum1994}; Sweedyk later obtained a $5/2$ guarantee~\cite{sweedyk1999}. Subsequent guarantees include $2+11/23$ due to Mucha~\cite{mucha2013} and $(14+\sqrt{67})/9<2.466$ due to Englert, Matsakis, and Vesel\'{y}~\cite{emv2023}. An August 2026 preprint of Chukhin, Kulikov, Mihajlin, and Smal reports a $7/3$-approximation~\cite{cycle2026}. \Cref{thm:main} answers affirmatively the question of whether a polynomial-time factor-$2$ approximation exists.
+
+Many earlier approximation algorithms use the distance graph on the input strings, whose edge costs are the lengths of prefixes left after maximum suffix--prefix overlap. A minimum-weight cycle cover gives a computable lower bound, but its cycles must then be opened and joined while controlling the extra length~\cite{blum1994}. Breslauer, Jiang, and Jiang use rotations of periodic strings to control overlaps~\cite{breslauer1997}; Mucha sharpens this analysis through lexicographically extremal primitive rotations~\cite{mucha2013}. These provide conceptual precedents for the periodic comparisons below; our lower bound and the graph to which we apply them are different.
+
+Our graph representation comes from another formulation. Golovnev, Kulikov, and Mihajlin introduced a hierarchical graph in the bounded-length setting~\cite{golovnev2014}, and the Collapsing framework developed its all-substrings form~\cite{collapsing2020}. In this graph, appending or deleting a letter gives an edge between substring vertices, and a superstring is represented by a closed walk visiting the required strings. The separation between a balanced graph and the operations that connect it also appears in the hierarchical cycle-cover discussion of that framework~\cite[revised version, Section~7.2]{collapsing2020}. We use this representation to turn forced occurrence counts into a balanced graph and then connect it within the same cost bound.
+
+The factor-$2$ question has also motivated conjectures about specific procedures. The classical Greedy conjecture concerns repeatedly merging a pair of maximum overlap, while the Collapsing conjecture concerns a prescribed normalization of doubled hierarchical-graph walks~\cite{collapsing2020}. A September 2026 preprint of Shibata reports a counterexample to the Greedy conjecture, with worst-case ratio at least $9/4$~\cite{greedy2026}. Our theorem concerns the algorithm proved here; neither of those prescribed procedures is assumed to attain factor $2$.
+
+\paragraph{Proof overview.}
+The vertices of our hierarchical graph are all substrings of the inputs, including the empty word. An up edge appends a letter and costs one; a down edge deletes the first letter and costs zero. A balanced multiset of edges has equal incoming and outgoing degree at each vertex. If its support is connected and visits the empty word and every required string, an Euler tour spells a common superstring whose length is its up-edge cost.
+
+We first construct integer counts $m(s)$ no larger than the number of occurrences of $s$ in any common superstring. Their one-letter sum $W$ is therefore a lower bound on the optimum. Counts for longer words constrain those of their prefixes and suffixes. A further rule handles periodic words. Relative to a fixed periodic text, a bracketing word consists of a matching interval with a mismatching letter at each end. When the assigned count of a long word exceeds the contribution of its bracketing words, a prefix shorter by $k$ periods must have count at least $k+1$ above its own bracketing contribution. The resulting counts determine a balanced, possibly disconnected graph of cost $W$.
+
+Next decompose that graph into closed walks. The letters appended during a walk, repeated in both directions, form a periodic text; each visited vertex is a finite interval of that text. Walks with the same periodic text can be rearranged, preserving every edge, into ordered walks that each traverse one least period. We call these walks \emph{layers}. A layer of period $p$ has cost $p$ and receives a further budget $p$, so all additional budgets together equal $W$. In the remaining case of the connection procedure, the count rule forces a shorter word to occur on a layer with a different aligned text. A visited interval agreeing with both texts gives an actual graph vertex at which to attach a connection.
+
+The final task is to arrange these connections without spending a budget twice. Some closed walks connect their source layers to a target using only the source budgets. Others require a contribution from the target, but these requests go only to a strictly larger period and are handled together when that period is processed. For each connection paid for only by its source layers, those layers can instead be joined internally while saving one period of budget. The remaining source-funded connections may form directed cycles. On each cycle, a maximal lexicographic rotation yields a contact word no longer than the target's period. Omitting that target's outgoing connection and joining its layers internally therefore pays for a round trip from the contact to the empty word. This connects the base graph at additional cost at most $W$ and completes the factor-$2$ bound.
+
+\paragraph{Organization.}
+\Cref{sec:counts} constructs the counts and their balanced graph. \Cref{sec:layers} develops the layer representation and the periodicity facts. \Cref{sec:connections} gives the connection operations, \Cref{sec:processing} processes groups of equal primitive text, and \Cref{sec:cycles} resolves the remaining cycles. \Cref{sec:algorithm} proves polynomial time and extracts the output string.
diff --git a/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/sections/02-counts.tex b/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/sections/02-counts.tex
new file mode 100644
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+++ b/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/sections/02-counts.tex
@@ -0,0 +1,107 @@
+\section{Forced occurrence counts}\label{sec:counts}
+
+Delete the empty string, duplicates, and every input string contained in another input string. This preserves the set of common superstrings and its optimum. If nothing remains, output the empty string. Henceforth the remaining instance is nonempty. Let $\Sigma$ be its alphabet and let $V$ be the set of its distinct substrings, including the empty word $\eps$. Thus $V$ is closed under taking substrings, $\len V=O(L^2)$, and every required string is maximal under extension in $V$.
+
+For a finite string $s$, indices start at zero. Write $s[i:j)$ for its substring on the indicated half-open interval. A \emph{periodic text} is a map $A:\ZZ\to\Sigma$ with $A(i+p)=A(i)$ for some positive integer $p$. The period need not be least unless this is specified.
+
+\subsection{Blocking an occurrence}
+
+Fix a text $A$ and a nonempty $s$ such that $A[0:\len s)=s$. For $r\in V$, let $b_{A,s}(r)$ be the number of integers $j$ with
+\[
+1\le j,\qquad j+\len s\le\len r-1,
+\]
+for which $r[j:j+\len s)=s$ and
+\begin{align}
+r(h)&=A(h-j) &&(1\le h\le\len r-2),\label{eq:block-interior}\\
+r(0)&\ne A(-j),\qquad
+r(\len r-1)\ne A(\len r-1-j).\label{eq:block-ends}
+\end{align}
+The word $r$ consists of a matching interval bracketed by one mismatching letter on each side of the specified $s$. Define the linear functional
+\begin{equation}\label{eq:block-functional}
+B_A(s;m)=\sum_{r\in V\setminus\{\eps\}}b_{A,s}(r)m(r).
+\end{equation}
+Only words with $\len r\ge\len s+2$ contribute.
+
+If an occurrence of $s$ in a finite word $T$ is aligned with $A$, its nearest mismatch on each side, when both exist, determines a unique bracketing word $r$. Call the occurrence \emph{blocked} if this word belongs to $V$, and \emph{unblocked} otherwise. In particular, an occurrence with no mismatch on one side is unblocked. When $N_T(r)$ denotes the number of occurrences of $r$ in $T$, uniqueness of the nearest mismatches gives
+\begin{equation}\label{eq:block-count}
+B_A(s;N_T)=\#\{\text{blocked occurrences of }s\text{ in }T\}
+\le N_T(s).
+\end{equation}
+The multiplicity in $b_{A,s}(r)$ is necessary: a single occurrence of $r$ may bracket several distinct occurrences of $s$.
+
+\subsection{The rules}
+
+Construct nonnegative integers $m(s)$ for $s\in V\setminus\{\eps\}$ in decreasing order of length. Missing words have count zero. At each $s$, take the smallest integer satisfying
+\begin{equation}\label{eq:extension-rules}
+m(s)\ge\sum_{c\in\Sigma}m(cs),\qquad
+m(s)\ge\sum_{c\in\Sigma}m(sc),
+\end{equation}
+and $m(s)\ge1$ if $s$ is required. Impose also the following rules for every period-$p$ text $A$, every $v=A[0:\len v)\in V$, and every nonempty prefix $w$ of $v$ with $\len v=\len w+kp$, $k\ge1$:
+\begin{equation}\label{eq:period-rule}
+m(v)>B_A(v;m)
+\quad\Longrightarrow\quad
+m(w)\ge B_A(w;m)+k+1.
+\end{equation}
+All quantities constraining $m(w)$ have already been determined: $v$ is longer than $w$, and every term of either blocking functional is longer than its inner word. Thus these are explicit lower bounds in a descending recursion.
+
+The apparent range over periodic texts is finite for this purpose. In a rule, $p<\len v$, and $A$ is determined by the first $p$ letters of $v$. One may enumerate $v$, $p$, and $k$, check that repetition of the first $p$ letters agrees with $v$, and retain the admissible rules. This includes nonleast periods.
+
+\begin{lemma}[Occurrence lower bound]\label{lem:counts}
+For every common superstring $T$ and every nonempty $s\in V$,
+\[
+m(s)\le N_T(s).
+\]
+Consequently, for
+\begin{equation}\label{eq:W}
+W=\sum_{c\in\Sigma}m(c),
+\end{equation}
+we have $W\le\OPT(\mathcal S)$ and $m(s)\le L$.
+\end{lemma}
+
+\begin{proof}
+Fix $T$ and abbreviate $N_T$ to $N$. Induct on decreasing word length. The counts $N$ satisfy \eqref{eq:extension-rules}: occurrences with different preceding letters are disjoint, as are those with different following letters. They also satisfy the requirement for input strings.
+
+Consider a rule \eqref{eq:period-rule} constraining $m(w)$ whose trigger holds. If $N(v)>B_A(v;N)$, choose an unblocked occurrence of $v$. It contains occurrences of $w$ at offsets $0,p,\ldots,kp$. Each is unblocked. Indeed, because $v$ matches $A$ and the offsets are multiples of $p$, a nearest mismatch bracketing one of these shorter occurrences must lie outside $v$. A blocking word in $V$ for it would therefore also block the chosen occurrence of $v$. The $k+1$ shorter occurrences are distinct, so \eqref{eq:block-count} and induction give
+\[
+N(w)\ge B_A(w;N)+k+1\ge B_A(w;m)+k+1.
+\]
+
+In the other case, \eqref{eq:block-count} implies
+\[
+B_A(v;N)=N(v)\ge m(v)>B_A(v;m).
+\]
+Every difference $N(r)-m(r)$ in this blocking sum is nonnegative by induction. Some contributing word $r$ therefore has $N(r)-m(r)\ge1$. Choose one of its counted placements of $v$. The $k+1$ period-spaced prefixes $w$ within that placement give distinct counted placements of $w$ in the same $r$. Hence $b_{A,w}(r)\ge k+1$. All other terms in the difference of blocking sums are nonnegative, and therefore
+\[
+N(w)\ge B_A(w;N)\ge B_A(w;m)+k+1.
+\]
+Every lower bound used to define $m(w)$ is thus at most $N(w)$, completing the induction.
+
+For an optimal $T$, summing over one-letter words gives $W\le\len T$. For a concatenation of the inputs, $N_T(s)\le\len T\le L$, giving the second assertion.
+\end{proof}
+
+\subsection{The balanced base graph}
+
+The \emph{hierarchical graph} on $V$ has, for each nonempty $s$, an up edge $\pref(s)\to s$ and a down edge $s\to\suf(s)$, where $\pref$ deletes the last letter and $\suf$ deletes the first. The up edge has cost one and the down edge cost zero. Golovnev, Kulikov, and Mihajlin introduced a hierarchical graph for bounded-length inputs~\cite{golovnev2014}; the all-substrings form and these edge costs appear in the Collapsing framework~\cite[Section~2.2]{collapsing2020}. We give every graph property needed here directly.
+
+Give the two edges at $s$ multiplicities
+\begin{equation}\label{eq:base-multiplicities}
+u(s)=m(s)-\sum_c m(cs),\qquad
+d(s)=m(s)-\sum_c m(sc),
+\end{equation}
+respectively. Write $G$ for the resulting directed multigraph, omitting unused edges and vertices.
+
+\begin{lemma}\label{lem:base}
+The graph $G$ is balanced, contains every required string as a used vertex, and has $W$ up edges and $W$ down edges, counted with multiplicity.
+\end{lemma}
+
+\begin{proof}
+The multiplicities are nonnegative by \eqref{eq:extension-rules}. At a nonempty $s$, the incoming and outgoing degrees are, respectively,
+\[
+u(s)+\sum_c d(cs),\qquad d(s)+\sum_c u(sc),
+\]
+and both simplify to $m(s)-\sum_{c,d}m(csd)$. Balance at $\eps$ follows by summing degrees over the other vertices. Telescoping \eqref{eq:base-multiplicities} over all nonempty words gives $\sum_s u(s)=\sum_s d(s)=\sum_c m(c)=W$. Finally, a remaining required string has no proper extension in $V$, so both its edge multiplicities equal its positive count.
+\end{proof}
+
+For example, suppose the only required word is $a^h$, with $h\ge2$. The extension inequalities alone allow $m(a^j)=1$ for $1\le j\le h$. Their base graph is only the two-edge walk between $a^{h-1}$ and $a^h$, with cost $1$. In the constant text $a$, every blocking functional is zero. Applying the period-one rule to $v=a^h$ and $w=a^j$ forces $m(a^j)\ge h-j+1$ for $1\le j<h$; the terminal requirement gives the same bound for $j=h$. The occurrence lower bound gives equality. The resulting graph contains both directions of the entire path from $\eps$ to $a^h$, and $W=h$. Thus the periodicity rules record the repeated letters that the extension inequalities alone miss.
+
+We retain all base edges throughout. Adding closed walks preserves balance. To prove \Cref{thm:main}, it remains to connect every component of $G$ to $\eps$ at additional up cost at most $W$.
diff --git a/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/sections/03-layers.tex b/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/sections/03-layers.tex
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--- /dev/null
+++ b/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/sections/03-layers.tex
@@ -0,0 +1,117 @@
+\section{Periodic layers}\label{sec:layers}
+
+Our goal is to replace the closed walks of the base graph by ordered layers without changing any edge. Their actual windows will later provide the occurrence records and same-text joins used to spend the second $W$ budget.
+
+\subsection{Lifting closed walks to windows}
+
+Decompose the balanced base graph into nonempty closed walks. Every such walk has the same positive number of up and down steps: the word length changes by $+1$ or $-1$ at each step and returns to its starting value. Read the letters appended on the up steps cyclically, and repeat this circular word in both directions to obtain a periodic text.
+
+On the infinite repetition of the walk, represent a vertex by a window $[x,e)$ of this text. An up step increases $e$ by one; a down step increases $x$ by one. The window spells the vertex word. To verify this last assertion, follow the repeated walk sufficiently far back: its down steps eventually remove every letter previously present, so each current letter must have entered at one of the indicated up steps. A closed walk with $P$ up steps advances both coordinates by $P$.
+
+An occurrence interval of a nonempty word $s$ is \emph{recorded} on the walk if some visited window contains it. Count occurrences modulo the walk's turn length $P$, so each starting-position residue contributes at most once. Here the modulus is $P$, even if the text has a smaller least period.
+
+\begin{lemma}\label{lem:recorded}
+The total number of recorded occurrences of $s$ on all the closed walks is $m(s)$.
+\end{lemma}
+
+\begin{proof}
+At the unique up step reaching the right endpoint of a given occurrence, the current window contains the occurrence if and only if it is recorded. Indeed, subsequent left endpoints only increase, so a later containing window implies containment at that step as well. The up step generates a word ending in $s$. Summing its multiplicity over all such words gives
+\[
+\sum_{r\in V:\ r\text{ ends in }s}u(r)=m(s)
+\]
+by telescoping over left extensions in \eqref{eq:base-multiplicities}. The argument is on the infinite lift and therefore also applies when $\len s$ exceeds a turn length.
+\end{proof}
+
+\subsection{Sorting primitive passages}
+
+Group the walks by their periodic texts up to integer translation, and align the texts within each group. Group membership is up to translation, while a later record compares a specified coordinate alignment of two layers; that alignment can differ even within one group. Write $p$ for the least positive period of a group. Every original turn length is a multiple of $p$: the shifts preserving a two-sided text form a subgroup of $\ZZ$, generated by its least positive member.
+
+\begin{lemma}[Ordered layers]\label{lem:layers}
+The walks of a group can be rearranged, without changing the multiset of their graph edges, into one-turn layers of period $p$. Each layer has an integer function $z:\ZZ\to\ZZ$ such that
+\begin{equation}\label{eq:layer-properties}
+z(x-1)\le z(x),\qquad z(x-1)\ge x,
+\qquad z(x+p)=z(x)+p.
+\end{equation}
+At start coordinate $x$, its windows are exactly
+\begin{equation}\label{eq:layer-vertical}
+[x,e),\qquad z(x-1)\le e\le z(x);
+\end{equation}
+it then takes a down step to start $x+1$. The functions of the layers in a group are pointwise ordered. Each layer has up cost $p$, and the sum of these costs over all groups is $W$.
+\end{lemma}
+
+\begin{proof}
+For original walk $j$, let $P_j=d_jp$ be its turn length and let $Z_j(x)$ be its exit end at start $x$ on the infinite lift. Thus $Z_j(x+P_j)=Z_j(x)+P_j$. At each integer start $x$, collect the normalized exit ends
+\[
+\mathcal E_x=
+\bigl\{Z_j(x+rp)-rp:\ j,\ 0\le r<d_j\bigr\},
+\]
+with multiplicity. These are the passages through the same start residue, translated to start $x$. Translation by $rp$ preserves their word labels. Sort $\mathcal E_x$ increasingly and call its order statistics $z_1(x),\ldots,z_h(x)$, where $h=\sum_j d_j$.
+
+The corresponding entering ends at $x$ form $\mathcal E_{x-1}$. Each original exit is at least its paired entry, so the increasing order statistics also satisfy $z_i(x-1)\le z_i(x)$. Every entry is at least $x$. Moreover, increasing $x$ by $p$ permutes the passages from each original walk and adds $p$ to their ends. Thus $\mathcal E_{x+p}=\mathcal E_x+p$, proving all of \eqref{eq:layer-properties}.
+
+The multiset of down steps is fixed by the exit-end multisets. At a fixed $x$, the number of copies of a vertical unit edge with lower end $e$ is the number of entries at or below $e$ minus the number of exits at or below $e$. Thus the up-edge multiset is unchanged as well. The sorted passages therefore use exactly the original graph edges, with multiplicity, and in particular only actual base vertices. Each layer advances both coordinates by $p$ in one turn and has $p$ up steps. Their total up cost remains $W$.
+\end{proof}
+
+We henceforth use this fixed layer decomposition. A layer of period $p$ receives an additional \emph{budget} $p$. Budgets belong to layers even when several layers happen to share graph vertices.
+
+For a layer $i$, write
+\begin{equation}\label{eq:first-last}
+f_i(x)=z_i(x-1),\qquad l_i(x)=z_i(x)
+\end{equation}
+for its first and last ends at start $x$. A higher layer has both functions at least those of a lower layer. When a collection of layers is under discussion, the subscripts $H$ and $L$ denote its highest and lowest layers; a scalar such as $H=f_H(t)$ denotes an end coordinate. \Cref{fig:layers} shows the first and last ends in start--end coordinates. Every actual window on a layer is a graph vertex in $V$; two windows that spell the same word are the same graph vertex, even when they have different coordinates.
+
+\begin{figure}[htbp]
+\centering
+\input{figures/layers}
+\caption{A layer in start--end coordinates. An up step appends a letter and increases the end; a down step deletes the first letter and increases the start. At start $x$, the layer visits every window $[x,e)$ between its first end $f_i(x)$ and its last end $l_i(x)$.}
+\label{fig:layers}
+\end{figure}
+
+\subsection{Unblocked occurrences on layers}
+
+Align a recorded occurrence of $s$ on a layer with a template $A$ containing $s$ at position zero. Call it blocked if the interval from the nearest left mismatch to the nearest right mismatch is itself recorded on that layer. Such a block automatically belongs to $V$, since it is a substring of a visited window.
+
+\begin{lemma}[Matching window]\label{lem:matching-window}
+Over all layers, the number of blocked occurrences of $s$ relative to $A$ is $B_A(s;m)$. Every unblocked occurrence is contained in an actual layer window that agrees with $A$ throughout.
+\end{lemma}
+
+\begin{proof}
+Each recorded occurrence of a word $r$ counted in \eqref{eq:block-functional}, together with a counted placement of $s$ in it, gives a blocked occurrence. Conversely, the nearest mismatch pair uniquely determines $r$ and its placement. This correspondence commutes with translation by one layer turn, so it is a bijection also for the occurrences counted modulo that turn. \Cref{lem:recorded} gives the asserted count. Sorting layers did not change that lemma, since it preserved all base edges.
+
+For the second assertion, write the selected occurrence as $[u,v)$. Its first containing window is reached by the up step with end $v$, so that window contains no right mismatch. If it has no left mismatch either, it suffices. Otherwise let $\ell<u$ be the nearest left mismatch. Follow the layer until a down step removes $\ell$; this happens while the windows still contain $[u,v)$, since their starts eventually reach $u$. No right mismatch can enter before that step: together with $\ell$ it would make the nearest-mismatch bracket a recorded interval, contrary to the occurrence being unblocked. The down step introduces no new letter. Immediately afterward no mismatch remains on either side, and the resulting actual window agrees throughout with $A$.
+\end{proof}
+
+In particular,
+\begin{equation}\label{eq:unblocked-count}
+m(s)-B_A(s;m)
+\end{equation}
+counts unblocked recorded occurrences. A layer whose entire text is the aligned template has no blocked occurrences relative to that template.
+
+\subsection{Two periodicity facts}
+
+Periodicity and extremal rotations also enter earlier superstring cycle-cover analyses~\cite{breslauer1997,mucha2013}. We prove the exact agreement and maximal-rotation statements needed here. Use the total order on $\Sigma$ induced by the encoded symbol labels. The lexicographic order of infinite forward words is determined by their first differing letter; equality means agreement at every nonnegative position. In a periodic group, a \emph{distinguished position} $t$ starts its lexicographically greatest forward rotation.
+
+The agreement bound below uses only the weaker $p+q$ consequence of
+Fine--Wilf periodicity~\cite{finewilf}; we retain its direct proof here.
+
+\begin{lemma}[Agreement bounds]\label{lem:agreement}
+Two distinct aligned periodic texts of periods $p,q$ agree on fewer than $p+q$ consecutive letters. If both have period $p$, they agree on fewer than $p$ consecutive letters.
+\end{lemma}
+
+\begin{proof}
+Suppose that a period-$p$ text and a period-$q$ text agree on an interval of length $p+q$. For a full set of $q$ consecutive residues, both positions $i$ and $i+p$ lie in that interval. Equality with the first text therefore makes the second invariant under shift by $p$ at those residues, and $q$-periodicity extends this invariance everywhere. Both texts now have period $p$, and agreement for $p$ consecutive letters makes them equal. The same final observation proves the equal-period bound.
+\end{proof}
+
+\begin{lemma}[Maximal rotation]\label{lem:max-rotation}
+Let $A$ have least period $p$, and let $t$ be distinguished. Let $D$ be a distinct aligned periodic text of period $q$, all of whose forward rotations are lexicographically at most the forward word of $A$ at $t$. Then $D$ and $A$ cannot agree on all $q$ positions starting at $t$.
+\end{lemma}
+
+\begin{proof}
+Set $t=0$ by translating coordinates, and let $S$ be the forward word of $A$. It is at least its shift by $q$. If they are equal, $A$ has period $q$, and any period-$q$ text agreeing on the first $q$ letters is the identical aligned text.
+
+Otherwise let $j\ge0$ be the first index at which $S(j)\ne S(j+q)$. Maximality gives $S(j)>S(j+q)$. Let $U$ repeat the first $q$ letters of $S$. The equalities $S(i)=S(i+q)$ for $i<j$ imply that $U$ and $S$ agree before index $j+q$. At that index,
+\[
+U(j+q)=U(j)=S(j)>S(j+q),
+\]
+so $U>S$. If $D$ agreed on the first $q$ letters, its forward word there would be $U$, contrary to the hypothesis on its rotations.
+\end{proof}
diff --git a/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/sections/04-connections.tex b/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/sections/04-connections.tex
new file mode 100644
index 0000000000000000000000000000000000000000..838889f88607396431549b4c9603612dabad358f
--- /dev/null
+++ b/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/sections/04-connections.tex
@@ -0,0 +1,131 @@
+\section{Connection operations}\label{sec:connections}
+
+We now construct closed walks that touch existing layers at a cost charged to specified period budgets. We first give joins paid for entirely by the layers being attached, then an operation that also spends the receiving layer's budget. Each cross-text contact uses an actual word common to the two alignments; the band operation below is restricted to layers of one primitive-text group.
+
+A collection of layers and added walks is \emph{rooted} if its graph support is connected to $\eps$. Connectivity is in the underlying undirected graph; every edge multiset we use is balanced. To \emph{touch} a layer means to share at least one of its graph vertices. Since all base edges remain, touching any vertex of a layer connects to every vertex of that layer.
+
+\subsection{Following ordered windows}
+
+\begin{lemma}[Ordered windows]\label{lem:ordered-windows}
+Let $[a,b)$ and $[c,d)$ be windows in a common text over $\Sigma$, each spelling a word in $V$. If $a\le c$ and $b\le d$, there is a hierarchical-graph walk from the first word to the second of cost $d-b$. It visits a word of length at most $\max(0,b-c)$.
+
+Consequently an ordered list of windows, with both starts and ends nondecreasing, can be followed at cost equal to the total increase in its ends. If its last window is a translate of the first by $h$ in a text of period dividing $h$, this is a closed walk of cost $h$.
+\end{lemma}
+
+\begin{proof}
+If $c\le b$, delete the first $c-a$ letters, reaching $[c,b)$, and append through $d$. The words on the down path are suffixes of the first word; those on the up path are prefixes of the second. All belong to $V$, and the up cost is $d-b$.
+
+If $c>b$, delete to the empty word. For each letter in the gap $[b,c)$, take an up step to that letter and a down step back to empty. Then build $[c,d)$. Single-letter words in the gap belong to $V$ because the text uses only $\Sigma$. The cost is $(c-b)+(d-c)=d-b$. Concatenating these constructions proves the assertions about lists and translates.
+\end{proof}
+
+In particular, a round trip from a vertex $s$ to $\eps$ and back costs $\len s$. Prescribed windows, rather than all intervals they sweep across, must belong to $V$; \Cref{lem:ordered-windows} supplies the intermediate vertices.
+
+\begin{lemma}[Joining an ordered band]\label{lem:band}
+Consider $k$ ordered layers from one primitive-text group of period $p$. At a start $t$, let $H$ be the top layer's first end and $E$ the bottom layer's last end. If $h$ is a nonnegative multiple of $p$ and $H-E\le h$, the layers can be joined together by a closed walk of cost at most $h$.
+\end{lemma}
+
+\begin{proof}
+If $E\ge H$, every layer contains $[t,E)$: its first end is at most $H$ and its last end is at least $E$. No added walk is needed. Otherwise grow at start $t$ from end $E$ to $H$. Every layer has a vertical range intersecting $[E,H]$, so the growth touches them all. The endpoints are actual bottom and top vertices, and every intermediate word is a prefix of the top vertex. Finish by following to $[t+h,E+h)$, which is a period translate of the starting window. Since $E+h\ge H$, \Cref{lem:ordered-windows} gives a closed walk of cost $h$.
+\end{proof}
+
+\subsection{A collective outgoing link}
+
+We call a chosen actual layer window, together with an alignment to a specified periodic text agreeing throughout it, a \emph{record}. A record on a receiving layer provides a vertex at which to attach layers following the other text. An ordinary \emph{link} spends only the attached layers' budgets. The following link attaches $k$ source layers at combined cost $kp$; it can therefore be planned without reserving any budget of the receiving layer. \Cref{fig:collective} illustrates its construction.
+
+\begin{lemma}[Collective link]\label{lem:collective}
+Consider $k$ ordered layers with a common period-$p$ text $A$, and put $K=kp$. Let $R=[a,b)$ spell a word in $V$ agreeing with $A$. Let $r_H=f_H(a)$ be the top layer's first end and $r_L=l_L(a)$ the bottom layer's last end. If
+\begin{equation}\label{eq:collective-conditions}
+r_H\le b+K,\qquad r_H-r_L\le K,
+\qquad r_L+K\ge b,
+\end{equation}
+there is a cost-$K$ closed walk based at the word of $R$ that touches all $k$ layers.
+\end{lemma}
+
+\begin{proof}
+Put $Y=\max(b,r_H)$. Then $b\le Y\le b+K$. For each constituent layer $i$,
+\[
+f_i(a)\le Y\le l_i(a)+K=l_i(a+K).
+\]
+Consequently its path meets height $Y$ at a start in $[a,a+K]$. More explicitly, choose the least such start $x_i$ with $z_i(x_i)\ge Y$. If $x_i=a$, then $z_i(a-1)=f_i(a)\le Y$; otherwise minimality gives $z_i(x_i-1)<Y$. Thus $[x_i,Y)$ is an actual vertex. Pointwise layer order makes the $x_i$ nondecreasing when layers are taken from top to bottom. The list from $R$ through these vertices to $R+K$ has nondecreasing starts and ends, and it closes because $K$ is a multiple of $p$. Apply \Cref{lem:ordered-windows}.
+\end{proof}
+
+\begin{figure}[tbp]
+\centering
+\input{figures/collective}
+\caption{A collective link in start--end coordinates. The common height $Y$ meets the ordered layers from top to bottom as the start increases. The endpoints $R$ and $R+K$ spell the same word in the source text, so the path closes at cost $K$. The illustration uses $Y=b$; the precise conditions are in \Cref{lem:collective}.}
+\label{fig:collective}
+\end{figure}
+
+\subsection{Several requests at one host}
+
+A \emph{request} reserves a source layer's budget for an operation at a receiving layer, called its \emph{host}. Unlike a link, this operation may also spend the host's budget. It handles all requests at that host together and roots every participating layer. In \Cref{sec:processing}, we will send requests only when the following hypotheses hold.
+
+\begin{lemma}[Host requests]\label{lem:host}
+Let a host layer have period $q$. On that layer choose actual windows $R_i=[a_i,b_i)$, indexed by a nonempty collection of distinct child groups. The text of child group $i$ has period $p_i$, is different up to translation from the host's text and from every other child group's text, and agrees with the host throughout $R_i$ in the chosen alignment.
+
+For each $i$, one or more child layers request a connection at this same record. Each has a specified actual window $[a_i,e)$ satisfying
+\begin{equation}\label{eq:request-condition}
+a_i+p_i<e\le b_i-p_i.
+\end{equation}
+Its period-$p_i$ budget is reserved for the request. If the host's period-$q$ budget is also available, the host and all requesting layers can be rooted using at most these budgets.
+\end{lemma}
+
+\begin{proof}
+We will spend the host budget on one closed excursion through contact words for the requesting layers. Requesters from all but one child group attach to those contacts using their own budgets. The requesters from the remaining group are touched directly by the host excursion, leaving one of their budgets to connect a contact shorter than their period to $\eps$.
+
+By \Cref{lem:agreement},
+\begin{equation}\label{eq:host-length}
+b_i-a_i<p_i+q.
+\end{equation}
+If record $i$ is before record $l$ along a lift of the host path, their starts and ends are ordered. When $b_i>a_l$, their overlap is an agreement interval between two distinct child texts, so
+\begin{equation}\label{eq:child-overlap}
+b_i-a_l<p_i+p_l.
+\end{equation}
+If there is no overlap, the same strict inequality is immediate.
+
+Choose an index $j$ of minimum $p_j$. Start at its record and order all other records before the next copy of it, translating coordinates by multiples of $q$ as needed. A shift by $\delta\in q\ZZ$ also transports that record's entire child coordinate system: replace its text by $A_i'(x)=A_i(x-\delta)$, its layer functions by $z_i'(x)=z_i(x-\delta)+\delta$, and its requester ends by $e+\delta$. This preserves all graph words, child periods, and budgets; the host text is unchanged because it is $q$-periodic. The same convention applies to the next copy of record $j$ when comparing overlaps. Put $T=a_j+q$. The order gives
+\begin{equation}\label{eq:host-order}
+a_j\le a_i\le T,\qquad b_j\le b_i\le b_j+q.
+\end{equation}
+
+To construct the host excursion, first visit all specified windows of the requesters at $j$ at start $a_j$, in increasing order of their ends $e$, and then visit $R_j$. For each later record $i$, the raw contact $[a_i+p_i,e)$ is obtained by deleting the first $p_i$ letters of the requester window and lies inside $R_i$. To place these contacts after $R_j$ and before the return at start $T$, we must keep their ends at least $b_j$ and their starts at most $T$. Accordingly, replace the raw contact by
+\begin{equation}\label{eq:host-target}
+Q_{i,e}=[s_i,h_{i,e})
+=[\min(a_i+p_i,T),\max(e,b_j)).
+\end{equation}
+Visit these targets in record order and, within each record, in increasing order of $e$. Finally return to the first requester window translated by $q$.
+
+For earlier $i$ and later $l$, \eqref{eq:request-condition} and \eqref{eq:child-overlap} give
+\[
+a_i+p_i<e_i\le b_i-p_i<a_l+p_l<e_l.
+\]
+Thus the raw targets have ordered starts and ends. The monotone operations in \eqref{eq:host-target} preserve this order. By \eqref{eq:host-order}, each modified target remains inside its record $R_i$, and it is nonempty because $s_i\le a_i+p_i<e\le h_{i,e}$. It therefore spells a word in $V$ and matches child text $i$.
+
+Applying \eqref{eq:child-overlap} to $i$ and the next occurrence of $j$ gives
+\[
+b_i-T<p_i+p_j.
+\]
+Together with \eqref{eq:host-length}, this implies
+\begin{equation}\label{eq:host-short-end}
+e\le b_i-p_i<T+p_j,\qquad b_j<T+p_j,
+\qquad h_{i,e}<T+p_j.
+\end{equation}
+If $e_0$ is the first requester end at $j$, then $e_0>a_j+p_j$, so the final translated window $[T,e_0+q)$ has end greater than $T+p_j$. The whole list is therefore ordered. Its closed excursion costs $q$ by \Cref{lem:ordered-windows}. On its final transition it reaches a vertex of length less than $p_j$. If there are no later records, the same conclusion follows using $b_j<T+p_j$.
+
+It remains to attach the other requesters and root the piece. For a requester at $i\ne j$, the target satisfies
+\begin{equation}\label{eq:target-child-range}
+a_i\le s_i\le a_i+p_i,
+\qquad e\le h_{i,e}\le e+p_i.
+\end{equation}
+Only the upper end bound needs an additional check. Here the choice of a minimum period $p_j$ is used: if $h_{i,e}=b_j$, then
+\[
+b_j<a_i+p_j+p_i\le a_i+2p_i<e+p_i.
+\]
+In child text $i$, follow the ordered list
+\[
+Q_{i,e},\quad[a_i+p_i,e+p_i),\quad Q_{i,e}+p_i.
+\]
+The middle word is a period translate of the actual requester vertex; the last word equals the first. The inequalities \eqref{eq:target-child-range} make the list ordered. Its cost-$p_i$ loop attaches this requester at the target. Do this separately for all requesters with $i\ne j$.
+
+The requesters at $j$ were already touched in the initial vertical part, so their budgets have not been spent. Use one of their budgets $p_j$ to connect the short contact from \eqref{eq:host-short-end} to $\eps$. The host is touched at $R_j$, all requesters are attached, and the expenditure is at most the host budget plus all requester budgets.
+\end{proof}
diff --git a/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/sections/05-processing.tex b/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/sections/05-processing.tex
new file mode 100644
index 0000000000000000000000000000000000000000..845c077d65dc33cb7f927b9b4fd2a4a396588a6f
--- /dev/null
+++ b/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/sections/05-processing.tex
@@ -0,0 +1,147 @@
+\section{Processing the period groups}\label{sec:processing}
+
+We now give the connection procedure. Process the primitive-text groups in increasing order of least period, breaking ties by the lexicographically least rotation of their primitive texts. Requests reserve budgets and are fulfilled when their host group is processed. Ordinary links are planned now and finalized in \Cref{sec:cycles}. Every record and every layer in these plans refers to the fixed base decomposition from \Cref{lem:layers}. For bookkeeping, mark a layer rooted when a stated operation explicitly roots it. Incidental connections to $\eps$ do not change these marks or budget assignments.
+
+\subsection{Incoming requests and a baseline}
+
+At entry to a group of period $p$, fulfill all requests received by each recipient layer in one application of \Cref{lem:host}. We will prove below that every requester has smaller period, and that a child group sends requests using only one record on one recipient layer. These are precisely the facts needed to apply that lemma with fresh recipient budgets.
+
+Every recipient has an actual vertex of length less than $2p$: its record with a child of period $r<p$ has length less than $p+r$ by \Cref{lem:agreement}. If there are recipients, choose their highest layer as a \emph{baseline}. They and their requesters have already been rooted.
+
+Each remaining layer below the baseline can be rooted with its own budget $p$. If it has a vertex of length at most $p$, take a round trip through $\eps$. Otherwise choose a baseline vertex $S=[a,h)$ of length less than $2p$. A lower layer has a vertex $[a,e)$ with
+\[
+p<e-a\le h-a<2p.
+\]
+For example, take $e=\min(l_i(a),h)$; pointwise ordering ensures this lies in its vertical range. The windows
+\[
+[a,h),\qquad[a+p,e+p),\qquad[a+p,h+p)
+\]
+are ordered because $h<a+2p<e+p$. Their cost-$p$ closed walk touches the lower layer and the rooted baseline. Recipient layers themselves need no further budget.
+
+Let $n$ be the number of layers above the baseline, or the entire number in the group when there is no baseline. Their budgets are all still available. If $n=0$, proceed to the next group.
+
+\subsection{Easy connections for the upper layers}
+
+Use a distinguished position $t$ of the group's text $A$, and put
+\[
+H=f_H(t),
+\]
+the top layer's first end at that start. If $H-t\le np$, a round trip from $\eps$ through $[t,H)$ costs at most $np$. Its up path touches every upper layer, since every such layer has first end at most $H$. This roots them all.
+
+If a baseline exists and its last end $E$ at $t$ satisfies $E\ge H-np$, \Cref{lem:band}, including the baseline as the bottom layer and taking $h=np$, joins it to all upper layers within their combined budget. This also roots them all.
+
+It remains to handle the case
+\begin{equation}\label{eq:hard-case}
+H>t+np,
+\qquad l_{\mathrm{base}}(t)<H-np
+\quad\text{when a baseline exists}.
+\end{equation}
+
+\subsection{Forcing a record on another aligned text}
+
+The count rule will supply $k+1$ unblocked occurrences of a word obtained by shortening the top vertex by $kp$. We choose $k$ so that at most $k$ of those occurrences can have the present full-text alignment. The same choice must keep the top $k$ layers close enough to join for at most $(k-1)p$, leaving one period of their combined budget available. The following threshold search gives both properties.
+
+Starting at $k=1$, increase $k$ while the number of layers of this group satisfying
+\[
+l_i(t)\ge H-kp
+\]
+exceeds $k$. Stop at the first failure of this test and set $K=kp$.
+
+\begin{lemma}\label{lem:choose-k}
+This procedure stops with $1\le k\le n$. If $L_*$ is the lowest of the top $k$ layers, then
+\begin{align}
+l_{L_*}(t)&\ge H-(k-1)p,\label{eq:sharp-band}\\
+f_H(x)-l_{L_*}(x)&\le K\qquad(x\in\ZZ).\label{eq:wide-band}
+\end{align}
+At most $k$ recorded occurrences of $w=A[t:H-K)$ have a full-text alignment identical to $A$.
+\end{lemma}
+
+\begin{proof}
+For $k\le n$, layers at or below the baseline do not qualify, by \eqref{eq:hard-case}. Thus the process stops by $n$. If $k=1$, the top last end is at least its first end $H$. If $k>1$, reaching $k$ means that at least $k$ layers qualified at the previous threshold $H-(k-1)p$. Pointwise ordering gives \eqref{eq:sharp-band}.
+
+For any $x$, choose its translate $x'$ modulo $p$ in $(t-p,t]$. Then
+\[
+f_H(x')\le H,
+\qquad l_{L_*}(x')\ge l_{L_*}(t)-p.
+\]
+Their difference is invariant under period translation, proving \eqref{eq:wide-band}.
+
+By \eqref{eq:hard-case}, $H-K>t$, so $w$ is nonempty. Each primitive layer has exactly one position residue whose full-text alignment agrees with $A$: the shifts preserving its text are exactly the multiples of $p$. At that alignment, $w$ is recorded exactly when $l_i(t)\ge H-K$. At most $k$ layers satisfy this inequality when the test stops. Other primitive-text groups have no identical full-text alignment.
+\end{proof}
+
+Apply the count rule \eqref{eq:period-rule}, with origin shifted to $t$, to the actual vertex word
+\[
+v=A[t:H),\qquad w=A[t:H-K).
+\]
+The top layer records an unblocked occurrence of $v$ in its own text. By \eqref{eq:unblocked-count}, the trigger holds, and the rule gives at least $k+1$ unblocked occurrences of $w$. \Cref{lem:choose-k} leaves at least one on a layer $D$ whose aligned text differs from $A$. A different phase of this same group is allowed.
+
+Choose one such occurrence. \Cref{lem:matching-window} gives an actual window $R=[a,b)$ on $D$, agreeing with $A$ throughout and containing the specified interval $[t,H-K)$. Let $q$ be the least period of $D$. Thus
+\begin{equation}\label{eq:chosen-record}
+a\le t,
+\qquad b\ge H-K,
+\qquad b-a<p+q.
+\end{equation}
+If $D$ belongs to this same primitive-text group, then
+\begin{equation}\label{eq:same-group-record}
+b-a<p.
+\end{equation}
+The strict length bounds follow from \Cref{lem:agreement}. Use this \emph{one} chosen record for every subsequent connection from this processed group.
+
+\subsection{Distributing the remaining budgets}
+
+At start $a$, put $r_H=f_H(a)$ and $r_L=l_{L_*}(a)$. Monotonicity, \eqref{eq:wide-band}, and \eqref{eq:chosen-record} give
+\begin{equation}\label{eq:block-preconditions}
+r_H-r_L\le K,
+\qquad r_H\le H\le b+K.
+\end{equation}
+
+\paragraph{The collective case.}
+If $r_L+K\ge b$, make the top $k$ layers one \emph{collective block}. Plan the link in \Cref{lem:collective} from these layers to $D$ at $R$, reserving their combined budget $K$. Here a collective block is allowed to have $k=1$; its plan is still the collective construction. If the test fails, do not make the block. In that event,
+\begin{equation}\label{eq:failed-collective}
+r_H\le r_L+K<b.
+\end{equation}
+
+\paragraph{The individual cases.}
+Handle every upper layer not placed in that collective block separately. Its first end at $a$ is at most $b$. When the collective test failed, this follows from \eqref{eq:failed-collective}; a layer below the top $k$ has
+\[
+f_i(a)\le l_i(t)<H-K\le b
+\]
+by the stopping rule and $a\le t$.
+
+For such a layer, do exactly one of the following.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item If some vertex has length at most $p$, root the layer by a round trip through $\eps$ using at most its budget.
+\item If all its vertex lengths exceed $p$ and
+\begin{equation}\label{eq:single-test}
+l_i(a)+p\ge b,
+\end{equation}
+make it an \emph{individual block} with a planned link to $D$ at $R$. At start $a+p$, its vertical range intersects $[b,b+p]$, since
+\[
+f_i(a+p)=f_i(a)+p\le b+p,
+\qquad l_i(a+p)=l_i(a)+p\ge b.
+\]
+Choose an actual end $e'\in[b,b+p]$ there and follow
+\begin{equation}\label{eq:individual-list}
+[a,b),\qquad[a+p,e'),\qquad[a+p,b+p).
+\end{equation}
+This is a cost-$p$ closed walk in $A$ touching the layer. Its incoming transition to start $a+p$ reaches a word of length at most
+\begin{equation}\label{eq:single-short}
+\max(0,b-a-p)\le q,
+\end{equation}
+by \eqref{eq:chosen-record} and \Cref{lem:ordered-windows}. Reserve this layer's budget for that link.
+\item Otherwise reserve the layer's budget $p$ and send a request to $D$ at the same record $R$. Its first end $e=f_i(a)$ satisfies
+\begin{equation}\label{eq:outgoing-request}
+a+p<e\le l_i(a)<b-p.
+\end{equation}
+This is the condition of \Cref{lem:host}. In particular $b-a>2p$; combining this with $b-a<p+q$ gives $q>p$. The recipient group will therefore be processed later.
+\end{enumerate}
+
+This verifies the promised hypotheses for incoming requests. They always go to strictly larger periods, and a child group uses just one record on one recipient layer, even if several of its layers request there. Ordinary planned links consume no recipient budget. When a recipient is processed, its own budget and all reserved requester budgets are available. After \Cref{lem:host} is applied, those layers are rooted and do not enter any outgoing block.
+
+\begin{proposition}\label{prop:partition}
+After all groups have been processed, every layer is either marked rooted or belongs to exactly one outgoing block. These two classes are disjoint. Each block consists of layers from one primitive-text group and targets a single actual layer. Its total budget remains reserved for its planned link. A collective block of $k$ layers and period $p$ also satisfies \eqref{eq:sharp-band} at its distinguished position; an individual block consists of one layer.
+\end{proposition}
+
+\begin{proof}
+Incoming requests use disjoint requester budgets, and each recipient budget is spent only when its group is processed. The baseline step uses only budgets of remaining lower layers. The easy upper-layer branches use at most the combined budgets of the upper layers and finish them. In the remaining branch, the collective case and the individual cases partition those upper layers. Each request is eventually fulfilled, because its recipient is processed later. The layers not marked rooted by a direct operation or a fulfilled request are precisely the outgoing blocks. Their links have only been planned, so their budgets are unspent. The asserted inequality for collective blocks is \eqref{eq:sharp-band}.
+\end{proof}
diff --git a/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/sections/06-cycles.tex b/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/sections/06-cycles.tex
new file mode 100644
index 0000000000000000000000000000000000000000..18026bdf02ee279f719dc6691a6c0427497ec9cd
--- /dev/null
+++ b/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/sections/06-cycles.tex
@@ -0,0 +1,68 @@
+\section{Opening cycles of planned links}\label{sec:cycles}
+
+It remains to execute the plans in \Cref{prop:partition}. Regard a block as one node and map its target layer to the block containing that layer, or to a rooted sink if the target is marked rooted. Each nonsink node has exactly one outgoing edge. A component therefore reaches a rooted sink or contains one directed cycle. Additional graph intersections between layers can only help; the following argument does not need them to be recorded by this bookkeeping graph.
+
+\begin{lemma}[One free period]\label{lem:free-period}
+A block of $k$ layers and period $p$ can be joined internally, in place of its outgoing link, for cost at most $(k-1)p$. Thus this replacement leaves at least $p$ of its total budget available.
+\end{lemma}
+
+\begin{proof}
+For an individual block, or any block with $k=1$, no internal addition is needed. For a collective block, \eqref{eq:sharp-band} gives
+\[
+f_H(t)-l_{L_*}(t)\le(k-1)p.
+\]
+Apply \Cref{lem:band} with $h=(k-1)p$.
+\end{proof}
+
+\begin{lemma}[A short contact on a maximal link]\label{lem:cycle-contact}
+On a directed cycle of blocks, choose a block whose group's distinguished forward rotation is lexicographically greatest. If its target layer has period $q$, its planned outgoing link can be realized within its reserved budget so that the resulting connected piece contains a vertex of length at most $q$.
+\end{lemma}
+
+\begin{proof}
+If the plan is an individual link, \eqref{eq:single-short} already gives the contact. For a collective plan with $k$ source layers of period $p$, put $K=kp$. Recall that $t$ is distinguished, $H=f_H(t)$, and the target record $R=[a,b)$ agrees with the source text and satisfies
+\[
+a\le t<H-K\le b,\qquad b-a<p+q.
+\]
+If any constituent layer already contains a vertex of length at most $q$, the ordinary collective link connects that vertex to the target and to all other constituents, so it suffices.
+
+Otherwise every vertex on every constituent layer has length greater than $q$. Every forward rotation of the target's text is at most its own distinguished rotation, and hence at most the selected block's distinguished rotation. The target alignment is different from the source alignment by construction. If $b-t\ge q$, agreement on $R$ would include all $q$ positions starting at $t$, contrary to \Cref{lem:max-rotation}. Thus
+\begin{equation}\label{eq:cycle-agreement}
+b-t<q.
+\end{equation}
+Put $x=\min(t,a+K)$. We have $a\le x\le a+K$ and $x\le t<b$. If $x=t$, \eqref{eq:cycle-agreement} gives $b-x<q$. If $x=a+K<t$, then
+\[
+b-x=b-a-K<p+q-K\le q.
+\]
+Thus $[x,b)$ is a suffix of $R$ of length less than $q$.
+
+At start $x$, every constituent layer's first end exceeds $b$, because its corresponding word has length greater than $q>b-x$. These first ends are all at most $H$: they are no greater than the top first end at $x$, which is at most its first end at $t$. Let $F=f_H(x)$. Then $b<F\le H\le b+K$. Descend from $R$ to $[x,b)$ and grow at start $x$ to $[x,F)$. This growth touches every constituent at its first end, and all intermediate words are prefixes of the actual top-layer vertex. Finish at $R+K$.
+
+In summary, the prescribed list is
+\begin{equation}\label{eq:cycle-list}
+[a,b),\qquad[x,b),\qquad[x,F),\qquad[a+K,b+K).
+\end{equation}
+Its starts and ends are nondecreasing. Every prescribed word is in $V$, and the last equals the first in the period-$p$ source text. \Cref{lem:ordered-windows} makes it a cost-$K$ closed walk, with the short contact $[x,b)$.
+\end{proof}
+
+\begin{theorem}[Connecting the base graph]\label{thm:connection}
+The base graph can be connected to $\eps$ by adding closed walks of total up cost at most $W$.
+\end{theorem}
+
+\begin{proof}
+Use the group procedure of \Cref{sec:processing}. Its fulfilled requests and direct connections have already rooted the layers outside outgoing blocks within their assigned budgets. Consider the remaining functional graph.
+
+For a cycle consisting of one block, its target layer belongs to the same block, hence to the same primitive-text group. The chosen record has a different alignment, so \eqref{eq:same-group-record} gives $\len R<p$. Replace the outgoing link by the internal joining of \Cref{lem:free-period}. The target layer, and therefore $R$, lies in this joined piece. The freed budget $p$ pays for a round trip between $R$ and $\eps$.
+
+For a cycle of at least two blocks, choose the outgoing link in \Cref{lem:cycle-contact}; write $q$ for its target block's period. Retain that selected link, with its contact of length at most $q$. Omit the \emph{target block's} outgoing link, replacing it by internal joining. This frees $q$ by \Cref{lem:free-period}. The selected link and the omitted link are different, since the cycle has length greater than one. All other cycle links, together with the target's internal joining, connect the entire cycle into one piece. Use the freed $q$ to root the short contact. \Cref{fig:cycles} illustrates this choice of the omitted edge.
+
+Distinct directed cycles have disjoint blocks, so these replacements do not compete for budgets. Finally execute the links leading into the rooted pieces, in increasing distance from a root or an opened cycle. Each attaches its source block using only that source's reserved budget. All layers are now rooted.
+
+Every addition is a closed walk. The initial layer budgets sum to $W$, and each budget was used only in its assigned operation or its cycle replacement. The total added up cost is therefore at most $W$.
+\end{proof}
+
+\begin{figure}[t]
+\centering
+\input{figures/cycles}
+\caption{Opening a directed cycle of planned block links. Keep a link whose source has maximal distinguished rotation and realize it with a contact of length at most $q$, the target's period. Omit the target block's outgoing link and join that block internally, freeing $q$ to connect the contact to the empty word. Arrows between blocks represent planned dependencies, each realized by a closed walk; the dot is a contact vertex, not an extra block.}
+\label{fig:cycles}
+\end{figure}
diff --git a/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/sections/07-algorithm.tex b/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/sections/07-algorithm.tex
new file mode 100644
index 0000000000000000000000000000000000000000..c6f7208c206192704c2e536b15d8442c84c26e51
--- /dev/null
+++ b/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/build/sections/07-algorithm.tex
@@ -0,0 +1,26 @@
+\section{Polynomial time and the output string}\label{sec:algorithm}
+
+\begin{proof}[Proof of \Cref{thm:main}]
+Parse and preprocess the explicit input in time polynomial in its encoded size $N$. This handles the empty instance and preserves the optimum otherwise. In the nonempty case, construct the counts and base graph of \Cref{sec:counts}. By \Cref{lem:counts,lem:base}, its cost is $W\le\OPT(\mathcal S)$ and it contains every required vertex. Apply \Cref{thm:connection}. The resulting graph is balanced, its used support is connected, it includes $\eps$, and its total up cost is at most $2W$.
+
+A finite balanced directed multigraph with connected used support has a closed Euler tour from any used vertex. One may obtain it by following unused edges until a closed walk is formed, and splicing further closed walks at visited vertices until all edges have been used. Balance prevents an unfinished walk from stopping away from its starting point, and connectivity ensures that any unused edges can eventually be reached for a splice.
+
+Start such a tour at $\eps$ and write down the appended letter at each up step. At every step, the current vertex word is a suffix of the string written so far: an up step appends a letter to both, and a down step removes the first letter of the vertex. Every required string is visited and therefore occurs as a substring of the output. Its length is the number of up steps, at most $2W\le2\OPT(\mathcal S)$.
+
+We finish by verifying that all preceding constructions are deterministic polynomial-time operations on explicit finite data. Order encoded symbols, input strings, substring vertices, layers, records, and edge copies by their explicit codes and stable indices. At every finite choice in the construction, including walk decompositions, eligible records, tied extrema, and Euler splices, take the first admissible item in this order. Comparisons and output of variable-width symbol codes are charged their bit cost; no unit-cost operation on an unbounded label or random choice is used.
+
+\paragraph{Counts and graph.}
+There are $O(L^2)$ words in $V$, each of length at most $L\le N$; store their explicit labels or interned identifiers in a dictionary. For each $v$, enumerate $1\le p<\len v$ and $k\ge1$ with $kp<\len v$. The period-$p$ template is specified by $v[0:p)$, and testing whether it matches $v$ takes polynomial time. Thus all rules \eqref{eq:period-rule} form a polynomial list. Each blocking functional is evaluated by enumerating $r\in V$ and the possible placements in \eqref{eq:block-interior}--\eqref{eq:block-ends}. The descending count recursion uses only such evaluations and integer maxima. By \Cref{lem:counts}, all counts are at most $L$, so their bit lengths are polynomial as well.
+
+By \Cref{lem:base}, the explicit base graph has exactly $2W\le2L$ edges with multiplicity; adjacency lists retain individual edge-copy IDs. Its closed-walk decomposition can be found by finite edge traversals. Primitive periods, alignments of texts, and sorted passage lists can be found by direct string comparisons and sorting. The total number of primitive layer phases is the sum of their periods, namely $W$. Store each layer by its primitive period word, one period of exit-end values $z(x)$, and a stable ID; \eqref{eq:layer-properties} recovers other coordinates without enumerating an infinite text.
+
+\paragraph{Records and connections.}
+The first and last ends of a layer, its minimum vertex length, and the distinguished rotation of its text can be computed from one period. Equality or lexicographic comparison of two periodic texts requires at most the sum of their periods to find a difference, by \Cref{lem:agreement}. Normalize the start of each considered periodic record to one turn, translating its aligned child coordinates with it as in \Cref{lem:host}. Each actual window labels a substring of length at most $L$, so the resulting coordinates, lengths, and end differences have polynomial bit length. A record is stored by layer ID, phase, normalized start and end, and its alignment.
+
+The search for a record in \Cref{sec:processing} is also finite. A period-$p$ layer has $p$ up steps and $p$ down steps, hence $2p$ window visits per turn, counting the initial vertex but not its repeated final copy. There are therefore $2W$ such visits over all layers. For the specified $w$ and template $A$, enumerate these actual windows and all placements of $w$ in them. Align each placement with $[t,H-K)$, reject a candidate whose complete aligned text equals $A$, and retain one whose entire window agrees with $A$. There are at most $2WL$ candidates. The count rule and \Cref{lem:choose-k,lem:matching-window} guarantee a covering window on a different aligned text, so this search succeeds.
+
+The host operations, collective links, and internal joins require only ordered lists of windows and the paths of \Cref{lem:ordered-windows}; materialize these as explicit edge lists. Finding the intersections used in \Cref{lem:collective} is a scan along at most $K=kp$ units of start coordinate, with $K$ no larger than the group's budget and hence at most $W$. Group processing and the final outgoing-block functional graph likewise use polynomially many layers and blocks.
+
+\paragraph{Materializing the answer.}
+Every added walk is closed, so its number of down steps equals its number of up steps. The total added up cost is at most $W$, and hence the final graph has at most $4W\le4L$ edges with multiplicity. Its explicit construction and Euler traversal are polynomial. This proves both the approximation guarantee and the running-time assertion.
+\end{proof}
diff --git a/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/paper.pdf b/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..ea5ca8ce7906cf672c2dfbcaf92eabc871e2f5c3
--- /dev/null
+++ b/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:45ef658ae322212b4c83626fd2bfa0ad8e459f6564b4bd0395e0c9ee8eb30dfe
+size 369674
diff --git a/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/README.md b/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..49114c97bb6cb8b46b24b7bc11ce45ddf03af824
--- /dev/null
+++ b/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/README.md
@@ -0,0 +1,20 @@
+# [A directional zero–one law under strict ellipticity](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 23, 2026
+
+## Citation
+
+To cite this manuscript:
+
+```bibtex
+@misc{OAI:A-directional-zero-one-law-under-strict-ellipticity-September-23-2026,
+  author = {{OpenAI}},
+  title = {{A directional zero--one law under strict ellipticity}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/paper.pdf}{OAI:A-directional-zero-one-law-under-strict-ellipticity-September-23-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/build/figures/opposed-arrangement.tex b/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/build/figures/opposed-arrangement.tex
new file mode 100644
index 0000000000000000000000000000000000000000..14cd4f78bc0520af20948810ab0a6359af6644f3
--- /dev/null
+++ b/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/build/figures/opposed-arrangement.tex
@@ -0,0 +1,48 @@
+\begin{tikzpicture}[
+  x=1cm,y=1cm,>=Stealth,
+  every node/.style={font=\small},
+  plus/.style={draw=blue!65!black,line width=1pt},
+  minus/.style={draw=red!65!black,line width=1pt},
+  guide/.style={draw=black!35,densely dashed,line width=.4pt}
+]
+  \node[anchor=west,font=\small\bfseries] at (0,.85)
+    {(a) Common departure bands};
+  \node[align=center] at (2.0,.35) {positive\\tape};
+  \node[align=center] at (3.9,.35) {negative\\tape};
+  \fill[black!4] (.9,-1.3) rectangle (5.1,-2.6);
+  \foreach \yy/\lab in {0/0,-1.3/W_1,-2.6/W_2,-3.9/W_3} {
+    \draw[guide] (.9,\yy)--(5.1,\yy);
+    \node[anchor=east] at (.75,\yy) {$\lab$};
+  }
+  \foreach \top/\bottom/\j in {0/-1.3/1,-1.3/-2.6/2,-2.6/-3.9/3} {
+    \draw[plus,->] (2.0,\bottom+.18)--(2.0,\top-.18);
+    \draw[minus,->] (3.9,\top-.18)--(3.9,\bottom+.18);
+    \node at (5.45,{(\top+\bottom)/2}) {$K_{\j}$};
+  }
+  \draw[->,black!70] (-.15,-.30)--(-.15,-3.60);
+  \node[rotate=90,anchor=south] at (-.45,-1.95) {progress $-x_1$};
+  \node[align=center,font=\footnotesize] at (2.95,-4.35)
+    {Arrows indicate orientation, not paths.\\Both tapes are placed downwards.};
+
+  \begin{scope}[xshift=7.0cm]
+    \node[anchor=west,font=\small\bfseries] at (-.15,.85)
+      {(b) The one-level offset};
+    \node at (1.4,.30) {$+$};
+    \node at (3.2,.30) {$-$};
+    \fill[black!6] (.7,0) rectangle (3.85,-2.7);
+    \foreach \yy/\lab in {0/s,-.8/{s+1},-2.7/{s+L},-3.5/{s+L+1}} {
+      \draw[guide] (.7,\yy)--(3.85,\yy);
+      \node[anchor=east] at (.55,\yy) {$\lab$};
+    }
+    \draw[plus,->] (1.4,-2.7)--(1.4,0);
+    \draw[minus,->] (3.2,-.8)--(3.2,-3.5);
+    \fill[blue!65!black] (1.4,-2.7) circle (2.1pt);
+    \draw[plus,fill=white] (1.4,0) circle (2.1pt);
+    \fill[red!65!black] (3.2,-.8) circle (2.1pt);
+    \draw[minus,fill=white] (3.2,-3.5) circle (2.1pt);
+    \node[align=center,font=\footnotesize] at (2.28,-1.72)
+      {departure\\heights};
+    \node[align=center,font=\footnotesize] at (2.1,-4.10)
+      {Both words depart only at integer progress\\in $(s,s+L]$; terminal arrivals are excluded.};
+  \end{scope}
+\end{tikzpicture}
diff --git a/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/build/figures/posterior-fresh-strip.tex b/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/build/figures/posterior-fresh-strip.tex
new file mode 100644
index 0000000000000000000000000000000000000000..92496189cff8860e78de9476be1a1bf25ae18132
--- /dev/null
+++ b/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/build/figures/posterior-fresh-strip.tex
@@ -0,0 +1,44 @@
+\begin{tikzpicture}[
+  x=1cm,y=1cm,>=Stealth,
+  every node/.style={font=\small},
+  guide/.style={draw=black!35,densely dashed,line width=.4pt},
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+  }
+  \draw[->,black!70] (-.8,.15)--(-.8,4.45);
+  \node[rotate=90,anchor=south] at (-1.0,2.3) {height $x_1$};
+
+  \draw[black!65,line width=1pt,->]
+    (.65,0)--(.65,.32)--(1.05,.32)--(1.05,.58)
+    --(1.45,.58)--(1.45,.88)--(1.75,.88)--(1.75,1.1);
+  \fill (1.75,1.1) circle (2pt);
+  \node[anchor=north west] at (1.85,1.04) {$z$};
+  \node[anchor=west,font=\footnotesize] at (2.65,-.35)
+    {lower prefix $a_0$: departures below $k$};
+
+  \draw[pathline,->]
+    (1.75,1.1)--(1.75,1.55)--(2.2,1.55)--(2.2,2.1)
+    --(2.65,2.1)--(2.65,1.85)--(3.2,1.85)--(3.2,2.55)
+    --(3.65,2.55)--(3.65,3.25)--(4.05,3.25)--(4.05,3.9)
+    --(4.45,3.9)--(4.45,4.6);
+  \draw[pathline,fill=white] (4.45,4.6) circle (2.4pt);
+  \node[anchor=south] at (4.45,4.77) {upper exit: stop at arrival};
+
+  \draw[pathline,densely dashed,->]
+    (3.2,2.55)--(4.6,2.55)--(4.6,2.15)--(5.0,2.15)
+    --(5.0,1.75)--(5.45,1.75)--(5.45,1.1)--(5.9,1.1)
+    --(5.9,.7);
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+
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+    {Rows at heights $k,\ldots,N-1$\\are fresh relative to $a_0$.};
+  \node[anchor=west,align=left,font=\footnotesize] at (6.3,2.25)
+    {The two exit alternatives are schematic.\\The stopped path laws agree;\\the original posteriors are retained.};
+\end{tikzpicture}
diff --git a/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/build/main.tex b/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/build/main.tex
new file mode 100644
index 0000000000000000000000000000000000000000..70ece40c8c85e6c370543317714fd428fe8f3f57
--- /dev/null
+++ b/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/build/main.tex
@@ -0,0 +1,54 @@
+\documentclass[11pt,a4paper]{article}
+\usepackage[margin=28mm]{geometry}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern,microtype}
+\input{math-glyph-unicode}
+\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs,bm}
+\usepackage{booktabs,graphicx,tikz}
+\usetikzlibrary{arrows.meta,calc,positioning}
+\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref}
+\hypersetup{pdftitle={A directional zero-one law under strict ellipticity},pdfauthor={OpenAI}}
+\pdfinfoomitdate=1
+\pdftrailerid{}
+\pdfsuppressptexinfo=15
+\emergencystretch=2em
+\newcommand{\PP}{\mathbb P}
+\newcommand{\EE}{\mathbb E}
+\newcommand{\ZZ}{\mathbb Z}
+\newcommand{\RR}{\mathbb R}
+\newcommand{\NN}{\mathbb N}
+\newcommand{\1}{\mathbf 1}
+\newcommand{\cF}{\mathcal F}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\title{A directional zero--one law\protect\\ under strict ellipticity}
+\author{OpenAI}
+\date{September 23, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We prove the directional zero--one conjecture for nearest-neighbor random
+walks in independent and identically distributed strictly elliptic environments
+on $\ZZ^d$, $d\ge3$: the probability of escape in each fixed nonzero real
+direction is zero or one. Only strict positivity of the transition probabilities
+is required; no uniform lower bound or moment assumption is imposed.
+\end{abstract}
+{\small
+\tableofcontents
+\par}
+\input{01-introduction}
+\input{02-regeneration}
+\input{03-radius}
+\input{04-arrangement}
+\input{05-entropy}
+\input{06-posterior-contact}
+\input{07-contradiction}
+\bibliographystyle{plain}
+\bibliography{refs}
+\end{document}
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diff --git a/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/build/refs.bib b/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/build/refs.bib
new file mode 100644
index 0000000000000000000000000000000000000000..3ea04a772b1a299d1254f6edeebaccd18a1a5a26
--- /dev/null
+++ b/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/build/refs.bib
@@ -0,0 +1,220 @@
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+}
+
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+
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+  title = {The zero-one law for planar random walks in {i.i.d.} random environments revisited},
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+  pages = {326--335},
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+  doi = {10.1214/ECP.v12-1314},
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+  archivePrefix = {arXiv}
+}
+
+@article{Simenhaus2007,
+  author = {Simenhaus, Fran{\c c}ois},
+  title = {Asymptotic direction for random walks in random environments},
+  journal = {Annales de l'Institut Henri Poincar{\'e}, Probabilit{\'e}s et Statistiques},
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+  year = {2007},
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+}
+
+@article{Berger2008,
+  author = {Berger, Noam},
+  title = {Limiting velocity of high-dimensional random walk in random environment},
+  journal = {The Annals of Probability},
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+}
+
+@article{DrewitzRamirez2010,
+  author = {Drewitz, Alexander and Ram{\'i}rez, Alejandro F.},
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+  pages = {212--225},
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+}
+
+@article{SabotTournier2011,
+  author = {Sabot, Christophe and Tournier, Laurent},
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+}
+
+@article{Bouchet2013,
+  author = {Bouchet, {\'E}lodie},
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+  archivePrefix = {arXiv}
+}
+
+@article{Tournier2015,
+  author = {Tournier, Laurent},
+  title = {Asymptotic direction of random walks in {Dirichlet} environment},
+  journal = {Annales de l'Institut Henri Poincar{\'e}, Probabilit{\'e}s et Statistiques},
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+  pages = {716--726},
+  year = {2015},
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+}
+
+@article{Slonim2024,
+  author = {Slonim, Daniel J.},
+  title = {Directional transience of random walks in random environments with bounded jumps},
+  journal = {ALEA, Latin American Journal of Probability and Mathematical Statistics},
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+  archivePrefix = {arXiv}
+}
+
+@article{AlexanderBerger2016,
+  author = {Alexander, Kenneth S. and Berger, Quentin},
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+  archivePrefix = {arXiv}
+}
+
+@article{Shannon1948,
+  author = {Shannon, Claude E.},
+  title = {A mathematical theory of communication},
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+  pages = {379--423, 623--656},
+  year = {1948},
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+}
+
+@article{KullbackLeibler1951,
+  author = {Kullback, Solomon and Leibler, Richard A.},
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+}
+
+@incollection{Kikuchi1992,
+  author = {Kikuchi, Masato},
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+}
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+  author = {Kesselheim, Thomas and Molinaro, Marco and Patton, Kalen and Singla, Sahil},
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+}
+
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+  author = {Ville, Jean},
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+}
+
+@article{Garsia1965,
+  author = {Garsia, Adriano M.},
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+}
diff --git a/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/paper.pdf b/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/paper.pdf
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@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:245e917113b3545b311c4c9eab739227d530f527081c1b0be4a92ea904559683
+size 540710
diff --git a/preprints/A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026/README.md b/preprints/A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..352f8d135eda49616687a2b10acf3eccf6d1732f
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+++ b/preprints/A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026/README.md
@@ -0,0 +1,20 @@
+# [A doubling Hilbert subset with no finite-dimensional bi-Lipschitz embedding](main.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 25, 2026
+
+## Citation
+
+To cite this manuscript:
+
+```bibtex
+@misc{OAI:A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026,
+  author = {{OpenAI}},
+  title = {{A doubling Hilbert subset with no finite-dimensional bi-Lipschitz embedding}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026/main.pdf}{OAI:A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026/build/main.tex b/preprints/A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026/build/main.tex
new file mode 100644
index 0000000000000000000000000000000000000000..4c45439184128f24b1bde5cbcf3dc56d8f95afe3
--- /dev/null
+++ b/preprints/A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026/build/main.tex
@@ -0,0 +1,560 @@
+\documentclass[11pt]{article}
+\pdftrailerid{}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern,microtype}
+\usepackage[margin=1in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta}
+\usepackage[colorlinks=true,linkcolor=blue!50!black,citecolor=blue!50!black,urlcolor=blue!50!black]{hyperref}
+\newtheorem{theorem}{Theorem}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\newtheorem{remark}[theorem]{Remark}
+\newcommand{\R}{\mathbb R}
+\newcommand{\N}{\mathbb N}
+\newcommand{\av}[2]{\frac{1}{|#1|}\int_{#1} #2}
+\newcommand{\norm}[1]{\left\lVert#1\right\rVert}
+\newcommand{\inner}[2]{\langle #1,#2\rangle}
+\DeclareMathOperator{\diam}{diam}
+\title{A doubling Hilbert subset with no finite-dimensional bi-Lipschitz embedding}
+\author{OpenAI}
+\hypersetup{pdftitle={A doubling Hilbert subset with no finite-dimensional bi-Lipschitz embedding},pdfauthor={OpenAI}}
+\date{September 25, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+Every infinite-dimensional real Banach space contains a compact doubling subset that admits no bi-Lipschitz embedding into any finite-dimensional real normed space. The doubling constant has a universal bound, independent of the ambient Banach space. This answers the Lang--Plaut problem negatively, even for compact subsets of Hilbert space.
+\end{abstract}
+
+\section{Introduction}
+\label{sec:introduction}
+A metric space $(X,d)$ is \emph{doubling with constant at most $\lambda$} if every ball in $X$ is covered by at most $\lambda$ balls of half its radius, with centers in $X$. A map $f:X\to\R^k$ is a \emph{bi-Lipschitz embedding with distortion at most $D$} if there is a scale $a>0$ such that
+\[
+ a\,d(x,y)\le\norm{f(x)-f(y)}\le Da\,d(x,y)
+ \qquad(x,y\in X).
+\]
+Both the target dimension $k$ and the distortion $D$ are required to be finite.
+
+Every subset of a finite-dimensional Euclidean space is doubling. Lang and Plaut asked whether every doubling subset of Hilbert space admits a bi-Lipschitz embedding into some finite-dimensional Euclidean space; see \cite[Question 2.4]{LP}, restated in \cite[Question 1]{NN}. Gupta, Krauthgamer, and Lee independently formulated the question in the context of algorithmic dimension reduction \cite[Section 6, Question 1]{GKL}. The hypothesis already supplies a Hilbert-space realization, so the issue is whether a bound on metric covering complexity permits a reduction to finitely many Euclidean coordinates while preserving all distances up to a fixed factor.
+
+Assouad's theorem gives a positive result after changing the metric: every doubling metric space admits such an embedding for the distance $d^\alpha$, where $0<\alpha<1$ \cite{Assouad}. This operation is called \emph{snowflaking}. Naor and Neiman proved that, when $\alpha$ is close to $1$, the target dimension can be bounded in terms of the doubling constant alone, independently of $\alpha$ \cite[Theorem 1.2]{NN}. The distortion bound still depends on the snowflake exponent. For finite Hilbert subsets, Gottlieb and Krauthgamer obtained embeddings of snowflakes with distortion arbitrarily close to one, with dimension depending on the doubling constant, exponent, and permitted distortion \cite[Theorem 1.2]{GK}. The Lang--Plaut question concerns the original distance $d$, even when arbitrary finite dimension and distortion are allowed.
+
+For exponents $p>2$, Lafforgue and Naor constructed doubling subsets of $L_p$ admitting no bi-Lipschitz embedding into any finite-dimensional Euclidean space \cite[Theorem 1.1]{LN}. Independently, Bartal, Gottlieb, and Neiman obtained dimension-reduction obstructions for doubling subsets of $\ell_p$ using thin Laakso configurations \cite{BGN}. Baudier, \'Swi\k{e}cicki, and Swift later gave an elementary proof of the corresponding fixed-set theorem for $\ell_q$, $q>2$ \cite[Theorem 1]{BSS}. These results concern ambient exponents strictly greater than two and do not settle the Hilbert case. Schioppa announced a negative answer in \cite{Schioppa}, but withdrew the preprint after reporting a possible issue with its duality argument.
+
+We give a negative answer to this question.
+\begin{theorem}\label{thm:main}
+There is a fixed subset $S$ of real $\ell_2$, equipped with its induced Hilbert distance, whose doubling constant is at most $76800$ and which admits no bi-Lipschitz embedding into $\R^k$ for any positive integer $k$, at any finite distortion.
+\end{theorem}
+In particular, dimension and distortion bounds depending only on the doubling constant cannot hold for all doubling Hilbert subsets.
+
+The same metric space has an isometric realization in $L_p([0,1];\R)$ for every $1\le p<\infty$, by a normalized Gaussian series; Section~\ref{sec:lp} gives this consequence and its relation to earlier $L_p$ obstructions. Section~\ref{sec:banach} constructs a compact obstruction in every infinite-dimensional real Banach space, including Hilbert space, with a universal doubling constant and with no bi-Lipschitz embedding into any finite-dimensional real normed space.
+
+\subsection{The construction and proof strategy}
+The proof couples a covering argument with a derivative obstruction. Differentiation has a substantial history as a method for ruling out Euclidean embeddings: Cheeger's framework gives such obstructions under doubling-measure and Poincar\'e-inequality hypotheses \cite[Section 14, Theorem 14.3]{Cheeger}. Laakso's nonembedding argument gives a related separation-versus-stretch principle: separated branches force an increase in stretch along a subsegment \cite[Section 4, Theorem 4.1]{Laakso}. Here the analytic step uses ordinary Euclidean differentiation of Lipschitz maps on open subsets of $\R^2$ and a quadratic variance identity. A lexicographic extremum treats the two derivative columns successively, and crossings turn their small variation into a packing obstruction. We describe the geometry first.
+
+\paragraph{Sparse scales and common sheets.}
+At each scale $r_j=1000^{-j}$, we color horizontal strips of width $r_j$ and vertical strips of width $W_jr_j$ periodically with $N_j$ colors. Here $N_j,W_j$ are positive integers, fixed in advance so that every pair occurs infinitely often. Choose unit vectors $e_{j,i}$, one for each level and color, mutually orthogonal and orthogonal to the base plane. For a base point $p\in\R^2$, color $i$ permits a displacement $r_je_{j,i}$ whenever $p$ is in a horizontal or a vertical strip of that color. Points of $S$ carry finitely many such displacements. Section~\ref{sec:construction} defines this set and proves the doubling bound: at any observation radius, the coarser coordinates are fixed, the finer coordinates have a small total diameter, and at most one intermediate level contributes labels to the cover.
+
+Now suppose that an embedding $f$ exists, normalized to have lower Lipschitz constant one and upper constant $D$. Fixing a finite-support tuple of displacement coordinates and varying $p$ gives a map from an open part of the base plane to $\R^k$; we call this map a \emph{sheet}. If $F$ is one sheet and $F_i$ is obtained by adding a color-$i$ displacement at an unused level of size $r$, then
+\[
+ \norm{F_i(p)-F(p)}\le Dr
+\]
+wherever the added displacement is permitted. The horizontal and vertical restrictions belong to the same map $F_i$. Our objective is to find one horizontal line and one vertical line of each color along which the relative offset $(F_i-F)/r$ is nearly constant.
+
+\paragraph{Two derivative extrema and two different error bounds.}
+The derivative of each sheet has horizontal and vertical columns. Section~\ref{sec:extrema} takes the closure of all pairs of their squared norms, chooses the largest first coordinate $A$, and then chooses the largest second coordinate $B$ among pairs whose first coordinate is $A$. Near this extremal pair, a mean derivative close to a reference vector forces small mean-square variation. This elementary rigidity is the analytic input.
+
+Fix the embedding parameters $k,D$, and then fix a color count $N$ larger than the number of points with pairwise distances greater than one that can fit in a radius-$D$ ball of $\R^k$. For each positive integer $n$, use a recurring level with $(N_j,W_j)=(N,n)$ inside a sufficiently small region of a nearly extremal sheet. Writing $r$ for its displacement scale, one period block contains one row and one column of each color and has width $Nnr$ and height $Nr$. In Section~\ref{sec:energy}, the first extremum gives horizontal derivative error whose mean square, multiplied by $n^2$, tends to zero. This factor compensates for the longer horizontal lines. The wide vertical strips first force the mean squared horizontal derivative norm of the vertically selected sheets toward $A$. The second extremum then gives vertical mean-square error tending to zero, with no growing factor required. Compactness suffices for this latter estimate; no quantitative modulus of the second extremum is needed.
+
+\paragraph{Crossings turn small variation into a contradiction.}
+Section~\ref{sec:crossings} selects one period block and a line of every color in each direction with the required small errors. Absolute continuity turns the derivative estimates into nearly constant relative offsets on those lines. At the crossing of a color-$i$ row and a color-$m$ column with $i\ne m$, both displacements are permitted, and the corresponding source points are separated by $\sqrt2\,r$. Transport to the selected same-color intersections therefore produces $N$ separated vectors in a radius-$D$ ball of $\R^k$. Euclidean volume comparison contradicts the choice of $N$. The fixed set is used throughout; only the sheets, local rectangles, and recurring level vary with $n$.
+
+\subsection{Conventions}
+Write $\N=\{1,2,\ldots\}$. In the proof of Theorem~\ref{thm:main}, all norms are Euclidean or Hilbert norms. For a Lipschitz map $F$ on an open subset of $\R^2$, write $F_x,F_y$ for its derivative columns, defined almost everywhere. For an integrable function $h$ on a set $E$ of positive finite area, write
+\[
+ \langle h\rangle_E=\frac{1}{|E|}\int_E h.
+\]
+The notation $\langle h\rangle$ omits $E$ when the averaging domain has been specified.
+
+\section{The subset and its doubling bound}
+\label{sec:construction}
+We first construct the set independently of any target map. The rapid separation of the displacement scales will give a uniform covering bound even though the number of colors is unbounded.
+
+Fix, once and for all, a sequence $(N_j,W_j)_{j\ge1}$ of pairs in $\N^2$ in which every pair occurs infinitely often. For example, concatenate the finite lists $\{1,\ldots,m\}^2$ for $m=1,2,\ldots$. Put $r_j=1000^{-j}$ and let
+\[
+ \mathcal H=\R^2\oplus\bigoplus_{j\ge1}\R^{N_j}
+\]
+be the Hilbert direct sum. This is isometric to real $\ell_2$. Let $e_{j,i}$, $1\le i\le N_j$, denote the coordinate vectors in its $j$th summand.
+
+At level $j$, color the horizontal strips
+\[
+ \{(x,y):qr_j<y<(q+1)r_j\},\qquad q\in\mathbb Z,
+\]
+by the residue $i\in\{1,\ldots,N_j\}$ determined by $q\equiv i-1\pmod{N_j}$. Give the vertical strips
+\[
+ \{(x,y):qW_jr_j<x<(q+1)W_jr_j\}
+\]
+the same periodic coloring. Denote their unions of color $i$ by $H_{j,i}$ and $V_{j,i}$, respectively, and put
+\[
+ U_{j,i}=H_{j,i}\cup V_{j,i}.
+\]
+In particular, $U_{j,i}$ is open.
+
+Define $S\subset\mathcal H$ to consist of the points
+\begin{equation}\label{eq:set}
+ (p,w),\qquad p\in\R^2,\quad w=(w_j)_{j\ge1},
+\end{equation}
+where $w$ has finite support and, independently at each level,
+\[
+ w_j=0\quad\text{or}\quad w_j=r_je_{j,i}\text{ for some }i\text{ with }p\in U_{j,i}.
+\]
+The zero choice is always allowed, including on strip boundaries. Neither $S$ nor its defining parameters will subsequently depend on a proposed embedding.
+
+\begin{proposition}\label{prop:doubling}
+The doubling constant of $S$ is at most $300\cdot16^2=76800$.
+\end{proposition}
+\begin{proof}
+Fix a ball of radius $t>0$ centered at a point of $S$. Distinct choices of $w_j$ have distance at least $r_j$. Thus at every level with $r_j>t$, all points of the ball have the same coordinate as its center.
+
+There is at most one intermediate level satisfying $t/64<r_j\le t$, since consecutive radii have ratio $1000$. The projection of the ball onto each base coordinate lies in an interval of length $2t$. Such an interval meets at most $\lfloor2t/r_j\rfloor+2\le129$ strips of width $r_j$, and the same upper bound applies to strips of width $W_jr_j\ge r_j$. Consequently, at the intermediate level there are fewer than $300$ available labels, including zero.
+
+For any two points of the ball, the fine-coordinate distance is bounded by
+\begin{equation}\label{eq:tail}
+ \sum_{r_j\le t/64}\norm{w_j-w'_j}^2
+ \le 2\sum_{r_j\le t/64}r_j^2
+ \le\frac{2}{1-10^{-6}}\left(\frac{t}{64}\right)^2
+ <\left(\frac{t}{32}\right)^2.
+\end{equation}
+Partition a base square of side $2t$ containing the projection into $16^2$ cells of side $t/8$, assigning their boundaries arbitrarily. Group the ball's points according to their cell and their intermediate coordinate, if one exists. In a group, the coarse and intermediate coordinates agree, the base diameter is at most $\sqrt2\,t/8$, and \eqref{eq:tail} bounds the fine coordinates. Each group therefore has diameter less than
+\[
+ t\sqrt{\frac{2}{64}+\frac{1}{1024}}=\frac{\sqrt{33}}{32}t<\frac t2.
+\]
+Choosing one point from each nonempty group as a center gives the required cover by balls in $S$.
+\end{proof}
+
+\section{Sheets and extremal derivatives}
+\label{sec:extrema}
+We now assume an embedding exists and extract one compact set of derivative data from all its sheets. Maximizing the two coordinates in order will constrain variation in the two directions separately.
+
+Since $S$ contains the base plane with all displacement coordinates zero, it cannot embed into $\R^0$. Suppose, for a contradiction, that for some finite $k\ge1$ and $D\ge1$ there is a map $f:S\to\R^k$ satisfying
+\begin{equation}\label{eq:bilip}
+ \norm{z-z'}\le\norm{f(z)-f(z')}\le D\norm{z-z'}
+ \qquad(z,z'\in S).
+\end{equation}
+For a fixed finite-support tuple $w$ of coordinate choices, let
+\[
+ \Omega_w=\{p:(p,w)\in S\}.
+\]
+If $w_j=r_je_{j,i_j}$ at its nonzero levels, then
+\[
+ \Omega_w=\bigcap_{j:w_j\ne0}U_{j,i_j}.
+\]
+This is a finite intersection of open sets; the empty intersection for the zero tuple is $\R^2$. Whenever the domain is nonempty, the map
+\[
+ F_w:\Omega_w\to\R^k,\qquad F_w(p)=f(p,w),
+\]
+is called a \emph{sheet}. It is $D$-Lipschitz, since the corresponding source points differ only in their base coordinates. This holds even when its domain is disconnected.
+
+Rademacher's theorem gives differentiability almost everywhere on each sheet domain \cite[Theorem 3.1]{Heinonen}. Its derivative columns have norm at most $D$. By Lebesgue differentiation, almost every point $p$ is also a Lebesgue point of these columns; see \cite[Theorem 32 and Exercise 33]{Tao}. At such a point, with $u=F_x(p)$ and $v=F_y(p)$, the mean-square error satisfies
+\begin{equation}\label{eq:good}
+ \lim_{l\downarrow0}\left\langle\norm{F_x-u}^2+\norm{F_y-v}^2\right\rangle_{Q(p,l)}=0,
+\end{equation}
+where $Q(p,l)$ is the centered square of side $l$. Indeed, boundedness upgrades the Lebesgue mean-norm conclusion to mean-square convergence, and squares may replace balls by area comparison. Call these differentiability and Lebesgue points \emph{good}; they form a set of full measure in each sheet domain.
+
+Let $K$ be the closure in $[0,D^2]^2$ of all pairs
+\[
+ (\norm{F_x(p)}^2,\norm{F_y(p)}^2)
+\]
+from all sheets and their good points. It is nonempty and compact. Define
+\begin{equation}\label{eq:AB}
+ A=\max\{s:(s,t)\in K\},\qquad
+ B=\max\{t:(A,t)\in K\}.
+\end{equation}
+Thus $(A,B)\in K$. This single compact set and these two numbers remain fixed throughout the argument. The next lemma explains the benefit of maximizing the coordinates in this order: if the mean deficit from $A$ tends to zero, then the mean second coordinate cannot exceed $B$ in the limit.
+
+\begin{lemma}[Near the lexicographic maximum]\label{lem:compact}
+Let $D>0$, let $K\subseteq[0,D^2]^2$ be nonempty and compact, and define
+$A=\max\{s:(s,t)\in K\}$ and $B=\max\{t:(A,t)\in K\}$.
+For every $\varepsilon>0$ there is $\delta>0$ such that
+\[
+ (s,t)\in K,\quad s>A-\delta\quad\Longrightarrow\quad t\le B+\varepsilon.
+\]
+Consequently, if measurable pairs $(s_n,t_n)$ take values in $K$ on probability spaces and $\mathbb E(A-s_n)\to0$, then
+$\limsup_n\mathbb E t_n\le B$.
+\end{lemma}
+\begin{proof}
+Failure of the first assertion would give a sequence in $K$ with first coordinate tending to $A$ and second coordinate greater than $B+\varepsilon$. A convergent subsequence would contradict the definition of $B$. For the second assertion, Markov's inequality gives
+\[
+ \mathbb P\{A-s_n\ge\delta\}\le\frac{\mathbb E(A-s_n)}{\delta},
+ \qquad
+ \mathbb E t_n\le B+\varepsilon+
+ D^2\frac{\mathbb E(A-s_n)}{\delta}.
+\]
+First let $n\to\infty$, and then $\varepsilon\downarrow0$.
+\end{proof}
+
+The lemma supplies an upper bound on a mean squared norm. To turn that bound into small variation around a reference vector, we will also control the mean vector and use the elementary identity
+\begin{equation}\label{eq:variance}
+ \left\langle\norm{G-a}^2\right\rangle
+ =\left\langle\norm{G}^2\right\rangle-\norm a^2
+       -2\inner{a}{\langle G\rangle-a}.
+\end{equation}
+We also use integration along line segments. A Lipschitz map on an interval is absolutely continuous, and its increments equal integrals of its derivative, componentwise \cite[Theorem 3.3]{Heinonen}. By Fubini, on almost every horizontal or vertical segment in an open sheet domain, these line derivatives agree almost everywhere with the corresponding partial derivatives. At excluded endpoints we take the unique one-sided limits of the Lipschitz restriction. Bounds for the map persist in those limits.
+
+\section{Near-extremal rectangles and two energy estimates}
+\label{sec:energy}
+The output of this section is small variation of the added-sheet offsets: a horizontal estimate strong enough for increasingly long rows, and a vertical estimate for columns of fixed normalized height. The compact set $K$ and the embedding are already fixed.
+
+Fix an integer
+\begin{equation}\label{eq:N}
+ N>(1+2D)^k.
+\end{equation}
+For each positive integer $n$, choose a sheet $F$ and a good point with derivative columns $u,v$ such that
+\begin{equation}\label{eq:nearmax}
+ 0\le A-\norm u^2<n^{-4},\qquad
+ |B-\norm v^2|<n^{-4}.
+\end{equation}
+This is possible since $(A,B)\in K$ is in the closure of actual good-point pairs. Choose a square $Q^0$ of side $l$, centered at that point, with closure in the sheet domain and
+\begin{equation}\label{eq:baseerror}
+ \left\langle\norm{F_x-u}^2+\norm{F_y-v}^2\right\rangle_{Q^0}<n^{-8}.
+\end{equation}
+The sheet, the vectors, and $l$ may all depend on $n$.
+
+Choose a level $j$ beyond the finite support of this sheet such that
+\begin{equation}\label{eq:fine}
+ (N_j,W_j)=(N,n),\qquad Nnr_j<\frac{l n^{-8}}{100}.
+\end{equation}
+Infinite recurrence of each pair permits this choice. Set $r=r_j$. The level-$j$ period blocks have width $Nnr$ and height $Nr$, with corners at $(aNnr,bNr)$ for integers $a,b$. Take inside $Q^0$ a rectangle $Q$ tiled, up to boundary lines, by complete period blocks. Each coordinate interval loses at most two periods by trimming to the grid, so its side lengths $L_x,L_y$ satisfy $L_x,L_y\ge l/2$. In particular,
+\begin{align}
+ \left\langle\norm{F_x-u}^2+\norm{F_y-v}^2\right\rangle_Q&<4n^{-8},\label{eq:Qerror}\\
+ \frac r{L_x},\frac r{L_y}&<\frac{n^{-9}}{50N}.\label{eq:ratio}
+\end{align}
+
+For each $1\le i\le N$, let $F_i$ be the sheet obtained from $F$ by adding $re_{j,i}$ at level $j$. Writing $w$ for the tuple of $F$, this is a valid finite-support tuple with domain exactly $\Omega_w\cap U_{j,i}$. In particular, the same map $F_i$ is defined on the portions of both $H_{j,i}$ and $V_{j,i}$ in $Q$. By \eqref{eq:bilip},
+\begin{equation}\label{eq:offset}
+ \norm{F_i-F}\le Dr
+ \quad\text{on }Q\cap(H_{j,i}\cup V_{j,i}).
+\end{equation}
+
+Off the grid boundaries, define $G_H^x$ by selecting $(F_i)_x$ according to the horizontal strip color $i$. Define $G_V^x,G_V^y$ by selecting $(F_i)_x,(F_i)_y$ according to the vertical strip color. Almost everywhere on $Q$,
+\begin{equation}\label{eq:Kbounds}
+ \norm{G_H^x}^2\le A,\qquad
+ (\norm{G_V^x}^2,\norm{G_V^y}^2)\in K.
+\end{equation}
+Indeed, the good points have full measure for each of the finitely many sheets in use. The map $f$, its target parameters $k,D$, the set $K$, and the numbers $A,B,N$ remain fixed throughout.
+
+\begin{proposition}[Anisotropic energy estimates]\label{prop:energy}
+For each $n\in\N$, let $F,F_i,Q,u,v,r$ and the selected derivative fields be chosen as above, satisfying \eqref{eq:nearmax}--\eqref{eq:ratio}. Then, as $n\to\infty$,
+\begin{align}
+ n^2\left\langle\norm{G_H^x-F_x}^2\right\rangle_Q&\longrightarrow0,\label{eq:horizontal}\\
+ \left\langle\norm{G_V^y-F_y}^2\right\rangle_Q&\longrightarrow0.\label{eq:vertical}
+\end{align}
+\end{proposition}
+The factor $n^2$ compensates for horizontal segments of length $Nnr$; the vertical segments have length $Nr$.
+\begin{proof}
+All averages in this proof are over $Q$.
+
+\paragraph{Horizontal selection.}
+On an interior horizontal strip of color $i$, the sheet $F_i$ spans the full width of $Q$. Integrating $(F_i-F)_x$ along a horizontal segment and using \eqref{eq:offset} gives an endpoint difference of norm at most $2Dr$. Averaging in $y$, and applying Cauchy--Schwarz to \eqref{eq:Qerror}, yields
+\begin{equation}\label{eq:Hmean}
+ \norm{\langle G_H^x\rangle-u}
+ \le\frac{2Dr}{L_x}+2n^{-4}.
+\end{equation}
+Equations \eqref{eq:variance}, \eqref{eq:nearmax}, and \eqref{eq:Kbounds} therefore imply
+\begin{align*}
+ \left\langle\norm{G_H^x-u}^2\right\rangle
+ &\le A-\norm u^2+2\norm u\norm{\langle G_H^x\rangle-u}\\
+ &\le(1+4D)n^{-4}+4D^2r/L_x.
+\end{align*}
+Using \eqref{eq:Qerror} and \eqref{eq:ratio} gives the explicit estimate
+\begin{equation}\label{eq:Hrate}
+ n^2\left\langle\norm{G_H^x-F_x}^2\right\rangle
+ \le 2(1+4D)n^{-2}+\frac{4D^2}{25N}n^{-7}+8n^{-6},
+\end{equation}
+which proves \eqref{eq:horizontal}.
+
+For the vertical selection, we first average the horizontal derivative over strips of width $nr$. Dividing the same endpoint error $2Dr$ by this wider strip width makes the selected horizontal mean approach $u$. The selected derivative has squared norm at most $A$ almost everywhere, so its mean squared norm must then approach $A$. Lemma~\ref{lem:compact} will control the mean vertical squared norm without requiring a rate.
+
+\paragraph{Vertical selection: the first derivative column.}
+Integrate in $x$ separately over the full vertical strips of width $nr$. Within each strip the endpoint error for $F_i-F$ is at most $2Dr$. Thus
+\begin{equation}\label{eq:Vxmean}
+ \norm{\langle G_V^x\rangle-u}\le\frac{2D}{n}+2n^{-4}.
+\end{equation}
+The estimate sums the separate strip integrals and therefore allows jumps in the selection at strip boundaries.
+
+Put $\eta_n=\langle A-\norm{G_V^x}^2\rangle$. By \eqref{eq:Kbounds}, it is nonnegative. The identity \eqref{eq:variance}, with its nonnegative left side discarded, gives
+\begin{align}
+ 0\le\eta_n
+ &\le A-\norm u^2+2\norm u\norm{\langle G_V^x\rangle-u}\notag\\
+ &\le (1+4D)n^{-4}+\frac{4D^2}{n}\longrightarrow0.\label{eq:eta}
+\end{align}
+Apply Lemma~\ref{lem:compact} to the pairs in \eqref{eq:Kbounds}, with normalized area measure on $Q$. We obtain
+\begin{equation}\label{eq:Vynorm}
+ \limsup_{n\to\infty}\left\langle\norm{G_V^y}^2\right\rangle\le B.
+\end{equation}
+
+\paragraph{Vertical selection: the second derivative column.}
+Each interior vertical strip spans the full height of $Q$. Integration in $y$, followed by \eqref{eq:Qerror}, gives
+\begin{equation}\label{eq:Vymean}
+ \norm{\langle G_V^y\rangle-v}\le\frac{2Dr}{L_y}+2n^{-4}\longrightarrow0.
+\end{equation}
+Since $\norm v\le D$ and $\norm v^2\to B$, Equations \eqref{eq:variance}, \eqref{eq:Vynorm}, and \eqref{eq:Vymean} imply
+\[
+ 0\le\left\langle\norm{G_V^y-v}^2\right\rangle
+ \le\left\langle\norm{G_V^y}^2\right\rangle-\norm v^2
+       +2D\norm{\langle G_V^y\rangle-v}.
+\]
+The upper bound has limsup at most zero, so the nonnegative variance tends to zero. In particular, convergence of the vectors $v$ themselves is unnecessary. Combining this with \eqref{eq:Qerror} proves \eqref{eq:vertical}.
+\end{proof}
+
+\section{Crossings force a packing contradiction}
+\label{sec:crossings}
+We use the two energy estimates to find one line of every color in each direction on a common period block. Their intersections will supply the separated vectors needed for the Euclidean packing bound.
+
+Choose a complete period block $P\subset Q$ on which the mean of the nonnegative combined energy
+\[
+ n^2\norm{G_H^x-F_x}^2+\norm{G_V^y-F_y}^2
+\]
+is no greater than its mean on $Q$. Denote its mean on $P$ by $\varepsilon_n$. Proposition~\ref{prop:energy} gives $\varepsilon_n\to0$. The block has width $Nnr$ and height $Nr$, and contains exactly one row and one column of each color.
+
+For every color $i$, its row has area $|P|/N$. Fubini therefore permits a horizontal segment $y=y_i$ in its interior, spanning $P$, such that
+\begin{equation}\label{eq:rowline}
+ \frac{1}{Nnr}\int_{\text{row segment}}
+ n^2\norm{(F_i-F)_x}^2\,dx\le2N\varepsilon_n.
+\end{equation}
+Similarly choose $x=x_i$ in the interior of column $i$ with
+\begin{equation}\label{eq:colline}
+ \frac{1}{Nr}\int_{\text{column segment}}
+ \norm{(F_i-F)_y}^2\,dy\le2N\varepsilon_n.
+\end{equation}
+These lines can simultaneously be chosen from the full-measure sets where partial derivatives agree with line derivatives. If $\varepsilon_n=0$, use lines of zero energy. The choices of the horizontal and vertical coordinates are independent.
+
+Write $h_i=(F_i-F)/r$ on its domain in $P$. Absolute continuity and Cauchy--Schwarz give, on each chosen horizontal segment,
+\[
+ \diam h_i(\text{segment})
+ \le\frac{Nnr}{r}\left(\frac{1}{Nnr}\int\norm{(F_i-F)_x}^2\,dx\right)^{1/2}
+ \le N\sqrt{2N\varepsilon_n}.
+\]
+The identical bound on each chosen vertical segment follows from \eqref{eq:colline}, with length $Nr$. Set
+\begin{equation}\label{eq:omega}
+ \omega_n=N\sqrt{2N\varepsilon_n}\longrightarrow0,
+ \qquad z_i=h_i(x_i,y_i).
+\end{equation}
+By \eqref{eq:offset}, $\norm{z_i}\le D$. Oscillation on a Lipschitz line is controlled at every point by its derivative integral, so no differentiability assertion is required at the crossings.
+
+\begin{figure}[!hb]
+\centering
+\begin{tikzpicture}[x=1cm,y=1cm,>=Latex]
+ \fill[blue!10] (0,0.7) rectangle (6,1.4);
+ \fill[orange!15] (3.7,0) rectangle (5.1,3.2);
+ \draw[gray!65] (0,0) rectangle (6,3.2);
+ \draw[gray!45,dashed] (0,0.7)--(6,0.7) (0,1.4)--(6,1.4);
+ \draw[gray!45,dashed] (3.7,0)--(3.7,3.2) (5.1,0)--(5.1,3.2);
+ \draw[blue!70!black,thick,->] (1.0,1.05)--(4.4,1.05);
+ \draw[orange!80!black,thick,->] (4.4,2.6)--(4.4,1.05);
+ \fill (1.0,1.05) circle (2pt) node[above,fill=white,inner sep=1.5pt] {$p_i=(x_i,y_i)$};
+ \fill (4.4,2.6) circle (2pt) node[above left] {$p_m=(x_m,y_m)$};
+ \fill (4.4,1.05) circle (2pt) node[below left] {$c=(x_m,y_i)$};
+ \node[blue!70!black,left] at (0,1.05) {row $i$};
+ \node[orange!80!black,above] at (4.4,3.2) {column $m$};
+ \draw[<->] (0,-.5)--node[below] {$Nnr$}(6,-.5);
+ \draw[<->] (6.5,0)--node[right] {$Nr$}(6.5,3.2);
+\end{tikzpicture}
+\caption{A schematic period block, for $i\ne m$. At $c$, both labels are allowed: $i$ by the row and $m$ by the column. Their normalized offsets are separated by at least $\sqrt2$. Transport along the arrows changes $h_i$ and $h_m$ by at most $\omega_n$ each, so $z_i=h_i(p_i)$ and $z_m=h_m(p_m)$ are separated by at least $\sqrt2-2\omega_n$. The block and strip proportions are schematic.}
+\label{fig:crossing}
+\end{figure}
+
+For $i\ne m$, both added labels $re_{j,i}$ and $re_{j,m}$ are permitted at $(x_m,y_i)$: the first by row $i$, the second by column $m$. The corresponding source points have distance exactly $\sqrt2\,r$, so \eqref{eq:bilip} gives
+\[
+ \norm{h_i(x_m,y_i)-h_m(x_m,y_i)}\ge\sqrt2.
+\]
+Transporting along row $i$ and column $m$, as in Figure~\ref{fig:crossing}, yields
+\[
+ \norm{z_i-z_m}\ge\sqrt2-2\omega_n>1
+\]
+for all pairs when $n$ is sufficiently large. This is simultaneous because $N$ is fixed.
+
+We have obtained $N$ points in the radius-$D$ ball of $\R^k$ with pairwise distances greater than one. Their open radius-$1/2$ balls are disjoint and contained in the radius-$(D+1/2)$ ball. Comparing Euclidean volumes gives
+\[
+ N(1/2)^k\le(D+1/2)^k,\qquad N\le(1+2D)^k,
+\]
+contrary to \eqref{eq:N}. No map satisfying \eqref{eq:bilip} exists. Together with Proposition~\ref{prop:doubling}, this proves Theorem~\ref{thm:main}.\qed
+
+\section{A consequence for finite-\texorpdfstring{$p$}{p} function spaces}
+\label{sec:lp}
+A normalized Gaussian series gives isometric realizations of the same fixed metric space in real $L_p[0,1]$ for every finite $p\ge1$.
+
+\begin{corollary}[Finite-$p$ function-space obstruction]\label{cor:lp}
+For every real $1\le p<\infty$, the fixed metric space $S$ of Theorem~\ref{thm:main} is isometric to a subset $S_p$ of $L_p([0,1];\R)$, where $[0,1]$ carries Lebesgue measure. With its induced $L_p$ metric, $S_p$ has doubling constant at most $76800$ and admits no bi-Lipschitz embedding into any finite-dimensional real normed space at any finite distortion.
+\end{corollary}
+\begin{proof}
+Fix $p$ and choose independent standard real Gaussian random variables $(g_i)_{i\ge1}$ on Lebesgue $[0,1]$. Put $c_p=(\mathbb E|g_1|^p)^{1/p}\in(0,\infty)$. For every finitely supported real sequence $a$, the sum $\sum_i a_i g_i$ is centered Gaussian with variance $\sum_i a_i^2$, and hence
+\[
+ \norm{\sum_i a_i g_i}_{L_p}=c_p\left(\sum_i a_i^2\right)^{1/2}.
+\]
+For $x\in\ell_2$, the same identity for tails makes the partial sums Cauchy in $L_p$. Completeness, including at $p=1$, therefore defines
+\[
+ J_p x=c_p^{-1}\lim_{m\to\infty}\sum_{i=1}^m x_i g_i
+ \quad\text{in }L_p([0,1];\R).
+\]
+Passing to limits proves that $J_p$ is linear and that
+$\norm{J_p x-J_p y}_{L_p}=\norm{x-y}_2$. Thus $S_p=J_p(S)$ is isometric to $S$, so its doubling constant is unchanged.
+
+A zero-dimensional target cannot contain an injective copy of $S_p$. If $f:S_p\to Y$ were a bi-Lipschitz embedding into a real normed space of finite positive dimension $k$, choose a linear isomorphism $T:Y\to\R^k$. Equivalence of finite-dimensional norms makes $T$ bi-Lipschitz with finite distortion. The composition $T\circ f\circ J_p|_S$ would then contradict Theorem~\ref{thm:main}.
+\end{proof}
+
+Lafforgue and Naor proved the qualitative fixed-set obstruction with Euclidean targets for $p>2$ and, in the paragraph following their Theorem~1.1, recorded the corresponding question for $1<p\le2$ as open at that time, while noting stronger known results for $p=1$ \cite[Theorem 1.1 and the following paragraph]{LN}. Theorem~\ref{thm:main} treats $p=2$, and Corollary~\ref{cor:lp} answers that qualitative fixed-set question with finite-dimensional Euclidean targets when $1<p<2$. This transfer adds no new metric construction. Corollary~\ref{cor:lp} concerns function-space sources $L_p[0,1]$, not coordinate $\ell_p$ sources; the compact result below is a separate transfer to arbitrary infinite-dimensional Banach sources.
+
+Finite-set dimension reduction in $L_p$, for $1<p<\infty$ and $p\ne2$,
+allows the target dimension to depend on cardinality~\cite[Theorem~1.1]{OpenAI2026LpReduction},
+unlike the fixed infinite-set obstruction of Corollary~\ref{cor:lp}.
+
+\section{Compact obstructions in infinite-dimensional Banach spaces}
+\label{sec:banach}
+Finite witnesses to Theorem~\ref{thm:main} can be placed in any infinite-dimensional real Banach space by the classical Dvoretzky theorem. A geometrically shrinking union of these finite clusters, with its limit point adjoined, is compact and retains a uniform doubling bound.
+
+\begin{corollary}[Compact Banach-space obstruction]\label{cor:banach}
+There is a universal constant $\Lambda$ such that every infinite-dimensional real Banach space $B$ contains a compact subset $K_B$ whose doubling constant is at most $\Lambda$ and which admits no bi-Lipschitz embedding into any finite-dimensional real normed space at any finite distortion. The subset $K_B$ may depend on $B$, but is chosen independently of the target dimension and distortion.
+\end{corollary}
+\begin{proof}
+Write $d_S$ for the induced Hilbert distance on $S$ and put $\lambda=76800$. We use closed balls below; the same estimates work for open balls.
+
+\paragraph{Finite witnesses.}
+For every $k\in\N$ and every finite $A\ge1$, there is a finite subset of $S$ admitting no embedding into $\R^k$ with distortion at most $A$. Indeed, suppose every finite subset admitted such an embedding, and fix $o\in S$. For each $x\in S$, let
+\[
+ Q_x=\{u\in\R^k:\norm u\le A d_S(o,x)\}.
+\]
+The product $\prod_{x\in S}Q_x$ is compact. In this product, each constraint
+\[
+ d_S(x,y)\le\norm{f(x)-f(y)}\le A d_S(x,y)
+\]
+defines a closed set. These closed sets have the finite-intersection property: finitely many constraints involve finitely many points, and an embedding of those points together with $o$, translated to send $o$ to zero and divided by its lower scale, satisfies the constraints and lies in the corresponding $Q_x$. Set all other coordinates to zero. Product compactness then supplies a map satisfying every constraint, contrary to Theorem~\ref{thm:main}.
+
+Choose, for each $j\in\N$, a finite $X_j\subset S$ that admits no embedding into $\R^j$ with distortion at most $j$. Each $X_j$ has at least two points. These choices are made before any target for $K_B$ is considered.
+
+\paragraph{Placement and cluster doubling.}
+Fix an infinite-dimensional real Banach space $B$. The classical finite-dimensional Dvoretzky theorem \cite[Section~7]{Dvoretzky}, in the formulation recalled in \cite[Theorem~1.1]{COO}, gives a linear embedding $T_j$ of the finite Hilbert span of $X_j$ into $B$ with distortion strictly below two; the one-dimensional case is immediate. After rescaling $T_j$, we may use
+\[
+ d_S(x,y)\le\norm{T_jx-T_jy}_B\le2d_S(x,y)
+ \qquad(x,y\in X_j).
+\]
+Only this qualitative finite-dimensional form of Dvoretzky is used.
+
+Each $Y_j=T_j(X_j)$ has doubling constant at most $\lambda_0=\lambda^4$, uniformly in $j$ and $B$. To see this, the preimage $E$ of a ball in $Y_j$ of radius $R$ centered at $T_jx$ lies in the $S$-ball of radius $R$ centered at $x$. Four halvings cover that $S$-ball by at most $\lambda^4$ $S$-balls of radius $R/16$. For each one meeting $E$, choose $x_0\in E$ in it. Every other $x'\in E$ in the same ball satisfies
+\[
+ \norm{T_jx'-T_jx_0}_B
+ \le2d_S(x',x_0)
+ \le2\left(\frac R{16}+\frac R{16}\right)
+ =\frac R4\le\frac R2.
+\]
+Thus the chosen points $T_jx_0$ give the required cover with centers in $Y_j$.
+
+\paragraph{Compact assembly and its covering bound.}
+Fix a unit vector $v\in B$, and put $a_j=2^{-j}$ and $\eta=1/100$. Since $Y_j$ is finite, a translation and positive rescaling place it as a cluster
+\[
+ C_j\subset B_B(a_jv,\eta a_j).
+\]
+This placement has distortion strictly below two as a map from $X_j$, and $C_j$ is still $\lambda_0$-doubling. Set
+\[
+ K_B=\{0\}\cup\bigcup_{j\ge1}C_j.
+\]
+For $z\in C_j$, one has $\norm z_B\le(1+\eta)a_j$. Every sequence in $K_B$ therefore has either a constant subsequence among $0$ and finitely many clusters, or a subsequence whose cluster indices tend to infinity and which converges to $0$. Hence $K_B$ is compact.
+
+Fix $x\in K_B$ and $r>0$. Every cluster with $a_j\le r/4$, together with $0$, is covered by the single $K_B$-ball $B_{K_B}(0,r/2)$, since
+\[
+ \norm z_B\le(1+\eta)a_j\le\frac{101}{400}r<\frac r2
+ \qquad(z\in C_j,\ a_j\le r/4).
+\]
+The center $0$ need not belong to the ball being covered. Let
+\[
+ I=\{j:C_j\cap B_{K_B}(x,r)\ne\varnothing,\ a_j>r/4\}.
+\]
+This is finite. If $i<j$, $p\in C_i$, and $q\in C_j$, then $a_j\le a_i/2$ and
+\[
+ \norm{p-q}_B
+ \ge(a_i-a_j)-\eta(a_i+a_j)
+ \ge\left(\frac12-\frac{3\eta}{2}\right)a_i
+ =\frac{97}{200}a_i.
+\]
+If $|I|\ge2$, take its least index $i$ and any other $j\in I$. Points from the two intersections with $B_{K_B}(x,r)$ are at distance at most $2r$, so $a_i\le400r/97$. All scales indexed by $I$ then lie in $(r/4,(400/97)r]$. Their largest-to-smallest ratio is strictly less than $1600/97<32$, so there are at most five such dyadic scales. If $|I|=1$, its one scale can be arbitrarily large, but no bound on that scale is needed.
+
+For each $j\in I$, choose $y_j\in C_j\cap B_{K_B}(x,r)$. The intersection is contained in the intrinsic $C_j$-ball $B_{C_j}(y_j,2r)$, which two applications of cluster doubling cover by at most $\lambda_0^2$ $C_j$-balls of radius $r/2$. The $K_B$-balls with these same centers and radii also cover the intersection. There are at most five clusters to cover, including the single-cluster case, so the tail ball and these covers use at most
+\[
+ 1+5\lambda_0^2\le1+6\lambda^8
+\]
+balls, all centered in $K_B$. This proves a universal doubling bound; one may take $\Lambda=1+6\lambda^8$.
+
+\paragraph{Obstruction for every finite-dimensional target.}
+Suppose $K_B$ admitted a finite-distortion bi-Lipschitz embedding into a finite-dimensional real normed space. The zero-dimensional case is impossible because $K_B$ has more than one point. Otherwise, choose a linear isomorphism from the target onto $\R^k$. Equivalence of finite-dimensional norms makes this isomorphism bi-Lipschitz, so composition gives an embedding $f:K_B\to\R^k$ with some finite distortion $D$. Choose an integer $j\ge k$ with $j\ge2D$. Composing the distortion-below-two placement of $X_j$ onto $C_j$, the restriction of $f$, and the isometric inclusion $\R^k\hookrightarrow\R^j$ gives an embedding of $X_j$ into $\R^j$ with distortion strictly below $2D\le j$. This contradicts the choice of $X_j$.
+\end{proof}
+
+\begin{thebibliography}{99}
+\bibitem{Assouad}
+P. Assouad, \emph{Plongements lipschitziens dans $\R^n$},
+Bulletin de la Soci\'et\'e Math\'ematique de France \textbf{111} (1983), 429--448.
+\href{https://doi.org/10.24033/bsmf.1997}{doi:10.24033/bsmf.1997}.
+\bibitem{BGN}
+Y. Bartal, L.-A. Gottlieb, and O. Neiman, \emph{On the impossibility of dimension reduction for doubling subsets of $\ell_p$},
+SIAM Journal on Discrete Mathematics \textbf{29} (2015), 1207--1222.
+\href{https://doi.org/10.1137/140977655}{doi:10.1137/140977655}.
+\bibitem{BSS}
+F. P. Baudier, K. \'Swi\k{e}cicki, and A. Swift, \emph{No dimension reduction for doubling subsets of $\ell_q$ when $q>2$ revisited},
+Journal of Mathematical Analysis and Applications \textbf{504} (2021), no.~2, 125407.
+\href{https://doi.org/10.1016/j.jmaa.2021.125407}{doi:10.1016/j.jmaa.2021.125407}.
+\bibitem{COO}
+F. Catrina, S. Ostrovska, and M. I. Ostrovskii, \emph{Dvoretzky-type theorem for locally finite subsets of a Hilbert space},
+Annales de l'Institut Fourier \textbf{75} (2025), no.~6, 2565--2607.
+\href{https://doi.org/10.5802/aif.3672}{doi:10.5802/aif.3672}.
+\bibitem{Cheeger}
+J. Cheeger, \emph{Differentiability of Lipschitz functions on metric measure spaces},
+Geometric and Functional Analysis \textbf{9} (1999), no.~3, 428--517.
+\href{https://doi.org/10.1007/s000390050094}{doi:10.1007/s000390050094}.
+\bibitem{Dvoretzky}
+A. Dvoretzky, \emph{Some results on convex bodies and Banach spaces},
+in Proceedings of the International Symposium on Linear Spaces
+(Jerusalem, 1960), Jerusalem Academic Press, Jerusalem;
+Pergamon Press, Oxford, 1961, 123--160.
+\bibitem{GK}
+L.-A. Gottlieb and R. Krauthgamer, \emph{A nonlinear approach to dimension reduction},
+Discrete \& Computational Geometry \textbf{54} (2015), no.~2, 291--315.
+\href{https://doi.org/10.1007/s00454-015-9707-9}{doi:10.1007/s00454-015-9707-9}.
+\bibitem{GKL}
+A. Gupta, R. Krauthgamer, and J. R. Lee, \emph{Bounded geometries, fractals, and low-distortion embeddings},
+Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science (2003), 534--543.
+\href{https://doi.org/10.1109/SFCS.2003.1238226}{doi:10.1109/SFCS.2003.1238226}.
+\bibitem{Heinonen}
+J. Heinonen, \emph{Lectures on Lipschitz Analysis},
+Report 100, Department of Mathematics and Statistics, University of Jyv\"askyl\"a, 2005.
+\bibitem{Laakso}
+T. J. Laakso, \emph{Ahlfors $Q$-regular spaces with arbitrary $Q>1$ admitting weak Poincar\'e inequality},
+Geometric and Functional Analysis \textbf{10} (2000), no.~1, 111--123.
+\href{https://doi.org/10.1007/s000390050003}{doi:10.1007/s000390050003}.
+Erratum: ibid.\ \textbf{12} (2002), no.~3, 650.
+\bibitem{LN}
+V. Lafforgue and A. Naor, \emph{A doubling subset of $L_p$ for $p>2$ that is inherently infinite dimensional},
+Geometriae Dedicata \textbf{172} (2014), 387--398.
+\href{https://doi.org/10.1007/s10711-013-9924-4}{doi:10.1007/s10711-013-9924-4}.
+\bibitem{LP}
+U. Lang and C. Plaut, \emph{Bilipschitz embeddings of metric spaces into space forms},
+Geometriae Dedicata \textbf{87} (2001), 285--307.
+\href{https://doi.org/10.1023/A:1012093209450}{doi:10.1023/A:1012093209450}.
+\bibitem{NN}
+A. Naor and O. Neiman, \emph{Assouad's theorem with dimension independent of the snowflaking},
+Revista Matem\'atica Iberoamericana \textbf{28} (2012), no.~4, 1123--1142.
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+\bibitem{OpenAI2026LpReduction}
+OpenAI, \emph{Subpolynomial dimension reduction in $L_p$},
+OpenAI Math Release preprint
+\href{https://github.com/openai/math/blob/main/preprints/Subpolynomial-dimension-reduction-in-Lp-September-23-2026/paper.pdf}{OAI:Subpolynomial-dimension-reduction-in-Lp-September-23-2026},
+2026.
+\bibitem{Schioppa}
+A. Schioppa, \emph{An example of a doubling ``inherently'' infinite-dimensional subset of $l_2$},
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+T. Tao, \emph{245A, Notes 5: Differentiation theorems}, 16 October 2010,
+\url{https://terrytao.wordpress.com/2010/10/16/245a-notes-5-differentiation-theorems/}.
+\end{thebibliography}
+\end{document}
diff --git a/preprints/A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026/main.pdf b/preprints/A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026/main.pdf
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+++ b/preprints/A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026/main.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:9c9b7e794d3a66305eda83409e40ca56eed2c09eb687c6abd5ec367f6e980b15
+size 362805
diff --git a/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/README.md b/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..8fbff40e5c7b797a462e3dbe7fc7189fad7e8b46
--- /dev/null
+++ b/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/README.md
@@ -0,0 +1,18 @@
+# [A finite Smith–Toda complex V(4) at the prime 1009](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 23, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026,
+  author = {{OpenAI}},
+  title = {{A finite Smith--Toda complex $V(4)$ at the prime $1009$}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/paper.pdf}{OAI:A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/build/main.tex b/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/build/main.tex
new file mode 100644
index 0000000000000000000000000000000000000000..0eb8fe5914d3af30d6a2db7e09e0e04b46da9bb0
--- /dev/null
+++ b/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/build/main.tex
@@ -0,0 +1,27 @@
+\documentclass[11pt]{article}
+\input{preamble}
+\title{A finite Smith--Toda complex $V(4)$ at the prime $1009$}
+\author{OpenAI}
+\date{September 23, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We construct a Smith--Toda complex $V(4)$ at the prime $1009$. It is an
+ordinary finite $1009$-local spectrum whose Brown--Peterson homology is
+$BP_*/(1009,v_1,v_2,v_3,v_4)$, with its canonical comodule structure and
+generator in degree zero.
+\end{abstract}
+\tableofcontents
+\clearpage
+\input{sections/00-introduction}
+\input{sections/01-framework}
+\input{sections/02a-cobar-preliminaries}
+\input{sections/02-lower-complexes}
+\input{sections/03-may-and-first-obstruction}
+\input{sections/04-frobenius-elimination}
+\input{sections/05-final-obstruction}
+\appendix
+\input{sections/appendix-resolution}
+\bibliographystyle{plainnat}
+\bibliography{references}
+\end{document}
diff --git a/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/build/preamble.tex b/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/build/preamble.tex
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+
+\usepackage[margin=1in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{microtype}
+\usepackage{enumitem}
+\usepackage{booktabs,longtable,array}
+\usepackage{graphicx,xcolor}
+\usepackage[numbers,sort&compress]{natbib}
+\definecolor{linkblue}{RGB}{28,64,104}
+\usepackage[colorlinks=true,linkcolor=linkblue,citecolor=linkblue,urlcolor=linkblue,breaklinks=true]{hyperref}
+\usepackage{bookmark}
+\providecommand{\doi}[1]{doi: \begingroup\urlstyle{rm}\href{https://doi.org/#1}{\nolinkurl{#1}}\endgroup}
+\renewcommand{\bibsection}{\section*{\refname}\addcontentsline{toc}{section}{\refname}}
+\hypersetup{pdftitle={A finite Smith-Toda complex V(4) at the prime 1009},pdfsubject={A finite Brown-Peterson quotient realization at the prime 1009},pdfauthor={OpenAI}}
+\numberwithin{equation}{section}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
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+\theoremstyle{remark}
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diff --git a/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/build/references.bib b/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/build/references.bib
new file mode 100644
index 0000000000000000000000000000000000000000..27f9e8827cde5003a43685c0bb27b9089ececd3c
--- /dev/null
+++ b/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/build/references.bib
@@ -0,0 +1,240 @@
+
+@article{BasterraMandell2013,
+  author = {Basterra, Maria and Mandell, Michael A.},
+  title = {The multiplication on {BP}},
+  journal = {Journal of Topology},
+  volume = {6},
+  number = {2},
+  year = {2013},
+  pages = {285--310},
+  doi = {10.1112/jtopol/jts032},
+  url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/basterra-mandell.pdf}
+}
+
+@article{CulverZhang2024,
+  author = {Culver, Dominic Leon and Zhang, Ningchuan},
+  title = {Exotic {Picard} groups and chromatic vanishing via the {Gross--Hopkins} duality},
+  journal = {Topology and its Applications},
+  volume = {341},
+  year = {2024},
+  pages = {108742},
+  doi = {10.1016/j.topol.2023.108742},
+  eprint = {2203.09455v3},
+  archivePrefix = {arXiv},
+  primaryClass = {math.AT},
+  url = {https://arxiv.org/abs/2203.09455v3},
+  note = {The cited Remark 3.29 is in the September 30, 2023 preprint version}
+}
+
+@article{Nave2010,
+  author = {Nave, Lee S.},
+  title = {The {Smith--Toda} complex {$V((p+1)/2)$} does not exist},
+  journal = {Annals of Mathematics},
+  series = {2},
+  volume = {171},
+  number = {1},
+  year = {2010},
+  pages = {491--509},
+  doi = {10.4007/annals.2010.171.491},
+  url = {https://annals.math.princeton.edu/2010/171-1/p10}
+}
+
+@book{Ravenel2004,
+  author = {Ravenel, Douglas C.},
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+  edition = {Second},
+  publisher = {AMS Chelsea Publishing},
+  address = {Providence, RI},
+  year = {2004},
+  note = {Author's revised second-edition PDF, uploaded April 29, 2026; accessed September 23, 2026},
+  url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/mybooks/ravenel.pdf}
+}
+
+@article{Toda1967,
+  author = {Toda, Hirosi},
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+  journal = {Proceedings of the Japan Academy},
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+  year = {1967},
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+  doi = {10.3792/pja/1195521423},
+  url = {https://www.jstage.jst.go.jp/article/pjab1945/43/9/43_9_839/_article}
+}
+
+@article{Toda1968,
+  author = {Toda, Hirosi},
+  title = {Extended {$p$}-th powers of complexes and applications to homotopy theory},
+  journal = {Proceedings of the Japan Academy},
+  volume = {44},
+  number = {4},
+  year = {1968},
+  pages = {198--203},
+  doi = {10.3792/pja/1195521243},
+  url = {https://www.jstage.jst.go.jp/article/pjab1945/44/4/44_4_198/_article/-char/ja/}
+}
+
+@article{Toda1971,
+  author = {Toda, Hirosi},
+  title = {On spectra realizing exterior parts of the {Steenrod} algebra},
+  journal = {Topology},
+  volume = {10},
+  number = {1},
+  year = {1971},
+  pages = {53--65},
+  doi = {10.1016/0040-9383(71)90017-6},
+  url = {https://www.sciencedirect.com/science/article/pii/0040938371900176}
+}
+
+@article{MathewNaumannNoel2017,
+  author = {Mathew, Akhil and Naumann, Niko and Noel, Justin},
+  title = {Nilpotence and descent in equivariant stable homotopy theory},
+  journal = {Advances in Mathematics},
+  volume = {305},
+  pages = {994--1084},
+  year = {2017},
+  doi = {10.1016/j.aim.2016.09.027},
+  url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/MMN17.pdf}
+}
+
+@article{Adams1966,
+  author = {Adams, J. F.},
+  title = {On the groups {$J(X)$}. {IV}},
+  journal = {Topology}, volume = {5}, year = {1966}, pages = {21--71},
+  doi = {10.1016/0040-9383(66)90004-8}
+}
+
+@article{Smith1970,
+  author = {Smith, Larry},
+  title = {On realizing complex bordism modules. {Applications} to the stable homotopy of spheres},
+  journal = {American Journal of Mathematics}, volume = {92}, number = {4},
+  year = {1970}, pages = {793--856}, doi = {10.2307/2373397}
+}
+
+@article{Novikov1967,
+  author = {Novikov, S. P.},
+  title = {The methods of algebraic topology from the viewpoint of cobordism theory},
+  journal = {Mathematics of the USSR-Izvestiya}, volume = {1}, number = {4},
+  year = {1967}, pages = {827--913}, doi = {10.1070/IM1967v001n04ABEH000591},
+  url = {https://www.mathnet.ru/eng/im2568}
+}
+
+@misc{HoveyShipleySmith1998v2,
+  author = {Hovey, Mark and Shipley, Brooke and Smith, Jeff},
+  title = {Symmetric spectra}, year = {1998},
+  eprint = {math/9801077v2}, archivePrefix = {arXiv}, primaryClass = {math.AT},
+  url = {https://arxiv.org/pdf/math/9801077v2},
+  note = {Version 2, March 4, 1998; cited theorem numbers and pagination refer to this version}
+}
+
+@book{ElmendorfKrizMandellMay1997,
+  author = {Elmendorf, A. D. and Kriz, I. and Mandell, M. A. and May, J. P.},
+  title = {Rings, Modules, and Algebras in Stable Homotopy Theory},
+  series = {Mathematical Surveys and Monographs}, volume = {47},
+  publisher = {American Mathematical Society}, year = {1997},
+  doi = {10.1090/surv/047},
+  url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/EKMM.pdf},
+  note = {Locators refer to the linked author-distributed version}
+}
+
+@book{SteenrodEpstein1962,
+  author = {Steenrod, N. E.}, title = {Cohomology Operations},
+  note = {Lectures written and revised by D. B. A. Epstein},
+  series = {Annals of Mathematics Studies}, number = {50},
+  publisher = {Princeton University Press}, address = {Princeton, NJ}, year = {1962},
+  url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/steenrod-epstein.pdf}
+}
+
+@article{LeeRavenel1994,
+  author = {Lee, Chun-Nip and Ravenel, Douglas C.},
+  title = {On the nilpotence order of {$\beta_1$}},
+  journal = {Mathematical Proceedings of the Cambridge Philosophical Society},
+  volume = {115}, number = {3}, year = {1994}, pages = {483--488},
+  doi = {10.1017/S0305004100072248},
+  url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/mypapers/lee.pdf},
+  note = {Linked author preprint dated April 1993}
+}
+
+@article{HopkinsSmith1998,
+  author = {Hopkins, Michael J. and Smith, Jeffrey H.},
+  title = {Nilpotence and stable homotopy theory {II}},
+  journal = {Annals of Mathematics}, series = {2}, volume = {148},
+  number = {1}, year = {1998}, pages = {1--49}, doi = {10.2307/120991},
+  url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/HS.pdf}
+}
+
+@article{Schwede2001,
+  author = {Schwede, Stefan}, title = {{$S$}-modules and symmetric spectra},
+  journal = {Mathematische Annalen}, volume = {319}, year = {2001},
+  pages = {517--532}, url = {https://www.math.uni-bonn.de/~schwede/comparison.pdf}
+}
+
+@article{MandellMaySchwedeShipley2001,
+  author = {Mandell, M. A. and May, J. P. and Schwede, S. and Shipley, B.},
+  title = {Model categories of diagram spectra},
+  journal = {Proceedings of the London Mathematical Society}, series = {3},
+  volume = {82}, number = {2}, year = {2001}, pages = {441--512},
+  doi = {10.1112/S0024611501012692},
+  url = {https://homepages.math.uic.edu/~bshipley/MMSS.pdf}
+}
+
+@article{BrownPeterson1966,
+  author = {Brown, Jr., Edgar H. and Peterson, Franklin P.},
+  title = {A spectrum whose {$Z_p$} cohomology is the algebra of reduced {$p$}th powers},
+  journal = {Topology}, volume = {5}, number = {2}, year = {1966},
+  pages = {149--154}, doi = {10.1016/0040-9383(66)90015-2}
+}
+
+@article{Quillen1969,
+  author = {Quillen, Daniel},
+  title = {On the formal group laws of unoriented and complex cobordism theory},
+  journal = {Bulletin of the American Mathematical Society}, volume = {75},
+  number = {6}, year = {1969}, pages = {1293--1298},
+  doi = {10.1090/S0002-9904-1969-12401-8}
+}
+
+@article{HazewinkelIII,
+  author = {Hazewinkel, Michiel},
+  title = {Constructing formal groups {III}: Applications to complex cobordism and {Brown--Peterson} cohomology},
+  journal = {Journal of Pure and Applied Algebra}, volume = {10}, number = {1},
+  year = {1977}, pages = {1--18}, doi = {10.1016/0022-4049(77)90026-3}
+}
+
+@article{May1966,
+  author = {May, J. P.},
+  title = {The cohomology of restricted {Lie} algebras and of {Hopf} algebras},
+  journal = {Journal of Algebra}, volume = {3}, year = {1966},
+  pages = {123--146}, doi = {10.1016/0021-8693(66)90009-3}
+}
+
+@article{McClureSmith2004,
+  author = {McClure, James E. and Smith, Jeffrey H.},
+  title = {Cosimplicial objects and little {$n$}-cubes. {I}},
+  journal = {American Journal of Mathematics}, volume = {126}, number = {5},
+  year = {2004}, pages = {1109--1153},
+  eprint = {math/0211368v2}, archivePrefix = {arXiv}, primaryClass = {math.QA},
+  url = {https://arxiv.org/pdf/math/0211368v2},
+  note = {Cited construction locators refer to arXiv version 2, February 7, 2004}
+}
+
+@techreport{Jantzen1985,
+  author = {Jantzen, Jens Carsten},
+  title = {Representations of Algebraic Groups {I}},
+  institution = {Max-Planck-Institut f{\"u}r Mathematik},
+  type = {SFB/MPI preprint},
+  number = {85-9},
+  year = {1985},
+  url = {https://archive.mpim-bonn.mpg.de/607/1/preprint_1985_9.pdf}
+}
+
+@misc{KatoShimomuraShimomura2025,
+  author = {Kato, Ryo and Shimomura, Katsumi and Shimomura, Mao-no-suke},
+  title = {A note on the {Greek} letter elements of the homotopy of the
+           {$L_n$}-local spheres},
+  year = {2025},
+  eprint = {2507.02155v1},
+  archivePrefix = {arXiv},
+  primaryClass = {math.AT},
+  url = {https://arxiv.org/abs/2507.02155v1},
+  note = {Version 1, July 2, 2025}
+}
diff --git a/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/build/sections/00-introduction.tex b/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/build/sections/00-introduction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..d8de958db2e72601a446bbd9d027b8787b284880
--- /dev/null
+++ b/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/build/sections/00-introduction.tex
@@ -0,0 +1,163 @@
+\section{Introduction}\label{intro:section}
+
+The Smith--Toda realization problem asks how far one can kill the chromatic
+parameters in Brown--Peterson homology, one at a time, while retaining a finite
+spectrum. At a prime $p$, write
+\[
+ BP_* = \mathbb Z_{(p)}[v_1,v_2,\ldots],
+ \qquad |v_i|=2(p^i-1),
+\]
+with Hazewinkel generators. A \emph{finite $p$-local spectrum} is a retract
+of the $p$-localization of a finite CW spectrum. A \emph{Smith--Toda
+complex $V(n)$} is such a spectrum with Brown--Peterson homology
+$BP_*/(p,v_1,\ldots,v_n)$, including the canonical $BP_*BP$-comodule
+structure and a generator in degree zero. In this quotient each listed
+generator is killed to its first power. The comodule records the
+Brown--Peterson cooperations, so its specification is part of the
+realization problem, in addition to the underlying graded module.
+The periodicity theorem gives generalized Moore spectra whose homology
+kills suitable powers of the initial generators
+\citep[Proposition~5.14]{HopkinsSmith1998}. Requiring every exponent
+to be one is a more restrictive realization problem.
+
+The mod-$p$ Moore spectrum gives $V(0)$ at every prime. The classical
+constructions of $V(1)$, $V(2)$, and $V(3)$ for $p>2$, $p>4$, and $p>6$
+are due to Adams, Smith, and Toda
+\citep{Adams1966,Smith1970,Toda1971}; see also the historical account in
+\citet[p.~493]{Nave2010}. Successive cofibers turn a suitable first-power
+$v_i$ self-map into the next quotient. The difficulty is producing the
+self-map on the finite spectrum: an algebraically primitive coefficient
+need not survive the Adams--Novikov spectral sequence (ANSS) to an actual map.
+These constructions thus connect the algebra of complex cobordism with
+specific attaching maps in the stable homotopy category.
+
+The next case remained outside those constructions. The September 2023
+version of \citet[Remark~3.29]{CulverZhang2024} records the existence of
+$V(4)$ as open at every prime. There are also height-dependent
+nonexistence results: \citet[Theorem~1.3]{Nave2010} proves that
+$V((p+1)/2)$ does not exist for $p\geq7$, and that $V((p-1)/2)$,
+if it exists, cannot be a ring spectrum. Those obstructions leave
+room for $V(4)$ at a sufficiently large prime.
+We establish the following specified-prime realization.
+
+\begin{theorem}\label{intro:main}
+At $p=1009$ there is a finite $p$-local spectrum $X$ and an isomorphism
+\[
+ BP_*X\ \cong\ BP_*/(p,v_1,v_2,v_3,v_4)
+\]
+of graded $BP_*BP$-comodules, with the canonical quotient comodule on the
+right and its generator in degree zero.
+\end{theorem}
+
+Theorem~\ref{intro:main} gives a positive answer to the Smith--Toda $V(4)$
+existence problem at the stated prime. Its conclusion concerns the ordinary
+$p$-local stable category; finiteness is established before any chromatic
+localization. For comparison,
+\citet[Theorem~1.7]{KatoShimomuraShimomura2025} construct at this prime
+an $L_5$-local spectrum $W_5$ with
+\[
+ BP_*W_5=v_5^{-1}BP_*/(p,v_1,v_2,v_3,v_4),
+\]
+where $L_5$ denotes localization with respect to $v_5^{-1}BP$.
+That realization has $v_5$ inverted and does not supply the ordinary
+finite realization of the unlocalized quotient required here.
+The choice $1009$ makes the finite inequalities in our calculation
+generous. We do not require a product on the final spectrum $X$.
+
+\subsection{Methods and proof strategy}
+
+The algebraic setting comes from the spectrum of Brown and Peterson
+\citep{BrownPeterson1966}, Quillen's formal-group description of
+cobordism and its $p$-typical summand \citep{Quillen1969}, and
+Hazewinkel's explicit $p$-typical generators \citep{HazewinkelIII}.
+Our main tool is the sphere-source Adams--Novikov spectral sequence,
+whose cobordism-theoretic construction goes back to Novikov
+\citep{Novikov1967}. We use its ordinary $BP$ resolution, its cobar
+$E_2$ term, and its compatibility with actual binary maps; see
+\citet[Sections~2.2--2.3]{Ravenel2004}. The algebraic estimates use
+filtered Hopf-algebra cohomology as developed by
+\citet[Section~4, Theorem~4]{May1966},
+with the odd-primary filtration of
+\citet[Theorem~3.2.5]{Ravenel2004} and an additional filtration
+adjustment constructed below.
+
+A closer predecessor is the work of
+\citet[author preprint, pp.~2--6]{LeeRavenel1994} on the nilpotence
+order of the first beta-family sphere class $\beta_1$.
+They use a modified May calculation on $V(3)$ and constrain
+Adams--Novikov differentials through multiplication by $\beta_1$
+and the $V(2)$-module structure of $V(3)$.
+Our final obstruction calculation requires an annihilation relation
+for products of a beta power with specific filtration-five classes,
+together with a bound on the page at which those products vanish.
+Toda's work on sphere relations \citep[p.~839, Corollary]{Toda1967}
+and his subsequent cyclic extended-power constructions
+\citep[Sections~1--3, especially Theorem~3]{Toda1968} provide the
+topological antecedents. We give the particular coefficient-cohomology
+calculations and finite-cell arguments needed for this bound below.
+
+The construction has three stages. First we construct $Y=V(3)$ with a
+binary product and a two-sided unit. These unit identities are enough
+to turn a sphere map realizing $v_4$ into a self-map of $Y$. We obtain
+the lower complexes and their products by finite cofiber constructions
+and explicit vanishing of their attaching obstructions; we do not need
+an associative product on $Y$.
+
+Second, we prove that the filtration-zero class $v_4$ survives in the
+Adams--Novikov spectral sequence for $Y$. To describe the obstruction
+calculation, put $q=2p-2$ and $u_i=1+p+\cdots+p^{i-1}$ for $i\geq1$.
+Divide the internal degree by $q$ and call the resulting integer the
+\emph{weight}. A cobar class of cohomological degree $s$ and weight $w$
+lies in stem $qw-s$. Every possible positive differential on $v_4$
+therefore has target
+\begin{equation}\label{intro:targets}
+ (s,w)=(qk+1,u_4+k),\qquad k\geq1.
+\end{equation}
+A finite May calculation leaves only $k=1,2,p-1,p$. For $k=1$, a
+relation supplied by the coboundary of $v_5$, together with an injective
+multiplication on the target, forces the differential to vanish. For
+$k=2,p-1$, we compute actual cohomology by taking the Frobenius kernel
+of the group of truncated additive coordinate changes. The quotient
+acts triangularly on kernel cohomology. Filtering this coefficient
+module by weight identifies the first possible differential and
+checks that both its source and its target are present. This step
+controls the polynomial generators as well as the exterior generators
+on the May page.
+
+The remaining case $k=p$ requires a topological argument. We write
+$\beta\in\pi_{pq-2}S^0_{(p)}$ for the first beta-family sphere class.
+The entire $E_2$ target consists of products of a large power of its
+detector with certain filtration-five classes of $Y$.
+We realize those low-filtration classes by maps
+$\gamma$ and construct an explicit relation
+\[
+ \beta^{\,p(p-2)+1}\gamma=0.
+\]
+The construction uses a cyclic extended power of a sphere and the
+averaging retract of a smash power of a two-cell spectrum. The corresponding
+Adams--Novikov product has filtration strictly smaller than the length of
+the prospective
+differential on $v_4$. Consequently the whole target has vanished
+before that differential could occur. The bound on the page is the
+point: eventual vanishing in homotopy, without such a bound, would
+not settle this obstruction.
+
+Finally the filtration-zero edge gives a map
+$S^{2(p^4-1)}\to Y$ with Brown--Peterson Hurewicz image $v_4$.
+Multiplication by the binary product on $Y$ extends it to
+$\Sigma^{2(p^4-1)}Y\to Y$. Its cofiber is finite, and the
+nonzerodivisor property of $v_4$ identifies its homology with the
+required quotient. The bottom sphere map identifies the quotient
+coaction and fixes the generator in degree zero.
+
+Section~\ref{fw:section} fixes the Brown--Peterson coordinates and
+the topological spectral-sequence conventions.
+Section~\ref{may:preliminaries} constructs the filtered cobar calculation,
+and Section~\ref{low:section} supplies the unital lower complexes.
+Section~\ref{may:section} lists the height-four targets and removes the
+first differential; Section~\ref{frob:section} proves the two Frobenius
+vanishings. Section~\ref{top:section} establishes the homotopy relation,
+its page bound, and the realizing cofiber. Appendix~\ref{app:resolution}
+provides the ordinary convergent resolution and the filtered binary
+pairing that connects the algebraic calculation to the actual
+homotopy product.
diff --git a/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/build/sections/01-framework.tex b/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/build/sections/01-framework.tex
new file mode 100644
index 0000000000000000000000000000000000000000..09bc6dbd31f634302b9ffb69323e52c7e733fbe6
--- /dev/null
+++ b/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/build/sections/01-framework.tex
@@ -0,0 +1,276 @@
+\section{Brown--Peterson coordinates and the Adams--Novikov resolution}
+\label{fw:section}
+
+Throughout the proof,
+\begin{equation}
+ p=1009,\qquad q=2p-2=2016,\qquad
+ u_i=\frac{p^i-1}{p-1}\quad(i\geq 0).
+ \label{fw:parameters}
+\end{equation}
+The integer $1009$ is prime: its square root is less than $32$, and none of
+$2,3,5,7,11,13,17,19,23,29,31$ divides it.  Put
+\[
+ R=BP_* =\mathbb Z_{(p)}[v_1,v_2,\ldots],\qquad
+ \Gamma=BP_*BP=R[t_1,t_2,\ldots],\qquad
+ I_m=(p,v_1,\ldots,v_{m-1}).
+\]
+Both $v_i$ and $t_i$ have internal degree $qu_i$.  We call internal degree
+divided by $q$ the \emph{weight}.  A cobar class of cohomological degree $s$
+and weight $w$ has stem $qw-s$.
+
+We use the ordinary Brown--Peterson spectrum, its polynomial cooperation
+algebra, and a strictly unital associative model of its multiplication.
+For the identification of these polynomial rings with the homotopy and
+cooperations of $BP$, see
+\citet[Theorems~4.1.18--4.1.19]{Ravenel2004}. The Hazewinkel recursion
+and the integrality of its polynomial generators are given in
+\citet[Equation~A2.2.1 and Theorem~A2.2.3]{Ravenel2004}; the formal-sum
+coordinate convention is explained in the proof of Theorem~A2.1.27 there.
+The formal-group foundations are due to
+\citet[Section~5, Theorem~4]{Quillen1969}; the explicit generators and
+cooperations are developed in \citet[Sections~3.1 and~4.4]{HazewinkelIII}.
+A coherent associative model exists, in particular, by forgetting structure
+from \citet[Theorem~1.1]{BasterraMandell2013}.  No commutative or
+$E_\infty$ model of $BP$ is required below.  We give the particular coordinate,
+convergence, and pairing arguments needed for the construction.
+
+\subsection{Coordinates and additive ranges}
+
+Let $a_i$ be the coefficients in the formal-group sum expression
+\[
+ [p]_F(x)=\mathop{\sum}\nolimits_F a_i x^{p^i},\qquad a_0=p.
+\]
+The strict coordinate change is written in the same convention as
+$\sum_F t_i x^{p^i}$, with $t_0=1$.  Reversing the convention for composition
+reverses the cobar conventions without changing any of the arguments.
+
+\begin{lemma}[Leading coefficients and low cooperations]
+\label{fw:coordinates}
+For each $m\geq 1$, $I_m$ is an invariant ideal and $v_m$ is primitive
+modulo $I_m$.  Modulo $I_m$, the coproducts of $t_i$, $i\leq m$, are the
+additive-composition coproducts.  At the specialization that kills $p$ and
+all $v_i$, the resulting Hopf algebra is
+\begin{equation}
+ P=\mathbb F_p[t_1,t_2,\ldots],\qquad
+ \Delta(t_i)=\sum_{\ell=0}^i t_\ell\otimes t_{i-\ell}^{p^\ell}.
+ \label{fw:additive-hopf}
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Write the logarithm as $\log_F(x)=\sum_{i\geq0}l_i x^{p^i}$, where $l_0=1$.
+The Hazewinkel recursion is
+\[
+ pl_n=v_n+\sum_{i=1}^{n-1}l_i v_{n-i}^{p^i}.
+\]
+Applying the logarithm to the formal sum for $[p]_F$ and comparing the
+coefficient of $x^{p^n}$ shows that the coefficient of the indecomposable
+$v_n$ in $a_n$ is
+\begin{equation}
+ c_n=1-p^{p^n-1}.
+ \label{fw:coordinate-unit}
+\end{equation}
+Indeed the two terms involving $l_n$ are $pl_n$ and $p^{p^n}l_n$, and the
+coefficient of $v_n$ in $l_n$ is $1/p$.  All other terms involve lower
+$v_i$.  The coefficients $a_n$ are integral, so the grading and polynomial
+independence of the $v_i$ give
+$a_n-c_nv_n\in(v_1,\ldots,v_{n-1})R$.  Each $c_n$ is a $p$-local unit.
+
+The invariance and primitivity follow inductively.  The element $p$ is
+invariant.  Once the lower coefficients vanish, the first surviving term
+of the $p$-series is $c_m v_m x^{p^m}$.  A strict coordinate change has
+leading term $x$, so comparison of this first surviving term on the two
+sides of the coordinate-change identity fixes $c_m v_m$, and hence $v_m$.
+This proves both invariance of the next ideal and the asserted primitivity.
+
+After all base generators have been killed, the formal group is additive,
+and composition of additive $p$-polynomials gives
+Equation~\eqref{fw:additive-hopf}.  Modulo $I_m$, a correction to this
+coproduct must have a positive base coefficient, of weight at least $u_m$.
+For $i<m$ its weight is already too large.  For $i=m$ the only possibility
+is a constant multiple of $v_m$ with no positive-weight cooperation factor.
+Applying either counit excludes that term.  This proves the stated range.
+\end{proof}
+
+\begin{lemma}[The relation supplied by $v_5$]
+\label{fw:v5-relation}
+In the cobar cohomology with coefficients $R/I_4$, there is a scalar
+$e\in\mathbb F_p$ such that
+\begin{equation}
+ v_4\bigl(h_{1,4}+e v_4^{p-1}h_{1,0}\bigr)=0,
+ \label{fw:v5-cohomology-relation}
+\end{equation}
+where $h_{1,j}$ is represented by the primitive $t_1^{p^j}$.
+\end{lemma}
+
+\begin{proof}
+Modulo $I_4$, the first two formal-sum coefficients are unit multiples of
+$v_4,v_5$.  There is no decomposable correction involving $v_4$ to the
+second of these coefficients: $u_5=pu_4+1$ is not a multiple of $u_4$.
+The formal group law has no nonlinear term of total ordinary degree less
+than $p^4$, because a coefficient of such a term would have positive weight
+less than $u_4$.  Thus, through exponent $p^5$, its $p$-series is the ordinary
+sum of its $x^{p^4}$ and $x^{p^5}$ terms.  The nonlinear terms in adding
+these two inputs have exponent greater than $p^5$.
+
+In the identity intertwining this $p$-series with a strict coordinate
+change, only $x+t_1x^p$ from that change can affect these two exponents.
+On the side where the change is applied after the $p$-series, the
+coefficient of $x^{p^5}$ acquires a term proportional to $v_4^p t_1$.
+On the other side it acquires a term proportional to
+$v_4t_1^{p^4}$.  All nonlinear corrections to the strict formal-sum change
+enter too late after taking the $p^4$-th power.  Consequently
+\[
+ \eta_R(v_5)-v_5=c\,v_4t_1^{p^4}+d\,v_4^p t_1
+ \quad\text{in }\Gamma/I_4\Gamma,
+\]
+where $c,d\in\mathbb F_p$ and $c\ne0$.  The nonzero coefficient follows
+also directly from the unit factors $c_4,c_5$ in
+Equation~\eqref{fw:coordinate-unit}; their reductions cannot annihilate
+the first displayed contribution.  Divide this coboundary by $c$ and put
+$e=d/c$.  Since $t_1$ is primitive modulo $I_4$, all its $p$-power iterates
+are primitive, and the resulting cohomology relation is
+Equation~\eqref{fw:v5-cohomology-relation}.
+\end{proof}
+
+For a comodule concentrated in weights that are nonnegative integers, let
+$C^{s,w}$ denote normalized cobar cochains of length $s$ and weight $w$.
+We use the convention in which a positive-length cochain has a coefficient
+followed by $s$ reduced cooperation factors.  This convention fixes the
+signs of the usual cobar differential and cup product.
+
+\begin{lemma}[Normalized positivity]
+\label{fw:normalized-bound}
+If the coefficient comodule has no negative weights, then
+\begin{equation}
+ C^{s,w}\ne0\quad\Longrightarrow\quad w\geq s.
+ \label{fw:positivity}
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Every reduced cooperation factor is a polynomial monomial containing at
+least one $t_i$, and hence has weight at least one.  The coefficient has
+nonnegative weight.  Adding these weights proves the claim.
+\end{proof}
+
+\begin{lemma}[A comparison in three cochain degrees]
+\label{fw:range-comparison}
+Suppose $s\geq1$, $w<u_{m+1}$, and
+$w-u_m<s-1$.  Additive specialization identifies the normalized cobar
+complexes for $R/I_m$ and $P$, in weight $w$, in all three degrees
+$s-1,s,s+1$.  It intertwines both adjacent differentials, and therefore
+induces an isomorphism on cohomology in degree $(s,w)$.
+The same conclusion in every cochain degree holds if $w<u_m$.
+\end{lemma}
+
+\begin{proof}
+In each of the three degrees, any positive coefficient monomial consumes
+at least $u_m$ weight.  It would leave at most $w-u_m<s-1$ for at least
+$s-1$ reduced bar factors, which is impossible.  A coordinate $t_i$ with
+$i\geq m+1$ exceeds the entire weight budget.  Thus these cochain groups
+have scalar coefficients and only $t_1,\ldots,t_m$ in their bars.  Their
+coproducts are additive by Lemma~\ref{fw:coordinates}.  This identifies
+the two groups adjacent to the desired cohomology group as well as the
+group itself, so both kernels and images agree.  If $w<u_m$, the same
+exclusion of coefficients and coordinates works without a cochain-length
+restriction.
+\end{proof}
+
+\subsection{Convergence, pairings, and the edge}
+
+The following two propositions specify the topological spectral sequence and
+the binary pairings used throughout the construction. The ordinary
+Adams--Novikov resolution and its convergence are treated in
+\citet[Theorems~2.2.3, 2.2.13--2.2.14]{Ravenel2004};
+\citet[Theorem~2.3.3]{Ravenel2004} gives the binary-pairing, Leibniz,
+and homotopy-compatibility statements in this form. Their complete proofs
+in the particular resolution used here,
+including the ordinary totalization model and its filtration, are given in
+Appendix~\ref{app:resolution}.
+
+\begin{proposition}[The bounded-below Adams--Novikov resolution]
+\label{fw:anss}
+For every bounded-below $p$-local spectrum $Z$, the sphere-source
+Adams--Novikov spectral sequence has
+\[
+ E_2^{s,t}(Z)=H^{s,t}\bigl(C^\bullet_\Gamma(BP_*Z)\bigr)
+ \quad\Longrightarrow\quad \pi_{t-s}Z.
+\]
+It converges strongly, and the induced filtration in each stem is finite.
+The differential $d_r$ changes $(s,t)$ by $(r,r-1)$.  The filtration-zero
+edge identifies $E_\infty^{0,*}$ with the image of the $BP$-Hurewicz map
+$\pi_*Z\longrightarrow BP_*Z$.
+\end{proposition}
+The proof is given in Appendix~\ref{app:convergence}.
+
+
+\begin{proposition}[Binary pairings of the resolution]
+\label{fw:anss-pairing}
+For bounded-below $p$-local spectra $Z_1,Z_2,Z_3$, an actual map
+$f:Z_1\wedge Z_2\to Z_3$ induces pairings of the
+Adams--Novikov spectral sequences, with the Leibniz rule on every page.
+The $E_2$ pairing is the cobar cup pairing induced by the exterior
+$BP$-homology pairing, and the $E_\infty$ pairing is the associated-graded
+pairing of the actual homotopy map $f$.  No associativity of $f$ is needed.
+If $Z_1=Z_2=Z_3=Z$ has a binary two-sided unital multiplication and
+$BP_*Z=R/I$ is cyclic on its unit, the $E_2$ pairing is the ordinary
+multiplication on cobar cohomology with quotient-ring coefficients.
+\end{proposition}
+The proof is given in Appendix~\ref{app:pairing}.
+
+
+\begin{corollary}[Sparsity]
+\label{fw:sparsity}
+If $BP_*Z$ is concentrated in internal degrees divisible by $q$, then all
+pages of its Adams--Novikov spectral sequence have that same internal
+sparsity.  Every possibly nonzero positive differential has length
+$r=qk+1$ for an integer $k\geq1$.  In weight notation it changes
+$(s,w)$ to $(s+qk+1,w+k)$.  In particular, a primitive coefficient of
+weight $u_n$ in filtration zero can have only targets
+\begin{equation}
+ (s,w)=(qk+1,u_n+k),\qquad k\geq1.
+ \label{fw:primitive-targets}
+\end{equation}
+\end{corollary}
+
+\begin{proof}
+The coefficient and cooperation gradings give the assertion on the cobar
+page.  A differential preserves the sparse support only if $q$ divides
+$r-1$.  Subsequent pages are subquotients of earlier pages.  Since $r\geq2$
+on or after the cohomology page, the first allowed length is $q+1$.
+\end{proof}
+
+\begin{lemma}[The additive comparison for the $v_4$ targets]
+\label{fw:additive-comparison}
+Let $s=qk+1$, $w=u_4+k$, $k\geq1$.  A nonzero normalized cobar group
+in degree $(s,w)$ with coefficients $R/I_4$ requires
+\begin{equation}
+ (q-1)k\leq u_4-1,\qquad
+ w\leq\frac{qu_4-1}{q-1}<2u_4<p^4<u_5.
+ \label{fw:preliminary-v4-bound}
+\end{equation}
+For every such $k$, additive specialization is an isomorphism on
+cohomology in degree $(s,w)$, by an identification of the cochain groups
+of lengths $s-1,s,s+1$.  No $v_4$ coefficient or $t_4$ bar occurs in
+any of those three cochain groups.
+\end{lemma}
+
+\begin{proof}
+The first two inequalities are the rearrangement of $w\geq s$.
+Here
+\[
+ u_4=1{,}028{,}262{,}820,\qquad
+ 2u_4=2{,}056{,}525{,}640<p^4=1{,}036{,}488{,}922{,}561,
+\]
+so the remaining inequalities in
+Equation~\eqref{fw:preliminary-v4-bound} hold at the fixed prime.
+For a cochain length $d\in\{s-1,s,s+1\}$, a coefficient containing
+$v_4$ would leave at most $k$ weight for $d\geq qk>k$ reduced factors.
+It therefore cannot occur.  A bar containing $t_4$ would leave at most
+$k$ weight for the other $d-1\geq qk-1>k$ bars, and likewise cannot occur.
+Higher generators exceed the weight bound.  The remaining coproducts
+are additive by Lemma~\ref{fw:coordinates}; equivalently, apply
+Lemma~\ref{fw:range-comparison}.  This identifies both adjacent
+differentials and proves the cohomology assertion.
+\end{proof}
diff --git a/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/build/sections/02-lower-complexes.tex b/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/build/sections/02-lower-complexes.tex
new file mode 100644
index 0000000000000000000000000000000000000000..f9eb26ea37a2e5fe9f819518bfd69937c4ed27d9
--- /dev/null
+++ b/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/build/sections/02-lower-complexes.tex
@@ -0,0 +1,315 @@
+\section{The unital complexes through height three}
+\label{low:section}
+
+We first construct the lower complexes, retaining a binary two-sided unit.
+The algebraic input is the May upper bound of Lemma~\ref{may:page}:
+a bidegree containing no initial-page monomial has zero cobar cohomology.
+The two vanishing families below serve different parts of the induction:
+the first makes each $v_n$ survive to a self-map, and the second permits
+the two unit maps to extend to a binary product over every pair of cells.
+
+\begin{lemma}[Lower obstruction groups]
+\label{low:algebraic-vanishing}
+The following cobar cohomology groups vanish.
+\begin{enumerate}
+ \item For $1\leq n\leq3$ and $k\geq1$, the group with coefficients
+ $R/I_n$ in bidegree
+ \[
+  (s,w)=(qk+1,u_n+k).
+ \]
+ \item For $0\leq n\leq3$, choose $e_i\in\{0,1,2\}$ for $0\leq i\leq n$
+ and put
+ \[
+  U=\sum_{i=0}^n e_i u_i,\qquad
+  \rho=\sum_{i=0}^n e_i\geq2.
+ \]
+ For every $k\geq1$, the group with coefficients $R/I_{n+1}$ in bidegree
+ \[
+  (s,w)=(qk+1-\rho,U+k)
+ \]
+ is zero.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+Include the first case in the notation of the second by setting
+$U=u_n$ and $\rho=0$, while retaining its indicated coefficient comodule.
+If the normalized cochain group in degree $(s,w)$ is nonzero,
+Lemma~\ref{fw:normalized-bound} gives
+\begin{equation}
+ 2015k\leq U+\rho-1.
+ \label{low:normalized-cutoff}
+\end{equation}
+The exact values
+\[
+ (u_0,u_1,u_2,u_3,u_4)=(0,1,1010,1{,}019{,}091,1{,}028{,}262{,}820)
+\]
+give Table~\ref{low:cutoffs}.  Each upper bound is obtained by substituting
+the displayed maxima in Equation~\eqref{low:normalized-cutoff} and taking
+the integer part.  A zero upper bound on $k$ excludes all $k\geq1$.
+
+\begin{table}[htbp]
+\centering
+\begin{tabular}{lrrrr}
+\toprule
+Case & Maximum $U$ & Maximum $\rho$ & Maximum $k$ & Maximum $w$\\
+\midrule
+Product, $n=0$ & $0$ & $2$ & $0$ & ---\\
+Product, $n=1$ & $2$ & $4$ & $0$ & ---\\
+Product, $n=2$ & $2022$ & $6$ & $1$ & $2023$\\
+Product, $n=3$ & $2{,}040{,}204$ & $8$ & $1012$ & $2{,}041{,}216$\\
+$v_1$ target & $1$ & $0$ & $0$ & ---\\
+$v_2$ target & $1010$ & $0$ & $0$ & ---\\
+$v_3$ target & $1{,}019{,}091$ & $0$ & $505$ & $1{,}019{,}596$\\
+\bottomrule
+\end{tabular}
+\caption{Preliminary bounds for all lower obstruction bidegrees.}
+\label{low:cutoffs}
+\end{table}
+
+These bounds also justify the additive comparison in the adjacent cochain
+degrees.  For either remaining product case, $w<u_{n+1}$, so no positive
+coefficient or coordinate $t_i$ with $i\geq n+1$ can occur.  The retained
+coproducts are additive modulo $I_{n+1}$, and the last assertion of
+Lemma~\ref{fw:range-comparison} applies in every degree.  For a remaining
+$v_n$ target, a positive coefficient costs at least $u_n$, leaving at most
+$k$ weight for $d\geq s-1=qk>k$ reduced bars when
+$d\in\{s-1,s,s+1\}$.  Such a coefficient cannot occur.  A bar containing
+$t_n$ or a higher coordinate leaves at most $k$ for the other
+$d-1\geq qk-1>k$ bars and cannot occur either.  Hence the same three
+cochain degrees are additive.  In all cases, both adjacent differentials
+are identified, so the desired cohomology is identified with that for $P$.
+
+Every remaining weight in Table~\ref{low:cutoffs} is at most $2{,}041{,}216$,
+whereas
+\[
+ p^3=1{,}027{,}243{,}729.
+\]
+A May letter covering position $3$ or above has weight at least $p^3$.
+Thus only positions $0,1,2$ can occur. We use the common exterior and
+polynomial alphabet of Equation~\eqref{may:low-alphabet}, with the interval
+coverage conventions of Section~\ref{may:preliminaries}.
+Allowing this entire alphabet only enlarges the upper bound if fewer
+coordinates were retained in the preceding comparison.
+
+Suppose a monomial of the desired bidegree exists.  Let $m\leq6$ be its
+number of exterior letters and $L=(s-m)/2$ its total polynomial exponent.
+Every polynomial letter has weight at least $p$, so
+\[
+ U+k\geq pL=p(p-1)k+\frac{p(1-\rho-m)}2.
+\]
+For all profiles under consideration,
+$U\leq2p^2+4p+6$ and $\rho\leq8$.  Consequently
+\begin{equation}
+ (p^2-p-1)k
+ \leq U+\frac{p(\rho+5)}2
+ \leq 2p^2+\frac{21}2p+6.
+ \label{low:polynomial-cutoff}
+\end{equation}
+If $k\geq3$, the left side exceeds the last expression by at least
+\[
+ p^2-\frac{27}2p-9=\frac{2{,}008{,}901}{2}>0.
+\]
+It follows that $1\leq k\leq2$.
+
+Write $U=F_0+pF_1+p^2F_2$.  In a product profile the digits are
+\[
+ F_0=e_1+e_2+e_3,\qquad F_1=e_2+e_3,\qquad F_2=e_3,
+\]
+with missing $e_i$ set to zero.  For a $v_n$ target they are initial
+strings of ones.  In every case,
+\begin{equation}
+ 6\geq F_0\geq F_1\geq F_2\geq0,
+ \qquad F_1\leq4,\quad F_2\leq2.
+ \label{low:digits}
+\end{equation}
+Let $H_r$ count the exterior letters covering position $r$, and let
+$n_B,n_C$ be the polynomial exponents of $B,C$.  Adding weights gives
+the exact identity
+\begin{equation}
+ F_0+k+pF_1+p^2F_2
+  =H_0+p(H_1+L-n_C)+p^2(H_2+n_B+n_C).
+ \label{low:weight-equation}
+\end{equation}
+We next extract its digits without an unproved no-carry assumption.
+
+First, $0\leq H_0\leq3$ and $1\leq F_0+k\leq8<p$.  Reduction modulo $p$
+therefore gives $H_0=F_0+k$.  Also $B,C$ each cost at least $p^2$, so
+\[
+ p^2(n_B+n_C)\leq w\leq2p^2+4p+8<3p^2;
+\]
+the last strict inequality has difference
+$p^2-4p-8=1{,}014{,}037>0$.  Hence $n_B+n_C\leq2$.
+
+Put $\delta=(m+\rho-1)/2$.  It is an integer because $s-m$ is even.
+For a target with $\rho=0$, that parity forces $m$ odd, hence $m\geq1$;
+for a product profile $\rho\geq2$.  Thus $\delta\geq0$.  The bounds
+$m\leq6$, $\rho\leq8$ and the same parity give $\delta\leq6$.
+Now $L=(p-1)k-\delta$.  Subtracting the position-zero equation from
+Equation~\eqref{low:weight-equation}, dividing by $p$, and rearranging gives
+\[
+ H_1-k-\delta-n_C-F_1
+   =p(F_2-k-H_2-n_B-n_C).
+\]
+Since $H_1\leq4$, the left side lies in $[-14,3]$.  Its only possible
+multiple of $p=1009$ is zero.  We have therefore proved the three exact
+relations
+\begin{equation}
+ H_0=F_0+k,\qquad
+ H_2+n_B+n_C=F_2-k,\qquad
+ H_1=F_1+k+\delta+n_C.
+ \label{low:coverage-relations}
+\end{equation}
+
+Among the exterior letters covering position zero, only $x,a$ fail to
+cover position two.  Thus $H_2\geq H_0-2$.  Equations~\eqref{low:digits}
+and~\eqref{low:coverage-relations} imply
+\[
+ F_0-F_2+2k\leq2.
+\]
+As $k\geq1$, equality is forced: $k=1$ and
+$F_0=F_1=F_2=f$.  All the intervening inequalities are equalities, giving
+\[
+ n_B=n_C=0,\qquad H_0=f+1,\qquad H_2=f-1.
+\]
+The bounds on $H_0,H_2$ leave only $f=1,2$.
+
+If $f=1$, the exterior letters $x,a$ must occur and $c,b,z$ must be
+absent.  Only $y$ is optional, so $H_1\leq2$ and $m\geq2$.  On the other
+hand, the last relation in Equation~\eqref{low:coverage-relations} gives
+$H_1=2+\delta>2$, since
+$\delta=(m+\rho-1)/2>0$.  If $f=2$, the letters $x,a,c$ must occur and
+$b,z$ must be absent.  Again only $y$ is optional, so $H_1\leq3$ and
+$m\geq3$, whereas the same relation gives $H_1=3+\delta>3$.
+Both possibilities are impossible.  There is no monomial on the May
+upper-bound page, and hence no cobar cohomology in any of the stated
+bidegrees.
+\end{proof}
+
+\begin{lemma}[Finite local cell attachments]
+\label{low:finite-localization}
+A spectrum obtained from finitely many $p$-local sphere cells is the
+$p$-localization of a finite CW spectrum.
+\end{lemma}
+
+\begin{proof}
+Proceed one attachment at a time.  If the previously constructed spectrum
+is the localization of a finite spectrum $Z$, an attaching map belongs to
+$\pi_d(Z_{(p)})=\pi_d(Z)\otimes\mathbb Z_{(p)}$ and can be written $a/m$
+with $p\nmid m$.  After localization, multiplication by $m$ on its sphere
+source is an equivalence.  The localized cone of the integral attaching
+map $a$ is therefore equivalent to the cone of $a/m$.  Induction over
+the finitely many cells proves the assertion.
+\end{proof}
+
+\begin{proposition}[Construction through $V(3)$]
+\label{low:V3}
+For $0\leq n\leq3$ there is a finite $p$-local cell spectrum $V(n)$ with
+a bottom-cell unit $\eta_n:S_{(p)}\to V(n)$ and a binary multiplication
+\[
+ \mu_n:V(n)\wedge V(n)\longrightarrow V(n)
+\]
+whose two unit identities hold.  Its cells are indexed by subsets of
+$\{0,\ldots,n\}$, in dimensions
+\begin{equation}
+ \sum_{i\in S}(qu_i+1),
+ \label{low:cell-dimensions}
+\end{equation}
+and its $BP$-homology is the canonical quotient comodule $R/I_{n+1}$,
+generated in degree zero by the bottom unit.  In particular, we may fix
+\begin{equation}
+ Y=V(3),\qquad BP_*Y=R/I_4,
+ \label{low:Y}
+\end{equation}
+with such a binary two-sided unital multiplication.
+\end{proposition}
+
+\begin{proof}
+Start with $V(0)$, the cofiber of $p:S_{(p)}\to S_{(p)}$.  Since
+multiplication by $p$ is injective on $R$, its $BP$-homology is $R/(p)$,
+with the canonical quotient coaction and bottom generator.  Its cells
+are in dimensions zero and one, as in
+Equation~\eqref{low:cell-dimensions}.
+
+We give first the product-extension argument, which applies to this
+initial complex and to every later complex once its asserted homology
+and cells have been obtained.  The subcomplex
+\[
+ U_n=(S_{(p)}\wedge V(n))
+       \mathop{\cup}_{S_{(p)}\wedge S_{(p)}}
+       (V(n)\wedge S_{(p)})
+       \ \subseteq\ V(n)\wedge V(n)
+\]
+has a map to $V(n)$ given by the two identity maps, which agree on the
+common bottom sphere.  Every remaining cell corresponds to a pair of
+nonempty subsets of $\{0,\ldots,n\}$.  If $e_i$ counts occurrences of $i$
+in that pair, its dimension is $qU+\rho$, with precisely the parameters
+in the second part of Lemma~\ref{low:algebraic-vanishing}.  The obstruction
+to extending across the cell lies in
+$\pi_{qU+\rho-1}V(n)$.
+
+An Adams--Novikov term in that stem has
+$qw-s=qU+\rho-1$, or
+\[
+ w=U+k,\qquad s=qk+1-\rho.
+\]
+Since $2\leq\rho\leq8<q$, nonnegative filtration requires $k\geq1$.
+Every corresponding $E_2$ group is zero by
+Lemma~\ref{low:algebraic-vanishing}.  Strong convergence and finite
+filtration, from Proposition~\ref{fw:anss}, give
+$\pi_{qU+\rho-1}V(n)=0$.  Extending cell by cell therefore produces
+$\mu_n$, agreeing with both identity maps on $U_n$.  This constructs the
+initial product on $V(0)$ as well.
+
+Suppose now that $V(n-1)$, including its product, has been constructed,
+where $1\leq n\leq3$.  By Lemma~\ref{fw:coordinates}, $v_n$ is a primitive
+element of $BP_*V(n-1)=R/I_n$.  It defines a filtration-zero $E_2$ class.
+Its possible positive differential targets are exactly
+Equation~\eqref{fw:primitive-targets}; the first part of
+Lemma~\ref{low:algebraic-vanishing} makes them all zero.  There is no
+incoming differential to filtration zero.  The edge assertion in
+Proposition~\ref{fw:anss} consequently realizes $v_n$ by a homotopy map
+\[
+ \widetilde v_n:S^{qu_n}_{(p)}\longrightarrow V(n-1).
+\]
+Use the product to form
+\[
+ f_n=\mu_{n-1}(\widetilde v_n\wedge\operatorname{id}):
+       \Sigma^{qu_n}V(n-1)\longrightarrow V(n-1),
+\]
+and define $V(n)$ as its cofiber.
+
+The induced map on $BP$-homology is multiplication by $v_n$: the
+coefficient pairing is $R$-bilinear, the homology is cyclic on the unit,
+and $\widetilde v_n$ has the specified Hurewicz image.  This multiplication is
+injective on the polynomial quotient $R/I_n$.  The cofiber exact sequence
+therefore gives precisely $R/I_{n+1}$, with no additional shifted kernel
+summand.  The bottom sphere map surjects onto this cyclic homology, so its
+kernel $I_{n+1}$ identifies the coaction with the canonical quotient
+coaction, as well as fixing the generator in degree zero.
+
+The mapping cone adds the old cells shifted by $qu_n+1$, giving exactly
+Equation~\eqref{low:cell-dimensions}.  The product-extension argument
+already proved then constructs $\mu_n$.  This completes the induction.
+All attachments are finite $p$-local cell attachments, and
+Lemma~\ref{low:finite-localization} gives finiteness in the sense of the
+stated realization problem.  No associativity of any $\mu_n$ has been
+asserted or used.
+\end{proof}
+
+\begin{corollary}[The action of $p$ on $Y$]
+\label{low:p-null}
+For the spectrum $Y$ of Proposition~\ref{low:V3}, $\pi_0Y=\mathbb F_p$ and
+$p\operatorname{id}_Y$ is nullhomotopic.
+\end{corollary}
+
+\begin{proof}
+The bottom two cells form the mod-$p$ Moore spectrum.  All other cells have
+dimension at least $qu_1+1=q+1>1$, so they do not change $\pi_0$.
+Thus $p\eta_3=0$ in $\pi_0Y$.  Applying the left unit identity of its
+binary multiplication gives
+\[
+ p\operatorname{id}_Y
+  =\mu_3\bigl((p\eta_3)\wedge\operatorname{id}_Y\bigr)=0.
+\]
+\end{proof}
diff --git a/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/build/sections/02a-cobar-preliminaries.tex b/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/build/sections/02a-cobar-preliminaries.tex
new file mode 100644
index 0000000000000000000000000000000000000000..50b0986968467af44f2300e7989a5b110d8e9afe
--- /dev/null
+++ b/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/build/sections/02a-cobar-preliminaries.tex
@@ -0,0 +1,124 @@
+\section{Filtered cobar preliminaries}
+\label{may:preliminaries}
+
+We use the filtered Hopf-algebra method of \citet{May1966}, with
+the odd-primary May-type filtration in the form of
+\citet[Theorem~3.2.5]{Ravenel2004}, and allow an additional digit-zero
+weight needed for the injectivity argument in Section~\ref{may:section}.
+For earlier applications of Ravenel's filtration to $V(3)$ and powers
+of $\beta_1$, see \citet{LeeRavenel1994}.
+We construct this modified filtration below, including its initial page
+and convergence. Throughout this section, cohomological degree and weight are denoted by
+$(s,w)$.  Unless an Adams--Novikov differential is named explicitly, the
+spectral sequences in this section are algebraic spectral sequences of
+cobar complexes.  Put
+\[
+ P=\mathbb F_p[t_1,t_2,\ldots],\qquad
+ |t_i|_w=u_i,
+ \qquad
+ \overline\Delta(t_i)=\sum_{0<l<i}t_l\otimes t_{i-l}^{p^l}.
+\]
+Reversing all coproducts reverses the cobar conventions and changes none of
+the conclusions.  We use the displayed convention.
+
+\begin{lemma}[The filtered cobar calculation]
+\label{may:page}
+For every integer $M\geq0$, the cobar cohomology of $P$ admits a
+multiplicative, degreewise finite spectral sequence whose initial
+cohomology page is
+\begin{equation}
+ \Lambda(h_{i,j}:i\geq1,j\geq0)
+ \otimes\mathbb F_p[b_{i,j}:i\geq1,j\geq0].
+ \label{may:initial-page}
+\end{equation}
+The bidegrees and auxiliary filtrations are
+\[
+\begin{array}{c|c|c}
+ & (s,w)& F_M\\ \hline
+ h_{i,j}&(1,p^ju_i)&2i-1+M[j=0]\\
+ b_{i,j}&(2,p^{j+1}u_i)&p(2i-1+M[j=0]).
+\end{array}
+\]
+Here $[j=0]$ is $1$ or $0$ according as $j=0$ or $j\neq0$.
+All differentials lower $F_M$.  If an exterior generator is written
+$h_{[a,b)}=h_{b-a,a}$, its first differential is
+\begin{equation}
+ d_1 h_{[a,b)}=\sum_{a<l<b}h_{[a,l)}h_{[l,b)},
+ \qquad d_1b_{i,j}=0.
+ \label{may:splitting}
+\end{equation}
+The same statements hold for the Hopf subalgebra
+$P_r=\mathbb F_p[t_1,\ldots,t_r]$, with $i\leq r$ in
+\eqref{may:initial-page}.
+\end{lemma}
+
+\begin{proof}
+Write every exponent in base $p$ and assign a digit factor $t_i^{p^j}$
+the filtration $2i-1+M[j=0]$.  For a monomial, add these filtrations with
+the digit multiplicities, without carrying.  Carrying $p$ copies of one
+digit to the next digit lowers filtration: its decrease is
+\[
+ p(2i-1+M[j=0])-(2i-1+M[j+1=0])\geq p-1>1.
+\]
+A nonprimitive coproduct summand of $t_i^{p^j}$ replaces it by
+$t_l^{p^j}\otimes t_{i-l}^{p^{j+l}}$ and decreases filtration by exactly
+one.  These observations show that this is a multiplicative coalgebra
+filtration, and that its associated graded Hopf algebra is the tensor
+product of primitive truncated polynomial Hopf algebras
+\[
+ \bigotimes_{i,j}\mathbb F_p[z_{i,j}]/(z_{i,j}^p).
+\]
+For one such factor the dual algebra is again a truncated polynomial
+algebra: factorial rescaling identifies its basis in exponents less than
+$p$ with the ordinary polynomial basis.  The free resolution alternating
+multiplication by its generator and its $(p-1)$st power computes its
+cohomology as one exterior generator in degree one and one polynomial
+generator in degree two.  Their weights are respectively the weight of
+$z_{i,j}$ and $p$ times that weight.  Tensoring these resolutions gives
+\eqref{may:initial-page}, including the stated auxiliary filtrations.
+The exterior square is zero since its cohomological degree is odd and
+$p$ is odd.
+
+The filtration-one part of the reduced coproduct gives the splitting
+formula for $h$.  For the polynomial generator associated to a digit
+$z$, a filtration-one contribution would replace one of its $p$ copies
+by a split pair and leave $p-1$ unsplit copies of $z$.  In the
+associated-graded cohomology of the $z$ factor, a nonzero multiplicity
+has the form $pa+\epsilon$, with $a\geq0$ and
+$\epsilon\in\{0,1\}$.  The multiplicity $p-1$ has neither form, so that
+component has zero cohomology.  Carries cannot contribute to this first
+differential by the strict inequality above.  This proves $d_1b=0$.
+
+In a fixed positive internal weight only finitely many digit factors
+occur, every monomial exponent is bounded, and a normalized bar has
+positive weight.  Its length is consequently bounded as well.  Thus all
+the filtrations under discussion are finite in each weight, and their
+spectral sequences converge to the indicated cobar cohomology.  The
+construction restricts to $P_r$ because its coproduct uses only the
+first $r$ coordinates.
+\end{proof}
+
+For later calculations we fix the following notation, always
+distinguishing the bare exterior letter $b$ from the indexed polynomial
+letters:
+\begin{equation}
+\begin{gathered}
+ x=h_{1,0},\quad a=h_{2,0},\quad c=h_{3,0},\qquad
+ y=h_{1,1},\quad b=h_{2,1},\quad z=h_{1,2},\\
+ A=b_{1,0},\qquad B=b_{2,0},\qquad C=b_{1,1}.
+\end{gathered}
+\label{may:low-alphabet}
+\end{equation}
+An exterior interval $h_{i,j}$ covers positions
+$j,\ldots,j+i-1$ in its weight, and a polynomial letter $b_{i,j}$
+covers positions $j+1,\ldots,j+i$.  Let $m$ be the number of exterior
+letters, let $H_r$ count those covering position $r$, and let
+$L=(s-m)/2$ be the number of polynomial letters, counted with
+multiplicity.  If only positions $0,1,2$ occur, the alphabet is precisely
+\eqref{may:low-alphabet}.  Writing $n_A,n_B,n_C$ for the polynomial
+multiplicities gives
+\begin{equation}
+ w=H_0+p(H_1+L-n_C)+p^2(H_2+n_B+n_C),
+ \qquad n_A=L-n_B-n_C.
+ \label{may:three-position-weight}
+\end{equation}
diff --git a/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/build/sections/03-may-and-first-obstruction.tex b/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/build/sections/03-may-and-first-obstruction.tex
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@@ -0,0 +1,520 @@
+\section{The height-four obstruction calculation}
+\label{may:section}
+
+The algebraic spectral sequences below use the filtered cobar complexes
+of Section~\ref{may:preliminaries}.
+
+\subsection{Possible differential targets}
+
+We now use the finite spectrum $Y$ furnished by
+Proposition~\ref{low:V3}.  Its homology is $R/I_4$, its bottom cell is a
+unit, and it has a binary two-sided unital multiplication.  The possible
+positive Adams--Novikov differentials from $v_4$ have targets
+\begin{equation}
+ s=qk+1,\qquad w=u_4+k,\qquad k\geq1.
+ \label{may:obstruction-degrees}
+\end{equation}
+
+\begin{lemma}[Explicit cutoffs]
+\label{may:target-bound}
+The groups in \eqref{may:obstruction-degrees} are identified with
+$H^{s,w}(P)$.  Their May page has only positions $0,1,2,3$, and it is zero
+unless $k\leq p+2$.  For $k\geq2$, only positions $0,1,2$ can occur.
+\end{lemma}
+
+\begin{proof}
+If $w<s$, the degree-$(s,w)$ cochain groups for both $Y$ and $P$
+are zero, as is the May page, and all the conclusions are immediate.
+We may therefore suppose $w\geq s$.  The normalized bound in
+Lemma~\ref{fw:normalized-bound} then gives
+$(q-1)k\leq u_4-1$.  Consequently
+\[
+ w\leq u_4+\frac{u_4-1}{q-1}
+ <\frac q{q-1}u_4<4p^3<p^4
+ \quad (p=1009),
+\]
+where $u_4<2p^3$ and $q/(q-1)<2$.  Thus no letter can reach position
+four.  Also $w<u_5$.  A coefficient containing $v_4$ leaves at most $k$
+weight for normalized bars, whereas the three lengths $s-1,s,s+1$ are
+all greater than $k$, because $s-1=qk>k$.  Higher coefficient generators
+are excluded by $w<u_5$.  The adjacent-chain comparison of
+Lemma~\ref{fw:additive-comparison} therefore identifies the required
+cohomology with that of $P$.
+
+There are ten possible exterior intervals on positions $0,1,2,3$.
+Since $s$ is odd, $m$ is odd and hence $m\leq9$.  Every polynomial letter
+has weight at least $p$, so
+\[
+ u_4+k\geq pL\geq p((p-1)k-4),
+ \qquad
+ (p^2-p-1)k\leq u_4+4p.
+\]
+But
+\[
+ (p^2-p-1)(p+3)-(u_4+4p)=p^2-9p-4>0
+ \quad (p=1009).
+\]
+It follows that $k\leq p+2$.
+
+For $k\geq2$ we have $L\geq2p-6$.  If any letter reaches position
+three, that letter has weight at least $p^3$, while all other polynomial
+letters have total weight at least $(L-1)p$.  This is a valid lower bound
+whether the distinguished letter is exterior or polynomial.  Thus
+\[
+ w\geq p^3+2p^2-7p
+ >p^3+p^2+2p+3=u_4+p+2\geq w,
+\]
+since the strict difference is $p^2-9p-3>0$ at $p=1009$.  This
+contradiction excludes position three in that range.
+\end{proof}
+
+\subsection{The first obstruction}
+
+The first possible differential is $d_{q+1}$, and its target is the
+actual cobar cohomology in degree $(q+1,u_4+1)$. We first explain why
+an injective multiplication on this group will rule out the differential.
+Lemma~\ref{fw:v5-relation} gives
+\[
+ v_4h_{1,4}+e v_4^p h_{1,0}=0
+\]
+in the $E_2$ algebra for $Y$. By $q$-sparsity no earlier positive
+differential occurs, so Proposition~\ref{fw:anss-pairing} makes
+$d_{q+1}$ a derivation on this algebra. Let $\sigma$ denote the
+algebraic specialization to $P$-cohomology. Differentiate the relation
+and then apply $\sigma$. All terms containing $v_4$ specialize to zero,
+and $d_{q+1}(v_4^p)=p v_4^{p-1}d_{q+1}(v_4)=0$. Hence
+\begin{equation}
+ \sigma\bigl(d_{q+1}(v_4)\bigr)h_{1,4}=0.
+ \label{may:specialized-obstruction}
+\end{equation}
+Specialization is applied after the topological differential, to an
+identity of cobar-cohomology classes; no spectrum realizing $\sigma$
+is required.
+
+It remains to prove that multiplication by $h_{1,4}$ is injective on
+$H^{q+1,u_4+1}(P)$. Its degree shift is $(1,p^4)$, so its product
+lies in degree $(q+2,u_5+1)$. An incoming May differential to that
+product must have source degree $(q+1,u_5+1)$. The next lemma lists
+the initial May monomials in the obstruction degree and in this
+incoming-source degree.
+We will use the digit-zero adjustment of Lemma~\ref{may:page} to
+separate the surviving source filtrations from the product filtration.
+
+In the alphabet of Equation~\eqref{may:low-alphabet}, the splitting
+differential is
+\begin{equation}
+ da=xy,\qquad db=yz,\qquad dc=xb+az,
+ \label{may:small-splitting}
+\end{equation}
+with the exterior order $x,a,c,y,b,z$ used for signs.
+
+\begin{lemma}[The first target and all incoming-source intervals]
+\label{may:first-list}
+In bidegree $(s,w)=(q+1,u_4+1)$ the complete initial May page consists of
+\begin{equation}
+ acb B^{p-2},\qquad acybz B^{p-3}.
+ \label{may:first-target-list}
+\end{equation}
+The first is not a splitting cycle and the second is a splitting cycle.
+In bidegree $(q+1,u_5+1)$ the complete list is given in
+Table~\ref{may:n5-table}.  The four entries of exterior length three or
+five have no nonzero splitting cycle in their span.
+\end{lemma}
+
+\begin{proof}
+We prove exhaustion simultaneously for $w=u_n+1$, with $n=4,5$.
+Since $u_n+1<p^n$, the available exterior intervals are
+$[a,b)$ with $0\leq a<b\leq n$, and the polynomial intervals cover
+positions $1,\ldots,n-1$.  The target weight has base-$p$ digits
+$(2,1,\ldots,1)$.  At position zero, $H_0\leq n<p$, so $H_0=2$ and
+there is no carry into position one.  As $s=2p-1$ is odd, write
+\[
+ m=2D-1,\qquad L=p-D.
+\]
+There are exactly two exterior intervals starting at zero; at most
+$n(n-1)/2$ others are available.  Hence $m\leq11$, $2\leq D\leq6$,
+and
+\begin{equation}
+ p-6\leq L\leq p-2.
+ \label{may:L-range}
+\end{equation}
+The lower bound on $D$ follows from $m$ being odd and at least two.
+
+Let $c_r$ be the carry into position $r$, with $c_1=c_n=0$, and let
+$B_r$ count polynomial intervals covering position $r$.  The largest
+possible exterior coverage at a position is nine.  Inductively, the
+quantity $H_r+B_r+c_r$ is at most $9+(p-2)+1=p+8<2p$;
+therefore every carry is zero or one.  The digit equations are
+\begin{equation}
+ B_r=1-H_r-c_r+pc_{r+1},\qquad 1\leq r\leq n-1.
+ \label{may:carry-equations}
+\end{equation}
+If there were no carry, then $B_r\leq1$ for every $r$, and
+$L\leq\sum B_r\leq4$, contrary to $L\geq p-6=1003$.
+
+Call a position high when $c_{r+1}=1$.  A high position has
+$B_r\geq p-9$, and a nonhigh position has $B_r\leq1$.
+There cannot be two blocks of high positions.  Indeed, choose high
+positions $a<b$ in different blocks and a nonhigh position $c$ between
+them.  An interval covering both $a$ and $b$ also covers $c$, whence
+\[
+ L\geq B_a+B_b-B_c\geq2(p-9)-1=2p-19>p-2.
+\]
+Here $2p-19=1999$ and $p-2=1007$.  This contradicts
+\eqref{may:L-range}.  Thus the high positions form a single block
+$[l,r]$ with $1\leq l\leq r\leq n-2$.
+At the next position \eqref{may:carry-equations} reads
+$B_{r+1}=-H_{r+1}$, so
+\begin{equation}
+ B_{r+1}=H_{r+1}=0.
+ \label{may:gap-after-block}
+\end{equation}
+At the beginning of the block, $B_l=p+1-H_l\leq L$, giving
+$H_l\geq D+1$.  If $l<r$, its end gives
+$B_r=p-H_r\leq L$, hence $H_r\geq D$.
+
+A singleton block is impossible.  Every exterior interval covering its
+position $l$ must end at $l+1$ by \eqref{may:gap-after-block}.  When
+$l=1$ there are only two such intervals.  When $l>1$, at most one begins
+before $l$, since $H_{l-1}\leq1$, and at most one begins at $l$.
+Thus in either case $H_l\leq2$, contradicting $H_l\geq D+1\geq3$.
+We henceforth have $l<r$.
+
+If $l>1$, at most two exterior intervals cover both $l$ and $r$: they
+must end at $r+1$, and at most one starts before $l$, while at most one
+starts at $l$.  At least one of the two intervals starting at zero
+misses both positions, since $H_{l-1}\leq1$.  Consequently
+\[
+ H_l+H_r
+ =m+\#\{\text{intervals meeting both}\}
+       -\#\{\text{intervals meeting neither}\}
+ \leq m+1.
+\]
+This contradicts $H_l+H_r\geq2D+1=m+2$.  Therefore $l=1$.
+There are now at most two exterior intervals covering both endpoints,
+namely $[0,r+1)$ and $[1,r+1)$.  Equality in the same counting identity
+is forced: both are present, every exterior interval meets position $1$
+or position $r$, and
+\[
+ H_1=D+1,\qquad H_r=D.
+\]
+It follows that $B_1=B_r=L$.  Every polynomial interval therefore covers
+both positions, and by \eqref{may:gap-after-block} every one is exactly
+$[1,r+1)$.  No exterior or polynomial interval can occur after the block.
+The carry supplies the target digit at $r+1$; all further digits would
+be zero.  Since the required remaining digits are all one, necessarily
+$r=n-2$.  Thus
+\begin{equation}
+ \text{polynomial factor }b_{n-2,0}^{p-D},\qquad
+ H_0=2,\quad H_1=D+1,\quad
+ H_2=\cdots=H_{n-2}=D,
+ \quad H_{n-1}=0.
+ \label{may:forced-profile}
+\end{equation}
+
+It remains to list a small, now forced set of exterior intervals.
+For $n=4$, the intervals $[0,3)$ and $[1,3)$ are mandatory.  The other
+start-zero interval must be $[0,2)$; $[0,1)$ would meet neither endpoint.
+The only optional intervals are $[1,2)$ and $[2,3)$.  The coverage
+requirements in \eqref{may:forced-profile} require both or neither.
+This is exactly \eqref{may:first-target-list}.
+
+For $n=5$, the mandatory intervals are $[0,4)$ and $[1,4)$.
+The remaining start-zero interval is $[0,t)$ with $t=2$ or $3$.
+The only optional intervals are $[1,2),[1,3),[2,4),[3,4)$;
+$[2,3)$ meets neither endpoint and is forbidden.  Write their inclusion
+indicators as $e_{12},e_{13},e_{24},e_{34}\in\{0,1\}$, and let their
+sum be $o$.  The three required coverage equations are
+\begin{equation}
+ e_{12}+e_{13}=o/2,\qquad
+ e_{24}+e_{34}=o/2,\qquad
+ e_{13}+e_{24}+[t=3]=o/2,
+ \label{may:four-indicators}
+\end{equation}
+where $o\in\{0,2,4\}$.  For $t=2$ the solutions are no optional
+intervals, the pair $\{12,24\}$, the pair $\{13,34\}$, and all four.
+For $t=3$ the only solution is the pair $\{12,34\}$.  This proves the
+exhaustive Table~\ref{may:n5-table}, including its length-seven entry.
+
+\begin{table}[htbp]
+\centering
+\begin{tabular}{c l l}
+\toprule
+$m$ & Exterior intervals (exclusive right endpoints) & Polynomial factor\\
+\midrule
+$3$ & $02,04,14$ & $b_{3,0}^{p-2}$\\
+$5$ & $03,04,12,14,34$ & $b_{3,0}^{p-3}$\\
+$5$ & $02,04,13,14,34$ & $b_{3,0}^{p-3}$\\
+$5$ & $02,04,12,14,24$ & $b_{3,0}^{p-3}$\\
+$7$ & $02,04,12,13,14,24,34$ & $b_{3,0}^{p-4}$\\
+\bottomrule
+\end{tabular}
+\caption{Every initial May monomial in bidegree $(q+1,u_5+1)$.
+The symbol $ab$ in this table denotes the interval $[a,b)$, not a
+product of the letters in \eqref{may:low-alphabet}.}
+\label{may:n5-table}
+\end{table}
+
+For completeness, the splitting calculation is also explicit.
+The length-three entry has, for example, a nonzero component on
+$01,04,12,14$.  For the three length-five entries in their displayed
+order, take the following four rows, ordering exterior intervals
+lexicographically:
+\[
+\begin{array}{c|rrr}
+01,04,12,13,14,34&1&-1&0\\
+02,03,12,14,24,34&1&0&1\\
+02,04,12,13,24,34&0&1&1\\
+02,04,12,14,23,34&-1&-1&1
+\end{array}
+\]
+The common polynomial factor is $b_{3,0}^{p-3}$.  The minor using the
+first, second, and fourth rows is $3$, which is nonzero in
+$\mathbb F_{1009}$.  Thus these three images are linearly independent.
+Finally, \eqref{may:small-splitting} shows that $acb$ has a nonzero
+splitting differential and that $acybz$ is closed: every term in the
+latter differential repeats an exterior factor.  This finishes the
+lemma.
+\end{proof}
+
+\begin{proposition}[An injective multiplication]
+\label{may:injectivity}
+Multiplication by the primitive class $h_{1,4}$ is injective on actual
+cobar cohomology in bidegree $(q+1,u_4+1)$:
+\[
+ H^{q+1,u_4+1}(P)\xrightarrow{\ h_{1,4}\ }
+ H^{q+2,u_5+1}(P).
+\]
+\end{proposition}
+
+\begin{proof}
+Use the filtration in Lemma~\ref{may:page} with the explicit choice
+$M=3p$.  If the source cohomology is nonzero, any nonzero class has
+leading detector a nonzero scalar multiple of
+$acybz B^{p-3}$, by Lemma~\ref{may:first-list}.  Its filtration is
+\[
+ F_0=3p^2-9p+13+M(2+p(p-3)).
+\]
+The cocycle $[t_1^{p^4}]$ represents $h_{1,4}$, with degree, weight, and
+filtration shifts $(1,p^4,1)$.  The proposed leading product therefore
+has filtration
+\begin{equation}
+ F_*=3p^2-9p+14+M(2+p(p-3)).
+ \label{may:product-filtration}
+\end{equation}
+
+Every possible incoming May differential to this product has source
+cohomological degree $q+1$ and weight $u_5+1$.
+Table~\ref{may:n5-table} is its complete initial-page source list.
+The length-three and length-five source filtrations are respectively
+\[
+ 5p^2-10p+15+M(2+p(p-2)),\qquad
+ 5p^2-15p+19+M(2+p(p-3));
+\]
+all three length-five sources have the latter filtration.
+The length-three and length-five entries disappear under $d_1$, since
+their displayed splitting maps are injective.  None can be a splitting
+boundary for the product itself, because its polynomial factor is a
+power of $B=b_{2,0}$, whereas their polynomial factor is a power of
+$b_{3,0}$.  The length-seven source has filtration
+\[
+ F_7=5p^2-20p+23+M(2+p(p-4)).
+\]
+For $M=3p$ the difference is
+\[
+ F_7-F_*=2p^2-11p+9-pM=-p^2-11p+9<0.
+\]
+As all May differentials lower filtration, this last source cannot hit
+the proposed product on any later page.  There are no other incoming
+sources.
+
+This argument does not require $B$ to represent a permanent class by
+itself.  Choose an actual cocycle for the given nonzero source
+cohomology class and multiply that entire cocycle by
+$[t_1^{p^4}]$.  The result is an actual cocycle.  Its leading
+associated-graded cup product is the nonzero monomial
+$acybz B^{p-3}h_{1,4}$, since the new exterior letter is distinct from
+all the old ones.  An actual cocycle has no outgoing May differential;
+its leading detector could disappear only by becoming a boundary,
+which the preceding incoming-source calculation excludes at every
+page.  Its product cohomology class is therefore nonzero.
+\end{proof}
+
+\begin{proposition}[The first Adams--Novikov differential]
+\label{may:first-differential}
+In the Adams--Novikov spectral sequence for $Y$, one has
+$d_{q+1}(v_4)=0$.
+\end{proposition}
+
+\begin{proof}
+Equation~\eqref{may:specialized-obstruction} and
+Proposition~\ref{may:injectivity} give
+$\sigma(d_{q+1}(v_4))=0$. Lemma~\ref{may:target-bound} identifies
+this specialization with the original target group. Hence
+$d_{q+1}(v_4)=0$.
+\end{proof}
+
+\subsection{The remaining targets}
+
+\begin{proposition}[All later obstruction candidates]
+\label{may:remaining-list}
+For $k\geq2$ the complete initial May page in
+\eqref{may:obstruction-degrees} is Table~\ref{may:remaining-table}; all
+other values of $k$ give an empty page.  Put
+$N_0=(p-1)^2$ and $N=p(p-1)-2$.
+\begin{table}[htbp]
+\centering
+\begin{tabular}{c l}
+\toprule
+$k$ & Initial May monomials\\
+\midrule
+$2$ & $xac B A^{p-2}C^{p-2}$,
+       $\quad xacyz A^{p-2}C^{p-2}$\\[2pt]
+$p-1$ & $y B^3 A^{N_0-3}$,
+         $\quad b B^2 A^{N_0-2}$,
+         $\quad ybz B A^{N_0-2}$\\[2pt]
+$p$ & $cyb A^{N+1}$,
+       $\quad ayb B A^N$\\
+\bottomrule
+\end{tabular}
+\caption{Every initial May monomial for the later $v_4$ obstruction
+bidegrees.}
+\label{may:remaining-table}
+\end{table}
+\end{proposition}
+
+\begin{proof}
+Lemma~\ref{may:target-bound} allows only positions $0,1,2$ and
+$2\leq k\leq p+2$.  Thus \eqref{may:three-position-weight} applies.
+There are six exterior letters; $m$ is odd, so write
+\[
+ m=2f+1,\quad 0\leq f\leq2,\qquad L=(p-1)k-f.
+\]
+Reducing the weight modulo $p$ gives
+$H_0\equiv1+k\pmod p$, with $0\leq H_0\leq3$.
+The only possibilities are $k=2$ or $k=p+d$ with
+$-1\leq d\leq2$.
+
+When $k=2$, $H_0=3$ forces all of $x,a,c$.  The other exterior letters
+are an even subset of $\{y,b,z\}$.  Put
+$T=H_2+n_B+n_C$.  Substitution in
+\eqref{may:three-position-weight} gives
+\begin{equation}
+ pT-n_C+H_1=p^2-p+3+f.
+ \label{may:k2-equation}
+\end{equation}
+For the four possible exterior subsets the value
+$e=H_1-3-f$ is $-2$ or $-1$.  Therefore
+$n_C\equiv e\pmod p$.  Since
+$0\leq n_C\leq L\leq2p-3$, the only possible lift is
+$n_C=p+e$: the lift $e$ is negative, and $2p+e\geq2p-2>L$.
+Equation~\eqref{may:k2-equation} now gives $T=p$.
+All the remaining counts follow as displayed:
+\[
+\begin{array}{c|c|c|c|c|c|c}
+\text{extra exterior letters}&f&H_1&H_2&n_C&n_B&n_A\\ \hline
+\varnothing&1&2&1&p-2&1&p-2\\
+y,b&2&4&2&p-1&-1&p-2\\
+y,z&2&3&2&p-2&0&p-2\\
+b,z&2&3&3&p-2&-1&p-1
+\end{array}
+\]
+The negative counts exclude the second and fourth rows.  The first and
+third are exactly the $k=2$ row of Table~\ref{may:remaining-table}.
+
+Now let $k=p+d$, $-1\leq d\leq2$.  Then $H_0=1+d$, and the weight
+equation becomes
+\begin{equation}
+ pT-n_C+H_1=(2-d)p+2+d+f.
+ \label{may:pd-equation}
+\end{equation}
+Since $n_C\leq T$ and $H_1\geq0$, it follows that
+\[
+ (p-1)T\leq(2-d)(p-1)+(4+f).
+\]
+Here $4+f\leq6<p-1$, so $T\leq2-d\leq3$ and $n_C\leq3$.
+Rearranging \eqref{may:pd-equation} now gives
+\[
+ p(T-(2-d))=2+d+f+n_C-H_1.
+\]
+The right side lies between $-3$ and $9$, strictly between $-p$ and
+$p$.  Both sides therefore vanish, proving the exact equations
+\begin{equation}
+ H_2+n_B+n_C=2-d,\qquad H_1=2+d+f+n_C.
+ \label{may:pd-exact}
+\end{equation}
+
+For $d\geq1$, these require
+$3+f\leq H_1\leq\min(2f+1,4)$, impossible for each of
+$f=0,1,2$.  For $d=0$, exactly one of $x,a,c$ occurs, so $H_1\leq3$.
+Equation~\eqref{may:pd-exact} forces $f=1$, $n_C=0$, and $H_1=3$.
+The only exterior triples are $ayb$ and $cyb$.  Their $H_2$ values are
+respectively $1$ and $2$, giving the two $k=p$ monomials in the table.
+
+For $d=-1$, only $y,b,z$ can occur.  If $m=1$, the choices $y,b,z$
+have $(H_1,H_2)$ equal to $(1,0),(1,1),(0,1)$.
+Equation~\eqref{may:pd-exact} excludes $z$ and gives respectively
+$(n_C,n_B)=(0,3),(0,2)$ for the other two.  If $m=3$, the exterior
+factor is $ybz$, with $f=1$, $H_1=H_2=2$; the same equations give
+$(n_C,n_B)=(0,1)$.  Since $L=N_0-f$, these are exactly the three
+$k=p-1$ monomials.  Every nonnegative count and every exterior subset
+has now been considered.
+\end{proof}
+
+The first two rows of Table~\ref{may:remaining-table} use only the first
+three or two coordinates, respectively. Every monomial in its last row
+has a large common power of $A$. The following comparisons turn these
+two observations into surjections on actual cobar cohomology, which are
+the inputs to the remaining obstruction arguments.
+
+\begin{lemma}[Two comparison surjections]
+\label{may:comparison}
+The following initial-page tests give surjections on actual cobar
+cohomology.
+\begin{enumerate}
+ \item If every monomial of \eqref{may:initial-page} in bidegree
+ $(s,w)$ uses only generators with $i\leq r$, then
+ $H^{s,w}(P_r)\longrightarrow H^{s,w}(P)$ is surjective.
+ \item Put $A=b_{1,0}$. If every initial-page monomial in bidegree
+ $(s,w)$ contains $A^a$, then multiplication by the actual cocycle
+ representing $A^a$ gives a surjection
+ \[
+ H^{s-2a,w-pa}(P)\longrightarrow H^{s,w}(P).
+ \]
+\end{enumerate}
+These conclusions also transport through additive comparisons that
+identify the source and target cohomology groups and commute with the
+indicated maps.
+\end{lemma}
+
+\begin{proof}
+The subalgebra map includes a subset of the exterior and polynomial
+generators in \eqref{may:initial-page}; its initial-page map is therefore
+injective in every bidegree. Likewise, multiplication by the polynomial
+$A^a$ is injective in every bidegree on that page. It is induced by a
+filtered cochain map: $t_1$ is primitive, and
+\begin{equation}
+ \sum_{i=1}^{p-1}\frac{\binom pi}{p}
+       [t_1^i\mid t_1^{p-i}]
+ \label{may:A-cocycle}
+\end{equation}
+is an actual cocycle, obtained by taking the integral divided reduced
+coproduct and then reducing modulo $p$. It has leading class $A$.
+For the second comparison shift the source cohomological degree,
+weight, and auxiliary filtration by $2a$, $pa$, and $ap(1+M)$.
+
+Take the filtered mapping cone of either cochain map. The long exact
+sequence of associated-graded cohomology, together with injectivity in
+\emph{all} bidegrees, identifies its initial cohomology page with the
+cokernel of the specified page map; there is no extra kernel from the
+next degree. That cokernel is zero in the stated target bidegree.
+Finite filtration therefore gives zero cone cohomology there. The
+ordinary cone long exact sequence gives the required surjection.
+Transport through the stated commuting comparisons preserves that
+surjectivity.
+\end{proof}
+
+Section~\ref{frob:section} removes the first two rows of
+Table~\ref{may:remaining-table} in actual cobar cohomology.  The last
+row is treated in Section~\ref{top:section} by a homotopy argument with an explicit
+Adams--Novikov page bound.
diff --git a/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/build/sections/04-frobenius-elimination.tex b/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/build/sections/04-frobenius-elimination.tex
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+\section{Eliminating two obstruction degrees by Frobenius}
+\label{frob:section}
+
+Throughout this section $p=1009$.  We eliminate the rows $k=2$ and
+$k=p-1$ of Table~\ref{may:remaining-table} in actual cobar cohomology.
+The Frobenius extension reduces this task to quotient-group cohomology
+with coefficients in kernel cohomology.  A filtration of those coefficients
+then reduces the calculation to trivial-coefficient quotient groups, which
+we compute by May methods.  We first establish these three steps and identify
+the coefficient modules that the two obstruction degrees require.
+
+\subsection{The Frobenius extension and its action}
+
+Let $G_r$, for $r=2,3$, be the affine group scheme over
+$\mathbb F_p$ with coordinate algebra
+\[
+ P_r=\mathbb F_p[t_1,\ldots,t_r],\qquad
+ \Delta t_i=t_i\otimes1+1\otimes t_i+
+       \sum_{0<l<i}t_l\otimes t_{i-l}^{p^l}.
+\]
+The weight of $t_i$ is $u_i$.  Thus, on points over a commutative
+$\mathbb F_p$-algebra,
+\begin{equation}\label{frob:group-law}
+ (g h)_i=g_i+h_i+\sum_{0<l<i}g_lh_{i-l}^{p^l}.
+\end{equation}
+We write $H^a(G_r;M)$ for rational group cohomology, equivalently the
+cohomology of the normalized coordinate cobar complex with coefficients
+in the $G_r$-module $M$.
+
+The spectral sequence below is the Lyndon--Hochschild--Serre spectral
+sequence for an affine group scheme and a normal subgroup; see
+\citet[Part~I, Proposition~6.6(3) and Section~6.7]{Jantzen1985} for
+the spectral sequence and the quotient action on kernel cohomology.
+We verify the required acyclicity and compute that action for this extension.
+
+\begin{lemma}[The Frobenius extension]\label{frob:extension}
+There is a finite faithfully flat exact sequence of affine group schemes
+\begin{equation}\label{frob:exact-sequence}
+ 1\longrightarrow K_r=(\alpha_p)^r\longrightarrow G_r
+   \xrightarrow{F} Q_r\longrightarrow1,
+ \qquad O(Q_r)=\mathbb F_p[t_1^p,\ldots,t_r^p].
+\end{equation}
+Its derived invariants give a first-quadrant spectral sequence
+\begin{equation}\label{frob:lhss}
+ {}^{\mathrm F}E_2^{a,v}
+   =H^a\bigl(Q_r;H^v(K_r;\mathbb F_p)\bigr)
+   \Longrightarrow H^{a+v}(G_r;\mathbb F_p).
+\end{equation}
+It preserves weight and converges in every weight and total
+cohomological degree.
+\end{lemma}
+
+\begin{proof}
+The inclusion of the Frobenius subring makes $P_r$ free of rank $p^r$,
+with basis $t_1^{e_1}\cdots t_r^{e_r}$, $0\leq e_i<p$.
+It is therefore faithfully flat.  Its kernel has coordinate algebra
+\[
+ O(K_r)=\mathbb F_p[e_1,\ldots,e_r]/(e_1^p,\ldots,e_r^p).
+\]
+All the mixed terms in its coproduct vanish, so each $e_i$ is primitive.
+This proves the description of the kernel and the exactness of
+\eqref{frob:exact-sequence}.
+
+Here is the acyclicity calculation needed to derive invariants.
+Right multiplication by a kernel point is, by
+\eqref{frob:group-law}, simply $t_i\mapsto t_i+e_i$: every mixed
+term contains a positive $p$-power of a kernel coordinate.  Hence
+multiplication gives an isomorphism of $K_r$-comodules
+\begin{equation}\label{frob:regular-decomposition}
+ O(Q_r)\otimes O(K_r)\xrightarrow{\ \cong\ }O(G_r),
+ \qquad
+ f\otimes e_1^{j_1}\cdots e_r^{j_r}
+ \longmapsto f t_1^{j_1}\cdots t_r^{j_r},
+ \quad 0\leq j_i<p.
+\end{equation}
+The first tensor factor has trivial $K_r$-coaction.  The second is the
+regular comodule.  In particular,
+$O(G_r)^{K_r}=O(Q_r)$, with its regular quotient coaction.
+
+A cofree comodule is injective: its cofree functor is right adjoint to
+the exact forgetful functor to vector spaces.  Equation
+\eqref{frob:regular-decomposition} shows that every cofree
+$G_r$-comodule is $K_r$-acyclic and that its $K_r$-invariants are
+cofree over $Q_r$.  Every injective is a summand of a cofree comodule,
+so these assertions also hold for injectives.  Finally,
+$M^{G_r}=(M^{K_r})^{Q_r}$.  Resolving these two invariants functors
+successively, using the indicated cofree resolutions, gives
+\eqref{frob:lhss}.  This argument derives $K_r$-invariants; it does
+not assert their exactness on arbitrary modules.  The spectral
+sequence is first quadrant, so each fixed total cohomological degree
+has a finite diagonal.  All coordinate weights are positive and each
+weight component is finite dimensional.  These observations give the
+asserted convergence, also weight by weight.
+\end{proof}
+
+\begin{lemma}[Conjugation on kernel cohomology]\label{frob:action}
+For $r=3$ there is an isomorphism of bigraded algebras
+\begin{equation}\label{frob:kernel-cohomology}
+ H^*(K_3;\mathbb F_p)
+   =\Lambda(x,a,c)\otimes\mathbb F_p[A,B,E].
+\end{equation}
+The exterior generators have cohomological degree one and weights
+$1,u_2,u_3$, and the polynomial generators have cohomological degree
+two and weights $p,pu_2,pu_3$, respectively.  Write
+$\xi=t_1^p$, $\eta=t_2^p$ for quotient coordinates.  With the
+right-conjugation convention, the quotient coaction is
+\begin{align}
+ x&\longmapsto x,&
+ a&\longmapsto a+x\xi,&
+ c&\longmapsto c+x\eta+a\xi^p,\label{frob:exterior-action}\\
+ A&\longmapsto A,&
+ B&\longmapsto B+A\xi^p,&
+ E&\longmapsto E+A\eta^p+B\xi^{p^2}.
+ \label{frob:polynomial-action}
+\end{align}
+Products in these formulas mean coefficient times quotient coordinate.
+In particular, $xac$ and $A$ are invariant.  Exterior degree and
+polynomial degree are separately preserved.  The corresponding
+statements for $r=2$ omit $c$ and $E$.
+\end{lemma}
+
+\begin{proof}
+The dual algebra of a primitive truncated coordinate
+$\mathbb F_p[e]/(e^p)$ is a truncated polynomial algebra after rescaling
+the dual basis by the invertible factorials $j!$, $j<p$.
+Its resolution alternates multiplication by its generator and by the
+$(p-1)$st power of that generator.  Applying the augmentation gives
+one class in each cohomological degree.  Its algebra of extensions is
+an exterior degree-one generator and a polynomial degree-two
+generator.  The shift by two of the periodic resolution represents
+the latter, and its iterates are nonzero in every even degree.
+The former has square $[e\mid e]=\tfrac12d[e^2]$, which is zero
+at this odd prime.
+Tensoring these resolutions gives \eqref{frob:kernel-cohomology},
+including the indicated weights.
+
+For completeness, the naturality of the polynomial generators is
+Frobenius semilinear.  A primitive coordinate $T$ has degree-two cocycle
+\[
+ \beta(T)=\sum_{j=1}^{p-1}\frac{1}{p}\binom pj
+                      [T^j\mid T^{p-j}],
+\]
+where the displayed coefficients are integers reduced modulo $p$.
+For a linear coordinate $T=\sum_i\lambda_i e_i$, the integral
+polynomial
+\[
+ \frac{(\sum_i\lambda_i e_i)^p-\sum_i\lambda_i^p e_i^p}{p}
+\]
+has reduced coproduct
+$\beta(\sum_i\lambda_i e_i)-\sum_i\lambda_i^p\beta(e_i)$.
+Thus a matrix $(\lambda_{ij})$ on degree-one coordinates induces
+$(\lambda_{ij}^p)$ on these degree-two cohomology classes.  This is a
+polynomial identity in the $\lambda_{ij}$, so it also applies when
+they are quotient coordinate functions.
+
+If $e$ is a kernel point, direct use of \eqref{frob:group-law} gives
+\[
+ (g^{-1}eg)_1=e_1,\qquad
+ (g^{-1}eg)_2=e_2+e_1g_1^p,\qquad
+ (g^{-1}eg)_3=e_3+e_1g_2^p+e_2g_1^{p^2}.
+\]
+These coefficients belong to $O(Q_r)$, so the action descends to
+the quotient.  The degree-one and degree-two naturality just proved
+gives \eqref{frob:exterior-action} and
+\eqref{frob:polynomial-action}.  The exterior matrix is triangular
+with diagonal entries one, so its top exterior power $xac$ is fixed.
+The formulas also prove separate preservation of the two degrees.
+\end{proof}
+
+\subsection{Coefficient filtration and quotient cohomology}
+
+\begin{lemma}[The coefficient-weight filtration]\label{frob:coefficients}
+Fix an exterior degree and a polynomial degree in
+\eqref{frob:kernel-cohomology}, obtaining an invariant $Q_r$-submodule $M$.
+At each total weight $w$, increasing coefficient weight filters the
+normalized quotient cobar complex $C^*(Q_r;M)_w$ by a finite
+filtration.  Its initial cohomology page is
+\begin{equation}\label{frob:coefficient-page}
+ {}^{\mathrm C}E_1^{a,\lambda}
+      =M_{\lambda}\otimes H^a(Q_r;\mathbb F_p)_{w-\lambda},
+\end{equation}
+where $M_\lambda$ is viewed with trivial quotient action.
+The coefficient differential with drop $d$ has the form
+$ {}^{\mathrm C}d_d:(a,\lambda)\mapsto(a+1,\lambda-d)$.
+If the nonidentity coefficient action has no weight drop smaller than
+$d$, all earlier coefficient differentials vanish, and the differential
+with drop $d$ is induced by that part of the coaction.
+
+For the module $\operatorname{Sym}^n(A,B)$ we may write
+\begin{equation}\label{frob:finite-D}
+ A^n\langle1,D,\ldots,D^n\rangle,
+ \qquad A^nD^j:=A^{n-j}B^j\quad(0\leq j\leq n).
+\end{equation}
+This is notation for a finite-dimensional module, with action
+$D\mapsto D+\xi^p$; no inverse of $A$ is introduced.
+\end{lemma}
+
+\begin{proof}
+In \eqref{frob:exterior-action} and
+\eqref{frob:polynomial-action}, every nonidentity term has lower
+coefficient weight, its difference being the positive weight of its
+quotient coordinate.  The cobar differential therefore preserves
+the increasing coefficient filtration.  In the associated graded
+complex only the trivial coefficient action remains, giving
+\eqref{frob:coefficient-page}.  At fixed total weight the coefficient
+weights lie between zero and $w$, so the filtration is finite.
+The stated description of its first possible differential follows by
+taking the leading nonzero weight-lowering component of the cobar
+differential.  Finally, expanding
+\[
+ (B+A\xi^p)^jA^{n-j}
+   =\sum_{l=0}^j\binom jl A^{n-l}B^l\xi^{p(j-l)}
+\]
+proves the stated action on \eqref{frob:finite-D}.
+\end{proof}
+
+For either obstruction bidegree $(s,w)$, our aim is to make every group
+$H^a(Q_r;H^{s-a}(K_r;\mathbb F_p))_w$ zero.  This makes the entire
+total-degree-$s$ diagonal of \eqref{frob:lhss} zero, and hence proves
+$H^{s,w}(G_r;\mathbb F_p)=0$.  We compute these groups by the coefficient
+filtration \eqref{frob:coefficient-page}, using May only to bound or identify
+its trivial-coefficient quotient groups.
+
+The resulting vector-space upper bound has exactly the exterior-polynomial
+letters used for $G_r$: kernel cohomology supplies the digit-zero letters,
+and the quotient's initial May page supplies all higher digits.  This description
+retains the kernel degree $v$.  Thus Table~\ref{may:remaining-table} bounds
+every coefficient summand on the relevant Frobenius diagonal.
+Lemma~\ref{may:comparison} reduces $k=2$ to $G_3$ and $k=p-1$ to $G_2$.
+Grouping the candidates by their invariant kernel modules gives the four
+computations in Table~\ref{frob:coefficient-obligations}.
+
+For each module $M$ in that table, let $\lambda_0$ be its least coefficient
+weight and put
+\begin{equation}\label{frob:residual-weight}
+ \mathcal R=\frac{w-\lambda_0}{p}.
+\end{equation}
+We call this the residual weight, and henceforth express quotient weights
+after division by $p$.  Recall that $N_0=(p-1)^2$.
+\begin{table}[htbp]
+\centering
+\begin{tabular}{ccccc}
+\toprule
+$k$ & $r$ & Kernel module $M$ & Quotient degree $a$ & $\mathcal R$\\
+\midrule
+$2$ & $3$ & $xac\operatorname{Sym}^{p-1}(A,B,E)$ & $2p-4$ & $p(p-1)$\\
+$2$ & $3$ & $xac\operatorname{Sym}^{p-2}(A,B,E)$ & $2p-2$ & $p^2-p+1$\\
+$p-1$ & $2$ & $\operatorname{Sym}^{N_0}(A,B)$ & $1$ & $3p+1$\\
+$p-1$ & $2$ & $\operatorname{Sym}^{N_0-1}(A,B)$ & $3$ & $3p+2$\\
+\bottomrule
+\end{tabular}
+\caption{The coefficient-cohomology groups that must vanish. Each row
+means $H^a(Q_r;M)_w$, with $w=u_4+k$ in original weights.}
+\label{frob:coefficient-obligations}
+\end{table}
+
+For the first two rows, the least weights are respectively
+$\operatorname{wt}(xacA^{p-1})=3+p+2p^2$ and
+$\operatorname{wt}(xacA^{p-2})=3+2p^2$; for the last two they are $np$,
+where $n$ is the indicated polynomial degree.  These give the displayed
+residual weights.  Replacing an $A$ by $E$ increases the coefficient weight,
+after division by $p$, by $p^2+p$.  This exceeds both residual weights in
+the $k=2$ rows, so no coefficient involving $E$ occurs in either fixed-weight
+complex, in any quotient degree.  All four rows therefore use the finite
+$D$-modules of \eqref{frob:finite-D}, multiplied by $xac$ in the first two
+rows.  Suppressing the common invariant factor $A^n$ or $xacA^n$, a term
+$D^j\zeta$ requires
+\begin{equation}\label{frob:D-weight}
+ \operatorname{wt}_Q(\zeta)=\mathcal R-jp.
+\end{equation}
+Their first possible coefficient differential lowers $j$ by one and raises
+the quotient degree by one: its weight drop is $p^2$ in original weights,
+or $p$ in these rescaled weights.
+
+We now calculate the trivial-coefficient quotient groups needed for these
+four computations and their adjacent degrees.
+The coordinates $\xi,\eta$ have rescaled weights $1,p+1$ and
+\begin{equation}\label{frob:quotient-coproduct}
+ \overline\Delta\xi=0,\qquad
+ \overline\Delta\eta=\xi\otimes\xi^p.
+\end{equation}
+In these rescaled weights, the letters
+\[
+ y=h_{1,0}^{Q},\quad b=h_{2,0}^{Q},\quad
+ z=h_{1,1}^{Q},\quad C=b_{1,0}^{Q}
+\]
+have weights $1,p+1,p,p$, respectively; these are exactly the
+letters $y,b,z,C$ used in Table~\ref{may:remaining-table}.
+
+\begin{lemma}[The small quotient groups]\label{frob:quotient-groups}
+For $Q_2$ or $Q_3$, at any positive rescaled weight
+$w<p^2$ all available May letters are $y,b,z,C$.
+Their monomials and bidegrees are
+\begin{equation}\label{frob:small-monomials}
+ C^j y^e z^f b^g,\quad
+ j\geq0,\quad e,f,g\in\{0,1\},\qquad
+ a=2j+e+f+g,\quad w=p(j+f+g)+e+g.
+\end{equation}
+The following lists are exhaustive, with a dash denoting the empty
+list:
+\begin{center}
+\begin{tabular}{lll}
+\toprule
+Rescaled weight & Cohomological degree & May monomials\\
+\midrule
+$p(p-1)$ & $2p-4$ & ---\\
+$p(p-1)$ & $2p-3$ & $zC^{p-2}$\\
+$p(p-2)$ & $2p-4$ & $C^{p-2}$\\
+$p(p-3)$ & $2p-5$ & ---\\
+$p+2$ & $1,3$ & ---\\
+$p+2$ & $2$ & $yb$\\
+$2p+2$ & $2$ & ---\\
+$2p+2$ & $3$ & $ybz$\\
+$3p+2$ & $3$ & ---\\
+$2$ & $1,3$ & ---\\
+\bottomrule
+\end{tabular}
+\end{center}
+The classes $C^{p-2}$ and $zC^{p-2}$ in the table are nonzero in actual
+quotient cohomology.  There is also a nonzero class $\upsilon$ of degree two
+and weight $p+2$, detected by $yb$, and $\upsilon z$ is nonzero
+in degree three and weight $2p+2$.
+\end{lemma}
+
+\begin{proof}
+The next exterior weights are $p^2$ and $p^2+p$; the third-coordinate
+weight, if present, is $p^2+p+1$.  The next polynomial weight is
+$p^2$.  All are outside the asserted range.  Lemma~\ref{may:page} therefore
+gives exactly \eqref{frob:small-monomials}.  To verify the table,
+first reduce its weight equation modulo $p$: its remainder is
+$e+g\in\{0,1,2\}$.  For weights divisible by $p$ it forces
+$e=g=0$, after which the degree equation fixes $j$ and $f$.
+For the listed weights congruent to two it forces $e=g=1$;
+substitution into both equations gives the entries shown.
+The weight-two entries follow in the same way, since $j=f=g=0$
+at that weight.  All weights in the table are within the asserted
+range, because $3p+2<p(p-1)$ at $p=1009$.
+
+The cocycles $z=[\xi^p]$ and $C=\beta(\xi)$ come from the
+primitive coordinate $\xi$.  Their product $zC^{p-2}$ is an
+actual cocycle with the indicated nonzero leading May monomial.
+The preceding cohomological degree at its weight has no May
+monomial in any filtration.  It consequently cannot become a
+boundary on any May page.  An actual cocycle has no outgoing May
+differential, proving its nonvanishing.  The factor $C^{p-2}$ is therefore
+nonzero as well.
+
+For the other two classes, use the cobar convention
+$d[u\mid v]=d[u]\mid v-u\mid d[v]$ with
+$d[\eta]=[\xi\mid\xi^p]$.  The corrected cocycle
+\begin{equation}\label{frob:corrected-cocycle}
+ U=[\xi\mid\eta]+\frac12[\xi^2\mid\xi^p]
+\end{equation}
+is closed: its two differentials are
+$-[\xi\mid\xi\mid\xi^p]$ and
+$[\xi\mid\xi\mid\xi^p]$, respectively.  In the ordinary
+May filtration its first summand has degree four and the correction
+has degree three.  Its leading detector is thus $yb$.  The table
+excludes both an incoming and an outgoing May differential, so
+$\upsilon=[U]\neq0$.  Similarly
+\[
+ U\mid\xi^p
+   =[\xi\mid\eta\mid\xi^p]
+       +\frac12[\xi^2\mid\xi^p\mid\xi^p]
+\]
+is a closed representative with leading detector $ybz$.
+Its correction has smaller May filtration.  The absence of a
+degree-two May monomial at weight $2p+2$ excludes every possible
+incoming differential, and the explicit cocycle excludes outgoing
+differentials.  Hence $\upsilon z\neq0$.
+\end{proof}
+
+\subsection{The row \texorpdfstring{$k=2$}{k=2}}
+
+\begin{proposition}[The row $k=2$]\label{frob:k2}
+At $s=4p-3$ and $w=u_4+2$, the cobar cohomology group in
+Table~\ref{may:remaining-table} is zero.
+\end{proposition}
+
+\begin{proof}
+We prove vanishing for $G_3$ by treating the first two rows of
+Table~\ref{frob:coefficient-obligations}.  Their kernel degrees are
+$2p+1$ and $2p-1$, so their coefficient differentials cannot mix.
+
+For $M=xac\operatorname{Sym}^{p-1}(A,B,E)$, we have
+$\mathcal R=p(p-1)$ and quotient degree $a=2p-4$.
+After the weight exclusion of $E$, the finite module is
+$xacA^{p-1}\langle1,D,\ldots,D^{p-1}\rangle$.
+The sole candidate from Table~\ref{may:remaining-table} is
+$D C^{p-2}$, suppressing the invariant factor $xacA^{p-1}$.
+Its quotient weight is $p(p-2)$ by \eqref{frob:D-weight}.
+The translation of $D$ gives, on the first possible coefficient page,
+\begin{equation}\label{frob:k2-differential}
+ D[C^{p-2}]\longmapsto \pm[zC^{p-2}].
+\end{equation}
+The target has quotient degree $2p-3$ and weight $p(p-1)$.
+Both source and target are nonzero actual quotient-cohomology classes
+by Lemma~\ref{frob:quotient-groups}; they are present on this page because
+all earlier coefficient differentials vanish.  Thus
+\eqref{frob:k2-differential} removes the candidate.
+
+We record the adjacent-degree checks.  A differential on this page
+entering its source would have $D$-degree two, quotient degree
+$2p-5$, and quotient weight $p(p-3)$; this group is zero
+by Lemma~\ref{frob:quotient-groups}.  The target, having $D$-degree
+zero, has no outgoing coefficient differential.  Its first-page
+incoming source can only have $D$-degree one, and
+\eqref{frob:small-monomials} in degree $2p-4$, weight $p(p-2)$,
+gives only $C^{p-2}$.  Thus no other source can cancel the nonzero
+map in \eqref{frob:k2-differential}.  In the original total degree,
+the same equation, or the exhaustive candidate table, leaves no
+other coefficient-leading term in this vertical summand.  Hence
+a change of representative cannot replace the removed class.
+
+Now take the full second summand
+$M=xac\operatorname{Sym}^{p-2}(A,B,E)$, with $\mathcal R=p^2-p+1$
+and quotient degree $2p-2$.
+Table~\ref{may:remaining-table} leaves only its $D$-degree-zero candidate,
+whose coefficient $xacA^{p-2}$ is invariant.
+Its quotient detector $yzC^{p-2}$ is already a
+splitting boundary, since $db=yz$ and $dC=0$ on the May page.
+This also has a direct cobar realization: $yz=d[\eta]$ and $C$
+is an actual cocycle, so $yzC^{p-2}$ is exact.  It is the only
+May monomial in this quotient bidegree, as follows by putting
+$a=2p-2$, $w=p^2-p+1$ in \eqref{frob:small-monomials}.
+It therefore contributes no actual quotient cohomology in
+\eqref{frob:coefficient-page}.  These are all initial candidates
+of total degree $4p-3$ and weight $u_4+2$.  The coefficient spectral
+sequences consequently vanish there, so all corresponding terms
+of \eqref{frob:lhss} are zero.  Its convergence proves the result.
+\end{proof}
+
+\subsection{The row \texorpdfstring{$k=p-1$}{k=p-1}}
+
+\begin{proposition}[The row $k=p-1$]\label{frob:kpminus1}
+Put $N_0=(p-1)^2$.  At $s=2N_0+1$ and
+$w=u_4+p-1$, the cobar cohomology group in
+Table~\ref{may:remaining-table} is zero.
+\end{proposition}
+
+\begin{proof}
+We treat the last two rows of Table~\ref{frob:coefficient-obligations}
+for $G_2$, using the finite module \eqref{frob:finite-D} in each case.
+For $n=N_0$, the residual weight is $\mathcal R=3p+1$ and the quotient
+degree is one.  We show directly that every one-cocycle vanishes.
+The leading candidates could survive only if lower powers of $D$
+supplied corrections.  The crossed-cocycle equation will rule these out:
+it requires unequal mixed coefficients where every available correction
+has equal ones.
+
+Write $\xi(g)=r$, $\eta(g)=u$,
+$\xi(h)=s$, $\eta(h)=v$, so that
+\[
+ \xi(gh)=r+s,\qquad\eta(gh)=u+v+rs^p.
+\]
+A normalized one-cocycle is a polynomial $f(D;g)$ satisfying
+\begin{equation}\label{frob:crossed-equation}
+ f(D;gh)=f(D;g)+f(D+r^p;h).
+\end{equation}
+Since $D$ has weight $p$ and $4p>3p+1$, it must have the form
+\[
+ f(D;g)=D^3f_3(g)+D^2f_2(g)+Df_1(g)+f_0(g).
+\]
+All these powers are in the finite module, since $N_0\geq3$.
+For a polynomial function $\varphi$ write
+$\delta\varphi(g,h)=\varphi(gh)-\varphi(g)-\varphi(h)$.
+Comparing the four coefficients in
+\eqref{frob:crossed-equation} gives
+\begin{align}
+ \delta f_3&=0,\nonumber\\
+ \delta f_2&=3r^pf_3(h),\nonumber\\
+ \delta f_1&=2r^pf_2(h)+3r^{2p}f_3(h),
+       \label{frob:crossed-coefficients}\\
+ \delta f_0&=r^pf_1(h)+r^{2p}f_2(h)+r^{3p}f_3(h).
+       \nonumber
+\end{align}
+Homogeneity forces
+\begin{align*}
+ f_3&=a_3\xi,\\
+ f_2&=b_2\eta+c_2\xi^{p+1},\\
+ f_1&=d_1\eta\xi^p+e_1\xi^{2p+1},\\
+ f_0&=r_0\xi^{3p+1}+s_0\eta\xi^{2p}
+                       +t_0\eta^2\xi^{p-1}.
+\end{align*}
+Here the coefficients $a_3,b_2,c_2,d_1,e_1,r_0,s_0,t_0$ lie in $\mathbb F_p$.
+The second equation of \eqref{frob:crossed-coefficients}, together
+with
+\[
+ \delta\eta=rs^p,\qquad
+ \delta(\xi^{p+1})=r^ps+rs^p,
+\]
+forces $c_2=3a_3$ and $b_2=-3a_3$.
+The next equation has coefficient $2b_2$ on $r^pv$ and zero
+on $us^p$.  On the other hand,
+\begin{equation}\label{frob:asymmetric-pair}
+ \delta(\eta\xi^p)
+       =us^p+vr^p+r^{p+1}s^p+rs^{2p},
+\end{equation}
+and $\delta(\xi^{2p+1})$ contains neither $u$ nor $v$.
+Its left side must therefore have equal coefficients on $r^pv$
+and $us^p$.  Since $2b_2=-6a_3$ and $6\neq0$ in
+$\mathbb F_{1009}$, we obtain $a_3=0$, and then
+$b_2=c_2=0$.  No term in $f_0$ can alter this
+coefficient-of-$D$ equation.
+
+Now $f_1$ is primitive.  Equation~\eqref{frob:asymmetric-pair}
+first forces $d_1=0$, and the coefficient of $r^{2p}s$
+in $\delta(\xi^{2p+1})$ then forces $e_1=0$.
+Finally $f_0$ is primitive.  In its reduced coproduct the monomial
+$u^2s^{p-1}$ has coefficient $t_0$, so $t_0=0$.
+The monomial $us^{2p}$ then has coefficient $s_0$, so
+$s_0=0$.  The coefficient of $r^{3p}s$ in
+$\delta(\xi^{3p+1})$ is one, giving $r_0=0$.
+Every normalized cocycle is thus zero, proving the required
+$H^1$ vanishing, including all lower-$D$ corrections.
+
+For $n=N_0-1$, the residual weight is $\mathcal R=3p+2$ and the desired
+quotient degree is three.  The first coefficient differential
+comes from the term $2D\xi^p$ in
+\[
+ (D+\xi^p)^2-D^2=2D\xi^p+\xi^{2p}.
+\]
+Using the actual class $\upsilon$ of
+Lemma~\ref{frob:quotient-groups}, it is
+\begin{equation}\label{frob:h3-differential}
+ D^2\upsilon\longmapsto\pm2D\upsilon z.
+\end{equation}
+The coefficient differential first prepends $z$. Its product
+$z\upsilon$ agrees with $\upsilon z$: both are actual cocycles
+with the same leading May monomial $ybz$, with coefficient $+1$
+after two exterior interchanges, and the list in Equation~\eqref{frob:small-monomials}
+leaves no other initial May monomial or lower-filtration cohomology
+in this bidegree. Thus the product order in
+Equation~\eqref{frob:h3-differential} is justified directly.
+Both powers of $D$ lie in the finite module, since $n\geq3$.
+Equation~\eqref{frob:D-weight} gives quotient degree and weight
+$(2,p+2)$ for the source and $(3,2p+2)$ for the target.
+Both therefore have the required residual weight $3p+2$.
+All earlier coefficient differentials vanish, so source and target are
+still their actual quotient-cohomology classes on this first possible
+page.  They are both nonzero by
+Lemma~\ref{frob:quotient-groups}.
+
+A same-page incoming differential to the source would require
+$D$-degree three and quotient $H^1$ of weight two,
+which is zero.  The only same-page incoming source for the target
+is $D^2H^2(Q_2)_{p+2}$, the one-dimensional span displayed in
+\eqref{frob:h3-differential}.  A same-page outgoing differential
+from the target is proportional to $\upsilon z^2=0$.
+Indeed $[\xi^p\mid\xi^p]=\tfrac12d[\xi^{2p}]$, so this
+last vanishing also follows directly in the cobar complex.
+The term $\xi^{2p}$ in the expansion of $D^2$ lowers two
+$D$-levels and cannot cancel its first differential.  Since
+$2\neq0$ in $\mathbb F_p$, \eqref{frob:h3-differential}
+removes the target.
+
+There is no other degree-three coefficient-leading term.  In fact,
+the possibilities for $D$-degrees $0,1,2,3$ require quotient
+degree three at respective weights
+$3p+2,2p+2,p+2,2$; by
+Lemma~\ref{frob:quotient-groups}, only the second has a
+candidate, namely $ybz$.  Larger $D$-degrees exceed the residual
+weight.  Thus there is no replacement in a lower coefficient
+filtration.  The initial candidate table excludes every other
+kernel degree or exterior summand at this total bidegree.
+Both possible terms of \eqref{frob:lhss} consequently vanish,
+and the proposition follows.
+\end{proof}
+
+\begin{corollary}\label{frob:vanishing}
+The obstruction groups in bidegrees
+\[
+ (s,w)=(2q+1,u_4+2),\qquad
+ (s,w)=(q(p-1)+1,u_4+p-1)
+\]
+vanish in the additive cobar calculation of
+Table~\ref{may:remaining-table}.  Among its rows with $k\geq2$,
+only the row $k=p$ remains to be addressed.
+\end{corollary}
+
+\begin{proof}
+Apply Propositions~\ref{frob:k2} and~\ref{frob:kpminus1},
+and the exhaustive list in Table~\ref{may:remaining-table}.
+\end{proof}
diff --git a/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/build/sections/05-final-obstruction.tex b/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/build/sections/05-final-obstruction.tex
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+\section{The final obstruction and the finite cofiber}
+\label{top:section}
+
+We retain the prime $p=1009$ and put
+\begin{equation}
+\label{top:constants}
+ N=p(p-1)-2,\qquad K=p(p-2)+1=(p-1)^2,
+ \qquad W=2p^2+4p+1.
+\end{equation}
+The spectrum $Y$ of Proposition~\ref{low:V3} has
+$BP_*Y=R/I_4$ and a binary two-sided unital multiplication
+$\mu:Y\wedge Y\longrightarrow Y$ with unit $\eta:S^0_{(p)}\to Y$.
+All spheres and smash products in this section are $p$-local.
+We shall use the product and convergence assertions of
+Proposition~\ref{fw:anss-pairing} and Proposition~\ref{fw:anss}.
+In this section the second superscript of $E_r^{s,w}$ is weight:
+it denotes the group of internal degree $qw$ in the earlier notation.
+
+The May comparison and the remaining-target calculation,
+Lemma~\ref{may:comparison} and Table~\ref{may:remaining-table},
+give a surjection
+\begin{equation}
+\label{top:An-surjection}
+ A^N\colon E_2^{5,W}(Y)\longrightarrow
+ E_2^{qp+1,u_4+p}(Y).
+\end{equation}
+Indeed, every initial May monomial in the target is divisible by
+$A^N$, and the two degree identities are
+\[
+ 5+2N=qp+1,\qquad W+pN=u_4+p.
+\]
+For the source, $W<u_4$, so specialization to the additive Hopf
+algebra identifies the normalized cobar groups in all three degrees
+$4,5,6$. For the target, $u_4+p<u_5$; a positive $v_4$ coefficient
+would leave weight $p$, less than the shortest adjacent tensor length
+$qp=(qp+1)-1$. Thus the target groups in degrees $qp,qp+1,qp+2$
+also agree with their additive specializations, by
+Lemma~\ref{fw:normalized-bound}. The two triples differ by $2N$.
+Cup multiplication by the actual carry cocycle representing $A^N$
+commutes with specialization, so the filtered surjection of
+Lemma~\ref{may:comparison} transports through this commuting square
+to the displayed surjection for $Y$.
+
+Let $\alpha\in\pi_{q-1}S^0$ and
+$\beta\in\pi_{pq-2}S^0$ be the sphere classes constructed in
+Lemma~\ref{top:sphere-classes}, detected by $[t_1]$ and $A$,
+respectively, and write $d=|\beta|=pq-2$.
+The two-cell spectrum that we use for lifts is
+\begin{equation}
+\label{top:J-definition}
+ J=\Sigma^{-q}C(\alpha),\qquad
+ \varepsilon:J\longrightarrow S^0,
+\end{equation}
+where $C(\alpha)$ is the cofiber of $\alpha:S^{q-1}\to S^0$
+and $\varepsilon$ is its desuspended top quotient. Thus $J$ has
+cells in degrees $-q$ and zero. Its connecting map is the corresponding
+suspension of $\alpha$, so the cofiber exact sequence gives
+\begin{equation}
+\label{top:lift-criterion}
+ x\in\pi_gY\text{ lifts to }\pi_g(J\wedge Y)
+ \quad\Longleftrightarrow\quad
+ \alpha x=0\in\pi_{g+q-1}Y.
+\end{equation}
+
+Put $n=p-1$. Since $n!$ is a $p$-local unit, the averaging idempotent
+\[
+ e=\frac1{n!}\sum_{\sigma\in\Sigma_n}\sigma
+ \quad\text{on }J^{\wedge n}
+\]
+has a finite retract $T=\operatorname{Sym}^{n}J$.
+Choose splitting maps $i:T\to J^{\wedge n}$ and
+$\operatorname{pr}:J^{\wedge n}\to T$, with
+$i\operatorname{pr}=e$, and put
+$\varepsilon_T=\varepsilon^{\wedge n}i:T\to S^0$.
+The key map will be
+\[
+ f=(\eta\beta)\circ\Sigma^d\varepsilon_T:\Sigma^dT\longrightarrow Y.
+\]
+The choice of $p-1$ factors has a concrete algebraic purpose:
+writing $DT$ for the Spanier--Whitehead dual of $T$, we will identify
+$BP_*(DT\wedge Y)$ as a free $R/I_4$-module with basis
+$1,r,\ldots,r^{p-1}$ of weights $0,\ldots,p-1$.
+These supply exactly the coefficient powers in the integral polynomial
+$((r+x)^p-r^p-x^p)/p$.
+Lemma~\ref{top:symmetric-comodule} determines the translated coaction,
+and Proposition~\ref{top:symmetric-null-map} shows that substituting
+$x=t_1$ gives a cochain whose cobar differential is the constant
+cocycle $A$. A filtration bound then proves that $f$ is an actual
+null map.
+
+We first show that every source class is represented by
+$\gamma\in\pi_{qW-5}Y$ and that
+$\alpha\gamma=0$ and $\alpha\cdot(\eta\beta^p)=0$ in $\pi_*Y$.
+One lift of $\gamma$ and $p-2$ lifts of $\eta\beta^p$ give, after
+multiplying the $Y$ factors and averaging the $J$ factors, a map to
+$T\wedge Y$ with top value $\beta^{p(p-2)}\gamma$.
+The null map $f$ then gives $\beta^K\gamma=0$.
+The strict inequality $5+2K<qp+1$, together with $K<N$, converts
+this homotopy relation into disappearance of the whole target before
+$d_{qp+1}(v_4)$.
+
+\subsection{The small stems used for the lifts}
+
+\begin{lemma}[Small-stem lifts]
+\label{top:small-stems}
+Every class of $E_2^{5,W}(Y)$ survives to homotopy. Moreover,
+\begin{equation}
+\label{top:alpha-gamma-vanishing}
+ \pi_{q(W+1)-6}Y=0.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+By $q$-sparsity, a positive differential out of $(5,W)$ has target
+\[
+ (s,w)=(6+qj,W+j),\qquad j\geq 1.
+\]
+All possible Adams--Novikov terms in the stem in
+\eqref{top:alpha-gamma-vanishing} have
+\[
+ (s,w)=(6+qj,W+1+j),\qquad j\geq 0.
+\]
+We treat both by writing
+\begin{equation}
+\label{top:small-stem-degrees}
+ (s,w)=(6+qj,W+\delta+j),\qquad
+ \begin{cases}
+ \delta=0,&j\geq1,\\
+ \delta=1,&j\geq0.
+ \end{cases}
+\end{equation}
+
+First we give explicit bounds permitting the additive calculation.
+The normalized-cobar inequality of Lemma~\ref{fw:normalized-bound}
+implies
+\[
+ (q-1)j\leq W+\delta-6.
+\]
+At the specified prime,
+\[
+ q-1=2015,\quad W=2{,}040{,}199,\quad
+ 2015\cdot1013=2{,}041{,}195>W+1-6.
+\]
+Consequently $j\leq1012$ and
+\begin{equation}
+\label{top:small-stem-cutoff}
+ w\leq2{,}041{,}212
+ <p^3=1{,}027{,}243{,}729
+ <u_4=1{,}028{,}262{,}820.
+\end{equation}
+Thus no nonconstant coefficient of $R/I_4$ occurs, in any cobar
+degree at these weights. The only possible digit positions are
+$0,1,2$, and the cooperation formulas are additive.
+Lemma~\ref{fw:range-comparison} therefore identifies the relevant
+cobar calculation with the indicated range of $P$.
+
+Use the six exterior letters $x,a,c,y,b,z$ and the three polynomial
+letters $A,B,C$ of Equation~\eqref{may:three-position-weight}.
+If $m$ is the number of exterior letters and $L=(s-m)/2$ the number
+of polynomial letters, then $m\leq6$ and
+\[
+ L\geq (p-1)j,\qquad w\geq pL.
+\]
+It follows that
+\[
+ \bigl(p(p-1)-1\bigr)j\leq W+\delta.
+\]
+Since $p(p-1)-1=1{,}017{,}071$ and
+$3\cdot1{,}017{,}071>W+1$, this improves the bound to $j\leq2$.
+
+For completeness, we enumerate this remaining finite calculation.
+Let $H_i$ count the exterior letters covering position $i$, and let
+$n_A,n_B,n_C$ be the polynomial exponents. The weight equation is
+\begin{equation}
+\label{top:weight-equation}
+ w=H_0+p(H_1+L-n_C)+p^2(H_2+n_B+n_C).
+\end{equation}
+In \eqref{top:small-stem-degrees}, $m$ is even and reduction modulo
+$p$ gives $H_0=1+\delta+j$, whenever a solution exists. Put
+$\kappa_2=H_2+n_B+n_C$. All position contributions are nonnegative;
+\eqref{top:small-stem-cutoff} also gives $w<3p^2$, so
+$\kappa_2\leq2$ and $n_C\leq2$. Substituting
+$L=3+(p-1)j-m/2$ into \eqref{top:weight-equation} yields
+\[
+ p(\kappa_2+j-2)=1+j+m/2+n_C-H_1.
+\]
+For $0\leq j\leq2$, its right side lies between $-3$ and $8$.
+Its only multiple of $p=1009$ is zero. Hence
+\begin{equation}
+\label{top:small-stem-coverage}
+ H_0=1+\delta+j,\qquad \kappa_2=2-j,\qquad
+ H_1=1+j+m/2+n_C.
+\end{equation}
+
+For $j=2$, either $H_0=4$, which is impossible, or $H_0=3$ forces
+$c$ to occur whereas $\kappa_2=0$. For $j=1,\delta=1$, all of $x,a,c$
+occur and $\kappa_2=1$ excludes $b,z$. Thus $H_1\leq3$, but the last
+equation of \eqref{top:small-stem-coverage}, with even $m\geq4$,
+requires $H_1\geq4$. For $j=1,\delta=0$, one has $H_0=2$.
+If $m=2$, the required inequality $H_1\geq3$ is impossible.
+If $m=4$, one must have $H_1=4$ and $n_C=0$; the four letters
+covering position one are exactly $a,c,y,b$, which have $H_2=2$,
+contrary to $\kappa_2=1$. Finally $m=6$ would give $H_0=3$.
+All positive-$j$ groups in \eqref{top:small-stem-degrees} therefore
+have empty initial May pages.
+
+It remains to examine $j=0,\delta=1$, where $H_0=2$ and $m$ is
+$2$ or $4$. For $m=2$ both letters start at zero, and
+\eqref{top:small-stem-coverage} permits only $ac$, followed by
+$n_B=1,n_C=0,n_A=1$. For $m=4$ choose two of $x,a,c$ and two of
+$y,b,z$. The equations become
+\[
+ n_C=H_1-3,\qquad n_B=2-H_2-n_C,\qquad
+ n_A=1-n_B-n_C.
+\]
+The following table lists all nine choices; a dash means that at
+least one exponent is negative.
+\begin{center}
+\begin{tabular}{c c c c}
+\toprule
+Start-zero pair & $yb$ & $yz$ & $bz$\\
+\midrule
+$xa$ & $xaybB$ & --- & ---\\
+$xc$ & $xcybA$ & --- & ---\\
+$ac$ & --- & $acyzA$ & ---\\
+\bottomrule
+\end{tabular}
+\end{center}
+Thus the complete initial-page list at $(6,W+1)$ is
+\begin{equation}
+\label{top:four-terms}
+ acAB,\qquad xaybB,\qquad xcybA,\qquad acyzA.
+\end{equation}
+With exterior order $x,a,c,y,b,z$ and
+$da=xy$, $db=yz$, $dc=xb+az$, their splitting differentials are
+\[
+\begin{aligned}
+ d(acAB)&=(xab-xcy)AB, &d(xaybB)&=0,\\
+ d(xcybA)&=-xaybzA, &d(acyzA)&=-xaybzA.
+\end{aligned}
+\]
+The first image has polynomial factor $AB$ and cannot cancel either
+of the other images. The two remaining cycle directions are
+boundaries, since
+\[
+ d(acyB)=-xaybB,\qquad
+ d(acbA)=-xcybA+acyzA.
+\]
+The splitting cohomology of \eqref{top:four-terms} is zero.
+The May spectral sequence, finite in each of these weights, proves
+all the claimed $E_2$ vanishings.
+
+For orientation, the same equations also enumerate the source
+$(5,W)$. Here $H_0=1$, $m$ is odd, and $L=(5-m)/2$. They force
+$H_2+n_B+n_C=2$ and $H_1+L-n_C=4$. If $m=1$, then
+$H_1+L\leq1+2<4$; if $m=5$, then at least two of $x,a,c$
+occur, contrary to $H_0=1$. Thus $m=3$, which forces
+$n_C=0$ and $H_1=3$. The only monomials are $aybB$ and $cybA$. We need no assertion that either
+one survives the May spectral sequence.
+
+No topological Adams--Novikov differential can hit filtration
+five: its length would have to be at least $q+1>5$.
+The outgoing target groups have just been proved zero. Thus every
+actual class of $E_2^{5,W}(Y)$ survives and is represented by a
+class $\gamma\in\pi_{qW-5}Y$. The vanishings for $\delta=1$ exhaust
+all filtrations in the stem $q(W+1)-6$; strong convergence proves
+\eqref{top:alpha-gamma-vanishing}.
+\end{proof}
+
+\subsection{Two sphere classes and an extended-power relation}
+
+\begin{lemma}[The sphere classes]
+\label{top:sphere-classes}
+There are sphere classes
+\[
+ \alpha\in\pi_{q-1}S^0_{(p)},\qquad
+ \beta\in\pi_{pq-2}S^0_{(p)}
+\]
+detected respectively by $[t_1]$ and the divided coproduct
+$b_{1,0}$. The former generates a cyclic group of order $p$, and
+\[
+ \pi_iS^0_{(p)}=0\qquad(0<i<q-1).
+\]
+Multiplication by $\beta$ on the Adams--Novikov spectral sequence
+of $Y$ induces multiplication by the actual cobar class $A$ used
+in \eqref{top:An-surjection}.
+\end{lemma}
+
+\begin{proof}
+The cooperation $t_1$ is primitive. In weight one the normalized
+sphere cobar is
+\[
+ \mathbb Z_{(p)}\{v_1\}\xrightarrow{\ p\ }
+ \mathbb Z_{(p)}\{t_1\},
+\]
+because $\eta_R(v_1)-v_1=pt_1$. Hence its degree-one cohomology is
+$\mathbb Z/p$ on $[t_1]$.
+
+The integral polynomial
+\begin{equation}
+\label{top:beta-cocycle}
+ b(x,y)=\frac{(x+y)^p-x^p-y^p}{p}
+ =\sum_{i=1}^{p-1}\frac{\binom pi}{p}x^iy^{p-i}
+\end{equation}
+defines the two-cobar cocycle $b_{1,0}$ by substituting the two
+primitive copies of $t_1$. Its differential is zero: the
+two-cocycle identity follows by expanding $(x+y+z)^p$, or by
+dividing the identity $d^2(t_1^p)=0$ in the torsion-free integral
+cobar. Its image in the additive specialization is nonzero in
+cohomology. At weight $p$ the only coordinate available is $t_1$;
+the only one-cochain of that weight is $t_1^p$, whose differential
+is zero in characteristic $p$, whereas \eqref{top:beta-cocycle}
+is a nonzero two-cochain. Thus it is not a boundary.
+
+The first positive topological differential length permitted by
+sparsity is $q+1$. From $(2,p)$ an outgoing differential has target
+$(qj+3,p+j)$ for $j\geq1$, violating $w\geq s$; no incoming
+differential can reach filtration two. The same argument applies
+to $(1,1)$. Therefore these cocycles survive to the stated sphere
+classes.
+
+For any positive stem $i$ with a nonzero normalized-cobar term,
+either $s=0$ and $i=qw\geq q$, or $s\geq1$ and
+\[
+ i=qw-s\geq(q-1)s\geq q-1.
+\]
+This proves the lower sphere vanishing. In stem $q-1$ the same
+inequality permits only $(s,w)=(1,1)$, so there is no extension
+ambiguity and $\pi_{q-1}S^0_{(p)}=\mathbb Z/p\{\alpha\}$.
+Finally, the sphere action and its cobar pairing in
+Proposition~\ref{fw:anss-pairing} identify the detector of
+multiplication by $\beta$ with $b_{1,0}=A$.
+\end{proof}
+
+We next prove that $\alpha\beta^p$ has zero image in $\pi_*Y$. We use
+the odd-primary reduced powers, including the Cartan formula and
+$\mathcal P^k(x)=x^p$ for a space class of degree $2k$, in
+\citet[Chapter~VI, Section~1, and Chapter~VIII, Theorem~2.2]{SteenrodEpstein1962}.
+Their compatibility with suspension gives operations on spectra; the
+instability normalization below is applied to a Thom space before
+desuspension. Toda proved the corresponding sphere relation
+\citep[p.~839, Corollary]{Toda1967} and subsequently treated such
+relations using cyclic extended powers
+\citep[Sections~1--3, especially Theorem~3]{Toda1968}.
+We give the image relation needed in $Y$ through the following
+finite-cell construction.
+
+\begin{lemma}[A finite three-cell model]
+\label{top:three-cell-model}
+Let $U_0$ be a connective $p$-local spectrum with
+\[
+ H_0(U_0;\mathbb Z_{(p)})=\mathbb Z_{(p)},\qquad
+ H_{q-1}(U_0;\mathbb Z_{(p)})=\mathbb Z/p,
+\]
+and with all other positive integral homology through degree $q$
+zero. Let $b_0:S^0\to U_0$ induce an isomorphism on $H_0$, and
+suppose the reduced power $\mathcal P^1$ of the bottom mod-$p$
+cohomology class is nonzero. Then there is a three-cell spectrum
+\[
+ Q=\bigl(S^0\vee S^{q-1}\bigr)
+       \cup_{(a\alpha,\,up)}e^q
+\]
+and a map $Q\to U_0$ whose restriction to $S^0$ is $b_0$.
+Here $u\in\mathbb Z_{(p)}^\times$ and
+$a\in\mathbb F_p^\times$.
+\end{lemma}
+
+\begin{proof}
+Let $C$ be the cofiber of $b_0:S^0\to U_0$.
+The assumed integral homology groups and successive applications
+of connective Hurewicz show that $C$ is $(q-2)$-connected and
+\[
+ \pi_{q-1}C=H_{q-1}(C;\mathbb Z_{(p)})=\mathbb Z/p.
+\]
+Choose a generator $S^{q-1}\to C$. Its obstruction to lifting to
+$U_0$ lies in $\pi_{q-2}S^0$, which is zero by
+Lemma~\ref{top:sphere-classes}. A lift, together with $b_0$, gives
+$S^0\vee S^{q-1}\to U_0$.
+Its cofiber $D$ is $(q-1)$-connected. The integral homology exact
+sequence gives
+\[
+ H_q(D;\mathbb Z_{(p)})
+ =\ker\bigl(\mathbb Z_{(p)}\longrightarrow\mathbb Z/p\bigr)
+ =p\mathbb Z_{(p)}.
+\]
+In particular this is a free rank-one module.
+Use Hurewicz to choose a generator of $\pi_qD$ and take its
+boundary in $\pi_{q-1}(S^0\vee S^{q-1})$. Attaching the
+corresponding cell gives a map $Q\to U_0$ extending the two lower
+cells. Its component on $S^{q-1}$ is multiplication by a unit
+times $p$, since its homology image is exactly
+$p\mathbb Z_{(p)}$. Its component on the bottom belongs to
+$\pi_{q-1}S^0=\mathbb Z/p\{\alpha\}$, so it is $a\alpha$ for
+some $a\in\mathbb F_p$.
+
+The map $Q\to U_0$ induces isomorphisms on integral homology in
+degrees zero and $q-1$. The universal coefficient sequence then
+gives isomorphisms on mod-$p$ cohomology in degrees zero and $q$;
+the latter is the class arising from the torsion in degree $q-1$.
+The assumed nonzero reduced power therefore pulls back to a
+nonzero reduced power of the bottom class of $Q$.
+If $a=0$, the actual attaching map would give a wedge
+\[
+ Q\simeq S^0\vee\operatorname{cofib}
+       \bigl(up:S^{q-1}\to S^{q-1}\bigr).
+\]
+The degree-zero class in this wedge is pulled back from $S^0$,
+so its positive reduced powers vanish by naturality. This is a
+contradiction. Hence $a\ne0$.
+The construction used an integral homology gap and relative
+Hurewicz in a finite range; no assertion that $U_0$ itself is
+finite is involved.
+\end{proof}
+
+\begin{proposition}[The extended-power relation]
+\label{top:extended-power-relation}
+The image of $\alpha\beta^p$ in $\pi_*Y$ is zero.
+\end{proposition}
+
+\begin{proof}
+Write $d=pq-2=2m$, where
+\[
+ m=p(p-1)-1\equiv-1\pmod p.
+\]
+Let $\rho_{\mathbb C}$ be the complex regular representation of
+$C_p$. The cyclic extended power of the even sphere $S^{2m}$ is
+the Thom spectrum of $m\rho_{\mathbb C}$ over $BC_p$:
+\[
+ (EC_p)_+\wedge_{C_p}(S^{2m})^{\wedge p}
+ \simeq\operatorname{Th}(m\rho_{\mathbb C}).
+\]
+Applying this functor to a sphere map representing $\beta$, then
+collapsing $BC_p$ in the extended power of $S^0$, gives a map
+\begin{equation}
+\label{top:extended-beta-map}
+ \Phi:\operatorname{Th}(m\rho_{\mathbb C})\longrightarrow S^0
+\end{equation}
+whose restriction to a fiber sphere $S^{pd}$ is $\beta^p$.
+Only the symmetric multiplication of the sphere spectrum is used
+in constructing this map.
+
+The group $\mathbb F_p^\times$ acts by permutations of the
+cyclically labelled smash coordinates and by the corresponding
+automorphisms of $C_p$. This normalizer action is compatible with
+\eqref{top:extended-beta-map}, on whose target it is trivial.
+Normalize the bottom degree by setting
+\[
+ U=\Sigma^{-pd}\operatorname{Th}(m\rho_{\mathbb C}),\qquad
+ e_0=\frac1{p-1}\sum_{a\in\mathbb F_p^\times}a:U\longrightarrow U.
+\]
+The idempotent $e_0$ splits in spectra. Denote its image by $U_0$,
+with maps $i_0:U_0\to U$, $r_0:U\to U_0$ satisfying
+$i_0r_0=e_0$ and $r_0i_0=1$.
+The projected Thom fiber $b_0:S^0\to U_0$ induces an isomorphism
+on $H_0(-;\mathbb Z_{(p)})$: the complex orientation makes every
+normalizer permutation act as one on the bottom class.
+Moreover, normalizer invariance of $\Phi$ gives a map
+\[
+ \Phi_0:\Sigma^{pd}U_0\longrightarrow S^0,
+ \qquad \Phi_0\Sigma^{pd}b_0=\beta^p.
+\]
+
+We describe the homology and the reduced power of this summand.
+The complex bundle is integrally oriented, so the integral Thom
+isomorphism identifies $H_*(U;\mathbb Z_{(p)})$ with
+$H_*(BC_p;\mathbb Z_{(p)})$. The periodic cyclic-group resolution,
+with maps $g-1$ and $1+g+\cdots+g^{p-1}$, gives
+\[
+ H_0(BC_p;\mathbb Z_{(p)})=\mathbb Z_{(p)},\quad
+ H_{2j-1}(BC_p;\mathbb Z_{(p)})=\mathbb Z/p\ (j\geq1),\quad
+ H_{2j}(BC_p;\mathbb Z_{(p)})=0\ (j\geq1).
+\]
+These groups are finitely generated degreewise, as are their
+direct summands for $U_0$. The same resolution, or the circle
+bundle of Chern class $p$ over infinite complex projective space,
+gives
+\[
+ H^*(BC_p;\mathbb F_p)=\Lambda(s)\otimes\mathbb F_p[t],
+ \qquad |s|=1,\quad |t|=2,\quad t=\operatorname{Bockstein}(s).
+\]
+An automorphism $a\in\mathbb F_p^\times$ multiplies both $s$ and
+$t$ by $a$. Equivalently it acts by $a^j$ on integral homology
+in degree $2j-1$. Averaging consequently retains, through degree
+$q$, exactly the mod-$p$ classes in degrees $0,q-1,q$ and the
+integral groups
+\begin{equation}
+\label{top:U0-homology}
+ H_0(U_0;\mathbb Z_{(p)})=\mathbb Z_{(p)},\quad
+ H_{q-1}(U_0;\mathbb Z_{(p)})=\mathbb Z/p,
+ \quad H_j(U_0;\mathbb Z_{(p)})=0\ (0<j\leq q,\ j\ne q-1).
+\end{equation}
+
+Let $u$ be the normalized Thom class. We claim
+\begin{equation}
+\label{top:Thom-P1}
+ \mathcal P^1u=-m\,t^{p-1}u\ne0.
+\end{equation}
+For a universal complex line bundle with first Chern class $l$
+and Thom class $u_l$, write
+$\mathcal P^1u_l=c\,l^{p-1}u_l$ by the Thom isomorphism.
+Pullback along the zero section and the identity
+$\mathcal P^1(l)=l^p$ imply $c=1$ in the polynomial cohomology of
+infinite complex projective space. Naturality gives the formula
+for every line bundle. The Cartan formula for a direct sum of
+lines therefore gives the sum of their $(p-1)$st Chern powers as
+the multiplier of $\mathcal P^1$ on its Thom class.
+The regular representation splits into the characters with first
+Chern classes $it$, $i\in\mathbb F_p$. Thus the multiplier here is
+\[
+ m\sum_{i\in\mathbb F_p}(it)^{p-1}
+ =-m\,t^{p-1}.
+\]
+Both this class and $u$ are normalizer-invariant. The reduced
+power commutes with the idempotent, so the same nonzero formula
+holds in $U_0$.
+
+Apply Lemma~\ref{top:three-cell-model} to
+\eqref{top:U0-homology} and \eqref{top:Thom-P1}. It supplies a
+three-cell spectrum
+\[
+ Q=(S^0\vee S^{q-1})\cup_{(a\alpha,\,up)}e^q\longrightarrow U_0
+\]
+with $a$ nonzero modulo $p$ and $u$ a $p$-local unit.
+
+Restrict $\Phi_0$ to $\Sigma^{pd}Q$. Its bottom value is
+$\beta^p$; let its value on the other lower cell be $\zeta$.
+The top cell gives the relation
+\[
+ a\alpha\beta^p+up\zeta=0
+ \quad\text{in }\pi_{pd+q-1}S^0.
+\]
+The action of $p$ on $Y$ is null by Corollary~\ref{low:p-null}.
+Apply the unit to this relation and use $a\ne0$ modulo $p$ to
+obtain $\alpha\beta^p=0$ in $\pi_*Y$.
+\end{proof}
+
+\subsection{The symmetric comodule and the null map}
+
+We now prove that the map $f:\Sigma^dT\to Y$ defined above is null.
+Finite duality turns it into the bottom image of $\beta$ in
+$\pi_d(DT\wedge Y)$. The next lemma identifies the coefficient
+comodule in which we will make this image a cobar boundary.
+
+\begin{lemma}[The symmetric coefficient comodule]
+\label{top:symmetric-comodule}
+For $T=\operatorname{Sym}^{p-1}(\Sigma^{-q}C(\alpha))$, the module
+$BP_*(DT\wedge Y)$ is free over $R/I_4$ with basis
+$1,r,\ldots,r^{p-1}$, where $r^k$ has weight $k$. Its coaction is
+\begin{equation}
+\label{top:translated-comodule}
+ r^k\longmapsto\sum_{j=0}^k\binom kjr^jt_1^{k-j}.
+\end{equation}
+The induced homology map of
+$D\varepsilon_T\wedge1_Y:Y\to DT\wedge Y$ is the inclusion
+of constants. The powers of $r$ denote basis elements; no
+multiplication on $DT$ is asserted.
+\end{lemma}
+
+\begin{proof}
+\smallskip\noindent\emph{The two-cell coaction.}
+We first compute the coefficient comodule needed for duality.
+The dual $DJ$ is the cofiber of
+$\alpha':S^{q-1}\to S^0$, where $\alpha'=\pm\alpha$ after fixing
+cell orientations, and its bottom inclusion is $D\varepsilon$.
+Put $\mathcal F=\operatorname{fib}(S^0\xrightarrow{\eta_{BP}}BP)$.
+A chosen $BP$-nullhomotopy of $\alpha'$ gives a lift
+$\widetilde\alpha:S^{q-1}\to\mathcal F$ and a map $H:DJ\to BP$
+extending the unit, in the following map of cofiber sequences:
+\[
+\begin{array}{ccccccc}
+ S^{q-1}&\xrightarrow{\alpha'}&S^0&\longrightarrow&DJ&
+       \longrightarrow&S^q\\[2pt]
+ \big\downarrow\mathrlap{\,\scriptstyle\widetilde\alpha}&&
+ \big\Vert&&\big\downarrow\mathrlap{\,\scriptstyle H}&&
+ \big\downarrow\mathrlap{\,\scriptstyle\Sigma\widetilde\alpha}\\[2pt]
+ \mathcal F&\longrightarrow&S^0&\xrightarrow{\eta_{BP}}&BP&
+       \longrightarrow&\Sigma\mathcal F .
+\end{array}
+\]
+The $BP$-module $BP\wedge DJ$ splits because $BP_{q-1}=0$.
+Let $e_0$ be its bottom generator and choose an upper generator
+$z\in BP_qDJ$ whose top quotient is $1$ and for which
+$\rho_*z=0$, where $\rho=\mu_{BP}(BP\wedge H):BP\wedge DJ\to BP$
+retracts the bottom inclusion. With $\epsilon:\Gamma\to R$
+the counit, the element $g=(BP\wedge H)_*z$ satisfies
+\[
+ \epsilon(g)=0,\qquad
+ \Gamma_q=\eta_L(R_q)\oplus\mathbb Z_{(p)}\{t_1\},
+ \qquad g=ct_1.
+\]
+Here $\eta_L$ is induced by the bottom map
+$BP\wedge\eta_{BP}:BP\to BP\wedge BP$.
+After smashing the cofiber diagram with $BP$, its connecting
+map $\partial:\Gamma_q\to\pi_{q-1}(BP\wedge\mathcal F)$ has kernel
+$\eta_L(R_q)$ and gives
+\[
+ \partial g=(BP\wedge\widetilde\alpha)_*(1).
+\]
+Thus $g$ represents the first Adams--Novikov detector of
+$\alpha'$. It is also the actual off-diagonal coaction:
+writing $\psi(z)=1\otimes z+\omega\otimes e_0$, naturality
+under $H$ gives
+$\Delta g=1\otimes g+\omega\otimes1$.
+Apply $\operatorname{id}\otimes\epsilon$ and use
+$\epsilon(g)=0$ to obtain $\omega=g$.
+By Lemma~\ref{top:sphere-classes} this detector is
+$\pm[t_1]$, so $c\not\equiv0\pmod p$.
+
+For completeness, fix the left-comodule convention
+$\psi(am)=\eta_L(a)\psi(m)$ with tensor product using $\eta_R$,
+so $\eta_R(v_1)-\eta_L(v_1)=pt_1$.
+Replacing $z$ by $z+av_1e_0$ changes its off-diagonal term to
+$g+a(\eta_L-\eta_R)(v_1)=g-pa\,t_1$.
+Changing $H$ by the composite
+$DJ\to S^q\xrightarrow{av_1}BP$ instead changes $g$ by
+$\eta_R(av_1)$; renormalizing the upper generator to
+$z-av_1e_0$ subtracts $\eta_L(av_1)$, giving $g+pa\,t_1$.
+Opposite conventions reverse these signs, and dual cell
+orientations change the detector by a sign; none changes its
+nonzero residue modulo $p$. The underlying $BP$-module splitting
+therefore need not split the comodule.
+Replacing $z$ by $c^{-1}z$ now makes the off-diagonal term
+exactly $t_1$. Its top quotient is then $c^{-1}$ rather than $1$;
+we no longer need the earlier top normalization. This change fixes
+$e_0$ and the bottom inclusion $D\varepsilon$, which are the data
+used in the following symmetric-power calculation.
+
+\smallskip\noindent\emph{The symmetric coefficient module.}
+Duality changes each permutation into its inverse, and hence
+preserves the average. Consequently
+\[
+ DT\simeq\operatorname{Sym}^{n}(DJ).
+\]
+Under this identification $D\varepsilon_T$ is the pure bottom
+tensor $e_0^{\otimes n}$. All the cell degrees of $DJ$ are even.
+If $z_S$ is the tensor with upper generator in positions $S$ and
+bottom generator elsewhere, define its normalized symmetric
+basis by
+\begin{equation}
+\label{top:symmetric-basis}
+ r_k=\binom nk^{-1}\sum_{|S|=k}z_S,
+ \qquad 0\leq k\leq n.
+\end{equation}
+Every denominator is a $p$-local unit. Expanding the translation
+of each tensor gives the coefficient
+\[
+ \frac{\binom{n-j}{k-j}\binom nj}{\binom nk}=\binom kj
+\]
+on $r_jt_1^{k-j}$ in the coaction of $r_k$.
+Thus we can denote this basis by $1,r,\ldots,r^{p-1}$, with
+coaction~\eqref{top:translated-comodule}. Each $r^k$ has weight $k$
+because it is represented by tensors with $k$ upper generators of
+degree $q$.
+
+The module $BP_*DT$ is free. Smashing its free $BP$-module
+splitting with $Y$ gives
+\[
+ BP_*(DT\wedge Y)=BP_*DT\otimes_R(R/I_4),
+\]
+with the same translated basis and with $D\varepsilon_T$ identified
+with constants. This argument uses a free factor, so it requires
+no flatness of $R/I_4$.
+\end{proof}
+
+\begin{proposition}[The symmetric null map]
+\label{top:symmetric-null-map}
+For $d=|\beta|=pq-2$, the actual stable map
+\begin{equation}
+\label{top:null-composite}
+ f:\Sigma^dT\xrightarrow{\ \Sigma^d\varepsilon_T\ }S^d
+       \xrightarrow{\ \beta\ }S^0
+       \xrightarrow{\ \eta\ }Y
+\end{equation}
+is nullhomotopic.
+\end{proposition}
+
+\begin{proof}
+Finite duality identifies
+\[
+ [\Sigma^dT,Y]\cong[S^d,DT\wedge Y]
+\]
+and carries $f$ to
+$(D\varepsilon_T\wedge1_Y)(\eta\beta)$.
+By Lemma~\ref{top:symmetric-comodule}, its induced $E_2$ class is
+represented by the constant-valued beta cocycle. We first show that this cocycle
+is a boundary in the translated coefficient module.
+Consider the integral polynomial
+\begin{equation}
+\label{top:carry-polynomial}
+ F(r,x)=\frac{(r+x)^p-r^p-x^p}{p}
+       =\sum_{i=1}^{p-1}\frac{\binom pi}{p}r^{p-i}x^i.
+\end{equation}
+The displayed sum defines it before reduction modulo $p$; it uses
+only the allowed coefficient powers $r,\ldots,r^{p-1}$.
+The exact integral identity
+\begin{equation}
+\label{top:carry-boundary}
+ F(r,x)+F(r+x,y)-F(r,x+y)
+       =\frac{(x+y)^p-x^p-y^p}{p}
+\end{equation}
+follows by cancellation of the four $p$th powers.
+After reduction, the coefficient coaction
+\eqref{top:translated-comodule} and primitivity of $t_1$ make the
+left side the cobar differential of $F(r,t_1)$, with the common
+overall sign determined by the cobar convention. The right side
+is the constant cocycle \eqref{top:beta-cocycle}. Thus the bottom
+image of $\beta$ has zero image on $E_2$.
+
+There is no possible higher-filtration homotopy value hidden by
+this boundary. The spectrum $DT\wedge Y$ is connective and its
+coefficient weights are nonnegative. In stem $pq-2$, its
+$q$-sparse terms have
+\[
+ (s,w)=(2+qj,p+j).
+\]
+For $j\geq1$ the inequality $w\geq s$ would give
+$p-2\geq(q-1)j$, which is impossible. Naturality places the
+image of $\beta$ in filtration at least two, and the boundary
+just found raises it to filtration at least three. All later
+associated-graded groups are zero, so convergence with finite
+filtration makes the actual image zero.
+
+The bottom beta image is therefore zero, and finite duality makes
+$f$ itself nullhomotopic.
+\end{proof}
+
+\subsection{The averaged lift and the page on which it vanishes}
+
+\begin{proposition}[Averaging and annihilation]
+\label{top:annihilation}
+If $\gamma\in\pi_{qW-5}Y$ represents a class of
+$E_2^{5,W}(Y)$, then
+\begin{equation}
+\label{top:beta-annihilation}
+ \beta^K\gamma=0,\qquad K=p(p-2)+1.
+\end{equation}
+Only the binary two-sided unit identities for $\mu$ are needed.
+\end{proposition}
+
+\begin{proof}
+Lemma~\ref{top:small-stems} gives $\alpha\gamma=0$ in precisely
+the stem $q(W+1)-6$. Proposition~\ref{top:extended-power-relation}
+gives $\alpha\beta^p=0$ after applying the unit to $Y$.
+By \eqref{top:lift-criterion}, choose lifts
+\[
+ \widetilde\gamma:S^{qW-5}\longrightarrow J\wedge Y,
+ \qquad
+ \widetilde b:S^{pd}\longrightarrow J\wedge Y
+\]
+of $\gamma$ and $\eta\beta^p$, respectively.
+Fix any parenthesization of the $n$-fold iteration of $\mu$.
+Smash one copy of $\widetilde\gamma$ with $p-2$ copies of
+$\widetilde b$, collect the $J$ factors, and apply that chosen
+iteration to the $Y$ factors. This gives
+\[
+ L:S^{qW-5+(p-2)pd}\longrightarrow J^{\wedge n}\wedge Y.
+\]
+Its top projection is $\beta^{p(p-2)}\gamma$.
+To verify this without an associativity assumption, for any
+sphere scalar $a$ use the two unit homotopies to identify
+\[
+ \mu(\eta a,x)=a x,\qquad
+ \mu(x,\eta a)=(-1)^{|a||x|}a x.
+\]
+The scalar $\beta^p$ is even. In the chosen binary tree, every
+subtree without $\gamma$ is a scalar times the unit, and the
+unique subtree containing $\gamma$ is the product of its scalars
+times $\gamma$. This proves the asserted top value by induction
+on the tree.
+
+Now project to the symmetric retract after those $Y$ factors
+have already been multiplied:
+\[
+ L_T=(\operatorname{pr}\wedge1_Y)L:
+ S^{qW-5+(p-2)pd}\longrightarrow T\wedge Y.
+\]
+For every permutation $\sigma$ of the $J$ factors, naturality of
+the symmetry and the zero-dimensional top sphere give an equality
+of stable maps
+\[
+ \varepsilon^{\wedge n}\sigma=\varepsilon^{\wedge n}.
+\]
+Consequently
+\begin{equation}
+\label{top:averaged-top-map}
+ (\varepsilon_T\wedge1_Y)L_T
+ = (\varepsilon^{\wedge n}e\wedge1_Y)L
+ = (\varepsilon^{\wedge n}\wedge1_Y)L
+ =\beta^{p(p-2)}\gamma.
+\end{equation}
+No permutation of the multiplied $Y$ factors occurs in this
+identity.
+
+The actual null map $f$ of Proposition~\ref{top:symmetric-null-map}
+now gives the following homotopy-commutative diagram:
+\begin{equation}
+\label{top:null-map-diagram}
+\begin{array}{ccc}
+\Sigma^d(T\wedge Y)&\xrightarrow{\ f\wedge1_Y\ }&Y\wedge Y\\[2pt]
+\mathllap{\scriptstyle\Sigma^d(\varepsilon_T\wedge1_Y)\,}\big\downarrow
+&&\big\downarrow\mathrlap{\,\scriptstyle\mu}\\[2pt]
+\Sigma^dY&\xrightarrow{\ \beta\wedge1_Y\ }&Y.
+\end{array}
+\end{equation}
+Commutativity is the left unit identity for $\mu$ and the
+definition of $f$. Its top arrow is null, so composing the
+diagram with $\Sigma^dL_T$ and using
+\eqref{top:averaged-top-map} proves
+$\beta^{p(p-2)+1}\gamma=0$.
+\end{proof}
+
+\begin{proposition}[Disappearance before the last differential]
+\label{top:early-disappearance}
+Every class in $E_2^{qp+1,u_4+p}(Y)$ is zero on the
+Adams--Novikov page $E_{qp+1}$.
+\end{proposition}
+
+\begin{proof}
+Let $\theta\in E_2^{5,W}(Y)$. By Lemma~\ref{top:small-stems},
+it is a permanent cycle represented by a homotopy class $\gamma$.
+The sphere class $\beta$ is also permanent. The pagewise product
+$A^K\theta$ therefore supports no nonzero outgoing differential;
+if either factor has zero image on a page, the product is already
+zero. By Proposition~\ref{top:annihilation} its filtered homotopy
+product is zero. Product compatibility and finite convergence
+then imply that $A^K\theta$ is zero at $E_\infty$.
+
+If it is nonzero at $E_2$, it must consequently be hit by an
+incoming differential. Its cohomological filtration is
+\[
+ S=5+2K=2p^2-4p+7.
+\]
+An incoming differential of length $r$ has source filtration
+$S-r\geq0$, so $r\leq S$. This bounds the page of disappearance,
+rather than merely its eventual abutment. The exact comparisons
+are
+\begin{equation}
+\label{top:strict-page-bound}
+ qp+1-S=2p-6>0,\qquad N-K=p-3>0.
+\end{equation}
+Numerically,
+\[
+ K=1{,}016{,}064,\quad N=1{,}017{,}070,\quad
+ S=2{,}032{,}133<qp+1=2{,}034{,}145.
+\]
+Thus $A^K\theta$ is zero before the page of the prospective
+$d_{qp+1}(v_4)$. Multiplication by the further power $A^{N-K}$ is
+defined on every page by the sphere permanent cycle $\beta$;
+it shows that $A^N\theta$ is zero there as well.
+The surjection \eqref{top:An-surjection} accounts for every
+$E_2$ class of the target, proving the assertion.
+\end{proof}
+
+\subsection{The realizing cofiber}\label{top:main-construction}
+
+All possible targets of a differential from $v_4$ have now been
+eliminated on the required pages. We finish by producing the ordinary
+finite spectrum and identifying its homology as a comodule.
+
+\begin{proof}[Proof of Theorem~\ref{intro:main}]
+The only possible positive differentials from $v_4$ have target
+$(s,w)=(qk+1,u_4+k)$. Proposition~\ref{may:first-differential}
+rules out $k=1$. Lemma~\ref{may:target-bound} and
+Table~\ref{may:remaining-table} leave, for $k\geq2$, only
+$k=2,p-1,p$. Corollary~\ref{frob:vanishing} eliminates the
+first two target groups already at $E_2$, and
+Proposition~\ref{top:early-disappearance} makes the last target
+zero before its possible differential. No differential can hit
+filtration zero. Hence $v_4\in E_2^{0,u_4}(Y)$ is permanent.
+
+The edge homomorphism of the convergent Adams--Novikov spectral
+sequence supplies a map
+$a:S^{qu_4}\to Y$ whose $BP$-Hurewicz image is exactly $v_4$.
+Extend it by the existing binary multiplication to
+\[
+ \widetilde v_4:\Sigma^{qu_4}Y
+ \xrightarrow{\ a\wedge1_Y\ }Y\wedge Y
+ \xrightarrow{\ \mu\ }Y,
+\]
+and take its cofiber $X$.
+The right unit identity shows that the bottom generator is sent
+to $v_4$. Since $BP_*Y=R/I_4$ is cyclic as an $R$-module, this
+determines the entire induced map as multiplication by $v_4$.
+That element is a nonzerodivisor in
+$R/I_4=\mathbb F_p[v_4,v_5,\ldots]$. The homology exact sequence
+of the cofiber therefore gives the short exact sequence
+\[
+ 0\longrightarrow\Sigma^{qu_4}R/I_4
+ \xrightarrow{\ v_4\ }R/I_4
+ \longrightarrow BP_*X\longrightarrow0.
+\]
+The composite bottom map $S^0\to Y\to X$ induces the quotient
+map $R\to R/(I_4,v_4)$. It is a comodule map and is surjective,
+so it identifies the induced coaction with the canonical quotient
+coaction, and identifies its generator in degree zero.
+
+Finally, $X$ is the cofiber of a map between finite $p$-local
+spectra. It is finite $p$-local in the sense defined in the introduction;
+equivalently, the finite cell construction can be realized by a
+finite spectrum before localization by clearing prime-to-$p$
+denominators in its attaching maps, as in
+Lemma~\ref{low:finite-localization}. More explicitly, the mapping cone
+adds to the cells of $Y$ their translates by $qu_4+1$.
+Together with Equation~\eqref{low:cell-dimensions}, this gives a finite
+$p$-local CW model for $X$ with one cell in each dimension
+\[
+ \sum_{i\in S}(qu_i+1)=\sum_{i\in S}(2p^i-1),
+ \qquad S\subseteq\{0,1,2,3,4\}.
+\]
+In particular, this model has $32$ cells.
+\end{proof}
diff --git a/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/build/sections/appendix-resolution.tex b/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/build/sections/appendix-resolution.tex
new file mode 100644
index 0000000000000000000000000000000000000000..f2a793c06e92c1a294aeb99e8b1fc0cdc774414c
--- /dev/null
+++ b/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/build/sections/appendix-resolution.tex
@@ -0,0 +1,340 @@
+\section{The ordinary resolution and its filtered pairing}
+\label{app:resolution}
+
+We use a symmetric monoidal stable model of spectra with the usual
+pushout-product axiom and tensors over based spaces. Such a model is
+provided by topological symmetric spectra:
+\citet[Section~6.2, Theorems~6.3.8 and~6.4.1, and
+Corollary~6.4.2]{HoveyShipleySmith1998v2}; its comparison with ordinary
+spectra follows from Theorems~6.3.8 and~4.3.2 there, in the cited
+preprint version.
+All smash products
+are derived; when point-set maps are used, their inputs are cofibrant.
+Rectify the coherent associative structure on $BP$ as an $S$-algebra by
+\citet[Part~II, Corollary~3.6 and the associative convention comparison
+in Section~4]{ElmendorfKrizMandellMay1997}. The equivalence for associative
+algebras in \citet[Main Theorem and Theorem~5.1(ii)]{Schwede2001} then
+places it in this same category of topological symmetric spectra.
+Choose a cofibrant associative symmetric ring representative using
+\citet[Theorem~12.1(iv)--(v)]{MandellMaySchwedeShipley2001}; its
+underlying spectrum is then stably cofibrant. Smash with such a spectrum
+preserves stable equivalences by Proposition~12.3 there, so the following
+point-set smash powers compute derived smash powers.
+The comparison uses the positive stable model structure, whose weak
+equivalences agree with the ordinary stable ones; we use the ordinary
+stable structure here. With this strictly unital associative model, write
+\[
+ T^n(Z)=BP^{\wedge(n+1)}\wedge Z.
+\]
+The cofaces insert units and the codegeneracies multiply adjacent $BP$
+factors.  Associativity and the unit identities give a cosimplicial
+spectrum.  Write $K_n(Z)=\operatorname{Tot}_n T^\bullet(Z)$ for its
+derived ordinary partial totalization, set $K_{-1}(Z)=*$, and put
+$K(Z)=\operatorname{Tot}T^\bullet(Z)$.  Ordinary totalization here uses
+all ordinal maps, including codegeneracies.
+
+\subsection{Convergence and the edge}\label{app:convergence}
+
+\begin{proof}[Proof of Proposition~\ref{fw:anss}]
+We first identify the ordinary totalization tower exactly.  Let
+$\mathcal P_0([n])$ be the poset of nonempty subsets of
+$\{0,\ldots,n\}$.  The functor
+\[
+ u_n:\mathcal P_0([n])\longrightarrow\Delta_{\leq n},
+ \qquad S\longmapsto[|S|-1],
+\]
+sends an inclusion to the corresponding order-preserving injection.
+It is homotopy-initial.  To prove this, for $m\leq n$ identify
+$(u_n\downarrow[m])$ with the nonempty face poset of the simplicial
+complex $L_{n,m}$ whose vertices are $(i,j)$, $0\leq i\leq n$,
+$0\leq j\leq m$, and whose simplices have strictly increasing first
+coordinates and weakly increasing second coordinates.  Its nerve is
+the barycentric subdivision of $L_{n,m}$.
+
+The complex $L_{n,m}$ is contractible for $n\geq m$, by induction on
+$n$.  The case $n=m=0$ is a point.  For the inductive step start with
+the cone with vertex $(n,m)$ on the entire subcomplex $L_{n-1,m}$.
+Adjoin the cones with vertices $(n,j)$, $0\leq j<m$, on their bases
+$L_{n-1,j}$.  Two distinct vertices with first coordinate $n$ cannot
+belong to a simplex.  Each new cone therefore intersects the preceding
+union in exactly its base.  That base is contractible by induction,
+since $n-1\geq j$.  Attachment along this simplicial subcomplex is a
+homotopy pushout and preserves contractibility.  This proves the claim.
+
+Homotopy-initiality, applied to the cosimplicial unit resolution, now
+gives, naturally in $n$ and $Z$,
+\begin{equation}
+ K_n(Z)\simeq
+ \operatorname*{holim}_{\varnothing\ne S\subseteq\{0,\ldots,n\}}
+       BP^{\wedge|S|}\wedge Z.
+ \label{fw:finite-cube-comparison}
+\end{equation}
+This is the standard cubical description of the partial Amitsur
+totalizations; compare \citet[Propositions~2.11 and~2.14]{MathewNaumannNoel2017}.
+Indeed derived partial totalization is the homotopy limit over
+$\Delta_{\leq n}$: the restricted standard simplex is a Reedy-cofibrant
+resolution of the constant point diagram.  Adjoining the empty vertex
+$Z$ on the right of Equation~\eqref{fw:finite-cube-comparison} gives
+the $(n+1)$-fold cube formed by smashing copies of the unit
+$S_{(p)}\to BP$.  If $\mathcal F$ is its fiber, iterated fibers and
+exactness of smash identify its total fiber with
+$\mathcal F^{\wedge(n+1)}\wedge Z$.  Thus
+\begin{equation}
+ \operatorname{fib}(Z\longrightarrow K_n(Z))
+       \simeq\mathcal F^{\wedge(n+1)}\wedge Z.
+ \label{fw:tot-fiber-index}
+\end{equation}
+The comparisons commute with restriction from $n+1$ to $n$: on the
+fibers this is the map induced by the last factor
+$\mathcal F\to S_{(p)}$.  There is no suspension in
+Equation~\eqref{fw:tot-fiber-index}.
+
+The unit induces an isomorphism on $\pi_0$, and $\pi_1BP=0$, so
+$\mathcal F$ has no homotopy in degrees at most zero.  If $Z$ is
+$c$-connective, $\mathcal F^{\wedge s}\wedge Z$ is at least
+$(c+s)$-connective.  Equation~\eqref{fw:tot-fiber-index} therefore
+identifies the coaugmentation $Z\to K(Z)$ as an equivalence, and
+identifies the relative totalization tower as
+\begin{equation}
+ G^s(Z):=\operatorname{fib}(K(Z)\longrightarrow K_{s-1}(Z))
+        \simeq\mathcal F^{\wedge s}\wedge Z
+        \quad(s\geq0).
+ \label{fw:relative-tower}
+\end{equation}
+Its homotopy groups vanish in each fixed stem for sufficiently large
+$s$.  The exact couple consequently has the asserted abutment, no
+inverse-limit ambiguity, and finite filtration in each stem.
+
+For completeness, the successive layer is
+\begin{equation}
+ \operatorname{cofib}(G^{s+1}(Z)\longrightarrow G^s(Z))
+ \simeq BP\wedge\mathcal F^{\wedge s}\wedge Z
+ \simeq\Sigma^{-s}BP\wedge\overline{BP}^{\wedge s}\wedge Z,
+ \label{fw:tot-layer-index}
+\end{equation}
+where $\overline{BP}=\operatorname{cofib}(S_{(p)}\to BP)$.
+This is also $\operatorname{fib}(K_s(Z)\to K_{s-1}(Z))$, with the
+usual normalization and the displayed $s$ desuspensions.
+
+The iterated extended terms compute the cobar complex.  Indeed
+$BP\wedge BP$, as a $BP$-module, is a wedge of suspended free
+$BP$-modules with basis the polynomial monomials in the $t_i$.
+Iteration identifies their homotopy with the iterated tensors of
+$\Gamma$ over $R$.  There are no derived tensor corrections, whether
+or not $BP_*Z$ is flat over $R$.  Cofaces give the cobar differential;
+the codegeneracy splittings give its normalized form.  The exact-couple
+grading gives the displayed differential shift.  Finally the first
+map is $Z\to BP\wedge Z$.  Its image is exactly the permanent
+filtration-zero part by the finite-filtration convergence just proved.
+\end{proof}
+
+\subsection{The filtered binary pairing}\label{app:pairing}
+
+\begin{proof}[Proof of Proposition~\ref{fw:anss-pairing}]
+We construct maps
+\[
+ G^s(Z_1)\wedge G^t(Z_2)\longrightarrow G^{s+t}(Z_3)
+\]
+which induce the stated cobar pairing and recover the actual map $f$.
+
+\smallskip\noindent\emph{The box pairing.}
+We use Batanin's cosimplicial box product in the form of
+\citet[Definition~2.1, Proposition~2.3, and Remark~2.4]{McClureSmith2004};
+they attribute its Kan-extension description to Cordier and Porter.
+We construct the pairing before any fibrant replacement and then transport
+the entire pairing. For cosimplicial spectra define $A\mathbin\square B$
+as the left Kan extension of $(a,b)\mapsto A^a\wedge B^b$ along ordinal
+concatenation
+\[
+ \Delta\times\Delta\longrightarrow\Delta,
+ \qquad([a],[b])\longmapsto[a+b+1].
+\]
+Equivalently, in degree $n$ it is the zigzag pushout of the terms
+$A^a\wedge B^b$ with $a+b=n$, with neighboring terms identified by
+the maps
+\[
+ A^a\wedge B^b\xrightarrow{d^{a+1}\wedge1}
+ A^{a+1}\wedge B^b,
+ \qquad
+ A^a\wedge B^b\xrightarrow{1\wedge d^0}
+ A^a\wedge B^{b+1},
+ \qquad a+b=n-1.
+\]
+The same definition applies to cosimplicial based spaces.
+For the original unit resolutions, concatenating the two $BP$ blocks
+and applying $f$ once gives maps
+\[
+ T^a(Z_1)\wedge T^b(Z_2)\longrightarrow T^{a+b+1}(Z_3).
+\]
+They are natural in both ordinals: cofaces and codegeneracies within a
+block insert a unit or multiply adjacent factors within that block.
+Kan-extension adjunction therefore gives an actual cosimplicial map
+\begin{equation}
+ m:T(Z_1)\mathbin\square T(Z_2)\longrightarrow T(Z_3).
+ \label{fw:original-box-pairing}
+\end{equation}
+On the degree-$n$ summand $a+b=n$ in the pushout description, this map
+multiplies the last factor of the first $BP$ block with the first factor
+of the second block.  This is the single overlapping slot.  The two
+descriptions agree by applying the codegeneracy that identifies the
+two junction vertices of $[a+b+1]$.
+
+\smallskip\noindent\emph{Deriving the pairing.}
+We record the model-category check that permits us to derive this
+construction.  The box product is a Quillen bifunctor for the Reedy
+model structures.  Let $h_a([n])=\Delta([a],[n])_+$, and let
+$\partial h_a$ be the subdiagram of noninjective maps.  These are the
+representables and boundaries defining the generating Reedy cells.
+Indeed, the boundary of the covariant representable consists of maps
+factoring through a nonidentity inverse map out of $[a]$, hence of
+noninjective maps by the unique surjection--injection factorization in
+$\Delta$. This representable boundary is distinct from the latching
+subobject in target degree $n$, which consists of maps factoring through
+a proper injection into $[n]$.
+Kan extension gives $h_a\square h_b=h_{a+b+1}$.  In the relative box
+boundary
+\[
+ \frac{h_{a+b+1}}
+ {\operatorname{im}(\partial h_a\square h_b)
+       \cup\operatorname{im}(h_a\square\partial h_b)},
+\]
+the remaining maps are exactly the monotone maps injective on each of
+the two blocks.  Such a map is either globally injective or identifies
+only the two junction vertices. A map that is not surjective onto its
+target lies in that target degree's latching subobject. The only
+remaining surjections are the junction identification onto $[a+b]$
+and the identity of $[a+b+1]$. Thus the junction-identifying and globally
+injective non-latching cells occur in degrees $a+b$ and $a+b+1$,
+respectively.  Attach the junction cell first;
+every further collapse of it is already in the boundary, since it
+collapses two vertices within one block.  Attach the globally injective
+cell second: its junction codegeneracy lands in the first cell, and
+its other codegeneracies land in the boundary. This gives a two-cell
+Reedy filtration of the relative box boundary. For two generating
+Reedy cofibrations, each of these attachments is tensored with the
+ordinary spectral pushout-product of their underlying generating
+cofibrations. The spectral pushout-product axiom therefore makes both
+attachments cofibrations, and makes them trivial cofibrations if either
+input is trivial. This proves the Reedy pushout-product axiom, including
+the acyclic case, for generating cofibrations.  Closure under pushouts,
+composition, and retracts proves the assertion for all cofibrations.
+
+Choose Reedy-cofibrant replacements $A_i\to T(Z_i)$, $i=1,2$, that
+are trivial fibrations, and trivial cofibrations $A_i\to R_i$ with
+$R_i$ Reedy fibrant.  For the target choose a Reedy fibrant replacement
+$T(Z_3)\to R_3$ directly.  Equation~\eqref{fw:original-box-pairing}
+gives $A_1\square A_2\to R_3$.  By the preceding Quillen check,
+$A_1\square A_2\to R_1\square R_2$ is a trivial Reedy cofibration.
+The lifting axiom extends the map to
+\begin{equation}
+ m_R:R_1\square R_2\longrightarrow R_3.
+ \label{fw:derived-box-pairing}
+\end{equation}
+The space of such extensions is contractible up to the usual relative
+homotopy.  This transports the original pairing; it places no condition
+on the literal $BP$ slots of the fibrant replacements.
+All resulting tower and spectral-sequence pairings are transported along
+the equivalences with the original diagrams.  In particular, when some
+$Z_i$ coincide, their source and target replacements need not be identical.
+
+\smallskip\noindent\emph{The filtered diagonal.}
+We use the simplex subdivision map given in
+\citet[Lemma~3.6]{McClureSmith2004}, credited to Grayson in their
+Remark~3.7. We record its formula to verify the relative filtration. For
+$t=(t_0,\ldots,t_n)\in\Delta^n$, put
+\[
+ I_i=\left[\sum_{j<i}t_j,\sum_{j\leq i}t_j\right],\qquad
+ x_i=2\,\operatorname{length}(I_i\cap[0,1/2]),\qquad
+ y_i=2t_i-x_i.
+\]
+For a cut index $k$, $x$ is supported in $0,\ldots,k$ and $y$ in
+$k,\ldots,n$.  The cut region is a copy of
+$\Delta^k\times\Delta^{n-k}$, with inverse $t=(x+y)/2$.
+Its boundary seams identify the last face of the first factor with
+the first face of the second factor, precisely as in the box pushout.
+Thus the cut defines
+\[
+ \delta:\Delta^\bullet_+\longrightarrow
+              \Delta^\bullet_+\square\Delta^\bullet_+.
+\]
+It is natural under all monotone maps: they aggregate consecutive
+intervals, and aggregation commutes with the cut.  In particular this
+is a map of ordinary cosimplicial objects, including codegeneracies.
+
+Define the relative simplex object
+\[
+ P_s^\bullet=\Delta^\bullet_+/\operatorname{sk}_{s-1}
+                      \Delta^\bullet_+\qquad(s\geq0),
+\]
+where the skeleton for $s=0$ is the basepoint.  The simplex, its
+skeleta, and these quotients are Reedy cofibrant: their latching maps
+are the corresponding boundary inclusions or identities.  Consequently
+\begin{equation}
+ G^s(Z_i)=\operatorname{Map}_\Delta(P_s,R_i)
+ \label{fw:relative-end-model}
+\end{equation}
+is a derived relative end and the actual fiber of the fibration
+$\operatorname{Tot}R_i\to\operatorname{Tot}_{s-1}R_i$.
+The degree-$n$ box summands have dimensions $k$ and $n-k$.
+If $n<s+t$, at least one is below its respective skeleton cutoff.
+Naturality applies the same observation to every face of a larger
+simplex, including faces with missing ambient vertices.  Hence the
+cut descends to maps
+\begin{equation}
+ \delta_{s,t}:P_{s+t}\longrightarrow P_s\square P_t.
+ \label{fw:relative-cut}
+\end{equation}
+Enriched evaluation, followed by $m_R$ and precomposition with
+$\delta_{s,t}$, gives the actual relative pairing
+\begin{align*}
+ \operatorname{Map}_\Delta(P_s,R_1)\wedge
+ \operatorname{Map}_\Delta(P_t,R_2)
+ &\longrightarrow
+ \operatorname{Map}_\Delta(P_s\square P_t,R_1\square R_2)\\
+ &\longrightarrow
+ \operatorname{Map}_\Delta(P_{s+t},R_3).
+\end{align*}
+It commutes with the tower inclusions in both inputs.  By
+Equation~\eqref{fw:relative-tower}, these are the exact relative fibers
+of the unit-resolution filtration, with filtration indices $s,t,s+t$.
+
+\smallskip\noindent\emph{The cobar and homotopy pairings.}
+Put $L_s=\operatorname{cofib}(G^{s+1}\to G^s)$.  The relative pairing
+induces $L_s(Z_1)\wedge L_t(Z_2)\to L_{s+t}(Z_3)$: raising either
+input filtration by one raises the output filtration by one.
+The boundary formula for the product of relative cells is
+$\partial(a\cdot b)=\partial a\cdot b+(-1)^{|a|}a\cdot\partial b$.
+It gives a pairing of the exact couples and the Leibniz rule on every
+derived page.  On the layer of dimension $s+t$, the only contributing
+cut cell has front and back dimensions $s,t$.  The inverse
+$t=(x+y)/2$ gives its usual oriented front-face/back-face cup map,
+with coefficient one.  The normalized cobar identification therefore
+gives exactly the asserted $E_2$ cup pairing.
+
+We finally verify the homotopy target without assuming that a replacement
+preserves units.  For cofibrant $Z_i$, the diagram
+$\Delta^\bullet_+\wedge Z_i$ is Reedy cofibrant and maps levelwise
+equivalently to the constant diagram $Z_i$.  Its original all-unit map
+to $T(Z_i)$ lifts to a map
+$\alpha_i:\Delta^\bullet_+\wedge Z_i\to A_i$ along the trivial
+fibration $A_i\to T(Z_i)$.  The composite of $\alpha_i$ with
+$A_i\to R_i$ represents the coaugmentation.  The box product of the
+$\alpha_i$, precomposed with $\delta$, lands in $A_1\square A_2$.
+Since $m_R$ extends the
+original map on that object, the composite is exactly the target
+coaugmentation composed with $f$, transported through $T(Z_3)\to R_3$.
+On totalizations the pairing therefore satisfies
+\[
+ m_{\mathrm{Tot}}\circ(e_1\wedge e_2)=e_3\circ f.
+\]
+The coaugmentations $e_i:Z_i\to K(Z_i)$ are the equivalences proved
+above.  Thus the limit pairing is the actual homotopy map $f$, and
+the $E_\infty$ pairing is its associated-graded pairing for the
+filtration by the images of $\pi_*G^s\to\pi_*K$.
+
+For the final assertion, the coefficient pairing is $R$-bilinear and
+sends the two units to the unit.  Every element of $R/I$ is a scalar
+multiple of that unit.  The pairing is consequently
+$(r\bmod I,r'\bmod I)\mapsto rr'\bmod I$.  This uses neither flatness
+of the quotient nor associativity of the target map.
+\end{proof}
diff --git a/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/paper.pdf b/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..e5bc93bcfb05f04fc868db97bd0325a8cb25fa9c
--- /dev/null
+++ b/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:a9cf0165a9ea294995b42fac2290fa90a519101e446be81331f7c31726c6d635
+size 579012
diff --git a/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/README.md b/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..0667d23bcc5a75e4f4191f14fb2de8f464548748
--- /dev/null
+++ b/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/README.md
@@ -0,0 +1,18 @@
+# [A finite-entropy separation of microstates and nonmicrostates free entropy](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 25, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026,
+  author = {{OpenAI}},
+  title = {{A finite-entropy separation of microstates and nonmicrostates free entropy}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/paper.pdf}{OAI:A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/build/paper.tex b/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/build/paper.tex
new file mode 100644
index 0000000000000000000000000000000000000000..dd6ba00ef19f6941b18d3aa15467d21f30ccf8b9
--- /dev/null
+++ b/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/build/paper.tex
@@ -0,0 +1,46 @@
+\documentclass[11pt]{article}
+\pdfinfoomitdate=1
+\pdftrailerid{}
+\pdfsuppressptexinfo=-1
+\usepackage[margin=1in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,microtype}
+\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
+\hypersetup{pdftitle={A finite-entropy separation of microstates and nonmicrostates free entropy},pdfauthor={OpenAI},pdfstartview={}}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{remark}[theorem]{Remark}
+\newcommand{\R}{\mathbb R}
+\newcommand{\C}{\mathbb C}
+\newcommand{\N}{\mathbb N}
+\newcommand{\E}{\mathbb E}
+\newcommand{\tr}{\operatorname{tr}}
+\newcommand{\Tr}{\operatorname{Tr}}
+\newcommand{\Vol}{\operatorname{Vol}}
+\newcommand{\rank}{\operatorname{rank}}
+\newcommand{\KL}{\operatorname{KL}}
+\newcommand{\NC}{\operatorname{NC}}
+\newcommand{\D}{\mathcal D}
+\title{A finite-entropy separation of microstates and\\nonmicrostates free entropy}
+\author{OpenAI}
+\date{September 25, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We answer the finite-entropy equality question for microstates and
+nonmicrostates free entropy negatively. We construct a bounded self-adjoint
+tuple $X$ in a von Neumann algebra with faithful normal tracial state such that
+$-\infty<\chi(X)\leq\chi^*(X)-\tfrac12<\infty$.
+Here $\chi$ is the original microstates entropy with an operator-norm
+cutoff and a limsup over matrix sizes. The counterexample uses a large but
+fixed number of variables.
+\end{abstract}
+
+\input{sections/01-introduction}
+\input{sections/02-conventions}
+\input{sections/03-fisher}
+\input{sections/04-signal}
+\input{sections/05-flow}
+\input{sections/06-blocks}
+\input{references}
+\end{document}
diff --git a/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/build/references.tex b/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/build/references.tex
new file mode 100644
index 0000000000000000000000000000000000000000..6553bac2cc2ecf091654f8381c5a31875779b9ad
--- /dev/null
+++ b/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/build/references.tex
@@ -0,0 +1,94 @@
+\clearpage
+\begin{thebibliography}{99}
+\raggedright
+\bibitem{BCG}
+P.~Biane, M.~Capitaine, and A.~Guionnet,
+\emph{Large deviation bounds for matrix Brownian motion},
+Invent. Math. \textbf{152} (2003), no.~2, 433--459.
+\href{https://doi.org/10.1007/s00222-002-0281-4}{doi:10.1007/s00222-002-0281-4}.
+
+\bibitem{Dab}
+Y.~Dabrowski,
+\emph{A Laplace principle for Hermitian Brownian motion and free entropy I:
+the convex functional case},
+\href{https://arxiv.org/abs/1604.06420v2}{arXiv:1604.06420v2},
+October 31, 2017.
+
+\bibitem{Ge}
+L.~Ge,
+\emph{Prime factors},
+Proc. Natl. Acad. Sci. USA \textbf{93} (1996), no.~23, 12762--12763.
+\href{https://doi.org/10.1073/pnas.93.23.12762}{doi:10.1073/pnas.93.23.12762}.
+
+\bibitem{Gsurvey}
+A.~Guionnet,
+\emph{Large deviations and stochastic calculus for large random matrices},
+Probab. Surv. \textbf{1} (2004), 72--172.
+\href{https://doi.org/10.1214/154957804100000033}{doi:10.1214/154957804100000033}.
+
+\bibitem{Jconv}
+D.~Jekel,
+\emph{An elementary approach to free entropy theory for convex potentials},
+Anal. PDE \textbf{13} (2020), no.~8, 2289--2374.
+\href{https://doi.org/10.2140/apde.2020.13.2289}{doi:10.2140/apde.2020.13.2289}.
+
+\bibitem{JP}
+D.~Jekel and J.~Pi,
+\emph{An elementary proof of the inequality $\chi\leq\chi^*$ for
+conditional free entropy},
+Doc. Math. \textbf{29} (2024), no.~5, 1085--1124.
+\href{https://doi.org/10.4171/DM/969}{doi:10.4171/DM/969}.
+
+\bibitem{NS}
+A.~Nica and R.~Speicher,
+\emph{Lectures on the Combinatorics of Free Probability},
+London Mathematical Society Lecture Note Series, vol.~335,
+Cambridge University Press, 2006.
+\href{https://doi.org/10.1017/CBO9780511735127}{doi:10.1017/CBO9780511735127}.
+
+\bibitem{FreeGroupFactors}
+OpenAI,
+\emph{An isomorphism of the free group factors},
+OpenAI Math Release preprint
+\href{https://github.com/openai/math/blob/main/preprints/An-isomorphism-of-the-free-group-factors-September-23-2026/An-isomorphism-of-the-free-group-factors-September-23-2026.pdf}{OAI:An-isomorphism-of-the-free-group-factors-September-23-2026},
+2026.
+
+\bibitem{Speicher}
+R.~Speicher,
+\emph{Multiplicative functions on the lattice of non-crossing partitions
+and free convolution},
+Math. Ann. \textbf{298} (1994), 611--628.
+\href{https://doi.org/10.1007/BF01459754}{doi:10.1007/BF01459754}.
+
+\bibitem{Szarek}
+S.~J.~Szarek,
+\emph{Metric entropy of homogeneous spaces},
+Banach Center Publ. \textbf{43} (1998), 395--410.
+\href{https://arxiv.org/abs/math/9701213}{arXiv:math/9701213}.
+
+\bibitem{Vgue}
+D.~Voiculescu,
+\emph{Limit laws for random matrices and free products},
+Invent. Math. \textbf{104} (1991), 201--220.
+\href{https://doi.org/10.1007/BF01245072}{doi:10.1007/BF01245072}.
+
+\bibitem{Vchi}
+D.~Voiculescu,
+\emph{The analogues of entropy and of Fisher's information measure in
+free probability theory. II},
+Invent. Math. \textbf{118} (1994), 411--440.
+\href{https://doi.org/10.1007/BF01231539}{doi:10.1007/BF01231539}.
+
+\bibitem{Vstar}
+D.~Voiculescu,
+\emph{The analogues of entropy and of Fisher's information measure in
+free probability theory. V. Noncommutative Hilbert transforms},
+Invent. Math. \textbf{132} (1998), 189--227.
+\href{https://doi.org/10.1007/s002220050222}{doi:10.1007/s002220050222}.
+
+\bibitem{Vsurvey}
+D.~Voiculescu,
+\emph{Free entropy},
+Bull. London Math. Soc. \textbf{34} (2002), no.~3, 257--278.
+\href{https://doi.org/10.1112/S0024609301008992}{doi:10.1112/S0024609301008992}.
+\end{thebibliography}
diff --git a/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/build/sections/01-introduction.tex b/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/build/sections/01-introduction.tex
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@@ -0,0 +1,117 @@
+\section{Introduction}
+
+Free entropy connects the joint moments of noncommuting variables with
+the geometry of their matrix approximations. For a bounded self-adjoint
+tuple $X=(X_1,\ldots,X_n)$ in a tracial von Neumann algebra, Voiculescu's
+microstates free entropy $\chi(X)$ measures the asymptotic volumes of
+matrix tuples whose normalized trace moments approximate those of $X$
+\cite{Vchi}. His nonmicrostates free entropy $\chi^*(X)$ instead uses
+conjugate variables, the free analogues of classical scores, and integrates
+free Fisher information along addition of freely independent semicircular
+noise \cite{Vstar}. We give both definitions, with their normalizations,
+in Section~2.
+
+The two quantities agree for a single self-adjoint variable, and their
+agreement for general tuples was one of the questions in Voiculescu's
+unification problem \cite[Sections~3.5 and~3.8]{Vsurvey}.
+The finite-entropy formulation asks whether
+\[
+ \chi(X)>-\infty\quad\Longrightarrow\quad\chi(X)=\chi^*(X).
+\]
+Guionnet stated this formulation explicitly
+\cite[Conjecture~7.1]{Gsurvey}, using the equivalent Gaussian-relative
+normalization. The finiteness hypothesis is substantial: as discussed
+by Jekel and Pi \cite[Introduction]{JP}, there are nonapproximable laws
+with $\chi=-\infty<\chi^*$. Here we construct a strict
+gap with both entropies finite.
+
+Biane, Capitaine, and Guionnet proved the general comparison
+$\chi\leq\chi^*$ through large-deviation bounds for matrix Brownian
+motion \cite{BCG}. Jekel and Pi obtained an elementary proof using
+classical entropy and Fisher information, and extended the comparison
+to conditional free entropy \cite{JP}. Equality is known in significant
+regular convex settings. Dabrowski proved it for a class of strictly
+convex potentials with Lipschitz derivatives and noncommutative polynomial
+approximations \cite[Theorem~A]{Dab}; Jekel established
+entropy and Fisher-information convergence for convex matrix potentials
+with appropriate regularity, concentration, and trace-polynomial
+approximation \cite[Theorem~7.1]{Jconv}. These results also identify
+conditions under which the upper and lower matrix-size limits agree.
+Our construction shows that finite entropy alone does not imply equality.
+
+\begin{theorem}\label{thm:main}
+There exist an integer $n\geq2$ and a bounded self-adjoint tuple
+$X=(X_1,\ldots,X_n)$ in a von Neumann algebra with faithful normal
+tracial state such that
+\[
+ -\infty<\chi(X)\leq\chi^*(X)-\tfrac12<\infty.
+\]
+Here $\chi$ is the original microstates entropy using an operator-norm
+cutoff and a limsup over matrix sizes.
+\end{theorem}
+
+Thus Theorem~\ref{thm:main} answers the finite-microstates-entropy
+equality question negatively. The construction uses a large but fixed number of
+variables. It does not address equality for every smaller prescribed
+number of variables.
+
+The proof compares two noise scales. Start with
+$C=(B_1,\ldots,B_p,Y)$, where $B_1,\ldots,B_p$ are free variance-one
+semicircular variables and $Y$, in a separate tensor factor, is uniformly
+distributed on $p$ equally spaced points in $[-1,1]$. Thus $Y$ commutes
+with every $B_i$. Let $S$ be a standard semicircular $(p+1)$-tuple free
+from $C$. A small perturbation $X=C+\sqrt h S$ has finite microstates
+entropy, while its last coordinate still determines spectral blocks
+containing much of the energy of the first $p$ coordinates.
+
+After an additional noise time $t$, we obtain two different estimates.
+The analytic estimate uses Speicher's noncrossing cumulant calculus
+\cite{Speicher,NS} and orthogonality of semicircular Wick polynomials.
+Proposition~\ref{prop:fisher} bounds the Fisher deficit by squared
+$\ell^2$ norms of the higher cumulant arrays. The sparsity of the signal's
+cumulants then makes the nonmicrostates entropy of
+$C+\sqrt{h+t}\,S$ close to the entropy of the semicircular family with
+the same coordinate variances.
+The matrix estimate shows that uniform laws on a selected sequence of
+increasingly accurate cutoff microstate sets of $X$, evolved by independent
+GUE noise of variance $t$, retain detectable simultaneous block structure.
+This event has probability bounded away from zero uniformly along
+the selected matrix sizes for those ensembles, and probability at most
+$e^{-c d^2}$ for size-$d$ independent GUE matrices with the coordinate
+variances of the limiting law, for a fixed $c>0$.
+Relative-entropy data processing therefore gives a deficit bounded away
+from zero in normalized classical entropy. The use of commutation constraints and projection
+geometry to control microstate volumes has antecedents in Ge's argument
+for prime factors \cite[Lemma~2 and its proof outline]{Ge}.
+The unitary-net estimate below uses the elementary volumetric covering
+method discussed by Szarek \cite{Szarek}.
+The event estimated here concerns the persistence of block energy after
+positive Gaussian time. The analytic entropy deficit becomes small when
+$t/\sqrt p\to\infty$, while the Gaussian block estimate still beats the
+unitary covering cost when $t\log p/p\to0$. The choice $t=p^{3/4}$ satisfies
+both conditions as $p\to\infty$. We first choose one sufficiently large
+finite $p$ and then $h$; all matrix limits are taken with these parameters fixed.
+
+The transfer between the scales uses the classical-to-free Fisher
+comparison of Jekel and Pi \cite[Proposition~6.1]{JP}.
+Lemma~\ref{lem:flow} proves the needed finite-time statement while retaining
+the entropies of this same sequence of terminal laws. The limsup of those
+normalized entropies need not equal the microstates entropy of the limiting
+law. Keeping this sequence in the comparison permits the positive-time
+structural deficit to produce the strict gap for $X$.
+
+The companion paper \emph{An isomorphism of the free group factors}
+\cite[Corollary~7.1]{FreeGroupFactors} proves that each of the four free
+entropy dimensions $\delta,\delta_0,\delta^*,\delta^\star$ attains every
+integer value $n\geq2$ on suitable finite self-adjoint $W^*$-generating
+tuples of $L(\mathbb F_2)$. That statement varies the generating tuple of
+a fixed algebra; Theorem~\ref{thm:main} compares $\chi$ and $\chi^*$ on one
+tuple, with both values finite.
+
+Section~2 fixes the conventions and establishes the matrix convergence
+facts, starting from Gaussian asymptotic freeness \cite{Vgue,NS}.
+Section~3 proves the cumulant/Fisher estimate. Section~4 constructs the
+signal, fixes the two noise scales, and proves $\chi(X)>-\infty$.
+Section~5 proves the specified-ensemble entropy-production
+lemma. Section~6 establishes the block event and combines the estimates
+to prove Theorem~\ref{thm:main}.
diff --git a/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/build/sections/02-conventions.tex b/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/build/sections/02-conventions.tex
new file mode 100644
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--- /dev/null
+++ b/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/build/sections/02-conventions.tex
@@ -0,0 +1,177 @@
+\section{Conventions and preliminary facts}\label{sec:conventions}
+
+We fix the entropy normalizations and matrix convergence facts used at
+both noise scales. The matrix inputs may be random tuples supported on
+bounded microstate sets.
+
+All noncommutative variables below belong to tracial von Neumann algebras
+with faithful normal tracial state $\tau$. Write $\|a\|_2^2=\tau(a^*a)$.
+A standard semicircular family has variance one in each coordinate and
+is free. Every noise family is free from the tuple it perturbs.
+
+For a bounded self-adjoint $n$-tuple $X$, let
+$\Gamma_R(X;m,d,\varepsilon)$ consist of self-adjoint $d\times d$ matrix
+tuples of coordinate operator norms at most $R$ whose normalized traces
+of all words of length at most $m$ differ from those of $X$ by less than
+$\varepsilon$. With volume taken in the real Euclidean space defined by
+unnormalized Hilbert--Schmidt inner product, set
+\[
+ \chi_R(X)=\inf_{m,\varepsilon}\limsup_{d\to\infty}
+ \left[d^{-2}\log\Vol\Gamma_R(X;m,d,\varepsilon)
+       +\frac n2\log d\right],\qquad
+ \chi(X)=\sup_R\chi_R(X).
+\]
+For a tuple $Z$, its conjugate variables $\xi_i\in L^2(W^*(Z),\tau)$
+satisfy
+\[
+ \tau(\xi_i P(Z))=(\tau\otimes\tau)((\partial_i P)(Z)),
+\]
+where $P$ is a noncommutative polynomial in formal indeterminates
+$x_1,\ldots,x_n$, and the difference quotient satisfies
+$\partial_i x_j=\delta_{ij}1\otimes1$ and obeys the
+Leibniz rule. Put $\Phi^*(Z)=\sum_i\|\xi_i\|_2^2$, or infinity if they
+do not exist. We will use the following form of the conjugate-variable
+definition.
+
+\begin{lemma}[Polynomial variational formula]\label{lem:variational}
+If a bounded self-adjoint tuple $Z$ has conjugate variables, then
+\begin{equation}\label{eq:fishervariational}
+ \Phi^*(Z)=\sup_{P_1=P_1^*,\ldots,P_n=P_n^*}
+ \sum_{i=1}^n\left[
+  2(\tau\otimes\tau)((\partial_iP_i)(Z))
+  -\tau(P_i(Z)^2)\right],
+\end{equation}
+where the supremum is over self-adjoint noncommutative polynomials.
+It suffices to take $P_i=(Q_i+Q_i^*)/2$ with coefficients of $Q_i$ in
+$\mathbb Q+i\mathbb Q$. Consequently, along a family with continuous
+moments and existing conjugate variables, free Fisher information is
+a measurable function of the parameter.
+\end{lemma}
+
+\begin{proof}
+The identity
+$\partial_i(P^*)=\operatorname{flip}((\partial_iP)^*)$
+and the symmetry of $\tau\otimes\tau$ show that $\xi_i^*$ is a
+conjugate variable whenever $\xi_i$ is. Uniqueness follows from polynomial
+$L^2$ density, so $\xi_i$ is self-adjoint. The expression under the
+supremum in \eqref{eq:fishervariational} equals
+\[
+ \sum_i\big(2\tau(\xi_iP_i(Z))-\|P_i(Z)\|_2^2\big)
+ =\sum_i\big(\|\xi_i\|_2^2-\|\xi_i-P_i(Z)\|_2^2\big).
+\]
+For completeness, the closed span of polynomial vectors is invariant
+under each bounded self-adjoint left multiplication operator $Z_j$.
+Its orthogonal projection therefore commutes with these operators and
+with $W^*(Z)$; since it fixes $1$, it fixes every bounded algebra vector.
+Those vectors are dense in $L^2$, proving polynomial density.
+Symmetrizing the approximants proves the claimed supremum. Approximating
+their finitely many coefficients proves the countable version. Each
+functional in that countable supremum is continuous in the moments,
+which proves measurability.
+\end{proof}
+
+Our nonmicrostates normalization is
+\begin{equation}\label{eq:chistar}
+ \chi^*(Z)=\frac n2\log(2\pi e)
+ +\frac12\int_0^\infty
+ \left[\frac n{1+s}-\Phi^*(Z+\sqrt s S)\right]ds.
+\end{equation}
+
+Write $\tr_d=d^{-1}\Tr$ and
+$\|D\|_{2,d}^2=\sum_i\tr_d(D_i^*D_i)$ for matrix tuples.
+A standard GUE tuple $G^{(d)}$ consists of independent centered Gaussian
+matrices whose real Hilbert--Schmidt coordinates have covariance $1/d$.
+For a random matrix tuple with density $f$, its normalized classical
+entropy is
+\[
+ H_d=-\frac1{d^2}\int f\log f+\frac n2\log d.
+\]
+Thus independent GUE coordinates of variances $w_i>0$ have normalized
+entropy $\frac12\sum_i\log(2\pi e w_i)$.
+
+Voiculescu's asymptotic-freeness theorem introduced the random-matrix
+input used here \cite{Vgue}. We use its arbitrary deterministic-tuple
+form in \cite[Theorem~22.35]{NS}: if deterministic matrix tuples
+converge in normalized trace moments, adjoining independent GUE matrices
+gives convergence of expected normalized traces to those of freely
+adjoined standard semicirculars. The following lemma supplies the
+additional probability and expectation statements needed in this paper.
+
+\begin{lemma}[Bounded matrix inputs]\label{lem:matrix-convergence}
+Let $K_d$ be sets of self-adjoint matrix tuples with a common coordinate
+operator-norm bound. Suppose their normalized trace moments converge
+uniformly over $K_d$ to the moments of a bounded tuple $T$. Let $A^{(d)}$
+be any random tuple supported on $K_d$, independent of a fixed finite
+standard GUE tuple $G^{(d)}$. Then every normalized polynomial trace in
+$(A^{(d)},G^{(d)})$ converges in probability to its value at $(T,S)$,
+where $S$ is a standard semicircular family free from $T$. Products of
+finitely many such traces converge in expectation as well. The dimensions
+may range over any sequence tending to infinity.
+\end{lemma}
+
+\begin{proof}
+We first record bounds independent of the deterministic input. A
+$1/4$-net of the unit sphere of $\C^d$, viewed in real dimension $2d$,
+has cardinality at most $9^{2d}$. For a Hermitian matrix $G$,
+$\|G\|\leq2\max_v|v^*Gv|$ over this net. For standard GUE, each
+$v^*Gv$ is a real centered Gaussian of variance $1/d$, since
+$\Tr((vv^*)^2)=1$. Hence
+\begin{equation}\label{eq:gue-norm-tail}
+ \mathbb P(\|G\|>L)
+ \leq 2\exp\big(d(2\log9-L^2/8)\big).
+\end{equation}
+In particular, the coordinate norms of any fixed finite GUE tuple are
+at most $8$ with probability tending to one. For sufficiently large $L$
+the right side of \eqref{eq:gue-norm-tail} is at most
+$2\exp(-dL^2/16)$. Integrating that tail shows that every fixed moment
+of the operator norm is bounded uniformly in $d$.
+
+Now take a deterministic input $a$ with coordinate norms at most $R$,
+and put $F=\tr_d P(a,G)$ for a fixed polynomial $P$. Let $(E_\alpha)$
+be a Hermitian basis orthonormal for the unnormalized Hilbert--Schmidt
+inner product, and write $G_j=\sum_\alpha g_{j\alpha}E_\alpha$.
+The real coordinates $g_{j\alpha}$ are independent Gaussians with
+variance $1/d$. Cyclicity of trace gives
+\[
+ \partial_{g_{j\alpha}}F=d^{-1}\Tr(E_\alpha B_j),\qquad
+ \sum_{j,\alpha}|\partial_{g_{j\alpha}}F|^2
+ =d^{-1}\sum_j\tr_d(B_j^*B_j),
+\]
+where $B_j$ is the sum of the cyclically ordered products obtained by
+deleting each occurrence of the $j$th GUE letter in $P$. The basis also
+complexifies to an orthonormal basis of all matrices, so the second
+identity applies to complex polynomials.
+
+For a polynomial in standard real Gaussian coordinates, expansion in
+orthonormal Hermite polynomials shows that its variance is at most the
+expected squared gradient: differentiation weights each nonconstant
+coefficient's squared modulus by its positive total degree. Rescaling
+the coordinates to variance $1/d$, and applying this to real and
+imaginary parts, gives
+\begin{equation}\label{eq:gue-trace-variance}
+ \E|F-\E F|^2
+ \leq d^{-2}\sum_j\E\tr_d(B_j^*B_j)
+ =O_{P,R}(d^{-2}).
+\end{equation}
+The final bound follows from the norm-moment estimate, since each $B_j$
+has fixed degree and the deterministic coordinates are uniformly bounded.
+Thus the cited expected-trace theorem yields convergence in probability
+for each deterministic moment-convergent input sequence.
+
+This convergence is uniform over the sets $K_d$. Otherwise, for some
+polynomial and positive error threshold, one could choose a subsequence
+and points $a_d\in K_d$ whose error probabilities stay bounded away
+from zero. These points form a bounded deterministic moment-convergent
+sequence, contradicting the preceding conclusion. Conditioning on
+$A^{(d)}$ proves convergence in probability for the random input.
+
+Finally, any fixed product $Q$ of normalized polynomial traces satisfies
+\[
+ |Q(A^{(d)},G^{(d)})|
+ \leq C(1+\max_j\|G_j^{(d)}\|)^D
+\]
+for constants $C,D$ depending only on the polynomials and the common
+cutoff. The same estimate squared has uniformly bounded expectation.
+The products are therefore uniformly integrable, so their convergence
+in probability implies convergence of their expectations.
+\end{proof}
diff --git a/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/build/sections/03-fisher.tex b/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/build/sections/03-fisher.tex
new file mode 100644
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+++ b/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/build/sections/03-fisher.tex
@@ -0,0 +1,172 @@
+\section{A cumulant bound for the Fisher deficit}\label{sec:fisher}
+
+We first estimate how far the Fisher information of a semicircularly
+perturbed tuple is from that of a semicircular family with the same
+covariance. The estimate will use the squared sizes of the higher
+cumulant arrays, which will allow us to exploit the sparse cumulants
+of the signal in the next section.
+
+For a centered bounded self-adjoint $n$-tuple $C$, write $\kappa_m(C)$
+for its order-$m$ free-cumulant array, with
+\[
+ \|\kappa_m(C)\|_2^2
+ =\sum_{i_1,\ldots,i_m=1}^n
+   |\kappa_m(C_{i_1},\ldots,C_{i_m})|^2.
+\]
+We use the moment--cumulant formula and its M\"obius inversion on
+noncrossing partitions, the vanishing of mixed cumulants between free
+algebras, and the fact that a standard semicircular family has only
+second cumulants, given by its covariance
+\cite[Proposition~11.4, Theorem~11.16, and Examples~11.21]{NS}.
+
+The following testing identity converts the higher-cumulant terms into
+an orthogonal series. For a bounded self-adjoint tuple $Z$, write
+$E[\,\cdot\mid W^*(Z)]$ for the trace-preserving conditional
+expectation, also viewed as the orthogonal
+projection onto $L^2(W^*(Z),\tau)$ when its argument belongs only to $L^2$.
+
+\begin{lemma}[Wick testing]\label{lem:wick}
+Let $C$ be a centered bounded self-adjoint $n$-tuple, let $S$ be a
+standard semicircular family free from $C$, and put $Z=C+\sqrt u S$,
+where $u>0$. For a multi-index $\mathbf j=(j_1,\ldots,j_r)$, $r\geq1$,
+and a monomial $P$ in the indeterminates $x_1,\ldots,x_n$, define
+\[
+ D_{\mathbf j}(P)
+ =\sum_{P=P_0x_{j_1}P_1\cdots x_{j_r}P_r}
+       \prod_{a=0}^r\tau(P_a(Z)).
+\]
+The sum selects increasing positions with the prescribed indices;
+empty gap words $P_a$ contribute one. Extend $D_{\mathbf j}$ linearly
+to polynomials. There is an orthonormal family of semicircular Wick
+polynomials $U_{\mathbf j}(S)$, indexed by all such multi-indices, with
+\begin{equation}\label{eq:wicktest}
+ D_{\mathbf j}(P)=u^{-r/2}\tau(U_{\mathbf j}(S)^*P(Z)).
+\end{equation}
+In particular, the conjugate variables of $Z$ exist and are
+\[
+ \xi_i=u^{-1/2}E[S_i\mid W^*(Z)].
+\]
+\end{lemma}
+
+\begin{proof}
+Define $U_{\mathbf j}$ from the word $S_{j_1}\cdots S_{j_r}$ by summing
+over sets of disjoint adjacent pairs in the \emph{original} index list:
+delete each chosen pair with factor $-\delta_{j_a,j_{a+1}}$.
+The empty set gives the original word. This definition also gives
+$U_{\mathbf j}^*=U_{(j_r,\ldots,j_1)}$.
+
+Expand $\tau(U_{\mathbf j}^*P(Z))$ by noncrossing cumulants. After
+multilinearly expanding $Z=C+\sqrt u S$, freeness and semicircularity
+show that every block meeting an initial $S$ position is a pair.
+Its covariance is $\delta_{ij}$ against another initial $S$ and
+$\sqrt u\,\delta_{ij}$ against a selected $S$ term from $Z$.
+
+To account for the deletion terms, fix a choice of $C$ or $S$ at every
+$P$-position in the multilinear expansion, and a noncrossing partition
+$\pi$ on the original positions. Let $A(\pi)$ be its initial-initial pair
+blocks whose endpoints are originally adjacent. For these fixed choices,
+contributions correspond exactly to subsets $F\subseteq A(\pi)$: delete the pairs
+in $F$ and restore them after expanding the remaining word. Restoring
+adjacent pairs preserves noncrossingness and the covariance factors,
+so their total coefficient is
+\[
+ \sum_{F\subseteq A(\pi)}(-1)^{|F|}=(1-1)^{|A(\pi)|}.
+\]
+If $\pi$ has any initial-initial pair, an innermost one is originally
+adjacent. Indeed, its intervening positions are all initial, and each
+must pair inside that interval to avoid crossing. Thus $A(\pi)$ is
+nonempty and the partition cancels. This argument includes nested
+internal pairs and does not require deleting newly adjacent positions.
+
+In each surviving partition all initial positions pair externally with
+positions of $P$, in reverse order. The initial indices in
+$U_{\mathbf j}^*$ are $j_r,\ldots,j_1$, so the selected indices in $P$
+occur as $j_1,\ldots,j_r$. The external pair arches prevent any remaining
+block from connecting distinct gaps. Summing the unrestricted partitions
+and the choices of $C$ or $S$ within each gap therefore gives its trace,
+even though the entries of $C$ need not be free. Each external pair contributes $\sqrt u$, proving
+\eqref{eq:wicktest}.
+
+Apply this calculation with zero signal and $u=1$. A Wick word tests
+to zero against a shorter word and gives the corresponding Kronecker
+delta against a word of its own degree. Since every other term of a
+Wick polynomial has smaller degree, this proves orthonormality against
+Wick words of equal or smaller degree; conjugate symmetry covers the
+remaining cases. For $r=1$, $D_i(P)=(\tau\otimes\tau)(\partial_iP(Z))$,
+so \eqref{eq:wicktest} and conditional expectation give the stated
+conjugate variables.
+\end{proof}
+
+\begin{proposition}[Squared-cumulant Fisher bound]\label{prop:fisher}
+Let $C$ be a centered bounded self-adjoint $n$-tuple with diagonal
+covariance $\tau(C_iC_j)=\delta_{ij}c_i$, where $c_i\geq0$, and let $S$ be a standard
+semicircular family free from $C$. Put $V_u=C+\sqrt u S$ and
+$v_i(u)=u+c_i$ for $u>0$. For every $u>0$,
+\[
+ \sum_i\frac1{v_i(u)}\leq\Phi^*(V_u)\leq\frac n u.
+\]
+Whenever
+$\sum_{r\geq2}u^{-r}\|\kappa_{r+1}(C)\|_2^2<\infty$,
+\begin{equation}\label{eq:fisherbound}
+ 0\leq\Phi^*(V_u)-\sum_i\frac1{v_i(u)}
+ \leq\frac1{u^2}\sum_{r\geq2}u^{-r}\|\kappa_{r+1}(C)\|_2^2.
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+Let $Z=V_u$ and let $\xi_i$ be supplied by Lemma~\ref{lem:wick}.
+Contraction of conditional expectation gives $\Phi^*(Z)\leq n/u$.
+Also $\tau(Z_i\xi_i)=1$ by testing the conjugate identity with $Z_i$.
+Since $\|Z_i\|_2^2=v_i(u)$, Cauchy--Schwarz gives
+$\|\xi_i\|_2^2\geq v_i(u)^{-1}$ for every $u>0$.
+
+For the upper deficit bound, expand the block containing the first
+position in $\tau(Z_iP(Z))$. Its other positions select the letters
+in $D_{\mathbf j}(P)$, while its gaps contribute their moments. Thus
+\[
+ \tau(Z_iP(Z))=\sum_{r\geq1,\mathbf j}
+ \kappa_{r+1}(Z_i,Z_{j_1},\ldots,Z_{j_r})D_{\mathbf j}(P).
+\]
+Only $r\leq\deg P$ contributes. Centering removes the order-one
+cumulant, and diagonal covariance makes the order-two term
+$v_i(u)D_i(P)$. Adding free semicircular noise leaves all higher
+cumulants equal to those of $C$.
+
+Define the ambient $L^2$ series
+\[
+ R_i=\sum_{r\geq2,\mathbf j}u^{-r/2}
+ \kappa_{r+1}(C_i,C_{j_1},\ldots,C_{j_r})U_{\mathbf j}(S)^*.
+\]
+The adjoint Wick family is orthonormal, and the assumed summability
+gives convergence in $L^2$ with
+\[
+ \sum_i\|R_i\|_2^2=\sum_{r\geq2}u^{-r}\|\kappa_{r+1}(C)\|_2^2.
+\]
+For a fixed polynomial $P$, all terms with $r>\deg P$ have zero trace
+pairing by \eqref{eq:wicktest}. Moreover, $A\mapsto\tau(AP(Z))$ is
+continuous on $L^2$. The finite cumulant expansion above therefore gives
+\[
+ \tau\big((Z_i-v_i(u)\xi_i)P(Z)\big)=\tau(R_iP(Z)).
+\]
+The coefficients in $R_i$ need no complex conjugation: this identity
+uses the bilinear trace pairing, rather than a Hilbert inner product.
+Testing with adjoint polynomials converts it into Hilbert-space
+orthogonality. Polynomial $L^2$ density then identifies
+\[
+ Z_i-v_i(u)\xi_i=E[R_i\mid W^*(Z)].
+\]
+
+The conjugate variables are self-adjoint by their conditional-expectation
+formula. Expanding the residual norm using $\tau(Z_i\xi_i)=1$ yields
+\[
+ \|Z_i-v_i(u)\xi_i\|_2^2
+ =v_i(u)^2\big(\|\xi_i\|_2^2-v_i(u)^{-1}\big).
+\]
+Summing after division by $v_i(u)^2$, using projection contraction and
+$v_i(u)\geq u$, proves \eqref{eq:fisherbound}.
+\end{proof}
+
+The proposition reduces the large-noise entropy estimate to control of
+the cumulant arrays of the signal. Its separate bound $\Phi^*(V_u)\leq n/u$
+will also ensure integrability on every compact interval of positive
+noise times.
diff --git a/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/build/sections/04-signal.tex b/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/build/sections/04-signal.tex
new file mode 100644
index 0000000000000000000000000000000000000000..33719adb5df30ca3afd63e7a708228a5235e1251
--- /dev/null
+++ b/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/build/sections/04-signal.tex
@@ -0,0 +1,170 @@
+\section{The signal and its two noise scales}\label{sec:signal}
+
+The signal must satisfy two competing requirements. Its higher cumulants
+must be sparse enough for the Fisher estimate, while many coordinates
+must share a block decomposition. Tensor independence supplies the latter
+without spoiling the former.
+
+Fix an integer $p\geq2$ and set $n=p+1$. Let $B_1,\ldots,B_p$ be free
+variance-one semicirculars. In a separate tensor factor take a variable
+$Y$ with uniform distribution on $p$ equally spaced points
+$y_l=-1+2(l-1)/(p-1)$, $1\leq l\leq p$. Thus $Y$ commutes with every
+$B_i$, but is not being taken free from them. Set
+\[
+ C=(B_1,\ldots,B_p,Y),\qquad V_a=C+\sqrt a S,\qquad
+ L(a)=\frac12\sum_{i=1}^n\log(2\pi e(a+c_i)),
+\]
+where $c_i=\tau(C_i^2)$. Thus $c_i=1$ for $i\leq p$, and $C$ is
+centered with diagonal covariance.
+The benchmark $L(a)$ is the nonmicrostates entropy of the centered
+semicircular family with coordinate variances $a+c_i$, and also the
+normalized classical entropy of the corresponding independent GUE law.
+
+The next construction provides a faithful normal tracial realization
+together with uniformly bounded deterministic matrix models for $C$.
+These models will later give the finite lower bound for $\chi(X)$.
+Choose uniformly bounded deterministic matrix models
+$B^{(q_k)}$ of the free semicircular $p$-tuple, using the convergence and
+norm bounds in Section~\ref{sec:conventions}. Tensor them with $I_p$ and
+take $I_{q_k}\otimes\operatorname{diag}(y_1,\ldots,y_p)$ for $Y$.
+In dimensions $d_k=pq_k$ these give models of the required tensor law.
+Adjoin an independent GUE $n$-tuple and select realizations satisfying
+successively more moment tests and one fixed norm bound. Their joint
+limiting moments give a positive tracial functional in the variables
+$(C,S)$ with the asserted freeness. In its GNS construction, left and
+right coordinate multiplication are bounded: the matrix inequality
+$\tr_d(Q^*D_i^2Q)\leq R_i^2\tr_d(Q^*Q)$ passes to the limit.
+The vacuum is cyclic for right polynomial multiplication, which commutes
+with the weak closure of the left algebra, so the vacuum is separating
+for that closure. Its vector state is therefore faithful and normal;
+the trace identity extends by separate weak continuity. This supplies
+the required tracial von Neumann algebra. Further independent noise
+families can be included in the same construction. Faithfulness and the
+semicircular distribution give $\|B_i\|=\|S_i\|=2$.
+
+\begin{lemma}\label{lem:cumulants}
+For $K=128$ and every $m\geq1$,
+\[
+ \|\kappa_m(C)\|_2\leq K^m p^{m/4}.
+\]
+\end{lemma}
+
+\begin{proof}
+There are at most $4^m$ noncrossing partitions and their M\"obius
+coefficients have magnitude at most $4^m$, by the Catalan product formula
+and the interval factorization of the noncrossing lattice
+\cite[Proposition~9.4, Theorem~9.29, and Propositions~10.14--10.15]{NS}.
+Each product of block moments has absolute value at most $2^m$, since
+$\|B_i\|=2$ and $\|Y\|\leq1$. Thus each cumulant is at most $32^m$
+in absolute value.
+
+Suppose $b$ of the $m$ positions have $B$ indices. A nonzero product of
+block moments in M\"obius inversion requires, after deleting the $Y$
+positions in each block, a semicircular word with a noncrossing pairing
+matching equal indices. Indeed, tensor independence factors each block
+moment into its $B$-word moment and its $Y$-moment. The union of these
+pairings is still noncrossing:
+crossing pairs from distinct blocks would cross the containing partition.
+For fixed $b$ positions, at most $4^b p^{b/2}$ labelings can therefore
+contribute. Summing over positions bounds the number of potentially
+nonzero entries by $(1+4\sqrt p)^m$. Consequently
+\[
+ \|\kappa_m(C)\|_2\leq32^m(1+4\sqrt p)^{m/2}
+ \leq(32\sqrt5)^m p^{m/4}\leq K^m p^{m/4}.
+\]
+\end{proof}
+
+By Proposition~\ref{prop:fisher} and a geometric series with ratio
+$K^2\sqrt p/u<1/2$, for
+$u>2K^2\sqrt p$ we have
+\begin{equation}\label{eq:specialfisher}
+ 0\leq\Phi^*(V_u)-\sum_i\frac1{u+c_i}
+ \leq 2K^6p^{3/2}u^{-4}.
+\end{equation}
+Adding independent free semicircular noise of variance $s$ to $V_a$
+gives the law of $V_{a+s}$. Equation~\eqref{eq:chistar} therefore yields
+\begin{equation}\label{eq:entropydeficit}
+ \chi^*(V_a)=L(a)-\frac12\int_a^\infty
+ \left[\Phi^*(V_u)-\sum_i\frac1{u+c_i}\right]du.
+\end{equation}
+All terms are finite for $a>0$: the integrand is measurable by
+\eqref{eq:fishervariational}, nonnegative, and bounded above by
+\[
+ \frac n u-\sum_i\frac1{u+c_i}
+ =\sum_i\frac{c_i}{u(u+c_i)}
+ \leq\frac{\sum_i c_i}{u^2},
+\]
+which is integrable on $[a,\infty)$. In deriving
+\eqref{eq:entropydeficit}, the covariance correction is
+$\int_0^\infty[(1+s)^{-1}-(a+c_i+s)^{-1}]\,ds=\log(a+c_i)$;
+thus no subtraction of divergent integrals is used.
+In particular,
+\begin{align}
+ \chi^*(V_a)&\geq L(a)-\frac{K^6p^{3/2}}{3a^3}>L(a)-\tfrac12
+ &&(a>2K^2\sqrt p),\label{eq:largeentropy}\\
+ \chi^*(V_{h+t})-\chi^*(V_h)
+ &=\frac12\int_0^t\Phi^*(V_{h+s})\,ds.
+ \label{eq:freeflow}
+\end{align}
+
+We now fix $p$ sufficiently large and put $t=p^{3/4}$ so that
+\begin{equation}\label{eq:parameters}
+ \begin{gathered}
+ t>2K^2\sqrt p,\qquad \frac{32t}{p}<\frac18,\\
+ \frac{p}{256(2+t)}>
+ 2\log(1+128\sqrt{2+t})+\log2+8.
+ \end{gathered}
+\end{equation}
+These inequalities are compatible because $p/t=p^{1/4}$ grows faster
+than $\log p$. An explicit choice is $p=2^{64}$ and $t=2^{48}$:
+the first threshold is $2^{47}$ and $32t/p=2^{-11}$.
+For the last inequality, its left side exceeds $128$ since
+$2+t<2^{49}$, whereas its right side is less than
+$67\log2+8<75$ since $1+128\sqrt{2+t}<2^{33}$.
+From this point on, $p$ and $t$ are fixed, and only matrix sizes will
+tend to infinity.
+
+For small $h>0$, partition the spectrum of $X_n=Y+\sqrt h S_n$ at the
+midpoints between the $y_l$ and denote its spectral projections by $e_l(h)$.
+Let $\Delta=2/(p-1)$ be the atom spacing. Since
+$\|\sqrt h S_n\|=2\sqrt h$, the condition $h<\Delta^2/64$ places
+the spectrum inside the disjoint $\Delta/4$-neighborhoods of the atoms,
+with every threshold in a spectral gap. Choose continuous functions
+agreeing with the interval indicators on these common neighborhoods.
+Continuous functional calculus, or uniform polynomial approximation
+there followed by norm continuity, gives $e_l(h)\to e_l(0)$ in norm.
+At $h=0$ these projections commute with every $B_i$, have trace $1/p$,
+and satisfy
+\[
+ \sum_{i=1}^p\sum_{l=1}^p\|e_l(0)B_i e_l(0)\|_2^2
+ =\sum_{i=1}^p\|B_i\|_2^2=p.
+\]
+We may therefore fix $0<h<\min\{1,\Delta^2/64\}$ and abbreviate
+$e_l=e_l(h)$ so that, for $X=V_h$,
+\begin{equation}\label{eq:smallblocks}
+ \tau(e_l)<\frac{3}{2p}\quad(1\leq l\leq p),\qquad
+ \sum_{i=1}^p\sum_{l=1}^p\|e_lX_i e_l\|_2^2>\frac{3p}{4}.
+\end{equation}
+
+\begin{lemma}\label{lem:finitechi}
+The tuple $X$ has $\chi(X)>-\infty$.
+\end{lemma}
+
+\begin{proof}
+Use the deterministic models for $C$ above, in dimensions $d_k=pq_k$,
+and add independent GUE noise of variance $h$.
+If $M$ bounds the coordinate norms of those models, take
+$R_0=M+8\sqrt h$. Lemma~\ref{lem:matrix-convergence} and
+\eqref{eq:gue-norm-tail} show that these noised models lie in every
+fixed moment neighborhood of $X$, within that cutoff, with probability
+tending to one. Their density is bounded above by
+$(d/(2\pi h))^{nd^2/2}$. If $q_d$ is the indicated probability, then
+\[
+ \Vol\Gamma_{R_0}(X;m,d,\varepsilon)
+ \geq q_d(2\pi h/d)^{nd^2/2}.
+\]
+Since $q_d\to1$ along the model dimensions, the normalized limsup
+logarithm is at least $\frac n2\log(2\pi h)$ for each $m,\varepsilon$.
+Take the infimum and then the supremum over cutoffs. The limsup definition
+makes this subsequence of matrix sizes sufficient.
+\end{proof}
diff --git a/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/build/sections/05-flow.tex b/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/build/sections/05-flow.tex
new file mode 100644
index 0000000000000000000000000000000000000000..aeb83137177f147abb6783cfea824c287378d4db
--- /dev/null
+++ b/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/build/sections/05-flow.tex
@@ -0,0 +1,130 @@
+\section{Entropy production for specified matrix ensembles}\label{sec:flow}
+
+We use the classical-to-free Fisher comparison of
+Jekel and Pi \cite[Proposition~6.1]{JP},
+keeping the terminal ensemble in the conclusion. The purpose is to
+transfer a later entropy bound for one evolved matrix law back to the
+initial microstates entropy. We prove the comparison for the exact
+laws that will enter the structural event in the next section.
+
+\begin{lemma}\label{lem:flow}
+For the fixed tuple $X=V_h$, fix a cutoff $R>0$ with
+$\chi_R(X)>-\infty$. There are dimensions tending to infinity and
+random tuples $A^{(d)}$, uniform on increasingly accurate
+positive-volume microstate sets at cutoff $R$, such that, for independent
+standard GUE tuples,
+\begin{equation}\label{eq:flowlower}
+ \limsup_d H_d(A^{(d)}+\sqrt t G^{(d)})
+ \geq\chi_R(X)+\frac12\int_0^t\Phi^*(V_{h+s})\,ds.
+\end{equation}
+The left side is the limsup of the normalized entropies of these particular ensembles.
+\end{lemma}
+
+\begin{proof}
+Every uniform law on a positive-volume microstate set at cutoff $R$
+has $\sum_i\E\tr_d(A_i^2)\leq nR^2$. Comparison with independent
+GUE matrices of variance $R^2$ bounds its normalized entropy above by
+$\frac n2\log(2\pi eR^2)$. Thus $\chi_R(X)$ is a finite real number.
+Choose a new increasing sequence of dimensions $d_k$, $k\geq1$, such that the normalized log-volume
+at parameters $(m,\varepsilon)=(k,1/k)$ is at least
+$\chi_R(X)-1/k$. Such arbitrarily large dimensions exist by the defining
+limsup, and the selected sets have positive volume. The pairs $(k,1/k)$
+are cofinal among all moment accuracies. Their uniform laws satisfy
+\begin{equation}\label{eq:initialentropy}
+ \liminf_d H_d(A^{(d)})\geq\chi_R(X).
+\end{equation}
+Here and below $d$ ranges over the selected dimensions $d_k$.
+Every tuple in the support of $A^{(d)}$ has the same fixed cutoff and
+increasingly accurate moments of $X$. The selection does not depend on
+the subsequent noise time or polynomial tests.
+
+Put $Z^{(d)}(s)=A^{(d)}+\sqrt s G^{(d)}$, and let $f_s$ be its density
+for $s>0$. In the $nd^2$ real Hilbert--Schmidt coordinates, the added
+Gaussian has covariance $(s/d)I$. Thus
+$\partial_s f_s=(1/(2d))\Delta f_s$. Differentiating the entropy and
+integrating by parts gives the normalized de Bruijn identity
+\begin{equation}\label{eq:matrix-debruijn}
+ \frac{d}{ds}H_d(Z^{(d)}(s))=\tfrac12 I_d(s),\qquad
+ I_d(s)=\frac1{d^3}\int\|\nabla\log f_s\|^2 f_s
+       =\sum_i\E\tr_d(F_i^2),
+\end{equation}
+where $F_i=-(1/d)\nabla_i\log f_s$ evaluated at $Z^{(d)}(s)$.
+Gaussian convolution of a compactly supported distribution justifies
+differentiation and integration by parts on compact positive-time
+intervals. Indeed, derivatives have Gaussian decay, a Gaussian lower
+bound on $f_s$ bounds $|\log f_s(x)|$ by $C(1+\|x\|^2)$, and
+\[
+ \nabla\log f_s(x)
+ =-\frac d s\big(x-\E[A^{(d)}\mid Z^{(d)}(s)=x]\big)
+\]
+has at most linear growth. Here the norm and gradient are Euclidean,
+and the conditional expectation is bounded at each fixed dimension.
+In particular the entropy and Fisher information are finite at positive
+times.
+Entropy concavity under mixtures of translations gives
+$H_d(Z^{(d)}(\varepsilon))\geq H_d(A^{(d)})$. Integrating first from
+$\varepsilon$ to $t$ and then letting $\varepsilon\downarrow0$ proves
+\begin{equation}\label{eq:classgain}
+ H_d(Z^{(d)}(t))\geq H_d(A^{(d)})+\tfrac12\int_0^t I_d(s)\,ds.
+\end{equation}
+No continuity of entropy at zero is needed.
+
+For fixed $s>0$, integration by parts against any complex
+noncommutative polynomial $P$ gives
+\begin{equation}\label{eq:matrixibp}
+ \E\tr_d(F_i P(Z^{(d)}(s)))
+ =\E(\tr_d\otimes\tr_d)((\partial_iP)(Z^{(d)}(s))).
+\end{equation}
+To verify this identity including its normalization, take an orthonormal
+Hermitian basis $(E_\alpha)$ for unnormalized Hilbert--Schmidt inner
+product. It satisfies
+$\sum_\alpha E_\alpha D E_\alpha=\Tr(D)1$ for every complex matrix $D$.
+Scalar integration by parts, separately on real and imaginary parts,
+turns the left side of \eqref{eq:matrixibp} into
+\[
+ d^{-2}\sum_\alpha\E\Tr(E_\alpha D_iP(Z^{(d)}(s))[E_\alpha]).
+\]
+Here $D_iP(Z)[E_\alpha]$ is the directional derivative in the $i$th
+matrix coordinate along $E_\alpha$.
+At an occurrence $P=LZ_iR$ in a monomial, its contribution is
+\[
+ d^{-2}\sum_\alpha\Tr(E_\alpha L E_\alpha R)
+ =d^{-2}\Tr(L)\Tr(R)=\tr_d(L)\tr_d(R).
+\]
+Summing over occurrences proves \eqref{eq:matrixibp} with the prefix
+and suffix in the same order as the free difference quotient.
+
+For self-adjoint polynomial tests $P_i$, completion of squares gives
+\[
+ I_d(s)\geq\sum_i\E\tr_d\big(2F_iP_i(Z^{(d)}(s))
+                                  -P_i(Z^{(d)}(s))^2\big).
+\]
+First use \eqref{eq:matrixibp} to replace every term involving the score
+by a product of normalized polynomial traces. Lemma~\ref{lem:matrix-convergence}
+then gives the limit of the right side as
+\[
+ \sum_i\left[2(\tau\otimes\tau)((\partial_iP_i)(V_{h+s}))
+                         -\tau(P_i(V_{h+s})^2)\right].
+\]
+The limiting law is $V_{h+s}$ because the new semicircular noise is
+free from $X=V_h$. Conjugate variables exist by Lemma~\ref{lem:wick}.
+Taking the supremum over these tests in Lemma~\ref{lem:variational} gives
+\begin{equation}\label{eq:matrix-free-fisher}
+ \liminf_d I_d(s)\geq\Phi^*(V_{h+s}).
+\end{equation}
+This holds for every fixed $s>0$ along the same selected dimensions.
+The moments of $V_{h+s}=C+\sqrt{h+s}S$ are continuous in $s$, so
+Lemma~\ref{lem:variational} also gives measurability of the free Fisher
+information. Applying Fatou's lemma to the nonnegative measurable
+functions $I_d$, and then using \eqref{eq:initialentropy} and
+\eqref{eq:classgain}, yields
+\[
+ \liminf_d H_d(Z^{(d)}(t))
+ \geq\chi_R(X)+\frac12\int_0^t\Phi^*(V_{h+s})\,ds.
+\]
+This proves \eqref{eq:flowlower}, even with liminf in place of limsup.
+The free Fisher integral is finite since
+$\Phi^*(V_{h+s})\leq n/(h+s)$ and $h>0$. Throughout the proof the
+terminal law is that of the same $A^{(d)}+\sqrt tG^{(d)}$ selected above;
+no entropy-maximizing law at the terminal distribution has been chosen.
+\end{proof}
diff --git a/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/build/sections/06-blocks.tex b/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/build/sections/06-blocks.tex
new file mode 100644
index 0000000000000000000000000000000000000000..b37077dceaacbb3046431e7b5e7e34f46790a161
--- /dev/null
+++ b/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/build/sections/06-blocks.tex
@@ -0,0 +1,209 @@
+\section{A persistent block-structure entropy deficit}\label{sec:blocks}
+
+We now bound the entropy of the particular terminal ensembles retained
+in Lemma~\ref{lem:flow}. Fix its cutoff $R$, dimension subsequence, and
+uniform microstate ensembles $A^{(d)}$. The parameters $p,h,t$ remain
+fixed as in Section~\ref{sec:signal}. For a decomposition of the identity
+into $p$ orthogonal projections $P=(P_l)$, write
+\[
+ \D_P D=\left(\sum_l P_lD_iP_l\right)_i,
+ \qquad D_{[p]}=(D_1,\ldots,D_p).
+\]
+Use the midpoint spectral partition of the last coordinate of $A^{(d)}$
+to obtain $P$. For every tuple in its support and all sufficiently large
+dimensions,
+\begin{equation}\label{eq:matrixblocks}
+ \rank(P_l)\leq\frac{2d}{p},\qquad
+ \|\D_P A^{(d)}_{[p]}\|_{2,d}^2>p/2.
+\end{equation}
+To prove this uniformly, first consider any deterministic sequence of
+tuples at cutoff $R$ whose joint moments converge to those of $X$.
+Write $I_l$ for the midpoint intervals, with any fixed convention at
+their endpoints, and let $\nu_d$ be the empirical spectral measure of
+$A_n^{(d)}$. These measures converge weakly to the law of $X_n$.
+Choose a union $N$ of small closed neighborhoods of the midpoint
+thresholds, disjoint from $\sigma(X_n)$. On a common compact interval containing
+$[-R,R]$ and $\sigma(X_n)$, choose continuous functions
+$0\leq f_l\leq1$ agreeing with $1_{I_l}$ outside $N$. Then
+$f_l(X_n)=e_l$, and, with $F_l^{(d)}=f_l(A_n^{(d)})$,
+\[
+ \|P_l-F_l^{(d)}\|_{2,d}^2\leq\nu_d(N)\longrightarrow0.
+\]
+Here weak convergence gives $\nu_d(N)\to0$ because $N$ is closed and
+has zero limiting mass. In particular, small spectral outliers and
+eigenvalues exactly at a threshold are allowed.
+
+The cutoff controls multiplication by every other coordinate:
+\begin{equation}\label{eq:block-outlier}
+ \|P_lA_i^{(d)}P_l-F_l^{(d)}A_i^{(d)}F_l^{(d)}\|_{2,d}
+ \leq2R\|P_l-F_l^{(d)}\|_{2,d}.
+\end{equation}
+The two compressed norms are at most $R$, so their squared norms also
+have difference tending to zero. Uniform polynomial approximation of
+$f_l$ and joint moment convergence now imply
+\[
+ \frac{\rank(P_l)}d\longrightarrow\tau(e_l),\qquad
+ \|P_lA_i^{(d)}P_l\|_{2,d}^2
+ \longrightarrow\|e_lX_i e_l\|_2^2.
+\]
+The block summands are orthogonal in Hilbert--Schmidt inner product;
+hence their squared norms add to
+$\|\D_P A^{(d)}_{[p]}\|_{2,d}^2$. The strict margins in
+\eqref{eq:smallblocks} give \eqref{eq:matrixblocks}.
+If uniformity over the selected supports failed, a violating tuple in
+each of a subsequence of dimensions would itself be a bounded
+moment-convergent sequence, contradicting this argument. Closure points
+of the microstate sets satisfy the same moment bounds with weak
+inequalities, whose errors still tend to zero.
+
+Put $v=1+h+t$, and let $\mathcal E_d$ be the following event for a matrix
+tuple $D$:
+\begin{equation}\label{eq:event}
+ \begin{gathered}
+ \|D_{[p]}\|_{2,d}\leq4\sqrt{vp},\qquad
+ \text{there exists a decomposition }P\text{ with}\\
+ \rank(P_l)\leq2d/p,\quad
+ \|\D_P D_{[p]}\|_{2,d}\geq\sqrt p/4.
+ \end{gathered}
+\end{equation}
+This is a deterministic Borel event. For each admissible rank list, the
+space of projection decompositions is a compact unitary orbit. The
+maximum of $\|\D_P D_{[p]}\|_{2,d}$ over that orbit is a continuous
+function of $D$: every compression is contractive, so the maximum is
+even Lipschitz in the tuple $2$-norm. The event is therefore a finite
+union of closed threshold sets intersected with a closed norm ball.
+Its definition does not depend on $A^{(d)}$; the spectral decomposition
+of $A_n^{(d)}$ will serve only as a witness for its probability under
+the evolved ensemble.
+
+\begin{lemma}\label{lem:eventprob}
+Let $\mu_d$ be the law of $A^{(d)}+\sqrt t G^{(d)}$, and let $\gamma_d$
+be the law of independent centered GUE coordinates of variances
+$v_i(h+t)=h+t+c_i$. For all sufficiently large $d$,
+\[
+ \mu_d(\mathcal E_d)\geq\tfrac12,
+ \qquad \gamma_d(\mathcal E_d)\leq e^{-4d^2}.
+\]
+\end{lemma}
+
+\begin{proof}
+For the first bound use the witness in \eqref{eq:matrixblocks}, before
+adding the noise. Conditionally on $A^{(d)}$, its projections are fixed
+and independent of $G^{(d)}$. Put $r_l=\rank(P_l)$. The block-diagonal
+Hermitian subspace has real dimension $\sum_l r_l^2$, so the GUE
+covariance $1/d$ and the normalized trace give
+\[
+ \E\big[\|\sqrt t\D_P G^{(d)}_{[p]}\|_{2,d}^2\mid A^{(d)}\big]
+ =tp\sum_l(\rank(P_l)/d)^2\leq2t.
+\]
+Markov's inequality at squared threshold $p/16$ bounds failure by
+$32t/p<1/8$. On success the projected total norm exceeds
+$\sqrt{p/2}-\sqrt p/4>\sqrt p/4$.
+Independence and centering of the noise also give
+\[
+ \E\|A^{(d)}_{[p]}+\sqrt t G^{(d)}_{[p]}\|_{2,d}^2
+ =\E\|A^{(d)}_{[p]}\|_{2,d}^2+tp\longrightarrow vp.
+\]
+Markov's inequality at the squared norm $16vp$ bounds the failure of
+the norm condition in \eqref{eq:event} by $1/16+o(1)<1/8$.
+The union bound shows that both conditions hold with probability at
+least $3/4$, proving the weaker claimed bound.
+
+For the Gaussian bound enumerate at most $(d+1)^p$ rank lists and choose
+an operator-norm $\delta$-net of the unitary group, with
+$\delta=1/(64\sqrt v)$. A maximal $\delta$-separated subset of $U(d)$
+has unitary centers and is a $\delta$-net. Its ambient operator-norm
+balls of radius $\delta/2$ are disjoint and lie in the ball of radius
+$1+\delta/2$ in the real $2d^2$-dimensional space $M_d(\C)$. Volume
+comparison gives cardinality at most $(1+2/\delta)^{2d^2}$.
+Every decomposition of a fixed rank list is a unitary conjugate of
+its standard coordinate blocks $P^0$, including lists with zero ranks.
+For unitaries $U,V$ with $\|U-V\|\leq\delta$,
+\[
+ \|U^*D_{[p]}U-V^*D_{[p]}V\|_{2,d}
+ \leq2\delta\|D_{[p]}\|_{2,d}.
+\]
+Fixed-block compression is contractive. Thus replacing a witnessing
+unitary by a net point loses at most
+$2\delta\cdot4\sqrt{vp}=\sqrt p/8$ on \eqref{eq:event}.
+Hence some rank list satisfying the bounds and some net unitary $U$
+must satisfy
+\[
+ \|\D_{P^0}(U^*D_{[p]}U)\|_{2,d}\geq\sqrt p/8.
+\]
+For a fixed list and unitary, unitary invariance preserves the independent
+variance-$v$ GUE law of the first $p$ coordinates. Their projected
+Hilbert--Schmidt vector has
+$q=p\sum_l\rank(P_l^0)^2\leq2d^2$ real coordinates of variance $v/d$.
+The further division by $d$ in the normalized squared norm gives the
+law $(v/d^2)Q$, where $Q$ is chi-squared of degree $q$.
+Since $\E e^{Q/4}=2^{q/2}$, its probability of reaching the squared-norm
+threshold $p/64$, equivalently $Q\geq pd^2/(64v)$, is at most
+\[
+ \exp\left[d^2\left(\log2-\frac{p}{256v}\right)\right].
+\]
+The full union bound consequently has logarithm at most
+\[
+ p\log(d+1)+d^2\left[2\log(1+128\sqrt v)+\log2-\frac{p}{256v}\right].
+\]
+Since $v<2+t$, Equation \eqref{eq:parameters} makes the bracket less
+than $-8$. The first term is $o(d^2)$ because $p$ is fixed. This proves
+the bound $e^{-4d^2}$ for large $d$.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+Retain the same cutoff, dimension subsequence, and ensembles throughout.
+Put $\alpha=\mu_d(\mathcal E_d)$ and
+$\beta=\gamma_d(\mathcal E_d)$. Binary data processing for relative
+entropy and the bound on binary entropy give
+\[
+ \KL(\mu_d\Vert\gamma_d)
+ \geq\alpha\log\frac\alpha\beta
+ +(1-\alpha)\log\frac{1-\alpha}{1-\beta}
+ \geq\alpha\log\frac1\beta-\log2.
+\]
+Thus Lemma~\ref{lem:eventprob} implies
+$\liminf_d d^{-2}\KL(\mu_d\Vert\gamma_d)\geq2$.
+We retain only one unit of this normalized deficit in the comparison below.
+At positive time the bounded-input Gaussian convolution has finite
+entropy and finite second moments, and its relative entropy with respect
+to $\gamma_d$ is finite. The Gaussian reference has density
+\[
+ g_d(D)=\prod_{i=1}^n
+ \left[
+ \left(\frac{d}{2\pi v_i(h+t)}\right)^{d^2/2}
+ \exp\left(-\frac{d^2\tr_d(D_i^2)}{2v_i(h+t)}\right)
+ \right].
+\]
+Using $-\int f\log f=-\int f\log g_d-\KL(f\Vert g_d)$ and the
+normalization of $H_d$ gives the exact identity
+\begin{align*}
+ H_d(Z^{(d)}(t))
+ &=\frac12\sum_i\left[
+ \log(2\pi v_i(h+t))+
+ \frac{\E\tr_d(Z_i^{(d)}(t)^2)}{v_i(h+t)}\right]
+ -d^{-2}\KL(\mu_d\Vert\gamma_d).
+\end{align*}
+The expected second moments equal those of $A_i^{(d)}$ plus $t$, and
+converge to $v_i(h+t)$. In particular,
+\[
+ \limsup_d H_d(Z^{(d)}(t))\leq L(h+t)-1.
+\]
+This upper bound concerns exactly the terminal law
+$Z^{(d)}(t)=A^{(d)}+\sqrt t G^{(d)}$ in Lemma~\ref{lem:flow}.
+Combining the two bounds on its entropy and then
+\eqref{eq:largeentropy}--\eqref{eq:freeflow} gives
+\begin{align*}
+ \chi_R(X)
+ &\leq L(h+t)-1-\frac12\int_0^t\Phi^*(V_{h+s})\,ds\\
+ &\leq\chi^*(V_{h+t})-\tfrac12
+       -\frac12\int_0^t\Phi^*(V_{h+s})\,ds\\
+ &=\chi^*(V_h)-\tfrac12.
+\end{align*}
+This holds for every fixed cutoff with $\chi_R(X)>-\infty$; for the
+remaining cutoffs it is automatic. Taking the supremum over cutoffs
+preserves the bound. Lemma~\ref{lem:finitechi} gives $\chi(X)>-\infty$,
+and \eqref{eq:entropydeficit} shows that $\chi^*(X)$ is a finite real number.
+The bounded matrix realization in Section~\ref{sec:signal} supplies
+the asserted faithful normal tracial state and bounded self-adjoint tuple.
+\end{proof}
diff --git a/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/paper.pdf b/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..4cc92b0db8d346f6c48b0be34e8c4acd64ef629d
--- /dev/null
+++ b/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:ab386b728e4dff538471550751e4ed18fbd6daf4fec6bc7eb0fa89943eccc5ea
+size 313363
diff --git a/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/README.md b/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..18d8b64eb5464621c9d53e150eda2ec73f112088
--- /dev/null
+++ b/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/README.md
@@ -0,0 +1,17 @@
+# [A zero-entropy system without a smooth positive-volume model](paper.pdf)
+
+OpenAI  
+September 25, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026,
+  author = {{OpenAI}},
+  title = {{A zero-entropy system without a smooth positive-volume model}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/paper.pdf}{OAI:A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/build/main.tex b/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/build/main.tex
new file mode 100644
index 0000000000000000000000000000000000000000..09b361a61c2eef50cbda7f33ec652a6ac16faf43
--- /dev/null
+++ b/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/build/main.tex
@@ -0,0 +1,28 @@
+\documentclass[11pt]{article}
+\input{preamble}
+\title{A zero-entropy system without a smooth positive-volume model}
+\author{OpenAI}
+\date{September 25, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We construct an ergodic invertible transformation of a standard nonatomic
+probability space with zero Kolmogorov--Sinai entropy that has no smooth
+positive-volume model. More precisely, it is not measurably conjugate to any
+$C^\infty$ diffeomorphism preserving a strictly positive smooth probability
+density on a compact finite-dimensional manifold. A single example excludes
+every finite dimension, including models on nonorientable manifolds and
+manifolds with smooth boundary.
+\end{abstract}
+\tableofcontents
+\clearpage
+\input{sections/01-introduction}
+\input{sections/02-stationary}
+\input{sections/03-linear}
+\input{sections/04-switching}
+\input{sections/05-words}
+\input{sections/06-process}
+\input{sections/07-obstruction}
+\bibliographystyle{plain}
+\bibliography{references}
+\end{document}
diff --git a/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/build/preamble.tex b/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/build/preamble.tex
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+++ b/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/build/preamble.tex
@@ -0,0 +1,48 @@
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage{lmodern}
+\usepackage[margin=28mm]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage[expansion=false]{microtype}
+\usepackage{enumitem,booktabs,array,graphicx,xcolor,tikz}
+\usetikzlibrary{arrows.meta,positioning,calc,decorations.pathreplacing}
+\usepackage[hyphens]{url}
+\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref}
+\usepackage{bookmark}
+\usepackage[capitalise,nameinlink,noabbrev]{cleveref}
+\hypersetup{pdftitle={A zero-entropy system without a smooth positive-volume model},pdfauthor={OpenAI}}
+\pdftrailerid{}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newcommand{\R}{\mathbb R}
+\newcommand{\Z}{\mathbb Z}
+\newcommand{\N}{\mathbb N}
+\newcommand{\F}{\mathbb F}
+\newcommand{\E}{\mathbb E}
+\newcommand{\Pp}{\mathbb P}
+\newcommand{\one}{\mathbf 1}
+\newcommand{\Id}{\mathrm{Id}}
+\DeclareMathOperator{\TV}{TV}
+\DeclareMathOperator{\supp}{supp}
+\DeclareMathOperator{\rank}{rank}
+\DeclareMathOperator{\diag}{diag}
+\DeclareMathOperator{\Span}{span}
+\DeclareMathOperator{\KL}{D}
+\DeclareMathOperator{\dist}{dist}
+\DeclareMathOperator{\diam}{diam}
+\DeclareMathOperator{\Lip}{Lip}
+\DeclareMathOperator{\tr}{tr}
+\DeclareMathOperator{\osc}{osc}
+\DeclarePairedDelimiter{\norm}{\lVert}{\rVert}
+\DeclarePairedDelimiter{\abs}{\lvert}{\rvert}
+\DeclarePairedDelimiter{\set}{\{}{\}}
+\newcommand{\ip}[2]{\langle #1,#2\rangle}
+\numberwithin{equation}{section}
+\setlist{itemsep=3pt,topsep=5pt}
+\setcounter{tocdepth}{1}
diff --git a/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/build/references.bib b/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/build/references.bib
new file mode 100644
index 0000000000000000000000000000000000000000..84332e356099a1334888088bad82c2142e19b017
--- /dev/null
+++ b/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/build/references.bib
@@ -0,0 +1,130 @@
+@article{KatokThouvenot1997,
+  author = {Katok, Anatole and Thouvenot, Jean-Paul},
+  title = {Slow entropy type invariants and smooth realization of commuting measure-preserving transformations},
+  journal = {Annales de l'Institut Henri Poincar{\'e}, Probabilit{\'e}s et Statistiques},
+  volume = {33},
+  number = {3},
+  year = {1997},
+  pages = {323--338},
+  note = {\url{https://www.numdam.org/item/AIHPB_1997__33_3_323_0/}}
+}
+
+@article{ForemanWeiss2022,
+  author = {Foreman, Matthew and Weiss, Benjamin},
+  title = {Measure preserving diffeomorphisms of the torus are unclassifiable},
+  journal = {Journal of the European Mathematical Society},
+  volume = {24},
+  number = {8},
+  year = {2022},
+  pages = {2605--2690},
+  note = {\href{https://doi.org/10.4171/JEMS/1151}{\nolinkurl{doi:10.4171/JEMS/1151}}}
+}
+
+@misc{DajaniDirksin2008,
+  author = {Dajani, Karma and Dirksin, Sjoerd},
+  title = {A Simple Introduction to Ergodic Theory},
+  year = {2008},
+  howpublished = {Lecture notes, Utrecht University, December 18},
+  note = {\url{https://www.staff.science.uu.nl/~kraai101/lecturenotes2009.pdf}}
+}
+
+@article{Ruelle1978,
+  author = {Ruelle, David},
+  title = {An inequality for the entropy of differentiable maps},
+  journal = {Boletim da Sociedade Brasileira de Matem\'atica},
+  volume = {9},
+  number = {1},
+  year = {1978},
+  pages = {83--87},
+  note = {\href{https://doi.org/10.1007/BF02584795}{\nolinkurl{doi:10.1007/BF02584795}}}
+}
+
+@article{Oseledets1968,
+  author = {Oseledets, V. I.},
+  title = {A multiplicative ergodic theorem. Characteristic {Ljapunov} exponents of dynamical systems},
+  journal = {Trudy Moskovskogo Matematicheskogo Obshchestva},
+  volume = {19},
+  year = {1968},
+  pages = {179--210},
+  note = {In Russian. English translation: Transactions of the Moscow Mathematical Society 19 (1968), 197--231. \url{https://www.mathnet.ru/eng/mmo214}}
+}
+
+@article{LindThouvenot1977,
+  author = {Lind, D. A. and Thouvenot, J.-P.},
+  title = {Measure-preserving homeomorphisms of the torus represent all finite entropy ergodic transformations},
+  journal = {Mathematical Systems Theory},
+  volume = {11}, year = {1977}, pages = {275--282},
+  note = {\href{https://doi.org/10.1007/BF01768481}{\nolinkurl{doi:10.1007/BF01768481}}}
+}
+
+@article{QuasSoo2016,
+  author = {Quas, Anthony and Soo, Terry},
+  title = {Ergodic universality of some topological dynamical systems},
+  journal = {Transactions of the American Mathematical Society},
+  volume = {368}, number = {6}, year = {2016}, pages = {4137--4170},
+  note = {\href{https://doi.org/10.1090/tran/6489}{\nolinkurl{doi:10.1090/tran/6489}}}
+}
+
+@article{Kunde2024,
+  author = {Kunde, Philipp},
+  title = {Anti-classification results for weakly mixing diffeomorphisms},
+  journal = {Mathematische Annalen},
+  volume = {390}, year = {2024}, pages = {5607--5668},
+  note = {\href{https://doi.org/10.1007/s00208-024-02890-0}{\nolinkurl{doi:10.1007/s00208-024-02890-0}}}
+}
+
+@misc{GerberKunde2025Mixing,
+  author = {Gerber, Marlies and Kunde, Philipp},
+  title = {Non-classifiability of mixing zero-entropy diffeomorphisms up to isomorphism},
+  year = {2025},
+  note = {Preprint, September 10, 2025. \url{https://arxiv.org/abs/2509.09003v1}}
+}
+
+@article{Smale1967,
+  author = {Smale, Stephen},
+  title = {Differentiable dynamical systems},
+  journal = {Bulletin of the American Mathematical Society},
+  volume = {73}, year = {1967}, pages = {747--817},
+  note = {\href{https://doi.org/10.1090/S0002-9904-1967-11798-1}{\nolinkurl{doi:10.1090/S0002-9904-1967-11798-1}}}
+}
+
+@article{ChowLinPalmer1989,
+  author = {Chow, S.-N. and Lin, X.-B. and Palmer, K. J.},
+  title = {A shadowing lemma with applications to semilinear parabolic equations},
+  journal = {SIAM Journal on Mathematical Analysis},
+  volume = {20}, year = {1989}, pages = {547--557},
+  note = {\href{https://doi.org/10.1137/0520038}{\nolinkurl{doi:10.1137/0520038}}}
+}
+
+@article{ForemanRudolphWeiss2011,
+  author = {Foreman, Matthew and Rudolph, Daniel J. and Weiss, Benjamin},
+  title = {The conjugacy problem in ergodic theory},
+  journal = {Annals of Mathematics},
+  volume = {173}, number = {3}, year = {2011}, pages = {1529--1586},
+  note = {\href{https://doi.org/10.4007/annals.2011.173.3.7}{\nolinkurl{doi:10.4007/annals.2011.173.3.7}}}
+}
+
+@article{Tonchev2003,
+  author = {Tonchev, Vladimir D.},
+  title = {Difference systems of sets and code synchronization},
+  journal = {Rendiconti del Seminario Matematico di Messina, Series II},
+  volume = {9}, year = {2003}, pages = {217--226},
+  note = {\url{https://pages.mtu.edu/~tonchev/dssn.pdf}}
+}
+
+@article{Sason2013,
+  author = {Sason, Igal},
+  title = {Entropy bounds for discrete random variables via maximal coupling},
+  journal = {IEEE Transactions on Information Theory},
+  volume = {59}, number = {11}, year = {2013}, pages = {7118--7131},
+  note = {\href{https://doi.org/10.1109/TIT.2013.2274515}{\nolinkurl{doi:10.1109/TIT.2013.2274515}}}
+}
+
+@techreport{Seidel1973,
+  author = {Seidel, J. J.},
+  title = {On two-graphs, and {Shult}'s characterization of symplectic and orthogonal geometries over {GF}(2)},
+  institution = {Technische Hogeschool Eindhoven},
+  type = {T.H.-Report}, number = {73-WSK-02},
+  year = {1973}, month = jan,
+  note = {\url{https://pure.tue.nl/ws/portalfiles/portal/2088237/252851.pdf}}
+}
diff --git a/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/build/sections/01-introduction.tex b/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/build/sections/01-introduction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..7e34958432713358e018cb3608a74343dae5c1a6
--- /dev/null
+++ b/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/build/sections/01-introduction.tex
@@ -0,0 +1,174 @@
+\section{Introduction}\label{sec:introduction}
+
+The smooth realization problem asks which measure-preserving systems can
+be represented by smooth dynamics on a compact manifold.
+Here a \emph{smooth positive-volume model} of an invertible
+probability-preserving transformation $(X,\mu,T)$ consists of a compact
+smooth manifold $M$ of finite dimension, a smooth positive probability
+volume $\nu$, and a $C^\infty$ diffeomorphism $S$ preserving $\nu$,
+together with a measure-preserving isomorphism
+\[
+ F:X_0\longrightarrow M_0,\qquad F\circ T=S\circ F,
+\]
+on invariant conull measurable sets. Both $F$ and $F^{-1}$ are measurable;
+no continuity is required. Here smooth positive volume means a probability
+measure with strictly positive smooth local densities,
+$d\nu=f(x)|dx_1\cdots dx_d|$, where $f>0$ is smooth.
+The manifold may have a smooth boundary, with these densities smooth up
+to the boundary. No orientation of $M$ or orientation preservation by
+$S$ is required. The manifold and its dimension may depend on the
+original system.
+
+A standard nonatomic probability space means a probability space
+isomorphic, modulo null sets, to the unit interval with Lebesgue
+probability.
+
+Finite entropy is a necessary restriction in the smooth realization
+problem. For example, Ruelle's differentiable entropy inequality
+\cite[Theorem~2]{Ruelle1978} gives finite entropy from compact derivative
+bounds. The question addressed here is
+whether this entropy restriction suffices when the invariant measure
+must be a smooth positive density.
+
+For invertible ergodic transformations of standard nonatomic
+probability spaces, two positive realization results help locate this
+restriction.
+Lind and Thouvenot realized finite-entropy ergodic transformations by
+Lebesgue-preserving homeomorphisms of the two-torus
+\cite{LindThouvenot1977}. They also proved that a hyperbolic
+two-torus automorphism represents every such system with measure entropy
+below the automorphism's topological entropy, using a suitable invariant
+Borel probability of full support; Quas and Soo recall this earlier result in
+\cite[\S1]{QuasSoo2016}. Quas and Soo's Theorem~2 extends the result to
+toral automorphisms with no root-of-unity eigenvalue.
+Choosing a sufficiently high iterate of a hyperbolic automorphism
+therefore gives a smooth model for any prescribed finite-entropy
+system, with an invariant Borel probability of full support.
+Full support does not require a smooth positive density.
+
+The requirement on the invariant measure is essential. A realization
+with an arbitrary invariant Borel probability, which may be singular,
+does not by itself give a smooth positive-volume model. This distinction is part of the
+classical formulation discussed by Katok and Thouvenot
+\cite[pp.~324--326]{KatokThouvenot1997}. They developed slow entropy
+invariants and constructed ergodic $\Z^2$ actions having no Lipschitz
+realization on a compact space of finite box dimension, even when the
+model may carry an arbitrary invariant Borel probability
+\cite[Proposition~4 and Corollary~2]{KatokThouvenot1997}.
+The commuting generators must be realized together in that problem;
+its obstruction does not by itself answer the question for one
+transformation. Foreman and Weiss
+\cite[p.~2607]{ForemanWeiss2022} also discuss finite-entropy realization,
+in a formulation allowing an invariant measure merely equivalent to
+volume. Our conclusion concerns the smooth positive-density version;
+we do not assert an obstruction for every equivalent-to-volume
+invariant measure. Their anti-classification theorems concern the complexity of the
+measure-isomorphism relation among smooth transformations, including
+an ergodic area-preserving torus subclass
+\cite[Theorems~1--2 and Corollary~4]{ForemanWeiss2022}.
+They do not assert that a particular abstract finite-entropy system
+lacks such a model. Subsequent anti-classification results treat
+weakly mixing smooth transformations \cite[Theorem~B]{Kunde2024} and mixing
+zero-entropy smooth transformations \cite[Theorem~2]{GerberKunde2025Mixing};
+these too study equivalence inside families of smooth models.
+
+We give a negative answer to the smooth-volume realization question,
+already among systems of zero entropy.
+For a finite probability vector $p$, its Shannon entropy is
+\[
+ H(p)=-\sum_a p(a)\log_2 p(a),\qquad 0\log_2 0=0.
+\]
+The entropy $H_\mu(\mathcal P)$ of a finite measurable partition is
+the entropy of its atom probabilities; the entropy of a finite random
+variable is likewise the entropy of its law.
+The Kolmogorov--Sinai entropy of $T$ is
+\[
+ h_\mu(T)=\sup_{\mathcal P}\lim_{n\to\infty}
+ \frac1n H_\mu\!\left(\bigvee_{j=0}^{n-1}T^{-j}\mathcal P\right),
+\]
+where $\mathcal P$ ranges over finite measurable partitions.
+
+\begin{theorem}\label{thm:main}
+There exists an ergodic invertible measure-preserving transformation
+$(X,\mu,T)$ of a standard nonatomic probability space such that
+$h_\mu(T)=0$ and $(X,\mu,T)$ is not measurably conjugate, on invariant conull sets, to
+any $C^\infty$ diffeomorphism of a compact finite-dimensional manifold
+preserving a strictly positive smooth probability density. The manifold
+may have smooth boundary and need not be orientable.
+\end{theorem}
+
+The proof isolates a property of smooth positive volume that remains
+useful after an arbitrary measurable change of coordinates.
+At a dyadic time $n=2^k$, partition the manifold into small cells,
+choose a cell with its volume as probability, and choose two independent
+points $x,y$ from its normalized volume. On most such pairs, one can
+find a point $z$ whose future shadows that of $x$ and whose past shadows
+that of $y$. More than the existence of this point is needed:
+the law of $z$ on the retained pairs is bounded by twice the original
+volume. The number of cells is at most $\exp(CB_kn)$ for a sequence
+with $\sum_k B_k^{16}<\infty$.
+The precise statement is \Cref{thm:switching}.
+
+This switch estimate is the main analytic ingredient. Its proof begins
+with a square-summable dyadic estimate for stationary diagonal
+derivative logs, rather than an assumed rate in an ergodic theorem.
+Finite-interval triangular changes of basis give the linear
+estimates needed to solve a boundary problem joining the two orbit
+halves. A relative exterior-power estimate controls the change in
+small endpoint minors, and a determinant identity and global
+injectivity control the law of the switch.
+The quantitative summability and the bound on the output measure are
+the features that will obstruct realization.
+
+We construct the opposing symbolic system from finite sets
+$\mathcal W_i$ of words of lengths $m_i$. Each word at the next level is
+a concatenation of words at the preceding level, and the system
+records their compatible alignment phases. Word labels are nonzero
+vectors over $\F_2$. Certain pairs of labels separated by a power-of-two
+number of blocks must be orthogonal for a fixed alternating form.
+These constraints coexist with high entropy of short label lists and
+with a positive symbol-distance separation between distinct or
+misaligned words.
+
+The entropy of the label lists makes two independent lists unlikely
+to satisfy most of the orthogonality tests. A smooth switch would
+force their phases and most of their labels to agree with the
+corresponding past and future lists of a single legal symbolic point.
+Those lists do satisfy the tests. Summability of the smooth partition
+cost, together with the growth of the word scales, supplies times at
+which conditioning on a cell loses too little entropy to reconcile
+the two conclusions.
+
+The order of choices is important: the symbolic system is fixed first.
+Only then is a hypothetical smooth model, with any one finite
+dimension, considered. The scale argument absorbs all constants
+depending on that model. Thus the construction does not select a
+different counterexample for each dimension.
+
+\paragraph{Organization.}
+\Cref{sec:stationary,sec:linear,sec:switching} prove the switching
+estimate. \Cref{sec:words} constructs the finite word families.
+\Cref{sec:process} supplies an ergodic probability on their phase
+extension and establishes the window entropy and graph constraints.
+\Cref{sec:obstruction} compares the two laws of one graph-passing
+statistic and proves \Cref{thm:main}.
+
+\paragraph{Conventions.}
+Logarithms in entropies and combinatorial estimates are to base $2$;
+natural logarithms are written $\ln$, and derivative growth is written
+with $\exp$ or $e$. Total variation distance between finite laws $p,q$
+is $\TV(p,q)=\tfrac12\sum_a|p(a)-q(a)|$.
+For finite random variables, conditional entropy means
+\[
+ H(Y\mid Z)=\sum_{z:\Pr(Z=z)>0}\Pr(Z=z)\,
+ H\bigl(\mathcal L(Y\mid Z=z)\bigr).
+\]
+Conditional partition entropy uses the same definition for atom labels.
+We write $H_{\mathrm{bin}}(s)=-s\log_2 s-(1-s)\log_2(1-s)$
+for binary entropy, with the same zero-mass convention at the endpoints.
+Constants denoted by $C$ may change from line to line.
+In the smooth part they depend only on the fixed smooth data,
+dimension, and a fixed exceptional-mass tolerance, unless otherwise
+specified. All asymptotic estimates there are uniform over the
+points, bins, and finite-interval frames satisfying the displayed
+hypotheses. The empty determinant and empty product equal one.
diff --git a/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/build/sections/02-stationary.tex b/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/build/sections/02-stationary.tex
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@@ -0,0 +1,269 @@
+\section{Stationary estimates for derivative growth}\label{sec:stationary}
+
+The switching construction requires control of derivatives on a long finite
+orbit segment.  The accuracy may depend on the length of the segment, but
+the squared errors must be summable over dyadic lengths.  We first obtain
+this summability for an arbitrary bounded stationary process.  Applying
+the result to a triangular representation of the derivative then gives
+large sets of orbit segments with the required control.
+
+\subsection{Deviation from a dyadic chord}
+
+Let $(Y_t)_{t\in\Z}$ be a real stationary process with
+$\E Y_0^2<\infty$.  Stationarity means that translating all indices in
+any finite list does not change its joint distribution.  For $h\geq 0$
+define the block average and normalized maximal chord deviation by
+\[
+ A_h=2^{-h}\sum_{t=0}^{2^h-1}Y_t,
+ \qquad
+ Z_h=2^{-h}\max_{0\leq u\leq 2^h}
+ \left|\sum_{t=0}^{u-1}Y_t-uA_h\right|.
+\]
+The maximum is over integer $u$, and an empty sum is zero.
+In particular $Z_0=0$.  The chord here uses the average on the
+particular block under consideration; it is not a line whose slope is
+the mean of the stationary process.
+
+\begin{lemma}[Dyadic chord estimate]\label{lem:dyadic}
+For the process above, put
+\[
+ v_l=\E A_{l-1}^2-\E A_l^2,\qquad l\geq 1.
+\]
+Then $v_l\geq 0$, $\sum_{l\geq 1}v_l\leq\E Y_0^2$, and
+\begin{equation}\label{eq:dyadic-convolution}
+ \|Z_h\|_{L^2}
+ \leq \frac12\sum_{a=0}^{h-1}2^{-a/2}\sqrt{v_{h-a}}
+ \qquad(h\geq 1).
+\end{equation}
+Consequently
+\begin{equation}\label{eq:dyadic-square-sum}
+ \sum_{h\geq 1}\|Z_h\|_{L^2}^2
+ \leq C\,\E Y_0^2
+\end{equation}
+for an absolute constant $C$.  The same estimates hold for a block
+beginning at any deterministic integer time.
+\end{lemma}
+
+\begin{proof}
+Consider a block of length $2^l$, and let $U$ and $V$ be the averages on
+its left and right halves.  Its own average is $W=(U+V)/2$.  The identity
+\[
+ \frac{(U-W)^2+(V-W)^2}{2}
+   =\frac{U^2+V^2}{2}-W^2
+\]
+and stationarity give
+\[
+ v_l
+ =\E\frac{(U-W)^2+(V-W)^2}{2}
+ =\E(U-W)^2.
+\]
+The last equality also follows pointwise from $V-W=-(U-W)$.
+Thus $v_l\geq0$, and telescoping the differences of second moments proves
+the asserted bound on their sum.  No independence of the two halves is
+used.
+
+We next compare the cumulative-sum polygon on $[0,2^h]$ with its chord.
+For $a=0,\ldots,h$, let $F_a$ be the piecewise-linear interpolant that
+agrees with the cumulative sum at the multiples of $2^{h-a}$.
+Then $F_0$ is the chord and $F_h$ agrees with every integer-time
+cumulative sum.  Passing from $F_a$ to $F_{a+1}$ inserts the midpoint
+of each of $2^a$ intervals of length $2^{h-a}$.
+If $U_j,W_j$ are respectively the left-child and parent averages on
+the $j$th such interval, the displacement at its midpoint is
+$2^{h-a-1}(U_j-W_j)$.
+The difference of the interpolants is linear between the old and new
+vertices, so
+\[
+ 2^{-h}\|F_{a+1}-F_a\|_\infty
+ \leq 2^{-a-1}\max_{1\leq j\leq 2^a}|U_j-W_j|.
+\]
+Bounding a squared maximum by the sum of squares and using stationarity
+of each interval gives
+\[
+ \left\|2^{-h}\|F_{a+1}-F_a\|_\infty\right\|_{L^2}
+ \leq 2^{-a-1}\bigl(2^a v_{h-a}\bigr)^{1/2}.
+\]
+The triangle inequality, first for the polygonal sup norm and then in
+$L^2$, proves \eqref{eq:dyadic-convolution}.
+
+Extend the sequence $(\sqrt{v_l})_{l\geq1}$ by zero to nonpositive
+indices.  Its squared $\ell^2$ norm is at most $\E Y_0^2$.
+The right side of \eqref{eq:dyadic-convolution} is its convolution
+with the summable sequence $(2^{-a/2-1})_{a\geq0}$.
+The triangle inequality for the $\ell^2$ norms of its translates
+therefore proves \eqref{eq:dyadic-square-sum}.
+Finally, stationarity identifies the distribution of each translated
+block with that of the block just considered.
+\end{proof}
+
+\subsection{A stationary triangular derivative representation}
+
+Fix a compact smooth manifold $M$ of dimension $d\geq1$, a smooth
+Riemannian metric, a smooth diffeomorphism $S:M\to M$, and an
+$S$-invariant probability measure $\nu$.  In the switching argument
+$\nu$ will have a smooth positive density.  That extra assumption is
+not needed in this section.  The constructions also apply to a compact
+manifold with smooth boundary.
+
+The compact frame extension and triangular cocycle have a classical
+precedent in Oseledets's proof of the multiplicative ergodic theorem
+\cite[\S4, pp.~208--209, equations~(6)--(10)]{Oseledets1968}.
+We use the full orthogonal frame bundle and a positive-diagonal upper
+triangular convention. The finite-time estimates below follow from the
+stationary chord lemma; we do not invoke a limiting Lyapunov splitting
+or a convergence rate in the multiplicative ergodic theorem.
+
+The orthonormal frame bundle $\mathcal F M$ consists of the linear
+isometries $e:\R^d\to T_xM$, with bundle projection $\pi(e)=x$.
+For each frame there is a unique positive-diagonal QR factorization
+\begin{equation}\label{eq:frame-qr}
+ D S_x\circ e=\widetilde S(e)\,R(e),
+\end{equation}
+where $\widetilde S(e)$ is an orthonormal frame at $Sx$ and $R(e)$
+is upper triangular with positive diagonal.
+This is ordinary Gram--Schmidt in the indicated order, with the length
+of each new perpendicular component chosen positive.
+It depends continuously, indeed smoothly, on the frame and the base
+point.
+
+The map $\widetilde S$ is invertible.  If
+$DS_x e=e'R$, then
+$(DS_x)^{-1}e'=D(S^{-1})_{Sx}e'=eR^{-1}$, where the inverse
+linear map has domain $T_{Sx}M$ and codomain $T_xM$.
+The matrix $R^{-1}$ is again upper triangular with
+positive diagonal.  Taking positive-diagonal QR after applying the
+inverse derivative therefore recovers $e$.  This also proves
+continuity of the inverse.  The construction uses the full orthonormal
+frame bundle and does not require a global frame or an orientation.
+
+There is a $\widetilde S$-invariant probability $\widetilde\nu$ whose
+projection is $\nu$.  To see this, first put the normalized
+orthogonal-group Haar measure on every fiber over $\nu$.
+Orthogonal changes of local trivialization preserve this fiber
+measure, so the resulting probability is defined without choosing a
+global section.  Each of its iterates under $\widetilde S$ projects
+to $\nu$.  Cesaro averages have a weakly convergent subsequence,
+because $\mathcal F M$ is compact.  The difference between the
+pushforward of an average and the average itself tends to zero
+against every continuous function, so its limit is invariant.
+Continuity of $\pi$ preserves the marginal $\nu$ in the limit.
+
+Compactness and invertibility provide a constant $A_*>1$ such that
+\begin{equation}\label{eq:qr-uniform-bounds}
+ \|R(e)\|,\ \|R(e)^{-1}\|\leq A_*
+ \qquad(e\in\mathcal F M).
+\end{equation}
+In particular $A_*^{-1}\leq R(e)_{jj}\leq A_*$.
+The functions
+\[
+ g_j(e)=\ln R(e)_{jj},\qquad 1\leq j\leq d,
+\]
+are continuous and bounded, and
+$(g_j(\widetilde S^t e))_{t\in\Z}$ is stationary under
+$\widetilde\nu$.
+
+\begin{proposition}[Good finite orbit segments]\label{prop:good-frames}
+For each $\eta>0$ there is a positive sequence $(b_k)_{k\geq1}$ with
+\begin{equation}\label{eq:b-summability}
+ b_k\geq 2^{-k/10},
+ \qquad
+ \sum_{k\geq1}b_k^2<\infty,
+\end{equation}
+and compact sets $\widetilde{\mathcal G}_k\subset\mathcal F M$ and
+$\mathcal G_k=\pi(\widetilde{\mathcal G}_k)\subset M$ such that
+\[
+ \widetilde\nu(\widetilde{\mathcal G}_k)\geq1-\eta/100,
+ \qquad \nu(\mathcal G_k)\geq1-\eta/100.
+\]
+For $n=2^k$, $b=b_k$, and $e\in\widetilde{\mathcal G}_k$, define
+\[
+ A_t=R(\widetilde S^t e)\quad(-n\leq t<n),
+ \qquad
+ \lambda_j=\frac1{2n}\sum_{t=-n}^{n-1}\ln(A_t)_{jj}.
+\]
+Then for every $j$ and every integer $-n\leq u\leq n$,
+\begin{equation}\label{eq:good-chord}
+ \left|\sum_{t=-n}^{u-1}\ln(A_t)_{jj}
+          -(u+n)\lambda_j\right|\leq bn.
+\end{equation}
+Consequently, on every subinterval $-n\leq u\leq v\leq n$,
+\begin{equation}\label{eq:good-subinterval}
+ \left|\sum_{t=u}^{v-1}\ln(A_t)_{jj}
+          -(v-u)\lambda_j\right|\leq2bn.
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+Apply \Cref{lem:dyadic} to each $g_j$.
+Let $c_{j,h}$ be the right side of
+\eqref{eq:dyadic-convolution} for that process, and put
+$\alpha_h^2=\sum_{j=1}^d c_{j,h}^2$.
+The lemma gives $\sum_h\alpha_h^2<\infty$.
+For example, take
+\[
+ b_k=40\eta^{-1/2}\alpha_{k+1}+2^{-k/10}.
+\]
+This sequence has the properties in \eqref{eq:b-summability}.
+The interval $[-n,n)$ has length $2^{k+1}$.
+Stationarity and the squared Markov inequality give
+\[
+ \widetilde\nu\!\left(
+ \max_{\substack{1\leq j\leq d\\-n\leq u\leq n}}
+ \left|\sum_{t=-n}^{u-1}g_j(\widetilde S^t e)
+                  -(u+n)\lambda_j\right|>b_kn
+ \right)
+ \leq \frac{4\alpha_{k+1}^2}{b_k^2}
+ \leq\frac{\eta}{400}.
+\]
+Define $\widetilde{\mathcal G}_k$ by the complementary non-strict
+inequalities.  These are finitely many continuous inequalities on a
+compact bundle, so this set and its projection are compact.
+Moreover
+\[
+ \pi^{-1}(M\setminus\mathcal G_k)
+   \subset\mathcal F M\setminus\widetilde{\mathcal G}_k.
+\]
+The pushforward identity $\pi_*\widetilde\nu=\nu$ gives the asserted
+base-measure estimate.  No measurable choice of a frame over each
+point of $\mathcal G_k$ is needed.
+Finally, subtracting the chord error at $u$ from the chord error at $v$
+gives \eqref{eq:good-subinterval}, with the same rates $\lambda_j$.
+\end{proof}
+
+We call a frame in $\widetilde{\mathcal G}_k$ good at scale $k$,
+and a base point in $\mathcal G_k$ good at that scale.
+Every good base point has a good witness frame.  The estimates below
+hold for every such witness; witnesses at different points or scales
+need not be chosen coherently.  We do not require an individual point
+to be good at all sufficiently large scales.
+
+The stationary argument has now supplied the summable quantity that
+will control the cost of switching.  Fix $\eta$ and its sequence
+$(b_k)$, and use the following parameters at scale $n=2^k$:
+\begin{equation}\label{eq:smooth-scales}
+ \begin{aligned}
+ b&=b_k,& B&=b^{1/8},& g&=b^{1/2},\\
+ L&=\lceil b^{1/4}n\rceil,&
+ h_0&=e^{-Bn},&r&=e^{-2Bn},&\delta&=e^{-3Bn/2}.
+ \end{aligned}
+\end{equation}
+The parameter $b$ controls the diagonal chord error $bn$;
+$g$ will be the separation margin for the finite-interval rates,
+$L$ is the short comparison time, and $B$ sets the spatial scales
+and the upper bound $CBn$ for the logarithm of the number of
+partition cells. Here $h_0$ and $r$ will be spatial lengths; $\delta$ will be a
+neighborhood radius for an orbit construction.
+Since $b_k\to0$ and $bn\geq n^{9/10}$,
+\begin{equation}\label{eq:smooth-hierarchy}
+ \ln n=o(bn),\quad bn=o(gL),\quad gL=o(gn),\quad
+ gn=o(L),\quad L=o(Bn).
+\end{equation}
+The ceiling in $L$ has no effect on these comparisons.
+We also have
+\[
+ \sum_k B_k^{16}=\sum_k b_k^2<\infty,
+ \qquad Bn\geq n^{79/80}.
+\]
+All subsequent asymptotic estimates are for $k\to\infty$.
+Constants may depend on the fixed smooth data, dimension, and $\eta$,
+but not on the point, its good witness, or the scale.
diff --git a/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/build/sections/03-linear.tex b/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/build/sections/03-linear.tex
new file mode 100644
index 0000000000000000000000000000000000000000..299ab5feeb5f59c78cdc024c8732699516d9e716
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+++ b/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/build/sections/03-linear.tex
@@ -0,0 +1,575 @@
+\section{Linear estimates on finite orbit segments}\label{sec:linear}
+
+We now turn the diagonal control of
+\Cref{prop:good-frames} into estimates for the full derivative.
+There are three steps.  A finite-interval change of basis separates
+groups of diagonal rates, short derivatives identify nearly the same
+spaces at nearby base points, and a product-expansion estimate controls
+small perturbations relative to the possibly small exterior volumes.
+These statements will supply the linear data for switching orbit
+segments.
+
+\subsection{Triangular products and separation of rates}
+
+For invertible matrices $A_t$ indexed by $-n\leq t<n$, write
+\[
+ A(t,s)=A_{t-1}\cdots A_s\quad(t>s),\qquad
+ A(s,s)=\Id,\qquad A(s,t)=A(t,s)^{-1}.
+\]
+All vector norms and exterior-power norms in the next proposition are
+Euclidean operator norms.  If $E$ is a $q$-dimensional subspace and
+$A$ is injective, let
+\[
+ J_q(A|E)=\|(\mathord{\bigwedge}^q A)\xi\|,
+\]
+where $\xi$ is either unit simple $q$-vector spanning $E$.
+This is the intrinsic $q$-dimensional volume expansion.  We set
+$J_0=1$, and the zeroth exterior power is the identity on $\R$.
+
+\begin{proposition}[Finite-interval splitting]\label{prop:finite-splitting}
+Fix $d\geq1$ and $A_*>1$.  There is a constant $C$, depending only on
+these two quantities, with the following properties.
+Let $n\geq2$, $b>0$, and $\ln(2n+1)\leq bn$.
+Suppose that $A_t$, $-n\leq t<n$, are upper triangular with positive
+diagonal and
+\[
+ \|A_t\|,\ \|A_t^{-1}\|\leq A_*,
+ \qquad
+ \left|\sum_{t=-n}^{u-1}\ln(A_t)_{jj}
+                  -(u+n)\lambda_j\right|\leq bn
+\]
+for $-n\leq u\leq n$ and $1\leq j\leq d$, where
+$\lambda_j=(2n)^{-1}\sum_{t=-n}^{n-1}\ln(A_t)_{jj}$.
+
+First, order the rates as
+$\lambda_{(1)}\geq\cdots\geq\lambda_{(d)}$.
+For every $0\leq q\leq d$ and every $-n\leq s\leq t\leq n$,
+\begin{equation}\label{eq:exterior-rate-comparison}
+ \left|\ln\|\mathord{\bigwedge}^q A(t,s)\|
+       -(t-s)\sum_{j=1}^q\lambda_{(j)}\right|\leq Cbn.
+\end{equation}
+The analogous estimate for $A(s,t)$ uses the $q$ largest members of
+$\{-\lambda_1,\ldots,-\lambda_d\}$.
+
+Next, suppose that real $\gamma$ and $g>0$ separate all the rates:
+each $\lambda_j$ is at most $\gamma-g$ or at least $\gamma+g$.
+Let $F$ be the set of indices in the second group, let
+$p=|F|$, and put
+\[
+ \Lambda=\sum_{j\in F}\lambda_j,\qquad
+ \Omega=-\sum_{j\notin F}\lambda_j.
+\]
+There are decompositions
+$\R^d=E_t^s\oplus E_t^f$, $-n\leq t\leq n$, invariant under each
+$A_t$, with $\dim E_t^f=p$, whose two projections have norm at most
+$e^{Cbn}$.  For $s\leq t$ and $l=t-s$ they satisfy
+\begin{align}
+ \|A(t,s)|E_s^s\|&\leq e^{(\gamma-g)l+Cbn},
+ &
+ \|A(s,t)|E_t^f\|&\leq e^{-(\gamma+g)l+Cbn},
+ \label{eq:split-propagation}\\
+ \|\mathord{\bigwedge}^p A(t,s)\|&\leq e^{\Lambda l+Cbn},
+ &
+ e^{\Lambda l-Cbn}\leq
+ J_p(A(t,s)|E_s^f)&\leq e^{\Lambda l+Cbn},
+ \label{eq:fast-volume}\\
+ \|\mathord{\bigwedge}^{d-p} A(s,t)\|&\leq e^{\Omega l+Cbn},
+ &
+ e^{\Omega l-Cbn}\leq
+ J_{d-p}(A(s,t)|E_t^s)&\leq e^{\Omega l+Cbn}.
+ \label{eq:slow-backward-volume}
+\end{align}
+If $0<p<d$ and at least one of $v_1,\ldots,v_p$ belongs to $E_s^s$,
+then
+\begin{equation}\label{eq:mixed-exterior}
+ \|A(t,s)v_1\wedge\cdots\wedge A(t,s)v_p\|
+ \leq e^{(\Lambda-2g)l+Cbn}\prod_{j=1}^p\|v_j\|.
+\end{equation}
+The backward analogue has exterior degree $d-p$, exponent
+$(\Omega-2g)l+Cbn$, and at least one factor in $E_t^f$.
+When a group is empty its space is $\{0\}$, all statements on that
+space are vacuous, and the exterior-degree-zero conventions apply.
+\end{proposition}
+
+\begin{proof}
+\emph{Diagonal normalization.}
+Write
+\[
+ e_j(u)=\sum_{t=-n}^{u-1}\bigl(\ln(A_t)_{jj}-\lambda_j\bigr),
+ \qquad
+ D_u=\diag(e^{e_1(u)},\ldots,e^{e_d(u)}).
+\]
+The chord hypothesis gives
+$\|D_u\|,\|D_u^{-1}\|\leq e^{bn}$.
+The matrices
+\[
+ \widehat A_t=D_{t+1}^{-1}A_tD_t
+\]
+are upper triangular with constant diagonal
+$(e^{\lambda_1},\ldots,e^{\lambda_d})$ and entries bounded by
+$e^{Cbn}$.  Since the rates are averages of logarithms of diagonal
+entries,
+\begin{equation}\label{eq:bounded-rates}
+ |\lambda_j|\leq\ln A_*.
+\end{equation}
+Their inverse matrices also have entries bounded by $e^{Cbn}$.
+
+We record how triangular products are estimated.  If $m$ is fixed and
+upper-triangular $m$-by-$m$ matrices have constant diagonal
+$e^{\alpha_1},\ldots,e^{\alpha_m}$, bounded $|\alpha_i|$, and
+off-diagonal entries at most $e^{Cbn}$, a term in any product entry
+can change its coordinate index strictly at most $m-1$ times.
+There are at most a fixed constant times $(2n+1)^{m-1}$ choices
+for those times and the intermediate indices.
+If the product length is $l$ and a term has $a\leq m-1$ strict
+changes, its diagonal factors contribute at most
+$e^{\alpha_{\max}(l-a)}$.
+The omitted $a$ diagonal factors cost at most a fixed factor
+$e^{a|\alpha_{\max}|}$ after $e^{\alpha_{\max}l}$ is factored out.
+The off-diagonal factors contribute $e^{C_mbn}$.
+The polynomial number of terms is absorbed by
+$\ln(2n+1)\leq bn$.  Thus the product norm is at most
+$e^{\alpha_{\max}l+C_mbn}$.
+
+Apply this argument to $\mathord{\bigwedge}^q\widehat A_t$ in the
+coordinate-wedge basis, ordered lexicographically.
+This matrix is upper triangular of fixed dimension $\binom dq$,
+and its diagonal entries are
+$\exp(\sum_{j\in I}\lambda_j)$ for $q$-element index sets $I$.
+The largest such sum is $\sum_{j=1}^q\lambda_{(j)}$.
+For the reverse inequality, the diagonal of the product indexed by
+a maximizing set $I$ is exactly
+$\exp(l\sum_{j\in I}\lambda_j)$; that matrix entry is no larger
+than the operator norm.  Converting back multiplies the exterior
+norms only by the two endpoint gauge bounds.
+This proves \eqref{eq:exterior-rate-comparison}.
+Applying the same argument to inverse normalized steps in reverse
+time proves its backward version.  The cases $q=0,d$ follow as well:
+the former is the identity, and the latter is the determinant line.
+
+\emph{Removing entries between the groups.}
+We next conjugate $\widehat A_t$ by time-dependent upper
+unitriangular matrices to make the two coordinate index groups
+invariant.  Process the superdiagonals in increasing order of their
+distance $j-i$ from the diagonal.
+At a position $i<j$ belonging to different groups, suppose the
+current entry is $h_t$.  Use an elementary change
+$Q_t=\Id+u_tE_{ij}$, where $E_{ij}$ has its only nonzero entry,
+equal to one, in position $(i,j)$.
+In $Q_{t+1}^{-1}\widehat A_tQ_t$ the new entry at this position is
+\[
+ h_t+e^{\lambda_i}u_t-e^{\lambda_j}u_{t+1}.
+\]
+To set it to zero, solve
+\begin{equation}\label{eq:shear-recurrence}
+ u_{t+1}=q u_t+f_t,\qquad
+ q=e^{\lambda_i-\lambda_j},\qquad f_t=e^{-\lambda_j}h_t.
+\end{equation}
+If $q<1$, prescribe $u_{-n}=0$ and solve forward:
+\[
+ u_t=\sum_{a=-n}^{t-1}q^{\,t-1-a}f_a.
+\]
+If $q>1$, prescribe $u_n=0$ and solve backward:
+\[
+ u_t=-\sum_{a=t}^{n-1}q^{\,t-1-a}f_a.
+\]
+In each formula the coefficients have modulus at most one, so
+$|u_t|\leq2n\max_a|f_a|$.
+Different groups cannot have $q=1$.
+This bound avoids dividing by the possibly small separation of rates.
+
+Right multiplication by $Q_t$ changes column $j$ by a multiple of
+column $i$, and left multiplication by $Q_{t+1}^{-1}$ changes row
+$i$ by a multiple of row $j$.
+Upper triangularity shows that every affected off-diagonal position
+has distance at least $j-i$; at that distance only $(i,j)$ changes.
+The possible quadratic cross term is zero because
+$E_{ij}\widehat A_tE_{ij}=(\widehat A_t)_{ji}E_{ij}=0$.
+Thus a later operation cannot restore an entry on an earlier
+superdiagonal.
+
+There are at most $d(d-1)/2$ operations.
+At each operation the bound above, the current entry bounds, and
+\eqref{eq:bounded-rates} bound the shear and its inverse by
+$e^{Cbn}$.  Multiplying or adding a fixed number of such quantities
+only changes $C$; the factor $2n$ is absorbed by $\ln(2n+1)\leq bn$.
+Induction over this fixed number of operations therefore gives
+matrices $C_t$ with
+\begin{equation}\label{eq:finite-gauge}
+ \|C_t\|,\ \|C_t^{-1}\|\leq e^{Cbn},
+ \qquad
+ B_t=C_{t+1}^{-1}A_tC_t,
+\end{equation}
+where $B_t$ is upper triangular, has diagonal $e^{\lambda_j}$,
+and has no entries between the two groups.  Its entries and those of
+$B_t^{-1}$ are bounded by $e^{Cbn}$.
+All constants are independent of the length of the interval.
+
+Let $\R^F$ be the coordinate span with indices in $F$ and
+$\R^{F^c}$ its coordinate complement.  Set
+\[
+ E_t^f=C_t\R^F,\qquad E_t^s=C_t\R^{F^c}.
+\]
+These spaces are invariant under the original matrices.
+Their projections are conjugates by $C_t$ of coordinate projections,
+and hence have norm at most $e^{Cbn}$ after changing $C$.
+The two groups may be interlaced in the original coordinate order.
+Each induced block is nevertheless upper triangular in its inherited
+order; a fixed permutation makes the two blocks consecutive if desired.
+The triangular-product estimate within these blocks proves
+\eqref{eq:split-propagation}.
+
+\emph{Exterior volumes and mixed factors.}
+The fast rates are exactly the $p$ largest rates.  The general exterior
+bound already proves the first inequality of \eqref{eq:fast-volume}.
+The normalized fast block has determinant $e^{\Lambda l}$ on an
+interval of length $l$.
+To account explicitly for its possibly oblique basis, define
+\[
+ V_F(t)=
+ \sqrt{\det\bigl((C_t|_{\R^F})^*(C_t|_{\R^F})\bigr)}.
+\]
+For $p=0$ this Gram determinant is one.  The endpoint bounds in
+\eqref{eq:finite-gauge} imply
+$e^{-Cbn}\leq V_F(t)\leq e^{Cbn}$.
+The intrinsic volume expansion is exactly
+\[
+ J_p(A(t,s)|E_s^f)
+       =e^{\Lambda l}\frac{V_F(t)}{V_F(s)}.
+\]
+This proves both volume bounds in \eqref{eq:fast-volume}.
+Applying the same argument to the inverse slow block gives
+\eqref{eq:slow-backward-volume}, including either sign of $\Omega$.
+
+The normalized block decomposition induces an invariant decomposition
+of each exterior power according to the number of fast and slow
+factors.  In degree $p$, every sector with a slow factor has diagonal
+rate sum at most $\Lambda-2g$: at least one fast rate has been
+replaced by a slow rate.
+The triangular-product estimate therefore bounds this entire sector
+by $e^{(\Lambda-2g)l+Cbn}$.
+If one original vector is slow, split each of the other vectors into
+its fast and slow parts.  There are only a fixed number of terms, each
+still having a slow factor.  The projection norms and endpoint gauges
+are absorbed into $e^{Cbn}$, proving \eqref{eq:mixed-exterior}.
+Reversing time and exchanging the groups gives the backward statement.
+The empty-group cases require none of these mixed-factor assertions
+and follow from the stated conventions.
+\end{proof}
+
+The estimates concern every subinterval of one good finite trajectory.
+The change of basis is made once on that trajectory; only its values
+at the two ends of a subinterval enter a product estimate.
+In particular, a gauge loss is not multiplied once for each time step.
+
+For a good frame in \Cref{prop:good-frames}, the matrices of the
+derivative in the moving orthonormal frames satisfy the hypotheses
+above for all sufficiently large $k$.
+The proposition therefore gives tangent spaces
+$E_t^s,E_t^f\subset T_{S^t x}M$ and the corresponding derivative
+estimates.  We use this notation also after expressing the spaces in
+chart coordinates.  Coordinate maps from a fixed uniformly regular
+finite atlas conjugate every product only at its endpoints, so their
+bounded norms and inverse norms change the constants in the estimates,
+but not their form.
+
+\subsection{Spaces that can be chosen uniformly in a spatial bin}
+
+We need spaces at nearby starting points to agree much more accurately
+than their possible small angle.  The comparison uses short derivative
+products and the parameters in \eqref{eq:smooth-scales}.
+Fix a finite family of coordinate charts with smaller relatively
+compact subcharts covering $M$.
+Their coordinate maps, inverse maps, and the local expressions of
+$S$ and $S^{-1}$ have uniform bounds through order two on the relevant
+compact sets.  Each spatial bin below is contained in one of the
+smaller subcharts, has coordinate diameter at most $C h_0$, and
+uses that specified chart to identify its tangent spaces with $\R^d$.
+At a boundary of $M$ use nested half-box charts.
+
+For equal-dimensional subspaces $E,F\subset\R^d$, measure their
+distance by $\|\Pi_E-\Pi_F\|$, where $\Pi_E,\Pi_F$ are Euclidean
+orthogonal projections.  This distance concerns the subspaces and
+does not require a choice of bases in them.
+
+\begin{lemma}[Common spaces in a bin]\label{lem:bin-spaces}
+For all sufficiently large $k$, each spatial bin $U$ meeting
+$\mathcal G_k$ has a number
+\[
+ \gamma_U\in[g,(8d+1)g]
+\]
+such that every good witness frame over $U$ has all its rates at most
+$\gamma_U-g$ or at least $\gamma_U+g$, with the same number $p_U$
+in the second group.
+Apply \Cref{prop:finite-splitting} using this threshold.
+There is $c>0$, independent of the bin and the witnesses, such that:
+\begin{enumerate}[label=\textup{(\roman*)}]
+ \item The spaces $E_0^s$ from any two good witnesses over $U$ are
+ within $e^{-c gL}$ of one another in its coordinate chart.
+ The same holds for the spaces $E_0^f$.
+ \item For good paths starting in $U$ whose time-$n$ endpoints lie
+ in a common endpoint bin $V$, their spaces $E_n^f$ are within
+ $e^{-c gL}$ of one another in the chart of $V$.
+ For paths whose time-$(-n)$ endpoints lie in a common bin,
+ the same assertion holds for $E_{-n}^s$.
+\end{enumerate}
+In particular, fix one representative space for each nonempty
+combination of bins in \textup{(ii)}.  Orthogonal projection onto it,
+restricted to any corresponding actual endpoint space, is an
+isomorphism with inverse norm at most $2$.
+If $p_U=0$ or $p_U=d$, the relevant zero or full spaces agree exactly.
+\end{lemma}
+
+\begin{proof}
+\emph{Short products in common charts.}
+Choose one reference point in a starting bin.
+Uniform Lipschitz bounds for $S$ and $S^{-1}$ give, for another point
+in that bin,
+\[
+ d_M(S^t x,S^t y)\leq C A_0^{|t|}h_0
+ \qquad(|t|\leq L)
+\]
+with a fixed $A_0>1$.
+Since $L=o(Bn)$, this tends to zero uniformly.
+At each time on either short reference orbit choose a chart whose
+smaller subchart contains the reference point.
+All nearby orbits then remain in the same larger chart at that time.
+Use the prescribed bin chart at the initial time.
+The one-step coordinate derivatives and their inverses are bounded by
+a fixed constant, and their differences are at most
+$C A_0^{|t|}h_0$.
+The telescoping product identity consequently gives
+\begin{equation}\label{eq:short-product-comparison}
+ \|P_x^\pm-P_y^\pm\|\leq e^{-Bn+CL}=:\zeta_k,\qquad
+ \sigma_{\min}(P_x^\pm)\geq e^{-CL}=:m_k.
+\end{equation}
+Here $P_x^+$ and $P_x^-$ denote $DS^L_x$ and $DS^{-L}_x$,
+respectively, in these common initial and terminal charts.
+For example, summing the $L$ terms in the product difference bounds
+it by $C L A_1^{2L}h_0$, which has the stated form.
+The inverse product bound gives the minimum stretch.
+On a manifold with boundary the same proof uses the nested half-box
+charts and uniform one-sided smooth bounds; equivalently the finitely
+many local expressions may be extended to their surrounding boxes.
+
+The singular values are Lipschitz with respect to operator norm.
+Since $\zeta_k/m_k\to0$, \eqref{eq:short-product-comparison} implies,
+after increasing the constants if necessary,
+\begin{equation}\label{eq:short-log-singular}
+ |\ln\sigma_j(P_x^+)-\ln\sigma_j(P_y^+)|
+       \leq 2\zeta_k/m_k .
+\end{equation}
+For completeness, the singular-value Lipschitz bound follows from
+the min--max characterization, because $\|Pv\|$ and $\|Qv\|$
+differ by at most $\|P-Q\|$ on unit vectors.
+The logarithmic inequality then follows from the lower bound $m_k$.
+
+\emph{A threshold for every witness.}
+Let $\lambda_{(j)}(e)$ be the sorted rates of a good witness $e$.
+Subtracting \eqref{eq:exterior-rate-comparison} for exterior degrees
+$j$ and $j-1$ gives
+\begin{equation}\label{eq:singular-rate}
+ \left|L^{-1}\ln\sigma_j(P_{\pi(e)}^+)
+                     -\lambda_{(j)}(e)\right|
+       \leq Cbn/L.
+\end{equation}
+This argument uses the no-gap part of
+\Cref{prop:finite-splitting}; a threshold has not yet been chosen.
+Fix one good witness $e_*$ in $U$.
+Equations \eqref{eq:short-log-singular} and \eqref{eq:singular-rate}
+show that every good witness in $U$, including every other witness
+over the same point, satisfies
+\[
+ \max_j|\lambda_{(j)}(e)-\lambda_{(j)}(e_*)|
+ \leq Cbn/L+2\zeta_k/(m_kL)=o(g).
+\]
+Consider the $d+1$ candidate thresholds $(1+8j)g$, $0\leq j\leq d$.
+A real number lies within distance $3g$ of at most one candidate.
+The $d$ representative rates therefore leave a candidate whose
+distance from all of them is greater than $3g$.
+For large $k$, all the other sorted rate lists differ by less than
+$g$, so this candidate separates every list with a margin greater
+than $2g$.  It has the weaker required margin $g$, and the number of
+rates above it is constant.  Fix this candidate as $\gamma_U$.
+
+\emph{Comparison of subspaces using one short map.}
+Assume first that $0<p_U<d$ and put $q=d-p_U$.
+Use the single forward short map $P_*=P_{\pi(e_*)}^+$.
+Let $W$ be its bottom $q$ right-singular subspace.
+Its top $p_U$ singular values are at least
+$\exp((\gamma_U+g)L-Cbn)$ by \eqref{eq:singular-rate}.
+If $v$ is a unit vector in the slow space of any good witness $e$,
+then \eqref{eq:split-propagation} and
+\eqref{eq:short-product-comparison} give
+\[
+ \|P_*v\|\leq e^{(\gamma_U-g)L+Cbn}+\zeta_k.
+\]
+Orthogonality of the images of right-singular vectors implies
+\[
+ \|\Pi_{W^\perp}v\|
+ \leq e^{-2gL+Cbn}
+       +\zeta_k e^{-(\gamma_U+g)L+Cbn}
+ \leq e^{-2gL+Cbn}+e^{-Bn+CL}.
+\]
+In the last bound, constants absorb
+$|\gamma_U|=O_d(g)$ and the other terms bounded by a constant times
+$L$.  This is at most $e^{-c_1gL}$ for some $c_1>0$ and large $k$.
+
+To pass from this directed estimate to a subspace estimate, observe
+that if $\|\Pi_{W^\perp}|E\|\leq\epsilon<1$ and
+$\dim E=\dim W$, then $\Pi_W|E$ is an isomorphism.
+The space $E$ is a graph over $W$ of norm at most
+$\epsilon/\sqrt{1-\epsilon^2}$.
+The corresponding orthogonal projectors differ by at most a fixed
+multiple of $\epsilon$ when $\epsilon\leq1/2$.
+Thus every slow space is within $C e^{-c_1gL}$ of this same $W$;
+the triangle inequality compares any two slow spaces.
+No continuity assertion about singular vectors or chosen good frames
+has been used.
+
+For fast spaces use the single backward short map
+$P_{\pi(e_*)}^-$.
+Its bottom $p_U$ right-singular space is the comparison space.
+The top $q$ singular values are at least
+$\exp(-(\gamma_U-g)L-Cbn)$, whereas a unit fast vector has backward
+stretch at most $\exp(-(\gamma_U+g)L+Cbn)$.
+The identical calculation gives the error bound
+\[
+ e^{-2gL+Cbn}
+       +\zeta_k e^{(\gamma_U-g)L+Cbn}
+ \leq e^{-2gL+Cbn}+e^{-Bn+CL}.
+\]
+This proves \textup{(i)} after reducing $c$ to absorb fixed factors.
+In particular, the error is $o(e^{-C_0bn})$ for every fixed $C_0$,
+by $bn=o(gL)$.
+
+\emph{Endpoint comparisons.}
+Keep the threshold of $U$.
+For paths starting in $U$ and ending at time $n$ in a common bin $V$,
+choose one representative path for this pair of bins.
+Apply the preceding short-product comparison from their time-$n$
+points backward for length $L$.
+Their initial closeness for this comparison is supplied by $V$,
+not by propagating the starting-bin closeness for $n$ steps.
+All intervals $[n-L,n]$ lie inside the good interval, and all paths
+use the same group dimension and threshold.
+In particular, \eqref{eq:exterior-rate-comparison} applies on each
+$[n-L,n]$ with the same full-segment rates, so the singular-value
+comparison and the backward argument just proved compare their fast
+spaces at $n$. At $-n$, the identical all-subinterval estimate for
+forward short maps on $[-n,-n+L]$ gives the slow-space assertion.
+
+Finally, if $E$ and its chosen representative $F$ have equal dimension
+and $\|\Pi_E-\Pi_F\|\leq\epsilon$, then for $v\in E$,
+\[
+ \|\Pi_F v\|\geq\sqrt{1-\epsilon^2}\,\|v\|.
+\]
+The restriction is onto $F$ by equality of dimensions, and its inverse
+has norm at most $(1-\epsilon^2)^{-1/2}\leq2$ for large $k$.
+If one group is empty, the associated spaces are identically zero or
+the full coordinate space, so all comparisons and projection
+conclusions follow directly.
+\end{proof}
+
+For later use, choose one good witness in each nonempty starting bin
+and orthonormal bases within each of its two spaces at time zero.
+The matrix having these two bases as columns has norm bounded by a
+dimension-only constant and inverse norm at most $e^{Cbn}$, by the
+projection bounds of \Cref{prop:finite-splitting}.
+It gives fixed product coordinates for that bin.
+Likewise \Cref{lem:bin-spaces} supplies fixed endpoint projections
+for each relevant pair of bins.
+All these choices are finite at a fixed scale.  Their constancy within
+the indicated bins, rather than any continuity of choices between
+bins, is what the switching construction will use.
+
+\subsection{Relative stability of exterior products}
+
+The bounds on every unperturbed subinterval also control a perturbed
+long product.  This must be a relative estimate: a leading volume can
+be exponentially small, so continuity with an absolute error alone is
+insufficient.
+
+\begin{lemma}[Exterior-product perturbation]\label{lem:relative-exterior}
+Fix $d\geq1$ and $K\geq1$.  Let $0\leq p\leq d$, let $m\geq1$,
+and let $A_0,\ldots,A_{m-1}$ and
+$\widehat A_0,\ldots,\widehat A_{m-1}$ be real $d$-by-$d$ matrices
+such that
+\[
+ \|A_t\|\leq K,\qquad
+ \|\widehat A_t-A_t\|\leq\epsilon\leq1.
+\]
+Use the same product notation as above for nonnegative indices.
+Suppose $a\geq0$, $\sigma\in\R$, and
+\[
+ \|\mathord{\bigwedge}^p A(t,s)\|
+       \leq e^{\sigma(t-s)+a}
+ \qquad(0\leq s\leq t\leq m).
+\]
+There is $C=C(d,K)$ such that
+\begin{equation}\label{eq:exterior-perturbation}
+ \left\|\mathord{\bigwedge}^p\widehat A(m,0)
+             -\mathord{\bigwedge}^p A(m,0)\right\|
+ \leq e^{\sigma m+a}
+       \left[(1+C\epsilon e^{a+|\sigma|})^m-1\right].
+\end{equation}
+For $p=0$ the left side is zero.
+
+In particular, use the scales \eqref{eq:smooth-scales}, take
+$m\leq2n$, $a\leq C_1bn$, $|\sigma|\leq C_2$, and
+$\epsilon\leq C_3e^{-3Bn/2}$, with fixed constants.
+Then for every fixed $C_0>0$ the left side of
+\eqref{eq:exterior-perturbation} is
+\begin{equation}\label{eq:relative-exterior-small}
+ o\!\left(e^{\sigma m-C_0bn}\right),
+\end{equation}
+uniformly for all matrices satisfying these hypotheses.
+\end{lemma}
+
+\begin{proof}
+The exterior power is multiplicative.  Multilinearity in its $p$
+columns, together with the one-step bound $K+1$ for the perturbed
+matrices, gives
+\[
+ E_t:=\mathord{\bigwedge}^p\widehat A_t
+             -\mathord{\bigwedge}^p A_t,
+ \qquad \|E_t\|\leq C\epsilon.
+\]
+Expand the product using either the unperturbed exterior matrix or
+$E_t$ at each time.  A term with $q\geq1$ insertions has $q+1$
+unperturbed intervals whose total length is $m-q$.
+Allowing empty intervals, the hypothesis bounds that term by
+\[
+ e^{\sigma(m-q)+(q+1)a}(C\epsilon)^q
+ =e^{\sigma m+a}(C\epsilon e^{a-\sigma})^q.
+\]
+There are $\binom mq$ choices of the inserted positions.
+Sum over $q=1,\ldots,m$ and use $-\sigma\leq|\sigma|$ to prove
+\eqref{eq:exterior-perturbation}.  This calculation explains why
+negative $\sigma$, as may occur for backward exterior volumes, is
+harmless: each removed time step contributes only the displayed
+factor $e^{-\sigma}$.
+
+For the final assertion put
+$u=C\epsilon e^{a+|\sigma|}$ and use
+$(1+u)^m-1\leq mu e^{mu}$.
+After division by $e^{\sigma m-C_0bn}$ the bound is at most
+\[
+ C m\epsilon e^{(2C_1+C_0)bn+C_2}
+       \exp\!\bigl(Cm\epsilon e^{C_1bn+C_2}\bigr).
+\]
+The logarithm of its decisive first factor is at most
+$-3Bn/2+O(bn)+\ln(2n)$, which tends to $-\infty$ by
+\eqref{eq:smooth-hierarchy}.  The quantity in the second exponential
+tends to zero for the same reason.
+All constants are fixed, proving the uniform little-oh assertion.
+For exterior degree zero the two products are both the identity, so
+the result holds without an expansion.
+\end{proof}
+
+The lemma applies to forward degree-$p$ products with
+$\sigma=\Lambda$ and to backward degree-$(d-p)$ products with
+$\sigma=\Omega$ from \Cref{prop:finite-splitting}.
+Together with \Cref{lem:bin-spaces}, it supplies uniform linear
+and relative-volume estimates for orbit segments whose starting
+points occupy one small spatial bin.
diff --git a/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/build/sections/04-switching.tex b/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/build/sections/04-switching.tex
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@@ -0,0 +1,679 @@
+\section{Switching a past and a future with controlled volume}
+\label{sec:switching}
+
+The finite-interval estimates now give a geometric operation on pairs of
+nearby points.  It joins the future of one point to the past of the other.
+We must control the distribution of the resulting point as well as its
+orbit: this requires constructing both complementary joins and estimating
+their joint Jacobian.
+
+The joining geometry is related to the canonical local-product
+coordinates of hyperbolic dynamics \cite[\S I.7]{Smale1967}.
+Solving a linear difference equation and then a nonlinear contraction
+also has a standard shadowing precedent
+\cite[Lemma~3.2 and Proposition~4.1]{ChowLinPalmer1989}.
+Here the slow group may include zero or positive rates; we prove the
+finite-interval boundary inverse directly. The measure inequality
+requires the further determinant and injectivity arguments below.
+
+\begin{theorem}[Smooth switching]\label{thm:switching}
+Let $M$ be a compact smooth manifold of positive finite dimension, let
+$\nu$ be a smooth positive probability volume in the density sense of
+\Cref{sec:introduction}, and let $S:M\to M$
+be a smooth diffeomorphism preserving $\nu$.  Fix a Riemannian distance
+$d_M$ and $\eta>0$.  There exist positive numbers $B_k$ with
+$\sum_k B_k^{16}<\infty$ and a constant $C$ such that the following
+holds for all sufficiently large $k$, with $n=2^k$.
+
+There is a finite measurable partition $\mathcal D_k$ of $M$ with
+$\#\mathcal D_k\le \exp(CB_kn)$.  Define the probability measure
+\begin{equation}\label{eq:pair-law}
+ \rho_k=\sum_{\substack{Q\in\mathcal D_k\\\nu(Q)>0}}
+       \frac{(\nu|_Q)\otimes(\nu|_Q)}{\nu(Q)}.
+\end{equation}
+Thus one first samples a cell with its $\nu$-weight and then samples
+two independent points in that cell.  There are a measurable set
+$G_k\subset M\times M$, with $\rho_k(G_k)\ge1-\eta$, and a measurable
+map $z_k:G_k\to M$ such that
+\begin{equation}\label{eq:switch-law}
+ (z_k)_*(\rho_k|_{G_k})\le 2\nu
+\end{equation}
+and
+\begin{equation}\label{eq:switch-shadow}
+ \sup_{(x,y)\in G_k}\left[
+ \max_{0\le t\le n}d_M(S^t z_k(x,y),S^t x)
+ +\max_{-n\le t\le0}d_M(S^t z_k(x,y),S^t y)
+ \right]\longrightarrow0.
+\end{equation}
+The measure on the left of \eqref{eq:switch-law} is not normalized
+after restriction to $G_k$.  The same conclusions hold if $M$ has a
+smooth boundary.
+\end{theorem}
+
+The proof occupies the rest of this section.  Its first two steps produce
+two complementary switches $(z,w)$.  The last two steps show that their
+pair map is injective and has volume Jacobian $1+o(1)$.  This is what
+will give \eqref{eq:switch-law}.
+
+\subsection{Grids, retained pairs, and fixed endpoint projections}
+
+We may assume $0<\eta<1$.  Apply \Cref{prop:good-frames} with parameter
+$\eta/2$.  It supplies $b=b_k\ge n^{-1/10}$,
+$\sum_k b_k^2<\infty$, and a compact set of good base points,
+denoted here by $\mathcal A_k$ to distinguish it from the retained pair set,
+with suitable frame paths on $[-n,n]$, such that
+$\nu(M\setminus\mathcal A_k)<\eta/100$.  Set
+\begin{equation}\label{eq:switch-scales}
+ B=b^{1/8},\quad h_0=e^{-Bn},\quad r=e^{-2Bn},\quad
+ L=\lceil b^{1/4}n\rceil,\quad g=b^{1/2},\quad
+ \delta=e^{-3Bn/2}.
+\end{equation}
+Here and below constants depend only on the fixed smooth data, dimension,
+and $\eta$.  They never depend on a point, bin, frame witness, or $k$.
+In particular,
+\begin{equation}\label{eq:switch-scale-order}
+ \log n=o(bn),\quad bn=o(gL),\quad gL=o(gn),\quad
+ gn=o(L),\quad L=o(Bn).
+\end{equation}
+These relations absorb every fixed power of a quantity bounded by
+$e^{Cbn}$ or $e^{Cgn}$ at the places where it is used below.
+
+Choose a finite system of coordinate patches with smaller coordinate
+boxes covering $M$, each smaller box having closure in its patch.
+Assign points to the first smaller box containing them.  This gives
+disjoint chart regions whose boundaries lie in finitely many smooth
+coordinate faces.  The coordinate maps, their inverses, and the chart
+density of $\nu$ have uniform bounds on slightly larger compact patches;
+the density has a uniform positive lower bound.  Within each region
+use the grid of coordinate cubes of side $h_0$, keeping only cubes whose
+closures lie in that region.  We call these cubes \emph{coarse bins}.
+The discarded part has volume $O(h_0)$: it is contained in a fixed
+multiple of the $h_0$-neighborhood of the finitely many region boundaries.
+All comparisons at a common starting or endpoint bin use its one fixed
+coordinate map.
+
+For a starting bin $Q$ meeting $\mathcal A_k$, \Cref{lem:bin-spaces}
+gives a common threshold $\gamma_Q\in[g,Cg]$, a fast dimension $p_Q$,
+and reference spaces $E_Q^s,E_Q^f$ at time zero.  The spaces of every
+good frame path beginning in $Q$ are within $e^{-cgL}$ of these spaces.
+Their complementary projections and the inverse of a basis orthonormal
+within each space have norm at most $e^{Cbn}$.  The threshold and the
+reference spaces are chosen once for this bin.  This involves finitely
+many choices at each scale, not a choice of a frame as a measurable
+function on $\mathcal A_k$.
+
+In the starting chart let $E_Q:\R^{d-p_Q}\times\R^{p_Q}\to\R^d$
+be the matrix with these orthonormal slow and fast columns, respectively.
+Tile the chart by the parallelepipeds $E_Q([0,r)^d+rj)$, $j\in\Z^d$,
+and keep the ones whose closures lie in $Q$.  Translations of the grid
+are immaterial.  These are the \emph{product cells}.  Although
+$\|E_Q^{-1}\|\le e^{Cbn}$, the forward norm satisfies
+$\|E_Q\|\le\sqrt2$: for $(a,b)$,
+$|E_Q(a,b)|\le |a|+|b|$.  Thus every cell has physical diameter at
+most $C_dr$ and volume at least $e^{-Cbn}r^d$ up to a fixed factor.
+The total number of full cells is consequently at most
+\begin{equation}\label{eq:cell-count}
+ C e^{Cbn}r^{-d}\le e^{CBn}.
+\end{equation}
+Every point of a coarse bin at distance greater than $C_dr$ from its
+walls belongs to a full product cell.  Hence the partial-cell loss,
+summed over all coarse bins, is $O(r/h_0)=o(1)$.
+Use one additional cell for all omitted points and for bins not meeting
+$\mathcal A_k$.  Assign grid boundaries arbitrarily; they are null.
+This completes the partition $\mathcal D_k$ without changing
+\eqref{eq:cell-count}.
+
+For two points $x=(x_s,x_f)$ and $y=(y_s,y_f)$ in the product
+coordinates of one cell, the ideal exchange
+\[
+ z_0=(y_s,x_f),\qquad w_0=(x_s,y_f)
+\]
+stays in that cell.  These points combine the slow coordinate of one
+input with the fast coordinate of the other.  For the pairs retained
+below, we will construct actual orbit joins $z,w$ whose product
+coordinates differ from $z_0,w_0$ by $o(r)$.
+Removing thin coordinate strips will therefore keep those
+joins in their original cell.  This is why the cells follow the two
+reference spaces, which need not be orthogonal; see
+\Cref{fig:ideal-cell-exchange}.
+
+\begin{figure}[htbp]
+\centering
+\begin{tikzpicture}[x=.9cm,y=.9cm,>=Stealth,
+ point/.style={circle,fill,inner sep=1.6pt}]
+ \draw[thick] (0,0)--(4.8,0)--(6.3,2.8)--(1.5,2.8)--cycle;
+ \draw[dashed,gray] (1.104,0)--(2.604,2.8);
+ \draw[dashed,gray] (3.504,0)--(5.004,2.8);
+ \draw[dashed,gray] (.39,.728)--(5.19,.728);
+ \draw[dashed,gray] (1.05,1.96)--(5.85,1.96);
+ \node[point,label=above left:{$x$}] at (2.154,1.96) {};
+ \node[point,label=below right:{$y$}] at (3.894,.728) {};
+ \node[point,label=above right:{$z_0$}] at (4.554,1.96) {};
+ \node[point,label=below left:{$w_0$}] at (1.494,.728) {};
+ \draw[->] (0,-.4)--(1.5,-.4) node[midway,below] {slow};
+ \draw[->] (-.45,0)--(.05,.933) node[midway,above left] {fast};
+\end{tikzpicture}
+\caption{A two-dimensional schematic of the ideal exchange in a
+product cell.  Dashed lines hold one product coordinate fixed.
+The two coordinate directions may be oblique.  For retained pairs, the
+actual joins are proved to differ from $z_0,w_0$ by $o(r)$ in
+\Cref{lem:switch-orbits}.}
+\label{fig:ideal-cell-exchange}
+\end{figure}
+
+Fix a small number $\tau>0$, independent of $k$.  The interior of a
+product cell retained at this stage consists of points whose every
+product coordinate is at least $\tau r$ from its two cell faces.
+The discarded coordinate strips have at most $2d\tau$ of its Euclidean
+volume.  The constant determinant of $E_Q$ cancels in this fraction.
+Moreover, the ratio between the largest and smallest smooth density on
+the physical cell is $1+O(r)$, uniformly in $Q$.  Thus the total smooth
+volume of these strips is at most $2d\tau+o(1)$.  The corresponding
+strips of width $\tau h_0$ in coarse bins have the same bound.  These
+are estimates for the original measure $\nu$, without conditioning on
+membership in $\mathcal A_k$.
+
+If $M$ has boundary, also exclude points having any iterate at an integer
+time in $[-n,n]$ at distance at most $c_0h_0$ from $\partial M$, where
+$c_0>0$ is fixed.  A smooth boundary collar has volume $O(h_0)$, so
+invariance bounds this loss by $O(nh_0)=o(1)$.  The latter limit follows
+from $Bn\ge n^{79/80}$.  Local coordinate extensions across the boundary
+may be used for uniform derivative bounds; all orbit tubes constructed
+below lie in the original manifold.
+
+Let $R_k$ be the Borel set of points in $\mathcal A_k$ that lie in the
+retained interiors of full product cells, whose two endpoints $S^{\pm n}u$
+lie in coarse bins at coordinate distance at least $\tau h_0$ from
+their faces, and that satisfy the boundary provision when needed.
+Bins not meeting $\mathcal A_k$ contain no such point.  The preceding
+estimates and invariance give
+\begin{equation}\label{eq:retained-point-loss}
+ \nu(M\setminus R_k)\le \eta/100+C_d\tau+o(1).
+\end{equation}
+Choose $\tau$ so that $C_d\tau<\eta/100$, and then increase $k$ so
+that the right-hand side is less than $\eta/4$.
+Both marginals of \eqref{eq:pair-law} are exactly $\nu$.  Consequently,
+for
+\begin{equation}\label{eq:retained-pairs}
+ G_k=(R_k\times R_k)\cap
+       \bigcup_{Q'\text{ a full product cell}}(Q'\times Q'),
+ \qquad \rho_k(G_k)\ge1-\eta/2\ge1-\eta.
+\end{equation}
+The measure $\rho_k$ is supported on pairs in the same partition cell,
+and $R_k$ meets only full product cells.  Hence
+$\rho_k(G_k)=\rho_k(R_k\times R_k)$, and the two marginal bounds give
+the displayed estimate.  No uniform distribution of good points inside
+individual cells is required.
+
+We next fix the maps that will specify the switch.  For each starting
+bin $Q$ and endpoint bin $U$ reached at time $n$ by a point of
+$\mathcal A_k\cap Q$, choose a representative endpoint fast space.
+Let $P^+_{Q,U}:\R^d\to\R^{p_Q}$ be its orthogonal projection, followed
+by a fixed orthonormal identification with $\R^{p_Q}$.  Similarly, for
+an endpoint bin $V$ reached at time $-n$, define
+$P^-_{Q,V}:\R^d\to\R^{d-p_Q}$ from a representative slow space.
+By the endpoint conclusion of \Cref{lem:bin-spaces}, these projections
+restrict to isomorphisms on the actual fast and slow endpoint spaces
+of every corresponding good path, with inverse norm at most $2$ for
+large $k$.  The maps depend only on the displayed bins and the starting
+threshold.  They do not depend on a later choice of input in those bins.
+Zero-dimensional projections and determinants have their usual empty
+space meanings.
+
+For a retained pair $(x,y)$, write $Q$ for its coarse starting bin and
+$U_x,V_x,U_y,V_y$ for the coarse bins of
+$S^nx,S^{-n}x,S^ny,S^{-n}y$, respectively.  If $\chi_U$ denotes the
+coordinate map attached to a bin, define the following smooth maps on
+the open sets where the coordinates make sense:
+\begin{equation}\label{eq:endpoint-maps}
+ \begin{aligned}
+ F_x(u)&=P^+_{Q,U_x}\chi_{U_x}(S^nu),&
+ P_x(u)&=P^-_{Q,V_x}\chi_{V_x}(S^{-n}u),\\
+ F_y(u)&=P^+_{Q,U_y}\chi_{U_y}(S^nu),&
+ P_y(u)&=P^-_{Q,V_y}\chi_{V_y}(S^{-n}u).
+ \end{aligned}
+\end{equation}
+The subscripts denote fixed bin labels.  We seek $z,w$ satisfying
+\begin{equation}\label{eq:four-switch-equations}
+ F_x(z)=F_x(x),\quad P_y(z)=P_y(y),\qquad
+ F_y(w)=F_y(y),\quad P_x(w)=P_x(x).
+\end{equation}
+The next step both solves these equations near the prescribed halves
+and gives a larger neighborhood in which each solution is unique.
+\Cref{fig:switch-data} displays the required exchange of endpoint data.
+
+\begin{figure}[htbp]
+\centering
+\begin{tikzpicture}[>=Stealth,
+ point/.style={circle,draw,minimum size=7mm,inner sep=1pt},
+ future/.style={->,thick,blue!65!black},
+ past/.style={->,thick,dashed,orange!75!black}]
+ \node[point] (x) at (-3,0.85) {$x$};
+ \node[point] (y) at (-3,-0.85) {$y$};
+ \node[point] (z) at (3,0.85) {$z$};
+ \node[point] (w) at (3,-0.85) {$w$};
+ \draw[future] (x) -- node[above] {$F_x$} (z);
+ \draw[future] (y) -- node[below] {$F_y$} (w);
+ \draw[past] (x) -- node[pos=0.27,above,sloped] {$P_x$} (w);
+ \draw[past] (y) -- node[pos=0.27,below,sloped] {$P_y$} (z);
+\end{tikzpicture}
+\caption{The endpoint equations for the complementary switches.
+Solid arrows copy projected future coordinates; dashed arrows copy
+projected past coordinates.  The arrows represent endpoint data,
+not trajectories in the manifold.}
+\label{fig:switch-data}
+\end{figure}
+
+\subsection{A boundary inverse and a nonlinear orbit}
+
+\begin{lemma}\label{lem:switch-orbits}
+For every sufficiently large $k$ and every $(x,y)\in G_k$, the first
+two equations of \eqref{eq:four-switch-equations} have a solution $z$
+with
+\begin{equation}\label{eq:sharp-shadow}
+ \max_{0\le t\le n}d_M(S^tz,S^tx)
+ +\max_{-n\le t\le0}d_M(S^tz,S^ty)
+ \le e^{Cgn}r.
+\end{equation}
+In fixed regular coordinates along these halves the solution is unique
+among displacement arrays of sup norm at most $\delta$.
+The complementary solution $w$ has the analogous properties with $x,y$
+exchanged.  In the product coordinates of the common cell, if
+$x=(x_s,x_f)$ and $y=(y_s,y_f)$, then
+\begin{equation}\label{eq:ideal-switch}
+ z=(y_s,x_f)+o(r),\qquad w=(x_s,y_f)+o(r),
+\end{equation}
+uniformly over all retained pairs.  Both outputs remain in the original
+product cell, and their relevant endpoints remain in the respective
+four original endpoint bins.
+\end{lemma}
+
+\begin{proof}
+\emph{Uniform coordinates.}
+At each reference location choose one of finitely many slightly larger
+regular patches containing a fixed-radius neighborhood of that location.
+At time zero both strings use $\chi_Q$; at the terminal times they use
+the endpoint charts already specified.  Shrinking the fixed radius once,
+the one-step coordinate maps
+$\chi_{t+1}\circ S\circ\chi_t^{-1}$ and their first two derivatives
+are uniformly bounded wherever they will be used.  The inverse
+one-step maps also have uniform first and second derivative bounds.
+For example, differentiating the inverse identity bounds their second
+derivatives by a fixed multiple of the forward second-derivative bound
+and the cube of the inverse first-derivative bound.  This
+follows from a finite atlas with nested compact patches and smoothness
+on compact $M$; it does not depend on the number of steps.  A patch
+can be chosen separately at each internal time.  No iterate $S^n$
+is assigned a uniform second derivative bound.
+
+Use displacement variables $v_t^+$ about $S^tx$, $0\le t\le n$,
+and $v_t^-$ about $S^ty$, $-n\le t\le0$.  Write $A_t^+,A_t^-$
+for the derivative steps in these charts.  Matching the two locations
+at zero and imposing the terminal equations gives the linear system
+\begin{equation}\label{eq:linear-boundary-system}
+ \begin{aligned}
+ v_{t+1}^{\pm}-A_t^{\pm}v_t^{\pm}&=f_t^{\pm},\\
+ v_0^+-v_0^-&=m,\\
+ P^+_{Q,U_x}v_n^+&=h_+,\qquad
+ P^-_{Q,V_y}v_{-n}^-=h_-.
+ \end{aligned}
+\end{equation}
+The matching datum for the actual problem is
+$m=\chi_Q(y)-\chi_Q(x)$; its norm is $O(r)$.
+The actual terminal data are zero.
+
+\emph{The inverse in weighted coordinates.}
+Let $\gamma=\gamma_Q$ and put $\widetilde v_t=e^{-\gamma t}v_t$.
+Use the same weighting on equation data, with the step datum at its
+output time $t+1$.  By \Cref{prop:finite-splitting}, the weighted
+propagators on slow spaces forward and fast spaces backward have norm
+at most $e^{Cbn}e^{-g|t-s|}$.  The associated projections have norm
+at most $e^{Cbn}$.  Geometric sums cost at most
+$C/g\le e^{Cbn}$, by \eqref{eq:switch-scale-order} and the floor on $b$.
+
+On the future half, prescribe a slow vector $a\in E_0^s(x)$.
+Variation of constants gives its slow part at every time by propagating
+$a$ forward and adding the projected step data.  If $s_n$ is the
+resulting slow value at time $n$, the fast value there is uniquely
+\[
+ c_n=(P^+_{Q,U_x}|_{E_n^f(x)})^{-1}
+                (\widetilde h_+-P^+_{Q,U_x}s_n).
+\]
+Propagate $c_n$ and the fast step data backward.  All forced terms have
+norm at most $e^{Cbn}$ times the sup norm of the weighted data.
+The dependence of the resulting fast value at zero on $a$ is the map
+\begin{equation}\label{eq:future-kernel-graph}
+ K_+=-\widetilde A_x(0,n)|_{E_n^f(x)}
+       (P^+_{Q,U_x}|_{E_n^f(x)})^{-1}
+       P^+_{Q,U_x}\widetilde A_x(n,0)|_{E_0^s(x)},
+ \qquad \|K_+\|\le e^{-2gn+Cbn}.
+\end{equation}
+Here $\widetilde A(t,s)=e^{-\gamma(t-s)}A(t,s)$.
+Both decaying factors are present in \eqref{eq:future-kernel-graph};
+there is no assumption that the endpoint projection annihilates the
+slow space.  Thus $\widetilde v_0^+=a+K_+a+q_+$, with $q_+$ bounded
+by the weighted data as above.
+
+On the past half prescribe $b'\in E_0^f(y)$, propagate fast backward,
+use the slow endpoint projection to fix the slow value at $-n$, and
+then propagate slow forward.  This gives
+$\widetilde v_0^-=b'+K_-b'+q_-$, where
+\begin{equation}\label{eq:past-kernel-graph}
+ \|K_-\|\le e^{-2gn+Cbn},\qquad
+ \|q_-\|\le e^{Cbn}\|\text{weighted data}\|_{\infty}.
+\end{equation}
+The map $(a,b')\mapsto a-b'$ from $E_0^s(x)\oplus E_0^f(y)$ to
+the starting coordinate space has inverse norm at most $e^{Cbn}$.
+Indeed the corresponding reference spaces are complementary with this
+bound, and their perturbation is $e^{-cgL}$; multiplication by the
+reference inverse still gives a perturbation tending to zero.
+The terms $K_+a-K_-b'$ are smaller still.  A Neumann series therefore
+solves the matching equation with inverse cost $e^{Cbn}$.
+This proves an $e^{Cbn}$ inverse bound for the full weighted system.
+Removing the weights costs at most $e^{2\gamma(n+1)}$, so its
+unweighted inverse $\mathcal L^{-1}$ satisfies
+\begin{equation}\label{eq:boundary-inverse}
+ \|\mathcal L^{-1}\|_{\infty\to\infty}\le Q_k,
+ \qquad Q_k=e^{Cgn}.
+\end{equation}
+If a group is empty, omit its variables, propagation formula, and
+terminal datum.  The matching map on the remaining spaces and the
+same bounds apply, including the cases $p_Q=0,d$.
+
+\emph{Contraction with a larger uniqueness radius.}
+Taylor expansion of each one-step map gives a residual
+$\mathcal R(v)$ with
+\begin{equation}\label{eq:one-step-remainder}
+ \|\mathcal R(v)\|_\infty\le C\|v\|_\infty^2,
+ \qquad
+ \|\mathcal R(v)-\mathcal R(u)\|_\infty
+       \le C\delta\|v-u\|_\infty
+\end{equation}
+on the sup norm ball of radius $\delta$.  Every residual coordinate
+depends on just its one step, so the constant has no factor $n$.
+The matching and terminal equations are affine and have zero nonlinear
+residual.  The nonlinear problem is
+$v=\mathcal L^{-1}(a_0+\mathcal R(v))$, where
+$\|a_0\|_\infty\le Cr$.
+The scale relations give
+\[
+ CQ_k\delta\longrightarrow0,\qquad Q_kr/\delta\longrightarrow0.
+\]
+For large $k$ the displayed map sends the $\delta$-ball into its
+$\delta/2$-ball and has Lipschitz constant at most $1/4$.
+It has a unique fixed point in the larger ball, with
+$\|v\|_\infty\le 2CQ_kr$.  The exact one-step equations and the
+matching equation make this array one genuine orbit; call its point
+at zero $z$.  Chart bounds give \eqref{eq:sharp-shadow}.
+In the boundary case its distance from each reference orbit is
+$o(h_0)$, so it stays inside $M$.
+
+\emph{Time-zero coordinates and wall margins.}
+For the linear solution with zero step and terminal data, the two
+initial displacement spaces are the graphs in
+\eqref{eq:future-kernel-graph} and \eqref{eq:past-kernel-graph}.
+In the reference product coordinates they differ from the slow and
+fast coordinate spaces by at most
+$e^{Cbn}(e^{-cgL}+e^{-2gn+Cbn})$.  Solving their matching equation
+therefore differs from ideal exchange of the fast and slow coordinates
+by at most
+\begin{equation}\label{eq:linear-switch-error}
+ e^{Cbn}\bigl(e^{-cgL}+e^{-2gn+Cbn}\bigr)r=o(r).
+\end{equation}
+All basis conversion and matching inverse factors are included in the
+constant $C$.  The nonlinear solution differs from the linear one by
+at most $CQ_k(2CQ_kr)^2=e^{Cgn}r^2$ in ordinary coordinates, and still
+by $e^{Cgn}r^2=o(r)$ in product coordinates.  This proves
+\eqref{eq:ideal-switch}.  The proof for $w$ is identical with the
+two inputs exchanged.
+
+Increase $k$ so that the error in each product coordinate is less than
+$\tau r/4$.  The ideal exchanged point has distance at least $\tau r$
+from every product face because both inputs do.  Thus $z,w$ have
+coordinate margins at least $3\tau r/4$ in their original cell.
+Similarly, \eqref{eq:sharp-shadow} in the fixed endpoint charts is
+less than $\tau h_0/4$, so each corresponding endpoint retains margin
+at least $3\tau h_0/4$ in its original coarse bin.  These strict
+margins will recover all labels from the outputs.
+The point obtained does not depend on the auxiliary internal charts or
+on the frame witnesses used to justify the estimates.  Two such
+constructions have the same endpoint equations and sharp radius
+$e^{Cgn}r=o(\delta)$.  In either one's charts both are therefore in
+its uniqueness ball, and hence coincide.
+\end{proof}
+
+\subsection{Relative minors and the joint determinant}
+
+We have constructed complementary orbit joins.  A single join need not
+have a convenient Jacobian, so we now keep both of them.  Their endpoint
+coordinates are permuted, which will cancel the large expansion and
+contraction factors.
+
+For the fixed labels of a retained pair define
+\begin{equation}\label{eq:joint-endpoint-map}
+ H_{ab}(u)=(F_a(u),P_b(u)),\qquad a,b\in\{x,y\}.
+\end{equation}
+Each target has dimension $p_Q+(d-p_Q)=d$ and uses the fixed
+orthonormal coordinates in the projection ranges.
+
+\begin{lemma}\label{lem:switch-jacobian}
+All four derivatives $DH_{xx}(x)$, $DH_{yy}(y)$,
+$DH_{xy}(z)$, and $DH_{yx}(w)$ are nonsingular.  Locally the equations
+\eqref{eq:four-switch-equations} define a smooth pair map
+$\Phi(x,y)=(z,w)$ with
+\begin{equation}\label{eq:pair-jacobian}
+ J_{\nu\otimes\nu}\Phi(x,y)=1+o(1),
+\end{equation}
+uniformly on the retained pairs.  The conclusion includes empty slow
+or fast groups.
+\end{lemma}
+
+\begin{proof}
+Put $p=p_Q$.  For the future map at $x$, use the actual fast and slow
+spaces first, with orthonormal bases within each.  The fast $p$-minor
+has absolute value at least $e^{\Lambda_x n-Cbn}$, where
+$\Lambda_x$ is the sum of its fast rates.  This follows from the
+fast volume lower bound in \Cref{prop:finite-splitting} and from the
+uniformly invertible restriction of the endpoint projection.
+Every other maximal minor contains a slow starting vector and is at
+most $e^{(\Lambda_x-2g)n+Cbn}$ when both groups occur.
+Replacing the actual bases with the common reference basis changes
+their spaces by $e^{-cgL}$.  After a change of orthonormal basis within
+each space, multilinearity and the full exterior norm bound show that
+the leading minor stays nonzero and that all other maximal minors
+are $o(1)$ times its absolute value.  Factors from the inverse of the
+possibly oblique starting basis are at most $e^{Cbn}$.  Call the
+resulting leading absolute minor $A_x$.  More explicitly, when
+$0<p<d$, every nonleading maximal future minor $\Delta$ in the common
+reference basis satisfies
+\[
+ \frac{|\Delta|}{A_x}
+ \le C\left(e^{-2gn+Cbn}+e^{-cgL+Cbn}\right)=o(1).
+\]
+The first term comes from the mixed-exterior estimate.  The second
+comes from the change of spaces, using the full exterior upper bound
+and the leading-minor lower bound.  Increasing $C$ includes the
+oblique-basis factors.  If $p=0$ or $p=d$, there are no nonleading
+maximal future minors, so this comparison is unnecessary.
+
+Define $A_y$ in the same way for the future of $y$.
+For the past maps the leading
+columns are slow, and the backward exterior rate is
+\[
+ \Omega_x=-\sum_{\text{slow at }x}\lambda_j(x).
+\]
+Denote their leading absolute minors by $B_x,B_y$.
+They have lower bounds $e^{\Omega_x n-Cbn}$ and
+$e^{\Omega_y n-Cbn}$, respectively, and all other maximal minors are
+relatively negligible.  No sign is prescribed for $\Lambda_x$ or
+$\Omega_x$ in a perturbation estimate.
+
+We must transfer the future minors from $x$ to $z$ relative to $A_x$,
+even if this leading quantity is small.  Use the reference charts
+along the future of $x$ for both nearby orbits.  By
+\eqref{eq:sharp-shadow}, their one-step derivatives differ by at most
+$C\delta$ for large $k$.  On every unperturbed subinterval the
+$p$-th exterior product has norm at most
+$e^{\Lambda_x\ell+Cbn}$.
+Applying \Cref{lem:relative-exterior} gives, for every fixed $C_0$,
+\begin{equation}\label{eq:relative-minor-transfer}
+ \left\|\bigwedge^pD(S^n)_z-
+                   \bigwedge^pD(S^n)_x\right\|
+       =o\bigl(e^{\Lambda_xn-C_0bn}\bigr),
+\end{equation}
+where the derivatives are represented in these common starting and
+endpoint coordinates.  To see the scale in this application, an
+expansion term with $q\ge1$ altered factors is bounded by
+\[
+ e^{\Lambda_x(n-q)+(q+1)C_1bn}(C\delta)^q.
+\]
+The binomial sum, divided by $e^{\Lambda_xn-C_0bn}$, tends to zero
+because $n\delta e^{Cbn}\to0$.  The bounded factor
+$e^{q|\Lambda_x|}$ covers either sign of the rate.  The same reasoning
+backward uses $\Omega_y$ and $e^{q|\Omega_y|}$.
+Fixed endpoint projections and the common starting basis preserve
+\eqref{eq:relative-minor-transfer}, after increasing $C_0$ if an
+inverse basis factor is needed.  Hence at $z$ the leading future and
+past minors are $A_x(1+o(1))$ and $B_y(1+o(1))$, with the same relative
+smallness of all the other minors.  At $w$ they are $A_y$ and $B_x$.
+
+Expand each full determinant along its $p$ future rows.  There is one
+leading product, from fast future columns and slow past columns.
+Every other term has a relatively negligible maximal minor.  In the
+common starting basis the four absolute determinants are therefore
+\begin{equation}\label{eq:four-leading-products}
+ \begin{array}{c|cccc}
+ \text{map and point}&H_{xx}(x)&H_{yy}(y)&H_{xy}(z)&H_{yx}(w)\\ \hline
+ \text{absolute determinant}&
+ A_xB_x(1+o(1))&A_yB_y(1+o(1))&
+ A_xB_y(1+o(1))&A_yB_x(1+o(1)).
+ \end{array}
+\end{equation}
+All leading factors are positive.  In particular these maps are
+locally invertible.  If $p=0$, future targets and minors are empty and
+$A_x=A_y=1$; if $p=d$, the past minors are $B_x=B_y=1$.  The same
+determinants and conclusions apply, without mixed minors of the
+missing kind.
+
+The defining equations say that
+$(H_{xy}(z),H_{yx}(w))$ is a fixed permutation of
+$(H_{xx}(x),H_{yy}(y))$.  Differentiation gives the exact identity
+\begin{equation}\label{eq:exact-switch-determinant}
+ |\det D\Phi(x,y)|=
+ \frac{|\det DH_{xx}(x)|\,|\det DH_{yy}(y)|}
+ {|\det DH_{xy}(z)|\,|\det DH_{yx}(w)|}.
+\end{equation}
+Each full derivative represented in the common product basis differs
+from its coordinate derivative by the factor $\det E_Q$.  There are
+two such factors in the numerator and two in the denominator, so
+they cancel exactly.  Substituting \eqref{eq:four-leading-products}
+cancels $A_x,B_x,A_y,B_y$ individually, with no comparison between the
+rates of $x$ and $y$.  Thus the coordinate Jacobian is $1+o(1)$.
+All four starting points lie in the same cell of diameter $O(r)$.
+The ratio of the two output smooth densities to the two input densities
+is therefore $1+o(1)$ as well.  This proves
+\eqref{eq:pair-jacobian}.
+\end{proof}
+
+\subsection{Measurability, recovered labels, and global injectivity}
+
+The Jacobian calculation is local.  To convert it into the desired
+measure inequality we must ensure that no multiplicity arises from
+different choices of bins or local branches.
+
+\begin{lemma}\label{lem:switch-injective}
+The map $\Phi:G_k\to M\times M$, $\Phi(x,y)=(z,w)$, is measurable
+and injective.  Its restriction to countably many measurable pieces
+agrees with smooth local diffeomorphisms having the Jacobian in
+\eqref{eq:pair-jacobian}.
+\end{lemma}
+
+\begin{proof}
+First fix a retained pair.  Its cell and four endpoint labels are
+locally constant among retained inputs near it, because of the positive
+wall margins.  By \Cref{lem:switch-jacobian}, the implicit equations
+define a smooth local pair branch near its outputs.  This branch
+agrees with the constructed switch on nearby retained inputs.
+Indeed, for this fixed $k$, continuity of the finitely many maps
+$S^t$, $|t|\le n$, makes their reference halves arbitrarily close
+when their inputs are close.  The constructed solutions have sharp
+radius $e^{Cgn}r=o(\delta)$ about those halves.  Both the old local
+implicit branch and the construction for the nearby pair therefore
+lie in the old $\delta$-tube, and satisfy the same fixed endpoint
+equations with the nearby right-hand sides.  The contraction uniqueness
+argument identifies them even for these changed data: two solutions
+$u,v$ in the old ball satisfy
+$\mathcal L(v-u)=\mathcal R(v)-\mathcal R(u)$, so
+$\|v-u\|_\infty\le CQ_k\delta\|v-u\|_\infty$ forces equality.
+Both candidate arrays here use the old reference halves. Their
+matching datum is therefore the same old coordinate difference;
+only the prescribed endpoint values change with the nearby inputs.
+The charts along the old pair may be kept fixed in this neighborhood.  In particular no regularity of a choice of good
+frame witnesses is needed.
+
+The retained set is Borel, and $M\times M$ is second countable.
+A countable subcollection of these relative neighborhoods covers it.
+Subtracting earlier members partitions it into countably many Borel
+pieces.  On each piece $\Phi$ is the restriction of the indicated
+smooth local diffeomorphism; this proves measurability and supplies
+the pieces for change of variables.
+
+For injectivity suppose
+$\Phi(x,y)=\Phi(x',y')=(z,w)$.  The interiors of product cells are
+disjoint, and both outputs stayed in the original cell.  Thus the
+two input pairs have the same product cell and hence the same starting
+bin $Q$, threshold, and dimensions.  The four endpoint-bin labels
+are recovered from the ordered output pair by
+\begin{equation}\label{eq:recovered-bin-labels}
+ (U_x,V_x,U_y,V_y)=
+ \bigl(\operatorname{bin}(S^nz),\operatorname{bin}(S^{-n}w),
+       \operatorname{bin}(S^nw),\operatorname{bin}(S^{-n}z)\bigr).
+\end{equation}
+All four endpoints have positive margins in their bins.  Therefore
+these labels, and exactly the four projections and coordinate maps in
+\eqref{eq:endpoint-maps}, are the same for the competing inputs.
+
+On the future half, both $x$ and $x'$ are within $e^{Cgn}r$ of $z$;
+on the past half both are within that distance of $w$.  Consequently
+their full orbits on $[-n,n]$ are within $2e^{Cgn}r=o(\delta)$ of
+each other.  Moreover the recovered equations give
+\[
+ F_x(x')=F_x(z)=F_x(x),\qquad
+ P_x(x')=P_x(w)=P_x(x).
+\]
+Apply the boundary uniqueness argument of \Cref{lem:switch-orbits}
+with both reference halves equal to the true good orbit of $x$.
+It uses $x$'s endpoint projections, zero matching datum, and zero
+terminal data.  The orbit of $x$ is one solution.  In the same charts
+the orbit of $x'$ is another solution in its $\delta$-ball, since its
+distance is $o(\delta)$ even after fixed chart conversions.  Uniqueness
+gives $x'=x$.  The same argument gives $y'=y$.
+Only the true input orbits were required to be good; neither $z$ nor
+$w$ was required to belong to $\mathcal A_k$ or to $R_k$.
+\end{proof}
+
+\begin{proof}[Proof of \Cref{thm:switching}]
+The construction above applies for all sufficiently large $k$.
+Equation \eqref{eq:switch-scales} gives
+$\sum_kB_k^{16}=\sum_kb_k^2<\infty$; any finitely many unused terms
+can be chosen positively.  The partition bound follows from
+\eqref{eq:cell-count}, retention from \eqref{eq:retained-pairs}, and
+uniform shadowing from \eqref{eq:sharp-shadow}.
+
+It remains to prove the output measure bound.  On the square of a
+positive-volume cell $Q'$, the original law $\rho_k$ has density
+$1/\nu(Q')$ with respect to $\nu\otimes\nu$.  The map $\Phi$ sends
+retained pairs in that square into the same square.  By
+\Cref{lem:switch-jacobian}, its inverse volume Jacobian is at most $2$
+for all sufficiently large $k$.  Apply change of variables on the
+countably many pieces of \Cref{lem:switch-injective}.  Their images
+are disjoint by global injectivity.  Summing therefore gives
+\[
+ \Phi_*(\rho_k|_{G_k})\le 2\rho_k.
+\]
+The factor $1/\nu(Q')$ is unchanged.  Restriction to endpoint-label
+pieces or to $G_k$ is never followed by normalization, and zero-volume
+cells contribute nothing.  Taking first marginals proves
+\eqref{eq:switch-law}, because the first marginal of $\rho_k$ is
+$\nu$.  The construction already handled the optional smooth boundary.
+\end{proof}
diff --git a/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/build/sections/05-words.tex b/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/build/sections/05-words.tex
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+\section{Words with prescribed relations and recognizable boundaries}
+\label{sec:words}
+
+We now construct finite families of words that will define the symbolic
+system.  At each level, short lists of constituent labels have large
+entropy, although specified pairs of labels satisfy a bilinear relation.
+We also require that a word can be recognized, together with its alignment,
+from a sufficiently accurate copy.  The construction in this section is
+entirely finite at each step; no invariant probability is chosen yet.
+
+Nested-word constructions and probabilistic control of frequencies and
+shifted comparisons have precedents in
+\cite[\S4.1 and Proposition~43]{ForemanRudolphWeiss2011}.
+Related word and phase frameworks appear in
+\cite[\S\S4.2.3--4.2.4]{ForemanWeiss2022}. Our graph constraints and the
+per-word condition below are proved separately; in particular, we do
+not require every lower-level word to occur in every parent or import
+unique ergodicity from those constructions.
+
+All positions in a finite word are numbered from zero.  For a word $u$,
+write $u[t,t+q)$ for its consecutive subword of length $q$ starting at $t$.
+For two words of the same length, $d_{\mathrm H}(u,v)$ denotes the number
+of coordinates at which they differ.  Thus this distance counts symbols
+or constituent labels according to the alphabet being used.  For
+probabilities $p,q$ on a finite set, we use
+\[
+  \TV(p,q)=\frac12\sum_a |p(a)-q(a)|.
+\]
+Write $\mathcal L(Y)$ for the distribution of a random variable $Y$.
+
+\subsection{Parameters and the four word properties}
+
+Each new word will concatenate words from the preceding family.
+We control two sampling experiments: choosing a word uniformly from
+its family, and fixing a parent at the next level and choosing one of
+its constituents uniformly. We compare the laws of windows at the
+same internal offset in these two chosen words. Requiring this
+approximation for every parent will allow arbitrary parent weights
+in the eventual invariant measure.
+
+The construction first samples constituent lists satisfying the
+relations inside independent groups. It then keeps lists satisfying
+the per-parent approximation, and finally selects a family with the
+required window distributions and distances. The parameters below
+provide enough groups and candidate words for these selections.
+
+Fix
+\begin{equation}\label{eq:word-parameters}
+  \varepsilon_i=2^{-i-2},\qquad
+  L_i=2^{2s_i}-1,\qquad \ell_i=\log L_i,\qquad
+  K_i=\lfloor\sqrt{\ell_i}\rfloor,
+\end{equation}
+and put
+\begin{equation}\label{eq:word-lengths}
+  N_i=2^{K_i},\qquad H_i=2^{\lfloor K_i/2\rfloor},\qquad
+  D_i^*=2^{\lfloor K_i/8\rfloor},\qquad
+  m_1=1,\qquad m_{i+1}=m_iN_i.
+\end{equation}
+Starting from the integer $s_1$, define $2s_{i+1}$ to be the greatest even
+integer not exceeding
+\begin{equation}\label{eq:word-recursion}
+  X_i:=\frac{\varepsilon_i}{100}N_i\ell_i.
+\end{equation}
+The proof below shows that this recursion is well defined at every level
+if $K_1\ge256$.  There is no need to optimize this initial size.
+
+Let $\mathcal A_i=\F_2^{2s_i}\setminus\{0\}$.  On the full vector space
+use the nondegenerate alternating form
+\[
+  [u,v]_i=\sum_{a=1}^{s_i}
+     \bigl(u_a v_{s_i+a}+u_{s_i+a}v_a\bigr)\in\F_2.
+\]
+The fixed symbol alphabet is $\mathcal A=\mathcal A_1$.
+Each family $\mathcal W_i\subset\mathcal A^{m_i}$ will have $L_i$ members.
+Once it has been constructed, choose a bijection with $\mathcal A_i$;
+this equips its word labels with the form $[\cdot,\cdot]_i$.
+In particular, a word label and its vector label are identified only
+through this fixed bijection.
+
+Whenever $w$ is a concatenation of $N_i$ members of $\mathcal W_i$, let
+\[
+  c_i(w)=(u_0,\ldots,u_{N_i-1})\in\mathcal W_i^{N_i}
+\]
+be its list of constituents.  The list is unambiguous because all members
+of $\mathcal W_i$ have the same length and are distinct.
+Here $m_i$ is the symbol length, $L_i$ counts distinct level-$i$
+words, and $N_i$ counts constituent occurrences in a level-$(i+1)$
+word, with repetitions allowed. In the properties below, $H_i$ is
+the group length and $D_i^*$ bounds the window lengths and prescribed
+power-of-two lags; these lengths and lags are measured in level-$i$
+constituent positions.
+We fix the approximation tolerance
+\begin{equation}\label{eq:word-tv-tolerance}
+  \vartheta=10^{-4}.
+\end{equation}
+
+\begin{proposition}[Finite word families]\label{prop:words}
+Choose $s_1$ so that $K_1\ge256$, and define the parameters by
+\eqref{eq:word-parameters}--\eqref{eq:word-recursion}.
+There are families $\mathcal W_i\subset\mathcal A^{m_i}$ of cardinality
+$L_i$, with $\mathcal W_1=\mathcal A$, such that every
+$w\in\mathcal W_{i+1}$ is a concatenation of $N_i$ members of
+$\mathcal W_i$ and the following properties hold.
+\begin{enumerate}[label=\textup{(W\arabic*)},ref=W\arabic*,leftmargin=*]
+\item\label{word:graph}\textbf{Relations within groups.}
+Write $c_i(w)=(u_0,\ldots,u_{N_i-1})$.
+For every $0\le b<N_i/H_i$, every
+$0\le k\le\lfloor K_i/8\rfloor$, and every
+$0\le r<r+2^k<H_i$,
+\[
+  [u_{bH_i+r},u_{bH_i+r+2^k}]_i=0.
+\]
+\item\label{word:distance}\textbf{Distance between words and shifted words.}
+If $v,w\in\mathcal W_{i+1}$ are distinct, then
+\[
+  d_{\mathrm H}\bigl(c_i(v),c_i(w)\bigr)
+     \ge(1-\varepsilon_i)N_i.
+\]
+For all $v,w,z\in\mathcal W_{i+1}$, with repetitions allowed, and all
+$1\le t<N_i$,
+\[
+  d_{\mathrm H}\bigl((c_i(v)c_i(w))[t,t+N_i),c_i(z)\bigr)
+     \ge(1-\varepsilon_i)N_i.
+\]
+\item\label{word:window}\textbf{Distributions of short windows.}
+Let $U$ be uniform on $\mathcal W_{i+1}$.
+For each $0\le t\le N_i-D_i^*$ there is a probability $Q_{i,t}$ on
+$\mathcal W_i^{D_i^*}$ such that
+\[
+  \TV\bigl(\mathcal L(c_i(U)[t,t+D_i^*)),Q_{i,t}\bigr)
+     \le\vartheta.
+\]
+Moreover, for $1\le d'\le D_i^*$, any distinct positions
+$0\le r_1<\cdots<r_{d'}<D_i^*$, and any prescribed labels
+$v_1,\ldots,v_{d'}\in\mathcal W_i$,
+\begin{equation}\label{eq:word-sublist}
+  Q_{i,t}\{u:u_{r_j}=v_j\text{ for }1\le j\le d'\}
+       \le 2\left(\frac{2^{2K_i}}{L_i}\right)^{d'}.
+\end{equation}
+\item\label{word:frequency}\textbf{Approximation in each individual word.}
+Suppose $i\ge2$.  For every fixed $w\in\mathcal W_{i+1}$, write
+$c_i(w)=(u_0,\ldots,u_{N_i-1})$ and let $J$ be uniform on
+$\{0,\ldots,N_i-1\}$.  If $U$ is uniform on $\mathcal W_i$, then for
+each $0\le t\le N_{i-1}-D_{i-1}^*$,
+\[
+ \TV\left(
+   \mathcal L\bigl(c_{i-1}(u_J)[t,t+D_{i-1}^*)\bigr),
+   \mathcal L\bigl(c_{i-1}(U)[t,t+D_{i-1}^*)\bigr)
+ \right)\le\vartheta.
+\]
+The first law averages the constituents of the single fixed word $w$;
+it does not average over $\mathcal W_{i+1}$.
+\end{enumerate}
+\end{proposition}
+
+Property~(\ref{word:frequency}) compares the two sampling experiments
+described above. The following elementary estimate controls empirical
+laws both within each parent and across the selected family.
+
+\begin{lemma}[Averages of probability vectors]\label{lem:word-tv}
+Let $Y_1,\ldots,Y_q$ be independent random probability vectors on a set
+of cardinality $A$, all with mean $p$.  Then
+\[
+  \E\,\TV\left(\frac1q\sum_{b=1}^qY_b,p\right)
+     \le\frac12\sqrt{\frac Aq}.
+\]
+Coordinates within an individual vector need not be independent.
+\end{lemma}
+\begin{proof}
+For $\widehat p=q^{-1}\sum_bY_b$, independence between vectors gives
+\[
+ \sum_a\operatorname{Var}(\widehat p(a))
+   =\frac1{q^2}\sum_{b,a}\operatorname{Var}(Y_b(a))
+   \le\frac1{q^2}\sum_b\E\sum_aY_b(a)^2
+   \le\frac1q.
+\]
+Apply the inequality $\E|Z|\le(\E|Z|^2)^{1/2}$ coordinatewise and then
+Cauchy--Schwarz to the sum over the $A$ coordinates.
+\end{proof}
+
+\begin{proof}[Proof of \Cref{prop:words}]
+\emph{Growth of the fixed parameters.}
+Whenever $X_i\ge6$, rounding in \eqref{eq:word-recursion} gives
+\begin{equation}\label{eq:word-rounding}
+  X_i-3<\ell_{i+1}\le X_i,\qquad
+  \ell_{i+1}\ge\frac{X_i}{2}
+       =\frac{\varepsilon_i}{200}2^{K_i}\ell_i.
+\end{equation}
+Indeed, $X_i-2<2s_{i+1}\le X_i$, and
+$2s_{i+1}-1<\log(2^{2s_{i+1}}-1)<2s_{i+1}$ when $2s_{i+1}\ge2$.
+We claim inductively that
+\begin{equation}\label{eq:word-growth}
+ K_i\ge256,\qquad
+ \varepsilon_iK_i\ge32\,2^{i-1},\qquad
+ K_{i+1}\ge2^{K_i/3}\ge K_i^2\ge4K_i.
+\end{equation}
+The first two inequalities hold at $i=1$.  If they hold at level $i$,
+then $\ell_i\ge K_i^2$ implies
+\[
+  X_i\ge\frac8{25}K_i2^{K_i}\ge6,
+  \qquad
+  \ell_{i+1}\ge\frac4{25}K_i2^{K_i}\ge2^{K_i}.
+\]
+Consequently
+$K_{i+1}\ge2^{K_i/2}-1\ge2^{K_i/3}\ge K_i^2\ge4K_i$.
+These elementary inequalities hold for every $K_i\ge256$;
+for example, $K/3-2\log K$ is increasing there and is positive at $256$.
+Since $\varepsilon_{i+1}=\varepsilon_i/2$, the second bound in
+\eqref{eq:word-growth} propagates as well.  In particular, all parameters
+are positive and defined, $H_i$ divides $N_i$, and $D_i^*\le H_i$.
+
+\smallskip\noindent
+\emph{Sampling the relations.}
+Suppose that $\mathcal W_i$ and its vector labeling have been fixed.
+Generate a list $(U_0,\ldots,U_{N_i-1})$ in independent consecutive
+groups of $H_i$ positions.  Within each group, proceed from left to
+right.  Choose each new label uniformly among the nonzero vectors
+orthogonal to all earlier labels at the lags
+$2^k$, $0\le k\le\lfloor K_i/8\rfloor$, that stay inside the group.  There are at most $\lfloor K_i/8\rfloor+1\le K_i$
+linear equations.  Their common kernel has codimension at most $K_i$,
+whether or not the equations are independent.  As $2s_i\ge\ell_i\ge K_i^2$,
+the number of nonzero choices is at least
+\begin{equation}\label{eq:word-choices}
+  2^{2s_i-K_i}-1
+    \ge2^{2s_i-K_i-1}
+    \ge\frac{L_i}{2^{2K_i}}.
+\end{equation}
+Denote this unconditioned law on $\mathcal W_i^{N_i}$ by $P_i^0$.
+Every sampled list satisfies the relations in Property~(\ref{word:graph}).
+
+For later use, set $\zeta_i=2^{2K_i}/L_i$ within this proof.
+The conditional probability of any specified label at any position,
+given all preceding positions, is at most $\zeta_i$.
+If $r_1<\cdots<r_{d'}$ are specified positions, conditioning first at
+$r_{d'}$ and then successively at the earlier specified positions gives
+\begin{equation}\label{eq:word-raw-sublist}
+  P_i^0\{U_{r_j}=v_j\text{ for all }j\}\le\zeta_i^{d'}.
+\end{equation}
+All unspecified labels are integrated out.  This argument does not
+require independence within a group.
+
+Each individual $U_r$ nevertheless has a uniform marginal on
+$\mathcal W_i$.  To see this, the group of linear maps preserving the
+alternating form acts transitively on its nonzero vectors.  Given such
+a vector $v$, nondegeneracy provides $w$ with $[v,w]_i=1$; the plane
+spanned by $v,w$ is nondegenerate, and its orthogonal complement carries
+a nondegenerate alternating form of dimension two less.  Induction
+extends $v,w$ to a symplectic basis, proving transitivity by mapping one
+such basis to another.  The sequential rule is equivariant under these
+maps, including its uniform choices, so each marginal is invariant
+under a transitive action and hence uniform.
+
+\smallskip\noindent
+\emph{Imposing the approximation in each word.}
+For $i\ge2$ and an offset $0\le t\le N_{i-1}-D_{i-1}^*$, let
+\[
+ f_t(u)=c_{i-1}(u)[t,t+D_{i-1}^*),\qquad u\in\mathcal W_i,
+\]
+and let $p_t$ be its law under uniform choice of $u$.
+For each of the $q_i=N_i/H_i$ independently sampled groups, form the
+empirical probability vector of the $H_i$ values $f_t(U_r)$ in that
+group.  Its mean is $p_t$ by the uniform one-label marginals just proved.
+The average of these independent vectors is exactly the law of
+$f_t(U_J)$ when $J$ is uniform over the entire sampled list.
+The alphabet of $f_t$ has cardinality at most
+$A_i=L_{i-1}^{D_{i-1}^*}$.
+Thus \Cref{lem:word-tv}, Markov's inequality, and a union over at most
+$N_{i-1}$ offsets bound the probability of violating any requirement in
+Property~(\ref{word:frequency}) by
+\begin{equation}\label{eq:word-filter-budget}
+  F_i:=\vartheta^{-1}N_{i-1}
+       \sqrt{\frac{L_{i-1}^{D_{i-1}^*}}{N_i/H_i}}.
+\end{equation}
+Below we verify $F_i<1/2$ at every level $i\ge2$.
+
+Let $\mathcal E_i$ be the event that all these requirements hold, and
+let $P_i=P_i^0(\,\cdot\mid\mathcal E_i)$.
+For $i=1$, take $\mathcal E_1$ to be the whole sample space and
+$P_1=P_1^0$.  In all cases $P_i^0(\mathcal E_i)\ge1/2$, so
+\eqref{eq:word-raw-sublist} yields
+\begin{equation}\label{eq:word-filtered-sublist}
+ P_i\{U_{r_j}=v_j\text{ for all }j\}\le2\zeta_i^{d'}.
+\end{equation}
+The factor two is paid once for the whole conditioning event.
+No uniform marginal or independence between groups is asserted for
+$P_i$.  The relations hold on its support, as does
+Property~(\ref{word:frequency}) for each individual list.
+
+\smallskip\noindent
+\emph{Selecting the finite family.}
+Put $M_i=L_{i+1}$ and sample $M_i$ independent complete lists with law
+$P_i$.  For each offset $0\le t\le N_i-D_i^*$, let $Q_{i,t}$ be the
+law under $P_i$ of the length-$D_i^*$ window at $t$.
+It satisfies \eqref{eq:word-sublist} by
+\eqref{eq:word-filtered-sublist}.
+The empirical window law of the $M_i$ samples is an average of
+independent point-mass probability vectors on an alphabet of size
+$L_i^{D_i^*}$.  A second application of \Cref{lem:word-tv} and Markov's
+inequality, now with a union over at most $N_i$ offsets, bounds the
+probability of violating any required window approximation by
+\begin{equation}\label{eq:word-window-budget}
+  E_i:=\vartheta^{-1}N_i
+        \sqrt{\frac{L_i^{D_i^*}}{L_{i+1}}}.
+\end{equation}
+We will have $E_i<1/4$ at every level.
+
+It remains to impose the distances, including comparisons in which
+some of the sampled-word indices coincide.  First use the reference
+law in which the $N_i$ labels in each distinct indexed list are
+independent and uniform.  Consider one pairwise comparison between
+distinct indices, or one comparison of a target list with a shifted
+window from two source lists.  In the latter case write $t$ for its
+offset, with $1\le t<N_i$; any of the three indices may coincide.
+
+For a specified set of matching positions, form a multigraph whose
+vertices are variables (sample index, position) and whose edges are
+the asserted equalities.  Every target position occurs once.  The
+source positions also occur once: they use $t,\ldots,N_i-1$ in the
+first source and $0,\ldots,t-1$ in the second, disjoint ranges even if
+the source indices agree.  The degree is therefore at most two,
+counting repeated edges.  There is no loop.  In a pairwise comparison
+the indices are distinct, and in a shifted comparison the source
+position differs from the target position by $t$ modulo $N_i$.
+
+A connected component with $v$ vertices requires $v-1$ independent
+equalities and has probability $L_i^{-(v-1)}$ under the reference law.
+A path has one fewer edge than vertices; a cycle has equally many,
+with at least two edges.  Thus every component imposes at least half
+as many independent equalities as edges, including a pair of parallel
+edges.  If $e$ positions are specified to match, their probability is
+at most $L_i^{-e/2}$.
+Failure of a distance requirement entails at least
+$\lceil\varepsilon_iN_i\rceil$ matching positions.  Summing over their
+possible sets, of which there are at most $2^{N_i}$, gives the bound
+\begin{equation}\label{eq:word-reference-distance}
+  2^{N_i}L_i^{-\varepsilon_iN_i/2}.
+\end{equation}
+
+By \eqref{eq:word-choices}, a complete list under $P_i^0$ has likelihood
+ratio at most $2^{2K_iN_i}$ relative to the reference law, and under
+$P_i$ at most $2\,2^{2K_iN_i}$.
+Each comparison uses at most three distinct sampled lists, so its
+probability under the actual sampling law is at most
+$(2\,2^{2K_iN_i})^3$ times \eqref{eq:word-reference-distance}.
+Repeated indices do not introduce an additional independent list.
+There are fewer than $4N_iM_i^3$ pairwise and shifted comparisons in
+total.  Their union failure probability is consequently at most
+\begin{equation}\label{eq:word-distance-budget}
+  T_i:=32N_iL_{i+1}^3\,
+       2^{(6K_i+1)N_i}L_i^{-\varepsilon_iN_i/2}.
+\end{equation}
+We will verify $T_i<1/4$.
+
+These estimates address separate requirements: the first sampling law
+enforces the relations, conditioning enforces the approximation in
+each individual word, and the final independent samples provide the
+window laws and distances.  We now check that the fixed initial size
+makes all three probability bounds hold at every step.
+
+\smallskip\noindent
+\emph{Uniform bounds from one initial choice.}
+For $K\ge256$, set
+\[
+  B_0(K)=14+K+\tfrac12\,2^{K/8}(K+1)^2.
+\]
+The elementary inequalities
+$14+K\le2^{K/8}$ and $(K+1)^2\le2^{K/8}$ hold at $256$ and persist
+thereafter, as the ratios of their left sides to their right sides
+decrease.  Hence
+\begin{equation}\label{eq:word-common-bound}
+ B_0(K)\le2^{K/4}\le\tfrac18\,2^{K/3}.
+\end{equation}
+We also have $\log(\vartheta^{-1})<14$ and
+$K_i^2\le\ell_i<(K_i+1)^2$.
+For the first probability bound, put $K=K_{i-1}$ and use
+$\log(N_i/H_i)\ge K_i/2$ to obtain
+\[
+ \log F_i
+   \le B_0(K)-\frac{K_i}{4}
+   \le-\frac18\,2^{K/3}.
+\]
+For the second, put $K=K_i$.  Since $\ell_{i+1}\ge2^K$,
+\[
+ \log E_i
+   \le B_0(K)-\frac{\ell_{i+1}}2
+   \le2^{K/4}-\frac{2^K}{2}
+   \le-\frac{2^K}{4}.
+\]
+Finally \eqref{eq:word-rounding} and
+$\varepsilon_i\ell_i\ge\varepsilon_iK_i^2\ge32K_i$ give
+\begin{align*}
+ \log T_i
+ &\le5+K_i+N_i
+       \left(6K_i+1-\frac{47}{100}\varepsilon_i\ell_i\right)\\
+ &\le5+K_i-9K_i2^{K_i}
+  \le-K_i2^{K_i}.
+\end{align*}
+It follows that
+\begin{equation}\label{eq:word-uniform-budgets}
+ F_i\le2^{-2^{K_{i-1}/3}/8}<\tfrac12\quad(i\ge2),\qquad
+ E_i\le2^{-2^{K_i}/4}<\tfrac14,\qquad
+ T_i\le2^{-K_i2^{K_i}}<\tfrac14.
+\end{equation}
+All right sides are decreasing functions of their $K$ argument.
+They apply simultaneously at every level by \eqref{eq:word-growth};
+no later adjustment of $s_1$ is made.
+
+The conditioning event thus has positive probability.  Under the
+conditioned law the two subsequent failures have total probability
+less than $1/2$, so a tuple satisfying every requirement exists.
+Its pairwise distance requirement is imposed on distinct sample
+indices and has a positive lower bound.  Equal sampled words would
+have distance zero, so the tuple has exactly $L_{i+1}$ distinct
+members.  Flatten their constituent lists to form $\mathcal W_{i+1}$.
+The empirical distribution is now exactly the uniform distribution
+on this set, without deduplication or a change of weights.
+It therefore has all four asserted properties.  Choose its bijection
+with $\mathcal A_{i+1}$ and continue the induction.
+\end{proof}
+
+\subsection{Recognition at every symbol offset}
+
+The distances in \Cref{prop:words} concern constituent labels.
+The next consequence converts them into distances in the fixed
+alphabet $\mathcal A$, including shifts that do not respect any
+lower-level boundary. In coding terminology the two conclusions below
+are a positive minimum distance and a positive \emph{comma-free index}:
+the latter measures distance from every nontrivial splice of two legal
+words. This is the synchronization notion described in
+\cite[\S1]{Tonchev2003}; the recursive lower bound here is proved directly.
+
+\begin{lemma}[Recognition of words and alignment]\label{lem:recognition}
+For the families in \Cref{prop:words}, put
+\[
+ d_i=\prod_{j=1}^{i-1}(1-\varepsilon_j),\qquad
+ d_*=\prod_{j=1}^{\infty}(1-\varepsilon_j)>0.
+\]
+Distinct $v,w\in\mathcal W_i$ satisfy
+\[
+ d_{\mathrm H}(v,w)\ge d_i m_i\ge d_*m_i.
+\]
+For all $v,w,z\in\mathcal W_i$ and every $1\le t<m_i$,
+\[
+ d_{\mathrm H}\bigl((vw)[t,t+m_i),z\bigr)
+     \ge d_i m_i\ge d_*m_i.
+\]
+These distances count symbols in $\mathcal A$, and repeated source
+words are allowed.
+\end{lemma}
+\begin{proof}
+The finite products are bounded below by
+$1-\sum_{j\ge1}\varepsilon_j=3/4$, proving that $d_*>0$.
+At level one, distinct one-symbol words have distance one and the
+offset assertion is vacuous.
+
+Suppose both assertions hold at level $i$.
+Distinct level-$(i+1)$ words differ in at least
+$(1-\varepsilon_i)N_i$ constituent labels by
+\Cref{prop:words}, Property~(\ref{word:distance}).
+Each unequal constituent pair contributes at least $d_i m_i$
+symbol mismatches, giving the bound $d_{i+1}m_{i+1}$.
+
+Now consider a nonzero offset $t<m_{i+1}$ in two concatenated
+level-$(i+1)$ words, compared with a third such word.
+If $m_i$ divides $t$, the offset $t/m_i$ lies between $1$ and $N_i-1$.
+The constituent-label offset requirement gives at least
+$(1-\varepsilon_i)N_i$ unequal pairs, and the preceding distinct-word
+bound again gives $d_{i+1}m_{i+1}$ symbol mismatches.
+If $m_i$ does not divide $t$, each of the $N_i$ aligned target
+constituents faces a length-$m_i$ source window with the same nonzero
+offset modulo $m_i$.  That source window crosses exactly two
+consecutive legal level-$i$ words.  The inductive offset assertion
+therefore gives at least $d_i m_i$ mismatches on each target
+constituent.  It applies also when the two source constituents lie
+on opposite sides of the join of the two larger source words.
+Adding over the disjoint target constituents gives
+$d_i m_{i+1}\ge d_{i+1}m_{i+1}$, completing the induction.
+\end{proof}
+
+\subsection{The number and size of the available scales}
+
+The word construction must retain enough label information across
+levels to compete with the summable smooth-switching costs.
+The following estimates are the quantitative input for that comparison.
+
+\begin{lemma}[Word rates]\label{lem:rates}
+Let $R_i=\ell_i/m_i$.  Every $m_i$ is a power of two, with
+\[
+  \log m_i=\sum_{j<i}K_j.
+\]
+The rates satisfy
+\begin{equation}\label{eq:word-rate-decay}
+  R_{i+1}\le\frac{\varepsilon_i}{100}R_i,
+  \qquad R_i\longrightarrow0.
+\end{equation}
+They also have the lower bound
+\begin{equation}\label{eq:word-rate-lower}
+ R_i\ge R_1\,200^{-(i-1)}
+          2^{-(i-1)(i+4)/2}
+     \ge R_1\,2^{-(i-1)(i+20)/2},
+\end{equation}
+and
+\begin{equation}\label{eq:word-rate-divergence}
+  \sum_{i=1}^{\infty}K_iR_i^{16}=\infty.
+\end{equation}
+\end{lemma}
+\begin{proof}
+The formula for $m_i$ follows from $m_1=1$ and $N_i=2^{K_i}$.
+Dividing \eqref{eq:word-rounding} by $m_{i+1}=N_i m_i$ gives
+\[
+ \frac{\varepsilon_i}{200}R_i
+ \le R_{i+1}\le\frac{\varepsilon_i}{100}R_i.
+\]
+Since $\varepsilon_i\le1/8$, the upper bound implies
+$R_i\le800^{-(i-1)}R_1\to0$, proving
+\eqref{eq:word-rate-decay}. Iterating the lower bound and using
+$\sum_{j<i}(j+2)=(i-1)(i+4)/2$ prove the first inequality in
+\eqref{eq:word-rate-lower}; the second uses $\log200<8$.
+Also \eqref{eq:word-growth} implies
+$K_i\ge K_1^{2^{i-1}}$.  Hence
+\[
+ \log(K_iR_i^{16})
+  \ge 2^{i-1}\log K_1+16\log R_1-8(i-1)(i+20)
+  \longrightarrow\infty.
+\]
+In particular the terms in \eqref{eq:word-rate-divergence} do not
+tend to zero, and the series diverges.
+\end{proof}
+
+The families are now fixed.  Their per-word approximation transfers
+short-window entropy to any invariant measure on the resulting
+sequence space, while \Cref{lem:recognition} recovers block labels
+and phases from small symbol errors.  We next construct that space
+and prove the required measure-theoretic statements.
diff --git a/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/build/sections/06-process.tex b/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/build/sections/06-process.tex
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+\section{An invariant process and its window laws}\label{sec:process}
+
+We now turn the finite word families of \Cref{prop:words} into one
+ergodic system. The probability on that system need not assign
+uniform frequencies to the different words. The per-word frequency
+condition in the construction is what will make the window estimates
+valid for every invariant probability.
+
+\subsection{The compact phase extension}
+
+Let $\mathcal A=\mathcal W_1$, regarded as an alphabet, and let
+\[
+ \Omega=\mathcal A^{\Z}\times\prod_{i\ge1}\Z/m_i\Z
+\]
+with its compact product topology. Write a point as $(\omega,a)$,
+where $\omega=(\omega_t)_{t\in\Z}$ and
+$a=(a_i)_{i\ge1}$. Define $X\subset\Omega$ by the following conditions:
+\begin{enumerate}[label=(\roman*)]
+\item $a_{i+1}\equiv a_i\pmod{m_i}$ for every $i$;
+\item whenever $t\equiv-a_i\pmod{m_i}$, the word
+$(\omega_t,\ldots,\omega_{t+m_i-1})$ belongs to $\mathcal W_i$.
+\end{enumerate}
+The coordinate $a_i\in\{0,\ldots,m_i-1\}$ records the index of the
+origin in its level-$i$ word. Let
+\[
+ T(\omega,a)=(\omega',a'),\qquad
+ \omega'_t=\omega_{t+1},\quad a'_i=a_i+1\pmod{m_i}.
+\]
+The defining conditions are closed, and $T$ is a homeomorphism of $X$.
+For each $j$, repeat a word of $\mathcal W_j$ bi-infinitely with its
+level-$j$ boundary at zero. Since higher-level words are concatenations
+of lower-level ones, this sequence with zero phases satisfies all
+conditions through level $j$. The finite-level closed sets are nested
+and nonempty. Compactness proves that $X$ is nonempty.
+
+The invariant-measure facts used here are the standard compact-system
+ones; see \cite[Theorems~6.1.4--6.1.6]{DajaniDirksin2008}.
+We give the construction together with the checks specific to the
+phase extension.
+
+\begin{proposition}\label{prop:system}
+There is a $T$-invariant ergodic Borel probability $\mu$ on $X$.
+Every $T$-invariant Borel probability on $X$ is nonatomic, is standard
+modulo null sets, and has zero Kolmogorov--Sinai entropy.
+\end{proposition}
+
+\begin{proof}
+For any $x\in X$, weak limits of
+$n^{-1}\sum_{j=0}^{n-1}\delta_{T^jx}$ are invariant probabilities.
+Indeed, the difference between the integrals of $f\circ T$ and $f$
+against these averages tends to zero for each continuous $f$.
+The set $\mathcal M_T(X)$ of invariant probabilities is nonempty,
+compact, and convex.
+
+Here is an explicit way to select an extreme point. Choose a
+countable uniformly dense family $(f_j)$ in the continuous real
+functions on $X$. Starting from $\mathcal M_T(X)$, successively
+restrict to the compact face on which $\int f_j\,d\mu$ is maximal.
+The nested intersection is nonempty and contains only one measure,
+because the integrals of every $f_j$ are then fixed and determine a
+probability. An intersection of faces is a face, so this measure is
+extreme. If it had a $T$-invariant measurable set of mass strictly
+between zero and one, normalized restriction to that set and its
+complement would express it as a nontrivial convex combination of
+invariant probabilities. Thus it is ergodic. The same argument applies
+to invariant sets modulo null sets after taking Borel representatives.
+
+For any invariant probability, $a_i$ is uniform on $\Z/m_i\Z$,
+because $T$ cyclically permutes its level sets. If a measurable atom
+had mass $c>0$, then for each $i$ it would be contained, modulo null
+sets, in one of these level sets. Hence $c\le1/m_i$ for every $i$,
+which is impossible since $m_i\to\infty$.
+
+The space $X$ is a compact metrizable subspace of a countable product
+of finite discrete sets. For completeness, its nonatomic
+probability is standard in the precise sense needed here. Encode the
+finite coordinates into successive disjoint blocks of ternary
+digits $0,2$; this gives a continuous injective map of $X$ into a
+compact subset of $[0,1]$. Let $\lambda$ be the image law and
+$F(t)=\lambda([0,t])$ its continuous distribution function. Define
+the measurable quantile
+\[
+ Q(u)=\inf\{t:F(t)\ge u\},\qquad 0<u<1.
+\]
+Continuity gives $F(Q(u))=u$, and
+$\{u:Q(u)\le t\}$ has Lebesgue measure $F(t)$, so
+$Q_*\operatorname{Leb}=\lambda$ and $F_*\lambda=\operatorname{Leb}$.
+Moreover $Q(F(x))\le x$ for $\lambda$-almost every $x$;
+both sides have law $\lambda$. Their nonnegative difference has zero
+integral, and therefore $Q(F(x))=x$ almost surely.
+After restricting to measurable conull sets, $F$ and $Q$ are
+inverse isomorphisms. Composing with the coordinate embedding
+establishes the assertion. Passing to completed sigma fields
+does not change it.
+
+It remains to check the entropy of the entire phase extension.
+A partition depending on finitely many coordinates is refined by
+one that records $\omega_{-r},\ldots,\omega_r$ and $a_I$ for
+some fixed $r,I$. Compatibility determines all the lower phases
+from $a_I$, and the phase evolves deterministically. Fix any $j\ge I$.
+The initial phase $a_j$ determines $a_I$ and the boundaries of all
+level-$j$ words. The symbol interval $[-r,n-1+r]$ used in an $n$-step
+name meets at most $\lceil(n+2r)/m_j\rceil+1$ such words, each in
+$\mathcal W_j$. Consequently the number of names is at most
+\[
+ m_j L_j^{\lceil(n+2r)/m_j\rceil+1}.
+\]
+Keeping $j$ fixed and sending $n\to\infty$ bounds the entropy rate
+by $R_j=(\log L_j)/m_j$. By \Cref{lem:rates}, $R_j\to0$.
+Only now let $j\to\infty$: every finite-coordinate partition has
+entropy rate zero. This count uses no assumption about word frequencies.
+
+Let $\mathcal P$ be any finite measurable partition with $A$ atoms.
+Cylinder sets generate the Borel sigma field, and their algebra is
+dense in measure. Thus the atom label of $\mathcal P$ can be
+approximated by an $A$-valued finite-coordinate label
+$\mathcal Q$ with mismatch probability $s$ arbitrarily small.
+Completion of the measure only adds null-set changes.
+The chain rule and conditioning give
+\[
+ H(\mathcal P^n)
+ \le H(\mathcal Q^n)
+     +\sum_{t=0}^{n-1}H(T^{-t}\mathcal P\mid T^{-t}\mathcal Q)
+ =H(\mathcal Q^n)+nH(\mathcal P\mid\mathcal Q),
+\]
+where $\mathcal P^n=\bigvee_{t=0}^{n-1}T^{-t}\mathcal P$.
+Recording first whether the two labels differ shows that
+\[
+ H(\mathcal P\mid\mathcal Q)\le H_{\rm bin}(s)+s\log A.
+\]
+Entropy rates exist by subadditivity. Divide by $n$, take the limit,
+and then let $s\downarrow0$. This proves $h_\mu(T)=0$.
+\end{proof}
+
+Fix henceforth an ergodic probability supplied by
+\Cref{prop:system}. The next estimates use only invariance, so their
+validity is unaffected by this choice.
+
+\subsection{What alignment does to the sampling law}
+
+For $x=(\omega,a)\in X$, let $V_i(x)\in\mathcal W_i$ be the word
+containing the origin, namely the symbol string beginning at
+$-a_i(x)$. Set
+\[
+ A_i=\{x:a_i(x)=0\},\qquad
+ \mu_i=m_i\,\mu|_{A_i}.
+\]
+The law $\mu_i$ is a probability. It describes a point observed at a
+level-$i$ boundary; it is not asserted to give uniform word
+frequencies.
+
+For an integer $t$, let
+\[
+ r_i(x,t)=(-a_i(T^t x))\bmod m_i\in\{0,\ldots,m_i-1\}.
+\]
+The first level-$i$ boundary at or after $t$ is $t+r_i(x,t)$.
+For an integer $D\ge2$, define the vector
+\begin{equation}\label{eq:aligned-list}
+ \mathcal L_{i,D}(x,t)=
+ \bigl(V_i(T^{\,t+r_i(x,t)+jm_i}x)\bigr)_{j=0}^{D-2}.
+\end{equation}
+All its words lie in $[t,t+Dm_i)$, since
+$r_i+(D-1)m_i\le Dm_i$.
+
+\begin{lemma}\label{lem:aligned-law}
+The law of \eqref{eq:aligned-list} under $\mu$ is the law under
+$\mu_i$ of the first $D-1$ level-$i$ words beginning at zero.
+Under $\mu_i$, the index
+$J=a_{i+2}/m_i$ is uniform on
+$\{0,\ldots,N_iN_{i+1}-1\}$ and is independent of $V_{i+2}$.
+More precisely, for each such $j$ and $v\in\mathcal W_{i+2}$,
+\begin{equation}\label{eq:parent-slot-law}
+ \mu_i(J=j,V_{i+2}=v)
+ =\frac{\mu_{i+2}(V_{i+2}=v)}{N_iN_{i+1}}.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Invariance first allows us to take $t=0$. On the phase class
+$a_i=r$, the map $T^{(-r)\bmod m_i}$ sends the normalized
+restriction of $\mu$ bijectively and measure-preservingly onto
+$\mu_i$. The first assertion follows by applying this separately
+on all $m_i$ phase classes.
+
+For the second assertion, $T^{jm_i}$ maps
+$\{a_{i+2}=0,V_{i+2}=v\}$ onto
+$\{a_{i+2}=jm_i,V_{i+2}=v\}$.
+It does not change the containing parent word because
+$0\le jm_i<m_{i+2}$. These sets have the same $\mu$-mass.
+Multiply by $m_i$, and use
+$m_{i+2}/m_i=N_iN_{i+1}$, to obtain
+\eqref{eq:parent-slot-law}.
+\end{proof}
+
+The identity is the reason that a deterministic condition on every
+parent word suffices. At a fixed internal offset in a parent,
+the slot being sampled is uniform even when the parent itself is
+not uniformly distributed.
+
+\subsection{Entropy of short label lists}
+
+We use two elementary entropy bounds. The second is the standard
+maximal-coupling and mismatch-flag entropy argument; compare
+\cite[Theorems~1--3]{Sason2013}. We include the weaker form needed here. If a law on a finite set
+has every atom at most $M$, its entropy is at least
+$\log(1/M)$. If two laws $P,Q$ on an $A$-point set have
+$\TV(P,Q)\le s$, then
+\begin{equation}\label{eq:tv-entropy}
+ H(P)\ge H(Q)-1-s\log A.
+\end{equation}
+Indeed, couple their labels with mismatch probability at most $s$.
+The mismatch flag and the chain rule bound either conditional
+entropy by $1+s\log A$. We use this simple bound rather than an
+optimized continuity estimate.
+
+\begin{proposition}\label{prop:window-entropy}
+For all sufficiently large $i$, for every power of two
+$2\le D\le D_i^*$, and every integer $t$,
+\begin{equation}\label{eq:window-entropy}
+ H_\mu(\mathcal L_{i,D}(\,\cdot\,,t))
+ \ge .996(D-1)\ell_i,\qquad \ell_i=\log L_i.
+\end{equation}
+The large-$i$ threshold is independent of $D,t$ and of the
+invariant probability on $X$.
+\end{proposition}
+
+\begin{proof}
+Write $q=D-1$. By \Cref{lem:aligned-law} we work under $\mu_i$.
+Decompose its uniform parent-slot index as
+\[
+ J=hN_i+u,\qquad 0\le h<N_{i+1},\quad 0\le u<N_i.
+\]
+Conditional on the parent $v=V_{i+2}$ and the internal offset $u$,
+the slot $h$ is uniform among the $N_{i+1}$ constituent
+$\mathcal W_{i+1}$ words of $v$.
+Call $u$ admissible if $u+D_i^*\le N_i$.
+Its failure probability is
+\[
+ \beta_i=\frac{D_i^*-1}{N_i}\le\frac{D_i^*}{N_i}.
+\]
+
+For every fixed $v$ and admissible $u$,
+\Cref{prop:words}, Property~(\ref{word:frequency}), applied at level $i+1$,
+puts the law of the $D_i^*$ labels at this offset within
+$10^{-4}$ of their law in a uniform word of $\mathcal W_{i+1}$.
+\Cref{prop:words}, Property~(\ref{word:window}), applied at level $i$,
+puts that law within a further $10^{-4}$ of a law whose
+every specified $q$-coordinate atom is at most
+\[
+ M_i=2\left(\frac{2^{2K_i}}{L_i}\right)^q.
+\]
+Restriction to the first $q$ entries cannot increase total
+variation. The comparison law may depend on $u$, but this
+atom bound does not.
+
+It follows from \eqref{eq:tv-entropy}, with $s=2\cdot10^{-4}$
+and alphabet size $L_i^q$, that the actual $q$-list conditional
+on $v,u$ has entropy at least
+\[
+ q\ell_i-2K_iq-1-(1+sq\ell_i)
+ =q\bigl((1-s)\ell_i-2K_i\bigr)-2.
+\]
+Entropy is concave. Mixing over all $v$ and admissible $u$
+preserves this lower bound; adding the remaining offsets loses
+at most their fraction, since entropy is nonnegative. Thus
+\begin{equation}\label{eq:window-entropy-explicit}
+ H(\mathcal L_{i,D})
+ \ge(1-\beta_i)
+       \left\{q\bigl((1-s)\ell_i-2K_i\bigr)-2\right\}.
+\end{equation}
+Here $\beta_i\to0$, $2K_i/\ell_i\to0$, and $2/\ell_i\to0$.
+Once each of these three quantities is at most $.0005$,
+division by $q\ell_i$, with $q\ge1$, gives a coefficient
+at least
+\[
+ (1-.0005)(1-.0002-.0005-.0005)>.996.
+\]
+This proves the uniform assertion.
+\end{proof}
+
+Only the two indicated levels of the construction enter this
+argument. There is no accumulation of total variation errors
+over all ancestors, and no independence assumption on the
+sequence of adjacent words.
+
+\subsection{The graph relation across two windows}
+
+Two level-$i$ labels $u,v\in\mathcal W_i$ pass the graph test
+when $[u,v]_i=0$, using their fixed vector identification.
+The same alignment law gives a quantitative bound on the one
+exception to the graph test: crossing a group boundary.
+
+\begin{lemma}\label{lem:graph-windows}
+Let $2\le D\le D_i^*$ be a power of two and set $n=m_iD$.
+For all sufficiently large $i$, with $\mu$-probability at least
+\[
+ 1-\alpha_{i,D},\qquad
+ \alpha_{i,D}=\frac{2D-2}{H_i},
+\]
+all corresponding entries of
+$\mathcal L_{i,D}(x,-n)$ and
+$\mathcal L_{i,D}(x,0)$ pass the level-$i$ graph test simultaneously.
+In particular $\sup_{D\le D_i^*}\alpha_{i,D}\to0$.
+\end{lemma}
+
+\begin{proof}
+Because $n$ is a multiple of $m_i$, the advance to alignment
+is the same in the two windows. Denote it by $q_0$.
+The entry starts are
+\begin{equation}\label{eq:two-list-starts}
+ -n+q_0+jm_i
+ \quad\hbox{and}\quad
+ q_0+jm_i,\qquad 0\le j\le D-2.
+\end{equation}
+Corresponding entries are separated by exactly $D$ level-$i$
+positions. The first start through the last start span
+$2D-1$ such positions, including the unused intervening position.
+\Cref{fig:lists} illustrates these positions and their common lag
+when $D=4$.
+
+By \Cref{lem:aligned-law}, the first boundary has the
+boundary-start law. Its position modulo $H_i$ is uniform:
+its position in a $\mathcal W_{i+1}$ parent is uniform modulo
+$N_i$, and $H_i$ divides $N_i$. Once $2D-1\le H_i$, exactly
+$2D-2$ residues have a span that crosses an $H_i$-group boundary.
+All parent boundaries are already group boundaries.
+On every other residue all the labels in \eqref{eq:two-list-starts}
+belong to one of the prescribed groups.
+\Cref{prop:words}, Property~(\ref{word:graph}), applies because $D$ is an
+allowed power-of-two lag.
+Finally $2D_i^*/H_i\to0$ by the definitions.
+\end{proof}
+
+\begin{figure}[tb]
+\centering
+\begin{tikzpicture}[x=.92cm,y=1cm,>=Stealth]
+ \foreach \j in {0,1,2,3,4,5,6}{
+   \draw (\j,0) rectangle ++(1,.55);
+ }
+ \foreach \j in {0,1,2}{
+   \node at (\j+.5,.275) {$U_{\j}$};
+   \pgfmathtruncatemacro{\v}{\j+4}
+   \node at (\v+.5,.275) {$V_{\j}$};
+   \draw[->] (\j+.5,.65) to[bend left=22] (\v+.5,.65);
+ }
+ \node at (3.5,.275) {$-$};
+ \draw[decorate,decoration={brace,mirror,amplitude=4pt}]
+   (0,-.12)--(3,-.12) node[midway,below=6pt] {past list};
+ \draw[decorate,decoration={brace,mirror,amplitude=4pt}]
+   (4,-.12)--(7,-.12) node[midway,below=6pt] {future list};
+ \node at (3.5,1.55) {each arrow has lag $D=4$};
+\end{tikzpicture}
+\caption{The two aligned lists of one legal symbolic point when $D=4$,
+writing $U_j,V_j$ for
+their corresponding past and future entries. Each has $D-1$ entries.
+The single unused block between them does not alter the lag between
+corresponding entries. Their span has $2D-1$ block positions, which
+explains the group-boundary error in \Cref{lem:graph-windows}.}
+\label{fig:lists}
+\end{figure}
+
+We now have a fixed ergodic system with zero entropy, uniformly
+large list entropy, and a graph relation across adjacent windows.
+These properties are compatible: the information per symbol in a
+level-$i$ list is of order $R_i$, which tends to zero. The many
+available test scales will compensate for this decreasing rate.
+The remaining task is to compare that relation with the
+conditional independence in a smooth switch.
diff --git a/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/build/sections/07-obstruction.tex b/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/build/sections/07-obstruction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..3f0941cd98680e74db9b45e2f70b99e2fb13e258
--- /dev/null
+++ b/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/build/sections/07-obstruction.tex
@@ -0,0 +1,380 @@
+\section{The obstruction to a smooth volume model}\label{sec:obstruction}
+
+Fix the ergodic system constructed in \Cref{sec:process}.
+The final argument concerns a single statistic of a pair of points:
+the fraction of corresponding past and future labels that pass the
+graph test. We define that statistic before restricting the pair
+experiment. Its upper and lower bounds will therefore refer to the
+same probability law.
+
+\subsection*{The quantitative interface}
+The smooth and symbolic constructions meet through the quantities in
+\Cref{tab:interfaces}. At $n=2^k$, write $\mathcal D_k$ for the smooth
+partition, $\rho$ for its cell-weighted pair law, $G$ for the retained
+pairs, and $z$ for the switch. The exceptional-mass tolerance $\eta>0$
+is fixed before these objects are chosen. On the symbolic side, $L_i=\#\mathcal W_i$,
+$R_i=(\log L_i)/m_i$, and $K_i=\lfloor\sqrt{\log L_i}\rfloor$.
+Every $m_i$ is a power of two. The symbolic tests have length $n=m_iD$,
+where $D$ is a power of two
+and $2^{\lceil K_i/16\rceil}\le D\le2^{\lfloor K_i/8\rfloor}$.
+In a window of $Dm_i$ symbols we use the $D-1$ full words beginning
+at the first level-$i$ boundary. Each input initially uses its own
+alignment. The compact system recording the symbol sequence and all
+compatible alignment phases has zero entropy for every invariant
+probability, by \Cref{prop:system}.
+
+\begin{table}[htbp]
+\centering\small
+\renewcommand{\arraystretch}{1.18}
+\begin{tabular}{@{}>{\raggedright\arraybackslash}p{.19\textwidth}>{\raggedright\arraybackslash}p{.37\textwidth}>{\raggedright\arraybackslash}p{.37\textwidth}@{}}
+\toprule
+Interface & Smooth conclusion & Symbolic use \\
+\midrule
+Dyadic cost &
+$\sum_k B_k^{16}<\infty$ and
+$\#\mathcal D_k\le\exp(CB_k2^k)$ at every sufficiently large scale &
+$\sum_i K_iR_i^{16}=\infty$ gives tests at which cell conditioning
+costs $o((D-1)\log L_i)$ bits. \\
+Original pair law &
+Choose a cell with its volume weight, then two independent normalized
+volume points; retain mass at least $1-\eta$ &
+Window entropy holds for the chosen invariant measure, with arbitrary
+parent-word weights. Independence is used only for the two inputs
+conditional on the cell and a fixed entry. \\
+Shadowing &
+The switch follows one input for $0\le t\le n$ and the other for
+$-n\le t\le0$, with error tending uniformly to zero &
+Positive symbol-distance recognition recovers the common phase and
+most labels, preserving the original entry pairing. \\
+Output measure &
+$z_*(\rho|_G)\le2\nu$, without normalizing the retained law &
+Compact symbol cores and the simultaneous graph-window law transfer
+to the switched point with controlled exceptional mass. \\
+\bottomrule
+\end{tabular}
+\caption{The exact internal interface between
+\Cref{thm:switching} and the fixed symbolic system.
+All smooth constants are chosen only after a model is proposed.
+The final comparison retains the original pair law $\rho$.}
+\label{tab:interfaces}
+\end{table}
+
+\subsection{Choosing times with a negligible partition cost}
+
+\begin{lemma}\label{lem:cheap-scales}
+Let $(B_k)_{k\ge1}$ be any positive sequence with
+$\sum_k B_k^{16}<\infty$. There exist levels $i\to\infty$ and
+powers of two $D$ satisfying
+\[
+ 2^{\lceil K_i/16\rceil}\le D\le2^{\lfloor K_i/8\rfloor}
+\]
+such that, for $n=m_iD$,
+\begin{equation}\label{eq:relative-cost}
+ \frac{B_{\log n}}{R_i}\longrightarrow0,
+ \qquad R_i=\frac{\ell_i}{m_i}.
+\end{equation}
+In particular any partitions with
+$\log\#\mathcal D_k\le C B_k2^k$ obey
+\begin{equation}\label{eq:cheap-partition}
+ \log\#\mathcal D_{\log n}=o((D-1)\ell_i)
+\end{equation}
+on this sequence, for every fixed $C$.
+\end{lemma}
+
+\begin{proof}
+The lengths $m_i$ are powers of two. Put $b_i'=\log m_i$,
+so $b_{i+1}'=b_i'+K_i$. The integer sets
+\[
+ J_i=\{b_i'+j:\lceil K_i/16\rceil\le j\le
+                         \lfloor K_i/8\rfloor\}
+\]
+are disjoint and, for large $i$, contain at least $K_i/32$
+integers. If the ratios in \eqref{eq:relative-cost} were bounded
+below by some $c>0$ on all sufficiently late tests, then
+\[
+ \sum_kB_k^{16}
+ \ge c^{16}\sum_{i\ {\rm large}}|J_i|R_i^{16}
+ \ge \frac{c^{16}}{32}
+        \sum_{i\ {\rm large}}K_iR_i^{16}=\infty,
+\]
+contrary to the hypothesis and \Cref{lem:rates}.
+Thus one can select successively later tests with ratio tending
+to zero. Since
+$B_{\log n}n=(B_{\log n}/R_i)D\ell_i$
+and $D/(D-1)\le2$, \eqref{eq:cheap-partition} follows.
+\end{proof}
+
+Suppose for a contradiction that a smooth volume model
+$(M,\nu,S)$ exists. Its dimension is positive because a
+zero-dimensional compact manifold is finite and cannot carry a
+nonatomic probability. Apply \Cref{thm:switching} with
+$\eta=.01$. The model is now fixed, so all its constants and
+its sequence $(B_k)$ are fixed.
+Use \Cref{lem:cheap-scales} for this sequence. In what follows,
+$i,D,n$ run through the selected tests and are taken sufficiently
+large for every stated eventual estimate.
+
+We use the measurable isomorphism to regard every finite symbolic
+observation as a measurable observation on $M$. The original domain
+can be replaced by the intersection of all its integer translates,
+so all orbit identities hold on an invariant conull set.
+Extend the observations arbitrarily on its null complement.
+For each test let $C$ be the cell sampled in the switching
+experiment, and let $\rho$ be the original law of $(x,y)$.
+Its two marginals are exactly $\nu$, and its conditional law
+given $C$ is the product of normalized volume on that cell.
+
+\subsection{The independent upper bound}
+
+Let
+\[
+ U=\mathcal L_{i,D}(y,-n),\qquad
+ V=\mathcal L_{i,D}(x,0),\qquad q=D-1.
+\]
+Each list uses its own input's alignment advance. Both are defined
+on every pair before a switch is selected.
+Under the identification of $\mathcal W_i$ with the nonzero
+vectors of $\F_2^{2s_i}$, write $[\,\cdot\,,\,\cdot\,]_i$ for
+the alternating form from the word construction, and set
+\begin{equation}\label{eq:passing-statistic}
+ Z(x,y)=\frac1q\sum_{j=0}^{q-1}
+          \one_{\{[U_j,V_j]_i=0\}}.
+\end{equation}
+For each fixed cell and each fixed $j$, the laws of $U_j$ and
+$V_j$ are independent.
+
+Apart from the diagonal pairs, the orthogonality relation gives the
+classical binary symplectic graph; see \cite[Definition~3.3 and Theorem~3.4]{Seidel1973} for its
+finite-geometric description and spectral scale. We include the
+character calculation, keeping the diagonal pairs, and pass from
+average Shannon entropy to the required graph bound by a heavy--light
+split of each law.
+
+\begin{lemma}\label{lem:graph-independent}
+Let $L=2^{2s}-1$ and equip the nonzero vectors of
+$\F_2^{2s}$ with a nondegenerate alternating form.
+Suppose a family of pairs of laws $(p_\xi,q_\xi)$ on these
+vectors is indexed by a probability space, and
+\[
+ \E_\xi H(p_\xi)\ge .99\log L,\qquad
+ \E_\xi H(q_\xi)\ge .99\log L.
+\]
+For all sufficiently large $L$, independent samples from the
+two laws, followed by averaging over $\xi$, pass the test
+$[u,v]=0$ with probability at most $.62$.
+\end{lemma}
+
+\begin{proof}
+For a law $p$, put
+$A_p=\{u:p(u)>L^{-.8}\}$ and $\theta_p=p(A_p)$.
+The set $A_p$ has at most $L^{.8}$ elements. Splitting entropy
+according to this set gives
+\begin{equation}\label{eq:heavy-entropy}
+ H(p)\le1+.8\theta_p\log L+(1-\theta_p)\log L
+      =1+(1-.2\theta_p)\log L.
+\end{equation}
+The same formula, with the usual zero-mass conventions, covers
+empty sets. The entropy hypotheses imply
+\[
+ \E_\xi\theta_{p_\xi},\ \E_\xi\theta_{q_\xi}
+ \le .05+\frac5{\log L}\le .055
+\]
+once $\log L\ge1000$.
+
+Let $A$ be the matrix $A_{uv}=(-1)^{[u,v]}$ on nonzero
+vectors. On the full vector space, the rows of the same matrix
+are orthogonal: for $u\ne w$, nondegeneracy makes
+$v\mapsto[u+w,v]$ a nonzero linear functional, so
+$\sum_v(-1)^{[u+w,v]}=0$. Every row has squared norm $L+1$.
+The full operator norm is therefore $\sqrt{L+1}$, and restriction
+to the nonzero rows and columns has no larger norm.
+
+The light subprobability $p^{\rm l}=p\,\one_{A_p^c}$
+satisfies
+\[
+ \|p^{\rm l}\|_2^2
+ \le L^{-.8}\sum_u p^{\rm l}(u)\le L^{-.8}.
+\]
+Hence
+\[
+ \left|\sum_{u,v}p^{\rm l}(u)A_{uv}q^{\rm l}(v)\right|
+ \le\sqrt{L+1}\,L^{-.8}=:\beta_L.
+\]
+The passing mass of the light product is at most
+$(1+\beta_L)/2$. All pairs with at least one heavy coordinate
+have total mass at most $\theta_p+\theta_q$.
+After averaging, and taking $L$ large enough that
+$\beta_L\le .02$, the passing probability is at most
+\[
+ .055+.055+\frac{1+.02}{2}=.62.
+\]
+The matrix includes its diagonal entries, so the automatic
+relations $[u,u]=0$ require no separate exclusion.
+\end{proof}
+
+By the two original marginals and \Cref{prop:window-entropy},
+\[
+ H_\rho(U),H_\rho(V)\ge .996q\ell_i.
+\]
+Conditioning each vector on the sampled cell loses at most
+$H(C)\le\log\#\mathcal D_{\log n}=o(q\ell_i)$. Conditional
+subadditivity consequently gives
+\begin{equation}\label{eq:entry-entropies}
+ \frac1q\sum_{j=0}^{q-1}H(U_j\mid C)\ge .99\ell_i,
+ \qquad
+ \frac1q\sum_{j=0}^{q-1}H(V_j\mid C)\ge .99\ell_i
+\end{equation}
+for late tests. Apply \Cref{lem:graph-independent} with index
+$\xi=(C,j)$, assigning $j$ uniform weight independently of the
+cell. Independence is used conditional on this full index:
+equivalently, first fix $j$, use the product law conditional
+on $C$, and then average. No product assertion conditional only
+on $C$ after mixing over a shared random $j$ is needed.
+We obtain
+\begin{equation}\label{eq:passing-upper}
+ \E_\rho Z\le .62.
+\end{equation}
+
+\subsection{From geometric shadowing to symbol agreement}
+
+Let $d_*>0$ be the symbol-distance constant in
+\Cref{lem:recognition}. It is fixed by the word construction.
+Let $f:M\to\mathcal A$ be the transported symbol at time zero.
+We first choose compact cores for this one fixed finite
+partition, rather than for the growing level-$i$ label sets.
+
+\begin{lemma}\label{lem:symbol-matching}
+For all sufficiently large dyadic times in the smooth switching
+estimate, the retained pairs for which
+\begin{align}
+ \#\{0\le t<n:f(S^tz)\ne f(S^tx)\}
+       &<d_*n/100,\label{eq:future-symbol-match}\\
+ \#\{-n\le t<0:f(S^tz)\ne f(S^ty)\}
+       &<d_*n/100\label{eq:past-symbol-match}
+\end{align}
+have original $\rho$-mass at least $.96$.
+\end{lemma}
+
+\begin{proof}
+Choose $\kappa<d_*/20000$. Inner regularity of the smooth
+probability gives disjoint compact sets
+$K_a\subset f^{-1}(a)$, for $a\in\mathcal A$, with
+\[
+ \nu\left(M\setminus\bigcup_{a\in\mathcal A}K_a\right)<\kappa.
+\]
+Measurable atoms may first be replaced by Borel representatives
+modulo null sets. There are finitely many cores, and the distance
+between any two nonempty distinct cores is positive. If there is
+only one nonempty core, the following equal-symbol implication is
+automatic. Otherwise take the positive minimum of these distances.
+Uniform shadowing is smaller than this fixed separation at all
+large scales. Therefore two compared iterates both in the cores
+have the same symbol.
+
+Write $G$ for the retained set and
+$\sigma=z_*(\rho|_G)$. The switching theorem gives
+$\sigma\le2\nu$. For every integer $t$,
+\[
+ (S^t)_*\sigma\le2(S^t)_*\nu=2\nu.
+\]
+The $x$ and $y$ time marginals under the original law are $\nu$.
+At each future comparison time, the $\rho$-mass within $G$
+where either $x$ or $z$ misses the cores is at most
+$\kappa+2\kappa=3\kappa$. The same holds for the past comparison
+with $y$.
+
+The integrals over $G$ of the fractions of mismatches on either
+half are therefore at most $3\kappa$. Markov's inequality at
+threshold $d_*/100$, on each half, bounds their combined failure
+mass by $600\kappa/d_*<.03$. Since $\rho(G)\ge .99$,
+the simultaneous success mass is at least $.96$.
+This is an estimate in the original law, without dividing
+by $\rho(G)$.
+
+The input marginals and $\sigma\le2\nu$ give zero mass to
+the complement of the invariant conjugacy domain. Thus the
+symbol strings and all orbit identities used in the argument
+are defined outside a $\rho$-null set.
+\end{proof}
+
+\subsection{Recognition preserves the original entry pairing}
+
+Consider a pair in the success event of \Cref{lem:symbol-matching}.
+If the phases of $z$ and $x$ modulo $m_i$ differed, take the
+$D-1$ full level-$i$ blocks of $x$ beginning at its first
+aligned boundary in $[0,n)$. At each such interval, the symbol
+string of $z$ is a window of length $m_i$ cut at a nonzero
+offset across two consecutive legal level-$i$ words. This
+remains true if those words are equal or if one extends beyond
+the observed window. By \Cref{lem:recognition}, each of the
+$D-1$ disjoint intervals contributes at least $d_*m_i$
+mismatches. Their total is at least
+\[
+ d_*(D-1)m_i\ge d_*n/2,
+\]
+contradicting \eqref{eq:future-symbol-match}.
+The same reasoning in the past gives
+\begin{equation}\label{eq:phase-recovery}
+ a_i(z)=a_i(x)=a_i(y).
+\end{equation}
+Here $a_i(S^{-n}u)=a_i(u)$ for every $u$ in the conjugacy
+domain, since $n$ is a multiple of $m_i$.
+
+Let $q_0$ be the common advance to alignment.
+The already defined entries $U_j,V_j$ occupy, in the symbolic
+orbit of $z$, the positions
+\[
+ -n+q_0+jm_i,\qquad q_0+jm_i,\qquad 0\le j\le D-2.
+\]
+Their difference is exactly $n=Dm_i$. The index $j$ has not
+been changed after selecting the switch.
+When one of these entries differs in label from the corresponding
+word of $z$, distinct-word separation in
+\Cref{lem:recognition} costs at least $d_*m_i$ mismatches.
+Consequently fewer than $D/100$ entries on each side can differ.
+
+Let $E_{i,D}\subset X$ be the group-boundary exception from
+\Cref{lem:graph-windows}, transported to $M$ and extended
+arbitrarily on its null complement. Its original measure is at most
+$\alpha_{i,D}\le2D_i^*/H_i\to0$, uniformly over the chosen
+tests. The retained switch has
+\[
+ \rho\{(x,y)\in G:z(x,y)\in E_{i,D}\}
+ \le2\alpha_{i,D}.
+\]
+Take a late test for which $2\alpha_{i,D}\le.01$.
+On a subset of the original pair space of mass at least
+$.96-.01$, the symbol-matching event holds and the two
+$z$-lists satisfy every graph relation. Removing the fewer
+than $D/100$ wrong entries on each side gives
+\[
+ Z(x,y)\ge 1-\frac{2D}{100(D-1)}\ge.96.
+\]
+The statistic is nonnegative elsewhere, so
+\begin{equation}\label{eq:passing-lower}
+ \E_\rho Z\ge(.96-.01)\cdot.96=.912.
+\end{equation}
+This contradicts \eqref{eq:passing-upper}.
+
+\begin{proof}[Completion of the proof of \Cref{thm:main}]
+\Cref{prop:system} supplies a standard nonatomic ergodic
+invertible probability-preserving system of zero entropy.
+If it had a smooth volume model, \Cref{thm:switching} and
+\Cref{lem:cheap-scales} would supply the tests above.
+The original pair law at each late test would satisfy both
+\eqref{eq:passing-upper} and \eqref{eq:passing-lower}, which is
+impossible. The dimension and all smooth constants were arbitrary
+but fixed after construction of the system. Thus no model in
+any finite dimension exists.
+\end{proof}
+
+\begin{remark}
+The role of smooth positive volume is quantitative.
+It gives a switch whose unnormalized output law is controlled
+by the original invariant measure. That control allows a
+measurable symbol observation to be compared along shadowing
+orbits, despite the absence of regularity in the conjugacy.
+The proof makes no analogous assertion for models carrying
+only a singular measure or a nonsmooth density equivalent
+to volume.
+\end{remark}
diff --git a/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/paper.pdf b/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..cd3022b35ed74dacec84f2679ca813ffbad96d2f
--- /dev/null
+++ b/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:67d6dd8619de6c6fc11eb88edd3791b180a2c3c9bc6d5a6a68f272a5ab445687
+size 565678
diff --git a/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/README.md b/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..0fb6b8ebb0a0bd89eee296320f102a79044fdf04
--- /dev/null
+++ b/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/README.md
@@ -0,0 +1,44 @@
+# [A finite-time singularity of Calabi flow on projective space](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 24, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026,
+  author = {{OpenAI}},
+  title = {{A finite-time singularity of Calabi flow on projective space}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/paper.pdf}{OAI:A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026}},
+  year = {2026}
+}
+```
+
+## Verification
+
+The exact certificate and supplementary rational checks require only Python 3's
+standard library. Run with assertions enabled: do not use `-O` or `-OO`, and
+unset `PYTHONOPTIMIZE` as below. These commands use a temporary copy because the
+allowances check writes `arithmetic_allowances.json` beside its source file.
+
+```sh
+(
+  set -eu
+  work_dir="$(mktemp -d)"
+  trap 'rm -rf "$work_dir"' 0 HUP INT TERM
+  cp -R verification "$work_dir/verification"
+  unset PYTHONOPTIMIZE
+  python3 -B "$work_dir/verification/certificate.py"
+  python3 -B "$work_dir/verification/arithmetic_allowances.py"
+  # Optional supplementary root and tail check:
+  python3 -B "$work_dir/verification/tail-check.py"
+)
+```
+
+The certificate is silent on success. The allowances check prints a report with
+`all_exact_rational_checks_pass` set to `true`, and the optional tail check prints
+a message that all checks passed exactly. Successful execution means the
+programs' assertions hold; it does not independently verify their analytic
+interpretation or the full theorem.
diff --git a/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/figures/cutoff-support.tex b/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/figures/cutoff-support.tex
new file mode 100644
index 0000000000000000000000000000000000000000..4e674c411fc0c0dd5e7b69344ff3940d382dce13
--- /dev/null
+++ b/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/figures/cutoff-support.tex
@@ -0,0 +1,28 @@
+\begin{tikzpicture}[x=.96cm,y=.6cm,font=\small]
+\definecolor{cutone}{RGB}{40,83,111}
+\tikzset{oneregion/.style={fill=cutone,draw=cutone},
+ transition/.style={pattern=north east lines,pattern color=black!55,draw=black!55}}
+\foreach \x/\lab in {0/0,1/{\beta/4},2/{\beta/2},3/{\beta},4.5/{2\beta},6/{4\beta}} {
+  \draw[black!18] (\x,.45)--(\x,4.6);
+  \node[above] at (\x,4.6) {$\lab$};
+}
+\draw[->] (0,4.6)--(6.8,4.6) node[right] {$r$};
+\foreach \yy/\lab in {4/{\chi_i},3/{\chi_b},2/{\chi_o},1/{\kappa}} {
+ \draw[black!30] (0,\yy-.18) rectangle (6.6,\yy+.18);
+ \node[left] at (-.25,\yy) {$\lab$};
+}
+\filldraw[oneregion] (0,3.82) rectangle (3,4.18);
+\filldraw[transition] (3,3.82) rectangle (4.5,4.18);
+\filldraw[oneregion] (0,2.82) rectangle (4.5,3.18);
+\filldraw[transition] (4.5,2.82) rectangle (6,3.18);
+\filldraw[transition] (2,1.82) rectangle (3,2.18);
+\filldraw[oneregion] (3,1.82) rectangle (6.6,2.18);
+\filldraw[transition] (1,.82) rectangle (2,1.18);
+\filldraw[oneregion] (2,.82) rectangle (6.6,1.18);
+\filldraw[oneregion] (-.3,-.13) rectangle (.15,.13);
+\node[right] at (.22,0) {identically $1$};
+\filldraw[transition] (2.35,-.13) rectangle (2.8,.13);
+\node[right] at (2.87,0) {may vary};
+\draw[black!30] (4.85,-.13) rectangle (5.3,.13);
+\node[right] at (5.37,0) {identically $0$};
+\end{tikzpicture}
diff --git a/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/main.tex b/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/main.tex
new file mode 100644
index 0000000000000000000000000000000000000000..27dc46402f7d54e801b0bacec1a28c0bf121b4d5
--- /dev/null
+++ b/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/main.tex
@@ -0,0 +1,41 @@
+\documentclass[11pt]{article}
+\input{preamble}
+\title{A finite-time singularity of Calabi flow\protect\\ on projective space}
+\author{OpenAI}
+\date{September 24, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We construct a smooth K\"ahler metric in the Fubini--Study class of
+\(\CP^{10}\) whose Calabi flow develops unbounded scalar curvature in finite time.
+This disproves Chen's smooth long-time existence conjecture, even in a class
+containing a constant-scalar-curvature metric.
+\end{abstract}
+\input{sections/introduction}
+\input{sections/geometry}
+\input{sections/profile}
+\input{sections/parabolic}
+\input{sections/flat}
+\input{sections/compact}
+\appendix
+\input{sections/certificate}
+\section{The exact-arithmetic certificate}\label{app:code}
+The following Python program is the certificate used in
+Appendix~\ref{sec:certificate}. It uses only Python's standard library and
+integer arithmetic. Run it with assertions enabled. Successful termination
+means that all its assertions hold; the analytic interpretation of those
+assertions is proved in Appendix~\ref{sec:certificate}.
+\lstinputlisting[style=certificate]{../verification/certificate.py}
+\begingroup
+\small
+\let\originalthebibliography\thebibliography
+\renewcommand{\thebibliography}[1]{%
+  \originalthebibliography{#1}%
+  \setlength{\itemsep}{1pt}%
+  \setlength{\parsep}{0pt}%
+  \setlength{\parskip}{0pt}%
+}
+\bibliographystyle{plain}
+\bibliography{references}
+\endgroup
+\end{document}
diff --git a/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/preamble.tex b/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/preamble.tex
new file mode 100644
index 0000000000000000000000000000000000000000..a07f638128729b81cc47b103ee0dd23921b46ee3
--- /dev/null
+++ b/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/preamble.tex
@@ -0,0 +1,57 @@
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage{lmodern}
+\usepackage[margin=1.05in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{microtype}
+\usepackage{booktabs,longtable,array}
+\usepackage{enumitem}
+\usepackage{xcolor}
+\usepackage{tikz}
+\usetikzlibrary{patterns}
+\usepackage{listings}
+\usepackage{needspace}
+\usepackage[unicode,hidelinks]{hyperref}
+\hypersetup{pdftitle={A finite-time singularity of Calabi flow on projective space},pdfauthor={OpenAI}}
+\ifdefined\pdfinfoomitdate\pdfinfoomitdate=1\fi
+\ifdefined\pdftrailerid\pdftrailerid{}\fi
+\ifdefined\pdfsuppressptexinfo\pdfsuppressptexinfo=-1\fi
+\numberwithin{equation}{section}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newcommand{\CP}{\mathbb{CP}}
+\newcommand{\R}{\mathbb{R}}
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+\DeclareMathOperator{\spec}{spec}
+\DeclareMathOperator{\diag}{diag}
+\setlist[enumerate]{itemsep=3pt,topsep=5pt}
+\setlist[itemize]{itemsep=3pt,topsep=5pt}
+\setlength{\emergencystretch}{1.5em}
+\lstdefinestyle{certificate}{language=Python,basicstyle=\ttfamily\footnotesize,
+  numbers=left,numberstyle=\tiny\color{black!60},numbersep=7pt,
+  breaklines=true,breakatwhitespace=false,columns=fullflexible,
+  keepspaces=true,showstringspaces=false,tabsize=4,
+  keywordstyle=\bfseries,commentstyle=\itshape\color{black!65},
+  frame=single,framerule=.3pt,xleftmargin=1em,xrightmargin=.2em,
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+  literate={²}{{$^2$}}1 {≤}{{$\le$}}1}
diff --git a/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/references.bib b/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/references.bib
new file mode 100644
index 0000000000000000000000000000000000000000..aedaadc3e0e1112278a9949e974d97f13f63e75d
--- /dev/null
+++ b/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/references.bib
@@ -0,0 +1,202 @@
+@incollection{Calabi1982,
+  author = {Calabi, Eugenio},
+  title = {Extremal {K}\"ahler metrics},
+  booktitle = {Seminar on Differential Geometry},
+  series = {Annals of Mathematics Studies},
+  volume = {102},
+  publisher = {Princeton University Press},
+  address = {Princeton, NJ},
+  year = {1982},
+  pages = {259--290}
+}
+@article{Chen2001,
+  author = {Chen, Xiuxiong},
+  title = {{Calabi} flow in {Riemann} surfaces revisited: a new point of view},
+  journal = {International Mathematics Research Notices},
+  year = {2001},
+  volume = {2001},
+  number = {6},
+  pages = {275--297},
+  note = {\href{https://doi.org/10.1155/S1073792801000149}{doi:10.1155/S1073792801000149}}
+}
+@article{ChenHe2008,
+  author = {Chen, Xiuxiong and He, Weiyong},
+  title = {On the {Calabi} flow},
+  journal = {American Journal of Mathematics},
+  volume = {130},
+  number = {2},
+  year = {2008},
+  pages = {539--570},
+  note = {\href{https://arxiv.org/abs/math/0603523}{arXiv:math/0603523}}
+}
+@article{HeZeng2021,
+  author = {He, Weiyong and Zeng, Yu},
+  title = {The {Calabi} flow with rough initial data},
+  journal = {International Mathematics Research Notices},
+  year = {2021},
+  volume = {2021},
+  number = {10},
+  pages = {7470--7551},
+  note = {\href{https://doi.org/10.1093/imrn/rnz050}{doi:10.1093/imrn/rnz050}}
+}
+@article{DongZhang2015,
+  author = {Dong, Hongjie and Zhang, Hong},
+  title = {Schauder estimates for higher-order parabolic systems with time irregular coefficients},
+  journal = {Calculus of Variations and Partial Differential Equations},
+  volume = {54},
+  number = {1},
+  year = {2015},
+  pages = {47--74},
+  note = {\href{https://doi.org/10.1007/s00526-014-0777-y}{doi:10.1007/s00526-014-0777-y}}
+}
+@article{Guan2007,
+  author = {Guan, Daniel},
+  title = {Extremal solitons and exponential {$C^\infty$} convergence of the modified {Calabi} flow on certain {$\mathbb{CP}^1$} bundles},
+  journal = {Pacific Journal of Mathematics},
+  volume = {233},
+  number = {1},
+  year = {2007},
+  pages = {91--124},
+  note = {\href{https://msp.org/pjm/2007/233-1/pjm-v233-n1-p05-s.pdf}{Publisher's full text}}
+}
+@article{BermanDarvasLu2017,
+  author = {Berman, Robert J. and Darvas, Tam{\'a}s and Lu, Chinh H.},
+  title = {Convexity of the extended {$K$}-energy and the large time behavior of the weak {Calabi} flow},
+  journal = {Geometry \& Topology},
+  volume = {21},
+  number = {5},
+  year = {2017},
+  pages = {2945--2988},
+  note = {\href{https://doi.org/10.2140/gt.2017.21.2945}{doi:10.2140/gt.2017.21.2945}}
+}
+@misc{LiZhangZheng2024,
+  author = {Li, Haozhao and Zhang, Linwei and Zheng, Kai},
+  title = {{Calabi} flow with bounded {$L^p$} scalar curvature},
+  year = {2024},
+  note = {\href{https://arxiv.org/abs/2312.05816v2}{arXiv:2312.05816v2}}
+}
+@misc{LiZhang2025,
+  author = {Li, Haozhao and Zhang, Linwei},
+  title = {{Calabi} flow with bounded {$L^p$} scalar curvature ({II})},
+  year = {2025},
+  note = {\href{https://arxiv.org/abs/2510.14302v2}{arXiv:2510.14302v2}}
+}
+
+@book{Kato1995,
+  author = {Kato, Tosio},
+  title = {Perturbation Theory for Linear Operators},
+  edition = {Second},
+  series = {Classics in Mathematics},
+  publisher = {Springer},
+  address = {Berlin},
+  year = {1995},
+  note = {\href{https://doi.org/10.1007/978-3-642-66282-9}{doi:10.1007/978-3-642-66282-9}}
+}
+
+@misc{Huang2026,
+  author = {Huang, Hong},
+  title = {The {Cauchy} problem for fully nonlinear parabolic systems on manifolds},
+  year = {2026},
+  note = {Version 8, 25 August 2026; \href{https://arxiv.org/abs/1506.05030v8}{arXiv:1506.05030v8}}
+}
+@article{Hormander1961,
+  author = {H{\"o}rmander, Lars},
+  title = {Hypoelliptic differential operators},
+  journal = {Annales de l'Institut Fourier},
+  volume = {11},
+  year = {1961},
+  pages = {477--492},
+  note = {\href{https://doi.org/10.5802/aif.117}{doi:10.5802/aif.117}}
+}
+
+@article{Calabi1979,
+  author = {Calabi, Eugenio},
+  title = {M\'etriques k\"ahl\'eriennes et fibr\'es holomorphes},
+  journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
+  series = {4},
+  volume = {12},
+  number = {2},
+  year = {1979},
+  pages = {269--294},
+  note = {\href{https://doi.org/10.24033/asens.1367}{doi:10.24033/asens.1367}}
+}
+@article{HwangSinger2002,
+  author = {Hwang, Andrew D. and Singer, Michael A.},
+  title = {A momentum construction for circle-invariant {K}\"ahler metrics},
+  journal = {Transactions of the American Mathematical Society},
+  volume = {354},
+  number = {6},
+  year = {2002},
+  pages = {2285--2325},
+  note = {\href{https://doi.org/10.1090/S0002-9947-02-02965-3}{doi:10.1090/S0002-9947-02-02965-3}}
+}
+@article{Guillemin1994,
+  author = {Guillemin, Victor},
+  title = {Kaehler structures on toric varieties},
+  journal = {Journal of Differential Geometry},
+  volume = {40},
+  number = {2},
+  year = {1994},
+  pages = {285--309},
+  note = {\href{https://doi.org/10.4310/jdg/1214455538}{doi:10.4310/jdg/1214455538}}
+}
+@article{Abreu1998,
+  author = {Abreu, Miguel},
+  title = {{K}\"ahler geometry of toric varieties and extremal metrics},
+  journal = {International Journal of Mathematics},
+  volume = {9},
+  year = {1998},
+  pages = {641--651},
+  note = {\href{https://doi.org/10.1142/S0129167X98000282}{doi:10.1142/S0129167X98000282}}
+}
+@article{Chrusciel1991,
+  author = {Chru{\'s}ciel, Piotr T.},
+  title = {Semi-global existence and convergence of solutions of the
+           {Robinson--Trautman} (2-dimensional {Calabi}) equation},
+  journal = {Communications in Mathematical Physics},
+  volume = {137},
+  year = {1991},
+  pages = {289--313}
+}
+@article{Struwe2002,
+  author = {Struwe, Michael},
+  title = {Curvature flows on surfaces},
+  journal = {Annali della Scuola Normale Superiore di Pisa, Classe di Scienze},
+  series = {5},
+  volume = {1},
+  number = {2},
+  year = {2002},
+  pages = {247--274},
+  note = {\href{https://www.numdam.org/item/ASNSP_2002_5_1_2_247_0/}{Publisher archive}}
+}
+@article{ChenCheng2021,
+  author = {Chen, Xiuxiong and Cheng, Jingrui},
+  title = {On the constant scalar curvature {K}\"ahler metrics ({I})---{A} priori estimates},
+  journal = {Journal of the American Mathematical Society},
+  volume = {34},
+  number = {4},
+  year = {2021},
+  pages = {909--936},
+  note = {\href{https://doi.org/10.1090/jams/967}{doi:10.1090/jams/967}}
+}
+
+@article{NedialkovJacksonCorliss1999,
+  author = {Nedialkov, N. S. and Jackson, K. R. and Corliss, G. F.},
+  title = {Validated solutions of initial value problems for ordinary differential equations},
+  journal = {Applied Mathematics and Computation},
+  volume = {105},
+  number = {1},
+  year = {1999},
+  pages = {21--68},
+  note = {\href{https://doi.org/10.1016/S0096-3003(98)10083-8}{doi:10.1016/S0096-3003(98)10083-8}}
+}
+
+@article{Streets2014,
+  author = {Streets, Jeffrey},
+  title = {Long time existence of minimizing movement solutions of {Calabi} flow},
+  journal = {Advances in Mathematics},
+  volume = {259},
+  year = {2014},
+  pages = {688--729},
+  note = {\href{https://doi.org/10.1016/j.aim.2014.03.027}{doi:10.1016/j.aim.2014.03.027}}
+}
diff --git a/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/sections/certificate.tex b/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/sections/certificate.tex
new file mode 100644
index 0000000000000000000000000000000000000000..0955d6d6f6b7f6563f451d385b86f14efd6bf882
--- /dev/null
+++ b/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/sections/certificate.tex
@@ -0,0 +1,701 @@
+\section{Certification of the finite shooting calculation}
+\label{cert:appendix}
+\label{sec:certificate}
+\label{cert:section}
+
+This Appendix proves the finite part of the shooting argument.  The calculation
+uses integer interval arithmetic, Taylor expansion on complex disks, and a
+separate estimate for the nonlinear dependence on the shooting parameters.
+Every terminating decimal in this Appendix denotes its exact rational value.
+
+\begin{proposition}[Finite shooting bounds]\label{cert:shoot}
+For every parameter \(z_*\in[-R,R]^2\), where \(R=5\cdot10^{-10}\), let
+\((k,d)\) be given by \eqref{prof:parameters}, and let \(p\) be the origin
+solution of \eqref{prof:ode} supplied by Lemma~\ref{prof:origin}.
+This solution extends to \(0\le s\le S_0=56\) and satisfies
+\[
+ 1+p(s)>0.
+\]
+For the coordinates of its jet relative to \(p_{\rm a}\), defined in
+\eqref{prof:coordinates}, one has
+\begin{equation}\label{cert:conclusion}
+ |q|\le .25R,\qquad
+ \big\|(\Re\xi,\Im\xi)-z_*\big\|_\infty\le .13R.
+\end{equation}
+\end{proposition}
+
+The proof compares the entire family of shots with a sequence of
+reference trajectories at the central parameter \(d=d_0\).  The state
+is \((p,p',p'',d)\), with \(d'=0\).  We divide \([1,56]\) into finitely
+many blocks.  On each block, the reference trajectory is the exact
+solution starting from a rounded three-component jet at its left
+endpoint.  At the right endpoint, an interval enclosure supplies the
+rounded starting jet for the next block.  Thus successive reference
+trajectories may have small jumps at block boundaries; their size is
+included in the error estimate.
+
+The interval calculation encloses the reference endpoint jets and
+the first-variation matrices of the four-component state across each
+block.  Products of these matrices give a linear prediction for the
+effect of the two shooting parameters through all blocks.  Rounding the reference jets
+and variation matrices introduces separate errors.  A perturbation
+estimate compares products of the rounded variation matrices with
+the exact transfers and bounds the propagation of those errors.
+The origin series supplies the initial central jet, its two parameter
+variations, and a bound for the nonlinear parameter remainder.
+
+It remains to compare these linear predictions with the actual
+solutions for every \(z_*\in[-R,R]^2\).  The only nonlinear term in the
+state equation depends on \(p\); its remainder after linearization
+is quadratic in the difference from the reference value.  A uniform
+bootstrap bounds this remainder using the certified positive lower
+bounds for \(1+p\) along the reference trajectories and the transfer estimates.
+It proves continuation and positivity for the whole shooting family and bounds its endpoint
+error.  Projecting the endpoint jets into the matching coordinates
+then gives \eqref{cert:conclusion}.
+
+We use the maximum norm on vectors and the induced maximum row-sum
+norm on matrices, including rectangular matrices.  For a
+\(4\times4\) matrix, its superscript \((3)\) denotes its leading
+\(3\times3\) block.  We first describe the interval arithmetic and
+the series recurrences, then justify their analytic enclosures and
+the separate nonlinear comparison.
+
+\subsection{Integer intervals and the two series recurrences}
+\label{cert:arithmetic}
+
+Put \(Q_{\rm int}=2^{256}\).  An integer pair \((a,b)\) represents the closed
+interval \([a/Q_{\rm int},b/Q_{\rm int}]\).  All arithmetic in
+\texttt{verification/certificate.py} is integer arithmetic.  Addition and
+multiplication by an integer are exact, with endpoint reversal for a negative
+multiplier.  For multiplication, let \(m\) and \(M\) be the minimum and maximum
+of the four products of integer endpoints.  The output pair is
+\[
+ \left(\left\lfloor\frac{m}{Q_{\rm int}}\right\rfloor,
+       \left\lceil\frac{M}{Q_{\rm int}}\right\rceil\right).
+\]
+For a same-sign interval \((a,b)\) excluding zero, the reciprocal pair is
+\[
+ \left(\left\lfloor\frac{Q_{\rm int}^2}{b}\right\rfloor,
+       \left\lceil\frac{Q_{\rm int}^2}{a}\right\rceil\right).
+\]
+These formulas are valid also for negative intervals.  Division by a positive
+integer rounds each endpoint outward in the same way.  Python's integer
+division is floor division, so the identity
+\(\lceil x\rceil=-\lfloor-x\rfloor\) implements the required upper roundings.
+In particular, \texttt{rat(a,b)} encloses the exact rational \(a/b\).
+The function \texttt{mag} bounds absolute value; the function \texttt{norm}
+bounds the matrix norm specified above.
+
+The operation \texttt{widen} enlarges an interval at each end by the upper
+endpoint of a nonnegative error allowance.  The operation \texttt{mid} instead
+selects the grid point
+\[
+ Q_{\rm int}^{-1}\left\lfloor\frac{a+b}{2}\right\rfloor.
+\]
+For an interval of real width \(w\), its distance from every enclosed number
+is at most \(w/2+Q_{\rm int}^{-1}\).  The origin and block estimates below
+explicitly charge this error; midpoint selection is never used as an
+enclosure operation.
+
+Here are the coefficient identities underlying the computation.
+If \(z(\theta)=\sum_{i\ge0}z_i\theta^i\) and
+\(b(\theta)=1/z(\theta)=\sum_{i\ge0}b_i\theta^i\), then
+\begin{equation}\label{cert:reciprocal}
+ b_0=z_0^{-1},\qquad
+ b_i=-z_0^{-1}\sum_{t=1}^i z_t b_{i-t}\quad(i\ge1).
+\end{equation}
+Writing a dot for any one parameter variation gives, for \(i\ge0\),
+\begin{equation}\label{cert:tangent-reciprocal}
+ \dot b_i=-z_0^{-1}
+ \left(\sum_{t=0}^i\dot z_t b_{i-t}
+             +\sum_{t=1}^i z_t\dot b_{i-t}\right).
+\end{equation}
+The second sum is empty at \(i=0\).  In the first sum the coefficient \(b_i\)
+has already been computed by \eqref{cert:reciprocal}.  This is exactly the
+ordering in \texttt{recip}; it includes the variation of the constant
+coefficient \(z_0\).
+
+At the origin, write \(r=s^{3/2}\) locally and
+\[
+ z=1+p=1+d+kr+\sum_{i\ge2}a_i r^i.
+\]
+Equation~\eqref{prof:ode} gives
+\begin{equation}\label{cert:origin-recurrence}
+ a_i=\frac{b_{i-2}-\mathbf1_{\{i=2\}}}
+ { (27/8)(i-1)(i+9)(i+10)},\qquad i\ge2,
+\end{equation}
+where now \(b_i\) are the coefficients of the reciprocal series in \(r\).
+The loop in \texttt{origin} computes \(a_2,\ldots,a_{60}\), together with
+their two variations under the parameter map \eqref{prof:parameters}.
+The weights for evaluation of \(p,p',p''\) at \(s=1\) are
+\[
+ 1,\qquad \frac{3i}{2},\qquad
+ \frac{3i}{2}\left(\frac{3i}{2}-1\right).
+\]
+The code subtracts the constant \(1\) from \(z\) only in the zeroth state
+entry, as required.
+
+For an ordinary Taylor step, let \(\ell=1/h\) be the block length,
+\(s_c\) the step's initial point, \(\theta=h(s-s_c)\), and
+\(\mathfrak r=hs_c+\theta\).  The equation for \(z=1+p\), with
+\(E=1+d\), is
+\begin{equation}\label{cert:scaled-ode}
+ 4\mathfrak r^3 z_{\theta\theta\theta}
+ +120\mathfrak r^2 z_{\theta\theta}
+ +751\mathfrak r z_\theta-1215(z-E)
+ =\frac{4}{h^3}\mathfrak r^3(z^{-1}-1).
+\end{equation}
+This identity also gives the four tangent equations upon differentiation.
+For completeness, put \(\mathfrak R=8hs_c\) and
+\(F_i=b_i-\mathbf1_{\{i=0\}}\), with negative-index coefficients zero.
+The scalar coefficient update used in \texttt{evolve} is
+\begin{align}
+ 4(i+1)(i+2)(i+3)\mathfrak R^3z_{i+3}
+ ={}&\frac4{h^3}\sum_{a=0}^3
+     c_a\mathfrak R^{3-a}F_{i-a}\notag\\
+ &-(96i+960)\mathfrak R^2(i+1)(i+2)z_{i+2}\notag\\
+ &-64\mathfrak R(i+1)
+       \bigl(12i(i-1)+240i+751\bigr)z_{i+1}\notag\\
+ &-512\bigl(4i(i-1)(i-2)+120i(i-1)+751i-1215\bigr)z_i\notag\\
+ &-1215\cdot512E\mathbf1_{\{i=0\}},
+ \qquad(c_0,c_1,c_2,c_3)=(1,24,192,512).
+ \label{cert:step-recurrence}
+\end{align}
+Indeed \(c_a\mathfrak R^{3-a}\) are the coefficients of
+\((\mathfrak R+8\theta)^3\).  The initial three coefficients are
+\(z_0,z_\theta(0),z_{\theta\theta}(0)/2\).
+For a tangent column one differentiates every term of
+\eqref{cert:step-recurrence}; the constant \(E\) has variation \(0,0,0,1\)
+in the four columns.  Thus the same update computes both state and tangent
+coefficients.  The values \(i=0,\ldots,43\) produce coefficients through
+degree \(46\).  Horner evaluation at \(\theta=1/8\), after the appropriate
+one or two differentiations, evaluates the three state entries through
+degrees \(46,45,44\), respectively.  The analytic justification for their
+common remainder allowance is given in Subsection~\ref{cert:disks}.
+
+\subsection{The complete error at the origin}\label{cert:origin-error}
+
+The parameter family of Lemma~\ref{prof:origin} is holomorphic on the
+complex polydisc
+\[
+ |k-k_0|\le .1,\qquad |d-d_0|\le .1,
+\]
+and its coefficients satisfy
+\begin{equation}\label{cert:origin-majorant}
+ \sum_{i\ge1}|a_i|3^i<.8,\qquad a_1=k.
+\end{equation}
+One can check the constants directly from
+\eqref{cert:origin-recurrence}: the nonfree denominators exceed \(440\),
+\(3|k|<.585\), and \(|1+d|>1.30\).  The majorant improves itself because
+\[
+ .585+\frac9{440}\left(1+\frac1{1.30-.8}\right)
+ =\frac{711}{1100}<.8.
+\]
+The reciprocal remains nonsingular throughout this construction.
+In particular the real solutions are positive up to \(s=1\).
+Also \(|d|<.501\), and \eqref{cert:origin-majorant} gives
+\[
+ |p(1)|<.501+\frac{.8}3<.768,\quad
+ |p'(1)|\le .8\max_{i\ge1}\frac{3i/2}{3^i}\le .4,\quad
+ |p''(1)|\le .8\max_{i\ge1}
+       \frac{(3i/2)(3i/2-1)}{3^i}<.534.
+\]
+In particular the uniform bound \(2\) for every jet entry is available on
+the parameter polydisc.
+
+Let
+\begin{equation}\label{cert:origin-tail}
+ E_{60}=.8\,\frac{(3\cdot61/2)^2}{3^{61}}
+       <5.267\cdot10^{-26}.
+\end{equation}
+The sequence \((3i/2)^2/3^i\) decreases for \(i\ge61\).
+Consequently \(E_{60}\) bounds the omitted tail of each of the three jet
+entries.  Cauchy's estimate in either parameter bounds the corresponding
+first derivative of this tail at the center by \(10E_{60}\).
+For the matrix in \eqref{prof:parameters}, the sum of the absolute values
+of all four entries, including the factor \(10^{-8}\), is
+\(.01169933\).  Thus the row norm of the omitted tail of the two
+\(z_*\)-variation columns is at most \(.12E_{60}\).
+
+Each of the two original parameter displacements has absolute value at most
+\(.011R\).  The multivariable Cauchy estimate, with
+\(t_*=.11R\), bounds the terms of total degree at least two in the parameter
+expansion of any jet entry by
+\begin{equation}\label{cert:parameter-tail}
+ E_{\rm par}
+ =2\sum_{j\ge2}(j+1)t_*^j
+ =2\bigl((1-t_*)^{-2}-1-2t_*\bigr)
+ <1.816\cdot10^{-20}.
+\end{equation}
+The factor \(j+1\) counts the monomials of total degree \(j\) in two
+parameters.
+
+The width assertion in \texttt{origin} bounds every computed state or
+variation interval by \(10^{-20}\).
+Let \(U_0\in\R^3\) be the three selected state midpoints.
+Let \(Z\) have their two selected variation columns in its first three
+rows, and in its fourth row the exact second row of the parameter map
+\eqref{prof:parameters}.  The code stores intervals containing that exact
+fourth row; its matrix estimates therefore apply to this choice of \(Z\).
+Combining the state midpoint error, the state and variation tails,
+the two variation midpoint errors, and \eqref{cert:parameter-tail}, gives
+\begin{align}
+ \left\|
+ \begin{pmatrix}p(1)\\p'(1)\\p''(1)\\d\end{pmatrix}
+ -\begin{pmatrix}U_0\\d_0\end{pmatrix}-Zz_*
+ \right\|_\infty
+ &\le 5\cdot10^{-21}+E_{60}+E_{\rm par}
+        +.12E_{60}R+10^{-20}R+\frac3{Q_{\rm int}}\notag\\
+ &<2.316\cdot10^{-20}<2\cdot10^{-19}.
+ \label{cert:initial-error}
+\end{align}
+The fourth component of the error is exactly zero.
+All uncertainty in the interval construction of the central rational
+parameters is already included in the width assertion.
+
+\subsection{Analytic continuation and Taylor remainder on a step}
+\label{cert:disks}
+
+The 288 blocks run from \(s=1\) to \(s=56\), with the following lengths:
+\[
+ \begin{array}{c|c|c}
+ \text{range}&\ell&\text{number of blocks}\\ \hline
+ {[1,2]}&1/32&32\\
+ {[2,4]}&1/16&32\\
+ {[4,8]}&1/8&32\\
+ {[8,56]}&1/4&192.
+ \end{array}
+\]
+Each block consists of eight Taylor steps of length \(\ell/8\).
+At every step start the interval assertions imply, for the exact reference
+state,
+\begin{equation}\label{cert:step-start}
+ -.76<p(0)<.4,\qquad |p_\theta(0)|<.06,\qquad
+ |p_{\theta\theta}(0)|<.01.
+\end{equation}
+Every entry in the first three rows of the four-column scaled variation
+matrix has modulus less than \(6\); its fourth row is \((0,0,0,1)\).
+The step schedule gives \(hs_c\ge32\).
+
+We now justify an analytic bound on the full complex disk
+\(|\theta|\le1/2\), using only these starting bounds.
+On a disk where \(|p(\theta)-p(0)|\le.1\), one has
+\[
+ |\mathfrak r|\ge31.5,\qquad |1+p|>.14,\qquad
+ |p|<.86,\qquad |p-d_0|<1.3.
+\]
+Let \(H_2\) be the supremum of \(|p_{\theta\theta}|\) on any smaller
+concentric disk.  Integrate the equation for \(p_{\theta\theta}\)
+along radial line segments and apply the integral Gronwall inequality.
+Since
+\[
+ \frac{30}{31.5}<.954,\qquad
+ \frac{B}{31.5^2}<.190,\qquad
+ \frac{|C|}{31.5^3}<.011,\qquad h^{-3}\le.015625,
+\]
+and \(|p_\theta|\le .06+H_2/2\), this gives
+\begin{equation}\label{cert:disk-h2}
+ H_2\le
+ e^{.954/2}\left[
+ .01+\frac12\left\{.190(.06+H_2/2)
+               +.011\cdot1.3+.015625\frac{.86}{.14}\right\}\right].
+\end{equation}
+The coefficient of \(H_2\) on the right is less than one.
+Solving this linear inequality yields \(H_2<.123602<.13\).
+Therefore
+\begin{equation}\label{cert:disk-displacement}
+ |p(\theta)-p(0)|\le .06/2+.13/8=.04625<.1.
+\end{equation}
+This strictly improves the bootstrap.  On every disk of radius at most
+\(1/2\), the state and its first two derivatives are bounded and both
+denominators \(\mathfrak r\) and \(1+p\) stay uniformly away from zero.
+The local holomorphic ODE theorem then continues the solution across any
+putative boundary of its disk of existence.  Uniqueness makes the
+continuations agree.  Thus the solution is holomorphic on a neighborhood
+of the closed disk of radius \(1/2\), with the displayed bounds.
+
+In the scaled state \((p,p_\theta,p_{\theta\theta},d)\), the Jacobian is
+\begin{equation}\label{cert:scaled-jacobian}
+ \mathcal J(\theta)=
+ \begin{pmatrix}
+ 0&1&0&0\\
+ 0&0&1&0\\
+ -C/\mathfrak r^3-h^{-3}/(1+p)^2&
+       -B/\mathfrak r^2&-30/\mathfrak r&C/\mathfrak r^3\\
+ 0&0&0&0
+ \end{pmatrix}.
+\end{equation}
+On the complex disk its norm is bounded by the maximum of \(1\) and
+\[
+ \frac{30}{31.5}+\frac{751/4}{31.5^2}
+       +\frac{1215/2}{31.5^3}+\frac{1/64}{.14^2}<2.1.
+\]
+Each actual variation column, with initial norm at most \(6\), consequently
+has norm at most \(6e^{2.1/2}<17.146<32\) throughout the disk.
+The state entries themselves also have modulus less than \(32\).
+These estimates apply to the actual block variation already propagated
+from the block's initial identity matrix; a new identity is not assumed
+at each substep.
+
+If a scalar holomorphic function is bounded by \(32\) on this disk,
+Cauchy's coefficient estimate gives a tail after degree \(44\), at
+\(\theta=1/8\), bounded by
+\begin{equation}\label{cert:step-tail}
+ 32\sum_{i\ge45}\left(\frac{1/8}{1/2}\right)^i
+ =\frac{128}{3}\,4^{-45}<48\,4^{-45}.
+\end{equation}
+Every state entry and every tangent entry is approximated to at least
+degree \(44\).  Hence the allowance \(48\,4^{-45}\) added by
+\texttt{evolve} encloses all their Taylor tails.  The interval recurrences
+then inductively enclose the exact reference solution and its exact
+first variation through all eight substeps.
+
+On a real substep, the sharper displacement bound is
+\begin{equation}\label{cert:real-step-displacement}
+ |p(\theta)-p(0)|
+ \le .06/8+\frac{.13}{2\cdot8^2}
+ =.008515625<.01.
+\end{equation}
+Accordingly, the smallest lower starting endpoint for \(1+p\), reduced
+by \((\lfloor Q_{\rm int}/100\rfloor+1)/Q_{\rm int}>.01\), is a valid
+lower bound \(m_j\) on the entire reference block.  This is the variable
+\texttt{mini} divided by \(Q_{\rm int}\).
+
+\subsection{The finite inequalities and their exact interpretation}
+\label{cert:ledger}
+
+Write the block endpoints as \(s_0=1,\ldots,s_N=56\), \(N=288\).
+On block \(j\), the reference solution has initial physical jet \(U_j\)
+and parameter \(d_0\); it is evolved exactly on that block.
+The rounded jet selected for the next block is \(U_{j+1}\).
+Let \(M_j\) be the exact \(4\times4\) transfer of the linearized
+four-component physical state along this reference block.
+
+The conversion from scaled derivatives to physical derivatives is essential.
+With
+\[
+ S_j=\diag(1,\ell_j,\ell_j^2,1),
+\]
+a scaled transfer \(T\) becomes \(S_j^{-1}TS_j\).
+Thus its entry in row \(a\), column \(b\), for \(a,b\in\{0,1,2\}\),
+is multiplied by \(h_j^a/h_j^b\); column \(3\) is multiplied by \(h_j^a\).
+The fourth row remains \((0,0,0,1)\).
+The code performs this conversion before asserting the widths.
+Consequently its assertions imply
+\begin{equation}\label{cert:local-errors}
+ \big\|U_{j+1}- (p_{\rm ref},p'_{\rm ref},p''_{\rm ref})
+                         (s_{j+1})\big\|_\infty<10^{-19},
+ \qquad
+ |(M_j-\widetilde M_j)_{ab}|<10^{-15},
+\end{equation}
+where \(\widetilde M_j\) is the selected matrix of transfer midpoints.
+The state and transfer widths are, respectively, less than
+\(10^{-19}\) and \(10^{-15}\); the stated looser midpoint bounds follow
+from the midpoint estimate in Subsection~\ref{cert:arithmetic}.
+All fourth-component errors in \eqref{cert:local-errors} are zero.
+
+Define the products of these point matrices by
+\[
+ \widetilde T_{i,j}=
+ \widetilde M_{i-1}\cdots\widetilde M_j\quad (i>j),
+ \qquad \widetilde T_{i,i}=I_4.
+\]
+The exact integer comparisons in \texttt{check} give
+\begin{align}
+ m_j&>.23,\qquad \|Z\|<.02,\notag\\
+ \max_{i\ge j}\|\widetilde T_{i,j}\|&\le8500,
+ &\max_i\sum_{j<i}\|\widetilde T_{i,j+1}^{(3)}\|&\le570000,\notag\\
+ \max_i\sum_{j<i}\frac{\ell_j}{m_j^3}
+                \|\widetilde T_{i,j+1}^{(3)}\|&\le246000,
+ &\max_i\|\widetilde T_{i,0}Z\|&\le2.7.
+ \label{cert:finite-products}
+\end{align}
+Here and below a maximum over endpoints includes endpoint \(0\);
+the assertion for it follows from \(\|Z\|<.02\).
+
+To explain the summation indices, for each endpoint \(i\) the inner loop
+starts with the identity and takes \(j=i-1,\ldots,0\).
+Immediately before multiplying by \(\widetilde M_j\), its current interval
+matrix encloses \(\widetilde T_{i,j+1}\).
+The update \texttt{T=mm(T,allm[j])} therefore produces the desired forward
+product in the correct order.  If \(r_j=Q_{\rm int}m_j\) and \(\mathfrak d\)
+is the integer interval bound for the leading \(3\times3\) norm, then
+\[
+ \frac{\mathfrak d Q_{\rm int}^{\,2}}{h_jr_j^3}
+ =\frac{\ell_j}{m_j^3}\frac{\mathfrak d}{Q_{\rm int}}.
+\]
+The expression used for \texttt{K} is its floor plus one, hence an upper
+bound even when the quotient is integral.  Thus the interval product
+arithmetic and the weighted sum both round in the required direction.
+
+Let \(P_{\rm a}(d)=(p_{\rm a},p'_{\rm a},p''_{\rm a})(56)\), and put
+\(Y=(\widetilde T_{N,0}Z)_{\mathrm{top}\,3}\).
+The final comparisons in \texttt{check} give
+\begin{align}
+ \|U_N-P_{\rm a}(d_0)\|_\infty&<.04R,&
+ \|\Lambda(U_N-P_{\rm a}(d_0))\|_\infty&<.006R,\notag\\
+ \|(\Lambda Y)_{1,\cdot}\|_{\rm row}&<.116,&
+ \|(\Lambda Y)_{\mathrm{bottom}\,2}-I_2\|&<.001.
+ \label{cert:finite-matching}
+\end{align}
+The jet factors used for the term \(d_m s^{-3m}\) are
+\(1,-3m,3m(3m+1)\), with denominators \(56^{3m}\),
+\(56^{3m+1}\), \(56^{3m+2}\), respectively.
+After multiplication by \(d_m\), these give its value and first two
+derivatives at \(s=56\).
+The index test in the last assertion subtracts the \(2\times2\) identity
+from the two nonreal-coordinate rows.
+
+The listed comparisons are finite integer statements: the displayed source
+constructs their operands and asserts them.  Subsection~\ref{cert:provenance} gives the command for checking them
+with assertions enabled.
+The preceding recurrence and disk arguments prove that these finite
+statements bound the specified analytic trajectories and variations.
+
+\subsection{Transfer errors and estimates within a block}
+\label{cert:transfer-errors}
+
+Let \(T_{i,j}=M_{i-1}\cdots M_j\) be the exact products.
+By \eqref{cert:local-errors},
+\(\|M_j-\widetilde M_j\|<4\cdot10^{-15}\).
+Expand a product of at most \(288\) factors, choosing the error factor at
+\(k\) positions.  Each intervening, possibly empty, product of approximate
+factors has norm at most \(8500\), by \eqref{cert:finite-products}.
+Summing the bounds for \(k\ge1\) gives
+\begin{equation}\label{cert:product-perturbation}
+ \|T_{i,j}-\widetilde T_{i,j}\|
+ \le8500\left[(1+4\cdot10^{-15}\cdot8500)^{288}-1\right]
+ <.001.
+\end{equation}
+In particular \(\|T_{i,j}\|<8501\).  Since the lengths of all the
+blocks sum to \(55\), \eqref{cert:finite-products} implies
+\begin{equation}\label{cert:exact-sums}
+ \max_i\sum_{j<i}\|T_{i,j+1}^{(3)}\|\le580000,\qquad
+ \max_i\sum_{j<i}\frac{\ell_j}{m_j^3}
+             \|T_{i,j+1}^{(3)}\|\le250000,
+\end{equation}
+using the elementary comparisons
+\[
+ 570000+288\cdot.001<580000,\qquad
+ 246000+\frac{55\cdot.001}{.23^3}<250000.
+\]
+Also
+\begin{equation}\label{cert:linear-size}
+ \|T_{i,0}Z\|\le2.7+.001\cdot.02<2.72.
+\end{equation}
+
+We need two estimates for transfers between any two ordered points of
+one reference block, including points in different Taylor substeps.
+The real scaled Jacobian \eqref{cert:scaled-jacobian} has norm at most
+\[
+ L_{\rm real}=
+ \max\left\{1,\frac{30}{32}+\frac{751/4}{32^2}
+                  +\frac{1215/2}{32^3}
+                  +\frac{1/64}{.23^2}\right\}.
+\]
+An interval of at most one block unit therefore has scaled transfer norm
+at most \(e^{L_{\rm real}}<4.3\).
+Let \(\Phi_j(s,t)\) denote its physical transfer.
+Conjugation by \(S_j\) does not increase either the first-row norm or the
+maximum norm of the third column: in the first row the scaling factors
+are \(1,\ell_j,\ell_j^2,1\), and in the third column they are
+\(\ell_j^2,\ell_j,1,\ell_j^2\).
+Thus, with \(e_3=(0,0,1,0)^t\),
+\begin{equation}\label{cert:partial-transfer}
+ \|(\Phi_j(s,t))_{1,\cdot}\|_{\rm row}\le4.3,\qquad
+ \|\Phi_j(s,t)e_3\|_\infty\le4.3
+ \quad(s_j\le t\le s\le s_{j+1}).
+\end{equation}
+Only these two parts of a physical transfer are needed.  Its full norm
+need not satisfy the same estimate.
+
+All exponential comparisons just used, and the one in
+\eqref{cert:disk-h2}, can be checked by rational arithmetic.
+For \(0\le x<82\), let
+\[
+ E_{80}(x)=\sum_{j=0}^{80}\frac{x^j}{j!}
+ +\frac{x^{81}}{81!}\frac1{1-x/82}.
+\]
+The positive exponential tail has successive ratios at most \(x/82\),
+so \(e^x\le E_{80}(x)\).
+Substituting the displayed rational values in this formula verifies all
+the strict numerical comparisons.  The supplementary
+\texttt{verification/arithmetic\_allowances.py} performs these rational
+checks, including \eqref{cert:initial-error},
+\eqref{cert:product-perturbation}, and the remaining comparisons below.
+
+\subsection{Uniform nonlinear error and closure of the bootstrap}
+\label{cert:nonlinear}
+
+Fix \(z_*\in[-R,R]^2\).  On each block compare its actual shot with the
+reference solution on that block, and write \(\delta p=p-p_{\rm ref}\).
+For the only nonlinear term in the physical third-component equation,
+\[
+ -\frac{p}{1+p}
+ +\frac{p_{\rm ref}}{1+p_{\rm ref}}
+ +\frac{\delta p}{(1+p_{\rm ref})^2}
+ =\frac{(\delta p)^2}{(1+p_{\rm ref})^2(1+p)}.
+\]
+It follows that the difference equation, beyond the reference
+linearization, has forcing \(e_3\mathcal N\), where
+\begin{equation}\label{cert:nonlinear-remainder}
+ |\mathcal N|=
+ \frac{|\delta p|^2}{|1+p_{\rm ref}|^2|1+p|}.
+\end{equation}
+The \(d\)-dependence is linear and is already included in the fourth
+variation component.
+
+Temporarily assume \(|\delta p|\le14R\) throughout the portions of the
+blocks reached so far.  Since \(m_j>.23\) and
+\(m_j/(m_j-14R)<1.001\), \eqref{cert:nonlinear-remainder} implies
+\[
+ |\mathcal N|\le\frac{1.001(14R)^2}{m_j^3}.
+\]
+The error accumulated by the end of block \(j\), before multiplication
+by subsequent complete-block transfers, has zero fourth component.
+Its norm is bounded by
+\[
+ 10^{-19}
+  +4.3\,\ell_j\frac{1.001(14R)^2}{m_j^3}.
+\]
+The first term is the reference jump
+\eqref{cert:local-errors}; the second follows by variation of constants
+and the third-column estimate in \eqref{cert:partial-transfer}.
+
+Define the endpoint error \(e_i\) by
+\[
+ \begin{pmatrix}p(s_i)\\p'(s_i)\\p''(s_i)\\d\end{pmatrix}
+ =\begin{pmatrix}U_i\\d_0\end{pmatrix}
+          +T_{i,0}Zz_*+e_i.
+\]
+Its fourth component is zero.  By \eqref{cert:initial-error},
+\eqref{cert:exact-sums}, and the preceding local error estimate,
+\begin{align}
+ \|e_i\|_\infty
+ &\le580000\cdot10^{-19}+8501\cdot(2\cdot10^{-19})
+       +4.3\cdot250000\cdot1.001(14R)^2\notag\\
+ &=.1055747504R<.11R.
+ \label{cert:endpoint-error}
+\end{align}
+For an interior point of a block, the first-row estimate in
+\eqref{cert:partial-transfer}, \eqref{cert:linear-size}, and the forcing
+estimate give
+\begin{align}
+ |\delta p|
+ &\le4.3(2.72+.11)R
+      +4.3\frac{1.001(14R)^2}{.23^3}\notag\\
+ &<12.169035R<14R.
+ \label{cert:bootstrap-improvement}
+\end{align}
+Here replacing the remaining block length by \(1\) is a harmless
+overestimate.  This improves the bootstrap strictly both at block starts
+and in block interiors, including the reference jumps.
+At \(s=1\) it starts from \(\|Z\|<.02\) and
+\eqref{cert:initial-error}.  Continuation over the finitely many blocks
+therefore proves \eqref{cert:endpoint-error} at every block endpoint.
+
+For clarity, existence is part of this continuation argument.
+As long as the bootstrap holds, \(1+p\ge .23-14R>0\).
+On the finite interval \(1\le s\le56\), the equations for \(p'\) and
+\(p''\) form a linear system with bounded coefficients and bounded forcing
+once \(p\) stays in this range.  Gronwall's inequality bounds these
+derivatives on every such interval.  Thus no other finite-time escape
+can interrupt continuation.  Positivity before \(s=1\) was established
+in Subsection~\ref{cert:origin-error}.  This proves the existence and
+positivity statements of Proposition~\ref{cert:shoot}.
+
+\subsection{Projection at the matching point}\label{cert:projection}
+
+Let \(\Gamma\) denote the exact real coordinate matrix of
+\eqref{prof:coordinates}.
+Lemma~\ref{prof:root-bounds} gives
+\[
+ |\Gamma_{ab}-\Lambda_{ab}|<.001,
+ \qquad
+ \|\Gamma_{1,\cdot}\|_{\rm row}<.92,\qquad
+ \|\Gamma_{a,\cdot}\|_{\rm row}<.68\quad(a=2,3).
+\]
+Thus \(\|\Gamma-\Lambda\|<.003\).
+The polynomial \(P_{\rm a}\) also satisfies
+\begin{equation}\label{cert:pa-variation}
+ \|P_{\rm a}(d)-P_{\rm a}(d_0)\|_\infty
+ \le .01|d-d_0|\le .01\cdot.011R;
+\end{equation}
+this follows by differentiating the four explicit coefficients in
+\eqref{prof:asymptotic-polynomial}, as detailed in the proof of
+Lemma~\ref{prof:tail-residual}.
+
+Set \(A_*=U_N-P_{\rm a}(d_0)\).
+The endpoint error and transfer error give
+\[
+ \begin{pmatrix}p\\p'\\p''\end{pmatrix}(56)-P_{\rm a}(d)
+ = A_*+Yz_*+\epsilon_*
+       -\bigl(P_{\rm a}(d)-P_{\rm a}(d_0)\bigr),
+ \qquad
+ \|\epsilon_*\|_\infty\le
+       \bigl(.11+.001\cdot.02\bigr)R.
+\]
+In fact \(\epsilon_*\) is the sum of the top three entries of \(e_N\)
+and \(((T_{N,0}-\widetilde T_{N,0})Zz_*)_{\mathrm{top}\,3}\).
+Apply one row of \(\Gamma\).
+The contribution of \(A_*\), using \eqref{cert:finite-matching}, is at
+most \((.006+.003\cdot.04)R\).
+For the first row the linear contribution is bounded by
+\((.116+.003\cdot2.7)R\).
+For row \(a=2,3\), after subtracting the corresponding coordinate of
+\(z_*\), it is bounded by \((.001+.003\cdot2.7)R\).
+The remaining contribution is bounded by the row norm of \(\Gamma\)
+times
+\[
+ \bigl(.11+.001\cdot.02+.01\cdot.011\bigr)R.
+\]
+Consequently, in units of \(R\), the two required estimates are
+\begin{align*}
+ |q|/R
+ &\le .006+.003\cdot.04+.116+.003\cdot2.7
+       +.92(.11+.001\cdot.02+.01\cdot.011)\\
+ &=.2315396<.25,\\
+ \big\|(\Re\xi,\Im\xi)-z_*\big\|_\infty/R
+ &\le .006+.003\cdot.04+.001+.003\cdot2.7
+       +.68(.11+.001\cdot.02+.01\cdot.011)\\
+ &=.0901084<.13.
+\end{align*}
+These are exact rational comparisons and establish
+\eqref{cert:conclusion}, completing the proof of
+Proposition~\ref{cert:shoot}.
+
+\subsection{Executable source and reproducibility}
+\label{cert:provenance}
+
+The complete source printed in Appendix~\ref{app:code} is
+\texttt{verification/certificate.py}, comprising \(6594\) UTF-8 bytes.
+Its SHA-256 digest is
+\begin{center}\small\ttfamily
+263389fd7217d84d0e5b66869797fc9f7998\\
+309d524a95f7ad7d3d0f538ea742
+\end{center}
+where the line break is for typesetting only.
+It uses no imported libraries, floating-point operations, external data,
+or platform-dependent numerical routines.  From the paper directory,
+the verification command is
+\begin{quote}\ttfamily
+python3 verification/certificate.py
+\end{quote}
+with Python assertions enabled (no \texttt{-O} or \texttt{-OO}, and no
+\texttt{PYTHONOPTIMIZE} setting that disables assertions).
+The program is silent when every assertion succeeds.
+
+The supplementary rational check
+\begin{quote}\ttfamily
+python3 verification/arithmetic\_allowances.py
+\end{quote}
+verifies the analytic allowances used above by exact fraction arithmetic;
+its decimal report is descriptive only.
+The optional script \texttt{verification/tail-check.py} checks the explicit
+root and tail estimates in Section~\ref{sec:profile} by exact rational
+arithmetic.
+The finite integer assertions, the analytic enclosure argument in
+Subsections~\ref{cert:origin-error}--\ref{cert:disks}, and the nonlinear
+argument in Subsections~\ref{cert:transfer-errors}--\ref{cert:projection}
+together prove the finite shooting proposition.
diff --git a/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/sections/compact.tex b/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/sections/compact.tex
new file mode 100644
index 0000000000000000000000000000000000000000..64410196d0f11fba5ac814ce325ccca24f5554e4
--- /dev/null
+++ b/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/sections/compact.tex
@@ -0,0 +1,666 @@
+\section{A concentrating solution on the compact manifold}
+\label{glue:section}
+\label{sec:compact}
+
+We now correct the flat solution on the auxiliary sphere.  All functions and
+operators in this Section are restricted to the
+$O(m)\times O(6)$ invariant subspace, where $m=24$ and
+$\Sigma=S^{29}\subset\R^{24}\times\R^6$.  We use the spaces of
+Definition~\ref{ana:spaces}, on $\Sigma\times[-T,0)$, with
+\[
+ r=\sqrt{x},\qquad \rho=\sqrt{-t},\qquad
+ \sigma=(r^2+\rho)^{1/2},\qquad p_0=-2+2\delta.
+\]
+When necessary, a subscript specifies the H\"older exponent or the scale:
+for example, $X_\alpha(s)$ has weight $p_0$ and scale $s$, and
+$Y_\alpha(s)$ has weight $p_0-4$.  Without a specified scale, it is
+$\sigma$.  The space $X'_\alpha$ includes the parabolic
+$C^\alpha$ norm of every spatial derivative of order at most four; only
+the time derivative is omitted.
+
+\begin{theorem}[Compact realization]\label{glue:solution}
+Let $G$ be the profile of Theorem~\ref{prof:profile}, put
+\[
+ h(y)=\frac{G(y)-1}{y},\qquad
+ F(t,x)=\rho^{-1}h(x/\rho),
+\]
+and fix $0<\alpha<\alpha'<1$.  Choose $\delta\in(0,1)$ and the ancient
+right inverse supplied by Theorem~\ref{flat:inverse}.  For every
+sufficiently small $\eps>0$, there are $T>0$ and a real invariant
+function $V\in X_\alpha$ on $\Sigma\times[-T,0)$ such that
+$\norm{V}_{X'_\alpha}\le\eps$ and $H=F+V$ satisfies
+\begin{equation}\label{glue:equation}
+ H_t+(1+v_*H)^2\A H=0,\qquad v_*=x(1-x).
+\end{equation}
+The function $H$ is smooth on $\Sigma\times(-T,0)$, and there are
+constants $0<c_1<c_2<\infty$, independent of $t$, such that
+\begin{equation}\label{glue:positive}
+ c_1\le 1+v_*H\le c_2.
+\end{equation}
+In particular, $H(t,\cdot)$ is smooth as a function of $x\in[0,1]$, and
+\begin{equation}\label{glue:center}
+ V(t,0)=O(\rho^{-1+\delta})\qquad(t\uparrow0).
+\end{equation}
+\end{theorem}
+
+We first establish the estimates and the right inverse needed in the
+proof.  All constants below may depend on the fixed profile, the
+exponents, the sphere, and the fixed box radius in
+Definition~\ref{ana:spaces}.  They do not depend on the localization
+radius $\beta$ or on $T$, subject to the restrictions explicitly stated.
+Changing that fixed box radius to another sufficiently small fixed
+radius changes the norms by uniform constants, by the covering property
+in Definition~\ref{ana:spaces}.
+
+\subsection{Background estimates and the compact error}
+
+Write
+\begin{align*}
+ g_c&=1+v_*F=x+(1-x)G(x/\rho),&
+ A_c&=g_c^2,& B_c&=2v_*g_c\A F,\\
+ g_f&=1+xF=G(x/\rho),&
+ A_f&=g_f^2,& B_f&=2xg_f\wh\A F.
+\end{align*}
+Here $B_f$ is in the original flat variables, before the similarity
+change of variables in Section~\ref{sec:flat}.  Set
+\begin{equation}\label{glue:linearizations}
+ \Pcal=\partial_t+A_c\A+B_c,\qquad
+ \wh\Pcal=\partial_t+A_f\wh\A+B_f,\qquad
+ E_c=F_t+A_c\A F.
+\end{equation}
+The profile equation gives $F_t+A_f\wh\A F=0$.  Moreover, if
+\[
+ \mu=\min(1,\inf G)>0,\qquad M=\max(1,\sup G),
+\]
+then
+\begin{equation}\label{glue:background-positive}
+ \mu\le g_c\le M.
+\end{equation}
+
+We make the meaning of all the scaled estimates explicit.  In a box
+with center $(z_0,t_0)$ and center scale $b=\sigma(z_0,t_0)$,
+$O_{\mathrm{sc}}(b^q)$ means that, for every pair of nonnegative integers
+$j,k$, the spatial derivatives of order $j$ and time derivatives of
+order $k$ are bounded by $C_{j,k}b^{q-j-4k}$ on that box.  The scale $b$
+is held fixed when differentiating.  These bounds also give the
+corresponding scaled parabolic H\"older bounds of any exponent less
+than one, on slightly smaller boxes.  The notation will only be used
+for smooth background quantities, not for arbitrary members of $X$
+or $Y$.
+
+\begin{lemma}[Scaled background and mismatch]\label{glue:background}
+For $0<T\le1$, on the compact domain one has
+\begin{equation}\label{glue:background-bounds}
+ F=O_{\mathrm{sc}}(b^{-2}),\qquad
+ \A F=O_{\mathrm{sc}}(b^{-6}),\qquad
+ A_c=O_{\mathrm{sc}}(1),\qquad
+ B_c=O_{\mathrm{sc}}(b^{-4}),\qquad
+ v_*=O_{\mathrm{sc}}(b^2).
+\end{equation}
+The analogous flat bounds hold in a fixed neighborhood of $r=0$.
+In the common coordinates there, write
+\[
+ \Pcal-\wh\Pcal=\sum_{|\nu|\le4}d_\nu(z,t)\partial_z^\nu.
+\]
+For each $j,k\ge0$, the rescaled coefficients satisfy
+\begin{equation}\label{glue:mismatch-coeff}
+ b^{4-|\nu|+j+4k}
+ \abs{\partial_z^j\partial_t^k d_\nu}\le C_{j,k}b^2.
+\end{equation}
+Finally, near $r=0$,
+\begin{equation}\label{glue:error-order}
+ E_c=O_{\mathrm{sc}}(b^{-4}).
+\end{equation}
+Away from a fixed neighborhood of $r=0$, all these background
+quantities extend smoothly to $t=0$ with bounds of every finite order.
+\end{lemma}
+
+\begin{proof}
+In the region $r^2\le C\rho$, the formula for $F$, smoothness of $h$ at
+zero, and the change $z=\sqrt\rho\,\zeta$ give
+\[
+ \abs{\partial_z^j\partial_t^k F}
+ \le C_{j,k}\rho^{-1-j/2-2k}
+ \le C'_{j,k}\sigma^{-2-j-4k}.
+\]
+In the region $r^2\ge\rho$, use the differentiable expansion from
+Theorem~\ref{prof:profile}.  For arbitrary $N$ it has the form
+\begin{equation}\label{glue:outer-expansion}
+ F(t,r^2)=\frac{c-1}{r^2}
+       +\sum_{\ell=1}^{N}a_\ell\frac{(-t)^\ell}{r^{4\ell+2}}
+       +\text{remainder}.
+\end{equation}
+The all-order remainder estimates allow $N$ to be chosen larger than
+any prescribed number of differentiations.  Differentiating the
+displayed terms and the remainder gives
+$\abs{\partial_z^j\partial_t^kF}\le C_{j,k}r^{-2-j-4k}$.
+This proves the first bound and also proves the asserted smooth
+extension away from the core.  In particular it supplies time
+derivative bounds up to the open terminal face, which do not follow
+from spatial decay alone.
+
+Near the core, write a point of the sphere as
+$(z,\sqrt{1-|z|^2}\,\theta)$, $\theta\in S^5$.  Invariant functions
+are independent of $\theta$, and on these functions
+\begin{equation}\label{glue:laplace-coordinate}
+ \Delta_\Sigma
+ =\Delta_z-\sum_{i,j}z_i z_j\partial_{ij}
+             -29\sum_i z_i\partial_i.
+\end{equation}
+The product coordinates and ordinary sphere coordinates have
+uniformly controlled transition maps on a fixed smaller
+neighborhood.  Thus their scaled norms are uniformly equivalent for
+the functions considered here.  Lemma~\ref{geo:operators} gives
+\[
+ \A=\frac1{16}\Delta_\Sigma^2
+       -\frac{37}{4}\Delta_\Sigma+312,
+ \qquad \wh\A=\frac1{16}\Delta_z^2.
+\]
+In $\A-\wh\A$, the coefficients of orders four, three, and two have
+sizes $O(r^2)$, $O(r)$, and $O(1)$, respectively; those of orders one
+and zero are smooth and bounded.  After multiplication by $b^4$ and
+rescaling space by $b$, every coefficient is $O_{\mathrm{sc}}(b^2)$.
+For example the first three sizes become $O(r^2)$, $O(br)$, and
+$O(b^2)$, and $r\le b$.  Differentiating these polynomial coefficients
+preserves the stated scaled bound.  The bounds for $\A F$ follow;
+away from this coordinate neighborhood they follow from the smooth
+extension and the positive lower bound for $\sigma$.
+
+The function $v_*=x-x^2$ has size $O_{\mathrm{sc}}(b^2)$, and
+$v_*-x=-x^2=O_{\mathrm{sc}}(b^4)$.  Hence
+\[
+ g_c-g_f=-x^2F=O_{\mathrm{sc}}(b^2),\qquad
+ A_c-A_f=O_{\mathrm{sc}}(b^2).
+\]
+Products give the remaining bounds in
+\eqref{glue:background-bounds}.  For the zeroth-order mismatch, use
+\[
+ B_c-B_f
+ =2(v_*g_c-xg_f)\wh\A F
+        +2v_*g_c(\A-\wh\A)F.
+\]
+The two terms have size $O_{\mathrm{sc}}(b^{-2})$, since
+$v_*g_c-xg_f=O_{\mathrm{sc}}(b^4)$ and
+$(\A-\wh\A)F=O_{\mathrm{sc}}(b^{-4})$.
+Together with the principal multiplier difference, this proves
+\eqref{glue:mismatch-coeff} in every derivative order.  Finally,
+\[
+ E_c=(A_c-A_f)\wh\A F+A_c(\A-\wh\A)F
+      =O_{\mathrm{sc}}(b^{-4}),
+\]
+using the exact flat equation, as required.
+\end{proof}
+
+One consequence that we will use repeatedly is that the local
+$Y_\vartheta$ size of $E_c$, for either $\vartheta=\alpha$ or
+$\vartheta=\alpha'$, is at most
+\begin{equation}\label{glue:error-weight}
+ C b^{6-2\delta}b^{-4}=C b^{2-2\delta}
+\end{equation}
+near the core.  The same statement holds with any fixed finite number
+of extra scaled derivatives.  Globally $\norm{E_c}_{Y_\vartheta}\le C$.
+If it is multiplied by a smooth cutoff supported in $r\le C_0\beta$
+and varying only where $r$ is comparable to $\beta$, then, whenever
+$T\le\beta^4$,
+\begin{equation}\label{glue:core-error-small}
+ \norm{\text{cutoff}\,E_c}_{Y_\vartheta}
+ \le C\beta^{2-2\delta}.
+\end{equation}
+Indeed every box meeting this support has $b\le C\beta$; cutoff
+derivatives have bounded scaled size.  The strict inequality
+$\delta<1$ is used here.
+
+\subsection{Localization of the ancient inverse}
+
+Multiplying a flat correction by a spatial cutoff produces a commutator
+on the transition annulus.  This error is bounded but need not be small.
+We will include its negative in the forcing for the outer forward
+problem, so it cancels in the combined equation.  The errors that remain
+are the geometric mismatch, made small by shrinking the core, and the
+outer cutoff commutator, made small by shortening the time interval.
+The nested cutoffs below ensure that these cancellations are exact.
+
+Choose once and for all a smooth function
+$\chi:[0,\infty)\to[0,1]$, equal to one on $[0,1]$ and zero on
+$[2,\infty)$.  All its compositions below are smooth at $r=0$
+because they are constant there.  Take $\beta>0$ so small that
+$r\le5\beta$ lies in the common coordinate neighborhood above, and
+assume $T\le\beta^4$.  Define
+\begin{equation}\label{glue:cutoffs}
+ \chi_i=\chi(r/\beta),\qquad
+ \chi_b=\chi(r/(2\beta)),\qquad
+ \chi_o=1-\chi(2r/\beta),\qquad
+ \kappa=1-\chi(4r/\beta).
+\end{equation}
+Thus the inner input cutoff $\chi_i$ is supported in $r\le2\beta$;
+the inner output cutoff $\chi_b$ is one there and supported in
+$r\le4\beta$.  The outer output cutoff $\chi_o$ is zero for
+$r\le\beta/2$ and one for $r\ge\beta$.  The coefficient cutoff
+$\kappa$ is already one for $r\ge\beta/2$ and is zero for
+$r\le\beta/4$.
+
+Figure~\ref{fig:cutoffs} records these nested regions. In particular,
+the inner output cutoff is one on the support of the inner input,
+and the modified outer coefficients agree with the compact coefficients
+where the outer output cutoff can be nonzero. The cutoff commutators are supported on the corresponding transition
+annuli.
+
+\begin{figure}[htbp]
+\centering
+\input{figures/cutoff-support}
+\caption{The cutoff regions in \eqref{glue:cutoffs}. Distances are
+schematic. Solid bands indicate the value one, blank bands the value
+zero, and hatched bands the intervals where a cutoff may vary in
+$[0,1]$; no shape or monotonicity of the transition is assumed.}
+\label{fig:cutoffs}
+\end{figure}
+
+\begin{lemma}[Extension of the inner source]\label{glue:extension}
+There is a real linear extension $\mathcal E_i:Y_\alpha\to
+Y_\alpha(\R^m\times(-\infty,0))$, uniformly bounded for
+$T\le\beta^4$, which equals $\chi_i f$ in flat coordinates for
+$-T\le t<0$, and is supported spatially in $r\le2\beta$.
+\end{lemma}
+
+\begin{proof}
+Put $f_i=\chi_i f$, view it in the flat coordinates, and extend it by
+zero beyond their neighborhood.  The spatial extension is bounded
+because the cutoff vanishes before the boundary of that neighborhood.
+The $Y_\alpha$ norm gives a spatial $C^\alpha$ trace $f_i(-T,z)$ at
+the lower time face.  Choose a smooth $\psi:[0,\infty)\to[0,1]$ equal
+to one near zero and zero for arguments at least one.  For $t<-T$ set
+\begin{equation}\label{glue:extension-formula}
+ (\mathcal E_i f)(t,z)
+ =f_i(-T,z)\,\psi\left(\frac{-t-T}{(|z|^2+\sqrt T)^2}\right).
+\end{equation}
+Let $b_0=(|z|^2+\sqrt T)^{1/2}$.  Wherever this expression is
+nonzero,
+\[
+ T\le-t\le T+b_0^4,
+ \qquad b_0^2\le |z|^2+\sqrt{-t}\le(1+\sqrt2)b_0^2.
+\]
+The ancient scale is therefore comparable to $b_0$.  The argument of
+$\psi$ has bounded derivatives in spatial units $b_0$ and temporal
+units $b_0^4$ on its transition region.  The trace norm, H\"older
+multiplication, and scale comparability prove the required estimate
+below $-T$.  Across that face the values agree.  Splitting a temporal
+increment at $-T$ bounds its $\alpha/4$ H\"older quotient by the sum
+of the two one-sided bounds; a spatial increment is handled by the
+trace bound on comparable boxes.  This proves the full parabolic
+$C^\alpha$ bound.  No time derivative of the source is required.
+The construction is real, linear, and radial.  Its continuation below
+$-T$ is in fact supported in $-26\beta^4\le t<-T$, though a temporal
+support restriction is unnecessary for the ancient inverse.
+\end{proof}
+
+Apply Theorem~\ref{flat:inverse} to obtain
+\[
+ W_f=\wh D\mathcal E_i f,\qquad V_i=\chi_b W_f,
+\]
+where $V_i$ is regarded as a function on the sphere and extended by
+zero.  Invariant norm equivalence and the cutoff bounds give
+$\norm{V_i}_{X_\alpha}\le C\norm{f}_{Y_\alpha}$.  Define
+\begin{equation}\label{glue:inner-errors}
+ R_i f=(\Pcal-\wh\Pcal)(\chi_bW_f),\qquad
+ J_i f=[\wh\Pcal,\chi_b]W_f.
+\end{equation}
+Both expressions are taken in the common coordinates and extended by
+zero.  Since $\chi_b f_i=f_i$, the exact identity is
+\begin{equation}\label{glue:inner-identity}
+ \Pcal V_i=f_i+R_i f+J_i f.
+\end{equation}
+Lemma~\ref{glue:background} and $b\le C\beta$ on the relevant boxes
+give
+\begin{equation}\label{glue:inner-error-bounds}
+ \norm{R_i f}_{Y_\alpha}\le C\beta^2\norm{f}_{Y_\alpha},
+ \qquad
+ \norm{J_i f}_{Y_\alpha}\le C\norm{f}_{Y_\alpha}.
+\end{equation}
+Here $R_i f$ is supported in $r\le4\beta$, and $J_i f$ in
+$2\beta\le r\le4\beta$.  The commutator has order at most three;
+on its support $b$ is comparable to $\beta$, so every derivative of
+$\chi_b$ is uniformly bounded in scale units.  The definition of
+$R_i$ includes the mismatch applied to the cutoff itself, and hence
+accounts for all mixed cutoff and mismatch terms.
+
+\subsection{The outer Cauchy problem and the exact right inverse}
+
+Set
+\[
+ s_\beta=\max(\beta,r),\qquad \eta=T/\beta^4,
+\]
+and use the nonsingular weighted spaces with scale $s_\beta$.
+Equivalently one can use the smooth comparable scale
+$(\beta^2+r^2)^{1/2}$; the norms and all following estimates are
+uniformly equivalent.  Define a modified operator on the entire sphere
+by
+\begin{equation}\label{glue:modified-operator}
+ \Pcal_\beta=\partial_t+A_\beta\A+B_\beta,
+ \qquad
+ A_\beta=\kappa A_c+(1-\kappa),\qquad B_\beta=\kappa B_c.
+\end{equation}
+Its principal multiplier is bounded above and below by positive
+constants, and $\Pcal_\beta=\Pcal$ for $r\ge\beta/2$.
+Lemma~\ref{glue:background} shows, for every $j,k$, uniform bounds for
+the derivatives of $A_\beta$ and $s_\beta^4B_\beta$ in units
+$(s_\beta,s_\beta^4)$, with the scale fixed at the center of each box.
+On the transition annulus $r\asymp\beta$ these are exactly its scaled
+background bounds and the cutoff bounds.  Inside $r\le\beta/4$ the
+coefficients are $1$ and $0$; outside the transition they are the
+original coefficients with $\sigma\asymp s_\beta$.  The sphere
+coefficients of $\A$ also have uniform bounds in these units.  These
+facts verify all coefficient and ellipticity hypotheses of
+Lemma~\ref{ana:short}, uniformly for $\eta\le1$.
+
+The outer source is
+\begin{equation}\label{glue:outer-source}
+ f_o=f-f_i-J_i f.
+\end{equation}
+It vanishes for $r<\beta$ and has
+\begin{equation}\label{glue:outer-source-bound}
+ \norm{f_o}_{Y_\alpha(s_\beta)}\le C\norm{f}_{Y_\alpha}.
+\end{equation}
+Indeed $s_\beta\asymp\sigma$ on its support and on every sufficiently
+small box meeting that support.  The latter assertion follows from
+the Lipschitz bound for $r$ and the box radius $a_0\sigma$; it also
+handles boxes crossing a support boundary.  The same comparison holds
+for outputs supported in $r\ge\beta/2$.
+
+For each fixed $\beta,T$, solve the closed-manifold Cauchy problem
+\begin{equation}\label{glue:outer-cauchy}
+ \Pcal_\beta Z_o=f_o,\qquad Z_o(-T)=0.
+\end{equation}
+The modified coefficients extend smoothly to $t=0$ by
+\eqref{glue:outer-expansion}, and $f_o$ extends H\"older continuously
+there: for fixed $\beta$ its nonsingular weighted norm is an ordinary
+H\"older bound on a compact cylinder.  Thus the solvability assertion
+in Lemma~\ref{ana:short} applies, or one may solve on truncated
+cylinders and use its uniform bounds.  Uniqueness makes $Z_o$ invariant
+and the solution operator real linear.
+
+Choose a fixed $\eta_0>0$ small enough for Lemma~\ref{ana:short}
+at both exponents $\alpha$ and $\alpha'$.  That Lemma gives, for
+$0<\eta\le\eta_0$,
+\begin{align}
+ \norm{Z_o}_{X_\alpha(s_\beta)}
+   &\le C\norm{f_o}_{Y_\alpha(s_\beta)},\label{glue:outer-bound}\\
+ \sup s_\beta^{-p_0}\abs{Z_o}
+   &\le C\eta\norm{f_o}_{Y_\alpha(s_\beta)}.\label{glue:outer-sup}
+\end{align}
+The constants are uniform in $\beta,T$ by the coefficient and scale
+bounds verified above.  The small parameter in Lemma~\ref{ana:short}
+is precisely $\eta=T/\beta^4$, since $\inf s_\beta=\beta$.
+That lemma applies to zero initial data even when $f_o(-T)$ is nonzero.
+
+Put $V_o=\chi_o Z_o$.  On the support of this product the coefficients
+have already become those of $\Pcal$, and $\chi_o f_o=f_o$.
+Consequently
+\begin{equation}\label{glue:outer-identity}
+ \Pcal V_o=f_o+J_o f,\qquad
+ J_o f=[\Pcal_\beta,\chi_o]Z_o.
+\end{equation}
+This commutator is supported in $\beta/2\le r\le\beta$ and involves
+spatial derivatives of $Z_o$ of order at most three.  For these
+derivatives, Lemma~\ref{ana:interpolation} and
+\eqref{glue:outer-bound}--\eqref{glue:outer-sup} give the scaled
+parabolic $C^\alpha$ bound
+$C\eta^{(1-\alpha)/4}\norm{f_o}_{Y_\alpha(s_\beta)}$.
+Since $s_\beta\asymp\sigma\asymp\beta$ on the commutator annulus,
+\begin{equation}\label{glue:outer-error-small}
+ \norm{J_o f}_{Y_\alpha}
+ \le C\eta^{(1-\alpha)/4}\norm{f}_{Y_\alpha}.
+\end{equation}
+In particular, no fourth derivative needs to be made small for this
+commutator estimate.
+
+\begin{proposition}[Uniform compact right inverse]\label{glue:inverse}
+There are $\beta_0,\eta_0,C_D>0$ such that, for
+$0<\beta\le\beta_0$ and $0<T\le\eta_0\beta^4$, the operator
+$\Pcal$ has a real invariant right inverse
+$D:Y_\alpha\to X_\alpha$ with $\norm D\le C_D$.
+No initial value is prescribed for $Df$.
+\end{proposition}
+
+\begin{proof}
+Define
+\[
+ D_{\mathrm{app}}f=V_i+V_o,\qquad \mathcal R=R_i+J_o.
+\]
+Equations~\eqref{glue:inner-identity}, \eqref{glue:outer-source}, and
+\eqref{glue:outer-identity} give the exact operator identity
+\begin{equation}\label{glue:parametrix}
+ \Pcal D_{\mathrm{app}}=I+\mathcal R,\qquad
+ \norm{D_{\mathrm{app}}}_{Y_\alpha\to X_\alpha}\le C,\qquad
+ \norm{\mathcal R}_{Y_\alpha\to Y_\alpha}
+ \le C\bigl(\beta^2+\eta^{(1-\alpha)/4}\bigr).
+\end{equation}
+Choose $\beta_0$ and then $\eta_0$ small enough that the last bound is
+at most $1/2$ throughout the asserted range, decreasing the earlier
+$\eta_0$ if necessary.  The convergent Neumann series defines
+\begin{equation}\label{glue:neumann}
+ D=D_{\mathrm{app}}(I+\mathcal R)^{-1}.
+\end{equation}
+It satisfies $\Pcal D=I$, is real and invariant, and has uniform norm
+at most twice the bound for $D_{\mathrm{app}}$.  This is a right inverse
+construction and requires no assertion of injectivity.
+\end{proof}
+
+\subsection{Smallness of the corrected error}
+
+The right inverse need not have small norm.  What is needed for the
+nonlinear problem is its small value on the particular error $E_c$ in
+the weaker space $X'_\alpha$.
+
+\begin{lemma}\label{glue:special-error}
+For the right inverses of Proposition~\ref{glue:inverse},
+\begin{equation}\label{glue:DE-small}
+ \norm{DE_c}_{X'_\alpha}
+ \le C\left(\beta^{2-2\delta}
+       +\eta^{(\alpha'-\alpha)/4}
+       +\eta^{(1-\alpha)/4}\right).
+\end{equation}
+In particular this norm becomes arbitrarily small by choosing $\beta$
+and then $\eta=T/\beta^4$ sufficiently small.
+\end{lemma}
+
+\begin{proof}
+Use the inner output cutoff, rather than the inner input cutoff, to
+split
+\[
+ E_c=f_c+f_e,\qquad f_c=\chi_bE_c,\qquad f_e=(1-\chi_b)E_c.
+\]
+Equation~\eqref{glue:core-error-small} gives
+$\norm{Df_c}_{X_\alpha}\le C\beta^{2-2\delta}$.
+The remaining source vanishes for $r\le2\beta$, so
+$\chi_i f_e=0$ exactly.  Its inner extension, flat correction,
+$R_i f_e$, and $J_i f_e$ all vanish.  Thus
+$D_{\mathrm{app}}f_e$ consists solely of the outer zero-data solution
+for source $f_e$, followed by multiplication by $\chi_o$, and
+$\mathcal R f_e=J_o f_e$.
+
+The extra regularity in Lemma~\ref{glue:background} gives
+\begin{equation}\label{glue:outer-error-high}
+ \norm{f_e}_{Y_{\alpha'}(s_\beta)}\le C.
+\end{equation}
+The constant is uniform in $\beta,T$: in the transition annulus all
+cutoff derivatives have bounded scaled size, and elsewhere this is
+\eqref{glue:error-weight} or the fixed exterior smooth bound.
+Apply Lemma~\ref{ana:short} at exponent $\alpha'$ to the outer
+zero-data problem with source $f_e$.  The coefficient bounds hold at
+that exponent, so its solution has uniformly bounded
+$X_{\alpha'}(s_\beta)$ norm.  Equation~\eqref{ana:fourth-small} of
+Lemma~\ref{ana:interpolation} gives an $X'_\alpha(s_\beta)$ bound
+of $C\eta^{(\alpha'-\alpha)/4}$.  Multiplication by $\chi_o$ is
+bounded in these scaled spaces, and $s_\beta$ is comparable to
+$\sigma$ on the output support.  This proves the first estimate below;
+the commutator bound \eqref{glue:outer-error-small} proves the second:
+\begin{equation}\label{glue:app-special-small}
+ \norm{D_{\mathrm{app}}f_e}_{X'_\alpha}
+ \le C\eta^{(\alpha'-\alpha)/4},\qquad
+ \norm{\mathcal Rf_e}_{Y_\alpha}
+ \le C\eta^{(1-\alpha)/4}.
+\end{equation}
+The first estimate includes the temporal H\"older seminorms of all
+spatial derivatives through order four, as supplied by
+\eqref{ana:fourth-small}.  It makes no claim that the first time
+derivative is small: the outer zero-data solution has time derivative
+$f_e$ at the initial face.
+
+Finally, \eqref{glue:neumann} gives
+$D(I+\mathcal R)=D_{\mathrm{app}}$, so
+\begin{equation}\label{glue:special-neumann-identity}
+ Df_e=D_{\mathrm{app}}f_e-D\mathcal Rf_e.
+\end{equation}
+The uniform bound for $D$ and
+\eqref{glue:app-special-small} prove the desired estimate for $f_e$.
+Combining it with the estimate for $f_c$ proves
+\eqref{glue:DE-small}.
+\end{proof}
+
+\subsection{The nonlinear correction and finite-time obstruction}
+
+\begin{proof}[Proof of Theorem~\ref{glue:solution}]
+Expanding \eqref{glue:equation} at $F$ gives
+\begin{equation}\label{glue:nonlinear-equation}
+ \Pcal V+E_c+N(V)=0,
+\end{equation}
+where
+\begin{equation}\label{glue:nonlinearity}
+ N(V)=v_*^2V^2\A F+
+       \bigl(2v_*g_cV+v_*^2V^2\bigr)\A V.
+\end{equation}
+This expression contains only spatial derivatives of $V$, so is
+defined on $X'_\alpha$.  For clarity, on a box of scale $b$, a
+function with $\norm{V}_{X'_\alpha}\le\eps$ has scaled sizes
+\[
+ V=O(\eps b^{-2+2\delta}),\qquad
+ \A V=O(\eps b^{-6+2\delta}),
+\]
+where these statements include the scaled parabolic $C^\alpha$
+norms.  By Lemma~\ref{glue:background}, the two quadratic terms in
+\eqref{glue:nonlinearity} therefore have size
+$C\eps^2b^{-6+4\delta}$, and the cubic term has size
+$C\eps^3b^{-6+6\delta}$.  Relative to the $Y$ weight these gain
+$b^{2\delta}$ and $b^{4\delta}$, respectively.  Since $b$ is bounded
+above on the compact domain, H\"older multiplication and the
+factorization of differences of products give uniform bounds
+\begin{align}
+ \norm{N(V)}_{Y_\alpha}&\le C_N(\eps^2+\eps^3),
+ \label{glue:nonlinear-size}\\
+ \norm{N(V)-N(W)}_{Y_\alpha}
+ &\le C_N(\eps+\eps^2)\norm{V-W}_{X'_\alpha}
+ \label{glue:nonlinear-lipschitz}
+\end{align}
+on the closed ball of radius $\eps$ in $X'_\alpha$.  The fixed sphere
+coefficients in $\A$ have bounded rescaled norms, so these estimates
+apply on the whole sphere, including the other endpoint $x=1$.
+
+We specify the parameter choices to avoid a dependence on an inverse
+norm that is still being chosen.  First fix the profile, exponents,
+flat inverse, and the constants $\beta_0,\eta_0,C_D$ of
+Proposition~\ref{glue:inverse}.  The nonlinear constant $C_N$ is
+uniform throughout this parameter range.  Choose the requested ball
+radius $\eps>0$ so small that
+\[
+ C_D C_N(\eps+\eps^2)\le\tfrac14,\qquad
+ C_*\eps\le\tfrac12\mu,
+\]
+where $C_*$ bounds $\abs{v_*V}/\norm{V}_{X'_\alpha}$ uniformly.
+Such a $C_*$ exists because $v_*V$ has scaled size
+$C\eps b^{2\delta}$ and $b\le\sqrt2$.
+Now take $\beta\le\beta_0$ sufficiently small, and next
+$\eta\le\min(\eta_0,a_0^4)$ sufficiently small, that
+Lemma~\ref{glue:special-error} gives
+$\norm{DE_c}_{X'_\alpha}\le\eps/2$; put $T=\eta\beta^4$.
+
+The map
+\[
+ \mathcal T(V)=-DE_c-DN(V)
+\]
+takes the closed $X'_\alpha$ ball of radius $\eps$ into itself by
+\eqref{glue:nonlinear-size}, and is a contraction by
+\eqref{glue:nonlinear-lipschitz}.  The real invariant subspace is
+closed, so the Banach fixed point theorem gives a real invariant
+fixed point there.  Its fixed point identity also shows that this
+same $V$ belongs to $X_\alpha$, because $E_c+N(V)\in Y_\alpha$ and
+$D:Y_\alpha\to X_\alpha$.  Applying $\Pcal D=I$ therefore yields
+\eqref{glue:nonlinear-equation} in the classical H\"older sense and
+proves \eqref{glue:equation}.  This recovers the time derivative after
+the contraction has been performed.
+
+By \eqref{glue:background-positive} and the radius choice,
+\[
+ \tfrac12\mu\le 1+v_*(F+V)\le M+\tfrac12\mu,
+\]
+which proves \eqref{glue:positive}.  On each compact time subinterval
+of $(-T,0)$, the weight is bounded above and below, and the equation
+for $H$ is a uniformly parabolic scalar equation with principal
+multiplier $(1+v_*H)^2$.  Its current regularity is
+$C^{4+\alpha,1+\alpha/4}$.  Interior spatial difference quotients in
+this equation have the same principal operator; their sources are
+bounded in $C^\alpha$ because the coefficient is a smooth function
+of $x,H$ and $H$ already has four spatial derivatives.  The interior
+estimate of Lemma~\ref{ana:schauder}, followed by passage to the
+difference-quotient limit on a smaller cylinder, gains one spatial
+derivative.  Repetition gains every spatial derivative.  The equation
+then gives corresponding spatial bounds for $H_t$; differentiating
+the equation in time and repeating the interior estimates gives all
+mixed derivatives.  Thus $H$ is smooth on $\Sigma\times(-T,0)$.
+Invariant smoothness gives smoothness in $x\in[0,1]$ by
+Lemma~\ref{geo:operators}.  At $x=0$, the $X'$ weight reads
+\[
+ \abs{V(t,0)}\le\eps\sigma(t,0)^{-2+2\delta}
+                =\eps\rho^{-1+\delta},
+\]
+up to the fixed equivalent-norm constant.  This proves
+\eqref{glue:center} and finishes the construction.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+Apply Theorem~\ref{glue:solution} to the nonconstant profile of
+Theorem~\ref{prof:profile}.  Proposition~\ref{geo:reduction} turns
+the smooth positive coefficient $H$ into a Calabi flow in the
+Fubini--Study class on $\CP^{10}$.  Choose its slice at $t=-T/2$ as
+the initial metric and translate time so that this slice has time
+zero.  It is a smooth K\"ahler metric in a class containing the
+Fubini--Study metric.  At the distinguished point,
+\[
+ H(t,0)=\rho^{-1}G'(0)+O(\rho^{-1+\delta}),\qquad G'(0)<0.
+\]
+Lemma~\ref{geo:obstruction} shows that the maximal smooth existence
+time is exactly $T_*=T/2$ and that, in the elapsed time $\tau$,
+\[
+ S(\omega(\tau))(p)\sim a(T_*-\tau)^{-1/2},\qquad
+ a=-110G'(0)>0.
+\]
+The metric is $U(10)$ invariant by the radial reconstruction.
+These are all the assertions of Theorem~\ref{thm:main}.
+\end{proof}
+
+\begin{proof}[Proof of Corollary~\ref{cor:product-singularities}]
+For a product metric the determinant factors and the inverse is block
+diagonal, so scalar curvatures add.  The product volume form also shows
+that their averages add.  Writing \(c_\eta=S(\eta)\), we obtain
+\[
+ S(\Omega(t))=\pi_X^*S(\omega(t))+c_\eta,
+ \qquad \bar S_\Omega=\bar S_\omega+c_\eta.
+\]
+Thus \(\pi_X^*\phi(t)\), initially zero, satisfies
+\eqref{eq:calabi-flow} for \(\Omega_0\).  Smooth local
+uniqueness~\cite[Theorem~3.2]{ChenHe2008} identifies this explicit
+solution with the unrestricted flow on \([0,T_*)\).  The displayed
+scalar-curvature identity gives the stated asymptotic.  A smooth
+extension across \(T_*\) would have bounded scalar curvature on the
+compact product, so the maximal time is exactly \(T_*\).
+
+For the final assertion put \(m=n-10\).  When \(m>0\), choose
+\(Y=\CP^m\) and \(\eta=\frac{m+1}{11}\omega_{\mathrm{FS},m}\).
+In our normalization,
+\(\operatorname{Ric}(\omega_{\mathrm{FS},j})
+=(j+1)\omega_{\mathrm{FS},j}\).  Constant scaling leaves the Ricci
+form unchanged, so \(\omega_{\mathrm{FS},10}\) and \(\eta\) both
+have Einstein constant \(11\).  Their product form is therefore
+K\"ahler--Einstein and lies in \([\Omega_0]\).
+For \(m=0\), use Theorem~\ref{thm:main} itself, with the second
+factor a point.
+\end{proof}
diff --git a/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/sections/flat.tex b/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/sections/flat.tex
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+\section{An ancient inverse for the flat linearization}
+\label{sec:flat}
+
+Fix the profile supplied by Theorem~\ref{prof:profile}, and put
+\[
+ \rho=\sqrt{-t},\qquad x=|z|^2,\qquad
+ h(y)=\frac{G(y)-1}{y},\qquad
+ F(z,t)=\rho^{-1}h(|z|^2/\rho),\qquad z\in\R^m,
+\]
+where $m=24$ and $h(0)=G'(0)$. Throughout this section the time
+domain is the entire interval $(-\infty,0)$. The flat linearization is
+\begin{equation}\label{flat:operator}
+ \widehat{\Pcal}V
+ =V_t+\frac{A(z,t)}{16}\Delta_z^2V+B(z,t)V,
+ \quad
+ A=G(|z|^2/\rho)^2,\quad
+ B=2|z|^2G(|z|^2/\rho)\widehat{\A}F,
+ \qquad \widehat{\A}=\frac1{16}\Delta_z^2.
+\end{equation}
+The spaces below are the radial versions of Definition~\ref{ana:spaces},
+with scale $\sigma=(|z|^2+\rho)^{1/2}$.
+
+\begin{theorem}[Ancient flat right inverse]\label{flat:inverse}
+There is a number $\delta\in(0,1)$ such that, for every fixed
+$\alpha\in(0,1)$, the operator \eqref{flat:operator} has a bounded real
+linear right inverse
+\[
+ \widehat D:Y\longrightarrow X,\qquad
+ \widehat{\Pcal}\widehat D f=f,\qquad
+ \|\widehat D f\|_X\le C\|f\|_Y,
+ \qquad p_0=-2+2\delta.
+\]
+Here $Y$ has weight $p_0-4$ and parabolic exponent $\alpha$, and $X$
+has weight $p_0$ and includes the parabolic $\alpha$ norms of all
+spatial derivatives through order four and of the first time
+derivative. The solution is radial and classical on $t<0$.
+Its initial values are selected by the construction. The constant
+depends only on the fixed profile, $\delta$, $\alpha$, and the fixed
+choices in the local norms.
+\end{theorem}
+
+\subsection{Similarity coordinates and coefficient bounds}
+
+The flat equation for $F$ in Theorem~\ref{prof:profile} and the
+radial operator identity of Lemma~\ref{geo:operators} yield
+\begin{equation}\label{flat:profile-identities}
+ \frac1{16}\Delta_\zeta^2\bigl(h(|\zeta|^2)\bigr)
+ =-\frac{G'(y)}{2G(y)^2},\qquad y=|\zeta|^2,
+ \qquad F_t+G(|z|^2/\rho)^2\widehat{\A}F=0.
+\end{equation}
+In particular, with
+\[
+ A_f(\zeta)=G(|\zeta|^2)^2,\qquad
+ B_f(\zeta)
+ =\frac{2|\zeta|^2G(|\zeta|^2)}{16}
+       \Delta_\zeta^2\bigl(h(|\zeta|^2)\bigr)
+ =-\frac{yG'(y)}{G(y)},
+\]
+one has $A(z,t)=A_f(z/\sqrt\rho)$ and
+$B(z,t)=\rho^{-2}B_f(z/\sqrt\rho)$.
+These coefficients are smooth at the origin. Positivity and the
+differentiated asymptotic expansion in Theorem~\ref{prof:profile} imply
+\begin{equation}\label{flat:symbols}
+ \inf A_f>0,\qquad
+ |\partial_\zeta^\nu(A_f-c^2)|+
+ |\partial_\zeta^\nu B_f|
+ \le C_\nu(1+|\zeta|)^{-4-|\nu|}.
+\end{equation}
+In particular all derivatives, including any fixed number of
+applications of $\zeta\cdot\nabla$, are bounded.
+
+Set
+\[
+ \tau=-\log(-t),\qquad \zeta=z/\sqrt\rho,
+ \qquad V(z,t)=\rho^{-1}v(\zeta,\tau),
+ \qquad f^\#(\zeta,\tau)=\rho^3f(\sqrt\rho\,\zeta,t).
+\]
+Since $\dd\tau/\dd t=\rho^{-2}$ and
+$\partial_t\zeta=\zeta/(4\rho^2)$, direct differentiation gives
+\[
+ V_t=\rho^{-3}
+   \left(v_\tau+\tfrac14\zeta\cdot\nabla v+\tfrac12v\right),
+ \qquad \Delta_z^2V=\rho^{-3}\Delta_\zeta^2v.
+\]
+Thus \eqref{flat:operator} becomes the autonomous equation
+\begin{equation}\label{flat:similarity-equation}
+ v_\tau+\frac{A_f}{16}\Delta^2v
+       +\frac14\zeta\cdot\nabla v+(\tfrac12+B_f)v=f^\#.
+\end{equation}
+We shall first solve it in
+$E=C_{0,\mathrm{rad}}(\R^m;\C)$, with the supremum norm. This denotes
+continuous radial functions tending to zero at spatial infinity;
+complex scalars are needed only for spectral theory.
+
+\subsection{Evolution on continuous functions vanishing at infinity}
+
+We give the evolution construction since the unbounded drift in
+\eqref{flat:similarity-equation} and the change in the principal
+coefficient both matter. On a time interval beginning at $\tau_0$, put
+\[
+ u(\eta,s)=v(e^{s/4}\eta,\tau_0+s),\qquad 0\le s\le1.
+\]
+Then
+\begin{equation}\label{flat:moving-equation}
+ u_s+a(\eta,s)\Delta_\eta^2u+b(\eta,s)u=l(\eta,s),
+ \qquad
+ a=\frac{e^{-s}}{16}A_f(e^{s/4}\eta),\quad
+ b=\tfrac12+B_f(e^{s/4}\eta).
+\end{equation}
+The right side is the corresponding dilation of $f^\#$.
+By \eqref{flat:symbols}, $a,b$ have bounded derivatives of every order
+on this slab, and $0<\kappa\le a\le\kappa^{-1}$ for a fixed
+$\kappa$. The following construction also applies without radial
+symmetry. It is the scalar whole-space version of the higher-order
+heat-kernel parametrix; see also \cite[Appendix~A]{HeZeng2021}.
+
+Let
+\[
+ k_\theta(w)=(2\pi)^{-m}\int_{\R^m}
+               e^{iw\cdot\xi-\theta|\xi|^4}\,\dd\xi,
+ \qquad \theta>0.
+\]
+Its integral is one. Scaling and integration by parts in the Fourier
+integral show that, uniformly for $\kappa\le a_*\le\kappa^{-1}$,
+\begin{equation}\label{flat:kernel-bounds}
+ |\partial_{a_*}^{\ell}\partial_w^\nu k_{a_*q}(w)|
+ \le C_{N,\ell,\nu}\,
+ q^{-(m+|\nu|)/4}
+       (1+|w|/q^{1/4})^{-N},\qquad 0<q\le1.
+\end{equation}
+Indeed after scaling $\xi=q^{-1/4}\xi'$ the multipliers and all their
+derivatives form a bounded set of Schwartz functions. No sign property
+of $k_\theta$ is used.
+
+Freeze the principal coefficient at the source point and time:
+\[
+ K_{s,r}(\eta,y)=k_{a(y,r)(s-r)}(\eta-y),\qquad r<s,
+\]
+and also denote by $K_{s,r}$ the associated integral operator. Applying
+the left side of \eqref{flat:moving-equation} at the evaluation point
+gives the residual kernel
+\begin{equation}\label{flat:residual}
+ J_{s,r}(\eta,y)
+ =\bigl(a(\eta,s)-a(y,r)\bigr)
+       \Delta_\eta^2 k_{a(y,r)(s-r)}(\eta-y)
+       +b(\eta,s)K_{s,r}(\eta,y).
+\end{equation}
+The global Lipschitz bound for $a$ and \eqref{flat:kernel-bounds} give
+\begin{equation}\label{flat:residual-bound}
+ \|K_{s,r}\|_{C_0\to C_0}\le C,
+ \qquad
+ \|J_{s,r}\|_{C_0\to C_0}
+ \le C\bigl((s-r)^{-3/4}+1\bigr)
+ \le C'(s-r)^{-3/4}.
+\end{equation}
+For the first residual term, the spatial difference $|\eta-y|$
+contributes $(s-r)^{1/4}$ against a fourth derivative of size
+$(s-r)^{-1}$; the temporal difference contributes $(s-r)$.
+The kernel bounds first show that both operators preserve $C_0$ for
+compactly supported continuous inputs, by decay in $\eta$, and then
+for all $C_0$ inputs by uniform approximation.
+
+These kernels give a strong approximate identity. To see this
+explicitly, replace $a(y,r)$ by $a(\eta,r)$ in $K_{s,r}$.
+The operator norm of the difference is $O((s-r)^{1/4})$, by the
+parameter derivative bound in \eqref{flat:kernel-bounds}. For the
+replacement kernel, its integral in $y$ is one at each $\eta$;
+uniform continuity of a $C_0$ function, followed by the uniform kernel
+tail bound, proves convergence to that function in supremum norm.
+Consequently
+\begin{equation}\label{flat:approximate-identity}
+ K_{s,r}d\longrightarrow d\quad\hbox{in }C_0
+ \quad\hbox{as }s-r\downarrow0,
+\end{equation}
+uniformly in the initial time $r$. This is strong convergence; only
+the difference between the two frozen kernels was estimated in
+operator norm.
+
+For $d\in C_0$ and $l\in C([0,1];C_0)$ solve
+\begin{equation}\label{flat:volterra}
+ \begin{split}
+ j(s)+\int_0^sJ_{s,r}j(r)\,\dd r&=l(s)-J_{s,0}d,\\
+ u(s)&=K_{s,0}d+\int_0^sK_{s,r}j(r)\,\dd r.
+ \end{split}
+\end{equation}
+All integrals are Bochner integrals. The inverse of the first equation
+is its Volterra series. For completeness, if
+$H(s)=Cs^{-3/4}$, its $k$-fold convolution on the positive half-line is
+\[
+ H^{*k}(s)
+ =\frac{C^k\Gamma(1/4)^k}{\Gamma(k/4)}s^{k/4-1}
+ \quad(k\ge1).
+\]
+This identity follows by the beta integral and induction. Since the
+right side of the first equation in \eqref{flat:volterra} is bounded
+by $Cs^{-3/4}(\|d\|_\infty+\|l\|_{C([0,1];C_0)})$, the same formula
+and the growth of the gamma function prove convergence of the series
+and the estimates
+\begin{equation}\label{flat:evolution-bound}
+ \|j(s)\|_\infty\le Cs^{-3/4}
+       (\|d\|_\infty+\|l\|_{C([0,1];C_0)}),\qquad
+ \sup_{0\le s\le1}\|u(s)\|_\infty
+ \le C(\|d\|_\infty+\|l\|_{C([0,1];C_0)}).
+\end{equation}
+The integral in the formula for $u$ tends to zero as $O(s^{1/4})$.
+Together with \eqref{flat:approximate-identity}, this proves
+$u\in C([0,1];C_0)$ and $u(0)=d$. Continuity at positive times follows
+by dominated convergence, splitting off a short interval adjacent to
+the diagonal.
+
+The kernel identity \eqref{flat:residual} proves that $u$ solves
+\eqref{flat:moving-equation} distributionally: apply the equation to
+the second line of \eqref{flat:volterra}, obtaining
+$J_{s,0}d+j(s)+\int_0^sJ_{s,r}j(r)\,\dd r=l(s)$.
+This calculation can be made first with a positive separation from
+the diagonal and smooth compactly supported tests; the diagonal
+term converges to $j$ by \eqref{flat:approximate-identity}, and
+\eqref{flat:residual-bound} justifies the remaining integrals.
+Lemma~\ref{ana:schauder} and interior regularity then give smoothness
+for positive times in the homogeneous case, and classical local
+regularity for locally parabolic H\"older sources.
+
+We also record uniqueness in the class just constructed. If $u$ is
+the difference of two such solutions, it solves the homogeneous
+equation with zero $C_0$ initial value. Let
+$W(\eta)=\exp(-\sqrt{1+|\eta|^2})$. Its derivatives are bounded by
+constant multiples of $W$. For $s\ge\varepsilon>0$, interior
+regularity supplies globally bounded spatial derivatives of $u$.
+Integration by parts, first with compact spatial cutoffs, gives
+\[
+ \Re\int W\overline u\,a\Delta^2u
+ =\Re\int\Delta(aW\overline u)\Delta u
+ \ge\frac\kappa2\|\Delta u\|_{L^2(W)}^2
+                  -C\|u\|_{L^2(W)}^2.
+\]
+Here the mixed terms are absorbed using
+\[
+ \|\nabla u\|_{L^2(W)}^2
+ \le\epsilon\|\Delta u\|_{L^2(W)}^2
+                 +C_\epsilon\|u\|_{L^2(W)}^2.
+\]
+This last inequality follows by integrating
+$\int W|\nabla u|^2$ once by parts and using
+$|\nabla W|\le CW$ and Young's inequality. Exponential decay of $W$
+allows removal of the cutoffs. The bounded zeroth order coefficient
+therefore gives
+\[
+ \frac{\dd}{\dd s}\|u(s)\|_{L^2(W)}^2
+ \le C\|u(s)\|_{L^2(W)}^2.
+\]
+Gronwall's inequality, followed by $\varepsilon\downarrow0$, proves
+$u=0$, because the zero $C_0$ initial trace implies a zero weighted
+$L^2$ trace. This also proves uniqueness for the inhomogeneous
+construction by subtraction.
+
+Undoing the dilation now gives a strongly continuous homogeneous
+semigroup $S(s)$ on $C_0(\R^m)$ for
+\eqref{flat:similarity-equation}; concatenation is justified by the
+uniqueness just proved. Dilation is strongly continuous on $C_0$,
+and the equation is autonomous. On $0\le s\le1$ one has
+$\|S(s)\|\le C$. Inhomogeneous evolution for a continuous $C_0$ source
+is
+\begin{equation}\label{flat:duhamel}
+ v(\tau_0+s)=S(s)v(\tau_0)
+       +\int_0^sS(s-r)f^\#(\tau_0+r)\,\dd r.
+\end{equation}
+The formula follows either from the distributional equation and
+uniqueness or by approximation with smooth sources. Rotation and
+complex conjugation preserve the equation and the construction.
+In particular $S(s)$ preserves $E$ and its real subspace.
+
+\subsection{Compactness at positive time}
+
+Write $P=S(1)|_E$, reserving this symbol here for a bounded operator,
+and let $P_c$ be the corresponding operator with $A_f=c^2$ and
+$B_f=0$. We prove
+\begin{equation}\label{flat:compact-difference}
+ P-P_c:E\longrightarrow E\quad\hbox{is compact}.
+\end{equation}
+Both time-one evolutions map their unit balls to locally
+equicontinuous functions by the interior estimates. It remains to
+prove uniform decay of their difference at spatial infinity.
+
+Suppose this decay fails. There are $d_j\in E$ with
+$\|d_j\|_\infty\le1$, points $\zeta_j$ with $|\zeta_j|\to\infty$,
+and $\epsilon_0>0$ such that
+$|(P-P_c)d_j(\zeta_j)|\ge\epsilon_0$.
+In the moving coordinates \eqref{flat:moving-equation}, translate
+by $\eta_j=e^{-1/4}\zeta_j$. The two solutions, denoted by
+$u_j$ and $u_{c,j}$ after translation, have the same initial value
+$d_j(\eta+\eta_j)$ and uniform supremum bounds. Their coefficients
+converge, smoothly on compact subsets of
+$\R^m\times[0,1]$, to
+\[
+ a_c(s)=\frac{c^2e^{-s}}{16},\qquad b_c=\tfrac12.
+\]
+This follows from \eqref{flat:symbols}; it also controls time
+derivatives in the moving coordinates.
+
+After passing to a subsequence, the translated initial functions
+converge weakly star in $L^\infty(\R^m)$ to some $d_\infty$.
+Sequential extraction is available because $L^1(\R^m)$ is separable.
+Interior estimates and a diagonal extraction give smooth local
+limits $u_\infty,u_{c,\infty}$ for $s>0$, including convergence of
+their values at $s=1$. Both limits are bounded on the whole slab.
+To identify their initial traces, for
+$\psi\in C_c^\infty(\R^m)$ integrate the translated equation:
+\begin{equation}\label{flat:translated-trace}
+ \left|\int u_j(s)\psi
+       -\int d_j(\eta+\eta_j)\psi\right|
+ \le Cs\,
+ \sup_{j,\,0\le r\le1}
+ \left(\|\Delta^2(a_j(\cdot,r)\psi)\|_{L^1}
+                    +\|b_j(\cdot,r)\psi\|_{L^1}\right)
+ \le C_\psi s.
+\end{equation}
+All four spatial derivatives have fallen on the coefficient times
+the test function. Uniform coefficient bounds make the last
+constant independent of $j$. The identity extends down to $s=0$
+using the individual $C_0$ initial traces. The same estimate holds
+for $u_{c,j}$. Passing to the limit proves that both limits have
+distributional initial value $d_\infty$.
+
+Here is the required whole-space uniqueness for these bounded
+limits; a $C_0$ initial trace of the limits is not needed.
+Their difference $w$ solves
+\[
+ w_s+a_c(s)\Delta^2w+\tfrac12w=0
+\]
+and has zero distributional initial value. For a fixed $t_1\in(0,1]$
+and $\psi\in\mathcal S(\R^m)$, use the backward test function
+\[
+ \phi(s)=e^{-(t_1-s)/2}
+       k_{\int_s^{t_1}a_c(r)\,\dd r}*\psi,
+ \qquad 0\le s\le t_1,
+\]
+with $\phi(t_1)=\psi$. It solves
+$-\phi_s+a_c(s)\Delta^2\phi+\tfrac12\phi=0$, is continuous in the
+Schwartz topology, and hence $\int w(s)\phi(s)$ is constant for
+$s>0$. One may justify the pairing by compact cutoffs and then use
+the Schwartz decay. Boundedness of $w$ extends its zero initial
+trace from compactly supported tests to every Schwartz test;
+$\phi(s)\to\phi(0)$ in $L^1$ also permits this changing test.
+The constant pairing is therefore zero. Since $\psi$ is arbitrary,
+$w(t_1)=0$. This contradicts
+$|u_\infty(0,1)-u_{c,\infty}(0,1)|\ge\epsilon_0$.
+
+We have proved uniform vanishing of the tails of
+$(P-P_c)$ on the unit ball. Together with local equicontinuity,
+the Arzel\`a--Ascoli criterion for $C_0$ proves
+\eqref{flat:compact-difference}: restrict to a large ball, extract
+there, and use the common small tails to obtain uniform convergence
+on the whole space. The closed radial subspace is preserved.
+
+\subsection{A spectral weight and the two-sided recursion}
+
+For the constant coefficient equation, the moving-coordinate formula
+on an interval of length $k$ is explicit:
+\begin{equation}\label{flat:constant-powers}
+ (P_c^kd)(\zeta)
+ =e^{-k/2}(k_{\theta_k}*d)(e^{-k/4}\zeta),
+ \qquad
+ \theta_k=\frac{c^2}{16}(1-e^{-k}),\qquad k\ge1.
+\end{equation}
+The $L^1$ norm of $k_\theta$ is independent of $\theta>0$ by
+scaling. Thus $\|P_c^k\|\le Ce^{-k/2}$, and the Neumann resolvent
+series converges for $|\lambda|>e^{-1/2}$.
+
+We spell out the spectral consequence of
+\eqref{flat:compact-difference}. On the connected domain
+$\Omega=\{|\lambda|>e^{-1/2}\}$ factor
+\[
+ \lambda I-P
+ =\left[I-(P-P_c)(\lambda I-P_c)^{-1}\right]
+                         (\lambda I-P_c).
+\]
+The operator subtracted from the identity is compact and analytic
+in $\lambda$. The analytic Fredholm alternative implies that
+$(\lambda I-P)^{-1}$ is meromorphic on $\Omega$, with finite rank
+principal parts, since it is invertible for large $|\lambda|$.
+One way to obtain this alternative is to approximate the compact
+operator at any one parameter by a finite rank operator, invert the
+remaining operator of norm less than one in a neighborhood, and
+reduce invertibility to a finite-dimensional analytic determinant.
+The required approximation is available here: identify radial $E$
+with $C_0([0,\infty))$, and use bounded polygonal interpolation
+operators with compact support, which converge strongly to the
+identity and hence uniformly on compact subsets of $E$.
+The determinant either vanishes identically or has isolated zeros;
+continuation through overlapping neighborhoods and invertibility
+at a large parameter exclude the first alternative. The same
+reduction makes every coefficient of a pole finite rank.
+It follows that the spectrum in $\Omega$ consists of isolated
+eigenvalues of finite algebraic multiplicity, with no accumulation
+inside $\Omega$; these are the usual compact-operator and resolvent
+arguments of \cite[Chapter~VII, Section~1.3, and the supplementary notes]{Kato1995}.
+
+Choose
+\begin{equation}\label{flat:weight-choice}
+ e^{-1/2}<q<1,\qquad
+ \{|\lambda|=q\}\cap\spec(P)=\varnothing,\qquad
+ \gamma=-\log q\in(0,\tfrac12),\qquad \delta=2\gamma.
+\end{equation}
+Such a circle exists by the discreteness just proved. There are only
+finitely many eigenvalues outside it: they lie in a bounded closed
+annulus contained in $\Omega$. Let $\Pi_u$ be their summed Riesz
+projection, defined by resolvent integration around these
+eigenvalues, and put $\Pi_s=I-\Pi_u$. The resolvent identity gives
+commuting projections and the invariant decomposition
+\[
+ E=E_s\oplus E_u,\qquad E_s=\Pi_sE,\quad E_u=\Pi_uE,
+ \qquad P_s=P|_{E_s},\quad P_u=P|_{E_u}.
+\]
+The space $E_u$ is finite dimensional, $P_u$ is invertible, and
+the two restricted spectra lie strictly inside and strictly outside
+the chosen circle, respectively. If $E_u=\{0\}$, all terms involving
+it below are omitted. Compactness of the restricted spectra and the
+spectral radius formula give numbers $r_s<q<r_u$ such that
+\begin{equation}\label{flat:dichotomy-bounds}
+ \|P_s^k\|\le C r_s^k,\qquad
+ \|P_u^{-k}\|\le C r_u^{-k},\qquad k\ge0.
+\end{equation}
+For example choose $r_s$ strictly between $r(P_s)$ and $q$, and
+$r_u$ strictly between $q$ and the smallest modulus of an eigenvalue
+of $P_u$; the spectral radius formula controls all sufficiently
+large powers, and their finitely many predecessors are absorbed
+in $C$. The projections commute with complex conjugation because
+the enclosed spectral sets are invariant under conjugation.
+
+Fix this $\delta$. If $M=\|f\|_Y$, the source weight gives the exact
+estimate
+\begin{equation}\label{flat:source-decay}
+ |f^\#(\zeta,\tau)|
+ \le M\rho^\delta(1+|\zeta|^2)^{-3+\delta}
+ =M e^{-\gamma\tau}(1+|\zeta|^2)^{-3+\delta}.
+\end{equation}
+Indeed $\sigma^2=\rho(1+|\zeta|^2)$ and
+$\rho^3\sigma^{p_0-4}=\rho^\delta(1+|\zeta|^2)^{-3+\delta}$.
+This proves that $f^\#(\tau)\in E$. It also proves continuity with
+values in $E$: on any compact $\tau$ interval its spatial tails are
+uniformly small by \eqref{flat:source-decay}, while on a fixed spatial
+ball local H\"older continuity of $f$ and smoothness of the change of
+coordinates give uniform continuity in $\tau$.
+
+Let the one-step source increments be
+\[
+ l_j=\int_0^1S(1-s)f^\#(j+s)\,\dd s,\qquad j\in\Z.
+\]
+Then $\|l_j\|_E\le CMq^j$. Define
+\begin{equation}\label{flat:recursion-solution}
+ v_j=\sum_{k=0}^\infty P_s^k\Pi_s l_{j-1-k}
+       -\sum_{k=0}^\infty P_u^{-k-1}\Pi_u l_{j+k}.
+\end{equation}
+Both series converge absolutely for every integer $j$. More
+explicitly, their norms are at most
+\[
+ CMq^j\left[
+   q^{-1}\sum_{k=0}^\infty(r_s/q)^k
+       +r_u^{-1}\sum_{k=0}^\infty(q/r_u)^k\right]
+ \le C'Mq^j.
+\]
+Shifting the summation indices proves
+$v_{j+1}=Pv_j+l_j$. On $[j,j+1]$ define $v$ by
+\eqref{flat:duhamel} with initial value $v_j$.
+The definitions agree at the endpoints and give
+\begin{equation}\label{flat:preliminary-bound}
+ \|v(\tau)\|_E\le CM e^{-\gamma\tau},\qquad
+ |V(z,t)|\le CM\rho^{-1+\delta}.
+\end{equation}
+The continuous joining creates no distributional time-boundary
+term, so \eqref{flat:similarity-equation} holds across every joined
+face. Interior regularity makes it a classical equation there.
+The construction is linear and real on real sources. The second
+sum in \eqref{flat:recursion-solution} selects the outside spectral
+components using future forcing; its use is permitted precisely
+because no initial values have been imposed.
+
+\subsection{The full weighted parabolic estimate}
+
+We finish by strengthening \eqref{flat:preliminary-bound} to the
+$X$ estimate. This step uses $\delta>0$ as well as $\delta<1$.
+For a box centered at $(z_0,t_0)$ with
+$r_0^2=|z_0|^2\le\rho_0$, one has
+$\sigma_0^2\asymp\rho_0$. On a fixed enlargement of this box,
+chosen small in units $(\sigma_0,\sigma_0^4)$, the time stays
+strictly negative, $\rho\asymp\rho_0$, and
+$|z|^2/\rho$ stays bounded. In these units the coefficients
+$A$ and $\sigma_0^4B$ have bounded derivatives of every fixed order,
+and the principal coefficient has a uniform positive lower bound.
+The source has scaled $C^\alpha$ norm at most $CM$, after division
+by $\sigma_0^{p_0-4}$. Also
+\[
+ |V|\le CM\rho_0^{-1+\delta}
+          \le C'M\sigma_0^{p_0}.
+\]
+Lemma~\ref{ana:schauder}, with the fixed preceding-time enlargement,
+therefore gives the required full $X$ bound on every such box.
+
+For the remaining boxes a separate estimate restores the spatial
+weight. Fix $a>0$ and write
+\[
+ z=ay,\qquad t=a^4s,\qquad
+ U(y,s)=a^{-p_0}V(ay,a^4s),\qquad
+ g(y,s)=a^{4-p_0}f(ay,a^4s).
+\]
+Use the fixed annuli
+\[
+ \mathcal U=\{\tfrac12<|y|<2\},\qquad
+ \mathcal U'=\{\tfrac34<|y|<\tfrac54\}.
+\]
+On $\mathcal U\times(-4,0)$ the rescaled equation is
+\begin{equation}\label{flat:annular-equation}
+ U_s+\frac{G(|y|^2/\sqrt{-s})^2}{16}\Delta_y^2U
+      +\mathfrak b(y,s)U=g,\qquad
+ \mathfrak b(y,s)=(-s)^{-1}
+                     B_f\bigl(y/(-s)^{1/4}\bigr).
+\end{equation}
+All coefficients are independent of $a$. Their smooth bounds remain
+uniform as $s\uparrow0$. To verify the assertion for the zeroth order
+term explicitly, put $Y=|y|^2/\sqrt{-s}$ and use
+\[
+ \mathfrak b(y,s)=-(-s)^{-1}\frac{YG'(Y)}{G(Y)}.
+\]
+The expansion of $G$ in powers of $Y^{-2}$ gives an expansion of
+$-YG'/G$ starting at $Y^{-2}$. Since
+$Y^{-2}=(-s)/|y|^4$, division by $-s$ leaves a smooth expansion in
+nonnegative integral powers of $-s$. Differentiated remainders of
+arbitrarily high order show that both this coefficient and the
+principal coefficient extend smoothly to $s=0$ on the annulus.
+Away from $s=0$ their bounds follow from ordinary smoothness of $G$.
+The principal coefficient stays uniformly positive throughout.
+
+Furthermore
+\[
+ \frac{\sigma(ay,a^4s)}a
+       =\bigl(|y|^2+\sqrt{-s}\bigr)^{1/2}
+\]
+is bounded above and below on the larger annular slab. A fixed
+covering by the local boxes of Definition~\ref{ana:spaces} therefore
+gives $\|g\|_{C^\alpha(\mathcal U\times(-4,0))}\le CM$.
+Here and below the norm is parabolic; pairs outside a common small
+box are controlled by the supremum bound and their fixed positive
+separation.
+
+The preliminary estimate becomes
+\[
+ |U(y,s)|\le CM(-s)^{(-1+\delta)/2},
+\]
+with no factor of $a$: indeed
+$a^{-p_0}(a^2\sqrt{-s})^{-1+\delta}
+=(-s)^{(-1+\delta)/2}$. Consequently
+\begin{equation}\label{flat:annular-l2}
+ \|U\|_{L^2(\mathcal U\times(-4,0))}^2
+ \le C M^2|\mathcal U|\int_0^4 q^{-1+\delta}\,\dd q
+ =C M^2|\mathcal U|\frac{4^\delta}{\delta}<\infty.
+\end{equation}
+The value of $\delta$ has already been fixed, so the dependence on
+$1/\delta$ is harmless.
+
+Apply Lemma~\ref{ana:l2} to \eqref{flat:annular-equation} on nested
+annuli, with the preceding-time buffer from $-4$ to $-2$. It gives
+\begin{equation}\label{flat:annular-full}
+ \sum_{j=0}^4\|D_y^jU\|_{C^\alpha(\mathcal U'\times(-2,0))}
+ +\|U_s\|_{C^\alpha(\mathcal U'\times(-2,0))}
+ \le CM.
+\end{equation}
+One first applies the estimate with terminal time $b<0$. On each
+such truncated slab $U$ has finite supremum norm by
+\eqref{flat:preliminary-bound}; the coefficient, source, and $L^2$
+bounds above are independent of $b$. Letting $b\uparrow0$ then
+proves \eqref{flat:annular-full}, including H\"older quotients between
+times arbitrarily close to zero. The estimate uses preceding times
+and requires no terminal value at $s=0$.
+
+Finally take an arbitrary remaining center with $r_0^2>\rho_0$ and
+choose $a=r_0$. Its rescaled time satisfies $-1<s_0<0$ and
+$1<\sigma_0/a<\sqrt2$. The entire defining $X$ box lies in
+$\mathcal U'\times(-2,0)$ when the fixed box radius $a_0$ is small:
+its spatial radius is at most $\sqrt2a_0$ and its time radius at
+most $4a_0^4$. In particular a box near the dividing time $s=-1$
+has the required preceding-time buffer. Rescaling
+\eqref{flat:annular-full} gives its weighted $X$ norm at most $CM$.
+The core and annular cases cover all centers. Thus
+$\|V\|_X\le C\|f\|_Y$, completing the proof of
+Theorem~\ref{flat:inverse}.
diff --git a/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/sections/geometry.tex b/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/sections/geometry.tex
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+\section{The geometric reduction and the finite-time obstruction}
+\label{geo:section}
+\label{sec:geometry}
+
+We use scalar curvature in the convention
+\(S(\omega)=\operatorname{tr}_{\omega}\operatorname{Ric}(\omega)\).
+Let \(\omega_*\) be the Fubini--Study form on \(\CP^n\) with affine
+potential \(\log(1+|Z|^2)\).  The geometric identities below hold for every
+\(n\geq1\); the construction will use \(n=10\).
+The radial and momentum descriptions follow the framework of
+Calabi~\cite{Calabi1979} and Hwang--Singer~\cite{HwangSinger2002};
+the Legendre-coordinate viewpoint is also central to the toric formulas
+of Guillemin and Abreu~\cite{Guillemin1994,Abreu1998}. We give the
+identities directly to fix the scalar convention and to verify
+smoothness at the point orbit as well as at the divisor.
+Set
+\begin{equation}\label{geo:reference}
+ q=|Z|^2,\qquad \ell=\log q,\qquad
+ v_*(x)=x(1-x),\qquad w(x)=x^{n-1}.
+\end{equation}
+For a radial potential \(\Phi(\ell)\), its momentum coordinate and
+momentum profile are
+\[
+ x=\Phi'(\ell),\qquad v(x)=\Phi''(\ell).
+\]
+The Legendre transform \(u(x)=x\ell-\Phi(\ell)\) satisfies
+\(u'=\ell\) and \(u''=1/v\).  The reference transform is
+\[
+ u_*(x)=x\log x+(1-x)\log(1-x),\qquad u_*''=v_*^{-1}.
+\]
+Primes on functions of \(x\) will denote differentiation in \(x\).
+
+\begin{proposition}[Exact fixed-class reduction]\label{geo:reduction}
+Let \(I\) be an open time interval, and let
+\(H\in C^\infty(I\times[0,1])\) satisfy
+\begin{equation}\label{geo:positivity}
+ b(t,x):=1+v_*(x)H(t,x)>0.
+\end{equation}
+Define
+\begin{equation}\label{geo:momentum}
+ v=v_*+v_*^2H=v_*b,\qquad
+ K(H)=w^{-1}(wv_*^2H)'',\qquad
+ \A H=\big[w^{-1}(wv_*^2H)''\big]''.
+\end{equation}
+At each time these data, together with an affine choice in the Legendre
+potential, determine a smooth \(U(n)\)-invariant K\"ahler form in
+\([\omega_*]\).  Its scalar curvature and average scalar curvature are
+\begin{equation}\label{geo:scalar}
+ S(t,x)=n(n+1)-K(H)(t,x),\qquad \bar S=n(n+1).
+\end{equation}
+If, in addition,
+\begin{equation}\label{geo:equation}
+ H_t+(1+v_*H)^2\A H=0,
+\end{equation}
+then for any \(t_0\in I\) and any initial affine choice there is a smooth
+family of these forms satisfying the Calabi flow exactly.  More
+precisely, its global potential relative to \(\omega(t_0)\) satisfies
+\begin{equation}\label{geo:potential-flow}
+ \phi(t_0)=0,\qquad
+ \omega(t)=\omega(t_0)+i\partial\bar\partial\phi(t),\qquad
+ \phi_t=S(\omega(t))-\bar S.
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+We first verify smoothness and positivity at both endpoints.  Suppress
+time for this part of the argument.  Since
+\begin{equation}\label{geo:smooth-legendre}
+ \frac1v-\frac1{v_*}=-\frac{H}{1+v_*H},
+\end{equation}
+we can choose \(\psi\in C^\infty[0,1]\) with second derivative equal to
+the right-hand side, and put \(u=u_*+\psi\).  The two integration
+constants are precisely its free affine part.  Write \(a=\psi'\).
+The function \(u'\) is strictly increasing from \(-\infty\) to
+\(+\infty\), so it has a smooth inverse on the open interval.  In terms
+of \(q=\exp\ell\), the inverse relation is
+\begin{equation}\label{geo:center-inverse}
+ q=\frac{x}{1-x}\exp a(x).
+\end{equation}
+At \(x=0\) its right-hand side vanishes and has derivative
+\(\exp a(0)>0\).  The inverse therefore extends smoothly to \(q=0\),
+with
+\[
+ x(q)=\exp(-a(0))q+O(q^2).
+\]
+Here and below a smooth remainder is meant: for example the last error
+is \(q^2\) times a smooth function of \(q\).  Since
+\(\dd\Phi/\dd q=x(q)/q\), integration shows that the Legendre dual
+extends smoothly as a function of \(q\), with expansion
+\begin{equation}\label{geo:center-expansion}
+ \Phi(\log q)=C_0+\exp(-a(0))q+O(q^2).
+\end{equation}
+Division by \(q\) here preserves smoothness because \(x(0)=0\), as is
+also clear from \(x(q)/q=\int_0^1x'(sq)\,\dd s\).
+Its complex Hessian at \(Z=0\) is
+\(\exp(-a(0))\) times the Euclidean Hermitian form, and is positive.
+
+At the other endpoint set \(\eta=q^{-1}\) and \(b_1=1-x\).  The same
+inverse relation becomes
+\[
+ \eta=\frac{b_1}{1-b_1}\exp\big(-a(1-b_1)\big),\qquad
+ b_1(\eta)=\exp a(1)\eta+O(\eta^2).
+\]
+If \(\Psi(\eta)=\Phi(\log q)-\log q\), then
+\(\Psi'(\eta)=b_1(\eta)/\eta\).  Consequently
+\begin{equation}\label{geo:infinity-expansion}
+ \Psi(\eta)=C_1+\exp a(1)\eta+O(\eta^2)
+\end{equation}
+is smooth at \(\eta=0\).  This expansion gives smoothness on the
+projective manifold, not merely along radial rays.  Indeed, on the
+projective chart meeting the divisor at infinity given by
+\[
+ W_0=1/Z_j,\qquad W_i=Z_i/Z_j\quad(i\ne j),\qquad
+ B=1+\sum_{i\ne j}|W_i|^2,
+\]
+we have \(q=B/|W_0|^2\).  After removing the pluriharmonic term
+\(-\log|W_0|^2\) from \(\Phi\), a local potential is
+\begin{equation}\label{geo:divisor-potential}
+ \log B+\Psi\!\left(\frac{|W_0|^2}{B}\right).
+\end{equation}
+On \(W_0=0\) its tangential complex Hessian is the Fubini--Study metric
+of the divisor, its normal coefficient is \(\exp a(1)/B>0\), and its
+mixed coefficients vanish.  This proves positivity there; for \(n=1\)
+only the normal coefficient is present.  On \(0<x<1\), direct
+differentiation gives the tangential and radial eigenvalues
+\(x/q\) and \(v/q\), respectively, so positivity holds everywhere.
+
+The difference
+\(\Phi(\log|Z|^2)-\log(1+|Z|^2)\) is a globally smooth function.
+At the origin this follows from \eqref{geo:center-expansion}; in the
+chart at infinity it equals
+\(\Psi(\eta)-\log(1+\eta)\), which is smooth by
+\eqref{geo:infinity-expansion}.  Thus its complex Hessian is a global
+exact form and the K\"ahler class is \([\omega_*]\).  All inversions and
+integrations above depend smoothly on time on compact subintervals of
+\(I\), by the positive endpoint derivatives and the inverse function
+theorem with parameters.
+
+We next compute scalar curvature.  The eigenvalues already found give
+\[
+ \det(g_{i\bar j})=q^{-n}x^{n-1}v=q^{-n}wv.
+\]
+For any radial function \(f\), the complex Laplacian is
+\begin{equation}\label{geo:complex-laplacian}
+ \Delta_\omega f
+ =\frac{f_{\ell\ell}}{v}+\frac{n-1}{x}f_\ell
+ =w^{-1}\partial_x(wf_\ell),
+\end{equation}
+where \(\partial_\ell=v\partial_x\) in the last expression.
+For \(f=-\log\det g\), this yields
+\[
+ f_\ell=n-\frac{(wv)'}w,\qquad
+ S=\frac{n(n-1)}x-w^{-1}(wv)''.
+\]
+Substituting \(v=v_*+v_*^2H\) gives
+\eqref{geo:scalar}.  The expression \(K(H)\) is smooth at zero:
+\(wv_*^2H=x^{n+1}(1-x)^2H\), whose second derivative is divisible by
+\(x^{n-1}\) as a smooth function.  It is also smooth at one.
+
+For completeness, the average can be read directly from this formula.
+The radial volume density is a constant multiple of \(w(x)\,\dd x\):
+the Euclidean polar factor \(q^{n-1}\dd q\), multiplied by the
+determinant, becomes \(wv\,\dd\ell=w\,\dd x\).  Moreover,
+\[
+ \int_0^1 K(H)w\,\dd x
+ =\big[(wv_*^2H)'\big]_{x=0}^{x=1}=0,
+\]
+because \(wv_*^2H\) vanishes to order at least two at both endpoints.
+This proves the asserted average, including when \(n=1\).
+
+Finally, fix \(t_0\in I\), choose \(u(t_0)\) as above, and define at
+fixed momentum coordinate
+\begin{equation}\label{geo:legendre-flow}
+ u(t,x)=u(t_0,x)+\int_{t_0}^{t}K(H)(s,x)\,\dd s.
+\end{equation}
+Equation \eqref{geo:equation} implies
+\[
+ \partial_t(1/v)=-\frac{v_*^2H_t}{v^2}
+ =\A H=\partial_x^2K(H)=\partial_tu''.
+\]
+Their initial values agree, so \(u''=1/v\) for all time.  In
+particular the preceding metric construction applies throughout \(I\).
+Differentiating the Legendre dual
+\(\Phi(t,\ell)=x(t,\ell)\ell-u(t,x(t,\ell))\) at fixed \(\ell\)
+cancels the terms containing \(x_t\), and gives
+\begin{equation}\label{geo:legendre-sign}
+ \Phi_t=-u_t=-K(H)=S-\bar S.
+\end{equation}
+The difference between \(\Phi(t)\) and \(\Phi(t_0)\) is globally smooth
+by the endpoint argument and therefore supplies \(\phi\) in
+\eqref{geo:potential-flow}.
+
+This also specifies the affine part of \(u\) during evolution.
+Replacing a fixed \(u\) by \(u+\alpha x+\beta\) changes its dual to
+\(\Phi(\ell-\alpha)-\beta\), corresponding to the projective
+dilation \(Z\mapsto\exp(-\alpha/2)Z\) and a potential constant.
+An arbitrary time-dependent choice of these terms would affect
+\eqref{geo:legendre-sign}; the prescription
+\eqref{geo:legendre-flow} fixes them and establishes that equation
+without an additional affine term.
+\end{proof}
+
+\subsection{A regular auxiliary sphere and its flat limit}
+
+The endpoints of the momentum interval can be handled by a smooth
+equation on a closed auxiliary manifold.  Set
+\begin{equation}\label{geo:sphere}
+ m=2n+4,\qquad
+ \Sigma=S^{m+5}\subset\R^m\times\R^6,\qquad x=|z|^2.
+\end{equation}
+We identify functions of \(x\) with
+\(O(m)\times O(6)\)-invariant functions on \(\Sigma\).
+
+\begin{lemma}[The compact and flat operators]\label{geo:operators}
+This identification is a bijection between \(C^\infty[0,1]\) and the
+smooth invariant functions on \(\Sigma\).  The pushforward of spherical
+volume under \(x\) is a positive constant multiple of
+\(wv_*^2\,\dd x\).  With the nonpositive Laplacian convention, put
+\(L=\frac14\Delta_\Sigma\).  On invariant functions,
+\begin{equation}\label{geo:sphere-operator}
+ L=x(1-x)\partial_x^2+\big(n+2-(n+5)x\big)\partial_x,
+ \qquad
+ \A=(-L+n+3)(-L+2(n+2)).
+\end{equation}
+The right-hand expression defines a smooth elliptic operator of order
+four on all of \(\Sigma\).
+
+In the core coordinates
+\((z,\theta)\mapsto(z,\sqrt{1-|z|^2}\,\theta)\), with
+\(\theta\in S^5\), its Laplacian on functions independent of \(\theta\)
+is
+\begin{equation}\label{geo:local-laplacian}
+ \Delta_\Sigma
+ =\Delta_z-\sum_{i,j=1}^m z_i z_j\partial_{z_i z_j}
+             -(m+5)\sum_{i=1}^m z_i\partial_{z_i}.
+\end{equation}
+The flat momentum operator obtained by replacing \(v_*\) by \(x\) in
+\eqref{geo:momentum} satisfies
+\begin{equation}\label{geo:flat-operator}
+ \wh\A H
+ :=\big[x^{1-n}(x^{n+1}H)''\big]''
+ =\big(x\partial_x^2+(n+2)\partial_x\big)^2H
+ =\frac1{16}\Delta_{\R^m}^2H,\qquad x=|z|^2.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+The only regularity issue in the identification occurs at the two
+singular orbits.  Here is an elementary smoothness argument.  If
+\(f(r)\) is smooth and even near zero, then the operator
+\[
+ \mathcal T f(r)=\frac{f'(r)}r
+              =\int_0^1f''(sr)\,\dd s
+\]
+defines another smooth even function.  Thus, for \(g(x)=f(\sqrt x)\),
+each derivative on \(x>0\) has the form
+\(g^{(j)}(x)=2^{-j}(\mathcal T^jf)(\sqrt x)\), and extends
+continuously to \(x=0\).  Induction gives smoothness up to zero of
+\(g\) and all its derivatives.  The same argument holds with smooth
+parameters.  Restricting a smooth invariant function on \(\Sigma\) to
+\((re_1,\sqrt{1-r^2}\,\theta_0)\) gives such an even \(f\), and
+invariance makes its value depend only on \(x=r^2\).  Interchanging the
+two factors proves smoothness in \(1-x\) at the other endpoint.
+Conversely a smooth function of \(x\) is smooth in these coordinates
+at both orbits, as well as on their complement.
+
+Write \(|z|=\sin\vartheta\), with \(0\leq\vartheta\leq\pi/2\).
+The spherical metric and volume density are
+\[
+ \dd\vartheta^2+\sin^2\vartheta\,g_{S^{m-1}}
+                 +\cos^2\vartheta\,g_{S^5},\qquad
+ \sin^{m-1}\vartheta\cos^5\vartheta\,\dd\vartheta\,
+       \dd\omega_{m-1}\,\dd\omega_5.
+\]
+Changing to \(x=\sin^2\vartheta\) gives density proportional to
+\[
+ M(x)\,\dd x:=x^{m/2-1}(1-x)^2\,\dd x
+              =wv_*^2\,\dd x.
+\]
+The radial Laplacian computed from this metric gives the displayed
+formula for \(L\), equivalently
+\(L=M^{-1}\partial_x(v_*M\partial_x)\).
+
+We verify the factorization by a polynomial argument that also checks
+the endpoint terms.  Both \(L\) and \(\A\) preserve polynomial degree
+and are symmetric for the inner product with density \(M\).  For
+\(\A\), twice integrating by parts gives
+\begin{equation}\label{geo:symmetry}
+ \int_0^1 a(\A b)M\,\dd x
+ =\int_0^1\frac{(Ma)''(Mb)''}{w}\,\dd x.
+\end{equation}
+There are no boundary terms: \(Ma\) and its first derivative vanish
+at each endpoint, and \(w^{-1}(Mb)''\) is smooth there.  For \(L\)
+symmetry follows from its divergence expression, since \(v_*M\)
+vanishes at both ends.  Orthogonalizing the monomials therefore gives
+common polynomial eigenfunctions.  On the polynomial of degree \(j\),
+the leading coefficients give respectively the eigenvalues
+\[
+ -j(j+n+4),\qquad
+ (j+1)(j+2)(j+n+2)(j+n+3).
+\]
+Indeed a symmetric degree-preserving operator maps the degree-\(j\)
+orthogonal polynomial to a multiple of itself, because its inner
+product with every lower-degree polynomial vanishes.  Finally,
+\[
+ \big(j(j+n+4)+n+3\big)
+ \big(j(j+n+4)+2(n+2)\big)
+ =(j+1)(j+n+3)(j+2)(j+n+2).
+\]
+This proves equality on every polynomial, hence equality of the
+differential expressions: at an interior point polynomials realize
+arbitrary jets through order four.  The sphere expression has leading
+term \(\Delta_\Sigma^2/16\), proving the claimed ellipticity.
+
+For \eqref{geo:local-laplacian}, the inverse metric in the \(z\)
+coordinates is \((\delta_{ij}-z_i z_j)\) and the volume density is
+\((1-|z|^2)^2\,\dd z\,\dd\omega_5\).  Applying the divergence formula
+therefore gives
+\[
+ (1-|z|^2)^{-2}\partial_{z_i}
+ \big((1-|z|^2)^2(\delta_{ij}-z_i z_j)\partial_{z_j}\big)
+ =(\delta_{ij}-z_i z_j)\partial_{z_i z_j}
+   -(m+5)z_i\partial_{z_i},
+\]
+with repeated indices summed.  This computation only requires
+independence of \(\theta\).  Finally both one-dimensional expressions
+for \(\wh\A\) in \eqref{geo:flat-operator} expand to
+\[
+ x^2H^{(4)}+2(n+3)xH^{(3)}+(n+2)(n+3)H'',
+\]
+and the radial Euclidean identity
+\(\frac14\Delta_{\R^m}=x\partial_x^2+\frac m2\partial_x\)
+proves the last equality.
+\end{proof}
+
+\subsection{The self-similar core}
+\label{geo:core-section}
+
+From now on set \(n=10\), so \(m=24\) and \(\Sigma=S^{29}\).  In the
+flat model the equation for the coefficient is
+\begin{equation}\label{geo:flat-flow}
+ F_t+(1+xF)^2\wh\A F=0.
+\end{equation}
+Let \(D=x\partial_x\).  For the function \(g=1+xF\),
+\eqref{geo:flat-operator} gives
+\begin{equation}\label{geo:euler-identity}
+ x\wh\A\big(x^{-1}(g-1)\big)
+ =x^{-2}P(D)(g-1),\qquad
+ P(D)=(D-2)(D-1)(D+9)(D+10).
+\end{equation}
+Here \(P\) denotes a polynomial, as distinct from the evolution
+operator denoted by \(P\) later.  One can check
+\eqref{geo:euler-identity} on \(g-1=x^j\): both sides equal
+\((j-2)(j-1)(j+9)(j+10)x^{j-2}\); equality of the differential
+expressions follows as in Lemma~\ref{geo:operators}.
+Consequently \eqref{geo:flat-flow} is equivalent, for \(x>0\), to
+\begin{equation}\label{geo:g-flow}
+ g_t=-g^2x^{-2}P(D)(g-1).
+\end{equation}
+
+Define the cubic polynomial operator
+\(Q(D)=(D-1)(D+9)(D+10)\).  Suppose that \(G\) is smooth and positive
+on \([0,\infty)\), with \(G(0)=1\), and for some \(c>0\) satisfies
+\begin{equation}\label{geo:profile-equation}
+ Q(y\partial_y)(G-1)=\frac{y^2}{2}\left(\frac1G-\frac1c\right).
+\end{equation}
+Then set
+\begin{equation}\label{geo:selfsimilar}
+ \rho=\sqrt{-t},\qquad y=x/\rho,\qquad
+ h(y)=\frac{G(y)-1}{y},\qquad
+ F(t,x)=\rho^{-1}h(x/\rho),\qquad t<0.
+\end{equation}
+The value \(h(0)=G'(0)\) makes \(h\) smooth at zero, since
+\(h(y)=\int_0^1G'(sy)\,\dd s\).  Thus \(F\) is smooth for \(t<0\)
+including at the flat origin.  Applying \(D-2\), now with
+\(D=y\partial_y\), to \eqref{geo:profile-equation} yields
+\[
+ P(D)(G-1)=-\frac{y^2DG}{2G^2}.
+\]
+On the other hand \(\rho_t=-1/(2\rho)\), so
+\(\partial_tG(x/\rho)=DG/(2\rho^2)\).  These identities prove
+\eqref{geo:g-flow}, and hence \eqref{geo:flat-flow}, first for \(x>0\)
+and then at zero by smoothness.  The existence of the required
+nonconstant positive \(G\) is the subject of
+Theorem~\ref{prof:profile}.
+
+The same expression \(F(t,x)\) is smooth on the compact sphere for
+every \(t<0\).  Its compact positivity multiplier obeys the useful
+identity
+\begin{equation}\label{geo:approximate-positivity}
+ 1+v_*F=x+(1-x)G(x/\rho).
+\end{equation}
+In particular, if \(G\) is bounded above and below by positive
+constants, this multiplier has the same property uniformly in
+\(t<0\) and \(0\leq x\leq1\).
+This is a bound in the momentum coordinate \(x\), whose relation
+to the fixed affine coordinate varies under the Legendre transform.
+It does not imply uniform equivalence to the Fubini--Study metric.
+
+\subsection{Curvature at the concentrating point}
+
+The final geometric implication uses only one value of the correction.
+It is convenient to state it before constructing that correction.
+
+\begin{lemma}[Finite-time obstruction]\label{geo:obstruction}
+Let \(G\in C^\infty[0,\infty)\) satisfy \(G(0)=1\) and
+\(G'(0)\ne0\), and define \(F\) by \eqref{geo:selfsimilar}.  Suppose
+that for some \(T>0\) and \(\delta>0\) there is a smooth invariant
+solution \(H=F+V\) of \eqref{geo:equation} on \((-T,0)\times\Sigma\)
+such that \(1+v_*H>0\) and
+\begin{equation}\label{geo:correction-bound}
+ V(t,0)=O(\rho^{-1+\delta})\qquad(t\uparrow0).
+\end{equation}
+Then a smooth \(U(10)\)-invariant metric \(\omega_0\in[\omega_*]\)
+has maximal smooth Calabi-flow existence interval \([0,T_*)\), where
+\(T_*=T/2<\infty\).  At the fixed affine origin \(p\), the scalar
+curvature of this flow satisfies
+\begin{equation}\label{geo:curvature-rate}
+ S(\tau,p)=a(T_*-\tau)^{-1/2}
+       +O\big(1+(T_*-\tau)^{(-1+\delta)/2}\big),\qquad
+ a=-110G'(0)\ne0.
+\end{equation}
+In particular \(S(\tau,p)\sim a(T_*-\tau)^{-1/2}\), and \(a>0\)
+when \(G'(0)<0\).
+\end{lemma}
+
+\begin{proof}
+By Lemma~\ref{geo:operators}, \(H\) is smooth in \(x\) at both
+endpoints for each time.  Proposition~\ref{geo:reduction}, with
+\(t_0=-T/2\), gives a smooth family of K\"ahler forms, and its slice
+\(\omega_0=\omega(-T/2)\) is a smooth positive form in the
+Fubini--Study class.  Put \(\tau=t+T/2\).  The corresponding global
+potential satisfies exactly the normalization and evolution in
+\eqref{geo:potential-flow} on \(0\leq\tau<T_*=T/2\).
+
+At \(x=0\), the coefficient of \(x^{n+1}\) in
+\(wv_*^2H\) is \(H(t,0)\).  Thus \eqref{geo:scalar} gives the exact
+identity
+\begin{equation}\label{geo:center-scalar}
+ S(t,p)=n(n+1)\big(1-H(t,0)\big).
+\end{equation}
+This evaluates scalar curvature at the same point \(p\) at every
+time: \eqref{geo:center-inverse} sends \(q=0\) to \(x=0\) for every
+affine choice.  Since
+\[
+ H(t,0)=\rho^{-1}G'(0)+O(\rho^{-1+\delta}),
+\]
+Equation~\eqref{geo:center-scalar} proves
+\eqref{geo:curvature-rate} and the asserted divergence.
+
+We recall why uniqueness applies to comparison with an unrestricted
+smooth flow; no symmetry hypothesis is needed for this step.  Smooth
+positive initial potentials have a unique local Calabi flow by
+\cite[Theorem~3.2]{ChenHe2008}.  The uniqueness mechanism can also be
+seen directly.  For two smooth solutions with the same initial
+potential, let \(\chi\) be their difference and interpolate their
+potentials by \(\varphi_s\), \(0\leq s\leq1\).  Their interpolated
+K\"ahler forms remain positive, and variation of scalar curvature
+gives
+\[
+ \chi_t=-\int_0^1
+ \left(\Delta_{\varphi_s}^2\chi
+       +\operatorname{Ric}_{\varphi_s}^{i\bar j}
+                            \partial_i\partial_{\bar j}\chi\right)
+ \,\dd s.
+\]
+On a compact time interval where both solutions are smooth, the
+coefficients are smooth and uniformly elliptic.  Relative to a fixed
+smooth volume form, integration twice by parts and interpolation give
+\[
+ \frac{\dd}{\dd t}\|\chi\|_{L^2}^2
+ \leq-c\|\chi\|_{H^2}^2+C\|\chi\|_{L^2}^2
+ \leq C\|\chi\|_{L^2}^2.
+\]
+For this estimate, the principal quadratic form is the integral of
+\(|\Delta_{\varphi_s}\chi|^2\); differentiating the smooth volume
+density introduces terms bounded by
+\(C\|\chi\|_{H^1}\|\chi\|_{H^2}\), and the Ricci term has the same
+or lower order.  The elliptic estimate for each uniformly elliptic
+Laplacian and
+\(\|\chi\|_{H^1}\leq\eps\|\chi\|_{H^2}+C_\eps\|\chi\|_{L^2}\)
+absorb those terms.  Gronwall's inequality proves uniqueness.
+
+It follows that any smooth flow from \(\omega_0\) agrees with the one
+constructed above on their common interval.  Hence its maximal
+interval contains \([0,T_*)\).  If it extended beyond \(T_*\), smooth
+positivity on the compact manifold across that time would bound its
+scalar curvature, contradicting \eqref{geo:curvature-rate}.  Its
+maximal smooth existence time is therefore exactly \(T_*\).
+\end{proof}
+
+The Fubini--Study class contains the constant-scalar-curvature form
+\(\omega_*\), so the
+hypotheses of Lemma~\ref{geo:obstruction} suffice to refute the
+smooth global-existence assertion.  Its requirement of smooth
+existence at every finite time has already failed.  In addition,
+\(\|S\|_{L^\infty}\) is unchanged by biholomorphic pullback, so the
+curvature divergence itself is unaffected by allowing
+automorphisms.
diff --git a/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/sections/introduction.tex b/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/sections/introduction.tex
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+\section{Introduction}\label{sec:introduction}
+Let \(\omega_0\) be a smooth K\"ahler form on a compact complex manifold \(M\).
+We use the scalar-curvature convention
+\(S(\omega)=\operatorname{tr}_\omega\operatorname{Ric}(\omega)\).
+The Calabi flow is
+\begin{equation}\label{eq:calabi-flow}
+ \omega(t)=\omega_0+i\partial\bar\partial\phi(t)>0,
+ \qquad \phi(0)=0,\qquad
+ \partial_t\phi=S(\omega(t))-\bar S,
+\end{equation}
+where \(\bar S\) is the average scalar curvature, determined by the K\"ahler class.
+Calabi introduced this flow to seek canonical metrics within a K\"ahler
+class~\cite{Calabi1982}. Chen's smooth long-time existence conjecture
+asserts that the flow exists for all positive times from arbitrary smooth
+K\"ahler initial data; see \cite[Conjecture~1.1]{HeZeng2021}.
+We give a counterexample on projective space. Write
+\(\omega_{\mathrm{FS}}\) for the Fubini--Study form whose potential
+on the standard affine chart is \(\log(1+|Z|^2)\).
+
+\begin{theorem}\label{thm:main}
+There is a smooth \(U(10)\)-invariant K\"ahler form \(\omega_0\) on
+\(\CP^{10}\), in the class \([\omega_{\mathrm{FS}}]\), whose maximal smooth
+Calabi flow has existence interval \([0,T_*)\) with \(0<T_*<\infty\).
+There are a point \(p\in\CP^{10}\) and a constant \(a>0\) such that
+\[
+ S(\omega(t))(p)\sim a(T_*-t)^{-1/2}\qquad\text{as }t\uparrow T_*.
+\]
+\end{theorem}
+
+The Fubini--Study metric has constant scalar curvature. Thus the example
+also answers negatively the smooth global-existence question for classes
+containing a constant-scalar-curvature K\"ahler metric, emphasized by
+Chen and Cheng~\cite[p.~912, following Conjecture~1.3]{ChenCheng2021}.
+In particular, no assertion of smooth global existence and subsequent
+smooth convergence modulo holomorphic automorphisms can hold for arbitrary
+smooth initial metrics in such a class: the obstruction already occurs
+at a finite time.
+
+The same singularity persists on products with a stationary factor.
+\begin{corollary}[Products with stationary factors]\label{cor:product-singularities}
+Let \(\omega(t)\), \(T_*\), \(p\), and \(a\) be as in
+Theorem~\ref{thm:main}, and let \((Y,\eta)\) be a compact connected
+K\"ahler manifold with \(\eta\) of constant scalar curvature.  With
+\(\pi_X\) and \(\pi_Y\) the projections from \(\CP^{10}\times Y\),
+the maximal smooth Calabi flow from
+\(\Omega_0=\pi_X^*\omega_0+\pi_Y^*\eta\) is
+\[
+ \Omega(t)=\pi_X^*\omega(t)+\pi_Y^*\eta,
+ \qquad 0\leq t<T_*.
+\]
+For every \(y\in Y\),
+\(S(\Omega(t))(p,y)\sim a(T_*-t)^{-1/2}\).
+In particular, finite-time smooth Calabi-flow singularities occur in
+every complex dimension \(n\geq10\), on
+\(\CP^{10}\times\CP^{n-10}\), in a class containing a
+K\"ahler--Einstein metric.
+\end{corollary}
+The corollary is proved after Theorem~\ref{thm:main} in
+Section~\ref{sec:compact}.
+
+\subsection{Background and significance}
+On compact Riemann surfaces, Chru\'sciel proved global forward existence
+and convergence to a constant-curvature metric~\cite[Theorem~2.1]{Chrusciel1991}.
+Chen gave a different proof using decreasing functionals and concentration
+compactness~\cite{Chen2001}, and Struwe developed a unified treatment of
+Calabi and normalized Ricci flows on closed surfaces~\cite{Struwe2002}.
+In higher dimensions, Chen and He proved local existence and uniqueness
+and established stability near a constant-scalar-curvature
+metric~\cite[Theorems~1.2 and~3.2]{ChenHe2008}. He and Zeng developed
+short-time theory for rougher initial metrics~\cite{HeZeng2021}.
+These higher-dimensional results concern either short times or restricted
+initial data.
+
+Curvature estimates give continuation criteria for the smooth flow.
+Chen and He proved extension under a uniform Ricci-curvature
+bound~\cite[Corollary~5.3]{ChenHe2008}, and Chen and Cheng established
+extension under a uniform scalar-curvature bound~\cite[Theorem~1.8]{ChenCheng2021}.
+Li, Zhang and Zheng state that a uniform spatial $L^p$ scalar-curvature
+bound, measured with respect to the evolving metric, suffices when
+$p>\dim_{\C}M$~\cite[Theorem~1.2]{LiZhangZheng2024}; Li and Zhang
+state a further criterion involving spacetime scalar and
+scalar-Laplacian integrability~\cite[Theorem~1.1]{LiZhang2025}.
+
+On a compact almost homogeneous manifold with two hypersurface ends,
+Guan proved exponential smooth convergence of the modified Calabi flow
+for initial metrics invariant under a maximal compact subgroup of the
+automorphism group~\cite[Theorem~A]{Guan2007}.
+The two special orbits of the action used here are instead a point and
+the hyperplane at infinity. The point has complex codimension ten, so
+the two-hypersurface hypothesis fails.
+
+Streets constructed global weak Calabi flows by minimizing
+movements~\cite{Streets2014}. Berman, Darvas and Lu identify these flows
+with trajectories in a space of finite-energy K\"ahler potentials and show
+that, when the class contains a constant-scalar-curvature metric, every
+weak trajectory converges to a smooth constant-scalar-curvature potential
+in the $d_1$ distance, the extension of the $L^1$ path-length distance on
+smooth K\"ahler potentials~\cite[Section~6 and Theorem~1.11]{BermanDarvasLu2017}.
+The weak and classical flows agree for as long as the latter exists
+smoothly~\cite[Proposition~6.1]{BermanDarvasLu2017}.
+Neither this weak continuation nor the large-time $d_1$ convergence implies
+smooth continuation through every finite time.
+
+\subsection{Structure of the construction}
+
+The construction first reduces the flow to a scalar equation, then builds
+an exact shrinking solution of its flat limit, and finally corrects that
+solution on the compact manifold. The geometric starting point is the
+radial ansatz of Calabi~\cite{Calabi1979} and the momentum formulation
+of Hwang and Singer~\cite{HwangSinger2002}. Abreu's toric
+scalar-curvature formula~\cite{Abreu1998} is expressed in Guillemin's
+symplectic-potential framework~\cite{Guillemin1994}.
+For a radial potential \(\Phi(\ell)\), where
+\(\ell=\log|Z|^2\), set \(x=\Phi'(\ell)\) and
+\(v=\Phi''(\ell)\). We use \(x\in(0,1)\) as the momentum
+coordinate and write
+\[
+ v=x(1-x)+x^2(1-x)^2H.
+\]
+Smoothness and positivity at both special orbits are encoded by
+\(H\in C^\infty[0,1]\) and \(1+x(1-x)H>0\).
+Section~\ref{sec:geometry} derives the scalar fourth-order equation
+for \(H\), including the affine choice in the Legendre potential
+needed to recover the Calabi flow with its prescribed normalization.
+It also realizes the fixed spatial operator on an auxiliary sphere,
+where the endpoint conditions become ordinary smooth invariance.
+
+The singularity is detected by one value of \(H\). At the fixed
+affine origin \(p\), the scalar-curvature identity is
+\[
+ S(t,p)=110\bigl(1-H(t,0)\bigr).
+\]
+We construct a smooth positive profile \(G\), bounded above and away from
+zero, with \(G(0)=1\) and \(G'(0)<0\), and set
+\[
+ F(t,x)=\rho^{-1}h(x/\rho),\qquad
+ h(y)=\frac{G(y)-1}{y},\qquad
+ \rho=\sqrt{-t},\quad t<0.
+\]
+Then \(F(t,0)=\rho^{-1}G'(0)\). Our task is to find an exact
+compact solution \(H=F+V\) for which positivity persists and
+\[
+ V(t,0)=O(\rho^{-1+\delta})\qquad\text{for some }\delta>0.
+\]
+The correction is therefore smaller than the leading term at the
+origin, preserving the desired scalar-curvature divergence. A smooth
+interior time slice of this solution will be the initial metric;
+no initial metric needs to be fixed before the construction.
+
+Section~\ref{sec:profile} constructs \(G\) so that \(F\) solves the
+scale-invariant flat equation exactly. The profile is obtained by
+matching a two-parameter analytic solution at the origin to a half-line
+solution approaching a positive constant.
+A finite exact-arithmetic calculation, with analytic enclosure and
+nonlinear error bounds, certifies the matching inequalities. This uses
+the framework of validated Taylor integration surveyed
+in~\cite{NedialkovJacksonCorliss1999}.
+The finite argument and its complete executable certificate appear in
+the appendices. The resulting profile also has a differentiable
+expansion at infinity; this expansion controls the compact error away
+from the shrinking core.
+
+The correction begins with a right inverse for the flat linearized
+equation. Section~\ref{sec:parabolic} introduces the weighted norms
+and local estimates, and Section~\ref{sec:flat} constructs this inverse
+on all negative times. In similarity variables its time-one evolution
+is a compact perturbation of a damped constant-coefficient evolution.
+A spectral split determines finitely many components from future
+forcing. This choice is available because the initial metric is still
+free; it requires no stability or nondegeneracy theorem for the
+profile. The compact-perturbation and spectral-projection tools are
+classical~\cite{Kato1995}. After selecting a spectral weight, an outer
+annulus estimate upgrades the temporal supremum bound to the full
+spatially weighted parabolic H\"older estimate. The decisive input
+there is a scale-invariant \(L^2\) bound that is integrable up to
+the singular time.
+
+Section~\ref{sec:compact} combines this ancient inverse near the core
+with a short-time forward solution outside it. The outer forcing
+cancels the inner cutoff commutator exactly; the remaining operator
+errors are small. For the nonlinear correction one further needs the
+compact residual to become small after applying the inverse. This
+holds in a norm controlling all four spatial derivatives and their
+temporal H\"older seminorms, with the time derivative omitted.
+The outer estimate uses a strictly higher H\"older exponent for the
+source, and the fixed-point equation then supplies the missing time
+derivative. Positivity and the pointwise correction bound above give
+the geometric hypotheses needed to finish Theorem~\ref{thm:main}.
diff --git a/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/sections/parabolic.tex b/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/sections/parabolic.tex
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+\section{Parabolic norms and uniform local estimates}\label{sec:parabolic}\label{ana:section}
+
+The correction to the profile must satisfy the lower-order bound in
+Lemma~\ref{geo:obstruction}.  To encode that bound together with its
+spatial derivatives, put
+\[
+ \rho=\sqrt{-t},\qquad r=\sqrt{x},\qquad
+ \sigma=(r^2+\rho)^{1/2},\qquad p_0=-2+2\delta,
+ \qquad 0<\delta<1.
+\]
+At the concentrating point, a bound $|V|\le\eps\sigma^{p_0}$ gives
+$|V(t,0)|\le\eps\rho^{-1+\delta}$, smaller than the profile's
+$\rho^{-1}$ term.  On the sphere, $v_*\le r^2\le\sigma^2$, so it
+also gives $|v_*V|\le\eps\sigma^{2\delta}$.  Since $\sigma$ is
+bounded on $\Sigma\times[-T,0)$ for fixed $T$, a small weighted norm
+will preserve positivity of the multiplier $1+v_*(F+V)$.
+Outside the shrinking core, $r^2\ge\rho$,
+the scale $\sigma$ is comparable to $r$ and the same weight records
+spatial decay.  The particular value of $\delta$ will be chosen in
+Theorem~\ref{flat:inverse}.
+
+The norms below measure four spatial derivatives and one time derivative
+in units $(\sigma,\sigma^4)$.  We also need estimates for an outer
+forward problem with zero initial data, uniform as the elapsed time
+tends to zero.  For this purpose we distinguish the full parabolic norm
+from the norm that omits the first time derivative.
+
+\subsection{Norms and scales}
+
+On a product $D=U\times I\subset\R^d\times\R$, put
+\[
+ d_{\rm p}((z,t),(w,s))=|z-w|+|t-s|^{1/4}.
+\]
+Write
+\[
+ [g]_{\alpha;D}=\sup_{P\ne Q\in D}
+       \frac{|g(P)-g(Q)|}{d_{\rm p}(P,Q)^\alpha},
+ \quad |g|_{\alpha;D}=\|g\|_{\infty;D}+[g]_{\alpha;D}.
+\]
+The joint seminorm is equivalent, with constants depending only on
+$\alpha$, to the sum of the spatial $\alpha$ and temporal $\alpha/4$
+seminorms.  We use the following explicit version of the fourth-order
+parabolic norm:
+\begin{equation}\label{ana:local-norm}
+ \mathcal N_\alpha(u;D)
+   =\sum_{j=0}^4|D_z^ju|_{\alpha;D}+|u_t|_{\alpha;D},\qquad
+ \mathcal N'_\alpha(u;D)=\sum_{j=0}^4|D_z^ju|_{\alpha;D}.
+\end{equation}
+Here and below norms of arrays mean sums of component norms.  Thus
+$\mathcal N'_\alpha$ still includes temporal H\"older seminorms of all
+four spatial derivatives.
+
+\begin{definition}[Weighted local spaces]\label{ana:spaces}
+Let $M$ be either $\R^m$ or the fixed sphere $\Sigma$, let $I$ be a time
+interval, and let $s(P)>0$ be a scale on $M\times I$.  Fix a sufficiently
+small number $a_0>0$.  In Euclidean coordinates, or normal coordinates
+on $\Sigma$, centered at $P=(z_0,t_0)$, write $s_0=s(P)$ and set
+\[
+ \Psi_P(y,\theta)=(\exp_{z_0}(s_0y),t_0+s_0^4\theta),\qquad
+ D_P=B_{a_0}\times
+       \bigl((-a_0^4,a_0^4)\cap s_0^{-4}(I-t_0)\bigr).
+\]
+In the Euclidean case the exponential map means translation.  The
+one-sided endpoints belonging to $I$ are included.  For a fixed weight
+$p$, define
+\begin{align}
+ \|u\|_{X_{\alpha,p}(s)}
+   &=\sup_P\mathcal N_\alpha(s_0^{-p}u\circ\Psi_P;D_P),\label{ana:X}\\
+ \|u\|_{X'_{\alpha,p}(s)}
+   &=\sup_P\mathcal N'_\alpha(s_0^{-p}u\circ\Psi_P;D_P),\label{ana:Xprime}\\
+ \|f\|_{Y_{\alpha,p}(s)}
+   &=\sup_P|s_0^{4-p}f\circ\Psi_P|_{\alpha;D_P}.
+                                                        \label{ana:Y}
+\end{align}
+The derivatives in these formulas are derivatives of the displayed
+pullbacks.  On the sphere $s$ is bounded above so that these charts,
+and their fixed enlargements used below, lie within the injectivity
+radius.  Restriction to radial functions on $\R^m$, or to invariant
+functions on $\Sigma$, gives closed subspaces with these same norms.
+
+Unless specified otherwise, $X,X',Y$ mean the spaces
+\eqref{ana:X}--\eqref{ana:Y} with $s=\sigma$, $p=p_0$, and the
+fixed exponent $0<\alpha<1$.  Subscripts $\alpha'$ indicate replacement
+of $\alpha$ by $\alpha'$, with $\alpha<\alpha'<1$.
+\end{definition}
+
+These are local weighted norms; no H\"older quotient between distant
+boxes is imposed.  Their completeness follows by convergence of the
+functions and indicated derivatives on every compact subset, followed
+by taking the supremum in the defining inequalities.  The derivative
+relations pass to the limit distributionally.  In particular $X$ and
+$X'$ are Banach spaces and $X\hookrightarrow X'$ has norm at most one.
+
+The radius function $r$ is Lipschitz on both spaces.  The elementary
+inequalities for Euclidean norms and fourth roots give
+\[
+ |\sigma(z,t)-\sigma(z_0,t_0)|
+ \le |r(z)-r(z_0)|+|t-t_0|^{1/4}.
+\]
+Consequently $\sigma$ is comparable to its center value on every box
+in Definition~\ref{ana:spaces}, including boxes truncated at $t=0$.
+Changing $a_0$ to another fixed sufficiently small radius gives
+equivalent norms: cover the enlarged box by finitely many smaller
+comparable-scale boxes, and split a H\"older quotient whose points
+are not in a common small box using the supremum norm.  The same
+argument applies to $s_\beta=\max(\beta,r)$, whose lack of smoothness
+at $r=\beta$ is harmless: the scale is frozen at each center, never
+differentiated.  Products satisfy the usual local H\"older inequality
+$|fg|_\alpha\le 2|f|_\alpha|g|_\alpha$, with their weights added.
+
+\subsection{Interpolation without a lower bound for the time length}
+
+We first isolate the interpolation used in localization.  For fixed
+nested balls $B_1\Subset B_2$ and any interval $I$ of length at most
+one, let
+\[
+ U=\|u\|_{\infty;B_2\times I},\qquad
+ M=U+\|D_z^4u\|_{\infty;B_2\times I}
+       +\|u_t\|_{\infty;B_2\times I}.
+\]
+Then, for $0\le j\le3$,
+\begin{equation}\label{ana:interp-basic}
+ |D_z^ju|_{\alpha;B_1\times I}
+ \le C U^{1-(j+\alpha)/4}M^{(j+\alpha)/4}
+ \le \eps M+C_\eps U.
+\end{equation}
+All constants are independent of $|I|$.  Spatial interpolation on
+nested balls gives the spatial assertion, including the supremum.
+For the temporal assertion apply that same interpolation to
+$w=u(\cdot,t)-u(\cdot,s)$.  With $b=|t-s|$,
+\[
+ \|w\|_\infty\le\min(2U,bM),\qquad
+ \|D^4w\|_\infty\le2M,
+\]
+and hence
+\[
+ \|D^jw\|_{\infty;B_1}
+ \le C\min(2U,bM)^{1-j/4}M^{j/4}.
+\]
+Divide by $b^{\alpha/4}$ and split at $b=U/M$; both resulting
+bounds give \eqref{ana:interp-basic}.  Restricting the available values
+of $b$ can only decrease their supremum.  The spatial interpolation
+inequalities themselves follow, for example, by subtracting a
+Taylor polynomial in a ball of radius $q$ and minimizing in $q$;
+fixed-radius remainders account for the $U$ term when the minimizer
+exceeds the gap between the balls.  The same argument at order four
+gives
+\begin{equation}\label{ana:fourth-sup-interp}
+ \|D^4u\|_{\infty;B_1\times I}
+ \le\eps[D^4u]_{\alpha;B_2\times I}+C_\eps U.
+\end{equation}
+Here it suffices to use the spatial part of the seminorm on the
+right.  Indeed, convolution on each spatial slice with a mollifier
+of radius $q$ gives
+$\|D^4u\|_{B_1}\le Cq^\alpha[D^4u]_{\alpha;B_2}
+ +Cq^{-4}U$; choose $q$ below the spatial gap and so that
+$Cq^\alpha\le\eps$.  Neither inequality uses a temporal seminorm
+of $u_t$.
+
+\subsection{Local fourth-order estimates}
+
+Consider in a coordinate cylinder the real scalar operator
+\begin{equation}\label{ana:operator}
+ \mathcal L u=u_t+\sum_{|\nu|\le4}a_\nu(z,t)D^\nu u,
+ \qquad
+ \sum_{|\nu|=4}a_\nu(z,t)\xi^\nu\ge\lambda|\xi|^4.
+\end{equation}
+Assume $\lambda>0$ and $\sum_{|\nu|\le4}|a_\nu|_\alpha\le K$.
+These assumptions are always imposed in the coordinates and units
+of the estimate.  On a fixed smooth manifold they include the
+coefficients arising from the coordinate expression of the operator.
+The leading symbols used later are positive multiples of squared
+positive definite quadratic forms, and satisfy \eqref{ana:operator}.
+
+\begin{lemma}[Local Schauder estimates]\label{ana:schauder}
+Under these assumptions the following estimates hold.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item If $Q_R=B_R\times(-R^4,0]$ and $\mathcal L u=f$ on $Q_2$, then
+\begin{equation}\label{ana:interior-schauder}
+ \mathcal N_\alpha(u;Q_1)
+ \le C\bigl(|f|_{\alpha;Q_2}+\|u\|_{\infty;Q_2}\bigr).
+\end{equation}
+\item If $0<h\le1$, $\mathcal L u=f$ on $B_2\times[0,h]$, and
+$u(\cdot,0)=0$, then
+\begin{equation}\label{ana:initial-schauder}
+ \mathcal N_\alpha(u;B_1\times[0,h])
+ \le C\bigl(|f|_{\alpha;B_2\times[0,h]}
+                   +\|u\|_{\infty;B_2\times[0,h]}\bigr).
+\end{equation}
+Here $C$ depends only on $d,\alpha,\lambda,K$ and the fixed spatial
+gaps, and in \textup{(ii)} is independent of $h$.
+\end{enumerate}
+These are a priori estimates for classical solutions in the indicated
+spaces; smooth approximation gives the same conclusions for limits
+of such solutions.  The usual interior regularity theorem also
+applies: with smooth coefficients, distributional solutions in
+$L^2_{\rm loc}$ and H\"older sources are locally in the space
+\eqref{ana:local-norm}, and are smooth when the source is smooth.
+The estimates are from preceding times and require no terminal data.
+\end{lemma}
+
+\begin{proof}
+We give the kernel and localization argument, particularly its
+uniformity in \textup{(ii)}.  For a frozen homogeneous principal
+operator $A_0$, with symbol $p(\xi)\ge\lambda|\xi|^4$, its heat kernel is
+\[
+ k_b(z)=(2\pi)^{-d}\int_{\R^d}e^{iz\cdot\xi-bp(\xi)}\,\dd\xi,
+ \qquad b>0.
+\]
+Fourier differentiation and scaling give, uniformly for the allowed
+symbols and every $N,j,\ell$,
+\begin{equation}\label{ana:kernel}
+ |D^j\partial_b^\ell k_b(z)|
+ \le C_{N,j,\ell}b^{-(d+j)/4-\ell}
+                  (1+|z|b^{-1/4})^{-N}.
+\end{equation}
+Moreover $\int k_b=1$ and $\int D^jk_b=0$ for $j>0$.  For
+$u(t)=\int_0^tk_{t-s}*f(s)\,\dd s$ and $j=4$, subtract $f(s,z)$
+inside the convolution.  Writing $F=\sup_s[f(s)]_{\alpha;\R^d}$,
+\begin{equation}\label{ana:kernel-moment}
+ \|D^4k_b*f(s)\|_\infty\le CF b^{-1+\alpha/4},\qquad
+ \|\partial_bD^4k_b*f(s)\|_\infty
+                    \le CF b^{-2+\alpha/4}.
+\end{equation}
+This proves $\|D^4u(t)\|_\infty\le CFt^{\alpha/4}$.
+For a spatial increment of size $q$, split the lag integral at $q^4$.
+The short-lag part is bounded by
+$CF\int_0^{q^4}b^{-1+\alpha/4}\,\dd b$.
+For the long-lag part use the mean value theorem and
+$\|D^5k_b*f(s)\|_\infty\le CFb^{(\alpha-5)/4}$.
+Its bound is $CFq\int_{q^4}^h b^{(\alpha-5)/4}\,\dd b$.
+Both are at most $CFq^\alpha$, with an empty integral omitted.
+
+For a temporal increment $e=t-s>0$, keep the source time unchanged:
+\begin{align*}
+ D^4u(t)-D^4u(s)
+ ={}&\int_s^tD^4k_{t-r}*f(r)\,\dd r\\
+ &+\int_0^s(D^4k_{t-r}-D^4k_{s-r})*f(r)\,\dd r.
+\end{align*}
+The first integral and the part of the second where $s-r<e$ are
+bounded by $CF e^{\alpha/4}$.  On the remaining part,
+\eqref{ana:kernel-moment} bounds the difference by
+$CF e(s-r)^{-2+\alpha/4}$, and
+\[
+ e\int_e^s b^{-2+\alpha/4}\,\dd b\le C e^{\alpha/4}.
+\]
+This proves the full parabolic $\alpha$ bound for $D^4u$, down to
+the initial face, without any requirement that $f(0)$ vanish.
+The bounds for $u$ and lower derivatives follow from the kernel
+integrals or \eqref{ana:interp-basic}; finally
+$u_t=f-A_0u$ supplies the full parabolic norm of $u_t$.
+We have proved the constant-coefficient zero-data estimate on every
+slab of length at most one, with one constant.
+
+Here are the localization details needed to preserve that constant.
+We apply the fourth-derivative frozen estimate in the original local
+units before localization; its constant is independent of the
+localization radius.
+In a spatial ball of radius $q$ and a time window of length at most
+$q^4$, freeze the principal coefficients at one point.  Use a spatial
+cutoff supported in that ball.  For windows meeting the initial
+face use no lower-time cutoff, so the localized function still has
+zero initial value.  For interior windows insert a time cutoff
+which vanishes at the bottom and equals one on the smaller window;
+its derivatives are bounded by fixed powers of $q^{-1}$.  The
+constant-coefficient estimate then applies in both cases.  The
+localized forcing consists of the original source, terms of spatial
+order at most three, a time-cutoff multiple of $u$, and
+$(a_\nu^0-a_\nu)D^\nu u$ for $|\nu|=4$.
+The last terms obey
+\begin{equation}\label{ana:freeze-bound}
+ |(a_\nu^0-a_\nu)D^\nu u|_\alpha
+ \le CKq^\alpha|D^4u|_\alpha
+             +CK\|D^4u\|_\infty,
+\end{equation}
+with fixed cutoff factors understood.  First make $q$ small enough
+to absorb the first coefficient.  Apply
+\eqref{ana:fourth-sup-interp} to the second term.  For the lower-order
+terms apply \eqref{ana:interp-basic}, and use the equation only in
+the form
+\begin{equation}\label{ana:ut-sup}
+ \|u_t\|_\infty\le\|f\|_\infty
+                   +K\sum_{j=0}^4\|D^ju\|_\infty.
+\end{equation}
+Thus no unknown temporal H\"older seminorm of $u_t$ is used to
+absorb a temporal seminorm of $D^4u$.
+
+For completeness, the resulting nested-domain absorption can be
+expressed as follows.  Let $H(r)$ be the supremum and parabolic
+$\alpha$ norm of $D^4u$ on the spatial ball of radius $r$, with the
+given time interval in the initial case; in the interior case also
+increase the preceding-time window with $r$.  For $1\le r<R\le3/2$,
+localization, interpolation, and covering give, for any fixed
+$\vartheta>0$,
+\[
+ H(r)\le\vartheta H(R)
+       +C_\vartheta(R-r)^{-k}
+                      (|f|_\alpha+\|u\|_\infty).
+\]
+Here $k$ is fixed; all losses are powers of spatial cutoff radii,
+and none is a power of $h^{-1}$.  For H\"older pairs not sharing a
+small localization window, the quotient is bounded by the
+corresponding inverse power of that window's radius times
+$\|D^4u\|_\infty$, which is handled by
+\eqref{ana:fourth-sup-interp}.  Iterate along radii increasing
+geometrically to $3/2$, choosing $\vartheta 2^k<1$.
+The last term tends to zero since the solution norm is finite on
+that compact cylinder.  This proves the estimate for $D^4u$.
+Equations~\eqref{ana:interp-basic} and \eqref{ana:ut-sup} then give
+the lower derivatives, and the equation in full H\"older norm gives
+$|u_t|_\alpha$.  This proves \textup{(i)} and \textup{(ii)}.
+
+For the distributional regularity assertion, extend the coefficients
+and a spatially localized source from a compact coordinate cylinder
+to a flat torus, combining the principal symbol with $|\xi|^4$
+outside the coordinate patch.  This convex combination preserves
+positivity.  The closed-manifold Cauchy theorem
+\cite[Theorem~2.4]{Huang2026}, with order $r=4$ and zero data at
+an earlier time, provides a particular solution
+$w\in C^{4+\alpha,1+\alpha/4}$ near the smaller cylinder.
+There $u-w$ is a homogeneous distributional solution.
+To apply \cite[Theorem~5.1]{Hormander1961}, write the full symbol as
+$p=i\tau+a_4(x,t,\xi)+a_{\le3}(x,t,\xi)$ and set
+$R=1+|\tau|+|\xi|^4$.  On a compact coordinate set, ellipticity
+and lower-order absorption give $|p|\ge cR$ at large frequencies.
+Smoothness and polynomial degree then give
+\[
+ |\partial_{x,t}^{B}\partial_{\xi,\tau}^{A}p|
+ \le C_{A,B}(1+|\xi|+|\tau|)^{-|A|/4}|p|.
+\]
+This verifies condition HE in
+\cite[Equations~(3.1)' and~(3.2)]{Hormander1961}
+with $d=1/4$ and $M_j=1$.  Bounded frequencies are covered by
+$\widetilde p=(\sum_A|\partial_{\xi,\tau}^Ap|^2)^{1/2}
+\ge|\partial_\tau p|=1$.
+Thus $u-w$ is smooth, proving local H\"older regularity.
+For smooth forcing the same theorem directly gives smoothness of $u$.
+
+The fixed-cylinder
+Schauder estimate, including an $L^2$ input for the solution, can
+also be taken directly from \cite[Theorem~2.2\textup{(I)}]{DongZhang2015}.
+Its estimate for the top spatial derivatives is parabolic;
+the full temporal norm of $u_t$ follows here from the equation
+and the assumed temporal H\"older regularity of the coefficients
+and source.  This invocation concerns fixed interior cylinders;
+the uniform initial-face assertion was proved above.
+\end{proof}
+
+\begin{lemma}[Interior estimate with an $L^2$ input]\label{ana:l2}
+Let $U_1\Subset U_2$ be fixed bounded spatial domains and
+$b_0<b_1<b$, with $b-b_1$ bounded above.  Suppose that the
+coefficient assumptions of Lemma~\ref{ana:schauder} hold on
+$U_2\times(b_0,b)$ and that $\mathcal Lu=f$ there.  Then
+\begin{equation}\label{ana:l2-estimate}
+ \mathcal N_\alpha(u;U_1\times[b_1,b))
+ \le C\bigl(|f|_{\alpha;U_2\times(b_0,b)}
+                  +\|u\|_{L^2(U_2\times(b_0,b))}\bigr).
+\end{equation}
+The constant depends on the spatial gap, $b_1-b_0$, the upper
+bound for $b-b_1$, and the coefficient constants, but requires
+no terminal trace.  The estimate is unchanged if the upper face
+is included and the solution is defined there.  In particular it
+holds on fixed nested annuli through an open upper time endpoint.
+\end{lemma}
+
+\begin{proof}
+We recall an interpolation proof of the replacement of the supremum
+term.  On three nested interior cylinders, with a gap of parabolic
+size $a$, the estimate already proved bounds $\|D_zu\|_\infty$
+and $\|u_t\|_\infty$ on the middle cylinder by
+$Ca^{-k}(\|u\|_{\infty;\mathrm{outer}}+|f|_\alpha)$.
+Average $u$ on a preceding cylinder of radius $q<a/4$ ending at
+the point being estimated.  Its volume is a fixed multiple of
+$q^{d+4}$, so
+\[
+ |u(z,t)|\le Cq^{-(d+4)/2}\|u\|_{L^2(\mathrm{outer})}
+           +C(q+q^4)a^{-k}
+                    (\|u\|_{\infty;\mathrm{outer}}+|f|_\alpha).
+\]
+Choose $q$ to be a sufficiently small fixed multiple of a suitable
+power of $a$.  The last supremum coefficient can be made any
+prescribed $\vartheta>0$, at the cost of another fixed inverse
+power of $a$ on the other terms.  Iteration over geometrically
+decreasing gaps, with $\vartheta$ smaller than the corresponding
+geometric growth factor, bounds the inner supremum by
+$C(\|u\|_{L^2}+|f|_\alpha)$.  Lemma~\ref{ana:schauder} on one
+further nested cylinder proves \eqref{ana:l2-estimate}; finite
+covering handles $U_1$ and the time interval.  All averaging
+cylinders use preceding times.  First carry this out up to any
+closed upper face $b'<b$ on which the local norms are finite,
+then let $b'\uparrow b$.  The constants use only the fixed lower
+time buffer.  Pairs of points approaching $b$ are included by
+this limiting argument.
+\end{proof}
+
+\subsection{The outer forward problem}
+
+The following formulation permits the localization scale to tend
+to zero.  Fix a compact smooth manifold without boundary and a
+time-independent positive Lipschitz scale $s(z)$.  Assume that on
+balls of radius $4a_0s(z)$ the scale is comparable to its center
+value, with uniform comparison constants and uniformly bounded
+coordinate changes in scaled units.  We impose the coefficient
+assumptions of \eqref{ana:operator} uniformly after scaling space
+by $s(z)$ and time by $s(z)^4$; an original coefficient of spatial
+order $j$ is thus multiplied by $s(z)^{4-j}$.  The ellipticity and
+H\"older constants in these units are denoted again by $\lambda,K$.
+These hypotheses hold for $s=s_\beta$ in the later application.
+
+\begin{lemma}[Uniform zero-data forward estimate]\label{ana:short}
+Fix $p\in\R$, $0<\alpha<1$, and the preceding scale and coefficient
+constants.  There exist $\eta_0>0$ and $C<\infty$, independent of
+$\inf s$, with the following property.  If
+\[
+ 0<T\le\eta_0(\inf s)^4,\qquad
+ \mathcal Lu=f\quad\hbox{on }M\times[0,T],\qquad u(\cdot,0)=0,
+\]
+then, with $\eta=T/(\inf s)^4$,
+\begin{equation}\label{ana:short-bounds}
+ \|u\|_{X_{\alpha,p}(s)}\le C\|f\|_{Y_{\alpha,p}(s)},\qquad
+ \sup_{z,t}s(z)^{-p}|u(z,t)|
+                         \le C\eta\|f\|_{Y_{\alpha,p}(s)}.
+\end{equation}
+For each fixed positive scale and smooth coefficients the zero-data
+Cauchy problem has a unique solution with this regularity for
+$f\in Y_{\alpha,p}(s)$.  The same estimates hold on every interval
+$[0,T')$, $T'\le T$, on which the coefficients and source obey
+the stated bounds.
+\end{lemma}
+
+\begin{proof}
+Choose $\eta_0<a_0^4$.  Every defining box meets the initial
+face, and its full time interval has length at most $\eta_0$
+in center-scale units.  Apply \eqref{ana:initial-schauder} on a
+fixed spatial enlargement of each such box, rescaling its fixed
+spatial radius if necessary.  Scale comparability and the covering
+observation after Definition~\ref{ana:spaces} give
+\[
+ \|u\|_{X_{\alpha,p}(s)}
+ \le C\bigl(\|f\|_{Y_{\alpha,p}(s)}+S\bigr),\qquad
+ S=\sup_{z,t}s(z)^{-p}|u(z,t)|.
+\]
+The pointwise time-derivative bound in this norm and zero initial
+data imply
+\[
+ S\le\sup_z\frac{T}{s(z)^4}\|u\|_{X_{\alpha,p}(s)}
+   \le\eta\|u\|_{X_{\alpha,p}(s)}.
+\]
+Decrease $\eta_0$ so that $C\eta_0\le1/2$, and absorb.  This
+proves both estimates in \eqref{ana:short-bounds} with constants
+independent of the number or size of the spatial boxes.
+
+Fixed-scale existence and uniqueness follow from
+\cite[Theorem~2.4]{Huang2026}, with the trivial real line bundle,
+$r=4$, exponent $\alpha$, $L_t=-A(t)$, and zero initial data,
+where $\mathcal L=\partial_t+A(t)$.
+Its solution is $C^{4+\alpha,1+\alpha/4}$ on the closed time slab,
+without requiring $f(\cdot,0)=0$; the equation gives
+$u_t(\cdot,0)=f(\cdot,0)$.
+Constants in this fixed-scale existence argument need not be
+uniform in $\inf s$; uniformity of the asserted estimates has
+already been established locally.  Truncation before an open
+terminal face and passage to the limit prove the last statement.
+\end{proof}
+
+\begin{lemma}[Small spatial derivatives on a short interval]
+\label{ana:interpolation}
+Under the hypotheses of Lemma~\ref{ana:short}, the local scaled
+parabolic $C^\alpha$ norms of the spatial derivatives of order
+$0\le j\le3$, with weight $p-j$, satisfy
+\begin{equation}\label{ana:lower-small}
+ \sup_P\bigl|D_y^j(s_0^{-p}u\circ\Psi_P)\bigr|_{\alpha;D_P}
+ \le C\eta^{(4-j-\alpha)/4}\|f\|_{Y_{\alpha,p}(s)}.
+\end{equation}
+If the coefficient and source bounds also hold at an exponent
+$\alpha'\in(\alpha,1)$, then
+\begin{equation}\label{ana:fourth-small}
+ \|u\|_{X'_{\alpha,p}(s)}
+ \le C\eta^{(\alpha'-\alpha)/4}
+                    \|f\|_{Y_{\alpha',p}(s)}.
+\end{equation}
+Constants may depend on $\alpha'$ but remain independent of the
+scale and elapsed time.
+\end{lemma}
+
+\begin{proof}
+On a fixed enlarged scaled box, \eqref{ana:short-bounds} gives
+$U\le C\eta\|f\|_Y$ and $M\le C\|f\|_Y$ in the notation of
+\eqref{ana:interp-basic}.  That inequality proves
+\eqref{ana:lower-small}.  In particular, all derivatives entering
+a commutator of a fourth-order spatial operator with a spatial
+cutoff have a common sufficient small factor
+$\eta^{(1-\alpha)/4}$.
+
+For the second claim apply Lemma~\ref{ana:short} at exponent
+$\alpha'$, decreasing $\eta_0$ if necessary.  Every spatial
+derivative of $u$ through order four has zero initial trace.
+Indeed the solution and these derivatives are continuous to the
+initial face, and the trace function $u(\cdot,0)$ is identically
+zero.  For each $g=D_y^j(s_0^{-p}u\circ\Psi_P)$, $0\le j\le4$,
+the temporal $\alpha'/4$ seminorm and the elapsed scaled time
+therefore give
+\[
+ \|g\|_\infty\le C\eta^{\alpha'/4}
+                            \|f\|_{Y_{\alpha',p}(s)},\qquad
+ [g]_{\alpha'}\le C\|f\|_{Y_{\alpha',p}(s)}.
+\]
+On any metric space, splitting distances where the estimates
+$|g(P)-g(Q)|\le2\|g\|_\infty$ and
+$|g(P)-g(Q)|\le[g]_{\alpha'}d(P,Q)^{\alpha'}$ agree proves
+\[
+ [g]_\alpha\le
+ C\|g\|_\infty^{1-\alpha/\alpha'}
+                   [g]_{\alpha'}^{\alpha/\alpha'}.
+\]
+Apply this with the parabolic metric and take the supremum over
+boxes.  It proves \eqref{ana:fourth-small}, including temporal
+H\"older quotients of $D^4u$ with one point at the initial face.
+\end{proof}
+
+\begin{remark}\label{ana:no-time-smallness}
+The exponent gap and omission of $u_t$ in
+\eqref{ana:fourth-small} are necessary.  On a circle the exact
+zero-data solution
+\[
+ u_N(z,t)=N^{-4-a}(1-e^{-N^4t})\cos(Nz),\qquad
+ f_N(z)=N^{-a}\cos(Nz),\qquad T_N=N^{-4},
+\]
+of $u_t+\partial_z^4u=f_N$ has
+$[\partial_z^4u_N]_{\alpha}$ of order $N^{\alpha-a}$ on
+$[0,T_N]$.  This need not tend to zero when $a=\alpha$, whereas
+$a=\alpha'$ gives precisely the power in
+\eqref{ana:fourth-small}.  Moreover $u_t(0)=f_N$.
+Thus the argument gives smallness of the full spatial norm $X'$,
+while making no smallness assertion for the time derivative.
+\end{remark}
diff --git a/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/sections/profile.tex b/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/sections/profile.tex
new file mode 100644
index 0000000000000000000000000000000000000000..a86a5109f271e09b945c5f90a01786550356898b
--- /dev/null
+++ b/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/build/sections/profile.tex
@@ -0,0 +1,684 @@
+\section{A positive shrinking profile}\label{prof:section}
+\label{sec:profile}
+
+We construct the profile used in the concentration argument.  Its construction
+has two parts: a finite shooting estimate, proved by rational enclosures in
+Appendix~\ref{cert:section}, and an analytic construction on a half-line.
+The finite calculation belongs to the established framework of
+validated Taylor integration for ordinary differential
+equations~\cite{NedialkovJacksonCorliss1999}. Here the integer interval
+recurrences and complex-disk remainder bounds are given explicitly.
+The half-line solution is constructed by a contraction, whereas the
+final matching argument uses only continuity of the shooting map.
+
+\begin{theorem}[Shrinking profile]\label{prof:profile}
+There exist a constant $c>0$ and a function $G\in C^\infty([0,\infty))$
+such that
+\begin{equation}\label{prof:profile-equation}
+ Q(y\partial_y)(G-1)=\frac{y^2}{2}\left(\frac1G-\frac1c\right),
+ \qquad Q(D)=(D-1)(D+9)(D+10).
+\end{equation}
+The function $G$ is bounded above and bounded away from zero, satisfies
+$G(0)=1$ and $G'(0)<0$, and is analytic near zero.  There are real
+coefficients $g_j$, $j\ge1$, such that, for every $J,\ell\ge0$,
+\begin{equation}\label{prof:expansion}
+ \frac{\dd^\ell}{\dd y^\ell}
+ \left(G(y)-c-\sum_{j=1}^{J}g_jy^{-2j}\right)
+ =O\bigl(y^{-2(J+1)-\ell}\bigr)\qquad(y\longrightarrow\infty).
+\end{equation}
+In particular, $u\mapsto G(u^{-1/2})$ extends smoothly to $u=0$ from
+the right, with value $c$.
+
+Set $h(y)=(G(y)-1)/y$, with its smooth value at zero.  For $t<0$, the
+function
+\begin{equation}\label{prof:flat-shrinker}
+ F(t,x)=\rho^{-1}h(x/\rho),\qquad \rho=\sqrt{-t},
+\end{equation}
+solves
+\[
+ F_t+(1+xF)^2\widehat{\A}F=0,
+ \qquad \widehat{\A}=(x\partial_x^2+12\partial_x)^2.
+\]
+\end{theorem}
+
+\subsection{Reduction and the analytic family at the origin}
+
+The calculation in Section~\ref{geo:core-section} shows that
+\eqref{prof:profile-equation} makes $F$ in
+\eqref{prof:flat-shrinker} an exact solution of the flat equation.
+It remains to construct a positive $G$ with the stated behavior at
+zero and infinity.
+
+Put
+\begin{equation}\label{prof:change-variables}
+ c=(1+d)^{-1},\qquad
+ s=\left(\frac{27}{16}\right)^{1/3}\left(\frac yc\right)^{2/3},
+ \qquad G(y)=c(1+p(s)).
+\end{equation}
+Since $y\partial_y=\frac23s\partial_s$ and $G-1=c(p-d)$,
+\eqref{prof:profile-equation} becomes
+\begin{equation}\label{prof:ode}
+ \mathcal E_d(p):=
+ p'''+\frac{30}{s}p''+\frac B{s^2}p'
+       +\frac C{s^3}(p-d)+\frac p{1+p}=0,
+ \qquad B=\frac{751}{4},\quad C=-\frac{1215}{4}.
+\end{equation}
+Primes in this section henceforth denote $s$-derivatives, except where
+another variable is explicitly displayed.  For example, the identity
+$Q(D)=D^3+18D^2+71D-90$ verifies all the coefficients in
+\eqref{prof:ode}.
+
+\Needspace{4\baselineskip}
+We will construct origin solutions
+\(p(s)=d+ks^{3/2}+O(s^3)\), with two real parameters: \(d\) fixes
+\(c=(1+d)^{-1}\), and \(k\) fixes \(G'(0)\).  These parameters must
+be chosen to match \(p\) to a tail tending to zero.  The limiting
+linearized equation \(p'''+p=0\) has roots \(-1\) and
+\((1\pm i\sqrt3)/2\): one decaying direction and two real growing
+directions.
+
+The tail argument uses a constant-coefficient comparison equation
+with the same splitting.  After subtracting an explicit decaying
+approximation, we express the residual jet at \(S_0=56\) in its
+eigenbasis.  For each allowed \(d\), a contraction constructs small
+bounded tails with one real coordinate free and the other two
+determined.  Choosing the free coordinate from the origin solution leaves two
+real matching conditions to enforce by varying \((k,d)\).
+
+The affine square below makes the shot's two remaining coordinates
+close to its parameter \(z_*\).  Together with the tail bound, the
+certified estimate yields a continuous self-map of the square;
+Brouwer's theorem supplies a matching point.  All terminating
+decimals in this section denote exact rational numbers.  Define
+\begin{equation}\label{prof:parameters}
+ \begin{gathered}
+ R=5\cdot10^{-10},\qquad
+ \mathcal Z=\{z_*\in\R^2:\norm{z_*}_\infty\le R\},\qquad S_0=56,\\
+ \begin{pmatrix}k\\d\end{pmatrix}
+ =\begin{pmatrix}-.094617116666041\\ .400682721856639\end{pmatrix}
+ +10^{-8}
+ \begin{pmatrix}25190&-63731\\721855&359157\end{pmatrix}z_*.
+ \end{gathered}
+\end{equation}
+Write $k_0,d_0$ for the central values.  The square implies
+\begin{equation}\label{prof:parameter-displacement}
+ |k-k_0|\le4.44605\cdot10^{-13},\qquad
+ |d-d_0|\le5.40506\cdot10^{-12}<.011R.
+\end{equation}
+
+\begin{lemma}[Origin family]\label{prof:origin}
+For $|k-k_0|\le .1$ and $|d-d_0|\le .1$ in $\C$, there is a unique
+solution germ of \eqref{prof:ode}, analytic in $r=s^{3/2}$, of the form
+\begin{equation}\label{prof:origin-series}
+ p=d+kr+\sum_{i\ge2}a_ir^i,\qquad r=s^{3/2}.
+\end{equation}
+The series is jointly holomorphic in $r,k,d$ for $|r|<3$ and satisfies
+\begin{equation}\label{prof:origin-majorant}
+ 3|k|+\sum_{i\ge2}|a_i|3^i<.8.
+\end{equation}
+Each of $p(1),p'(1),p''(1)$ has modulus at most $2$ on this parameter
+polydisc.  For the real parameters in \eqref{prof:parameters}, $1+p>0$
+on $0\le s\le1$.
+\end{lemma}
+
+\begin{proof}
+In the variable $r=s^{3/2}$, \eqref{prof:ode} is equivalent to
+\[
+ Q(r\partial_r)(p-d)=\frac8{27}r^2\bigl((1+p)^{-1}-1\bigr).
+\]
+Consequently the coefficients of an analytic solution must satisfy
+\begin{equation}\label{prof:origin-recurrence}
+ a_i=
+ \frac{[r^{i-2}]\bigl((1+p)^{-1}-1\bigr)}
+ {(27/8)(i-1)(i+9)(i+10)},\qquad i\ge2.
+\end{equation}
+The constant and linear equations leave $d,k$ free, and this recurrence
+determines every subsequent coefficient uniquely.
+
+For convergence, use the Banach algebra of power series with norm
+$\sum_i|b_i|3^i$.  On the closed ball of series $a$ with no constant or
+linear term and norm at most $.2$, one has
+\[
+ \norm{kr+a}<.585+.2<.8,\qquad |1+d|>1.30.
+\]
+Thus inversion of $1+d+kr+a$ is given by a convergent geometric series
+in that Banach algebra and has norm at most $2$.  The map specified by
+\eqref{prof:origin-recurrence} sends this ball into the ball of radius
+$9(2+1)/440<.062$, since every denominator in that formula exceeds $440$.
+Its Lipschitz constant is at most $9\cdot 2^2/440<1$.  Iteration therefore
+gives a unique fixed point, jointly holomorphic in the parameters.  The
+strict inequalities also hold on a neighborhood of the closed parameter
+polydisc.  Moreover
+\[
+ 3|k|+\norm{a}
+ \le 3(.094617116666041+.1)+\frac{27}{440}<.8,
+\]
+which proves \eqref{prof:origin-majorant}.
+
+For $0\le r\le1$, the nonconstant sum is at most $.8r/3$.  In particular
+the real parameters in \eqref{prof:parameters} give $1+p>1.4-.8/3>0$.
+At $s=1$, a term $a_is^{3i/2}$ acquires factors $3i/2$ and
+$(3i/2)(3i/2-1)$ under one and two derivatives.  The elementary bounds
+\[
+ \sup_{i\ge1}\frac{3i/2}{3^i}=\frac12,
+ \qquad
+ \sup_{i\ge1}\frac{(3i/2)(3i/2-1)}{3^i}=\frac23
+\]
+together with \eqref{prof:origin-majorant} and $|d|<.501$ give the stated
+jet bounds.  Termwise differentiation is valid since $1<3$.
+\end{proof}
+
+\subsection{The matching coordinates}
+
+Define the four-term approximation
+\begin{equation}\label{prof:asymptotic-polynomial}
+ \begin{gathered}
+ p_{\rm a}(s;d)=\sum_{j=1}^4d_js^{-3j},\qquad
+ a=d_1=Cd,\qquad b=d_2=a^2+567a,\\
+ d_3=2ab-a^3+\frac{2025}{4}b,\qquad
+ d_4=b^2+2ad_3-3a^2b+a^4+\frac{567}{2}d_3.
+ \end{gathered}
+\end{equation}
+We choose the constant coefficients in $D_0$ at the midpoints of the
+corresponding linearized tail ranges on $[S_0,\infty)$, giving the
+uniform difference bounds used in Lemma~\ref{prof:tail}.  Let
+\begin{equation}\label{prof:coordinates}
+ \begin{gathered}
+ D_0(\lambda)=\lambda^3+\frac{15}{S_0}\lambda^2
+       +\frac{B}{2S_0^2}\lambda+1+\frac{C}{2S_0^3},\\
+ v_i=(1,\lambda_i,\lambda_i^2)^t,\qquad
+ \begin{pmatrix}p-p_{\rm a}\\p'-p_{\rm a}'\\p''-p_{\rm a}''\end{pmatrix}_{s=S_0}
+       =qv_0+\xi v_1+\bar\xi v_2.
+ \end{gathered}
+\end{equation}
+The roots are ordered so that $\lambda_0$ is real,
+$\Im\lambda_1>0$, and $\lambda_2=\bar\lambda_1$.  The exact real
+projection matrix $\Gamma$ takes a real jet to $(q,\Re\xi,\Im\xi)^t$.
+Its rational approximation used in the finite certificate is
+\begin{equation}\label{prof:projection-approximation}
+ \Lambda=10^{-5}
+ \begin{pmatrix}
+ 30756&-27384&33447\\
+ 34622&13692&-16724\\
+ -2925&-34040&-28850
+ \end{pmatrix}.
+\end{equation}
+
+\begin{lemma}[Root and convolution bounds]\label{prof:root-bounds}
+The roots in \eqref{prof:coordinates} are distinct and satisfy
+\begin{equation}\label{prof:root-boxes}
+ \begin{split}
+ -1.08659&<\lambda_0<-1.08655,\\
+ .40933&<\Re\lambda_1<.40938,\qquad
+ .8671<\Im\lambda_1<.86721.
+ \end{split}
+\end{equation}
+Every entry of $\Gamma-\Lambda$ has modulus less than $.001$.
+The absolute row sums of $\Gamma$ are less than $.92,.68,.68$,
+respectively.  With
+\begin{equation}\label{prof:convolution-constants}
+ M_i=\sum_{j=0}^2
+ \frac{|\lambda_j|^i}{|\Re\lambda_j|\,|D_0'(\lambda_j)|},\qquad 0\le i\le2,
+\end{equation}
+one has
+\[
+ M_0<1.96,\qquad M_1<1.92,\qquad M_2<1.88,
+ \qquad \frac1{\Re\lambda_1|D_0'(\lambda_1)|}<.82.
+\]
+\end{lemma}
+
+\begin{proof}
+Write $\alpha=15/56$, $\beta=B/(2\cdot56^2)$ and
+$H_0=1+C/(2\cdot56^3)$.  The exact calculations
+\[
+ \alpha^2-3\beta=-\frac{453}{25088}<0,\qquad
+ D_0(-1.08659)=-\frac{17566501197647}{343000000000000000}<0,
+\]
+\[
+ D_0(-1.08655)=\frac{187622583137}{2744000000000000}>0
+\]
+show that $D_0$ is strictly increasing on the real line and has exactly
+one real root in the indicated interval.  Write $\lambda_0=-r_0$ and
+$\lambda_1=u+iv$ with $v>0$.  Vieta's identities give
+\begin{equation}\label{prof:root-arithmetic}
+ u=\frac{r_0-\alpha}{2},\qquad
+ v^2=\frac{H_0}{r_0}-u^2,\qquad |\lambda_1|^2=\frac{H_0}{r_0}.
+\end{equation}
+In particular, if $u_-=114617/280000$ and $u_+=573113/1400000$, then
+$u_-<u<u_+$, with $.40933<u_-$ and $u_+<.40938$.  The rational inequalities
+\[
+ .8671^2<\frac{H_0}{1.08659}-u_+^2
+ \le v^2\le\frac{H_0}{1.08655}-u_-^2<.86721^2,
+ \qquad \frac{H_0}{1.08655}<.959^2
+\]
+give the other root bounds and $|\lambda_1|<.959$.
+
+For each root, the corresponding complex projection row is
+\begin{equation}\label{prof:projection-row}
+ \frac{(\lambda_j\lambda_k,-\lambda_j-\lambda_k,1)}
+ {D_0'(\lambda_i)},\qquad \{i,j,k\}=\{0,1,2\}.
+\end{equation}
+Taking the real row for $i=0$ and the real and imaginary parts for $i=1$
+gives $\Gamma$.  Put $E=(r_0+u)^2+v^2$.  The three real rows are
+\[
+ \begin{gathered}
+ \left(\frac{u^2+v^2}{E},\frac{-2u}{E},\frac1E\right),\\
+ \left(\frac{r_0(r_0+2u)}{2E},\frac uE,\frac{-1}{2E}\right),\\
+ \left(\frac{r_0[u(r_0+u)-v^2]}{2vE},
+       \frac{-(v^2+r_0^2-u^2)}{2vE},\frac{-(r_0+u)}{2vE}\right).
+ \end{gathered}
+\]
+Substituting $r_0\in[1.08655,1.08659]$ and the intervals just obtained
+for $u,v,v^2$ gives the following rational enclosures.  For each entry,
+products are enclosed by the minimum and maximum of the four endpoint
+products, and positive reciprocals by reversing the reciprocal endpoints.
+\begin{center}
+\begin{tabular}{c c c c}
+\toprule
+ & First entry & Second entry & Third entry\\
+\midrule
+$q$ & $(.3074,.3078)$ & $(-.2740,-.2737)$ & $(.3344,.3346)$\\
+$\Re\xi$ & $(.3461,.3464)$ & $(.1368,.1370)$ & $(-.1673,-.1672)$\\
+$\Im\xi$ & $(-.0294,-.0291)$ & $(-.3406,-.3402)$ & $(-.2887,-.2883)$\\
+\bottomrule
+\end{tabular}
+\end{center}
+The entrywise error from $\Lambda$ is therefore less than $.001$, and
+the row sums are less than $.9164,.6507,.6587$, respectively.
+
+Finally,
+\[
+ E>(1.08655+u_-)^2+.8671^2
+   =\frac{46876434629}{15680000000}>2.9895>1.729^2.
+\]
+Since $D_0'(-r_0)=E$ and $|D_0'(u+iv)|=2v\sqrt E$, for $i=0,1,2$ one has
+\[
+ M_i<\frac{1.08659^i}{1.08655\cdot2.9895}
+      +\frac{.959^i}{.40933\cdot.8671\cdot1.729}.
+\]
+These three rational expressions are less than $1.96,1.92,1.88$,
+respectively, by multiplication by their positive denominators.  Similarly,
+\[
+ \frac1{\Re\lambda_1|D_0'(\lambda_1)|}
+ <\frac1{2\cdot.40933\cdot.8671\cdot1.729}
+ =\frac{500000000000}{613674044347}<.82.
+\]
+\end{proof}
+
+The finite part of the construction is the following consequence of
+Proposition~\ref{cert:shoot}.  The origin solution of
+Lemma~\ref{prof:origin}, for every $z_*\in\mathcal Z$, continues to
+$s=S_0$ and satisfies
+\begin{equation}\label{prof:forward-bounds}
+ 1+p(s)>0\quad(0\le s\le S_0),\qquad
+ |q_{\rm shot}(z_*)|\le .25R,
+ \qquad
+ \norm{(\Re\xi_{\rm shot},\Im\xi_{\rm shot})-z_*}_\infty\le .13R.
+\end{equation}
+The proof in the Appendix includes the analytic interpretation of its
+rational Taylor enclosures, the propagation of all rounding and truncation
+errors, and the uniform nonlinear continuation estimate.  In particular,
+\eqref{prof:forward-bounds} is a statement about the exact solutions.
+Their endpoint jets depend continuously on $z_*$: this follows first from
+Lemma~\ref{prof:origin} at $s=1$, and then from ordinary continuous
+dependence for \eqref{prof:ode} where $s>0$ and $1+p>0$.
+The estimates from this section used in Appendix~\ref{sec:certificate}
+are the origin family and the explicit root and tail-polynomial estimates;
+all are independent of the matching step.
+
+\subsection{Construction on the half-line}
+
+\begin{lemma}[Tail residual]\label{prof:tail-residual}
+Uniformly for the parameters in \eqref{prof:parameters},
+\[
+ \sup_{s\ge56}|\mathcal E_d(p_{\rm a})(s)|<3\cdot10^{-13}.
+\]
+Each derivative of this residual tends to zero at infinity.
+If $P_{\rm a}(d)=(p_{\rm a},p_{\rm a}',p_{\rm a}'')(56;d)$, then
+\begin{equation}\label{prof:tail-parameter-bound}
+ \norm{P_{\rm a}(d)-P_{\rm a}(d_0)}_\infty\le .01|d-d_0|.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Put $t=s^{-3}$.  For a monomial $d_js^{-3j}$, the differential part
+$p'''+30s^{-1}p''+Bs^{-2}p'+Cs^{-3}p$ contributes
+$L(-3j)d_jt^{j+1}$, where
+\[
+ L(\nu)=\nu(\nu-1)(\nu-2)+30\nu(\nu-1)+B\nu+C.
+\]
+Its first four relevant values are
+\[
+ L(-3)=-567,\quad L(-6)=-2025/4,\quad
+ L(-9)=-567/2,\quad L(-12)=-243/4.
+\]
+Using $p/(1+p)=p-p^2/(1+p)$ shows that the coefficients through $t^4$
+cancel exactly for \eqref{prof:asymptotic-polynomial}.
+
+The parameter interval \eqref{prof:parameter-displacement} is contained
+in $[.40068272185,.40068272187]$.  Substituting this rational interval
+in \eqref{prof:asymptotic-polynomial} and its $d$-derivative gives
+\[
+\begin{array}{c|rrrr}
+ j&1&2&3&4\\ \hline
+ |d_j|<&122&54200&12500000&5100000000\\
+ |\partial_d d_j|<&304&100000&65000000&100000000000.
+\end{array}
+\]
+These interval bounds use just the endpoint operations described in
+Lemma~\ref{prof:root-bounds}; the coefficient formulas are polynomials.
+The first row in particular implies
+\begin{equation}\label{prof:tail-term-bounds}
+ |d_1|56^{-3}<.00070,\quad |d_2|56^{-6}<1.8\cdot10^{-6},\quad
+ |d_3|56^{-9}<8.1\cdot10^{-9},\quad |d_4|56^{-12}<3.2\cdot10^{-11}.
+\end{equation}
+Here is a finite positive-coefficient way to check the residual bound.
+Let $b=(.00070,1.8\cdot10^{-6},8.1\cdot10^{-9},3.2\cdot10^{-11})$
+and $B_*(z)=\sum_{i=1}^4b_iz^i$.  If $T_*=\sum b_i$, the sums of the
+coefficients of degrees greater than four in its second, third, and fourth
+powers are, respectively,
+\[
+ T_*^2-b_1^2-2b_1b_2-2b_1b_3-b_2^2,\qquad
+ T_*^3-b_1^3-3b_1^2b_2,\qquad T_*^4-b_1^4.
+\]
+Substitution of the displayed four rational numbers gives
+\begin{equation}\label{prof:residual-arithmetic}
+ \sum_{j>4}[z^j]B_*(z)^2<8\cdot10^{-14},\qquad
+ \sum_{j>4}[z^j]B_*(z)^3<2\cdot10^{-14},\qquad
+ \sum_{j>4}[z^j]B_*(z)^4<3\cdot10^{-15}.
+\end{equation}
+Since $B_*(1)<.001$, all powers at least five contribute at most
+$.001^5/(1-.001)$.  The remaining linear term contributes at most
+$(243/4)(3.2\cdot10^{-11})/56^3$.  Thus the residual is bounded by
+\[
+ 8\cdot10^{-14}+2\cdot10^{-14}+3\cdot10^{-15}
+ +\frac{.001^5}{1-.001}
+ +\frac{(243/4)(3.2\cdot10^{-11})}{56^3}
+ <3\cdot10^{-13}.
+\]
+These bounds hold for all $s\ge56$ because every remaining absolute
+monomial decreases there.  The residual is a rational function of
+$s^{-3}$, analytic at zero with vanishing constant term, which also
+proves the assertion about its derivatives.  For the parameter estimate,
+the factors $3j/56$ and $3j(3j+1)/56^2$ are at most one for $1\le j\le4$.
+Thus the second row of the coefficient table gives
+\[
+ \norm{\partial_dP_{\rm a}}_\infty
+ \le\frac{304}{56^3}+\frac{100000}{56^6}
+       +\frac{65000000}{56^9}+\frac{100000000000}{56^{12}}
+ <.001735<.01.
+\]
+The mean value theorem yields \eqref{prof:tail-parameter-bound}.
+\end{proof}
+
+\begin{lemma}[Tail solutions]\label{prof:tail}
+For each $d$ arising in \eqref{prof:parameters} and each real
+$|q|\le .26R$, there is a solution $p_{\rm tail}$ of \eqref{prof:ode}
+on $[56,\infty)$ such that
+\[
+ |p_{\rm tail}|<.001,
+ \qquad q_{\rm tail}=q,\qquad |\xi_{\rm tail}|<.25R.
+\]
+Its first three jet entries tend to zero at infinity.  The solution,
+as a bounded jet, and its initial coordinates depend continuously on $q,d$.
+\end{lemma}
+
+\begin{proof}
+Let $b_i=D_0'(\lambda_i)^{-1}$ and let $j$ be a bounded continuous
+real function on $[S_0,\infty)$.  Define a vector $\mathcal V=(V,V',V'')^t$
+by the convergent integrals
+\begin{equation}\label{prof:green-formula}
+ \begin{split}
+ \mathcal V(s)={}&q\e^{\lambda_0(s-S_0)}v_0
+  +b_0v_0\int_{S_0}^s\e^{\lambda_0(s-u)}j(u)\,\dd u\\
+ &-\sum_{i=1}^2 b_iv_i\int_s^\infty\e^{\lambda_i(s-u)}j(u)\,\dd u.
+ \end{split}
+\end{equation}
+Indeed this vector solves the companion system for
+$D_0(\partial_s)V=j$: the identities
+$\sum b_i=\sum\lambda_ib_i=0$ and $\sum\lambda_i^2b_i=1$ follow
+by partial fractions of $1/D_0$.  They show both the derivative relations
+between its entries and the source in the last component.  The vector is
+real by complex conjugation.  It is the unique bounded solution with the
+prescribed real-mode coordinate at $S_0$, since a bounded homogeneous
+solution has no growing nonreal modes.
+
+For $0\le i\le2$, Lemma~\ref{prof:root-bounds} gives
+\begin{equation}\label{prof:green-bound}
+ \norm{V^{(i)}}_\infty\le |\lambda_0|^i|q|+M_i\norm j_\infty.
+\end{equation}
+In particular, on the closed ball $\norm j_\infty\le .3R$, the first
+entry satisfies
+$\norm V_\infty\le(.26+1.96\cdot.3)R=.848R$.
+Together with \eqref{prof:tail-term-bounds}, this ensures
+$|p_{\rm a}+V|<.001$ throughout the half-line.
+
+Write $N(p)=p^2/(1+p)$ and $e_{\rm a}=\mathcal E_d(p_{\rm a})$.
+Then $p=p_{\rm a}+V$ solves \eqref{prof:ode} exactly when
+\begin{equation}\label{prof:tail-map}
+ \begin{split}
+ j=\mathcal T_{q,d}j:={}&-e_{\rm a}
+ -\left(\frac{30}{s}-\frac{15}{S_0}\right)V''
+ -\left(\frac B{s^2}-\frac B{2S_0^2}\right)V'\\
+ &-\left(\frac C{s^3}-\frac C{2S_0^3}\right)V
+ +N(p_{\rm a}+V)-N(p_{\rm a}).
+ \end{split}
+\end{equation}
+For $|p|\le.001$,
+\[
+ |N'(p)|=|1-(1+p)^{-2}|<.0021.
+\]
+The three coefficient differences in \eqref{prof:tail-map} have absolute
+bounds $15/S_0$, $B/(2S_0^2)$ and $|C|/(2S_0^3)$ on the entire half-line.
+It follows from \eqref{prof:green-bound} that the Lipschitz constant of
+$\mathcal T_{q,d}$ on the ball is at most
+\begin{equation}\label{prof:tail-contraction-constant}
+ \frac{15}{56}(1.88)+\frac{B}{2\cdot56^2}(1.92)
+ +\left(\frac{|C|}{2\cdot56^3}+.0021\right)(1.96)<.57.
+\end{equation}
+The corresponding coefficient of the homogeneous term is bounded by
+\begin{equation}\label{prof:tail-homogeneous-constant}
+ \frac{15}{56}(1.08659)^2+\frac{B}{2\cdot56^2}(1.08659)
+ +\frac{|C|}{2\cdot56^3}+.0021<.36.
+\end{equation}
+Consequently
+\[
+ \norm{\mathcal T_{q,d}j}_\infty
+ \le3\cdot10^{-13}+.36|q|+.57\norm j_\infty
+ \le(.0006+.36\cdot.26+.57\cdot.3)R=.2652R<.3R.
+\]
+The contraction theorem gives a unique fixed point in this ball.
+Equation~\eqref{prof:green-formula} then gives a $C^3$ solution, and
+\eqref{prof:ode} bootstraps it to a smooth solution for $s>0$.
+Its nonreal initial coordinate satisfies
+\[
+ \xi_{\rm tail}=-b_1\int_{S_0}^\infty
+                  \e^{\lambda_1(S_0-u)}j(u)\,\dd u,
+ \qquad |\xi_{\rm tail}|<.82(.3R)=.246R<.25R.
+\]
+
+All coefficients in \eqref{prof:tail-map} depend continuously on $d$
+in the relevant uniform norms, and the homogeneous vector depends
+continuously on $q$.  The common contraction constant therefore gives
+continuous dependence of the fixed point, the bounded jet, and its
+initial coordinates.  Finally, exponential convolutions of continuous
+functions tending to zero again tend to zero.  Thus
+\eqref{prof:tail-map} preserves the closed subspace of such functions;
+the residual tends to zero and all its other coefficients are bounded.
+Iteration from $j=0$ shows that the fixed point belongs to this subspace.
+Formula~\eqref{prof:green-formula} proves $p,p',p''\to0$.
+\end{proof}
+
+\subsection{Matching the two solutions}
+
+For a parameter $z_*\in\mathcal Z$, apply Lemma~\ref{prof:tail} using
+the value $d(z_*)$ and the coordinate $q_{\rm shot}(z_*)$ supplied by
+the finite shot.  This is allowed by $|q_{\rm shot}|\le .25R<.26R$.
+Define a map on the closed square by
+\[
+ \mathcal B(z_*)=z_*
+ -\bigl(\Re(\xi_{\rm shot}-\xi_{\rm tail}),
+        \Im(\xi_{\rm shot}-\xi_{\rm tail})\bigr).
+\]
+It is continuous by the continuous dependence established above and
+satisfies
+\[
+ \norm{\mathcal B(z_*)}_\infty\le(.13+.25)R=.38R<R.
+\]
+Brouwer's fixed point theorem gives $z_*\in\mathcal Z$ with
+$\xi_{\rm shot}=\xi_{\rm tail}$.  The real coordinate and the parameter
+$d$ already agree.  The three jet entries in \eqref{prof:coordinates}
+therefore agree, and ordinary uniqueness for \eqref{prof:ode} joins
+the shot and tail to one solution on $[0,\infty)$.
+
+The finite shot has $1+p>0$, and the tail has $|p|<.001$.
+Compactness of the finite interval and the uniform tail bound imply
+that $1+p$ is bounded and bounded away from zero globally.  Since
+$1+d>0$, the function $G$ in \eqref{prof:change-variables} has the same
+properties.  Near zero,
+\[
+ s^{3/2}=\frac{\sqrt{27}}{4c}y,
+ \qquad G(y)=1+\frac{\sqrt{27}}4ky+O(y^2).
+\]
+It is therefore analytic there, with
+$G'(0)=k\sqrt{27}/4<0$ by \eqref{prof:parameter-displacement}.
+This proves all assertions of Theorem~\ref{prof:profile} except for the
+differentiable expansion, which we now establish independently of the
+finite shooting estimates.
+
+\subsection{Differentiable asymptotics at every order}
+
+The distinction between a bounded remainder and a remainder in a prescribed
+polynomial weight matters here.  The following argument first proves
+uniqueness among bounded solutions and then identifies the actual
+remainder with the weighted solution.
+
+\begin{lemma}[Weighted remainder]\label{prof:weighted-remainder}
+Let $p$ be the matched solution above.  There are uniquely determined
+coefficients $d_j$, extending \eqref{prof:asymptotic-polynomial}, such that
+for every $J,\ell\ge0$,
+\begin{equation}\label{prof:s-expansion}
+ \frac{\dd^\ell}{\dd s^\ell}
+ \left(p(s)-\sum_{j=1}^{J}d_js^{-3j}\right)
+ =O\bigl(s^{-3(J+1)-\ell}\bigr).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+The limits $p,p',p''\to0$ and \eqref{prof:ode} imply that every derivative
+of $p$ is bounded on a sufficiently late half-line.  To see this without
+assuming any asymptotic expansion, solve \eqref{prof:ode} for $p'''$.
+Its right-hand side is bounded, since $p$ is separated from $-1$.
+Successively differentiating this formula proves the claim by induction:
+the coefficient functions $s^{-j}$ have bounded derivatives and all
+derivatives of $p\mapsto p/(1+p)$ are bounded on the range in question.
+
+Formal substitution of $p=\sum_{j\ge1}d_jt^j$, $t=s^{-3}$, in
+\eqref{prof:ode} determines the coefficients successively.  Indeed the
+coefficient of $t^j$ contains the new coefficient $d_j$ with coefficient
+one from $p/(1+p)$; every other term involves only previous coefficients.
+The source $-Cd/s^3$ determines $d_1=Cd$.  For
+$p_J=\sum_{j=1}^Jd_js^{-3j}$, including $p_0=0$, one consequently has
+\begin{equation}\label{prof:formal-residual}
+ e_J:=\mathcal E_d(p_J),\qquad
+ e_J^{(\ell)}(s)=O\bigl(s^{-N-\ell}\bigr),
+ \quad N=3(J+1),\quad \ell\ge0.
+\end{equation}
+This follows directly from analyticity of the finite expression in
+$s^{-3}$ near zero; no convergence of the infinite formal series is asserted.
+
+Let $W=p-p_J$.  The exact divided-difference identity
+\[
+ \frac p{1+p}-\frac{p_J}{1+p_J}
+ =\frac{W}{(1+p)(1+p_J)}
+\]
+gives the linear equation
+\begin{equation}\label{prof:remainder-equation}
+ W'''+W=-e_J-a_2W''-a_1W'-a_0W,
+\end{equation}
+where
+\[
+ a_2=\frac{30}{s},\qquad a_1=\frac B{s^2},\qquad
+ a_0=\frac C{s^3}+\frac1{(1+p)(1+p_J)}-1.
+\]
+These are now treated as given coefficient functions.  Their absolute
+sum tends to zero because $p,p_J\to0$, and all their derivatives are
+bounded on a sufficiently late half-line.
+
+The polynomial $\lambda^3+1$ has roots $-1$ and
+$(1\pm i\sqrt3)/2$.  The analogue of \eqref{prof:green-formula} on
+$[S,\infty)$, with these roots, writes the bounded solution of
+$W'''+W=j$ as a stable homogeneous vector plus a convolution vector
+$\mathcal K_Sj$.  Every root has modulus one and
+$|(\lambda^3+1)'|=3$ at the roots.  Hence
+\begin{equation}\label{prof:weighted-green}
+ \max_{0\le i\le2}\norm{(\mathcal K_Sj)_i}_\infty
+ \le\frac53\norm j_\infty.
+\end{equation}
+It also acts in the Banach space with norm
+$\norm j_N=\sup_{s\ge S}s^N|j(s)|$.  For $S\le u\le s$,
+\[
+ (s/u)^N\le\exp\bigl(N(s-u)/S\bigr),
+\]
+while for $u\ge s$ this ratio is at most one.  If $S\ge2N$, the
+forward kernel is therefore bounded by an exponential with decay rate
+$1/2$, and the two backward kernels already have decay rate $1/2$.
+It follows that
+\begin{equation}\label{prof:weighted-green-2}
+ \max_{0\le i\le2}\sup_{s\ge S}s^N| (\mathcal K_Sj)_i(s)|
+ \le2\norm j_N.
+\end{equation}
+The stable homogeneous vector $H(s)=q_S\e^{-(s-S)}(1,-1,1)^t$
+belongs to both spaces.  Here $q_S$ is chosen to be the stable coordinate
+of the actual jet $(W,W',W'')(S)$.
+
+Choose $S\ge2N$ so late that
+$2\sup_{s\ge S}(|a_0|+|a_1|+|a_2|)<1$.
+The affine equation for the source,
+\begin{equation}\label{prof:weighted-source}
+ j=-e_J-\sum_{i=0}^2a_i\bigl(H_i+(\mathcal K_Sj)_i\bigr),
+\end{equation}
+is a contraction both in the bounded norm and in the norm $\norm\cdot_N$,
+by \eqref{prof:weighted-green}--\eqref{prof:weighted-green-2}.
+It has a unique solution in each space.  The weighted solution is bounded,
+so uniqueness in the bounded space identifies the two solutions.
+On the other hand, the actual remainder has a bounded jet and bounded
+source $j=W'''+W$.  The growing-mode integrals from infinity are forced
+by boundedness, so its jet has the representation
+$H+\mathcal K_Sj$ with the chosen $q_S$.  It satisfies
+\eqref{prof:weighted-source} and hence equals the weighted solution.
+We have proved
+\begin{equation}\label{prof:preliminary-remainder}
+ W,W',W''=O(s^{-N}).
+\end{equation}
+
+Equation~\eqref{prof:remainder-equation} gives $W'''=O(s^{-N})$.
+Differentiating it repeatedly, using the bounded coefficient derivatives
+and \eqref{prof:formal-residual}, proves
+$W^{(\ell)}=O(s^{-N})$ for every fixed $\ell$.
+To obtain the extra decay for each derivative in
+\eqref{prof:s-expansion}, fix $J,\ell$ and apply this last conclusion to
+the deeper truncation $p_K$, where $K\ge J+\ell$.  Then
+\[
+ (p-p_J)^{(\ell)}
+ =(p-p_K)^{(\ell)}+
+   \sum_{j=J+1}^{K}d_j(s^{-3j})^{(\ell)}
+ =O\bigl(s^{-3(J+1)-\ell}\bigr).
+\]
+The starting point $S$ and constants may depend on the finite orders
+$J,\ell$, which is exactly what the asserted asymptotic statement requires.
+\end{proof}
+
+\begin{proof}[Completion of the proof of Theorem~\ref{prof:profile}]
+Finally $s^{-3}=(16/27)c^2y^{-2}$, so
+\eqref{prof:s-expansion} and the chain rule give
+\eqref{prof:expansion}, with
+$g_j=c\,d_j((16/27)c^2)^j$.  Repeated differentiation of
+$u\mapsto G(u^{-1/2})$ expresses each derivative as a finite sum of
+powers of $y$ times derivatives of $G$.  Subtracting sufficiently many
+terms in \eqref{prof:expansion} shows that these derivatives have their
+Taylor limits at $u=0$.  The resulting extension is $C^\infty$.
+The origin expansion gives the smooth extension of $h$, and the
+calculation in Section~\ref{geo:core-section} proves the flat
+equation in Theorem~\ref{prof:profile}, also at $x=0$ by smoothness.
+\end{proof}
diff --git a/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/paper.pdf b/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..dca7a7c630c24f391180271bf2a6defe0dd7d3bc
--- /dev/null
+++ b/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:486e9e5ff47e92a4db90c7904eb5842085de78f02f825f5b68bcc5bd2719f25d
+size 749917
diff --git a/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/verification/arithmetic_allowances.py b/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/verification/arithmetic_allowances.py
new file mode 100644
index 0000000000000000000000000000000000000000..15369caf0b01087db5025dcc9f615d52068ce058
--- /dev/null
+++ b/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/verification/arithmetic_allowances.py
@@ -0,0 +1,95 @@
+"""Independent exact-rational checks of the proof's analytic allowances.
+
+These checks are separate from, and do not replace, certificate.py.
+All comparisons use Fraction. Decimal output is descriptive only.
+Positive exponentials are bounded above by a rational Taylor sum and
+geometric bound for its positive remainder.
+"""
+from fractions import Fraction as F
+from decimal import Decimal, localcontext
+import json
+from pathlib import Path
+
+
+def exp_upper(x, degree=80):
+    x = F(x)
+    term = total = F(1)
+    for j in range(1, degree + 1):
+        term *= x / j
+        total += term
+    first_omitted = term * x / (degree + 1)
+    ratio_bound = x / (degree + 2)
+    assert 0 <= ratio_bound < 1
+    return total + first_omitted / (1 - ratio_bound)
+
+
+def dec(x):
+    with localcontext() as c:
+        c.prec = 22
+        return str(Decimal(x.numerator) / Decimal(x.denominator))
+
+
+R = F(5, 10**10)
+out = {}
+majorant = F('0.585') + F(9, 440) * (1 + 1 / (F('1.30') - F('.8')))
+assert majorant < F('.8')
+out['origin_series_majorant'] = majorant
+origin_tail = F('.8') * (F('1.5') * 61) ** 2 / 3**61
+t = F('.11') * R
+parameter_tail = 2 * (1 / (1 - t) ** 2 - 1 - 2 * t)
+# Cauchy differentiated origin tails contribute at most .12*tail per
+# parameter-row sum: 10 times the sum of the two parameter-map row norms.
+# Include both rounded first-variation columns and midpoint grid units.
+origin_error = (F(5, 10**21) + origin_tail + parameter_tail
+                + F('.12') * origin_tail * R + F(1, 10**20) * R
+                + F(3, 2**256))
+assert origin_error < F(2, 10**19)
+out.update(origin_jet_tail=origin_tail, parameter_taylor_tail=parameter_tail,
+           initial_error_upper=origin_error)
+
+e = exp_upper(F('.954') / 2)
+base = F('.01') + (F('.190') * F('.06') + F('.011') * F('1.3')
+                         + F('.015625') * F('.86') / F('.14')) / 2
+linear = F('.190') / 4
+h2 = e * base / (1 - e * linear)
+assert h2 < F('.13')
+out['H2_upper'] = h2
+disk_displacement = F('.06') / 2 + F('.13') / 8
+real_displacement = F('.06') / 8 + F('.13') / (2 * 8**2)
+assert disk_displacement < F('.1') and real_displacement < F('.01')
+out.update(disk_displacement=disk_displacement, real_substep_displacement=real_displacement)
+
+complex_jacobian = F(30) / F('31.5') + F(751, 4) / F('31.5') ** 2 + F(1215, 2) / F('31.5') ** 3 + F(1, 64) / F('.14') ** 2
+assert complex_jacobian < F('2.1')
+variation = 6 * exp_upper(F('2.1') / 2)
+assert variation < 32
+tail = F(32) * F(1, 4)**45 / (1 - F(1, 4))
+allowance = F(48) * F(1, 4)**45
+assert tail < allowance
+out.update(complex_jacobian_norm=complex_jacobian, variation_entry_upper=variation,
+           cauchy_tail_upper=tail, code_tail_allowance=allowance)
+
+real_jacobian = F(30, 32) + F(751, 4) / 32**2 + F(1215, 2) / 32**3 + F(1, 64) / F('.23')**2
+partial_transfer = exp_upper(real_jacobian)
+assert partial_transfer < F('4.3')
+product_error = 8500 * ((1 + F(4, 10**15) * 8500)**288 - 1)
+assert product_error < F('.001')
+assert 570000 + 288 * F('.001') < 580000
+assert 246000 + F(55) * F('.001') / F('.23')**3 < 250000
+out.update(real_jacobian_norm=real_jacobian, partial_transfer_upper=partial_transfer,
+           all_product_error_upper=product_error)
+
+endpoint_error = 580000 * F(1, 10**19) + 8501 * F(2, 10**19) + F('4.3') * 250000 * F('1.001') * (14 * R)**2
+intermediate = F('4.3') * (F('2.72') + F('.11')) * R + F('4.3') * F('1.001') * (14 * R)**2 / F('.23')**3
+assert endpoint_error < F('.11') * R and intermediate < 14 * R
+out.update(endpoint_error_in_R=endpoint_error / R, intermediate_error_in_R=intermediate / R)
+for name, b0, b1, limit in [('real', F('.116'), F('.92'), F('.25')), ('complex_rows', F('.001'), F('.68'), F('.13'))]:
+    value = F('.006') + F('.003') * F('.04') + b0 + F('.003') * F('2.7') + b1 * (F('.11') + F('.001') * F('.02') + F('.01') * F('.011'))
+    assert value < limit
+    out[name + '_projection_error_in_R'] = value
+
+report = {'all_exact_rational_checks_pass': True,
+          'values_decimal_descriptive_only': {k: dec(v) for k, v in out.items()}}
+dest = Path(__file__).with_suffix('.json')
+dest.write_text(json.dumps(report, indent=2) + '\n')
+print(json.dumps(report, indent=2))
diff --git a/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/verification/certificate.py b/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/verification/certificate.py
new file mode 100644
index 0000000000000000000000000000000000000000..92522eed0813f689890afd1b82674370dadae72a
--- /dev/null
+++ b/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/verification/certificate.py
@@ -0,0 +1,158 @@
+# interval recurrence
+scale=2**256
+def pt(a): return (a*scale,a*scale)
+O,I=pt(0),pt(1)
+def neg(x):return (-x[1],-x[0])
+def plus(x,y):return (x[0]+y[0],x[1]+y[1])
+def timesn(x,n):
+    return (n*x[0],n*x[1]) if n>=0 else (n*x[1],n*x[0])
+def sums(L):
+    a=b=0
+    for x,y in L:a+=x;b+=y
+    return (a,b)
+def mul(x,y):
+    p=x[0]*y[0];q=x[1]*y[0];r=x[0]*y[1];s=x[1]*y[1]
+    return (min(p,q,r,s)//scale, -(min(-p,-q,-r,-s)//scale))
+def inv(x):
+    assert x[0]*x[1]>0
+    return (scale*scale//x[1], -(-scale*scale//x[0]))
+def divn(x,n):
+    assert n>0
+    return (x[0]//n,-(-x[1]//n))
+def rat(a,b): return divn(pt(a),b)
+def mag(x):return max(x[1],-x[0])
+def mid(x):
+    a=(x[0]+x[1])//2
+    return (a,a)
+def widen(x,z):
+    a=z[1] # nonnegative
+    return (x[0]-a,x[1]+a)
+def width(x):return x[1]-x[0]
+# p and linear P
+A,B,C=pt(30),rat(751,4),rat(-1215,4)
+k=rat(-94617116666041,10**15)
+d=rat(400682721856639,10**15)
+ks=[rat(25190,10**8),rat(-63731,10**8)]
+ds=[rat(721855,10**8),rat(359157,10**8)]
+
+def origin():
+    # z stores series for W=1+p; return p,p',p'' at s=1 and variation cols
+    z=[[plus(I,d)]+ds,[k]+ks]; b=[]
+    for i in range(59):   # b_i coeff reciprocal W
+        q=recip(z,b,i)
+        den=pt(27*(i+1)*(i+11)*(i+12))
+        z.append([mul(inv(den),timesn(plus(x,O if j or i else pt(-1)),8)) for j,x in enumerate(q)])
+    ans=[]
+    for a in range(3):
+        ans.append([sums(divn(timesn(plus(v[j],O if i or j else pt(-1)),(4,6*i,3*i*(3*i-2))[a]),4) for i,v in enumerate(z))
+                    for j in range(3)])
+    assert max(width(v) for t in ans for v in t)*10**20<scale
+    return [mid(x[0]) for x in ans], [[mid(y) for y in x[1:]] for x in ans]+[ds]
+
+def recip(z,b,i):
+    # z list 1+p and tangent cols; b reciprocal
+    jlen=len(z[0]);f=inv(z[0][0])
+    first=neg(mul(f,sums(mul(z[t][0],b[i-t][0]) for t in range(1,i+1)))) if i else f
+    q=[first]
+    q+= [neg(mul(f,sums([mul(z[t][j],(first if t==0 else b[i-t][0])) for t in range(i+1)]+
+                        [mul(z[t][0],b[i-t][j]) for t in range(1,i+1)]))) for j in range(1,jlen)]
+    b.append(q)
+    return q
+
+def horner(c):
+    # step=1/8
+    x=O
+    for y in reversed(c): x=plus(y,divn(x,8))
+    return x
+
+degree=46
+def evolve(V,T,R,h):
+    # h positive integer inverse block length ell=1/h; current s h=R/8; theta=(s-sc)*h, z_j=coeff
+    # V p,p_theta,p_thetatheta; T corresponding variations, 4 cols (vs scaled and d)
+    assert all(x[0]>rat(-76,100)[1] and x[1]<rat(4,10)[0] for x in V[:1]) and mag(V[1])*100<6*scale and mag(V[2])*1000<10*scale
+    assert max(mag(v) for t in T for v in t)<6*scale
+    z=[([plus(x,I) if i==0 else x]+T[i]) if i!=2 else [divn(y,2) for y in [x]+T[i]] for i,x in enumerate(V)]
+    E=[plus(d,I),O,O,O,I] # z variable = p+1, term C(z-(1+d)), tangent
+    b=[]
+    # ratio s/ell = R/8 (could exact integer); multiply equation by 4*(ratio+theta)^3:
+    # 4 r^3 z''' +120 r^2 z'' +751 r z' -1215 (z-E) = 4/h^3 * r^3(b-1)
+    for i in range(degree-2):
+        recip(z,b,i)
+        new=[]
+        for j in range(5):
+            def F(t):
+                return plus(b[t][j],pt(-1) if j==t==0 else O) if t>=0 else O
+            # multiply equation terms by 8^3
+            x=divn(sums(timesn(F(i-p),4*(1,24,192,512)[p]*R**(3-p)) for p in range(4)),h**3)
+            # (R+8theta)^3, coefficients 3R²*8; etc
+            x=sums([x,
+                timesn(z[i+2][j],-((4*i*24+120*8)*R**2*(i+1)*(i+2))),
+                timesn(z[i+1][j],-((4*i*(i-1)*3+120*i*2+751)*64*R*(i+1))),
+                timesn(z[i][j],-512*(4*i*(i-1)*(i-2)+120*i*(i-1)+751*i-1215)),
+                timesn(E[j] if i==0 else O,-1215*512)])
+            new.append(divn(x,4*(i+1)*(i+2)*(i+3)*R**3))
+        z.append(new)
+    z[0][0]=plus(z[0][0],pt(-1))
+    # Cauchy remainder derived for all coefficients of STATE at degree degree-2 (differentiate z twice)
+    # assert analytic sup all state <32 on radius .5; ratio=.25; err ≤ 32*(4/3)*.25^(degree-1)
+    err=rat(48,4**(degree-1))
+    res=[]
+    for a in range(3):
+        res.append([widen(horner([timesn(z[i][j],(1,i,i*(i-1))[a]) for i in range(a,degree+1)]),err) for j in range(5)])
+    return [x[0] for x in res], [x[1:] for x in res]
+
+def makeflows():
+    U,Z=origin()
+    hlist=sum(([h]*32 for h in (32,16,8)),[])+[4]*(4*(56-8))
+    allm=[];mins=[]
+    for index,h in enumerate(hlist):
+        V=[divn(x,h**i) for i,x in enumerate(U)]
+        T=[[I if i==j else O for j in range(4)] for i in range(3)]
+        # scaled variations vs scaled input
+        if index==0:s32=32
+        R=(s32*h)//4; assert R*4==s32*h
+        mini=2*scale
+        for t in range(8):
+            mini=min(mini,V[0][0]+scale-(scale//100+1))  # variation across substep < .01 analytically
+            V,T=evolve(V,T,R+t,h)
+        V=[timesn(x,h**i) for i,x in enumerate(V)]
+        assert max(map(width,V))*10**19<scale
+        U=[mid(x) for x in V]
+        M=[[divn(timesn(x,h**i),h**j if j<3 else 1) for j,x in enumerate(row)] for i,row in enumerate(T)]+[[O,O,O,I]]
+        assert max(width(x) for row in M for x in row)*10**15<scale
+        allm.append([[mid(x) for x in row] for row in M])
+        mins.append((h,mini));s32+=32//h
+    return allm,mins,U,Z
+
+def mm(x,y):
+ return [[sums(mul(w,z) for w,z in zip(t,c)) for c in zip(*y)] for t in x]
+def norm(x):return max(sum(mag(z) for z in t) for t in x)
+
+def check(allm,mins,U,Z):
+ assert norm(Z)<scale//50
+ for i in range(1,len(allm)+1):
+    T=[[I if m==n else O for n in range(4)] for m in range(4)]
+    D=K=0
+    for j in reversed(range(i)):
+        h,r=mins[j];assert r>scale*23//100
+        dT=norm([t[:3] for t in T[:3]])
+        D+=dT
+        K+=dT*scale**2//(h*r**3)+1  # rounds up to integer
+        T=mm(T,allm[j])
+        assert norm(T)<=8500*scale
+    z=mm(T,Z)
+    assert norm(z)<=scale*27//10
+    assert D<=scale*570000 and K<=246000
+ d1=mul(C,d);d2=mul(d1,plus(pt(567),d1))
+ d3=sums([mul(rat(2025,4),d2),timesn(mul(d1,d2),2),neg(mul(mul(d1,d1),d1))])
+ d4=sums([mul(rat(567,2),d3),timesn(mul(d1,d3),2),mul(d2,d2),timesn(mul(mul(d1,d1),d2),-3),mul(mul(d1,d1),mul(d1,d1))])
+ pr=[[divn(timesn(x,(1,-3*m,3*m*(3*m+1))[a]),56**(3*m+a)) for a in range(3)] for m,x in enumerate([d1,d2,d3,d4],1)]
+ res=[[plus(u,neg(sums(c)))] for u,c in zip(U,zip(*pr))]
+ l=[[rat(v,10**5) for v in row] for row in [[30756,-27384,33447],[34622,13692,-16724],[-2925,-34040,-28850]]]
+ assert norm(res)<rat(1,5*10**10)[0]
+ assert norm(mm(l,res))<rat(3,10**12)[0]
+ z=mm(l,z[:3])
+ assert norm(z[:1])*1000<116*scale
+ assert norm([[plus(w,neg(I) if i==j else O) for i,w in enumerate(t)] for j,t in enumerate(z[1:])])*10**3<scale
+
+check(*makeflows())
diff --git a/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/verification/tail-check.py b/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/verification/tail-check.py
new file mode 100644
index 0000000000000000000000000000000000000000..a1da1f550befafbcf66fd20f201c0e99710ff096
--- /dev/null
+++ b/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/verification/tail-check.py
@@ -0,0 +1,180 @@
+"""Supplementary exact rational checks for the half-line profile construction.
+
+This is independent of certificate.py. All decimal strings below denote exact
+rational numbers, and every asserted inequality is an exact Fraction comparison.
+No floating point or transcendental evaluation is used.
+"""
+from fractions import Fraction as F
+
+
+class I:
+    def __init__(self, lo, hi=None):
+        self.lo = F(lo)
+        self.hi = F(lo if hi is None else hi)
+        assert self.lo <= self.hi
+
+    def __add__(self, other):
+        other = other if isinstance(other, I) else I(other)
+        return I(self.lo + other.lo, self.hi + other.hi)
+
+    __radd__ = __add__
+
+    def __neg__(self):
+        return I(-self.hi, -self.lo)
+
+    def __sub__(self, other):
+        return self + (-other if isinstance(other, I) else -F(other))
+
+    def __rsub__(self, other):
+        return -self + other
+
+    def __mul__(self, other):
+        other = other if isinstance(other, I) else I(other)
+        vals = [a * b for a in (self.lo, self.hi)
+                for b in (other.lo, other.hi)]
+        return I(min(vals), max(vals))
+
+    __rmul__ = __mul__
+
+    def __truediv__(self, other):
+        other = other if isinstance(other, I) else I(other)
+        assert not other.lo <= 0 <= other.hi
+        return self * I(1 / other.hi, 1 / other.lo)
+
+    def __rtruediv__(self, other):
+        return I(other) / self
+
+    def __pow__(self, power):
+        assert power >= 0
+        answer = I(1)
+        for _ in range(power):
+            answer = answer * self
+        return answer
+
+    def within(self, lo, hi):
+        return F(lo) < self.lo <= self.hi < F(hi)
+
+    def abs_upper(self):
+        return max(abs(self.lo), abs(self.hi))
+
+
+S = 56
+A = F(15, S)
+B = F(751, 4)
+C = -F(1215, 4)
+H = 1 + C / (2 * S**3)
+beta = B / (2 * S**2)
+D0 = lambda t: t**3 + A*t**2 + beta*t + H
+assert A*A - 3*beta == -F(453, 25088)
+assert D0(-F('1.08659')) == -F(17566501197647, 343000000000000000)
+assert D0(-F('1.08655')) == F(187622583137, 2744000000000000)
+r = I('1.08655', '1.08659')
+u = (r - A) / 2
+assert u.within('.40933', '.40938')
+v2 = H/r - u*u
+assert v2.within(F('.8671')**2, F('.86721')**2)
+v = I('.8671', '.86721')
+rho2 = H/r
+assert rho2.hi < F('.959')**2
+E = (r+u)**2 + v2
+assert E.lo > F('2.9895')
+assert F('1.729')**2 < F('2.9895')
+
+# Rows are q, Re(xi), Im(xi), acting on (V,V',V'').
+rows = [
+    [rho2/E, -2*u/E, 1/E],
+    [r*(r+2*u)/(2*E), u/E, -1/(2*E)],
+    [r*(u*(r+u)-v2)/(2*v*E),
+     -(v2+r*r-u*u)/(2*v*E), -(r+u)/(2*v*E)],
+]
+boxes = [
+    [('.3074', '.3078'), ('-.2740', '-.2737'), ('.3344', '.3346')],
+    [('.3461', '.3464'), ('.1368', '.1370'), ('-.1673', '-.1672')],
+    [('-.0294', '-.0291'), ('-.3406', '-.3402'), ('-.2887', '-.2883')],
+]
+Lambda = [[F(n, 10**5) for n in row] for row in
+          [[30756, -27384, 33447], [34622, 13692, -16724],
+           [-2925, -34040, -28850]]]
+for row, rowboxes, ref, limit in zip(rows, boxes, Lambda, ['.92', '.68', '.68']):
+    for entry, (lo, hi), center in zip(row, rowboxes, ref):
+        assert entry.within(lo, hi)
+        assert (entry-center).abs_upper() < F('.001')
+    assert sum(entry.abs_upper() for entry in row) < F(limit)
+
+M = [r.hi**i/(r.lo*F('2.9895'))
+     + F('.959')**i/(F('.40933')*F('.8671')*F('1.729'))
+     for i in range(3)]
+for value, limit in zip(M, ['1.96', '1.92', '1.88']):
+    assert value < F(limit)
+xi = 1/(2*F('.40933')*F('.8671')*F('1.729'))
+assert xi == F(500000000000, 613674044347) < F('.82')
+
+R = F(5, 10**10)
+d0 = F('.400682721856639')
+delta_d = F(721855+359157, 10**8)*R
+d = I(d0-delta_d, d0+delta_d)
+assert d.within('.40068272185', '.40068272187')
+# Use the wider manuscript interval for all coefficient and derivative bounds.
+d = I('.40068272185', '.40068272187')
+a = C*d
+b = a*a + 567*a
+c3 = 2*a*b - a**3 + F(2025, 4)*b
+c4 = b*b + 2*a*c3 - 3*a*a*b + a**4 + F(567, 2)*c3
+da = I(C)
+db = (2*a + 567)*da
+dc3 = 2*da*b + 2*a*db - 3*a*a*da + F(2025, 4)*db
+dc4 = (2*b*db + 2*da*c3 + 2*a*dc3 - 6*a*da*b - 3*a*a*db
+       + 4*a**3*da + F(567, 2)*dc3)
+coefficient_bounds = [122, 54200, 12500000, 5100000000]
+derivative_bounds = [304, 100000, 65000000, 100000000000]
+for coefficient, derivative, bound, dbound in zip(
+        [a,b,c3,c4], [da,db,dc3,dc4], coefficient_bounds, derivative_bounds):
+    assert coefficient.abs_upper() < bound
+    assert derivative.abs_upper() < dbound
+derivative_sum = sum(F(bound, S**(3*j))
+                     for j, bound in enumerate(derivative_bounds, 1))
+assert derivative_sum < F('.001735')
+terms = [F('.00070'), F('.0000018'), F('.0000000081'), F('.000000000032')]
+for j, (coefficient, coarse_bound, bound) in enumerate(
+        zip([a,b,c3,c4], coefficient_bounds, terms), 1):
+    assert coefficient.abs_upper()/S**(3*j) < bound
+    assert F(coarse_bound, S**(3*j)) < bound
+
+def shifted_linear(j):
+    t = -3*j
+    return t*(t-1)*(t-2)+30*t*(t-1)+B*t+C
+
+assert [shifted_linear(j) for j in range(5)] == [
+    C, -F(567), -F(2025,4), -F(567,2), -F(243,4)]
+t1, t2, t3, t4 = terms
+T = sum(terms)
+tails = [T*T-t1*t1-2*t1*t2-2*t1*t3-t2*t2,
+         T**3-t1**3-3*t1*t1*t2, T**4-t1**4]
+for value, limit in zip(tails, [F(8,10**14), F(2,10**14), F(3,10**15)]):
+    assert 0 < value < limit
+assert T < F('.001')
+higher = F('.001')**5/(1-F('.001'))
+linear = F(243,4)*t4/S**3
+assert higher < F(2,10**15)
+assert linear < F(12,10**15)
+assert sum(tails)+higher+linear < F(3,10**13)
+
+assert T + (F('.26')+F('1.96')*F('.3'))*R < F('.001')
+nonlinear_derivative = F('.001')*(2+F('.001'))/(1-F('.001'))**2
+assert nonlinear_derivative < F('.0021')
+lip = A*F('1.88') + beta*F('1.92') + (abs(C)/(2*S**3)+F('.0021'))*F('1.96')
+stable = A*r.hi**2 + beta*r.hi + abs(C)/(2*S**3) + F('.0021')
+assert lip == F(1777663401, 3136000000) < F('.57')
+assert stable == F(3860747328989, 10976000000000) < F('.36')
+assert F(3,10**13)/R+F('.36')*F('.26')+F('.57')*F('.3') == F('.2652') < F('.3')
+assert F('.82')*F('.3') == F('.246') < F('.25')
+
+print('All root, projection, convolution, coefficient, derivative, residual, and contraction checks passed exactly.')
+print('Projection entry enclosures:', boxes)
+print('Convolution rational bounds:', [str(value) for value in M])
+print('Nonreal response bound:', xi)
+print('Coefficient bounds on the wider d interval:', coefficient_bounds)
+print('Parameter-derivative bounds on the wider d interval:', derivative_bounds)
+print('Asymptotic parameter-derivative sum bound:', derivative_sum)
+print('Lipschitz bound:', lip)
+print('Stable-mode bound:', stable)
diff --git a/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/README.md b/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..60b875ee3e318157a8d46c19d93934fa51d597ce
--- /dev/null
+++ b/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/README.md
@@ -0,0 +1,18 @@
+# [A finitely generated counterexample to the Eilenberg–Ganea conjecture](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 23, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026,
+  author = {{OpenAI}},
+  title = {{A finitely generated counterexample to the Eilenberg--Ganea conjecture}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/paper.pdf}{OAI:A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/build/glyph-mappings.tex b/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/build/glyph-mappings.tex
new file mode 100644
index 0000000000000000000000000000000000000000..7691a5823767e8917ac719e9def2dc917506fcf5
--- /dev/null
+++ b/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/build/glyph-mappings.tex
@@ -0,0 +1,27 @@
+\input{glyphtounicode}
+\pdfglyphtounicode{arrowhookleft}{21AA}
+\pdfglyphtounicode{mapsto}{007C}
+\pdfglyphtounicode{parenleftbig}{0028}
+\pdfglyphtounicode{parenrightbig}{0029}
+\pdfglyphtounicode{vextendsingle}{007C}
+\pdfglyphtounicode{vextenddouble}{2225}
+\pdfglyphtounicode{parenleftbigg}{0028}
+\pdfglyphtounicode{parenrightbigg}{0029}
+\pdfglyphtounicode{parenleftBigg}{0028}
+\pdfglyphtounicode{parenrightBigg}{0029}
+\pdfglyphtounicode{summationtext}{2211}
+\pdfglyphtounicode{summationdisplay}{2211}
+\pdfglyphtounicode{productdisplay}{220F}
+\pdfglyphtounicode{tildewide}{0303}
+\pdfgentounicode=1
+
+\let\EGoriginalhookrightarrow\hookrightarrow
+\renewcommand{\hookrightarrow}{\mathrel{%
+  \pdfliteral direct{/Span << /ActualText <FEFF21AA> >> BDC}%
+  \EGoriginalhookrightarrow
+  \pdfliteral direct{EMC}}}
+\let\EGoriginallongmapsto\longmapsto
+\renewcommand{\longmapsto}{\mathrel{%
+  \pdfliteral direct{/Span << /ActualText <FEFF27FC> >> BDC}%
+  \EGoriginallongmapsto
+  \pdfliteral direct{EMC}}}
diff --git a/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/build/main.tex b/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/build/main.tex
new file mode 100644
index 0000000000000000000000000000000000000000..a983da7ee751afd8d58d575010547f11e84ffd5c
--- /dev/null
+++ b/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/build/main.tex
@@ -0,0 +1,51 @@
+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage[margin=1.1in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs}
+\usepackage{microtype}
+\usepackage{enumitem}
+\usepackage{tikz}
+\usepackage{float}
+\usetikzlibrary{arrows.meta,calc,positioning}
+\usepackage[hidelinks,pdfauthor={OpenAI},pdftitle={A finitely generated counterexample to the Eilenberg--Ganea conjecture}]{hyperref}
+\usepackage{bookmark}
+\input{glyph-mappings}
+\AddToHook{env/thebibliography/begin}{\phantomsection\addcontentsline{toc}{section}{References}}
+\numberwithin{equation}{section}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newcommand{\Z}{\mathbb Z}
+\newcommand{\Q}{\mathbb Q}
+\newcommand{\R}{\mathbb R}
+\newcommand{\C}{\mathbb C}
+\newcommand{\SU}{\operatorname{SU}}
+\newcommand{\cdim}{\operatorname{cd}_{\Z}}
+\newcommand{\gdim}{\operatorname{gd}}
+\newcommand{\one}{\mathbf 1}
+\newcommand{\norm}[1]{\left\lVert#1\right\rVert}
+\newcommand{\abs}[1]{\left\lvert#1\right\rvert}
+\setlist{topsep=5pt,itemsep=3pt}
+\title{A finitely generated counterexample to the\protect\\ Eilenberg--Ganea conjecture}
+\author{OpenAI}
+\date{September 23, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We construct a finitely generated residually finite group of integral cohomological dimension two and geometric dimension three, disproving the Eilenberg--Ganea conjecture.
+\end{abstract}
+\input{sections/introduction}
+\input{sections/degree}
+\input{sections/presentation}
+\input{sections/geometry}
+\input{sections/transport}
+\input{sections/conclusion}
+\bibliographystyle{plain}
+\bibliography{references}
+\end{document}
diff --git a/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/build/references.bib b/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/build/references.bib
new file mode 100644
index 0000000000000000000000000000000000000000..a9e5bf20704a658e7d97d1a689849949e58139ba
--- /dev/null
+++ b/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/build/references.bib
@@ -0,0 +1,212 @@
+
+@article{EG1957,
+  author = {Eilenberg, Samuel and Ganea, Tudor},
+  title = {On the {Lusternik--Schnirelmann} category of abstract groups},
+  journal = {Annals of Mathematics},
+  series = {Second Series},
+  volume = {65},
+  number = {3},
+  pages = {517--518},
+  year = {1957},
+  doi = {10.2307/1970062},
+  note = {\href{https://doi.org/10.2307/1970062}{doi:\nolinkurl{10.2307/1970062}}},
+  url = {https://www.jstor.org/stable/1970062}
+}
+
+@article{BB1997,
+  author = {Bestvina, Mladen and Brady, Noel},
+  title = {Morse theory and finiteness properties of groups},
+  journal = {Inventiones Mathematicae},
+  volume = {129},
+  number = {3},
+  pages = {445--470},
+  year = {1997},
+  doi = {10.1007/s002220050168},
+  note = {\href{https://doi.org/10.1007/s002220050168}{doi:\nolinkurl{10.1007/s002220050168}}},
+  url = {https://doi.org/10.1007/s002220050168}
+}
+
+@article{GR1962,
+  author = {Gerstenhaber, Murray and Rothaus, Oscar S.},
+  title = {The solution of sets of equations in groups},
+  journal = {Proceedings of the National Academy of Sciences of the United States of America},
+  volume = {48},
+  number = {9},
+  pages = {1531--1533},
+  year = {1962},
+  doi = {10.1073/pnas.48.9.1531},
+  note = {\href{https://doi.org/10.1073/pnas.48.9.1531}{doi:\nolinkurl{10.1073/pnas.48.9.1531}}},
+  url = {https://pmc.ncbi.nlm.nih.gov/articles/PMC220992/}
+}
+
+@article{Froyshov2013,
+  author = {Fr{\o}yshov, Kim A.},
+  title = {On the existence of representations of finitely presented groups in compact {Lie} groups},
+  journal = {Topology and its Applications},
+  volume = {160},
+  number = {15},
+  pages = {1987--2002},
+  year = {2013},
+  doi = {10.1016/j.topol.2013.08.003},
+  note = {\href{https://doi.org/10.1016/j.topol.2013.08.003}{doi:\nolinkurl{10.1016/j.topol.2013.08.003}}},
+  eprint = {1304.0936},
+  archivePrefix = {arXiv},
+  primaryClass = {math.GT},
+  url = {https://arxiv.org/abs/1304.0936v4}
+}
+
+@book{Hatcher2002,
+  author = {Hatcher, Allen},
+  title = {Algebraic Topology},
+  publisher = {Cambridge University Press},
+  address = {Cambridge},
+  year = {2002},
+  note = {\href{https://pi.math.cornell.edu/~hatcher/AT/ATpage.html}{Author's online edition}},
+  url = {https://pi.math.cornell.edu/~hatcher/AT/ATpage.html}
+}
+
+@book{Milnor1965,
+  author = {Milnor, John W.},
+  title = {Topology from the Differentiable Viewpoint},
+  publisher = {University Press of Virginia},
+  address = {Charlottesville},
+  year = {1965},
+  note = {Based on notes by David W. Weaver}
+}
+
+@misc{LP2026,
+  author = {Leary, Ian J. and Petrosyan, Nansen},
+  title = {Universal structure of graph product kernels},
+  year = {2026},
+  eprint = {2605.07853},
+  archivePrefix = {arXiv},
+  primaryClass = {math.GR},
+  note = {Preprint, \href{https://arxiv.org/abs/2605.07853v1}{arXiv:2605.07853v1}, 8 May 2026},
+  url = {https://arxiv.org/abs/2605.07853v1}
+}
+
+@article{Fox1954,
+  author = {Fox, Ralph H.},
+  title = {Free differential calculus. {II}. The isomorphism problem of groups},
+  journal = {Annals of Mathematics},
+  series = {Second Series},
+  volume = {59},
+  number = {2},
+  pages = {196--210},
+  year = {1954},
+  doi = {10.2307/1969686},
+  note = {\href{https://doi.org/10.2307/1969686}{doi:\nolinkurl{10.2307/1969686}}}
+}
+
+@book{Munkres2000,
+  author = {Munkres, James R.},
+  title = {Topology},
+  edition = {Second},
+  publisher = {Prentice Hall},
+  address = {Upper Saddle River, NJ},
+  year = {2000},
+  isbn = {0-13-181629-2}
+}
+
+@article{DicksLeary1999,
+  author = {Dicks, Warren and Leary, Ian J.},
+  title = {Presentations for subgroups of {Artin} groups},
+  journal = {Proceedings of the American Mathematical Society},
+  volume = {127},
+  number = {2},
+  pages = {343--348},
+  year = {1999},
+  doi = {10.1090/S0002-9939-99-04873-X},
+  eprint = {1210.6585},
+  archivePrefix = {arXiv},
+  primaryClass = {math.GR},
+  url = {https://arxiv.org/abs/1210.6585v1},
+  note = {Preprint available as \href{https://arxiv.org/abs/1210.6585v1}{arXiv:1210.6585v1}; \href{https://doi.org/10.1090/S0002-9939-99-04873-X}{doi:\nolinkurl{10.1090/S0002-9939-99-04873-X}}}
+}
+
+@article{Howie1999,
+  author = {Howie, James},
+  title = {{Bestvina--Brady} groups and the plus construction},
+  journal = {Mathematical Proceedings of the Cambridge Philosophical Society},
+  volume = {127},
+  number = {3},
+  pages = {487--493},
+  year = {1999},
+  doi = {10.1017/S0305004199003928},
+  url = {https://doi.org/10.1017/S0305004199003928},
+  note = {\href{https://doi.org/10.1017/S0305004199003928}{doi:\nolinkurl{10.1017/S0305004199003928}}}
+}
+
+@article{Whitehead1941,
+  author = {Whitehead, J. H. C.},
+  title = {On adding relations to homotopy groups},
+  journal = {Annals of Mathematics},
+  series = {Second Series},
+  volume = {42},
+  number = {2},
+  pages = {409--428},
+  year = {1941},
+  doi = {10.2307/1968907},
+  url = {https://doi.org/10.2307/1968907},
+  note = {\href{https://doi.org/10.2307/1968907}{doi:\nolinkurl{10.2307/1968907}}}
+}
+
+@phdthesis{Green1990,
+  author = {Green, Elisabeth Ruth},
+  title = {Graph products of groups},
+  school = {University of Leeds},
+  year = {1990},
+  month = mar,
+  url = {https://etheses.whiterose.ac.uk/id/eprint/236/},
+  note = {\href{https://etheses.whiterose.ac.uk/id/eprint/236/}{University of Leeds thesis}}
+}
+
+@article{HsuWise1999,
+  author = {Hsu, Tim and Wise, Daniel T.},
+  title = {On linear and residual properties of graph products},
+  journal = {Michigan Mathematical Journal},
+  volume = {46},
+  number = {2},
+  pages = {251--259},
+  year = {1999},
+  doi = {10.1307/mmj/1030132408},
+  url = {https://doi.org/10.1307/mmj/1030132408},
+  note = {\href{https://doi.org/10.1307/mmj/1030132408}{doi:\nolinkurl{10.1307/mmj/1030132408}}}
+}
+
+@book{CartierFoata1969,
+  author = {Cartier, Pierre and Foata, Dominique},
+  title = {Probl{\`e}mes combinatoires de commutation et r{\'e}arrangements},
+  series = {Lecture Notes in Mathematics},
+  volume = {85},
+  publisher = {Springer-Verlag},
+  address = {Berlin},
+  year = {1969},
+  doi = {10.1007/BFb0079468},
+  note = {Authorized electronic reedition with appendices, 2006; \href{https://doi.org/10.1007/BFb0079468}{doi:\nolinkurl{10.1007/BFb0079468}}},
+  url = {https://irma.math.unistra.fr/~foata/paper/pub11.html}
+}
+
+@article{Stallings1968,
+  author = {Stallings, John R.},
+  title = {On torsion-free groups with infinitely many ends},
+  journal = {Annals of Mathematics},
+  series = {Second Series},
+  volume = {88},
+  number = {2},
+  pages = {312--334},
+  year = {1968},
+  doi = {10.2307/1970577},
+  note = {\href{https://doi.org/10.2307/1970577}{doi:\nolinkurl{10.2307/1970577}}}
+}
+
+@article{Swan1969,
+  author = {Swan, Richard G.},
+  title = {Groups of cohomological dimension one},
+  journal = {Journal of Algebra},
+  volume = {12},
+  pages = {585--610},
+  year = {1969},
+  doi = {10.1016/0021-8693(69)90030-1},
+  note = {\href{https://doi.org/10.1016/0021-8693(69)90030-1}{doi:\nolinkurl{10.1016/0021-8693(69)90030-1}}}
+}
diff --git a/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/build/sections/conclusion.tex b/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/build/sections/conclusion.tex
new file mode 100644
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+\subsection*{Completion of the proof of Theorem~\ref{thm:main}}
+\addcontentsline{toc}{subsection}{Completion of the proof}
+
+Choose the finite acyclic flag complex and the nontrivial representation from Section~\ref{sec:geometry}. Proposition~\ref{prop:model} gives a finitely generated residually finite group $G$ of integral cohomological dimension two, together with finite lists $S,\mathcal D$ satisfying the cycle-basis hypothesis of Theorem~\ref{thm:presentation}. If a two-dimensional classifying space for $G$ existed, that theorem would supply a positive threshold $\eta$. Proposition~\ref{prop:small-labels}, applied below this threshold, supplies labels with trivial product around every translate of every word in $\mathcal D$ but nontrivial product on a closed path. This is a contradiction.
+
+The three-dimensional contractible complex $E$ of Section~\ref{sec:geometry} carries a free cellular action of $G$, so $E/G$ is a three-dimensional classifying space. Therefore $\gdim G=3$, whereas $\cdim G=2$. This completes the proof of Theorem~\ref{thm:main}.\qed
diff --git a/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/build/sections/degree.tex b/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/build/sections/degree.tex
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+\section{Degree on a conjugation-invariant part of a word fiber}
+\label{sec:degree}
+
+The comparison argument will need a solution that remains in a specified
+part of a word map's identity fiber as its coefficients vary. A global
+degree calculation does not locate that solution. When the exponent-sum matrix is nonsingular, we prove nonvanishing
+for any conjugation-invariant isolating part that contains the identity.
+
+Write $H=\SU(2)$ as the group of unit quaternions. We orient $H$ once and give $H^m$ the product orientation. A \emph{word map} $f:H^m\to H^m$ has coordinates that are words in its $m$ arguments and their inverses, without coefficients. Its exponent-sum matrix $B=(B_{ij})$ records the total exponent of the $j$th argument in the $i$th word.
+
+We use ordinary mapping degree on an isolating open set. Thus, if $M$ is a closed oriented $n$-manifold, $f:M\to M$ is smooth, and $y\notin f(\partial\mathcal O)$, then $\deg(f,\mathcal O,y)$ counts the part of $f^{-1}(y)$ in $\mathcal O$. It is defined even when $\partial\mathcal O$ is not smooth. For the compact selected fiber $K=f^{-1}(y)\cap\mathcal O$, map the local orientation class in $H_n(\mathcal O,\mathcal O\setminus K;\Z)$ to $H_n(M,M\setminus\{y\};\Z)\cong\Z$. This relative-homology construction gives excision and homotopy invariance while the boundary avoids $y$; when $y$ is regular it gives the signed count of preimages. We use the relative orientation class over a compact subset from \cite[Lemma~3.27(a)]{Hatcher2002}, together with excision. Regular fibers and the signed-degree and homotopy-invariance facts are supplied by \cite[Section~2, Lemmas~1--2; Section~5, Theorems~A--B]{Milnor1965}.
+
+The global degree formula for compact connected Lie groups goes back to Gerstenhaber and Rothaus \cite[Theorem~1]{GR1962}; for $H=\SU(2)$ it gives $\deg f=\det B$. A coefficient-free formulation also appears in \cite[Proposition~2.1]{Froyshov2013}. We need a statement about an individual invariant part of the fiber. The following proof provides that localization directly.
+
+\begin{samepage}
+\begin{lemma}[Localized word-map degree]
+\label{lem:localized-degree}
+Let $m\ge1$, and let $f:H^m\to H^m$ be a word map with $\det B\ne0$. Suppose that $\mathcal O\subset H^m$ is open, invariant under simultaneous conjugation, contains $\one$, and satisfies
+\[
+ \partial\mathcal O\cap f^{-1}(\one)=\varnothing.
+\]
+Then $\deg(f,\mathcal O,\one)\ne0$.
+\end{lemma}
+\end{samepage}
+
+\begin{proof}
+Let $J=\{e^{i\theta}:\theta\in\R\}$ be the standard maximal torus. On $J^m$, the map $f$ is the torus homomorphism with matrix $B$. In particular, its fiber over $\one$ consists of $\abs{\det B}$ points, each regular for the torus restriction, each with tangent sign $\operatorname{sign}(\det B)$. Let $b$ be the number of these points in $\mathcal O$. The identity is one of them, so $b\ge1$.
+
+Fix an odd prime $p>b$. Simultaneous conjugation by $e^{2\pi i/p}$ generates a cyclic group $C_p$ acting on $H^m$. Its fixed set is $J^m$, and the action is free off $J^m$: since $p$ is prime, a nontrivial stabilizer is the whole group. The word map is equivariant. We shall perturb it equivariantly, without changing its restriction to $J^m$, until $\one$ is regular. The fixed preimages will each contribute $\operatorname{sign}(\det B)$, while the others will contribute in multiples of $p$.
+
+\smallskip
+\noindent\emph{Regularizing the fixed preimages.}
+At a torus root $x$, use left-translated exponential coordinates in the source and exponential coordinates at $\one$ in the target. Since $x$ commutes with $J$, the coordinate actions are the same adjoint representation:
+\[
+ \R^m\oplus\C^m,
+ \qquad (u,v)\longmapsto (u,e^{4\pi i/p}v).
+\]
+Here the real directions are the $i$-directions, and each complex plane is spanned by $j,k$. An equivariant derivative has no mixed blocks, because the normal rotation has no fixed vector. Its normal block $N$ is complex linear: commuting with the nonreal scalar $e^{4\pi i/p}$ is equivalent to commuting with multiplication by $i$. The tangent block is $B$.
+
+Choose mutually disjoint invariant coordinate neighborhoods of all torus roots. Each can be chosen entirely inside $\mathcal O$ or entirely outside its closure, because no root lies on the boundary. In one such chart add
+\[
+ (u,v)\longmapsto s\chi(u,v)(0,v)
+\]
+to the coordinate expression of $f$, where $\chi$ is an invariant smooth bump function equal to one near the root. Choose $s\in\R$ arbitrarily small with $\det_{\C}(N+sI)\ne0$; only finitely many real values are excluded. This changes neither the torus restriction nor the tangent block. The resulting derivative is invertible, and its sign is
+\[
+ \operatorname{sign}(\det B)\,
+ \operatorname{sign}\det_{\R}(N+sI)
+ =\operatorname{sign}(\det B),
+\]
+since the real determinant of an invertible complex linear map is positive. The coordinate orientations can be chosen compatibly in source and target. Each fixed preimage is now isolated and regular.
+
+\smallskip
+\noindent\emph{Regularizing the remaining preimages.}
+Choose an invariant open neighborhood of each regular fixed root, with closure contained in a coordinate neighborhood where that root is the only zero. Keep the map unchanged near the closures of the chosen neighborhoods. The zero set outside their union is closed in $H^m$, hence compact. It lies in the free-action locus: a sequence of these remaining zeros approaching $J^m$ would limit to a torus root, already inside one of the chosen neighborhoods.
+
+Cover this compact set by finitely many smaller coordinate neighborhoods, each inside a larger neighborhood whose $p$ translates are disjoint. Different families of translates need not be disjoint from one another. Choose the closures of the bump supports to map into a smaller target logarithm ball, and keep the parameters small enough that every perturbed value stays in the larger exponential coordinate ball. In these target coordinates, add a vector parameter times a bump function in each larger neighborhood, and propagate the change to its translates by equivariance. Choose the bumps positive on the smaller neighborhoods. Near every remaining zero at least one parameter block therefore gives a surjective derivative to the target. By compactness, sufficiently small perturbations introduce no zeros elsewhere outside the protected fixed-root neighborhoods.
+
+The joint map of source point and parameters is consequently transverse to $\one$ near all its zeros. The regular-value theorem and Sard's theorem, applied to the projection of its zero manifold onto parameter space, give arbitrarily small parameter values for which the slice has only regular zeros. These are standard finite-dimensional results; no equivariant transversality theorem is needed on the fixed set. All perturbations can be kept uniformly small enough that $\partial\mathcal O$ stays disjoint from the fiber over $\one$, since this boundary is compact. Homotopy invariance preserves the selected degree.
+
+Every nonfixed zero of the resulting equivariant map occurs in an orbit of $p$ points. The local signs in an orbit agree: conjugation preserves the orientations of both source and target. The signed-count formula thus gives
+\[
+ \deg(f,\mathcal O,\one)
+ \equiv b\,\operatorname{sign}(\det B)\pmod p.
+\]
+The right side is nonzero modulo $p$, since $0<b<p$. The degree is therefore nonzero.
+\end{proof}
+
+\begin{remark}
+The same argument works for arbitrarily large odd primes, so it actually gives
+$\deg(f,\mathcal O,\one)=b\,\operatorname{sign}(\det B)$.
+Only nonvanishing is needed below. When there are no variables and no equations, the corresponding map is the identity of a point, with degree one.
+\end{remark}
diff --git a/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/build/sections/geometry.tex b/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/build/sections/geometry.tex
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+\section{An acyclic level model for a height kernel}\label{sec:geometry}
+
+We use the height-kernel construction of Bestvina and Brady
+\cite{BB1997}.  We give the geometric details, including the integral
+acyclicity statements that will supply the boundary basis required in
+Section~\ref{sec:presentation}.
+
+\subsection{A finite seed with quaternion monodromy}
+
+\begin{lemma}\label{lem:geometric-seed}
+There is a finite, connected, two-dimensional flag simplicial complex
+$L$ with $\widetilde H_*(L;\Z)=0$ and a nontrivial homomorphism
+$\alpha:\pi_1(L)\longrightarrow H=\SU(2)$.
+\end{lemma}
+
+\begin{proof}
+Start with the presentation complex $K$ of
+\begin{equation}\label{eq:seed-presentation}
+ \langle x,y\mid x^2y^{-5},\ x^2(xy^{-1})^{-3}\rangle.
+\end{equation}
+The presented group also occurs in Hatcher
+\cite[Example~2.38]{Hatcher2002}, with relations
+$a^5=b^3=(ab)^2$: take $a=y$ and $b=y^{-1}x$.
+Indeed, $x^2=y^5$ is central and $b$ is conjugate to $xy^{-1}$.
+We verify the acyclicity of $K$ and construct the required
+$\SU(2)$ representation directly.
+
+Its cellular boundary $C_2(K;\Z)\to C_1(K;\Z)$ has exponent-sum
+rows $(2,-5)$ and $(-1,3)$.  Their determinant is $1$, so this boundary
+is an isomorphism.  Since $K$ has one vertex and no cells above
+dimension two, it is integrally acyclic.
+
+Identify $H$ with the unit quaternions.  Put $\theta=\pi/5$ and choose
+imaginary unit quaternions $u,v$ with
+\[
+ u\mathbin{\cdot}v=\frac{1}{2\sin\theta};
+\]
+this is possible because $\sin\theta>1/2$.  Send $x$ to $u$ and $y$ to
+$\cos\theta+v\sin\theta$.  Then $x^2$ and $y^5$ both map to $-1$.
+Moreover, the real part of $u(\cos\theta-v\sin\theta)$ is $1/2$.
+A unit quaternion with real part $1/2$ has cube $-1$, so both relations
+in \eqref{eq:seed-presentation} hold.  The homomorphism is nontrivial
+because the image of $x$ is $u\ne1$.
+
+Here is a flag triangulation that avoids any assumption that a
+presentation complex is regular.  Subdivide each circle of its
+one-skeleton into at least three edges.  Express the two attaching
+maps as polygonal edge paths, and cone the boundary of each polygon
+to a new interior vertex before making the boundary identifications.
+Keep the radial edges indexed by the boundary occurrences: distinct
+occurrences of the same graph vertex give distinct radial edges.
+Every sector is now a closed triangle with three distinct vertices,
+and its closure is embedded.  Indeed, its one outer edge is embedded
+in the subdivided graph, while its two radial edges have disjoint
+interiors.  Thus this is a finite regular CW structure on $K$.
+
+Take the order complex of its nonempty cells, ordered by inclusion of
+closures.  This triangulates $K$: inductively triangulate cell
+boundaries and cone each such triangulation from an interior point
+of the cell.  Regularity makes these constructions compatible on
+common faces.  The result is two-dimensional, and it is flag, since
+pairwise comparable cells form a chain.  Call it $L$.  The resulting
+homeomorphism transfers both acyclicity and the nontrivial
+homomorphism to $L$.
+\end{proof}
+
+Fix this $L$ and $\alpha$ for the rest of the paper.  Write $V$ for
+its finite vertex set and put
+\begin{equation}\label{eq:height-kernel}
+ \begin{gathered}
+ P=A_L=
+ \left\langle a_q\ (q\in V)\ \middle|\
+ [a_q,a_r]=1\text{ if }\{q,r\}\in L\right\rangle,
+ \\
+ \lambda:P\longrightarrow\Z,\quad \lambda(a_q)=1,
+ \qquad G=\ker\lambda.
+ \end{gathered}
+\end{equation}
+The map $\lambda$ is well defined and surjective.  We shall prove all
+the properties of $G$ needed below directly.
+
+\subsection{Residual finiteness and the cubical universal cover}
+
+Residual finiteness is classical for graph products of residually finite
+groups \cite[Corollary~5.4]{Green1990}; see also
+\cite[Theorem~3.7]{HsuWise1999}. The following induction includes the
+retract-separation argument in the form needed here; compare
+\cite[Lemma~3.9(1)]{HsuWise1999}.
+
+\begin{lemma}\label{lem:graph-group-rf}
+The group associated to any finite commutation graph is residually
+finite.  In particular, $P$ and $G$ are residually finite.
+\end{lemma}
+
+\begin{proof}
+We induct on the number of vertices.  Delete one vertex, with
+generator $t$, and let $P'$ be the group on the remaining graph.
+The neighbors of the deleted vertex generate the graph group $C$.
+Both inclusions into $P'$ and into the original group are injective:
+killing the other generators gives retractions.  Let
+$r:P'\to C$ be the first of these retractions.  There is an isomorphism
+\begin{equation}\label{eq:graph-group-splitting}
+ P\cong F(P'/C)\rtimes P',
+\end{equation}
+where $P'/C$ denotes left cosets, $F(P'/C)$ is the free group on
+symbols $b_{qC}$, and $P'$ acts by left translation of their indices.
+To verify the formula, map $P'$ identically to the second factor and
+$t$ to $b_C$.  This respects the relations $[t,c]=1$ for $c\in C$.
+Conversely, send $b_{qC}$ to $qtq^{-1}$.  This is independent of the
+representative of $qC$ and respects the semidirect-product action.
+The maps are inverse on the indicated generators.
+
+By induction $P'$ is residually finite.  For $g\notin C$, we have
+$g\ne r(g)$, so some finite quotient $p$ of $P'$ satisfies
+$p(g)\ne p(r(g))$.  The map $(p,p\circ r)$ separates $g$ from $C$:
+every element of $C$ has equal coordinates, whereas $g$ does not.
+Taking a product of finitely many such maps shows that any finite
+list of distinct left cosets in $P'/C$ remains distinct in
+$D/\overline C$ for a suitable finite quotient $D$ of $P'$, where
+$\overline C$ is the image of $C$.
+
+Consider a nonidentity element $(w,q)$ of
+\eqref{eq:graph-group-splitting}.  If $q\ne1$, a finite quotient of
+$P'$ separates it from the identity.  If $q=1$, choose $D$ preserving
+the distinct indices appearing in the nonempty reduced free word
+$w$.  Its image is still nontrivial in
+$F(D/\overline C)\rtimes D$.  A free group is residually finite:
+for a nonempty reduced word of length $n$, prescribe its successive
+steps on distinct points $0,\ldots,n$.  For each free generator these
+steps prescribe an injective partial permutation, because the word
+has no adjacent inverse letters.  Complete each partial permutation
+to a permutation of this finite set.  The resulting finite
+permutation representation sends $0$ to $n$ along the word.
+Apply this fact to obtain a finite-index normal subgroup $N$ of
+$F(D/\overline C)$ not containing the image of $w$.  Intersect the
+finitely many translates of $N$ under $D$.  The intersection is
+normal, has finite index, is $D$-invariant, and still omits $w$.
+Its quotient, semidirect $D$, is the required finite quotient.
+This proves the induction.  Residual finiteness passes to subgroups,
+so it also holds for $G$.
+\end{proof}
+
+We now use the cubical construction in
+\cite[Theorem~5.12]{BB1997}, giving a direct proof of its needed
+properties. Let $S_L$ be the union, inside the coordinate torus $(S^1)^V$, of
+the coordinate subtori corresponding to the simplices of $L$,
+including the common basepoint.  Use the usual one-vertex cell
+structure on each circle.  This gives $S_L$ a cubical cell structure,
+with one $k$-cube for each clique of size $k$.  Its two-skeleton gives
+the presentation of $P$ in \eqref{eq:height-kernel}.  Since $L$ is flag
+and two-dimensional, $S_L$ has dimension three.  Let $E$ be its
+universal cover.  Identify the vertices of $E$ with $P$ so that an
+edge in the positive $q$-direction runs from $p$ to $pa_q$; deck
+transformations act by multiplication on the left.
+
+\begin{lemma}\label{lem:cubical-cover}
+The space $E$ is a locally finite, contractible, three-dimensional
+cubical complex.  Its cubes are embedded, and two intersecting cubes
+intersect in a common face.
+\end{lemma}
+
+\begin{proof}
+First we prove acyclicity of the universal cover for every finite
+commutation graph, by induction on its number of vertices.  Use
+$t,P',C$ as in the preceding proof.  Downstairs, the cubical complex
+is obtained from $S_{P'}$ by attaching
+$S_C\times[0,1]$, identifying both ends with the coordinate
+subcomplex $S_C\subset S_{P'}$.  Here $S_{P'}$ and $S_C$ denote the
+same coordinate-torus construction for those graphs.
+
+Upstairs, the components over $S_{P'}$ are its universal covers,
+indexed by left cosets $P/P'$, and the lifted cylinders are products
+of the universal cover of $S_C$ with $[0,1]$, indexed by $P/C$.
+These covering assertions follow from the injectivity of the
+subgroup inclusions.  The cylinder indexed by $gC$ joins the vertex
+spaces indexed by $gP'$ and $gtP'$.  Its incidence graph is a tree.
+Indeed, under \eqref{eq:graph-group-splitting}, write $g=wq$ with
+$w\in F(P'/C)$ and $q\in P'$.  The vertex cosets are identified with
+$w$, and this cylinder runs from $w$ to $wb_{qC}$.  The incidence
+graph is therefore the Cayley tree of the free group $F(P'/C)$.
+
+By induction every vertex space and every cylinder cross-section
+is acyclic.  The union over a finite subtree is acyclic: adjoining
+a leaf amounts to gluing in its vertex space and incident cylinder
+along one acyclic cylinder end, and the assertion follows from
+Mayer--Vietoris.  Any cellular cycle in $E$ meets only finitely many
+vertex spaces and cylinders, hence lies in such a finite-subtree
+union.  It bounds there.  This proves acyclicity of $E$.  The base
+case, the empty graph, is a point.
+
+As a universal cover, $E$ is simply connected.  A simply connected
+acyclic CW complex is contractible: a first nonzero homotopy group
+would, by the Hurewicz theorem, give a nonzero homology group, and
+the Whitehead theorem then applies to the map to a point.  These
+are the usual CW forms of the two theorems; see
+\cite[Sections~4.1--4.2]{Hatcher2002}.
+
+For completeness, consider the cubical structure itself.  A cube
+is specified by its initial vertex $p$ and a clique $\sigma$; its
+vertices are
+\[
+ p\prod_{q\in\tau}a_q,\qquad \tau\subseteq\sigma.
+\]
+They are distinct by the abelianization map
+$P\to\Z^V$.  Each lifted face is determined by its initial vertex
+and its direction subset; distinct faces of one cube have distinct
+such data, again by abelianization.  Thus every closed cube is
+embedded.  Suppose two cubes contain a vertex $p$.  Seen from $p$,
+each has a set of distinct signed directions $a_q^{\varepsilon_q}$,
+with mutually commuting underlying generators.  Abelianization
+shows that any common vertex is obtained from $p$ using only the
+equally signed directions shared by the two cubes.  These directions
+span the same face in each cube, by uniqueness of lifts.  It contains
+all common vertices and hence every common cell.  Since the
+intersection is a union of common cells, it is exactly this face.
+Every nonempty intersection contains such a vertex.  Finally, only
+finitely many signed cliques are available at a vertex, proving
+local finiteness.  The three-dimensional assertion follows from
+the existence of two-simplices in $L$.
+\end{proof}
+
+Extend $\lambda$ affinely over every cube of $E$.  On a cube with
+initial vertex $p$ and coordinates $x_1,\ldots,x_k\in[0,1]$, it is
+\begin{equation}\label{eq:affine-height}
+ \lambda(p)+x_1+\cdots+x_k.
+\end{equation}
+These formulas agree on faces, and
+$\lambda(gx)=\lambda(g)+\lambda(x)$.  In particular, $G$ preserves
+the level
+\begin{equation}\label{eq:zero-level}
+ X=\lambda^{-1}(0).
+\end{equation}
+
+The ambient complex $E$ is contractible, and its quotient by $G$ already
+gives the three-dimensional classifying space. To obtain the required
+length-two resolution, we must instead prove that the two-dimensional
+level $X$ is acyclic. This is a homological assertion about the level;
+we do not assert that the level is contractible.
+
+\subsection{Acyclicity of the level}
+
+\begin{lemma}\label{lem:acyclic-level}
+The level $X$ is a connected, integrally acyclic, two-dimensional
+simplicial complex.  Its vertices are $G$, its edges are the integer
+level diagonals of squares, and its two-simplices are integer level
+sections of three-cubes.
+\end{lemma}
+
+\begin{proof}
+Triangulate each cube of $E$ by chains in its Boolean poset of
+vertices: a simplex is a chain under the order of increasing
+coordinates.  These triangulations agree on faces.  Heights are
+strictly increasing along every such chain; in particular, no edge
+of this triangulation joins vertices of equal height.
+
+At a vertex $p$, the full subcomplex of its link spanned by vertices
+of height greater than $\lambda(p)$ is the barycentric subdivision
+of $L$.  Indeed, a higher vertex adjacent to $p$ in the triangulation
+is obtained by multiplying by the positive generators of a nonempty
+clique, and a simplex of these vertices corresponds exactly to a
+chain of such cliques.  This description remains valid when $p$ is
+an intermediate corner of a cube, since any comparable higher corner
+changes only positive directions.  We call this the ascending link.
+The descending link is described identically using negative
+directions, and is also a subdivision of $L$.  Both links are
+integrally acyclic.
+
+Let $K_n$ be the full subcomplex of the triangulation spanned by
+vertices of height at least $-n$, for $n\ge0$.  To pass from $K_n$ to
+$K_{n+1}$, add simultaneously the vertices $p$ of height $-n-1$ and
+the cones on their ascending links.  No simplex contains two of
+these new vertices.  Consequently the relative cellular chain
+complex is the direct sum, over the new vertices, of
+\[
+ C_*\bigl(\operatorname{cone}(\operatorname{Asc}(p)),
+                   \operatorname{Asc}(p);\Z\bigr).
+\]
+Each summand has zero homology, since its base is acyclic and
+nonempty.  Hence $K_n\hookrightarrow K_{n+1}$ induces an isomorphism
+on all homology groups.  This argument explicitly allows infinitely
+many vertices at each height.  The union of the $K_n$ is $E$, and
+cellular chains have finite support, so homology commutes with this
+union.  Thus $K_0\hookrightarrow E$ is a homology isomorphism.
+Using descending links in the same way, the full subcomplex on
+vertices of height at most zero is also acyclic.
+
+The closed halfspace $E_+=\lambda^{-1}([0,\infty))$ deformation
+retracts onto $K_0$.  To see this directly, write a point in a
+triangulated simplex as $\sum_i t_iv_i$ and discard the weights of
+vertices of negative height, renormalizing the others:
+\[
+ R_+\left(\sum_i t_iv_i\right)
+   =\frac{\sum_{\lambda(v_i)\ge0}t_iv_i}
+          {\sum_{\lambda(v_i)\ge0}t_i}.
+\]
+The denominator is positive on $E_+$: otherwise all vertices with
+positive weight would have negative height, contrary to the affine
+height of the point.  The linear homotopy from the point to $R_+$
+stays in the same simplex and in $E_+$.  It fixes $K_0$, and its
+formulas agree on faces.  Local finiteness gives a continuous global
+homotopy.  The corresponding construction retracts
+$E_-=\lambda^{-1}(( -\infty,0])$ onto the full subcomplex on
+nonpositive-height vertices.  Thus both closed halfspaces are
+acyclic.
+
+Cut the triangulation along height zero, and subdivide the resulting
+polytopes if desired.  The spaces $E_+$, $E_-$, and their intersection
+$X$ are then subcomplexes, so the reduced Mayer--Vietoris sequence
+applies to $E=E_+\cup E_-$.  Since $E$ and the two halfspaces are
+acyclic and $X$ contains the vertex $1$, that sequence proves
+$\widetilde H_*(X;\Z)=0$, including connectedness.
+
+For the final cell structure on $X$, return to the original cubes.
+The integer slices of $[0,1]^2$ by $x_1+x_2=m$ are a vertex or,
+for $m=1$, the diagonal joining $(1,0)$ to $(0,1)$.  The nondegenerate
+integer slices of $[0,1]^3$ are the triangles at sums $1$ and $2$.
+Slices of edges introduce no new vertices.  Formula
+\eqref{eq:affine-height} therefore gives exactly the asserted
+vertices, edges, and triangles.  The face-intersection property in
+Lemma~\ref{lem:cubical-cover} makes them a simplicial complex.
+Triangles occur by taking a three-cube whose initial vertex has
+height $-1$, so its dimension is two.  Figure~\ref{fig:level-slices}
+illustrates the positive-dimensional slices.
+\end{proof}
+
+\begin{figure}[H]
+\centering
+\begin{tikzpicture}[
+  font=\small, line join=round,
+  ambient/.style={draw=black!55,line width=.65pt},
+  level/.style={draw=blue!75!black,line width=1.3pt},
+  dot/.style={circle,fill=black,inner sep=1.5pt},
+  zero/.style={circle,fill=blue!75!black,inner sep=1.8pt}]
+  \begin{scope}[shift={(0,.4)}]
+    \coordinate (A) at (0,0);
+    \coordinate (B) at (2.6,0);
+    \coordinate (C) at (0,2.6);
+    \coordinate (D) at (2.6,2.6);
+    \draw[ambient] (A)--(B)--(D)--(C)--cycle;
+    \draw[level] (B)--(C);
+    \node[dot,label=below left:$-1$] at (A) {};
+    \node[zero,label=below right:$0$] at (B) {};
+    \node[zero,label=above left:$0$] at (C) {};
+    \node[dot,label=above right:$1$] at (D) {};
+    \node[below] at (1.3,0) {$a_q$};
+    \node[left] at (0,1.3) {$a_r$};
+  \end{scope}
+  \node at (1.3,3.7) {(a) Commuting square};
+  \node at (1.3,-.7) {$\lambda=-1+x_1+x_2$};
+  \begin{scope}[shift={(6.8,0)}]
+    \coordinate (O) at (0,0);
+    \coordinate (Q) at (2.6,0);
+    \coordinate (R) at (.8,.55);
+    \coordinate (S) at (0,2.5);
+    \coordinate (QR) at (3.4,.55);
+    \coordinate (QS) at (2.6,2.5);
+    \coordinate (RS) at (.8,3.05);
+    \coordinate (QRS) at (3.4,3.05);
+    \fill[blue!13] (Q)--(R)--(S)--cycle;
+    \draw[ambient,dashed] (O)--(R)--(QR) (R)--(RS);
+    \draw[ambient] (O)--(Q)--(QR)--(QRS)--(RS)--(S)--cycle
+      (Q)--(QS)--(S) (QS)--(QRS);
+    \draw[level] (Q)--(R)--(S)--cycle;
+    \node[dot,label=below left:$-1$] at (O) {};
+    \node[zero,label=below right:$0$] at (Q) {};
+    \node[zero,label={[xshift=2pt,yshift=1pt]above right:$0$}]
+      at (R) {};
+    \node[zero,label=above left:$0$] at (S) {};
+    \node[dot,label=right:$1$] at (QR) {};
+    \node[dot,label={[xshift=-2pt]below left:$1$}] at (QS) {};
+    \node[dot,label=above left:$1$] at (RS) {};
+    \node[dot,label=above right:$2$] at (QRS) {};
+    \node[below] at (1.3,0) {$a_q$};
+    \node[left] at (0,1.25) {$a_s$};
+    \path (O)--node[above,sloped,pos=.48] {$a_r$} (R);
+  \end{scope}
+  \node at (8.5,3.7) {(b) Commuting three-cube};
+  \node at (8.5,-.7) {$\lambda=-1+x_1+x_2+x_3$};
+  \draw[ambient] (2,-1.25)--(2.65,-1.25);
+  \node[right] at (2.75,-1.25) {ambient cube edges};
+  \draw[level] (6.8,-1.25)--(7.45,-1.25);
+  \node[right] at (7.55,-1.25) {level set $X$};
+\end{tikzpicture}
+\caption{Integer slices of cubes; vertex labels are heights. The blue
+diagonal and triangle are an edge and a two-simplex of $X$. Each
+generator increases height by one. Initial height $-2$ gives the other
+triangular slice, at $x_1+x_2+x_3=2$.}
+\label{fig:level-slices}
+\end{figure}
+
+The edge and triangular-word descriptions belong to the presentation
+framework of Dicks and Leary \cite[Theorem~1 and Proposition~2]{DicksLeary1999}.
+We now pass from the level geometry to the precise algebraic input for
+Theorem~\ref{thm:presentation}. The finite cell orbits will give finite
+alphabets, and acyclicity in both dimensions one and two will give an
+isomorphism onto the cycle module, not merely a collection of generators.
+
+\subsection{The group and its boundary basis}
+
+\begin{proposition}\label{prop:model}
+The finite flag complex $L$, its nontrivial representation
+$\alpha:\pi_1(L)\to H$, and the group $G$ of
+\eqref{eq:height-kernel} have the following properties.
+\begin{enumerate}
+\item $G$ is countable, finitely generated, and residually finite,
+      with $\cdim(G)=2$ and $\gdim(G)\le3$.
+\item $G$ acts freely on the integrally acyclic two-dimensional
+      simplicial complex $X$ of \eqref{eq:zero-level}.  The action
+      has finitely many cell orbits and one vertex orbit, identified
+      with $G$ acting on itself on the left.
+\item There is a finite generating alphabet $S$ for $G$ and a
+      finite indexed list $\mathcal D$ of words trivial in $G$ such
+      that, for $\Lambda=\Z[G]$, the graph of $X$ has edges
+      $g\to gs$ and
+      \begin{equation}\label{eq:model-resolution}
+       0\longrightarrow\Lambda^{(\mathcal D)}
+       \xrightarrow{\ \partial_2\ }\Lambda^{(S)}
+       \xrightarrow{\ \partial_1\ }\Lambda
+       \xrightarrow{\ \mathrm{aug}\ }\Z\longrightarrow0
+      \end{equation}
+      is exact, with $\partial_1(e_s)=s-1$ and
+      $\partial_2(e_z)=[z]$.  Here $[z]$ is the oriented edge chain
+      of the triangular boundary word $z$ based at $1$.
+      In particular, the chains $[z]$, $z\in\mathcal D$, are a
+      free $\Lambda$-basis of $\ker\partial_1$.
+\end{enumerate}
+\end{proposition}
+
+\begin{proof}
+The first two lemmas supply $L$, $\alpha$, and residual finiteness.
+The cubical deck action restricts to $G$ on $X$.  It preserves the
+cell structure just described, and no nonidentity element stabilizes
+a cell: such an element would preserve its unique ambient open cube
+in $E$, hence also that cube's initial vertex.  The action is free,
+being the restriction of a deck action.  Vertices of $X$ are exactly
+the elements of $G$, so there is one vertex orbit.
+
+There are only finitely many cell orbits.  An ambient cube is
+specified by its clique of directions and its initial vertex; a
+nonempty integer slice has only finitely many possible relative
+heights.  Two initial vertices of the same height differ by left
+multiplication by an element of $G$.  Hence the finite list of
+cliques and possible relative heights accounts for all the orbits.
+The quotient $Y=X/G$ is thus a finite complex with one vertex, and
+$X\to Y$ is a regular $G$-cover.  One can obtain this covering by
+restricting $E\to E/G$.
+
+Choose orientations for the edge orbits.  For each orbit choose its
+oriented lift starting at $1$, and call its endpoint $s\in G$.
+Keep different edge orbits as different letters even if their
+values coincide; this defines $S$.  Left translation identifies
+every oriented edge with $g\to gs$.  Connectedness of $X$ implies
+that its graph is connected, so $S$ generates $G$.
+
+For each triangle orbit choose an orientation, a starting corner,
+and the lift having that corner at $1$.  Reading its boundary gives
+a word $z$ on $S$, necessarily trivial in $G$.  These indexed words
+form $\mathcal D$.  The cellular chains of $X$, with the chosen cell
+orbit representatives, are precisely the free modules in
+\eqref{eq:model-resolution}.  Acyclicity makes the augmented chain
+complex exact.  In particular, $H_2(X;\Z)=0$ and the absence of
+three-cells make $\partial_2$ injective, which gives the asserted
+basis rather than only a spanning set.
+
+This free resolution proves $\cdim(G)\le2$.  To obtain the
+opposite inequality, choose a two-simplex of $L$ with vertices
+$q,r,s$.  The three corresponding commuting generators form an
+embedded $\Z^3$ in $P$: their map from $\Z^3$ is injective after
+abelianization in $\Z^V$.  Its height kernel is a subgroup
+$\Z^2\subset G$.  Cohomological dimension cannot increase upon
+restriction to a subgroup, because a free group-ring module remains
+free over the subgroup ring, and consequently projective modules
+remain projective.  The torus resolution gives
+$H^2(\Z^2;\Z)\cong\Z$, so $\cdim(G)\ge2$.
+
+Finally, $P$ is finitely generated, hence countable, and so is $G$.
+The contractible three-dimensional complex $E$ is also a free
+$G$-complex; the quotient $E/G$ is a three-dimensional $K(G,1)$.
+Thus $\gdim(G)\le3$, completing the proof.
+\end{proof}
diff --git a/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/build/sections/introduction.tex b/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/build/sections/introduction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..87ad72bcbf1b1a04b84fe59de6f4dc88d1358957
--- /dev/null
+++ b/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/build/sections/introduction.tex
@@ -0,0 +1,87 @@
+\section{Introduction}
+\label{sec:introduction}
+
+For a discrete group $\Gamma$, its integral cohomological dimension $\cdim\Gamma$ is the projective dimension of the trivial left $\Z[\Gamma]$-module $\Z$: it measures the length of the shortest projective resolution of $\Z$. Its geometric dimension $\gdim\Gamma$ is the least dimension of a connected CW complex with fundamental group $\Gamma$ and contractible universal cover. Such a complex is a \emph{classifying space}, or a $K(\Gamma,1)$. These are ordinary dimensions; no family of subgroups or proper-action variant is involved. The cellular chains of a contractible universal cover give a free resolution, so $\cdim\Gamma\le\gdim\Gamma$.
+
+The Eilenberg--Ganea theorem of 1957 identifies these dimensions when $\cdim\Gamma\ge3$. In dimension one, Stallings proved that a finitely generated group of integral cohomological dimension one is free, and Swan removed the finite-generation hypothesis \cite{Stallings1968,Swan1969}; a free group has a one-dimensional classifying space. The remaining dimension-two conjecture asks whether every group of integral cohomological dimension two has a two-dimensional classifying space \cite{EG1957}. We answer this question negatively with a finitely generated residually finite group. Here \emph{residually finite} means that every nonidentity element remains nonidentity in some finite quotient.
+
+Our group comes from a finite simplicial complex. A simplicial complex is \emph{flag} if every finite set of pairwise adjacent vertices spans a simplex. For such a complex $L$ with vertex set $V$, the associated \emph{right-angled Artin group} is
+\[
+ A_L=\left\langle a_q\ (q\in V)\ \middle|\
+ [a_q,a_r]=1\text{ if }\{q,r\}\text{ is an edge of }L\right\rangle.
+\]
+The \emph{height homomorphism} $\lambda:A_L\to\Z$ sends every standard generator $a_q$ to $1$. Its kernel is a Bestvina--Brady group. Acyclicity below always means vanishing reduced integral homology.
+
+\begin{theorem}
+\label{thm:main}
+There is a finitely generated residually finite group $G$ such that
+\[
+ \cdim G=2,\qquad \gdim G=3.
+\]
+One may take $G$ to be the height kernel in the right-angled Artin group associated to a finite acyclic flag triangulation of the presentation complex
+\begin{equation}
+\label{eq:seed-presentation-intro}
+ \langle x,y\mid x^2=y^5,\ x^2=(xy^{-1})^3\rangle.
+\end{equation}
+No two-dimensional classifying space for this $G$ exists, regardless of the number of its cells.
+\end{theorem}
+
+Section~\ref{sec:geometry} constructs the triangulation and proves the stated algebraic and geometric properties of its height kernel. Finite generation also makes $G$ countable, so restricting the conjecture to countable groups does not restore it.
+
+Proposition~\ref{prop:model} also supplies a length-two resolution of the trivial $\Z[G]$-module $\Z$ by finitely generated free $\Z[G]$-modules. A group admitting such a finite-length free resolution is said to be of \emph{type $\mathrm{FL}$}. Consequently, $G$ is of \emph{type $\mathrm{FP}_\infty$}: it admits a projective resolution with finitely generated terms in every degree. Here the length-two free resolution extends to such a projective resolution by taking all higher terms to be zero. The group is not finitely presented: $L$ has nontrivial fundamental group by Lemma~\ref{lem:geometric-seed}, and the Bestvina--Brady finite-presentability criterion applies \cite[Main Theorem, part~(3)]{BB1997}.
+
+\subsection*{History and the obstruction}
+
+Bestvina and Brady used height kernels to distinguish homological from homotopical finiteness conditions \cite[Main Theorem]{BB1997}. Their acyclic two-dimensional examples provide candidates for the dimension-two problem \cite[Example~6.3(3)]{BB1997}. For suitable flag triangulations of a spine of the Poincar\'e homology sphere, they proved that at least one of the Eilenberg--Ganea and Whitehead conjectures fails \cite[Theorem~8.7]{BB1997}. Whitehead's asphericity question asks whether every connected subcomplex of an aspherical two-dimensional CW complex is aspherical \cite{Whitehead1941}; see also \cite[Section~8]{BB1997}. The classical alternative does not itself determine the geometric dimension of a height kernel. Leary and Petrosyan give a recent conditional comparison between Coxeter and Bestvina--Brady candidates for the ordinary problem \cite[Section~5.6, Conjecture~5.20 and Corollary~5.21]{LP2026}.
+
+Dicks and Leary describe height kernels by edge generators and power relators associated to cycles \cite[Theorem~1 and Proposition~2]{DicksLeary1999}. Howie studied related Bestvina--Brady examples through plus constructions and obtained finite-rank free relation modules \cite{Howie1999}. These works provide the construction and relation-module setting for the present paper. The interface needed here is the particular free basis represented by triangular boundary chains; its exact form is displayed below and proved directly in Proposition~\ref{prop:model}.
+
+The obstacle is that a free action on an acyclic two-complex supplies a short free resolution, whereas a classifying space requires a contractible universal cover. An acyclic complex need not be contractible. Proving that one acyclic model is noncontractible therefore leaves open the possibility of a different two-dimensional classifying space. We compare the cycle module of the acyclic model with the presentation arising from \emph{any} hypothetical two-dimensional classifying space. The comparison permits arbitrary sets of generators and relators in that hypothetical model.
+
+\subsection*{The geometric model and the proof interfaces}
+
+For the complex $L$ in Theorem~\ref{thm:main}, form the union of the coordinate subtori in $(S^1)^V$ indexed by the simplices of $L$, using the common basepoint in all other coordinates. This is a cubical model for $A_L$ with a contractible three-dimensional universal cover $E$. Identify its vertices with $A_L$. The height homomorphism extends affinely over its cubes, and
+\[
+ X=\lambda^{-1}(0)\subset E
+\]
+is a two-dimensional level complex with vertex set $G$. The group $G=\ker\lambda$ acts freely on $X$ by left translation, with one vertex orbit and finitely many cell orbits. Proposition~\ref{prop:model} proves that $X$ is acyclic and that $G$ is finitely generated and residually finite, with $\cdim G=2$ and $\gdim G\le3$. These are the geometric inputs to the obstruction; their proofs are included in Section~\ref{sec:geometry}.
+
+Here is the exact algebraic information supplied by $X$. Choose the lift based at $1$ of one oriented edge in each $G$-orbit. Its endpoint is the value of the corresponding letter in a finite generating alphabet $S$ for $G$; distinct edge orbits retain distinct letters. Reading the boundaries of representatives of the triangular two-cell orbits gives a finite indexed list $\mathcal D$ of words on $S$. Write parenthesized superscripts for direct sums. For $\Lambda=\Z[G]$, let $\partial_1:\Lambda^{(S)}\to\Lambda$ be the cellular edge boundary and let $[z]$ be the edge chain of a word $z\in\mathcal D$. The canonical map
+\[
+ \Lambda^{(\mathcal D)}\longrightarrow\ker\partial_1,
+ \qquad e_z\longmapsto[z],
+\]
+is an isomorphism. Thus the specified boundary chains form a free basis, with both spanning and independence. The words in $\mathcal D$ are trivial in $G$, but they need not be a presentation of $G$; this distinction is the reason for the comparison argument.
+
+We use labels in $H=\SU(2)$, identified with the unit quaternions. An edge labeling assigns an element of $H$ to each oriented edge and the inverse element to its reverse. A \emph{path product} multiplies the labels in traversal order; for a closed path we also call this its \emph{holonomy}. A labeling is \emph{flat on the triangles} when every triangular boundary has product $1$. The two main interfaces give incompatible conclusions about these labels:
+\begin{enumerate}
+\item Theorem~\ref{thm:presentation} shows that a hypothetical two-dimensional $K(G,1)$ would force every sufficiently small labeling, flat on all translates of the words in $\mathcal D$, to have trivial product along every closed path.
+\item Proposition~\ref{prop:small-labels} constructs, for every $\epsilon>0$, labels of distance less than $\epsilon$ from $1$ that are flat on every triangle of $X$ and have nontrivial product along some closed path.
+\end{enumerate}
+Choosing the second labeling below the threshold in the first statement excludes every two-dimensional classifying space. The quotient $E/G$ supplies a three-dimensional one.
+
+\subsection*{How the comparison and transport work}
+
+Boundary chains record signed edge traversals but forget their order, so equal chains need not give equal path products in $H$. Theorem~\ref{thm:presentation} overcomes this loss of information using an actual aspherical presentation. After adding generators with defining relators, we change the relator words so that a finite distinguished list has exactly the boundary chains $[z]$ supplied by $X$; all additional generators represent the identity of $G$ and therefore give loop edges. The word identities in this presentation yield exact translated signed counts. Once the remaining relator equations hold, these counts cancel the linear discrepancy between the distinguished products and the triangular products. Commutators of elements close to $1$ contribute only a quadratic error. The construction works even if the hypothetical presentation has infinitely or uncountably many generators and relators.
+
+This estimate prevents the distinguished products from reaching a fixed positive distance from $1$ while the prescribed edge labels remain sufficiently small. On a finite quotient, each finite collection of the remaining relator equations has independent vectors recording the total exponents of the unknown labels. The word-equation degree method of Gerstenhaber and Rothaus \cite{GR1962}, with related representation-theoretic applications in Fr\o yshov \cite{Froyshov2013}, supplies the starting point for solving these systems. At the coefficient-free starting system, Lemma~\ref{lem:localized-degree} gives nonzero degree on the conjugation-invariant region where the distinguished products are small. The quadratic barrier preserves this degree as the prescribed labels move to their required values.
+
+Residual finiteness allows us to copy any finite collection of prescribed labels to a suitable quotient. Compactness then gives labels on all additional edges of the original graph. Since the original triangular products are exactly $1$, the same quadratic estimate forces the distinguished products to be exactly $1$ as well. All relators of an actual presentation now vanish, so every closed path has trivial product. The original labeling need not descend to any single finite quotient.
+
+The transport construction contradicts this conclusion. Reduced words in $A_L$ give maps from the graph of $X$ to $L$ whose edge and triangle-boundary images are uniformly small, while special long paths map exactly around prescribed loops of $L$. The seed in Equation~\eqref{eq:seed-presentation-intro} supplies a nontrivial representation $\pi_1(L)\to\SU(2)$. Pulling its transport back along these maps gives small edge labels: the small triangle images force trivial triangular products, and an exactly traced loop detected by the representation retains nontrivial holonomy. Section~\ref{sec:transport} constructs the maps and labels explicitly.
+
+\subsection*{Conventions and organization}
+
+We measure distance in $H$ by the quaternion norm $\abs{a-b}$; this metric is bi-invariant. Our traversal-order convention makes the product along a path followed by another path the product of their respective labels in that order. We use
+\begin{equation}
+\label{eq:quaternion-basic}
+ \abs{a_1\cdots a_k-1}\le\sum_{j=1}^k\abs{a_j-1},
+ \qquad
+ \abs{ab-ba}\le2\abs{a-1}\abs{b-1}.
+\end{equation}
+The first inequality follows by telescoping; the second follows by expanding
+$ab-ba=(a-1)(b-1)-(b-1)(a-1)$.
+
+Group rings act from the left. Generator edges run from $g$ to $gs$, and translating a path by $g$ means left translation. We work in ordinary set theory with choice, including compactness of products indexed by arbitrary sets.
+
+Section~\ref{sec:degree} proves the localized degree statement. Section~\ref{sec:presentation} derives the presentation obstruction, including its arbitrary-cardinality comparison and compactness steps. Section~\ref{sec:geometry} constructs the seed, the group and the acyclic level model, and Section~\ref{sec:transport} constructs the small labels and completes the contradiction. Standard CW topology, covering-space theory, and finite-dimensional degree theory enter with their stated hypotheses at the relevant steps.
diff --git a/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/build/sections/presentation.tex b/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/build/sections/presentation.tex
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+++ b/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/build/sections/presentation.tex
@@ -0,0 +1,510 @@
+\section{Comparison with an arbitrary aspherical presentation}
+\label{sec:presentation}
+
+Let $\Gamma$ be a group with a finite generating alphabet $S$.
+Distinct letters are kept distinct, even if they have the same value in
+$\Gamma$.  Its generator graph has vertices $g\in\Gamma$ and an oriented
+edge $(g,s)$ from $g$ to $gs$ for each $s\in S$.
+Put $\Lambda=\Z[\Gamma]$, acting on chains on the left.  For a word $w$
+on $S$, write $\bar w$ for its value in $\Gamma$ and $[w]$ for the chain
+of the path starting at $1$.  Thus
+\[
+ \partial_1:\Lambda^{(S)}\longrightarrow\Lambda,
+ \qquad \partial_1(e_s)=s-1,
+ \qquad \partial_1[w]=\bar w-1.
+\]
+Here and throughout this section, a parenthesized superscript denotes a
+\emph{direct sum}.  All words and all chains have finite support.
+
+\begin{theorem}[The presentation comparison theorem]
+\label{thm:presentation}
+Suppose that $\Gamma$ is residually finite and that $\mathcal D$ is a
+finite indexed list of words on $S$, trivial in $\Gamma$, whose chains
+form a free $\Lambda$-basis of $\ker\partial_1$.  Thus the canonical map
+\begin{equation}\label{eq:cycle-basis}
+ \Lambda^{(\mathcal D)}\longrightarrow\ker\partial_1,
+ \qquad e_z\longmapsto[z],
+\end{equation}
+is an isomorphism.
+If $\Gamma$ admits a $K(\Gamma,1)$ of dimension at most two, there is an
+$\eta>0$ with the following property.  Assign a label in $H=\SU(2)$ to
+every oriented generator edge, assign the inverse label to its reverse,
+and assume that every label has distance less than $\eta$ from $1$.
+If the product along every translate of every $z\in\mathcal D$ is $1$,
+then the product along every closed edge path is $1$.
+\end{theorem}
+
+The words in $\mathcal D$ need not present $\Gamma$.  This distinction is
+the reason a comparison argument is necessary.  We first arrange an
+arbitrary aspherical presentation so that its relators include a finite
+list with chains exactly $[z]$.  The remaining relators give independent
+word equations in additional loop labels.  A uniform quadratic estimate
+isolates the small values of the distinguished relators.  We solve finite
+systems on finite group quotients by Lemma~\ref{lem:localized-degree},
+and then use compactness to obtain an extension of the original labels.
+The presentation used in this argument can have any cardinality.
+
+\subsection{A presentation with distinguished boundary chains}
+
+The correspondence between relator operations and group-ring boundary
+operations is classical; compare Fox's transformation~(III) in
+\cite[Section~2]{Fox1954}. We give the word realization explicitly
+because the presentation here may have arbitrary index sets.
+
+\begin{lemma}\label{lem:aligned-presentation}
+Under the hypotheses of Theorem~\ref{thm:presentation}, including the
+existence of the two-dimensional $K(\Gamma,1)$, there is a presentation
+\[
+ \Gamma=\langle S,\mathcal T\mid
+        (\rho_z)_{z\in\mathcal D},\ (r)_{r\in\mathcal R}\rangle
+\]
+with the following properties:
+\begin{enumerate}
+\item every letter of $\mathcal T$ has value $1$ in $\Gamma$;
+\item the complete relator boundary list is a $\Lambda$-basis of the
+cycle module of the generator graph on $S\amalg\mathcal T$;
+\item $[\rho_z]=[z]$ as edge chains, for each $z\in\mathcal D$;
+\item projecting the ordinary boundaries $[r]$, $r\in\mathcal R$, to
+the $\mathcal T$-edge coordinates gives an isomorphism
+\begin{equation}\label{eq:ordinary-projection}
+ \Lambda^{(\mathcal R)}\xrightarrow{\ \cong\ }
+ \Lambda^{(\mathcal T)}.
+\end{equation}
+\end{enumerate}
+No finiteness assumption is imposed on $\mathcal T$ or $\mathcal R$.
+\end{lemma}
+
+\begin{proof}
+Start with any two-dimensional $K(\Gamma,1)$.  Collapse a maximal tree
+in its connected one-skeleton and replace the two-cell attaching loops
+by finite edge words.  These operations preserve the homotopy type:
+a contractible CW subcomplex can be collapsed, and homotopic attaching
+maps give homotopy equivalent adjunction spaces
+\cite[Propositions~0.17 and~0.18]{Hatcher2002}.  These facts apply to
+infinite CW complexes as well; a loop in a graph is homotopic to a finite
+edge path.  The resulting presentation complex has one vertex, with
+some sets of generators and relators.  Its universal cover is acyclic
+and has no cells above dimension two.  Consequently its relator boundary
+map is an isomorphism onto the cycle module.  The relator words also
+normally generate the kernel of the map from the free generator group
+to $\Gamma$.
+
+Adjoin the prescribed letters $S$, with one definition relator for each
+letter.  Each new relator boundary has coefficient $1$ on its own new
+edge and coefficient $0$ on the other new edges.  Subtracting these
+boundaries reduces any cycle uniquely to an old cycle, so the boundary
+basis property persists.  For each former generator $a$, choose a word
+$w_a(S)$ with value $a$ in $\Gamma$, and replace $a$ by $t_a w_a(S)$.
+This is an invertible change of free generators: its inverse sends
+$t_a$ to $a w_a(S)^{-1}$ and fixes $S$.  The new letters $t_a$ all have
+value $1$.  On edge chains the substitution fixes the $S$-edges and
+sends $e_a$ to $e_{t_a}+[w_a]$, so it is a chain isomorphism and preserves
+the boundary basis property.  Let $T$ be the set of these new letters,
+$R$ the current relator index set, and $I=\mathcal D\amalg T$.
+Equation~\eqref{eq:cycle-basis} identifies the cycle module with
+$\Lambda^{(I)}$.  In these coordinates the relator boundary map is an
+isomorphism
+\[
+ A:\Lambda^{(R)}\longrightarrow\Lambda^{(I)}.
+\]
+
+We next arrange that the prescribed cycles themselves occur as relator
+boundaries. An auxiliary copy of the cycle module allows the three
+shears below to move this basis into a new relator block.
+Adjoin letters $U=(u_i)_{i\in I}$ and defining relators $u_i$.
+The stabilized boundary map is
+\[
+ D_0=\begin{pmatrix}A&0\\0&1\end{pmatrix}:
+ \Lambda^{(R)}\oplus\Lambda^{(I)}
+ \longrightarrow\Lambda^{(I)}\oplus\Lambda^{(I)},
+\]
+where the second target summand consists of the $U$-edges.  On its
+domain put
+\[
+ P=\begin{pmatrix}1&A^{-1}\\0&1\end{pmatrix},
+ \qquad Q=\begin{pmatrix}1&0\\-A&1\end{pmatrix}.
+\]
+These are blocks of homomorphisms of left modules; all products below
+mean composition.  Direct multiplication gives
+\begin{equation}\label{eq:presentation-shears}
+ PQP=\begin{pmatrix}0&A^{-1}\\-A&0\end{pmatrix},
+ \qquad
+ D_0PQP=\begin{pmatrix}0&1\\-A&0\end{pmatrix}.
+\end{equation}
+
+We realize each of the successive precompositions $P,Q,P$ by actual
+relator words.  A shear leaves one block of words unchanged and adds to
+each boundary in the other block a finite sum
+$\sum_k n_k g_k[b_k]$ of translates of unchanged boundaries.
+Choose a generator word $v_k$ representing $g_k$ and multiply the
+changing relator by copies of $v_k b_k^{\pm1}v_k^{-1}$, with the indicated
+multiplicities and signs.  All these words represent $1$ in $\Gamma$;
+the added boundary is precisely the stated sum.  The unchanged words
+belong to both old and new relator lists, so the old changing word is
+recovered from the new one by a finite inverse multiplication.  Thus
+the old and new lists have exactly the same normal closure.
+Every column of $A$ and $A^{-1}$ has finite support because their domains
+and targets are direct sums.  Hence this construction uses finite words
+for every relator even when the index sets are uncountable.  It does
+not require an infinite product of words or a limiting sequence of
+presentation moves.
+
+In the final second block, the relator indexed by $i\in I$ has boundary
+$(e_i,0)$.  Designate those indexed by $z\in\mathcal D$ as $\rho_z$.
+Set $\mathcal T=T\amalg U$.  The remaining second-block relators are
+indexed by $T$, and the first-block relators are indexed by $R$; these
+are the ordinary relators, with $\mathcal R=T\amalg R$.  Their projected
+boundary map is, in these orders,
+\begin{equation}\label{eq:ordinary-diagonal}
+ \begin{pmatrix}1_T&0\\0&-A\end{pmatrix}:
+ \Lambda^{(T)}\oplus\Lambda^{(R)}
+ \xrightarrow{\ \cong\ }
+ \Lambda^{(T)}\oplus\Lambda^{(I)}.
+\end{equation}
+Equation~\eqref{eq:presentation-shears} also shows that the complete
+boundary list remains a basis.  Normal closure was preserved at every
+step, so these are the relators of an actual presentation of $\Gamma$.
+\end{proof}
+
+The relators now form an actual presentation, and their boundaries are
+aligned with the chosen cycle basis. The next step extracts information
+that survives evaluation in the nonabelian group $H$: exact signed
+counts will cancel the linear error, leaving a quadratic estimate.
+
+\subsection{Exact counts and a uniform quadratic estimate}
+
+Fix the presentation supplied by Lemma~\ref{lem:aligned-presentation}.
+For every $z\in\mathcal D$, choose a finite normal-closure expression
+in the free group on $S\amalg\mathcal T$:
+\begin{equation}\label{eq:relator-expression}
+ z=\prod_{j=1}^{n_z}u_{z,j}\,r_{z,j}^{\sigma_{z,j}}u_{z,j}^{-1},
+ \qquad \sigma_{z,j}\in\{1,-1\}.
+\end{equation}
+Each $r_{z,j}$ is an indexed final relator, ordinary or distinguished.
+Let $\mathcal T_0\subset\mathcal T$ consist of the finitely many extra letters
+appearing in the words $u_{z,j}$.  For each $t\in\mathcal T_0$, also fix a finite
+expression of $t$ as a product of conjugates of final relators and their
+inverses.  Finally choose a finite set $\mathcal A\subset\mathcal R$
+containing every ordinary relator used in these expressions, both
+those for $z$ and those for $t$.
+
+\begin{lemma}\label{lem:exact-counts}
+In the free module on all final relator indices,
+\begin{equation}\label{eq:translated-counts}
+ \sum_{j=1}^{n_z}\sigma_{z,j}\bar u_{z,j}e_{r_{z,j}}
+       =e_{\rho_z}.
+\end{equation}
+\end{lemma}
+\begin{proof}
+The boundary of a conjugate $u r^{\sigma}u^{-1}$ is
+$\sigma\bar u[r]$: the two conjugating paths cancel as chains because
+$r$ has group value $1$.  Every factor in
+Equation~\eqref{eq:relator-expression} also has value $1$, so multiplying
+the factors introduces no further prefix translations in its chain.
+Applying the complete relator boundary isomorphism to the left side of
+Equation~\eqref{eq:translated-counts} therefore gives $[z]$.
+The right side has the same image, namely $[\rho_z]=[z]$.
+Injectivity gives the asserted equality, coefficient by coefficient.
+\end{proof}
+
+We will use labels either on $\Gamma$ or on any finite quotient $F$ of
+$\Gamma$.  In the latter case the graph is formed with the prescribed
+images of the generators, retaining distinct edges for distinct alphabet
+letters even when their images coincide.  All extra-generator edges are loops in
+either graph.  Write $L(w,v)$ for the product of the labels along the
+word $w$ starting at $v$, in traversal order.  Endpoints depend only on
+the group values of the letters, not on their labels.  Consequently
+free word identities can be evaluated on these graphs.  In particular,
+\[
+ L(u r^{\sigma}u^{-1},v)
+   =L(u,v)L(r,v\bar u)^{\sigma}L(u,v)^{-1}
+ \quad\text{if }\bar r=1,
+\]
+with group values projected to $F$ when appropriate.
+
+\begin{lemma}[Uniform estimate]\label{lem:quadratic-estimate}
+There is a constant $C\ge1$, depending only on the finite choices above,
+with the following property on $\Gamma$ and on every finite quotient.
+Suppose that all $S$-labels have distance at most $\epsilon$ from $1$,
+and that all translates of the ordinary relators in $\mathcal A$
+have product $1$.  Put
+\[
+ h=\sup_{z\in\mathcal D,\ v}|L(\rho_z,v)-1|,
+\]
+where the supremum is $0$ if $\mathcal D$ is empty.  If
+$0\le h,\epsilon\le1$, then, for every $z\in\mathcal D$ and vertex $v$,
+\begin{align}
+ |L(z,v)-L(\rho_z,v)|&\le C(h+\epsilon)h,
+                      \label{eq:quadratic-estimate}\\
+ |L(z,v)-1|&\le C\epsilon.\label{eq:short-word-estimate}
+\end{align}
+\end{lemma}
+
+\begin{proof}
+For unit quaternions, telescoping products and bi-invariance give
+\begin{equation}\label{eq:quaternion-estimates}
+ \left|\prod_{j=1}^n a_j-\prod_{j=1}^n b_j\right|
+       \le\sum_{j=1}^n|a_j-b_j|,
+ \qquad
+ |ab-ba|\le2|a-1|\,|b-1|.
+\end{equation}
+The second inequality follows by expanding
+$ab-ba=(a-1)(b-1)-(b-1)(a-1)$.
+Inverses and conjugation preserve distance from $1$.
+
+Evaluate the fixed expression for each $t\in\mathcal T_0$ at any vertex.
+Ordinary relators disappear, and every distinguished factor has
+distance at most $h$ from $1$, regardless of its conjugator.  Thus
+\[
+ |L(t,v)-1|\le M h\qquad(t\in\mathcal T_0)
+\]
+for one finite constant $M$, independent of the vertex and the group
+quotient.  No bound on the conjugators in these auxiliary expressions
+is required.  Each $u_{z,j}$ uses only letters of $S\amalg\mathcal T_0$, so another
+fixed constant $K$ gives
+\[
+ |L(u_{z,j},v)-1|\le K(\epsilon+h).
+\]
+On evaluating Equation~\eqref{eq:relator-expression}, discard its
+ordinary factors.  Removing the conjugation from a remaining factor
+costs at most
+\[
+ |aba^{-1}-b|=|ab-ba|
+       \le 2K(\epsilon+h)h,
+\]
+including when the relator occurs with exponent $-1$.
+There are at most $n_z$ such factors.
+
+The resulting factors are values of distinguished relators, or their
+inverses, at specified vertices.  Any interchange of two adjacent
+factors costs at most $2h^2$, by
+Equation~\eqref{eq:quaternion-estimates}.  Group together occurrences
+having the same relator index and the same vertex; at most
+$n_z(n_z-1)/2$ interchanges are needed.  Their signed multiplicities
+are prescribed exactly by Equation~\eqref{eq:translated-counts},
+translated from $1$ to $v$.  On a finite quotient, coincident vertices
+merely add the corresponding coefficients, so the same identity
+still applies.  After cancellation within each group, the product is
+exactly $L(\rho_z,v)$.  The total error is bounded by
+\[
+ 2n_zK(\epsilon+h)h+n_z(n_z-1)h^2.
+\]
+Since $\mathcal D$ is finite, this proves
+Equation~\eqref{eq:quadratic-estimate} with one $C$.
+The word $z$ contains only $S$-letters, so
+$|L(z,v)-1|\le\operatorname{length}(z)\epsilon$.
+Increasing $C$ proves Equation~\eqref{eq:short-word-estimate} as well.
+If the distinguished list is empty, the two assertions are vacuous
+and any $C\ge1$ suffices.
+\end{proof}
+
+Fix from now on numbers $0<\delta<1$ and $0<\eta<1$ such that
+\begin{equation}\label{eq:small-thresholds}
+ C(\delta+\eta)<\tfrac12,
+ \qquad C\eta<\tfrac{\delta}{2}.
+\end{equation}
+Under the hypotheses of Lemma~\ref{lem:quadratic-estimate}, with
+$\epsilon\le\eta$, the value $h=\delta$ is impossible.  Indeed, taking
+the supremum in its two inequalities gives
+\begin{equation}\label{eq:no-threshold-crossing}
+ \begin{aligned}
+ h&\le C\epsilon+C(h+\epsilon)h;\\
+ h=\delta&\ \Longrightarrow\
+ \delta\le C\eta+C(\delta+\eta)\delta<\delta.
+ \end{aligned}
+\end{equation}
+The supremum need not be attained for this argument.
+
+The constants $C$, $\delta$, and $\eta$ are now fixed, independently of
+any finite quotient or finite collection of equations. The exclusion of
+$h=\delta$ will keep the degree calculation inside the region of small
+distinguished products while the prescribed labels move.
+
+\subsection{Solving finite systems by degree}
+
+\begin{lemma}\label{lem:finite-quotient-solvability}
+Let $F$ be any finite quotient of $\Gamma$, and prescribe arbitrary
+$S$-edge labels on $F$ of distance less than $\eta$ from $1$.
+For every finite set $\mathcal B\subset\mathcal R$ containing
+$\mathcal A$, there are labels on all $\mathcal T$-edges such that
+\[
+ L(r,v)=1\quad(r\in\mathcal B,\ v\in F),
+ \qquad
+ \max_{z\in\mathcal D,\ v\in F}|L(\rho_z,v)-1|<\delta.
+\]
+The maximum over an empty distinguished list is interpreted as $0$.
+\end{lemma}
+
+\begin{proof}
+Tensor the isomorphism~\eqref{eq:ordinary-projection} with the right
+$\Lambda$-module $\Z[F]$.  This gives an isomorphism
+\begin{equation}\label{eq:quotient-basis}
+ \Z[F]^{(\mathcal R)}\xrightarrow{\ \cong\ }
+ \Z[F]^{(\mathcal T)}.
+\end{equation}
+No flatness claim is involved: tensoring the inverse gives an inverse
+to the displayed map.  As an abelian-group basis, the source has indices
+$(v,r)\in F\times\mathcal R$.  Its image consists of the projected
+boundary chains of $r$ based at $v$.  Thus the chains for
+$F\times\mathcal B$ are linearly independent over $\Q$.
+
+Give an unknown $x_{v,t}\in H$ to each extra edge $(v,t)$.
+In the equation $L(r,v)=1$, the exponent count of each unknown is
+exactly its coefficient in the projected boundary chain.  This holds
+even though the fixed $S$-labels need not commute: the path traversed
+is determined by the generator values in $F$, and all $t\in\mathcal T$
+have value $1$.  Write the $m=|F|\,|\mathcal B|$ exponent vectors as
+rows.  Their union of supports is finite, and their row rank is $m$.
+Choose $m$ unknowns giving a nonsingular $m\times m$ integer minor
+$B$, and set every other extra label equal to $1$.
+
+Move all the prescribed $S$-labels from $1$ to their final values
+along the short quaternion arcs, with common parameter $\tau\in[0,1]$.
+These arcs stay at distance less than $\eta$ from $1$.
+The selected equations now define a continuous family of smooth maps
+\[
+ f_\tau:H^m\longrightarrow H^m.
+\]
+For an assignment $x\in H^m$, let
+\[
+ h(\tau,x)=\max_{z\in\mathcal D,\ v\in F}
+       |L_\tau(\rho_z,v;x)-1|,
+ \qquad \mathcal O_\tau=\{x:h(\tau,x)<\delta\}.
+\]
+Only finitely many words occur here, so $h$ is continuous.
+Every root $f_\tau(x)=\one$ satisfies all equations from $\mathcal A$.
+Equation~\eqref{eq:no-threshold-crossing} therefore implies
+\begin{equation}\label{eq:root-isolation}
+ f_\tau(x)=\one\quad\Longrightarrow\quad h(\tau,x)\ne\delta.
+\end{equation}
+If $m=0$, the domain and target are points.  The unique assignment
+starts with $h=0$, and continuity together with
+Equation~\eqref{eq:root-isolation} keeps $h<\delta$ throughout.
+This proves the assertion in that case; henceforth assume $m\ge1$.
+
+We justify degree invariance with these moving open sets.  For fixed
+$\tau_0$, the roots with $h(\tau_0,x)<\delta$ form a compact set:
+by~\eqref{eq:root-isolation} they are also the roots with
+$h(\tau_0,x)\le\delta$.  Choose an open neighborhood $V$ containing
+exactly these roots, with
+$\overline V\subset\mathcal O_{\tau_0}$.
+If there are no such roots, take $V=\varnothing$.
+Compactness of $H^m$ and continuity show that, for all $\tau$ sufficiently
+close to $\tau_0$, every root with $h(\tau,x)<\delta$ lies in $V$,
+every root in $\overline V$ has $h(\tau,x)<\delta$, and there is no
+root on $\partial V$.  For example, a sequence of selected roots
+outside $V$ with parameters tending to $\tau_0$ would have a limit
+root with $h\le\delta$, hence $h<\delta$, outside $V$, a contradiction.
+The other assertions follow in the same way, or from
+$\overline V\subset\mathcal O_{\tau_0}$.
+Excision and homotopy invariance of ordinary degree now give
+\[
+ \deg(f_\tau,\mathcal O_\tau,\one)
+       =\deg(f_\tau,V,\one)
+       =\deg(f_{\tau_0},V,\one)
+\]
+near $\tau_0$.  Thus the selected degree is locally constant, and hence
+constant on $[0,1]$.
+
+At $\tau=0$, the map $f_0$ is a pure word map with exponent matrix $B$:
+all fixed coefficients are $1$.  The set $\mathcal O_0$ is invariant
+under simultaneous conjugation, contains the identity assignment,
+and its boundary contains no root.  Lemma~\ref{lem:localized-degree}
+therefore gives
+\[
+ \deg(f_0,\mathcal O_0,\one)\ne0.
+\]
+The same holds at $\tau=1$, so a root exists in $\mathcal O_1$, as
+required.
+\end{proof}
+
+Finite quotients now provide solutions for every finite set of ordinary
+relators, with the distinguished products uniformly bounded. To return
+to the original graph, we must preserve its prescribed $S$-labels exactly.
+The following compactness argument does this one finite collection of
+constraints at a time.
+
+\subsection{Finite-coordinate compactness and the conclusion}
+
+We use compactness of arbitrary products of compact spaces
+\cite[Theorem~37.3]{Munkres2000}. Its unrestricted index set is essential:
+the set of extra generators in the hypothetical presentation need not
+be countable.
+
+\begin{proof}[Proof of Theorem~\ref{thm:presentation}]
+Let $a$ be any prescribed labeling of the $S$-edges on $\Gamma$ satisfying
+the theorem's hypotheses, with the fixed $\eta$ of
+Equation~\eqref{eq:small-thresholds}.  In the compact product
+\[
+ \mathscr P=H^{\Gamma\times\mathcal T}
+\]
+consider all the following conditions on the unknown extra-edge labels:
+\begin{align}
+ L(r,g)&=1 &&(r\in\mathcal R,\ g\in\Gamma),
+                           \label{eq:compact-ordinary}\\
+ |L(\rho_z,g)-1|&\le\delta
+             &&(z\in\mathcal D,\ g\in\Gamma).
+                           \label{eq:compact-distinguished}
+\end{align}
+Each condition is closed and depends on finitely many coordinates.
+We verify the finite intersection property, keeping the original
+labeling $a$ fixed throughout.
+
+First choose an arbitrary finite collection $\mathscr C$ of the
+conditions~\eqref{eq:compact-ordinary}--\eqref{eq:compact-distinguished}.
+Let $P\subset\Gamma$ be the finite set of starting vertices of all
+\emph{positively oriented} $S$-edges used when evaluating the words
+in $\mathscr C$.  An inverse occurrence traversing the edge from
+$g$ to $gs^{-1}$ uses the positively oriented edge starting at
+$gs^{-1}$, and this starting vertex is included in $P$.
+Let $\mathcal B$ contain $\mathcal A$ and all ordinary relator indices
+appearing in $\mathscr C$.  These choices are made before choosing a
+finite quotient.
+
+Residual finiteness supplies a finite quotient
+$q:\Gamma\twoheadrightarrow F$ that is injective on $P$:
+separate the finitely many nonidentity elements $p^{-1}p'$ for distinct
+$p,p'\in P$, and let $F$ be the image of $\Gamma$ in the product
+of the resulting finite quotients.
+For each $S$-edge actually used in $\mathscr C$, assign to its image
+edge in $F$ its prescribed label from $a$.  Injectivity on $P$ ensures
+that no two inconsistent prescriptions are placed on the same edge.
+Give every remaining $S$-edge of $F$ label $1$.  All these labels have
+distance less than $\eta$ from $1$.
+
+Apply Lemma~\ref{lem:finite-quotient-solvability} to this finite quotient,
+this assignment, and $\mathcal B$.  Pull the resulting extra-edge labels
+back by $q$, and combine them with the \emph{original} $S$-labels $a$
+on $\Gamma$.  In every word from $\mathscr C$, each traversed $S$-edge
+has the same label as its image in $F$, by construction; the same is
+true for each extra edge by pullback.  Its path product therefore
+agrees with the quotient path product.  The selected ordinary equations
+hold, and the selected distinguished products even have distance
+strictly less than $\delta$.  This supplies a point of $\mathscr P$
+satisfying $\mathscr C$.
+
+We have proved: for each finite collection of constraints, one can
+choose a finite quotient, copy the finitely many required prescribed
+labels, solve a finite system there, and pull back a solution to that
+collection.  The quotient may depend on the collection; the original
+assignment $a$ is never assumed to descend to any one finite quotient.
+Compactness of $\mathscr P$ now gives an extension satisfying
+all of~\eqref{eq:compact-ordinary}--\eqref{eq:compact-distinguished}
+simultaneously.
+
+For this extension, $h\le\delta$ and all the equations in $\mathcal A$
+hold.  The original hypothesis says $L(z,g)=1$ for every $z$ and $g$.
+Taking suprema in Equation~\eqref{eq:quadratic-estimate}, with
+$\epsilon=\eta$, gives
+\[
+ h\le C(h+\eta)h\le C(\delta+\eta)h<\tfrac12h
+ \quad\text{if }h>0.
+\]
+Thus $h=0$.  All distinguished relators, as well as all ordinary
+relators, have product $1$ at every vertex.
+Finally, a closed edge path spells a word representing $1$ in $\Gamma$.
+Because Lemma~\ref{lem:aligned-presentation} gives an actual presentation,
+that word is a finite product of conjugates of final relators and their
+inverses.  Evaluating at the starting vertex makes every factor $1$.
+Hence the original closed path has product $1$, as claimed.
+\end{proof}
diff --git a/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/build/sections/transport.tex b/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/build/sections/transport.tex
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+\section{Small transport with nontrivial holonomy}\label{sec:transport}
+
+Retain the finite flag complex $L$, the homomorphism
+$\alpha:\pi_1(L)\to H=\SU(2)$, and the level complex $X$ of
+Proposition~\ref{prop:model}.  Write $V$ for the vertex set of $L$ and
+$a_v$ for the corresponding standard generator of $P=A_L$.
+We realize $|L|$ in $\R^V$ by barycentric coordinates and give it the
+metric induced by the $\ell^1$ norm.  Our aim is the following.
+
+\begin{proposition}\label{prop:small-labels}
+For every $\epsilon>0$, the oriented edges of $X$ admit labels
+$h_e\in H$ such that
+\[
+ h_{\bar e}=h_e^{-1},\qquad |h_e-1|<\epsilon,
+\]
+the product around every triangular two-cell is $1$, and the product
+around some closed edge path is different from $1$.
+\end{proposition}
+
+For an abelian target, products along closed paths would factor through
+$H_1(X;\Z)=0$ and would all be trivial.  The nonabelian target is
+therefore essential.
+
+The construction has two parts.  A direction map on reduced words of
+$P$ changes slowly far from the identity.  Translating the level $X$
+far from the identity therefore gives a map of its graph to $L$ with
+small edge images.  Its image nevertheless contains prescribed loops
+of $L$, allowing $\alpha$ to detect nontrivial holonomy.
+
+The long commuting-power paths below lie along expanded copies of $L$
+in level sets, formed from commuting coordinate directions as in
+Bestvina and Brady
+\cite[Definition~8.8 and the following paragraph]{BB1997}.
+
+\subsection{Reduced traces and their direction vectors}
+
+Two signed standard generators are called \emph{independent} when
+their underlying vertices are distinct and adjacent in $L$.
+In particular, two occurrences of the same generator are dependent,
+regardless of their signs.  A \emph{trace} is a word modulo exchanges
+of adjacent independent letters, as in the theory of partially
+commutative monoids \cite[Chapter~I, Section~2]{CartierFoata1969}.  Occurrences can be followed through
+these exchanges.  The \emph{occurrence poset} of a word orders every
+dependent pair in its order of appearance and takes the transitive
+closure.  It records the trace: its linear extensions are precisely
+the representatives of that trace; see also Krattenthaler's appendix
+to the 2006 reedition of \cite[Definition~2.2 and Section~3 of the appendix]{CartierFoata1969}.
+Indeed, exchanges preserve this
+poset, and any two linear extensions of a finite poset are related by
+exchanges of adjacent incomparable elements.  Here incomparable
+occurrences have independent directions.
+
+The following signed-letter version of the graph-product normal form
+\cite[Theorem~3.9]{Green1990} records individual occurrences rather than
+vertex-group syllables. We include its reduction proof because that
+occurrence information is used in the direction-vector estimate.
+
+\begin{lemma}\label{lem:trace-normal-form}
+Every element of $P$ has a unique reduced trace, obtained by deleting
+inverse pairs that can be made adjacent by exchanges.  Its length is
+the word length in the standard generators.  Right multiplication by
+a signed standard generator changes this occurrence poset by either
+adjoining a maximal occurrence or deleting a maximal occurrence.
+\end{lemma}
+
+\begin{proof}
+An inverse pair can be deleted exactly when every occurrence between
+its endpoints is independent of its direction.  This criterion is
+independent of the representative: the order of an occurrence
+dependent on that direction relative to either endpoint cannot change
+under exchanges.  Deletion gives the trace represented by removing the
+two occurrences.
+
+We check the local confluence of deletion.  For two disjoint deletable
+pairs, either deletion leaves the other available, and both orders
+remove exactly the same four occurrences.  If two pairs share an
+occurrence, they cannot share their first endpoint or their last
+endpoint, because an intervening occurrence of the same direction
+would prevent deletion.  The relevant subword consequently has the
+form
+\[
+ x B x^{-1} C x
+\]
+or the same form with $x$ replaced by $x^{-1}$, where every letter of
+$B$ and $C$ is independent of $x$.  The two deletions leave $BCx$ and
+$xBC$, which are the same trace.  Since deletion decreases length,
+induction on length using these local diamonds proves uniqueness of
+the terminal trace.  Exchanges and adjacent inverse insertions or
+deletions preserve that terminal trace.  These operations generate
+equality in the defining presentation of $P$, so the terminal trace
+depends only on the group element.  Every word representing that
+element reduces to it, proving the assertion about word length.
+
+Write $|g|_P$ for word length in the signed standard generators.
+Let $w$ be reduced of length $n$, and append a signed generator $x$.
+If the result is reduced, its new occurrence is maximal.  Otherwise
+an available deletion must involve the new last occurrence, since a
+deletion involving only old occurrences would already have been
+available in $w$.  Its inverse mate $y$ has only independent
+occurrences after it in $w$, and is therefore maximal in the old
+occurrence poset.  Deleting $y$ and the new $x$ leaves a word of length
+$n-1$.  If $g$ is the element represented by $w$, the word-metric
+inequality gives
+\[
+ |gx|_P\geq |g|_P-1=n-1.
+\]
+Thus the remaining word is reduced.  Deleting a maximal occurrence
+also leaves exactly the induced order on the retained occurrences:
+no dependency chain between two retained occurrences can pass through
+a maximal occurrence.  This proves the last assertion.
+\end{proof}
+
+Fix a total order on $V$.  In a nonempty reduced trace, the minimal
+occurrences have distinct pairwise independent directions, hence
+their directions span a simplex of $L$.  Assign each occurrence $p$
+to the first direction, in our fixed order, among the minimal
+occurrences below or equal to $p$.  This set is nonempty because the
+poset is finite.  Let $c_v(g)$ be the number assigned to $v$, and set
+\begin{equation}\label{eq:direction-map}
+ b(g)=\frac{1}{|g|_P}\sum_{v\in V}c_v(g)e_v\in |L|,
+ \qquad g\ne 1.
+\end{equation}
+Here $e_v$ is the barycentric vertex corresponding to $v$.
+Each minimal occurrence is assigned to its own direction, so the
+support of $b(g)$ is exactly the set of minimal directions.
+
+\begin{lemma}\label{lem:trace-direction}
+Suppose $g$ and $gx$ are nonidentity, where $x$ is a signed standard
+generator.  Their direction vectors belong to a common simplex of
+$L$.  If their word lengths are $n$ and $n+1$, in either order, then
+\begin{equation}\label{eq:trace-direction-bound}
+ \norm{b(gx)-b(g)}_1\leq \frac{2}{n+1}.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+By Lemma~\ref{lem:trace-normal-form}, one occurrence poset is obtained
+from the other by adjoining a maximal occurrence.  Every retained
+occurrence has the same principal downset before and after this
+operation: a maximal occurrence cannot lie on a dependency chain
+ending at a different occurrence.  The globally minimal predecessors
+of a retained occurrence are precisely the minimal elements of its
+principal downset.  Its assigned direction is therefore unchanged.
+This also covers a changed occurrence that is both minimal and
+maximal: it is isolated and has no retained successors.
+
+Consequently, if $c$ is the count vector for the shorter trace, the
+longer trace has count vector $c+e_v$ for some $v\in V$.  Since
+$\norm{c/n}_1=1$,
+\[
+ \norm{\frac{c+e_v}{n+1}-\frac{c}{n}}_1
+ =\frac{\norm{e_v-c/n}_1}{n+1}\leq\frac{2}{n+1}.
+\]
+For the common-simplex assertion, adjoining a maximal occurrence
+preserves all old minimal occurrences.  If the new occurrence is
+not minimal, the minimal set is unchanged.  If it is also
+minimal, it is isolated, and its direction is independent of every
+old occurrence: dependent occurrences would be comparable.  Thus
+the union of the two sets of minimal directions is a clique.
+Flagness of $L$ completes the proof.
+\end{proof}
+
+The direction vector changes by at most the reciprocal word-length
+bound in Equation~\eqref{eq:trace-direction-bound} under a generator
+step. We next apply this estimate on a translate of the level graph
+where all relevant words are long. At the same time, we retain exact
+control of certain paths, so that making every edge image small does
+not erase the loops detected by $\alpha$.
+
+\subsection{A slowly varying map of the level graph}
+
+Let $N\geq2$ and choose $d\in P$ with $\lambda(d)=N$.  Define a map on
+the vertices of $X$ by
+\begin{equation}\label{eq:translated-direction-map}
+ f_N(g)=b(d^{-1}g),\qquad g\in G.
+\end{equation}
+It is defined because $\lambda(d^{-1}g)=-N$.  Every edge of $X$ is
+the diagonal of a commuting square in $E$ with corner heights
+$-1,0,0,1$.  If its endpoints are $g,h$, write $u$ for the square's
+unique corner of height $1$.  The two steps $g,u,h$ are standard
+generator steps in $E$.  Their translates by $d^{-1}$ have heights
+$-N,1-N,-N$, hence word lengths at least $N,N-1,N$.
+The larger word length in each step is therefore at least $N$.
+Lemma~\ref{lem:trace-direction} gives
+\[
+ \norm{b(d^{-1}g)-b(d^{-1}u)}_1\leq 2/N,
+ \qquad
+ \norm{b(d^{-1}u)-b(d^{-1}h)}_1\leq 2/N.
+\]
+Each pair belongs to a common simplex, so the two straight segments
+joining them lie in $|L|$.
+
+Map the edge from $g$ to $h$ to this two-segment path, with the
+following convention: whenever $f_N(g)$ and $f_N(h)$ themselves lie
+in a common simplex, use their single straight segment instead.
+This segment is unambiguous even if several simplices contain the
+endpoints: it is the same segment in barycentric coordinates and lies
+in their common face.  Choose the reverse path for the reverse
+orientation.  These choices define a continuous map
+$f_N:X^{(1)}\to |L|$, since they agree at vertices and the domain is a
+CW complex.  Every edge image has diameter at most $4/N$.  Moreover,
+any two of the three edge images of a triangle meet at their common
+vertex image.  Therefore
+\begin{equation}\label{eq:small-triangle-image}
+ \operatorname{diam} f_N(\partial\sigma)\leq 8/N
+ \qquad\text{for every triangular two-cell $\sigma$ of $X$.}
+\end{equation}
+
+There is also exact control on selected long paths.  If $\{q,r\}$ is
+an edge of $L$, put
+\begin{equation}\label{eq:commuting-power-path}
+ p_j=d\,a_q^{-(N-j)}a_r^{-j},\qquad 0\leq j\leq N.
+\end{equation}
+These vertices have height zero.  Consecutive vertices are joined by
+an edge of $X$: the square based at $p_j a_r^{-1}$ in the positive
+$q,r$ directions has level-zero corners $p_j$ and $p_{j+1}$.
+The trace of $d^{-1}p_j$ consists of two mutually independent chains
+of negative occurrences, one in each direction, with lengths $N-j$
+and $j$; a chain of length zero is omitted.  Every occurrence is
+assigned to its own direction.  Hence
+\begin{equation}\label{eq:exact-subdivision}
+ f_N(p_j)=\left(1-\frac{j}{N}\right)e_q+\frac{j}{N}e_r.
+\end{equation}
+Our straight-segment convention means that this level path maps
+exactly to the edge $[q,r]$, subdivided into $N$ equal parts; see
+Figure~\ref{fig:commuting-path}.
+
+\begin{figure}[ht]
+ \centering
+ \begin{tikzpicture}[x=0.72cm,y=0.72cm,>=Stealth]
+  \draw[step=1,gray!35,thin] (0,0) grid (4,4);
+  \draw[blue!70!black,very thick] (0,4)--(4,0);
+  \foreach \j in {0,...,4}{
+   \fill[blue!70!black] (\j,{4-\j}) circle (2.2pt);
+  }
+  \node[above left] at (0,4) {$p_0=d a_q^{-N}$};
+  \node[below right] at (4,0) {$p_N=d a_r^{-N}$};
+  \node[below] at (1.6,-0.65) {Level path in a commuting flat};
+  \draw[->,thick] (4.6,2.2)--node[above] {$f_N$}(6.3,2.2);
+  \draw[blue!70!black,very thick] (7,2)--(11,2);
+  \foreach \j in {0,...,4}{
+   \fill[blue!70!black] ({7+\j},2) circle (2.2pt);
+  }
+  \node[below] at (7,1.9) {$q$};
+  \node[below] at (11,1.9) {$r$};
+  \node[above] at (8,2.15) {$\frac14$};
+  \node[above] at (9,2.15) {$\frac12$};
+  \node[above] at (10,2.15) {$\frac34$};
+  \node[below] at (9,0.65) {The edge $[q,r]\subset L$};
+ \end{tikzpicture}
+ \caption{The special path~\eqref{eq:commuting-power-path}, shown for
+ $N=4$, joins $d a_q^{-N}$ to $d a_r^{-N}$ across level-cut squares.
+ Formula~\eqref{eq:exact-subdivision} sends it to the exact subdivision
+ of $[q,r]$.  Here $a_q,a_r$ commute and $\lambda(d)=N$.
+ Each blue diagonal step multiplies by $a_q a_r^{-1}$, preserving
+ height zero; $f_N$ is shown on this level path.}
+ \label{fig:commuting-path}
+\end{figure}
+
+We have obtained both properties needed from $f_N$: every triangle
+boundary has small image, and selected closed paths trace prescribed
+loops of $L$ exactly. The representation $\alpha$ will turn these two
+properties into flatness on triangles and nontrivial closed-path
+holonomy, respectively.
+
+\subsection{Pulling back transport}
+
+\begin{proof}[Proof of Proposition~\ref{prop:small-labels}]
+Choose a base vertex $q_0$ for $\alpha$ and a base lift of $q_0$ in
+the universal cover $\widetilde L\to L$.  Choose the left deck
+action convention so that lifting a loop representing $\gamma$ from
+that base lift ends at $\gamma$ times the lift.  There is a
+continuous map
+\begin{equation}\label{eq:developing-map}
+ D:\widetilde L\longrightarrow H,
+ \qquad D(\gamma\xi)=\alpha(\gamma)D(\xi).
+\end{equation}
+To construct it, choose images for representatives of the vertex
+orbits and extend equivariantly.  On representatives of the edge
+orbits, extend between the prescribed endpoints using path
+connectedness of $H$.  Extend over representatives of the two-cell
+orbits using $\pi_1(H)=0$.  The deck action is free on cells, so
+equivariance introduces no ambiguity; $L$ has dimension two, so there
+are no further extensions.  The CW topology gives continuity.
+
+Fix $\epsilon>0$.  There is a finite open cover $(U_i)$ of $L$ such
+that each $U_i$ is evenly covered and $D$ has image of diameter less
+than $\epsilon$ on each sheet over $U_i$.  Indeed, around any lift
+of a point, continuity of $D$ gives a sufficiently small such
+neighborhood on one sheet.  All other sheets are its deck translates,
+and left multiplication by $\alpha(\gamma)$ preserves quaternion
+distance.  Compactness of $L$ then gives a finite subcover.  Let
+$\ell>0$ be a Lebesgue number in the diameter form: every subset of
+$L$ of diameter less than $\ell$ lies in some $U_i$.  Choose $N\geq2$
+so large that $8/N<\ell$, and construct $f_N$ as above.
+
+For an oriented edge $e$ of $X$, lift its assigned path $f_N(e)$ from
+$\xi$ to $\xi'$ in $\widetilde L$, and define
+\begin{equation}\label{eq:edge-transport}
+ h_e=D(\xi)^{-1}D(\xi').
+\end{equation}
+Changing the lift multiplies both $D$ values on the left by the same
+$\alpha(\gamma)$, so the label is independent of the lift.  Reversing
+the path inverts the label.  Its image lies in some $U_i$ because its
+diameter is at most $4/N<\ell$.  The lift stays in one sheet, and
+therefore
+\[
+ |h_e-1|=|D(\xi')-D(\xi)|<\epsilon.
+\]
+
+The whole image of the boundary of a triangular two-cell lies in one
+$U_i$ by~\eqref{eq:small-triangle-image}.  The inverse homeomorphism
+of a sheet over $U_i$ lifts this boundary to a closed loop in that
+sheet.  In multiplying the three edge
+labels, choose these consecutive lifts; the intermediate $D$ values
+cancel, giving product $1$.
+
+Finally, choose a finite edge loop based at $q_0$,
+$q_0,q_1,\ldots,q_k=q_0$ in $L$ whose class $\gamma$ has
+$\alpha(\gamma)\ne1$.  Such a loop exists because $\alpha$ is
+nontrivial and every element of the fundamental group of a simplicial
+complex is represented by a finite edge loop.  For each successive
+pair, concatenate the path~\eqref{eq:commuting-power-path}.  The
+endpoint of each piece is $d a_{q_{i+1}}^{-N}$, the initial vertex of
+the next piece, so this is a closed edge path in $X$ based at
+$d a_{q_0}^{-N}$.  By~\eqref{eq:exact-subdivision}, its image under
+$f_N$ is precisely the chosen loop with each edge subdivided.
+Lift that image from the chosen base lift $\xi$ of $q_0$.  The terminal lift is
+$\gamma\xi$, and telescoping~\eqref{eq:edge-transport} gives the
+path product
+\[
+ D(\xi)^{-1}D(\gamma\xi)
+ =D(\xi)^{-1}\alpha(\gamma)D(\xi)\ne1.
+\]
+This proves all assertions of the proposition.
+\end{proof}
diff --git a/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/paper.pdf b/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..b13bc5ed63b310702b0a33227c26db80e8552030
--- /dev/null
+++ b/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:bbc781ac6f1739388fb537680e6597ddcfdce94978578b2e413666e592d1875d
+size 454858
diff --git a/preprints/A-group-without-fixed-price-October-5-2026/INPUTS.md b/preprints/A-group-without-fixed-price-October-5-2026/INPUTS.md
new file mode 100644
index 0000000000000000000000000000000000000000..0ee82f7f5ff1ba90a67e3e31f01ebdb39f270f23
--- /dev/null
+++ b/preprints/A-group-without-fixed-price-October-5-2026/INPUTS.md
@@ -0,0 +1,10 @@
+# Mathematical input and scope
+
+This manuscript concerns Gaboriau's general fixed-price problem for countably
+infinite groups, using a rank-100 amalgam construction. This is distinct from
+the fixed-price-one question restricted to groups with property (T).
+
+The manuscript was developed from an intermediate source argument. The
+available source record does not establish that it was the final revision of
+that argument. The article and its mathematical claims are listed in
+[README.md](README.md).
diff --git a/preprints/A-group-without-fixed-price-October-5-2026/README.md b/preprints/A-group-without-fixed-price-October-5-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..bc88f1c71189786e5e2e52a444396d8f276e4193
--- /dev/null
+++ b/preprints/A-group-without-fixed-price-October-5-2026/README.md
@@ -0,0 +1,18 @@
+# [A group without fixed price](unequal-costs-rank-100-amalgam.pdf)
+
+**Author:** OpenAI
+
+**Date:** October 5, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-group-without-fixed-price-October-5-2026,
+  author = {{OpenAI}},
+  title = {{A group without fixed price}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-group-without-fixed-price-October-5-2026/unequal-costs-rank-100-amalgam.pdf}{OAI:A-group-without-fixed-price-October-5-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-group-without-fixed-price-October-5-2026/build/compression.tex b/preprints/A-group-without-fixed-price-October-5-2026/build/compression.tex
new file mode 100644
index 0000000000000000000000000000000000000000..846dbf072ac27c5c911a2de215c0cf6808916270
--- /dev/null
+++ b/preprints/A-group-without-fixed-price-October-5-2026/build/compression.tex
@@ -0,0 +1,296 @@
+\section{Compressing a graphing after enlarging a supplied relation}
+\label{compression:section}
+
+This section proves a cost estimate for replacing part of a graphing by a
+larger relation whose edges are supplied without charge.  The measure on
+the underlying space is finite and need not be a probability measure.
+
+Let $(Z,\lambda)$ be a standard Borel space with a finite Borel measure.
+Let $\mathcal R$ be a Borel equivalence relation covered by the graphs of
+countably many measure-preserving Borel partial isomorphisms
+$(\psi_j)_{j\geq 1}$.  A partial isomorphism means a Borel bijection between
+Borel subsets with Borel inverse.  For a Borel subrelation $\mathcal U$ of
+$\mathcal R$ and a Borel set $W\subseteq Z$, put
+\[
+ \kappa(\mathcal U;W)
+ =\int_W\frac{1}{|[z]_{\mathcal U}|}\,d\lambda(z),
+ \qquad
+ \kappa(\mathcal U)=\kappa(\mathcal U;Z),
+\]
+where the reciprocal of an infinite class size is zero.  A set is
+$\mathcal U$-complete if it meets every $\mathcal U$-class.
+All relations in this section are pointwise Borel relations.  When an
+application starts with identities holding almost everywhere, it may
+first remove the $\mathcal R$-saturation of the exceptional null set.
+That saturation is Borel and null because the maps $\psi_j$ are
+measure-preserving.
+
+\subsection{Borel selections and small complete sets}
+
+We record the elementary measurability facts used in the construction.
+Restricting $\psi_j$ to the set where
+$(z,\psi_j(z))\in\mathcal U$ gives an enumeration of any Borel
+subrelation $\mathcal U$.  Its saturations of Borel sets are therefore
+countable unions of Borel images and are Borel.  Its class-size function
+is Borel as well: having at least $k$ class members is a countable union
+over $k$-tuples of enumeration values that are defined and pairwise
+distinct.
+
+Every Borel partial isomorphism $h$ with graph in $\mathcal R$ preserves
+measure.  Indeed, partition its domain according to the first index $j$
+for which $h(z)=\psi_j(z)$.  Each restriction preserves measure, and the
+images of the pieces are disjoint because $h$ is injective.
+
+Fix a Borel linear order on $Z$, obtained from a Borel injection into the
+real line.  Nonempty finite subsets have least points.  For a Borel
+relation with finite classes, selecting this least point is Borel: among
+the countably enumerated candidates, the condition that none is smaller
+is Borel.  The same observation applies to nonempty Borel subsets of
+finite classes and to the successive points in their ordered lists.
+
+\begin{lemma}[Finite-class counting]\label{compression:counting}
+Let $W\subseteq Z$ be a Borel $\mathcal U$-invariant set all of whose
+$\mathcal U$-classes are finite.  If $B\subseteq W$ is Borel and meets
+each such class exactly once, then
+\[
+ \lambda(B)=\kappa(\mathcal U;W).
+\]
+\end{lemma}
+
+\begin{proof}
+On the part $W_k$ consisting of classes of size $k$, order each class
+into $k$ Borel sheets.  The map from $B\cap W_k$ to any one sheet that
+stays within each class is a Borel partial isomorphism inside
+$\mathcal R$, so all $k$ sheets have measure $\lambda(B\cap W_k)$.
+Thus $\lambda(W_k)=k\lambda(B\cap W_k)$.  Summing over $k$ proves the
+identity.
+\end{proof}
+
+\begin{lemma}[Small complete sets]\label{compression:complete-set}
+For every Borel subrelation $\mathcal T\subseteq\mathcal R$ and every
+$\varepsilon>0$, there is a Borel $\mathcal T$-complete set
+$M\subseteq Z$ such that
+\[
+ \lambda(M)\leq\kappa(\mathcal T)+\varepsilon.
+\]
+\end{lemma}
+
+\begin{proof}
+On the union $Z_{\mathrm{fin}}$ of finite $\mathcal T$-classes, choose
+the least point of each class.  The resulting Borel set
+$M_{\mathrm{fin}}$ has measure $\kappa(\mathcal T)$ by
+Lemma~\ref{compression:counting}.
+
+On $Z_{\mathrm{inf}}=Z\setminus Z_{\mathrm{fin}}$, choose increasing
+finite Borel partitions $\mathcal P_n$ that separate all points, for
+example those generated by the first $n$ members of a countable
+separating Borel family.  Let $k_n(z)$ be the number of atoms of
+$\mathcal P_n$ meeting $[z]_{\mathcal T}$.  This function is Borel, and
+$k_n(z)\longrightarrow\infty$: any prescribed finite collection of
+distinct points in the infinite class is separated by some partition.
+
+Fix $0<p<1$ with $p\lambda(Z_{\mathrm{inf}})<\varepsilon/2$.
+At a fixed level $n$, mark each atom independently with probability $p$,
+and let $A$ be the union of the marked atoms.  The Borel set
+\[
+ M_{\mathrm{inf}}
+ =A\cup\bigl(Z_{\mathrm{inf}}\setminus[A]_{\mathcal T}\bigr)
+\]
+meets every infinite $\mathcal T$-class for every marking.  The displayed
+union is disjoint, and
+\[
+ \mathbb E\lambda(M_{\mathrm{inf}})
+ =p\lambda(Z_{\mathrm{inf}})
+   +\int_{Z_{\mathrm{inf}}}(1-p)^{k_n(z)}\,d\lambda(z).
+\]
+The integral tends to zero by dominated convergence.  For sufficiently
+large $n$ the expectation is less than $\varepsilon$, so one of the
+finitely many markings has measure less than $\varepsilon$.  Fix that
+marking and set $M=M_{\mathrm{fin}}\cup M_{\mathrm{inf}}$.
+\end{proof}
+
+\subsection{Compression and its cost}
+
+For a countable family $\mathcal E=(\varphi_i:D_i\to E_i)$ of
+measure-preserving Borel partial isomorphisms, write
+$C(\mathcal E)=\sum_i\lambda(D_i)$.  The join
+$\mathcal S\vee\mathcal E$ denotes the equivalence relation generated
+by $\mathcal S$ and all pairs $(x,\varphi_i(x))$.
+
+\begin{lemma}[Compression]\label{compression:main}\label{compression:lemma}
+On the finite-measure space $(Z,\lambda)$ with the ambient relation
+$\mathcal R$ described above, suppose
+$\mathcal S\subseteq\mathcal T\subseteq\mathcal Q\subseteq
+\mathcal R$ are Borel equivalence relations and
+$\mathcal Q=\mathcal S\vee\mathcal E$, where $\mathcal E$ is a
+countable family of measure-preserving Borel partial isomorphisms with
+finite total cost.  For every $\varepsilon>0$ there is a countable
+family $\mathcal F$ of measure-preserving Borel partial isomorphisms
+inside $\mathcal Q$ such that
+\begin{equation}\label{compression:bound}
+ \mathcal Q=\mathcal T\vee\mathcal F,
+ \qquad
+ C(\mathcal F)\leq C(\mathcal E)-\kappa(\mathcal S)
+                          +\kappa(\mathcal T)+\varepsilon.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Choose a Borel $\mathcal T$-complete set $M$ from
+Lemma~\ref{compression:complete-set}, and put
+\[
+ Y=[M]_{\mathcal S}
+     \cup\{z:|[z]_{\mathcal S}|=\infty\}.
+\]
+This set is Borel and $\mathcal S$-invariant.  It is
+$\mathcal T$-complete and hence $\mathcal Q$-complete, since it contains
+$M$.  Every $\mathcal S$-class outside $Y$ is finite.  In each finite
+$\mathcal S$-class in $Y$, choose a point of its intersection with $M$.
+Lemma~\ref{compression:counting} gives
+\begin{equation}\label{compression:retained-classes}
+ \kappa(\mathcal S;Y)\leq\lambda(M).
+\end{equation}
+
+\paragraph{Choosing and deleting one edge per class outside $Y$.}
+Let $d(z)$ be the least number of edges from $\mathcal E$, in either
+direction, needed to reach $Y$, allowing $\mathcal S$-steps for free.
+This distance is finite because $Y$ is $\mathcal Q$-complete.  Its
+sublevel sets are Borel: starting with $Y$, successively take images
+under the maps of $\mathcal E$ and their inverses, and take
+$\mathcal S$-saturations.  It is constant on $\mathcal S$-classes and
+vanishes precisely on $Y$.
+
+For every $\mathcal S$-class $K$ outside $Y$, choose an oriented edge
+instance from $y_K\in K$ to $y'_K$ with $d(y'_K)=d(K)-1$.
+Such an instance exists by a shortest path.  To make the choice Borel,
+first choose the least eligible pair consisting of a map index and a
+direction, then choose the least eligible point in the finite class
+$K$.  Eligibility of a map index and direction is detected by the
+$\mathcal S$-saturation of a Borel set.  Thus all these choices are
+Borel.  The set $V=\{y_K\}$ is a Borel transversal of
+$\mathcal S$ on $Z\setminus Y$, so
+\begin{equation}\label{compression:selector-cost}
+ \lambda(V)=\kappa(\mathcal S;Z\setminus Y).
+\end{equation}
+
+For each original map $\varphi_i:D_i\to E_i$, let $V_i^+$ be the
+chosen origins whose edge is traversed forward under $\varphi_i$, and
+let $V_i^-$ be those whose edge is traversed under $\varphi_i^{-1}$.
+Delete the domain set
+\[
+ H_i=V_i^+\cup\varphi_i^{-1}(V_i^-)
+\]
+from $D_i$.  The union is disjoint: a labeled edge instance chosen in
+both directions would make each endpoint class have smaller distance
+than the other.  The sets $V_i^+,V_i^-$ partition $V$, and inverse
+images under $\varphi_i$ preserve measure.  Consequently the residual
+family $\mathcal E'=(\varphi_i|_{D_i\setminus H_i})$ satisfies the
+exact identity
+\begin{equation}\label{compression:deleted-cost}
+ C(\mathcal E')
+ =C(\mathcal E)-\sum_i\lambda(H_i)
+ =C(\mathcal E)-\kappa(\mathcal S;Z\setminus Y).
+\end{equation}
+Different original indices count as different edge instances, even
+when their geometric edges coincide.  Thus this accounting also covers
+graphings with repetitions.
+
+\paragraph{Moving the remaining edges into $Y$.}
+Define $f:Z\to Y$ by $f(x)=x$ for $x\in Y$, and, for $x$ in an
+$\mathcal S$-class $K$ outside $Y$, by
+\[
+ f(x)=f(y'_K).
+\]
+The recursion decreases the integer distance at every step.  Defining
+$f$ successively on the distance levels proves it is Borel.  Each
+$(x,f(x))$ lies in $\mathcal Q$.  Because $Y$ is
+$\mathcal S$-invariant, $f$ fixes each class inside $Y$ pointwise and
+is constant on each class outside $Y$.  It therefore sends
+$\mathcal S$-steps to $\mathcal S$-steps.  Each deleted edge has equal
+images at its two endpoints.
+
+The map $f$ can be many-to-one.  To obtain partial isomorphisms after
+moving the remaining edges, we must split their domains at both
+endpoints.  Fix a residual edge map $\varphi_i$ with domain $D'_i$.
+Partition $D'_i$ into Borel pieces $P_{i,r,s}$ according to the least
+indices $r,s$ for which
+\[
+ f(x)=\psi_r(x),\qquad
+ f(\varphi_i(x))=\psi_s(\varphi_i(x)).
+\]
+These indices exist because the graph of $f$ is contained in
+$\mathcal Q\subseteq\mathcal R$.  On this piece the pushed edge is
+the partial isomorphism
+\[
+ \widehat\varphi_{i,r,s}
+ =\psi_s\circ\varphi_i\circ\psi_r^{-1}:
+ \psi_r(P_{i,r,s})\longrightarrow
+ \psi_s(\varphi_i(P_{i,r,s})).
+\]
+Its domain and range are Borel, it preserves measure, and its graph is
+contained in $\mathcal Q|Y$.  Its cost equals
+$\lambda(P_{i,r,s})$.  Let $\mathcal F$ consist of all these maps.
+Although the images of distinct pieces may overlap, graphing cost is
+the sum over the indexed family, and the original pieces partition
+each $D'_i$.  Hence
+\begin{equation}\label{compression:pushed-cost}
+ C(\mathcal F)=\sum_{i,r,s}\lambda(P_{i,r,s})=C(\mathcal E').
+\end{equation}
+One may discard identity maps or redundant instances, in which case
+the equality becomes an inequality in the needed direction.
+
+\paragraph{Recovering the relation.}
+For two $\mathcal Q$-related points in $Y$, take a finite path with
+steps in $\mathcal S$ and $\mathcal E$ and apply $f$ to its vertices.
+The endpoints remain fixed.  Each $\mathcal S$-step remains such a
+step, each deleted edge becomes stationary, and each surviving edge
+becomes an edge of $\mathcal F$.  Therefore
+\[
+ \mathcal Q|Y=(\mathcal S|Y)\vee\mathcal F.
+\]
+For arbitrary $x\mathrel{\mathcal Q}x'$, choose
+$y\in[x]_{\mathcal T}\cap Y$ and
+$y'\in[x']_{\mathcal T}\cap Y$ using completeness of $Y$.
+Then $y\mathrel{\mathcal Q}y'$, so the preceding identity joins them
+using $\mathcal S\subseteq\mathcal T$ and $\mathcal F$.
+Together with the first and last $\mathcal T$-steps this proves
+$\mathcal Q=\mathcal T\vee\mathcal F$.
+
+Finally, by \eqref{compression:retained-classes},
+\eqref{compression:deleted-cost}, and
+\eqref{compression:pushed-cost},
+\[
+ \begin{aligned}
+ C(\mathcal F)
+ &\leq C(\mathcal E)-\kappa(\mathcal S;Z\setminus Y)\\
+ &=C(\mathcal E)-\kappa(\mathcal S)+\kappa(\mathcal S;Y)\\
+ &\leq C(\mathcal E)-\kappa(\mathcal S)+\lambda(M)\\
+ &\leq C(\mathcal E)-\kappa(\mathcal S)
+                       +\kappa(\mathcal T)+\varepsilon.
+ \end{aligned}
+\]
+This is \eqref{compression:bound}.
+\end{proof}
+
+\begin{corollary}\label{compression:cost-one}
+A countable p.m.p. Borel equivalence relation on a standard probability
+space, all of whose classes are infinite, has cost at least one.
+\end{corollary}
+\begin{proof}
+Let $\mathcal E$ be a finite-cost generating graphing of the relation
+$\mathcal Q$. Apply Lemma~\ref{compression:main} with
+$\mathcal S=\mathrm{id}$ and $\mathcal T=\mathcal Q$. Since
+$\kappa(\mathcal S)=1$ and $\kappa(\mathcal Q)=0$, it gives
+$0\le C(\mathcal F)\le C(\mathcal E)-1+\varepsilon$ for every
+$\varepsilon>0$. Thus $C(\mathcal E)\ge1$. Infinite-cost graphings
+satisfy this bound as well; taking the infimum proves the assertion.
+\end{proof}
+
+\begin{remark}\label{compression:normalization}
+The estimate uses the original finite measure throughout; in
+particular $\kappa(\mathrm{id}_Z)=\lambda(Z)$.  The output graphing
+stays inside the same relation $\mathcal Q$, but no inclusion between
+the original and replacement edge families is asserted.  These are
+the two features needed when applying the lemma repeatedly on an
+enlarged space.
+\end{remark}
diff --git a/preprints/A-group-without-fixed-price-October-5-2026/build/conclusion.tex b/preprints/A-group-without-fixed-price-October-5-2026/build/conclusion.tex
new file mode 100644
index 0000000000000000000000000000000000000000..b8bc28c8e1633077ebdcd3968b0b400099a124e6
--- /dev/null
+++ b/preprints/A-group-without-fixed-price-October-5-2026/build/conclusion.tex
@@ -0,0 +1,50 @@
+\section{Completing the cost comparison}
+\label{conclusion:section}
+
+We now combine the measurable and finite arguments on the same group
+and the same Bernoulli action.
+
+\begin{proof}[Proof of Theorem~\ref{intro:main}]
+Lemma~\ref{actions:normal-form} shows that $\Gamma$ is finitely
+generated and that $J$ is infinite. The actions of
+Section~\ref{actions:section} are essentially free and p.m.p., and
+conull restriction does not change their costs.
+
+Fix $\alpha$ with the inequalities in the theorem, and set
+$\eta=\alpha/100>0$. Proposition~\ref{models:sequence} gives exact
+finite right $A$-sets $V_k$ of sizes $n_k\to\infty$, with vanishing
+fixed-point fractions for every nonidentity $A$-element, expansion
+\eqref{models:expansion}, and $b(V_k)=o(n_k)$. For all sufficiently
+large $k$, we have $\alpha n_k\ge1$ and
+$100b(V_k)\le\alpha n_k$. The row systems defining $D_{V_k}$ then
+satisfy every hypothesis of Theorem~\ref{rank:main}. Hence
+\[
+ \operatorname{rk}(D_{V_k})>\eta n_k
+ \quad\text{for all sufficiently large }k.
+\]
+By Proposition~\ref{models:transfer},
+\[
+ \delta_X\ge\limsup_{k\to\infty}
+       \frac{\operatorname{rk}(D_{V_k})}{n_k}\ge\eta.
+\]
+Proposition~\ref{deployment:main} therefore gives
+\[
+ \operatorname{Cost}(\mathcal R_{\Gamma\curvearrowright X})
+ \ge1+\delta_X\ge1+\eta.
+\]
+For every integer $M>100$, Proposition~\ref{actions:low} gives
+\[
+ \operatorname{Cost}(\mathcal R_{\Gamma\curvearrowright Y_M})
+ \le1+99/M.
+\]
+The $Y_M$-orbits are infinite, because $\Gamma$ is infinite and its
+action is free. Corollary~\ref{compression:cost-one} therefore gives
+\[
+ 1\le\operatorname{Cost}(\mathcal R_{\Gamma\curvearrowright Y_M})
+       \le1+99/M,
+\]
+so these costs tend to one as $M\to\infty$.
+An integer $M>\max\{100,99/\eta\}$ exists and satisfies
+$1+99/M<1+\eta$. The two costs are therefore strictly different,
+which proves the theorem.
+\end{proof}
diff --git a/preprints/A-group-without-fixed-price-October-5-2026/build/deployment.tex b/preprints/A-group-without-fixed-price-October-5-2026/build/deployment.tex
new file mode 100644
index 0000000000000000000000000000000000000000..c026ea5987116e5baeb15033de655b73dac57de9
--- /dev/null
+++ b/preprints/A-group-without-fixed-price-October-5-2026/build/deployment.tex
@@ -0,0 +1,257 @@
+\section{Deployment and a relative cost lower bound}
+\label{deployment:section}
+
+We next turn a graphing of an amalgamated product action into a graphing
+of one factor with the amalgamating relation supplied for free.
+The construction is Gaboriau's deployment of a graphing
+\cite[\S\,IV.28--IV.32]{gaboriau-2000}, specialized to finitely
+many sheets. The compression lemma accounts for its cost.
+Throughout this section,
+\(\Gamma=A*_{J}B\) is finitely generated, \(J\) is infinite, and
+\(\Gamma\) acts essentially freely by measure-preserving Borel
+automorphisms on a standard probability space \((X,\mu)\).
+We use right actions and write \(\mathcal R_L\) for the
+orbit relation of a subgroup \(L\leq\Gamma\).  Define
+\[
+ \delta_X=\inf\bigl\{C(\mathcal E):
+   \mathcal E\text{ is a graphing in }\mathcal R_A,
+   \quad\mathcal R_J\vee\mathcal E=\mathcal R_A\bigr\}.
+\]
+Thus only the edges added to \(\mathcal R_J\) are charged in
+\(\delta_X\).
+
+\begin{proposition}
+\label{deployment:main}
+Let $\Gamma=A*_J B$ be finitely generated, let $J$ be infinite, and
+let $\Gamma$ act essentially freely and p.m.p. on a standard probability
+space $(X,\mu)$. If $\delta_X$ is the infimum cost of graphings that
+generate $\mathcal R_A$ together with the supplied relation
+$\mathcal R_J$, then
+\[
+ \operatorname{Cost}(\mathcal R_\Gamma)\geq 1+\delta_X.
+\]
+\end{proposition}
+
+We prove the proposition using graphings at finite stages.  The only
+limit will concern the sizes of increasing equivalence classes.
+We first pass to an invariant conull Borel set on which the action is
+free.  Generation identities may also be made exact after removing
+countably many null sets and their saturations under the ambient
+countable p.m.p. relation.  These restrictions leave all costs unchanged.
+
+\subsection{A finite graphing and its subdivision}
+\label{deployment:finite}
+
+Fix \(\rho>0\).  There is a finite generating graphing of
+\(\mathcal R_\Gamma\), consisting of restrictions of nonidentity group
+translations, whose cost \(c\) satisfies
+\begin{equation}
+\label{deployment:near-optimal}
+ c<\operatorname{Cost}(\mathcal R_\Gamma)+\rho.
+\end{equation}
+Here is the approximation argument.  Start with a countable generating
+graphing of cost less than
+\(\operatorname{Cost}(\mathcal R_\Gamma)+\rho/2\), and split each map
+into disjoint Borel pieces according to its group displacement.  This
+does not change its cost; discard identity restrictions, which can only
+lower it.  Let \(\Psi_N\) be the first \(N\) maps of
+the resulting graphing and let \(T_N\) be their generated relation.
+For a fixed finite generating set \(K\) of \(\Gamma\) not containing
+the identity, put
+\[
+ D_{N,s}=\{x\in X:(x,xs)\notin T_N\},\qquad s\in K.
+\]
+Each set is Borel because membership in \(T_N\) is witnessed by a
+finite path.  For each \(s\), these sets decrease to a null set, so
+\(\sum_{s\in K}\mu(D_{N,s})\to0\).  Adding the translation by \(s\)
+on \(D_{N,s}\), for every \(s\in K\), gives a finite generating
+graphing and proves \eqref{deployment:near-optimal} for large \(N\).
+
+Write its maps as \(x\mapsto xg_h\) on Borel domains \(D_h\), with
+\(h\) in a finite index set.  Choose a factor
+word
+\[
+ g_h=g_{h,1}\cdots g_{h,\ell_h},\qquad
+ g_{h,i}\in A\cup B,\quad \ell_h\geq1.
+\]
+For letters in \(J\), choose either factor type.  Form a space \(Z\)
+by adjoining \(\ell_h-1\) disjoint copies of \(D_h\) to \(X\), one
+for each internal vertex of this word.  Give each copy its inherited
+measure and denote the resulting, unnormalized measure by \(\lambda\).
+Then
+\[
+ \lambda(Z)=1+\sum_h(\ell_h-1)\mu(D_h)<\infty.
+\]
+Define \(P:Z\to X\) to be the identity on \(X\), and on the copy
+indexed by \((h,i)\) define
+\[
+ P(x,h,i)=xg_{h,1}\cdots g_{h,i}.
+\]
+The restriction of \(P\) to each sheet is a measure-preserving Borel
+bijection onto its image.  For \(L=\Gamma,A,B,J\), let
+\[
+ \overline{\mathcal R}_L
+   =\{(z,z')\in Z^2:(Pz,Pz')\in\mathcal R_L\}.
+\]
+These are countable p.m.p. Borel equivalence relations.  Indeed, if
+\(P_r\) and \(P_s\) denote the restrictions to two sheets, the partial
+maps \(P_s^{-1}\circ(x\mapsto x\ell)\circ P_r\), for \(\ell\in L\),
+enumerate their pairs and preserve measure on their natural domains.
+
+Replace each original edge by the chain through its new vertices, and
+divide the successive edges according to their factor types into
+\(\mathcal E_A^0\) and \(\mathcal E_B^0\).  Every chain edge belonging
+to the \(h\)-th map has cost \(\mu(D_h)\), so
+\begin{equation}
+\label{deployment:initial-cost}
+ C(\mathcal E_A^0)+C(\mathcal E_B^0)
+   =c+\lambda(Z)-1.
+\end{equation}
+Their join is \(\overline{\mathcal R}_\Gamma\).  To verify this,
+first connect base-sheet points by replacing paths in the original
+graphing by their subdivided paths.  Every additional point is attached
+along its chain to a base-sheet point in the same projected orbit.
+This also connects arbitrary pairs in the pullback relation.
+
+\subsection{Increasing the supplied relation}
+\label{deployment:iteration}
+
+Set \(S_0=\mathrm{id}_Z\).  We construct relations
+\(S_n\subseteq\overline{\mathcal R}_J\) and graphings
+\(\mathcal E_i^n\subseteq\overline{\mathcal R}_i\), for \(i=A,B\),
+and put
+\[
+ Q_i^n=S_n\vee\mathcal E_i^n.
+\]
+Fix \(\epsilon>0\), and choose positive errors \(e_n\) with
+\(\sum_{n\geq0}e_n<\epsilon\).
+At stage \(n\), choose \(i=A\) or \(B\) alternately and set
+\[
+ S_{n+1}=Q_i^n\cap\overline{\mathcal R}_J.
+\]
+We have \(S_n\subseteq S_{n+1}\subseteq Q_i^n\).  Apply
+Lemma~\ref{compression:main} with
+\(S=S_n\), \(T=S_{n+1}\), \(Q=Q_i^n\), and error \(e_n\).
+It replaces \(\mathcal E_i^n\) by \(\mathcal E_i^{n+1}\) such that
+\[
+ Q_i^n=S_{n+1}\vee\mathcal E_i^{n+1}
+\]
+and
+\[
+ C(\mathcal E_i^{n+1})
+ \leq C(\mathcal E_i^n)-\kappa(S_n)+\kappa(S_{n+1})+e_n.
+\]
+Leave the other edge family unchanged.  The processed relation
+\(Q_i^{n+1}\) equals \(Q_i^n\); the other relation grows because its
+supplied relation grows.  Thus both sequences \(Q_i^n\) increase,
+remain inside their respective factor relations, and have join
+\(\overline{\mathcal R}_\Gamma\) at every stage.  The edge families
+themselves need not be increasing.
+
+All costs and all values of \(\kappa\) here use \(\lambda\), without
+normalizing its total mass.  Summing the compression inequalities and
+using \(\kappa(S_0)=\lambda(Z)\) and
+\eqref{deployment:initial-cost} gives
+\begin{equation}
+\label{deployment:finite-bound}
+ C(\mathcal E_A^n)+C(\mathcal E_B^n)
+  \leq c-1+\kappa(S_n)+\epsilon
+  \qquad(n\geq0).
+\end{equation}
+In particular all graphings used in subsequent applications of the
+compression lemma have finite cost.
+
+\subsection{Exhausting the amalgamating relation}
+\label{deployment:limit}
+
+Let \(S_\infty=\bigcup_nS_n\) and
+\(Q_i^\infty=\bigcup_nQ_i^n\).  These are Borel equivalence relations,
+and their monotonicity gives
+\begin{equation}
+\label{deployment:intersections}
+ Q_A^\infty\vee Q_B^\infty=\overline{\mathcal R}_\Gamma,
+ \qquad Q_i^\infty\cap\overline{\mathcal R}_J=S_\infty.
+\end{equation}
+For the nontrivial inclusion in the second identity, a pair in
+\(Q_i^n\cap\overline{\mathcal R}_J\) enters \(S_{m+1}\) at the next
+later stage \(m\) at which type \(i\) is processed.
+
+We claim that \(S_\infty=\overline{\mathcal R}_J\).  Take a pair in
+\(\overline{\mathcal R}_J\) and a shortest chain between its endpoints
+with steps in \(Q_A^\infty\) or \(Q_B^\infty\).  If the chain has at
+least two steps, its types alternate: adjacent steps of the same type
+could be composed.  Moreover, no step lies in
+\(\overline{\mathcal R}_J\).  By
+\eqref{deployment:intersections}, such a step would belong to
+\(S_\infty\), hence to both factor relations, and could be composed
+with a neighboring step.
+
+Projecting the chain to \(X\), freeness gives unique group
+displacements for its steps.  These form an alternating word with
+letters outside \(J\), whose product belongs to \(J\) because the
+projected endpoints are \(J\)-related.  This contradicts amalgam
+normal form.  The shortest chain therefore has length at most one;
+\eqref{deployment:intersections} puts its endpoint pair in
+\(S_\infty\), as claimed.
+
+Every \(\overline{\mathcal R}_J\)-class is infinite: the class of
+\(z\) contains the distinct base-sheet points \(P(z)j\), for
+\(j\in J\).  Since \([z]_{S_n}\) increases with this infinite union,
+\[
+ \frac{1}{|[z]_{S_n}|}\downarrow0.
+\]
+The integrands are bounded by \(1\) on the fixed finite-measure space
+\(Z\).  Dominated convergence consequently gives
+\begin{equation}
+\label{deployment:kappa-limit}
+ \kappa(S_n)\longrightarrow0.
+\end{equation}
+
+\subsection{Returning to the original space}
+\label{deployment:projection}
+
+Fix a finite stage \(n\).  For each map \(\varphi\) in
+\(\mathcal E_i^n\), partition its domain according to both its source
+sheet \(r\) and its target sheet \(s\).  On such a piece \(D\), the
+map
+\[
+ P_s\circ\varphi\circ P_r^{-1}:
+ P_r(D)\longrightarrow P_s(\varphi(D))
+\]
+is a p.m.p. Borel partial isomorphism on \(X\), belongs to
+\(\mathcal R_i\), and has cost \(\lambda(D)\).  These maps form a
+graphing \(\mathcal F_i^n\) with
+\begin{equation}
+\label{deployment:projection-cost}
+ C(\mathcal F_i^n)\leq C(\mathcal E_i^n).
+\end{equation}
+This construction uses the sheet restrictions of \(P\); it requires
+no injectivity of \(P\) on all of \(Z\).
+
+Put \(T_i=\mathcal R_J\vee\mathcal F_i^n\).  Projecting paths from
+the base sheet shows that
+\(T_A\vee T_B=\mathcal R_\Gamma\): every \(S_n\)-step projects into
+\(\mathcal R_J\), and all remaining steps project into the indicated
+graphings.  We show that \(T_A=\mathcal R_A\).
+For a pair in \(\mathcal R_A\), choose a shortest chain with steps
+from \(T_A,T_B\).  A chain of length at least two would alternate and
+contain no \(\mathcal R_J\)-step, since that relation is contained in
+both \(T_A\) and \(T_B\).  Its displacement word would alternate
+outside \(J\), have length at least two, and belong to \(A\), again
+contradicting amalgam normal form.  A single step of type \(B\)
+between \(A\)-related endpoints has displacement in
+\(A\cap B=J\), so it already belongs to \(T_A\).  This proves the
+claim.
+
+Thus \(\mathcal F_A^n\) is admissible in the definition of
+\(\delta_X\).  Equations \eqref{deployment:finite-bound} and
+\eqref{deployment:projection-cost} give, at every finite stage,
+\[
+ \delta_X\leq C(\mathcal F_A^n)
+ \leq C(\mathcal E_A^n)+C(\mathcal E_B^n)
+ \leq c-1+\kappa(S_n)+\epsilon.
+\]
+Let \(n\to\infty\) using \eqref{deployment:kappa-limit}, then let
+\(\epsilon\) and \(\rho\) tend to zero in
+\eqref{deployment:near-optimal}.  This proves
+Proposition~\ref{deployment:main}.
diff --git a/preprints/A-group-without-fixed-price-October-5-2026/build/finite-models.tex b/preprints/A-group-without-fixed-price-October-5-2026/build/finite-models.tex
new file mode 100644
index 0000000000000000000000000000000000000000..00ce9e11dab55f6e572b69b4a28f437d8d7cfefc
--- /dev/null
+++ b/preprints/A-group-without-fixed-price-October-5-2026/build/finite-models.tex
@@ -0,0 +1,327 @@
+\section{Bernoulli graphings and finite permutation models}
+\label{models:section}
+
+We next turn the relative graphing quantity into a generator bound for
+explicitly presented groups. The finite models used here are actions of
+$A$ alone. We then construct models whose defining rows have the
+expansion and overlap properties needed for the rank argument.
+
+A finite exact right $A$-set is a finite set equipped with a right
+action of $A$ by permutations; the action laws hold as equalities.
+For such a set $V$, put
+\begin{equation}
+ D_V=\left\langle x_v\ (v\in V)\ \middle|\
+ x_vx_{vu_1}\cdots x_{vu_{99}}=1\quad(v\in V)\right\rangle.
+ \label{models:presentation}
+\end{equation}
+Here $u_0=1$ and $u_i=u_{i-1}ab_i$, as in the definition of $J$.
+The notation $\operatorname{rk}(D)$ means the minimum size of a
+generating set of a group $D$. The groups $D_V$ need not be finite.
+We write $\operatorname{Fix}_V(g)=\{v\in V:vg=v\}$.
+
+\subsection{The transfer inequality}
+
+\begin{proposition}\label{models:transfer}
+Let $V_k$ be finite exact right $A$-sets, with $n_k=|V_k|\to\infty$,
+such that for every fixed $g\in A\setminus\{1\}$,
+\begin{equation}
+ \frac{|\{v\in V_k:vg=v\}|}{n_k}\longrightarrow0.
+ \label{models:fixed-points}
+\end{equation}
+For the free Bernoulli action on $X$ from
+Section~\ref{actions:section}, whose cost agrees with the full product
+action on $X_0=[0,1]^\Gamma$, one has
+\[
+ \delta_X\ge\limsup_{k\to\infty}
+ \frac{\operatorname{rk}(D_{V_k})}{n_k}.
+\]
+\end{proposition}
+
+\begin{proof}
+We use right actions, so that $(xg)(h)=x(gh)$. Choose representatives
+$h_c$ for the cosets $Ah_c$ in $\Gamma$, and let
+$\Omega=[0,1]^{A\backslash\Gamma}$ with its product probability measure.
+The coordinate rearrangement
+\[
+ x\longmapsto \bigl((x(gh_c))_c\bigr)_{g\in A}
+\]
+identifies the restriction of $X_0$ to $A$ with $\Omega^A$, with independent
+base coordinates and right action $(\xi q)(g)=\xi(qg)$.
+We write $\mu$ for the product measure on this space. All graphing
+identities are considered on invariant conull subsets, so the
+restriction to $X$ does not change the argument.
+
+For every finite exact $A$-set $V$, define
+$\theta:V\times A\to D_V$ on the free basis of $A$ by
+\[
+ \theta(v,a)=x_v,\qquad \theta(v,b_i)=1,
+ \qquad
+ \theta(v,s^{-1})=\theta(vs^{-1},s)^{-1}
+\]
+for each basis letter $s$. Extend by multiplying these values along a
+word. Cancellation of adjacent inverse letters cancels their two
+values at the corresponding sources. Since $A$ is free, this gives a
+well-defined function satisfying
+\begin{equation}
+ \theta(v,gh)=\theta(v,g)\theta(vg,h).
+ \label{models:cocycle}
+\end{equation}
+The word $w=u_{99}a=ab_1ab_2\cdots ab_{99}a$ gives
+\[
+ \theta(v,w)=x_vx_{vu_1}\cdots x_{vu_{99}}=1.
+\]
+Also $\theta(v,b_i)=1$ at every source. The inverse rule and
+\eqref{models:cocycle} therefore give
+\begin{equation}
+ \theta(v,j)=1\qquad(v\in V,\ j\in J).
+ \label{models:j-trivial}
+\end{equation}
+Notice that $\theta(v,g)$ depends on the group label $g$, not merely on
+the pair of endpoints $(v,vg)$.
+
+Fix a finite-cost graphing $\mathcal E$ such that
+$\mathcal R_J\vee\mathcal E=\mathcal R_A$, and fix $\epsilon>0$.
+Such finite-cost graphings suffice for the infimum defining $\delta_X$:
+the full $a$-translation, together with $\mathcal R_J$, already
+generates $\mathcal R_A$.
+On an invariant conull free set, split each map in $\mathcal E$
+according to its unique displacement in $A$. This produces countably
+many restrictions
+\[
+ \varphi_\ell:L_\ell\longrightarrow L_\ell g_\ell,
+ \qquad \varphi_\ell(x)=xg_\ell,
+ \qquad g_\ell\in A,
+\]
+with $\sum_\ell\mu(L_\ell)=C(\mathcal E)$.
+
+A path type is a finite list of full $J$-translations and signed
+indices $\ell$ of these paid maps. Its label is the product in $A$ of
+its successive step labels. Almost every $x$ has a valid path to $xa$.
+Freeness implies that every such path has label exactly $a$ in $A$.
+There are countably many path types. Choose finitely many, each with
+label $a$, whose validity sets cover a set of measure at least
+$1-\epsilon$. Let $I$ be their finite set of paid indices, and let
+$m_\ell$ count all occurrences of index $\ell$ among these types.
+
+Choose cylinder-measurable sets $L'_\ell$, depending on finitely many
+$A$-coordinates, such that
+\[
+ \sum_{\ell\in I}(1+m_\ell)
+       \mu(L_\ell\mathbin{\triangle}L'_\ell)<\epsilon.
+\]
+This approximation is available because finite-coordinate measurable
+sets form an algebra generating the product sigma-algebra. Its closure
+for the metric $\mu(E\mathbin{\triangle}F)$ is a sigma-algebra, so
+every measurable set is approximable in this metric.
+
+For clarity, a forward step of index $\ell$ after a prefix with label
+$h$ tests $xh\in L_\ell$. An inverse step after that prefix tests
+$xhg_\ell^{-1}\in L_\ell$. Replace these tests by the corresponding
+tests in $L'_\ell$, keeping all path labels unchanged. Each occurrence
+changes its validity predicate on a set of measure at most
+$\mu(L_\ell\mathbin{\triangle}L'_\ell)$, by invariance of $\mu$.
+If $P'$ is the union of the modified path validity sets, then
+\begin{equation}
+ \sum_{\ell\in I}\mu(L'_\ell)\le C(\mathcal E)+\epsilon,
+ \qquad \mu(P')\ge1-2\epsilon.
+ \label{models:cylinder-bounds}
+\end{equation}
+All sets $L'_\ell$ and $P'$ depend on a single finite coordinate set
+$F\subset A$: for example, a test of an $F_\ell$-cylinder after prefix
+$h$ uses coordinates $hF_\ell$, or $hg_\ell^{-1}F_\ell$ for an inverse
+step. This finite set is fixed before $k$ tends to infinity.
+
+We have reduced generation to finitely many cylinder tests. We now
+evaluate these same tests on $V_k$.
+For independent base labels $(z_v)_{v\in V_k}$ with law $\Omega$, put
+\[
+ \xi_v=(z_{vg})_{g\in A}.
+\]
+Exactness of the action gives $\xi_{vq}=\xi_vq$. If $vg\ne vh$ for
+all distinct $g,h\in F$, the $F$-coordinates of $\xi_v$ have exactly
+the product distribution. The fraction of exceptional sources is at
+most
+\[
+ \rho_k(F):=
+ \sum_{\substack{g,h\in F\\g\ne h}}
+ \frac{|\{v\in V_k:vgh^{-1}=v\}|}{n_k},
+\]
+which tends to zero by \eqref{models:fixed-points}. Consequently, for
+every measurable $F$-cylinder $E$,
+\begin{equation}
+ \left|\frac1{n_k}\sum_{v\in V_k}
+       \Pr(\xi_v\in E)-\mu(E)\right|\le\rho_k(F).
+ \label{models:local-law}
+\end{equation}
+This uses no independence between different sources $v$.
+
+For any realization of the labels, make a list containing
+$\theta(v,g_\ell)$ for every $\ell\in I$ and every $v$ with
+$\xi_v\in L'_\ell$. If $\xi_v\in P'$, follow a modified valid path
+from $v$. A forward paid factor occurs on this list. For an inverse
+paid step after prefix $h$, the factor is
+\[
+ \theta(vh,g_\ell^{-1})
+   =\theta(vhg_\ell^{-1},g_\ell)^{-1},
+\]
+and its inverse-domain test puts
+$\theta(vhg_\ell^{-1},g_\ell)$ on the list.
+Each $J$-step contributes $1$ by \eqref{models:j-trivial}. Since the
+path label is exactly $a$, \eqref{models:cocycle} expresses
+$x_v=\theta(v,a)$ as a product from this list and its inverses.
+The finite configuration $\xi_v$ need not be free; freeness was used
+only to select the original path labels.
+
+Add $x_v$ for every source with $\xi_v\notin P'$. The resulting list
+generates $D_{V_k}$, so for every labeling,
+\[
+ \operatorname{rk}(D_{V_k})\le
+ \sum_{\ell\in I}\sum_{v\in V_k}\mathbf1_{L'_\ell}(\xi_v)
+ +\sum_{v\in V_k}\mathbf1_{(P')^c}(\xi_v).
+\]
+Taking expectations, using \eqref{models:local-law}, and then letting
+$k\to\infty$ gives
+\[
+ \limsup_k\frac{\operatorname{rk}(D_{V_k})}{n_k}
+ \le\sum_{\ell\in I}\mu(L'_\ell)+1-\mu(P')
+ \le C(\mathcal E)+3\epsilon.
+\]
+Let $\epsilon$ tend to zero and then take the infimum over
+$\mathcal E$. All finite path and cylinder tests were fixed before
+taking the limit in $k$.
+\end{proof}
+
+\subsection{Expanding rows with few overlaps}
+
+Regard the defining relator for $v$ as an ordered row with columns
+$vu_0,vu_1,\ldots,vu_{99}$. For $T\subseteq V$, write
+\[
+ N(T)=\bigcup_{v\in T}\{vu_0,vu_1,\ldots,vu_{99}\}.
+\]
+A row is called bad if it has a repeated column or shares at least
+two distinct columns with a different row. Let $b(V)$ be the number
+of bad rows.
+
+\begin{proposition}\label{models:sequence}
+Put $K=e^{96}96^{99}/95^{95}$. Fix
+\[
+ 0<\alpha<1/200,\qquad K\alpha^3<1/2.
+\]
+There exists a sequence of finite exact right $A$-sets $V_k$, with
+$n_k=|V_k|\to\infty$, satisfying \eqref{models:fixed-points} and
+\begin{align}
+ |N(T)|&>96|T|
+ &&(T\subseteq V_k,\ 1\le |T|\le\alpha n_k),
+ \label{models:expansion}\\
+ b(V_k)&=o(n_k).
+ \label{models:bad-rows}
+\end{align}
+\end{proposition}
+
+\begin{proof}
+First, $a,u_1,\ldots,u_{99}$ is a free basis of $A$. Indeed, the
+substitutions defining $u_i$ have the inverse substitutions
+\[
+ b_i=a^{-1}u_{i-1}^{-1}u_i.
+\]
+Between abstract free groups on the two proposed bases, these formulas
+give inverse homomorphisms: they recover $b_i$ directly and recover
+$u_i$ by induction. Assign independent uniform permutations of
+$\{1,\ldots,n\}$ to the 100 elements of this basis. This defines an
+exact right $A$-action.
+
+Fix an integer $s$ with $1\le s\le\lfloor\alpha n\rfloor$. If a set
+$T$ of size $s$ violates \eqref{models:expansion}, then $N(T)$ is
+contained in a set $C$ of size $96s$ containing $T$, since $u_0=1$.
+Such a set exists because $96s<n$. For fixed $T$ there are
+$\binom{n-s}{95s}$ possible sets $C$. For one uniform permutation
+$\sigma$, the probability that $T\sigma\subseteq C$ is
+\[
+ \frac{(96s)(96s-1)\cdots(96s-s+1)}{n(n-1)\cdots(n-s+1)}
+ \le \left(\frac{96s}{n}\right)^s.
+\]
+Using independence of the 99 permutations assigned to $u_1,\ldots,u_{99}$,
+followed by the bounds $\binom nr\le(en/r)^r$, gives
+\begin{align*}
+ \Pr\bigl(\exists T:\ |T|=s,\ |N(T)|\le96s\bigr)
+ &\le\binom ns\binom n{95s}
+                 \left(\frac{96s}{n}\right)^{99s}\\
+ &\le\left[K\left(\frac{s}{n}\right)^3\right]^s.
+\end{align*}
+The exponent is $99-1-95=3$. Set $q=K\alpha^3<1/2$.
+For every fixed integer $R\ge1$, the sum of these bounds over all
+allowed sizes is at most
+\[
+ \sum_{s=1}^{\min(R,\lfloor\alpha n\rfloor)}
+       [K(s/n)^3]^s+\frac{q^{R+1}}{1-q}.
+\]
+First let $n\to\infty$ and then $R\to\infty$. Thus
+\eqref{models:expansion} holds with probability tending to one.
+This includes $s=1$; if $\lfloor\alpha n\rfloor=0$, the assertion is
+vacuous.
+
+We also need \eqref{models:fixed-points}. Fix a nonempty freely reduced
+word of length $L$ in the random basis, and a starting vertex $v_0$.
+Follow the word, exposing permutation entries only when needed. Until
+the first repeated vertex, the current vertex has not occurred earlier
+except as the endpoint of its incoming edge. A positive next letter
+could request an already exposed domain entry only by immediately
+inverting that incoming edge. A negative next letter could request an
+already exposed range entry only in the same way. Both are excluded
+by free reduction. At step $t$, therefore, the requested partner is
+uniform among at least $n-t+1$ available vertices. At most $t$ of
+these vertices have already been visited. For $n\ge L$,
+\[
+ \Pr(v_0\text{ is fixed by the word})
+ \le\sum_{t=1}^L\frac{t}{n-t+1}
+ \le\frac{L(L+1)}{2(n-L+1)}.
+\]
+The same bound holds for the expected fixed-point fraction. It tends
+to zero for each fixed nonidentity element of $A$.
+
+Here is an explicit simultaneous choice. Enumerate the nonidentity
+elements of $A$ as $g_1,g_2,\ldots$. At stage $k$, take $n_k>n_{k-1}$
+so large that expansion fails with probability at most $1/4$, and
+\[
+ \mathbb E\frac{|\operatorname{Fix}_{V}(g_j)|}{n_k}
+       \le\frac1{4k^2}\qquad(1\le j\le k).
+\]
+Markov's inequality and a union bound show that the probability that
+some $j\le k$ has fixed-point fraction greater than $1/k$ is at most
+$1/4$. Choose an outcome with expansion and all these fixed-point
+bounds. This gives \eqref{models:fixed-points} for every fixed element,
+without requiring estimates uniform over words of increasing length.
+
+Finally we deduce \eqref{models:bad-rows}. A repeated column in row
+$v$ gives $vu_i=vu_j$ for some $i\ne j$, so $v$ is fixed by the
+nonidentity word $u_i u_j^{-1}$. If two distinct columns are shared
+between rows $v\ne y$, choose indices with
+\[
+ vu_i=yu_k,\qquad vu_j=yu_l.
+\]
+Then $i\ne j$ and $k\ne l$ by distinctness of the columns, while
+$i\ne k$ and $j\ne l$ follow from injectivity of each permutation.
+Consequently $v$ is fixed by
+\[
+ u_i u_k^{-1}u_lu_j^{-1}.
+\]
+This word is nontrivial. In the cyclic order $(i,k,l,j)$, neighboring
+indices differ. If all indices are nonzero, the displayed word is
+freely reduced. Removing one occurrence of $u_0=1$ joins letters of
+the same sign and leaves a nonempty reduced word. Two zero indices
+can occur only in opposite positions; the remaining two letters then
+have the same sign. Three zero indices are impossible. Thus every
+possible overlap gives a fixed point of a nonidentity word from a
+finite list. The union of the fixed sets in this list, together with
+the finite list for repeated columns, contains every bad row.
+Equation~\eqref{models:fixed-points} now gives $b(V_k)=o(n_k)$.
+\end{proof}
+
+For later use, the sequence in Proposition~\ref{models:sequence}
+eventually satisfies
+\[
+ \alpha n_k\ge1,\qquad 100b(V_k)\le\alpha n_k.
+\]
+These are the finite size and exceptional-row bounds used in the
+deterministic rank argument. Proposition~\ref{models:transfer} applies
+to this same sequence.
diff --git a/preprints/A-group-without-fixed-price-October-5-2026/build/group-actions.tex b/preprints/A-group-without-fixed-price-October-5-2026/build/group-actions.tex
new file mode 100644
index 0000000000000000000000000000000000000000..13911e03884386eda3bfb1e1158c63240e849a70
--- /dev/null
+++ b/preprints/A-group-without-fixed-price-October-5-2026/build/group-actions.tex
@@ -0,0 +1,254 @@
+\section{The group and an action of small cost}
+\label{actions:section}
+
+We construct the group and two free probability-measure-preserving actions
+that will be compared.  The purpose of this section is to prove the upper
+bound for the action with an additional finite height coordinate.
+All actions are right actions: $(zg)h=z(gh)$.
+
+\subsection{The amalgam}
+
+Let $A=F(a,b_1,\ldots,b_{99})$ be the free group on the
+displayed generators.  Define words in $A$ by
+\[
+ u_0=1,\qquad u_i=u_{i-1}ab_i\quad(1\leq i\leq99),
+ \qquad w=u_{99}a.
+\]
+Let
+\begin{equation}
+ \label{actions:group}
+ J=\langle b_1,\ldots,b_{99},w\rangle\leq A,
+ \qquad B=J\times\langle t\rangle,
+ \qquad \Gamma=A*_J B,
+\end{equation}
+where $\langle t\rangle$ is infinite cyclic and the embedding of $J$ in
+$B$ is $j\mapsto(j,1)$.
+
+\begin{lemma}
+\label{actions:normal-form}
+The factors $A$ and $B$ embed in $\Gamma$ with intersection $J$.
+If $g_1,\ldots,g_n$ lie alternately in $A\setminus J$ and
+$B\setminus J$, then $g_1\cdots g_n\notin J$ for $n\geq1$, and
+$g_1\cdots g_n$ belongs to neither factor for $n\geq2$.
+The group $J$ is free on $b_1,\ldots,b_{99},w$; in particular it is
+infinite.  The group $\Gamma$ is countably infinite and generated by
+$a,b_1,\ldots,b_{99},t$.
+\end{lemma}
+
+\begin{proof}
+Choose, in each factor, representatives for the cosets $gJ$, with $1$
+representing $J$.  Consider formal strings
+\[
+ r_1\cdots r_n j,
+ \qquad j\in J,
+\]
+where the $r_i$ are nonidentity representatives, alternately from the
+two factors; the case $n=0$ is allowed.  Define right multiplication by
+$g$ in one factor as follows.  If the last representative is from that
+factor, combine it with the final $j$ and $g$, and write their product
+uniquely as $rj'$ in that factor.  Otherwise write $jg=rj'$ and append
+$r$.  Omit $r$ when $r=1$.
+
+This gives a right action of each factor on the strings.  To check the
+action law for two successive multipliers in the same factor, retain
+the prefix ending in the other factor and simply multiply the remaining
+suffix in the chosen factor.  This description continues to apply when
+the suffix representative becomes $1$.  The two actions agree for
+multipliers in $J$, which only multiply the final $J$-entry.  They
+therefore define an action of the amalgam on the strings.
+
+Acting on the identity string by a factor element recovers its coset
+decomposition, so the factors embed.  Their possible strings intersect
+exactly in the strings with no representatives, proving $A\cap B=J$.
+Successively multiplying by an alternating sequence outside $J$
+adds one nonidentity representative at every step: for $j\in J$ and
+$g\notin J$, one has $jg\notin J$.  The resulting string has length
+$n$, whereas elements of $J$ have length zero and factor elements have
+length at most one.  This proves the stated normal-form assertions.
+
+For $J$, expand any reduced nonempty word in the abstract generators
+$b_1,\ldots,b_{99},w$ in the free basis of $A$.
+The word $w=ab_1ab_2\cdots ab_{99}a$ starts and ends with $a$;
+$w^{-1}$ starts and ends with $a^{-1}$.  No boundary between a
+$b_i^{\pm1}$ and $w^{\pm1}$ cancels.  Two successive $w$-symbols
+either have equal signs, in which case their boundary does not cancel,
+or would be inverse symbols, which reducedness excludes.  Boundaries
+between $b$-symbols do not cancel either.  Thus the expansion is a
+nonempty reduced word in $A$, proving that the displayed generators of
+$J$ are a free basis.
+
+Finally, the displayed generators of $\Gamma$ generate both factors;
+the embedded copy of $A$ makes $\Gamma$ infinite, and finite generation
+makes it countable.  The embedded copy of $B$ also shows that $t$ has
+infinite order in $\Gamma$.
+\end{proof}
+
+\subsection{Probability spaces and the two actions}
+
+We use the definitions of graphing, cost, and p.m.p. action from the
+introduction. A standard Borel space is a measurable space isomorphic
+to a Borel subset of a complete separable metric space. For a subgroup
+$L\leq\Gamma$, write $\mathcal R_L$ for its orbit relation on the
+space being considered.
+
+For an essentially free action of a countable group, the set of points with trivial stabilizer is a Borel invariant
+conull set.  Restricting
+to this set makes the action pointwise free and preserves the measure
+and orbit-relation cost.  More generally, a Borel conull set can be
+made invariant by intersecting its countably many translates.
+To see that an invariant conull restriction preserves cost, restrict
+any orbit graphing to that set; invariance keeps its edges inside
+the set, and domain measures do not change.  Conversely, a graphing
+on the restricted set generates the original relation on a conull
+set and has the same cost when viewed on the original space.
+
+Let
+\[
+ X_0=[0,1]^\Gamma,\qquad
+ \mu_0=\operatorname{Leb}^{\Gamma},\qquad
+ (xg)(h)=x(gh)\quad(g,h\in\Gamma).
+\]
+This is a standard Borel probability space, since it is a countable
+product of compact metrizable spaces, with its product measure.
+Coordinate permutations are Borel automorphisms preserving $\mu_0$,
+and the displayed formula satisfies the right-action law.  For
+$g\ne1$, the equality $xg=x$ implies $x(g)=x(1)$.  These are two
+independent uniform coordinates, so this equality has probability zero.
+Consequently
+\[
+ X=\{x\in X_0:xg\ne x\text{ for every }g\ne1\}
+\]
+is an invariant Borel conull set.  Write $\mu$ for the restricted
+probability measure.  We henceforth use the free Bernoulli action on
+$(X,\mu)$; it has the same orbit-relation cost as the original action
+on $(X_0,\mu_0)$.
+
+\begin{lemma}
+\label{actions:height}
+There is a homomorphism $\chi:\Gamma\to\mathbb Z$ with
+\[
+ \chi(a)=1,\qquad \chi(b_i)=\chi(t)=0.
+\]
+It satisfies $\chi(u_i)=i$ and $\chi(w)=100$.
+For every positive integer $M$, the action on
+\[
+ Y_M=X\times\mathbb Z/M\mathbb Z,
+ \qquad \nu_M=\mu\times\text{uniform measure},
+\]
+defined by
+\begin{equation}
+ \label{actions:product-action}
+ (x,r)g=(xg,r+\chi(g)\bmod M)
+\end{equation}
+is free and p.m.p. on a standard Borel probability space.
+\end{lemma}
+
+\begin{proof}
+The exponent sum of $a$ defines a homomorphism $\chi_A:A\to\mathbb Z$.
+Define $\chi_B(jt^n)=\chi_A(j)$ on $B=J\times\langle t\rangle$.
+This is a homomorphism and agrees with $\chi_A$ on $J$, so the
+amalgam's universal property gives $\chi$.  The formulas for $u_i$
+and $w$ follow by their definitions.  In particular, the restriction
+of $\chi$ to $J$ need not be zero: it sends $w$ to $100$.
+The homomorphism law verifies the right-action law in
+\eqref{actions:product-action}.  Both coordinate actions preserve
+their measures.  Freeness follows from freeness on the first
+coordinate, and the finite product remains standard Borel.
+\end{proof}
+
+\subsection{A graphing on the height extension}
+
+The direct-product factor $B$ can be generated at arbitrarily small
+cost above one by retaining all $t$-edges and using the generators of
+$J$ on a small set meeting every $t$-orbit.  This is the product
+construction in Gaboriau's Proposition~VI.23
+\cite[Proposition~VI.23]{gaboriau-2000}.  For our Bernoulli action the
+small set has the following explicit description.
+
+\begin{lemma}
+\label{actions:small-marker}
+For every $0<p<1$ and every positive integer $M$, there is a Borel
+set $C_p\subseteq Y_M$ of measure $p$ meeting every
+$\langle t\rangle$-orbit.  The graphing consisting of $t$ on all
+of $Y_M$ and $b_1,\ldots,b_{99},w$ restricted to $C_p$ generates
+$\mathcal R_B$ and has cost $1+100p$.
+\end{lemma}
+
+\begin{proof}
+In $X$, put
+\[
+ N_p=\{x\in X:x(t^n)\geq p\text{ for every }n\in\mathbb Z\},
+ \qquad
+ C_p=(\{x\in X:x(1)<p\}\cup N_p)\times\mathbb Z/M\mathbb Z.
+\]
+The set $N_p$ is Borel, invariant under $t$, and null: the coordinates
+$x(t^n)$ are independent, and the group elements $t^n$ are distinct.
+Indeed, the probability that the condition holds for $0\leq n<N$
+is $(1-p)^N$.  Thus $\nu_M(C_p)=p$.
+If a $t$-orbit does not meet the first part of $C_p$, all its points
+belong to $N_p\times\mathbb Z/M\mathbb Z$.  Hence $C_p$ meets every
+$t$-orbit, including the exceptional ones.
+
+Let $j$ be one of $b_1,\ldots,b_{99},w$, and let $z\in Y_M$.
+Choose $n\in\mathbb Z$ with $zt^n\in C_p$.  The graphing contains
+the path
+\[
+ z\longrightarrow zt^n\longrightarrow zt^nj
+   \longrightarrow zt^njt^{-n}=zj,
+\]
+where the first and last portions use $t$ or $t^{-1}$.  The final
+identity uses $jt=tj$ in $B$.  Thus every generator of $B$ is
+available everywhere in the generated relation.  Conversely, all
+the graphing's maps are restrictions of elements of $B$.
+Each of the $100$ restricted generators costs $p$, and the full
+$t$-map costs one.
+\end{proof}
+
+\begin{proposition}
+\label{actions:low}
+For every integer $M>100$, the free p.m.p. action
+$\Gamma\curvearrowright(Y_M,\nu_M)$ of
+\eqref{actions:product-action} satisfies
+\[
+ \operatorname{Cost}(\mathcal R_{\Gamma\curvearrowright Y_M})
+ \leq1+\frac{99}{M}.
+\]
+\end{proposition}
+
+\begin{proof}
+Fix $0<p<1$ and use the graphing of
+Lemma~\ref{actions:small-marker}.  Add the restriction of $a$ to
+the union of the height fibers $0,1,\ldots,98$.  Its domain has
+measure $99/M$, so the resulting graphing has cost
+\[
+ 1+100p+\frac{99}{M}.
+\]
+Let $E$ be the equivalence relation it generates.  It contains
+$\mathcal R_B$, and we will prove that it contains every $a$-edge.
+
+Suppose that $(z,za)\in E$ whenever the height of $z$ is one of
+$k,k+1,\ldots,k+98$, interpreted modulo $M$.  Given $z$ at height
+$k+99$, set $y=zu_{99}^{-1}$, which has height $k$.
+The sequence
+\[
+ y=yu_0,\quad yu_1,\quad\ldots,\quad yu_{99}=z
+\]
+is connected by $E$: at its $i$th step apply $a$ to $yu_{i-1}$,
+of height $k+i-1$, and then apply $b_i$.  The $a$-edge is among
+those assumed available, and the $b_i$-edge belongs to
+$\mathcal R_B$.  The $w$-edge also belongs to $\mathcal R_B$, and
+\[
+ yw=yu_{99}a=za.
+\]
+Thus $z$ and $za$ are $E$-equivalent, proving the implication from
+$99$ consecutive known source heights to the next one.
+
+The heights $0,\ldots,98$ are known initially.  Apply this
+implication for the ordinary integers
+$k=0,1,\ldots,M-100$.  It successively supplies source heights
+$99,100,\ldots,M-1$; the last $a$-edges end at height $0$ modulo
+$M$, as required.  Hence all $a$-edges lie in $E$.
+Since $B$ and $a$ generate $\Gamma$, this graphing generates
+$\mathcal R_\Gamma$.  Taking the infimum over graphings and then
+letting $p\downarrow0$ proves the bound.
+\end{proof}
diff --git a/preprints/A-group-without-fixed-price-October-5-2026/build/introduction.tex b/preprints/A-group-without-fixed-price-October-5-2026/build/introduction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..8fe123350cfe90d0793f7ecfefbae6ed1af4dee3
--- /dev/null
+++ b/preprints/A-group-without-fixed-price-October-5-2026/build/introduction.tex
@@ -0,0 +1,152 @@
+\section{Introduction}
+
+The cost of an action measures how many edges, on average, are needed
+to connect its orbits. More precisely, let a countable group $\Gamma$
+act by measure-preserving Borel automorphisms on a standard probability
+space $(X,\mu)$. We use right actions, so $(xg)h=x(gh)$.
+Its orbit relation is
+\[
+ \mathcal R=\{(x,xg):x\in X,\ g\in\Gamma\}.
+\]
+A \emph{graphing} $\mathcal E$ of $\mathcal R$ is a countable family of
+measure-preserving Borel bijections between Borel subsets of $X$,
+with graphs contained in $\mathcal R$. It generates $\mathcal R$ if
+finite paths in these maps and their inverses connect all related
+points on a conull set. Its cost $C(\mathcal E)$ is the sum of its domain measures;
+$\operatorname{Cost}(\mathcal R)$ is the infimum of these sums over
+generating graphings. An action is \emph{essentially free} if each
+nonidentity group element fixes a null set. A group has \emph{fixed
+price} if all its essentially free probability-measure-preserving
+(p.m.p.) actions have the same cost.
+
+Levitt introduced cost \cite{levitt-1995}. Gaboriau developed its
+connections with group theory and the fixed-price problem
+\cite{gaboriau-2000}. His work shows, in particular, that free groups
+have fixed price equal to their rank, and establishes fixed price one
+for many groups with commuting or amenable structure. Recent results
+include the fixed-price-one theorem for products of two infinite
+countable groups \cite{khezeli-2026-v4}, metric and infinite-measure
+action criteria \cite{bevilacqua-bowen-2025-v1}, and a criterion based
+on small product neighborhoods of finite subsets
+\cite{slutsky-2026-v1}. We give a group for which fixed price fails.
+
+\subsection{The group and the two actions}
+
+Let
+\[
+ A=F(a,b_1,\ldots,b_{99}),\qquad
+ w=ab_1ab_2\cdots ab_{99}a,
+ \qquad J=\langle b_1,\ldots,b_{99},w\rangle\le A.
+\]
+Let $\langle t\rangle$ be infinite cyclic and set
+\begin{equation}\label{intro:group}
+ \Gamma=A*_J(J\times\langle t\rangle).
+\end{equation}
+The homomorphism $\chi:\Gamma\to\mathbb Z$ given by
+$\chi(a)=1$ and $\chi(b_i)=\chi(t)=0$ is well defined; note that
+$\chi(w)=100$. On the Bernoulli
+probability space $X_0=[0,1]^\Gamma$ with product Lebesgue measure, let
+\[
+ (xg)(h)=x(gh).
+\]
+Let $X\subseteq X_0$ be the invariant conull set of points with trivial
+stabilizer, equipped with the restricted probability measure. For an
+integer $M>100$, give $Y_M=X\times\mathbb Z/M\mathbb Z$
+product measure with uniform height coordinate, and define
+\[
+ (x,r)g=(xg,r+\chi(g)\bmod M).
+\]
+Both actions are free; restriction from $X_0$ to $X$ preserves cost.
+Their elementary group-theoretic
+and measure-theoretic properties are proved in Section~\ref{actions:section}.
+
+\begin{theorem}\label{intro:main}
+Put $K=e^{96}96^{99}/95^{95}$, choose
+\[
+ 0<\alpha<1/200,\qquad K\alpha^3<1/2,
+ \qquad \eta=\alpha/100.
+\]
+For the group and actions just defined,
+\[
+ \operatorname{Cost}(\mathcal R_{\Gamma\curvearrowright X})
+      \ge1+\eta,
+ \qquad
+ \operatorname{Cost}(\mathcal R_{\Gamma\curvearrowright Y_M})
+      \le1+\frac{99}{M}\quad(M>100).
+\]
+In particular, choosing an integer $M>\max\{100,99/\eta\}$ gives
+two essentially free p.m.p. actions of the same finitely generated
+group with strictly different costs. Thus the general fixed-price
+assertion is false.
+\end{theorem}
+
+\subsection{How the proof works}
+
+The upper bound uses the finite height coordinate. All $t$-edges,
+together with the generators of $J$ on a small set meeting every
+$t$-orbit, generate the $J\times\langle t\rangle$ relation at cost
+arbitrarily close to one. It remains to add $a$-edges. The relation
+$w=ab_1\cdots ab_{99}a$ propagates $a$-edges from any $99$
+consecutive heights to the next height, so an initial cost $99/M$
+is enough.
+
+For the lower bound, write $\mathcal R_A$ and $\mathcal R_J$ for
+the restricted orbit relations on $X$, and define the relative
+quantity
+\[
+ \delta_X=\inf\{C(\mathcal E):
+       \mathcal R_J\vee\mathcal E=\mathcal R_A,
+       \ \mathcal E\subseteq\mathcal R_A\}.
+\]
+Here $C(\mathcal E)$ is the graphing cost, and the join denotes the
+relation generated by the supplied $\mathcal R_J$-steps and the
+edges of $\mathcal E$. Three steps give the desired bound:
+\begin{enumerate}
+\item \emph{Deploy and compress graphings.}
+Subdivide a nearly optimal $\Gamma$-graphing into factor edges on a
+larger finite-measure space. Repeated compression makes more of the
+$J$-relation available and recovers its exact saved cost. Amalgam
+normal form then gives
+$\operatorname{Cost}(\mathcal R_\Gamma)\ge1+\delta_X$.
+The construction follows Gaboriau's deployment method
+\cite[\S\,IV.28--IV.32 and Proposition~IV.35]{gaboriau-2000};
+Sections~\ref{compression:section}--\ref{deployment:section}
+prove the required relative estimate directly.
+\item \emph{Pass to finite permutation models.}
+Put $u_0=1$ and $u_i=u_{i-1}ab_i$ for $1\le i\le99$. A finite set $V$ with a right
+$A$-action by permutations defines a group whose symbols record
+$a$-steps, while all $J$-steps are assigned the identity:
+\[
+ D_V=\langle x_v\ (v\in V)\mid
+              x_vx_{vu_1}\cdots x_{vu_{99}}=1\ (v\in V)\rangle.
+\]
+Write $\operatorname{rk}(D_V)$ for its minimum number of generators.
+Local approximation of Bernoulli graphings gives
+$\delta_X\ge\limsup\operatorname{rk}(D_V)/|V|$ along any
+asymptotically free sequence of such models, meaning that each fixed
+nonidentity element of $A$ fixes a fraction of points tending to zero.
+The displayed row makes $w$ trivial in this encoding. In
+Section~\ref{models:section}, random permutations supply models with
+strong expansion and few overlapping relators.
+\item \emph{Bound the rank by shortening a labelled graph.}
+The expansion property forces
+$\operatorname{rk}(D_V)>\eta|V|$ for large models. A minimal
+labelled graph representing a generating set would otherwise contain
+a long relator segment that could be replaced by a shorter path.
+This uses the Arzhantseva--Ol'shanskii method
+\cite{arzhantseva-olshanskii-1996,kapovich-schupp-2002}.
+The additional ingredients are a saturation procedure for exceptional
+letters and a planar word lemma over an arbitrary coefficient group,
+proved in Sections~\ref{planar:section} and~\ref{rank:section}.
+\end{enumerate}
+
+The proof isolates three estimates beyond this example: compression
+relative to a supplied equivalence relation, a transfer from Bernoulli
+graphings to ranks of finitely presented groups, and a deterministic
+rank bound from expansion with sparse overlaps. Their statements
+specify their own inputs and conclusions. The coefficient group is the actual subgroup generated by the letters
+absorbed during saturation. It need not be finitely presented: all
+relations true in that subgroup are retained, so its injection into the
+presented group is part of the construction. The planar lemma keeps
+track of these coefficient values while counting only the remaining
+letters.
diff --git a/preprints/A-group-without-fixed-price-October-5-2026/build/paper.tex b/preprints/A-group-without-fixed-price-October-5-2026/build/paper.tex
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+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage{amsmath,amssymb,amsthm}
+\usepackage[margin=1.1in]{geometry}
+\usepackage{microtype}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta}
+\usepackage[hidelinks]{hyperref}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\pdfinfoomitdate=1
+\pdftrailerid{}
+\pdfsuppressptexinfo=15
+\title{A group without fixed price}
+\author{OpenAI}
+\date{October 5, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We construct a finitely generated group with two essentially free
+probability-measure-preserving actions of different costs. This gives
+a negative answer to Gaboriau's general fixed-price problem. The group is an
+amalgam of a free group of rank $100$ with a direct product. Its
+Bernoulli action has cost bounded away from one, whereas finite height
+extensions have costs tending to one.
+\end{abstract}
+\tableofcontents
+\input{introduction}
+\input{group-actions}
+\input{compression}
+\input{deployment}
+\input{finite-models}
+\input{planar}
+\input{rank-surgery}
+\input{conclusion}
+\clearpage
+\bibliographystyle{plain}
+\bibliography{references}
+\end{document}
diff --git a/preprints/A-group-without-fixed-price-October-5-2026/build/planar.tex b/preprints/A-group-without-fixed-price-October-5-2026/build/planar.tex
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@@ -0,0 +1,257 @@
+\section{A planar word lemma with coefficients}
+\label{planar:section}
+
+This section proves the word-matching statement used in the rank argument.
+The coefficient group may have torsion and need not be finitely presented.
+The conclusion records equality of coefficient elements, as well as matching
+letters. The proof follows the long-boundary principle of classical
+small-cancellation theory associated with Greendlinger's lemma
+\cite{greendlinger-1960}. We prove the coefficient-sensitive statement
+in full.
+
+Let $H$ be a group generated by a finite alphabet $S$, let $U$ be a disjoint
+finite alphabet, and put $L=H*F(U)$.  Words will be written in
+$S^{\pm1}\cup U^{\pm1}$.  An occurrence of a letter of $U^{\pm1}$ is called
+an \emph{exterior occurrence}.  Between successive exterior occurrences
+there is a word in $S^{\pm1}$; its value in $H$ is the corresponding
+\emph{coefficient gap}.  For a cyclic word, this convention includes the
+gap from the last exterior occurrence to the first.
+
+A cyclic word is \emph{cyclically $H$-reduced} if no two successive
+exterior occurrences are inverse letters separated by a coefficient gap
+equal to $1$ in $H$.  An interval from one exterior occurrence through
+another \emph{matches} a relator interval if their signed exterior letters
+agree in order and all intervening coefficient gaps have equal values in
+$H$.  The interval uses no position of the relator more than once.  A
+\emph{full cyclic match} means that, after choosing cyclic starting
+positions, the complete signed exterior lists have the same length and
+agree, and all corresponding coefficient gaps, including the closing
+gap, agree in $H$.
+
+\begin{lemma}[Planar word lemma]
+\label{planar:main}
+Let $H$ be generated by a finite alphabet $S$, let $U$ be a disjoint
+finite alphabet, and put $L=H*F(U)$. Let $\mathcal R$ be a family of
+words in $L$ with the following properties.
+Each $r\in\mathcal R$ has $m(r)\geq 7$ exterior occurrences, all positive
+and with pairwise distinct underlying generators.  Distinct relators
+share at most one generator from $U$.  Suppose that the natural map
+\[
+ H\longrightarrow D
+ :=L/\langle\!\langle\mathcal R\rangle\!\rangle
+\]
+is injective.  If a cyclically $H$-reduced word $W$ has an exterior
+occurrence and represents $1$ in $D$, then at least one of the following
+conclusions holds:
+\begin{enumerate}
+\item $W$ has a full cyclic match with a relator or its inverse;
+\item $W$ has two intervals disjoint in their exterior occurrences, each
+  matching an interval of a cyclic relator or its inverse with at least
+  $m(r)-3$ exterior occurrences, where $r$ is the relator matched by that
+  interval.
+\end{enumerate}
+\end{lemma}
+
+The two intervals in the second conclusion need not come from different
+relators.  Disjointness refers to occurrences in $W$, and matching refers
+to equality in $H$ of every gap strictly between the first and last
+matched exterior letters.  In particular, no bound on the lengths of the
+$S$-words spelling those gaps is asserted or needed.
+
+\begin{proof}
+We first turn a least-relator expression for $W$ into a planar graph.
+The edges of this graph pair exterior occurrences.  Bigon faces give
+consecutive matches, and a corner count will produce either one full
+match or two long strings of bigons.
+
+\paragraph{A finite filling over an arbitrary coefficient group.}
+Choose an expression for $W$ in $L$ as a product of conjugates of words
+from $\mathcal R^{\pm1}$ using the least possible number $N$ of those
+words.  Free-product reduction shows that a cyclically $H$-reduced word
+with an exterior occurrence is nontrivial in $L$, so $N>0$.
+
+Give $H$ the presentation on $S$ with all words trivial in $H$ as
+relations.  An equality in this presentation uses finitely many
+relations: membership in their normal closure means a finite product
+of conjugates.  Thus the chosen expression for $W$ can be witnessed
+using a finite set of true $H$-relations.  Let $K_H$ be a finite
+presentation complex for this finite set of relations, and wedge it
+with one circle for each $u\in U$.  The resulting complex $K$ has a
+natural homomorphism $\pi_1(K)\to L$.  We do not assume this homomorphism
+is injective.
+
+Realize the expression by a map from a sphere with $N+1$ open disks
+removed to $K$.  Denote the distinguished boundary disk by $O$, and the
+other disks by $B_1,\ldots,B_N$.  Read every disk boundary using its disk
+orientation in the oriented sphere.  The boundary of $O$ reads $W^{-1}$,
+and each $B_i$ reads one chosen relator or its inverse.  To construct the
+map, cut the punctured sphere along disjoint spokes carrying the
+conjugating paths.  Its remaining boundary reads the equality in $L$;
+the finitely many chosen $H$-relations and free cancellations fill this
+boundary.  Conversely, cutting any such mapped punctured sphere along
+spokes expresses $W$ as a product of $N$ conjugates in $L$.
+
+We may make the complex and map piecewise linear.  For example, realize
+each relation with a mapping cylinder and a disk, triangulate, and use
+the finite filling just described.  During later replacements we may
+enlarge the finite set of true $H$-relations: any loop trivial in $L$
+becomes fillable after adding the finitely many such relations used by
+its equality.  Minimality throughout concerns $N$ in the actual group
+$L$, not a possibly larger finitely presented approximation to it.
+
+\paragraph{Tracks and connectedness.}
+For every $u\in U$, choose a point $q_u$ in the interior of its circle
+edge, avoiding the images of all vertices of a subdivision for the
+piecewise linear map.  Near $q_u$ the target is an interval.  Its inverse
+image consists of line segments across triangles, joined in pairs
+across their sides.  These form disjoint cooriented arcs and circles in
+the punctured sphere.  Each exterior occurrence on a boundary is the
+endpoint of exactly one arc.  Each arc and each circle maps constantly
+to its chosen point.  The two endpoints of an arc have opposite
+crossing signs when read in the disk orientations.  This follows from
+coorientation and the orientation of the surface; all disk-boundary
+orientations are opposite the corresponding orientations as boundaries
+of the punctured surface.
+
+Regard the boundary disks as vertices and the arc tracks as edges of
+an embedded graph $G$ on the sphere.  Closed tracks are not edges of
+$G$; we retain them as part of the map.  We claim that $G$ is connected.
+If it were not, choose a component $C$ not containing $O$ and a small
+closed regular neighborhood of its disks and arc tracks.  This
+neighborhood can be chosen disjoint from all other disks and tracks,
+including the finitely many closed tracks.  Select the boundary curve
+$\gamma$ facing the complementary component that contains $O$.
+The disk side of $\gamma$ away from $O$ contains $C$ and at least one
+inner disk; it may also contain other components, which causes no
+problem.
+
+The image of $\gamma$ avoids every $q_u$.  Cutting the exterior circles
+at these points leaves a space retracting to $K_H$.  Consequently the
+class of $\gamma$, after a change of basepoint, is an element of $H$
+inside $L$.  Cap all relator holes on the side away from $O$ in a
+presentation space for $D$.  The resulting mapped disk shows that this
+element is trivial in $D$.  Injectivity of $H\to D$ makes it trivial in
+$H$, hence $\gamma$ is trivial in $L$.  Replace that entire disk side
+by a filling using only true $H$-relations and free cancellations.
+At least one relator disk disappears, contradicting the choice of $N$.
+Thus $G$ is connected.  This argument neither removes individual closed
+tracks nor supposes that such tracks bound disks in the target.
+
+\paragraph{Inner loops and dipoles.}
+There is no graph loop at an inner vertex.  All exterior letters on
+that disk have the same sign, whereas an arc pairs opposite signs.
+Nor can an edge join two copies of the same relator.  Such copies would
+have opposite orientations and meet the unique occurrence of the same
+exterior generator $u$.  Base their boundary loops at the point $q_u$
+on that occurrence.  Writing the positive occurrence as $u=ps$, a
+positive based boundary has the form $sTp$, while the negative one is
+$p^{-1}T^{-1}s^{-1}$; these are inverse loops.  The joining track is
+constant and inserts no conjugating coefficient.  A small disk
+neighborhood of the two disks and this track therefore has null
+boundary in $L$.  Replacing its interior removes both relator disks,
+again contradicting minimality.  Other tracks crossing the boundary of
+this neighborhood impose no restriction on the replacement map.
+
+For copies of different relators there is at most one joining edge:
+the relators share at most one exterior generator, used once on each
+disk.  Thus between any two inner vertices there is at most one edge.
+Every inner vertex corresponding to $r$ has degree exactly $m(r)$.
+
+\paragraph{Face equations and bigon matches.}
+The complement of a small regular neighborhood of the connected
+disk-and-arc graph consists of disks.  Their boundary curves follow the
+face walks of $G$, even when a walk repeats vertices or edges.  Collapse
+the exterior circles of $K$ to its basepoint and then take values in
+$H$.  Each track becomes constant, and a sector between successive
+track endpoints on a disk contributes its coefficient gap.  The map
+on each complementary disk proves that the ordered product of its
+sector values is $1$ in $H$.  Closed tracks inside that disk do not
+affect this argument.
+
+There are no monogon faces.  An inner loop has already been excluded;
+a monogon at $O$ would pair two inverse exterior occurrences across a
+gap trivial in $H$, contrary to cyclic $H$-reduction.  Any bigon incident
+to an inner vertex consists of two edges joining that vertex to $O$.
+Indeed, there are no parallel edges between inner vertices.  A face
+walk of length two that traverses one edge twice would require degree
+one at its inner endpoint, which is impossible since $m(r)\geq7$.
+
+Such an inner--outer bigon matches consecutive exterior letters and
+their intervening coefficient gap.  The exterior signs match because
+$O$ is labelled $W^{-1}$ in its disk orientation.  Following $W$ along
+the outer sector follows the corresponding inner relator in its disk
+orientation.  The face equation, with this convention, is the inner
+gap times the inverse of the outer gap equal to $1$.  It therefore gives
+equality of the two elements of $H$, rather than equality merely up to
+conjugacy.
+
+We have now reduced the desired word matches to strings of bigons.
+The remaining argument forces either a full cyclic string or two long
+strings at different inner vertices.
+
+\paragraph{The corner count.}
+Give weight zero to every corner at $O$ and to every corner of a bigon.
+For any other corner at an inner vertex, assign weight
+\[
+ \begin{cases}
+  1/3,&\text{if both incident edges go to inner vertices},\\
+  1/2,&\text{if exactly one incident edge goes to }O,\\
+  1,&\text{if both incident edges go to }O.
+ \end{cases}
+\]
+Occurrences are counted with multiplicity along a face walk.  A face
+of length $\ell$ has total weight at most $\ell-2$.  For a bigon this
+is equality.  For $\ell\geq3$ with no occurrence of $O$, the total is
+$\ell/3\leq\ell-2$.  With exactly one occurrence of $O$, the two
+neighboring corners have weight $1/2$ and the remaining $\ell-3$
+corners weight $1/3$, again totaling $\ell/3$.  With at least two
+occurrences of $O$, there are at most $\ell-2$ inner corners, each of
+weight at most one.  These cases include repeated face walks and loops
+at $O$.
+
+Write $E$ and $F$ for the numbers of edges and faces of $G$, and $c(v)$
+for the total corner weight at an inner vertex $v$.  Summing over faces
+and using Euler's formula for the connected sphere graph with $N+1$
+vertices gives
+\begin{equation}
+\label{planar:curvature}
+ \sum_{v\ne O}\bigl(2-c(v)\bigr)
+ \geq 2N-2E+2F=2.
+\end{equation}
+
+Call $2-c(v)$ the contribution of $v$.  A vertex of degree $m\geq7$
+with no edge to $O$ has $c(v)=m/3>2$, so a positive vertex has an edge
+to $O$.  At such a vertex, consider the cyclic gaps between successive
+edges to $O$.  A gap that bounds a bigon costs zero.  A gap containing
+no edge to an inner vertex, but not bounding a bigon, costs one.  A
+gap containing $h\geq1$ edges to inner vertices costs
+\[
+ \frac12+\frac{h-1}{3}+\frac12
+ =1+\frac{h-1}{3}.
+\]
+Each gap not bounding a bigon therefore costs at least one.
+
+If every gap bounds a bigon, all edges at $v$ go to $O$.  Successive
+edges at $v$ are also successive at $O$, in the opposite rotation.
+They exhaust the cyclic rotation at $O$: otherwise a further half-edge
+would interrupt one of these bigons.  Connectedness leaves no other
+vertex.  The bigon equations then match every exterior occurrence of
+$W$ with this relator, including all cyclic gaps.  This is the first
+conclusion.
+
+Otherwise positivity means that exactly one gap does not bound a
+bigon, and that gap contains at most three edges to inner vertices.
+Indeed, two such gaps cost at least two, while a gap containing
+$h\geq4$ inner edges also costs at least two.  The other gaps form one
+consecutive string of bigons joining at least $m-3$ edges to $O$.
+Its bigon equations give a matching interval with at least $m-3$
+exterior occurrences.  The contribution of this vertex is at most one.
+
+Unless the first conclusion has already occurred, every positive
+vertex contributes at most one.  The total contribution is at least
+two, so there are at least two distinct positive vertices.  Take a
+matching interval from each.  Each exterior occurrence of $W$ is the
+endpoint of exactly one arc track, so intervals arising from different
+inner vertices have disjoint exterior occurrences.  This proves the
+second conclusion.
+\end{proof}
diff --git a/preprints/A-group-without-fixed-price-October-5-2026/build/rank-surgery.tex b/preprints/A-group-without-fixed-price-October-5-2026/build/rank-surgery.tex
new file mode 100644
index 0000000000000000000000000000000000000000..8b4e6fa0906b61758012dadc858bca1dbcedc913
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+++ b/preprints/A-group-without-fixed-price-October-5-2026/build/rank-surgery.tex
@@ -0,0 +1,288 @@
+\section{A deterministic rank bound}\label{rank:section}
+
+We now turn expansion of a finite system of relators into a lower bound
+for the number of generators of its presented group.  The proof represents
+a generating family by a labelled graph.  After absorbing a small set of
+letters into a coefficient group, Lemma~\ref{planar:main} finds a long
+relator segment in that graph.  Replacing this segment by the short
+complement of the relator contradicts minimality of the graph.
+This is an Arzhantseva--Ol'shanskii shortening argument
+\cite[\S~Marked graphs]{arzhantseva-olshanskii-1996}; see also
+\cite[Definitions~2.6--2.9 and Proposition~2.11]{kapovich-schupp-2002}.
+The coefficient group and the saturation procedure below allow us to
+use the move for the present system of relators.
+
+\begin{theorem}\label{rank:main}
+Let $V$ and $\mathcal T$ be sets of cardinality $n$.  For each
+$t\in\mathcal T$, let
+\[
+ r_t=x_{c(t,0)}x_{c(t,1)}\cdots x_{c(t,99)},\qquad c(t,i)\in V,
+ \qquad
+ D=\langle x_v\ (v\in V)\mid r_t=1\ (t\in\mathcal T)\rangle.
+\]
+For $T\subseteq\mathcal T$, put
+$N(T)=\{c(t,i):t\in T,\ 0\le i\le99\}$.
+Call a row $t$ bad if it has a repeated column, or if it shares at least
+two distinct columns with another row, and let $b$ be the number of bad
+rows.  Suppose that
+\[
+ 0<\alpha<\frac1{200},\qquad \alpha n\ge1,
+ \qquad 100b\le\alpha n,
+\]
+and that
+\begin{equation}\label{rank:expansion}
+ |N(T)|>96|T|\qquad
+ \text{whenever }1\le |T|\le\alpha n.
+\end{equation}
+Then the minimum size of a generating set of $D$ satisfies
+\[
+ \operatorname{rk}(D)>\frac{\alpha n}{100}.
+\]
+\end{theorem}
+
+\begin{proof}
+Suppose instead that $D$ has at most $\alpha n/100$ generators.
+We proceed in four steps: choose a minimal graph, saturate a small
+coefficient alphabet, locate a long relator segment, and shorten the graph.
+
+\subsection*{A minimal labelled graph}
+Give each edge of a finite graph an orientation and a label $x_v$ or
+$x_v^{-1}$; reading the edge backwards inverts its label.  Path labels
+are words in these letters, evaluated in $D$.  A generating set written
+as words in the $x_v$ gives a subdivided rose whose based loops map onto
+$D$.  Among all finite connected labelled graphs $Y$ with
+\[
+ \beta(Y)=|E(Y)|-|V(Y)|+1\le\frac{\alpha n}{100}
+\]
+and surjective loop-label map to $D$, choose one with the fewest edges.
+Write $\beta=\beta(Y)$.  Changing basepoint conjugates the loop image in
+$D$, so it preserves surjectivity.  Consequently $Y$ has no vertex of
+degree one: pruning such a vertex and its edge, and moving the basepoint
+if necessary, would give a smaller graph with the same property.
+
+Choose a basepoint $p$.  Decompose $Y$ into arcs by splitting at $p$ and
+at every vertex of degree at least three.  An arc is an edge path with
+distinct interior vertices, all of degree two and different from $p$;
+its endpoints may coincide.  If $Y$ is a point, it has no arcs.  If $Y$
+is a circle, then $\beta=1$ and marking $p$ gives one arc.  In all other
+cases, the identity
+\[
+ \sum_{v\in V(Y)}(\deg(v)-2)=2\beta-2
+\]
+bounds the number of vertices of degree at least three by $2\beta-2$.
+Suppressing all degree-two vertices except possibly $p$ leaves at most
+$2\beta-1$ vertices and therefore at most $3\beta-2$ edges.
+In particular, in every case the number of arcs is at most $3\beta$.
+
+\subsection*{Saturating the coefficient alphabet}
+Start with $S\subseteq V$ equal to the set of columns occurring in bad
+rows.  Apply either of the following rules whenever it adds a new column:
+\begin{enumerate}
+\item If an arc has at most $30$ occurrences of labels with index
+outside $S$, put every index occurring on that arc into $S$.
+\item If a row has at least $10$ occurrences with index in $S$, put
+all its columns into $S$.
+\end{enumerate}
+The process terminates, since each step enlarges $S\subseteq V$.
+An arc can contribute at most $30$ new columns and is used at most once.
+Thus the initial set and all arc additions together contribute at most
+\begin{equation}\label{rank:seed}
+ 100b+30(3\beta)
+ \le\alpha n+\frac{90\alpha n}{100}<2\alpha n.
+\end{equation}
+Every nontrivial row addition contributes at most $90$ new columns.
+
+Put $s=\lfloor\alpha n\rfloor\ge1$.  If $s$ rows were processed by
+the second rule, let $T$ consist of the first $s$ such rows.  At that
+moment,
+\[
+ N(T)\subseteq S,
+ \qquad |S|\le2\alpha n+90s<96s.
+\]
+For the strict inequality, use
+$\alpha n<s+1\le2s<3s$.  This contradicts
+\eqref{rank:expansion}.  Thus fewer than $s$ rows are processed, and the
+final set satisfies
+\begin{equation}\label{rank:saturation}
+ |S|\le2\alpha n+90(s-1)\le92\alpha n<n.
+\end{equation}
+Let $U=V\setminus S$, which is nonempty.  At termination, every arc
+has either no $U$-labelled edge or more than $30$ such occurrences.
+Moreover, every row not entirely in $S$ is good and has at most nine
+$S$-occurrences.  Such a row has at least $91$ distinct positive
+$U$-letters, and distinct such rows share at most one $U$-letter.
+
+Let $H=\langle x_s:s\in S\rangle$ be the actual subgroup of $D$
+generated by these elements.  There is an isomorphism
+\begin{equation}\label{rank:relative-presentation}
+ D\cong
+ \bigl(H*F(x_u:u\in U)\bigr)
+ \big/\langle\!\langle r_t:\text{$r_t$ is not entirely in $S$}
+ \rangle\!\rangle.
+\end{equation}
+Indeed, the group on the right maps to $D$ because every defining
+relation of $H$ and every displayed row hold in $D$.  Conversely, its
+generators satisfy all the original rows: an omitted row is a true
+relation of $H$, and every other row is retained.  The resulting maps
+are inverse on the generators, hence are inverse isomorphisms.
+In particular, the coefficient group in
+\eqref{rank:relative-presentation} embeds in the quotient.  No finite
+presentation of $H$ is being assumed.
+
+\subsection*{Finding a long segment in a shortest loop}
+Call an occurrence of $x_u^{\pm1}$ with $u\in U$ exterior.
+Choose $j\in U$ and a shortest edge loop $P$ based at $p$ whose label
+represents $x_j$ in $D$.  This includes the possibility that $P$ is
+empty.  It has no immediate backtracks and therefore traverses complete
+arcs.  Its number of exterior occurrences is consequently either zero
+or greater than $30$.
+
+The label of $P$ is linearly $H$-reduced: two consecutive exterior
+occurrences in its linear list cannot be inverse letters separated by
+an $S$-word trivial in $H$.  To prove this, suppose that the relevant
+portion of $P$ is $eRf$, where $R$ uses only $S$-edges and has value
+$1$ in $H$.  If $f=e^{-1}$ as oriented edges, the closed detour $eRf$
+has value $1$ in $D$, so deleting it shortens $P$ without changing its
+value.  Otherwise $e$ and $f$ are different unoriented edges, and $R$
+uses neither of them.  Slide the initial incidence of $f$ backwards
+along $R$ to the terminal vertex of $e$.  This keeps the numbers of
+vertices and edges unchanged and keeps the graph connected: the path
+$R$ is still present.  Each old traversal of $f$ can be replaced by
+$R^{-1}f$ in the slid graph, and a reverse traversal by its inverse.
+Since $R$ has value $1$ in $D$, these substitutions preserve the values
+of all based loops, so the loop-label image still surjects onto $D$.
+Now $e^{-1}$ and $f$ leave the same vertex with the same label.  Fold
+them together.  This removes one edge and identifies either zero or
+one pair of distinct vertices; the graph remains connected, its rank
+does not increase, and the quotient preserves all loop labels.
+It is therefore a smaller admissible graph, contrary to the choice of
+$Y$.  This proves linear $H$-reduction.
+
+Put $L=H*F(x_u:u\in U)$.  The normal form in this free product shows
+that a linearly $H$-reduced word representing $x_j$ in $L$ has exactly
+one exterior occurrence: after evaluating its $H$-gaps, adjacent free
+letters separated by trivial gaps form freely reduced blocks.
+Since $P$ has zero or more than $30$ exterior occurrences, its label
+does not equal $x_j$ in $L$.
+Thus $\operatorname{lab}(P)x_j^{-1}$ is nontrivial in $L$ but trivial
+in $D$.
+
+Cyclically $H$-reduce this word by cancelling adjacent inverse exterior
+letters across gaps trivial in $H$.  Let $W$ be the resulting cyclic
+word.  It still has an exterior occurrence.  Otherwise the original
+word would be conjugate in $L$ to an element of $H$ trivial in $D$;
+the embedding of $H$ would make that element, and hence the original
+word, trivial in $L$.
+
+There is just one possible disruption of the original path when
+passing to $W$.  If no cancellation occurs, it is the appended letter
+$x_j^{-1}$, which we call the seam letter.  If cancellation occurs,
+the first cancellation must involve that appended letter, by linear
+$H$-reduction of $P$.  All later cancellations occur between the two
+ends of the remaining exterior list of $P$.  The surviving exterior
+occurrences form a contiguous block of that list, all its intervening
+gaps are unchanged, and only its closing gap, called the seam gap,
+may have changed.
+
+Apply Lemma~\ref{planar:main} to
+\eqref{rank:relative-presentation} and $W$.  Each mixed row has at
+least $91$ exterior occurrences.  If the lemma gives two
+exterior-disjoint intervals, at least one avoids the seam letter,
+when present.  When there is a seam gap, at most one interval can
+cross it, because a crossing uses both adjacent exterior occurrences.
+In either case an interval with at least $91-3=88$ exterior
+occurrences remains intact in $P$.  If the lemma instead gives an
+entire cyclic relator, remove the seam letter, when present, or cut at
+the seam gap.  This leaves at least $90$ exterior occurrences intact
+in $P$.
+
+We have therefore obtained a subpath $Q$ of $P$, beginning and ending
+with exterior edges, that matches a consecutive portion of a cyclic
+row or its inverse and has at least $88$ exterior occurrences.
+Its exterior indices are distinct.  Its intervening $S$-words have
+the same values in $H$ as the corresponding row segments, although
+their lengths need not agree.  The matching row segment has at least
+$88$ of the original $100$ letters.  Its complementary segment has
+length at most $12$, so the inverse complementary word gives a word
+$z$ in the original generators such that
+\begin{equation}\label{rank:short-word}
+ [z]_D=[\operatorname{lab}(Q)]_D,
+ \qquad 1\le |z|\le12.
+\end{equation}
+If the complementary word is empty, use $x_jx_j^{-1}$ for $z$.
+
+\subsection*{Shortening the graph}
+Decompose $Q$ according to its arc traversals, allowing partial
+traversals at its two ends.  Some such traversal contains more than
+$30$ exterior edges of $Q$.  Otherwise no complete traversal could
+have exterior edges, by saturation, and the two partial traversals
+would contribute at most $60$ exterior occurrences in total.
+In the chosen traversal, let $I$ run from its first exterior edge
+through its last exterior edge, inclusive.  Then
+\[
+ Q=Q_1 I Q_2,\qquad \ell:=|E(I)|\ge31.
+\]
+The segment $I$ has distinct interior vertices of degree two, none
+equal to $p$; its endpoints may coincide.
+
+The paths $Q_1,Q_2$ avoid both the edges and the interior vertices of
+$I$.  First, neither endpoint of $Q$ is interior to $I$: at such a
+vertex the first or last edge of $Q$ would be one of the edges of $I$,
+because that vertex has degree two.  That edge is exterior, so its
+two occurrences in $Q$ would repeat an exterior index.  Next, any
+path entering the interior of $I$ from outside must use its first
+or last edge.  Both are exterior and already occur in $I$, so neither
+$Q_1$ nor $Q_2$ can enter this way.  These arguments also exclude a
+second traversal of an edge of $I$; when $I$ is closed, the same two
+boundary incidences give the identical conclusion.
+
+Figure~\ref{rank:shortening-schematic} illustrates the construction.
+
+\input{surgery-figure}
+
+Attach a new path $Z$ between the endpoints of $Q$ with label $z$,
+using new interior vertices and $k=|z|\le12$ new edges.  Then delete
+the edges and interior vertices of $I$.  The old endpoints of $I$
+are still connected by the path
+\[
+ Q_1^{-1} Z Q_2^{-1},
+\]
+which avoids everything deleted.  Thus the new graph is connected.
+The inserted path has $k-1$ new interior vertices, and the deleted
+segment has $\ell-1$ interior vertices, also when either path has
+coincident endpoints.  Hence the changes in the numbers of edges
+and vertices are both $k-\ell$, and the graph rank is unchanged.
+The basepoint survives, since it is not an interior vertex of $I$.
+
+Equation~\eqref{rank:short-word} gives
+\[
+ [\operatorname{lab}(Q_1^{-1}ZQ_2^{-1})]_D
+   =[\operatorname{lab}(I)]_D.
+\]
+Every reduced loop at $p$ in the old graph that uses an edge of $I$
+must traverse all of $I$, in one direction or the other: its interior
+vertices have degree two and do not contain $p$.  Replacing each
+such traversal by the displayed bypass, or its inverse, gives a loop
+in the new graph with the same value in $D$.  The loop-label map of
+the new graph therefore remains surjective.  Its edge count has
+decreased by at least $31-12=19$, contradicting minimality of $Y$.
+This proves the theorem.
+\end{proof}
+
+\begin{corollary}\label{rank:asymptotic}
+Suppose that a sequence of length-$100$ row systems satisfies
+\eqref{rank:expansion} for a fixed $0<\alpha<1/200$, their sizes
+$n$ tend to infinity, and their bad-row counts satisfy $b=o(n)$.
+Then their presented groups satisfy
+\[
+ \operatorname{rk}(D)>\frac{\alpha n}{100}
+ \quad\text{for all sufficiently large }n,
+ \qquad
+ \liminf_{n\to\infty}\frac{\operatorname{rk}(D)}n
+ \ge\frac\alpha{100}.
+\]
+\end{corollary}
+\begin{proof}
+Eventually $\alpha n\ge1$ and $100b\le\alpha n$, so
+Theorem~\ref{rank:main} applies.
+\end{proof}
diff --git a/preprints/A-group-without-fixed-price-October-5-2026/build/references.bib b/preprints/A-group-without-fixed-price-October-5-2026/build/references.bib
new file mode 100644
index 0000000000000000000000000000000000000000..11e2dae35e81a0942d1fe687a6cac4b6d54aea39
--- /dev/null
+++ b/preprints/A-group-without-fixed-price-October-5-2026/build/references.bib
@@ -0,0 +1,88 @@
+@article{levitt-1995,
+  author = {Levitt, Gilbert},
+  title = {On the cost of generating an equivalence relation},
+  journal = {Ergodic Theory and Dynamical Systems},
+  volume = {15},
+  number = {6},
+  pages = {1173--1181},
+  year = {1995},
+  doi = {10.1017/S0143385700009846}
+}
+
+@article{gaboriau-2000,
+  author = {Gaboriau, Damien},
+  title = {Co{\^u}t des relations d'{\'e}quivalence et des groupes},
+  journal = {Inventiones mathematicae},
+  volume = {139},
+  pages = {41--98},
+  year = {2000},
+  doi = {10.1007/s002229900019},
+  url = {https://perso.ens-lyon.fr/gaboriau/Travaux-Publi/Cout/Cout.pdf}
+}
+
+@article{arzhantseva-olshanskii-1996,
+  author = {Arzhantseva, G. N. and Ol'shanskii, A. Yu.},
+  title = {The class of groups all of whose subgroups with lesser number of generators are free is generic},
+  journal = {Mathematical Notes},
+  volume = {59},
+  pages = {350--355},
+  year = {1996},
+  doi = {10.1007/BF02308683},
+  url = {https://www.mat.univie.ac.at/~arjantseva/Abs/arzholsh.pdf}
+}
+
+@misc{kapovich-schupp-2002,
+  author = {Kapovich, Ilya and Schupp, Paul},
+  title = {Genericity, the {Arzhantseva--Ol'shanskii} method and the Isomorphism Problem for One-Relator Groups},
+  year = {2002},
+  eprint = {math/0210307},
+  howpublished = {\href{https://arxiv.org/abs/math/0210307v1}{arXiv:math/0210307}},
+  archivePrefix = {arXiv},
+  primaryClass = {math.GR},
+  note = {Version 1, 19 October 2002; numbered statements cited from this version},
+  url = {https://arxiv.org/abs/math/0210307v1}
+}
+
+@misc{khezeli-2026-v4,
+  author = {Khezeli, Ali},
+  title = {Products of Infinite Countable Groups Have Fixed Price One},
+  year = {2026},
+  eprint = {2509.08325},
+  howpublished = {\href{https://arxiv.org/abs/2509.08325v4}{arXiv:2509.08325}},
+  archivePrefix = {arXiv},
+  note = {Version 4, 4 January 2026},
+  url = {https://arxiv.org/abs/2509.08325v4}
+}
+
+@misc{slutsky-2026-v1,
+  author = {Slutsky, Raz},
+  title = {A Product-Neighbourhood Criterion for Fixed Price One},
+  year = {2026},
+  eprint = {2607.20273},
+  howpublished = {\href{https://arxiv.org/abs/2607.20273v1}{arXiv:2607.20273}},
+  archivePrefix = {arXiv},
+  note = {Version 1, 22 July 2026},
+  url = {https://arxiv.org/abs/2607.20273v1}
+}
+
+@misc{bevilacqua-bowen-2025-v1,
+  author = {Bevilacqua, Erin and Bowen, Lewis},
+  title = {Metric criteria for fixed price of countable groups},
+  year = {2025},
+  eprint = {2510.05459},
+  howpublished = {\href{https://arxiv.org/abs/2510.05459v1}{arXiv:2510.05459}},
+  archivePrefix = {arXiv},
+  note = {Version 1, submitted 6 October 2025; PDF title page dated 8 October 2025},
+  url = {https://arxiv.org/abs/2510.05459v1}
+}
+
+@article{greendlinger-1960,
+  author = {Greendlinger, Martin},
+  title = {Dehn's algorithm for the word problem},
+  journal = {Communications on Pure and Applied Mathematics},
+  volume = {13},
+  number = {1},
+  pages = {67--83},
+  year = {1960},
+  doi = {10.1002/cpa.3160130108}
+}
diff --git a/preprints/A-group-without-fixed-price-October-5-2026/build/surgery-figure.tex b/preprints/A-group-without-fixed-price-October-5-2026/build/surgery-figure.tex
new file mode 100644
index 0000000000000000000000000000000000000000..f252dfede9a12af673731fa4d43c7f1681b7258d
--- /dev/null
+++ b/preprints/A-group-without-fixed-price-October-5-2026/build/surgery-figure.tex
@@ -0,0 +1,49 @@
+\begin{figure}[ht]
+\centering
+\begin{tikzpicture}[>=Stealth, line width=.7pt,
+  every node/.style={font=\small},
+  endpoint/.style={circle,fill=black,inner sep=1.5pt},
+  pathlabel/.style={fill=white,inner sep=2pt}]
+  \begin{scope}
+    \node at (2.45,2.15) {Before};
+    \coordinate (a) at (0,0);
+    \coordinate (b) at (1.35,0);
+    \coordinate (c) at (3.55,0);
+    \coordinate (d) at (4.90,0);
+    \draw[->] (a) -- node[pathlabel,above] {$Q_1$} (b);
+    \draw[->,very thick] (b) -- node[pathlabel,above] {$I$} (c);
+    \draw[->] (c) -- node[pathlabel,above] {$Q_2$} (d);
+    \foreach \q in {a,b,c,d} \node[endpoint] at (\q) {};
+    \node[below=5pt] at (a) {$q_-$};
+    \node[below=5pt] at (b) {$i_-$};
+    \node[below=5pt] at (c) {$i_+$};
+    \node[below=5pt] at (d) {$q_+$};
+    \node at (2.45,-.88) {$Q=Q_1IQ_2,\qquad |E(I)|\ge31$};
+  \end{scope}
+  \begin{scope}[xshift=6.2cm]
+    \node at (2.45,2.15) {After};
+    \coordinate (a) at (0,0);
+    \coordinate (b) at (1.35,0);
+    \coordinate (c) at (3.55,0);
+    \coordinate (d) at (4.90,0);
+    \draw[->] (a) -- node[pathlabel,above] {$Q_1$} (b);
+    \draw[->] (c) -- node[pathlabel,above] {$Q_2$} (d);
+    \draw[->,very thick] (a) to[out=65,in=115]
+      node[pathlabel,above] {$Z,\quad |E(Z)|\le12$} (d);
+    \foreach \q in {a,b,c,d} \node[endpoint] at (\q) {};
+    \node[below=5pt] at (a) {$q_-$};
+    \node[below=5pt] at (b) {$i_-$};
+    \node[below=5pt] at (c) {$i_+$};
+    \node[below=5pt] at (d) {$q_+$};
+    \node at (2.45,-.88) {Bypass from $i_-$ to $i_+$: $Q_1^{-1}ZQ_2^{-1}$};
+  \end{scope}
+\end{tikzpicture}
+\caption{The shortening move: add $Z$ between the endpoints of $Q$,
+then delete the edges and interior vertices of $I$. The retained paths
+$Q_1,Q_2$ and the new path give a bypass with the same value in $D$
+as $I$. The drawing is schematic: displayed endpoints may coincide,
+and $Q_1,Q_2$ may overlap or be empty. The proof establishes that both
+avoid the edges and interior vertices of $I$. Each line denotes a path,
+and the rest of the graph is omitted.}
+\label{rank:shortening-schematic}
+\end{figure}
diff --git a/preprints/A-group-without-fixed-price-October-5-2026/unequal-costs-rank-100-amalgam.pdf b/preprints/A-group-without-fixed-price-October-5-2026/unequal-costs-rank-100-amalgam.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..7dbc18db13b617ee2184c16940081cbac161f800
--- /dev/null
+++ b/preprints/A-group-without-fixed-price-October-5-2026/unequal-costs-rank-100-amalgam.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:aa2ee4de0db26fc8aa62f95ec695e2ef11db6c9b326d2d354465aa50bee4c078
+size 434930
diff --git a/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/README.md b/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..61349820a8c2fb34ed455de6f5f1a23e4732f0c2
--- /dev/null
+++ b/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/README.md
@@ -0,0 +1,17 @@
+# [A high-arity counterexample to Pixton completeness in Chow](paper.pdf)
+
+OpenAI · September 24, 2026
+
+## Citation
+
+To cite this manuscript:
+
+```bibtex
+@misc{OAI:A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026,
+  author = {{OpenAI}},
+  title = {{A high-arity counterexample to Pixton completeness in Chow}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/paper.pdf}{OAI:A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026}},
+  year = {2026}
+}
+```
diff --git a/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/build/figures/bridge-cuts.tex b/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/build/figures/bridge-cuts.tex
new file mode 100644
index 0000000000000000000000000000000000000000..6cf077385ad4f1151e8cc99f4d4bbdc4c01e2db6
--- /dev/null
+++ b/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/build/figures/bridge-cuts.tex
@@ -0,0 +1,39 @@
+\begin{figure}[tb]
+\centering
+\begin{tikzpicture}[x=.85cm,y=.85cm]
+  \draw[thick] (-4.5,.65) .. controls (-3.8,1.2) and (-3,.9) .. (-2,.25);
+  \draw[thick] (-4.5,-.35) .. controls (-3.8,-.9) and (-3,-.4) .. (-2,.25);
+  \draw[thick] (2,.25) .. controls (3,.9) and (3.8,1.2) .. (4.5,.65);
+  \draw[thick] (2,.25) .. controls (3,-.4) and (3.8,-.9) .. (4.5,-.35);
+  \draw[thick] (-2,.25) -- (2,.25);
+  \fill (-2,.25) circle (2pt);
+  \fill (2,.25) circle (2pt);
+  \foreach \x/\lab in {-1/1,0/2,1/3}{
+    \fill (\x,.25) circle (1.6pt);
+    \node[above] at (\x,.36) {$\lab$};
+  }
+  \draw[dashed] (-2,-.7) -- (-2,1.2);
+  \draw[dashed] (2,-.7) -- (2,1.2);
+  \node[above] at (-3.7,1.0) {left core};
+  \node[above] at (3.7,1.0) {right core};
+  \node[below] at (0,.02) {rational bridge};
+  \node[below] at (-2,-.75) {first cut};
+  \node[below] at (2,-.75) {second cut};
+  \draw[->] (0,-1.1) -- (0,-2.0);
+  \node[right] at (.15,-1.55) {forget the three lights};
+  \draw[thick] (-3.3,-2.1) .. controls (-2.5,-1.7) and (-1,-2.1) .. (0,-2.55);
+  \draw[thick] (-3.3,-2.95) .. controls (-2.5,-3.35) and (-1,-3) .. (0,-2.55);
+  \draw[thick] (0,-2.55) .. controls (1,-2.1) and (2.5,-1.7) .. (3.3,-2.1);
+  \draw[thick] (0,-2.55) .. controls (1,-3) and (2.5,-3.35) .. (3.3,-2.95);
+  \fill (0,-2.55) circle (2pt);
+  \node[below] at (0,-3.45) {one persistent node};
+\end{tikzpicture}
+\caption{The two bounding nodes of a light-supported bridge.
+After forgetting the lights the bridge contracts.  On a cut chart the
+two node conditions contribute two normal cotangent factors.  The same
+calculation applies when the two core branches belong to one component.
+On the full exceptional surface, coincidence and the bridge class
+have integrals $1$ and $2$, respectively, giving
+$1-\tfrac12\cdot2=0$.}
+\label{fig:bridge-cuts}
+\end{figure}
diff --git a/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/build/figures/proof-map.tex b/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/build/figures/proof-map.tex
new file mode 100644
index 0000000000000000000000000000000000000000..fa958706c3adf3c8de6408396c8298aa9c38985a
--- /dev/null
+++ b/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/build/figures/proof-map.tex
@@ -0,0 +1,38 @@
+\begin{figure}[tb]
+\centering
+\begin{tikzpicture}[
+ box/.style={draw,rounded corners=2pt,align=center,text width=5.7cm,
+             minimum height=1.0cm,inner sep=5pt,font=\small},
+ result/.style={box,fill=black!5},
+ flow/.style={-{Stealth[length=2mm]},semithick},
+ node distance=6mm]
+\node[box] (candidate) {Corrected three-slot factors\\ and their alternation \(Y\)\\
+ Definition~\ref{def:candidate}};
+\node[box,below=8mm of candidate,xshift=-3.2cm] (formal) {Formal Frobenius algebra\\ over an exterior ring\\
+ Rational-tail detector (Section~\ref{sec:detector})};
+\node[box,below=8mm of candidate,xshift=3.2cm] (chow) {Faithful model of relative Chow correspondences\\
+ Finite color tests (Section~\ref{sec:tensors})};
+\node[result,below=8mm of formal] (outside) {\(t_n(\Pspan_{g,n})=0\), \(t_n(Y)\ne0\)\\
+ Hence \(Y\notin\Pspan_{g,n}\)};
+\node[box,below=8mm of chow] (surface) {Marked Quot relation \(aT_\gamma^{t_0}=0\)\\
+ Section~\ref{sec:quot}; \(a\ne0\) in Section~\ref{sec:coefficient}};
+\node[box,below=8mm of surface] (colors) {Arbitrary color products\\
+ Lemma~\ref{lem:large-vertex-test} (Section~\ref{sec:coefficient})};
+\node[box,below=8mm of colors] (boundary) {Controlled compactification\\ and support induction\\
+ Sections~\ref{sec:compactification}--\ref{sec:support}};
+\node[result,below=8mm of boundary] (zero) {\(q_{g,n}(Y)=0\) in rational Chow};
+\draw[flow] (candidate.south) -- ++(0,-.25) -| (formal.north);
+\draw[flow] (candidate.south) -- ++(0,-.25) -| (chow.north);
+\draw[flow] (formal) -- (outside);
+\draw[flow] (chow) -- (surface);
+\draw[flow] (surface) -- (colors);
+\draw[flow] (colors) -- (boundary);
+\draw[flow] (boundary) -- (zero);
+\end{tikzpicture}
+\caption{The two assertions about the same class \(Y\). The formal
+branch is independent of Chow vanishing. In the geometric branch,
+the nonzero scalar completes the isotropic surface bound. The
+exterior-ideal argument then gives arbitrary-color vanishing modulo base
+support before the passage to the compactification.}
+\label{fig:proof-map}
+\end{figure}
diff --git a/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/build/main.tex b/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/build/main.tex
new file mode 100644
index 0000000000000000000000000000000000000000..25c7d642ac8c9306f52e66dca68956fedc096cee
--- /dev/null
+++ b/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/build/main.tex
@@ -0,0 +1,78 @@
+\documentclass[11pt]{article}
+\pdfinfoomitdate=1
+\pdftrailerid{}
+\pdfsuppressptexinfo=15
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage{lmodern}
+\usepackage[margin=1in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{microtype,enumitem,booktabs,array}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta,positioning,calc,decorations.pathreplacing}
+\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=green!35!black,urlcolor=blue!55!black]{hyperref}
+\hypersetup{
+  pdftitle={A high-arity counterexample to Pixton completeness in Chow},
+  pdfauthor={OpenAI},
+  pdfsubject={Tautological Chow relations on the moduli space of stable curves}
+}
+\setcounter{tocdepth}{2}
+\numberwithin{equation}{section}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\newtheorem{notation}[theorem]{Notation}
+\newtheorem{example}[theorem]{Example}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newcommand{\Q}{\mathbb Q}
+\newcommand{\C}{\mathbb C}
+\newcommand{\Mbar}{\overline{\mathcal M}}
+\newcommand{\M}{\mathcal M}
+\newcommand{\Salg}{S}
+\newcommand{\Pspan}{\mathcal P}
+\newcommand{\CH}{A}
+\DeclareMathOperator{\Aut}{Aut}
+\DeclareMathOperator{\Sym}{Sym}
+\DeclareMathOperator{\Sp}{Sp}
+\DeclareMathOperator{\Spec}{Spec}
+\DeclareMathOperator{\Pic}{Pic}
+\DeclareMathOperator{\Quot}{Quot}
+\DeclareMathOperator{\Ext}{Ext}
+\DeclareMathOperator{\Hom}{Hom}
+\DeclareMathOperator{\End}{End}
+\DeclareMathOperator{\im}{im}
+\DeclareMathOperator{\id}{id}
+\DeclareMathOperator{\rk}{rk}
+\DeclareMathOperator{\ch}{ch}
+\DeclareMathOperator{\td}{td}
+\DeclareMathOperator{\coker}{coker}
+\DeclareMathOperator{\Supp}{Supp}
+\setlist{itemsep=3pt,topsep=5pt}
+\emergencystretch=2em
+\title{A high-arity counterexample to\\Pixton completeness in Chow}
+\author{OpenAI}
+\date{September 24, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We disprove the Chow and rational-cohomological forms of Pixton's completeness
+conjecture for his original relation system. We construct a formal tautological
+class outside the original Pixton relation span whose image is zero in the Chow
+ring and in rational cohomology. The example has genus \(10^{60}\) and
+\(3\binom{10^{60}}3\) markings.
+\end{abstract}
+\tableofcontents
+\input{sections/01-introduction}
+\input{sections/02-strata}
+\input{sections/03-detector}
+\input{sections/04-tensors}
+\input{sections/05-quot}
+\input{sections/06-coefficient}
+\input{sections/07-compactification}
+\input{sections/08-support}
+\input{references}
+\end{document}
diff --git a/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/build/references.tex b/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/build/references.tex
new file mode 100644
index 0000000000000000000000000000000000000000..278525ee50f9022922b93c39e65ace652c9f5c9c
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@@ -0,0 +1,367 @@
+\begin{thebibliography}{Stacks}
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+Numbered statements refer to
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+Ann.\ Sci.\ \'Ecole Norm.\ Sup.\ (4) \textbf{50} (2017), no.~1, 239--267.
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+\href{https://arxiv.org/abs/1507.00688v2}{arXiv:1507.00688v2}.
+
+\bibitem[MOP2]{MOPCombinatorics}
+A.~Marian, D.~Oprea, and R.~Pandharipande,
+\emph{The combinatorics of Lehn's conjecture},
+J.\ Math.\ Soc.\ Japan \textbf{71} (2019), 299--308.
+\href{https://arxiv.org/abs/1708.08129v2}{arXiv:1708.08129v2}.
+Numbered statements refer to the December 2017 author manuscript,
+\href{https://people.math.ethz.ch/~rahul/LehnF.pdf}{author version}.
+
+\bibitem[Mil]{Milne2022}
+J.~S.~Milne,
+\emph{Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field},
+Cambridge Studies in Advanced Mathematics \textbf{170}, Cambridge University Press, 2017.
+Numbered statements refer to the revised 2022 author version,
+\href{https://www.jmilne.org/math/Books/iAG2022.pdf}{Algebraic Groups (2022)}.
+
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+Nagoya Math. J. \textbf{81} (1981), 153--175.
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+in \emph{Arithmetic and Geometry, Vol.~II}, Progress in Mathematics
+\textbf{36}, Birkh\"auser, 1983, 271--328.
+\href{https://doi.org/10.1007/978-1-4757-9286-7_12}{doi:10.1007/978-1-4757-9286-7\_12}.
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+\href{https://arxiv.org/abs/math/0504590v1}{arXiv:math/0504590v1}, 2005.
+
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+D.~Oprea and R.~Pandharipande,
+\emph{Quot schemes of curves and surfaces: virtual classes, integrals,
+Euler characteristics},
+Geom.\ Topol.\ \textbf{25} (2021), 3425--3505.
+\href{https://doi.org/10.2140/gt.2021.25.3425}{doi:10.2140/gt.2021.25.3425}.
+\href{https://arxiv.org/abs/1903.08787v2}{arXiv:1903.08787v2}.
+
+\bibitem[OS]{OSullivan2011}
+P.~O'Sullivan,
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+J.\ Reine Angew.\ Math.\ \textbf{654} (2011), 1--81.
+\href{https://doi.org/10.1515/crelle.2011.025}{doi:10.1515/crelle.2011.025}.
+Numbered statements refer to
+\href{https://arxiv.org/abs/0908.0626v1}{arXiv:0908.0626v1}.
+
+\bibitem[PP]{PandharipandePixton2021}
+R.~Pandharipande and A.~Pixton,
+\emph{Relations in the tautological ring of the moduli space of curves},
+Pure Appl.\ Math.\ Q.\ \textbf{17} (2021), no.~2, 717--771.
+\href{https://doi.org/10.4310/PAMQ.2021.v17.n2.a7}{doi:10.4310/PAMQ.2021.v17.n2.a7}.
+\href{https://arxiv.org/abs/1301.4561v2}{arXiv:1301.4561v2}.
+
+\bibitem[PPZ]{PPZ}
+R.~Pandharipande, A.~Pixton, and D.~Zvonkine,
+\emph{Relations on $\overline{\mathcal M}_{g,n}$ via $3$-spin structures},
+J.\ Amer.\ Math.\ Soc.\ \textbf{28} (2015), no.~1, 279--309.
+Numbered statements and formulas refer to the corrected version
+\href{https://arxiv.org/abs/1303.1043v3}{arXiv:1303.1043v3}.
+
+\bibitem[PZ]{PZ2019}
+R.~Pandharipande and D.~Zvonkine, with an appendix by D.~Petersen,
+\emph{Cohomological field theories with non-tautological classes},
+Ark. Mat. \textbf{57} (2019), no.~1, 191--213.
+\href{https://doi.org/10.4310/ARKIV.2019.v57.n1.a10}{doi:10.4310/ARKIV.2019.v57.n1.a10}.
+
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+D.~Petersen,
+\emph{Poincar\'e duality of wonderful compactifications and tautological rings},
+Int.\ Math.\ Res.\ Not.\ \textbf{2016} (2016), no.~17, 5187--5201.
+Numbered statements refer to
+\href{https://arxiv.org/abs/1501.04742v2}{arXiv:1501.04742v2}.
+
+\bibitem[Pix]{Pixton}
+A.~Pixton,
+\emph{Conjectural relations in the tautological ring of
+$\overline{\mathcal M}_{g,n}$},
+\href{https://arxiv.org/abs/1207.1918v1}{arXiv:1207.1918v1}, 2012.
+
+\bibitem[Pix26]{Pixton2026}
+A.~Pixton,
+\emph{The tautological ring of $\mathcal M_{25}$ is not Gorenstein},
+\href{https://arxiv.org/abs/2609.15949v1}{arXiv:2609.15949v1}, 2026.
+
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+\href{https://arxiv.org/abs/1411.2955v1}{arXiv:1411.2955v1}, 2014.
+
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+\emph{Combinatorics of symplectic invariant tensors},
+Discrete Math.\ Theor.\ Comput.\ Sci.\ Proc., FPSAC 2015 (2015), 285--296.
+\href{https://doi.org/10.46298/dmtcs.2508}{doi:10.46298/dmtcs.2508}.
+
+\bibitem[STV]{STV}
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+\emph{Derived algebraic geometry, determinants of perfect complexes,
+and applications to obstruction theories for maps and complexes},
+J.\ Reine Angew.\ Math.\ \textbf{702} (2015), 1--40.
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+Numbered statements follow the
+\href{https://flore.unifi.it/retrieve/e398c37f-a432-179a-e053-3705fe0a4cff/crelle.pdf}{accepted-manuscript galley}.
+
+\bibitem[Stacks]{Stacks}
+The Stacks Project Authors,
+\emph{The Stacks Project}, 2026.
+\url{https://stacks.math.columbia.edu}.
+Specific results are cited by their stable tags.
+
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+$R^*(\mathcal M_{g,n}^{\mathrm{rt}})$},
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+Preprint version, titled \emph{A conjectural connection between
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+\href{https://arxiv.org/abs/1412.5036v2}{arXiv:1412.5036v2}.
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+Numbered statements follow the
+\href{https://www.math.univ-toulouse.fr/~btoen/ttt.pdf}{April 2007 author manuscript}.
+
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+
+\end{thebibliography}
diff --git a/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/build/sections/01-introduction.tex b/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/build/sections/01-introduction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..bfa29bfa7ea899b14b27360ac697e65c220890b4
--- /dev/null
+++ b/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/build/sections/01-introduction.tex
@@ -0,0 +1,313 @@
+\section{Introduction}\label{sec:introduction}
+
+The tautological ring of the moduli space of stable curves is generated
+by cotangent and kappa classes, together with the classes obtained by
+forgetting markings and gluing nodes. Describing all relations among
+these natural classes is a central problem in the intersection theory
+of moduli spaces. Pixton proposed a uniform system of stable-graph
+relations and conjectured that it is complete. We construct a relation
+outside that system: an alternating product of corrected small diagonals
+that vanishes in rational Chow, and therefore in rational cohomology.
+
+\subsection{The relation problem}
+
+Mumford initiated the systematic study of natural classes in the Chow
+ring of the moduli space of curves \cite[Sections~4--5]{Mumford1983}.
+Faber's calculations and geometric constructions on powers of the
+universal curve led to a proposed description of the tautological ring
+of the smooth unpointed space \(\mathcal M_g\), including its Gorenstein
+property \cite[Conjectures~1--2]{Faber1999}. Faber and Zagier later gave
+explicit hypergeometric formulas for relations among the kappa classes.
+Pandharipande--Pixton proved these relations in Chow using stable
+quotients and formulated their completeness on \(\mathcal M_g\)
+\cite[Theorem~1 and Conjecture~2]{PandharipandePixton2021}.
+
+On the compactification, a presentation must also account for boundary
+strata and their intersections. Arbarello--Cornalba and
+Graber--Pandharipande developed the forgetful and boundary calculus
+\cite[Section~1]{ArbarelloCornalba1996},
+\cite[Appendix~A]{GraberPandharipande2003}; in genus zero, Keel gave a
+presentation of the entire Chow ring by boundary divisors
+\cite[Section~4, Theorem~1]{Keel1992}.
+Pixton's 2012 graph formula extends the Faber--Zagier system to
+\(\Mbar_{g,n}\) and proposes a complete set of relations
+\cite[Sections~1--3, Conjecture~2]{Pixton}.
+Pandharipande--Pixton--Zvonkine proved the cohomological validity of this
+system using the \(3\)-spin CohFT \cite{PPZ}; Janda proved its Chow
+validity using equivariant Gromov--Witten theory of \(\mathbb P^1\)
+\cite{Janda}. These results establish relations, while completeness asks
+whether any further relations exist.
+
+We fix the original relation space precisely. Let \(\Salg_{g,n}\) be the
+strata algebra of formal decorated stable graphs, with decoration degree
+truncated at the dimension of each vertex factor. Its realization is
+\[
+q_{g,n}\colon\Salg_{g,n}\longrightarrow\CH^*(\Mbar_{g,n};\Q),
+\qquad [\Gamma,\gamma]\longmapsto\xi_{\Gamma*}\gamma.
+\]
+The gluing pushforward generators are unnormalized, as in
+\cite[Section~0.3]{PPZ}. Let \(\Pspan_{g,n}\) be the span obtained by
+placing one of Pixton's graph-formula relations at a vertex of an outer
+stable graph, placing arbitrary strata classes at the other vertices,
+and gluing. Pixton's Proposition~1 shows that this span is already an
+ideal, stable under the gluing and forgetting pushforwards and pullbacks
+\cite{Pixton}. Thus it includes these formal consequences from the
+outset. The smaller \(0/1\)-marking seed system recovers the same span
+under the operations described in \cite[Section~3.5]{PPZ}.
+Section~\ref{sec:strata} gives the full recipe and all conventions.
+
+Writing \(R^*(\Mbar_{g,n})=\im q_{g,n}\), the Chow completeness question is
+whether
+\[
+\ker q_{g,n}=\Pspan_{g,n}
+\]
+for every stable pair. The known validity theorem gives the inclusion
+\(\Pspan_{g,n}\subseteq\ker q_{g,n}\). Our example makes it strict.
+
+\subsection{The counterexample}
+
+The construction uses \(g\) colors and one three-label factor for each
+increasing triple of colors. Each factor is a product of two
+rational-tail divisor sums, corrected by a class supported on two-node
+rational bridges. We multiply these factors and alternate over
+relabelings that preserve the color of each slot.
+Definition~\ref{def:candidate} specifies the resulting formal class \(Y\).
+
+\begin{theorem}\label{thm:main}
+Set
+\[
+g=10^{60},\qquad D_*=\binom g3,\qquad n=3D_*.
+\]
+The explicitly defined class \(Y\in \Salg_{g,n}^{2D_*}\) of
+Definition~\ref{def:candidate} satisfies
+\[
+q_{g,n}(Y)=0,\qquad Y\notin\Pspan_{g,n}.
+\]
+Consequently Pixton's original relation span does not exhaust the
+tautological Chow relations: the equality
+\(\ker q_{g,n}=\Pspan_{g,n}\) fails for this stable pair.
+\end{theorem}
+
+The same class gives the rational-cohomological consequence. To specify
+that formulation, write \(q^{\mathrm{CH}}_{g,n}=q_{g,n}\) and define, in
+formal degree \(d\),
+\[
+q^H_{g,n}=\operatorname{cl}\circ q^{\mathrm{CH}}_{g,n}
+\colon\Salg^d_{g,n}\longrightarrow
+H^{2d}\!\left(\Mbar_{g,n}(\C);\Q\right),
+\]
+where \(\operatorname{cl}\) is the rational cycle-class map for the complex
+Deligne--Mumford stack. Its image is the cohomological tautological ring
+\(RH^*=\operatorname{cl}(R^*)\). Completeness in both \(R^*\) and \(RH^*\)
+is explicitly recorded as Pixton's conjecture in
+\cite[Section~1, Conjecture~1]{Canning2025}.
+
+\begin{corollary}[Rational-cohomological incompleteness]
+\label{cor:cohomological-incompleteness}
+For the stable pair and class \(Y\) of Theorem~\ref{thm:main},
+\[
+q^H_{g,n}(Y)=0
+\quad\text{in }H^{4D_*}\!\left(\Mbar_{g,n}(\C);\Q\right),
+\qquad
+Y\notin\Pspan_{g,n}.
+\]
+In particular, \(\Pspan_{g,n}\subsetneq\ker q^H_{g,n}\).
+\end{corollary}
+
+\begin{proof}
+Cycle class commutes with Chern classes, products, and proper
+pushforward. In particular,
+\(\operatorname{cl}(\xi_{\Gamma*}\gamma)
+=\xi_{\Gamma*}\operatorname{cl}(\gamma)\), and the same compatibility
+applies to the forgetting-pushforward definition of kappa classes.
+Thus \(q^H_{g,n}\) is the cohomological graph realization of
+\cite[Section~0.3]{PPZ}, with the same vertexwise truncation and
+unnormalized generators. Applying cycle class to
+Theorem~\ref{thm:main} gives \(q^H_{g,n}(Y)=0\); nonmembership remains
+an assertion in the same formal strata algebra. The strict inclusion
+follows from
+\[
+\Pspan_{g,n}\subseteq\ker q^{\mathrm{CH}}_{g,n}
+\subseteq\ker q^H_{g,n}.
+\]
+Passing from Chow vanishing to cohomological vanishing uses the second
+inclusion; it requires no injectivity of cycle class.
+\end{proof}
+
+The theorem and corollary give negative answers to the Chow and
+rational-cohomological completeness questions for the full original
+system. The large parameters leave ample room in the support and color
+estimates; they are not proposed as minimal parameters.
+
+Completeness is distinct from the Gorenstein property. The latter asks
+for nondegenerate multiplication pairings in complementary degrees,
+whereas completeness asks for all relations among the generators.
+Pixton proved that \(R^*(\mathcal M_{25})\) is not Gorenstein while
+establishing Faber--Zagier completeness there
+\cite[Theorem~1.1 and Section~2.2]{Pixton2026}.
+Canning--Larson--Schmitt proved completeness together with
+non-Gorenstein behavior on the compact-type spaces
+\(\mathcal M^{\mathrm{ct}}_6\), \(\mathcal M^{\mathrm{ct}}_{5,2}\), and
+\(\mathcal M^{\mathrm{ct}}_7\)
+\cite[Theorems~1 and~7]{CanningLarsonSchmitt2026}.
+Our relation lies on the full stable compactification.
+
+There is also a distinction between completeness and comparison of
+methods for producing relations. Janda's comparison theorem identifies
+the systems arising from cancellation of reconstruction singularities
+for convergent semisimple CohFTs with nonempty discriminant
+\cite[Definition~3.3.1 and Theorem~3.3.6]{JandaDiscriminant}.
+It concerns the relations generated by those reconstructions; it does
+not identify the entire Chow relation kernel. The comparison throughout
+this paper is with the explicitly fixed span \(\Pspan_{g,n}\).
+
+\subsection{Proof and methods}
+
+The proof compares two algebras of cubic coefficients. A formal
+super Frobenius algebra can carry independent odd cubic coefficients
+while satisfying the identities needed to annihilate the original
+Pixton span. The actual Chow correspondences of a curve satisfy an
+additional nilpotence identity obtained from a surface Quot scheme.
+The first algebra proves nonmembership; the second proves vanishing.
+Figure~\ref{fig:proof-map} separates these two assertions about \(Y\).
+
+\paragraph{The formal test.}
+Section~\ref{sec:detector} constructs a linear map on decorated graphs
+by discarding graphs outside the rational-tails locus and evaluating
+rational branches by Frobenius multiplication and coproduct. This uses
+the graph calculus familiar from two-dimensional topological field
+theory \cite[Section~2 and Theorem~3]{Abrams1996}.
+The odd symplectic curve pairing and the vanishing of contractions
+between isotropically supported corrections have precedents in
+Pandharipande--Zvonkine's CohFT constructions
+\cite[Sections~1.1--1.5 and Remark~11]{PZ2019}.
+Here the coefficient ring is the exterior algebra on independent
+\(\theta_{ijk}\), one for every color triple; we verify the graph
+identities directly. Every element of \(\Pspan_{g,n}\) has zero test,
+while the test of \(Y\) has a nonzero coefficient proportional to
+\(\prod_{i<j<k}\theta_{ijk}\).
+
+\paragraph{The smooth-family Chow identity.}
+Section~\ref{sec:tensors} splits the correspondence object of a smooth
+curve into two even summands and an odd symplectic summand. The
+coefficient-algebra construction is a concrete form of O'Sullivan's
+semisimple tensor reconstruction
+\cite[Lemma~2.3.1]{OSullivan2011}. It retains all actual relative Chow
+correspondences, so its faithfulness is an assertion about Chow, not
+about cohomological realization. The small diagonal projected to three
+odd summands gives an alternating cubic \(\varphi\) with odd
+coefficients. Over a pointed curve this is the modified diagonal of
+Gross--Schoen \cite[Equation~(0.1) and Sections~2--3]{GrossSchoen1995};
+the family construction also retains the necessary base corrections.
+
+Sections~\ref{sec:quot}--\ref{sec:coefficient} prove the additional
+nilpotence. The surface is the square of the curve. A relative
+Quot scheme with fixed determinant produces a relation for products
+of the cubic on isotropic vector sets. Its obstruction theory uses
+derived perfect-complex moduli and determinant maps \cite{TV,STV},
+virtual pullbacks \cite{BF,Manolache}, and virtual localization
+\cite{GP}. Surface Quot localization and relative Chow identities
+appear in Marian--Oprea--Pandharipande \cite{MOPSegre}; the point
+obstruction factors have the antecedent in
+\cite[Lemma~34 and Section~5.2]{OPQuot}.
+Here the calculation keeps the determinant constraint, arbitrary
+irregularity, and every marked surface output.
+
+The nested-Hilbert recursion of Ellingsrud--G\"ottsche--Lehn
+\cite{EGL} reduces that marked calculation to decorated equality
+diagonals. Kong develops relative forms of this recursion and, for
+curve families, formulas retaining one and two marked outputs
+\cite[Section~2.1 and Lemma~3.4]{Kong2026}.
+We prove the version retaining an arbitrary finite set of surface
+outputs. Only bare triple blocks survive the relevant codimension
+bound. Section~\ref{sec:coefficient} computes their common scalar and
+proves it is nonzero, using symmetric-power and residue methods related
+to \cite[Equation~(9), Theorem~2, and Lemma~3]{MOPCombinatorics}.
+It then converts the isotropic identity into vanishing tests for
+sufficiently long products with arbitrary color inputs
+(Lemma~\ref{lem:large-vertex-test}). Finite faithful polarization returns
+a Chow identity modulo classes supported over the prescribed positive
+codimension in the base.
+
+\paragraph{Passage to the compactification.}
+Section~\ref{sec:compactification} compares the corrected stable cubic
+with these projected smooth-curve cubics. The framework uses Hassett's
+weighted stable curves \cite{Hassett} and configuration-space blowups
+\cite{FultonMacPherson,Routis}; Petersen's proof of Tavakol's
+presentation conjecture treats the corresponding rational-tails
+comparison over a smooth core \cite{TavakolRT,PetersenWonderful}.
+The nodal comparison here requires a rational-bridge correction with
+coefficient \(-1/2\). A three-position calculation shows that it
+cancels the remaining triple-exceptional contribution. The errors are
+then either a two-slot operation times a one-slot operation from a
+bounded span, or operations supported over positive base codimension.
+
+Section~\ref{sec:support} uses these errors to lower the dimension of
+the image of the supporting cycle in the base. Chow-localization
+support methods occur in Graber--Vakil's filtration by rational
+components \cite[Theorem~1.1 and Section~4.2]{GraberVakil}; the induction
+here acts on arbitrary cycle multipliers over the base parametrizing
+the curve components.
+When one component carries almost all the genus, the surface identity
+advances support. Otherwise a repeated-color identity supplies a
+smaller vanishing factor, using the curve algebra centered at a marked
+node branch,
+Schur--Weyl alternation, and symplectic contraction. The final color
+budget leaves enough entries for every advancement, until the support
+is empty. Pullback from the weighted space then proves \(q_{g,n}(Y)=0\).
+
+Thus Sections~\ref{sec:strata}--\ref{sec:detector} construct and detect
+the formal candidate; Sections~\ref{sec:tensors}--\ref{sec:coefficient}
+prove the smooth-family identity; and
+Sections~\ref{sec:compactification}--\ref{sec:support} extend its
+vanishing to the compactification. Table~\ref{tab:notation} collects
+the notation used across these parts.
+
+\input{figures/proof-map}
+
+\subsection{Conventions}
+
+All varieties, stacks, and families are over \(\C\), and Chow groups
+have rational coefficients. On a smooth space we identify Chow classes
+with operational classes when pulling them to a test space.
+Products of stacks are understood in rational Chow theory
+\cite[Theorem~2.1.12]{Kresch}.
+We use proper rational pushforward for relative Deligne--Mumford maps
+and its compatibilities from \cite[Theorem~B.17 and
+Propositions~B.18--B.19]{BaeSchmitt}. Smooth Deligne--Mumford stacks have
+the operational identification in \cite[Theorem~C.6 and
+Corollary~C.9]{BaeSchmitt}; for smooth quotient stacks, see also
+\cite[Propositions~4 and~19]{EdidinGraham}.
+Every formal series calculation is
+truncated at a specified coefficient, so only finitely many terms occur.
+
+Parity is distinct from Chow codimension. In a super tensor product,
+interchanging homogeneous factors of parities \(p,q\) gives
+\((-1)^{pq}\). In particular, on an odd vector space the
+super-symmetric tensors are ordinary alternating tensors.
+All finite averaging and polarization arguments take place in
+characteristic zero.
+
+
+\begin{table}[tb]
+\centering
+\small
+\begin{tabular}{@{}p{.20\textwidth}p{.53\textwidth}p{.19\textwidth}@{}}
+\toprule
+Notation & Meaning & Definition\\
+\midrule
+\(\Salg_{g,n},q_{g,n},\Pspan_{g,n}\) & Formal strata algebra, Chow realization, original Pixton span & Section~\ref{sec:strata}\\
+\(D_*,n,Y\) & Number of cubic factors, number of labels, candidate relation & Definition~\ref{def:candidate}\\
+\(R,V,t_m\) & Exterior coefficient ring, formal Frobenius algebra, detector & Section~\ref{sec:detector}\\
+\(e'_C,\eta,H_C\) & Corrected degree-one class, pure projector, complementary curve object & Section~\ref{sec:tensors}\\
+\(\mathfrak B,W_b,\varphi\) & Faithful Chow coefficients, odd symplectic space, projected cubic & Proposition~\ref{prop:tensor-test}\\
+\(t_0,T_\gamma,a_{N,t_0}\) & Nilpotence exponent, isotropic cubic contraction, localization scalar & Sections~\ref{sec:quot}--\ref{sec:coefficient}\\
+\(F^0_{h,B},G_{h,B}\) & Corrected weighted cubic and its chosen pure lift & Section~\ref{sec:compactification}\\
+\(X^\Gamma,\mathcal B_\Gamma,\mathcal S_\Gamma\) & Cut chart, compact heavy base, smooth-core open & Section~\ref{sec:compactification}\\
+\bottomrule
+\end{tabular}
+\caption{Notation used across the proof. The formal coefficient ring
+\(R\) and the faithful Chow coefficient algebra \(\mathfrak B\) have
+different roles.}
+\label{tab:notation}
+\end{table}
diff --git a/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/build/sections/02-strata.tex b/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/build/sections/02-strata.tex
new file mode 100644
index 0000000000000000000000000000000000000000..7873afaafa5c622a57e3580f5960560972c3d85c
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+++ b/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/build/sections/02-strata.tex
@@ -0,0 +1,245 @@
+\section{The strata algebra and the explicit class}\label{sec:strata}
+
+This section fixes the relation space. The distinction between an
+unnormalized gluing generator and an automorphism-weighted graph sum
+will be used in both parts of the proof.
+
+\subsection{Decorated graphs}
+
+A stable graph \(\Gamma\) of type \((g,n)\) is a connected graph with
+\(n\) labeled legs and genera \(g_v\geq0\) at its vertices, satisfying
+\[
+g=\sum_{v}g_v+|E(\Gamma)|-|V(\Gamma)|+1,
+\qquad 2g_v-2+n_v>0.
+\]
+Here \(n_v\) counts all incident half-edges, including legs. Its gluing
+map is
+\[
+\xi_\Gamma\colon\prod_{v\in V(\Gamma)}
+\Mbar_{g_v,n_v}\longrightarrow\Mbar_{g,n}.
+\]
+At every half-edge let \(\psi\) denote the cotangent class. At a vertex
+define
+\[
+\kappa_j=\pi_*(\psi_{n_v+1}^{j+1}),\qquad
+\kappa_0=2g_v-2+n_v,
+\]
+where \(\pi\) forgets an additional marking.
+
+The vector space \(\Salg_{g,n}\) is spanned by formal symbols
+\([\Gamma,\gamma]\), modulo graph isomorphism, with \(\gamma\) a monomial
+in the half-edge \(\psi\)'s and vertex \(\kappa_j\)'s for \(j\geq1\).
+At each vertex its decoration degree is at most \(3g_v-3+n_v\).
+The total degree is \(|E(\Gamma)|+\deg\gamma\), and
+\[
+q_{g,n}[\Gamma,\gamma]=\xi_{\Gamma*}(\gamma).
+\]
+There is no automorphism denominator in this definition of \(q_{g,n}\).
+The per-vertex degree bound is the explicit convention in
+\cite[Section~0.3]{PPZ}.
+
+We use the usual decorated-graph intersection product. More explicitly,
+take common refinements with specified contractions to fixed
+representatives of the input graphs, requiring every refined edge to
+survive in at least one contraction. Divide by the source symmetries
+preserving those contractions. For each additional use of an overlapping
+edge insert its normal factor \(-\psi-\psi'\); pull the leg and flag
+decorations to their components and split kappa classes additively.
+Specified contractions record the lifts to the gluing parameter spaces,
+so this convention gives the product of the \emph{unnormalized}
+pushforward generators. The construction is the transverse and
+self-intersection rule for node cuts
+\cite[Appendix~A, Proposition~9 and Equation~(11)]{GraberPandharipande2003}.
+
+Terms exceeding any vertex dimension are set to zero. For multiplication
+and restriction to a further graph splitting, an excessive decoration
+remains excessive: the combined dimensions on the pieces decrease,
+so at least one piece still exceeds its dimension.
+Forgetting pullback of an undecorated graph is obtained by adding
+the new leg successively at each vertex. These conventions lift the
+corresponding Chow operations and do not impose any unproved relation.
+The preceding dimension argument is not an assertion about forgetting,
+which raises a vertex dimension. Compatibility of forgetting with the
+formal detector is checked separately in Section~\ref{sec:detector}.
+
+\subsection{The exact Pixton span}
+
+Define
+\[
+A(T)=\sum_{m\geq0}\frac{(6m)!}{(3m)!(2m)!}T^m,\qquad
+B(T)=\sum_{m\geq0}
+\frac{6m+1}{6m-1}\frac{(6m)!}{(3m)!(2m)!}T^m.
+\]
+For a commuting parity variable \(\zeta^2=1\), put
+\[
+\widehat C_{3j}(T,\zeta)=T^jA(\zeta T),\qquad
+\widehat C_{3j+1}(T,\zeta)=\zeta T^jB(\zeta T).
+\]
+Indices congruent to \(2\) modulo \(3\) are not used. For a series \(F\)
+write
+\[
+\{F\}=\sum_{m\geq0,\ \epsilon\in\{0,1\}}
+[F]_{T^m\zeta^\epsilon}K_{m,\epsilon}T^m.
+\]
+Give every vertex of \(\Gamma\) its own variable \(\zeta_v\), with
+\(\zeta_v^2=1\). Define the operator
+\[
+\widehat\kappa_\Gamma
+\left(\prod_{i=1}^{\ell}K_{e_i,\epsilon_i}\right)
+=
+\sum_{\tau\in\mathfrak S_\ell}
+\prod_{\substack{c\text{ a cycle}\\\text{of }\tau}}
+\left(
+\sum_{v\in V(\Gamma)}
+\kappa^{(v)}_{\sum_{i\in c}e_i}
+\zeta_v^{\sum_{i\in c}\epsilon_i}
+\right).
+\]
+The empty product maps to \(1\), and the operator acts coefficientwise
+in \(T\). The permutation-cycle expression is Faber's formula, as
+recorded in \cite[Equations~(1.12)--(1.13)]{ArbarelloCornalba1996}.
+
+For an edge whose two half-edges lie at \(v,w\), with cotangent symbols
+\(x,y\), set
+\[
+\Delta_e=
+\frac{
+A(\zeta_vxT)\zeta_wB(\zeta_wyT)
++\zeta_vB(\zeta_vxT)A(\zeta_wyT)
++\zeta_v+\zeta_w
+}{(x+y)T}.
+\]
+This is a power series by
+\[
+A(T)B(-T)+A(-T)B(T)+2=0.
+\]
+The same rule applies to a loop.
+
+\begin{definition}[Pixton's original span]\label{def:pixton-span}
+Let \(\sigma=(\sigma_1,\ldots,\sigma_\ell)\) be a partition with no
+part congruent to \(2\) modulo \(3\), possibly empty, and let
+\(a_1,\ldots,a_n\geq0\) also avoid that residue class.
+For \(0\leq r\leq3g-3+n\), require
+\begin{equation}\label{eq:pixton-admissibility}
+3r\geq g+1+|\sigma|+\sum_i a_i,\qquad
+3r\equiv g+1+|\sigma|+\sum_i a_i\pmod2.
+\end{equation}
+Set
+\begin{align}
+P_\Gamma={}&
+\left[
+2^{-h^1(\Gamma)}
+\widehat\kappa_\Gamma
+\left(
+\exp\{1-\widehat C_0\}
+\prod_{j=1}^{\ell}\{\widehat C_{\sigma_j}\}
+\right)\right.\notag\\
+&\left.\hspace{12mm}\cdot
+\prod_{i=1}^n\widehat C_{a_i}(\psi_iT,\zeta_{v(i)})
+\prod_{e\in E(\Gamma)}\Delta_e
+\right]_{T^{r-|E(\Gamma)|}\prod_v\zeta_v^{g_v+1}},
+\label{eq:pixton-graph}
+\end{align}
+where \(h^1(\Gamma)=|E|-|V|+1\), \(v(i)\) carries the \(i\)-th leg,
+and parity exponents are read modulo \(2\). In braces, the arguments
+of \(\widehat C_i\) are \(T,\zeta\). A negative requested \(T\)-degree
+means zero. The relation is
+\[
+\mathcal R(g,n,r;\sigma,a)=
+\sum_\Gamma\frac{[\Gamma,P_\Gamma]}{|\Aut\Gamma|}.
+\]
+
+The space \(\Pspan_{g,n}\subseteq\Salg_{g,n}\) is the following
+linear span in each degree. Choose an outer stable graph of type
+\((g,n)\); put one such relation at one vertex, and arbitrary decorated
+strata elements at all other vertices; then glue, using the
+unnormalized gluing operation. Take the span over every choice,
+including the graph with one vertex and no edges.
+\end{definition}
+
+\begin{proposition}[Known validity and stability]\label{prop:known-relations}
+The span in Definition~\ref{def:pixton-span} is an ideal and is stable
+under tautological gluing and forgetting pushforwards and pullbacks.
+For gluing pullback to a product, the target relation space is the sum
+of the spaces with a relation in one factor.
+Moreover
+\[
+\Pspan_{g,n}\subseteq\ker q_{g,n}.
+\]
+\end{proposition}
+
+\begin{proof}
+The ideal and operation statements are
+\cite[Proposition~1]{Pixton} for precisely this original span.
+Chow validity of the \(a_i\in\{0,1\}\) seed relations follows from
+\cite[Theorem~1 and its reconstruction]{Janda}.
+The construction in \cite[Section~3.5]{PPZ} generates the higher
+marking indices by psi multiplication, the partition insertions by
+forgetting extra markings, and the outer-graph relations by gluing.
+These are formal identities, so they apply to the Chow-valid seeds.
+The parity condition in \eqref{eq:pixton-admissibility} converts the
+strict seed inequality to the displayed admissibility inequality.
+In Janda's reconstructed normalization, a degree-\(r\) seed is
+\((-1)^{\sum_i a_i}(-1/1728)^r\) times the empty-partition recipe
+above. This nonzero scalar does not change its vanishing or its span.
+\end{proof}
+
+No converse to Proposition~\ref{prop:known-relations} is used.
+In particular, same-vertex multiplication of a relation does not enlarge
+the span, whereas an arbitrary new Chow relation is not assumed to
+belong to it.
+
+\subsection{The corrected cubic and its alternation}
+
+For the rest of the construction fix
+\begin{equation}\label{eq:global-parameters}
+g=10^{60},\qquad D_*=\binom g3,\qquad n=3D_*.
+\end{equation}
+Form a word \(\mathbf i\) in the colors \(1,\ldots,g\) by concatenating
+all increasing triples of colors in lexicographic order. Its consecutive
+three-slot blocks will be denoted \(J\). Each color occurs
+\(\binom{g-1}{2}\) times.
+
+For a set of labels \(T\) of size at least two, let \(D_T\) be the
+one-edge class cutting off a genus-zero tail with precisely those
+labels. For distinct labels \(a,b\), put
+\[
+\bar D_{ab}=\sum_{T\supseteq\{a,b\}}D_T.
+\]
+We also need a class supported on rational bridges, not tails.
+
+\begin{definition}[The explicit counterexample]\label{def:candidate}
+On three labeled slots \(J=(a,b,c)\), take every one-edge
+\emph{unmarked} stable graph of genus \(g\), subdivide its edge by
+a genus-zero vertex, and put all three labels at that new vertex.
+Sum the resulting undecorated graphs, dividing each by its automorphism
+order. Pull this sum back by forgetting all labels outside \(J\);
+call the resulting class \(\beta_J^{\mathrm{st}}\).
+
+Define
+\begin{equation}\label{eq:candidate}
+F_J^{\mathrm{st}}=
+\bar D_{ab}\bar D_{ac}-\frac12\beta_J^{\mathrm{st}},
+\qquad
+Y=\sum_{\substack{\tau\in\mathfrak S_n\\\tau\mathbf i=\mathbf i}}
+\operatorname{sgn}(\tau)\,
+\tau\left(\prod_J F_J^{\mathrm{st}}\right).
+\end{equation}
+Here \(\tau\) relabels slots, and products and forgetting pullbacks
+are the formal graph operations fixed above.
+\end{definition}
+
+Every operation in Definition~\ref{def:candidate} is finite.
+Each cubic factor has codimension two, so
+\[
+\deg Y=2D_*<3g-3+n.
+\]
+The pair \((g,n)\) is stable. The automorphism weights in
+\(\beta_J^{\mathrm{st}}\) specify coefficients of this particular
+class; they do not change the unnormalized generators of
+\(\Salg_{g,n}\).
+
+The proof of Theorem~\ref{thm:main} now divides into two independent
+claims. Proposition~\ref{prop:detector} proves formal nonmembership.
+The remaining sections establish the geometric vanishing
+in Theorem~\ref{thm:chow-vanishing}.
diff --git a/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/build/sections/03-detector.tex b/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/build/sections/03-detector.tex
new file mode 100644
index 0000000000000000000000000000000000000000..d461136e4660d32b0efb04c38e745b6e72ce10ad
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+++ b/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/build/sections/03-detector.tex
@@ -0,0 +1,763 @@
+\section{A formal detector}
+\label{sec:detector}
+
+We first separate the candidate from the relation space.  Throughout this
+section the genus \(g\) is fixed as in Definition~\ref{def:candidate},
+but the number of labels is allowed to vary, including zero.
+No Chow relation on a positive-genus vertex will be imposed in the
+construction.
+
+\begin{proposition}
+\label{prop:detector}
+There are a supercommutative \(\Q\)-algebra \(R\), a finite free
+supermodule \(V\) over \(R\), and \(\Q\)-linear maps
+\[
+ t_m:\Salg_{g,m}\longrightarrow V^{\otimes_R m}
+ \qquad(m\geq 0)
+\]
+such that \(t_m(\Pspan_{g,m})=0\) for every \(m\), whereas
+\(t_n(Y)\neq0\).  In particular \(Y\notin\Pspan_{g,n}\).
+\end{proposition}
+
+The maps are tests on formal strata.  In particular, they are not
+asserted to factor through the Chow ring.
+
+\subsection{An isotropic cubic Frobenius algebra}
+
+The state space has the curve pairing familiar from Frobenius-algebra
+and CohFT constructions \cite{Abrams1996,PZ2019}. The exterior
+coefficient ring below allows an independent cubic on every triple of
+colors. We check associativity and the handle formula directly, since
+these particular identities govern the graph test.
+
+Let
+\[
+ R=\Lambda_{\Q}\bigl(\theta_{ijk}:1\leq i<j<k\leq g\bigr),
+\]
+with each generator odd.  Let \(W\) be a purely odd symplectic vector
+space of dimension \(2g\), with complementary Lagrangians
+\(W_+\) and \(W_-\).  Choose bases \(a_1,\ldots,a_g\) and
+\(b_1,\ldots,b_g\) such that
+\(\langle a_i,b_j\rangle=\delta_{ij}\).
+Extend the symbols \(\theta_{ijk}\) alternately to all triples of
+indices.  There is an alternating, odd-coefficient three-form
+\(\Phi\) on \(W\) defined by
+\[
+ \Phi(a_i,a_j,a_k)=\theta_{ijk},
+ \qquad
+ \Phi(w_1,w_2,w_3)=0
+ \quad\text{if some }w_i\in W_-.
+\]
+
+Consider the free supermodule
+\[
+ V=R\otimes_{\Q}\bigl(\Q1\oplus W\oplus\Q e\bigr),
+\]
+where \(1,e\) are even.  Its trace, denoted by \(\int\), takes \(e\)
+to \(1\) and is zero on \(1,W\).  Give \(V\) the even pairing which
+is hyperbolic on \(\Q1\oplus\Q e\), the prescribed symplectic pairing
+on \(W\), and zero between these two summands.  Declare \(1\) to be
+the unit, \(e^2=eW=0\), and determine the remaining products by
+\[
+ \int w_1w_2=\langle w_1,w_2\rangle,
+ \qquad
+ \int w_1w_2w_3=\Phi(w_1,w_2,w_3).
+\]
+Equivalently, the product of two standard odd vectors is its
+symplectic pairing times \(e\), plus the vector dual to the cubic
+functional in the third input.  The latter vector has odd
+coefficients and belongs to \(R\otimes W_-\).
+
+Write \(\mu\) for multiplication and \(\Delta\) for its adjoint
+coproduct.
+
+\begin{lemma}
+\label{lem:detector-algebra}
+These rules give a supercommutative associative Frobenius algebra.
+The product of three standard odd vectors belongs to \(Re\), and
+the product of four such vectors is zero.  Its handle element is
+\[
+ \mu\Delta(1)=(2-2g)e.
+\]
+\end{lemma}
+
+\begin{proof}
+The pairing is nondegenerate, so the rules determine the product.
+Supercommutativity follows from alternation of the symplectic form
+and of \(\Phi\), with coefficients moved using the super sign rule.
+For example,
+\[
+ a_i a_j=-\sum_k\theta_{ijk}b_k,
+ \qquad a_i b_j=\delta_{ij}e,
+\]
+and therefore \((a_i a_j)a_k=\theta_{ijk}e
+=a_i(a_j a_k)\).
+
+More generally, pair the associator of three vectors from \(W\)
+with a fourth basis vector.  Pairing with \(1\) gives the cyclic
+symmetry of the alternating cubic, and pairing with \(e\) gives
+zero.  If the fourth vector is in \(W\), both expressions are
+contractions of two pair-product tensors.  Their \(W\)-channel
+contractions vanish because the cubic is supported on a
+Lagrangian; their hyperbolic-channel contractions vanish because
+\(e^2=eW=0\).  Nondegeneracy proves associativity.  Cases involving
+\(1\) or \(e\) are immediate.  The same product description proves
+the assertions about three and four odd inputs.
+
+The hyperbolic channels contribute \(2e\) to the handle element,
+and the \(2g\) odd channels contribute \(-2ge\).  The cubic
+perturbation has zero internal trace, because a symplectic
+contraction pairs the two different Lagrangians.  This gives the
+displayed formula.
+\end{proof}
+
+Set
+\[
+ \lambda=(2g-2)e.
+\]
+For a nonempty finite label set \(L\), write
+\(\Delta_L:V\to V^{\otimes L}\) for the coproduct adjoint to
+iterated multiplication; for a singleton it is the identity.
+Coproducts are coassociative and satisfy the projection formula.
+In particular, writing \(\Delta_{ij}(1)\) with units in all other
+slots, we have
+\begin{align}
+ \Delta_{ij}(1)\Delta_{ik}(1)&=\Delta_{ijk}(1), \label{eq:detector-diagonals}\\
+ \Delta_{ij}(1)^2&=-\Delta_{ij}(\lambda). \label{eq:detector-self}\\
+ \int_j\Delta_{ij}(v)&=v. \label{eq:detector-counit}
+\end{align}
+The minus sign in \eqref{eq:detector-self} will match the normal
+bundle of a rational-tail boundary divisor.
+
+\subsection{Compression of rational tails}
+
+Call a graph a \emph{rational-tails graph} if it is a tree with one
+genus-\(g\) vertex, called its root, and all other vertices rational.
+Every branch away from the root has nonempty labeled descendants.
+Such a graph has no automorphisms fixing the labels: the descendant
+label sets distinguish its branches, recursively.
+
+We initially work without the degree cutoff, discard all graphs
+which are not rational-tails graphs, and allow genuine genus-zero
+Chow identities on rational parameter factors.  This is only an
+auxiliary computation; we will prove that its output respects the
+cutoff in \(\Salg_{g,m}\). We use the genus-zero Chow boundary
+presentation of \cite[Section~4, Theorem~1]{Keel1992}.
+
+For each label \(i\), put
+\[
+ \bar\psi_i=\psi_i-\sum_{\substack{T\ni i\\ |T|\geq2}}D_T.
+\]
+If the labels are ordered \(1,\ldots,m\), let \(L_x\) be the
+pullback of \(\psi_x\) from the space retaining just
+\(1,\ldots,x\).  Thus
+\[
+ L_x=\psi_x-
+ \sum_{\substack{T\ni x,\ |T|\geq2\\T\setminus\{x\}\subseteq\{x+1,\ldots,m\}}}
+ D_T.
+\]
+For \(j\geq1\), define
+\[
+ \bar\kappa_j=\kappa_j-\sum_{x=1}^m L_x^j.
+\]
+The leading, graphless change of variables is
+\(\bar\psi_i=\psi_i\) and
+\(\bar\kappa_j=\kappa_j-\sum_i\psi_i^j\), which is an invertible
+polynomial change.  Hence the bar monomials form a complement to
+the boundary subspace before truncation.
+
+Define \(t_m\) on that complement by
+\begin{equation}
+ t_m\left(\prod_i\bar\psi_i^{d_i}
+                 \prod_{j\geq1}\bar\kappa_j^{c_j}\right)
+ =
+ \begin{cases}
+   \lambda_1^{d_1}\cdots\lambda_m^{d_m},
+      &\text{if every }c_j=0,\\
+   0,&\text{otherwise}.
+ \end{cases}
+ \label{eq:detector-prescription}
+\end{equation}
+Here the tensor on the right has a unit in every unused slot.
+For a nontrivial rational-tails graph, integrate each non-root
+decoration on its genus-zero parameter space, retaining only its
+top-degree integral.  Apply \(t_a\) to the decorated root with its
+\(a\) incident slots.  Finally duplicate each root slot into the
+descendant labels of its branch by \(\Delta_L\).
+Every branch attached to the root has at least two descendant
+labels, so \(a<m\); this is a recursive definition.  For \(m=0\),
+the empty tensor is \(R\), the unit maps to \(1\), and positive
+bar-kappa monomials map to zero.
+
+\begin{lemma}
+\label{lem:detector-cutoff}
+The prescription is well defined modulo genuine relations on
+rational parameter factors and induces a map on \(\Salg_{g,m}\).
+For a homogeneous expression of degree \(r\), its image is a sum
+of tensors indexed by partitions of the labels into nonempty
+blocks, of the form
+\begin{equation}
+ \bigotimes_{L}\Delta_L(\lambda^{d_L}),
+ \qquad
+ r=\sum_L\bigl(|L|-1+d_L\bigr).
+ \label{eq:detector-blocks}
+\end{equation}
+In particular \(t_m\) is zero in degrees \(r>m\).
+\end{lemma}
+
+\begin{proof}
+Grafting a genus-zero graph amounts to composing coproducts and
+multiplying the corresponding integrals.  Coassociativity shows
+that the result is independent of the internal decomposition of
+a rational parameter class.  In particular a zero class on such
+a factor contributes zero.
+
+For a rational subtree with \(k\) descendant labels, top
+decoration degrees together with its edges, including the edge
+attaching it to the root, add to \(k-1\).  Formula \eqref{eq:detector-blocks} follows by
+induction, starting from the bar prescription.  Since
+\(\lambda^2=0\), a nonzero term has \(d_L\leq1\), and hence \(r\leq m\).
+
+An excessive decoration on a rational vertex has zero integral.
+At a root with \(a\) incident slots, a decoration exceeding
+\(3g-3+a\) also exceeds \(a\), and its recursive test is zero.
+This applies since \(g\geq2\).  Finally, under refinement the sum
+of the dimensions of the pieces of a vertex decreases by the
+number of new edges.  Splitting a decoration additively between
+those pieces cannot make an excessive total degree admissible
+on every piece.  Thus discarded terms also remain invisible
+under multiplication and refinement.  Forgetting operations
+require a separate argument, supplied below.
+\end{proof}
+
+\subsection{Boundary operations and the test}
+
+We first show how the test treats multiplication by bar classes and
+rational-tail divisors.  We then prove the forgetting identities
+that will reduce the number of labels in the relation argument.
+All tensor products, contractions, and permutations below use
+the super sign convention.
+
+\begin{lemma}[Boundary operations]
+\label{lem:detector-operations}
+The maps \(t_m\) are equivariant for relabeling.  For every
+strata expression \(b\), they satisfy
+\begin{align}
+ t(\bar\psi_i b)&=\lambda_i t(b),
+ &t(\bar\kappa_j b)&=0, \label{eq:detector-bars}\\
+ t(\bar D_{ij}b)&=\Delta_{ij}(1)t(b). \label{eq:detector-pair}
+\end{align}
+More generally, gluing along \(D_T\) integrates the entire
+genus-zero input and duplicates the root-side attaching slot
+into the labels of \(T\).
+\end{lemma}
+
+\begin{proof}
+We first establish the boundary restrictions and the graph
+compatibilities used in the calculation. The descendant and kappa
+identities are the forgetting and gluing formulas in
+\cite[Equations~(1.7)--(1.10)]{ArbarelloCornalba1996}.  At the gluing for
+\(D_T\), let \(p\) be the root-side flag and \(p'\) the
+rational-side flag.  The normal class is
+\(-\psi_p-\psi_{p'}\).  On
+\(\Mbar_{0,T\cup\{p'\}}\), the sum of boundary splits separating
+\(i\) from \(p'\) equals \(\psi_i+\psi_{p'}\).  The sum whose
+side away from \(p'\) contains two fixed labels \(i,j\) equals
+\(\psi_{p'}\).  These are the genus-zero cotangent boundary
+formulas; they also follow by successively forgetting to three
+labels and applying the cotangent pullback formula.
+
+Consequently, \(\bar\psi_i\) restricts to the modified cotangent
+of its ancestor slot on the root side, using \(p\) if \(i\in T\).
+The restriction of \(\bar D_{ij}\) is the analogous pair class
+on the root side, except when both labels lie in \(T\), when it
+is \(-\bar\psi_p\).
+
+The class \(\bar\kappa_j\) is independent of the chosen ordering.
+Indeed, for two consecutive labels made last, write
+\(D=D_{\{i,j\}}\).  The identities
+\(\psi_i|_D=\psi_j|_D=0\) and
+\(D|_D=-\psi_p\) give, for \(d\geq1\),
+\begin{equation}
+  (\psi_i-D)^d-\psi_i^d=-D\psi_p^{d-1}
+  =(\psi_j-D)^d-\psi_j^d.
+ \label{eq:detector-order}
+\end{equation}
+After pulling to the remaining labels, this is precisely the
+invariance under a consecutive transposition.  To restrict
+\(\bar\kappa_j\) to \(D_T\), put the labels of \(T\) last.
+The kappa class splits additively between the two vertices.
+The first label of \(T\) gives the root-side cotangent at \(p\);
+the remaining partial cotangents on the rational factor cancel
+its kappa class by the forgetful kappa formula.  The result is
+exactly \(\bar\kappa_j\) on the root side.
+
+These restrictions and the coproduct projection formula prove
+\eqref{eq:detector-bars} recursively, beginning with \eqref{eq:detector-prescription}.
+For \eqref{eq:detector-pair}, only the two-label tail has a
+nonzero undecorated integral.  If \(i,j\) are in different
+branches, the root-side pair class joins their ancestor slots.
+If they are in the same branch, its restriction is
+\(-\bar\psi_p\), and the required identity is
+\eqref{eq:detector-self}.  This proves the assertion for an
+arbitrary boundary expression as well as for a bar monomial.
+The gluing assertion follows directly by coassociativity and
+the factorization of genus-zero integrals.
+
+These calculations take place in the auxiliary formal graph space
+modulo relations on rational parameter factors.  Simultaneous
+compatible splits produce the same tree independently of their
+order, with one normal factor for every repeated edge.  Hence the
+polynomial multiplications used here commute and associate.
+A genuine zero on a rational factor remains zero: restriction
+there is a genuine genus-zero operation, and operations at another
+factor retain that zero with relabeled flags.
+Lemma~\ref{lem:detector-cutoff} permits the dimension truncations.
+Graphs outside the rational-tails locus remain outside it under
+refinement, so their products contribute zero as well.
+
+Finally, independence of the bar-kappa ordering, the restriction
+formulas, and the symmetric coproduct construction give
+equivariance under arbitrary relabeling.
+\end{proof}
+
+\subsection{Forgetting markings}
+
+The forgetting identities are needed in both directions: pullback
+will remove a zero-index leg from the Pixton recipe, and pushforward
+will test a tensor slot against the two even basis vectors.
+
+\begin{lemma}[Forgetting operations]
+\label{lem:detector-forgetting}
+For every strata expression \(b\), the maps \(t_m\) satisfy
+\begin{equation}
+ t(\pi_y^*b)=t(b)\otimes1,
+ \qquad
+ t(\pi_{y*}b)=\int_y t(b).
+ \label{eq:detector-forget}
+\end{equation}
+Here \(\pi_y\) forgets the label \(y\); the identities hold
+in any order of the remaining labels.
+\end{lemma}
+
+\begin{proof}
+We first calculate before the degree cutoff, modulo genuine
+genus-zero relations, and check at the end that every discarded
+term has zero test after forgetting.  No relation on a
+positive-genus parameter factor enters the calculation.
+
+\paragraph{Formal pullback and projection.}
+Forget pullback adds
+\(y\) at every possible vertex.  At that vertex it replaces
+\begin{equation}
+ \psi_h\longmapsto\psi_h-D_{\{h,y\}},
+ \qquad
+ \kappa_j\longmapsto\kappa_j-\psi_y^j.
+ \label{eq:detector-vertex-pull}
+\end{equation}
+A tail divisor pulls back as \(D_T+D_{T\cup\{y\}}\).
+Against a tree, an incompatible split remains incompatible;
+a compatible new edge splits a vertex.  If the split edge
+already occurs, the two pullback terms give its pulled normal
+class: adding \(y\) at an endpoint adds the boundary term
+cutting off that half-edge together with \(y\).  Thus pullback
+commutes with multiplication by the indicated polynomial
+classes.
+
+The push-projection calculation is equally local.  At a stable
+vertex, put
+\[
+ X_i=\pi_y^*\psi_i,\qquad k_j=\pi_y^*\kappa_j,\qquad
+ d_i=D_{\{i,y\}},
+\]
+so that \(\psi_i=X_i+d_i\) and
+\(\kappa_j=k_j+\psi_y^j\).  The relevant rules are
+\begin{equation}
+ d_i d_l=0\ (i\neq l),\qquad
+ d_i^2=-X_i d_i,\qquad
+ \psi_y d_i=0,
+ \label{eq:detector-forget-products}
+\end{equation}
+and
+\begin{equation}
+ \pi_{y*}1=0,\qquad
+ \pi_{y*}\psi_y^d=\kappa_{d-1}\ (d\geq1),\qquad
+ \pi_{y*}d_i=1.
+ \label{eq:detector-forget-push}
+\end{equation}
+They prove projection with graphless cotangent and kappa
+polynomials by substitution.  For a tail multiplier, its two
+pullback splits distribute the \(\kappa_{d-1}\) in \eqref{eq:detector-forget-push}
+additively over the two vertices; for \(d_i\), exactly the
+compatible split pushes to the original tail.  On a tree with
+a stable \(y\)-vertex this reduces to the same vertex
+calculation.  If forgetting contracts a three-point vertex
+with a surviving terminal label, the compatible split places
+\(y\) with that label.  If it contracts a bridge, the two
+normal contributions add to the normal of the combined edge.
+All other compatible splits lift uniquely.  This proves the
+needed push projection on boundary graphs as well.
+
+Forgetting also preserves the rational-factor relations used in
+the auxiliary computation: its pullbacks and pushforwards on
+those factors are actual genus-zero operations.  Operations at
+another factor retain a rational-factor zero with relabeled flags.
+
+\paragraph{Pullback of the test.}
+Both bar classes
+pull to themselves.  On a boundary graph, adding \(y\) to a
+rational factor increases its dimension without increasing
+the pulled decoration degree, so its integral is zero.
+Adding \(y\) at the root uses the smaller recursive pull
+identity.  This proves the pull assertion.
+
+\paragraph{Pushforward of the test.}
+For pushforward on a boundary graph, forgetting on a rational
+factor commutes with its integral; a three-point contraction
+uses the counit identity \eqref{eq:detector-counit}.
+Forgetting at the root again uses the smaller recursive
+assertion.  On the bar-monomial complement, projection reduces
+the question to powers of \(\bar\psi_y\), since all other bar
+classes are pulled back.  The zeroth power pushes to zero.
+For \(d>0\), telescope
+\begin{equation}
+ \psi_y^d-\bar\psi_y^d
+ =
+ \sum_{\varnothing\neq S}
+ D_{S\cup\{y\}}\sum_{a=0}^{d-1}
+        \psi_y^{d-1-a}\bar\psi_y^a.
+ \label{eq:detector-telescope}
+\end{equation}
+On the rational tail with labels \(S\cup\{y\}\), the dimension
+is \(|S|-1\), and only the term \(a=d-|S|\) integrates.
+Its integral is
+\(\int_{\Mbar_{0,|S|+2}}\psi_y^{|S|-1}=1\).
+After pushforward and application of the test, \eqref{eq:detector-telescope} is
+therefore
+\begin{equation}
+ \sum_{1\leq |S|\leq d}\Delta_S(\lambda^{d-|S|}),
+ \label{eq:detector-tail-sum}
+\end{equation}
+with units in all unused slots.
+
+On the other hand,
+\begin{equation}
+ t_m(\kappa_{d-1})
+ =
+ \int\lambda^d+
+ \sum_{1\leq |S|\leq d}\Delta_S(\lambda^{d-|S|}).
+ \label{eq:detector-kappa}
+\end{equation}
+For \(d=1\) this is the scalar identity
+\(\kappa_0=2g-2+m\).
+For \(d>1\), expand
+\(\kappa_{d-1}=\bar\kappa_{d-1}+\sum_xL_x^{d-1}\).
+Before pulling from the stage with labels \(1,\ldots,x\),
+write the exact polynomial identity
+\begin{equation}
+ \psi_x^{d-1}-\bar\psi_x^{d-1}
+ =
+ \sum_{\substack{T\subseteq\{1,\ldots,x\}\\x\in T,\ |T|\geq2}}
+ D_T\sum_{a=0}^{d-2}
+       \psi_x^{d-2-a}\bar\psi_x^a .
+ \label{eq:detector-partial-telescope}
+\end{equation}
+On the tail for \(T\), the dimension is \(|T|-2\).
+Thus its integral selects exactly \(a=d-|T|\), when that
+exponent lies between \(0\) and \(d-2\), and the integral
+of \(\psi_x^{|T|-2}\) is \(1\).  The remaining bar cotangent
+is the root-side cotangent, so this term contributes
+\(\Delta_T(\lambda^{d-|T|})\), with coefficient \(+1\).
+The term \(\bar\psi_x^{d-1}\) gives the singleton \(T=\{x\}\).
+Pulling through the later labels adds only unit tensor
+slots, by the pull identity already proved.  Every block
+therefore occurs once, indexed by its largest label, with
+no binomial coefficient or normal self-intersection sign.
+Subtracting \eqref{eq:detector-tail-sum}
+from \eqref{eq:detector-kappa} proves
+\(t_m(\pi_{y*}\bar\psi_y^d)=\int\lambda^d\), as required.
+
+\paragraph{Compatibility with the cutoff.}
+A non-rational-tails graph stays outside the test under forgetting:
+its nontrivial unmarked stabilization persists.
+
+We must also check forgetting on terms discarded by the
+vertexwise cutoff; this does not follow from the refinement
+dimension argument alone.  An excessive decoration on a
+rational factor is an actual zero in its genus-zero Chow
+ring.  Its genuine forget pullback or pushforward is still
+zero, including the integral used when a three-point factor
+contracts.  Operations on another factor preserve this
+rational-factor zero.
+For an excessive root decoration of degree \(d\) with
+\(a\) root slots, we have \(d\geq3g-2+a\).  Adding a mark
+at that root gives an expression of degree \(d>a+1\),
+so the block bound kills its test with \(a+1\) slots.
+Forgetting a root mark gives degree \(d-1>a-1\), killed
+by the same bound with \(a-1\) slots.  The root stays stable
+because \(g\geq2\).  Adding or forgetting on a rational
+factor leaves the root decoration unchanged; a contracted
+rational three-point vertex merely replaces an attaching
+flag without changing the root arity.
+Thus discarded terms have zero test before and after each
+forget operation.  Together with
+Lemma~\ref{lem:detector-cutoff}, this permits every cutoff
+in the displayed test identities.  It does not assert that
+untruncated pullback descends unchanged to the truncated
+formal algebra.
+Equivariance from Lemma~\ref{lem:detector-operations} allows
+the forgotten label to be placed last in any chosen order.
+\end{proof}
+
+\subsection{The zero-leg identity}
+
+We require one precise identity for the recipe in
+Definition~\ref{def:pixton-span}.  Package the partition
+insertions with formal variables \(s_\ell\), one for each
+allowed positive part, using exponential coefficients for
+repeated parts, and put
+\[
+ H(T,\zeta)=\widehat C_0(T,\zeta)
+             -\sum_\ell s_\ell\widehat C_\ell(T,\zeta).
+\]
+The variables \(s_\ell\) are purely formal; all subsequent
+coefficient extractions involve finitely many of them. The logarithm
+below is expanded jointly in \(T\) and the \(s_\ell\), so its constant
+term at \(T=s_\ell=0\) is defined even though \(H(0,\zeta)\) depends
+on the partition variables.
+The cycle exponential formula for \(\widehat\kappa_\Gamma\)
+rewrites the vertex characteristic as
+\begin{equation}
+ \exp\{-\log H\}_{\kappa,\zeta_v}.
+ \label{eq:detector-characteristic}
+\end{equation}
+Here a monomial \(T^d\zeta^\epsilon\) in \(-\log H\) is
+replaced by \(\kappa_dT^d\zeta_v^\epsilon\), including \(d=0\).
+For the edges use \(U(x,x')=T\Delta_e\), so the coefficient
+of total degree \(T^r\) is the degree-\(r\) recipe.
+Denote this packaged series, with its usual parity
+extractions, by \(\mathcal R(s;a)\).
+
+\begin{lemma}
+\label{lem:detector-unit}
+If all preceding leg indices \(a_j\) are zero or one, then,
+in the untruncated rational-tails computation,
+\begin{equation}
+ \mathcal R(s;a,0)
+ -\sum_\ell s_\ell\mathcal R(s;a,\ell)
+ =\pi_y^*\mathcal R(s;a).
+ \label{eq:detector-unit}
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+The kappa pullback formula, including
+\(\kappa_0^{\mathrm{old}}=\kappa_0^{\mathrm{new}}-1\),
+multiplies the new characteristic \eqref{eq:detector-characteristic} by
+\(H(\psi_yT,\zeta_v)\).  This is exactly the linear
+combination of new leg factors on the left.
+
+It remains to check bubbles introduced in pulling the old
+leg and half-edge cotangents.  For a function \(f(x)\), this
+pull gives the boundary correction
+\((f(0)-f(x))/x\), where \(x\) is the stable-side cotangent.
+At a new rational three-point vertex, its characteristic is
+\(H(0,\epsilon)^{-1}\); the new \(H\)-leg cancels it.
+Let \(\eta,\eta'\) be stable-side parity variables, and write
+\[
+ A_x=A(\eta xT),\quad B_x=B(\eta xT),
+ \qquad
+ A_{x'}=A(\eta'x'T),\quad B_{x'}=B(\eta'x'T).
+ \]
+The edge adjoining that bubble has factor
+\begin{equation}
+ \frac{\eta(B_x+1)+\epsilon(1-A_x)}{x}.
+ \label{eq:detector-bubble-edge}
+\end{equation}
+If the bubble carries an old leg, its constant factor is
+\(1\) or \(-\epsilon\).  Extracting \(\epsilon\) in \eqref{eq:detector-bubble-edge}
+then gives \((1-A_x)/x\) or
+\(-\eta(B_x+1)/x\), respectively.  These are exactly the
+two divided differences.
+
+If the bubble subdivides an old edge, the coefficient of
+\(\epsilon\) in the product of its two edge factors is
+\begin{equation}
+ \frac{\eta(B_x+1)(1-A_{x'})
+       +\eta'(B_{x'}+1)(1-A_x)}{xx'}.
+ \label{eq:detector-bubble-product}
+\end{equation}
+Substituting the original edge numerator shows that \eqref{eq:detector-bubble-product}
+equals
+\begin{equation}
+ \frac{U(0,x')-U(x,x')}{x}
+ +\frac{U(x,0)-U(x,x')}{x'}.
+ \label{eq:detector-edge-difference}
+\end{equation}
+Thus it also gives precisely the two cotangent pullback
+corrections.  Bubbles for distinct slots at a vertex are
+disjoint, and these are all possibilities for adding one
+mark.  Finally, rational-tails graphs have \(h^1=0\) and
+trivial automorphism groups, so neither the factor
+\(2^{-h^1}\) nor unnormalized gluing introduces another
+coefficient.  This proves the identity.
+\end{proof}
+
+\subsection{Annihilating the relation span}
+
+\begin{lemma}
+\label{lem:detector-annihilation}
+For every \(m\geq0\), \(t_m(\Pspan_{g,m})=0\).
+\end{lemma}
+
+\begin{proof}
+We induct first on arity.  At a fixed arity, the induction already
+kills every nontrivial outer gluing and every relation multiplied
+by a tail divisor.  With those terms disposed of, degree and
+partition length reduce the basic recipes to the all-ones case,
+where an alternating-form calculation finishes the argument.
+
+Use the precise span and operation stability in
+Proposition~\ref{prop:known-relations}, as established for the
+original relation family in \cite[Proposition~1]{Pixton}.
+For a basic recipe relation, use the lexicographic order
+\((m,r,\ell(\sigma))\in\mathbb Z_{\geq0}^3\):
+first the number of labels, then degree, then the number
+of partition parts.  At a fixed arity, all relation spaces
+at smaller arities, in every degree, have already been
+treated.  At zero labels, every admissible basic relation
+has positive degree, whereas \(t_0\) kills positive degrees.
+
+Consider a generator obtained by a nontrivial outer gluing.
+If that outer graph is not a rational-tails graph, all its
+refinements have zero test.  Otherwise, a relation on a
+rational factor has zero top integral, by its established
+Chow validity.  A relation on the root is killed by the
+smaller-arity induction, because that root has fewer labels.
+Integration and coproducts then give zero for the entire
+generator.
+
+We will also use the following consequence, valid for any
+\(b\in\Pspan_{g,m}\):
+\begin{equation}
+ t_m(D_Tb)=0.
+ \label{eq:detector-tail-ideal}
+\end{equation}
+Indeed, write this product as the gluing pushforward of the
+gluing pullback of \(b\).  The latter belongs to the sum of
+the relation ideals in the two factors.  Integrate the
+rational factor and use the smaller-arity induction on the
+root factor, as above.  The root arity is
+\(m-|T|+1<m\).  This argument needs no induction on the
+degree of \(b\).
+
+It remains to consider a basic recipe relation.  If
+\(a_i=3k+h\), with \(k>0\) and \(h=0,1\), its defining
+leg factor shows that the relation is \(\psi_i^k\) times
+the degree-\((r-k)\) relation with \(a_i=h\).
+The admissibility inequality and parity remain unchanged
+after subtracting \(3k\) on both sides.  Expand
+\(\psi_i=\bar\psi_i+\sum_{T\ni i}D_T\).
+The term containing only \(\bar\psi_i\) uses the lower
+triple \((m,r-k,\ell(\sigma))\) and is killed by
+\eqref{eq:detector-bars}.  Every other term is \(D_Tb'\)
+with \(b'\in\Pspan_{g,m}\), by the ideal property, and is
+killed by \eqref{eq:detector-tail-ideal}.  That identity
+uses smaller arity only, regardless of the degree of
+the remaining multiplier.  We may therefore suppose that all
+leg indices are zero or one.  We may also suppose \(r\leq m\),
+by Lemma~\ref{lem:detector-cutoff}.
+
+If a leg has index zero, extract its degree and partition
+coefficient in Lemma~\ref{lem:detector-unit}.  The pullback
+term uses \((m-1,r,\ell(\sigma))\) and is killed by
+\eqref{eq:detector-forget}.  A remaining term moves a part
+\(\ell\) to the zero leg and has order
+\((m,r,\ell(\sigma)-1)\).  If \(\ell=1\), this is already
+the required strict decrease.  Every other allowed positive
+part is \(\ell=3k+h\) with \(k>0\) and \(h=0,1\).
+Reducing that leg first gives a bar-multiplied relation
+of order \((m,r-k,\ell(\sigma)-1)\), while its tail terms
+are again covered by \eqref{eq:detector-tail-ideal}.
+Thus every invocation is well founded.  These are legitimate recipe
+relations: moving a part preserves its total weight, hence
+both admissibility conditions, and removing the zero leg
+does not change them.  Moreover
+\(r\leq m\leq3g-3+(m-1)\), since \(g\geq2\), so the
+pullback term fits the lower-arity dimension bound.
+The identity can be used before cutoff, because the test
+commutes with pullback and kills the excessive terms.
+
+Only the all-ones case remains.  Its recipe is symmetric
+under permutations of the labels, and the same is true of
+its test tensor.  Contracting any one slot with \(1\)
+gives the test of a forget pushforward, which vanishes by
+operation stability and smaller arity.  Contracting with
+\(e\) does also: multiply first by
+\(\bar\psi_y/(2g-2)\), use the ideal property, and then
+forget that slot.
+
+It suffices, therefore, to test all slots on standard
+vectors of \(W\).  In the block expansion \eqref{eq:detector-blocks}, a singleton
+has zero pairing with an odd input, a positive power of
+\(\lambda\) kills odd inputs, and a block of at least four
+odd inputs has zero product.  Only undecorated pair and
+triple blocks survive.  If there are \(s\) pairs and \(u\)
+triples, then
+\begin{equation}
+ m=2s+3u,\qquad r=s+2u,\qquad
+ s+3u=3r-m\geq g+1+|\sigma|>g.
+ \label{eq:detector-lagrangian-count}
+\end{equation}
+The permutation symmetry of the tensor makes its pairing
+an ordinary alternating form on the odd input vectors.
+Each symplectic pair uses one dual coordinate of \(W_+\);
+each cubic uses three such coordinates.  Thus every
+alternated term in \eqref{eq:detector-lagrangian-count} uses more than \(g\) coordinates
+from the \(g\)-dimensional space \(W_+^\vee\) and is zero.
+All standard input tests vanish, so nondegeneracy of the
+pairing proves that the tensor itself is zero.
+\end{proof}
+
+\subsection{The nonzero value on the candidate}
+
+We finish the proof of Proposition~\ref{prop:detector}.
+Every graph in a bridge correction
+\(\beta_J^{\mathrm{st}}\) has nontrivial unmarked
+stabilization and is not a rational-tails graph.
+Its refinements, pullbacks, and products remain outside
+the support of the test.  By
+\eqref{eq:detector-pair} and \eqref{eq:detector-diagonals},
+the remaining part of each factor of \(Y\) tests as
+\[
+ \Delta_{ab}(1)\Delta_{ac}(1)=\Delta_{abc}(1).
+\]
+Pair slot \(j\) with the first-Lagrangian vector
+\(a_{i_j}\) prescribed by the color word in
+Definition~\ref{def:candidate}.  A block with increasing
+colors \(i<j<k\) gives \(\theta_{ijk}\).  The product of
+the unsymmetrized factors therefore evaluates, up to one
+fixed super sign, to
+\[
+ \Theta=\prod_{1\leq i<j<k\leq g}\theta_{ijk}\neq0,
+\]
+with the product ordered as the blocks.
+
+Let \(G_{\mathbf i}\) be the stabilizer of the word.
+For \(\tau\in G_{\mathbf i}\), transporting its super
+permutation across the pairing of odd input slots
+contributes \(\operatorname{sgn}(\tau)\); the input word
+itself is unchanged.  This cancels the explicit sign of
+that summand in \(Y\).  No reordering of the exterior
+generators is required in this comparison: all summands
+are compared with the same ordered evaluation.
+Every color occurs \(\binom{g-1}{2}\) times, and hence
+\begin{equation}
+ \bigl\langle t_n(Y),
+     a_{i_1}\otimes\cdots\otimes a_{i_n}\bigr\rangle
+ =
+ \pm\left(\binom{g-1}{2}!\right)^g\Theta\neq0.
+ \label{eq:detector-candidate-value}
+\end{equation}
+Together with Lemma~\ref{lem:detector-annihilation}, this
+proves the proposition.
diff --git a/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/build/sections/04-tensors.tex b/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/build/sections/04-tensors.tex
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+\section{Faithful tensor tests in relative Chow}
+\label{sec:tensors}
+
+The formal detector and the Chow-vanishing argument use different
+coefficient algebras.  For the latter we must retain all Chow
+correspondences, including those not generated by the pairing.  We
+construct such a tensor model in this section.  Its coefficient algebra
+is generally large; the point is its faithfulness, not a presentation of
+its generators.
+
+Let
+\[
+ \pi\colon C\longrightarrow S
+\]
+be a smooth projective family of connected curves of genus \(b\), with a
+section, over an irreducible smooth quasi-projective complex variety.
+Fix a rational divisor class \(e_C\in\CH^1(C)\) of relative degree one.
+The class \(e_C\) need not be the class of the section.  Set
+\begin{equation}
+ \label{eq:curve-projector}
+ a_C=\pi_*(e_C^2),\qquad
+ e'_C=e_C-\frac12\pi^*a_C,\qquad
+ \eta=[\Delta_{C/S}]-e'_{C,1}-e'_{C,2}.
+\end{equation}
+Here and below products of curve families are over \(S\).
+
+For an integer \(a>0\), we may work modulo classes that vanish after
+restriction to \(S\setminus Z\), for some closed subset \(Z\subset S\)
+of codimension at least \(a\).  This convention does not mean taking
+numerical equivalence, nor does it mean testing individual geometric
+fibers.
+
+\begin{proposition}[Faithful Chow tensor test]
+ \label{prop:tensor-test}
+In the setup above, either in relative rational Chow or in the indicated
+quotient by base support, there is a supercommutative rational algebra
+\(\mathfrak B\) with the following properties.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item
+The curve correspondence algebra has a faithful tensor model on
+\[
+ \mathfrak B\otimes
+ \bigl(\Q 1\oplus W_b\oplus\Q e\bigr),
+\]
+where \(1,e\) are even and \(W_b\) is an odd symplectic space of dimension
+\(2b\).  The classes \(1_C,e'_C\) correspond to \(1,e\), and \(\eta\)
+projects onto \(W_b\).  Integration gives the hyperbolic pairing
+\(\langle1,e\rangle=1\), \(\langle1,1\rangle=\langle e,e\rangle=0\)
+on the two even summands and the symplectic pairing on \(W_b\).
+All actual Chow correspondences between curve powers, not only pairing
+diagrams, are included faithfully in this model.
+\item
+The projected small diagonal is an alternating three-form
+\[
+ \varphi\colon W_b^3\longrightarrow\mathfrak B_{\mathrm{odd}}.
+\]
+Cup product, integration, tensor products, permutations, and contractions
+are computed in the model with the Koszul signs.
+\item
+Let \(U\) be a finite-dimensional ordinary rational color space.  A pure
+colored tensor in
+\begin{equation}
+ \label{eq:colored-chow-space}
+ \left(
+   \eta^{\otimes k}\CH^*(C^k)
+   \otimes U^{\otimes k}\otimes\mathrm{sgn}_k
+ \right)_{\mathfrak S_k}
+\end{equation}
+is zero if and only if all its evaluations using the same ordinary
+linear map \(z\colon U\to W_b\) in every slot are zero in \(\mathfrak B\).
+Finitely many rational maps \(z\) suffice for each fixed \(U,k\).
+The assertion holds in the base-support quotient as well.
+\end{enumerate}
+In particular, a product of projected cubic entries is tested by
+products of
+\(\varphi(z(i),z(j),z(k))\), with the fixed Koszul sign coming from the
+ordering of the factors.  A vanishing established by these tests is a
+vanishing in Chow, with the stated base-support bound.
+\end{proposition}
+
+\subsection{Correspondences and the curve projector}
+
+We use the rational correspondence category whose objects are smooth
+projective \(S\)-families and whose morphisms are
+\[
+ \Hom(P,Q)=\CH^*(P\times_S Q),
+\]
+with all codimensions allowed.  Composition is pullback to the triple
+product, intersection, and pushforward.  Tensor product is relative
+product.  Equivalently, this is the usual correspondence calculus with
+Tate twists forgotten.  We take finite direct sums and split
+idempotents.  The diagonal identifies every generating object with its
+dual; the evaluation for the curve is its intersection pairing followed
+by \(\pi_*\).
+
+The morphisms that vanish away from a closed base subset of codimension
+at least \(a\) form a tensor ideal.  Indeed, a finite union of such
+subsets has the same codimension bound, and restriction to its
+complement commutes with every operation just described.  We can
+therefore first quotient by this ideal and then split idempotents.
+Morphisms between the original objects are unchanged by the latter
+completion.  Denote either the unquotiented category or this completed
+quotient by \(\mathcal C\).
+Its tensor unit is the base \(S\).  The endomorphisms of that unit are
+\(\CH^*(S)\), or their base-support quotient; positive-degree base
+classes remain part of the correspondence calculus unless the chosen
+quotient kills them.
+
+The projection formula gives
+\begin{equation}
+ \label{eq:hyperbolic-normalization}
+ \pi_*1_C=0,\qquad
+ \pi_*e'_C=1,\qquad
+ \pi_*((e'_C)^2)=a_C-a_C=0.
+\end{equation}
+Thus \(1_C,e'_C\) span a nondegenerate hyperbolic plane in the curve
+object.  The two corresponding projectors sum to
+\(e'_{C,1}+e'_{C,2}\), so their orthogonal complement has the
+self-transpose projector \(\eta\).  Write this complementary object as
+\(H_C\).
+
+The categorical trace of the curve identity is
+\(\pi_*c_1(T_{C/S})=2-2b\), by diagonal self-intersection.  Each of the
+two unit summands has trace one.  Consequently
+\begin{equation}
+ \label{eq:odd-curve-dimension}
+ \dim_{\mathcal C}H_C=-2b.
+\end{equation}
+The remaining relation needed to model \(H_C\) by an odd symplectic
+space holds already in relative Chow.
+
+\begin{lemma}[Relative pairing relation]
+ \label{lem:relative-pfaffian}
+On \(C^{2b+2}\), one has
+\begin{equation}
+ \label{eq:relative-pfaffian}
+ \sum_{\mathcal M}
+   \prod_{\{i,j\}\in\mathcal M}\eta_{ij}=0,
+\end{equation}
+where \(\mathcal M\) ranges over the perfect matchings of
+\(\{1,\ldots,2b+2\}\).
+\end{lemma}
+
+\begin{proof}
+Let \(A\to S\) be the relative Jacobian. The section of \(C\) permits a
+universal degree-zero line bundle on \(C\times_S A\), normalized along
+that section \cite[Section~9.4, Proposition~4; Section~8.2,
+Proposition~4]{BLR}. We use the Poincar\'e bundle and biduality for
+abelian schemes in \cite[Section~8.4, Theorem~5]{BLR}.  If \(\ell_1,\ell_2\) are its two first Chern classes on
+\(C\times_S A\times_S A\), put
+\[
+ P=(\pi_{A\times A})_*(\ell_1\ell_2)\in\CH^1(A\times_S A),
+ \qquad u=\frac12\Delta_A^*P\in\CH^1(A).
+\]
+Base twists of the universal line do not affect \(P\), because its
+relative degree is zero.  The class \(P\) is symmetric and biadditive.
+In particular
+\begin{equation}
+ \label{eq:jacobian-eigenclass}
+ [m]^*u=m^2u.
+\end{equation}
+We first prove \(u^{b+1}=0\) in relative rational Chow.
+
+The Fourier method originates in the work of Mukai and Beauville
+\cite[Theorem~2.2]{Mukai1981}, \cite[Section~2, Proposition~3]{Beauville1983}.
+We give the argument with its relative base factor.
+For the normalized Poincare line on \(A\times_S\widehat A\), write its
+first Chern class as \(c\), and define on total Chow
+\[
+ \mathcal F(\alpha)=
+ p_{\widehat A*}\bigl(p_A^*\alpha\,\exp(c)\bigr).
+\]
+All exponential expressions are finite in each Chow ring.  Convolution
+with the analogous transform for \(\widehat A\) has kernel
+\[
+ (+)^*K,\qquad
+ K=p_{A*}\exp(c),
+\]
+on \(A\times_S A\).  This follows by the biadditivity of the Poincare
+class and flat base change.
+
+The class \(K\) is supported on the zero section \(0_A\).  To see this,
+apply Grothendieck--Riemann--Roch \cite[Section~7]{BorelSerre} to the Poincare line under
+\(p_A\).  The relative tangent bundle of an abelian scheme is pulled
+back from the base, so its Todd factor is an invertible base class.
+The derived pushforward of the Poincare line vanishes away from
+\(0_A(S)\): on a geometric fiber, a nontrivial degree-zero line on an
+abelian variety has zero cohomology, and perfect direct image and derived base change give the assertion
+\cite[Tags~0B91 and~068V]{Stacks}.  The fiber statement can
+also be seen from the flat unitary-character description of
+\(\Pic^0\): a nontrivial character has no harmonic Dolbeault forms on a
+flat torus.
+
+Chow localization now gives \(K=0_{A*}\delta\) for
+\(\delta\in\CH^*(S)\).  This coefficient is unique: if
+\(\pi_A\colon A\to S\), then \(\pi_{A*}0_{A*}=\id\), so
+\(\delta=\pi_{A*}K\).  Its degree-zero component is nonzero.  It is
+the fiber integral
+\[
+ \int_{A_s\times\widehat A_s}\frac{c^{2b}}{(2b)!},
+\]
+which is nonzero because the Poincare class is the full-rank evaluation
+pairing between the two first cohomology spaces.  This computation
+determines only a scalar, not a Chow class by its realization.
+The positive-degree ideal of \(\CH^*(S)\) is nilpotent, so \(\delta\)
+is invertible.  Flat base change along addition identifies
+\((+)^*0_{A*}\delta\) with the graph of inversion carrying the
+pullback of \(\delta\).  Convolution therefore gives inversion on
+\(A\), times this invertible base class.  In particular
+\(\mathcal F\) is injective.
+
+The same biadditivity and projection formula show
+\[
+ \mathcal F\circ[m]_*
+   =[m]^*\circ\mathcal F.
+\]
+Furthermore its component
+\[
+ \mathcal F_j(\alpha)=
+ p_{\widehat A*}\left(p_A^*\alpha\,\frac{c^j}{j!}\right)
+\]
+has \([m]^*\)-eigenvalue \(m^j\), where \(j\ge0\).
+On the other hand, \([m]\) is finite flat of degree \(m^{2b}\), and
+\eqref{eq:jacobian-eigenclass} gives
+\[
+ [m]_*u^{b+1}=m^{-2}u^{b+1}.
+\]
+For a fixed integer \(m>1\), the number \(m^{-2}\) is distinct from
+every eigenvalue \(m^j\) occurring in the finite sum
+\(\mathcal F(u^{b+1})\).  Applying the product of the corresponding
+linear polynomials in \([m]^*\) shows that this transform is zero.
+Injectivity proves the claimed vanishing.
+
+Choose \(N>0\) clearing the denominators of \(e_C\).  The maps
+\[
+ a_N\colon C\longrightarrow A,\qquad
+ x\longmapsto[Nx-Ne_C]
+\]
+are defined relatively.  On an additional curve coordinate, the
+pullback of the universal degree-zero line has class
+\(N[\Delta]-Ne_C\), up to a twist from the parameter curve.
+Integrating the product for two parameter positions gives
+\begin{equation}
+ \label{eq:abel-jacobi-cross-term}
+ (a_N\times a_N)^*P
+ =N^2\bigl([\Delta]-e_{C,1}-e_{C,2}+\pi^*a_C\bigr)
+ =N^2\eta.
+\end{equation}
+The possible base twists contribute zero because the other line has
+relative degree zero.
+
+Pull back \(u^{b+1}=0\) along
+\[
+ (x_1,\ldots,x_{2b+2})
+ \longmapsto\sum_i t_i a_N(x_i),
+\]
+where the \(t_i\) are independent integers.  Biadditivity gives a
+polynomial identity in these integers.  The coefficient of
+\(t_1\cdots t_{2b+2}\) is
+\[
+ (b+1)!\,N^{2b+2}
+ \sum_{\mathcal M}\prod_{\{i,j\}\in\mathcal M}\eta_{ij}.
+\]
+Indeed, the self terms have a squared multiplier and cannot
+contribute to this coefficient.  Rational polynomial interpolation
+and division by the displayed nonzero scalar prove
+\eqref{eq:relative-pfaffian}.
+\end{proof}
+
+\subsection{Retaining arbitrary Chow correspondences}
+
+Let \(\mathcal R_b=\operatorname{Rep}^{\mathrm{odd}}(\Sp_{2b})\).
+This is the usual semisimple rational representation category of the
+split symplectic group, with parity prescribed by its central
+involution.  Its standard representation \(W_b\) is odd.  The ordinary
+alternating symplectic form is therefore a symmetric pairing in the
+super sense; its closed loop has value \(-2b\).
+
+The first and second fundamental theorems for symplectic tensors
+present this category, after finite direct sums and idempotent
+completion, by pairing diagrams with that loop value and the
+sum-of-matchings relation on \(2b+2\) slots.  In this formulation
+crossings on the standard odd space are minus the ordinary swap.
+See \cite[Theorems~2.2 and~2.13]{RW15}, or
+\cite[Theorems~4.6, 4.8, and~5.9]{LZ15}.
+The diagram spaces and relations are rational, so the complex
+formulation descends by faithful scalar extension to \(\Q\).
+Complete reducibility, absolute simplicity, and tensor generation over
+\(\Q\) follow from \cite[Theorems~4.14, 22.2, and~22.42, and
+Paragraph~22.4]{Milne2022}. In particular, the irreducibles occur as
+rational direct summands of tensor powers of the standard representation.
+Equations
+\eqref{eq:odd-curve-dimension} and \eqref{eq:relative-pfaffian} thus
+give a symmetric tensor functor
+\begin{equation}
+ \label{eq:symplectic-chow-functor}
+ F\colon\mathcal R_b\longrightarrow\mathcal C,
+ \qquad F(W_b)=H_C.
+\end{equation}
+For \(b=0\), \eqref{eq:relative-pfaffian} says \(\eta=0\), which gives
+the same conclusion with \(W_0=0\).
+
+It is essential not to assume that \eqref{eq:symplectic-chow-functor}
+is full.  The target contains the small diagonal and many other
+correspondences.  We record them by enlarging the coefficient algebra.
+The following reconstruction is a concrete split semisimple form of
+O'Sullivan's tensor realization
+\cite[Lemma~2.3.1\textup{(i)}]{OSullivan2011}.  It applies to a
+symmetric tensor functor with this semisimple source and an arbitrary
+rational tensor target; the endomorphism ring of the target unit need
+not be a field.  We give its coefficient construction explicitly.
+
+\begin{lemma}[Coefficient algebra]
+ \label{lem:coefficient-algebra}
+Let \(F\) be the functor in \eqref{eq:symplectic-chow-functor}, and put
+\(H(X)=\Hom_{\mathcal C}(1,F(X))\).  There is a supercommutative
+\(\Sp_{2b}\)-algebra \(\mathfrak B\) such that, naturally in \(X\),
+\begin{equation}
+ \label{eq:coefficient-invariants}
+ H(X)\simeq(\mathfrak B\otimes X)^{\Sp_{2b}}.
+\end{equation}
+Under these identifications, tensor products of target morphisms
+become multiplication of coefficients.  Arbitrary morphisms
+\(F(X)\to F(Y)\) are represented faithfully by equivariant
+\(\mathfrak B\)-linear maps
+\(\mathfrak B\otimes X\to\mathfrak B\otimes Y\), and their composition
+is the usual tensor contraction.
+\end{lemma}
+
+\begin{proof}
+Choose one representative \(X_\lambda\) of each irreducible rational
+symplectic representation and set
+\begin{equation}
+ \label{eq:coefficient-algebra-definition}
+ \mathfrak B=
+ \bigoplus_\lambda H(X_\lambda)\otimes X_\lambda^\vee.
+\end{equation}
+The multiplicity spaces \(H(X_\lambda)\) are ordinary vector spaces;
+the parity on each summand comes from \(X_\lambda^\vee\).
+Semisimplicity gives
+\[
+ H(X)\simeq
+ \bigoplus_\lambda
+ H(X_\lambda)\otimes\Hom_{\Sp_{2b}}(X_\lambda,X),
+\]
+which is exactly \eqref{eq:coefficient-invariants}.
+Only finitely many summands enter for any fixed \(X\).
+
+An element \(s\in H(X)\) corresponds equivalently to an equivariant
+map \(i_s\colon X^\vee\to\mathfrak B\).  For
+\(s\in H(X)\) and \(t\in H(Y)\), the target tensor product gives
+\[
+ s\otimes t\in H(X\otimes Y).
+\]
+Define multiplication on \(\mathfrak B\) by requiring the product of
+\(i_s\) and \(i_t\) to be the map corresponding to this element,
+using the natural super identification
+\(X^\vee\otimes Y^\vee\simeq(X\otimes Y)^\vee\).
+Explicitly, for homogeneous dual vectors \(f,g\) this identification
+sends \(f\otimes g\) to
+\[
+ x\otimes y\longmapsto(-1)^{|g||x|}f(x)g(y).
+\]
+On the irreducible summands of
+\eqref{eq:coefficient-algebra-definition} this specifies every
+product; naturality and semisimplicity give the same rule for all
+\(X,Y\).  Associativity, the unit, and supercommutativity follow from
+the associativity, unit, and symmetry constraints for the target
+tensor product.  Thus \eqref{eq:coefficient-invariants} respects
+tensor products, including their Koszul signs.
+
+Finally rigidity identifies
+\[
+ \Hom_{\mathcal C}(F(X),F(Y))
+ \simeq H(X^\vee\otimes Y).
+\]
+Apply \eqref{eq:coefficient-invariants} and the usual tensor--Hom
+identification for finite free supermodules:
+\[
+ H(X^\vee\otimes Y)
+ \simeq(\mathfrak B\otimes X^\vee\otimes Y)^{\Sp_{2b}}
+ \simeq
+ \Hom^{\mathrm{even}}_{\mathfrak B,\Sp_{2b}}
+   (\mathfrak B\otimes X,\mathfrak B\otimes Y).
+\]
+This gives the asserted faithful representation of every target
+morphism.  Evaluation and
+coevaluation are images of the corresponding representation
+morphisms, so composition is precisely coefficient multiplication
+and contraction.  Faithfulness here is the injectivity of an actual
+vector-space isomorphism, not faithfulness of a cohomological
+realization.
+\end{proof}
+
+Apply the lemma to
+\[
+ V_b=\Q1\oplus W_b\oplus\Q e,
+ \qquad F(V_b)\simeq C.
+\]
+In particular, a pure Chow class on \(C^k\) is a morphism
+\(1\to H_C^{\otimes k}\), so it is identified with an invariant in
+\(\mathfrak B\otimes W_b^{\otimes k}\).  This identification retains
+the class itself, including any component invisible to cohomology.
+The small diagonal and the structure map supply multiplication,
+unit, and integration.  Their correspondence identities give a
+supercommutative Frobenius algebra on \(\mathfrak B\otimes V_b\).
+The pairing is the one fixed by
+\eqref{eq:hyperbolic-normalization}.  In particular
+\begin{equation}
+ \label{eq:actual-cubic-form}
+ \varphi(w_1,w_2,w_3)=\int_C w_1w_2w_3
+ \quad (w_i\in W_b)
+\end{equation}
+is the tensor of the small diagonal after projection by \(\eta\) in
+all three slots.  Its coefficients are odd, and it is alternating
+in the ordinary underlying vectors \(w_i\). For a pointed curve over a
+field, this projection is the modified diagonal of Gross and Schoen
+\cite[Equation~(0.1) and Sections~2--3]{GrossSchoen1995}. The normalization
+here also incorporates the base corrections required for a family.
+
+\begin{lemma}[Centering at a section]
+ \label{lem:point-centered-algebra}
+Work modulo positive-codimension base support, and suppose \(e_C\)
+is the class of a section.  Then \(e'_C=e_C\) in the quotient.
+In the model of Proposition~\ref{prop:tensor-test}, the functional
+\[
+ \epsilon(x)=\int_C e_Cx
+\]
+is multiplicative and satisfies
+\[
+ \epsilon(1)=1,\qquad
+ \epsilon(e)=0,\qquad
+ \epsilon(W_b)=0.
+\]
+Consequently the component of a product \(w_1w_2\), with
+\(w_1,w_2\in W_b\), in the two even summands is
+\(\langle w_1,w_2\rangle e\).
+\end{lemma}
+
+\begin{proof}
+Every positive-degree base class vanishes after removing a suitable
+positive-codimension subset.  In particular \(a_C\) vanishes, so
+\(e'_C=e_C\).  Pairing with \(e_C\) is pullback along the section, by
+the projection formula, and this pullback is multiplicative.
+The displayed values follow from the hyperbolic splitting and the
+orthogonality of \(H_C\).
+
+The coefficient of \(1\) in \(w_1w_2\) is its pairing with \(e\),
+namely \(\epsilon(w_1w_2)=0\).  The coefficient of \(e\) is its
+pairing with \(1\), namely \(\langle w_1,w_2\rangle\).  These are
+identities of actual correspondences, hence also identities in the
+faithful coefficient model.
+\end{proof}
+
+\subsection{Colored operations and finite polarization}
+
+For an ordinary rational color space \(U\), define
+\[
+ \mathcal A_k(U)=
+ \left(
+  \CH^*(C^k)\otimes U^{\otimes k}\otimes\mathrm{sgn}_k
+ \right)_{\mathfrak S_k}.
+\]
+The product is pullback to disjoint curve slots followed by the
+coinvariant projection.  With parity \(k\bmod2\) in arity \(k\), the
+direct sum of these spaces is supercommutative: interchanging blocks
+of \(k\) and \(\ell\) slots has sign \((-1)^{k\ell}\).
+The same definition applies to a collection of configuration spaces
+with equivariant forgetting maps, using their forgetting pullbacks
+for the product.  We will use that version for weighted pointed
+curves.
+
+A symmetric three-slot class gives an odd alternating entry
+\(f(i,j,k)\), multilinear in its colors.  For the pure small diagonal,
+its scalar-color evaluation is
+\(\varphi(z(i),z(j),z(k))\).  We use a fixed ordering of tensor factors
+throughout; reordering several entries introduces exactly their
+Koszul signs.
+
+\begin{lemma}[Finite color polarization]
+ \label{lem:color-polarization}
+In the pure subspace
+\(\eta^{\otimes k}\mathcal A_k(U)\), evaluations using the same scalar
+map \(z\colon U\to W_b\) in every slot are jointly injective.
+For fixed \(k,U\), a finite collection of rational maps is jointly
+injective.  These statements hold after quotienting by base support
+of any prescribed positive codimension.
+\end{lemma}
+
+\begin{proof}
+By Lemma~\ref{lem:coefficient-algebra}, before taking color
+coinvariants a pure class is an invariant tensor with coefficients
+in \(\mathfrak B\) and \(k\) odd standard factors.  The super
+permutation of those factors is the ordinary permutation multiplied
+by \(\mathrm{sgn}_k\).  It cancels the explicit sign representation
+in the definition of \(\mathcal A_k(U)\).
+Writing \(W_{b,\mathrm{ord}}\) for the underlying ordinary vector
+space, the invariant-tensor inclusion therefore induces an injection
+\[
+ \eta^{\otimes k}\mathcal A_k(U)
+ \lhook\joinrel\longrightarrow
+ \mathfrak B\otimes
+ \Sym^k(W_{b,\mathrm{ord}}\otimes U).
+\]
+The symmetric group acts trivially on the single coefficient factor
+\(\mathfrak B\) in this formula.
+
+Using the nondegenerate symplectic pairing to identify a standard
+factor with its dual, the colored tensor is therefore an ordinary
+homogeneous polynomial tensor of degree \(k\) on
+\(\Hom_\Q(U,W_b)\), with coefficients in \(\mathfrak B\).
+The invariant-tensor identification is injective, and taking finite
+symmetric-group coinvariants is exact over \(\Q\).  Hence no
+additional kernel was introduced in obtaining this polynomial
+tensor.
+
+Evaluation at \(z\) is precisely polynomial evaluation.  A
+homogeneous polynomial with coefficients in any rational vector
+space is determined by its values on rational points.  More
+explicitly, after choosing bases, interpolation on a finite rational
+grid recovers every degree-\(k\) coefficient.  The interpolation
+matrix has rational entries and a rational left inverse.  It
+therefore works without requiring the coefficients to be reduced,
+even, or finite-dimensional.  This proves both injectivity
+assertions.
+
+The proof can be carried out from the start in the quotient
+correspondence category.  Alternatively, use the same finite
+interpolation: take the union of the finitely many closed base
+supports occurring in the evaluations and coefficient
+identifications.  Its codimension is at least the prescribed bound.
+Restriction to its complement makes every evaluation zero, and the
+interpolation identity makes the original colored class zero
+there.  This explains why a separate exceptional support for each
+test does not weaken the conclusion.
+\end{proof}
+
+\begin{proof}[Proof of Proposition~\ref{prop:tensor-test}]
+The hyperbolic decomposition, Lemma~\ref{lem:relative-pfaffian}, and
+the symplectic diagram presentation construct the functor \(F\).
+Lemma~\ref{lem:coefficient-algebra} supplies the faithful model for
+all curve correspondences and therefore for the Frobenius algebra.
+Equation~\eqref{eq:actual-cubic-form} identifies the projected small
+diagonal.  Lemma~\ref{lem:color-polarization} supplies the
+same-map test and its base-support assertion.  Tensor products
+represent products of disjoint-slot operations, so the final
+statement follows with their Koszul signs.
+\end{proof}
diff --git a/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/build/sections/05-quot.tex b/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/build/sections/05-quot.tex
new file mode 100644
index 0000000000000000000000000000000000000000..02db5f6821a0b577a8871bc6407b03619c4f0eb0
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+++ b/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/build/sections/05-quot.tex
@@ -0,0 +1,956 @@
+\section{A surface relation from determinant-fixed Quot schemes}
+\label{sec:quot}
+
+The tensor formalism of Section~\ref{sec:tensors} does not by itself bound
+products of the odd coefficients of a projected small diagonal. We obtain
+the additional relation we need from the surface square of the curve.
+The decisive features are a determinant constraint, which removes the
+surface obstruction directions, and a marked Hilbert-scheme calculation,
+which retains the Chow classes of all output positions.
+
+\subsection{The surface bound}
+
+Let \(\pi_C:C\to S\) be a smooth projective family of connected curves of
+genus \(b\), with a section, over an irreducible smooth quasi-projective
+complex variety. Choose a rational divisor \(e_C\) of relative degree one,
+and use the corrected divisor \(e'_C\), projector \(\eta\), and odd
+alternating cubic form \(\varphi\) of Proposition~\ref{prop:tensor-test}.
+Set
+\begin{equation}
+\label{eq:surface-parameters}
+ \chi=(b-1)^2,\qquad s_K=8\chi,\qquad d=3,\qquad
+ t_0=\chi+\frac{d(d-1)}2s_K-1=25\chi-1,\qquad m=3t_0.
+\end{equation}
+Throughout this section, the support quotient is the quotient by Chow
+classes that vanish after deleting a closed subset of \(S\) of codimension
+at least \(b-1\). Its coefficient superalgebra will be denoted by
+\(\mathfrak B=\mathfrak B_{S,b-1}\).
+
+\begin{theorem}[Surface bound]
+\label{thm:surface}
+Suppose \(b\geq5\) and \(\dim S<\chi-1\).
+Let \((w_\alpha)_{\alpha\in I}\) be any finite collection of vectors in the
+odd symplectic standard space \(W_b\) whose span is isotropic. For any
+matrix of scalars \(\gamma=(\gamma_{\alpha\beta})_{\alpha,\beta\in I}\),
+put
+\begin{equation}
+\label{eq:surface-tensor}
+ T_\gamma=
+ \sum_{\substack{\alpha_1,\alpha_2,\alpha_3\in I\\
+                  \beta_1,\beta_2,\beta_3\in I}}
+ \left(\prod_{j=1}^3\gamma_{\alpha_j\beta_j}\right)
+ \varphi(w_{\alpha_1},w_{\alpha_2},w_{\alpha_3})
+ \varphi(w_{\beta_1},w_{\beta_2},w_{\beta_3}).
+\end{equation}
+Then
+\begin{equation}
+\label{eq:surface-bound}
+ T_\gamma^{\,t_0}=0\qquad\text{in }\mathfrak B.
+\end{equation}
+The assertion remains valid after any characteristic-zero scalar
+extension of the tensor formalism.
+\end{theorem}
+
+Isotropic tests alone do not separate arbitrary Chow tensors.
+In Lemma~\ref{lem:large-vertex-test}, we first fix an arbitrary color
+map, find a subspace of its domain with isotropic image, and use the
+exterior-ideal argument to annihilate the entire sufficiently long
+product, throughout modulo base support of codimension at least
+\(b-1\). Only after doing this for every color map do we apply the
+faithful test of Proposition~\ref{prop:tensor-test}.
+
+We prove the localization relation underlying this theorem in the present
+section. Section~\ref{sec:coefficient} proves that its scalar coefficient
+is nonzero and thereby completes the proof.
+
+Write
+\[
+ X=C\times_S C,\qquad \pi_X:X\longrightarrow S,\qquad
+ K=K_{X/S}.
+\]
+We write line bundles additively. On every fiber, \(K\) is ample,
+\(K^2=s_K\), and \(\chi(\mathcal O_X)=\chi\).
+The curve section gives a section \(o:S\to X\).
+The two summands of the target bundle and their torus weights are
+\begin{equation}
+\label{eq:quot-target}
+ V=L_1\oplus L_2=K\oplus\mathcal O_X,\qquad
+ (w_1,w_2)=(0,1),\qquad A_0=L_1-L_2=K.
+\end{equation}
+Here the use of \(w_1,w_2\) for weights is confined to formulas concerning
+the target bundle; the tensor vectors retain their indices \(\alpha\).
+Let \(\mathcal Q\) be the relative Quot scheme of rank-zero quotients
+\[
+ 0\longrightarrow I'\longrightarrow V\longrightarrow Q
+ \longrightarrow0
+\]
+with the following determinant and numerical data:
+\begin{equation}
+\label{eq:quot-data}
+ D=dK,\qquad
+ \det Q=D\text{ in }\Pic(X/S),\qquad
+ \int_{X_s}\ch_2(Q_s)=\nu+L_1D-\frac{D^2}{2},
+ \qquad \nu=t_0-1.
+\end{equation}
+All intersections in this display are fiber intersections.
+One may use the relative polarization \(K\) to specify the Hilbert
+polynomial. Projective representability and the Grassmannian construction
+are supplied by Grothendieck's Quot theorem
+\cite[Theorem~5.1 and Section~5]{Nitsure}. The determinant condition is a closed fiber of the separated relative
+Picard scheme \cite[Theorem~4.8(1)]{KleimanPicard}, so \(\mathcal Q\)
+is projective over \(S\).
+
+\subsection{The derived obstruction theories}
+
+The determinant constraint changes both the obstruction space and the
+virtual dimension of \(\mathcal Q\).  We first construct its tangent
+complex as the fiber of the determinant differential.  The surjectivity
+of that differential on obstruction spaces will give a two-term theory.
+The same construction on the torus-fixed factors will then express their
+virtual classes as virtual pullbacks from one rank-one Quot scheme.
+
+\subsubsection{Arrows and the Quot tangent}
+
+The perfect-complex moduli theorem of
+\cite[Corollaries~3.15, 3.17, and~3.19, Definition~3.18,
+Lemma~3.27, and Corollary~3.29]{TV}
+applies over every affine open of \(S\).
+Indeed, the dg-category of perfect complexes on a smooth proper scheme
+over an affine base is proper by perfection of relative derived Hom
+and smooth by perfection of its diagonal. These smooth proper categories are saturated in the sense of
+\cite[Lemma~3.27 and Corollary~3.29]{TV}. Their moduli of perfect objects
+and perfect arrows are locally geometric and locally of finite
+presentation. These constructions glue over \(S\).
+The tangent of the object moduli at a perfect complex \(F\) is
+\(\mathbf R\Hom(F,F)[1]\).
+All Hom complexes in what follows are derived Hom on the surface,
+followed by derived pushforward to the parameter scheme. Perfect direct
+images and derived base change are used in the form of
+\cite[Tag~0B91]{Stacks}, and relative duality in the form of
+\cite[Tags~0BRT, 0A9Q, and~0B6I]{Stacks}.
+
+\begin{lemma}[The open arrow enhancement]
+\label{lem:quot-arrow}
+The derived stack of arrows with target \(V\) has an open derived
+substack whose classical truncation is the ordinary relative Quot scheme
+with the prescribed Hilbert polynomial, before fixing determinant.
+On this truncation its relative tangent complex is
+\[
+ \operatorname{Cone}\bigl(
+ \mathbf R\Hom(I',I')\longrightarrow
+ \mathbf R\Hom(I',V)\bigr)
+ \simeq \mathbf R\Hom(I',Q).
+\]
+This is a perfect complex of Tor-amplitude \([0,1]\), compatible with
+base change. The differential of forgetting the arrow and keeping its
+source is the boundary map
+\[
+ \mathbf R\Hom(I',Q)\longrightarrow\mathbf R\Hom(I',I')[1].
+\]
+\end{lemma}
+
+\begin{proof}
+In the perfect-arrow stack take the locus where the source and cone are
+degree-zero sheaves on surface fibers, with the prescribed Hilbert data.
+For a perfect complex, relative Tor-amplitude zero is open.
+Concretely, a locally free resolution can be truncated first at its
+positive end; at its negative end, fiberwise injectivity and the local
+flatness criterion successively give flat cokernels.
+Properness of the surface family makes the condition open on the
+parameter scheme. Thus on a classical base the selected arrows are
+injections with flat sheaf quotients.
+
+Conversely, the kernel and quotient of a flat quotient family are perfect
+on the total surface family, including over a singular parameter scheme.
+In a local free resolution the successive syzygies remain flat over the
+parameters. Regularity of the surface fibers makes the second syzygy
+fiberwise locally free; the local flatness criterion then makes it
+locally free on the total space. Hence the resolution terminates.
+These arguments commute with base change. An automorphism of the source
+over its inclusion in \(V\) is the identity, and negative Ext groups of
+the sheaves vanish. The classical moduli problem is therefore the Quot
+scheme, not a gerbe over it.
+
+For an arrow \(F\to G\) with both vertices variable, the endomorphism
+complex of the diagram is
+\[
+ \operatorname{fib}\bigl(
+ \mathbf R\Hom(F,F)\oplus\mathbf R\Hom(G,G)
+ \longrightarrow\mathbf R\Hom(F,G)\bigr),
+\]
+where the map is the difference of pre- and postcomposition.
+Its shift by \([1]\) is the tangent of the arrow moduli.
+Taking the tangent fiber over the fixed target gives the displayed
+cone and its boundary map. These identifications are identifications of
+perfect relative complexes: they can be computed with a locally free
+resolution of the one-arrow diagram and commute with derived base
+change.
+
+At a geometric point, surface Serre duality gives
+\[
+ \Ext^2(I',Q)^\vee=\Hom(Q,I'(K))=0,
+\]
+since \(Q\) is torsion and \(I'\) is torsion-free.
+The other possible Ext degrees lie in \([0,1]\).
+The geometric-fiber criterion for a perfect complex proves the asserted
+Tor-amplitude.
+\end{proof}
+
+\subsubsection{Fixing the relative determinant}
+
+Let \(\mathcal P_o^{\mathrm{der}}\) be the derived line stack
+rigidified along \(o\), namely the fiber of evaluation along \(o\) over
+the trivial line. For any line \(M\) on a base-changed surface, its
+normalization is
+\[
+ M\otimes\pi_X^*(o^*M)^{-1}.
+\]
+It has its canonical trivialization along \(o\).
+Because \((\pi_X)_*\mathcal O_X=\mathcal O_S\) universally, all
+line automorphisms come from the base, and the normalization removes
+them. The classical truncation of \(\mathcal P_o^{\mathrm{der}}\) is
+therefore the relative Picard scheme, parametrizing lines modulo base
+twists. Its tangent is
+\begin{equation}
+\label{eq:based-picard-tangent}
+ \mathbf R\pi_{X*}\mathcal I_o[1],
+\end{equation}
+where constant functions split evaluation
+\(\mathbf R\pi_{X*}\mathcal O_X\to\mathcal O_S\).
+This complex has geometric-fiber cohomology \(H^1(\mathcal O_X)\)
+in degree zero and \(H^2(\mathcal O_X)\) in degree one.
+
+The determinant of perfect complexes is a functorial derived morphism
+\cite[Section 3.1]{STV}. Its differential is trace
+\cite[Proposition 3.2]{STV}. After normalization it maps the open
+arrow enhancement to \(\mathcal P_o^{\mathrm{der}}\).
+Take its derived fiber at the normalized line \(L_1+L_2-D\).
+Its truncation is exactly \(\mathcal Q\): it fixes the Picard class
+of \(\det I'\), not a trivialization of the unnormalized determinant
+on a parameter scheme. The relative tangent restricted to \(\mathcal Q\)
+is
+\begin{equation}
+\label{eq:quot-fixed-tangent}
+ T_{\mathcal Q/S}^{\mathrm{vir}}=
+ \operatorname{fib}\left\{
+ \mathbf R\Hom(I',Q)
+ \xrightarrow{\ \partial,\ \mathrm{tr},\ \mathrm{based}\ }
+ \mathbf R\pi_{X*}\mathcal I_o[1]\right\}.
+\end{equation}
+Here the trace identification is an identity of relative complexes,
+not an inference from equality at closed points.
+Locally represent a perfect family by a bounded complex \(P^\bullet\)
+of finite free modules. Its determinant is
+\(\bigotimes_j(\det P^j)^{(-1)^j}\).
+Differentiation on infinitesimal automorphisms gives the supertrace
+\(\sum_j(-1)^j\mathrm{tr}_{P^j}\).
+The same calculation with square-zero complex coefficients gives
+the chain map \(\mathbf R\Hom(P^\bullet,P^\bullet)\to\mathcal O\);
+it respects homotopies and change of perfect resolution.
+Using the identification of object tangents with shifted loop
+tangents gives its shift by \([1]\) as the determinant differential.
+This construction glues and commutes with base change, and therefore
+gives the relative trace in \eqref{eq:quot-fixed-tangent}, including
+over nonreduced parameter schemes.
+
+\begin{lemma}[The determinant obstruction map]
+\label{lem:det-surjective}
+Let \(0\to F\to V_0\to Q_0\to0\) be an exact sequence on a smooth
+projective surface, where \(V_0\) is a vector bundle of rank \(r\)
+and \(Q_0\) is torsion.
+If \(K\) is ample and \(Kc_1(Q_0)>rK^2\), the map
+\[
+ \Ext^1(F,Q_0)\longrightarrow\Ext^2(F,F)
+ \xrightarrow{\mathrm{tr}}H^2(\mathcal O_X)
+\]
+is surjective.
+\end{lemma}
+
+\begin{proof}
+By duality, a canonical form \(\omega\) annihilating the image has the
+property that \(\omega\id_F:F\to F(K)\) extends to a map
+\(a:V_0\to F(K)\).
+If \(i:F\to V_0\), the maps \(i(K)a\) and
+\(\omega\id_{V_0}\) agree where \(F=V_0\), hence everywhere.
+It follows that \(\omega Q_0=0\).
+For nonzero \(\omega\), write \((\omega)=\sum a_jC_j\).
+At the generic point of \(C_j\) a quotient of
+\((\mathcal O/(\omega))^r\) has length at most \(ra_j\).
+These generic lengths are the multiplicities of \(c_1(Q_0)\).
+Thus \(c_1(Q_0)\leq r(\omega)\) as divisorial cycles, and ampleness
+gives \(Kc_1(Q_0)\leq rK^2\), a contradiction.
+The argument permits embedded points and nonreduced curve support.
+\end{proof}
+
+Applying Lemma~\ref{lem:det-surjective} with \(r=2\) and \(D=3K\)
+shows that the only possible degree-two cohomology of
+\eqref{eq:quot-fixed-tangent} vanishes. The complex is perfect of
+Tor-amplitude \([0,1]\).
+The canonical map from the cotangent of a derived enhancement to that
+of its classical truncation has cone in degrees at most \(-2\)
+\cite[Proposition 1.2 and Corollary 1.4]{STV}.
+Consequently the dual of \eqref{eq:quot-fixed-tangent} is a relative
+perfect obstruction theory in the sense of \cite[Definitions 4.4 and
+5.1]{BF}. Adding the tangent of the smooth base by transitivity gives
+the absolute theory. All constructions are torus equivariant.
+
+\subsubsection{Virtual rank and compatible pullbacks}
+
+Riemann--Roch gives the relative virtual rank
+\begin{equation}
+\label{eq:quot-rank}
+ v=\chi(I',Q)+\chi-1
+   =2\nu+(A_0-K)D+\chi-1
+   =2t_0+\chi-3>2t_0.
+\end{equation}
+Indeed, if \(q_2=\int\ch_2Q\), then
+\(\chi(V,Q)=2q_2-(K+L_1+L_2)D\) and
+\(\chi(Q,Q)=-D^2\), while the rank of
+\eqref{eq:based-picard-tangent} is \(1-\chi\).
+Thus fixing determinant contributes the extra \(\chi-1\) in
+\eqref{eq:quot-rank}.  The inequality \(v>2t_0\) will force the
+marked localization relation.
+
+To compare the fixed virtual class with simpler classes, we need
+morphisms of obstruction theories, rather than only tangent classes
+in K-theory.  The derived enhancements supply these morphisms as follows.
+For a morphism of these derived enhancements, transitivity maps to the
+transitivity triangle on truncations. If the two absolute comparison
+cones are \(C_Y,C_Z\), the relative comparison cone fits into
+\[
+ Lf^*C_Z\longrightarrow C_Y\longrightarrow C_{Y/Z}
+ \longrightarrow Lf^*C_Z[1].
+\]
+The absolute cones lie in degrees at most \(-2\); derived pullback
+preserves this upper bound and \([1]\) lowers it.
+Therefore \(C_{Y/Z}\) also lies in degrees at most \(-2\).
+Whenever the relative derived cotangent has perfect amplitude
+\([-1,0]\), this diagram supplies the compatible triple of obstruction
+theories. Virtual-pullback functoriality then applies
+\cite[Definition 4.5, Theorem 4.8, and Corollary 4.9]{Manolache};
+see also \cite[Proposition 1.5]{STV}.
+
+\subsection{Fixed loci and their virtual classes}
+
+The equivariant embedding and resolution hypotheses of virtual
+localization \cite[Section 1]{GP} hold here.
+The Grassmannian construction of relative Quot is equivariant and
+embeds the determinant fiber in a smooth scheme over the smooth base
+\(S\). One can construct a global equivariant two-term presentation of
+its perfect obstruction theory using a linearized ample line.
+For completeness, over an affine base sufficiently positive twists
+generate its degree-zero cohomology and kill the higher cohomology
+of its degree-minus-one cohomology. Lifting generators gives a bundle
+mapping to the complex and surjective on degree-zero cohomology;
+its derived fiber is a bundle by the Tor-amplitude criterion.
+Further Serre vanishing lifts the map to the two-term embedding
+cotangent presentation. Torus linear reductivity makes the choices
+equivariant.
+For a nonaffine \(S\), the same argument may be made after a
+Jouanolou affine-bundle pullback
+\cite[Proposition~4.1.4(1)]{ADOJouanolou}. The resulting identity descends by
+equivariant Chow homotopy invariance and smooth pullback of the
+base-changed obstruction theories. Subsequent dimension arguments
+are on the original base \(S\).
+
+A torus-fixed subsheaf of \(L_1\oplus L_2\) splits into its two weight
+subsheaves, also in families. Kernels and quotients in the two weights
+are flat, since they are direct summands of flat families.
+Thus a fixed point is a pair of rank-one arrows, subject to the total
+determinant constraint. Their direct-sum morphism of derived
+enhancements identifies its tangent triangle with the weight-zero part
+of \eqref{eq:quot-fixed-tangent}, compatibly with the cotangent
+comparison. Normalizing a determinant cancels all character lines
+pulled from the base, so the based condition does not acquire a moving
+term. The moving virtual normal complex is
+\begin{equation}
+\label{eq:moving-normal}
+ \bigoplus_{i\ne j}\mathbf R\Hom(I'_i,Q_j)
+\end{equation}
+with weight \(w_j-w_i\).
+
+We use the divisor-and-residual-point factorization established in the
+relative setting by \cite[Theorem~1.4]{Fogarty}. We spell out its
+compatibility with the determinant and nonreduced bases.
+Each rank-one kernel has a unique factorization
+\begin{equation}
+\label{eq:fixed-factorization}
+ I'_i=L_i(-\mathcal C_i)\mathcal I_{Z_i},
+ \qquad D_i=[\mathcal C_i]\text{ in }\Pic(X/S),
+ \qquad D_1+D_2=D.
+\end{equation}
+Here \(\mathcal C_i\) is a relative effective Cartier divisor and \(Z_i\)
+is a finite-flat point family.
+This holds over nonreduced parameter schemes. To see this, let \(T\)
+be such a parameter scheme and untwist the target line.
+The \(T\)-flat kernel ideal \(I\) has a length-one locally free
+resolution, by the fiberwise projective-dimension bound.
+Its determinant is a line bundle on \(X_T\), commuting with arbitrary
+base change. The maximal-minor map \(I\to\det I\) restricts on every
+surface fiber to the inclusion of a rank-one torsion-free sheaf into
+its double dual. It is therefore fiberwise injective with
+zero-dimensional cokernel. Since both its source and target are
+\(T\)-flat, the local flatness criterion makes the map injective
+and its cokernel \(\mathcal T\) \(T\)-flat. Properness and the fiber
+support make \(\mathcal T\) finite over \(T\).
+
+The inclusion into the original target line extends uniquely across
+that finite support: on a smooth relative surface a finite-flat
+module has grade at least two, as follows by lifting a fiber regular
+sequence with flat successive quotients. Thus its Hom and first Ext
+into the structure sheaf vanish. Applying Hom to
+\(0\to I\to\det I\to\mathcal T\to0\) gives the extension.
+The extension is fiberwise an injection of lines with Cartier
+quotient; the local flatness criterion makes its divisor relative
+Cartier. It gives the required point ideal and divisor, uniquely and
+compatibly with base change.
+
+Let the point lengths be \(r,s\). The Chern character of the two
+quotients gives
+\begin{equation}
+\label{eq:fixed-lengths}
+ r+s=\nu+(A_0-D_1)D_2.
+\end{equation}
+The fixed scheme has only finitely many numerical choices.
+Its summand Hilbert polynomials are locally constant.
+In addition, the determinant line of each perfect kernel is a
+base-change-compatible line family, so its fiber Hilbert polynomial
+is locally constant. The residual point length is the difference
+between the Euler characteristics of this determinant and its
+kernel, and is therefore locally constant as well.
+Finite type now gives finiteness of these choices.
+All divisor and Picard moduli
+used in a choice may be restricted to the required finite-type
+quasi-projective pieces. For instance, lines of a fixed Hilbert
+polynomial form an open in the moduli of stable rank-one sheaves;
+rigidification at \(o\) represents their Picard classes.
+
+Choose a \emph{high} index with \(KD_i>K^2\), which exists since
+\(KD_1+KD_2=3K^2\), and call the other index \emph{low}.
+Write \(P=D_{\mathrm{low}}\).  For this numerical choice, denote the fixed locus by \(\mathcal F\)
+and the unrestricted low rank-one Quot scheme by
+\(\mathcal Q_{\mathrm{low}}\).  Forgetting the high arrow gives a map
+\[
+ f:\mathcal F\longrightarrow\mathcal Q_{\mathrm{low}}.
+\]
+Its relative derived tangent is
+\[
+ \operatorname{fib}\left\{
+ \mathbf R\Hom(I'_{\mathrm{high}},Q_{\mathrm{high}})
+ \longrightarrow\mathbf R\pi_{X*}\mathcal I_o[1]\right\}.
+\]
+Lemma~\ref{lem:det-surjective}, now with rank one, gives amplitude
+\([0,1]\). The transitivity comparison above is therefore a compatible
+triple.  Consequently
+\begin{equation}
+\label{eq:fixed-low-pullback}
+ [\mathcal F]^{\mathrm{vir}}
+   =f^![\mathcal Q_{\mathrm{low}}]^{\mathrm{vir}}.
+\end{equation}
+Here \(f^!\) is a bivariant virtual pullback.  We will use its action
+on cycle representatives of the low virtual class, including cycles
+supported on special divisor loci.
+
+Obstruction Euler factors of point Hilbert schemes also enter the
+surface Quot calculations of Oprea and Pandharipande
+\cite[Lemma~34 and Section~5.2]{OPQuot}. We establish the relative
+comparison with varying determinant and arbitrary irregularity needed
+here.
+
+\begin{lemma}[Point excess]
+\label{lem:point-excess}
+The unrestricted low rank-one Quot scheme is classically
+\[
+ \operatorname{Div}_{P}(X/S)\times_S X^{[r_{\mathrm{low}}]}_{/S},
+\]
+where the divisor scheme permits its line to vary in the prescribed
+Picard data. Its virtual class is the divisor virtual class times the
+smooth relative point-Hilbert class, capped with
+\begin{equation}
+\label{eq:point-obstruction}
+ e(F_{\mathrm{low}}),\qquad
+ F_i=\left(\pi_*\mathcal O_{Z_i}(K-\mathcal C_i)\right)^\vee.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+The divisor enhancement parametrizes invertible-source arrows.
+Its tangent is
+\(\mathbf R\pi_*\mathcal O_{\mathcal C}(\mathcal C)\), of amplitude
+\([0,1]\).
+Tensoring its universal source with the universal point ideal gives
+a morphism from this enhancement times the smooth relative Hilbert
+scheme to the rank-one arrow enhancement, with identity truncation.
+The smoothness of the relative point Hilbert scheme follows from the
+smooth surface Hilbert theorem \cite[Theorem 2.4]{Fogarty}.
+Indeed, choose disjoint etale surface charts around the finite support,
+with etale maps to the relative affine plane. Finite-subscheme
+deformations lift uniquely through these etale maps. Thus locally on
+the Hilbert functor the question is the product of the corresponding
+plane Hilbert functors over \(S\), which is smooth over \(S\).
+
+Compare first the source objects of the arrows. The scalar trace
+summand of \(\mathbf R\Hom(\mathcal I_Z,\mathcal I_Z)\) splits off.
+Writing \(A=\mathbf R\pi_*\mathcal O_X\) and denoting the trace-free
+summand by \(B_Z\), the source-object tangent is therefore
+\(A[1]\oplus B_Z[1]\). The following calculation identifies
+\(B_Z[1]\) with the relative Hilbert tangent, while retaining the
+entire Picard summand \(A[1]\).
+The trace-free Ext groups in degrees zero and two vanish: an
+endomorphism extends to the determinant line, and surface duality
+gives the degree-two assertion. Riemann--Roch leaves a trace-free
+degree-one space of dimension \(2r_{\mathrm{low}}\).
+The point-ideal boundary injects
+\(\Hom(\mathcal I_Z,\mathcal O_Z)\) into this space, since every
+map \(\mathcal I_Z\to\mathcal O_X\) is a scalar inclusion with zero
+composite into \(\mathcal O_Z\).
+This boundary is the Hilbert Kodaira--Spencer map; its trace is zero
+because the point ideal has trivial determinant. Smoothness and the
+dimension \(2r_{\mathrm{low}}\) make it an isomorphism.
+The family complexes are perfect, so the fiberwise isomorphism is an
+isomorphism of complexes after including the scalar line-moduli
+tangent.
+
+For the arrow directions, the comparison is
+\[
+ \mathbf R\Hom(\mathcal O_X,\mathcal O_X(\mathcal C))
+ \longrightarrow
+ \mathbf R\Hom(\mathcal I_Z,\mathcal O_X(\mathcal C)).
+\]
+Its cone is
+\[
+ \mathbf R\Hom(\mathcal O_Z,\mathcal O_X(\mathcal C))[1]
+ \simeq F_{\mathrm{low}}[-1]
+\]
+by relative Serre duality. The latter is a genuine vector bundle in
+degree one: \(Z\) is finite flat and its direct image has no higher
+cohomology.
+We have therefore obtained a morphism of tangent theories whose cone
+is this excess obstruction bundle.
+
+Because it comes from the morphism of enhancements, the dual
+comparison commutes with both obstruction morphisms to the same
+classical cotangent complex.
+The smaller obstruction bundle stack embeds in the larger with
+quotient \(F_{\mathrm{low}}\), and the intrinsic-cone embedding
+factors through the smaller stack. Zero-section excess intersection
+therefore multiplies the smaller virtual class by
+\(e(F_{\mathrm{low}})\). This proves the assertion, including when
+the divisor scheme is singular.
+\end{proof}
+
+Let
+\[
+ \rho:\mathcal Q_{\mathrm{low}}
+   =\operatorname{Div}_{P}(X/S)\times_S X^{[r_{\mathrm{low}}]}_{/S}
+   \longrightarrow\operatorname{Div}_{P}(X/S)
+\]
+be the smooth projection.  Combining point excess with
+\eqref{eq:fixed-low-pullback} gives the fixed-cycle formula
+\begin{equation}
+\label{eq:fixed-cycle-formula}
+ [\mathcal F]^{\mathrm{vir}}
+ =f^!\left(
+   e(F_{\mathrm{low}})\cap
+   \rho^*[\operatorname{Div}_{P}(X/S)]^{\mathrm{vir}}
+       \right).
+\end{equation}
+This formula separates the two remaining tasks.  We first determine
+where the low divisor virtual cycle can be supported.  Over the
+surviving support, the high map \(f\) becomes smooth, and its virtual
+pullback will supply the second point Euler factor.
+
+\subsection{Support of the low divisor class}
+
+Put \(y=KP/s_K\).
+The virtual relative dimension of the divisor scheme is
+\begin{equation}
+\label{eq:divisor-virtual-dimension}
+ \frac{P(P-K)}2.
+\end{equation}
+If \(0<y<1\), this is at most \(-(b-1)\).
+To check the integer bound, put \(q=b-1\).
+The positive intersection \(p=KP\) is a multiple of \(2q\):
+each of the two canonical divisors on the product has this fiber
+degree. Thus
+\[
+ 2q\leq p\leq8q^2-2q,\qquad
+ \frac{P(P-K)}2
+ \leq\frac{p^2/(8q^2)-p}{2}
+ \leq-q+\frac14
+\]
+by Hodge index and convexity in \(p\).
+The left side is integral, proving the claim.
+The absolute divisor virtual cycle consequently has a representative
+of dimension at most \(\dim S-(b-1)\).  The image of its support in
+\(S\) has codimension at least \(b-1\).  In
+\eqref{eq:fixed-cycle-formula}, smooth pullback by \(\rho\) and cap
+product with \(e(F_{\mathrm{low}})\) preserve this base support.
+So does \(f^!\): it is bivariant and commutes with restriction and
+proper pushforward \cite[Theorems 4.1 and 4.4]{Manolache}.
+Although these pullbacks can increase the dimension of a cycle, they
+do not enlarge the locus over which it lies in \(S\).  Every marked
+fixed contribution above this low cycle therefore vanishes in our
+support quotient.
+
+On the Picard open where \(H^2(P)=0\), consider instead the morphism
+from the divisor enhancement to the derived based Picard scheme.
+The divisor exact sequence shows that its obstruction map surjects
+onto \(H^2(\mathcal O_X)\). Hence its relative tangent has amplitude
+\([0,1]\), and the transitivity construction again gives a compatible
+virtual-pullback triple.
+The absolute virtual dimension of that Picard theory is
+\[
+ \dim S+1-\chi<0.
+\]
+Its virtual class is zero, so the restricted divisor virtual class
+is zero as well.
+For \(y>1\), the condition \(H^2(P)=0\) is automatic: otherwise
+duality makes \(K-P\) effective, contrary to ampleness.
+For \(y=1\), a nonzero section of \(K-P\) has zero intersection with
+\(K\), and therefore has empty divisor. Thus the exceptional line is
+exactly \(K\). Chow localization expresses the divisor virtual class
+as a pushforward from the canonical linear system \(|K|\).
+For \(y=0\), an effective divisor is empty; its tangent complex is
+zero and its class is the base fundamental class.
+
+We have reduced the low divisor support to \(0\) or \(K\), and hence
+the high line to \(3K\) or \(2K\). On a neighborhood of either support
+the high direction is smooth: Kodaira vanishing gives its projective
+bundle of sections, together with its smooth point Hilbert scheme.
+The relative virtual tangent class there is
+\begin{equation}
+\label{eq:high-tangent}
+ [\mathbf R\pi_*\mathcal O(\mathcal C_{\mathrm{high}})]
+ -1-[F_{\mathrm{high}}]+[T_{X^{[r_{\mathrm{high}}]}/S}].
+\end{equation}
+Indeed, the unrestricted rank-one arrow class subtracts
+\(\mathbf R\pi_*\mathcal O_X\), while fixing the based determinant
+subtracts \(\mathbf R\pi_*\mathcal I_o[1]\); together these subtract
+exactly one. The remaining point-ideal calculation is that in
+Lemma~\ref{lem:point-excess}.
+The projective Euler sequence identifies actual tangent minus
+\eqref{eq:high-tangent} with \([F_{\mathrm{high}}]\).
+
+For a smooth morphism with a relative perfect obstruction theory,
+the obstruction sheaf is locally free and its virtual pullback is
+smooth pullback capped with its top Chern class. This follows by
+locally splitting the cotangent comparison onto the locally free
+relative cotangent; equivalently, the intrinsic cone is
+\([0/T_{\mathrm{rel}}]\) and zero-section excess gives the Euler
+factor. The argument works on arbitrary base cycles, not just the
+fundamental cycle; compare \cite[Proposition 7.3]{BF} and
+\cite[Theorem 4.4]{Manolache}.
+Here the obstruction bundle has K-class \([F_{\mathrm{high}}]\),
+so its Euler class is \(e(F_{\mathrm{high}})\).  Let
+\(\mathcal D\subset\operatorname{Div}_{P}(X/S)\) be the empty-divisor
+locus \(S\) or the canonical linear system \(|K|\).
+Base-change \(f\) to
+\(\mathcal D\times_S X^{[r_{\mathrm{low}}]}_{/S}\), and use the
+same letters \(f,\rho\) for the resulting maps.  For every cycle
+\(\alpha\) on this product, the high virtual pullback is therefore
+\begin{equation}
+\label{eq:high-smooth-pullback}
+ f^!(\alpha)=e(F_{\mathrm{high}})\cap f^*(\alpha).
+\end{equation}
+In particular, apply it to
+\(\alpha=e(F_{\mathrm{low}})\cap\rho^*\beta\), where \(\beta\)
+is a cycle on \(\mathcal D\) supplied by Chow localization for the
+low divisor virtual class.  Bivariant proper-pushforward compatibility
+then returns the result to \(\mathcal F\).  All operational
+insertions restrict to the actual linear systems on those cycles;
+no formula for the low cycle supported on \(|K|\) is needed.
+These linear systems are smooth over \(S\): for \(K\), constant
+cohomology dimensions follow from the product of curves and base
+change, and for \(2K,3K\) from Kodaira vanishing.
+Their universal divisor classes are \(D_i+x_i\), where \(x_i\) is
+the hyperplane class.
+
+\subsection{Marked localization and the short-length exclusion}
+
+Adjoin \(m=3t_0\) independent surface positions to \(\mathcal Q\).
+At each position insert \(\ch_2^{\mathbb G_m}(Q)\), cap with the
+smoothly pulled virtual class, and push to \(X^m_{/S}\).
+This push is proper. The torus acts trivially on the output.
+Before localization the total codimension is \(2m-v\).
+An ordinary output component of codimension \(4t_0\) would therefore
+have equivariant exponent
+\[
+ (2m-v)-4t_0=2t_0-v<0.
+\]
+It is absent. Thus the ordinary codimension-\(4t_0\) part of the
+localized fixed sum is zero after setting the equivariant parameter
+equal to one.
+
+On a fixed branch the insertion is a sum, for \(i=1,2\), of
+\begin{equation}
+\label{eq:fixed-insertion}
+ [Z_i]+(L_i+w_i)(D_i+x_i)-\frac{(D_i+x_i)^2}{2}.
+\end{equation}
+Here \([Z_i]\) denotes the incidence cycle on the corresponding
+surface position.
+Test every surface slot by the even tensor
+\begin{equation}
+\label{eq:surface-test}
+ z_\gamma=\sum_{\alpha,\beta}\gamma_{\alpha\beta}
+ w_\alpha\otimes w_\beta.
+\end{equation}
+This is a tensor test in the coefficient algebra, not a numerical
+intersection test alone.
+Since a unit curve class pairs to zero with a pure \(W_b\) input,
+a line-only insertion that survives this test uses a positive divisor
+degree on both curve axes. Hence each such position supplies at least
+two ordinary codimensions, from powers of the two curve canonical
+classes.
+
+Suppose \(k\) of the positions use line-only insertions.
+Extract these \(2k\) divisor codimensions by the projection formula.
+The remaining \(m-k\) incidence positions lie in the union of the two
+point subschemes, of total length \(r+s\). Their pushed support
+therefore has at most \(r+s\) distinct surface positions and has
+codimension at least
+\[
+ 2(m-k-r-s)
+\]
+in those factors. Operational incidence classes vanish away from
+their incidence supports, so this support assertion applies to
+successive caps with arbitrary low cycle multipliers.
+If \(r+s<t_0\), the displayed codimension is strictly larger than
+the available \(4t_0-2k\), and the tested contribution is zero.
+
+By \eqref{eq:fixed-lengths}, the four remaining ordered divisor pairs
+have lengths
+\[
+\begin{array}{c|cccc}
+ (D_1,D_2)&(0,3K)&(3K,0)&(K,2K)&(2K,K)\\ \hline
+ r+s&\nu+3s_K&\nu&\nu&\nu-s_K .
+\end{array}
+\]
+Only \((D_1,D_2)=(0,D)\) survives.
+On this branch the low divisor is empty and its virtual class is the
+base class. The calculation is on the product of the two point
+Hilbert schemes and \(|D|\), with both point Euler factors and the
+ordinary fundamental cycle. We have
+\begin{equation}
+\label{eq:surviving-branch}
+ N=r+s=\nu+ds_K,\qquad
+ \dim(|D|/S)=\chi+\frac{d(d-1)}2s_K-1=t_0,\qquad
+ x_1=0,\quad x_2=x.
+\end{equation}
+
+\subsection{A relative marked Hilbert-scheme recursion}
+
+The surviving fixed contribution must be expanded as a Chow class on
+\(X^m_{/S}\), with every output retained.  Relative EGL recursion for
+surface pushforwards to a base appears in
+\cite[Section~2.1, Lemma~2.1 and Proposition~2.2]{Kong2026};
+the same work also treats one and two marked curve outputs
+\cite[Section~3.3, Lemma~3.4]{Kong2026}.
+The construction below retains an arbitrary finite set of surface
+outputs.  This is the form that lets the codimension bound identify
+the Chow tensor left by localization.
+
+\begin{lemma}[Marked universality]
+\label{lem:marked-universality}
+Fix finitely many marked output surface factors.
+Push forward, over relative point Hilbert schemes and any additional
+surface factors, a finite polynomial in Chern characters of universal
+point ideals, their derived duals, line classes, relative tangent Chern
+classes, and equality diagonals.
+The resulting Chow class is a universal finite sum of decorated
+equality diagonals among the outputs, multiplied by relative
+characteristic pushes from closed surface components.
+One fixed polynomial procedure chooses the expression, independently
+of the surface family. No marked output is integrated in the procedure.
+\end{lemma}
+
+\begin{proof}
+For a positive point length \(r\), use the consecutive nested scheme
+\[
+ \mathcal N_r=X^{[r-1,r]}_{/S}
+ \xrightarrow{\ \psi\ }X^{[r]}_{/S},\qquad
+ \sigma=(\phi,\rho):\mathcal N_r\longrightarrow
+ X^{[r-1]}_{/S}\times_S X .
+ \]
+It is the quotient projectivization of the smaller universal ideal
+\(\mathcal I_{r-1}\). The universal quotient line \(\mathcal L\)
+and the residual graph \(j\) give
+\begin{equation}
+\label{eq:nested-ideal-sequence}
+ 0\longrightarrow\psi_X^*\mathcal I_r
+ \longrightarrow\phi_X^*\mathcal I_{r-1}
+ \longrightarrow j_*\mathcal L\longrightarrow0.
+\end{equation}
+The consecutive nested scheme and point Hilbert schemes are smooth
+over \(S\). These are the smooth-surface facts underlying
+\cite[Section 1]{EGL}. For the relative assertion, choose disjoint
+etale charts about the finite support. Deformations of a finite
+subscheme, and of an inclusion of two such subschemes, lift uniquely
+under an etale map to the relative plane. The resulting local
+deformation functors are products of the plane functors, base-changed
+from the ground field to \(S\); their smoothness gives the assertion.
+Over the distinct-point locus \(\psi\) has degree \(r\), so
+\(\psi_*[\mathcal N_r]=r[X^{[r]}_{/S}]\).
+Pullback followed by pushforward and division by \(r\) therefore
+preserves the original calculation with all other factors present.
+
+Put \(\zeta=c_1(\mathcal L)\).
+In every surface slot, replace the ideal by
+\eqref{eq:nested-ideal-sequence} in K-theory.
+GRR for the residual graph gives
+\[
+ \ch(j_*\mathcal L)
+ =j_*\bigl(e^\zeta\rho^*\td(T_{X/S})^{-1}\bigr).
+\]
+The graph is a regular codimension-two embedding with the indicated
+normal bundle. Thus its only new ingredients are an equality
+diagonal to the residual surface coordinate, \(\zeta\), and relative
+tangent classes.
+Derived duals give no new ingredients, since
+\(\ch_j(E^\vee)=(-1)^j\ch_j(E)\).
+
+Choose a two-term locally free resolution
+\[
+ 0\longrightarrow V_-\longrightarrow V_+
+ \longrightarrow\mathcal I_{r-1}\longrightarrow0 .
+\]
+The resolution exists by the fiberwise projective-dimension bound
+and an ample locally free presentation.
+The quotient projectivization is the zero locus of
+\(V_-^\vee(1)\) on \(\mathbb P(V_+)\).
+Its codimension is \(\rk V_-\), by the dimension of the consecutive
+nested scheme, so the section is regular.
+The projective bundle formula then gives
+\begin{equation}
+\label{eq:nested-segre}
+ \sigma_*(\zeta^k\cap[\mathcal N_r])
+ =\left[\frac{c(V_-^\vee)}{c(V_+^\vee)}\right]_k
+ \cap[X^{[r-1]}_{/S}\times_S X].
+\end{equation}
+This is the resolution Segre rule of
+\cite[Section 1, Lemma 1.1]{EGL}.
+After substituting the universal sequence, the integrand is a
+polynomial in \(\zeta\) with coefficients pulled from the smaller
+Hilbert scheme, the residual point, and the unchanged factors.
+Formula~\eqref{eq:nested-segre} lowers the total Hilbert length.
+Repeated application ends at length zero.
+
+Products of surface pushes are first rewritten using separate surface
+slots, by flat base change and the projection formula.
+The same applies to GRR expressions for point-ideal Hom complexes
+and tautological bundles, so it covers all the Euler factors that
+will occur here after the finite Chow-degree truncation.
+At length zero only equality diagonals and the specified decorations
+remain. An equality edge connecting two previously distinct
+components identifies their coordinates transversely.
+An edge within an already connected component contributes the excess
+class \(c_2(T_{X/S})\).
+Each connected component meeting outputs gives their equality
+diagonal with all its decorations on the common coordinate;
+a component meeting no output is integrated relatively over \(X\).
+Fixing the order of reductions and equality edges gives one universal
+polynomial procedure throughout.
+\end{proof}
+
+Apply Lemma~\ref{lem:marked-universality} to the surviving fixed
+branch. For the surface product of curves,
+all the allowed output decorations expand into powers of the two
+curve canonical classes.
+Consider a connected equality block containing \(k\) outputs.
+For \(k=1\), pure-slot testing requires at least two decorating
+codimensions, one on each axis.
+For \(k=2\), a bare diagonal on either curve axis pairs two isotropic
+inputs and is zero. Decorations are therefore needed on both axes,
+in addition to the codimension-two surface diagonal.
+For \(k\geq3\), the equality block already costs \(2(k-1)\)
+codimensions.
+These lower bounds are summarized in Table~\ref{tab:block-costs}.
+Every surviving block costs at least \(4/3\) codimensions per output,
+with equality only for an undecorated three-output block.
+
+\begin{table}[tb]
+\centering
+\begin{tabular}{@{}clc@{}}
+\toprule
+Outputs & Requirement under isotropic testing & Codimension at least\\
+\midrule
+\(1\) & A divisor on each curve axis & \(2\)\\
+\(2\) & Equality diagonal and a divisor on each axis & \(4\)\\
+\(k\geq3\) & Surface equality diagonal & \(2(k-1)\)\\
+\bottomrule
+\end{tabular}
+\caption{The codimension budget with all output positions retained.
+Only a bare triple attains cost \(4/3\) per output.}
+\label{tab:block-costs}
+\end{table}
+
+The total selected codimension is \(4t_0\) on \(3t_0\) outputs.
+It follows that only undecorated triple blocks occur.
+No line-only insertion survives.
+There is no remaining positive horizontal codimension: every closed
+surface component must have top relative degree, and every base
+bundle contributes only its scalar characteristic term.
+Integration over \(|D|\) therefore extracts exactly \(x^{t_0}\).
+In particular, the scalar multiplying the triple blocks can be
+computed over a point with the same numerical characteristic data,
+using the fixed polynomial procedure of
+Lemma~\ref{lem:marked-universality}.
+
+\subsection{The scalar interface}
+
+To specify this scalar without an independence assumption on diagonal
+classes, use the polynomial procedure just fixed.
+For a formal surface input \(z\), write a connected component carrying
+\(k\) marked inputs and decoration \(\alpha\) as the moment
+\[
+ M_k(\alpha)=\int_X z^k\alpha.
+\]
+Give unmarked moments their actual top intersection numbers.
+Give marked moments value zero except for \(M_3(1)=1\).
+Thus this evaluation retains precisely bare triple blocks.
+In the generating series of Section~\ref{sec:coefficient}, the
+variable \(q\) records total point length, \(p\) is the exponential
+variable for marked positions, and \(x\) is the high hyperplane class.
+Let \(a_{N,t_0}\) be the coefficient defined in
+\eqref{eq:scalar-coefficient}, with the moments evaluated in this
+way and \(N\) as in \eqref{eq:surviving-branch}.
+Equivalently, \((3t_0)!\,a_{N,t_0}\) is the sum of the coefficients
+of all bare triple partitions in the labeled marked pushforward.
+The polynomial procedure is fixed before imposing relations among
+the diagonal classes of a particular surface.  Section~\ref{sec:coefficient}
+evaluates this same representative: its moment-additivity argument
+is what permits numerical surface calculations to determine the
+selected scalar.  Thus the passage to those calculations does not
+replace the marked Chow identity by numerical universality.
+
+\begin{proposition}[Localization relation]
+\label{prop:localization}
+Under the hypotheses of Theorem~\ref{thm:surface}, the scalar
+\(a_{N,t_0}\) satisfies
+\[
+ a_{N,t_0}\,T_\gamma^{\,t_0}=0
+ \qquad\text{in }\mathfrak B_{S,b-1}.
+\]
+\end{proposition}
+
+\begin{proof}
+The codimension-\(4t_0\) marked localization sum is zero.
+The low-divisor and short-length arguments discard every fixed
+contribution except the branch \((0,D)\), in the stated support
+quotient. Marked universality and the output-degree bound leave only
+its bare triple partitions, with total coefficient
+\((3t_0)!\,a_{N,t_0}\).
+
+Test each surface position with \(z_\gamma\) from
+\eqref{eq:surface-test}. A bare surface triple diagonal is the
+product of the two curve triple diagonals, hence evaluates to
+\(T_\gamma\) up to the Koszul sign from grouping the two curve axes.
+This sign is the same for every block and every triple partition:
+each whole surface input is even, so permuting those inputs
+introduces no further sign. Thus every partition evaluates to the
+same signed \(T_\gamma^{t_0}\).
+Dividing the resulting equality by its common sign and by
+\((3t_0)!\) proves the assertion.
+All equalities were Chow equalities or tensor tests in the faithful
+coefficient formalism of Proposition~\ref{prop:tensor-test}.
+\end{proof}
diff --git a/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/build/sections/06-coefficient.tex b/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/build/sections/06-coefficient.tex
new file mode 100644
index 0000000000000000000000000000000000000000..c5b85bd4432ed995cde221690f2298bfcd9e13b6
--- /dev/null
+++ b/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/build/sections/06-coefficient.tex
@@ -0,0 +1,643 @@
+\section{The triple coefficient and arbitrary color products}
+\label{sec:coefficient}
+
+We first compute the scalar left by Proposition~\ref{prop:localization},
+completing the surface bound. We then convert that bound on isotropic
+tests into vanishing for sufficiently long products with arbitrary color
+inputs. For the coefficient calculation, put
+\[
+ t=t_0,\qquad N=t-1+ds_K,\qquad
+ A_0=K,\qquad D=dK,\qquad d=3,
+ \qquad B=-A_0-D.
+\]
+The letter \(q\) in the generating series below is a formal variable.
+It is unrelated to the realization maps \(q_{g,n}\).
+
+\begin{proposition}\label{prop:scalar}
+The bare-triple scalar \(a_{N,t_0}\) in
+Proposition~\ref{prop:localization} is nonzero.
+Consequently Theorem~\ref{thm:surface} holds.
+\end{proposition}
+
+The calculation belongs to the study of tautological Hilbert-scheme
+integrals developed by Lehn and Ellingsrud--G\"ottsche--Lehn
+\cite[Section~4.3]{Lehn99}, \cite{EGL}. For the symmetric-power and
+residue methods used below, compare
+\cite[Equation~(9), Theorem~2, and Lemma~3]{MOPCombinatorics}.
+The coupling of two Hilbert factors, all marked moments, and the
+Euler-shift derivative will be retained explicitly.
+
+We first identify a generating series for this scalar. We then compute
+its first nonzero hyperplane coefficient and prove that the resulting
+one-variable coefficient cannot vanish.
+
+\subsection{The moving factor}
+
+For a vector bundle \(E\), write \(e_a(E)\) for its Euler class after
+adding \(a\) to each Chern root. If \(a\) has invertible constant term,
+this convention extends multiplicatively to virtual bundles and
+perfect complexes. Inverses are expanded in positive Chow degree.
+Let
+\[
+ M_{r,s}=X^{[r]}\times X^{[s]},
+\]
+with universal finite subschemes \(Z_1,Z_2\) on \(M_{r,s}\times X\).
+Derived Hom expressions in this subsection include pushforward along
+the surface \(X\).
+
+On the surviving fixed component, the two kernels are
+\[
+ I'_1=L_1\mathcal I_{Z_1},\qquad
+ I'_2=L_2(-D-x)\mathcal I_{Z_2},
+\]
+where \(x\) is the hyperplane class of \(|D|\).
+Apart from the factor \((1-x)^{\chi(B)}\), the inverse moving Euler
+class is
+\begin{equation}\label{eq:moving-ratio}
+ \begin{aligned}
+ R(x)={}&
+ \frac{e_1\bigl(((K+A_0)^{[r]})^\vee\bigr)}
+ {e_{1-x}\bigl(((K-B)^{[r]})^\vee\bigr)\,
+  e_{1-x}(B^{[s]})\,e_{-1+x}((-B)^{[r]})}\\
+ &{}\times
+ e_{1-x}\bigl(R\Hom(\mathcal O_{Z_1},\mathcal O_{Z_2}(B))\bigr)\,
+ e_{1-x}\bigl(R\Hom(\mathcal O_{Z_1},\mathcal O_{Z_2}(K+B))\bigr).
+ \end{aligned}
+\end{equation}
+Here and below only scalar terms of bundles pulled from the original
+base can contribute, by Proposition~\ref{prop:localization}.
+In particular the omitted inverse Euler factor of
+\(R\pi_*(-A_0)\), at weight one, has scalar term one.
+
+To check \eqref{eq:moving-ratio}, the forward moving arrow has
+virtual tangent
+\[
+ R\Hom(\mathcal I_{Z_1},-A_0)
+ -
+ R\Hom(\mathcal I_{Z_1},\mathcal I_{Z_2}(B)),
+\]
+at weights \(1\) and \(1-x\), respectively.
+Expand each point ideal as \(\mathcal O-\mathcal O_Z\).
+The constant term of the second Hom gives
+\((1-x)^{\chi(B)}\). Surface duality identifies
+\[
+ R\Hom(\mathcal O_{Z_1},B)
+ \quad\text{with}\quad
+ ((K-B)^{[r]})^\vee[-2]
+\]
+in the convention relevant to the Euler class, and likewise for
+\(-A_0\). This gives the first numerator and the first two denominator
+factors of \eqref{eq:moving-ratio}.
+The reverse moving arrow is
+\[
+ R\Hom(\mathcal I_{Z_2},\mathcal O_{Z_1}(-B))
+\]
+at weight \(-1+x\). Its point term gives the final denominator.
+Duality changes its cross term to the remaining numerator term.
+The cross Hom has virtual rank zero, so changing both its dual and
+the sign of its weight introduces no rank sign.
+
+The fixed obstruction Euler classes on this component are
+\[
+ e\bigl((K^{[r]})^\vee\bigr),
+ \qquad
+ e_x\bigl(((K-D)^{[s]})^\vee\bigr).
+\]
+The sign of the second shift is worth recording: the line in the
+obstruction bundle is \(K-D-x\) before taking the dual, and hence
+has twist \(+x\) afterwards.
+
+\subsection{Marked moments and shifted additivity}
+
+Temporarily replace \(K\) and \(K-D\) in these two fixed Euler
+classes by independent divisor variables \(L,H\). Also separate the
+Euler shift \(x_e\) from the moving shift \(x_m\). For a surface
+class \(z\), let
+\[
+ \mu_r(z)=
+ (\operatorname{pr}_{X^{[r]}})_*
+ \bigl([Z_1]\operatorname{pr}_X^*z\bigr),
+\]
+and define \(\mu_s(z)\) similarly. Consider
+\begin{equation}\label{eq:theta-series}
+ \Theta(q,p,x_e,x_m)
+ =
+ \sum_{r,s\geq0}q^{r+s}
+ \int_{M_{r,s}}
+ e\bigl((L^{[r]})^\vee\bigr)\,
+ e_{x_e}\bigl((H^{[s]})^\vee\bigr)\,
+ R(x_m)\,
+ \exp\bigl(p(\mu_r(z)+\mu_s(z))\bigr).
+\end{equation}
+The exponential records labeled output insertions with their usual
+factorial denominators.
+
+We use the fixed marked recursion of
+Proposition~\ref{prop:localization} to regard each finite coefficient
+of \eqref{eq:theta-series} as a polynomial in decorated moments
+\[
+ M_k(\alpha)=\int_X z^k\alpha.
+\]
+Here \(\alpha\) is a monomial in
+\(L,H,A_0,D,K,c_2(T_X)\).
+More precisely, carry out the nested-Hilbert recursion while retaining
+the marked output slots. Each connected component of the final
+equality graph is integrated over one copy of \(X\); if it contains
+\(k\) marked slots and decoration \(\alpha\), replace that integration
+by \(M_k(\alpha)\). This fixes a polynomial representative before
+imposing numerical relations between moments. The recursion and its
+projective-bundle signs are those of
+\cite[Sections 1 and 3]{EGL}.
+The version with arbitrary retained outputs used here is proved in
+Lemma~\ref{lem:marked-universality}. This is the interface that keeps
+the calculation in relative Chow throughout.
+
+Let \(\mathcal E\) denote the following evaluation:
+\begin{equation}\label{eq:bare-moments}
+ \mathcal E(M_0(\alpha))=\int_X\alpha,\qquad
+ \mathcal E(M_3(1))=1,\qquad
+ \mathcal E(M_k(\alpha))=0
+ \quad(k>0,\ (k,\alpha)\neq(3,1)).
+\end{equation}
+Unmarked integrations in this formula are their top-degree numerical
+integrals; other degrees give zero.
+The scalar normalization agreed with
+Proposition~\ref{prop:localization} is
+\begin{equation}\label{eq:scalar-coefficient}
+ a_{N,t_0}
+ =
+ [q^N p^{3t}x^t]\,
+ (1-x)^{\chi(B)}
+ \mathcal E\bigl(\Theta(q,p,x,x)\bigr)
+ \big|_{L=K,\ H=K-D}.
+\end{equation}
+Indeed, \(\mathcal E\) retains precisely the partitions into bare
+triples. Passing from the exponential series to labeled outputs
+multiplies by the nonzero common factor \((3t)!\).
+Every such partition has the same tested value, up to the common
+Koszul sign for grouping the two curve axes. Those factors have
+already been divided out in Proposition~\ref{prop:localization}.
+
+The next observation allows us to compute \eqref{eq:scalar-coefficient}
+using ordinary numerical surface calculations, without assuming that
+all formal moments are independent.
+
+\begin{lemma}\label{lem:moment-additivity}
+There is a series \(c(q,x_e,x_m)\) such that
+\begin{equation}\label{eq:theta-exponential}
+ \mathcal E(\Theta)
+ =
+ \mathcal E(\Theta|_{p=0})
+ \exp\bigl(c(q,x_e,x_m)p^3/3!\bigr).
+\end{equation}
+Moreover, let \(c_H(q)\) be the coefficient of \(M_3(H)\), with the
+factor \(p^3/3!\) removed, in the one-moment part of
+\(\log\Theta(q,p,0,0)\). Then
+\begin{equation}\label{eq:shift-derivative}
+ [x_e]c(q,x_e,0)=-c_H(q).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Fix a finite coefficient cutoff, so that all polynomials and moment
+spaces in this argument are finite-dimensional. On genuine numerical
+data, allowing any even inhomogeneous cohomology class \(z\),
+\(\Theta\) is multiplicative under disjoint union of surfaces.
+The Hilbert schemes split according to the lengths on the components;
+all bundles and Hom complexes split; shifted Euler classes are
+multiplicative on direct sums; and the incidence exponentials
+multiply. These statements hold with both \(x_e\) and \(x_m\)
+retained.
+
+Consequently the polynomial for \(\log\Theta\) is additive on
+nonnegative integral combinations of genuine moment vectors.
+For any finite collection of those vectors, polynomial interpolation
+on the nonnegative integer grid extends additivity to their rational
+linear span. In particular, its homogeneous components of
+moment-count at least two vanish on that span. This is the
+disjoint-union argument underlying \cite[Theorem 4.2]{EGL}, applied
+coefficientwise to the specified marked polynomial.
+
+The vector in \eqref{eq:bare-moments} belongs to this span.
+Start with the genuine unmarked vector obtained by setting \(z=0\).
+On a fixed connected surface compare \(z=u+c_0[\mathrm{pt}]\)
+with \(z=u\), for rational \(u,c_0\).
+The difference changes only the undecorated marked moments, by
+\[
+ M_k(1)\longmapsto k u^{k-1}c_0.
+\]
+For every positive-degree decoration the change is zero.
+Varying \(u\) and using a finite Vandermonde matrix isolates
+\(M_3(1)\) among all the moments within the cutoff.
+The logarithm therefore has, at this vector, only an unmarked term
+and a term proportional to \(p^3\), proving
+\eqref{eq:theta-exponential}.
+
+We next justify differentiating in the Euler shift.
+Use the universal polynomial before discarding decoration degrees
+larger than the dimension of the surface. In this representative,
+the substitution
+\[
+ H\longmapsto H-x_e
+\]
+computes the shifted Euler class, since
+\[
+ e_{x_e}\bigl((H^{[s]})^\vee\bigr)
+ =(-1)^s e\bigl((H-x_e)^{[s]}\bigr).
+\]
+This notation means a base-line twist, not subtraction of a
+numerical surface divisor. To realize it, use the constant surface
+family over \(\mathbb P^a\), with \(a\) larger than the coefficient
+cutoff, and twist \(H\) by \(\mathcal O(-x_e)\).
+GRR for the tautological bundle uses \(\exp(H-x_e)\);
+the nested-Hilbert recursion and every projection formula commute
+with this twist and with moving powers of \(x_e\) out of the
+pushforward. It thus gives exactly the substitution in the
+ unsimplified moment polynomial. The moving factor is independent of
+ the auxiliary \(H\), so its degree cutoff is unaffected.
+ More explicitly, an output consisting of \(j\) bare triples requires
+ total intrinsic Euler-and-ratio degree \(2(r+s)-2j\) on the Hilbert
+ factors. The Euler factors have nonnegative intrinsic degrees.
+ Thus moving terms of degree greater than \(2(r+s)\) contribute
+ neither to genuine integrals nor to the desired formal coefficient,
+ before or after the twist.
+
+It is important to apply additivity to the shifted invariant itself.
+The substitution is linear on the moment symbols and preserves their
+number in a monomial. For each \(x_e\), the resulting polynomial
+still computes the genuine shifted invariant and is additive after
+taking its logarithm. Hence every shifted component of moment-count
+at least two vanishes on the original genuine span, and in
+particular at \(\mathcal E\).
+There is no claim that shifting the evaluation vector preserves
+that span. For example, an individual term involving \(M_0(H^3)\)
+can acquire a nonzero derivative after shifting, but the nonlinear
+terms cancel collectively by this additivity.
+
+In the one-moment part of \(p\)-degree three, the derivative followed
+by \(\mathcal E\) can only come from
+\[
+ M_3(H-x_e)=M_3(H)-x_eM_3(1).
+\]
+Other positive decorations remain positive after a single derivative,
+unless the decoration was \(H\) itself. Thus the derivative is
+\(-c_H(q)\), proving \eqref{eq:shift-derivative}.
+\end{proof}
+
+\subsection{Restriction to two curves}
+
+We first claim that
+\begin{equation}\label{eq:zero-unshifted-c}
+ c(q,0,x_m)=0.
+\end{equation}
+Represent very ample \(L,H\) by smooth transverse curves.
+At \(x_e=0\), the two Euler classes in
+\eqref{eq:theta-series} restrict the integral to
+\(L^{(r)}\times H^{(s)}\), with sign \((-1)^{r+s}\).
+These are the regular zero loci of the tautological sections:
+the zero schemes are the Hilbert schemes of points on the two smooth
+curves, of the expected codimensions \(r,s\).
+A point-class perturbation of the constant \(z=u\) restricts to zero
+on both curves. The Vandermonde test in the preceding proof therefore
+annihilates the bare marked coefficient.
+The coefficient of the single moment \(M_3(1)\) is a universal scalar,
+independent of the numerical line classes, so this proves
+\eqref{eq:zero-unshifted-c} identically.
+
+To determine \(c_H\), set both shifts to zero and take
+\[
+ z=u+\tau Z
+\]
+for a divisor \(Z\) on the surface. Write \(x',y'\) for the numerical
+point divisors on \(L^{(r)},H^{(s)}\).
+After the Euler restrictions, the one-body factors of
+\eqref{eq:moving-ratio}, including their signs, are
+\begin{equation}\label{eq:curve-one-body}
+ c_{-1}\bigl((A_0+2D)^{[r]}\bigr)^{-1}
+ \,(-1)^s c_{+1}(B^{[s]})^{-1}.
+\end{equation}
+Here \(c_a(E)=\sum_j a^j c_j(E)\).
+For verification, on the ordered cover of a symmetric power,
+a tautological line bundle \(M^{[r]}\) has a filtration with roots
+\[
+ M_i-\delta_i,\qquad
+ \delta_i=\sum_{j<i}[\Delta_{ij}].
+\]
+Two line classes pulled from the same curve position multiply to
+zero. Applying this to the low ratio in
+\eqref{eq:moving-ratio} reduces its factor at position \(i\) to
+\[
+ \frac{1}{1+\delta_i-(A_0+2D)_i},
+\]
+which gives the first term in \eqref{eq:curve-one-body}.
+
+Put \(I=LH\). At each of the \(I\) transverse intersections, the two
+universal finite subschemes meet by independently imposing their
+point-containment divisors. Their derived Hom has the class of the
+structure sheaf of the product of those divisors, twisted by the
+determinant normal of the first universal subscheme.
+Numerically its class is
+\[
+ e^{x'}(1-e^{-x'})(1-e^{-y'}).
+\]
+Its Euler class at weight one is therefore
+\[
+ \frac{(1+x')(1-y')}{1+x'-y'}.
+\]
+The two cross Hom terms give the combined factor
+\begin{equation}\label{eq:curve-cross}
+ \left(\frac{(1+x')(1-y')}{1+x'-y'}\right)^{2I}.
+\end{equation}
+This calculation also follows by resolving the two transverse
+containment divisors and taking their tensor-product resolution;
+the twists by surface lines have numerical degree zero at the
+fixed intersection point.
+The incidence exponential becomes
+\begin{equation}\label{eq:curve-incidence}
+ \exp\bigl(pu(r+s)+p\tau(ZL\,x'+ZH\,y')\bigr).
+\end{equation}
+
+We recall the symmetric-power integral needed to sum these terms.
+For a curve of genus \(h\), a line of degree \(d_1\), and its
+tautological bundle on the \(r\)-th symmetric power,
+\begin{equation}\label{eq:symmetric-chern}
+ c_a(\mathrm{line}^{[r]})
+ =(1-a x')^{r-d_1-1+h}
+ \exp\left(\frac{a\vartheta}{1-a x'}\right),
+ \qquad
+ \int (x')^{r-j}\frac{\vartheta^j}{j!}=\binom hj.
+\end{equation}
+Here \(\vartheta\) is the pulled-back Jacobian polarization.
+For completeness, the universal divisor sequence and GRR give
+\[
+ \ch(\mathrm{line}^{[r]})
+ =d_1+1-h+
+   (r-d_1-1+h+\vartheta)\exp(-x').
+\]
+Indeed the universal divisor class has terms
+\(r[\mathrm{pt}]+x'+\ell\), where the mixed class satisfies
+\(\int_{\mathrm{curve}}\ell^2/2=-\vartheta\).
+Converting this Chern character to the total Chern class gives the
+first formula in \eqref{eq:symmetric-chern}.
+For the second, pull to the ordered cover and expand the polarization
+in a symplectic basis. Intersecting the point divisors first leaves
+\(j\) factors; choosing \(j\) distinct symplectic pairs contributes
+one after the factorials, giving \(\binom hj\).
+
+Let
+\[
+ d_1=(A_0+2D)L,\qquad d_2=BH,\qquad
+ 2h_1-2=L(L+K),\qquad 2h_2-2=H(H+K).
+\]
+By \eqref{eq:symmetric-chern}, low-curve integration of the first
+factor in \eqref{eq:curve-one-body}, times any series in \(x'\),
+is coefficient extraction against
+\begin{equation}\label{eq:curve-integration}
+ [ (x')^r]\,
+ (1+x')^{d_1+1-2h_1-r}(1+2x')^{h_1}.
+\end{equation}
+High-curve integration has the analogous expression after replacing
+\(y'\) by \(-y'\), which absorbs the sign \((-1)^s\).
+Accordingly \eqref{eq:curve-cross} becomes
+\[
+ \left(\frac{(1+x')(1+y')}{1+x'+y'}\right)^{2I},
+\]
+and the nonconstant part of \eqref{eq:curve-incidence} becomes
+\(\exp(p\tau(ZL\,x'-ZH\,y'))\).
+
+The elementary Lagrange identity used for each variable is
+\begin{equation}\label{eq:lagrange-hilbert}
+ \sum_{r\geq0}Q^r [x^r]\frac{F(x)}{(1+x)^r}
+ =F(w)\frac{1+w}{1+2w},
+ \qquad w(1+w)=Q,\quad w(0)=0.
+\end{equation}
+One proof is to express coefficient extraction as the formal residue
+at zero: summing the geometric series gives
+\(F(x)(1+x)/(x(1+x)-Q)\), whose small root is \(w\);
+its residue there is the right side.
+The identity holds with other formal variables in the coefficients,
+so it can be applied successively to \(x'\) and \(y'\).
+Using \(Q=q\exp(pu)\), both small roots equal \(w\).
+Equations \eqref{eq:curve-integration}--\eqref{eq:lagrange-hilbert}
+therefore give
+\begin{align}
+ \log\Theta(q,p,0,0)
+ ={}&
+ (d_1+d_2+4-2h_1-2h_2+4I)\log(1+w) \notag\\
+ &+(h_1+h_2-2-2I)\log(1+2w)
+   +p\tau(ZL-ZH)w .
+ \label{eq:theta-curves}
+\end{align}
+This equation is the genuine numerical evaluation with
+\(z=u+\tau Z\). It determines the universal coefficients used below.
+
+\subsection{The first hyperplane coefficient}
+
+Let \(w_0(1+w_0)=q\), with \(w_0(0)=0\), and let
+\(\mathcal D=q\partial_q\).
+By Lemma~\ref{lem:moment-additivity}, the numerical evaluation of
+\(\log\Theta\) on a genuine moment vector equals that of its
+one-moment part: every component with at least two moment factors
+vanishes there. Its term of \(p\)-degree three is a universal scalar
+combination of the moments \(M_3(\alpha)\). The term involving
+\(M_3(H)\) is \(c_H(q)M_3(H)p^3/3!\).
+Since
+\[
+ [u^2\tau]M_3(H)=3ZH,
+\]
+we can identify \(c_H\) by varying the very ample \(H\), with the
+other divisor classes fixed. Only degree-one decorations can
+contribute to \(u^2\tau\), and \(H\) is the only varying such
+decoration. For example, one may take arbitrarily large integral
+multiples of a fixed ample line for \(H\), with \(Z\) fixed ample.
+The first two terms of \eqref{eq:theta-curves} contain no \(\tau\).
+The coefficient of \(p^3u^2\tau(ZH)\) in its last term is
+\(-\mathcal D^2w_0/2\). Comparing factorials yields
+\[
+ c_H(q)=-\mathcal D^2w_0.
+\]
+Together with \eqref{eq:shift-derivative} and
+\eqref{eq:zero-unshifted-c}, this proves
+\begin{equation}\label{eq:first-c-coefficient}
+ [x]c(q,x,x)
+ =\mathcal D^2w_0
+ =\frac{q(1+2q)}{(1+4q)^{3/2}}.
+\end{equation}
+There is no contribution from differentiating \(x_m\), because
+\(c(q,0,x_m)\) is identically zero.
+
+The unmarked calculation is equally explicit.
+Set \(p=0\) in \eqref{eq:theta-curves}.
+Every finite coefficient is polynomial in the intersection numbers
+of \(L,H,A_0,D,K\), so the formula continues from very ample
+\(L,H\) to \(L=K,\ H=K-D\).
+This can also be seen by varying sufficiently large multiples of
+\(K\) in both variables and then using polynomial identity.
+The adjunction expressions for \(h_1,h_2\) are polynomial
+expressions in this continuation, not assertions that a negative
+line has a smooth effective representative.
+Substitution gives
+\begin{equation}\label{eq:unmarked-theta}
+ \mathcal E(\Theta(q,0,0,0))
+ \big|_{L=K,\ H=K-D}
+ =
+ (1+w_0)^{ds_K}
+ (1+2w_0)^{d(d+1)s_K/2}.
+\end{equation}
+
+\subsection{A residue with a definite sign}
+
+We now prove Proposition~\ref{prop:scalar}.
+By \eqref{eq:theta-exponential}, the \(p^{3t}\)-coefficient is
+\(c(q,x,x)^t/(6^t t!)\) times the unmarked series.
+The series \(c(q,x,x)\) has zero constant term in \(x\).
+Thus the coefficient of \(x^t\) uses its linear term in every factor,
+and uses only the constant terms in \(x\) of the unmarked series and
+of \((1-x)^{\chi(B)}\).
+Equations \eqref{eq:scalar-coefficient},
+\eqref{eq:first-c-coefficient}, and \eqref{eq:unmarked-theta} give
+\begin{equation}\label{eq:scalar-final-series}
+ a_{N,t_0}
+ =
+ \frac{1}{6^t t!}
+ [q^{ds_K-1}]
+ (1+w_0)^{ds_K}(1+2q)^t(1+4q)^E,
+ \qquad
+ E=\frac{d(d+1)s_K}{4}-\frac{3t}{2}.
+\end{equation}
+
+Put \(v=1+2w_0\), so that \(q=(v^2-1)/4\).
+The coefficient residue in \eqref{eq:scalar-final-series} becomes
+\begin{equation}\label{eq:scalar-residue}
+ 2^{ds_K-t-1}
+ \operatorname*{Res}_{v=1}
+ \frac{(1+v^2)^t v^{2E+1}}{(v-1)^{ds_K}}\,dv.
+\end{equation}
+For the present parameters,
+\[
+ ds_K=24\chi,\qquad
+ t=25\chi-1,\qquad
+ E=\frac{3-27\chi}{2},\qquad
+ \chi=(b-1)^2\geq16.
+\]
+Hence the rational differential in \eqref{eq:scalar-residue} has
+finite poles only at zero and one.
+Its rational coefficient has degree \(2-\chi\leq-14\) at infinity,
+so its residue at infinity is zero.
+The residue at zero is
+\[
+ [v^{27\chi-5}]
+ (1+v^2)^{25\chi-1}(1-v)^{-24\chi}.
+\]
+This is strictly positive: the second factor has positive
+coefficients in every nonnegative degree, the first has nonnegative
+coefficients and constant term one, and \(27\chi-5\geq0\).
+The residue at one is its negative and therefore is nonzero.
+Both scalar prefactors in
+\eqref{eq:scalar-final-series}--\eqref{eq:scalar-residue} are nonzero.
+This proves \(a_{N,t_0}\neq0\).
+
+Finally, Proposition~\ref{prop:localization} gives
+\(a_{N,t_0}T_\gamma^{t_0}=0\) for the isotropic cubic contraction
+\(T_\gamma\) appearing in Theorem~\ref{thm:surface}.
+The scalar is a nonzero rational number and is invertible in the
+coefficient algebra. Thus \(T_\gamma^{t_0}=0\), which completes
+the proof of that theorem.
+
+\subsection{From isotropic tests to arbitrary color products}
+
+The surface theorem now gives nilpotence for every isotropic tensor
+test. To use it in the product defining the
+counterexample, we need a vanishing statement for arbitrary color
+inputs. We return to a smooth genus-$b$ curve family, choosing a rational
+divisor of relative degree one and its pure projector as in
+Proposition~\ref{prop:tensor-test}. We use the operation algebra $\mathcal A_k(E)$ of
+Section~\ref{sec:tensors} for an ordinary rational color space $E$.
+A \emph{pure cubic entry} is the class of the projected small diagonal
+in $\mathcal A_3(E)$ with a specified triple of color inputs; products
+use disjoint curve slots.
+
+We will fix an arbitrary color map $E\to W_b$, apply the surface theorem
+to a subspace with isotropic image, and use the following exterior-algebra
+lemma to annihilate every sufficiently long product. The final
+application of faithful polarization will then give a Chow identity
+with the same base-support bound.
+
+\begin{lemma}[An exterior ideal]
+\label{lem:exterior-ideal}
+Let \(F\subset H\) be vector spaces of dimensions \(2r\) and \(D\), and
+let \(\Omega\in\bigwedge^2F\) be nondegenerate.  If \(0\le t\le r\), then
+the ideal generated by \(\Omega^t\) in \(\bigwedge H\) contains
+\(\bigwedge^qH\) for every \(q\ge D-r+t\).
+\end{lemma}
+
+\begin{proof}
+The linear symplectic Lefschetz decomposition gives surjectivity of
+\[
+ \Omega^t\wedge-:
+ \bigwedge^{k-2t}F\longrightarrow\bigwedge^kF
+ \qquad(k\ge r+t).
+\]
+Equivalently, decompose the exterior algebra into primitive Lefschetz
+strings; every term in those degrees is at least \(t\) positions up its
+string.  Split \(H=F\oplus F'\).  A nonzero summand of
+\(\bigwedge^qH\) has \(F'\)-degree at most \(D-2r\), hence \(F\)-degree
+at least \(q-(D-2r)\ge r+t\).  It therefore lies in the stated ideal.
+\end{proof}
+
+\begin{lemma}[Arbitrary color products]
+\label{lem:large-vertex-test}
+Let a smooth curve family of genus \(b\ge5\), with a section, be over a
+smooth irreducible quasi-projective base \(S\) satisfying
+\(\dim S<(b-1)^2-1\).  Put
+\[
+ t_b=25(b-1)^2-1.
+\]
+Let \(E\) be an \(m\)-dimensional color space, and let \(e\) be an even
+integer with \(e\ge m/2-2\) and \(e\le m/2\).
+Write \(D_E=\binom m3\) and \(d_e=\binom e3\).
+If \(t_b\le d_e/2\), every product of
+\[
+ q\ge D_E-d_e/2+t_b
+\]
+pure cubic entries whose colors lie in \(E\) vanishes modulo support in
+base codimension at least \(b-1\).
+\end{lemma}
+
+\begin{proof}
+Fix a color test \(z:E\to W_b\).  The pullback to \(E\) of the symplectic
+form has a totally isotropic subspace of dimension at least \(m/2\):
+its radical together with a Lagrangian in its symplectic quotient has
+that dimension.  Choose an \(e\)-dimensional subspace \(E'\) of it.
+Its image under \(z\) is isotropic.
+
+Choose an invertible skew matrix \(\gamma\) in a basis of \(E'\).
+Grouping the ordered triples in Theorem~\ref{thm:surface}, the coefficients
+of the resulting quadratic expression in cubic generators are a
+nonzero common scalar times the three-by-three minors of \(\gamma\).
+Thus this expression is a nondegenerate bivector
+\[
+ \Omega\in\bigwedge^2\bigl(\bigwedge^3E'\bigr)
+\]
+up to a nonzero scalar.  Indeed, the third compound matrix is invertible,
+and its transpose is its negative because three is odd.
+In particular \(d_e\) is even.  Theorem~\ref{thm:surface} says that the
+map from the exterior algebra on \(\bigwedge^3 E\) to the odd cubic
+coefficients sends \(\Omega^{t_b}\) to zero in the specified support
+quotient.
+
+Lemma~\ref{lem:exterior-ideal}, with
+\(F=\bigwedge^3E'\) and \(H=\bigwedge^3E\), now kills every element of
+the asserted degree.  This holds for every \(z\); although \(E'\) can
+depend on \(z\), the degree threshold does not.
+Proposition~\ref{prop:tensor-test} therefore proves the Chow statement.
+Its finite coefficient test also ensures that the resulting support
+still has codimension at least \(b-1\).
+\end{proof}
+
+
+The smooth-family argument now gives vanishing of long products of
+pure cubic entries for arbitrary color inputs, modulo the stated
+base-support bound. The remaining task is
+to compare the compact candidate with these pure entries and control
+the errors along the boundary.
diff --git a/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/build/sections/07-compactification.tex b/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/build/sections/07-compactification.tex
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@@ -0,0 +1,511 @@
+\section{A cubic class on weighted compactifications}
+\label{sec:compactification}
+
+Lemma~\ref{lem:large-vertex-test} gives vanishing of long products of
+projected small diagonals over smooth curve families, modulo the stated
+base support.  To apply it on the entire moduli space, we construct a
+compact cubic operation whose restriction to a boundary chart differs
+from these pure diagonals by controlled terms.  Weighted stable curves
+and reduction morphisms are due to Hassett
+\cite{Hassett}. The smooth-core configurations below are related to the
+blowup compactifications of Fulton--MacPherson \cite{FultonMacPherson}
+and their weighted relative versions \cite{Routis}. For rational tails,
+Tavakol's proposed comparison with powers of the universal curve
+\cite{TavakolRT} was established by Petersen
+\cite[Proposition~2.11]{PetersenWonderful}.
+Here we also need control under cuts of nodal heavy cores.
+The correction at a rational bridge is essential: without it, a codimension-two exceptional class would
+remain on a three-light-point configuration space.
+
+We distinguish three cubic objects.  The formal class
+$F_J^{\mathrm{st}}$ belongs to the strata algebra and enters the
+definition of $Y$.  For the empty heavy set, the corrected weighted
+cubic $F^0_{g,\varnothing}$ pulls back under weight reduction to
+$q(F_J^{\mathrm{st}})$, with the corresponding label-forgetting pull.
+We also use its flagged versions $F^0_{h,B}$, called raw only to
+distinguish them from the auxiliary pure lifts $G_{h,B}$.
+Each $G_{h,B}$ is obtained from $F^0_{h,B}$ by controlled corrections:
+it agrees with the
+projected small diagonal over the smooth-core open, not globally
+on the compactification.
+
+\subsection{Weighted configurations and cut charts}
+
+Let $(h,B)$ be stable heavy data: $B$ is a finite ordered set and
+$2h-2+|B|>0$.  Write
+\[
+ X_{h,B}(k)=
+ \overline{\mathcal M}_{h,(1^B,\epsilon^k)},
+ \qquad 0<k\epsilon\leq1,
+\]
+with $X_{h,B}(0)=\Mbar_{h,B}$.  Here and below any positive rational
+weights of total at most one give the same moduli problem for the light
+labels.  Lights can coincide with one another, but not with a heavy
+marking.  A rational component with one node and only light markings
+cannot be stable; a rational component with two nodes and positive light
+mass is stable.  These observations also show the independence from the
+individual light weights.
+
+The stacks $X_{h,B}(k)$ are smooth and proper, and reduction of weights
+and forgetting labels give the usual stabilization morphisms
+\cite[Theorem~2.1, Proposition~3.8, Theorems~4.1 and~4.3]{Hassett}.
+The one-light space is the universal curve over $\Mbar_{h,B}$:
+when its coarse position is a node or a heavy marking, a unique
+three-special-point rational component is inserted.  Thus sections and
+the relative dualizing line define natural one-light classes.
+
+Fix a stable graph $\Gamma$ of the heavy type.  At its vertex $v$ let
+$h_v$ be the genus and let $B_v$ comprise the original heavy markings
+and the incident half-edges, all ordered.  Put
+\[
+ \mathcal B_\Gamma=\prod_v\Mbar_{h_v,B_v},
+ \qquad
+ \mathcal S_\Gamma=\prod_v\M_{h_v,B_v}.
+\]
+The \emph{cut chart} with $k$ lights is the disjoint union
+\begin{equation}\label{eq:cut-chart}
+ X^\Gamma(k)=
+ \coprod_{a:\{1,\ldots,k\}\to V(\Gamma)}
+ \prod_{v\in V(\Gamma)}X_{h_v,B_v}\bigl(a^{-1}(v)\bigr).
+\end{equation}
+It has a proper gluing map to the weighted space of the original heavy
+type.  Assignments are part of the indexing, so a class supported on
+the assignment of specified labels to a vertex means zero on the other
+components of \eqref{eq:cut-chart}.
+
+Refining a graph gives further proper cut charts.  These charts cover
+the preimage of the corresponding heavy boundary.  Indeed, a node of
+the light-forgotten curve may have a chain above it before stabilization.
+Choose a node on that chain and cut it, replacing its branches by heavy
+flags.  The resulting pieces remain weighted stable; gluing recovers
+the original curve.  Several prescribed nodes can be cut independently.
+The map on heavy bases is finite, as for the ordinary gluing of stable
+curves.
+
+Cutting and light forgetting commute.  The heavy flag replacing a
+node branch retains exactly the stability contribution of that branch.
+Thus stabilizing on the separate pieces and then gluing gives the same
+curve as forgetting and stabilizing after gluing.  This compatibility
+will permit operations on unconsumed light labels to pass through any
+later cut.
+
+We use rational Chow classes operationally on these smooth stacks.
+For an ordinary color space $U$, the arity-indexed operation algebra is
+\begin{equation}\label{eq:weighted-operation-algebra}
+ \bigoplus_{k\geq0}
+ \left(
+   \CH^*(X^\Gamma(k))\otimes U^{\otimes k}
+                    \otimes\operatorname{sgn}_k
+ \right)_{\mathfrak S_k}.
+\end{equation}
+The product pulls operations to disjoint sets of labels by forgetting
+and then multiplies them.  Successive stabilization proves
+associativity; exchanging the two sets of labels gives the sign
+$(-1)^{k\ell}$.  In particular, cubic and singleton operations are odd
+and pair operations are even.  All expressions below are first
+identities of labeled classes and then identities in
+\eqref{eq:weighted-operation-algebra}.
+
+\subsection{The corrected cubic}
+
+For three light labels define
+\begin{equation}\label{eq:weighted-cubic}
+ F^0_{h,B}
+   =[1=2=3]-\frac12\beta_{h,B}
+             +\frac12\sum_{p\in B}\delta_{p,123}\psi_p
+       \ \in\CH^2(X_{h,B}(3)).
+\end{equation}
+The coincidence class is the product of the divisors $[1=2]$ and
+$[1=3]$.  The class $\beta_{h,B}$ is defined by taking each stable
+one-edge graph of the heavy type, subdividing its edge by a genus-zero
+vertex carrying exactly the three lights, and summing the resulting
+gluing classes divided by their graph automorphism orders.
+The divisor $\delta_{p,123}$ has a tail containing exactly the heavy
+mark $p$ and the three lights; $\psi_p$ is the heavy cotangent class on
+the weighted space.  Each term in \eqref{eq:weighted-cubic} is symmetric
+in the three light labels.
+
+\input{figures/bridge-cuts}
+
+
+\begin{lemma}\label{lem:weighted-pullback}
+Let $g\geq2$, let the heavy set be empty, and pull the three-light
+operations to any larger number of light labels by forgetting.
+Under weight reduction from ordinary stable pointed curves,
+$[i=j]$ pulls back to $\bar D_{ij}$ and $\beta_{g,\varnothing}$
+pulls back to the normalized two-edge bridge sum used to define
+$\beta_J^{\mathrm{st}}$.
+Consequently the coefficient of the prescribed color word in the
+signed product of all $F^0_{g,\varnothing}$ pulls back to $q(Y)$,
+up to a nonzero rational scalar.
+\end{lemma}
+
+\begin{proof}
+In codimension one, the inverse image of coincidence consists of
+exactly the rational tails containing both labels.  At the generic
+point of each tail, the difference of the two coarse positions is a
+node-smoothing parameter times a nonzero difference of coordinates on
+the tail.  Every multiplicity is therefore one.
+
+Consider next the two bounding nodes of a weighted bridge.  Weight
+reduction retaining all labels contracts light-only tails, not a bridge
+with positive light mass.  Thus both bounding nodes exist already in
+the source, and all the light labels assigned to the bridge originate
+between those cuts.  The inverse image is supported on the
+corresponding ordinary strata of codimension at least two.  At the
+generic two-edge stratum the three labels are distinct on the bridge
+and no component is contracted, so the pullback multiplicity is one.
+Further collisions add edges and have higher codimension.  Normalized
+graph sums count these local cut choices once, including for a loop.
+This proves the supported codimension-two pullback assertion.
+
+Forgetting commutes with reduction.  Finally, extracting a fixed color
+word from signed averaging retains exactly the permutations preserving
+that word, with their signs.  Depending on the averaging convention
+there is a positive factorial denominator, but no zero scalar.
+\end{proof}
+
+\subsection{The local three-position calculation}
+
+Over $\mathcal S_\Gamma$ let $C_v$ denote the smooth core curve at
+vertex $v$.  For a fixed assignment of three labels, there is a coarse
+position map
+\begin{equation}\label{eq:coarse-positions}
+ c:X^\Gamma(3)|_{\mathcal S_\Gamma}
+       \longrightarrow
+       C_{a(1)}\times_{\mathcal S_\Gamma}
+       C_{a(2)}\times_{\mathcal S_\Gamma} C_{a(3)}.
+\end{equation}
+All products in this subsection are over the indicated heavy base.
+
+\begin{lemma}\label{lem:three-position-blowup}
+The map \eqref{eq:coarse-positions} is the iterated blowup obtained,
+at each flag $p$, by first blowing up the locus where all three
+assigned positions equal $p$, when present, and then the strict
+transforms of the loci where two assigned positions equal $p$.
+The latter centers are disjoint after the first blowup.
+\end{lemma}
+
+\begin{proof}
+Extra components above a smooth pointed core are chains attached at
+its flags.  A branch with no heavy endpoint and only light markings
+would have a terminal light-only tail and is impossible.  Every level
+of a chain carries positive light mass.
+
+Choose a local parameter at a flag and write $x_i$ for the corresponding
+coarse coordinate of label $i$.  The zero divisor of $x_i$ on the
+configuration space is the sum, with multiplicity one, of the cuts
+moving that label onto the expanded chain.  The multiplicity follows
+from a one-node smoothing $xy=t$.  At a fixed point of the
+configuration space these cuts form an initial segment of the
+ordered chain.  Hence the products of smoothing parameters occurring
+in the different $x_i$ are ordered by divisibility.  In particular,
+each relevant pair or triple ideal is principal after pullback.
+This gives a morphism to the first blowup and then to the successive
+pair blowups; after the first step the pulled pair ideal is its strict
+transform ideal times the exceptional line.
+
+The blowup ratios recover the expanded configuration.  On a level
+put its outward endpoint at infinity and its inward endpoint at zero.
+The leading ratios of coarse coordinate differences are the ratios
+of the light coordinates on that level, modulo common scaling.
+Labels on separate levels give the limiting ratios zero or infinity.
+Forgetting to pairs gives all these ratios compatibly, so the ratios
+determine the levels and their configurations.  Automorphisms fixing
+the core and the marked lights are trivial.  The resulting
+representable map to the iterated blowup is therefore quasi-finite.
+It is also proper and birational.  The target is smooth, and the
+source is smooth by weighted deformation theory; working on
+an etale chart of the core data, a proper birational quasi-finite map
+to this normal target is an isomorphism
+\cite[Tags~02LS and~0AB1]{Stacks}.
+
+In the first exceptional plane, the strict pair centers meet at the
+three distinct coordinate points, proving their disjointness.
+The constructions at distinct flags have disjoint centers.
+\end{proof}
+
+Denote the total first exceptional divisor at a triple flag locus by
+$E_{p,123}$.  Denote pair exceptional divisors, preferably pulled back
+from the corresponding two-light spaces, by $E_{p,ij}$.
+The blowup formula for Chow groups \cite[Proposition~6.7(e)]{Fulton} shows that a
+codimension-two class on \eqref{eq:coarse-positions} has, in addition to
+its main pullback, terms of the following forms:
+\begin{equation}\label{eq:blowup-components}
+ E_{p,ij}\,A_{p,ij},\qquad
+ E_{p,123}\,D_p,\qquad
+ a_p E_{p,123}^{\,2}.
+\end{equation}
+Here $A_{p,ij}$ is a divisor class on the remaining curve over the
+base, $D_p$ is a base divisor class, and $a_p$ is a scalar on each
+connected component of the base.  For clarity, if \(E_{p,ij}^{\mathrm{str}}\) is the strict pair
+exceptional divisor and \(E_{p,ij}^{\mathrm{pull}}\) is pulled back
+from two labels, then
+\[
+ E_{p,ij}^{\mathrm{pull}}
+   =E_{p,ij}^{\mathrm{str}}+E_{p,123}.
+\]
+Multiplying the difference by a divisor on the remaining curve
+restricts that divisor, on \(E_{p,123}\), to the flag section.
+It is therefore a pulled base divisor. Replacing the strict version
+by the pulled version changes only the
+triple-exceptional-times-base-divisor term, and introduces no new
+scalar multiple of \(E_{p,123}^2\).  Formula
+\eqref{eq:blowup-components} includes assignments to different
+vertices, in which some of the centers are simply absent.
+
+When a raw class $F^0$ is pulled from an ancestor chart, its main
+component is the small diagonal if all three labels are assigned to
+one relevant descendant vertex, and is zero otherwise.  Away from
+flags this is immediate.  The coincidence, bridge, and heavy-tail
+corrections not seen there project to loci where all three coarse
+positions are flags, of codimension three; they cannot alter a main
+codimension-two class.
+
+For the same pulled raw class, the divisor coefficients $A_{p,ij}$ in
+\eqref{eq:blowup-components} are supported on flag sections of the
+remaining curve, up to base divisor terms.  To see this, remove all
+flags from that remaining curve.  Neither the raw class nor its main
+diagonal pullback meets the pair exceptional locus over this open.
+Intersecting with the exceptional divisor and pushing to its center
+recovers its coefficient, with the usual negative self-intersection
+sign for a codimension-two blowup center.  The coefficient is
+therefore zero on this open.  Chow localization on the curve over the
+base gives the assertion.
+
+\begin{lemma}\label{lem:triple-exceptional-cancellation}
+For the pullback of a raw cubic class $F^0$ from an ancestor chart, the
+coefficient $a_p$ in \eqref{eq:blowup-components} is zero at every
+triple flag locus.
+\end{lemma}
+
+\begin{proof}
+Work first over a field point of the smooth heavy base.  The full
+fiber over three positions at one flag is the weighted chain space
+with two heavy endpoints and three lights, the Losev--Manin surface
+\cite[Theorems~2.2 and~2.6.3]{LosevManin},
+\cite[\S6.4]{Hassett}. By
+Lemma~\ref{lem:three-position-blowup}, it is the first exceptional
+$\mathbb P^2$ blown up at its three pair-collision points.
+The total class $E_{p,123}$ restricts to the pullback of
+$\mathcal O_{\mathbb P^2}(-1)$; hence
+\begin{equation}\label{eq:exceptional-square}
+ \int E_{p,123}^{\,2}=1.
+\end{equation}
+Main pullbacks, pair terms multiplied by a remaining-position divisor,
+and base terms have integral zero on this fiber.  Thus this integral
+detects exactly $a_p$.
+
+The coincidence term gives one: its locus is the single transverse
+configuration in which the three lights coincide away from the
+endpoints.  We distinguish new and inherited flags relative to the
+ancestor supplying the raw cubic.  Suppose first that $p$ is a new cut
+flag.  There is one pair of bounding nodes isolating the chain carrying
+the three lights.
+This remains true on the boundary of the chain space.  Indeed, a
+subcurve contracted upon forgetting the lights has two endpoint flags
+and no persistent heavy marking; every component has at least two
+incident branches before counting lights, so the component tree must
+be a chain, with positive light mass on every level.  Its extreme
+bounding nodes are uniquely determined.
+
+With these cuts distinguished, as in Figure~\ref{fig:bridge-cuts},
+the bridge locus is a regular
+two-node condition in a versal deformation chart.  Its two normal
+lines are the tensor products of the corresponding branch tangent
+lines.  The core factors are constant on our fiber, so its
+self-intersection restricts to
+\begin{equation}\label{eq:bridge-normal}
+ (-\psi_0)(-\psi_\infty)=\psi_0\psi_\infty.
+\end{equation}
+The normalized bridge sum counts this local branch once.  This
+argument also treats a self-node: one distinguishes the two branches
+before applying the normal formula, and the graph automorphism factor
+removes the duplicate ordering.
+
+Reduction from $\Mbar_{0,5}$ to the chain space preserves both heavy
+cotangent lines.  A rational component carrying a heavy mark cannot
+be contracted when the other three weights are decreased but remain
+positive.  The genus-zero string recurrence gives
+\begin{equation}\label{eq:chain-cotangent-integral}
+ \int_{\Mbar_{0,5}}\psi_0\psi_\infty
+   =\int_{\Mbar_{0,4}}\psi_0
+     +\int_{\Mbar_{0,4}}\psi_\infty
+   =2.
+\end{equation}
+The last equality follows by forgetting once more to
+$\Mbar_{0,3}$.  Equivalently,
+\eqref{eq:chain-cotangent-integral} is the degree-two case of the
+genus-zero cotangent integral formula.  The bridge contribution to
+$F^0$ is therefore $-\tfrac12\cdot2=-1$, canceling coincidence.
+
+If instead $p$ is an inherited heavy flag, the relevant term is the
+unique outer tail with that heavy marking and all three lights.
+Its normal factor is $-\psi_0$ on the chain and the inserted
+$\psi_p$ is $\psi_\infty$.  Thus
+$\delta_{p,123}\psi_p$ has integral $-2$, and its coefficient
+$+\tfrac12$ again cancels coincidence.  No other bridge or heavy-tail
+type meets this fiber: after the lights are forgotten its contracted
+locus is this specified node or this specified heavy mark.
+
+We have shown that the scalar coefficient in
+\eqref{eq:blowup-components} is zero on every connected smooth-core
+base component.  This proves the result over that base.
+\end{proof}
+
+\subsection{Controlled pure lifts}
+
+For a smooth heavy core $\pi:C\to S$, choose a degree-one rational
+class
+\[
+ e_C=
+ \begin{cases}
+ [p],&\text{if a heavy flag $p$ has been chosen},\\
+ c_1(\omega_{C/S})/(2h-2),&\text{if $B=\varnothing$}.
+ \end{cases}
+\]
+The second case occurs only for $h\geq2$.  As in the curve tensor
+construction, put
+\begin{equation}\label{eq:compact-centering}
+ e_C'=e_C-\frac12\pi^*\pi_*(e_C^2),
+ \qquad
+ \eta_C=[\Delta_C]-e'_{C,1}-e'_{C,2}.
+\end{equation}
+Write $\Delta_C^{\mathrm{pure}}$ for the small diagonal projected by
+$\eta_C$ in each of its three slots.  Expanding the three projectors
+shows that
+\begin{equation}\label{eq:nonpure-pair-singleton}
+ \begin{gathered}
+ [\Delta_{123}]-\Delta_C^{\mathrm{pure}}
+ \quad\text{is a sum of pair--singleton products},\\
+ \text{with singleton factors }1_C\text{ or }e_C'.
+ \end{gathered}
+\end{equation}
+No numerical equivalence is used in this expansion: it is the
+decomposition of the diagonal correspondence into the pure projector
+and the two hyperbolic projectors.
+
+\begin{proposition}[Controlled pure lifts]
+\label{prop:controlled-lifts}
+For every stable heavy type $(h,B)$ one can choose a symmetric class
+$G_{h,B}\in\CH^2(X_{h,B}(3))$ with the following properties.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item
+Over the smooth heavy base, $G_{h,B}$ is exactly the pullback of
+$\Delta_C^{\mathrm{pure}}$ by the coarse-position map.
+\item
+Pull a raw cubic $F^0$, or one of these chosen pure lifts $G$, from any
+ancestor chart to a compact descendant cut chart.  Its operation is
+a sum of the chosen pure lifts at the relevant descendant vertices,
+with all three labels assigned to that vertex, together with
+pair-times-singleton operations and a residual cubic operation.
+\item
+The singleton operations in \textup{(ii)}, and all their further
+pullbacks through cuts and light forgetting, are controlled as
+follows.  Over fixed smooth core data their restrictions lie
+componentwise in the span of the unit, the canonical class, and the
+flag points.  On any smooth test family the same statement holds
+modulo classes supported over positive base codimension.
+\item
+On the current smooth-core open the residual operation in
+\textup{(ii)} is a finite sum of classes containing a pulled base
+divisor factor.  The base divisor may come from the current heavy
+base or from a compact ancestor base.  Consequently its pullback to
+every smooth test family is zero operationally after removing a
+positive-codimension subset of that test base.
+\end{enumerate}
+All expansions preserve the color multiset.  They may always use one
+and the same chosen $G$ and centering at a given current vertex,
+regardless of which ancestor supplied the operation.
+\end{proposition}
+
+\begin{proof}
+First work on the smooth heavy base of the class itself.  The blowup
+decomposition \eqref{eq:blowup-components} and
+Lemma~\ref{lem:triple-exceptional-cancellation} express $F^0$ as the
+pullback of the small diagonal, plus pair-exceptional-times-singleton
+terms and triple-exceptional-times-base-divisor terms.  Subtract these
+terms, and also subtract \eqref{eq:nonpure-pair-singleton}.
+
+These subtractions have compact extensions with the required
+structure.  A pair exceptional divisor extends as the heavy-flag
+tail boundary on the two-light space; a triple exceptional divisor
+extends as its three-light counterpart.  Divisor coefficients on
+the heavy base extend by Chow localization.  Pair coefficients may
+also be extended by Chow localization on the smooth compact
+two-light stack.  The singleton factors have specific extensions:
+use units, the heavy sections, the relative dualizing line on the
+one-light universal curve, and base classes.  In particular we do
+not choose an arbitrary compact extension of the pure diagonal.
+Define $G$ by making these structured subtractions from $F^0$, and
+average over the three labels.  This gives \textup{(i)}.
+
+Now pull to a deeper cut chart.  The raw $F^0$ has the decomposition
+just proved there.  Its main component is the sum of the small
+diagonals of the relevant descendant vertices.  Replace each by the
+corresponding chosen $G$ plus its structured corrections.
+Inherited pair-times-singleton subtractions retain that form:
+forgetting commutes with cutting, so the pair coefficient remains a
+two-label operation and the singleton remains a one-label operation.
+The former is allowed to be arbitrary.  For the latter, the
+normalization formula for the dualizing line gives the canonical
+class plus the new branch points on each component.  A section
+either lies on the relevant component or has zero restriction there;
+base terms remain base terms.  This proves the asserted span and its
+stability under further cuts.
+
+The only other corrections are exceptional divisors multiplied by
+base divisor classes.  On pulling an inherited correction, the
+exceptional factor need not have a special form, but its base
+divisor factor remains pulled from the ancestor heavy base.
+The same description applies to the new corrections on the current
+chart.  Thus the residual over its smooth-core open is of the form
+\begin{equation}\label{eq:controlled-residual}
+ \sum_j H_j\cdot\rho_j^*D_j,
+\end{equation}
+where $H_j$ is a one-codimensional three-label operation and $D_j$
+is a divisor class on a current or ancestor heavy base, with the
+natural induced map $\rho_j$.
+Extending this decomposition to the compact chart can also leave a
+difference supported over its heavy boundary, by Chow localization.
+Include that difference in the residual operation.  Its restriction to
+the smooth-core open is zero, so the residual there still has precisely
+the form \eqref{eq:controlled-residual}.
+
+To verify the last assertion after an arbitrary smooth test base
+change $S$, represent each pulled rational divisor class by the
+first Chern class of a rational line bundle.  A rational
+trivialization makes it zero off a divisor of $S$.  After removing
+the finite union of these divisors, every summand of
+\eqref{eq:controlled-residual} is zero as an operational class.
+This remains true if the image of $S$ lies inside a divisor on an
+ancestor base: it is the pulled line bundle on $S$, not a presumed
+transverse pullback of its original supporting cycle, that is used.
+The base corrections in $e_C'$ are covered by the same argument.
+
+Finally, if a previous operation used a different centering or a
+different inherited presentation, expand it through its raw $F^0$
+and the structured subtractions defining that presentation.  The
+preceding argument again leaves the same allowed pair and singleton
+terms, and a residual of the stated form on the smooth-core open.
+We may therefore re-expand to
+the fixed choice of $G$ at every current vertex.  Assignment
+conditions can be imposed separately on the pair and singleton
+factors by making their operations zero on the other components of
+the disjoint union \eqref{eq:cut-chart}.  All steps occur before
+inserting colors, and so preserve their multiset.
+\end{proof}
+
+\begin{remark}\label{rem:family-not-fiber}
+The conclusion in Proposition~\ref{prop:controlled-lifts}\textup{(iv)}
+is stronger than vanishing on individual geometric fibers.  It gives
+operational vanishing on an open of every smooth test family,
+including families mapping into a special locus of the heavy base.
+This is the form needed when the remaining operation is multiplied
+by an arbitrary cycle and Chow localization is used to lower its
+base support.  Likewise, on the smooth-core open the equality in
+\textup{(i)} is an equality of relative Chow correspondences.
+Smooth proper base change for their pull, intersection, and push
+operations identifies it with the centered tensor construction on
+any such test family.
+\end{remark}
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+\section{Support induction and Chow vanishing}
+\label{sec:support}
+
+We now combine the arbitrary-color bound of
+Lemma~\ref{lem:large-vertex-test} with the compact lifts of
+Proposition~\ref{prop:controlled-lifts}.  We will make successive parts
+of the complete product of three-color entries vanish over dense opens
+of its base support, until no support remains.  Throughout this section,
+\[
+ g=10^{60},\qquad D_*=\binom g3,\qquad n=3D_*,
+ \qquad U_0=\Q^g.
+\]
+The basis of \(U_0\) is the set of colors.  An \emph{entry} is one factor
+with three specified color inputs.  Initially there is one entry for each
+increasing triple of distinct basis colors.  Products and their
+factorizations are taken in the sign-coinvariant operation algebra of the
+weighted spaces from the preceding section.
+
+The graph charts, their compact heavy bases, and their smooth-core opens
+will be denoted by
+\[
+ X^\Gamma(k),\qquad \mathcal B_\Gamma,\qquad
+ \mathcal S_\Gamma\subset\mathcal B_\Gamma,
+\]
+respectively.  Every graph in this section has unmarked type \((g,0)\);
+its vertex flags are heavy marks.  An operation supported on a vertex is
+zero on assignment components on which its active labels are not all at
+that vertex.
+
+\subsection{Advancing support with an arbitrary multiplier}
+
+Boundary-support methods also appear in tautological vanishing results
+such as \cite[Theorem~1.1]{GraberVakil}, where support is measured by
+rational components. Here we lower the dimension of the image in the
+heavy base, using operational vanishing over smooth test families.
+We first make this induction precise.  A term in it consists of
+active operations, pulled to disjoint sets of light labels, acting on a
+cycle on the full labeled chart.  The cycle need not factor over the
+labels or over the vertices.
+
+\begin{lemma}[Support advancement]
+\label{lem:support-advancement}
+Let a cycle multiplier on \(X^\Gamma(n)\) have base image contained in an
+irreducible closed subspace \(Z\subset\mathcal B_\Gamma\).  After passing
+to a refinement if necessary, suppose that its generic base image lies
+in \(\mathcal S_\Gamma\).  Let \(L\) be a factor involving a specified
+subset of the active entries.
+
+Suppose that the pull of \(L\) vanishes operationally over a dense open
+of a smooth test family dominating \(Z\cap\mathcal S_\Gamma\).  Then its
+action can be written as a finite sum of cycles with strictly smaller
+base-image dimension.  Only the entries in \(L\) are consumed.
+
+More precisely, take a smooth test base \(S\), proper and generically
+finite over \(Z\cap\mathcal S_\Gamma\).  If the vanishing holds off base
+codimension at least \(a\), the terms remaining over \(\mathcal S_\Gamma\)
+have base-image dimension at most \(\dim Z-a\).  The other terms lie over
+\(Z\setminus\mathcal S_\Gamma\) and require a strict graph refinement.
+If one first replaces \(S\) by the total space of one of its curve
+families to obtain a section, the corresponding bound is
+\(\dim Z-a+1\).
+\end{lemma}
+
+\begin{proof}
+The proper cut-chart maps cover the preimages of boundary strata.
+Consequently a cycle whose generic heavy curve has a finer dual graph
+can be lifted to a chart of that graph.  These lifts do not increase the
+dimension of its base image: heavy gluing is finite.
+
+We use the following elementary property of rational Chow groups.  If
+\(T'\to T\) is proper and surjective, every integral cycle on \(T\) has,
+after multiplication by a nonzero rational number, a proper lift
+generically finite over it.  Indeed, choose a closed point of the fiber
+over its generic point, take its finite residue-field extension, and
+close up that point.  The pushforward is the original cycle multiplied
+by the generic degree.  The same property holds for the finite-type Deligne--Mumford charts
+here with rational coefficients \cite[Proposition~B.19]{BaeSchmitt}.
+
+For the test base, take a finite level cover of the smooth heavy-curve
+stacks and resolve the appropriate component over
+\(Z\cap\mathcal S_\Gamma\)
+\cite[Theorems~1.0.2 and~1.0.3]{Wlodarczyk}.  This gives a smooth quasi-projective scheme
+proper and generically finite over that space.  A full level of order at least three removes automorphisms in positive
+genus \cite[Sections~5.3--5.4 and~5.14]{DeligneMumford},
+\cite[IV.2.7--2.9 and IV.3.1--3.2]{DeligneRapoport}; the genus-one
+vertices here have flags, and smooth genus-zero factors are already
+schemes.  Lift the cycle multiplier properly to the resulting family by
+the preceding argument.
+
+Let \(A\subset S\) have codimension at least \(a\), and suppose that \(L\)
+is zero operationally over \(S\setminus A\).  Its action on the lifted
+multiplier restricts to zero there.  Chow localization expresses the
+result as a pushforward of cycles lying over \(A\).  Projection formula
+and proper pushforward return this equality to the original family.
+Because \(\dim S=\dim Z\), these cycles have base-image dimension at most
+\(\dim Z-a\).  Closing up inside the inverse image of \(Z\) can introduce
+additional cycles only over \(Z\setminus\mathcal S_\Gamma\), a proper
+closed subset of \(Z\).  Refining their generic dual graphs places them
+on the required cut charts.
+
+To obtain a section of a smooth curve family \(C\to S\), pull that
+family back along \(C\to S\), where it has its diagonal section.
+The new test base is smooth and has dimension \(\dim Z+1\).
+Although this base change is not generically finite, the cycle multiplier
+still admits a proper lift generically finite over itself: apply the
+same generic-point argument to its pullback.  A support of codimension
+at least \(a\) in the new base has image dimension at most
+\(\dim Z+1-a\), proving the last assertion.
+
+These operations concern the chosen factor \(L\).  All remaining
+operations are carried along by pullback and projection formula and can
+be applied afterwards to the resulting cycle multipliers.
+\end{proof}
+
+The operational hypothesis in this lemma is essential.  In its
+applications below, it follows from identities over smooth test
+families, rather than from identities separately on geometric fibers.
+For a pure operation, Proposition~\ref{prop:controlled-lifts} identifies
+its restriction with a pullback from a smooth curve power.  Vanishing
+of its Chow class on that power is therefore operational vanishing and
+can be pulled further to any cycle multiplier.  The formation of the
+projector and the projected diagonal commutes with these base changes:
+the relative diagonals and the integrations along smooth proper curve
+powers have the requisite flat or refined-Gysin base-change identities.
+For residual operations, the explicit base-divisor factors in
+Proposition~\ref{prop:controlled-lifts} give the required open vanishing.
+The same is true of the controlled singleton relations below.
+
+There is also no restriction on the multiplier hidden in our use of
+sign coinvariants.  In characteristic zero, signed averaging identifies
+coinvariants with the corresponding signed invariant subspace.  A
+factorization \(L*H\) of operations becomes a sum over shuffles of
+products of the separately averaged \(L\) and \(H\), pulled to disjoint
+slot sets.  In each summand, apply \(L\) first to the multiplier.
+The factor \(H\) remains an averaged operation on its own active slots;
+it is pulled equivariantly through later cuts before it acts.
+This proves that the lemma applies repeatedly, even after the multiplier
+has become an arbitrary cycle produced by an earlier step.
+More explicitly, every later factorization of \(H\) is first an equality
+of its signed-average representatives on its own active slots.  Pull
+that equality to the full chart and only then cap with the multiplier.
+No permutation is applied to the multiplier, and no invariance of the
+multiplier is required.  Restriction to an assignment component or
+pullback through a further cut preserves this equality.
+
+Initially the heavy base has dimension \(3g-3\).  Every advancement
+strictly lowers the base-image dimension.  Thus on any branch of the
+finite expansions there can be at most
+\begin{equation}
+\label{eq:advancement-number}
+ 3g-2
+\end{equation}
+advancements before the support is empty.
+
+\subsection{The elementary vanishing tests}
+
+On a current chart, Proposition~\ref{prop:controlled-lifts} expands an
+ancestor entry into pure entries at current vertices, products of a pair
+operation with a controlled singleton, and residual cubic operations.
+Call the pair-times-singleton terms \emph{noise entries}.
+Perform these expansions one entry at a time.  If a residual summand is
+chosen, apply Lemma~\ref{lem:support-advancement} to it immediately, at
+the cost of that one entry.  We therefore never retain residual entries
+while counting the pure and noise entries.
+
+The pure entries at any one vertex always use the same chosen symmetric
+class \(G_{h,B}\).  This is important both for the finite coefficient
+bound and for the common-color test below.
+
+\begin{lemma}[Noise and small-genus tests]
+\label{lem:elementary-entry-tests}
+Set
+\[
+ R_\Gamma=2|V(\Gamma)|+2|E(\Gamma)|.
+\]
+If \(R_\Gamma+1\) noise entries have their singleton at the same color,
+their product advances support while consuming those entries.
+If a genus-\(h\) vertex has more than \(\binom{2h}{3}\) pure entries,
+the product of any \(\binom{2h}{3}+1\) of them advances support.
+\end{lemma}
+
+\begin{proof}
+Over smooth generic core data, the one-light space at a vertex is its
+curve.  The permitted singleton classes lie in the span of its unit,
+canonical class, and flag points.  Over all vertices this gives a space
+of dimension at most
+\[
+ \sum_v(2+|B_v|)=2|V|+2|E|=R_\Gamma.
+\]
+A one-slot operation is odd in the sign-coinvariant algebra.
+At a fixed color, a product of \(R_\Gamma+1\) such elements is zero by
+multilinearity and alternation.  The accompanying pair operations have
+even arity and can be grouped separately.  This proves the first
+vanishing, and it holds on an open of each smooth test family by the
+controlled singleton descriptions and Chow localization.
+
+For the second assertion apply Proposition~\ref{prop:tensor-test}.
+Every tested pure entry is a value of the same odd-coefficient form
+\(\varphi\in\mathfrak B\otimes\bigwedge^3 W_h^\vee\).
+Its values belong to the span of at most \(\binom{2h}{3}\) odd
+coefficients.  A longer product vanishes by supercommutativity.
+The faithful same-map tensor test proves vanishing of the operation
+modulo positive base support.  Only finitely many coefficient tests are
+needed in a fixed arity, so their supports can be combined without
+losing positive codimension.  Apply
+Lemma~\ref{lem:support-advancement}.
+\end{proof}
+
+\subsection{A repeated-color test at a flagged vertex}
+
+The surface test of Lemma~\ref{lem:large-vertex-test} is needed only
+while one vertex has almost all the genus.  On every other chart a less expensive advancement follows
+from the flag-centered curve algebra.
+
+\begin{lemma}[Repeated-color advancement]
+\label{lem:repeated-color}
+Suppose that a vertex of genus \(h\) has a flag and that
+\(k>\binom{h+1}{2}\) pure entries contain one common color \(i\).
+Their product is a finite linear combination of products \(L*H\) with
+the following properties:
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item \(L\) involves at most \(h+2\) entries and vanishes modulo positive
+base support;
+\item \(H\) is still a product of individual pure entries at that vertex;
+\item before removing the entries in \(L\), each term has the same color
+multiset as the original product.
+\end{enumerate}
+Consequently each term admits a support advancement consuming at most
+\(h+2\) entries.
+\end{lemma}
+
+\begin{proof}
+A symmetric cubic class defines an alternating three-color operation.
+Put the common color first in each entry.  The product, as a function of
+the other \(2k\) inputs, factors through
+\[
+ \bigwedge^k\bigl(\bigwedge^2 U_0\bigr).
+\]
+This holds already on the compact chart: alternation inside a pair comes
+from cubic symmetry, and interchange of two cubic entries has sign
+minus because their arity is odd.
+
+Let \(p\in\Q[\mathfrak S_{2k}]\) be the corresponding projector on the
+input slots.  The exterior functor displayed above is zero on a vector
+space of dimension \(h+1\).  By the Schur--Weyl decomposition and its row cutoff
+\cite[Theorems~5.12.2, 5.18.4, and~5.22.1]{EtingofEtAl2011},
+\(p\) lies in
+the two-sided ideal generated by the alternator on \(h+2\) slots:
+the quotient of the group algebra by that ideal acts faithfully on
+tensor powers in dimension \(h+1\).  Indeed, its simple blocks are
+indexed by partitions; precisely those with at most \(h+1\) rows
+survive in that tensor power, and the \((h+2)\)-alternator generates
+all the other blocks.  Thus \(p\) is a finite sum of
+operators of the form
+\[
+ c\,\sigma\,\operatorname{Alt}_{h+2}\,\tau,
+ \qquad c\in\Q,\quad \sigma,\tau\in\mathfrak S_{2k}.
+\]
+In one such term, first permute the color inputs by \(\tau\), and
+interpret \(\sigma\) as their assignment to the pairs of entries.
+Only entries meeting the \(h+2\) alternated positions vary during the
+alternation.  There are at most \(h+2\) of them.  Group their alternated
+product into \(L\), leaving the other entries in \(H\).
+This proves the asserted factorization and the color-multiset statement.
+
+It remains to prove the vanishing of \(L\).  Work on a smooth test base
+modulo positive base support and center the curve projector at the chosen
+flag section.  By Lemma~\ref{lem:point-centered-algebra}, its class \(e\)
+has
+\[
+ \int e=1,\qquad \int e^2=0.
+\]
+Evaluation at the section is pairing with \(e\), is multiplicative, and
+kills the pure odd space \(W_h\).  Hence for standard odd vectors \(u,v\),
+their product in the coefficient Frobenius algebra has the form
+\begin{equation}
+\label{eq:point-centered-pair}
+ uv=\omega(u,v)e+(uv)_{\mathrm{pure}}.
+\end{equation}
+There is no unit component, since its coefficient is the value of \(uv\)
+at the section.
+
+Test the colors by \(z:U_0\to W_h\), and write \(a=z(i)\).  Each entry
+becomes
+\[
+ T_a(u,v)=\varphi(a,u,v),
+\]
+an alternating two-form with odd coefficients.  Contracting a variable
+slot between two such entries gives zero.  To see this, the full
+Frobenius contraction is, up to its fixed Koszul sign,
+\[
+ \int (au)(av)=0,
+\]
+because \(a\) is odd and \(a^2=0\).
+The removed hyperbolic contraction in
+\eqref{eq:point-centered-pair} is a multiple of \(\int e^2=0\);
+orthogonality eliminates its mixed contractions.
+The remaining symplectic contraction of the pure products is exactly
+the stated contraction of the two cubic tensors.
+
+An internal contraction of the two variable slots in a single \(T_a\)
+is one and the same odd scalar, say \(\ell(a)\), up to the orientation
+sign.  Two internal contractions in distinct entries therefore give
+zero, because \(\ell(a)^2=0\).
+
+After alternation, \(L\) is an ordinary alternating
+\((h+2)\)-form on \(W_h\), with coefficients in the supercommutative
+algebra.  Every summand of its double symplectic contraction is zero:
+if one contraction joins distinct entries, perform that external
+contraction first; otherwise the two contractions are internal in
+distinct entries and give the square just considered.  All other
+variables can be held arbitrary in the external identity, so this
+reasoning also covers chains of contractions.
+
+For \(h\ge2\), double contraction
+\[
+ \bigwedge^{h+2}W_h^\vee
+ \longrightarrow \bigwedge^{h-2}W_h^\vee
+\]
+is injective by symplectic Lefschetz decomposition.  For \(h=0,1\),
+the source alternation is already zero.  Thus every color test of \(L\)
+vanishes.  Proposition~\ref{prop:tensor-test} and
+Lemma~\ref{lem:support-advancement} finish the proof.
+\end{proof}
+
+\subsection{The finite color budget}
+
+Set
+\begin{equation}
+\label{eq:color-constants}
+ K=10^5,\qquad m_0=2000g^{2/3},\qquad
+ C_K=\binom{2K}{3}.
+\end{equation}
+These are integers.  Reserve \(4K+20\) disjoint sets of \(m_0\) basis
+colors each, and let \(S_0\) be all the remaining colors.  This is possible
+because
+\[
+ \frac{(4K+20)m_0}{g}
+   =8.0004\cdot10^{-12}<0.1;
+ \qquad |S_0|>0.9g.
+\]
+For a current graph put
+\[
+ b=\max_v h_v,\qquad l=g-b.
+\]
+Under refinement the maximum vertex genus cannot increase, so \(l\)
+cannot decrease.  The number of edges cannot decrease either.
+Unmarked stability gives
+\begin{equation}
+\label{eq:global-noise-bound}
+ |V|\le2g-2,\qquad |E|\le3g-3,\qquad R_\Gamma\le10g.
+\end{equation}
+
+An advancement by a residual entry, by a noise test, by a small-genus
+test, or by Lemma~\ref{lem:repeated-color} will be called
+\emph{ordinary}.  These are descriptions of the cost in entries, not
+additional geometric assumptions.
+
+The two regimes below use different information about the remaining
+colors.  While \(l<K\), we remove whole entries without changing the
+three colors of any retained entry; the reserved blocks can therefore
+be counted by their original triples.  We use the repeated-color
+rewrites only once \(l\ge K\).  They can change which colors occur
+together in an entry, but preserve the total multiset of color
+occurrences before the advancing factor is removed.  In that regime
+we count occurrences in \(S_0\).  Since \(l\) cannot decrease, we never
+need an original-triple count after making such a rewrite.
+
+\paragraph{Small genus deficit: \(l<K\).}
+There is a unique vertex of genus \(b\), since \(b>g-K>g/2\).
+The total genus of all the others is at most \(l\), and stability gives
+\begin{equation}
+\label{eq:small-graph-size}
+ |V|-1\le2l,\qquad |E|\le3l,\qquad R_\Gamma\le10K+2.
+\end{equation}
+For completeness, sum \(2h_v-2+\operatorname{val}(v)\ge1\) over all
+vertices except the large one.  The sum equals
+\(2l-\operatorname{val}(v_{\mathrm{large}})\), proving the first
+inequality.  The genus formula then gives the second.
+
+After expanding the active entries, use a residual immediately.
+If a color occurs as a singleton in more than \(R_\Gamma\) noise entries,
+use Lemma~\ref{lem:elementary-entry-tests}.  Otherwise there are at most
+\(g(10K+2)\) noise entries.  If a small vertex has more than \(C_K\)
+pure entries, use the same lemma there.  Otherwise the number of pure
+entries off the large vertex is at most \(2K C_K\).
+Each ordinary advancement in this regime costs at most
+\[
+ M_K=\max\{10K+3,C_K+1\}=C_K+1.
+\]
+As specified above, retained entries in this regime still have their
+original triples, including those internal to each unused reserved
+block.
+
+Suppose none of these ordinary tests applies.  Choose a reserved color
+block \(E_0\) that has not been used for a large-vertex test.  Let
+\(D_0=\binom{m_0}{3}\).  Among its original \(D_0\) entirely internal
+entries, the number that are no longer active pure entries at the large
+vertex is at most
+\begin{equation}
+\label{eq:small-loss}
+ 3gM_K+g(10K+2)+2K C_K
+ <10^{17}g<0.01D_0.
+\end{equation}
+Here \eqref{eq:advancement-number} bounds all preceding ordinary losses.
+Earlier large-vertex tests used only entries internal to different,
+disjoint reserved blocks, so they removed none of these entries.
+The numerical inequalities follow from
+\[
+ C_K<1.334\cdot10^{15},
+ \qquad D_0>10^9g^2,
+ \qquad g=10^{60}.
+\]
+Thus more than \(0.99D_0\) internal entries remain pure at the large
+vertex.
+
+Consume the product of all those entries.  On a smooth test base,
+obtain a section if necessary as in
+Lemma~\ref{lem:support-advancement}.  Its dimension is at most \(3g-2\);
+since \(b>g-K\),
+\[
+ 3g-2<(b-1)^2-1.
+\]
+We may therefore use Lemma~\ref{lem:large-vertex-test}.
+Choose an even integer \(e\) between \(m_0/2-2\) and \(m_0/2\).
+For \(d_e=\binom e3\) and \(t_b=25(b-1)^2-1\), the explicit bounds are
+\[
+ d_e>D_0/10,\qquad t_b<25g^2,\qquad t_b<d_e/2,
+\]
+and consequently
+\begin{equation}
+\label{eq:large-test-threshold}
+ D_0-d_e/2+t_b
+   <(0.95+2.5\cdot10^{-8})D_0<0.99D_0.
+\end{equation}
+For the first bound, \(m_0\ge128\) and
+\[
+ \frac{\binom e3}{\binom{m_0}3}
+ \ge \left(\frac12-\frac4{m_0}\right)^3>\frac1{10}.
+\]
+The selected product thus vanishes modulo base support of codimension
+at least \(b-1\).  Applying
+Lemma~\ref{lem:support-advancement}, it either lowers the base-image
+dimension by at least \(b-2>g-K-2\), or gives a boundary remainder
+requiring a strict refinement.  Call this an \emph{expensive}
+advancement and mark \(E_0\) as used.
+
+There are enough fresh blocks for every such advancement.
+Those that require a refinement start with fewer than \(3K\) edges by
+\eqref{eq:small-graph-size}; along a branch there are at most \(3K+1\)
+of them.  Every other expensive advancement lowers the base-image
+dimension by at least \(g-K-2\).  There can be at most three such
+advancements with nonempty resulting support, because
+\[
+ 4(g-K-2)>3g-3.
+\]
+A fourth can be terminal, making the support empty.
+In total at most \(3K+5<4K+20\) blocks are needed.
+No expensive advancement consumes an occurrence of a color in \(S_0\).
+
+\paragraph{Large genus deficit: \(l\ge K\).}
+Once this regime is reached it persists.  We continue to expand
+ancestors using the same pure lift at each current vertex and peel
+residuals immediately.  A noise test consumes at most \(10g+1\) entries.
+If no such test applies, at most \(10g^2\) noise entries remain.
+
+All ordinary advancements on the entire preceding path, including
+those in the small-deficit regime, cost at most \(11g\) entries each:
+\[
+ C_K+1<11g,\qquad 10g+1<11g,\qquad h_v+2<11g.
+\]
+There have been at most \(3g-2\) advancements in total.
+Each consumed entry has three color occurrences.  The cumulative
+ordinary loss of occurrences in \(S_0\) is therefore less than
+\(99g^2\).  The unpeeled noise accounts for at most another \(30g^2\).
+Expensive advancements used no \(S_0\) colors.  Hence the number of
+pure occurrences of colors from \(S_0\) is at least
+\begin{equation}
+\label{eq:remaining-pure-occurrences}
+ |S_0|\binom{g-1}{2}-140g^2.
+\end{equation}
+The allowance \(140g^2\) is larger than the sum \(129g^2\) of the
+two losses just obtained.
+The permutations used in Lemma~\ref{lem:repeated-color} preserve the
+color multiset before its advancing factor is removed, so the same
+bound remains valid after any number of those rewrites.
+
+Discard any identically zero product, in particular a product containing
+a pure entry with a repeated color.  Suppose that every color
+\(i\in S_0\), at every vertex \(v\), belongs to at most
+\(\binom{h_v+1}{2}\) pure entries.  The total number of pure occurrences
+would then be at most
+\[
+ |S_0|\sum_v\binom{h_v+1}{2}
+ \le \frac{|S_0|}{2}(b+1)\sum_v h_v
+ \le \frac{|S_0|g(b+1)}2.
+\]
+This contradicts \eqref{eq:remaining-pure-occurrences}.  Indeed, the
+lower bound minus this upper bound is
+\[
+ \frac{|S_0|}{2}\bigl(g(l-4)+2\bigr)-140g^2
+ >\bigl(0.45(K-4)-140\bigr)g^2>0.
+\]
+There is therefore a color and vertex to which
+Lemma~\ref{lem:repeated-color} applies.
+Every vertex in this regime has a flag: a connected graph without any
+flags at one of its vertices could only be the one-vertex, no-edge
+graph, for which \(l=0\).
+The lemma supplies another ordinary advancement, at the claimed cost.
+
+\subsection{Completion}
+
+\begin{theorem}
+\label{thm:chow-vanishing}
+The class \(Y\in\Salg_{g,n}\) constructed above satisfies
+\[
+ q_{g,n}(Y)=0
+ \quad\text{in }\CH^*(\Mbar_{g,n};\Q).
+\]
+\end{theorem}
+
+\begin{proof}
+Begin on \(X_{g,\varnothing}(n)\) with the product of the cubic
+operations \(F^0_{g,\varnothing}\), one for each increasing color triple,
+and with the fundamental cycle as multiplier.  Apply
+Proposition~\ref{prop:controlled-lifts} and the preceding induction.
+At every stage, each nonzero branch admits an advancement: use a
+residual or noise test when available, the small-vertex and reserved-block
+tests if \(l<K\), and otherwise the repeated-color test.
+The counts show that all required active entries and unused blocks are
+available at the moment they are needed.
+
+Each individual expansion and localization involves finitely many
+cycles.  Each branch has length at most \(3g-2\), by
+\eqref{eq:advancement-number}.  Thus the process ends with empty
+support and proves that the full product is zero in the sign-coinvariant
+Chow group of the weighted space.
+
+The weighted-to-unweighted comparison in
+Lemma~\ref{lem:weighted-pullback} identifies the pullback of the
+raw cubic with \(q(F_J^{\mathrm{st}})\).  After signed averaging, the
+coefficient of the ordered color word \(\mathbf i\) is a nonzero
+rational scalar times
+\[
+ q_{g,n}\!\left(
+ \sum_{\tau:\,\tau\mathbf i=\mathbf i}
+   \operatorname{sgn}(\tau)\,
+   \tau\!\left(\prod_J F_J^{\mathrm{st}}\right)
+ \right)=q_{g,n}(Y).
+\]
+Pulling back the zero weighted class proves the theorem.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+Theorem~\ref{thm:chow-vanishing} places \(Y\) in the Chow kernel.
+Proposition~\ref{prop:detector} supplies a linear map annihilating the
+specified Pixton span and taking a nonzero value on \(Y\).  Thus
+\[
+ Y\in\ker q_{g,n}\setminus\Pspan_{g,n}.
+\]
+The stable pair \(g=10^{60}\), \(n=3\binom g3\) is a counterexample to
+the proposed equality for all stable pairs.
+\end{proof}
diff --git a/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/paper.pdf b/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..4e78eb3dda23f76137cb9e321cbff06b24d726ee
--- /dev/null
+++ b/preprints/A-high-arity-counterexample-to-Pixton-completeness-in-Chow-September-24-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:e52ae0e71ab4c7b5edb9bf47269fc79c674a93161bb1709920990dceccf33a53
+size 714470
diff --git a/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/README.md b/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..0b6c773afc1dd6eb0461d51a102a942d441ae991
--- /dev/null
+++ b/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/README.md
@@ -0,0 +1,18 @@
+# [A hyperbolic group with no geometric CAT(0) action](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 25, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026,
+  author = {{OpenAI}},
+  title = {{A hyperbolic group with no geometric CAT(0) action}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/paper.pdf}{OAI:A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/figures/fano-template.tex b/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/figures/fano-template.tex
new file mode 100644
index 0000000000000000000000000000000000000000..8c14aa14763e7de6bfe527b04eebba2b263343d5
--- /dev/null
+++ b/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/figures/fano-template.tex
@@ -0,0 +1,35 @@
+\begin{figure}[htbp]
+\centering
+\begin{tikzpicture}[x=1.25cm,y=1.25cm,
+  fano edge/.style={draw=black!75,line width=.65pt},
+  fano point/.style={circle,fill=black,inner sep=1.65pt},
+  fano label/.style={font=\small,fill=white,inner sep=1.5pt}]
+  \coordinate (A) at ({-sqrt(3)},-1);
+  \coordinate (B) at ({sqrt(3)},-1);
+  \coordinate (C) at (0,2);
+  \coordinate (D) at (0,0);
+  \coordinate (AB) at (0,-1);
+  \coordinate (AC) at ({-sqrt(3)/2},.5);
+  \coordinate (BC) at ({sqrt(3)/2},.5);
+  \draw[fano edge] (A)--(B)--(C)--cycle;
+  \draw[fano edge] (A)--(BC) (B)--(AC);
+  \draw[fano edge,densely dashed] (D) circle[radius=1];
+  \draw[draw=orange!75!black,line width=1.35pt] (C)--(AB);
+  \foreach \P in {A,B,C,D,AB,AC,BC}
+    \node[fano point] at (\P) {};
+  \node[fano label,anchor=north east] at (A) {$A=e_1$};
+  \node[fano label,anchor=north west] at (B) {$B=e_2$};
+  \node[fano label,anchor=south] at (C) {$C=e_3$};
+  \node[fano label,anchor=north west] at (.07,-.03) {$D$};
+  \node[fano label,anchor=north] at (0,-1.08) {$A+B$};
+  \node[fano label,anchor=east] at ({-sqrt(3)/2-.07},.5) {$A+C$};
+  \node[fano label,anchor=west] at ({sqrt(3)/2+.07},.5) {$B+C$};
+  \node[fano label,anchor=west,text=orange!75!black] at (.05,.65) {$CD$};
+\end{tikzpicture}
+\caption{The seven points and seven lines of the Fano plane.  The three sides,
+three medians, and dashed circle each represent one projective line; only
+marked dots represent projective points.  The
+points $A,B,C,D=A+B+C$ form a quadrangle: no three are collinear.  The thick median
+is the line $CD=\{C,D,A+B\}$, at which the incidence graph is modified.}
+\label{fig:fano-template}
+\end{figure}
diff --git a/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/figures/fans.tex b/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/figures/fans.tex
new file mode 100644
index 0000000000000000000000000000000000000000..e370f8c529d5e9cf2663c94a9d989267d0adbcb3
--- /dev/null
+++ b/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/figures/fans.tex
@@ -0,0 +1,59 @@
+\begin{figure}[htbp]
+\centering
+\begin{tikzpicture}[x=1.05cm,y=1.05cm,
+  fan edge/.style={draw=blue!60!black,line width=.85pt},
+  fan vertex/.style={circle,fill=black,inner sep=1.65pt},
+  fan label/.style={font=\small,fill=white,inner sep=1.5pt},
+  fan region/.style={rounded corners=7pt,draw=blue!45!black,
+    densely dashed,fill=blue!3,line width=.5pt}]
+  \begin{scope}[xshift=-3cm]
+    \draw[fan region] (-1.75,-1.65) rectangle (1.75,1.65);
+    \node[font=\small] at (0,2) {$F_C=K_{2,3}$};
+    \coordinate (XC) at (-1.3,0);
+    \coordinate (YC) at (1.3,0);
+    \coordinate (Cac) at (0,1.2);
+    \coordinate (Cbc) at (0,0);
+    \coordinate (Ccd) at (0,-1.2);
+    \foreach \L in {Cac,Cbc,Ccd}
+      \draw[fan edge] (XC)--(\L)--(YC);
+    \foreach \P in {XC,YC,Cac,Cbc,Ccd}
+      \node[fan vertex] at (\P) {};
+    \node[fan label,anchor=east] at (-1.36,0) {$X_C$};
+    \node[fan label,anchor=west] at (1.36,0) {$Y_C$};
+    \node[fan label,anchor=south] at (0,1.24) {$AC$};
+    \node[fan label,anchor=south] at (0,.04) {$BC$};
+    \node[fan label,anchor=north] at (0,-1.24) {$CD$};
+  \end{scope}
+  \begin{scope}[xshift=3cm]
+    \draw[fan region] (-1.75,-.38) rectangle (1.75,1.65);
+    \node[font=\small] at (0,2) {$F_D=K_{2,2}$};
+    \coordinate (XD) at (-1.3,0);
+    \coordinate (YD) at (1.3,0);
+    \coordinate (Dad) at (0,1.2);
+    \coordinate (Dbd) at (0,0);
+    \coordinate (Dcd) at (0,-1.2);
+    \foreach \L in {Dad,Dbd}
+      \draw[fan edge] (XD)--(\L)--(YD);
+    \draw[draw=orange!75!black,line width=1.25pt] (XD)--(Dcd);
+    \draw[draw=black!40,densely dashed,line width=.5pt] (YD)--(Dcd);
+    \draw[draw=black!65,line width=.7pt] (.53,-.48)--(.77,-.72)
+                                              (.53,-.72)--(.77,-.48);
+    \foreach \P in {XD,YD,Dad,Dbd,Dcd}
+      \node[fan vertex] at (\P) {};
+    \node[fan label,anchor=east] at (-1.36,0) {$X_D$};
+    \node[fan label,anchor=west] at (1.36,0) {$Y_D$};
+    \node[fan label,anchor=south] at (0,1.24) {$AD$};
+    \node[fan label,anchor=south] at (0,.04) {$BD$};
+    \node[fan label,anchor=north] at (0,-1.24) {$CD$};
+    \node[fan label,text=orange!75!black,anchor=east] at (-.8,-.7) {$e_0$};
+    \node[font=\scriptsize,anchor=west,text=black!65] at (.91,-.73) {omitted};
+  \end{scope}
+\end{tikzpicture}
+\caption{A full fan and the exceptional fan.  The blue edges and their
+endpoints form the indicated fans.  The loose edge
+$e_0=X_D\text{--}CD$ belongs to $\Gamma$ but not to $F_D$; the crossed dashed
+segment $Y_D\text{--}CD$ is absent from $\Gamma$.  In particular, $CD$ is
+not a vertex of $F_D$.  The two occurrences of $CD$ denote the same line
+vertex of $\Gamma$.}
+\label{fig:fans}
+\end{figure}
diff --git a/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/figures/transverse-square.tex b/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/figures/transverse-square.tex
new file mode 100644
index 0000000000000000000000000000000000000000..0f3aed6bec4966142843068ab93fa7ef257f10a1
--- /dev/null
+++ b/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/figures/transverse-square.tex
@@ -0,0 +1,58 @@
+\begin{figure}[htbp]
+\centering
+\begin{minipage}[c]{.43\textwidth}
+\centering
+\begin{tikzpicture}[x=3.3cm,y=3.3cm,
+  square vertex/.style={circle,fill=black,inner sep=1.65pt},
+  square label/.style={font=\small,fill=white,inner sep=1.5pt}]
+  \coordinate (u) at (-.5,.5);
+  \coordinate (v) at (.5,.5);
+  \coordinate (up) at (.5,-.5);
+  \coordinate (vp) at (-.5,-.5);
+  \coordinate (m) at (0,.5);
+  \coordinate (mp) at (0,-.5);
+  \coordinate (q) at (0,0);
+  \fill[blue!3] (u)--(v)--(up)--(vp)--cycle;
+  \draw[draw=black!40,densely dashed] (u)--(up) (v)--(vp);
+  \draw[line width=.85pt] (u)--(v)--(up)--(vp)--cycle;
+  \draw[draw=blue!65!black,line width=1.2pt] (m)--(mp);
+  \foreach \P in {u,v,up,vp,m,mp,q}
+    \node[square vertex] at (\P) {};
+  \node[square label,anchor=south east] at (u) {$u$};
+  \node[square label,anchor=south west] at (v) {$v$};
+  \node[square label,anchor=north west] at (up) {$u'$};
+  \node[square label,anchor=north east] at (vp) {$v'$};
+  \node[square label,anchor=south] at (0,.52) {$m$};
+  \node[square label,anchor=north] at (0,-.52) {$m'$};
+  \node[square label,anchor=east] at (-.025,0) {$q_0$};
+  \node[square label,anchor=west] at (.025,.25) {$\frac12$};
+  \node[square label,anchor=west] at (.025,-.25) {$\frac12$};
+  \node[square label,anchor=west] at (.54,0) {$1$};
+  \node[font=\small] at (0,.85) {Transverse factor $Q$};
+\end{tikzpicture}
+\end{minipage}\hfill
+\begin{minipage}[c]{.53\textwidth}
+\centering
+\begin{tikzpicture}[x=2.05cm,y=1cm,
+  axis label/.style={font=\small,fill=white,inner sep=1.5pt}]
+  \draw[{Stealth[length=4pt]}-{Stealth[length=4pt]},line width=.7pt]
+    (-1.22,0)--(1.22,0);
+  \foreach \x in {-1,0,1}\fill (\x,0) circle[radius=1.65pt];
+  \node[axis label,anchor=south] at (-1,.07) {$M_D$};
+  \node[axis label,anchor=south] at (1,.07) {$M_C$};
+  \node[axis label,anchor=north] at (0,-.07) {$0_T$};
+  \node[font=\small] at (0,.9) {Axis $\ell=\mathbb R\times\{q_0\}$};
+\end{tikzpicture}
+\par\smallskip
+$P(\ell)\cong\mathbb R\times Q$\par\medskip
+$M_A=2(0,m),\qquad M_B=2(0,m')$
+\end{minipage}
+\caption{The final contradiction takes place in the transverse factor after
+splitting off the $M_C,M_D$ axis.  The cyclically ordered points $u,v,u',v'$
+form a flat unit square in $Q$, with common diagonal midpoint $q_0$.
+The opposite-side midpoints $m,m'$ are distance $1/2$ from $q_0$ and distance
+$1$ apart.  Dilation in the original tangent cone $T$ therefore makes
+$M_A=2(0,m)$ and $M_B=2(0,m')$ antipodal unit vectors.  The panels display
+the two distinct factors of the parallel set.}
+\label{fig:transverse-square}
+\end{figure}
diff --git a/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/main.tex b/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/main.tex
new file mode 100644
index 0000000000000000000000000000000000000000..d1d9b3d5ddc6a96dca1e6bba83c389a4dd67d2c7
--- /dev/null
+++ b/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/main.tex
@@ -0,0 +1,23 @@
+\documentclass[11pt]{article}
+\input{preamble}
+\title{A hyperbolic group with no geometric CAT(0) action}
+\author{OpenAI}
+\date{September 25, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We construct a hyperbolic group with a finite classifying space that admits no geometric action on a proper complete CAT(0) space. Consequently, a finite aspherical simplicial complex with a linear combinatorial disk-filling inequality need not have a finite locally CAT(0) homotopy model, and hence need not have a finite locally CAT($-1$) model. Here metric models carry geodesic length metrics inducing the given complex topology.
+\end{abstract}
+\setcounter{tocdepth}{1}
+\tableofcontents
+\clearpage
+\input{sections/01-introduction}
+\input{sections/02-model}
+\input{sections/03-diagrams}
+\input{sections/04-sampling}
+\input{sections/05-energy}
+\input{sections/06-obstruction}
+\input{sections/07-completion}
+\bibliographystyle{plain}
+\bibliography{references}
+\end{document}
diff --git a/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/preamble.tex b/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/preamble.tex
new file mode 100644
index 0000000000000000000000000000000000000000..67c1256a0ccc27c991212a16006572d3ea797478
--- /dev/null
+++ b/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/preamble.tex
@@ -0,0 +1,42 @@
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage{mathpazo}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage[margin=1.05in]{geometry}
+\usepackage{microtype}
+\usepackage{enumitem}
+\usepackage{booktabs,array}
+\usepackage{graphicx,xcolor}
+\usepackage{placeins}
+\usepackage{tikz}
+\usetikzlibrary{calc,arrows.meta,positioning,decorations.pathreplacing}
+\usepackage{xurl}
+\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref}
+\usepackage[capitalise,noabbrev]{cleveref}
+\hypersetup{pdftitle={A hyperbolic group with no geometric CAT(0) action},pdfauthor={OpenAI},bookmarksdepth=2}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newcommand{\CAT}{\operatorname{CAT}}
+\newcommand{\E}{\mathbb E}
+\newcommand{\Prob}{\mathbb P}
+\newcommand{\F}{\mathbb F}
+\newcommand{\R}{\mathbb R}
+\newcommand{\Z}{\mathbb Z}
+\newcommand{\ip}[2]{\langle #1,#2\rangle}
+\newcommand{\norm}[1]{\lVert #1\rVert}
+\DeclareMathOperator{\barc}{bar}
+\DeclareMathOperator{\supp}{supp}
+\DeclareMathOperator{\diam}{diam}
+\DeclareMathOperator{\TV}{TV}
+\setlist{topsep=4pt,itemsep=3pt,parsep=0pt}
+\setlength{\emergencystretch}{1.5em}
+\numberwithin{equation}{section}
+\pdfminorversion=5
+\ifdefined\pdfinfoomitdate\pdfinfoomitdate=1\fi
+\ifdefined\pdftrailerid\pdftrailerid{}\fi
diff --git a/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/references.bib b/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/references.bib
new file mode 100644
index 0000000000000000000000000000000000000000..4db07b71d0e444840388a568292fde238c70ff6e
--- /dev/null
+++ b/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/references.bib
@@ -0,0 +1,288 @@
+@book{Hatcher2002,
+  author    = {Hatcher, Allen},
+  title     = {Algebraic Topology},
+  publisher = {Cambridge University Press},
+  year      = {2002},
+  isbn      = {0-521-79540-0},
+  url       = {https://pi.math.cornell.edu/~hatcher/AT/AT.pdf},
+  note      = {\href{https://pi.math.cornell.edu/~hatcher/AT/AT.pdf}{Author PDF}}
+}
+
+@book{BH1999,
+  author    = {Bridson, Martin R. and Haefliger, Andr{\'e}},
+  title     = {Metric Spaces of Non-Positive Curvature},
+  series    = {Grundlehren der mathematischen Wissenschaften},
+  volume    = {319},
+  publisher = {Springer-Verlag},
+  address   = {Berlin},
+  year      = {1999},
+  isbn      = {978-3-540-64324-1},
+  doi       = {10.1007/978-3-662-12494-9},
+  url       = {https://doi.org/10.1007/978-3-662-12494-9},
+  note      = {\href{https://doi.org/10.1007/978-3-662-12494-9}{DOI}}
+}
+
+@book{Ballmann1995,
+  author    = {Ballmann, Werner},
+  title     = {Lectures on Spaces of Nonpositive Curvature},
+  series    = {DMV Seminar},
+  volume    = {25},
+  publisher = {Birkh{\"a}user},
+  address   = {Basel},
+  year      = {1995},
+  isbn      = {978-3-7643-5242-4},
+  doi       = {10.1007/978-3-0348-9240-7},
+  url       = {https://doi.org/10.1007/978-3-0348-9240-7},
+  note      = {With an appendix by Misha Brin}
+}
+
+@article{LiptonTarjan1979,
+  author  = {Lipton, Richard J. and Tarjan, Robert Endre},
+  title   = {A separator theorem for planar graphs},
+  journal = {SIAM Journal on Applied Mathematics},
+  volume  = {36},
+  number  = {2},
+  pages   = {177--189},
+  year    = {1979},
+  doi     = {10.1137/0136016},
+  url     = {https://doi.org/10.1137/0136016},
+  note      = {\href{https://doi.org/10.1137/0136016}{DOI}}
+}
+
+@misc{Stark2025,
+  author        = {Stark, Emily},
+  title         = {Visual metrics on boundaries of hyperbolic spaces},
+  year          = {2025},
+  eprint        = {2506.10108},
+  archivePrefix = {arXiv},
+  primaryClass  = {math.GT},
+  doi           = {10.48550/arXiv.2506.10108},
+  url           = {https://arxiv.org/abs/2506.10108v1},
+  note      = {\href{https://arxiv.org/abs/2506.10108v1}{arXiv:2506.10108v1}}
+}
+
+@misc{Wilton2011,
+  author       = {Wilton, Henry},
+  title        = {Non-positively curved cube complexes},
+  year         = {2011},
+  howpublished = {Lecture notes},
+  note      = {Winter 2011; revised June 29, 2012; \href{https://www.dpmms.cam.ac.uk/~hjrw2/Notes/cubenotes.pdf}{Author PDF}},
+  url          = {https://www.dpmms.cam.ac.uk/~hjrw2/Notes/cubenotes.pdf}
+}
+
+@misc{Cashen2023,
+  author       = {Cashen, Christopher H.},
+  title        = {Notes on {Coxeter} groups},
+  year         = {2023},
+  howpublished = {Lecture notes, University of Vienna},
+  note      = {Version of October 31, 2023; \href{https://www.mat.univie.ac.at/~cashen/Notes_on_Coxeter_groups.pdf}{Author PDF}},
+  url          = {https://www.mat.univie.ac.at/~cashen/Notes_on_Coxeter_groups.pdf}
+}
+
+@article{Toyoda2016,
+  author        = {Toyoda, Tetsu},
+  title         = {Fixed point property for a {CAT(0)} space which admits a proper cocompact group action},
+  journal       = {Kodai Mathematical Journal},
+  volume        = {39},
+  number        = {1},
+  pages         = {129--153},
+  year          = {2016},
+  doi           = {10.2996/kmj/1458651696},
+  url           = {https://doi.org/10.2996/kmj/1458651696},
+  eprint        = {1102.0729},
+  archivePrefix = {arXiv},
+  primaryClass  = {math.MG},
+  note      = {\href{https://doi.org/10.2996/kmj/1458651696}{DOI}}
+}
+
+@article{AntoniukFriedgutLuczak2017,
+  author        = {Antoniuk, Sylwia and Friedgut, Ehud and {\L}uczak, Tomasz},
+  title         = {A sharp threshold for collapse of the random triangular group},
+  journal       = {Groups, Geometry, and Dynamics},
+  volume        = {11},
+  number        = {3},
+  pages         = {879--890},
+  year          = {2017},
+  doi           = {10.4171/GGD/417},
+  url           = {https://doi.org/10.4171/GGD/417},
+  eprint        = {1403.3516},
+  archivePrefix = {arXiv},
+  primaryClass  = {math.GR},
+  note      = {\href{https://doi.org/10.4171/GGD/417}{DOI}}
+}
+
+@incollection{Gromov1987,
+  author    = {Gromov, Mikhail},
+  title     = {Hyperbolic groups},
+  editor    = {Gersten, S. M.},
+  booktitle = {Essays in Group Theory},
+  series    = {Mathematical Sciences Research Institute Publications},
+  volume    = {8},
+  pages     = {75--263},
+  publisher = {Springer-Verlag},
+  address   = {New York},
+  year      = {1987},
+  doi       = {10.1007/978-1-4613-9586-7_3},
+  url       = {https://doi.org/10.1007/978-1-4613-9586-7_3},
+  note      = {\href{https://doi.org/10.1007/978-1-4613-9586-7_3}{DOI}}
+}
+
+@article{BradyCrisp2007,
+  author  = {Brady, Noel and Crisp, John},
+  title   = {{CAT}(0) and {CAT}(-1) dimensions of torsion free hyperbolic groups},
+  journal = {Commentarii Mathematici Helvetici},
+  volume  = {82},
+  number  = {1},
+  pages   = {61--85},
+  year    = {2007},
+  doi     = {10.4171/CMH/85},
+  url     = {https://ems.press/journals/cmh/articles/1290},
+  note    = {\href{https://doi.org/10.4171/CMH/85}{DOI}}
+}
+
+@article{Ollivier2004,
+  author  = {Ollivier, Yann},
+  title   = {Sharp phase transition theorems for hyperbolicity of random groups},
+  journal = {Geometric and Functional Analysis},
+  volume  = {14},
+  number  = {3},
+  pages   = {595--679},
+  year    = {2004},
+  doi     = {10.1007/s00039-004-0470-y},
+  url     = {https://doi.org/10.1007/s00039-004-0470-y},
+  eprint  = {math/0301187},
+  archivePrefix = {arXiv},
+  note    = {\href{https://doi.org/10.1007/s00039-004-0470-y}{DOI}}
+}
+
+@article{Gromov2003,
+  author  = {Gromov, Mikhail},
+  title   = {Random walk in random groups},
+  journal = {Geometric and Functional Analysis},
+  volume  = {13},
+  number  = {1},
+  pages   = {73--146},
+  year    = {2003},
+  doi     = {10.1007/s000390300002},
+  url     = {https://doi.org/10.1007/s000390300002},
+  note    = {\href{https://doi.org/10.1007/s000390300002}{DOI}}
+}
+
+@article{IzekiNayatani2005,
+  author  = {Izeki, Hiroyasu and Nayatani, Shin},
+  title   = {Combinatorial harmonic maps and discrete-group actions on {Hadamard} spaces},
+  journal = {Geometriae Dedicata},
+  volume  = {114},
+  pages   = {147--188},
+  year    = {2005},
+  doi     = {10.1007/s10711-004-1843-y},
+  url     = {https://doi.org/10.1007/s10711-004-1843-y},
+  eprint  = {math/0410019},
+  archivePrefix = {arXiv},
+  note    = {\href{https://doi.org/10.1007/s10711-004-1843-y}{DOI}}
+}
+
+@incollection{Sturm2003,
+  author    = {Sturm, Karl-Theodor},
+  title     = {Probability measures on metric spaces of nonpositive curvature},
+  booktitle = {Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces},
+  series    = {Contemporary Mathematics},
+  volume    = {338},
+  pages     = {357--390},
+  publisher = {American Mathematical Society},
+  address   = {Providence, RI},
+  year      = {2003},
+  doi       = {10.1090/conm/338/06080},
+  url       = {https://doi.org/10.1090/conm/338/06080},
+  note      = {\href{https://doi.org/10.1090/conm/338/06080}{DOI}}
+}
+
+@article{GigliNobili2021,
+  author  = {Gigli, Nicola and Nobili, Francesco},
+  title   = {A differential perspective on gradient flows on {CAT}($\kappa$)-spaces and applications},
+  journal = {Journal of Geometric Analysis},
+  volume  = {31},
+  pages   = {11780--11818},
+  year    = {2021},
+  doi     = {10.1007/s12220-021-00701-5},
+  url     = {https://doi.org/10.1007/s12220-021-00701-5},
+  eprint  = {2012.12952},
+  archivePrefix = {arXiv},
+  note    = {\href{https://doi.org/10.1007/s12220-021-00701-5}{DOI}}
+}
+
+@article{KleinerLeeb1997,
+  author  = {Kleiner, Bruce and Leeb, Bernhard},
+  title   = {Rigidity of quasi-isometries for symmetric spaces and {Euclidean} buildings},
+  journal = {Publications Math{\'e}matiques de l'IH{\'E}S},
+  volume  = {86},
+  pages   = {115--197},
+  year    = {1997},
+  doi     = {10.1007/BF02698902},
+  url     = {https://www.numdam.org/articles/10.1007/BF02698902/},
+  note    = {\href{https://doi.org/10.1007/BF02698902}{DOI}}
+}
+
+@article{RaussenSkau2010,
+  author  = {Raussen, Martin and Skau, Christian},
+  title   = {Interview with {Mikhail Gromov}},
+  journal = {Notices of the American Mathematical Society},
+  volume  = {57},
+  number  = {3},
+  pages   = {391--403},
+  year    = {2010},
+  url     = {https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/rtx100300391p.pdf},
+  note    = {Interview of May 18, 2009; \href{https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/rtx100300391p.pdf}{Author-hosted PDF}}
+}
+
+@incollection{BaumslagMyasnikovShpilrain1999,
+  author    = {Baumslag, Gilbert and Myasnikov, Alexei G. and Shpilrain, Vladimir},
+  title     = {Open problems in combinatorial group theory},
+  booktitle = {Groups, Languages and Geometry},
+  series    = {Contemporary Mathematics},
+  volume    = {250},
+  pages     = {1--27},
+  publisher = {American Mathematical Society},
+  address   = {Providence, RI},
+  year      = {1999},
+  url       = {https://web.stevens.edu/algebraic/alexeim/Publications/All_files_new/Openproblem_final_40.pdf},
+  note      = {\href{https://web.stevens.edu/algebraic/alexeim/Publications/All_files_new/Openproblem_final_40.pdf}{Author manuscript}}
+}
+
+@misc{DuchesneSimon2026,
+  author        = {Duchesne, Bruno and Simon, Christopher-Lloyd},
+  title         = {The variety of group actions on all algebraic real hyperbolic spaces},
+  year          = {2026},
+  eprint        = {2603.03863},
+  archivePrefix = {arXiv},
+  primaryClass  = {math.MG},
+  doi           = {10.48550/arXiv.2603.03863},
+  url           = {https://arxiv.org/abs/2603.03863v1},
+  note          = {Version submitted March 4, 2026; \href{https://arxiv.org/abs/2603.03863v1}{arXiv:2603.03863v1}}
+}
+
+@article{OllivierWise2011,
+  author  = {Ollivier, Yann and Wise, Daniel T.},
+  title   = {Cubulating random groups at density less than {$1/6$}},
+  journal = {Transactions of the American Mathematical Society},
+  volume  = {363},
+  number  = {9},
+  pages   = {4701--4733},
+  year    = {2011},
+  doi     = {10.1090/S0002-9947-2011-05197-4},
+  url     = {https://www.ams.org/tran/2011-363-09/S0002-9947-2011-05197-4/S0002-9947-2011-05197-4.pdf},
+  note    = {\href{https://doi.org/10.1090/S0002-9947-2011-05197-4}{DOI}}
+}
+
+@article{IzekiKondoNayatani2012,
+  author  = {Izeki, Hiroyasu and Kondo, Takefumi and Nayatani, Shin},
+  title   = {{N}-step energy of maps and the fixed-point property of random groups},
+  journal = {Groups, Geometry, and Dynamics},
+  volume  = {6},
+  number  = {4},
+  pages   = {701--736},
+  year    = {2012},
+  doi     = {10.4171/GGD/171},
+  url     = {https://ems.press/content/serial-article-files/29632},
+  note    = {\href{https://doi.org/10.4171/GGD/171}{DOI}}
+}
diff --git a/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/sections/01-introduction.tex b/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/sections/01-introduction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..bce4febb3ac625f07d482abee5f5ec15112b2ff7
--- /dev/null
+++ b/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/sections/01-introduction.tex
@@ -0,0 +1,180 @@
+\section{Introduction}\label{sec:introduction}
+A linear isoperimetric inequality is a large-scale expression of negative curvature: for finitely presented groups, it characterizes word hyperbolicity. Local comparison geometry is more restrictive. The geometric realization question asks whether every hyperbolic group admits a geometric action on a proper $\CAT(0)$ space, or on a proper $\CAT(-1)$ space. We construct a group for which neither action exists.
+
+Throughout the paper, a \emph{geometric action} is a proper, cocompact action by isometries, and metric models of finite complexes carry geodesic length metrics inducing the given complex topology.
+
+\begin{theorem}\label{thm:main}
+There exist a finite connected two-dimensional aspherical simplicial complex $K$ and a constant $C<\infty$ with the following properties.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item Every combinatorial edge loop of length $\ell$ in the universal cover of $K$ bounds a disk diagram with at most $C\ell$ triangular faces.
+\item The group $\pi_1K$ admits no geometric action on a proper complete $\CAT(0)$ space.
+\end{enumerate}
+In particular, $\pi_1K$ is word-hyperbolic, and no finite simplicial complex homotopy equivalent to $K$ admits a locally $\CAT(0)$ geodesic length metric inducing its given topology. Consequently, no such model admits a compatible locally $\CAT(-1)$ geodesic length metric.
+\end{theorem}
+
+The proof is an existence argument and does not give an effective size for $K$. The filling constant may depend on the selected finite presentation.
+
+\subsection{Context}
+The realization problem goes back to the development of hyperbolic groups.
+In his account of that development, Gromov recalls introducing linear
+isoperimetry as a group-theoretic form of negative curvature and then
+spending years seeking geometric realizations by negatively curved
+spaces~\cite[p.~396]{RaussenSkau2010}.
+The equivalence between linear filling and word hyperbolicity became
+part of the foundational theory~\cite{Gromov1987}; see also
+\cite[Theorem~III.$\Gamma$.2.6 and Remark~III.$\Gamma$.2.7(2)]{BH1999}.
+The $\CAT(-1)$ realization question appears as Problem~H11 in
+Baumslag--Myasnikov--Shpilrain's 1999 problem
+list~\cite{BaumslagMyasnikovShpilrain1999}.
+Both the $\CAT(-1)$ and $\CAT(0)$ questions appear as open problems
+in Stark's 2025 survey~\cite[Open Problem~2.15]{Stark2025} and in the
+March 2026 preprint of Duchesne--Simon~\cite[Question~W]{DuchesneSimon2026}.
+
+There are several successful ways to obtain such geometric actions.
+Finitely generated free groups act on trees and closed hyperbolic manifold groups act on
+hyperbolic space. More generally, a finite complex carrying a locally
+$\CAT(-1)$ or locally $\CAT(0)$ geodesic length metric inducing its topology has a universal cover
+with the corresponding global curvature bound; its deck action is
+geometric~\cite[Theorem~II.4.1]{BH1999}.
+Cubulation provides another route to $\CAT(0)$ models.
+For example, Ollivier--Wise prove that random groups in the density
+model at density less than $1/6$ act freely and cocompactly on
+$\CAT(0)$ cube complexes with probability tending to one as relator
+length grows, with the number of generators fixed~\cite[Theorem~10.4]{OllivierWise2011}.
+Thus random presentations can produce geometric $\CAT(0)$ actions as
+well as hyperbolicity; the relations used here must enforce an
+additional obstruction.
+
+Dimension restrictions already yield obstructions.
+Brady--Crisp~\cite[Theorem~2]{BradyCrisp2007} construct infinitely many
+torsion-free hyperbolic groups of geometric dimension two which admit
+no proper isometric action on a proper two-dimensional $\CAT(0)$
+space. Their Theorem~1 gives a hyperbolic group of $\CAT(0)$ dimension
+two and $\CAT(-1)$ dimension three. These conclusions rely on
+constraints on translation lengths in two-dimensional targets.
+Theorem~\ref{thm:main} excludes geometric $\CAT(0)$ actions in every
+dimension. Its finite-model consequence also allows the entire
+simplicial complex to change within its homotopy type: any compatible
+locally $\CAT(0)$ geodesic length metric on such a model would recover a geometric
+action by passing to the universal cover.
+
+Random-presentation proofs of hyperbolicity combine counting for
+bounded diagrams with a local-to-global argument.
+Our treatment of repeated relator occurrences is related to the
+multiplicity and constraint methods in
+Ollivier~\cite[Section~2.2]{Ollivier2004} and
+Antoniuk--Friedgut--\L uczak~\cite[Section~3]{AntoniukFriedgutLuczak2017}.
+Within each finite presentation we first sample small labelled graphs,
+then larger labelled graphs whose required small graphs have already
+been selected. The same prerequisite can support several larger
+graphs, so its probability cost must be charged only once.
+Section~\ref{sec:diagrams} proves the resulting estimate directly.
+
+The obstruction to actions uses the harmonic-energy framework of
+Gromov~\cite[Sections~3.3 and~3.7]{Gromov2003} and
+Izeki--Nayatani~\cite{IzekiNayatani2005}.
+In this framework, expansion in a finite graph can force a fixed point
+when the target satisfies a suitable nonlinear spectral inequality.
+The target hypothesis matters: Izeki--Kondo--Nayatani obtain random
+hyperbolic groups with fixed-point properties for target classes whose
+tangent-cone invariant is bounded by a prescribed constant strictly
+less than one~\cite[Corollary~3.8]{IzekiKondoNayatani2012}.
+Toyoda~\cite[Theorem~1.4]{Toyoda2016} supplies such spectral bounds
+for a prescribed finite family of geodesically complete $\CAT(0)$
+spaces admitting proper cocompact actions, with constants depending
+on that family.
+Here a hypothetical target and its action can be chosen after the
+finite presentation is known. We obtain the required uniformity from
+bounds on every generator displacement and total-variation couplings
+of table matches. Sturm's inductive
+means~\cite[Definition~4.6 and Theorem~4.7, proof~(a)]{Sturm2003}
+then turn those couplings into the averaging estimate used in the
+energy argument.
+
+\subsection{Structure of the proof}
+The construction must satisfy two requirements: its relations must be sparse
+enough to preserve hyperbolicity and asphericity, yet impose enough metric
+constraints to exclude geometric actions on proper $\CAT(0)$ spaces.
+We organize the relations using a fixed modification of the incidence graph
+of the projective plane over $\mathbb F_2$.  Four point vertices are doubled.
+At one of them we omit one incidence and keep its counterpart at the other
+representative separate.  The remaining edges over each point form a small
+connected graph, called a fan.  We call the resulting incidence graph
+the core.  Edges receive signed generator labels,
+and every closed walk in a selected labelled graph gives a relation.
+Fans are selected first; a labelled copy of the whole graph is selected only
+when its seven fan restrictions have already been selected.  We study
+copies for which further four-letter relations close every nonbacktracking two-edge core
+path to a square.  An independent collection of six-letter relations will
+control individual generator displacements.  A labelled core copy together
+with one chosen square completion for every nonbacktracking two-edge core path is a
+\emph{match}.
+
+We choose a sequence of such finite presentations, with progressively stronger
+finite counting requirements.  At each stage, a finite list of statistical
+tests is fixed before choosing the sampling parameters; those parameters then
+remain fixed while the alphabet grows.  If infinitely many of their groups admitted
+geometric actions on proper complete $\CAT(0)$ spaces, choose one
+action for each of those groups.
+The closed-walk relations realize the fixed graph by orbit points in every
+chosen action.  After normalizing the average squared generator displacement,
+products of these configurations have a limiting configuration in a complete
+$\CAT(0)$ space.  The counting requirements force exact metric identities in
+that limit, and a finite geometric contradiction excludes it.  Thus every
+sufficiently late group in the chosen sequence admits no such action.  The argument uses the following three ingredients.
+
+\begin{enumerate}
+\item \textbf{A positive deficit in diagram counting.}
+A fan shared by several copies of the whole graph supplies only one random
+table entry, so its probability cost must be counted once.  At the same time,
+a disk or sphere diagram may use the same relation many times.  We reconcile
+these two counts by assigning multiplicity weights to distinct table entries.
+Local incidence estimates at vertices then leave a positive multiple
+of the diagram's area to be paid by its boundary.  This is first proved for
+bounded fragments.  Planar separators extend it to all diagrams, yielding
+both a linear filling inequality and asphericity in
+Section~\ref{sec:diagrams}.
+
+\item \textbf{Sampling that is uniform over later actions.}
+The metric argument compares a bounded function of one or two edge labels
+with a further edge label.  The relevant tuples need not have dense
+projections in the sampled tables.  Instead, we couple several matches
+so that the conditioning labels agree, each match is approximately
+uniform among all matches, and the further labels are approximately independent
+uniform generators.  Section~\ref{sec:sampling} obtains this coupling by
+gluing copies along an enlarged conditioning tuple and proving concentration
+of the numbers of extensions.  The enlargement is chosen by counting
+exponents so that every further collection of label positions has a strictly positive
+margin.  The conclusions concern total variation of finite probability laws.  They therefore apply to bounded tests from any action
+chosen after the tables are revealed.  The common conditioning tuple may still depend on the further labels; the
+averaging estimate allows this dependence.  The six-letter relations supply
+the uniform displacement bound that makes these tests bounded.
+
+\item \textbf{A rigid configuration and incompatible scalar products.}
+Energy minimization and the coupling yield lower bounds for squared orbit
+distances along short paths.  In the limit, the decoration squares force
+squared two-step distances to equal two.  The six-cycles through three noncollinear point vertices then have a
+common center, and their central triangles are flat
+(Section~\ref{sec:energy}).  The doubled points give four unit vectors obtained
+by doubling certain midpoints in the tangent cone at that center.  The line
+incidences force two of these vectors to have nonnegative scalar product.
+A three-field test involving the separated incidence edge forces the other
+two vectors to be antipodal.  Splitting the parallel set of the resulting line exposes a flat
+square in the transverse factor, which makes the first pair antipodal as well.
+Their scalar product is consequently $-1$, the contradiction in
+Section~\ref{sec:obstruction}.
+\end{enumerate}
+
+Section~\ref{sec:model} defines the finite graph and its random tables.
+Section~\ref{sec:completion} makes the finite-test selection precise and
+transfers one resulting finite classifying complex to the simplicial model
+in Theorem~\ref{thm:main}.  Its filling constant may depend on that chosen
+presentation.
+
+\subsection{Conventions}
+A disk diagram may have boundary spurs or edges collapsed to the target's $1$-skeleton; such features have zero area and can be removed. Its area is the number of its noncollapsed $2$-cells. Constants in filling estimates may depend on a fixed finite presentation. A statement holds \emph{with high probability} if its probability tends to one as the alphabet size tends to infinity, with all parameters in that statement fixed. Finite-test limits and the final diagonal selection will always be distinguished.
+
+We use completed tangent cones and completed spaces of directions. The latter are $\CAT(1)$ and the former are $\CAT(0)$~\cite[Theorem~II.3.19]{BH1999}. For the tangent-cone construction in this generality, see also
+Kleiner--Leeb~\cite[Section~2.1.3]{KleinerLeeb1997}, who credit
+Nikolaev's tangent-cone theorem.
+Subsequent products and limits need not be proper; properness is used only for the original geometric actions. Standard comparison, covering-space and topological results are cited with the hypotheses needed here. All estimates specific to the new construction are proved below.
diff --git a/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/sections/02-model.tex b/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/sections/02-model.tex
new file mode 100644
index 0000000000000000000000000000000000000000..1684bfbe68d1ad508eb36bd92a28ebdeeaa5e01c
--- /dev/null
+++ b/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/sections/02-model.tex
@@ -0,0 +1,137 @@
+\section{The incidence template and the nested tables}\label{sec:model}
+We begin with the fixed finite combinatorics. All randomness will be in the labels, not in the template.
+
+\subsection{Points, lines, and fans}
+\begin{definition}[The incidence template]\label{def:template}
+Identify the points of the projective plane over $\F_2$ with the seven nonzero vectors of $\F_2^3$. A line consists of the three nonzero vectors in a two-dimensional subspace. Fix
+\[
+ A=e_1,\qquad B=e_2,\qquad C=e_3,\qquad D=e_1+e_2+e_3.
+\]
+No three of these four points are collinear. For distinct points $i,j$, write $ij$ for their line.
+
+The bipartite graph $\Gamma$ has a vertex $X_i$ for every point, an additional vertex $Y_i$ for $i\in\{A,B,C,D\}$, and a vertex for every line. Both representatives over a point are joined to each incident line, with one exception: the edge $Y_D$--$CD$ is omitted. The edge
+\[
+ e_0=X_D\text{--}CD
+\]
+is called the \emph{loose edge}. Every other edge belongs to the \emph{fan} $F_i$ at its point $i$. At $D$, the fan uses only the two lines other than $CD$, so $CD$ is not a vertex of $F_D$.
+\end{definition}
+
+Each $F_i$ is connected. At $A,B,C$ it is $K_{2,3}$, at $D$ it is $K_{2,2}$, and at each of the other three points it is a three-edge star. Let $g=7$ and $k_i=|E(F_i)|$. Thus
+\begin{equation}\label{eq:model-counts}
+ (k_i)_i=(6,6,6,4,3,3,3),\qquad
+ \sum_i k_i=31,\qquad |E(\Gamma)|=32.
+\end{equation}
+The graph has $18$ vertices. Its cycle rank is $15$; the sum of the
+cycle ranks of the fans is $7$. We will first impose the cycles within
+the fans, then use a smaller graph of cycle rank $8$ to impose the
+remaining core relations. The cone construction below makes this
+decomposition explicit.
+
+Figures~\ref{fig:fano-template} and~\ref{fig:fans} show the incidence
+pattern and the exceptional fan.
+
+\input{figures/fano-template}
+\input{figures/fans}
+
+Choose an orientation for each edge once and for all. A \emph{field} is an edge position in a template or a position in a polygon word. An assignment gives one signed letter to each field. A walk reads that letter in the specified orientation and its inverse in the other orientation. The fields are bookkeeping coordinates; they all take values in the same alphabet.
+
+A \emph{core hexagon} is the six-cycle obtained from three noncollinear points, the three lines joining them, and one representative over each point for which all six incidences are present. In particular, the all-$X$ choice is always permitted. Any three successive edges of a core hexagon contain at most two fields from each fan; the loose field, if present, belongs to no fan.
+
+\FloatBarrier
+\subsection{The independent tables}
+For $n\ge1$, let
+\[
+ S_n=\{a_1,a_1^{-1},\ldots,a_n,a_n^{-1}\},\qquad N=2n.
+\]
+These are formal signed letters; assignments are not required to use distinct letters or to produce freely reduced words. Let
+\begin{equation}\label{eq:model-parameters}
+ f=2-\alpha,\qquad p=3-\beta,\qquad
+ 0<\alpha,\delta<1,\qquad 7\alpha<\beta<1.
+\end{equation}
+Additional smallness requirements will be imposed when we choose finitely many sampling tests.
+
+\begin{definition}[The tables]\label{def:tables}
+Use the following mutually independent Bernoulli arrays.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item For each point $i$, an array indexed by $S_n^{E(F_i)}$ includes every tuple with probability $N^{f-k_i}$. An included tuple, together with its type $i$, is a \emph{fan occurrence} $b$.
+\item An array indexed by $S_n^{E(\Gamma)}$ marks every full core tuple with probability $N^{p-gf-1}$. A \emph{top occurrence} $a$ is a marked tuple whose restriction to every $F_i$ is an included fan occurrence. We write $a\to b$ when $b$ is the fan in one of the seven slots of $a$.
+\item For each unoriented nonbacktracking two-edge path $\tau$ in $\Gamma$, choose one order and direction. Give $\tau$ a separate four-field array: its first two coordinates are the two core fields on $\tau$, and its last two are new coordinates. Include each row with probability $N^{-2-\delta}$. Its relator reads $\tau$, with the required orientation signs, followed by the two new letters around a closing path.
+\item A separate six-field array includes each word in $S_n^6$ with probability $N^{-3-\delta}$. Its word is a relator.
+\end{enumerate}
+All included rows in \textup{(iii)} and \textup{(iv)} are imposed, not only rows matching some top occurrence. We call them \emph{decorations}; their lengths are $2h$, with $h\in\{2,3\}$.
+\end{definition}
+
+There is no ambiguity from opposite path directions: reversing one chosen convention simply inverts or reorders the corresponding coordinates, preserving the Bernoulli law. There are $90$ tables in Definition~\ref{def:tables}\textup{(iii)}. Indeed, the eleven point-representative vertices contribute $10\binom32+\binom22=31$ two-edge paths; the line degrees are $5,5,4,5,5,5,3$, contributing $59$ more.
+
+The exponents in Definition~\ref{def:tables} are negative. In particular,
+\[
+ p-gf-1=-12-(\beta-7\alpha)<0.
+\]
+Each is therefore a valid probability. Distinct numerical tuples of a fixed table are independent. Top \emph{occurrences} are correlated because they may share fan slots; top \emph{marks} remain independent.
+
+The expected counts explain the normalization of these probabilities.
+For each fan type there are $N^{k_i}$ candidate tuples, so the expected
+number of included fans is $N^f$. A fixed core tuple requires its mark
+and seven independent fan entries. Since the fans contain $31$ fields
+in total, the expected number of top occurrences is
+\[
+ N^{32}\,N^{p-7f-1}\prod_i N^{f-k_i}=N^p.
+\]
+Each square table has expected size $N^{2-\delta}$, and the six-letter
+table has expected size $N^{3-\delta}$. Thus the expected exponents
+are just below $2$ for fans and squares, and just below $3$ for tops
+and hexagons. Section~\ref{sec:diagrams} explains how these deficits,
+together with $\beta>7\alpha$, leave a positive margin in the diagram
+estimates. The correlation between top occurrences is the
+reason that their shared fan costs must be counted only once.
+
+\subsection{The group and its two-complex}
+The letter presentation has the generators $a_1,\ldots,a_n$. For each included fan and each top occurrence, impose the labels of all closed walks in its graph. Impose also the decoration words. This is a finite presentation: in each connected graph a fixed spanning tree supplies a finite basis of closed walks that suffices.
+
+For the diagram arguments we use a specific finite two-dimensional CW complex $P$. Start with the letter bouquet, with vertex $o$. For each fan occurrence $b$ of type $i$, attach the cone on $F_i$ along its labelled map to the bouquet. The cone apex is a new vertex, also denoted $b$. For each vertex $v$ of $F_i$, its radial edge $r_{b,v}$ joins $b$ to the image $o$ of $v$. Every fan edge gives one triangle with one letter side and two radial sides. Different radial edges can have the same endpoints; they retain their distinct labels $v$.
+
+The fan cones already fill all fan cycles. To impose the remaining
+relations of a top occurrence $a$, take the ordinary Fano incidence
+graph and subdivide the edge $D$--$CD$ once. Call the resulting graph
+$B$. It has $15$ vertices and $22$ edges, hence cycle rank $8$,
+matching the difference $15-7$ computed above. Map a point vertex $i$ to the apex
+of the fan in slot $i$ of $a$, and map all line vertices and the
+subdivision vertex to $o$. An ordinary incidence at $i$ maps to the
+radial edge indexed by its line. The subdivided exceptional incidence maps to
+\[
+ r_{b_D,X_D}\quad\text{followed by the loose letter edge,}
+\]
+with the orientation giving the path from $X_D$ to $CD$. Attach the cone on this mapped graph, with its own new apex. We call its radial triangles \emph{top triangles}. Finally attach every decoration polygon. All these attachments are finite, and $P$ has dimension two.
+
+\begin{lemma}[Presentation equivalence]\label{lem:model-presentation}
+The complex $P$ has the fundamental group of the letter presentation above. In particular, it has a quotient isomorphic to $C_2$.
+\end{lemma}
+\begin{proof}
+For a connected graph $H$ and a map $H\to X$ into a connected complex, attaching its cone changes $\pi_1X$ by the normal closure of the image of $\pi_1H$. It introduces no extra generator. This follows from van Kampen, or by choosing a spanning tree of $H$ and eliminating all but one of the new radial edges successively. The statement does not require distinct graph vertices to have distinct images.
+
+It remains to check that the top macro graph imposes the required core relations after the fan relations. Each fan triangle replaces a fan edge by the two radial edges from its endpoints to the fan apex. A path in a fan can therefore be replaced, relative its endpoints, by the radial path through that apex. Apply these replacements to a closed core walk. Excursions through $X_i$ or $Y_i$ then cancel, leaving a closed walk in the mapped macro graph. At $D$, the only unreplaced edge is the loose edge; together with $r_{b_D,X_D}$ it gives exactly the exceptional subdivided incidence. Conversely, every macro turn can be replaced by a path in its connected fan. The two sets of relations have the same normal closure modulo the fan relations. This proves the presentation assertion.
+
+The graph $\Gamma$, and each fan, is bipartite, so every closed-walk word has even length. The decorations have lengths four and six. Sending each original generator to the nonidentity element of $C_2$ respects all the defining relations and is surjective.
+\end{proof}
+
+\begin{remark}\label{rem:model-finite}
+The cone description records occurrences, not just relator words. If the same fan occurs in several tops, its cone is attached only once. A top occurrence has its own cone, and a decoration occurrence has its own disk. These distinctions are essential in the multiplicity count of Section~\ref{sec:diagrams}.
+\end{remark}
+
+\subsection{The master pattern and admissible tests}
+Define the finite \emph{master pattern} $\mathcal M$ by taking one top occurrence with its seven fan slots and, for each of the $90$ two-edge paths, one matching row of its square table. The two core coordinates of that row are identified with their core fields; its two closing coordinates are new. The separate six-letter array is not part of $\mathcal M$.
+
+A \emph{match} is an assignment to the fields of $\mathcal M$ for which every required table entry is present, including the top mark. We denote the set of matches by $\Omega_n$. When it is nonempty, it carries its uniform probability measure. Its $32+2\cdot90$ fields include the shared core fields only once.
+
+\begin{definition}[Admissible fields]\label{def:admissible}
+A set of fields is \emph{admissible} if it is either
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item a set of at most three distinct core fields containing at most two fields of each fan, or
+\item a set of at most two fields of a single square decoration in $\mathcal M$.
+\end{enumerate}
+The loose core field does not count against any fan quota.
+\end{definition}
+
+The first case includes all three-edge half-paths of core hexagons. It also includes two $F_C$ fields together with the loose field, the combination that will produce the decisive midpoint identity. The second case includes every two-edge half-path in a decoration square.
+
+Every match has a labelled realization of the core graph and the closing paths of its square decorations in the group: fix an anchor, and label a vertex by the word read along any path from the anchor. All closed-path labels are trivial by Lemma~\ref{lem:model-presentation} and the square relators. Consequently this group-valued vertex assignment is independent of the chosen path. For any action and basepoint, it gives a corresponding configuration of orbit points.
diff --git a/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/sections/03-diagrams.tex b/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/sections/03-diagrams.tex
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+\section{Diagrams and linear filling}\label{sec:diagrams}
+
+This section proves the combinatorial properties of the random complex
+$P$ of Section~\ref{sec:model}.  An \emph{occurrence} means an included
+tuple, together with its table type; copies of a cell in a diagram need
+not be distinct occurrences.  All constants in this section may depend
+on the fixed incidence template and on the fixed parameters, but not on
+$n$.
+
+\begin{theorem}\label{thm:diagram}
+Fix sufficiently small positive $\alpha,\beta,\delta$ with
+$\beta>7\alpha$, and put $f=2-\alpha$ and $p=3-\beta$.
+As $n\longrightarrow\infty$, the probability tends to one that $P$ is
+aspherical and its letter presentation satisfies a linear disk
+isoperimetric inequality.
+\end{theorem}
+
+The central estimate will bound the number of faces in a bounded
+piece of a normalized diagram by the number of its exposed sides. The expected
+occurrence counts in Section~\ref{sec:model} have exponents
+$2-\alpha$, $3-\beta$, and $h-\delta$ for fans, tops, and decorations
+of length $2h$. The local incidence geometry compares these with the
+values $2$, $3$, and $h$. After shared fan costs have been accounted
+for, the differences $\alpha$, $\beta-7\alpha$, and $\delta$ supply the
+strictly positive margin in the boundary estimate.
+
+We first obtain a probability bound for every bounded collection of
+distinct occurrences. We then normalize disk and sphere pictures so
+that this bound also controls repeated copies of their cells. The
+resulting estimate applies to arbitrary sets of faces, which lets us
+partition a large picture by planar separators and sum the bounds on
+its pieces. The same estimate proves linear filling for disks and
+rules out nonzero two-cycles in the universal cover.
+
+We retain every fan slot of a top occurrence when making a collection of
+occurrences, even if a diagram uses only some of those slots.  This
+convention is essential to the counting argument.
+
+\subsection{Coincidence ranks}
+
+Counting constraints while retaining repeated relator occurrences is
+standard in random-group diagram arguments; see
+Ollivier~\cite[Section~2]{Ollivier2004} and
+Antoniuk--Friedgut--\L uczak~\cite[Section~3]{AntoniukFriedgutLuczak2017}.
+We count the shared fan entries explicitly in the estimate below.
+
+For a finite collection of top, fan, and decoration occurrences, its
+\emph{field positions} are the following: the loose field of each top,
+the $k_i$ fields of each fan occurrence of type $i$, and all fields of
+each decoration occurrence.  A fan field is included only once when the
+same fan occurrence occupies slots in several tops.  The
+\emph{unsigned coincidence rank} $R$ is the number of positions minus
+the number of distinct underlying generators represented at those
+positions.  Thus a letter and its inverse have the same unsigned value.
+
+\begin{lemma}\label{diagram:rank}
+For each fixed integer $M$, with probability tending to one the
+following holds simultaneously for every collection of at most $M$
+distinct occurrences that contains all fan slots of its tops.  If the
+collection has $s$ tops and $u$ fans, then
+\begin{equation}\label{diagram:rank-bound}
+ R\le (p-7f)s+fu+\sum_e(h_e-\delta),
+\end{equation}
+where the sum is over its decorations and $2h_e\in\{4,6\}$ is the
+length of decoration $e$.
+\end{lemma}
+
+\begin{proof}
+Fix the table types, the incidence of the fan slots, and a partition of
+the field positions into unsigned-value classes.  There are only
+finitely many such choices for fixed $M$.  Write $J$ for the number of
+positions.  Assigning letters to a pattern of rank $R$ has at most
+$C_M N^{J-R}$ possibilities: first choose one underlying generator per
+class, then choose the signs of the positions.  The latter contributes
+only a factor bounded in terms of $M$.
+
+We count only choices whose occurrences are distinct within their
+respective tables.  For such choices all the required Bernoulli trials
+are independent.  Their product contributes the power of $N$ with
+exponent
+\[
+ (p-7f-1)s+\sum_b(f-k_b)
+       +\sum_e(h_e-\delta-2h_e).
+\]
+Here a shared fan occurrence appears once in the second sum, not once
+per top using it.  Since
+$J=s+\sum_b k_b+\sum_e2h_e$, the expected number of realizations of the
+chosen pattern is at most
+\[
+ C_M N^{(p-7f)s+fu+\sum_e(h_e-\delta)-R}.
+\]
+If Equation~\eqref{diagram:rank-bound} fails, this exponent is negative.
+There are finitely many exponents to consider for the fixed size bound,
+so summing these expectations proves the assertion.
+\end{proof}
+
+\subsection{Spherical pictures and their reductions}
+
+Lemma~\ref{diagram:rank} counts distinct occurrences, whereas a filling
+diagram may use the same occurrence many times. We normalize the
+diagram in two ways: the triangles incident to each top apex will be
+grouped into polygons tracing simple macro cycles, and the cyclic
+link at a fan apex will have no repeated radial label.
+These properties will bound how often a field can appear near each
+apex. For the asphericity argument, the normalization must also
+preserve a prescribed lifted two-cycle.
+
+An \emph{elementary picture} is a finite collection of oriented polygons with
+orientation-reversing pairings of their sides, such that the resulting
+closed oriented surfaces are spheres. Vertex links are taken to be
+circles, with a separate source vertex for each link circle.
+An interior face maps to a fan triangle, a triangle of a top cone, or a
+decoration disk of $P$, with either orientation.  In the disk problem
+there is one distinguished outside polygon whose boundary reads the
+prescribed letter word; no extension of its interior to $P$ is required.
+In the sphere problem there is no outside polygon.  Collapsed regions
+mapping to vertices and free cancellations of edge paths have zero
+face cost.  They can equivalently be recorded by thickening the paired
+sides to strips and capping the boundary circles of the resulting
+vertex regions.
+
+Every cell has a specified map to $P$, including the index of each of
+its sides.  Equal edge labels alone do not identify two side indices.
+For a picture mapped to $\widetilde P$, its \emph{cellular chain} is the
+sum of its signed lifted two-cells; the outside polygon is omitted.
+
+\begin{lemma}\label{diagram:existence}
+Every null letter word has a disk picture.  If
+$c\ne0$ is a cellular two-cycle of $\widetilde P$, there is a finite
+collection of spherical pictures with total cellular chain exactly
+$c$.
+\end{lemma}
+
+\begin{proof}
+The disk assertion is the cellular van Kampen construction, applied to
+the finite cone and polygon complex.  Here is also a description that
+applies to the sphere assertion.  Since $\widetilde P$ is simply
+connected, the Hurewicz map
+$\pi_2(\widetilde P)\longrightarrow H_2(\widetilde P;\mathbb Z)$ is an
+isomorphism~\cite[Theorem~4.32]{Hatcher2002}.  There are no three-cells.
+Consequently a sphere map
+representing the class of $c$ has cellular two-chain equal to $c$, not
+merely congruent to it modulo boundaries of three-cells.
+
+Put this map in cellular general position.  One way to do so is to
+choose an interior point in each two-cell, make its inverse image a
+finite set of regular points, and take small disks about those
+preimages.  Each such disk maps, with a sign, across that two-cell.
+Retracting each punctured target cell to its boundary moves the
+complement of these disks into the one-skeleton.  Inverse images of
+interior edge points now give strips pairing the corresponding cell
+sides; the remaining regions map to vertices.  Circular strips not
+meeting cell sides may be cut and capped.  A vertex region with several
+boundary components may likewise be replaced by a separate cap for
+each component.  These operations are compressions along curves in an
+oriented sphere.  They therefore produce a collection of oriented
+spheres, not surfaces of positive genus.  They change none of the
+signed cell disks or their lifts.  Components with no cell disks may
+be discarded.  This gives the claimed spherical pictures and preserves
+the chain $c$ exactly.
+
+For a disk map, adjoin the distinguished outside polygon before the
+same construction.  Keep that polygon as a single face throughout.
+The construction preserves its word and gives one picture containing
+it, possibly together with sphere components.  All the required
+pictures are finite.
+\end{proof}
+
+At a vertex mapping to the apex of a top cone, every incident face is
+a triangle of that cone.  Grouping its incident triangles gives a
+polygon whose boundary is a closed walk in the macro graph of that top
+occurrence.  We call it a \emph{top polygon}; its filling in $P$ is
+always the indicated radial cone filling.  Its perimeter is the number
+of these cone triangles.  The other interior faces remain fan triangles
+and decoration disks.
+
+The macro graph is the Fano incidence graph with one subdivided edge.
+It has fifteen vertices.  A simple macro cycle has at most fifteen
+sides, visits each point at most once, and visits at least three
+points.  At a point it turns between two distinct \emph{ports}, namely
+the radial labels of the fan there.  The ports are the incident line
+vertices, except that the exceptional port at $D$ is $X_D$.
+
+\begin{lemma}\label{diagram:normal-form}
+A null letter word has a picture of minimum total top perimeter and,
+subject to this, minimum number of remaining interior faces, with the
+following properties.  The same assertion holds for a fixed nonzero
+cellular two-cycle $c$, minimizing among finite collections of sphere
+pictures whose total lifted cellular chain is exactly $c$.
+\begin{enumerate}
+\item Every top polygon is a simple macro cycle.
+\item Adjacent top polygons do not use the same side of the same
+top occurrence.  Two copies of one fan triangle cannot be adjacent
+along equally indexed sides, and two copies of one decoration cannot
+be adjacent along the same position of that decoration.
+\item At each fan-apex instance, the cyclic list of radial labels has
+no repetition.  A circuit with two distinct parallel steps is
+permitted.  Each such circuit has at most five steps.  If it contains
+no top turn, it has at least four fan triangles.
+\end{enumerate}
+The modifications establishing these properties preserve the outside
+word and, in the sphere problem, the exact lifted cellular chain.
+\end{lemma}
+
+\begin{proof}
+The nonempty class of finite pictures in
+Lemma~\ref{diagram:existence} has a lexicographic minimum for the stated
+pair of nonnegative integer costs.  In the sphere case fix the chain
+$c$ once and for all in taking this minimum.
+
+\emph{Simple macro cycles.}  Consider a top polygon whose macro walk repeats a template
+vertex.  Its two boundary points at that repetition have the same
+lifted value: both are joined to the same lifted top apex by the same
+indexed cone spoke.  Split the polygon by a chord collapsed to that
+vertex.  The two resulting macro walks have the same total perimeter
+and the same total cone chain.  A free backtrack cancels two oppositely
+oriented copies of a single lifted cone triangle and reduces total
+perimeter by two.  Delete any resulting empty polygons.  Repeating
+splitting and removal of backtracks therefore gives simple macro
+cycles.  Collapsed chords may be put in
+the picture form of Lemma~\ref{diagram:existence}; they carry no
+two-chain.  This proves the first property.
+
+\emph{Cancellation.}  Suppose two top polygons meet at the same
+macro side of one occurrence.  The two elementary cone triangles
+incident to that side are copies of the same triangle with opposite
+orientations.  Their lifts agree, because the common lifted macro side
+determines the lifted triangle and its apex.  Cancel the pair and
+merge the remaining cone walks.  This reduces total top perimeter by
+two.  It is an ordinary diagram cancellation: one uses the two face
+interiors and a thin strip across the specified common side, so other
+contacts between their boundaries cause no obstruction.  Any vertex
+regions produced by the cancellation are capped as above.  A
+same-template contact cannot occur between two sides of a single
+simple top polygon, because a simple macro cycle uses each edge at
+most once.  The identical cancellation of two fan triangles or two
+decoration disks along equally indexed sides reduces the second cost
+without increasing the first.  These cancellations preserve the
+cellular chain, so they are forbidden by minimality in both problems.
+
+\emph{Fan-apex links.}  Fix a source vertex $v$
+mapping to a fan apex $b$.  A step in its link is either a fan triangle
+or a corner of a top polygon.  Suppose two different outgoing spokes
+at $v$ have the same radial label.  Each spoke is an edge from $v$ to a
+vertex mapping to the letter-bouquet vertex.  Cut the side pairings of
+these two edges and pair them crosswise, with the outgoing orientations
+at $v$ agreeing.  This preserves the orientation of every face and all
+face maps.  In terms of vertex-link cycles, it splits the link cycle of
+$v$ into two.  At the far ends it either merges two distinct vertex
+cycles or splits one vertex cycle.
+
+For completeness, this switch preserves the genus-zero condition.
+The numbers of faces and edges do not change.  When the two far
+vertices were distinct, the total vertex count does not change, and
+the affected surface remains connected: both switched edges are
+incident to the merged far vertex.  When the far vertices were the
+same, the total vertex count increases by two. There can be at most
+two affected components after either switch. To see this, cut the two
+corresponding edges of the connected dual graph before re-pairing
+their four exposed half-edges. Every component of the cut graph
+contains an exposed half-edge: otherwise it was already disconnected
+before the cut. After re-pairing, every component therefore contains
+one of the two switched edges. This argument includes loops and
+parallel edges in the dual graph.
+In the first case the connected oriented surface consequently has
+Euler characteristic two.  In the second case the total Euler
+characteristic is four, which forces exactly two oriented spheres.
+All unaffected spheres are retained, and the outside polygon remains
+one unchanged face.
+
+The switch is also valid for the fixed lifted chain.  The repeated
+radial label and the lifted initial fan vertex determine the same
+lifted radial edge and far endpoint.  Thus the new vertex
+identifications agree in $\widetilde P$, and every lifted face is
+unchanged.  Each switch increases the number of fan-apex instances
+while keeping the finite set of polygon corners fixed.  Repetition
+therefore terminates.  The two minimized costs are unchanged, so a
+newly created forbidden adjacency would still contradict their
+minimality by the cancellation just described.
+
+The resulting link uses each radial label at most once.  There are at
+most five labels in any fan.  A two-step link made only of fan
+triangles is the reverse traversal of one edge of the simple fan graph;
+the two triangles would cancel along an equally indexed radial side.
+Every other nonempty triangle-only link is a circuit in the simple
+bipartite fan graph and has length at least four.  This proves the last
+property.
+\end{proof}
+
+The outside word uses only letter edges, so an outside polygon never
+meets a fan-apex instance.  In particular each full link considered in
+Lemma~\ref{diagram:normal-form} is a cyclic link of interior faces.
+
+\subsection{Weights on a bounded fragment}
+
+Fix a normal picture.  A \emph{fragment} is any selection of its
+interior faces; the selection need not be connected or simply
+connected.  Keep a pairing of sides precisely when both incident faces
+are selected.  All other selected sides are fragment boundary sides.
+Let $T,Q,E$ be the numbers of selected top polygons, fan triangles, and
+decoration disks, respectively, and put $A=T+Q+E$.
+
+For a top occurrence $a$ let $t_a$ be its number of selected top
+polygons, and for a decoration occurrence $e$ let $t_e$ be its selected
+multiplicity.  At a fan-apex instance $z$ represented in the fragment,
+let $d_z$ count selected top turns and $q_z$ count selected fan
+triangles.  It is \emph{partial} if not every step of its full cyclic
+link is selected.
+
+For a fan occurrence $b$, two multiplicities must be distinguished.
+The number of represented apex instances bounds the actual use of
+each fan field. We must also retain this fan whenever we retain a top
+of sufficiently large multiplicity that requires it. The fan weight
+must therefore dominate the largest such top multiplicity. Write
+\[
+ r_b=\#\{z:z\text{ is represented over }b\},\qquad
+ m_b=\max_{a\to b}t_a,\qquad
+ S_b=\sum_{a\to b}t_a,\qquad
+ w_b=\max\{r_b,m_b\}.
+\]
+Empty maxima are zero.  All fan slots of every selected top are
+included in these definitions, whether or not a polygon of that top
+visits the fan.  In particular
+\begin{equation}\label{diagram:slot-sum}
+ \sum_b S_b=7T.
+\end{equation}
+Write $J_b$ for the number of partial instances over $b$, and let
+$P_b^*$ count the selected loose top sides belonging to turns at this
+fan.  This number is zero unless $b$ has type $D$.  Each such loose side
+records one incidence with the exceptional radial port $X_D$.
+
+Let $I$ count internal contacts between two letter sides of the
+fragment.  Radial side contacts are not counted in $I$.  Denote the
+numbers of letter and radial fragment boundary sides by $B_{\rm raw}$
+and $B_{\rm rad}$, and put $B=B_{\rm raw}+B_{\rm rad}$.
+
+We first use the coincidence bound to control internal letter contacts.
+The remaining estimate is local at fan apices: full links contribute
+no error, while partial links contribute an error paid by radial
+boundary sides.  Combining the two estimates will bound the fragment's
+area by its total boundary length.
+
+\begin{lemma}\label{diagram:weighted-rank}
+Suppose Equation~\eqref{diagram:rank-bound} holds for collections of at
+most $8A$ occurrences.  Then
+\begin{equation}\label{diagram:contact-bound}
+ I\le pT+f\sum_b(w_b-S_b)
+              +\sum_e(h_e-\delta)t_e.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Give each fan field of $b$ weight $w_b$, each loose field of $a$ weight
+$t_a$, and each field of a decoration $e$ weight $t_e$.  These are
+upper bounds for their side multiplicities in the fragment.  Indeed a
+fan field is used at most once at a given normalized fan-apex link, so
+its multiplicity is at most $r_b$.  A simple top polygon uses its loose
+field at most once, and a copy of a decoration uses each indexed field
+once.
+
+For each integer $j\ge1$, apply
+Equation~\eqref{diagram:rank-bound} to the positions of weight at least
+$j$.  This is an eligible collection: $t_a\ge j$ implies
+$w_b\ge m_b\ge t_a\ge j$ for every slot $a\to b$.  Its size is at most
+$8A$, since there are at most $T$ distinct tops, at most $7T+Q$ fans,
+and at most $E$ decorations.  Sum the resulting inequalities over $j$.
+Within an unsigned-value class, the sum of its ranks is its total
+weight minus its largest position weight.  Thus, writing
+$\mathcal C$ for the unsigned-value classes, we obtain
+\begin{equation}\label{diagram:weighted-coincidence}
+ \sum_{C\in\mathcal C}
+ \left(\sum_{v\in C}\operatorname{wt}(v)
+                  -\max_{v\in C}\operatorname{wt}(v)\right)
+ \le (p-7f)T+f\sum_b w_b+\sum_e(h_e-\delta)t_e.
+\end{equation}
+
+Every internal letter contact joins two different field positions.
+For fan triangles and decorations this follows from the cancellations
+in Lemma~\ref{diagram:normal-form}; for two loose top sides it follows
+from the same-occurrence top cancellation.  A contact between different
+kinds of cells automatically has different positions.  In each class
+$C$, choose one maximum-weight position.  Charge each internal contact
+to one of its endpoints other than that position.  Such an endpoint
+always exists.  Different contacts use different actual side copies,
+so the charge to any position is at most its side multiplicity and
+hence at most its weight.  This proves that $I$ is bounded by the left
+side of Equation~\eqref{diagram:weighted-coincidence}.  Using
+Equation~\eqref{diagram:slot-sum} on its right side gives
+Equation~\eqref{diagram:contact-bound}.
+\end{proof}
+
+This argument allows arbitrary cycles and repetitions in the contact
+graph.  It charges side copies, and does not require the contacts to
+form a forest.
+
+\begin{lemma}\label{diagram:apex}
+For every fan occurrence $b$ in a fragment,
+\begin{equation}\label{diagram:apex-bound}
+ \sum_{z\mapsto b}d_z+2(w_b-S_b)
+ \le \frac12\left(\sum_{z\mapsto b}q_z+P_b^*\right)+2J_b.
+\end{equation}
+Moreover
+\begin{equation}\label{diagram:partial-bound}
+ 2\sum_bJ_b\le B_{\rm rad}.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+First suppose $w_b=r_b$.  A simple top polygon has at most one turn at
+this fan, so $S_b\ge\sum_{z\mapsto b}d_z$.  Consequently the left side
+of Equation~\eqref{diagram:apex-bound} is at most
+$\sum_{z\mapsto b}(2-d_z)$.
+For a full link with $d_z=0$, at least four fan triangles give the
+required inequality.  For a full link with $d_z=1$, the triangles form
+a path in the fan between the two distinct ports of that top turn.
+Ordinary ports are distinct line vertices, so this path has at least
+two edges.  The only exception is a path incident to the port $X_D$;
+it may have one edge, in which case its top turn has the loose side
+counted by $P_b^*$.  Thus in either case its triangle count plus its
+loose-side payment is at least two.  For $d_z\ge2$ the expression
+$2-d_z$ is nonpositive.  At a represented partial link with $d_z=0$
+there is at least one selected fan triangle, so its deficit is at most
+$3/2$.  With $d_z=1$ its deficit is at most one, and with $d_z\ge2$
+it is nonpositive.  Thus the error $2J_b$ suffices in this case.
+
+Now suppose $w_b=m_b>r_b$, and choose a top occurrence $a_0$ with
+$t_{a_0}=m_b$.  Call its turns dominant.  Let $x$ and $y$ be the total
+numbers of selected dominant and nondominant turns over $b$.
+Nondominant selected faces have at most one turn each at this fan,
+including when other selected faces omit the fan entirely; hence
+$y\le S_b-m_b$.  It follows that
+\begin{equation}\label{diagram:dominant-algebra}
+ \sum_{z\mapsto b}d_z+2(m_b-S_b)
+       =x+y-2(S_b-m_b)\le x-y.
+\end{equation}
+
+Consider a full link.  In its cyclic list of turns the number of
+consecutive dominant-to-dominant interfaces is at least the number of
+dominant turns minus the number of other turns.  To see this, assign to
+each dominant turn its following turn: at most as many dominant turns
+can be followed by a nondominant one as there are nondominant turns.
+A single turn is followed by itself.  Between a counted pair of turns
+there is a path of fan triangles.  This path cannot have length zero,
+because then the two top polygons use the same radial side of the
+same occurrence $a_0$, forbidden by
+Lemma~\ref{diagram:normal-form}.  If the path has length one, it must
+have an exceptional port, since the other ports lie in the line part
+of the bipartite fan graph.  That port has its loose-side payment.
+Thus the path length plus the available payments is at least two for
+each counted interface.  The triangle paths are disjoint portions of
+the link.  A port incidence is an endpoint of just one such intervening
+path, so no loose-side payment is used twice.  The triangle counts and
+payments at full links therefore bound twice their total
+dominant-minus-nondominant count from below.
+
+At a partial link, the selected steps form a nonempty proper subset
+of a cyclic list of at most five steps.  Split this subset into maximal
+consecutive selected runs.  Each run and each intervening unselected
+gap contains a step, so there are at most two runs.  In a run containing
+turns, let $k$ and $l$ be the numbers of dominant and nondominant turns.
+Their linear turn list has at most $l+1$ blocks of dominant turns.
+Consequently, if $H$ counts its consecutive dominant-to-dominant
+interfaces, then
+\[
+ H\ge k-l-1,\qquad k-l\le H+1.
+\]
+The fan-triangle path at each of these interfaces is entirely selected.
+The same indexed-side cancellation and exceptional-port argument used
+for a full link shows that its length plus its available loose-side
+payments is at least two.  Different interfaces, including those in
+different runs, use disjoint triangle paths and distinct port incidences,
+so no payment is used twice.  A run without turns contributes zero to
+the dominant-minus-nondominant count.  Summing over the at most two runs
+therefore bounds that count by half the selected triangle count and
+loose-side payments, plus two.
+
+Choose the dominant occurrence $a_0$ once for the entire fan occurrence
+$b$, and apply these estimates separately at all its apex instances.
+They do not require the fragment to be connected, or every selected
+polygon of $a_0$ to visit this fan.  Summing the partial- and full-link
+estimates, and then using
+Equation~\eqref{diagram:dominant-algebra}, proves
+Equation~\eqref{diagram:apex-bound} also in the second case.
+
+Finally, a represented partial cyclic link has at least two
+transitions between selected and unselected steps.  Each transition
+is a radial edge with exactly one selected incident face, and hence
+is a radial fragment boundary side.  A radial side belongs to only one
+fan-apex instance.  This proves
+Equation~\eqref{diagram:partial-bound}.
+\end{proof}
+
+\begin{lemma}\label{diagram:fragment}
+Put
+\begin{equation}\label{diagram:kappa}
+ \kappa=\min\{\beta-7\alpha,\alpha/5,\delta\}>0.
+\end{equation}
+If Equation~\eqref{diagram:rank-bound} holds for collections of at most
+$8A$ occurrences, then every fragment of area $A$ satisfies
+\begin{equation}\label{diagram:fragment-bound}
+ A\le C_0 B,\qquad C_0=\frac{1}{\kappa}.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Write $W=\sum_b w_b$ and $P^*=\sum_bP_b^*$.  Every simple top polygon
+has at least three turns.  Summing
+Equation~\eqref{diagram:apex-bound}, and using
+Equation~\eqref{diagram:slot-sum}, gives
+\begin{equation}\label{diagram:nominal-apex}
+ 3T+2(W-7T)\le \frac{Q+P^*}{2}+2\sum_bJ_b.
+\end{equation}
+The right side of Equation~\eqref{diagram:contact-bound} can be written
+as
+\[
+ 3T+2(W-7T)+\sum_eh_et_e-\Delta,
+ \qquad
+ \Delta=(\beta-7\alpha)T+\alpha W+\delta E.
+\]
+There are at most five selected triangles at a represented fan-apex
+instance, so
+$Q\le5\sum_b r_b\le5W$.  Therefore
+\begin{equation}\label{diagram:defect}
+ \Delta\ge\kappa(T+Q+E)=\kappa A.
+\end{equation}
+The total number of raw side copies in the fragment is
+\[
+ R_{\rm raw}=Q+P^*+\sum_e2h_et_e.
+\]
+Equations~\eqref{diagram:contact-bound} and
+\eqref{diagram:nominal-apex} imply
+\[
+ I\le \frac{R_{\rm raw}}2-\Delta+2\sum_bJ_b.
+\]
+Since $B_{\rm raw}=R_{\rm raw}-2I$,
+Equations~\eqref{diagram:partial-bound} and
+\eqref{diagram:defect} give
+\[
+ 2\kappa A\le 2\Delta
+       \le B_{\rm raw}+4\sum_bJ_b
+       \le B_{\rm raw}+2B_{\rm rad}\le2B.
+\]
+This is Equation~\eqref{diagram:fragment-bound}.
+\end{proof}
+
+\subsection{From bounded fragments to all pictures}
+
+The probability estimate controls only bounded collections of
+occurrences. To use it for an arbitrarily large picture, we partition
+the faces into bounded sets while cutting few contacts. A fragment
+was allowed to be disconnected and to have holes, so any partition
+of the dual graph gives eligible fragments. The following separator
+estimate makes their total new boundary small enough to absorb into
+the area bound.
+
+\begin{lemma}\label{diagram:separator}
+For every $D<\infty$ and $\eta>0$ there is an integer $K=K(D,\eta)$
+such that the vertices of any finite planar multigraph of maximum
+degree at most $D$ can be partitioned into sets of size at most $K$,
+with at most $\eta$ times the total number of vertices edges crossing
+between sets.
+\end{lemma}
+
+\begin{proof}
+Ignore loops and apply the planar separator theorem to the underlying
+simple graph.  In a graph with $q$ vertices it gives a set of at most
+$c\sqrt q$ vertices whose removal leaves two sets, with no edges
+between them, each of size at most $2q/3$; one may take
+$c=2\sqrt2$ \cite[Corollary~2]{LiptonTarjan1979}.  Repeat inside every set larger
+than $K$, and make every removed separator vertex a singleton part.
+
+Charge $c/\sqrt q$ to every vertex present in a recursive set of size
+$q$.  These charges pay for all removed separator vertices.  For any
+fixed original vertex, the successive set sizes containing it decrease
+by a factor at most $2/3$ while they exceed $K$.  Reading that sequence
+backward shows that its total charge is at most
+\[
+ \frac{c}{\sqrt K}\sum_{j\ge0}(\sqrt{2/3})^j
+       =\frac{c}{(1-\sqrt{2/3})\sqrt K}.
+\]
+If the original graph has $A$ vertices, the number of removed
+separator vertices is at most $C A/\sqrt K$, where
+$C=c/(1-\sqrt{2/3})$.  Every edge crossing between final parts has a
+removed separator vertex as an endpoint.  The degree bound therefore
+limits their number, counting multiplicity, to $DC A/\sqrt K$.
+Choose $K$ so large that $DC/\sqrt K\le\eta$.  Disconnected graphs
+are treated component by component.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{thm:diagram}]
+Fix the parameters, so that $\kappa$ and $C_0$ in
+Lemma~\ref{diagram:fragment} are positive constants.  Choose
+$\eta>0$ with $2C_0\eta\le1/2$, and apply
+Lemma~\ref{diagram:separator} with $D=15$.  This gives a fixed finite
+threshold $K$.  By Lemma~\ref{diagram:rank}, with probability tending
+to one Equation~\eqref{diagram:rank-bound} holds for all collections of
+at most $8K$ occurrences.  Work with any realization having that
+property.  The remainder of the proof is deterministic and applies
+simultaneously to every picture for this realization.
+
+For a normal picture, the dual graph of its interior faces and their
+contacts is planar of degree at most fifteen.  Fan triangles have
+three sides, decorations have at most six, and simple top polygons
+have at most fifteen.  Parallel contacts and self-contacts cause no
+problem in Lemma~\ref{diagram:separator}.  Partition its $A$ interior
+faces into fragments of size at most $K$, cutting at most $\eta A$
+contacts.  Let $B_0$ count contacts with the outside polygon; in the
+sphere problem set $B_0=0$.  The sum of fragment boundary lengths is
+at most $B_0+2\eta A$.  Lemma~\ref{diagram:fragment} applies to every
+fragment, and summing it gives
+\[
+ A\le C_0(B_0+2\eta A),
+ \qquad\text{hence}\qquad A\le2C_0 B_0.
+\]
+In a disk picture whose outside word has length $\ell$, one has
+$B_0\le\ell$; outside-to-outside pairings represent free boundary
+cancellations and do not add an interior contact.  Thus the normal
+picture uses at most $2C_0\ell$ interior faces.
+
+If a nonzero cellular two-cycle $c$ existed in $\widetilde P$, use its
+normal spherical picture supplied by
+Lemmas~\ref{diagram:existence} and \ref{diagram:normal-form}.
+For it $B_0=0$, forcing $A=0$.  Its cellular chain would then be zero,
+contrary to its construction.  Therefore
+$H_2(\widetilde P;\mathbb Z)=0$.  The universal cover is a simply
+connected two-dimensional CW complex, so all its reduced homology
+groups vanish.  If it had a first nonzero homotopy group, the
+Hurewicz theorem would identify it with a nonzero homology
+group~\cite[Theorem~4.32]{Hatcher2002}.
+It is consequently weakly contractible, and the Whitehead theorem
+makes it contractible~\cite[Theorem~4.5]{Hatcher2002}.
+This proves asphericity.
+
+We finish by making explicit the claimed filling inequality over
+letters.  Choose a vertex of every fan template and, for each of its
+vertices, a fixed path from the chosen vertex to that vertex.  These
+paths have length at most two.  Send every fan-apex vertex of a normal
+picture to the bouquet vertex, and replace its indexed radial edge by
+the label word of the corresponding fixed fan path, with the
+appropriate orientation.  A fan triangle boundary becomes a closed
+walk in that fan of length at most five.  A top polygon boundary
+becomes a closed walk in its copy of $\Gamma$: at an ordinary point,
+its two radial paths concatenate to a fan path between the two ports;
+at the exceptional point the path to $X_D$ is followed by the loose
+edge.  Its expanded length is at most thirty.  A decoration boundary
+is unchanged.
+
+Use as a finite letter presentation all simple cycle relators in each
+included fan and top graph, together with the decoration rows.  This
+presents the same group by the cone construction in
+Section~\ref{sec:model}.  Any closed graph walk of length at most
+thirty splits at repeated vertices into at most thirty simple cycles,
+after free backtracks are erased.  Each expanded face above thus has
+a filling with at most thirty relator disks in this presentation.
+Replacing the faces of the normal picture by these fillings leaves
+the outside letter word unchanged and produces a disk diagram, with
+free folds allowed, of area at most
+$60C_0\ell$.  Sphere components can be discarded.  Any other fixed
+finite choice of defining cycle relators gives the same linear
+inequality after bounded relator substitutions.  This proves the
+theorem.
+\end{proof}
+
+\begin{remark}\label{diagram:quantifiers}
+The separator threshold is chosen after the positive parameters are
+fixed and before $n$ tends to infinity.  No uniform bound as
+$\alpha,\beta,\delta\longrightarrow0$ is asserted or required.
+For each fixed parameter choice, the single finite-collection event in
+Lemma~\ref{diagram:rank} controls all disk words and all cellular
+two-cycles simultaneously.
+\end{remark}
diff --git a/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/sections/04-sampling.tex b/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/sections/04-sampling.tex
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@@ -0,0 +1,533 @@
+\section{Conditional sampling from the nested tables}\label{sec:sampling}
+
+The relations will be tested against an action chosen after all the tables
+have been sampled.  We therefore prove statements about total variation of
+finite distributions on table matches.  These statements do not refer to a
+target metric space.  In particular, they will apply simultaneously to all
+uniformly bounded functions on the relevant assignments, including functions obtained
+from a subsequently chosen action.
+
+\subsection{The master pattern and its critical exponents}
+
+Recall the master pattern $\mathcal M$ from Section~\ref{sec:model},
+and let $\mathcal F$ be its field set.  Its required entries are the
+seven fan entries, the top-mark entry, and one entry from each of the
+$90$ four-letter tables.  In each four-letter entry its first two fields are
+identified with the corresponding core fields; its other two fields are
+new and are used in no other entry.  The six-letter table is not part of
+$\mathcal M$.  Thus $|\mathcal F|=212$.  Write $\mathcal R$ for the set of
+its $98$ array entries and $E_\rho\subseteq\mathcal F$ for the field
+support of $\rho\in\mathcal R$.  Each $\rho$ belongs to a different
+Bernoulli array.  The order of its coordinates is fixed throughout.
+
+A \emph{match} is a map $\sigma:\mathcal F\to S_n$ for which every
+required array entry is present.  No injectivity is required of a single
+match.  We write $\Omega_n$ for the random set of matches and, when it is
+nonempty, write $\mu_N$ for its uniform probability measure.
+
+Recall from Definition~\ref{def:admissible} that a set of fields is
+admissible if it consists either of at most three core fields, at most
+two in each fan, or of at most two fields of one square decoration.
+The loose field belongs to no fan.
+\begin{definition}\label{sample:def-test}
+A \emph{conditional test} is a triple $(I,e,t)$ with
+$I\subseteq\mathcal F$, $e\in\mathcal F\setminus I$,
+$I\cup\{e\}$ admissible, and $t$ a positive integer.
+\end{definition}
+
+For $\theta=(\alpha,\beta,\delta)$ define $c_\rho(\theta)$ so that
+the inclusion probability of entry $\rho$ is $N^{-c_\rho(\theta)}$.
+Its value and its critical value $c_\rho=c_\rho(0)$ are
+\begin{equation}\label{sample:eq-costs}
+\begin{array}{c|c|c}
+\text{entry}&c_\rho(\theta)&c_\rho\\ \hline
+\text{fan }i&k_i-2+\alpha&k_i-2\\
+\text{top mark}&12+\beta-7\alpha&12\\
+\text{four-letter entry}&2+\delta&2.
+\end{array}
+\end{equation}
+For a set of fields $W$ put
+\begin{equation}\label{sample:eq-h}
+h_\theta(W)=|W|-\sum_{\rho:E_\rho\subseteq W}c_\rho(\theta),
+\qquad h(W)=h_0(W).
+\end{equation}
+Thus $N^{h_\theta(W)}$ is the expected number of assignments to $W$
+for which all required entries supported in $W$ are present.
+The function $h$ is submodular.  Indeed, for each fixed $E_\rho$,
+the indicator of $E_\rho\subseteq W$ is supermodular, and all
+$c_\rho$ are nonnegative.  Notice also that
+\begin{equation}\label{sample:eq-master-mean}
+h(\mathcal F)=212-(17+12+180)=3,
+\qquad
+\E|\Omega_n|=N^{h_\theta(\mathcal F)}
+             =N^{3-\beta-90\delta}.
+\end{equation}
+The expectation is exact: each assignment uses one entry from each of
+the distinct arrays, so all of its inclusion requirements are independent.
+
+\begin{lemma}\label{sample:lem-budget}
+For every $W\subseteq\mathcal F$,
+\begin{equation}\label{sample:eq-budget}
+h(W)\geq\min\{2,|W|\}.
+\end{equation}
+If $S$ is admissible and $S\subseteq W$, then
+\begin{equation}\label{sample:eq-admissible-budget}
+h(W)\geq |S|.
+\end{equation}
+\end{lemma}
+\begin{proof}
+First consider only the core part of $W$.  Let $a_i$ be its number of
+fields in $F_i$, and let $\ell\in\{0,1\}$ indicate the loose field.
+Before charging the top mark, the exponent is
+\[
+\sum_i\bigl(a_i-(k_i-2)\mathbf 1_{\{a_i=k_i\}}\bigr)+\ell
+\ \geq\ \sum_i\min\{a_i,2\}+\ell.
+\]
+The top-mark cost is charged only for the complete core.  For that set
+the exponent is $32-17-12=3$.  Consequently every core subset has
+exponent at least the size of any admissible core subset it contains,
+and at least the smaller of two and its own cardinality.
+
+Now add the fresh fields of the four-letter entries.  Adding one fresh
+field of an entry adds one to the exponent.  Adding both adds two and
+can charge the cost two only if that entry's two core fields are also
+present.  The net contribution of each pair of fresh fields is therefore
+nonnegative.  This proves Equation~\eqref{sample:eq-admissible-budget}
+when $S$ consists of core fields, and proves
+Equation~\eqref{sample:eq-budget} when $W$ has at least two core fields.
+If $W$ has fewer than two core fields, no four-letter entry is charged,
+and no fan or top entry is charged either.  In that case $h(W)=|W|$.
+This proves Equation~\eqref{sample:eq-budget} in all cases.  Finally,
+an admissible subset of one four-letter entry has size at most two, so
+its required bound follows from Equation~\eqref{sample:eq-budget}.
+\end{proof}
+
+Fix a conditional test $(I,e,t)$.  To couple $t$ matches agreeing on
+$I$, we will require them to agree on a possibly larger set $U$.
+The purpose of this enlargement is to make every further extension
+increase the critical exponent.  For example, if $I$ consists of two
+fields in one fan $F_i$, then completing that fan leaves the exponent
+unchanged:
+\[
+h(I)=2=h(E(F_i)).
+\]
+Once the fan fields have been included, adjoining both fresh fields of
+any square whose two core fields lie in that fan also leaves the exponent
+unchanged.  These are examples of the enlargements that the choice of
+$U$ must absorb.
+
+In general, $|I|\leq2$, and every entry support has at least three
+fields, so $h(I)=|I|$.  Choose an inclusion-maximal set
+$U\supseteq I$ such that
+\begin{equation}\label{sample:eq-closure}
+h(U)=|I|.
+\end{equation}
+Such a set exists because the field set is finite.  The next two
+properties are the reason for this choice:
+\begin{equation}\label{sample:eq-closure-gap}
+e\notin U,
+\qquad
+h(U\cup V)-h(U)\geq1
+\quad\text{for every nonempty }V\subseteq\mathcal F\setminus U.
+\end{equation}
+The first follows from Equation~\eqref{sample:eq-admissible-budget}
+applied to $I\cup\{e\}$.  For the second, every superset of $U$
+has exponent at least $|I|$ by the same lemma applied to $I$.
+Maximality rules out equality for a strict superset, and all critical
+exponents are integers.
+
+\subsection{Stars and their overlap estimates}
+
+Form a star $\mathcal M_U^{(t)}$ from $t$ ordered copies of
+$\mathcal M$ by identifying all copies of each field in $U$.
+Entries supported entirely in $U$ are also identified.  Each other
+entry retains one copy in each branch.  We require assignments to this
+star to be \emph{injective in each field}: for each
+$s\in\mathcal F\setminus U$, its $t$ branch values must be pairwise
+distinct.  Values in different fields may coincide.  All fields continue
+to take values in the same alphabet $S_n$.
+
+This injectivity has two consequences.  First, copies of an entry not
+supported in $U$ are distinct tuples in their array, since some one of
+their coordinates lies outside $U$.  Second, when one branch is fixed,
+an entry in another branch that is not supported in $U$ cannot equal
+the corresponding entry of the fixed branch.  Thus all random variables
+counted below are products of distinct Bernoulli trials, after explicitly
+removing any trials on which we have conditioned.
+
+We give the overlap count in detail because the coordinates are not
+disjoint letter domains.  An \emph{overlap type} between two star
+assignments specifies precisely which entries of each array coincide
+across the two stars.  There are finitely many such types for fixed $t$.
+Each equality of array entries equates corresponding coordinates.  For
+a given field $s\notin U$, injectivity implies that these equalities
+give a partial matching between the $t$ branch variables of the first
+star and those of the second.  Different arrays may use different
+partial matchings; their union must still be a partial matching whenever
+the overlap type is realizable.  Otherwise two different branch values
+of the same field would be forced equal.  For $s\in U$ there is just
+one variable in each star.  In particular, an equality never compares
+two different coordinate fields, even if their numerical letter values
+happen to be equal.
+
+Here is the resulting counting rule.  Reveal the first assignment.
+Suppose the equalities of an overlap type prescribe the values of $q$
+distinct, previously unpinned variables in the second assignment.
+For each field with unpinned variables, there are a bounded number of
+those variables, and they are chosen uniformly without replacement,
+possibly avoiding one already pinned value.  A field whose variables
+are all pinned contributes a factor one.  Prescribing $q_s$ of them has probability at most
+$C_t N^{-q_s}$: the number of remaining choices is at most
+$N^{r-q_s}$, whereas the number of unrestricted choices is a falling
+factorial $(N-a)_r$, with $r\leq t$ and $a\in\{0,1\}$.
+Multiplying over the finitely many fields gives
+\begin{equation}\label{sample:eq-assignment-penalty}
+\text{fraction of pairs having the specified overlap}
+\ \leq\ C_tN^{-q}.
+\end{equation}
+Incompatible specifications simply contribute zero.
+
+If the coincident, unconditioned entries have total actual cost $C$,
+their joint inclusion probability divided by the product of the two
+individual inclusion probabilities is $N^C$.  All other trials are
+independent.  Consequently this overlap type contributes at most
+\begin{equation}\label{sample:eq-variance-rule}
+C_t N^{-q+C}
+\end{equation}
+to the relative variance of the count in question.  Pairs with no
+coincident unconditioned entry have covariance zero, including when
+they have coincident values that do not complete any common entry.
+To justify the relative-variance normalization directly, all candidate
+assignments in any one of our counting problems have the same
+probability of inclusion, and Equation~\eqref{sample:eq-assignment-penalty}
+bounds the fraction of pairs of such candidates.  Summing their
+covariances and dividing by the square of the expectation gives
+Equation~\eqref{sample:eq-variance-rule}, summed over the finitely many
+nonempty overlap types.
+
+We first apply this rule to a single copy of $\mathcal M$.  If $W$ is
+the set of fields belonging to its shared entries, then $q=|W|$ and
+the recovered cost is at most the sum of the costs of all entries
+supported in $W$.  Its critical excess is therefore at least
+$h(W)>0$.  For sufficiently small actual parameters, every one of
+these finitely many positive excesses stays positive.  It follows that
+\begin{equation}\label{sample:eq-single-concentration}
+\frac{\operatorname{Var}|\Omega_n|}{(\E|\Omega_n|)^2}=o(1).
+\end{equation}
+In particular the master count is asymptotic to its expectation in
+probability.  We always choose the parameters small enough that the
+exponent in Equation~\eqref{sample:eq-master-mean} is positive.
+
+For the two star counts, write
+\[
+C_U(\theta)=\sum_{\rho:E_\rho\subseteq U}c_\rho(\theta),
+\qquad
+C_{\mathrm{out}}(\theta)
+ =\sum_{\rho:E_\rho\not\subseteq U}c_\rho(\theta),
+\qquad d=|\mathcal F\setminus U|.
+\]
+Let $Z_N$ be the number of present injective stars.  With
+$(N)_t=N(N-1)\cdots(N-t+1)$,
+\begin{equation}\label{sample:eq-star-mean}
+\E Z_N
+ =N^{|U|}(N)_t^d
+  N^{-C_U(\theta)-tC_{\mathrm{out}}(\theta)}.
+\end{equation}
+
+\begin{lemma}\label{sample:lem-conditional-counts}
+For each fixed conditional test, sufficiently small actual parameters
+have the following properties as $N\to\infty$.
+\begin{enumerate}
+\item Fix any assignment $\sigma:\mathcal F\to S_n$, place it in a
+specified branch, and condition on it being a match.  Let $A_\sigma$
+count its extensions to a present injective star.  Its conditional
+mean is independent of $\sigma$ and equals
+\begin{equation}\label{sample:eq-extension-mean}
+a_N=((N-1)_{t-1})^d
+       N^{-(t-1)C_{\mathrm{out}}(\theta)}.
+\end{equation}
+Uniformly in $\sigma$, its conditional relative variance is $o(1)$.
+\item Fix any ordered list $b=(b_1,\ldots,b_t)$ of distinct letters
+as the values of field $e$ in the branches.  Let $B_b$ count the present
+injective stars having those values.  Its mean is independent of $b$
+and equals
+\begin{equation}\label{sample:eq-leaf-mean}
+b_N=N^{|U|}(N)_t^{d-1}
+       N^{-C_U(\theta)-tC_{\mathrm{out}}(\theta)}.
+\end{equation}
+Uniformly in $b$, its relative variance is $o(1)$.
+\end{enumerate}
+\end{lemma}
+\begin{proof}
+For the first assertion, condition only on the Bernoulli trials used by
+$\sigma$.  All other trials retain their original independent laws.
+Entries supported in $U$ have thereby been fixed present, as have the
+other entries of the specified branch.  Every other branch uses new
+entries distinct from those conditioned trials.  For each field outside
+$U$, its other $t-1$ values are distinct and avoid the value in
+$\sigma$, giving $(N-1)_{t-1}$ choices.  Each extra branch requires
+all the entries not supported in $U$, giving exactly
+Equation~\eqref{sample:eq-extension-mean}.
+
+Consider an overlap of the unconditioned entries of two such extensions.
+In extra branch $j$ of the first extension let $V_j$ consist of the
+outside-$U$ fields involved in its shared entries.  The assignment
+penalty in Equation~\eqref{sample:eq-assignment-penalty} is
+$\sum_j|V_j|$: all $U$-values are fixed, whereas every involved
+outside-$U$ variable is free and, by injectivity, cannot match a value
+in the pinned branch.  Charge each shared entry to its branch in the
+first extension.  The recovered critical cost there is at most
+\[
+\sum_{\rho:E_\rho\subseteq U\cup V_j,\ E_\rho\not\subseteq U}
+c_\rho.
+\]
+Thus the critical excess of assignment penalty over recovered cost is
+at least
+\[
+\sum_{j:V_j\ne\varnothing}
+ \bigl[h(U\cup V_j)-h(U)\bigr]\geq1
+\]
+for every nonempty overlap, by
+Equation~\eqref{sample:eq-closure-gap}.  The finitely many bounds remain
+strict for small actual parameters.  Equations~\eqref{sample:eq-assignment-penalty}
+and~\eqref{sample:eq-variance-rule} prove the conditional relative
+variance claim, uniformly in the values of $\sigma$.  If $t=1$, the
+extension count is identically one and there is no variance to estimate.
+
+For the second assertion, no Bernoulli trials are conditioned on;
+only letter values are pinned.  Since $e\notin U$, its $t$ pinned
+values satisfy precisely its field-injectivity constraint.  The other
+fields have $N^{|U|}(N)_t^{d-1}$ choices, proving
+Equation~\eqref{sample:eq-leaf-mean}.
+
+For an overlap let $W$ be the set of variables in the first star
+belonging to shared entries.  Write $A=W\cap U$, and let $V_j$ be
+its outside-$U$ fields in branch $j$.  Let $m(W)$ be the number of its
+$e$-variables.  In a realizable overlap an $e$-variable can only match
+the variable with the same pinned value; those equalities impose no
+further assignment penalty.  All other involved variables do impose
+one.  Thus $q=|W|-m(W)$ in
+Equation~\eqref{sample:eq-assignment-penalty}.
+
+Define $h_\star(W)$ using the critical entry costs in the star, counting
+entries entirely in $U$ once and the other entries separately in each
+branch.  The recovered cost is at most the cost of all star entries
+supported in $W$.  Therefore the critical excess is at least
+\begin{equation}\label{sample:eq-leaf-excess}
+h_\star(W)-m(W).
+\end{equation}
+If $A\ne\varnothing$, separating entries entirely in $U$ from the
+entries in each branch gives
+\begin{align}
+h_\star(W)
+&\geq h(A)+
+ \sum_{j:V_j\ne\varnothing}
+       \bigl[h(U\cup V_j)-h(U)\bigr].
+\label{sample:eq-star-submodular}
+\end{align}
+Indeed the exact contribution of branch $j$ is $|V_j|$ minus the
+costs of entries supported in $A\cup V_j$ but not entirely in $U$;
+replacing $A$ by $U$ can only increase those costs.  This proves the
+displayed inequality directly.  Its first term is at least one by
+Lemma~\ref{sample:lem-budget}, and each nonempty summand is at least
+one by Equation~\eqref{sample:eq-closure-gap}.  Each such branch
+contains at most one pinned $e$-variable.  Hence the excess in
+Equation~\eqref{sample:eq-leaf-excess} is at least one.
+
+If $A=\varnothing$, no shared entry meets $U$, and entries in different
+branches use disjoint variables.  Consequently
+$h_\star(W)=\sum_j h(V_j)$.  An active branch contains the full
+support of at least one shared entry.  Every entry support has at least
+three fields, so $|V_j|\geq3$ in such a branch and
+Lemma~\ref{sample:lem-budget} gives $h(V_j)\geq2$.
+Subtracting at most one pinned leaf per branch again leaves a positive
+excess.  This argument is applied to supports of shared entries, not
+to arbitrary nonempty field subsets.
+
+All nonempty overlap types thus have critical excess at least one.
+Perturbing to sufficiently small actual parameters preserves a positive
+excess for every type.  Equation~\eqref{sample:eq-variance-rule}, summed
+over those types, proves the second relative variance assertion.
+The bounds are uniform in the pinned letters because all assignment
+counts and Bernoulli probabilities used above are independent of their
+particular values.
+\end{proof}
+
+\subsection{Normalization to probability couplings}
+
+For probability measures $\lambda,\lambda'$ on a finite set we use
+\[
+d_{\mathrm{TV}}(\lambda,\lambda')
+ =\frac12\sum_z|\lambda(z)-\lambda'(z)|.
+\]
+All convergence statements about the random tables below are in
+probability as $N\to\infty$; equivalently one can choose deterministic
+error bounds tending to zero for which the corresponding assertions
+hold with probability tending to one.
+
+\begin{lemma}\label{sample:lem-coupling}
+For each fixed conditional test and sufficiently small actual
+parameters, with probability tending to one there is a probability
+law on $t$ matches $(\sigma_1,\ldots,\sigma_t)$ such that:
+\begin{enumerate}
+\item their restrictions to $I$ agree identically;
+\item every branch marginal has total-variation distance $o(1)$ from
+$\mu_N$;
+\item the law of $(\sigma_1(e),\ldots,\sigma_t(e))$ has
+total-variation distance $o(1)$ from the uniform product law on $S_n^t$.
+\end{enumerate}
+\end{lemma}
+\begin{proof}
+Use the uniform law on present injective stars.  We verify that its
+normalization is nonzero and has the stated properties.
+
+Put $L_N=|\Omega_n|$ and $\lambda_N=\E L_N$.
+For a specified branch let
+\[
+D_N=\sum_{\sigma\in\Omega_n}|A_\sigma-a_N|.
+\]
+The conditional variance bound of
+Lemma~\ref{sample:lem-conditional-counts}, followed by the
+Cauchy--Schwarz inequality, gives uniformly in each possible assignment
+$\sigma$
+\[
+\E\bigl[|A_\sigma-a_N|\mid \sigma\in\Omega_n\bigr]=o(a_N).
+\]
+Multiplying by the probability that $\sigma$ is present and summing
+over all assignments shows
+$\E D_N=o(\lambda_Na_N)$.  Markov's inequality yields
+$D_N=o(\lambda_Na_N)$ in probability.  By
+Equation~\eqref{sample:eq-single-concentration},
+$L_N/\lambda_N\to1$ in probability.  Since
+\[
+Z_N=\sum_{\sigma\in\Omega_n}A_\sigma,
+\qquad |Z_N-L_Na_N|\leq D_N,
+\]
+we have $Z_N/(L_Na_N)\to1$ in probability, and in particular
+$Z_N>0$ with probability tending to one.  Moreover,
+\begin{align*}
+\sum_{\sigma\in\Omega_n}
+ \left|\frac{A_\sigma}{Z_N}-\frac1{L_N}\right|
+&\leq \frac{D_N}{Z_N}
+      +\left|\frac{L_Na_N}{Z_N}-1\right|
+\longrightarrow0.
+\end{align*}
+The same argument applies to each of the finitely many branches.
+This proves the marginal assertion.  Agreement on $I$ is exact because
+the branches agree on $U\supseteq I$.
+
+For the leaves, let $\mathcal B_N$ be the set of ordered distinct
+letter lists of length $t$, so $|\mathcal B_N|=(N)_t$.  By the second
+part of Lemma~\ref{sample:lem-conditional-counts},
+\[
+\E\sum_{b\in\mathcal B_N}|B_b-b_N|
+       =o((N)_t b_N).
+\]
+Markov's inequality and $\sum_bB_b=Z_N$ imply that the total variation
+between the leaf distribution $B_b/Z_N$ and the uniform distribution
+on $\mathcal B_N$ tends to zero, by the same displayed normalization
+estimate with $(N)_t,b_N$ in place of $L_N,a_N$.
+Finally, an iid uniform list in $S_n^t$ has a repeated value with
+probability at most $\binom t2/N$.  Its distribution conditioned on
+distinctness is uniform on $\mathcal B_N$.  This proves the final
+assertion.
+\end{proof}
+
+\subsection{Finite tests, parameter choices, and the six-letter table}
+
+The two marginal conclusions of the coupling serve different purposes.
+The branch comparison transfers averages of bounded tests between the
+coupled branches and a uniform match.  In
+Lemma~\ref{lem:conditional-test}, a pointwise norm estimate bounds the
+average over the $t$ branches by a function of the $e$-labels alone; the common
+$I$-labels enter only through a uniform bound.  The comparison with
+independent uniform letters then controls that function of the leaves.
+Thus the argument needs no independence between the common $I$-tuple
+and the test labels.
+
+All critical overlap margins above are at least one.  Each shared trial
+is charged once to its occurrence in the first star, which has at most
+$t$ copies of each array entry.  Since $\beta>7\alpha$, all cost
+increments are positive.  Replacing critical costs by actual costs
+therefore increases any recovered-cost sum by at most
+\[
+t\bigl[7\alpha+(\beta-7\alpha)+90\delta\bigr]
+=t(\beta+90\delta).
+\]
+Conditioning removes trials and can only decrease this bound; a single
+master match is covered by $t=1$.  For a finite family of tests, let
+$t_{\max}$ be the largest requested value of $t$, or one if the family
+is empty.  Choosing $t_{\max}(\beta+90\delta)<1/2$ preserves every
+overlap margin at a value at least $1/2$, including the singleton
+tests used below.  This is compatible with $\beta>7\alpha>0$: take
+$\beta=8\alpha$ and then take $\alpha,\delta$ sufficiently small.
+The parameters are held fixed while $N$ tends to infinity.  In
+particular, none of the variance arguments asserts uniformity for
+unbounded $t$ at a single fixed choice of parameters.
+
+Let $\mathcal D_N\subseteq S_n^6$ be the separate six-letter table.
+For positions $1\leq r<q\leq6$ and letters $u,v$, write
+\[
+Q_{rq}(u,v)
+ =\#\{(s_1,\ldots,s_6)\in\mathcal D_N:s_r=u, s_q=v\}.
+\]
+Each such count is a sum of $N^4$ independent Bernoulli variables
+of parameter $N^{-3-\delta}$, and so has mean $N^{1-\delta}$.
+For fixed $0<\delta<1$, put
+$\varepsilon_N=N^{-(1-\delta)/4}$.  The binomial Chernoff bound gives
+\[
+\Prob\bigl(|Q_{rq}(u,v)-N^{1-\delta}|>
+       \varepsilon_NN^{1-\delta}\bigr)
+ \leq 2\exp\bigl(-\varepsilon_N^2N^{1-\delta}/3\bigr).
+\]
+A union bound over the $15N^2$ choices of positions and letters
+therefore proves
+\begin{equation}\label{sample:eq-six-pair}
+\max_{r<q}\max_{u,v\in S_n}
+ \left|\frac{Q_{rq}(u,v)}{N^{1-\delta}}-1\right|=o(1)
+\end{equation}
+with probability tending to one.  No independence between these
+different counts is needed.
+
+We collect the conclusions in the form used below.
+\begin{proposition}\label{prop:sampling}
+For any finite family $\mathcal T$ of conditional tests, there is a
+neighborhood of the critical parameters such that every sufficiently
+small fixed positive $(\alpha,\beta,\delta)$ in that neighborhood
+with $\beta>7\alpha$ has the following properties as $N\to\infty$.
+With probability tending to one, the master pattern has a nonempty
+set of matches, its count is
+\[
+|\Omega_n|=(1+o(1))N^{3-\beta-90\delta},
+\]
+every single-field marginal of its uniform law $\mu_N$ is $o(1)$
+in total variation from the uniform law on $S_n$, and every test in
+$\mathcal T$ has the coupling of Lemma~\ref{sample:lem-coupling}.
+The six-letter counts also satisfy Equation~\eqref{sample:eq-six-pair}.
+All these properties concern only the tables and are independent of
+any choice of action or metric space.
+\end{proposition}
+\begin{proof}
+Adjoin to $\mathcal T$ the finitely many tests
+$(\varnothing,e,1)$, one for each field $e$.  For such a test
+Equation~\eqref{sample:eq-budget} forces $U=\varnothing$, and its
+one-branch star is exactly a master match.  The leaf assertion in
+Lemma~\ref{sample:lem-coupling} therefore gives single-field
+uniformity.  Choose parameters satisfying all the finite overlap
+margins just described, $\beta+90\delta<3$, and $\delta<1$.
+Equation~\eqref{sample:eq-single-concentration},
+Lemma~\ref{sample:lem-coupling}, and
+Equation~\eqref{sample:eq-six-pair}, combined over this finite set of
+events, prove the proposition.
+\end{proof}
+
+\begin{remark}\label{sample:rem-diagonal}
+Proposition~\ref{prop:sampling} is compatible with
+Theorem~\ref{thm:diagram}: for each finite family of tests, choose
+positive parameters sufficiently small for that family and the theorem,
+then the diagram fragment threshold, and finally $N$.
+The diagram and sampling events then hold simultaneously with probability
+tending to one.  Lemma~\ref{lem:completion-diagonal} constructs the
+deterministic diagonal sequence used below, with parameters tending to
+zero and every fixed test valid in the limit.
+\end{remark}
diff --git a/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/sections/05-energy.tex b/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/sections/05-energy.tex
new file mode 100644
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+++ b/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/sections/05-energy.tex
@@ -0,0 +1,591 @@
+\section{Energy and a limiting configuration}\label{sec:energy}
+
+The sampling statements in Proposition~\ref{prop:sampling} concern finite
+tables alone.  In this section they will be applied to functions obtained
+from an isometric action chosen after those tables are known.  The order
+is important: we first obtain a bound on every generator displacement
+using the pointwise relative counts in the six-letter table.  Only then
+do we use total variation estimates for action-dependent functions.
+
+We use the following standard notation for a complete $\CAT(0)$ space
+$X$.  Its completed tangent cone at $x$ is denoted by $T_xX$, with vertex
+$0$.  For $y\in X$, the vector $\log_x y$ has length $d(x,y)$ and the
+direction of the geodesic from $x$ to $y$.  In a Euclidean metric cone set
+\[
+ \ip{u}{v}=\frac{\norm{u}^2+\norm{v}^2-d(u,v)^2}{2}.
+\]
+Thus $\ip{u}{v}=\norm{u}\norm{v}\cos\angle(u,v)$, where the angle is
+truncated at $\pi$.  Multiplication of a vector by a nonnegative real
+number always means radial dilation.  No additive vector-space structure
+on a cone is assumed.
+
+We will use strong convexity of squared distance, first variation, the
+flat-triangle equality theorem, and the tangent-cone theorem for
+$\CAT(0)$ spaces; see \cite[Chapters~II.1--II.3, in particular
+II.2.9--II.2.10, II.3.6, and II.3.19]{BH1999}.  These results do not require
+properness.  In particular, comparison of angles with comparison angles
+gives
+\begin{equation}\label{energy:log-comparison}
+ d(y,z)^2\ge
+ d(x,y)^2+d(x,z)^2
+ -2\ip{\log_x y}{\log_x z}.
+\end{equation}
+All the tangent cones below are completed cones.
+
+\subsection{A minimizing basepoint and uniform displacement bounds}
+
+The first-variation characterization of an energy minimizer in a
+tangent cone is part of the harmonic-map framework of
+Gromov~\cite[Sections~3.3 and~3.7]{Gromov2003} and
+Izeki--Nayatani~\cite{IzekiNayatani2005}.
+We give the argument together with the existence of a minimizing
+basepoint under the action hypotheses used here.
+
+\begin{lemma}\label{energy:minimizer}
+Let an infinite finitely generated group $G$ act properly and
+cocompactly by isometries on a proper complete $\CAT(0)$ space $X$.
+For a finite symmetric generating alphabet $S$, counted with its given
+multiplicities, the function
+\[
+ E(x)=\frac1{|S|}\sum_{s\in S}d(x,sx)^2
+\]
+has a minimizer and its minimum is positive.  At any minimizing
+basepoint, the vectors $v_s=\log_x(sx)$ satisfy
+\begin{equation}\label{energy:zero-barycenter}
+ \frac1{|S|}\sum_{s\in S}\ip{h}{v_s}\le0
+ \quad(h\in T_xX).
+\end{equation}
+Equivalently, $0$ is their barycenter, and
+\begin{equation}\label{energy:variance}
+ \frac1{|S|}\sum_{s\in S}d(h,v_s)^2
+ \ge \norm{h}^2+\frac1{|S|}\sum_{s\in S}\norm{v_s}^2.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Choose a minimizing sequence $x_j$ and a compact set $C$ with $GC=X$.
+There are $g_j\in G$ such that $y_j=g_jx_j\in C$.  Bounded energy bounds
+each individual displacement $d(x_j,sx_j)$, since $S$ is finite.  It
+therefore bounds
+\[
+ d(y_j,(g_jsg_j^{-1})y_j).
+\]
+The closed neighborhood of $C$ of any fixed finite radius is compact,
+by properness of $X$.  Properness of the action implies that only
+finitely many group elements can carry a point of $C$ into that
+neighborhood.  For each $s$ there are consequently only finitely many
+possibilities for $g_jsg_j^{-1}$.  Passing to a subsequence, the entire
+conjugated generating tuple is constant and $y_j$ converges to a point
+$y\in C$.  Its energy for that tuple is the original infimum.  Fixing
+one element $g_{j_0}$ in the subsequence, the point $g_{j_0}^{-1}y$
+attains the infimum for the original tuple.  An energy of zero would
+make all generators, and hence $G$, fix a point.  A proper action has
+finite point stabilizers, so the minimum is positive.
+
+Fix a minimizing point $x$.  First let $h$ be a tangent vector
+represented by a geodesic $x_t$ issuing from $x$, parameterized with
+initial speed $\norm h$.  For sufficiently small $t>0$, convexity at
+the midpoints of the crossed geodesics gives
+\[
+ d(x_t,sx_t)^2\le
+ \frac12d(x_{2t},sx)^2+\frac12d(x,sx_{2t})^2.
+\]
+After averaging over the symmetric alphabet, the two terms on the
+right have the same average: apply $s^{-1}$ to the second distance and
+replace $s$ by $s^{-1}$.  Minimality and first variation now imply
+\[
+ 0\le
+ \left.\frac{d}{dt}\right|_{0+}
+ \frac1{|S|}\sum_s d(x_{2t},sx)^2
+ =-\frac4{|S|}\sum_s\ip{h}{v_s}.
+\]
+Represented vectors are dense in the completed tangent cone, so
+continuity gives Equation~\eqref{energy:zero-barycenter} for every $h$.
+The cone cosine formula immediately gives
+Equation~\eqref{energy:variance}; in particular $0$ minimizes the mean
+squared distance to the $v_s$.
+\end{proof}
+
+For each aspherical instance $P$ of Theorem~\ref{thm:diagram}, its group
+$G$ is infinite.  Indeed, the parity quotient in Lemma~\ref{lem:model-presentation}
+makes $G$ nontrivial.  If $G$ were finite, the universal cover would be a
+finite contractible complex and
+\[
+ 1=\chi(\widetilde P)=|G|\chi(P),
+\]
+which forces $|G|=1$.  Thus Lemma~\ref{energy:minimizer} applies to any
+geometric action considered here.  Rescale its metric so that at the
+minimizing basepoint
+\begin{equation}\label{energy:normalization}
+ \frac1N\sum_{s\in S_n}\norm{v_s}^2=1.
+\end{equation}
+Rescaling preserves completeness, properness, and the $\CAT(0)$
+condition.
+
+\begin{lemma}\label{energy:displacement-bound}
+Suppose the six-letter table has pointwise relative error at most
+$\varepsilon<1$ in the counts for positions $1$ and $k$, for every
+$2\le k\le6$.  Then every action normalized as in
+Equation~\eqref{energy:normalization} satisfies
+\begin{equation}\label{energy:max-displacement}
+ \max_{s\in S_n}\norm{v_s}^2
+ \le25\frac{1+\varepsilon}{1-\varepsilon}.
+\end{equation}
+The bound is independent of the action.
+\end{lemma}
+
+\begin{proof}
+Write $Q_{1k}(a,b)$ for the number of included rows whose first
+coordinate is $a$ and whose $k$th coordinate is $b$.  The asserted
+relative estimate is
+\[
+ (1-\varepsilon)N^{1-\delta}
+ \le Q_{1k}(a,b)\le
+ (1+\varepsilon)N^{1-\delta}.
+\]
+Summing over $b$ shows that rows beginning with any fixed $a$ are
+nonempty and that their conditional $k$th-coordinate distribution is
+bounded above, pointwise, by
+\[
+ \frac{1+\varepsilon}{1-\varepsilon}\frac1N.
+\]
+The relation attached to a row $(a,s_2,\ldots,s_6)$ gives
+$a^{-1}=s_2\cdots s_6$.  Since
+$d(x,a^{-1}x)=d(x,ax)$, the triangle inequality and Cauchy--Schwarz give
+\[
+ \norm{v_a}^2\le5\sum_{k=2}^6\norm{v_{s_k}}^2.
+\]
+Average over the rows beginning with $a$ and use the preceding
+pointwise bound and Equation~\eqref{energy:normalization}.  Each of the
+five conditional averages is at most
+$(1+\varepsilon)/(1-\varepsilon)$, proving the assertion.  This uses
+only nonnegative weights, and therefore holds even when the weights
+were chosen from an action after the table was revealed.
+\end{proof}
+
+\subsection{Conditional tests in tangent cones}
+
+For a geodesic from $u$ to $v$ in a $\CAT(0)$ Euclidean cone, write
+$[u,v]_t$ for its point at parameter $t$.  Its squared
+radial norm satisfies the exact identity
+\begin{equation}\label{energy:radial-identity}
+ \norm{[u,v]_t}^2
+ =(1-t)\norm u^2+t\norm v^2-t(1-t)d(u,v)^2.
+\end{equation}
+If either endpoint is the cone vertex, the identity is immediate.
+Otherwise the segment is a Euclidean segment in the sector between
+its two rays when their angular distance is less than $\pi$.  When
+their angular distance is at least $\pi$, it passes through the cone vertex, and
+$\norm{[u,v]_t}=|(1-t)\norm u-t\norm v|$ gives the same identity.
+These two cases follow directly from the Euclidean-cone metric;
+see \cite[Definition~I.5.6 and Proposition~I.5.10(1)--(2)]{BH1999}.
+
+The scalar-product concavity used next is recorded by
+Gigli--Nobili~\cite[Proposition~2.6, equation~(2.7f), and the proof of
+Proposition~3.8]{GigliNobili2021}. The inductive-mean estimate is the
+unit-variance case of Sturm's bound~\cite[Definition~4.6 and
+Theorem~4.7, proof~\textup{(a)}]{Sturm2003}. We include the short proofs in the present
+normalization.
+
+\begin{lemma}\label{energy:cone-mean}
+In a complete $\CAT(0)$ metric cone, $u\mapsto\ip{h}{u}$ is concave
+along every geodesic, for fixed $h$.  Suppose a finite probability
+distribution of vectors $V$ has barycenter $0$ and
+$\mathbb E\norm V^2=1$.  If $V_1,\ldots,V_t$ are independent samples
+and
+\[
+ m_1=V_1,\qquad
+ m_j=[m_{j-1},V_j]_{1/j},
+\]
+then
+\begin{equation}\label{energy:inductive-mean}
+ \mathbb E\norm{m_t}^2\le\frac1t.
+\end{equation}
+For any deterministic list $V_1,\ldots,V_t$ one also has
+\begin{equation}\label{energy:mean-concavity}
+ \frac1t\sum_{j=1}^t\ip{h}{V_j}\le\ip{h}{m_t}.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Subtract strong convexity of $d(h,\cdot)^2$ from
+Equation~\eqref{energy:radial-identity} in the identity
+$2\ip{h}{u}=\norm h^2+\norm u^2-d(h,u)^2$.  The quadratic terms
+cancel, proving concavity.  Induction gives
+Equation~\eqref{energy:mean-concavity}.
+
+For the independent samples, condition on $m_{j-1}=u$.
+The barycenter variance inequality gives
+$\mathbb E[d(u,V_j)^2\mid u]\ge\norm u^2+1$.  Strong convexity
+of squared distance from the cone vertex therefore gives, with
+$a_j=\mathbb E\norm{m_j}^2$,
+\[
+ a_j\le(1-1/j)^2a_{j-1}+1/j^2.
+\]
+Starting with $a_1=1$, induction yields $a_j\le1/j$.
+\end{proof}
+
+Consider now a sequence of table instances indexed by $\nu$ for which
+the conclusions of Proposition~\ref{prop:sampling} hold with errors
+tending to zero, for every fixed test and every fixed positive integer
+$t$.  Suppose their groups have geometric actions as above, and choose
+the normalized minimizing basepoints $x_\nu$.  Write
+$\Omega_\nu=\Omega_{n_\nu}$ for the nonempty finite set of master
+matches and $\mu_\nu=\mu_{N_\nu}$ for its uniform distribution.
+Lemma~\ref{energy:displacement-bound} gives an
+absolute constant $M$ such that
+\begin{equation}\label{energy:uniform-bound}
+ \norm{v_s}\le M
+ \quad\text{for all letters in all sufficiently large instances.}
+\end{equation}
+All statements about the limit are unchanged by discarding finitely
+many instances.  The actions, their spaces, and the functions in the
+next lemma may depend on the complete table instances.
+
+\newpage
+\begin{lemma}[Uniform conditional test]\label{lem:conditional-test}
+Let $I$ and $\{e\}$ be disjoint sets of fields with admissible union.
+For each $\nu$, let $h_I$ be a function of the labels on $I$ with values
+in $T_{x_\nu}X_\nu$, and suppose $\norm{h_I}\le B$ for a constant
+independent of $\nu$ and the match.  For either fixed choice
+$\sigma(s)=s$ or $\sigma(s)=s^{-1}$,
+\begin{equation}\label{energy:conditional-test}
+ \limsup_{\nu\to\infty}
+ \mathbb E_{\mu_\nu}
+ \ip{h_I}{v_{\sigma(e)}}\le0.
+\end{equation}
+Here $e$ in a subscript denotes the letter occupying that field.
+\end{lemma}
+
+\begin{proof}
+Fix $t$.  Use the coupling in Proposition~\ref{prop:sampling}: it
+gives $t$ matches agreeing on $I$, each branch marginal at total
+variation distance at most $\eta_{\nu,t}$ from $\mu_\nu$, and their
+$e$-labels at distance at most $\eta_{\nu,t}$ from independent uniform
+letters, where $\eta_{\nu,t}\to0$.  Increasing the error if necessary
+allows one bound for all these comparisons.  Applying the fixed
+inversion $\sigma$ preserves this assertion.
+
+The vector $h_I$ has the same value in every branch.  Form the
+inductive mean $m_t$ of the branch vectors $v_{\sigma(e_j)}$.
+Equation~\eqref{energy:mean-concavity} gives, for every coupled tuple,
+\[
+ \frac1t\sum_{j=1}^t\ip{h_I}{v_{\sigma(e_j)}}
+ \le\ip{h_I}{m_t}\le B\norm{m_t}.
+\]
+The last step uses only the norm bound; it does not assume that $h_I$
+is independent of the leaves.
+
+Under independent uniform leaves,
+Equation~\eqref{energy:inductive-mean} applies by
+Equations~\eqref{energy:variance} and~\eqref{energy:normalization}.
+Under the coupling, $\norm{m_t}\le M$, by convexity of the ball of
+radius $M$.  Consequently the total variation comparison changes the
+expected squared norm by at most $2M^2\eta_{\nu,t}$.  The branch
+comparisons change the expectation of each scalar-product test by at
+most $2BM\eta_{\nu,t}$.  It follows that
+\[
+ \mathbb E_{\mu_\nu}\ip{h_I}{v_{\sigma(e)}}
+ \le B\sqrt{1/t+2M^2\eta_{\nu,t}}+2BM\eta_{\nu,t}.
+\]
+First take the upper limit in $\nu$ with $t$ fixed, and then let
+$t\to\infty$.  All error bounds used only $B$ and $M$, so the
+conclusion is uniform over the action-dependent functions in the
+statement.
+\end{proof}
+
+\subsection{Products of orbit configurations}
+
+Enlarge $\Gamma$ by adding, for each of its prescribed two-edge paths,
+the complementary two-edge path of its decoration square.  Denote
+this fixed finite graph by $\widehat\Gamma$; its additional midpoint
+vertices are distinct abstract vertices.  A master match gives a
+consistent orbit realization of this graph in $X_\nu$: choose one
+anchor at $x_\nu$, and send a vertex reached by the successive letters
+$s_1,\ldots,s_k$ to $s_1\cdots s_kx_\nu$.  All core closed walks are
+relations, and the decoration square relations guarantee consistency
+on the added paths.  Fixing one path for each vertex shows that these
+orbit points are at distance at most $LM$ from $x_\nu$, where $L$
+depends only on $\widehat\Gamma$.
+
+Take the finite product over the matches with metric
+\begin{equation}\label{energy:product-metric}
+ d_\nu((y_m),(z_m))^2
+ =\frac1{|\Omega_\nu|}\sum_{m\in\Omega_\nu}d(y_m,z_m)^2,
+\end{equation}
+based at the tuple of minimizing basepoints.  It is a complete
+$\CAT(0)$ space.  The vertex tuples have bounded distance from this
+basepoint.  Fix a nonprincipal ultrafilter on the sequence index set
+$\mathbb N$ and take the pointed metric ultralimit $Z$.  This space is
+complete by \cite[Lemma~I.5.53]{BH1999}, and it is $\CAT(0)$ by
+\cite[Corollary~II.3.10(2)]{BH1999}.  The bounded vertex tuples give a
+configuration in $Z$ indexed by the vertices of $\widehat\Gamma$.
+Midpoints
+and fixed-fraction points on segments in the products are formed
+coordinatewise, and their ultralimits are the corresponding points
+in $Z$ by uniqueness of geodesics.  We identify the abstract vertex
+names with the resulting points.  Neither properness nor a dimension
+bound for $Z$ is needed.
+
+\newpage
+\begin{lemma}\label{energy:path-bounds}
+Every edge in $\widehat\Gamma$ has length one in $Z$.  Every two-step
+path using two distinct fields of one decoration square has squared
+endpoint distance at least two.  More generally, a path of $k\le3$
+distinct core fields whose field set is admissible has squared
+endpoint distance at least $k$.
+\end{lemma}
+
+\begin{proof}
+For an edge with letter $s$, translation at its initial orbit vertex
+identifies its squared length with $\norm{v_s}^2$, or with the same
+quantity for $s^{-1}$.  Single-field total variation convergence,
+Equation~\eqref{energy:uniform-bound}, and the normalization show that
+its squared product length tends to one.
+
+For a path with oriented readings $a_1,\ldots,a_k$, translate its
+penultimate orbit vertex to $x_\nu$.  Its earlier endpoint then becomes
+$a_{k-1}^{-1}\cdots a_1^{-1}x_\nu$, and its next endpoint becomes
+$a_kx_\nu$.  Set
+\[
+ h_I=\log_{x_\nu}(a_{k-1}^{-1}\cdots a_1^{-1}x_\nu).
+\]
+This is a function of precisely the preceding labels, and
+$\norm{h_I}\le(k-1)M$.  Equation~\eqref{energy:log-comparison} gives
+\[
+ d(u_0,u_k)^2\ge d(u_0,u_{k-1})^2+\norm{v_{a_k}}^2
+       -2\ip{h_I}{v_{a_k}}
+\]
+in every factor.  Average, apply Lemma~\ref{lem:conditional-test},
+and take the ultralimit.  Induction adds at least one at each step.
+Every prefix of the field sets in the statement is admissible, so
+the conditional test is available at each step.  Distinct fields
+here need not carry distinct numerical letter values.
+\end{proof}
+
+\subsection{Quadrilateral equality and hexagon centers}
+
+The path estimates give lower bounds on squared distances.  We now
+compare them with a $\CAT(0)$ upper bound on the sum of the two
+diagonal squares of a quadrilateral.  Equality will first determine
+the diagonals of every decoration square, and then force the three
+opposite pairs in each core hexagon to share a midpoint.
+
+\begin{lemma}\label{energy:quadrilateral}
+For four points $z_1,z_2,z_3,z_4$ in a $\CAT(0)$ space, let
+$m=\operatorname{mid}(z_1,z_3)$ and
+$n=\operatorname{mid}(z_2,z_4)$.  Then
+\begin{equation}\label{energy:quadrilateral-inequality}
+ d(z_1,z_3)^2+d(z_2,z_4)^2+4d(m,n)^2
+ \le\sum_{i=1}^4d(z_i,z_{i+1})^2,
+ \qquad z_5=z_1.
+\end{equation}
+In particular, equality between the two diagonal-square sum and the
+side-square sum forces the diagonal midpoints to coincide.
+\end{lemma}
+
+\begin{proof}
+Apply midpoint convexity to $d(m,z_2)^2$ and $d(m,z_4)^2$ using the
+segment $z_1z_3$, and then to $d(m,n)^2$ using $z_2z_4$.  Substituting
+the first two inequalities into the third and multiplying by four
+gives Equation~\eqref{energy:quadrilateral-inequality}.
+\end{proof}
+
+Each decoration square has unit sides.  Both of its diagonals have
+squared length at least two by Lemma~\ref{energy:path-bounds}.
+Lemma~\ref{energy:quadrilateral} forces both squared lengths to equal
+two.  In particular,
+\begin{equation}\label{energy:two-step-equality}
+ d(u,w)^2=2
+ \quad\text{for every nonbacktracking two-edge core path }u,v,w.
+\end{equation}
+Every such path was assigned a decoration table, including paths
+whose two fields lie in the same fan.
+
+Recall that a core hexagon is obtained from three
+noncollinear Fano points $i,j,k$, their joining lines, and a choice of
+$X$ or an available $Y$ representative at each of the three points,
+with all six required edges present.  The squared distances of its
+vertices $z_0,\ldots,z_5$ at cyclic separation one and two are,
+respectively, one and two.  Each three-edge half-path uses at most
+two fields of any one fan, so its opposite endpoints have squared
+distance at least three.
+
+Apply Lemma~\ref{energy:quadrilateral} to
+$(z_0,z_1,z_3,z_4)$.  The side-square sum is $1+2+1+2=6$, whereas each
+diagonal-square is at least three.  Both are therefore three and
+their midpoints coincide.  Applying the same argument to
+$(z_1,z_2,z_4,z_5)$ identifies the third opposite-pair midpoint with
+the first two.  Denote the resulting center of this hexagon by $c$.
+Every vertex has distance
+\begin{equation}\label{energy:radius}
+ r=\frac{\sqrt3}{2}
+\end{equation}
+from $c$.
+
+\begin{lemma}\label{energy:hexagon-flatness}
+For a core hexagon, every triangle formed by its center and two of
+its vertices is Euclidean flat, including the degenerate case of
+opposite vertices.  The corresponding tangent vectors satisfy
+\begin{equation}\label{energy:hexagon-products}
+ \ip{\log_c z_i}{\log_c z_j}
+ =r^2-\frac12d(z_i,z_j)^2.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+The opposite pairs are geodesic segments through $c$, so their
+directions in $T_cZ$ are antipodal and their radial lengths are $r$.
+If $v^+$ and $v^-$ are any such opposite vectors, the angular
+triangle inequality implies
+\begin{equation}\label{energy:antipodal-products}
+ \ip{u}{v^+}+\ip{u}{v^-}\le0
+ \quad(u\in T_cZ).
+\end{equation}
+Indeed, the two angles from $u$ to their directions have sum at
+least $\pi$, and the sum of their cosines is nonpositive.
+
+Set $u=\log_c z_i$, $v^+=\log_c z_j$, and
+$v^-=\log_c z_{j+3}$, with indices modulo six.  Log comparison gives
+\[
+ \ip{u}{v^+}\ge r^2-\tfrac12d(z_i,z_j)^2,
+ \qquad
+ \ip{u}{v^-}\ge r^2-\tfrac12d(z_i,z_{j+3})^2.
+\]
+The two prescribed squared distances sum to $3=4r^2$, so the lower
+bounds sum to zero.  Equation~\eqref{energy:antipodal-products}
+forces equality in each bound.  Thus the angle at $c$ equals its
+Euclidean comparison angle.  For a nondegenerate triangle the
+flat-triangle equality theorem gives a flat triangle; for an
+opposite or repeated pair the statement is a segment or a point.
+\end{proof}
+
+\begin{lemma}\label{energy:common-center}
+All core hexagons have the same center $c$.  Every core vertex lies
+in a core hexagon, and hence is at distance $r$ from $c$.
+\end{lemma}
+
+\begin{proof}
+First use only $X$ representatives.  A noncollinear ordered triple
+of Fano points is a basis of $\mathbb F_2^3$.  Reordering a basis
+does not change its hexagon, and replacing one basis vector $i$ by
+$i+j$ preserves the line $ij$ and the third point $k$.  The old and
+new hexagons therefore share the opposite pair $(X_k,ij)$, and their
+centers are its common midpoint.  Permutations and elementary
+additions generate all changes of basis over $\mathbb F_2$, so the
+centers of all $X$ hexagons agree.
+
+Replace one $Y$ representative in a core hexagon by its $X$
+representative.  The new hexagon is a core hexagon, since every corresponding
+$X$ edge exists.  Either of the unchanged point representatives,
+together with its opposite line, is an opposite pair shared by the
+two hexagons.  Their centers agree.  Repeating this replacement
+connects every core hexagon to one using only $X$ representatives.
+
+Every $X_i$ lies in a hexagon obtained by extending $i$ to a basis.
+Every line lies in one by choosing two points spanning that line
+and a third point outside it.  Each $Y_i$ has two distinct incident
+lines available.  Choose a point other than $i$ on each of them;
+these two points together with $i$ are noncollinear and yield a
+core hexagon containing $Y_i$.  This includes $Y_D$, using its two
+full lines.  The radius assertion follows.
+\end{proof}
+
+\subsection{The ternary midpoint test}
+
+The square and hexagon comparisons have supplied a common center.
+We now use three admissible core fields to control one further angle: the angle
+at $CD$ between the segment to the midpoint of $X_C,Y_C$ and the loose
+edge to $X_D$.
+
+Put
+\[
+ H_0=CD,\qquad p=X_C,\qquad q=Y_C,\qquad z=X_D,
+ \qquad J=\operatorname{mid}(p,q)
+\]
+in $Z$.  The following argument gives an angle statement in $Z$
+without any assumption that angles commute with ultralimits.
+
+Choose another line $H\ne H_0$ through $C$.  The core four-cycle
+$(p,H_0,q,H)$ has unit sides and both diagonals of squared length
+two, by Equation~\eqref{energy:two-step-equality}.  Its diagonal
+midpoints coincide by Lemma~\ref{energy:quadrilateral}, so
+\begin{equation}\label{energy:ternary-radii}
+ J=\operatorname{mid}(H_0,H),\qquad
+ d(H_0,J)^2=\frac12,\qquad d(H_0,z)=1.
+\end{equation}
+
+In each orbit factor let $J_m$ be the midpoint of the realizations
+of $p$ and $q$.  Translate the realization of $H_0$ to $x_\nu$.
+The realizations of $p,q$ are then given by the two letters on the
+adjacent $F_C$ edges, with their fixed orientation signs.  Therefore
+the translated vector toward $J_m$ is a function $h_I$ of just
+those two fields, with norm at most $M$.  The translated vector
+toward $z$ is $v_{\sigma(e_0)}$, where $e_0$ is the loose field.
+These three fields are admissible.  Lemma~\ref{lem:conditional-test}
+shows that the upper limit of the averaged scalar product of these
+two vectors is at most zero.
+
+Fix $a,b\in(0,1]$.  In each factor take the points at fractions
+$a,b$ along the segments from $H_0$ to $J_m,z$.  Their tangent
+vectors are the corresponding scalar multiples of the two vectors
+just considered.  Apply Equation~\eqref{energy:log-comparison},
+average over factors, and take the ultralimit.  Denoting the
+limiting fractional points by $J_a,z_b$, respectively, gives
+\begin{equation}\label{energy:fractional-angle}
+ d(J_a,z_b)^2\ge
+ a^2d(H_0,J)^2+b^2d(H_0,z)^2
+ =\frac{a^2}{2}+b^2.
+\end{equation}
+Thus every such comparison angle at $H_0$ is at least $\pi/2$.
+Letting both fractions decrease to zero in $Z$ proves
+\begin{equation}\label{energy:ternary-angle}
+ \angle_{H_0}(J,z)\ge\frac\pi2.
+\end{equation}
+Only distances of fixed-fraction points passed through the
+ultralimit; the angle was computed afterwards in the limiting
+space.
+
+\subsection{The resulting finite configuration}
+
+We collect the conclusions in the form needed in
+Section~\ref{sec:obstruction}.
+
+\begin{proposition}\label{prop:configuration}
+Let $P_\nu$ be a sequence of aspherical instances whose master-match
+statistics and six-letter relative counts satisfy the conclusions
+of Proposition~\ref{prop:sampling}, with errors tending to zero for
+every fixed test and every fixed $t$.  If every $\pi_1(P_\nu)$ has
+a proper cocompact isometric action on a proper complete $\CAT(0)$
+space, then there are a complete $\CAT(0)$ space $Z$, a
+configuration indexed by the core vertices of $\Gamma$, and a
+point $c\in Z$ with the following properties.
+\begin{enumerate}
+\item Every core edge has length one.  Every nonbacktracking
+two-edge core path has squared endpoint distance two.  Every
+three-edge core path with distinct admissible fields has squared
+endpoint distance at least three.
+\item Every core hexagon has its three opposite-pair midpoints
+equal to $c$.  All core vertices have distance $r=\sqrt3/2$ from
+$c$.  For any two vertices occurring in one core hexagon, their
+central triangle with $c$ is flat and their tangent scalar
+product is given by Equation~\eqref{energy:hexagon-products}.
+\item With $H_0=CD$ and $J=\operatorname{mid}(X_C,Y_C)$, one has
+\[
+ d(H_0,J)^2=\frac12,\qquad d(H_0,X_D)=1,
+ \qquad \angle_{H_0}(J,X_D)\ge\frac\pi2.
+\]
+More precisely, all the fractional-point inequalities in
+Equation~\eqref{energy:fractional-angle} hold.
+\end{enumerate}
+\end{proposition}
+
+\begin{proof}
+Use the normalized minimizing actions of
+Lemma~\ref{energy:minimizer} and the uniform bound of
+Lemma~\ref{energy:displacement-bound}.  The product construction,
+Lemma~\ref{energy:path-bounds}, and the decoration-square argument
+give the first assertion.  Lemmas~\ref{energy:hexagon-flatness}
+and~\ref{energy:common-center} give the second.  The fixed-fraction
+argument above gives the third.
+\end{proof}
diff --git a/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/sections/06-obstruction.tex b/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/sections/06-obstruction.tex
new file mode 100644
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@@ -0,0 +1,477 @@
+\section{The geometric obstruction}
+\label{sec:obstruction}
+
+We now show that the configuration obtained in
+Proposition~\ref{prop:configuration} cannot occur in a complete $\CAT(0)$
+space.  The argument takes place first in that space and then in its
+completed tangent cone at the common hexagon center.  The tangent cone
+is not assumed to be a vector space, so we specify the meaning of its
+sums and use concavity whenever an additive inequality is required.
+
+\subsection{Cone notation and square rigidity}
+
+For a $\CAT(0)$ Euclidean cone $T$ with vertex $0$, let $\delta_a$ denote
+radial dilation by $a\geq 0$.  Write $a u=\delta_a(u)$ and
+\[
+ \norm{u}=d_T(0,u),\qquad
+ \ip{u}{v}=\frac{\norm{u}^2+\norm{v}^2-d_T(u,v)^2}{2}.
+\]
+For nonzero vectors this last expression is
+$\norm{u}\norm{v}\cos\angle(u,v)$, where the angular distance is
+truncated at $\pi$.  Define
+\begin{equation}
+ u\oplus v=2\operatorname{mid}(u,v).
+ \label{geom:cone-addition}
+\end{equation}
+This operation is commutative, but no associativity or cancellation law
+will be used.  If the directions of $u,v$ are joined by a geodesic of
+length less than $\pi$, its cone is a Euclidean sector, and
+$u\oplus v$ is the ordinary vector sum in that sector.  More generally,
+all two-vector calculations below are either made in such a sector or
+follow from the metric definition in Equation~\eqref{geom:cone-addition}.
+
+The scalar-product concavity established in Lemma~\ref{energy:cone-mean}
+implies
+\begin{equation}
+ \ip{w}{u\oplus v}\geq \ip{w}{u}+\ip{w}{v}.
+ \label{geom:sum-concavity}
+\end{equation}
+Indeed, apply concavity at the midpoint of $u,v$ and use homogeneity
+under radial dilation.  This is an inequality in a general cone; we
+will not replace it by an equality without establishing rigidity.
+
+We use two established equality principles for $\CAT(0)$ spaces.  A
+triangle is flat if one vertex angle equals its Euclidean comparison
+angle, or if a nontrivial interior comparison inequality is an
+equality; see \cite[Proposition~II.2.9 and
+Exercise~II.2.10(1)]{BH1999} and
+\cite[Proposition~I.3.13(ii)]{Ballmann1995}.  Here and below, flat means that the
+convex hull is isometric to the corresponding Euclidean triangle.
+
+\begin{lemma}[Square rigidity]
+\label{geom:square}
+Let $p_0,p_1,p_2,p_3$ be points of a $\CAT(0)$ space, with indices
+taken modulo four.  Suppose
+\[
+ d(p_i,p_{i+1})=1,\qquad
+ d(p_0,p_2)=d(p_1,p_3)=\sqrt{2}.
+\]
+Their convex hull is a unit Euclidean square, in the indicated cyclic
+order.  In particular, the two diagonals have the same midpoint.
+\end{lemma}
+
+\begin{proof}
+Let $o$ be the midpoint of $p_0,p_2$.  Squared-distance convexity gives
+\[
+ d(o,p_1)^2\leq\frac12,\qquad d(o,p_3)^2\leq\frac12.
+\]
+Since
+\[
+ \sqrt2=d(p_1,p_3)\leq d(p_1,o)+d(o,p_3)\leq\sqrt2,
+\]
+both inequalities are equalities and $o$ is also the midpoint of
+$p_1,p_3$.  The equality for $d(o,p_1)$ is an interior comparison
+equality in the triangle $p_1,p_0,p_2$, so that triangle is flat.  The
+same argument, with each of the four vertices as the vertex opposite
+a diagonal midpoint, makes each triangle on three consecutive vertices
+flat.  The angle at each corner of the quadrilateral is therefore
+$\pi/2$.  The flat quadrilateral theorem
+\cite[Theorem~II.2.11]{BH1999} applies: the sum of these four angles is
+$2\pi$.  Its Euclidean quadrilateral has the prescribed sides and
+diagonals, and hence is a unit square.
+\end{proof}
+
+\subsection{The sums at the four doubled points}
+
+Suppose that $Z$ is a complete $\CAT(0)$ space containing the
+configuration from Proposition~\ref{prop:configuration}.  Use the graph
+vertex names for the corresponding points of $Z$.  Let $c$ be the common
+hexagon center and put
+\[
+ r=\frac{\sqrt3}{2},\qquad T=T_c Z.
+\]
+The completed tangent cone $T$ is $\CAT(0)$, and its completed space of
+directions is $\CAT(1)$ by \cite[Theorem~II.3.19]{BH1999}.  Write
+\[
+ x_i=\log_c X_i,\qquad y_i=\log_c Y_i,\qquad
+ h_H=\log_c H
+\]
+for the named radial vectors, and abbreviate $h_{ij}=h_H$ when
+$H=ij$.  All these vectors have norm $r$.
+
+Any two distinct line vertices $H,H'$ occur together in a core
+hexagon: take their intersection point $i$, choose a point $j\ne i$
+on $H$ and a point $k\ne i$ on $H'$, and use the $X$ representatives
+of the noncollinear triple $i,j,k$.  Their central triangle is flat
+by Proposition~\ref{prop:configuration}, and their distance is
+$\sqrt2$.  Consequently
+\begin{equation}
+ \ip{h_H}{h_{H'}}=-\frac14
+ \quad\text{for distinct lines }H,H'.
+ \label{geom:line-products}
+\end{equation}
+
+A line through a doubled point $i\in\{A,B,C,D\}$ is called
+\emph{full at $i$} if it is adjacent to both $X_i$ and $Y_i$.
+All lines through $A,B,C$ are full; the full lines at $D$ are $AD,BD$.
+
+\begin{lemma}[Fan identities]
+\label{geom:fan-sums}
+For $i\in\{A,B,C,D\}$ there is a unit vector $M_i\in T$ such that
+\begin{equation}
+ M_i=x_i\oplus y_i=h_H\oplus h_{H'}
+ \quad\text{whenever }H,H'\text{ are distinct full lines at }i.
+ \label{geom:fan-identities}
+\end{equation}
+Each sum displayed here is computed in a nondegenerate Euclidean
+sector.  Moreover,
+\begin{equation}
+ \ip{x_i}{y_i}=-\frac14,
+ \qquad
+ \ip{M_i}{h_H}=\ip{M_i}{h_{H'}}=\frac12.
+ \label{geom:fan-products}
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+The four-cycle $X_i,H,Y_i,H'$ has unit sides and diagonals of length
+$\sqrt2$, by the two-step assertions of
+Proposition~\ref{prop:configuration}.  Lemma~\ref{geom:square} makes it
+a flat square.  Let $J_i$ be its center, which is the midpoint of
+$X_i,Y_i$ and also of $H,H'$.
+
+The triangle $c,H,H'$ is flat.  Its median therefore gives
+\[
+ d(c,J_i)^2=r^2-\frac14 d(H,H')^2=\frac14.
+\]
+On the other hand, squared-distance convexity in the triangle
+$c,X_i,Y_i$ has right-hand side
+\[
+ \frac12 d(c,X_i)^2+\frac12 d(c,Y_i)^2
+       -\frac14 d(X_i,Y_i)^2
+ =\frac14.
+\]
+Thus this is also an equality, and $c,X_i,Y_i$ is a flat triangle.
+Taking its midpoint in the tangent cone proves
+\[
+ x_i\oplus y_i=2\log_c J_i=h_H\oplus h_{H'}.
+\]
+Define $M_i=2\log_c J_i$; it has norm one.  The left-hand side makes
+it independent of the full-line pair chosen.  The cosine law gives
+$\ip{x_i}{y_i}=-1/4$.  The pair $h_H,h_{H'}$ has angular separation
+$\arccos(-1/3)<\pi$, and in its Euclidean sector
+\[
+ \ip{h_H\oplus h_{H'}}{h_H}
+ =\norm{h_H}^2+\ip{h_{H'}}{h_H}
+ =\frac34-\frac14=\frac12.
+\]
+The same computation applies to $h_{H'}$.
+\end{proof}
+
+We will obtain a contradiction by evaluating the scalar product of
+the unit vectors $M_A,M_B$ in two ways.  One bound is already
+available.  At $A,B$, the fan identities give
+\[
+ M_A=h_{AB}\oplus h_{AC},\qquad
+ M_B=h_{AB}\oplus h_{BC}.
+\]
+Apply Equation~\eqref{geom:sum-concavity} in both entries.  The three
+lines $AB,AC,BC$ are distinct, so
+Equation~\eqref{geom:line-products} yields
+\begin{equation}
+ \ip{M_A}{M_B}
+ \geq\norm{h_{AB}}^2+\ip{h_{AB}}{h_{BC}}
+       +\ip{h_{AC}}{h_{AB}}+\ip{h_{AC}}{h_{BC}}
+ =0.
+ \label{geom:nonnegative}
+\end{equation}
+The remaining argument will force $M_A,M_B$ to be antipodal, with
+scalar product $-1$.  We first use the exceptional incidence to show
+that $M_C,M_D$ are antipodal.  Their axis will let us place
+$h_{AC},h_{BC},h_{AD},h_{BD}$ in a product $\mathbb R\times Q$,
+where the $\CAT(0)$ factor $Q$ contains a flat square formed by
+their second coordinates.  The opposite side midpoints of that square
+will then force $M_A,M_B$ to be antipodal through the fan identities
+at $A,B$.
+
+\subsection{The identity at the exceptional incidence}
+
+Set
+\[
+ H_0=CD,\qquad h_0=h_{CD},\qquad
+ p=X_C,\quad q=Y_C,\quad z=X_D,
+\]
+and let $J$ be the midpoint of $p,q$.  The missing edge from $Y_D$
+to $H_0$ is responsible for the asymmetry used in the next lemma.
+
+\begin{lemma}[Exceptional-incidence identity]
+\label{geom:exceptional}
+The vectors $M_C,x_D$ lie in a Euclidean sector in $T$, and
+\begin{equation}
+ h_0=M_C\oplus x_D.
+ \label{geom:exceptional-sum}
+\end{equation}
+Within this sector,
+\begin{equation}
+ \ip{M_C}{x_D}=-\frac12,\qquad
+ \ip{M_C}{h_0}=\frac12,\qquad
+ \ip{h_0}{x_D}=\frac14.
+ \label{geom:exceptional-products}
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Apply the square construction in Lemma~\ref{geom:fan-sums} at $C$,
+using $H_0$ and any other line through $C$.  Since $J$ is its center,
+\begin{equation}
+ d(H_0,J)^2=\frac12,\qquad d(c,J)^2=\frac14.
+ \label{geom:exceptional-distances}
+\end{equation}
+Also $d(H_0,z)=1$, $d(c,z)=d(c,H_0)=r$, and the two paths
+$p,H_0,z$ and $q,H_0,z$ give
+\[
+ d(p,z)^2=d(q,z)^2=2.
+\]
+The ternary midpoint conclusion of
+Proposition~\ref{prop:configuration} states that the angle at $H_0$
+between the segments to $J,z$ is at least $\pi/2$.  Angle comparison
+and Equation~\eqref{geom:exceptional-distances} therefore imply
+\[
+ d(J,z)^2\geq d(H_0,J)^2+d(H_0,z)^2=\frac32.
+\]
+The reverse inequality follows from the fact that $J$ is the midpoint
+of $p,q$:
+\[
+ d(J,z)^2\leq\frac12d(p,z)^2+\frac12d(q,z)^2
+                    -\frac14d(p,q)^2=\frac32.
+\]
+The angle and distance comparisons are consequently equalities.
+The triangle $H_0,J,z$ is flat, with a right angle at $H_0$.
+
+Let $B_0$ be the point of $[J,z]$ at parameter $1/3$ from $J$.
+Calculating in this right triangle gives
+\[
+ d(H_0,B_0)^2
+ =\left(\frac23\right)^2\frac12
+  +\left(\frac13\right)^2=\frac13.
+\]
+Squared-distance convexity toward $c$ gives
+\begin{align*}
+ d(c,B_0)^2
+ &\leq\frac23 d(c,J)^2+\frac13d(c,z)^2
+                 -\frac29d(J,z)^2\\
+ &=\frac23\frac14+\frac13\frac34-\frac29\frac32
+ =\frac1{12}.
+\end{align*}
+The triangle inequality now reads
+\[
+ \frac{\sqrt3}{2}=d(H_0,c)
+ \leq d(H_0,B_0)+d(B_0,c)
+ \leq\frac1{\sqrt3}+\frac1{2\sqrt3}
+ =\frac{\sqrt3}{2}.
+\]
+Hence every inequality is an equality.  In particular, $B_0$ lies on
+$[H_0,c]$, in that order, and the triangle $c,J,z$ is flat by its
+interior comparison equality.  Put $j=\log_c J$.  In the Euclidean
+sector containing $j,x_D$, the vector to $B_0$ is
+$(2/3)j+(1/3)x_D$.  The collinearity just proved and the distances
+$d(c,H_0)=3d(c,B_0)$ therefore imply
+\[
+ h_0=3\log_c B_0=2j+x_D=M_C\oplus x_D.
+\]
+Here the two ordinary additions take place in that Euclidean sector,
+and $M_C=2j$ by Lemma~\ref{geom:fan-sums}.
+
+The norm identity in this sector gives
+\[
+ \frac34=\norm{h_0}^2
+ =1+\frac34+2\ip{M_C}{x_D},
+\]
+so $\ip{M_C}{x_D}=-1/2$.  Pairing the sector identity with either
+summand yields the remaining two products in
+Equation~\eqref{geom:exceptional-products}.
+\end{proof}
+
+\subsection{A semicircle in the space of directions}
+
+\begin{lemma}
+\label{geom:antipodal}
+The unit vectors $M_C,M_D$ are antipodal in $T$:
+\[
+ d_T(M_C,M_D)=2,\qquad \ip{M_C}{M_D}=-1.
+\]
+\end{lemma}
+
+\begin{proof}
+Take either full line $H$ at $D$.  The path
+$Y_D,H,X_D,H_0$ uses two fields of $F_D$ and the loose field, so it is
+one of the admissible three-step paths.  Thus
+\[
+ d(Y_D,H_0)\geq\sqrt3=d(Y_D,c)+d(c,H_0).
+\]
+Equality follows from the triangle inequality.  The segment from
+$Y_D$ to $H_0$ passes through $c$, and the directions of $y_D,h_0$
+are antipodal.
+
+Let $\Sigma$ be the completed space of directions of $Z$ at $c$, and
+use a hat for the direction of a nonzero vector.  Define
+\[
+ a=\arccos(1/\sqrt3),\qquad b=\arccos(1/3).
+\]
+Both lie strictly between zero and $\pi/2$, and $2a+b=\pi$.
+Equation~\eqref{geom:exceptional-products} gives
+\[
+ d_\Sigma(\widehat M_C,\widehat h_0)=a,
+ \qquad d_\Sigma(\widehat h_0,\widehat x_D)=b.
+\]
+The two arcs occur consecutively in the Euclidean sector of
+Lemma~\ref{geom:exceptional}; their concatenation is its geodesic
+direction arc.  By Equation~\eqref{geom:fan-products},
+\[
+ d_\Sigma(\widehat x_D,\widehat y_D)=\arccos(-1/3)=\pi-b.
+\]
+The direction of $M_D=x_D\oplus y_D$ bisects this last arc, because
+its two summands have equal length.  Consequently
+$d_\Sigma(\widehat x_D,\widehat M_D)=a$.
+
+The concatenation
+$\widehat h_0,\widehat x_D,\widehat y_D$ has length $\pi$ and
+antipodal endpoints.  It is therefore minimizing, so its two arcs
+continue geodesically through $\widehat x_D$.  It follows that
+\[
+ \widehat M_C,\quad\widehat h_0,\quad\widehat x_D,
+ \quad\widehat M_D
+\]
+is a local geodesic in $\Sigma$: continuation at $\widehat h_0$
+holds inside the first sector, and continuation at $\widehat x_D$
+holds inside the minimizing semicircle just described.  Its length
+is $a+b+a=\pi$.  The space $\Sigma$ is $\CAT(1)$, so
+\cite[Proposition~II.1.4(2)]{BH1999} makes this local geodesic
+globally minimizing, including the endpoint case of length exactly
+$\pi$.  Equivalently, apply the short local-geodesic assertion to
+every strict initial subarc and pass to the endpoint by continuity
+of distance.  Its endpoints are antipodal.  Since $M_C,M_D$ have norm
+one, the cone distance and scalar product have the stated values.
+\end{proof}
+
+\subsection{Splitting the axis and obtaining a flat square}
+
+\begin{proposition}
+\label{prop:no-configuration}
+No complete $\CAT(0)$ space contains the configuration asserted in
+Proposition~\ref{prop:configuration}.
+\end{proposition}
+
+\begin{proof}
+Suppose otherwise and retain the notation above.  By
+Equation~\eqref{geom:nonnegative}, $\ip{M_A}{M_B}\geq0$.
+We will contradict this bound by splitting the axis supplied by
+Lemma~\ref{geom:antipodal}.  Use the fan identities
+\[
+ M_C=h_{AC}\oplus h_{BC},\qquad
+ M_D=h_{AD}\oplus h_{BD}.
+\]
+The four lines in these expressions are pairwise distinct.  Therefore
+Equation~\eqref{geom:sum-concavity} implies
+\[
+ \ip{M_C}{h_{AD}}\geq-\frac12,
+ \qquad \ip{M_C}{h_{BD}}\geq-\frac12.
+\]
+Lemma~\ref{geom:antipodal} forces equality in the chain
+\[
+ -1=\ip{M_C}{M_D}
+ \geq\ip{M_C}{h_{AD}}+\ip{M_C}{h_{BD}}\geq-1.
+\]
+Interchanging the roles of the two sums gives the symmetric
+equalities.  Combining these with
+Equation~\eqref{geom:fan-products}, we obtain
+\begin{equation}
+\begin{array}{c|rrrr}
+ &h_{AC}&h_{BC}&h_{AD}&h_{BD}\\ \hline
+ \ip{M_C}{\,\cdot\,}& 1/2&1/2&-1/2&-1/2\\
+ \ip{M_D}{\,\cdot\,}&-1/2&-1/2& 1/2& 1/2.
+\end{array}
+\label{geom:axis-products}
+\end{equation}
+
+The opposite rays $M_C,M_D$ form a complete geodesic line
+$\ell\subset T$ through the cone vertex.  Put
+$a=\arccos(1/\sqrt3)$.  For each vector in the
+four columns of Equation~\eqref{geom:axis-products}, the angles to
+the two ends of $\ell$ are $a$ and $\pi-a$, in one order or the
+other.  The two corresponding direction arcs concatenate to a
+minimizing semicircle.  Its Euclidean cone is an isometric flat
+half-plane with boundary $\ell$ and containing the vector.  In
+particular, that vector lies on a complete line parallel to $\ell$.
+
+The parallel-set product decomposition
+\cite[Theorem~II.2.14]{BH1999} now applies in $T$; it requires no
+properness.  Write its convex parallel set as
+\[
+ P(\ell)=\R\times Q,\qquad
+ \ell=\R\times\{q_0\},\qquad
+ 0=(0,q_0),\quad M_C=(1,q_0),\quad M_D=(-1,q_0).
+\]
+The factor $Q$ is a $\CAT(0)$ space.  Within each of the flat
+half-planes above, projection onto $\ell$ shows that the axial
+coordinate is the scalar product with the unit vector $M_C$.
+Thus Equation~\eqref{geom:axis-products} gives
+\begin{equation}
+ h_{AC}=(1/2,u),\quad h_{BC}=(1/2,u'),\quad
+ h_{AD}=(-1/2,v),\quad h_{BD}=(-1/2,v')
+ \label{geom:split-coordinates}
+\end{equation}
+for points $u,u',v,v'\in Q$.  Their distances to $q_0$ are all
+$1/\sqrt2$, since each original vector has squared norm $3/4$.
+
+Distinct line vectors have squared cone distance two, by
+Equation~\eqref{geom:line-products}.  The product metric in
+Equation~\eqref{geom:split-coordinates} therefore gives
+\[
+ d_Q(u,u')=d_Q(v,v')=\sqrt2,
+\]
+and
+\[
+ d_Q(u,v)=d_Q(u,v')=d_Q(u',v)=d_Q(u',v')=1.
+\]
+Furthermore, $h_{AC}\oplus h_{BC}=M_C$ means that the midpoint
+of these two line vectors is $(1/2,q_0)$: the midpoint is the
+half-dilation of $M_C$ on the axis.  Product midpoints consequently
+give $q_0=\operatorname{mid}(u,u')$.  The identity at $D$ likewise
+gives $q_0=\operatorname{mid}(v,v')$.
+
+In cyclic order $u,v,u',v'$, these points satisfy
+Lemma~\ref{geom:square}.  They therefore span a unit flat square
+in $Q$, centered at $q_0$.  Let
+\[
+ m=\operatorname{mid}(u,v),\qquad
+ m'=\operatorname{mid}(u',v').
+\]
+They are the midpoints of opposite sides of that square, and hence
+\begin{equation}
+ d_Q(q_0,m)=d_Q(q_0,m')=\frac12,
+ \qquad d_Q(m,m')=1.
+ \label{geom:opposite-midpoints}
+\end{equation}
+This square lies in the transverse factor only; no planar realization
+of the original incidence configuration has been assumed.
+Figure~\ref{fig:transverse-square} records the square and the axis that
+has been split off.
+
+\input{figures/transverse-square}
+
+Finally, the full-line choices $AC,AD$ at $A$ and $BC,BD$ at $B$
+give
+\[
+ M_A=h_{AC}\oplus h_{AD}=\delta_2(0,m),\qquad
+ M_B=h_{BC}\oplus h_{BD}=\delta_2(0,m').
+\]
+Here $\delta_2$ is dilation in the original cone $T$; no cone
+structure on $Q$ is needed.  It doubles all distances.  By
+Equation~\eqref{geom:opposite-midpoints},
+\[
+ \norm{M_A}=\norm{M_B}=1,\qquad d_T(M_A,M_B)=2,
+\]
+so $\ip{M_A}{M_B}=-1$.  This contradicts
+Equation~\eqref{geom:nonnegative} and proves the proposition.
+\end{proof}
diff --git a/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/sections/07-completion.tex b/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/sections/07-completion.tex
new file mode 100644
index 0000000000000000000000000000000000000000..bd2ce3e35d6d06e2367cecf92925be92b6ec05f2
--- /dev/null
+++ b/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/build/sections/07-completion.tex
@@ -0,0 +1,70 @@
+\section{Selection and the finite-model conclusion}\label{sec:completion}
+We now assemble the arguments, keeping the order of all choices explicit.
+
+\begin{lemma}[A deterministic diagonal sequence]\label{lem:completion-diagonal}
+There is a sequence of realizations of the tables, with $n_j\to\infty$ and positive parameters $\alpha_j,\beta_j,\delta_j\to0$, such that
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item the associated complexes $P_j$ are aspherical and have a linear letter disk-filling inequality;
+\item the master match sets are nonempty and their single-field marginals tend to uniformity in total variation;
+\item for every admissible pair $I,\{e\}$ and every fixed integer $t$, the common-$I$ coupling in Proposition~\ref{prop:sampling} has both of its total variation errors tending to zero along the sequence;
+\item the six-letter table pair counts have uniformly vanishing relative error.
+\end{enumerate}
+The groups $G_j=\pi_1P_j$ are infinite.
+\end{lemma}
+\begin{proof}
+At stage $j$, request all the admissible tests in the fixed master pattern with $1\le t\le j$. This is a finite list. Choose $\varepsilon_j>0$ small enough for Proposition~\ref{prop:sampling} for this list and for Theorem~\ref{thm:diagram}, with $\varepsilon_j<1/(400j)$, and set
+\[
+ \alpha_j=\delta_j=\varepsilon_j,
+ \qquad \beta_j=8\varepsilon_j.
+\]
+These parameters satisfy $\beta_j>7\alpha_j$, and the exponent $3-\beta_j-90\delta_j$ of the master count is positive. Moreover,
+\[
+ j(\beta_j+90\delta_j)=98j\varepsilon_j<\frac{98}{400}<\frac12,
+\]
+so the explicit bound in Section~\ref{sec:sampling} preserves every overlap margin required for the finite list.
+
+Fix these parameters. The proof of Theorem~\ref{thm:diagram} selects a finite fragment threshold, possibly depending on $j$. Its coincidence inequalities hold with probability tending to one as $n\to\infty$. For the same fixed parameters, Proposition~\ref{prop:sampling} gives the requested finite collection of sampling properties with probability tending to one. Choose $n_j>n_{j-1}$ so large that there is positive probability that all these events hold and all requested sampling errors are at most $1/j$. Select one realization in their intersection. No comparison between the fragment threshold and $n$ is needed before fixing the parameters and threshold. This proves \textup{(i)}--\textup{(iv)}.
+
+Each $G_j$ has a quotient $C_2$ by Lemma~\ref{lem:model-presentation}. If it were finite, its universal cover would be a finite contractible CW complex, and multiplicativity of Euler characteristic would give
+\[
+ 1=\chi(\widetilde P_j)=|G_j|\chi(P_j).
+\]
+Since $\chi(P_j)$ is an integer, this forces $|G_j|=1$, contrary to the quotient. Hence $G_j$ is infinite.
+\end{proof}
+
+\begin{proposition}\label{prop:completion-no-action}
+For all sufficiently large $j$, the group $G_j$ in Lemma~\ref{lem:completion-diagonal} admits no geometric action on any proper complete $\CAT(0)$ space.
+\end{proposition}
+\begin{proof}
+Suppose, more generally, that infinitely many $G_j$ admitted such actions. Restrict to that subsequence and choose one action for each of its groups. All fixed-test limits in Lemma~\ref{lem:completion-diagonal} remain valid. The action spaces, their dimensions, and their local geometries are allowed to vary.
+
+Normalize the minimizing generator energy for each action as in Equation~\eqref{energy:normalization}. The infinitude just proved makes this energy positive. The displacement estimate and Lemma~\ref{lem:conditional-test} hold uniformly for these choices of actions. Proposition~\ref{prop:configuration} therefore supplies the limiting configuration. Proposition~\ref{prop:no-configuration} excludes it. Thus only finitely many of the selected groups can admit such actions; in particular, an instance with no action exists.
+\end{proof}
+
+\subsection{Passing to a finite simplicial model}
+We record the filling transfer because the conclusion is about edge loops in a finite simplicial complex, rather than just words in a presentation.
+
+\begin{lemma}[Finite-model filling transfer]\label{lem:completion-transfer}
+Let $P$ be a finite connected CW complex with contractible universal cover. Suppose a finite presentation of $\pi_1P$ has a linear disk-filling inequality. Then $P$ has a finite connected simplicial homotopy model $K$, whose universal cover is contractible and satisfies a linear combinatorial disk-filling inequality.
+\end{lemma}
+\begin{proof}
+A finite CW complex has a finite simplicial homotopy model of the same dimension~\cite[Theorem~2C.5]{Hatcher2002}. One construction proceeds over its finitely many cells: approximate attaching maps by simplicial maps after finite subdivisions, and use simplicial analogues of their mapping cylinders for the attachments. Let $K$ be the resulting finite simplicial model. A homotopy equivalence lifts to a homotopy equivalence of universal covers, so $\widetilde K$ is contractible.
+
+For completeness, the linear estimate is independent of the chosen finite presentation. Choose substitutions $\phi$ from new generators to old words and $\psi$ from old generators to new words, inducing inverse group identifications. Let $L$ bound the lengths of the words $\phi(b)$, let $M$ bound the new-presentation filling areas of the finitely many substituted old relators $\psi(r)$, and let $D$ bound the filling areas of the words $b\psi(\phi(b))^{-1}$. The latter two lists consist of null words, and all three lists are finite, so these bounds exist. A new null word of length $\ell$ substitutes to an old word of length at most $L\ell$. If the old filling constant is $C_{\rm old}$, filling that word and replacing its relator cells costs at most $MC_{\rm old}L\ell$ new cells. The boundary substitutions cost at most $D\ell$ more. Thus the new filling constant is at most $MC_{\rm old}L+D$.
+
+Apply this to the maximal-tree presentation of the finite complex $K$. Its generators correspond to the non-tree edges, and its relators to its triangular faces. A null-homotopic combinatorial edge loop in $K$ of length $\ell$ gives a null word of length at most $\ell$ in these generators after inserting paths in the fixed maximal tree. The tree has finitely many edges, so the inserted paths have uniformly bounded lengths. Fill the word linearly in the maximal-tree presentation; each relator cell is replaced by the corresponding triangular face and tree-path cancellations. The boundary collar likewise cancels the inserted tree paths with linear overhead. Thus the original loop bounds a diagram over $K$ with at most $C\ell$ triangular faces for a fixed finite $C$.
+
+An edge loop in $\widetilde K$ projects to a null loop in $K$. Its disk diagram lifts with the chosen boundary lift, giving the required estimate in the universal cover.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+Choose an instance $P_j$ supplied by Proposition~\ref{prop:completion-no-action}. Its finite letter presentation has a linear disk bound by Theorem~\ref{thm:diagram}. Apply Lemma~\ref{lem:completion-transfer} to obtain $K$. Since $P_j$ has dimension two, the same-dimension construction gives a two-dimensional $K$. This proves the asserted asphericity and linear filling estimate. Its fundamental group is $G_j$, so it has the no-action property. The equivalence between a linear Dehn function and word hyperbolicity gives the final group-theoretic assertion~\cite[Exercise~III.H.2.5(5), Proposition~III.H.2.7, and Theorem~III.H.2.9]{BH1999}.
+
+Suppose that a finite simplicial complex $L$ homotopy equivalent to $K$ admitted a compatible locally $\CAT(0)$ geodesic length metric. The metric space $L$ is compact, hence complete and locally compact. Give its universal cover the induced length metric, for which the covering map is a local isometry~\cite[Proposition~I.3.25]{BH1999}. This cover is complete: a Cauchy sequence upstairs projects to a convergent sequence downstairs; sufficiently short connecting paths force a tail into one sheet over an evenly covered neighborhood of the limit, where it converges. The Cartan--Hadamard theorem makes the cover globally $\CAT(0)$~\cite[Theorem~II.4.1(2)]{BH1999}. It is locally compact, and the length-space Hopf--Rinow theorem makes it proper~\cite[Proposition~I.3.7]{BH1999}. Its deck transformations act freely, properly and cocompactly by isometries.
+
+The isomorphism $\pi_1L\cong\pi_1K=G_j$ therefore gives a geometric action of $G_j$ on a proper complete $\CAT(0)$ space, a contradiction. This also excludes compatible locally $\CAT(-1)$ geodesic length metrics, since a $\CAT(-1)$ space is $\CAT(0)$~\cite[Theorem~II.1.12(1)]{BH1999}.
+\end{proof}
+
+\begin{remark}
+The argument makes no curvature assumption on the random complexes $P_j$. Curvature comparison is used only in a hypothetical action space, after the combinatorial filling and asphericity statements have been established. Nor is a common filling constant needed for the diagonal sequence: only the constant for the single chosen finite complex appears in Theorem~\ref{thm:main}.
+\end{remark}
diff --git a/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/paper.pdf b/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..172de3b1ad51dac6e63a1d3023e47ea535e73381
--- /dev/null
+++ b/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:124f55b0821cc7f18c1b64600b5e5157d939bbc4a6f66294dc1456fc633eb264
+size 331299
diff --git a/preprints/A-latest-anchor-induction-with-spectrally-compact-masks-for-worst-case-trace-reconstruction-October-5-2026/README.md b/preprints/A-latest-anchor-induction-with-spectrally-compact-masks-for-worst-case-trace-reconstruction-October-5-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..005bb672ce31cd8737eae0959f46ef7666d06984
--- /dev/null
+++ b/preprints/A-latest-anchor-induction-with-spectrally-compact-masks-for-worst-case-trace-reconstruction-October-5-2026/README.md
@@ -0,0 +1,18 @@
+# [A latest-anchor induction with spectrally compact masks for worst-case trace reconstruction](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** October 5, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-latest-anchor-induction-with-spectrally-compact-masks-for-worst-case-trace-reconstruction-October-5-2026,
+  author = {{OpenAI}},
+  title = {{A latest-anchor induction with spectrally compact masks for worst-case trace reconstruction}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-latest-anchor-induction-with-spectrally-compact-masks-for-worst-case-trace-reconstruction-October-5-2026/paper.pdf}{OAI:A-latest-anchor-induction-with-spectrally-compact-masks-for-worst-case-trace-reconstruction-October-5-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-latest-anchor-induction-with-spectrally-compact-masks-for-worst-case-trace-reconstruction-October-5-2026/build/main.tex b/preprints/A-latest-anchor-induction-with-spectrally-compact-masks-for-worst-case-trace-reconstruction-October-5-2026/build/main.tex
new file mode 100644
index 0000000000000000000000000000000000000000..a35fa7bce62082169c3b7c6d1bb82fa7991bd6ab
--- /dev/null
+++ b/preprints/A-latest-anchor-induction-with-spectrally-compact-masks-for-worst-case-trace-reconstruction-October-5-2026/build/main.tex
@@ -0,0 +1,9 @@
+\documentclass{article}
+\usepackage{pdfpages}
+\usepackage[unicode,hidelinks]{hyperref}
+\hypersetup{pdftitle={A latest-anchor induction with spectrally compact masks for worst-case trace reconstruction},pdfauthor={OpenAI},pdfsubject={Stated improvement; optimal complexity open}}
+\pagestyle{empty}
+\begin{document}
+\includepdf[pages=-,pagecommand={},addtotoc={1,section,1,{Summary},s-summary}]{summary.pdf}
+\includepdf[pages=-,pagecommand={},addtotoc={1,section,1,{Report},s-part-1}]{part-1.pdf}
+\end{document}
diff --git a/preprints/A-latest-anchor-induction-with-spectrally-compact-masks-for-worst-case-trace-reconstruction-October-5-2026/build/part-1.tex b/preprints/A-latest-anchor-induction-with-spectrally-compact-masks-for-worst-case-trace-reconstruction-October-5-2026/build/part-1.tex
new file mode 100644
index 0000000000000000000000000000000000000000..c4d578a0ea3ba54e43632963b9e9ef0508829aee
--- /dev/null
+++ b/preprints/A-latest-anchor-induction-with-spectrally-compact-masks-for-worst-case-trace-reconstruction-October-5-2026/build/part-1.tex
@@ -0,0 +1,488 @@
+\documentclass[11pt]{article}
+\usepackage{amsmath,amssymb,amsthm}
+\usepackage[margin=1in]{geometry}
+\usepackage[hidelinks]{hyperref}
+\newtheorem{thm}{Theorem}
+
+\title{A latest-anchor induction with spectrally compact masks\\for worst-case trace reconstruction}
+\author{OpenAI}
+\date{}
+
+\emergencystretch=3em
+\allowdisplaybreaks
+\begin{document}
+\maketitle
+
+\begin{abstract}
+We give an improved worst-case sample bound for reconstructing a string
+from independent deletion traces, with its length and retention
+probability known. For each fixed retention probability, the number of
+traces is quasipolynomial: the logarithm of the sample budget is
+$O((\log n)^3(1+\log\log(2n))^6)$. If the deletion probability is at most
+$n^{-\varepsilon}$ for fixed $\varepsilon>0$, polynomially many traces
+suffice. These bounds apply to binary strings and to strings of general
+symbols observed exactly. They concern sample complexity and do not
+assert an efficient reconstruction algorithm or matching optimality.
+\end{abstract}
+
+\section*{Bound and comparison}
+Consider the i.i.d.\ deletion channel on an unknown string $x$ of length $n$:
+a trace consists of the symbols retained, in order with no labels on their
+original positions, when each position is retained independently with
+probability $p$. Different traces are independent. The length and retention
+probability (and the channel model) are known to the reconstructor. All logarithms without
+specified base are natural. The standard worst-case reconstruction task is to
+recover the entire string with success probability at least $2/3$ for
+each possible $x$.
+\begin{thm}
+For integer $n\ge 2$ and $0<p\le1$, worst-case trace reconstruction of
+binary strings is possible in this channel with at most
+\[
+\exp\!\left(O\!\left(p^{-1}(\log n)^3(1+\log\log(2n))^6\right)\right)
+\]
+traces, with an absolute constant. Write $q=1-p$. More finely, when $q>0$, setting
+$\mathcal X=1+\log n/(1+\log(1/q))$, the logarithm of the required sample
+budget is at most
+\[
+ O\big(p^{-1}(\log n)\mathcal X^2 (1+\log\mathcal X)^6\big)
+\]
+with an absolute constant. In particular for fixed $\varepsilon>0$ and
+$0\le q\le n^{-\varepsilon}$ the number of traces can be polynomial in $n$
+(with parameters of the polynomial allowed to depend on $\varepsilon$).
+The same asymptotic bounds also suffice for
+strings with general symbols (assuming exact observation of retained symbols).
+This concerns sample complexity, not computational efficiency.
+\end{thm}
+
+This improves the order of the upper estimate with the parameter prescriptions
+of the quasipolynomial-bound proof of Burudgunte, Valiant and Wang \cite{BVW}
+(BVW). For clarity, their Theorem 43 statement uses an unspecified log-power
+$c$ and dependence $p^{-7/3}$ in the exponent. The proof in Section 8 uses
+$b_{\rm old}=\lceil\log_{1.1}\log_2 n\rceil$ as its depth bound,
+$p_{\rm base}=3^{-b_{\rm old}}\min(p,1/4)$, and a pair-test bound
+$4\tau_{\rm base}^{-2C^{b_{\rm old}}}$ (some $C\ge 1$), with
+$\log(1/\tau_{\rm base})$ given by a constant times
+$d_{\rm base}^{1/3}p_{\rm base}^{-1/3}$ for
+$d_{\rm base}=\max(p_{\rm base}^{-6},O(1))$. Thus even ignoring the
+$C^{b_{\rm old}}$ multiplier, the order from the inverse retention term of
+that estimate uses a log-power at least $(7/3)\log_{1.1}(3)>26$ for fixed $p$.
+Our improvement is relative to the proved quantitative estimate with those
+prescriptions, not a claim about all possible optimizations of BVW's approach.
+In particular our bound does not go below quasipolynomial order for fixed $0<p<1$.
+The method here is inspired by BVW's idea of testing phase relations via
+Fourier products; we give a different induction entirely on source-string
+generating functions, including a weighted, up-to-four-factor phase argument.
+
+We keep the period-avoiding-word idea from Chase's analysis and its relatives
+(see \cite{Chase,Rub}), using an exact contiguous pattern at the base. A new
+ingredient in the bound is to propagate additional low-dimensional factors
+with spectral compactification, enabling partial angle-budget relaxation
+during the induction before expanding at the last pivot. The polynomial bound
+for low deletion concerns sample complexity only; for polynomial-time bounds
+for the binary problem under e.g.\ deletion probability $q\le n^{-1/3-\varepsilon}$
+for fixed $\varepsilon>0$, see \cite{CDLS}.
+
+\section*{Gap probes with level masks}
+We first work with binary strings. Throughout assume the original length
+sufficiently large by an absolute constant. Smaller $n\ge2$ can be handled by
+collecting $\lceil p^{-n}\log 3\rceil$ traces to see one of full length with
+probability at least $2/3$; for bounded $n\ge2$ use $\log(1/p)\le1/p$ to absorb
+this budget. For $n=1$ separately $O(p^{-1})$ suffice. If $n\ge2$ and $p<1/n$ the
+full-length-trace strategy costs at most $\exp(O(p^{-1}\log n))$, since
+with $s=np\le1$, $n\log(1/p)=p^{-1}s(\log n+\log(1/s))$. Also if $n\ge2$ and
+$q=1-p<1/n$, then $p^n\ge(1-1/n)^n\ge1/4$,
+so a constant number of traces suffices via the same strategy. Hence for the rest
+write $q=1-p$ and suppose $p,q\ge1/n$.
+Prepend a known block of zeros of length equal to the original length to both candidates when comparing. Traces of the extended inputs can be simulated by prepending independently sampled deletion traces of the zero block. In the binary-case proof from here, \(n\) will now denote this doubled length so still \(p,q\ge1/n\).
+
+Fix any two distinct extended binary candidate strings \(x,y\) (sharing the known all-zero prefix), use \(0\)-based source indices, and write \(d\ge n/2\) for their first mismatch. Put
+\[
+\begin{split}
+\ell_*&=1+\log(1/q),\qquad b=\lceil\log_2(1+\log n/\ell_*)\rceil,\\
+M_h&=4^{b-h},\qquad L_h=2^{b-h}\ (0\le h\le b),\\
+D&=10^{14},\qquad \epsilon=10^{-5},\qquad a_0=\exp(-10^{16})/(D L_0),\\
+t_h&=a_{h+1}=D q L_h a_h^2\ (0\le h<b),\qquad P_0=\lceil 1/a_0\rceil,\\
+R_h&=D q(h+1)^2 M_h a_h^2,\qquad B_j=10^{100}\ell_*8^j(b+1)^3 .
+\end{split}
+\]
+All estimates below using big-O or little-o are uniform over the remaining
+parameters (including channel probabilities in the above range and induction
+levels); little-o refers to \(n\to\infty\), and in particular \(n\) may be
+taken larger than huge absolute constants throughout. We have \(b\ge1\);
+\(b\) need not tend to infinity, but \(\ell_*(2^b-1)\ge\log n\) gives
+\(\ell_*(b+1)\to\infty\), hence \(B_0\to\infty\), uniformly. Put \(v_h=D q L_h a_h\);
+then \(v_0=q e^{-10^{16}},\ v_{h+1}=v_h^2/2\), so \(v_h=2(v_0/2)^{2^h}\).
+Hence \(a_b n<.0001\), since
+\[
+ a_b=\frac{2}{D q}(q e^{-10^{16}}/2)^{2^b}
+ \le \exp(-10^{16})\exp[-\ell_*(2^b-1)].
+\]
+Moreover \(a_{h+1}/a_h=v_h\le v_0\ll .001\),
+\(\log(1/a_h)\le10^{18}(b+2^h)\ell_*\), and \(P_0=O(2^b)\)
+(in particular \(\log(2P_0)=O(b)\) independently of the size of \(\ell_*\)).
+
+A tuple probe of order \(k\ge1\) uses factors \(\chi_e(x_{i_e})\), \(e=1,\ldots,k\), where \(\chi_e:\{0,1\}\to\{0,1\}\), at strictly increasing selected indices \(i_1,\ldots,i_k\). Some edges \((i_e,i_{e+1})\) are designated exact, meaning the factor \(\mathbf{1}_{i_{e+1}-i_e=1}\) must also be inserted. All other edges are called loose and do not directly get additional factors. The block count \(A\) is 1 plus the number of loose edges. All chosen positions satisfying the exact constraints split into \(A\) contiguous blocks, not necessarily separated by gaps \(>1\).
+
+A probe at stage \(j\) also has finitely many admissible masks of levels \(h<j\) as factors; each level \(h\) mask is specified by \(r-1\) (with \(2\le r\le4\)) differences \(g_s=i_{b_s}-i_{c_s}\) for specified \(1\le c_s\le b_s\le k\), each nonnegative by the ordering (0 for a repeated endpoint). It has the form
+\[
+K(g)\prod_{s=1}^{r-1} e^{i\beta_s g_s},\qquad \beta_s\in\mathbb R,\quad |\beta_s|\le10^4 a_h .
+\]
+Here \(K\) may be complex, with two types of control on the nonnegative quadrant:
+\[
+\begin{gathered}
+ |K(g)|\le 2 e^{-\epsilon t_h\sum_s g_s},\qquad
+ K(g)=e^{-R_h\sum_s g_s}\int e^{i\xi\cdot g}\,\mu(d\xi),\\
+ |\xi|_\infty\le C R_h,\qquad \|\mu\|\le \exp(C (R_h/t_h) B_h).
+\end{gathered}
+\]
+The integral means that \(\mu\) is a complex measure supported in the stated real frequency cube, \(\|\cdot\|\) its total variation. Throughout \(C\) in these mask conditions denotes some sufficiently large universal constant; existence of such a constant follows below from all absolute constants in the construction, and the same one can be used for every stage. A difference \(g_s\) covers precisely the edges inside its interval and none if \(g_s=0\).
+
+Write \(W_x(i_1,\ldots,i_k)\) for the product of all bit-predicate, exact-gap and mask factors of a probe (including the phases with coefficients \(\beta_s\)). Its latest-anchor array and difference polynomial are
+\[
+ s_x(m)=\sum_{0\le i_1<\cdots<i_k<n:\ i_k=d+m} W_x(i_1,\ldots,i_k),
+\quad H(z)=\sum_m(s_x(m)-s_y(m)) z^m ,
+\]
+with \(s_y\) defined similarly and invalid latest positions giving zero. All negative-offset terms vanish in the difference: for \(m<0\) the summands query only indices \(<d\) (if valid). Thus \(H\) is a polynomial with no negative powers.
+
+Our induction goal at stage $j$ is the existence of some probe with
+\begin{itemize}
+\item \(1\le k\le 2P_0\cdot 4^j, 1\le A\le4^j\);
+\item At most \(4^{j-h-1}\) masks of each level \(h<j\); every loose edge is covered by at least one of their differences.
+\item For some real \(\theta\) with \(|\theta|\le5a_j\), \(|H(e^{-a_j+i\theta})|\ge e^{-B_j/f(\theta/a_j)}\), where \(f(u)=e^{u^2/20}\).
+\end{itemize}
+Stop as soon as \(a_j n\le .001\), certainly by \(j=b\). In particular the root parameter at stopping has much smaller attenuation than \(p\).
+
+Any probe satisfying the first two conditions has \(\sup_m |s_x(m)|,\,\sup_m |s_y(m)|\le Z_j\), where
+\[
+ Z_j=2^{4^j}\left(\frac{2}{\epsilon a_j}\right)^{4^j}.
+\]
+Indeed every loose edge receives decay \(e^{-\epsilon a_j\Delta}\) or better by coverage and \(t_h\ge a_j\); exact distances are fixed. Sum the loose distances independently in \(\{1,2,\ldots\}\) going backwards from the fixed latest position, use bounded bit predicates, and bound the sum of mask counts by \(4^j\). The logs of auxiliary magnitudes are tiny compared to the budget: for \(0\le j<b\),
+\[
+\begin{split}
+ \log Z_j+\log Z_{j+1} &+ \log(1/a_j)+\log(1/t_j)\\
+ &\le 10^{21}\ell_* 4^j(b+2^j) \le10^{-60} B_j,
+\end{split}
+\]
+by the formulas for \(Z_j,Z_{j+1}\) and the bound on \(\log(1/a_h)\).
+
+\section*{Separated-word base} Say a positive integer is a period of a word if symbols agree at all pairs of positions whose difference equals it. By the padding we have \(d\ge 2P_0\) for all sufficiently large \(n\). Let \(w'=x_{d-2P_0+1}\cdots x_{d-1}=y_{d-2P_0+1}\cdots y_{d-1}\). At least one of \(w'0,w'1\) has no period of length \(<P_0\). Indeed if both had such periods \(p_1,p_2\), then \(w'\) would have both periods, hence period \(\gcd(p_1,p_2)\) (by the Fine--Wilf phenomenon \cite{FW}). Indeed \(|w'|\ge p_1+p_2\); on positions \(1,\ldots,p_1+p_2\), cyclic addition of \(p_1\) preserves the symbol by either period \(p_1\) or (when wrapping around, moving by \(-p_2\)) period \(p_2\). This operation connects all positions in each residue class mod \(\gcd(p_1,p_2)\) there. Positions beyond this range can be related to ones within it by repeatedly subtracting \(p_1\). Positions \(2P_0-p_1\) and \(2P_0-p_2\) in \(w'\) -- numbered from 1 -- would agree, contradicting opposite required final bits. Choose such a period-avoiding extension \(w\in\{w'0,w'1\}\) and take the tuple with \(2P_0\) slots, all edges exact, no masks, and predicates testing the bits of \(w\). The indicators of occurrences ending at \(d\) differ. Occurrence endings in each individual string are \(P_0\)-separated, by the period condition. Hence up to sign,
+\[
+ H(z)=1-\eta z^u +\sum_{m\ge P_0} c_m z^m
+\]
+with \(1\le u<P_0,\ \eta\in\{0,1\},\ |c_m|\le1\), with arbitrary valid \(u\) if \(\eta=0\). Let \(v=10\log(P_0)/P_0\). At the positive real \(z=e^{-v}\), the first two terms have magnitude \(\ge1-e^{-v}\ge v/2\) and the remaining tail is \(\le e^{-vP_0}/(1-e^{-v})\le 2P_0^{-10}/v\le v/6\); thus \(|H(e^{-v})|\ge v/3\). On \(|z|=e^{-a_0}\), \(|H|\le 2/a_0\). The subharmonic Poisson inequality for \(\log|H|\) on this disk centered at 0, evaluated at \(e^{-v}\), gives
+\[
+ \max_{|\theta|\le a_0}|H(e^{-a_0+i\theta})|\ge
+ \exp(-O((v/a_0)\log(1/a_0))) .
+\]
+Indeed the harmonic weight of that arc is \(\gtrsim a_0/v\) by the Poisson kernel, since \(v\ge 2a_0, v<.01\); bound the log modulus by the arc maximum log on the arc and by \(\log(2/a_0)\) elsewhere. Boundary zeros cause no issue (e.g.\ apply the Poisson inequality first to the subharmonic regularization \(\log(|H|+\delta)\) for \(\delta>0\) then let \(\delta\downarrow0\)). Since \((v/a_0)\log(1/a_0)=O(b^2)=o(B_0)\), this gives the desired pivot.
+
+\section*{Weighted phase step} For the induction step (not yet at stopping) abbreviate \(a=a_j,\ t=a_{j+1},\ B=B_j,\ E=B/f(\theta/a_j)\); thus \(t/a\le.01\), \(a\le a_0\ll .001\), and during a step \(a>.001/n\). Use a polynomial-Gaussian mask
+\[
+ \sigma^2=2.4 B/t^2,\quad \ell=\lfloor10^{-10}E\rfloor,\quad
+ v(m)=e^{-y^2/2}\sum_{k=0}^{\ell}(y^2/2)^k/k!,\qquad y=m/\sigma .
+\]
+We have \(0\le v\le1\), and for \(0\le m\le3E/a\), \(1-v(m)\le e^{-2E}\): putting \(w=y^2/2\le2 E(t/a)^2\le2 E\exp(-2\cdot10^{16})\), the exponential Taylor remainder times \(e^{-w}\) is at most \((e w/(\ell+1))^{\ell+1}\) by the Taylor remainder formula and the factorial bound. Choose either \(c(s)=\operatorname{Re}s\) or \(c(s)=\operatorname{Im}s\), so that replacing \(s_x,s_y\) by \(c(s_x),c(s_y)\) retains pivot magnitude at least \(\frac12 e^{-E}\) (the original complex pivot is a sum of the two pivots, the latter times \(i\)). For every \(m\) with \(0\le d+m<n\) (including when an old-tuple sum is empty), form nonnegative arrays by the following formula, and set both arrays to zero elsewhere:
+\[
+f_x(m)=(2+c(s_x(m)/Z_j))v(m)/Y,\qquad f_y(m)=(2+c(s_y(m)/Z_j))v(m)/Y.
+\]
+Here \(Y=3\sum_{m\in\mathbb Z}v(m)\), so \(Y\ge3, \sum f_x,\sum f_y\le1\). We have \(\log Y\le10^{-8}E\) for all sufficiently large \(n\): split at \(|m|=H_*=\lceil1000B/t\rceil\), bound \(v\le1\) within and \(v(m)\le 2^\ell\exp[-m^2/(4\sigma^2)]\) outside by exponential series. This also shows the omitted tail beyond \(H_*\) has \(\ell_1\) norm \(<e^{-200 E}\) for \(f_x,f_y\), by \(\log(1/t)\le10^{-60}B,\ B\le4E,\ \log B=o(E)\) and a Gaussian tail sum estimate.
+
+The polynomial \(J(z)=\sum_m(f_x(m)-f_y(m))z^m\) has \(\sup_{|z|=1}|J|\le2\) and at the pivot radius the unmasked difference of real projections loses at most \(o(e^{-E})\) by masking, using \(1-v(m)\le e^{-2E}\) on \(0\le m\le3E/a\), the exponential damping beyond, and \(|s_x|,|s_y|\le Z_j\) with \(\log Z_j+\log(1/a)\le 10^{-60} B\). Thus \(|J(e^{-a+i\theta})|\ge e^{-1.0005E}\) after division by \(Z_j Y\). Its truncation \(J_\diamond\) to \(|m|\le H_*\) has \(|J_\diamond(e^{-a+i\theta})|\ge e^{-1.0006E}\) by the same exponential damping or the tail bound since all negative powers vanish. The grid below keeps the full arrays; truncation of the difference polynomial is only used for controlling a deficit.
+
+\textbf{Phase test on the arc.} Put \(N=\lceil (1000B/t)^6\rceil,\ h=2\pi/(Na),\ v_0^*=\theta/a\), \(\gamma=\lfloor 10^4/h\rfloor\), \(F(l)=\sum_m f_x(m)e^{2\pi i lm/N},\ G(l)=\sum_m f_y(m)e^{2\pi i lm/N}\). We claim, putting \(\Lambda=e^{-6B}\), that for some \(r\in\{2,3,4\}\) there are integers \(l_i\), \(i=1,\ldots,r\) (allowing repeats), with \(|l_i|\le\gamma\) and ordinary, unwrapped sum satisfying \(|\sum_{i=1}^r l_i|\le1\), for which
+\[
+ \left|\prod_i F(l_i)-\prod_i G(l_i)\right|>\Lambda.
+\]
+Use grid weights \(k_l=h/(\pi(1+(lh-v_0^*)^2))\) for \(|l|\le\gamma\), zero elsewhere. With \(m_l=\max(|F(l)|,|G(l)|)\le1\), define
+\[
+ g_l=\min(50,E^{-1}\log(1/m_l)),\quad q_l=\min(50,E^{-1}\log(2 m_l/|F(l)-G(l)|)).
+\]
+Set \(g_l=50\) when \(m_l=0\), and \(q_l=50\) when the denominator vanishes. Thus \(g_l,q_l\) are nonnegative and symmetric by reality, with
+\[
+ \mathcal G:=\sum_l k_l g_l,\qquad Q:=\sum_l k_l q_l,\qquad \mathcal G+Q\le1.003.
+\]
+Indeed \(g_l+q_l\le\min(100,\log(2/|J(e^{ialh})|)/E)\le\min(100,\log(3/|J_\diamond(e^{ialh})|)/E)\) for all large \(n\): when the latter capped upper bound from \(J_\diamond\) does not give 100 use \(|J_\diamond(e^{ialh})|\ge 3 e^{-100E}\) and \(\sup_{|z|=1}|J_\diamond-J|\le 2e^{-200E}\). The Poisson inequality applied at the truncated pivot bounds the continuous integral of this last capped upper bound on \([-10^4,10^4]\) against \(dx/(\pi(1+(x-v_0^*)^2))\) by \(1.0025\) since the Poisson density on the unit circle at angle \(ax\), times \(a\), is at least \((1-2a)/(\pi(1+(x-v_0^*)^2))\), the truncated circle supremum is \(\le2\), and the full-circle deficit is nonnegative; also \(10^4 a<\pi\). Indeed the Poisson density in the rescaled variable \(x\) is
+\[
+ \frac{a(1-e^{-2a})}
+ {2\pi((1-e^{-a})^2+2e^{-a}(1-\cos(a x-\theta)))}
+ \ge \frac{1-2a}{\pi(1+(x-v_0^*)^2)}.
+\]
+Logarithmic zeros cause no issue by regularization as in the base. For discretization, uniformly for thresholds in \([0,100)\) the strict upper level set of the cap on this interval is a union of \(O(H_*)\) intervals by counting zeros of squared modulus of the truncated polynomial minus a constant (the constant-modulus case is immediate). The smooth-kernel integral and grid sum differ by \(O(H_* h)=o(1)\), uniformly by bounded variation and bounded supremum. Apply layer cake.
+
+For the rest of the phase proof, \(h\) denotes the grid mesh above and \(j\) when indexing edge separations or references denotes a nonnegative shift, not the induction stage. Take parameters from the following table by rounding \(|v_0^*|\) up to the next positive integer \(w\) (allow equality, with \(0\mapsto1\)). Here \(S\) is a real interval radius parameter and \(M,J_*\) are conservative lower bounds:
+\[
+\begin{array}{c|ccc}
+w&S&M&J_*\\\hline
+1&1.5&.419&.524\\
+2&2.1&.357&.453\\
+3&2.8&.297&.380\\
+4&3.6&.246&.335\\
+5&4.4&.208&.292
+\end{array}
+\]
+The simple estimates used below are
+\[
+\begin{gathered}
+ M< \tfrac12-\tfrac2\pi \arctan(\exp(-\pi/S))-.0002,\\
+ J_* < \tfrac1\pi(\arctan(S+w)+\arctan(S-w))-.0002,\\
+ T_{\rm lim}(v):=6 f(v)-.1,\\
+ T_{\rm lim}(w-1)> \max\left\{2.006/M,\,1.005/(J_*-(1+2J_*/M)^2 M/16)\right\}
+\end{gathered}
+\]
+with positive denominator. For example, these rational bounds may be checked without sensitive numerical evaluations as follows. Use \(3.14159<\pi<3.14160\); replace \(\exp(-\pi/S)\) by the upper bound \(1/\sum_{i=0}^{12}(3.14159/S)^i/i!\). Write \(\arctan(S+w)=\pi/2-\arctan(1/(S+w))\). The remaining arctangent arguments have modulus at most \(.6\) and evaluating \(\sum_{i=0}^6(-1)^i x^{2i+1}/(2i+1)\) incurs error at most \(.6^{15}/15\). Lower bound \(\exp(u)\) in \(f(w-1)\) by \(1+u+u^2/2\). More explicitly these substitutions (taking the worst endpoints when dividing by a $\pi$ interval) yield respectively lower bounds on the first two right sides and upper bounds on the maximum on the last right side as follows:
+\[
+\begin{array}{c|ccc}
+w &\text{first lower}&\text{second lower}&\text{last upper}\\\hline
+1&.4217&.5262&4.96\\
+2&.3594&.4553&5.79\\
+3&.2993&.3825&6.94\\
+4&.2478&.3370&8.26\\
+5&.2098&.2940&9.71
+\end{array}
+\]
+Here the last column just uses the rational $M,J_*$ in the parameter table and rounds up.
+
+We record why these capture the two relevant masses. First, for every integer \(1\le j\le S/h\), every (binary or fractional) vertex cover on \([-\gamma,\gamma]\cap\mathbb Z\) of all edges \((l,l+j)\) in this interval has total \(k_l\) weight at least \(M\), uniformly for large \(n\). To see this extend the weights to the untruncated Cauchy kernel \(h K_{\rm Cau}(lh-v_0^*)\), \(K_{\rm Cau}(x)=1/[\pi(1+x^2)]\), and extend the cover by weight 1 at all vertices outside the interval (cost \(<.0001\) for large \(n\), by e.g. the tail bound and \(O(h)\) Riemann error). A fractional cover means vertex numbers in \([0,1]\) with endpoint sums \(\ge1\); on each residue chain mod \(j\) these can be rounded to a random binary cover preserving expected weight (select where the vertex number is at least its threshold), by a uniform threshold \(U_0\in[0,1]\) at even vertices (center counted as even in each chain) and threshold \(1-U_0\) at odd vertices (use non-strict comparison). In each chain take a central vertex \(l\) nearest to \(v_0^*/h\). All weights are nonincreasing outward on each of its two arms. For binary covers the minimum weight is the lesser of the two alternating parity cover weights: if center is omitted, both first neighbors are forced, then pair disjoint edges at offsets \((2,3),(4,5),\ldots\) on each arm require at least the odd weights there; if center is included, edges \((1,2),(3,4),\ldots\) on each arm require the even weights. This reasoning works for the infinite chain by summability.
+
+Put \(d'=jh\), \(x=lh-v_0^*\) for the central vertex chosen with \(-d'/2\le x<d'/2\). Even parity has total weight \(h\Xi(x)\) where
+\[
+\Xi(x)=\sum_{\nu\in\mathbb Z} K_{\rm Cau}(x+2\nu d') = \frac1{2d'}\left(1+2\sum_{m\ge1}e^{-m\pi/d'}\cos(m\pi x/d')\right),
+\]
+by the Fourier transform of the Cauchy kernel (the absolutely convergent Fourier series obtained by computing coefficients of the continuous periodic sum termwise; $\int K_{\rm Cau}(x)e^{i\xi x}\,dx=e^{-|\xi|}$ over the real line, e.g.\ for $\xi>0$ by a semicircle contour closed above with residue at $i$, similarly by symmetry and continuity otherwise). Odd parity similarly has weight \(h\Xi(x+d')\). The latter is no larger since summing the geometric Fourier series gives \(\Xi(x)=(1-r_0^2)/(2d'(1+r_0^2-2r_0\cos(\pi x/d')))\) with \(r_0=e^{-\pi/d'}\). Summing odd parity masses over the \(j\) central representatives gives up to \(O(h)\) error the integral
+\[
+\int_{-d'/2}^{d'/2}\Xi(x+d')\,dx =\tfrac12-\tfrac2\pi\arctan(e^{-\pi/d'}),
+\]
+where the identity follows by integrating the series. The error is uniform: the variation of \(x\mapsto\Xi(x+d')\) on this interval is at most \(\int_{\mathbb R}|K_{\rm Cau}'|\); the representatives are one per subinterval in the equal partition of mesh \(h\). This parity mass is minimized at the largest possible \(d'\), yielding the cover bound after the tail subtraction. Second, \(\sum_{|l|\le S/h}k_l>J_*\) directly by Riemann-summing the integral on \([-S,S]\), centered at \(v_0^*\), whose translated Cauchy mass there is minimized at the furthest center \(\pm w\).
+
+Write now \(R=\mathcal G/M\), \(T_0=T_{\rm lim}(v_0^*)\ge T_{\rm lim}(w-1)\), so \(2R<T_0\). There exists an integer \(s\) with \(|s|\le S/h\) and both
+\[
+ 2R+q_s<T_0,\qquad 2g_s+q_s<T_0.
+\]
+Otherwise on this interval \(q_s\ge (T_0-2R)\max(0,1-2g_s/T_0)\). Hence \(Q\ge(T_0-2\mathcal G/M)(J_*-2\mathcal G/T_0)\), giving by quadratic minimization
+\[
+ \mathcal G+Q\ge T_0[J_*-(1+2J_*/M)^2 M/16]>1.004
+\]
+(a contradiction). Here and below \(T_0<50\) since \(|v_0^*|\le5\). By symmetry choose this witness \(s\ge0\). For each integer \(0\le j\le s\) choose a reference pair \((e_j,e_j+j)\) in the large arc grid with sum of \(g\)-deficits at its endpoints at most \(R\). Indeed if the minimum sum on such edges at \(j\ge1\) is \(R'>R\), the clipped deficits \(\min(1,g_l/R')\) give a fractional cover costing at most \(\mathcal G/R'<M\). For \(j=0\) simply minimize \(2g_l\) and use the total kernel mass \(>.999\) together with \(M<.499\).
+
+Suppose the claim fails for contradiction. The failed pairs \((l,-l)\) bound the modulus difference at every endpoint or witness \(l\) by \(\Lambda/m_l\), since they bound the difference of squares by \(\Lambda\). At all these indices \(2g_l<T_0\) by construction, so \(m_l=e^{-g_l E}\) with both magnitudes at least \(m_l/2\) for large \(n\). Write \(U(l)=\arg F(l)-\arg G(l)\) modulo \(2\pi\) there and at their negatives (using odd symmetry). Set \(d_j=U(e_j+j)-U(e_j)\); \(d_0=0\). For integers $1\le j\le s$, the failed quadruple product at frequencies \((e_j+j,-e_j,-e_{j-1}-j+1,e_{j-1})\) (sum 1) bounds the distance of \(d_j-d_{j-1}\) to zero mod \(2\pi\) by \(100\Lambda e^{2R E}\): both products have magnitude at least \(e^{-2RE}/16\) so their angles are this close (modulo $2\pi$; in general the angle distance of two numbers differing by at most $\Lambda$ and both of modulus at least $v>0$ is at most $\pi\Lambda/v$ by normalization and the chord bound). Hence \(|d_s|\le 100N\Lambda e^{2R E}\) modulo \(2\pi\). Similarly the triple with frequencies \((e_s+s,-e_s,-s)\) gives \(|U(s)-d_s|\le100\Lambda e^{(R+g_s)E}\) modulo \(2\pi\). But \(R+g_s+q_s\le\max(2R+q_s,2g_s+q_s)<T_0\). Therefore
+\[
+ |F(s)-G(s)|/m_s\le \Lambda/m_s^2 +100\Lambda[e^{(R+g_s)E}+N e^{2R E}]<e^{-q_s E}
+\]
+using \(\log N\le 10^{-8} E\) for sufficiently large \(n\), from \(\log(1/t)\le10^{-60} B\), \(\log B=o(E)\) and the formula for \(T_0\). This contradicts uncapped \(q_s<T_0<50\). The claim follows.
+
+\section*{Shuffle and center-mask transfer} The claim gives \(r\in\{2,3,4\}\) and angles \(\theta_s\) with
+\[
+ |\theta_s|\le10^4 a,\quad |\sum_{s=1}^r\theta_s|\le2\pi/N<.01t,
+\]
+with product discrepancy \(>\Lambda\). Expand the products. Each array copy is a combination of a constant latest valid probe (one wildcard, value 1), the stage-\(j\) probe, and its complex conjugate (including any mask factors), all with scalar coefficients of modulus \(\le1\) from the floor, \(Y,Z_j\) and projection. Each copy also carries the Gaussian-polynomial mask \(v(m_s)\) and phase \(e^{i\theta_s m_s}\). Group the expansion both by factor types per tuple and by the total weak ordering of all selected source positions in all copies, including all equality ties (same source index) as part of the ordering. The number \(P\) of nonempty order categories (also counting component types) satisfies \(\log P=o(B)\) uniformly. To see this, take only assignments already satisfying all exact adjacencies within copies. They consist of at most \(4^{j+1}\) ordered-within-copy contiguous blocks. Specify the weak order of all block starting positions with equality ties included, costing at most \(m!\,2^m\) for \(m\le4^{j+1}\) blocks. After sorting starts and breaking start ties by a fixed label order for counting, process blocks in that order; a new start is either beyond the maximum end index of all prior blocks, or specify its offset from that maximum. The latter offset has magnitude at most \(k-1\) since the new start never precedes the start of the prior block attaining that maximum (of length at most \(k\)). This determines the full total weak position order (lengths known); whenever a start lies beyond the maximum end, its cluster lies strictly after all prior blocks. We only use this encoding to count, not to impose extra distance constraints on a class. There are at most \(81(4k+1)^{4^{j+1}}\) additional specifications including factor types. Thus \(\log P=O(4^j\log(4k+4^{j+1}+2))=O(4^j b)=o(B)\), including the case \(k=1\).
+
+Fix any resulting category. All its exact adjacency constraints act across adjacent distinct slots of the merged sorted order, since an old exact edge spanning an intervening distinct slot would make the category empty. Collect them as exact edges of the merged tuple and designate its remaining edges loose. All assignments of remaining distinct slots satisfying the merged exact constraints and strict increase give precisely the assignments in the category (ordering and equalities incorporate every constraint from distinct copies). The merged order \(k'\le4k\); its block count is at most the sum of old block counts because each old block stays within a single merged block, and each resulting block contains at least one old block. Tie bit-predicates are multiplied. Retain all old mask factors, unchanged including their interval endpoints (and conjugations under prior projection; conjugate measures just negate frequencies and conjugate the scalars). They preserve their level bounds and each creates at most 4 copies here. The wildcard component has no old masks.
+
+Choose a component whose latest selected index is last in the fixed category (break ties deterministically); relabel as \(s=r\). Denote the overall latest offset by \(Y_0=m_r\), set \(g_r=0,\ g_s=Y_0-m_s\ge0\) for \(s<r\), and \(\bar g=\sum_s g_s/r,\ \mathcal{Q}(g)=\sum_{s=1}^r(g_s-\bar g)^2\). Each old frequency phase gives
+\[
+ \prod_s e^{i\theta_s m_s}=e^{i(\sum_s\theta_s)Y_0}\prod_{s<r}e^{-i\theta_s g_s}.
+\]
+We will attach a level-\(j\) mask with the \(r-1\) differences \(g_s\), possibly including some identically zero. Then every loose merged edge is covered either by an inherited or the new mask. Indeed consider a copy with a slot at the left endpoint of a loose edge. If it is that copy's latest slot, it is not the overall latest and contributes a new covering interval. If it is not latest, the copy has a consecutive pair straddling the edge (i.e. across a merged adjacent slot pair, with copy slots at or before the left and at or after the right respectively). This pair was loose (else merged edge exact or impossible) and hence some old mask covers it.
+
+\textbf{Center Fourier expansion.} For fixed real \(g\) (free coordinates \(g_s,\,s<r\), always appending \(g_r=0\)) use Fourier inversion momentarily varying \(Y_0\) as a real variable:
+\[
+ W_g(Y_0):=e^{t Y_0}\prod_{s=1}^r v(Y_0-g_s) = e^{1.203 B}\int_{\mathbb R} e^{i\omega Y_0} K(g,\omega)\,\nu(d\omega),
+\]
+where \(\omega\) ranges over the real line and \(\nu\) is the centered Gaussian probability with variance \(r/((1-\zeta)\sigma^2)\), \(\zeta=10^{-5}\). Here \(K(g,\omega)=e^{-1.203B}\widehat W_g(\omega)/(d\nu/d\omega)\) with convention \(\widehat W_g(\omega)=(2\pi)^{-1}\int W_g(Y_0)e^{-i\omega Y_0}dY_0\). Constants use \(\sigma^2=2.4 B/t^2\) as above. For all \(\omega\in\mathbb R\), \(g_s\ge0\), one has
+\[
+ |K(g,\omega)|\le e^{-\epsilon t\sum_s g_s}.
+ \tag{1}
+\]
+To verify, scale \(z=(Y_0-\bar g)\sqrt r/\sigma,\ \rho=t\sigma/\sqrt r,\ u'=\omega\sigma/\sqrt r,\ p_\ell(x)=\sum_{k=0}^{\ell}(x^2/2)^k/k!\), \(d_s=(g_s-\bar g)/\sigma\). Shift the \(z\) contour by \(-i u'\), justified by the polynomial times Gaussian decay. This gives, with the normalization for \(du'\) instead of \(d\omega\) (equivalently multiply the actual \(\widehat W\) by \(\sqrt r/\sigma\)),
+\[
+ (2\pi)^{-1/2}
+ e^{(t-i\omega)\bar g-\mathcal{Q}(g)/(2\sigma^2)+\rho^2/2-(u')^2/2-i\rho u'}
+ \ \mathbb E_{X\sim\mathcal N(\rho,1)}
+ \prod_s p_\ell((X-i u')/\sqrt r-d_s).
+ \tag{2}
+\]
+The scaled formula (2) continues to complex \(g\) (or defines an entire continuation), with \(g_r=0\) and quadratic form \(\mathcal{Q}\) still using squares, not absolute squares. For real \(g\), \(\sum_s|(X-i u')/\sqrt r-d_s|^2=X^2+(u')^2+\mathcal{Q}(g)/\sigma^2\). Bound the product of polynomials by \(\zeta^{-r\ell}\) times the exponential of \(\zeta/2\) times this sum, then integrate \(X\). The ratio of \(|\widehat W_g(\omega)|\) to the Gaussian density at \(\omega\) is at most
+\[
+ \frac{\zeta^{-r\ell}}{1-\zeta}
+ \exp\!\left[t\bar g-(1-\zeta)\frac{\mathcal{Q}(g)}{2\sigma^2}
+            +\frac{2.4B}{2r(1-\zeta)}\right].
+ \tag{3}
+\]
+Now \(\mathcal{Q}(g)\ge r\bar g^2/(r-1)\), since \(g_r=0\). Thus
+\[
+ t\bar g+\epsilon t \sum_s g_s-(1-\zeta)\frac{\mathcal{Q}(g)}{2\sigma^2}
+ \ \le\ \frac{(1+r\epsilon)^2(2.4 B)(r-1)}{2(1-\zeta)r}.
+\]
+The right side plus \(2.4B/(2r(1-\zeta))\) and \(\log[\zeta^{-r\ell}/(1-\zeta)]\) is \(<1.203B\) for all sufficiently large \(B\), proving (1).
+
+Writing \(\theta_{\rm sum}=\sum_s\theta_s\), each fixed-\(\omega\) discrepancy contribution after this expansion compares unnormalized merged sums with new outer factor \(e^{-t Y_0+i(\omega+\theta_{\rm sum})Y_0}\), the \(r-1\) phases \(e^{-i\theta_sg_s}\), and \(K(g,\omega)\). Its modulus is at most \(2Z_{j+1}/(1-e^{-t})\le e^{10^{-8} B}\) for large \(n\), apart from the exterior \(e^{1.203B}\) factor. Indeed every merged loose edge has the stated coverage and (1) applies, even if \(K\) is not yet an admissible final-frequency-representable mask; bit factors, old masks and the strict ordered position sum therefore obey the same fixed-latest bound as before. Latest \(Y_0<0\) terms cancel exactly, leaving only \(Y_0\ge0\) and thus a geometric sum from the outer radial damping.
+
+Uniformly over categories the part where \(|\omega+\theta_{\rm sum}|>5t\) then contributes at most
+\[
+ \exp(1.2031B)\cdot 2\exp[-(4.99)^2(1-\zeta)2.4 B/8] = o(e^{-6 B})
+\]
+still true after multiplication by \(P\). We will replace \(K(g,\omega)\) for retained \(\omega\) by an admissible \(K^{\rm c}(g,\omega)\) satisfying on \(g_s\ge0\)
+\[
+ |K-K^{\rm c}|\le e^{-20 B} e^{-\epsilon t \sum_s g_s}.
+ \tag{4}
+\]
+The constants here and in admissibility are absolute, uniform on \(|\omega|\le5.01t\), for all stages if \(n\) sufficiently large. The replacement below can be taken measurably in \(\omega\). The resulting replacement error is at most \(\exp(1.2031B-20 B+o(B))\) over all categories by the same coverage and discrepancy-cancellation bound.
+
+Given these estimates, suppose for contradiction every merged category with a retained \(\omega\), using \(K^{\rm c}\), fails the next target. Let \(u=(\omega+\theta_{\rm sum})/t\) and \(v=\omega/t\), so \(|u-v|<.01,\ |u|\le5\). The retained discrepancy integrated for any category is then bounded by \(e^{1.203 B}\) times the integral on this interval of \(\exp(-8 B / f(u))\) under \(\nu\). The density in \(v\) is at most \(C\sqrt{B}\exp[-.3(1-\zeta)v^2 B]\). For these \(u,v\),
+\[
+ 8/f(u)+.3(1-\zeta)v^2
+ \ge 7.8\exp(-.05 u^2)+.3u^2-.04 > 7.4 ,
+\]
+since the minimum of \(7.8e^{-x}+6x\) over \(x\ge0\) is \(6(1+\log(1.3))>7.5\). Hence even after integration and multiplication by \(P\) and adding tails and (4) errors we would get full product discrepancy \(o(e^{-6B})\), contradiction. This proves existence of the required next pivot using the admissible masks if (4) holds.
+
+\section*{Radial compactification proving (4)} Fix \(|\omega|\le 5.01t\) and suppress it in \(K(g,\omega)\). Put \(R=R_j\), so \(R/t=(j+1)^2\,M_j/L_j\ge1\). Take \(H_0\) a sufficiently large fixed absolute constant (unrelated to the polynomial \(H\)), \(W=H_0 B/t,\ \tau^2=W/R\), and
+\[
+ U(x)=\frac1{\tau\sqrt{2\pi}}\int_{-W}^W e^{-(x-v)^2/(2\tau^2)}\,dv .
+\]
+For sufficiently large fixed \(H_0\),
+\[
+ \left|K(g)-K(g)\prod_{s<r}U(g_s)\right|
+ \le \tfrac14 e^{-20 B} e^{-\epsilon t\sum_s g_s}\qquad(g_s\ge0).
+ \tag{5}
+\]
+If all \(0\le g_s\le W/2\), use \(1-U(g_s)\le 2e^{-W^2/(8\tau^2)}=2e^{-RW/8}\) and (1). Otherwise some \(g_s>W/2\); then \(\mathcal{Q}(g)\ge W^2/(4r)\) by \(g_r=0\) and Cauchy (in fact \(\mathcal{Q}(g)\ge |g|_2^2/r\) on these \(r-1\) free coordinates). In (3) expend half the negative quadratic on \(t\bar g+\epsilon t\sum g_s\) by completing the square as above (cost \(O(B)\)), leaving decay \(\exp(-c W^2/\sigma^2)\); use \(0\le U\le1\). This yields (5) since \(R W\ge H_0 B\) and \(W^2/\sigma^2=(H_0^2/2.4) B\).
+
+Consider the entire function of \(d'=r-1\) variables
+\[
+ \mathcal J(g)=e^{R\sum_s g_s} K(g)\prod_{s<r}U(g_s)
+\]
+and its inverse-transform density \(\widehat{\mathcal J}(\lambda)=(2\pi)^{-d'}\int_{\mathbb R^{d'}} \mathcal J(g)e^{-i\lambda\cdot g}\,d g\). There are absolute \(C_2,C_3\), allowed to depend on \(H_0\), such that
+\[
+ \int|\widehat{\mathcal J}(\lambda)|d\lambda\le \exp(C_2 RW),\qquad
+ \int_{|\lambda|_\infty>C_3 R}|\widehat{\mathcal J}(\lambda)|d\lambda \le \tfrac14 e^{-20 B}.
+ \tag{6}
+\]
+Here and in the following estimates \(s\) index sums may include the coordinate \(g_r=0\). To check this explicitly, from the entire formula (2) valid for \(g=x-i y\) (here \(x,y\in\mathbb R^{d'}\) are real vectors unrelated to the source strings, with zero appended as last coordinate in \(s\) sums), using \(|\omega|\le5.01t\), one obtains
+\[
+ |K(x-i y)|\le \exp(C_4 B+C_4 t\textstyle\sum_{s<r}(|x_s|+|y_s|)+C_4|y|_2^2/\sigma^2)
+\]
+with an absolute \(C_4\). Indeed \(\Re \mathcal{Q}(x-i y)=\mathcal{Q}(x)-\mathcal{Q}(y)\), and
+\[
+ \sum_s |(X-i u')/\sqrt r-(x_s-i y_s-\bar x+i\bar y)/\sigma|^2
+ = X^2+(u')^2 + [\mathcal{Q}(x)+\mathcal{Q}(y)]/\sigma^2 .
+\]
+Thus the real polynomial-exponential bound used in (3) applies with an extra \(C_4 \mathcal{Q}(y)/\sigma^2\) and \(C_4 t\sum|y_s|\) in the log, and with the favorable \(-(1-\zeta)\mathcal{Q}(x)/(2\sigma^2)\) dropped. Also
+\[
+ |U(x_s-i y_s)|\le
+ \frac{2W}{\tau\sqrt{2\pi}}\exp\!\left[\frac{y_s^2-\max(|x_s|-W,0)^2}{2\tau^2}\right].
+\]
+Hence shifting \(g=x-i y\) in \(\widehat{\mathcal J}(\lambda)\), for fixed \(y\), is justified by Gaussian tail decay in every real direction (iterated contour shifts and absolute convergence). Bounding the \(x\) integrals by splitting each at \(|x_s|=W\), or just using \(\max(|x_s|-W,0)^2\ge x_s^2/2 - W^2\) and completing squares, yields
+\[
+ |\widehat{\mathcal J}(\lambda)|\le \tau^{d'}\exp[C_5 RW + C_5 |y|_2^2/\tau^2-\lambda\cdot y]
+\]
+for some absolute \(C_5\). Here \(t\le R,\ R^2\tau^2=RW,\ W/\tau=\sqrt{RW}\), and \(\tau^2/\sigma^2\le H_0/2.4\); extra polynomial prefactors in \(W/\tau\) are absorbed into \(\exp(C_5 RW)\), while \(C_4 t\sum|y_s|\) is absorbed by both positive terms via Young's inequality. With \(y=\tau^2\lambda/(2C_5)\) the Gaussian envelope in \(\lambda\) proves (6) by Gaussian tail integration (take sufficiently large constants in (6)) and \(RW\ge H_0 B\). Finally define
+\[
+ K^{\rm c}(g)= e^{-R\sum_sg_s}\!\int_{|\lambda|_\infty\le C_3 R} e^{i\lambda\cdot g}\widehat{\mathcal J}(\lambda)\,d\lambda .
+\]
+Here the formula depends measurably on \(\omega\), e.g.\ by continuity of the integrals on the indicated frequency range with the above Gaussian domination. Fourier inversion (both the continuous function \(\mathcal J\) on real \(g\) and its transform are integrable), (5) and (6) yield (4) (as \(R\ge t\)). Also (1), (4) show the required quadrant modulus bound for \(K^{\rm c}\); (6) gives the complex-measure representation with \(C\) enlarged absolutely since \(RW=H_0(R/t)B\). This finishes the inductive construction.
+
+\section*{Selection and interpolation at the stopping stage} Now \(a_j n\le .001\), \(j\le b\), and write \(z_0=e^{-a_j+i\theta}\) for the last pivot. Expand all the retained masks by their compact representations. The total variation of the product measure is at most
+\[
+ \exp(C\sum_{h<j} 4^{j-h-1}(R_h/t_h) B_h)
+ \le \exp(O(\ell_*8^b(b+1)^6)).
+ \tag{7}
+\]
+Indeed \(R_h/t_h=(h+1)^2 2^{b-h}\), \(B_h=10^{100}\ell_*8^h(b+1)^3\). At each set of frequencies in this product integral the difference contribution is still bounded as a finite sum, so by the large pivot discrepancy and (7) some frequency choice has the resulting unintegrated discrepancy of magnitude \(\ge \exp(-O(\ell_*8^b(b+1)^6))\). Fix it. A mask now contributes the product over its differences of numbers
+\(\exp[-R_h g_s+i(\beta_s+\xi_s)g_s]\) with \(|\xi_s|\le C R_h,\ |\beta_s|\le10^4 a_h\). Each \(g_s\) is a sum of consecutive gap distances in the merged tuple. Assign these factors edgewise, dropping factors on exact edges since they are scalars of modulus \(\le1\) (and nonzero) there. Thus there is a pure product probe with the same bit predicates and exact constraints, unnormalized, whose loose-edge parameters \(z^*=\exp(-\eta+i\varphi)\) now obey
+\[
+ \kappa(z^*):=\Re\frac{1+z^*}{1-z^*}=\frac{1-|z^*|^2}{|1-z^*|^2}\ \ge c_0 q>0
+ \tag{8}
+\]
+for an absolute constant \(c_0\le1\). In fact for each such edge let the sums over its covering differences be taken with multiplicity. Then \(\eta=\sum R_h>0\) by coverage. The frequency perturbations sum to at most \(C\eta\) in absolute value. The sum of the \(\beta_s\)'s has square bounded by
+\[
+ \eta\sum_{\rm coverings}\frac{(10^4 a_h)^2}{R_h}
+ \le C_6(\eta/q)\sum_{h\ge0}\frac{1}{(h+1)^2} = O(\eta/q),
+\]
+using at most \(3\cdot4^{j-h-1}\le M_h\) coverings per level \(h\). Hence when \(\eta\le1\), \(1-|z^*|^2=1-e^{-2\eta}\ge(1-e^{-2})\eta\) and \(|1-z^*|^2\le C_7\eta/q\) by \(|\varphi|=O(\sqrt{\eta/q})+O(\eta)\). When \(\eta>1\), (8) follows from \(|1-z^*|\le2,\ 1-|z^*|^2\ge1-e^{-2}\). For the root \(z_0\) itself one has \(\kappa(z_0)\ge 4/p\): use \(p\ge1/n,\ 0<a_j\le.001/n,\ |\theta|\le5 a_j\) to get \(|1-z_0|\le6a_j,\ 1-|z_0|^2\ge a_j\), giving \(\kappa(z_0)\ge1/(36a_j)>4/p\).
+
+We move all loose-edge arguments further into the retention disk and also interpolate the exact constraints by the same analytic parameter. Let \(S=4/(p c_0)\) and \(\varsigma\) now denote a complex variable on the strip \(1/2\le\Re\varsigma\le S\). For each loose edge \(e\), let \(\alpha_e+i\delta_e=(1+z_e^*)/(1-z_e^*)\); for each exact edge set its target \(z_e^*=0\), so \(\alpha_e=1,\delta_e=0\). Define
+\[
+ z_e(\varsigma)=\frac{\alpha_e\varsigma+i\delta_e-1}{\alpha_e\varsigma+i\delta_e+1}
+\]
+so \(z_e(1)=z_e^*\), \(\kappa(z_e(\varsigma))=\alpha_e\Re\varsigma\). Use \(K_{\rm ex}=k-A\) for the number of exact edges in the selected tuple of order \(k\), with \(A\) blocks. Let \(G(\varsigma)\) be its difference sum at \(z_0\) with all exact-edge selectors on gap distance \(\Delta\) replaced by \(z_e(\varsigma)^{\Delta-1}\) (including \(0^0=1\)), loose edges using \(z_e(\varsigma)^\Delta\), and normalize the entire sum by \(n^{-A}(1+\varsigma)^{-2K_{\rm ex}}\). All negative latest-offset terms cancel. On this strip the analytic function \(G\) is bounded by 2: for every exact edge sum over \(\Delta\ge1\) with geometric bound \(1/(1-|z_e|)\), where \(1-|z_e|^2=4\Re\varsigma/|1+\varsigma|^2\) (since \(\alpha=1,\delta=0\)); this costs at most 1 per exact edge against the normalization \(|1+\varsigma|^{-2}\). For other edges use at most \(n\) choices per gap and similarly \(n\) choices for the latest (cancellation if latest before \(d\)); their factors have modulus at most 1. At \(\varsigma=1\) we have
+\[
+ \log |G(1)|\ge -A\log n-2K_{\rm ex}\log 2-O(\ell_*8^b(b+1)^6)
+             = -O(\ell_*8^b(b+1)^6).
+\]
+Here \(k\le2P_0\,4^b=O(8^b)\), \(A\le4^b\), \(\log n\le\ell_*2^b\). Hadamard's three-lines theorem on \(\Re\varsigma\in[1/2,S]\) gives some \(\varsigma\) on \(\Re\varsigma=S\) with
+\[
+ |G(\varsigma)|\ge \exp(-O(p^{-1}\ell_*8^b(b+1)^6)).
+ \tag{9}
+\]
+Indeed the right-line weight at 1 is \((1/2)/(S-1/2)\). All constructions up to now may depend on the given pair but the implied constants are absolute (if needed take a point within a factor 2 of the right-line supremum). To recall the strip argument: take positive numbers bounding the suprema on both lines strictly from above, divide the analytic function by the exponential of the complex affine interpolant of the logs of these two numbers, and apply the maximum principle to the quotient times \(\exp(\delta\varsigma^2)\), for \(\delta>0\), on increasingly tall rectangles in the strip. The horizontal-side bound tends to 0 by boundedness and the damping, and on the vertical sides the extra factor has modulus at most \(\exp(\delta S^2)\). Letting \(\delta\downarrow0\) and then taking limits of the two bounds proves log-convex interpolation of the suprema.
+
+For this line \(\kappa(z_e(\varsigma))\ge 4q/p\) by (8) and \(\kappa(0)=1\), so \(z_0 z_e(\varsigma)\) and \(z_0\) lie in the closed disk of center \(q=1-p\), radius \(p\). To see the product, for any two inputs \(u,v\) strictly in the unit disk, with \(\kappa(u),\kappa(v)>0\),
+\[
+ \frac{|1-u v|^2}{1-|u v|^2}
+ \le \frac{|1-u|^2}{1-|u|^2}+\frac{|1-v|^2}{1-|v|^2}
+\]
+by \(1-uv=(1-u)+u(1-v)\) and Cauchy--Schwarz. Hence for the product \(y_e=z_0 z_e(\varsigma)\) this ratio is at most \(p/(2q)\). Also
+\[
+ p^2-|y_e-q|^2=p(1-|y_e|^2)-q|1-y_e|^2
+ \ge \frac p2(1-|z_e|^2).
+\]
+The root disk inclusion follows similarly. Put \(\lambda_e=(y_e-q)/p,\lambda_0=(z_0-q)/p\) indexing edges here by \(e=1,\ldots,k-1\). They have modulus \(\le1\), and \(1-|\lambda_e|\ge (1-|z_e|^2)/(4p)\).
+
+On a trace \(U\) of the extended input consider
+\[
+ T(U)=n^{-A}(1+\varsigma)^{-2K_{\rm ex}}
+ \sum_{0\le s_1<\cdots<s_k<|U|}
+  \lambda_0^{s_1}\prod_{e=1}^{k-1}\lambda_e^{s_{e+1}-s_e-1}
+  \prod_{e=1}^k\chi_e(U_{s_e}).
+\]
+It is bounded by 1 using geometric sums over the exact-edge gaps: each has \((1-|z_e|^2)|1+\varsigma|^2=4 S\), so contributes at most \(p/S\) after its share of normalization. The loose edges and the latest cost at most \(n^A\). As usual a zeroth power here is 1 even for base 0. The loose \(z_e(\varsigma)\) are nonzero at the chosen right-line point, since otherwise every term in \(G(\varsigma)\) has a loose factor \(z_e(\varsigma)^\Delta=0\), contradicting (9). Conditioning on survival of the source bits at each tuple's positions, the counts before and between them have independent binomial laws, so
+\[
+ \mathbb E_x T-\mathbb E_y T =
+ p^k z_0^{d-k+1}\Big(\prod_{e:\ \rm loose} z_e(\varsigma)^{-1}\Big)\,G(\varsigma),
+\]
+using \(z_0^{i_1}\prod_{e=1}^{k-1}(z_0 z_e)^{i_{e+1}-i_e-1}=z_0^{i_k-k+1}\prod_e z_e^{i_{e+1}-i_e-1}\). Therefore using \(d-k+1\le n,\ a_j n\le.001,\ \log(1/p)\le1/p\), the discrepancy magnitude by (9) is \(\ge\exp(-O(p^{-1}\ell_*8^b(b+1)^6))\). A bounded rotated real-part comparison thus distinguishes this pair (average over traces and threshold halfway between expectations) with error \(\le e^{-2n}\) by Hoeffding's inequality \cite{Hoeffding} using \(\exp(O(p^{-1}\ell_*8^b(b+1)^6+\log n))\) traces, uniformly over pairs. Use all pair comparisons on the same traces; with high probability the true extended input wins against all \(<2^n\) challengers by union bound ($2^n e^{-2n}<1/3$) and so can be identified as the unique candidate winning against all competitors (output any default on failure; restrict to extended inputs with the known zero prefix and then discard it in the output). Equivalently, the expectation gap of a bounded rotated real test implies a gap at least half as large for an indicator by total variation, without explicitly locating these particular complex parameters. For instance the binary pair indicator attaining total variation on padded traces can just compare subsequence embedding counts for the two candidates: each trace word's probability is its embedding count times the same length-dependent factor for either candidate. Sampling the needed padded traces only uses the original traces and independent randomness. The comparisons can depend on the candidates and known parameters, not on the hidden truth. This information-theoretic union argument has no assertion of efficiency and does not rely on computational checking or search as a proof. With \(\mathcal X\) defined as in the theorem but using the doubled length \(n\),
+we have \(2^b\le2\mathcal X\) and \(\ell_*\mathcal X=\ell_*+\log n\le1+2\log n\)
+since \(q\ge1/n\). Thus the sample exponent is at most
+\(O(p^{-1}(\log n)\mathcal X^2(1+\log \mathcal X)^6)\), and switching in
+this estimate back to the original length rather than its double changes it
+by at most an absolute factor. This is at most
+\(O(p^{-1}(\log n)^3(1+\log\log(2n))^6)\) using original-length notation.
+In original-length notation, if \(q\le n^{-\varepsilon}\), \(\varepsilon>0\) fixed, then either \(q=0\)
+(trivial) or \(\mathcal X\le1+1/\varepsilon\);
+also \(p\) is bounded below for large \(n\); hence polynomial samples suffice.
+
+\section*{Extension to general symbols} For clarity use $n$
+here for the original length. Collect $\lceil p^{-1}\log(12n)\rceil$
+preliminary samples; except with probability at most $1/12$ by a union
+bound, every source symbol is in the observed set $\mathcal S$, and
+$|\mathcal S|\le n$ always. Given success, on independent fresh traces
+run tests for pairs in $\mathcal S^n$, mapping symbols for each pair to bits by
+the indicator of one of the distinct symbols at its first differing position,
+then using the binary pair argument (including padding as needed) for the two
+projected candidates. For large $n$, $p,1-p\ge1/n$ the Hoeffding sample size can
+be multiplied by $O(\log n)$ for tail $\le e^{-3n\log n}$ per pair and a
+union bound over rivals. Output a candidate winning all its comparisons or
+declare failure if there is none. The increase does not affect the stated asymptotic
+budget. The bounded-size or $\min(p,1-p)<1/n$ cases can again directly use the
+whole-trace strategy instead. This extension also concerns sample complexity
+only.
+
+\begin{thebibliography}{9}
+\bibitem{BVW}
+A.~Burudgunte, P.~Valiant and H.~Wang, Quasipolynomial Trace Reconstruction, arXiv:2607.04073v1.
+\bibitem{Chase}
+Z.~Chase, New upper bounds for trace reconstruction, arXiv:2009.03296.
+\bibitem{CDLS}
+X.~Chen, A.~De, C.~H.~Lee, R.~A.~Servedio and S.~Sinha,
+Polynomial-time trace reconstruction in the low deletion rate regime,
+ITCS 2021.
+\bibitem{FW}
+N.~J. Fine and H.~S. Wilf, Uniqueness theorems for periodic functions, \emph{Proc.\ AMS} 16 (1965), 109--114.
+\bibitem{Hoeffding}
+W.~Hoeffding,
+Probability inequalities for sums of bounded random variables.
+\emph{Journal of the American Statistical Association} 58 (1963), 13--30.
+\bibitem{Rub}
+I.~Rubinstein, Average-Case to (shifted) Worst-Case Reduction for the Trace Reconstruction Problem, arXiv:2207.11489.
+\end{thebibliography}
+\end{document}
diff --git a/preprints/A-latest-anchor-induction-with-spectrally-compact-masks-for-worst-case-trace-reconstruction-October-5-2026/build/summary.tex b/preprints/A-latest-anchor-induction-with-spectrally-compact-masks-for-worst-case-trace-reconstruction-October-5-2026/build/summary.tex
new file mode 100644
index 0000000000000000000000000000000000000000..224a44ac33053c702c6dc8f026aa1ded45d762c3
--- /dev/null
+++ b/preprints/A-latest-anchor-induction-with-spectrally-compact-masks-for-worst-case-trace-reconstruction-October-5-2026/build/summary.tex
@@ -0,0 +1,117 @@
+\PassOptionsToPackage{unicode}{hyperref}
+\PassOptionsToPackage{hyphens}{url}
+\documentclass[
+  11pt,
+]{article}
+\usepackage{amsmath,amssymb}
+\usepackage{iftex}
+\ifPDFTeX
+  \usepackage[T1]{fontenc}
+  \usepackage[utf8]{inputenc}
+  \usepackage{textcomp} %
+\else %
+  \usepackage{unicode-math} %
+  \defaultfontfeatures{Scale=MatchLowercase}
+  \defaultfontfeatures[\rmfamily]{Ligatures=TeX,Scale=1}
+\fi
+
+\ifPDFTeX\else
+  \IfFontExistsTF{DejaVu Serif}{\setmainfont{DejaVu Serif}}{
+    \setmainfont{DejaVuSerif}[
+      Extension=.ttf,UprightFont=*,BoldFont=*-Bold,
+      ItalicFont=*-Italic,BoldItalicFont=*-BoldItalic]}
+  \IfFontExistsTF{DejaVu Sans Mono}{\setmonofont{DejaVu Sans Mono}}{
+    \setmonofont{DejaVuSansMono}[
+      Extension=.ttf,UprightFont=*,BoldFont=*-Bold,
+      ItalicFont=*-Oblique,BoldItalicFont=*-BoldOblique]}
+  \setmathfont{latinmodern-math.otf}
+\fi
+\IfFileExists{upquote.sty}{\usepackage{upquote}}{}
+\IfFileExists{microtype.sty}{%
+  \usepackage[]{microtype}
+  \UseMicrotypeSet[protrusion]{basicmath} %
+}{}
+\makeatletter
+\@ifundefined{KOMAClassName}{%
+  \IfFileExists{parskip.sty}{%
+    \usepackage{parskip}
+  }{%
+    \setlength{\parindent}{0pt}
+    \setlength{\parskip}{6pt plus 2pt minus 1pt}}
+}{%
+  \KOMAoptions{parskip=half}}
+\makeatother
+\usepackage{xcolor}
+\usepackage[margin=1in]{geometry}
+\setlength{\emergencystretch}{3em} %
+\providecommand{\tightlist}{%
+  \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
+\setcounter{secnumdepth}{-\maxdimen} %
+
+\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs}
+\usepackage{unicode-math}
+\setmathfont{latinmodern-math.otf}
+\usepackage{newunicodechar,fvextra,adjustbox}
+\newunicodechar{≫}{\ensuremath{\gg}}
+\newunicodechar{≪}{\ensuremath{\ll}}
+\newunicodechar{ℓ}{\ensuremath{\ell}}
+\newunicodechar{ℝ}{\ensuremath{\mathbb R}}
+\newunicodechar{〈}{\ensuremath{\langle}}
+\newunicodechar{〉}{\ensuremath{\rangle}}
+\newunicodechar{𝜖}{\ensuremath{\epsilon}}
+\newunicodechar{≲}{\ensuremath{\lesssim}}
+\RecustomVerbatimEnvironment{verbatim}{Verbatim}{breaklines,breakanywhere,fontsize=\small}
+\newtheorem{theorem}{Theorem}
+\newtheorem{lemma}{Lemma}
+\newtheorem{proposition}{Proposition}
+\newtheorem{claim}{Claim}
+\newtheorem{thm}{Theorem}
+\newtheorem{lem}{Lemma}
+\providecommand{\HH}{\mathbb H}
+\providecommand{\RR}{\mathbb R}
+\providecommand{\CC}{\mathbb C}
+\providecommand{\CP}{\mathbb {CP}}
+\providecommand{\SL}{\operatorname{SL}}
+\providecommand{\PSL}{\operatorname{PSL}}
+\providecommand{\Ric}{\operatorname{Ric}}
+\providecommand{\vol}{\operatorname{vol}}
+\emergencystretch=3em
+\sloppy
+\allowdisplaybreaks
+\ifLuaTeX
+  \usepackage{selnolig}  %
+\fi
+\IfFileExists{bookmark.sty}{\usepackage{bookmark}}{\usepackage{hyperref}}
+\IfFileExists{xurl.sty}{\usepackage{xurl}}{} %
+\urlstyle{same}
+\hypersetup{
+  hidelinks,
+  pdfcreator={LaTeX via pandoc}}
+
+\title{A latest-anchor induction with spectrally compact masks\\for worst-case trace reconstruction}
+\author{OpenAI}
+\date{October 5, 2026}
+
+\begin{document}
+
+\hypertarget{trace-reconstruction-sample-upper-bound}{%
+\maketitle\label{trace-reconstruction-sample-upper-bound}}
+
+\textbf{Summary. Stated improvement; optimal complexity open}
+
+\hypertarget{question}{%
+\subsection{Question}\label{question}}
+
+How many independently deleted copies of an unknown binary string suffice to reconstruct every length-n string?
+
+\hypertarget{result}{%
+\subsection{Result}\label{result}}
+
+For retention probability \(p\), an upper bound \(\exp(O(p^{-1}(\log n)^3(1+\log\log(2n))^6))\) on the number of traces. The report also gives a polynomial sample bound when deletion probability is at most \(n^{-\varepsilon}\) for fixed \(\varepsilon>0\).
+
+\hypertarget{scope-and-remaining-questions}{%
+\subsection{Scope and remaining questions}\label{scope-and-remaining-questions}}
+
+Provides the requested improved sample upper bound. It does not prove matching optimality or polynomial reconstruction running time.
+
+\end{document}
diff --git a/preprints/A-latest-anchor-induction-with-spectrally-compact-masks-for-worst-case-trace-reconstruction-October-5-2026/paper.pdf b/preprints/A-latest-anchor-induction-with-spectrally-compact-masks-for-worst-case-trace-reconstruction-October-5-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..78c654429773a319eb8f6e57bbb195db03d77c88
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+++ b/preprints/A-latest-anchor-induction-with-spectrally-compact-masks-for-worst-case-trace-reconstruction-October-5-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:56113d1f2019792bc3fe96cccde891a44de1c2e5bea1447ddea254051219bece
+size 373650
diff --git a/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/README.md b/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..5ef81c80feb35a48233afdf60ddaa9f54b23a4cd
--- /dev/null
+++ b/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/README.md
@@ -0,0 +1,20 @@
+# [A linear cycle-and-edge decomposition of every graph](main.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 24, 2026
+
+## Citation
+
+To cite this manuscript:
+
+```bibtex
+@misc{OAI:A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026,
+  author = {{OpenAI}},
+  title = {{A linear cycle-and-edge decomposition of every graph}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/main.pdf}{OAI:A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/build/figures/lifting.tex b/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/build/figures/lifting.tex
new file mode 100644
index 0000000000000000000000000000000000000000..f83688ef536c598d77ea0bb8c5f0fe6f5580dd6a
--- /dev/null
+++ b/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/build/figures/lifting.tex
@@ -0,0 +1,49 @@
+\begin{figure}[tbp]
+\centering
+\begin{tikzpicture}[x=0.88cm,y=0.88cm,
+  original/.style={line width=0.9pt},
+  route/.style={line width=1.1pt,blue!65!black},
+  vertex/.style={circle,fill=black,inner sep=1.8pt},
+  inside/.style={circle,fill=blue!65!black,inner sep=1.6pt}]
+\coordinate (z1) at (0,1.8);
+\coordinate (z2) at (1.3,-0.2);
+\coordinate (z3) at (-1.3,-0.2);
+\draw[original] (z1)--node[right] {$e_1$}(z2)--node[below] {$e_2$}(z3)--node[left] {$e_3$}(z1);
+\node[vertex,label=above:$z_1$] at (z1) {};
+\node[vertex,label=right:$z_2$] at (z2) {};
+\node[vertex,label=left:$z_3$] at (z3) {};
+\node at (0,-1.1) {a quotient cycle};
+\draw[->,gray,thick] (1.8,0.8)--(3.2,0.8);
+\node[above] at (2.5,0.8) {lift};
+\coordinate (u1) at (4.4,1.8);
+\coordinate (v1) at (5.6,1.8);
+\coordinate (u2) at (6.5,0.4);
+\coordinate (v2) at (5.9,-0.3);
+\coordinate (u3) at (4.1,-0.3);
+\coordinate (v3) at (3.5,0.4);
+\draw[original] (v1)--node[right] {$\widehat e_1$}(u2);
+\draw[original] (v2)--node[below] {$\widehat e_2$}(u3);
+\draw[original] (v3)--node[left] {$\widehat e_3$}(u1);
+\draw[route] (u1)--(5,2.3)--(v1);
+\draw[route] (u2)--(6.5,-0.25)--(v2);
+\draw[route] (u3)--(3.5,-0.25)--(v3);
+\node[inside] at (5,2.3) {};
+\node[inside] at (6.5,-0.25) {};
+\node[inside] at (3.5,-0.25) {};
+\node[vertex,label=above left:$u_1$] at (u1) {};
+\node[vertex,label=above right:$v_1$] at (v1) {};
+\node[vertex,label=right:$u_2$] at (u2) {};
+\node[vertex,label=below right:$v_2$] at (v2) {};
+\node[vertex,label=below left:$u_3$] at (u3) {};
+\node[vertex,label=left:$v_3$] at (v3) {};
+\node at (5,-1.1) {original edges and switch paths};
+\end{tikzpicture}
+\caption{Lifting a quotient cycle, schematically. The fiber at \(z_j\)
+contains \(u_j,v_j\); when they differ, a reserved path (blue) joins
+them. The black edges are the unique original representatives of the
+quotient edges. Every path interior avoids all black-edge endpoints,
+and the interiors are pairwise disjoint for this cycle. Each blue
+path is drawn with one internal vertex only for illustration; its
+actual length may differ. A visit with \(u_j=v_j\) requires no path.}
+\label{fig:cycle-lifting}
+\end{figure}
diff --git a/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/build/main.tex b/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/build/main.tex
new file mode 100644
index 0000000000000000000000000000000000000000..e5d0fa423e8bf1ad7dd973e79ab8930644e9a1f1
--- /dev/null
+++ b/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/build/main.tex
@@ -0,0 +1,260 @@
+\documentclass[11pt]{article}
+\usepackage[OT1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage[margin=1in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{enumitem,booktabs,array}
+\usepackage{microtype}
+\usepackage{xcolor}
+\usepackage{tikz}
+\usepackage{hyperref}
+\usepackage[nameinlink,noabbrev]{cleveref}
+\hypersetup{
+  colorlinks=true,
+  linkcolor=blue!45!black,
+  citecolor=green!35!black,
+  urlcolor=blue!55!black,
+  pdftitle={A linear cycle-and-edge decomposition of every graph},
+  pdfauthor={OpenAI}
+}
+\numberwithin{equation}{section}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{corollary}[theorem]{Corollary}
+\newtheorem{claim}[theorem]{Claim}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newcommand{\abs}[1]{\left|#1\right|}
+\newcommand{\ceil}[1]{\left\lceil#1\right\rceil}
+\newcommand{\floor}[1]{\left\lfloor#1\right\rfloor}
+\newcommand{\E}{\mathbb E}
+\newcommand{\N}{\mathbb N}
+\newcommand{\R}{\mathbb R}
+\newcommand{\eps}{\varepsilon}
+\setlist{itemsep=3pt,topsep=5pt}
+\setlist[itemize]{label=\(\bullet\)}
+\setlength{\emergencystretch}{2em}
+\title{A linear cycle-and-edge decomposition of every graph}
+\author{OpenAI}
+\date{September 24, 2026}
+
+\begin{document}
+\maketitle
+
+\begin{abstract}
+We prove that every finite simple undirected graph on \(n\) vertices
+has an edge partition into at most \(Cn\) simple cycles and single
+edges, for an absolute constant \(C\). This resolves the
+Erd\H{o}s--Gallai cycle decomposition conjecture positively.
+\end{abstract}
+
+\section{Introduction}\label{sec:introduction}
+
+How efficiently can the edges of a graph be partitioned into cycles?
+Single edges must also be allowed: a forest contains no cycle.
+The Erd\H{o}s--Gallai cycle decomposition conjecture asserts that this
+necessary allowance suffices for a universal linear bound.
+Here and throughout, a cycle is simple and has length at least three.
+
+\begin{theorem}\label{thm:main}
+There is an absolute constant \(C>0\) such that, for every finite
+simple undirected graph \(G\) with \(n\) vertices, \(E(G)\) has a
+partition into at most \(Cn\) parts, each of which is the edge set of
+a cycle or consists of a single edge.
+\end{theorem}
+
+The parts in this theorem may share vertices. They share no edges,
+and every edge of \(G\) occurs in exactly one part. Isolated vertices
+need not be covered, and the empty partition is permitted when
+\(E(G)=\varnothing\).
+
+The conjecture appears in the 1966 paper of Erd\H{o}s, Goodman and
+P\'osa~\cite[Section~5]{EGP1966}, which explicitly distinguishes
+unrestricted covers from edge-disjoint covers and records an
+\(O(n\log n)\) upper bound. Conlon, Fox and
+Sudakov~\cite{CFS2014} improved the general bound to
+\(O(n\log\log n)\) in 2014. Buci\'c and
+Montgomery~\cite{BM2024} subsequently proved the bound
+\(O(n\log^*n)\); their paper appeared in journal form in 2024.
+Here \(\log^*n\) is the number of logarithm iterations needed to
+reach at most one. \Cref{thm:main} establishes the conjectured
+linear bound. Its order is optimal, since a tree on \(n\) vertices
+requires \(n-1\) single-edge parts. More sharply, complete bipartite
+examples require \((3/2-o(1))n\) parts; see
+\cite[Section~6]{BM2024} for this lower bound and its history.
+
+Linear bounds were already known for important classes. Conlon, Fox
+and Sudakov~\cite[Theorems~1.3 and~1.4]{CFS2014} proved them with high
+probability for binomial random graphs, and deterministically for
+graphs of linear minimum degree.
+Kor\'andi, Krivelevich and Sudakov~\cite{KKS2015} obtained
+asymptotically sharp counts over a broad range of random-graph
+probabilities. For every fixed \(\alpha,\delta>0\), Gir\~ao,
+Granet, K\"uhn and Osthus~\cite[Theorem~1.10(iii)]{GGKO2021}
+proved that sufficiently large graphs of minimum degree at least
+\(\alpha n\) admit at most \((3/2+\delta)n\) cycles and edges.
+Akbari, Aloni, Beikmohammadi and Clow~\cite[Theorem~1.3]{AABC2025}
+proved the bound \(n-1\) for graphs of positive order and maximum
+degree at most four.
+The general problem allows both sparse regions and vertices of
+widely differing degrees.
+
+The edge-disjoint requirement is essential to the problem. If cycles
+and edges may overlap, Pyber~\cite{Pyber1985} proved that \(n-1\)
+parts suffice to cover every graph of positive order. Akbari, Aloni,
+Beikmohammadi and Clow~\cite[Theorem~1.7]{AABC2025} improved this
+covering bound to \(n-2\) for graphs containing a cycle.
+A cover of that kind does not supply a partition: deleting repeated
+edges may break its cycles into many paths. Our construction keeps a
+separate assignment for every original edge throughout.
+
+\Cref{thm:main} also yields two cycle-only consequences. Here a graph is
+\emph{Eulerian} if all its degrees are even; connectivity is not required.
+
+\begin{corollary}\label{cor:eulerian-cycles}
+Every finite simple undirected Eulerian graph on \(n\) vertices has an
+edge partition into at most \(Cn\) cycles, with \(C\) from \Cref{thm:main}.
+
+For every fixed \(\delta,p>0\), there exist \(\eps>0\) and an integer
+\(n_0\ge1\) such that every finite simple undirected Eulerian graph \(G\)
+on \(n\ge n_0\) vertices has an edge partition into at most
+\(\Delta(G)/2+\delta n\) cycles, provided that every partition
+\(V(G)=A\mathbin{\dot\cup}B\) with \(|A|,|B|\ge\eps n\) satisfies
+\[
+ e_G(A,B)\ge p|A||B|.
+\]
+Here \(\Delta(G)\) denotes the maximum degree, and \(e_G(A,B)\)
+is the number of edges joining \(A\) to \(B\).
+\end{corollary}
+
+Any edge partition of \(G\) into cycles needs at least \(\Delta(G)/2\)
+cycles, since a cycle uses at most two edges incident with a given
+vertex. The second assertion attains this lower bound up to an
+arbitrarily small linear error under the stated large-cut condition.
+
+\begin{proof}
+For the first claim, let the partition from \Cref{thm:main} have \(r\)
+cycle parts and \(s\) single-edge parts, with \(r+s\le Cn\). The graph
+formed by the latter edges has even degrees, since deleting cycles
+preserves degree parity. Repeatedly removing a cycle preserves this
+property and ends with no edges: every even graph with an edge contains
+a cycle. This uses at most \(s\) cycles, for a total of at most
+\(r+s\le Cn\). The equivalence of this \(O(n)\) Eulerian cycle bound
+with the Erd\H{o}s--Gallai conjecture is recorded in
+\cite[Section~1.1]{GGKO2021}.
+
+The partition-cut condition in the second claim is exactly the
+weakly-\((\eps,p)\)-quasirandom condition of
+\cite[Section~1.2]{GGKO2021}. Their Proposition~6.3 proves that
+Conjecture~1.14, the stated bound with these quantifiers, is equivalent
+to their Erd\H{o}s--Gallai Conjecture~1.4, supplied by \Cref{thm:main}.
+\end{proof}
+
+Our argument builds on the expansion, hypergraph matching and
+path-closing methods of Buci\'c and Montgomery~\cite{BM2024}, with
+earlier reservoir and sparse-cut methods of Conlon, Fox and
+Sudakov~\cite{CFS2014}.
+We use Lov\'asz's path-and-cycle decomposition
+theorem~\cite{Lovasz1968} and the Aharoni--Haxell matching
+theorem~\cite{AH2000}, in the precise forms stated below.
+The additional mechanism is an induction that saves vertices
+across different scales while reserving enough edges to lift
+cycles from the smaller graphs. A decomposition that repeatedly pays
+a cost proportional to the full graph order can accumulate a factor
+from the number of scales. Here the work over a suitable prefix is
+charged to one expanding layer, and pair identifications save a fixed
+fraction of that layer's order. The savings pay for the work outside
+the recursive calls.
+
+\subsection*{Outline of the proof}
+
+All logarithms below are to base two. A graph has cut expansion
+\(h\) when every cut has at least \(h\) times the size of its
+smaller side in crossing edges.
+We split the graph at successively smaller scales
+\(D_j=n^{(0.95)^j}\). At scale \(D\), a small collection of long cycles
+is removed, and the remaining edges are assigned to boxes of order at
+most \(D^{1.02}\). Their vertex sets may overlap, but the total order
+increases by only a factor \(1+O(1/\log D)\). Within each box, we
+extract vertex-disjoint pieces with cut expansion \(D^{0.90}\).
+Each residual box graph has average degree at most \(D^{0.95}\), so
+the construction continues at the next scale.
+
+Processing every expanding piece at every scale would accumulate too
+much cost. Instead, we choose a layer whose total piece order pays
+for an entire prefix, including a fixed number of further scales.
+The pieces of this layer have two roles. First, an expanding graph
+can be partitioned into linearly many completed parts and residual
+graphs whose total order is an arbitrarily small fraction of its
+own order (\Cref{lem:small-residue}). Second, part of its edge set
+can be reserved for routing without destroying that conclusion.
+The sparse graphs passed to the next scale retain their full box
+vertex sets; the small-residue graphs produced when pieces are
+processed are eventually handled by induction.
+
+We describe the nonterminal case; the terminal cases are treated
+separately in \Cref{sec:induction}. After the fixed gap in scales,
+the unprocessed edges are sparse and
+live on very small descendant boxes. We gather those boxes into
+batches. Within each earlier box \(Y\), all but
+\(O(\abs{Y}/\log D)\) vertices have small degree in the union of
+that box's batch graphs. We pair these vertices in sufficiently large
+intersections between a batch and a selected expanding piece, and
+identify each pair. Edges between a folding set and vertices having
+too many neighbors in that set are first partitioned into paths; reserved
+routes close the trimmed remnants into cycles at linear cost. The
+remaining graph can then be simplified after pair identification by
+setting aside only linearly many edges as single-edge parts
+(\Cref{lem:pair-folding,lem:batch-resolution}).
+
+The quotient graphs have fewer vertices, so induction partitions
+their edges into cycles and single edges. Each quotient edge retains
+one original representative. At a paired vertex, the representatives
+of a quotient cycle may enter through one member of the pair and leave
+through the other. We join these two members by a path in the reserved
+expanding piece. The routing lemma chooses these paths edge-disjointly,
+with interiors that avoid the representative endpoints and are pairwise
+disjoint within each cycle (\Cref{lem:routing}). Consequently each
+quotient cycle lifts to one simple cycle. The degree bound is imposed
+on the union of the batch graphs within each
+earlier box, so the routing load does not grow with the number of batches.
+
+In this case, write \(S\) for the total order of the selected layer.
+All work outside the recursive calls costs \(O(S+n/\log D)\) parts.
+The quotient orders sum to at most \(n-S/3+O(n/\log D)\), and the
+small-residue graphs have total order at most \(S/100\). In the
+resulting cost estimate, choosing \(C\) sufficiently large makes
+the coefficient of \(S\) nonpositive. To absorb the remaining
+\(O(Cn/\log D)\) overhead, we prove the stronger inductive bound
+\(C\phi(n)n\), where
+\[
+ \phi(n)=\max\left\{\frac12,\,
+                 1-\frac1{\sqrt{\log n}}\right\}
+ \quad(n>1),\qquad \phi(1)=\frac12.
+\]
+The quotient graphs have order at most \(D^{0.30}\). Their smaller
+value of \(\phi\) supplies a gain of order \(1/\sqrt{\log D}\),
+which dominates the \(O(1/\log D)\) overlap cost.
+All recursive graphs are strictly smaller, and all scale
+requirements are fixed before the final constant \(C\) is chosen.
+
+Section~\ref{sec:preliminaries} fixes the graph conventions and the two
+external theorem inputs. Sections~\ref{sec:routing} and~\ref{sec:residue}
+establish the routing
+and small-residue tools. Section~\ref{sec:splitting} constructs the
+scale layers, Section~\ref{sec:folding} combines folding with those
+tools, and Section~\ref{sec:induction} completes the induction.
+
+\input{sections/preliminaries}
+\input{sections/routing}
+\input{sections/residue}
+\input{sections/splitting}
+\input{sections/folding}
+\input{sections/induction}
+
+\bibliographystyle{plain}
+\bibliography{references}
+\end{document}
diff --git a/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/build/references.bib b/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/build/references.bib
new file mode 100644
index 0000000000000000000000000000000000000000..c6bafcc28d51e664fce62259bd9063a7d432cdc3
--- /dev/null
+++ b/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/build/references.bib
@@ -0,0 +1,134 @@
+@article{AH2000,
+  author = {Aharoni, Ron and Haxell, Penny},
+  title = {{Hall}'s theorem for hypergraphs},
+  journal = {Journal of Graph Theory},
+  volume = {35},
+  number = {2},
+  year = {2000},
+  pages = {83--88},
+  doi = {10.1002/1097-0118(200010)35:2<83::AID-JGT2>3.0.CO;2-V},
+  url = {https://doi.org/10.1002/1097-0118(200010)35:2%3C83::AID-JGT2%3E3.0.CO;2-V}
+}
+
+@article{BM2024,
+  author = {Buci\'c, Matija and Montgomery, Richard},
+  title = {Towards the {Erd\H{o}s--Gallai} cycle decomposition conjecture},
+  journal = {Advances in Mathematics},
+  volume = {437},
+  year = {2024},
+  pages = {109434},
+  doi = {10.1016/j.aim.2023.109434},
+  eprint = {2211.07689},
+  archivePrefix = {arXiv},
+  primaryClass = {math.CO},
+  url = {https://arxiv.org/abs/2211.07689v2},
+  note = {Theorem numbering refers to arXiv:2211.07689v2, 14 November 2023}
+}
+
+@article{CFS2014,
+  author = {Conlon, David and Fox, Jacob and Sudakov, Benny},
+  title = {Cycle packing},
+  journal = {Random Structures \& Algorithms},
+  volume = {45},
+  number = {4},
+  year = {2014},
+  pages = {608--626},
+  doi = {10.1002/rsa.20574},
+  eprint = {1310.0632},
+  archivePrefix = {arXiv},
+  primaryClass = {math.CO},
+  url = {https://people.math.ethz.ch/~sudakovb/cycle-packing.pdf}
+}
+
+@article{EGP1966,
+  author = {Erd\H{o}s, Paul and Goodman, A. W. and P\'osa, Louis},
+  title = {The representation of a graph by set intersections},
+  journal = {Canadian Journal of Mathematics},
+  volume = {18},
+  year = {1966},
+  pages = {106--112},
+  doi = {10.4153/CJM-1966-014-3},
+  url = {https://www.renyi.hu/~p_erdos/1966-21.pdf}
+}
+
+@incollection{Lovasz1968,
+  author = {Lov\'asz, L\'aszl\'o},
+  title = {On covering of graphs},
+  booktitle = {Theory of Graphs (Proceedings of the Colloquium, Tihany, 1966)},
+  editor = {Erd\H{o}s, P. and Katona, G.},
+  publisher = {Academic Press; Akad\'emiai Kiad\'o},
+  address = {New York--London; Budapest},
+  year = {1968},
+  pages = {231--236},
+  url = {https://www.researchgate.net/profile/Laszlo-Lovasz/publication/268652766_On_covering_of_graphs/links/5e1465b7299bf10bc3967731/On-covering-of-graphs.pdf}
+}
+
+@article{Krivelevich2019,
+  author = {Krivelevich, Michael},
+  title = {Long cycles in locally expanding graphs, with applications},
+  journal = {Combinatorica},
+  volume = {39},
+  number = {1},
+  year = {2019},
+  pages = {135--151},
+  doi = {10.1007/s00493-017-3701-1},
+  eprint = {1609.06851},
+  archivePrefix = {arXiv},
+  primaryClass = {math.CO},
+  url = {https://www.math.tau.ac.il/~krivelev/long_cycles_expanders.pdf}
+}
+
+@article{BKS2012,
+  author = {Ben-Eliezer, Ido and Krivelevich, Michael and Sudakov, Benny},
+  title = {Long cycles in subgraphs of (pseudo)random directed graphs},
+  journal = {Journal of Graph Theory},
+  volume = {70},
+  number = {3},
+  year = {2012},
+  pages = {284--296},
+  doi = {10.1002/jgt.20616},
+  eprint = {1009.3721},
+  archivePrefix = {arXiv},
+  primaryClass = {math.CO},
+  url = {https://arxiv.org/abs/1009.3721}
+}
+
+@article{Pyber1985,
+  author = {Pyber, L\'aszl\'o},
+  title = {An {Erd\H{o}s--Gallai} conjecture},
+  journal = {Combinatorica},
+  volume = {5},
+  number = {1},
+  year = {1985},
+  pages = {67--79},
+  doi = {10.1007/BF02579444},
+  url = {https://doi.org/10.1007/BF02579444}
+}
+
+@article{KKS2015,
+  author = {Kor{\'a}ndi, D{\'a}niel and Krivelevich, Michael and Sudakov, Benny},
+  title = {Decomposing random graphs into few cycles and edges},
+  journal = {Combinatorics, Probability and Computing},
+  volume = {24}, number = {6}, year = {2015}, pages = {857--872},
+  doi = {10.1017/S0963548314000844},
+  url = {https://people.math.ethz.ch/~sudakovb/cycle-packing-random-graphs.pdf}
+}
+@article{GGKO2021,
+  author = {Gir{\~a}o, Ant{\'o}nio and Granet, Bertille and K{\"u}hn, Daniela and Osthus, Deryk},
+  title = {Path and cycle decompositions of dense graphs},
+  journal = {Journal of the London Mathematical Society},
+  volume = {104}, number = {3}, year = {2021}, pages = {1085--1134},
+  doi = {10.1112/jlms.12455},
+  url = {https://web.mat.bham.ac.uk/D.Osthus/path_cycle_dec_dense.pdf}
+}
+
+@misc{AABC2025,
+  author = {Akbari, Saieed and Aloni, Jonny and Beikmohammadi, Arash and Clow, Alexander},
+  title = {Tight Bounds for Cycle-Edge Decompositions and Covers},
+  year = {2025},
+  eprint = {2509.01901},
+  archivePrefix = {arXiv},
+  primaryClass = {math.CO},
+  url = {https://arxiv.org/abs/2509.01901v2},
+  note = {arXiv:2509.01901v2, 7 September 2025}
+}
diff --git a/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/build/sections/folding.tex b/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/build/sections/folding.tex
new file mode 100644
index 0000000000000000000000000000000000000000..24077533b9d42604522f201b77edb3de48dd3b39
--- /dev/null
+++ b/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/build/sections/folding.tex
@@ -0,0 +1,453 @@
+\section{Pair folding and resolving batches}
+\label{sec:folding}
+
+Identifying pairs of vertices reduces the order of the graph to which
+induction will be applied. After setting aside loops and all but one edge
+in each parallel class, each quotient edge has a unique original
+representative. A quotient cycle may enter a pair at one member and leave
+through the other; a reserved path joins them. To obtain a simple cycle,
+the interiors of these paths must avoid all representative endpoints and
+be pairwise disjoint within that cycle.
+
+The first lemma bounds the number of edges set aside in this cleanup.
+A degree bound at the paired vertices alone is insufficient: for positive
+integers $a$ and $m$ with $m$ even, pairing the $m$-vertex side of
+$K_{m,a}$ forces $am/2$ edges to be set aside, although each paired vertex
+has degree $a$. The lemma therefore also bounds the number of neighbors
+in each folding set at every vertex, including vertices outside those sets.
+
+\begin{lemma}[Pair folding]\label{lem:pair-folding}
+Let $J$ be a finite simple graph and let $X_1,\ldots,X_t$ be pairwise
+disjoint subsets of its vertex set.  Suppose that $a\ge1$, that
+$m_i=\abs{X_i}\ge\max\{8,a^2\}$ for every $i$, and that
+\begin{align}
+ d_J(x)&\le a &&\text{for every }x\in\bigcup_i X_i,\label{eq:fold-degree}\\
+ d_i(v):=\abs{N_J(v)\cap X_i}&\le a
+     &&\text{for every }v\in V(J)\text{ and every }i.
+     \label{eq:fold-neighbors}
+\end{align}
+There are pairings within the $X_i$, each leaving exactly $m_i\bmod2$
+vertices unpaired, and a set $E_{\rm err}\subseteq E(J)$ with
+\[
+ \abs{E_{\rm err}}\le 2\sum_i m_i,
+\]
+such that identifying each pair in $J-E_{\rm err}$ gives a simple graph
+$Q$.  Every edge of $Q$ has exactly one retained original edge as its
+representative.  Retaining all quotient vertices, including isolated ones,
+we have
+\begin{equation}\label{eq:fold-order}
+ \abs{V(Q)}=\abs{V(J)}-\sum_i\floor{m_i/2}.
+\end{equation}
+The assertion also holds when the family of folding sets is empty.
+\end{lemma}
+
+\begin{proof}
+Choose a uniform perfect pairing when $m_i$ is even.  When $m_i$ is odd,
+adjoin a dummy vertex, choose a uniform perfect pairing on the $m_i+1$
+vertices, and delete the dummy vertex and its incident pairing edge,
+leaving its original partner unpaired.  Each pairing with
+one unpaired original vertex has exactly one extension of this kind, so
+the resulting distribution is uniform.  Make the choices for different
+$i$ independently.
+
+For distinct specified vertices $u,v$ of a set of order $m$, symmetry of
+the partner of $u$ gives the following exact probabilities.  For two
+prescribed disjoint pairs, conditioning on the first pair leaves a uniform
+perfect pairing on the remaining vertices, including the dummy when
+present.  Thus
+\begin{equation}\label{eq:fold-probabilities}
+\begin{array}{c|cc}
+ &m\text{ even}&m\text{ odd}\\ \hline
+ \Pr(\{u,v\}\text{ is a pair})
+     &\dfrac1{m-1}&\dfrac1m\\[6pt]
+ \Pr(\text{two prescribed disjoint pairs occur})
+     &\dfrac1{(m-1)(m-3)}&\dfrac1{m(m-2)}
+\end{array}
+\end{equation}
+In particular the even-column expressions are valid upper bounds for
+both parities.
+
+Let $\pi$ send each vertex to its fiber after the identifications.  Every
+fiber has size one or two.  Before removing any edges, let $Z$ count the
+original edges that become loops, plus the unordered pairs of distinct
+original edges that become the same nonloop edge.  We bound $\E Z$ by
+classifying all possible errors.
+
+\paragraph{Loops and collisions with a common original endpoint.}
+An original edge becomes a loop precisely when its endpoints form a pair
+in some $X_i$.  There are at most $a m_i/2$ original edges inside $X_i$,
+so its expected loop contribution is at most $a m_i/(2(m_i-1))$.
+
+Two distinct edges sharing an original endpoint have the form $vu,vw$
+with $u\ne w$.  If they become the same nonloop edge, then
+$\pi(u)=\pi(w)\ne\pi(v)$, so $u,w$ must be paired in some $X_i$.
+No other identification is possible: identifying all three original
+vertices would require a fiber of size at least three.  Consequently the
+contribution charged to $X_i$ is at most
+\[
+ \frac1{m_i-1}\sum_{v\in V(J)}\binom{d_i(v)}2
+ \le \frac{a(a-1)m_i}{2(m_i-1)}.
+\]
+Here $\binom d2\le(a-1)d/2$ for every integer $0\le d\le a$, and
+\[
+ \sum_{v\in V(J)}d_i(v)=\sum_{x\in X_i}d_J(x)\le a m_i.
+\]
+The equality counts both incidences of every internal edge on both sides.
+Combining the loop and common-endpoint bounds, the expectation is at most
+\begin{equation}\label{eq:fold-first-errors}
+ \sum_i\frac{a^2m_i}{2(m_i-1)}
+ \le\frac47\sum_i a^2,
+\end{equation}
+since $m_i\ge8$.
+
+\paragraph{Collisions with four distinct original endpoints.}
+Write the two edges as $uv$ and $xy$, with four different vertices.
+For them to become the same nonloop edge, the two fibers must be either
+$\{u,x\},\{v,y\}$ or $\{u,y\},\{v,x\}$.  All four vertices therefore
+belong to folding sets: a singleton fiber cannot contain two distinct
+original endpoints.  If the two paired fibers lie in one $X_i$, then both
+original edges are internal to $X_i$.  Writing $e_i=\abs{E(J[X_i])}$,
+there are at most $\binom{e_i}2\le e_i^2/2$ possible vertex-disjoint edge
+pairs, and the two possible
+pairing patterns give probability at most
+$2/((m_i-1)(m_i-3))$.  As $e_i\le a m_i/2$, their total contribution is
+at most
+\begin{equation}\label{eq:fold-internal-errors}
+ \sum_i\frac{a^2m_i^2}{4(m_i-1)(m_i-3)}
+ \le\frac{16}{35}\sum_i a^2.
+\end{equation}
+For the last inequality use $m_i/(m_i-1)\le8/7$ and
+$m_i/(m_i-3)\le8/5$.
+
+If the two paired fibers instead lie in different sets $X_i,X_j$, then
+both original edges run between those sets.  There is just one possible
+pairing pattern, since each pair must remain within its own set.  Put
+$e_{ij}=e_J(X_i,X_j)$; there are at most $\binom{e_{ij}}2\le e_{ij}^2/2$
+vertex-disjoint edge pairs.  Independence between the two pairings and
+\eqref{eq:fold-probabilities} bound this contribution by
+\begin{align}
+ \sum_{i<j}\frac{e_{ij}^2}{2(m_i-1)(m_j-1)}
+ &\le\frac{32}{49}\sum_{i<j}\frac{e_{ij}^2}{m_i m_j}
+ \le\frac{16}{49}\sum_i a^2.\label{eq:fold-cross-errors}
+\end{align}
+Indeed, for each fixed $i$,
+\[
+ \sum_{j\ne i}\frac{e_{ij}^2}{m_i m_j}
+ \le\frac a{m_i}\sum_{j\ne i}e_{ij}\le a^2:
+\]
+the first inequality uses $e_{ij}\le a m_j$, and the second follows from
+the total degree bound on $X_i$.  Summing over $i$ counts each unordered
+pair of sets twice.  The preceding cases exhaust distinct original
+edges, which either have one common endpoint or have four distinct
+endpoints in a simple graph.
+
+Equations \eqref{eq:fold-first-errors}--\eqref{eq:fold-cross-errors} give
+\[
+ \E Z\le
+ \left(\frac47+\frac{16}{35}+\frac{16}{49}\right)\sum_i a^2
+ =\frac{332}{245}\sum_i a^2
+ \le\frac{332}{245}\sum_i m_i<2\sum_i m_i
+\]
+when $t>0$.  Some choice of pairings has no greater error count than its
+expectation.  For this choice delete each original loop edge and, in each
+nonloop parallel class of size $s$, delete all but one original edge.
+The latter requires $s-1\le\binom s2$ deletions, so the number of deleted
+edges is at most $Z$.  The remaining graph is simple after identification,
+and the retained representative in each class is unique.  Deletions do
+not remove vertices, giving \eqref{eq:fold-order}.  If $t=0$, use the
+identity map and delete no edges.
+\end{proof}
+
+The next lemma resolves all batches within a single earlier box.  Its
+hypotheses keep the degree bound global over those batches.  Different
+batches may contain the same vertex and may pair that vertex differently;
+their original edge sets are disjoint.
+
+\begin{lemma}[Batch resolution]\label{lem:batch-resolution}
+For every sufficiently large real $D$, set
+\[
+ a=D^{0.01},\qquad L_0=D^{0.03},\qquad
+ B_0=D^{0.30},\qquad \sigma=D^{0.90}.
+\]
+Let $Y$ be a finite vertex set.  Let $\mathcal W$ be a finite indexed
+family of subsets $W\subseteq Y$, with $\abs W\le B_0$, and let $J_W$
+be a simple graph on $W$ for each index.  Suppose these graphs have
+pairwise disjoint edge sets, and let $F=F(Y)$ be their union, taken on
+$Y$.
+
+Let $\mathcal R$ be a family of simple graphs on pairwise disjoint vertex
+subsets of $Y$.  Suppose that each $R\in\mathcal R$ has order
+$\sigma\le r_R\le D^{1.02}$, that
+$E(R)\cap E(F)=\varnothing$, and that its edges are partitioned into
+two spanning graphs $P_R,T_R$, both of cut expansion at least
+$\sigma/4$.  Call $P_R$ the router.
+
+For some pairs $(W,R)$ let an active set
+$X_{W,R}\subseteq W\cap V(R)$ be specified, satisfying
+\begin{equation}\label{eq:batch-active-hypotheses}
+ \abs{X_{W,R}}\ge L_0,
+ \qquad d_F(x)\le a\quad(x\in X_{W,R}).
+\end{equation}
+Sums over $(W,R)$ below include only active sets, and write
+$\Sigma_Y=\sum_{W,R}\abs{X_{W,R}}$.
+
+There are finite simple quotient graphs $Q_W$, with all quotient vertices
+retained, whose orders satisfy
+\begin{align}
+ q_W&=\abs W-\sum_{R:\,(W,R)\text{ active}}
+                 \floor{\abs{X_{W,R}}/2}\notag\\
+ &\le\abs W-\frac13
+            \sum_{R:\,(W,R)\text{ active}}\abs{X_{W,R}}
+ \le B_0,\label{eq:batch-quotient-order}\\
+ \sum_W q_W&\le\sum_W\abs W-\frac13\Sigma_Y.
+ \label{eq:batch-total-order}
+\end{align}
+For every choice of cycle-and-edge partitions of these $Q_W$, having
+$t_W$ parts, there are sets $U_R\subseteq E(P_R)$ and a cycle-and-edge
+partition of
+\[
+ E(F)\mathbin{\dot\cup}\mathop{\dot\bigcup}_{R\in\mathcal R}U_R
+\]
+with at most
+\begin{equation}\label{eq:batch-parts}
+ \sum_W t_W+8\Sigma_Y
+\end{equation}
+parts.  The graphs $R'=R-U_R$, kept on $V(R)$, contain the untouched
+spanning graphs $T_R$, so each still has a spanning subgraph of cut
+expansion at least $\sigma/4$.  The completed parts and these remaining
+graphs together partition
+$E(F)\mathbin{\dot\cup}\mathop{\dot\bigcup}_{R\in\mathcal R}E(R)$.
+All thresholds in this lemma are absolute.
+\end{lemma}
+
+\begin{proof}
+We first choose the quotient graphs and prepare some paths to be closed.
+No router edges are used until all quotient partitions have also been
+specified.
+
+\paragraph{Removing edges to heavy neighbors.}
+The path trimming below is the elementary operation used by
+Conlon, Fox and Sudakov~\cite[Lemma~6.3]{CFS2014}: decompose a
+bipartite graph into paths and remove terminal edges so that the
+remaining endpoints lie on one prescribed side. We keep the endpoint
+loads under the single degree bound on the union of all batches.
+Fix a batch $W$.  Its active sets are pairwise disjoint because the pieces
+$R$ have disjoint vertex sets.  For an active set $X=X_{W,R}$ put
+\[
+ H_X=\{v\in W:\abs{N_{J_W}(v)\cap X}>a\},
+ \qquad E_X=E_{J_W}(X,H_X).
+\]
+Every vertex in any active set has degree at most $a$ in $J_W$, since
+$J_W\subseteq F$.  Thus $H_X$ is disjoint from every active set in this
+batch, including $X$.  If $H_X$ is nonempty, counting edges gives
+\[
+ a\abs{H_X}<\abs{E_X}
+ \le\sum_{x\in X}d_{J_W}(x)\le a\abs X;
+\]
+in all cases $\abs{H_X}\le\abs X$.  The sets $E_X$ for different active
+$X$ are disjoint.  In fact each of their edges has one endpoint in its
+specified active set and its other endpoint outside every active set,
+so it cannot belong to the set specified by a different active set.
+
+Apply \Cref{cor:endpoint-paths} to the bipartite graph with parts
+$X,H_X$ and edge set $E_X$.  It gives at most
+$\abs X+\abs{H_X}\le2\abs X$ positive-length simple paths, since their
+total number of endpoint occurrences is at most
+$2(\abs X+\abs{H_X})$.  From each path delete its terminal
+edge at each endpoint belonging to $H_X$, counting those deleted edges
+singly.  At most two distinct edges are deleted.  A one-edge path has
+only one endpoint in $H_X$, and if both endpoints lie in $H_X$, the
+path has at least two edges, with distinct terminal edges.
+
+Ignore any remnant with no edges.  Every other remnant is a simple path
+with both endpoints in $X$: this follows directly by removing the
+$H_X$ endpoints from the alternating bipartite path.  Its length is
+even and at least two, so its endpoints are distinct.  For each such
+remnant $L$ record a demand between its endpoints in $P_R$, forbid
+internally the set $V(L)\cap V(R)$, and give this demand its own team.
+Once this demand is routed, its remnant will form one cycle.  Each
+original path therefore accounts for at most two single edges and one
+cycle, for a total of at most $6\abs X$ parts.  We do not require an
+endpoint-load-two bound for the trimmed paths; their endpoint loads will
+be bounded using their incident original edges in the global load
+argument below.
+
+\paragraph{Choosing the quotients.}
+Remove all the sets $E_X$ from $J_W$ and call the resulting graph $J_W^0$,
+retaining the vertex set $W$.  Each active vertex still has degree at
+most $a$.  For each active $X$ and every $v\in W$, either $v$ was in
+$H_X$, in which case all its edges into $X$ were removed, or it originally
+had at most $a$ neighbors in $X$.  Hence
+$\abs{N_{J_W^0}(v)\cap X}\le a$ in both cases.
+For large $D$ we have $L_0\ge\max\{8,a^2\}$.  Apply
+\Cref{lem:pair-folding} to $J_W^0$ and its active sets.  Count its error
+edges singly in their original form and call the simple resulting
+quotient $Q_W$.  Denote its quotient map by $\pi_W$; each edge $e$ of
+$Q_W$ has a unique retained original representative $\widehat e$.
+The number of these single-edge parts is at most
+$2\sum_R\abs{X_{W,R}}$.
+
+Every identified pair removes exactly one vertex.  Since
+$\floor{m/2}\ge m/3$ for every integer $m\ge2$, the exact order formula
+from \Cref{lem:pair-folding} yields
+\eqref{eq:batch-quotient-order}, and summing yields
+\eqref{eq:batch-total-order}.  This construction of all $Q_W$ is
+independent of their subsequent partitions.
+
+\paragraph{The switch demands of a quotient cycle.}
+Now fix arbitrary cycle-and-edge partitions of all the $Q_W$.  Restore
+each single-edge part as its unique original representative.  For a
+cycle $C$ in $Q_W$, list its distinct vertices in cyclic order as
+$z_1,\ldots,z_s$, where $s\ge3$, and write $e_j=z_jz_{j+1}$, with
+indices interpreted cyclically.  At $z_j$ let $u_j$ be the endpoint of
+$\widehat e_{j-1}$ in the fiber $\pi_W^{-1}(z_j)$, and let $v_j$ be
+the endpoint of $\widehat e_j$ in that fiber.  If $u_j=v_j$, the two
+edges join without any added path.  Otherwise that fiber is a folded
+pair $\{u_j,v_j\}$ in one active set $X_{W,R}$; record a demand from
+$u_j$ to $v_j$ in $P_R$.
+
+For every demand from this cycle forbid internally
+\[
+ Z_C\cap V(R),\qquad
+ Z_C=\bigcup_{j=1}^s V(\widehat e_j).
+\]
+Thus every original endpoint on the entire cycle, including the
+endpoints of all its other switches, is forbidden internally.  The
+demands belonging to this cycle and this router form one team.  Teams
+for different cycles, including cycles from different batch indices,
+are distinct.  The single-demand teams recorded during the heavy-neighbor
+work are also distinct from these teams.
+Figure~\ref{fig:cycle-lifting} shows the switch construction. The
+next two steps verify that all these requested paths can be chosen
+simultaneously and that the resulting closed walks are simple.
+
+\input{figures/lifting}
+
+
+\paragraph{The load bound over all batches.}
+Fix one piece $R$.  Gather every demand for $P_R$ from all batches and
+both constructions.  Each endpoint is in an active set, hence has
+degree at most $a$ in the one graph $F(Y)$.  To bound its total demand
+load, associate each occurrence of an endpoint $x$ with an incident
+edge of $F(Y)$ as follows:
+\begin{itemize}
+ \item For the demand closing a remnant $L$, use the terminal edge of
+       $L$ at $x$.
+ \item At a switch $u_j\ne v_j$, associate the occurrence of $u_j$
+       with $\widehat e_{j-1}$ and that of $v_j$ with $\widehat e_j$.
+\end{itemize}
+At a fixed vertex these associated incident edges are distinct.  For
+the first construction this follows from the edge-disjoint path
+partition and from the disjointness of the designated edge sets.
+For the second construction a quotient edge belongs to exactly one
+part, has exactly one original representative, and its incidence at a
+given original vertex can be used only at the unique visit to its
+quotient fiber in that cycle.  The cycles are simple, and a switch uses
+different original vertices for its two incidences.  The two constructions
+use disjoint edges because all designated edges were removed before
+folding.  Finally, edges from different batches are distinct by
+hypothesis, regardless of overlap of their vertex sets or differences
+in their pairings.  This proves the injection into incident edges of
+$F(Y)$ and hence
+\begin{equation}\label{eq:batch-global-load}
+ \text{number of demand-end occurrences at }x
+ \le d_{F(Y)}(x)\le a.
+\end{equation}
+Vertices that are never endpoints have load zero.
+
+\paragraph{Supplying all the connections.}
+A team from a cycle $C$ has at most $s\le q_W\le B_0$ demands, and
+$\abs{Z_C}\le2s\le2B_0$.  A heavy-neighbor remnant has at most
+$\abs W\le B_0$ vertices; its team has size one.  Thus in
+\Cref{lem:routing} we may use
+\[
+ r=r_R,\qquad h=\sigma/4,\qquad
+ k=a,\qquad b=B_0,\qquad z=2B_0.
+\]
+With that lemma's definitions of $g$ and $\ell$, the range
+$\sigma\le r_R\le D^{1.02}$ gives, uniformly in $R$,
+\[
+ \ell=O(D^{0.12}\log^2 D),
+ \qquad
+ \ell^2(a+B_0)+2B_0
+       =O(D^{0.54}\log^4 D)=o(D^{0.90}).
+\]
+For all sufficiently large $D$ this is at most $\sigma/256=h/64$,
+which is exactly enough for \eqref{eq:routing-condition}.  Apply the
+routing lemma once to this full collection in $P_R$; if the collection
+is empty, use no edges.  The paths supplied are pairwise edge-disjoint
+over all teams and all batches for this $R$, and their internal vertex
+sets are pairwise disjoint within each team.  Do this for every piece
+$R$.  The constants are absolute, and do not depend on the number of
+batches or demands.
+
+\paragraph{Simplicity of the completed cycles.}
+For a heavy-neighbor remnant $L$, its supplied path is simple and has
+distinct endpoints equal to those of $L$.  Its internal vertices avoid
+$V(L)$: the forbidden set contains $V(L)\cap V(R)$, and all supplied
+path vertices lie in $V(R)$.  The union
+is therefore a simple cycle.  It has at least two remnant edges and at
+least one supplying edge, so its length is at least three.
+
+For a quotient cycle $C$, the fibers of $z_1,\ldots,z_s$ are mutually
+disjoint.  At a visit with $u_j=v_j$, its original representative edges
+use one vertex.  At a switch, they use the two different vertices of
+that fiber and the supplied simple path joins them.  Thus before
+adding path interiors, the vertices at different visits are distinct.
+Every supplying path interior avoids $Z_C$, which includes all these
+original vertices.  Two supplying paths for this cycle in the same
+router have disjoint interiors by their common team; paths in different
+routers lie in the disjoint vertex sets of different pieces $R$.
+Consequently interiors of any two switches are disjoint, and none
+contains an endpoint of another switch or any other original vertex
+used by the cycle.  Traversing the representatives and these connections
+in cyclic order therefore visits no vertex twice, except the start
+when closing the cycle.
+
+An unused twin deserves explicit mention.  If a visited fiber is a pair
+but $u_j=v_j$, its other vertex need not belong to $Z_C$ and may occur
+inside a supplying path.  It is absent from all original representative
+edges of this cycle, since those edges use distinct fibers.  If it is
+used internally, no second supplying path of this cycle can use it:
+the team condition excludes this in its router, and the other routers
+have disjoint vertex sets.  This therefore creates no repeated vertex.
+The same reasoning covers vertices in fibers not visited by the cycle.
+
+The resulting cycle contains its $s$ distinct original representative
+edges and possibly additional edges, so its length is at least
+$s\ge3$.  A quotient cycle with no switches is already a simple cycle
+of original edges.  Each quotient part thus yields exactly one permitted
+part.
+
+\paragraph{Edge accounting and cost.}
+For each batch, its original edges have exactly four destinations:
+\begin{enumerate}
+ \item designated edges trimmed from paths are single-edge parts;
+ \item designated edges in positive-length remnants occur in their
+       closed cycles;
+ \item folding error edges are single-edge parts, taken before
+       identification;
+ \item retained edges have unique representatives in $Q_W$ and occur
+       once in the restored quotient parts.
+\end{enumerate}
+These classes partition $E(J_W)$.  Empty remnants leave no unaccounted
+edge, and the deletion rule for parallel classes retains exactly one
+representative for each quotient edge.  Since the $J_W$ have disjoint
+edge sets, every edge of $F$ occurs once in this list over all batches.
+
+Let $U_R$ be the set of edges used by all supplying paths in $P_R$.
+Each is used once by the routing lemma.  These edges are disjoint from
+$E(F)$ by hypothesis, and different routers have disjoint edge sets.
+Thus all completed parts form an edge partition of the edge set in the
+statement.  The heavy-neighbor work contributes at most $6\Sigma_Y$
+parts, folding errors at most $2\Sigma_Y$, and the quotient partitions
+contribute exactly $\sum_W t_W$.  Adding a supplying path changes no
+part count.  This proves \eqref{eq:batch-parts}.
+
+No edge of $T_R$ was used: $U_R\subseteq E(P_R)$ and $E(P_R)$ is
+disjoint from $E(T_R)$.  Therefore $R'=R-U_R$ contains the entire
+spanning graph $T_R$ on its original vertex set.  The edges of $R'$
+are precisely the edges of this assigned piece not already used in a
+completed part.  This proves both the remaining expansion assertion
+and the final edge partition.
+\end{proof}
diff --git a/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/build/sections/induction.tex b/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/build/sections/induction.tex
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+\section{The induction}\label{sec:induction}
+
+The preceding lemmas will be applied on a succession of scales. We
+first select a prefix whose total piece order is controlled by one
+of its layers. A fixed window beyond that layer makes the surviving
+edges suitable for batching and folding.
+
+Fix the absolute parameters
+\begin{equation}\label{eq:induction-parameters}
+ \theta=0.95,\qquad K=200,\qquad
+ P_0=4(K+1),\qquad \eta=\frac1{100P_0}.
+\end{equation}
+In particular,
+\begin{equation}\label{eq:scale-window}
+ 1.02\theta^K<0.001,\qquad \theta^{K+1}<0.001,
+\end{equation}
+since \(\theta^K\le e^{-10}\).
+
+\begin{lemma}[Selection of a prefix]\label{lem:prefix-choice}
+Let \((S_j)_{j\ge0}\) be a nonnegative, finitely supported sequence
+which is not identically zero. There is an index \(i\) with
+\(S_i>0\) such that
+\begin{equation}\label{eq:prefix-charge}
+ \sum_{j=0}^{i+K}S_j\le P_0S_i.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Start at the first positive term and maintain
+\(\sum_{j=0}^iS_j\le2(K+1)S_i\). If
+\eqref{eq:prefix-charge} fails, the sum of the next \(K\) terms
+is greater than \(2(K+1)S_i\). Their largest value \(M\) is
+therefore greater than \(2S_i\). Move to an index \(i'\) in
+that window with \(S_{i'}=M\). Then
+\[
+ \sum_{j=0}^{i'}S_j
+ \le 2(K+1)S_i+KM
+ <(2K+1)M\le2(K+1)S_{i'}.
+\]
+The invariant persists. Each move increases the index and lands
+at a positive term. Finite support therefore forces the procedure
+to stop, at which point \eqref{eq:prefix-charge} holds.
+\end{proof}
+
+For real \(x\ge1\), define the nondecreasing function
+\begin{equation}\label{eq:induction-potential}
+ \phi(1)=\frac12,\qquad
+ \phi(x)=\max\left\{\frac12,
+                1-\frac1{\sqrt{\log x}}\right\}
+ \quad(x>1).
+\end{equation}
+The small improvement over a linear bound will absorb the overlap
+introduced when boxes are split.
+
+\begin{proposition}\label{prop:induction-bound}
+There is an absolute constant \(C\) such that every finite simple
+graph on \(n\ge1\) vertices has an edge partition into at most
+\(C\phi(n)n\) simple cycles and single edges.
+\end{proposition}
+
+\begin{proof}
+We use strong induction on \(n\). The lower scale cutoff \(D_*\)
+is an absolute constant, chosen to satisfy the uniform requirements
+below. Only after fixing it will we choose \(C\). All auxiliary
+constants introduced before that final choice are independent of
+\(C\) and remain valid if the cutoff is increased.
+
+Edgeless graphs have the empty partition. For \(1\le n<D_*\),
+using every edge singly proves the assertion when \(C\ge D_*\),
+since \(\phi(n)\ge1/2\). Now let \(n\ge D_*\), assume the
+assertion for every smaller positive order, and put
+\[
+ D_j=n^{\theta^j},\qquad
+ m=\min\{j\ge0:D_j<D_*\}.
+\]
+Take \(D_*>2\). Then \(m\) is finite and positive, and the
+scales \(D_0,\ldots,D_{m-1}\) all satisfy the cutoff.
+
+\paragraph{Prospective layers.}
+Starting with the given graph \(G\), perform both operations of
+\Cref{lem:scale-split} at each successive scale
+\(D_0,\ldots,D_{m-1}\). The inputs at stage \(j\) have average
+degree at most \(D_j\): this holds initially because
+\(\overline d(G)\le n-1<D_0\), and the residual graph in each
+new box has average degree at most
+\(D_j^{0.95}=D_{j+1}\).
+
+Let \(\mathcal Y_j\) be the indexed family of level-\(j\)
+boxes and \(\mathcal R_j\) the indexed family of pieces removed
+inside them. The residual graph passed to the next stage retains
+the entire vertex set of its box, including isolated vertices.
+Every box has a unique indexed parent, and its vertex set is
+contained in its parent's vertex set. Descendants of any box
+cover that box's vertex set. Distinct indices are retained even
+when their vertex sets overlap or coincide.
+
+These operations define prospective layers: we will subsequently
+keep only a prefix of them. At the end of every prefix, the long
+cycles removed so far, the pieces removed so far, and the remaining
+input graphs partition \(E(G)\). In particular, discarding the
+prospective operations beyond a chosen prefix means retaining
+their input graphs, with all their assigned edges, at that
+prefix's boundary.
+
+Write
+\[
+ N_{-1}=n,\qquad N_j=\sum_{Y\in\mathcal Y_j}|Y|,
+ \qquad S_j=\sum_{R\in\mathcal R_j}|V(R)|,
+\]
+and set \(S_j=0\) for \(j\ge m\). Increase the absolute
+constant \(C_0\) in \Cref{lem:scale-split} if necessary so that
+\(C_0\ge1\), and put \(C_2=40C_0\). For \(0\le j<m\),
+\[
+ \sum_{l=0}^j\frac1{\log D_l}
+ =\frac1{\log D_j}\sum_{t=0}^j\theta^t
+ \le\frac{20}{\log D_j}.
+\]
+Provided \(\log D_*\ge C_2\), the product of the order
+increases is consequently bounded by
+\begin{equation}\label{eq:layer-order}
+ n\le N_j
+ \le n\prod_{l=0}^j\left(1+\frac{C_0}{\log D_l}\right)
+ \le n\left(1+\frac{C_2}{\log D_j}\right)
+ \le2n.
+\end{equation}
+Here we used \(1+x\le e^x\) and \(e^x\le1+2x\) for
+\(0\le x\le1/2\). The lower bound follows from preservation
+of the vertex cover at every split. The same product calculation,
+using the relevant subproduct, shows that the orders of all
+level-\(j\) descendants of a box of order \(y\) sum to at most
+\((1+C_2/\log D_j)y\). Their union still covers its full
+vertex set.
+
+We also require \(D^{-0.01}\le1/\log D\) for every
+\(D\ge D_*\). If \(\mathcal L_j\) is the family of long
+cycles removed through level \(j\), then
+\begin{equation}\label{eq:prefix-cycle-cost}
+ |\mathcal L_j|
+ \le\sum_{l=0}^j\frac{N_{l-1}}{2D_l^{0.01}}
+ \le n\sum_{l=0}^j\frac1{\log D_l}
+ \le\frac{20n}{\log D_j}.
+\end{equation}
+These bounds have no dependence on the number of layers.
+Piece disjointness within each box gives
+\(S_j\le N_j\le2n\).
+
+\paragraph{Terminal cases.}
+If every \(S_j\) vanishes, keep all prospective stages. Use the
+long cycles and take the remaining edges singly. The remaining
+graphs, on the level-\((m-1)\) boxes, have average degree at most
+\(D_m<D_*\), so their total number of edges is at most
+\(N_{m-1}D_m/2<D_*n\). By
+\eqref{eq:prefix-cycle-cost}, the total cost is at most
+\((20+D_*)n\).
+
+Otherwise choose \(i\) by \Cref{lem:prefix-choice}. If
+\(i+K\ge m-1\), again keep every stage. Retain all long cycles as
+completed parts and take every edge of the final boundary graphs
+singly, at cost at most \((20+D_*)n\) as above. The zero extension of
+\((S_j)\) ensures that
+\(\sum_{j=0}^{m-1}S_j\le P_0S_i\). Apply
+\Cref{lem:small-residue} to each piece at its own scale, with
+\(c=1\) and the fixed \(\eta\). This completes at most
+\(9P_0S_i\) parts and leaves graphs of total order at most
+\[
+ \eta P_0S_i=\frac{S_i}{100}\le\frac n{50}<\frac n4.
+\]
+Each individual residual graph has order at most \(\eta r<n\),
+where \(r\) is the order of its parent piece: the piece's
+vertex set is a subset of \(V(G)\), so \(r\le n\), and
+\(\eta<1\). Apply the induction hypothesis to each nonempty
+residual graph; their total additional cost is at most \(Cn/4\),
+using \(\phi\le1\).
+
+Both terminal cases are therefore covered by the bound
+\[
+ T(D_*)n+\frac{Cn}{4},\qquad
+ T(D_*)=20+D_*+18P_0.
+\]
+For \(C\ge4T(D_*)\), this is at most
+\(Cn/2\le C\phi(n)n\). This choice also covers the base cases.
+
+\paragraph{A prefix with a scale gap.}
+It remains to consider \(i+K<m-1\). Keep the stages through
+\(t=i+K\), and put
+\begin{equation}\label{eq:batch-parameters}
+ \begin{gathered}
+ D=D_i,\qquad w=\log D,\qquad H=C_2\theta^{-K},\\
+ B_0=D^{0.30},\qquad a=D^{0.01},\qquad
+ L_0=D^{0.03},\qquad \sigma=D^{0.90}.
+ \end{gathered}
+\end{equation}
+Since \(\log D_t=\theta^K w\),
+\eqref{eq:layer-order} and \eqref{eq:prefix-cycle-cost} imply
+\begin{equation}\label{eq:gap-global-bounds}
+ N_t\le(1+H/w)n,\qquad |\mathcal L_t|\le Hn/w.
+\end{equation}
+The latter uses \(C_2\ge40>20\).
+
+Fix a level-\(i\) box \(Y\), of order \(y\le D^{1.02}\).
+Let \(T_Y\) be the sum of the orders of its level-\(t\)
+descendants. Each such descendant carries its remaining graph
+after stage \(t\); it has order at most
+\(D^{1.02\theta^K}<D^{0.001}\) and that graph has average
+degree at most \(D^{\theta^{K+1}}\). Moreover,
+\begin{equation}\label{eq:descendant-order}
+ y\le T_Y\le(1+H/w)y\le2y.
+\end{equation}
+For the last inequality, note that
+\(H/w=C_2/\log D_t\le1\).
+
+Let \(F(Y)\) be the graph on \(Y\) whose edges are exactly
+the remaining edges in these descendants. Their assigned edge
+sets are disjoint, so \(F(Y)\) is simple and
+\begin{equation}\label{eq:descendant-degree}
+ 2|E(F(Y))|\le T_YD^{\theta^{K+1}}
+ \le2yD^{\theta^{K+1}}.
+\end{equation}
+
+Each descendant has fewer than \(L_0\) vertices, so we first group
+descendants to make larger folding sets possible. The number of
+batches will bound the loss when an intersection of a batch with a
+selected piece contains fewer than \(L_0\) vertices of degree at
+most \(a\) in \(F(Y)\). The batch order cap \(B_0\) keeps quotient
+cycles within the routing team bound.
+
+Partition the indexed descendants into batches whose sums of
+orders are at most \(B_0\). This can be done with at most
+\(1+4y/B_0\) batches: require the cutoff to ensure
+\(D^{0.001}\le B_0/2\), fill a batch until the next descendant
+would exceed \(B_0\), and then start a new one. Every batch
+except possibly the last has order sum greater than \(B_0/2\),
+so their number is at most \(1+2T_Y/B_0\).
+
+For each batch let \(W\) be the union of its vertex sets, and
+let \(J_W\) contain exactly its remaining assigned edges. The
+batch index is retained in this notation. Denote the family by
+\(\mathcal W(Y)\). Thus the \(J_W\) partition \(E(F(Y))\),
+the sets \(W\) cover \(Y\), and
+\begin{equation}\label{eq:batch-orders}
+ |W|\le B_0,\qquad
+ \sum_{W\in\mathcal W(Y)}|W|\le T_Y,\qquad
+ \sum_{Y\in\mathcal Y_i}\sum_{W\in\mathcal W(Y)}|W|
+ \le N_t.
+\end{equation}
+The order sums used in forming batches include all descendant
+multiplicities; passing to a union within a batch can only
+decrease them.
+
+Since each level-\(t\) box has a unique indexed level-\(i\)
+ancestor, the batch edge sets over all \(Y\) partition the boundary
+remainder. Thus, at the boundary of the committed prefix, the
+assigned edge sets give the disjoint union
+\begin{equation}\label{eq:global-edge-ledger}
+ E(G)=
+ \mathop{\dot\bigcup}_{L\in\mathcal L_t}E(L)
+ \;\dot\cup\;
+ \mathop{\dot\bigcup}_{j=0}^{t}\ 
+             \mathop{\dot\bigcup}_{R\in\mathcal R_j}E(R)
+ \;\dot\cup\;
+ \mathop{\dot\bigcup}_{Y\in\mathcal Y_i}\ 
+             \mathop{\dot\bigcup}_{W\in\mathcal W(Y)}E(J_W).
+\end{equation}
+
+\paragraph{Active sets and multiplicities.}
+Call a vertex of \(Y\) good if its degree in \(F(Y)\) is at
+most \(a\). By \eqref{eq:descendant-degree} and
+\eqref{eq:scale-window}, the number of other vertices is at most
+\[
+ 2yD^{\theta^{K+1}-0.01}\le2yD^{-0.009}\le y/w,
+\]
+where the final inequality is another uniform large-scale
+requirement. For each \(W\in\mathcal W(Y)\) and each
+level-\(i\) piece \(R\) belonging to \(Y\), consider all
+good vertices of \(W\cap V(R)\). If their number is at least
+\(L_0\), retain this set as \(X_{W,R}\) and call it active;
+otherwise skip it. Active sets in a fixed batch are disjoint,
+because the pieces in \(Y\) have disjoint vertex sets.
+
+There are at most \(y/\sigma\) such pieces. Consequently the
+total size of all skipped sets in \(Y\), counting occurrences
+in different batches separately, is at most
+\begin{align}
+ L_0(1+4y/B_0)y/\sigma
+ &\le y\bigl(D^{-0.87}+4D^{-0.15}\bigr)
+ \le y/w.\label{eq:skipped-occurrences}
+\end{align}
+Here the exponent \(-0.15\) uses \(y\le D^{1.02}\), and the
+last inequality is imposed for all scales above the cutoff.
+
+Let \(s(Y)\) be the sum of the orders of the level-\(i\)
+pieces in \(Y\), and define
+\[
+ \Sigma_Y=\sum_{\substack{W\in\mathcal W(Y),\ R\text{ in }Y\\
+                         X_{W,R}\text{ active}}}|X_{W,R}|,
+ \qquad \Sigma=\sum_{Y\in\mathcal Y_i}\Sigma_Y.
+\]
+Every good piece vertex occurs in at least one batch, since the
+batches cover \(Y\). Choose one such occurrence per vertex.
+At most \(y/w\) piece vertices are not good, and at most
+\(y/w\) of the chosen occurrences can lie in skipped sets by
+\eqref{eq:skipped-occurrences}. Hence
+\(\Sigma_Y\ge s(Y)-2y/w\).
+
+For the upper bound, let \(u_Y(v)\ge1\) be the number of
+batches in \(Y\) containing \(v\). Then
+\[
+ \sum_{v\in Y}\bigl(u_Y(v)-1\bigr)
+ =\sum_{W\in\mathcal W(Y)}|W|-y.
+\]
+Since the pieces in \(Y\) are vertex-disjoint, their excess
+occurrences are bounded by this full excess. Thus
+\(\Sigma_Y\le s(Y)+\sum_W|W|-y\). Summing the two bounds,
+using \(N_i\le2n\), \(N_i\ge n\), and
+\eqref{eq:gap-global-bounds}--\eqref{eq:batch-orders}, gives
+\begin{equation}\label{eq:active-order}
+ S_i-\frac{4n}{w}\le\Sigma
+ \le S_i+N_t-N_i\le S_i+\frac{Hn}{w}.
+\end{equation}
+This counts a nongood vertex only once per indexed box in the
+lower bound; no bound on its number of batch occurrences is
+needed. Overlaps between different boxes are already included
+in both \(S_i\) and \(N_i\).
+
+\paragraph{Resolving the batches and the pieces.}
+In every level-\(i\) piece \(R\), apply
+\Cref{lem:edge-split} with two classes to partition its assigned
+edges into spanning graphs \(P_R\) and \(T_R\), each with
+cut expansion at least \(\sigma/4\). Reserve \(P_R\) as a
+router and leave \(T_R\) untouched during the batch work.
+
+All hypotheses of \Cref{lem:batch-resolution} now hold in each
+\(Y\). The \(J_W\) have order at most \(B_0\) and disjoint
+assigned edges; their union is \(F(Y)\). The pieces have
+orders between \(\sigma\) and \(D^{1.02}\), disjoint vertex
+sets within \(Y\), and edges disjoint from \(F(Y)\).
+Every vertex of an active \(X_{W,R}\) has
+degree at most \(a\) in the whole \(F(Y)\), and its active
+set has order at least \(L_0\). In particular the degree bound
+is shared across all batches, as required by that lemma.
+
+The lemma produces simple quotient graphs \(Q_W\), with orders
+\begin{equation}\label{eq:individual-quotient-order}
+ q_W=|W|-\sum_{R:X_{W,R}\text{ active}}
+                       \floor{|X_{W,R}|/2}
+ \le |W|-\frac13\sum_{R:X_{W,R}\text{ active}}|X_{W,R}|.
+\end{equation}
+Every batch has positive order and pair identifications leave
+at least one vertex, so \(1\le q_W\le B_0\). Also
+\(B_0=D^{0.30}\le n^{0.30}<n\). Apply the induction
+hypothesis to these quotients, retaining isolated quotient
+vertices in their order counts. The batch lemma lifts all the
+resulting quotient parts simultaneously. In each \(Y\), the
+additional number of completed parts is at most \(8\Sigma_Y\),
+and the remaining graph on every \(R\) still contains \(T_R\).
+In particular, lifting a quotient cycle produces one simple
+cycle and has no additional part charge.
+
+Apply \Cref{lem:small-residue}, with the fixed \(\eta\), to
+the remaining graph of every level-\(i\) piece using \(c=1/4\).
+Apply it to every other piece of the committed prefix at its
+own scale using \(c=1\). The total number of parts supplied
+by these applications is at most \(9P_0S_i\). If
+\(\mathcal H\) is their family of residual graphs, then
+\begin{equation}\label{eq:piece-residual-order}
+ \sum_{H'\in\mathcal H}|V(H')|
+ \le\eta\sum_{j=0}^tS_j\le\eta P_0S_i.
+\end{equation}
+Every nonempty member has order at most \(\eta r<n\), for
+the same reason as in the terminal case. Induction applies to
+all of them.
+
+Within each \(Y\), \Cref{lem:batch-resolution} resolves exactly
+the batch category in \eqref{eq:global-edge-ledger} together
+with the router edges it consumes.
+Every consumed router edge comes from exactly one level-\(i\)
+piece and is used once. The piece lemmas are applied only after
+those edges have been removed, and partition all remaining
+piece edges into completed parts and the members of
+\(\mathcal H\). Distinct indexed boxes have disjoint assigned
+edge sets even when their vertex sets overlap. Thus the only
+unresolved original edges after lifting the quotient partitions
+are precisely the assigned edges of \(\mathcal H\), each
+appearing once. Recursively partitioning them completes an edge
+partition of \(G\).
+
+\paragraph{Order savings and the final estimate.}
+Define the absolute constants
+\[
+ A_1=H+\frac43,\qquad
+ A_2=\max\{9P_0+8,\,9H\}.
+\]
+By \eqref{eq:individual-quotient-order},
+\eqref{eq:batch-orders}, and \eqref{eq:active-order},
+\begin{equation}\label{eq:total-quotient-order}
+ \sum_{Y,W}q_W\le N_t-\frac{\Sigma}{3}
+ \le n-\frac{S_i}{3}+\frac{A_1n}{w}.
+\end{equation}
+The parts completed apart from the quotient partitions and
+the recursive work on \(\mathcal H\) number at most
+\begin{align}
+ |\mathcal L_t|+8\Sigma+9P_0S_i
+ &\le (9P_0+8)S_i+9Hn/w\notag\\
+ &\le A_2(S_i+n/w).\label{eq:nonrecursive-cost}
+\end{align}
+Monotonicity of \(\phi\), \(q_W\le B_0\), and
+\eqref{eq:piece-residual-order} therefore bound the complete
+partition size by
+\begin{equation}\label{eq:inductive-cost}
+ A_2(S_i+n/w)+C\eta P_0S_i
+ +C\phi(B_0)\left(n-\frac{S_i}{3}+\frac{A_1n}{w}\right).
+\end{equation}
+The coefficient of \(S_i\) in this expression is at most
+\[
+ A_2+\frac C{100}-\frac C6
+ =A_2-\frac{47C}{300}\le0
+ \qquad\text{if }C\ge7A_2.
+\]
+It remains to pay for the order-overlap term. Set
+\(\delta=1/\sqrt{0.30}-1>0\). For \(w\ge40/3\), both
+\(\phi(n)\) and \(\phi(B_0)\) take their nonconstant
+values, and \(\log n\ge w\) gives
+\begin{equation}\label{eq:potential-gain}
+ \phi(n)-\phi(B_0)
+ =\frac1{\sqrt{0.30w}}-\frac1{\sqrt{\log n}}
+ \ge\frac{\delta}{\sqrt w}.
+\end{equation}
+For \(C\ge7A_2\), the remaining overhead in
+\eqref{eq:inductive-cost}, beyond \(C\phi(B_0)n\), is at
+most \(C(A_1+1)n/w\). It is absorbed by
+\eqref{eq:potential-gain} whenever
+\begin{equation}\label{eq:cutoff-absorption}
+ \log D_*\ge\left(\frac{A_1+1}{\delta}\right)^2.
+\end{equation}
+No choice made here requires the total quotient order in
+\eqref{eq:total-quotient-order} to be at most \(n\); the
+potential gain pays for its possible excess.
+
+We finally specify the order of constant choices. The parameters
+in \eqref{eq:induction-parameters}, the scale-split constant
+\(C_0\ge1\), and hence \(C_2,H,A_1,A_2,\delta\), are
+already absolute and fixed. Choose \(D_*\) large enough for
+\Cref{lem:scale-split,lem:batch-resolution}, for
+\Cref{lem:edge-split} with two classes at expansion
+\(D^{0.90}\), and for \Cref{lem:small-residue} with the
+fixed \(\eta\) and \(c\in\{1,1/4\}\). Also impose
+\(\log D_*\ge\max\{C_2,40/3\}\),
+\eqref{eq:cutoff-absorption}, and, for every \(D\ge D_*\),
+\[
+ \begin{gathered}
+ D^{-0.01}\le\frac1{\log D},\qquad
+ D^{0.001}\le\tfrac12D^{0.30},\\
+ 2D^{-0.009}\le\frac1{\log D},\qquad
+ D^{-0.87}+4D^{-0.15}\le\frac1{\log D}.
+ \end{gathered}
+\]
+These are uniform eventual inequalities between fixed powers
+and logarithms. Such a cutoff exists independently of \(C\).
+After fixing it, choose
+\[
+ C\ge\max\{7A_2,\,4T(D_*)\}.
+\]
+The base and terminal cases then hold, and
+\eqref{eq:inductive-cost}--\eqref{eq:potential-gain} give at
+most \(C\phi(n)n\) parts in the remaining case. This closes
+the strong induction.
+\end{proof}
+
+Since \(\phi(n)\le1\), \Cref{prop:induction-bound} proves
+\Cref{thm:main} for positive order. The zero-vertex graph has
+the empty partition, completing all cases.
diff --git a/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/build/sections/preliminaries.tex b/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/build/sections/preliminaries.tex
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+\section{Conventions and preliminary results}\label{sec:preliminaries}
+
+\subsection{Graphs, partitions and scales}
+
+Unless explicitly stated otherwise, graphs are finite, simple and
+undirected. A path is simple and has at least one edge. The
+\emph{order} of a graph is its number of vertices, including any
+isolated vertices that have been retained. Temporary quotients
+formed by identifying vertices may have loops or parallel edges;
+these are removed before any decomposition theorem is applied.
+
+An indexed family of subgraphs partitions a graph when their
+assigned edge sets are pairwise disjoint and have the required
+union. Each subgraph carries only its assigned edges, rather than
+all edges induced by its vertex set. Different indices may have
+overlapping, or even equal, vertex sets. Order sums count those
+indices separately. Whenever edges from different members are
+combined, we specify the assignment and account for the edges
+that remain.
+
+For disjoint vertex sets \(A,B\), write \(E_H(A,B)\) for the set
+of edges of \(H\) between them and \(e_H(A,B)=|E_H(A,B)|\).
+Write \(\partial_H U\) for
+the edge boundary of \(U\subseteq V(H)\). An \(r\)-vertex graph
+\(H\) has \emph{cut expansion at least \(h\)} if
+\begin{equation}\label{eq:cut-expansion}
+ |\partial_H U|\ge h\min\{|U|,r-|U|\}
+ \qquad\text{for every }U\subseteq V(H).
+\end{equation}
+If \(r\ge2\), singleton cuts show that its minimum degree is at
+least \(h\), and consequently \(r\ge h+1\).
+
+All logarithms denoted by \(\log\) are to base two; \(\ln\)
+denotes the natural logarithm. Scales \(D\) are real, and decimal
+exponents denote exact rational numbers. Inequalities comparing
+integer orders with real bounds have their literal meaning.
+Asymptotic notation always refers to the scale tending to
+infinity. Once the explicitly fixed parameters of a lemma have
+been chosen, its implicit constants and lower scale threshold
+are uniform over the graphs and orders allowed by that lemma.
+In the final induction the finite list of parameters is fixed
+first, a common lower cutoff \(D_*\) is fixed next, and only then
+is \(C\) chosen.
+
+\subsection{A path decomposition with controlled endpoints}
+
+We use the following form of Lov\'asz's decomposition theorem.
+Disjointness in its original statement means
+edge-disjointness~\cite[Theorem~1]{Lovasz1968}; see also
+\cite[Theorem~21]{BM2024}.
+
+\begin{theorem}[Lov\'asz]\label{thm:lovasz}
+The edges of a finite simple graph on \(n\) vertices can be
+partitioned into at most \(\floor{n/2}\) paths and cycles.
+\end{theorem}
+
+The following consequence is recorded in
+\cite[Corollary~22]{BM2024}. We include the deduction because
+endpoint multiplicities are used in two different constructions.
+
+\begin{corollary}\label{cor:endpoint-paths}
+Every finite simple graph on \(n\) vertices has an edge
+partition into paths such that each vertex is an endpoint of at
+most two paths. In particular, the partition has at most \(n\)
+paths.
+\end{corollary}
+
+\begin{proof}
+For \(n=0\) there is nothing to prove. Otherwise add a vertex
+\(v_0\) and join it to precisely the even-degree vertices of
+the original graph, including the isolated vertices. Every old
+vertex now has odd degree. In a path-and-cycle partition its
+number of path-endpoint occurrences must therefore be odd and
+positive.
+
+Apply \Cref{thm:lovasz} to the augmented graph, and let \(p\)
+be the number of path parts. The old vertices force \(2p\ge n\),
+whereas the theorem gives
+\[
+ \ceil{n/2}\le p
+ \le \text{number of parts}
+ \le \floor{(n+1)/2}=\ceil{n/2}.
+\]
+Thus every part is a path. Each old vertex is an endpoint
+exactly once: an old vertex with at least three endpoint
+occurrences, together with the other old vertices, would force
+at least \(n+2\) occurrences, although \(2p\le n+1\).
+
+Delete \(v_0\) from these paths and discard any edgeless
+remnants. Every old vertex is incident to at most one deleted
+edge, so it gains at most one new endpoint occurrence. The
+remaining paths partition the original edge set and have
+endpoint load at most two. Each positive-length path contributes
+two endpoint occurrences, proving the final count.
+\end{proof}
+
+\subsection{Disjoint representatives for hypergraphs}
+
+A hypergraph here is a finite family of nonempty subsets of a
+finite ground set. A matching is a subfamily of pairwise
+disjoint edges. We use the following sufficient condition,
+from the hypergraph Hall theorem of Aharoni and
+Haxell~\cite{AH2000}, in the bounded-rank form stated in
+\cite[Theorem~6]{BM2024}.
+
+\begin{theorem}[Aharoni--Haxell]\label{thm:aharoni-haxell}
+Let \(s\) be a positive integer and let
+\((\mathcal H_i)_{i\in J}\) be a finite indexed family of
+hypergraphs on a common ground set. Suppose every edge is
+nonempty and has size at most \(s\). If, for every nonempty
+\(I\subseteq J\), the union
+\(\bigcup_{i\in I}\mathcal H_i\) has a matching of size
+strictly greater than \(s(|I|-1)\), then there are edges
+\(E_i\in\mathcal H_i\), one for each \(i\in J\), that are
+pairwise disjoint.
+\end{theorem}
+
+One may pass from the uniform-size version to this statement
+by padding each edge occurrence with private fresh vertices.
+Every matching before padding remains a matching after padding,
+and disjoint representatives after padding have disjoint
+original projections. The hypergraphs need not be distinct, and equal
+edges in different indexed hypergraphs cause no difficulty.
+
+\subsection{Elementary probabilistic estimates}
+
+We use linearity of expectation, Markov's inequality, the union
+bound, and the following binomial estimates. They are stated
+explicitly to fix constants for the sampling arguments.
+
+\begin{lemma}\label{lem:chernoff}
+If \(X\) is binomial with mean \(\mu\), then
+\[
+ \Pr(X\le\mu/2)\le e^{-\mu/8},
+ \qquad
+ \Pr(X\ge2\mu)\le e^{-\mu/3}.
+\]
+\end{lemma}
+
+\begin{proof}
+For \(X\) a sum of independent Bernoulli variables,
+\(\E e^{tX}\le\exp(\mu(e^t-1))\).
+Apply Markov's inequality with \(t=-\ln2\) for the lower
+tail and \(t=\ln2\) for the upper tail. The resulting exponents
+are \(-\mu(1-\ln2)/2\) and \(-\mu(2\ln2-1)\),
+respectively. They are at most \(-\mu/8\) and \(-\mu/3\).
+\end{proof}
diff --git a/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/build/sections/residue.tex b/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/build/sections/residue.tex
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+\section{Expansion in small reservoirs}\label{sec:residue}
+
+We next process a graph containing a spanning expanding subgraph.
+Reserving vertices and edges to connect and close paths is already
+central to the cycle decomposition argument of Conlon, Fox and
+Sudakov~\cite[Section~3]{CFS2014}. Our aim here is to leave residual
+graphs with arbitrarily small total vertex order, so that their
+recursive cost can be paid by the later folding savings. At a scale $D$, put
+\[
+  \sigma=D^{0.90},\qquad \tau=D^{0.74}.
+\]
+Throughout this section the order $r$ lies in
+$\sigma\le r\le D^{1.02}$.  All assertions for sufficiently large $D$
+are uniform over the graphs and orders in this range.  Their thresholds
+may depend on the fixed positive constants in the statements.
+
+\subsection{Splitting the expanding edges}
+
+\begin{lemma}[Edge splitting]\label{lem:edge-split}
+Fix $c>0$ and a positive integer $l$.  For all sufficiently large $D$,
+if an $r$-vertex graph $P$ has cut expansion at least
+$h_0\ge cD^{0.90}$ and $D^{0.90}\le r\le D^{1.02}$, then its edge set
+can be partitioned into $l$ spanning subgraphs, each with cut expansion
+at least $h_0/(2l)$.
+\end{lemma}
+
+\begin{proof}
+Assign each edge independently and uniformly to one of $l$ classes.
+Fix a cut whose smaller side has size $u\ge1$, and let $e$ be its
+number of edges in $P$.  In any fixed class its size is binomial with
+mean $e/l\ge h_0u/l$.  By \Cref{lem:chernoff}, the probability that
+this class has fewer than $h_0u/(2l)$ cut edges is at most
+\[
+  \exp\bigl(-h_0u/(8l)\bigr).
+\]
+There are at most $2\binom ru\le2r^u$ choices of a cut with smaller
+side of size $u$.  Thus the probability of any failure is at most
+\[
+  2l\sum_{u=1}^{\floor{r/2}}
+       \bigl(r\exp(-h_0/(8l))\bigr)^u=o(1).
+\]
+Indeed, $h_0\ge cD^{0.90}$ while $\log r\le1.02\log D$, so the base
+of this geometric sum tends to zero uniformly.  A successful
+assignment gives the required edge partition.  Each class is taken on
+the full vertex set of $P$.
+\end{proof}
+
+\subsection{Expansion after vertex sampling}
+
+The issue is to control every sampled cut without a union bound over
+all vertex subsets. Buci\'c and Montgomery establish connectivity by
+short paths through a random vertex set, using successive exposures
+and well-expanding subsets~\cite[Section~4]{BM2024}. In the polynomial cut-expansion
+range used here, we instead fix a small family of neighborhood unions
+and show that it approximates every possible failing sampled cut.
+
+\begin{lemma}[Vertex sampling]\label{lem:vertex-sampling}
+Fix $c'>0$ and $p\in(0,1)$.  Let $P$ have order
+$D^{0.90}\le r\le D^{1.02}$ and cut expansion at least
+$h=c'D^{0.90}$.  Select each vertex independently with probability $p$
+to form $V'$.  With probability $1-o(1)$ as $D\to\infty$, all of the
+following hold:
+\begin{equation}\label{eq:residue-sampling-properties}
+\begin{gathered}
+  |V'|\le2pr,\qquad
+  |N_P(v)\cap V'|\ge ph/2\quad(v\in V(P)),\\
+  P[V']\text{ has cut expansion at least }\tau=D^{0.74}.
+\end{gathered}
+\end{equation}
+The error term is uniform over the stated graphs and orders, with
+$c'$ and $p$ fixed.
+\end{lemma}
+
+\begin{proof}
+The minimum degree of $P$ is at least $h$.  The order bound and all
+the sampled-degree bounds fail with total probability at most
+\begin{equation}\label{eq:residue-size-degree-error}
+  \exp(-pr/3)+r\exp(-ph/8)=o(1)
+\end{equation}
+by \Cref{lem:chernoff}.  We prove the expansion assertion by first
+controlling a deterministic family of cuts and then approximating
+any hypothetical failing sampled cut by a member of that family.
+
+\paragraph{A deterministic witness family.}
+In a failing sampled cut, the sampled-degree bound forces most vertices
+on the smaller side to have many neighbors on that side. A small
+selection from that side can hit their neighborhoods while exposing few
+vertices across the cut; this is why we enumerate unions of a small
+number of original neighborhoods.
+Set
+\[
+  M=C_p\log r,\qquad C_p=16/p,\qquad
+  s=\ceil{4rM/h}.
+\]
+For sufficiently large $D$, we have $M/h<1$ and $s\le r$.
+Before sampling $V'$, form the family
+\[
+  \mathcal U=
+  \left\{\ \bigcup_{v\in S}N_P(v):
+                   S\subseteq V(P),\ |S|\le s\ \right\}.
+\]
+Its members are unions of open neighborhoods in the original graph,
+and its size satisfies
+\begin{equation}\label{eq:residue-witness-count}
+  |\mathcal U|
+  \le\sum_{t=0}^s\binom rt
+  \le(s+1)r^s
+  =\exp\!\bigl(O((1+rM/h)\log r)\bigr).
+\end{equation}
+Repeated neighborhood unions only decrease this count.
+
+We claim that, with probability $1-o(1)$, simultaneously for every
+$U\in\mathcal U$ satisfying
+$\min\{|U|,r-|U|\}\ge ph/8$, one has
+\begin{equation}\label{eq:residue-witness-cut}
+  e_P(U\cap V',V'\setminus U)
+     \ge\frac{p^2}{4}|\partial_P U|.
+\end{equation}
+Fix such a $U$, and put $e=|\partial_P U|$.  Expansion gives
+$e\ge ph^2/8$.  Partition its cut edges into at most $2r$ matchings:
+greedily color the cut edges, observing that an edge has fewer than
+$2r$ other edges incident with its endpoints.  Fix one such coloring
+for each cut before sampling.  The matching classes of size less than
+$e/(4r)$ contain fewer than $e/2$ edges in total.  In a remaining
+matching of size $m$, the indicators that both endpoints of an edge
+belong to $V'$ are independent, since its edges have disjoint
+endpoints.  Its retained size is therefore binomial with mean $p^2m$.
+The probability of retaining fewer than $p^2m/2$ edges is at most
+\[
+  \exp(-p^2m/8)
+  \le\exp\bigl(-p^2e/(32r)\bigr)
+  \le\exp\bigl(-p^3h^2/(256r)\bigr).
+\]
+When none of these matching events fails, the large classes retain
+at least $(p^2/2)(e/2)=p^2e/4$ edges.  Independence between different
+matching classes or between different cuts is unnecessary: a union
+bound bounds the probability of any failure of
+\eqref{eq:residue-witness-cut} by
+\begin{equation}\label{eq:residue-witness-error}
+  2r|\mathcal U|\exp\bigl(-p^3h^2/(256r)\bigr)=o(1).
+\end{equation}
+To verify this uniformly, the scale restrictions give
+\[
+  \frac{h^2}{r}\ge(c')^2D^{0.78},\qquad
+  \frac rh\le(c')^{-1}D^{0.12},\qquad
+  M=O_p(\log D).
+\]
+Consequently the positive term in the exponent coming from
+\eqref{eq:residue-witness-count} is
+$O_{p,c'}(D^{0.12}(\log D)^2)$, while the negative term in
+\eqref{eq:residue-witness-error} has magnitude at least a fixed
+positive multiple of $D^{0.78}$.
+
+\paragraph{Approximating a failing cut.}
+Fix an outcome satisfying the size and degree bounds and all of
+\eqref{eq:residue-witness-cut}.  Suppose, for a contradiction, that
+some $A\subseteq V'$ satisfies
+\begin{equation}\label{eq:residue-bad-cut}
+  0<a=|A|\le |V'|/2,
+  \qquad e_P(A,V'\setminus A)<\tau a.
+\end{equation}
+We may assume $D$ is large enough that $\tau<ph/4$.  If $a\le ph/4$,
+every vertex of $A$ has more than $ph/4$ neighbors in $V'\setminus A$,
+because its sampled degree is at least $ph/2$ and it has at most
+$a-1$ neighbors in $A$.  This contradicts
+\eqref{eq:residue-bad-cut}.  Thus
+\begin{equation}\label{eq:residue-bad-cut-size}
+  a>ph/4.
+\end{equation}
+Let
+\[
+  A_{\mathrm{good}}=
+    \{v\in A:|N_P(v)\cap A|\ge ph/4\}.
+\]
+Each vertex in $A\setminus A_{\mathrm{good}}$ has more than $ph/4$
+neighbors across the sampled cut.  Hence
+\begin{equation}\label{eq:residue-exceptional-vertices}
+  |A\setminus A_{\mathrm{good}}|
+       \le\frac{4\tau a}{ph}.
+\end{equation}
+
+We now make a separate auxiliary random choice solely to prove the
+existence of a suitable set $S\subseteq A$.  Include each vertex of
+$A$ independently with probability $q=M/h$.  For every
+$v\in A_{\mathrm{good}}$, the probability that $S$ misses all its
+neighbors in $A$ is at most
+\[
+  (1-q)^{ph/4}\le\exp(-pM/4).
+\]
+Thus the probability of any required miss is at most
+$r\exp(-pM/4)\le r^{-3}<1/4$.  Here the chosen value of $C_p$
+is more than sufficient, since all logarithms are base two.
+Also $\E|S|=aM/h$.  If $T(S)$ denotes the number of vertices of
+$V'\setminus A$ with a neighbor in $S$, then a union bound for each
+such vertex gives
+\[
+  \E T(S)
+  \le\frac Mh\sum_{v\in V'\setminus A}|N_P(v)\cap A|
+  =\frac Mh e_P(A,V'\setminus A)
+  <\frac{\tau Ma}{h}.
+\]
+Markov's inequality shows that each of the events
+$|S|>4aM/h$ and $T(S)>4\tau Ma/h$ has probability at most $1/4$.
+The sum of these three failure probabilities is less than one.
+We can therefore choose $S$ satisfying all three desired properties:
+\begin{equation}\label{eq:residue-hitting-set}
+\begin{gathered}
+  |S|\le4aM/h,\qquad T(S)\le4\tau Ma/h,\\
+  S\cap N_P(v)\ne\varnothing
+      \quad(v\in A_{\mathrm{good}}).
+\end{gathered}
+\end{equation}
+The auxiliary choice is made after $V'$ and $A$ have been fixed; the
+uniform event \eqref{eq:residue-witness-cut} was already established
+for every possible resulting neighborhood union.
+
+Put $U=\bigcup_{v\in S}N_P(v)$.  Since
+$|S|\le4aM/h\le4rM/h$, we have $U\in\mathcal U$.  The vertices of
+$A$ absent from $U$ are all exceptional in
+\eqref{eq:residue-exceptional-vertices}, and the vertices of
+$U\cap V'$ outside $A$ are counted by $T(S)$.  Consequently
+\begin{equation}\label{eq:residue-cut-approximation}
+  \delta:=|(U\cap V')\mathbin{\triangle}A|
+   \le\frac{4\tau a}{ph}+\frac{4\tau Ma}{h}
+   =O_p\!\left(\frac{\tau Ma}{h}\right)=o(a).
+\end{equation}
+In the last step we used
+$\tau M/h=O_{p,c'}(D^{-0.16}\log D)=o(1)$.
+Both sides of the cut in the \emph{original} graph are large:
+\[
+  |U|\ge |A|-\delta=a-\delta,
+  \qquad
+  r-|U|\ge |V'\setminus U|
+           \ge |V'\setminus A|-\delta\ge a-\delta.
+\]
+For sufficiently large $D$, these are at least $a/2>ph/8$.
+Applying \eqref{eq:residue-witness-cut} and the expansion of $P$
+therefore gives
+\begin{equation}\label{eq:residue-approximating-cut-lower}
+  e_P(U\cap V',V'\setminus U)\ge \frac{p^2ha}{8}.
+\end{equation}
+On the other hand, changing the side of one vertex changes the size
+of a cut in a simple $r$-vertex graph by at most $r-1$.  Changing
+the $\delta$ vertices in \eqref{eq:residue-cut-approximation} yields
+\[
+  e_P(U\cap V',V'\setminus U)
+       \le e_P(A,V'\setminus A)+r\delta
+       <\tau a+r\delta.
+\]
+The ratio of this upper bound to $ha$ is at most
+\begin{equation}\label{eq:residue-cut-perturbation}
+  \frac\tau h+
+       O_p\!\left(\frac{\tau Mr}{h^2}\right)
+  =O_{p,c'}(D^{-0.16})+
+       O_{p,c'}(D^{-0.04}\log D)=o(1),
+\end{equation}
+contradicting \eqref{eq:residue-approximating-cut-lower}.
+Thus no cut \eqref{eq:residue-bad-cut} exists.  Together with
+\eqref{eq:residue-size-degree-error} and
+\eqref{eq:residue-witness-error}, this proves the lemma.
+
+All uses of $o(1)$ above occur after $p$ and $c'$ have been fixed.
+In particular, an arbitrarily small fixed positive sampling
+probability changes the cutoff but does not change any of the
+strict exponent inequalities.
+\end{proof}
+
+\subsection{Closing paths through three reservoirs}
+
+Buci\'c and Montgomery use three vertex classes and reserved connecting
+subgraphs to close paths through a class avoided by the original
+path~\cite[Section~2.2 and Lemma~23]{BM2024}. We use three disjoint reservoirs of
+arbitrarily small fixed relative order. Every edge has both endpoints
+outside at least one of them. After reserving the connecting edges,
+we can therefore assign each remaining edge to such a complement,
+partition that graph into paths, and close the paths through its
+reservoir. The preceding sampling lemma supplies the internal expansion
+needed for these routes. Unused reserved edges between a reservoir and
+its complement are taken singly, leaving only unused reserved edges
+inside reservoirs for recursion.
+
+\begin{lemma}[Small residue]\label{lem:small-residue}
+Fix $\eta>0$ and $c>0$.  For all sufficiently large $D$, let $G$ be
+an $r$-vertex graph with
+$D^{0.90}\le r\le D^{1.02}$ containing a spanning subgraph of cut
+expansion at least $cD^{0.90}$.  Then $E(G)$ has a partition into at
+most $9r$ completed parts, each a simple cycle or a single edge,
+and the edge sets of at most three residual simple graphs.  The
+residual graphs have pairwise disjoint vertex sets contained in
+$V(G)$, their total order is at most $\eta r$, and each has order
+strictly less than $r$.
+\end{lemma}
+
+\begin{proof}
+Write $c_0=\min\{c,1\}$, and fix
+\[
+  p=\min\{\eta/12,1/12\},\qquad
+  h=(c_0/6)\sigma.
+\]
+In particular, $3p<1$, $6p\le\eta$, and $2p<1$.
+By \Cref{lem:edge-split}, the edges of the given spanning expanding
+subgraph can be partitioned into three spanning subgraphs
+$P_1,P_2,P_3$, each of cut expansion at least $h$.
+
+\paragraph{Disjoint reservoirs and bounded representative loads.}
+Label the vertices independently, using each label $i\in\{1,2,3\}$
+with probability $p$ and a fourth label with probability $1-3p$.
+Let $V_i$ be the vertices with label $i$.  These three sets are
+pairwise disjoint, and each marginal $V_i$ is a Bernoulli-$p$
+sample for the fixed graph $P_i$.  By \Cref{lem:vertex-sampling}
+and a union bound over the three marginal failure probabilities,
+we can choose the labels so that, for each $i$,
+\begin{equation}\label{eq:residue-three-reservoirs}
+\begin{gathered}
+  |V_i|\le2pr,\qquad
+  |N_{P_i}(v)\cap V_i|\ge ph/2\quad(v\in V(G)),\\
+  P_i[V_i]\text{ has cut expansion at least }\tau.
+\end{gathered}
+\end{equation}
+Independence between these three success events is not required.
+The neighbor assertion makes $V_i$ nonempty and gives every vertex
+of $V_i$ a neighbor in $V_i$, so $|V_i|\ge2$.
+
+For each $i$, reserve all edges
+$E_i^{\mathrm{in}}=E(P_i[V_i])$.  Also reserve two edges from each
+$v\in V(G)\setminus V_i$ to two distinct representatives in $V_i$.
+The representatives can be chosen so that at most
+\begin{equation}\label{eq:residue-representative-capacity}
+  k_0=\ceil{8r/(ph)}
+\end{equation}
+of these edges are incident with any one vertex of $V_i$.
+Indeed, process the two choices for every outside vertex in turn.
+At any point fewer than $2r$ edges have been chosen, so the number
+of representatives already at capacity $k_0$ is at most
+$2r/k_0\le ph/4$.  There are at least $ph/2$ potential
+representatives for the next choice by
+\eqref{eq:residue-three-reservoirs}.  Excluding the saturated ones
+and, for the second choice at a vertex, its first representative,
+leaves at least $ph/4-1>0$ choices for sufficiently large $D$.
+This greedy procedure proves the claim.  Denote the selected
+outside-to-representative edges by $E_i^{\mathrm{out}}$.
+
+For a fixed $i$, the internal and outside reservations are disjoint.
+Reservations for different values of $i$ also have disjoint edge
+sets, because they belong to the edge-disjoint graphs $P_i$.
+Let
+\[
+  E_{\mathrm{res}}=
+     \bigcup_{i=1}^3(E_i^{\mathrm{in}}\cup E_i^{\mathrm{out}})
+\]
+be the entire reserved edge set.
+
+\paragraph{The complementary path partitions.}
+Every edge in $E(G)\setminus E_{\mathrm{res}}$ has both endpoints
+outside at least one $V_i$, since its two endpoints meet at most two
+of the three disjoint reservoirs.  Assign each such edge to one
+such index $i$, choosing just one index if several are available.
+This gives edge-disjoint graphs $G_i$ on $V(G)\setminus V_i$ whose
+edge sets partition all nonreserved edges.
+By \Cref{cor:endpoint-paths}, each $G_i$ has a partition into
+positive-length simple paths, with each vertex an endpoint at most
+twice.  Since each path has two endpoint occurrences, its path
+family $\mathcal P_i$ has size at most $|V(G)\setminus V_i|\le r$.
+An empty edge set contributes no paths.
+
+For each endpoint occurrence of $v$ in $\mathcal P_i$, assign one
+of its two reserved edges in $E_i^{\mathrm{out}}$, using different
+edges for different occurrences.  This is possible by the endpoint
+load bound.  For a path $Q\in\mathcal P_i$ with distinct endpoints
+$x,y$, let $x',y'\in V_i$ be their assigned representatives.
+When $x'\ne y'$, issue one demand between them in $P_i[V_i]$.
+When $x'=y'$, no routing demand is necessary.
+Every demand-end occurrence uses a different reserved edge at its
+representative.  Hence the total demand load at each vertex of
+$V_i$ is at most $k_0$.
+
+\paragraph{Routing and simple cycles.}
+Apply \Cref{lem:routing} separately in each $P_i[V_i]$, with
+expansion parameter $\tau$, endpoint load $k=k_0$, singleton teams
+($b=1$), and no forbidden vertices ($z=0$).  We check its numerical
+hypothesis using the actual order $r_i=|V_i|$.  This graph has order
+at least two and cut expansion at least $\tau$, and therefore
+$\tau\le r_i-1$.  The value $\ell_i$ in \Cref{lem:routing} satisfies
+\[
+  \ell_i=O\!\left(\frac r\tau(\log r)^2\right),
+  \qquad k_0=O_{p,c}\!\left(\frac rh\right).
+\]
+Here the ceiling in \eqref{eq:residue-representative-capacity}
+is absorbed because $r/h\ge6$.  Consequently
+\begin{equation}\label{eq:residue-routing-margin}
+\begin{aligned}
+  \ell_i^2(k_0+1)
+   &=O_{p,c}\!\left(
+        \frac{r^3}{\tau^2h}(\log r)^4\right)\\
+   &=O_{p,c}(D^{0.68}(\log D)^4)
+     =o(D^{0.74})=o(\tau).
+\end{aligned}
+\end{equation}
+Thus \eqref{eq:routing-condition} holds for all sufficiently large
+$D$.  The lemma supplies pairwise edge-disjoint simple routing
+paths for all the distinct-representative demands in this reservoir.
+All their edges are in the reserved set $E_i^{\mathrm{in}}$.
+
+For $Q$ as above with $x'\ne y'$, join its endpoints by the edge
+$xx'$, the routing path from $x'$ to $y'$, and the edge $y'y$.
+Every new internal vertex lies in $V_i$, while $Q$ is entirely
+outside $V_i$.  The resulting closed walk is therefore a simple
+cycle.  When $x'=y'$, the two reserved edges themselves form a
+simple two-edge connection from $x$ to $y$ through that
+representative, and again close $Q$ to a simple cycle.  Its two
+reserved edges are distinct because $x\ne y$.  In this case even a
+one-edge path $Q$ produces a triangle; with different
+representatives the resulting cycle has at least four edges.
+Thus every completed cycle has length at least three.
+
+The cycles are edge-disjoint.  Their original path edges partition
+the nonreserved edges; each assigned representative edge is used
+once; and the routing paths are edge-disjoint within each $P_i[V_i]$.
+These three types of edges are disjoint by reservation.  The
+reservations in different $P_i$ are also edge-disjoint, regardless
+of vertex overlaps among different cycles.
+
+\paragraph{Counting and the exact remaining edge sets.}
+We have produced at most
+$\sum_i|\mathcal P_i|\le3r$ cycles.  Take every unused edge in
+$\bigcup_iE_i^{\mathrm{out}}$ as a single-edge part, using at most
+\[
+  \sum_{i=1}^3 2|V(G)\setminus V_i|\le6r
+\]
+further parts.  The number of completed parts is therefore at most
+$9r$.
+
+For each $i$, let $H_i$ be the graph on $V_i$ consisting of exactly
+the unused edges of $E_i^{\mathrm{in}}$.  These are the only
+remaining edges: every nonreserved edge was placed in a path,
+every reserved outside edge was used in a cycle or taken singly,
+and every reserved internal edge was used by a routing path or
+placed in $H_i$.  In particular, $H_i$ is not defined by taking
+all remaining edges of the original induced graph $G[V_i]$;
+nonreserved edges, including ones with both endpoints in a
+reservoir, were already assigned to the appropriate complementary
+graphs.  This accounts for every original edge exactly once.
+
+The graphs $H_i$ are simple, have the pairwise disjoint vertex sets
+$V_i$, and satisfy
+\begin{equation}\label{eq:residue-final-order}
+  \sum_{i=1}^3|V(H_i)|
+     =\sum_{i=1}^3|V_i|\le6pr\le\eta r,
+  \qquad |V(H_i)|\le2pr\le r/6<r.
+\end{equation}
+Isolated vertices may be retained; a residual graph with no edges
+may alternatively be omitted.  The only parameters fixed before
+increasing $D$ were $\eta,c$ and the constants determined from
+them.  This completes the proof.
+\end{proof}
diff --git a/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/build/sections/routing.tex b/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/build/sections/routing.tex
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+++ b/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/build/sections/routing.tex
@@ -0,0 +1,244 @@
+\section{Routing with team constraints}
+\label{sec:routing}
+
+A demand specifies two distinct vertices to be joined.  Demands are indexed,
+so several demands may specify the same pair.  Besides edge-disjointness of
+all routes, we will need internal vertex-disjointness for specified groups
+of routes.  The following lemma supplies both requirements.  Its proof uses
+the hypergraph formulation and expansion-to-connectivity method of
+Buci\'c and Montgomery~\cite[Proposition~8 and Lemma~9]{BM2024}.
+The vertex tokens introduced in the proof impose a separate
+internal-disjointness constraint for each team; different teams
+compete only for graph edges.
+
+\begin{lemma}[Routing]\label{lem:routing}
+Let $P$ be a graph of order $r\ge 2$ with cut expansion $h>0$, and put
+\begin{equation}\label{eq:routing-radii}
+  g=\ceil{8(r/h)\log(2r)},\qquad
+  \ell=2g\bigl(1+\ceil{\log r}\bigr).
+\end{equation}
+Let $\mathcal D$ be a finite set of demands, where demand $p$ has distinct
+endvertices $x_p,y_p\in V(P)$.  Suppose that each vertex occurs as an
+endvertex in at most $k$ demands, where $k\ge 1$.  Partition $\mathcal D$
+into teams of size at most $b$, where $b\ge 1$, and give each demand a set
+$Z_p\subseteq V(P)$ of at most $z$ vertices, where $z\ge 0$.
+If
+\begin{equation}\label{eq:routing-condition}
+  h\ge 64\bigl(\ell^2(k+b)+z\bigr),
+\end{equation}
+then there are simple $x_p$--$y_p$ paths $Q_p$ in $P$, one for each
+$p\in\mathcal D$, with the following properties:
+\begin{enumerate}[label=\textup{(\roman*)}]
+  \item each $Q_p$ has length at most $\ell$ and no internal vertex in $Z_p$;
+  \item the paths $Q_p$ are pairwise edge-disjoint;
+  \item the internal vertex sets of paths in the same team are pairwise
+    disjoint.
+\end{enumerate}
+The sets $Z_p$ may contain $x_p$ or $y_p$: their restrictions concern only
+internal vertices.  If a route must also avoid another demand's endvertices
+internally, those vertices may be included in its set $Z_p$.
+\end{lemma}
+
+\begin{proof}
+There is nothing to prove when $\mathcal D$ is empty.  Suppose it is
+nonempty, and write $c(p)$ for the team of demand $p$.
+
+\paragraph{A hypergraph obstruction.}
+Take a resource set that is the tagged disjoint union of $E(P)$ and the
+tokens $(v,c)$, where $v\in V(P)$ and $c$ is a team.  For each demand $p$,
+form a hypergraph $\mathcal H_p$ on this resource set.  Its hyperedges
+correspond to all simple $x_p$--$y_p$ paths of length at most $\ell$ whose
+internal vertices avoid $Z_p$.  A candidate path contributes its graph
+edges and the token $(v,c(p))$ for each of its internal vertices.  A path
+of length $t\ge 1$ therefore contributes $2t-1\le 2\ell$ resources.
+In particular all hyperedges are nonempty, and all the hypergraphs are
+finite.  A choice of pairwise disjoint representatives from the
+$\mathcal H_p$ gives exactly the three required properties.
+
+We verify the sufficient condition of \Cref{thm:aharoni-haxell} with
+$s=2\ell$.  Suppose, for a contradiction, that it fails for a nonempty
+$I\subseteq\mathcal D$.  Let $\mathcal M$ be a maximum matching in
+$\bigcup_{p\in I}\mathcal H_p$.  Then
+$\abs{\mathcal M}\le 2\ell(\abs I-1)$.  Let $F\subseteq E(P)$ be the
+graph-edge resources used by $\mathcal M$, and put
+\[
+  Y_c=\{v\in V(P):(v,c)\text{ is used by }\mathcal M\}.
+\]
+Each matching member uses at most $\ell$ graph edges and at most $\ell$
+tokens, so
+\begin{equation}\label{eq:routing-obstruction-budgets}
+  \abs F\le 2\ell^2\abs I,\qquad
+  \sum_c\abs{Y_c}\le 2\ell^2\abs I.
+\end{equation}
+These estimates allow several matching members to come from one demand;
+no restriction on their demand indices is being imposed.
+
+For each team let $m_c=\abs{\{p\in I:c(p)=c\}}\le b$.  Thus
+\[
+  \sum_{p\in I}\abs{Y_{c(p)}}
+  =\sum_c m_c\abs{Y_c}
+  \le 2b\ell^2\abs I.
+\]
+Consequently the set
+$I_1=\{p\in I:\abs{Y_{c(p)}}\le 4b\ell^2\}$ has size at least
+$\abs I/2$.  Choose an inclusion-maximal set $I_2\subseteq I_1$ whose
+endpoint pairs are pairwise vertex-disjoint.  Every demand in $I_1$ meets
+one of these selected pairs.  The two vertices of any selected pair
+occur in at most $2k$ demands in total, so
+\begin{equation}\label{eq:routing-pair-count}
+  \abs{I_2}\ge\frac{\abs{I_1}}{2k}
+  \ge\frac{\abs I}{4k}.
+\end{equation}
+We will also use the global endpoint count
+\begin{equation}\label{eq:routing-endpoint-count}
+  2\abs I\le kr.
+\end{equation}
+
+\paragraph{A small ball for each selected demand.}
+For $p\in I_2$, define
+\[
+  W_p=(Z_p\cup Y_{c(p)})\setminus\{x_p,y_p\},\qquad
+  P_p=(P-F)-W_p.
+\]
+Here the first deletion removes edges and the second removes vertices.
+Both endpoints remain in $P_p$.  By \eqref{eq:routing-condition},
+\begin{equation}\label{eq:routing-forbidden-budget}
+  \abs{W_p}\le z+4b\ell^2
+  \le 4\bigl(z+(k+b)\ell^2\bigr)
+  \le h/16\le h/4.
+\end{equation}
+Set $T=\ell/2=g(1+\ceil{\log r})$, an integer.  If the balls of
+radius $T$ about both $x_p$ and $y_p$ in $P_p$ had size greater than
+$r/2$, they would intersect.  Joining their paths to an intersection
+vertex gives an $x_p$--$y_p$ walk of length at most $2T=\ell$; deleting
+closed subwalks gives a simple path of no greater length.  It uses no
+edge of $F$ and no vertex of $W_p$.  Its internal vertices are also
+different from $x_p,y_p$, and hence avoid both $Z_p$ and $Y_{c(p)}$.
+Its hyperedge in $\mathcal H_p$ is therefore disjoint from $\mathcal M$,
+contrary to maximality of that matching.
+
+For every $p\in I_2$ we may consequently choose an endpoint $s_p$ whose
+radius-$T$ ball in $P_p$ has size at most $r/2$.  Let
+$S=\{s_p:p\in I_2\}$.  These roots are distinct because the selected
+endpoint pairs are disjoint.  Each root retains its demand label: write
+$p(s)$ for the unique $p\in I_2$ with $s=s_p$.  In particular,
+\begin{equation}\label{eq:routing-small-root-balls}
+  \abs{\{v:\operatorname{dist}_{P_{p(s)}}(s,v)\le T\}}\le r/2
+  \qquad(s\in S).
+\end{equation}
+
+\paragraph{Growth with different forbidden sets.}
+The deleted-edge budget may be too large to grow a ball from one root.
+We first grow unions of balls, retaining each root's graph, and then
+alternate halving the root set with further growth until one root remains.
+For a nonempty $A\subseteq S$ and an integer $t\ge 0$, put
+\[
+  B_t(A)=\bigcup_{s\in A}
+  \{v\in V(P_{p(s)}):\operatorname{dist}_{P_{p(s)}}(s,v)\le t\}.
+\]
+These are unions of balls in their respective graphs; the forbidden set
+is kept with its root throughout.  We claim that
+\begin{equation}\label{eq:routing-union-growth}
+  \abs{B_{t+1}(A)}
+  \ge\left(1+\frac{h}{2r}\right)\abs{B_t(A)}
+\end{equation}
+whenever
+\begin{equation}\label{eq:routing-growth-threshold}
+  0<\abs{B_t(A)}\le r/2,
+  \qquad \abs F\le h\abs{B_t(A)}/4.
+\end{equation}
+
+To prove this, let $B=B_t(A)$ and assign to each $v\in B$ one root
+$s(v)\in A$ that reaches $v$ within $t$ steps in $P_{p(s(v))}$.
+Consider a boundary edge $vw$ of $B$, with $v\in B$ and $w\notin B$.
+It extends this particular root's path whenever $vw\notin F$ and
+$w\notin W_{p(s(v))}$.  In that event $w\in B_{t+1}(A)\setminus B$.
+Because $P$ is simple, at most $\abs{W_{p(s(v))}}\le h/4$ boundary
+edges at $v$ fail the latter condition.  Cut expansion and
+\eqref{eq:routing-growth-threshold} therefore leave at least
+\[
+  h\abs B-(h/4)\abs B-\abs F\ge(h/2)\abs B
+\]
+usable boundary edges.  Each new vertex receives at most $r$ such
+edges.  Thus at least $h\abs B/(2r)$ new vertices are reached, proving
+\eqref{eq:routing-union-growth}.
+
+It follows that, for every integer $t\ge 0$,
+\begin{equation}\label{eq:routing-growth-block}
+  \abs{B_t(A)}\ge 1\ \text{ and }\quad
+  \abs F\le h\abs{B_t(A)}/4
+  \quad\Longrightarrow\quad
+  \abs{B_{t+g}(A)}>r/2.
+\end{equation}
+Indeed, the conclusion is immediate by monotonicity if the starting
+union already has size greater than $r/2$.  Otherwise, if it still had
+size at most $r/2$ after $g$ more steps, all intervening unions would
+have size at most $r/2$, while their monotonicity would preserve the
+edge-deletion threshold.  We could apply
+\eqref{eq:routing-union-growth} at all $g$ steps.  Singleton cuts in
+the simple graph $P$ give $0<h\le r-1<r$, and the elementary inequality
+$\log(1+h/(2r))\ge h/(4r)$ then yields
+\[
+  \abs{B_{t+g}(A)}
+  \ge \left(1+\frac{h}{2r}\right)^g\abs{B_t(A)}
+  \ge 2^{gh/(4r)}
+  \ge (2r)^2>r/2,
+\]
+a contradiction.  This also verifies the growth time with the ceiling
+in the definition of $g$.
+
+\paragraph{Restarting after halving the roots.}
+Initially $B_0(S)=S$.  By \eqref{eq:routing-obstruction-budgets},
+\eqref{eq:routing-pair-count}, and \eqref{eq:routing-condition},
+\[
+  \abs F\le 2\ell^2\abs I
+  \le 8k\ell^2\abs S\le h\abs S/4.
+\]
+The growth block \eqref{eq:routing-growth-block} gives
+$\abs{B_g(S)}>r/2$.
+
+Suppose now that a nonempty root set $A$ has
+$\abs{B_t(A)}>r/2$ and $\abs A\ge 2$.  Partition $A$ into two
+nonempty sets whose sizes differ by at most one.  Their radius-$t$
+ball unions together cover $B_t(A)$, so one of them, say $A'$, satisfies
+\[
+  \abs{A'}\le\ceil{\abs A/2},\qquad
+  \abs{B_t(A')}>r/4.
+\]
+The original matching has a bound independent of the number of retained
+roots: by \eqref{eq:routing-obstruction-budgets},
+\eqref{eq:routing-endpoint-count}, and \eqref{eq:routing-condition},
+\begin{equation}\label{eq:routing-restart-budget}
+  \abs F\le 2\ell^2\abs I
+  \le \ell^2kr\le hr/64
+  < h\abs{B_t(A')}/4.
+\end{equation}
+Thus another growth block gives $\abs{B_{t+g}(A')}>r/2$.
+
+Start with $A=S$ at radius $g$ and repeat this operation until only one
+root remains.  After $j$ halvings the number of roots is at most
+$\ceil{\abs S/2^j}$; this follows inductively from rounding each half
+upward.  Hence at most $\ceil{\log\abs S}$ halvings are needed,
+including zero if $\abs S=1$.  At termination a single root has a ball
+larger than $r/2$ at an integer radius at most
+\[
+  g\bigl(1+\ceil{\log\abs S}\bigr)
+  \le g\bigl(1+\ceil{\log r}\bigr)=T.
+\]
+This contradicts \eqref{eq:routing-small-root-balls} and monotonicity of
+that root's balls.
+
+There is therefore no set $I$ for which the matching condition fails.
+Applying \Cref{thm:aharoni-haxell} gives disjoint hypergraph
+representatives, and their paths prove the lemma.
+\end{proof}
+
+For later estimates, the exact radii also give the useful uniform bound
+\begin{equation}\label{eq:routing-length-bound}
+  \ell\le 108(r/h)(\log r)^2
+  =O\bigl((r/h)(\log r)^2\bigr).
+\end{equation}
+Indeed, $h<r$ and $r\ge 2$ imply
+$g\le 9(r/h)\log(2r)$,
+$\log(2r)\le 2\log r$, and
+$1+\ceil{\log r}\le 3\log r$.
diff --git a/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/build/sections/splitting.tex b/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/build/sections/splitting.tex
new file mode 100644
index 0000000000000000000000000000000000000000..7b8b80f92a5fafc14370bb6a534c1c8948fa7cb4
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+++ b/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/build/sections/splitting.tex
@@ -0,0 +1,353 @@
+\section{Splitting at a scale}
+\label{sec:splitting}
+
+The first operation below splits the graph into small boxes, allowing a
+small amount of vertex overlap.  The second operation extracts expanding
+pieces on disjoint vertex sets within each box.  Only edges leave when a
+piece is extracted: the residual graph retains every box vertex.  Thus
+every earlier piece vertex still occurs in a descendant box when the
+operations are iterated.  The overlapping split
+and its inverse-logarithmic order potential adapt the expander
+splitting argument of Buci\'c and Montgomery~\cite[Lemma~14]{BM2024}.
+Here we stop at a prescribed scale and obtain a total box order
+of \(N(1+O(1/\log D))\), which will be charged in the final induction.
+
+\begin{lemma}[Splitting at a scale]\label{lem:scale-split}
+There are absolute constants $D_{\mathrm{sp}}$ and $C_0$ such that, for
+every real $D\ge D_{\mathrm{sp}}$, every graph $G$ on a nonempty vertex
+set of order $N$ and with average degree at most $D$ admits the following
+two successive operations.  One may take $C_0=2$.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item\label{item:scale-boxes}
+Remove a family $\mathcal C$ of edge-disjoint cycles, each of length at
+least $D^{1.01}$, where
+\begin{equation}\label{eq:scale-cycle-cost}
+  \abs{\mathcal C}\le \frac{N}{2D^{0.01}}.
+\end{equation}
+Partition the remaining edges into graphs $G_Y$ on an indexed family
+$\mathcal Y$ of vertex sets, called boxes.  These satisfy
+\begin{equation}\label{eq:scale-box-order}
+\begin{gathered}
+  \bigcup_{Y\in\mathcal Y}Y=V(G),\qquad
+  1\le \abs Y\le D^{1.02},\\
+  \sum_{Y\in\mathcal Y}\abs Y
+    \le \left(1+\frac{C_0}{\log D}\right)N.
+\end{gathered}
+\end{equation}
+The graphs $G_Y$ contain only their assigned edges; their vertex sets
+may overlap.
+
+\item\label{item:scale-expanders}
+Put $\sigma=D^{0.90}$.  For each box $Y$, partition $E(G_Y)$ into
+the edge sets of a family $\mathcal R_Y$ of graphs $R$ and a graph $H_Y$.
+The vertex sets of the graphs in $\mathcal R_Y$ are pairwise disjoint
+subsets of $Y$.  Each $R$ has cut expansion at least $\sigma$ and order
+\begin{equation}\label{eq:scale-piece-order}
+  \sigma\le \abs{V(R)}\le D^{1.02}.
+\end{equation}
+The residual graph $H_Y$ is taken on all of $Y$, including isolated
+vertices, and satisfies
+\begin{equation}\label{eq:scale-residual-degree}
+  \frac{2\abs{E(H_Y)}}{\abs Y}\le D^{0.95}.
+\end{equation}
+\end{enumerate}
+Thus the cycles in $\mathcal C$, the pieces $R$, and the residual graphs
+$H_Y$ together partition $E(G)$.
+\end{lemma}
+
+We first give the elementary long-cycle argument used to find the
+overlapping splits. It is the depth-first-search and bypass argument
+of Buci\'c and Montgomery~\cite[Lemma~25]{BM2024}, with constants
+suited to the present scale cutoff. Related depth-first-search balance
+arguments appear in Ben-Eliezer, Krivelevich and
+Sudakov~\cite[Lemma~2.3]{BKS2012}; Krivelevich~\cite[Theorem~1]{Krivelevich2019}
+gives a stronger general criterion for long cycles from local expansion.
+For a vertex set $U$ in a graph $H$, write
+$N_H(U)=\bigcup_{v\in U}N_H(v)$; its external neighbor set is
+$N_H(U)\setminus U$.
+
+\begin{lemma}[Vertex expansion gives a long cycle]
+\label{lem:split-vertex-expansion}
+Fix $\eps=1/32$.  For every sufficiently large integer $u$, put
+$\alpha=\eps/\log^2 u$.  If a graph $H$ of order $u$ satisfies
+\begin{equation}\label{eq:split-vertex-expansion}
+  \abs{N_H(U)\setminus U}\ge\alpha\abs U
+  \qquad\left(0<\abs U\le\frac u2\right),
+\end{equation}
+then $H$ contains a simple cycle of length greater than
+\begin{equation}\label{eq:split-long-cycle-length}
+  q(u)=\frac{\alpha^2u}{100}
+      =\frac{\eps^2u}{100\log^4u}.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+The expansion condition implies that $H$ is connected: in a
+disconnected graph some component has order between $1$ and $u/2$ and
+has no external neighbors.
+
+Run a depth first search.  Its state consists of the unexplored
+vertices, a stack whose vertices in stack order form a simple path, and the finished
+vertices.  Initially the stack consists of one vertex, all other
+vertices are unexplored, and none are finished.  If the top vertex of
+the stack has an unexplored neighbor, push such a neighbor onto the
+stack.  Otherwise pop the top vertex and mark it finished.  At all
+times there are no edges between finished and unexplored vertices:
+when a vertex is finished it has no unexplored neighbor, and the
+unexplored set subsequently only shrinks.  Connectivity ensures that
+the stack cannot become empty while an unexplored vertex remains, so
+this search finishes all vertices.
+
+The number of unexplored vertices minus the number of finished
+vertices starts at $u-1$, ends at $-u$, and decreases by exactly one at
+every push or pop.  Consequently there is a state at which these two
+numbers are exactly equal, say to $t$.  Let $P$ be the stack path at
+that state, and let $L=\abs{V(P)}=u-2t$.  If $t\ge u/3$, then
+$0<t\le u/2$, and all external neighbors of the unexplored set lie on
+$P$.  Applying \eqref{eq:split-vertex-expansion} gives
+$L\ge\alpha t\ge\alpha u/3$.  If $t<u/3$, then
+$L>u/3\ge\alpha u/3$, since $\alpha\le1$ for the values of $u$ under
+consideration.  Thus in either case
+\begin{equation}\label{eq:split-stack-order}
+  L\ge\frac{\alpha u}{3}.
+\end{equation}
+
+Write $q=q(u)$ and $j=\floor q$.  We may assume $q\ge1$, since
+$u/\log^4u$ tends to infinity.  In particular $j\ge1$, and
+\[
+  j+1\le2q=\frac{\alpha^2u}{50}
+       \le\frac{\alpha u}{6}\le\frac L2.
+\]
+Partition $P$ into three consecutive vertex segments $X,J,Z$, with
+$\abs J=j$, distributing the other $L-j$ vertices as evenly as
+possible between $X$ and $Z$.  Both outer segments have order at least
+\[
+  \floor{\frac{L-j}{2}}
+    \ge\frac{L-j-1}{2}\ge\frac L4.
+\]
+These inequalities include the integer rounding needed to make the
+three segments nonempty.
+
+Suppose first that $H-J$ has a path from $X$ to $Z$.  Choose one of
+minimum length among all such paths, with endpoints $x\in X$ and
+$z\in Z$.  A shortest path is simple.  Its interior contains no vertex
+of $X$ or $Z$, because encountering either set internally would give a
+shorter path between the two sets.  It also avoids $J$.  Since
+$X,J,Z$ partition $V(P)$, the chosen path meets $P$ only at $x,z$.
+Its union with the $x$--$z$ subpath of $P$ is therefore a simple cycle.
+That subpath contains all $j$ vertices of $J$ and has at least $j+1$
+edges; the new path has at least one edge.  The cycle consequently has
+at least $j+2\ge3$ edges, and its length is greater than $q$.
+
+It remains to show that such a path must exist.  Otherwise $X$ and
+$Z$, each connected by its segment of $P$, lie in distinct components
+of $H-J$.  Separate the component containing $X$ from the union of all
+other components, which contains $Z$, and let $A$ be the smaller of
+these two sets.  Then
+\[
+  \frac L4\le\abs A\le\frac{u-j}{2}\le\frac u2,
+  \qquad N_H(A)\setminus A\subseteq J.
+\]
+Expansion and \eqref{eq:split-stack-order} imply
+\[
+  j\ge\alpha\abs A\ge\frac{\alpha L}{4}
+    \ge\frac{\alpha^2u}{12}
+    >\frac{\alpha^2u}{100}=q,
+\]
+contrary to $j=\floor q$.  This proves the lemma.
+\end{proof}
+
+\begin{proof}[Proof of \Cref{lem:scale-split}]
+We prove the two operations in order.  All lower bounds imposed on $D$
+in this proof are absolute and hold simultaneously for every
+$D\ge D_{\mathrm{sp}}$ after one choice of $D_{\mathrm{sp}}$.
+
+\paragraph{Removing long cycles.}
+Choose a maximal family $\mathcal C$ of edge-disjoint cycles in $G$
+whose lengths are at least $D^{1.01}$.  Since
+$\abs{E(G)}\le ND/2$, we have
+\[
+  \abs{\mathcal C}D^{1.01}\le\abs{E(G)}\le\frac{ND}{2},
+\]
+which proves \eqref{eq:scale-cycle-cost}.  Integral cycle lengths are
+compared with the real threshold as stated, equivalently with
+$\ceil{D^{1.01}}$.  Delete these cycle edges and retain the entire
+vertex set.  By maximality the remaining graph has no cycle of length
+at least $D^{1.01}$.  Every graph subsequently formed in the first
+operation is a subgraph of this remainder, so it inherits this
+prohibition.
+
+\paragraph{A sparse external neighborhood.}
+Put $T=D^{1.02}$ and $\eps=1/32$.  We claim that every current graph
+$H$ of order $u>T$ has a nonempty vertex set $U$ such that
+\begin{equation}\label{eq:split-sparse-neighborhood}
+  \abs U\le\frac u2,\qquad
+  \abs{N_H(U)\setminus U}
+    <\frac{\eps\abs U}{\log^2u}.
+\end{equation}
+Otherwise \Cref{lem:split-vertex-expansion} supplies a cycle longer
+than $q(u)$ as defined in \eqref{eq:split-long-cycle-length}.
+To obtain a contradiction uniformly for every $u>T$, including $u$
+arbitrarily large relative to $D$, observe that $x/\log^4x$ is
+increasing for $\ln x>4$, as follows by differentiating.  Moreover,
+\begin{equation}\label{eq:split-uniform-margin}
+  \frac{q(T)}{D^{1.01}}
+    =\frac{\eps^2}{100(1.02)^4}
+       \frac{D^{0.01}}{(\log D)^4}
+    \longrightarrow\infty\qquad(D\longrightarrow\infty).
+\end{equation}
+Choose $D_{\mathrm{sp}}$ large enough that $T$ is in the range of
+\Cref{lem:split-vertex-expansion}, that $T>e^4$, and that
+$q(T)>D^{1.01}$.  Then $q(u)>D^{1.01}$ for every $u>T$,
+contradicting the inherited cycle prohibition.  This proves
+\eqref{eq:split-sparse-neighborhood}.
+
+\paragraph{The overlapping split and its edge assignment.}
+For such a set $U$, put $B=N_H(U)\setminus U$ and form two children
+on vertex sets
+\[
+  V_1=U\cup B,\qquad V_2=V(H)\setminus U.
+\]
+Assign to the first child every edge of $H$ with both endpoints in
+$V_1$, and assign to the second child every other edge of $H$.
+Every edge incident with $U$ has both endpoints in $U\cup B$.
+Consequently every edge assigned to the second child has both endpoints
+in $V_2$.  This is an edge partition into two simple subgraphs; in
+particular, edges with both endpoints in the overlap $B$ are assigned
+only to the first child.  The child vertex sets cover the parent and
+have intersection exactly $B$.  All vertices of each specified child
+set are retained, even when isolated in its assigned graph.
+
+Writing $s=\abs U$, $b=\abs B$, $u_1=s+b$ and $u_2=u-s$, we have
+$1\le s\le u/2$ and $b<\eps s/\log^2u$.  For sufficiently large
+$D$, therefore,
+\begin{equation}\label{eq:split-child-orders}
+\begin{gathered}
+  1\le u_1\le\frac{3u}{4},\qquad 1\le u_2<u,\\
+  u_1+u_2=u+b
+      \le u+\frac{\eps u_1}{\log^2u}.
+\end{gathered}
+\end{equation}
+Indeed $\eps/\log^2u\le1/2$ gives
+$u_1<(3/2)s\le3u/4$, and $u_2<u$ follows from $s\ge1$.
+Repeat this split whenever the current order exceeds $T$.
+Both child orders are positive integers strictly smaller than their
+parent order, so the resulting binary splitting tree is finite.
+Its leaves give the graphs $G_Y$ on boxes of order at most $T$.
+The child cover property gives $\bigcup_Y Y=V(G)$, and the edge
+assignments at each split show that the leaf graphs partition exactly
+the edges left after the cycle removal.
+
+\paragraph{Controlling the sum of box orders.}
+The sum of child orders can exceed the parent order by $b$.  The
+following potential pays for this overlap.  For every real $x\ge1$
+define
+\begin{equation}\label{eq:split-potential}
+  f(x)=1+\frac1{\log(T/2)}
+         -\frac1{\log(\max\{x,T/2\})}.
+\end{equation}
+For $T\ge4$, this is nondecreasing and
+\begin{equation}\label{eq:split-potential-range}
+  1\le f(x)\le1+\frac1{\log(T/2)}\le2.
+\end{equation}
+At a split of a graph of order $u>T$, put
+$M=\max\{u_1,T/2\}$.  By \eqref{eq:split-child-orders} and $u>T$,
+we have $M\le3u/4$.  Both logarithms below are positive, and
+\begin{equation}\label{eq:split-potential-decrement}
+\begin{aligned}
+  f(u)-f(u_1)
+    &=\frac1{\log M}-\frac1{\log u}\\
+    &=\frac{\log(u/M)}{\log M\,\log u}
+      \ge\frac{\log(4/3)}{\log^2u}.
+\end{aligned}
+\end{equation}
+Monotonicity also gives $f(u_2)\le f(u)$.  Hence, using
+\eqref{eq:split-child-orders},
+\begin{equation}\label{eq:split-potential-subadditivity}
+\begin{aligned}
+  u_1f(u_1)+u_2f(u_2)
+    &\le (u_1+u_2)f(u)
+         -\frac{u_1\log(4/3)}{\log^2u}\\
+    &\le uf(u)
+         +\frac{u_1}{\log^2u}
+                 \bigl(\eps f(u)-\log(4/3)\bigr)\\
+    &\le uf(u).
+\end{aligned}
+\end{equation}
+For the last step, \eqref{eq:split-potential-range} gives
+$\eps f(u)\le1/16<\log(4/3)$.  Repeatedly applying
+\eqref{eq:split-potential-subadditivity} in the finite splitting tree,
+and using $f\ge1$ on its leaves, proves
+\[
+  \sum_{Y\in\mathcal Y}\abs Y
+    \le\sum_{Y\in\mathcal Y}\abs Y f(\abs Y)
+    \le Nf(N)
+    \le N\left(1+\frac1{\log(T/2)}\right).
+\]
+This argument also applies if $N\le T$, when the tree has just its
+root.  Since $T=D^{1.02}$ and $D\ge4$,
+\[
+  \frac1{\log(T/2)}
+    =\frac1{1.02\log D-1}\le\frac2{\log D}.
+\]
+This proves \eqref{eq:scale-box-order} with $C_0=2$ and finishes
+operation~\ref{item:scale-boxes}.
+
+\paragraph{Extracting pieces of cut expansion $\sigma$.}
+This repeated removal of sparse cuts follows the small-side charging
+method used by Conlon, Fox and Sudakov~\cite[Lemma~3.1]{CFS2014}.
+We give the accounting for the precise cut expansion and residual
+average degree required here.
+Fix a box $Y$, write $y=\abs Y$, and start with its assigned graph
+$G_Y$.  Whenever a current graph on a vertex set $A$ of order at least
+two fails to have cut expansion $\sigma=D^{0.90}$, choose a partition
+$A=A_1\mathbin{\dot\cup}A_2$ into nonempty sets such that its cut has
+fewer than $\sigma\min\{\abs{A_1},\abs{A_2}\}$ edges.  Put those
+cut edges into the residual graph $H_Y$, and continue separately with
+the edges internal to each of $A_1,A_2$.  Both child orders strictly
+decrease, so this procedure terminates.  Its terminal vertex sets
+partition $Y$.  The nonsingleton terminal graphs form
+$\mathcal R_Y$; by the stopping rule each has cut expansion at least
+$\sigma$.  If such a graph has order $r$, its singleton cuts show that
+its minimum degree is at least $\sigma$.  Simplicity then gives
+$r-1\ge\sigma$, and in particular \eqref{eq:scale-piece-order}.
+Terminal singletons carry no edges and are omitted from
+$\mathcal R_Y$.
+
+To bound the edges put into $H_Y$, at each split charge their number
+equally to the vertices in a smaller side, choosing either side in
+case of equality.  Each charged vertex receives less than $\sigma$
+at that split.  Whenever a given vertex is charged, the current set
+containing it decreases in size by a factor of at least two; any
+intervening splits can only decrease its size further.  If the vertex
+is charged $k$ times, after its last charge its set has order at most
+$y/2^k$ and at least one.  Thus $k\le\log y$.
+
+All residual cut edges are distinct, because each subsequent split
+uses only edges internal to a previous child.  Summing the charges
+therefore gives
+\begin{equation}\label{eq:split-cut-charge}
+  \abs{E(H_Y)}\le\sigma y\log y.
+\end{equation}
+Every edge not set aside ends in exactly one terminal graph, and
+singleton terminal graphs contain no edges.  Thus $H_Y$ and the
+graphs in $\mathcal R_Y$ partition $E(G_Y)$ exactly.  Define
+$V(H_Y)=Y$, retaining also the vertices of all terminal graphs.
+For $y=1$ the residual has no edges.  For every $1\le y\le D^{1.02}$,
+\eqref{eq:split-cut-charge} yields
+\[
+  \frac{2\abs{E(H_Y)}}{y}
+    \le2\sigma\log y
+    \le2.04D^{0.90}\log D
+    \le D^{0.95},
+\]
+where the last inequality holds for all sufficiently large $D$.
+This proves operation~\ref{item:scale-expanders} and the lemma.
+\end{proof}
+
+In particular, when the two operations are repeated on the residual
+graphs, the boxes descending from any fixed box continue to cover
+every vertex of that box.  This follows from the cover property of
+each overlapping split and the fact that $H_Y$ always has vertex set
+$Y$, irrespective of which incident edges were removed as cycles or
+expanding pieces.
diff --git a/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/main.pdf b/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/main.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..e19f88bfe92cf344928a429263d7a3ed9e9adc33
--- /dev/null
+++ b/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026/main.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:4aa44187578ea59b64c78258eb8c5dd3601ee979a1a402ca9ed26696b30b515f
+size 387380
diff --git a/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/INPUTS.md b/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/INPUTS.md
new file mode 100644
index 0000000000000000000000000000000000000000..41c4741c1d57515566519c17ba9bc6af003d81e1
--- /dev/null
+++ b/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/INPUTS.md
@@ -0,0 +1,15 @@
+# Mathematical input interfaces
+
+The complete locally authored proof is in `main.tex` and `sections/01-introduction.tex` through `sections/08-recursion.tex`. The ordinary halving scaffold is integrated in Section 3; there is no dependency on an internal ordinary-coloring manuscript. Empty auxiliary graphs have `h = ell = chi = f = 0`.
+
+Three nontrivial graph-theoretic statements are invoked without proof:
+
+1. Reed–Seymour's fractional minor theorem: `f(P) <= 4 h(P)`, a weakened form of their statement (1.3), which gives factor 2. Used to initialize thinning and obtain private clique models. Finite simple graph hypotheses hold in both uses.
+2. Ordinary-chromatic connectivity extraction: for integer `k >= 1`, `chi(P) >= 7k` gives a `k`-connected subgraph `J` with `chi(J) >= chi(P) - 6k`. The exact form is Delcourt–Postle arXiv:2108.01633v5, Theorem 5.1, credited there to Girão–Narayanan. Used on auxiliary clique blow-ups; no minor-exclusion hypothesis is imposed on those blow-ups.
+3. Terminal grouping: there is an absolute integer `gamma` such that `gamma*m` connectivity realizes every partition of `m` distinct terminals into disjoint connected sets. This is Delcourt–Postle v5, Definition 5.7 and Theorem 5.8, second knitting parameter one, based on Bollobás–Thomason. All uses count distinct representatives or proxies and verify their connectivity budget.
+
+Finite linear-programming duality is a standard additional input, cited to Gale–Kuhn–Tucker, for the fractional-coloring primal/dual equality. The paper verifies finite feasibility and boundedness; it does not claim to prove LP duality. This corrects an inherited input inventory that described a local convex-hull proof absent from the canonical source.
+
+Stable-set sampling, the bounded-differences estimate, integral vertex path packing, list connectivity extraction, color reservation, coarse density and incidence bounds, the small-district construction, unpaired path minimization, path-history accounting, separability, and the complete three-district recursion are proved locally. Attribution to predecessors remains attached to those arguments.
+
+The clean build is a document build, not a mathematical certificate. Author and component review records live in the coordination directory; fresh final whole-proof reviews must be bound to the final article sources and PDF.
diff --git a/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/README.md b/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..5e5874d03676d8b083c5bd01a86a292dd811cc50
--- /dev/null
+++ b/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/README.md
@@ -0,0 +1,18 @@
+# [A linear list-coloring bound in terms of the Hadwiger number](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 23, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026,
+  author = {{OpenAI}},
+  title = {{A linear list-coloring bound in terms of the Hadwiger number}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/paper.pdf}{OAI:A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/figures/separability.tex b/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/figures/separability.tex
new file mode 100644
index 0000000000000000000000000000000000000000..356550c170022da2b58c4149b721dda8ab364dfd
--- /dev/null
+++ b/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/figures/separability.tex
@@ -0,0 +1,69 @@
+\begin{figure}[tbp]
+\centering
+\begin{tikzpicture}[x=1cm,y=1cm,font=\footnotesize]
+  \node[anchor=west,font=\small\bfseries] at (0,6.55)
+    {(a) Original minimizing systems};
+  \draw[rounded corners,fill=black!3] (0.1,3.85) rectangle (2.35,5.75);
+  \node at (1.22,5.38) {\(F\)};
+  \node at (1.22,5.02) {old footprint};
+  \fill (2.35,4.45) circle (2pt);
+  \node[below left] at (2.32,4.43) {\(W\)};
+
+  \draw[rounded corners,fill=blue!4] (4.1,5.2) rectangle (6.35,6.1);
+  \node at (4.6,5.77) {\(J\)};
+  \fill (6.35,5.52) circle (2pt);
+  \node[above left] at (6.32,5.53) {\(R_J\)};
+
+  \draw[rounded corners,fill=blue!4] (7.6,3.85) rectangle (9.8,5.75);
+  \node at (8.7,5.4) {\(D\)};
+  \fill (7.6,4.45) circle (2pt);
+  \fill (7.6,5.02) circle (2pt);
+  \fill (9.8,4.45) circle (2pt);
+
+  \draw[rounded corners,fill=orange!5] (11.35,3.85) rectangle (13.6,5.75);
+  \node at (12.47,5.38) {\(H'\)};
+  \node at (12.47,5.02) {new region};
+  \fill (11.35,4.45) circle (2pt);
+  \node[below right] at (11.38,4.43) {\(T'\), \(|T'|=2t\)};
+
+  \draw[->,very thick,blue!70!black] (2.35,4.45) -- (7.6,4.45);
+  \node[above,text=blue!70!black] at (4.5,4.45)
+    {\(\mathcal Q_1\): two paths per start};
+  \draw[->,very thick,blue!70!black]
+    (6.35,5.52) to[out=-15,in=155] (7.6,5.02);
+  \draw[->,very thick,dashed,orange!85!black]
+    (11.35,4.45) -- (9.8,4.45);
+  \node[above,text=orange!85!black] at (10.6,4.45)
+    {\(\mathcal Q_2\)};
+
+  \node[anchor=west,font=\small\bfseries] at (0,3.38)
+    {(b) One label after combination, rerouting, and grouping};
+  \draw[rounded corners,fill=green!8] (0.1,0.25) rectangle (2.6,1.6);
+  \node at (1.05,1.21) {old bag \(B_j\)};
+  \fill[green!40!black] (2.6,0.75) circle (2pt);
+  \node[below left] at (2.57,0.72) {\(w_j\)};
+
+  \draw[rounded corners,fill=green!12] (6.45,0.25) rectangle (8.95,1.6);
+  \node at (7.7,0.9) {\(E_j\subseteq D\)};
+
+  \draw[rounded corners,fill=green!8] (6.3,2.0) rectangle (9.1,2.85);
+  \node at (7.7,2.57) {\(C_j\subseteq J\), if \(j\in I\)};
+  \fill[green!40!black] (7.7,2.0) circle (2pt);
+  \node[above right] at (7.76,2.03) {\(u_j\)};
+  \draw[very thick,green!40!black] (7.7,2.0) -- (7.7,1.6);
+
+  \draw[rounded corners,fill=orange!5] (11.35,0.25) rectangle (13.6,1.6);
+  \node at (12.47,1.21) {\(H^+\)};
+  \fill[green!40!black] (11.35,0.75) circle (2pt);
+  \node[right] at (11.46,0.65) {\(w_j^+\)};
+  \draw[very thick,green!40!black] (2.6,0.75) -- (6.45,0.75);
+  \draw[very thick,green!40!black] (8.95,0.75) -- (11.35,0.75);
+\end{tikzpicture}
+\caption{A separability round, shown schematically.
+The upper arrows represent families of paths; the two families may
+intersect, and their entire vertex sets remain in the footprint.
+Below, \(E_j\) joins label \(j\)'s path ends in \(D\).
+The region \(H^+\) is obtained from \(H'\) by removing unused starts;
+it meets the footprint exactly in the new tips \(W^+\).}
+\label{fig:separability-round}
+\end{figure}
diff --git a/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/figures/three-districts.tex b/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/figures/three-districts.tex
new file mode 100644
index 0000000000000000000000000000000000000000..563fd74d1b53d55c5e66e803a8ee9157e83fb8c6
--- /dev/null
+++ b/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/figures/three-districts.tex
@@ -0,0 +1,17 @@
+\begin{tikzpicture}[font=\small,
+ block/.style={draw,rounded corners,minimum width=2.15cm,minimum height=.75cm,fill=blue!4},
+ district/.style={draw,rounded corners,minimum width=2.15cm,minimum height=.75cm,fill=green!5},
+ every path/.style={line width=.6pt}]
+\node[block] (b1) at (0,2) {$B_1$};
+\node[block] (b2) at (3.8,2) {$B_2$};
+\node[block] (b3) at (7.6,2) {$B_3$};
+\node[district] (j12) at (0,0) {$J_{12}: B_1\cup B_2$};
+\node[district] (j13) at (3.8,0) {$J_{13}: B_1\cup B_3$};
+\node[district] (j23) at (7.6,0) {$J_{23}: B_2\cup B_3$};
+\draw[-{Latex[length=2mm]}] (b1.south) -- (j12.north);
+\draw[-{Latex[length=2mm]}] (b1.south east) -- (j13.north west);
+\draw[-{Latex[length=2mm]}] (b2.south west) -- (j12.north east);
+\draw[-{Latex[length=2mm]}] (b2.south east) -- (j23.north west);
+\draw[-{Latex[length=2mm]}] (b3.south west) -- (j13.north east);
+\draw[-{Latex[length=2mm]}] (b3.south) -- (j23.north);
+\end{tikzpicture}
diff --git a/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/glyph-mappings.tex b/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/glyph-mappings.tex
new file mode 100644
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+++ b/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/glyph-mappings.tex
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diff --git a/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/main.tex b/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/main.tex
new file mode 100644
index 0000000000000000000000000000000000000000..8cc7d0a2cfb5603c08cc873476748bb5ec854656
--- /dev/null
+++ b/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/main.tex
@@ -0,0 +1,48 @@
+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\input{glyphtounicode}
+\pdfgentounicode=1
+\input{glyph-mappings}
+\pdfinfoomitdate=1
+\pdftrailerid{}
+\pdfsuppressptexinfo=15
+\usepackage[margin=1in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{booktabs,enumitem,graphicx,tikz}
+\usetikzlibrary{arrows.meta,positioning,fit,calc,backgrounds}
+\usepackage{microtype}
+\usepackage[numbers,sort&compress]{natbib}
+\usepackage{xurl}
+\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref}
+\hypersetup{pdftitle={A linear list-coloring bound in terms of the Hadwiger number},pdfauthor={OpenAI},pdfsubject={Linear List Hadwiger conjecture}}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\title{A linear list-coloring bound in terms\\of the Hadwiger number}
+\author{OpenAI}
+\date{September 23, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We prove that every finite nonempty graph $G$ satisfies
+$\chi_{\mathrm{list}}(G)\le C h(G)$ for an absolute integer $C$, where
+$h(G)$ is the largest order of a clique minor. This resolves the Linear
+List Hadwiger conjecture affirmatively.
+\end{abstract}
+\input{sections/01-introduction}
+\input{sections/02-preliminaries}
+\input{sections/03-small-graphs}
+\input{sections/04-list-transfer}
+\input{sections/05-density}
+\input{sections/06-routing}
+\input{sections/07-separability}
+\input{sections/08-recursion}
+\bibliographystyle{plainnat}
+\bibliography{references}
+\end{document}
diff --git a/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/references.bib b/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/references.bib
new file mode 100644
index 0000000000000000000000000000000000000000..bac0bae38aa3c894cdbef113890fded2db85092e
--- /dev/null
+++ b/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/references.bib
@@ -0,0 +1,244 @@
+
+@article{ReedSeymour1998,
+  author  = {Reed, Bruce and Seymour, Paul},
+  title   = {Fractional Colouring and {Hadwiger}'s Conjecture},
+  journal = {Journal of Combinatorial Theory, Series B},
+  volume  = {74},
+  pages   = {147--152},
+  year    = {1998},
+  url     = {https://cgm.cs.mcgill.ca/~breed/SummerNSERC04/frachad.pdf}
+}
+
+@article{DelcourtPostle2021,
+ author={Delcourt, Michelle and Postle, Luke},
+ title={Reducing linear {Hadwiger}'s conjecture to coloring small graphs},
+ journal={Journal of the American Mathematical Society}, volume={38}, number={2},
+ pages={481--507}, year={2025}, doi={10.1090/jams/1047},
+ note={Theorem locators refer to arXiv:2108.01633v5, March 5, 2024},
+ url={https://arxiv.org/abs/2108.01633v5}
+}
+
+@misc{DelcourtPostle2021v1,
+  author        = {Delcourt, Michelle and Postle, Luke},
+  title         = {Reducing linear {Hadwiger}'s conjecture to coloring small graphs},
+  year          = {2021},
+  eprint        = {2108.01633v1},
+  archivePrefix = {arXiv},
+  primaryClass  = {math.CO},
+  note          = {First version, 3 August 2021; Section 6 outlines the list-coloring extension},
+  url           = {https://arxiv.org/abs/2108.01633v1}
+}
+
+@misc{GuXu2026,
+  author        = {Gu, Yangyan and Xu, Rongxing},
+  title         = {Improved upper bounds on the list chromatic number of {$K_t$}-minor-free graphs},
+  year          = {2026},
+  eprint        = {2610.01946v1},
+  archivePrefix = {arXiv},
+  primaryClass  = {math.CO},
+  note          = {arXiv:2610.01946v1, version of 1 October 2026},
+  url           = {https://arxiv.org/abs/2610.01946v1}
+}
+
+@misc{Postle2020,
+  author        = {Postle, Luke},
+  title         = {Further Progress towards the List and Odd Versions of {Hadwiger}'s Conjecture},
+  year          = {2020},
+  eprint        = {2010.05999v1},
+  archivePrefix = {arXiv},
+  primaryClass  = {math.CO},
+  note          = {arXiv:2010.05999v1, version of 12 October 2020},
+  url           = {https://arxiv.org/abs/2010.05999v1}
+}
+
+@article{GiraoNarayanan2022,
+  author  = {Gir{\~a}o, Ant{\'o}nio and Narayanan, Bhargav},
+  title   = {Subgraphs of large connectivity and chromatic number},
+  journal = {Bulletin of the London Mathematical Society},
+  volume  = {54},
+  number  = {3},
+  pages   = {868--875},
+  year    = {2022},
+  doi     = {10.1112/blms.12569},
+  url     = {https://doi.org/10.1112/blms.12569}
+}
+
+@article{BollobasThomason1996,
+  author  = {Bollob{\'a}s, B{\'e}la and Thomason, Andrew},
+  title   = {Highly linked graphs},
+  journal = {Combinatorica},
+  volume  = {16},
+  number  = {3},
+  pages   = {313--320},
+  year    = {1996},
+  doi     = {10.1007/BF01261316},
+  url     = {https://doi.org/10.1007/BF01261316}
+}
+
+@misc{LiuLuo2026,
+  author        = {Liu, Chun-Hung and Luo, Jason},
+  title         = {Beyond halfway to {Hadwiger}'s conjecture},
+  year          = {2026},
+  eprint        = {2609.06867v2},
+  archivePrefix = {arXiv},
+  primaryClass  = {math.CO},
+  note          = {arXiv:2609.06867v2, version of 9 September 2026},
+  url           = {https://arxiv.org/abs/2609.06867v2}
+}
+
+
+@article{Hadwiger1943,
+  author  = {Hadwiger, Hugo},
+  title   = {{\"U}ber eine {Klassifikation} der {Streckenkomplexe}},
+  journal = {Vierteljahrsschrift der Naturforschenden Gesellschaft in Z{\"u}rich},
+  volume  = {88},
+  pages   = {133--142},
+  year    = {1943},
+  url     = {https://ngzh.ch/wp-content/uploads/2024/08/88_17.pdf}
+}
+
+@article{KawarabayashiMohar2007,
+  author  = {Kawarabayashi, Ken-ichi and Mohar, Bojan},
+  title   = {A relaxed {Hadwiger}'s conjecture for list colorings},
+  journal = {Journal of Combinatorial Theory, Series B},
+  volume  = {97},
+  number  = {4},
+  pages   = {647--651},
+  year    = {2007},
+  doi     = {10.1016/j.jctb.2006.11.002},
+  url     = {https://doi.org/10.1016/j.jctb.2006.11.002}
+}
+
+@article{fordfulkerson1956,
+  author = {Ford, Jr., L. R. and Fulkerson, D. R.},
+  title = {Maximal Flow Through a Network},
+  journal = {Canadian Journal of Mathematics}, volume = {8},
+  pages = {399--404}, year = {1956}, doi = {10.4153/CJM-1956-045-5}
+}
+
+@article{menger1927,
+  author = {Menger, Karl}, title = {Zur allgemeinen Kurventheorie},
+  journal = {Fundamenta Mathematicae}, volume = {10},
+  pages = {96--115}, year = {1927}, doi = {10.4064/fm-10-1-96-115}
+}
+
+@incollection{galekuhntucker1951,
+  author = {Gale, David and Kuhn, Harold W. and Tucker, Albert W.},
+  title = {Linear Programming and the Theory of Games},
+  booktitle = {Activity Analysis of Production and Allocation},
+  editor = {Koopmans, Tjalling C.},
+  series = {Cowles Commission Monographs}, volume = {13},
+  chapter = {19}, pages = {317--329},
+  publisher = {John Wiley \& Sons}, year = {1951}
+}
+
+@article{azuma1967,
+  author = {Azuma, Kazuoki},
+  title = {Weighted sums of certain dependent random variables},
+  journal = {Tohoku Mathematical Journal, Second Series},
+  volume = {19}, number = {3}, pages = {357--367}, year = {1967},
+  doi = {10.2748/tmj/1178243286}
+}
+
+@incollection{mcdiarmid1989,
+  author = {McDiarmid, Colin}, title = {On the method of bounded differences},
+  booktitle = {Surveys in Combinatorics, 1989}, editor = {Siemons, J.},
+  series = {London Mathematical Society Lecture Note Series}, volume = {141},
+  pages = {148--188}, publisher = {Cambridge University Press}, year = {1989},
+  doi = {10.1017/CBO9781107359949.008}
+}
+
+@article{Vizing1976,
+ author={Vizing, V. G.},
+ title={Coloring the vertices of a graph in prescribed colors},
+ journal={Diskret. Analiz., no. 29}, year={1976}, pages={3--10},
+ note={In Russian}
+}
+
+@inproceedings{ErdosRubinTaylor1980,
+ author={Erd{\H o}s, Paul and Rubin, Arthur L. and Taylor, Herbert},
+ title={Choosability in graphs},
+ booktitle={Proceedings of the West Coast Conference on Combinatorics, Graph Theory and Computing (Arcata, 1979)},
+ series={Congressus Numerantium}, volume={26}, pages={125--157},
+ publisher={Utilitas Mathematica}, address={Winnipeg}, year={1980},
+ url={https://www.renyi.hu/~p_erdos/1980-07.pdf}
+}
+
+@article{Kostochka1982,
+ author={Kostochka, A. V.},
+ title={The minimum {Hadwiger} number for graphs with a given mean degree of vertices},
+ journal={Metody Diskret. Analiz., no. 38}, pages={37--58}, year={1982},
+ note={In Russian; English translation in American Mathematical Society Translations, Series 2, 132 (1986), 15--32}
+}
+
+@article{Kostochka1984,
+ author={Kostochka, A. V.},
+ title={Lower bound of the {Hadwiger} number of graphs by their average degree},
+ journal={Combinatorica}, volume={4}, number={4}, pages={307--316}, year={1984},
+ doi={10.1007/BF02579141}, url={https://kostochk.web.illinois.edu/docs/old/cca84.pdf}
+}
+
+@article{Thomason1984,
+ author={Thomason, Andrew},
+ title={An extremal function for contractions of graphs},
+ journal={Mathematical Proceedings of the Cambridge Philosophical Society},
+ volume={95}, number={2}, year={1984}, pages={261--265},
+ doi={10.1017/S0305004100061521}
+}
+
+@article{Voigt1993,
+ author={Voigt, Margit}, title={List colourings of planar graphs},
+ journal={Discrete Mathematics}, volume={120}, number={1--3},
+ pages={215--219}, year={1993}, doi={10.1016/0012-365X(93)90579-I}
+}
+
+@article{BaratJoretWood2011,
+ author={Bar{\'a}t, J{\'a}nos and Joret, Gwena{\"e}l and Wood, David R.},
+ title={Disproof of the List {Hadwiger} Conjecture},
+ journal={Electronic Journal of Combinatorics}, volume={18}, number={1},
+ pages={P232}, year={2011}, doi={10.37236/719},
+ eprint={1110.2272}, archivePrefix={arXiv}
+}
+
+@article{Steiner2022List,
+ author={Steiner, Raphael},
+ title={Improved lower bound for the list chromatic number of graphs with no {$K_t$} minor},
+ journal={Combinatorics, Probability and Computing}, volume={31}, number={6},
+ pages={1070--1075}, year={2022}, doi={10.1017/S0963548322000116},
+ eprint={2110.09403}, archivePrefix={arXiv}
+}
+
+@article{NorinPostleSong2023,
+ author={Norin, Sergey and Postle, Luke and Song, Zi-Xia},
+ title={Breaking the degeneracy barrier for coloring graphs with no {$K_t$} minor},
+ journal={Advances in Mathematics}, volume={422}, pages={109020}, year={2023},
+ doi={10.1016/j.aim.2023.109020},
+ eprint={1910.09378}, archivePrefix={arXiv}, url={https://arxiv.org/abs/1910.09378v2}
+}
+
+@article{NorinPostle2023,
+ author={Norin, Sergey and Postle, Luke},
+ title={Connectivity and choosability of graphs with no {$K_t$} minor},
+ journal={Journal of Combinatorial Theory, Series B}, volume={158},
+ pages={283--300}, year={2023}, doi={10.1016/j.jctb.2021.02.001},
+ eprint={2004.10367}, archivePrefix={arXiv}, url={https://arxiv.org/abs/2004.10367v1}
+}
+
+@misc{OpenAIHadwigerCounterexample2026,
+  author = {{OpenAI}},
+  title = {{A counterexample to Hadwiger's conjecture}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Hadwigers-conjecture-September-23-2026/paper.pdf}{OAI:A-counterexample-to-Hadwigers-conjecture-September-23-2026}},
+  year = {2026}
+}
+
+@article{Mader1972,
+  author = {Mader, W.},
+  title = {Existenz {$n$}-fach zusammenh{\"a}ngender Teilgraphen in Graphen gen{\"u}gend gro{\ss}er Kantendichte},
+  journal = {Abhandlungen aus dem Mathematischen Seminar der Universit{\"a}t Hamburg},
+  volume = {37},
+  pages = {86--97},
+  year = {1972},
+  doi = {10.1007/BF02993903},
+  url = {https://doi.org/10.1007/BF02993903}
+}
diff --git a/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/sections/01-introduction.tex b/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/sections/01-introduction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..a32e4703d47498bdefb1d1afebee93b8766ff550
--- /dev/null
+++ b/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/sections/01-introduction.tex
@@ -0,0 +1,177 @@
+\section{Introduction}\label{sec:introduction}
+
+A clique minor records a form of complete interaction inside a graph:
+its vertices are represented by disjoint connected sets, and every two
+sets have an edge between them. For a finite nonempty simple graph $G$,
+let $h(G)$ be the largest order of a clique minor in $G$. Hadwiger
+conjectured that the chromatic number always satisfies
+$\chi(G)\le h(G)$ \cite{Hadwiger1943}. The companion manuscript
+\cite[Corollary~1.2]{OpenAIHadwigerCounterexample2026} constructs finite
+simple graphs with $\chi(G)>h(G)$, disproving this ordinary coefficient-one
+assertion. Its linear relaxation asks whether
+an absolute constant can replace the coefficient one. We prove a bound
+of this form even when allowable colors vary from vertex to vertex.
+
+A \emph{list assignment} $\mathcal L$ assigns a finite set of colors to
+each vertex. An $\mathcal L$-coloring is a proper coloring $c$ satisfying
+$c(v)\in\mathcal L(v)$ at every vertex. The list chromatic number
+$\ell(G)=\chi_{\mathrm{list}}(G)$ is the least integer $k$ for which every
+assignment with $|\mathcal L(v)|\ge k$ is colorable. List coloring was
+introduced independently by Vizing \cite{Vizing1976} and by Erd\H{o}s,
+Rubin, and Taylor \cite{ErdosRubinTaylor1980}. The linear list
+version of Hadwiger's conjecture was formulated by Kawarabayashi and
+Mohar \cite[Conjecture~1.3]{KawarabayashiMohar2007}.
+
+\begin{theorem}\label{thm:main}
+There is an absolute integer $C\ge1$ such that every finite nonempty
+simple graph $G$ satisfies
+\[
+ \ell(G)\le C h(G).
+\]
+\end{theorem}
+
+Thus the Linear List Hadwiger conjecture has an affirmative resolution.
+The constant is independent of the graph order, the list assignment, and
+the number of distinct colors appearing in its union. Taking the same
+list at every vertex immediately gives the ordinary conclusion.
+
+\begin{corollary}\label{cor:ordinary}
+Every finite nonempty simple graph $G$ satisfies $\chi(G)\le C h(G)$,
+with the constant $C$ in Theorem~\ref{thm:main}.
+\end{corollary}
+\begin{proof}
+A coloring from the common list $\{1,\ldots,\ell(G)\}$ shows that
+$\chi(G)\le\ell(G)$.
+\end{proof}
+
+\subsection{Prior work and the two obstacles}
+
+The density theorems of Kostochka \cite{Kostochka1982,Kostochka1984}
+and Thomason \cite{Thomason1984} show that every $K_t$-minor-free graph
+has a vertex of degree $O(t\sqrt{\log t})$, as does each of its
+subgraphs. Greedy coloring in reverse deletion order therefore gives
+this bound for arbitrary lists. Norin, Postle, and Song
+\cite{NorinPostleSong2023} went beyond that degeneracy threshold for
+ordinary coloring, proving $O(t(\log t)^\beta)$ for every fixed
+$\beta>1/4$. Norin and Postle \cite{NorinPostle2023} extended such
+bounds to list coloring using the structure of highly connected
+minor-free graphs.
+
+The coefficient in a linear list bound has a different role from the
+coefficient in ordinary Hadwiger's conjecture. Voigt
+\cite{Voigt1993} constructed planar graphs that are not $4$-choosable.
+Bar\'at, Joret, and Wood \cite{BaratJoretWood2011} constructed graphs
+with no $K_{3s+2}$ minor that are not $4s$-choosable for every integer
+$s\ge1$, and Steiner
+\cite{Steiner2022List} proved that for every $\varepsilon>0$ and all
+sufficiently large $t$ there are $K_t$-minor-free graphs with list
+chromatic number at least $(2-\varepsilon)t$. Thus a universal constant
+in Theorem~\ref{thm:main} must be at least two. Our constants are not
+optimized.
+
+The fractional coloring theorem of Reed and Seymour
+\cite{ReedSeymour1998} gives $\chi_f(G)\le2h(G)$, where fractional
+coloring permits weighted stable sets in place of color classes. It
+provides a linear supply of stable sets but does not directly give a
+linear ordinary or list coloring: naive repeated sampling introduces a
+factor depending on the graph order. Delcourt and Postle
+\cite{DelcourtPostle2021} obtained the ordinary bound
+$O(t\log\log t)$ for graphs excluding a $K_t$ minor and reduced a linear
+ordinary bound to coloring suitably small subgraphs. Liu and Luo
+\cite{LiuLuo2026} subsequently proved the ordinary bound
+$O(t\log\log\log t)$ and developed bounded stable-set sampling that we
+use below.\footnote{Liu and Luo explicitly acknowledge assistance from
+ChatGPT Codex; their declaration does not identify an underlying model
+version.} These ordinary-coloring statements do not by themselves give
+colorings from arbitrary vertex lists.
+
+For list coloring, Postle proved an $O(t(\log\log t)^6)$ bound for
+$K_t$-minor-free graphs \cite[Theorem~1.8]{Postle2020}. His separability
+and woven-minor constructions provide the structural framework for our
+passage from small graphs to arbitrary orders. Delcourt and Postle outlined
+an $O(t(\log\log t)^2)$ list bound in the first version of their paper
+\cite[Section~6]{DelcourtPostle2021v1}, leaving the details to the reader.
+In subsequent work posted on October~1, 2026, Gu and Xu
+\cite[Theorem~1.3]{GuXu2026} proved an $O(t\log\log t)$ list bound.
+Connectivity extraction by the constraint-budget method of Gir\~ao and Narayanan
+\cite{GiraoNarayanan2022}, and the highly linked graph theorem of
+Bollob\'as and Thomason \cite{BollobasThomason1996}, are further key
+antecedents. We reproduce the list transfer, density, path accounting,
+and recursion arguments in the forms required here.
+
+Two difficulties remain after these reductions. Even on a small graph,
+a sampled stable set cannot be given a single color unless that color is
+available at every vertex receiving it. In a graph of unrestricted order,
+the paths used to assemble a minor may themselves contain arbitrarily
+many vertices. Their deletion must therefore be charged through their
+structure, rather than through their order. The proof addresses these
+two difficulties separately.
+
+\subsection{Proof strategy}
+
+First, Proposition~\ref{prop:small-list} proves that a $K_t$-minor-free
+graph on at most $t^{11/10}$ vertices is $Lt$-choosable, for an absolute
+$L$ and all sufficiently large $t$. We partition the union of all lists
+into bins, assigning each color to a single bin throughout the graph.
+A vertex is eligible for a bin precisely when its own list contains a
+color in that bin. A random partition gives every vertex enough eligible
+bins at every stage, before any remaining graph is chosen.
+Lemma~\ref{small:eligible-thinning} then repeatedly halves an upper bound
+on the fractional chromatic number using sampled stable sets restricted to eligible
+vertices. Vertices assigned to the same bin lie in one stable set, so
+each can choose its own allowable color from that bin; colors in different
+bins cannot conflict.
+
+To prove the thinning lemma, suppose every sampling outcome leaves a
+fractional chromatic number greater than half the current bound. A deterministic separator
+decomposition finds a fixed highly connected subgraph in which a
+remainder still has large expected fractional chromatic number. Independent
+repetitions produce disjoint private vertex sets, each supporting a clique
+model. A small, fixed family of matching tests shows that the complement
+of the union of the sampled remainders remains highly connected.
+Connecting equally labeled bags through this complement produces the
+excluded clique minor. The proof uses bounded stable-set sampling and
+ordinary-chromatic connectivity extraction, with the full argument given
+in Section~\ref{small:section}. Summing the costs of successive halving
+stages requires only $O(t)$ bins.
+
+Second, Lemma~\ref{lem:list-extraction} extracts highly connected
+subgraphs with a linear loss of list chromatic number. The additive
+list-deletion estimate, Lemma~\ref{lem:additive-deletion}, reserves
+colors for a small set and controls its interaction with all remaining
+vertices. Its probabilistic estimate tests only polynomially many
+exceptional vertices, independently of the ambient graph order and
+palette size. Minimum-length path systems give the sparsity needed to
+apply this estimate even when the paths are long. The density argument
+also supplies small highly connected subgraphs in which clique models
+can be built while connecting paths are rerouted.
+
+The sequential construction in Lemma~\ref{lem:separability} then shows
+that a graph of sufficiently large list chromatic number either contains
+the desired clique minor or splits into two disjoint subgraphs retaining
+nearly all of that list chromatic number. At each step, one small district
+supplies new adjacencies between model bags and a separate junction joins
+the paths carrying those bags forward. The deletion estimate applies to
+the complete history of original minimizing paths, including vertices
+no longer used by the growing model.
+
+Finally, Theorem~\ref{thm:recursive-woven} builds a clique model whose
+bags contain prescribed roots, together with paths for prescribed
+terminal pairs. After reserving the paths for the terminal pairs, the
+remaining task is a rooted model inside a fresh region. Separability
+supplies three disjoint districts for recursive calls. Partitioning the
+root labels into three blocks and assigning one district to each pair
+of blocks reduces each call to about two thirds as many roots, while
+every pair of labels still occurs together in one district. Connectivity
+extraction and the case with no prescribed pairs complete the theorem.
+
+Section~\ref{sec:preliminaries} fixes the external inputs and
+probabilistic tools. Section~\ref{small:section} proves the small-graph
+list bound, and Section~\ref{sec:list-transfer} establishes list
+extraction and deletion. Section~\ref{sec:density} constructs small
+highly connected districts; Section~\ref{sec:routing} develops their
+routing interfaces. Sections~\ref{sec:separability} and
+\ref{sec:recursion} prove separability and the final recursive theorem.
+The only structural results used without proof are the fractional minor
+bound, ordinary-chromatic connectivity extraction, and the terminal
+grouping theorem, whose precise statements follow.
diff --git a/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/sections/02-preliminaries.tex b/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/sections/02-preliminaries.tex
new file mode 100644
index 0000000000000000000000000000000000000000..06576b1fa8a35b6ffa0cd39e889056a2a28e5f37
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@@ -0,0 +1,174 @@
+\section{Preliminaries and probabilistic tools}\label{sec:preliminaries}
+
+All graphs are finite, simple, and undirected. A \emph{stable set} is a set
+of pairwise nonadjacent vertices. For a vertex set $X$, write $P[X]$ for
+the induced subgraph and $P-X=P[V(P)\setminus X]$. A clique model consists
+of disjoint nonempty connected vertex sets, called \emph{bags}, with an
+edge between every two bags. Models lift through edge contractions and
+vertex deletions, so $h$ is nonincreasing under taking minors. Whenever
+a subgraph is specified by its vertices, it is induced. Adding edges to
+a connected or highly connected subgraph preserves the corresponding
+property, so the subgraphs produced below may be replaced by induced
+subgraphs on their vertices.
+
+A graph is $k$-connected if it has more than $k$ vertices and remains
+connected after deletion of fewer than $k$ vertices. Such a graph has
+minimum degree at least $k$. A separator split is a partition
+$V(P)=A\mathbin{\dot\cup}Z\mathbin{\dot\cup}A'$, with $A,A'$ nonempty
+and anticomplete. If $|V(P)|>k$ and $P$ is not $k$-connected, there is such
+a split with $|Z|<k$. Deleting $z<k$ vertices from a $k$-connected graph
+leaves a $(k-z)$-connected graph. All logarithms are natural. The null
+graph has $h=\ell=\chi=f=0$. Integer parameters include minor orders,
+connectivity requirements, and numbers of paths; powers bounding graph
+orders need not be integers.
+
+\subsection{Fractional coloring and established structural results}
+
+The fractional chromatic number $f(P)$ is the minimum total weight of
+nonnegative weights on the stable sets of $P$ that cover every vertex
+with weight at least one. Finite linear-programming duality
+\cite{galekuhntucker1951} gives
+\begin{equation}\label{eq:fractional-dual}
+ f(P)=\max\left\{\sum_{v\in V(P)}w(v):
+ w(v)\ge0,\quad \sum_{v\in I}w(v)\le1
+ \text{ for every stable set }I\right\}.
+\end{equation}
+The primal is feasible, using singleton stable sets, and has a finite
+optimum; thus this use of duality has no feasibility or boundedness
+exception. For $r>0$, $f(P)\le r$ is equivalent to a distribution on stable
+sets with every vertex included with probability at least $1/r$.
+Indeed, divide a fractional coloring by $r$ and put any missing mass on
+the empty set; conversely multiply such a distribution by $r$.
+Restriction to an induced subgraph cannot increase $f$, and adding one
+vertex increases it by at most one.
+
+\begin{theorem}[Fractional minor bound]\label{thm:fractional}
+Every graph $P$ satisfies $f(P)\le4h(P)$.
+\end{theorem}
+This follows from the stronger bound $f(P)\le2h(P)$ of Reed and Seymour
+\cite[Statement~(1.3)]{ReedSeymour1998}. We keep the factor four throughout.
+
+\begin{theorem}[Ordinary-chromatic connectivity extraction]
+\label{thm:ordinary-extraction}
+If $k\ge1$ is an integer and $\chi(P)\ge7k$, then $P$ has a
+$k$-connected subgraph $J$ with $\chi(J)\ge\chi(P)-6k$.
+\end{theorem}
+We use the form stated in \cite[Theorem~5.1]{DelcourtPostle2021},
+attributed there to Gir\~ao and Narayanan. There is no minor-exclusion
+hypothesis. In particular the theorem applies to auxiliary clique
+blow-ups, which need not inherit any minor exclusion.
+
+\begin{theorem}[Grouping terminals]\label{thm:grouping}
+There is an absolute integer $\gamma\ge1$ such that the following holds.
+In a $\gamma m$-connected graph, every partition of $m\ge1$ distinct
+terminals into nonempty groups can be realized by pairwise disjoint
+connected vertex sets, one containing each group.
+\end{theorem}
+This is a consequence of the highly linked graph theorem of
+Bollob\'as and Thomason \cite{BollobasThomason1996}; we use the
+case with second knitting parameter one in
+\cite[Definition~5.7 and Theorem~5.8]{DelcourtPostle2021}. The constant
+$\gamma$ is uniform in the number and sizes of the groups. For a group
+of size two, a path within its connected set joins its terminals. We
+will apply this observation to any prescribed pairing of distinct
+terminals.
+
+\subsection{Bounded sampling and concentration}
+
+The following real-parameter form of bounded stable-set sampling is
+convenient. Its integer-parameter form appears in
+\cite[Lemma~4.1]{LiuLuo2026}; we include the short proof.
+
+\begin{lemma}\label{lem:bounded}
+Suppose $|V(P)|\le n$, where $n\ge1$, and $f(P)\le r$, where $r\ge1$.
+There is a distribution on stable sets $I$ such that
+\[
+ |I|\le\lceil n/r\rceil,
+ \qquad \Pr(v\in I)=\frac1{2r}\quad(v\in V(P)).
+\]
+\end{lemma}
+\begin{proof}
+Start with a stable-set distribution having inclusion probabilities at
+least $1/r$. Conditional independent deletion of each included vertex
+reduces every marginal to exactly $1/r$ and preserves stability.
+Put $M=\lceil n/r\rceil$. Partition each resulting stable set $I$ into
+parts of size at most $M$, assigning its probability mass to each part.
+This gives a finite measure of total mass at most
+\[
+ \mathbb E\bigl(1+|I|/M\bigr)
+ =1+|V(P)|/(rM)\le2.
+\]
+Every vertex still has mass $1/r$. Add mass on the empty set to make the
+total exactly two and divide by two.
+\end{proof}
+
+We also record the standard bounded-martingale-difference estimate
+\cite{azuma1967,mcdiarmid1989}, with its short exponential-moment proof.
+
+\begin{lemma}[Bounded differences]\label{lem:concentration}
+Let $Z$ be a function of $s\ge1$ independent finite random variables.
+If changing one variable changes $Z$ by at most $b>0$, then
+\[
+ \Pr\bigl(|Z-\mathbb EZ|\ge u\bigr)
+ \le2\exp\bigl(-u^2/(2sb^2)\bigr)\qquad(u>0).
+\]
+\end{lemma}
+\begin{proof}
+Expose the variables in order. The martingale of conditional
+expectations has differences of conditional mean zero and absolute
+value at most $b$. Convexity bounds the conditional exponential moment
+of a difference at $\lambda$ by
+$\cosh(\lambda b)\le\exp(\lambda^2b^2/2)$.
+Multiplying these bounds and applying Markov's inequality with
+$\lambda=u/(sb^2)$ gives the upper tail. Apply the same argument to $-Z$
+and add the two bounds.
+\end{proof}
+
+\subsection{A small family of connectivity tests}
+
+The next test replaces an enumeration of arbitrary vertex cuts by an
+enumeration of unions of a few open neighborhoods. The family is fixed
+before the random deleted set is chosen.
+
+\begin{lemma}[Cut test]\label{lem:cut-test}
+Let $P$ be a $k$-connected graph on at most $n$ vertices, where $k$ is a
+sufficiently large integer. Set
+\[
+ s=\left\lceil10(n/k)\log(2n)\right\rceil.
+\]
+There is a family of at most $(n+1)^{s+1}$ matchings, each of size
+$\lfloor k/8\rfloor$, with the following property. If $U\subseteq V(P)$,
+every vertex of $P$ has at least $k/2$ neighbors outside $U$, and fewer
+than $k/32$ endpoints of each matching belong to $U$, then $P-U$ is
+$\lfloor k/32\rfloor$-connected.
+\end{lemma}
+\begin{proof}
+For every set $R$ of at most $s$ vertices, put
+$B_R=\bigcup_{v\in R}N_P(v)$. Whenever both $B_R$ and its complement
+have size at least $k/3$, fix a matching of size $\lfloor k/8\rfloor$
+across this cut. Such a matching exists: otherwise the endpoints of a
+maximal smaller matching form a vertex cover of the crossing edges,
+of size less than $k/4$. Both sides remain nonempty after their removal,
+contradicting $k$-connectivity. The number of possible sets $R$, and
+hence of matchings, is at most $(n+1)^{s+1}$.
+
+The neighborhood assumption ensures that $P-U$ has more than
+$\lfloor k/32\rfloor$ vertices. If it fails the asserted connectivity,
+there are a set $Z\subseteq V(P)\setminus U$ with $|Z|<k/32$ and
+nonempty anticomplete sides $A,A'$ partitioning $V(P)\setminus(U\cup Z)$.
+Every vertex of either side has more than $k/3$ neighbors in that same
+side, since it has at least $k/2$ neighbors outside $U$ and loses fewer
+than $k/32$ to $Z$. In particular both sides have size at least $k/3$.
+
+Make $s$ independent uniform draws from $A$. For each vertex $v\in A$,
+the probability that none of the draws is an open neighbor of $v$ is
+at most $\exp(-ks/(3n))$. The union bound over $A$ is less than one by
+the choice of $s$. Thus the set $R$ of drawn vertices, after repetitions
+are removed, satisfies $A\subseteq B_R$. Since $A,A'$ are anticomplete,
+$A'\cap B_R=\varnothing$. The matching fixed for this cut has every edge
+incident with $U\cup Z$, for otherwise an edge would join $A$ to $A'$.
+But $U$ contains fewer than $k/32$ of its endpoints, and $Z$ contains
+fewer than $k/32$ vertices. They cannot cover its
+$\lfloor k/8\rfloor$ disjoint edges. This contradiction proves the
+lemma.
+\end{proof}
diff --git a/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/sections/03-small-graphs.tex b/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/sections/03-small-graphs.tex
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@@ -0,0 +1,534 @@
+\section{List coloring small graphs}
+\label{small:section}
+
+\begin{proposition}\label{prop:small-list}
+There are an absolute positive integer $L$ and an absolute integer $t_0$
+such that, for every integer $t\ge t_0$ and every graph $G$,
+\[
+ h(G)<t,\qquad |V(G)|\le t^{11/10}
+ \quad\Longrightarrow\quad \ell(G)\le Lt.
+\]
+\end{proposition}
+
+We will color the graph in stages, halving an upper bound on its fractional
+chromatic number at each stage. For each sampled stable set, only a specified
+subset of vertices is eligible to be removed. The next lemma allows
+these eligibility restrictions to be fixed in advance. In the final
+application they will come from one partition of the actual colors
+appearing in the lists, chosen before the successive remaining graphs
+are determined.
+
+\begin{lemma}[Eligible thinning]\label{small:eligible-thinning}
+There is an absolute positive integer $B$ such that the following holds
+for all sufficiently large integers $t$. Let $n,r$ be real parameters with
+\[
+ 1\le n\le t^{11/10},\qquad \frac{t}{\log t}\le r\le4t,
+\]
+and let $P$ be a graph with $|V(P)|\le n$, $h(P)<t$, and $f(P)\le r$.
+Set
+\[
+ N=\left\lceil Br\log\bigl(2\max\{1,t/r\}\bigr)\right\rceil.
+\]
+Suppose $E_1,\ldots,E_N\subseteq V(P)$ contain every vertex at least
+$N/2$ times. There are stable sets $I_1,\ldots,I_N$ of $P$, each of size
+at most $\lceil n/r\rceil$, such that
+\[
+ f\left(P-\bigcup_{j=1}^N(I_j\cap E_j)\right)\le r/2.
+\]
+The cutoff for $t$ is uniform over $P,n,r$ and the eligibility sets.
+\end{lemma}
+
+We prove the lemma by contradiction. If every sampled remainder has
+fractional chromatic number greater than $r/2$, we first find a fixed
+highly connected subgraph $H$ where a remainder still has large expected
+fractional chromatic number. Fresh independent remainders then contain
+disjoint subsets, each carrying a clique model. The complement in $H$
+of the union of the remainders remains highly connected. To assemble a
+$K_t$ model, partition its $t$ labels into small blocks and use one of
+these disjoint models for each
+block and each pair of blocks. The complement connects all bags assigned
+to the same label. Every pair of labels is represented together in one
+model, so the resulting label sets are adjacent. The estimates below
+provide the models and the connectivity for this construction.
+
+\subsection{Eligible samples and uniform concentration}
+
+We prove Lemma~\ref{small:eligible-thinning} with the following fixed
+choice of $B$. Let $\gamma$ be the integer in
+Theorem~\ref{thm:grouping}. Fix
+\begin{equation}\label{small:constants}
+ D_0=10^6\gamma,\qquad Q_0=401^2,
+\end{equation}
+and choose a positive integer $B$ such that
+\begin{equation}\label{small:B-choice}
+ B\ge16,\qquad
+ 2^{B/4}\ge
+ \max\bigl\{2000(D_0+1),\,10^4Q_0^2\bigr\}.
+\end{equation}
+These constants remain fixed throughout this section.
+
+Consider an integer $t$ and real parameters $n,r$ satisfying
+\begin{equation}\label{small:range}
+ 1\le n\le t^{11/10},\qquad
+ \frac{t}{\log t}\le r\le4t.
+\end{equation}
+Let $P$ be a graph on at most $n$ vertices with $h(P)<t$ and $f(P)\le r$.
+Set
+\begin{equation}\label{small:parameters}
+ \begin{gathered}
+ y=\max\{1,t/r\},\qquad x=\lfloor r/200\rfloor,
+ \qquad q=\lceil t/x\rceil^2,\qquad k=\lceil D_0ty\rceil,\\
+ N=\lceil Br\log(2y)\rceil,\qquad M=\lceil n/r\rceil.
+ \end{gathered}
+\end{equation}
+Here $x$ bounds the size of a label block, $q$ bounds the number of
+disjoint clique models, and $k$ is the connectivity required of $H$.
+The parameters $N$ and $M$ are the number of trials and the maximum
+size of one sampled stable set.
+We always take $t$ above an absolute cutoff large enough that $r\ge400$.
+In particular $x\ge1$. The following bounds hold uniformly in
+\eqref{small:range}:
+\begin{equation}\label{small:parameter-bounds}
+ \begin{gathered}
+ 1\le y\le\log t,\qquad \frac{t}{y}\le r\le\frac{4t}{y},\qquad
+ x\ge\frac r{400},\\
+ q\le Q_0y^2,\qquad xq\le403ty,\qquad
+ D_0ty\le k\le(D_0+1)ty.
+ \end{gathered}
+\end{equation}
+Indeed, $t/x\le400t/r\le400y$, so
+$\lceil t/x\rceil\le401y$. Also
+\[
+ x\lceil t/x\rceil^2
+ \le\frac{t^2}{x}+2t+x
+ \le400\frac{t^2}{r}+2t+\frac r{200}
+ \le403ty.
+\]
+Our choice of $D_0$ consequently gives, for all sufficiently large $t$,
+\begin{equation}\label{small:attachment-budget}
+ k\ge r,\qquad 2xq<\frac{k}{64},\qquad
+ \left\lfloor\frac{k}{32}\right\rfloor\ge\gamma(2xq).
+\end{equation}
+
+Fix sets $E_1,\ldots,E_N\subseteq V(P)$ such that each vertex belongs
+to at least $N/2$ of them. These are the eligibility sets and are fixed
+before sampling. Independently for each $j\in\{1,\ldots,N\}$, sample a
+stable set $I_j$ from the distribution in Lemma~\ref{lem:bounded}, with
+\[
+ |I_j|\le M,\qquad \Pr(v\in I_j)=\frac1{2r}.
+\]
+The covered vertices at trial $j$ are $I_j\cap E_j$, and the remainder is
+\begin{equation}\label{small:remainder}
+ W=V(P)\setminus\bigcup_{j=1}^N(I_j\cap E_j).
+\end{equation}
+For every vertex $v$, independence of the trials and the eligibility
+condition imply
+\begin{equation}\label{small:survival}
+ \Pr(v\in W)
+ \le\left(1-\frac1{2r}\right)^{N/2}
+ \le p:=\exp\left(-\frac{N}{4r}\right)
+ \le(2y)^{-B/4}.
+\end{equation}
+In particular, \eqref{small:B-choice} and \eqref{small:parameter-bounds}
+give
+\begin{equation}\label{small:sparseness}
+ 2kp\le\frac r{1000},\qquad q^2p\le10^{-4}.
+\end{equation}
+For the first inequality, divide by $r$ and use
+\[
+ \frac{2kp}{r}
+ \le2(D_0+1)y^2(2y)^{-B/4}
+ \le2(D_0+1)2^{-B/4}.
+\]
+For the second, use
+$q^2p\le Q_0^2y^4(2y)^{-B/4}\le Q_0^2 2^{-B/4}$.
+
+\begin{lemma}\label{small:concentration}
+Take at most $q$ independent copies of the experiment
+\eqref{small:remainder}. Let $Z$ be any one of the following quantities,
+where the vertex set $A\subseteq V(P)$ is fixed before sampling:
+\[
+ |A\cap W_i|,\qquad
+ \left|A\cap\bigcup_iW_i\right|,\qquad
+ f(P[A\cap W_i]).
+\]
+For each fixed $\delta>0$, and all sufficiently large $t$,
+\begin{equation}\label{small:uniform-tail}
+ \Pr\bigl(|Z-\mathbb EZ|\ge\delta r\bigr)
+ \le2\exp(-t^{7/10}),
+\end{equation}
+uniformly over all the data in \eqref{small:range} and all eligibility
+sets satisfying the stated condition.
+\end{lemma}
+
+\begin{proof}
+Changing one sampled stable set changes $W_i$ on at most $2M$ vertices.
+It changes a union of remainders on at most the same number of vertices.
+Adding or deleting a vertex changes fractional chromatic number by at
+most one, so all the quantities in the statement have bounded differences
+at most $2M$ per trial.
+
+For sufficiently large $t$, $r\log(2y)\ge1$, and hence
+$N\le(B+1)r\log(2y)$. With
+$A_0=4Q_0(B+1)$, we obtain
+\[
+ qN\le A_0ty\log(2y),\qquad M\le2t^{1/10}y.
+\]
+Lemma~\ref{lem:concentration} therefore gives
+\begin{align*}
+ \Pr\bigl(|Z-\mathbb EZ|\ge\delta r\bigr)
+ &\le2\exp\left(-\frac{\delta^2r^2}{2(qN)(2M)^2}\right)\\
+ &\le2\exp\left(
+ -\frac{\delta^2t^{4/5}}{32A_0y^5\log(2y)}\right).
+\end{align*}
+Since $1\le y\le\log t$, the exponent in the last line has magnitude at
+least $t^{7/10}$ for all sufficiently large $t$. The cutoff depends only
+on $\delta$ and the fixed constants. Independence between different
+vertices is not used.
+\end{proof}
+
+All asymptotic probability estimates below are uniform over these data.
+Only finitely many fixed positive values of $\delta$ will be used in
+Lemma~\ref{small:concentration}, so their cutoffs can be combined.
+
+\subsection{Localizing a failure of thinning}
+
+\begin{lemma}\label{small:fixed-region}
+Suppose that every outcome of the experiment
+\eqref{small:remainder} having positive probability satisfies
+$f(P[W])>r/2$. Then there is a fixed induced $k$-connected subgraph
+$H\subseteq P$ such that
+\begin{equation}\label{small:region-expectation}
+ \mathbb E f(H[W\cap V(H)])>r/4.
+\end{equation}
+The subgraph $H$ is chosen before any fresh copies of the experiment
+are sampled.
+\end{lemma}
+
+\begin{proof}
+Build a binary decomposition of $P$, making all choices before sampling.
+A node is an induced subgraph. Stop at a node of order at most $2k$, and
+also at a larger node which is $k$-connected. Otherwise choose a separator
+of size less than $k$, partition the vertices outside it into two
+nonempty anticomplete sides, and form two children by adding the
+separator to each side.
+
+There are $O(n^2+1)$ nodes. To see this, give a node of order $u>2k$
+potential $u-k$. Each large child's potential is strictly smaller than
+its parent's, since the other exclusive side is nonempty. If both
+children are large, their potentials sum to at most their parent's:
+their orders sum to $u+|S|<u+k$, where $S$ is the separator. If just one
+child is large, its potential alone is at most the parent's. Thus at
+each depth the sum of the large-node potentials is at most $n$, while
+the depth of a large-node branch is at most $n$. The small leaves
+contribute at most two children per large node, in addition to a possible
+small root.
+
+Every separator and every small leaf has order at most $2k$. Its expected
+intersection size with $W$ is at most $2kp\le r/1000$.
+Lemma~\ref{small:concentration} and a union bound over the decomposition
+show that, with probability $1-o(1)$, all these intersections have fewer
+than $r/100$ vertices. If every $k$-connected leaf had expected fractional
+chromatic number at most $r/4$ after restriction to $W$, the same lemma
+would give, again with probability $1-o(1)$, fractional chromatic number
+less than $3r/10$ in every such leaf. A small leaf has this latter bound
+as well, since its fractional chromatic number is at most its order.
+
+Fix an outcome satisfying all these conclusions. Write $F=P[W]$.
+The assumption of the lemma gives $f(F)>r/2$, whereas every leaf
+restriction has fractional chromatic number less than $3r/10$.
+We compare these bounds by replacing every vertex of $F$ with a clique
+of the same large order. Ordinary-chromatic connectivity extraction in
+this larger graph will produce a subgraph of large chromatic number that
+cannot cross any of the blown-up separators. It must lie in one
+blown-up leaf, giving the contradiction.
+
+For a positive integer $m$, form $F^{(m)}$ by replacing each vertex of
+$F$ with a clique of order $m$, and replacing each edge with a complete
+join between its two replacement cliques. Every color class of $F^{(m)}$ projects to a stable
+set of $F$ and uses at most one vertex of each replacement clique.
+Dividing the multiplicities of these projected stable sets by $m$
+therefore gives a fractional coloring of $F$. Consequently
+\begin{equation}\label{small:blowup-lower}
+ \chi(F^{(m)})\ge mf(F)>mr/2.
+\end{equation}
+
+For each leaf restriction, choose a fractional coloring of weight less
+than $3r/10$. Round its finitely many stable-set weights up to multiples
+of $1/m$. Give a distinct color slot to each unit of weight $1/m$.
+Every original vertex belongs to at least $m$ slots, so its $m$ copies
+can be assigned distinct slots containing that vertex. A slot is a
+stable set upstairs: it contains at most one copy of any original
+vertex, and its original vertices form a stable set. The rounding adds
+only a bounded number of slots, independent of $m$ for this fixed leaf.
+There are finitely many leaves, so for all sufficiently large $m$, every
+blown-up leaf restriction has a coloring with fewer than $31mr/100$
+colors.
+
+Apply Theorem~\ref{thm:ordinary-extraction} to $F^{(m)}$ with the integer
+parameter $k'=\lceil mr/50\rceil$. For sufficiently large $m$,
+\eqref{small:blowup-lower} exceeds $7k'$. We obtain a $k'$-connected
+subgraph $J$ with
+\[
+ \chi(J)\ge\chi(F^{(m)})-6k'
+ >\frac{19mr}{50}-6
+ >\frac{31mr}{100}.
+\]
+Each restricted separator of the decomposition has fewer than $r/100$
+vertices, so its blow-up has fewer than $mr/100<k'$ vertices. Thus $J$
+cannot meet both exclusive sides of any such split: deleting its
+vertices in the separator would disconnect $J$ with fewer than $k'$
+deletions. Nor can $J$ be contained in the separator, because its order
+is greater than $k'$. Descending the decomposition places $J$ inside a
+single blown-up leaf restriction, contradicting that leaf's coloring
+bound.
+
+It follows that some $k$-connected leaf satisfies
+\eqref{small:region-expectation}. This argument also covers the case
+where $P$ is too small to be $k$-connected: if all leaves were small, the
+same contradiction would apply. Finally, the blow-up is only an
+auxiliary graph for the extraction theorem, which requires no minor
+exclusion; no bound on its clique-minor number was used.
+\end{proof}
+
+\subsection{Private models and a connected complement}
+
+Assume the failure hypothesis of Lemma~\ref{small:fixed-region}, and fix
+the resulting subgraph $H$. Sample $q$ fresh independent copies
+$W_1,\ldots,W_q$ of \eqref{small:remainder}. Put
+\[
+ T_i=V(H)\cap\left(W_i\setminus\bigcup_{j\ne i}W_j\right),
+ \qquad
+ U=V(H)\cap\bigcup_{i=1}^qW_i.
+\]
+The sets $T_i$ are pairwise disjoint subsets of $U$.
+
+\begin{lemma}\label{small:private}
+With failure probability at most $4q(q-1)p+o(1)$, every $H[T_i]$ has
+fractional chromatic number greater than $r/8$, and hence contains a
+$K_{2x}$ model.
+\end{lemma}
+
+\begin{proof}
+By \eqref{small:region-expectation} and
+Lemma~\ref{small:concentration}, the events
+\[
+ \mathcal A_i=\{f(H[W_i\cap V(H)])>r/5\}
+\]
+all hold with probability $1-o(1)$: the individual failure probability
+is at most $2\exp(-t^{7/10})$, and $q\le Q_0(\log t)^2$.
+
+For each outcome of $W_i$ satisfying $\mathcal A_i$, choose optimal dual
+weights on $H[W_i\cap V(H)]$ of total weight $a_i>r/5$.
+Choose these weights as a function of $W_i$ alone. Conditional on this
+one copy, each other copy still satisfies the vertex survival bound
+\eqref{small:survival}. Therefore the expected weight on vertices
+belonging to some other copy is at most $(q-1)pa_i$. Markov's inequality
+shows that the probability of $\mathcal A_i$ together with the loss of
+more than $a_i/4$ is at most $4(q-1)p$.
+
+Taking a union bound over $i$, we conclude that, except with probability
+$4q(q-1)p+o(1)$, all private sets retain weight at least
+$3a_i/4>3r/20>r/8$. Restriction to $T_i$ preserves dual feasibility:
+each stable set in $H[T_i]$ is also a stable set in
+$H[W_i\cap V(H)]$. Hence $f(H[T_i])>r/8$. In this calculation we have
+conditioned only on a single copy at a time, not on the simultaneous
+events $\mathcal A_1,\ldots,\mathcal A_q$.
+
+Theorem~\ref{thm:fractional} now gives
+\[
+ h(H[T_i])>r/32>2\lfloor r/200\rfloor=2x,
+\]
+so each private set contains the required model. The models are
+vertex-disjoint because the private sets are disjoint.
+\end{proof}
+
+\begin{lemma}\label{small:complement}
+With probability $1-o(1)$, every vertex of $H$ has at least $k/2$
+neighbors in $H-U$, and $H-U$ is $\lfloor k/32\rfloor$-connected.
+\end{lemma}
+
+\begin{proof}
+Each fixed vertex belongs to $U$ with probability at most
+$qp\le10^{-4}$ by \eqref{small:sparseness}. Before sampling, fix $k$
+neighbors of each vertex of $H$, possible since $H$ is $k$-connected.
+The expected number of those neighbors in $U$ is at most $k/10000$.
+Having at least $k/2$ of them in $U$ thus requires a deviation greater
+than $k/3\ge r/3$. Lemma~\ref{small:concentration} and a union bound
+over at most $n$ vertices prove the neighborhood assertion with
+probability $1-o(1)$. This assertion holds for all vertices of $H$,
+including those in $U$.
+
+For the connectivity assertion, use the family of test matchings in
+Lemma~\ref{lem:cut-test} for the fixed graph $H$, with order bound $n$
+and connectivity parameter $k$. Fix this family before sampling the
+fresh copies, and set
+\[
+ s=\left\lceil10\frac nk\log(2n)\right\rceil.
+\]
+The number of tests is at most
+\begin{align*}
+ (n+1)^{s+1}
+ &=\exp\bigl(O((1+n/k)\log^2(2n))\bigr)\\
+ &\le\exp\bigl(O(t^{1/10}\log^2t)\bigr).
+\end{align*}
+Here the constants are absolute, since $D_0$ is fixed.
+
+The endpoint set of each test matching is a fixed set to which
+Lemma~\ref{small:concentration} applies. It has size at most $k/4$, so
+its expected intersection size with $U$ is at most $k/40000$. A count of
+at least $k/32$ therefore requires a deviation greater than
+$k/64\ge r/64$. Lemma~\ref{small:concentration} and the displayed bound
+on the number of tests show that, with probability $1-o(1)$, every test
+matching has fewer than $k/32$ endpoints in $U$. Together with the
+neighborhood assertion, these are the hypotheses of
+Lemma~\ref{lem:cut-test}, which gives the claimed connectivity.
+\end{proof}
+
+\begin{proof}[Proof of Lemma~\ref{small:eligible-thinning}]
+If $P$ is empty, take all the stable sets empty. Otherwise use the
+experiment \eqref{small:remainder} and suppose that every outcome of
+positive probability has $f(P[W])>r/2$.
+Lemma~\ref{small:fixed-region} supplies a fixed $H$. By
+\eqref{small:sparseness}, the failure probability in
+Lemma~\ref{small:private} is at most $4\cdot10^{-4}+o(1)$.
+Thus Lemmas~\ref{small:private} and~\ref{small:complement} hold
+simultaneously with positive probability for sufficiently large $t$.
+Fix such an outcome, including its $q$ disjoint $K_{2x}$ models.
+
+Partition the $t$ labels of a prospective $K_t$ model into
+$b=\lceil t/x\rceil$ nonempty blocks of size at most $x$. Assign a
+different private model to each single block and each unordered pair
+of distinct blocks. There are
+$b+\binom b2=b(b+1)/2\le b^2=q$ assignments. In a model assigned to
+one block, retain one bag for each label of that block. In a model
+assigned to two blocks, retain one bag for each label in their union.
+At most $2x$ bags are retained from each model, and every pair of
+distinct labels appears together in one assigned model.
+
+Choose a port vertex in every retained bag. There are at most $2xq$
+ports, and each has at least $k/2$ neighbors in $H-U$. By
+\eqref{small:attachment-budget}, we can choose a distinct such neighbor
+for every port, greedily. Let $m$ be the actual number of chosen
+neighbors, so $1\le m\le2xq$. Their labels partition them into $t$
+nonempty groups. The graph $H-U$ has connectivity at least
+\[
+ \left\lfloor\frac{k}{32}\right\rfloor
+ \ge\gamma(2xq)\ge\gamma m,
+\]
+so Theorem~\ref{thm:grouping} provides pairwise disjoint connected sets
+in $H-U$ containing these groups.
+
+For each label, unite its connected set in $H-U$ with all its assigned
+private bags and their port edges. This set is nonempty and connected.
+Different labels give disjoint sets: their connected sets in $H-U$
+are disjoint, all private bags are disjoint, and the bags lie in $U$.
+Every two label sets are adjacent because their labels occur together
+in an assigned clique model. These sets form a $K_t$ model in $P$,
+contrary to $h(P)<t$. Some outcome of the sampling experiment must
+therefore have fractional chromatic number at most $r/2$, as required.
+\end{proof}
+
+\subsection{Assigning the colors to bins}
+
+\begin{proof}[Proof of Proposition~\ref{prop:small-list}]
+Fix a sufficiently large integer $t$, and let $G$ satisfy the hypotheses
+of the proposition. The empty graph needs no colors, so suppose that
+$G$ is nonempty. Set
+\[
+ r_j=4t2^{-j},\qquad y_j=\max\{1,t/r_j\},
+\]
+and let $J$ be the largest nonnegative integer for which
+$r_J\ge t/\log t$. For each $j\in\{0,\ldots,J\}$, allocate a separate
+block of
+\[
+ N_j=\lceil Br_j\log(2y_j)\rceil
+\]
+bins. Allocate a further $16t$ bins for cleanup. All these bins are
+distinct. Their total number $n_{\mathrm{bin}}$ is at most a fixed
+constant times $t$. Indeed, $y_j\le2^j$, so
+\begin{align*}
+ n_{\mathrm{bin}}
+ &\le4Bt\log2\sum_{j\ge0}(j+1)2^{-j}+(J+1)+16t\\
+ &=16B(\log2)t+(J+1)+16t.
+\end{align*}
+Also $J+1=O(\log\log t)$, since $2^J\le4\log t$.
+Choose an integer
+\[
+ D_{\mathrm{bin}}\ge16B\log2+18.
+\]
+For all sufficiently large $t$, we have
+$n_{\mathrm{bin}}\le D_{\mathrm{bin}}t$.
+Finally choose a positive integer $L$ such that
+\begin{equation}\label{small:L-choice}
+ \frac{L}{2D_{\mathrm{bin}}}>\log2+1.
+\end{equation}
+
+Let $\mathcal L$ be any finite list assignment on $G$ with
+$|\mathcal L(v)|\ge Lt$ at every vertex. Map each color in the finite
+union $\bigcup_v\mathcal L(v)$ independently and uniformly to one of
+the $n_{\mathrm{bin}}$ bins. We claim that a map can be fixed such that
+every vertex has an allowable color in at least half the bins of
+every block, including cleanup.
+
+For a fixed vertex and a block of $u$ bins, failure of this property
+implies that a subset of $\lceil u/2\rceil$ bins is missed by its list.
+For a specified such subset, the probability of being missed is at most
+\[
+ \left(1-\frac{u}{2n_{\mathrm{bin}}}\right)^{|\mathcal L(v)|}
+ \le\exp\left(-\frac{Lt\,u}{2n_{\mathrm{bin}}}\right)
+ \le\exp\left(-\frac{Lu}{2D_{\mathrm{bin}}}\right).
+\]
+There are at most $2^u$ choices for the missed subset, so
+\eqref{small:L-choice} bounds the failure probability by $e^{-u}$.
+Every nonfinal block has
+$u\ge B(\log2)t/\log t$, and the cleanup block has $u=16t$.
+Thus, with $c_B=\min\{B\log2,16\}>0$ and $\log t\ge1$, a union bound
+over all vertices and blocks gives failure probability at most
+\[
+ (J+2)t^{11/10}\exp\left(-\frac{c_Bt}{\log t}\right)=o(1).
+\]
+For sufficiently large $t$, the desired map therefore exists; fix one.
+This probability estimate depends on the list sizes, but not on the
+number of distinct colors in their union.
+
+Call a vertex eligible for a bin if its list has a color assigned to
+that bin. It remains eligible whenever it survives to a later stage.
+At the start, Theorem~\ref{thm:fractional} gives $f(G)\le4t=r_0$.
+At stage $j$, let $P$ be the current induced remainder. If it is empty,
+stop. Otherwise $h(P)<t$, $|V(P)|\le t^{11/10}$, and inductively
+$f(P)\le r_j$. For each bin of that stage, take its eligible vertex set
+inside $P$. Every vertex belongs to at least half these sets.
+Lemma~\ref{small:eligible-thinning}, with $n=t^{11/10}$, supplies a
+stable set for each bin such that deleting the eligible covered
+vertices leaves fractional chromatic number at most
+$r_j/2=r_{j+1}$. Retain one covering bin for every vertex removed at
+this stage.
+
+After stage $J$, the remaining graph has fractional chromatic number
+at most $r_{J+1}<t/\log t$. If it is nonempty, put $\rho=t/\log t$ and
+independently sample a stable set in each cleanup bin from a fractional
+coloring distribution having every vertex inclusion probability at
+least $1/\rho$. Such a distribution exists because $f\le\rho$.
+Every remaining vertex is eligible in at least $8t$ cleanup bins, so
+its probability of being missed by all eligible samples is at most
+\[
+ (1-1/\rho)^{8t}\le\exp(-8t/\rho)=t^{-8}.
+\]
+The probability that any remaining vertex is missed is therefore at
+most $t^{11/10}t^{-8}=t^{-69/10}<1$. Choose an outcome covering every
+remaining vertex and again retain one covering bin for each vertex.
+
+Every vertex now has a retained bin and belongs to the stable set
+associated with that bin. Choose any color in its original list which
+maps to that bin. If two vertices receive the same color, the global
+map sends that color to their common retained bin. Both vertices lie
+in that bin's stable set and are therefore nonadjacent. This is a
+proper $\mathcal L$-coloring of $G$.
+
+The choices of constants were made in the order $\gamma$, $B$,
+$D_{\mathrm{bin}}$, and $L$. All subsequent cutoffs depend only on these
+constants, and every thinning estimate is uniform over the intermediate
+graphs and eligibility sets. Enlarging one absolute $t_0$ to satisfy
+the finitely many cutoff requirements proves the proposition.
+\end{proof}
diff --git a/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/sections/04-list-transfer.tex b/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/sections/04-list-transfer.tex
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+\section{Extraction and additive deletion for lists}
+\label{sec:list-transfer}
+
+We first establish a list-coloring form of connectivity extraction. We then
+show that deleting a small set together with a suitably sparse set costs only
+linearly many colors. Throughout this section, list assignments may have
+arbitrary finite palettes.
+
+\begin{lemma}\label{list:elementary}
+The following hold for every finite graph $P$.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item If $H$ is a subgraph of $P$, then $\ell(H)\le \ell(P)$.
+\item For every $Z\subseteq V(P)$,
+\[
+\ell(P-Z)\ge \ell(P)-|Z|.
+\]
+\item If $P$ is $q$-degenerate, where $q$ is a nonnegative integer, then
+$\ell(P)\le q+1$.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+For the first assertion, extend any list assignment on $H$ with lists of
+size at least $\ell(P)$ to the other vertices of $P$, retaining the given
+lists on $V(H)$. A list coloring of $P$ restricts to one of $H$; additional
+edges among $V(H)$ cause no difficulty.
+
+For the second assertion, write $s=\ell(P-Z)$ and give every vertex of $P$
+a list of size at least $s+|Z|$. Greedily color $P[Z]$. At any such step
+there are at most $|Z|-1$ colors to avoid. Delete from each remaining list
+the colors used on its neighbors in $Z$. At most $|Z|$ entries are removed,
+so $P-Z$ can be colored from the remaining lists. This proves
+$\ell(P)\le s+|Z|$; the empty-graph cases hold as well under our convention.
+
+For the third assertion, repeatedly remove a vertex of degree at most $q$
+from the current graph. In the reverse order, every vertex has at most $q$
+already colored neighbors, so lists of size at least $q+1$ suffice.
+\end{proof}
+
+We use the constraint-budget method of Gir\~ao and
+Narayanan~\cite{GiraoNarayanan2022}, with slack chosen to obtain the
+extraction estimate needed here; see also \cite[Section~3.4]{Postle2020}.
+The precoloring invariant pays for the
+constraints introduced when a separator is colored. A proper list
+precoloring on $A$ means a proper coloring of $P[A]$ assigning to each
+$v\in A$ a color in its given list.
+
+\begin{lemma}\label{list:constrained-coloring}
+Let $k\ge1$ and $m\ge0$ be integers. Suppose every induced $k$-connected
+subgraph of $P$ has list chromatic number at most $m$. Give each vertex $v$
+a list $\mathcal L(v)$ of size at least $m+6k$. Suppose also that:
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item a set $A\subseteq V(P)$ has a proper list precoloring $\varphi$;
+\item for each $v\notin A$, a finite set $B_v$ of forbidden colors is
+specified, with $|B_v|<k$;
+\item the constraints satisfy
+\begin{equation}\label{list:eq-budget}
+k|A|+\sum_{v\notin A}|B_v|\le2k^2.
+\end{equation}
+\end{enumerate}
+Then $P$ has a proper $\mathcal L$-coloring extending $\varphi$ and avoiding
+$B_v$ at every $v\notin A$.
+\end{lemma}
+
+\begin{proof}
+We induct on $|V(P)|$. The hypothesis about induced $k$-connected subgraphs
+is inherited by induced subgraphs of $P$. The budget gives $|A|\le2k$.
+
+If $|V(P)|\le2k$, extend $\varphi$ greedily. At each vertex there are at
+most $2k-1$ colors used by already colored vertices and at most $k-1$
+forbidden colors. Since the lists have size at least $m+6k$, a choice is
+available. This includes the empty graph.
+
+If $P$ is $k$-connected, then $\ell(P)\le m$ by hypothesis. For every
+$v\notin A$, remove $B_v$ and the colors on its precolored neighbors from
+$\mathcal L(v)$. The remaining list has size at least
+\[
+m+6k-|A|-|B_v|\ge m+3k+1\ge m.
+\]
+By Lemma~\ref{list:elementary}, $\ell(P-A)\le m$, so these lists color
+$P-A$. Combining this coloring with $\varphi$ proves the assertion in
+this case. If $P-A$ is empty there is nothing to extend.
+
+It remains to consider $|V(P)|>2k$ when $P$ is not $k$-connected. Choose
+a partition
+\[
+V(P)=V_1\,\dot\cup\,Z\,\dot\cup\,V_2,
+\]
+where $V_1,V_2$ are nonempty, there are no edges between them, and
+$|Z|<k$. Assign to each vertex its constraint cost
+\[
+w(v)=
+\begin{cases}
+k,&v\in A,\\
+|B_v|,&v\notin A,
+\end{cases}
+\qquad
+w(U)=\sum_{v\in U}w(v).
+\]
+Interchange the sides if necessary so that $w(V_1)\ge w(V_2)$.
+We color the side with the larger constraint cost first, leaving the
+other side to absorb the cost of precoloring the whole separator.
+In particular,
+\begin{equation}\label{list:eq-second-side-cost}
+w(V_2)\le k^2.
+\end{equation}
+
+We first color $P_1=P[V_1\cup Z]$. Retain the precoloring on
+$A_1=A\cap(V_1\cup Z)$. At its other vertices use forbidden sets
+\[
+B'_v=
+\begin{cases}
+B_v\cup\{\varphi(x):x\in A\cap V_2,\ xv\in E(P)\},&v\in Z\setminus A,\\
+B_v,&v\in V_1\setminus A.
+\end{cases}
+\]
+These extra prohibitions make the eventual separator coloring compatible
+with the precolored vertices on the other side. Their total cost obeys
+\begin{align}
+k|A_1|+\sum_{v\in V(P_1)\setminus A_1}|B'_v|
+&\le w(V(P))-w(V_2)+|Z|\,|A\cap V_2|\notag\\
+&\le w(V(P))-k|A\cap V_2|+(k-1)|A\cap V_2|\notag\\
+&\le2k^2.
+\label{list:eq-first-side-cost}
+\end{align}
+Repeated occurrences of the same color can only reduce the left side.
+
+The sets $B'_v$ need not all have size less than $k$, but each satisfies
+\[
+|B'_v|\le(k-1)+|A\cap V_2|\le3k-1.
+\]
+We now convert every vertex with $|B'_v|\ge k$ into a precolored vertex.
+Process these vertices one at a time. Choose a color from $\mathcal L(v)$
+avoiding $B'_v$ and all colors on its currently precolored neighbors in
+$P_1$, and then discard the forbidden set at this newly precolored vertex.
+Its contribution to the constraint cost changes from $|B'_v|$ to $k$, so
+the total cost does not increase. In particular, throughout this procedure
+there are at most $2k$ precolored vertices. Each choice therefore avoids
+at most
+\[
+(3k-1)+2k=5k-1<m+6k
+\]
+colors and is possible. Every promoted vertex lies in $Z\setminus A$,
+since the original forbidden sets on $V_1\setminus A$ had size less than
+$k$. Its chosen color also avoids all precolored neighbors in $V_2$, by
+the definition of $B'_v$.
+
+After the conversions, the constraints on $P_1$ satisfy all the hypotheses
+of the lemma: its precoloring is proper, its remaining forbidden sets have
+size less than $k$, and its total cost is at most $2k^2$.
+Since $V_2$ is nonempty, $|V(P_1)|<|V(P)|$, and induction supplies a
+coloring $\psi_1$ of $P_1$. It respects all the original constraints in
+$P_1$. It also avoids every color $\varphi(x)$ at a separator neighbor of
+$x\in A\cap V_2$: this was enforced either by an original precoloring,
+by a conversion, or by a retained forbidden set.
+
+Next consider $P_2=P[V_2\cup Z]$. Precolor $Z$ according to $\psi_1$ and
+retain $\varphi$ on $A\cap V_2$. The resulting precoloring on
+$Z\cup(A\cap V_2)$ is proper by the preceding compatibility property.
+For each vertex of $V_2\setminus A$, retain its original forbidden set
+$B_v$. The new constraint cost is
+\[
+k|Z|+k|A\cap V_2|+
+\sum_{v\in V_2\setminus A}|B_v|
+=k|Z|+w(V_2)
+\le k(k-1)+k^2<2k^2,
+\]
+using \eqref{list:eq-second-side-cost}. All its remaining forbidden sets
+have size less than $k$. Since $V_1$ is nonempty, induction applies to
+$P_2$. Its coloring agrees with $\psi_1$ on $Z$. The two colorings therefore
+combine into a proper coloring of $P$, because there are no edges between
+$V_1$ and $V_2$. They extend the original precoloring and respect every
+original forbidden set.
+\end{proof}
+
+\begin{lemma}[List connectivity extraction]\label{lem:list-extraction}
+For every integer $k\ge1$, if $\ell(P)>6k$, then some induced
+$k$-connected subgraph $H$ of $P$ satisfies
+\[
+\ell(H)\ge\ell(P)-6k.
+\]
+\end{lemma}
+
+\begin{proof}
+Suppose otherwise and put $m=\ell(P)-6k-1$, a nonnegative integer.
+Every induced $k$-connected subgraph then has list chromatic number at
+most $m$. Apply Lemma~\ref{list:constrained-coloring} with no precolored
+vertices and no forbidden colors. It says that every list assignment of
+size at least $m+6k=\ell(P)-1$ is colorable, contradicting the definition
+of $\ell(P)$.
+\end{proof}
+
+The deletion estimate will reserve colors for a small vertex set. Only
+vertices with many neighbors in that set need a probabilistic upper bound
+on the number of reserved colors in their lists. Minor exclusion gives a
+bound on their number independent of the order of the whole graph.
+
+\begin{lemma}\label{list:high-neighbor-count}
+Let $t\ge2$ be an integer, let $h(P)<t$, and let $X\subseteq V(P)$. Then
+\[
+\bigl|\{v\in V(P)\setminus X:|N_P(v)\cap X|\ge t\}\bigr|
+\le\binom{|X|}{2}.
+\]
+\end{lemma}
+
+\begin{proof}
+Process the indicated outside vertices in any order. Assign each one an
+as yet unused unordered pair of its neighbors in $X$. Suppose this fails
+when processing a vertex $v$. Choose $t$ distinct vertices in
+$N_P(v)\cap X$. Every pair of the chosen vertices has already been
+assigned to a different previously processed outside vertex adjacent to
+both. The corresponding paths of length two have mutually distinct
+internal vertices, all outside $X$. They form a subdivision of $K_t$ and
+hence give a $K_t$ minor, a contradiction. Thus the assignment never fails
+and is an injection into the unordered pairs from $X$.
+\end{proof}
+
+\begin{lemma}[Additive list-coloring loss]\label{lem:additive-deletion}
+Let $L\ge1$ be the integer constant in Proposition~\ref{prop:small-list}.
+For each fixed integer $d\ge0$, put
+\[
+c_d=4L+d+5.
+\]
+For every sufficiently large integer $t$, the following holds. Suppose
+$h(P)<t$ and $X,Y\subseteq V(P)$ are disjoint sets such that
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item $|X|\le t^{11/10}$;
+\item $P[Y]$ is $dt$-degenerate;
+\item every vertex outside $X\cup Y$ has at most $dt$ neighbors in $Y$.
+\end{enumerate}
+Then
+\[
+\ell(P-X-Y)\ge\ell(P)-c_dt.
+\]
+The cutoff is independent of $P$, its order, and all list assignments and
+their palettes.
+\end{lemma}
+
+\begin{proof}
+Write $R=P-X-Y$ and $s=\ell(R)$, and set
+\[
+u=s+(4L+d+5)t.
+\]
+We show that $P$ is $u$-choosable. Given any lists of size at least $u$,
+first restrict each list to exactly $u$ entries. Denote the resulting
+assignment by $\mathcal L$. Its total palette
+$\mathcal C=\bigcup_{v\in V(P)}\mathcal L(v)$ is finite, but its size is
+otherwise unrestricted.
+
+Independently for each color in $\mathcal C$, reserve that color with
+probability
+\[
+p=\frac{2Lt}{u}.
+\]
+Since $u\ge(4L+d+5)t>2Lt$, this is a probability strictly between zero and
+one. Reservation is global: all occurrences of a color are reserved
+together. For each vertex $v$, the number $Z_v$ of reserved entries in
+$\mathcal L(v)$ has distribution $\operatorname{Bin}(u,p)$ and mean
+$\mu=2Lt$. This mean is independent of $s$, and overlaps between
+different lists do not affect this marginal distribution.
+
+For completeness, for every real $\lambda$ its exponential moment obeys
+\[
+\mathbb E e^{\lambda Z_v}
+=(1-p+pe^\lambda)^u
+\le\exp\bigl(\mu(e^\lambda-1)\bigr).
+\]
+Markov's inequality with $\lambda=-\log2$ and $\lambda=\log2$,
+respectively, gives
+\begin{align}
+\Pr(Z_v<\mu/2)
+&\le\exp\bigl(-\tfrac{1-\log2}{2}\mu\bigr),
+\label{list:eq-lower-tail}\\
+\Pr(Z_v>2\mu)
+&\le\exp\bigl(-(2\log2-1)\mu\bigr).
+\label{list:eq-upper-tail}
+\end{align}
+
+Let
+\[
+T=\{v\in V(P)\setminus X:|N_P(v)\cap X|\ge t\}.
+\]
+By Lemma~\ref{list:high-neighbor-count}, $|T|\le\binom{|X|}{2}$. Put
+$\eta=\min\{(1-\log2)/2,\,2\log2-1\}>0$. A union bound using
+\eqref{list:eq-lower-tail} and \eqref{list:eq-upper-tail} shows that the
+probability of either of the failures
+\[
+Z_x<Lt\quad\text{for some }x\in X,
+\qquad
+Z_v>4Lt\quad\text{for some }v\in T
+\]
+is at most
+\[
+(|X|+|T|)e^{-2\eta Lt}
+\le2t^{11/5}e^{-2\eta Lt}<1
+\]
+for sufficiently large $t$. There are only polynomially many tested
+vertices even if $P$ has arbitrarily many vertices. No independence
+between different vertices' failure events is required. Fix a reservation
+for which neither failure occurs.
+
+Use the reserved entries to color $P[X]$. They provide at least $Lt$
+colors at each of its vertices, and Proposition~\ref{prop:small-list}
+applies because $|X|\le t^{11/10}$ and $h(P[X])<t$. If $X$ is empty this
+step is vacuous. Write the coloring as $\varphi_X$.
+
+For each vertex $v\notin X$, delete from its list the colors on its
+neighbors in $X$:
+\[
+\mathcal L_1(v)=\mathcal L(v)\setminus
+\{\varphi_X(x):x\in N_P(v)\cap X\}.
+\]
+If $v\in T$, every deleted entry is reserved, so at most $4Lt$ entries
+are deleted. If $v\notin T$, fewer than $t$ entries are deleted because
+$v$ has fewer than $t$ neighbors in $X$. In particular, the uniform bound
+\begin{equation}\label{list:eq-first-pruning}
+|\mathcal L_1(v)|\ge u-(4L+1)t=s+(d+4)t
+\end{equation}
+holds. We have removed only colors actually used on neighbors in $X$;
+there is no need to discard all reserved entries at the other vertices.
+
+The bound in \eqref{list:eq-first-pruning} is at least $dt+1$. By
+Lemma~\ref{list:elementary}, the $dt$-degenerate graph $P[Y]$ therefore
+has a coloring $\varphi_Y$ from these lists. This coloring has no conflict
+with $\varphi_X$.
+
+Finally, at each vertex of $R$ delete from $\mathcal L_1(v)$ the colors
+used by its neighbors in $Y$. There are at most $dt$ such neighbors, so
+its resulting list has size at least
+\[
+u-(4L+1)t-dt=s+4t\ge s.
+\]
+The definition of $s=\ell(R)$ supplies a coloring of $R$ from these
+lists, with nothing to do if $R$ is empty. The three colorings combine
+properly, since both pruning steps excluded colors used on neighbors in
+the previously colored parts. We have colored every $u$-list assignment
+on $P$, proving
+\[
+\ell(P)\le u=\ell(P-X-Y)+c_dt.
+\]
+Only the fixed cutoff in Proposition~\ref{prop:small-list} and the last
+exponential-versus-polynomial estimate impose a sufficiently-large-$t$
+requirement. Neither depends on the graph order or palette size.
+\end{proof}
diff --git a/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/sections/05-density.tex b/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/sections/05-density.tex
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+\section{Density and small connected districts}
+\label{sec:density}
+
+The sharp order of growth of the density threshold for clique minors
+was established by Kostochka \cite{Kostochka1982,Kostochka1984} and
+Thomason \cite{Thomason1984}. We prove below a coarser estimate sufficient for
+our purpose. We need two consequences of density. The incidence estimate of
+Lemma~\ref{lem:incidence} will control exceptional vertices in the
+path-history bound, Lemma~\ref{separate:footprint}, after paths are
+contracted. Lemma~\ref{lem:district} supplies small highly connected
+districts in which the later constructions can route paths and build
+clique models.
+
+Write \(e(R)=|E(R)|\) and \(d(R)=e(R)/|V(R)|\) for a nonempty graph \(R\).
+Throughout this section, fix
+\[
+ \epsilon=\frac1{200}.
+\]
+All assertions involving sufficiently large \(t\) have a cutoff independent of
+the graph. When a fixed parameter \(K\) is present, the cutoff may depend on
+\(K\). An upper bound on an integer order by a real power of \(t\) is understood
+as an ordinary real inequality.
+
+\begin{lemma}\label{lem:coarse-density}
+There is an absolute constant \(C_*>0\), for example \(C_*=10^4\),
+such that, for every integer \(t\ge2\)
+and every nonempty graph \(P\) with \(h(P)<t\),
+\[
+ d(P)\le C_*t\log(2t).
+\]
+Consequently, every graph \(P\) with \(h(P)<t\) satisfies
+\[
+ \ell(P)\le 2C_*t\log(2t)+1.
+\]
+\end{lemma}
+
+\begin{proof}
+Choose a nonempty minor \(R\) of \(P\) whose density \(d\) is as large as
+possible. If \(d<10^4\), the claimed density estimate holds with
+\(C_*=10^4\). We may therefore assume \(d\ge10^4\).
+Put \(n=|V(R)|\). The graph \(R\) has no isolated vertex, since deleting
+one would increase its positive density.
+Contracting an edge \(xy\) with \(c_{xy}\) common neighbors
+loses exactly \(1+c_{xy}\) edges. The maximality of \(d\) gives
+\[
+ \frac{dn-1-c_{xy}}{n-1}\le d,
+ \qquad\text{and hence}\qquad c_{xy}\ge d-1.
+\]
+There is a vertex \(v\) of degree at most \(2d\). Therefore
+\(J=R[N_R(v)]\) has at most \(2d\) vertices and minimum degree at least \(d-1\).
+
+We first record a bound for connected dominating sets in any connected graph
+\(C\) of order at most \(2d\) and minimum degree at least \(d/2\).
+Take
+\[
+ m=\lceil4\log(2d)\rceil+1
+\]
+independent uniform samples from \(V(C)\). A fixed vertex has a neighbor in
+each draw with probability at least \(1/4\), so the probability that some
+vertex has no sampled neighbor is at most
+\[
+ 2d(3/4)^m\le 2d\exp(-m/4)<1.
+\]
+Consequently, some choice of at most \(m\) sampled vertices dominates \(C\).
+The diameter of \(C\) is at most \(11\): along a shortest path, the open
+neighborhoods of the vertices at positions \(0,3,6,\ldots\) are pairwise
+disjoint, each has at least \(d/2\) vertices, and the graph has at most \(2d\)
+vertices. There are at most four such positions. Joining each sampled
+vertex to one fixed sampled vertex along a shortest path therefore gives
+a connected dominating set of size at most
+\(11m\le66\log(2d)\), which is at most
+\[
+ M=\lceil100\log(2d)\rceil.
+\]
+We have \(M\le101\log(2d)\) and \(d\ge6M\).
+For the latter inequality, \(x/\log(2x)\) is increasing for \(x\ge10^4\),
+and \(10^4>606\log(20000)\).
+
+We now construct
+\[
+ q=\left\lfloor\frac{d}{3M}\right\rfloor
+\]
+pairwise adjacent, disjoint connected sets. Start with \(J_0=J\).
+At step \(i\), choose a component \(C_i\) of \(J_i\), take a connected
+dominating set \(B_i\) of \(C_i\) of size at most \(M\), and put
+\(J_{i+1}=C_i-B_i\).
+For every retained vertex, passing to a component deletes no neighbor.
+Its neighbors in the original graph \(J\) that have disappeared are
+therefore all in previously chosen bags. As long as at most \(qM\le d/3\)
+bag vertices have been deleted, each retained vertex has degree at least
+\[
+ d-1-qM\ge d/2.
+\]
+In particular, each chosen component has at least \(d/2+1\) vertices,
+whereas \(M\le d/6\); its deletion of \(B_i\) leaves a nonempty graph.
+The construction consequently continues for all \(q\) steps.
+
+Every later bag lies inside \(C_i-B_i\), so it has an edge to \(B_i\)
+by domination. The bags form a \(K_q\) model. Since \(d/(3M)\ge2\),
+\[
+ h(P)\ge q\ge\frac{d}{6M}
+ \ge\frac{d}{606\log(2d)}.
+\]
+The hypothesis \(h(P)<t\) implies \(d/\log(2d)<606t\).
+The elementary inequality \(\log(2d)\le2\sqrt d\) for \(d\ge1\) first
+gives \(d<(1212t)^2\). Since \(2\cdot1212^2<2^{22}\) and \(t\ge2\),
+\[
+ \log(2d)<22\log2+2\log t\le12\log(2t).
+\]
+It follows that \(d<7272t\log(2t)\). Together with the case \(d<10^4\)
+and the inequality \(d(P)\le d\), this proves the density claim with
+\(C_*=10^4\).
+
+Every nonempty subgraph of \(P\) has average degree at most
+\(2C_*t\log(2t)\), and hence has a vertex of degree at most
+\(\lfloor2C_*t\log(2t)\rfloor\).
+Successively deleting such vertices and coloring in the reverse order
+proves the asserted list-coloring bound. The empty graph causes no exception.
+\end{proof}
+
+\begin{lemma}\label{lem:incidence}
+There is an absolute positive integer \(B_*\) such that, for all sufficiently
+large integers \(t\), every graph \(P\) with \(h(P)<t\) and disjoint vertex
+sets \(A,T\) satisfies
+\begin{equation}\label{density:incidence-bound}
+ \sum_{\substack{v\in T\\ |N_P(v)\cap A|\ge B_*t}}
+       |N_P(v)\cap A|
+ \le t^{1+2\epsilon}|A|.
+\end{equation}
+In particular,
+\[
+ \bigl|\{v\in T:|N_P(v)\cap A|\ge B_*t\}\bigr|
+ \le t^{2\epsilon}|A|.
+\]
+\end{lemma}
+
+\begin{proof}
+Choose a fixed positive even integer \(s\) with \(\epsilon s/2>4\), and
+put \(B_*=2s\). We first show that, in any graph without a \(K_t\) minor,
+every set \(U\) of size \(b\ge B_*t\) contains a vertex with at least
+\(t^{-\epsilon}b\) nonneighbors in \(U\) distinct from itself.
+Suppose otherwise, and set \(\delta=t^{-\epsilon}\).
+Sample an ordered sequence of \(st\) distinct vertices of \(U\) uniformly,
+and divide its positions into \(t\) batches of size \(s\).
+
+After any previously exposed sampled positions have been fixed, a further
+position is uniform among at least \(b-st\ge b/2\) remaining vertices.
+For any already exposed vertex, the conditional probability that this
+new vertex is a nonneighbor is consequently at most \(2\delta\).
+Consider a fixed nontrivial cut of one batch. Expose a fixed position on
+its smaller side first, and then the positions on its larger side.
+If the cut has no crossing edge, at least \(s/2\) successive exposures
+must all be nonneighbors of the first vertex. The probability is at most
+\((2\delta)^{s/2}\). Taking the union over at most \(2^s\) cuts, the probability
+that the batch is disconnected is at most \(2^s(2\delta)^{s/2}\).
+For two fixed batches to be anticomplete, every position of the second
+batch must be a nonneighbor of one fixed position of the first; this
+has probability at most \((2\delta)^s\).
+Thus the probability that some batch is disconnected or some two batches
+are anticomplete is at most
+\[
+ t2^s(2t^{-\epsilon})^{s/2}
+ +\binom t2(2t^{-\epsilon})^s=o(1).
+\]
+For large \(t\), the batches can therefore be chosen connected and pairwise
+adjacent, giving a \(K_t\) model. This contradiction proves the claim.
+
+Let \(T_0\) consist of the vertices appearing in the sum in
+\eqref{density:incidence-bound}. If \(A\) or \(T_0\) is empty, there is
+nothing to prove. Otherwise, delete all vertices outside \(A\cup T_0\)
+and all edges with both ends in \(T_0\).
+Process the vertices of \(T_0\) one at a time. Throughout the procedure,
+the vertices representing \(A\) carry a graph that only gains edges.
+Each unprocessed vertex retains precisely its original neighborhood in
+\(A\), because there were no edges between vertices of \(T_0\).
+
+For an unprocessed vertex \(v\), put \(b_v=|N_P(v)\cap A|\).
+Apply the preceding claim to its neighborhood in the current graph on
+\(A\), which is a subgraph of a minor of \(P\). Choose there a vertex
+with at least \(t^{-\epsilon}b_v\) missing neighbors in that neighborhood,
+and contract \(v\) onto this vertex. This operation adds at least
+\(t^{-\epsilon}b_v\) new edges to the graph on \(A\).
+After all vertices have been processed, the resulting graph on \(A\)
+is a minor of \(P\). By Lemma~\ref{lem:coarse-density},
+\[
+ t^{-\epsilon}\sum_{v\in T_0}b_v
+ \le e_{\mathrm{final}}-e(P[A])
+ \le C_*t\log(2t)|A|.
+\]
+For sufficiently large \(t\), \(C_*\log(2t)\le t^\epsilon\), proving
+\eqref{density:incidence-bound}. Dividing by \(B_*t\), and then using
+\(B_*\ge1\), proves the last assertion.
+\end{proof}
+
+The following density-to-connectivity lemma is Mader's theorem
+\cite{Mader1972}; see also \cite[Lemma~4.4]{DelcourtPostle2021}.
+We include the elementary proof needed here.
+
+\begin{lemma}\label{density:connectivity}
+For every positive integer \(k\), a graph of density greater than \(2k\)
+contains an induced \(k\)-connected subgraph.
+\end{lemma}
+
+\begin{proof}
+Choose an induced subgraph \(H\) of smallest order \(v\ge2k\) such that
+\[
+ e(H)>2k(v-k).
+\]
+Such a subgraph exists: the given graph has density greater than \(2k\)
+and, being simple, has more than \(4k+1\) vertices.
+The equality \(v=2k\) is impossible because
+\(\binom{2k}{2}<2k^2\).
+Deleting any one vertex therefore leaves at least \(2k\) vertices,
+and minimality implies
+\[
+ \deg_H(x)
+ =e(H)-e(H-x)
+ >2k(v-k)-2k(v-1-k)=2k
+\]
+for every \(x\in V(H)\).
+
+Suppose that \(H\) is not \(k\)-connected. There is a separator \(Z\)
+of size \(z<k\) and nonempty anticomplete sides \(A,B\) partitioning
+\(V(H)\setminus Z\).
+Every neighbor of a vertex of \(A\) lies in \(A\cup Z\), and likewise
+for \(B\). Both \(H[A\cup Z]\) and \(H[B\cup Z]\) therefore have at
+least \(2k\) vertices and are proper induced subgraphs of \(H\).
+Their minimality bounds give
+\[
+ \begin{aligned}
+ e(H)
+ &=e(H[A\cup Z])+e(H[B\cup Z])-e(H[Z])\\
+ &\le2k\bigl(|A|+z-k+|B|+z-k\bigr)-e(H[Z])\\
+ &\le2k(v+z-2k)
+ <2k(v-k),
+ \end{aligned}
+\]
+a contradiction. Thus \(H\) is \(k\)-connected.
+\end{proof}
+
+The next argument uses the maximal-contraction method of Delcourt and
+Postle~\cite{DelcourtPostle2021}. The edge-loss estimates below give the
+particular order and connectivity bounds needed here.
+
+\begin{lemma}\label{lem:district}
+For every fixed positive integer \(K\), there is an integer
+\[
+ D_K>\max(400B_*,12800K)
+\]
+such that, for all sufficiently large integers \(t\), every nonempty
+graph \(P\) with
+\[
+ h(P)<t,\qquad d(P)\ge D_Kt
+\]
+contains an induced \((Kt)\)-connected subgraph of order at most
+\(t^{1.06}\).
+\end{lemma}
+
+\begin{proof}
+Choose an integer \(D_K\) satisfying the displayed inequality, and keep
+it fixed. Write \(n=|V(P)|\). Delete edges to obtain a spanning subgraph
+\(P_0\) with exactly \(dn\) edges, where \(d=D_Kt\).
+This is possible because \(D_Ktn\) is an integer.
+Restoring deleted edges at the end preserves the connectivity of any
+induced subgraph. Put
+\[
+ p_0=\lfloor t^{5\epsilon}\rfloor,\qquad
+ \lambda=\frac d{100}.
+\]
+We take \(t\) large enough that \(p_0\ge2\).
+
+We will find a small dense subgraph on vertices untouched by the
+contractions. First we contract a maximum family of connected
+\(p_0\)-vertex sets within an edge-loss budget. We then retain
+untouched vertices whose degrees have not fallen too far. A second
+maximal choice of a connected set will stop before reaching \(p_0\),
+since reaching that order would enlarge the first family. The failure
+of its one-vertex extensions will force a dense neighborhood, even
+after the contracted vertices are removed.
+
+Choose a family \(\mathcal C\) of as many pairwise disjoint connected
+vertex sets of \(P_0\), each of size \(p_0\), as possible subject to
+\begin{equation}\label{density:family-budget}
+ e(P_0)-e(P_0/\mathcal C)
+ \le \lambda\sum_{C\in\mathcal C}(|C|-1).
+\end{equation}
+Here \(P_0/\mathcal C\) denotes the simple graph obtained by contracting
+each set in the family and suppressing loops and parallel edges.
+The empty family is allowed, so a maximum exists.
+Let \(R=P_0/\mathcal C\), and let \(X\) be the set of contracted vertices.
+The vertices outside \(X\) are identified with their original vertices
+in \(P_0\), and edges between two such vertices are unchanged. We have
+\begin{equation}\label{density:quotient-bounds}
+ |X|\le\frac n{p_0},
+ \qquad
+ (d-\lambda)n\le e(R)\le dn.
+\end{equation}
+Indeed the total vertex loss in the contractions is at most \(n\),
+so \eqref{density:family-budget} gives the lower edge bound.
+
+Let \(B\) be the union of the following three sets:
+\[
+ X,\qquad
+ \{v\in V(R)\setminus X:|N_R(v)\cap X|\ge\lambda/4\},
+ \qquad
+ \{v\in V(R):\deg_R(v)\ge dt^{4\epsilon}\}.
+\]
+The first exclusion ensures that the vertices retained below are
+original vertices. The second limits how many neighbors a retained
+vertex can lose when \(X\) is removed from the eventual neighborhood.
+The third bounds the order of that neighborhood by bounding the degrees
+of the vertices that define it.
+Since \(\lambda/4=D_Kt/400\ge B_*t\), Lemma~\ref{lem:incidence} bounds
+the second set by \(t^{2\epsilon}|X|\).
+The degree sum and \eqref{density:quotient-bounds} bound the third
+set by \(2n/t^{4\epsilon}\).
+For sufficiently large \(t\), \(p_0\ge t^{5\epsilon}/2\), and hence
+\begin{equation}\label{density:bad-size}
+ |B|\le(1+t^{2\epsilon})\frac n{p_0}+\frac{2n}{t^{4\epsilon}}
+ \le6nt^{-3\epsilon}.
+\end{equation}
+
+By Lemma~\ref{lem:coarse-density}, the number of edges within \(B\) is
+at most
+\[
+ C_*t\log(2t)|B|
+ \le6C_*nt^{1-3\epsilon}\log(2t).
+\]
+To bound the edges from \(B\) to its complement, first count the
+vertices outside \(B\) with fewer than \(B_*t\) neighbors in \(B\):
+their contribution is at most \(B_*tn\).
+Lemma~\ref{lem:incidence} bounds all remaining crossing edges by
+\[
+ t^{1+2\epsilon}|B|\le6nt^{1-\epsilon}.
+\]
+The same bounds hold if \(B\) is empty. It follows that
+\[
+ \begin{aligned}
+ e(R-B)
+ &\ge (d-\lambda)n
+       -6C_*nt^{1-3\epsilon}\log(2t)
+       -B_*tn-6nt^{1-\epsilon}\\
+ &\ge dn\left(
+ \frac{99}{100}-\frac{B_*}{D_K}
+ -\frac{6C_*}{D_K}t^{-3\epsilon}\log(2t)
+ -\frac6{D_K}t^{-\epsilon}\right)
+ >\frac{dn}{2}
+ \end{aligned}
+\]
+for sufficiently large \(t\), since \(D_K>400B_*\).
+
+Starting with \(R-B\), repeatedly delete a vertex \(v\) whose degree
+in the current graph is less than \(d/4\), or less than
+\(\deg_R(v)/8\). Charge each deleted edge to the endpoint deleted first.
+The total charge is at most
+\[
+ \frac d4n+\frac18\sum_{v\in V(R)}\deg_R(v)
+ \le\frac{dn}{2}.
+\]
+The process cannot delete every vertex, because \(R-B\) initially
+has more than \(dn/2\) edges. The surviving nonempty set \(S\) satisfies
+\begin{equation}\label{density:surviving-degrees}
+ \deg_{R[S]}(v)\ge d/4,
+ \qquad
+ \deg_{R[S]}(v)\ge\deg_R(v)/8
+ \quad(v\in S).
+\end{equation}
+The second degree bound will let us compare a neighborhood in \(R\)
+with the part remaining in \(S\), once the loss caused by contracting
+its defining set is controlled.
+
+For a nonempty connected vertex set \(U\) in \(R\), let
+\(\mathcal L(U)\) be the number of edges lost when \(U\) is contracted
+to one vertex. If \(N_R(U)\) denotes its external neighborhood, then
+the exact loss is
+\begin{equation}\label{density:loss-identity}
+ \mathcal L(U)
+ =e(R[U])+
+ \sum_{w\in N_R(U)}\bigl(|N_R(w)\cap U|-1\bigr).
+\end{equation}
+The first term counts the internal edges that become loops.
+At each outside neighbor, all but one of its edges into \(U\)
+become parallel edges and are suppressed, giving the second term.
+
+Choose a connected set \(H\subseteq S\) of largest possible order
+\(u\le p_0\) subject to
+\begin{equation}\label{density:new-budget}
+ \mathcal L(H)\le\lambda(u-1).
+\end{equation}
+A singleton is admissible. We have \(u<p_0\).
+Indeed \(H\) contains only untouched original vertices, so it is
+connected in \(P_0\) as well. If \(u=p_0\), append it to
+\(\mathcal C\). Contracting the old family and then \(H\) gives
+the same graph as contracting the enlarged family.
+The actual edge losses add, as do the vertex losses; combining
+\eqref{density:family-budget} and \eqref{density:new-budget}
+would make this a larger admissible family, a contradiction.
+
+Write \(N=N_R(H)\), and put
+\[
+ a=\sum_{v\in H}\deg_{R[S]}(v),
+ \qquad
+ b=\sum_{v\in H}\deg_R(v).
+\]
+We first show that at least one sixteenth of \(N\) lies in \(S\).
+By \eqref{density:loss-identity},
+\[
+ \begin{aligned}
+ a-|N\cap S|
+ &=2e(R[H])
+   +\sum_{w\in N\cap S}\bigl(|N_R(w)\cap H|-1\bigr)\\
+ &\le2\mathcal L(H)\le2\lambda(u-1)\le2\lambda u.
+ \end{aligned}
+\]
+By \eqref{density:surviving-degrees},
+\(a\ge du/4=25\lambda u\) and \(a\ge b/8\).
+As \(|N|\le b\), this gives
+\begin{equation}\label{density:neighborhood-ratio}
+ |N\cap S|
+ \ge a-2\lambda u
+ \ge\frac{23}{25}a
+ \ge\frac{23}{200}b
+ \ge\frac{|N|}{16}.
+\end{equation}
+In particular \(N\cap S\) is nonempty, since \(a\ge25\lambda u>0\).
+
+Fix \(w\in N\cap S\), and contract \(H\) to a vertex \(h\).
+Contracting the edge \(hw\) next loses exactly
+\[
+ 1+|N_R(w)\cap N|
+\]
+edges: the common neighbors of \(h\) and \(w\) are precisely
+\(N_R(w)\cap N\).
+If this loss were at most \(\lambda\), the connected set
+\(H\cup\{w\}\), whose order is at most \(p_0\), would satisfy
+the budget \(\mathcal L(H\cup\{w\})\le\lambda u\).
+This would contradict the maximality of \(u\).
+Thus \(w\) has more than \(\lambda-1\) neighbors in \(N\).
+Since \(w\in S\) is not in \(B\), it has fewer than \(\lambda/4\)
+neighbors in \(X\). For \(\lambda\ge4\), it follows that
+\[
+ \deg_{R[N\setminus X]}(w)
+ >\frac{3\lambda}{4}-1
+ \ge\frac{\lambda}{2}.
+\]
+Using \(N\cap S\subseteq N\setminus X\) and
+\eqref{density:neighborhood-ratio}, we obtain
+\[
+ 2e(R[N\setminus X])
+ \ge\frac{\lambda}{2}|N\cap S|
+ \ge\frac{\lambda}{32}|N|.
+\]
+Consequently,
+\begin{equation}\label{density:small-neighborhood-density}
+ d(R[N\setminus X])
+ \ge\frac{\lambda|N|}{64|N\setminus X|}
+ \ge\frac{\lambda}{64}.
+\end{equation}
+
+Every vertex of \(H\subseteq S\) has degree less than \(dt^{4\epsilon}\)
+in \(R\). Therefore
+\[
+ |N\setminus X|
+ \le |N|
+ \le\sum_{v\in H}\deg_R(v)
+ <u\,dt^{4\epsilon}
+ \le D_Kt^{1+9\epsilon}
+ =D_Kt^{1.045}.
+\]
+Since \(D_K\) is fixed, this is at most \(t^{1.06}\) once
+\(t^{3\epsilon}\ge D_K\).
+Also,
+\[
+ \frac{\lambda}{64}=\frac{D_Kt}{6400}>2Kt.
+\]
+Lemma~\ref{density:connectivity}, applied to \(R[N\setminus X]\),
+now gives an induced \((Kt)\)-connected subgraph of order at most
+\(t^{1.06}\). All vertices in \(N\setminus X\) are untouched original
+vertices, and the edges between them in \(R\) are exactly their edges
+in \(P_0\). Restoring the deleted edges on the same vertex set gives
+the required induced subgraph of \(P\).
+\end{proof}
+
+\begin{corollary}\label{density:list-district}
+Fix \(K\) and \(D_K\) as in Lemma~\ref{lem:district}.
+For all sufficiently large integers \(t\), if
+\[
+ h(P)<t,\qquad \ell(P)>2D_Kt+2,
+\]
+then \(P\) contains an induced \((Kt)\)-connected subgraph of order
+at most \(t^{1.06}\).
+\end{corollary}
+
+\begin{proof}
+If every nonempty subgraph had a vertex of degree at most \(2D_Kt+1\),
+reverse deletion order would color arbitrary lists of size \(2D_Kt+2\).
+Thus some subgraph has minimum degree at least \(2D_Kt+2\).
+Passing to the induced subgraph on its vertices preserves this degree
+bound, so that induced subgraph has density at least \(D_Kt+1\).
+Apply Lemma~\ref{lem:district} there.
+\end{proof}
diff --git a/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/sections/06-routing.tex b/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/sections/06-routing.tex
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+\section{Routing, rooted models, and minimizing path systems}
+\label{sec:routing}
+
+The minor constructions in the next two sections require two kinds of
+control over their connecting paths. We must insert rooted clique models
+inside districts while preserving the paths' prescribed ends. We must also
+bound the list-coloring cost of deleting long path systems. We first
+establish the packing and replacement tools, then prove the sparsity bounds
+for minimum-length systems and show how to combine two families sharing
+a target set.
+
+All paths in this section are simple; a path with one vertex is allowed when
+its two prescribed ends coincide.  The length of a path is its number of
+edges.  A family of paths is vertex-disjoint unless explicitly permitted to
+share prescribed starting vertices.  When a path is oriented from its source
+to its target, its first and last vertices refer to that orientation.
+
+\subsection{Packing paths and rooting a clique minor}
+
+We begin with the vertex-capacity form of Menger's theorem
+\cite{menger1927}, proved here by the integral max-flow/min-cut argument
+of Ford and Fulkerson \cite{fordfulkerson1956}.
+
+\begin{lemma}[Vertex path packing]\label{lem:packing}
+Let $A,B$ be disjoint vertex sets in a graph $P$, and let $j\ge1$ be an
+integer.  Suppose that for every $X\subseteq V(P)$ with $|X|<j$, the graph
+$P-X$ contains a path from $A\setminus X$ to $B\setminus X$.
+Then there are $j$ pairwise vertex-disjoint paths from distinct vertices of
+$A$ to distinct vertices of $B$.  Each path can be chosen to meet $A\cup B$
+only at its two ends.
+\end{lemma}
+
+\begin{proof}
+Replace each vertex $v$ by two vertices $v^-,v^+$ and an arc
+$v^-v^+$ of capacity one.  For each edge $uv$, add arcs $u^+v^-$ and
+$v^+u^-$, both of capacity $j$.  Add a source $s$ with arcs $sa^-$ of
+capacity $j$ for $a\in A$, and a sink $z$ with arcs $b^+z$ of capacity $j$
+for $b\in B$.
+
+Start with zero flow.  Whenever the residual directed graph has an
+$s$--$z$ path, augment by one unit along it.  Residual capacities remain
+integral.  Stop upon reaching value $j$, or if no residual $s$--$z$ path
+exists.  In the latter case let $R$ be the vertices reachable from $s$ in
+the residual graph.  Every original arc from $R$ to its complement is
+saturated, and every original arc entering $R$ from its complement has
+zero flow.  Flow conservation therefore shows that the capacity of the
+outgoing cut equals the current flow value.
+
+If this value is less than $j$, no capacity-$j$ arc crosses the cut.
+Its outgoing arcs are consequently unit arcs $v^-v^+$, for a set $X$ of
+fewer than $j$ original vertices.  Any $A$--$B$ path in $P-X$ would give
+an $s$--$z$ path avoiding those cut arcs, which is impossible.  This
+contradicts the hypothesis.  Thus an integral flow of value $j$ is reached.
+
+Decompose that flow into unit $s$--$z$ paths and directed cycles, and
+discard the cycles.  Each original vertex is used by at most one of the
+unit paths because its unit arc has capacity one.  The corresponding
+paths in $P$ are therefore vertex-disjoint.  On each such path, take its
+first visit to $B$ and its last preceding visit to $A$, and retain the
+segment between them.  This gives the asserted endpoint convention
+without changing the number or disjointness of the paths.
+\end{proof}
+
+\begin{lemma}[Two paths from every source]\label{lem:double-packing}
+Let $A,B$ be disjoint vertex sets in a graph $P$, where $r=|A|\ge1$.
+If $P$ is $2r$-connected and $|B|\ge2r$, there are two paths starting at
+each vertex of $A$, ending at $2r$ distinct vertices of $B$, with no
+intersections other than the two paths sharing their prescribed start.
+Their internal vertices lie outside $A\cup B$.
+\end{lemma}
+
+\begin{proof}
+Replace each vertex of $A$ by two copies.  For every original edge, join
+all copies of one endpoint to all copies of the other, treating a vertex
+outside $A$ as having a single copy.  Write $A^*$ for the $2r$ source
+copies.  The target set $B$ is unchanged.
+
+Let $X^*$ be a set of fewer than $2r$ vertices of this new graph.
+At least one source copy and at least one target survive.  In the
+original graph delete the set $X$ of vertices all of whose copies were
+deleted.  Then $|X|\le |X^*|<2r$, so the surviving original source and
+target are connected in $P-X$.  A simple path between them lifts to a
+path in the copy graph minus $X^*$ by choosing a surviving copy at each
+vertex.  Thus $X^*$ cannot separate $A^*$ from $B$.
+
+Lemma~\ref{lem:packing} now gives $2r$ vertex-disjoint paths from $A^*$
+to $B$.  Every source copy is an endpoint of one path, so no source copy
+is internal to another path.  Identifying the two copies of each source
+therefore introduces exactly the permitted common starts and no other
+overlaps.  The endpoint convention in Lemma~\ref{lem:packing} also
+excludes internal vertices in $B$.
+\end{proof}
+
+A stronger rooted-minor theorem of Kawarabayashi is recorded in
+\cite[Lemma~5.14]{DelcourtPostle2021}. We include a proof of the form
+needed here.
+
+\begin{lemma}[Rooting a large clique minor]\label{lem:rooting}
+Let $k\ge1$.  If $P$ is $2k$-connected and $h(P)\ge4k$, then any given
+$k$ distinct vertices are roots of a $K_k$ model: its bags can be indexed
+by the given vertices so that each bag contains its indexing root.
+\end{lemma}
+
+\begin{proof}
+We use two successive path packings. The first sends two paths from each
+root towards the given clique model and leaves some model bags untouched.
+Its path--bag unions may overlap, but each vertex lies in at most two of
+them. This bound will certify a second packing to untouched bags, producing
+the disjoint rooted model.
+
+Let $R$ be the prescribed root set.  Start with $4k$ disjoint bags of a
+clique model.  At most $k$ of these bags meet $R$, so choose $3k$ bags
+disjoint from $R$ and call them eligible.  Choose one representative in
+each eligible bag.  Lemma~\ref{lem:double-packing}, applied to these
+representatives as targets, gives two paths from each root, with their
+$2k$ ends in distinct eligible bags.
+
+Among all such double systems, now allowing arbitrary endpoints within
+the eligible bags but still requiring distinct end bags, choose one of
+minimum total length.  No path meets an eligible bag that is not an end
+bag of the system: its first such visit would allow that path to be
+truncated and assigned the unused bag, strictly reducing total length.
+Consequently at least $k$ eligible bags are untouched by every path.
+Choose $k$ of them and contract each to a single target vertex; let $T$
+be the resulting set of $k$ targets.
+
+For each of the $2k$ paths of the double system, take its union with its
+own end bag.  These $2k$ connected sets are unchanged by the contractions
+just made.  Every vertex belongs to at most two of them.  Indeed, a
+vertex outside $R$ belongs to at most one path and at most one eligible
+bag, whereas a root belongs to its two paths and to no eligible bag.
+Paths may pass through other used end bags; this observation already
+allows for those intersections.
+
+In the contracted graph, delete fewer than $k$ vertices.  They meet
+fewer than $2k$ of the connected sets just described, so at least one
+such set survives intact, including its root.  At least one vertex of
+$T$ also survives.  The surviving set has an edge to every surviving
+vertex of $T$, because its end bag was adjacent to every one of the
+contracted eligible bags.  Hence these deletions cannot separate $R$
+from $T$.
+
+Apply Lemma~\ref{lem:packing} with $j=k$.  All $k$ roots and all $k$
+targets are used as endpoints, and none is internal on another path.
+Expand each contracted target bag and add to it the vertices of its
+root-to-target path.  The resulting bags are connected and disjoint.
+They remain pairwise adjacent through the original edges between the
+target bags, and each contains its assigned root.
+\end{proof}
+
+\subsection{Wovenness and replacement inside districts}
+
+Wovenness records the simultaneous rooted-model and path property used in
+Postle's framework \cite[Section~3.3]{Postle2020}; see also
+\cite[Definition~5.12]{DelcourtPostle2021}. We specify the endpoint
+coincidences because the recursive construction uses them.
+
+\begin{definition}\label{def:woven}
+For integers $a\ge1$ and $b\ge0$, a graph $J$ is \emph{$(a,b)$-woven}
+if the following holds.  Prescribe $a$ distinct roots
+$R=\{r_1,\ldots,r_a\}$ and at most $b$ pairs
+$(x_1,y_1),\ldots,(x_p,y_p)$ of vertices of $J$.  The endpoint sets
+$\{x_i,y_i\}$ must be disjoint for different $i$, but $x_i=y_i$ is
+allowed, and endpoints may belong to $R$.  Then there are a $K_a$ model
+with bags $M_1,\ldots,M_a$ satisfying $r_j\in M_j$, and pairwise
+vertex-disjoint $x_i$--$y_i$ paths $Q_i$, such that
+\[
+ V(Q_i)\cap\bigcup_{j=1}^a M_j
+ \ \subseteq\ R\cap\{x_i,y_i\}\qquad(1\le i\le p).
+\]
+For a coincident pair the path is the single prescribed vertex.
+Because the model bags are disjoint and contain all the roots, any
+allowed intersection at $r_j$ belongs to the designated bag $M_j$.
+\end{definition}
+
+We use the woven-subgraph replacement argument of
+\cite[Lemma~5.15]{DelcourtPostle2021}, recording its intersection
+conclusions at the first and last district visits of each path.
+
+\begin{lemma}[Rerouting through a woven district]\label{lem:rerouting}
+Let $J$ be an $(a,b)$-woven subgraph of a graph $P$, and prescribe $a$
+distinct roots in $J$.  Let $Q_1,\ldots,Q_q$ be vertex-disjoint paths in $P$,
+where $q\le b$.  Orient each path arbitrarily.  For every path meeting
+$J$, let $x_i,y_i$ be its first and last vertices in $J$.
+There are vertex-disjoint paths $Q'_1,\ldots,Q'_q$ with the same
+respective endpoints, and a model rooted at the prescribed roots in $J$,
+such that the following properties hold:
+\begin{enumerate}[label=(\roman*)]
+\item every vertex added to a path belongs to $J$; more precisely,
+      $V(Q'_i)\setminus V(J)\subseteq V(Q_i)\setminus V(J)$;
+\item a path originally disjoint from $J$ is unchanged;
+\item the intersection of $Q'_i$ with the model is contained in the
+      prescribed roots among $x_i,y_i$.
+\end{enumerate}
+The construction can be applied successively in disjoint districts.
+If some prescribed roots are endpoints of the original path system,
+the bags for these roots meet the resulting paths only at their own
+roots, including after all subsequent replacements in other districts.
+\end{lemma}
+
+\begin{proof}
+The pairs $(x_i,y_i)$ are admissible in Definition~\ref{def:woven}:
+their endpoint sets are disjoint across paths, and a path meeting $J$
+only once gives a permitted coincident pair.  Apply wovenness to all
+these pairs.  On each path meeting $J$, replace the entire segment from
+$x_i$ to $y_i$ by the path supplied inside $J$.  Its retained prefix and
+suffix have no vertices in $J$ except their attachment endpoints.
+They are disjoint from the interiors of all the replacement paths.
+The retained pieces of different original paths are themselves
+disjoint, and the replacement paths are disjoint by wovenness.
+The resulting concatenations are therefore simple, vertex-disjoint
+paths with the required endpoints and intersections.  Notice that a
+replaced segment may have made excursions outside $J$; those excursions
+are removed, explaining the inclusion rather than equality in (i).
+
+For successive replacements, the new vertices at a later step lie in
+a district disjoint from every earlier district and its model.  Thus
+no new intersection with an earlier model is introduced.  A prescribed
+root that is an original path endpoint remains that endpoint at every
+step.  In its own district it is a first or last visit of that path,
+and no other path contains it.  Its bag therefore has precisely the
+allowed path intersection.
+\end{proof}
+
+\begin{remark}\label{routing:padding-roots}
+The last conclusion does not require every prescribed root to be a
+path endpoint.  An auxiliary root can be a first or last visit of an
+unrelated path, and its bag can then meet that path at this root.
+If only bags whose roots are designated path endpoints are retained,
+discarding the auxiliary bags removes this issue.  A root that was
+strictly between the first and last district visits need not remain
+on that path after rerouting.
+\end{remark}
+
+Fix the positive integer $\gamma$ from Theorem~\ref{thm:grouping}, and set
+\begin{equation}\label{routing:K0}
+ K_0=10^5\gamma.
+\end{equation}
+
+\begin{lemma}[Small districts are woven]\label{lem:small-woven}
+For all sufficiently large integers $t$, every $(K_0t)$-connected graph
+$J$ of order at most $t^{1.06}$ is $(s,3t)$-woven whenever
+\[
+ 1\le s\le 2\left\lceil\frac{t}{(\log t)^2}\right\rceil.
+\]
+The cutoff is absolute and uniform in $s$ and $J$.  No minor-exclusion
+hypothesis on $J$ is required.
+\end{lemma}
+
+\begin{proof}
+We set aside the prescribed roots and pair endpoints, then split the
+remaining vertices into one part containing an unrooted clique model
+and a second, still highly connected part in which to join its bags to
+the roots and route all prescribed pairs.
+
+Fix an admissible root set and a system of at most $3t$ pairs.
+Let $Z$ be the union of the roots and all pair endpoints.  For sufficiently
+large $t$ the displayed upper bound on $s$ is at most $t$, so $|Z|\le7t$.
+Put $k'=\lfloor K_0t/2\rfloor$.  Deleting $Z$ leaves a $k'$-connected
+graph: deleting a further set of fewer than $k'$ vertices still deletes
+fewer than $K_0t$ vertices in total, and the order condition also follows
+from $|V(J)|>K_0t$.  Every vertex of $J$, including each vertex of $Z$,
+has at least $K_0t-|Z|\ge k'$ neighbors outside $Z$.
+
+Independently place each vertex of $J-Z$ in $A$ with probability $1/1000$,
+and write $B'=V(J)\setminus(Z\cup A)$.  We show that with probability
+tending to one both of the following hold:
+\begin{align}
+ |N_J(v)\cap A|&\ge k'/2000,
+ &|N_J(v)\cap B'|&\ge k'/2 &&(v\in V(J)),
+ \label{routing:split-degrees}\\
+ J[B']&\text{ is }\lfloor k'/32\rfloor\text{-connected}.
+ \label{routing:split-connectivity}
+\end{align}
+For each vertex of $J$, fix $k'$ of its neighbors outside $Z$.
+The number of these neighbors placed in $A$ is a sum of $k'$
+independent indicators with mean $k'/1000$.  By
+Lemma~\ref{lem:concentration}, the probability that it is below
+$k'/2000$ or above $k'/2$ is $\exp(-\Omega(k'))$.
+A union bound over at most $t^{1.06}$ vertices proves
+\eqref{routing:split-degrees} with probability tending to one.
+
+For \eqref{routing:split-connectivity}, form in advance the cut-test
+matchings of Lemma~\ref{lem:cut-test} for $J-Z$, with connectivity
+parameter $k'$ and order bound $n=t^{1.06}$.  The number of tests is at most
+\[
+ \exp\!\bigl(O(t^{0.06}(\log t)^2)\bigr).
+\]
+A test matching has at most $k'/4$ endpoints.  The expected number of
+its endpoints in $A$ is at most $k'/4000$, whereas the cut test allows
+fewer than $k'/32$.  This leaves a fixed positive multiple of $k'$ as
+a deviation margin.  Lemma~\ref{lem:concentration}, applied to at most
+$k'/4$ independent indicators, bounds each failure probability by
+$\exp(-\Omega(k'))$.  Their union has probability tending to zero.
+On the degree event \eqref{routing:split-degrees}, every vertex of
+$J-Z$ has at least $k'/2$ neighbors outside $A$.  Thus all hypotheses
+of Lemma~\ref{lem:cut-test} hold simultaneously with positive probability,
+and it gives \eqref{routing:split-connectivity}.  Fix such a split.
+
+The graph $J[A]$ is nonempty and has minimum degree at least $k'/2000$;
+hence its density is at least $k'/4000=\Omega(t)$.  Uniformly for the
+allowed values of $s\ge2$,
+\[
+ s\log(2s)=O\!\left(\frac{t}{\log t}\right)=o(t).
+\]
+Lemma~\ref{lem:coarse-density} therefore implies that $J[A]$ contains
+a $K_s$ model, for all sufficiently large $t$.  If $s=1$, a single
+vertex of $A$ supplies the model.  Denote the model bags by
+$M_1^0,\ldots,M_s^0$ and select a port $p_j\in M_j^0$ from each bag.
+
+Choose distinct representatives in $B'$ for all the following roles:
+each port, each root, and each of the two endpoints of every
+noncoincident prescribed pair.  A representative is required to be a
+neighbor of the vertex it represents.  When a root is also a pair
+endpoint, its two roles receive different representatives.  There are
+at most $2s+6t\le8t$ roles.  By \eqref{routing:split-degrees}, each has
+at least $k'/2$ available neighbors in $B'$, so greedy distinct selection
+is possible.
+
+Partition the representatives into groups of size two: the port and
+root representatives for each $j$, and the two endpoint representatives
+for each noncoincident pair.  By \eqref{routing:K0}, for sufficiently
+large $t$,
+\[
+ \left\lfloor\frac{k'}{32}\right\rfloor\ge \gamma(2s+6t).
+\]
+Theorem~\ref{thm:grouping} in $J[B']$ gives disjoint connected sets for
+these groups.  Write $C_j$ for the set joining the representatives of
+$p_j$ and $r_j$.  The set
+\[
+ M_j=M_j^0\cup C_j\cup\{r_j\}
+\]
+is connected, using the two representative attachment edges.  These
+sets are disjoint and remain pairwise adjacent through the original
+model on $A$.
+
+In each other grouping set choose a path joining its two representatives,
+and extend it by their attachment edges to the original pair endpoints.
+Use the trivial path for every coincident pair.  The nontrivial paths
+are disjoint inside $B'$, all their original endpoints lie in $Z$, and
+different pairs have disjoint endpoint sets.  The model bags contain
+no vertices of $Z$ except their own roots.  Consequently the only
+model--path intersections are roots that are endpoints of the respective
+pairs, exactly as in Definition~\ref{def:woven}.
+\end{proof}
+
+\subsection{Minimum-length systems with free target assignments}
+
+The bounds below use the unpaired geodesic-path argument of Postle
+\cite[Lemma~4.14]{Postle2020}, extended here to two paths at each source.
+The freedom to reassign targets is essential: the proof shortens a
+system by exchanging its path suffixes.
+
+The graph in which a path system is minimized matters: all shortcut and
+degree statements below concern edges of that same graph.  We shall use
+two kinds of systems with disjoint source and target sets $A,B$:
+\begin{enumerate}[label=\textup{(\arabic*)}]
+\item two paths from each vertex of $A$, ending at distinct vertices of
+      $B$, with intersections allowed only at their prescribed repeated
+      starts;
+\item a specified number $q$ of vertex-disjoint paths from arbitrary
+      distinct vertices of $A$ to arbitrary distinct vertices of $B$.
+\end{enumerate}
+In either case there is no prescribed pairing between sources and targets.
+For type~\textup{(1)}, the total number of paths is $q=2|A|$.
+
+\begin{lemma}[Bounds from unpaired path minimization]
+\label{lem:path-minimization}
+Let $A,B$ be disjoint vertex sets in a graph $P$.  Suppose that a system
+of one of the two types above is feasible, and choose one of minimum
+total length.  Let $q$ be its number of paths and $U$ its vertex union.
+Then:
+\begin{enumerate}[label=(\roman*)]
+\item the paths have no internal vertices in $A\cup B$;
+\item each vertex of $V(P)\setminus U$ has at most three neighbors
+      on any one path;
+\item $P[U\setminus A]$ is $2q$-degenerate for type~\textup{(1)},
+      and $P[U]$ is $2q$-degenerate for type~\textup{(2)}.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+An internal visit to $B$ would allow truncation at that visit, strictly
+shortening a path.  Its new target cannot be the target of another path,
+by disjointness.  For type~\textup{(2)}, an internal visit to $A$ likewise
+allows deletion of the initial segment, making that visit the new source.
+For type~\textup{(1)}, all sources are already prescribed starts, and
+the allowed-intersection rule excludes any internal visit to $A$.
+This proves (i).
+
+For (ii), let $v\notin U$ have at least four neighbors on a path
+$x_0x_1\cdots x_l$, ordered from source to target.  Its first and last
+neighbor indices $i<j$ satisfy $j-i\ge3$.  If $v\notin A\cup B$,
+replace the segment from $x_i$ to $x_j$ by $x_i v x_j$.  This saves at
+least one edge and introduces no intersection with another path.
+If $v\in B$, truncate the path after $x_i$ and use $v$ as its target;
+here $i+1\le l-2<l$.  If $v\in A$, the system must have type~\textup{(2)},
+since all type~\textup{(1)} sources belong to $U$.  Start instead at $v$
+and follow the suffix from $x_j$ to $x_l$; its length is
+$1+l-j\le l-2$.  These are valid systems with new free endpoint choices,
+contradicting minimality in every case.
+
+To prove (iii), put $U_0=U\setminus A$ for type~\textup{(1)} and
+$U_0=U$ for type~\textup{(2)}.  Suppose that some nonempty induced
+subgraph $P[S]$, $S\subseteq U_0$, has minimum degree greater than $2q$.
+For every path meeting $S$, mark its first two vertices in $S$, or its
+only such vertex if there is just one.  At most $2q$ vertices are marked.
+Each vertex of $S$ belongs to exactly one path: the only possible
+overlaps were the repeated starts, which were removed in type~\textup{(1)}.
+
+For a path $Q_i$ meeting $S$, let $u_i$ be its first vertex in $S$.
+Its degree in $P[S]$ is greater than $2q$, so it has a neighbor $v_i$
+in $S$ which is not marked.  This vertex belongs to some path $Q_j$
+and is at least its third vertex in $S$.  Direct an arrow from $i$ to
+$j$, recording the edge $u_i v_i$.  Every participating path has one
+outgoing arrow, so these arrows contain a directed cycle, possibly a loop.
+
+For each arrow $i\to j$ on this cycle, form a new path from the prefix
+of $Q_i$ ending at $u_i$, the edge $u_i v_i$, and the suffix of $Q_j$
+starting at $v_i$.  Leave all paths outside the cycle unchanged.
+On each cycle path $Q_j$, the retained prefix ends at $u_j$, while the
+retained suffix begins at the incoming arrow's vertex, at least the
+third vertex of $S$ on $Q_j$.  The two retained pieces are disjoint and
+separated by at least two original path edges.  Pieces lying on
+different original paths are disjoint except for the permitted repeated
+starts.  Thus the new concatenations are simple and satisfy precisely
+the original disjointness conditions.
+
+Every original source prefix is retained, so the required sources and
+their multiplicities are unchanged.  This remains true if both paths
+with the same source belong to the directed cycle: only their suffixes
+are exchanged, and their shared initial vertex remains their only
+intersection.  The target suffixes are permuted, preserving distinct
+targets.  Such a permutation is allowed because no pairing was prescribed.
+
+If the cycle has $c$ paths, at least $2c$ original edges are removed
+between their retained prefixes and suffixes, and precisely $c$ joining
+edges are inserted.  The total length decreases by at least $c>0$.
+This final contradiction proves (iii).
+\end{proof}
+
+\subsection{Combining two doubled families}
+
+\begin{lemma}[Combining paths by pairing temporary sources]
+\label{lem:combine}
+Let $Z,D,V(H)$ be pairwise disjoint vertex sets in a graph $P$, and let
+$m\ge1$.  Suppose there are:
+\begin{enumerate}[label=(\roman*)]
+\item two paths from each vertex of $Z$ to distinct vertices of $D$,
+      with all target ends distinct, with intersections only at their
+      prescribed repeated starts, and avoiding $V(H)$;
+\item $2m$ vertex-disjoint paths from distinct vertices of $H$ to
+      distinct vertices of $D$, avoiding $Z$ and meeting $H$ only at
+      their starting vertices.
+\end{enumerate}
+Then, using only vertices of these two families, one can obtain
+$|Z|+m$ vertex-disjoint paths to distinct vertices of $D$, one starting
+at each vertex of $Z$ and one starting at each of $m$ distinct vertices
+of $H$.  These paths meet $D$ and $H$ only at their appropriate ends.
+Moreover, after any partition of the $2m$ second-family starts into $m$
+pairs, the $m$ selected starts in $H$ can be required to include exactly
+one vertex from each pair.  No unselected second-family start is visited
+by any of the resulting paths.
+\end{lemma}
+
+\begin{proof}
+First truncate every input path at its first visit to $D$ if necessary.
+Within either family this preserves all distinctness and disjointness
+requirements.  Work in the graph consisting only of the vertices and
+edges of the two resulting path families; ambient edges are unnecessary.
+For each prescribed pair of second-family starts, identify its two
+vertices to form one temporary source.  The paths of that family have
+no other vertices in $H$, and the first family avoids $H$, so each
+original path still gives a path after identification.
+
+Write $N=|Z|+m$.  There are now $N$ sources, namely $Z$ together with
+the $m$ temporary sources, and there are two given routes from each
+source to $D$.  Across these $2N$ routes, every vertex has congestion
+at most two, where congestion counts the number of routes containing
+the vertex.  At a vertex of $Z$ there are only its two first-family
+routes, since the second family avoids $Z$.  At a temporary source
+there are only its two second-family routes, since the first family
+avoids $H$.  Elsewhere there is at most one route from each family.
+This includes vertices of $D$, whose ends are distinct within each
+family but need not be distinct between the two families.
+
+Deleting fewer than $N$ vertices meets fewer than $2N$ routes.
+Some entire source-to-$D$ route therefore survives.  By
+Lemma~\ref{lem:packing}, there are $N$ disjoint paths from the full
+source set to $D$, with no internal source or target vertices.
+Every source is used, since the source set has cardinality $N$.
+
+Each temporary source occurs on exactly one packed path and only as
+its starting endpoint.  Lift the first edge of that path to whichever
+one of its two original starts supplied this edge, and keep the rest
+of the path unchanged.  This selects exactly one original start from
+the pair.  No switching between the two original starts is needed,
+because the identified vertex was not internal.  Different temporary
+sources came from disjoint pairs, so all the lifted paths remain
+vertex-disjoint.
+
+The only vertices of $H$ in the graph of the original families were
+the $2m$ second-family starts.  All their images were source vertices
+and could not be internal in the packing.  Thus the lifted paths meet
+$H$ exactly at the $m$ selected starts; all unselected starts are
+absent.  The target-end convention survives lifting, and every used
+vertex belongs to an original path family, as required.
+\end{proof}
diff --git a/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/sections/07-separability.tex b/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/sections/07-separability.tex
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+\section{Separating two subgraphs of large list chromatic number}
+\label{sec:separability}
+
+We adapt the list-chromatic separability framework of Postle
+\cite[Definition~2.4 and Lemmas~2.5--2.6]{Postle2020}.
+The next lemma allows the construction to branch. Its proof treats the
+alternative in which two disjoint subgraphs cannot both retain nearly all
+of the list chromatic number. In that alternative, successive highly
+connected regions must overlap substantially. We use these overlaps to
+extend a clique model through \(O(\log^4t)\) small districts.
+Figure~\ref{fig:separability-round} illustrates one extension.
+
+\begin{lemma}[Separability]
+\label{lem:separability}
+There is an absolute positive integer \(S\) such that, for all sufficiently
+large integers \(t\), every graph \(G\) satisfying
+\[
+ h(G)<t,\qquad r=\ell(G)\ge 2St
+\]
+has two vertex-disjoint induced subgraphs \(G_1,G_2\) with
+\[
+ \ell(G_1),\ell(G_2)\ge r-St.
+\]
+\end{lemma}
+
+To understand the construction behind the lemma, suppose that no such
+pair of subgraphs exists. We carry \(t\) disjoint connected label bags,
+each with one tip in a highly connected region of large list chromatic
+number. Alongside the bags we keep a \emph{footprint}: the set of all
+vertices recorded so far, including unused path vertices. A round adds
+adjacencies among a selected group of labels. Inside the current region,
+choose two disjoint small districts outside the footprint: one will
+supply a rooted clique model on these labels, while the other serves
+as a junction. Choose a minimum-total-length system of two paths from
+every old tip and every selected model root to distinct junction vertices.
+
+Deleting the accumulated footprint, the districts, and these paths
+leaves enough list chromatic number to extract a fresh highly connected
+region. The supposition on \(G\) forces the old and fresh regions to
+overlap. This overlap makes a second family of \(2t\) paths to the
+junction feasible, with distinct freely chosen starts in the fresh
+region; again minimize their total length. The two families may
+intersect. Lemma~\ref{lem:combine} combines them into disjoint paths
+from every old tip and model root, and from \(t\) selected new tips.
+Rerouting through the model district installs the clique model;
+grouping the path ends in the junction joins each old bag to its new
+tip and, for a selected label, to its assigned model bag. Repeating the
+construction for suitably chosen groups makes every two label bags
+adjacent.
+
+Every district and both original minimizing systems remain in the
+footprint. The paths actually extending the bags may have been rerouted
+and need not still minimize length; all deletion estimates use the
+recorded originals. These paths can have arbitrarily many vertices.
+The next lemma bounds the list-coloring cost of the whole footprint.
+By Lemma~\ref{lem:additive-deletion}, it suffices to put a small set of
+vertices in \(X\) and show that the remaining footprint \(Y\) is
+\(O(t)\)-degenerate and that every vertex outside \(X\cup Y\) has
+\(O(t)\) neighbors in \(Y\).
+
+Throughout this section put
+\[
+ m_0=\left\lceil(\log t)^2\right\rceil,\qquad
+ R_t=\frac{m_0(m_0+1)}2.
+\]
+We take \(t\) sufficiently large that \(m_0\le t\).
+For a family of paths \(\mathcal Q\), write \(V(\mathcal Q)\) for the
+union of their vertex sets.
+
+\begin{lemma}[Deleting a history of paths]
+\label{separate:footprint}
+Let \(h(G)<t\). Start with \(F_0=W_0\), where \(|W_0|=t\), and perform
+at most \(R_t\) rounds with the following rules. At the start of a round
+there is a footprint \(F\) and a set \(W\subseteq F\) of \(t\) tips.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item Choose disjoint sets \(J,D\subseteq V(G)\setminus F\), each of
+size at most \(t^{1.06}\), and a set \(R\subseteq J\) of at most \(t\)
+vertices. Put \(Z=W\cup R\).
+\item In the induced graph \(G-(F\setminus W)\), choose a
+minimum-total-length double system \(\mathcal Q_1\) from \(Z\) to \(D\):
+two paths from each vertex of \(Z\), otherwise vertex-disjoint, with
+distinct target ends. Set
+\[
+ F^- =F\cup J\cup D\cup V(\mathcal Q_1).
+\]
+\item Choose a source set \(A\subseteq V(G)\setminus F^-\). In the
+induced graph \(G-(F\cup R)\), choose a minimum-total-length system
+\(\mathcal Q_2\) of \(2t\) vertex-disjoint paths from arbitrary distinct
+vertices of \(A\) to arbitrary distinct vertices of \(D\). Set
+\[
+ F^+=F^-\cup V(\mathcal Q_2),
+\]
+and choose \(t\) tips \(W^+\subseteq F^+\) for the next round.
+\end{enumerate}
+Only histories in which the indicated systems exist are considered.
+The final round may end after step \textup{(ii)}.
+
+There is an absolute integer
+\begin{equation}
+ \label{separate:cf}
+ c_f=c_{6B_*+20}=4L+6B_*+25
+\end{equation}
+such that, for all sufficiently large \(t\), every final footprint
+\(F_{\rm last}\) of such a history satisfies
+\[
+ \ell(G-F_{\rm last})\ge\ell(G)-c_ft.
+\]
+The same assertion holds for the initial footprint.
+\end{lemma}
+
+\begin{proof}
+We use \(c_d\) from Lemma~\ref{lem:additive-deletion} and \(B_*\) from
+Lemma~\ref{lem:incidence}. All constants and cutoffs in this proof are
+independent of the history and of the lengths of its paths.
+
+\paragraph{The two path collections.}
+Remove the starting vertex from each first-system path, and collect all
+the resulting nonempty paths in a family \(\mathcal A\). These paths
+are pairwise vertex-disjoint. Within one round the only intersections
+of first-system paths are their prescribed starts, which have been
+removed. Across rounds, a new first system avoids the old footprint
+except at its tips \(W\). Every tip is a prescribed start, and no such
+start is internal on any path of the system. After stripping starts,
+the new paths therefore avoid the entire old footprint.
+
+Let \(\mathcal B\) consist of the full second-system paths from all
+rounds. These too are pairwise vertex-disjoint: each new second system
+avoids the entire old footprint, which contains every earlier
+second-system path. No disjointness between \(\mathcal A\) and
+\(\mathcal B\) is asserted or needed.
+There are at most \(4t\) first paths and \(2t\) second paths per round,
+so, if \(q\le R_t\) rounds have been started,
+\begin{equation}
+ \label{separate:path-count}
+ |\mathcal A|+|\mathcal B|\le6tq\le6tR_t.
+\end{equation}
+
+\paragraph{Small sets and exceptional vertices.}
+Let \(X_0\) contain all districts \(J,D\), all tip sets appearing in
+the history, and the ends of every path in both systems. Counting
+repetitions only increases the following bound:
+\begin{equation}
+ \label{separate:x0}
+ |X_0|\le 2R_t t^{1.06}+(13R_t+1)t.
+\end{equation}
+Indeed, there are at most \(R_t+1\) tip sets; the two systems together
+have at most \(12t\) path-end occurrences per round.
+
+Consider the footprints at the start of each round and also
+\(F_{\rm last}\); call these at most \(R_t+1\) sets the snapshots.
+The final snapshot will control neighbors of vertices outside the
+final footprint. Earlier snapshots will control the neighbors that a
+newly entering vertex has on older paths.
+At a snapshot \(T\), let \(\mathcal A_T,\mathcal B_T\) be the two
+collections formed by systems already included in \(T\).
+They use the original recorded paths, with only each first path's
+own start removed; no vertices of the set \(X\) defined below have
+yet been removed from these collections.
+For either collection \(\mathcal C\), declare a vertex
+\(v\in V(G)\setminus T\) exceptional if it has a neighbor on at least
+\(B_*t\) distinct members of \(\mathcal C\).
+
+To count these vertices, contract the disjoint connected paths of
+\(\mathcal C\) to a set of \(|\mathcal C|\) vertices. Vertices outside
+\(T\) are not contracted. Their numbers of distinct neighbors in the
+contracted set are exactly the numbers of paths they see. The
+resulting graph is a minor of \(G\), so Lemma~\ref{lem:incidence}
+implies that the number of exceptional vertices is at most
+\[
+ \frac{t^{1+2\epsilon}|\mathcal C|}{B_*t}
+ \le t^{2\epsilon}|\mathcal C|.
+\]
+Apply this separately to \(\mathcal A_T\) and \(\mathcal B_T\).
+By \eqref{separate:path-count}, the number at one snapshot is at most
+\(6R_t t^{1+2\epsilon}\). Let \(X\) be \(X_0\) together with the
+exceptional vertices from all snapshots, including exceptional
+vertices outside \(F_{\rm last}\). Then
+\begin{align}
+ \label{separate:x-bound}
+ |X|
+ &\le 2R_t t^{1.06}+(13R_t+1)t
+       +6R_t(R_t+1)t^{1+2\epsilon}\notag\\
+ &=O(t^{1.06}\log^4t)+O(t^{1.01}\log^8t)
+ \le t^{11/10}
+\end{align}
+for sufficiently large \(t\), since \(\epsilon=1/200\).
+Thus every required snapshot has been counted.
+
+Put \(Y=F_{\rm last}\setminus X\). The sets \(X,Y\) are disjoint,
+and every vertex of \(Y\) lies on a recorded path.
+
+\paragraph{Neighbors from outside the footprint.}
+Let \(v\notin X\cup Y\). Then \(v\notin F_{\rm last}\), so it was
+outside every recorded path union. Moreover, it belonged to the
+graph in which each system was minimized: every excluded set
+\(F\setminus W\) or \(F\cup R\) is contained in \(F_{\rm last}\).
+By Lemma~\ref{lem:path-minimization}, it has at most three neighbors
+on each path. Nonexceptionality at the final snapshot bounds the
+number of paths it sees by fewer than \(B_*t\) in each collection.
+All stripped starts lie in \(X_0\). Consequently
+\begin{equation}
+ \label{separate:external-degree}
+ |N_G(v)\cap Y|\le6B_*t.
+\end{equation}
+
+\paragraph{Degeneracy inside the footprint.}
+Order the vertices of \(Y\) by their round of first entry into the
+footprint. Within a round, put vertices on \(\mathcal Q_1\) first,
+and then the remaining new vertices on \(\mathcal Q_2\).
+This partitions \(Y\) into at most two phases per round, because
+district vertices and all initial tips lie in \(X_0\).
+
+A vertex \(v\) entering in a given round was outside that round's
+old footprint \(F\). If \(v\in Y\), it was not exceptional at the
+snapshot \(F\). For every earlier system, its forbidden vertices
+belong to \(F\), while \(v\notin F\); hence \(v\) was eligible in
+the graph of that minimization and was outside its path union.
+The preceding argument therefore gives at most \(6B_*t\) neighbors
+of \(v\) in \(Y\) from earlier rounds.
+
+A first phase is contained in the union of at most \(4t\)
+first-system paths with their starts removed. Lemma~
+\ref{lem:path-minimization} makes its induced graph \(8t\)-degenerate.
+A second phase is contained in the union of \(2t\) minimizing
+second-system paths, so its induced graph is \(4t\)-degenerate,
+and in particular \(8t\)-degenerate. These are assertions about
+the induced graphs in \(G\): all vertices of the relevant phase
+belong to the graph used for that minimization, which was an
+induced subgraph of \(G\).
+
+Finally, a vertex in the second phase of a round lies outside
+the entire first-system union and outside the old footprint.
+It therefore belongs to \(G-(F\setminus W)\).
+Lemma~\ref{lem:path-minimization} bounds its neighbors in the first
+phase by \(3(4t)=12t\).
+
+In any nonempty subgraph of \(G[Y]\), select the latest represented
+phase and a vertex of degree at most \(8t\) within that phase.
+It has no neighbors in a later represented phase, at most \(12t\)
+neighbors in an earlier phase of the same round, and at most
+\(6B_*t\) neighbors in earlier rounds. Hence \(G[Y]\) is
+\((6B_*+20)t\)-degenerate.
+
+Equations~\eqref{separate:x-bound} and
+\eqref{separate:external-degree} now verify every hypothesis of
+Lemma~\ref{lem:additive-deletion}, with \(d=6B_*+20\).
+It follows that
+\[
+ \ell(G-X-Y)\ge\ell(G)-c_ft.
+\]
+Since \(F_{\rm last}\subseteq X\cup Y\), the graph
+\(G-F_{\rm last}\) contains \(G-X-Y\) as an induced subgraph.
+Monotonicity of list chromatic number proves the assertion.
+For the initial footprint the same proof has \(Y=\varnothing\)
+and \(X=W_0\).
+\end{proof}
+
+\begin{proof}[Proof of Lemma~\ref{lem:separability}]
+Let \(K_0=10^5\gamma\) be the integer fixed in
+Lemma~\ref{lem:small-woven}, and put \(k=K_0t\).
+Let \(D_{K_0}\) be supplied by Lemma~\ref{lem:district}.
+Use \(c_f\) from \eqref{separate:cf}, put
+\begin{equation}
+ \label{separate:ch}
+ c_h=c_f+6(K_0+1)+2,
+\end{equation}
+and let \(c_0=4L+5\) be the deletion constant for \(d=0\).
+Choose a positive integer \(S\) satisfying
+\begin{align}
+ \label{separate:S}
+ S&>c_h+K_0+2,\notag\\
+ 2S-c_f&>6(K_0+1)+2,\\
+ 2S-c_h-1-c_0&>2D_{K_0}+3.\notag
+\end{align}
+These choices involve only constants already fixed. We then take
+\(t\) sufficiently large for Lemmas~\ref{lem:district},
+\ref{lem:small-woven}, \ref{lem:additive-deletion}, and
+\ref{separate:footprint} with the fixed parameters used below,
+and for all displayed size inequalities in the construction.
+
+Suppose that \(r=\ell(G)\ge2St\), \(h(G)<t\), but no two
+vertex-disjoint induced subgraphs of \(G\) both have list chromatic
+number at least \(r-St\). We will construct a \(K_t\) model.
+
+\paragraph{Labels, rounds, and invariants.}
+Partition the label set \(\{1,\ldots,t\}\) into \(m_0\) nonempty
+blocks, each of size at most \(\lceil t/m_0\rceil\).
+Schedule one round for each individual block and one for each
+unordered pair of distinct blocks. There are \(R_t\) rounds.
+The labels treated in a round form a set \(I\), which is either
+a block or the union of its two assigned blocks. Its size \(s\)
+satisfies, for sufficiently large \(t\),
+\begin{equation}
+ \label{separate:s}
+ 1\le s\le2\left\lceil\frac{t}{m_0}\right\rceil
+ \le2\left\lceil\frac{t}{(\log t)^2}\right\rceil
+ \le t.
+\end{equation}
+Every pair of distinct labels occurs together in a scheduled round.
+
+At the start of a round we maintain the following data.
+\begin{enumerate}[label=\textup{(\arabic*)}]
+\item Pairwise disjoint nonempty connected vertex sets
+\(B_1,\ldots,B_t\), with all adjacencies obtained in earlier rounds.
+\item A footprint \(F\) containing these sets and arising from the
+history rules of Lemma~\ref{separate:footprint}. It retains all
+vertices of every district and of both original minimizing path
+systems from every completed round.
+\item A \(k\)-connected induced region \(H\) with
+\[
+ \ell(H)\ge r-c_ht,\qquad V(H)\cap F=W,
+\]
+where \(W=\{w_1,\ldots,w_t\}\) consists of distinct tips with
+\(w_j\in B_j\).
+\end{enumerate}
+In particular, no vertex of \(H-W\) belongs to an earlier footprint.
+The distinction between the footprint and the sets \(B_j\) is
+useful: unused path vertices remain in the footprint and are
+accounted for by Lemma~\ref{separate:footprint}.
+
+The inequalities in \eqref{separate:S} imply \(r>6k\).
+Lemma~\ref{lem:list-extraction} supplies a \(k\)-connected induced
+subgraph \(H\) with
+\(\ell(H)\ge r-6k\ge r-c_ht\).
+It has more than \(k\ge t\) vertices. Choose distinct tips \(w_j\)
+there and initialize \(B_j=\{w_j\}\) and \(F=W\).
+
+\paragraph{Two districts and the first minimizing system.}
+Consider a round with label set \(I\) of size \(s\). Deleting its
+current tips gives
+\[
+ \ell(H-W)\ge r-(c_h+1)t.
+\]
+Corollary~\ref{density:list-district} states that
+if \(\ell(P)>2D_{K_0}t+2\) and \(h(P)<t\), then \(P\) contains
+an induced \(k\)-connected district of order at most \(t^{1.06}\).
+
+First obtain such a district \(J\subseteq H-W\). Delete it using
+Lemma~\ref{lem:additive-deletion} with \(X=V(J)\) and
+\(Y=\varnothing\), losing at most \(c_0t\).
+The remaining list chromatic number is at least
+\[
+ r-(c_h+1+c_0)t
+ \ge(2S-c_h-1-c_0)t
+ >(2D_{K_0}+3)t>2D_{K_0}t+2.
+\]
+The same district consequence supplies a second district
+\(D\subseteq H-W-J\). Both \(J,D\) are \(k\)-connected and have
+order at most \(t^{1.06}\), and both are disjoint from \(F\).
+Choose distinct roots \(R_J=\{u_j:j\in I\}\) in \(J\), and put
+\[
+ Z=W\cup R_J,\qquad |Z|=t+s\le2t.
+\]
+
+The graph \(H\) is a subgraph of \(G-(F\setminus W)\),
+is \(2|Z|\)-connected, and contains the disjoint source and target
+sets \(Z,D\). Also \(|D|>k\ge2|Z|\).
+Lemma~\ref{lem:double-packing} therefore gives a feasible double
+system. Among all such systems in the full induced graph
+\begin{equation}
+ \label{separate:first-domain}
+ G-(F\setminus W)
+\end{equation}
+choose a minimum-total-length system \(\mathcal Q_1\).
+Its only intersections with \(F\) are its prescribed starts in \(W\).
+Set
+\[
+ F^-=F\cup V(J)\cup V(D)\cup V(\mathcal Q_1).
+\]
+The history ending with this first system satisfies
+Lemma~\ref{separate:footprint}, and hence
+\begin{equation}
+ \label{separate:intermediate-loss}
+ \ell(G-F^-)\ge r-c_ft.
+\end{equation}
+
+\paragraph{The next region and the second minimizing system.}
+By \eqref{separate:S}, the right side of
+\eqref{separate:intermediate-loss} is greater than
+\(6(K_0+1)t\).
+Lemma~\ref{lem:list-extraction} gives a
+\((K_0+1)t\)-connected induced region \(H'\subseteq G-F^-\)
+with
+\begin{equation}
+ \label{separate:new-region-list}
+ \ell(H')\ge r-\bigl(c_f+6(K_0+1)\bigr)t.
+\end{equation}
+
+We claim that \(|V(H')\cap V(H)|\ge k\).
+Otherwise deleting this intersection from \(H'\) gives an induced
+subgraph disjoint from \(H\), of list chromatic number at least
+\[
+ r-\bigl(c_f+6(K_0+1)+K_0\bigr)t\ge r-St.
+\]
+The other graph \(H\) has list chromatic number at least
+\(r-c_ht\ge r-St\). This contradicts the supposition on \(G\).
+
+It follows that the graph induced by \(V(H)\cup V(H')\) is
+\(k\)-connected. After deleting fewer than \(k\) vertices, each
+of its two regions is connected and at least one vertex of their
+intersection survives. This union meets \(F\) only in \(W\).
+Deleting \(Z=W\cup R_J\) leaves a graph of connectivity at least
+\[
+ k-|Z|\ge(K_0-2)t\ge2t,
+\]
+contained in the induced graph
+\begin{equation}
+ \label{separate:second-domain}
+ G-(F\cup R_J).
+\end{equation}
+Neither \(H'\) nor \(D\) meets \(Z\), these two endpoint sets are
+disjoint, and each has more than \(2t\) vertices.
+Lemma~\ref{lem:packing} supplies \(2t\) disjoint paths between them.
+
+Choose \(\mathcal Q_2\) of minimum total length among all
+\(2t\)-path systems from arbitrary distinct vertices of \(H'\)
+to arbitrary distinct vertices of \(D\), in the full graph
+\eqref{separate:second-domain}. This is an unpaired minimization.
+Its paths meet \(H'\) and \(D\) only at their respective ends.
+Let \(T'\subseteq V(H')\) be their \(2t\) starting vertices, and put
+\[
+ F^+=F^-\cup V(\mathcal Q_2).
+\]
+In particular,
+\begin{equation}
+ \label{separate:region-footprint}
+ V(H')\cap F^+=T'.
+\end{equation}
+This completes another round of the history in
+Lemma~\ref{separate:footprint}.
+
+\paragraph{Combining the systems and inserting the district model.}
+The sets \(Z,V(H'),V(D)\) are pairwise disjoint.
+The first system avoids \(H'\), since it lies in \(F^-\);
+the second avoids \(Z\), by its domain \eqref{separate:second-domain}.
+Lemma~\ref{lem:combine}, with \(m=t\), gives
+\[
+ q=|Z|+t=2t+s\le3t
+\]
+vertex-disjoint paths to distinct vertices of \(D\), one from
+each vertex of \(Z\) and one from each vertex of a selected
+set \(W^+\subseteq T'\) of size \(t\).
+They meet \(D,H'\) only at their designated ends, use only vertices
+of the two systems, and meet \(F\) only in the \(W\)-ends.
+In particular, they do not visit the unused starts \(T'\setminus W^+\).
+Assign \(W^+=\{w_1^+,\ldots,w_t^+\}\) bijectively to the labels.
+
+By \eqref{separate:s} and Lemma~\ref{lem:small-woven}, the district
+\(J\) is \((s,3t)\)-woven. Apply Lemma~\ref{lem:rerouting} to
+the \(q\) combined paths and the roots \(R_J\). It produces in \(J\)
+a rooted \(K_s\) model with bags \(C_j\), \(j\in I\), and
+reroutes the combined paths without changing their ends.
+Each root \(u_j\) was already the endpoint of its own path.
+Thus the model meets the paths only at these roots, each on
+its corresponding path and in its corresponding bag.
+
+The new vertices of the rerouted paths lie in \(J\subseteq F^+\).
+Since \(J\) is disjoint from \(F,H',D\), all the previous
+endpoint-only intersection statements with these sets persist.
+All rerouted paths and all bags \(C_j\) lie in \(F^+\).
+We continue to record the original minimizing systems in the
+history; the rerouted paths are used only to extend the label bags.
+
+\paragraph{Grouping in \(D\) and restoring the invariants.}
+For each label \(j\), group the terminals in \(D\) belonging to
+the paths from \(w_j\), from \(w_j^+\), and, if \(j\in I\),
+from \(u_j\). These groups partition the \(q\) distinct terminals.
+Since
+\[
+ \gamma q\le3\gamma t\le K_0t,
+\]
+Theorem~\ref{thm:grouping} supplies pairwise disjoint connected
+sets \(E_1,\ldots,E_t\) in \(D\) containing their respective groups.
+For the next region put
+\[
+ H^+=H'-(T'\setminus W^+).
+\]
+Figure~\ref{fig:separability-round} now shows how these pieces fit
+into one extended label bag.
+
+\input{figures/separability}
+
+Extend \(B_j\) by the two or three corresponding paths, by \(E_j\),
+and by \(C_j\) when \(j\in I\). The resulting set \(B_j^+\) is
+connected: its old bag contains \(w_j\); the paths join every
+attached piece to \(E_j\); and it contains the new tip \(w_j^+\).
+The sets \(B_j^+\) are pairwise disjoint. Indeed, the old bags
+are disjoint and meet the added paths only at their own \(W\)-ends;
+the paths are disjoint; the sets \(E_j\) are disjoint and meet paths
+only at the designated terminals; and the bags \(C_j\) are
+disjoint, lie outside \(F\cup D\), and meet paths only at their
+own roots. All these sets lie in \(F^+\).
+Old adjacencies persist because \(B_j\subseteq B_j^+\), and every
+pair of labels in \(I\) gains an adjacency through its bags in
+the district model.
+
+It remains to check the next-region invariant. Exactly \(t\) vertices
+were deleted from the \((K_0+1)t\)-connected
+graph \(H'\), so \(H^+\) is \(K_0t\)-connected. By
+\eqref{separate:new-region-list} and \eqref{separate:ch},
+\[
+ \ell(H^+)
+ \ge r-\bigl(c_f+6(K_0+1)+1\bigr)t
+ \ge r-c_ht.
+\]
+Equation~\eqref{separate:region-footprint} gives
+\[
+ V(H^+)\cap F^+=W^+.
+\]
+Thus \(F^+,H^+,W^+\), and the bags \(B_j^+\) restore every
+invariant for the next round.
+
+After all \(R_t\) rounds, every pair of labels has an adjacency.
+The \(t\) disjoint connected label bags therefore form a \(K_t\)
+model in \(G\), contradicting \(h(G)<t\).
+This proves Lemma~\ref{lem:separability}.
+\end{proof}
diff --git a/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/sections/08-recursion.tex b/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/sections/08-recursion.tex
new file mode 100644
index 0000000000000000000000000000000000000000..b822b901592aa8fbbba65cfb64f91f634afcf68d
--- /dev/null
+++ b/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/build/sections/08-recursion.tex
@@ -0,0 +1,253 @@
+\section{Recursive weaving and the universal bound}\label{sec:recursion}
+
+The recursive use of woven districts follows the framework of
+Postle \cite[Section~6]{Postle2020} and Delcourt and Postle
+\cite[Section~7]{DelcourtPostle2021}. We now use separability to replace
+the successive small steps of the preceding section by a recursion with
+a fixed shrinkage factor.
+
+\begin{theorem}\label{thm:recursive-woven}
+There are absolute positive integers $K,Q$ such that, for every integer
+$a\ge1$, every $Ka$-connected graph $P$ with $\ell(P)\ge Qa$ is
+$(a,3a)$-woven.
+\end{theorem}
+\begin{proof}
+The main inductive step concerns $h(P)<64a$ and sufficiently large $a$.
+It separates the paths for prescribed pairs from the clique-model
+construction. Give each root and each end of a noncoincident pair its
+own neighboring proxy outside the original terminal set, with all proxies
+distinct, and delete the original terminals. Choose a minimum-total-length
+system of two paths per proxy to a small highly connected district $D$.
+Deleting the proxies, $D$, and these paths costs only $O(a)$ in list
+chromatic number. We can therefore extract a highly connected region
+$H'$ disjoint from the first path system. Further connections
+from $H'$ to $D$ and Lemma~\ref{lem:combine} produce disjoint paths from
+all proxies and from a set $W$ of $a$ vertices of $H'$ to $D$, meeting
+$H'$ only at $W$. It then suffices to build a clique model rooted at $W$
+inside $H'$: grouping the path ends in $D$ attaches the model to the
+root proxies and completes the prescribed pair paths.
+
+For this internal model, divide the labels of $W$ into three blocks.
+Separability and connectivity extraction supply three disjoint
+districts for recursive calls, one for each pair of blocks. A call
+needs at most $2\lceil a/3\rceil$ roots. Each label receives two bags
+to be joined to its vertex of $W$. The $2a$ joining paths may pass through
+the recursive districts; wovenness lets us reroute them while constructing
+the child models, keeping the retained bags disjoint from the paths
+except at their designated roots. Every pair of labels shares a district
+and hence receives an adjacency.
+
+We specify the order of the constants first. Fix all constants in the
+preceding sections, including $L,C_*,B_*,K_0,D_{K_0}$, the deletion
+constants $c_d$, and the separability constant $S$. Choose an integer
+$K\ge100$ such that
+\begin{equation}\label{recursive:K}
+ K-14>128D_{K_0}.
+\end{equation}
+Choose an absolute integer $a_0\ge24$ so large that all sufficiently-large
+parameter conclusions used below hold with $t=64a$ whenever $a>a_0$.
+Enlarge $a_0$ if necessary so that
+\begin{equation}\label{recursive:shrink}
+ a'=2\lceil a/3\rceil\le3a/4\qquad(a>a_0),
+\end{equation}
+and $t^{1.06}+7a\le t^{11/10}$ for $t=64a$, $a>a_0$.
+All these requirements involve only constants already fixed, and none
+involves $Q$. Finally choose an integer $Q$ satisfying
+\begin{align}
+ Q/4&>10+64(c_{42}+6K_0+2S)+(9/2)K,
+       \label{recursive:Q-induction}\\
+ Q&>10+64(c_{42}+6K_0+3S),
+       \label{recursive:Q-separation}\\
+ Q&>128C_*\log(128a_0)+1.
+       \label{recursive:Q-base}
+\end{align}
+We prove the assertion by strong induction on $a$.
+
+\paragraph{Separating the terminal roles.}
+Fix $a$ distinct roots and at most $3a$ pairs as in
+Definition~\ref{def:woven}. Let $Z_0$ be their union, so $|Z_0|\le7a$.
+For each root, and for each end of each noncoincident pair, choose a
+neighbor outside $Z_0$ as its \emph{proxy}. All proxies are distinct,
+even for different roles of the same original vertex. This is possible
+greedily because $\delta(P)\ge Ka$ and fewer than $14a$ vertices are
+forbidden at any choice. Coincident pairs will ultimately use trivial
+paths at their original vertices and need no proxies.
+
+Put $G=P-Z_0$ and let $Z$ be the proxy set. Then
+\begin{equation}\label{recursive:proxy-bounds}
+ a\le|Z|\le7a,\qquad
+ G\text{ is }(K-7)a\text{-connected},\qquad
+ \ell(G)\ge Qa-7a.
+\end{equation}
+It suffices to find in $G$ a clique model rooted at the root proxies and
+paths for the other proxy pairs, with all model bags and all these paths
+pairwise disjoint. Adding the proxy edges and the original vertices then
+restores precisely the root--path coincidences allowed by wovenness.
+Different prescribed pairs have no common original endpoint; a
+coincident pair at a root meets only that root's expanded bag.
+
+\paragraph{A large minor and the finite base range.}
+Set $t=64a$. If $h(P)\ge t$, delete from a $K_t$ model every bag meeting
+$Z_0$. This leaves a $K_{57a}$ model in $G$. Since
+$57a\ge4|Z|$ and $(K-7)a\ge2|Z|$, Lemma~\ref{lem:rooting} gives a
+clique model rooted at all of $Z$. Retain the root-proxy bags. For each
+other proxy pair, a path through its two bags and an edge between them
+joins the pair. These paths and the retained bags are disjoint because
+all proxies and their bags are distinct. This proves the required
+conclusion in the large-minor case, for every $a$.
+
+Suppose $h(P)<64a$. Lemma~\ref{lem:coarse-density} gives
+\[
+ \ell(P)\le128C_*a\log(128a)+1.
+\]
+For $1\le a\le a_0$ this is strictly less than $Qa$ by
+\eqref{recursive:Q-base}. Thus the minor-free case is impossible
+throughout the base range. We may henceforth assume $a>a_0$.
+
+\paragraph{A junction district and a fresh region.}
+Every vertex of $G-Z$ has degree at least $(K-14)a$, which is greater
+than $2D_{K_0}t$ by \eqref{recursive:K}. In particular this graph is
+nonempty and has density greater than $D_{K_0}t$. Apply
+Lemma~\ref{lem:district} to find a $K_0t$-connected district
+$D\subseteq G-Z$ with $|V(D)|\le t^{1.06}$.
+
+The connectivity of $G$ and the order of $D$ permit two paths per proxy
+from $Z$ to distinct vertices of $D$, with only the prescribed repeated
+starts in common, by Lemma~\ref{lem:double-packing}. Choose such a system
+$\mathcal Q_1$ of minimum total length in $G$. It has at most $14a$ paths.
+Put
+\[
+ X=Z\cup V(D),\qquad Y=V(\mathcal Q_1)\setminus X.
+\]
+We have $|X|\le t^{11/10}$ by the choice of $a_0$.
+Lemma~\ref{lem:path-minimization} gives $28a$-degeneracy of $G[Y]$ and
+at most $42a$ neighbors in $Y$ for every vertex outside $X\cup Y$.
+These bounds are at most $42t$, so Lemma~\ref{lem:additive-deletion}
+with parameter $42$ gives
+\begin{equation}\label{recursive:after-paths}
+ \ell(G-X-Y)\ge Qa-7a-c_{42}t.
+\end{equation}
+By \eqref{recursive:Q-separation} the right-hand side exceeds $6K_0t$.
+Lemma~\ref{lem:list-extraction} therefore supplies a $K_0t$-connected
+induced subgraph $H'\subseteq G-X-Y$ such that
+\begin{equation}\label{recursive:Hprime}
+ \ell(H')\ge Qa-7a-(c_{42}+6K_0)t.
+\end{equation}
+
+In $G-Z$, the sets $H'$ and $D$ have order greater than $K_0t$, and the
+connectivity is at least $(K-14)a>2a$. Hence there are $2a$ disjoint
+$H'$-to-$D$ paths. Truncate them at their last visit to $H'$ and their
+first subsequent visit to $D$, so they have no internal vertex in either
+set. They avoid $Z$. Lemma~\ref{lem:combine}, applied to these paths and
+$\mathcal Q_1$, gives disjoint paths to $D$ from all proxies in $Z$ and
+from a set $W$ of $a$ distinct vertices of $H'$. These paths meet $H'$ and
+$D$ only at their appropriate ends.
+
+It is now enough to construct a clique model rooted at $W$ entirely
+inside $H'$. Indeed, in $D$ group the terminal ends of the outside paths
+as follows: pair each root proxy with a different member of $W$, and
+pair the other proxies as prescribed. There are at most $|Z|+a\le8a$
+terminals, so Theorem~\ref{thm:grouping} applies because
+$K_0t\ge8\gamma a$. These disjoint groups and the outside paths attach
+the rooted model to the root proxies and supply all other required
+paths. The model inside $H'$ has no unwanted intersection with the
+outside paths, since their only vertices in $H'$ are precisely $W$.
+
+\paragraph{Three recursive districts.}
+For each $w\in W$ choose two neighbors in $H'-W$, all $2a$ choices
+distinct. Minimum degree at least $K_0t$ permits this greedy choice.
+Let $N$ be their set and put $H''=H'-W-N$. Then
+\begin{equation}\label{recursive:r0}
+ \ell(H'')\ge r_0:=Qa-10a-(c_{42}+6K_0)t.
+\end{equation}
+Equation~\eqref{recursive:Q-separation} gives $r_0>3St$.
+Apply Lemma~\ref{lem:separability} to $H''$, and then to one of the two
+resulting subgraphs. The first application is legal because
+$\ell(H'')\ge r_0>2St$. Each of its subgraphs has list chromatic number
+at least $r_0-St>2St$, making the second application legal as well.
+We obtain three disjoint subgraphs each of list chromatic number at
+least $r_0-2St$.
+
+Take $a'=2\lceil a/3\rceil$. After the loss of $2St$ to separability,
+each subgraph must still pay $6Ka'$ for connectivity extraction and
+retain $Qa'$ for the recursive call. If
+$A=10+64(c_{42}+6K_0+2S)$, then
+\begin{align*}
+ r_0-2St-6Ka'-Qa'
+ &=Q(a-a')-Aa-6Ka'\\
+ &\ge\bigl[Q/4-A-(9/2)K\bigr]a>0
+\end{align*}
+by \eqref{recursive:shrink} and \eqref{recursive:Q-induction}.
+In particular $r_0-2St>6Ka'$, and list extraction in each subgraph
+produces disjoint $Ka'$-connected induced districts $J_1,J_2,J_3$
+with list chromatic number at least $Qa'$. Since $a'<a$, induction
+makes each district $(a',3a')$-woven.
+
+Partition the labels of $W$ into three nonempty blocks $B_1,B_2,B_3$,
+each of size at most $\lceil a/3\rceil$. Assign one district to each
+pair of blocks, relabeling them $J_{12},J_{13},J_{23}$ accordingly.
+In its district, designate a distinct root for each
+label of the corresponding two blocks, and add arbitrary distinct
+padding roots to bring the total to $a'$. The districts have enough
+vertices for this choice. Every label has exactly two designated roots,
+in different districts; every pair of labels occurs together in at
+least one district, including two labels in the same block; see
+Figure~\ref{fig:three-districts}.
+
+\begin{figure}[tb]
+\centering
+\input{figures/three-districts}
+\caption{The three recursive districts. The blocks $B_1,B_2,B_3$ partition
+the labels, and a label has one designated root in each of the two
+districts receiving its block. Arrows indicate label assignments, not
+graph edges. Every pair of labels is represented together in a district.}
+\label{fig:three-districts}
+\end{figure}
+
+Inside $H'-W$, join the two chosen neighbors of each $w$ to its two
+designated roots, respectively, using disjoint paths. This is an
+application of Theorem~\ref{thm:grouping} to $4a$ distinct terminals:
+the neighbors lie in $N$, while the districts were chosen in $H''$
+and hence avoid $N$. The connectivity is at least $K_0t-a\ge4\gamma a$.
+There are $2a$ paths. Apply Lemma~\ref{lem:rerouting} successively in the
+three districts to incorporate their rooted clique models while
+retaining these path ends. Each district has sufficient weaving
+capacity, since
+\[
+ 2a\le3a'=6\lceil a/3\rceil.
+\]
+Discard all padding bags. Each designated root remains an endpoint
+of its path; the retained bags meet the rerouted paths only at their
+own designated roots. Since the districts are disjoint, later rerouting
+does not introduce an intersection in a previously retained bag.
+
+For each $w$, unite its two designated bags with their respective paths,
+the two edges to $w$, and $w$ itself. The resulting $a$ sets are connected
+and disjoint, rooted at $W$. Two labels have an edge between their sets
+because they occur together in one of the district clique models.
+This constructs the required rooted $K_a$ model inside $H'$ and
+completes the induction.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+Let $K,Q$ be as in Theorem~\ref{thm:recursive-woven}. We first show that,
+for every integer $t\ge2$,
+\begin{equation}\label{recursive:final-minor-free}
+ h(G)<t\quad\Longrightarrow\quad\ell(G)\le(Q+6K)t.
+\end{equation}
+Otherwise Lemma~\ref{lem:list-extraction}, with connectivity parameter
+$Kt$, gives a $Kt$-connected induced subgraph $P$ with
+$\ell(P)\ge\ell(G)-6Kt>Qt$. Theorem~\ref{thm:recursive-woven} makes
+$P$ $(t,3t)$-woven. Its order is greater than $Kt\ge t$, so we may choose
+$t$ roots and no pairs. Wovenness then supplies a $K_t$ model,
+contradicting $h(G)<t$.
+
+For a nonempty graph, $h(G)\ge1$. Taking $t=h(G)+1\le2h(G)$ in
+\eqref{recursive:final-minor-free} proves
+\[
+ \ell(G)\le2(Q+6K)h(G).
+\]
+Thus the absolute integer $C=2(Q+6K)$ has all the claimed uniformity.
+The finite base range was covered before the induction began, so no
+small-parameter exception remains.
+\end{proof}
diff --git a/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/paper.pdf b/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..14896f933ab476cffc810d7beacc442d69494f3a
--- /dev/null
+++ b/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:1becff24453872c5c7973bdc72bc2b12120048580ac71fbac816b523f910163e
+size 587502
diff --git a/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/README.md b/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..57ad67ad38db02aac63385886cee57de858d049b
--- /dev/null
+++ b/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/README.md
@@ -0,0 +1,18 @@
+# [A local Penrose inequality for conformal perturbations of Schwarzschild–anti-de Sitter data](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** October 5, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026,
+  author = {{OpenAI}},
+  title = {{A local Penrose inequality for conformal perturbations of Schwarzschild--anti-de Sitter data}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/paper.pdf}{OAI:A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/build/paper.tex b/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/build/paper.tex
new file mode 100644
index 0000000000000000000000000000000000000000..03a2c6e7431d28ede4fc5de644a72f0e2d4018f4
--- /dev/null
+++ b/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/build/paper.tex
@@ -0,0 +1,84 @@
+\documentclass[11pt,a4paper]{article}
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage{lmodern}
+\usepackage{microtype}
+\usepackage[margin=30mm]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{mathrsfs}
+\usepackage{enumitem}
+\usepackage{needspace}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta,calc,positioning}
+\usepackage[nottoc,notlof,notlot]{tocbibind}
+\usepackage{xcolor}
+\definecolor{linkblue}{RGB}{24,62,107}
+\usepackage[colorlinks=true,linkcolor=linkblue,citecolor=linkblue,urlcolor=linkblue,
+  pdftitle={A local Penrose inequality for conformal perturbations of Schwarzschild--anti-de Sitter data},
+  pdfauthor={OpenAI}]{hyperref}
+\usepackage[nameinlink,noabbrev]{cleveref}
+\usepackage{bookmark}
+\numberwithin{equation}{section}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\newtheorem{remark}[theorem]{Remark}
+\crefname{theorem}{Theorem}{Theorems}
+\crefname{proposition}{Proposition}{Propositions}
+\crefname{lemma}{Lemma}{Lemmas}
+\crefname{corollary}{Corollary}{Corollaries}
+\crefname{definition}{Definition}{Definitions}
+\crefname{remark}{Remark}{Remarks}
+\crefname{section}{Section}{Sections}
+\crefname{equation}{Equation}{Equations}
+\newcommand{\eps}{\epsilon}
+\newcommand{\dd}{\mathrm{d}}
+\newcommand{\AH}{\mathrm{AH}}
+\newcommand{\II}{\mathrm{II}}
+\newcommand{\tf}{\circ}
+\newcommand{\R}{\mathbb R}
+\newcommand{\N}{\mathbb N}
+\DeclareMathOperator{\Ric}{Ric}
+\DeclareMathOperator{\Hess}{Hess}
+\DeclareMathOperator{\Div}{div}
+\DeclareMathOperator{\tr}{tr}
+\DeclareMathOperator{\scal}{Scal}
+\DeclareMathOperator{\grad}{grad}
+\DeclareMathOperator{\sym}{sym}
+\DeclareMathOperator{\supp}{supp}
+\setlist[enumerate]{label=\textup{(\roman*)},leftmargin=2em}
+\setlength{\emergencystretch}{2em}
+\allowdisplaybreaks[1]
+\title{A local Penrose inequality for conformal perturbations\\
+of Schwarzschild--anti-de Sitter data}
+\author{OpenAI}
+\date{October 5, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We prove the asymptotically hyperbolic Penrose inequality for sufficiently
+small maximal vacuum conformal perturbations of a positive-mass
+Schwarzschild--anti-de Sitter exterior, for every fixed decaying
+transverse-traceless seed and every solution branch satisfying the stated
+decay and mass assumptions. The bound uses the area of the marginally
+outer trapped boundary itself; it requires neither outermostness nor
+outer area-minimization, and no bulk-versus-boundary domination condition.
+For sufficiently small positive parameters, equality holds for radial
+seeds and the inequality is strict otherwise.
+\end{abstract}
+\tableofcontents
+\clearpage
+\input{sections/00-introduction}
+\input{sections/01-setting}
+\input{sections/02-analysis}
+\input{sections/03-cmc}
+\input{sections/04-quadratic}
+\input{sections/05-quartic}
+\input{sections/06-conclusion}
+\clearpage
+\bibliographystyle{support/alpha-linked}
+\bibliography{refs}
+\end{document}
diff --git a/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/build/refs.bib b/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/build/refs.bib
new file mode 100644
index 0000000000000000000000000000000000000000..03bcc9e72958caad3ee8a901a0ab23a2a2129b54
--- /dev/null
+++ b/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/build/refs.bib
@@ -0,0 +1,110 @@
+
+@article{Penrose1973,
+  author  = {Penrose, Roger},
+  title   = {Naked singularities},
+  journal = {Annals of the New York Academy of Sciences},
+  volume  = {224},
+  number  = {1},
+  year    = {1973},
+  pages   = {125--134},
+  doi     = {10.1111/j.1749-6632.1973.tb41447.x},
+  url     = {https://doi.org/10.1111/j.1749-6632.1973.tb41447.x}
+}
+
+@article{Hawking1968,
+  author  = {Hawking, S. W.},
+  title   = {Gravitational radiation in an expanding universe},
+  journal = {Journal of Mathematical Physics},
+  volume  = {9},
+  year    = {1968},
+  pages   = {598--604},
+  doi     = {10.1063/1.1664615},
+  url     = {https://doi.org/10.1063/1.1664615}
+}
+
+@article{HuiskenIlmanen2001,
+  author  = {Huisken, Gerhard and Ilmanen, Tom},
+  title   = {The inverse mean curvature flow and the {Riemannian Penrose} inequality},
+  journal = {Journal of Differential Geometry},
+  volume  = {59},
+  number  = {3},
+  year    = {2001},
+  pages   = {353--437},
+  doi     = {10.4310/jdg/1090349447},
+  url     = {https://doi.org/10.4310/jdg/1090349447}
+}
+
+@article{Bray2001,
+  author        = {Bray, Hubert L.},
+  title         = {Proof of the {Riemannian Penrose} inequality using the positive mass theorem},
+  journal       = {Journal of Differential Geometry},
+  volume        = {59},
+  number        = {2},
+  year          = {2001},
+  pages         = {177--267},
+  doi           = {10.4310/jdg/1090349428},
+  url           = {https://doi.org/10.4310/jdg/1090349428},
+  eprint        = {math/9911173},
+  archivePrefix = {arXiv},
+  primaryClass  = {math.DG}
+}
+
+@article{ChruscielHerzlich2003,
+  author        = {Chru{\'s}ciel, Piotr T. and Herzlich, Marc},
+  title         = {The mass of asymptotically hyperbolic {Riemannian} manifolds},
+  journal       = {Pacific Journal of Mathematics},
+  volume        = {212},
+  number        = {2},
+  year          = {2003},
+  pages         = {231--264},
+  doi           = {10.2140/pjm.2003.212.231},
+  url           = {https://doi.org/10.2140/pjm.2003.212.231},
+  eprint        = {math/0110035},
+  archivePrefix = {arXiv},
+  primaryClass  = {math.DG}
+}
+
+@article{NevesTian2009,
+  author        = {Neves, Andr{\'e} and Tian, Gang},
+  title         = {Existence and uniqueness of constant mean curvature foliation of asymptotically hyperbolic 3-manifolds},
+  journal       = {Geometric and Functional Analysis},
+  volume        = {19},
+  number        = {3},
+  year          = {2009},
+  pages         = {910--942},
+  doi           = {10.1007/s00039-009-0019-1},
+  url           = {https://doi.org/10.1007/s00039-009-0019-1},
+  eprint        = {math/0610767},
+  archivePrefix = {arXiv},
+  primaryClass  = {math.DG}
+}
+
+@article{Ambrozio2015,
+  author        = {Ambrozio, Lucas C.},
+  title         = {On perturbations of the {Schwarzschild anti-de Sitter} spaces of positive mass},
+  journal       = {Communications in Mathematical Physics},
+  volume        = {337},
+  number        = {2},
+  year          = {2015},
+  pages         = {767--783},
+  doi           = {10.1007/s00220-015-2360-6},
+  url           = {https://doi.org/10.1007/s00220-015-2360-6},
+  eprint        = {1402.4317},
+  archivePrefix = {arXiv},
+  primaryClass  = {math.DG}
+}
+
+@article{KhuriKopinski2023,
+  author        = {Khuri, Marcus and Kopi{\'n}ski, Jaros{\l}aw},
+  title         = {Asymptotically hyperbolic {Einstein} constraint equations with apparent horizon boundary and the {Penrose} inequality for perturbations of {Schwarzschild-AdS}},
+  journal       = {Classical and Quantum Gravity},
+  volume        = {40},
+  number        = {4},
+  year          = {2023},
+  pages         = {045007},
+  doi           = {10.1088/1361-6382/acb24b},
+  url           = {https://doi.org/10.1088/1361-6382/acb24b},
+  eprint        = {2209.01234},
+  archivePrefix = {arXiv},
+  primaryClass  = {gr-qc}
+}
diff --git a/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/build/sections/00-introduction.tex b/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/build/sections/00-introduction.tex
new file mode 100644
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--- /dev/null
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@@ -0,0 +1,116 @@
+\section{Introduction}\label{sec:introduction}
+
+The Penrose inequality relates the total mass of gravitational initial
+data to the area of a black-hole boundary. Its original formulation arose
+from the relation between gravitational collapse and cosmic
+censorship~\cite{Penrose1973}. In the asymptotically flat,
+time-symmetric setting it was proved by Huisken and Ilmanen for a
+connected horizon and by Bray for the total area of possibly disconnected
+horizons~\cite{HuiskenIlmanen2001,Bray2001}. For negative cosmological
+constant, the Schwarzschild--anti-de Sitter family suggests the
+three-dimensional inequality
+\begin{equation}\label{eq:intro-penrose}
+ m_{\AH}\geq
+ \sqrt{\frac{A}{16\pi}}\left(1+\frac{A}{4\pi}\right),
+\end{equation}
+when the cosmological constant is normalized to $-3$.
+Here $m_{\AH}$ is the Lorentz norm of the hyperbolic mass covector.
+In the outermost, outer area-minimizing formulation of
+Corollary~\ref{cor:penrose}, $A$ is the least enclosing area of the
+future marginally outer trapped boundary.
+
+We prove \eqref{eq:intro-penrose} for a local conformal class of
+maximal vacuum data with a marginally outer trapped boundary.
+The background is a time-symmetric Schwarzschild--anti-de Sitter
+exterior of arbitrary positive mass. One fixes a smooth decaying
+transverse-traceless (TT) tensor $q$ and considers any solution branch of
+the conformal constraint equation with conformal second fundamental
+form $\eps q$. The parameter interval in the result may depend on this
+fixed seed and branch. Theorem~\ref{thm:main} imposes neither symmetry
+nor an additional comparison between the size of the seed in the
+interior and at the boundary.
+
+This setting is closely related to the perturbative theorem of
+Khuri and Kopi\'nski~\cite[Theorem~2]{KhuriKopinski2023}.
+Their sufficient condition requires the static-lapse-weighted integral
+of the squared seed norm to dominate a multiple of the squared $H^1$
+norm of its normal component on a domain containing the boundary;
+see their Equation~(2.5). Both sides scale quadratically when the seed is
+multiplied by $\eps$, so shrinking $\eps$ alone does not remove that
+condition. The argument below replaces it by a sharp TT boundary
+identity and treats the equality directions of the resulting
+quadratic estimate. It thereby proves the fixed-seed local form
+of the Penrose inequality in this conformal class without the
+domination hypothesis. In particular, it applies when the boundary
+is outermost and outer area-minimizing, as in the usual local
+Penrose formulation.
+
+Two related geometric ingredients have established precedents.
+The hyperbolic mass covector and its covariance are studied by
+Chru\'sciel and Herzlich~\cite{ChruscielHerzlich2003}.
+Neves and Tian construct constant mean curvature (CMC) foliations near
+infinity under
+positive-mass Schwarzschild--anti-de Sitter asymptotic
+hypotheses~\cite{NevesTian2009}; Ambrozio uses CMC foliations to prove
+a local Riemannian Penrose inequality near
+Schwarzschild--anti-de Sitter~\cite{Ambrozio2015}.
+We derive directly the mass and CMC identities needed here.
+Only finite Taylor families of surfaces are used, so the proof
+does not require an actual global CMC foliation for a nonzero
+parameter.
+
+\subsection*{The mechanism of the proof}
+
+Write $\mathfrak D_q(\eps)$ for the difference between the two
+sides of \eqref{eq:intro-penrose}, with $A$ the boundary area.
+In the notation of Section~\ref{sec:setting}, $g_0$ is the background
+metric of mass $m_0$, $s$ is proper radial distance, $r(s)$ is the
+area radius, and $N=r'(s)$ is the static lapse.
+The constant and linear deficit coefficients vanish. The quadratic
+calculation combines a CMC
+Hawking-mass identity with the static TT identity
+\[
+ NQ(v)=\sym\nabla(N\,\dd v-v\,\dd N),
+ \qquad Q(v)=\nabla^2v-v(\Ric_{g_0}+3g_0).
+\]
+After the boundary area correction, the quadratic coefficient is a
+sum of nonnegative terms (Proposition~\ref{prop:quadratic}). Its kernel
+is the four-dimensional space of tensors $Q(v)$, where
+$(\Delta_{g_0}-3)v=0$, $v$ decays, and $v|_S$ contains only constant
+and degree-one spherical harmonics (Corollary~\ref{cor:kernel}).
+
+The degree-one directions are a real obstruction to concluding from
+second order. They have zero first-order shear and lapse fluctuation,
+and a radial component can make their boundary normal component
+nonnegative. We remove their first extrinsic-curvature coefficient
+by a change of spacetime slice at the level of finite Taylor expansions. The
+resulting quartic coefficient is again a sum of nonnegative terms.
+If it vanishes, the changed metric must be a round warped-product
+jet through second order. Conformality of the original branch then
+forces the radial profile $V(s)$ of the degree-one part of $v$ to satisfy
+(with primes denoting $s$-derivatives)
+\[
+ \left(V'-\frac{r'}rV\right)^2=\frac{6m_0}{r^3}V^2.
+\]
+This is incompatible with a nonzero decaying profile. Thus the
+quartic coefficient is strictly positive in every nonradial
+quadratic-null direction (Proposition~\ref{prop:quartic}). The remaining
+radial directions satisfy an exact conserved-mass identity
+(Proposition~\ref{prop:radial}).
+
+Three parts of the method may be useful beyond this application:
+the sharp static TT boundary square; the passage from CMC surface
+jets to a positive mass-coefficient identity by integrating first
+at finite radius; and the use of a null boundary transport together
+with a change of slice to detect a quartic obstruction. Analytically,
+an exact Green formula converts the derivative-defined mass into
+convergent integrals. It allows actual remainder estimates in a
+weighted function norm, and rotation averaging then covers seeds
+with arbitrary angular dependence. Proposition~\ref{prop:expansions}
+then identifies the finite Taylor calculations with the actual deficit.
+
+The result concerns this fixed-seed local conformal problem.
+No existence of a solution branch for every seed is asserted.
+The proof does not use a general spacetime Penrose inequality,
+a constrained extension through the boundary, or a global
+spacetime development.
diff --git a/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/build/sections/01-setting.tex b/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/build/sections/01-setting.tex
new file mode 100644
index 0000000000000000000000000000000000000000..aa2dc7e1983ab5cd70b56d0323bc9c4c429a0a39
--- /dev/null
+++ b/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/build/sections/01-setting.tex
@@ -0,0 +1,248 @@
+\section{Geometric setting and the theorem}\label{sec:setting}
+
+Fix $m_0>0$, and let $a>0$ be the unique positive root of
+$1+a^2-2m_0/a=0$. On
+\[
+ M=[0,\infty)\times S^2,\qquad S=\{0\}\times S^2,
+\]
+let
+\begin{equation}\label{eq:background}
+ g_0=\dd s^2+r(s)^2\sigma,\qquad
+ r(0)=a,\qquad r'(s)>0\quad(s>0),\qquad
+ (r')^2=1+r^2-\frac{2m_0}{r},
+\end{equation}
+where $\sigma$ is the unit round metric. The coordinate $s$ is smooth
+at the boundary, and $n=\partial_s$ points from $S$ toward infinity.
+On the end, $r=r(s)$ is a coordinate and
+\begin{equation}\label{eq:hyperbolic-reference}
+ b=\frac{\dd r^2}{1+r^2}+r^2\sigma
+   =\dd\eta^2+\sinh^2\eta\,\sigma,\qquad
+ \eta=\operatorname{arsinh}r.
+\end{equation}
+In particular, $r$ is asymptotic to a positive constant times $e^s$.
+
+Our Laplacian is $\Delta=\Div\grad$, with nonpositive eigenvalues
+on a closed manifold. The spaces $C^{k,\alpha}_\tau$ use the
+hyperbolic metric $b$ in the end: the tensor and its covariant
+derivatives through order $k$ are $O(e^{-\tau s})$, and the
+correspondingly weighted $\alpha$-H\"older seminorms on unit
+background balls are bounded. On a compact set we use ordinary
+$C^{k,\alpha}$ regularity. This definition is equivalent to one
+using hyperbolic distance.
+
+Fix $0<\alpha<1$, $3/2<\tau<3$, and a smooth symmetric tensor
+$q\in C^{1,\alpha}_\tau$, smooth up to $S$, satisfying
+\begin{equation}\label{eq:tt-seed}
+ \tr_{g_0}q=0,\qquad \Div_{g_0}q=0.
+\end{equation}
+Throughout the paper the seed $q$ is independent of $\eps$.
+Suppose a smooth branch of positive solutions, defined for
+$0\leq\eps<\eps_*$, satisfies
+\begin{align}
+ \Delta_{g_0}\phi_\eps
+ &=\frac34(\phi_\eps^5-\phi_\eps)
+   -\frac{\eps^2}{8}|q|_{g_0}^2\phi_\eps^{-7}
+       &&\text{on }M,\label{eq:conformal-equation}\\
+ \partial_s\phi_\eps
+ &=\frac{\eps}{4}q_{ss}\phi_\eps^{-3}
+       &&\text{on }S,\label{eq:conformal-boundary}\\
+ \phi_0&=1,\qquad
+ \phi_\eps-1\in C^{2,\alpha}_\tau,\qquad
+ \|\phi_\eps-1\|_{C^{2,\alpha}_\tau}\longrightarrow0
+       &&\text{as }\eps\longrightarrow0.\label{eq:branch}
+\end{align}
+No differentiability in stronger weighted spaces is assumed.
+Set
+\begin{equation}\label{eq:actual-data}
+ g_\eps=\phi_\eps^4g_0,\qquad
+ K_\eps=\eps\phi_\eps^{-2}q,\qquad
+ A_\eps=|S|_{g_\eps}.
+\end{equation}
+
+\subsection{Mass and boundary conventions}
+
+Let $x_i$, $1\leq i\leq3$, be the coordinate functions on the unit
+sphere. Define $V_0=\sqrt{1+r^2}$, $V_i=rx_i$, and
+$e_\eps=g_\eps-b$. We use the flux normalization
+\begin{equation}\label{eq:mass-flux}
+\begin{split}
+ p_\mu(\eps)=\frac1{16\pi}\lim_{R\to\infty}
+ \int_{\{r=R\}}\big[
+ &V_\mu(\Div_b e_\eps-\dd\tr_b e_\eps)\\
+ &+(\tr_b e_\eps)\dd V_\mu
+      -e_\eps(\grad_bV_\mu,\cdot)
+ \big](\nu_b)\,\dd A_b ,
+\end{split}
+\end{equation}
+where $\nu_b$ points toward increasing $r$. All operators and the
+area form in this formula are those of $b$. Assume that these
+limits are finite and that the covector is future timelike for
+all sufficiently small positive $\eps$. Thus
+\[
+ m_{\AH}(\eps)=\sqrt{p_0(\eps)^2-\sum_{i=1}^3p_i(\eps)^2}
+\]
+is the positive mass norm. This normalization gives $p_0(0)=m_0$
+and $p_i(0)=0$. The $p_i$ are spatial components of the hyperbolic
+mass covector, not angular momentum charges.
+
+For a two-sided surface with unit normal $\nu$ toward the chosen
+end, put
+\[
+ H=\Div_\Sigma\nu,\qquad
+ \theta_+=H+\tr_\Sigma K.
+\]
+For a spacetime realization our convention is
+$K(X,Y)=\bar g(\bar\nabla_Xn_{\mathrm{future}},Y)$;
+in Gaussian normal coordinates this means $\partial_tg=2K$.
+A future marginally outer trapped surface, or MOTS, has
+$\theta_+=0$.
+
+The standard conformal identities give
+\begin{equation}\label{eq:constraints}
+ \tr_{g_\eps}K_\eps=0,\qquad
+ \Div_{g_\eps}K_\eps=0,\qquad
+ R_{g_\eps}+6=|K_\eps|_{g_\eps}^2.
+\end{equation}
+Indeed, for a tracefree $q$ the divergence identity is
+$\Div_{\phi^4g_0}(\phi^{-2}q)=\phi^{-6}\Div_{g_0}q$.
+Also
+\[
+ R_{\phi^4g_0}
+ =\phi^{-5}(-8\Delta_{g_0}\phi-6\phi)
+ =-6+\eps^2\phi^{-12}|q|_{g_0}^2.
+\]
+The boundary $S$ is totally geodesic for $g_0$.
+Its normal in $g_\eps$ is $\phi_\eps^{-2}\partial_s$, and
+\begin{equation}\label{eq:boundary-mots}
+ H_{g_\eps}(S)=4\phi_\eps^{-3}\partial_s\phi_\eps
+       =\eps\phi_\eps^{-6}q_{ss},\qquad
+ \tr_SK_\eps=-\eps\phi_\eps^{-6}q_{ss}.
+\end{equation}
+Consequently it is a future MOTS. Its mean curvature need not vanish.
+
+\subsection{Statement}
+
+Define
+\begin{equation}\label{eq:deficit}
+ b_*(A)=\sqrt{\frac A{16\pi}}\left(1+\frac A{4\pi}\right),
+ \qquad
+ \mathfrak D_q(\eps)=m_{\AH}(\eps)-b_*(A_\eps).
+\end{equation}
+A tensor is called \emph{radial} if it is invariant under every
+rotation of the $S^2$ factor.
+
+\begin{theorem}\label{thm:main}
+For every background, fixed TT seed, and solution branch satisfying
+\eqref{eq:background}--\eqref{eq:branch} and the mass assumptions
+above, there is $0<\eps_0\leq\eps_*$ such that
+\[
+ m_{\AH}(\eps)\geq
+ \sqrt{\frac{A_\eps}{16\pi}}\left(1+\frac{A_\eps}{4\pi}\right)
+ \qquad(0\leq\eps<\eps_0).
+\]
+For all sufficiently small positive $\eps$, equality holds if the
+seed is radial, and the inequality is strict otherwise.
+\end{theorem}
+
+The theorem makes no sign assumption on $q_{ss}|_S$.
+It also does not require outermostness or outer area-minimization.
+In particular it implies the following formulation with the usual
+geometric boundary hypotheses. Here an enclosing surface separates $S$
+from the end and bounds with $S$ a compact region. Write $A_{\min}(S)$
+for the infimum of areas of smooth enclosing surfaces, allowing
+disconnected competitors and $S$ itself.
+
+\begin{corollary}\label{cor:penrose}
+In addition to the hypotheses of Theorem~\ref{thm:main}, suppose
+$q_{ss}\geq0$ on $S$. Assume, for every sufficiently small positive
+$\eps$, that no compact smooth embedded two-sided surface in the
+interior enclosing $S$ has $\theta_+\leq0$ everywhere, and that
+every smooth enclosing surface has area at least $A_\eps$.
+Disconnected enclosing competitors are allowed, and $S$ itself is
+allowed in the area infimum. Then the exact local Penrose inequality
+holds with $A_{\min}(S)=A_\eps$.
+\end{corollary}
+
+The outermostness condition excludes
+all weakly future outer trapped enclosing surfaces, not only
+additional MOTS. These additional assumptions are relevant to the
+geometric formulation of the conjecture but are not needed in the
+coefficient argument below.
+
+\subsection{Static operators and Taylor notation}\label{sec:taylor}
+
+We use
+\begin{equation}\label{eq:static-operators}
+ \begin{gathered}
+ N=r',\qquad
+ \kappa=N'(0)=\frac{1+3a^2}{2a},\qquad
+ B=\Ric_{g_0}+3g_0,\\
+ L=\Delta_{g_0}-3,\qquad
+ D=-\Delta_{a^2\sigma}+\frac{2\kappa}{a}.
+ \end{gathered}
+\end{equation}
+The letter $D$ denotes the boundary operator, while
+$\mathfrak D_q$ denotes the mass deficit.
+
+\begin{lemma}\label{lem:static}
+The tensor $B$ and the static lapse satisfy
+\begin{gather}
+ B=B_s\,\dd s^2+B_t r^2\sigma,\qquad
+ B_s=1-\frac{2m_0}{r^3},\qquad
+ B_t=1+\frac{m_0}{r^3},\label{eq:B-components}\\
+ \nabla^2N=NB,\qquad
+ \tr_{g_0}B=3,\qquad \Div_{g_0}B=0,\qquad
+ LN=0.\label{eq:static-identities}
+\end{gather}
+Moreover $D$ is positive on all spherical harmonics and
+\begin{equation}\label{eq:boundary-constants}
+ b_*(4\pi a^2)=m_0,\qquad
+ b_*'(4\pi a^2)=\frac{\kappa}{8\pi}.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+The radial and tangential sectional curvatures of
+$\dd s^2+r^2\sigma$ are $-r''/r$ and $(1-(r')^2)/r^2$.
+Differentiating \eqref{eq:background} gives
+$N'=r+m_0/r^2$ and $N''=N(1-2m_0/r^3)$.
+The Ricci eigenvalues are therefore $-2-2m_0/r^3$ and
+$-2+m_0/r^3$, giving \eqref{eq:B-components} and $R_{g_0}=-6$.
+The radial and tangential Hessian eigenvalues of $N$ are
+$N''$ and $N'N/r$, respectively, proving the first identity
+in \eqref{eq:static-identities}. Its trace gives $LN=0$.
+The divergence identity follows from the contracted Bianchi
+identity and the constant scalar curvature.
+At $S$, $B_t=\kappa/a$ and $2\kappa/a>0$, so $D>0$.
+Finally $2m_0=a(1+a^2)$ gives \eqref{eq:boundary-constants}
+by direct differentiation.
+\end{proof}
+
+We write $[\mathcal F]_j$ for the coefficient of $\eps^j$ in a
+Taylor expansion; coefficients do not include an extra factorial.
+Thus $u_j[q]=[\phi_\eps-1]_j$ and $c_j[q]=[\mathfrak D_q]_j$
+when the indicated actual expansions have been established.
+For a tensor $T$ we also use the weighted function norm
+\[
+ \|T\|_\beta=\sup_M e^{\beta s}|T|_{g_0}.
+\]
+On the end this is equivalent to using $b$.
+Finite angular type means that the span of all rotational
+pullbacks of the function or tensor is finite-dimensional.
+For functions this is precisely a finite sum of spherical-harmonic
+spaces, with arbitrary smooth radial coefficients.
+
+Some geometric arguments use Taylor families of metrics and
+surfaces only to order $p$, with identities read in the finite Taylor
+algebra $\mathbb R[\eps]/(\eps^{p+1})$. On any compact radial interval,
+their coefficients can be represented by a smooth family and
+all differential-geometric identities then hold to the retained
+order. At $S$, a formal displacement may have negative first
+coefficient. It is evaluated using the given smooth one-sided
+jets: extend sufficiently many coefficients smoothly across
+$s=0$, perform the finite Taylor calculation, and retain only
+the prescribed order. Every boundary derivative of an identity
+holding on $s\geq0$ is fixed by those jets. This convention
+does not assume a constrained extension at a fixed point with
+$s<0$. The time-recursion argument below will specify how the
+constraint identities are used after such a displacement.
diff --git a/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/build/sections/02-analysis.tex b/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/build/sections/02-analysis.tex
new file mode 100644
index 0000000000000000000000000000000000000000..c061066fd5b934dadf6eea7d378af8b96e9db3c4
--- /dev/null
+++ b/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/build/sections/02-analysis.tex
@@ -0,0 +1,616 @@
+\section{Weighted coefficients and actual expansions}
+\label{ana:section}
+
+We first justify the passage between finite Taylor calculations and the
+given solution branch.  Throughout this section, fix
+\begin{equation}
+  \frac32<\beta<\min\{\tau,\sqrt3\}.
+  \label{ana:beta}
+\end{equation}
+We use the weighted norms and Taylor conventions of
+Section~\ref{sec:taylor}. Boundary norms without a weight are uniform
+norms on $S$.
+
+\subsection{A scalar inverse and its asymptotics}
+
+Equivariant linear differential operations preserve finite angular type.  Products and contractions preserve
+it after enlarging the angular space by a finite tensor product.  On
+functions, the round Laplacian preserves these spaces: it is the sum of
+squares of the three infinitesimal rotation fields.  Its restriction is
+self-adjoint, so we may split into finitely many eigenspaces with
+eigenvalues $\lambda\geq0$ for $-\Delta_\sigma$.
+
+\Needspace{9\baselineskip}
+\begin{lemma}[Comparison and finite angular inverse]
+\label{lem:inverse}
+There are constants $c_\beta,C_\beta>0$ with the following properties.
+Suppose $z=o(e^{-\beta s})$ uniformly at infinity and
+\[
+  (L-V)z=F,\qquad \|V\|_{C^0(M)}\leq c_\beta.
+\]
+If either $z|_S=b$ or $\partial_s z|_S=b$, then
+\begin{equation}
+  \|z\|_\beta\leq C_\beta
+       \bigl(\|F\|_\beta+\|b\|_{C^0(S)}\bigr).
+  \label{ana:comparison}
+\end{equation}
+The constants do not depend on an angular cutoff.  The same estimate
+holds for $\partial_s z-a_*z=b$ if
+$\|a_*\|_{C^0(S)}\leq\beta/2$.
+
+If $F$ and $b$ are smooth and of finite angular type, and
+$F=O(e^{-\gamma s})$ for some $\gamma>3$, there is a unique decaying
+finite angular type solution of $Lu=F$ with either of the above
+Dirichlet or Neumann data.  It satisfies
+\begin{equation}
+  u=r^{-3}\zeta(x)+o(r^{-3}),\qquad
+  \partial_s\bigl(u-r^{-3}\zeta(x)\bigr)=o(r^{-3})
+  \label{ana:inverse-asymptotic}
+\end{equation}
+for a smooth finite angular type function $\zeta$.  These statements
+hold after any fixed number of angular differentiations.  If
+$\partial_s^jF=O(e^{-\gamma s})$ for $0\leq j\leq k$, the remainder
+in \eqref{ana:inverse-asymptotic} can also be differentiated in $s$
+through order $k+2$, with each resulting remainder $o(r^{-3})$.
+\end{lemma}
+
+\begin{proof}
+Let $h=e^{-\beta s}$.  Since $N/r\geq0$,
+\[
+  Lh=(\beta^2-2\beta N/r-3)h
+       \leq-(3-\beta^2)h.
+\]
+Choose $c_\beta<(3-\beta^2)/2$.  A sufficiently large multiple
+$Ah$ satisfies $(L-V)(Ah)\leq-|F|$.  For Dirichlet data, enlarge
+$A$ to dominate $|b|$.  For Neumann data, enlarge it so that
+$-\beta A-b<0$ and $-\beta A+b<0$ on $S$.
+The functions $Ah-z$ and $Ah+z$ are positive sufficiently far out.
+A negative interior minimum contradicts their differential inequalities,
+because the zeroth-order coefficient of $L-V$ is negative.
+A negative minimum on $S$ is impossible in the Neumann case: the
+inward derivative at such a minimum is nonnegative, whereas the
+chosen derivative is negative.  This proves \eqref{ana:comparison}.
+For the Robin condition, apply the maximum argument to $z/h$.
+At a positive boundary maximum its inward derivative is nonpositive,
+whereas
+$\partial_s(z/h)=(\beta+a_*)(z/h)+b$ on $S$.
+Since $\beta+a_*\geq\beta/2$, a sufficiently large positive maximum
+is impossible; apply the same argument to $-z/h$.
+
+For the existence assertion, work in each angular eigenspace.  The
+radial equation is
+\begin{equation}
+ y''+2\frac Nr y'-\left(3+\frac\lambda{r^2}\right)y=F_\lambda.
+ \label{ana:radial-ode}
+\end{equation}
+First impose the required condition at $0$ and $y(T)=0$ on a finite
+interval.  The homogeneous problem has trivial kernel: multiplication
+by $r^2y$ and integration gives
+\[
+  0=-\int_0^T r^2(y')^2\,\dd s
+    -\int_0^T(3r^2+\lambda)y^2\,\dd s,
+\]
+with zero boundary terms.  Thus the finite-interval linear problem is
+solvable.  The barrier estimate is independent of $T$.  On each compact
+interval, the equation bounds $y'$ and then $y''$: the mean value
+theorem supplies a point with bounded $y'$, and the first-order equation
+for $y'$ propagates that bound.  A subsequence therefore converges on
+compact intervals to a solution with $y=O(e^{-\beta s})$.
+The same local argument on unit intervals gives
+$y'=O(e^{-\beta s})$.
+
+Now $N/r=1+O(e^{-2s})$ and $r^{-2}=O(e^{-2s})$.  Hence
+\[
+  y''+2y'-3y=G,\qquad
+  G=O(e^{-\gamma' s}),\qquad
+  \gamma'=\min\{\gamma,\beta+2\}>3.
+\]
+Variation of constants for the roots $1,-3$ eliminates the growing
+homogeneous term and gives
+\[
+ y=c e^{-3s}+O(e^{-\gamma' s}),\qquad
+ y'=-3c e^{-3s}+O(e^{-\gamma' s}).
+\]
+For example the particular solution is a linear combination of
+$e^s\int_s^\infty e^{-t}G(t)\,\dd t$ and
+$e^{-3s}\int_s^\infty e^{3t}G(t)\,\dd t$; both integrals converge
+with the claimed bounds.  Since
+$r=C e^s(1+O(e^{-2s}))$, this is
+\eqref{ana:inverse-asymptotic}.  The equation gives the second
+differentiated remainder, and differentiated equations give the rest.
+Angular differentiations are bounded operations on the fixed finite
+spaces.  Any other solution tending uniformly to zero also satisfies
+the initial weighted bound: compare on sufficiently long intervals
+with $Ah+\delta$, where $\delta>0$ dominates the far boundary value,
+and then let $\delta\downarrow0$.  The same ODE improvement applies.
+The improved decay permits comparison for a difference of two
+solutions, proving uniqueness.
+\end{proof}
+
+\subsection{Coefficient equations and the required spatial derivatives}
+
+Put $\psi=\phi-1$, $h_q=q_{ss}|_S$, and $Q_q=|q|_{g_0}^2$.  Define
+\begin{align}
+ \mathcal A(z)&=\frac{15}{2}z^2+\frac{15}{2}z^3
+                    +\frac{15}{4}z^4+\frac34z^5,\notag\\
+ \mathcal N_{\eps,q}(z)&=\mathcal A(z)
+                    -\frac{\eps^2}{8}Q_q(1+z)^{-7},\notag\\
+ \mathcal B_{\eps,q}(z)&=\frac{\eps}{4}h_q(1+z)^{-3}.
+ \label{ana:nonlinearities}
+\end{align}
+The actual equations become
+\begin{equation}
+ L\psi=\mathcal N_{\eps,q}(\psi),\qquad
+ \partial_s\psi|_S=\mathcal B_{\eps,q}(\psi|_S).
+ \label{ana:actual-equations}
+\end{equation}
+
+For a finite angular type seed, define $u_j=u_j[q]$ recursively.
+Set $P_{j-1}=\sum_{i<j}\eps^iu_i$, and let
+\begin{equation}
+ \begin{gathered}
+ F_j=[\mathcal N_{\eps,q}(P_{j-1})]_j,
+ \qquad B_j=[\mathcal B_{\eps,q}(P_{j-1}|_S)]_j,\\
+ Lu_j=F_j,\qquad \partial_su_j|_S=B_j,
+ \qquad u_j\longrightarrow0.
+ \end{gathered}
+ \label{ana:coefficient-recursion}
+\end{equation}
+Here $[\cdot]_j$ extracts the coefficient of $\eps^j$; inverse powers
+are Taylor expanded at $1$.  The first two equations are explicitly
+\begin{align}
+ F_1&=0,& B_1&=\tfrac14h_q,\notag\\
+ F_2&=\tfrac{15}{2}u_1^2-\tfrac18Q_q,
+ & B_2&=-\tfrac34h_qu_1|_S.
+ \label{ana:first-two}
+\end{align}
+Only preceding coefficients occur on the right of
+\eqref{ana:coefficient-recursion}.  The construction does not require
+an actual solution branch for the seed.
+
+\begin{lemma}[Finite coefficient asymptotics]
+\label{ana:coefficients}
+For every smooth finite angular type seed in $C^{1,\alpha}_\tau$,
+the coefficients through degree two are well defined and
+\[
+  u_j=r^{-3}\zeta_j(x)+o(r^{-3}),\qquad j=1,2.
+\]
+The first coefficient has this asymptotic with arbitrarily many
+spatial derivatives.  The second has it with the first spatial
+derivatives, in particular with every derivative needed at degree
+two below.
+
+For the fourth-order construction, suppose specifically that
+\begin{equation}
+ \begin{gathered}
+ q=\Hess v-Bv,\qquad Lv=0,\qquad v\longrightarrow0,\\
+ v|_S\text{ has only constant and degree-one spherical modes}.
+ \end{gathered}
+ \label{ana:kernel-seed}
+\end{equation}
+Corollary~\ref{cor:kernel} will show that every seed with zero quadratic
+deficit has this form, so fourth-order estimates are needed only in
+this class. Then $q$ and all its derivatives have $O(r^{-3})$ scaled
+components.
+The coefficients $u_1,\ldots,u_4$ exist and have the above asymptotic
+with arbitrarily many spatial derivatives.
+
+Let $p=2$ in the first case and $p=4$ in the second.  In the coordinate
+$\eta=\operatorname{arsinh}r$, the conformal metric jet
+$G=(1+\sum_{j=1}^p\eps^ju_j)^4g_0$, taken through degree $p$, satisfies
+\begin{equation}
+ \begin{gathered}
+ G-b=r^{-3}\bigl(C\,\dd\eta^2+E r^2\sigma\bigr)+o(r^{-3}),\\
+ C=2m_0+4\sum_{j=1}^p\eps^j\zeta_j,\qquad
+ E=4\sum_{j=1}^p\eps^j\zeta_j.
+ \end{gathered}
+ \label{ana:metric-asymptotic}
+\end{equation}
+For coefficient $j$, the remainder has this decay after angular and
+$\eta$ derivatives through order $p-j+1$, in scaled coframes
+$\dd\eta,r\,\dd x$.
+\end{lemma}
+
+\begin{proof}
+Lemma~\ref{lem:inverse} constructs $u_1$ with homogeneous interior
+equation.  Its derivatives have the asserted asymptotics by
+\eqref{ana:radial-ode}.  In degree two the interior source is a sum
+of $u_1^2=O(r^{-6})$ and $|q|^2=O(r^{-2\tau})$, with
+$\min\{6,2\tau\}>3$.  Its first radial derivative has the same
+bound because $q\in C^{1,\alpha}_\tau$.  Finite angular type supplies
+any fixed number of angular derivatives.  The inverse lemma therefore
+gives the claimed second coefficient and, in particular, its first
+differentiated asymptotic.  No bound on higher radial derivatives of a
+general seed is asserted here.
+
+For \eqref{ana:kernel-seed}, solve the homogeneous Dirichlet problem
+separately in its four boundary modes.  The homogeneous ODE gives
+$v=O(r^{-3})$ with all differentiated versions.  The scaled components
+of the connection, $B$, and all their derivatives are bounded at
+infinity.  It follows that $\Hess v-Bv$ has $O(r^{-3})$ components
+with all derivatives.  Inductively, every interior source in
+\eqref{ana:coefficient-recursion} is a sum of products containing at
+least two factors among lower solution coefficients and $q$.
+It therefore has $O(r^{-6})$ decay with all derivatives.  Another
+application of the inverse lemma completes the induction through
+degree four.
+
+Finally $g_0-b=2m_0r^{-3}\dd\eta^2+O(r^{-5})$ in scaled components,
+with all derivatives.  In each positive degree the only $r^{-3}$
+term in $(1+\sum\eps^ju_j)^4-1$ is $4r^{-3}\zeta_j$;
+products of two solution coefficients are $O(r^{-6})$.
+This proves \eqref{ana:metric-asymptotic}.  At $p=2$, the derivative
+count is two for coefficient one and one for coefficient two, both
+already established.  At $p=4$ all the needed derivatives are
+available from the stronger case.
+\end{proof}
+
+\subsection{An exact Green formula for the mass}
+
+Define the test functions
+\begin{equation}
+ W_0=N,\qquad W_i=rx_i\quad(1\leq i\leq3).
+ \label{ana:tests}
+\end{equation}
+The background identities and the radial equation give
+\begin{equation}
+ LW_0=0,\qquad LW_i=-3m_0r^{-2}x_i.
+ \label{ana:tests-equations}
+\end{equation}
+
+\begin{lemma}[Green representation of the mass]
+\label{lem:green}
+For every sufficiently small member of the actual branch,
+\begin{equation}
+ p_\mu(\eps)-p_\mu(0)
+ =\frac1{2\pi}\left\{
+   \int_S\bigl(\psi\partial_sW_\mu-W_\mu\partial_s\psi\bigr)\,\dd A_{g_0}
+   -\int_M\bigl(W_\mu L\psi-\psi LW_\mu\bigr)\,\dd V_{g_0}
+                 \right\}.
+ \label{ana:green}
+\end{equation}
+In particular the right-hand side is an absolutely convergent integral
+expression.  More generally the linear functional
+\begin{equation}
+ \mathcal G_\mu(u,F,B)
+ =\frac1{2\pi}\left\{
+ \int_S(u\partial_sW_\mu-W_\mu B)\,\dd A_{g_0}
+ -\int_M(W_\mu F-uLW_\mu)\,\dd V_{g_0}\right\}
+ \label{ana:green-functional}
+\end{equation}
+is bounded in the norms
+$\|u\|_\beta+\|F\|_{2\beta}+\|B\|_{C^0(S)}$.
+For finite angular type coefficient jets, this same functional computes
+the coefficient of their metric mass.
+\end{lemma}
+
+\begin{proof}
+First fix $\eps$; no parameter limit is taken in this argument.
+The assumed weighted regularity gives
+$\psi,\nabla^b\psi=O(r^{-\tau})$.  The difference between the actual
+metric perturbation $\phi^4g_0-g_0$ and $4\psi b$ consists of terms
+with scaled components and first derivatives
+\[
+ O(r^{-2\tau})+O(r^{-\tau-3}).
+\]
+Indeed $\phi^4-1-4\psi=O(\psi^2)$ and
+$g_0-b=O_1(r^{-3})$.  Since a mass test function and its derivative
+have size $O(r)$ and the sphere area has size $O(r^2)$, these terms
+give flux errors
+\begin{equation}
+ O(r^{3-2\tau})+O(r^{-\tau})\longrightarrow0.
+ \label{ana:flux-errors}
+\end{equation}
+For the tensor $4\psi b$ in dimension three,
+\[
+ \Div_b(4\psi b)-\dd\tr_b(4\psi b)=-8\dd\psi,
+\]
+and the remaining two terms of the prescribed mass integrand add
+$8\psi\,\dd V_\mu$.  Thus the flux difference is $1/(2\pi)$ times
+the flux of $\psi\grad_bV_\mu-V_\mu\grad_b\psi$.
+Replacing the hyperbolic normal by $\partial_s$ produces a vanishing
+error since the relative normal change is $O(r^{-3})$.
+For the time test, replacing $\sqrt{1+r^2}$ by $N$ also produces a
+vanishing error, because their difference and its first radial
+derivative are $O(r^{-2})$.  The sphere area forms are the same.
+
+The $g_0$ divergence of
+$\psi\grad W_\mu-W_\mu\grad\psi$ is
+$\psi LW_\mu-W_\mu L\psi$.  Apply the divergence theorem to
+$[0,T]\times S^2$.  The outward normal of this integration domain
+at its inner boundary is $-\partial_s$, yielding exactly the sign of
+the boundary term in \eqref{ana:green}.  Finally
+$L\psi=O(r^{-2\tau})$ by \eqref{ana:actual-equations}; hence the
+bulk integrals converge, and $T\to\infty$ proves the identity.
+
+For the asserted boundedness, $W_\mu=O(e^s)$ and
+$\dd V_{g_0}=O(e^{2s})\,\dd s\,\dd\omega$.
+The $W_\mu F$ term is dominated by a constant times
+\[
+ \|F\|_{2\beta}e^{(3-2\beta)s},
+\]
+which is integrable by \eqref{ana:beta}.  The $uLW_i$ term is
+dominated by $C\|u\|_\beta e^{-\beta s}$, and $LW_0=0$.
+The boundary terms are bounded by the stated norms.
+For finite coefficient jets, apply the same finite-radius identity
+coefficientwise.  Lemma~\ref{ana:coefficients} supplies their
+differentiated asymptotics, and all nonlinear coefficient products
+discarded from the flux are $O(r^{-6})$ with first derivatives.
+Thus their coefficient flux is also
+\eqref{ana:green-functional}.
+\end{proof}
+
+\subsection{Approximation on the full space of seeds}
+
+Let $\mathscr T_{\tau,\alpha}$ denote the real vector space of smooth
+TT tensors in $C^{1,\alpha}_\tau$.  No boundary sign is imposed on
+this space.  We shall construct coefficient functionals on it even
+when no actual branch is being considered.
+
+\begin{lemma}[Rotation approximation and coefficient continuity]
+\label{lem:rotation}
+Every $q\in\mathscr T_{\tau,\alpha}$ is a limit, in $\|\cdot\|_\beta$,
+of finite angular type tensors $q_\nu\in\mathscr T_{\tau,\alpha}$.
+The approximants may be chosen as averages of rotated copies of $q$.
+Their weighted norms are uniformly bounded, and the construction
+preserves a nonnegative boundary value $q_{ss}|_S$ when that condition
+holds.
+
+The maps $q\mapsto u_1[q]$ and $q\mapsto u_2[q]$ defined for finite
+angular type seeds extend uniquely by these approximations to
+continuous, respectively linear and quadratic, maps on
+$\mathscr T_{\tau,\alpha}$ with values in the weighted uniform norm.
+The corresponding pairs $(F_j,B_j)$ in \eqref{ana:first-two}
+extend continuously in the norms $\|\cdot\|_{2\beta}$ and
+$\|\cdot\|_{C^0(S)}$.
+\end{lemma}
+
+\begin{proof}
+Let $U(R)q$ denote the pullback by a rotation $R\in SO(3)$, and
+let $\dd R$ be normalized invariant volume.  Define
+\[
+ k_\nu(R)=c_\nu\left(\frac{1+\tr R}{4}\right)^\nu,
+ \qquad \int_{SO(3)}k_\nu\,\dd R=1,
+ \qquad q_\nu=\int_{SO(3)}k_\nu(R)U(R)q\,\dd R.
+\]
+The expression raised to the power $\nu$ lies in $[0,1]$ and attains
+one only at the identity.  The normalized densities therefore
+concentrate there.  To verify this directly, outside any fixed
+neighborhood their base is at most $1-\delta$, whereas on a smaller
+neighborhood of positive volume it is at least $1-\delta/2$;
+the ratio of the corresponding integrals tends to zero.
+
+The map $R\mapsto U(R)q$ is continuous in $\|\cdot\|_\beta$.
+On a compact part of $M$ this follows from smoothness.  On the tail,
+the bound $Ce^{-(\tau-\beta)s}$ is uniform in $R$, so the strict
+weight margin controls it.  Concentration gives
+$\|q_\nu-q\|_\beta\to0$.
+Rotations preserve $g_0$, the end background, and $s$.  They commute
+with trace and divergence and preserve the weighted regularity.
+Consequently each average is smooth and TT, with the asserted uniform
+bounds.  They also preserve the radial normal, so positivity of
+$q_{ss}|_S$ is retained by the nonnegative average.
+
+For fixed $\nu$, the density is a polynomial in matrix entries.
+After a change of integration variable, the rotation orbit of $q_\nu$
+depends on translates of this polynomial.  These translates lie in a
+finite-dimensional polynomial space; their coefficients multiply
+finitely many fixed averaged tensors.  Thus $q_\nu$ has finite angular
+type.
+
+It remains to check that the coefficient limits are independent of
+the approximation.  On a bounded set in $\|q\|_\beta$, comparison
+applied to \eqref{ana:first-two} gives, for finite type seeds $q,\hat q$,
+\begin{align*}
+ \|u_1[q]-u_1[\hat q]\|_\beta
+     &\leq C\|q-\hat q\|_\beta,\\
+ \|F_2[q]-F_2[\hat q]\|_{2\beta}
+  +\|B_2[q]-B_2[\hat q]\|_{C^0(S)}
+     &\leq C\|q-\hat q\|_\beta,\\
+ \|u_2[q]-u_2[\hat q]\|_\beta
+     &\leq C\|q-\hat q\|_\beta.
+\end{align*}
+Here the elementary product estimate is
+\[
+ \bigl\||q|^2-|\hat q|^2\bigr\|_{2\beta}
+ \leq(\|q\|_\beta+\|\hat q\|_\beta)\|q-\hat q\|_\beta.
+\]
+The comparison constant has no angular-cutoff dependence.
+These estimates prove existence, uniqueness, and continuity of the
+limits in the complete weighted uniform space.  Formula
+\eqref{ana:first-two} preserves their linear and quadratic dependence.
+No derivative convergence of the limiting coefficients is needed in
+this construction or in the mass functional.
+\end{proof}
+
+\subsection{Actual expansions and mass coefficients}
+
+\begin{proposition}[Expansions of the given branch]
+\label{prop:expansions}
+For every prescribed seed and branch in the theorem, set $p=2$ and
+take $u_j,F_j,B_j$ from Lemma~\ref{lem:rotation}.  If the seed has the
+form \eqref{ana:kernel-seed}, we may instead take $p=4$ with the
+finite coefficient construction.  In both cases,
+\begin{align}
+ \left\|\psi-\sum_{j=1}^p\eps^ju_j\right\|_\beta
+     &=O(\eps^{p+1}),\notag\\
+ \left\|L\psi-\sum_{j=1}^p\eps^jF_j\right\|_{2\beta}
+     &=O(\eps^{p+1}),\notag\\
+ \left\|\partial_s\psi|_S-\sum_{j=1}^p\eps^jB_j\right\|_{C^0(S)}
+     &=O(\eps^{p+1}).
+ \label{ana:three-remainders}
+\end{align}
+The area and all four mass components have actual expansions to the
+same order.  Their mass coefficients are
+\begin{equation}
+ a_{\mu j}[q]=\mathcal G_\mu(u_j,F_j,B_j),\qquad
+ p_\mu(\eps)=p_\mu(0)+\sum_{j=1}^p\eps^j a_{\mu j}[q]
+                      +O(\eps^{p+1}).
+ \label{ana:mass-coefficients}
+\end{equation}
+
+In particular, the actual deficit satisfies
+\begin{equation}
+ \mathfrak D_q(\eps)=c_2[q]\eps^2+O(\eps^3)
+ \label{ana:deficit-second}
+\end{equation}
+for every seed under consideration.  The coefficient $c_2$ is a
+continuous quadratic form on $\mathscr T_{\tau,\alpha}$ in the
+$\|\cdot\|_\beta$ topology, invariant under rotations.  For seeds
+of the form \eqref{ana:kernel-seed} there is additionally the actual
+expansion
+\begin{equation}
+ \mathfrak D_q(\eps)
+     =c_2[q]\eps^2+c_3[q]\eps^3+c_4[q]\eps^4+O(\eps^5).
+ \label{ana:deficit-fourth}
+\end{equation}
+The subsequent geometric arguments determine the signs and the
+vanishing of these coefficients.
+\end{proposition}
+
+\begin{proof}
+\emph{Initial estimate.}
+The given convergence of $\psi$ in $C^{2,\alpha}_\tau$ implies that
+$\|\psi\|_{C^0}$ is small.  Write
+$\mathcal A(\psi)=V_\psi\psi$, extending the quotient by zero at
+$\psi=0$.  Then $\|V_\psi\|_{C^0}=o(1)$.  Moving this term to the
+potential in \eqref{ana:actual-equations} leaves an interior source
+bounded by $C\eps^2e^{-2\tau s}$ and boundary derivative bounded
+by $C\eps$.  Since $\psi=o(e^{-\beta s})$, Lemma~\ref{lem:inverse}
+gives
+\begin{equation}
+ \|\psi\|_\beta=O(\eps).
+ \label{ana:initial-bound}
+\end{equation}
+
+\emph{Finite angular type seeds.}
+Suppose first that the needed coefficients have been constructed
+by \eqref{ana:coefficient-recursion}.  The first bound
+\eqref{ana:initial-bound} starts an induction.  If
+\[
+ \|\psi-P_{j-1}\|_\beta=O(\eps^j),
+\]
+then $\psi$ and $P_{j-1}$ are both $O(\eps e^{-\beta s})$.
+For such arguments, the mean value theorem applied to
+\eqref{ana:nonlinearities} gives
+\[
+ |\mathcal N_{\eps,q}(\psi)
+       -\mathcal N_{\eps,q}(P_{j-1})|
+ \leq C\eps e^{-\beta s}|\psi-P_{j-1}|
+ =O(\eps^{j+1}e^{-2\beta s}).
+\]
+Taylor substitution into the finite polynomial $P_{j-1}$ leaves
+an error of this same order after its coefficients through degree $j$
+are extracted.  Every interior term contains at least two decaying
+factors, counting the factor $Q_q$ as two.  On $S$,
+\[
+ |\mathcal B_{\eps,q}(\psi)
+       -\mathcal B_{\eps,q}(P_{j-1})|
+ \leq C\eps|\psi-P_{j-1}|=O(\eps^{j+1}),
+\]
+and the boundary Taylor error has the same order.  Thus
+$\psi-P_j$ has interior source
+$O(\eps^{j+1}e^{-2\beta s})$ and boundary derivative
+$O(\eps^{j+1})$.  It is $o(e^{-\beta s})$ at infinity, so comparison
+improves its weighted norm to $O(\eps^{j+1})$.
+This proves all three assertions in \eqref{ana:three-remainders}
+inductively through the required degree.  In particular, it proves
+the fourth-order assertions in the special case
+\eqref{ana:kernel-seed}.
+
+\emph{Arbitrary seeds through degree two.}
+Let $q_\nu$ be the approximants of Lemma~\ref{lem:rotation}, and
+put $d_\nu=\|q-q_\nu\|_\beta$.  Write $u_{j,\nu},F_{j,\nu},B_{j,\nu}$
+for their coefficients.  These have uniform bounds in the norms of
+that lemma.  We compare them directly with the given branch for $q$;
+no branch for $q_\nu$ is introduced.
+Using \eqref{ana:initial-bound}, the equation for
+$\psi-\eps u_{1,\nu}$ has source
+$O(\eps^2e^{-2\beta s})$ and boundary derivative
+$O(\eps d_\nu+\eps^2)$.  Hence
+\begin{equation}
+ \|\psi-\eps u_{1,\nu}\|_\beta
+       \leq C(\eps d_\nu+\eps^2).
+ \label{ana:approx-first}
+\end{equation}
+Expanding the nonlinearities once more, formula
+\eqref{ana:first-two} and \eqref{ana:approx-first} imply
+\begin{align}
+ \|L\psi-\eps^2F_{2,\nu}\|_{2\beta}
+   &\leq C(\eps^2d_\nu+\eps^3),\notag\\
+ \|\partial_s\psi|_S-\eps B_{1,\nu}-\eps^2B_{2,\nu}\|_{C^0(S)}
+   &\leq C\bigl((\eps+\eps^2)d_\nu+\eps^3\bigr).
+ \label{ana:approx-sources}
+\end{align}
+For clarity, the only quadratic interior discrepancies are
+$\psi^2-\eps^2u_{1,\nu}^2$ and
+$\eps^2(Q_q-Q_{q_\nu})$; their weighted norms are bounded by the
+first right-hand side.  At the boundary the linear discrepancy is
+$\eps(h_q-h_{q_\nu})/4$, and the next one contains
+$\eps h_q\psi-\eps^2h_{q_\nu}u_{1,\nu}$, producing exactly the
+second bound.  All remaining terms are cubic with the stated weights.
+Comparison applied to
+$\psi-\eps u_{1,\nu}-\eps^2u_{2,\nu}$ now gives
+\[
+ \|\psi-\eps u_{1,\nu}-\eps^2u_{2,\nu}\|_\beta
+ \leq C\bigl((\eps+\eps^2)d_\nu+\eps^3\bigr).
+\]
+The constants are independent of $\nu$.  Letting $\nu\to\infty$
+and using coefficient continuity proves the full three estimates
+\eqref{ana:three-remainders} for $p=2$ and the original arbitrary seed.
+
+\emph{Mass, area, and the deficit.}
+Apply the bounded functional in Lemma~\ref{lem:green} to the three
+remainders.  This proves \eqref{ana:mass-coefficients} directly.
+The area is the compact-boundary integral
+\[
+ A_\eps=\int_S(1+\psi)^4\,\dd A_{g_0},
+\]
+so its expansion follows from the first remainder alone.  If
+$A_\eps=A_0+\eps A_1+\eps^2 A_2+O(\eps^3)$, then
+\[
+ A_1=4\int_Su_1\,\dd A_{g_0},\qquad
+ A_2=\int_S(4u_2+6u_1^2)\,\dd A_{g_0}.
+\]
+The mass norm is smooth near $(m_0,0,0,0)$ and $b_*$ is smooth near
+$A_0=4\pi a^2$.  Their expansions therefore have the same controlled
+remainders.  The constant deficit is zero.  Also $F_1=0$,
+$N|_S=0$, $\partial_sN|_S=\kappa$, and $LN=0$, so
+\[
+ a_{01}=\frac\kappa{2\pi}\int_Su_1\,\dd A_{g_0}
+        =b_*'(A_0)A_1.
+\]
+The spatial mass components have zero base value and first enter
+the mass norm quadratically.  This proves that the linear deficit
+coefficient vanishes.
+
+In degree two the scalar expression is explicitly
+\begin{equation}
+ c_2[q]=a_{02}
+       -\frac1{2m_0}\sum_{i=1}^3a_{i1}^2
+       -b_*'(A_0)A_2-\frac12b_*''(A_0)A_1^2.
+ \label{ana:quadratic-functional}
+\end{equation}
+Lemma~\ref{lem:rotation} and the bounded Green functional show that
+this is a continuous quadratic form on the entire formal seed space
+$\mathscr T_{\tau,\alpha}$.  Formula \eqref{ana:quadratic-functional}
+uses only the coefficient functions and absolutely convergent Green
+integrals; it does not assume a geometric integral formula for
+arbitrary seeds.  Rotations leave the area invariant, fix the time
+mass component, and rotate the three spatial components.  Thus $c_2$
+is rotation invariant.  The identical finite coefficient argument
+through degree four proves \eqref{ana:deficit-fourth}.
+\end{proof}
+
+\begin{remark}
+\label{ana:order-of-limits}
+The derivative control used in \eqref{ana:flux-errors} comes from each
+actual member's assumed $C^{2,\alpha}_\tau$ decay.  It is not a
+consequence of the weighted uniform Taylor estimates.  The radius
+limit is taken for each fixed member to obtain the exact identity
+\eqref{ana:green}; only then are its integrals expanded.  Similarly,
+each finite angular type coefficient is identified with its own flux
+before passing to an approximant limit in the bounded functional
+\eqref{ana:green-functional}.  No uniform convergence of discarded
+derivative flux errors in either parameter or angular cutoff is used.
+\end{remark}
diff --git a/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/build/sections/03-cmc.tex b/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/build/sections/03-cmc.tex
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+\section{CMC sphere jets and their Hawking mass}
+\label{sec:cmc}
+
+We use the finite Taylor conventions of Section~\ref{sec:taylor}.
+Taylor expansions of inverses, compositions, and other smooth geometric
+operations at a rotation-invariant background preserve finite angular
+type at each fixed order: each coefficient uses only finitely many
+linear differential operations and tensor products.
+
+\subsection{The sphere construction}
+
+In the end put \(\eta=\operatorname{arsinh}r\), so that
+\[
+ b=\dd\eta^2+\sinh^2\eta\,\sigma.
+\]
+An estimate for a tensor below refers to its components in scaled
+coframes \(\dd\eta,r\,\dd x^A\), in fixed smooth angular coordinate
+charts.  A differentiated little-oh estimate includes all mixed
+radial and angular derivatives of the stated total order.
+
+\begin{proposition}[CMC sphere jets]\label{prop:cmc-jets}
+Let \(p\geq1\), and let \(G\) be a smooth Riemannian metric jet of order
+\(p\), with base \(G_0=g_0\), whose coefficients have finite angular type.
+Suppose that there are angular scalar jets \(C,E\), with
+\(C_0=2m_0\) and \(E_0=0\), such that
+\begin{equation}\label{cmc:metric-asymptotics}
+ G-b=r^{-3}\bigl(C\,\dd\eta^2+E r^2\sigma\bigr)+o(r^{-3}).
+\end{equation}
+For the coefficient of degree \(j\), \(1\leq j\leq p\), suppose that
+the remainder satisfies this estimate after every mixed radial and
+angular derivative of total order at most \(p-j+1\).
+
+There is a smooth family of sphere jets
+\(\Sigma_s\), \(0\leq s<\infty\), given by graphs over
+\(\{s\}\times S^2\), with the following properties:
+\begin{enumerate}
+ \item \(\Sigma_s\) has constant outward mean curvature through order
+ \(p\), and \(\Sigma_0\) is minimal through that order.
+ \item The graph coefficients have finite angular type.  The normal
+ lapse \(f\) for variation in \(s\) has base value \(f_0=1\).
+ \item If \(A(s)\) and \(H(s)\) are the area and mean curvature jets,
+ then the CMC Hawking mass
+ \begin{equation}\label{cmc:hawking-definition}
+ m_H(\Sigma_s)=
+ \sqrt{\frac{A(s)}{16\pi}}
+ \left(1-\frac{(H(s)^2-4)A(s)}{16\pi}\right)
+ \end{equation}
+ satisfies, coefficientwise,
+ \begin{equation}\label{cmc:hawking-limit}
+ \lim_{s\to\infty}m_H(\Sigma_s)
+ =m_{\AH}(G)
+ :=\sqrt{p_0(G)^2-\sum_{i=1}^3p_i(G)^2}.
+ \end{equation}
+ Here \(p_\mu(G)\) is the coefficientwise flux \eqref{eq:mass-flux},
+ and the square root is its formal branch with base value
+ \(m_0>0\).
+\end{enumerate}
+\end{proposition}
+
+The expression \eqref{cmc:hawking-definition} is the CMC specialization
+of the asymptotically hyperbolic Hawking mass used in
+\cite[Equation~(2)]{Ambrozio2015}.  The proof below checks its limiting
+normalization against \eqref{eq:mass-flux} directly.
+
+\begin{proof}
+We first solve the Jacobi equations on finite radial intervals.
+At infinity, hyperbolic boosts account for the order-one degree-one
+displacement. The asymptotic CMC equation selects a frame in which the
+spatial mass components vanish. We then choose the free graph means
+to join those spheres to the minimal first leaf.
+
+\paragraph{The graph equation on finite radial intervals.}
+For a normal variation with speed \(w\), the mean-curvature variation is
+\begin{equation}\label{cmc:jacobi-general}
+ Jw=-\Delta_{\Sigma}w
+ -\bigl(\Ric_G(\nu,\nu)+|\II|^2\bigr)w.
+\end{equation}
+Indeed, the induced metric varies by \(2w\II\), the variation of the
+unit normal is \(-\grad_\Sigma w\), and differentiation of the second
+form gives the Hessian term, the ambient curvature term, and the
+quadratic second-form term.  Tracing, including the variation of the
+inverse induced metric, gives \eqref{cmc:jacobi-general}.  A tangential
+velocity adds the directional derivative of \(H\).
+
+For a background coordinate sphere, \(H_0=2N/r\) and
+\[
+ \Ric_{g_0}(\partial_s,\partial_s)=-2-\frac{2m_0}{r^3},
+ \qquad |\II_0|^2=\frac{2N^2}{r^2}.
+\]
+Thus its Jacobi operator is
+\begin{equation}\label{cmc:background-jacobi}
+ J_s^0=-\Delta_{r^2\sigma}-\frac2{r^2}+\frac{6m_0}{r^3}.
+\end{equation}
+On degree-\(\ell\) spherical harmonics its eigenvalue is
+\begin{equation}\label{cmc:jacobi-eigenvalues}
+ \lambda_\ell(s)
+ =\frac{\ell(\ell+1)-2}{r^2}+\frac{6m_0}{r^3}.
+\end{equation}
+It is strictly positive on every nonconstant mode.  In particular,
+the degree-one eigenvalue is \(6m_0/r^3>0\).
+At \(s=0\), using \(2m_0=a(1+a^2)\), the constant eigenvalue is
+\[
+ -\frac2{a^2}+\frac{6m_0}{a^3}
+ =3+\frac1{a^2}=\frac{2\kappa}{a},
+\]
+and hence \(J_0^0=D\), positive on all modes.
+
+Write a graph as \(s+\rho(s,x;\eps)\), with \(\rho_0=0\).
+At degree \(j\), its mean-curvature coefficient is
+\[
+ J_s^0\rho_j+\mathcal R_j(s,x),
+\]
+where \(\mathcal R_j\) is known from the metric coefficients and the
+lower graph coefficients.  To make the first sphere minimal, solve
+this equation at \(s=0\) with right-hand side zero, using \(D^{-1}\).
+At general \(s\), prescribe any smooth round mean for \(\rho_j\) that
+agrees with this value at zero, and solve the projection of the
+equation onto the nonconstant modes.
+The operator \eqref{cmc:background-jacobi} preserves the splitting
+into constants and nonconstants, so the latter equation has a unique
+solution.  At each order it is a finite-dimensional linear equation
+with a smooth invertible coefficient matrix at every finite radius.
+This produces smooth coefficients on each finite interval.
+Induction also shows that the resulting graph at zero is exactly the
+minimal graph jet already constructed there.
+The constant eigenvalue need not be inverted away from zero.
+
+\paragraph{The flux aspect.}
+For an angular scalar \(F\), write
+\(\langle F\rangle_\sigma=(4\pi)^{-1}\int_{S^2}F\,\dd A_\sigma\).
+The asymptotic form \eqref{cmc:metric-asymptotics} gives
+\begin{equation}\label{cmc:mass-aspect}
+ (p_0(G),p_1(G),p_2(G),p_3(G))
+ =\left\langle\mu(x)(1,x)\right\rangle_\sigma,
+ \qquad \mu=\frac{C+3E}{2}.
+\end{equation}
+Here \(\mu\) denotes only a flux aspect.
+To verify the formula directly, let \(e=G-b\).  Its leading trace is
+\((C+2E)r^{-3}\), and its radial divergence-minus-trace derivative is
+\begin{align*}
+ (\Div_b e-\dd\tr_b e)(\partial_\eta)
+ &=2\coth\eta(C-E)r^{-3}
+       -2\partial_\eta(Er^{-3})+o(r^{-3})\\
+ &=(2C+4E)r^{-3}+o(r^{-3}).
+\end{align*}
+For \(V_0=\sqrt{1+r^2}\) or \(V_i=rx_i\), let \(v=1\) or \(x_i\),
+respectively.  Both \(V\) and \(\partial_\eta V\) have leading term
+\(rv\).  The terms \((\tr_b e)\dd V-e(\grad_bV,\cdot)\) add
+\(2Ev r^{-2}+o(r^{-2})\) to the normal flux integrand.
+The total is therefore
+\((2C+6E)v r^{-2}+o(r^{-2})\).
+Multiplying by \(r^2\dd A_\sigma/(16\pi)\) proves
+\eqref{cmc:mass-aspect}.  The first differentiated remainder assumed
+in \eqref{cmc:metric-asymptotics} makes every omitted flux integral
+tend to zero.
+
+\paragraph{Boosted coordinates.}
+Realize hyperbolic space as
+\[
+ (\cosh\eta,\sinh\eta\,x)\subset\mathbb R^{3,1}.
+\]
+Let \(\mathcal B(d)\) be the exponential of the Lorentz Lie-algebra
+matrix with time-space entries \(d\in\mathbb R^3\) and zero spatial
+rotation part.  Thus \(\mathcal B(0)\) is the identity.
+Write its action on future null vectors as
+\begin{equation}\label{cmc:boost-null}
+ \mathcal B(d)(1,x)=k(x)(1,y(x)).
+\end{equation}
+For small \(d\), \(k>0\).
+In pullback coordinates the old radius is \(kr+O(r^{-1})\), the old
+radial distance is \(\eta+\log k+O(r^{-2})\), and the old radial unit
+covector agrees with the new one up to \(O(r^{-1})\) in hyperbolic
+norm.  These statements, including all their needed derivatives,
+follow by applying the matrix \(\mathcal B(d)\) to the displayed
+hyperboloid coordinates.  Since \(\mathcal B(d)\) preserves \(b\),
+the leading coefficients of its pullback of \(G\) are
+\begin{equation}\label{cmc:boost-aspect}
+ C_d=k^{-3}C\circ y,\qquad E_d=k^{-3}E\circ y.
+\end{equation}
+For example, the angular part of \(\dd\log k\) has hyperbolic norm
+\(O(r^{-1})\); it therefore changes neither leading scaled diagonal
+coefficient in \eqref{cmc:metric-asymptotics}.
+
+Differentiating \eqref{cmc:boost-null} tangentially and taking
+Minkowski inner products gives \(y^*\sigma=k^{-2}\sigma\).
+Consequently \(\dd A_\sigma(y)=k^{-2}\dd A_\sigma(x)\), and
+\((1,x)=k\mathcal B(d)^{-1}(1,y)\).  Changing angular variables proves
+the exact leading-flux transformation
+\begin{equation}\label{cmc:boost-mass}
+ \left\langle k^{-3}\mu(y)(1,x)\right\rangle_\sigma
+ =\mathcal B(d)^{-1}\left\langle\mu(x)(1,x)\right\rangle_\sigma.
+\end{equation}
+This establishes the needed covariance directly for the specified
+flux.
+
+\paragraph{The asymptotic CMC equation.}
+For large \(s\), put \(R=r(s)\) and \(\eta_R=\operatorname{arsinh}R\).
+In the coordinates obtained from \(\mathcal B(d)\), seek a graph
+\begin{equation}\label{cmc:far-graph}
+ \eta=\eta_R+\frac{h(x)}{R},
+\end{equation}
+where \(d\) and \(h\) are jets with zero base values, and every
+coefficient of \(h\) has zero constant and degree-one components.
+The parameters \(d,h\) are held fixed when computing an individual
+leaf.  We claim, coefficientwise for bounded input jets in fixed
+finite angular spaces, that
+\begin{equation}\label{cmc:mean-curvature-asymptotic}
+ R^3\bigl(H-2\coth\eta_R\bigr)
+ =-(\Delta_\sigma+2)h-(C_d+3E_d)+o(1).
+\end{equation}
+
+Here are the estimates underlying this claim.
+In \(b\), the unit normal to \eqref{cmc:far-graph} is
+\[
+ \frac{\partial_\eta-r^{-2}\grad_\sigma(h/R)}
+ {\sqrt{1+r^{-2}|\dd(h/R)|_\sigma^2}}.
+\]
+Its divergence gives
+\[
+ H_b=2\coth\eta_R
+       -R^{-3}(\Delta_\sigma+2)h+O(R^{-4}).
+\]
+Indeed the slope has hyperbolic norm \(O(R^{-2})\), its normalization
+differs from one by \(O(R^{-4})\), and the second Taylor term in
+\(2\coth(\eta_R+h/R)\) is \(O(R^{-4})\).
+These estimates hold for every coefficient of the finite Taylor
+expansion when the input coefficients are bounded.
+
+On an unshifted sphere the perturbation in
+\eqref{cmc:metric-asymptotics}, after boosting, changes the radial
+lapse by \(C_dR^{-3}/2+o(R^{-3})\).  This contributes
+\(-C_dR^{-3}+o(R^{-3})\) to \(H\).
+The relative area density changes by \(E_dR^{-3}+o(R^{-3})\);
+its radial logarithmic derivative contributes
+\(-3E_dR^{-3}+o(R^{-3})\).
+The mixed components in the remainder contribute \(o(R^{-3})\),
+including their tangential-divergence term.
+For completeness, on a fixed surface the comparison between two
+metrics \(b\) and \(\widehat g\) is
+\[
+ \II_{\widehat g}(X,Y)
+ =\widehat g(\nu_{\widehat g},\nu_b)\II_b(X,Y)
+ -\widehat g\bigl(\nu_{\widehat g},
+       (\nabla^{\widehat g}-\nabla^b)_XY\bigr).
+\]
+It follows by splitting \(\nabla_X^bY\) into its tangential and normal
+parts and using normal orthogonality.
+After tracing, the linear metric correction depends smoothly on
+the tangent plane, the background shape operator, and the scaled
+components of \(\widehat g-b,\nabla^b(\widehat g-b)\).
+The slope of \eqref{cmc:far-graph} tends to zero, and its background
+shape operator differs by \(o(1)\) from that of a coordinate sphere.
+Its displacement is \(O(R^{-1})\) in radial distance.
+The leading metric correction on that graph is consequently still
+\(-(C_d+3E_d)R^{-3}\).
+Quadratic metric corrections are \(O(R^{-6})\).
+This proves \eqref{cmc:mean-curvature-asymptotic}.
+
+The estimate just proved also justifies its use at every Taylor
+order through \(p\).
+A coefficient of \(G\) of degree \(j\) can undergo at most \(p-j\)
+spatial differentiations from Taylor composition with the graph and
+the boost.  Mean curvature requires one additional derivative of
+the metric.  Thus the mixed derivative allowance \(p-j+1\) in
+\eqref{cmc:metric-asymptotics} controls the coefficient of every
+remainder after multiplication by \(R^3\).
+The base \(g_0\) has the required bounds to all orders.
+Angular differential operators cause no further restriction within
+a fixed finite-dimensional angular space.
+More explicitly, denote the degree-\(j\) residual in
+\eqref{cmc:metric-asymptotics} by \(\mathcal E_j\), and set
+\[
+ \omega_j(R)=
+ \max_{a+b\le p-j+1}\ 
+ \sup_{r\ge R/2}
+ \left\|\partial_\eta^a\nabla_\sigma^b
+              (r^3\mathcal E_j^{\mathrm{scaled}})\right\|_\infty.
+\]
+Then \(\omega_j(R)\to0\).
+For every \(n\le p\), bounded coefficients of \(d,h\) in fixed finite
+angular spaces give the estimate
+\[
+ \left\|[\,
+ R^3(H-2\coth\eta_R)+(\Delta_\sigma+2)h+C_d+3E_d\,]_n
+       \right\|_\infty
+ \le C_{p,\mathcal V,M}
+       \left(R^{-1}+\sum_{j=1}^n\omega_j(R)\right).
+\]
+Here \(\mathcal V\) denotes the finitely many angular spaces and \(M\)
+a bound for the input coefficients; the fixed metric coefficients
+may also enter the constant.  The explicit base residual is
+\(O(r^{-5})\) with all derivatives and is included in the
+\(R^{-1}\) bound.  The preceding graph estimates, the product rule,
+and the derivative count prove this estimate term by term.
+It is an estimate for Taylor coefficients, exactly as needed for
+the recursion.
+
+\paragraph{Solving the rescaled equation.}
+Project \(R^3(H-2\coth\eta_R)\) onto the nonconstant modes and set the
+projection equal to zero, through order \(p\).
+At the background, the finite-radius linearization preserves the
+splitting between the \(h\)-modes, of degrees at least two, and the
+three boost modes.
+Set
+\[
+ \alpha_R=\frac{\sqrt{1+R^2}}{N(R)}.
+\]
+A graph variation \(h/R\) in \(\eta\) has physical normal
+displacement \(\alpha_Rh/R\).  Equations
+\eqref{cmc:background-jacobi}--\eqref{cmc:jacobi-eigenvalues} give its
+exact degree-\(\ell\) multiplier in the rescaled equation:
+\begin{equation}\label{cmc:scaled-inverse-blocks}
+ \alpha_R\bigl(\ell(\ell+1)-2+6m_0/R\bigr),
+ \qquad \ell\ge2.
+\end{equation}
+An infinitesimal boost has \(\eta\)-displacement \(d\cdot x\).
+Its physical normal displacement is \(\alpha_Rd\cdot x\), so its
+multiplier is \(6m_0\alpha_R\).
+These blocks converge respectively to \(\ell(\ell+1)-2\) and
+\(6m_0\), both invertible.
+Equivalently, the latter sign and coefficient follow by
+differentiating \(2m_0k^{-3}\) in
+\eqref{cmc:mean-curvature-asymptotic}, since
+\(k=1+d\cdot x+O(|d|^2)\).
+
+At each Taylor order only fixed finite angular spaces are involved.
+In particular, the Taylor coefficients at \(d=0\) of the boost map
+and its compositions are finite sums of coordinate functions and
+angular derivatives of the given profiles.  This assertion concerns
+the finite Taylor expansion: it does not require a finite harmonic
+cutoff to be preserved by a finite nonzero boost.
+One can see the closure directly from the infinitesimal boost field
+\[
+ \mathcal K_a=(a\cdot x)\partial_\eta
+               +\coth\eta\,\grad_\sigma(a\cdot x).
+\]
+It follows by differentiating the hyperboloid coordinates.
+Its Lie derivative, repeated finitely many times, uses only radial
+derivatives, angular derivatives, and multiplication by degree-one
+profiles, also for tensor fields.
+It follows that the inverses of the finite-dimensional blocks above
+are bounded for all sufficiently large \(R\) and converge as
+\(R\to\infty\).
+
+Now solve successively for the coefficients of \(h,d\).
+At each order their linear operator is the displayed background
+linearization.  The remaining source depends only on lower solved
+coefficients and the prescribed metric coefficients.
+Equation \eqref{cmc:mean-curvature-asymptotic}, with its uniform
+coefficientwise remainder estimate, shows inductively that this
+source converges once the lower coefficients do.
+The converging inverse then gives bounded coefficients of \(h,d\)
+with limits.  This starts at degree one and proves the assertion
+through order \(p\).
+Smooth dependence on \(s\) follows at every finite radius from the
+smooth finite-dimensional equations.
+
+In original coordinates these spheres are graph jets over the
+background sphere: the angular map has identity as base and can be
+inverted formally.  Their round mean graph heights are smooth scalar
+functions of \(s\).  Choose the free means in the finite-radius
+recursion to agree with these functions for all sufficiently large
+\(s\), and to agree with the minimal graph's means at zero, using a
+smooth interpolation.  Inductive uniqueness of the nonconstant
+graph coefficients makes the resulting recursion coincide with the
+far construction.  This gives the family on the whole half-line.
+Its base is the coordinate-sphere family, so its lapse has base one.
+
+\paragraph{The Hawking mass limit.}
+Let \(d_\infty\) be the coefficientwise limiting boost.
+The degree-one projection of
+\eqref{cmc:mean-curvature-asymptotic} gives
+\[
+ \left\langle\mu_{d_\infty}x_i\right\rangle_\sigma=0,
+ \qquad
+ \mu_d=(C_d+3E_d)/2.
+\]
+Writing
+\[
+ H_b(R)=2\coth\eta_R,\qquad
+ B_R=R^3(H-H_b(R)),
+\]
+the round average of that same equation gives
+\[
+ B_R=-2\langle\mu_d\rangle_\sigma+o(1).
+\]
+The area of \eqref{cmc:far-graph} satisfies
+\begin{equation}\label{cmc:area-error}
+ \frac{A}{4\pi R^2}=1+\delta_R,\qquad \delta_R=O(R^{-2})
+\end{equation}
+coefficientwise.  To see the relevant cancellation, the relative
+hyperbolic area density is
+\[
+ 1+2\coth\eta_R\,\frac hR+O(R^{-2}),
+\]
+and the metric correction is \(O(R^{-3})\).  The sole possible
+relative term of order \(R^{-1}\) integrates to zero because every
+coefficient of \(h\) has zero round mean.
+
+Since \(H_b(R)^2-4=4/R^2\), direct substitution into
+\eqref{cmc:hawking-definition} gives
+\begin{align*}
+ m_H={}&-\frac{R\delta_R}{2}\sqrt{1+\delta_R}
+ -\frac{H_b(R)B_R}{4}(1+\delta_R)^{3/2}\\
+ &-\frac{B_R^2}{8R^3}(1+\delta_R)^{3/2}.
+\end{align*}
+Therefore
+\[
+ \lim_{s\to\infty}m_H(\Sigma_s)
+ =\langle\mu_{d_\infty}\rangle_\sigma.
+\]
+By \eqref{cmc:boost-mass}, this last scalar is the time component of
+\(\mathcal B(d_\infty)^{-1}p(G)\), whose spatial components vanish.
+Lorentz invariance and its positive base value \(m_0\) identify it
+with the formal square root in \eqref{cmc:hawking-limit}.
+\end{proof}
+
+\begin{remark}\label{cmc:regularity-budget}
+For the conformal jets used in the quadratic argument, the coefficient
+of degree one solves a homogeneous finite-mode equation \(Lu=0\);
+Lemma~\ref{lem:inverse} gives all its differentiated asymptotics.
+The degree-two coefficient needs only the first differentiated
+asymptotic in Proposition~\ref{prop:cmc-jets}, which is supplied by the
+given \(C^{1,\alpha}_\tau\) decay of a finite-angular-type seed and the
+same radial equation.
+The fourth-order construction will be used only after the seed has
+been reduced to \(Q(v)=\Hess v-Bv\), with \(v\) a homogeneous solution
+containing only constant and degree-one angular modes.
+Then all spatially differentiated decay estimates needed above hold.
+These are distinct uses of the proposition; higher radial decay
+derivatives of an arbitrary seed are not required.
+\end{remark}
+
+\subsection{The Hawking variation identity}
+
+On the sphere jets of Proposition~\ref{prop:cmc-jets}, write
+\[
+ \langle u\rangle=\frac1A\int_{\Sigma_s}u\,\dd A_G,
+ \qquad \bar f=\langle f\rangle,\qquad \delta f=f-\bar f.
+\]
+Here averages are with respect to the induced metric; the round
+average retains the subscript \(\sigma\).
+Let \(J\) be the full Jacobi operator
+\eqref{cmc:jacobi-general}, and let \(\II^\circ\) be the tracefree
+second fundamental form.
+
+\begin{lemma}[CMC Hawking variation]\label{lem:hawking-variation}
+The following identity holds through the order of the sphere and
+metric jets:
+\begin{equation}\label{cmc:hawking-variation}
+ \frac{\dd}{\dd s}m_H
+ =\sqrt{\frac A{16\pi}}\frac{HA}{8\pi}
+ \left[
+ \bar f\left\langle
+      \frac{\scal_G+6+|\II^\circ|^2}{2}
+       \right\rangle
+ +\frac1{\bar f}\langle\delta f\,J\delta f\rangle
+ \right].
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+The area and mean-curvature variation formulas give
+\[
+ A'=AH\bar f,\qquad H'=Jf.
+\]
+The latter is a constant on each leaf; any tangential transport
+term vanishes since \(H\) is constant there.
+For brevity put
+\[
+ c=\frac{4\pi}{A}+3-\frac{3H^2}{4},
+ \qquad
+ Q_H=\frac{\scal_G+6+|\II^\circ|^2}{2}.
+\]
+Differentiation of \eqref{cmc:hawking-definition} gives
+\begin{equation}\label{cmc:raw-variation}
+ m_H'=\sqrt{\frac A{16\pi}}\frac{HA}{8\pi}
+          (c\bar f-H').
+\end{equation}
+The Gauss equation and
+\(|\II|^2=H^2/2+|\II^\circ|^2\) imply
+\[
+ J1=K_\Sigma+3-\frac{3H^2}{4}-Q_H,
+\]
+where \(K_\Sigma\) is intrinsic Gauss curvature.
+The sphere jets have \(\int K_\Sigma\,\dd A_G=4\pi\)
+coefficientwise.
+One can verify this without a limiting argument: the variation of
+integrated scalar curvature on a closed two-dimensional metric is
+the integral of a divergence minus its variation tensor paired with
+\(\Ric-\frac12\scal\,g\).  The latter tensor is zero in dimension
+two, so that integral is constant under every variation and equals
+its round value.  Applying the same identity to the Taylor
+coefficients proves the assertion for jets.
+Consequently
+\[
+ \langle J1\rangle=c-\langle Q_H\rangle.
+\]
+
+The operator \(J\) is self-adjoint on the closed leaf.
+Since \(\langle\delta f\rangle=0\) and \(Jf=H'\) is constant,
+\begin{align*}
+ H'&=\bar f\bigl(c-\langle Q_H\rangle\bigr)
+                  +\langle\delta f\,J1\rangle,\\
+ 0&=\langle\delta f\,Jf\rangle
+   =\bar f\langle\delta f\,J1\rangle
+                  +\langle\delta f\,J\delta f\rangle.
+\end{align*}
+The scalar jet \(\bar f\) is invertible because its base value is one.
+Eliminating the common term gives
+\[
+ c\bar f-H'
+ =\bar f\langle Q_H\rangle
+       +\bar f^{-1}\langle\delta f\,J\delta f\rangle.
+\]
+Substitution into \eqref{cmc:raw-variation} proves the identity.
+All calculations may first be made on a compact radial interval.
+Smooth representatives of the metric and graph jets realize the
+ordinary geometric variation identities there, and the imposed CMC
+identities hold to the required Taylor order.
+\end{proof}
+
+\subsection{The first nonzero coefficient}
+
+\begin{proposition}[Leading coefficient as a positive integral]
+\label{prop:leading-energy}
+Use the construction of Proposition~\ref{prop:cmc-jets} through order
+\(2l\), where \(l\ge1\), and let \(A_{\mathrm{leaf}}=A(0)\) be the area
+jet of its minimal first leaf.
+Suppose, as ambient and leafwise coefficient identities, respectively,
+that
+\begin{equation}\label{cmc:leading-assumptions}
+ \scal_G+6=\eps^{2l}|k|_{g_0}^2+O(\eps^{2l+1}),
+ \qquad
+ \II^\circ=\eps^lU+O(\eps^{l+1}),
+ \qquad
+ \delta f=\eps^lF+O(\eps^{l+1}).
+\end{equation}
+Here \(k\) is a smooth tensor on \(M\); the fields \(U,F\) on the
+background spheres are identified with fields on \(M\).
+Then every coefficient below degree \(2l\) of
+\(m_{\AH}(G)-b_*(A_{\mathrm{leaf}})\) vanishes, and
+\begin{equation}\label{cmc:leading-integral}
+ \begin{split}
+ &\bigl[m_{\AH}(G)-b_*(A_{\mathrm{leaf}})\bigr]_{2l}\\
+ &\quad=\frac1{16\pi}\int_0^\infty N(s)
+   \int_{\{s\}\times S^2}
+    \left(|k|^2+|U|^2+2FJ_s^0F\right)\dd A_{g_0}\,\dd s
+ \ge0.
+ \end{split}
+\end{equation}
+All norms in this formula use the background metrics.
+The displayed iterated integral is finite.
+If it vanishes, then \(k,U,F\) vanish for \(s>0\), and also at the
+boundary by continuity.
+\end{proposition}
+
+\begin{proof}
+The first nonzero coefficient \(F\) has zero round mean on each
+background sphere: extract degree \(l\) from
+\(\int_{\Sigma_s}\delta f\,\dd A_G=0\), using the vanishing of all
+lower coefficients of \(\delta f\).
+Every term in the bracket in \eqref{cmc:hawking-variation} starts at
+degree \(2l\).  At that degree, all exterior factors can therefore
+be evaluated at the background:
+\[
+ A_0=4\pi r^2,\quad H_0=2N/r,\quad \bar f_0=1,
+ \quad\sqrt{\frac{A_0}{16\pi}}H_0=N.
+\]
+The same reasoning evaluates the norms, the induced measure, and
+the Jacobi operator inside the leading terms at the background.
+Composing the ambient scalar coefficient with a graph displacement
+changes only higher degrees.
+It follows that
+\begin{equation}\label{cmc:leading-derivative}
+ [m_H']_{2l}
+ =\frac{N(s)}{16\pi}\int_{\{s\}\times S^2}
+        \bigl(|k|^2+|U|^2+2FJ_s^0F\bigr)\dd A_{g_0},
+\end{equation}
+and all lower coefficient derivatives vanish.
+
+The integrand in \eqref{cmc:leading-derivative} is nonnegative after
+sphere integration.  Indeed, on the mean-zero field \(F\),
+\[
+ \int_{\{s\}\times S^2}FJ_s^0F\,\dd A_{g_0}
+ =\sum_{\ell\ge1}
+  \left(\frac{\ell(\ell+1)-2}{r^2}+\frac{6m_0}{r^3}\right)
+       \|F_\ell\|_{L^2(r^2\sigma)}^2.
+\]
+Only finitely many modes occur, and every displayed eigenvalue is
+strictly positive at a finite radius.
+Also \(N(s)>0\) for \(s>0\).
+
+For any finite \(T\), integrate the coefficient identities on
+\([0,T]\).
+The first leaf is minimal to the required order, so
+\(m_H(\Sigma_0)=b_*(A_{\mathrm{leaf}})\) through that order.
+The coefficientwise endpoint limits exist by
+\eqref{cmc:hawking-limit}.
+For degree \(2l\), the right-hand side of the integrated
+\eqref{cmc:leading-derivative} is an increasing function of \(T\)
+and its limit equals the finite limiting endpoint coefficient.
+This proves both its convergence and
+\eqref{cmc:leading-integral}.
+For lower degrees the integrated derivative is zero, proving their
+claimed vanishing.
+
+This order of operations uses only smooth jets on finite radial
+intervals and coefficientwise endpoint limits.
+In particular, convergence of the derivatives of the asymptotic
+graph coefficients or of their lapse is not needed to obtain the
+integral's convergence.
+If the integral is zero, each of its continuous nonnegative
+sphere-integrated summands is zero for every \(s>0\).
+The positivity just established forces \(k=U=F=0\) there.
+Smoothness gives their boundary values.
+\end{proof}
diff --git a/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/build/sections/04-quadratic.tex b/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/build/sections/04-quadratic.tex
new file mode 100644
index 0000000000000000000000000000000000000000..4ff0c59b7d5e8733b8ae94218236a69ec1ea3c97
--- /dev/null
+++ b/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/build/sections/04-quadratic.tex
@@ -0,0 +1,341 @@
+\section{The quadratic estimate and its complete kernel}
+\label{sec:quadratic}
+
+All contractions in this section, unless a different metric is indicated, use
+$g_0$.  We write $\dd V_0$ and $\dd A_0$ for its volume form and its induced
+area form.  A function appearing in an integral over $S$ is understood to be
+restricted to $S$.  Introduce
+\[
+ Q(v)=\Hess_{g_0}v-vB.
+\]
+The boundary normal $n=\partial_s$ points into $M$; the outward normal of
+the domain $M$ along its inner boundary is therefore $-n$.
+
+\Needspace{9\baselineskip}
+\begin{lemma}[The weighted TT square identity]
+\label{lem:tt-square}
+Let $k$ be a smooth TT tensor of finite angular type, with decay
+$k=O(r^{-\gamma})$ for some $\gamma>3/2$.  Let $v$ be the decaying solution of
+\begin{equation}
+ Lv=0,
+ \qquad v|_S=D^{-1}(k_{ss}|_S).
+ \label{eq:tt-dirichlet}
+\end{equation}
+Then $Q(v)$ is TT, $Q(v)_{ss}|_S=k_{ss}|_S$, and
+\begin{equation}
+ \int_M N|k|^2\,\dd V_0
+ -\kappa\int_S vDv\,\dd A_0
+ =\int_M N|k-Q(v)|^2\,\dd V_0.
+ \label{eq:tt-square}
+\end{equation}
+All integrals in this identity converge.  In particular, no sign assumption
+on $k_{ss}|_S$ is needed.
+\end{lemma}
+
+\begin{proof}
+The operator $D=-\Delta_S+2\kappa/a$ is strictly positive on all spherical
+harmonics.  Lemma~\ref{lem:inverse}, applied to the finitely many modes of
+the boundary datum, gives the unique solution of
+\eqref{eq:tt-dirichlet}; it has $v=O(r^{-3})$ with the differentiated
+asymptotics needed below.
+
+The background identities $\tr B=3$, $\Div B=0$, and
+$B=\Ric_{g_0}+3g_0$ imply, by commuting the derivatives of a function,
+\[
+ \tr Q(v)=\Delta v-3v=Lv,
+ \qquad
+ (\Div Q(v))_i
+ =\nabla_i\Delta v+(\Ric_{ij}-B_{ij})\nabla^jv
+ =\nabla_iLv.
+\]
+Thus $Q(v)$ is TT.  Since $S$ is totally geodesic, the tangential trace of
+$\Hess v$ on $S$ is $\Delta_Sv$.  Moreover $B_t(0)=\kappa/a$.
+Taking the tangential trace of $Q(v)$ and using its vanishing full trace
+therefore yields
+\begin{equation}
+ Q(v)_{ss}|_S=-\Delta_Sv+2(\kappa/a)v=Dv.
+ \label{eq:q-normal-boundary}
+\end{equation}
+
+Set $\omega=N\,\dd v-v\,\dd N$.  With the averaged convention for
+the symmetric gradient, $\operatorname{sym}\nabla\omega
+=(\nabla_i\omega_j+\nabla_j\omega_i)/2$, the mixed products cancel and
+the static identity $\Hess N=NB$ gives
+\[
+ \operatorname{sym}\nabla\omega
+ =N\Hess v-v\Hess N=NQ(v).
+\]
+Symmetry and vanishing divergence of $k$ consequently give, on
+$M_R=\{0\leq s\leq s(R)\}$,
+\[
+ \int_{M_R}N\langle k,Q(v)\rangle\,\dd V_0
+ =\int_{\partial M_R}k(\nu,\omega^\sharp)\,\dd A_0.
+\]
+On $S$, $N=0$ and $\dd N=\kappa\,\dd s$, so that
+$\omega^\sharp=-\kappa vn$.  Since $\nu=-n$ there, the inner contribution
+is $+\kappa\int_Svk_{ss}\,\dd A_0$.  At infinity,
+$|\omega|=O(r^{-2})$, and hence the outer contribution is
+$O(r^{-\gamma})$ and tends to zero.  We obtain
+\begin{equation}
+ \int_M N\langle k,Q(v)\rangle\,\dd V_0
+ =\kappa\int_Svk_{ss}\,\dd A_0
+ =\kappa\int_SvDv\,\dd A_0.
+ \label{eq:tt-pairing}
+\end{equation}
+The same calculation with $k$ replaced by $Q(v)$ gives
+$\int_MN|Q(v)|^2\,\dd V_0=\kappa\int_SvDv\,\dd A_0$.
+Expanding the square proves \eqref{eq:tt-square}.
+Finally, $N\dd V_0$ has radial size $O(r^3)\,\dd s\,\dd A_\sigma$,
+so the integral of $N|k|^2$ converges because $2\gamma>3$.
+The stronger decay of $Q(v)$ gives all remaining convergence assertions.
+\end{proof}
+
+\subsection{The boundary area correction}
+
+For the moment let $q$ have finite angular type, and use the conformal
+metric jet $g=\phi^4g_0$ through order two.  Write $u=u_1[q]$, so that
+$Lu=0$ and $u_s|_S=q_{ss}/4$.  Apply
+Proposition~\ref{prop:cmc-jets}, and let $A_{\mathrm{leaf}}$ denote the
+area of its first, minimal Taylor leaf.  This notation concerns that
+formal leaf alone; it is not an enclosing-area infimum.  Let
+\[
+ U=[\II^{\circ}]_1,
+ \qquad F=[f-\bar f]_1,
+\]
+where $f$ is the lapse and $\bar f$ its area average on each leaf.
+Thus $F$ has zero round mean on each background sphere.
+
+Let $v$ solve \eqref{eq:tt-dirichlet} with $k=q$.  At the fixed boundary,
+the conformal boundary equation gives
+\[
+ H_g(S)=\eps\phi^{-6}q_{ss}.
+\]
+If $h_1$ is the first normal height of the minimal Taylor leaf, its
+mean-curvature equation is
+\[
+ Dh_1+q_{ss}=0,
+ \qquad h_1=-v|_S.
+\]
+The height difference from that leaf to $S$ is therefore
+$\eps v|_S+O(\eps^2)$.  Area has zero first variation at a minimal
+leaf, and its Hessian at the base sphere is $D$.  Taylor expansion about
+the minimal leaf gives
+\begin{equation}
+ A(S)-A_{\mathrm{leaf}}
+ =\frac{\eps^2}{2}\int_SvDv\,\dd A_0+O(\eps^3)
+ \quad\text{as an identity of jets}.
+ \label{eq:quadratic-area-gap}
+\end{equation}
+Indeed, a first-order perturbation of the area Hessian would multiply
+two first-order displacements and first enter order three.  The first
+variation at the minimal Taylor leaf vanishes to the order needed.
+This reasoning also applies when its leading displacement points across
+the boundary: it uses only the smooth boundary jets and their Taylor
+extension, as in the construction of the formal leaves.
+
+\begin{proposition}[The quadratic deficit]
+\label{prop:quadratic}
+On the real linear space of all smooth TT seeds with the prescribed
+decay, the first deficit coefficient is zero and the second coefficient
+$c_2[q]$ is a continuous, rotation-invariant, nonnegative quadratic form.
+For a finite-angular-type seed it is given by
+\begin{equation}
+ \begin{gathered}
+ c_2[q]=\frac1{16\pi}\int_M N\left(
+ |q-Q(v)|^2+|U|^2+2FJ_s^0F\right)\,\dd V_0,\\
+ J_s^0=-\Delta_{r^2\sigma}-\frac2{r^2}+\frac{6m_0}{r^3}.
+ \end{gathered}
+ \label{eq:quadratic-positive-form}
+\end{equation}
+The last term is integrated on the spheres before radial integration.
+For every actual branch in Theorem~\ref{thm:main},
+\begin{equation}
+ \mathfrak D_q(\eps)=c_2[q]\eps^2+o(\eps^2).
+ \label{eq:actual-quadratic-deficit}
+\end{equation}
+In particular, $c_2[q]>0$ implies a strictly positive deficit for all
+sufficiently small positive $\eps$.
+\end{proposition}
+
+\begin{proof}
+For a finite-angular-type seed the scalar constraint gives
+$\scal_g+6=\eps^2|q|^2+O(\eps^3)$.
+Proposition~\ref{prop:leading-energy}, with $l=1$, shows that the
+constant and first coefficients of the mass norm minus
+$b_*(A_{\mathrm{leaf}})$ vanish, and that its second coefficient is
+\[
+ \frac1{16\pi}\int_M N\bigl(|q|^2+|U|^2+2FJ_s^0F\bigr)\,\dd V_0.
+\]
+The areas in \eqref{eq:quadratic-area-gap} agree through order one,
+so the first coefficient of the original deficit is also zero.
+Since $b_*'(4\pi a^2)=\kappa/(8\pi)$, the contribution to its second
+coefficient from that area difference is
+$-\kappa(16\pi)^{-1}\int_SvDv\,\dd A_0$.
+Lemma~\ref{lem:tt-square} now proves
+\eqref{eq:quadratic-positive-form} with precisely the displayed factor
+and sign.
+
+On a spherical harmonic with $-\Delta_\sigma$ eigenvalue $\lambda\geq2$,
+the eigenvalue of $J_s^0$ is
+\[
+ \frac{\lambda-2}{r^2}+\frac{6m_0}{r^3}>0.
+\]
+Thus its quadratic form is strictly positive on mean-zero functions at
+every finite $s$, and $N>0$ for $s>0$.
+This proves nonnegativity for finite-angular-type seeds without any
+boundary sign restriction.
+
+Proposition~\ref{prop:expansions} defines \(c_2\) on the full real
+space \(\mathscr T_{\tau,\alpha}\) of smooth TT seeds as a continuous,
+rotation-invariant quadratic form, independently of branch existence
+and boundary sign, and proves \(c_1=0\). For an arbitrary seed \(q\),
+take the finite-angular-type TT approximants \(q_\nu\) from
+Lemma~\ref{lem:rotation}. The formula proved above gives
+\(c_2[q_\nu]\geq0\), and continuity in \(\|\cdot\|_\beta\) gives
+\(c_2[q]\geq0\). Proposition~\ref{prop:expansions} also supplies
+\eqref{eq:actual-quadratic-deficit} for every prescribed actual branch.
+\end{proof}
+
+\subsection{All null directions}
+
+Define the fixed vector space
+\begin{equation}
+ \mathcal K=\left\{Q(v):Lv=0,\ v\text{ decays},\quad
+ v|_S\in\operatorname{span}_{\mathbb R}\{1,x_1,x_2,x_3\}\right\}.
+ \label{eq:kernel-space}
+\end{equation}
+The decaying solutions in this definition have finite angular type and
+$O(r^{-3})$ decay with all the differentiated bounds obtained from
+their homogeneous radial equations.
+
+\begin{corollary}[The complete quadratic kernel]
+\label{cor:kernel}
+The kernel of $c_2$ is exactly $\mathcal K$, a four-dimensional real
+vector space.  In particular, the nonzero degree-one directions are
+genuine quadratic null directions.
+\end{corollary}
+
+\begin{proof}
+First suppose $q$ has finite angular type and $c_2[q]=0$.
+Equation~\eqref{eq:quadratic-positive-form} and the strict positivity
+just proved imply
+\begin{equation}
+ q=Q(v),\qquad U=0,\qquad F=0
+ \quad\text{on }s>0.
+ \label{eq:quadratic-equality-conditions}
+\end{equation}
+Smoothness extends these equalities to the boundary.  Coordinate
+spheres are umbilic under conformal changes of $g_0$.  At the base
+minimal sphere, the tracefree second-form variation due to a normal
+height $h_1$ is $-(\Hess_Sh_1)^{\circ}$: the radial curvature term is
+pure trace.  Since $h_1=-v|_S$, the vanishing of $U|_S$ implies
+$(\Hess_\sigma(v|_S))^{\circ}=0$.
+For completeness, the divergence of this equation on the unit sphere
+gives $\dd(\Delta_\sigma v+2v)=0$.  Subtracting the mean reduces it to
+$\Hess_\sigma(v-\bar v)=-(v-\bar v)\sigma$.
+This equation determines its solution along every geodesic from its
+value and differential at one point, and its solutions are precisely
+the linear coordinate functions.  Thus $v|_S$ is a constant plus a
+linear harmonic.
+
+Uniqueness of the decaying Dirichlet problem for $L$ then excludes
+every other angular mode throughout $M$.  Each degree-one radial
+coefficient is a multiple of the same decaying solution of
+\begin{equation}
+ V''+2\frac NrV'-\left(3+\frac2{r^2}\right)V=0.
+ \label{eq:dipole-radial-ode}
+\end{equation}
+Hence $q\in\mathcal K$.  Conversely, every element of $\mathcal K$
+is null, as we now check; the conditions $U=F=0$ do not impose a further
+restriction on its degree-one part.
+
+Take $q=Q(v)\in\mathcal K$.  The first conformal coefficient $u$
+also has only constant and linear modes, by its Neumann equation and
+decaying uniqueness.  Let $h(s,x)$ be the first height coefficient
+of the CMC spheres, and put $p=N/r$.  Conformal variation of mean
+curvature and first variation of the lapse give
+\begin{equation}
+ [H]_1=J_s^0h+4(u_s-pu),
+ \qquad [f]_1=h_s+2u.
+ \label{eq:first-cmc-conformal-variations}
+\end{equation}
+Here the second identity follows by taking the normal component of
+the velocity of the graph $s+\eps h(s,x)$ in
+$g_0+4\eps u g_0$.
+On the degree-one part, the CMC equation therefore fixes
+\begin{equation}
+ h=-\frac{2r^3}{3m_0}(u_s-pu).
+ \label{eq:dipole-cmc-height}
+\end{equation}
+All formulas in the next calculation are componentwise on that part.
+The background equations and \eqref{eq:dipole-radial-ode} give
+\[
+ p'=\frac{3m_0}{r^3}-\frac1{r^2},\qquad
+ p^2=1+\frac1{r^2}-\frac{2m_0}{r^3},\qquad
+ u''=-2pu'+\left(3+\frac2{r^2}\right)u.
+\]
+Consequently,
+\begin{align}
+ \frac1{r^3}\partial_s\bigl[r^3(u'-pu)\bigr]
+ &=3p(u'-pu)+u''-p'u-pu'\notag\\
+ &=\left(3+\frac2{r^2}-p'-3p^2\right)u
+ =\frac{3m_0}{r^3}u.
+ \label{eq:dipole-cancellation}
+\end{align}
+It follows that $h_s=-2u$ on the degree-one part, so its lapse
+fluctuation vanishes.  The freely chosen radial mean of $h$ changes
+$[f]_1$ only by a radial function, which disappears upon subtracting
+the mean.  Thus $F=0$ independently of the radial labeling.
+
+At every background sphere, the tracefree second-form variation of
+these graphs is $-(\Hess_{r^2\sigma}h)^{\circ}$; the conformal
+variation and radial curvature terms are pure trace.
+The constant and linear modes of $h$ have vanishing tracefree
+spherical Hessian, so $U=0$ as well.
+At $S$ there is also exact agreement with the prescribed first minimal
+height: on degree one,
+\[
+ D_1=\frac{6m_0}{a^3},\qquad
+ h(0)=-\frac{4u_s(0)}{D_1}=-v(0).
+\]
+We already have $q=Q(v)$, and hence
+\eqref{eq:quadratic-positive-form} proves $c_2[q]=0$.
+Existence and uniqueness for each of the four boundary modes show
+that $\mathcal K$ has dimension at most four; it has dimension
+exactly four because \eqref{eq:q-normal-boundary} and invertibility
+of $D$ make the map from these boundary data to $Q(v)$ injective.
+
+It remains to rule out additional null directions of arbitrary angular
+type.  Let $\mathcal Z=\{q:c_2[q]=0\}$ in the full smooth TT space.
+A nonnegative quadratic form has a linear kernel: if $c_2[q]=0$,
+nonnegativity of $c_2[q+th]$ for all real $t$ forces its polarized
+bilinear form to vanish on $(q,h)$ for every $h$.
+By Proposition~\ref{prop:quadratic}, $\mathcal Z$ is also closed in
+$\|\cdot\|_\beta$ and invariant under rotations.
+For $q\in\mathcal Z$, let $q_j$ be its polynomial rotational averages
+from Lemma~\ref{lem:rotation}.  Each such average is a norm limit of
+finite linear combinations of rotations of $q$; therefore
+$q_j\in\mathcal Z$.  Each $q_j$ has finite angular type, so the
+preceding argument puts it in $\mathcal K$.
+Since $q_j\to q$ in norm and the fixed finite-dimensional space
+$\mathcal K$ is closed in that norm, $q\in\mathcal K$.
+This proves the full assertion without applying a pointwise geometric
+limit to the CMC fields $U$ and $F$.
+\end{proof}
+
+\begin{remark}[The boundary sign and the null space]
+\label{rem:kernel-sign}
+If $v|_S=c+d\cdot x$, then
+\[
+ Q(v)_{ss}|_S=D_0c+D_1d\cdot x,
+ \qquad D_0=\frac{2\kappa}{a},\quad
+ D_1=\frac2{a^2}+\frac{2\kappa}{a}=\frac{6m_0}{a^3}.
+\]
+Thus $Q(v)_{ss}|_S\geq0$ is equivalent to $D_0c\geq D_1|d|$.
+A nonzero pure dipole fails that condition, but a sufficiently large
+positive radial component makes a mixed radial--dipole seed satisfy
+it.  This is a compatibility statement about seeds, not an existence
+claim for branches satisfying outermostness and outer
+area-minimization.  In particular, the boundary sign cannot be used
+to discard the degree-one null directions from the proof.
+\end{remark}
diff --git a/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/build/sections/05-quartic.tex b/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/build/sections/05-quartic.tex
new file mode 100644
index 0000000000000000000000000000000000000000..33e833a6472cd8e1dd2d6eb4ba65d63fcaafd07b
--- /dev/null
+++ b/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/build/sections/05-quartic.tex
@@ -0,0 +1,876 @@
+\section{Fourth order in the quadratic kernel}
+\label{sec:quartic}
+
+We now treat the kernel left by Proposition~\ref{prop:quadratic}.
+A statement about a fourth-order jet is an identity modulo \(\eps^5\),
+in the conventions of Section~\ref{sec:taylor}; it does not assert the
+existence of the corresponding geometric object at a nonzero parameter.
+
+\begin{proposition}[The quartic coefficient]
+\label{prop:quartic}
+Suppose that \(c_2[q]=0\).  Write the representation furnished by
+Corollary~\ref{cor:kernel} as
+\[
+ q=Q(v):=\Hess_{g_0}v-Bv,\qquad Lv=0,
+ \qquad v=v_0(s)+v_1(s,x),
+\]
+where \(v_1\) has only degree-one spherical modes and \(v\) decays at
+infinity.  Then
+\[
+ c_0[q]=c_1[q]=c_2[q]=c_3[q]=0,
+ \qquad c_4[q]\geq 0.
+\]
+If \(v_1\not\equiv0\), then \(c_4[q]>0\).  In particular, in that
+case the actual deficit \(\mathfrak D_q(\eps)\) is strictly positive
+for all sufficiently small positive \(\eps\).
+\end{proposition}
+
+The proof has three steps. Lemmas~\ref{lem:slice} and
+\ref{lem:null-cut} change the slice and compare its boundary area with
+a minimal leaf, giving a quartic TT square. If that square vanishes,
+Sections~\ref{quart:round-section}--\ref{quart:obstruction-section}
+force the degree-one profile to vanish. The actual expansion from
+Proposition~\ref{prop:expansions} then proves the assertion for the
+given branch.
+
+Throughout the proof we use the stronger decay supplied by the kernel
+representation.  The homogeneous separated equations for \(v\), and
+Lemma~\ref{lem:inverse}, give \(O(r^{-3})\) decay with every required
+derivative for \(v\) and \(q=Q(v)\).  The conformal coefficients through
+degree four consequently have differentiated \(O(r^{-3})\) decay and
+finite angular type.  Put
+\[
+ u=[\phi_\eps]_1,\qquad Lu=0.
+\]
+We use the fourth-order jets of the original data
+\(g_\eps=\phi_\eps^4g_0\) and
+\(K_\eps=\eps\phi_\eps^{-2}q\).  Their initial constraints and their
+boundary expansion equation hold as Taylor identities.
+
+\subsection{A change of slice at the level of jets}
+
+\begin{lemma}[The corrected slice]
+\label{lem:slice}
+There is a decaying finite-angular-type function \(w\), with
+\(w|_S=0\), for which the following construction is defined through
+degree four.  Formally evolve the initial jets in Gaussian time and
+take the graph
+\begin{equation}
+ T=-\eps v+\eps^2w.
+ \label{quart:time-graph}
+\end{equation}
+The induced metric and future second fundamental form, denoted by
+\(\widetilde g\) and \(\widetilde K\), satisfy
+\begin{align}
+ \widetilde K&=\eps^2k+O(\eps^3),
+ &\tr_{g_0}k&=0,&\Div_{g_0}k&=0,
+ \label{quart:leading-k}\\
+ \scal_{\widetilde g}+6
+   &=\eps^4|k|_{g_0}^2+O(\eps^5).
+ \label{quart:scalar-four}
+\end{align}
+Here \(k=O(r^{-3})\), with all derivatives needed for the fourth-order
+CMC construction.  Coefficientwise through degree four,
+\begin{equation}
+ \widetilde g-g_\eps=O(r^{-6})
+ \label{quart:metric-decay}
+\end{equation}
+with the required differentiated bounds.  Thus the mass covectors and
+their Lorentz norms agree through that degree.  Moreover,
+\begin{equation}
+ [\widetilde g-g_\eps]_2
+   =-2vq-Bv^2-\dd v\otimes\dd v
+   =-\Hess_{g_0}(v^2)+\dd v\otimes\dd v+Bv^2.
+ \label{quart:metric-two}
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+We give the formal evolution and its constraint justification before
+performing the graph calculation.  Write
+\[
+ \overline g=-\dd t^2+\gamma(t),\qquad
+ \gamma(0)=g_\eps,\qquad \mathcal K(0)=K_\eps,
+\]
+and determine a full formal time series recursively from
+\begin{align}
+ \partial_t\gamma&=2\mathcal K,\nonumber\\
+ \partial_t\mathcal K
+   &=-\Ric_\gamma-3\gamma
+     -(\tr_\gamma\mathcal K)\mathcal K
+     +2\mathcal K\circ_\gamma\mathcal K.
+ \label{quart:gaussian-evolution}
+\end{align}
+The coefficients take values in the ring of smooth spatial jets modulo
+\(\eps^5\).  At each time order the right side is a known spatial
+differential expression in previously determined coefficients, so the
+recursion is well defined.  A substitution \(t=T=O(\eps)\) uses only
+finitely many of these coefficients.  Keeping a full time series here
+ensures that taking time derivatives never treats an omitted time
+coefficient as zero.
+
+Let \(H=\tr_\gamma\mathcal K\).  The time-index Christoffel symbols
+are
+\[
+ \overline\Gamma^t_{ij}=\mathcal K_{ij},\qquad
+ \overline\Gamma^i_{tj}=\mathcal K^i{}_j,
+\]
+and direct contraction of the curvature gives
+\begin{align}
+ \overline\Ric_{ij}
+  &=\Ric_{\gamma,ij}+\partial_t\mathcal K_{ij}
+       +H\mathcal K_{ij}
+       -2(\mathcal K\circ_\gamma\mathcal K)_{ij},\nonumber\\
+ \overline\Ric_{tt}
+  &=-\tr_\gamma(\partial_t\mathcal K)+|\mathcal K|_\gamma^2,
+ &\overline\Ric_{ti}
+  &=(\Div_\gamma\mathcal K)_i-\partial_iH.
+ \label{quart:gaussian-ricci}
+\end{align}
+Consequently \(E:=\overline\Ric+3\overline g\) has \(E_{ij}=0\).
+The initial Hamiltonian and momentum constraints give
+\(E_{tt}=E_{ti}=0\) at \(t=0\).  For completeness, the Hamiltonian
+constraint is
+\(\scal_\gamma+H^2-|\mathcal K|_\gamma^2=-6\); tracing
+\eqref{quart:gaussian-evolution} at \(t=0\) therefore gives
+\(\tr_\gamma\partial_t\mathcal K=-3+|\mathcal K|_\gamma^2\),
+which proves the asserted \(tt\) equation with the required sign.
+
+The contracted Bianchi identity propagates these remaining equations
+formally.  Indeed, if \(e=E_{tt}\) and \(j_i=E_{ti}\), the trace
+reversal of \(E\) has components
+\[
+ \widehat E_{tt}=e/2,\qquad
+ \widehat E_{ti}=j_i,\qquad
+ \widehat E_{ij}=e\gamma_{ij}/2.
+\]
+Its vanishing divergence, evaluated with the displayed Christoffel
+symbols, is
+\begin{equation}
+ \partial_t e=2\Div_\gamma j-2He,
+ \qquad
+ \partial_tj_i=\tfrac12\partial_i e-Hj_i.
+ \label{quart:bianchi-system}
+\end{equation}
+This homogeneous system determines each next time coefficient of
+\((e,j)\) from the preceding ones.  All coefficients vanish by
+induction.  Thus
+\begin{equation}
+ \overline\Ric=-3\overline g
+ \label{quart:formal-einstein}
+\end{equation}
+as a formal time series modulo \(\eps^5\).
+
+There is no extension hypothesis hidden in this construction.  The
+original coefficient fields are smooth up to \(S\), and their
+constraint residuals and all one-sided spatial derivatives vanish
+there.  The recursion and \eqref{quart:bianchi-system} therefore also
+hold in their spatial Taylor jets at \(S\).  Each finite calculation
+below involves only finitely many such derivatives.  They may, if
+desired, be realized in a collar across \(S\) by their Taylor
+polynomials in \(s\), to an order larger than the differential order
+of that calculation.  Curvature and its residual are formed before
+substituting a displacement of order \(\eps\).  Their retained
+boundary jets vanish, so the substitution is valid even when the
+leading displacement is negative.  This neither constructs nor uses
+an actual Einstein extension on the other side of \(S\).
+
+Gauss--Codazzi applied as a differential identity to any spacelike
+graph now gives
+\begin{equation}
+ \scal_{\rm slice}+6
+   =|K_{\rm slice}|^2-(\tr K_{\rm slice})^2,
+ \qquad
+ \Div\bigl(K_{\rm slice}-(\tr K_{\rm slice})g_{\rm slice}\bigr)=0.
+ \label{quart:slice-constraints}
+\end{equation}
+Equivalently one can form Gaussian coordinates about that graph by
+the Taylor recursion for its normal geodesics and use
+\eqref{quart:gaussian-ricci}.  Either interpretation needs only
+differential identities of finite jets.
+
+We next compute the graph second form explicitly.  With
+\(T_i=\partial_iT\), its tangent fields are
+\(X_i=\partial_i+T_i\partial_t\), and its future unit normal is
+\[
+ n_T=\frac{\partial_t+\grad_{\gamma(t)}T}
+             {\sqrt{1-|\dd T|_{\gamma(t)}^2}}\bigg|_{t=T}.
+\]
+Using \(-\overline g(n_T,\overline\nabla_{X_i}X_j)\) gives
+\begin{equation}
+ \widetilde K_{ij}
+  =\left.
+    \frac{\mathcal K_{ij}(t)+(\Hess_{\gamma(t)}T)_{ij}
+       -T_k\bigl(T_i\mathcal K^k{}_j(t)
+                 +T_j\mathcal K^k{}_i(t)\bigr)}
+     {\sqrt{1-|\dd T|_{\gamma(t)}^2}}
+   \right|_{t=T}.
+ \label{quart:graph-k}
+\end{equation}
+The Hessian connection in this formula is taken at fixed time before
+evaluation at \(t=T\).  The induced metric is
+\(\widetilde g=\gamma(T)-\dd T\otimes\dd T\).
+
+At the background initial slice, \(\mathcal K=0\) and
+\(\partial_t\mathcal K=-B\).  A first-order time displacement \(b\)
+therefore changes the second form by
+\(\Hess b-Bb=Q(b)\).  Equation~\eqref{quart:time-graph} gives
+\[
+ [\widetilde K]_1=q-Q(v)=0.
+\]
+We choose \(w\) to remove the trace at the next degree.  This can be
+done with an explicit scalar source.  Since
+\([\gamma(0)]_1=4ug_0\), the conformal Ricci variation and \(Lu=0\)
+give
+\[
+ [\partial_t\mathcal K(0)]_1
+   =2\Hess u-6ug_0,
+ \qquad [\mathcal K(0)]_2=-2uq.
+\]
+At the background \(\partial_t^2\mathcal K(0)=0\): differentiating
+\eqref{quart:gaussian-evolution} there uses
+\(\partial_t\gamma(0)=0\) and \(\mathcal K(0)=0\).
+Expanding \eqref{quart:graph-k} to degree two thus gives
+\begin{align}
+ k={}&Q(w)-2uq-2v\Hess u+6uvg_0\nonumber\\
+     &\quad+2\dd u\otimes\dd v+2\dd v\otimes\dd u
+          -2\langle\dd u,\dd v\rangle_{g_0}g_0.
+ \label{quart:k-explicit}
+\end{align}
+In particular,
+\[
+ \tr_{g_0}k=Lw+12uv-2\langle\dd u,\dd v\rangle_{g_0}.
+\]
+Choose the unique decaying solution of
+\begin{equation}
+ Lw=2\langle\dd u,\dd v\rangle_{g_0}-12uv,
+ \qquad w|_S=0.
+ \label{quart:w-equation}
+\end{equation}
+Its right side is \(O(r^{-6})\), with differentiated bounds, so
+Lemma~\ref{lem:inverse} applies and gives \(w=O(r^{-3})\) with the
+corresponding derivatives.  Formula~\eqref{quart:k-explicit} gives the
+asserted decay of \(k\).  The degree-two momentum equation in
+\eqref{quart:slice-constraints}, together with \(\tr_{g_0}k=0\),
+gives \(\Div_{g_0}k=0\).
+
+The full transformed trace is \(O(\eps^3)\).  Its square consequently
+begins in degree six, whereas
+\[
+ |\widetilde K|_{\widetilde g}^2
+    =\eps^4|k|_{g_0}^2+O(\eps^5).
+\]
+This proves \eqref{quart:scalar-four}; maximality at higher orders on
+the changed slice is not needed.
+
+Finally,
+\[
+ \gamma(T)-\gamma(0)
+  =2\mathcal K(0)T+\tfrac12\partial_t^2\gamma(0)T^2+\cdots.
+\]
+In every retained coefficient the first term is \(O(r^{-6})\), and
+every subsequent term has at least two decaying time-displacement
+factors multiplying bounded background time coefficients.  The
+recursion preserves these bounded differentiated estimates: in the
+scaled coframe \(\dd s,r\,\dd x\), the background connection and
+curvature, with their required derivatives, are bounded; each
+coefficient is obtained by finitely many covariant differentiations,
+contractions, and inverse-metric Taylor operations from the smooth
+initial coefficients.  The
+same bounds hold for \(\dd T\otimes\dd T\), proving
+\eqref{quart:metric-decay}.  An \(O(r^{-6})\) metric coefficient with
+its first covariant derivative contributes \(O(r^{-3})\) to the
+integrated mass flux, since the test potentials and their gradients
+are \(O(r)\) and the sphere area is \(O(r^2)\).  Thus the leading
+asymptotic data, every mass component, and the mass norm are unchanged
+through degree four.
+
+Since \(\partial_t^2\gamma(0)=-2B\) at the background, the
+degree-two difference is
+\(-2vq-Bv^2-\dd v\otimes\dd v\).  Substituting
+\(q=\Hess v-Bv\) and using
+\(\Hess(v^2)=2v\Hess v+2\dd v\otimes\dd v\) proves
+\eqref{quart:metric-two}.
+\end{proof}
+
+\subsection{Transport of the boundary and its area}
+
+\begin{lemma}[The null cut]
+\label{lem:null-cut}
+For the changed slice of Lemma~\ref{lem:slice} there is a sphere jet
+\(S^\sharp\), based at \(S\), such that
+\begin{align}
+ |S^\sharp|_{\widetilde g}&=|S|_{g_\eps}+O(\eps^5),
+ \label{quart:null-area}\\
+ H_{\widetilde g}(S^\sharp)
+      &=\eps^2k_{ss}|_S+O(\eps^3).
+ \label{quart:null-h}
+\end{align}
+Let \(S_{\min}\) be the minimal Taylor leaf of the CMC construction
+for \(\widetilde g\).  Their height jets agree through degree one.  If
+\begin{equation}
+ z=D^{-1}(k_{ss}|_S),
+ \label{quart:z}
+\end{equation}
+then the height of \(S^\sharp\) minus that of \(S_{\min}\) is
+\(\eps^2z+O(\eps^3)\), and
+\begin{equation}
+ |S^\sharp|_{\widetilde g}-|S_{\min}|_{\widetilde g}
+     =\tfrac12\eps^4\int_S zDz\,\dd A_{g_0}+O(\eps^5).
+ \label{quart:area-hessian}
+\end{equation}
+These statements remain valid when some leading boundary
+displacements are negative.
+\end{lemma}
+
+\begin{proof}
+On the original initial slice take the future outward null normal
+\(\ell=n_{\rm future}+\nu\) to \(S\).  Its affinely parametrized
+null geodesics determine a null hypersurface jet, with parameter
+\(\lambda\) zero at \(S\).  These are Taylor solutions of the
+geodesic and Jacobi equations for the formal metric
+\(\overline g\) constructed above.  At the initial sphere the
+expansion is identically zero by the original boundary equation.
+
+Write \(\chi\) for the null second form and \(\theta=\tr\chi\)
+for its expansion.  The initial sphere is umbilic in
+\(\phi_\eps^4g_0\). Moreover the coefficient of \(\eps\) in the
+tracefree tangential part of \(K_\eps\) equals the tracefree tangential
+part of \(Q(v)\).
+Since \(S\) is totally geodesic in \(g_0\), that part is
+\((\Hess_{a^2\sigma}(v|_S))^{\tf}\).  It vanishes for the constant
+and degree-one modes of \(v|_S\).  The vanishing initial expansion
+and this shear calculation together give
+\begin{equation}
+ \chi(0,\eps)=O(\eps^2).
+ \label{quart:initial-chi}
+\end{equation}
+
+The screen space is the positive-definite quotient
+\(\ell^\perp/\operatorname{span}\{\ell\}\). In a parallel orthonormal
+frame of this space the null shape operator obeys the optical Riccati
+equation
+\[
+ \partial_\lambda\chi=-\chi^2-\mathcal R_\ell,
+ \qquad
+ \partial_\lambda\theta
+       =-|\chi|^2-\overline\Ric(\ell,\ell).
+\]
+These equations follow directly from the Jacobi equation: angular
+Jacobi fields remain orthogonal to \(\ell\), and their derivative
+matrix times their inverse is the shape operator.  At
+\((\lambda,\eps)=(0,0)\), spherical symmetry makes
+\(\mathcal R_\ell\) a scalar multiple of the screen identity.
+Its trace is \(\overline\Ric(\ell,\ell)=0\) by
+\eqref{quart:formal-einstein} and nullness, so this curvature
+endomorphism is zero.  Thus
+\(\partial_\lambda\chi(0,0)=0\).
+
+It follows from \eqref{quart:initial-chi} that every retained monomial
+of \(\chi\) has total degree at least two in \((\lambda,\eps)\).
+Raychaudhuri and the identically zero initial expansion then imply
+\begin{equation}
+ \chi=O_{\rm tot}(2),\qquad
+ \theta=O_{\rm tot}(5).
+ \label{quart:optical-orders}
+\end{equation}
+Here $O_{\rm tot}(j)$ means that each monomial $\lambda^a\eps^b$
+has $a+b\geq j$. The order statements are made in the formal series modulo
+\(\eps^5\).  Equivalently one can choose any sufficiently high
+finite Taylor lift: an Einstein residual of order \(\eps^5\)
+integrated along a segment \(\lambda=O(\eps)\) contributes only
+beyond every degree used here.  Since the logarithmic screen area
+density has derivative \(\theta\), its change along such a segment
+vanishes through degree four.
+
+Intersect this null hypersurface jet with \(t=T\).  At the background
+initial point, the derivative of \(t-T\) along its generator is one.
+Successive Taylor substitution therefore solves for the intersection
+parameter, which is \(O(\eps)\).  The angular projection has identity
+background differential and likewise has a formal inverse.  The
+resulting cut \(S^\sharp\) is a graph over \(S\) in the changed
+slice.
+
+The area conclusion also holds for this variable-parameter cut.  If
+\(J_A\) are the transported angular Jacobi fields, its tangent fields
+are \(J_A+(\partial_A\lambda)\ell\).  Nullness and
+\(\overline g(J_A,\ell)=0\) show that the additional terms do not
+change the induced metric.  Its area density is just the transported
+screen density evaluated at the variable parameter.  This proves
+\eqref{quart:null-area}.
+
+The outward future null normal obtained from the new slice differs
+from \(\ell\) by a scalar with positive, nonzero background value.
+Expansion is multiplied by that scalar, so it still vanishes through
+the needed orders.  At leading order
+\[
+ \tr_{S^\sharp,\widetilde g}\widetilde K
+       =-\eps^2k_{ss}|_S+O(\eps^3),
+\]
+because \(\tr_{g_0}k=0\).  Its zero null expansion gives
+\eqref{quart:null-h}.
+
+Both \(S^\sharp\) and the minimal Taylor leaf have zero mean-curvature
+coefficients through degree one.  The difference of their first height
+coefficients is therefore annihilated by the background operator
+\(D\), which is invertible.  At degree two, their already equal first
+coefficients make all common nonlinear terms cancel, leaving
+\(D(h_2^\sharp-h_2^{\min})=k_{ss}|_S\).  This proves
+\eqref{quart:z}.  The first variation of area at the minimal Taylor
+leaf vanishes to the retained order.  A height difference beginning
+with \(\eps^2z\) consequently has, at degree four, only the base
+second-variation contribution
+\(\frac12\int_S zDz\,\dd A_{g_0}\).  This is
+\eqref{quart:area-hessian}.
+
+All of these assertions are identities in the joint spatial,
+\(\lambda\), and \(\eps\) Taylor jets at the original boundary.
+The boundary-jet justification in Lemma~\ref{lem:slice} permits their
+evaluation at negative leading displacements.  Neither the sign of an
+affine segment nor that of a height coefficient changes the order
+calculation or the area Hessian.  No area inequality for an actually
+extended manifold is being applied.
+\end{proof}
+
+\subsection{The quartic square and its vanishing consequences}
+
+The two boundary comparisons in Lemma~\ref{lem:null-cut} are
+summarized in Figure~\ref{fig:boundary-comparisons}.  Null transport
+preserves the boundary area to the degree being studied; the
+subsequent minimal-graph correction produces the boundary term
+that the TT identity will absorb.
+
+\begin{figure}[ht]
+\centering
+\begin{tikzpicture}[
+  surface/.style={draw,rounded corners=2pt,align=center,
+    minimum width=2.3cm,minimum height=1.25cm},
+  >=Latex,font=\small]
+ \node[surface] (original) at (0,0)
+   {$S$ in $g_\eps$\\[3pt]$H=O(\eps)$};
+ \node[surface] (cut) at (4.4,0)
+   {$S^\sharp$ in $\widetilde g$\\[3pt]$H=O(\eps^2)$};
+ \node[surface] (minimal) at (8.8,0)
+   {$S_{\min}$ in $\widetilde g$\\[3pt]$H=0$};
+ \draw[->] (original.east) -- (cut.west)
+   node[midway,above,align=center]{null\\transport};
+ \draw[->] (cut.east) -- (minimal.west)
+   node[midway,above,align=center]{minimal\\graph};
+ \node[align=center,text width=3.7cm] at (2.2,-1.2)
+   {area unchanged\\through degree four};
+ \node[align=center,text width=3.7cm] at (6.6,-1.2)
+   {height gap $\eps^2z$;\\quartic area gap};
+\end{tikzpicture}
+\caption{The boundary comparisons in the changed slice.  The diagram
+describes finite Taylor families, including the possibility of negative
+leading displacements. The signed height gap is
+$h^\sharp-h^{\min}=\eps^2z+O(\eps^3)$, and the corresponding area gap is
+$|S^\sharp|_{\widetilde g}-|S_{\min}|_{\widetilde g}
+=\frac12\eps^4\int_S zDz\,\dd A_{g_0}+O(\eps^5)$;
+see \eqref{quart:area-hessian}. All displayed geometric statements are
+understood through the required Taylor order.}
+\label{fig:boundary-comparisons}
+\end{figure}
+
+Apply Proposition~\ref{prop:cmc-jets} to \(\widetilde g\) through
+degree four.  Its hypotheses follow from
+Lemma~\ref{lem:slice}, including the unchanged leading asymptotic
+data.  Denote by \(f\) the lapse, by \(\bar f\) its area average on
+each leaf, and set
+\[
+ \delta f=f-\bar f,\qquad
+ U_j=[\II^{\tf}]_j,\qquad \mathcal F_j=[\delta f]_j.
+\]
+At degree two, the scalar contribution is zero by
+\eqref{quart:scalar-four}.  Equations~\eqref{quart:null-area} and
+\eqref{quart:area-hessian}, and the preserved mass, identify the
+quadratic coefficient of mass minus \(b_*\) of the minimal-leaf area
+with the original zero coefficient \(c_2[q]\).  The leading-energy
+identity, Proposition~\ref{prop:leading-energy}, with its scalar
+tensor equal to zero, gives
+\[
+ 0=\frac1{16\pi}\int_M N
+       \bigl(|U_1|^2+2\mathcal F_1J_s^0\mathcal F_1\bigr)\,\dd V_{g_0}.
+\]
+The lapse coefficient is mean free on each background sphere, and
+\(J_s^0\) is strictly positive on that space.  Since \(N>0\) for
+\(s>0\), sphere integration and smoothness imply
+\begin{equation}
+ U_1=0,\qquad \mathcal F_1=0.
+ \label{quart:first-geometric-zero}
+\end{equation}
+
+We can now use Proposition~\ref{prop:leading-energy} at order four:
+the scalar-curvature defect starts with
+\(\eps^4|k|^2\), and both geometric square terms start in degree
+four.  It follows first that mass minus \(b_*\) of the minimal-leaf
+area has no coefficients below degree four.  The area correction
+\eqref{quart:area-hessian} also starts in that degree.  Thus
+\(c_3[q]=0\), as well as the previously known lower vanishings.
+
+Let \(v_k\) be the decaying solution
+\begin{equation}
+ Lv_k=0,\qquad v_k|_S=z=D^{-1}(k_{ss}|_S).
+ \label{quart:vk}
+\end{equation}
+It is supplied by Lemma~\ref{lem:inverse}.  The TT tensor \(k\) has
+the decay and finite angular type required in
+Lemma~\ref{lem:tt-square}.  Since
+\(b_*'(4\pi a^2)=\kappa/(8\pi)\), subtracting the area correction
+from the leading-energy identity and applying that lemma gives
+\begin{equation}
+ c_4[q]=\frac1{16\pi}\int_M N
+   \left(
+      |k-Q(v_k)|_{g_0}^2+|U_2|^2+2\mathcal F_2J_s^0\mathcal F_2
+   \right)\,\dd V_{g_0}.
+ \label{quart:square}
+\end{equation}
+Indeed the subtracted boundary term is exactly
+\(\kappa(16\pi)^{-1}\int_S zDz\,\dd A_{g_0}\).
+The integrals are understood with sphere integration first, as in
+Proposition~\ref{prop:leading-energy}; no pointwise sign is asserted
+for \(\mathcal F_2J_s^0\mathcal F_2\).  Its sphere integral is nonnegative.  This proves
+\(c_4[q]\geq0\), and equality implies
+\begin{equation}
+ \delta f=O(\eps^3),\qquad
+ \II^{\tf}=O(\eps^3)
+ \label{quart:second-geometric-zero}
+\end{equation}
+on every leaf.  Boundary values follow by smoothness.  We show next
+that \eqref{quart:second-geometric-zero} excludes a nonzero degree-one
+part of \(v\).
+
+\subsection{Round angular jets and conformal coordinates}
+\label{quart:round-section}
+
+We now assume equality in the quartic square. The vanishing in
+\eqref{quart:second-geometric-zero} makes the changed metric a warped
+product through degree two. The following lemma will round its angular
+metric before we compare it with the conformal class of \(g_0\).
+
+\Needspace{7\baselineskip}
+\begin{lemma}[Rounding a two-jet on the sphere]
+\label{lem:round-jet}
+Let
+\(\gamma_\eps=\sigma+\eps P_1+\eps^2P_2\) be a smooth metric
+two-jet on \(S^2\) of finite angular type.  Suppose its scalar
+curvature is angularly constant through degree two.  Then there are
+a positive scalar two-jet \(a_\eps\), with \(a_0=1\), and a
+diffeomorphism two-jet \(\Phi_\eps\), based at the identity, such that
+\[
+ \gamma_\eps=a_\eps\Phi_\eps^*\sigma+O(\eps^3).
+\]
+All coefficients can be chosen of finite angular type.
+\end{lemma}
+
+\begin{proof}
+We first prove the infinitesimal decomposition used twice below.  For
+a symmetric tensor \(P\) on the round sphere, there are a vector
+field \(Y\) and a function \(h\) such that
+\begin{equation}
+ P=\mathcal L_Y\sigma+h\sigma.
+ \label{quart:sphere-splitting}
+\end{equation}
+For finite-angular-type \(P\), the choices may retain that property.
+
+Let \(P^{\tf}=P-\frac12(\tr_\sigma P)\sigma\), and identify vector
+fields and one-forms using \(\sigma\).  Define
+\[
+ \mathcal C(Y)_{ij}
+   =\nabla_iY_j+\nabla_jY_i-(\Div Y)\sigma_{ij}.
+\]
+On the curvature-one sphere, commuting covariant derivatives gives
+\begin{equation}
+ \Div\mathcal C(Y)=(\Delta_\sigma+1)Y,
+ \qquad
+ (\Delta_\sigma+1)\dd f=\dd(\Delta_\sigma+2)f.
+ \label{quart:conformal-divergence}
+\end{equation}
+Here the Laplacian on one-forms is the rough Laplacian
+\(\nabla^i\nabla_i\).
+The second identity also holds after applying the parallel Hodge star
+\(*\) to one-forms.
+
+Put \(\alpha=\Div P^{\tf}\).  To decompose it, solve the scalar
+equations
+\[
+ \Delta_\sigma\varphi=\Div\alpha,\qquad
+ \Delta_\sigma\psi=-\Div(*\alpha)
+\]
+with zero means.  Both right sides have zero integral.  In finite
+angular spaces these equations are solved by inversion on the
+nonconstant spherical modes.  The residual
+\(\omega=\alpha-\dd\varphi-*\dd\psi\) is closed and co-closed.
+Commuting derivatives of a closed one-form of zero divergence gives
+\(\nabla^i\nabla_i\omega=\omega\); integration against \(\omega\)
+forces \(\omega=0\).  We have therefore obtained
+\[
+ \alpha=\dd\varphi+*\dd\psi.
+\]
+
+The one-forms \(\dd x_i\) and \(*\dd x_i\) correspond to conformal
+Killing vector fields.  Integration by parts against the tracefree
+tensor \(P^{\tf}\) consequently shows that \(\alpha\) is
+orthogonal to all of them.  It follows that both \(\varphi\) and
+\(\psi\) have zero degree-one components.  Thus we may solve
+\[
+ (\Delta_\sigma+2)A=\varphi,\qquad
+ (\Delta_\sigma+2)C=\psi
+\]
+on their angular modes, and set \(Y^\flat=\dd A+*\dd C\).
+Equation~\eqref{quart:conformal-divergence} gives
+\(\Div\mathcal C(Y)=\Div P^{\tf}\).
+
+The tracefree tensor \(P_*=P^{\tf}-\mathcal C(Y)\) is consequently
+divergence free.  Such a tensor vanishes on the round sphere; here is
+a direct verification.  In an orthonormal frame write its components
+as
+\(\left(\begin{smallmatrix}a&b\\b&-a\end{smallmatrix}\right)\).
+Its two divergence equations are precisely the two independent
+Codazzi equations
+\(\nabla_i(P_*)_{jk}=\nabla_j(P_*)_{ik}\).
+Contracting and commuting these equations on the curvature-one sphere
+gives
+\[
+ \nabla^i\nabla_i(P_*)_{jk}
+      =2(P_*)_{jk}-\sigma_{jk}\tr_\sigma P_*
+      =2(P_*)_{jk}.
+\]
+Integration yields
+\(-\int|\nabla P_*|^2=2\int|P_*|^2\), hence \(P_*=0\).
+Taking
+\(h=\frac12\tr_\sigma P-\Div Y\) proves
+\eqref{quart:sphere-splitting}.  Every operation used above commutes
+with rotations or inverts a scalar operator within finitely many
+spherical eigenspaces, so it preserves finite angular type.
+
+The scalar-curvature derivative in the direction \(h\sigma\) is
+\[
+ (D\scal)_\sigma(h\sigma)=-\Delta_\sigma h-2h.
+\]
+The derivative in a Lie direction is zero because the background
+scalar curvature is constant.  If the scalar-curvature variation of
+\(P\) is constant, \eqref{quart:sphere-splitting} therefore implies
+that \(h\) consists only of a constant and degree-one modes.  A
+degree-one function \(h_1\) satisfies
+\(\Hess_\sigma h_1=-h_1\sigma\), so
+\[
+ h_1\sigma
+   =\mathcal L_{-\frac12\grad_\sigma h_1}\sigma.
+\]
+Thus \(P\) is a Lie derivative plus a constant multiple of \(\sigma\).
+
+Apply this conclusion to \(P_1\).  A formal inverse flow and a scalar
+scaling remove its first coefficient.  Both operations preserve
+the property that curvature is angularly constant.  The resulting
+metric two-jet has the form \(\sigma+\eps^2\widehat P_2\), so its
+degree-two curvature equation is the same linearized equation, with
+no quadratic contribution from a first coefficient.  The argument
+just given removes \(\widehat P_2\) by a degree-two flow and scaling.
+Undoing the two operations proves the stated rounding.  This proves
+only an identity of two-jets, which is exactly what is required.
+\end{proof}
+
+\paragraph{Local conformal tangency forced by equality.}
+Suppose now that \(c_4[q]=0\). Fix a compact interval
+\(I\Subset(0,\infty)\).  In coordinates of the CMC sphere jets we can
+remove tangential shift through degree two by integrating its
+tangential reparametrization equation on \(I\).  Its background
+value is zero, so the coordinate change is based at the identity.
+In these coordinates \eqref{quart:second-geometric-zero} says that the
+lapse is a function of the leaf parameter only and that
+\(\II=(H/2)g_{\rm leaf}\) through degree two.  Since \(H\) is
+constant on each leaf, the normal metric variation is
+\[
+ \partial_s g_{\rm leaf}=2f\II=fH g_{\rm leaf}.
+\]
+Consequently on \(I\times S^2\) the metric has the form
+\begin{equation}
+ f_*(s)^2\dd s^2+b(s)^2\gamma_\eps(x)+O(\eps^3),
+ \label{quart:warped-two}
+\end{equation}
+where all quantities are two-jets, with backgrounds \(f_*=1\),
+\(b=r\), and \(\gamma_0=\sigma\).  Angular finiteness is preserved
+by these Taylor coordinate changes and integrations.
+
+For this warped metric the scalar curvature through degree two is
+\begin{equation}
+ b^{-2}\scal_{\gamma_\eps}
+   -\frac4{f_*b}\frac{\dd}{\dd s}
+                      \left(\frac{b'}{f_*}\right)
+   -2\left(\frac{b'}{f_*b}\right)^2.
+ \label{quart:warped-scalar}
+\end{equation}
+Equation~\eqref{quart:scalar-four} makes it the constant \(-6\)
+through degree two.  The last two terms in
+\eqref{quart:warped-scalar} are radial, so
+\(\scal_{\gamma_\eps}\) is angularly constant to that order.
+Lemma~\ref{lem:round-jet} makes \(\gamma_\eps\) round up to
+diffeomorphism and scaling.  Absorbing the scaling into \(b\), we
+obtain \(f_*^2\dd s^2+b^2\sigma\).
+
+This last two-jet is locally conformal to a pullback of \(g_0\).
+Indeed solve, coefficientwise on \(I\),
+\begin{equation}
+ \frac{y'}{r(y)}=\frac{f_*}{b},
+ \label{quart:radial-coordinate}
+\end{equation}
+with background \(y=s\) and a fixed identity background initial value
+at an interior point of \(I\).  This is a nonsingular scalar ODE,
+and it gives
+\[
+ f_*^2\dd s^2+b^2\sigma
+   =\frac{b^2}{r(y)^2}
+       \bigl(\dd y^2+r(y)^2\sigma\bigr)
+       \quad\bmod\eps^3.
+\]
+Composing the coordinate changes already made shows, in the original
+coordinates on the interval, that
+\begin{equation}
+ \widetilde g=\omega_\eps\Psi_\eps^*g_0+O(\eps^3),
+ \qquad \omega_0=1,\quad\Psi_0=\mathrm{id}.
+ \label{quart:conformal-pullback}
+\end{equation}
+Only this local statement is used.
+
+There is a useful consequence of the known first coefficient in
+\eqref{quart:conformal-pullback}.  Write
+\(\omega_\eps=1+\eps a_1+\eps^2a_2\) and express the
+diffeomorphism two-jet as
+\[
+ \Psi_\eps^*g_0
+  =g_0+\eps\mathcal L_Xg_0
+      +\eps^2\bigl(\mathcal L_Zg_0
+                      +\tfrac12\mathcal L_X^2g_0\bigr).
+\]
+The graph calculation shows that \(\widetilde g-g_\eps\) starts
+in degree two, so \([\widetilde g]_1=4ug_0\).  Hence
+\(\mathcal L_Xg_0=\lambda g_0\) for some function \(\lambda\).
+It follows that
+\[
+ \mathcal L_X^2g_0=(X\lambda+\lambda^2)g_0,
+\]
+and the mixed scalar term \(a_1\mathcal L_Xg_0\) is pure trace as
+well.  Therefore
+\begin{equation}
+ [\widetilde g]_2=\mathcal L_Zg_0+c g_0
+ \label{quart:second-tangent}
+\end{equation}
+for a function \(c\) on \(I\times S^2\).  The original metric
+coefficient \([g_\eps]_2\) is itself pure trace.  Thus the same
+Lie-derivative-plus-trace conclusion holds for their difference.
+Since \(\Hess(v^2)=\frac12\mathcal L_{\grad(v^2)}g_0\),
+\eqref{quart:metric-two} gives the necessary condition
+\begin{equation}
+ \dd v\otimes\dd v+Bv^2=\mathcal L_Yg_0+c g_0
+ \quad\hbox{on }I\times S^2
+ \label{quart:necessary-tangent}
+\end{equation}
+for some local \(Y,c\).  Their choices may depend on \(I\).
+
+\subsection{The degree-one obstruction}\label{quart:obstruction-section}
+
+\begin{lemma}[A nonzero degree-one mode is obstructed]
+\label{lem:dipole-obstruction}
+Let \(v=v_0(s)+V(s)x\) solve \(Lv=0\), where \(x\) is one fixed
+unit-axis linear coordinate on \(S^2\) and \(V=O(r^{-3})\).
+If \eqref{quart:necessary-tangent} holds on every compact radial
+interval in the interior, with possibly different local vector fields
+and functions, then \(V\equiv0\).
+\end{lemma}
+
+\begin{proof}
+Set \(Y_2=x^2-1/3\), a degree-two spherical eigenfunction.  Project
+\eqref{quart:necessary-tangent} onto its scalar, gradient, and
+tangential Hessian modes.  Write the contributing radial vector-field
+coefficient as \(p(s)Y_2\partial_s\), its tangential coefficient as
+\(d(s)\grad_\sigma Y_2\), and the scalar-multiple coefficient as
+\(c(s)Y_2\).  These are the only generator components contributing to
+the tested modes.  This follows either from the one-form decomposition
+in Lemma~\ref{lem:round-jet} or by integration by parts, using
+\(\Delta_\sigma Y_2=-6Y_2\).  In particular, the coexact component
+is orthogonal to both the gradient test and the electric Hessian
+test, and the other spherical eigenspaces are orthogonal as well.
+
+For clarity, the angular identity
+\[
+ \dd x\otimes\dd x
+       =\tfrac12\Hess_\sigma Y_2+x^2\sigma
+\]
+shows all terms of the left side in these modes.  With
+\(B=B_s\dd s^2+B_t r^2\sigma\), equality of the radial, mixed,
+Hessian, and tangential scalar coefficients respectively gives
+\begin{align}
+ V'^2+B_sV^2&=2p'+c,
+ &\frac{VV'}2&=p+r^2d',\nonumber\\
+ \frac{V^2}2&=2r^2d,
+ &V^2+r^2B_tV^2&=2rNp+r^2c.
+ \label{quart:projection-system}
+\end{align}
+Terms involving \(v_0\) have only degrees zero or one and do not
+enter these equations.  Solving the last three equations yields
+\[
+ d=\frac{V^2}{4r^2},\qquad
+ p=\frac{NV^2}{2r},\qquad
+ c=\frac{3m_0V^2}{r^3}.
+\]
+Substitute these expressions in the first equation and use
+\[
+ B_s=1-\frac{2m_0}{r^3},\qquad
+ B_t=1+\frac{m_0}{r^3},\qquad
+ N'=r+\frac{m_0}{r^2},\qquad
+ N^2=1+r^2-\frac{2m_0}{r}.
+\]
+After cancellation one obtains
+\begin{equation}
+ \left(V'-\frac NrV\right)^2
+       =\frac{6m_0}{r^3}V^2.
+ \label{quart:dipole-identity}
+\end{equation}
+The local generators have disappeared from this relation.  Because
+the radial interval was arbitrary, it holds throughout the end even
+though their choices need not agree on overlapping intervals.
+
+For \(z=V/r\), \eqref{quart:dipole-identity} reads
+\[
+ |z'|=\sqrt{6m_0}\,r^{-3/2}|z|.
+\]
+The coefficient on the right is integrable toward infinity, and
+\(z\to0\).  The integral differential inequality, applied backwards
+between \(s\) and \(S>s\), gives
+\[
+ |z(s)|\leq |z(S)|
+       \exp\left(\int_s^S\sqrt{6m_0}\,r(t)^{-3/2}\,\dd t\right).
+\]
+Letting \(S\to\infty\) proves \(z=0\) on the end.  The homogeneous
+degree-one ODE obtained from \(Lv=0\) then gives \(V\equiv0\) by
+uniqueness.
+\end{proof}
+
+\begin{proof}[Completion of the proof of Proposition~\ref{prop:quartic}]
+The coefficient vanishings and nonnegativity were proved in
+\eqref{quart:first-geometric-zero}--\eqref{quart:square}.  Suppose
+that the degree-one part of \(v\) is nonzero and that \(c_4[q]=0\).
+All three degree-one radial coefficients solve the same decaying
+homogeneous ODE.  Decaying uniqueness makes them proportional, so
+after a fixed rotation their sum is \(V(s)x\) for one axis and a
+nonzero radial function \(V\).  The vanishing consequences of
+\eqref{quart:square} give \eqref{quart:necessary-tangent} on each
+interior radial interval.  Lemma~\ref{lem:dipole-obstruction} forces
+\(V=0\), a contradiction.  Thus \(c_4[q]>0\).
+
+Finally this is a statement about the actual deficit, not only a
+formal inequality.  The kernel representation gives precisely the
+finite angular type and differentiated decay needed for the
+fourth-order expansion in Proposition~\ref{prop:expansions}.
+Lemma~\ref{lem:green} converts the weighted function and equation
+remainders to the mass remainder.  Consequently
+\[
+ \mathfrak D_q(\eps)=c_4[q]\eps^4+O(\eps^5)
+\]
+in the case under consideration.  Its positive coefficient gives
+strict positivity for all sufficiently small positive \(\eps\),
+with the interval allowed to depend on the fixed data.  The radial
+kernel, including the zero seed, is handled by the exact argument in
+the next section.
+\end{proof}
diff --git a/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/build/sections/06-conclusion.tex b/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/build/sections/06-conclusion.tex
new file mode 100644
index 0000000000000000000000000000000000000000..d6cc10eaaf8688efd3b3d81425734e2655fd8d27
--- /dev/null
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@@ -0,0 +1,222 @@
+\section{Radial equality and the nonlinear conclusion}
+\label{sec:conclusion}
+
+\begin{proposition}[Exact radial equality]
+\label{prop:radial}
+Suppose the fixed TT seed $q$ is invariant under rotations of $S^2$.
+Every branch in Theorem~\ref{thm:main} is radial for all sufficiently
+small $\eps$, and on that parameter interval
+\[
+ m_{\AH}(\eps)=b_*(A_\eps).
+\]
+This includes the zero seed, and requires no sign for $q_{ss}|_S$.
+\end{proposition}
+
+\begin{proof}
+We first prove radiality of the actual solution, rather than merely
+of its Taylor coefficients.  Fix a rotation $\mathcal R$, and set
+$\widehat\phi=\phi_\eps\circ\mathcal R$ and
+$w=\widehat\phi-\phi_\eps$.
+Because $q$ is radial, $\widehat\phi$ satisfies the same equation and
+boundary condition as $\phi_\eps$.  Their difference satisfies
+\begin{equation}
+ (\Delta_{g_0}-c_\eps)w=0,
+ \qquad w_s|_S=c_\eps^*w|_S,
+ \label{eq:radial-uniqueness-equation}
+\end{equation}
+where divided differences of the nonlinearities give, uniformly on
+$M$ and uniformly in $\mathcal R$,
+\[
+ c_\eps=3+o(1),\qquad c_\eps^*=O(\eps).
+\]
+Indeed, the derivatives of the interior and boundary nonlinearities
+with respect to their positive scalar argument $z$ are respectively
+\[
+ \frac34(5z^4-1)+\frac{7\eps^2}{8}|q|^2z^{-8},
+ \qquad -\frac{3\eps}{4}q_{ss}z^{-4}.
+\]
+The asserted bounds follow from boundedness of $q$ and the given
+uniform convergence $\phi_\eps\to1$.
+
+Choose $3/2<\beta<\min(\tau,\sqrt3)$. Then
+$w=o(e^{-\beta s})$, and \eqref{eq:radial-uniqueness-equation} has the
+form of Lemma~\ref{lem:inverse} with $V=c_\eps-3$, Robin coefficient
+$a_*=c_\eps^*$, and zero interior and boundary data. For sufficiently
+small $\eps$, uniformly in the rotation, $\|V\|_\infty\leq c_\beta$
+and $\|a_*\|_\infty\leq\beta/2$. The comparison estimate gives $w=0$,
+so the actual $\phi_\eps$ is radial.
+
+Write the invariant seed as
+$q=A_q(s)\,\dd s^2+B_q(s)r^2\sigma$.
+Its trace and divergence equations give
+\[
+ B_q=-A_q/2,
+ \qquad A_q'+3(N/r)A_q=0.
+\]
+Consequently, for a constant $C_q\in\mathbb R$,
+\begin{equation}
+ q=C_qr^{-3}\left(\dd s^2-\frac12r^2\sigma\right).
+ \label{eq:radial-tt-tensor}
+\end{equation}
+Fix a sufficiently small parameter.  Define proper radial distance
+$\ell$ and the actual area radius $R$ by
+\[
+ \dd\ell=\phi_\eps^2\,\dd s,
+ \qquad R=r\phi_\eps^2.
+\]
+Then the actual metric and second-form data are
+\[
+ g_\eps=\dd\ell^2+R^2\sigma,
+ \qquad K_\eps=-2k\,\dd\ell^2+kR^2\sigma,
+ \qquad k=-\frac{\eps C_q}{2R^3}.
+\]
+A dot denotes an $\ell$-derivative.  The momentum and scalar constraints
+are therefore
+\begin{equation}
+ \dot k=-3\frac{\dot R}{R}k,
+ \qquad
+ -\frac{4\ddot R}{R}
+ +\frac{2(1-\dot R^2)}{R^2}+6=6k^2.
+ \label{eq:radial-constraints}
+\end{equation}
+The scalar-curvature formula here follows from the sectional curvatures
+$-\ddot R/R$ and $(1-\dot R^2)/R^2$ of this warped product.
+
+Define
+\begin{equation}
+ \mathcal M
+ =\frac R2\left(1+R^2-\dot R^2+R^2k^2\right).
+ \label{eq:radial-conserved-mass}
+\end{equation}
+Direct differentiation, using the momentum equation, gives
+\[
+ \dot{\mathcal M}
+ =\frac{\dot R}{2}
+ \left(1+3R^2-\dot R^2-2R\ddot R-3R^2k^2\right)=0
+\]
+by the scalar equation in \eqref{eq:radial-constraints}.
+No division by $\dot R$ has been used, so conservation is valid at
+turning points of $R$ as well.
+At the boundary the MOTS equation reads
+\[
+ \frac{2\dot R}{R}+2k=0.
+\]
+Substitution in \eqref{eq:radial-conserved-mass} yields
+\begin{equation}
+ \mathcal M|_S
+ =\frac{R|_S}{2}\bigl(1+(R|_S)^2\bigr)
+ =b_*\bigl(4\pi(R|_S)^2\bigr)=b_*(A_\eps).
+ \label{eq:radial-mass-at-boundary}
+\end{equation}
+
+We identify the same constant with the specified mass flux at infinity.
+Put $\psi=\phi_\eps-1$.  Its actual radial equation gives
+\[
+ \psi''+2\frac Nr\psi'-3\psi
+ =O(\psi^2)+O(r^{-6}),
+\]
+where \eqref{eq:radial-tt-tensor} was used in the source estimate.
+The assumed weighted decay gives
+$\psi,\psi'=O(e^{-\tau s})$, and $N/r=1+O(e^{-2s})$.
+It follows that
+\[
+ \psi''+2\psi'-3\psi=O(e^{-\gamma s}),
+ \qquad \gamma=\min(2\tau,\tau+2,6)>3.
+\]
+Variation of constants for the roots $1,-3$ excludes the growing
+solution by the prescribed decay and yields, as in
+Lemma~\ref{lem:inverse}, a constant $c=c(\eps)$ with
+\begin{equation}
+ \phi_\eps=1+cr^{-3}+o(r^{-3}),
+ \qquad
+ \partial_s\phi_\eps=-3cNr^{-4}+o(r^{-3}).
+ \label{eq:radial-actual-asymptotic}
+\end{equation}
+In particular,
+\[
+ R=r+2cr^{-2}+o(r^{-2}),\qquad
+ \dot R=N+2r\frac{\partial_s\phi_\eps}{\phi_\eps}
+ =N-6cNr^{-3}+o(r^{-2}).
+\]
+Using $N^2=1+r^2-2m_0/r$, we obtain
+\[
+ 1+R^2-\dot R^2
+ =\frac{2m_0+16c}{r}+o(r^{-1}).
+\]
+Also $R^3k^2=O(R^{-3})$, so
+\begin{equation}
+ \lim_{\ell\to\infty}\mathcal M=m_0+8c.
+ \label{eq:radial-mass-at-infinity}
+\end{equation}
+
+For a direct flux normalization check, rotational invariance implies
+$p_i=0$ for $i=1,2,3$.
+Equation~\eqref{eq:radial-actual-asymptotic} implies
+\[
+ g_\eps-g_0=4cr^{-3}b+o(r^{-3})
+\]
+in scaled components, with its first differentiated asymptotic.
+For a perturbation $fb$ in dimension three, the mass-flux covector
+integrand with test function $V$ simplifies to
+$2(f\,\dd V-V\,\dd f)$.
+With $f=4cr^{-3}$, $V=V_0=\sqrt{1+r^2}$, and
+$\nu_b=\sqrt{1+r^2}\,\partial_r$, its normal component is
+\[
+ 2\bigl(f\partial_{\nu_b}V_0
+       -V_0\partial_{\nu_b}f\bigr)
+ =32cr^{-2}+24cr^{-4}.
+\]
+The error terms give zero limiting flux.  Integrating over a coordinate
+sphere of area $4\pi r^2$ and dividing by $16\pi$ gives
+$p_0-m_0=8c$.
+Since $p_0\to m_0>0$ as $\eps\to0$,
+\[
+ m_{\AH}=p_0=m_0+8c=\mathcal M=b_*(A_\eps)
+\]
+for sufficiently small $\eps$, by
+\eqref{eq:radial-mass-at-boundary}--\eqref{eq:radial-mass-at-infinity}.
+This argument includes $C_q=0$.  In that case one can also apply the
+comparison estimate of Lemma~\ref{lem:inverse} to $\phi_\eps-1$
+and conclude $\phi_\eps\equiv1$ directly.
+\end{proof}
+
+\begin{proof}[Completion of the proof of Theorem~\ref{thm:main}]
+Fix the seed and the given solution branch.  If $q$ is radial,
+Proposition~\ref{prop:radial} gives exact equality on a sufficiently
+small parameter interval.
+Suppose henceforth that $q$ is not radial.
+Proposition~\ref{prop:quadratic} gives the actual expansion
+\[
+ \mathfrak D_q(\eps)=c_2[q]\eps^2+o(\eps^2),
+ \qquad c_2[q]\geq0.
+\]
+If $c_2[q]>0$, this proves strict positivity of the deficit for every
+sufficiently small positive $\eps$.
+
+If $c_2[q]=0$, Corollary~\ref{cor:kernel} gives
+$q=Q(v)$ with $Lv=0$ and with only constant and linear angular modes.
+Its linear part is nonzero, for otherwise $q$ would be radial.
+These homogeneous modes have the stronger differentiated decay needed
+for the actual fourth-order expansion in
+Proposition~\ref{prop:expansions}.
+Proposition~\ref{prop:quartic} then gives
+\[
+ \mathfrak D_q(\eps)=c_4[q]\eps^4+o(\eps^4),
+ \qquad c_4[q]>0,
+\]
+with all coefficients below order four equal to zero.
+Again the actual deficit is strictly positive for every sufficiently
+small positive $\eps$.
+
+At $\eps=0$ the background identity
+$m_0=b_*(4\pi a^2)$ gives equality.
+The preceding alternatives exhaust all fixed seeds.  Choose
+$\eps_0>0$ small enough for the applicable alternative and the given
+branch, with $\eps_0\leq\eps_*$.
+This proves the exact inequality on $0\leq\eps<\eps_0$, equality for
+radial seeds, and strict inequality for nonradial seeds at positive
+parameters in that interval.
+The size of the interval is allowed to depend on the fixed seed and
+branch; no uniform positive lower bound for $c_2$ or $c_4$ is asserted
+or required.
+\end{proof}
diff --git a/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/paper.pdf b/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..618f60556e6e4cf9361525413df1f44ae44de8cb
--- /dev/null
+++ b/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:624b287b6675a96256bf93ce5eefca9e08e04d6655f6808fd8529e6001b75012
+size 551930
diff --git a/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/README.md b/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..f8b54e594ee78f7fcee2d537f8253746b9affd0b
--- /dev/null
+++ b/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/README.md
@@ -0,0 +1,18 @@
+# [A marked tensor obstruction to four-dimensional disk embedding](paper.pdf)
+
+OpenAI · September 24, 2026
+
+## Citation
+
+To cite this manuscript:
+
+```bibtex
+@misc{OAI:A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026,
+  author = {{OpenAI}},
+  title = {{A marked tensor obstruction to four-dimensional disk embedding}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/paper.pdf}{OAI:A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/appendices/formal-trace-comparison.tex b/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/appendices/formal-trace-comparison.tex
new file mode 100644
index 0000000000000000000000000000000000000000..3d1c445c72471a0dd048261c4522534c626939b1
--- /dev/null
+++ b/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/appendices/formal-trace-comparison.tex
@@ -0,0 +1,370 @@
+\section{The full relative trace comparison}
+\label{app:formal-trace-comparison}
+
+We prove Proposition~\ref{prop:formal-trace-comparison}, including its
+compatibility with the relative integration homotopy.  This is the
+comparison used in Section~\ref{sec:algebraicity} to identify the
+shifted symplectic structure on the punctured filling with the integrated
+trace on the edge complex.
+
+We retain the closed-form complex and total signs of
+Section~\ref{subsec:formal-trace-statement}.  There are three
+differentials: the source differential \(d_X\), the coefficient
+cdga differential \(Q\), and coordinate de Rham differentiation
+\(\delta_R\).  On coordinate forms \(Q\) acts by
+\(\mathcal L_Q\).  Source and simplex forms contribute to internal
+degree but have exterior weight zero; the coordinate forms supply
+exterior weight.  Coefficient completion and the product over exterior
+weights are both retained throughout the argument.
+
+\begin{proof}[Proof of Proposition~\ref{prop:formal-trace-comparison}]
+\noindent\textbf{1. The target cocycle.}\quad
+We first construct the target form on a presentation of the formal
+classifying stack.  Let \(\widehat G\) be the formal group of \(G\)
+at the identity.  Here the pointed completion of \(BG\) means the
+deformation functor with a specified identification of its reduction
+with the trivial object; its automorphisms reduce to the identity.
+This pointed completion is \(B\widehat G\), presented by the group nerve
+\(\widehat G^\bullet\).  Its term \(\widehat G^n\) is formally smooth,
+with completed coordinate algebra
+\[
+                  R_n=\mathbb C[[x_1,\ldots,x_{n\dim G}]].
+\]
+Continuous coordinate forms on these algebras therefore compute
+derived forms on the nerve.
+
+We use backward transition elements for the group nerve.  More explicitly,
+if \(U_{ij}\) is the forward transport from vertex \(i\) to vertex \(j\)
+for the connection \(d+a\), then
+\(U_{ik}=U_{jk}U_{ij}\).  The nerve labels
+\(h_{ij}=U_{ij}^{-1}\) therefore satisfy
+\(h_{ik}=h_{ij}h_{jk}\).  For a left Maurer--Cartan form
+\(a=\gamma^{-1}d\gamma\), these labels are
+\(h_{ij}=\gamma(i)^{-1}\gamma(j)\), whereas the forward transport is
+\(U_{ij}=\gamma(j)^{-1}\gamma(i)\).  Thus Getzler's interval labeled
+by \(\exp(x)\) has form \(x\,dt\), and its forward holonomy is
+\(\exp(-x)\), as in the edge construction.  This is the opposite-arrow
+convention for describing the same local system; inversion is not being
+regarded as a homomorphism from a noncommutative group to itself.  All
+based holonomy markings below remain the forward transport matrices.
+
+For an ordinary nilpotent ideal \(\mathfrak m\), the Lie algebra
+\(\mathfrak g\otimes\mathfrak m\) is nilpotent.  Getzler's simplicial
+Maurer--Cartan space is
+\[
+ \MC_\bullet(\mathfrak g\otimes\mathfrak m)
+ =\MC\bigl(\mathfrak g\otimes\mathfrak m
+                     \otimes\Omega_{\mathrm{poly}}^*(\Delta^\bullet)\bigr).
+\]
+Its Dupont-gauge subspace \(\gamma_\bullet\), defined using the
+simplicial de Rham contraction of \cite{Dupont1976}, is the nerve of the
+Campbell--Hausdorff group in this degree-zero case, and its inclusion
+in \(\MC_\bullet\) is a natural homotopy equivalence
+\cite[\S1, Theorem~5.4, and Corollary~5.11]{Getzler2009}.
+Applying this construction to the universal \(n\)-tuple, first at
+each finite augmentation order and then taking the inverse limit,
+gives compatible flat simplices
+\[
+ a_n\in
+ \bigl(\mathfrak g\otimes\mathfrak m_{R_n}
+          \widehat\otimes\Omega_{\mathrm{poly}}^*(\Delta^n)\bigr)^1.
+\]
+The compatibility here is that of Getzler's simplicial construction,
+including every face and degeneracy.
+
+Differentiate only in the \(R_n\) directions and set
+\begin{equation}
+\label{eq:nerve-trace-cocycle}
+                       \beta_n
+                  =\tfrac12\kappa(\delta_{R_n}a_n,\delta_{R_n}a_n).
+\end{equation}
+Polynomial simplex forms have internal degree but exterior weight
+zero in this expression.  Differentiating the Maurer--Cartan equation
+gives the covariant-closedness equation for \(\delta_{R_n}a_n\).
+Invariance of \(\kappa\) cancels the two bracket terms and yields
+\[
+                         d_\Delta\beta_n=0,
+                         \qquad \delta_{R_n}\beta_n=0.
+\]
+These identities and simplicial compatibility make \(\beta_\bullet\)
+a Thom--Whitney descent cocycle of internal degree two and exterior
+weight two on the smooth formal nerve.  All its higher closed-form
+components are zero.  Equivalently, integration over the simplices
+gives a normalized descent cocycle; its descent differential is zero
+by Stokes' formula.
+
+\smallskip\noindent\textbf{2. The canonical closed-form class.}\quad
+We identify its class with the restriction of \(\omega_\kappa\).
+The cotangent complex of \(B\widehat G\) is the coadjoint bundle
+\(\mathfrak g^*[-1]\).  Consequently
+\[
+ R\Gamma(B\widehat G,\mathop{\bigwedge}\nolimits^p
+                  \mathbb L_{B\widehat G})\in D^{\geq p}.
+\]
+Indeed, \(\bigwedge^p\mathbb L\) is the bundle
+\(\Sym^p(\mathfrak g^*)[-p]\), and derived invariants of a bundle in
+degree zero have no negative cohomology.  Applied to the exterior
+weight filtration of the closed two-form complex, this connectivity
+gives
+\begin{equation}
+\label{eq:formal-bg-form-uniqueness}
+ H^2\operatorname{Cl}^2(B\widehat G)
+       \simeq \Sym^2(\mathfrak g^*)^{\widehat G},
+ \qquad H^1\operatorname{Cl}^2(B\widehat G)=0.
+\end{equation}
+For the \(j\)-th higher component, the relevant internal degree would
+be \(2-j\), whereas weight \(2+j\) starts in degree \(2+j\).
+This is a statement about the derived form complexes and their
+classes, independent of the terms in a chosen descent resolution.
+
+Linearizing the nerve at the identity gives the Dold--Kan nerve of
+\(\mathfrak g[1]\).  Formula~\eqref{eq:nerve-trace-cocycle}
+linearizes to its quadratic two-form with associated symmetric
+bilinear form \(\kappa\).  Here we use the usual normalized simplex
+integration and shuffle convention: the two shuffles in the
+polarization cancel the factor \(1/2\).
+This is precisely the underlying bilinear form of
+\(\omega_\kappa\) \cite[\S1.2, classifying stacks]{PTVV2013}.
+Equation~\eqref{eq:formal-bg-form-uniqueness} identifies their full
+closed-form classes, and makes the comparison homotopy unique up
+to homotopy.
+
+\smallskip\noindent\textbf{3. Derived coefficients.}\quad
+We now give the comparison on derived coefficients, including its map on
+linear fibers.  By an Artin test algebra here we mean a nonpositive local
+augmented cdga \(B\) over \(\mathbb C\), with nilpotent augmentation
+ideal \(\mathfrak m_B\).  Set
+\[
+ L_B=\mathfrak g\otimes\mathfrak m_B,
+ \qquad \Omega_q=\Omega^*_{\mathrm{poly}}(\Delta^q),
+ \qquad H_q(B)=\exp Z^0(L_B\otimes\Omega_q).
+\]
+The exponential uses the finite Campbell--Hausdorff series.  These groups
+form a simplicial group \(H_\bullet(B)\).  They compute the derived
+evaluation of \(\widehat G\) on \(B\): in exponential coordinates a
+map from its completed smooth coordinate algebra sends each coordinate
+to a degree-zero cycle in \(\mathfrak m_B\otimes\Omega_q\).  Equivalently
+one uses the nonpositive truncation of this polynomial simplicial frame.
+Consequently the diagonal of the bisimplicial nerve
+\(N_nH_q(B)\) computes \(B\widehat G(B)\).
+
+Apply the ordinary degree-zero Getzler construction to an \(n\)-tuple
+in \(H_q(B)\).  The inclusion of the degree-zero Lie algebra
+\(Z^0(L_B\otimes\Omega_q)\) into \(L_B\otimes\Omega_q\) gives a
+flat element
+\[
+ A_{n,q}\in
+ \MC\bigl(L_B\otimes\Omega_q\otimes\Omega_n\bigr).
+\]
+The coefficient cycles have zero total differential, so this is the
+Maurer--Cartan equation for the full displayed tensor product.
+Getzler's compatibility in \(n\) and functoriality in the coefficient
+Lie algebra give compatibility in both simplicial directions, including
+all faces and degeneracies.  Restriction to the diagonal simplex gives
+a natural simplicial map
+\begin{equation}
+\label{eq:explicit-dg-nerve-comparison}
+ \Phi_B:\operatorname{diag}N_\bullet H_\bullet(B)
+                  \longrightarrow\MC_\bullet(L_B),
+ \qquad
+ (\Phi_B)_q=\Delta_q^*A_{q,q}.
+\end{equation}
+
+Here is a direct verification that this particular map is an equivalence.
+Consider a central step \(B\twoheadrightarrow B'\) in the filtration
+by powers of \(\mathfrak m_B\), with kernel \(I\) satisfying
+\(\mathfrak m_BI=0\), and write \(V=\mathfrak g\otimes I\).
+The map \(Z^0(L_B\otimes\Omega_q)\to
+Z^0(L_{B'}\otimes\Omega_q)\) is surjective.  Indeed, lift a closed
+element to \(w\).  Its differential \(Dw\) is a closed degree-one
+element of \(V\otimes\Omega_q\); the polynomial Poincar\'e lemma and
+nonpositivity give
+\(H^1(V\otimes\Omega_q)=H^1(V)=0\).  Subtracting a degree-zero
+primitive of \(Dw\) gives a closed lift.  Thus the induced map of
+simplicial groups is degreewise surjective, with additive kernel
+\[
+                         K_q=Z^0(V\otimes\Omega_q).
+\]
+Its classifying-space homotopy fiber is \(BK_\bullet\).
+On the Maurer--Cartan side the surjection is a Kan fibration with fiber
+\(\MC_\bullet(V)\), by \cite[Proposition~4.7]{Getzler2009}.
+We use homotopy fibers of the classifying spaces here; no Kan-fibration
+assertion about an arbitrary diagonal map is needed.
+
+The restriction of \(\Phi_B\) to these fibers can be checked explicitly.
+For the abelian complex \(V\), put
+\[
+ C_q=(V\otimes\Omega_q)^0,
+ \qquad M_q=Z^1(V\otimes\Omega_q)=\MC_q(V).
+\]
+There is a degreewise exact sequence of simplicial vector spaces
+\[
+                    0\longrightarrow K_\bullet
+                    \longrightarrow C_\bullet
+                    \xrightarrow{D}M_\bullet\longrightarrow0.
+\]
+Let \((EK)_q=K_q^{q+1}\) be the homogeneous bar construction, with
+diagonal translation action of \(K_q\).  Its quotient is the diagonal
+inhomogeneous nerve \(BK\).  The map
+\[
+ \Psi_q:(EK)_q\longrightarrow C_q,
+ \qquad
+       (k_0,\ldots,k_q)\longmapsto\sum_{j=0}^q t_j k_j
+\]
+is simplicial and restricts to the identity on the diagonal copy of
+\(K_q\).  Since \(Dk_j=0\), its quotient map is
+\(D\Psi_q=\sum_jdt_j k_j\).  In the inhomogeneous coordinates
+\(x_i=k_i-k_{i-1}\), this is exactly the abelian Getzler simplex
+after the diagonal restriction in
+\eqref{eq:explicit-dg-nerve-comparison}.  Thus we have a commuting
+diagram of exact simplicial vector spaces
+\[
+\begin{array}{ccccccccc}
+0&\longrightarrow&K&\longrightarrow&EK&\longrightarrow&BK&\longrightarrow&0\\
+ &&\Vert&&\downarrow\Psi&&\downarrow\Phi_V\\
+0&\longrightarrow&K&\longrightarrow&C&\xrightarrow{D}&M&\longrightarrow&0.
+\end{array}
+\]
+Both middle terms are contractible.  This can be seen without choosing
+a contraction of \(V\): for a simplicial interval parameter
+\(\sigma:[q]\to[1]\), multiplication by
+\(\sum_{\sigma(j)=1}t_j\) gives a simplicial homotopy from zero to
+the identity on \(C_q\).  On \((EK)_q\), use
+\((k_j)_j\mapsto(\mathbf1_{\sigma(j)=1}k_j)_j\).
+The connecting maps of the two exact sequences therefore identify
+\(\Phi_V\) with the identity suspension of \(K\).  In particular it
+is a weak equivalence, and under the compatible integration and
+suspension identifications it is the canonical map from the bar model
+of \(V\) to the model of \(V[1]\).  This statement uses those
+connecting-map conventions, not an unsigned equality of raw simplex
+integrals.  Induction over the finite central filtration, starting at
+\(B=\mathbb C\), now proves that \(\Phi_B\) is an equivalence.
+The construction is natural in \(B\); the usual nilpotent
+Maurer--Cartan invariance then also gives its compatibility with
+quasi-isomorphisms.  This is a proof of the dg extension, rather than
+an assertion that the ordinary coefficient-ring statement of
+\cite[Proposition~4.9]{Getzler2009} already states that extension.
+
+\smallskip\noindent\textbf{4. Compatibility of the closed cocycle.}\quad
+We finally check the closed cocycle under this very comparison.  Do this
+before simplex integration, in the Thom--Whitney mixed complexes, and
+use a completed cofibrant coefficient resolution when computing derived
+forms.  A tuple in \(H_q(B)\) gives the substitution of the universal
+nerve coordinates by closed elements of
+\(\mathfrak m_B\otimes\Omega_q\).  Pull back
+\(\beta_n=\tfrac12\kappa(\delta a_n,\delta a_n)\) by that
+substitution, and apply the coefficient morphism
+\[
+ \operatorname{DR}(B\otimes\Omega_q)
+             \longrightarrow\operatorname{DR}(B)\otimes\Omega_q.
+\]
+As throughout this proof, this notation means derived de Rham objects
+computed on the chosen resolutions.  The morphism puts the polynomial
+simplex forms in exterior weight zero.  In particular coordinate
+differentiation still differentiates every \(B\)-dependent coefficient
+of the logarithms of the tuple; those coefficients are not held
+constant.  Naturality of coordinate differentiation gives precisely
+\[
+                \frac12\kappa(\delta_B A_{n,q},
+                                      \delta_B A_{n,q}).
+\]
+Both simplex directions have exterior weight zero.  Diagonal restriction
+commutes with their internal differential and with \(\delta_B\), with
+the total signs already specified.  Consequently
+\[
+ \Delta_q^*\!\left(\frac12\kappa(\delta_B A_{q,q},
+                                         \delta_B A_{q,q})\right)
+       =\frac12\kappa\bigl(\delta_B(\Phi_B)_q,
+                          \delta_B(\Phi_B)_q\bigr)
+\]
+as full mixed cocycles.  The higher closed-form components of these
+representatives are all zero; no exterior weights have been dropped.
+Faces, degeneracies, coefficient maps, and their homotopies commute with
+this equality.  The standard Thom--Whitney-to-normalized comparison
+therefore gives equality of the full closed-form classes.  We do not
+identify integration over a product of simplices with integration over
+its diagonal.  This establishes the derived pullback of the class
+already constructed on the smooth nerve, together with its natural
+comparison homotopy.  Passing continuously through the nilpotent
+coefficient quotients gives the same assertion for a completed semifree
+\(R\), where continuous coordinate forms compute the derived forms.
+
+\smallskip\noindent\textbf{5. Source evaluation and relative integration.}\quad
+We next carry out evaluation on the source.  Choose a finite
+triangulation \(K\) of \(X\).  A compatible family of flat simplices
+\[
+ a_\sigma\in
+ \MC\bigl(\mathfrak g\otimes\mathfrak m_R
+                    \widehat\otimes\Omega_{\mathrm{poly}}^*(\Delta^{\dim\sigma})
+       \bigr),\qquad \sigma\in K,
+\]
+is a simplicial map from \(K\) to the Maurer--Cartan model of
+\(B\widehat G\).  Including simplicial test parameters gives its
+families and all their homotopies.  Evaluation is the evaluation
+of these simplicial maps.  Pulling back the target cocycle therefore
+assigns to each \(\sigma\) the expression
+\(\tfrac12\kappa(\delta_Ra_\sigma,\delta_Ra_\sigma)\).
+
+Before totalizing this diagram, apply the coefficient map
+\[
+ \kappa_{\Spf R,X_B}:
+ \operatorname{DR}(\Spf R\times X_B)
+       \longrightarrow
+ \operatorname{DR}(\Spf R)\widehat\otimes
+                         R\Gamma(X_B,\mathcal O_{X_B})
+\]
+used in \cite[\S2.1, before Definition~2.3]{PTVV2013}.
+It keeps source functions and cochains in exterior weight zero.
+Thus the polynomial simplex forms above belong to the source
+cochain factor, while \(\delta_R\) supplies every exterior weight
+in the coefficient form.  Totalizing afterwards produces
+exactly \(\Theta_X\) in
+\eqref{eq:formal-trace-representative}.  We have proved the equality
+of full closed-form classes
+\[
+     \kappa_{\Spf R,X_B}
+                  \operatorname{ev}_{a_X}^*[\omega_\kappa]
+                          =[\Theta_X].
+\]
+All these operations are performed weight by weight and are
+continuous in the coefficient ideal.  They then define the
+displayed completed closed-form complex.
+
+For a smooth triangulated manifold, the same statement uses smooth
+forms through the multiplicative comparison
+\[
+ \Omega_{\mathrm{sm}}^*(X)
+       \longrightarrow\Omega_{\mathrm{ps}}^*(K)
+       \longleftarrow\Omega_{\mathrm{poly}}^*(K),
+\]
+where the middle term consists of compatible piecewise smooth forms.
+Both arrows are quasi-isomorphisms and commute with restriction.
+Integration of forms over the oriented simplices is the same on
+all three models, including their boundaries.  The construction
+therefore commutes with the orientation cochain maps, as well as
+with any multiplicative replacement carrying those maps.
+
+Finally let \(\partial Z=S\).  Write \(I_Z\) and \(I_S\) for
+integration over compatible fundamental chains.  With
+\(\partial[Z]=[S]\) and the suspension convention for transgression,
+the equations in~\eqref{eq:formal-trace-representative} give
+\[
+ h_Z=I_Z\Theta_Z,\qquad
+ \mathcal L_Qh_Z=I_S\Theta_S,\qquad \delta_Rh_Z=0.
+\]
+Thus \(h_Z\), of internal degree \(-1\), is the relative
+closed-form homotopy from zero to the pulled-back boundary form.
+This is the homotopy obtained by applying the orientation
+construction to evaluation
+\cite[Definition~2.6 and Claim~2.7]{Calaque2015}.
+The comparison homotopy on \(B\widehat G\) is carried through the
+same evaluation and relative-chain maps.  It consequently compares
+these relative homotopies together with their endpoints.
+
+On the sphere condition \(a_S=\nu b\), every term of
+\(\kappa(\delta_Ra_S,\delta_Ra_S)\) contains \(\nu\wedge\nu=0\).
+Its full closed-form representative and its chosen path are
+therefore zero.  Subtracting this zero path in the shifted
+Lagrangian intersection construction leaves \(h_Z\), now closed
+of degree \(-1\), as asserted.
+\end{proof}
diff --git a/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/figures/geometry-rim-collar.tex b/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/figures/geometry-rim-collar.tex
new file mode 100644
index 0000000000000000000000000000000000000000..3ae770e58aa6727e781a240ce8e411a3a590bbe8
--- /dev/null
+++ b/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/figures/geometry-rim-collar.tex
@@ -0,0 +1,57 @@
+\begin{figure}[tbp]
+\centering
+\begin{tikzpicture}[x=1cm,y=1cm,
+ every node/.style={font=\small},
+ collar/.style={line width=1pt},
+ guide/.style={gray!70,densely dashed},
+ annotation/.style={font=\footnotesize,align=center}]
+ \fill[gray!15] (0,0) circle (0.64);
+ \draw[gray!55] (0,0) circle (0.64);
+ \node[annotation,text width=1.2cm] at (0,0)
+  {bottom\\tube};
+
+ \draw[line width=1.15pt] (20:1.55)
+  arc[start angle=20,end angle=340,radius=1.55];
+ \draw[guide] (340:1.55)
+  arc[start angle=340,end angle=380,radius=1.55];
+ \node[annotation] at (-0.3,2.03) {retained rim arc};
+ \draw[gray!70] (-0.15,1.84)--(-0.15,1.55);
+ \node[annotation] at (-0.25,-1.95) {\(r=\rho\)};
+ \draw[gray!70] (-0.17,-1.76)--(-0.17,-1.54);
+
+ \draw[collar] (145:0.64)--(145:2.65);
+ \node[annotation,anchor=east] at (-2.25,1.64)
+  {\(a\)-cap collar\\\(\theta=\theta_A\)};
+ \fill (145:1.55) circle (2pt);
+ \draw[gray!70] (-2.22,0.36)--(-1.75,0.36)--(145:1.55);
+ \node[annotation,anchor=east,text width=2.8cm] at (-2.28,0.36)
+  {distinguished \(f_j\)--\(g_j\)\\collar intersection};
+
+ \draw[guide] (0.64,0)--(3.1,0);
+ \draw[gray!70] (3.1,0)--(3.1,-0.35);
+ \node[annotation,anchor=west] at (3.18,-0.35)
+  {upper attachment\\\(\theta=\theta_B\)};
+ \draw[collar,-{Stealth[length=2mm]}] (20:1.55)--(20:2.85);
+ \draw[collar,-{Stealth[length=2mm]}] (340:1.55)--(340:2.85);
+ \node[annotation,anchor=west] at (2.77,1.02)
+  {collar of \(C_j^+\)};
+ \node[annotation,anchor=west] at (2.77,-1.02)
+  {collar of \(C_j^-\)};
+ \draw[gray!70] (1.55,0.13)--(1.85,0.44)--(2.35,0.44);
+ \node[annotation,anchor=west] at (2.40,0.44) {deleted rim arc};
+
+ \node[annotation] at (0,-2.55)
+  {Normal-disk cross-section; the coordinates \(s\) and \(t\) are suppressed.};
+\end{tikzpicture}
+\caption{The collar of the compressed rim sphere near the crossing of
+\(a_j\) and \(b_j\) on the bottom surface. Restoring the coordinate
+\(s\) along \(b_j\) turns the solid rim arc into the retained annulus,
+the dashed rim arc near \(\theta_B\) into the removed compression band,
+and the two outgoing segments into collars of the compression disks.
+The marked point locates the distinguished collar intersection of
+\(f_j\) with \(g_j\); the suppressed transverse surface coordinate is
+\(t\). Only local collar directions are shown. The interiors of
+\(C_j^\pm\), the other caps, and their possible mutual intersections
+are not depicted.}
+\label{fig:geometry-rim-collar}
+\end{figure}
diff --git a/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/figures/geometry-sheets.tex b/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/figures/geometry-sheets.tex
new file mode 100644
index 0000000000000000000000000000000000000000..7530756173f293b272adcaf15b52fd7a3faef4ec
--- /dev/null
+++ b/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/figures/geometry-sheets.tex
@@ -0,0 +1,52 @@
+\begin{figure}[tbp]
+\centering
+\begin{minipage}[c]{0.40\textwidth}
+\centering
+\begin{tikzpicture}[x=1cm,y=1cm,
+ body/.style={draw,rounded corners,minimum width=9mm,minimum height=6mm},
+ gropecap/.style={draw,circle,inner sep=2pt},
+ every node/.style={font=\small}]
+ \node[body] (s) at (0,0) {$S_j$};
+ \node[gropecap] (a) at (-1.2,1.05) {$D_{x,j}$};
+ \node[body] (u) at (1.0,1.05) {$U_j$};
+ \node[gropecap] (y) at (0.2,2.2) {$D_{y,j}$};
+ \node[gropecap] (z) at (1.8,2.2) {$D_{z,j}$};
+ \draw (s)--node[left] {$a$}(a);
+ \draw (s)--node[right] {$b$}(u);
+ \draw[very thick] (u)--node[left] {$y$}(y);
+ \draw[dashed] (u)--node[right] {$z$}(z);
+ \node[align=center,text width=4.4cm] at (0.3,-0.85)
+  {Compress \(U_j\) using \(D_{y,j}\);\\
+   \(D_{z,j}\) remains available in \(E\).};
+\end{tikzpicture}
+\end{minipage}
+\hfill
+\begin{minipage}[c]{0.57\textwidth}
+\centering
+\begin{tikzpicture}[x=1cm,y=1cm,
+ sheet/.style={draw,rounded corners,minimum width=1.45cm,minimum height=8mm},
+ every node/.style={font=\small}]
+ \node[sheet] (pp) at (0,1.1) {$D_y^{+,+}\ (+)$};
+ \node[sheet] (pm) at (3.1,1.1) {$D_y^{+,-}\ (-)$};
+ \node[sheet] (mp) at (0,-0.25) {$D_y^{-,+}\ (-)$};
+ \node[sheet] (mm) at (3.1,-0.25) {$D_y^{-,-}\ (+)$};
+ \draw[<->] (pp)--node[above] {$z_j=1$}(pm);
+ \draw[<->] (mp)--node[above] {$z_j=1$}(mm);
+ \node[anchor=east] at (-0.85,1.1) {$C_j^+$};
+ \node[anchor=east] at (-0.85,-0.25) {$C_j^-$};
+ \node[align=center,text width=5.6cm] at (1.2,-1.3)
+  {Cancel within each row: hold \(\varepsilon\) fixed,\\
+   vary the inner sheet index \(\eta\).};
+\end{tikzpicture}
+\end{minipage}
+\caption{Combinatorial schematics, not projections of the four-dimensional
+embedding. Left: the two-stage body and its one-sided upper contraction.
+Right: the four occurrences of its surgery cap in the compressed rim
+sphere; parentheses record their orientation coefficients
+\(\varepsilon\eta\). The right panel suppresses the cap's subscript \(j\).
+The labels \(z_j=1\) are equalities in \(\pi_1(E)\), proved using the upper
+dual cap in \(E\).
+No comparison between the two rows, which could involve a bottom
+normal-circle meridian, is used.}
+\label{fig:geometry-sheets}
+\end{figure}
diff --git a/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/paper.tex b/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/paper.tex
new file mode 100644
index 0000000000000000000000000000000000000000..5772dbbe678b502f235255e6f61f776ef6a073ba
--- /dev/null
+++ b/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/paper.tex
@@ -0,0 +1,117 @@
+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\input{glyphtounicode}
+\pdfgentounicode=1
+\pdfglyphtounicode{parenleftbig}{0028} %
+\pdfglyphtounicode{parenrightbig}{0029} %
+\pdfglyphtounicode{parenleftBig}{0028} %
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+\pdfglyphtounicode{parenleftbigg}{0028} %
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+\hypersetup{colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black,
+  pdftitle={A marked tensor obstruction to four-dimensional disk embedding},
+  pdfauthor={OpenAI},pdfsubject={Four-dimensional topology and a finite tensor obstruction}}
+\usepackage{bookmark}
+\numberwithin{equation}{section}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{corollary}[theorem]{Corollary}
+\newtheorem{assumption}[theorem]{Assumption}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\newtheorem{construction}[theorem]{Construction}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newtheorem{convention}[theorem]{Convention}
+\DeclareMathOperator{\tr}{tr}
+\DeclareMathOperator{\Ad}{Ad}
+\DeclareMathOperator{\ad}{ad}
+\DeclareMathOperator{\im}{im}
+\DeclareMathOperator{\coker}{coker}
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+\DeclareMathOperator{\Spf}{Spf}
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+\newcommand{\kk}{\Bbbk}
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+\newcommand{\ideal}[1]{\left(#1\right)}
+\setlist{itemsep=2pt,topsep=5pt}
+\setlength{\emergencystretch}{1em}
+\title{A marked tensor obstruction to\\four-dimensional disk embedding}
+\author{OpenAI}
+\date{September 24, 2026}
+\begin{document}
+\maketitle
+\input{sections/00-introduction}
+\clearpage
+\tableofcontents
+\clearpage
+\input{sections/01-geometry}
+\input{sections/02-tensor}
+\input{sections/03-edge-germs}
+\input{sections/04-algebraicity}
+\input{sections/05-vertices}
+\input{sections/06-conclusion}
+\appendix
+\input{appendices/formal-trace-comparison}
+\phantomsection
+\addcontentsline{toc}{section}{References}
+\bibliographystyle{plain}
+\bibliography{references}
+\end{document}
diff --git a/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/references.bib b/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/references.bib
new file mode 100644
index 0000000000000000000000000000000000000000..296a8996b2e3a707aec9fe01aa783809d2a5c547
--- /dev/null
+++ b/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/references.bib
@@ -0,0 +1,190 @@
+@article{PRT2025,
+  author  = {Powell, Mark and Ray, Arunima and Teichner, Peter},
+  title   = {The 4-dimensional disc embedding theorem and dual spheres},
+  journal = {Selecta Mathematica. New Series},
+  volume  = {31},
+  number  = {4},
+  year    = {2025},
+  pages   = {80},
+  note    = {25 pp. \href{https://doi.org/10.1007/s00029-025-01069-y}{DOI}},
+  doi     = {10.1007/s00029-025-01069-y},
+  url     = {https://www.maths.gla.ac.uk/~mpowell/s00029-025-01069-y.pdf}
+}
+
+@article{KQ2000,
+  author  = {Krushkal, Vyacheslav S. and Quinn, Frank},
+  title   = {Subexponential groups in 4-manifold topology},
+  journal = {Geometry \& Topology},
+  volume  = {4},
+  number  = {1},
+  year    = {2000},
+  pages   = {407--430},
+  doi     = {10.2140/gt.2000.4.407},
+  url     = {https://arxiv.org/abs/math/0001063},
+  note    = {\href{https://doi.org/10.2140/gt.2000.4.407}{DOI}},
+}
+
+@article{FK2016,
+  author  = {Freedman, Michael and Krushkal, Vyacheslav},
+  title   = {{Engel} relations in 4-manifold topology},
+  journal = {Forum of Mathematics, Sigma},
+  volume  = {4},
+  year    = {2016},
+  pages   = {e22},
+  note    = {57 pp. \href{https://doi.org/10.1017/fms.2016.20}{DOI}},
+  doi     = {10.1017/fms.2016.20},
+  url     = {https://www.cambridge.org/core/journals/forum-of-mathematics-sigma/article/engel-relations-in-4manifold-topology/5B02E521E997E9E89E7FD19906DA684B}
+}
+
+@article{FK2020,
+  author  = {Freedman, Michael and Krushkal, Vyacheslav},
+  title   = {{Engel} groups and universal surgery models},
+  journal = {Journal of Topology},
+  volume  = {13},
+  number  = {3},
+  year    = {2020},
+  pages   = {1302--1316},
+  doi     = {10.1112/topo.12155},
+  url     = {https://uva.theopenscholar.com/files/slava-krushkal/files/universal.pdf},
+  note    = {\href{https://doi.org/10.1112/topo.12155}{DOI}},
+}
+
+@book{FQ1990,
+  author    = {Freedman, Michael H. and Quinn, Frank},
+  title     = {Topology of 4-Manifolds},
+  series    = {Princeton Mathematical Series},
+  volume    = {39},
+  publisher = {Princeton University Press},
+  address   = {Princeton, NJ},
+  year      = {1990},
+  isbn      = {0-691-08577-3},
+  doi       = {10.1515/9781400861064},
+  url       = {https://archive.mpim-bonn.mpg.de/id/eprint/4789/1/FreedmanQuinn-TopologyOf4Manifolds-PMS39.pdf},
+  note    = {\href{https://doi.org/10.1515/9781400861064}{DOI}},
+}
+
+@article{PTVV2013,
+  author  = {Pantev, Tony and To{\"e}n, Bertrand and Vaqui{\'e}, Michel and Vezzosi, Gabriele},
+  title   = {Shifted symplectic structures},
+  journal = {Publications Math{\'e}matiques de l'IH{\'E}S},
+  volume  = {117},
+  year    = {2013},
+  pages   = {271--328},
+  doi     = {10.1007/s10240-013-0054-1},
+  url     = {https://www.dma.unifi.it/~vezzosi/papers/PTVV.pdf},
+  note    = {\href{https://doi.org/10.1007/s10240-013-0054-1}{DOI}},
+}
+
+@incollection{Calaque2015,
+  author    = {Calaque, Damien},
+  title     = {Lagrangian structures on mapping stacks and semi-classical {TFTs}},
+  editor    = {Pantev, Tony and Simpson, Carlos and To{\"e}n, Bertrand and Vaqui{\'e}, Michel and Vezzosi, Gabriele},
+  booktitle = {Stacks and Categories in Geometry, Topology, and Algebra},
+  series    = {Contemporary Mathematics},
+  volume    = {643},
+  publisher = {American Mathematical Society},
+  address   = {Providence, RI},
+  year      = {2015},
+  pages     = {1--23},
+  doi       = {10.1090/conm/643/12894},
+  url       = {https://arxiv.org/abs/1306.3235v2},
+  note    = {Theorem and Definition locators refer to \href{https://arxiv.org/abs/1306.3235v2}{arXiv:1306.3235v2}},
+}
+
+@article{BBJ2019,
+  author  = {Brav, Christopher and Bussi, Vittoria and Joyce, Dominic},
+  title   = {A {Darboux} theorem for derived schemes with shifted symplectic structure},
+  journal = {Journal of the American Mathematical Society},
+  volume  = {32},
+  number  = {2},
+  year    = {2019},
+  pages   = {399--443},
+  doi     = {10.1090/jams/910},
+  url     = {https://www.ams.org/jams/2019-32-02/S0894-0347-2018-00910-5/S0894-0347-2018-00910-5.pdf},
+  note    = {\href{https://doi.org/10.1090/jams/910}{DOI}. Theorem and Example locators refer to \href{https://arxiv.org/abs/1305.6302v4}{arXiv:1305.6302v4}},
+}
+
+@article{Getzler2009,
+  author  = {Getzler, Ezra},
+  title   = {Lie theory for nilpotent {$L_\infty$}-algebras},
+  journal = {Annals of Mathematics. Second Series},
+  volume  = {170},
+  number  = {1},
+  year    = {2009},
+  pages   = {271--301},
+  doi     = {10.4007/annals.2009.170.271},
+  url     = {https://annals.math.princeton.edu/2009/170-1/p07},
+  note    = {\href{https://doi.org/10.4007/annals.2009.170.271}{DOI}},
+}
+
+@article{Artin1969,
+  author  = {Artin, Michael},
+  title   = {Algebraic approximation of structures over complete local rings},
+  journal = {Publications Math{\'e}matiques de l'IH{\'E}S},
+  volume  = {36},
+  year    = {1969},
+  pages   = {23--58},
+  doi     = {10.1007/BF02684596},
+  url     = {https://www.numdam.org/item/PMIHES_1969__36__23_0/},
+  note    = {\href{https://doi.org/10.1007/BF02684596}{DOI}},
+}
+
+@article{Freedman1982,
+  author  = {Freedman, Michael H.},
+  title   = {The topology of four-dimensional manifolds},
+  journal = {Journal of Differential Geometry},
+  volume  = {17},
+  number  = {3},
+  year    = {1982},
+  pages   = {357--453},
+  doi     = {10.4310/jdg/1214437136},
+  url     = {https://doi.org/10.4310/jdg/1214437136}
+}
+
+@article{FT1995,
+  author  = {Freedman, Michael H. and Teichner, Peter},
+  title   = {{4}-manifold topology. {I}. {Subexponential} groups},
+  journal = {Inventiones Mathematicae},
+  volume  = {122},
+  year    = {1995},
+  pages   = {509--529},
+  doi     = {10.1007/BF01231454},
+  url     = {https://doi.org/10.1007/BF01231454}
+}
+
+@article{ChernSimons1974,
+  author  = {Chern, Shiing-Shen and Simons, James},
+  title   = {Characteristic forms and geometric invariants},
+  journal = {Annals of Mathematics. Second Series},
+  volume  = {99},
+  number  = {1},
+  year    = {1974},
+  pages   = {48--69},
+  doi     = {10.2307/1971013},
+  url     = {https://annals.math.princeton.edu/1974/99-1/p03}
+}
+
+@article{GM1988,
+  author  = {Goldman, William M. and Millson, John J.},
+  title   = {The deformation theory of representations of fundamental groups of compact {K\"ahler} manifolds},
+  journal = {Publications Math\'ematiques de l'IH\'ES},
+  volume  = {67},
+  year    = {1988},
+  pages   = {43--96},
+  doi     = {10.1007/BF02699127},
+  url     = {https://math.umd.edu/~wmg/PMIHES_1988__67__43_0.pdf},
+  note    = {\href{https://doi.org/10.1007/BF02699127}{DOI}}
+}
+
+@article{Dupont1976,
+  author  = {Dupont, Johan L.},
+  title   = {Simplicial de {Rham} cohomology and characteristic classes of flat bundles},
+  journal = {Topology},
+  volume  = {15},
+  number  = {3},
+  year    = {1976},
+  pages   = {233--245},
+  doi     = {10.1016/0040-9383(76)90038-0},
+  url     = {https://doi.org/10.1016/0040-9383(76)90038-0},
+  note    = {\href{https://doi.org/10.1016/0040-9383(76)90038-0}{DOI}}
+}
diff --git a/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/sections/00-introduction.tex b/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/sections/00-introduction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..487d7350cd1b086eda3802fcee6789ad92afd8ce
--- /dev/null
+++ b/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/sections/00-introduction.tex
@@ -0,0 +1,275 @@
+\begin{abstract}
+We disprove the unrestricted four-dimensional disk-embedding conjecture.
+We construct immersed disks in a compact oriented smooth four-manifold with
+framed algebraic dual spheres satisfying the usual equivariant intersection
+and reduced self-intersection conditions, but with no pairwise disjoint locally
+flat replacements that preserve the boundary maps and induced normal framings.
+The obstruction holds even when the replacement disks' relative homotopy classes
+are not prescribed.
+\end{abstract}
+
+\section{Introduction}\label{sec:introduction}
+
+The disk-embedding problem asks when immersed disks in a four-manifold can
+be replaced by disjoint locally flat embedded disks with the same boundary
+data. Its role in four-dimensional topological surgery is to turn
+algebraic cancellation data into embedded disks.
+In dimension four, the auxiliary disks used to carry out cancellations
+can themselves intersect the original surfaces and one another.
+Dual spheres provide additional control. An algebraic dual records one
+intersection with its designated disk and zero with the others, counted
+with signs and fundamental-group labels; a geometric dual realizes these
+counts by meeting its designated disk once transversely and missing the
+other disks.
+
+Casson's construction replaces the finite task of removing intersections
+by an infinite tower of immersed disks. Freedman's recognition theorem
+identifies the resulting Casson handle, relative to its attaching region,
+with the standard open topological two-handle. Together these results
+established the simply connected foundation of disk embedding
+\cite[Introduction and Theorem~1.1]{Freedman1982}.
+Freedman and Quinn developed the capped-grope version of this method and
+organized the disk theorem and its surgery and cobordism applications
+\cite[Theorem~5.1A and Chapter~11]{FQ1990}.
+The group restriction enters through the double-point loops of the caps:
+their null-homotopies supply further stages of the construction.
+The class of \emph{good groups} is defined by the availability of the
+required \(\pi_1\)-null disk operations
+\cite[Definition~3.2]{PRT2025}.
+Freedman and Teichner proved that groups of subexponential growth are good
+\cite[Theorem~0.1]{FT1995}. Krushkal and Quinn supplied another proof
+\cite[Theorem, p.~408]{KQ2000}. Their paper also contains
+Freedman and Teichner's correction of the earlier height-raising
+argument \cite[Appendix, pp.~424--430]{KQ2000}.
+
+The modern formulation of Powell, Ray, and Teichner supplies framed
+geometric dual spheres as well as embedded disks, and preserves the
+homotopy classes of the input dual spheres
+\cite[Theorem~A]{PRT2025}. Their modification supplies the
+geometric-dual step missing from the earlier argument; the disk
+conclusion without these output spheres was already proved
+\cite[Remark~1.3]{PRT2025}.
+For generically immersed input disks, their theorem preserves the induced
+boundary normal framings without prescribing relative homotopy classes
+for the replacement disks.
+The \emph{unrestricted disk-embedding conjecture} considered here is the
+extension of that theorem obtained by removing only the good-group
+hypothesis. It would in particular give framed disk replacements for
+compact oriented smooth input manifolds, even after the output-dual
+requirement is discarded. We resolve this geometric conjecture negatively
+by obstructing that disk conclusion.
+
+We use the following intersection conventions in stating the result.
+After choosing basepoints and whiskers, \(\lambda\) denotes the
+equivariant intersection pairing with values in \(\mathbb Z[\pi_1M]\).
+For an oriented four-manifold \(M\), the reduced self-intersection invariant
+\(\widetilde\mu\) takes values in the additive quotient
+\[
+ \mathbb Z[\pi_1M]\big/
+ \big\langle h-h^{-1}\ (h\in\pi_1M),\ \mathbb Z\cdot1\big\rangle .
+\]
+A framed immersed sphere has a trivialized normal bundle. A boundary
+normal framing is the framing induced by the normal bundle of the
+specified immersed disk.
+
+\begin{theorem}\label{thm:main}
+There exist a compact connected oriented smooth four-manifold \(M\) with boundary,
+an integer \(k\geq1\), generically immersed proper disks
+\[
+ f_1,\ldots,f_k:(D^2,S^1)\longrightarrow(M,\partial M)
+\]
+with disjoint embedded boundary circles, and generically immersed framed spheres
+\(g_1,\ldots,g_k:S^2\longrightarrow M\), such that
+\[
+ \lambda(f_i,g_j)=\delta_{ij},\qquad
+ \lambda(g_i,g_j)=0\quad\text{for all }i,j,\qquad
+ \widetilde\mu(g_i)=0\quad\text{for all }i,
+\]
+but there are no pairwise disjoint locally flat embedded disks in \(M\)
+with the same boundary maps and the boundary normal framings induced by the
+\(f_i\).
+\end{theorem}
+
+The diagonal intersection equations are part of the statement.
+The conclusion of Theorem~\ref{thm:main} requires no homotopy constraint on a
+hypothetical replacement disk.
+The markings used in the proof specify based curves and their paths in
+the fixed construction; they impose no further condition on hypothetical
+replacement disks. The result concerns geometric disk replacement.
+The relation to independently formulated surgery assertions is discussed
+after the proof in Section~\ref{sec:conclusion}.
+
+\subsection{Idea of the proof}
+
+We first construct a fixed immersed disk problem in the exterior of a
+finite collection of capped gropes. A capped grope here is a surface tree
+with disks attached at its terminal curves; its boundary records an
+iterated commutator in the terminal curves. The construction installs
+finitely many such commutator relations, with independently chosen based
+conjugations, in a smooth plumbing model. It also supplies the algebraic
+dual spheres required in Theorem~\ref{thm:main}. These data are fixed
+before any hypothetical embedded replacements are chosen.
+
+Suppose this particular disk problem has replacements with the required
+boundary maps and framings. They allow the plumbing model to be cut into
+two vertex pieces \(A,B\), joined by compact three-manifolds
+\(Y_1,\ldots,Y_{2m}\) with torus boundary, for an integer \(m\geq5\)
+chosen in the construction. The cuts are smooth and miss
+the locally flat disks. The installed boundary words are now relations
+in the vertex groups, and restriction in cohomology gives complementary
+linear data on the two sides. Proposition~\ref{prop:geometric-reduction}
+records this marked geometry; Lemma~\ref{lem:exterior-duals} verifies
+the equivariant disk-and-sphere input.
+
+For each edge \(Y_i\), we study based flat \(SL_2\)-connections with one
+torus holonomy fixed. Basing records holonomy as a matrix, with no quotient
+by simultaneous conjugation. A \emph{potential} is a function in finitely many
+formal coordinates whose gradient equations describe these flat
+connections. The \emph{central potential} fixes this torus
+holonomy to the identity. Proposition~\ref{prop:edge-potential} constructs it from
+the Chern--Simons functional, including a first-order variation of the
+fixed boundary holonomy. Theorem~\ref{thm:edge-algebraicity} then replaces
+the central potential by an algebraic function through a formal
+coordinate change. The comparison preserves based holonomy on the full
+critical scheme: its coordinate ring is the power-series ring modulo
+the actual gradient ideal, with its nilpotent elements retained.
+The geometric reduction selects \(m\) edges for the first-order
+boundary variations; we call these edges active.
+
+The two vertex pieces determine formal graphs in the product of the edge
+coordinate spaces. Their central tangent spaces are complementary, and
+suitable sums of the edge potentials vanish identically on the graphs.
+The boundary relations imply more than vanishing at common critical
+points: products associated to every three distinct active edges belong
+to the restricted gradient ideals. On one side these products are
+\(t_it_jt_k\), where each \(t_i^2=0\) and mixed products remain in the
+parameter ring. On the other they are products of three algebraic
+holonomy functions. Proposition~\ref{prop:vertex-data} establishes the
+exact vanishing and ideal memberships. The graph parametrizations may
+mix all edge coordinates, while each fixed potential depends only on
+its own edge block.
+
+Theorem~\ref{thm:tensor-obstruction} rules out precisely these algebraic
+data when at least five edges are active. Artin approximation and finite
+specialization first reduce them to large positive characteristic.
+Truncating each internal coordinate by its characteristic power makes
+the associated Koszul complexes finite. The two transverse graphs
+determine classes with nonzero pairing. The target triple relations,
+together with ordinary exterior differentiation, decompose one class
+into tensors admitting three independent first-order parameter lifts.
+The source triple relations force each resulting pairing to vanish,
+giving the contradiction.
+
+Three features are useful beyond the final contradiction. First, the geometry
+imposes a finite family of independently conjugated laws before the cuts are
+chosen. Second, the algebraicity argument compares full relative closed shifted
+forms and preserves the marked holonomies; an algebraic model of the reduced
+representation locus would be insufficient. Third, the tensor argument uses
+actual gradient ideals over nonreduced parameter rings and makes no
+isolated-critical-point or regular-sequence assumption.
+
+\subsection{The geometric and algebraic antecedents}
+
+Capped-grope contraction and transverse grope constructions provide the
+local geometric operations \cite[\S\S1.2 and 1.4]{KQ2000}.
+Their use to relate surface intersections and commutator words also
+appears in the work of Freedman and Krushkal. Using the repeated-variable
+2-Engel relation, they constructed homotopy solutions to the A--B slice
+problem and new universal surgery models
+\cite[Theorems~1 and~2]{FK2016}; their later work established universality
+for half-\(\pi_1\)-null surgery models, with a capped grope on one side
+and a \(\pi_1\)-null sphere on the other
+\cite[Theorem~1 and Section~3]{FK2020}.
+These results concern particular surgery or slicing formulations, with
+their own embedding and complement conditions. They explain the value
+of commutator calculus in this setting without supplying the marked
+geometric reduction used here.
+
+Our relations use three distinct group inputs and become trivial when
+any one is the identity. Their logarithmic expansion begins with the
+trilinear bracket \([Z_1,[Z_2,Z_3]]\), where \(Z_a\) is the logarithm
+of the \(a\)-th input. The deletion property is the Brunnian property defined in
+Section~\ref{sec:geometry}. We install a finite family of such words
+with independently chosen conjugations, before knowing the cut pieces.
+The later argument uses those independent choices to recover every
+triple coefficient. It neither imposes a universal Engel relation nor
+requires the vertex groups to be nilpotent. The simultaneous placement,
+framing, and algebraic-dual calculations needed for this finite family
+are given in Section~\ref{sec:geometry}.
+
+The deformation-theoretic part has a different origin. Goldman and
+Millson describe Artin-algebra families of flat connections by differential
+graded Lie algebras and identify their based holonomies with representation
+deformations \cite[Corollary~6.4 and Proposition~6.6]{GM1988}.
+Their quadraticity results use additional formality hypotheses in the
+compact K\"ahler setting. The edge manifolds here are arbitrary compact
+three-manifolds, so higher deformation terms remain.
+Chern and Simons constructed the classical transgression form whose
+exterior derivative is the corresponding curvature characteristic form
+\cite[Section~3, especially Proposition~3.2]{ChernSimons1974}.
+Section~\ref{sec:edge-germs} uses this form to construct the potential,
+computes its boundary adjustment, and proves equality of its full
+curvature and gradient ideals.
+
+The algebraic model comes from shifted symplectic geometry. The Betti
+transgression theorem of Pantev, To\"en, Vaqui\'e, and Vezzosi produces
+shifted symplectic forms on local-system stacks; their Lagrangian
+intersection theorem lowers the shift by one
+\cite[Theorems~2.5 and~2.9]{PTVV2013}.
+Calaque's relative orientation construction supplies the Lagrangian
+restriction map associated to an oriented manifold with boundary
+\cite[Theorem~2.9 and \S3.1]{Calaque2015}.
+After a punctured filling of the edge, these results put us in the
+setting of Brav, Bussi, and Joyce's algebraic Darboux theorem in
+shift \(-1\), which supplies an algebraic critical chart
+\cite[Example~5.15 and Theorem~5.18(i)]{BBJ2019}.
+
+To use that chart, we must identify its full relative closed form with
+the form underlying the particular Chern--Simons potential already
+constructed. Proposition~\ref{prop:formal-trace-comparison}, proved in
+Appendix~\ref{app:formal-trace-comparison}, carries out this comparison
+on derived coefficients and source cochain models. Its inputs include
+Getzler's simplicial Maurer--Cartan comparison for nilpotent Lie algebras
+\cite[Theorem~5.4 and Corollary~5.11]{Getzler2009} and the simplicial
+de Rham contraction of Dupont \cite{Dupont1976}.
+Section~\ref{sec:algebraicity} then turns the closed-form comparison
+into a coordinate change fixing the critical scheme and its based
+holonomies. Only after this step, and the polynomial replacement of the
+holonomy logarithms, do we apply ordinary Artin approximation
+\cite[Theorem~1.10]{Artin1969}. The approximation is applied to the
+final graph equations with their fixed, separately algebraic edge
+functions.
+
+\subsection{Conventions and organization}
+
+We assume familiarity with basic four-manifold topology, cohomology with
+local coefficients, and formal deformation theory. The algebraicity
+argument additionally uses derived mapping stacks and shifted symplectic
+forms; its required comparison is stated in the body and proved in full
+in Appendix~\ref{app:formal-trace-comparison}.
+
+All manifolds and maps in the initial model are smooth, with corners rounded
+when necessary. Hypothetical output disks are only locally flat. The subsequent
+cuts are smooth hypersurfaces in the original smooth manifold and miss those
+disks. Thus the vertex and edge calculations use ordinary smooth forms and
+Stokes' theorem.
+
+Unless a field is specified, cohomology has characteristic-zero coefficients.
+We use \(\g=\mathfrak{sl}_2(\mathbb C)\) and the pairing
+\(\kappa(X,Y)=\tr(XY)\). Connections and coordinate changes are formal power
+series, coefficient by coefficient. A statement about an ideal is a statement
+in the complete coordinate ring, including its nilpotents.
+In Sections~\ref{sec:edge-germs}--\ref{sec:vertices}, \(d\) denotes the
+differential on source forms and \(\delta\) differentiation in formal
+coordinates. In the tensor argument of Section~\ref{sec:tensor}, \(d\) is
+the coordinate de Rham differential and \(\delta\) its induced operator on
+Koszul homology. We specify total signs when using both differentials.
+
+Section~\ref{sec:geometry} proves the geometric reduction.
+Section~\ref{sec:tensor} states and proves the finite obstruction before
+we construct its inputs, making the target algebraic interface explicit.
+Sections~\ref{sec:edge-germs} and \ref{sec:algebraicity}
+construct the edge germs and their algebraic models.
+Section~\ref{sec:vertices} constructs the vertex graphs and checks every
+hypothesis of the tensor obstruction. Section~\ref{sec:conclusion} completes
+the proof of Theorem~\ref{thm:main}.
diff --git a/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/sections/01-geometry.tex b/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/sections/01-geometry.tex
new file mode 100644
index 0000000000000000000000000000000000000000..f2d21b06c7cf0db1fe46eff7fc692fd32141c1ae
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+++ b/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/sections/01-geometry.tex
@@ -0,0 +1,744 @@
+\section{The geometric reduction}
+\label{sec:geometry}
+
+We use the disk assertion only to compress a finite collection of surfaces.
+The following restricted version of its disk conclusion is therefore sufficient.
+All manifolds in the constructions before that application are smooth and
+oriented; corners are rounded when necessary.
+The output will be a fixed disk problem with algebraic duals, together
+with a description of the cut manifolds that its hypothetical embedded
+solutions would produce. The marked boundary words and their based paths
+are chosen as part of the initial disk problem.
+
+\begin{assumption}[Universal framed disk replacement]
+\label{ass:disk}
+Let \(M\) be a compact connected oriented smooth four-manifold with boundary.
+Suppose that a finite collection of proper generic immersions
+\[
+ f_i\colon(D^2,S^1)\longrightarrow(M,\partial M),\qquad 1\leq i\leq k,
+\]
+has pairwise disjoint embedded boundary circles, and admits generically immersed
+framed spheres \(g_1,\ldots,g_k\) such that
+\[
+ \lambda(f_i,g_j)=\delta_{ij},\qquad
+ \lambda(g_i,g_j)=0,\qquad
+ \widetilde\mu(g_i)=0
+ \quad\text{for all }i,j.
+\]
+Here intersections are counted in \(\mathbb Z[\pi_1M]\), with fixed whiskers,
+the diagonal equations are included, and
+\[
+ \widetilde\mu(g_i)\in
+ \mathbb Z[\pi_1M]\big/
+ \big\langle h-h^{-1},\,\mathbb Z\cdot1\big\rangle
+\]
+uses the additive quotient. Then the \(f_i\) can be replaced by pairwise disjoint
+locally flat embedded disks with the same boundary maps and the same induced
+normal framings on those boundary circles.
+\end{assumption}
+
+No relative homotopy class for the replacement disks is prescribed in
+Assumption~\ref{ass:disk}. Nor does it require output dual spheres. The
+unrestricted version of \cite[Theorem~A]{PRT2025} would imply this assumption,
+including its framing clause. We will contradict the assumption itself.
+The capped-surface contractions used below are the usual one-sided
+contractions; see \cite[\S1.2]{KQ2000} and
+\cite[Lemma~3.4 and its preceding discussion]{PRT2025}.
+The relation between grope trees and commutator words is also used in
+\cite{FK2016,FK2020}. We give the particular finite construction needed here.
+
+\subsection{A finite list of relations}
+
+Our commutator convention is \([a,b]=aba^{-1}b^{-1}\).
+A word \(R\) in three free generators \(X_1,X_2,X_3\) will be called a
+\emph{three-slot grope word} if it is the boundary word of the rooted surface
+tree consisting of a punctured torus with a single tip on its first branch
+and a second punctured torus on its other branch. Paths to its three tips
+are part of its marking. Changing those paths inside the body can conjugate
+a tip by words in the three based tips, and can conjugate the inner
+commutator by such a word. Overall conjugation of the boundary word is
+irrelevant to its being a relation. In particular,
+\begin{equation}
+\label{eq:grope-word-properties}
+ \left.R\right|_{X_a=1}=1\quad(a=1,2,3),
+ \qquad
+ R\equiv[X_1,[X_2,X_3]]
+       \pmod{\Gamma_4F(X_1,X_2,X_3)}.
+\end{equation}
+Here \(\Gamma_rF\) denotes the lower central series of the free group \(F\).
+The first property is the \emph{Brunnian property in the three slots}:
+setting any one input equal to the identity trivializes the word.
+The orientations of the two surfaces can be fixed to give the displayed
+sign. Indeed a tip conjugation changes its image modulo \(\Gamma_2\)
+by zero, and hence changes this weight-three commutator only modulo
+\(\Gamma_4\). The first assertion follows directly by deleting a branch
+of the rooted tree. Equivalently, in the completed free associative algebra
+over \(\mathbb Q\),
+\begin{equation}
+\label{eq:grope-word-leading}
+ \log R(e^{Z_1},e^{Z_2},e^{Z_3})
+   =[Z_1,[Z_2,Z_3]]+\text{terms of total degree at least four};
+\end{equation}
+each term contains a letter from every slot. These two properties,
+rather than a claim that an entire vertex group is nilpotent, will be used.
+
+Fix \(m\geq5\) and \(p\geq9\). Each plumbing block will have \(2m\)
+solid-torus boundary regions, called \emph{ports}, through which it
+joins other blocks. Removing these regions leaves \(2m\) boundary
+tori. Write \(x_1,\ldots,x_{2m}\) for their meridians and
+\(l_1,\ldots,l_p\) for the extra one-handle generators, and put
+\[
+ \mathcal C=\{1,l_1,\ldots,l_p\}.
+\]
+Before making any cut, install one marked grope body for every ordered
+triple of distinct port indices, every independent choice
+\(c_1,c_2,c_3\in\mathcal C\), and every independent choice
+\(\delta_1,\delta_2,\delta_3\in\{1,-1\}\). Its three based tips represent
+\begin{equation}
+\label{eq:three-slot-inputs}
+ X_a=c_a x_{r_a}^{\delta_a}c_a^{-1},
+ \qquad a=1,2,3.
+\end{equation}
+There are at most
+\[
+ 8(2m)(2m-1)(2m-2)(p+1)^3
+\]
+such bodies. Any additional fixed finite list of words in the extra letters
+could be included in \(\mathcal C\) by the same construction.
+In particular, neither the list of bodies nor their markings depends on
+the slopes selected by the cuts below.
+
+\begin{proposition}[Geometric reduction]
+\label{prop:geometric-reduction}
+For the integers \(m,p\) and the finite list just specified, there is a
+compact connected oriented smooth four-manifold \(X\), constructed without
+Assumption~\ref{ass:disk}, with the following properties.
+
+It has a regular cover with deck group \(F_m\), whose block graph is the
+Cayley tree of \(F_m\). Each block is a four-dimensional one-handlebody on
+the \(p\) extra letters. Its exposed boundary piece is
+\[
+ K_v\cong
+ \bigl(\#^p(S^1\times S^2)\bigr)
+ \setminus\operatorname{int}\nu(L_{2m}),
+\]
+where \(L_{2m}\) is a \(2m\)-component unlink in a ball. The port meridians
+and the extra letters freely generate \(\pi_1K_v\).
+At every join, the meridian on one side is the longitude on the other.
+There is a finite family of framed grope bodies with boundary in these
+pieces, equivariant in the cover, realizing the list
+\eqref{eq:three-slot-inputs}.
+
+If Assumption~\ref{ass:disk} holds, there are equivariant cuts, with compact
+quotient, having connected vertex pieces \(V_v\) and connected edge
+three-manifolds \(Y_e\). Each \(Y_e\) has as its entire boundary the prescribed
+join torus. Every corresponding three-slot grope word is trivial in
+\(\pi_1V_v\).
+
+In the index-two parity quotient, these cuts give two connected vertex
+pieces \(A,B\) and \(2m\) connected oriented edge pieces \(Y_1,\ldots,Y_{2m}\).
+Their ambient manifold \(X_2\) is homotopy equivalent to the graph with two
+vertices and \(2m\) edges, with \(p\) additional circles at each vertex.
+For any characteristic-zero field \(k\), difference of restrictions gives
+\begin{equation}
+\label{eq:geometry-mv-degree-two}
+ H^2(A;k)\oplus H^2(B;k)
+       \xrightarrow{\;\cong\;}
+       \bigoplus_{i=1}^{2m}H^2(Y_i;k).
+\end{equation}
+In degree one it is surjective, and restricts to an isomorphism on the
+kernel of evaluation on the \(p\) extra loops at each vertex.
+
+At each boundary torus choose the two original coordinate loops
+\(\alpha_i,\beta_i\) so that \(\alpha_i\) has nonzero image in
+\(H_1(Y_i;k)\) and \(\beta_i\) has zero image. Such a choice is possible:
+one of the original coordinate directions, not just an arbitrary slope,
+is killed. Call the endpoint where \(\beta_i\) is a longitude the
+\emph{plus endpoint}; there \(\alpha_i\) is the port meridian.
+There are exactly \(m\) edges with plus endpoint \(A\).
+After numbering these as \(1,\ldots,m\), the finite relation list includes,
+at \(A\), all the required three-slot relations on any three distinct
+\(\alpha_i\), and, at \(B\), those on the corresponding \(\beta_i\),
+with independent conjugations by \(\mathcal C\).
+\end{proposition}
+
+The proof constructs a particular exterior \(E\) and a finite disk
+family in it before making any cut. A simultaneous framed replacement
+of that family alone implies all the cut conclusions of the proposition;
+the universal Assumption~\ref{ass:disk} is used only to provide that
+replacement. Lemma~\ref{lem:exterior-duals} specifies the fixed family
+and verifies its algebraic duals. This local implication will give the
+fixed counterexample in Theorem~\ref{thm:main}.
+
+\subsection{The plumbing blocks}
+\label{subsec:plumbing-blocks}
+
+Start with a standard four-dimensional one-handlebody with \(2m+p\)
+one-handles. Attach a cancelling two-handle to each of the first \(2m\)
+one-handles, using the product framing and disjoint standard attaching
+regions. Before plumbing, the resulting block \(B\) is a one-handlebody
+on the remaining \(p\) handles. The belt circles of the cancelling
+two-handles are a standard unlink in a ball in \(\partial B\).
+One can see each component separately from the cancelling pair:
+the complement of its attaching solid torus in \(S^1\times S^2\) is a
+solid torus, which becomes the complement of the belt unknot after
+cancellation. The framed attaching pushoff is its belt meridian.
+The cancellation regions for different pairs are disjoint.
+
+Pair the \(2m\) two-handles. In each handle choose a small product patch
+\(D^2_{\mathrm{base}}\times D^2_{\mathrm{fibre}}\), and plumb the patches
+of its pair by interchanging the two factors. This is mutual plumbing;
+there are no self-plumbings. To specify the local manifold structure,
+the two branches at a plumbing can be represented by
+\[
+ \bigl(D^2_R\times D^2_r\bigr)
+ \ \cup\
+ \bigl(D^2_r\times D^2_R\bigr)
+ \subset\mathbb R^2\times\mathbb R^2,\qquad 0<r<R.
+\]
+Their overlap is the product ball \(D^2_r\times D^2_r\). Rounding its
+corners gives the usual smooth plumbing chart. Distinct patches are
+disjoint. The factor interchange preserves the orientation in dimension
+four, and the product framings fix the signs of the resulting transverse
+core intersections.
+
+Equivalently, take one copy \(B_v\) for each vertex of the Cayley tree of
+\(F_m=\langle a_1,\ldots,a_m\rangle\), and identify the designated patches
+of \(B_v\) and \(B_{va_j}\) by this chart. Neighbouring blocks intersect in
+one product ball, and there are no other or triple intersections.
+Left translation acts freely. The quotient is the finite plumbing \(X\),
+and the union is its regular cover \(\widetilde X\).
+Since all attaching intersections are contractible cofibrations, the
+homotopy type of the union is obtained by joining the block homotopy
+types along the tree. Thus
+\begin{equation}
+\label{eq:plumbing-homotopy}
+ \widetilde X\simeq
+ \text{the Cayley tree with \(p\) circles attached at each vertex}.
+\end{equation}
+This is not an assertion that \(\widetilde X\) is the universal cover:
+the extra circles remain.
+
+At a product patch the removed part of the unplumbed block boundary is a
+solid-torus neighbourhood of the belt circle. Its boundary is
+\(S^1_{\mathrm{base}}\times S^1_{\mathrm{fibre}}\).
+Consequently the exposed boundary is exactly the piece \(K_v\) in
+Proposition~\ref{prop:geometric-reduction}, with a torus collar at each join.
+The factor interchange exchanges the two peripheral slopes.
+An unlink longitude bounds a disk in \(K_v\), whereas its meridian is
+one of its free generators. Adding the \(p\) extra one-handles gives the
+remaining free generators \(l_s\).
+
+Fix boundary whiskers for these generators. In the cover, fix the boundary
+map to the tree which is constant on each \(K_v\) core and crosses the
+corresponding tree edge on every torus collar. The middle of each collar
+maps to the midpoint of that edge. The construction is equivariant.
+It descends both to the one-vertex graph and to the parity quotient.
+
+\subsection{Marked grope bodies and their framings}
+
+The plumbing fixes the port meridians and peripheral slopes. We next
+place the finite list of surface bodies with their prescribed based tip
+curves, and match their framings to the handle cores. These choices must
+be made before the exterior disk problem is defined.
+
+\begin{lemma}[The marked body model]
+\label{lem:marked-body}
+The two-stage grope body used in \eqref{eq:grope-word-properties} has a
+three-dimensional handlebody thickening which, with its three marked
+tip annuli, is a boundary join of three longitudinal-annulus collars.
+Finitely many copies can be placed disjointly in the boundary of the
+initial one-handlebody, with prescribed parallel tip tracks to the
+cancelling-handle attaching regions and with the external conjugations
+\eqref{eq:three-slot-inputs}. Their surface framings can be matched to the
+handle-core framings.
+\end{lemma}
+
+\begin{proof}
+We first identify the thickening together with its tip annuli, then
+place the finite marked family, and finally match the surface framings
+to the cap framings while keeping the terminal attaching circles fixed.
+
+\smallskip\noindent\textbf{1. Primitive tip annuli.}\quad
+Thicken a punctured torus \(S\) to \(S\times[-1,1]\).
+Put its two selected basis curves on opposite faces of this product.
+There are disjoint proper cutting arcs in \(S\), each dual to one basis
+curve and disjoint from the other. For an explicit model, puncture the
+square torus near \((1/2,1/2)\), take the basis circles at \(y=1/4\)
+and \(x=1/4\), and take the cutting arcs at \(x=1/2\) and \(y=1/2\),
+with their ends on the puncture. The only intersection of the latter
+two arcs was removed by the puncture. Their products with the interval
+are disjoint meridian disks, each meeting just its own marked tip once.
+
+For the lower stage, view the primitive \(b\)-annulus as the longitudinal
+annulus of a solid torus. That solid torus is an annular collar, up to
+rounding, and the rest of the lower handlebody is the spare \(a\)-handle.
+Glue the annular boundary of the upper punctured-torus product to this
+\(b\)-annulus. Gluing the collar only thickens that annulus on the upper
+handlebody. Extend the two upper meridian disks across the collar.
+Their traces occupy finitely many strips there. Place the feet and return
+arc of the spare \(a\)-handle in a complementary interval. Its meridian
+disk is then disjoint from both extended upper disks. We have obtained
+three disjoint disk duals for the three marked tips.
+
+This is a statement about the marked annuli, not just about a free basis.
+Cut the handlebody along the three dual disks. Each marked annulus becomes
+a rectangle on the boundary of the resulting ball, joining only the two
+faces of its own cutting disk. The rectangles are disjoint. A neighbourhood
+of each rectangle together with its two disk faces is a disk in the
+boundary sphere, disjoint from the other two such disks. Smooth their
+corners. A smooth isotopy of the boundary sphere carries these three
+labelled disk neighbourhoods to standard ones; within each neighbourhood,
+choose it also to standardize the two disk faces and their connecting
+rectangle. This last choice is an isotopy of a disk containing two
+subdisks joined by a band. Extend the boundary isotopy over the ball
+by smooth isotopy extension. Reattaching the handles therefore identifies
+the \emph{marked} smooth handlebody with a boundary join of the three
+longitudinal-annulus collars.
+
+\smallskip\noindent\textbf{2. Finite placement with based paths.}\quad
+Reserve, on the attaching regions in the initial boundary, disjoint
+annuli of product parallels for every tip in the finite list. Join their
+thin collars by thin tubes along the desired based graph paths, avoiding
+the reserved approaching strips. Finitely many arcs in a three-manifold
+can be made disjoint, apart from the prescribed common ends of a graph;
+parallel copies separate repeated portions. Regular neighbourhoods
+therefore give the required disjoint marked handlebodies. A path winding
+along an extra handle changes a based tip by the prescribed external
+conjugation while leaving its free attaching curve a parallel of the
+same cancelling-handle attaching loop. Paths internal to the marked
+body can add only the core-word changes already allowed in
+\eqref{eq:grope-word-properties}.
+
+\smallskip\noindent\textbf{3. Matching the cap framings.}\quad
+The placement fixes the external paths and the terminal product circles
+at the two-handles. We now vary the winding of each surface ribbon and
+its return collar to match the cap's normal framing, without moving
+those terminal circles or paths. These choices precede the definition
+of the exterior disk problem.
+
+In an oriented three-dimensional body crossed with an interval, frame a
+surface normal bundle by its
+oriented normal in the body and the interval vector. At a tip, a cap collar
+departing in the first normal direction has normal frame consisting of
+the in-surface direction transverse to the tip and the interval vector.
+Transport this ordered pair with the collar to the attaching region.
+The discrepancy with the two-handle core's product frame is an integer.
+It can be changed on the relevant solid-torus handle by
+\[
+ (\theta,z)\longmapsto(\theta,e^{in\theta}z),\qquad n\in\mathbb Z.
+\]
+The core stays fixed and the marked longitude changes by \(n\) meridians.
+The three disk duals make these choices independent. To realize the
+winding change with the same terminal product circle, choose the return
+collar after making the twist. Use coordinates
+\(S^1_\theta\times D^2_z\times[0,1]_u\) around a
+separated tip and its return collar.  A twisted surface ribbon has local
+form \((\theta,t e^{in\theta},0)\).  Its outgoing normal in the
+three-dimensional body is \(i e^{in\theta}\).  Choose an embedded
+plane arc \((a(s),b(s))\), \(0\leq s\leq1\), with
+\(a(0)=a(1)=b(0)=0\), \(b(1)=1\), \(a'(0)>0\),
+\(b'(0)=0\), and \(a=0\), \(b'>0\) near \(s=1\).
+We may require \(b'>0\) for \(0<s\leq1\).  Then
+\[
+ (\theta,s)\longmapsto
+       \bigl(\theta,a(s)i e^{in\theta},b(s)\bigr)
+\]
+is an embedded cap collar.  It leaves the ribbon in the required
+outgoing direction and ends at the same product circle \(z=0,u=1\),
+independently of \(n\).  In its normal quotient bundle a frame is given by
+\[
+ v_n=e^{in\theta},\qquad
+ w_n=a'(s)\partial_u-b'(s)i e^{in\theta}.
+\]
+The two vectors are independent modulo the tangent plane; orthogonal
+projection gives a normal frame if desired.  At the initial circle this
+is the surface transverse vector together with the collar vector.  At
+the terminal circle it is, up to a constant change of frame,
+\((e^{in\theta},i e^{in\theta})\), whose winding relative to the
+product core frame is \(n\).  Thus the framing discrepancy changes by an
+arbitrary integer while the terminal cap circle stays fixed.  If a handle
+foot is to remain fixed, replace \(e^{in\theta}\) by a degree-\(n\)
+circle map constant near that foot.  The disjoint tip neighbourhoods
+allow all these choices simultaneously.  The construction uses the
+separated tip representatives on the opposite faces of the surface
+products, not disjointness of the unpushed symplectic basis curves.
+
+Choose the integers to remove the framing discrepancies. This transports
+the body and its marked annuli together; it does not add a relative Euler
+term to a previously fixed cap. Meridians are null in the handlebody, so
+the core words and their leading commutator terms are unchanged.
+The twists are fixed near the connector feet, so the external paths
+realizing the \(c_a\) are unchanged.  Any internal change of path is a
+word \(W(X_1,X_2,X_3)\) in the three abstract tips of the marked
+handlebody.  Under the external marking
+\(X_a=c_a x_{r_a}^{\delta_a}c_a^{-1}\), this word specializes to
+\(W(1,1,1)=1\) when the three port meridians are trivial, even if the
+extra-letter holonomies are not trivial.  Thus such path changes do not
+alter the degree-three term in \eqref{eq:grope-word-leading}.
+
+Finally put the bodies at an inner level of a short boundary collar, with
+their initial boundary and three tip tracks returning to the outer level
+in disjoint collars of the four marked curves. The initial boundary avoids
+the attaching regions. The two directions at a symplectic pair depart on
+opposite normal sides. Continue the tip tracks into separate product
+parallels of the two-handle cores. These caps miss every body away from
+their own attachments. Their only possible intersections are the core
+intersections in the specified plumbing charts.
+\end{proof}
+
+In particular the finite family is constructed without knowing the future
+cut pieces. Each cap is individually embedded: a core has no self-plumbing.
+Caps of different labels may intersect, and no disjointness of their
+spanning surfaces or of all their interiors is asserted.
+
+\subsection{The disk problem in the exterior}
+
+The marked bodies have now been placed with framed caps. Removing only
+their bottom surfaces leaves an exterior in which one cap per body is
+the input disk. The remaining upper stage will supply its algebraic
+dual sphere; the unused upper cap will identify the group-ring labels
+needed for cancellation.
+
+Denote the disjoint bottom punctured tori by \(S_1,\ldots,S_k\).
+Choose their open product tubes, including their proper boundary collars,
+and put
+\[
+ E=X\setminus\bigcup_{j=1}^k\operatorname{int}\nu(S_j).
+\]
+The tubes are chosen after the finite placements of
+Lemma~\ref{lem:marked-body}. They can be sufficiently thin that they avoid
+all the upper caps and meet the upper bodies only in their short attaching
+collars. The exterior is connected: a path can be moved off finitely many
+proper codimension-two surfaces and then off their sufficiently small
+tubes. It is a compact oriented smooth four-manifold after rounding.
+
+\begin{lemma}[Algebraic duals in the exterior]
+\label{lem:exterior-duals}
+There are proper generically immersed disks \(f_j\) in \(E\), with
+disjoint boundary circles and with the bottom-surgery framings, and framed
+spheres \(g_j\) in \(E\), such that
+\[
+ \lambda(f_i,g_j)=\delta_{ij},\qquad
+ \lambda(g_i,g_j)=0,\qquad
+ \widetilde\mu(g_j)=0 .
+\]
+All equalities hold over \(\mathbb Z[\pi_1E]\).
+\end{lemma}
+
+\begin{proof}
+Write \(D_{x,j}\) for the single cap on the \(a\)-side of \(S_j\), and use
+as \(f_j\) its proper truncation on \(\partial\nu(S_j)\).
+Its product framing is the framing
+required for compression of the bottom.
+
+Let \(U_j\) be the upper punctured torus. Write \(D_{y,j}\) for its chosen
+surgery cap and \(D_{z,j}\) for the other cap. Compress \(U_j\) along
+\(D_{y,j}\). Cutting \(U_j\) along its \(y\)-curve gives a pair of pants;
+fill its two new boundary components by opposite parallel copies of
+\(D_{y,j}\). The result is a disk
+\begin{equation}
+\label{eq:one-sided-contraction}
+ C_j=U_j^{\mathrm{cut}}\cup D_{y,j}^{+}\cup(-D_{y,j}^{-})
+\end{equation}
+with boundary a parallel of \(b_j\). Its collar is on the outgoing side
+of the upper attachment. The unused \(D_{z,j}\) is not part of \(C_j\):
+it will instead give a null-homotopy in \(E\) identifying the group-ring
+labels of the two surgery-cap sheets.
+
+Take the normal-circle torus over a parallel of \(b_j\) near
+\(\partial\nu(S_j)\), and compress it with two parallels of \(C_j\).
+Here is a collar model fixing the relevant separations. Use coordinates
+\((s,t,r,\theta)\), where \(s\) is along \(b_j\), \(t\) is transverse to it
+on \(S_j\), and \((r,\theta)\) are polar coordinates normal to \(S_j\).
+The torus is \(t=0,\ r=\rho\). The \(a\)-cap collar and the upper attachment
+depart at distinct angles \(\theta_A,\theta_B\).
+Remove the thin compression band at \(\theta_B\), keep the annulus
+containing \(\theta_A\), and place the interiors of the two compression
+disks outside radius \(\rho\). Thus
+\begin{equation}
+\label{eq:compressed-rim-sphere}
+ g_j=R_j^{\mathrm{cut}}\cup C_j^+\cup(-C_j^-).
+\end{equation}
+The retained annulus meets the \(a_j\)-collar in one transverse point.
+Orient and base it so that this point contributes \(1\).
+All other body pieces, approach tracks and collars are in the separated
+neighbourhoods of Lemma~\ref{lem:marked-body}. In particular no collar
+point occurs for \(f_i,g_j\) with \(i\ne j\).
+Figure~\ref{fig:geometry-rim-collar} shows the normal-disk cross-section
+of this compression. Restoring the \(s\)-coordinate turns the retained
+arc into the annulus and the deleted arc into the compression band.
+
+\input{figures/geometry-rim-collar}
+
+We count all remaining intersections. They occur in cap--cap plumbing
+charts, because the parent bodies and the collar pieces just described
+are disjoint away from their prescribed attachments. Each occurrence of
+a \(D_{y,j}\) sheet in \eqref{eq:compressed-rim-sphere} has two indices:
+\(\varepsilon\in\{+1,-1\}\) specifies the copy of \(C_j\), and
+\(\eta\in\{+1,-1\}\) specifies the cap sheet inside that copy.
+Its orientation coefficient is \(\varepsilon\eta\); the two indices are
+shown in Figure~\ref{fig:geometry-sheets}.
+At a plumbing chart with coordinates \((u,v)\in D^2\times D^2\), these
+sheets are small parallel coordinate disks \(v=v_{\varepsilon,\eta}\);
+the opposite core sheets are \(u=u_{\varepsilon',\eta'}\).
+For any fixed opposing sheet, their local intersection signs differ by
+the factor \(\eta\).
+
+Fix \(\varepsilon\). The two paths to the \(\eta\)-sheets have the same
+stem from the rim torus to that copy of the upper body. Their difference
+is, up to this common conjugation, the upper basis loop \(z_j\).
+That loop bounds \(D_{z,j}\), together with its tip-annulus collar, in
+\(E\). Indeed those upper caps were disjoint from all the bottom tubes;
+intersections with other caps do not remove them from \(E\).
+An infinitesimal normal turn around the upper body also bounds a normal
+fibre disk in \(E\), since that body was not removed.
+It follows that the two intersection labels in \(\pi_1E\) agree.
+The paired contributions are therefore
+\[
+ \varepsilon\eta\, h+\varepsilon(-\eta)\,h=0
+ \quad\text{in }\mathbb Z[\pi_1E].
+\]
+The potential winding around the \emph{bottom} normal circle distinguishes
+the outer \(\varepsilon\)-copies. We never compare those copies in this
+cancellation: \(\varepsilon\) is held fixed throughout.
+
+\input{figures/geometry-sheets}
+
+This accounts for every cap contribution to \(\lambda(f_i,g_j)\).
+For \(g_i,g_j\), fix a sheet of \(g_i\) and the outer index of \(g_j\),
+and cancel the inner index of \(g_j\) by the same argument. It also
+applies when all parallel sheets at a plumbing point are counted
+simultaneously. Hence the off-diagonal pairings of the \(g_j\) vanish.
+
+The diagonal and framing admit a direct geometric check. The disk
+\(C_j\) uses just the one core type \(D_{y,j}\), which has no self-plumbing.
+That cap and its parallels are embedded and miss the parent body off their
+attachments. Thus \(C_j\), and then the sphere in
+\eqref{eq:compressed-rim-sphere}, can each be individually embedded.
+Mutual intersections with other \(g_i\) do not affect this assertion.
+The normal frame of the rim torus is
+\((\partial_t,\partial_r)\). Along the outgoing radial compression collar
+the cap normal frame is \((\partial_t,\partial_\theta)\).
+Rounding the surgery corner rotates the second vector from
+\(\partial_r\) to \(\partial_\theta\), while the first remains fixed.
+The product framing matched in Lemma~\ref{lem:marked-body} extends this
+ordered pair. The earlier compression \eqref{eq:one-sided-contraction}
+is the same framed operation, so its outgoing \(b_j\)-annulus has exactly
+the required inherited framing. Consequently \(g_j\) has trivial normal
+bundle. Its normal push-off is disjoint from it, proving the diagonal
+\(\lambda(g_j,g_j)=0\); its embedded representative also gives
+\(\widetilde\mu(g_j)=0\).
+
+Combining this with the distinguished collar point proves all the stated
+equalities. Small perturbations make the finite collection generic while
+preserving the calculation and the framings.
+\end{proof}
+
+Assumption~\ref{ass:disk}, applied to \(E\) and
+Lemma~\ref{lem:exterior-duals}, now gives pairwise disjoint locally flat
+embedded replacement caps with the specified boundary framings.
+Compress each bottom \(S_j\) with two parallels of its replacement cap,
+using the returning annuli in the removed tube. The result is a collection
+of pairwise disjoint properly embedded disks in \(X\), bounded by the
+initial grope boundaries. The framing is what makes this compression
+locally flat with the required product collar. No relative homotopy of
+the replacement cap to \(f_j\) is used. The output dual-sphere part of a
+disk embedding theorem is not used either.
+
+\subsection{Equivariant cuts with connected pieces}
+
+The preceding compression uses the assumed replacement disks to make
+the initial grope boundaries bound disjoint disks. We now keep those
+disks in their prescribed vertex regions while cutting the ambient
+plumbing into smooth pieces. Connectedness of these pieces will be
+needed for the restriction maps in the final part of this section.
+
+\begin{lemma}[Connected tree cuts]
+\label{lem:connected-tree-cuts}
+Suppose the initial grope boundaries bound the disjoint disks just
+constructed. Then the boundary map to the plumbing tree extends
+equivariantly to a map whose cuts can be chosen to have the connected
+vertex and edge pieces asserted in
+Proposition~\ref{prop:geometric-reduction}. Every lifted disk lies in
+its prescribed vertex piece.
+\end{lemma}
+
+\begin{proof}
+Lift the disks to \(\widetilde X\). Each is labelled by the vertex
+containing its boundary. Distinct translates and distinct members have
+disjoint images, because the disks were embedded and disjoint in \(X\).
+Prescribe the map to be constant at the label vertex on a neighbourhood
+of every such disk, and retain the fixed boundary map. These prescriptions
+agree on their overlaps.
+
+An equivariant extension is equivalently a section of the associated
+bundle over \(X\) with fibre the tree. The prescribed section extends first
+to a neighbourhood of the prescribed closed set, since the tree is an
+absolute neighbourhood retract, and then over the remaining cells,
+since the fibre is contractible. One may take a finite triangulation of
+the smooth ambient manifold for the latter extension after shrinking the
+neighbourhood. The section formulation ensures equivariance. This step
+does not require extending to the noncontractible quotient graph by
+contractibility of that graph.
+
+On the quotient, approximate the map smoothly over the open middle
+intervals of its finitely many graph edges, keeping it fixed on the
+specified boundary collars and away from the disk neighbourhoods.
+Choose regular values there. Their inverse images are compact smooth
+proper hypersurfaces, disjoint from all disks. After rescaling an edge
+interval we call its chosen value the midpoint. The lift of the level
+corresponding to a tree edge \(e\) has exactly one boundary component:
+the specified torus \(T_e\). More precisely, the unique component meeting
+the boundary contains that torus as its entire boundary; every other
+component of that level is closed.
+
+Consider the component graph \(\mathcal T'\) of this cut in
+\(\widetilde X\): its vertices are connected complementary pieces, and
+its edges are connected cut hypersurfaces. It is connected. The collapse
+to \(\mathcal T'\) is surjective on fundamental groups: a finite graph
+loop is realised by choosing crossing points and joining successive
+points by paths in the connected pieces. By
+\eqref{eq:plumbing-homotopy}, the fundamental group of \(\widetilde X\)
+is generated by the extra loops in its boundary pieces, with whiskers.
+Each such loop maps to a vertex of \(\mathcal T'\). Thus
+\(\pi_1\mathcal T'=0\), and \(\mathcal T'\) is a tree.
+
+The boundary pieces and their torus collars determine a principal copy
+\(\mathcal T\) of the original tree inside \(\mathcal T'\).
+To check injectivity directly, remove the chosen midpoints from the
+target tree. A connected component of their inverse-image complement
+maps into just one resulting vertex-star component. It cannot contain
+both \(K_v\) and \(K_w\) for \(v\ne w\). Similarly distinct principal
+edge components have distinct midpoint labels. The boundary collars
+give exactly the required incidences. Hence the principal copy is
+indeed an embedded subtree.
+
+Every remaining branch of \(\mathcal T'\) has a unique nearest vertex in
+\(\mathcal T\). A deck transformation stabilising a vertex or edge of
+such a branch fixes that nearest principal vertex and is therefore the
+identity. Thus all these stabilisers are trivial. The finite compact
+quotient cut has only finitely many components, so \(\mathcal T'\)
+has finite quotient. Trivial stabilisers imply finite valence.
+Moreover distance to \(\mathcal T\) is invariant under translation and
+is bounded on the finite set of component orbits. A finite-valence tree
+of bounded depth rooted at one principal vertex is finite.
+
+Ignore the closed cuts in each such finite hanging branch, merging it
+with its principal vertex piece. This operation is equivariant, preserves
+compact quotient and the principal hypersurfaces, and gives connected
+vertex pieces \(V_v\) and a single connected edge piece \(Y_e\) with
+\(\partial Y_e=T_e\). All prescribed disks remain in their own pieces:
+they lay in the component containing their boundary before the merging.
+The \(Y_e\) and \(V_v\) are smooth manifolds with corners; the topological
+disks themselves were never required to become smooth.
+\end{proof}
+
+Each initial boundary is therefore null-homotopic in its vertex piece.
+Its actual boundary word has the three-slot properties
+\eqref{eq:grope-word-properties} with inputs
+\eqref{eq:three-slot-inputs}. This proves precisely the relation statement
+in Proposition~\ref{prop:geometric-reduction}.
+
+\subsection{The parity quotient and its peripheral homology}
+
+Send every plumbing generator of \(F_m\) to \(1\in\mathbb Z/2\).
+The quotient of \(\widetilde X\) by the kernel has two principal vertex
+pieces \(A,B\) and two lifts of each of the \(m\) plumbing edges.
+Thus there are \(2m\) edges. The free translation interchanging parity
+gives a deck involution interchanging \(A\) and \(B\), and pairing these
+edge lifts. Equation~\eqref{eq:plumbing-homotopy}, or the same
+contractible-overlap argument in this quotient, gives
+\begin{equation}
+\label{eq:parity-homotopy}
+ X_2\simeq\Gamma\ \text{with \(p\) extra circles at each vertex},
+ \qquad
+ |V\Gamma|=2,\quad |E\Gamma|=2m.
+\end{equation}
+In particular \(H^q(X_2;k)=0\) for \(q\geq2\), and
+\(\dim_kH^1(X_2;k)=2m-1+2p\).
+The extra loops at the two vertices can be assigned independently in
+the free ambient fundamental group. All port meridians are trivial
+there, since they bound the corresponding handle-core disks in \(X_2\).
+
+\begin{lemma}[Restriction maps and coordinate slopes]
+\label{lem:geometric-edge-cohomology}
+The cohomology and slope conclusions of
+Proposition~\ref{prop:geometric-reduction} hold.
+For each edge piece one also has
+\begin{equation}
+\label{eq:edge-relative-cohomology}
+ H^1(Y_i,\partial Y_i;k)
+  \cong\ker\bigl(H^1(Y_i;k)\longrightarrow H^1(\partial Y_i;k)\bigr),
+ \qquad b_2(Y_i)=b_1(Y_i)-1.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Insert small edge collars so that Mayer--Vietoris applies to the two
+vertex pieces with intersection the disjoint union of the \(Y_i\).
+Its degree-one portion is
+\[
+\begin{aligned}
+ H^0(A;k)\oplus H^0(B;k)
+ &\longrightarrow\bigoplus_iH^0(Y_i;k)
+ \longrightarrow H^1(X_2;k)\\
+ &\longrightarrow H^1(A;k)\oplus H^1(B;k)
+ \xrightarrow{\,R\,}\bigoplus_iH^1(Y_i;k)
+ \longrightarrow0 .
+\end{aligned}
+\]
+Connectedness makes the first cokernel have dimension \(2m-1\);
+these are the graph classes of \(\Gamma\).
+They restrict to zero on the vertex pieces. The remaining kernel of
+\(R\) has dimension \(2p\). Evaluation on the \(p\) extra loops at each
+vertex identifies this kernel with \(k^{2p}\): their ambient classes in
+\eqref{eq:parity-homotopy} give surjectivity, and an ambient class with
+zero extra evaluations is a graph class and restricts to zero.
+Consequently \(R\) restricts to an isomorphism on the kernel of the
+extra evaluations. The next part of the same sequence, using
+\(H^2(X_2;k)=H^3(X_2;k)=0\), gives
+\eqref{eq:geometry-mv-degree-two}.
+
+Let \(Y\) be one of the connected oriented three-manifolds with single
+torus boundary \(T\). Poincaré--Lefschetz duality and
+\(\chi(Y)=\chi(T)/2=0\) give \(b_2(Y)=b_1(Y)-1\).
+The exact sequence of the pair, using connectedness of \(Y,T\), gives
+the isomorphism in \eqref{eq:edge-relative-cohomology}.
+Equivalently the image of \(H_1(T;k)\) in \(H_1(Y;k)\) has dimension
+one, the half-lives/half-dies statement.
+
+Write \(u,v\) for the original two coordinate directions on \(T\).
+One is a longitude in the boundary piece on one side and the other
+is a longitude on the other side. Suppose both had nonzero image in
+the one-dimensional image in \(H_1(Y;k)\). They would then be nonzero
+scalar multiples of each other. A cohomology class restricted from
+the first side annihilates its longitude and hence both directions;
+the same is true of a class restricted from the second side.
+Surjectivity of \(R\) would imply that every class of \(H^1(Y;k)\)
+annihilates both directions, contradicting the one-dimensional image.
+Thus one original coordinate direction has zero image and the other
+has nonzero image. Denote them by \(\beta,\alpha\), respectively.
+In cohomology the boundary restriction image is the line spanned by
+the coordinate form dual to \(\alpha\).
+
+The deck involution preserves the transported slope choice and swaps the
+two vertices. For the two lifts of each plumbing edge it therefore
+interchanges which one has its plus endpoint at \(A\).
+Exactly one lift in each pair is active, giving exactly \(m\) active
+edges. At \(A\) their \(\alpha\)'s are distinct port meridians, and at
+\(B\) their \(\beta\)'s are distinct port meridians. Their three-slot
+relations, with every independent choice of the stipulated conjugators,
+were all included before any slope was selected.
+\end{proof}
+
+\begin{proof}[Proof of Proposition~\ref{prop:geometric-reduction}]
+The plumbing construction supplies \(X\), its boundary pieces, and its
+regular cover. Lemma~\ref{lem:marked-body} supplies the finite marked
+family. Lemma~\ref{lem:exterior-duals} checks every input hypothesis of
+Assumption~\ref{ass:disk} in the exterior of the bottoms.
+The replacement disks and framed bottom compressions then give the
+disks required by Lemma~\ref{lem:connected-tree-cuts}, and hence the
+relations in connected vertex pieces. Finally
+Lemma~\ref{lem:geometric-edge-cohomology} gives the claimed parity-quotient data.
+\end{proof}
diff --git a/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/sections/02-tensor.tex b/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/sections/02-tensor.tex
new file mode 100644
index 0000000000000000000000000000000000000000..d090f81c804bb314f77c6fc9862da0074c49ddcc
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+++ b/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/sections/02-tensor.tex
@@ -0,0 +1,585 @@
+\section{A finite tensor obstruction}\label{sec:tensor}
+
+This section proves an algebraic obstruction to the formal data that will
+arise from the two vertex pieces.  The formal parametrizations may involve
+every edge block; each potential depends only on its own block.
+All parameter rings below retain every mixed square-free parameter monomial.
+
+For a characteristic-zero field \(K\), write \(K\langle h\rangle\) for the
+ring of algebraic power series at the origin, identified with the
+henselization of \(K[h]_{(h)}\) inside \(K[[h]]\).  An algebraic germ can
+therefore be represented by a regular function on a pointed étale
+neighborhood.  Gradients in this section always differentiate internal
+coordinates, keeping the external parameters fixed.
+
+\begin{theorem}[Finite tensor obstruction]\label{thm:tensor-obstruction}
+Let \(h=(h_1,\ldots,h_n)\) be a finite list of coordinate blocks over a
+characteristic-zero field \(K\), of total dimension \(N\).  Let
+\(I\subset\{1,\ldots,n\}\) have cardinality \(m\geq5\), and let
+\[
+ F_i\in K\langle h_i\rangle\quad(1\leq i\leq n),
+ \qquad b_i\in K\langle h_i\rangle\quad(i\in I).
+\]
+Put
+\[
+ T=K[t_i:i\in I]/(t_i^2:i\in I),\qquad
+ S_0(h)=\sum_{i=1}^nF_i(h_i),\qquad
+ S_t(h)=S_0(h)+\sum_{i\in I}t_i b_i(h_i).
+\]
+Suppose there are formal parametrizations
+\[
+ H_L(x,t)\in (x,t)\,T[[x]]^N,\qquad
+ H_R(y)\in (y)\,K[[y]]^N,
+\]
+where \(x\) has \(a\) coordinates and \(y\) has \(N-a\) coordinates, with
+the following properties.
+\begin{enumerate}[label=\textup{(\roman*)}]
+ \item The central tangent maps
+ \[
+  \partial_xH_L(0,0):K^a\longrightarrow K^N,\qquad
+  \partial_yH_R(0):K^{N-a}\longrightarrow K^N
+ \]
+ are injective and have complementary images.
+ \item The identities
+ \[
+     S_t(H_L(x,t))=0,\qquad S_0(H_R(y))=0
+ \]
+ hold in \(T[[x]]\) and \(K[[y]]\), respectively.
+ \item For every three distinct \(i,j,k\in I\),
+ \begin{align}
+ t_i t_j t_k
+ &\in\bigl((\partial_{h_\nu}S_t)(H_L(x,t)):
+                    1\leq\nu\leq N\bigr)\subset T[[x]],
+       \label{eq:tensor-source-membership}\\
+ (b_i b_j b_k)(H_R(y))
+ &\in\bigl((\partial_{h_\nu}S_0)(H_R(y)):
+                    1\leq\nu\leq N\bigr)\subset K[[y]].
+       \label{eq:tensor-target-membership}
+ \end{align}
+\end{enumerate}
+Then these data do not exist.
+\end{theorem}
+
+Condition (i) makes the parametrizations smooth formal graph embeddings,
+including over the full ring \(T\).  Nilpotent constant displacements of
+\(H_L\) are allowed.  The ideals in
+\eqref{eq:tensor-source-membership}--\eqref{eq:tensor-target-membership}
+are the actual gradient ideals; no passage to their radicals is made.
+There is no flatness assumption on their quotients.
+
+The later construction supplies the hypotheses as follows; all triples
+in the table have distinct active indices.
+\begin{center}
+\begin{tabular}{@{}
+ >{\raggedright\arraybackslash}p{0.25\linewidth}
+ >{\raggedright\arraybackslash}p{0.30\linewidth}
+ >{\raggedright\arraybackslash}p{\dimexpr0.45\linewidth-4\tabcolsep\relax}
+ @{}}
+\toprule
+Input & Source of the data & Later verification \\
+\midrule
+Separate algebraic \(F_i(h_i)\) and \(b_i(h_i)\)
+ & Individual marked edge models and polynomial perturbations
+ & Theorem~\ref{thm:edge-algebraicity} and
+   Section~\ref{subsec:vertex-polynomial} \\[4pt]
+Complementary central tangents
+ & Restriction maps after fixing the prescribed evaluations
+ & Lemma~\ref{lem:vertex-linear} \\[4pt]
+Exact vanishing \(S_t|_L=0\) and \(S_0|_R=0\)
+ & Stokes and section corrections preserving the residual quotient
+ & Lemmas~\ref{lem:vertex-offshell-identities},
+   \ref{lem:vertex-correction}, and
+   \ref{lem:tensor-nilpotent-correction} \\[4pt]
+Every source triple \(t_i t_j t_k\) in the restricted gradient ideal on \(L\)
+ & Boundary laws at \(A\)
+ & Lemma~\ref{lem:vertex-triple-ideals} \\[4pt]
+Every target triple \(b_i b_j b_k\) in the restricted gradient ideal on \(R\)
+ & Boundary laws at \(B\) and polynomial holonomy weights
+ & Lemma~\ref{lem:vertex-triple-ideals} and
+   Section~\ref{subsec:vertex-polynomial} \\
+\bottomrule
+\end{tabular}
+\end{center}
+
+The threshold \(m\geq5\) comes from two counts.  The target triple
+identities put the right graph's Koszul class in a sum of tensor patterns
+with at most two active factors outside the multiplication kernels.
+A compatible projection for the induced exterior differential then gives
+pure tensors of classes with at least \(m-2\geq3\) active factors both
+closed for that differential and killed by \(b_i\).
+These give three independent parameter lifts, and the source triple
+membership forces their pairing to vanish.  The closure statements concern
+Koszul classes; the chosen form representatives need not be closed for
+ordinary \(\dd\).
+
+\subsection{Algebraic approximation and finite specialization}
+
+\begin{lemma}\label{lem:tensor-reduction}
+If the data in Theorem~\ref{thm:tensor-obstruction} existed, data satisfying
+the same conditions would exist over a field of positive characteristic
+\(p\), for arbitrarily large \(p\).  The edge functions would still be
+separated in their original blocks.
+\end{lemma}
+
+\begin{proof}
+First put the parametrizations into graph form.  Choose complementary
+linear projections adapted to their central tangent spaces.  For \(L\),
+the chosen projection of \(H_L\) has invertible \(x\)-Jacobian modulo
+\((x,t)\).  It therefore has a formal inverse over \(T\), with the parameters
+fixed.  Its constant displacement is in \((t)\), so this is a legitimate
+substitution in the complete local ring \(T[[x]]\).
+For \(R\) the same assertion is the ordinary formal inverse function
+theorem.  After these reparametrizations each graph has a fixed linear
+projection equal to its independent variables.
+
+We describe explicitly the approximation problem for \(L\).
+For \(A\subset I\), let \(t_A=\prod_{i\in A}t_i\), including \(t_\varnothing=1\).
+Expand its graph functions and all the coefficients witnessing
+\eqref{eq:tensor-source-membership} in this finite basis:
+\[
+ H_L(x,t)=\sum_{A\subset I}t_A H_A(x),\qquad
+ c(x,t)=\sum_{A\subset I}t_A c_A(x).
+\]
+All unknown coefficient functions have the same list \(x\) of independent
+variables.
+
+The fixed algebraic functions \(F_i,b_i\) admit finite pointed étale
+presentations.  In such a presentation, differentiation in \(h_\nu\)
+extends uniquely from the polynomial coordinate ring: implicit
+differentiation expresses the derivative of every represented function
+using the inverse étale Jacobian.  Thus the functions and their ambient
+first derivatives can be handled in one finite presentation.  Introduce
+auxiliary unknowns for its coordinates and its inverse units, and expand
+them too in the basis \(t_A\).  Equating coefficients in the following
+identities gives a finite polynomial system:
+\begin{itemize}
+ \item the étale presentation equations and the inverse-unit equations;
+ \item the fixed graph-projection equation;
+ \item \(S_t(H_L)=0\);
+ \item each equation expressing \(t_i t_j t_k\) as a linear combination
+       of the restricted ambient gradient entries.
+\end{itemize}
+No derivative of an unknown graph function occurs.  The only derivatives
+are the fixed ambient derivatives of the given potentials, represented
+as just described.  The existing formal graph and membership witnesses
+give a formal solution of this polynomial system.
+
+Artin approximation over the henselization \(K[x]_{(x)}^h\)
+\cite[Theorem~1.10]{Artin1969} supplies an algebraic solution preserving
+the desired finite jets.  Preserve the centers, the branch centers of
+the étale presentations, and the central first graph jet.
+Do the same independently for \(R\), encoding
+\eqref{eq:tensor-target-membership} in its own independent variables \(y\).
+These two systems share only fixed functions.  Their compatibility
+condition is the open condition of complementary central tangents, which
+the preserved jets retain.  There is consequently no nested approximation
+problem.  If one independent-variable list is empty, the corresponding
+system is already a finite system over \(K\).
+
+We now have finitely many algebraic functions and finitely many exact
+identities in henselizations.  Each function is represented on a finite
+pointed étale neighborhood.  An identity in a henselization holds at some
+finite stage of the filtered system of these neighborhoods.  Enlarging
+the stages finitely many times represents all the functions, compositions,
+membership identities, branch centers, and inverse units together.
+Their finite presentations descend to a finitely generated integral
+\(\mathbb Z\)-subalgebra \(A\subset K\), after adjoining the required
+inverse units.  In particular we retain invertibility of the étale
+Jacobians and of the transverse tangent determinant.  The implicit
+differentiation formulas descend as well, so the specialized gradient
+entries remain the derivatives of the specialized potentials.
+The original presentations for the edge functions involve only their
+own blocks \(h_i\), and retain that property after descent.
+
+For completeness, this finite localization still has closed points in
+arbitrarily large positive characteristics.  The nonzero finite-type
+\(\mathbb Q\)-algebra \(A\otimes_{\mathbb Z}\mathbb Q\) has a maximal
+ideal with residue field a number field \(E\).  The images of the finitely
+many generators of \(A\), including all inverted elements, belong to
+\(\mathcal O_E[1/d]\) for some positive integer \(d\).
+Reduction at any prime of \(\mathcal O_E\) over a rational prime not
+dividing \(d\) gives a homomorphism from \(A\) to a finite field.
+All the declared units remain units.  The reduced pointed étale
+presentations give formal germs over that residue field, satisfying the
+same identities and the same tangent condition.
+\end{proof}
+
+\begin{remark}\label{rem:tensor-coefficient-field}
+The lemma also applies when \(K=\mathbb C((\sigma))\).
+Only finitely many elements of \(K\) occur as coefficients of the finite
+presentations, and each is treated as a single coefficient.
+Specialization is performed on \(A\), not coefficientwise on Laurent
+series or on the whole field \(K\).
+In particular, no boundedness of the negative \(\sigma\)-orders of the
+coefficients of an Artin solution is required.
+The algebraicity of the fixed separated potentials is essential:
+the argument does not assert that ordinary Artin approximation can
+algebraize arbitrary unknown formal potentials while preserving separate
+variable restrictions.
+\end{remark}
+
+\subsection{Truncated forms and transverse graph classes}
+
+The approximation step preserves all graph identities and actual ideal
+memberships. We now turn their positive-characteristic specializations
+into identities in finite-dimensional complexes, where transverse
+graphs can be tested by a nonzero pairing.
+
+Fix a specialization supplied by Lemma~\ref{lem:tensor-reduction} in
+characteristic \(p>2\), and denote its field by \(\kk\).
+Continue to use the same letters for the specialized germs.
+Write \(z_1,\ldots,z_N\) for all internal coordinates and set
+\[
+ A_p=\kk[z_1,\ldots,z_N]/(z_1^p,\ldots,z_N^p),\qquad
+ \Omega_p=A_p\otimes_\kk\bigwedge\nolimits_\kk
+                   \langle \dd z_1,\ldots,\dd z_N\rangle .
+\]
+This is the ordinary algebra of differential forms of \(A_p\).
+Its differential is well defined because \(\dd(z_\nu^p)=0\).
+With external parameters, use
+\[
+ T_p=\kk[t_i:i\in I]/(t_i^2:i\in I),
+ \qquad \Omega_{p,T}=\Omega_p\otimes_\kk T_p,
+\]
+and differentiate only in the internal coordinates.
+All these spaces are finite over the indicated coefficient rings.
+
+\begin{lemma}\label{lem:tensor-frobenius}
+Every formal change of internal coordinates over \(T_p\) that is centered
+modulo \((t)\) preserves the ideal
+\((z_1^p,\ldots,z_N^p)\).  Hence it induces an automorphism of the truncated
+algebra and of its relative differential forms.
+\end{lemma}
+
+\begin{proof}
+If \(c\in(z,t)T_p[[z]]\), Frobenius shows that \(c^p\) belongs to
+\((z_1^p,\ldots,z_N^p)\): every nonconstant internal monomial acquires an
+exponent divisible by \(p\), and every positive parameter monomial has
+zero \(p\)-th power.  Apply this to each new coordinate.
+The inverse coordinate change has the same property, giving equality of
+the ideals and the asserted automorphisms.
+\end{proof}
+
+Define a \(\kk\)-linear functional on forms, zero outside exterior degree
+\(N\), by
+\[
+ \int f(z)\,\dd z_1\wedge\cdots\wedge\dd z_N
+   =[z_1^{p-1}\cdots z_N^{p-1}]f .
+\]
+Extend it \(T_p\)-linearly in the presence of parameters.
+It vanishes on exact forms: a derivative contributing exponent \(p-1\)
+in its differentiated variable would have to come from exponent \(p\),
+whose derivative is zero.  Consequently, for homogeneous forms,
+\begin{equation}\label{eq:tensor-integration-by-parts}
+ \int\dd\alpha\wedge\beta
+       =-(-1)^{|\alpha|}\int\alpha\wedge\dd\beta .
+\end{equation}
+
+Let \(c_1,\ldots,c_r\) be graph equations for a smooth formal graph over
+the parameter ring.  Their joint center is zero modulo \((t)\).
+Its graph form is
+\begin{equation}\label{eq:tensor-thom-form}
+ U_c=\bigl(c_1^{p-1}\dd c_1\bigr)\wedge\cdots\wedge
+                       \bigl(c_r^{p-1}\dd c_r\bigr)
+       \in\Omega_{p,T}^{r}.
+\end{equation}
+This expression involves only a finite jet after truncation.
+
+\begin{lemma}\label{lem:tensor-graph-form}
+The form \(U_c\) is \(\dd\)-closed and is annihilated by the graph ideal.
+Every internal one-form whose restriction to the graph is zero wedges
+to zero with \(U_c\).
+For two central graphs with complementary tangents and graph forms
+\(U_0,V\),
+\begin{equation}\label{eq:tensor-nonzero-pairing}
+                     \int U_0\wedge V\ne0 .
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Each factor \(c_j^{p-1}\dd c_j\) is closed, and \(c_j^p=0\) in the
+truncated ring by Lemma~\ref{lem:tensor-frobenius}.
+This proves the first two assertions.
+In graph coordinates, the kernel of restriction of internal one-forms is
+the sum of graph-ideal multiples of one-forms and the span of the
+\(\dd c_j\).  Both summands wedge to zero with \(U_c\).
+This description remains valid after truncation because the graph
+coordinate change preserves the truncation ideal.
+
+For the pairing, concatenate the central graph equations for the two
+graphs, and let \(M\) be their linear coefficient matrix in the original
+coordinates.  Complementarity says that \(M\) is invertible.
+The coefficient part of \(U_0\wedge V\) has degree at least \(N(p-1)\);
+all terms of higher degree vanish in \(A_p\).
+Thus only the linear terms of the graph equations and the constant terms
+of their differentials contribute.  A linear coordinate change \(M\)
+acts on the polynomial socle
+\(\kk z_1^{p-1}\cdots z_N^{p-1}\) by \(\det(M)^{p-1}\), and on the
+exterior top form by \(\det(M)\).  To check the socle assertion one may
+use diagonal matrices, permutations, and elementary transvections:
+for a transvection every nonconstant binomial term has an exponent at
+least \(p\) and vanishes.  These matrices generate the general linear
+group.  The pairing is therefore \(\det(M)^p\), up to the choice of order
+of the graph equations, and is nonzero.
+\end{proof}
+
+Let \(U_t\) be the graph form for \(L\), and let \(V\) be the graph form
+for \(R\).  Define the degree-one operators
+\[
+       D_t=\dd S_t\wedge(-),\qquad D=\dd S_0\wedge(-).
+\]
+Both square to zero.  The graph identities and
+Lemma~\ref{lem:tensor-graph-form} give
+\begin{equation}\label{eq:tensor-graph-cycles}
+ D_tU_t=0,\qquad DV=0,\qquad
+ \dd U_t=0,\qquad\dd V=0 .
+\end{equation}
+In particular \(DU_0=0\) and the nonzero pairing
+\eqref{eq:tensor-nonzero-pairing} is defined.
+
+\begin{lemma}\label{lem:tensor-membership-boundary}
+For every three distinct active indices \(i,j,k\), there are actual forms
+\(B_{ijk}\) and \(C_{ijk}\) with
+\begin{equation}\label{eq:tensor-actual-boundaries}
+ t_i t_j t_k U_t=D_t B_{ijk},\qquad
+ b_i b_j b_k V=D C_{ijk}.
+\end{equation}
+The first equality is over the full parameter algebra \(T_p\).
+\end{lemma}
+
+\begin{proof}
+Lift the graph membership coefficients to the ambient ring.
+The difference between the asserted scalar and the resulting ambient
+gradient combination belongs to the graph ideal, so it annihilates the
+graph form.  If \(\iota_\nu\) is contraction with the coordinate vector
+\(\partial/\partial z_\nu\), then
+\[
+ \iota_\nu D_t+D_t\iota_\nu
+                =(\partial_{z_\nu}S_t)\,\id.
+\]
+For a \(D_t\)-cycle \(U_t\), multiplication by
+\(\sum_\nu a_\nu\partial_{z_\nu}S_t\) is consequently
+\[
+ \left(\sum_\nu a_\nu\partial_{z_\nu}S_t\right)U_t
+       =D_t\left(\sum_\nu a_\nu\iota_\nu U_t\right).
+\]
+The operator \(D_t\) does not differentiate the coefficients \(a_\nu\).
+This proves the first equality, and the same argument with \(D,V\)
+proves the second.
+\end{proof}
+
+We will use two elementary facts about this pairing.  If
+\(\theta\) is a one-form and \(D_\theta=\theta\wedge(-)\), then, for
+homogeneous \(\alpha\),
+\begin{equation}\label{eq:tensor-wedge-adjoint}
+ (D_\theta\alpha)\wedge\beta
+          =(-1)^{|\alpha|}\alpha\wedge D_\theta\beta.
+\end{equation}
+Thus a \(D_\theta\)-boundary wedges to zero with a
+\(D_\theta\)-cycle, even before integration.
+Also \(\dd D+D\dd=0\), since \(\dd^2S_0=0\).
+Ordinary differentiation therefore induces a differential on the
+homology of \(D\); Equation~\eqref{eq:tensor-integration-by-parts}
+continues to test its boundaries against actual closed representatives.
+
+\subsection{Two homologies and a compatible projection}
+
+We have graph cycles with a nonzero pairing and actual boundary
+identities for every triple. The separate edge blocks now let us
+analyze the target graph class one factor at a time. We first take
+Koszul homology, then use the differential induced on it by ordinary
+exterior differentiation.
+
+Let \(\Omega_i\) be the truncated form algebra in the block \(h_i\), and
+write
+\[
+ D_i=\dd F_i\wedge(-),\qquad H_i=H(\Omega_i,D_i).
+\]
+The tensor products of graded spaces and operators in what follows use
+the usual Koszul signs.  The block separation gives
+\[
+ (\Omega_p,D)=\bigotimes_{i=1}^n(\Omega_i,D_i),\qquad
+ H(\Omega_p,D)=\bigotimes_{i=1}^n H_i .
+\]
+The second equality is the finite-dimensional Künneth isomorphism over
+the field \(\kk\).
+Let \(\delta_i\) denote the differential induced on \(H_i\) by ordinary
+\(\dd\), and let \(\delta\) be their tensor differential.
+For \(i\in I\), multiplication by \(b_i\) induces a degree-zero
+endomorphism \(B_i\) of \(H_i\), since it commutes with \(D_i\).
+Set
+\[
+                         K_i=\ker B_i .
+\]
+There is no assertion that \(K_i\) is a \(\delta_i\)-subcomplex.
+
+\begin{lemma}[Kernel support]\label{lem:tensor-kernel-support}
+Let \(v\in\bigotimes_{i=1}^nH_i\) be killed by
+\(B_iB_jB_k\) for every three distinct active indices.
+Then \(v\) belongs to the sum of tensor subspaces having all but at most
+two active factors in the corresponding \(K_i\).
+\end{lemma}
+
+\begin{proof}
+Choose a graded complement \(J_i\) to \(K_i\) in each active factor.
+The restriction of \(B_i\) to \(J_i\) is injective.
+For a fixed triple, the kernel of its tensor operator is the sum of
+subspaces having a \(K_i\) factor in at least one of the three specified
+positions: the induced tensor of injections on the quotient by those
+kernels is injective.
+Expand the whole tensor product using \(H_i=K_i\oplus J_i\).
+Intersecting the kernel conditions for all triples leaves exactly the
+summands with at most two active \(J_i\) positions.
+\end{proof}
+
+\begin{lemma}[Projection preserving kernel representatives]
+\label{lem:tensor-kernel-projection}
+Let \((H,\delta)\) be a finite-dimensional graded complex over a field,
+and let \(K\subset H\) be any graded subspace.
+There is a chain projection \(P:H\to H\), homotopic to the identity,
+whose image is a space of representatives for \(H(H,\delta)\) and which
+satisfies
+\[
+                    P(K)\subset K\cap\ker\delta .
+\]
+\end{lemma}
+
+\begin{proof}
+Write \(Z=\ker\delta\) and \(B=\im\delta\).
+Choose a complement \(A_K\) to \(K\cap B\) in \(K\cap Z\), and extend
+\(A_K\) to a complement \(A\) to \(B\) in \(Z\).
+Choose a complement \(W_K\) to \(K\cap Z\) in \(K\).
+It is disjoint from \(Z\), so extend it to a complement \(W\) to \(Z\)
+in \(H\).  All these choices are made degree by degree.
+Then
+\[
+                       H=B\oplus A\oplus W .
+\]
+Let \(P\) be projection onto \(A\).  It is a chain map, and
+\(P(K)\subset A_K\subset K\cap Z\).
+The map \(\delta:W\to B\) is an isomorphism with the appropriate shift
+of degrees.  Define a degree-minus-one map \(s\) to be its inverse on
+\(B\), and zero on \(A\oplus W\).  Then
+\[
+                        \id-P=\delta s+s\delta .
+\]
+\end{proof}
+
+Apply Lemma~\ref{lem:tensor-kernel-projection} to each active factor
+\((H_i,\delta_i,K_i)\), and choose an ordinary homology projection in
+each inactive factor.  Denote these maps by \(P_i\), and put
+\(\Pi=\bigotimes_iP_i\).  This tensor projection is chain-homotopic
+to the identity.  Indeed, the telescoping identity for
+\(\id-\bigotimes_iP_i\), together with the individual homotopies and
+the graded tensor convention, supplies a homotopy for the total
+differential \(\delta\).
+
+Let \(v=[V]\in H(\Omega_p,D)\).  Equations
+\eqref{eq:tensor-actual-boundaries} imply the hypotheses of
+Lemma~\ref{lem:tensor-kernel-support}, and \(\delta v=0\) by
+\eqref{eq:tensor-graph-cycles}.  Thus \(\Pi v\) is a sum of homogeneous
+pure tensors
+\begin{equation}\label{eq:tensor-pure-cycles}
+                   v_1\otimes\cdots\otimes v_n
+\end{equation}
+such that every \(v_i\) is a \(\delta_i\)-cycle and at least \(m-2\)
+active factors lie in \(K_i\cap\ker\delta_i\).
+This conclusion follows by applying \(\Pi\) to the tensor subspaces in
+Lemma~\ref{lem:tensor-kernel-support}; the projection of a kernel factor
+is still a kernel factor.
+Since \(m\geq5\), every nonzero pure tensor in this sum has at least three
+such active factors.
+
+Choose \(D_i\)-cycle representatives for all the factors in
+\eqref{eq:tensor-pure-cycles}, and let \(V'\) be the resulting sum of
+actual forms.  The chain homotopy for \(\Pi\) shows that
+\([V']-[V]=\delta[Z]\) for some class in \(H(\Omega_p,D)\).
+Choose a \(D\)-cycle representative \(Z\).
+There is then an actual form \(E\) such that
+\begin{equation}\label{eq:tensor-chain-replacement}
+                         V'-V=\dd Z+D E .
+\end{equation}
+Pairing with \(U_0\), equations
+\eqref{eq:tensor-integration-by-parts},
+\eqref{eq:tensor-graph-cycles}, and
+\eqref{eq:tensor-wedge-adjoint} give
+\begin{equation}\label{eq:tensor-pairing-preserved}
+                    \int U_0\wedge V'=\int U_0\wedge V .
+\end{equation}
+The representatives composing \(V'\) need not be closed for ordinary
+\(\dd\).  Their classes are \(\delta_i\)-cycles, which is precisely what
+was used to obtain \eqref{eq:tensor-chain-replacement}.
+
+\subsection{Lifts over a three-parameter cube}
+
+The replacement class has the same pairing with the source graph as
+the original target class. Each of its tensor summands has at least
+three active factors lying in both the multiplication kernel and the
+kernel of the induced differential. Those three factors supply the
+parameter lifts used to force that summand's pairing to vanish.
+
+\begin{proof}[Proof of Theorem~\ref{thm:tensor-obstruction}]
+Suppose the stated data exist and perform the reduction and constructions
+above.  Consider one pure tensor \eqref{eq:tensor-pure-cycles} contributing
+to \(V'\).  Choose three active positions \(i,j,k\) in which
+\[
+                   B_\nu v_\nu=0,\qquad\delta_\nu v_\nu=0 .
+\]
+On \(H_\nu\), the commutator identity is
+\[
+       [\delta_\nu,B_\nu]
+         =\delta_\nu B_\nu-B_\nu\delta_\nu
+         =[\,\dd b_\nu\wedge(-)\,].
+\]
+It follows that \([\,\dd b_\nu\wedge(-)\,]v_\nu=0\).
+For a homogeneous \(D_\nu\)-cycle representative \(a_\nu\) of \(v_\nu\),
+there is therefore a form \(w_\nu\) of the same exterior degree with
+\[
+                    D_\nu w_\nu=-\dd b_\nu\wedge a_\nu .
+\]
+Consequently
+\begin{equation}\label{eq:tensor-first-order-lift}
+ (D_\nu+t_\nu\dd b_\nu\wedge)
+                      (a_\nu+t_\nu w_\nu)=0
+             \quad\text{in }\Omega_\nu[t_\nu]/(t_\nu^2).
+\end{equation}
+Only the vanishing of the indicated classes in \(H_\nu\) was required;
+no ordinary \(\dd\)-closed representative is being assumed.
+
+Set all parameters outside this triple equal to zero, and put
+\[
+ T_{ijk}=\kk[t_i,t_j,t_k]/(t_i^2,t_j^2,t_k^2).
+\]
+Tensor the three lifts \eqref{eq:tensor-first-order-lift} with the chosen
+unchanged \(D_\nu\)-cycle representatives in every other block.
+The resulting form \(W_t\) is a cycle for
+\[
+              \dd(S_0+t_i b_i+t_j b_j+t_k b_k)\wedge(-).
+\]
+This is an exact cycle equation over \(T_{ijk}\): each perturbation acts
+in its own block, so the three separate first-order lifts require no
+additional mixed corrections.
+
+Specialize the actual first identity in
+\eqref{eq:tensor-actual-boundaries} to these three parameters.
+It remains an actual boundary identity in forms.  In particular, this
+step does not assert that homology commutes with the possibly nonflat
+parameter specialization.
+By \eqref{eq:tensor-wedge-adjoint}, pairing that identity with \(W_t\)
+gives
+\[
+ t_i t_j t_k\,P(t)=0\quad\text{in }T_{ijk},\qquad
+ P(t)=\int U_t\wedge W_t ,
+\]
+where \(U_t\) is specialized in the same way.
+The square-free monomials form a \(\kk\)-basis of \(T_{ijk}\).
+Multiplication by \(t_i t_j t_k\) kills all its positive-degree
+parameter monomials, so the displayed equation implies
+\[
+  P(0)=\int U_0\wedge(a_1\otimes\cdots\otimes a_n)=0 .
+\]
+The chosen triple may vary with the pure tensor; the argument applies
+to each term separately.  Summing gives \(\int U_0\wedge V'=0\).
+This contradicts \eqref{eq:tensor-pairing-preserved} and the nonzero
+transverse graph pairing \eqref{eq:tensor-nonzero-pairing}.
+\end{proof}
diff --git a/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/sections/03-edge-germs.tex b/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/sections/03-edge-germs.tex
new file mode 100644
index 0000000000000000000000000000000000000000..3a5457f08e3348fe10801576eacc157769dfa95e
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+++ b/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/sections/03-edge-germs.tex
@@ -0,0 +1,506 @@
+\section{Formal potentials for a single edge}
+\label{sec:edge-germs}
+
+Let $Y$ be a compact connected oriented smooth three-manifold with
+connected boundary $T\cong S^1\times S^1$.  Fix based coordinate circles
+$\alpha,\beta$ on $T$.  We assume that the image of
+$H^1(Y;\mathbb C)\to H^1(T;\mathbb C)$ is the line evaluating on
+$\alpha$ and vanishing on $\beta$.  These are the boundary data furnished
+by the edge pieces.  All cohomology in this section has complex
+coefficients.  Put $\g=\mathfrak{sl}_2(\mathbb C)$, with its invariant
+nondegenerate pairing $(X,Y)\mapsto\tr(XY)$.
+Our goal is a formal potential whose actual gradient ideal describes
+based flat systems with one peripheral holonomy fixed. We also need
+the first-order change in this potential when that holonomy varies.
+
+Choose closed one-forms $\zeta,\xi$ on $T$ with periods
+\[
+ \int_\alpha\zeta=\int_\beta\xi=1,
+ \qquad
+ \int_\beta\zeta=\int_\alpha\xi=0.
+\]
+They may be taken to be bump densities in the two circle coordinates.
+Their supports are thin strips; choose their common basepoint outside
+both strips, and choose the coordinate circles through that basepoint.
+Thus $\xi$ vanishes along $\alpha$, and $\zeta$ vanishes along $\beta$.
+Write
+\[
+ \epsilon=\int_T\zeta\wedge\xi\in\{1,-1\},
+\]
+where $T$ has its boundary orientation.  Choose a two-form $\eta$ of
+integral one supported in a small disk disjoint from the strips.  The
+restriction hypothesis gives a closed extension
+$\widehat\zeta\in\Omega^1(Y)$ of $\zeta$: first choose a closed
+representative with the required boundary cohomology class, and then
+correct its boundary trace by the differential of an extended function.
+
+\subsection{The boundary complex and its pairing}
+
+Define a nonunital commutative differential graded algebra $C$ by
+\begin{equation}
+\label{eq:edge-boundary-complex}
+ C^j=\left\{a\in\Omega^j(Y):
+ a|_T\in
+ \begin{cases}
+  0,&j=0,\\
+  \mathbb C\xi,&j=1,\\
+  \mathbb C\eta,&j=2,\\
+  0,&j\geq3.
+ \end{cases}\right\}.
+\end{equation}
+In degrees at least three the boundary condition is automatic.
+Multiplication is closed because degree-zero boundary values vanish,
+$\xi\wedge\xi=0$, and all other positive-degree products in question
+vanish on the two-dimensional boundary.
+
+\begin{lemma}
+\label{lem:edge-cohomology}
+There are natural isomorphisms
+\[
+ H^1(Y,T)\cong H^1(C),\qquad H^2(C)\cong H^2(Y),
+\]
+and $H^0(C)=H^3(C)=0$.  Both middle groups have dimension
+$n=b_1(Y)-1=b_2(Y)$.  Every closed one-form in $C$ has zero actual
+boundary pullback.  Integration gives a perfect pairing
+\[
+ H^1(C)\otimes H^2(C)\longrightarrow\mathbb C.
+\]
+\end{lemma}
+
+\begin{proof}
+The quotient of $C$ by the relative de Rham complex has just the two
+terms $\mathbb C\xi$ in degree one and $\mathbb C\eta$ in degree two,
+with zero differential.  The connecting map
+\[
+ \mathbb C\xi\longrightarrow H^2(Y,T)
+\]
+is injective, since $[\xi]$ does not lie in the image of $H^1(Y)$.
+The connecting map
+\[
+ \mathbb C\eta\longrightarrow H^3(Y,T)
+\]
+is an isomorphism: under the relative orientation it is boundary
+integration.  The long exact sequence consequently identifies $H^1(C)$
+with $H^1(Y,T)$ and $H^2(C)$ with the quotient of $H^2(Y,T)$ by the
+connecting $\xi$-line.  The ordinary relative sequence identifies this
+last quotient with $H^2(Y)$.  It also gives the asserted vanishing in
+degrees zero and three.  Poincar\'e--Lefschetz duality gives
+$\dim H^1(Y,T)=b_2(Y)$, and
+$\chi(Y)=\tfrac12\chi(T)=0$ gives $b_2(Y)=b_1(Y)-1$.
+
+If a closed one-form in $C$ has trace $v\xi$, its boundary cohomology
+class is both a multiple of $[\xi]$ and an element of the $[\zeta]$-line.
+Hence $v=0$.  Under the displayed isomorphisms, the integration pairing
+is the relative/absolute Poincar\'e--Lefschetz pairing in degrees one
+and two, so it is perfect.
+\end{proof}
+
+We keep $C$ for the scalar complex.  The tensor product $C\otimes\g$
+is a dg Lie algebra, with bracket given by the wedge product combined
+with the Lie bracket.  For homogeneous $a,b\in C\otimes\g$ whose
+degrees sum to three, put
+\[
+ \langle a,b\rangle=\int_Y\tr(a\wedge b).
+\]
+This pairing satisfies cyclic Stokes.  Indeed, when the degrees of
+$a,b$ sum to two, their boundary product is zero: either one has degree
+zero or both boundary traces are multiples of $\xi$.  Thus
+\begin{equation}
+\label{eq:edge-cyclic-stokes}
+ \langle da,b\rangle+(-1)^{|a|}\langle a,db\rangle=0.
+\end{equation}
+Invariance of the matrix trace gives the corresponding cyclic identity
+for the bracket.  No assertion that integration of a single
+degree-two element commutes with the differential is needed; its
+boundary trace may be a multiple of $\eta$.
+
+Choose representatives $i:H^*(C)\to C$, a projection
+$\operatorname{pr}:C\to H^*(C)$, and a degree-minus-one map $K$ such that
+\begin{equation}
+\label{eq:edge-contraction}
+ dK+Kd=\id-i\operatorname{pr},\qquad
+ K^2=Ki=\operatorname{pr}K=0,\qquad
+ \operatorname{pr}i=\id.
+\end{equation}
+These maps exist by splitting the vector spaces into boundaries,
+cohomology representatives, and complements on which the differential
+is an isomorphism onto boundaries.  Extend them by the identity on
+$\g$.  In particular, the degree-one representatives have zero
+boundary pullback.  Since $H^0(C)=0$, we have $Kd=\id$ on $C^0$.
+
+All series below are formal.  Applying one of these linear maps means
+applying it separately to each coefficient, which is an actual smooth
+form.  No boundedness or analytic convergence of $K$ is required.
+
+\subsection{The central slice and its potential}
+
+Set
+\[
+ H=H^1(C)\otimes\g,\qquad N=\dim H=3n,
+ \qquad S=\mathbb C[[h_1,\ldots,h_N]],
+\]
+and let $h$ denote the universal element of $H\otimes S$.  Its
+coordinates generate the maximal ideal $\m$ of $S$.  The recursion
+\begin{equation}
+\label{eq:edge-central-recursion}
+ s=i(h)-\tfrac12K[s,s]
+\end{equation}
+has a unique solution in $C^1\otimes\g\widehat\otimes\m$: its linear
+term is $i(h)$, and each higher homogeneous term is determined by
+previous terms.  It satisfies
+\[
+ \operatorname{pr}s=h,\qquad Ks=0,
+ \qquad K P(s)=0,
+ \qquad P(a)=da+\tfrac12[a,a].
+\]
+Conversely these three slice conditions imply
+\eqref{eq:edge-central-recursion}, by
+\eqref{eq:edge-contraction}.  Write $s|_T=v(h)\xi$.
+
+We use the Chern--Simons transgression functional
+\cite[Section~3]{ChernSimons1974}. For a Lie-algebra-valued one-form $a$,
+define
+\begin{equation}
+\label{eq:edge-cs}
+ \mathcal C(a)=\int_Y\tr\left(
+        \tfrac12a\wedge da+\tfrac16a\wedge[a,a]\right),
+ \qquad f(h)=\mathcal C(s(h)).
+\end{equation}
+For any coefficient variation $\dot a$, integration by parts gives
+\begin{equation}
+\label{eq:edge-variation}
+ \dot{\mathcal C}(a)
+   =\int_Y\tr(\dot a\wedge P(a))
+       -\tfrac12\int_T\tr(a\wedge\dot a).
+\end{equation}
+In the central slice the boundary term is zero.
+
+For a formal form $P$, its \emph{curvature ideal} means the ideal
+generated by all its scalar coefficient functions, equivalently by
+applying arbitrary linear functionals on the vector space of
+Lie-algebra-valued forms coefficientwise.  The next calculation in
+particular shows that the ideals occurring here are finitely generated.
+
+\begin{lemma}
+\label{lem:edge-central-ideal}
+The curvature ideal of $s(h)$ is the Jacobian ideal
+\[
+ J=\Jac(f)=\left(\frac{\partial f}{\partial h_1},\ldots,
+                 \frac{\partial f}{\partial h_N}\right).
+\]
+Moreover $f\in\m^3$.
+\end{lemma}
+
+\begin{proof}
+Put $P=P(s)$ and $r=\operatorname{pr}P\in H^2(C)\otimes\g
+\widehat\otimes S$.  The boundary trace of $P$ is zero, so $P$ lies coefficientwise in $C^2\otimes\g$.
+Bianchi and the projected slice equation give
+\begin{equation}
+\label{eq:edge-central-curvature}
+ dP=-[s,P],\qquad KP=0,
+ \qquad P=i(r)-K[s,P].
+\end{equation}
+The operator $K\ad_s$ raises maximal-ideal order.  Therefore
+\[
+ P=(1+K\ad_s)^{-1}i(r).
+\]
+Every curvature coefficient lies in the ideal of the finitely many
+components of $r$, and the converse holds by applying
+$\operatorname{pr}$.  This is an equality of ideals, without passage to
+their radicals.
+
+By \eqref{eq:edge-variation}, the gradient entries are
+\[
+ \frac{\partial f}{\partial h_j}
+    =\left\langle\frac{\partial s}{\partial h_j},P\right\rangle.
+\]
+Consequently the gradient is a square matrix of formal functions times
+the vector $r$.  Its constant matrix is the perfect pairing between
+$H^1(C)\otimes\g$ and $H^2(C)\otimes\g$ from
+Lemma~\ref{lem:edge-cohomology}; it is invertible.  This proves the ideal
+equality.  Finally $s=i(h)+O(\m^2)$ and $di(h)=0$.  Both terms in
+\eqref{eq:edge-cs} therefore have order at least three.
+\end{proof}
+
+\subsection{A prescribed first-order boundary holonomy}
+
+The central slice fixes the surviving peripheral holonomy to the
+identity. The vertex construction will vary that holonomy by a
+square-zero parameter. Its curvature may then have a boundary
+component, so the next calculation includes that component in the
+gradient ideal.
+
+Fix $T_0\in\g$ and a scalar parameter $\tau$ with $\tau^2=0$.  Put
+\[
+ S_\tau=S[\tau]/(\tau^2),\qquad u=\tau T_0.
+\]
+The formal topology is the one at $(h,\tau)=0$.  Choose any linear
+extension
+\[
+ K_{\mathrm{ext}}:\Omega^2(Y)\longrightarrow
+       \im\bigl(K:C^2\to C^1\bigr)
+\]
+of the degree-two part of $K$, and tensor it with $\g$.  Define
+$s=s(h,\tau)$ by
+\begin{equation}
+\label{eq:edge-affine-recursion}
+ s=i(h)+u\widehat\zeta-\tfrac12K_{\mathrm{ext}}[s,s].
+\end{equation}
+The recursion again determines a unique formal series.  It reduces to
+the central slice when $\tau=0$, and
+\[
+ s-u\widehat\zeta\in C^1\otimes\g\widehat\otimes S_\tau,
+ \quad \operatorname{pr}(s-u\widehat\zeta)=h,
+ \quad K(s-u\widehat\zeta)=0.
+\]
+Its boundary value has the form
+\[
+ s|_T=u\zeta+v(h,\tau)\xi.
+\]
+The recursion implies $K_{\mathrm{ext}}P(s)=0$.  To see this directly,
+write its correction as an element $\ell$ of the original one-form
+image of $K$.  The contraction identities give $Kd\ell=\ell$;
+also $di(h)=d\widehat\zeta=0$.  Applying $K_{\mathrm{ext}}$ to the
+curvature then gives exactly the recursion.
+
+The boundary curvature is generally nonzero off the critical scheme:
+\begin{equation}
+\label{eq:edge-boundary-curvature}
+ P(s)|_T=c\,\zeta\wedge\xi,\qquad c=[u,v].
+\end{equation}
+In particular it need not satisfy the degree-two boundary condition
+defining $C$.  The following argument keeps this additional component.
+
+\begin{lemma}
+\label{lem:edge-affine-ideal}
+The function
+\begin{equation}
+\label{eq:edge-adjusted-potential}
+ f_\tau(h)=\mathcal C(s(h,\tau))
+                   +\tfrac{\epsilon}{2}\tr(u v(h,\tau))
+\end{equation}
+has internal Jacobian ideal equal to the full curvature ideal of the
+affine slice.  Here the internal derivatives are only the derivatives
+with respect to the $h_j$.
+\end{lemma}
+
+\begin{proof}
+Choose a scalar two-form $\rho$ on $Y$ with boundary trace
+$\zeta\wedge\xi$, and put
+\[
+ P=P(s),\qquad P_0=P-c\rho,\qquad
+ r=\operatorname{pr}P_0.
+\]
+Now $P_0$ has zero boundary trace and lies coefficientwise in $C^2\otimes\g$.  Applying Stokes
+to the Lie-algebra-valued Bianchi identity yields
+\begin{equation}
+\label{eq:edge-boundary-bianchi}
+ \epsilon c=\int_T P=\int_Y dP=-\int_Y[s,P].
+\end{equation}
+Thus $c$ is a linear expression in $P$ whose coefficients have positive
+formal order.  Since
+\[
+ KP_0=-cK_{\mathrm{ext}}\rho,
+ \qquad dP_0=-[s,P]-c\,d\rho,
+\]
+the contraction identity for $P_0$ gives the exact formula
+\begin{equation}
+\label{eq:edge-affine-curvature}
+ P=i(r)-K[s,P]
+       +c\bigl(\rho-dK_{\mathrm{ext}}\rho-Kd\rho\bigr).
+\end{equation}
+The original degree-three part of $K$ is defined on both $[s,P]$ and
+$d\rho$, since all three-forms satisfy the boundary condition.
+Substituting \eqref{eq:edge-boundary-bianchi} into
+\eqref{eq:edge-affine-curvature} writes this as
+\[
+ P=i(r)+L_s(P),
+\]
+where $L_s$ raises $(h,\tau)$-adic order.  Hence
+\[
+ P=(1-L_s)^{-1}i(r).
+\]
+The residual components generate every curvature coefficient.  In the
+other direction, $c=\epsilon^{-1}\int_TP$ and
+$r=\operatorname{pr}(P-c\rho)$ are linear expressions in curvature
+coefficients.  The residual and curvature ideals are therefore equal,
+also over the nonreduced ring $S_\tau$.  This argument is valid for every
+allowed extension $K_{\mathrm{ext}}$.
+
+For a variation with $u$ fixed, the boundary term in
+\eqref{eq:edge-variation} is
+$-\epsilon\tr(u\dot v)/2$.  The added term in
+\eqref{eq:edge-adjusted-potential} cancels it, so
+\begin{equation}
+\label{eq:edge-fixed-u-gradient}
+ \frac{\partial f_\tau}{\partial h_j}
+    =\left\langle\frac{\partial s(h,\tau)}{\partial h_j},P\right\rangle.
+\end{equation}
+As in the central case, this is a square matrix times $r$, with the
+same perfect constant pairing.  The matrix is invertible over
+$S_\tau$.  It follows that the internal Jacobian, residual, and full
+curvature ideals coincide.  In particular the boundary commuting
+equation $[u,v]=0$ is included in that ideal; it was not discarded.
+\end{proof}
+
+\subsection{Based gauge and marked holonomies}
+
+The curvature and Jacobian ideals have been identified for both slices.
+We now identify their quotient schemes with based representations and
+track the two torus holonomies. Working on Artinian coefficient algebras
+retains the nilpotent structure used by the ideal calculations.
+
+We use the connection $d+a$ and parallel transport solving
+$\dot U=-a(\dot\gamma)U$.  With the chosen boundary paths this gives
+\begin{equation}
+\label{eq:edge-holonomy-sign}
+ G_\alpha=\exp(-u),\qquad G_\beta=\exp(-v)
+\end{equation}
+for a flat boundary connection $u\zeta+v\xi$.  Reversing the transport
+convention changes the corresponding signs throughout.
+
+Let $A$ be a local Artinian complex algebra with residue field
+$\mathbb C$ and maximal ideal $\m_A$.  All connections and gauge
+transformations in this paragraph reduce to the trivial ones modulo
+$\m_A$.  The based gauge group consists of transformations equal to
+the identity at the fixed point of $T$.  On a connection already in
+the strip boundary form, we subsequently use gauge transformations
+equal to the identity on all of $T$.
+
+\begin{lemma}
+\label{lem:edge-based-slice}
+The central slice identifies its curvature-zero scheme with the formal
+scheme of based representations of $\pi_1Y$ satisfying $G_\alpha=1$.
+Over the dual numbers, the affine slice identifies its curvature-zero
+scheme with the based representations satisfying
+$G_\alpha=\exp(-\tau T_0)$.  These statements hold on all local
+Artinian coefficient algebras, not only on reduced points.
+\end{lemma}
+
+\begin{proof}
+Flat small connections modulo based gauge are equivalent to based
+small representations by parallel transport.  This equivalence can
+be constructed coefficientwise.  A representation defines the flat
+bundle with its given framing; its small transition functions can be
+trivialized smoothly by induction over the nilpotent filtration.
+At each central square-zero stage this is the trivialization of an
+additive cocycle of smooth functions, which follows from a partition
+of unity.  Conversely, parallel transport is a finite iterated-integral
+expression at each nilpotent order.  Flatness makes it invariant under
+based homotopies.  Two flat connections with the same based monodromy
+are related by the unique based gauge transformation obtained by
+comparing their parallel transports.
+
+Suppose first that a based representation has
+$G_\alpha=\exp(-u)$, where $u=0$ or $u=\tau T_0$.  Put
+$v=-\log G_\beta$.  The two boundary monodromies commute.  In the
+affine case this implies $[u,v]=0$: conjugating the square-zero
+exponential gives $(\exp(\ad_v)-1)u=0$, and
+\[
+ \exp(\ad_v)-1
+    =\ad_v\left(1+\frac{\ad_v}{2!}
+                   +\frac{(\ad_v)^2}{3!}+\cdots\right),
+\]
+whose factor in parentheses is invertible for nilpotent coefficients.
+The assertion is automatic when $u=0$.  Hence
+$u\zeta+v\xi$ is a flat torus connection with exactly the given based
+monodromy.  Based parallel transport gives a gauge transformation
+normalizing the original boundary connection to this strip form.
+Its logarithm extends smoothly from the boundary, so the normalization
+extends over $Y$.
+
+Now use gauge transformations equal to the identity on $T$ to impose
+\[
+ K(a-u\widehat\zeta)=0.
+\]
+Their infinitesimal action is $a\mapsto a+d\phi$, with
+$\phi\in C^0\otimes\g\otimes\m_A$, and $Kd\phi=\phi$.
+Thus the exact slice coordinate can be killed uniquely, successively
+in each nilpotent order.  For a flat connection in this gauge, setting
+$h=\operatorname{pr}(a-u\widehat\zeta)$ and applying $K$ or
+$K_{\mathrm{ext}}$ to its zero curvature gives exactly the corresponding
+recursion.  Therefore the normalized connection is $s(h)$ or
+$s(h,\tau)$.
+
+There is no additional quotient by boundary conjugation.  If two
+normalized boundary connections have the same based monodromy, their
+coefficients $u,v$ agree by the formal logarithm.  A based gauge
+transformation between those identical boundary connections is the
+identity everywhere on $T$, again by parallel transport.  The subsequent
+slice gauge is unique by $Kd=\id$ on $C^0$.  These constructions are
+natural under maps of Artinian algebras, proving the assertions about
+formal schemes, including their nilpotent structure.
+\end{proof}
+
+\Needspace{14\baselineskip}
+We collect the conclusions, including the first-order coefficient
+needed later.
+
+\begin{proposition}[The marked edge potential]
+\label{prop:edge-potential}
+With the boundary polarization fixed above, there is a formal potential
+$f\in\mathbb C[[h_1,\ldots,h_N]]$, where
+$N=3(b_1(Y)-1)$, and a slice $s(h)$ such that:
+\begin{enumerate}[label=\textup{(\roman*)}]
+ \item $f=\mathcal C(s)$ has order at least three.  Its full Jacobian
+ ideal equals the curvature ideal of $s$.  The formal critical scheme
+ $\Crit(f)=\Spf(S/\Jac(f))$ is the based flat-representation scheme
+ with $G_\alpha=1$, and carries its marked based matrix $G_\beta$.
+ \item For each fixed $T_0\in\g$ there is an affine slice with boundary
+ $\tau T_0\zeta+v(h,\tau)\xi$ and adjusted potential
+ \[
+  f_\tau=f+\tau b
+    =\mathcal C(s(h,\tau))
+             +\tfrac{\epsilon}{2}\tr(\tau T_0v(h,\tau)),
+  \qquad \tau^2=0.
+ \]
+ Its internal Jacobian ideal equals its full curvature ideal.
+ Accordingly $\Crit_h(f_\tau)$ is the based flat-representation scheme
+ with $G_\alpha=\exp(-\tau T_0)$.  No derivative with respect to
+ $\tau$ is imposed in this relative critical scheme.
+ \item On the entire, possibly nonreduced, central critical scheme,
+ \begin{equation}
+ \label{eq:edge-log-coefficient}
+  b=\epsilon\tr(T_0v(h,0))
+      =-\epsilon\tr\bigl(T_0\log G_\beta\bigr)
+       \quad\bmod\Jac(f).
+ \end{equation}
+ The function $b$ has order at least two.  Every flat connection with
+ the indicated normalized boundary value is carried uniquely to the
+ corresponding slice by gauge transformations equal to the identity
+ on $T$.
+\end{enumerate}
+\end{proposition}
+
+\begin{proof}
+The preceding lemmas prove all assertions except the identification
+of $b$ and its order.  For a general first variation of the two
+boundary coefficients, the boundary part of
+\eqref{eq:edge-variation} is
+\[
+ -\tfrac{\epsilon}{2}\tr(u\dot v-v\dot u).
+\]
+After adding the variation of $\epsilon\tr(uv)/2$, this becomes
+$\epsilon\tr(v\dot u)$.  Thus the adjusted variation is
+\begin{equation}
+\label{eq:edge-adjusted-variation}
+ \dot f_\tau
+    =\int_Y\tr(\dot s\wedge P(s))
+                      +\epsilon\tr(v\dot u).
+\end{equation}
+Take the coefficient derivative in $\tau$ at $\tau=0$, so that
+$\dot u=T_0$.  On $S/\Jac(f)$ the central curvature vanishes as an
+actual coefficientwise formal form, by
+Lemma~\ref{lem:edge-central-ideal}.  The integral term therefore
+vanishes in that quotient.  Equation~\eqref{eq:edge-holonomy-sign}
+then proves \eqref{eq:edge-log-coefficient}.  This uses equality of
+ideals and does not pass to the reduced critical locus.
+
+The linear part of the affine slice is
+$i(h)+\tau T_0\widehat\zeta$, which is closed and has no $\xi$
+boundary coefficient.  Hence its Chern--Simons expression and its
+boundary adjustment have joint $(h,\tau)$-order at least three.
+The coefficient of $\tau$ consequently has $h$-order at least two.
+All formal exponentials and logarithms used here are defined by their
+nilpotent truncations on Artinian algebras and their formal limits.
+\end{proof}
diff --git a/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/sections/04-algebraicity.tex b/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/sections/04-algebraicity.tex
new file mode 100644
index 0000000000000000000000000000000000000000..0fcdc0eb17bc8bea3bd5c13d61a59eb655633432
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+++ b/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/sections/04-algebraicity.tex
@@ -0,0 +1,515 @@
+\section{Algebraicity with based holonomy markings}
+\label{sec:algebraicity}
+
+An algebraic power series will mean an element of
+\(\mathbb C\langle x_1,\ldots,x_N\rangle\), the henselization of
+\(\mathbb C[x_1,\ldots,x_N]_{(x)}\) inside \(\mathbb C[[x]]\).
+We use the edge manifold, boundary coordinates, complex \(C\), and potential
+\(f\) of Proposition~\ref{prop:edge-potential}.  All ideals in this section
+are scheme-theoretic ideals; no passage to radicals is understood.
+
+\Needspace{12\baselineskip}
+\begin{theorem}[Algebraic edge potential with markings]
+\label{thm:edge-algebraicity}
+Put \(N=\dim(H^1(C)\otimes\mathfrak{sl}_2)\),
+\(R=\mathbb C[[h_1,\ldots,h_N]]\), and \(J=\Jac(f)\).
+There are an invertible formal coordinate change \(x=\varphi(h)\) and an
+algebraic power series \(F(x)\in(x)^3\) such that
+\[
+                         f(h)=F(\varphi(h)).
+\]
+Fix a torus basepoint and any finite collection of based loops in \(Y\).
+For every matrix entry of their holonomies on the central critical scheme
+\(\Spf(R/J)\), there is an algebraic power series in the \(x\)-coordinates
+whose pullback is that entry.  In particular this holds for the based
+\(\beta\)-holonomy.  The coordinate change can be chosen to identify these
+markings on the entire, possibly nonreduced, critical scheme.
+\end{theorem}
+
+The proof has five steps.  First, the punctured filling provides an
+algebraic Darboux chart.  Second, the full relative closed-form comparison
+identifies its symplectic class with the integrated trace on the cochain
+model.  Third, transfer gives the Hamiltonian identity for the given
+potential \(f\).  Fourth, the closed-form homotopy expresses the
+difference of the two potentials as an element of \(\Jac(f)^2\).
+Fifth, a formal coordinate correction removes that difference while
+fixing the critical scheme and every marked based holonomy.
+
+\subsection{The punctured filling and its algebraic critical chart}
+
+Let \(M\) be a solid torus with an open ball removed.  Glue its torus boundary
+to \(\partial Y\), with the meridian glued to \(\alpha\), and write
+\[
+                   Z=Y\cup_{T^2}M,\qquad \partial Z=S^2.
+\]
+Choose a point on this sphere and a path to the chosen torus basepoint.
+The filling kills \(\alpha\) in the fundamental group.  Thus its based
+representations are the based representations of \(\pi_1Y\) with
+\(G_\alpha=1\), and the path identifies their torus markings.
+The sphere boundary will supply the relative orientation used below.
+Set \(G=SL_2\).  For a finite CW complex \(K\), write
+\(\Loc_G(K)=\operatorname{Map}(K_B,BG)\); a superscript \(\mathrm{fr}\)
+means that the local system is trivialized at the chosen basepoint.
+These are derived mapping stacks.  In particular, framing is a derived
+fiber of evaluation at the basepoint, not a quotient by conjugation.
+
+The invariant pairing \(\operatorname{tr}(XY)\) is nondegenerate on
+\(\mathfrak g=\mathfrak{sl}_2\) and defines a \(2\)-shifted symplectic form
+on \(BG\).  Betti transgression gives a \(0\)-shifted symplectic form on
+\(\Loc_G(S^2)\) \cite[Theorem~2.5 and Corollary~2.6(4)]{PTVV2013}.
+Restriction
+\[
+                     \Loc_G(Z)\longrightarrow\Loc_G(S^2)
+\]
+is Lagrangian \cite[Definition~2.6, Claim~2.7, and Theorem~2.9]{Calaque2015}.
+Here the hypotheses of these constructions have concrete meanings.
+The manifolds have finite CW type, so the Betti sources are
+\(\mathcal O\)-compact and their mapping stacks are derived Artin stacks
+locally of finite presentation.  Integration over the fundamental class
+is the orientation map.  Poincar\'e--Lefschetz duality supplies the required nondegeneracy for
+\emph{every} locally constant perfect complex \(E\) over every base
+cdga \(D\), including homotopy-coherent coefficient systems:
+\[
+ C^*(Z,S^2;E)\simeq
+ \mathbf R\!\operatorname{Hom}_D\bigl(C^*(Z;E^\vee),D\bigr)[-3].
+\]
+To see the coefficient scope, local derived duality on contractible
+stars identifies the oriented interior dualizing complex with \(D[3]\)
+and takes \(E\) to \(E^\vee\otimes D[3]\).  These natural local
+identifications descend with all higher restriction coherences;
+relative boundary cells give the displayed equivalence.  The finite
+constructions in perfect fibers commute with derived base change.
+This is the compact-manifold application of
+\cite[\S\S3.1.1--3.1.2]{Calaque2015}, and includes adjoint coefficients.
+The isotropic homotopy is integration over the relative fundamental chain
+of \((Z,S^2)\).
+
+The second Lagrangian is
+\[
+       \Loc_G(S^2)^{\mathrm{fr}}\longrightarrow\Loc_G(S^2).
+\]
+Indeed, the based sphere stack is
+\(\{1\}\mathbin{\times^{\mathbf R}_G}\{1\}\), the derived
+self-intersection of the identity point in the smooth group \(G\).
+It is the affine odd vector space with functions
+\(\Sym(\mathfrak g^*[1])\), whose generators have degree \(-1\).
+On this scheme a form of exterior degree \(2+j\) has internal degree at
+most \(-2-j\).  Thus neither internal degree \(-j\), used by a
+\(0\)-shifted closed two-form, nor internal degree \(-j-1\), used by its
+nullhomotopy, occurs.  The pulled-back form is zero, and its isotropic
+path is unique up to homotopy.  At the unique classical point, the tangent
+complex of the unframed sphere stack is
+\(C^*(S^2;\mathfrak g)[1]\); the framed tangent retains exactly the
+degree-two cohomology summand.  Integration and trace pair this summand
+perfectly with the omitted degree-zero summand.  This proves the
+Lagrangian nondegeneracy condition.  There are no other classical points
+at which to check it.
+
+The shifted intersection theorem therefore equips
+\begin{equation}
+\label{eq:framed-filling-intersection}
+ \mathcal X
+ =\Loc_G(Z)\mathbin{\times^{\mathbf R}_{\Loc_G(S^2)}}
+                   \Loc_G(S^2)^{\mathrm{fr}}
+ =\Loc_G(Z)^{\mathrm{fr}}
+\end{equation}
+with a \((-1)\)-shifted symplectic structure
+\cite[Theorem~2.9]{PTVV2013}.  This is a derived scheme.  To see the
+finite-presentation assertion directly, a compact connected
+three-manifold with nonempty boundary has a finite spine of dimension at
+most two: a handle decomposition relative to a collar can be chosen
+with no absolute three-handles.  Collapsing a maximal tree of its
+one-skeleton gives one vertex, say \(r\) edges, and \(s\) two-cells.
+Consequently its framed local-system scheme is the derived fiber
+\[
+                    G^r\mathbin{\times^{\mathbf R}_{G^s}}\{1\},
+\]
+where the map records the attaching words.  This description also shows
+that every fixed based holonomy is a regular \(G\)-valued function on
+the classical scheme.
+
+The algebraic Darboux theorem applies over \(\mathbb C\), at the trivial
+classical point of \(\mathcal X\).  More precisely,
+\cite[Theorem~5.18(i)]{BBJ2019} supplies a Zariski local algebraic
+Darboux chart minimal at that point; in shift \(-1\),
+\cite[Example~5.15]{BBJ2019} presents its algebra as
+\begin{equation}
+\label{eq:algebraic-darboux-chart}
+ A^0[p_1,\ldots,p_N],\qquad |p_i|=-1,
+ \qquad Qp_i=\frac{\partial F}{\partial x_i},
+\end{equation}
+where \(A^0\) is smooth and the \(x_i\) are \(\acute{e}\)tale coordinates.
+Minimality means that the cotangent differential vanishes at the point
+\cite[Definition~2.13]{BBJ2019}; in this presentation it is the Hessian
+of \(F\).  Subtracting the constant term therefore gives \(F\in(x)^3\).
+In the completed \(x\)-coordinates this is an algebraic power series.
+Each of the finitely many regular holonomy entries on the classical
+critical chart lifts through the quotient \(A^0\to A^0/\Jac(F)\).
+After shrinking the smooth chart, the lifts are regular, and hence give
+algebraic power series in the same coordinates.  This assertion concerns
+the matrix entries of holonomy; it makes no assertion that their formal
+logarithms are algebraic.
+
+The Darboux chart supplies an algebraic potential and regular holonomy
+functions on its critical scheme.  To identify that potential with the
+particular Chern--Simons function \(f\), we need more than an
+isomorphism of critical schemes or equality of tangent pairings.
+The full relative closed-form comparison below will give a difference
+of potentials in \(\Jac(f)^2\).  A coordinate correction can then remove
+that difference while acting trivially on the marked critical scheme.
+We state the comparison here and give its proof in
+Appendix~\ref{app:formal-trace-comparison}, so that we can first follow
+its application to the edge complex.
+
+\input{sections/04a-formal-comparison}
+
+\subsection{A model preserving the integrated trace}
+
+We identify the cochain model of
+\eqref{eq:framed-filling-intersection} with the edge complex \(C\).
+On \(S^2\), choose a closed area form \(\nu\).  The inclusion
+\(\mathbb C\nu\) into the augmented-kernel forms on the based sphere is
+a quasi-isomorphism.  Thus the sphere boundary condition may be imposed
+by requiring its trace to lie in \(\mathbb C\nu\), with zero trace in
+the other degrees.  Restriction of unrestricted forms to a boundary is
+surjective, so this strict fiber computes the required homotopy fiber.
+
+The representatives on the punctured solid torus can be chosen as
+follows.  Extend \(\xi\) using a bump density in its longitudinal
+circle coordinate.  Put the puncture and a tube from the support disk
+of \(\eta\) to the puncture in the complement of that longitudinal
+strip.  A transverse Thom form of the tube extends \(\eta\) to a
+closed two-form \(\widetilde\eta\) whose sphere trace is the appropriately
+oriented \(\nu\).  Denote the extended one-form by
+\(\widetilde\xi\).  These choices give
+\begin{equation}
+\label{eq:zero-added-pairing}
+ d\widetilde\xi=d\widetilde\eta=0,
+ \qquad \widetilde\xi\wedge\widetilde\eta=0.
+\end{equation}
+The span of these two forms, with zero multiplication, is a
+quasi-isomorphic subalgebra of the forms on \(M\) with the sphere
+boundary condition.  In fact that complex has one-dimensional
+cohomology in degrees one and two and no other cohomology, as follows
+from the relative sequence for \((M,S^2)\); the displayed forms
+represent the two generators.
+
+For gluing, use forms smooth on each piece with matching tangential
+traces.  This piecewise smooth de Rham model computes the cochains of
+\(Z\); its integration is the sum over the two pieces, and their torus
+boundary terms cancel by Stokes.  Now glue the small model to the
+unrestricted forms on \(Y\).
+Surjectivity of restriction from \(Y\) to the torus shows that replacing
+the model on \(M\) by this quasi-isomorphic subalgebra preserves the
+homotopy pullback.  The resulting algebra is exactly \(C\): its torus
+traces are zero in degree zero, multiples of \(\xi\) in degree one,
+and multiples of \(\eta\) in degree two.  Moreover,
+\eqref{eq:zero-added-pairing} shows that every complementary-degree
+pairing on the added piece vanishes before integration.  The full
+integrated pairing on the filling is consequently
+\begin{equation}
+\label{eq:edge-cyclic-pairing}
+                \langle u,v\rangle
+                  =\int_Y\operatorname{tr}(u\wedge v)
+\end{equation}
+on \(C\otimes\mathfrak g\).  The comparison of full closed forms and
+of the relative integration homotopy in
+Proposition~\ref{prop:formal-trace-comparison} applies to this
+replacement.  It identifies the shifted symplectic structure of
+\(\mathcal X\), rather than just its value on tangent cohomology, with
+this strict integrated form.
+
+Choose the basing path through \(M\) outside the longitudinal strip.
+The degree-one connection on \(M\) is a multiple of
+\(\widetilde\xi\), so its transport along this path is the identity.
+The resulting comparison uses the fixed torus framing and preserves
+based holonomies as matrices, including the chosen \(\beta\)-holonomy.
+
+\subsection{Transfer with an arbitrary cochain contraction}
+
+The preceding model identifies the integrated trace form and the based
+holonomies. We next transfer it to the finite coordinate space used by
+the edge slice, keeping the original contraction. The required output
+is the Hamiltonian identity for the particular potential $f$.
+
+Write \(H^j=H^j(C)\otimes\mathfrak g\), and retain the splitting
+\((i,\operatorname{pr},K)\) from the edge construction, so
+\[
+ dK+Kd=1-i\operatorname{pr},\qquad
+ K^2=Ki=\operatorname{pr}K=0.
+\]
+No compatibility of \(K\) with the pairing is required.  Choose bases
+of \(H^1,H^2\), and let \(h\) and \(q\) be their corresponding
+coordinates of degrees zero and \(-1\).  In the completed semifree
+algebra \(R[q_1,\ldots,q_N]\), set
+\begin{equation}
+\label{eq:transfer-v}
+ s=i_1h-\tfrac12K[s,s],\qquad
+ P=ds+\tfrac12[s,s],\qquad r=\operatorname{pr}P,
+ \qquad V=(1+K\operatorname{ad}_s)^{-1}i_2.
+\end{equation}
+The inverse exists in the \((h)\)-adic topology.  The contraction
+identity and Bianchi identity give
+\(P=i_2r-K[s,P]\), and hence
+\begin{equation}
+\label{eq:transfer-mc}
+                 P=Vr,\qquad a=s+Vq,\qquad Qh=0,
+                 \qquad Qq=-r.
+\end{equation}
+Here and below the suspension signs in the integrated symplectic form
+are fixed compatibly with the last convention.
+
+For completeness, these formulas give the entire transferred
+Maurer--Cartan map.  Put \(W=dV+[s,V]\).  The recursion for \(V\) gives
+\(KV=0\), \(\operatorname{pr}V=1\), and
+\[
+ KW=KdV+K[s,V]=V-i_2+K[s,V]=0.
+\]
+The columns of \(W\) are three-forms in \(C\otimes\mathfrak g\).
+They have zero differential by dimension and zero cohomology projection
+because \(H^3(C)=0\); the contraction identity therefore gives \(W=0\).
+The bracket of two columns of \(Vq\) is a physical four-form and is
+zero.  Equations~\eqref{eq:transfer-mc} now imply
+\[
+                    (d+Q)a+\tfrac12[a,a]=0.
+\]
+In particular there are no omitted terms quadratic in \(q\).
+The induced map of formal moduli problems has the identity on tangent
+cohomology, and is a formal equivalence.  One can also see this directly
+by induction over nilpotent coefficient ideals: each square-zero
+extension has the same cochain deformation and obstruction groups, and
+the contraction identifies those groups.
+
+Pull back the strict integrated closed form by this map and denote it
+by \(\omega\).  It consists of \(dh\,dq\) terms and
+\(q\,dh\,dh\) terms.  Its constant \(dh\,dq\) matrix is the perfect
+pairing \(H^1\otimes H^2\to\mathbb C\), so it is formally
+nondegenerate.  Explicitly, give coordinate forms total parity equal
+to internal degree plus exterior degree, and extend coordinate
+differentiation across a physical form \(c\) by
+\(\delta(c\otimes u)=(-1)^{|c|}c\otimes\delta u\).
+The mixed part is then
+\[
+ \omega_{\mathrm{mixed}}
+   =-\sum_{i,\mu}\left\langle\partial_i s,V_\mu\right\rangle
+                                                  dh_i\,dq_\mu.
+\]
+The contraction \(\iota_Q\) is total-even.  Since
+\(\iota_Qdq_\mu=-r_\mu\), contraction removes the \(dq\) and gives
+the pairing with \(Vr=P\); its value on the \(q\,dh\,dh\) terms
+is zero.  The variation formula for the central
+Chern--Simons functional, whose boundary term vanishes, consequently
+gives the exact identity
+\begin{equation}
+\label{eq:transferred-hamiltonian}
+                 \iota_Q\omega
+                   =\int_Y\operatorname{tr}(\delta s\wedge P)
+                   =\delta f.
+\end{equation}
+The symbol \(\delta\) differentiates only the formal coordinates.
+This proves the Hamiltonian identity for the chosen, possibly
+noncyclic, contraction.  Reversing the global symplectic sign reverses
+both algebraic and Chern--Simons Hamiltonians and has no effect on the
+statement of Theorem~\ref{thm:edge-algebraicity}.
+
+\subsection{A closed-form homotopy gives a squared-Jacobian error}
+
+We now have two potentials on equivalent minimal formal models: the
+algebraic Darboux potential and the Chern--Simons potential.  The
+comparison already identifies their critical schemes and based
+holonomies.  We use its additional full closed-form homotopy to place
+the difference of potentials in the square of the Jacobian ideal.
+This stronger conclusion permits a coordinate correction that fixes
+the critical scheme.
+
+Compare the two minimal completed semifree models, the transferred model
+and the Darboux model \eqref{eq:algebraic-darboux-chart}, over the formal
+completion of \(\mathcal X\).  A map between these models is represented
+by a formal power-series map: equivalently one uses homotopy transfer
+and maps of minimal Lie models, or lifts maps of completed semifree
+algebras order by order.  An equivalence has invertible linear part on
+generators, since the cotangent differentials at the point are zero.
+It therefore has a formal inverse.  Degree considerations make its
+shape particularly simple:
+\begin{equation}
+\label{eq:minimal-model-isomorphism}
+                       x=x(h),\qquad p=B(h)q,
+\end{equation}
+where the Jacobian of \(x(h)\) and the matrix \(B(h)\) are invertible.
+Let \(f'=F(x(h))\), and let \(\omega'\) be the pullback of the
+canonical Darboux form.  Then
+\[
+ \iota_Q\omega'=\delta f',\qquad
+ (Qq_1,\ldots,Qq_N)=\Jac(f')=\Jac(f)=J\subset(h)^2.
+\]
+The first equality is naturality of the Darboux Hamiltonian identity;
+the equality of ideals follows from the two invertible matrices in
+\eqref{eq:minimal-model-isomorphism} and from the edge curvature
+calculation.  The comparison is over \(\mathcal X\), so it also
+identifies all the marked classical holonomy functions.
+
+Both strict forms represent the same shifted closed-form class.
+We record explicitly the consequence for the Hamiltonians; equality
+of the ordinary leading two-form alone would not suffice.
+
+\begin{lemma}[Completed Cartan calculation]
+\label{lem:cartan-jacobian-square}
+Suppose that \(Qh=0\), \(Qq\in(h)^2\), and that strict closed
+\((-1)\)-shifted two-forms \(\omega,\omega'\) on \(R[q]\) are
+homotopic as closed forms.  If
+\(\iota_Q\omega=\delta f\) and
+\(\iota_Q\omega'=\delta f'\), then, after normalizing constants,
+\[
+                       f-f'\in(Qq_1,\ldots,Qq_N)^2.
+\]
+\end{lemma}
+
+\begin{proof}
+In this proof put \(J=(Qq_1,\ldots,Qq_N)\subset R\).
+Use total signs for which \(I=\iota_Q\) is even and
+\[
+                 [I,\delta]=\mathcal L_Q,
+                 \qquad [I,\mathcal L_Q]=0.
+\]
+The closed-form homotopy has components \(\psi_j\), of exterior
+degree \(j\) and internal degree \(-j\), for \(j\geq2\).
+Changing the component signs if necessary, its equations are
+\[
+ \omega-\omega'=\mathcal L_Q\psi_2,
+                  \qquad \delta\psi_j=\mathcal L_Q\psi_{j+1}.
+\]
+Set
+\[
+             S_j=I^j\psi_j,\qquad
+             T_j=I^{j-1}\mathcal L_Q\psi_j.
+\]
+Commuting \(\delta\) past the \(j\) contractions gives
+\(\delta S_j=T_{j+1}-jT_j\), while
+\(T_2=\delta(f-f')\).  Thus, for every \(L\geq2\),
+\begin{equation}
+\label{eq:cartan-factorial-tail}
+ \delta(f-f')=-\delta\sum_{j=2}^{L}\frac{S_j}{j!}
+                                      +\frac{T_{L+1}}{L!}.
+\end{equation}
+Every contraction replaces a \(dq_i\) by \(Qq_i\) and kills a
+\(dh_i\).  A monomial of \(\psi_j\) containing \(a\) factors \(q\),
+\(b\) factors \(dq\), and \(c\) factors \(dh\) satisfies
+\(a+b=j=b+c\), by internal and exterior degree.  Hence \(a=c\).
+Survival after \(j\) contractions requires \(c=0\), so also \(a=0\)
+and \(b=j\).  Consequently \(S_j\) is an ordinary function in \(J^j\).
+For the remainder, the surviving monomials contain either \(L+1\)
+factors from \(J\), or \(L\) such factors and one differentiated
+factor \(\delta(Qq_i)\).  Since
+\(Qq_i\in(h)^2\) and \(\delta(Qq_i)\in(h)\,dh\), their coefficient
+order is at least \(2L+1\).  Hence the remainder tends to zero.
+The sum converges in the complete power-series ring, and
+\eqref{eq:cartan-factorial-tail} implies
+\[
+                 f-f'=-\sum_{j\geq2}\frac{S_j}{j!}+\text{constant}.
+\]
+The ideal \(J^2\) is adically closed.  The normalized constant is
+zero, proving the claim without an isolated-critical-point hypothesis.
+\end{proof}
+
+\begin{lemma}[Removing the error while fixing the critical scheme]
+\label{lem:jacobian-square-flow}
+Let \(f\in(h)^3\) and \(e\in\Jac(f)^2\).
+There is a formal automorphism \(\Phi\) of \(R\), equal to the
+identity modulo \(J=\Jac(f)\), such that
+\[
+                           (f+e)\circ\Phi=f.
+\]
+\end{lemma}
+
+\begin{proof}
+Write \(f_i=\partial_i f\) and
+\(e=\sum_i v_i f_i\), with \(v_i=\sum_j a_{ij}f_j\in J\).
+Since the Hessian of \(f\) has entries in \((h)\), differentiating
+this expression gives a matrix \(D\) with entries in \((h)\) such
+that
+\[
+                   \nabla e=D\nabla f.
+\]
+Explicitly,
+\(D_{kj}=\partial_kv_j+\sum_i a_{ij}\partial_k f_i\), which has
+the stated order.  Put \(f_t=f+te\).  The vector field
+\[
+                  X_t=-(1+tD)^{-\mathsf T}v
+\]
+has entries in \(J\subset(h)^2\) and satisfies
+\(df_t(X_t)=-e\).  Its formal flow \(\Phi_t\), with
+\(\Phi_0=\id\), exists coefficient by coefficient: each fixed
+\((h)\)-adic order involves finitely many powers of \(t\), and
+integration of those polynomials is defined over \(\mathbb C\).
+It obeys
+\[
+                 \frac{d}{dt}\Phi_t^*f_t
+                    =\Phi_t^*(e+df_t(X_t))=0.
+\]
+Every \(X_t\) preserves \(J\), since its coefficients belong to
+\(J\); its induced derivation on \(R/J\) is zero.  Therefore the
+flow preserves \(J\) and is the identity on \(R/J\).
+Taking \(\Phi=\Phi_1\) proves both assertions.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{thm:edge-algebraicity}]
+Lemma~\ref{lem:cartan-jacobian-square} gives \(f'-f\in J^2\).
+Apply Lemma~\ref{lem:jacobian-square-flow} with \(e=f'-f\).
+Then
+\[
+                    f= f'\circ\Phi=F\circ x\circ\Phi,
+\]
+so \(\varphi=x\circ\Phi\) is the required coordinate change.
+The original comparison \(x\) identifies the critical scheme with
+the algebraic critical chart and carries its regular holonomy
+markings to the specified based holonomies.  For each such entry
+\(H\), the additional correction obeys
+\[
+                      \Phi^*x^*H=x^*H\quad\text{in }R/J,
+\]
+because \(\Phi\) is the identity on that quotient.  The algebraic
+lifts constructed after \eqref{eq:algebraic-darboux-chart} consequently
+remain the required lifts after this correction.  This proves the
+marking assertion scheme-theoretically.
+\end{proof}
+
+\subsection{Polynomial replacement of the active logarithm}
+
+Make the fixed coordinate change of the theorem separately on each
+edge, and again write \(f\) for its now algebraic central potential.
+For an active edge choose an algebraic matrix lift \(G_\beta\) of
+the based \(\beta\)-holonomy, with \(G_\beta(0)=1\).  It suffices to
+lift its entries; all identities with the actual holonomy below are
+identities in \(R/\Jac(f)\).  Set
+\[
+ I_\beta=((G_\beta-1)_{ij})\subset R,
+ \qquad
+ L_d(G_\beta)=\sum_{k=1}^{d}\frac{(-1)^{k+1}}{k}(G_\beta-1)^k.
+\]
+With the connection convention \(d+a\) of the edge construction,
+\(G_\beta=\exp(-v)\) on the central critical scheme.  Its active
+coefficient satisfies
+\[
+             b\equiv-\epsilon\operatorname{tr}
+                           (T_0\log G_\beta)\pmod{\Jac(f)}.
+\]
+Define the algebraic germ
+\(b_{\mathrm{alg}}=-\epsilon\operatorname{tr}(T_0 L_d(G_\beta))\).
+For every \(d\geq2\), the \((h)\)-adically convergent logarithm series gives
+\begin{equation}
+\label{eq:polynomial-log-error}
+             b-b_{\mathrm{alg}}\in\Jac(f)+I_\beta^{d+1}
+                                      \subset\Jac(f)+I_\beta^3.
+\end{equation}
+If \(b-b_{\mathrm{alg}}=\sum_i c_i\partial_i f+r\), with
+\(r\in I_\beta^{d+1}\), then the square-zero coordinate change
+\(x_i\mapsto x_i-\tau c_i\) transforms
+\(f+\tau b\) into
+\[
+                         f+\tau(b_{\mathrm{alg}}+r),
+                         \qquad \tau^2=0.
+\]
+Thus the Jacobian part is removed exactly.  The remaining error has
+at least cubic order in the multiplicative holonomy entries; it is
+this explicit error, rather than an algebraicity assertion about the
+formal logarithm, that will be used in the vertex matching.
diff --git a/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/sections/04a-formal-comparison.tex b/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/sections/04a-formal-comparison.tex
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+++ b/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/sections/04a-formal-comparison.tex
@@ -0,0 +1,70 @@
+\subsection{Comparison of the formal trace forms}
+\label{subsec:formal-trace-statement}
+
+The following comparison identifies the full closed form on the cochain
+model, including the relative integration homotopy.  Write
+\(\kappa(u,v)=\operatorname{tr}(uv)\), and denote its \(2\)-shifted
+closed form on \(BG\) by \(\omega_\kappa\).
+For a nonpositive augmented completed semifree coefficient cdga \((R,Q)\), let
+\(\Omega_R^p\) mean continuous coordinate forms, with their internal
+grading.  These compute the exterior powers of the cotangent complex.
+If \(A_X\) is a commutative cochain model of a finite Betti source, put
+\[
+ \operatorname{Cl}^2_R(A_X)^n
+   =\prod_{j\geq0}
+        \bigl(A_X\widehat\otimes\Omega_R^{2+j}\bigr)^{n-j}.
+\]
+Its differential combines \(d_X+\mathcal L_Q\) and coordinate de Rham
+differentiation \(\delta_R\), with the total-complex signs.
+The factors of \(A_X\) have exterior weight zero.
+Coordinate differentiation is extended by
+\(\delta_R(c\otimes u)=(-1)^{|c|}c\otimes\delta_Ru\).
+In particular a strict representative has only its \(j=0\) component.
+Completion in this notation is in the coefficient augmentation ideal;
+the displayed product is the separate completion in exterior weight.
+
+\Needspace{10\baselineskip}
+\begin{proposition}[Formal trace comparison]
+\label{prop:formal-trace-comparison}
+Let \(X\) be a finite triangulated space, and let
+\[
+ a_X\in
+  \bigl(A_X\otimes\mathfrak g\widehat\otimes\mathfrak m_R\bigr)^1,
+ \qquad
+ (d_X+Q)a_X+\tfrac12[a_X,a_X]=0,
+\]
+represent a family of local systems formally near the trivial one.
+For its evaluation map
+\(\operatorname{ev}_{a_X}:X_B\times\Spf R\to BG\), the image of
+\(\operatorname{ev}_{a_X}^*\omega_\kappa\) under the Betti
+coefficient map of \cite[\S2.1]{PTVV2013} is the class of
+\begin{equation}
+\label{eq:formal-trace-representative}
+ \Theta_X=\tfrac12\kappa(\delta_Ra_X,\delta_Ra_X),
+ \qquad
+ (d_X+\mathcal L_Q)\Theta_X=0,\qquad \delta_R\Theta_X=0,
+\end{equation}
+in the full complex \(\operatorname{Cl}^2_R(A_X)\).
+This identification is natural for source restrictions, derived
+coefficient changes, and integration of source cochains.
+
+For an oriented compact three-manifold \(Z\) with boundary \(S\), it
+also identifies the relative-orientation isotropic homotopy with
+integration of \(\Theta_Z\) over \(Z\).  If \(S=S^2\) and the sphere
+condition is represented by \(a_S=\nu b\), where \(\nu\) is a
+two-form and \(b\) has internal degree \(-1\), then
+\(\Theta_S=0\) strictly.  Using the zero isotropic path on this sphere
+model, the resulting intersection form is represented by
+\[
+                 \tfrac12\int_Z
+                     \kappa(\delta_Ra_Z,\delta_Ra_Z)
+\]
+as a closed two-form of degree \(-1\).
+\end{proposition}
+
+The proof is given in Appendix~\ref{app:formal-trace-comparison}.
+It constructs the comparison on the formal group nerve, extends it to
+derived coefficient algebras, and checks source evaluation and relative
+integration in the full closed-form complex.  Throughout the comparison,
+the marked holonomies are the forward transports fixed in
+Equation~\eqref{eq:edge-holonomy-sign}.
diff --git a/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/sections/05-vertices.tex b/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/sections/05-vertices.tex
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+++ b/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/sections/05-vertices.tex
@@ -0,0 +1,873 @@
+\section{Matching the edge potentials at the vertices}
+\label{sec:vertices}
+
+We now pass from the cut manifolds to the exact algebraic data of
+Theorem~\ref{thm:tensor-obstruction}.  All residual ideals in this section are
+ideals of formal rings, without passage to their radicals.
+
+\begin{proposition}[Vertex data]
+\label{prop:vertex-data}
+Suppose that the data of Proposition~\ref{prop:geometric-reduction} exist,
+with $m\geq 5$, $p\geq9$ extra letters, and the finite family of
+independently conjugated boundary laws described there.  Their Brunnian
+property means that omitting any slot makes the law the identity, as in
+Equation~\eqref{eq:grope-word-properties}.
+There are a characteristic-zero field $K$, internal coordinate blocks $h_i$
+for the $2m$ edges, and algebraic power-series germs $F_i(h_i)$ and $b_i(h_i)$,
+the latter for the $m$ active edges, with the following properties.
+Put
+\[
+ T=K[t_i:i\text{ active}]/(t_i^2:i\text{ active}),\qquad
+ S_0=\sum_i F_i,
+ \qquad S_t=S_0+\sum_{i\text{ active}}t_i b_i.
+\]
+There are formal parametrizations $H_L(x,t)$ over $T$ and $H_R(y)$ over $K$
+whose central tangent maps are injective and have complementary images in
+the full internal coordinate space.  They are smooth graph germs over $T$
+and $K$, respectively, and
+\begin{equation}
+\label{eq:vertex-final-vanishing}
+ S_t(H_L)=0,\qquad S_0(H_R)=0.
+\end{equation}
+For every three distinct active indices $i,j,k$,
+\begin{align}
+ t_i t_j t_k
+ &\in\left((\partial_{h_a}S_t)(H_L):a=1,\ldots,N\right)
+       \subset T[[x]],                                      \label{eq:vertex-final-source}\\
+ (b_i b_j b_k)(H_R)
+ &\in\left((\partial_{h_a}S_0)(H_R):a=1,\ldots,N\right)
+       \subset K[[y]],                                      \label{eq:vertex-final-target}
+\end{align}
+where $N$ is the total number of internal coordinates.  The field can be
+taken to be $K=\mathbb C((\sigma))$.  The individual functions can be
+normalized so that $F_i(0)=b_i(0)=0$.
+\end{proposition}
+
+We first construct formal sections from the two vertex connection spaces.
+Their curvature ideals will remain fixed as we normalize the boundary
+restrictions and correct the sections to make the sums of potentials
+vanish.  We then pass to algebraic edge coordinates and use the boundary
+laws to obtain the triple memberships.  The final step replaces the
+logarithmic perturbations by algebraic functions and corrects the source
+section once more.
+
+Orient each $Y_i$ as a boundary face of $A$.  Its orientation as a face
+of $B$ is the opposite one.  Write
+$\alpha_i$ for its surviving peripheral direction and $\beta_i$ for the
+other direction.  The plus endpoint is the one at which $\beta_i$ bounds
+in the free boundary piece; the minus endpoint is the one at which
+$\alpha_i$ bounds.  An edge is \emph{active} when its plus endpoint is at
+$A$.  There are $m$ active edges.
+
+Lemma~\ref{lem:connected-tree-cuts} gives smooth cuts that miss the locally
+flat disks.  Their complementary pieces are smooth manifolds with corners,
+so ordinary smooth forms and Stokes' formula apply, with matching
+pullbacks on their smooth faces at corners.
+
+\subsection{The linear restriction maps}
+
+Put $\g=\mathfrak{sl}_2(\mathbb C)$ and use the invariant pairing
+$\kappa(U,V)=\tr(UV)$.  The internal edge space is
+\[
+ \mathcal H=\bigoplus_i
+ H^1(Y_i,\partial Y_i;\mathbb C)\otimes\g.
+\]
+For a connected oriented three-manifold with connected torus boundary,
+the natural map from relative degree-one cohomology identifies it with
+the kernel of restriction to the boundary.  Thus these are precisely the
+internal spaces in Proposition~\ref{prop:edge-potential}.
+
+At this linear stage, fixing an evaluation means requiring its tangent
+variation to vanish.  For \(P=A,B\), let \(\mathcal V_P\) be the common
+kernel in \(H^1(P;\mathbb C)\otimes\g\) of evaluation on the extra
+letters at \(P\) and on the \(\alpha_i\) whose plus endpoint is \(P\).
+The degree-one assertion below is the isomorphism
+\(\mathcal V_A\oplus\mathcal V_B\to\mathcal H\).  Later prescribed
+holonomies need not be the identity; it is their variations that are
+fixed in this tangent-space statement.
+
+\begin{lemma}
+\label{lem:vertex-linear}
+After fixing the evaluations on the extra letters and on all plus
+meridians, the direct sum of the two vertex degree-one spaces maps
+isomorphically, by difference of restrictions, to $\mathcal H$.
+Moreover,
+\begin{equation}
+\label{eq:vertex-h2-isomorphism}
+ H^2(A;\mathbb C)\oplus H^2(B;\mathbb C)
+       \longrightarrow\bigoplus_i H^2(Y_i;\mathbb C)
+\end{equation}
+is an isomorphism.  In particular each vertex summand maps injectively.
+\end{lemma}
+
+\begin{proof}
+Lemma~\ref{lem:geometric-edge-cohomology} gives
+Equation~\eqref{eq:vertex-h2-isomorphism} and the degree-one restriction
+isomorphism after the extra evaluations are fixed.
+
+For each edge the image of $H^1(Y_i)$ in the cohomology of its boundary
+torus is a line.  Restrictions from the two vertex sides span $H^1(Y_i)$,
+and their torus restrictions lie on the two coordinate axes, because the
+two respective longitudes bound in the boundary pieces.  This line must
+be one coordinate axis.  Evaluation on its surviving loop $\alpha_i$
+comes only from the plus vertex.  Consequently, after the extra
+evaluations have been fixed, the inverse image of
+$\bigoplus_i H^1(Y_i,\partial Y_i)$ is exactly the subspace on which all
+plus evaluations vanish.  The asserted restricted map is therefore an
+isomorphism.  Tensoring these statements with $\g$ proves the result.
+\end{proof}
+
+The pairing
+\begin{equation}
+\label{eq:vertex-perfect-edge-pairing}
+ \bigoplus_i H^1(Y_i,\partial Y_i)\otimes\g
+ \ \times\ \bigoplus_i H^2(Y_i)\otimes\g
+ \longrightarrow\mathbb C,
+ \qquad (u,v)\longmapsto\sum_i\int_{Y_i}\kappa(u\wedge v)
+\end{equation}
+is perfect by Poincar\'e--Lefschetz duality.  We will use both the
+degree-one isomorphism and this degree-two pairing.
+
+\subsection{Marked formal backgrounds and vertex slices}
+
+The boundary laws will be tested at a finite family of conjugating
+matrices whose adjoint operators span every endomorphism of $\g$.
+Introduce a formal parameter $\sigma$ and set $K=\mathbb C((\sigma))$.
+Choose nine matrices in $SL_2(\mathbb C[[\sigma]])$, one the identity,
+all equal to the identity at $\sigma=0$, whose adjoint operators span
+$\End_K(\g\otimes K)$. Equation~\eqref{eq:vertex-explicit-background}
+gives an explicit family, and Lemma~\ref{lem:vertex-adjoint-span} verifies
+its spanning property. The family is fixed independently of the cuts.
+Its independent actions on the three slots will detect every triple
+coefficient in the boundary laws. Assign the
+eight nonidentity values to eight extra letters at each vertex and set
+the remaining extra holonomies to the identity.  Write
+$g_s(\sigma)\in SL_2(\mathbb C[[\sigma]])$, with $g_s(0)=1$, for these
+assigned holonomies.  These choices extend to a flat background
+on the whole ambient bipartite model with trivial meridian holonomies.
+To see the marking explicitly, choose the paths through the joins as
+edge paths of the free graph groupoid.  The extra loops are its vertex
+generators, and edge transports can be chosen freely.  Choose those
+transports so that the two torus frames at every join agree with the
+frames determined by the boundary basing paths.  This simultaneously
+fixes the markings used for holonomy and for the boundary laws.
+
+Initially all constructions take place jointly formally at
+$\sigma=0$.  The restricted flat backgrounds have coordinates of positive
+$\sigma$ order in the central vertex and edge charts.  We do not yet
+invert $\sigma$.
+
+Fix $P=A$ or $P=B$, with its basepoint in $K_P$.  In degree zero use smooth
+$\g$-valued functions vanishing at that point, and in positive degrees
+use unrestricted forms on $P$.  Choose a contraction onto cohomology,
+denoted by inclusion $\iota$, projection $\pi$ and homotopy $\mathsf K_P$, with
+\[
+ \dd\mathsf K_P+\mathsf K_P\dd=1-\iota\pi,
+ \qquad \mathsf K_P^2=\mathsf K_P\iota=\pi\mathsf K_P=0.
+\]
+Degree-zero cohomology is zero.  For
+$y_{\rm full}\in H^1(P;\mathbb C)\otimes\g$, the recursion
+\[
+ a_P=\iota y_{\rm full}-\tfrac12\mathsf K_P[a_P,a_P]
+\]
+defines a unique formal one-form.  Write
+\[
+ \mathcal F_P=\dd a_P+\tfrac12[a_P,a_P],\qquad r_P=\pi\mathcal F_P.
+\]
+The list $r_P$ has $q_P=\dim(H^2(P;\mathbb C)\otimes\g)$ entries, of
+order at least two.  The slice equation and Bianchi identity give
+\begin{equation}
+\label{eq:vertex-residual-curvature}
+ \mathcal F_P=\iota r_P-\mathsf K_P[a_P,\mathcal F_P]
+       =(1+\mathsf K_P\ad_{a_P})^{-1}\iota r_P.
+\end{equation}
+In particular the curvature ideal is exactly the ideal generated by
+$r_P$, and its leading coefficient forms are the selected representatives
+of $H^2(P)\otimes\g$.  Successively removing exact one-form coordinates
+by based gauges shows that every based flat formal connection is
+represented by this slice modulo $r_P$.  These constructions are
+functorial over nilpotent coefficient rings and pass to their inverse
+limits.  No regular-sequence assertion about $r_P$ is made.
+
+Fix nonzero $T_i\in\g$ for the active edges.  We use the connection
+convention of Proposition~\ref{prop:edge-potential}, in which the strip
+connection $u\zeta$ has holonomy $\exp(-u)$.  Impose on the vertex slice
+the exact logarithmic holonomies
+\begin{itemize}
+\item $\log g_s(\sigma)$ on the extra letters;
+\item $-t_iT_i$ on an active plus meridian at $A$;
+\item zero on every plus meridian at $B$.
+\end{itemize}
+There are no parameters on the other plus ends.  Logarithmic holonomy
+along a fixed based loop is a formal function of the off-shell
+connection, whose linear term is minus integration of the connection along
+that loop.  Lemma~\ref{lem:vertex-linear} therefore makes these conditions
+a smooth formal parameter slice.  Keep its other coordinates $y_P$
+free.  The resulting coefficient rings are
+\[
+ R_A=\mathbb C[[\sigma,y_A]][t_i:i\text{ active}]/(t_i^2),
+ \qquad R_B=\mathbb C[[\sigma,y_B]],
+\]
+completed at their displayed joint origins.  Write $I_P=(r_P)$ for the
+substituted residual ideal.  It is contained in the square of the joint
+maximal ideal.  Modulo $I_P$ there is an actual based flat system with
+exactly the prescribed extra and plus holonomies.  We keep this list
+$r_P$ as functions on the parameter domain throughout the subsequent
+normalizations and section corrections.  Thus the flat quotient
+$R_P/I_P$ stays fixed even when its chosen lifts to forms and edge
+coordinates change.
+
+For an active edge, the induced holonomies in the flat quotient
+$R_A/I_A$ satisfy
+\[
+ G_{\alpha_i}=\exp(-t_iT_i),\qquad G_{\beta_i}=1.
+\]
+In the flat quotient $R_B/I_B$ they satisfy $G_{\alpha_i}=1$, while the
+marked $G_{\beta_i}$ varies.  The central edge potential there contributes
+$-f_i$ because the boundary orientation is reversed, where $f_i$ is the
+central potential of Proposition~\ref{prop:edge-potential}.  These
+holonomy identities concern the respective flat quotients, not the
+off-shell lifts.
+
+\subsection{Boundary normalization and lifts of the flat quotient}
+
+The quotient $R_P/I_P$ carries the flat systems and their group laws.
+The tensor theorem, however, requires sections in the ambient edge
+coordinates, where the curvature need not vanish.  We first normalize
+the restrictions on the flat quotient and then lift them to these
+ambient, or off-shell, coordinates.  Changes in the lifts will be
+measured against the fixed list $r_P$.
+
+Recall that $K_P$ is a connected sum of $p$ copies of $S^1\times S^2$
+with tubes about an unlink removed.  In the unlink exterior choose
+disjoint proper disks dual to the port meridians, with boundaries the
+unlink longitudes.  In the connected-sum pieces choose disjoint
+nonseparating spheres dual to the extra $S^1$ generators.  Thin mutually
+disjoint collars of these disks and spheres carry closed Thom
+one-forms $\rho_\nu$, normalized to evaluate to one on the corresponding
+generator.  The disk strips meet the port tori in the coordinate
+densities dual to the port meridians; the sphere strips miss those tori.
+Choose the basing arcs off the strips.  Since the port meridians and
+extra letters freely generate $\pi_1K_P$, a flat system with their
+specified holonomies has the representative
+\[
+ a_{K_P}=\sum_\nu M_\nu\rho_\nu,
+\]
+where the $M_\nu$ are the corresponding signed logarithms.  Each strip
+form is closed and their cross wedges vanish, so this connection is
+flat for arbitrary matrix coefficients.
+At a plus torus its trace is $u_i\zeta_i$, and at a minus torus its trace
+is a multiple of $\xi_i$.  The two strip choices are the compatible
+torus choices of Proposition~\ref{prop:edge-potential}.  The area
+representatives can be supported away from both strips.  In particular,
+\begin{equation}
+\label{eq:vertex-k-cs-zero}
+ \mathcal C_{K_P}(a_{K_P})=0.
+\end{equation}
+
+Normalize first over $R_P/I_P$.  Based parallel transport gives a gauge
+from the restriction of $a_P$ to this strip representative.  After this gauge,
+the flat restriction on every $Y_i$ can be put into the edge slice
+$s_i(h_i,t_i)$, or $s_i(h_i,0)$ as appropriate, by a gauge equal to the
+identity on its torus.  This gives coordinates $h_i^P(y_P)$ over
+$R_P/I_P$.  The normalizations are compatible on intersections of
+boundary faces: the second gauges are the identity on the tori.
+
+To lift these quotient-ring data to actual formal forms, choose a
+power-series presentation of the Noetherian complete ring $R_P$ and
+fix continuous linear division operators
+for the finite list $r_P$ and for passage to $R_P/I_P$.  Such operators
+are obtained by formal division, or by choosing compatible filtered
+linear splittings.  Applied coefficient by coefficient, they lift
+scalar series and write a form-valued discrepancy in $I_P$ as
+$\sum_a r_{P,a}\gamma_a$ with form-valued series $\gamma_a$.
+Each output coefficient involves only finitely many input coefficients.
+Thus the same operators apply to the vector spaces of smooth functions
+and differential forms.  This is the actual finite-list ideal
+membership used below.
+
+Lift the near-identity gauges by first lifting their Lie-algebra-valued
+logarithms.  Smooth collar extension extends each logarithm off the
+relevant boundary face; the compatibility at the tori allows the
+extensions to be combined.  Lift the coordinates $h_i^P$ as well.
+Finally change the connection by $I_P$-multiples of one-forms, extended
+over a collar of $K_P$, so that its restriction to $K_P$ is
+\emph{exactly} the strip representative with the constrained plus
+coefficients.  Denote this trial connection by $\widehat a_P$.  It has
+the following properties:
+\begin{enumerate}
+\item its curvature belongs to $I_P$;
+\item its restriction to $Y_i$ equals the specified edge slice modulo
+      $I_P$;
+\item to first order in the actual residual list, modulo the joint
+      maximal ideal times that list, its curvature representatives are
+      those in Equation~\eqref{eq:vertex-residual-curvature}, changed only by
+      exact two-forms.
+\end{enumerate}
+For the last statement, write an added one-form as
+$\eta=\sum_a r_{P,a}\gamma_a$ and put
+$\mathcal F(a)=\dd a+\tfrac12[a,a]$ for an intermediate connection $a$.
+The source differential treats the
+functions $r_{P,a}$ of the formal parameters as constants.  Hence
+\[
+\begin{aligned}
+ \mathcal F(a+\eta)
+ &=\mathcal F(a)+\sum_a r_{P,a}\,\dd\gamma_a
+                  +[a,\eta]+\tfrac12[\eta,\eta]\\
+ &\equiv\mathcal F(a)+\sum_a r_{P,a}\,\dd\gamma_a
+                  \pmod{\mathfrak m_P I_P}.
+\end{aligned}
+\]
+The congruence is coefficientwise, and $\mathfrak m_P$ is the joint
+maximal ideal: $a$ has positive
+order and $I_P\subset\mathfrak m_P^2$.  A gauge with identity value
+at the joint origin also changes curvature only modulo
+$\mathfrak m_PI_P$.  Thus the leading coefficient forms of the fixed
+residual list change by exact two-forms.  This calculation uses a
+chosen expression for $\eta$ and requires no independence of the
+$r_{P,a}$.  The lifts have the same first derivatives as the original
+quotient coordinates because $I_P$ has order at least two.
+
+\subsection{The square-ideal and gradient identities}
+
+We have lifted the flat boundary restrictions to ambient edge
+coordinates without changing them modulo the fixed residual ideal.
+The next calculation measures the error in the sum of their
+potentials and identifies the restricted gradient ideal. These two
+statements will permit correction of the sections while preserving
+their flat-system markings.
+
+Let $f_i$ and $b_i^{\rm form}$ be the edge functions before algebraizing
+the perturbations.  With the orientations already chosen, put
+\[
+ S_A=\sum_i f_i+\sum_{i\text{ active}}t_i b_i^{\rm form},
+ \qquad S_B=-\sum_i f_i.
+\]
+Write $H_P$ for the lifted parametrization consisting of all $h_i^P$.
+
+\begin{lemma}
+\label{lem:vertex-offshell-identities}
+At the joint origin the tangent images of $H_A$ and $H_B$, with the
+external parameters fixed, are complementary.  For each vertex,
+\begin{equation}
+\label{eq:vertex-square-and-gradient}
+ S_P(H_P)\in I_P^2,
+ \qquad (\partial_h S_P)(H_P)=M_P r_P,
+\end{equation}
+where $M_P$ is an actual coefficient matrix whose reduction at the
+joint origin has full column rank $q_P$.  Consequently the restricted
+gradient ideal equals $I_P$.
+\end{lemma}
+
+\begin{proof}
+The assertion about tangent images is Lemma~\ref{lem:vertex-linear}:
+the linearization of restriction to the edge slices is the restriction
+map into the relative degree-one edge spaces.  Thus each parametrization
+is an immersion and can be written as a graph after a linear projection.
+
+For the first identity, Stokes' formula gives
+\[
+ \mathcal C_{\partial P}(\widehat a_P|_{\partial P})
+       =\tfrac12\int_P\tr(\mathcal F_{\widehat a_P}
+                              \wedge\mathcal F_{\widehat a_P})
+       \in I_P^2.
+\]
+The $K_P$ contribution is zero by Equation~\eqref{eq:vertex-k-cs-zero}.  On an
+edge interpolate between the trial restriction and its chosen slice.
+Their difference is in $I_P$, and every curvature along the interpolation
+is in $I_P$.  The interior variation of the action is therefore in
+$I_P^2$.  On a minus end both boundary traces lie in the $\xi_i$
+polarization, so the boundary variation vanishes.  The same holds at a
+frozen plus end, where $u_i=0$.  At an active plus end the trial trace
+is $u_i\zeta_i$, the slice trace is $u_i\zeta_i+v_i\xi_i$, and the
+boundary variation is
+\[
+ -\tfrac{\epsilon_i}{2}\tr(u_i v_i).
+\]
+It is canceled exactly by the adjustment in
+Proposition~\ref{prop:edge-potential}.  This proves $S_P(H_P)\in I_P^2$
+with an actual quadratic expression in the residual list.
+
+The fixed-$u$ variation formula in Equation~\eqref{eq:edge-fixed-u-gradient} expresses the
+internal gradient entries as integrations of the slice curvatures
+against derivatives of the slices.  Since those curvatures vanish
+modulo $I_P$, all entries belong to $I_P$.  To determine a coefficient
+matrix, start with Equation~\eqref{eq:vertex-residual-curvature} and the explicit
+$I_P$-multiple differences just constructed.  At the joint origin, the
+coefficient of $r_{P,a}$ in the $j$th gradient entry is, with its
+boundary-orientation sign,
+\begin{equation}
+\label{eq:vertex-leading-matrix}
+ \int_{Y_i}\kappa\bigl(\theta_{i,j}\wedge
+                     \omega_{P,a}|_{Y_i}\bigr).
+\end{equation}
+Here $\theta_{i,j}$ is the closed relative one-form representing the
+$j$th edge tangent and $\omega_{P,a}$ is the chosen closed two-form
+representative of the residual coordinate.  Added leading exact
+two-forms do not change this number: integrate by parts and use the
+zero boundary pullback of $\theta_{i,j}$.  By
+Equation~\eqref{eq:vertex-h2-isomorphism} and the perfect pairing in
+Equation~\eqref{eq:vertex-perfect-edge-pairing}, the matrix in
+Equation~\eqref{eq:vertex-leading-matrix} has full column rank.
+
+Choose such a maximal minor.  It is a unit in $R_P$, so the corresponding
+gradient entries recover the list $r_P$ by its inverse.  This proves
+equality of ideals.  We used the matrix obtained from the curvature
+expansion, not an inference that the residuals are independent modulo
+some ideal.  Possible syzygies among them have no effect on the argument.
+\end{proof}
+
+We record the formal correction argument used at the joint origin.
+
+\begin{lemma}[Corrections with a fixed residual list]
+\label{lem:vertex-correction}
+Let $R$ be a complete Noetherian local $\mathbb Q$-algebra with maximal
+ideal $\mathfrak m_R$, let $I=(r_1,\ldots,r_q)$, let
+$S\in R[[h_1,\ldots,h_N]]$, and let $H\in\mathfrak m_R^N$.
+Internal derivatives are taken over $R$, and substitution at $H$ is
+$\mathfrak m_R$-adically continuous.  Suppose
+$S(H)\in I^2$ and $(\partial_h S)(H)=Mr$, with a unit full-column minor
+in $M$.  Assume also that $I\subset\mathfrak m_R^2$ and the internal
+Hessian of $S$ evaluated at $H$ has entries in $\mathfrak m_R$.
+Then there is a convergent correction of $H$ by $I$-multiples making
+$S(H)=0$.  It is unchanged modulo $I$ and to first order at the origin;
+the gradient still generates $I$.  The functions $r_a$ on the parameter
+domain are held fixed throughout the correction.
+\end{lemma}
+
+\begin{proof}
+If the current error is $E=\sum_{a,b}c_{ab}r_ar_b$, use the inverse
+minor of $M$ to choose a vector $\delta H$ of $I$-multiples with
+$(\partial_h S)(H)\cdot\delta H=-E$.  More generally, if the coefficients
+$c_{ab}$ lie in an ideal $Q$, choose $\delta H\in QI$.  This is a linear
+calculation using the displayed list and its matrix; it does not require
+unique expressions for elements of $I$ or $I^2$.
+
+An error in $\mathfrak m_R^n I^2$ gives a
+correction in $\mathfrak m_R^n I$.  The quadratic Taylor term is
+$\frac12\delta H^{\mathsf T}(\operatorname{Hess}S)(H)\delta H$;
+the factor $1/2$ and all higher Taylor coefficients are defined because
+$R$ is a $\mathbb Q$-algebra.  This term belongs to
+$\mathfrak m_R^{2n+1}I^2$ by the Hessian hypothesis, and higher terms
+belong to the same ideal because $I\subset\mathfrak m_R^2$.
+At the $k$th step write
+$\delta H_k=D_kr$, with every entry of $D_k$ in
+$\mathfrak m_R^{n_k}$, where $n_0=0$ and $n_{k+1}=2n_k+1$.
+The Hessian remains in $\mathfrak m_R$, so the gradient coefficient
+matrix changes by a matrix in $\mathfrak m_R^{n_k+1}$.
+Thus both $\sum_kD_k$ and the sequence of gradient coefficient matrices
+converge.  The total correction is $(\sum_kD_k)r\in I$, and the limiting
+gradient coefficient matrix has the same reduction as the original
+one, hence retains the chosen unit full-column minor.  Continuity now
+gives $S(H_\infty)=0$ and $(\partial_h S)(H_\infty)=M_\infty r$.  The
+assumption $I\subset\mathfrak m_R^2$ preserves the tangent map.
+
+\end{proof}
+
+Apply Lemma~\ref{lem:vertex-correction} to the identities of
+Lemma~\ref{lem:vertex-offshell-identities}.
+The central edge functions have order at least three; the other terms
+carry a parameter $t_i$.  Hence the internal Hessian vanishes at the
+joint origin.  We obtain corrected sections, still denoted by $H_A,H_B$,
+with
+\begin{equation}
+\label{eq:vertex-exact-formal-sections}
+ S_A(H_A)=S_B(H_B)=0,
+ \qquad ((\partial_h S_P)(H_P))=I_P.
+\end{equation}
+They retain their complementary tangents and their values modulo $I_P$.
+In particular they retain exactly the marked edge holonomies on the
+flat residual schemes.
+
+\subsection{Algebraic central coordinates and the order of completion}
+\label{rem:tensor-recentering}
+
+The corrected sections already satisfy exact vanishing and retain the
+marked holonomies modulo their residual ideals. We now make the
+central edge potentials algebraic and move the common background to
+the origin. The order of this recentering and the subsequent passage
+to Laurent coefficients is part of the construction.
+
+Apply Theorem~\ref{thm:edge-algebraicity} separately on the edges,
+initially at the trivial system.  Thus, after separate invertible formal
+changes of the $h_i$, the central functions are algebraic germs $F_i$.
+On their critical schemes the marked $\beta_i$ holonomies have algebraic
+lifts $G_i$ in the smooth ambient germs.  Choose these lifts with
+$G_i(0)=1$.  The functions $b_i^{\rm form}$ satisfy
+\begin{equation}
+\label{eq:vertex-log-mod-jacobian}
+ b_i^{\rm form}\equiv -\epsilon_i\tr(T_i\log G_i)
+                                      \pmod{\Jac(F_i)}.
+\end{equation}
+Here the sign is the one fixed by $\operatorname{Hol}_{\beta_i}=\exp(-v_i)$.
+Write the Jacobian-ideal difference as
+$\sum_a c_{i,a}\partial_{h_{i,a}}F_i$.  A separate coordinate change
+linear in $t_i$ absorbs it: the identity
+\[
+ F_i(h_i+t_i c_i(h_i))
+     =F_i(h_i)+t_i\sum_a c_{i,a}(h_i)\partial_{h_{i,a}}F_i(h_i)
+\]
+is exact because $t_i^2=0$.  Choosing the sign according to the direction
+of the change replaces the coefficient by
+\[
+ \ell_i(G_i)=-\epsilon_i\tr(T_i\log G_i).
+\]
+The changes are separate in the edge blocks and preserve all equations
+and ideal memberships on the transformed sections.  At parameter zero
+the changes linear in $t_i$ are the identity.
+
+Let $y_P^0(\sigma)$ and $h_i^0(\sigma)$ be the coordinates of the fixed
+flat background in these charts.  They have positive $\sigma$ order.
+Both vertex sections contain the same edge background, because all
+earlier corrections vanish on their residual schemes.  Recenter by
+$y_P=y_P^0(\sigma)+z_P$ and $h_i=h_i^0(\sigma)+z_i$ \emph{before}
+passing to Laurent coefficients.  Substitution gives continuous maps
+\[
+ \mathbb C[[\sigma,y_P]]\longrightarrow\mathbb C[[\sigma,z_P]],
+ \qquad
+ \mathbb C[[\sigma,h_i]]\longrightarrow\mathbb C[[\sigma,z_i]].
+\]
+For each coefficient in the new internal variables, the defining sum
+converges $\sigma$-adically.  We then include the coefficient ring in
+$K=\mathbb C((\sigma))$ and complete in the new internal variables.
+We rename them $y_P,h_i$.
+
+This order also describes the changes already used: if $\phi$ is a
+separate original formal coordinate change, its recentered expression is
+\[
+ z\longmapsto\phi(h_i^0(\sigma)+z)-\phi(h_i^0(\sigma)),
+\]
+formed over $\mathbb C[[\sigma,z]]$ first.  Later inverses over $K$ may
+have coefficients with unbounded negative $\sigma$ orders.  They are
+only composed with centered series, or with series whose constant terms
+are nilpotent parameter multiples; these substitutions are well defined
+in $K[[z]][t_i]/(t_i^2)$.  In particular, an absorbed Jacobian term gives
+a separate square-zero change
+\[
+ z_i\longmapsto z_i+t_i c_i(z_i),\qquad
+ z_i\longmapsto z_i-t_i c_i(z_i)
+ \quad\text{for its inverse}.
+\]
+Its constant displacement is nilpotent.  No later substitution shifts
+an arbitrary Laurent-coefficient series by a nonzero
+$\sigma$-dependent constant, and no operation introduces relations
+among the external parameters.  Artin approximation is applied only
+after the final graph identities have been obtained, over the field
+$K$ as in Lemma~\ref{lem:tensor-reduction}.
+
+The central functions and the $G_i$ are still algebraic germs over $K$:
+substitute the background values, which are elements of $K$, into their
+finite algebraic presentations.  The selected background is critical
+on every central edge and $G_i(h_i^0)=1$.  Moreover $F_i(h_i^0)=0$.
+Indeed $\partial F_i$ vanishes along the formal background curve, so its
+derivative with respect to $\sigma$ is zero; its value at $\sigma=0$
+was zero.  We retain these normalizations after recentering.
+
+The complementary tangent determinant and the gradient minors have
+nonzero constant terms at the original joint origin.  Their evaluations
+at the background are units of $\mathbb C[[\sigma]]$, and hence are
+nonzero over $K$.  Thus the two recentered sections are still transverse
+with complementary dimensions, and their gradient ideals still equal
+the recentered residual ideals.  The latter may now have linear terms;
+no later argument requires their original quadratic order.
+
+\subsection{Conjugated laws on the full residual schemes}
+
+The prescribed extra holonomies can be chosen explicitly.  In the basis
+$(e,h,f)$ of $\mathfrak{sl}_2$, put
+\[
+ E=\ad(e)=\begin{pmatrix}0&-2&0\\0&0&1\\0&0&0\end{pmatrix},
+ \qquad
+ F=\ad(f)=\begin{pmatrix}0&0&0\\-1&0&0\\0&2&0\end{pmatrix}.
+\]
+For $a,b\in\{0,1,2\}$ prescribe
+\begin{equation}
+\label{eq:vertex-explicit-background}
+ g_{ab}(\sigma)=\exp(a\sigma e)\exp(b\sigma f).
+\end{equation}
+The identity conjugator supplies $g_{00}$ and eight extra letters supply
+the other eight matrices.  These choices can be made at both vertices.
+They require only a finite list of conjugations in the geometric
+construction, chosen before the cuts are made.
+
+\begin{lemma}
+\label{lem:vertex-adjoint-span}
+The nine matrices $\Ad(g_{ab}(\sigma))$ span $\End_K(\g\otimes K)$.
+\end{lemma}
+
+\begin{proof}
+Since $E^3=F^3=0$, the matrices are
+\[
+ (1+a\sigma E+(a\sigma)^2E^2/2)
+ (1+b\sigma F+(b\sigma)^2F^2/2).
+\]
+Order $(a,b)$ lexicographically and vectorize matrices row by row.  The
+determinant of the resulting nine columns is
+\[
+ 512\sigma^{18}.
+\]
+For a direct calculation, expand in the nine ordered matrices
+$E^pF^q/(p!q!)$, $0\leq p,q\leq2$.  Their vectorization determinant is
+$8$.  The two evaluation matrices on $0,1,2$ are Vandermonde matrices
+of determinant $2$, contributing $2^6$ to the tensor determinant.  The
+sum of the exponents $p+q$ is $18$.  The displayed determinant is
+nonzero in $K$.
+\end{proof}
+
+\begin{lemma}
+\label{lem:vertex-triple-ideals}
+For any three distinct active edges, the product of their three source
+parameters belongs to $I_A$.  At $B$, every product of one matrix entry
+from each of their three logarithmic $\beta$ holonomies belongs to
+$I_B$.  The same conclusion holds with entries of $G_i-1$ in place of
+the logarithmic holonomies.
+\end{lemma}
+
+\begin{proof}
+Work in the recentered complete local ring over $K$ of one section,
+again denoted by $R_P$, retaining the full square-free parameter
+algebra at $A$.  Choose off-shell lifts
+$X_i\in\g\otimes R_P$ of its marked meridian logarithms on
+$R_P/I_P$, and write $\overline X_i$ for their quotient images.
+At $A$ take $X_i=-t_iT_i$; at $B$ take the logarithmic holonomy of
+the original vertex slice along $\beta_i$.  Each $X_i$ vanishes at
+the background.  For three specified edges let $J_{ijk}$ be the ideal
+in $R_P$ generated by all coordinate products containing one entry
+from each of $X_i,X_j,X_k$.
+
+Evaluate the marked boundary words at $\exp(X_i),\exp(X_j),\exp(X_k)$
+and the fixed extra holonomies.  Modulo $I_P$ these are the actual
+based holonomies of the flat vertex system, so the boundary laws of
+Proposition~\ref{prop:geometric-reduction} make every scalar entry of a
+law minus the identity lie in $I_P$.  The same entries lie in $J_{ijk}$:
+the words are Brunnian in their three core meridians, so omitting any
+one of the three makes the word the identity.  This latter assertion
+is first an identity for independent formal matrix logarithms and is
+then substituted into $R_P$; it does not require the chosen off-shell
+connection to be flat.
+
+Their cubic terms, up to the fixed signs and an invertible output
+conjugation, are the scalar tests of
+\begin{equation}
+\label{eq:vertex-cubic-laws}
+ [\Ad(g_a)X_i,[\Ad(g_b)X_j,\Ad(g_c)X_k]],
+\end{equation}
+with the three conjugators chosen independently.  The additional
+core-word conjugations allowed by the geometric construction equal
+the identity when the core logarithms vanish, and only change higher
+terms.  In particular, their presence does not assert or require that
+the globally based word is literally a fixed nested commutator.
+
+By Lemma~\ref{lem:vertex-adjoint-span}, each input of the nonzero
+trilinear double bracket can independently be acted on by the full
+endomorphism algebra.  To see the spanning assertion directly, choose
+a nonzero coefficient of this trilinear tensor and use rank-one
+endomorphisms in its three inputs to select any three desired input
+coordinates; choose an output functional detecting that coefficient.
+Every scalar triple coordinate product is therefore a linear
+combination of the cubic expressions in Equation~\eqref{eq:vertex-cubic-laws}.
+
+All higher terms of a Brunnian law lie in $\mathfrak m J_{ijk}$, where
+$\mathfrak m$ is the recentered maximal ideal, including the external
+parameters.  If $Q$ is the ideal generated by the scalar law entries,
+we have
+\[
+ Q\subset J_{ijk}\cap I_P,
+ \qquad J_{ijk}=Q+\mathfrak m J_{ijk}.
+\]
+The module $J_{ijk}/Q$ is finitely generated over the complete
+Noetherian local ring, so Nakayama's lemma gives $J_{ijk}=Q\subset I_P$.
+This proves the assertion scheme-theoretically, including all
+nilpotents and any syzygies in $r_P$.
+
+At $A$ the quotient logarithms satisfy $\overline X_i=-t_iT_i$.
+Choose a nonzero coordinate of each $T_i$ to deduce
+$t_it_jt_k\in I_A$.  At $B$ use the corresponding $\beta$ logarithms.
+On $B$'s flat quotient the image of $G_i(H_B)$ is
+$\overline G_i=\exp(\overline X_i)$.  Every entry of
+$\overline G_i-1$ is therefore in the ideal generated by the entries
+of $\overline X_i$.  The triple products of the lifted entries
+$G_i-1$ consequently belong to $I_B$ as well.  All earlier section
+corrections were the
+identity modulo the residual ideals, so the same conclusions hold on
+the corrected sections.
+\end{proof}
+
+\subsection{Polynomial perturbations and the final graph families}
+\label{subsec:vertex-polynomial}
+
+The triple-ideal identities are now established on the full residual
+schemes. The remaining task is to replace the logarithmic perturbation
+by an algebraic function without losing exact vanishing or gradient
+generation. Algebraic lifts of the holonomy matrices suffice: a
+quadratic polynomial in those matrices has a controlled cubic error,
+which can be removed on the source graph by a nilpotent correction.
+
+For an active edge replace the logarithmic function by the polynomial
+\begin{equation}
+\label{eq:vertex-polynomial-weight}
+ b_i(h_i)=-\epsilon_i\tr\left(
+ T_i\left((G_i(h_i)-1)-\tfrac12(G_i(h_i)-1)^2\right)\right).
+\end{equation}
+These are algebraic germs over $K$, with zero constant term at the
+recentered background.  We check the exact effects of this replacement.
+
+On the plus section, setting $t_i=0$ gives, modulo $I_A$, a central
+critical system on edge $i$ with both peripheral holonomies trivial:
+$\alpha_i$ is constrained to be trivial and $\beta_i$ bounds in $K_A$.
+The marked algebraic holonomy therefore satisfies
+\begin{equation}
+\label{eq:vertex-holonomy-ideal}
+ G_i(H_A)-1\in (I_A,t_i)
+\end{equation}
+entry by entry.  This is equality in the quotient ring, not just
+evaluation at its closed points.  The coordinate change used to absorb the difference in
+Equation~\eqref{eq:vertex-log-mod-jacobian} is immaterial after setting $t_i=0$.
+
+Let $R_i=\ell_i(G_i)-b_i$.  It has order at least three in the entries
+of $G_i-1$, and each internal derivative has order at least two in
+those entries.  Since $t_i^2=0$, Equation~\eqref{eq:vertex-holonomy-ideal} gives
+\begin{align}
+ t_iR_i(H_A)&\in t_i(I_A,t_i)^3=t_iI_A^3,\label{eq:vertex-taylor-value}\\
+ t_i(\partial_hR_i)(H_A)&\in t_i(I_A,t_i)^2=t_iI_A^2.
+                                                        \label{eq:vertex-taylor-gradient}
+\end{align}
+Thus the new total function on the existing plus section lies in
+$\mathfrak t I_A^2$, where $\mathfrak t=(t_i:i\text{ active})$, and
+its restricted gradient still has the form
+$(M_A+\mathfrak t Q)r_A$ with a unit full-column minor.
+
+The correction at the joint origin used the vanishing internal Hessian
+and gained powers of the joint maximal ideal.  After recentering, the
+Hessian may be nonzero.  Here the error instead contains a factor from
+the nilpotent parameter ideal, and the following correction gains powers
+of that ideal while keeping the same residual list.
+
+\begin{lemma}[Correction over the full parameter algebra]
+\label{lem:tensor-nilpotent-correction}
+Let \(I\) be a finite set of cardinality \(m\), put
+\(T=K[t_i:i\in I]/(t_i^2:i\in I)\), let \(J=(t_i:i\in I)\), and
+work in \(T[[x]]\).  Fix a finite residual list \(r\), with ideal
+\(\mathcal I=(r)\).
+Suppose a formal section \(H\) and a formal potential \(S\) satisfy
+\[
+ S(H)\in J\mathcal I^2,\qquad
+               (\partial_h S)(H)=M r,
+\]
+where \(M\) has a unit full-column minor.
+Then a displacement of \(H\) in \(J\mathcal I\) makes \(S(H)=0\)
+identically and keeps the restricted gradient ideal equal to
+\(\mathcal I\).
+If the original central parametrization has an injective tangent map,
+the corrected section is still a smooth formal graph over \(T\).
+\end{lemma}
+
+\begin{proof}
+Keep the functions \(r\) fixed on the parameter domain.
+Suppose at one step the error is in \(J^q\mathcal I^2\), \(q\geq1\).
+Write it as \(\sum_{a,b}c_{ab}r_a r_b\), with \(c_{ab}\in J^q\).
+Using the inverse full-column minor of the restricted gradient matrix,
+choose a displacement \(\Delta H\in J^q\mathcal I\) whose linear Taylor
+term cancels this error.
+The Taylor remainder belongs to
+\[
+       (\Delta H)^2\subset J^{2q}\mathcal I^2
+                         \subset J^{q+1}\mathcal I^2 .
+\]
+This uses no vanishing of the internal Hessian.
+The gradient changes by an element of \(J^q\mathcal I\), so its new
+coefficient matrix is \(M+J^qQ\) for some matrix \(Q\).
+Its chosen minor remains a unit because \(J\) is nilpotent.
+Iterating terminates, since \(J^{m+1}=0\), and preserves gradient
+generation throughout.
+Every correction vanishes at \(t=0\).
+A fixed linear projection that was invertible on the central tangent
+therefore still has a unit relative Jacobian.  Formal inversion over
+\(T\), fixing all parameters, gives the asserted graph.
+The inverse reparametrization also pulls back every residual membership
+witness.
+\end{proof}
+
+Apply Lemma~\ref{lem:tensor-nilpotent-correction} over $T[[y_A]]$, with
+$J=\mathfrak t$, $\mathcal I=I_A$, and the fixed residual list $r_A$.
+The preceding estimates give an error in
+$\mathfrak t I_A^3\subset\mathfrak t I_A^2$ and a unit full-column
+gradient minor against that same list.  The section is centered modulo
+$\mathfrak t$: $H_A(0,0)=0$, so $H_A\in(y_A,\mathfrak t)^N$.
+Formal substitution is continuous even with its nilpotent constants:
+write $H_A=H_A(y_A,0)+\eta$, where $\eta\in\mathfrak t T[[y_A]]^N$;
+expansion in $\eta$ is finite and substitution of $H_A(y_A,0)$ is
+centered.  The central tangent remains injective by
+Lemma~\ref{lem:vertex-linear} and the preceding coordinate changes.
+Thus the lemma gives a corrected source section $H_L$ with exact
+vanishing for the polynomial perturbations.  The corrections lie in
+$\mathfrak t I_A$ and preserve the gradient ideal $I_A$.  The ideal
+$\mathfrak t$ is nilpotent of exponent at most $m+1$; its square is not
+set to zero.  In particular, mixed parameter monomials are retained at
+every step.  A nonzero Hessian of a recentered central potential causes
+no difficulty because the Taylor remainders gain powers of
+$\mathfrak t$.
+
+The minus function is still $S_0$ up to its harmless overall sign, so
+the minus section needs no additional correction.  Denote it by $H_R$.
+Every weight in Equation~\eqref{eq:vertex-polynomial-weight} has zero constant term
+in the entries of $G_i-1$.  Lemma~\ref{lem:vertex-triple-ideals} therefore
+gives $(b_ib_jb_k)(H_R)\in I_B$ for distinct active indices.  The source
+parameter products remain in $I_A$.  Since the final restricted
+gradients generate these same ideals, these are exactly
+Equations~\eqref{eq:vertex-final-source} and~\eqref{eq:vertex-final-target}.
+
+For completeness, the corrected source is a smooth graph over the
+\emph{full} ring $T$, rather than just on its reduced central fiber.
+Choose a linear projection $\rho$ on the internal coordinate space
+which is invertible on the central tangent of the source.  The Jacobian
+of $y_A\mapsto\rho H_L(y_A,t)$ is invertible modulo $(t)$ and hence is
+a unit over $T[[y_A]]$.  The constant term of $\rho H_L$ can lie in the nilpotent ideal
+$(t)$; translation by that constant is continuous for the
+$(y_A,t)$-adic topology, which here agrees with the $y_A$-adic topology.
+The formal inverse theorem over $T$ reparametrizes this map as
+\[
+ H_L(x,t)=(x,\Psi(x,t)),
+\]
+while leaving the parameters fixed.  All memberships pull back under
+this same $T$-algebra automorphism.  At $t=0$ the last corrections
+vanish, so complementarity with $H_R$ is preserved.  No flatness of
+$T[[y_A]]/I_A$ over $T$ is required.
+
+\begin{proof}[Proof of Proposition~\ref{prop:vertex-data}]
+Use the smooth cut data and choose the finite extra-letter background
+in Equation~\eqref{eq:vertex-explicit-background}.  Lemma~\ref{lem:vertex-linear},
+the based connection slices, and Lemma~\ref{lem:vertex-offshell-identities}
+give complementary formal sections, square-ideal errors, and gradient
+generation by the actual residual lists.
+Lemma~\ref{lem:vertex-correction} gives exact vanishing.  Apply the
+separate marked algebraic edge-coordinate comparisons, absorb the
+Jacobian terms, and recenter before extending to $K=\mathbb C((\sigma))$.
+Lemma~\ref{lem:vertex-triple-ideals} supplies the exact source and target
+triple-product memberships.  Finally
+Equations~\eqref{eq:vertex-polynomial-weight}--\eqref{eq:vertex-taylor-gradient}
+and the nilpotent correction produce algebraic $F_i,b_i$ and the
+required final graphs.  The preceding graph reparametrization supplies
+precisely the stated smooth families.  This proves all assertions.
+\end{proof}
+
+\begin{corollary}
+\label{cor:vertex-tensor-interface}
+The cut data of Proposition~\ref{prop:geometric-reduction}, with
+$m\geq5$ and the indicated finite law list, supply all the hypotheses
+of Theorem~\ref{thm:tensor-obstruction}.
+\end{corollary}
+
+\begin{proof}
+Take the field, separate algebraic edge functions and graph families
+of Proposition~\ref{prop:vertex-data}.  Equations
+\eqref{eq:vertex-final-vanishing}--\eqref{eq:vertex-final-target} are
+the exact equations and memberships required there.  The graphs may
+mix the edge coordinates, while the fixed potentials remain separated
+by edge.  Their external parameter ring is the complete square-free
+algebra $T$, and their central tangent spaces are complementary.
+Ordinary algebraic approximation and subsequent spreading out are
+applied only at this final stage, as in the proof of
+Theorem~\ref{thm:tensor-obstruction}; they are not applied to the earlier
+arbitrary formal potentials or to a Laurent series term by term.
+\end{proof}
diff --git a/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/sections/06-conclusion.tex b/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/sections/06-conclusion.tex
new file mode 100644
index 0000000000000000000000000000000000000000..9ec16bd9c6165b5b4662995b7397c223d13d0605
--- /dev/null
+++ b/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/build/sections/06-conclusion.tex
@@ -0,0 +1,54 @@
+\section{The obstruction}\label{sec:conclusion}
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+Choose \(m\geq5\) and the finite extra-letter and law data used in
+Proposition~\ref{prop:vertex-data}. Before invoking
+Assumption~\ref{ass:disk}, the geometric construction produces the fixed
+immersed disk problem of Lemma~\ref{lem:exterior-duals} in
+\[
+ E=X\setminus\bigcup_j\operatorname{int}\nu(S_j).
+\]
+Suppose that this particular family admits simultaneous pairwise disjoint
+locally flat replacements with the same boundary maps and induced boundary
+framings. The proof of Proposition~\ref{prop:geometric-reduction} then
+constructs the required smooth vertex and edge pieces.
+Proposition~\ref{prop:edge-potential} and
+Theorem~\ref{thm:edge-algebraicity} give their separate algebraic central
+potentials and marked holonomy germs.
+
+Proposition~\ref{prop:vertex-data} then produces complementary formal graphs
+over the full parameter algebra
+\[
+ T=K[t_1,\ldots,t_m]/(t_1^2,\ldots,t_m^2),
+\]
+whose parameters are individually square-zero, with the source and target
+triple-product gradient-ideal memberships of
+Theorem~\ref{thm:tensor-obstruction}. There are at least five active factors.
+That theorem gives a contradiction.
+
+Consequently this fixed disk problem has no such embedded replacement.
+Its ambient manifold \(E\) is compact, connected, oriented, and smooth.
+Lemma~\ref{lem:exterior-duals} verifies all the displayed equivariant
+intersection, self-intersection, and framing conditions for its immersed
+disk and sphere inputs. Taking \(M=E\) gives the asserted example.
+\end{proof}
+
+\begin{remark}[Scope of the geometric conclusion]
+The examples lie in the compact orientable smooth class. They therefore
+refute the corresponding universal assertion for all topological
+four-manifolds, including versions allowing noncompact and nonorientable
+manifolds. The markings used to construct the example impose no additional
+condition on the hypothetical replacements: only their boundary maps and
+induced boundary normal framings have been prescribed, with no relative
+homotopy requirement.
+
+The contradiction already uses the disk conclusion of the unrestricted
+assertion. Allowing self-intersections and mutual intersections among the
+hypothetical output dual spheres does not remove it. We make no separate
+counterexample claim here for the ordinary surgery sequence, the simple
+surgery sequence on finite simple Poincar\'e pairs, or five-dimensional
+topological \(s\)-cobordism. Such a conclusion would require an additional
+implication from the independently formulated assertion to the disk
+replacement problem treated here. In particular, finiteness of a
+Poincar\'e complex is not being substituted for simple Poincar\'e duality.
+\end{remark}
diff --git a/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/paper.pdf b/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..9e3323a02e8284074bfb54b941b31b99d55d1397
--- /dev/null
+++ b/preprints/A-marked-tensor-obstruction-to-four-dimensional-disk-embedding-September-24-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:6453a6d9ab0cf56629cf67b7a90191b6846933691faf1960c9be2fad15e17d28
+size 679568
diff --git a/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/README.md b/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..4bfdff8627c81b351c3b45c3f10a71b9dbd23df0
--- /dev/null
+++ b/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/README.md
@@ -0,0 +1,17 @@
+# [A negatively pinched Kähler threefold without bounded holomorphic coordinates](paper.pdf)
+
+OpenAI  
+September 25, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026,
+  author = {{OpenAI}},
+  title = {{A negatively pinched K{\"a}hler threefold without bounded holomorphic coordinates}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/paper.pdf}{OAI:A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/main.tex b/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/main.tex
new file mode 100644
index 0000000000000000000000000000000000000000..c1ae218aa118cca54d4c3696c9aa92831e489ae9
--- /dev/null
+++ b/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/main.tex
@@ -0,0 +1,64 @@
+\newcommand{\DoNotLoadEpstopdf}{}
+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage[a4paper,margin=29mm]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{booktabs,array}
+\PassOptionsToPackage{hyphens}{url}
+\usepackage[hidelinks]{hyperref}
+\hypersetup{unicode=true,
+  pdftitle={A negatively pinched K\"ahler threefold without bounded holomorphic coordinates},
+  pdfauthor={OpenAI}}
+\usepackage{microtype}
+\ifdefined\pdfinfoomitdate\pdfinfoomitdate=1\fi
+\ifdefined\pdftrailerid\pdftrailerid{}\fi
+\ifdefined\pdfsuppressptexinfo\pdfsuppressptexinfo=15\fi
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newcommand{\B}{\mathbb B}
+\newcommand{\Sph}{S}
+\newcommand{\C}{\mathbb C}
+\newcommand{\R}{\mathbb R}
+\newcommand{\dd}{\mathrm d}
+\DeclareMathOperator{\supp}{supp}
+\DeclareMathOperator{\dist}{dist}
+\DeclareMathOperator{\osc}{osc}
+\numberwithin{equation}{section}
+\allowdisplaybreaks[1]
+\setlength{\emergencystretch}{1em}
+\title{A negatively pinched K\"ahler threefold\\without bounded holomorphic coordinates}
+\author{OpenAI}
+\date{September 25, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We construct a contractible domain in complex dimension three with a
+complete K\"ahler metric whose real sectional curvatures lie between two
+finite negative constants. It admits no bounded holomorphic map to
+$\mathbb C^3$ with nowhere-vanishing Jacobian and is therefore not
+biholomorphic to a bounded domain. This gives a negative answer to the
+negatively pinched K\"ahler uniformization question.
+\end{abstract}
+\setcounter{tocdepth}{1}
+\tableofcontents
+\clearpage
+\input{sections/01-introduction}
+\input{sections/02-reduction}
+\input{sections/03-peaks}
+\input{sections/04-densities-construction}
+\input{sections/04-densities-estimates}
+\input{sections/05-logarithmic-tests}
+\input{sections/06-base-potential}
+\input{sections/07-hartogs-metric}
+\input{sections/08-curvature}
+\input{sections/09-pinching}
+\bibliographystyle{plain}
+\bibliography{references}
+\end{document}
diff --git a/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/references.bib b/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/references.bib
new file mode 100644
index 0000000000000000000000000000000000000000..4b9df568de9dae821f9d1756c48c9aa4ee34402a
--- /dev/null
+++ b/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/references.bib
@@ -0,0 +1,177 @@
+@article{WuNormal1967,
+  author = {Wu, H.},
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+}
+
+@article{GrauertLevi1958,
+  author = {Grauert, Hans},
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+  journal = {Annals of Mathematics},
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+  year = {1958},
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+  url = {https://doi.org/10.2307/1970257}
+}
+
+@article{MostowSiu1980,
+  author = {Mostow, G. D. and Siu, Yum-Tong},
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+  pages = {321--360},
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+  url = {https://doi.org/10.2307/1971149}
+}
+
+@article{Deraux2005,
+  author = {Deraux, Martin},
+  title = {A negatively curved {K\"ahler} threefold not covered by the ball},
+  journal = {Inventiones Mathematicae},
+  volume = {160},
+  number = {3},
+  year = {2005},
+  pages = {501--525},
+  doi = {10.1007/s00222-004-0414-z},
+  url = {https://doi.org/10.1007/s00222-004-0414-z}
+}
+
+@book{GreeneWu1979,
+  author = {Greene, Robert E. and Wu, Hung-Hsi},
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+  volume = {699},
+  publisher = {Springer-Verlag},
+  address = {Berlin--Heidelberg},
+  year = {1979},
+  doi = {10.1007/BFb0063413},
+  url = {https://doi.org/10.1007/BFb0063413}
+}
+
+@misc{WuYauSurvey2019,
+  author = {Wu, Damin and Yau, Shing-Tung},
+  title = {Some negatively curved complex geometry},
+  year = {2019},
+  howpublished = {\href{https://dwu.math.uconn.edu/wp-content/uploads/sites/1743/2019/08/Wu-Yau_Some-negatively-curved-complex-geometry_ICCM_Aug-15-2019.pdf}{Author manuscript}},
+  url = {https://dwu.math.uconn.edu/wp-content/uploads/sites/1743/2019/08/Wu-Yau_Some-negatively-curved-complex-geometry_ICCM_Aug-15-2019.pdf}
+}
+
+@article{WuYauInvariant2020,
+  author = {Wu, Damin and Yau, Shing-Tung},
+  title = {Invariant metrics on negatively pinched complete {K\"ahler} manifolds},
+  journal = {Journal of the American Mathematical Society},
+  volume = {33},
+  number = {1},
+  year = {2020},
+  pages = {103--133},
+  doi = {10.1090/jams/933},
+  url = {https://doi.org/10.1090/jams/933}
+}
+
+@article{Calabi1979,
+  author = {Calabi, Eugenio},
+  title = {M{\'e}triques k{\"a}hl{\'e}riennes et fibr{\'e}s holomorphes},
+  journal = {Annales scientifiques de l'{\'E}cole Normale Sup{\'e}rieure},
+  series = {4},
+  volume = {12},
+  number = {2},
+  year = {1979},
+  pages = {269--294},
+  doi = {10.24033/asens.1367},
+  url = {https://www.numdam.org/articles/10.24033/asens.1367/}
+}
+
+@article{HwangSinger2002,
+  author = {Hwang, Andrew D. and Singer, Michael A.},
+  title = {A momentum construction for circle-invariant {K{\"a}hler} metrics},
+  journal = {Transactions of the American Mathematical Society},
+  volume = {354},
+  number = {6},
+  year = {2002},
+  pages = {2285--2325},
+  doi = {10.1090/S0002-9947-02-02965-3},
+  eprint = {math/9811024},
+  archivePrefix = {arXiv},
+  primaryClass = {math.DG},
+  url = {https://doi.org/10.1090/S0002-9947-02-02965-3}
+}
+
+@article{RyllWojtaszczyk1983,
+  author = {Ryll, J. and Wojtaszczyk, P.},
+  title = {On homogeneous polynomials on a complex ball},
+  journal = {Transactions of the American Mathematical Society},
+  volume = {276},
+  number = {1},
+  year = {1983},
+  pages = {107--116}
+}
+
+@article{Hedenmalm1989,
+  author = {Hedenmalm, H{\aa}kan},
+  title = {Closed ideals in the ball algebra},
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+  volume = {21},
+  number = {5},
+  year = {1989},
+  pages = {469--474},
+  doi = {10.1112/blms/21.5.469},
+  url = {https://doi.org/10.1112/blms/21.5.469}
+}
+
+@misc{HaoChenWang2025,
+  author = {Hao, Yihong and Chen, Mingming and Wang, An},
+  title = {On the negatively pinched properties of the disc bundles over negatively pinched {K{\"a}hler} manifolds},
+  howpublished = {\href{https://arxiv.org/abs/2509.04179v1}{arXiv:2509.04179v1 [math.CV]}},
+  year = {2025},
+  eprint = {2509.04179},
+  archivePrefix = {arXiv},
+  primaryClass = {math.CV},
+  doi = {10.48550/arXiv.2509.04179},
+  url = {https://arxiv.org/abs/2509.04179v1},
+  note = {Version 1}
+}
+
+@misc{HaoChenWangZhang2026,
+  author = {Hao, Yihong and Chen, Mingming and Wang, An and Zhang, Ben},
+  title = {The curvature estimation of the complete {K{\"a}hler--Einstein} metrics on the disk bundles},
+  howpublished = {\href{https://arxiv.org/abs/2609.14482v1}{arXiv:2609.14482v1 [math.CV]}},
+  year = {2026},
+  eprint = {2609.14482},
+  archivePrefix = {arXiv},
+  primaryClass = {math.CV},
+  doi = {10.48550/arXiv.2609.14482},
+  url = {https://arxiv.org/abs/2609.14482v1},
+  note = {Version 1, September 13, 2026}
+}
+
+@article{Aleksandrov1983,
+  author = {Aleksandrov, A. B.},
+  title = {The existence of inner functions in the ball},
+  journal = {Mathematics of the USSR-Sbornik},
+  volume = {46},
+  number = {2},
+  year = {1983},
+  pages = {143--159},
+  doi = {10.1070/SM1983v046n02ABEH002759},
+  url = {https://doi.org/10.1070/SM1983v046n02ABEH002759},
+  note = {Russian original: Matematicheskii Sbornik 118(160) (1982), no. 2(6), 147--163}
+}
+
+@misc{OpenAIOneSided2026,
+  author = {{OpenAI}},
+  title = {{One-sided negative sectional curvature and the holomorphic Liouville property}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026/paper.pdf}{OAI:One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026}},
+  year = {2026},
+  note = {Theorem 1.1}
+}
diff --git a/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/sections/01-introduction.tex b/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/sections/01-introduction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..934a63766aad7376931fdf62f63ee36169b56220
--- /dev/null
+++ b/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/sections/01-introduction.tex
@@ -0,0 +1,178 @@
+\section{Introduction}
+\label{sec:introduction}
+
+A bounded domain in $\C^n$ has bounded holomorphic coordinate functions
+whose differentials are independent everywhere. The negatively curved
+uniformization problem asks whether curvature and completeness can
+force a complex manifold to possess such coordinates. In complex
+dimension one, a complete simply connected surface whose curvature is
+bounded above by a negative constant is biholomorphic to the disk. In higher
+dimension, the existence of bounded holomorphic coordinates is a
+separate global analytic requirement.
+
+An early formulation appears in H.~Wu's work on normal families
+\cite[p.~195, question~(1)]{WuNormal1967}. He asks whether a complete
+simply connected K\"ahler manifold with nonpositive real sectional
+curvature and holomorphic sectional curvature bounded above by a
+negative constant is biholomorphic to a bounded domain. Wu and Yau
+state the two-sided real-sectional version with a bounded pseudoconvex
+domain as conclusion \cite[Conjecture~4.3, p.~18]{WuYauSurvey2019}:
+if a simply connected complete K\"ahler manifold has every real
+sectional curvature between two negative constants, must it be
+biholomorphic to a bounded domain in its complex dimension?
+
+The unit ball is already too restrictive a candidate in higher
+dimension. Mostow and Siu constructed compact negatively curved
+K\"ahler surfaces not covered by the ball \cite{MostowSiu1980}.
+Deraux reviews their construction and obtains three-dimensional
+examples \cite{Deraux2005}.
+Their lifted metrics on the universal covers have two-sided negative
+sectional bounds by compactness. These non-ball results, however, do
+not exclude a biholomorphism to some other bounded domain; that is
+the stronger obstruction sought here.
+
+Here and throughout, sectional curvature refers to the underlying real
+Riemannian metric, and completeness means geodesic completeness. A
+bounded holomorphic map is a holomorphic map with bounded image in its
+Euclidean target.
+
+\begin{theorem}\label{main:theorem}
+There exist a contractible domain $M\subset\C^3$, a smooth complete
+K\"ahler metric $g$ on $M$, and finite constants $0<A\le B$ such that
+\[
+ -B\le K_g(\sigma)\le -A<0
+\]
+for every real two-plane $\sigma\subset T_pM$ and every $p\in M$.
+No bounded holomorphic map $F:M\to\C^3$ has a nowhere-vanishing Jacobian
+determinant. In particular, $M$ is not biholomorphic to a bounded domain
+in $\C^3$.
+\end{theorem}
+
+Theorem~\ref{main:theorem} gives a negative answer to the stated
+uniformization question, and its curvature hypotheses also place the
+example within Wu's earlier question. The example has two bounded
+nonconstant holomorphic functions, supplied by its base coordinates.
+Its analytic obstruction concerns a fixed triple of bounded functions
+with independent differentials at every point.
+The domain is nevertheless Stein: the construction supplies a smooth
+strictly plurisubharmonic exhaustion (Lemma~\ref{metric:complete}), so Grauert's solution
+of the Levi problem applies \cite[Theorem~2 and Section~3.4]{GrauertLevi1958}.
+Thus the obstruction is to bounded holomorphic coordinates, not to
+holomorphic convexity.
+
+\subsection{Intrinsic metrics and bounded functions}
+
+Two-sided negative real sectional curvature has strong consequences for
+intrinsic complex geometry. Greene and Wu established the lower
+comparison for the Bergman metric \cite{GreeneWu1979}; Wu and Yau recall that result and
+prove the complementary upper bound
+\cite[Theorems~4 and~6]{WuYauInvariant2020}. Under completeness and
+simple connectivity, Wu and Yau then compare the Bergman,
+Kobayashi--Royden, complete negative K\"ahler--Einstein, and original
+metrics \cite[Corollary~7]{WuYauInvariant2020}. The Bergman metric
+uses square-integrable holomorphic top-degree forms. These comparisons
+do not supply a fixed tuple of bounded holomorphic functions with
+independent differentials everywhere. Applied to the example in
+Theorem~\ref{main:theorem}, they show that all these intrinsic metrics
+can be complete and mutually comparable while no such tuple exists.
+
+The existence of even one nonconstant bounded holomorphic function is
+a related question with a different conclusion. The companion paper
+\cite[Theorem~1.1]{OpenAIOneSided2026} constructs, in a sufficiently
+large fixed finite complex dimension, a domain diffeomorphic to
+Euclidean space with a complete K\"ahler metric of real sectional
+curvature at most $-1$ and only constant bounded holomorphic functions.
+Its curvature is unbounded below. Here both curvature bounds are
+finite and uniform, while the two bounded base coordinates remain.
+The two constructions and their proofs are independent.
+
+\subsection{Construction and proof strategy}
+
+We construct $M$ as a family of discs over the unit ball
+$\B=\{z\in\C^2:|z|<1\}$. A smooth real function $\phi$ on $\B$ determines
+\[
+ M_\phi=\{(z,w)\in\B\times\C:e^{\phi(z)}|w|^2<1\}.
+\]
+Thus the disc above $z$ has radius $e^{-\phi(z)/2}$. Put
+$\psi(z)=-\log(1-|z|^2)$. Our metric is the complex Hessian of
+\[
+ \lambda\psi(z)-\log\bigl(1-e^{\phi(z)}|w|^2\bigr),
+\]
+with the real-metric convention specified in
+Section~\ref{sec:reduction}. This is a specialization of Calabi's
+construction from a base potential and a radial function of a Hermitian
+fiber norm \cite[Section~3, equations~(3.1)--(3.2)]{Calabi1979}.
+The momentum formulation of Hwang and Singer develops this framework
+for general base and bundle data \cite{HwangSinger2002}.
+
+For the logarithmic radial potential used here, Hao, Chen, and Wang
+prove holomorphic and real sectional pinching results when the bundle
+curvature determines the base metric
+\cite[Theorems~1 and~2]{HaoChenWang2025}. Hao, Chen, Wang, and Zhang
+obtain related results for complete K\"ahler--Einstein disk-bundle
+metrics under Einstein and curvature hypotheses on the base
+\cite[Sections~4.2--4.3]{HaoChenWangZhang2026}.
+The estimate needed here concerns all real two-planes when the base
+form $\lambda\partial\bar\partial\psi$ and the fiber-weight curvature
+$\partial\bar\partial\phi$ are independent. We assume upper and lower
+bounds for the latter form and bounds for the component derivatives
+needed for curvature, expressed in centered ball coordinates. No curvature sign for the metric
+$\partial\bar\partial\phi$ is assumed. We derive and estimate the
+full curvature tensor under precisely these hypotheses.
+
+The analytic task is to choose $\phi$ so that it has these derivative
+bounds but cannot dominate a holomorphic real part, even after adding
+the logarithmic boundary allowance forced by Cauchy's estimate for a
+Jacobian. To do this, we construct radii $r_N\uparrow1$ and positive
+probabilities $\mu_N$ on the unit sphere $S\subset\C^2$ such that
+\[
+ \int_S\left(\phi(r_N\xi)+4\log\frac1{1-r_N}\right)
+                  \,d\mu_N(\xi)\longrightarrow-\infty,
+\]
+while $\int_S\operatorname{Re}H(r_N\xi)\,d\mu_N(\xi)
+=\operatorname{Re}H(0)$ for every holomorphic $H$ on $\B$.
+The two averages exclude any holomorphic real part bounded above by
+the tested function plus a constant. The probabilities preserve
+holomorphic evaluation at the origin, but such
+measures need not preserve the logarithmic mean inequality for
+holomorphic functions: Hedenmalm gives an explicit example of this
+distinction \cite[Theorem~3.7]{Hedenmalm1989}.
+
+The construction proceeds at degrees $k_j=Q^j$ for one fixed large
+integer $Q$. At each degree we sum homogeneous peaks centered on a
+separated set of complex lines. The projective packing and power-sum
+estimate have a direct antecedent in Ryll and Wojtaszczyk
+\cite[Section~2]{RyllWojtaszczyk1983}. A radial correction around each
+peak annihilates every holomorphic moment exactly. Separation of the
+degrees makes the corrections compatible with positivity at every
+stage. Regularized logarithms built from the peak polynomials then have a
+fixed negative average, which survives all subsequent corrections.
+Summing these logarithms with a sufficiently large fixed coefficient
+makes their negative averages dominate the boundary allowance.
+Low degrees vary little
+in a centered chart, while high degrees are exponentially small;
+this gives the uniform derivatives through order four needed for
+curvature.
+
+The geometric task has its own delicate point. To prove negative
+curvature, we must show that its negated numerator is positive on every
+real two-plane. After rescaling the base directions, the leading
+expression is nonnegative but can vanish on planes with nonzero base
+and fiber projections. On these limiting planes the contribution of the
+potentially largest error term vanishes as well. This cancellation gives stronger error
+control near the degenerate planes and proves strict negativity for
+one sufficiently large base parameter $\lambda$.
+With $\lambda$ fixed, compactness on bounded
+fiber ranges and a uniform limiting tensor at the fiber boundary give
+both finite curvature bounds.
+
+Section~\ref{sec:reduction} states the two precise ingredients and derives
+Theorem~\ref{main:theorem} from them by Cauchy estimates for a putative
+Jacobian. Sections~\ref{sec:peaks}--\ref{sec:densities} construct the
+peaks and positive representing densities;
+Sections~\ref{sec:tests}--\ref{sec:base-potential} establish their
+logarithmic tests and the controlled potential.
+Section~\ref{sec:hartogs} proves completeness.
+Sections~\ref{sec:curvature}--\ref{sec:pinching} compute the curvature
+and prove the two-sided estimate, including the limiting planes just
+described.
diff --git a/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/sections/02-reduction.tex b/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/sections/02-reduction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..3e3c3f8beb7643fb698c94f1315114c20548b972
--- /dev/null
+++ b/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/sections/02-reduction.tex
@@ -0,0 +1,177 @@
+\section{Two inputs and the bounded-map obstruction}\label{sec:reduction}
+
+The construction separates a problem about functions on the ball from a
+curvature estimate.  The first input is a strictly plurisubharmonic
+function with controlled derivatives and an obstruction to a prescribed
+holomorphic real-part bound.  The second turns its complex Hessian bounds
+into a complete metric with uniformly negative real sectional curvature.
+We state both inputs precisely and then prove their implication for
+Theorem~\ref{main:theorem}.
+
+\subsection{Centered charts and the two propositions}
+
+Let $\B=\{z\in\C^2:|z|<1\}$ and put
+\[
+ \psi(z)=-\log(1-|z|^2),\qquad
+ I=(\psi_{i\bar j}).
+\]
+Here a complex Hessian is also viewed as a Hermitian form, linear in its
+first argument.  To center coordinates at a point $z_0\in\B$, let
+$r=|z_0|$, choose a unitary map $U$ with $U(r,0)=z_0$, and set
+\begin{equation}\label{base:charts}
+ T=U\circ T_r,\qquad
+ T_r(\zeta)=\left(
+ \frac{r+\zeta_1}{1+r\zeta_1},
+ \frac{\sqrt{1-r^2}\,\zeta_2}{1+r\zeta_1}
+ \right).
+\end{equation}
+These maps are ball automorphisms with $T(0)=z_0$.  The identity
+\[
+ 1-|T_r(\zeta)|^2
+ =\frac{(1-r^2)(1-|\zeta|^2)}{|1+r\zeta_1|^2}
+\]
+shows that $T^*I$ has the same expression in $\zeta$ as $I$ has in $z$:
+the additional terms in $\psi\circ T$ are constant or pluriharmonic.
+In particular, this matrix is the identity at $0$ and has zero first
+derivatives there.  We call \eqref{base:charts} the centered ball charts.
+
+For a smooth real function $\phi$ on $\B$, write
+$h=\partial\bar\partial\phi$ and, in a centered chart $T$, write
+\[
+ h^T_{i\bar j}(\zeta)
+ =\partial_{\zeta_i}\partial_{\bar\zeta_j}(\phi\circ T)(\zeta).
+\]
+The uniform control needed below is the existence of constants
+$0<c\le C<\infty$ such that, for every centered chart,
+\begin{equation}\label{base:control}
+ c\,\mathrm{Id}\le h^T(0)\le C\,\mathrm{Id}
+\end{equation}
+and
+\begin{equation}\label{base:jets}
+ \left|\partial_k h^T_{i\bar j}(0)\right|
+ +\left|\partial_{\bar\ell}h^T_{i\bar j}(0)\right|
+ +\left|\partial_k\partial_{\bar\ell}h^T_{i\bar j}(0)\right|
+ \le C
+ \qquad(i,j,k,\ell\in\{1,2\}).
+\end{equation}
+The derivatives here are ordinary derivatives of the displayed component
+functions.  In invariant form, \eqref{base:control} says $cI\le h\le CI$.
+
+\begin{proposition}[A controlled potential]\label{base:existence}
+There exist $\phi\in C^\infty(\B,\R)$ and a finite constant $C$ for
+which \eqref{base:control} and \eqref{base:jets} hold with $c=1$, and
+such that no $H\in\mathcal O(\B)$ and $C_H\in\R$ satisfy
+\begin{equation}\label{base:no-minorant}
+ \operatorname{Re}H(z)
+ \le C_H+4\log\frac1{1-|z|}+\phi(z)
+ \qquad(z\in\B).
+\end{equation}
+\end{proposition}
+
+For any smooth real $\phi$ on $\B$, define its Hartogs domain by
+\begin{equation}\label{metric:domain}
+ M_\phi=\{(z,w)\in\B\times\C:t(z,w)<1\},
+ \qquad t(z,w)=e^{\phi(z)}|w|^2.
+\end{equation}
+For $\lambda>0$ define the potential and its complex Hessian by
+\begin{equation}\label{metric:potential}
+ \mathcal K_\lambda=\lambda\psi-\log(1-t),\qquad
+ \widehat g_\lambda=\partial\bar\partial\mathcal K_\lambda.
+\end{equation}
+Our real metric convention is
+$g_\lambda=2\operatorname{Re}\widehat g_\lambda$: if a real vector $X$
+has $(1,0)$ part $u$, then
+$g_\lambda(X,X)=2\widehat g_\lambda(u,u)$.
+
+\begin{proposition}[Hartogs metric transfer]\label{metric:transfer}
+Let $\phi\in C^\infty(\B,\R)$ satisfy \eqref{base:control} and
+\eqref{base:jets} for fixed $0<c\le C<\infty$.  There is a finite
+$\lambda_0=\lambda_0(c,C)>0$ such that, for every
+$\lambda\ge\lambda_0$, the domain $M_\phi$ in
+\eqref{metric:domain} is contractible and the tensor
+$g_\lambda=2\operatorname{Re}\widehat g_\lambda$ defined by
+\eqref{metric:potential} is a smooth geodesically complete K\"ahler
+metric.  For this fixed $\lambda$ there are constants
+$0<A_\lambda\le B_\lambda<\infty$ such that
+\[
+ -B_\lambda\le K_{g_\lambda}(\sigma)\le-A_\lambda<0
+\]
+for every point of $M_\phi$ and every real two-plane $\sigma$ in its
+tangent space.
+\end{proposition}
+
+The proof of Proposition~\ref{base:existence} occupies the analytic
+construction below.  The global geometry and curvature calculation in
+Sections~\ref{sec:hartogs}--\ref{sec:pinching} prove
+Proposition~\ref{metric:transfer}.  We first establish the short argument
+that connects their conclusions to bounded holomorphic maps.
+
+\subsection{The Jacobian on the zero section}
+
+\begin{lemma}\label{map:obstruction}
+Let $\phi\in C^\infty(\B,\R)$ have the nonexistence property in
+Proposition~\ref{base:existence}: no holomorphic $H$ and real constant
+$C_H$ satisfy \eqref{base:no-minorant}.  Then no bounded holomorphic
+map $F:M_\phi\to\C^3$ has a nowhere-vanishing complex Jacobian
+determinant.
+\end{lemma}
+
+\begin{proof}
+Suppose that $F=(F_1,F_2,F_3)$ is such a map, and choose $B\ge1$ so
+that $|F_\ell|\le B$ on $M_\phi$ for every component.  Since
+$M_\phi$ is open in $\C^3$, its coordinates $(z_1,z_2,w)$ are global.
+The function
+\[
+ d(z)=\det\frac{\partial(F_1,F_2,F_3)}{\partial(z_1,z_2,w)}(z,0)
+\]
+is holomorphic and nowhere zero on $\B$.
+
+Fix $z\in\B$ and put $\delta=1-|z|$.  The restriction
+$F_\ell(\,\cdot\,,0)$ is bounded and holomorphic on the entire base
+ball.  For $j=1,2$, the disk
+$\{z+\zeta e_j:|\zeta|<\delta\}$ is contained in that ball.  Cauchy's
+estimate, applied on disks with radii tending to $\delta$, gives
+\begin{equation}\label{map:base-cauchy}
+ |\partial_{z_j}F_\ell(z,0)|\le B\delta^{-1}.
+\end{equation}
+For this fixed $z$, the fiber in $M_\phi$ is the disk of radius
+$e^{-\phi(z)/2}$.  Cauchy's estimate on this disk gives
+\begin{equation}\label{map:fiber-cauchy}
+ |\partial_wF_\ell(z,0)|\le B e^{\phi(z)/2}.
+\end{equation}
+In particular, the base estimate requires no uniform fiber radius near
+$z$; it is taken entirely on the zero section.
+
+The six terms in the determinant expansion, together with
+\eqref{map:base-cauchy} and \eqref{map:fiber-cauchy}, yield
+\[
+ |d(z)|\le6B^3\delta^{-2}e^{\phi(z)/2},
+\]
+and hence
+\begin{equation}\label{map:determinant-bound}
+ \log|d(z)|^2
+ \le 2\log6+6\log B+4\log\frac1{1-|z|}+\phi(z).
+\end{equation}
+Because $\B$ is simply connected and $d$ has no zeros, it has a global
+holomorphic logarithm $\ell$.  For example, a primitive of the closed
+holomorphic one-form $d^{-1}\partial d$, with a suitable constant,
+satisfies $e^\ell=d$.  The holomorphic function $H=2\ell$ has
+$\operatorname{Re}H=\log|d|^2$, so \eqref{map:determinant-bound}
+contradicts \eqref{base:no-minorant}.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{main:theorem}]
+Choose $\phi$ from Proposition~\ref{base:existence}, and then choose
+$\lambda$ as in Proposition~\ref{metric:transfer}.  The domain
+$M=M_\phi\subset\C^3$ is contractible and carries the asserted smooth
+complete K\"ahler metric with two uniform negative real-sectional
+bounds.  Lemma~\ref{map:obstruction} excludes a bounded holomorphic
+map to $\C^3$ with nowhere-vanishing Jacobian.  A biholomorphism to a
+bounded domain would be such a map: the chain rule applied to its
+holomorphic inverse makes its Jacobian invertible everywhere.
+\end{proof}
+
+The logarithm argument uses nonvanishing along the entire zero section.
+It gives no assertion that a bounded map's determinant vanishes
+identically.  Nor are all bounded holomorphic functions constant:
+$z_1$ and $z_2$ are nonconstant bounded holomorphic functions on $M_\phi$.
diff --git a/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/sections/03-peaks.tex b/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/sections/03-peaks.tex
new file mode 100644
index 0000000000000000000000000000000000000000..a862e0043f07788076a3a44cb4f97455c7a45b3d
--- /dev/null
+++ b/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/sections/03-peaks.tex
@@ -0,0 +1,277 @@
+\section{Separated homogeneous peaks}\label{sec:peaks}
+
+We begin the construction of the potential in
+Proposition~\ref{base:existence}.  Its analytic requirement will follow
+from increasingly negative averages against positive measures that
+preserve holomorphic means.  The following observation explains the
+role of those measures before we construct them.
+
+Write $S=\{\xi\in\mathbb C^2:|\xi|=1\}$, and let $\sigma$ be its
+surface measure normalized to have mass one.  A probability measure
+$\mu$ on $S$ \emph{represents holomorphic evaluation at zero} if
+\[
+ \int_S p(\xi)\,d\mu(\xi)=p(0)
+ \quad\text{for every holomorphic polynomial }p.
+\]
+
+\begin{lemma}[Representing-mean criterion]\label{peaks:mean-criterion}
+Let $u$ be a continuous real function on $\mathbb B$.  Suppose there
+are radii $0<r_N<1$ and representing probabilities $\mu_N$ on $S$
+such that
+\[
+ \int_S u(r_N\xi)\,d\mu_N(\xi)\longrightarrow-\infty.
+\]
+Then no holomorphic function $H$ on $\mathbb B$ and constant $C_H$
+satisfy $\operatorname{Re}H\le C_H+u$ on $\mathbb B$.
+\end{lemma}
+
+\begin{proof}
+For each fixed $r_N<1$, the Taylor polynomials of $H$ converge
+uniformly on $r_NS$.  The representing identities therefore give
+\[
+ \int_S\operatorname{Re}H(r_N\xi)\,d\mu_N(\xi)
+ =\operatorname{Re}H(0).
+\]
+Integrating the proposed inequality contradicts the assumed limit.
+\end{proof}
+
+We will apply this criterion to
+$u(z)=\phi(z)+4\log(1/(1-|z|))$.  The first ingredient is a supply
+of homogeneous polynomials with many separated peaks.  Their uniform
+bounds will also control the derivatives of the eventual potential.
+
+\subsection{Projective packing and polynomial bounds}
+
+The projective distance and separated-power estimate used here occur
+in the direct construction of Ryll and Wojtaszczyk
+\cite[p.~112 and Lemma~2.7]{RyllWojtaszczyk1983}.
+We give the packing argument with a Gaussian tail and an adjustable
+spacing parameter, since the density corrections will require both.
+Related constructions with peaks localized in nonisotropic sphere
+balls appear in Aleksandrov \cite[Lemmas~2--3]{Aleksandrov1983};
+the caps used here are invariant under a common phase rotation.
+
+Identify complex lines in $\mathbb C^2$ with points of
+$\mathbb {CP}^1$.  For unit representatives define
+\[
+ d([\xi],[p])=\sqrt{1-|\langle\xi,p\rangle|^2},
+\]
+where the Hermitian inner product is linear in its first argument.
+This equals the operator norm distance between the rank-one
+orthogonal projections onto the two lines: their difference has
+trace zero and eigenvalues
+$\pm\sqrt{1-|\langle\xi,p\rangle|^2}$.  The triangle inequality
+for the operator norm therefore proves that $d$ is a metric.  We also write
+$d(\xi,p)$ when the representatives are understood.
+
+In unitary coordinates with $\xi_1=\langle\xi,p\rangle$, write
+\[
+ \xi=(e^{i\theta}\cos v,e^{i\gamma}\sin v),
+ \qquad 0\le v\le\pi/2.
+\]
+The round volume element is
+$\cos v\sin v\,dv\,d\theta\,d\gamma$.  After normalization and
+the change of variable $u=\cos^2v$, this gives
+\begin{equation}\label{peaks:sphere-measure}
+ d\sigma=du\,\frac{d\theta}{2\pi}\frac{d\gamma}{2\pi},
+ \qquad 0\le u\le1.
+\end{equation}
+In particular, a projective cap $d(\xi,p)<s$ has measure $s^2$
+for $0\le s\le1$.
+
+Fix $D_0\ge1$.  For an integer $k\ge D_0^2$, choose a maximal
+$D_0/\sqrt{k}$-separated set of lines, and choose one unit
+representative $p$ for each line.  Let $\mathcal P_k$ denote this
+set of representatives and $N_k=|\mathcal P_k|$.  Define the
+homogeneous holomorphic polynomial
+\begin{equation}\label{peaks:polynomial}
+ P_k(z)=\sum_{p\in\mathcal P_k}\langle z,p\rangle^k.
+\end{equation}
+The representatives may be chosen arbitrarily; the estimates below
+hold for every such choice.
+
+\begin{lemma}[Peak bounds]\label{peaks:bounds}
+There are absolute constants $C,c>0$, independent of $D_0\ge1$,
+such that
+\begin{equation}\label{peaks:count}
+ \frac{k}{D_0^2}\le N_k\le\frac{4k}{D_0^2}
+\end{equation}
+and, for every $\xi\in S$ and $u\ge0$,
+\begin{equation}\label{peaks:gaussian}
+ \sum_{\substack{p\in\mathcal P_k\\\sqrt{k}d(\xi,p)\ge u}}
+       |\langle\xi,p\rangle|^k\le Ce^{-cu^2}.
+\end{equation}
+Consequently,
+\begin{equation}\label{peaks:global}
+ |P_k(z)|\le C|z|^k\quad(z\in\mathbb C^2),
+ \qquad
+ \int_S|P_k|\,d\sigma\le\frac{2N_k}{k+2}.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Put $\rho=D_0/\sqrt{k}\le1$.  Maximality gives a covering by caps
+of radius $\rho$, and separation gives disjoint interiors for caps
+of radius $\rho/2$.  Their measures yield
+$1\le N_k\rho^2$ and $N_k\rho^2/4\le1$, proving
+\eqref{peaks:count}.
+
+We also need a local count whose constant does not depend on $D_0$.
+About the centers within distance $u/\sqrt{k}$ of a fixed line,
+place caps of radius $1/(3\sqrt{k})$.  Their interiors are disjoint,
+and they lie inside the cap of radius $(u+1/3)/\sqrt{k}$ about that
+line.  Comparing measures bounds their number by $9(u+1/3)^2$ when
+the latter radius is at most one.  Otherwise their number is at
+most $9k\le9(u+1/3)^2$, by comparison with the whole space.
+Thus the local count is at most $C(1+u)^2$ in all cases.
+
+Since
+\[
+ |\langle\xi,p\rangle|^k
+ =(1-d(\xi,p)^2)^{k/2}
+ \le e^{-kd(\xi,p)^2/2},
+\]
+the contribution from the annulus
+$n\le\sqrt{k}d(\xi,p)<n+1$ is at most
+$C(n+2)^2e^{-n^2/2}$.  Summing these bounds from $n=\lfloor u\rfloor$
+onward proves \eqref{peaks:gaussian}: the polynomial factor is
+absorbed by a weaker Gaussian exponent, and the range $0\le u\le2$
+is absorbed by the constant.  Setting $u=0$ and using homogeneity
+gives the first assertion in \eqref{peaks:global}, including points
+outside the unit ball.  Finally, \eqref{peaks:sphere-measure} gives
+\[
+ \int_S|\langle\xi,p\rangle|^k\,d\sigma
+ =\int_0^1 u^{k/2}\,du=\frac2{k+2}.
+\]
+The triangle inequality proves the stated integral bound.
+\end{proof}
+
+\subsection{Patches and a radial profile}
+
+For a fixed $R_0>0$ and a center $p\in\mathcal P_k$, use the
+adapted coordinates above and put
+\begin{equation}\label{peaks:patch-definition}
+ y=-k\log|\xi_1|,\qquad
+ \mathcal U_{k,p}=\{\xi\in S:0\le y<R_0\}.
+\end{equation}
+The set $y=0$ is the phase circle through $p$.  Every patch is
+invariant under multiplication of $\xi$ by a common phase.
+
+\begin{lemma}[Patch estimates]\label{peaks:patches}
+Fix $R_0>0$.  If $D_0>\max\{1,2\sqrt{2R_0}\}$ and
+$k\ge\max\{D_0^2,2R_0\}$, the patches $\mathcal U_{k,p}$
+are pairwise disjoint.  On each patch,
+\begin{equation}\label{peaks:patch-measure}
+ d\sigma=\frac2k e^{-2y/k}\,dy\,
+             \frac{d\theta}{2\pi}\frac{d\gamma}{2\pi},
+\end{equation}
+and
+\begin{equation}\label{peaks:single-peak}
+ |P_k(\xi)-e^{-y}e^{ik\theta}|
+ \le C e^{-c(D_0-\sqrt{2R_0})^2}.
+\end{equation}
+Outside their union,
+\begin{equation}\label{peaks:off-patch}
+ |P_k(\xi)|\le Ce^{-cR_0}.
+\end{equation}
+The constants $C,c$ can be chosen independently of $R_0,D_0,k$.
+\end{lemma}
+
+\begin{proof}
+The projective radius of a patch is
+$\sqrt{1-e^{-2R_0/k}}\le\sqrt{2R_0/k}$, so the separation of
+the centers makes the patches disjoint.  Formula
+\eqref{peaks:patch-measure} follows from
+$|\xi_1|^2=e^{-2y/k}$ in \eqref{peaks:sphere-measure}.
+
+If $\xi\in\mathcal U_{k,p}$ and $p'\ne p$ is another center,
+the metric triangle inequality gives
+\[
+ \sqrt{k}d(\xi,p')\ge D_0-\sqrt{2R_0}.
+\]
+The peak from $p$ is exactly $e^{-y}e^{ik\theta}$.
+Summing all other peaks by \eqref{peaks:gaussian} proves
+\eqref{peaks:single-peak}.
+Outside every patch, each center satisfies
+\[
+ kd(\xi,p)^2\ge k(1-e^{-2R_0/k})\ge R_0.
+\]
+The final inequality uses $k\ge2R_0$ and
+$1-e^{-v}\ge v/2$ for $0\le v\le1$.  Another application of
+\eqref{peaks:gaussian} proves \eqref{peaks:off-patch}.
+\end{proof}
+
+The density correction will use a radial profile with an exact
+weighted cancellation.  The following choice also supplies the
+strict inequality needed for the logarithmic tests later.
+
+\begin{lemma}[Radial profiles]\label{peaks:profiles}
+Fix $1<a<2$ and put $\alpha=\log a$.  One can choose $0<s_1<1$,
+smooth real functions $f,b$ on $[0,\infty)$, and $R_0>0$ such that
+\begin{equation}\label{peaks:profile-negativity}
+ \alpha^2-2s_1(\cosh\alpha-1)<0,
+\end{equation}
+\begin{equation}\label{peaks:profile-identities}
+ \|f\|_\infty<1,\quad f=-s_1\ \text{on }[0,\alpha],\quad
+ \int_0^\infty e^{-y}f(y)\,dy=0,\quad
+ b'-b=f,\quad b(0)=0.
+\end{equation}
+Both functions vanish on a neighborhood of $[R_0,\infty)$.
+Moreover, $R_0$ can be chosen so that $|aP_k|\le1/2$ off the
+patches whenever the hypotheses of Lemma~\ref{peaks:patches} hold.
+\end{lemma}
+
+\begin{proof}
+The strict inequality $2(\cosh\alpha-1)>\alpha^2$ allows a choice
+\[
+ \frac{\alpha^2}{2(\cosh\alpha-1)}<s_1<1.
+\]
+Since $a<2$, this choice also satisfies
+$s_1(1-e^{-\alpha})<e^{-\alpha}$.
+Choose a small $\delta>0$ and a smooth cutoff $\chi$, equal to one
+on $[0,\alpha]$, between zero and one, and zero for
+$y\ge\alpha+\delta$.  For sufficiently small $\delta$,
+\[
+ N=s_1\int_0^\infty e^{-y}\chi(y)\,dy
+ <e^{-(\alpha+2\delta)}.
+\]
+Choose a smooth compactly supported function $\nu$, supported
+strictly after $\alpha+2\delta$, with $0\le\nu\le1$, and with
+\[
+ P=\int_0^\infty e^{-y}\nu(y)\,dy>N.
+\]
+Such a function is obtained by approximating the indicator of a
+sufficiently long interval beginning just after $\alpha+2\delta$.
+Then $f=-s_1\chi+(N/P)\nu$ has the required plateau and weighted
+integral.  The two supports are disjoint, so
+$\|f\|_\infty\le\max\{s_1,N/P\}<1$.
+
+Define
+\[
+ b(y)=e^y\int_0^y e^{-v}f(v)\,dv.
+\]
+It satisfies $b'-b=f$ and $b(0)=0$.  Beyond the support of $f$,
+the integral is exactly zero by the weighted cancellation, so $b$
+also has compact support.  Near zero,
+$b(y)=-s_1(e^y-1)$, which is smooth up to the boundary of the
+half-line.  Smoothness here does not require extending $f$ by zero
+to negative arguments.  Finally choose $R_0$ larger than both
+supports, and enlarge it further until
+$aCe^{-cR_0}\le1/2$ in \eqref{peaks:off-patch}.
+\end{proof}
+
+The plateau $0\le y\le\alpha$ is exactly the region where the
+model peak $ae^{-y}$ has modulus at least one.  For the profiles
+just constructed, the integral of
+$\log|1-ae^{-y}e^{it'}|^2$ against
+$(1+f(y)\cos t')\,dy\,dt'/(2\pi)$ on
+$[0,R_0]\times[0,2\pi]$ equals
+$\alpha^2-2s_1(\cosh\alpha-1)$, as calculated in
+\eqref{tests:model-sign}.  Thus the strict profile inequality
+prescribes a negative logarithmic integral, while the weighted
+cancellation will preserve holomorphic moments.
+
+Fix these profiles and $R_0$ from now on.  The spacing $D_0$ and
+the degrees will remain available to be chosen sufficiently large.
+In particular, enlarging $D_0$ later improves the single-peak
+approximation without altering the profiles.
diff --git a/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/sections/04-densities-construction.tex b/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/sections/04-densities-construction.tex
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@@ -0,0 +1,212 @@
+\section{Positive representing densities}\label{sec:densities}
+
+We now construct the representing probabilities required by
+Lemma~\ref{peaks:mean-criterion}.  Each step adds an oscillation on
+the current patches.  A radial integration by parts makes every
+holomorphic moment of that oscillation vanish exactly.  The
+separation between consecutive degrees will ensure that the new
+density stays positive.
+
+Keep the profiles $f,b,R_0$ from Lemma~\ref{peaks:profiles} fixed,
+and put
+\[
+ c_0=\frac{1-\|f\|_\infty}{4}>0.
+\]
+Fix any $D_0>\max\{1,2\sqrt{2R_0}\}$.  For an integer $Q\ge2$
+large enough that every degree below meets the patch thresholds,
+put
+\begin{equation}\label{density:degrees}
+ k_j=Q^j,\qquad \ell_0=0,\qquad
+ \ell_j=\sum_{l=1}^j k_l\quad(j\ge1).
+\end{equation}
+Thus $\ell_{j-1}<k_j$ and $\ell_j<2k_j$.  At each degree use any
+of the separated sets and patches constructed in Section~\ref{sec:peaks}.
+
+Start with $W_0=1$.  At stage $j$, set $k=k_j$.  For each
+$p\in\mathcal P_k$ let
+\[
+ S_p(\theta)=W_{j-1}(e^{i\theta}p).
+\]
+To determine the correction, consider a phase mode $e^{is\theta}$
+with $|s|\le\ell_{j-1}$, the frequency range maintained below.
+Multiplication by $e^{ik\theta}$ shifts it to frequency
+$m=k+s>0$; its conjugate can pair with the holomorphic monomial
+$\xi_1^m=e^{-my/k}e^{im\theta}$.  After canceling the surface-measure factor
+$e^{-2y/k}$, its radial moment will vanish if its coefficient is
+$b'-(m/k)b$, because
+\[
+ e^{-my/k}\left(b'-\frac{m}{k}b\right)
+ =\frac{d}{dy}\left(e^{-my/k}b\right),
+ \qquad b(0)=b(R_0)=0.
+\]
+The identity $f=b'-b$ therefore prescribes the coefficient
+$f-(s/k)b$ for this mode.  Summing the modes leads to the following
+definition in the adapted coordinates of $\mathcal U_{k,p}$:
+\begin{equation}\label{density:recursion}
+ \begin{split}
+ A_{j,p}(\xi)
+ &=e^{2y/k}\operatorname{Re}\left[
+     e^{ik\theta}\left(f(y)S_p(\theta)
+               -b(y)\frac{S_p'(\theta)}{ik}\right)\right],\\
+ W_j&=W_{j-1}+\sum_{p\in\mathcal P_k}A_{j,p},
+ \end{split}
+\end{equation}
+where each correction is extended by zero off its patch.  The
+prime on $S_p$ denotes differentiation in $\theta$.  This recursion
+is defined without assuming positivity; that property will be
+proved for one fixed sufficiently large $Q$.
+
+\begin{lemma}[Smoothness of the corrections]\label{density:smooth}
+Every $W_j$ defined by \eqref{density:recursion} is a smooth real
+function on $S$.
+\end{lemma}
+
+\begin{proof}
+Proceed by induction, beginning with $W_0=1$.  The restriction
+$S_p$ is a smooth periodic function.  To check the central circle
+of a patch, use the smooth coordinates
+\[
+ (\theta,\zeta)\longmapsto
+       (e^{i\theta}\sqrt{1-|\zeta|^2},\zeta),
+ \qquad y=-\frac{k}{2}\log(1-|\zeta|^2).
+\]
+All terms of \eqref{density:recursion} are smooth in these
+coordinates, including at $\zeta=0$.  They do not depend on the
+transverse polar angle $\gamma$, which would be singular there.
+Their dependence on $\theta$ is periodic, so it defines a smooth
+function along the full central circle.  At the outer boundary,
+$f$ and $b$ vanish in a collar of $R_0$; hence the correction is
+identically zero in a collar of the patch boundary.  Its extension
+by zero is therefore smooth.  Summing the finitely many corrections
+proves the induction.
+\end{proof}
+
+Let $T$ denote the phase derivative on $S$,
+\[
+ TW(\xi)=\left.\frac{d}{d\omega}W(e^{i\omega}\xi)
+                  \right|_{\omega=0}.
+\]
+A horizontal vector $Z\in T_\xi S$ will always mean a
+\emph{Euclidean unit} real tangent vector satisfying the complex
+equation $\langle Z,\xi\rangle=0$.
+
+A function has phase bandwidth at most $\ell$ if, on every orbit
+$\omega\mapsto e^{i\omega}\xi$, it is a trigonometric polynomial
+with frequencies in $[-\ell,\ell]$.  Its orbitwise mean is the
+constant coefficient of this polynomial.
+
+\begin{theorem}[Uniform representing densities]\label{density:main}
+There is a constant $K_*>0$, depending only on the fixed profiles
+$f,b$ and $R_0$, with the following property.  For every fixed
+$D_0>\max\{1,2\sqrt{2R_0}\}$ there is an integer threshold $Q_0$
+such that, for every integer $Q\ge Q_0$ and every choice of the
+separated sets above, the recursion \eqref{density:recursion}
+has the following properties for all $j\ge1$:
+\begin{enumerate}
+\item $W_j$ is smooth and strictly positive, and $W_j\,d\sigma$
+represents holomorphic evaluation at zero.  In particular it is
+a probability measure.
+\item Under global phase rotation, $W_j$ has frequencies only
+between $-\ell_j$ and $\ell_j$, and its mean on every phase orbit
+is exactly one.
+\item At every point of $S$,
+\begin{equation}\label{density:uniform-bounds}
+ \begin{gathered}
+ c_0W_{j-1}\le W_j\le2W_{j-1},\\
+ |ZW_j|\le K_*\sqrt{k_j}\,W_j,
+ \qquad |TW_j|\le K_*k_jW_j
+ \end{gathered}
+\end{equation}
+for every unit horizontal vector $Z$.
+\end{enumerate}
+The threshold $Q_0$ may also be enlarged to impose any fixed lower
+threshold on the degrees $k_j$.
+\end{theorem}
+
+We first prove the exact moment and frequency assertions; they
+hold before positivity is known.  The remainder of the theorem,
+including the order of choices of $K_*$ and $Q_0$, is proved in
+the next subsection.
+
+\subsection{Exact moments and phase frequencies}
+
+\begin{lemma}[Moment preservation]\label{density:moments}
+For every stage of \eqref{density:recursion}, the phase bandwidth
+of $W_j$ is at most $\ell_j$, its orbitwise mean is one, and
+\begin{equation}\label{density:representing}
+ \int_S p(\xi)W_j(\xi)\,d\sigma(\xi)=p(0)
+ \quad\text{for every holomorphic polynomial }p.
+\end{equation}
+Each individual correction has zero integral against every such
+polynomial, including the constant polynomial.
+\end{lemma}
+
+\begin{proof}
+At stage zero the phase assertions are immediate.  The polynomial
+identity for $\sigma$ follows by global phase averaging: every
+nonconstant homogeneous holomorphic polynomial has nonzero
+positive phase frequency.
+
+Assume the assertions through stage $j-1$, and write $k=k_j$.
+The central restriction has a finite expansion
+\[
+ S_p(\theta)=\sum_{|s|\le\ell_{j-1}}\widehat S_p(s) e^{is\theta},
+ \qquad \widehat S_p(0)=1.
+\]
+Since $b'-b=f$, the bracket in \eqref{density:recursion} is
+\[
+ \sum_s \widehat S_p(s)\left(b'-\frac{k+s}{k}b\right)e^{is\theta}.
+\]
+All integers $m=k+s$ are positive because $\ell_{j-1}<k$.
+The patch and $y$ are invariant under global phase rotation,
+whereas $\theta$ increases by the rotation angle.  Hence the
+correction has frequencies in
+\begin{equation}\label{density:bandwidth}
+ [k-\ell_{j-1},k+\ell_{j-1}]
+ \ \cup\ [-k-\ell_{j-1},-k+\ell_{j-1}].
+\end{equation}
+They omit zero.  Multiplying by a function of $y$ and extending
+by zero across the invariant patch boundary do not alter these
+frequencies.  The bandwidth of $W_j$ is thus at most
+$k+\ell_{j-1}=\ell_j$, and its orbitwise mean remains one.
+
+For the full moment calculation, use an adapted monomial
+$\xi_1^n\xi_2^l$, with nonnegative integers $n,l$.  If $l>0$,
+averaging in the transverse phase $\gamma$ gives zero.  If
+$l=n=0$, averaging in $\theta$ gives zero because all $m=k+s$
+are positive.  Suppose then that $l=0$ and $n\ge1$.  As $b$ is
+real, the correction is
+\[
+ A_{j,p}=\frac{e^{2y/k}}2
+ \sum_s\left(\widehat S_p(s) e^{i(k+s)\theta}
+              +\overline{\widehat S_p(s)}e^{-i(k+s)\theta}\right)
+       \left(b'-\frac{k+s}{k}b\right).
+\]
+Since $\xi_1^n=e^{-ny/k}e^{in\theta}$, only the conjugate
+frequency with $k+s=n$ can survive the $\theta$ integration.
+Formula~\eqref{peaks:patch-measure} gives exactly
+\[
+ \begin{split}
+ \int_{\mathcal U_{k,p}}\xi_1^n A_{j,p}\,d\sigma
+ &=\frac{\overline{\widehat S_p(n-k)}}{k}
+   \int_0^{R_0}e^{-ny/k}\left(b'-\frac nk b\right)dy\\
+ &=\frac{\overline{\widehat S_p(n-k)}}{k}
+   \left[e^{-ny/k}b(y)\right]_0^{R_0}=0.
+ \end{split}
+\]
+Here a coefficient outside the previous bandwidth is zero.  The
+factor $2/k$ in surface measure combines with the $1/2$ from the
+real part to leave $1/k$.  Both endpoint terms vanish because
+$b(0)=b(R_0)=0$.
+The argument applies to every degree $n$, without a degree cutoff.
+Unitary changes of coordinates preserve holomorphic polynomials,
+so each correction annihilates every polynomial in the original
+coordinates.  Adding the corrections proves
+\eqref{density:representing} and completes the induction.
+\end{proof}
+
+In particular, every $W_j$ has total integral one.  What remains
+for Theorem~\ref{density:main} is to prove that these exact
+representing densities are positive with the uniform derivative
+bounds.  The next estimates accomplish this with a single choice
+of $Q$, independent of the number of stages.
diff --git a/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/sections/04-densities-estimates.tex b/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/sections/04-densities-estimates.tex
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@@ -0,0 +1,285 @@
+\subsection{Uniform positivity and derivative estimates}
+
+We now prove the remaining assertions of Theorem~\ref{density:main}.
+The main point is to choose one derivative constant \(K_*\), followed
+by one threshold for \(Q\), that works at every stage.  A phase
+derivative of the correction in \eqref{density:recursion} involves
+\(S_p''\).  We obtain a bound for this derivative from the frequency
+support and the estimates already proved at earlier stages.
+
+\begin{lemma}[Growth along phase orbits]\label{density:orbit-growth}
+Suppose \(Q\ge2\), \(k_j=Q^j\), and the positive functions
+\(W_0=1,W_1,\ldots,W_n\), with \(n\ge1\), satisfy
+\[
+ c_0W_{j-1}\le W_j\le2W_{j-1},
+ \qquad |TW_j|\le K_*k_jW_j
+ \quad(1\le j\le n).
+\]
+Put
+\[
+ R=\frac2{c_0},\qquad \beta=\log_2R.
+\]
+For every \(p\in S\), the function
+\(S_p(\theta)=W_n(e^{i\theta}p)\) satisfies
+\begin{equation}\label{density:orbit-growth-bound}
+ S_p(\theta+u)
+ \le e^{K_*}R(1+k_n|u|)^\beta S_p(\theta),
+ \qquad |u|\le\pi.
+\end{equation}
+In particular, the constants in this bound are independent of \(Q,n,p\).
+\end{lemma}
+
+\begin{proof}
+Choose the largest \(1\le l\le n\) such that \(k_l|u|\le1\),
+or put \(l=0\) if no such index exists.  The comparisons between
+successive densities give
+\[
+ \frac{W_n(e^{i(\theta+u)}p)}{W_n(e^{i\theta}p)}
+ \le R^{n-l}
+ \frac{W_l(e^{i(\theta+u)}p)}{W_l(e^{i\theta}p)}.
+\]
+When \(l\ge1\), logarithmic differentiation and the phase derivative
+bound show that the last ratio is at most
+\(\exp(K_*k_l|u|)\le e^{K_*}\).
+For \(l=0\) it equals one.
+If \(l=n\), this already proves the assertion.
+Otherwise the maximality of \(l\), including the case \(l=0\), gives
+\[
+ k_n|u|>Q^{n-l-1}\ge2^{n-l-1}.
+\]
+It follows that
+\(R^{n-l}\le R(1+k_n|u|)^\beta\), proving
+\eqref{density:orbit-growth-bound}.
+\end{proof}
+
+\begin{lemma}[A weighted derivative bound]\label{density:weighted-derivative}
+Fix \(A\ge1\) and \(\beta\ge0\).  Let \(q\ge1\) be an integer,
+and let \(s\) be a positive trigonometric polynomial of degree at most
+\(2q\).  Suppose that
+\[
+ s(\theta+u)\le A(1+q|u|)^\beta s(\theta)
+ \qquad(\theta\in\mathbb R,\ |u|\le\pi).
+\]
+There is a constant \(C(A,\beta)\), independent of \(q,s\), such that
+\[
+ |s''(\theta)|\le C(A,\beta)q^2s(\theta).
+\]
+\end{lemma}
+
+\begin{proof}
+Choose a fixed smooth compactly supported function
+\(\chi:\mathbb R\to\mathbb R\) which equals one on \([-2,2]\), and put
+\[
+ K_q(u)=\sum_{h\in\mathbb Z}\chi(h/q)e^{ihu}.
+\]
+For normalized convolution on the circle, the frequency assumption
+implies \(s''=s*K_q''\).  We claim that, for every positive integer \(N\),
+\begin{equation}\label{density:kernel-decay}
+ |K_q''(u)|\le C_Nq^3(1+q|u|)^{-N},
+ \qquad |u|\le\pi,
+\end{equation}
+where \(C_N\) depends only on \(N,\chi\).
+
+Indeed the coefficients of \(K_q''\) are
+\[
+ a_h=-h^2\chi(h/q)=q^2g(h/q),
+ \qquad g(v)=-v^2\chi(v).
+\]
+There are \(O(q)\) nonzero coefficients, each of size \(O(q^2)\),
+so direct summation gives \(|K_q''|\le Cq^3\).
+An \(N\)-th finite difference of \(a_h\) is an iterated integral of
+the \(N\)-th derivative of \(q^2g(v/q)\), and hence has size at most
+\(C_Nq^{2-N}\).  Only \(O_N(q)\) such differences can be nonzero,
+since \(g\) is compactly supported and \(q\ge1\).  Their absolute sum
+is therefore at most \(C_Nq^{3-N}\).
+Multiplication of the Fourier series by \((1-e^{iu})^N\) produces
+these differences, up to an index shift.  Using
+\(|1-e^{iu}|\ge2|u|/\pi\) on \([-\pi,\pi]\) gives the second estimate
+\[
+ |K_q''(u)|\le C_Nq^{3-N}|u|^{-N}\qquad(u\ne0).
+\]
+Combining it with the direct estimate proves
+\eqref{density:kernel-decay}.
+
+Choose an integer \(N>\beta+1\).
+Positivity, the assumed growth bound, and
+\eqref{density:kernel-decay} now give
+\[
+ \begin{split}
+ |s''(\theta)|
+ &\le\int_{-\pi}^{\pi}s(\theta-u)|K_q''(u)|
+                         \,\frac{du}{2\pi}\\
+ &\le AC_Nq^3s(\theta)
+       \int_{-\pi}^{\pi}(1+q|u|)^{\beta-N}\,\frac{du}{2\pi}\\
+ &\le C(A,\beta)q^2s(\theta).
+ \end{split}
+\]
+\end{proof}
+
+Combining these two lemmas with the bandwidth assertion of
+Lemma~\ref{density:moments} gives the following consequence.
+If the estimates of Theorem~\ref{density:main} hold through stage
+\(j-1\), then for \(k=k_j\) and the central function \(S_p\) in
+\eqref{density:recursion},
+\begin{equation}\label{density:central-derivatives}
+ \frac{|S_p'|}{kS_p}\le\frac{K_*}{Q},
+ \qquad
+ \frac{|S_p''|}{k^2S_p}\le\frac{C_1}{Q^2},
+\end{equation}
+where \(C_1\) depends only on \(K_*,c_0\) and the fixed cutoff
+\(\chi\).  It is independent of \(Q,j,p\).
+For \(j\ge2\), apply the lemmas with \(q=k_{j-1}\), noting that
+\(\ell_{j-1}\le2k_{j-1}\); the first inequality follows directly
+from the phase derivative bound.  For \(j=1\), both derivatives vanish
+because \(S_p=1\).
+
+We next estimate a single correction using only normalized bounds.
+This will allow us to choose \(K_*\) before requiring
+\eqref{density:central-derivatives} to be small.
+All horizontal vectors \(Z\) below have Euclidean length one.
+Enlarge a constant \(D_R\ge1\), depending only on \(R_0\), so that
+on every patch at every sufficiently large degree \(k\),
+\begin{equation}\label{density:patch-directions}
+ |Yy|\le D_R,\qquad |Y\theta|\le D_R/k,\qquad
+ |Y(e^{ik\theta})|\le D_R,
+ \quad Y=Z/\sqrt{k}\ \hbox{or}\ Y=T/k.
+\end{equation}
+Here is a direct verification.  With \(c=\langle\xi,p\rangle\),
+complex horizontality gives
+\[
+ |Zc|\le\sqrt{1-|c|^2}
+ \le\sqrt{2R_0/k},\qquad |c|\ge e^{-R_0/k}.
+\]
+Differentiate \(y=-k\log|c|\) and \(\theta=\arg c\) to obtain
+the first two bounds for \(Z/\sqrt{k}\); the third follows from
+the second.  For \(T/k\), one has \(Yy=0\) and \(Y\theta=1/k\).
+The fixed lower threshold for \(k\) makes \(e^{-R_0/k}\ge1/2\).
+We also choose \(D_R\) large enough to bound the length of the radial
+arc from \(e^{i\theta}p\) to each patch point by \(D_R/\sqrt{k}\).
+In adapted coordinates this arc is
+\[
+ u\longmapsto(e^{i\theta}\cos u,e^{i\gamma}\sin u),
+ \qquad 0\le u\le \arccos(e^{-y/k}).
+\]
+It has unit speed and is complex-horizontal.  The length bound follows
+from \(0\le y\le R_0\) and \(1-e^{-y/k}\le R_0/k\).
+Fix this enlarged \(D_R\) throughout the rest of the proof.
+
+\begin{lemma}[Normalized correction estimate]
+\label{density:normalized-patch}
+Write
+\[
+ F=\|f\|_\infty,\quad B=\|b\|_\infty,\quad
+ F_1=\|f'\|_\infty,\quad B_1=\|b'\|_\infty.
+\]
+On a degree-\(k\) patch suppose \(W_{j-1}>0\), \(S_p>0\), and
+\[
+ \frac{e^{2y/k}S_p}{W_{j-1}}\le2,\qquad
+ r_1=\frac{|S_p'|}{kS_p}\le1,\qquad
+ r_2=\frac{|S_p''|}{k^2S_p}\le1.
+\]
+For the correction \(A_{j,p}\) in \eqref{density:recursion},
+\[
+ |YA_{j,p}|\le C_2W_{j-1},
+ \quad Y=Z/\sqrt{k}\ \hbox{or}\ T/k,
+ \qquad
+ C_2=2D_R\bigl[4(F+B)+F_1+B_1\bigr].
+\]
+In particular, \(C_2\) is independent of \(K_*,Q,j,p\).
+\end{lemma}
+
+\begin{proof}
+Besides \eqref{density:patch-directions}, use
+\[
+ |Y(e^{2y/k})|\le(2D_R/k)e^{2y/k},\qquad
+ |YS_p|\le(D_R/k)|S_p'|,\qquad
+ |YS_p'|\le(D_R/k)|S_p''|.
+\]
+Differentiating \eqref{density:recursion} and taking absolute values gives
+\[
+ |YA_{j,p}|
+ \le D_Re^{2y/k}S_p
+ \left[(1+2/k)(F+Br_1)+F_1+B_1r_1+Fr_1+Br_2\right].
+\]
+Since \(k\ge1\), the assumed normalized bounds imply the stated estimate.
+\end{proof}
+
+\begin{proof}[Completion of the proof of Theorem~\ref{density:main}]
+Fix \(K_*>2C_2/c_0\).  The constant \(C_1\) in
+\eqref{density:central-derivatives} is now fixed as well.
+We give a finite list of requirements on \(Q\), all independent of
+the induction stage.
+
+The horizontal derivative estimate at stage \(j-1\), integrated along
+the radial arc just described, will imply
+\begin{equation}\label{density:central-comparison}
+ e^{-K_*D_R/\sqrt Q}
+ \le\frac{S_p(\theta)}{W_{j-1}(\xi)}
+ \le e^{K_*D_R/\sqrt Q}.
+\end{equation}
+Indeed the logarithmic derivative along a unit horizontal arc is at most
+\(K_*\sqrt{k_{j-1}}\), and the arc length is at most \(D_R/\sqrt{k_j}\).
+At the initial stage the ratio is exactly one.
+Since \(k_j\ge Q\), put
+\[
+ H(Q)=\exp\left(K_*D_R/\sqrt Q+2R_0/Q\right).
+\]
+
+Choose a threshold \(Q_0\) such that every integer \(Q\ge Q_0\)
+satisfies all earlier degree and patch requirements, and also
+\begin{equation}\label{density:Q-conditions}
+ \begin{gathered}
+ Q\ge2,\qquad H(Q)\le2,\qquad
+ K_*/Q\le1,\qquad C_1/Q^2\le1,\qquad
+ Q^{-1/2}\le c_0/2,\\
+ H(Q)(F+BK_*/Q)\le1-2c_0.
+ \end{gathered}
+\end{equation}
+These conditions are compatible: \(H(Q)\to1\) and
+\(F=1-4c_0<1-2c_0\).
+
+We now fix any integer \(Q\ge Q_0\) and induct on \(j\).
+Assume the claimed estimates through stage \(j-1\); at \(j=1\)
+the preceding density is the constant one.
+Equation~\eqref{density:central-derivatives}, obtained from earlier
+stages only, and the horizontal comparison
+\eqref{density:central-comparison} establish the normalized hypotheses
+of Lemma~\ref{density:normalized-patch}.  Moreover,
+\[
+ |A_{j,p}|
+ \le e^{2y/k}S_p(F+Br_1)
+ \le H(Q)(F+BK_*/Q)W_{j-1}
+ \le(1-2c_0)W_{j-1}.
+\]
+At any point at most one patch correction is nonzero.  Hence
+\[
+ 2c_0W_{j-1}\le W_j\le(2-2c_0)W_{j-1},
+\]
+which proves positivity and the required multiplicative comparison.
+
+For \(Y=Z/\sqrt{k_j}\), the previous derivative bound gives
+\(|YW_{j-1}|\le(K_*/\sqrt Q)W_{j-1}\).
+For \(Y=T/k_j\), it gives the smaller coefficient \(K_*/Q\).
+Together with Lemma~\ref{density:normalized-patch},
+\eqref{density:Q-conditions}, and the choice of \(K_*\), these yield
+\[
+ |YW_j|\le(C_2+K_*/\sqrt Q)W_{j-1}
+ \le K_*c_0W_{j-1}\le K_*W_j.
+\]
+Outside the patches the correction vanishes.  Its smooth zero
+extension at their boundaries gives the same estimate there.
+Multiplying by \(\sqrt{k_j}\) or \(k_j\) proves the two derivative
+bounds and completes the induction.
+
+Lemma~\ref{density:moments} supplies the exact moments, rotation mean
+and bandwidth independently of these inequalities.  In particular
+\(\int_SW_j\,d\sigma=1\); positivity therefore makes
+\(W_j\,d\sigma\) a probability measure.
+\end{proof}
+
+We emphasize the order of the choices.  The normalized correction
+constant \(C_2\) depends only on the fixed patch data.  It determines
+\(K_*\); this in turn determines \(C_1\); finally \(Q_0\) is chosen.
+No lower bound for \(W_j\) uniform in \(j\) is required.  The positive
+comparison at each stage suffices for logarithmic differentiation,
+and all later integral estimates will use the probability normalization.
diff --git a/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/sections/05-logarithmic-tests.tex b/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/sections/05-logarithmic-tests.tex
new file mode 100644
index 0000000000000000000000000000000000000000..7f712ec17944572c6c5581135b1869a275cd37fc
--- /dev/null
+++ b/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/sections/05-logarithmic-tests.tex
@@ -0,0 +1,275 @@
+\section{Regularized logarithmic tests}\label{sec:tests}
+
+Keep the profile parameters $a,\alpha,s_1,f,b,R_0$ from
+Lemma~\ref{peaks:profiles}.  We shall construct a smooth
+plurisubharmonic function at each degree whose average against its
+representing density is uniformly negative.  The same negative bound
+will hold against all later densities.  The estimates are uniform for
+all sufficiently large integers $Q$, so the final value of $Q$ can be
+chosen after the regularization and spacing parameters.
+
+A positive measure representing holomorphic evaluation need not satisfy
+the logarithmic inequality
+$\log|f(0)|\le\int_S\log|f|\,d\mu$ for every holomorphic $f$
+continuous on the closed ball. Hedenmalm gives an explicit example of
+this distinction between representing and Jensen measures
+\cite[Theorem~3.7]{Hedenmalm1989}.
+Here we need smooth logarithmic tests and negative bounds that persist
+through all later density corrections. We prove these properties
+directly for the densities of Theorem~\ref{density:main}.
+
+Fix a constant $C_P$ in the bound
+$|P_k(z)|\le C_P|z|^k$ of Lemma~\ref{peaks:bounds}.
+
+For $\epsilon>0$, define
+\begin{equation}\label{tests:regularized}
+ F_\epsilon(v)=
+ \log\frac{|1-av|^2+\epsilon^2}{1+\epsilon^2},
+ \qquad U_k(z)=F_\epsilon(P_k(z)).
+\end{equation}
+In particular $F_\epsilon(0)=U_k(0)=0$.  The function $U_k$ is smooth
+and plurisubharmonic on $\mathbb C^2$, since
+\begin{equation}\label{tests:hessian}
+ (U_k)_{i\bar j}
+ =\frac{a^2\epsilon^2}
+        {(|1-aP_k|^2+\epsilon^2)^2}
+   (P_k)_i\overline{(P_k)_j}.
+\end{equation}
+For fixed $\epsilon$ and $A<\infty$, all real derivatives of
+$F_\epsilon$ of any fixed order are bounded on $|v|\le A$.
+The mean-value estimate also gives
+\begin{equation}\label{tests:lipschitz}
+ |F_\epsilon(v)|\le C_{\epsilon,A}|v|
+ \qquad (|v|\le A).
+\end{equation}
+All subsequent regularization constants may depend on the fixed
+$\epsilon$.
+
+\begin{lemma}[Exact averaging]\label{tests:averaging}
+Let $k=k_j=Q^j$, where $Q$ satisfies the hypotheses of
+Theorem~\ref{density:main}.  For $0\le r\le1$,
+\begin{equation}\label{tests:averaging-identity}
+ \int_S U_k(r\xi)W_j(\xi)\,d\sigma
+ =
+ \int_S U_k(r\xi)\,d\sigma
+ +\sum_p\int_{\mathcal U_{k,p}}
+     U_k(r\xi)e^{2y/k}f(y)\cos(k\theta)\,d\sigma.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Under the global phase rotation $\xi\mapsto e^{i\omega}\xi$,
+homogeneity gives $P_k(e^{i\omega}\xi)=e^{ik\omega}P_k(\xi)$.
+Thus $U_k(r\xi)$ has only phase frequencies in $k\mathbb Z$.
+By Lemma~\ref{density:moments}, the phase bandwidth of $W_{j-1}$
+is $\ell_{j-1}<k$ and its constant coefficient on each orbit is one.
+Averaging the product over the phase therefore gives
+\[
+ \int_S U_k(r\xi)W_{j-1}(\xi)\,d\sigma
+ =\int_S U_k(r\xi)\,d\sigma.
+\]
+
+Each patch is phase-invariant.  Its radial variable $y$ is fixed
+under rotation, while $\theta$ and $\gamma$ both increase by $\omega$.
+Write the central restriction as
+$S_p(\theta)=W_{j-1}(e^{i\theta}p)
+ =\sum_s\widehat S_p(s)e^{is\theta}$.
+In the patch increment of~\eqref{density:recursion}, the frequencies
+are $\pm(k+s)$ with $|s|\le\ell_{j-1}<k$.  Pairing with a frequency in
+$k\mathbb Z$ is possible only for $s=0$.
+Since $\widehat S_p(0)=1$ and $S_p'$ has zero constant coefficient,
+the surviving part of the increment
+is exactly $e^{2y/k}f(y)\cos(k\theta)$.  Integrating this phase average
+proves~\eqref{tests:averaging-identity}.
+\end{proof}
+
+The next estimate chooses the regularization before the spacing.
+This order matters: the single-peak approximation will be applied
+to a fixed smooth logarithm, and its error can then be made small by
+increasing $D_0$.
+
+\begin{proposition}[Negative averages at one scale]\label{tests:negative}
+There exist $\epsilon>0$, $\eta\in(0,1/2)$ and $D_*\ge1$, depending
+only on the fixed profile data, with the following property.
+For every $D_0\ge D_*$ there are $k_*>0$ and $d_2>0$ such that, for
+every integer $Q$ satisfying the density theorem and $Q\ge k_*$,
+\begin{equation}\label{tests:one-scale-negative}
+ \int_S U_{k_j}(r\xi)W_j(\xi)\,d\sigma\le-d_2
+ \qquad
+ \bigl(j\ge1,\quad 1-\eta\le r^{k_j}\le1\bigr).
+\end{equation}
+The constants $k_*,d_2$ may depend on $D_0$, but are independent of
+$j$ and of further increases in $Q$.
+\end{proposition}
+
+\begin{proof}
+Put
+\[
+ \Delta=2s_1(\cosh\alpha-1)-\alpha^2>0.
+\]
+We first compute the integral for a single peak:
+\begin{equation}\label{tests:model}
+ I_\epsilon(\rho)=
+ \int_0^{R_0}\int_0^{2\pi}
+ \log\frac{|1-a\rho e^{-y}e^{it'}|^2+\epsilon^2}
+               {1+\epsilon^2}
+ (1+f(y)\cos t')\,\frac{dt'}{2\pi}\,dy.
+\end{equation}
+At $\rho=1$, the unregularized logarithm has only one singularity,
+at $(y,t')=(\alpha,0)$ modulo $2\pi$.  Near that point,
+\[
+ |1-e^{\alpha-y+it'}|^2
+ \asymp (y-\alpha)^2+(t')^2.
+\]
+Its logarithm is locally integrable.  For $0<\epsilon\le1$, the
+regularized logarithm is bounded above uniformly, and its negative
+part is bounded by the negative part of the unregularized logarithm
+plus $\log2$.  Dominated convergence therefore applies in
+\eqref{tests:model}.
+
+For $q<1$, the mean and cosine moment of
+$\log|1-qe^{it'}|^2$ are $0$ and $-q$, respectively, by the logarithm
+series.  For $q>1$, factoring out $q^2$ gives mean $2\log q$ and
+cosine moment $-q^{-1}$.  Using the cancellation
+$\int_0^{R_0}e^{-y}f(y)\,dy=0$, we obtain
+\begin{align}
+ I_0(1)
+ &=\alpha^2-\int_0^\alpha f(y)e^{y-\alpha}\,dy
+             -\int_\alpha^{R_0}f(y)e^{\alpha-y}\,dy \notag\\
+ &=\alpha^2+\int_0^\alpha
+       f(y)\bigl(e^{\alpha-y}-e^{-(\alpha-y)}\bigr)\,dy \notag\\
+ &=\alpha^2-2s_1(\cosh\alpha-1)=-\Delta.
+ \label{tests:model-sign}
+\end{align}
+The last equality uses $f=-s_1$ on $[0,\alpha]$.
+
+We now compare this model with~\eqref{tests:averaging-identity}.
+For $k=k_j$ put $\rho=r^k$, so homogeneity gives
+$U_k(r\xi)=F_\epsilon(\rho P_k(\xi))$.
+Off the patches, $|aP_k|\le1/2$ by Lemma~\ref{peaks:profiles}.
+If $v=\rho P_k(\xi)$, $0\le\rho\le1$, and $s=|1-av|^2$, then
+$1/4\le s\le9/4$ and
+\[
+ F_\epsilon(v)-\log|1-av|^2
+ =\log(1+\epsilon^2/s)-\log(1+\epsilon^2).
+\]
+The derivative in $s$ of the first term has magnitude at most
+$16\epsilon^2$, and $|s-1|\le C|v|$.  Hence
+\[
+ |F_\epsilon(v)-\log|1-av|^2|
+ \le C\epsilon^2|P_k(\xi)|.
+\]
+The off-patch set is phase-invariant, and the phase mean of the
+unregularized logarithm is zero.  By the $L^1$ estimate of
+Lemma~\ref{peaks:bounds}, its total contribution is thus at most
+\begin{equation}\label{tests:off-patch}
+ C_{\rm off}\epsilon^2\,\frac{N_k}{k}.
+\end{equation}
+Here $C_{\rm off}$ is independent of $D_0,k,\rho$ and
+$0<\epsilon\le1$.
+
+On a patch, the measure formula of Lemma~\ref{peaks:patches} writes
+the combined contribution as
+\[
+ \frac2k\int_0^{R_0}\int_{\mathbb T^2}
+ F_\epsilon(\rho P_k(\xi))
+       \bigl(e^{-2y/k}+f(y)\cos(k\theta)\bigr)
+       \,\frac{d\theta\,d\gamma}{(2\pi)^2}\,dy.
+\]
+For fixed $\epsilon,R_0$, replace $P_k(\xi)$ by
+$e^{-y}e^{ik\theta}$ using the single-peak estimate, and replace
+$e^{-2y/k}$ by one.  The bounded derivative of $F_\epsilon$ and the
+bound $|e^{-2y/k}-1|\le2R_0/k$ show that the result differs from
+$(2/k)I_\epsilon(\rho)$ by at most
+\begin{equation}\label{tests:patch-error}
+ \frac2k C_{\epsilon,R_0}
+ \left(e^{-c(D_0-\sqrt{2R_0})^2}+\frac1k\right).
+\end{equation}
+All constants are uniform over patches and $0\le\rho\le1$.
+The change from $k\theta$ to $t'$ in the angular average is exact,
+as $k$ is an integer.
+
+Choose $\epsilon\in(0,1]$ so small that
+\[
+ |I_\epsilon(1)+\Delta|\le\Delta/8,
+ \qquad C_{\rm off}\epsilon^2\le\Delta/8.
+\]
+By continuity of~\eqref{tests:model} in $\rho$, choose
+$\eta\in(0,1/2)$ such that
+$I_\epsilon(\rho)\le-3\Delta/4$ for $1-\eta\le\rho\le1$.
+Next choose $D_*$ large enough for patch disjointness and for the
+first error in~\eqref{tests:patch-error} to be at most $\Delta/16$
+inside its factor $2/k$.  For each $D_0\ge D_*$, take $k_*$ large
+enough for all the geometric estimates and for the remaining error
+inside this factor to be at most $\Delta/16$.
+Summing the patches and adding~\eqref{tests:off-patch} gives, in
+particular,
+\[
+ \int_S U_k(r\xi)W_j\,d\sigma
+ \le-\frac{\Delta}{2}\frac{N_k}{k}
+ \le-\frac{\Delta}{2D_0^2}.
+\]
+Thus $d_2=\Delta/(2D_0^2)$ is a valid choice.
+\end{proof}
+
+\begin{lemma}[Negative averages against later densities]\label{tests:later}
+Fix $\epsilon,\eta,D_0,k_*,d_2$ as in
+Proposition~\ref{tests:negative}, and put $d_3=d_2/2$.
+There is a threshold $Q_*$ such that, for every integer $Q\ge Q_*$,
+the densities satisfy
+\begin{equation}\label{tests:later-negative}
+ \int_S U_{k_j}(r\xi)W_N(\xi)\,d\sigma\le-d_3
+ \qquad
+ \bigl(N\ge j\ge1,\quad1-\eta\le r^{k_j}\le1\bigr).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Include the density and degree thresholds in $Q_*$, and require
+$Q\ge4$.  By Lemma~\ref{density:moments}, the increment
+$W_\nu-W_{\nu-1}$ has frequencies within $\ell_{\nu-1}$ of
+$\pm k_\nu$.  Since $\ell_{\nu-1}\le2k_{\nu-1}$, each such
+frequency has absolute value at least $(Q-2)k_{\nu-1}$.
+Summing the increments with $\nu>j$ shows that every frequency in
+$W_N-W_j$ has absolute value at least $(Q-2)k_j$.
+
+For fixed $\xi,r$, write the logarithmic profile as
+\[
+ V(t')=\log\frac{|1-Ae^{it'}|^2+\epsilon^2}{1+\epsilon^2},
+ \qquad A=a r^{k_j}P_{k_j}(\xi).
+\]
+The uniform polynomial bound gives $|A|\le aC_P$.  Differentiating
+the logarithm shows that
+\[
+ \|V''\|_\infty\le
+ B_\epsilon:=
+ \frac{2aC_P}{\epsilon^2}
+ +\frac{4a^2C_P^2}{\epsilon^4}.
+\]
+Two integrations by parts bound its $m$th Fourier coefficient by
+$B_\epsilon/m^2$ for $m\ne0$. To track these coefficients as functions
+of $\xi$, define
+\[
+ c_m(\xi)=\int_0^{2\pi}
+ F_\epsilon\bigl(r^{k_j}P_{k_j}(\xi)e^{it'}\bigr)
+ e^{-imt'}\,\frac{dt'}{2\pi}.
+\]
+Homogeneity and a change of variable give
+$c_m(e^{i\omega}\xi)=e^{imk_j\omega}c_m(\xi)$.
+Thus $c_m$ has global phase frequency $mk_j$, and
+$U_{k_j}(r\xi)=\sum_{m\in\mathbb Z}c_m(\xi)$.
+The coefficient bound makes this series uniformly convergent.
+Its truncation to $|m|\le Q-3$ has uniform error at most
+$2B_\epsilon/(Q-3)$ and only frequencies of absolute value less
+than $(Q-2)k_j$. Phase averaging therefore makes its integral
+against $W_N-W_j$ vanish.
+Both densities are positive probabilities by
+Theorem~\ref{density:main}; hence
+$\|W_N-W_j\|_{L^1(\sigma)}\le2$.  It follows that
+\[
+ \left|\int_S U_{k_j}(r\xi)(W_N-W_j)\,d\sigma\right|
+ \le\frac{4B_\epsilon}{Q-3}.
+\]
+Choose $Q_*$ so that this is at most $d_2/2$, and apply
+\eqref{tests:one-scale-negative}.
+\end{proof}
diff --git a/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/sections/06-base-potential.tex b/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/sections/06-base-potential.tex
new file mode 100644
index 0000000000000000000000000000000000000000..44034747a844737b03d0fa16ebbcfe5e7d93ac0a
--- /dev/null
+++ b/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/sections/06-base-potential.tex
@@ -0,0 +1,266 @@
+\section{The controlled base potential}\label{sec:base-potential}
+
+We now sum the logarithmic tests to prove
+Proposition~\ref{base:existence}.  Fix parameters and an integer $Q$
+for which Theorem~\ref{density:main} and Lemma~\ref{tests:later}
+hold, and write $k_j=Q^j$.  For a finite amplitude $L>0$, set
+\begin{equation}\label{base:potential}
+ \phi(z)=\psi(z)+L\sum_{j=1}^\infty U_{k_j}(z).
+\end{equation}
+We first justify this definition.  We then choose $L$ to exclude
+the required pluriharmonic minorant, and finally prove the uniform
+chart estimates needed by the metric construction.
+
+\begin{lemma}[Smooth convergence]\label{base:convergence}
+For every finite $L>0$, the series in~\eqref{base:potential}
+converges locally in $C^\infty(\mathbb B)$.  Its sum is smooth,
+satisfies $\phi(0)=0$, and has complex Hessian $h\ge I$.
+\end{lemma}
+
+\begin{proof}
+Fix $0<r_0<r_1<1$.  Around each point of $|z|\le r_0$, the complex
+polydisk of radius $(r_1-r_0)/(2\sqrt2)$ lies in $|z|<r_1$.
+Cauchy estimates there, together with Lemma~\ref{peaks:bounds}, give
+\[
+ \sup_{|z|\le r_0}|D^mP_k(z)|
+ \le C_{m,r_0,r_1}r_1^k
+\]
+for every fixed derivative order $m$.  Here and below $D^m$ denotes
+any real partial derivative of total order $m$.
+The values of $P_k$ remain in a fixed bounded disk.
+Every positive-order derivative of $F_\epsilon(P_k)$ is a sum of
+products containing at least one positive-order derivative of
+$P_k$ or its conjugate.  The bounded derivatives of $F_\epsilon$
+therefore give the same exponential estimate for $D^mU_k$.
+For $m=0$, use~\eqref{tests:lipschitz}.
+Since $\sum_j r_1^{Q^j}<\infty$, the series converges in every local
+$C^m$ norm.
+
+Each summand vanishes at zero and is plurisubharmonic by
+\eqref{tests:hessian}.  Termwise differentiation is justified by
+the local $C^2$ convergence, and hence
+\[
+ \partial\bar\partial\phi
+ =I+L\sum_j\partial\bar\partial U_{k_j}\ge I.
+\]
+\end{proof}
+
+\begin{lemma}[Absence of a pluriharmonic minorant]\label{base:minorant}
+If
+\begin{equation}\label{base:amplitude}
+ L>\frac{5\log Q}{d_3},
+\end{equation}
+there are no $H\in\mathcal O(\mathbb B)$ and $C_H\in\mathbb R$
+satisfying~\eqref{base:no-minorant}.
+\end{lemma}
+
+\begin{proof}
+Put
+\begin{equation}\label{base:radii}
+ r_N=1-\frac{\eta}{k_N}.
+\end{equation}
+For every $j\le N$, Bernoulli's inequality gives
+\[
+ r_N^{k_j}
+ =\left(1-\frac{\eta}{k_N}\right)^{k_j}
+ \ge1-\eta\frac{k_j}{k_N}\ge1-\eta.
+\]
+Thus all of the first $N$ modes satisfy the hypothesis of
+\eqref{tests:later-negative} at this same radius.
+For $j=N+m>N$, on the other hand,
+\[
+ r_N^{k_j}\le e^{-\eta k_j/k_N}=e^{-\eta Q^m}.
+\]
+By the uniform polynomial bound and~\eqref{tests:lipschitz},
+\begin{equation}\label{base:tail}
+ \sum_{j>N}\sup_{\xi\in S}|U_{k_j}(r_N\xi)|
+ \le C_\epsilon\sum_{m\ge1}e^{-\eta Q^m}
+ =:C_{\rm tail}<\infty,
+\end{equation}
+independently of $N$.  Because $W_N$ is a positive probability,
+the same bound controls the weighted tail, without requiring a
+uniform pointwise bound on $W_N$.
+
+The base potential contributes only one logarithm:
+\[
+ \psi(r_N\xi)=-\log(1-r_N^2)
+ \le\log\frac1{1-r_N}=\log(k_N/\eta).
+\]
+Combining this with~\eqref{tests:later-negative} and
+\eqref{base:tail} yields
+\begin{align}
+ &\int_S\left(\phi(r_N\xi)
+                  +4\log\frac1{1-r_N}\right)W_N\,d\sigma \notag\\
+ &\hspace{1cm}\le
+ 5\log(k_N/\eta)+L(-d_3N+C_{\rm tail}) \notag\\
+ &\hspace{1cm}=
+ N(5\log Q-Ld_3)+5\log(1/\eta)+LC_{\rm tail}
+ \longrightarrow-\infty.
+ \label{base:negative-means}
+\end{align}
+
+By Theorem~\ref{density:main}, the measures
+$\mu_N=W_N\,d\sigma$ are positive representing probabilities.
+Apply Lemma~\ref{peaks:mean-criterion} to these measures, the radii
+$r_N$, and the continuous function
+$u(z)=\phi(z)+4\log(1/(1-|z|))$.
+The limit~\eqref{base:negative-means} is exactly its hypothesis,
+so the required minorant cannot exist.
+\end{proof}
+
+It remains to control the metric jets as the chart centers
+approach the boundary.  The value of $\phi$ need not be bounded.
+For its derivatives, the low degrees become almost constant in a
+centered chart, while the high degrees are exponentially small.
+
+\begin{lemma}[Uniform chart estimates]\label{base:chart-estimates}
+For every fixed finite $L>0$, the potential~\eqref{base:potential}
+has uniformly bounded positive-order derivatives through order four
+at the origins of all centered ball charts.  Consequently its
+complex Hessian satisfies~\eqref{base:control} and
+\eqref{base:jets}, with $c=1$ and a finite constant $C$.
+\end{lemma}
+
+\begin{proof}
+After a unitary change of coordinates, put the center at $(r,0)$,
+where $0\le r<1$, and let $\delta=1-r$.  Rotating $P_k$ preserves its
+degree and its uniform homogeneous bound.  Use the chart $T_r$ of
+\eqref{base:charts}, on the fixed complex polydisk
+\[
+ D_\rho=\{|\zeta_1|<\rho,\ |\zeta_2|<\rho\},
+ \qquad \rho=\frac1{16}.
+\]
+The denominators in $T_r$ have modulus at least $1-\rho$.
+We estimate the polynomial in the two ranges of $k\delta$.
+
+\paragraph{High degrees: $k\delta>1$.}
+The ball automorphism identity gives, on $D_\rho$,
+\[
+ 1-|T_r(\zeta)|^2
+ =\frac{(1-r^2)(1-|\zeta|^2)}{|1+r\zeta_1|^2}
+ \ge \frac{1-2\rho^2}{(1+\rho)^2}\,\delta.
+\]
+It follows that $1-|T_r(\zeta)|\ge c\delta$, with a fixed $c>0$.
+Thus
+\[
+ |P_k(T_r(\zeta))|\le C_P e^{-ck\delta}.
+\]
+Cauchy estimates in this fixed polydisk bound every fixed-order
+holomorphic derivative at zero, including order zero, by
+$C_m e^{-ck\delta}$.
+
+\paragraph{Low degrees: $k\delta\le1$.}
+Consider the polydisk in the original coordinates
+\[
+ A_k=\left\{|z_1-r|<\frac1k,\quad
+                 |z_2|<\frac1{\sqrt k}\right\}.
+\]
+It may extend outside the ball, but $P_k$ is an entire homogeneous
+polynomial.  The bound of Lemma~\ref{peaks:bounds} therefore holds
+there as well.  In fact
+\[
+ |z|^2\le(1+1/k)^2+1/k\le1+4/k,\qquad
+ |P_k(z)|\le C_P(1+2/k)^k\le C_Pe^2.
+\]
+Cauchy estimates on the concentric half-polydisk give
+\[
+ |\partial_1P_k|\le Ck,\qquad
+ |\partial_2P_k|\le C\sqrt k.
+\]
+Meanwhile the chart displacements satisfy
+\[
+ |T_r^1(\zeta)-r|
+ \le\frac{2\rho}{1-\rho}\delta,\qquad
+ |T_r^2(\zeta)|
+ \le\frac{\sqrt2\rho}{1-\rho}\sqrt\delta.
+\]
+Both numerical coefficients are less than $1/2$.  As
+$\delta\le1/k$, the chart image lies in the half-polydisk just
+used.  Integrating the gradient along the straight segment from
+$(r,0)$ to $T_r(\zeta)$ gives
+\[
+ |P_k(T_r(\zeta))-P_k(r,0)|
+ \le C(k\delta+\sqrt{k\delta})
+ \le2C\sqrt{k\delta}.
+\]
+This segment stays in the half-polydisk by convexity.  Apply
+Cauchy estimates on $D_\rho$ to the holomorphic difference
+$P_k\circ T_r-P_k(r,0)$.  Every positive-order derivative at zero
+is bounded by $C_m\sqrt{k\delta}$.  We make no such assertion for
+the zeroth-order value in this range.
+
+\paragraph{Composition and summation.}
+In both ranges the values of $P_k\circ T_r$ stay in a fixed bounded
+disk.  Each positive-order real derivative of
+$U_k\circ T_r=F_\epsilon(P_k\circ T_r)$ is a sum of products
+containing differentiated factors of $P_k\circ T_r$ or its
+conjugate.  For $1\le m\le4$, the preceding bounds and the smooth
+composition estimate give
+\begin{equation}\label{base:scale-jets}
+ |D^m(U_k\circ T_r)(0)|
+ \le C_{\epsilon,m}
+ \begin{cases}
+  \sqrt{k\delta},&k\delta\le1,\\
+  e^{-ck\delta},&k\delta>1.
+ \end{cases}
+\end{equation}
+Indeed every term contains at least one differentiated factor,
+and a product of several factors obeys the same bound because the
+displayed scale factor is at most one.  This proves the estimate
+for all real derivatives, hence for all pure and mixed complex
+derivatives of these orders.
+
+Let $J$ be the largest index with $Q^J\delta\le1$, or put $J=0$
+if there is no such index.  The low-degree contribution is bounded
+by the geometric sum
+\[
+ \sum_{j\le J}\sqrt{Q^j\delta}
+ \le\frac1{1-Q^{-1/2}}.
+\]
+For the remaining degrees, the first value of $Q^j\delta$ exceeds
+one, so
+\[
+ \sum_{j>J}e^{-cQ^j\delta}
+ \le\sum_{m\ge0}e^{-cQ^m}<\infty.
+\]
+Both bounds are uniform in the chart center.  Termwise
+differentiation is legitimate by Lemma~\ref{base:convergence}.
+Multiplication by the fixed amplitude $L$ preserves these bounds.
+
+Finally, the base potential transforms as
+\[
+ \psi\circ T_r
+ =-\log(1-|\zeta|^2)-\log(1-r^2)
+      +\log|1+r\zeta_1|^2.
+\]
+Its positive-order derivatives through order four at zero are
+uniformly bounded.  The last term is pluriharmonic, so the complex
+Hessian of $\psi\circ T_r$ is exactly the standard ball Hessian in
+these coordinates.  The derivatives of $\phi\circ T_r$ needed for
+the pulled-back Hessian components and their first and second
+derivatives have orders two, three and four.  The bounds just
+proved, together with $h\ge I$ from
+Lemma~\ref{base:convergence}, establish the assertion.
+\end{proof}
+
+\begin{proof}[Proof of Proposition~\ref{base:existence}]
+Choose $a,\alpha,s_1,f,b,R_0$ as in Lemma~\ref{peaks:profiles}.
+Proposition~\ref{tests:negative} first chooses $\epsilon,\eta$ and
+then permits any sufficiently large spacing $D_0$.  Fix such a
+$D_0$ and its degree threshold $k_*$, and fix
+$d_3=d_2/2>0$.  The density theorem holds for every $Q$ above a
+threshold depending only on the fixed patch data and spacing.
+Its derivative constant $K_*$ is fixed before that threshold is
+chosen.  We may therefore choose one integer $Q$ satisfying the
+density threshold, $Q\ge k_*$, and the Fourier-error threshold of
+Lemma~\ref{tests:later}.
+
+Now fix a finite amplitude $L$ satisfying~\eqref{base:amplitude}.
+Lemma~\ref{base:convergence} constructs the smooth potential with
+$h\ge I$; Lemma~\ref{base:minorant} proves the required obstruction;
+and Lemma~\ref{base:chart-estimates} supplies its finite uniform
+chart bounds.  The chart constant is allowed to depend on these
+fixed choices, including $L$.  The metric parameter $\lambda$ in
+Proposition~\ref{metric:transfer} is chosen only afterward from
+these bounds, so none of the choices above depends on $\lambda$.
+\end{proof}
diff --git a/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/sections/07-hartogs-metric.tex b/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/sections/07-hartogs-metric.tex
new file mode 100644
index 0000000000000000000000000000000000000000..b9147ba0a5be15c8fa34f96237e7e7055ba0a0e9
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+++ b/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/sections/07-hartogs-metric.tex
@@ -0,0 +1,173 @@
+\section{The global Hartogs metric}\label{sec:hartogs}
+
+We now turn to Proposition~\ref{metric:transfer}.  In this section we
+prove all its global geometric assertions.  The real sectional-curvature
+bounds will follow from the calculation in Section~\ref{sec:curvature}
+and the estimates in Section~\ref{sec:pinching}.
+
+The potential specializes Calabi's construction on a Hermitian line
+bundle \cite[Section~3]{Calabi1979}. We verify its positivity,
+smoothness, and completeness directly, including at the zero section.
+
+\begin{lemma}\label{metric:complete}
+Let $\phi$ be a smooth real plurisubharmonic function on $\B$, and let
+$\lambda>0$.  The domain $M_\phi$ in \eqref{metric:domain} is
+contractible, and the complex Hessian of
+$\mathcal K_\lambda=\lambda\psi-\log(1-e^\phi|w|^2)$ defines a smooth
+geodesically complete K\"ahler metric
+$g=2\operatorname{Re}\widehat g$ on $M_\phi$.
+Moreover, $\mathcal K_\lambda$ is a smooth strictly plurisubharmonic
+exhaustion of $M_\phi$.
+\end{lemma}
+
+\begin{proof}
+\emph{The domain and a smooth positive metric.}
+The function $t=e^\phi|w|^2$ is smooth on $\B\times\C$, so
+$M_\phi=\{t<1\}$ is open in $\C^3$.  The map
+\[
+ (z,w)\longmapsto (z,e^{\phi(z)/2}w)
+\]
+is a smooth real diffeomorphism from $M_\phi$ to $\B\times\mathbb D$,
+where $\mathbb D=\{u\in\C:|u|<1\}$.  Its inverse is
+$(z,u)\mapsto(z,e^{-\phi(z)/2}u)$.  Thus $M_\phi$ is contractible;
+no holomorphic trivialization is needed for this conclusion.
+
+The potential $\mathcal K_\lambda$ is smooth at every point of
+$M_\phi$, including its zero section.  Write
+\[
+ h=\partial\bar\partial\phi,\qquad
+ x=\frac{t}{1-t},\qquad G=\lambda I+xh,
+ \qquad \vartheta=dw+w\partial\phi.
+\]
+The base covectors together with $\vartheta$ form a global smooth
+complex coframe.  Differentiating the potential gives
+\begin{equation}\label{metric:smooth-form}
+ \widehat g
+ =G+\frac{e^\phi}{(1-t)^2}\,
+       \vartheta\otimes\overline{\vartheta}.
+\end{equation}
+For a complex tangent vector $u=(u_z,u_w)$ this means
+\[
+ \widehat g(u,u)
+ =G(u_z,u_z)
+  +\frac{e^\phi}{(1-t)^2}
+      |u_w+w\partial\phi(u_z)|^2.
+\]
+Since $h\ge0$, the first term controls every nonzero horizontal
+projection through $G\ge\lambda I$; when $u_z=0$, the second term is
+positive unless $u_w=0$.  Thus $\widehat g$ is positive definite.
+At the zero section its matrix is particularly simple:
+\begin{equation}\label{metric:axis}
+ \widehat g\big|_{(z,0)}
+ =\lambda I(z)+e^{\phi(z)}\,dw\otimes d\bar w.
+\end{equation}
+The mixed blocks vanish.  Being a positive complex Hessian of a smooth
+real potential, $\widehat g$ defines the stated K\"ahler metric.
+
+Away from the zero section one can write $s=\log t$ and
+$q=x(1+x)=t/(1-t)^2$.  Then $\partial s=\vartheta/w$, so
+\eqref{metric:smooth-form} also reads
+\begin{equation}\label{metric:log-form}
+ \widehat g=\lambda I+xh+
+             q\,\partial s\otimes\overline{\partial s}.
+\end{equation}
+We will use this expression for the curvature computation.  Formula
+\eqref{metric:smooth-form}, rather than the coordinate $s$, defines the
+metric across $w=0$.
+
+\emph{Two functions with bounded gradient.}
+For a real smooth function $f$, the convention
+$g=2\operatorname{Re}\widehat g$ gives
+\begin{equation}\label{metric:gradient-convention}
+ |\nabla f|_g^2=2|\partial f|_{\widehat g^{-1}}^2.
+\end{equation}
+Indeed, for a real vector $X$ with $(1,0)$ part $u$,
+$g(X,X)=2\widehat g(u,u)$ and
+$df(X)=2\operatorname{Re}\partial f(u)$, and the identity follows
+by taking dual norms.
+
+Globally on $M_\phi$,
+\[
+ \partial[-\log(1-t)]
+ =\frac{e^\phi\bar w}{1-t}\,\vartheta.
+\]
+In the coframe of \eqref{metric:smooth-form}, the Hermitian metric is
+block diagonal.  Therefore
+\[
+ |\partial[-\log(1-t)]|_{\widehat g^{-1}}^2
+ =\frac{|e^\phi\bar w/(1-t)|^2}{e^\phi/(1-t)^2}=t.
+\]
+This calculation includes $w=0$, where both sides are zero.  For the
+base potential, direct inversion of its complex Hessian gives
+\[
+ |\partial\psi|_{I^{-1}}^2=|z|^2.
+\]
+The covector $\partial\psi$ has only base components in the same
+coframe, and $G\ge\lambda I$.  Combining these facts with
+\eqref{metric:gradient-convention} yields
+\begin{equation}\label{metric:gradient-bounds}
+ |\nabla[-\log(1-t)]|_g^2=2t<2,
+ \qquad
+ |\nabla\psi|_g^2\le\frac{2|z|^2}{\lambda}\le\frac2\lambda.
+\end{equation}
+
+\emph{Compact containment of metric balls.}
+Fix $p\in M_\phi$ and a finite radius $R\ge0$.  Integrating
+\eqref{metric:gradient-bounds} along arbitrary piecewise smooth paths
+and taking the infimum of their lengths shows that
+$d_g(p,q)\le R$ implies
+\[
+ -\log(1-t(q))\le a_R:=-\log(1-t(p))+\sqrt2R,
+ \qquad
+ \psi(q)\le b_R:=\psi(p)+\sqrt{2/\lambda}\,R.
+\]
+This argument does not require a minimizing geodesic.  Set
+\[
+ \tau_R=1-e^{-a_R}<1,\qquad
+ \rho_R=\sqrt{1-e^{-b_R}}<1.
+\]
+Every such $q=(z,w)$ satisfies $t(q)\le\tau_R$ and
+$|z|\le\rho_R$.  Let $m_R$ be the minimum of $\phi$ on the compact
+base ball $\{|z|\le\rho_R\}$.  Then
+\[
+ |w|^2=e^{-\phi(z)}t(q)\le e^{-m_R}\tau_R.
+\]
+Consequently the metric ball is contained in
+\begin{equation}\label{metric:compact-containment}
+ K_R=\{(z,w):|z|\le\rho_R,
+                  \ e^{\phi(z)}|w|^2\le\tau_R\}.
+\end{equation}
+This set is closed and bounded in $\C^3$, hence compact, and the
+strict bounds $\rho_R<1$ and $\tau_R<1$ place it entirely inside
+$M_\phi$.  In particular, a bounded-length curve cannot escape through
+the base boundary, through a fiber boundary, or through unbounded fiber
+coordinates.  The zero section is an interior locus with the smooth
+positive metric \eqref{metric:axis}.
+
+The same compact-containment argument applies to sublevel sets of
+$\mathcal K_\lambda$.  Both $\psi$ and $-\log(1-t)$ are nonnegative,
+so $\mathcal K_\lambda\le R$ implies
+$\psi\le R/\lambda$ and $-\log(1-t)\le R$.  These bounds place the
+sublevel set in a compact set of the form
+\eqref{metric:compact-containment}; the sublevel set is closed there,
+hence compact.  Positivity of its complex Hessian was proved above,
+so $\mathcal K_\lambda$ is a strictly plurisubharmonic exhaustion.
+
+\emph{Geodesic completeness.}
+Suppose a unit-speed geodesic has a finite maximal forward time $T$.
+Its image lies in the compact set $K_T$.  On this set the smooth
+positive metric has a uniform positive Euclidean lower bound, so the
+geodesic's Euclidean velocity is bounded.  The Christoffel symbols are
+also bounded on $K_T$, and the geodesic equation then bounds its
+Euclidean acceleration.  Both position and velocity have limits as
+time tends to $T$, with limiting position in $K_T\subset M_\phi$.
+Local existence for the smooth geodesic equation extends the geodesic
+beyond $T$, a contradiction.  The same argument in reverse time proves
+geodesic completeness.
+\end{proof}
+
+The hypotheses of Proposition~\ref{metric:transfer} imply $h\ge cI>0$,
+so Lemma~\ref{metric:complete} applies for every $\lambda>0$ under
+consideration.  It remains to choose $\lambda$ large enough for the two
+uniform real-sectional bounds.  We do this in the next two sections,
+keeping the convention $g=2\operatorname{Re}\widehat g$ throughout.
diff --git a/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/sections/08-curvature.tex b/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/sections/08-curvature.tex
new file mode 100644
index 0000000000000000000000000000000000000000..893bc0f13522edc6f680a03c9f79cfe546975c0f
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+++ b/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/sections/08-curvature.tex
@@ -0,0 +1,419 @@
+\section{Curvature on real two-planes}\label{sec:curvature}
+
+We retain the base potential and Hartogs metric of
+Proposition~\ref{metric:transfer}.  Section~\ref{sec:pinching} will prove
+the two uniform sectional-curvature bounds.  Here we compute the full
+curvature tensor and reduce its value on an arbitrary real two-plane
+to a scalar expression.  In particular, the calculation includes real
+planes that are not complex lines.
+
+Throughout this section, constants in tensor estimates depend only on
+the uniform centered-chart bounds for $h=(\phi_{i\bar j})$ in
+Proposition~\ref{metric:transfer}; the base dimension is two.
+We take $\lambda\ge1$.  The base form remains
+$I=(\psi_{i\bar j})$, with $\psi=-\log(1-|z|^2)$.
+
+\subsection{A pointwise holomorphic gauge and the tensor blocks}
+
+Fix a point with $w\ne0$, choose centered ball coordinates at its base
+point, and choose a local branch of $\log w$.  Put
+\[
+ s=\log t=\phi+\log w+\overline{\log w},\qquad
+ x=\frac{t}{1-t},\qquad q=x(1+x).
+\]
+In these coordinates the centered base point is zero.  Remove the first
+and pure second derivatives of the weight by the holomorphic fiber change
+\[
+ P(z)=\sum_i\phi_i(0)z_i+
+       \frac12\sum_{i,k}\phi_{ik}(0)z_i z_k,
+ \qquad v=\log w+P(z).
+\]
+More precisely, $s=v+\bar v+\widetilde\phi$, where
+$\widetilde\phi=\phi-P-\bar P$.  At the point under consideration,
+\begin{equation}\label{curv:gauge}
+ s_i=s_{ik}=0,\qquad s_v=s_{\bar v}=1,\qquad
+ s_{i\bar j}=h_{i\bar j}.
+\end{equation}
+All higher derivatives of $s$ involving a vertical index vanish.
+Subtracting $P+\bar P$ changes neither $h$ nor its component
+derivatives in the centered base coordinates.
+
+The metric at the point consequently has blocks
+\[
+ \widehat g=\begin{pmatrix}G&0\\0&q\end{pmatrix},
+ \qquad G=\lambda I+xh.
+\]
+At the center, $I$ is the identity matrix.  Normalize the vertical
+vector to $q^{-1/2}\partial_v$ and denote its index by $0$.
+Horizontal indices $i,j,k,l$ range over $1,2$.
+In this pointwise frame the metric is $G+V$, where $V$ is the
+Hermitian form whose only nonzero entry is $V_{0\bar0}=1$.
+Horizontal forms are extended by zero on the vertical direction.
+For any Hermitian form $P$, define
+\begin{equation}\label{curv:symmetric-tensor}
+ (S_P)_{a\bar b c\bar d}
+   =P_{a\bar b}P_{c\bar d}+P_{a\bar d}P_{c\bar b}.
+\end{equation}
+
+Our complex curvature components use the convention
+\begin{equation}\label{curv:coordinate-formula}
+ R_{a\bar b c\bar d}
+ =-\mathcal K_{ac\bar b\bar d}
+   +\widehat g^{p\bar e}
+       \mathcal K_{ac\bar e}\mathcal K_{p\bar b\bar d}.
+\end{equation}
+Repeated indices in this formula are summed over all coordinates;
+repeated horizontal indices below are summed over $1,2$.
+The inverse coefficients satisfy
+$\sum_p\widehat g^{p\bar e}\widehat g_{p\bar j}
+=\delta_{ej}$, so their numerical array is the transpose of the
+ordinary inverse of the displayed metric matrix.
+We relate \eqref{curv:coordinate-formula} to real sectional curvature
+below, with the fixed convention $g=2\operatorname{Re}\widehat g$.
+
+\begin{proposition}\label{curv:tensor}
+In the frame just described, define the horizontal Hermitian form
+\[
+ D=(1+2x)h-qhG^{-1}h.
+\]
+The full curvature tensor is
+\begin{equation}\label{curv:tensor-identity}
+ R=-\lambda S_I-qS_h-S_V
+    -(S_{D+V}-S_D-S_V)+T_4+T_3,
+\end{equation}
+where $T_j$ has exactly $j$ horizontal indices.  Its nonzero
+components, up to the K\"ahler symmetries and conjugation, are
+\begin{align}
+ (T_4)_{i\bar j k\bar l}
+   &=-x h_{i\bar j,k\bar l}
+      +x^2G^{p\bar e}h_{i\bar e,k}h_{p\bar j,\bar l},
+       \label{curv:four-horizontal-error}\\
+ (T_3)_{i\bar j k\bar0}
+   &=\sqrt q\bigl(-h_{i\bar j,k}
+           +xG^{p\bar e}h_{i\bar e,k}h_{p\bar j}\bigr).
+       \label{curv:three-horizontal-error}
+\end{align}
+Measured by the Euclidean component norms in this frame, uniformly
+for $\lambda\ge1$,
+\begin{equation}\label{curv:component-errors}
+ \|T_4\|\le C_4x,\qquad \|T_3\|\le C_3\sqrt q.
+\end{equation}
+If $\lambda$ is at least the uniform upper bound for the eigenvalues
+of $h$ at centered-chart origins, then
+\begin{equation}\label{curv:D-bounds}
+ (1+x)h\le D\le(1+2x)h.
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+Write $F(s)=-\log(1-e^s)$.  Direct differentiation gives
+\[
+ F'=x,\qquad F''=q,\qquad F'''=q(1+2x),\qquad
+ F''''=q(1+6x+6x^2).
+\]
+The centered-chart expression for $\psi$ differs from
+$-\log(1-|z|^2)$ by a pluriharmonic function and a constant.
+Its derivatives of type $(2,1)$ therefore vanish at the center,
+and its derivatives of type $(2,2)$ are $S_I$.
+
+For brevity in the derivative tables, put
+\[
+ A_{ik\bar j}=h_{i\bar j,k},\qquad
+ B_{ik\bar j\bar l}=h_{i\bar j,k\bar l}.
+\]
+Before normalizing the vertical vector, the only possibly nonzero
+third derivatives of type $(2,1)$, up to symmetry, are
+\begin{equation}\label{curv:third-derivatives}
+ \begin{aligned}
+ \mathcal K_{ik\bar j}&=xA_{ik\bar j},\qquad
+ \mathcal K_{iv\bar j}=qh_{i\bar j},\qquad
+ \mathcal K_{vv\bar v}=q(1+2x).
+ \end{aligned}
+\end{equation}
+The fourth derivatives of type $(2,2)$, up to symmetry and
+conjugation, are
+\begin{equation}\label{curv:fourth-derivatives}
+ \begin{aligned}
+ \mathcal K_{ik\bar j\bar l}
+   &=\lambda(S_I)_{i\bar j k\bar l}
+       +xB_{ik\bar j\bar l}+q(S_h)_{i\bar j k\bar l},\\
+ \mathcal K_{ik\bar j\bar v}&=qA_{ik\bar j},\\
+ \mathcal K_{iv\bar j\bar v}&=q(1+2x)h_{i\bar j},\\
+ \mathcal K_{vv\bar v\bar v}&=q(1+6x+6x^2).
+ \end{aligned}
+\end{equation}
+To check that these lists are complete, apply the chain rule to
+$F(s)$ and use \eqref{curv:gauge}.  In the four-horizontal
+derivative, the two pairings of mixed second derivatives of $s$
+give $qS_h$; the third pairing contains
+$s_{ik}s_{\bar j\bar l}=0$.  All other terms contain a vanished
+first or pure second horizontal derivative.  In particular,
+$\mathcal K_{ik\bar v}=\mathcal K_{vv\bar j}=0$,
+$\mathcal K_{ik\bar v\bar v}=0$, and the fourth derivatives
+with one horizontal and three vertical indices vanish.
+No vanishing of pure third horizontal derivatives is needed.
+
+Substitution into \eqref{curv:coordinate-formula} gives
+\eqref{curv:four-horizontal-error} and
+\eqref{curv:three-horizontal-error}.  For clarity, all remaining
+curvature types in the normalized frame are
+\begin{equation}\label{curv:remaining-blocks}
+ \begin{aligned}
+ R_{i\bar j0\bar0}
+   &=-(1+2x)h_{i\bar j}
+       +q(hG^{-1}h)_{i\bar j}=-D_{i\bar j},\\
+ R_{i\bar0k\bar0}&=0,\qquad
+ R_{i\bar00\bar0}=0,\qquad
+ R_{0\bar00\bar0}=-2.
+ \end{aligned}
+\end{equation}
+The last identity follows because its unnormalized value is
+\[
+ -q(1+6x+6x^2)+q(1+2x)^2=-2q^2,
+\]
+and the four vertical normalization factors multiply to $q^{-2}$.
+The tensors $S_V$ and $S_{D+V}-S_D-S_V$ give precisely the
+vertical and balanced two-horizontal blocks in
+\eqref{curv:remaining-blocks}.  The K\"ahler symmetries and
+conjugation now account for every component, proving
+\eqref{curv:tensor-identity}.
+
+The lower bound $h\ge cI$ implies
+\[
+ \|G^{-1}\|\le\frac1{\lambda+cx},\qquad
+ x^2\|G^{-1}\|\le\frac{x}{c}.
+\]
+The assumed component-derivative bounds for $h$, applied to
+\eqref{curv:four-horizontal-error} and
+\eqref{curv:three-horizontal-error}, prove
+\eqref{curv:component-errors} with constants independent of
+$\lambda\ge1$ and $x>0$.
+
+Finally, $G=\lambda I+xh$ commutes with $h$ at the center.
+On an eigendirection of $h$ with eigenvalue $d$, the ratio of $D$
+to $h$ is
+\[
+ 1+2x-\frac{x(1+x)d}{\lambda+xd}.
+\]
+The subtracted fraction is nonnegative, and it is at most $x$
+when $\lambda\ge d$.  This proves \eqref{curv:D-bounds}.
+\end{proof}
+
+Although the logarithmic fiber coordinate was used only for $w\ne0$,
+the tensor at the zero section can be checked directly in regular
+coordinates.  A holomorphic fiber change $W=e^{P(z)+a}w$, with
+constant $a$ and a polynomial of degree at most two, makes the new
+weight have value zero and vanishing first and pure second
+derivatives at the base point.  Its mixed Hessian remains $h$.
+Through total degree four, the terms relevant to the curvature are
+\[
+ \lambda\psi+
+ |W|^2\left(1+\sum_{i,j}h_{i\bar j}z_i\bar z_j\right)
+ +\frac12|W|^4.
+\]
+At the origin the metric is $\operatorname{diag}(\lambda I,1)$,
+all relevant third derivatives vanish, and direct differentiation
+gives
+\begin{equation}\label{curv:zero-section}
+ R_{i\bar j k\bar l}=-\lambda(S_I)_{i\bar j k\bar l},
+ \qquad R_{i\bar j0\bar0}=-h_{i\bar j},
+ \qquad R_{0\bar00\bar0}=-2,
+\end{equation}
+with all other types zero.  In the original fiber coordinate the
+normalized vertical vector is $e^{-\phi/2}\partial_w$.
+Thus the formulas above hold at the zero section as well by setting
+$x=q=0$, $G=\lambda I$, $D=h$, and $T_3=T_4=0$, and using this
+regular normalized vertical vector.
+
+\subsection{Real curvature and Gram determinants}
+
+Let $U_1,U_2$ be real linearly independent tangent vectors and let
+$u_1,u_2$ be their $(1,0)$ parts, so $U_j=u_j+\bar u_j$.
+For a Hermitian form $P$, linear in its first argument, define
+\begin{equation}\label{curv:gram-quantities}
+ \begin{aligned}
+ a_P&=P(u_1,u_1)P(u_2,u_2)
+             -(\operatorname{Re}P(u_1,u_2))^2,\\
+ b_P&=\operatorname{Im}P(u_1,u_2),\qquad
+ m_P=a_{P+V}-a_P-a_V\quad\text{when $P$ is horizontal}.
+ \end{aligned}
+\end{equation}
+Thus $a_P$ is the determinant of the real Gram matrix for
+$\operatorname{Re}P$, $b_P$ is the imaginary Hermitian pairing, and
+$m_P$ is the cross term in that determinant after adding $V$.
+For the metric form $\widehat g$, the real squared area is
+$4a_{\widehat g}$ because $g=2\operatorname{Re}\widehat g$.
+For a tensor $T$ with the K\"ahler symmetries, also put
+\[
+ \mathcal H_T=T_{u_1\bar u_1u_2\bar u_2}
+       -\operatorname{Re}T_{u_1\bar u_2u_1\bar u_2}.
+\]
+Here contraction with complex vectors has its usual multilinear
+meaning.  These definitions allow the horizontal or vertical projection
+of the real plane to be degenerate.
+
+\begin{lemma}\label{curv:real-plane}
+For the metric convention $g=2\operatorname{Re}\widehat g$ and
+the curvature convention \eqref{curv:coordinate-formula},
+\begin{equation}\label{curv:sectional}
+ K_g(\operatorname{span}_{\mathbb R}\{U_1,U_2\})
+       =\frac{\mathcal H_R}{2a_{\widehat g}}.
+\end{equation}
+Define the matrix
+\begin{equation}\label{curv:xi-definition}
+ \Xi^{a\bar b}=u_1^a\overline{u_2^b}
+                   -u_2^a\overline{u_1^b}.
+\end{equation}
+For every tensor $T$ with the K\"ahler symmetries and Hermitian
+conjugation symmetry,
+\begin{equation}\label{curv:xi-contraction}
+ \mathcal H_T=-\frac12
+ T_{a\bar b c\bar d}\Xi^{a\bar b}\Xi^{c\bar d}.
+\end{equation}
+For every Hermitian form $P$,
+\begin{equation}\label{curv:S-contraction}
+ \mathcal H_{S_P}=a_P+3b_P^2.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+The real squared area of the pair is $4a_{\widehat g}$, because
+$g(U_i,U_j)=2\operatorname{Re}\widehat g(u_i,u_j)$.
+In particular $a_{\widehat g}>0$ for a real linearly independent
+pair.  Write
+\[
+ A=R_{u_1\bar u_1u_2\bar u_2},\qquad
+ B=R_{u_1\bar u_2u_1\bar u_2}.
+\]
+The K\"ahler symmetries in the expansion on
+$U_j=u_j+\bar u_j$ give the real numerator
+\[
+ R^{\mathbb R}(U_1,U_2,U_2,U_1)
+      =A-B-\bar B+A=2\mathcal H_R.
+\]
+Division by the real squared area proves \eqref{curv:sectional}.
+Expanding the right-hand side of \eqref{curv:xi-contraction}
+gives the same expression $A-\operatorname{Re}B$ for $T$.
+Finally, if $c_P=P(u_1,u_2)$, direct contraction of
+\eqref{curv:symmetric-tensor} gives
+\[
+ \mathcal H_{S_P}
+   =P(u_1,u_1)P(u_2,u_2)+|c_P|^2-2\operatorname{Re}(c_P^2)
+   =a_P+3b_P^2.
+\]
+\end{proof}
+
+For example, the disk potential $-\log(1-|w|^2)$ has normalized
+complex curvature component $-2$ by
+\eqref{curv:remaining-blocks}.  Taking $u_2=iu_1$ in
+\eqref{curv:sectional} gives real sectional curvature $-2$.
+This fixes the normalization also on complex lines.
+
+\begin{lemma}\label{curv:gram}
+For a positive semidefinite Hermitian form $P$,
+$a_P\ge b_P^2$.  For the vertical form $V$ one has $a_V=b_V^2$.
+If $P$ is horizontal and positive semidefinite, $m_P$ is nonnegative,
+linear and monotone in $P$, and
+\begin{equation}\label{curv:mixed-gram}
+ m_P\ge2\sqrt{a_Pa_V}.
+\end{equation}
+When $a_Pa_V>0$, equality in \eqref{curv:mixed-gram} holds exactly
+when the two real Gram matrices of the pair for $P$ and $V$ are
+proportional.
+\end{lemma}
+
+\begin{proof}
+Complex Cauchy--Schwarz gives
+$P(u_1,u_1)P(u_2,u_2)\ge|P(u_1,u_2)|^2$, proving
+$a_P\ge b_P^2$.  Equality holds for $V$ because its complex rank
+is one.
+Let $A$ and $B$ be the two-by-two real Gram matrices with entries
+$\operatorname{Re}P(u_i,u_j)$ and
+$\operatorname{Re}V(u_i,u_j)$, respectively.  Then
+\[
+ a_P=\det A,\qquad a_V=\det B,\qquad
+ m_P=\operatorname{tr}(\operatorname{adj}(A)B).
+\]
+For positive definite $A$, put $C=A^{-1/2}BA^{-1/2}$.
+The last expression equals $(\det A)\operatorname{tr}C$,
+whereas $\sqrt{a_Pa_V}=(\det A)\sqrt{\det C}$.
+The inequality $\operatorname{tr}C\ge2\sqrt{\det C}$ proves
+\eqref{curv:mixed-gram}, and continuity gives the semidefinite
+case.  If both determinants are positive, equality means that
+$C$ is a scalar matrix, which is exactly the claimed proportionality.
+The displayed formula for $m_P$ is linear in the entries of $P$.
+Its nonnegativity for positive semidefinite $P$ then also proves
+monotonicity.
+\end{proof}
+
+\subsection{Error terms and the curvature numerator}
+
+Decompose $u_j=(u_{j,H},u_{j,0})$ in the pointwise frame.
+Write $\Xi_{HH}$ for the horizontal two-by-two block of $\Xi$
+and $\Xi_{H\bar0}$ for its horizontal-to-vertical column.
+The other mixed block is the negative conjugate transpose of this
+column.  Write $\Xi_{\mathrm{mix}}$ for the matrix consisting of
+both mixed blocks, with all other entries zero.
+All norms of these blocks below are Euclidean Frobenius norms.
+
+Direct expansion of \eqref{curv:xi-definition} gives
+\begin{equation}\label{curv:xi-norms}
+ \begin{aligned}
+ \|\Xi_{HH}\|^2&=2(a_I+b_I^2)\le4a_I,\\
+ \|\Xi_{H\bar0}\|^2&=m_I+2b_Ib_V,\qquad
+ \|\Xi_{\mathrm{mix}}\|^2=2(m_I+2b_Ib_V)\le4m_I.
+ \end{aligned}
+\end{equation}
+For the second identity, explicitly expand
+$\|u_{1,H}\overline{u_{2,0}}
+      -u_{2,H}\overline{u_{1,0}}\|^2$.
+The last inequality follows from
+$|b_Ib_V|\le\sqrt{a_Ia_V}\le m_I/2$,
+by Lemma~\ref{curv:gram}.
+
+\begin{lemma}\label{curv:errors}
+For every real linearly independent tangent pair,
+\begin{equation}\label{curv:contracted-errors}
+ |\mathcal H_{T_4}|\le Cx a_I,\qquad
+ |\mathcal H_{T_3}|\le C\sqrt q\,\sqrt{a_I m_I}.
+\end{equation}
+More explicitly, the three-horizontal error satisfies
+\begin{equation}\label{curv:mixed-error}
+ |\mathcal H_{T_3}|
+   \le C\sqrt q\,\|\Xi_{HH}\|\,
+                         \|\Xi_{\mathrm{mix}}\|.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+In \eqref{curv:xi-contraction}, the tensor $T_4$ pairs two
+horizontal blocks of $\Xi$.  The tensor $T_3$ pairs one horizontal
+block and one mixed block, since it has exactly three horizontal
+indices.  Cauchy--Schwarz and \eqref{curv:component-errors} give
+\eqref{curv:mixed-error} and
+$|\mathcal H_{T_4}|\le Cx\|\Xi_{HH}\|^2$.
+Now use \eqref{curv:xi-norms}.  These estimates remain valid when
+one of the projected real areas vanishes.
+\end{proof}
+
+We can now express the entire real curvature numerator.
+By \eqref{curv:S-contraction} and $a_V=b_V^2$,
+\[
+ \mathcal H_{S_V}=4a_V,\qquad
+ \mathcal H_{S_{D+V}-S_D-S_V}=m_D+6b_Db_V.
+\]
+Consequently \eqref{curv:tensor-identity} gives
+\begin{equation}\label{curv:negative-numerator}
+ \begin{aligned}
+ -\mathcal H_R={}&\lambda(a_I+3b_I^2)+q(a_h+3b_h^2)
+        +4a_V+m_D+6b_Db_V\\
+       &-\mathcal H_{T_4}-\mathcal H_{T_3}.
+ \end{aligned}
+\end{equation}
+Its denominator in \eqref{curv:sectional} is positive for every real
+two-plane.  It remains to prove that
+\eqref{curv:negative-numerator} is uniformly positive relative to
+that denominator for sufficiently large fixed $\lambda$, and that
+the quotient is also uniformly bounded above.  The mixed term
+$6b_Db_V$ can have either sign; Section~\ref{sec:pinching} treats
+it together with the possible degeneration of the projected planes.
diff --git a/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/sections/09-pinching.tex b/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/build/sections/09-pinching.tex
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+\section{Uniform pinching on real two-planes}\label{sec:pinching}
+
+We now prove the curvature bounds in Proposition~\ref{metric:transfer}.
+The identity of Section~\ref{sec:curvature} reduces the problem to
+finite-dimensional algebra.  The limiting main expression is nonnegative,
+but it can vanish on certain real planes.
+We identify those planes and show that the error with three horizontal
+indices vanishes to the additional order needed there.
+
+Throughout this section, horizontal forms are extended by zero on the
+vertical factor of \(\mathbb C^2\oplus\mathbb C\).
+We retain \(I,V,S_P,a_P,b_P,m_P,\Xi\) from
+Section~\ref{sec:curvature}.  Unless otherwise indicated, constants depend
+only on the fixed bounds \(cI\le h\le CI\) and the component bounds for
+the error tensors.
+
+\subsection{The mixed real-plane algebra}
+
+The leading expression below can vanish even when both projections
+of the real plane are nonzero.  For an explicit algebraic model, take
+$h=I$, $E=2I$, and
+$u_1=(e_1,1)$, $u_2=(ie_1,-i)$, where $e_1=(1,0)\in\mathbb C^2$.
+Direct substitution gives
+\[
+ a_h+3b_h^2+4a_V+m_E+6b_Eb_V=4+4+4-12=0,
+ \qquad \Xi_{\mathrm{mix}}=0.
+\]
+The horizontal and vertical projections have opposite complex
+orientations.  The next lemma proves nonnegativity for all planes in
+the stated range of $E$ and shows that every zero has this mixed-block
+vanishing.
+
+\begin{lemma}\label{pinch:algebra}
+Let \(h\) be a positive definite horizontal Hermitian form, let \(E\) be
+another horizontal Hermitian form with \(E\ge h\), and let \(u_1,u_2\)
+be real-linearly independent vectors in \(\mathbb C^2\oplus\mathbb C\).
+Define
+\[
+ P_h(E)=a_h+3b_h^2+4a_V+m_E+6b_Eb_V.
+\]
+If \(a_ha_V=0\), then \(P_h(E)>0\).
+If \(a_ha_V>0\), put \(\tau=|b_h|/\sqrt{a_h}\).
+For every real \(\beta\) such that \(E\le(2-\beta)h\),
+\begin{equation}\label{pinch:square}
+ P_h(E)\ge
+ \left(\sqrt{a_h+3b_h^2}-2\sqrt{a_V}\right)^2
+ +2\beta(1+\tau)\sqrt{a_ha_V}.
+\end{equation}
+In particular, \(h\le E\le2h\) implies \(P_h(E)\ge0\).
+If equality holds in this last conclusion, then
+\[
+ a_h=a_V>0,\qquad |b_h|=\sqrt{a_h},\qquad b_Eb_V<0,
+\]
+and both mixed blocks \(\Xi^{i\bar0}\) and \(\Xi^{0\bar i}\) vanish.
+\end{lemma}
+
+\begin{proof}
+If \(a_V=0\), then \(b_V=0\), because the vertical factor has complex
+dimension one.
+If \(a_h=0\), the two horizontal projections are linearly dependent
+over \(\mathbb R\), since \(h\) is positive definite.
+Their pairing under every Hermitian form is therefore real, so
+\(b_E=0\).
+Thus \(a_ha_V=0\) implies \(b_Eb_V=0\), and
+\[
+ P_h(E)\ge a_h+4a_V+m_h\ge a_{h+V}>0.
+\]
+Here the last inequality uses the positive definiteness of \(h+V\)
+and the real linear independence of the pair.
+
+Suppose now that \(A=a_h>0\) and \(U=a_V>0\).
+The Hermitian Gram determinant gives \(0\le\tau\le1\).
+Make a real change of basis in the plane so that its horizontal
+projections, denoted \(u_H,v_H\), are real-orthonormal for \(h\).
+Both ratios \(|b_h|/\sqrt{a_h}\) and \(|b_E|/\sqrt{a_h}\) are unchanged
+by this operation.
+The squared \(h\)-norms of \(u_H+i v_H\) and \(u_H-i v_H\) are
+\(2(1+\tau)\) and \(2(1-\tau)\), in some order.
+The absolute difference of their squared \(E\)-norms is \(4|b_E|\)
+in the new basis.  Comparing one norm from above and the other from
+below gives
+\begin{equation}\label{pinch:imaginary-bound}
+ \frac{|b_E|}{\sqrt A}
+ \le\frac{(2-\beta)(1+\tau)-(1-\tau)}2
+ =\frac{1+3\tau}{2}-\frac{\beta(1+\tau)}2
+ \le\sqrt{1+3\tau^2}-\frac{\beta(1+\tau)}2.
+\end{equation}
+For either order of the two norms, the displayed right side is an
+upper bound, since \(2-\beta\ge1\).
+The last inequality follows from
+\[
+ 1+3\tau^2-\left(\frac{1+3\tau}{2}\right)^2
+ =\frac34(1-\tau)^2;
+\]
+it is strict unless \(\tau=1\).
+This argument does not require \(\beta\ge0\).
+
+The mixed Gram inequality \eqref{curv:mixed-gram} and the Hermitian
+Gram inequality yield
+\[
+ m_E\ge2\sqrt{a_EU}\ge2|b_E|\sqrt U.
+\]
+Since \(U=b_V^2\), it follows that
+\[
+\begin{aligned}
+ P_h(E)
+ &\ge A+3b_h^2+4U-4|b_E|\sqrt U\\
+ &\ge
+ \left(\sqrt{A+3b_h^2}-2\sqrt U\right)^2
+ +2\beta(1+\tau)\sqrt{AU},
+\end{aligned}
+\]
+which proves \eqref{pinch:square}.
+
+It remains to examine equality when \(h\le E\le2h\).
+The zero-area case already treated shows that \(A,U>0\).
+Set \(s=\sqrt{A+3b_h^2}\) and \(B=|b_E|\).
+The inequalities above, with \(\beta=0\), give
+\[
+ P_h(E)\ge(s-2\sqrt U)^2+4\sqrt U(s-B)\ge0.
+\]
+Equality forces \(s=2\sqrt U\), \(B=s\), and equality in the bounds
+used to reach this expression.
+The strictness in \eqref{pinch:imaginary-bound} gives \(\tau=1\).
+Consequently
+\[
+ U=A,\qquad B=2\sqrt A,\qquad b_Eb_V<0,\qquad
+ m_E=2B\sqrt U.
+\]
+Since \(a_E\ge B^2\) and \(m_E\ge2\sqrt{a_EU}\), these identities force
+\(a_E=B^2\) and equality in the mixed Gram inequality.
+The horizontal real \(E\)-Gram matrix and the vertical real Gram matrix
+are therefore proportional.
+
+In the real \(h\)-orthonormalized basis used above, \(\tau=1\) says
+that the horizontal projections have the form
+\[
+ u_H,\quad \sigma i u_H,\qquad \sigma\in\{1,-1\}.
+\]
+Their real \(E\)-Gram matrix is a positive scalar multiple of the
+identity.  Proportionality implies that the vertical projections have
+equal positive norms and real inner product zero.
+The sign \(b_Eb_V<0\) selects the opposite complex orientation, so these
+projections have the form
+\[
+ y,\quad-\sigma i y.
+\]
+Direct substitution now gives
+\[
+ \Xi^{i\bar0}
+ =u_H^i\overline{-\sigma i y}-(\sigma i u_H^i)\bar y=0.
+\]
+The other mixed block vanishes as well.
+Under any real change of basis in the plane, \(\Xi\) is multiplied by
+the determinant of that change.  The vanishing therefore holds in the
+original basis.
+\end{proof}
+
+\subsection{Uniform negativity for large \texorpdfstring{\(\lambda\)}{lambda}}
+
+The next statement allows every error tensor with the specified bounds
+and component types.  This uniformity will also give the compactness
+argument needed after \(\lambda\) has been fixed.
+
+\begin{lemma}\label{pinch:positivity}
+Fix \(0<c\le C<\infty\) and \(0\le C_{\mathrm e}<\infty\).
+There is a finite \(\lambda_0=\lambda_0(c,C,C_{\mathrm e})>0\) with the following
+property.
+Let \(\lambda\ge\lambda_0\), \(x\ge0\), and let \(h\) be a horizontal
+Hermitian form with \(cI\le h\le CI\).  Put
+\[
+ q=x(1+x),\qquad G=\lambda I+xh,\qquad
+ D=(1+2x)h-qhG^{-1}h.
+\]
+Let \(T_4,T_3\) be tensors with the Kähler symmetries and reality
+condition, supported respectively on components with four and with
+three horizontal indices, and satisfying
+\[
+ \|T_4\|\le C_{\mathrm e}x,\qquad
+ \|T_3\|\le C_{\mathrm e}\sqrt q.
+\]
+Define
+\[
+ R=-\lambda S_I-qS_h-S_V-(S_{D+V}-S_D-S_V)+T_4+T_3.
+\]
+Then \(\mathcal H_R<0\) on every real two-plane.
+\end{lemma}
+
+\begin{proof}
+Increase the prospective threshold so that \(\lambda\ge\max(1,C)\).
+Then \eqref{curv:D-bounds} gives
+\[
+ (1+x)h\le D\le(1+2x)h.
+\]
+If the lemma were false, there would be a sequence
+\(\lambda\to\infty\) of permitted data and real two-planes for which
+\(F=-\mathcal H_R\le0\).
+We suppress the sequence index.
+By \eqref{curv:negative-numerator},
+\begin{equation}\label{pinch:leading-numerator}
+ F=\lambda(a_I+3b_I^2)+q(a_h+3b_h^2)+4a_V
+ +m_D+6b_Db_V-\mathcal H_{T_4}-\mathcal H_{T_3}.
+\end{equation}
+The contracted estimates \eqref{curv:contracted-errors} apply to the
+allowed error tensors by their component types and symmetries.
+
+\paragraph{The regime \(q/\lambda\to0\).}
+Suppose first that a subsequence has this property.
+Monotonicity and linearity of the mixed Gram term give
+\(m_D\ge c(1+x)m_I\), while
+\[
+ |b_D|\le\sqrt{a_D}\le C'(1+x)\sqrt{a_I}.
+\]
+Discarding nonnegative terms in \eqref{pinch:leading-numerator}, we get
+\[
+ F\ge\lambda a_I+4a_V+c(1+x)m_I
+ -C'\bigl((1+x)\sqrt{a_Ia_V}+x a_I+
+             \sqrt q\,\sqrt{a_Im_I}\bigr).
+\]
+Absorb the first product into \(2a_V\), and the last into
+\(\frac c2(1+x)m_I\).  Since \(q/(1+x)=x\), this gives
+\[
+ F\ge[\lambda-C''((1+x)^2+x)]a_I+
+ 2a_V+\frac c2(1+x)m_I.
+\]
+Here \((1+x)^2+x=O(1+q)=o(\lambda)\), because \(\lambda\to\infty\).
+Every coefficient is eventually positive, whereas
+\(a_I+a_V+m_I=a_{I+V}>0\).  This contradicts \(F\le0\).
+
+\paragraph{The regime with \(\lambda/q\) bounded.}
+If the preceding regime has no subsequence, we may pass to one on
+which \(q/\lambda\) is bounded below by a positive constant.
+Then \(\lambda/q\) is bounded and \(x\to\infty\).
+The term \(qS_h\) has four horizontal slots, whereas the vertical
+term \(S_V\) already has coefficient one.  To balance these two blocks,
+multiply horizontal vector components by \(q^{1/4}\).
+In the resulting plane choose a real-orthonormal pair for
+\(\operatorname{Re}(I+V)\).
+The inverse transformation and a real basis change give a basis of
+the original plane; the latter change multiplies \(F\) by a positive
+factor.
+Thus its sign is unchanged.
+The rescaled pairs belong to a fixed compact Stiefel manifold, and
+their quantities satisfy \(a_{I+V}=1\).
+
+In the rescaled variables, the leading part of
+\eqref{pinch:leading-numerator} is
+\begin{equation}\label{pinch:rescaled-leading}
+ L_x=\frac{\lambda}{q}(a_I+3b_I^2)+P_h(E),
+ \qquad E=\frac D{\sqrt q}.
+\end{equation}
+The transformed tensors, denoted \(\widetilde T_4,\widetilde T_3\),
+satisfy
+\begin{equation}\label{pinch:rescaled-errors}
+ \|\widetilde T_4\|\le C'\frac{x}{q}=O(x^{-1}),
+ \qquad
+ \|\widetilde T_3\|\le C'q^{-1/4}=O(x^{-1/2}).
+\end{equation}
+Indeed, each horizontal slot supplies a factor \(q^{-1/4}\).
+
+The lower bound on \(D\) gives \(E\ge h\).
+For an eigenvalue \(d\in[c,C]\) of \(h\), the corresponding ratio is
+\[
+ \frac Eh=\frac{1+2x}{\sqrt q}
+           -\frac{\sqrt q\,d}{\lambda+xd}.
+\]
+Since
+\[
+ \frac{1+2x}{\sqrt q}=\sqrt{4+\frac1q}
+ \le2+\frac{C_1}{x^2},
+ \qquad
+ \frac{\sqrt q\,d}{\lambda+xd}
+ \ge c_1\frac{x}{\lambda+x},
+\]
+we have the uniform bound
+\begin{equation}\label{pinch:E-bounds}
+ h\le E\le
+ \left(2+\frac{C_1}{x^2}
+          -c_1\frac{x}{\lambda+x}\right)h.
+\end{equation}
+The constants \(c_1>0,C_1<\infty\) depend only on \(c,C\).
+
+Pass to a subsequence on which \(h,E,\lambda/q\), and the normalized
+pair converge.
+The limiting forms satisfy \(h\le E\le2h\).
+The error contributions tend to zero by
+\eqref{pinch:rescaled-errors}.
+Lemma~\ref{pinch:algebra} shows that the two limiting summands in
+\eqref{pinch:rescaled-leading} are nonnegative.
+Since \(F\le0\), both must vanish.
+In particular \(P_h(E)=0\) at the limit, so \(a_h,a_V>0\) there and
+the mixed blocks of the limiting \(\Xi\) vanish.
+The comparison
+\[
+ c^2a_I\le a_h\le C^2a_I
+\]
+follows from the corresponding real Gram-matrix bounds.
+Hence \(a_I,a_h,a_V\) are bounded below by positive constants along
+this subsequence.
+
+We now improve the error estimate on these particular planes.
+By the contraction formula \eqref{curv:xi-contraction}, a tensor with exactly three
+horizontal indices can pair only one horizontal-horizontal block
+with one mixed block.  Thus
+\begin{equation}\label{pinch:vanishing-error}
+ |\mathcal H_{\widetilde T_3}|
+ \le C'q^{-1/4}\|\Xi_{HH}\|\,\|\Xi_{\mathrm{mix}}\|
+ =o(x^{-1/2}).
+\end{equation}
+Here \(\Xi_{HH}\) denotes the block \((\Xi^{i\bar j})\), and
+\(\Xi_{\mathrm{mix}}\) comprises the blocks
+\((\Xi^{i\bar0})\) and \((\Xi^{0\bar i})\).
+The first block is bounded because the normalized pairs form a
+compact set, and the mixed blocks tend to zero by the equality case
+just proved.
+The explicit tensor factor \(q^{-1/4}\) is what makes this an
+\(o(x^{-1/2})\) estimate for the evaluated contribution.
+No rate of convergence of the planes is being assumed.
+
+Apply \eqref{pinch:square} before taking the limit, with
+\[
+ \beta=c_1\frac{x}{\lambda+x}-\frac{C_1}{x^2}.
+\]
+This choice is permitted by \eqref{pinch:E-bounds}, including when
+\(\beta<0\).
+The positive lower bounds for \(a_I,a_h,a_V\), and boundedness of the
+normalized pairs, imply
+\begin{equation}\label{pinch:positive-gap}
+ L_x\ge c_2\left(\frac{\lambda}{q}
+                  +\frac{x}{\lambda+x}\right)-\frac{C_2}{x^2}
+\end{equation}
+along the subsequence, for some \(c_2>0,C_2<\infty\).
+For \(x\ge1\), if \(\lambda\ge x^{3/2}\), then
+\[
+ \frac{\lambda}{q}\ge\frac{x^{3/2}}{2x^2}
+ =\frac1{2\sqrt x}.
+\]
+If instead \(\lambda\le x^{3/2}\), then
+\[
+ \frac{x}{\lambda+x}\ge\frac1{\sqrt x+1}
+ \ge\frac1{2\sqrt x}.
+\]
+Thus \eqref{pinch:positive-gap} is bounded below by a positive
+constant times \(x^{-1/2}\) for large \(x\).
+The evaluated \(\widetilde T_4\) contribution is
+\(O(x^{-1})=o(x^{-1/2})\), and
+\eqref{pinch:vanishing-error} gives the same little-o comparison for
+\(\widetilde T_3\).
+It follows that \(F>0\) eventually, a contradiction.
+
+The two regimes exhaust the possible failure sequences.
+This proves the asserted finite threshold \(\lambda_0\), uniformly
+over all permitted matrices, error tensors, and real planes.
+\end{proof}
+
+\subsection{Both curvature bounds at a fixed parameter}
+
+\begin{proof}[Proof of Proposition~\ref{metric:transfer}]
+The geometric conclusions concerning the domain and completeness were
+proved in Lemma~\ref{metric:complete}.
+It remains to obtain both uniform real sectional-curvature bounds.
+By Proposition~\ref{curv:tensor}, the errors of the actual metric have
+the types and bounds in Lemma~\ref{pinch:positivity}, with
+\(C_{\mathrm e}\) depending only on the chart bounds of
+Proposition~\ref{metric:transfer}.
+Choose any finite \(\lambda\ge\lambda_0\).
+Then \(-\mathcal H_R>0\) for every point and every real two-plane.
+We show that the sectional-curvature quotient is bounded away from
+zero and from negative infinity.
+
+First restrict \(x\) to a bounded interval \([0,X]\).
+Normalize the pairs for \(\operatorname{Re}(I+V)\).
+The data consisting of \(x\), the matrix \(h\), the component arrays
+of \(T_4,T_3\), and the normalized pair range over a closed bounded
+subset of a finite-dimensional space: impose
+\(cI\le h\le CI\), the stated component types and symmetries, and
+the closed norm bounds \(C_{\mathrm e}x,C_{\mathrm e}\sqrt q\).
+This set is compact.
+The formulas for \(G,D,R\) are continuous on it.
+Since Lemma~\ref{pinch:positivity} applies to this entire set,
+\(-\mathcal H_R\) has a strictly positive minimum and a finite maximum.
+The metric \(G+V\) is uniformly positive definite and bounded there,
+so \(a_{G+V}\) is bounded above and away from zero.
+Formula~\eqref{curv:sectional} therefore gives both a strictly negative
+upper curvature bound and a finite lower curvature bound on
+\(0\le x\le X\).
+
+To cover the end \(x\to\infty\), use the horizontal rescaling from
+the preceding proof.
+Here \(\lambda\) remains fixed: the mixed form \(D/\sqrt q\) will
+converge to \(h\), rather than to an arbitrary form between \(h\)
+and \(2h\) as allowed in the preceding contradiction argument.
+The rescaled Hermitian metric is
+\[
+ \widetilde g_x=G/\sqrt q+V.
+\]
+For the fixed finite \(\lambda\),
+\begin{equation}\label{pinch:metric-limit}
+ \frac G{\sqrt q}-h
+ =\frac{\lambda}{\sqrt q}I+
+   \left(\frac{x}{\sqrt q}-1\right)h
+ =O_\lambda(x^{-1})
+\end{equation}
+uniformly in the permitted data.
+Because \(G\) commutes with \(h\), the formula for \(D\) also gives
+\[
+ D=xh+\lambda(1+x)hG^{-1}.
+\]
+Consequently
+\begin{equation}\label{pinch:D-limit}
+ \frac D{\sqrt q}-h
+ =\left(\frac{x}{\sqrt q}-1\right)h+
+   \frac{\lambda(1+x)}{\sqrt q}hG^{-1}
+ =O_\lambda(x^{-1}).
+\end{equation}
+The rescaled tensor identity now implies
+\begin{equation}\label{pinch:tensor-limit}
+ \widetilde R_x=-S_{h+V}+O_\lambda(x^{-1/2})
+\end{equation}
+in component norm, uniformly over all allowed data.
+Indeed, the term \(\lambda S_I\) is divided by \(q\), the term
+\(qS_h\) becomes exactly \(S_h\), the mixed term converges by
+\eqref{pinch:D-limit}, and the error tensors satisfy
+\eqref{pinch:rescaled-errors}.
+
+For the limiting metric \(h+V\) and tensor \(-S_{h+V}\),
+\[
+ K=-\frac12\left(1+
+     3\frac{b_{h+V}^2}{a_{h+V}}\right)
+ \in[-2,-1/2].
+\]
+For an \((I+V)\)-real-orthonormal pair,
+\(a_{h+V}\ge\min(c,1)^2\), so these area denominators are uniformly
+bounded away from zero.  The forms \(h+V\) also range over a compact
+bounded family.  The metric and tensor estimates
+\eqref{pinch:metric-limit}--\eqref{pinch:tensor-limit} therefore imply
+uniform convergence of the corresponding sectional-curvature
+quotients over all real planes.
+For sufficiently large \(X_\lambda<\infty\), they give, for example,
+\[
+ -3\le K_g\le-1/4\qquad(x\ge X_\lambda).
+\]
+Combining this with the compact-range bounds at \(X=X_\lambda\)
+produces finite constants \(0<A\le B\) for which
+\[
+ -B\le K_g(\sigma)\le-A<0
+\]
+for every point and every real two-plane.
+The tensor identity at \(x=0\), given in
+\eqref{curv:zero-section}, is computed in a smooth normalized
+fiber frame; equivalently the same bounds extend to the zero section
+by smoothness of the metric.
+This completes Proposition~\ref{metric:transfer}.
+\end{proof}
diff --git a/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/paper.pdf b/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..144c42abc604f8506504e498c715ac1c3d9981c8
--- /dev/null
+++ b/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:59e7bcf19b96d5d90e787622473764011de3be9d93854b39fc065f019667e022
+size 486089
diff --git a/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/README.md b/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..0e86c962afd1aed80fdcd9e905dcae1901e1bc97
--- /dev/null
+++ b/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/README.md
@@ -0,0 +1,18 @@
+# [A nine-dimensional counterexample to Borsuk's covering assertion](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 23, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026,
+  author = {{OpenAI}},
+  title = {{A nine-dimensional counterexample to Borsuk's covering assertion}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/paper.pdf}{OAI:A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/figures/partner-blocks.tex b/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/figures/partner-blocks.tex
new file mode 100644
index 0000000000000000000000000000000000000000..4f0edf92867d0694abe9b6197fbc862a28837f45
--- /dev/null
+++ b/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/figures/partner-blocks.tex
@@ -0,0 +1,38 @@
+\begin{figure}[tbp]
+\centering
+\begin{tikzpicture}[
+  x=1cm,y=1cm,font=\small,
+  present/.style={circle,fill=black,inner sep=1.7pt},
+  omitted/.style={circle,draw=black,fill=white,inner sep=1.6pt,line width=.7pt},
+  every path/.style={line width=.6pt}
+]
+\draw[rounded corners=3pt] (0,.35) rectangle (3,1.8);
+\draw[rounded corners=3pt] (3.55,.35) rectangle (5.55,1.8);
+\draw[rounded corners=3pt] (6.1,.35) rectangle (8.1,1.8);
+\draw[rounded corners=3pt,densely dashed] (8.8,.35) rectangle (11.8,1.8);
+\node at (1.5,2.15) {$B_1$};
+\node at (4.55,2.15) {$B_2$};
+\node at (7.1,2.15) {$B_3$};
+\node at (10.3,2.15) {$R$};
+\node[present,label=below:$u_1$] at (.5,1.15) {};
+\node[present,label=below:$u_2$] at (1.5,1.15) {};
+\node[omitted,label=below:$q_1$] at (2.5,1.15) {};
+\node[present,label=below:$v$] at (4.05,1.15) {};
+\node[omitted,label=below:$q_2$] at (5.05,1.15) {};
+\node[present,label=below:$w$] at (6.6,1.15) {};
+\node[omitted,label=below:$q_3$] at (7.6,1.15) {};
+\node[omitted,label=below:$r$] at (9.3,1.15) {};
+\node[omitted,label=below:$s$] at (10.3,1.15) {};
+\node[omitted,label=below:$r_3$] at (11.3,1.15) {};
+\node at (5.9,-.2) {$S=\{u_1,u_2,v,w\},\qquad [10]\setminus S=\{q_1,q_2,q_3,r,s,r_3\}$};
+\node at (5.9,-1) {$\{r,s,q_2\}\subseteq T\quad\Longrightarrow\quad T=\{r,s,q_2,v\}$};
+\end{tikzpicture}
+\caption{The only remaining block sizes are $3,2,2$, with three
+labels in $R$ outside the blocks. Filled dots are the vertices of the
+fixed tetrahedron $S$; the omitted block elements are $q_1,q_2,q_3$.
+Every tetrahedron must contain two vertices of one block. After
+relabeling the two-element blocks, the final argument selects a triangle
+$\{r,s,q_2\}$ with $r,s\in R$. Its only possible tetrahedral extension
+is obtained by adjoining $v$, contradicting the two required extensions.}
+\label{fig:partner-blocks}
+\end{figure}
diff --git a/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/figures/partner-links.tex b/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/figures/partner-links.tex
new file mode 100644
index 0000000000000000000000000000000000000000..a079792224a7736de2da1322144e24d977144709
--- /dev/null
+++ b/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/figures/partner-links.tex
@@ -0,0 +1,31 @@
+\begin{figure}[tbp]
+\centering
+\begin{tikzpicture}[
+  x=1cm,y=1cm,font=\small,
+  vertex/.style={circle,fill=black,inner sep=1.6pt},
+  every path/.style={line width=.8pt}
+]
+\node at (1.25,2.6) {(a) $a=b$};
+\node at (7.25,2.6) {(b) $a\ne b$};
+\draw (0,0)--(2.5,0)--(1.25,1.65)--cycle;
+\node[vertex,label=below:$v$] at (0,0) {};
+\node[vertex,label=below:$w$] at (2.5,0) {};
+\node[vertex,label=above:{$a=b$}] at (1.25,1.65) {};
+\node at (1.25,-.8) {$p'(w)=v$};
+\draw (6,1.65)--(6,0)--(8.5,0)--(8.5,1.65);
+\draw[blue!55!black,densely dashed,line width=1pt] (8.5,1.65)--(6,1.65);
+\node[vertex,label=below:$v$] at (6,0) {};
+\node[vertex,label=below:$w$] at (8.5,0) {};
+\node[vertex,label=above:$b$] at (6,1.65) {};
+\node[vertex,label=above:$a$] at (8.5,1.65) {};
+\node at (7.25,-.8) {$p'(w)=b$};
+\end{tikzpicture}
+\caption{The two partner updates in the link of
+$D=S\setminus\{v,w\}$. Put $a=p(v)$ and $b=p(w)$, and replace
+$v$ by $a$ in the tetrahedron $S$. The new partner map is $p'$.
+The known edges $\{v,w\}$, $\{w,a\}$ and $\{v,b\}$ form a triangle
+when $a=b$. When $a\ne b$, they form a path on four vertices; the
+short-link condition forces the closing edge $\{a,b\}$ (dashed).
+In both cases $p'(a)=v$, and the blocks retain their elements.}
+\label{fig:partner-links}
+\end{figure}
diff --git a/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/figures/six-label-paths.tex b/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/figures/six-label-paths.tex
new file mode 100644
index 0000000000000000000000000000000000000000..9944d53d60d78b25314fbfd2d8759af918468b6a
--- /dev/null
+++ b/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/figures/six-label-paths.tex
@@ -0,0 +1,45 @@
+\begin{figure}[tbp]
+\centering
+\begin{tikzpicture}[
+  x=1cm,y=1cm,font=\small,
+  vertex/.style={circle,fill=black,inner sep=1.6pt},
+  every path/.style={line width=.7pt},
+  matching/.style={draw=blue!55!black,line width=1pt}
+]
+\node at (1.7,2.2) {(a) A complementary pair};
+\node at (5.85,2.2) {(b) Link at $a$};
+\node at (9.8,2.2) {(c) Link at $b$};
+\draw (0,0)--(.7,1.25)--(1.4,0)--cycle;
+\node[vertex,label=above:$a$] at (.7,1.25) {};
+\node[vertex,label=below:$c$] at (0,0) {};
+\node[vertex,label=below:$e$] at (1.4,0) {};
+\draw[densely dashed,draw=black!55] (2,0)--(2.7,1.25)--(3.4,0)--cycle;
+\node[vertex,label=above:$b$] at (2.7,1.25) {};
+\node[vertex,label=below:$d$] at (2,0) {};
+\node[vertex,label=below:$f$] at (3.4,0) {};
+\node at (.7,-.7) {present};
+\node at (2.7,-.7) {absent};
+\draw (5,0)--(6.7,0);
+\draw[matching] (5,1.25)--(5,0) (6.7,0)--(6.7,1.25);
+\node[vertex,label=above:$c$] at (5,1.25) {};
+\node[vertex,label=above:$d$] at (6.7,1.25) {};
+\node[vertex,label=below:$e$] at (5,0) {};
+\node[vertex,label=below:$f$] at (6.7,0) {};
+\node at (5.85,-.7) {$c-e-f-d$};
+\draw (8.95,0)--(10.65,0);
+\draw[matching] (8.95,1.25)--(10.65,0) (8.95,0)--(10.65,1.25);
+\node[vertex,label=above:$c$] at (8.95,1.25) {};
+\node[vertex,label=above:$d$] at (10.65,1.25) {};
+\node[vertex,label=below:$e$] at (8.95,0) {};
+\node[vertex,label=below:$f$] at (10.65,0) {};
+\node at (9.8,-.7) {$c-f-e-d$};
+\end{tikzpicture}
+\caption{The complementary-triple rule forces opposite matchings.
+Suppose the two triangles through $\{a,b\}$ are $\{a,b,c\}$ and
+$\{a,b,d\}$. The links at $a$ and $b$, restricted to
+$\{c,d,e,f\}$, share the edge $\{e,f\}$ and use opposite matchings
+between $\{c,d\}$ and $\{e,f\}$ (shown in blue).
+One of the two possible choices is drawn. The crossing in (c) is not a
+vertex. Panel (a) shows one complementary pair for this choice.}
+\label{fig:six-label-paths}
+\end{figure}
diff --git a/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/main.tex b/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/main.tex
new file mode 100644
index 0000000000000000000000000000000000000000..7f8e2cd358acaaed419d9ea9959e123adee1894e
--- /dev/null
+++ b/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/main.tex
@@ -0,0 +1,29 @@
+\pdfinfoomitdate=1
+\pdftrailerid{}
+\pdfsuppressptexinfo=15
+\documentclass[11pt]{article}
+\input{preamble}
+\title{A nine-dimensional counterexample to Borsuk's covering assertion}
+\author{OpenAI}
+\date{September 23, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+The compact set of rank-one orthogonal projectors on $\mathbb R^4$,
+with the Frobenius metric, cannot be covered by ten sets of strictly
+smaller diameter. It therefore gives a counterexample to Borsuk's
+conjecture in dimension nine.
+\end{abstract}
+\setcounter{tocdepth}{1}
+\tableofcontents
+\clearpage
+\input{sections/introduction}
+\input{sections/geometry}
+\input{sections/extension}
+\input{sections/topology}
+\input{sections/six-labels}
+\input{sections/ten-labels}
+\input{sections/conclusion}
+\bibliographystyle{plain}
+\bibliography{references}
+\end{document}
diff --git a/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/preamble.tex b/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/preamble.tex
new file mode 100644
index 0000000000000000000000000000000000000000..e49fd1bb06ad59b57ac748bc3a518337f4bcd58d
--- /dev/null
+++ b/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/preamble.tex
@@ -0,0 +1,27 @@
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage[margin=1.05in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{microtype}
+\usepackage{enumitem}
+\usepackage{xcolor}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta,positioning,calc,fit}
+\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=green!35!black,urlcolor=blue!55!black]{hyperref}
+\hypersetup{pdftitle={A nine-dimensional counterexample to Borsuk's covering assertion},pdfauthor={OpenAI}}
+\usepackage[nameinlink,noabbrev]{cleveref}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\numberwithin{equation}{section}
+\DeclareMathOperator{\tr}{tr}
+\DeclareMathOperator{\rank}{rank}
+\DeclareMathOperator{\Sym}{Sym}
+\DeclareMathOperator{\supp}{supp}
+\setlist{nosep}
+\setlength{\emergencystretch}{1em}
diff --git a/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/references.bib b/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/references.bib
new file mode 100644
index 0000000000000000000000000000000000000000..ce92bf89811a95d64bd37339bb6a874dfd5da808
--- /dev/null
+++ b/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/references.bib
@@ -0,0 +1,212 @@
+@article{FranklWilson1981,
+  author = {Frankl, Peter and Wilson, Richard M.},
+  title = {Intersection theorems with geometric consequences},
+  journal = {Combinatorica},
+  volume = {1},
+  number = {4},
+  year = {1981},
+  pages = {357--368},
+  doi = {10.1007/BF02579457}
+}
+@incollection{Nilli1994,
+  author = {Nilli, A.},
+  title = {On {Borsuk}'s problem},
+  booktitle = {Jerusalem Combinatorics '93},
+  series = {Contemporary Mathematics},
+  volume = {178},
+  publisher = {American Mathematical Society},
+  year = {1994},
+  pages = {209--210}
+}
+@misc{GreyWeissbach1997,
+  author = {Grey, J{\"o}rn and Wei{\ss}bach, Bernulf},
+  title = {{Ein weiteres Gegenbeispiel zur Borsukschen Vermutung}},
+  year = {1997},
+  howpublished = {17th Kolloquium {\"u}ber Kombinatorik, Braunschweig,
+                  November 14--15, 1997},
+  note = {Conference announcement},
+  url = {https://www.kolkom.de/download/program/1997-Braunschweig-program.pdf}
+}
+@article{Raigorodskii1997,
+  author = {Raigorodskii, A. M.},
+  title = {On the dimension in {Borsuk}'s problem},
+  journal = {Russian Mathematical Surveys},
+  volume = {52},
+  number = {6},
+  year = {1997},
+  pages = {1324--1325},
+  doi = {10.1070/RM1997v052n06ABEH002184}
+}
+@article{Weissbach2000,
+  author = {Wei{\ss}bach, Bernulf},
+  title = {Sets with Large {Borsuk} Number},
+  journal = {Beitr{\"a}ge zur Algebra und Geometrie},
+  volume = {41},
+  number = {2},
+  year = {2000},
+  pages = {417--423},
+  url = {https://ftp.gwdg.de/pub/misc/EMIS/journals/BAG/vol.41/no.2/11.html}
+}
+@article{Hinrichs2002,
+  author = {Hinrichs, Aicke},
+  title = {Spherical codes and {Borsuk}'s conjecture},
+  journal = {Discrete Mathematics},
+  volume = {243},
+  number = {1--3},
+  year = {2002},
+  pages = {253--256},
+  doi = {10.1016/S0012-365X(01)00202-3}
+}
+@misc{Pikhurko2002,
+  author = {Pikhurko, Oleg},
+  title = {{Borsuk}'s Conjecture Fails in Dimensions 321 and 322},
+  year = {2002},
+  howpublished = {\href{https://arxiv.org/abs/math/0202112v1}{arXiv:\allowbreak math/0202112v1}},
+  eprint = {math/0202112v1},
+  archivePrefix = {arXiv},
+  primaryClass = {math.CO},
+  url = {https://arxiv.org/abs/math/0202112v1}
+}
+@article{HinrichsRichter2003,
+  author = {Hinrichs, Aicke and Richter, Christian},
+  title = {New sets with large {Borsuk} numbers},
+  journal = {Discrete Mathematics},
+  volume = {270},
+  number = {1--3},
+  year = {2003},
+  pages = {137--147},
+  doi = {10.1016/S0012-365X(02)00833-6}
+}
+
+@article{Borsuk1933,
+  author = {Borsuk, Karol},
+  title = {{Drei S{\"a}tze {\"u}ber die $n$-dimensionale euklidische Sph{\"a}re}},
+  journal = {Fundamenta Mathematicae},
+  volume = {20},
+  year = {1933},
+  pages = {177--190},
+  doi = {10.4064/fm-20-1-177-190}
+}
+
+@article{ConwayHardinSloane1996,
+  author = {Conway, John H. and Hardin, Ronald H. and Sloane, Neil J. A.},
+  title = {Packing lines, planes, etc.: Packings in {Grassmannian} spaces},
+  journal = {Experimental Mathematics},
+  volume = {5},
+  number = {2},
+  year = {1996},
+  pages = {139--159},
+  doi = {10.1080/10586458.1996.10504585},
+  eprint = {math/0208004v1},
+  archivePrefix = {arXiv}
+}
+
+@article{Walkup1970,
+  author = {Walkup, David W.},
+  title = {The lower bound conjecture for 3- and 4-manifolds},
+  journal = {Acta Mathematica},
+  volume = {125},
+  year = {1970},
+  pages = {75--107},
+  doi = {10.1007/BF02392331}
+}
+
+@article{KahnKalai1993,
+  author = {Kahn, Jeff and Kalai, Gil},
+  title = {A counterexample to {Borsuk}'s conjecture},
+  journal = {Bulletin of the American Mathematical Society (N.S.)},
+  volume = {29},
+  number = {1},
+  year = {1993},
+  pages = {60--62},
+  doi = {10.1090/S0273-0979-1993-00398-7},
+  eprint = {math/9307229},
+  archivePrefix = {arXiv}
+}
+
+@incollection{Kalai2015,
+  author = {Kalai, Gil},
+  title = {Some old and new problems in combinatorial geometry {I}: Around {Borsuk}'s problem},
+  booktitle = {Surveys in Combinatorics 2015},
+  series = {London Mathematical Society Lecture Note Series},
+  volume = {424},
+  publisher = {Cambridge University Press},
+  year = {2015},
+  pages = {147--174},
+  doi = {10.1017/CBO9781316106853.005},
+  eprint = {1505.04952v1},
+  archivePrefix = {arXiv}
+}
+
+@article{Bondarenko2014,
+  author = {Bondarenko, Andriy V.},
+  title = {On {Borsuk}'s conjecture for two-distance sets},
+  journal = {Discrete \& Computational Geometry},
+  volume = {51},
+  number = {3},
+  year = {2014},
+  pages = {509--515},
+  doi = {10.1007/s00454-014-9579-4},
+  eprint = {1305.2584v3},
+  archivePrefix = {arXiv}
+}
+
+@article{JenrichBrouwer2014,
+  author = {Jenrich, Thomas and Brouwer, Andries E.},
+  title = {A 64-dimensional counterexample to {Borsuk}'s conjecture},
+  journal = {Electronic Journal of Combinatorics},
+  volume = {21},
+  number = {4},
+  year = {2014},
+  pages = {P4.29},
+  doi = {10.37236/4069},
+  note = {3 pp.}
+}
+
+@misc{Grinsztajn2026,
+  author = {Grinsztajn, Max},
+  title = {A 63-dimensional counterexample to {Borsuk}'s conjecture},
+  year = {2026},
+  month = may,
+  howpublished = {Author manuscript, \href{https://github.com/maaxgrin/borsuk-63-counterexample/blob/cdcdbeac2e692b8641218c70ce9f414522e125e5/borsuk_63_counterexample.pdf}{author's repository}},
+  note = {6 pp.}
+}
+
+@article{ArnouxMarin1991,
+  author = {Arnoux, Pierre and Marin, Alexis},
+  title = {The {K{\"u}hnel} triangulation of the complex projective plane from the view point of complex crystallography. {II}},
+  journal = {Memoirs of the Faculty of Science, Kyushu University. Series A, Mathematics},
+  volume = {45},
+  number = {2},
+  year = {1991},
+  pages = {167--244},
+  doi = {10.2206/kyushumfs.45.167}
+}
+
+@book{Hatcher2002,
+  author = {Hatcher, Allen},
+  title = {Algebraic Topology},
+  publisher = {Cambridge University Press},
+  year = {2002},
+  note = {Corrected 2015 reprinting},
+  url = {https://pi.math.cornell.edu/~hatcher/AT/AT.pdf}
+}
+
+@book{Lee2013,
+  author = {Lee, John M.},
+  title = {Introduction to Smooth Manifolds},
+  edition = {Second},
+  series = {Graduate Texts in Mathematics},
+  volume = {218},
+  publisher = {Springer},
+  year = {2013},
+  doi = {10.1007/978-1-4419-9982-5}
+}
+
+@misc{HatcherVB2017,
+  author = {Hatcher, Allen},
+  title = {Vector Bundles and {$K$}-Theory},
+  year = {2017},
+  howpublished = {Version 2.2, \href{https://pi.math.cornell.edu/~hatcher/VBKT/VB.pdf}{author's notes}},
+  note = {November 2017}
+}
diff --git a/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/sections/conclusion.tex b/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/sections/conclusion.tex
new file mode 100644
index 0000000000000000000000000000000000000000..b396e98c0a26b7d6b5ffa7692f05654ec1b8461b
--- /dev/null
+++ b/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/sections/conclusion.tex
@@ -0,0 +1,58 @@
+\section{The diameter obstruction}
+\label{sec:conclusion}
+
+All ingredients now apply to the projector set in Theorem~\ref{thm:main}.
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+Lemma~\ref{lem:projector-geometry}, with $k=4$, proves compactness and
+identifies the trace-one affine space with Euclidean $\mathbb R^9$.
+It also gives the exact diameter $\sqrt2$. If ten subsets of strictly
+smaller diameter covered $X$, Lemma~\ref{lem:cover-to-map} would supply
+an admissible map $\mathbb{RP}^3\to\Delta^9$. This contradicts
+Theorem~\ref{thm:no-ten-labels}.
+\end{proof}
+
+The spherical geometry of this witness also gives counterexamples in
+all higher dimensions. We use the familiar construction of adjoining
+points at diameter distance; see
+\cite[Lemma~9]{HinrichsRichter2003} and
+\cite[proof of Corollary~1]{Bondarenko2014}. The following Gram-matrix
+description adds all the new points at once.
+
+\begin{corollary}\label{cor:higher-dimensions}
+For every integer $d\geq9$, there is a compact subset of $\mathbb R^d$
+of diameter $\sqrt2$ that cannot be covered by $d+1$ subsets of
+strictly smaller diameter.
+\end{corollary}
+
+\begin{proof}
+The case $d=9$ is Theorem~\ref{thm:main}. For $d=9+s$ with $s\geq1$,
+translate its projector set to
+\[
+ Y=\{P_x-\tfrac14 I:x\in\mathbb{RP}^3\}
+\]
+in the nine-dimensional space of trace-zero symmetric matrices.
+Every $y\in Y$ has $\|y\|_F^2=3/4$. In an orthogonal copy of
+$\mathbb R^s$, choose vectors $v_1,\ldots,v_s$ with Gram matrix
+\[
+ (\langle v_i,v_j\rangle)_{i,j=1}^s
+ =I_s+\tfrac14\mathbf1\mathbf1^{\mathsf T},
+ \qquad \mathbf1=(1,\ldots,1)^{\mathsf T}.
+\]
+Such vectors exist because the displayed matrix is positive definite.
+They satisfy $\|v_i\|^2=5/4$ and
+$\|v_i-v_j\|^2=2$ for $i\ne j$. Therefore the compact set
+\[
+ Z=(Y\times\{0\})\cup\{(0,v_i):1\leq i\leq s\}
+ \subset\mathbb R^9\oplus\mathbb R^s
+\]
+has diameter $\sqrt2$: each new point is at that distance from every
+point of $Y\times\{0\}$ and from every other new point.
+
+A subset of $Z$ of strictly smaller diameter containing a new point
+can contain neither a second new point nor a point of $Y\times\{0\}$.
+Thus a cover of $Z$ by at most $d+1=10+s$ such subsets would leave
+at most ten subsets to cover $Y\times\{0\}$. This contradicts
+Theorem~\ref{thm:main}, since translation preserves diameter and the
+number of covering sets.
+\end{proof}
diff --git a/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/sections/extension.tex b/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/sections/extension.tex
new file mode 100644
index 0000000000000000000000000000000000000000..039529b72d230c33ff07444320fba5b190d9fe64
--- /dev/null
+++ b/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/sections/extension.tex
@@ -0,0 +1,185 @@
+\section{An extension to symmetric matrices}
+\label{sec:extension}
+
+Fix an admissible map $f:\mathbb P(V)\to\mathbb R^m$, where
+$\dim V=k$.  We extend $f$ first to positive semidefinite operators and
+then to all symmetric operators.  Orthogonal spectral subspaces will give
+disjoint sets of labels, producing an odd map between spheres.
+
+Define the continuous degree-two homogeneous function
+$H:V\to\mathbb R^m$ by
+\[
+ H(v)=\begin{cases}\|v\|^2f([v]),&v\ne0,\\0,&v=0.\end{cases}
+\]
+Continuity at zero follows from $\|H(v)\|_1=\|v\|^2$.
+For a positive semidefinite operator $Q$, let
+\begin{equation}
+ E_V(Q)=k\int_{\mathbb S(V)}H(Q^{1/2}u)\,d\mu_V(u),
+ \label{eq:extension}
+\end{equation}
+where $\mu_V$ is rotation-invariant probability measure on the unit sphere.
+The positive square root is meant throughout.  We usually write $E$
+when the ambient space is clear.
+
+\begin{lemma}[Properties of the positive extension]
+\label{lem:positive-extension}
+The map $E_V$ is continuous on the full positive semidefinite cone,
+including its singular boundary.  For $t\geq0$ it satisfies
+\[
+ E_V(tQ)=tE_V(Q),\qquad E_V(Q)_i\geq0,\qquad
+ \sum_{i=1}^m E_V(Q)_i=\operatorname{tr}Q.
+\]
+Its positive coordinates are exactly the labels used on the range of $Q$:
+\begin{equation}
+ \{i:E_V(Q)_i>0\}
+ =\bigcup_{x\in\mathbb P(\operatorname{ran}Q)}S(x),
+ \label{eq:extension-support}
+\end{equation}
+where the right side is empty for $Q=0$.  Moreover:
+\begin{enumerate}
+\item $E_V(P_x)=f(x)$ for every line $x$;
+\item if $Q$ is supported on a nonzero subspace $W\subseteq V$, then
+$E_V(Q)=E_W(Q|_W)$, where the latter construction uses $f|_{\mathbb P(W)}$
+and the factor $\dim W$;
+\item this last expression is smooth when $W$ varies smoothly with fixed
+dimension and $Q|_W$ varies smoothly through positive definite operators.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+The positive square root depends continuously on a positive semidefinite
+matrix.  For example, approximate the square-root function uniformly by
+polynomials on a compact interval containing the spectra under
+consideration, and apply those polynomials to the matrices.
+Thus the integrand in \eqref{eq:extension} is jointly continuous in
+$(Q,u)$; integration over the compact sphere proves continuity of $E_V$.
+Its homogeneity and nonnegativity follow from the corresponding properties
+of $H$.  Since $\sum_i H(v)_i=\|v\|^2$ and
+$\int uu^{\mathsf T}\,d\mu_V(u)=k^{-1}I$, we have
+\[
+ \sum_i E_V(Q)_i=k\int\langle Qu,u\rangle\,d\mu_V(u)
+ =\operatorname{tr}Q.
+\]
+
+Every nonzero $Q^{1/2}u$ lies in $\operatorname{ran}Q$.
+Conversely, every line in that range is spanned by some $Q^{1/2}u$ with
+$\|u\|=1$, because $Q^{1/2}$ is invertible on its range.
+If $f_i$ is positive on such a line, the $i$th integrand is positive at
+that $u$, and hence on a nonempty open set of the sphere.  Such a set has
+positive measure.  This proves \eqref{eq:extension-support} in both
+directions.  For $Q=P_x$, every nonzero integrand is a scalar multiple
+of $f(x)$, and the scalars integrate to $\operatorname{tr}P_x=1$.
+Thus $E_V(P_x)=f(x)$.
+
+To prove the subspace identity, write $r=\dim W$ and let $\pi_W$ be
+orthogonal projection.  The finite measure on $\mathbb S(W)$ defined by
+\[
+ B\longmapsto\int_{\mathbb S(V)}
+ \|\pi_Wu\|^2\,\mathbf1_{\{\pi_Wu\ne0,\;
+                  \pi_Wu/\|\pi_Wu\|\in B\}}\,d\mu_V(u)
+\]
+is invariant under all orthogonal transformations of $W$.  Its total mass is
+$\int\|\pi_Wu\|^2\,d\mu_V(u)=r/k$, so it equals
+$(r/k)\mu_W$.  Because $Q$ vanishes on $W^\perp$ and $H$ is homogeneous
+of degree two, this gives
+\[
+ k\int_{\mathbb S(V)}H(Q^{1/2}u)\,d\mu_V(u)
+ =r\int_{\mathbb S(W)}H((Q|_W)^{1/2}w)\,d\mu_W(w),
+\]
+as claimed.
+
+Finally choose, locally in the parameters, a smooth orthonormal frame
+$T:\mathbb R^r\to W$.  Write $Q|_W$ in that frame as the positive
+definite matrix $C$.  The subspace formula becomes
+\[
+ E_V(Q)=r\int_{\mathbb S^{r-1}}H(TC^{1/2}w)\,d\mu(w).
+\]
+The matrix square root is smooth on positive definite matrices, and
+$TC^{1/2}w$ never vanishes.  The function $H$ is smooth away from zero.
+Differentiating this expression over the compact sphere proves the
+asserted local smoothness.
+\end{proof}
+
+For a symmetric operator $A$, let $|A|=(A^2)^{1/2}$ and
+$A_+=(|A|+A)/2$, $A_-=(|A|-A)/2$.  These are its positive and negative
+parts, so $A=A_+-A_-$ and their ranges are orthogonal.  Define
+\begin{equation}
+ F(A)=E_V(A_+)-E_V(A_-).
+ \label{eq:signed-extension}
+\end{equation}
+
+\begin{lemma}[The odd sphere map]
+\label{lem:signed-extension}
+The map $F:\operatorname{Sym}(V)\to\mathbb R^m$ is continuous, odd,
+and homogeneous of degree one for nonnegative scalars.  It satisfies
+\begin{equation}
+ \|F(A)\|_1=\operatorname{tr}|A|,
+ \qquad \sum_iF(A)_i=\operatorname{tr}A.
+ \label{eq:extension-norms}
+\end{equation}
+More precisely, its positive and negative label sets are
+\begin{equation}
+\begin{aligned}
+ \{i:F(A)_i>0\}&=\bigcup_{x\in\mathbb P(\operatorname{ran}A_+)}S(x),\\
+ \{i:F(A)_i<0\}&=\bigcup_{x\in\mathbb P(\operatorname{ran}A_-)}S(x).
+\end{aligned}
+ \label{eq:spectral-support}
+\end{equation}
+If every coordinate of $F(A)$ is nonzero, then $A$ is nonsingular.
+If all coordinates of $F(A)$ are strictly negative, then $A$ is negative
+definite; the analogous statement holds with both signs reversed.
+The map $F$ is smooth on the open set of nonsingular symmetric operators.
+In particular, it restricts to a continuous odd map
+\begin{equation}
+ \{A:\operatorname{tr}|A|=1\}\longrightarrow
+ \{y\in\mathbb R^m:\|y\|_1=1\}.
+ \label{eq:norm-sphere-map}
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Continuity and homogeneity follow from Lemma~\ref{lem:positive-extension}
+and the continuous formulas for $A_\pm$.  Replacing $A$ by $-A$ exchanges
+$A_+$ and $A_-$, so $F$ is odd.  Any line in
+$\operatorname{ran}A_+$ is orthogonal to every line in
+$\operatorname{ran}A_-$.  Admissibility and
+\eqref{eq:extension-support} therefore imply that $E_V(A_+)$ and
+$E_V(A_-)$ have disjoint positive coordinates.  There is no cancellation
+in their difference.  This proves \eqref{eq:spectral-support} and, by
+summing coordinates, \eqref{eq:extension-norms}.
+
+If $\ker A$ contains a line $x$, choose $i\in S(x)$, which is possible
+because $\sum_i f_i(x)=1$.  The line $x$ is orthogonal to both spectral
+ranges.  Thus label $i$ is absent from both unions in
+\eqref{eq:spectral-support}, so $F(A)_i=0$.  This proves the
+nonsingularity assertion.  If all coordinates of $F(A)$ are negative,
+then $E_V(A_+)=0$ by disjointness of the two positive supports.  Hence
+$\operatorname{tr}A_+=0$, so $A_+=0$, and nonsingularity makes $A$
+negative definite.  Oddness gives the positive case.
+
+For local smoothness, fix nonsingular $A$.  In a neighborhood of $A$,
+the positive and negative spectral subspaces have constant dimensions
+and depend smoothly on the matrix.  Explicitly their projections are
+\[
+ \Pi_\pm(A)=\frac12\bigl(I\pm A(A^2)^{-1/2}\bigr),
+\]
+which are smooth there because $A^2$ is positive definite.
+Each projection admits a local smooth orthonormal frame: project a fixed
+basis for its range and apply Gram--Schmidt after shrinking the
+neighborhood.  In these frames, $A_+$ and $A_-$ restrict to positive
+definite operators on their respective ranges.  The last assertion of
+Lemma~\ref{lem:positive-extension} proves smoothness of $E_V(A_+)$ and
+$E_V(A_-)$, with a zero term if the corresponding range has dimension
+zero.  This proves smoothness of $F$.
+\end{proof}
+
+The two norm spheres in \eqref{eq:norm-sphere-map} are not being regarded
+as globally smooth hypersurfaces.  Radial normalization identifies them
+antipodally with the ordinary Euclidean spheres of dimensions $N_k-1$
+and $m-1$.  These identifications are homeomorphisms everywhere and
+smooth diffeomorphisms locally at nonsingular matrices and label vectors
+with every coordinate nonzero.  Indeed, $\operatorname{tr}|A|$ is smooth
+near a nonsingular $A$, while the $\ell^1$ norm is a fixed linear form
+within each open sign orthant.  Consequently ordinary sphere degree
+applies to \eqref{eq:norm-sphere-map} when $m=N_k$, and local differential arguments
+are valid at every preimage of a full-support value.
diff --git a/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/sections/geometry.tex b/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/sections/geometry.tex
new file mode 100644
index 0000000000000000000000000000000000000000..be19e1b78a734f1615427084a69672370815b4ca
--- /dev/null
+++ b/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/sections/geometry.tex
@@ -0,0 +1,139 @@
+\section{From diameter covers to projective maps}
+\label{sec:geometry}
+
+The geometric part of the proof converts a smaller-diameter cover into a
+smooth map whose positive coordinates separate orthogonal lines.  We first
+make this conversion explicit, so that the remaining obstruction can be
+formulated entirely on projective space.
+
+The projection-matrix metric is the rank-one case of the standard
+Euclidean embedding of real Grassmannians
+\cite[Section~5, Theorem~2]{ConwayHardinSloane1996}. Their chordal
+distance is $1/\sqrt2$ times the Frobenius distance used here. We give
+the short computation to fix the normalization throughout the proof.
+
+Let $V=\mathbb R^k$ have its usual inner product, with $k\geq2$, and let
+$\mathbb P(V)$ denote its space of one-dimensional linear subspaces.
+For a line $x=[u]$, represented by a unit vector $u$, write $P_x=uu^{\mathsf T}$
+for the orthogonal projector onto $x$.  Let $\operatorname{Sym}(V)$ be the
+space of real symmetric operators, equipped with the Frobenius inner product
+$\langle A,B\rangle_F=\operatorname{tr}(AB)$, and put
+\[
+ N_k=\dim\operatorname{Sym}(V)=\frac{k(k+1)}2,
+ \qquad X_k=\{P_x:x\in\mathbb P(V)\}.
+\]
+
+\begin{lemma}[Projector geometry]
+\label{lem:projector-geometry}
+The map $x\mapsto P_x$ identifies $\mathbb P(V)$ homeomorphically with the
+compact set $X_k$ in the trace-one affine subspace of
+$\operatorname{Sym}(V)$.  This affine subspace is Euclidean of dimension
+$N_k-1$.  If $u,v$ are unit representatives of $x,y$, then
+\begin{equation}
+ \|P_x-P_y\|_F^2=2-2\langle u,v\rangle^2.
+ \label{eq:projector-distance}
+\end{equation}
+Consequently $\operatorname{diam}(X_k)=\sqrt2$, and two projectors are at
+this distance exactly when their lines are orthogonal.
+\end{lemma}
+
+\begin{proof}
+The formula $uu^{\mathsf T}$ is unchanged when $u$ is replaced by $-u$,
+so it gives a continuous map on projective space.  Its range as an operator
+is exactly $x$, which proves injectivity.  A continuous injection from the
+compact space $\mathbb P(V)$ into a Hausdorff space is a homeomorphism onto
+its image.  Every $P_x$ has trace one, and the trace-one hyperplane has
+codimension one in $\operatorname{Sym}(V)$.  Finally,
+$\operatorname{tr}(P_x^2)=\operatorname{tr}(P_y^2)=1$ and
+$\operatorname{tr}(P_xP_y)=\langle u,v\rangle^2$, proving
+\eqref{eq:projector-distance}.  Orthogonal lines exist because $k\geq2$.
+\end{proof}
+
+\begin{remark}[Euclidean coordinates]
+An explicit isometry from the trace-one symmetric $4\times4$
+matrices to $\mathbb R^9$ is
+\begin{equation}
+\begin{split}
+ A\longmapsto\bigg(&\frac{a_{11}-a_{22}}{\sqrt2},\;
+ \frac{a_{11}+a_{22}-2a_{33}}{\sqrt6},\;
+ \frac{a_{11}+a_{22}+a_{33}-3a_{44}}{\sqrt{12}},\\
+ &\sqrt2a_{12},\sqrt2a_{13},\sqrt2a_{14},
+   \sqrt2a_{23},\sqrt2a_{24},\sqrt2a_{34}\bigg).
+\end{split}
+\label{eq:euclidean-coordinates}
+\end{equation}
+The first three coordinates use an orthonormal basis of the trace-zero
+diagonal matrices.  The factors $\sqrt2$ in the remaining six coordinates
+account for the two equal off-diagonal entries in the Frobenius norm.
+Thus the nine-dimensional witness will be $X_4$ with exactly its stated
+metric.
+\end{remark}
+
+Write $[m]=\{1,\ldots,m\}$ and
+$\Delta^{m-1}=\{(t_1,\ldots,t_m):t_i\ge0,\ \sum_i t_i=1\}$.
+
+\begin{definition}
+\label{def:admissible}
+A smooth map $f=(f_1,\ldots,f_m):\mathbb P(V)\to\mathbb R^m$ is
+\emph{admissible} if each $f_i$ is nonnegative, $\sum_i f_i=1$, and,
+writing $S(x)=\{i:f_i(x)>0\}$,
+\begin{equation}
+ x\perp y\quad\Longrightarrow\quad S(x)\cap S(y)=\varnothing.
+ \label{eq:admissibility}
+\end{equation}
+Here smoothness means smoothness of the real coordinate functions on the
+usual smooth manifold $\mathbb P(V)$.
+\end{definition}
+
+Restricting an admissible map to $\mathbb P(W)$ for a nonzero linear
+subspace $W\subseteq V$ preserves admissibility.  Coordinates that vanish
+identically on this subspace may be omitted.
+
+\begin{lemma}[The strict-gap reduction]
+\label{lem:cover-to-map}
+If $X_k$ is covered by at most $m$ subsets, each of diameter strictly less
+than $\sqrt2$, then there is an admissible map
+$f:\mathbb P(V)\to\mathbb R^m$.
+\end{lemma}
+
+\begin{proof}
+Pad the cover by empty sets to write it as $A_1,\ldots,A_m$.
+For every nonempty $A_i$, set $d_i=\operatorname{diam}(A_i)$ and
+$\varepsilon_i=(\sqrt2-d_i)/4>0$, and define the relatively open set
+\[
+ U_i=\{P\in X_k:\operatorname{dist}_F(P,A_i)<\varepsilon_i\}.
+\]
+Set $U_i=\varnothing$ if $A_i=\varnothing$.  The sets $U_i$ cover $X_k$.
+For any $P,Q\in U_i$, points of $A_i$ within $\varepsilon_i$ of $P,Q$
+and the triangle inequality give
+\[
+ \|P-Q\|_F<d_i+2\varepsilon_i
+ =\frac{d_i+\sqrt2}{2}<\sqrt2.
+\]
+In particular $\operatorname{diam}(U_i)\leq(d_i+\sqrt2)/2<\sqrt2$.
+
+Pull these sets back to a finite open cover $V_i$ of $\mathbb P(V)$.
+A smooth partition of unity subordinate to this cover
+\cite[Theorem~2.23]{Lee2013} gives smooth
+nonnegative functions $f_i$, with sum one and $f_i(x)>0$ only if
+$x\in V_i$.  Empty sets receive the zero function.  If $i$ belonged to
+both $S(x)$ and $S(y)$ for orthogonal lines $x,y$, then $P_x,P_y\in U_i$
+would be at distance $\sqrt2$ by Lemma~\ref{lem:projector-geometry},
+a contradiction.
+\end{proof}
+
+The strict diameter inequalities are what permit the open neighborhoods
+and the smooth partition of unity; merely excluding individual orthogonal
+pairs from arbitrary color classes does not supply this reduction.
+
+For an admissible map, define its \emph{support complex} $K$ on the label
+set $\{1,\ldots,m\}$ by
+\[
+ I\in K\quad\Longleftrightarrow\quad
+ I\subseteq S(x)\ \text{for some }x\in\mathbb P(V).
+\]
+It is a finite simplicial complex, possibly with unused labels, and $f$
+takes values in its geometric realization inside the standard simplex.
+No triangulation of $\mathbb P(V)$ is assumed.  The next sections derive
+the properties of $K$ that will rule out an admissible map when $k=4$ and
+$m=10$.
diff --git a/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/sections/introduction.tex b/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/sections/introduction.tex
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@@ -0,0 +1,121 @@
+\section{Introduction}
+\label{sec:introduction}
+
+Borsuk's partition problem asks whether every bounded set of positive
+diameter in $\mathbb R^d$ can be covered by $d+1$ subsets of strictly
+smaller diameter. For a bounded set $Y$ of positive diameter, write $b(Y)$ for the least
+number of such subsets needed to cover $Y$. Covers and partitions give
+the same number: overlaps may be removed without increasing diameters.
+We prove that $b(Y)>10$ for a compact set in $\mathbb R^9$.
+
+Let $\operatorname{Sym}_4(\mathbb R)$ be the space of real symmetric
+$4\times4$ matrices, with Frobenius norm
+$\|A\|_F^2=\operatorname{tr}(A^2)$. Its trace-one affine hyperplane is
+a nine-dimensional Euclidean space. For a unit vector $u\in\mathbb R^4$,
+the matrix $uu^{\mathsf T}$ is the orthogonal projector onto the line
+spanned by $u$.
+
+\begin{theorem}\label{thm:main}
+The compact set
+\[
+ X=\{uu^{\mathsf T}:u\in\mathbb R^4,\ \|u\|=1\}
+ \subset\{A\in\operatorname{Sym}_4(\mathbb R):\operatorname{tr}A=1\},
+\]
+with the Frobenius metric, has diameter $\sqrt2$ and cannot be covered
+by ten subsets of diameter strictly less than $\sqrt2$.
+\end{theorem}
+
+Thus Borsuk's assertion fails in dimension nine. The theorem does not
+determine the smallest dimension in which it fails
+or the exact value of $b(X)$. Its witness is the full compact projector
+image of real projective three-space. Corollary~\ref{cor:higher-dimensions}
+extends the construction to compact counterexamples in every dimension
+$d\geq9$.
+
+\subsection*{History and the method}
+Borsuk posed the partition question in 1933 \cite{Borsuk1933}.
+Kahn and Kalai disproved it in high dimensions \cite{KahnKalai1993},
+using a forbidden-intersection theorem of Frankl and Wilson
+\cite{FranklWilson1981} to force many smaller-diameter parts.
+Early reductions in counterexample dimension include $946$
+by Nilli \cite{Nilli1994}, the announced bound $903$ of Grey and
+Weissbach \cite{GreyWeissbach1997}, $561$ by Raigorodskii
+\cite{Raigorodskii1997}, and $560$ by Weissbach \cite{Weissbach2000}.
+
+Spherical codes led to a further series of reductions. Hinrichs obtained
+dimension $323$ using vectors of the Leech lattice
+\cite{Hinrichs2002}. Pikhurko reduced the bound to $321$ and credited
+an independent discovery by Hinrichs and Richter \cite{Pikhurko2002};
+Hinrichs and Richter published their bound $298$ in 2003
+\cite{HinrichsRichter2003}.
+
+A different series of advances used strongly regular graphs.
+Bondarenko constructed a two-distance counterexample in dimension $65$
+\cite{Bondarenko2014}; Jenrich and Brouwer obtained dimension $64$
+\cite{JenrichBrouwer2014}. Grinsztajn's public proof note gives a
+finite counterexample in dimension $63$ \cite{Grinsztajn2026}.
+Theorem~\ref{thm:main} improves this dimension bound using the full
+compact image of projective space rather than a finite configuration.
+
+The projector embedding itself is classical. Kalai describes the map
+$x\mapsto x\otimes x$ and the full rank-one positive semidefinite
+configuration in his account of the Kahn--Kalai construction
+\cite[Section~2.3]{Kalai2015}. Conway, Hardin, and Sloane use the
+same projection-matrix model for Grassmannian packing and its Euclidean
+metric \cite[Section~5, Theorem~2]{ConwayHardinSloane1996}.
+Our contribution is the covering obstruction for the projector set
+associated with $\mathbb R^4$.
+
+The relevant topological antecedents include Walkup's eleven-vertex
+minimum for triangulations of $\mathbb{RP}^3$
+\cite[Theorem~3]{Walkup1970} and the projective-space vertex bounds
+of Arnoux and Marin \cite{ArnouxMarin1991}. These results cannot be
+applied directly to the complex here: its faces are positive coordinate
+supports of a map, not simplices in a triangulation of projective space.
+The central difficulty is to derive enough of its face structure from
+the map itself. We do this through an odd map on symmetric matrices,
+mod-two degree, and local preimage counts, and then give the finite
+combinatorial obstruction in full.
+
+\subsection*{Proof overview}
+Two projectors in $X$ are at diameter distance precisely when their
+underlying lines are orthogonal. A hypothetical cover by ten sets of
+strictly smaller diameter can be enlarged to an open cover. A smooth
+partition of unity then gives nonnegative functions $f_1,\ldots,f_{10}$
+on $\mathbb{RP}^3$, with sum one, such that orthogonal lines have disjoint
+sets of positive coordinates. Section~\ref{sec:geometry} proves this
+reduction for arbitrary covering sets.
+
+We record these sets of positive coordinates in a finite simplicial
+complex: a set of labels is a face if all its coordinates are positive
+at some line. To constrain this complex, Section~\ref{sec:extension}
+extends the map $f$ to positive semidefinite matrices by integration,
+then subtracts its values on the positive and negative parts of a
+symmetric matrix. For the resulting odd map, the sum of the absolute output coordinates
+equals the sum of the absolute eigenvalues of the input matrix.
+Normalizing these two sums gives an odd map between spheres. In Section~\ref{sec:topology}, its mod-two
+degree forces every maximal support to have four labels and the sum of
+these tetrahedral faces to be a cycle.
+
+The main topological step is the equality argument in
+Section~\ref{sec:topology}. At the threshold of $k(k+1)/2$ labels on
+$\mathbb{RP}^{k-1}$, a maximal support with fewer than $k$ labels
+would give a positive-dimensional regular level set inside one affine
+projective chart. Applying the matrix extension on the orthogonal
+complement of each line gives a map on a sphere bundle over this level
+set. The odd degree of the original extension forces an odd preimage
+count for this map, whereas the chart trivialization makes the same
+count even through the induced map on the projective bundle. Once every maximal support has
+$k$ labels, the remaining odd local counts determine the coefficients
+of the facet cycle.
+
+The last stage converts this topological information into a finite
+obstruction. The same degree argument in $\mathbb R^3$ produces a rigid triangle system on six
+labels, described in Section~\ref{sec:six-labels}. Each tetrahedral face
+of the ten-label complex forces such a system on its six-label
+complement. Once these systems have been derived,
+Section~\ref{sec:ten-labels} follows them across adjacent tetrahedra by
+purely combinatorial arguments. Their compatibility restricts the cycles
+around edges and organizes a subset of the labels into small blocks.
+The blocks and the remaining labels give the final contradiction.
+Section~\ref{sec:conclusion} returns to the diameter cover.
diff --git a/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/sections/six-labels.tex b/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/sections/six-labels.tex
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+++ b/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/sections/six-labels.tex
@@ -0,0 +1,102 @@
+\section{Triangle systems on six labels}\label{sec:six-labels}
+
+We now extract the finite combinatorial consequences of the equality case.
+We call a face with three vertices a \emph{triangle}, and a face with four
+vertices a \emph{tetrahedron}. These names refer to faces of the support
+complex, not to geometric simplices in projective space.
+Our eventual application is to the six labels complementary to a
+tetrahedral facet of the ten-label support complex.
+
+\begin{definition}\label{def:six-system}
+A \emph{six-label triangle system} on a set $C$ of cardinality six is a
+collection $\mathcal T\subseteq\binom{C}{3}$ such that
+\begin{enumerate}
+\item exactly one of $T$ and $C\setminus T$ belongs to $\mathcal T$ for
+      each $T\in\binom{C}{3}$;
+\item every pair of elements of $C$ belongs to exactly two members of
+      $\mathcal T$.
+\end{enumerate}
+\end{definition}
+
+\begin{lemma}\label{lem:six-from-map}
+The triangles of the support complex of an admissible map
+$f:\mathbb{RP}^2\to\Delta^5$ form a six-label triangle system.
+\end{lemma}
+
+\begin{proof}
+By Theorem~\ref{thm:equality-complex}, the complex is pure of
+dimension two, has all fifteen edges, and each edge belongs to a positive
+even number of triangles. Two complementary triples cannot both be
+triangles: choose witness lines $x,y\subset\mathbb R^3$ whose supports
+contain the respective triples. A line perpendicular to both exists and,
+by admissibility, could use none of the six labels.
+
+There are ten complementary pairs of triples, so there are at most ten
+triangles. Counting incidences between triangles and edges gives at least
+$2\cdot15/3=10$ triangles. Equality holds throughout: there is one triangle
+in each complementary pair, and every edge occurs twice.
+\end{proof}
+
+For a triangle system $\mathcal T$ on $C$ and $a\in C$, its \emph{link
+graph at $a$} has vertex set $C\setminus\{a\}$ and an edge $\{u,v\}$
+precisely when $\{a,u,v\}\in\mathcal T$.
+
+\begin{lemma}\label{lem:six-properties}
+Let $\mathcal T$ be a six-label triangle system on $C$.
+\begin{enumerate}
+\item Every vertex link is a cycle of length five.
+\item Every four-element subset of $C$ contains a triangle, but does not
+      contain all four of its triples as triangles.
+\item If $a,b\in C$ are distinct and $R\subset C\setminus\{a,b\}$ has
+      cardinality three, the link graphs at $a$ and $b$, restricted to $R$,
+      are different. In particular, no transposition of two labels
+      preserves $\mathcal T$.
+\item The triangles contained in any five-element subset of $C$
+      determine $\mathcal T$ uniquely.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+Every vertex of the link at $a$ has degree two, since the corresponding
+pair with $a$ lies in exactly two triangles. A simple graph in which every
+vertex has degree two is a disjoint union of cycles of length at least
+three. On five vertices it must therefore be a single five-cycle.
+
+Let $A\subset C$ have four elements and choose $a\in A$. A five-cycle
+has no independent set of size three: if three vertices were independent,
+the three gaps following them around the cycle would each contain another
+vertex, requiring at least six vertices. Thus the other three elements of
+$A$ contain an edge of the link at $a$, giving a triangle in $A$. If all
+four triples in $A$ were triangles, the link at $a$ would contain the
+three-cycle on $A\setminus\{a\}$, which a five-cycle cannot contain.
+
+For the third assertion, write
+$C=\{a,b,c,d,e,f\}$, where the two triangles containing $\{a,b\}$ are
+$\{a,b,c\}$ and $\{a,b,d\}$. Deleting $b$ from the five-cycle at $a$
+leaves a path on $\{c,d,e,f\}$ with endpoints $c,d$. Its middle edge is
+$\{e,f\}$, so its other two edges are one of the matchings
+\[
+ M_1=\bigl\{\{c,e\},\{d,f\}\bigr\},\qquad
+ M_2=\bigl\{\{c,f\},\{d,e\}\bigr\}.
+\]
+Deleting $a$ from the link at $b$ gives a path with the same endpoints
+and middle edge. The two paths must use opposite matchings, as in
+Figure~\ref{fig:six-label-paths}. For example,
+if the first uses $M_1$, the triangles $\{a,c,e\}$ and $\{a,d,f\}$
+exclude their complementary triples $\{b,d,f\}$ and $\{b,c,e\}$,
+forcing the second path to use $M_2$.
+
+The symmetric difference of the two paths is the four-cycle with edges
+$M_1\cup M_2$. Every three of its four vertices retain an edge, so the
+restrictions of the paths to any such triple differ. A transposition
+$(a\ b)$ preserving the triangle system would make the two restricted
+link graphs identical, which is impossible.
+
+Finally, if $z$ is the omitted vertex of a five-element subset, every
+triple containing $z$ has its complementary triple in that subset.
+The complementary-triple rule therefore recovers its membership in
+$\mathcal T$.
+\end{proof}
+
+
+\input{figures/six-label-paths}
diff --git a/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/sections/ten-labels.tex b/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/sections/ten-labels.tex
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+++ b/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/sections/ten-labels.tex
@@ -0,0 +1,216 @@
+\section{The obstruction on ten labels}\label{sec:ten-labels}
+
+We prove that an admissible map $\mathbb{RP}^3\to\Delta^9$ cannot exist.
+Its support complex would have tetrahedral facets. Each facet forces a
+six-label triangle system on its complement; transporting these systems
+between adjacent facets will constrain the whole complex.
+
+\begin{lemma}\label{lem:facet-complement}
+Suppose $f:\mathbb{RP}^3\to\Delta^9$ is admissible, let $K$ be its support
+complex, and let $S$ be a tetrahedron of $K$. The triangles of $K$
+contained in the six-element set $C=[10]\setminus S$ form a six-label
+triangle system. In particular, two tetrahedra of $K$ cannot be disjoint.
+\end{lemma}
+
+\begin{proof}
+Theorem~\ref{thm:equality-complex} says that $K$ is pure of
+dimension three. A witness line $x$ for the facet $S$ has support exactly
+$S$. The restriction of $f$ to $\mathbb P(x^\perp)\cong\mathbb{RP}^2$
+uses only labels in $C$. It uses all six: otherwise the lower bound in
+Proposition~\ref{thm:label-bound}, applied in dimension three,
+would be violated. By Lemma~\ref{lem:six-from-map}, this restriction
+supplies a six-label triangle system contained among the triangles of
+$K$ on $C$.
+
+There are no additional triangles on $C$. Otherwise some complementary
+triples $T,C\setminus T$ would both be triangles of $K$. Take witness
+lines for $S,T,C\setminus T$. In $\mathbb R^4$ there is a line
+perpendicular to all three; their supports together contain all ten
+labels, leaving that line no admissible label.
+
+If a tetrahedron $S'$ were disjoint from $S$, its four triples would all
+be triangles on $C$. This contradicts
+Lemma~\ref{lem:six-properties}(2).
+\end{proof}
+
+The rest of the proof is purely combinatorial. We retain the notation
+$K$ and write $\mathcal T_C$ for the triangle system on a facet
+complement $C$. By Theorem~\ref{thm:equality-complex}, each
+triangle of $K$ lies in a positive even number of tetrahedra.
+
+\begin{lemma}\label{lem:two-tetrahedra}
+Every triangle of $K$ belongs to exactly two tetrahedra.
+\end{lemma}
+
+\begin{proof}
+First observe that if two tetrahedra differ by exchanging labels $a,b$,
+the same exchange carries their complementary triangle systems to one
+another. Their complements share five labels, on which both systems are
+the triangles of $K$. After the exchange these triangles remain fixed,
+so the systems agree everywhere by
+Lemma~\ref{lem:six-properties}(4).
+
+Suppose three tetrahedra $T\cup\{a\}$, $T\cup\{b\}$,
+$T\cup\{c\}$ contain the same triangle $T$. Put
+$R=[10]\setminus(T\cup\{a,b,c\})$, which has four elements. Transporting
+the complementary systems successively by
+\[
+ R\cup\{b,c\}
+ \xrightarrow{(a\ b)}R\cup\{a,c\}
+ \xrightarrow{(b\ c)}R\cup\{a,b\}
+ \xrightarrow{(c\ a)}R\cup\{b,c\}
+\]
+preserves the initial system. The composite fixes $R$ and exchanges $b$
+and $c$, contradicting Lemma~\ref{lem:six-properties}(3). Thus at most
+two tetrahedra contain any triangle. Positive even multiplicity gives
+exactly two.
+\end{proof}
+
+For an edge $D$ of $K$, define its \emph{link graph} to have as vertices
+the labels $v\notin D$ for which $D\cup\{v\}$ is a triangle, and as
+edges the pairs $\{v,w\}$ for which $D\cup\{v,w\}$ is a tetrahedron.
+
+\begin{lemma}\label{lem:short-links}
+Every connected component of an edge link is a cycle of length three
+or four.
+\end{lemma}
+
+\begin{proof}
+Every vertex of an edge link has degree two by
+Lemma~\ref{lem:two-tetrahedra}, so each component is a simple cycle.
+We show that the link contains no path on five distinct vertices.
+
+Suppose such a path has consecutive vertices $v_0,v_1,v_2,v_3,v_4$.
+Let $R$ be the three labels outside $D\cup\{v_0,v_1,v_2,v_3,v_4\}$.
+The four successive edges give tetrahedra
+$D\cup\{v_i,v_{i+1}\}$, whose complementary systems are transported by
+\[
+\begin{aligned}
+ R\cup\{v_2,v_3,v_4\}
+ &\xrightarrow{(v_0\ v_2)} R\cup\{v_0,v_3,v_4\}
+ \xrightarrow{(v_1\ v_3)} R\cup\{v_0,v_1,v_4\}\\
+ &\xrightarrow{(v_2\ v_4)} R\cup\{v_0,v_1,v_2\}.
+\end{aligned}
+\]
+The composite fixes $R$ and sends $v_4$ to $v_2$. Hence for every pair
+$\{r,s\}\subset R$, the first system contains $\{v_4,r,s\}$ if and
+only if the last contains $\{v_2,r,s\}$. The latter triple belongs to
+both complementary sets, so its membership in either system is simply
+its membership in $K$. The first system therefore has identical link
+graphs at $v_4$ and $v_2$ when restricted to $R$. This contradicts
+Lemma~\ref{lem:six-properties}(3). Every cycle of length at least five
+contains such a path, proving the assertion.
+\end{proof}
+
+The short links let us describe every facet obtained by repeatedly
+crossing a triangular face of one fixed tetrahedron. The description
+will produce enough facets to contradict the complementary triangle
+system.
+
+\begin{lemma}\label{lem:partner-blocks}
+Fix a tetrahedron $S$ of $K$. For $v\in S$, let $p(v)$ be the unique
+label outside $S$ such that
+$(S\setminus\{v\})\cup\{p(v)\}$ is a tetrahedron. For each distinct
+value $y$ of $p$, set
+\[
+ B_y=p^{-1}(y)\cup\{y\}.
+\]
+The sets $B_y$ are disjoint and have at least two elements. For every
+choice $q_y\in B_y$, the set
+\[
+ \bigcup_{y\in p(S)}(B_y\setminus\{q_y\})
+\]
+is a tetrahedron. Moreover, every tetrahedron of $K$ contains at least two
+elements of one of these blocks.
+\end{lemma}
+
+\begin{proof}
+The partner $p(v)$ exists and is unique by
+Lemma~\ref{lem:two-tetrahedra}, applied to $S\setminus\{v\}$. It lies
+outside $S$, since the only tetrahedron on the vertices of $S$ is $S$
+itself. The stated disjointness and lower size bound are immediate.
+
+We first prove that replacing $v$ by $a=p(v)$ leaves the collection of
+blocks unchanged. Put $S'=(S\setminus\{v\})\cup\{a\}$ and let $p'$
+be its partner map. We have $p'(a)=v$. For $w\in S\setminus\{v\}$,
+put $D=S\setminus\{v,w\}$ and $b=p(w)$. The link of $D$ contains
+the edges
+\[
+ \{v,w\},\qquad \{w,a\},\qquad \{v,b\}.
+\]
+If $a=b$, these form a three-cycle. Since every link vertex has degree
+two, that cycle is a whole component, and the other tetrahedron on
+$D\cup\{a\}$ is $D\cup\{a,v\}$. Thus $p'(w)=v$.
+If $a\ne b$, the displayed edges form the path $a,w,v,b$ on four
+distinct vertices. Lemma~\ref{lem:short-links} forces it to complete
+to a four-cycle, with the edge $\{a,b\}$. Thus $p'(w)=b=p(w)$.
+The two cases are illustrated in Figure~\ref{fig:partner-links}.
+We have proved
+\[
+ p'(a)=v,\qquad
+ p'(w)=
+ \begin{cases}
+ v,&p(w)=a,\\
+ p(w),&p(w)\ne a.
+ \end{cases}
+\]
+\input{figures/partner-links}
+
+The block formerly indexed by $a$ is now indexed by $v$ and has the
+same elements; all other blocks are unchanged.
+
+The tetrahedron $S$ contains all elements of each block except its
+indexing partner. The same is true after any such exchange, by block
+invariance. To change the omitted element of a block from $q$ to $u$,
+replace the present element $u$ by its partner $q$. Doing this once
+in each block where a change is wanted realizes every possible choice
+of omissions. These sets have cardinality four, since
+$\sum_y(|B_y|-1)=|S|=4$.
+
+Finally, suppose a tetrahedron $T$ met each block in at most one
+element. In each block omit that element if present, and otherwise
+omit any element. The resulting tetrahedron lies in the union of the
+blocks and is disjoint from $T$, contrary to
+Lemma~\ref{lem:facet-complement}.
+\end{proof}
+
+\begin{theorem}\label{thm:no-ten-labels}
+There is no admissible map $\mathbb{RP}^3\to\Delta^9$.
+\end{theorem}
+
+\begin{proof}
+Assume such a map exists and fix a tetrahedron $S$ of its support
+complex. Let $t$ be the number of blocks in
+Lemma~\ref{lem:partner-blocks}, and let $R$ be the set of labels outside
+their union. The blocks contain $4+t$ labels in total, so
+\[
+ 1\le t\le4,\qquad |R|=6-t.
+\]
+If $t=4$, each block has size two and $S$ meets each in just one
+element, contradicting the last assertion of
+Lemma~\ref{lem:partner-blocks}. Thus $t\le3$.
+
+There is no triangle entirely in $R$. By purity, such a triangle would
+extend to a tetrahedron; its one remaining vertex could not supply two
+vertices in a block. On the other hand, $R$ is contained in
+$[10]\setminus S$, where every four labels contain a triangle by
+Lemmas~\ref{lem:facet-complement} and~\ref{lem:six-properties}(2).
+Consequently $|R|\le3$, and therefore $t\ge3$.
+
+It follows that $t=3$, $|R|=3$, and the block sizes are $3,2,2$,
+as in Figure~\ref{fig:partner-blocks}.
+\input{figures/partner-blocks}
+
+The complement of $S$ consists of $R$ and one omitted element from
+each block. Choose two distinct labels $r,s\in R$. In the triangle
+system on this complement, $\{r,s\}$ belongs to two distinct triangles.
+Their third vertices cannot be in $R$, because $R$ contains no
+triangle. They are therefore two distinct omitted block elements.
+Only one block has size three, so at least one of these triangles is
+$\{r,s,z\}$ with $z$ in a block $B$ of size two.
+
+Every tetrahedron containing $\{r,s,z\}$ must contain two vertices of
+some block. As $r,s$ lie outside all blocks, its fourth vertex must be
+the other element of $B$. Thus this triangle belongs to at most one
+tetrahedron, contradicting Lemma~\ref{lem:two-tetrahedra}.
+\end{proof}
diff --git a/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/sections/topology.tex b/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/sections/topology.tex
new file mode 100644
index 0000000000000000000000000000000000000000..b68e9e528ac8218c76adc7734dfdcac158d8edf0
--- /dev/null
+++ b/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/build/sections/topology.tex
@@ -0,0 +1,312 @@
+\section{The equality case gives a simplicial cycle}
+\label{sec:topology}
+
+The symmetric-matrix extension gives more than a lower bound on the
+number of labels.  At equality, its degree controls every maximal
+support of the admissible map.  We prove that these supports all have
+the same size and that their sum is a cycle.  No triangulation of
+projective space is assumed.
+
+Throughout this section, homology and cohomology have coefficients in
+$\mathbb F_2$.  For a Euclidean space $V$ of
+dimension $k$, write
+\[
+ \Sigma(V)=\{A\in\operatorname{Sym}(V):\operatorname{tr}|A|=1\},
+ \qquad
+ \Sigma_m=\{y\in\mathbb R^m:\textstyle\sum_i|y_i|=1\}.
+\]
+Radial projection identifies these spaces antipodally with
+$S^{N_k-1}$ and $S^{m-1}$, respectively.  We give them the smooth
+structures transported by these identifications.  Near nonsingular
+operators, respectively full-support label vectors, these structures
+agree with their usual smooth hypersurface structures: the relevant
+norm functions are smooth there.  Let
+$f:\mathbb P(V)\to\Delta^{m-1}$ be a smooth admissible map, and let
+$F:\Sigma(V)\to\Sigma_m$ be its continuous odd extension constructed
+in the preceding section.
+
+\begin{proposition}[The number of labels]\label{thm:label-bound}
+If $k\ge2$, every smooth admissible map
+$f:\mathbb P(\mathbb R^k)\to\Delta^{m-1}$ satisfies $m\ge N_k$.
+If $m=N_k$, its extension $F$ has mod-two degree one and is
+surjective.  In this equality case, every pair of labels is an edge
+of the support complex of $f$.
+\end{proposition}
+
+\begin{proof}
+An odd map $S^n\to S^d$ requires $n\le d$, and an odd self-map
+of $S^n$ has odd degree; see \cite[Proposition~2B.6 and
+Corollary~2B.7]{Hatcher2002}.  Applying these facts to $F$ proves the
+bound and the degree assertion.  A map of nonzero degree cannot omit
+a point of its target sphere, since the complement of a point is
+contractible.
+
+Suppose $m=N_k$ and fix distinct labels $i,j$.  Choose a vector in
+$\Sigma_m$ with every coordinate nonzero, with precisely the
+$i$th and $j$th coordinates positive.  At a preimage $A$ under $F$,
+every line in the positive spectral subspace of $A$ uses only these
+two labels.  That subspace cannot have dimension at least two:
+restriction to a two-dimensional subspace would contradict the bound
+$N_2=3$ just proved.  Its dimension is therefore one.  The exact
+spectral-support property of $F$ says that this line has both
+labels $i,j$ in its support, so $\{i,j\}$ is an edge.
+\end{proof}
+
+We will count regular preimages of maps which need not be smooth
+everywhere.  The following local form of the degree formula explains
+why this causes no problem.
+
+\begin{lemma}[Local counting]\label{lem:local-degree}
+Let $M$ be a closed smooth $n$-manifold, possibly disconnected, and
+let $q:M\to Y$ be continuous, with $Y$ Hausdorff.  Suppose a point $y\in Y$ has a
+neighborhood homeomorphic to an open subset of $\mathbb R^n$, and
+that $q$ is smooth in the corresponding coordinates near
+$q^{-1}(y)$, with invertible derivative at each point of that fiber.
+Then the fiber is finite, and the image of the total fundamental
+class of $M$ in
+\[
+ H_n(Y,Y\setminus\{y\})\cong\mathbb F_2
+\]
+is $|q^{-1}(y)|$ modulo two.  In particular, if $Y$ is a connected
+closed $n$-manifold, this parity is the mod-two degree of $q$.
+\end{lemma}
+
+\begin{proof}
+The inverse function theorem makes every point of the fiber
+isolated.  The fiber is closed in compact $M$, hence finite.
+By excision,
+\[
+ H_n(M,M\setminus q^{-1}(y))
+ \cong\bigoplus_{x\in q^{-1}(y)}H_n(M,M\setminus\{x\}).
+\]
+The fundamental class has coordinate one in each summand.  A local
+diffeomorphism sends each local generator to the generator at $y$.
+Naturality of relative homology now gives the assertion.  This
+argument uses no smoothness away from the selected fiber.
+\end{proof}
+
+\begin{theorem}[Supports at equality]\label{thm:equality-complex}
+Let $k\ge3$ and $m=N_k$.  Suppose
+$f:\mathbb P(\mathbb R^k)\to\Delta^{m-1}$ is a smooth admissible map.
+Its support complex $K$ has every pair of labels as an edge, every
+facet has $k$ vertices, and
+\[
+ \sum_{I\text{ a facet of }K} I
+\]
+is a simplicial cycle over $\mathbb F_2$.  Consequently every
+$(k-2)$-dimensional face lies in a positive even number of facets.
+\end{theorem}
+
+\begin{proof}
+The edge assertion is Proposition~\ref{thm:label-bound}.  We prove
+the remaining assertions by examining an arbitrary facet $I$.
+Set
+\[
+ s=|I|,\qquad J=\{1,\ldots,m\}\setminus I,\qquad
+ a=|J|,\qquad h=N_{k-1}.
+\]
+Since $I$ is a maximal support face, there is a line $x_0$ with
+$S(x_0)=I$.  Every line in $x_0^\perp$ uses only labels in $J$.
+Proposition~\ref{thm:label-bound}, applied in $x_0^\perp$, gives
+\begin{equation}\label{eq:facet-size-upper}
+ a\ge h,\qquad s=m-a\le N_k-N_{k-1}=k.
+\end{equation}
+
+\paragraph{A compact level set over one affine chart.}
+Define
+\[
+ U_I=\{x\in\mathbb P(V):f_i(x)>0\text{ for every }i\in I\}.
+\]
+Maximality of $I$ implies $S(x)=I$ for all $x\in U_I$.  Thus the
+coordinate map $f_I$ takes $U_I$ smoothly to the open simplex
+$\mathring\Delta_I$ on $I$.  Moreover, $U_I$ lies in the affine
+projective chart
+\[
+ \mathbb P(V)\setminus\mathbb P(x_0^\perp),
+\]
+because a line orthogonal to $x_0$ cannot use any label in $I$.
+Choose a regular value $r\in\mathring\Delta_I$ of $f_I$, using
+Sard's Theorem \cite[Theorem~6.10]{Lee2013}, and put
+\[
+ L=f_I^{-1}(r)\subset U_I.
+\]
+For now $L$ is allowed to be empty.  If $\widetilde r$ denotes $r$
+extended by zero on $J$, then
+$L=f^{-1}(\widetilde r)$: positivity of the coordinates of $r$
+puts every such preimage in $U_I$.  Hence $L$ is compact.  It is a
+smooth manifold without boundary \cite[Corollary~5.14]{Lee2013} of dimension
+\begin{equation}\label{eq:level-dimension}
+ \dim L=(k-1)-(s-1)=k-s=a-h.
+\end{equation}
+
+If $s<k$, the level-set dimension in \eqref{eq:level-dimension} is positive.
+We will construct a map on a sphere bundle over $L$ whose regular
+preimages have odd cardinality, by comparison with $F$. The affine-chart
+trivialization will force the induced map on the projective bundle to
+have degree zero when $\dim L>0$, contradicting that count. When $s=k$,
+the same count will determine the coefficient of $I$ in the desired cycle.
+
+For $x\in U_I$, admissibility ensures that $f$ on
+$\mathbb P(x^\perp)$ uses only labels in $J$.  Apply the extension
+construction in this $(k-1)$-dimensional space to obtain an odd map
+\[
+ G_x:\Sigma(x^\perp)\longrightarrow\Sigma_a.
+\]
+The subspace-restriction and continuity properties of the extension
+make these maps a continuous map on the corresponding sphere bundle.
+Restrict that bundle to $L$ and write
+\[
+ M=\{(x,B):x\in L,\ B\in\operatorname{Sym}(x^\perp),
+                     \ \operatorname{tr}|B|=1\},
+ \qquad G(x,B)=G_x(B).
+\]
+
+The bundle $x\mapsto x^\perp$ is trivial on the affine chart above.
+Indeed, choose the unit representative $u$ of $x$ having positive
+inner product with a fixed unit representative of $x_0$.  Orthogonal
+projection from $x_0^\perp$ onto $u^\perp$ is invertible and varies
+smoothly with $x$; orthonormalizing its images of a fixed basis gives
+a smooth orthonormal trivialization.  The induced trivialization
+of symmetric operators, followed by radial change from trace norm
+to a Euclidean norm, gives
+\begin{equation}\label{eq:product-sphere-bundle}
+ M\cong L\times S^{h-1},\qquad
+ (x,B)\longmapsto(x,-B)\quad\leftrightarrow\quad(x,z)\longmapsto(x,-z).
+\end{equation}
+Equip $M$ with the smooth structure given by the Euclidean sphere
+bundle.  The trace-norm parametrization is smooth near every
+definite $B$, which is all that will be needed.  In particular,
+\begin{equation}\label{eq:bundle-dimension}
+ \dim M=(a-h)+(h-1)=a-1.
+\end{equation}
+
+\paragraph{Odd parity on the bundle.}
+Consider the open negative orthant of $\Sigma_a$, where all $J$
+coordinates are strictly negative.  At a preimage of this orthant,
+$B$ has no positive spectral subspace.  It has no kernel either:
+the exact support property says that all $J$ labels occur on its
+range, leaving no label for a kernel line.  Thus $B$ is negative
+definite on $x^\perp$.  Consequently $G$ is smooth on the preimage
+of the negative orthant.  Sard's Theorem allows us to choose a
+regular value $g$ in that orthant.  Its preimage is finite by
+compactness, including the possibility of an empty preimage.
+
+Fix $0<\lambda<1$ and form the full-support vector
+\[
+ y=(\lambda r,(1-\lambda)g)\in\Sigma_m,
+\]
+with the two coordinate blocks indexed by $I$ and $J$.  We claim
+that $F^{-1}(y)$ corresponds exactly to $G^{-1}(g)$.
+Every $A\in F^{-1}(y)$ is nonsingular by the full-support property.
+All lines in its nonzero positive spectral subspace use only
+labels in $I$.  That subspace must be one-dimensional: a subspace
+of dimension at least two intersects the hyperplane $x_0^\perp$,
+where no label in $I$ is available.  Call the resulting positive
+line $x$.  The positive output block has every $I$ coordinate
+nonzero, so $S(x)=I$ and $x\in U_I$.  Since the sum of the positive
+coordinates of $F(A)$ is $\lambda$, the positive eigenvalue is
+$\lambda$.  Therefore
+\begin{equation}\label{eq:preimage-splitting}
+ A=\lambda P_x+(1-\lambda)B,
+ \qquad B<0\text{ on }x^\perp,\quad \operatorname{tr}|B|=1.
+\end{equation}
+The subspace formula for the extension now gives
+$f_I(x)=r$ and $G_x(B)=g$.  Conversely, every $(x,B)\in G^{-1}(g)$
+gives a preimage by \eqref{eq:preimage-splitting}.
+
+The value $y$ is regular in the local sense of
+Lemma~\ref{lem:local-degree}.  To see this precisely, near a
+preimage use $(\lambda',x',B')$ as coordinates for symmetric
+operators of signature $(1,k-1)$, with $B'$ negative definite and
+normalized on $x'^\perp$.  The single positive eigenvalue and its
+line vary smoothly, so these are smooth coordinates on the
+trace-norm sphere near that preimage.  In the target orthant, use
+the positive mass followed by the two normalized coordinate
+blocks.  In these coordinates $F$ is
+\begin{equation}\label{eq:triangular-derivative}
+ (\lambda',x',B')\longmapsto
+       (\lambda',f_I(x'),G_{x'}(B')).
+\end{equation}
+The first derivative block is the identity.  The second is
+surjective because $r$ is regular for $f_I$.  On the kernel of
+these two blocks, the third block is the derivative of
+$G|_M$, which is surjective because $g$ is regular.  Thus the full
+derivative is surjective, and is an isomorphism since source and
+target both have dimension $m-1$.  By the odd degree of $F$ and
+Lemma~\ref{lem:local-degree},
+\begin{equation}\label{eq:bundle-odd-count}
+ |G^{-1}(g)|=|F^{-1}(y)|\equiv1\pmod2.
+\end{equation}
+This proves, in particular, that $L$ is nonempty.
+
+\paragraph{The double cover forces maximal supports to have size $k$.}
+The oddness of $G$ in the sphere variable gives a quotient map
+\[
+ \overline G:L\times\mathbb{RP}^{h-1}
+                  \longrightarrow\mathbb{RP}^{a-1}.
+\]
+Write $w\in H^1(\mathbb{RP}^{a-1})$ and
+$u\in H^1(\mathbb{RP}^{h-1})$ for the first Stiefel--Whitney classes
+of the real line bundles associated with their antipodal double covers.
+Equivariance identifies the pullback of the target
+sphere cover with
+\[
+ L\times S^{h-1}\longrightarrow L\times\mathbb{RP}^{h-1}.
+\]
+Indeed, the identification sends $(x,z)$ to
+$((x,[z]),G(x,z))$ and is an isomorphism on each two-point fiber.
+This cover isomorphism induces an isomorphism of the associated real
+line bundles. Their first Stiefel--Whitney classes are natural under
+pullback \cite[Theorem~3.1(a)]{HatcherVB2017}.
+The source cover is pulled back from the second factor, so
+\begin{equation}\label{eq:cover-class}
+ \overline G^*w=\operatorname{pr}_2^*u.
+\end{equation}
+There is no contribution from $H^1(L)$ in this identity.
+
+Suppose $a>h$.  The projective-space cohomology calculation
+\cite[Theorem~3.19]{Hatcher2002} gives
+\[
+ u^h=0,\qquad
+ \overline G^*(w^{a-1})=
+       \operatorname{pr}_2^*(u^{a-1})=0.
+\]
+Both source and target of $\overline G$ are closed manifolds of
+dimension $a-1$, and $w^{a-1}$ is the target's top cohomology
+generator.  Evaluation on the total fundamental class therefore
+gives mod-two degree zero.  On the other hand, $[g]$ is a regular
+value in local quotient coordinates, and every preimage of $[g]$
+has exactly one representative mapping to $g$.  Its preimage count
+is therefore the odd number in \eqref{eq:bundle-odd-count}, a
+contradiction.  Together with \eqref{eq:facet-size-upper}, this
+proves $a=h$ and $s=k$ for every facet.
+
+\paragraph{The coefficients of the facet cycle.}
+Now $\dim L=0$, so $L$ is a finite set.  For each $x\in L$, the
+map $G_x$ is odd between spheres of the same dimension $h-1$.
+Regularity of $g$ for $G$ implies its regularity for each such
+fiber map.  Each fiber therefore contributes an odd number to
+\eqref{eq:bundle-odd-count}, and hence
+\begin{equation}\label{eq:facet-odd-count}
+ |f^{-1}(\widetilde r)|=|L|\equiv1\pmod2.
+\end{equation}
+
+Every facet has $k$ vertices, so $K$ has dimension $k-1$.
+Using the identification of simplicial and singular homology
+\cite[Theorem~2.27]{Hatcher2002}, consider the homology class
+\[
+ f_*[\mathbb P(V)]\in H_{k-1}(|K|).
+\]
+The interior point $\widetilde r$ of the facet $I$ has a
+neighborhood in $|K|$ contained in that facet.  Passing to
+$H_{k-1}(|K|,|K|\setminus\{\widetilde r\})$ reads the coefficient
+of $I$ in a simplicial top cycle.  By the regularity of $r$,
+Lemma~\ref{lem:local-degree} and \eqref{eq:facet-odd-count}, this
+coefficient is one.  The same holds for every facet.  Since a
+$(k-1)$-dimensional complex has no simplicial $k$-chains, its
+top homology classes are exactly its top cycles.  Thus
+$f_*[\mathbb P(V)]$ is represented by the sum of all facets.
+Its boundary vanishes.  Each codimension-one face consequently
+occurs in an even number of facets, and purity makes that number
+positive.
+\end{proof}
diff --git a/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/paper.pdf b/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..4425db56493a920a9bc93452b8e15c5629f809fe
--- /dev/null
+++ b/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:e7ffa6f812d97395aa7794f4dc0f394ea36f24b00ced0acad29157c4c3bcca87
+size 413156
diff --git a/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/README.md b/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..2752a41df1be934019f0928412fdad0e745397a9
--- /dev/null
+++ b/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/README.md
@@ -0,0 +1,18 @@
+# [A nondegenerate counterexample to the Morse-number Arnold bound](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 23, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026,
+  author = {{OpenAI}},
+  title = {{A nondegenerate counterexample to the Morse-number Arnold bound}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/paper.pdf}{OAI:A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/build/main.tex b/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/build/main.tex
new file mode 100644
index 0000000000000000000000000000000000000000..cc82bed2ef91979e69f6c06da6a646efe8248d8e
--- /dev/null
+++ b/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/build/main.tex
@@ -0,0 +1,57 @@
+\documentclass[11pt]{article}
+\usepackage[a4paper,margin=28mm]{geometry}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern,microtype}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{booktabs,array}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta,positioning}
+\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref}
+\pdfinfoomitdate=1
+\pdftrailerid{}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newcommand{\R}{\mathbb R}
+\newcommand{\C}{\mathbb C}
+\newcommand{\Z}{\mathbb Z}
+\newcommand{\Q}{\mathbb Q}
+\newcommand{\F}{\mathbb F}
+\DeclareMathOperator{\Fix}{Fix}
+\DeclareMathOperator{\Crit}{Crit}
+\DeclareMathOperator{\Morse}{Morse}
+\DeclareMathOperator{\aug}{aug}
+\DeclareMathOperator{\Hom}{Hom}
+\DeclareMathOperator{\End}{End}
+\DeclareMathOperator{\im}{im}
+\DeclareMathOperator{\Spec}{Spec}
+\DeclareMathOperator{\Jac}{Jac}
+\title{A nondegenerate counterexample to the Morse-number Arnold bound}
+\author{OpenAI}
+\date{September 23, 2026}
+\hypersetup{pdftitle={A nondegenerate counterexample to the Morse-number Arnold bound},pdfauthor={OpenAI}}
+\begin{document}
+\maketitle
+\begin{abstract}
+We construct a smooth one-periodic Hamiltonian on a closed symplectic
+twelve-manifold whose time-one map has fewer fixed points than the
+manifold's ordinary Morse number. Every fixed point is nondegenerate and
+has a contractible Hamiltonian trajectory. This disproves the Morse-number
+form of the Arnold conjecture. The deficit can be arbitrarily large among
+twelve-dimensional examples.
+\end{abstract}
+\tableofcontents
+\input{sections/01-introduction}
+\input{sections/02-algebra}
+\input{sections/03-relative-model}
+\input{sections/04-cancellation}
+\input{sections/05-hamiltonian}
+\input{sections/06-counting}
+\bibliographystyle{plain}
+\bibliography{refs}
+\end{document}
diff --git a/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/build/refs.bib b/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/build/refs.bib
new file mode 100644
index 0000000000000000000000000000000000000000..ecf38ef296540fbe4056ce8d587f0d4e024e75f0
--- /dev/null
+++ b/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/build/refs.bib
@@ -0,0 +1,307 @@
+@techreport{Gompf1992,
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+}
+
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+}
+
+@article{Damian2002,
+  author = {Damian, Mihai},
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+  doi = {10.1112/S0024609301008955},
+  url = {https://doi.org/10.1112/S0024609301008955}
+}
+
+@book{Milnor1965,
+  author = {Milnor, John},
+  title = {Lectures on the {$h$}-Cobordism Theorem},
+  publisher = {Princeton University Press},
+  address = {Princeton, NJ},
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+  note = {Notes by L. Siebenmann and J. Sondow},
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+}
+
+@book{Ranicki2002,
+  author = {Ranicki, Andrew},
+  title = {Algebraic and Geometric Surgery},
+  publisher = {Oxford University Press},
+  year = {2002},
+  note = {Citations use the March 2014 electronic version; its pagination differs from the printed book},
+  url = {https://webhomes.maths.ed.ac.uk/~v1ranick/books/surgery.pdf}
+}
+
+@article{Laudenbach2013,
+  author = {Laudenbach, Fran\c{c}ois},
+  title = {A proof of {Morse}'s theorem about the cancellation of critical points},
+  journal = {Comptes Rendus Math\'ematique},
+  volume = {351},
+  number = {11--12},
+  year = {2013},
+  pages = {483--488},
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+  url = {https://comptes-rendus.academie-sciences.fr/mathematique/item/10.1016/j.crma.2013.06.009.pdf}
+}
+
+@misc{Morgan2020,
+  author = {Morgan, John},
+  title = {Lecture {IV}: {$K_0$}, {$K_1$}, {Whitehead} torsion, and the {$s$}-cobordism theorem},
+  howpublished = {Lecture notes, Columbia University},
+  year = {2020},
+  note = {Dated 15 October 2020},
+  url = {https://www.math.columbia.edu/~jmorgan/Lecture4scobordism.pdf}
+}
+
+@article{Bass,
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+  title = {{$K$}-theory and stable algebra},
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+  volume = {22},
+  year = {1964},
+  pages = {5--60},
+  doi = {10.1007/BF02684689},
+  url = {https://www.numdam.org/article/PMIHES_1964__22__5_0.pdf}
+}
+
+@article{GruenbergLinnell,
+  author = {Gruenberg, Karl W. and Linnell, Peter A.},
+  title = {Generation gaps and abelianized defects of free products},
+  journal = {Journal of Group Theory},
+  volume = {11},
+  number = {5},
+  year = {2008},
+  pages = {587--608},
+  doi = {10.1515/JGT.2008.036},
+  eprint = {math/0701225},
+  archivePrefix = {arXiv},
+  note = {Preprint version 2, 26 December 2007},
+  url = {https://arxiv.org/abs/math/0701225v2}
+}
+
+@article{CGK1974,
+  author = {Cossey, John and Gruenberg, Karl W. and Kov{\'a}cs, L. G.},
+  title = {The presentation rank of a direct product of finite groups},
+  journal = {Journal of Algebra},
+  volume = {28},
+  number = {3},
+  year = {1974},
+  pages = {597--603},
+  doi = {10.1016/0021-8693(74)90062-3},
+  url = {https://archives.maths.anu.edu.au/people/Kovacs/K034.pdf}
+}
+
+@incollection{DRG2017,
+  author = {{Dimitroglou Rizell}, Georgios and Golovko, Roman},
+  title = {The number of {Hamiltonian} fixed points on symplectically aspherical manifolds},
+  booktitle = {Proceedings of the G\"okova Geometry-Topology Conference 2016},
+  publisher = {International Press},
+  year = {2017},
+  pages = {138--150},
+  eprint = {1609.04776},
+  archivePrefix = {arXiv},
+  url = {https://gokovagt.org/proceedings/2016/04ggt16-dimigolo.pdf}
+}
+
+@article{DRG2018,
+  author = {{Dimitroglou Rizell}, Georgios and Golovko, Roman},
+  title = {The stable {Morse} number as a lower bound for the number of {Reeb} chords},
+  journal = {Journal of Symplectic Geometry},
+  volume = {16},
+  number = {5},
+  year = {2018},
+  pages = {1209--1248},
+  doi = {10.4310/JSG.2018.v16.n5.a2},
+  eprint = {1510.08838},
+  archivePrefix = {arXiv},
+  url = {https://doi.org/10.4310/JSG.2018.v16.n5.a2}
+}
+
+@incollection{OnoPajitnov2016,
+  author = {Ono, Kaoru and Pajitnov, Andrei},
+  title = {On the fixed points of a {Hamiltonian} diffeomorphism in presence of fundamental group},
+  booktitle = {Essays in Mathematics and its Applications: In Honor of Vladimir Arnold},
+  editor = {Rassias, Themistocles M. and Pardalos, Panos M.},
+  publisher = {Springer},
+  address = {Cham},
+  year = {2016},
+  pages = {199--228},
+  doi = {10.1007/978-3-319-31338-2_10},
+  eprint = {1405.2505},
+  archivePrefix = {arXiv},
+  note = {Citations use preprint version 2, 5 November 2014},
+  url = {https://arxiv.org/abs/1405.2505v2}
+}
+
+@article{Barraud2018,
+  author = {Barraud, Jean-Fran{\c c}ois},
+  title = {A {Floer} fundamental group},
+  journal = {Annales scientifiques de l'{\'E}cole normale sup{\'e}rieure},
+  series = {4},
+  volume = {51},
+  number = {3},
+  year = {2018},
+  pages = {773--809},
+  doi = {10.24033/asens.2366},
+  url = {https://www.numdam.org/item/10.24033/asens.2366.pdf}
+}
+
+@misc{BaiXu2022,
+  author = {Bai, Shaoyun and Xu, Guangbo},
+  title = {{Arnold} conjecture over integers},
+  year = {2022},
+  eprint = {2209.08599},
+  archivePrefix = {arXiv},
+  primaryClass = {math.SG},
+  note = {Version 1, 18 September 2022},
+  url = {https://arxiv.org/abs/2209.08599v1}
+}
+
+@misc{BaiXu2026,
+  author = {Bai, Shaoyun and Xu, Guangbo},
+  title = {Integral {Hamiltonian} {Floer} theory: Foundations},
+  year = {2026},
+  eprint = {2606.15847},
+  archivePrefix = {arXiv},
+  primaryClass = {math.SG},
+  note = {Version 1, 14 June 2026},
+  url = {https://arxiv.org/abs/2606.15847v1}
+}
+
+@incollection{EliashbergGromov1998,
+  author = {Eliashberg, Yakov and Gromov, Mikhail},
+  title = {Lagrangian intersection theory: finite-dimensional approach},
+  booktitle = {Geometry of Differential Equations},
+  series = {American Mathematical Society Translations, Series 2},
+  volume = {186},
+  year = {1998},
+  pages = {27--118},
+  publisher = {American Mathematical Society},
+  doi = {10.1090/trans2/186/02},
+  note = {Section locators refer to the authors' version dated 24 September 2004},
+  url = {https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/998.pdf}
+}
+
+@article{Yorke1969,
+  author = {Yorke, James A.},
+  title = {Periods of periodic solutions and the {Lipschitz} constant},
+  journal = {Proceedings of the American Mathematical Society},
+  volume = {22},
+  number = {2},
+  year = {1969},
+  pages = {509--512},
+  doi = {10.1090/S0002-9939-1969-0245916-7},
+  url = {https://doi.org/10.1090/S0002-9939-1969-0245916-7}
+}
+
+@article{Arnold1986,
+  author = {Arnol'd, Vladimir I.},
+  title = {First steps in symplectic topology},
+  journal = {Russian Mathematical Surveys},
+  volume = {41},
+  number = {6},
+  year = {1986},
+  pages = {1--21},
+  doi = {10.1070/RM1986v041n06ABEH004221},
+  url = {https://doi.org/10.1070/RM1986v041n06ABEH004221}
+}
+
+@article{Golovko2020,
+  author = {Golovko, Roman},
+  title = {On variants of {Arnold} conjecture},
+  journal = {Archivum Mathematicum},
+  volume = {56},
+  number = {5},
+  year = {2020},
+  pages = {277--286},
+  doi = {10.5817/AM2020-5-277},
+  url = {https://doi.org/10.5817/AM2020-5-277}
+}
+
+@article{ConleyZehnder1983,
+  author = {Conley, Charles C. and Zehnder, Eduard},
+  title = {The {Birkhoff--Lewis} fixed point theorem and a conjecture of {V. I. Arnold}},
+  journal = {Inventiones Mathematicae},
+  volume = {73},
+  year = {1983},
+  pages = {33--49},
+  doi = {10.1007/BF01393824},
+  url = {https://doi.org/10.1007/BF01393824}
+}
+
+@article{Floer1989,
+  author = {Floer, Andreas},
+  title = {Symplectic fixed points and holomorphic spheres},
+  journal = {Communications in Mathematical Physics},
+  volume = {120},
+  number = {4},
+  year = {1989},
+  pages = {575--611},
+  doi = {10.1007/BF01260388},
+  url = {https://doi.org/10.1007/BF01260388}
+}
+
+@article{FukayaOno1999,
+  author = {Fukaya, Kenji and Ono, Kaoru},
+  title = {{Arnold} conjecture and {Gromov--Witten} invariant},
+  journal = {Topology},
+  volume = {38},
+  number = {5},
+  year = {1999},
+  pages = {933--1048},
+  doi = {10.1016/S0040-9383(98)00042-1},
+  url = {https://doi.org/10.1016/S0040-9383(98)00042-1}
+}
+
+@article{LiuTian1998,
+  author = {Liu, Gang and Tian, Gang},
+  title = {{Floer} homology and {Arnold} conjecture},
+  journal = {Journal of Differential Geometry},
+  volume = {49},
+  number = {1},
+  year = {1998},
+  pages = {1--74},
+  doi = {10.4310/jdg/1214460936},
+  url = {https://doi.org/10.4310/jdg/1214460936}
+}
+
+@article{Ruan1999,
+  author = {Ruan, Yongbin},
+  title = {Virtual neighborhoods and pseudo-holomorphic curves},
+  journal = {Turkish Journal of Mathematics},
+  volume = {23},
+  number = {1},
+  year = {1999},
+  pages = {161--232},
+  url = {https://journals.tubitak.gov.tr/math/vol23/iss1/10/}
+}
+
+@misc{Ma2013,
+  author = {Ma, Renyi},
+  title = {Proofs on {Arnold} conjectures},
+  year = {2013},
+  eprint = {0808.0613},
+  archivePrefix = {arXiv},
+  primaryClass = {math.SG},
+  note = {Version 7, 5 July 2013},
+  url = {https://arxiv.org/abs/0808.0613v7}
+}
diff --git a/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/build/sections/01-introduction.tex b/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/build/sections/01-introduction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..0a383c0012e193fd5410c9511f27af4f6598167d
--- /dev/null
+++ b/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/build/sections/01-introduction.tex
@@ -0,0 +1,207 @@
+\section{The Morse-number question}\label{sec:introduction}
+Arnold's fixed-point conjectures compare Hamiltonian fixed points with
+critical points of functions on the underlying symplectic manifold
+\cite[Section~2]{Arnold1986}. For a sufficiently $C^2$-small autonomous Hamiltonian,
+the fixed points of its time-one map are precisely its critical points.
+The question is how much of this finite-dimensional lower bound survives
+for an arbitrary Hamiltonian path. We address the formulation using the
+ordinary Morse number.
+
+Let $(M,\omega)$ be a closed connected symplectic manifold and let
+$H\in C^\infty((\R/\Z)\times M,\R)$. We use the convention
+\[
+ \omega(X_H(t,\cdot),\cdot)=dH_t,
+ \qquad \phi_H^0=\mathrm{id}_M
+\]
+for the Hamiltonian vector field and its flow. A fixed point $x$ of
+$\phi_H^1$ is \emph{nondegenerate} if
+$1\notin\Spec(d_x\phi_H^1)$. Write $\Fix_0(\phi_H^1;H)$ for the fixed
+points whose loops $t\mapsto\phi_H^t(x)$ are contractible in $M$.
+Contractibility here refers to this Hamiltonian path.
+
+The \emph{ordinary Morse number} is
+\[
+ \Morse(M)=\min\{\#\Crit(f):f\in C^\infty(M,\R)\text{ is Morse}\}.
+\]
+The Morse-number form of the Arnold conjecture asks whether
+$\#\Fix_0(\phi_H^1;H)\ge\Morse(M)$ whenever the time-one map is
+nondegenerate \cite[Conjecture~1.1]{Golovko2020}.
+We construct examples in which even the total number of fixed points is
+smaller than $\Morse(M)$, and every fixed-point trajectory is contractible.
+
+\begin{theorem}\label{thm:main}
+There exist a closed connected symplectic manifold $(M,\omega)$ of real
+dimension twelve and a smooth one-periodic Hamiltonian $H$ such that every
+fixed point of $\phi_H^1$ is nondegenerate and
+\[
+\#\Fix_0(\phi_H^1;H)=\#\Fix(\phi_H^1)
+                   =\chi(M)+128<\Morse(M).
+\]
+\end{theorem}
+
+The same construction gives a quantitative separation in fixed dimension.
+\begin{corollary}\label{cor:deficit}
+For every real number $D>0$, there is an example as in
+Theorem~\ref{thm:main} satisfying
+\[
+ \Morse(M)-\#\Fix(\phi_H^1)>D.
+\]
+The manifold is allowed to depend on $D$.
+\end{corollary}
+
+\subsection{From homological bounds to ordinary Morse theory}
+Conley and Zehnder proved the nondegenerate bound on the standard
+symplectic torus, where the ordinary Morse number equals the total Betti
+number~\cite{ConleyZehnder1983}.
+Floer's pseudoholomorphic-curve approach established homological Morse
+inequalities for Hamiltonian fixed points, including the monotone
+setting~\cite{Floer1989}. Virtual-moduli constructions of Fukaya--Ono,
+Liu--Tian and Ruan extended the rational Betti-number bound to general
+closed symplectic manifolds
+\cite{FukayaOno1999,LiuTian1998,Ruan1999}.
+The integral bounds of Bai--Xu retain torsion information as well
+\cite[Theorem~A]{BaiXu2022}, \cite[Corollary~1.2]{BaiXu2026}.
+These homological bounds do not identify the number of generators of a
+Floer complex with the minimum number of critical points of an ordinary
+Morse function.
+
+Stabilization explains one source of this distinction. The
+\emph{stable Morse number} minimizes the number of critical points of
+Morse functions $F:M\times\R^r\to\R$ that are almost quadratic at
+infinity, allowing the integer $r\geq0$ to vary. Here this means that $\|dF-dQ\|$ is
+uniformly bounded in a product metric for some nondegenerate quadratic
+form $Q$ on $\R^r$~\cite[Section~2.1]{DRG2017}.
+Arnold already considered a related stable critical-point minimum and
+asked whether it equals the minimum on the original manifold
+\cite[Section~5]{Arnold1986}.
+Eliashberg and Gromov developed the relationship with simple chain
+homotopy over the fundamental-group ring and augmentation-ideal
+generation~\cite[Sections~0.2.2 and~0.2.5]{EliashbergGromov1998}.
+Damian proved that ordinary and stable Morse numbers of closed manifolds
+can differ~\cite{Damian2002}. A relative realization theorem after
+sufficiently large stabilization is given in
+\cite[Proposition~2.9]{DRG2018}.
+
+The algebraic distinction used here is classical and particularly sharp
+for direct powers of nontrivial perfect finite groups. Cossey, Gruenberg and
+Kov\'acs proved that the augmentation-ideal generator number remains
+constant under these direct powers, while the group generator number
+is unbounded~\cite[Theorems~1--2]{CGK1974}; see also
+\cite[Sections~1--2]{GruenbergLinnell}.
+Fundamental-group refinements of Floer theory reflect this distinction:
+Ono--Pajitnov obtain augmentation-ideal bounds under weak monotonicity
+\cite[Theorem~5.7]{OnoPajitnov2016}, whereas, in the symplectically
+aspherical or monotone setting, Barraud's reconstruction of group
+generators counts Floer trajectories with multiplicities
+\cite[Theorem~1.3 and Remark~3]{Barraud2018}.
+The trajectory multiplicities are part of the bound; these results do
+not yield the unrestricted ordinary Morse-number inequality.
+
+For closed symplectic manifolds on which both $[\omega]$ and $c_1(TM)$
+vanish on $\pi_2(M)$, Dimitroglou Rizell and Golovko prove the stable
+Morse number as a lower bound for nondegenerate Hamiltonian fixed
+points~\cite[Corollary~1.3]{DRG2017}.
+The sphere factors in our examples have positive symplectic area, so
+this hypothesis does not hold. The construction below instead realizes
+a short relative Morse complex in exactly eight additional variables,
+with the function equal to a prescribed quadratic form outside a compact
+set. Its small-time return map is then inserted into a closed product
+of spheres. The relative handle realization and the exclusion of all
+additional fixed points are the two geometric steps needed to turn the
+ordinary/stable distinction into the stated closed Hamiltonian example.
+
+The integral homology of our examples is torsion-free and concentrated
+in even degrees, with total rank $\chi(M)$, as verified in
+Section~\ref{sec:counting}. Their fixed-point count therefore exceeds
+the homological quantities above.
+
+Ma's preprint asserts that the minimum number of fixed points of a
+nondegenerate Hamiltonian diffeomorphism always equals the ordinary
+Morse number~\cite[Theorem~1.5]{Ma2013}. Theorem~\ref{thm:main}
+contradicts that unrestricted assertion.
+
+\subsection{The construction and its two counts}
+For a finitely generated group $G$, let $d(G)$ be the least number of group
+generators. Set $b=2$ and consider the finite groups $G=A_5^k$, where
+$A_5$ has order $60$. Two elementary algebraic facts will drive the proof:
+$d(A_5^k)$ is unbounded as $k$ increases, whereas the right augmentation
+ideal
+\[
+ I_G=\ker(\aug:\Z[G]\longrightarrow\Z)
+\]
+has a generating list of length $b$, independent of $k$.
+The distinction is between generation of a group and generation of a
+module over its group ring.
+
+Gompf's existence theorem supplies a closed connected symplectic
+four-manifold $X$ with $\pi_1(X)=G$~\cite[Theorem~3.1]{Gompf1992};
+see also \cite{Gompf1995}.
+We construct a Morse function on $X\times\R^8$ with precisely
+$\chi(X)+4b$ critical points, equal outside a compact set to the quadratic
+form of index four
+\[
+ Q(u,v)=\frac{|v|^2-|u|^2}{2},\qquad u,v\in\R^4.
+\]
+The eight variables are fixed throughout: the proof performs handle moves
+on one twelve-dimensional cobordism, rather than appealing to a
+stabilization of unspecified dimension.
+
+The closed symplectic manifold is
+$M=X\times(S^2)^4$. Sixteen disjoint product-cap neighborhoods correspond
+to the north/south choices on the four sphere factors. We place the same
+small-time return map of the constructed Morse function in each cap.
+The main gluing issue is to match the local \emph{endpoint} with product
+rotations at both poles while excluding fixed points outside the caps.
+The resulting smooth one-periodic Hamiltonian satisfies
+\begin{equation}\label{eq:overview-fixed-count}
+ \#\Fix(\phi_H^1)=16\bigl(\chi(X)+4b\bigr)
+                   =\chi(M)+128.
+\end{equation}
+
+On the other hand, the fundamental group forces every Morse function on
+$M$ to have at least $d(G)$ critical points in each of indices one and
+eleven. Euler's identity gives
+\begin{equation}\label{eq:overview-morse-bound}
+ \Morse(M)\ge\chi(M)+4d(G).
+\end{equation}
+Since $d(A_5^k)$ is unbounded while $b$ is fixed, these two estimates
+give the strict inequality and the arbitrary deficits. Figure~\ref{fig:construction}
+separates the two roles of the fundamental group.
+
+\begin{figure}[ht]
+\centering
+\begin{tikzpicture}[
+ every node/.style={align=center,font=\small},
+ box/.style={draw,inner sep=6pt,text width=58mm},
+ >={Stealth}]
+\node[box,text width=90mm] (group) at (0,0)
+{$G=A_5^k$\\$I_G$ has two generators; $d(G)$ is unbounded};
+\node[box] (morse) at (-3.4,-1.8)
+ {Relative handle cancellation\\$X^4\times\R^8$\\$\chi(X)+8$ critical points};
+\node[box] (bound) at (3.4,-1.8)
+ {Ordinary Morse lower bound\\$M^{12}=X\times(S^2)^4$\\$\Morse(M)\ge\chi(M)+4d(G)$};
+\node[box] (closed) at (-3.4,-3.6)
+ {Sixteen cap insertions\\$\#\Fix(\phi_H^1)=\chi(M)+128$};
+\node[box] (gap) at (3.4,-3.6)
+ {Unbounded deficit\\$\Morse(M)-\#\Fix(\phi_H^1)$};
+\draw[->] (group)--(morse);
+\draw[->] (group)--(bound);
+\draw[->] (morse)--(closed);
+\draw[->] (bound)--(gap);
+\draw[->] (closed)--(gap);
+\end{tikzpicture}
+\caption{The augmentation-ideal bound controls the constructed fixed-point
+count. The group generator bound supplies the larger ordinary Morse
+lower bound on the same closed manifold.}
+\label{fig:construction}
+\end{figure}
+
+\subsection{Proof organization}
+Section~\ref{sec:algebra} proves the two group-ring facts and the exact
+row operation needed for geometric cancellation.
+Section~\ref{sec:relative-model} constructs the relative Morse model and
+its handle complex. Section~\ref{sec:cancellation} reduces the handle
+counts while preserving the quadratic collars.
+Section~\ref{sec:hamiltonian} realizes the local map on the closed product.
+Section~\ref{sec:counting} compares the two counts and proves the theorem
+and its deficit corollary.
diff --git a/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/build/sections/02-algebra.tex b/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/build/sections/02-algebra.tex
new file mode 100644
index 0000000000000000000000000000000000000000..d9769c2bd8d858da6df7984bc0604cb0dbb5ef74
--- /dev/null
+++ b/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/build/sections/02-algebra.tex
@@ -0,0 +1,278 @@
+\section{Group generators and augmentation ideals}
+\label{sec:algebra}
+
+We need finite groups whose group generators are numerous but whose
+augmentation ideals have uniformly short generating lists.  The distinction
+between these two numbers is classical; see
+\cite[Sections~1--2]{GruenbergLinnell}.  We give the estimates needed here,
+then prove the elementary row reduction used in the handle construction.
+All modules in this section are right modules.
+
+For a finitely generated group $G$, let $d(G)$ be its minimum number of
+group generators.
+
+\begin{lemma}\label{lem:group-gap}
+Let $S$ be a nontrivial finite group, let $k$ be a finite positive
+integer, and let $m\ge0$ be an integer.
+If $k>|S|^m$, then $d(S^k)>m$.
+\end{lemma}
+
+\begin{proof}
+Suppose that $g_1,\ldots,g_m$ generate $S^k$, padding a shorter list with
+identities.  Each factor coordinate $i$ gives a tuple
+$((g_1)_i,\ldots,(g_m)_i)\in S^m$.  Since $k>|S|^m$, two coordinates
+$i\ne j$ have the same tuple.  Every listed generator then belongs to the
+proper subgroup
+\[
+  \{g\in S^k:g_i=g_j\},
+\]
+a contradiction.
+\end{proof}
+
+We shall use
+\begin{equation}\label{eq:group-choice}
+  b=2,\qquad S=A_5,\qquad G=S^k.
+\end{equation}
+For now $k\ge1$ is arbitrary.  The group $S$ is nonabelian simple, hence
+perfect, and the direct product $G$ is also perfect.  Every finite group is finitely presented: one may
+take symbols $t_g$ for its elements and relations $t_1=1$ and
+$t_gt_h=t_{gh}$.  Thus these groups satisfy the finite-presentation
+condition needed for the symplectic realization below.
+
+Write $\Lambda=\Z[G]$ and let
+\[
+  \operatorname{aug}:\Lambda\longrightarrow\Z,
+  \qquad \sum_{g\in G}a_gg\longmapsto\sum_{g\in G}a_g
+\]
+be the augmentation map.  Its kernel $I$ is the augmentation ideal,
+regarded as a right $\Lambda$-module.
+
+The uniform bound below is a special case of the direct-product theorem
+of Cossey, Gruenberg and Kov\'acs~\cite[Theorem~2]{CGK1974}.
+For finite groups, that theorem expresses the augmentation-ideal
+generator number of a direct product as the maximum of the corresponding
+factor numbers and the generator number of its abelianization.
+We give an elementary proof for the present family, including the
+passage from modular to integral generators.
+
+\begin{lemma}\label{lem:augmentation}
+For every finite positive integer $k$, the right augmentation ideal of
+$\Z[A_5^k]$ can be generated by two elements.
+\end{lemma}
+
+\begin{proof}
+We first prove that two generators suffice modulo every prime.
+The group $A_5$ is generated by
+$a=(1\,2\,3\,4\,5)$ and $c=(1\,2\,3)$.
+Indeed, $aca^{-1}=(2\,3\,4)$, and the subgroup generated by
+$c$ and $aca^{-1}$ is transitive on $\{1,2,3,4\}$, and the
+stabilizer of $4$ contains the order-three group generated by $c$. It is therefore all of the copy
+of $A_4$ fixing $5$. The subgroup generated by $a,c$ is transitive
+on five letters and has a stabilizer containing this $A_4$, so it
+has order at least $5\cdot12=60$ and equals $A_5$.
+
+Put $\Lambda=\Z[G]$ and $I=\ker(\aug:\Lambda\to\Z)$.
+The splitting $\Lambda=I\oplus\Z\cdot1$ as abelian groups identifies
+\[
+ R_p=\F_p[G],\qquad I_p=I/pI=\ker(R_p\xrightarrow{\aug}\F_p).
+\]
+For each simple right $R_p$-module $V$, we claim
+\begin{equation}\label{eq:augmentation-hom-bound}
+ \dim_{\F_p}\Hom_{R_p}(I_p,V)\le 2\dim_{\F_p}V.
+\end{equation}
+If $G$ acts trivially on $V$, every homomorphism $\ell:I_p\to V$
+defines a group homomorphism $g\mapsto\ell(g-1)$ to the additive
+group of $V$. Since $G$ is perfect, this map vanishes, and thus
+$\ell=0$.
+
+Otherwise some factor $S_i\cong A_5$ acts nontrivially on $V$.
+Normality of $S_i$ makes $V^{S_i}$ a submodule, so simplicity gives
+$V^{S_i}=0$. Write $a_i,c_i$ for the two displayed generators in this
+factor. We show that $\ell$ is determined by
+$\ell(a_i-1)$ and $\ell(c_i-1)$. The identity
+\[
+ \ell(gh-1)=\ell(g-1)h+\ell(h-1)
+\]
+shows that if both values vanish, then $\ell(s-1)=0$ for every
+$s\in S_i$. If $h$ lies in the product of the other factors, then
+for every $s\in S_i$,
+\[
+ \ell(h-1)(s-1)
+ =\ell((h-1)(s-1))
+ =\ell((s-1)(h-1))=0.
+\]
+Thus $\ell(h-1)\in V^{S_i}=0$. Every element of $G$ has the form
+$sh$, so the same identity gives $\ell=0$. Evaluation at the two
+generators therefore embeds $\Hom_{R_p}(I_p,V)$ into $V^2$, proving
+\eqref{eq:augmentation-hom-bound}.
+
+Let $J=\Jac(R_p)$. We use the elementary finite-dimensional algebra
+facts that $J$ annihilates every simple module, $J$ is nilpotent,
+and $R_p/J$ is semisimple. For completeness, $J$ is the intersection
+of the maximal right ideals, equivalently the annihilators of the
+simple right modules. Multiplication by $J$ lowers a composition
+series of the regular module, proving nilpotence. Finite dimension
+allows finitely many maximal right ideals with intersection $J$;
+thus $R_p/J$ embeds in a finite direct sum of simple quotients and is
+semisimple, as are its finite-dimensional modules.
+
+Let $a_V,t_V$ denote the multiplicities of a simple module $V$ in
+$I_p/I_pJ$ and $R_p/J$, respectively, and let
+$e_V=\dim_{\F_p}\End_{R_p}(V)$. Schur's lemma and evaluation at $1$
+give
+\[
+ \dim_{\F_p}\Hom_{R_p}(I_p,V)=a_Ve_V,
+ \qquad
+ \dim_{\F_p}\Hom_{R_p}(R_p,V)=t_Ve_V=\dim_{\F_p}V.
+\]
+Consequently $a_V\le2t_V$ for every $V$. There is a surjection
+$(R_p/J)^2\to I_p/I_pJ$. Lift the two generator images to $I_p$
+and let $N_p$ be their right span. Then
+$I_p/N_p=(I_p/N_p)J$, and nilpotence gives $N_p=I_p$.
+Hence $I_p$ has a generating pair for every prime $p$.
+
+It remains to lift to two integral generators. We first choose one
+element whose right ideal has finite additive index in $I$, while
+retaining a residue that extends to a generating pair at each prime
+dividing $|G|$. A second element will then remove the finite quotient.
+Put $n=|G|=60^k$.
+For each $p\in\{2,3,5\}$ choose a generating pair
+$(u_p,v_p)$ of $I_p$. By the Chinese remainder theorem on the free
+abelian group $I$, choose $y_0\in I$ with residue $u_p$ at each of
+these three primes. Define
+\[
+ z=n\cdot1-\sum_{g\in G}g\in I.
+\]
+For $x\in I$, we have $zx=nx$. Thus left multiplication on $I$ by
+$y_0+30t z$ has matrix $L_{y_0}+30tn\,\mathrm{id}$. Its determinant
+is a nonzero polynomial in $t$, so there is an integer $t$ for which
+it is nonzero. Fix this $t$ and put
+\[
+ y_1=y_0+30t z,\qquad D=|\det(L_{y_1}:I\to I)|>0.
+\]
+The adjugate identity gives $DI\subset y_1I\subset y_1\Lambda$.
+Moreover, $y_1$ still reduces to $u_p$ for $p=2,3,5$.
+
+For each prime $p\mid D$, the right module $I_p/y_1R_p$ is cyclic.
+For $p\mid n$, it is generated by the image of $v_p$.
+For $p\nmid n$, the element
+\[
+ e_p=1-n^{-1}\sum_{g\in G}g\in R_p
+\]
+satisfies $I_p=e_pR_p$, so its image generates every quotient of
+$I_p$, including $I_p/y_1R_p$. Apply the Chinese remainder theorem on $I$
+to choose $y_2\in I$ reducing to $v_p$ when $p\mid D$ and $p\mid n$,
+and to $e_p$ when $p\mid D$ and $p\nmid n$. When $D=1$,
+take $y_2=0$.
+
+The quotient $C=I/(y_1\Lambda+y_2\Lambda)$ is a finite abelian group
+killed by $D$, and the residue choices give $C=pC$ for every prime
+$p\mid D$. Iterating these equalities through the prime factorization
+of $D$ yields $C=DC=0$. Therefore $y_1,y_2$ generate $I$ as a right
+$\Lambda$-module.
+\end{proof}
+
+In particular, choosing any finite $k>60^{16b}$ gives
+\begin{equation}\label{eq:generator-gap}
+  d(G)>16b,
+  \qquad I\text{ has a generating list of length }b.
+\end{equation}
+We always pad such a list by zeros if necessary.  Increasing $k$ makes
+$d(G)$ unbounded by Lemma~\ref{lem:group-gap}, while
+Lemma~\ref{lem:augmentation} keeps the same bound $b$.
+
+\subsection{An elementary row reduction}
+
+The next lemma supplies an entry equal to $1$ using only elementary
+basis additions.  This is the form needed for handle slides.  It is a
+special case of Bass's stable-range results applied to the opposite
+ring~\cite[Lemma~6.4 and Theorems~4.2(a),~11.1]{Bass}; we include a direct
+proof to retain control of the multiplication sides and the row length.
+
+\begin{lemma}\label{lem:row}
+Let $G$ be a finite group and $\Lambda=\Z[G]$.  If $m\ge3$ and
+\[
+  x_1\Lambda+\cdots+x_m\Lambda=\Lambda,
+\]
+then finitely many elementary additions
+\[
+  x_j\longleftarrow x_j+x_iq,
+  \qquad i\ne j,\quad q\in\Lambda,
+\]
+can make an entry of the row equal to $1$.
+\end{lemma}
+
+\begin{proof}
+We first record how to correct a designated entry over a
+finite-dimensional semisimple algebra $R$.  Suppose $xR+U=R$, where
+$U$ is a right ideal.  Let $T=L_x:R\to R$ be left multiplication by
+$x$, let $\pi:R\to R/U$ be the quotient map, and put
+$K=\ker(\pi T)$.  The map $\pi T$ is onto, and the split exact
+sequences give
+\[
+  R\cong K\oplus R/U,
+  \qquad R\cong U\oplus R/U.
+\]
+Comparison of simple-module multiplicities therefore gives $K\cong U$.
+Choose an isomorphism $h:K\to U$.  Extend
+$h-T|_K:K\to U$ to a right-module map $c:R\to U$ using a complement
+of $K$.  The map $T+c$ is an isomorphism: it restricts to $h$ on $K$
+and induces the isomorphism $R/K\to R/U$ supplied by $\pi T$.
+Every right-module endomorphism of $R$ is left multiplication by its
+value at $1$.  Hence $T+c=L_{x+u}$ for $u=c(1)\in U$, and $x+u$ is
+a unit.  If $U=\sum_{i\ne j}x_iR$, this corrects only the designated
+entry $x_j$ by additions of right multiples of the other entries.
+
+Apply this observation first to $R=\Q[G]$, which is semisimple by
+Maschke's theorem, with designated entry $x_1$.  In group-basis
+coordinates for the correcting coefficients $q_i$, the determinant
+\[
+  \det_{\Q} L_{x_1+\sum_{i\ne1}x_iq_i}
+\]
+is a polynomial with integer coefficients.  The unit correction over
+$\Q[G]$ shows that it is nonzero.  A nonzero polynomial over $\Q$
+does not vanish at every integer tuple, by induction on its number of
+variables.  We may therefore choose all $q_i\in\Lambda$ with this
+determinant nonzero.  After these elementary additions, rename the
+first entry $x_1$.  Its principal right ideal has finite additive index
+\[
+  D=[\Lambda:x_1\Lambda]=|\det L_{x_1}|,
+  \qquad D\Lambda\subset x_1\Lambda.
+\]
+
+We keep this entry fixed and correct only $x_2$.  For each prime
+$p\mid D$, let $R_p=\Lambda/p\Lambda$ and
+$J_p=\operatorname{Jac}(R_p)$.  The current row generates the unit
+right ideal in the semisimple algebra $R_p/J_p$.  The preceding
+observation gives coefficients $\bar q_{i,p}\in R_p$, $i\ne2$, such
+that
+\[
+  \bar x_2+\sum_{i\ne2}\bar x_i\bar q_{i,p}
+\]
+is a unit modulo $J_p$, and hence a unit in $R_p$.  Here units lift
+across the nilpotent ideal $J_p$, since $1-j$ has a finite geometric
+series inverse for $j\in J_p$.  Choose $q_i\in\Lambda$, $i\ne2$,
+with these residues for all $p\mid D$ by coefficientwise Chinese
+remaindering, and make the additions
+\[
+  x_2\longleftarrow x_2+\sum_{i\ne2}x_iq_i.
+\]
+All source entries remain fixed during these additions, so the new
+$x_2$ is a unit modulo every prime dividing the unchanged integer $D$.
+
+The additive quotient
+$A=\Lambda/(x_1\Lambda+x_2\Lambda)$ is finite and killed by $D$.
+For each $p\mid D$, the invertibility of $x_2$ modulo $p$ gives
+$x_2\Lambda+p\Lambda=\Lambda$, and hence $A=pA$.  Iteration gives
+$A=DA=0$.  This also covers $D=1$, when $A$ is already zero.
+We have proved $x_1\Lambda+x_2\Lambda=\Lambda$.
+
+Finally put $t=x_3$ and choose $a,c\in\Lambda$ with $x_1a+x_2c=1$.  The two
+elementary additions to the third entry with coefficients
+$a(1-t)$ and $c(1-t)$ give
+\[
+  t+x_1a(1-t)+x_2c(1-t)=1.
+\]
+Neither source entry changes, and all products retain the required
+order.
+\end{proof}
diff --git a/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/build/sections/03-relative-model.tex b/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/build/sections/03-relative-model.tex
new file mode 100644
index 0000000000000000000000000000000000000000..d4ddf1fb0754d8fde0df7724fb6b7a81e03dbeb9
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+++ b/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/build/sections/03-relative-model.tex
@@ -0,0 +1,264 @@
+\section{A relative Morse model in twelve dimensions}
+\label{sec:relative-model}
+
+We now turn a short generating list for an augmentation ideal into a Morse
+function with few critical points.  Let $G$ be a finite group, put
+$\Lambda=\Z[G]$, and let $I\subset\Lambda$ be its right augmentation ideal.
+Assume that $I$ has a generating list of length $b\geq2$.
+We will use this list and the elementary-row lemma established above.
+Let $X$ be a closed connected oriented smooth four-manifold with
+$\pi_1(X)\cong G$.  Since $G$ is finite, rational homology and Poincar\'e
+duality give
+\[
+ H_1(X;\Q)=H_3(X;\Q)=0,
+ \qquad \chi(X)=2+\dim_{\Q}H_2(X;\Q)\geq 2.
+\]
+No symplectic structure is needed for this part of the argument.
+The relation between quadratic stabilization, group-ring complexes and
+augmentation generators is developed by Eliashberg and Gromov
+\cite[Sections~0.2.2 and~0.2.5]{EliashbergGromov1998}.
+We carry out the required realization on a specified relative cobordism,
+so that the number of variables and the exterior function remain fixed.
+
+For $u,v\in\R^4$, put
+\begin{equation}
+ Q(u,v)=\frac{|v|^2-|u|^2}{2},
+ \qquad r^2=|u|^2+|v|^2.
+ \label{eq:quadratic}
+\end{equation}
+We also write $Q$ for its pullback to $X\times\R^8$.
+
+\begin{proposition}
+\label{prop:relative-morse}
+Let $G$ be a finite group whose right augmentation ideal in $\Z[G]$
+has a generating list of length $b\geq2$.
+Let $X$ be a closed connected oriented smooth four-manifold with
+$\pi_1(X)\cong G$.
+There is a smooth Morse function $F:X\times\R^8\to\R$ such that
+\[
+ F=Q\quad\text{for }r\geq 3,
+ \qquad
+ \#\Crit(F)=\chi(X)+4b.
+\]
+Its critical points have indices between four and eight.
+\end{proposition}
+
+The number of added variables is part of the assertion.  We will prove the
+proposition by changing handles on one fixed compact twelve-dimensional
+cobordism.  This section constructs the cobordism and its handle complex;
+Section~\ref{sec:cancellation} performs the reduction and completes the proof.
+
+\subsection{The domain and its collars}
+
+Define
+\begin{equation}
+ \begin{gathered}
+ E=\{(u,v):r^2\leq 9,\ -1\leq Q\leq 1\},
+ \qquad E_{\pm}=E\cap\{Q=\pm1\},\\
+ W=X\times E,\qquad A=X\times E_-,\qquad B=X\times E_+.
+ \end{gathered}
+ \label{eq:relative-domain}
+\end{equation}
+The incoming and outgoing faces are $A$ and $B$; the side is the face
+$r=3$.  At its corners, both $u$ and $v$ are nonzero and $dr,dQ$ are
+linearly independent.  In a neighborhood of the side, the coordinates
+\[
+ (r,Q,u/|u|,v/|v|)
+\]
+give a product collar with interval coordinate $Q$.  Since $dQ\neq0$ on
+$A\cup B$, these faces also have collars with coordinate $Q$.  To make
+the collars compatible, choose a vector field with $dQ$-value one which
+is the product coordinate field near the side, and extend it over the
+ends by a partition of unity.  All handle constructions below take place
+away from a fixed smaller side collar.
+
+Radial contraction preserves $E$, so $E$ is contractible.  On $E_-$,
+the parametrization
+\[
+ (\xi,v)\longmapsto
+ \bigl(\sqrt{2+|v|^2}\,\xi,v\bigr),
+ \qquad \xi\in S^3,\quad |v|^2\leq\frac72,
+\]
+identifies $E_-$ with $S^3\times D^4$.  Interchanging $u$ and $v$ gives
+the same description of $E_+$.  Consequently both inclusions
+$A\hookrightarrow W$ and $B\hookrightarrow W$ induce isomorphisms on
+fundamental groups.
+
+Write $\widetilde X$ for the universal cover of $X$ and use tildes for
+the corresponding covers of $W,A,B$.  The relative homology of
+$(E,E_-)$ is $\Z$ in degree four and zero in every other degree.  The
+relative K\"unneth formula therefore gives, equivariantly for the deck
+action,
+\begin{equation}
+ H_{4+i}(\widetilde W,\widetilde A;\Z)
+ \cong H_i(\widetilde X;\Z).
+ \label{eq:relative-homology}
+\end{equation}
+There is no torsion term because the relative homology of $(E,E_-)$ is
+free.  The same formula holds with $\widetilde B$ in place of
+$\widetilde A$.  In particular,
+\begin{equation}
+ H_4(\widetilde W,\widetilde A;\Z)\cong\Z,
+ \qquad H_5(\widetilde W,\widetilde A;\Z)=0,
+ \label{eq:low-relative-homology}
+\end{equation}
+where the action on $\Z$ is trivial.  Here we used connectedness and
+simple connectedness of $\widetilde X$.
+
+\subsection{An initial height function}
+
+Choose a Morse function $f:X\to\R$ with one minimum and one maximum.
+For completeness, this follows from the usual handle description of a
+connected manifold.  After ordering the handles, the zero- and
+one-handles form a connected graph.  Cancelling the edges of a spanning
+tree against all but one zero-handle leaves one minimum.  Attachments
+above each cancelling pair are transferred through the resulting
+boundary identification.  Applying the same procedure to the dual
+handle decomposition removes the extra maxima without changing the
+minimum.  This uses the handle decomposition theorem and the geometric
+cancellation criterion, in their elementary zero/one-handle cases
+\cite[Theorem~2.22 and Lemma~8.24]{Ranicki2002}.
+
+Let $\eta:[0,\infty)\to[0,1]$ be smooth, equal to one near zero and
+supported in $[0,1)$.  For $\sigma>0$ sufficiently small, set
+\begin{equation}
+ h_0(x,u,v)=Q(u,v)+\sigma\eta(r^2)f(x).
+ \label{eq:initial-height}
+\end{equation}
+Choose $\sigma$ so that
+\[
+ \sigma\|f\|_{\infty}<\frac12,
+ \qquad
+ 2\sigma\|f\|_{\infty}\|\eta'\|_{\infty}<1.
+\]
+The perturbation is supported where $r<1$ and hence $|Q|<1/2$.
+Thus $h_0(W)\subset[-1,1]$, and $h_0=Q$ near every face of $W$.
+For $r>0$, its derivative in the extra variables has norm at least
+\[
+ r\bigl(1-2\sigma\|f\|_{\infty}\|\eta'\|_{\infty}\bigr)>0.
+\]
+At $r=0$, the critical-point equation is $df=0$, and the Hessian is
+the direct sum of $\sigma\operatorname{Hess}f$ and
+$\operatorname{Hess}Q$.  Therefore $h_0$ is Morse, with exactly the
+critical points of $f$, each shifted upward by four in index.  In
+particular it has one critical point of index four and one of index
+eight.
+
+\subsection{Levels and the relative handle complex}
+
+Use gradient-like fields which are product fields on the side collar.
+The Morse--handle correspondence then describes $(W;A,B)$ by handles
+attached in the interiors of its regular levels.  The standard local
+handle traces and their product regions give this correspondence
+relative to the side collar; see
+\cite[Proposition~2.20 and Theorem~2.22]{Ranicki2002} for the handle
+construction.  Only handles of indices $4,\ldots,8$ will occur.
+
+We may order these handles by index.  Indeed, if an index-$j$ handle
+is followed by an index-$\ell$ handle with $\ell\leq j$, the latter
+attaching sphere has dimension $\ell-1$, and the former belt sphere
+has dimension $11-j$ in the intervening eleven-dimensional level.
+Their dimension sum is
+\[
+ (\ell-1)+(11-j)=10+\ell-j<11.
+\]
+General position makes the attaching sphere disjoint from that belt
+sphere.  After shrinking its framed neighborhood, it can be moved
+back through the surgery, interchanging the two handles.  All later
+attachments are transferred through the induced identification of
+the upper level.  The same argument permits arbitrary ordering
+within an equal-index block, with all moves supported in the interior.
+
+Every regular level $L$ is connected, and its inclusion into $W$
+induces
+\begin{equation}
+ \pi_1(L)\cong\pi_1(W)\cong G.
+ \label{eq:level-fundamental-group}
+\end{equation}
+To see this directly, surgery for an index-$j$ handle removes a
+neighborhood of $S^{j-1}$ and inserts one of $S^{11-j}$.  For
+$4\leq j\leq8$, both spheres have dimension at least three and
+codimension at least four in the level.  General position and van
+Kampen show that these surgeries preserve connectedness and the
+fundamental group.  Start with $A$, whose inclusion into $W$ is a
+fundamental-group isomorphism.  Alternatively, from any intermediate
+level both the upper part and the reversed lower part are built with
+handles of index at least four, so neither part changes the
+fundamental group.  The interior of $L$ has the same fundamental
+group as $L$.
+
+Choose an orientation and a lift of each handle.  Regard the deck
+action as a right action: if it was originally written on the left
+as $D_g$, use $c\cdot g=D_{g^{-1}}c$.  Let $C_i$ be the based free
+right $\Lambda$-module generated by the handles of index $4+i$, and
+let $r_i$ be their number.  The resulting relative handle complex is
+\begin{equation}
+ 0\longrightarrow C_4\xrightarrow{\partial_4}C_3
+ \xrightarrow{\partial_3}C_2\xrightarrow{\partial_2}C_1
+ \xrightarrow{\partial_1}C_0\longrightarrow0,
+ \qquad C_i\cong\Lambda^{r_i}.
+ \label{eq:handle-complex}
+\end{equation}
+Its degree-$i$ homology is
+$H_{4+i}(\widetilde W,\widetilde A;\Z)$.  One way to obtain the
+complex is to filter the handle decomposition by index; each
+successive relative group is freely generated by the lifted handles
+in that index.  Boundary coefficients are signed intersections of
+lifted attaching spheres with lifted belt spheres one index lower.
+Thus, writing
+\[
+ \partial e_j=\sum_i e_i a_{ij},
+\]
+the entries $a_{ij}$ lie in $\Lambda$ and multiply the basis elements
+on the right.  Replacing a basis vector $e_j$ by $e_j+e_i q$, with $i\neq j$, adds
+column$_i\,q$ to column$_j$ in its outgoing differential.  If $T$
+is the elementary matrix of this basis change, the adjacent
+differentials change coherently as $D\mapsto DT$ and
+$D'\mapsto T^{-1}D'$.  These conventions will be used for the handle
+slides in Section~\ref{sec:cancellation}.
+
+\begin{lemma}
+\label{lem:handle-complex}
+For any relative handle presentation of this same $(W;A,B)$ with
+indices in $4,\ldots,8$ and $r_0=r_4=1$, its based right-module
+complex satisfies
+\begin{equation}
+ \operatorname{im}\partial_1=I,
+ \qquad
+ \operatorname{im}\partial_2=\ker\partial_1,
+ \qquad
+ r_2=\chi(X)-2+r_1+r_3\geq r_1,
+ \label{eq:handle-identities}
+\end{equation}
+where $C_0$ is identified with $\Lambda$ using its chosen basis.
+The same assertions hold for the reversed cobordism relative to $B$.
+\end{lemma}
+
+\begin{proof}
+By \eqref{eq:low-relative-homology}, the quotient
+$C_0/\operatorname{im}\partial_1$ is the trivial right module $\Z$.
+A surjective right-module map $\Lambda\to\Z$ sends $1$ to $1$ or
+$-1$, so its kernel is $I$.  The vanishing of relative homology in
+degree five gives the second equality.
+
+Since $E$ is contractible and $E_-$ has the homotopy type of $S^3$,
+we have $\chi(W,A)=\chi(X)$.  All handle indices have been shifted
+by the even number four, and hence
+\[
+ \chi(X)=r_0-r_1+r_2-r_3+r_4
+         =2-r_1+r_2-r_3.
+\]
+This gives the rank identity; its inequality follows from
+$\chi(X)\geq2$ and $r_3\geq0$.  Finally, the same relative covering
+homology computation holds with $B$ in place of $A$.  Reversal sends
+index $4+i$ to index $8-i$ and preserves the endpoint counts, so the
+argument applies unchanged.
+\end{proof}
+
+We have now constructed a height on the required fixed cobordism
+and identified the algebra that its handles must satisfy.  To finish
+Proposition~\ref{prop:relative-morse}, it remains to reduce the
+index-five and index-seven counts to $b$ while retaining the single
+index-four and index-eight handles and the prescribed collars.
+Then Lemma~\ref{lem:handle-complex} will force
+$r_2=\chi(X)-2+2b$ and the total count $\chi(X)+4b$.
diff --git a/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/build/sections/04-cancellation.tex b/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/build/sections/04-cancellation.tex
new file mode 100644
index 0000000000000000000000000000000000000000..e774c4edc60b0ff8f3ed79c0ed66824280bb45ed
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+++ b/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/build/sections/04-cancellation.tex
@@ -0,0 +1,249 @@
+\section{Geometric reduction of the handles}
+\label{sec:cancellation}
+
+We continue with the fixed cobordism $(W;A,B)$ and the conventions of
+Section~\ref{sec:relative-model}.  The aim is to retain just $b$ handles
+in each of indices five and seven.  There are two distinct steps:
+elementary operations produce a coefficient exactly equal to $1$, and
+Whitney moves turn that coefficient into one geometric intersection.
+All moves below occur in the interior of $W$.
+
+\subsection{Introducing the augmentation generators}
+
+A slide of one index-$j$ handle over another replaces a basis vector
+$e_j$ by $e_j+e_iq$ in the associated free right $\Lambda$-module.
+To realize this, first take $q=\pm g$, with $g\in G$: choose a band
+whose path represents $g$, and choose its orientation to give the sign.
+The slide arcs can be chosen embedded and disjoint from the other
+attaching cores in an eleven-dimensional level; their tubular
+neighborhoods can then be shrunk as needed.  Repeated slides
+realize an arbitrary finite integral combination of group elements.
+The framing is carried through the band construction.  This is the
+usual labelled handle slide; see \cite[proof of Theorem~4.2,
+pp.~9--10]{Morgan2020}.  The two adjacent differentials change together
+as described in Section~\ref{sec:relative-model}.
+
+Write $e_1,\ldots,e_s$ for the existing index-five handles and
+$a_i=\partial_1e_i\in I$.  By Lemma~\ref{lem:handle-complex}, the $a_i$
+generate $I$ as a right ideal.  Fix a generating list
+$y_1,\ldots,y_b$ for $I$, padded with zeros if necessary.
+
+Introduce $b$ cancelling pairs of indices five and six, with new lower
+handles $n_1,\ldots,n_b$ satisfying $\partial_1n_j=0$.
+Here is a relative placement of the births.  In the level just after
+the index-four handle, choose disjoint small balls missing its belt
+sphere and all old index-five attaching cores.  These cores and the
+belt sphere have dimensions four and seven, respectively, so their
+finite union has empty interior.  After choosing the balls, shrink the
+old attaching neighborhoods to miss them.  The balls then survive the
+old index-five attachments.  Their product traces support ordinary
+local five/six births.  When ordering the resulting handles by index,
+pull the new lower attaching spheres back along these product traces.
+They still miss the index-four belt sphere, so their boundary columns
+are zero as asserted.
+
+Choose coefficients in $\Lambda$ such that
+\[
+ y_j=\sum_{i=1}^s a_iq_{ij},
+ \qquad
+ a_i=\sum_{j=1}^b y_jt_{ji}.
+\]
+First slide each new handle to replace $n_j$ by
+$n_j+\sum_i e_iq_{ij}$.  Its new column is $y_j$.
+Next replace each old handle by $e_i-\sum_j n_jt_{ji}$, using the
+new bases.  Every old column is now zero.  The new $b$ columns still
+generate $I$.
+
+\subsection{Producing a cancellable pair}
+
+\noindent\emph{A coefficient equal to one.}
+Select an old index-five handle $e$, and place it last among the
+index-five handles.  Since $\partial_1e=0$, exactness in
+Lemma~\ref{lem:handle-complex} gives $e\in\operatorname{im}\partial_2$.
+If $h_1,\ldots,h_m$ are the index-six handles and $x_j$ is the
+$e$-coefficient in $\partial_2h_j$, it follows that
+\[
+ x_1\Lambda+\cdots+x_m\Lambda=\Lambda.
+\]
+While an old handle remains, $r_1\geq b+1$; hence
+$m=r_2\geq r_1\geq b+1\geq3$ by \eqref{eq:handle-identities}.
+Lemma~\ref{lem:row} therefore applies.  Slides among the index-six
+handles produce one whose coefficient into $e$ is exactly $1$.
+Place this handle first among the index-six handles, immediately
+after $e$.
+
+\smallskip
+\noindent\emph{Removing the extra intersections.}
+Let $L$ be the intervening regular level, $T\cong S^5$ the upper
+handle's attaching core sphere, and $D\cong S^6$ the lower handle's
+belt sphere.  Their signed, group-labelled intersection is $1$.
+For each group label, pair intersection points with opposite signs,
+leaving exactly one positive point with identity label.  This pairing
+is possible because all nonidentity coefficients vanish and the
+identity coefficient is one.  In particular, the argument uses the
+specific coefficient $1$, rather than an arbitrary unit of $\Lambda$.
+
+For each pair, choose arcs joining its points on $T$ and on $D$.
+The two spheres are simply connected.  Equality of the group labels
+means that the resulting Whitney loop is null-homotopic in $W$.
+By \eqref{eq:level-fundamental-group}, it is null-homotopic in $L$.
+It therefore bounds a disk in the interior of $L$.  General position
+makes this disk embedded, with interior disjoint from $T$, $D$,
+the other disks, and the other attaching or belt cores that must be
+retained.  The relevant inequalities are
+\[
+ 2+5<11,\qquad 2+6<11,\qquad 2+2<11,
+ \qquad 2+7<11.
+\]
+The last inequality covers the largest other core that occurs in
+our index range.  Boundary arcs can likewise be chosen to miss the
+other cores and intersection points.  All of these are finite,
+compact constructions in $\operatorname{int}L$.
+
+The high-dimensional Whitney theorem now removes each paired
+intersection by an isotopy of $T$.  Its hypotheses are complementary
+sphere dimensions at least three, opposite intersection signs, and
+the null-homotopic Whitney loop just established; simple connectedness
+of $L$ is not required.  The framing step is part of this theorem:
+the normal splitting along the boundary of the Whitney disk extends
+across it.  In the present dimensions the frame-extension space is
+$V_4(\R^9)=O(9)/O(5)$, which is simply connected.
+See \cite[Theorem~6.6 and Lemma~6.13]{Milnor1965}, or the group-ring
+formulation in \cite[Theorem~7.27(i) and Corollary~7.30]{Ranicki2002}.
+Transport the existing normal framing of $T$ through its isotopy.
+We do not require that this entire handle framing extend over the
+Whitney disk.
+
+For support, first arrange the Whitney disks and their compact move
+neighborhoods to miss the other cores; then shrink the other tubular
+attaching neighborhoods to miss those supports.  A dimension argument
+alone would not give disjointness from preassigned full-dimensional
+neighborhoods.  The selected belt sphere remains the reference sphere
+against which intersections are counted; the isotopy of the attaching
+sphere removes intersections with it.  After the finite sequence of
+Whitney moves, $T$ meets $D$ transversely in exactly one point.
+
+\smallskip
+\noindent\emph{Cancellation relative to the boundary.}
+The geometric handle cancellation criterion now applies
+\cite[Lemma~8.24]{Ranicki2002}.  We spell out its relative support
+here because the quadratic collars must survive.  Isolate the two
+adjacent critical points in a slab containing no others, and use a
+descending gradient-like field that is a product on the side collar.
+The single transverse intersection gives precisely one connecting
+trajectory from the index-six point to the index-five point.  Choose
+a regular level $c$ between the slab's bottom and the lower critical
+value.  Every other nonconstant trajectory in the upper point's unstable manifold
+crosses $c$.  Indeed, a trajectory that remained above $c$ could not
+escape through the product side; compactness and strict decrease of
+the height would force convergence to the only lower critical point,
+contrary to uniqueness of the connecting trajectory.
+
+A product side collar has an invariant inner boundary, which the
+unstable manifold cannot cross.  Retain a strictly smaller side collar
+outside that boundary.  The closure of the unstable manifold above
+$c$ then lies in the compact part of the slab between $c$ and the
+upper critical value, away from the retained collar and the slab's ends.
+This closure can include limiting trajectories through the lower
+critical point; it need not be a smooth disk.
+The local Morse cancellation theorem gives a deformation supported
+in any sufficiently small neighborhood of this closure, fixed elsewhere
+\cite[Theorem and Corollary~1]{Laudenbach2013}.
+One can apply its closed-manifold formulation by first extending the
+height and field to a closed enlargement; they agree on the slab,
+and the deformation restricts to the same $W$.
+Thus the cancellation changes neither the earlier handles nor the
+side and end collars.  The upper attaching maps are transferred
+through the induced identification of levels.
+
+For clarity, the chosen index-six column may also have coefficients
+into other lower handles.  In bases beginning with the selected pair,
+its differential has the block form
+\[
+ \partial_2=
+ \begin{pmatrix}1&a\\ v&B\end{pmatrix},
+ \qquad \partial_1=(0\ \ d),\qquad dv=0.
+\]
+Cancellation leaves $B-va$ as the next differential and leaves $d$
+unchanged.  This records algebraically why one coefficient equal to
+one suffices and why the other old zero columns of $\partial_1$
+survive.  Geometrically this is the preservation of the lower handles
+and transfer of the upper attachments just described.
+
+\smallskip
+\noindent\emph{Iteration.}
+We can therefore repeat the construction until every old index-five
+handle has been removed.  Each step cancels one handle of index five
+and one of index six.  The identities of Lemma~\ref{lem:handle-complex}
+continue to hold for the same cobordism, and the $b$ new lower handles
+remain.  At the end,
+\[
+ r_1=b.
+\]
+
+\subsection{The reverse reduction and the quadratic collars}
+
+Turn the cobordism around.  Near its faces the new height is $-Q$,
+and an original handle of index $4+i$ has reversed index $8-i$.
+The relative covering homology for $(W,B)$ is the same as for $(W,A)$,
+so the preceding construction applies without changing the dimension
+or adding any variables.  It reduces the reversed index-five count
+to $b$, that is, the original index-seven count to $b$.
+Reversed five/six births and cancellations are original seven/six
+births and cancellations.  They preserve the number $r_1=b$ of original
+index-five handles; the upper attaching data in the reversed
+presentation are transferred as needed.  Both endpoint counts remain
+one.  Lemma~\ref{lem:handle-complex} now gives
+\begin{equation}
+ (r_0,r_1,r_2,r_3,r_4)
+   =\bigl(1,b,\chi(X)-2+2b,b,1\bigr).
+ \label{eq:final-handle-counts}
+\end{equation}
+
+The handle traces and their intervening product regions supply an
+actual smooth Morse height on the same $W$.  Since only finitely many
+compactly supported moves were used, a common smaller side collar
+still has height $Q$.  We may also prescribe height $-1$ and $1$ on
+the two end faces and make its inward derivative strictly positive
+at the incoming end and strictly negative at the outgoing end.
+It remains to ensure equality with $Q$ on whole end collars, compatible
+with the side collar.
+
+At $A$, use the inward coordinate $s=Q+1$.  On a sufficiently small
+collar the height has the form
+\[
+ -1+s\,a(y,s),\qquad a(y,s)>0,
+\]
+and $a=1$ near the side.  Replace it by
+\[
+ -1+s\bigl[1+\theta(s)(a(y,s)-1)\bigr],
+\]
+where $0\leq\theta\leq1$, with $\theta=0$ close to $s=0$ and
+$\theta=1$ farther inside.
+Put $c_0=\min(1,\inf a)>0$.  Its normal derivative is
+\[
+ 1+\theta(a-1)+s\theta\,\partial_sa+s\theta'(a-1).
+\]
+The first two terms together are at least $c_0$.  Choose the collar
+so small that $|s\partial_sa|<c_0/4$, and choose the transition so
+slowly in $\log s$ that
+$|s\theta'|\,\|a-1\|_\infty<c_0/4$.
+For example, a fixed smooth transition applied to
+$\log(s/s_0)/L$, with $L$ large and then $s_0$ sufficiently small,
+has these properties and is constant near both endpoints.
+The normal derivative is then at least $c_0/2$.
+This introduces no critical points and leaves the side collar fixed.
+Apply the identical argument at $B$ to one minus the height with
+inward coordinate $1-Q$.
+
+The final height equals $Q$ near every face and corner of $W$.
+Extend it by $Q$ to $X\times\R^8$ and call the result $F$.
+This extension is smooth across all seams.  Its exterior has no
+critical points because $dQ=0$ only on the zero section, which lies
+in the interior of $W$.  It equals $Q$ for $r\geq3$ and, by
+\eqref{eq:final-handle-counts}, has exactly
+\[
+ 1+b+\bigl(\chi(X)-2+2b\bigr)+b+1=\chi(X)+4b
+\]
+critical points.  This completes the proof of
+Proposition~\ref{prop:relative-morse}.
diff --git a/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/build/sections/05-hamiltonian.tex b/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/build/sections/05-hamiltonian.tex
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+\section{Closing the Hamiltonian model}
+\label{sec:hamiltonian}
+
+We now turn a Morse function with eight quadratic variables into a
+Hamiltonian map on a closed manifold.  Each of the sixteen products of
+north and south sphere caps will contain one copy of its critical set.
+The rotations outside the caps must be chosen with care: their endpoint
+maps agree with the quadratic model at both poles, although their paths
+can differ by full turns. We track this difference to identify the
+homotopy classes of the resulting fixed-point loops as well.
+
+\begin{proposition}\label{prop:cap-transfer}
+Let $(X,\omega_X)$ be a closed connected symplectic four-manifold. Write
+$(u,v)\in\R^4\times\R^4$ and set
+\[
+ Q(u,v)=\frac{|v|^2-|u|^2}{2},
+ \qquad r^2=|u|^2+|v|^2.
+\]
+Suppose that $F\colon X\times\R^8\to\R$ is a smooth Morse function
+such that $F=Q$ whenever $r\ge3$, and let $N=\#\Crit(F)$.
+Let $\sigma$ be the standard area form of total area $4\pi$ on the unit
+sphere. For every $\lambda>25/2$, the closed symplectic manifold
+\[
+ (M,\omega_M)=
+ \left(X\times(S^2)^4,
+ \omega_X+\lambda\sum_{j=1}^4\sigma_j\right)
+\]
+admits a Hamiltonian $H\in C^\infty((\R/\mathbb Z)\times M,\R)$ such
+that every fixed point of $\phi_H^1$ is nondegenerate, every loop
+$t\mapsto\phi_H^t(p)$ based at a fixed point $p$ is contractible in
+$M$, and
+\[
+ \#\Fix_0(\phi_H^1;H)=\#\Fix(\phi_H^1)=16N.
+\]
+Here $\sigma_j$ denotes the pullback of $\sigma$ from the $j$th sphere,
+and Hamiltonian vector fields satisfy $\omega_M(X_H,\cdot)=dH$.
+\end{proposition}
+
+We first record the local short-period estimate used below. Yorke proved
+the sharp lower bound $2\pi/L$ for a nonconstant periodic solution of a
+Euclidean vector field with Lipschitz constant $L$~\cite{Yorke1969}.
+The weaker coordinate estimate here suffices on a compact invariant set.
+
+\begin{lemma}\label{lem:short-period}
+Let $V$ be a smooth vector field on a smooth manifold, with flow
+$\Phi^t$, and let $K$ be a compact set on which this flow is defined
+for all times and which it preserves. There is a number $\tau_0>0$
+such that, for $p\in K$ and $0<\tau<\tau_0$,
+\[
+ \Phi^\tau(p)=p\quad\Longleftrightarrow\quad V(p)=0.
+\]
+\end{lemma}
+
+\begin{proof}
+Choose finitely many charts with image the Euclidean ball of radius
+$3$, whose inverse images of the unit ball cover $K$. Let $C\ge1$
+bound both the coordinate vector fields and their first derivatives
+on the closed balls of radius $2$, and put $\tau_0=1/(2C)$.
+If $0<\tau<\tau_0$, a trajectory
+$t\mapsto\Phi^{\tau t}(p)$, $0\le t\le1$, with $p\in K$ stays
+in the radius-$2$ ball of a chart whose unit ball contains $p$:
+a first exit would require coordinate displacement greater than $1$,
+whereas the speed bound gives displacement at most $\tau C<1/2$.
+If the trajectory is a loop, write $w$ for its coordinate expression
+and $\widehat V$ for the coordinate vector field. Then
+\[
+ \dot w(t)=\tau\widehat V(w(t)),\qquad
+ \int_0^1\widehat V(w(t))\,dt=0.
+\]
+Set $S=\max_t|\widehat V(w(t))|$. Since
+$|w(t)-w(a)|\le\tau S$ for $a,t\in[0,1]$, subtracting the zero
+average and using the Lipschitz bound on this convex ball gives
+\[
+ |\widehat V(w(t))|
+ \le\int_0^1|\widehat V(w(t))-\widehat V(w(a))|\,da
+ \le C\tau S.
+\]
+Thus $S=0$, so the loop is constant and $V(p)=0$. The converse follows
+from uniqueness of solutions. No homotopy assumption on the loop is used.
+\end{proof}
+
+\begin{proof}[Proof of Proposition~\ref{prop:cap-transfer}]
+\emph{The local flows.}
+Identify $\R^8$ with $\C^4$ by writing
+\[
+ u=(x_1,y_1,x_2,y_2),\qquad
+ v=(x_3,y_3,x_4,y_4),\qquad z_j=x_j+iy_j,
+\]
+and equip $X\times\C^4$ with
+$\Omega=\omega_X+\sum_jdx_j\wedge dy_j$.
+Put $s_1=s_2=-1$ and $s_3=s_4=1$, so that
+$Q=\frac12\sum_js_j|z_j|^2$.
+Let $\Phi_F^\tau$ and $\Phi_Q^\tau$ denote the autonomous Hamiltonian
+flows for this form and the stated sign convention.  In a standard
+plane, $X_H=(H_y,-H_x)$; consequently
+\begin{equation}\label{eq:quadratic-rotation}
+ \Phi_Q^\tau(x,z_1,\ldots,z_4)
+ =\bigl(x,e^{-is_1\tau}z_1,\ldots,e^{-is_4\tau}z_4\bigr).
+\end{equation}
+
+The vector fields of $F$ and $Q$ agree on $r\ge3$, including its
+boundary.  That boundary is invariant, so uniqueness of solutions in
+both time directions makes $r<3$ invariant as well.  Trajectories there
+remain in the compact set $X\times\{r\le3\}$, whereas exterior
+trajectories are the explicit rotations in
+\eqref{eq:quadratic-rotation}.  Thus both flows are complete.  They
+preserve the open set
+\[
+ U_0=X\times\{z\in\C^4:|z_j|<4\text{ for all }j\}
+\]
+and its compact closure $K$: points with $r<3$ have every $|z_j|<3$,
+and exterior trajectories preserve each $|z_j|$.
+Since $dQ\ne0$ on $r\ge3$, all critical points of $F$ lie in $r<3$.
+They form a closed discrete subset of $X\times\{r\le3\}$ and hence
+are finite.
+
+\smallskip
+\noindent\emph{One small time for all fixed points.}
+We choose $\epsilon>0$ so that the fixed points of
+$\Phi_F^\epsilon$ in $K$ are precisely the critical points of $F$,
+all nondegenerate. Lemma~\ref{lem:short-period}, applied to $X_F$ on
+the compact invariant set $K$, supplies $\tau_0>0$ for the first
+assertion. The zero set of $X_F$ equals $\Crit(F)$ because $\Omega$
+is nondegenerate. We next choose the time small enough to obtain
+nondegeneracy of the return map as well.
+
+At a critical point $p$, put $A_p=D_pX_F$.  Differentiating
+$\Omega(X_F,\cdot)=dF$ at $p$, where $X_F(p)=0$, shows that $A_p$
+is the composition of the nonsingular Hessian of $F$ with a symplectic
+identification, and is therefore invertible.  Moreover,
+\[
+ D_p\Phi_F^\epsilon=\exp(\epsilon A_p).
+\]
+Let $R\ge1$ bound the moduli of all complex eigenvalues of the finitely
+many matrices $A_p$.  If $0<\epsilon R<2\pi$, no nonzero eigenvalue
+$\mu$ of any $A_p$ satisfies $\epsilon\mu\in2\pi i\mathbb Z$.
+The eigenvalues of $\exp(\epsilon A_p)$ therefore all differ from one.
+Fix, once and for all,
+\begin{equation}\label{eq:small-epsilon}
+ 0<\epsilon<\min\{\pi,\tau_0,2\pi/R\}.
+\end{equation}
+This proves the required fixed-point and nondegeneracy assertions for
+the local time-$\epsilon$ map.
+
+\smallskip
+\noindent\emph{Sphere caps and matching rotations.}
+On each sphere use polar coordinates $(\vartheta,\varphi)$ and write
+$h=\cos\vartheta$.  Its symplectic form is
+\[
+ \lambda\sigma
+ =\lambda\sin\vartheta\,d\vartheta\wedge d\varphi
+ =-\lambda\,dh\wedge d\varphi.
+\]
+The north and south cap coordinates are, respectively,
+\begin{equation}\label{eq:pole-coordinates}
+ z_N=\sqrt{2\lambda(1-h)}\,e^{i\varphi},
+ \qquad
+ z_S=\sqrt{2\lambda(1+h)}\,e^{-i\varphi}.
+\end{equation}
+These are smooth coordinates at their poles.  Indeed, in the usual
+embedding coordinates $(a,b,h)$ of the unit sphere, they are
+\[
+ z_N=\sqrt{\frac{2\lambda}{1+h}}(a+ib),
+ \qquad
+ z_S=\sqrt{\frac{2\lambda}{1-h}}(a-ib).
+\]
+To check the area forms, write $z=R_ze^{i\psi}$.  At the north pole
+$R_zdR_z=-\lambda dh$ and $d\psi=d\varphi$; at the south pole
+$R_zdR_z=\lambda dh$ and $d\psi=-d\varphi$.  In both cases
+$dx\wedge dy=R_zdR_z\wedge d\psi=-\lambda dh\wedge d\varphi$.
+
+The two radius-$5$ caps are disjoint because $\lambda>25/2$:
+their height intervals are
+\[
+ h>1-\frac{25}{2\lambda}
+ \quad\text{and}\quad
+ h<-1+\frac{25}{2\lambda}.
+\]
+For each choice $\nu\in\{N,S\}^4$, let $U_\nu\subset M$ be the
+product of $X$ with the corresponding four radius-$4$ caps, and let
+$\iota_\nu\colon U_0\to U_\nu$ be the symplectic identification
+given by \eqref{eq:pole-coordinates}.  The sixteen sets $U_\nu$ are
+pairwise disjoint.
+
+For each $j$ choose a smooth function
+$\alpha_j\colon[-1,1]\to(0,2\pi)$, constant on the radius-$5$ caps,
+with the following values:
+\[
+\begin{array}{c|cc}
+ &\text{north}&\text{south}\\ \hline
+ s_j=-1&\epsilon&2\pi-\epsilon\\
+ s_j=1&2\pi-\epsilon&\epsilon
+\end{array}
+\]
+Smooth interpolation is possible within
+$[\epsilon,2\pi-\epsilon]\subset(0,2\pi)$.
+Define the Hamiltonian on $M$
+\[
+ P=\sum_{j=1}^4P_j(h_j),\qquad
+ P_j(h)=\lambda\int_0^h\alpha_j(\xi)\,d\xi,
+\]
+and denote its flow by $\Psi_t$.  Since
+\[
+ \iota_{\alpha_j(h)\partial_\varphi}
+       (-\lambda\,dh\wedge d\varphi)
+ =\lambda\alpha_j(h)\,dh=dP_j,
+\]
+this flow rotates $\varphi_j$ with positive speed $\alpha_j(h_j)$
+and fixes the $X$ coordinate.  A north coordinate rotates by
+$e^{i\alpha_jt}$ and a south coordinate by $e^{-i\alpha_jt}$.
+The table therefore gives the endpoint multiplier $e^{-is_j\epsilon}$
+in either chart, and hence
+\begin{equation}\label{eq:cap-endpoint-match}
+ \iota_\nu^{-1}\circ\Psi_1\circ\iota_\nu=\Phi_Q^\epsilon
+ \quad\text{on }U_0.
+\end{equation}
+Only the endpoints are equal: for some cap choices the rotation paths
+differ by a full turn. Figure~\ref{fig:pole-rotations} illustrates this
+distinction in one complex coordinate.
+
+\begin{figure}[ht]
+\centering
+\begin{tikzpicture}[>=Stealth,every node/.style={font=\small},
+                    x=1cm,y=1cm]
+\begin{scope}[xshift=-3.1cm]
+ \node[align=center] at (0,1.95)
+   {North chart: $z_N=R_z e^{i\varphi}$\\
+    chart angle $+\epsilon t$};
+ \draw[gray!55] (0,0) circle (1.15);
+ \draw[gray!65,dashed] (0,0)--(1.15,0);
+ \draw[gray!65,dashed] (0,0)--(35:1.15);
+ \fill (1.15,0) circle (1.5pt);
+ \fill (35:1.15) circle (1.5pt);
+ \node[right] at (1.15,0) {$1$};
+ \node[above right] at (35:1.15) {$e^{i\epsilon}$};
+ \draw[blue!60!black,very thick,->]
+   (1.15,0) arc[start angle=0,end angle=35,radius=1.15];
+ \node[align=center] at (0,-1.65)
+   {$e^{i\epsilon t}$\\short counterclockwise path};
+\end{scope}
+\begin{scope}[xshift=3.1cm]
+ \node[align=center] at (0,1.95)
+   {South chart: $z_S=R_z e^{-i\varphi}$\\
+    chart angle $-(2\pi-\epsilon)t$};
+ \draw[gray!55] (0,0) circle (1.15);
+ \draw[gray!65,dashed] (0,0)--(1.15,0);
+ \draw[gray!65,dashed] (0,0)--(35:1.15);
+ \fill (1.15,0) circle (1.5pt);
+ \fill (35:1.15) circle (1.5pt);
+ \node[right] at (1.15,0) {$1$};
+ \node[above right] at (35:1.15) {$e^{i\epsilon}$};
+ \draw[blue!60!black,very thick,->]
+   (1.15,0) arc[start angle=0,end angle=-325,radius=1.15];
+ \node[align=center] at (0,-1.65)
+   {$e^{-i(2\pi-\epsilon)t}=e^{i\epsilon t}e^{-2\pi it}$\\
+    long clockwise path};
+\end{scope}
+\end{tikzpicture}
+\caption{The case $s_j=-1$, shown as multiplier paths in the two local
+complex charts with angles drawn schematically. Both sphere longitudes increase,
+but the south chart reverses their angular orientation. The paths
+start at $1$ and end at the same multiplier $e^{i\epsilon}$; the
+south path makes one additional negative turn. For $s_j=1$, the
+north path makes the additional turn and both endpoints are
+$e^{-i\epsilon}$.}
+\label{fig:pole-rotations}
+\end{figure}
+
+Every $\Psi_t$ preserves all cap radii, so it preserves each $U_\nu$
+and the complement of their union.  A nonpolar sphere point cannot be
+fixed by $\Psi_1$, because its rotation angle lies strictly between
+zero and $2\pi$.  Consequently
+\begin{equation}\label{eq:rotation-fixed-set}
+ \Fix(\Psi_1)=X\times\{N,S\}^4
+ \subset\bigcup_\nu U_\nu.
+\end{equation}
+
+\smallskip
+\noindent\emph{A correction supported inside the caps.}
+We use the elementary composition rule that if Hamiltonian paths
+$a_t,b_t$ have generators $A_t,B_t$, then $a_t\circ b_t$ has generator
+$A_t+B_t\circ a_t^{-1}$.  Its vector field is
+$X_{A_t}+(a_t)_*X_{B_t}$, and symplectic transport gives
+$(a_t)_*X_{B_t}=X_{B_t\circ a_t^{-1}}$, proving the rule with our
+sign convention.
+
+On the model space define
+\begin{equation}\label{eq:local-correction}
+ \delta_t=\Phi_F^{\epsilon t}\circ\Phi_Q^{-\epsilon t},
+ \qquad
+ K_t=\epsilon\bigl(F-Q\circ\Phi_F^{-\epsilon t}\bigr),
+ \qquad 0\le t\le1.
+\end{equation}
+The composition rule shows that $K_t$ generates $\delta_t$.
+When $r\ge3$, the backward $F$ flow equals the backward $Q$ flow and
+preserves $Q$.  Thus $K_t=0$ and $\delta_t=\operatorname{id}$ there.
+Both constituent flows preserve $r<3$, so $\delta_t$ preserves $U_0$.
+Uniformly in $t$, the support of $K_t$ is contained in the compact
+set $X\times\{r\le3\}$, strictly inside $U_0$.
+
+Define a Hamiltonian $\widetilde K_t$ on $M$ by
+\[
+ \widetilde K_t|_{U_\nu}=K_t\circ\iota_\nu^{-1},
+ \qquad
+ \widetilde K_t=0\quad\text{off }\bigcup_\nu U_\nu.
+\]
+This is smooth jointly in time and space.  In particular it vanishes
+on a whole neighborhood of every cap boundary, since a point there
+has some $|z_j|$ close to $4$ and therefore $r>3$.
+Let $\Delta_t$ be its Hamiltonian flow.  By uniqueness,
+\[
+ \Delta_t|_{U_\nu}
+ =\iota_\nu\circ\delta_t\circ\iota_\nu^{-1},
+ \qquad
+ \Delta_t=\operatorname{id}\quad\text{off }\bigcup_\nu U_\nu.
+\]
+Set $\Gamma_t=\Delta_t\circ\Psi_t$.  Combining
+\eqref{eq:cap-endpoint-match} and \eqref{eq:local-correction} yields
+\begin{equation}\label{eq:inserted-endpoint}
+ \iota_\nu^{-1}\circ\Gamma_1\circ\iota_\nu
+ =\Phi_F^\epsilon\circ\Phi_Q^{-\epsilon}\circ\Phi_Q^\epsilon
+ =\Phi_F^\epsilon.
+\end{equation}
+On the complement of the cap union, $\Psi_1$ remains in that
+complement and $\Delta_1$ acts trivially.  Hence $\Gamma_1=\Psi_1$
+there, including at the cap boundaries.  By
+\eqref{eq:rotation-fixed-set}, this exterior has no fixed points.
+Inside each cap, \eqref{eq:inserted-endpoint} gives precisely the
+$N$ critical points of $F$, all lying in $r<3$ and all nondegenerate
+by \eqref{eq:small-epsilon}.  Thus every fixed point has been counted,
+and $\Gamma_1$ has exactly $16N$ nondegenerate fixed points.
+
+\smallskip
+\noindent\emph{The fixed-point loops.}
+For each cap choice $\nu$, define the path on $U_0$
+\[
+ R_{\nu,t}=\Phi_Q^{-\epsilon t}\circ
+             \iota_\nu^{-1}\circ\Psi_t\circ\iota_\nu.
+\]
+The rotation table shows that $R_{\nu,t}$ fixes the $X$ coordinate
+and multiplies each $z_j$ by $e^{2\pi i m_{\nu,j}t}$ for an integer
+$m_{\nu,j}\in\{-1,0,1\}$. In particular,
+$R_{\nu,0}=R_{\nu,1}=\operatorname{id}$, and these rotations preserve
+$r$. For a critical point $p=(x,z)$ of $F$, the fixed-point loop of
+$\Gamma_t$, expressed in this cap, is
+\[
+ \iota_\nu^{-1}\Gamma_t\iota_\nu(p)
+   =\Phi_F^{\epsilon t}\bigl(R_{\nu,t}(p)\bigr).
+\]
+Write $R_{\nu,t}(p)=(x,z_\nu(t))$. The fibre loop contracts to $p$
+by
+\[
+ q_s(t)=\bigl(x,(1-s)z_\nu(t)+sz\bigr),
+ \qquad 0\le s,t\le1.
+\]
+Convexity keeps $q_s(t)$ in $r<3$, since $r(p)<3$ and the rotations
+preserve $r$. Therefore
+\[
+ C(s,t)=\Phi_F^{\epsilon t}\bigl(q_s(t)\bigr)
+\]
+is a homotopy inside $r<3$ from the displayed fixed-point loop to
+the constant loop at $p$. It is based at $p$: the endpoint identities
+$q_s(0)=q_s(1)=p$ and the fact that $p$ is an equilibrium of $F$
+give $C(s,0)=C(s,1)=p$ and $C(1,t)=p$. Thus every fixed-point loop
+of $\Gamma_t$ is contractible within its cap.
+
+\smallskip
+\noindent\emph{A smooth periodic Hamiltonian.}
+The composition rule gives the generator
+\[
+ L_t=\widetilde K_t+P\circ\Delta_t^{-1}
+\]
+for $\Gamma_t$.  Choose a smooth nondecreasing map
+$\rho\colon[0,1]\to[0,1]$ equal to zero near zero and to one near
+one.  The reparametrized path $\Gamma_{\rho(t)}$ is generated by
+\[
+ H_t=\rho'(t)L_{\rho(t)}
+ =\rho'(t)\bigl(\widetilde K_{\rho(t)}
+                  +P\circ\Delta_{\rho(t)}^{-1}\bigr).
+\]
+This Hamiltonian vanishes on neighborhoods of both time endpoints,
+so its periodic extension belongs to
+$C^\infty((\R/\mathbb Z)\times M,\R)$.  Its time-one map is exactly
+$\Gamma_1$, with the same derivatives at all fixed points.
+Composing each based contraction with $t\mapsto\rho(t)$ proves
+contractibility for the reparametrized fixed-point loops as well.
+This proves the proposition.
+\end{proof}
diff --git a/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/build/sections/06-counting.tex b/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/build/sections/06-counting.tex
new file mode 100644
index 0000000000000000000000000000000000000000..5c4149f44998c64e353db835f1950f46dbd631a1
--- /dev/null
+++ b/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/build/sections/06-counting.tex
@@ -0,0 +1,110 @@
+\section{The Morse lower bound and the deficit}
+\label{sec:counting}
+
+We finish by comparing the count from Proposition~\ref{prop:cap-transfer}
+with the ordinary Morse number.  The lower bound uses the number of
+generators of the fundamental group, not the augmentation ideal.
+
+\begin{lemma}
+\label{lem:morse-lower}
+Let $Y$ be a closed connected smooth manifold of even dimension
+$n\geq4$, and let $d(\pi_1Y)$ denote the minimum number of generators
+of its fundamental group.  Then
+\[
+ \Morse(Y)\geq\chi(Y)+4d(\pi_1Y).
+\]
+\end{lemma}
+
+\begin{proof}
+For a Morse function on $Y$, let $m_i$ be its number of index-$i$
+critical points.  The zero- and one-handles form a connected graph:
+handles of index at least two cannot join distinct components.
+This graph has $m_0$ vertices and $m_1$ edges, so its fundamental
+group is free of rank $m_1-m_0+1$.  Adding the higher handles gives
+a surjection from this group onto $\pi_1Y$.  Thus
+\[
+ d(\pi_1Y)\leq m_1-m_0+1\leq m_1.
+\]
+Apply the same argument to the negative of the Morse function to
+obtain $m_{n-1}\geq d(\pi_1Y)$.  Since $n$ is even, the Euler
+identity gives
+\[
+ \sum_{i=0}^{n}m_i
+ =\chi(Y)+2\sum_{i\text{ odd}}m_i
+ \geq\chi(Y)+2(m_1+m_{n-1})
+ \geq\chi(Y)+4d(\pi_1Y).
+\]
+Taking the minimum over all Morse functions proves the claim.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+Choose $b=2$ and $G=A_5^k$ with $k>60^{16b}$ as in
+Lemma~\ref{lem:group-gap}.  In particular $d(G)>16b$.
+Every finite group is finitely presented.  Gompf's existence theorem
+therefore provides a closed connected symplectic four-manifold
+$(X,\omega_X)$ with $\pi_1(X)\cong G$ \cite[Theorem~3.1]{Gompf1992}.
+The finiteness of $G$ implies $\chi(X)\geq2$, as explained in
+Section~\ref{sec:relative-model}.  Lemmas~\ref{lem:augmentation}
+and~\ref{lem:row}, followed by Proposition~\ref{prop:relative-morse},
+give a Morse function on $X\times\R^8$, equal to $Q$ for $r\geq3$,
+with
+\[
+ N=\chi(X)+4b
+\]
+critical points.
+
+Proposition~\ref{prop:cap-transfer} produces a smooth one-periodic
+Hamiltonian $H$ on the closed connected symplectic manifold
+$M=X\times(S^2)^4$.  Its time-one map is nondegenerate at every
+fixed point and has exactly
+\begin{equation}
+ \#\Fix(\phi_H^1)=16N
+ =16\chi(X)+64b=\chi(M)+128.
+ \label{eq:final-fixed-count}
+\end{equation}
+Here $\dim_\R M=12$, $\pi_1(M)\cong G$, and
+$\chi(M)=16\chi(X)$.  Lemma~\ref{lem:morse-lower} yields
+\[
+ \Morse(M)\geq\chi(M)+4d(G)
+ >\chi(M)+64b=\#\Fix(\phi_H^1).
+\]
+The cap-transfer proposition also proves that every fixed-point
+trajectory of this Hamiltonian is contractible. Hence
+\[
+ \#\Fix_0(\phi_H^1;H)=\#\Fix(\phi_H^1)<\Morse(M),
+\]
+which completes the proof.
+\end{proof}
+
+\begin{proof}[Proof of Corollary~\ref{cor:deficit}]
+Given a real number $D>0$, choose an integer
+$m>16b+D/4$ and then a finite integer $k>60^m$.
+Lemma~\ref{lem:group-gap} gives $d(A_5^k)>m$.
+The bound $b=2$ in Lemma~\ref{lem:augmentation} is independent of
+$k$, and all handle operations above take place in the same
+dimension twelve with eight added variables.  Applying the
+construction for this group and using \eqref{eq:final-fixed-count}
+gives
+\[
+ \Morse(M)-\#\Fix(\phi_H^1)
+ \geq4d(A_5^k)-64b>D.
+\]
+The manifold $X$, and hence $M$, may depend on $D$, as may the
+small time and sphere-area parameters used in the construction.
+The dimension, quadratic index, and bound $b$ do not depend on $D$.
+\end{proof}
+
+In these examples the integral homology is also easy to identify
+to the extent needed for the comparison with homological bounds.
+The group $G$ is perfect, so $H_1(X;\Z)=0$.
+Poincar\'e duality gives $H_3(X;\Z)=H^1(X;\Z)=0$;
+the universal coefficient theorem and duality give
+$H_2(X;\Z)\cong H^2(X;\Z)\cong\Hom(H_2(X;\Z),\Z)$,
+so $H_2(X;\Z)$ is free abelian.  Hence the integral homology of
+$M=X\times(S^2)^4$ is torsion-free and concentrated in even degrees.
+Its total rank is $\chi(M)$, and the fixed-point count in
+\eqref{eq:final-fixed-count} exceeds this number.  Thus the total
+fixed-point count exceeds both the rational Betti number and the
+rank-and-torsion quantity discussed in the introduction.
+The obstruction detected by the
+ordinary Morse number is the additional cost of the fundamental group.
diff --git a/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/paper.pdf b/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..02d7e09eece9116fa109a49f0b6c1cac5ca01a67
--- /dev/null
+++ b/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:f7980995971b08dfbdcf0a1dcc34d1669c2bcc24648dc3782bc111463af0ce9e
+size 424654
diff --git a/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/README.md b/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..62722b952738c61890e4f887de949fef1a561fd8
--- /dev/null
+++ b/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/README.md
@@ -0,0 +1,18 @@
+# [A pointwise 2-converse for elliptic curves with rational two-torsion](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 24, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026,
+  author = {{OpenAI}},
+  title = {{A pointwise $2$-converse for elliptic curves with rational two-torsion}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/paper.pdf}{OAI:A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/build/paper.tex b/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/build/paper.tex
new file mode 100644
index 0000000000000000000000000000000000000000..58717023df4626c2dfa5761e3441fecd5253d9f3
--- /dev/null
+++ b/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/build/paper.tex
@@ -0,0 +1,80 @@
+\documentclass[11pt]{article}
+\pdftrailerid{}
+\input{glyphtounicode}
+\input{glyphtounicode-cmex}
+\pdfgentounicode=1
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage{lmodern}
+\usepackage[a4paper,margin=28mm]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs}
+\usepackage{booktabs,array,longtable,enumitem}
+\usepackage{microtype}
+\usepackage{xcolor}
+\usepackage[nottoc]{tocbibind}
+\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black,bookmarksnumbered=true,bookmarksdepth=2]{hyperref}
+\usepackage{bookmark}
+\numberwithin{equation}{section}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{corollary}[theorem]{Corollary}
+\newtheorem{claim}[theorem]{Claim}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\newtheorem{hypothesis}[theorem]{Hypothesis}
+\newtheorem{assumption}[theorem]{Assumption}
+\newtheorem{conjecture}[theorem]{Conjecture}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newtheorem{example}[theorem]{Example}
+\newcommand{\Q}{\mathbb Q}
+\newcommand{\Z}{\mathbb Z}
+\newcommand{\F}{\mathbb F}
+\newcommand{\C}{\mathbb C}
+\newcommand{\R}{\mathbb R}
+\newcommand{\Sha}{\operatorname{Sha}}
+\DeclareMathOperator{\Sel}{Sel}
+\DeclareMathOperator{\Gal}{Gal}
+\DeclareMathOperator{\Hom}{Hom}
+\DeclareMathOperator{\End}{End}
+\DeclareMathOperator{\rank}{rank}
+\DeclareMathOperator{\corank}{corank}
+\DeclareMathOperator{\length}{length}
+\DeclareMathOperator{\ord}{ord}
+\DeclareMathOperator{\im}{im}
+\DeclareMathOperator{\coker}{coker}
+\DeclareMathOperator{\Tr}{Tr}
+\DeclareMathOperator{\Nm}{Nm}
+\DeclareMathOperator{\adj}{adj}
+\DeclareMathOperator{\id}{id}
+\setlist{nosep,topsep=4pt}
+\setlength{\emergencystretch}{1.5em}
+\hypersetup{pdftitle={A pointwise 2-converse for elliptic curves with rational two-torsion},pdfsubject={Selmer coranks, integral interpolation, and pointwise quadratic-twist constructions},pdfauthor={OpenAI}}
+\title{A pointwise $2$-converse\\for elliptic curves with rational two-torsion}
+\author{OpenAI}
+\date{September 24, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We prove a pointwise $2$-converse for elliptic curves over $\mathbf Q$
+with nonzero rational two-torsion: if the $2^\infty$-Selmer corank is
+zero or one, then the analytic rank and Mordell--Weil rank equal that
+corank, and the Shafarevich--Tate group is finite. The result allows
+arbitrary reduction at $2$.
+\end{abstract}
+\setcounter{tocdepth}{1}
+\tableofcontents
+\clearpage
+\input{sections/introduction}
+\input{sections/preliminaries}
+\input{sections/cyclotomic}
+\input{sections/ring-limits}
+\input{sections/ring-determinants}
+\input{sections/nonvanishing}
+\input{sections/coefficients}
+\input{sections/graphs}
+\input{sections/pointwise}
+\bibliographystyle{plain}
+\bibliography{references}
+\end{document}
diff --git a/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/build/references.bib b/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/build/references.bib
new file mode 100644
index 0000000000000000000000000000000000000000..86d03180a99880a3856718fc89455f7a5436995b
--- /dev/null
+++ b/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/build/references.bib
@@ -0,0 +1,355 @@
+@article{BCDT2001,
+ author={Breuil, Christophe and Conrad, Brian and Diamond, Fred and Taylor, Richard},
+ title={On the modularity of elliptic curves over {$\mathbf Q$}: wild 3-adic exercises},
+ journal={Journal of the American Mathematical Society},volume={14},number={4},pages={843--939},year={2001},
+ note={\href{https://doi.org/10.1090/S0894-0347-01-00370-8}{doi:10.1090/S0894-0347-01-00370-8}}
+}
+@article{BirchSwinnertonDyer1965,
+ author={Birch, Bryan J. and Swinnerton-Dyer, H. P. F.},
+ title={Notes on elliptic curves. {II}},
+ journal={Journal f{\"u}r die reine und angewandte Mathematik},volume={218},pages={79--108},year={1965},
+ note={\href{https://doi.org/10.1515/crll.1965.218.79}{doi:10.1515/crll.1965.218.79}}
+}
+@incollection{Tate1966,
+ author={Tate, John},
+ title={On the conjectures of {Birch} and {Swinnerton-Dyer} and a geometric analog},
+ booktitle={S{\'e}minaire Bourbaki},volume={9},pages={415--440},year={1966},
+ publisher={Soci{\'e}t{\'e} math{\'e}matique de France},
+ note={Expos{\'e} 306; \href{https://www.numdam.org/item/SB_1964-1966__9__415_0/}{primary text}}
+}
+@article{GrossZagier1986,
+ author={Gross, Benedict H. and Zagier, Don B.},
+ title={Heegner points and derivatives of {$L$}-series},
+ journal={Inventiones Mathematicae},volume={84},number={2},pages={225--320},year={1986},
+ note={\href{https://doi.org/10.1007/BF01388809}{doi:10.1007/BF01388809}}
+}
+@incollection{Kolyvagin1990,
+ author={Kolyvagin, Viktor A.},title={Euler systems},
+ booktitle={The Grothendieck Festschrift, Volume II},
+ series={Progress in Mathematics},volume={87},pages={435--483},
+ publisher={Birkh{\"a}user},year={1990},
+ note={\href{https://doi.org/10.1007/978-0-8176-4575-5_11}{chapter reprint}}
+}
+@article{Skinner2020,
+ author={Skinner, Christopher},
+ title={A converse to a theorem of {Gross}, {Zagier}, and {Kolyvagin}},
+ journal={Annals of Mathematics},volume={191},number={2},pages={329--354},year={2020},
+ note={\href{https://annals.math.princeton.edu/wp-content/uploads/annals-v191-n2-p01-s.pdf}{primary text}}
+}
+@article{Kim2023SoftConverse,
+ author={Kim, Chan-Ho},
+ title={On the soft {$p$}-converse to a theorem of {Gross--Zagier} and {Kolyvagin}},
+ journal={Mathematische Annalen},volume={387},pages={1961--1968},year={2023},
+ note={\href{https://doi.org/10.1007/s00208-022-02511-8}{doi:10.1007/s00208-022-02511-8};
+ text at \href{https://arxiv.org/abs/2109.12344v3}{arXiv:2109.12344v3}}
+}
+@article{Zhang2014Heegner,
+ author={Zhang, Wei},
+ title={Selmer groups and the indivisibility of {Heegner} points},
+ journal={Cambridge Journal of Mathematics},volume={2},number={2},
+ pages={191--253},year={2014},
+ note={\href{https://doi.org/10.4310/CJM.2014.v2.n2.a2}{doi:10.4310/CJM.2014.v2.n2.a2}}
+}
+@article{CGLS2022,
+ author={Castella, Francesc and Grossi, Giada and Lee, Jaehoon and Skinner, Christopher},
+ title={On the anticyclotomic {Iwasawa} theory of rational elliptic curves at {Eisenstein} primes},
+ journal={Inventiones Mathematicae},volume={227},number={2},
+ pages={517--580},year={2022},
+ note={\href{https://doi.org/10.1007/s00222-021-01072-y}{doi:10.1007/s00222-021-01072-y};
+ text at \href{https://arxiv.org/abs/2008.02571v2}{arXiv:2008.02571v2}}
+}
+@article{BurungaleTian2026,
+ author={Burungale, Ashay A. and Tian, Ye},
+ title={A rank zero {$p$}-converse to a theorem of {Gross--Zagier}, {Kolyvagin} and {Rubin}},
+ journal={Annals of Mathematics},volume={203},number={1},pages={1--13},year={2026},
+ note={\href{https://doi.org/10.4007/annals.2026.203.1.1}{doi:10.4007/annals.2026.203.1.1};
+ Theorem~2.6 numbering in \href{https://arxiv.org/abs/2506.03465v1}{arXiv:2506.03465v1}}
+}
+@misc{KellerYin2024,
+ author={Keller, Timo and Yin, Mulun},
+ title={{$p$}-converse theorems for elliptic curves of potentially good ordinary reduction at {Eisenstein} primes},
+ year={2024},note={\href{https://arxiv.org/abs/2410.23241v1}{arXiv:2410.23241v1}}
+}
+@misc{Smith2025,
+ author={Smith, Alexander},title={The {Birch} and {Swinnerton-Dyer} conjecture implies {Goldfeld}'s conjecture},
+ year={2025},note={\href{https://arxiv.org/abs/2503.17619v1}{arXiv:2503.17619v1}}
+}
+@article{DokchitserDokchitser2010,
+ author={Dokchitser, Tim and Dokchitser, Vladimir},
+ title={On the {Birch--Swinnerton-Dyer} quotients modulo squares},
+ journal={Annals of Mathematics},volume={172},number={1},pages={567--596},year={2010},
+ note={\href{https://doi.org/10.4007/annals.2010.172.567}{doi:10.4007/annals.2010.172.567}}
+}
+@article{Kato2004,
+ author={Kato, Kazuya},
+ title={{$p$}-adic {Hodge} theory and values of zeta functions of modular forms},
+ journal={Ast{\'e}risque},volume={295},pages={117--290},year={2004},
+ note={\href{https://www.numdam.org/item/AST_2004__295__117_0/}{primary text}}
+}
+@article{FerreroWashington1979,
+ author={Ferrero, Bruce and Washington, Lawrence C.},
+ title={The {Iwasawa} invariant {$\mu_p$} vanishes for abelian number fields},
+ journal={Annals of Mathematics},volume={109},number={2},pages={377--395},year={1979},
+ note={\href{https://doi.org/10.2307/1971116}{doi:10.2307/1971116}}
+}
+@article{Chevalley1935,
+ author={Chevalley, Claude},
+ title={D{\'e}monstration d'une hypoth{\`e}se de {M. Artin}},
+ journal={Abhandlungen aus dem Mathematischen Seminar der Universit{\"a}t Hamburg},
+ volume={11},pages={73--75},year={1935},
+ note={\href{https://doi.org/10.1007/BF02940714}{doi:10.1007/BF02940714}}
+}
+@article{Warning1935,
+ author={Warning, Ewald},
+ title={Bemerkung zur vorstehenden Arbeit von {Herrn Chevalley}},
+ journal={Abhandlungen aus dem Mathematischen Seminar der Universit{\"a}t Hamburg},
+ volume={11},pages={76--83},year={1935},
+ note={\href{https://doi.org/10.1007/BF02940715}{doi:10.1007/BF02940715}}
+}
+@book{Nekovar2006,
+ author={Nekov{\'a}{\v r}, Jan},title={Selmer complexes},
+ series={Ast{\'e}risque},volume={310},publisher={Soci{\'e}t{\'e} math{\'e}matique de France},year={2006},
+ note={\href{https://www.numdam.org/item/AST_2006__310__R1_0/}{primary text}}
+}
+@article{Howard2012,
+ author={Howard, Benjamin},title={The {Heegner} point {Kolyvagin} system},
+ journal={Compositio Mathematica},volume={140},number={6},pages={1439--1472},year={2004},
+ note={\href{https://doi.org/10.1112/S0010437X04000569}{doi:10.1112/S0010437X04000569};
+ text at \href{https://arxiv.org/abs/1202.6340v1}{arXiv:1202.6340v1}}
+}
+@article{CaiShuTian2014,
+ author={Cai, Li and Shu, Jie and Tian, Ye},
+ title={Explicit {Gross--Zagier} and {Waldspurger} formulae},
+ journal={Algebra \& Number Theory},volume={8},number={10},pages={2523--2572},year={2014},
+ note={\href{https://doi.org/10.2140/ant.2014.8.2523}{doi:10.2140/ant.2014.8.2523}}
+}
+@article{Manin1972,
+ author={Manin, Yu. I.},
+ title={Parabolic points and zeta-functions of modular curves},
+ journal={Mathematics of the USSR-Izvestiya},volume={6},number={1},
+ pages={19--64},year={1972},
+ note={\href{https://doi.org/10.1070/IM1972v006n01ABEH001867}{doi:10.1070/IM1972v006n01ABEH001867}}
+}
+@article{Drinfeld1973,
+ author={Drinfeld, V. G.},
+ title={Two theorems on modular curves},
+ journal={Functional Analysis and Its Applications},volume={7},number={2},
+ pages={155--156},year={1973},
+ note={English translation of Funktsional'nyi Analiz i ego Prilozheniya 7 (1973), no.~2, 83--84;
+ \href{https://doi.org/10.1007/BF01078890}{doi:10.1007/BF01078890}}
+}
+@article{Cassels1965,
+ author={Cassels, J. W. S.},
+ title={Arithmetic on curves of genus 1. {VIII}. On conjectures of {Birch} and {Swinnerton-Dyer}},
+ journal={Journal f{\"u}r die reine und angewandte Mathematik},volume={217},pages={180--199},year={1965},
+ note={\href{https://doi.org/10.1515/crll.1965.217.180}{doi:10.1515/crll.1965.217.180}}
+}
+@book{MilneADT2006,
+ author={Milne, J. S.},title={Arithmetic Duality Theorems},
+ edition={Second},publisher={BookSurge},year={2006},
+ note={\href{https://www.jmilne.org/math/Books/ADTnot.pdf}{author's electronic edition}}
+}
+@book{Serre1968,
+ author={Serre, Jean-Pierre},
+ title={Abelian {$\ell$}-adic Representations and Elliptic Curves},
+ publisher={W. A. Benjamin},
+ address={New York},
+ year={1968},
+ note={\href{https://www.math.mcgill.ca/darmon/courses/18-19/gs/serre-mcgill.pdf}{1989 reprint of the 1968 text}}
+}
+
+@article{Serre1972,
+ author={Serre, Jean-Pierre},
+ title={Propri{\'e}t{\'e}s galoisiennes des points d'ordre fini des courbes elliptiques},
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+}
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+ \href{https://arxiv.org/abs/1403.7067}{arXiv:1403.7067}}
+}
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+}
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+}
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+ author={Friedlander, John and Iwaniec, Henryk},title={Opera de cribro},
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+}
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+}
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+ author={Gelbart, Stephen and Jacquet, Herv{\'e}},
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+ note={\href{https://doi.org/10.24033/asens.1355}{doi:10.24033/asens.1355}}
+}
+@article{Waldspurger1981,
+ author={Waldspurger, Jean-Loup},
+ title={Sur les coefficients de {Fourier} des formes modulaires de poids demi-entier},
+ journal={Journal de Math{\'e}matiques Pures et Appliqu{\'e}es},volume={60},number={4},pages={375--484},year={1981},
+ note={\href{https://mate.dm.uba.ar/~nsirolli/files/waldspurger.pdf}{complete text}}
+}
+@incollection{Waldspurger1983,
+ author={Waldspurger, Jean-Loup},title={Correspondances de {Shimura}},
+ booktitle={Proceedings of the International Congress of Mathematicians, Warszawa, 1983},
+ volume={1},pages={525--531},year={1984},publisher={PWN and Elsevier},
+ note={\href{https://www.imj-prg.fr/wp-content/uploads/2020/prix/waldspurger1983.pdf}{primary text}}
+}
+@article{JetchevKane2010,
+ author={Jetchev, Dimitar and Kane, Ben},
+ title={Equidistribution of {Heegner} points and ternary quadratic forms},
+ journal={Mathematische Annalen},volume={350},number={3},pages={501--532},year={2011},
+ note={\href{https://doi.org/10.1007/s00208-010-0568-5}{doi:10.1007/s00208-010-0568-5};
+ section numbering in \href{https://arxiv.org/abs/0908.3905v1}{arXiv:0908.3905v1}}
+}
+@article{CornutJetchev2013,
+ author={Cornut, Christophe and Jetchev, Dimitar},
+ title={Liftings of reduction maps for quaternion algebras},
+ journal={Bulletin of the London Mathematical Society},volume={45},
+ pages={370--386},year={2013},
+ note={\href{https://doi.org/10.1112/blms/bds098}{doi:10.1112/blms/bds098};
+ section numbering in the \href{https://webusers.imj-prg.fr/~christophe.cornut/papers/liftings.pdf}{authors' version}}
+}
+@article{Weil1964,
+ author={Weil, Andr{\'e}},title={Sur certains groupes d'op{\'e}rateurs unitaires},
+ journal={Acta Mathematica},volume={111},pages={143--211},year={1964},
+ note={\href{https://doi.org/10.1007/BF02391012}{doi:10.1007/BF02391012}}
+}
+@article{Faltings1983,
+ author={Faltings, Gerd},
+ title={Endlichkeitss{\"a}tze f{\"u}r abelsche {Variet{\"a}ten} {\"u}ber {Zahlk{\"o}rpern}},
+ journal={Inventiones Mathematicae},volume={73},number={3},pages={349--366},year={1983},
+ note={\href{https://doi.org/10.1007/BF01388432}{doi:10.1007/BF01388432}}
+}
+@article{Bogomolov1981Points,
+ author={Bogomolov, F. A.},
+ title={Points of finite order on an {Abelian} variety},
+ journal={Mathematics of the USSR-Izvestiya},volume={17},number={1},pages={55--72},year={1981},
+ note={English translation of Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 4, 782--804;
+ \href{https://doi.org/10.1070/IM1981v017n01ABEH001329}{doi:10.1070/IM1981v017n01ABEH001329}}
+}
+@incollection{Deligne1969,
+ author={Deligne, Pierre},title={Formes modulaires et repr{\'e}sentations {$\ell$}-adiques},
+ booktitle={S{\'e}minaire Bourbaki, 1968/69},series={Lecture Notes in Mathematics},
+ volume={179},pages={139--172},publisher={Springer},year={1971},
+ note={Expos{\'e} 355 (1969); \href{https://www.numdam.org/item/SB_1968-1969__11__139_0/}{primary text}}
+}
+@article{Carayol1986,
+ author={Carayol, Henri},
+ title={Sur les repr{\'e}sentations {$\ell$}-adiques associ{\'e}es aux formes modulaires de {Hilbert}},
+ journal={Annales scientifiques de l'{\'E}cole Normale Sup{\'e}rieure},
+ series={4},volume={19},number={3},pages={409--468},year={1986},
+ note={\href{https://doi.org/10.24033/asens.1512}{doi:10.24033/asens.1512}}
+}
+
+@article{Monsky1996,
+ author={Monsky, Paul},
+ title={Generalizing the {Birch--Stephens} theorem. {I}. {Modular} curves},
+ journal={Mathematische Zeitschrift},volume={221},number={3},pages={415--420},year={1996},
+ note={\href{https://doi.org/10.1007/PL00004518}{doi:10.1007/PL00004518}}
+}
+
+@misc{BurungaleSkinner2024Appendix,
+  author = {Burungale, Ashay and Skinner, Christopher},
+  title = {A {$p$}-converse theorem for {CM} elliptic curves},
+  year = {2024},
+  howpublished = {Appendix A to Levent Alp{\"o}ge, Manjul Bhargava and Ari Shnidman, \emph{Integers expressible as the sum of two rational cubes}},
+  eprint = {2210.10730v3},
+  archivePrefix = {arXiv},
+  primaryClass = {math.NT},
+  url = {https://arxiv.org/abs/2210.10730v3},
+  note = {Version 3, 19 October 2024; Theorems A.1, A.3 and Remark A.5; \href{https://arxiv.org/abs/2210.10730v3}{arXiv:2210.10730v3}}
+}
+
+@article{BurungaleCastellaSkinnerTian2022,
+  author = {Burungale, Ashay and Castella, Francesc and Skinner, Christopher and Tian, Ye},
+  title = {{$p^{\infty}$}-{Selmer} groups and rational points on {CM} elliptic curves},
+  journal = {Annales math{\'e}matiques du Qu{\'e}bec},
+  volume = {46},
+  pages = {325--346},
+  year = {2022},
+  doi = {10.1007/s40316-022-00203-y},
+  url = {https://link.springer.com/article/10.1007/s40316-022-00203-y},
+  note = {\href{https://doi.org/10.1007/s40316-022-00203-y}{doi:10.1007/s40316-022-00203-y}}
+}
+
+@misc{Kriz2022CM,
+  author = {Kriz, Daniel},
+  title = {Supersingular main conjectures, {Sylvester}'s conjecture and {Goldfeld}'s conjecture},
+  year = {2022},
+  eprint = {2002.04767v5},
+  archivePrefix = {arXiv},
+  primaryClass = {math.NT},
+  url = {https://arxiv.org/abs/2002.04767v5},
+  note = {Preprint, version 5, 19 October 2022; \href{https://arxiv.org/abs/2002.04767v5}{arXiv:2002.04767v5}}
+}
+
+@article{HeathBrownChevalleyWarning,
+ author={Heath-Brown, D. R.},
+ title={A note on the {Chevalley--Warning} theorems},
+ journal={Russian Mathematical Surveys},volume={66},number={2},
+ pages={427--436},year={2011},
+ note={\href{https://doi.org/10.1070/RM2011v066n02ABEH004744}{doi:10.1070/RM2011v066n02ABEH004744};
+ equation numbering in \href{https://arxiv.org/abs/1009.3764v2}{arXiv:1009.3764v2}}
+}
+
+@article{Chaiken1982,
+ author={Chaiken, Seth},
+ title={A combinatorial proof of the all minors matrix tree theorem},
+ journal={SIAM Journal on Algebraic and Discrete Methods},
+ volume={3},number={3},pages={319--329},year={1982},
+ note={\href{https://doi.org/10.1137/0603033}{doi:10.1137/0603033}}
+}
+
+@incollection{Gross1987Heights,
+ author={Gross, Benedict H.},
+ title={Heights and the special values of {$L$}-series},
+ booktitle={Number theory (Montreal, Que., 1985)},
+ editor={Kisilevsky, H. and Labute, J.},
+ series={CMS Conference Proceedings},volume={7},
+ publisher={American Mathematical Society},address={Providence, RI},
+ year={1987},pages={115--187},
+ note={\href{https://wstein.org/papers/bib/Gross-Heights_and_the_Special_Values_of_L-series.pdf}{primary text}}
+}
+@article{Shimura1973HalfIntegral,
+ author={Shimura, Goro},
+ title={On modular forms of half integral weight},
+ journal={Annals of Mathematics},series={2},volume={97},number={3},
+ year={1973},pages={440--481},
+ note={\href{https://doi.org/10.2307/1970831}{doi:10.2307/1970831}}
+}
+@article{Ramsey2006HalfIntegral,
+ author={Ramsey, Nick},
+ title={Geometric and {$p$}-adic modular forms of half-integral weight},
+ journal={Annales de l'Institut Fourier},volume={56},number={3},
+ pages={599--624},year={2006},
+ note={\href{https://doi.org/10.5802/aif.2195}{doi:10.5802/aif.2195}}
+}
+
+@misc{OpenAIGoldfeldTwoConverse2026,
+ author={{OpenAI}},
+ title={{Goldfeld's analytic density conjecture and the $2$-converse for elliptic curves}},
+ howpublished={OpenAI Math Release preprint
+               \href{https://github.com/openai/math/blob/main/preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-September-23-2026/paper.pdf}{OAI:Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-September-23-2026}},
+ year={2026}
+}
diff --git a/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/build/sections/coefficients.tex b/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/build/sections/coefficients.tex
new file mode 100644
index 0000000000000000000000000000000000000000..f157e7a7c87f49f28fa7b96e1b34b75e999272b7
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+++ b/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/build/sections/coefficients.tex
@@ -0,0 +1,1028 @@
+\section{Coefficient systems and their contraction identities}
+\label{co:section}
+
+We construct Fourier coefficients detecting central values and Heegner
+points, then use arithmetic lower bounds to normalize their least nonzero
+valuation.  The resulting binary tests satisfy the contraction and isolation
+identities used in Section~\ref{gr:section}.  The even lower bound is already
+available from Section~\ref{cy:section}; the odd detector is constructed
+without one, and its normalization remains conditional until
+Proposition~\ref{pw:odd-lower}.
+
+Throughout, $S$ contains $2$ and the primes dividing the
+conductor $N$.  A filter prescribes local unit squareclasses at $S$, together
+with the infinite sign, and excludes finitely many further primes.  Its
+indices are $h=\sigma D$, where $D$ is positive and squarefree.  All primes
+adjoined subsequently are odd, good, and outside the current support.
+Put $F=\mathbb Q(E[2])$.  In the present residual range, $F$ is trivial or
+quadratic.  A prime split in $F$ has weight $1$, and a prime inert in $F$,
+called simple, has weight $1/2$.  We write $w(D)$ for the sum of these weights.
+All valuations satisfy $v(2)=1$.
+
+We use the theta multiplier cubed for weight $3/2$.  Local unit characters
+and their associated Dirichlet characters are related by the usual inverse
+idele convention.  Thus, if a finite Schwartz function satisfies
+$f(ux)=\alpha(u)f(x)$, its scalar classical character on the lower-right
+entry is the Dirichlet character associated to $\alpha^{-1}$.
+This convention removes a possible inverse ambiguity in the formulas below.
+
+\subsection{Even coefficients and exact comparison of ratios}
+
+\begin{proposition}[Even coefficient system]\label{co:even}
+Fix an even-sign filter and a nonvanishing index $D_0$ in it.
+There is a holomorphic cuspidal form $g=\sum_{n\geq1}a(n)q^n$ of weight
+$3/2$, orthogonal to unary theta series, of level supported on a fixed
+enlargement of $S$, and characters $\eta,\psi$ of $2$-power order such that
+$\psi/\eta=\chi_{-\sigma}$ and
+\begin{equation}\label{co:recurrence}
+ a(p^2n)=\bigl(\eta(p)a_p(E)-\psi(p)(-n/p)\bigr)a(n)
+              -p\eta(p)^2a(n/p^2)\qquad(p\notin S).
+\end{equation}
+Its coefficients are zero off the unit-squareclass filter.  After
+normalization at $D_0$, every squarefree coefficient is rational up to a
+$2$-power root of unity.  For two nonzero squarefree coefficients,
+\begin{equation}\label{co:exact-ratio}
+ 2v\!\left(\frac{a(D)}{a(D_0)}\right)
+   =v\!\left(\frac{L_*(\sigma D)}{L_*(\sigma D_0)}\right).
+\end{equation}
+Moreover $a(D)\neq0$ exactly when $L(E^{(\sigma D)},1)\neq0$.
+Consequently any previously established bound
+$v(L_*(\sigma D))\geq2w(D)+s(\sigma D)-C$ gives
+\begin{equation}\label{co:even-bound}
+ v(a(D))\geq w(D)+\tfrac12s(\sigma D)-C_g
+\end{equation}
+with $C_g$ fixed independently of $D$.
+\end{proposition}
+
+\begin{proof}
+Use the Shimura packet of $E^{(-\sigma)}$.  Its positive index $D$ tests
+the twist by $\chi_{-D}$, which is $E^{(\sigma D)}$.
+The nonvanishing at $D_0$ selects a globally occurring member admitting
+that Fourier functional.  Take its spherical vector outside $S$ and its
+holomorphic vector of weight $3/2$ at infinity.
+
+Here is the support-local vector selection.  Start with a vector on which
+the $D_0$ Fourier functional is nonzero.  Projection under a sufficiently
+large compact group of upper unipotents to additive frequency $D_0$
+restricts the potentially nonzero unit translates to arbitrarily small
+neighborhoods of the two solutions of $u^2=1$.  Choose a unit character
+agreeing with the forced action at $-1$.  It can have $2$-power order:
+the image of $-1$ is detected in the $2$-primary quotient of the unit
+group.  The neighborhoods of $1$ and $-1$ then contribute with the same
+sign, so this projection preserves the nonzero functional.  Upper
+integral translations act trivially, and smoothness gives sufficiently
+deep lower-unipotent invariance.  The vector therefore defines a
+classical form at a level supported on $S$, allowing arbitrary depth at
+$2$ and at the bad primes.  The unit-squareclass and coprimality sieves
+commute with good $T(p^2)$, since their conditions are unchanged by
+multiplication of the index by $p^2$.  The classical Hecke formula
+\cite{Shimura1973HalfIntegral} gives
+\eqref{co:recurrence}, with $\eta=\psi\chi_{-\sigma}$.
+
+At the level just chosen, let $V$ be the full space of weight-$3/2$
+cusp forms orthogonal to unary theta series, with scalar character
+$\psi$ and good $T(p^2)$ eigenvalues $\eta(p)a_p(E)$.  It contains
+the constructed form, whose coefficient at $D_0$ is nonzero.  For
+$D/D_0$ a unit square at every support prime, Waldspurger's comparison
+\cite[Corollary~2, pp.~483--484]{Waldspurger1981} gives, for every
+$f\in V$,
+\[
+ a_f(D)^2=
+ \psi(D/D_0)\frac{L_*(\sigma D)}{L_*(\sigma D_0)}a_f(D_0)^2.
+\]
+Its hypothesis (H1) follows from the nonzero eigenspace
+\cite[\S I.2, Proposition~2]{Waldspurger1981}; the final paragraph of
+the comparison proof removes (H2) for these local squareclasses.
+Factoring the displayed quadratic identity in the two coefficient
+functionals shows that they are proportional on all of $V$, with one
+scalar independent of $f$.  This includes oldvectors and their linear
+combinations; if the central value at $D$ vanishes, the first functional
+is simply zero.
+
+Choose an algebraic form $g$ in the filtered subspace of $V$ with
+$a(D_0)\ne0$.  To justify this choice, multiplication by ordinary theta
+embeds the space into an integral-weight space, and divisibility by
+theta is specified by algebraic cusp conditions
+\cite[\S\S3--4]{Ramsey2006HalfIntegral}.  The character, good-Hecke
+and filter conditions are algebraic linear conditions as well.
+Since evaluation at $D_0$ is nonzero after extension to $\mathbb C$,
+it is nonzero on some algebraic form.
+
+Let $u$ be an automorphism of $\overline{\mathbb Q}$ acting on the
+character values by an odd power $e$, and put $\nu=\eta^{(e-1)/2}$.
+The forms $g^u$ and $g\otimes\nu$ have the same scalar character
+$\psi^e=\psi\nu^2$ and the same good eigenvalues
+$\eta(p)^e a_p(E)=\nu(p)^2\eta(p)a_p(E)$; here $\psi/\eta$ is
+quadratic.  They belong to one good-Hecke eigenspace at a common level
+still supported on $S$.  Applying the preceding proportionality in
+that nonzero space and normalizing at $D_0$ gives
+\[
+ u(\xi)=\nu(D/D_0)\xi,\qquad \xi=a(D)/a(D_0).
+\]
+Local square roots of $D/D_0$ at the conductor of $\eta$ give a value
+$z$ of $\eta$ with $\eta(D/D_0)=z^2$.  Then $u(z)=z^e$ and
+$\nu(D/D_0)=z^{e-1}$, so $\xi/z$ is fixed by every algebraic
+automorphism and is rational.  Moreover the same local-square condition
+gives $\psi(D/D_0)=\eta(D/D_0)=z^2$.  The comparison therefore reads
+\[
+ (\xi/z)^2=\frac{L_*(\sigma D)}{L_*(\sigma D_0)},
+\]
+which proves \eqref{co:exact-ratio} since $v(z)=0$.
+There is no product over the new prime factors: the unramified
+squarefree formula includes both local valuations zero and one.
+No exceptional local representation at $2$ is excluded by the
+same-local-class comparison.  The vanishing assertion and
+\eqref{co:even-bound} follow.
+\end{proof}
+
+\subsection{A detecting Kummer functional}
+
+Fix now the positive odd-sign filter and its original support, including
+the auxiliary class-group tests needed for arithmetic realization.
+Choose a fixed negative odd
+fundamental discriminant $h_*$ with $L(E^{(h_*)},1)\neq0$, whose local
+squareclasses agree with those of the filter at $2N$.  Adjoin its prime
+factors to $S$, exclude them from the variable indices, and refine the
+fixed unit tests there as required, all before varying $D$.
+Choose it so that $|h_*|>4$ and one of its ramified primes
+excludes the CM field of $E$, if there is one.  Thus
+\[
+ K_D=\mathbb Q(\sqrt{Dh_*})
+\]
+is imaginary, split at $2N$, and has only the units $\{1,-1\}$.
+Fix a modular parametrization $\varphi$
+with $\varphi(\infty)=0$ throughout.  With this normalization the
+cuspidal constants in its Hecke and Atkin--Lehner relations are torsion;
+see \cite{Manin1972,Drinfeld1973}.  An arbitrary translation by a
+rational point is not permitted.
+
+Let $H_D$ be the Hilbert class field of $K_D$.  Choose an
+$\mathcal O_{K_D}$-ideal $\mathfrak n_D$ with
+$\mathcal O_{K_D}/\mathfrak n_D\simeq\mathbb Z/N\mathbb Z$,
+fixing one orientation at each level prime, and let
+$x_D\in X_0(N)(H_D)$ be its conductor-one CM point.  The unramified
+genus character $\chi_D$ attached to
+$K_D(\sqrt{h_*})/K_D$ satisfies
+$\chi_D([\mathfrak a])=\chi_{h_*}(\operatorname{Norm}\mathfrak a)$
+on ideals prime to $Dh_*$.  Define the unaveraged character sum
+\[
+ P_D=\sum_{[\mathfrak a]\in\operatorname{Pic}(\mathcal O_{K_D})}
+       \chi_D([\mathfrak a])\varphi(x_D^{[\mathfrak a]}).
+\]
+These conductor-one genus sums over the varying fields $K_D$ are
+distinct from the conductor-$|h|$ ring-class sums over a fixed field
+$K$ defined in Section~\ref{rl:section}.
+The point $P_D$ lies in $E(\mathbb Q(\sqrt D,\sqrt{h_*}))$, with the quadratic
+interpretation when $D=1$.  Its genus-character component over $K_D$
+is the sum of the $D$- and $h_*$-twist components over $\mathbb Q$.
+The latter has rank zero because $L(E^{(h_*)},1)\ne0$.
+Thus the free part of $P_D$ lies in the $D$-twist component, a line
+when $E^{(D)}$ has a simple zero.  We retain the unaveraged sum in the
+theta comparison; a fixed integer $L$ kills bounded-degree torsion
+and makes its comparison with a rational twist point integral.
+At a simple zero write $j(P_D)$ for its index valuation in the free
+rational twist line, allowing the fixed restriction and projection
+indices.  Put $j(P_D)=+\infty$ if the product derivative vanishes.
+
+\begin{lemma}[Kummer detection, including CM]\label{co:kummer}
+Let $C$ act on $T=T_2E$ as complex conjugation, and choose a nonzero
+integral row $b:T\longrightarrow\mathbb Z_2$ with $bC=b$.
+For a fixed positive simple-zero witness $D_w$, there are arbitrarily
+large precisions $M$ and good primes $\ell$ such that
+\begin{equation}\label{co:ell-conditions}
+ \rho_T(\operatorname{Fr}_\ell)\equiv C\pmod {2^M},\qquad
+ (h_*/\ell)=-1,\qquad(D_w/\ell)=1.
+\end{equation}
+Evaluation at Frobenius squared defines an integral homomorphism
+\[
+ \lambda_M:E(\mathbb F_{\ell^2})\longrightarrow\mathbb Z/2^M\mathbb Z,
+ \qquad \lambda_M=b\kappa_M,
+\]
+of Frobenius sign $+1$.  The valuation of
+$\lambda_M(\operatorname{red}_\ell(LP_{D_w}))$ is bounded independently
+of $M$, for a fixed nonzero integer $L$ making the twist comparison
+integral.  Extra fixed finite-field restrictions can be retained when they agree
+with the chosen complex-conjugation coset, or when a compatible lift
+has been specified.
+\end{lemma}
+
+\begin{proof}
+For every elliptic curve $A/\mathbb Q$, the rational Tate module $V_2A$
+is irreducible.  Indeed Faltings' semisimplicity and endomorphism theorem
+\cite[\S1, assertions (a),(b)]{Faltings1983} would turn a proper
+summand into a nontrivial idempotent in
+$\operatorname{End}_{\mathbb Q}(A)\otimes\mathbb Q_2=\mathbb Q_2$.
+The equality of this endomorphism ring with $\mathbb Z$ includes CM
+curves: a nonreal CM endomorphism is not defined over $\mathbb Q$.
+The Tate image contains a nonidentity central scalar, by the homothety
+theorem \cite[\S2, Theorem~3 and paragraph~7]{Bogomolov1981Points}.
+
+Let $Q\in A(\mathbb Q)$ be nontorsion and let $c_Q$ be its division
+cocycle.  On the kernel of the Tate action its image is a closed
+$\mathbb Z_2$-submodule of $T_2A$, stable under the full Galois image.
+If its rational span were zero, the rational Kummer class would factor
+through the Tate image.  A central scalar $z\neq1$ annihilates its
+first cohomology: for a cocycle $c$ the equality of $c(zg)$ and $c(gz)$
+gives $(z-1)c(g)=(g-1)c(z)$, so $c$ is a coboundary over $\mathbb Q_2$.
+This contradicts Kummer injectivity and the finite generation of
+$A(\mathbb Q)$.  Irreducibility consequently makes the translation
+image contain $2^aT_2A$ for some $a$.  Intersecting with the kernel of
+additional finite data retains a full lattice.
+
+Apply this to the rational twist point supplied by $LP_{D_w}$ on
+$A=E^{(D_w)}$.  Since $D_w>0$, conjugation acts on its Tate module as
+$C$.  Modify a conjugation lift by an element of the simultaneous
+kernel whose translation is $v$.  The square of the modified element
+has translation $(1+C)v$, and $b(1+C)v=2b(v)$.  The full-lattice
+property allows a choice with a fixed finite nonzero valuation.
+Conjugation acts trivially on $\mathbb Q(\sqrt{D_w})$ and nontrivially
+on $\mathbb Q(\sqrt{h_*})$.  Chebotarev in each finite torsion,
+division, and character extension gives \eqref{co:ell-conditions}
+and the chosen finite nonzero detection.
+
+For good $\ell$, Lang's theorem and finite-field Kummer theory give
+\[
+ E(\mathbb F_{\ell^2})/2^M E(\mathbb F_{\ell^2})
+ \simeq T/\bigl(2^MT+(\operatorname{Fr}_\ell^2-1)T\bigr)=T/2^MT.
+\]
+This is evaluation of a cocycle at arithmetic Frobenius squared.
+Its equivariance and $bC=b$ prove the sign assertion.  Reduction of
+the division cocycle gives precisely the detection already arranged.
+Finally $\det C=-1$ implies $\ell\equiv-1\pmod {2^M}$.
+\end{proof}
+
+\subsection{The finite Schwartz function and its transformation}
+
+\begin{lemma}[Explicit weighted theta construction]\label{co:theta-bridge}
+For $\ell$ as in Lemma~\ref{co:kummer}, the reduced modular
+parametrization followed by $\lambda_M$ admits an integral weighted
+ternary theta lift.  After extraction of the coefficients at multiples
+of $|h_*|$ it gives a classical holomorphic weight-$3/2$ form
+$g_M=\sum a_M(n)q^n$, of level supported on $S\cup\{\ell\}$, with
+\eqref{co:recurrence} modulo $2^M$.  The associated characters have
+$2$-power order and quadratic ratio.  For squarefree admissible $D$,
+\begin{equation}\label{co:weighted-comparison}
+ a_M(D)=c_0\,2^{\omega(N)}\zeta_M(D)
+       \lambda_M\bigl(\operatorname{red}_\ell(LP_D)\bigr)
+                     \pmod {2^M},
+\end{equation}
+where $\zeta_M(D)$ is a $2$-power root of unity and $c_0,L$ are fixed
+integers independent of $D,M,\ell$.  The extra index restriction at
+$\ell$ is $(D/\ell)=1$.
+\end{lemma}
+
+\begin{proof}
+Let $B$ be ramified at $\ell$ and infinity, $R$ an Eichler order of
+level $N$, and $V=B^0$ with $Q(x)=-x^2=\operatorname{Nrd}(x)$.
+The root characters at the fixed level primes below are chosen once.
+Let $\mathcal O$ be the valuation ring at a place above $2$ of the
+field generated by their values.  The other local weights are
+quadratic, so this field is independent of $M$ and $\ell$.
+In this construction integrality means membership in $\mathcal O$,
+and congruences mean congruences modulo $2^M\mathcal O$.
+\par\smallskip\noindent\emph{The supersingular coefficient function.}
+The supersingular pairs on $X_0(N)$ are the finite set
+\[
+ X=B^\times\backslash\widehat B^\times/\widehat R^\times.
+\]
+This description permits every $N$ prime to $\ell$, including arbitrary
+depth at $2$; see \cite[\S\S1.1--1.2,2.4--2.6]{JetchevKane2010}.
+For $z\in X$ put $R_z=B\cap z\widehat Rz^{-1}$.
+Choose $L$ killing the finite subgroup generated by the images of the
+cuspidal constants in the Atkin--Lehner relations, and also all the
+bounded-degree torsion used below.  Then
+\[
+ P_M(z)=\lambda_M(L\varphi_\ell(z)),
+\]
+where $\varphi_\ell$ is the specialization of $\varphi$ to the
+good-reduction special fiber.  This function has Brandt eigenvalues
+$a_p(E)$ away from $N\ell$, Atkin--Lehner
+eigenvalue $w_{p^e}$ at $p^e\Vert N$, and eigenvalue $+1$ under the
+right $\ell$-uniformizer.  These identities are modulo $2^M$.
+Choose any integer lifts of this finite list of values.
+
+\par\smallskip\noindent\emph{The local Schwartz weights.}
+We now define the local functions, rather than appealing to an
+unspecified weighted theta theorem.  Off $2Nh_*\ell$ take the
+characteristic function of $R_p^0$.  At $p^e\Vert N$, use a model
+\[
+ R_p=\left\{\begin{pmatrix}a&b\\c&d\end{pmatrix}
+       :a,b,d\in\mathbb Z_p, c\in p^e\mathbb Z_p\right\}.
+\]
+Require the split fundamental unit squareclass and an integral
+embedding.  Its action on the cyclic level quotient selects one root
+$r\in\mathbb Z_p^\times$ of $x^2$.  Give $x$ weight $\alpha_p(r)$,
+where $\alpha_p$ is a finite unit character of $2$-power order with
+\begin{equation}\label{co:root-parity}
+ \alpha_p(-1)=w_{p^e}\chi_{h_*}(p)^e.
+\end{equation}
+Such a character exists for either prescribed sign.  The root is a
+locally constant function to any fixed character precision: the two
+idempotents of the split integral quadratic algebra distinguish the
+two summands of the cyclic quotient.  Unit scaling $x\mapsto ux$
+multiplies this weight by $\alpha_p(u)$, and order-unit conjugation
+preserves it.
+
+At $2$ require $(1+x)/2\in R_2$ and $x^2\in\mathbb Z_2^{\times2}$.
+If $2\mid N$, the preceding selected-root rule is imposed using
+$e_x=(1+x)/2$.  Its two eigenvalues $(1+r)/2,(1-r)/2$ differ by the
+odd unit $r$; hence the two idempotents remain distinct modulo $2$.
+This verifies the construction at every exponent of $2$ in $N$.
+Scaling by an odd $u$ preserves integrality, since
+$(1+ux)/2=u(1+x)/2+(1-u)/2$.  This is an invertible affine change of
+the operator $e_x$, so it preserves its two summands individually.
+The selected summand is selected by the cyclic level quotient, not
+by declaring its eigenvalue to be $0$ or $1$ modulo $2$.  On that same
+summand $x$ acts as $r$ and $ux$ acts as $ur$.  Even when
+$u\equiv-1\pmod4$ interchanges the two eigenvalue parities, the root
+weight therefore changes by exactly $\alpha_2(u)$.
+If $2\nmid N$ there is no root weight at $2$.
+
+For $p\mid h_*$ require $x\in M_2(\mathbb Z_p)^0$ and
+$v_p(Q(x))=1$.  Its reduction is a nonzero nilpotent.  Fix
+$X_0=\left(\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\right)$ and
+define
+\[
+ \epsilon_p(gX_0g^{-1})=\left(\frac{\det g}{p}\right).
+\]
+The centralizer consists of
+$\left(\begin{smallmatrix}a&b\\0&a\end{smallmatrix}\right)$, with
+determinant $a^2$, so this is well-defined on the entire nonzero
+nilpotent orbit.  It obeys
+\[
+ \epsilon_p(uyu^{-1})=(\det u/p)\epsilon_p(y),\qquad
+ \epsilon_p(ay)=(a/p)\epsilon_p(y).
+\]
+The second equality follows by conjugating $X_0$ by
+$\operatorname{diag}(a,1)$.  Set $f_p(x)=\epsilon_p(\bar x)$ on the
+specified support.  Thus the right conjugation character is precisely
+the local component of $\chi_{h_*}\circ\operatorname{Nrd}$.
+
+At $\ell$, let $\mathfrak P$ be the maximal ideal of the maximal
+order.  Its residue field is $\mathbb F_{\ell^2}$, and the reduction of
+an integral trace-zero element lies on its one-dimensional trace-zero
+line.  Fix a nonzero basis vector $\beta$ on that line and put
+\[
+ f_\ell(x)=\left(\frac{t}{\ell}\right)
+   \quad\text{if }\bar x=t\beta\neq0,
+ \qquad f_\ell(x)=0\quad\text{otherwise}.
+\]
+Order units act trivially by conjugation on the residue field.
+A uniformizer acts by its nontrivial automorphism and sends
+$\beta$ to $-\beta$.  Therefore its multiplier is $-1$, since
+$\ell\equiv-1\pmod4$; scaling has character $(u/\ell)$.
+Moreover $\beta^2$ is a nonsquare and $-1$ is a nonsquare modulo
+$\ell$, so $Q(x)=-t^2\beta^2$ is a nonzero square.  Because
+$(|h_*|/\ell)=1$, the extracted indices automatically satisfy
+$(D/\ell)=1$.  No average over $\mathbb F_\ell^\times$ is used.
+
+All these functions have compact support and are locally constant.
+Let $f=\prod f_p$, incorporating any additional fixed unit-squareclass
+conditions by their characteristic functions.  Those conditions are
+stable under unit squares and change none of the scaling characters.
+For $u\in\widehat R^\times$,
+\begin{equation}\label{co:order-equivariance}
+ f(u^{-1}xu)=\chi_{h_*}(\operatorname{Nrd}u)^{-1}f(x).
+\end{equation}
+The product scaling character is even.  Indeed its value at $-1$ is
+\[
+ \left(\prod_{p^e\Vert N}w_{p^e}\chi_{h_*}(p)^e\right)
+       \chi_{h_*}(-1)(-1)=1;
+\]
+the parenthesized factor is the even functional sign of $E^{(h_*)}$.
+Denote its associated classical character by $\eta_B$.  In the
+inverse-idele convention fixed above, if $\alpha$ is this finite
+scaling character, then $\eta_B(d)=\alpha(d)$ for positive integers
+$d$ prime to the support: the inverse idele character becomes the
+product of the finite unit values at the lower-right entry.
+
+\par\smallskip\noindent\emph{The scalar transformation law.}
+Here are all the Weil-representation checks needed for scalar
+modularity.  For an additive character of conductor $\mathbb Z_p$,
+write $n(b)=\left(\begin{smallmatrix}1&b\\0&1\end{smallmatrix}\right)$,
+$m(u)=\operatorname{diag}(u,u^{-1})$, and
+$n^-(c)=\left(\begin{smallmatrix}1&0\\c&1\end{smallmatrix}\right)$.
+Use a Weyl lift $w$, its compatible self-dual Fourier transform, and
+the Weil factors $\gamma_p(u),\gamma_p$ for this additive character.
+The defining formulas are
+\[
+ \omega(n(b))f_p(x)=\mathbf e_p(bQ(x))f_p(x),\quad
+ \omega(m(u))f_p(x)=\gamma_p(u)|u|_p^{3/2}f_p(ux),\quad
+ \omega(w)f_p=\gamma_p\widehat f_p.
+\]
+They follow from the Weil construction and Poisson summation
+\cite[I, no.~13]{Weil1964}.  Integral $b$ fixes $f_p$, since $Q$ is integral on
+its support.  If $f_p$ is invariant under translations by a lattice
+$L_p'$, its Fourier transform is supported on the dual lattice
+$(L_p')^\vee$.  Choose $A_p$ so that
+$p^{A_p}Q((L_p')^\vee)\subset\mathbb Z_p$.  Then $n(p^{A_p}b)$ fixes
+$\widehat f_p$, and conjugating by $w$ proves that every
+$n^-(p^{A_p}b)$ fixes $f_p$.  This argument works without change at
+$2$ and for all finite character depths.
+
+Relative to the ordinary theta multiplier cubed, the diagonal Weil
+factor is $(\det Q,u)_p$.  Indeed diagonalizing $Q$ and multiplying
+the one-dimensional Weil-index identities
+\cite[II, nos.~25, 28]{Weil1964} gives the Hilbert symbol
+of the product of its three coefficients.  In a presentation
+$B=(-a,-b)$ that product is the determinant of
+$aX^2+bY^2+abZ^2$, namely $(ab)^2$.  Thus this additional factor is
+$1$.  The remaining diagonal action is exactly the scaling character
+already computed.  Increasing $A_2$ to the depth needed for the
+standard metaplectic splitting, the factorization
+\[
+ \begin{pmatrix}a&b\\c&d\end{pmatrix}
+    =n^-(c/a)m(a)n(b/a),\qquad c\in p^{A_p}\mathbb Z_p, a\in\mathbb Z_p^\times,
+\]
+proves scalar transformation on the required local $\Gamma_0$.
+Conjugation by $z$ commutes with these Weil actions and preserves $Q$.
+Consequently the same level and scalar character work for every
+summand.  The Gaussian at infinity gives holomorphic weight $3/2$;
+positive definiteness gives nonnegative exponents at every cusp.
+
+Choose an integer $c_0$ divisible by every possible $|R_z^\times|$.
+Such a uniform choice exists because these unit groups embed in
+$\operatorname{GL}_4(\mathbb Z)$ and their orders are bounded.
+The theta series is
+\begin{equation}\label{co:theta-sum}
+ H_M(n)=c_0\sum_{z\in X}
+  \frac{P_M(z)\chi_{h_*}(\operatorname{Nrd}z)}{|R_z^\times|}
+  \sum_{\substack{x\in B^0\\Q(x)=n}} f(z^{-1}xz).
+\end{equation}
+Equation~\eqref{co:order-equivariance} proves independence of right
+representatives; rational reduced norms are positive, so the norm
+character is trivial on the left rational action.  The preceding Weil
+calculation proves modularity \cite[III, Theorem~6]{Weil1964}, and
+the choice of $c_0$ proves integral Fourier coefficients.  More
+precisely, each $c_0/|R_z^\times|$ is an integer, so changing a lift
+of $P_M(z)$ by $2^M$ changes every coefficient by an element of
+$2^M\mathcal O$.  No precision is lost to a stabilizer denominator.
+The chosen integer lifts need satisfy no characteristic-zero
+eigenrelation.
+
+\par\smallskip\noindent\emph{The good-prime Hecke relation.}
+For completeness, the good-prime relation can be checked in a matrix
+order.  Neighbors correspond to the lines in $\mathbb F_p^2$.
+A nonzero primitive nilpotent reduction has exactly one invariant
+line.  For an integral element of nonzero reduction and norm $n$,
+the number of invariant lines is $1+(-n/p)$; zero reduction contributes
+the additional $p$ lines.  If a primitive $y$ has $p^2\mid Q(y)$,
+choose a basis in which its reduction is $X_0$.  Its lower-left entry
+is divisible by $p^2$, and exactly the neighbor for that invariant
+line makes $y/p$ integral.  These counts partition the three terms
+of the theta Hecke operator.  Scaling by $p$ at the support contributes
+$\eta_B(p)$, and the reduced norm of a Brandt neighbor contributes
+$\chi_{h_*}(p)$.  Thus, with $A(n)=H_M(n)$,
+\[
+ A(p^2n)+\eta_B(p)(-n/p)A(n)+p\eta_B(p)^2A(n/p^2)
+       =\eta_B(p)\chi_{h_*}(p)a_p(E)A(n)\pmod {2^M}.
+\]
+The orbit-stabilizer weights in \eqref{co:theta-sum} are exactly the
+weights in this neighbor count.  Extracting
+$a_M(n)=H_M(|h_*|n)$ gives
+$\eta=\eta_B\chi_{h_*}$ and
+$\psi=\eta_B(|h_*|/\cdot)$, proving \eqref{co:recurrence}.
+Fixed-support coefficient sieves are exact zero-or-retain operations;
+they preserve coefficient integrality and commute with this relation.
+
+\par\smallskip\noindent\emph{The weighted coefficient comparison.}
+It remains to prove the weighted comparison; modularity alone does
+not supply it.  At $n=|h_*|D$ the counted elements are precisely the
+optimal embeddings $\mathcal O_{K_D}\hookrightarrow R_z$, sending
+$\sqrt{Dh_*}$ to $x$.  The condition at $2$ is exactly the condition
+on the integral generator $(1+\sqrt{Dh_*})/2$.
+At a level prime there are two orientations.  Indeed a stable lattice
+splits into its two integral idempotent summands, and a cyclic quotient
+of order $p^e$ lies wholly in one summand.  At $\ell$ there are two
+orientations, the two residue-field embeddings.  At all remaining
+primes there is a single local optimal-embedding orbit.
+
+Fix a rational embedding $x_0$ and an initial orientation tuple $z_0$.
+The orbits with these orientations are indexed by
+$\operatorname{Pic}(\mathcal O_{K_D})$, with representatives $az_0$
+for $a\in\widehat K_D^\times$.  The equality
+\[
+ (az_0)^{-1}x_0(az_0)=z_0^{-1}x_0z_0
+\]
+shows that the Schwartz weight is constant along that orbit.  Its
+remaining relative weight is
+$\chi_{h_*}(\operatorname{Norm}_{K_D/\mathbb Q}a)$, exactly the
+unramified genus character.  The reduction correspondence commutes
+with this ideal-class action and remembers the cyclic level subgroup
+\cite[\S2]{JetchevKane2010}; a Galois-equivariant description of the
+lift, class action and reduction is given in
+\cite[\S3.2, \S\S4.6, 4.9--4.10 and Corollary~4.1]{CornutJetchev2013}.
+We use these correspondences only for their geometry; the weight
+identity has just been computed, and the multiplicity is computed next.
+
+An embedding stabilizer is $\mathcal O_{K_D}^\times=\{\pm1\}$.
+An orbit in $R_z$ therefore contains $|R_z^\times|/2$ elements, giving
+the factor $1/2$ in \eqref{co:theta-sum}.  At $p^e\Vert N$, reversing
+orientation multiplies its Schwartz factor by
+$\alpha_p(-1)=w_{p^e}\chi_{h_*}(p)^e$, and its norm-weighted
+parametrization value by the same sign.  At $\ell$ both multipliers
+are $-1$: one is the residue-line calculation and the other is
+$\chi_{h_*}(\ell)$ times the Frobenius eigenvalue $+1$.
+Every one of the $2^{\omega(N)+1}$ orientation tuples consequently
+has the same contribution.  The initial tuple has a $2$-power unit
+phase.  Combining this count with its factor $1/2$ gives exactly
+\eqref{co:weighted-comparison}.  There is no division by a class
+number, and none of its rational multipliers grows with $\ell$ or
+the number of factors of $D$.
+\end{proof}
+
+\begin{proposition}[Odd coefficient detection]\label{co:odd}
+The forms in Lemma~\ref{co:theta-bridge}, after a fixed adjustment of
+precision, satisfy
+\[
+ v(a_M(D))\geq\min\{M,j(P_D)-C\},\qquad
+ v(a_M(D))<M\ \Longrightarrow\ j(P_D)\leq v(a_M(D))+C.
+\]
+A visible coefficient detects $a(E^{(D)})=1$, and a fixed positive
+witness has bounded coefficient valuation as $M\to\infty$.
+All constants are independent of $M,\ell,D$.
+\end{proposition}
+
+\begin{proof}
+Gross--Zagier identifies the height of the genus sum with the product
+derivative \cite{GrossZagier1986}.  Since $L(E^{(h_*)},1)\neq0$,
+vanishing of the derivative makes $P_D$ torsion, and a nontorsion
+$P_D$ is equivalent to a simple zero of $E^{(D)}$.
+The genus sum lies in a biquadratic field, so a fixed integer kills
+all possible torsion: reduce prime-to-residue-characteristic torsion
+at two fixed good primes of different residue characteristic; the
+residue degrees are at most four.  Include that integer in $L$.
+On the free twist line, divisibility by $2^{j(P_D)}$ remains divisibility
+after the integral maps of reduction and $\lambda_M$.
+Restriction and the two quadratic projections have composite
+multiplication by $2$, so their integral indices have a fixed bound.
+Equation~\eqref{co:weighted-comparison} therefore proves both
+valuation assertions, with fixed losses only.  Lemma~\ref{co:kummer}
+gives the witness.
+\end{proof}
+
+\subsection{Minimum depth and integral coefficient operators}
+
+We now pass from detection to normalization.  For the even source,
+the cyclotomic lower bound gives \eqref{co:even-bound} at every
+nonzero squarefree coefficient.  For the odd source, assume for now
+that every simple-zero parameter satisfies
+\[
+ j(P_D)\geq w(D)+s(D)/2-C_*.
+\]
+Proposition~\ref{pw:odd-lower} will prove this using only the even
+construction.  Proposition~\ref{co:odd} then gives
+\[
+ v(a_M(D))\geq\min\{M,w(D)+s(D)/2-C'\}
+\]
+at simple zeros, where $s(D)$ is defined.  At every other odd-sign
+parameter the product derivative vanishes and $a_M(D)=0$ modulo
+$2^M$.  In particular the coarser bound
+$v(a_M(D))\geq\min\{M,w(D)-C'\}$ holds at every admissible $D$.
+The remaining coefficient calculus is conditional on these lower
+bounds; it does not establish them.
+
+For the operator and graph arguments, at each actual odd stage we
+write $S$ for the original fixed support together with the primes of
+$h_*$ and the current reduction prime $\ell$.  The latter is stationary
+during that stage and is never an element of a newly adjoined set
+$U$.  Thus every statement below about ramification outside $S\cup U$
+includes $\ell$.  This notation does not change the original fixed
+data controlling the constants, or choose a new minimum when a vertex
+is omitted.
+
+\begin{lemma}[Stabilized minimum depth]\label{co:minimum}
+Assume the uniform arithmetic lower bounds just stated.  Each coefficient source
+defines a binary symbol
+\begin{equation}\label{co:symbol}
+ A(D)=\begin{cases}
+ 1,&v(a(D))=\delta+w(D),\\
+ 0,&v(a(D))>\delta+w(D)\text{ or }D\text{ is off the filter},
+ \end{cases}
+\end{equation}
+with a fixed minimum normalized depth $\delta$ and at least one
+bounded-weight witness.  For the odd source $a=a_M$ at an actual stage:
+the numerical minimum and its divisibility statements stabilize at
+every fixed finite weight bound.  The finite symbol tables on arithmetic
+labels will be stabilized separately in Section~\ref{pw:odd-stages}.
+At a unit symbol,
+\[
+ v(L_*(\sigma D))\leq2w(D)+O(1)
+ \quad\text{or}\quad j(P_D)\leq w(D)+O(1),
+\]
+respectively, and the analytic order is respectively zero or one.
+\end{lemma}
+
+\begin{proof}
+For the even source normalize $a(D_0)=1$; the phase-removed squarefree
+coefficients are rational by Proposition~\ref{co:even}.  For the odd
+source, \eqref{co:weighted-comparison} makes every visible squarefree
+coefficient an integer times a root of unity modulo $2^M$.
+Their finite valuations are therefore integers in both cases, not
+arbitrary elements of a ramified coefficient field's value group.
+The fractional part of $w(D)$ is fixed on the filter,
+because the parity of the number of simple primes is its fixed
+quadratic discriminant character.  The lower bounds and a nonzero
+witness therefore give a minimum for the even source.
+
+For the odd source choose a nonprincipal ultrafilter on the increasing
+precisions.  Minimize $v(a_M(D_M))-w(D_M)$ over sequences with bounded
+weight and visible coefficient on an ultrafilter-large set.  The
+uniform lower bound, discrete values, and fixed witness give a finite
+attained minimum.  At each fixed weight cutoff $W$, failure of its
+divisibility on an ultrafilter-large set would allow a choice of a
+violating $D_M$ with $w(D_M)\leq W$, contrary to the minimum.
+Hence every prescribed finite list of these divisibilities holds at
+a sufficiently advanced actual stage.  If larger weights arise in a
+coefficient-integrality check, the coarse bound
+$\min\{M,w(D)-C\}$ already pays any fixed normalization once both
+$M$ and $w(D)$ exceed its fixed threshold.  This also explains why
+no infinite-precision modular lift is asserted or needed.
+The last conclusions follow from the two coefficient comparisons.
+\end{proof}
+
+Write $U_p\sum c(n)q^n=\sum c(pn)q^n$ and
+$V_p\sum c(n)q^n=\sum c(n)q^{pn}$.  For a squarefree set $U$ put
+\begin{equation}\label{co:BU}
+ B_U=2^{-\lceil\delta+w(U)\rceil}
+       \prod_{p\in U}(U_p-\eta(p)V_p)g,\qquad
+ e_U=\lceil\delta+w(U)\rceil-\delta-w(U).
+\end{equation}
+We call $U$ matching if $e_U=0$ and opposite if $e_U=1/2$.
+These operators introduce each new prime to level exponent one:
+$V_p$ changes the character by $(p/\cdot)$ and raises the level by
+$p$, while $U_p=U_{p^2}V_p$ on that raised level.
+Thus the new character is $\psi_U=\psi\prod_{p\in U}(p/\cdot)$.
+
+\begin{lemma}[Coefficient bounds and unary removal]\label{co:operators}
+For the even source let $U$ be arbitrary.  For the odd source, fix
+an integer $R\ge1$ and a weight bound $W$; all assertions below are
+made at a sufficiently advanced actual stage, simultaneously for every
+$U$ with $w(U)\le W$ and every Fourier index.
+In these respective settings the coefficients of $B_U$ are integral.
+Write uniquely
+$n=Jrm^2$, with $J\subseteq U$, $r$ squarefree, and $(r,U)=1$.
+The coefficient vanishes unless $D=(U/J)r$ belongs to the filter,
+and for the even source otherwise
+\begin{equation}\label{co:BU-bound}
+ v(B_U[n])\geq w(r)+\tfrac12\#\{p\in J:p\text{ simple}\}-e_U.
+\end{equation}
+At that stage the odd source satisfies this inequality for every
+$U$ with $w(U)\le W$ and every $n$, with its right side replaced
+by its minimum with $R$.
+In particular the untruncated bound holds on any prescribed bounded
+range of squarefree weights, after choosing $R$ and the stage.
+For a split $p\in J$ the difference factor, after its unit phase and
+one factor of $2$ are removed, has residue
+\begin{equation}\label{co:cp}
+ c_p(D)=\frac{a_p(E)-(\psi/\eta)(p)(-D/p)-1}{2}\pmod2
+\end{equation}
+at every square shell.  For a simple $p$ this formula holds on the
+first shell $p\nmid m$.
+
+Choose $i\in\{0,1\}$ such that $\eta\chi_{-4}^i$ is odd, and set
+\[
+ \Theta_J=\sum_{\substack{m>0\\(m,S)=1}}
+              \eta(m)\chi_{-4}(m)^i m q^{Jm^2}.
+\]
+For matching $U$ the modular form
+\begin{equation}\label{co:HU}
+ H_U=\frac12\left(B_U-
+       \sum_{\substack{J\subseteq U\\J\text{ split only}}}
+                 B_U[J]\Theta_J\right)
+\end{equation}
+has integral coefficients, with the same stage and weight restriction
+in the odd case.  For opposite $U$ put $H_U=B_U$.
+The new primes in the level of $\Theta_J$ have exponent one, and its
+character is $\eta\chi_{-4}^i(-J/\cdot)$.
+\end{lemma}
+
+\begin{proof}
+After removing the powers of $\eta(p)$, the square-index multipliers
+in \eqref{co:recurrence} are
+\[
+ b_0=1,\quad b_1=a_p-\kappa,\quad
+ b_{s+1}=a_pb_s-pb_{s-1},\qquad
+ \kappa=(\psi/\eta)(p)(-D/p).
+\]
+If $p\mid D$, then $\kappa=0$ and $b_s$ alternates between $1$ and
+$0$ modulo $2$, because $a_p$ is even.  Thus the difference from
+$p\in U/J$ is odd.  If $p\nmid D$, all $b_s$ are odd and the
+difference from $p\in J$ is even.  Subtracting the normalization in
+\eqref{co:BU} gives exactly
+\[
+ \delta+w((U/J)r)+\#J-\lceil\delta+w(U)\rceil
+ =w(r)+\tfrac12\#\{p\in J:p\text{ simple}\}-e_U.
+\]
+This is an integer on the allowed filter.  Its only potential
+negative value before imposing parity is $-1/2$, so it is nonnegative.
+
+For the odd source this calculation first applies on a fixed bounded
+range of squarefree weights, with enough precision to pay the fixed
+normalizing power of $2$.  To obtain the global truncated assertion,
+fix $R$ and a bound $W$ for $w(U)$.  The normalizing exponents
+$\lceil\delta+w(U)\rceil$ are uniformly bounded.  Above a sufficiently large weight
+threshold, the coarse bound $\min\{M,w(D)-C\}$ from
+Lemma~\ref{co:minimum} pays that normalizing power and at least $R$
+further factors of $2$, once $M$ is large enough.  Below the threshold
+use the stabilized minimum and increase the precision again if needed.
+The recurrence has integral multipliers, so errors modulo $2^M$
+remain divisible by $2^M$ at every square shell before normalization.
+Thus at the chosen stage the truncated bound holds at every
+coefficient for every $U$ with $w(U)\le W$, giving integrality
+without asserting an unbounded valuation estimate for a fixed
+finite-precision lift.
+
+For split $p$, rationality of $E[2]$ over $\mathbb F_p$ gives
+$a_p\equiv p+1\pmod4$.  With $\Delta_s=b_{s+1}-b_s$, every
+$\Delta_s$ is even and
+$\Delta_{s+1}=a_p\Delta_s-p\Delta_{s-1}$, so
+$\Delta_{s+1}\equiv\Delta_{s-1}\pmod4$.
+Direct calculation gives
+\[
+ \Delta_1-\Delta_0
+  =a_p^2-a_p\kappa-2a_p+2\kappa+1-p\equiv0\pmod4.
+\]
+Consequently every $\Delta_s/2$ has the residue in \eqref{co:cp}.
+At a simple prime only $\Delta_0$ is used.
+
+In matching parity the right side of \eqref{co:BU-bound} can vanish
+only when $r=1$ and $J$ is split only.  The same recurrence, retaining
+the character phases, gives
+$B_U[Jm^2]\equiv B_U[J]\eta(m)\pmod {2}$.
+The corresponding coefficient of $\Theta_J$ has exactly that
+residue, and different squarefree $J$ have disjoint supports.
+The numerator in \eqref{co:HU} is therefore divisible by the integer
+$2$, not just by the maximal ideal of a ramified character field.
+The ordinary odd-character unary theta transformation and dilation
+by $J$ give the asserted character and level.  For the odd source
+these calculations use the same sufficiently advanced stage,
+uniformly for $w(U)\le W$, with precision paying the fixed
+divisions as in Lemma~\ref{co:minimum}.
+\end{proof}
+
+\subsection{Trace operators and omission of a whole rational prime}
+
+The next step turns changing a Frobenius label into a difference of
+coefficient tests.  Fix a coefficient valuation ring $\mathcal O$,
+enlarging its field if needed for the finite eigensystem span below,
+and write $\mathfrak m$ for its maximal ideal.  Divisibility by $2^a$
+means divisibility in $2^a\mathcal O$; reduced traces and expansions
+mean coefficientwise reduction of Fourier expansions modulo $\mathfrak m$.
+These are different operations; no saturation of the integral module
+below is assumed.
+The normalized squarefree tests are binary because their phase-removed
+values are rational $2$-adic integers (integer congruences at an odd
+stage), and every $2$-power root of unity reduces to $1$.
+
+Multiply the form under consideration by
+$\theta=1+2\sum_{m>0}q^{m^2}$.  This gives a weight-two form with the
+same reduced expansion.  Work in the finite-dimensional sum of its
+characteristic-zero good Hecke eigensystems, retaining old
+multiplicity spaces and its diamond transforms.  The associated
+two-dimensional representations are the usual modular Galois
+representations for cusp forms \cite{Deligne1969}, and the sums of
+the two defining characters for Eisenstein series.  Their commuting
+trace and determinant operators are denoted by $t(u),d(u)$.
+Arithmetic Frobenius at a good prime $q$ has trace $T_q$ and
+determinant $q\langle q\rangle$.
+Let $\mathcal M$ be the $\mathcal O$-module generated by this
+weight-two vector and the undivided terms $\theta B_U[J]\Theta_J$,
+together with all their good-Hecke and diamond translates.
+
+\begin{lemma}[Integral traces]\label{co:traces}
+The module $\mathcal M$ is an integral Fourier lattice preserved by
+the trace and determinant operators.  On the coefficientwise reduced
+trace orbit of the indicated vector,
+\begin{equation}\label{co:trace-identities}
+ d(u)t(v)=t(v),\qquad
+ t(u)t(v)=t(uv)+t(uv^{-1}),\qquad t(1)=0.
+\end{equation}
+Every coefficient test of a finite product of traces can be performed
+using distinct fresh primes avoiding the tested index.  In particular
+testing $t(u_1)\cdots t(u_a)$ at $n$ becomes testing the original
+reduced coefficient at $nq_1\cdots q_a$.
+\end{lemma}
+
+\begin{proof}
+We first check integrality of the generators.  Diamonds act by scalars
+on $B_U$ and on each unary term.  For \eqref{co:HU}, the only apparent
+denominator is removed by
+\[
+ \langle u\rangle H_U=\psi_U(u)H_U+
+ \frac12\sum_J B_U[J]
+ \bigl(\psi_U(u)-\eta(u)\chi_{-4}(u)^i(-J/u)\bigr)\Theta_J;
+\]
+the difference of the two characters is zero or twice a unit.
+The same formula applies after multiplication by theta.
+Good Hecke operators commute with diamonds, and their coefficient
+formula preserves integrality.  Thus all generators of $\mathcal M$
+are integral.  Finitely many Fourier coefficients inject the ambient
+finite-dimensional space into a finite-dimensional coefficient space,
+so $\mathcal M$ embeds in $\mathcal O^r$ for some finite $r$.
+It is consequently a finite, closed $\mathcal O$-module.
+Continuity and simultaneous Chebotarev approximation on the finitely
+many eigensystems approximate $t(u)$ and $d(u)$ by $T_q$ and
+$q\langle q\rangle$ at every prescribed precision.  These operators
+preserve $\mathcal M$, so their limits do as well.  They act by scalars
+on each good eigensystem and by the identity on its old multiplicity
+space, which retains the asserted commutation with level operators.
+
+The formula
+$(T_qf)[n]=f[qn]+q(\langle q\rangle f)[n/q]$ has no second term
+when $q\nmid n$.  Successive distinct fresh primes therefore give
+the coefficient assertion.  On a single character system,
+determinants reduce to $1$ because all characters have $2$-power
+order and the cyclotomic character is odd.  On a subtracted vector
+the displayed diamond formula leaves unary corrections.  A fresh
+trace kills each correction modulo the maximal ideal: the tested
+index $nq$ cannot have its prescribed squarefree part $J$ when
+$q\nmid nJ$.  This continues to hold after further fresh tests.
+Thus $d(u)t(v)=t(v)$ on the reduced trace orbit.  Apply the matrix
+identity
+\[
+ \operatorname{tr}(A)\operatorname{tr}(B)
+  =\operatorname{tr}(AB)+\det(B)\operatorname{tr}(AB^{-1})
+\]
+to each characteristic-zero system and use that determinant
+identity.  Finally $t(1)=2$ before reduction.
+\end{proof}
+
+\begin{lemma}[The level-prime inertia identity]\label{co:inertia}
+For $p\in U$, choose inertia $\tau_p$ on which the new ramified
+quadratic character is $-1$, and an arithmetic Frobenius lift $\phi_p$
+in the same decomposition group.  In characteristic zero,
+\begin{equation}\label{co:inertia-identity}
+ U_p^{(2)}\bigl(t(g\tau_p\phi_p)-t(g\phi_p)\bigr)
+   =d(\phi_p)\bigl(t(g\tau_p)-t(g)\bigr).
+\end{equation}
+Here $U_p^{(2)}$ extracts $p$-multiples in weight two.  The identity
+commutes with further good traces.
+\end{lemma}
+
+\begin{proof}
+A constituent old from a level omitting $p$ has trivial inertia, so
+both differences are zero, whatever $U_p^{(2)}$ does on its old
+multiplicity space.  Every other constituent has conductor exponent
+one and primitive ramified quadratic determinant.  The conductor
+formula excludes monodromy and identifies it as principal series
+with one unramified and one ramified character.  Local compatibility
+\cite[Theorem~(A)]{Carayol1986}, after converting its geometric
+Frobenius convention to the arithmetic convention fixed above,
+identifies its $U_p^{(2)}$ eigenvalue
+with the Frobenius eigenvalue $\lambda$ on the unramified line.
+Write
+\[
+ \rho(\tau_p)=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\qquad
+ \rho(\phi_p)=\begin{pmatrix}\lambda&0\\0&\mu\end{pmatrix}.
+\]
+If $z$ is the lower-right entry of $\rho(g)$, the two trace
+differences are $-2z\mu$ and $-2z$.  Both sides of
+\eqref{co:inertia-identity} equal $-2z\lambda\mu$.
+The Eisenstein character pair has the same calculation.
+A unary term is old if $p\notin J$, and has precisely this ramified
+quadratic character if $p\in J$.  This exhausts the constituents.
+\end{proof}
+
+For the interface below a \emph{label} is the restriction of a
+Frobenius element to an elementary $2$-extension unramified outside
+the indicated support.  Its coordinates are quadratic characters.
+When $F$ is quadratic, choosing one of the two places at a split prime specifies an
+orientation; changing its ramification bit means multiplying by the
+image of inertia at that place.  Product labels are the restrictions
+of products of Frobenius elements, so their quadratic coordinates add.
+Simple primes use rational quadratic labels.  Section~\ref{gr:section}
+will realize these labels simultaneously by primes and interpret the
+independent inertia changes as graph edges.
+
+\begin{proposition}[Symbol interface]\label{co:symbol-interface}
+Under the arithmetic lower bounds in Lemma~\ref{co:minimum}, the
+symbols of either coefficient source have the following properties.
+\begin{enumerate}
+\item For matching $U$, the trace on split elements factors through
+the maximal elementary $2$-extension of $F$ unramified outside
+$S\cup U$.  For opposite $U$, traces factor through the corresponding
+rational elementary extension.  Adding a split vertex $q$ is read
+by coefficient $1$ of $t(\operatorname{Fr}_q)H_U$; adding a simple
+vertex is read on the opposite $B_U$.
+\item Let $p\in U$ and $g=\operatorname{Fr}_q$.  Changing one
+oriented ramification bit at $p$ changes the symbol by
+\begin{equation}\label{co:contraction}
+ A\bigl((U\setminus\{p\})\cup\{q'\}\bigr),\qquad
+ \operatorname{Fr}_{q'}\sim g\phi_p,
+\end{equation}
+unless $p,q$ are both simple, in which case the change is zero.
+The replacement has the product labels at all retained places.
+Its coefficient system has support $S\cup(U\setminus\{p\})$:
+the entire rational prime $p$, including both places above it when
+it splits in $F$, is omitted.
+\item A split vertex with trivial label in the elementary extension
+for the other vertices has zero symbol.
+\item In matching parity, insertion of two simple primes with equal
+rational labels on the fixed support and at the existing split
+vertices preserves the symbol if
+$\#E(\mathbb F_q)/2$ is odd for that label.  Their mutual simple
+bits and their incidences with other simple vertices do not affect
+this statement.
+\end{enumerate}
+These are coefficient identities.  Simultaneous realization of
+their labels by primes is a separate arithmetic construction.
+\end{proposition}
+
+\begin{proof}
+For an odd source choose the stage in Lemma~\ref{co:operators} with
+cutoff $R\ge2$; only these fixed divisibilities and finitely many
+bounded-weight coefficient tests are needed below.
+Two fresh traces in matching parity, at least one split, add weight
+at least $3/2$.  The allowed valuation in \eqref{co:BU-bound} is an
+integer, so it is at least $2$ before the unary division and the
+result vanishes after it.  In opposite parity any two fresh traces
+add weight at least $1$; subtracting $e_U=1/2$ and using integral
+valuation again gives vanishing.  The estimates persist under
+further tests.  Hence respectively
+$t(g)t(u)=0$ for split $u$, or $t(g)t(u)=0$ for every $u$.
+Applying \eqref{co:trace-identities} to $t(gu)t(u)$ gives
+$t(gu^2)=t(g)$.  The quotient by the subgroup generated by squares
+is elementary abelian, and the Galois systems are unramified off
+their level.  This proves the stated factorization.  At coefficient
+$q$ the fresh split weight $1$ pays exactly the division in $H_U$;
+the fresh simple weight $1/2$ pays exactly $e_U$ in opposite parity.
+No unary term survives.  This proves the symbol-reading assertion.
+
+For contraction, apply \eqref{co:inertia-identity} at coefficient
+$1$, so that its left side is a coefficient-$p$ test.  Replace its
+trace arguments by fresh primes.  The first-shell calculation in
+Lemma~\ref{co:operators} gives, in each of the split--split,
+split--simple, and simple--split cases,
+\[
+ c_p(D')A(D'),\qquad D'=(U/p)q',\quad
+ \operatorname{Fr}_{q'}\sim g\phi_p.
+\]
+Here the omitted source is exactly $B_V$ or $H_V$, with
+$V=U\setminus\{p\}$, constructed from the same original modular form
+and the same $\eta,\psi,\delta$;
+no new minimum or normalization is chosen.  To check the integer
+powers explicitly, put $n_U=\lceil\delta+w(U)\rceil$ and let
+$m_U$ be $1$ in matching parity and $0$ otherwise.  After fresh
+tests kill the unary terms, the first shell relative to the
+omitted-source test has multiplier
+\[
+ 2^{1+n_V-n_U-m_U+m_V}c_p(D').
+\]
+For split $p$ the exponent is zero.  For simple $p$ and matching
+$U$ it is also zero.  For simple $p$ and opposite $U$ it is one;
+this is precisely the simple--simple case.  The character change
+$\psi_U/\psi_V=(p/\cdot)$ is already the Legendre factor in this
+shell calculation.  The phase is a unit and disappears on reduction.
+Let $q_0',q_1'$ realize $g\phi_p,g\tau_p\phi_p$, respectively,
+and put $D_i'=Vq_i'$.  Applying inertia changes the Legendre term
+in \eqref{co:cp} by a sign, so $c_p(D_1')=c_p(D_0')+1$.
+Although these Frobenius elements may first be specified in the full
+current elementary extension, their restrictions to the omitted-source
+extension agree: inertia
+at either place above $p$ is killed in that extension.  Its original
+characters are unramified at $p$, and its only new character factors
+come from $V$.  Thus it is unramified at the whole rational prime
+$p$, giving $A(D_1')=A(D_0')$.  If $\Delta_{\tau_p}A(Uq)$ denotes
+the difference of the original tests with Frobenius labels $g\tau_p$
+and $g$, the reduced local identity is now explicitly
+\[
+ \Delta_{\tau_p}A(Uq)
+   =c_p(D_1')A(D_1')+c_p(D_0')A(D_0')=A(D_0'),
+\]
+which is \eqref{co:contraction}.  When both vertices are simple the product
+Frobenius is split and the first-shell valuation has a positive
+excess; its reduction is zero.  The entire-prime assertion follows
+from the actual support of the omitted-$p$ coefficient form, rather
+than from any convention about a chosen orientation.  Isolation
+is the same symbol-reading test with $t(1)=0$.
+
+For the final assertion first use two copies of a single simple
+Frobenius trace on matching $H_U$.  Put
+$b=B_U[1]$, $\epsilon=\chi_{-4}(q)$, and
+$\kappa=(\psi_U/\eta)(q)$.  Substitution of
+\eqref{co:recurrence} and of the unary character gives the two terms
+in the coefficient-$1$ test of $T_q^2$, modulo $\mathfrak m$:
+\begin{align*}
+ H_U[q^2]&=\frac{b\eta(q)}2
+        (a_q-\kappa\epsilon-q\epsilon^i),\\
+ q(\langle q\rangle H_U)[1]&=\frac{b\eta(q)}2
+        q(\kappa-\epsilon^{i+1}).
+\end{align*}
+Their sum is
+\[
+ \frac{b\eta(q)}2
+ \left(a_q+\kappa(q-\epsilon)
+                    -q\epsilon^i(1+\epsilon)\right).
+\]
+For $q\equiv3\pmod4$ the parenthesized half has residue $a_q/2$;
+for $q\equiv1\pmod4$ it has residue $a_q/2+1$.
+In both cases it is $\#E(\mathbb F_q)/2$ modulo $2$.
+Since $b$ reduces to $A(U)$, the squared trace has the required
+coefficient.  Fresh-prime testing replaces its two copies by
+distinct primes.  To pass to any pair with the specified labels,
+vary one prime at a time.  After that simple prime is omitted the
+opposite-parity symbol depends only on rational elementary labels;
+simple--simple changes vanish by the proved contraction rule.
+Thus only the stated support and split-vertex labels matter.
+\end{proof}
diff --git a/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/build/sections/cyclotomic.tex b/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/build/sections/cyclotomic.tex
new file mode 100644
index 0000000000000000000000000000000000000000..a8765b85855763d84971b10275891bad7d4b4ff2
--- /dev/null
+++ b/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/build/sections/cyclotomic.tex
@@ -0,0 +1,788 @@
+\section{Cyclotomic interpolation for extensions of trivial residual modules}
+\label{cy:section}
+
+Throughout this section, $E/\mathbb Q$ is fixed, its conductor is $N$, and
+$E[2]$ is an extension of two trivial $\mathbb F _2$-modules.  Write
+$T=T_2E$ and $V=T\otimes_{\mathbb Z _2}\mathbb Q _2$.  The word
+``ordinary'' applied to global cohomology below means that the real-place
+modification has not been made.  It imposes no ordinary-reduction hypothesis
+at~$2$.
+
+The discriminants considered here are products of distinct signed primes
+$q^*=(-1/q)q$, with $q\nmid 2N$.  Fix their sign and local squareclasses at
+$2N$.  For such a discriminant $h$, put
+\[
+ t_q=\dim_{\mathbb F _2}E[2]^{\operatorname{Fr}_q},\qquad
+ 2w(h)=\sum_{q\mid h}t_q,\qquad
+ \mathcal L(h)=\frac{\sqrt{|h|}\,L(E^{(h)},1)}{\Omega_E^{\operatorname{sgn}(h)}}.
+\]
+Here fixed nonzero periods in the two period lines have been chosen once.
+Changing these choices changes the bounds below by a constant depending
+only on~$E$.  Valuations of algebraic central values mean their $2$-adic
+valuations, with $v_2(2)=1$.  When $s_2(E^{(h)})=0$, the group
+$\operatorname{Sha}(E^{(h)}/\mathbb Q)[2^\infty]$ is finite; set
+\[
+ s(h)=\operatorname{length}_{\mathbb Z _2}
+       \operatorname{Sha}(E^{(h)}/\mathbb Q)[2^\infty],\qquad
+ D(h)=v_2(\mathcal L(h))-2w(h)-s(h)
+\]
+when the central value is nonzero.  By Lemma~\ref{pr:ranks}, $s(h)$ is
+defined at a corank-zero base even before its central value is known to
+be nonzero.  Only the definition of $D(h)$ requires that nonvanishing.
+
+We prove the two assertions needed later.
+
+\begin{theorem}[Cyclotomic lower bound]\label{cy:lower-bound}
+There is a constant $C_E$ such that, for every allowed $h$ with
+$L(E^{(h)},1)\ne0$,
+\[
+ v_2(\mathcal L(h))\ \geq\ 2w(h)+s(h)-C_E.
+\]
+The constant is independent of the number and sizes of the primes dividing
+$h$.
+\end{theorem}
+
+\begin{theorem}[Transfer to the missing vertex]\label{cy:missing-vertex}
+Let $h_0$ be an allowed discriminant with $s_2(E^{(h_0)})=0$.  Suppose that
+for arbitrarily large $b$ there are a finite set $Q$ of distinct primes
+$q\nmid 2Nh_0$ and linear forms
+$\lambda_q:\mathbb F _2^b\longrightarrow\mathbb F _2$ giving
+\[
+ h_x=h_0\prod_{q\in Q}(q^*)^{\lambda_q(x)},\qquad x\in\mathbb F _2^b,
+\]
+all in the fixed local and sign classes.  Suppose that a number $B$,
+independent of the family, satisfies
+\[
+ L(E^{(h_x)},1)\ne0,\qquad D(h_x)\leq B\qquad(x\ne0).
+\]
+Then $L(E^{(h_0)},1)\ne0$ and $D(h_0)\leq B+C_E$.
+Here $C_E$ is independent of $h_0$, $Q$, and $b$; the dimension needed in
+the proof is allowed to depend on the fixed precision being tested.
+\end{theorem}
+
+\subsection{The integral scalar family and two algebraic steps}
+
+Put
+\[
+ \Lambda=\mathbb Z _2[[t]],\qquad
+ \mathcal O=\widehat{\Lambda_{(2)}},\qquad
+ \kappa=\mathcal O/2\mathcal O=\mathbb F _2((t)).
+\]
+The completion defining $\mathcal O$ is $2$-adic.  An element of $\mathcal O$
+has, modulo any $2^M$, a Laurent expansion with a finite negative part.
+There is no evaluation homomorphism $\mathcal O\to\mathbb Q _2$ given by
+$t=0$.  All central evaluations below are made on elements of
+$\Lambda[1/2]$ before extending coefficients to~$\mathcal O$.
+
+For $G=(\mathbb Z/2\mathbb Z)^b$, index its characters by
+$x\in\mathbb F _2^b$, and denote character evaluation by a subscript~$x$.
+
+The arithmetic construction below will produce a scalar family
+$u\in\Lambda[1/2][G]$ with one uniform denominator:
+\[
+ 2^{C_{\mathrm{int}}}u\in\Lambda[G].
+\]
+At a constituent with $s_2(E^{(h_x)})=0$, its central value $u_x(0)$
+will vanish exactly when $L(E^{(h_x)},1)$ does; in the nonvanishing case,
+\[
+ \bigl|v_2(u_x(0))-D(h_x)\bigr|\leq C_{\mathrm{loc}}.
+\]
+Both constants will be independent of the number and sizes of all varying
+primes, including those of the base $h_0$.  Integrality gives the lower
+bound, while a congruence between the constant coefficients at $0$ and at
+some nonzero address gives the missing-vertex theorem.
+
+Two different arguments establish the uniform denominator.  Rational
+divisibility will show $u_x\in\Lambda[1/2]$ character by character.
+A fixed integral Kato construction will instead give
+$2^{C_{\mathrm{int}}}u\in\mathcal O[G]$ before taking characters.
+The first lemma combines these assertions without a loss depending on
+$|G|$; the second supplies the congruence used at the end.
+
+\begin{lemma}[Integral Fourier intersection]\label{cy:fourier}
+Suppose $u\in\mathcal O[1/2][G]$, all $u_x$ belong to $\Lambda[1/2]$, and
+$2^C u\in\mathcal O[G]$.  Then $2^C u\in\Lambda[G]$.
+\end{lemma}
+\begin{proof}
+Fourier inversion in characteristic zero expresses every group-basis
+coefficient of $u$ as a sum of the $u_x$ divided by $|G|$.  Thus it belongs
+to $\Lambda[1/2]$.  The corresponding coefficient of $2^C u$ also belongs
+to~$\mathcal O$.  For a power series with bounded $2$-denominator,
+nonnegative $\mathcal O$-valuation means that every coefficient is integral.
+Consequently $\mathcal O\cap\Lambda[1/2]=\Lambda$, which proves the assertion
+with the original exponent~$C$.
+\end{proof}
+
+The common-zero parity step below is the characteristic-two case of the
+Chevalley--Warning principle; see
+\cite[Section~1, equation~(1)]{HeathBrownChevalleyWarning}.
+We retain the elementary proof of this step.
+
+\begin{lemma}[Binary congruences]\label{cy:binary}
+Let $z_1,\ldots,z_r\in\mathbb Z _2[G]$.  Given $M\geq1$, if
+$b>r(2^M-1)$, there is an $x\ne0$ such that
+\[
+ (z_j)_x\equiv(z_j)_0\pmod {2^M}\qquad(1\leq j\leq r).
+\]
+The bound depends on the number of tests and their precision, and not on
+the group-ring coefficients.
+\end{lemma}
+\begin{proof}
+Choose generators of $G$.  Every character-value function $z(x)$ has an
+expansion
+\[
+ z(x)=a_\varnothing+
+       \sum_{\varnothing\ne I\subseteq\{1,\ldots,b\}}
+           2^{|I|}a_I\prod_{i\in I}x_i,
+       \qquad a_I\in\mathbb Z _2.
+\]
+This follows by expanding each sign character as a product of factors
+$1-2x_i$.  The $j$th binary digit of a $2$-adic integer $a$ is
+$\binom a{2^j}\bmod2$, equivalently the coefficient of $Y^{2^j}$ in
+$(1+Y)^a$ over $\mathbb F _2$.  On the Boolean cube, the contribution of
+the monomial $x_I=\prod_{i\in I}x_i$ to this generating function is
+\[
+ 1+x_I\bigl((1+Y)^{2^{|I|}a_I}-1\bigr).
+\]
+Its nonconstant terms have $Y$-degree at least $2^{|I|}$.  In a contribution
+to $Y^{2^j}$, the sum of the quantities $2^{|I|}$ used is at most $2^j$.
+Since $|I|\leq2^{|I|}$, the resulting Boolean polynomial has degree at
+most~$2^j$.  These formulas for $2$-adic exponents follow either by
+continuity from nonnegative integers or in each finite coefficient
+quotient.
+
+Express all $rM$ digit equalities as Boolean equations $P_{ij}(x)=0$.
+Their total degree is at most $r(2^M-1)$.  The indicator of their common
+zero set is $\prod_{i,j}(1+P_{ij})$.  The remaining count is the
+Chevalley--Warning degree argument \cite{Chevalley1935,Warning1935}.
+A Boolean polynomial of degree less than $b$ has even sum over
+$\mathbb F _2^b$: every monomial omits a variable,
+and summing over that variable gives zero in $\mathbb F _2$.  Hence the
+number of common zeros is even.  Since $0$ is a common zero, another one
+exists.
+\end{proof}
+
+\subsection{The positive global complex}
+
+The scalar $u$ will be the coordinate of two cohomology classes in an
+integral determinant line.  One class comes from the real place and the
+other from Kato's construction.  We first construct the complex and show
+that its localization at $(2)$ is free of rank two in degree one.
+
+Let $\Gamma=\operatorname{Gal}(\mathbb Q_\infty/\mathbb Q)$ be the real
+cyclotomic $\mathbb Z _2$-group and identify its generator with $1+t$.
+For a binary family, let $\psi:G_{\mathbb Q}\to G$ be its universal
+quadratic character: evaluation at $x$ is $\chi_{h_x/h_0}$.  Put
+\[
+ R=\Lambda[G],\qquad
+ \mathbb T=T\otimes_{\mathbb Z _2}R,
+\]
+with the fixed twist $\chi_{h_0}$, the universal twist $\psi$, and the
+inverse cyclotomic action.  One may replace $G$ by the actual image of
+$\psi$ and subsequently extend scalars.  In a family in
+Theorem~\ref{cy:missing-vertex}, a nonzero character trivial on that image
+already gives the conclusion directly.
+
+Let $S$ contain $2N$, the prime factors of $h_0$, and all the new primes.
+Use continuous cochains with their completed coefficient topology, and set
+\begin{equation}\label{cy:positive}
+ C_S^+(\mathbb T)=
+ \operatorname{Cone}\bigl(R\Gamma(G_{\mathbb Q,S},\mathbb T)
+              \longrightarrow R\Gamma(\mathbb R,\mathbb T)\bigr)[-1].
+\end{equation}
+The real complex here uses ordinary cochains in nonnegative degrees.
+There are no imposed local conditions at the finite places.
+
+\begin{lemma}[Perfection and the real-place class]\label{cy:perfect}
+The complex \eqref{cy:positive} is perfect over $R$ and computes derived
+coefficient specializations.  There is a class $a\in H^1(C_S^+(\mathbb T))$
+whose rational span is the real-place summand.  Its integral index in
+that summand depends only on $E$ and the fixed real sign.
+\end{lemma}
+\begin{proof}
+Here are the finiteness and coefficient issues in the use of global
+cochains.  For a finite coefficient quotient of $R$, continuous cochains
+are filtered unions of finite products of free coefficient modules.
+Refinement of finite partitions makes these modules flat; over an Artin
+local coefficient ring they are free.  For finite residual coefficients,
+global cohomology is finite in each degree, and restriction to the real
+places is an isomorphism in degrees greater than two.  The latter is the
+real-place clause of global duality.  Thus the residual fiber has bounded
+amplitude and finite-dimensional cohomology.  The same amplitude bound
+holds for every finite coefficient module, by a composition series and
+the cohomology long exact sequence.
+
+This gives a bounded finite free model as follows.  Over the residue
+field split off the acyclic disks, leaving its finitely many cohomology
+spaces.  Lift each invertible disk differential over an Artin coefficient
+quotient and cancel it.  Invertibility of a lifted inverse follows from
+nilpotence of the maximal ideal.  Equivalently one can perform this on a
+free resolution; a minimal free model cannot have an additional term
+whose reduction has zero cohomology, because its residual differential
+is zero.  Hence the remaining ranks are the finite residual ranks and
+are bounded independently of the coefficient quotient.  Minimal models
+identified by a homotopy equivalence over an Artin local ring are
+isomorphic.  Lifting their changes of basis therefore makes these models
+compatible along $R/\mathfrak m^n$.  Taking the inverse limit gives a
+bounded finite free model over~$R$.  The cochain construction with finite
+free coefficients and the same cancellation also proves derived base
+change.  This is a fixed-$S$ argument; it does not assert uniform bounds
+on the number of generators as $S$ varies.
+
+All the characters of the family have the same real sign, so the action
+at the real place is the fixed action on $T\chi_{h_0}$ extended to $R$.
+Choose a nonzero vector in its integral invariant line.  Its connecting
+image in the fiber is $a$.  There are no global invariant vectors in the
+rational elliptic representation, and consequently this connecting map
+is injective rationally.  The choice and its index are fixed.  Notice also
+that $H^1(\mathbb R,\mathbb T)$ is killed by~$2$; thus twice any ordinary
+cohomology class lifts to the positive complex.  These observations use
+compact global duality, including the real-place extension
+\cite[Sections~5.7 and~6.9.1]{Nekovar2006} of the complex formulation
+in \cite[Theorem~6.3.4]{Nekovar2006}.
+\end{proof}
+
+\begin{lemma}[Residual concentration]\label{cy:residual}
+There is an isomorphism in the derived category
+\[
+ C_S^+(\mathbb T)\otimes_R\mathcal O[G]
+       \simeq \mathcal O[G]^2[-1].
+\]
+\end{lemma}
+\begin{proof}
+The ring $\mathcal O[G]$ is local, with residue field $\kappa$ obtained
+by setting $2=0$ and every $g=1$.  Thus the quadratic characters disappear.
+Since $E[2]$ has a filtration with trivial factors, it is enough first to
+compute the positive residual complex for the trivial representation.
+
+Let $K_n$ be the real cyclotomic layer of degree $d_n=2^n$.  The number
+of finite primes of $K_n$ above a fixed $S$ is bounded with $n$: at $2$
+there is total ramification, and the Frobenius of each fixed odd prime
+has nonzero image in $\Gamma$.  The $2$-ranks of the class groups of
+$K_n$ are also bounded.  Indeed the cyclotomic $\mu$-invariant is zero
+for this abelian tower \cite{FerreroWashington1979}; the structure theorem
+and class-group control identify their $2$-ranks, up to bounded kernels
+and cokernels, with those of the coinvariants of a finitely generated
+$\mathbb Z _2$-module.  The same bound holds for $S$-class groups.
+
+The signature map of units in $K_n$ is surjective.  For $n\geq1$, let
+$\zeta$ be a primitive $2^{n+2}$th root and put
+$\epsilon=1+\zeta+\zeta^{-1}$.  If $f_1(X)=X^2-2$ and
+$f_{n+1}(X)=f_n(X^2-2)$, then $f_n$ is the minimal polynomial of
+$\zeta+\zeta^{-1}$ and $f_n(-1)=-1$.  Hence
+$N_{K_n/\mathbb Q}(\epsilon)=-1$.  Its signature vector has odd
+augmentation.  The embeddings form a regular set for the cyclic
+$2$-group $\operatorname{Gal}(K_n/\mathbb Q)$, whose group algebra over
+$\mathbb F _2$ is local.  A vector with odd augmentation is a unit in
+this group algebra, so its translates span every signature.  For $n=0$
+the assertion follows from the unit~$-1$.
+
+We now use Kummer theory to isolate the one dimension that survives
+after inverting $t$.
+Writing $s_n$ for the number of finite support places and $c_n$ for the
+$2$-rank of the $S$-class group, one has
+\[
+ \dim H^1(G_{K_n,S},\mu_2)=d_n+s_n+c_n.
+\]
+The real degree-one restriction is surjective by the signature result.
+In degree two, the Kummer sequence has class-group term of dimension
+$c_n$ and Brauer term.  The Brauer invariants at the real and finite
+support places have their single sum relation.  Since a finite support
+place above $2$ is present, restriction of the Brauer term to the real
+places is surjective and has kernel of dimension $s_n-1$.  Thus the
+degree-two kernel of restriction to infinity is bounded.  Restriction
+is an isomorphism in higher degrees by global duality.  The degree-zero
+restriction is injective.  The cohomology of the positive fiber is
+therefore bounded in every degree except one; its degree-one dimension
+is $d_n+O_S(1)$.
+
+Apply Lemma~\ref{cy:perfect} modulo $2$.  Derived specialization to the
+$n$th layer is specialization modulo $t^{2^n}$.  For a perfect complex
+over $\mathbb F _2[[t]]$, the universal-coefficient exact sequence shows
+that a rank-$r$ cohomology module contributes $r2^n+O(1)$ to these
+dimensions; the torsion terms contribute only $O(1)$.  It follows that
+after inverting $t$ the trivial-constituent fiber has dimension one in
+degree one and vanishes in every other degree.  Extensions preserve
+this concentration, so the fiber for $E[2]$ has dimension two in degree
+one.  A minimal finite free complex over the local ring $\mathcal O[G]$
+has zero residual differentials.  The concentration just proved forces
+all its terms except the rank-two term in degree one to vanish.
+No denominator is introduced by this exact freeness statement.
+\end{proof}
+
+\subsection{Fixed-form zeta classes and the integral coefficient map}
+
+We need a global class whose denominator stays bounded as the tame
+support grows.  For this purpose every auxiliary trace will land in
+the lattice of one fixed modular form.  Let $f$ be the fixed weight-two
+form of $E$, and choose its fixed integral cohomological lattice, with
+the Tate twist making its
+rational representation~$V$.  Kato's classes (8.1.2)--(8.1.3), for a fixed
+symbol $\xi\in\mathrm{SL}_2(\mathbb Z)$, exist for arbitrary $m$ and any
+$S$ containing the prime factors of $mN$.  The trace in Section~8.9 is
+a homomorphism into the lattice
+\[
+ \mathbb Z _2[\operatorname{Gal}(\mathbb Q(\mu_m)/\mathbb Q)]
+       \otimes V_{2,\mathbb Z _2}(Y_1(N))(1).
+\]
+Section~8.11 then projects to the lattice of the fixed form.
+Proposition~8.12 supplies its norm relations.  These assertions hold at
+the coefficient prime~$2$ \cite[Sections~5.1, 8.1, 8.9--8.12]{Kato2004}.
+In particular, the auxiliary modular-curve level used to define a trace
+does not become the level of the projected newform.
+
+Choose an isomorphism of the resulting fixed rational representation with
+$V$.  A single power $2^{c_E}$ sends its integral lattice into~$T$.
+Choose fixed symbols with nonzero projections on the required period
+lines; their existence and the construction with both signs are made
+explicit in \cite[Sections~13.6 and 13.9]{Kato2004}.  Taking a fixed linear
+combination, if needed, costs another fixed denominator.
+
+Here is the coefficient map in this application.  Let $m$ be odd and
+divisible by the conductors of all the tame characters in the family,
+including the odd part of $\chi_{h_0}$.  At a sufficiently large
+cyclotomic level write $A_{m,n}=\operatorname{Gal}(\mathbb Q(\mu_{m2^n})/
+\mathbb Q)$.  Shapiro identifies the cohomology of the cyclotomic field
+with global cohomology with induced group-ring coefficients.  Push the
+class along the ring homomorphism
+\begin{equation}\label{cy:coefficient-map}
+ \mathbb Z _2[A_{m,n}]\longrightarrow
+       \mathbb Z _2[\Gamma_n\times G],\qquad
+ [\sigma]\longmapsto
+       \chi_{h_0}(\sigma)[\sigma|_{\Gamma_n}][\psi(\sigma)].
+\end{equation}
+Use inverse actions on both sides if this is required by the chosen
+Shapiro convention.  This is a homomorphism of integral coefficient
+modules and acts on actual cohomology classes.  It requires neither
+division by $|G|$ nor surjectivity of a cohomological specialization map.
+The prime $2$ belongs to the fixed omitted support, so Proposition~8.12
+gives exact norm compatibility as $n$ increases.  The pushed classes
+therefore give an integral smoothed Iwasawa class with coefficients in
+$\mathbb T$, up to the single fixed power~$2^{c_E}$.
+
+We next remove smoothing.  In weight two and with $r=r'=1$, its factors
+are
+\begin{equation}\label{cy:smoothing}
+ (c^2-c\sigma_c)(d^2-d\sigma_d).
+\end{equation}
+This is the specialization of
+\cite[Theorem~6.6 and (4.2.4)]{Kato2004}.  Choose a fixed
+$e\geq\max(2,v_2(N))$, increasing it to accommodate the finitely many
+fixed symbol levels.  The Chinese remainder theorem permits $c,d>1$,
+prime to $6Nm$, such that
+\[
+ c,d\equiv1\pmod{3N_{\mathrm{odd}}m},\qquad
+ c,d\equiv1+2^e\pmod{2^{e+1}}.
+\]
+Their tame actions in \eqref{cy:coefficient-map} are trivial.  If
+$\sigma_c$ acts in $\Gamma$ as $(1+t)^{\kappa(c)}$, then
+$v_2(\kappa(c))=e-2$.  Modulo the maximal ideal of $\mathcal O[G]$ its
+smoothing factor is
+\[
+ 1-(1+t)^{\kappa(c)}\ne0\quad\hbox{in }\mathbb F _2((t)).
+\]
+It is therefore a unit in $\mathcal O[G]$.  Its central value is
+$c(c-1)$, of valuation exactly~$e$.  The same statements hold for $d$
+and for inverse-translate conventions.  Thus unsmoothing introduces
+neither a $2$-adic denominator at $(2)$ nor a growing central denominator.
+The integers $c,d$ vary with $m$; the congruences fix the needed valuations,
+not their exact cyclotomic group elements.
+
+Denote the resulting rational ordinary class by $z_S^{(0)}$.  Theorem~9.7 of
+\cite{Kato2004} applies to $k=2,r=r'=1$.  In the character constituent
+$x$, its central reciprocity formula reads
+\begin{equation}\label{cy:reciprocity}
+ v_2\bigl(\exp_2^*z_{S,x}^{(0)}(0)\bigr)
+ =v_2(\mathcal L(h_x))+
+  \sum_{\substack{q\in Q\,,\ q\nmid h_x}}
+       v_2\bigl(\#E^{(h_x)}(\mathbb F _q)\bigr)+O_E(1)
+\end{equation}
+when the value is nonzero, and a zero central value gives a zero dual
+exponential.  Fixed omitted primes are included in the error.
+To check the normalization, the character sum in
+\eqref{cy:coefficient-map} is the unnormalized sum, as in Kato's trace.
+Its tame quadratic Gauss sum supplies $\sqrt{|h_x|}$ up to a root of
+unity.  Tame conductors and their Gauss sums are $2$-adic units.  The
+period vector is the fixed symbol vector of $f$.  More explicitly,
+in the required sign line write the period represented by
+$\delta(f,1,\xi)^{\pm}$ as $c_\xi^{\pm}\Omega_E^{\pm}$.
+The nonzero scalar $c_\xi^{\pm}$ is fixed: the character-sum formula
+of \cite[Theorem~6.6]{Kato2004} uses this same $\delta(f,1,\xi)^{\pm}$
+for every tame character.  The tame character enters through its
+$L$-value and Gauss sum, not through a newly chosen period vector.
+The comparison with the minimal differential of $E^{(h_x)}$ at $2$ has bounded valuation,
+because there are only finitely many local twisting classes there.
+At a good unused prime the omitted factor is
+\[
+ 1-\frac{a_q(E)\chi_{h_x}(\operatorname{Fr}_q)}q+\frac1q
+       =\frac{\#E^{(h_x)}(\mathbb F _q)}q;
+\]
+at a good ramified prime the local $L$-factor is~$1$.  These account for
+every varying factor in \eqref{cy:reciprocity}.
+
+\subsection{Rational divisibility and the local Euler corrections}
+
+The fixed-form construction controls the denominator at $(2)$.  We now
+prove the complementary assertion: the determinant coordinate has no
+pole at any other height-one prime.  This step is characterwise and uses
+only rational divisibility.  Write $A=\Lambda[1/2]$.  Theorem~12.5(3) of
+\cite{Kato2004} is a height-one divisibility over~$A$; its possible local
+$H^2$ correction comes from Galois cohomology over $\mathbb Q_2$.
+The global groups there use $j_*$ outside that prime.  The stronger integral assertion
+12.5(4) is not used.  In the non-CM case, the large-image input in Kato's
+proof follows from the open image theorem \cite{Serre1972}; see
+\cite[Remark~12.8, (12.8.2), Theorem~13.4(v), and Section~13.13]{Kato2004}.
+This is not an additional residual-image hypothesis in Theorem~12.5(3).
+For CM the required rational statement at every prime, including~$2$, is supplied by
+\cite[Theorem~2.6 and Remark~2.7]{BurungaleTian2026}.  Neither statement
+asserts uniform integrality in a quadratic group ring.
+
+Fix a character constituent $h$ and write $V_h=V\chi_h$.  Let $C_{j_*}$
+be the rational Iwasawa complex in Kato's formulation and $C_S$ the
+rational full global complex.  At an odd prime put
+\[
+ W_q=H^1(I_q,V_h).
+\]
+The localization triangle has local cone
+\begin{equation}\label{cy:local-block}
+ Q_q=R\Gamma\bigl(G_{\mathbb F _q},
+                   W_q\otimes A(\Gamma^\iota)\bigr)[-1]
+   =\left[W_q\otimes A
+       \xrightarrow{\ 1-\gamma_q\operatorname{Fr}_{W_q}\ }
+             W_q\otimes A\right]
+\end{equation}
+in degrees one and two.  Here $\gamma_q$ denotes the scalar cyclotomic
+Frobenius in the chosen convention.  Its exponent is nonzero.  Consequently
+$D_q=\det(1-\gamma_q\operatorname{Fr}_{W_q})$ is a nonzero element of~$A$,
+and the map in \eqref{cy:local-block} is injective.  This remains true
+after localization at a height-one prime dividing $D_q$: localization
+does not mean reduction modulo that prime.  Thus
+\[
+ H^1(Q_q)=0,\qquad
+ \operatorname{length}_{A_{\mathfrak p}}H^2(Q_q)_{\mathfrak p}
+       =v_{\mathfrak p}(D_q).
+\]
+The long exact sequence gives
+\begin{equation}\label{cy:local-sequence}
+ \begin{gathered}
+ H^1(C_{j_*})\simeq H^1(C_S),\\
+ 0\longrightarrow H^2(C_{j_*})\longrightarrow H^2(C_S)
+       \longrightarrow\bigoplus_{q\in S\setminus\{2\}}H^2(Q_q)
+       \longrightarrow H^3(C_{j_*}).
+ \end{gathered}
+\end{equation}
+In particular, the length of $H^2(C_S)$ is at most the length of
+$H^2(C_{j_*})$ plus the sum of the local lengths.  No assertion that
+$H^3(C_{j_*})$ vanishes is required here.
+
+At a good unused prime, $W_q=V_h(-1)$, so, on including the quadratic
+Frobenius sign in $Z$,
+\begin{equation}\label{cy:good-euler}
+ D_q=P_q(Z),\qquad P_q(Z)=1-\frac{a_q(E)}q Z+\frac1q Z^2.
+\end{equation}
+At a good active prime, inertia acts by $-1$ on $V_h$.  Its rational
+inertia cohomology is zero, so $W_q=0$ and $D_q=1$.
+
+At fixed bad primes the general expression is $H^1(I_q,V_h)$ in
+\eqref{cy:local-block}.  Replacing it indiscriminately by $V_h^{I_q}(-1)$
+would be incorrect.  For example, at split multiplicative reduction,
+\[
+ V_h^{I_q}=\mathbb Q _2(1),\qquad
+ H^1(I_q,V_h)=\mathbb Q _2(-1),
+\]
+and the correct determinant is $1-\gamma_q/q$, with nonzero central
+value.  The incorrect replacement would give $1-\gamma_q$.
+In general $D_q(0)\ne0$ follows from local duality:
+$H^2(\mathbb Q _q,V_h)=H^0(\mathbb Q _q,V_h)^*=0$, since elliptic torsion
+over a local field is finite.  The fixed bad-prime corrections therefore
+have nonzero central values for every reduction type.
+
+Let $z_{\mathrm{BK},h}$ denote a generator of the rational Kato zeta
+line in the real cyclotomic component with the fixed sign.  Compare it
+with the fixed-form class only after taking the character constituent.
+In the chosen Euler convention the comparison is
+\begin{equation}\label{cy:rational-comparison}
+ z_{S,h}^{(0)}
+  =c_h\epsilon_h(t)E_h^{\mathrm{fix}}(t)
+    \prod_{\substack{q\in Q\,,\ q\nmid h}}
+       P_q\bigl(\chi_h(\operatorname{Fr}_q)\gamma_q\bigr)
+       z_{\mathrm{BK},h},
+\end{equation}
+where $c_h\in\mathbb Q_2^\times$ and $\epsilon_h$ is a group-like
+unit of~$A$.  The factor $E_h^{\mathrm{fix}}$ consists of the omitted
+local $L$-polynomials away from $2$ at primes of $N$ and the fixed
+auxiliary support.  It ranges over a finite collection, because the
+local twisting classes there are fixed.  The primes of $h_0$ are good
+ramified primes and contribute~$1$, regardless of their number.
+These factors have nonzero central values: they are good-prime point
+counts divided by the prime, multiplicative factors $1\mp1/q$, or~$1$.
+They are not being identified with the local cohomology determinants
+$D_q$ at bad primes; any discrepancy there will be cleared by a fixed
+multiplier below.
+
+Here is the derivation of \eqref{cy:rational-comparison}.  Both global
+Iwasawa $H^1$ modules have rank one.  The reciprocity formulas
+\cite[Theorems~6.6, 9.7, 12.5(1), and Lemma~13.10(2)]{Kato2004}
+compare the same fixed period vector and the relevant imprimitive
+$L$-values.  Their quotient supplies the displayed omitted Euler
+factors; a good ramified prime contributes~$1$.  Coprime Gauss-sum
+factorization and the fixed period comparison supply $c_h$ and
+$\epsilon_h$; restricting conductor exponents to either fixed parity
+removes any factor $\chi_h(2)^n$.  Infinitely many of these tests have
+nonzero central value by cyclotomic nonvanishing, as used in
+\cite[Theorem~13.5(2)]{Kato2004}.  After clearing the finitely many
+denominators of the scalar ratio, the difference is a power series
+vanishing at infinitely many finite-order characters.  Weierstrass
+preparation forces that difference to be zero.  This proves the generic
+comparison without identifying the integral lattices of varying
+twisted newforms.  The scalar $c_h$ need not have a uniformly bounded
+$2$-valuation: every such scalar is a unit of $A$ and does not affect
+height-one divisibility over~$A$.  The uniform integral and central bounds instead concern
+the actual fixed-form class and the fixed period vector in
+\eqref{cy:reciprocity}.
+
+Equations \eqref{cy:local-sequence}--\eqref{cy:good-euler} now show exactly
+how the imprimitive factors enter divisibility.  At any height-one
+$\mathfrak p$ of $A$, enlarging the global complex increases its
+degree-two length by at most $\sum_qv_{\mathfrak p}(D_q)$, whereas
+multiplying the rank-one zeta generator by $\prod_qD_q$ adds exactly
+that quantity to its index.  To cover Kato's local correction at $2$,
+multiply the class also by a fixed element $F_2\in\Lambda$ divisible over $A$ by the
+characteristic ideal of the relevant local $H^2_{\mathrm{Iw}}$.
+It may be chosen with $F_2(0)\ne0$.  Indeed derived local control gives
+\[
+ H^2_{\mathrm{Iw}}(\mathbb Q _2,T\chi_h)_\Gamma\otimes\mathbb Q _2
+       =H^2(\mathbb Q _2,V_h)=0,
+\]
+so the characteristic ideal has no factor $t$.  Local twisting classes
+at $2$ are fixed, and one element covers their finite set.
+
+Likewise any additional fixed Euler factors in the rational comparison
+can be cleared by a fixed product with nonzero central value, using the
+actual blocks \eqref{cy:local-block}.  Denote the total fixed multiplier
+by~$F_E$.  Multiplying by it only changes the fixed constants in
+\eqref{cy:reciprocity} and the integrality bound.  Set
+$z_S=F_Ez_S^{(0)}$.  For this corrected class we have at every
+height-one $\mathfrak p$ of~$A$
+\begin{equation}\label{cy:divisibility}
+ \operatorname{length}_{A_{\mathfrak p}}H^2(C_S)_{\mathfrak p}
+ \leq
+ \operatorname{length}_{A_{\mathfrak p}}
+       \bigl(H^1(C_S)/Az_{S,h}\bigr)_{\mathfrak p}.
+\end{equation}
+
+We finish by checking that the chosen Euler convention can be imposed
+integrally before character evaluation.  If the original coefficient
+map uses inverse Frobenius, the varying unused factors change as follows.
+In $\mathcal O[G]$, both $P_q(Z)$ and $Z^2P_q(Z^{-1})$ are units: their
+reductions are nonzero polynomials evaluated at the nonconstant
+cyclotomic Frobenius.  The identity
+\[
+ P_q(Z)-Z^2P_q(Z^{-1})=\frac{q-1}{q}(1-Z^2)
+\]
+shows that their ratio $r_q$ is $1$ modulo~$2$.  If $g_q$ is the universal
+inertia element, then
+\begin{equation}\label{cy:euler-convention}
+ U_q=1+\frac{r_q-1}{2}(1+g_q)\in\mathcal O[G]^\times
+\end{equation}
+is integral.  At an active character it is $1$; at an unused character
+it is $r_q$.  Its unused central value is $1$, since $Z=\pm1$ there.
+Thus inverse Frobenius conventions can be reconciled before character
+evaluation, without a denominator for each prime.  Products of these
+units have the same property.  The resulting rational character
+coordinates have the Euler convention in \eqref{cy:divisibility}.
+We use this convention for $z_S$ from now on; these units alter neither
+its uniform integrality bound nor its central reciprocity values.
+
+\subsection{The determinant and its central specialization}
+
+Choose a basis $\mathbf e$ of the invertible $R$-module
+$\det_R^{-1}C_S^+(\mathbb T)$; $R$ is local, so such a basis exists.
+Lift $2z_S$ to positive cohomology and take its wedge with $a$.
+The factor $2$ is fixed, and two lifts differ by a real-place class,
+which does not change this wedge.  In generic cohomology, the inverse
+determinant identifies with the exterior square of the rank-two
+degree-one space.  Define its scalar coordinate $u$ by
+\[
+ a\wedge\widetilde{2z_S}=u\mathbf e.
+\]
+Here $z_S$ already includes the fixed multiplier $F_E$ above.
+The real-place summand introduces no height-one torsion over~$A$.  Thus
+\eqref{cy:divisibility} says that $u_x$ has no pole at any height-one
+prime of $\Lambda$ except possibly $(2)$.  Since $\Lambda$ is normal,
+\begin{equation}\label{cy:rational-membership}
+ u_x\in\Lambda[1/2]\qquad\hbox{for every }x.
+\end{equation}
+On the other hand, the fixed integral trace,
+\eqref{cy:coefficient-map}, unsmoothing over $\mathcal O[G]$, and
+Lemma~\ref{cy:residual} give
+\[
+ 2^{C_{\mathrm{int}}}u\in\mathcal O[G]
+\]
+with $C_{\mathrm{int}}$ independent of $S$, $h_0$, $G$, and~$b$.
+The determinant identification in a free complex introduces no further
+index.  Applying Lemma~\ref{cy:fourier} yields the crucial uniform statement
+\begin{equation}\label{cy:uniform-integrality}
+                  2^{C_{\mathrm{int}}}u\in\Lambda[G].
+\end{equation}
+
+\begin{proposition}[Central determinant formula]\label{cy:central}
+There is a constant $C_{\mathrm{loc}}$, depending only on the fixed curve
+and local classes, with the following property.  If $s_2(E^{(h_x)})=0$,
+then $u_x(0)=0$ when $L(E^{(h_x)},1)=0$.  Otherwise
+\begin{equation}\label{cy:central-formula}
+ \bigl|v_2(u_x(0))-D(h_x)\bigr|\leq C_{\mathrm{loc}}.
+\end{equation}
+The rational positive complex at $t=0$ has cohomology only in degree one,
+of dimension two, so these are specializations of the same determinant.
+\end{proposition}
+\begin{proof}
+Abbreviate $T_h=T_2E^{(h_x)}$ and $A_h=E^{(h_x)}[2^\infty]$.
+Use the integral Kummer subgroup
+\[
+ K_v=\operatorname{im}\bigl(E^{(h_x)}(\mathbb Q _v)^{\wedge}_2
+                              \longrightarrow H^1(\mathbb Q _v,T_h)\bigr)
+\]
+at every finite place.  For $v\ne2$ it is all of $H^1(\mathbb Q _v,T_h)$.
+One way to see this integrally is the inverse-limit Kummer exact sequence:
+its quotient is $T_2H^1(\mathbb Q _v,E^{(h_x)})$, which is zero because
+local Tate duality identifies the $2$-primary local Weil--Ch\^atelet group
+with the dual of the finite $2$-completion of the local point group.
+At $2$ the quotient
+\[
+ L_2=H^1(\mathbb Q _2,T_h)/K_2
+\]
+is a free rank-one $\mathbb Z _2$-module.  The same Kummer sequence and
+local Tate duality identify it with the $\mathbb Z _2$-dual of the free
+part of $E^{(h_x)}(\mathbb Q _2)^{\wedge}_2$.
+
+The dual exponential identifies $L_2\otimes\mathbb Q _2$ with the
+one-dimensional de Rham target.  Integrally its index depends only on
+the local curve and the chosen differential.  This follows from the
+compatibility of the Kummer map and local Tate duality with the formal
+elliptic logarithm: on a sufficiently small open formal subgroup that
+logarithm is an isomorphism, and its adjoint is the dual exponential.
+Every elliptic curve over $\mathbb Q _2$ has such a formal subgroup,
+irrespective of reduction type.  Since only finitely many local twists
+occur, the indices are uniformly bounded.
+
+The assumption on the Selmer corank implies that the compact finite
+Selmer group is torsion and that the discrete finite Selmer group has
+length $s(h_x)$.  Rational Poitou--Tate consequently identifies ordinary
+global $H^1$ with the singular line at $2$, and gives ordinary global
+$H^2=0$ rationally.  Adding the real invariant line gives the claimed
+dimension two in degree one.  This equals the generic rank, so derived
+base change of the perfect complex specializes its determinant without
+a change of rational cohomological rank.
+
+The integral calculation explains the two subtractions in $D(h_x)$.
+The index of global localization will cancel; the remaining local terms
+will cancel the unused Euler factors and contribute $2w(h_x)$ at the
+active primes.  Let $J$ be the image of the free part of global
+$H^1(G_{\mathbb Q,S},T_h)$ in $L_2$, and put
+$i=\operatorname{length}(L_2/J)$.  The compact/discrete Poitou--Tate
+sequence for the mutually orthogonal Kummer conditions contains
+\[
+\begin{aligned}
+ 0\longrightarrow L_2/J
+ &\longrightarrow \operatorname{Sel}_{2^\infty}(E^{(h_x)}/\mathbb Q)^\vee
+ \longrightarrow H^2(G_{\mathbb Q,S},T_h)\\
+ &\longrightarrow \bigoplus_{q\in S}H^2(\mathbb Q _q,T_h)
+ \longrightarrow H^0(\mathbb Q,A_h)^\vee\longrightarrow0,
+\end{aligned}
+\]
+with the usual real Tate-cohomology terms inserted.  Those terms are
+killed by $2$ and have dimensions bounded by the fixed dimension of~$T$.
+Taking lengths and using local duality gives
+\begin{equation}\label{cy:pt-length}
+ \operatorname{length}H^2(G_{\mathbb Q,S},T_h)
+   =s(h_x)-i+\sum_{q\in S}\operatorname{length}H^0(\mathbb Q _q,A_h)
+      +O_E(1).
+\end{equation}
+This sequence is the cohomology consequence of integral
+Poitou--Tate duality with the finite local conditions; see
+\cite[Theorem~6.3.4 and Section~6.9.1]{Nekovar2006}, with the real
+Tate-cochain convention of \cite[Section~5.7.1.6]{Nekovar2006}.
+No unramified condition replaces
+the Kummer condition at a bad prime.  Global torsion is bounded by its
+injection into $E^{(h_x)}(\mathbb Q _2)[2^\infty]$, whose order is bounded
+by the fixed local twisting classes.  The finite terms at $2$ and at
+fixed bad primes are bounded for the same reason.  Passing between
+ordinary and positive cohomology changes only these bounded real and
+global torsion terms.
+
+Suppose first that the central value is nonzero.  The coordinate of
+$z_{S,x}(0)$ in the free global line has valuation
+\[
+ v_2(\exp_2^*z_{S,x}(0))-i+O_E(1).
+\]
+For the inverse determinant, torsion in degree two subtracts its length
+and torsion in degree one adds its length.  This sign can be checked on
+the complex $[\mathbb Z _2\xrightarrow{2^a}\mathbb Z _2]$ in degrees one
+and two: its inverse determinant lattice maps to $2^a\mathbb Z _2$ in
+the rational determinant.  A vector's coordinate thus loses~$a$.
+Using \eqref{cy:pt-length}, the two appearances of $i$ cancel, leaving
+\[
+ v_2(u_x(0))=
+ v_2(\exp_2^*z_{S,x}(0))-s(h_x)
+       -\sum_{q\in S}\operatorname{length}H^0(\mathbb Q _q,A_h)+O_E(1).
+\]
+
+At an active good prime, quadratic inertia acts by $-1$ on the Tate
+module.  Its invariants on $A_h$ are therefore exactly $E[2]$, and taking
+Frobenius invariants gives length~$t_q$.  At an unused good prime, reduction
+identifies the local $2$-primary torsion with
+$E^{(h_x)}(\mathbb F _q)[2^\infty]$, of length
+$v_2(\#E^{(h_x)}(\mathbb F _q))$.  These unused terms cancel exactly
+those in \eqref{cy:reciprocity}.  The active terms, including the primes
+of $h_0$, sum to $2w(h_x)$.  All other terms have already been bounded
+independently of the growing support.  This proves
+\eqref{cy:central-formula}.
+
+If the central value is zero, reciprocity makes the dual exponential
+zero.  The rational ordinary specialization then satisfies the finite
+condition at $2$, and it satisfies the finite conditions elsewhere
+automatically.  It lies in the rational finite Selmer group, which is
+zero.  Its wedge with the real-place class is therefore zero.  The
+constant-rank determinant specialization proves $u_x(0)=0$.
+\end{proof}
+
+\subsection{Proofs of the interpolation theorems}
+
+\begin{proof}[Proof of Theorem~\ref{cy:lower-bound}]
+Use the preceding construction for the single discriminant~$h$.
+The forward analytic rank-zero theorem gives $s_2(E^{(h)})=0$ and
+finiteness of its Shafarevich--Tate group.  By
+\eqref{cy:uniform-integrality}, $v_2(u_h(0))\geq-C_{\mathrm{int}}$.
+Proposition~\ref{cy:central} then gives
+$D(h)\geq-C_{\mathrm{int}}-C_{\mathrm{loc}}$.  Every constant in the
+construction came from the fixed form, a fixed real period line, or
+the finite set of local twists at $2N$.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{cy:missing-vertex}]
+At every nonzero address the forward rank-zero theorem gives Selmer
+corank zero.  Proposition~\ref{cy:central} gives
+\[
+ v_2(u_x(0))\leq B+C_{\mathrm{loc}}\qquad(x\ne0).
+\]
+Choose an integer $M$ such that
+$M-C_{\mathrm{int}}>B+C_{\mathrm{loc}}$.  By
+\eqref{cy:uniform-integrality}, the constant coefficient of
+$2^{C_{\mathrm{int}}}u$ belongs to $\mathbb Z _2[G]$.  For sufficiently
+large $b$, Lemma~\ref{cy:binary} gives an $x\ne0$ such that
+\[
+ v_2(u_x(0)-u_0(0))\geq M-C_{\mathrm{int}}.
+\]
+It follows that $u_0(0)$ is nonzero and
+$v_2(u_0(0))\leq B+C_{\mathrm{loc}}$: either failure would contradict
+the displayed upper bound at this $x$.  The corank-zero hypothesis
+at the base and Proposition~\ref{cy:central} imply
+$L(E^{(h_0)},1)\ne0$ and
+\[
+ D(h_0)\leq B+2C_{\mathrm{loc}}.
+\]
+Taking $C_E$ large enough proves both theorems with a common constant.
+In particular no loss proportional to $b$, $|Q|$, or the prime count of
+$h_0$ has entered this argument.
+\end{proof}
diff --git a/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/build/sections/graphs.tex b/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/build/sections/graphs.tex
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+\section{Arithmetic graphs and simultaneous addressing}\label{gr:section}
+
+Fix $b\ge1$.  An \emph{address} is a point $x\in\mathbf F_2^b$;
+a vertex with activation $\lambda\in(\mathbf F_2^b)^*$ is present at $x$
+when $\lambda(x)=1$.  We will adjoin fresh primes with nonzero activations
+to a given collection of primes.  The goal is to make one or two binary
+coefficient tests equal to $1$ at every $x\ne0$, while retaining all the
+original primes and their residue data.  No new prime is present at $0$.
+The hypothesis is that each test, separately at each nonzero address,
+is a nonzero polynomial in the available residue bits.
+
+The arithmetic identities are those of
+Proposition~\ref{co:symbol-interface}, conditional on its coefficient
+lower bounds.  We first describe and realize their graph data, then state
+the simultaneous-testing result.  Its proof uses contraction operators
+and two finite networks that isolate a prescribed derivative at one address.
+
+\subsection{Labels and arithmetic realization}
+
+Fix a finite set $S$ of rational primes, containing $2$ and all the ramified
+primes in the fixed data.  Let $F$ be either $\mathbf Q$ or a quadratic field.
+For the rational two-torsion application $F=\mathbf Q(E[2])$; the construction
+with full rational two-torsion also uses a quadratic auxiliary field in place
+of $F$.  Include in $S$ primes whose prime ideals generate the class group of
+$F$, and include the infinite places in $S_F$.  Put
+\[
+ V_S(F)=\{\alpha\in F^\times/F^{\times2}:
+           v_{\mathfrak p}(\alpha)\equiv0\pmod2
+           \text{ for }\mathfrak p\notin S_F\}.
+\]
+This is a finite-dimensional $\mathbf F_2$-space: the valuation sequence
+expresses it as an extension of the finite squareclass group of $S_F$-units
+by a subgroup of the $2$-torsion of the $S_F$-class group.  Its Kummer extension
+is denoted $F_S^{(2)}=F(\sqrt{V_S(F)})$.
+
+A \emph{split vertex} is a rational prime $p\notin S$ split in $F$, together
+with a choice $\mathfrak p\mid p$ if $F$ is quadratic.  This choice is its
+\emph{orientation}.  Choose a generator $\pi_{\mathfrak p}$ with divisor
+$\mathfrak p$ outside $S_F$; class-group generation makes this possible.
+Conjugation supplies the generator for the other orientation, and
+\begin{equation}\label{gr:norm}
+ N_{F/\mathbf Q}(\pi_{\mathfrak p})=p u_{\mathfrak p},
+ \qquad u_{\mathfrak p}\text{ a signed rational }S\text{-unit}.
+\end{equation}
+For $F=\mathbf Q$ use the rational generator $p$.  A \emph{simple vertex}
+is an inert rational prime in the quadratic case, with rational generator
+$p$.  The fixed label of a split vertex is its Frobenius on $F_S^{(2)}$;
+simple vertices have rational fixed-support labels.  A \emph{type} may
+additionally record the generators' local squareclasses, valuation parities,
+real signs, and prescribed finite ray conditions at $S_F$.  Only compatible,
+individually realizable types will be prescribed.  A \emph{filter} fixes the
+total rational labels and sign of the signed-prime product
+$\prod_p p^*$, where $p^*=(-1/p)p$.
+
+For a pair of split vertices in the quadratic case there are two independent
+forward bits: the residues of $\pi_{\mathfrak p}$ and
+$\overline{\pi_{\mathfrak p}}$ at the chosen place of the other vertex.
+Write quadratic residues additively in $\mathbf F_2$.  Reverse bits are
+determined by Hilbert reciprocity and the fixed types.  Rational norm bits
+are determined by summing the two oriented bits and the fixed correction
+in \eqref{gr:norm}.  When at least one endpoint is simple, or $F=\mathbf Q$,
+there is one rational forward bit.  Self-residues are never graph data.
+An \emph{arithmetic graph} is this collection of vertices, types, and bits.
+A split vertex is \emph{primary} when its generator is a square unit at
+every finite place of $S_F$ and positive at all real places.
+Different symbols may read compatible projections of the same graph, for
+example its rational norm graph.  Here compatibility means that the
+projection forgets fixed-label coordinates or takes rational norm data,
+carries primary vertices to primary vertices, and commutes with the
+omitted-vertex product substitution.  For the norm projection a single
+oriented-bit toggle changes the rational bit once.  We use these arithmetic
+projections throughout.
+
+\begin{lemma}[Finite realization and product labels]\label{gr:realization}
+Fix finitely many old vertices, with all their mutual data, and
+individually realizable new types.  Every prescription of the independent
+forward bits involving at least one new vertex has a realization by
+distinct fresh primes, avoiding an arbitrary finite set.  The old vertices
+and their mutual data remain unchanged.  The same holds
+with further fixed local tests included in $S$.  A contracted split cluster
+may be represented by the product of its oriented generators, with
+conjugations prescribed along its contraction tree; its support types and
+external residue bits are the corresponding sums.  A cluster containing
+one simple vertex has the corresponding rational product data.
+
+For a symbol satisfying the omitted-vertex dependence in
+Proposition~\ref{co:symbol-interface}, its value depends only on the graph
+data.  Products in the preceding assertion may therefore be replaced by
+fresh prime representatives without changing the prescribed contracted
+symbol.
+\end{lemma}
+
+\begin{proof}
+First note that the listed generators account for all the relevant radicals.
+If $U$ is a finite set of new primes and a squareclass has even valuations
+outside $S_F\cup U_F$, divide by the generators at the primes of $U_F$
+where its valuation is odd.  The result belongs to $V_S(F)$.  At an inert
+rational prime its rational generator has valuation one at the unique place
+of $F$.  Thus fixed-support labels and the displayed residue bits determine
+every Frobenius test in an extension with this allowed ramification.
+
+We describe the prime selection carefully.  For a new split vertex take a
+template $\pi_0$ of its prescribed type, with prime ideal
+$\mathfrak p_0$ away from the old vertices.  Weak approximation gives
+$a\in F^\times$ which is sufficiently close to $1$ at $S_F$, positive at
+the prescribed real places, and has any prescribed nonzero residues at the
+old prime ideals.  Choose those residues so that $a\pi_0$ has the desired
+quadratic residues there.  Use the ray modulus consisting of sufficiently
+deep powers of the primes of $S_F$, all old prime ideals, and the real
+places.  Choose a prime ideal $\mathfrak q$ in the ray class of
+$(a)\mathfrak p_0$.  Then
+\[
+ \mathfrak q=(a\beta)\mathfrak p_0,
+ \qquad \beta\equiv1\pmod{\text{ray modulus}},
+\]
+and $\pi_{\mathfrak q}=a\beta\pi_0$ is the required generator outside $S_F$.
+There are infinitely many such degree-one prime ideals.  Indeed, the
+prime-ideal theorem in a ray class gives order $X/\log X$ prime ideals of
+norm at most $X$, whereas prime ideals of residue degree greater than one
+contribute only $O(X^{1/2})$ in a quadratic field.  Degree one gives a split
+rational prime, and finite exclusions remove only finitely many choices.
+
+At an old inert prime with residue field $\mathbf F_{p^2}$, prescribe the
+norm of the nonzero residue.  The norm onto $\mathbf F_p^\times$ is
+surjective, so either rational residue bit is possible.  For a new simple
+vertex the requirements are rational congruences, together with the inert
+class in $F$.  The Chinese remainder theorem and Dirichlet's theorem supply
+fresh primes.  Successively adjoining vertices proves simultaneous finite
+realization.  At every step reciprocity determines the reverse bits:
+the local Hilbert symbols at the support are fixed, and the only other
+contributions are the two endpoint residues.  Taking norms gives exactly
+\eqref{gr:norm}, so it imposes no additional independent bit.
+
+The same argument applies to a product template.  Its valuations and local
+squareclasses at $S_F$ are those of the product, and its residues at the
+remaining vertices multiply.  The ray-class construction replaces that
+template by a degree-one prime with those data.  In a simple cluster we
+instead use rational congruences for the norm labels.  No residue at a prime
+inside the cluster is imposed on its replacement.
+
+In particular this preserves the rational filter, including its norm
+correction.  Write $\pi=\prod_i\pi_i$ for the oriented product template
+and $\pi'=a\beta\pi$ for its fresh representative; $a\beta$ is a square unit at
+$S_F$ and positive at its real places.  The support valuations of $\pi'$
+and $\pi$ agree exactly.  Hence
+\[
+ N\pi=(\prod_i p_i)u,\qquad N\pi'=q u
+\]
+with the same signed rational $S$-unit $u$: all its valuations and its
+sign agree.  The quotient $q/\prod_i p_i=N(a\beta)$ is a local square at
+the rational support.  Its squareclass at $2$ also identifies
+$q^*$ with $\prod_i p_i^*$ at those fixed tests.  Thus no free choice
+of an untracked $S$-unit has entered a product replacement.
+
+Finally, compare two disjoint realizations with identical graph data.
+Introduce fresh bridge vertices with matching bits to both realizations,
+and replace vertices one at a time.  At each replacement the omitted-vertex
+elementary Frobenius tests agree, by the generator description in the first
+paragraph.  The symbol interface gives equal values.  A preliminary bridge
+removes any overlap between the original realizations.  The same reasoning
+applies to the product replacements.
+\end{proof}
+
+Primary types are realized by the preceding construction, starting with the principal
+ray class.  Reciprocity makes their fixed radical labels trivial.  Their
+signed rational contributions are squares at the fixed rational tests.
+If stationary vertices are also to have neutral incidences with an
+auxiliary, include their primes among these local tests.  In particular,
+including the support of a fixed quadratic discriminant $k$ ensures that
+every such primary auxiliary splits in $\mathbf Q(\sqrt{k})$.
+
+\subsection{The simultaneous-testing statement}
+
+Separate the given vertices into \emph{terminals} and, if present,
+stationary external vertices.  Terminals may be always present or may
+have nonzero linear activations; external vertices have fixed types and
+incidences throughout.  At each nonzero address the symbol
+is a function of the independent bits between the active terminals, with
+those types and external data fixed.  We call its unique multilinear
+polynomial over $\mathbf F_2$ the \emph{terminal polynomial}.  Its variables
+describe formal changes of the terminal bits, while the prescribed prime
+realization supplies a fixed evaluation point.
+
+For a bit $z$ define the finite difference
+$\Delta_z f=f|_{z=0}+f|_{z=1}$.  On a multilinear polynomial it is the
+formal partial derivative.  In particular, differences commute and
+$\Delta_z^2=0$.
+
+\begin{definition}[Witnesses and permitted partitions]\label{gr:partitionable}
+A \emph{witness} is a finite graph on which the
+symbol is $1$.  A graph is \emph{partitionable} for the symbol if its vertices
+can be partitioned into nonempty clusters, each with at most one simple
+vertex, with a choice of oriented contraction trees whose contracted graph
+is a witness.
+This definition includes a specified contraction certificate; it makes no
+assertion about a Selmer group.
+\end{definition}
+
+\begin{theorem}[Address lemma]\label{gr:address}
+Suppose the binary symbols under consideration depend only on the
+arithmetic graph data and satisfy the contraction, whole-rational-prime
+omission, and primary-isolation rules of
+Proposition~\ref{co:symbol-interface}, with the realization freedoms
+of Lemma~\ref{gr:realization}.  Contraction includes the product labels
+and external incidences, and the zero rule for two simple vertices.
+These identities are required only on nonempty configurations and for
+contractions with nonempty output.
+Fix $b\ge1$ and either of the following terminal arrangements.
+\begin{enumerate}
+\item One symbol, a nonempty set of always active terminals, and a
+nonzero polynomial in their independent pair bits.
+\item One or two symbols on terminals with nonzero linear activations,
+including at least one terminal for every nonzero linear form.  At each
+$x\ne0$ each symbol separately is a nonzero polynomial in the active
+terminal bits.  The symbols may read different compatible projections
+of a common graph alphabet and may have fixed stationary external data.
+\end{enumerate}
+Suppose also that at each nonzero address the total active terminal data,
+together with any stationary external data, obey the required filter.
+Starting from any fixed realization of the terminal and stationary
+vertices, one can adjoin finitely many fresh primary auxiliary vertices,
+activated by nonzero linear forms, and choose only incidences involving
+these new vertices so that every symbol equals $1$ at every $x\ne0$.
+All data on the original vertices remain unchanged.  No auxiliary is
+present at $0$.  All auxiliaries may satisfy the fixed splitting and
+neutral external-incidence conditions incorporated into the primary type.
+
+It is enough that a terminal polynomial have a nonzero evaluation or
+a nonzero iterated edge difference.  In particular a partitionability
+certificate suffices for the stationary hypothesis.
+\end{theorem}
+
+The proof has two algebraic steps.  Forests of auxiliary vertices act as
+finite-difference operators on the original terminal polynomial.  We then
+build networks whose degree-one terms select a chosen derivative at a
+single address.  Applying them to top-degree coefficients will give the
+simultaneous value $1$, independently of the fixed terminal bits.
+
+\subsection{From arithmetic contraction to forest operators}
+
+\begin{lemma}[Oriented forests]\label{gr:forest}
+Assume the graph-symbol rules stated in Theorem~\ref{gr:address}.
+Differentiation in a tree of graph
+bits contracts the tree, adding product labels and external incidences;
+its value is zero if the tree contains two simple vertices.  Differentiation
+in any collection of edge variables containing a cycle is zero, even when
+different edges use incompatible relative orientations.  Two different
+oriented variables on the same pair count as a cycle of length two.
+\end{lemma}
+
+\begin{proof}
+The one-edge assertion is precisely the arithmetic contraction rule,
+with product replacements justified by Lemma~\ref{gr:realization}.
+Induction proves the tree assertion: a split--split merger is split, a
+split--simple merger is simple, and the first merger of two components
+containing simple vertices has zero derivative.  Along a tree there is
+a unique choice of relative orientations once one root orientation is
+fixed.  Its product template depends on support types and external bits,
+not on the internal bits being differentiated.
+
+For clarity, consider first the two variables on a split pair $p,q$.
+Differentiation in the inertia bit at one place over $p$ gives the symbol
+formed with the \emph{entire rational prime $p$ omitted}.  Its elementary
+extension is unramified at both places over $p$.  Differentiating in the
+other oriented bit is therefore zero.  Repetition of the same variable
+is zero by $\Delta_z^2=0$.  This argument would not follow from omission
+of only one oriented prime ideal.
+
+For a longer cycle, first contract a spanning path.  By the product-label
+rule the closing variable has become internal to the merged cluster.
+Its change affects neither the product's support data nor any incidence
+to a remaining vertex.  Equivalently it is an inertia change at an omitted
+endpoint, covered by the preceding unramifiedness argument.  Thus its
+derivative vanishes.  Possible conjugation of the product generator only
+changes the induced orientation and does not reintroduce that endpoint.
+Since there is no self-residue among the inputs to the replacement symbol,
+there is no extra closing-edge datum to retain.  Additional derivatives
+commute with this zero derivative.
+
+For example, contract the aligned edge $12$ in a triangle.  With
+$\pi_{12}=\pi_1\pi_2$, the two bits toward vertex $3$ are
+$t_0=z_{13}^0+z_{23}^0$ and $t_1=z_{13}^1+z_{23}^1$, up to fixed
+type corrections.  An aligned edge $23$ then differentiates $t_0$;
+a conjugately aligned closing edge $31$ differentiates $t_1$.
+Their mixed derivative is the omitted-endpoint zero just proved.
+If the closing edge is aligned instead, both become the same
+difference and its square is zero.  Contracting a longer spanning
+path reduces to exactly these two alternatives.
+\end{proof}
+
+Fix a realizable baseline terminal graph and let $P$ be its terminal
+polynomial.  A \emph{network} consists of additional primary
+vertices and programmed edges, with no direct terminal--terminal edges.
+All unprogrammed incidences of its primary vertices are neutral.  Within
+one network choose one orientation at each terminal and consistently use
+the corresponding edge alignment.  Networks may choose different
+alignments.  The \emph{degree} of a contraction is the number of terminal
+vertices it removes by merging them.
+
+\begin{lemma}[Network expansion and its first two degrees]\label{gr:network}
+A network acts on $P$ by a sum of finite-difference operators.  A surviving
+term is a forest in which every component containing a primary vertex
+meets a terminal.  A component meeting $t$ terminals acts by a contraction
+of degree $t-1$; a component containing two simple terminals is zero.
+Networks on disjoint sets of internal vertices, with no edges between
+those sets, multiply as operators on the same terminal polynomial.
+Consequently an operator of degree greater than $\deg P$ annihilates $P$.
+
+Let $Q$ be the grounded Laplacian on the internal vertices: $Q_{ij}$ for
+$i\ne j$ records the internal edge $ij$, while $Q_{ii}$ is the total
+incidence at $i$, including its terminal edges.  Let $(c_a)_i$ record the
+incidence from internal vertex $i$ to terminal $a$.  All entries are in
+$\mathbf F_2$.  The constant and pair-contraction coefficients of the
+operator are respectively
+\begin{equation}\label{gr:forest-coefficients}
+ \det Q,\qquad c_a^{\mathsf t}\operatorname{adj}(Q)c_b\quad(a\ne b).
+\end{equation}
+The second coefficient multiplies the terminal difference in the alignment
+selected by the network.  A prohibited pair multiplies the zero operator.
+\end{lemma}
+
+\begin{proof}
+Expand the evaluation at the programmed incidences from the graph with
+all primary incidences zero.  For a set $E_0$ of programmed bits this is
+the elementary identity
+\[
+ f(\mathbf1)=\sum_{T\subseteq E_0}(\Delta_T f)(\mathbf0).
+\]
+Lemma~\ref{gr:forest} kills cyclic $T$.  A tree component meeting no
+terminal contracts to an isolated primary vertex, and the isolation rule
+kills it.  This includes an untouched primary vertex.  In any remaining
+component all primary labels are trivial and all external primary bits
+at the expansion baseline are zero.  Its contracted label and its
+incidences are exactly those obtained by contracting its terminal
+vertices in the original graph.  Choose any terminal tree inducing its
+relative orientations; the corresponding terminal differences therefore
+give the same value.  Its degree is the number of lost terminal vertices,
+namely $t-1$.  This establishes the operator interpretation, rather than
+merely a count of possible contraction degrees.
+
+Expanding two disjoint networks at once gives the composition of these
+operators: the product-label rule gives the same final labels in either
+order, and any cycles created between networks vanish by
+Lemma~\ref{gr:forest}.  Ordinary polynomial differentiation now proves
+the asserted degree bound.
+
+This argument also explains the compatible projections.  Taking norms
+commutes with every oriented product in a forest; primary factors have
+neutral norm corrections, and each programmed oriented toggle projects
+to its one rational toggle.  Thus the projected forest acts by the
+corresponding differences on the projected \emph{original} terminal
+polynomial at every degree.  If projected differences repeat an edge
+or merge two simple clusters, that term is zero.  Forgetting fixed-label
+coordinates plainly commutes with the same products.  Each projected
+network therefore acts on the original projected terminal polynomial.
+
+The determinant counts are instances of the all-minors matrix-tree
+theorem \cite[\S1]{Chaiken1982}.  We give the Cauchy--Binet proof
+in the form needed here, including singular $Q$.  Let $I$
+be the internal vertex set and $A$ the terminal set.  Write the full
+Laplacian as $L=DD^{\mathsf t}$, where a column of the incidence matrix
+$D$ has ones at its edge's endpoints.  Cauchy--Binet applied to
+$L_{I,I}=Q$ counts edge sets whose incidence matrix with the terminal
+rows deleted is invertible.  Such an edge set is a forest with exactly
+one terminal in each component: a cycle gives dependent columns; a
+component with no deleted row gives dependent rows; and for a tree with
+one row deleted, leaf removal proves invertibility.  These are exactly
+the degree-zero terms.
+
+For the pair coefficient apply Cauchy--Binet with row set $I\cup\{a\}$
+and column set $I\cup\{b\}$.  Each counted forest has one root from
+$A\setminus\{a\}$ and one from $A\setminus\{b\}$ in each component.
+This requires that $a,b$ lie together and that every other terminal lie
+in its own component.  The corresponding Laplacian minor is
+\[
+ \det\begin{pmatrix}Q&c_b\\c_a^{\mathsf t}&0\end{pmatrix}
+       =c_a^{\mathsf t}\operatorname{adj}(Q)c_b.
+\]
+The corner is zero because the network has no direct terminal edge;
+signs disappear in characteristic two.  The determinant identity uses
+the adjugate and so remains valid when $Q$ is singular.
+\end{proof}
+
+\subsection{Two networks which select one address}
+
+Fix a nonzero target $x_*$ and a terminal bit $z$ joining two terminals
+$a_0,a_1$ active at $x_*$.  In the
+moving case assume that some terminal has each nonzero activation form;
+in the stationary case all terminals are always present.
+
+Choose a basis $\lambda_1,\ldots,\lambda_b$ of the dual space with
+$\lambda_i(x_*)=1$ for every $i$.  To obtain it, choose $\lambda_1(x_*)=1$
+and replace a basis $\mu_2,\ldots,\mu_b$ of the annihilator of $x_*$ by
+$\lambda_1+\mu_2,\ldots,\lambda_1+\mu_b$.
+Make $b$ groups of three primary vertices, group $i$ activated by
+$\lambda_i$.  Since these forms are a basis, all groups are present
+exactly at $x_*$.  At every other address at least one group is absent.
+In each physical three-dimensional vertex space choose
+a basis $(e_i,u_i,v_i)$ with $e_i=(1,1,1)$.  Thus the physical coordinate
+change is block diagonal by groups.
+
+Prescribe a symmetric bilinear form $Q$ by making the $e_i$ orthonormal
+and orthogonal to all $u_j,v_j$, and putting on the remaining coordinates
+\begin{equation}\label{gr:H}
+ H(T,B)=\begin{pmatrix}0&T\\T^{\mathsf t}&B\end{pmatrix}.
+\end{equation}
+Matrix entries in the physical coordinates are denoted by the same
+letter $Q$.  For a subset $J$ of active groups, $Q_J$ means its physical
+principal restriction.  Define the column $h_i=Qe_i$.
+It is supported on group $i$: in adapted coordinates it is the covector
+dual to $e_i$, and the coordinate change does not mix groups.  Its value
+is independent of $T$ and $B$.
+
+Attach $h_i$ to an anchor terminal with activation $\lambda_i$.  In the
+stationary case attach it to any fixed terminal instead.  The support
+property gives, on every address,
+\[
+ \sum_{i\in J}h_i|_J=Q_J\mathbf1_J.
+\]
+Program internal edges using the off-diagonal
+physical entries of $Q$.  A matrix is their grounded Laplacian precisely
+when the sum of the terminal incidence columns is its row-sum vector,
+so these anchor columns give exactly $Q_J$, including its diagonal.
+
+We shall also attach the same additional column $c$ to $a_0$ and $a_1$.
+In the moving case with distinct activation forms $\alpha,\beta$, attach
+$c$ also to a terminal of activation $\alpha+\beta$; this terminal is
+inactive at $x_*$.  If $\alpha=\beta$, no third column is needed.  The
+sum of the additional active columns is zero at every address, since
+$\alpha+\beta+(\alpha+\beta)=0$.  In the stationary case the two columns
+cancel directly.  Coincident terminal roles mean addition of their
+columns.  This preserves the prescribed grounded matrix after every
+restriction.
+
+For a concrete instance, take $b=1$, target $x_*=1$, and two always-active
+split terminals $a_0,a_1$ with selected bit $z$.  Label the three primary
+vertices $1,2,3$, all activated by $x$.  Choose
+$e=(1,1,1)^{\mathsf t}$, $u=(1,0,0)^{\mathsf t}$,
+$v=(0,1,0)^{\mathsf t}$, and $T=(0)$, $B=(1)$.
+The form in this adapted basis is $\operatorname{diag}(1,0,1)$.
+In physical coordinates $h=Qe=(0,0,1)^{\mathsf t}$ and the covector
+dual to $u$ is $c=(1,0,1)^{\mathsf t}$.  Using $a_0$ as the anchor gives
+\[
+ Q=\begin{pmatrix}0&0&0\\0&1&1\\0&1&0\end{pmatrix},\qquad
+ c_{a_0}=h+c=\begin{pmatrix}1\\0\\0\end{pmatrix},\qquad
+ c_{a_1}=c=\begin{pmatrix}1\\0\\1\end{pmatrix}.
+\]
+Thus the programmed graph is the path
+$a_0\! -\! 1\! -\! a_1\! -\! 3\! -\! 2$.
+Here $\det Q=0$ and $\operatorname{adj}(Q)=uu^{\mathsf t}$, so
+$c_{a_0}^{\mathsf t}\operatorname{adj}(Q)c_{a_1}=1$.
+There are only two terminals, so no higher contraction degree occurs:
+the network acts exactly by $\Delta_z$ at $1$.  At $0$ all three primary
+vertices are absent, and the network acts by the identity.
+
+Write $O_{\ge d}$ for a sum of contraction operators of degrees at least
+$d$; it is a filtration notation, not an estimate.
+
+\begin{lemma}[The two address programs]\label{gr:programs}
+With the preceding grounding convention one has the following programs
+for every $b\ge1$.
+\begin{enumerate}
+\item A network $\mathcal S$ with
+\[
+ \mathcal S_{x_*}=\Delta_z+O_{\ge2},\qquad
+ \mathcal S_x=1+O_{\ge1}\quad(x\ne x_*).
+\]
+\item Two networks on the same activated sites, indexed by
+$\varepsilon\in\mathbf F_2$, with
+\[
+ \mathcal N_x^{(\varepsilon)}=1+O_{\ge1},\qquad
+ \mathcal N_x^{(1)}+\mathcal N_x^{(0)}
+ =\begin{cases}\Delta_z+O_{\ge2},&x=x_*,\\
+ O_{\ge2},&x\ne x_*.
+ \end{cases}
+\]
+\end{enumerate}
+All nonzero-activation sites are absent at $0$.  Unspecified incidences,
+including those to stationary external vertices, may remain neutral.
+\end{lemma}
+
+\begin{proof}
+For the first program take $T=I+P$, where $P$ is the cyclic permutation
+matrix on $b$ coordinates, and $B$ diagonal with exactly one nonzero
+entry.  A proper principal group restriction of $P$ is a union of
+directed paths, hence nilpotent.  Thus $T_J$ is invertible for every
+proper $J$, and so is $H(T_J,B_J)$: its kernel equations first give
+$v=0$ and then $u=0$.  This includes the empty restriction, whose
+determinant is $1$.
+
+On the full group set, both $\ker T$ and $\ker T^{\mathsf t}$ are
+spanned by the all-one vector.  A radical vector of $H$ has
+$v=t\mathbf1$; summing the equation $T^{\mathsf t}u+Bv=0$ gives $t=0$
+because $\mathbf1^{\mathsf t}B\mathbf1=1$.  Thus the full $Q$ has
+one-dimensional radical spanned by $r=\sum_i u_i$.  In characteristic
+two its adjugate is $rr^{\mathsf t}$.  Indeed a rank-$n-1$ matrix has a
+nonzero rank-one adjugate with image its kernel; symmetry identifies
+the two factors, and the only nonzero scalar is $1$.
+Choose $c$ to be the covector dual to $u_1$, so $c^{\mathsf t}r=1$.
+All anchors pair to zero with $r$.  At $x_*$ the only columns pairing
+nontrivially with $r$ are consequently those of $a_0,a_1$.  Formula
+\eqref{gr:forest-coefficients} gives zero constant and exactly
+$\Delta_z$ in degree one.  At every other address the group subset is
+proper and the constant is $1$.  When $b=1$, $I+P=0$ and $B=(1)$,
+so the same radical calculation applies without an exception.
+
+For the second program let $T=I+L$, where $L$ is the consecutive lower
+shift, and take $B_\varepsilon=\varepsilon E_{bb}$.  Every group
+restriction of $T$ is invertible, and block multiplication gives
+\begin{equation}\label{gr:inverse}
+ H(T,B)^{-1}=
+ \begin{pmatrix}T^{-\mathsf t}BT^{-1}&T^{-\mathsf t}\\
+ T^{-1}&0\end{pmatrix}.
+\end{equation}
+Again take $c$ dual to $u_1$.  On a group subset $J$, the change of
+$c^{\mathsf t}Q_J^{-1}c$ is
+\[
+ \bigl((T_J^{-1})_{b1}\bigr)^2,
+\]
+interpreted as zero if $1$ or $b$ is absent.  Since
+$(I+L_J)^{-1}=I+L_J+L_J^2+\cdots$, that entry is $1$ precisely when
+the entire path $1,2,\ldots,b$ is present.  This is exactly the target
+address.  Pairings involving an anchor do not change:
+$Q_J^{-1}h_i=e_i$, and the vectors $e_i,h_i,c$ do not depend on
+$\varepsilon$.  At the target the balancing terminal is absent, so
+the only changed pair coefficient is the one for $a_0,a_1$.
+Every constant is $1$.  Lemma~\ref{gr:network} proves the claims,
+including $b=1$, when the path consists of its single vertex.
+\end{proof}
+
+\subsection{Proof of the address lemma}
+
+\begin{proof}[Proof of Theorem~\ref{gr:address}]
+Use the prescribed realization of the terminal and stationary vertices
+as a common baseline for all terminal bits.  Regard
+their toggles as common variables, omitting variables invisible to a
+given symbol.  A nonzero iterated difference implies that the polynomial
+is nonzero.  Conversely, the unique multilinear polynomial of a
+nonzero function has a top monomial with nonzero coefficient.  These
+observations also explain the last assertion using
+Lemma~\ref{gr:forest}.
+Every nonzero address has an active terminal, and contractions never
+remove the last vertex.  Thus only nonempty configurations are used.
+
+First consider one symbol.  At each $x\ne0$ let $P_x$ have degree $d_x$
+and choose a monomial with coefficient $1$ in its top homogeneous part.
+For each variable of this monomial add a separate singular program
+$\mathcal S$ targeted at $x$.  Give all these networks disjoint internal
+vertices.  At address $x$, its own $d_x$ networks each have zero constant,
+so every surviving term spends at least $d_x$ contraction degrees.
+An extra degree from any network kills $P_x$.  Thus only the designated
+degree-one contractions of its own networks and the constant terms
+of all other networks survive.  Their value is the chosen top
+coefficient, namely $1$.  A nonzero constant polynomial needs no
+network and already has value $1$.  This argument applies equally to
+always active terminals, since Lemma~\ref{gr:programs} supplies their
+stationary anchors.
+
+For two symbols write their polynomials at address $x\ne0$ as
+$P_x,R_x$, of degrees $d_x,e_x$, and let $P_x^{\rm top},R_x^{\rm top}$
+be their top homogeneous parts.  Multiply these parts in the ordinary
+polynomial ring
+$\mathbf F_2[z_1,\ldots,z_m]$, \emph{before} imposing any Boolean
+relations.  That ring is an integral domain, so choose a coefficient
+equal to $1$ at a monomial
+\begin{equation}\label{gr:chosen-monomial}
+ M_x=\prod_z z^{\nu_{xz}},\qquad
+ \nu_{xz}\in\{0,1,2\},\qquad
+ \sum_z\nu_{xz}=d_x+e_x.
+\end{equation}
+For $\nu_{xz}=2$ add one singular program for $z$ targeted at $x$.
+For $\nu_{xz}=1$ add one nonsingular program for $z$ targeted at $x$,
+with its own choice $\varepsilon_{xz}\in\mathbf F_2$.  All internal
+vertices of different programs are disjoint.  Let
+\[
+ \Phi(\varepsilon)=\prod_{x\ne0}
+       \bigl(\text{final evaluation of }P_x\bigr)
+       \bigl(\text{final evaluation of }R_x\bigr).
+\]
+We prove that the sum of $\Phi$ over all choices of $\varepsilon$ is $1$.
+This proves that some single choice has all the required evaluations
+equal to $1$.
+
+Expand every program in contraction degree using
+Lemma~\ref{gr:network}.  A singular program targeted at $x$ has zero
+constant in both factors there, so it costs at least one degree in
+$P_x$ and one in $R_x$.  A nonsingular program has constant $1$
+independent of its choice.  In the sum over that choice, a term using
+only its constants in all factors cancels.  Every remaining term must
+therefore spend at least one degree somewhere for this program.  The
+sum of these minimum costs is exactly
+\begin{equation}\label{gr:budget}
+ D=\sum_{x\ne0}(d_x+e_x),
+\end{equation}
+by \eqref{gr:chosen-monomial}.  A nonzero term cannot exceed this total:
+if it did, some factor would receive more differences than its degree.
+Hence every surviving term uses every minimum cost exactly.
+
+It follows that a singular program spends exactly one degree in each
+factor at its target and zero elsewhere.  Its degree-one term there
+is precisely $\Delta_z$.  Each nonsingular program spends exactly one
+degree in exactly one factor.  After summing its choice, that
+degree-one contribution is zero away from its target, and is
+$\Delta_z$ at its target.  All other occurrences of these programs
+contribute their common constant $1$.  Thus no unknown higher forest
+coefficient enters the surviving calculation.
+
+At a fixed address, let $Z_2$ be the variables with exponent two and
+$Z_1$ those with exponent one.  The local allocation sum is
+\[
+ \sum_{J\subseteq Z_1}
+ [z_{Z_2}z_J]P_x^{\rm top}\,
+ [z_{Z_2}z_{Z_1\setminus J}]R_x^{\rm top}
+   =[M_x](P_x^{\rm top}R_x^{\rm top})=1.
+\]
+Coefficients of the wrong degree in this display are zero.  Every
+factor must in fact receive its full degree, since the total is $D$
+and none can exceed its own degree.  Therefore its final derivative
+is exactly the displayed top coefficient, independent of the chosen
+baseline values.  Allocations at different addresses are independent,
+so their product is $1$, as asserted.  This also covers a constant
+factor, whose degree is zero.  For example, when both top forms are
+$z$, their ordinary product is $z^2$ and the singular program
+differentiates each factor once; Boolean reduction would lose this
+allocation.
+
+In the two-symbol case, choose the program parameters for which
+$\Phi=1$.  In either case, apply Lemma~\ref{gr:realization} to adjoin
+the resulting finite networks to
+the fixed original vertices.  Only new vertices and incidences involving
+them have been prescribed.  The primary types preserve the filters.
+All activations are nonzero linear forms,
+so no new vertex appears at $0$, and the stipulated neutral and
+splitting conditions persist.
+\end{proof}
+
+The theorem is finite for each $b$; it does not assert a single infinite
+array of primes.  For a stabilized symbol, first choose $b$, its finite
+terminal data, and all the networks above.  Their total weight and the
+list of coefficient tests are then finite.  A sufficiently advanced
+stage at which the symbol interface holds for that weight range
+realizes the construction.  Required precision and the number of
+auxiliaries may grow with $b$.  The arithmetic valuation constant
+attached to a unit symbol must still be supplied by the coefficient
+construction independently of $b$; no determinant or forest count in
+this section proves that uniformity.
diff --git a/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/build/sections/introduction.tex b/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/build/sections/introduction.tex
new file mode 100644
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+++ b/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/build/sections/introduction.tex
@@ -0,0 +1,173 @@
+\section{Introduction}\label{sec:introduction}
+
+The Birch--Swinnerton-Dyer conjecture predicts that the order of vanishing
+of the $L$-function of an elliptic curve equals its Mordell--Weil rank,
+and that its Shafarevich--Tate group is finite; its refined form also
+specifies the leading coefficient \cite{BirchSwinnertonDyer1965,Tate1966}.
+For analytic order zero or one, the rank equality and finiteness are
+theorems of Gross--Zagier and Kolyvagin
+\cite{GrossZagier1986,Kolyvagin1990}.
+The converse problem begins with algebraic information and seeks the
+corresponding analytic order.
+
+The appropriate algebraic hypothesis here is the corank of the full
+$2$-power Selmer group. It records a possible divisible contribution
+from the Shafarevich--Tate group as well as the Mordell--Weil rank.
+With the usual local Kummer conditions, put
+\[
+s_2(E)=\corank_{\Z_2}\Sel_{2^\infty}(E/\Q).
+\]
+Our main result is the following pointwise statement.
+
+\begin{theorem}[The rational-two-torsion $2$-converse]\label{thm:main}
+Let $E/\Q$ be an elliptic curve with $E(\Q)[2]\ne0$ and
+$s_2(E)\in\{0,1\}$. Then
+\[
+\ord_{s=1}L(E,s)=\rank_{\Z}E(\Q)=s_2(E),
+\qquad
+\Sha(E/\Q)\text{ is finite}.
+\]
+\end{theorem}
+
+The statement includes one rational two-torsion line and full rational
+two-torsion, CM and non-CM curves, every rational isogeny configuration,
+and every reduction type at $2$ and at the bad primes. It is a
+$2$-converse in this residual range. It does not assert the full
+Birch--Swinnerton-Dyer leading-coefficient formula, nor does a bound
+$\rank E(\Q)\le1$ alone imply its Selmer hypothesis.
+
+The broader companion \cite[Theorem~1.1]{OpenAIGoldfeldTwoConverse2026}
+proves the $2$-converse without the rational-two-torsion hypothesis.
+Here we give a separate full treatment of the rational-two-torsion case,
+developing its integral interpolation and finite twist constructions
+within this paper; the broader converse is not an input to the proof.
+
+The converse with the stronger hypotheses
+$\rank E(\Q)=1$ and $\Sha(E/\Q)$ finite is known for every
+elliptic curve over $\Q$; see \cite[\S1]{Kim2023SoftConverse}.
+The fixed-prime problem asks instead whether a low Selmer corank
+alone forces the analytic order. Skinner and Wei Zhang established
+general rank-one converses at good ordinary odd primes under residual
+and local hypotheses \cite{Skinner2020,Zhang2014Heegner}.
+For odd Eisenstein primes, Castella, Grossi, Lee and Skinner prove
+the good-reduction converse under a local-character restriction
+\cite[Theorem~E]{CGLS2022}; Keller and Yin treat potentially good
+ordinary reduction without that restriction
+\cite[Theorem~B]{KellerYin2024}. Smith's theorem gives the $50/50$
+distribution of $2^\infty$-Selmer coranks in every fixed quadratic-twist
+family, counting twist parameters of both signs \cite{Smith2025};
+this density statement does not decide a prescribed member.
+
+The CM case includes pointwise results at the prime $2$.
+Burungale and Tian prove the rank-zero $p$-converse for CM curves at
+every prime \cite{BurungaleTian2026}. In rank one, Burungale and
+Skinner treat good ordinary primes and, with a localization hypothesis,
+good supersingular primes
+\cite[Theorems~A.1 and~A.3, Remark~A.5]{BurungaleSkinner2024Appendix}.
+The earlier ordinary CM converse of Burungale, Castella, Skinner and
+Tian retains a Hecke-conductor condition
+\cite{BurungaleCastellaSkinnerTian2022}.
+Kriz's preprint gives rank-zero and rank-one converses for maximal-order
+CM curves at primes ramified in the CM field, including the pointwise
+$2$-converse for the congruent-number family
+\cite[Theorems~1.1 and~10.17(2)]{Kriz2022CM}.
+These results already treat individual curves in their stated ranges;
+the theorem here includes non-CM curves with rational two-torsion and
+imposes no reduction condition at $2$.
+
+The proof brings together two arithmetic interpolation constructions and a
+finite construction with quadratic twists.  Kato's zeta classes provide
+the cyclotomic input \cite{Kato2004}; Ferrero--Washington's
+vanishing theorem controls the residual cyclotomic cohomology
+\cite{FerreroWashington1979}.  Heegner points and their derivative
+classes provide the ring-class input, following the Euler-system method
+of Kolyvagin and its cohomological formulation by Howard
+\cite{Kolyvagin1990,Howard2012}.  We retain integral local conditions and
+duality maps in the framework of Selmer complexes \cite{Nekovar2006}.
+The coefficient tests use the half-integral weight theory of Shimura
+and Waldspurger's relation between coefficients and central values
+\cite{Shimura1973HalfIntegral,Waldspurger1981}.  Their ternary theta
+construction lies in the definite quaternion framework developed by
+Gross \cite{Gross1987Heights}; we use the Weil representation and the
+reduction description of Jetchev and Kane for the weighted Heegner-point
+tests \cite{Weil1964,JetchevKane2010}.
+Each of these inputs has a precise role; none supplies the pointwise
+conclusion by itself.
+
+Here is the common interpolation problem.  Fix the curve that is to be
+tested, and index a finite family of quadratic twists by
+$x\in\mathbb F_2^b$, with $x=0$ denoting that curve.  Each auxiliary
+prime divides the twist parameter according to a linear form in $x$.
+We construct these families so that every nonzero address has the
+required analytic order and a uniform normalized valuation bound.
+An integral interpolation argument then recovers the value or derivative
+at the missing address $0$.  The number and sizes of the auxiliary
+primes may grow with $b$; the valuation bound may not.
+
+For Selmer corank zero, the required nonzero values are central
+$L$-values.  Their normalized valuations are measured by coefficients
+of half-integral weight forms.  The cyclotomic determinant makes these
+values part of one integral family, and a congruence on a sufficiently
+large binary cube recovers the missing value.  This also supplies the
+even estimate needed to normalize the rank-one construction.
+
+The rank-one construction requires a further preparation.  Fix a negative
+quadratic discriminant $h_*$ with $L(E^{(h_*)},1)\ne0$.
+For positive $h$ in the prescribed local squareclasses, the
+genus-character Heegner sum $P_h$ attached to $\Q(\sqrt{hh_*})$
+detects a simple zero of $E^{(h)}$: its height measures
+$L'(E^{(h)},1)L(E^{(h_*)},1)$.  To normalize the corresponding coefficient tests
+uniformly, we first bound the $2$-divisibility of $P_h$ from below
+in terms of the prime factors of $h$, with a constant independent
+of their number.
+For each parameter $h$ of known analytic rank one, a different negative companion
+$k'(h)$, with a bounded number of prime factors, combines the even
+estimate with a ring-class lower bound to give this uniform bound.
+Only after that argument do we normalize the odd coefficients and use
+their minimum-depth values as binary nonvanishing tests.
+
+For the return to a prescribed corank-one curve, choose one final
+negative discriminant $k$ with $L(E^{(k)},1)\ne0$, and fix
+$K=\Q(\sqrt{k})$.  This companion does not vary with $h$ or with the
+cube dimension.  The quadratic corank identity gives $s_2(E/K)=1$.
+We construct families with every new prime split in $K$ for which,
+at every nonzero address, $E^{(h)}$ has a simple zero and $E^{(hk)}$
+has nonzero central value.  The two coefficient tests together bound
+the normalized index of the ring-class Heegner sum over $K$.
+Ring-class interpolation then recovers the derivative at $h=1$.
+Thus $h_*$ defines the detector, $k'(h)$ proves its lower bound,
+and the fixed $k$ permits the final transfer.
+
+The arithmetic difficulty is uniform integral control at the prime $2$.
+Residual irreducibility is unavailable here.  On the cyclotomic side we
+use Kato classes from one fixed modular-form lattice, separating rational
+divisibility from uniform integrality.  On the ring-class side we retain
+the full integral Kummer conditions at the bad places and actual maps
+from finite free models to Galois cochains.  Heegner derivative operations
+preserve the determinant valuation even when a localized class is
+divisible by $2$.  These constructions give the missing-vertex
+Theorems~\ref{cy:missing-vertex} and~\ref{rd:missing-vertex}.
+
+The finite construction has a different task: it must make the required
+coefficient tests nonzero at every nonzero address simultaneously.
+The variables are quadratic residue symbols between auxiliary primes.
+Changing one such symbol contracts two prime labels; repeated changes
+are governed by forests.  Finite networks of new primes then isolate
+one address at a time.  For rank one, the derivative test and its
+central-value partner may be separately nonzero without taking nonzero
+values on the same assignment of residue symbols.  Theorem~\ref{gr:address}
+adjoins prime networks making both tests equal to $1$ at every nonzero
+address.
+
+Figure~\ref{fig:proof-roadmap} records the order in which the valuation
+bounds are obtained and used.  Section~\ref{sec:preliminaries} fixes the
+Selmer and local conventions.  Section~\ref{cy:section} proves the
+cyclotomic interpolation statements; Sections~\ref{rl:section}
+and~\ref{rd:section} construct the ring-class deformation and prove its
+bounds and transfer theorem.  Section~\ref{sec:nonvanishing} supplies
+auxiliary twists with prescribed local behavior and a bounded number of
+prime factors where needed.  Sections~\ref{co:section}
+and~\ref{gr:section} give the coefficient tests and finite networks.
+Section~\ref{pw:section} assembles these results at the prescribed curve.
+
+\input{sections/proof-roadmap-figure}
diff --git a/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/build/sections/nonvanishing.tex b/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/build/sections/nonvanishing.tex
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@@ -0,0 +1,425 @@
+\section{Auxiliary twists with prescribed local behavior}\label{sec:nonvanishing}
+
+We need auxiliary twists with specified local behavior and nonzero
+central value or derivative. For a fixed witness, no bound on the prime
+count is needed. When the curve itself varies, we also need central-value
+witnesses with a uniformly bounded number of prime factors. In both cases
+their primes can be chosen beyond any previously specified finite set.
+
+\begin{proposition}[Local witnesses]\label{nv:local-witness}
+Fix an elliptic curve $A/\Q$ and a nonempty filter of fundamental
+discriminants prime to $2N_A$, with specified real sign and unit
+squareclasses at finitely many primes. If the root number in the filter
+is $+1$, it contains $d$ of arbitrarily large absolute value with
+$L(A^{(d)},1)\ne0$.
+If the root number is $-1$, it contains such $d$ with
+$L'(A^{(d)},1)\ne0$. Any additional finite set of primes may be excluded.
+\end{proposition}
+
+\begin{proposition}[Bounded-factor completion]\label{nv:rough}
+There is an absolute integer $A_0$ with the following property.
+In the positive-root-number case of Proposition~\ref{nv:local-witness},
+one can require $\omega(d)\le A_0$ and require every prime divisor of
+$d$ to exceed any fixed bound.
+The size threshold for the resulting $d$ may depend on the curve, the
+filter, and the excluded primes; $A_0$ does not.
+\end{proposition}
+
+The use of a nonnegative first moment and a lower linear sieve to obtain
+bounded-factor twists goes back to Hoffstein--Luo
+\cite[\S2]{HoffsteinLuo1997}. We retain an arbitrary fixed progression
+and track an absolute factor-count bound without optimizing it.
+The central moment follows the model of
+\cite[Proposition~2]{RadziwillSoundararajan2015}, which has a sharper
+divisor error; the derivative calculation adapts the first-moment
+argument of \cite{Iwaniec1990}. The enlarged modulus and the absolute
+exponents needed here are retained in the proof below.
+
+A nonzero first moment alone supplies the derivative witness. For the
+central-value completion, positivity allows us to sieve the first
+moment, so we also need an error term for each prescribed squarefree
+divisor of the twist parameter.
+
+Write $f$ for the weight-two newform of $A$, normalized so that its
+central point is $1/2$, and write $\lambda(n)$ for its normalized Hecke
+eigenvalues. Thus
+\[
+L(s,f\otimes\chi)=\sum_{n\ge1}\lambda(n)\chi(n)n^{-s},
+\qquad |\lambda(n)|\le d(n).
+\]
+Choose $H$ divisible by $8N_A$, a reduced residue class $a\bmod H$, and
+$\sigma\in\{1,-1\}$ such that $\sigma a\equiv1\bmod4$.
+The required twists have discriminant $\sigma n$ with
+$n>0$ squarefree and $n\equiv a\bmod H$. A filter is a finite union of
+such progressions, so it suffices to use one of them. Fix a nonnegative
+smooth nonzero $\Phi$ supported in $(1,2)$.
+
+\begin{lemma}[A fixed-progression moment]\label{nv:moment}
+There are absolute constants $\eta>0$ and $B>0$ such that the
+following assertions hold. For squarefree $b$ prime to $H$, put
+\[
+S_{j,b}(X)=
+\sum_{\substack{n>0\ {\rm squarefree}\\ n\equiv a\ (H)\\b\mid n}}
+ L^{(j)}(1/2,f\otimes\chi_{\sigma n})\Phi(n/X).
+\]
+If the common root number is $+1$, then
+\[
+S_{0,b}(X)=cX\prod_{p\mid b}\beta_p+
+ O_{f,H,\Phi}(b^B X^{1-\eta}),
+\qquad
+c>0,\quad 0<\beta_p<1,\quad
+\beta_p=\frac1p+O(p^{-2}).
+\]
+The last implied constant is absolute. If the common root number is
+$-1$, then
+\[
+S_{1,1}(X)=cX\log X+c_1X+
+ O_{f,H,\Phi}(X^{1-\eta}),\qquad c>0.
+\]
+\end{lemma}
+
+\begin{proof}
+We first control the square-divisor tail, then use Poisson summation to
+separate a positive main term from a power-saving error. Tracking the
+dependence on the imposed divisor $b$ will give the absolute exponents.
+Let $G(z)=e^{z^2}$ and
+\[
+ V_j(y)=\frac1{2\pi i}\int_{(2)}
+       G(z)\Gamma(1+z)y^{-z}\frac{dz}{z^{j+1}}.
+\]
+Writing $Q_A=\sqrt{N_A}/(2\pi)$, the symmetric approximate functional
+equation in the required sign is
+\begin{equation}\label{nv:afe}
+L^{(j)}(1/2,f\otimes\chi_{\sigma n})
+=2\sum_{u\ge1}\frac{\lambda(u)\chi_{\sigma n}(u)}{\sqrt u}
+ V_j\!\left(\frac{u}{Q_A n}\right).
+\end{equation}
+For $j=1$, differentiating the completion contributes no central-value
+term, because that value is zero.
+
+\par\smallskip\noindent\emph{Square-divisor truncation.}
+The quadratic large sieve
+\cite[Theorem~1 and Corollary~1]{HeathBrown1995}, with the approximate functional equation,
+gives for fixed $\delta>0$
+\begin{equation}\label{nv:second}
+ \sum_{\substack{0<t\le Z\ {\rm squarefree}\\
+                 \sigma t\ {\rm fundamental}\\(t,2N_A)=1}}
+ |L(1/2+\delta+i\tau,f\otimes\chi_{\sigma t})|^2
+ \ll_{f,\rho,\delta} Z^{1+\rho}(1+|\tau|)^{C_\rho}.
+\end{equation}
+Here the polynomial exponent can be chosen independently of the level.
+To see that square indices cause no extra power, write each coefficient
+index as $ur^2$ with $u$ squarefree. Minkowski's inequality and the
+quadratic large sieve bound the square root of the moment of a polynomial
+of length $U$ by
+\[
+ (ZU)^{O(\rho')}
+ \sum_{r\le\sqrt U}\frac{\sqrt{Z+U/r^2}}r
+ \ll (ZU)^{O(\rho')}(\sqrt Z+\sqrt U).
+\]
+Coprimality restrictions at $r$ are discarded only after taking absolute
+squares. To include both parts of the approximate functional equation,
+work first with $t\asymp T$. Their lengths are
+$U\ll_f T(1+|\tau|)^C$, with rapidly decreasing tails.
+The main coefficients have the additional factor $n^{-\delta}$,
+which improves the partial-summation bound. The dual coefficients
+have $n^{\delta}$ and lose at most $U^{\delta}$, whereas their
+functional-equation multiplier contributes $T^{-2\delta}$, up to
+a fixed power of $1+|\tau|$. The resulting factor is at most
+$T^{-\delta}(1+|\tau|)^{C_\delta}$ and causes no positive power
+of $T$. Mellin inversion separates the smooth $t$-weights; the
+gamma factors control the resulting integrals and tails.
+Taking the auxiliary large-sieve exponents sufficiently small and
+summing dyadic intervals proves Equation~\eqref{nv:second}.
+
+Let $\mathcal A_j(n)$ denote the right side of
+Equation~\eqref{nv:afe}, also for nonsquarefree $n\equiv a\bmod H$.
+Expanding $\mu^2(n)$ and retaining divisors $d\le Y$ leaves an error
+\begin{equation}\label{nv:tail}
+ O_{f,H,\Phi,\rho}(X^{1+\rho}/Y).
+\end{equation}
+Indeed write $n=tv^2$, with $t$ squarefree. The omitted terms have $v>Y$,
+and their multiplicity is at most the divisor count of $v$.
+Let $L^{(v)}(s,f\otimes\chi_{\sigma t})$ be the primitive-twist
+$L$-function with its Euler factors at primes dividing $v$ removed.
+For $0<\delta_{\mathrm M}<\rho/8$, Mellin inversion gives
+\[
+ \mathcal A_j(tv^2)=\frac{2}{2\pi i}
+ \int_{(\delta_{\mathrm M})}
+ G(z)\Gamma(1+z)(Q_Atv^2)^z
+ L^{(v)}(1/2+z,f\otimes\chi_{\sigma t})
+ \frac{dz}{z^{j+1}}.
+\]
+Choose the large-sieve and divisor exponents smaller still.
+On this line the removed Euler
+factors cost an arbitrarily small fixed power of $v$. Cauchy's
+inequality and Equation~\eqref{nv:second} give, for suitably small
+$\rho'>0$,
+\[
+ \sum_{t\le2X/v^2}^{\!*}
+ |L(1/2+\delta_{\mathrm M}+i\tau,f\otimes\chi_{\sigma t})|
+ \ll_{f,\rho',\delta_{\mathrm M}}
+ (X/v^2)^{1+\rho'}(1+|\tau|)^C,
+\]
+where the star retains the primitive discriminant conditions.
+The gamma factors control the integral. The divisor count, the removed
+Euler factors, and $X^{\delta_{\mathrm M}}$ are absorbed in $X^\rho$,
+leaving $X^{1+\rho}\sum_{v>Y}v^{-2}\ll X^{1+\rho}/Y$.
+The factor $z^{-2}$ changes only the constant.
+
+\par\smallskip\noindent\emph{Poisson summation and the main term.}
+Only $(d,H)=1$ occurs, because $n\equiv a\bmod H$ is a reduced
+class. For $d\le Y$ put $J=[d^2,b]$ and
+$c_J\equiv aJ^{-1}\bmod H$, which is therefore defined.
+Write $u=u_1q$ with $u_1\mid H^\infty$ and $(q,H)=1$, and put
+$\kappa(u_1)=\chi_{\sigma a}(u_1)$. Define
+\[
+ G_\nu(q)=\sum_{y\bmod q}\left(\frac yq\right)e(\nu y/q),
+\quad
+ W_j(z,R)=\int_0^\infty
+ \Phi(x)V_j\!\left(\frac z{Q_Ax}\right)e(-Rx)\,dx,
+\]
+where $e(x)=e^{2\pi ix}$. Put
+\[
+ T_j(J)=\sum_{\substack{n>0,\ n\equiv a\ (H)\\J\mid n}}
+             \mathcal A_j(n)\Phi(n/X).
+\]
+Poisson summation in the fixed progression gives the exact identity
+\begin{align}\label{nv:poisson}
+ T_j(J)
+ &=\frac{2X}{HJ}
+ \sum_{u_1\mid H^\infty}\frac{\lambda(u_1)\kappa(u_1)}{\sqrt{u_1}}
+ \sum_{(q,H)=1}\frac{\lambda(q)}{q^{3/2}}
+       \left(\frac{\sigma JH}{q}\right) \\
+ &\hspace{14mm}\cdot
+ \sum_{\nu\in\Z}G_\nu(q)e(\nu c_J\bar q/H)
+ W_j\!\left(\frac{u_1q}{X},\frac{\nu X}{HJq}\right).
+ \nonumber
+\end{align}
+Here $q\bar q\equiv1\bmod H$. The arithmetic factor follows directly by
+applying the Chinese remainder theorem to
+\[
+\sum_{\substack{x\bmod Hq\\x\equiv c_J\ (H)}}
+ \left(\frac xq\right)e(\nu x/(Hq))
+=e(\nu c_J\bar q/H)\left(\frac Hq\right)G_\nu(q).
+\]
+When $(q,J)>1$, the additional Jacobi symbol is zero.
+
+At zero frequency only square $q$ occur. Put
+\[
+ H_p(z)=1+(1-p^{-1})
+     \sum_{r\ge1}\lambda(p^{2r})p^{-r(1+2z)},\qquad H_p=H_p(0),
+\]
+\[
+ L_H(z)=\sum_{u_1\mid H^\infty}
+             \lambda(u_1)\kappa(u_1)u_1^{-1/2-z},
+ \qquad F_J(z)=\prod_{p\nmid HJ}H_p(z).
+\]
+The zero-frequency term is
+\[
+\frac{2X}{HJ}\frac1{2\pi i}\int_{(2)}
+G(z)\Gamma(1+z)(Q_AX)^z\widehat\Phi_0(z)L_H(z)F_J(z)
+\frac{dz}{z^{j+1}},
+\quad
+\widehat\Phi_0(z)=\int_0^\infty\Phi(x)x^z\,dx .
+\]
+For the Satake parameters $\alpha_p,\alpha_p^{-1}$, with
+$|\alpha_p|=1$, we have
+\[
+ \sum_{r\ge0}\lambda(p^{2r})x^r
+   =\frac{1+x}{(1-\alpha_p^2x)(1-\alpha_p^{-2}x)}.
+\]
+Thus $F_J$ factors through $L(1+2z,\operatorname{sym}^2f)$ and a
+correction product convergent near $\Re z\ge-1/16$. Omitting a prime
+dividing $J$ removes its correction factor and inserts an inverse
+symmetric-square Euler polynomial; it creates no pole even when
+$H_p(z)$ has zeros. The omitted factors cost at most $J^{B_1}$ for an
+absolute $B_1$. Symmetric-square continuation, including the CM case
+\cite[Theorem~9.3 and Remark~9.9]{GelbartJacquet1978}, and the
+polynomial strip bounds from its functional equation therefore give an error
+$O_{f,H,\Phi}(J^{B_1}X^{1-1/16})$ when shifting to $\Re z=-1/16$.
+
+The residues have a positive leading coefficient. Indeed
+\[
+ F_p^{\rm ev}=\sum_{r\ge0}\lambda(p^{2r})p^{-r}
+ =\frac{1+p^{-1}}{(1-\alpha_p^2/p)(1-\alpha_p^{-2}/p)}>0,\quad
+ H_p=p^{-1}+(1-p^{-1})F_p^{\rm ev}>p^{-1}.
+\]
+The fixed Euler factors are positive at the center, and
+$L(1,\operatorname{sym}^2f)>0$ by the Rankin--Selberg residue formula.
+This also holds for CM forms. Summing the residue over the square
+divisor gives, at $p\nmid Hb$, the factor $H_p-p^{-2}$, and at
+$p\mid b$ the factor $p^{-1}-p^{-2}$. Consequently
+\[
+ \beta_p=\frac{p-1}{p^2H_p-1}
+        =\frac1{1+pF_p^{\rm ev}}
+        =p^{-1}+O(p^{-2}).
+\]
+For $j=1$, the double pole gives $cX\log X+c_1X$ with the same
+positive leading coefficient. Extending the residue sum from $d\le Y$
+to all $d$ costs $O(b^{B_2}X^{1+\rho}/Y)$; the $d^{-2+\rho}$ tail
+also permits differentiation of that sum.
+
+\par\smallskip\noindent\emph{The nonzero frequencies.}
+It remains to obtain a power saving with polynomial dependence on $J$.
+Put $\epsilon_q=1$ or $i$ according as $q\equiv1$ or $3\bmod4$
+and $\widetilde G_\nu(q)=\epsilon_q^{-1}G_\nu(q)$. This is multiplicative
+in $q$. Since $4\mid H$, expand
+$\epsilon_q e(\nu c_J\bar q/H)$ in characters $\psi\bmod H$.
+For $\nu\ne0$ the resulting Dirichlet series is
+\[
+ D_{\psi,J,\nu}(s)=
+ \sum_{(q,H)=1}\frac{\lambda(q)}{q^s}
+ \frac{\widetilde G_\nu(q)}{\sqrt q}
+ \left(\frac{\sigma JH}{q}\right)\psi(q)
+ =L(s,f\otimes\xi_{\rm prim})C_{\psi,J,\nu}(s),
+\]
+where $\xi(q)=\psi(q)(\sigma JH\nu/q)$.
+At $p\nmid2HJ\nu$ its local factor is
+$1+\lambda(p)\xi(p)p^{-s}$; dividing by the $\mathrm{GL}_2$ Euler
+factor leaves a polynomial $1+O(p^{-2\Re s})$.
+The bound at primes dividing $\nu$ has an absolute exponent.
+Indeed, for an odd prime $p$, put $a=v_p(\nu)$. Directly summing
+over the lifts of a residue modulo $p$ gives
+\[
+G_\nu(p^r)=
+\begin{cases}
+ \varphi(p^r),&r\text{ even},\ r\le a,\\
+ -p^{r-1},&r\text{ even},\ r=a+1,\\
+ \epsilon_p p^{r-1/2}
+       \left(\dfrac{\nu/p^{r-1}}p\right),
+       &r\text{ odd},\ r=a+1,\\
+ 0,&\text{otherwise}.
+\end{cases}
+\]
+For even $r$ this is the Ramanujan sum; for odd $r$ the inner
+sum of lifts vanishes unless $p^{r-1}\mid\nu$, after which
+the primitive quadratic Gauss sum gives the displayed value.
+In particular
+$|\widetilde G_\nu(p^r)/p^{r/2}|\le p^{r/2}$, and the local
+polynomial on $\Re s\ge3/4$ is bounded by
+$\sum_{r\ge0}(r+1)p^{-r/4}\le C_0$ with $C_0$ absolute.
+The inverse degree-two Euler polynomial is bounded by another
+absolute constant $C_1$. Primes dividing $J$ simply remove the
+local sum, and have the same bound after multiplication by that
+inverse Euler polynomial. Thus all primes dividing $J\nu$ cost
+at most $(C_0C_1)^{\omega(J\nu)}\le (2J|\nu|)^C$, for an
+absolute $C$. The remaining correction product converges
+absolutely on $\Re s\ge3/4$, since its local error is
+$O(p^{-3/2})$. Hence the correction is holomorphic on
+$\Re s>1/2$ and has the asserted polynomial bound with absolute exponent.
+The conductor of $\xi_{\rm prim}$ is at most $4HJ|\nu|$, and the
+twisted cuspidal $L$-function is entire. Thus, with absolute $C$,
+\[
+ |D_{\psi,J,\nu}(3/4+i\tau)|
+ \ll_{f,H}(J|\nu|)^C(1+|\tau|)^C.
+\]
+
+Integration by parts in both weight variables gives, for fixed $r,s$
+and any $K>0$,
+\[
+ |(z\partial_z)^r(R\partial_R)^sW_j(z,R)|
+ \ll_{f,\Phi,r,s,K}(1+z)^{-K}(1+|R|)^{-K}(1+|\log z|).
+\]
+The logarithm is needed only for $j=1$ and is $O(\log X)$ on the
+actual sums, where $z\ge X^{-1}$.
+For an absolute small $\rho>0$ the sums can therefore be restricted to
+\[
+ u_1q\ll X^{1+\rho},\qquad
+ 0<|\nu|\ll_H JX^{2\rho},\qquad
+ q\gg_H X^{1-\rho}/J,
+\]
+at a negligible cost with polynomial dependence on $J$.
+A trivial bound $|G_\nu(q)|\le q$ gives a polynomial majorant for the
+discarded tails. To identify the sum to be shifted, write
+$D_{\psi,J,\nu}(s)=\sum_q\gamma(q)q^{-s}$. On a dyadic range
+$q\asymp T$, take a fixed smooth cutoff $\vartheta$ and put
+\[
+ w_T(y)=\vartheta(y)
+ W_j\!\left(\frac{u_1Ty}{X},\frac{\nu X}{HJTy}\right).
+\]
+Each dyadic $q$-sum after the character expansion is therefore
+\[
+ \sum_q\frac{\gamma(q)}q w_T(q/T)
+ =\frac1{2\pi i}\int_{(1)}
+       \widehat w_T(s)T^sD_{\psi,J,\nu}(1+s)\,ds,
+\]
+where $\widehat w_T(s)=\int_0^\infty w_T(y)y^{s-1}\,dy$.
+The weight bounds permit a contour shift to $\Re s=-1/4$, giving
+\[
+ O_{f,H,\Phi}\bigl((J|\nu|)^C X^{C'\rho}T^{-1/4}\bigr).
+\]
+There is no residue. Summing over $\nu$, over the dyadic ranges,
+and over $u_1$ (for which
+$\sum_{u_1\mid H^\infty}|\lambda(u_1)|/\sqrt{u_1}<\infty$)
+is bounded, up to a further $X^{O(\rho)}$, by
+\[
+ \frac XJ\,J^C(JX^{2\rho})^{C+1}
+       (X^{1-\rho}/J)^{-1/4}
+ =J^{2C+1/4}X^{3/4+O(\rho)}.
+\]
+This gives $O(J^{B_3}X^{3/4+C''\rho})$. Choose $\rho$ so that this is
+$O(J^{B_3}X^{7/8})$.
+
+\par\smallskip\noindent\emph{Choosing the truncation.}
+Since $J\le bd^2$, summing the two contour errors for $d\le Y$,
+and adjoining Equation~\eqref{nv:tail}, gives
+\[
+ O_{f,H,\Phi}\bigl(
+ b^{B_4}Y^{2B_4+1}X^{1-\kappa}
+ +b^{B_4}X^{1+\rho}/Y\bigr)
+\]
+for absolute $\kappa>0$ and $B_4$.
+Take $Y=X^\delta$ with $(2B_4+1)\delta<\kappa/2$, and then
+$\rho<\delta/4$. Both errors have the form in the statement,
+with absolute positive saving. This proves the lemma.
+\end{proof}
+
+\begin{proof}[Proof of Propositions~\ref{nv:local-witness} and~\ref{nv:rough}]
+For negative root number, Lemma~\ref{nv:moment} gives a nonzero sum of
+derivatives in every sufficiently large dyadic interval. One derivative
+is nonzero, and its central zero is already forced by the sign.
+
+For positive root number, use the nonnegative sequence
+\[
+ a_n=\mu^2(n)\mathbf1_{n\equiv a\ (H)}
+      L(1/2,f\otimes\chi_{\sigma n})\Phi(n/X).
+\]
+Its nonnegativity follows from the Waldspurger formula
+\cite{Waldspurger1981}. The lemma gives a dimension-one sieve density
+$\beta_p$ and an absolute exponent of distribution:
+choose $\theta>0$ with $(B+1)\theta<\eta$ and put $D=X^\theta$.
+The total remainder for squarefree $b\le D$ is
+\[
+ O_{f,H,\Phi}(D^{B+1}X^{1-\eta})=o(X/\log X).
+\]
+The density estimate also gives the interval product condition
+\[
+ \prod_{\substack{w\le p<z\\p\nmid H}}(1-\beta_p)^{-1}
+ \le \frac{\log z}{\log w}
+       \left(1+\frac{C_{f,H}}{\log w}\right)
+ \qquad(2\le w<z).
+\]
+Indeed, comparison with $(1-p^{-1})^{-1}$ contributes a logarithmic
+error $O(\sum_{p\ge w}p^{-2})$, and Mertens' product estimate gives
+the displayed bound; finitely many small primes are absorbed in
+$C_{f,H}$.  The same comparison gives
+\[
+ \prod_{\substack{p<z\\p\nmid H}}(1-\beta_p)\asymp_{f,H}(\log z)^{-1}.
+\]
+The lower linear sieve \cite[Theorem~11.13]{FriedlanderIwaniec2010},
+with $z=D^{1/s}$ and any fixed $s>2$, therefore gives positive
+remaining weight. Since $z=X^{\theta/s}$ tends to infinity, its
+surviving parameters eventually avoid all primes up to any prescribed bound.
+Every surviving parameter is prime to $H$ and has all other prime
+factors at least $z$. Since $n<2X$, its number of prime factors is
+bounded by any fixed integer greater than $2s/\theta$ for sufficiently
+large $X$. This proves Proposition~\ref{nv:rough} and the positive-sign
+part of Proposition~\ref{nv:local-witness}. Enlarging $H$ and choosing
+a reduced CRT lift excludes any additional finite prime set.
+\end{proof}
+
+The constants in the size threshold may change when the curve or
+the support is changed. The exponent $\theta$ and the factor-count
+bound do not. In particular, a previously chosen twist can be
+absorbed into the curve before this proposition is applied.
diff --git a/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/build/sections/pointwise.tex b/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/build/sections/pointwise.tex
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+\section{Assembly at a fixed curve}\label{pw:section}
+
+Fix $E/\mathbb Q$ with nonzero rational two-torsion.  The even construction
+has two uses: it proves the corank-zero case and supplies the uniform
+estimate needed to normalize the odd coefficients.  After that normalization,
+the corank-one argument fixes a nonvanishing companion and constructs the
+two simultaneous coefficient tests required for ring-class transfer.
+
+\subsection{Interfaces and conventions}\label{pw:interfaces-section}
+
+We use the fixed-support conventions of Section~\ref{sec:preliminaries}.
+Thus all variable prime factors are odd primes of good reduction,
+$p^*=(-1/p)p$, and a filter fixes the real sign and unit squareclasses
+at a finite support containing $2N$, with finitely many further exclusions.
+Recall
+\[
+ t_p=\dim_{\mathbb F_2}E[2]^{\operatorname{Fr}_p},\qquad
+ w(h)=\frac12\sum_{p\mid h}t_p.
+\]
+Whenever the two-primary group is finite, we write
+\[
+s(h)=\log_2|\Sha(E^{(h)}/\mathbb Q)[2^\infty]|.
+\]
+This includes a base of Selmer corank zero and every parameter of analytic
+order zero or one.  It is distinct from $s_2(E^{(h)})$, the Selmer corank.
+The normalized central value is denoted $L_*(h)$, and $v=v_2$ with
+$v(2)=1$.
+
+Here are the preceding results in the form used below.  Their roles are
+different: coefficient units give upper bounds, interpolation returns those
+bounds to an unknown base, and the ring-class lower bound makes the odd
+normalization possible.  The latter step is carried out in
+Proposition~\ref{pw:odd-lower}.
+
+\begin{proposition}[Collected interfaces]\label{pw:interfaces}
+The following statements hold for the fixed curve and fixed local data.
+\begin{enumerate}
+\item The cyclotomic missing-vertex statement of
+Theorem~\ref{cy:missing-vertex} applies to any fixed base $h_0$ of Selmer
+corank zero.  If arbitrarily large binary cubes based at $h_0$, with no old
+prime removed, have nonzero central values at every other address and satisfy
+\[
+ v(L_*(h_x))-2w(h_x)-s(h_x)\le B\quad(x\ne0),
+\]
+then the base central value is nonzero and satisfies the same inequality
+with $B$ replaced by $B+C$.  The constant $C$ depends only on the fixed curve
+and local data, not on $h_0$, its number of prime factors, or the cube
+dimension.  The dimension required for the argument may depend on the base.
+The cyclotomic lower bound at an analytically nonvanishing parameter is also
+available.
+\item Corollary~\ref{rd:ring-lower} has the following uniform form.  If
+$k<0$ is an odd fundamental discriminant with at most $A$ prime factors,
+$K=\mathbb Q(\sqrt{k})$ is neither an extra-unit field nor the CM field,
+$k$ is a square at $2Nh$, and the twists $h,hk$ have respective analytic
+orders one and zero, then their ring-class Heegner sum has free-line index
+$j_h(k)$ satisfying
+\begin{equation}\label{pw:ring-lower-interface}
+ 2j_h(k)\ge 4w(h)+s(h)+s(hk)-C_{E,A}.
+\end{equation}
+For a fixed such $k$, Theorem~\ref{rd:missing-vertex} applies to
+cubes based at $h=1$ whose new primes all split in $K$.  If
+$s_2(E/K)=1$ and the nonzero addresses have product analytic order one and
+uniformly bounded
+$2j_h(k)-4w(h)-s(h)-s(hk)$, the base product has analytic order one.
+\item The even coefficient system, exact same-filter square comparison,
+divided operators, and trace identities of
+Proposition~\ref{co:even} and Lemmas~\ref{co:operators},
+\ref{co:traces}, and \ref{co:inertia} are available.
+After a uniform lower bound for the relevant point indices has been proved,
+the moving odd coefficient system and its stabilized minimum are available
+as in Proposition~\ref{co:odd} and Lemma~\ref{co:minimum}.  The resulting unit symbols
+satisfy the contraction, good-twin, and primary-isolation rules of
+Proposition~\ref{co:symbol-interface}.  An even unit gives
+\begin{equation}\label{pw:even-unit-interface}
+ L_*(h)\ne0,\qquad v(L_*(h))\le 2w(h)+C;
+\end{equation}
+an odd unit gives analytic order one and
+\begin{equation}\label{pw:odd-unit-interface}
+ j(P_h)\le w(h)+C.
+\end{equation}
+The constants are independent of the number and sizes of the variable
+primes.  The fixed genus partner used to define $P_h$ is denoted $h_*$.
+\item Arithmetic realization and the forest and address statements of
+Lemmas~\ref{gr:realization} and \ref{gr:forest} and
+Theorem~\ref{gr:address} hold.
+In particular, separate nonzero terminal polynomials may be addressed
+simultaneously.  Primary auxiliaries preserve all fixed local classes,
+have neutral incidence to specified stationary external vertices, and may
+be required to split in any compatible fixed fields.  They contribute no
+prime at address zero.
+\end{enumerate}
+\end{proposition}
+
+\begin{proof}
+These are the cited results, with the arithmetic ring-class hypotheses
+verified by Proposition~\ref{rd:assembly}.  The even coefficient minimum
+uses the cyclotomic lower bound; the odd detector of
+Proposition~\ref{co:odd} still needs the arithmetic lower bound for
+Lemma~\ref{co:minimum}.  Neither interpolation statement nor the even
+symbol uses that odd normalization.
+\end{proof}
+
+The analytic witnesses and the rough completion used below are supplied by
+Propositions~\ref{nv:local-witness} and \ref{nv:rough}; the latter has an absolute
+prime-count bound.  The bounds in Proposition~\ref{pw:interfaces}, unlike the
+eventual sizes of the primes chosen by nonvanishing or Chebotarev, must be
+uniform in the parameters indicated there.
+
+\begin{lemma}[Isogeny orientation and coranks]\label{pw:isogeny-corank}
+Recall that a rational isogeny preserves analytic rank, Mordell--Weil rank,
+and $2^\infty$-Selmer corank.  If $k$ is a quadratic discriminant and
+$K=\mathbb Q(\sqrt{k})$, then
+\begin{equation}\label{pw:quadratic-corank}
+ s_2(E/K)=s_2(E/\mathbb Q)+s_2(E^{(k)}/\mathbb Q).
+\end{equation}
+For every $E/\mathbb Q$ one has, separately,
+\begin{equation}\label{pw:kummer-corank}
+ s_2(E)=r(E)+
+ \operatorname{corank}_{\mathbb Z_2}\Sha(E/\mathbb Q)[2^\infty].
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+These are Lemmas~\ref{pr:invariance} and \ref{pr:ranks}.  In particular,
+the quadratic identity is an identity of coranks, not an integral
+direct-sum decomposition.
+\end{proof}
+
+If the isogeny class contains a curve with full rational two-torsion, we use
+that curve.  Otherwise choose a rational two-isogeny $E\to E'$ and write
+$F=\mathbb Q(E[2])$, $F'=\mathbb Q(E'[2])$; both fields are quadratic.
+A prime split in $F$ is called split, and one inert in $F$ is called simple.
+Their weights are respectively $1$ and $1/2$.  A simple prime is good if it
+is also inert in $F'$.  At such a prime
+\begin{equation}\label{pw:good-simple}
+ \#E(\mathbb F_p)\equiv2\pmod4.
+\end{equation}
+Indeed there is exactly one nonzero rational two-torsion point in the
+reduction.  If the group order were divisible by four, its two-primary
+subgroup would contain a rational point of order four.  Its image under the
+two-isogeny would supply on $E'$ a rational two-torsion point different from
+the dual kernel, contradicting inertness in $F'$.  Thus the good-twin rule
+applies.  When $F=F'$, every simple prime is good.
+A prime split in $F$ and inert in $F'$ is a \emph{donor}: attaching its
+label to a bad simple label makes that label good.  This terminology
+describes finite label operations; the required prime supplies will be
+constructed below.
+
+\subsection{Seeds and finite supplies}\label{pw:seeds-section}
+
+The word ``label'' below means an individually realizable fixed-support
+elementary Frobenius label, with orientations when required.  It does not
+include a self-residue of a prime generator.  All contractions and
+comparisons are those of Proposition~\ref{co:symbol-interface} and
+Lemma~\ref{gr:forest}.
+
+\begin{lemma}[Minimum seeds]\label{pw:minimum-seeds}
+\begin{enumerate}
+\item In an all-split alphabet, suppose a unit symbol has a nonempty
+witness.  At the least positive vertex count $r$ of such a witness, all
+mutual edge bits are immaterial.  The symbol is determined by the individual
+fixed-support labels.  This remains true if the empty configuration is a
+unit, provided no contraction to the empty configuration is used.
+\item In the quadratic case, suppose the symbol has a nonzero witness,
+the alphabet is closed under the allowed contractions, and it contains
+the good twin operations.  There are
+$r\ge0$ and $s_1\ge1$ such that for every $s\ge s_1$ of the permitted simple
+parity, the minimum split count of a unit with $s$ simple vertices is $r$.
+At these counts all edge bits are immaterial.  A chosen minimum seed may be
+padded by arbitrarily many good twin pairs.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+For the first assertion, a derivative in an edge of an $r$-vertex graph is a
+symbol on $r-1$ vertices.  If $r\ge2$, this graph remains nonempty and has
+value zero by minimality.  If $r=1$, there is no edge.  The omitted-vertex
+description then leaves exactly the fixed-support labels.
+
+For the second assertion let $r(s)$ be the minimum split count at simple
+count $s$, with value $+\infty$ when no witness exists.  Once a finite value
+occurs, good-twin insertion gives $r(s+2)\le r(s)$.  A nonincreasing sequence
+of nonnegative integers stabilizes.  Choose $s_1\ge1$ in that range.  A
+derivative of an edge with a split endpoint reduces the split count and
+preserves the simple count; it is therefore zero at the minimum.  A
+simple--simple derivative is zero by the symbol rule.  Twin insertion gives
+the last assertion.
+\end{proof}
+
+Call a list partitionable if some permitted contraction forest has a unit
+symbol as its output.  A contraction certificate proves that its terminal
+polynomial is nonzero: the indicated iterated finite difference is nonzero.
+
+\begin{lemma}[Finite label reserves]\label{pw:reserves}
+Fix a minimum seed and a finite label alphabet.
+\begin{enumerate}
+\item In the all-split rational alphabet, a list with the same total label
+as the seed and sufficiently many copies of its labels is partitionable.
+\item In the quadratic alphabet, retain $r$ split anchors with the seed
+labels, and suppose the list and seed have the same total rational label
+on the projection used by the symbol.  Suppose there are sufficiently
+many simple vertices of the required parity and sufficiently many split
+reserves of each realizable rational shift.  If $F\ne F'$, also suppose
+\begin{equation}\label{pw:donor-balance}
+ \Delta=\#\{p:\ p\text{ split in }F\text{ and inert in }F'\}
+       -\#\{p:\ p\text{ inert in }F\text{ and split in }F'\}>C_0.
+\end{equation}
+Here $C_0$ depends only on the anchors and the reserves.  Then all other
+split vertices can be attached to simple vertices so that the result is
+the minimum seed and good twin pairs.  The twins may match a prescribed
+finite enrichment of the rational labels.  The base simple labels may be
+specified on any projection on which the required total agrees.
+\end{enumerate}
+All reserve requirements depend only on the finite alphabets, seed, and
+projection.  They do not depend on the original list length.
+\end{lemma}
+
+\begin{proof}
+In the first case retain the first $r-1$ seed labels and merge all other
+vertices to the last cluster.  Sufficient copies ensure that this cluster
+is nonempty.  Its label is forced by the total, hence is the last seed
+label.  Lemma~\ref{pw:minimum-seeds} makes its unknown edges immaterial.
+
+For the second case, the rational labels of split attachments form the
+subgroup with trivial $F$-coordinate.  They act transitively on the simple
+coset, so every prescribed change of a simple label has a realizable split
+shift.  Reserve sufficiently many representatives of every such shift.
+Remove these reserves and the anchors from the donor pool.  If $F\ne F'$,
+\eqref{pw:donor-balance}, with $C_0$ exceeding the number removed, leaves a
+distinct donor for each bad simple vertex.  Attach these donors; every
+simple vertex is now good.  If $F=F'$, this operation was unnecessary.
+
+Pair equal enriched good labels.  By withholding a bounded number of pairs,
+leave between $s_1$ and $s_1+C_1$ unpaired vertices, with the correct parity;
+$C_1$ depends only on the number of enriched labels.  Use reserved shifts
+to turn the excess unpaired vertices into good pairs and to set the first
+$s_1-1$ base positions to the desired seed labels.  The seed labels
+themselves need not all be good.  Attach every remaining nonanchor split
+vertex to the last base position.  Its projected label is forced by the
+total, the anchors, the other base positions, and the twin pairs.  It is
+the required last seed label.  Each cluster contains one simple vertex,
+so every contraction is permitted.  Only a bounded number of reserve
+shifts was used.
+
+When incidences to a retained split anchor are needed only to remove twins,
+they can be prescribed after the contracted labels are fixed, by the
+independent terminal-edge realization.  Fixed stationary vertices instead
+enter the finite label enrichment from the outset.  These two uses do not
+increase the number of label types with the original list length.
+\end{proof}
+
+\begin{lemma}[Neutral packets]\label{pw:packets}
+For any of the finite reserve problems above with prescribed rational total
+zero, one can choose a finite packet of terminal labels with rational total
+zero such that every nonempty union of copies of the packet has the required
+supplies.  In the quadratic case with \(F\ne F'\), its donor surplus
+may be made arbitrarily large.  When \(F=F'\), every simple label is
+good and no donor surplus is required.  Compatible fixed splitting
+conditions may be included whenever
+the needed individual labels and shifts remain realizable.
+\end{lemma}
+
+\begin{proof}
+Choose ample copies of every required label, including anchors and shifts.
+In the quadratic case with \(F\ne F'\), add enough donors to exceed
+all the losses from the
+other labels and the prescribed reserve margin.  Duplicate every full
+label in the resulting list.  Duplication gives total zero in every
+rational elementary coordinate and preserves any strict donor surplus.
+Choose the margins so that after reserving the finitely many anchors in
+any one packet, the union of any positive number of packets still meets
+the reserve requirements.  All labels are realized by distinct fresh
+primes; duplication never means repeating an actual prime factor.
+\end{proof}
+
+\subsection{The even estimate and the corank-zero case}
+\label{pw:lower-section}
+
+\begin{proposition}[Partitionable even estimate]\label{pw:partitionable-even}
+Fix an even symbol furnished by Proposition~\ref{pw:interfaces}.  For every
+partitionable parameter $h$ on its filter with $s_2(E^{(h)})=0$,
+\begin{equation}\label{pw:partitionable-bound}
+ L_*(h)\ne0,\qquad v(L_*(h))-2w(h)-s(h)\le C.
+\end{equation}
+The constant depends only on the symbol and fixed local data.
+\end{proposition}
+
+\begin{proof}
+If the parameter has no prime factors, a contraction certificate means
+that the empty symbol itself is a unit.  Equation~\eqref{pw:even-unit-interface}
+then proves the assertion directly.  Otherwise the stationary terminal
+set is nonempty.
+Keep the prime factors of $h$ stationary.  Their terminal polynomial is
+nonzero by its contraction certificate.  The stationary one-symbol case
+of Theorem~\ref{gr:address} gives arbitrarily large binary cubes based at
+$h$, with only primary new primes, whose other addresses have unit symbol.
+Equation~\eqref{pw:even-unit-interface} and $s(h_x)\ge0$ give the uniform
+upper bound required by the cyclotomic transfer.  The base corank is zero,
+so that transfer gives \eqref{pw:partitionable-bound}.  A base-dependent
+dimension threshold does not affect the asserted constant.
+\end{proof}
+
+The original corank-zero base can now be treated without any odd
+coefficient system or companion field.
+
+\begin{proposition}[Even original base]\label{pw:even-base}
+If $s_2(E)=0$, then $a(E)=0$.
+\end{proposition}
+
+\begin{proof}
+Lemma~\ref{pr:invariance} gives even functional sign.  Fix an even symbol
+on the filter of $1$, including all auxiliary class-group tests.  If its
+value at $1$ is a unit, analytic nonvanishing is already detected.
+Otherwise use a nonempty minimum seed in the full-torsion case, or a
+stabilized quadratic seed in the other case.
+
+For each $b$, assign a neutral packet from Lemma~\ref{pw:packets} to
+every nonzero linear form on $\mathbb F_2^b$, and activate its primes
+by that form.  At every nonzero address at least one packet is active;
+their union satisfies the reserve problem and has a nonzero terminal
+polynomial.  At zero no terminal is active.  The single-symbol case of
+Theorem~\ref{gr:address} adds primary auxiliaries and makes the symbol a
+unit at every nonzero address.  All local classes remain those of $1$.
+Equation~\eqref{pw:even-unit-interface}, followed by cyclotomic transfer
+at the original corank-zero base, gives $a(E)=0$.
+\end{proof}
+
+\subsection{The lower bound needed by the odd coefficients}
+\label{pw:odd-lower-section}
+
+For the odd construction, first fix the original coefficient and
+class-group support and its unit tests.  Then choose the genus partner
+$h_*$ as in Section~\ref{co:section}, adjoin its prime factors to the
+support, and refine the unit tests there.  All variable $h$ avoid this
+support.  The positive filter below is this prepared filter, fixed before
+$h$ varies or the odd coefficients are normalized.
+
+We next use the even estimate at analytically nonvanishing parameters,
+not the corank-zero conclusion just proved.  The companion $k'=k'(h)$
+in this step may vary with $h$, but its number of prime factors will be
+bounded.  It is distinct from both the detector's fixed genus partner
+$h_*$ and the fixed companion used in the final ring-class transfer.
+
+At a parameter already known to have analytic rank zero, the estimate
+\eqref{pw:partitionable-bound} transfers between either orientation of a
+fixed two-isogeny.  The isogeny formula for the BSD quotient gives a bounded
+change in $s(h)+2w(h)$.  At an active odd good prime the component-group
+valuation is $t_p$, and the two active contributions are included in the
+two weights.  At the fixed support only finitely many local twisting
+classes occur.  Differential, period, and torsion indices are bounded by
+the fixed isogeny and its reverse, and there is no regulator at rank zero.
+This uses an isogeny identity for known finite groups, not BSD or converse
+finiteness at an unknown parameter.
+
+\begin{lemma}[A bounded companion for a varying odd parameter]
+\label{pw:rough-companion}
+Fix a positive odd-sign filter, including the coefficient and class-group
+support used below, and a negative genus partner $h_*$ with
+$L(E^{(h_*)},1)\ne0$, chosen using Proposition~\ref{nv:local-witness} and then
+excluded from the variable parameters.  There are constants $A,C$ such
+that every $h$ on this filter of analytic order one admits a negative odd
+fundamental discriminant $k'$ for which
+\begin{enumerate}
+\item $(k',2Nh)=1$, $k'$ is a square at every place of $2Nh$, and
+$\mathbb Q(\sqrt{k'})$ is nonexceptional;
+\item $\omega(k')\le A$;
+\item $L(E^{(hk')},1)\ne0$ and
+\begin{equation}\label{pw:rough-bound}
+ v(L_*(hk'))\le 2w(hk')+s(hk')+C.
+\end{equation}
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+At the fixed class-group support and the other fixed label tests, prepare
+an even negative-index symbol for each possible total type at this
+support, and for both isogeny orientations if needed.
+There are finitely many such preparations.  The positive index $|hk'|$
+has nonzero total rational type: its negative-discriminant parity is
+already nontrivial among the tests at $2$.  Thus in the full-torsion case
+a unit seed has positive count.  In the quadratic case use the stabilized
+seed of Lemma~\ref{pw:minimum-seeds}.
+
+When $F\ne F'$, interchanging the isogeny orientation interchanges the two
+counts in \eqref{pw:donor-balance}.  Select the orientation with initial
+balance at least zero.  The existing donors can then pay for arbitrarily
+many existing bad simples.  They do not have to be supplied by new primes.
+
+Let $R$ be the absolute prime-count bound in Proposition~\ref{nv:rough}.  Insert
+in $k'$ a finite dummy supply that meets the anchors, minimum simple count,
+and reserve requirements for every one of the finitely many prepared
+symbols.  When \(F\ne F'\), add donor slack paying for every reserved
+loss, the new dummy bad simples, and a further possible loss of $R$
+from the later rough factor.  When \(F=F'\), all simple primes,
+including the later rough primes, are already good; only the simple
+and split-reserve supplies are needed.  Add a fresh ramified prime if
+needed to exclude the exceptional
+imaginary fields.  The number of dummy primes is bounded independently
+of $h$.  Their labels and initial incidences may be chosen by
+Lemma~\ref{gr:realization}.
+
+Absorb $h$ and the dummy product into the curve.  Apply
+Proposition~\ref{nv:rough} to complete $k'$ by a squarefree factor coprime to all
+primes already used.  Prescribe its sign and its local unit classes so
+that the final $k'$ is an odd negative fundamental discriminant, is a
+square at $2Nh$, and has the required original local classes at $2N$.
+These conditions give a nonempty reduced progression by the Chinese
+remainder theorem.  Its root number is even: since $E^{(h)}$ has odd
+sign and $k'$ is negative and square at every prime of $2Nh$,
+the quadratic twist formula multiplies that sign by
+$\chi_{k'}(-N_{E^{(h)}})=-1$.  Thus this is the positive-root-number
+case of Proposition~\ref{nv:rough} for the absorbed curve.
+Additional fixed primes may be excluded.  The proposition makes the
+central value at $hk'$ nonzero with at most $R$ new factors.  Its size
+threshold may depend on $h$ and all these congruences; $R$ does not.
+
+No particular resulting unit class at the auxiliary class-group support
+was assumed in advance: dummy supplies were prepared for every possible
+total there.  For the actual total, Lemma~\ref{pw:reserves} makes $hk'$
+partitionable.  Analytic nonvanishing gives Selmer corank zero by the
+forward theorem.  Proposition~\ref{pw:partitionable-even} therefore gives
+\eqref{pw:rough-bound}, with the isogeny comparison above if the other
+orientation was used.  The finite preparation and the bound on dummy and
+rough factors give $A,C$ independent of $h$.
+\end{proof}
+
+\begin{lemma}[Height comparison]\label{pw:height-comparison}
+Here $j_h(k)$ is the index of the conductor-$|h|$ ring-class sum over
+$\mathbb Q(\sqrt{k})$, whereas $j(P_h)$ is the index of the conductor-one
+genus sum over $\mathbb Q(\sqrt{hh_*})$.
+With the fixed modular parametrization and the fixed genus partner $h_*$,
+the explicit Gross--Zagier formula gives, whenever the relevant factors
+have analytic orders one and zero,
+\begin{equation}\label{pw:height-ratio}
+ 2j_h(k)-2j(P_h)=v(L_*(hk))+O(1).
+\end{equation}
+For $k$ with bounded prime count as in
+Lemma~\ref{pw:rough-companion}, the error is uniform in $h,k$; for a fixed
+$k$ it is uniform in $h$.
+\end{lemma}
+
+\begin{proof}
+Use the character-sum normalization of the explicit Gross--Zagier formula
+in \cite[Theorem~1.1]{CaiShuTian2014}.  The restriction of $\chi_h$ to
+$\mathbb Q(\sqrt{k})$ has exact ring-class conductor $|h|$: each odd
+prime dividing $h$ splits and retains its ramified quadratic character.
+The genus character over $\mathbb Q(\sqrt{hh_*})$ is unramified, so the
+sum defining $P_h$ has conductor one.  Their canonical
+heights are fixed modular factors times, respectively,
+\[
+ |h|\sqrt{|k|}\,L'(E^{(h)},1)L(E^{(hk)},1),
+ \qquad
+ \sqrt{|hh_*|}\,L'(E^{(h)},1)L(E^{(h_*)},1).
+\]
+In that theorem the multiplier is $c\sqrt{|D|}$, the level-discriminant
+factor is $2^{\mu(N,D)}$, and the unit factor is
+$[\mathcal O_c^\times:\mathbb Z^\times]^2$.  Here every level prime
+splits, so $(N,D)=1$ and $\mu(N,D)=0$; the excluded extra-unit fields
+make the unit factor $1$ in both applications.
+Take their ratio.  The derivative and the rank-one regulator cancel;
+the remaining varying real factor is $\sqrt{|hk|}$, precisely the factor
+in $L_*(hk)$.  Comparison of the rational free line with its two quadratic
+projections has bounded index because their composites are multiplication
+by $2$.  The modular parametrization and the factor involving $h_*$ are
+fixed.  The level is split and the discriminant and conductor are prime
+to it; no growing conductor-prime product occurs in this normalization.
+Local differential and period comparisons have finitely many possibilities
+at the fixed support.  The extra-unit fields have been excluded, and the
+number of additional places in the varying $k$ case is bounded.  These
+observations give the asserted bounded valuation error.
+\end{proof}
+
+\begin{proposition}[Uniform lower bound for the odd detector]
+\label{pw:odd-lower}
+Every analytically rank-one $h$ in the prepared positive filter of
+Lemma~\ref{pw:rough-companion} satisfies
+\begin{equation}\label{pw:odd-lower-bound}
+ 2j(P_h)\ge 2w(h)+s(h)-C.
+\end{equation}
+In particular $j(P_h)\ge w(h)-C$ uniformly in $h$ and its prime count.
+\end{proposition}
+
+\begin{proof}
+Choose $k'$ by Lemma~\ref{pw:rough-companion}.  Combine
+\eqref{pw:ring-lower-interface}, \eqref{pw:height-ratio}, and
+\eqref{pw:rough-bound}.  Since the factors are coprime,
+$w(hk')=w(h)+w(k')$, and $w(k')\le\omega(k')\le A$.  Hence
+\begin{align*}
+ 2j(P_h)
+ &\ge 4w(h)+s(h)+s(hk')-v(L_*(hk'))-C\\
+ &\ge 2w(h)+s(h)-2w(k')-C\\
+ &\ge 2w(h)+s(h)-C'.
+\end{align*}
+Only even symbols and independently nonvanishing companions were used.
+Thus this proof precedes, and does not assume, the odd minimum-depth
+normalization.
+\end{proof}
+
+\subsection{From the moving odd coefficients to finite graphs}
+\label{pw:odd-stages}
+
+Proposition~\ref{pw:odd-lower} now supplies the arithmetic premise of
+Lemma~\ref{co:minimum}.  We obtain an odd symbol with fixed minimum
+normalized depth and the uniform unit bound
+\eqref{pw:odd-unit-interface}.  To use the finite seed and packet lemmas,
+we also need to stabilize its graph labels and values, not just this
+numerical depth.
+
+Keep the ultrafilter used in Lemma~\ref{co:minimum}.  The intrinsic fixed
+support includes the genus partner $h_*$; at an actual stage it also
+includes the detecting prime $\ell$.  That prime adds at most two places
+in any fixed quadratic field.  The unit and class-group description of
+$V_S(F)$ therefore bounds the dimension of the elementary label space
+independently of the stage.  At $\ell$ the Schwartz function in
+Lemma~\ref{co:theta-bridge} uses only the quadratic residue-line character:
+increasing the Kummer precision does not impose a character of growing
+order on the variable primes.  The old-support characters are fixed.
+
+Choose bases and restrict to an ultrafilter-large set of stages on which
+the finite label spaces, their addition tables, conjugation and projection
+maps have one abstract form.  At every fixed weight bound there are only
+finitely many typed graphs and symbol truth tables.  Select the table
+occurring on an ultrafilter-large set.  These selected tables are
+compatible under restriction and define one limit symbol.  Its contraction
+and twin identities follow from the finite-stage identities.  The witness
+sequence of Lemma~\ref{co:minimum} has bounded weight, hence takes only
+finitely many graph types in this alphabet; one type is a witness in the
+selected limit table.
+
+If the empty configuration is a unit, a sufficiently advanced actual stage
+already detects analytic order one at $h=1$.  Otherwise the limit witness
+is nonempty, so Lemma~\ref{pw:minimum-seeds} supplies the minimum seeds
+and stabilized simple count needed below.  Fix them in this remaining
+case; Lemmas~\ref{pw:reserves} and \ref{pw:packets} apply to their finite
+label alphabets.  Every
+particular contraction or twin comparison uses only a bounded graph.
+Its selected truth table agrees with the actual symbol on an
+ultrafilter-large set of stages.  Thus any prescribed finite collection
+of such comparisons holds at one sufficiently advanced stage.
+
+Later we will add the fixed support of a separate companion and the label
+projection read by its even test.  Once compatible lifts have been
+verified, we may stabilize this finite enrichment by restricting the
+\emph{same} ultrafilter.  This preserves the intrinsic odd symbol, its
+minimum, and its chosen seeds.  The actual stage is selected only after
+the cube dimension and all its finite networks have been chosen: their
+maximum weight gives a finite list of table and precision requirements.
+No intersection over all weight bounds, and no single stage working for
+every cube dimension, is required.
+
+\subsection{A nonempty split-field symbol with full two-torsion}
+\label{pw:split-section}
+
+We give the extra argument required when $E[2]$ is rational.  It addresses
+the possibility that the usual even minimum is attained only at the empty
+configuration after restricting all new primes to split in the companion
+field.
+
+\begin{lemma}[Recurrence congruences]\label{pw:recurrence}
+Let $p$ be odd, $a\equiv p+1\pmod4$, and let
+\[
+ b_0=1,\quad b_1=a-\kappa,\quad
+ b_{s+1}=ab_s-pb_{s-1}.
+\]
+If $\kappa=\pm1$, then $b_{s+1}-b_s$ is even and
+\[
+ \frac{b_{s+1}-b_s}{2}\equiv\frac{a-\kappa-1}{2}\pmod2
+ \quad(s\ge0).
+\]
+If $\kappa=0$, put $b_{-1}=0$; then $b_s-b_{s-1}$ is odd for every
+$s\ge0$.  Finally, for $\kappa=1$, if $a\equiv p+1\pmod{2^m}$ then
+$b_s\equiv p^s\pmod{2^m}$ for every $s\ge0$.
+\end{lemma}
+
+\begin{proof}
+Modulo $2$ the recurrence is $b_{s+1}=b_{s-1}$.  Its initial values prove
+the parity assertions.  For $d_s=b_{s+1}-b_s$, the recurrence for $d_s$
+is the same.  Directly,
+$d_1-d_0\equiv(p-1)(p-\kappa)\equiv0\pmod4$ when $\kappa=\pm1$.
+Thus $d_s/2$ is constant modulo $2$.  The last assertion follows by
+substituting the sequence $p^s$ in the recurrence and checking its first
+two terms.
+\end{proof}
+
+\begin{lemma}[Nonempty $K$-split witness]\label{pw:split-witness}
+Suppose $E[2]$ is rational.  Choose a negative unit $k$ of a fixed even
+symbol for $E$, prime to all previous fixed support and to the ramification
+support of $E[8]$.  Put $K=\mathbb Q(\sqrt{k})$, chosen nonexceptional.
+Let $S_0$ contain the complete fixed support of the coefficient system,
+$2N$, the primes of $k$, and primes generating the class group of $K$.
+On the positive filter of $1$ square at $S_0$, there is an
+even unit test on nonempty products of primes split in $K$ with these
+properties:
+\begin{enumerate}
+\item it has a nonempty witness;
+\item it satisfies the oriented split contraction and primary-isolation
+rules over $K$, for contractions with nonempty output;
+\item a unit at $D$ implies
+\begin{equation}\label{pw:split-unit-bound}
+ L(E^{(kD)},1)\ne0,\qquad v(L_*(kD))\le 2\omega(D)+C_k.
+\end{equation}
+\end{enumerate}
+This conclusion uses the integral trace and local identities in
+Proposition~\ref{pw:interfaces}(3).
+\end{lemma}
+
+\begin{proof}
+If the ordinary coefficient test already has a nonempty split witness,
+there is nothing to construct.  Otherwise we first obtain an additional
+trace congruence modulo $8$.  That congruence makes one further division
+of the coefficient form integral.  We verify its contraction identities,
+then use a prime inert in $K$ to force a nonzero test at a split prime.
+
+\emph{Extra depth if the ordinary test has no witness.}
+Use the ordinary even coefficient form for $E^{(k)}$, normalized by
+$a(1)=1$.  The positive filter is chosen so that $kD$ lies in the original
+negative filter.  Since $k$ attains its least normalized depth, the exact
+ratio comparison of Proposition~\ref{co:even} gives
+\begin{equation}\label{pw:split-basic-depth}
+ v(a(D))\ge\omega(D)
+\end{equation}
+at every allowed squarefree $D$.  The factor $|k|$ cancels from this ratio;
+one is not subtracting unrelated bounded-error inequalities.
+
+If equality holds at a nonempty product of $K$-split primes, use the usual
+even bit, projected from oriented $K$-labels to rational labels.  It has
+all the required properties by Proposition~\ref{co:symbol-interface}.  We
+therefore assume for the rest of the proof that equality never occurs
+there.  By the rational-phase assertion in Proposition~\ref{co:even},
+normalization by $a(1)=1$ makes each nonzero squarefree coefficient a
+rational number times a root of unity.  Its valuation is an integer,
+not just an element of the possibly ramified coefficient field's
+discrete value group.  Thus strict inequality implies
+\begin{equation}\label{pw:split-extra-depth}
+ v(a(D))\ge\omega(D)+1
+ \quad\text{for every nonempty allowed all-$K$-split }D.
+\end{equation}
+
+Write $g=\sum a(n)q^n$ for the form.  Its characters satisfy
+$\psi/\eta=\chi_{-4}$, so $\eta$ is odd.  For a set $U$ of $K$-split
+primes put
+\[
+ B_U=2^{-|U|}\prod_{p\in U}(U_p-\eta(p)V_p)g,
+ \qquad
+ \Theta_J=\sum_{m>0,\ (m,S_0)=1}\eta(m)m q^{Jm^2}.
+\]
+All expansions retain the prescribed support sieves.  The ordinary divided
+form is
+\[
+ H_U=\frac12\left(B_U-\sum_{J\subseteq U}B_U[J]\Theta_J\right).
+\]
+Here $B_U[J]$ denotes its coefficient at the squarefree index $J$.
+
+\emph{The resulting trace congruence.}
+We first prove
+\begin{equation}\label{pw:split-mod8}
+ a_p(E^{(k)})\equiv p+1\pmod8
+ \quad\text{for every allowed good }p\text{ split in }K.
+\end{equation}
+Apply the ordinary $H_{\{p\}}$ trace at a Frobenius lift $\phi_p$,
+testing coefficient $p^2$ after theta multiplication.  The form
+$H_{\{p\}}$ is already integral, and $\theta\equiv1\pmod2$;
+hence theta multiplication does not alter any of its reduced
+coefficients after these divisions.  On the
+conductor-one ramified principal-series part the identity is
+\[
+ U_p^2t(\phi_p)=U_p^3+d(\phi_p)U_p.
+\]
+On the unraised unary part it is the ordinary good-prime coefficient
+identity.  These are the only two types in this expression.  Consequently
+\[
+ (t(\phi_p)f)[p^2]=f[p^3]+(d(\phi_p)f)[p].
+\]
+Replace the trace by a fresh split prime $p'$ with the same finite data,
+including every support-unit test.  At each rational quadratic character
+unramified outside $S_0$, a Frobenius lift at $p$ has the character value
+of $p$.  Hence the matched $p'$ has the same value, and $pp'$ has value
+$1$.  Equivalently, $p'$ has the same unit squareclasses as $p$ at all
+places of $S_0$, so their product is square at all of them, including $2$.
+Since $\phi_p\in G_K$, the new prime is also split in $K$.
+Its coefficient on the left tests $pp'$; by
+\eqref{pw:split-extra-depth} it is zero modulo the maximal ideal after
+the two divisions.  The determinant term on the right is zero: the
+unary summand at $p$ has the same character and cancels there.
+
+On the right, the indices $p$ and $p^3$ in $B_{\{p\}}$ have underlying
+squarefree index $1$; they involve only $a(1),a(p^2),a(p^4)$.  The
+$J=\varnothing$ unary term has no coefficient at either nonsquare
+index.  Thus no other squarefree coefficient contributes there.
+For $a=a_p(E^{(k)})$, the phase-free square coefficients are
+$1,a-1,a(a-1)-p$.  Hence the remaining coefficient is, up to a unit,
+\[
+ H_{\{p\}}[p^3]
+ =\frac{a^2-2a-p+1-p(a-2)}4
+ =\frac{(a-1)(a-p-1)}4.
+\]
+It is zero modulo $2$.  Full rational two-torsion makes $a$ even, so
+$a-1$ is odd; this proves \eqref{pw:split-mod8}.  No individual total
+filter condition on $p$ was needed: its matched prime $p'$ makes $pp'$
+lie on the filter, and the square-index terms on the right come from $1$.
+
+\emph{One further integral division.}
+Now define
+\begin{equation}\label{pw:four-divided-form}
+ \widetilde H_U=\frac14
+ \left(B_U-\sum_{J\subseteq U}B_U[J]\Theta_J\right).
+\end{equation}
+We verify integrality at every coefficient.  Write a test index uniquely
+as $Jrm^2$, where $J\subseteq U$ and $r$ is squarefree and disjoint from
+$U$.  The underlying squarefree coefficient is at
+$D=(U/J)r$.  Removing the unit phases, a prime of $U/J$ contributes an odd
+factor $b_s-b_{s-1}$ with $\kappa=0$, and a prime of $J$ contributes
+$b_{s+1}-b_s$ with $\kappa=(D/p)$.  Thus
+\begin{equation}\label{pw:shell-valuation}
+ v(B_U[Jrm^2])\ge\omega(r)
+\end{equation}
+by \eqref{pw:split-basic-depth}, with one additional unit of valuation
+when $D$ is nonempty and all its primes split in $K$.
+
+If $r\ne1$ and all primes of $D$ split in $K$, the last inequality is
+at least $\omega(r)+1\ge2$.  If $D$ contains an inert prime, then $r$
+contains at least two: the total rational $K$-character is trivial on
+the filter, and all primes of $U$ split.  Then
+\eqref{pw:shell-valuation} is again at least $2$.  No unary term has
+this squarefree part.  This proves divisibility by $4$ for $r\ne1$.
+
+Suppose $r=1$ and $J\ne U$.  Then $D=U/J$ is nonempty and all split,
+so $B_U[J]$ is even.  This cancellation can be checked without dividing
+by a leading multiplier, which may be zero.  Factor from both shell
+coefficients the common phase and the scalar
+$2^{-|U|+|J|}a(U/J)$, whose valuation is at least $1$ by
+\eqref{pw:split-extra-depth}.  The remaining factors are the odd
+differences at $U/J$, the integral divided differences at $J$, and
+the ordinary square-index factors outside $U$.  By
+Lemma~\ref{pw:recurrence}, their product is congruent modulo $2$ to
+the product of their leading values, even when one of those values
+is zero modulo $2$.  Since $m$ is odd, multiplication by $m$ preserves
+this congruence.  The common scalar supplies the second factor $2$.
+Retaining the phase $\eta(m)$ therefore gives
+\[
+ B_U[Jm^2]\equiv B_U[J]\eta(m)m\pmod4.
+\]
+This is exactly the cancellation with $B_U[J]\Theta_J$.
+
+Finally let $J=U$, so $D=1$.  At a prime outside $U$ the recurrence gives
+$b_s\equiv p^s\pmod4$.  At a prime of $U$,
+\eqref{pw:split-mod8} gives $b_s\equiv p^s\pmod8$, and hence
+\[
+ \frac{b_{s+1}-b_s}{2}
+ \equiv \frac{a_p-2}{2}\,p^s\pmod4.
+\]
+Multiplying these congruences, again retaining the phases, gives
+$B_U[Um^2]\equiv B_U[U]\eta(m)m\pmod4$.  This proves integrality in
+the last case and establishes \eqref{pw:four-divided-form}.
+
+For the trace argument we specify the integral module being used.
+Write $\psi_U$ for the character of $B_U$.  The character of $\Theta_U$
+is also $\psi_U$, so its diamond correction is zero.  For the proper
+subsets the diamond formula is
+\[
+ \langle u\rangle\widetilde H_U
+ =\psi_U(u)\widetilde H_U+
+ \sum_{J\subsetneq U}\frac{B_U[J]}2
+ \frac{\psi_U(u)-\eta(u)(-J/u)}2\Theta_J.
+\]
+Each $B_U[J]/2$ here is integral, and each divided character difference
+is zero or a unit.  Let $\widetilde{\mathcal M}_U$ be the $\mathcal O$-module
+generated by $\theta\widetilde H_U$ and
+$\theta(B_U[J]/2)\Theta_J$ for $J\subsetneq U$, together with all their
+good-Hecke and diamond translates.  Its generators are integral by this
+formula and the good-Hecke coefficient formula.  The finite-module and
+Chebotarev-continuity argument of Lemma~\ref{co:traces} therefore applies
+to $\widetilde{\mathcal M}_U$.  A fresh trace kills every displayed unary
+correction after coefficientwise reduction, also after further fresh
+tests.  Thus the same coefficientwise reduced determinant and trace
+identities hold on the required orbit.
+
+\emph{The contraction and isolation identities.}
+On $\theta\widetilde H_U$, two fresh traces in $G_K$ give zero after
+reduction at every coefficient.  To see this, start with an arbitrary
+test index $Jrm^2$ and choose the two fresh primes outside that index
+and the level.  The filter imposes $\chi_K(D)=1$; since $U$ is all split,
+the number of inert factors in $r$ is either zero or at least two.
+The fresh traces replace $r$ by $rq_1q_2$, with both $q_i$ split.
+In the first case the valuation after division by $4$ is at least
+$\omega(r)+2+1-2=\omega(r)+1\ge1$.  In the second case it is at least
+$\omega(r)+2-2=\omega(r)\ge2$.  The unary terms disappear under these
+fresh tests.  Additional fresh $G_K$ traces only improve these estimates,
+so vanishing holds on the entire reduced trace orbit.  The trace identity
+therefore gives
+\[
+ t(gu^2)=t(g)\qquad(g,u\in G_K)
+\]
+on this reduced trace orbit.  Thus the tests factor through the elementary
+extension of $K$ unramified off the level.
+
+A single fresh $K$-split trace at coefficient $1$ reads the bit
+\begin{equation}\label{pw:deeper-bit}
+ \widetilde A(D)=
+ \left(2^{-\omega(D)-1}a(D)\text{ with its phase removed}\right)\bmod2
+\end{equation}
+for a nonempty all-split configuration.  The contraction rule follows
+from the same integral local identity as in Lemma~\ref{co:inertia}.
+For completeness its first shell, with $p\in U$ and a fresh trace $q$,
+has underlying configuration $D'=(U/p)q$, which is still nonempty, and
+its reduction is
+\[
+ c_p(D')\widetilde A(D'),\qquad
+ c_p(D')=\frac{a_p-(D'/p)-1}{2}\bmod2.
+\]
+Changing the oriented inertia bit changes $c_p$ once.  The omitted-prime
+identity gives the product label and removes the entire rational prime
+$p$ from the new ramification support.  Consequently its repeated
+contraction has precisely the forest interpretation of
+Lemma~\ref{gr:forest}.  An omitted-vertex label equal to the identity has
+trace zero, which is primary isolation.  No contraction to the empty
+configuration has occurred.
+
+\emph{A nonempty witness for the deeper test.}
+It remains to prove that this deeper test has a witness.  Fresh
+ramification gives $K\not\subseteq\mathbb Q(E[8])$.  By Chebotarev there
+is a good prime $p$ inert in $K$ and trivial on the original $E[8]$.
+Then
+\[
+ p\equiv1\pmod8,\qquad a_p(E^{(k)})\equiv-2\pmod8.
+\]
+On $\widetilde H_\varnothing$, whose coefficient at $1$ is zero, test
+$t(u)^2$ at coefficient $1$, where $u=\operatorname{Fr}_p$.  The square
+coefficient and matching unary character give, up to a unit,
+\[
+ \frac{a_p(E^{(k)})-1-p}{4}\bmod2=1.
+\]
+The determinant term at coefficient $1$ cancels.  In characteristic two,
+$t(u^2)=t(u)^2$, so a single trace in $G_K$ is nonzero.  A fresh prime
+realizing this finite trace test is split in $K$.  Moreover $u^2$ is
+trivial on every rational quadratic character, so the prime lies on the
+required total rational filter.  It is a nonempty witness for
+\eqref{pw:deeper-bit}.  Finally the exact coefficient ratio gives
+\eqref{pw:split-unit-bound}, with the additional depth at most one.
+\end{proof}
+
+\begin{lemma}[Orientation correction]\label{pw:coinvariants}
+Let $K$ be imaginary quadratic, let $S_0$ contain $2$ and its ramification
+support, and let $V_0$ be the elementary label space over $K$ unramified
+off $S_0$.  Complex conjugation acts on $V_0$ by $c$.  If the rational
+total of an oriented prime configuration is zero on every rational
+quadratic character unramified off $S_0$, then its oriented total belongs
+to $(1+c)V_0$.  A reserved supply containing every label allows its
+orientations to be flipped to realize any prescribed change in
+$(1+c)V_0$.
+\end{lemma}
+
+\begin{proof}
+Complex conjugation has order two outside $G_K$, so
+$G_{\mathbb Q}=G_K\rtimes\langle c\rangle$.  A $c$-invariant quadratic
+character of $G_K$ extends by giving $c$ value zero.  At a rational prime
+outside $S_0$, inertia lies in $G_K$ and the character is trivial there;
+thus its extension remains unramified outside $S_0$.  The assumed total
+therefore annihilates $(V_0^*)^c$.
+
+For a finite vector space with involution, the annihilator of
+$\ker(1+c^*)=(V_0^*)^c$ is $\operatorname{im}(1+c)$.  This proves the
+first assertion.  Flipping a vertex of label $v$ changes its oriented
+label from $v$ to $cv$, and hence changes the total by $(1+c)v$.
+Reserve one representative for every preimage needed to express the
+desired element of the image.  Their independent flips give the second
+assertion.  These flips preserve every rational norm label.
+\end{proof}
+
+\subsection{The corank-one original base}\label{pw:assembly-section}
+
+\begin{proposition}[Odd original base]\label{pw:odd-base}
+If $s_2(E)=1$, then $a(E)=1$.
+\end{proposition}
+
+\begin{proof}
+Parity gives odd functional sign.  Use the normalized odd limit symbol
+on the positive filter of $1$ from Section~\ref{pw:odd-stages}, with
+its fixed support, genus partner $h_*$, and chosen ultrafilter.  If it
+detects the base, a sufficiently advanced actual stage proves the result.
+Otherwise choose its nonempty minimum seed.  The moving detecting prime
+$\ell$ is not an extra fixed support requirement.
+
+\emph{Fixing the companion.}
+Choose a separate even symbol for negative discriminants square at $2N$,
+excluding the fixed odd support, the relevant torsion ramification, and
+the exceptional imaginary fields.  Choose a unit $k$ of that symbol.
+Then $L(E^{(k)},1)\ne0$.  Fix this $k$ and
+$K=\mathbb Q(\sqrt{k})$ for the rest of the proof, before choosing any
+cube dimension.  Its fresh ramification makes it independent of the old
+fixed label extensions.  At sufficiently advanced stages $\ell$ avoids
+$k$.  The intrinsic odd label tests continue to omit the primes of $k$.
+
+We construct common terminal packets, individually split in $K$, such
+that the odd terminal polynomial and an even terminal polynomial testing
+$hk$ are separately nonzero at each nonzero address.  They need not have
+a common unit evaluation yet; the address lemma will supply that step.
+
+\emph{The quadratic residual case.}
+First suppose $F,F'$ are quadratic.  Keep the original even symbol
+evaluated on $hk$, treating the vertices of $k$ as stationary external
+data.  Enrich the rational labels of the $h$-vertices by the primes of
+$k$.  Their total there and at the even support is required to be zero.
+Let $V$ be the resulting rational elementary label space, including the
+$F,F'$ and $K$ coordinates and all relations among these characters.
+The allowed simple labels form the affine set
+$\{v\in V:\chi_F(v)=1,\ \chi_K(v)=0\}$, in additive notation.
+The difference of any two of its elements belongs to
+$\ker\chi_F\cap\ker\chi_K$, precisely the realizable rational labels
+of the permitted split attachments.  Translation by this subgroup is
+therefore transitive.  The fields $F,F'$ are unramified outside the old
+support, whereas $K$ has a fresh ramified prime; thus $K$ is independent
+of their compositum.  When $F\ne F'$, this realizes both the good-simple
+pattern $(1,1,0)$ and the donor pattern $(0,1,0)$ in their three
+coordinates.  When $F=F'$, all simple labels are good.
+
+For the old odd projection, take a normal closure of its finite label
+extension at any stage with $\ell$ avoiding $k$.  It is unramified outside
+the old support and $\ell$.  Once $\ell$ avoids
+$k$, a prime ramified in $K$ remains outside that set.  Hence $K$
+has trivial intersection with this normal closure, and every old odd
+label can be realized together with splitting in $K$.  This applies
+also to oriented split anchors over $F$.  The added stationary $k$
+coordinates may have relations, but they are already included in $V$;
+they do not impose extra requirements on the omitted odd projection.
+These compatible lifts let us stabilize the common alphabet and its
+projection maps on the same ultrafilter, as in Section~\ref{pw:odd-stages}.
+The intrinsic odd minimum and seed are unchanged.  Choose the following
+packets in this fixed abstract alphabet, before selecting an actual stage.
+
+Use packets with sufficiently large simple supply and reserves in this
+kernel, and with donor slack when \(F\ne F'\).  For the even test, take
+no split anchors.  If the fields are distinct, correct all bad simples
+to good ones; if they coincide, every simple is already good.  Pair
+equal enriched labels, and
+retain a nonempty good base pair.  Use reserves to match the pair and
+the remaining good pairs, and attach every leftover split vertex to the
+last base position.  Its label is forced by total zero.  The output is
+a collection of good twin pairs, each matching also at every split
+vertex of $k$.  Twin insertion into the unit at $k$ proves that this
+output has even symbol one.  Thus the even terminal polynomial is
+nonzero.
+
+For the odd test, project away the newly added tests at $k$.  The same
+packets contain enough split anchors, simple vertices and reserve shifts
+to contract to the stabilized odd seed and good pairs, by
+Lemma~\ref{pw:reserves}.  Its total on that projection is the prescribed
+zero total.  Incidences between the retained anchors and the good pairs
+can be set after contraction, by independent edge choices.  The odd
+polynomial is therefore nonzero as well.  This odd test does not include
+the stationary vertices of $k$.
+
+\emph{Full rational two-torsion.}
+Now suppose the curve has full rational two-torsion.  Use the even test
+of Lemma~\ref{pw:split-witness}, with $K$-oriented labels on its support
+$S_0$, and let $v_1,\ldots,v_r$ be a minimum nonempty witness.  Its
+oriented total $v_1+\cdots+v_r$ lies in $(1+c)V_0$, by
+Lemma~\ref{pw:coinvariants}.  This even test uses the oriented elementary
+extension of $K$, whereas the odd test reads only its rational projection;
+the even contraction rule is needed only for nonempty outputs.
+Every odd seed label lifts to a $K$-split prime because $K$ was freshly
+ramified outside its intrinsic support.  Each even label also has a
+compatible lift to the common alphabet.  Stabilize the finite common
+alphabet and projections on the same ultrafilter, preserving the
+intrinsic odd symbol as in Section~\ref{pw:odd-stages}.
+
+Choose packets containing ample copies of every needed even label, all
+odd seed labels on their rational projection, and orientation-shifting
+reserves.  Arbitrary pairs of projected labels need not lift: take a
+union of sufficiently many compatible lifts for each projection
+separately.  Duplicate each full label, so the packet has rational
+total zero.
+
+For the even contraction, retain anchors with labels
+$v_1,\ldots,v_{r-1}$.  After reserving these anchors, use orientation
+flips to make the total equal to $\sum v_i$; this is possible by
+Lemma~\ref{pw:coinvariants}.  In a contraction certificate these flips
+are choices of conjugations along the forest, as permitted by
+Lemma~\ref{gr:realization}; they do not change the common terminal data
+separately at different addresses.  Merge all remaining vertices into the
+last cluster.  Its label is $v_r$.  The cluster is nonempty, and the
+minimum nonempty count makes the output a unit independently of its
+mutual edges.  For the odd contraction, use its rational projection.
+Retain all but the last odd seed label and merge the remainder.  Total
+zero gives the last label,
+so this is also a unit.  Orientation corrections for the even test did
+not change these rational labels.
+
+\emph{Simultaneous addressing and an actual stage.}
+Now fix $b$ and put a sufficient common packet at every nonzero linear
+form on $\mathbb F_2^b$.  Each nonzero address has two separately
+nonzero terminal polynomials.  In the fixed abstract alphabet,
+Theorem~\ref{gr:address} gives finite primary-auxiliary networks making
+both symbols one there.  Require their primary types to split in $K$
+(and $F$ in the quadratic case) and to be neutral at the stationary
+primes of $k$.  They are primary at every support coordinate used by
+either projection, so both symbol identities apply to the same common
+graph data.
+
+Let $W_b$ bound all graph weights required by these finite networks.
+A finite intersection of ultrafilter-large sets supplies an actual stage
+with the same truth table through $W_b$, all
+required divisibilities and trace identities, and sufficient precision
+for the prescribed divisions.  The moving prime also avoids the fixed
+companion $k$.  Realize the resulting finite graph at that stage using
+Lemma~\ref{gr:realization}.  All active products $h$ are positive and
+square at the fixed tests, and all their primes split in $K$.  No
+auxiliary appears at address zero.  The depth constant and fixed
+companion remain unchanged as $b$ grows.
+
+\emph{Returning to the original curve.}
+At every nonzero address the analytic orders of $E^{(h)}$ and $E^{(hk)}$
+are respectively one and zero.  The odd unit bound, the even unit bound
+(or \eqref{pw:split-unit-bound}), and
+Lemma~\ref{pw:height-comparison} with fixed $k$ give
+\[
+ 2j_h(k)\le 4w(h)+C.
+\]
+The finite Sha lengths at these addresses are nonnegative, so this also
+bounds $2j_h(k)-4w(h)-s(h)-s(hk)$ from above.
+
+At address zero, $L(E^{(k)},1)\ne0$ gives $s_2(E^{(k)})=0$ by the
+forward theorem.  Lemma~\ref{pw:isogeny-corank} therefore gives
+$s_2(E/K)=1$.  Ring-class missing-vertex transfer now makes the base
+product $L(E,s)L(E^{(k)},s)$ have a simple zero.  Its second factor is
+nonzero at $1$, and hence $a(E)=1$.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+Use Lemma~\ref{pw:isogeny-corank} to choose the indicated isogeny
+orientation before constructing either coefficient system.
+The even coefficient system and cyclotomic transfer give both
+Proposition~\ref{pw:partitionable-even} and the corank-zero conclusion
+of Proposition~\ref{pw:even-base}.  The bounded varying companion,
+ring-class lower bound, and height comparison then establish
+Proposition~\ref{pw:odd-lower}.  This discharges the arithmetic premise
+before the odd normalization and finite-graph construction of
+Section~\ref{pw:odd-stages}.
+
+For the corank-one case, Proposition~\ref{pw:odd-base} fixes a nonvanishing
+even companion before the cube dimension, constructs simultaneous unit
+vertices, and applies ring-class transfer at the original base.  This gives
+$a(E)=s_2(E)$ in both cases.  The established Gross--Zagier--Kolyvagin implication in
+analytic rank zero or one gives $r(E)=a(E)$ and finiteness of the whole
+Shafarevich--Tate group.  Equation~\eqref{pw:kummer-corank} remains the
+underlying corank identity throughout.  In particular, neither a finite
+two-Selmer dimension nor Mordell--Weil rank was substituted for the
+original Selmer corank hypothesis.
+\end{proof}
+
+\begin{remark}[The prescribed base]\label{pw:status}
+The final companion has Selmer corank zero because it was chosen with
+nonzero central value.  Thus \eqref{pw:quadratic-corank} supplies the
+required corank over $K$; it is not an additional companion hypothesis.
+Only the finite realizations and their precisions change as the cube
+grows.  The valuation constants remain fixed, so transfer applies to the
+prescribed original curve without deleting a density-zero set of bases.
+\end{remark}
diff --git a/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/build/sections/preliminaries.tex b/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/build/sections/preliminaries.tex
new file mode 100644
index 0000000000000000000000000000000000000000..073b1469233eebf281bcda64303b60b26f084079
--- /dev/null
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@@ -0,0 +1,151 @@
+\section{Selmer coranks and fixed local data}\label{sec:preliminaries}
+
+For an elliptic curve $A/\Q$, put
+\[
+ r(A)=\rank_{\Z}A(\Q),\qquad
+ a(A)=\ord_{s=1}L(A,s),\qquad
+ s_2(A)=\corank_{\Z_2}\Sel_{2^\infty}(A/\Q).
+\]
+The Selmer group uses the local Kummer image at every place.
+The analytic continuation and functional equation are supplied by modularity
+\cite{BCDT2001}. Write $\varepsilon(A)\in\{1,-1\}$ for the root number.
+
+\begin{lemma}[The three ranks]\label{pr:ranks}
+There is an exact sequence
+\[
+0\longrightarrow A(\Q)\otimes\Q_2/\Z_2
+\longrightarrow \Sel_{2^\infty}(A/\Q)
+\longrightarrow \Sha(A/\Q)[2^\infty]\longrightarrow0.
+\]
+Consequently
+\[
+s_2(A)=r(A)+\corank_{\Z_2}\Sha(A/\Q)[2^\infty].
+\]
+If $a(A)\in\{0,1\}$, then $r(A)=a(A)$ and the whole group
+$\Sha(A/\Q)$ is finite. In that case $s_2(A)=a(A)$.
+\end{lemma}
+
+\begin{proof}
+The Kummer sequences for multiplication by $2^n$, with the local
+Kummer conditions imposed, give
+\[
+0\longrightarrow A(\Q)/2^n A(\Q)
+\longrightarrow \Sel_{2^n}(A/\Q)
+\longrightarrow \Sha(A/\Q)[2^n]\longrightarrow0.
+\]
+Their direct limit is the displayed sequence. Taking $\Z_2$-coranks proves
+the identity. The analytic-rank-zero and analytic-rank-one assertions are
+the Gross--Zagier--Kolyvagin theorem
+\cite{GrossZagier1986,Kolyvagin1990}; see also the precise unrestricted
+forward statement in \cite[\S1]{Skinner2020}.
+\end{proof}
+
+No finite $2$-Selmer dimension will be substituted for $s_2$.
+In particular, $s_2(A)=1$ does not supply a rational point of infinite order:
+the exact sequence allows a contribution from the divisible part of $\Sha$.
+
+\begin{lemma}[Parity and change of curve]\label{pr:invariance}
+For every elliptic curve $A/\Q$,
+\[
+(-1)^{s_2(A)}=\varepsilon(A).
+\]
+A rational isogeny preserves $a$, $r$, and $s_2$. If
+$K=\Q(\sqrt{k})$ is quadratic, then
+\[
+s_2(A/K)=s_2(A/\Q)+s_2(A^{(k)}/\Q).
+\]
+\end{lemma}
+
+\begin{proof}
+The $2$-parity assertion is due to Monsky \cite{Monsky1996};
+see also \cite[Theorems~1.4 and~4.19]{DokchitserDokchitser2010},
+which states the Selmer-parity result for every prime and credits
+Monsky for the case $2$.
+For an isogeny and its dual, both composites are multiplication by the
+degree. The corresponding maps on rational points and on Selmer groups
+respect the Kummer conditions. After rationalizing their Tate modules,
+the maps are inverse up to a nonzero scalar; hence the ranks and coranks
+agree. The equality of $L$-functions is isogeny invariance.
+
+Over $K$, identify $A^{(k)}$ with $A$, and let $c$ be the nontrivial
+element of $\Gal(K/\Q)$. Restriction and the two conjugation
+projections $1+c$ and $1-c$ define maps between the sum of the two
+Selmer groups over $\Q$ and the Selmer group over $K$ whose composites are
+multiplication by $2$. Functoriality of local Kummer images gives these
+maps at every place. Rationalization therefore gives the stated corank
+identity. This argument does not assert an integral direct-sum
+decomposition of the Selmer groups.
+\end{proof}
+
+Fix a curve $E/\Q$ of conductor $N$. A finite support $S$ always contains
+the primes dividing $2N$. It may be enlarged by finitely many primes when
+the fixed construction is prepared. A \emph{local filter} consists of a
+real sign and unit squareclasses at $S$ for the variable fundamental
+discriminant. The filter is imposed on every relevant Fourier index,
+not just the squarefree indices. Its root number is constant.
+All variable primes lie outside $S$.
+
+For an odd prime $p$, set
+\[
+p^*=\left(\frac{-1}{p}\right)p.
+\]
+Nonempty products of distinct $p^*$ are odd fundamental discriminants.
+We also admit the empty product $h=1$, which denotes the untwisted
+curve rather than a quadratic field. For a squarefree product $h$ of
+such factors, set
+\[
+ t_p=\dim_{\F_2} E[2]^{\operatorname{Fr}_p},
+ \qquad
+ w(h)=\frac12\sum_{p\mid h}t_p.
+\]
+In the rational-two-torsion range, $t_p$ is $1$ or $2$.
+Whenever the $2$-primary group below is finite, write
+\[
+s(h)=\length_{\Z_2}\Sha(E^{(h)}/\Q)[2^\infty].
+\]
+Thus $s(h)$ is a finite length, whereas $s_2(E^{(h)})$ is a corank.
+In particular $s(h)$ is defined when $s_2(E^{(h)})=0$: the exact
+sequence in Lemma~\ref{pr:ranks} and finite generation of the dual
+Selmer group make its $2$-primary quotient finite. This conclusion
+concerns the $2$-primary group; the low-analytic-rank theorem supplies
+finiteness of the whole group after analytic nonvanishing is proved.
+Let $v$ be a valuation above $2$, normalized by $v(2)=1$.
+After the coefficient field has been fixed, its value group is discrete.
+
+The notation $L_*(h)$ denotes $L(E^{(h)},1)\sqrt{|h|}$ divided by the
+fixed real or imaginary period appropriate to the filter.
+The notation $\mathcal L(h)$ in Section~\ref{cy:section} denotes
+this same normalized value.
+Every modular parametrization used to form Heegner points is normalized
+to send the cusp $\infty$ to the identity of the elliptic curve.
+Changing among fixed nonzero modular parametrizations, isogenous curves,
+or period conventions contributes only bounded constants to valuations.
+Every such dependence is fixed before a binary cube is allowed to grow.
+Dependence on a growing number of prime factors will always be stated
+explicitly.
+
+The following ledger records the order of choices used in the proof.
+It distinguishes bounds from the sizes and precisions needed to realize
+the auxiliary primes.
+\begin{center}
+\small
+\begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.29\textwidth}>{\raggedright\arraybackslash}p{0.65\textwidth}@{}}
+\toprule
+Data or bound & Allowed dependence and order of choice \\
+\midrule
+Fixed arithmetic constants & The curve, its parametrization and periods,
+the fixed part of the support, and the chosen local classes; these are fixed before
+the varying prime support or cube dimension. \\
+Varying quadratic companions & The same data and a bound $A$ for
+the number of prime factors of the companion discriminant; its prime
+sizes do not enter. \\
+Fixed-companion transfer & The companion field is fixed before the
+cube dimension. Constants may depend on its fixed base Selmer data,
+as specified in Section~\ref{rd:section}. \\
+Finite realization & Choose the cube dimension and finite networks,
+then the required coefficient precision and a sufficiently late
+arithmetic stage. Prime sizes and stage thresholds may depend on
+these choices. \\
+\bottomrule
+\end{tabular}
+\end{center}
diff --git a/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/build/sections/proof-roadmap-figure.tex b/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/build/sections/proof-roadmap-figure.tex
new file mode 100644
index 0000000000000000000000000000000000000000..043e4957ca626620f2835c3af8a5c1c938941e37
--- /dev/null
+++ b/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/build/sections/proof-roadmap-figure.tex
@@ -0,0 +1,44 @@
+\begin{figure}[htbp]
+\centering
+\small
+\setlength{\tabcolsep}{6pt}
+\begin{tabular}{c}
+\fbox{\parbox{0.86\textwidth}{\centering
+Even coefficients, finite prime networks and cyclotomic transfer\\
+\begin{tabular}{@{}c@{\qquad}c@{}}
+$\swarrow$ & $\searrow$\\
+Uniform even estimate & Corank-zero conclusion\\
+(Proposition~\ref{pw:partitionable-even}) & (Proposition~\ref{pw:even-base})
+\end{tabular}}}
+\\[5pt]
+Even estimate $\Downarrow$\\[5pt]
+\fbox{\parbox{0.86\textwidth}{\centering
+Fix $h_*$ to define $P_h$; choose a bounded-factor companion $k'(h)$\\
+Even estimate + ring-class lower bound\\
+$\Downarrow$\\
+Uniform lower bound for the $2$-divisibility of $P_h$\\
+and odd coefficient normalization}}
+\\[5pt]
+$\Downarrow$\\[5pt]
+\fbox{\parbox{0.86\textwidth}{\centering
+Fix $k$ with $L(E^{(k)},1)\ne0$ and put $K=\Q(\sqrt{k})$\\
+Odd and even tests + finite prime networks split in $K$\\
+$\Downarrow$\\
+Simple zeros for $E^{(h_x)}$ and nonzero values for $E^{(h_xk)}$\\
+with a common normalized index bound, for every $x\ne0$}}
+\\[5pt]
+$\Downarrow$\\[5pt]
+\fbox{\parbox{0.86\textwidth}{\centering
+Fixed-$K$ ring-class transfer and $s_2(E/K)=1$\\
+$\Downarrow$\\
+Analytic rank one at the original vertex $x=0$}}
+\end{tabular}
+\caption{The even construction proves the corank-zero case and a uniform
+estimate; the latter supplies the bound needed to normalize the odd coefficients.
+The detector partner $h_*$ is fixed, the bounded-factor companion $k'(h)$
+varies only in the lower-bound step, and the final companion $k$ is fixed
+before the simultaneous families are constructed.  The families and
+auxiliary primes may grow with the cube dimension; the normalized
+index bound may not.}
+\label{fig:proof-roadmap}
+\end{figure}
diff --git a/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/build/sections/ring-determinants.tex b/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/build/sections/ring-determinants.tex
new file mode 100644
index 0000000000000000000000000000000000000000..7ec6e3040f458cbd24369e8ed756d64a624d3cdf
--- /dev/null
+++ b/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/build/sections/ring-determinants.tex
@@ -0,0 +1,1769 @@
+\section{Ring-class switches, determinant lattices, and transfer}
+\label{rd:section}
+
+This section constructs the derivative classes and proves the determinant
+and transfer statements.  The arithmetic models, the character \(\Psi\), and their
+evaluation maps are those of Lemmas~\ref{rl:finite-model},
+\ref{rl:finite-limits}, \ref{rl:evaluation}, and \ref{rl:kernel},
+and Propositions~\ref{rl:characters} and \ref{rl:outer-evaluation}.
+There are two outputs.  Corollary~\ref{rd:ring-lower} bounds a
+Heegner-point index at a known simple zero, uniformly in the conductor
+and with dependence only on a bound for the number of companion primes.
+Theorem~\ref{rd:missing-vertex} instead fixes the companion field and
+recovers an unknown base derivative from a family of known simple zeros.
+The first step is an integral switch: adding a derivative prime replaces
+a finite local condition by its singular counterpart without changing
+the determinant valuation.  Repeated switches then give the uniform
+lower bound.  For the missing vertex we need more: the determinants
+must occur in one group-ring family with a uniformly bounded clearing
+factor.  We construct the arithmetic diagrams before applying these two
+arguments, and verify their specialization and family properties where
+they are used.
+
+Fix \(E/\mathbb Q\), put \(T=T_2(E)\), and normalize \(v(2)=1\).  Let
+\[
+ \Lambda=\mathbb Z_2[[t]],\qquad
+ \mathcal O=\widehat{\Lambda_{(2)}},\qquad
+ \Omega=\mathbb F_2((t)).
+\]
+For a perfect cohomological complex \(C\), use the inverse determinant line
+\begin{equation}
+ \mathcal D(C)=\bigotimes_i(\det C^i)^{(-1)^{i+1}}.
+ \label{rd:det-convention}
+\end{equation}
+In particular, for a complex in degrees one and two this is
+\(\det C^1\otimes(\det C^2)^{-1}\).  A basis of this integral line is fixed
+whenever a determinant coordinate is mentioned.  Changing it multiplies
+the coordinate by an integral unit.
+
+The ring data consist of an imaginary quadratic field \(K=\mathbb Q(\sqrt k)\)
+split at \(2N\), a product \(h\) of signed good odd primes split in \(K\),
+and \((h,k)=1\).  The fields with extra units and the CM field of \(E\)
+are excluded.  Write
+\begin{equation}
+ t_p=\dim_{\mathbb F_2} E[2]^{\operatorname{Fr}_p},\qquad
+ w(h)=\frac12\sum_{p\mid h}t_p.
+\label{rd:weights}
+\end{equation}
+Choose the modular parametrization \(\pi:X_0(N)\to E\)
+normalized by \(\pi(\infty)=0\), translating it by the rational
+point \(-\pi(\infty)\) first if necessary.  By the Manin--Drinfeld theorem
+\cite{Manin1972,Drinfeld1973}, images of cusp
+differences are torsion, so one fixed integer kills every
+cuspidal ambiguity.  No assertion that an arbitrary translate
+has torsion value at \(\infty\) is used.
+The scalar deformation is trivial at \(2N\), has unit exponent on the
+auxiliary conductor inertia, and odd exponent at the indicated split
+conductor Frobenii.  Its auxiliary prime \(r\) is allowed to change with
+the precision.  The trace bound in Proposition~\ref{rl:characters} is part of
+the data.
+
+\subsection{The local switch and its integral determinant}
+\label{rd:local-subsection}
+
+\begin{lemma}[The split local model]\label{rd:local-model}
+Let a sequence of inert good primes \(\ell\) have rational Frobenius
+lifts tending on \(T\) to a fixed complex conjugation \(J\).
+At the unique prime of \(K\) above \(\ell\), the limiting local complex
+has
+\begin{equation}
+ H^0=M,\qquad H^1=F\oplus S=M\oplus M,\qquad H^2=M,
+ \label{rd:local-cohomology}
+\end{equation}
+with zero differentials in a free split model.  Here \(M=T\otimes\Lambda\)
+with the relevant scalar coefficient structure.  The two degree-one
+coordinates of a cocycle are
+\begin{equation}
+ f(c)=c(\gamma^2),\qquad s(c)=c(\sigma),
+ \label{rd:coordinates}
+\end{equation}
+where \(\gamma\) is the chosen rational Frobenius lift and
+\(\sigma\) is an oriented pro-\(2\) tame generator.  The plane
+conditions \(F=\{s=0\}\) and \(S=\{f=0\}\) include \(H^0\) and exclude
+\(H^2\).  Each is its own orthogonal complement for conjugate-Weil
+duality.  The strict condition \(H^0\) and the relaxed condition
+\(H^0\oplus H^1[-1]\) are mutually dual.  All four inclusions can be
+equipped with compatible integral duality triangles.
+\end{lemma}
+
+\begin{proof}
+The prime-to-\(\ell\) local cochains are computed by the tame inertia and
+residue-field directions.  In these two directions the differentials
+are formed from inertia minus one and Frobenius minus one, with the
+usual Frobenius action on tame inertia.  On \(M\), inertia is trivial at
+the derivative prime.  The square of a rational Frobenius is trivial
+on every ring-class character.  In the limit it is also the identity on
+\(T\), while \(\ell^2\) tends to one.  Thus both differentials become zero,
+giving \eqref{rd:local-cohomology}.
+
+Here is a finite-precision check of the pairing, needed at \(2\).
+At a retained quotient killed by \(2^a\), impose also
+\(q=\ell^2\equiv1\pmod{2^{a+1}}\), as well as trivial
+Frobenius-square action on the coefficient module at this quotient.
+The tame two-generator complex then has zero differentials,
+with the coordinates in \eqref{rd:coordinates}.  Its degree-one
+scalar generators are the unramified and tame characters.
+More explicitly, use the presentation
+\(\phi\sigma\phi^{-1}\sigma^{-q}=1\).
+After augmentation with trivial coefficient action its Fox
+resolution has terms of ranks \(1,2,1\), with
+\(d^0=0\) and \(d^1=(0,1-q)\), in the ordered finite and
+tame coordinates.  The diagonal from the same resolution
+has zero pure-finite term, unit mixed terms, and pure-tame
+coefficient \(\pm\binom q2\).  The latter coefficient is
+the augmented second derivative of \(\sigma^{-q}\)
+together with its adjacent \(\sigma\) in the relation.
+Both \(1-q\) and \(\binom q2\) vanish modulo \(2^a\)
+under the imposed congruence.  Thus this is an actual
+split complex and an actual pure-plane zero pairing
+at the retained precision.
+The field contains the \(2^{a+1}\)-st roots of unity, so
+\(-1\) is a \(2^a\)-th power.  The norm-residue identity
+\((x,x)=(x,-1)\) for the \(2^a\)-Hilbert symbol therefore
+makes the square of each scalar generator zero.
+The two generators form a basis; local Tate duality makes
+their cross pairing a unit.  The same calculation applies
+coefficientwise to the free Artin coefficient module, including
+its nilpotent \(t\) and quadratic variables.  Thus the pure
+plane restrictions are already zero at every retained quotient.
+All these congruences can be imposed simultaneously with the
+derivative Frobenius conditions by increasing their precision.
+Taking the compatible limit proves integral vanishing of the
+pure squares.  It is not deduced from skew symmetry in
+characteristic two.
+
+Equivalently, in the limiting tame presentation
+\(\phi\sigma\phi^{-1}=\sigma^q\) with \(q\to1\), the
+rank-\((1,2,1)\) resolution specializes to the exterior
+resolution for two procyclic directions.  The finite
+Hilbert-symbol calculation specifies the cup product on
+its split cohomology and rules out a hidden square term.
+Conjugation transports the finite Tate coordinate by \(J\) and the
+singular coordinate by \(-J\), with possible scalar units coming from
+the orientations.  Consequently the cross pairing is
+\begin{equation}
+ \beta(x,y)=\epsilon\,\langle x,Jy\rangle,\qquad
+ \epsilon\in\Lambda^\times,\quad x\in F,\ y\in S,
+ \label{rd:cross-pairing}
+\end{equation}
+where \(\langle\, ,\,\rangle\) is the perfect alternating Weil form.
+This proves the assertions about orthogonal complements.  Since the
+restrictions of this explicit pairing to either pure plane are zero as
+maps of the split complexes, their isotropy homotopies may be chosen to
+be zero.  Only a product of two degree-one terms could map to the
+degree-two local invariant target; the degree-zero terms introduce
+no additional restriction.  These zero homotopies agree on the
+strict subcomplex.  These are the actual
+orthogonality data required in the Selmer duality construction
+\cite[Theorem~6.3.4]{Nekovar2006}; the compatibility assertion is therefore
+an assertion about these displayed complexes and maps.  Lemma~
+\ref{rl:finite-limits} is used to retain this finite diagram together
+with the unchanged local data.
+\end{proof}
+
+We now construct global classes with these local coordinates, using
+the CM points \(y_m\in E(H_m)\) fixed in Section~\ref{rl:section}.
+For the fixed twist put
+\(c=|h|\).  If \(I\) is a finite set of derivative primes, put
+\(n=\prod_{\ell\in I}\ell\); the points used for this set have
+conductor \(crn\).
+
+Let \(C_I\) denote the limiting Selmer complex with coefficients
+\(T\chi_h\Psi\).  Its local conditions are Kummer at \(2N\), full
+at the active primes dividing \(h\) and at \(r\), singular at the
+primes in \(I\), and unramified elsewhere.  At a new derivative
+prime \(\ell\notin I\), the unramified condition is the finite
+plane \(F_\ell\) of Lemma~\ref{rd:local-model}.  Both plane
+conditions retain local \(H^0\) and omit \(H^2\), as in that
+lemma.  The next result constructs the class \(P_I\) in this
+complex; its two displayed localizations are taken at the newly added
+prime \(\ell\).
+
+\begin{lemma}[Descent and the switch at \(2\)]\label{rd:derivative}
+Perform each finite ring-class construction and its weighted transfer
+before retaining its target-side data by the construction following
+Lemma~\ref{rl:finite-limits}.  There is one
+integer \(L\ne0\), depending on the fixed curve, parametrization and
+fixed local types, such that the classes obtained from the Heegner
+derivatives of \(L\) times the point satisfy, for every finite switched
+set \(I\),
+\begin{equation}
+ P_I\in H^1(C_I),\qquad
+ f(P_{I\cup\{\ell\}})=0,\qquad
+ s(P_{I\cup\{\ell\}})=J f(P_I).
+ \label{rd:switch}
+\end{equation}
+The same \(L\) works for every number of active and derivative primes.
+The previously imposed singular conditions are retained.
+\end{lemma}
+
+\begin{proof}
+We use Kolyvagin's derivative operators \cite{Kolyvagin1990}, keeping
+their integral descent explicit at $2$.  Choose the orientations
+coherently.  The relative ring-class group at
+an inert prime \(\ell\) is cyclic of order \(\ell+1\).  For distinct new
+primes these groups form a product: the ring-class exact sequence has
+kernel the product of the local residue-unit quotients, and the only
+global units here are \(1,-1\), whose images in these quotients are
+trivial.  Choose a generator \(\sigma_\ell\) compatibly with the
+tame orientation in Lemma~\ref{rd:local-model}; its pro-\(2\)
+component gives the coordinate \(s\) at the retained precision.
+Set
+\[
+ D_\ell=\sum_{i=1}^{\ell} i\sigma_\ell^i,\qquad
+ D_I=\prod_{\ell\in I}D_\ell.
+\]
+Direct multiplication gives
+\begin{equation}
+ (\sigma_\ell-1)D_\ell=(\ell+1)-
+             \sum_{i=0}^{\ell}\sigma_\ell^i.
+ \label{rd:derivative-identity}
+\end{equation}
+The Hecke correspondence and reduction of its CM isogenies give, for
+every class-group translate,
+\begin{equation}
+ \operatorname{Tr}(y_{c r n})=a_\ell y_{c r n/\ell},
+ \qquad
+ \widetilde y_{c r n}=\operatorname{Fr}_\ell
+                       \widetilde y_{c r n/\ell}.
+ \label{rd:heegner-relations}
+\end{equation}
+These are the point relations, not an application of a residual
+irreducibility theorem.  A fixed multiple of the parametrization removes
+cuspidal terms.
+
+There is a uniform bound on the \(2\)-primary torsion over the union of
+the ring-class fields in use.  To see this, choose a fixed nonidentity
+scalar homothety in the \(2\)-adic Tate image.  Square a lift so that it
+belongs to every quadratic subgroup under consideration.  Multiply it
+by its conjugate under complex conjugation.  Its action on each
+ring-class field is trivial, because conjugation inverts the abelian
+ring-class action, while its action on \(T\) is still a fixed
+nonidentity scalar.  Write that scalar minus one as \(2^e\) times
+a unit.  Then \(2^e\) kills the torsion in question.  The existence of
+this fixed scalar, including the CM case, is the Tate-image input
+used in Lemma~\ref{rl:kernel}.
+
+Work first with coefficients modulo \(2^a\).  Require \(a_\ell\) and
+\(\ell+1\) to vanish to a precision substantially larger than \(a\).
+Equations \eqref{rd:derivative-identity} and
+\eqref{rd:heegner-relations} then make the derivative Kummer class
+invariant under the relative ring-class group.  In the
+inflation--restriction sequence, both the obstruction to descent and
+the ambiguity of descent are killed by \(2^e\), since their
+coefficient module is killed by \(2^e\).  Multiplication by \(2^{2e}\)
+therefore gives a descent independent of this ambiguity: descend
+\(2^e\) times the class and multiply its lift by \(2^e\).
+There is no repetition of this factor for individual primes; descent
+is performed once from the field with the complete conductor.
+
+The target coefficient module is obtained by an integral transfer.
+For a finite Artin coefficient ring \(A\), let
+\(\Delta=\operatorname{Gal}(H_{cr}/K)\), through which the
+finite scalar and quadratic characters factor.  Shapiro's
+cochain map and the coefficient homomorphism
+\[
+ A[\Delta]\longrightarrow A,\qquad
+ [\sigma]\longmapsto\chi_h(\sigma)\Psi(\sigma)
+ \]
+send the descended class to coefficients \(T\chi_h\Psi\).
+For a universal quadratic family, replace the quadratic
+value by its group element in \(A[G]\), also including the
+fixed base character.  Choose inverse actions on both
+sides when using that Shapiro convention.  Explicitly,
+these maps are finite coset sums with the indicated scalar
+weights; there is no division by \(\#\Delta\).
+These chain maps, in all degrees, are used at this finite stage.
+Naturality of restriction and Kummer corestriction supplies their local
+lifts and comparison homotopies.  Apply the maps to the classes and
+these witnesses before passing to bounded target models.  Neither
+the complex over \(H_{cr}\) nor its induced coefficient module
+\(A[\Delta]\) is retained in the limiting diagram.  We check the
+required target local conditions next.
+
+At a place over \(2N\) these descent extensions are unramified.
+For the identity component of a Neron model, the obstruction to
+unramified descent vanishes.  Indeed, Lang's theorem gives the assertion
+on the connected special fiber, and the successive formal-group
+quotients are additive residue-field modules, whose unramified cyclic
+cohomology vanishes by the normal-basis and trace calculation.
+Passing to the inverse limit of these quotients preserves the assertion.
+The remaining obstruction is killed by the geometric component-group
+order.  Those orders have a uniform bound over unramified extensions
+of the finitely many fixed local twisting types.  Include their product,
+the descent factor, and the parametrization factor in \(L\).
+Thus the descended class has the integral Kummer condition at \(2N\).
+No good-ordinary or semistability condition was used.  Transfer with
+the scalar character preserves this condition because \(\Psi=1\)
+there.
+
+Here is the local calculation, including the division by two.  Apply
+the derivatives at all primes other than \(\ell\), and write \(X,Y\)
+for the resulting lower and upper points.  Put \(F_\ell=\operatorname{Fr}_\ell\).
+The lower field splits at the \(K\)-prime \((\ell)\), since that ideal
+is principal.  Reduction therefore gives \(F_\ell^2\widetilde X=\widetilde X\).
+Good reduction identifies the prime-to-\(\ell\) torsion with the
+torsion of the reduced curve.  In the following coordinate formulas,
+\(F_\ell\) acts on reduced points, where it satisfies its
+Frobenius polynomial.  If \(Q\) is a \(2^a\)-division point
+of \(LX\), the finite coordinate
+of its Kummer class is
+\begin{equation}
+ f=(F_\ell^2-1)\widetilde Q=
+       \bigl(a_\ell F_\ell-(\ell+1)\bigr)\widetilde Q.
+ \label{rd:finite-formula}
+\end{equation}
+We give the descent-cocycle calculation of the other coordinate.
+Choose \(Z\) with \(2^aZ=LD_\ell Y\), and choose the descended
+cocycle \(b\) so that on the upper-field Galois group it is
+\(b(g)=(g-1)Z\).  A change by the already bounded descent ambiguity
+has no effect on the eventual limit.  The crossed identity gives
+\[
+ U=(\sigma_\ell-1)Z-b(\sigma_\ell)
+                  \in E(H_{crn}).
+ \]
+For \(g\) in the upper-field Galois group, compare the two
+expansions of \(b(g\sigma_\ell)\); they give \(gU=U\).
+Moreover,
+\[
+ 2^aU=L\bigl((\ell+1)Y-a_\ell X\bigr).
+ \]
+Both integers \(\ell+1,a_\ell\) are divisible by \(2^a\).
+Consequently
+\[
+ U_0=L\frac{\ell+1}{2^a}Y-
+                      L\frac{a_\ell}{2^a}X
+ \]
+is an actual upper-field point with the same \(2^a\)-multiple
+as \(U\).  Thus \(U-U_0\) is upper-field torsion of bounded
+exponent.  This comparison with an actual integer quotient
+is what bounds the ambiguity; arbitrary division points
+would not give this bound.  Reduction makes
+\(\sigma_\ell\) act trivially and
+\(\widetilde Y=F_\ell\widetilde X\).  Therefore
+reduction of \(b(\sigma_\ell)=-U\) gives its singular coordinate
+\begin{equation}
+ s=\bigl(a_\ell-(\ell+1)F_\ell\bigr)\widetilde Q
+ \label{rd:singular-formula}
+\end{equation}
+up to the uniformly bounded descent torsion already described.
+The Frobenius polynomial yields the exact operator identity
+\begin{equation}
+ (F_\ell-a_\ell)
+     \bigl(a_\ell F_\ell-(\ell+1)\bigr)
+       =a_\ell-(\ell+1)F_\ell.
+ \label{rd:exact-frobenius}
+\end{equation}
+Thus \(s=(F_\ell-a_\ell)f\); when \(a_\ell\) vanishes at the
+retained torsion precision, this is \(s=F_\ell f\).  Taking the
+Frobenius limit gives \(s=Jf\).
+
+For the new finite coordinate, reduction of the derivative point is
+\begin{equation}
+ \sum_{i=1}^{\ell}iF_\ell\widetilde X
+       =\frac{\ell(\ell+1)}2F_\ell\widetilde X.
+ \label{rd:half-factor}
+\end{equation}
+The conductor extension at \(\ell\) is totally ramified, so a
+Frobenius lift may be chosen in the upper-field local Galois group.
+There the descended cocycle restricts to the Kummer cocycle of the
+derivative point.  If \(v_2(\ell+1)\ge a+1\), its displayed
+reduction is \(2^a\) times an \(F_\ell^2\)-fixed point.
+Its finite Kummer coordinate is therefore zero modulo \(2^a\).
+The additional \(1\) is essential at
+\(2\).  It is imposed in the choice of the next precision, and does
+not multiply the class by a new denominator.  Bounded torsion
+ambiguities disappear on passing from \(E[2^a]\) to \(E[2^m]\) with
+\(a-m\ge e\).  Choose \(a\) and the Frobenius precision successively
+faster than \(m\).
+
+The calculation is compatible with every translate in the transfer:
+transport the place and its Frobenius before applying that translate.
+The scalar weight on the lower and upper class is identical.  Applying
+the same calculation at each \(\ell\in I\) proves the earlier
+singular conditions as well.
+
+Finally complete the weighted transfer at each stage and retain the
+resulting target Selmer-cone cycles and their local lifts.  An equality
+of the transferred local classes gives a boundary witness in the
+target local complex; retain that witness after contraction as well.
+For instance, retain the degree-zero vectors whose boundaries are the
+differences in the two switch identities.  Include the split local Fox
+complexes, their zero isotropy homotopies, and the target comparison
+matrices.  The target-side construction in Section~\ref{rl:section} proves
+that this is a bounded diagram for each fixed active support and
+finite switched set.  Its bound is independent of the degrees of
+the ring-class fields.
+When a further derivative prime is selected, extend the
+same original finite-stage diagram and its contraction
+maps, using the same inner ultrafilter.  Do not construct
+an unrelated cohomologically equivalent old complex.
+The earlier evaluations, including all degree-two
+cocycle identities, are consequently retained by the
+same contraction maps after each new switch.  Lemma~
+\ref{rl:finite-limits} applies to these extended diagrams.
+Their limiting classes give
+\eqref{rd:switch}.  The displayed local formulas agree with the
+calculation underlying \cite[Proposition~1.7.4]{Howard2012}; Howard's
+odd-prime global hypotheses are not being asserted at \(2\).
+\end{proof}
+
+\paragraph{The paired arithmetic determinant.}
+All the target classes and comparison maps are constructed in the same
+finite diagram.
+At each Artin precision, use the character of
+Proposition~\ref{rl:characters}, the global Selmer cone, and the
+Kummer complexes at \(2N\).  At a derivative prime impose enough
+Frobenius precision for Lemma~\ref{rd:derivative} and also
+\(\ell^2\equiv1\pmod{2^{a+1}}\) at every retained quotient killed
+by \(2^a\).  Lemma~\ref{rd:local-model} then gives the finite,
+singular, strict and relaxed inclusions with their common zero
+isotropy homotopies.  Applying the actual global duality morphism
+\eqref{rl:global-duality} constructs their integral localization
+and duality triangles.  Perform the descent and weighted transfers of
+Lemma~\ref{rd:derivative} at each finite stage.  Include their target
+cycles, local Kummer lifts and target boundary witnesses in this
+diagram before applying Lemmas~\ref{rl:finite-model} and
+\ref{rl:finite-limits}, using only the target-side data.
+All remaining maps have bounded target models as both source and
+target; the ring-class source complexes are not among these objects.
+
+A further switch extends these original finite stages, their
+contractions and their fixed inner ultrafilter.  Theorem~
+\ref{rl:limit-evaluation-package} therefore retains the same
+degree-zero, degree-one and degree-two evaluations and the
+exact-kernel restrictions.  In particular, the old evaluations
+are not replaced by maps on an independently chosen equivalent
+complex.
+
+Here is how a class produces its paired functional.  Work over a
+height-one localization where the full local complexes at the active
+primes and \(r\) are acyclic.  By Lemma~\ref{rl:local-acyclic}
+this includes \(\mathcal O\) and every characteristic-zero test
+used below.  These complexes have integral contractions.  They
+identify the relaxed Selmer complex with its orthogonal complex;
+at switched primes the plane condition is already self-dual, and
+at \(2N\) the integral Kummer condition is self-dual.  Lift the
+class through this comparison and compose it with the retained
+global duality map.  The result is a closed degree-two functional
+\(\lambda_I\), integral up to the single initial multiplier and
+any fixed old-support correction.  Thus it is not an arbitrarily
+chosen rational cohomology dual.
+
+When the generic cohomology has dimension one in degrees one and
+two and is zero elsewhere, define \(u_I\) to be the coordinate of
+\[
+ [P_I]\otimes[\lambda_I]\in
+ H^1(C_I)\otimes H^2(C_I)^*
+       \simeq\mathcal D(C_I)
+\]
+over the fraction field, in the fixed integral determinant basis.
+For an acyclic constituent set this class-dual coordinate to zero.
+The generic functional is determined by the same derived duality
+at all localizations; different local contractions represent that
+functional, not one cochain asserted to be integral everywhere.
+For \(I=\varnothing\) write \(u_h\).  The rank-one assertions
+needed for the arithmetic applications are verified below by
+rational central specialization.
+
+The arithmetic switch must now be compared in these determinant
+lattices.  A localized class may be divisible by \(2\); the next
+lemma shows that such a division does not cost a factor at each switch.
+
+\begin{lemma}[Exact determinant switch over a DVR]\label{rd:det-switch}
+Let \(O\) be a DVR with fraction field \(F_O\).  Suppose that integral
+perfect complexes \(C_{\rm str},C_F,C_S,C_{\rm rel}\), their
+localization triangles, and their dualities are the base changes of
+the diagram in Lemma~\ref{rd:local-model}.  Suppose the generic cohomology of these four global complexes is
+concentrated in degrees one and two, that
+\(H^1(C_F\otimes F_O)=F_OA\), and
+\[
+ f(A)=v_0\ne0,\qquad s(B)=Jv_0,\qquad f(B)=0
+\]
+for a class \(B\) in the switched complex.  The determinant tensors
+are formed from each class and its functional under these same
+dualities.  Then the switched generic cohomology has rank one,
+is spanned by \(B\), and
+\begin{equation}
+ v_O(u_F)=v_O(u_S).
+ \label{rd:valuation-switch}
+\end{equation}
+Neither localization is required to be primitive.
+\end{lemma}
+
+\begin{proof}
+For this proof only, cohomology without a coefficient ring means
+cohomology over \(F_O\).  Nonzero localization of \(A\) gives
+\(H^1(C_{\rm str})=0\).  The relaxed localization image is
+self-annihilating in \(F\oplus S\), so it has dimension two.
+It contains the two independent vectors \((v_0,0)\) and
+\((0,Jv_0)\) in the opposite planes.  Its intersections with either plane are therefore
+exactly one-dimensional.  This proves the generic rank assertions.
+
+Put \(V=H^1(C_{\rm rel})\), with ordered basis \(A,B\).
+Duality identifies \(H^2(C_{\rm str})\) with \(V^*\); denote the
+dual basis by \(A^*,B^*\).  The two integral triangles are
+\begin{equation}
+ C_{\rm str}\longrightarrow C_F\longrightarrow F[-1]\longrightarrow,
+ \qquad
+ C_{\rm str}\longrightarrow C_S\longrightarrow S[-1]\longrightarrow.
+ \label{rd:triangles}
+\end{equation}
+Their fraction-field exact sequences are
+\[
+ 0\longrightarrow F_OA\longrightarrow F
+       \xrightarrow{\delta_F}V^*
+       \longrightarrow F_OA^*\longrightarrow0
+\]
+and the same sequence with \(A,F,A^*\) replaced by \(B,S,B^*\).
+The boundary is the dual of the relaxed localization.  In view of
+\eqref{rd:cross-pairing} and \(s(B)=Jv_0\), it satisfies
+\begin{equation}
+ \delta_F(x)=\epsilon_F\langle x,v_0\rangle B^*,
+ \qquad \epsilon_F\in O^\times.
+ \label{rd:boundary}
+\end{equation}
+Choose \(x\in F\otimes F_O\) with \(\delta_F(x)=B^*\).
+Then \(v_0\wedge x\) has unit volume in the integral determinant
+lattice \(\det_O F\), because the alternating form on \(F\) is
+unimodular and \(\langle x,v_0\rangle\) is a unit.  Although \(x\)
+can be nonintegral, this assertion about its exterior product is
+an equality of lattices.  Similarly choose \(y\in S\otimes F_O\)
+mapping to \(A^*\).  The volume of \(Jv_0\wedge y\) is a unit in
+\(\det_O S\).
+
+The determinant isomorphisms of the two exact sequences send
+\[
+ A\otimes(A^*)^*
+ \quad\hbox{and}\quad B\otimes(B^*)^*
+\]
+respectively, up to units and the irrelevant determinant sign, to
+\begin{equation}
+ (A^*\wedge B^*)^*\otimes(v_0\wedge x),
+ \qquad
+ (A^*\wedge B^*)^*\otimes(Jv_0\wedge y).
+ \label{rd:common-tensor}
+\end{equation}
+These comparisons take place in
+\(\mathcal D(C_{\rm str})\otimes\det F\) and
+\(\mathcal D(C_{\rm str})\otimes\det S\).
+The two plane determinants are identified by a unimodular map.
+Hence the two coordinates have the same valuation.
+
+This also explains exactly where the finite indices go.  For a
+two-term elementary summand
+\([O\xrightarrow{\pi^d}O]\) in degrees one and two, its contribution
+to the coordinate in \(\mathcal D\) is \(\pi^{-d}\).
+Such summands, including those arising from nonprimitive
+localization, occur inside the integral isomorphisms
+\eqref{rd:triangles}.  They are not discarded when writing
+\eqref{rd:common-tensor}.  All strict torsion contributions are common
+to the two sides.  The argument does not ask for an integral
+complement to \(v_0\); for example \(v_0=2^de_1\) is paired with
+the fractional vector \(2^{-d}e_2\), whose exterior product with
+\(v_0\) has unit volume.
+\end{proof}
+
+\subsection{Rank reduction and the outer DVR}
+\label{rd:outer-subsection}
+
+The local switch preserves a determinant valuation.  We now use it to
+reduce the dimension of any larger specialized Selmer space until a
+nonzero minor controls that valuation.  The first argument handles each
+characteristic-zero height-one prime.  A second, outer limit turns a
+hypothetical sequence of unbounded losses at $(2)$ into the same rank
+reduction problem.
+
+\begin{lemma}[Selecting a rank-two evaluation]\label{rd:rank-two}
+Let \(\mathcal G\) be a group, \(H\) a normal subgroup acting trivially
+on an absolutely irreducible representation \(W\) over a field \(F_O\),
+and suppose that restriction of the relevant cohomology classes to
+\(H\) is injective.  For two independent classes, their joint
+evaluations on \(H\) span \(W\oplus W\).  If the finite localization
+is the rank-two projection supplied by \(1+\tau\) on the induced
+representation, some actual evaluation has rank two.  One of the
+two classes may be any prescribed nonzero class.
+\end{lemma}
+
+\begin{proof}
+On \(H\) a crossed homomorphism is an additive homomorphism, and its
+image is equivariant for \(\mathcal G\).  A proper submodule of
+\(W\oplus W\) dominating both factors would impose a nonzero
+scalar linear relation on the two evaluations: the direct sum of
+two simple modules is semisimple, and
+\(\operatorname{End}_{\mathcal G}(W)=F_O\).
+That relation, followed by injectivity of restriction, would
+contradict independence of the classes.  Thus the joint span is
+the whole direct sum.  Projecting by \(1+\tau\) gives all pairs
+in the two-dimensional finite plane.
+
+It remains to pass from a span to a single element of the additive
+evaluation image \(U\).  If the determinant quadratic polynomial
+\(Q\) vanished on \(U\), then its polar form
+\(Q(u+v)-Q(u)-Q(v)\) would vanish for \(u,v\in U\).
+Consequently
+\[
+ Q\Bigl(\sum_i a_i u_i\Bigr)
+ =\sum_i a_i^2 Q(u_i)
+  +\sum_{i<j}a_i a_j
+       \bigl(Q(u_i+u_j)-Q(u_i)-Q(u_j)\bigr)
+ =0.
+ \]
+It would vanish on the entire field span, where it is not the zero
+polynomial.  This contradiction proves the assertion.  This
+polarization argument is valid in characteristic two as well.
+\end{proof}
+
+\begin{proposition}[Nonnegative valuations away from \((2)\)]
+\label{rd:away-two}
+Assume the simultaneous diagrams, derivative classes, and exact-kernel
+evaluation maps of Lemmas~\ref{rl:evaluation}, \ref{rl:kernel},
+\ref{rd:local-model}, and \ref{rd:derivative}.  For a primitive
+constituent whose generic cohomology has rank at most one in degrees
+one and two and zero elsewhere, its class-dual determinant
+coordinate belongs to \(\Lambda[1/2]\).
+This includes a simple-zero constituent and a base with
+\(2^\infty\)-Selmer corank one.
+\end{proposition}
+
+\begin{proof}
+The zero coordinate needs no argument.  Otherwise the generic
+rank is one and the class spans it.  Localize at a height-one prime
+\(\mathfrak p\ne(2)\), and denote the residue field by \(F_O\).
+Its characteristic is zero.  Lemma~\ref{rl:local-acyclic}, the
+degree-zero local conditions, and duality give zero fiber
+cohomology in degrees zero and three.  Canceling unit blocks
+therefore gives a free DVR model in degrees one and two.
+A primitive vector in its generic kernel has nonzero image in its
+\(\mathfrak p\)-fiber: the kernel of a map of free modules over a
+DVR is saturated.  Include this vector in each localization test.
+The two-sign reduction below has the same pattern as
+\cite[Lemmas~1.5.3 and~1.6.4]{Howard2012}; the integral switch
+and the evaluation hypotheses needed here at \(2\) have been
+constructed above.
+
+Suppose first that \(\Psi^2\ne1\) on \(F_O\).  Inertia at \(r\)
+distinguishes the two summands of the induced representation.
+The Tate representation over \(K\) is absolutely irreducible in
+characteristic zero: this follows from the open Tate image, or
+from its CM normalizer when \(K\) is not the CM field.
+Thus the induced representation is absolutely irreducible.
+If the test Selmer dimension is \(s>1\), Lemma~\ref{rd:rank-two}
+and the finite Chebotarev realization in Lemma~\ref{rl:chebotarev-evaluation}
+give a derivative prime with rank-two finite localization,
+nonzero on the chosen primitive vector.
+
+Poitou--Tate duality now determines the effect on the fiber.
+The relaxed localization image is a self-annihilating plane.
+It contains the whole finite plane, and hence equals that plane.
+The strict kernel has dimension \(s-2\), and its replacement by
+the singular condition adds no new class.  The fiber dimension
+therefore drops by two.  On the generic field, the localization
+is nonzero, so Lemmas~\ref{rd:derivative} and
+\ref{rd:det-switch} preserve generic rank one, the nonzero
+derivative class, and the determinant valuation.
+
+If \(\Psi^2=1\), use conjugation, including its action
+\((1+t)\mapsto(1+t)^{-1}\), to split the fiber into two signs.
+All prior plane conditions are stable under this involution in
+the split local model.  The two Galois summands are nonisomorphic
+absolutely irreducible representations; the only CM quadratic
+self-twist has been excluded.  Each sign has one finite and one
+singular coordinate.  The local pairing in a sign is a
+nondegenerate symmetric hyperbolic pairing, whose only isotropic
+lines in characteristic zero are the finite and singular axes.
+If both signs occur globally, choose one class from each, including
+a nonzero component of the primitive vector, to obtain the same
+rank-two reduction.
+
+If only one sign occurs and \(s>1\), choose a rank-one finite
+localization nonzero on that vector.  In this sign the relaxed
+image is the finite line.  In the other sign its intersection
+with the finite line is zero, and self-duality forces it to be
+the singular line.  A switch retains the old strict kernel of
+dimension \(s-1\ge1\), introduces the other sign, and preserves
+the total dimension.  The next rank-two switch reduces it.
+After finitely many such operations the fiber dimension is one.
+It cannot fall below one, since a primitive vector from the
+generic rank-one kernel survives after each switch.
+
+At fiber dimension one, a minimal self-dual model over the DVR
+has one free term in each of degrees one and two and zero
+differential.  The class and paired functional are integral at
+this height-one prime.  Their determinant coordinate is
+therefore integral.  Reversing
+\eqref{rd:valuation-switch} proves nonnegative valuation at
+\(\mathfrak p\).  Applying this at every height-one prime other
+than \((2)\), and using that the normal ring \(\Lambda\) is the
+intersection of its height-one valuation rings inside its fraction
+field, gives membership in \(\Lambda[1/2]\).
+
+For the stated arithmetic instances, the rational central
+specialization has Selmer dimension one and no invariants.
+Upper semicontinuity bounds the generic dimension by one.
+If the generic class is nonzero, it spans this space.  This
+uses Selmer corank at an unknown base and does not replace that
+corank by Mordell--Weil rank.
+\end{proof}
+
+\begin{lemma}[The terminal minor estimate]\label{rd:minor}
+Let
+\[
+ C_i=[\,O_i^n\xrightarrow{D_i}O_i^n\,]
+\]
+be based complexes over DVRs with uniformizer \(2\), with generic
+rank-one kernel.  Let \(z_i\in\ker D_i\) be integral, and let
+\(\lambda_i\in 2^{-a}(O_i^n)^*\) annihilate \(\operatorname{im}D_i\),
+where \(a\) is fixed.  If an \((n-1)\)-minor \(\Delta_i\) has
+\(v(\Delta_i)\le b\), the determinant coordinate \(u_i\) of
+\(z_i\otimes\lambda_i\) satisfies
+\begin{equation}
+ v(u_i)\ge-a-b.
+ \label{rd:minor-bound}
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Permute rows and columns so that the minor is the upper-left block
+\(A_i\).  Write \(D_i=\left(\begin{smallmatrix}A_i&b_i\\c_i&d_i
+\end{smallmatrix}\right)\).  Since its rank is \(n-1\), its Schur
+complement is zero.  Over the fraction field, use the triangular
+basis changes with entries \(A_i^{-1}b_i,c_iA_i^{-1}\) to leave
+\(A_i\) and a zero one-by-one block.  These triangular changes
+have determinant one.  If \(\alpha_i\) is the last coordinate
+of \(z_i\) and \(\beta_i\) the value of \(\lambda_i\) on the last
+target basis vector, cancellation gives exactly
+\begin{equation}
+ u_i=\pm\frac{\alpha_i\beta_i}{\det A_i}.
+ \label{rd:minor-formula}
+\end{equation}
+Here \(\alpha_i\in O_i\) and \(\beta_i\in2^{-a}O_i\).
+This proves \eqref{rd:minor-bound}.  If \(n=1\), the empty
+minor is \(1\) and the same formula applies.
+\end{proof}
+
+\begin{proposition}[Uniform residual valuation from outer rank reduction]
+\label{rd:outer-bound}
+Assume all of the following properties of a collection of primitive
+ring instances:
+\begin{enumerate}
+\item after any fixed number of switches, minimal \(\mathcal O\)-models
+have size bounded independently of \(h\) and of \(K\) with
+\(\omega(k)\le A\);
+\item the all-sequence evaluation and exact-kernel restriction of
+Proposition~\ref{rl:outer-evaluation}, and the finite Chebotarev
+realization of Lemma~\ref{rl:chebotarev-evaluation}, hold for these
+models, retaining their degree-two cocycle identities;
+\item the integral localization and duality diagrams and the
+derivative classes of Lemma~\ref{rd:derivative} survive this same
+outer limit; the class and its closed dual functional have a common
+bounded clearing exponent.
+\end{enumerate}
+Then there is \(C(E,A)\) such that
+\begin{equation}
+ v_{(2)}(u)\ge-C(E,A)
+ \label{rd:residual-bound}
+\end{equation}
+for every simple-zero ring instance.  The hypotheses refer to a
+single simultaneous construction.  In particular, separate
+abstract models realizing its individual cohomology groups do
+not establish this proposition's arithmetic applicability.
+\end{proposition}
+
+\begin{proof}
+The preceding paired-diagram construction supplies the common
+clearing exponent in the arithmetic application.  The bound on
+minimal sizes is Lemma~\ref{rl:residual-bound}; in particular,
+active conductor primes do not contribute to that size after
+residual localization.
+
+Suppose that \(v_{(2)}(u_i)\to-\infty\).  All these coordinates
+are nonzero.  First put the complexes into square two-term form.
+At the residual \(\Omega\)-test, moving-prime inertia excludes
+global invariants, and the local degree-zero conditions identify
+Selmer invariants with those global invariants.  Thus \(H^0=0\).
+The retained self-duality gives \(H^3=0\) and equal dimensions
+in degrees one and two.  A minimal \(\mathcal O\)-model therefore
+has only two free terms, of equal rank, in those degrees; all other
+terms cancel in integral disks.  Fix a nonprincipal ultrafilter
+\(\mathcal U\) and fix the bounded rank on an ultrafilter-large
+set.  Write these models as
+\[
+ C_i=[\,O_i^n\xrightarrow{D_i}O_i^n\,].
+\]
+
+Transport the class, the uniformly cleared functional, all evaluation
+matrices, and the finitely many retained comparison maps and boundary
+witnesses through these integral disk contractions.  Their entries
+remain integral and their identities remain exact.  The individual
+active-place complexes, whose number may grow with the outer instance,
+are not retained as an outer list of objects.  Their effect has already
+been incorporated in the target cone and its determinant; only the
+bounded minimal models and the fixed number of switch diagrams enter
+this limit.
+
+Apply Proposition~\ref{rl:outer-evaluation} to these models and
+their retained integral diagrams.  It gives a DVR \(O_\infty\)
+with uniformizer \(2\), characteristic-zero fraction field
+\(F_\infty\), and injective all-sequence evaluation, including
+restriction to the product of the exact inner kernels.
+The induced representation uses one simultaneous complex
+conjugation, so the sequence group over the quadratic fields has
+index two in the rational sequence group.  Lemma~
+\ref{rl:irreducibility} gives absolute irreducibility over
+\(F_\infty\): the moving inertia character has infinite order,
+and the finite Tate-matrix witness survives the outer quotient.
+
+The outer complex may have more cohomology than any inner generic
+complex.  Its new cycles can be represented by vectors \(w_i\)
+for which \(v(D_iw_i)\to+\infty\), without exact inner lifts.
+The retained degree-two evaluation matrices are integral, so the
+evaluations of \(D_iw_i\) also tend to zero.  This is the part of
+Proposition~\ref{rl:outer-evaluation} that makes these new classes
+available for the following rank reduction.
+
+Choose in each inner generic kernel a primitive vector \(v_i\).
+Some coordinate is a unit on a set in \(\mathcal U\), so its
+outer limit is nonzero.  Thus the outer kernel has dimension
+at least one.  If that dimension is \(s>1\), apply
+Lemma~\ref{rd:rank-two} to two outer classes, one the surviving
+primitive vector.  Realize their rank-two evaluation by inner
+derivative primes.  This realization only asks for finite
+quotients at each stage.  For an approximate inner cocycle
+one chooses the retained precision below the valuation of
+its coboundary; no exact inner lift of the extra outer class
+is being assumed.
+
+More explicitly, retain the original inner index \(i\)
+for outer instance \(j\), its ultrafilter \(\mathcal U_j\),
+and the outer ultrafilter \(\mathcal U\).  Represent the
+selected kernel element by \(u_j=(u_{j,i})_i\).
+At stage \((j,i)\), apply finite Chebotarev to this actual
+element and the old finite data to choose \(\ell_{j,i}\).
+Adjoin its local complexes, the derivative cocycle and
+the pairing homotopies to the same original stage
+diagram with its old contractions.  Take the same
+\(\mathcal U_j\) and then the same \(\mathcal U\).
+The retained coefficient precision may increase more
+slowly, but cofinally; the earlier maps and their
+arbitrary evaluations are not reconstructed by an
+independent choice of subsequence or model.
+
+The localization of \(v_i\) is generically nonzero on a set
+in \(\mathcal U\): its outer image is nonzero.  Therefore
+the original inner spanning class also localizes nontrivially,
+even if its multiple of \(v_i\) has large valuation.
+Lemma~\ref{rd:det-switch} preserves each inner determinant
+valuation exactly.  On the outer field, the relaxed image
+is the finite plane and the singular switch lowers \(s\)
+by two, by the same localization exact sequence as in
+Proposition~\ref{rd:away-two}.  The new inner generic kernels
+again supply primitive vectors, so the outer rank remains
+at least one.
+
+Only finitely many switches are necessary: the initial outer
+rank is bounded and drops by two.  After each fixed number
+of switches the required model-size bound remains available.
+At the terminal outer rank one, the outer differential has
+a nonzero maximal boundary minor.  Its inner determinants
+\(\Delta_i\) have one finite valuation on a set in
+\(\mathcal U\), by the definition of \(O_\infty\).
+Lemma~\ref{rd:minor} then gives a fixed lower bound on
+\(v(u_i)\), a contradiction.
+
+This proof uses the minor only after rank reduction.  Bounded
+matrix size alone would not suffice: the matrices \([2^i]\)
+have bounded size and unbounded torsion length.  In that
+example the entire matrix becomes zero in the outer quotient;
+the rank-reduction and evaluation hypotheses would have to
+supply additional arithmetic operations before a nonzero
+terminal minor could be used.
+\end{proof}
+
+\subsection{Central specialization and the local indices}
+\label{rd:specialization-subsection}
+
+The preceding bounds concern the deformed determinant.  We next identify
+its value at $t=0$ with the index of the original Heegner point and the
+finite Shafarevich--Tate lengths.  The comparison retains the whole
+integral complex; in particular, every active split conductor prime has
+an explicit local contribution.
+
+\begin{lemma}[Derived specialization at constant rational rank]
+\label{rd:derived-specialization}
+Let \(C\) be a perfect \(\Lambda\)-complex and put
+\(C_0=C\otimes_\Lambda^{\mathbf L}\mathbb Z_2\), where \(t=0\).
+Suppose its generic rational cohomology and its central rational
+cohomology both have dimension one in degrees one and two and
+are zero in other degrees.  Put \(A=\Lambda_{(t)}\), with residue field \(\mathbb Q_2\).
+Let \(z\) be a degree-one cycle and \(\lambda\) a closed
+degree-two functional in a finite free model of \(C\otimes_\Lambda A\),
+and let \(z_0,\lambda_0\) be their images under \(A\to\mathbb Q_2\).
+Then the generic determinant tensor \([z]\otimes[\lambda]\) is
+regular at \((t)\) and specializes, under the determinant base-change
+isomorphism, to \([z_0]\otimes[\lambda_0]\).  Its central valuation is
+computed from the entire integral complex \(C_0\), including
+its torsion cohomology.  It cannot be computed by substituting
+the naive quotient of \(H^1(C)\) by \(t\).
+\end{lemma}
+
+\begin{proof}
+Represent \(C\) by a bounded free complex.  Derived reduction
+is its termwise reduction, so the determinant line itself
+commutes with specialization.  For clarity, the cohomology
+base-change sequence is
+\begin{equation}
+ 0\longrightarrow H^i(C)/tH^i(C)
+ \longrightarrow H^i(C_0)
+ \longrightarrow H^{i+1}(C)[t]\longrightarrow0.
+ \label{rd:tor-basechange}
+\end{equation}
+These last terms are the relevant \(\operatorname{Tor}_1\)
+terms.  Diagonalize the complex over the DVR \(A\), or over its
+completion \(\mathbb Q_2[[t]]\).  A nonunit pivot
+would create an additional central cohomology class in its
+two adjacent degrees.  The assumed constant dimensions
+exclude such pivots.  Thus over \(\mathbb Q_2[[t]]\) there
+are only invertible disks and the two free cohomology lines.
+The rational class-dual tensor consequently specializes
+to the stated central tensor.  Over \(\mathbb Z_2\), all
+elementary divisors of \(C_0\), including the torsion supplied
+by \eqref{rd:tor-basechange}, remain in its determinant
+isomorphism.  This proves the assertion.
+\end{proof}
+
+\begin{lemma}[A ramified good odd place]\label{rd:active-local}
+Let \(F_v/\mathbb Q_p\) be unramified, \(p\ne2\), and let
+\(T'=T_2(E)\otimes\chi\), where \(E\) has good reduction
+and \(\chi\) is ramified quadratic.  Then
+\begin{equation}
+ H^0(F_v,T')=0,\qquad
+ H^1(F_v,T')=E^\chi(F_v)^{\wedge}_2,\qquad
+ \operatorname{length}_{\mathbb Z_2}H^2(F_v,T')
+ =\dim_{\mathbb F_2}E[2]^{\operatorname{Fr}_v}.
+ \label{rd:local-length}
+\end{equation}
+The quotient of the full local complex by the integral Kummer
+condition therefore contributes precisely this last length
+in degree two.
+\end{lemma}
+
+\begin{proof}
+The pro-\(2\) tame inertia generator acts by \(-1\).
+Its complex has differential \(-2\), so its cohomology is
+zero in degree zero and \(T'/2T'\) in degree one.
+Taking residue-field cohomology gives
+\[
+ H^1(F_v,T')\simeq E[2]^{\operatorname{Fr}_v},\qquad
+ H^2(F_v,T')\simeq E[2]/(\operatorname{Fr}_v-1)E[2].
+ \]
+The possible residue-cardinality factor on inertia cohomology
+is one modulo \(2\).  Kernel and cokernel of an endomorphism
+of a finite vector space have the same dimension, proving
+the length assertion.
+
+The integral local Kummer exact sequence identifies the
+\(2\)-adic completion of the points with all of \(H^1(F_v,T')\).
+Indeed the inverse Tate module of the local Weil--Chatelet
+group is zero here: local duality identifies its
+\(2\)-primary part with the dual of the finite completion
+of the point group.  The point completion is finite because
+the residue characteristic is odd.  Equivalently, inertia
+\(-1\) implies that the local \(2\)-primary torsion is
+exactly the Frobenius-fixed \(2\)-torsion.  The Kummer
+complex includes the derived invariants of this finite
+module on reduction, as in Lemma~\ref{rl:kummer}.
+The quotient therefore has exactly the degree-two
+cohomology displayed above.
+\end{proof}
+
+\begin{proposition}[The central determinant formula]
+\label{rd:central-formula}
+Suppose the product
+\[
+ L(E^{(h)},s)L(E^{(hk)},s)
+\]
+has a simple zero at \(s=1\).
+Use the primitive Heegner sum \(P_h\) and its free-line index
+\(j_h\) defined at the start of Section~\ref{rl:section}, and put
+\[
+ s(h)=\operatorname{length}_{\mathbb Z_2}
+             \Sha(E^{(h)}/\mathbb Q)[2^\infty].
+ \]
+Use the primitive Selmer determinant coordinate \(u_h\)
+with the fixed class multiplier of Lemma~\ref{rd:derivative}.
+For the primitive Selmer diagrams constructed above, we have
+\begin{equation}
+ \begin{split}
+ v(u_h(0))
+  &=2j_h-
+    \operatorname{length}_{\mathbb Z_2}
+                  \Sha(E^{(h)}/K)[2^\infty]
+    -2\sum_{p\mid h}t_p+O_E(1)\\
+  &=2j_h-s(h)-s(hk)-4w(h)+O_{E,A}(1),
+       \qquad \omega(k)\le A .
+ \end{split}
+ \label{rd:central-value}
+\end{equation}
+The first error includes the auxiliary prime and the fixed
+parametrization multiplier.  The second has no dependence
+on the number of primes dividing \(h\).
+\end{proposition}
+
+\begin{proof}
+The two rational twists have analytic ranks zero and one.
+The forward low-analytic-rank theorems give finiteness of
+their whole Shafarevich--Tate groups; equivalently the
+Heegner simple-zero theorem over \(K\) gives rank one
+and finite Shafarevich--Tate group there.  Thus all finite
+lengths in this proof refer to known finite groups.
+
+The specialization comparison is made at each finite stage, before
+the target data are retained.  Put \(c=|h|\).  At the split auxiliary
+prime \(r_i\), the point distribution relation is
+\[
+ \operatorname{Tr}_{H_{cr_i}/H_c}(y_{cr_i})
+       =(a_{r_i}-\sigma_{r_i}-\sigma_{\bar r_i})y_c,
+\]
+where the two Artin symbols have inverse ring-class actions.  At
+\(t=0\), the scalar character is trivial, while the rational quadratic
+character has value \(\chi_h(r_i)\) on both symbols.  The completed
+weighted sum therefore gives exactly
+\(L m_i P_h\), where \(m_i=a_{r_i}-2\chi_h(r_i)\).
+
+This point identity is retained on bounded target complexes as follows.
+Let \(C_h^{\rm Kum}\) be a fixed free model of the ordinary Kummer Selmer complex
+over \(K\), and let \(b_h\) represent the Kummer class of \(P_h\).
+At the retained coefficient precision, enlargement of the ramification
+set and relaxation at the active and auxiliary primes give a target
+comparison \(j_i:C_h^{\rm Kum}/2^i\to C_i/(t)\).  Each retained local
+condition is an isomorphism on degree zero and an injection on degree
+one, so the Selmer-cone exact sequence injects its degree-one
+cohomology into global cohomology.  Kummer naturality and the point
+relation therefore give, after contraction, a target degree-zero vector
+\(v_i\) with
+\[
+ z_i(0)-L m_i j_i(b_h)=d v_i.
+\]
+Include this vector, the comparison matrix, and the associated local
+localization triangles in the same target diagram.  Their ranks are
+bounded for this fixed \(h,K\), independently of \(r_i\) and of
+\([H_{cr_i}:K]\).  All statements are reduced to the same slower
+cofinal precision if needed.  Lemma~\ref{rl:finite-limits} now retains
+the displayed boundary identity.  The trace bound gives
+\(v_2(m_i)\le c_E\); hence their limit \(m\in\mathbf Z_2\) on the
+same ultrafilter is nonzero and has \(v_2(m)\le c_E\).
+The active local quotients specialize
+to Lemma~\ref{rd:active-local}, and the Kummer conditions at
+\(2N\), including their differential lattices, are unchanged.
+The central point is therefore nonzero, so generic
+and central rational cohomology both have rank one.
+
+To specialize the paired functional, choose its representative
+over \(\Lambda_{(t)}\).  At an active ramified prime, inertia
+minus one is \(-2\), a unit in this ring.  At \(r\), the
+central Frobenius determinants are nonzero by the trace bound.
+Their local complexes thus contract over \(\Lambda_{(t)}\).
+Composing these contractions with retained global duality gives
+the same generic functional as the contractions over
+\(\mathcal O\), and specializes to rational central Kummer
+duality.  Lemma~\ref{rd:derived-specialization} applies to this
+representative; no regularity is claimed for an arbitrary
+\(\mathcal O\)-cochain representing the functional.
+
+The ordinary integral Kummer complex has compact Selmer
+cohomology in degree one.  Its degree-two torsion has
+length equal to that of the finite quotient of the
+discrete Selmer group by its maximal divisible subgroup.
+This follows by derived base change to
+\(\mathbb Q_2/\mathbb Z_2\) and the exact local Kummer
+conditions; the base-change terms must be retained.
+Here that quotient is \(\Sha(E^{(h)}/K)[2^\infty]\).
+The free parts of degrees one and two are paired
+unimodularly by integral duality.  Torsion in degree
+one and its dual contribution in degree three have
+bounded length, by the fixed-scalar bound on global
+torsion.
+
+For completeness, an elementary divisor of size \(2^d\)
+whose cokernel is in degree \(i\) contributes
+\((-1)^{i+1}d\) to the coordinate in the inverse
+determinant.  Consequently the degree-two
+Shafarevich--Tate torsion subtracts its length.  On
+the free line, a point of index \(2^{j_h}\) and its
+functional each contribute \(j_h\), giving \(2j_h\).
+
+At an active prime \(p\mid h\), both primes of \(K\)
+above \(p\) are governed by
+Lemma~\ref{rd:active-local}, each with length \(t_p\).
+Relaxation therefore subtracts exactly \(2t_p\).
+There is no bound hiding an error at every such prime.
+At \(r\), the central scalar character is trivial;
+the local degree-two determinant has the valuation
+of \(r+1-a_r\chi_h(r)\) at each split place.
+The trace condition
+\[
+ v_2(2-a_r),\ v_2(2+a_r)\le c_E,\qquad
+ v_2(r-1)>c_E
+ \]
+makes both valuations at most \(c_E\).  It also bounds
+the trace multiplier
+\(a_r-2\chi_h(r)\) on the central point.
+At primes over \(2N\) the integral Kummer condition
+is retained.  The local twisting types form a fixed
+finite set and \(K\) is split there; all comparison
+indices, including those of Neron differential
+lattices at \(2\), are therefore bounded in terms
+of \(E\).  This proves the first equality, using
+Lemma~\ref{rd:derived-specialization} to compare
+the generic tensor with the central tensor.
+
+We spell out the quadratic comparison needed for
+the second equality.  Set
+\[
+ A_h=\operatorname{Res}_{K/\mathbb Q}(E^{(h)}_K),
+ \qquad B_h=E^{(h)}\times E^{(hk)}.
+ \]
+There are isogenies in both directions whose
+composites are multiplication by \(2\).
+The isogeny-invariance identity for the BSD quotient,
+with finite Shafarevich--Tate groups, is
+\begin{equation}
+ \frac{\#\Sha(A_h)\operatorname{Reg}(A_h)
+             \operatorname{Per}(A_h)\prod_p c_p(A_h)}
+      {\#A_h(\mathbb Q)_{\rm tors}
+       \#A_h^\vee(\mathbb Q)_{\rm tors}}
+ =
+ \frac{\#\Sha(B_h)\operatorname{Reg}(B_h)
+             \operatorname{Per}(B_h)\prod_p c_p(B_h)}
+      {\#B_h(\mathbb Q)_{\rm tors}
+       \#B_h^\vee(\mathbb Q)_{\rm tors}} .
+ \label{rd:isogeny-identity}
+\end{equation}
+Regulators and periods are taken with compatible
+dual and Neron normalizations.  This is the
+Cassels--Tate isogeny identity.  The elliptic-curve local-measure
+formula is due to Cassels \cite[Theorem~1.1]{Cassels1965}.
+For the general abelian-variety identity used here, see the proof of
+\cite[I, Theorem~7.3, especially (7.3.1), pp.~97--100]{MilneADT2006}
+and \cite[Section~4.1]{DokchitserDokchitser2010}.  This identity requires
+finite Shafarevich--Tate groups, as already established at these
+noncentral twists, but does not assume the BSD conjecture.
+The fixed-degree composites bound the rank-one
+regulator indices, the torsion ratios, and the
+differential and period indices by fixed powers
+of \(2\).  The identity is essential: a statement
+that the maps on Shafarevich--Tate groups have
+kernel and cokernel killed by \(2\) would not
+bound their lengths uniformly.
+
+At every split prime of \(K/\mathbb Q\) the component
+factors of \(A_h\) and \(B_h\) agree.  In particular
+this cancels all primes dividing \(h\).
+At an odd prime \(p\mid k\) outside \(2N\), the curve
+over \(K\) has good reduction.  Its Weil restriction
+has connected special fiber: the reduction of
+Weil restriction over the ramified quadratic
+extension is an extension of the good special
+fiber by a connected additive group.  Thus its
+component factor is \(1\).  Of the two rational
+twists, one is good and the other is ramified
+quadratic with good original curve.  The latter
+has \(2\)-part of its component number equal to
+\(2^{t_p}\).  This follows as well from the
+ramified tame description in
+Lemma~\ref{rd:active-local}: its connected
+reduction is unipotent, and its local
+\(2\)-primary torsion is the fixed
+\(2\)-torsion.  At other good nonsplit places
+the component factors are \(1\); at \(2N\)
+there are only the fixed local comparisons.
+Taking valuations in \eqref{rd:isogeny-identity}
+therefore gives
+\begin{equation}
+ \operatorname{length}\Sha(E^{(h)}/K)[2^\infty]
+ =s(h)+s(hk)+\sum_{p\mid k}t_p+O_E(1).
+ \label{rd:sha-comparison}
+\end{equation}
+Since \(0\le t_p\le2\), the sum over \(k\) is at
+most \(2A\).  Substituting this equality and
+\(2\sum_{p\mid h}t_p=4w(h)\) proves the result.
+\end{proof}
+
+\begin{proof}[Proof of Corollary~\ref{rd:ring-lower}]
+The arithmetic diagrams constructed with Lemma~\ref{rd:derivative},
+their paired functionals, and the model bound of
+Lemma~\ref{rl:residual-bound} give the hypotheses of
+Propositions~\ref{rd:away-two} and \ref{rd:outer-bound}.
+Those propositions give nonnegative height-one valuations
+away from \((2)\) and a lower bound \(-C(E,A)\) at \((2)\).
+Normality of \(\Lambda\) gives \(2^{C(E,A)}u_h\in\Lambda\).
+Its constant coefficient has valuation at least \(-C(E,A)\).
+Apply \eqref{rd:central-value}.
+\end{proof}
+
+\subsection{The universal correction and bounded clearing}
+\label{rd:clearing-subsection}
+
+For transfer to a missing vertex, the individual determinant coordinates
+must belong to one quadratic group-ring family.  We first remove the
+Euler factors at primes unused by a character.  We then clear the
+remaining denominators with a factor whose central valuation and
+negative Laurent support are bounded independently of the cube size.
+The base curve and the companion field are fixed in this argument.
+
+Fix \(K\) and a binary family \eqref{rd:binary-data}, based at
+\(h_0=1\).  Put
+\[
+ G=(\mathbb Z/2\mathbb Z)^b,\qquad R=\Lambda[G].
+\]
+Let \(g_q\in G\) be the inertia bit:
+\(g_q(x)=(-1)^{\lambda_q(x)}\).
+Augmentation is denoted by subscript \(0\).
+
+The universal class is constructed before making any character
+evaluation.  Fix this finite family and \(G\).  At precision \(i\)
+take the Heegner point of conductor \(r_i\prod_{q\in Q}q\).
+Let \(\varepsilon:G_K\to G\) be the universal quadratic character.
+In the finite Shapiro transfer of Lemma~\ref{rd:derivative}, use
+the coefficient map
+\[
+ A_i[\operatorname{Gal}(H_{r_i\prod q}/K)]
+       \longrightarrow A_i[G],\qquad
+ [\sigma]\longmapsto\Psi_i(\sigma)[\varepsilon(\sigma)],
+\]
+with the same inverse-action convention on both sides.  Complete
+this weighted coset sum to obtain a target Selmer-cone cycle over
+\(A_i[G]\), with local lifts and boundary witnesses, and then
+contract these target data.  The target-side rank bound applies
+for this fixed family and gives one trace class over \(\Lambda[G]\).
+Evaluation at \(x\) commutes with the finite coefficient map and
+every target identity, so gives the full trace for that character.
+If scalar models have already been chosen, compare the evaluated
+model with them through the original target cochains and retain the
+resulting comparison matrices and contraction homotopies.  There are
+only finitely many characters for this fixed \(G\).  At \(t=0\), the
+universal split-prime multiplier is
+\[
+ a_{r_i}-2[\varepsilon(\operatorname{Fr}_{\mathfrak r_i})].
+\]
+The conjugate Frobenius values agree because the character values
+have exponent two.  Its evaluation at \(x\) is
+\(a_{r_i}-2\chi_{h_x}(r_i)\), so the central boundary comparison
+used in Proposition~\ref{rd:central-formula} is retained in this
+same group-ring diagram before character evaluation.
+It includes Euler factors at primes unused by \(x\), which the next
+lemma removes.  No inverse limit over growing \(G\) or growing active
+support is used in this construction.
+
+The accompanying dual functional is constructed over
+\(\mathcal O[G]\), not separately at its characters.  At a new
+prime and at \(r\), the full local complex is acyclic modulo
+the maximal ideal by Lemma~\ref{rl:local-acyclic}, hence has an
+integral contraction over this local ring.  Use these contractions
+to lift the class to the orthogonal Selmer complex, and then apply
+the retained integral global duality.  This gives a closed integral
+functional, up to one fixed old-support power of \(2\) when such
+a support is included.  The present base is \(h_0=1\).
+
+\begin{lemma}[Removing unused primes in the group ring]
+\label{rd:unused}
+Suppose the universal relaxed trace class and its duality
+diagram have been constructed over \(R\).
+There is \(\eta\in\mathcal O[G]\) such that
+\(\eta u_{\rm rel}\) has each character coordinate
+equal to the primitive determinant coordinate, up
+to a \(\Lambda\)-unit.  The correction has no
+denominator accumulating with \(\#Q\).
+\end{lemma}
+
+\begin{proof}
+At an unused split prime \(q\), write \(z\) for the
+full scalar Frobenius at one orientation.  Conjugation
+gives \(z^{-1}\) at the other orientation.  The
+point trace contributes
+\[
+ e_q=a_q-z-z^{-1},
+ \]
+whereas relaxation of the singular quotient contributes
+\begin{equation}
+ l_q=
+ \det(1-q^{-1}z\operatorname{Fr}_q\mid T)
+ \det(1-q^{-1}z^{-1}\operatorname{Fr}_q\mid T).
+ \label{rd:unused-factors}
+\end{equation}
+Using reciprocal Frobenius conventions changes only
+the corresponding unit conventions in the paired
+determinant.  The localization triangle gives
+\(u_{\rm rel}=e_q^2u_{\rm prim}/l_q\) at this
+unused prime.  The square is required because the
+point and its dual both occur.
+
+Both factors are units in the local ring
+\(\mathcal O[G]\).  To check this it suffices to
+reduce modulo its maximal ideal, generated by \(2\)
+and the augmentation ideal.  There the scalar
+is \((1+t)^m\) with \(m\) odd, and the residual
+Frobenius polynomial is nonzero at this nonconstant
+scalar.  More precisely, before character
+evaluation the following identity holds modulo \(2\):
+\begin{equation}
+ (1+a_qz+z^2)(1+a_qz^{-1}+z^{-2})
+       =a_q^2+z^2+z^{-2}
+       =(a_q+z+z^{-1})^2 .
+ \label{rd:residual-euler}
+\end{equation}
+Thus \(l_q/e_q^2-1\in2\mathcal O[G]\), and
+\begin{equation}
+ \eta=
+ \prod_{q\in Q}
+ \left(1+
+       \frac{l_q/e_q^2-1}{2}(1+g_q)\right)
+ \in\mathcal O[G].
+ \label{rd:eta}
+\end{equation}
+Its factor evaluates to \(l_q/e_q^2\) when \(q\)
+is unused and to \(1\) when \(q\) is active.
+This proves the claim.  The apparent inertia
+projector denominator is canceled inside each
+factor before taking the product.
+\end{proof}
+
+\begin{lemma}[Laurent expansion and simultaneous elimination]
+\label{rd:schur}
+There is a natural identification
+\begin{equation}
+ \mathcal O/2^M\mathcal O
+       =(\mathbb Z/2^M\mathbb Z)((t)).
+ \label{rd:laurent}
+\end{equation}
+Suppose \(A=\operatorname{diag}(A_1,\ldots,A_m)\)
+has entries in \(\Lambda\), each residual determinant
+is nonzero, and
+\(\operatorname{ord}_t(\det\overline A_j)\le h\).
+If \(B=A+2E\) with \(E\) over \(\Lambda\), then
+every entry of \(B^{-1}\bmod2^M\) has no term below
+\(t^{-Mh}\).  The same bound holds for a Schur
+complement \(P-QB^{-1}S\), where \(P,Q,S\)
+have power-series entries.  The bound is independent
+of \(m\).
+\end{lemma}
+
+\begin{proof}
+For a denominator \(s\in\Lambda\setminus(2)\),
+write \(s=t^d v+2w\), with \(v\) a power-series
+unit.  Its inverse modulo \(2^M\) is its finite
+geometric expansion about \(t^dv\), which has
+finite negative support.  Conversely all Laurent
+series of finite negative support occur after
+inverting \(t\).  This proves \eqref{rd:laurent}.
+It also proves that an element of \(\mathcal O\)
+has a unique series \(\sum a_nt^n\), \(a_n\in\mathbb Z_2\),
+with \(v(a_n)\to+\infty\) as \(n\to-\infty\),
+and that its valuation is \(\min_n v(a_n)\).
+
+Write \(\det A_j=t^{d_j}v_j+2w_j\), with
+\(d_j\le h\), and put
+\[
+ C_j=\frac{\operatorname{adj}(A_j)}{t^{d_j}v_j},
+ \qquad C=\operatorname{diag}(C_j).
+ \]
+Each entry of \(C\) has pole at most \(h\), and
+\(BC=1+2F\), where entries of \(F\) have the same
+pole bound.  Therefore
+\begin{equation}
+ B^{-1}\equiv C\sum_{j=0}^{M-1}(-2F)^j
+                                  \pmod{2^M}.
+ \label{rd:inverse-expansion}
+\end{equation}
+The \(j\)-th summand has pole at most \((j+1)h\).
+Adding more entries in a matrix product cannot
+worsen this pole bound.  Multiplication by
+power-series matrices proves the Schur assertion.
+
+For a block over \(R\) whose augmentation is \(A\),
+the block is invertible over \(\mathcal O[G]\):
+its residual determinant in the local ring is a
+unit.  Each sign evaluation is \(A+2E_x\) with
+power-series entries, so the same estimate holds
+at all characters.  If \(S_0=0\), its Schur
+complement has augmentation exactly \(P_0\).
+\end{proof}
+
+\begin{lemma}[Clearing a bounded complex]\label{rd:clearing}
+Let \(D\) be a based complex of finite free
+\(\mathcal O[G]\)-modules of total graded rank
+at most \(r_0\), with \(D_0\) defined over
+\(\Lambda\).  Suppose \(D_0\otimes_\Lambda^{\mathbf L}\mathbb Z_2\)
+has rational cohomology of dimension one in degrees
+one and two, zero elsewhere, and total integral
+torsion length at most \(h_1\).
+Let \(z\) be an integral cycle and let
+\(\lambda\in2^{-a_0}(D^2)^*\) be a closed functional.
+Assume every character constituent is either
+generically acyclic or has the same generic rank-one pattern;
+in the generically acyclic case its determinant coordinate
+is defined to be zero.  Multiplication of these
+coordinates by any \(\eta\in\mathcal O[G]\)
+is permitted.  Then there are \(f,d\in\mathcal O[G]\)
+such that
+\begin{equation}
+ f_xu_x=d_x,\qquad
+ f_0\in\Lambda,\qquad
+ v(f_0(0))\le C(r_0,h_1,a_0)<\infty.
+ \label{rd:clearing-equations}
+\end{equation}
+If all entries of \(D_x\) have bounded negative
+Laurent support at each precision, uniformly
+in \(b\), the same is true of \(f_x\).
+No such bound on \(d_x,z,\lambda,\eta\) is required.
+\end{lemma}
+
+\begin{proof}
+Put the central integral complex into elementary-divisor
+bases over \(\mathbb Z_2\).  Its positive pivot
+valuations sum to at most \(h_1\).  Lift these
+constant integral basis changes.  Formally cancel
+the same pivot positions upstairs, recording the
+pivot numerators and all denominators in the
+resulting rational basis changes and determinant
+comparison.  There are at most \(r_0\) such
+operations.  A product \(P\) of sufficiently
+large, but bounded, powers of their numerators
+clears every denominator.  Its degree in the
+differential entries is bounded in terms of
+\(r_0\); its augmentation belongs to \(\Lambda\)
+and has central valuation bounded in terms of
+\(r_0,h_1\).
+
+On constituents where \(P_x\ne0\), the remaining
+complex is one-by-one in degrees one and two.
+If its differential is nonzero, the cycle
+identity forces the projected cycle to be zero,
+so the same rational determinant expression
+gives the stipulated zero coordinate.  Otherwise
+the expression is the product of the projected
+cycle and functional, with the canceled-pivot
+factor.  Clearing denominators gives
+\[
+ u_x=2^{-a_0}\eta_x Q_x/P_x,\qquad
+ Q\in\mathcal O[G].
+ \]
+Take \(f=2^{a_0}P^2\) and \(d=\eta P Q\).
+On a constituent where a pivot first vanishes,
+\(P_x=0\), so both sides of \(f_xu_x=d_x\)
+are zero.  Thus the identity holds on every
+character.  Since \(f\) is a polynomial of
+bounded degree in the bounded differential
+entries, its central valuation and pole
+properties follow.  All unused-prime
+corrections and eliminated-block units occur
+in \(d\); none is inserted into \(f\).
+\end{proof}
+
+\begin{proposition}[The bounded clearing factor for the family]
+\label{rd:family-clearing}
+Assume the compatible universal Selmer and Heegner
+diagrams and their primitive specializations.
+Suppose every character constituent of the universal Selmer complex
+is generically acyclic or has generic
+cohomology of dimension one in degrees one and two and zero
+elsewhere.  If the fixed base has \(2^\infty\)-Selmer corank
+one over \(K\), the corrected coordinates of
+Lemma~\ref{rd:unused} admit the factorization
+\eqref{rd:clearing-equations}, with central
+valuation and fixed-precision pole bounds
+independent of \(b\) and \(\#Q\).
+\end{proposition}
+
+\begin{proof}
+Use the integral universal class and paired functional just
+constructed.  To apply Lemma~\ref{rd:clearing}, it remains to
+reduce the complex to bounded size without losing uniform control
+of its negative Laurent powers.
+
+At augmentation, the localization triangles
+present the complex as the old complex extended
+by a direct sum of singular local blocks in
+degrees one and two, one block for each new
+prime.  Its differential is upper triangular,
+with the old complex a subcomplex.  This graded
+model lifts over \(R\): split disks over the
+augmentation local ring, identify the minimal
+parts, and lift the invertible basis changes.
+This is an application of the actual finite
+diagram construction, rather than just an
+equality of Euler characteristics.
+
+Each new block has determinant \(l_{q,0}\).
+Its residual \(t\)-order is uniformly bounded.
+In the reducible residual range the residual
+Frobenius polynomial is \((X+1)^2\), and an
+odd scalar exponent gives order two at each
+split place; thus order four suffices for the
+two-place block.  Eliminate the entire new
+block over \(\mathcal O[G]\).  By
+Lemma~\ref{rd:schur}, the resulting complex
+\(D\) has the bounded old size, has augmentation
+equal to the old complex, and has a uniform
+pole bound at every fixed precision.
+
+The central cohomology of the old complex is
+the fixed base Kummer cohomology, modified only
+by the bounded auxiliary-prime terms.  Its
+rational dimension is one in degrees one and
+two.  The quotient of the discrete Selmer
+group by its maximal divisible subgroup is
+finite because its Pontryagin dual is a
+finitely generated \(\mathbb Z_2\)-module.
+For this fixed \(K,E\), its length is a finite
+constant, even when the Shafarevich--Tate
+group itself is not known to be finite.
+Together with global torsion and the trace
+bound at \(r\), this bounds all central
+torsion lengths of \(D_0\).
+
+Apply Lemma~\ref{rd:clearing}.  The determinant
+of the eliminated block is a unit over
+\(\mathcal O[G]\), so its inverse is an
+allowed integral multiplier.  Include it
+and \(\eta\) from \eqref{rd:eta} in the
+numerator multiplier of that lemma.
+This proves the assertion.
+
+The dependence here is on fixed \(K,E\);
+it is different from the uniform
+\(C(E,A)\) of Proposition~\ref{rd:outer-bound}.
+No uniform bound on the finite Selmer quotient
+as \(K\) varies is asserted or needed for
+the missing-vertex argument.
+\end{proof}
+
+\begin{proposition}[Assembly of the arithmetic diagrams]
+\label{rd:assembly}
+For the ring data stated at the start of this section, the actual
+finite Heegner and Selmer constructions satisfy the simultaneous
+diagram hypotheses of Propositions~\ref{rd:away-two},
+\ref{rd:outer-bound}, and \ref{rd:central-formula}.
+The bounds needed for the lower-bound argument depend only on
+\(E\), the fixed local data, and a bound \(A\) on \(\omega(k)\).
+For the binary data \eqref{rd:binary-data} with fixed \(E,K\)
+and base Selmer corank one, assume in addition that every
+character constituent of the universal Selmer complex is generically acyclic
+or has generic cohomology of
+dimension one in degrees one and two and zero elsewhere.
+Then the constructions satisfy all the hypotheses of
+Proposition~\ref{rd:family-clearing}, with
+bounds independent of the cube dimension and of its
+new-prime support.
+\end{proposition}
+
+\begin{proof}
+The paired arithmetic diagram constructed after
+Lemma~\ref{rd:derivative} contains the actual weighted target classes,
+local Kummer lifts, target comparison and boundary witnesses, and
+simultaneous duality triangles.  The target-side construction
+gives its inner rank bound for each fixed finite conductor support;
+no ring-class source complex is passed to the limit.
+Theorem~\ref{rl:limit-evaluation-package} retains
+their evaluations and exact-kernel restrictions.  The same-stage
+extension in Proposition~\ref{rd:outer-bound} shows explicitly
+that these data remain compatible after each further switch.
+
+Lemma~\ref{rl:residual-bound} bounds the minimal models in terms
+of \(E,A\) and the number of retained derivative primes, not the
+active conductor support.  Outer rank reduction uses only a bounded
+number of switches, hence only finitely many of these bounds.
+The paired functional has the common clearing exponent established
+above.  These are the three hypotheses of
+Proposition~\ref{rd:outer-bound}; the characteristic-zero
+evaluations and local acyclicity give those of
+Proposition~\ref{rd:away-two}.  The actual central trace and
+\(\Lambda_{(t)}\)-representative, together with the local
+quotients of Lemma~\ref{rd:active-local}, were verified in the
+proof of Proposition~\ref{rd:central-formula}.
+
+For a binary family, Section~\ref{rd:clearing-subsection}
+constructs the universal trace and its dual functional over the
+group ring before evaluating characters.  Lemma~\ref{rd:unused}
+then removes the unused-prime factors integrally.  The bounded
+complex in Proposition~\ref{rd:family-clearing} comes from the
+actual localization triangles and simultaneous Schur elimination,
+with the order-four residual local bound.  Its central torsion is
+controlled by the finite quotient of the fixed base Selmer group
+and the bounded auxiliary-prime terms.  Together with the stated
+constituent-rank assumption, these give all the hypotheses of
+Lemma~\ref{rd:clearing}.
+
+For varying \(K\), only its bounded ramification count enters
+the residual-model and local-comparison constants.  For the final
+binary argument \(K\) is fixed, so its finite base Selmer
+quotient may enter the clearing constants.  At \(2N\) the
+original Kummer conditions have been retained throughout; the
+construction imposes no ordinary or semistable hypothesis.
+\end{proof}
+
+\subsection{Coefficient comparison and the missing vertex}
+\label{rd:transfer-subsection}
+
+The bounded clearing factor reduces the last step to finitely many
+coefficient congruences.  A parity argument on a sufficiently large
+binary cube makes a nonzero address agree with the base to the required
+precision.  If the base derivative were zero, that congruence would
+contradict the upper bound already supplied at every nonzero address.
+
+The common-zero count in the following proof is the characteristic-two
+Chevalley--Warning principle
+\cite[Section~1, equation~(1)]{HeathBrownChevalleyWarning}; its short parity
+proof is included here.
+
+\begin{lemma}[A nonzero address with prescribed coefficient bits]
+\label{rd:bits}
+Let \(A_1(x),\ldots,A_q(x)\) be specified Laurent
+coefficients of sign evaluations of elements of
+\(\mathcal O[G]\).  For \(M\ge1\), if
+\begin{equation}
+ b>q(2^M-1),
+ \label{rd:bit-threshold}
+\end{equation}
+there is \(x\ne0\) with
+\(A_i(x)\equiv A_i(0)\pmod{2^M}\) for every \(i\).
+\end{lemma}
+
+\begin{proof}
+Write the element supplying the \(i\)-th test as
+\(a_i=\sum_{g\in G}a_{i,g}(t)g\), and let \(n_i\) be the Laurent
+degree being tested.  Lemma~\ref{rd:schur} gives integral Laurent
+coefficients, so
+\[
+ z_i=\sum_{g\in G}[t^{n_i}]a_{i,g}(t)\,g\in\mathbb Z_2[G].
+\]
+The group \(G\) is finite, hence coefficient extraction commutes
+with character evaluation: \((z_i)_x=A_i(x)\).  Apply
+Lemma~\ref{cy:binary} to these \(q\) integral group-ring elements.
+Its bound is exactly \eqref{rd:bit-threshold}; no bound on the
+Laurent degrees or on the coefficients enters this argument.
+\end{proof}
+
+\begin{lemma}[Transfer through the bounded factor]
+\label{rd:binary-transfer}
+Suppose \(f,d\in\mathcal O[G]\),
+\(u_x\in\Lambda[1/2]\), and \(f_xu_x=d_x\)
+for every character.  Suppose
+\(f_0\in\Lambda\) and
+\(c=v(f_0(0))<\infty\).
+If every \(f_x\bmod2^M\) has no powers below
+\(-L(M)\), then for
+\[
+ M>c,\qquad
+ b>(L(M)+2)(2^M-1)
+ \]
+there is \(x\ne0\) satisfying
+\begin{equation}
+ v\bigl(u_x(0)-u_0(0)\bigr)\ge M-2c.
+ \label{rd:binary-congruence}
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Apply Lemma~\ref{rd:bits} to the coefficients
+of \(f_x\) in degrees \(-L(M),\ldots,0\)
+and the constant coefficient of \(d_x\).
+At its nonzero solution, all negative
+coefficients of \(f_x\) are divisible by
+\(2^M\), and its constant coefficient is
+congruent to \(f_0(0)\).  Hence the latter
+has valuation \(c\), and
+\(v_{\mathcal O}(f_x)\le c\).
+Since \(d_x\) is integral over \(\mathcal O\),
+\(v_{\mathcal O}(u_x)\ge-c\), and the same
+holds for \(u_0\).  The Laurent expansion
+in Lemma~\ref{rd:schur} then shows that all
+their coefficients lie in \(2^{-c}\mathbb Z_2\).
+They have no negative coefficients because
+they belong to \(\Lambda[1/2]\).
+
+The constant-term convolution converges
+\(2\)-adically, and gives
+\[
+ [t^0]d_x\equiv [t^0]f_x\,u_x(0)
+                          \pmod{2^{M-c}},
+ \qquad
+ [t^0]d_0=f_0(0)u_0(0).
+ \]
+Use the imposed congruences and divide by
+\(f_0(0)\).  This proves
+\eqref{rd:binary-congruence}.  The possibly
+large negative support of \(d_x\) was never
+tested; only its constant coefficient was used.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{rd:missing-vertex}]
+For every character the primitive generic
+cohomology has rank at most one.  At nonzero
+vertices this follows from the simple-zero
+theorem; at zero it follows from
+\eqref{rd:base-corank} and rational
+specialization.  There are no generic invariants, and duality
+pairs degrees one and two; thus each primitive constituent is acyclic
+or has exactly the rank-one pattern.  The unused-prime local blocks
+are acyclic over the fraction field, by the nonzero factors
+\eqref{rd:unused-factors}.  The universal Selmer complex has the
+same generic pattern, as required by
+Proposition~\ref{rd:family-clearing}.  With this hypothesis
+verified, Proposition~\ref{rd:assembly} supplies the compatible
+universal arithmetic construction.  By Proposition~\ref{rd:away-two}
+and Lemma~\ref{rd:unused}, the corrected
+coordinates belong to \(\Lambda[1/2]\).
+Proposition~\ref{rd:central-formula} gives
+\[
+ v(u_x(0))\le B_1\quad(x\ne0)
+ \]
+for a constant \(B_1\) depending only on
+\(B,E,K\) and the fixed local data.
+
+If the base derivative vanished, the
+Gross--Zagier formula would make the
+central Heegner sum torsion: its canonical
+height is a fixed nonzero multiple of
+that derivative.  Here only the conductor-one formula at
+the fixed base \(h=1\) is used, with the corrected primitive
+sum and the split Heegner data of
+\cite[Theorem~1.1]{CaiShuTian2014}.
+At the base this implies \(u_0(0)=0\).
+Indeed, if the generic rank is zero the
+coordinate was defined to be zero.  If the
+generic rank is one, its central rank is
+also one by \eqref{rd:base-corank}, so
+Lemma~\ref{rd:derived-specialization}
+identifies its specialized class with
+the zero rational Kummer class of that
+torsion point.
+
+Proposition~\ref{rd:family-clearing} supplies
+\(f,d\) with a common bound \(c_*\) on
+\(v(f_0(0))\) and common pole bounds
+\(L(M)\).  Choose
+\[
+ M>\max\{c_*,B_1+2c_*\}
+ \]
+and then take \(b\) above the threshold
+of Lemma~\ref{rd:binary-transfer}.
+That lemma gives a nonzero vertex with
+\(v(u_x(0))\ge M-2c_*>B_1\), a
+contradiction.  Thus the base derivative
+is nonzero.  The classical split Heegner
+sign is odd, so the base product has
+a simple zero.
+\end{proof}
+
+\begin{remark}[The simultaneous arithmetic interface]
+\label{rd:interface-status}
+The determinant argument can be reused only with
+the simultaneous arithmetic construction verified
+in Proposition~\ref{rd:assembly} and
+Theorem~\ref{rl:limit-evaluation-package}.
+Neither the published duality theorem nor the
+classical Heegner formulas alone supplies that
+interface.
+\end{remark}
diff --git a/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/build/sections/ring-limits.tex b/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/build/sections/ring-limits.tex
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+\section{Ring-class interpolation and integral deformations}
+\label{rl:section}
+
+The ring-class argument has two outputs: a lower bound for the index of
+a Heegner point at a known simple zero, and a transfer of nonvanishing to
+an unknown base.  The first is uniform as the companion field varies
+with bounded ramification; the second fixes that field before a binary
+family grows.  We state both outputs before constructing their cochain
+and evaluation maps.  Their determinant proofs occupy
+Section~\ref{rd:section}.
+
+Fix $E/\mathbf Q$ of conductor $N$, and write $T=T_2E$ and
+$\Lambda=\mathbf Z_2[[t]]$.  The imaginary quadratic field
+$K=\mathbf Q(\sqrt k)$ satisfies the split Heegner hypothesis at $2N$.
+We omit the two fields with extra units and, when $E$ has complex
+multiplication, its CM field.  Let $h$ be a product of distinct signed
+good odd primes, all split in $K$, with $(h,k)=1$; $h=1$ is allowed.
+The primes dividing $h$ are called the \emph{active conductor primes}.
+The character $\chi=\chi_h$ is the corresponding quadratic character
+over $\mathbf Q$.
+
+Fix a modular parametrization $\pi:X_0(N)\to E$ with $\pi(\infty)=0$.
+Write $H_c$ for the ring-class field of $K$ of conductor $c$, and $y_c$
+for the image under $\pi$ of a conductor-$c$ CM point, with fixed CM
+orientations.  The restriction of $\chi_h$ to $G_K$ factors through
+$\operatorname{Gal}(H_{|h|}/K)$.  The primitive weighted Heegner sum is
+\[
+ P_h=\sum_{\sigma\in\operatorname{Gal}(H_{|h|}/K)}
+             \chi_h(\sigma)\,\sigma(y_{|h|}).
+\]
+Through the quadratic-twist identification, it is a point of
+$E^{(h)}(K)$.  When the product of $L(E^{(h)},s)$ and $L(E^{(hk)},s)$
+has a simple zero,
+the Heegner-point theorem makes $P_h$ nontorsion and
+$E^{(h)}(K)$ has rank one.  Let $j_h$ be the $2$-adic index valuation of $P_h$
+in the free rank-one lattice
+$E^{(h)}(K)^\wedge_2/\mathrm{torsion}$.  The quantities $w(h)$,
+$s(h)$, and $s(hk)$ have the meanings fixed in
+Section~\ref{sec:preliminaries}; the two Shafarevich--Tate lengths are
+finite in this simple-zero case.
+
+\begin{corollary}[Ring-class lower bound]
+\label{rd:ring-lower}
+Every simple-zero ring instance in the stated data satisfies
+\begin{equation}
+ 2j_h\ge 4w(h)+s(h)+s(hk)-C(E,A),
+       \qquad \omega(k)\le A.
+ \label{rd:ring-lower-equation}
+\end{equation}
+\end{corollary}
+
+For the second output, fix $K$ and the base $h_0=1$.  A binary family
+has the form
+\begin{equation}
+ h_x=\prod_{q\in Q}(q^*)^{\lambda_q(x)},\qquad
+ \lambda_q:\mathbb F_2^b\longrightarrow\mathbb F_2\text{ linear},
+ \quad x\in\mathbb F_2^b.
+ \label{rd:binary-data}
+\end{equation}
+All primes in $Q$ are good, odd, disjoint from the fixed support, and
+split in $K$.  All vertices have the same local squareclasses at $2N$
+and the same required real sign.
+
+\begin{theorem}[The fixed-\(K\) missing-vertex implication]
+\label{rd:missing-vertex}
+Fix the ring data \(E,K\), with
+\begin{equation}
+ \operatorname{corank}_{\mathbb Z_2}
+       \operatorname{Sel}_{2^\infty}(E/K)=1.
+ \label{rd:base-corank}
+\end{equation}
+Suppose binary families \eqref{rd:binary-data}
+exist for arbitrarily large \(b\), and for
+every nonzero \(x\) the product
+\(L(E^{(h_x)},s)L(E^{(h_xk)},s)\) has a simple zero and
+\begin{equation}
+ 2j_{h_x}-4w(h_x)-s(h_x)-s(h_xk)\le B
+ \label{rd:nonzero-bound}
+\end{equation}
+with \(B\) independent of the family.  Then
+\begin{equation}
+ \operatorname{ord}_{s=1}
+      \bigl(L(E,s)L(E^{(k)},s)\bigr)=1.
+ \label{rd:missing-conclusion}
+\end{equation}
+This implication does not assume Mordell--Weil
+rank one or finite Shafarevich--Tate group
+at the base.
+\end{theorem}
+
+To prove these assertions, we compare the Heegner-point indices with
+integral Selmer determinants.  The present section constructs the
+limiting complexes and the maps needed to test their classes at new
+primes.  Taking limits of differential matrices alone would not suffice:
+a cycle in the limit may lift only to approximate cycles at finite
+precision, and its localization must still be evaluation of a crossed
+cocycle.
+
+There are two passages to a limit.  In the inner construction $K$ and a
+finite active support are fixed while an auxiliary scalar-conductor
+prime $r_i$ and the coefficient precision vary.  Additional inert
+\emph{derivative primes} will realize evaluations and switch local
+conditions.  In the outer construction $K$ and the active support may
+vary, but $\omega(k)$ and the number of retained derivative primes are
+bounded.  This second passage gives the uniform lower bound; the final
+fixed-$K$ transfer instead uses the bounded clearing factor of
+Proposition~\ref{rd:family-clearing}.
+
+We use
+\[
+ A_i=(\mathbf Z/2^i\mathbf Z)[t]/(t^i).
+\]
+The same constructions work over $A_i[G]$ and $\Lambda[G]$, where $G$ is a
+fixed finite elementary abelian $2$-group and the additional character has
+values in $G$.  Statements uniform in the number of generators of $G$ will
+be identified separately; finite-diagram compactness by itself is a statement
+for fixed $G$.
+
+\subsection{Free models with maps to the original cochains}
+
+\begin{lemma}[Finite Artin models]\label{rl:finite-model}
+Let $(A,\mathfrak m,k)$ be an Artin local ring, and let $D$ be a complex of
+flat $A$-modules.  Suppose $H^*(D\otimes_A k)$ is finite dimensional and
+supported in $[a,b]$.  There is a bounded finite free complex $C$, with
+\[
+ \operatorname{rank}_A C^j=\dim_k H^j(D\otimes_A k),
+\]
+and a strong deformation retraction
+\[
+ i:C\longrightarrow D,\qquad p:D\longrightarrow C,\qquad
+ pi=1,\qquad 1-ip=dh+hd.
+\]
+Thus maps to the original complex, as well as maps between finite models,
+are available.  Any fixed finite collection of chain maps, homotopies,
+triangles, and perfect pairings can be transported to these models.
+\end{lemma}
+
+\begin{proof}
+Flat modules over an Artin local ring are free, including modules of infinite
+rank.  Here is a useful version of the argument.  Lift a basis of $M/\mathfrak
+mM$ to a flat module $M$, and map the corresponding free module to $M$.
+Its cokernel is zero because $\mathfrak m$ is nilpotent.  Flatness shows that
+the kernel has zero reduction, and nilpotence again makes the kernel zero.
+
+Split the residual complex as its cohomology, with zero differential, plus
+contractible two-term disks.  Lift all the graded bases.  These lifts are
+graded isomorphisms: an inverse modulo $\mathfrak m$ lifts to an inverse by
+a finite geometric series.  Let $d_0$ be the lifted disk differential and
+write $d=d_0+\epsilon$.  All coefficients of $\epsilon$ belong to
+$\mathfrak m$.  For the disk contraction $(i_0,p_0,h_0)$, arrange
+$h_0^2=h_0i_0=p_0h_0=0$ and
+$1-i_0p_0=d_0h_0+h_0d_0$.  The operators
+$1+h_0\epsilon$ and $1+\epsilon h_0$ have finite geometric inverses.
+The elementary perturbation identities give the differential
+\[
+ d_C=p_0\epsilon(1+h_0\epsilon)^{-1}i_0
+\]
+on the lifted residual cohomology, and give
+\[
+ i=(1+h_0\epsilon)^{-1}i_0,\quad
+ p=p_0(1+\epsilon h_0)^{-1},\quad
+ h=h_0(1+\epsilon h_0)^{-1}.
+\]
+Substitution, using $(d_0+\epsilon)^2=0$, proves the asserted contraction
+identities.  These calculations are valid for infinite disk sums: each
+operator is a module homomorphism and every geometric series terminates
+at the nilpotence exponent of $\mathfrak m$.
+
+For a map between original complexes use $p'fi$; insert the contraction
+homotopies when composing maps.  A quasi-isomorphism between bounded free
+complexes has a contractible cone and hence a homotopy inverse.  This gives
+the stated finite-diagram assertion, including the homotopies expressing
+triangles and perfect duality.
+\end{proof}
+
+For a profinite group and a finite free coefficient module over $A$,
+continuous inhomogeneous cochains are flat: locally constant functions are
+the filtered union of the modules of functions on finite clopen partitions,
+and these modules are finite free.  Their reduction is the cochain complex
+with reduced coefficients.  The same observations apply to restrictions,
+mapping cones, and local conditions given by finite free complexes.  In the
+applications below the groups are Galois groups of totally imaginary number
+fields or nonarchimedean local fields.  Their $2$-cohomological dimension and
+the usual finiteness theorems give the bounded residual cohomology required
+in Lemma~\ref{rl:finite-model}.
+
+\begin{lemma}[Limits of specified diagrams]\label{rl:finite-limits}
+Suppose diagrams over $A_i$, or over $A_i[G]$, have uniformly bounded cohomological amplitudes and graded free
+ranks, and include the chain homotopies and inverses for every asserted
+identity or equivalence.  After taking a cofinal subsequence, or a
+nonprincipal ultrafilter limit, their matrices define the same diagram of
+perfect complexes over $\Lambda$, or $\Lambda[G]$.
+
+Moreover, the contractions into the original cochains may be retained for
+the evaluation construction in Lemma~\ref{rl:evaluation}.  Adding finitely
+many new objects or maps does not require changing already chosen models
+for the old objects.
+\end{lemma}
+
+\begin{proof}
+There are finitely many possible graded ranks.  Fix them on an
+ultrafilter-large set.  At a fixed precision $j$, all matrix entries belong
+to the finite ring $A_j$ (or $A_j[G]$), so have unique ultrafilter limits.
+These limits are compatible as $j$ varies.  Matrix identities, including
+the homotopy-inverse identities, are finite polynomial identities and pass
+to the limit.  New models and maps may be mapped to old models by their
+already fixed contractions.  Passing to any slower cofinal precision has
+the same old limits.  For the all-sequence evaluation statements below,
+fix this ultrafilter once and for all, and keep the original contraction
+at every stage.  Later arithmetic diagrams must extend these same
+stages.  Restriction to an ultrafilter-large set and passage to slower
+cofinal coefficient precision preserve every old evaluation on every
+old group sequence.  An arbitrary later subsequence or a different
+ultrafilter is not allowed in this assertion.  Notice that this argument preserves specified
+arithmetic maps; it does not create an arithmetic map from a formal
+description of its expected properties.
+\end{proof}
+
+\paragraph{Target-side data.}
+The ring-class fields used to construct a class need not have bounded
+degree.  Their cochain complexes are not objects of the limiting diagram.
+At each finite stage, first perform descent, Shapiro transfer, the
+coefficient weighting, and every coset sum.  These operations give
+cochains on the fixed target field \(K\), with their local lifts.  Only
+then apply Lemma~\ref{rl:finite-model} to the target global and local
+complexes and their Selmer cones.
+
+Here is the required inner rank bound.  Fix \(K\), the active support,
+a finite set of derivative slots, and, when used, \(G\).  The permitted
+ramification set \(S_i\) consists of this fixed support and a bounded
+number of moving primes.  Modulo the maximal ideal of \(A_i\) or
+\(A_i[G]\), all scalar characters are trivial and the representation is
+\(E[2]\).  Put \(F=K(E[2])\), a fixed field.  Kummer theory bounds
+\(H^1(G_{F,S_i},\mathbf F_2)\) by the rank of the \(S_i\)-unit group
+and the dimension of the fixed group \(\operatorname{Cl}(F)[2]\).
+Both bounds are uniform: the number of primes above \(S_i\) is bounded
+and passing to the \(S_i\)-class group only decreases the required
+class-group bound.  The fixed finite group
+\(\operatorname{Gal}(F/K)\) and inflation--restriction then bound
+\(H^1(G_{K,S_i},E[2])\).  Degree zero is bounded by two, and the global
+Euler characteristic bounds degree two; higher degrees vanish because
+\(K\) is totally imaginary.  Local cohomology has uniform bounds as
+well: the local degrees over \(\mathbf Q\) are at most two, there are
+only finitely many fixed local conditions at \(2N\), and the moving
+good-prime and derivative complexes have fixed ranks.  The Selmer-cone
+exact sequence gives the same conclusion for every target local
+condition used below.  Thus Lemma~\ref{rl:finite-model} gives uniformly
+bounded target ranks.  This inner bound may depend on the fixed active
+support and is not a bound for a sequence of growing binary families.
+The separate uniform bound after completion at \((2)\) is
+Lemma~\ref{rl:residual-bound}; binary-family uniformity is obtained only
+after the elimination in Proposition~\ref{rd:family-clearing}.
+
+For clarity, let \(D_i\) be a target cochain complex and write its
+contraction as \((\iota_i,p_i,h_i)\).  A transferred cycle \(z_i\)
+is retained as the vector \(p_i z_i\).  Its representative in the
+original target cochains differs from \(z_i\) by the boundary
+\(d h_i z_i\).  If an arithmetic identity is
+\(z_i-z'_i=d b_i\), retain the target vectors
+\(p_i z_i,p_i z'_i,p_i b_i\); their identity is exact.  For a target
+comparison map \(f_i:D_i\to D'_i\), retain
+\(p'_i f_i\iota_i\), together with the contraction homotopy.  For
+example,
+\[
+ p'_i f_i z_i-p'_i f_i\iota_i p_i z_i
+       =d\bigl(p'_i f_i h_i z_i\bigr).
+\]
+Apply this also to local lifts in the Selmer cone, specialization
+comparisons, and the homotopies expressing duality and local switches.
+Each specified cycle or boundary witness is a vector in a bounded-rank
+target term, so adds only finitely many coordinates.  Boundary witnesses
+are compressed after transfer and have target rank; no list indexed by
+the ring-class cosets is retained.  The lemma therefore retains all
+these identities without retaining a source cochain complex or a
+growing transfer matrix.  The original target contractions remain
+available in degrees zero, one, and two for all-sequence evaluation.
+All operations use the same original stages and ultrafilter; a loss of
+precision is accommodated by one common slower cofinal precision for
+the entire finite diagram.  Adding derivative slots means extending
+each old stage by finitely many target objects, not taking a limit with
+an unbounded number of objects.
+
+\subsection{Kummer conditions, including their derived degree zero}
+
+\begin{lemma}[Integral Kummer local condition]\label{rl:kummer}
+Let $F$ be a finite extension of a nonarchimedean local field of
+characteristic zero, and let $B/F$ be any elliptic curve.  Put
+$M_F=B(F)^\wedge_2=\varprojlim_n B(F)/2^nB(F)$.  There is a perfect local
+condition
+\[
+ U_F=M_F[-1]\longrightarrow R\Gamma(F,T_2B).
+\]
+For every $n\geq1$, derived reduction has
+\[
+ H^0(U_F\otimes^{\mathbf L}\mathbf Z/2^n)=B(F)[2^n],\qquad
+ H^1(U_F\otimes^{\mathbf L}\mathbf Z/2^n)=B(F)/2^nB(F).
+\]
+Its map in degree zero is the isomorphism onto
+$H^0(F,B[2^n])$, and its map in degree one is the finite Kummer injection.
+The condition is its own exact orthogonal complement under local Tate
+duality.  These assertions hold at $F\mid2$, for all reduction types, and
+without a condition on $B[2]$.
+\end{lemma}
+
+\begin{proof}
+The local analytic description of $B(F)$ shows that $M_F$ is a finitely
+generated $\mathbf Z_2$-module.  Its torsion is the finite group
+$B(F)[2^\infty]$.  Consequently $M_F$ has a free resolution of length at
+most one.  Also $H^0(F,T_2B)=0$, since $B(F)[2^\infty]$ has bounded
+exponent.  The integral Kummer injection into $H^1(F,T_2B)$ therefore
+defines the displayed morphism using the truncation in degrees at most
+one of $R\Gamma(F,T_2B)$.
+
+Tensor the free resolution of $M_F$ with $\mathbf Z/2^n$.  Its two
+cohomology groups are $M_F[2^n]$ and $M_F/2^nM_F$.  The inverse limits of
+the ordinary Kummer exact sequences identify their maps with the two
+maps in the statement.  In particular finite torsion invariants have not
+been discarded.
+
+For orthogonality, compose cup product and the Weil pairing with the local
+invariant map.  A morphism
+\[
+ M_F[-1]\otimes^{\mathbf L}M_F[-1]
+       \longrightarrow\mathbf Z_2[-2]
+\]
+is determined by its homomorphism $M_F\otimes M_F\to\mathbf Z_2$:
+the possible $\operatorname{Tor}_1$ is in degree one and gives no extra
+degree-zero morphism to this target.  The indicated homomorphism vanishes,
+because the finite Kummer image annihilates itself at every level by local
+Tate duality \cite[I, Theorem~3.2 and Corollary~3.4]{MilneADT2006}.
+Choose a nullhomotopy in free models.  The induced morphism
+\[
+ R\Gamma(F,T_2B)/U_F
+       \longrightarrow R\operatorname{Hom}_{\mathbf Z_2}
+                         (U_F,\mathbf Z_2)[-2]
+\]
+is a quasi-isomorphism.  This can be checked after reduction modulo $2$:
+degree one is the perfect pairing of the finite Kummer quotient with the
+Kummer image, and degree two is dual to $B(F)[2]$.  Its perfect cone is
+therefore zero by derived Nakayama.  This proves exact orthogonality and
+also its compatibility with derived reduction.  The derived duality of
+the ambient local cochains is \cite[Proposition~5.2.4]{Nekovar2006};
+the Kummer condition and its orthogonality have been constructed above.
+No part of the proof uses an ordinary filtration or a classification of
+reduction types.
+\end{proof}
+
+Where the scalar deformation is trivial, extend this local condition by
+derived scalar extension to $A_i$ or $A_i[G]$.  For a finite free
+coefficient representation $M$ unramified at a good prime $v$, use the
+inflated residue-field complex
+\[
+ U_v=[\,M\xrightarrow{\operatorname{Fr}_v-1}M\,]
+ \quad\text{in degrees }0,1.
+\]
+Its quotient in full local cochains is the singular local complex.
+The full local condition is the identity on local cochains, and its
+orthogonal strict condition is zero.  The conditions at a derivative
+prime are different: once its local cochains have the split model
+constructed in Lemma~\ref{rd:local-model}, they are the truncations
+specified below.  Write that model as
+\[
+ L_v=D^0\oplus(F_v\oplus S_v)[-1]\oplus D^2[-2],
+\]
+where the differential is zero, $D^0$ is dual to $D^2$, and the finite
+and transverse planes $F_v,S_v$ are each isotropic.  Put
+\[
+ U_{\rm str}=D^0,\qquad
+ U_{\rm rel}=D^0\oplus(F_v\oplus S_v)[-1],\qquad
+ U_F=D^0\oplus F_v[-1],\qquad U_S=D^0\oplus S_v[-1].
+\]
+Then $U_{\rm str}^{\perp}=U_{\rm rel}$ and each plane condition is
+its own orthogonal complement.  Every displayed condition maps
+isomorphically onto local $H^0$ and injectively into local $H^1$.
+Thus they satisfy precisely the detection hypotheses used below.
+Lemma~\ref{rd:local-model} proves their finite arithmetic realization
+and supplies the isotropic-plane nullhomotopies.  These truncated
+conditions are not identified with the full local condition.
+
+For a finite permitted set $S$ the Selmer complex is, by definition,
+\begin{equation}\label{rl:selmer-definition}
+ C_S=\operatorname{Cone}\left(
+ R\Gamma(G_{K,S},M)\oplus\bigoplus_{v\in S}U_v
+ \longrightarrow\bigoplus_{v\in S}R\Gamma(K_v,M)
+ \right)[-1].
+\end{equation}
+At $2N$ take the Kummer conditions just constructed.  Restriction and
+localization triangles are the triangles of this cone.  The finite rings
+$A_i$ and $A_i[G]$ are Frobenius coefficient rings; alternatively the
+duality maps may be checked after reduction to their residue field.
+Exact local orthogonality and global duality therefore give
+\begin{equation}\label{rl:global-duality}
+ C_S\simeq R\operatorname{Hom}(C_S^{\perp},A_i)[-3],
+\end{equation}
+with the appropriate dual representation.  This application uses the
+orthogonality hypotheses of
+\cite[Theorem~6.3.4]{Nekovar2006}; the theorem is not being used to assert
+the existence of a varying-prime limit.  Complex places contribute zero
+Tate complexes.  If the character is inverted by the nontrivial
+automorphism of $K/\mathbf Q$, conjugate transport followed by the Weil
+pairing identifies the two representations over the same scalar ring.
+All maps and nullhomotopies in this paragraph are finite diagrams to
+which Lemma~\ref{rl:finite-limits} applies.
+
+\subsection{The moving conductor character}
+
+\begin{proposition}[Finite-precision ring-class characters]\label{rl:characters}
+Fix finitely many split primes of $K$ above odd rational primes, outside
+$2N$, and orient one prime in each conjugate pair.  There are distinct
+split rational primes $r_i$ avoiding all the fixed data and characters
+\[
+ \xi_i:\operatorname{Gal}(H_{r_i}/K)\longrightarrow
+                 \mathbf Z/2^{n_i}\mathbf Z,
+ \qquad n_i\longrightarrow\infty,
+\]
+such that $\xi_i$ is surjective on the inertia quotient at $r_i$, is zero
+at the primes above $2N$, and has odd value at each prescribed oriented
+prime.  We can require $r_i\to1$ in $\mathbf Z_2$ and
+\begin{equation}\label{rl:trace-bound}
+ v_2(2+a_{r_i}(E)),\quad v_2(2-a_{r_i}(E)),\quad
+ v_2\det(1\pm\operatorname{Fr}_{r_i}\mid T)\ \leq c_E.
+\end{equation}
+The constant $c_E$ is independent of the list and of $K$.
+\end{proposition}
+
+\begin{proof}
+We give the compatibility argument before imposing the Tate condition.
+Choose prime ideals $\mathfrak p_1,\ldots,\mathfrak p_s$ with pairwise
+disjoint conjugate pairs, including the oriented prescribed primes and
+the primes above $2N$, and enough additional split prime ideals to
+generate $\operatorname{Cl}(K)$.  Such additional ideals may avoid any
+fixed finite set.  Put weight $w_j=1$ at a prescribed odd prime and
+$w_j=0$ at every additional prime and every prime above $2N$.
+The conjugate weight is $-w_j$.  Let
+\[
+ R=\ker\bigl(\mathbf Z^s\longrightarrow\operatorname{Cl}(K)\bigr),
+\]
+choose a basis $m^{(a)}$ of $R$, and choose $\beta_a\in K^\times$ with
+$(\beta_a)=\prod_j\mathfrak p_j^{m_j^{(a)}}$.
+
+Work first over $F_\infty=K(\mu_{2^\infty})$ and adjoin $2^n$-th roots
+of the finitely many $\beta_a/\overline\beta_a$.  Prescribe on these
+roots the multipliers
+\begin{equation}\label{rl:kummer-prescription}
+ \zeta_{2^n}^{\,-\sum_j m_j^{(a)}w_j}.
+\end{equation}
+The minus sign fixes the choice of connecting map in the ring-class
+sequence and could be reversed throughout.  These prescriptions respect
+every Kummer relation.  Indeed, if
+\[
+ \prod_a(\beta_a/\overline\beta_a)^{b_a}
+                         \in(F_\infty^\times)^{2^n},
+\]
+take its valuation at a prime above each $\mathfrak p_j$ of nonzero
+weight.  These odd primes are unramified in $F_\infty/K$.  Thus
+$2^n$ divides $\sum_a b_am_j^{(a)}$ for each such $j$, proving the
+required compatibility of~\eqref{rl:kummer-prescription}.  Primes over
+$2$ have weight zero and impose no division by a ramification index.
+Units impose no additional weighted relation.  Kummer duality supplies
+the prescribed automorphism of the radical extension over $F_\infty$.
+
+At a rational prime $r$ split in a sufficiently large finite cyclotomic
+subfield, the power-residue character of $\mathbf F_r^\times$ is
+surjective onto $\mu_{2^n}$.  The split ring-class exact sequence is
+\[
+ 1\longrightarrow
+  (\mathcal O_K/r\mathcal O_K)^\times/(\mathbf Z/r\mathbf Z)^\times
+ \longrightarrow\operatorname{Pic}(\mathbf Z+r\mathcal O_K)
+ \longrightarrow\operatorname{Cl}(K)\longrightarrow1.
+\]
+There is no unit correction under our exclusion of the extra-unit
+fields.  The first group is $\mathbf F_r^\times$, with a principal
+relation measured by $\beta/\overline\beta$ at the oriented prime over
+$r$.  Consequently~\eqref{rl:kummer-prescription} says exactly that its
+primitive power-residue character extends to the class group with the
+specified values $w_j$.  Since the ring-class extension is unramified at
+$2N$, value zero on Frobenius means that the resulting local character
+is trivial there.  Conjugation acts by inversion.
+
+We now show that imposing these finitely many radical conditions cannot
+force either forbidden Tate trace.  In the non-CM case the image of
+$G_{K(\mu_{2^\infty})}$ is an open subgroup of
+$\operatorname{SL}_2(\mathbf Z_2)$, by the open-image theorem
+\cite{Serre1972}.  Its index in the corresponding fixed-$E$ image changes
+by at most two when $K$ changes.  Its commutator subgroup contains a
+fixed open subgroup $U_E$ of $\operatorname{SL}_2(\mathbf Z_2)$.
+One may see uniformity by starting with a principal congruence subgroup
+in the fixed image: every index-two subgroup contains its squares, and
+the commutators of the resulting fixed congruence subgroup are open.
+Because the radical extension over $F_\infty$ is abelian, these
+commutators act trivially on it.  Hence every permitted Tate coset
+contains a translate of $U_E$.
+
+For completeness, in the CM case let $M$ be the CM field.  The image over
+$KM(\mu_{2^\infty})$ contains a fixed open subgroup of the norm-one
+Cartan; the index changes by a bounded factor, as $[KM:\mathbf Q]\leq4$.
+Choose $\gamma\in G_{KM}$ with
+$\chi_{\rm cyc}(\gamma)=5^4=625$.  This is possible uniformly: the
+cyclotomic image of $G_{KM}$ has index at most four in
+$\mathbf Z_2^\times$, so contains fourth powers.  On the radical Galois
+group, conjugation by $\gamma$ is raising to $625$, because all the
+radicands lie in $K$.  On the Tate image over $KM$, conjugation is
+trivial.  Therefore
+\[
+ \gamma u\gamma^{-1}u^{-625},\qquad
+ u\in G_{KM(\mu_{2^\infty})},
+\]
+fixes every prescribed radical and acts on $T$ as $\rho_T(u)^{-624}$.
+Raising to the fixed nonzero power $624$ carries an open subgroup of the
+one-dimensional norm-one Cartan onto an open subgroup.  This proves the
+required uniform kernel statement also in the CM case.  The open Cartan
+image used here is the CM part of the usual $\ell$-adic image theorem
+\cite[Chapter~IV, \S2.2, Remark]{Serre1968}.
+
+On a coset of an open subgroup of $\operatorname{SL}_2$, trace is not
+identically $2$ or $-2$.  The same holds on a coset of an open norm-one
+Cartan subgroup; on the other Cartan-normalizer component trace is zero.
+The relevant determinant-one sets are compact and have finitely many
+cosets modulo the fixed open subgroup.  Choose in each coset an element
+whose trace is neither forbidden value.  The largest of the finitely
+many resulting valuations is a constant depending only on $E$.
+
+Finally restrict the prescribed radical automorphism and such a Tate
+lift to a finite Galois extension, including as many roots of unity as
+required.  Chebotarev gives infinitely many rational primes with this
+Frobenius class; choose the place over the prime to realize the chosen
+representative.  They split in $K$ and have arbitrarily high prescribed
+congruence to $1$.  Since
+$\det(1\pm\operatorname{Fr}_r)=1\pm a_r+r$, a congruence precision
+larger than the fixed trace bound gives all of~\eqref{rl:trace-bound}.
+This is a finite Chebotarev condition at each stage, not a demand that
+one prime split in an infinite extension.
+\end{proof}
+
+Choose $n_i$ large enough that $1+t$ has order dividing $2^{n_i}$ in
+$A_i^\times$, and put $\Psi_i=(1+t)^{\xi_i}$.  For example $n_i\geq2i$
+is sufficient.  After the matrix limits, the inertia exponent at the
+moving prime is a unit of $\mathbf Z_2$, and a prescribed oriented
+Frobenius exponent is odd.  We can orient the generator of inertia so
+its action is exactly $1+t$.
+
+\begin{lemma}[Local height-one tests]\label{rl:local-acyclic}
+For the individual coefficient system $T\chi_h\Psi$, use full local
+conditions at the moving prime $r$ and the active primes dividing $h$,
+without adding primes from a larger binary family.  The limiting full
+local complexes are acyclic over every height-one residue field of
+$\Lambda$ of characteristic zero.  At $(2)$, with residue field
+$\Omega=\mathbf F_2((t))$, the $r$-complex is acyclic, and so are the
+complexes at active split primes whose prescribed exponent is odd.
+These full conditions therefore agree with their orthogonal conditions
+on the stated tests.
+\end{lemma}
+
+\begin{proof}
+Use tame inertia and unramified Frobenius, or the Hochschild--Serre
+complex for these two procyclic directions.  At $r$, inertia minus one
+is multiplication by $t$ times a unit.  It is invertible on all the
+indicated fields except the characteristic-zero fiber $t=0$.  There,
+$r\to1$, and the two remaining Frobenius differentials have nonzero
+determinant by~\eqref{rl:trace-bound}, including either sign of the
+quadratic twist.  The bound also ensures that the integral local
+torsion at this central fiber has bounded length.
+
+At an active odd conductor prime, inertia of the quadratic character
+acts by $-1$.  Its differential is thus $-2$ and is invertible in
+characteristic zero.  In characteristic two the quadratic character
+disappears.  At a split prime the Frobenius action is $zB$, where
+$B\in\operatorname{GL}_2(\mathbf F_2)$ and
+$z=(1+t)^a$ has odd $a\in\mathbf Z_2$.  Its coefficient of $t$ is one,
+so $z$ is nonconstant.  No nonzero polynomial over the finite constant
+field vanishes at $z$.  Thus both Frobenius complexes, with operators
+$zB-1$ and $q^{-1}zB-1$, are invertible over $\Omega$; here $q=1$ in
+the residue field.  This argument applies also to a reducible residual
+representation.
+
+The orthogonal full and strict conditions agree when full local cochains
+are acyclic.  Equation~\eqref{rl:global-duality} then gives self-duality.
+The global degree-zero assertion will follow from the evaluation maps
+in Lemma~\ref{rl:evaluation}.
+\end{proof}
+
+\begin{lemma}[Uniform residual sizes]\label{rl:residual-bound}
+Suppose $\omega(\operatorname{disc}K)\leq A$, the local twisting classes
+at $2N$ belong to a fixed finite list, and all active conductor primes
+are split with odd exponent.  After localization and completion at
+$(2)$, the minimal free ranks of the Selmer complexes are bounded in
+terms of $E,A$, and the number of retained derivative primes.  They do
+not depend on the number of active conductor primes.
+\end{lemma}
+
+\begin{proof}
+Reduce first over $\mathbf F_2[[t]]$.  The quadratic character is now
+trivial.  Removing the active conductor primes from the permitted
+ramification set changes the global complex by their singular local
+complexes.  These, and their unramified local complexes, are acyclic
+after inverting $t$, by Lemma~\ref{rl:local-acyclic}.  The resulting
+smaller-support complex, including its comparison maps, can be
+constructed at the original Artin stages before taking the limit.
+Its support consists of $2N,r$, and the retained derivative primes.
+Its $\mathbf F_2((t))$ cohomology dimensions are bounded by those at
+$t=0$: choose a minimal finite free model over $\mathbf F_2[[t]]$.
+
+Here is a uniform bound for the latter residual cohomology.  Put
+$F=\mathbf Q(E[2])$, a fixed field.  Then $FK/F$ is at most quadratic.
+Enlarge the allowed set of primes of $F$ by those ramified in $K$.
+Its cardinality is bounded by a constant depending on $E,A$ and the
+number of derivative primes.  Kummer theory bounds the number of
+generators of the maximal pro-$2$ Galois group over $F$ unramified
+outside this set, using the fixed unit group, the fixed $2$-class group,
+and the allowed valuations.  The subgroup defining $FK$ has index at
+most two; the elementary Schreier bound bounds its generator number.
+Every finite Galois $2$-extension of $FK$ with this ramification has
+Galois closure of $2$-power degree over $F$, so this subgroup indeed
+controls the required pro-$2$ extensions of $FK$.
+
+Over $FK$ the residual Tate module is trivial.  Restriction and
+inflation--restriction therefore bound the residual $H^1$ over $K$.
+The $H^0$ bound is immediate, and the global Euler characteristic or
+global duality bounds $H^2$.  The local terms at the remaining bounded
+set have bounded dimensions, with degrees of the local fields at most
+two over the fixed rational local fields.  At $2N$ there are only
+finitely many local quadratic twisting classes.  The same bounds
+apply to the Selmer cone.  Lemma~\ref{rl:finite-model} now bounds the
+minimal ranks.  No bound on a torsion exponent is inferred from these
+rank bounds.
+\end{proof}
+
+\subsection{Evaluation on every sequence, rather than only a chosen list}
+
+\begin{lemma}[All-sequence evaluation]\label{rl:evaluation}
+Fix $K$ and a sequence of the finite cochain constructions above.  Keep
+the contraction maps into the original global cochains.  Let
+$\Gamma_K=\prod_i G_K$ and
+$\Gamma=\Gamma_K\rtimes\langle\tau\rangle\subset\prod_iG_{\mathbf Q}$,
+with simultaneous complex conjugation $\tau$.  Thus $\Gamma_K$ has
+index two in $\Gamma$; the larger unrestricted product of rational
+Galois groups is not used for induction.  The matrix limit defines a
+coefficient action of $\Gamma_K$ on $\Lambda^2$.  For every field $L$
+receiving a map from $\Lambda$, there is a natural injection
+\[
+ H^1(C\otimes_\Lambda L)\longrightarrow
+ H^1_{\rm abs}(\Gamma_K,L^2),
+\]
+where the right side denotes abstract crossed-homomorphism classes.
+Degree-zero evaluation likewise injects
+$H^0(C\otimes_\Lambda L)$ into $(L^2)^{\Gamma_K}$.
+Here $C$ may be the global complex or a Selmer complex whose local
+condition maps induce isomorphisms on local $H^0$ and injections on
+local $H^1$.  The same statement holds for the induced representation
+of $\Gamma$.
+
+There are evaluation maps on arbitrary sequences, not just on the
+finite list used to detect cohomology.  Their degree-two identities
+remain valid after arbitrary coefficient extension.
+\end{lemma}
+
+\begin{proof}
+Let $C_i$ be a finite model and let $\iota_i$ be its chosen map to the
+original global cochains, using the projection from the Selmer cone
+when appropriate.  For every $g=(g_i)$ and $h=(h_i)$ define matrices
+\[
+ e^0_i:C_i^0\to A_i^2,\quad
+ e^1_{g,i}:C_i^1\to A_i^2,\quad
+ e^2_{g,h,i}:C_i^2\to A_i^2
+\]
+by evaluating $\iota_i$ in degrees zero, one, and two.  Their source
+ranks are fixed and their entries belong to $A_i$.  Thus each has a
+coefficientwise ultrafilter limit, for every choice of $g,h$.
+No diagonal selection over the set of group elements is necessary:
+one fixed ultrafilter defines all these limits.  Likewise
+$\rho(g)=\lim_i\rho_i(g_i)$ is a representation.  The original
+cochain identities imply
+\begin{align}
+ e^1_g d&=(\rho(g)-1)e^0,\label{rl:eval-boundary}\\
+ e^2_{g,h}d&=\rho(g)e^1_h-e^1_{gh}+e^1_g.
+                         \label{rl:eval-cocycle}
+\end{align}
+All identities pass to the limit because their matrix expressions
+have finitely many terms.  A cycle $x\in C^1\otimes L$ consequently
+gives the crossed homomorphism $g\mapsto e^1_g(x)$, and a boundary
+gives a coboundary.
+
+We prove injectivity in a way that also controls base change.  At each
+finite stage let $k=\mathbf F_2$ be the residual field.  Choose a
+space of representatives for residual $H^1$.  Evaluate pairs $(c,m)$,
+where $c$ is in this space and $m\in k^2$, by
+$b_c(g)-(\rho(g)-1)m$.  The intersection of the kernels of all
+these evaluations consists precisely of pairs $(0,m)$ with $m$ an
+invariant.  Since the vector space is finite dimensional, a list of
+at most $\dim H^1+2$ elements already has this intersection.  For
+these elements evaluation is a chain map
+\[
+ C_i\longrightarrow D_i=
+ [\,A_i^2\xrightarrow{((\rho_i(g_j)-1))_j}(A_i^2)^d\,]
+\]
+in degrees zero and one, inducing an isomorphism on residual $H^0$
+and an injection on residual $H^1$.  For Selmer complexes this uses
+the exact sequence of~\eqref{rl:selmer-definition}: the indicated
+local degree-zero isomorphisms give $H^0(C_i)=H^0_{\rm glob}$ and
+$H^1(C_i)\hookrightarrow H^1_{\rm glob}$ residually.
+
+Retain these matrices and their chosen group-element sequences.  The
+cone of the limiting map $C\to D$ has zero residual cohomology in
+degrees at most zero.  Successively cancel its unit differential
+blocks.  The resulting bounded free cone starts in degree one.
+After every coefficient extension it still starts in degree one;
+the long exact sequence therefore makes $H^1(C\otimes L)\to
+H^1(D\otimes L)$ injective.  A crossed cocycle that is a coboundary
+on all of $\Gamma_K$ is in particular a coboundary on the detecting
+list, proving the desired injection.  The same cone gives an
+isomorphism on $H^0$ after coefficient extension.  Since $H^0(D\otimes L)$
+is a subspace of $L^2$, degree-zero evaluation is injective; by
+\eqref{rl:eval-boundary}, its image is fixed by every element of
+$\Gamma_K$, not merely by the detecting list.  For induction the same proof
+has coefficient rank four and the detecting bound is $\dim H^1+4$.
+\end{proof}
+
+We can now exclude global invariants on the tests of
+Lemma~\ref{rl:local-acyclic}.  In characteristic two the nontrivial
+$r$-inertia scalar excludes coefficient invariants.  In characteristic
+zero the same argument works unless that scalar is trivial.  In the
+latter case $t=0$, and $T\otimes\mathbf Q_2$ over $G_K$ is absolutely
+irreducible: for non-CM curves this follows from the open image, and
+for CM curves from the open Cartan and its nontrivial normalizer
+component, since $K$ is not the CM field.  Twisting by $\chi_h$ does
+not alter its invariant subspaces.  The degree-zero injection just
+proved therefore gives $H^0=0$ for the limiting Selmer complex.
+Together with \eqref{rl:global-duality}, this leaves degrees one and
+two paired and kills degree three as well.
+
+\begin{lemma}[The exact common kernel]\label{rl:kernel}
+In the notation of Lemma~\ref{rl:evaluation}, let $H$ be the group of
+sequences each of whose entries acts trivially on the full Tate module,
+on all ring-class fields of $K$, and on the quadratic characters in
+use.  Restriction from the crossed classes in that Lemma to $H$ is
+injective at every characteristic-zero test.  It is also injective
+at the residual $(2)$ test.  The characteristic-zero annihilator
+needed for this assertion can be chosen with a fixed finite
+$2$-valuation depending only on $E$, also as $K$ varies.
+\end{lemma}
+
+\begin{proof}
+The Tate image contains an open subgroup of scalar homotheties, both
+in the CM and non-CM cases
+\cite[\S2, Theorem~3 and paragraph~7]{Bogomolov1981Points}.  Choose once and for all
+$g\in G_{\mathbf Q}$ with Tate action $u\,1$, where
+$u\in1+4\mathbf Z_2$ and $u\ne1$.  Its square belongs to every
+quadratic subgroup.  Put
+\[
+ z=g^2\tau g^2\tau^{-1}.
+\]
+The generalized dihedral property of every ring-class extension
+implies that $z$ is ring-class trivial.  It is also trivial on every
+quadratic character, and its Tate action is the fixed scalar $u^4$.
+For every sequence in $\Gamma_K$, its commutator with the constant
+sequence $z$ is in the exact group $H$, since its Tate commutator and
+all its abelian character images are trivial.  Thus $z$ is central in
+$\Gamma_K/H$ and acts on the coefficients by $u^4$.
+
+Here the elementary central-scalar argument suffices.  If a crossed
+cocycle $b$ is defined on a group with such a central element, comparison
+of $b(zg)$ and $b(gz)$ gives
+\[
+ (u^4-1)b(g)=(g-1)b(z).
+\]
+In characteristic zero this makes $b$ a coboundary.  Inflation--restriction
+then gives injectivity on $H$.  The valuation of $u^4-1$ is fixed.
+
+In characteristic two use instead, at the $i$-th stage, an inertia
+element at $r_i$ mapping to the chosen generator of $\xi_i$.  Its
+Tate action is trivial and its ring-class image commutes with all
+ring-class images.  The associated sequence is therefore central
+modulo $H$, and its coefficient action is $1+t$.  Since $t$ is
+invertible in $\mathbf F_2((t))$, the same calculation applies.
+It is essential here that $H$ is defined by the exact actions at each
+stage, not merely by an action converging to the identity.
+\end{proof}
+
+\subsection{The outer characteristic-zero test}
+
+\begin{proposition}[Integral evaluation through the outer limit]
+\label{rl:outer-evaluation}
+Let $C_j$ be any family of the preceding limiting complexes, completed
+over $\mathcal O_j=\widehat{\Lambda_{(2)}}$, possibly for different
+quadratic fields $K_j$.  Suppose their minimal ranks are bounded.
+Keep their actual evaluation matrices, and transport them through the
+integral contractions to these minimal models.  For a nonprincipal
+ultrafilter $\mathcal U$, put
+\[
+ \mathcal O_\infty=
+ \left(\prod_{\mathcal U}\mathcal O_j\right)/\mathfrak a,
+ \qquad
+ \mathfrak a=\{(x_j):
+    \{j:v_2(x_j)\geq n\}\in\mathcal U\text{ for every }n\}.
+\]
+Then $\mathcal O_\infty$ is a DVR with uniformizer $2$ and a
+characteristic-zero fraction field $L_\infty$.  The outer complex has
+an injection of its $H^1$ over $L_\infty$ into crossed classes on the
+product of the inner sequence groups.  Its restriction to the product
+of their exact common kernels is injective.  All arbitrary-sequence
+degree-two evaluation identities survive.  These conclusions also
+hold after retaining any fixed finite collection of additional
+integral diagrams.
+\end{proposition}
+
+\begin{proof}
+A nonzero element of the quotient has a representative whose
+valuations are bounded on an ultrafilter-large set.  A finite
+partition then makes its valuation equal to a single integer $n$
+on such a set.  It is $2^n$ times a unit.  Products add these finite
+valuations, so the quotient is a domain, and every nonzero ideal has
+an element of smallest valuation.  It is consequently a DVR.  The
+integer $2^n$ is nonzero for every $n$, so its fraction field has
+characteristic zero.
+
+For each $j$, all-sequence evaluation maps originally have entries
+in $\Lambda$.  Extension to $\mathcal O_j$ and composition with an
+integral contraction still give integral matrices.  Thus evaluation
+on any sequence of inner group elements has an entrywise image in
+$\mathcal O_\infty$.  Equations~\eqref{rl:eval-boundary} and
+\eqref{rl:eval-cocycle} survive for all such sequences.  In particular,
+if an outer cycle is represented by vectors $x_j$ for which
+$d_jx_j$ has valuation tending to infinity, the evaluation of
+$d_jx_j$ does too: the degree-two evaluation matrix is integral.
+This proves the crossed identity for an outer cycle without replacing
+the $x_j$ by genuine inner cycles.
+
+Over the inner residue field $\Omega_j$, choose a detecting list as
+in Lemma~\ref{rl:evaluation}; its length is at most
+$\dim_{\Omega_j}H^1(C_j\otimes\Omega_j)+\operatorname{rank}M$.
+These lengths are bounded.  Include in the integral diagrams the
+evaluation map to its two-term complex and a contraction of its cone
+to a complex starting in degree one.  Such a contraction uses only
+unit pivots over $\mathcal O_j$ and has bounded graded sizes.
+All its matrices, including inverses and homotopies, are integral
+and pass to the quotient.  Its cone still starts in degree one
+after extension to $L_\infty$.  This proves injectivity just as in
+Lemma~\ref{rl:evaluation}.
+
+Let $H_j$ be the inner exact kernel.  The product of the central
+elements of Lemma~\ref{rl:kernel} is central modulo $\prod_jH_j$.
+Its scalar is the same fixed $u^4$ in every component, and
+$u^4-1$ remains nonzero in $\mathcal O_\infty$.  The explicit
+central-scalar calculation and inflation--restriction give the
+claimed restriction injection.  Additional fixed finite diagrams
+are treated by the same entrywise operations.
+\end{proof}
+
+\begin{lemma}[Irreducibility on the characteristic-zero tests]
+\label{rl:irreducibility}
+On an inner or outer characteristic-zero test where $\Psi^2\ne1$,
+the four-dimensional representation induced from $T\chi\Psi$ is
+absolutely irreducible.  If $\Psi^2=1$ on an inner test, it splits
+as two inequivalent absolutely irreducible quadratic twists.  In
+the outer test $\Psi$ has infinite order.
+\end{lemma}
+
+\begin{proof}
+The two induced summands are distinguished by $r$-inertia, whose
+scalars are $\Psi$ and $\Psi^{-1}$.  On either summand, multiplying
+the Tate matrices by scalars does not alter invariant subspaces.
+The Tate representation restricted to $G_K$ is absolutely
+irreducible under the exclusions already specified.  These facts
+give absolute irreducibility of the induction when the two inertia
+characters differ.
+
+For the outer argument this assertion requires a uniform finite
+witness.  The images of the quadratic subgroups have index at most
+two in the fixed compact $2$-adic Tate image.  A compact $2$-adic
+analytic group is topologically finitely generated and hence has
+only finitely many closed subgroups of index at most two.  Choose
+one on an ultrafilter-large set.  Its absolutely irreducible
+representation has finitely many matrices spanning
+$M_2(\mathbf Q_2)$; the determinant expressing this spanning
+property is a fixed nonzero $2$-adic number.  Every one of these
+Tate matrices is realized by choices of Galois elements in each
+component.  Their accompanying character scalars are units and
+do not change the spanning assertion.  Its determinant remains
+nonzero in $L_\infty$.  This proves absolute irreducibility in
+that field.  Also $1+t$ has infinite order already in every
+$\Omega_j$: for a nonzero integer $n$, the first nonzero term of
+$(1+t)^n-1$ occurs in finite degree.  It remains a unit modulo
+$2$ in every component, so no such power becomes one in the
+outer field.
+
+If $\Psi^2=1$, extension across conjugation gives the two quadratic
+twists.  They are inequivalent, since a quadratic self-twist of the
+Tate representation here would make its restriction to $G_K$
+reducible over an algebraic closure.  In the CM case this would
+force $K$ to be the CM field, which was excluded.
+\end{proof}
+
+\begin{lemma}[Simultaneous finite Chebotarev realization]
+\label{rl:chebotarev-evaluation}
+For any element $u$ of an exact kernel in Lemma~\ref{rl:kernel},
+derivative primes can be chosen, at increasing finite precision,
+whose rational Frobenius is $\tau u$ on all retained data.  Their
+finite localization is evaluation on $(\tau u)^2$.  The assertion
+is compatible with the outer construction of
+Proposition~\ref{rl:outer-evaluation} and with any fixed finite
+list of previous derivative operations.
+\end{lemma}
+
+\begin{proof}
+At one Artin stage, a coordinate of a cochain in the image of a
+finite model is a locally constant function.  The finitely many
+coordinates being used factor through one finite quotient of the
+absolute Galois group.  Enlarge it to contain the Tate action at
+the required precision, the finitely many ring-class fields and
+quadratic characters currently in use, and all old finite data.
+Chebotarev realizes the conjugacy class of $\tau u$ on this finite
+quotient, with a chosen place realizing the desired representative.
+It gives a rational prime inert in $K$, outside the old finite set.
+Because $u$ is exactly Tate-trivial, its Frobenius tends to
+$\rho_T(\tau)$; in particular $a_\ell\to0$ and $\ell\to-1$.
+Because $u$ is exactly ring-class trivial, the square Frobenius on
+the $K$-prime is the identity on every retained ring-class field.
+The unramified local coordinate is therefore the stated
+Frobenius-square evaluation.
+
+Repeat at each finite stage with increasing precision.  All
+conditions remain finite, and no density bound or prime-size
+bound is required.  For an outer sequence do this separately
+for each inner sequence, retaining the relevant finite evaluation
+matrices and degree-two identities before taking the outer
+quotient.  Approximate inner cycles cause no problem, by the
+integrality calculation in Proposition~\ref{rl:outer-evaluation}.
+Previously imposed finite diagrams are unchanged.
+\end{proof}
+
+In induced-module notation, restriction to the exact kernel is additive.
+For a crossed cocycle $b$ and an exact-kernel element $u$,
+\[
+ b((\tau u)^2)=(1+\tau)b(u).
+\]
+Indeed $\tau^2=1$, $u$ acts trivially on the coefficients, and the
+cocycle identity cancels the two occurrences of $b(\tau)$.  Projection
+to either induced summand identifies $\operatorname{im}(1+\tau)$
+with a two-dimensional coefficient module, also in characteristic two.
+
+\begin{theorem}[The limit and evaluation interface]
+\label{rl:limit-evaluation-package}
+For the ring data fixed at the start of this section, retain the finite
+target-side arithmetic diagrams with their maps to original cochains, homotopies,
+and dualities as in Lemma~\ref{rl:finite-limits}.  The moving characters,
+local conditions, and evaluation constructions then have the following
+simultaneous properties:
+\begin{enumerate}
+\item Every specified finite target-side arithmetic diagram admits a bounded free
+model together with maps to its original cochains; the specified
+maps, homotopies and dualities survive the inner limit.
+\item At the stated height-one tests the relaxed active local
+complexes are acyclic, and the Selmer duality pairs degrees one
+and two.  Integral Kummer conditions at $2N$ include their full
+derived degree-zero invariants.
+\item The residual sizes needed for the outer construction are
+bounded independently of the active conductor support, with
+dependence only on $E$, the bound on $\omega(\operatorname{disc}K)$,
+and the number of retained derivative primes.
+\item Inner and outer $H^1$ are detected by actual crossed
+cocycles, including on the exact common kernel.  The evaluations
+and their degree-two identities can be realized by finite
+Chebotarev conditions without disturbing earlier finite data.
+\end{enumerate}
+These assertions do not assert an Euler-system reciprocity law or
+an integral derivative class that has not first been constructed
+at finite level.  Once those finite arithmetic diagrams are
+constructed, this Theorem supplies their limit compatibility.
+\end{theorem}
+
+\begin{proof}
+Combine Lemmas~\ref{rl:finite-model} and~\ref{rl:finite-limits},
+Lemma~\ref{rl:kummer}, Proposition~\ref{rl:characters},
+Lemmas~\ref{rl:local-acyclic} and~\ref{rl:residual-bound},
+Lemmas~\ref{rl:evaluation} and~\ref{rl:kernel},
+Proposition~\ref{rl:outer-evaluation}, and
+Lemma~\ref{rl:chebotarev-evaluation}.  Every passage uses actual
+integral matrices of bounded size; arbitrary evaluations are
+retained by their original contraction maps, not by an inference
+from a finite list of identities.
+\end{proof}
diff --git a/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/paper.pdf b/preprints/A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-September-24-2026/paper.pdf
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+size 893281
diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/boundary.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/boundary.tex
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@@ -0,0 +1,168 @@
+\section{Preserving the global boundary invariants}\label{sec:boundary}
+
+We verify Invariants~\ref{inv:past} and~\ref{inv:swap}
+for a child \(W'=(a',b')\) of \(W=(a,b)\).
+All notation is in the working frame of the child edge.
+In a same update this is also the parent's tag frame.
+In a switch update the old list is not carried; the newly constructed
+upsets are used instead.
+
+Preparatory changes to the schedule of \(W\) preserve the parent
+invariants. Its boundary triples and all earlier exterior jobs
+stay fixed, so Invariant~\ref{inv:past} is unchanged.
+Invariant~\ref{inv:swap} already quantifies over every feasible
+reordering of \(W\).
+
+\subsection{Excluding distant qualifiers}
+
+\begin{lemma}\label{lem:past}
+The updated child list satisfies Invariant~\ref{inv:past}.
+\end{lemma}
+\begin{proof}
+It suffices to prove the assertion before pruning, since pruning only
+removes qualifiers. Let \(z\) qualify for the pre-pruning child list and be scheduled
+at or before \(a'\).
+
+If \(a<\operatorname{time}(z)\le a'\), the job belongs to the normalized
+open side \(E\) of the parent. By \Cref{lem:update-labels}, it is
+high at the edge cutoff \(S\). The local separator at \(a'\) puts
+it in \(P_{a'}\) when strictly earlier, and membership in the
+triple puts it in \(\wP{a'}\) when its time equals \(a'\).
+
+Now suppose \(z\) is at or before \(a\).
+The new injection \(D_a\setminus P_b\) cannot contain such a job.
+Nor can either switch entry, by \Cref{lem:G} and the strictness
+of \(D_a\). Thus the only remaining case is a same-direction
+carry qualifier.
+The parent's Invariant~\ref{inv:past} supplies a job \(y\in a\)
+with \(z=y\) or \(z\prec y\).
+The carry is a global upset, so \(y\) also qualifies by carry.
+In particular \(a\) is actual. If \(a'=a\), we are done.
+
+Assume \(a'>a\), and write \(a'=s_i\), with edge cutoff \(S=S_i\).
+Follow all the strict advances of the walk from \(a\) through
+promotion \(i\), including intermediate advances not retained as marks.
+Every cutoff used in this portion of the walk is contained in \(S_i\).
+On \(W\), the final carry is exactly \(O\setminus S_i\).
+Consequently every carry qualifier in \(E\) is high, and has worse
+priority than the low exceptions, at every one of these advances.
+
+The first advance needs the parent exchange invariant, because
+\(a\) lies outside the normalized open side \(E\).
+Consider that first actual advance off \(a\), with earliest exception
+\(x\) at its target \(t\).
+All \emph{global} predecessors of \(x\) are before \(a\).
+To verify this stronger readiness assertion, every predecessor is
+temporally before \(t\). If one were at \(a\), then \(x\in D_a\).
+If one were strictly between \(a\) and \(t\), it would belong to
+the contiguous side \(E\), be low, and descend from \(a\) by
+earliestness of the exception. Again \(x\in D_a\), a contradiction.
+There is no other possible predecessor at or after \(a\).
+
+Also \(x\in X=W\setminus D_a\), so Invariant~\ref{inv:cones}
+puts it among the old qualifiers, outside the starting low set.
+It has already been promoted by stage \(i\), so its old key is
+strictly below the least unpromoted key at \(S_i\).
+By \Cref{lem:carry}, the boundary job \(y\) has old key at least
+that least unpromoted key.
+The parent's Invariant~\ref{inv:swap}, applied after the current
+preparatory reorderings of \(W\), therefore prohibits exchanging
+\(x\) and \(y\) in the complete schedule.
+
+If \(y\) had no global successor at or before \(t\), that exchange
+would be feasible. All global predecessors of \(x\) lie strictly before \(a\);
+moving it earlier preserves its successor constraints, and delaying
+\(y\) to \(t\) preserves its predecessor constraints and, under
+the supposition, its successor constraints. Thus \(y\) must have
+a strict successor by \(t\).
+Such a successor is in \(E\) and is still a carry qualifier.
+
+Any carry job strictly between \(a\) and \(t\) must likewise have
+a successor by \(t\): otherwise exchange it with \(x\) within
+\(E\), contradicting its lexicographic normalization.
+The carry is an upset, so these successors remain in the carry.
+Iterating along strictly increasing times reaches a carry job
+at \(t\). It cannot be \(x\), which is low and lies outside
+the final carry.
+
+At every subsequent advance, the local readiness and exchange argument
+from \Cref{lem:shared} applies inside \(E\). All carry jobs remain
+high at that advance, so each carry job at its starting slot
+has a carry descendant at its target. Tracing from the descendant
+of \(y\) at the first target eventually gives a descendant of
+\(y\) at \(a'\).
+Hence \(z\in\wP{a'}\), as required.
+\end{proof}
+
+\subsection{Exchanges after future child reorderings}
+
+\begin{lemma}\label{lem:swap}
+The pruned child list satisfies Invariant~\ref{inv:swap},
+with its quantifier over all feasible reorderings of \(W'\).
+\end{lemma}
+\begin{proof}
+Take any feasible reordering of \(W'\) in its slots.
+Suppose \(x\in W'\setminus D_{a'}\), that \(y\in a'\)
+qualifies for the child list, and that the new key of \(x\)
+is smaller than the new key of \(y\).
+Invariant~\ref{inv:cones}, already proved for the child in
+\Cref{lem:update-labels}, ensures that \(x\) qualifies.
+By \Cref{lem:prune}, the same comparison holds in the new list
+before pruning.
+
+First let \(a'>a\). Both \(x\) and \(y\) are in the parent's
+normalized side \(E\), and both are high at its edge cutoff \(S\),
+by \Cref{lem:update-labels}. The pre-pruning key order agrees with
+the priority order used to normalize \(E\). Thus \(x\) has
+better priority than \(y\).
+The child reordering changes only slots strictly after \(a'\).
+A feasible exchange of \(x,y\) in the complete schedule would
+therefore give a feasible full schedule of the same \(E\)
+with a better prefix at \(a'\), contrary to
+\Cref{lem:lex-prefix}.
+
+Now let \(a'=a\).
+In the switch case there is no new qualifier at \(a\), so no
+such \(y\) exists.
+In the same case neither \(x\) nor \(y\) belongs to the new
+injection: \(x\notin D_a\), and no job at \(a\) is a strict
+descendant of \(a\).
+Their pre-pruning comparison therefore uses only carry entries.
+By \Cref{lem:carry}, it implies the same old key comparison
+in the parent's list.
+Moreover \(x\in W\setminus D_a\).
+The preparatory reorderings and the arbitrary subsequent reordering
+of \(W'\) together form a feasible reordering confined to \(W\).
+The parent's Invariant~\ref{inv:swap} rules out the proposed
+exchange.
+\end{proof}
+
+\begin{proposition}\label{prop:construction-valid}
+Starting from any feasible full schedule, the marking and list
+construction is defined at every node. Its pruned lists are global
+upset lists of length at most five, with fewer than \(40\) leaves
+per entry, and satisfy Invariants~\ref{inv:cones}--\ref{inv:swap}.
+Every branch terminates.
+\end{proposition}
+\begin{proof}
+Proceed by depth from the root data of \Cref{sec:construction}.
+Given valid ancestor and parent data, the center and the two
+normalized sides exist by \Cref{cor:any-ideal,lem:reorder}.
+The shared walk and \Cref{lem:edge-labels} define the children.
+For a same update the new list is direct. For a switch,
+\Cref{lem:old-low,lem:G} construct the needed global upset
+using only earlier data, as explained at the end of
+\Cref{sec:lists}.
+Then \Cref{lem:update-labels,lem:list-length,lem:entry-size}
+give the first invariant and the bounds.
+\Cref{lem:past,lem:swap} give the other two invariants without
+using a claim about any deeper child.
+This completes the simultaneous induction.
+Every child omits the center, so its size decreases strictly.
+\end{proof}
+
+Invariant~\ref{inv:past} will let a bounded predicate distinguish
+the child from qualifiers arbitrarily far in the past.
+Invariant~\ref{inv:swap} is what allows that first invariant to
+continue propagating after recursive reorderings. Neither is
+replaced by an assertion only about local separators.
diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/conclusion.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/conclusion.tex
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@@ -0,0 +1,13 @@
+\section{Conclusion}\label{sec:conclusion}
+
+The finite search succeeds because every interval in a suitable recursive
+schedule decomposition has a global description of bounded size,
+independent of its depth. Local separator descriptions alone do not give
+that guarantee: the short upset lists, the exterior boundary invariants,
+and the simplification at direction changes prevent the accumulation of
+ancestor information.
+
+The resulting algorithm constructs an exact optimum for the stated
+three-machine model. The fixed leaf allowance $10000$ and the explicit
+large exponent establish polynomial time for the stated model.
+The proof supplies no practical performance guarantee.
diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/construction.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/construction.tex
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+++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/construction.tex
@@ -0,0 +1,239 @@
+\section{The interval construction and its invariants}\label{sec:construction}
+
+We now construct the hierarchy from a feasible full schedule of \(J\).
+Each node has a \emph{tag}, one of the two frames, and a witnessing
+complete schedule. Descending to a child changes only the jobs inside
+that child; its boundary triples and exterior schedule remain fixed.
+Branches can retain separate witnesses, as allowed in
+\Cref{def:hierarchy}.
+
+\subsection{Global lists and boundary conditions}
+
+In the tag frame of a node \(W=(a,b)\), attach an ordered list
+\(\K=(K_1,\ldots,K_\ell)\) of global upsets.
+Define the global qualifier set and the local old-high set by
+\[
+ \Qual(\K)=\bigcup_{i=1}^{\ell}K_i,
+ \qquad O=W\cap\Qual(\K).
+\]
+For every globally qualifying job \(x\), set
+\begin{equation}
+ \key_{\K}(x)=
+ \left(\max\{i:x\in K_i\},\,\rk(x)\right),
+ \label{eq:key}
+\end{equation}
+with lexicographic comparison. Entries may overlap. Their largest index,
+not their smallest one, is used.
+Because each entry is an upset, \(x\prec y\) with \(x\) qualifying
+implies that \(y\) qualifies and
+\(\key_{\K}(x)<\key_{\K}(y)\).
+
+We maintain the following three invariants in the node's tag frame.
+\begin{enumerate}[label=\textup{(I\arabic*)},ref=I\arabic*]
+ \item\label{inv:cones}
+ \(O\subseteq P_b\), and \(W\setminus O\subseteq D_a\).
+ \item\label{inv:past}
+ Every global qualifier scheduled at or before \(a\) belongs to
+ \(\wP a\).
+ \item\label{inv:swap}
+ After \emph{any} feasible reordering of \(W\) in its existing slots,
+ there is no feasible direct exchange between
+ \(x\in W\setminus D_a\) and a qualifying job \(y\) at \(a\) when
+ \(\key_{\K}(x)<\key_{\K}(y)\).
+\end{enumerate}
+Invariant~\ref{inv:cones} controls the jobs inside the interval;
+Invariant~\ref{inv:past} controls qualifiers outside its left boundary,
+and Invariant~\ref{inv:swap} makes that control persist after later
+reorderings.
+
+The exchange in Invariant~\ref{inv:swap} is tested for feasibility in
+the complete schedule. The hypothetical exchanged schedule need not
+preserve any decomposition or any of the list invariants.
+The key of \(x\) is defined because Invariant~\ref{inv:cones} implies
+\(W\setminus D_a\subseteq O\). A sentinel has no boundary jobs, so
+the corresponding exchange condition is vacuous.
+
+Lexicographic minimality within an interval controls exchanges of its
+interior jobs; it gives no such control over exchanges with an exterior
+boundary job. Invariant~\ref{inv:swap} supplies the latter control.
+Suppose a qualifying job \(x\in W\setminus D_a\) has all its
+predecessors strictly before \(a\), and its key is smaller than
+that of a qualifying \(y\in a\). If \(y\) had no successor at or
+before the slot of \(x\), their exchange would be feasible.
+Invariant~\ref{inv:swap} therefore forces such a successor.
+The proof in \Cref{sec:boundary} uses this first successor to
+connect exterior qualifiers to the local separator walk.
+
+All predicates are frozen global subsets of \(J\) once constructed.
+In particular, a cone used in an old predicate continues to refer to
+its fixed triple of jobs; it is not redefined after later reorderings.
+We shall prove, simultaneously with the three invariants, that each
+pruned list has at most five entries and each entry has fewer than
+40 leaves.
+
+At the root, use the original frame, the two sentinels, and the one-entry
+list \((J)\). Invariant~\ref{inv:cones} follows from the sentinel cones,
+and Invariants~\ref{inv:past} and~\ref{inv:swap} are vacuous.
+
+\subsection{A center and two open sides}
+
+In either working frame, let \(a,b\) now mean the left and right
+boundaries of \(W\) in that frame, and write
+\[
+ X=W\setminus D_a,\qquad Y=W\setminus P_b.
+\]
+Invariant~\ref{inv:cones} gives
+\begin{equation}
+ W\subseteq D_a\cup P_b,\qquad X\subseteq P_b,\qquad Y\subseteq D_a.
+ \label{eq:cover}
+\end{equation}
+This cover is unchanged by reversal: the two boundary cones exchange
+roles. In either frame \(X\) is an ideal in \(W\), and \(Y\) an
+upset. The list, \(O\), and the other two invariants are used in their
+stated form only in the tag frame.
+
+In the tag frame choose an actual center \(v\) separating
+the ideal \(W\cap P_b\), using \Cref{cor:any-ideal} and, if needed,
+reordering \(W\). Mark \(v\). At the root use the last slot as
+center; it separates the ideal \(J\).
+The center triple and the job sets on its two sides are now fixed.
+
+Treat each side as an open prefix \(E=(a,v)\), in the frame that
+makes it a prefix. Call that side \(\same\) if its working frame is
+the node's tag and \(\switch\) otherwise. Recompute all boundary
+notation in that working frame. The center is a separator at cutoff
+\begin{equation}
+ S_*=
+ \begin{cases}
+ W\cap P_b,&\same,\\
+ X=W\setminus D_a,&\switch.
+ \end{cases}
+ \label{eq:center-cutoff}
+\end{equation}
+Indeed the second formula is the complement of the original center-low
+set after reversing order and time, so
+\Cref{lem:separator-description} applies.
+
+Use the following priority orders on \(W\), and the following nested
+low cutoffs.
+\begin{description}[style=nextline,leftmargin=1.5em]
+ \item[Same direction.]
+ Begin with \(S_0=(W\setminus O)\cap P_b\).
+ Order \(S_0\) first by increasing rank, then \(O\) by increasing
+ \(\key_{\K}\), and then \(Y\) by increasing rank.
+ Promote the jobs of \(O\) in that key order.
+ \item[Switched direction.]
+ Begin with \(S_0=\varnothing\).
+ Order \(X\) first and \(W\cap D_a\) second, each by increasing
+ rank in the switched frame.
+ Promote the jobs of \(X\) in that order.
+\end{description}
+These are total orders extending precedence.
+For the same case, \(S_0\) is an ideal, \(Y\) is an upset,
+and the key order on the middle group extends precedence.
+The groups partition \(W\) since \(O\subseteq P_b\).
+For the switch case use idealness of \(X\).
+Every cutoff lies in \(P_b\) on \(W\), by \eqref{eq:cover},
+and the initial low set lies in \(D_a\).
+
+Normalize the open side \(E\) using the restriction of its priority
+order to \(E\). This can be done independently on the two sides
+before constructing children. It preserves feasibility by
+\Cref{lem:reorder} and preserves the center separator, since neither
+side's job set changes.
+The resulting sequence of low subsets of \(W\) is
+\[
+ S_0,S_1,\ldots,S_q=S_* .
+\]
+Some promoted jobs may lie outside \(E\).
+Starting at its exterior left boundary \(s_0=a\), use
+\Cref{lem:shared} for every promotion, taking \(s_i\) to be the
+\emph{final} position of that promotion. Thus
+\begin{equation}
+ a=s_0\le s_1\le\cdots\le s_q<s_{q+1}=v.
+ \label{eq:positions}
+\end{equation}
+The starting boundary is a separator on \(E\) because
+\(S_0\cap E\subseteq D_a\).
+For each \(i<q\), both \(s_i,s_{i+1}\) separate the cutoff \(S_i\)
+on \(E\). The same assertion for \(i=q\) uses the center separator
+at \(v\).
+
+Mark the distinct actual walk positions in \(E\), in addition to \(v\).
+For each strict step \(s_i<s_{i+1}\), use
+\begin{equation}
+ a'=s_i,\qquad b'=s_{i+1},\qquad S=S_i
+ \label{eq:edge-data}
+\end{equation}
+as its edge data; Figure~\ref{fig:shared-cutoff} shows one such step.
+Removing repetitions from \eqref{eq:positions}
+does not create an unassigned gap: consecutive distinct positions
+are exactly the endpoints of one of these strict steps.
+If the gap \(W'=(a',b')\) is nonempty, make it a child and tag it
+with this working frame.
+
+\begin{lemma}\label{lem:edge-labels}
+At every child edge, \(b'\) is an actual triple in \(W\), while
+\(a'\) is either \(a\) or an actual triple of \(E\). On its child,
+\begin{equation}
+ W'\cap S\subseteq D_{a'}\cap P_b,
+ \qquad
+ W'\setminus S\subseteq P_{b'}.
+ \label{eq:edge-labels}
+\end{equation}
+Every child excludes the center and is strictly smaller than its parent.
+The root has no nonempty switched child.
+\end{lemma}
+\begin{proof}
+The shared cutoff at \(a',b'\) gives the two required separator
+implications on the open side. All cutoffs lie in \(P_b\) on \(W\).
+The remaining assertions follow from the positions of the marks,
+and from the root's last-slot center.
+\end{proof}
+
+\begin{figure}[t]
+\centering
+\input{figures/shared-cutoff}
+\caption{One side of a node, viewed in its working frame.
+Each nonempty gap uses the cutoff belonging to its strict walk step.
+The endpoints separate that same cutoff, although adjacent gaps may
+use different cutoffs. The horizontal arrow denotes schedule time,
+not a precedence relation.}
+\label{fig:shared-cutoff}
+\end{figure}
+
+It remains to give each child a short list selecting exactly
+its high jobs, and to preserve the global boundary invariants.
+The following sections prove these claims by a top-down induction.
+When a child of \(W\) is constructed, all data for \(W\) and its
+ancestors have already been constructed. No claim about a deeper
+descendant is used.
+
+\begin{table}[t]
+\centering
+\small
+\begin{tabular}{@{}p{.25\textwidth}p{.69\textwidth}@{}}
+\toprule
+Proof component & Interface used by the next step \\
+\midrule
+Shared cutoffs & A normalized side, a starting separator, and successive
+priority prefixes give two boundaries separating the same cutoff
+(\Cref{lem:shared}). \\[4pt]
+Child intervals & A valid parent gives a center and strictly smaller gaps
+with their shared cutoffs (\Cref{lem:edge-labels}). \\[4pt]
+Global lists & Each child receives its high-job set and first invariant
+through at most five upset entries, each with fewer than \(40\) leaves
+(\Cref{lem:update-labels,lem:list-length,lem:entry-size}). \\[4pt]
+Boundary stability & Parent invariants and side normalization give child
+Invariants~\ref{inv:past}--\ref{inv:swap}, including all future feasible
+child reorderings (\Cref{lem:past,lem:swap}). \\[4pt]
+Finite descriptions & Valid lists and boundaries give global child sets
+and relative splits with fewer than \(500\) leaves
+(\Cref{lem:global-child,lem:relative-splits,lem:final-count}). \\
+\bottomrule
+\end{tabular}
+\caption{The contracts connecting the structural proof. The executable
+dynamic program consumes the final node sets and split sets; lists and
+normalized schedules are used to prove that these sets suffice.}
+\label{tab:contracts}
+\end{table}
diff --git a/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/descriptions.tex b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/descriptions.tex
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index 0000000000000000000000000000000000000000..88919361b0f76d493e95297ad3fe58de1253eb0d
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+++ b/preprints/A-polynomial-time-algorithm-for-three-machine-unit-job-scheduling-September-24-2026/build/descriptions.tex
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+\section{Global interval predicates and completion}\label{sec:descriptions}
+
+We now describe each child as a global set without intersecting
+with a recursively described parent. This is the step that converts
+the construction into a polynomial family of subproblems.
+
+\subsection{Global tests for local cutoffs}
+
+The cutoff predicates in this section are used only in the final
+interval and split formulas. The list construction and its entry-size
+bound are now complete, so these formulas may retain the existing
+bounded entries. The historical extension in \Cref{lem:old-low},
+by contrast, had to flatten an earlier run before a new switch
+entry could be constructed.
+
+For every walk cutoff \(S\) in a parent \(W\), choose a global
+predicate \(\widetilde S\) satisfying
+\begin{equation}
+ \widetilde S\cap W=S.
+ \label{eq:cutoff-agreement}
+\end{equation}
+In the same case, let \(\K_{\mathrm{carry}}\) be the carry determined
+by \(O\setminus S\), before the new injection. Use
+\begin{equation}
+ \widetilde S=P_b\setminus\Qual(\K_{\mathrm{carry}}).
+ \label{eq:direct-same-cutoff}
+\end{equation}
+On \(W\), the carry selects precisely \(O\setminus S\), while
+the low jobs are the initial \((W\setminus O)\cap P_b\) and the
+promoted portion of \(O\). Thus \eqref{eq:cutoff-agreement} holds.
+In the switch case use
+\begin{equation}
+ \widetilde S=(J\setminus D_a)\cap\{\rk\le r\},
+ \label{eq:direct-switch-cutoff}
+\end{equation}
+where \(r\) is the rank of the last promoted member of \(X\),
+or \(r=0\) before any promotion.
+These formulas are available at every walk cutoff, whether or not
+the associated open gap is nonempty.
+They are not required to have any prescribed meaning outside \(W\).
+
+\subsection{A child as a global set}
+
+Figure~\ref{fig:gap} summarizes the localization steps in the following
+formulas.
+
+\begin{lemma}\label{lem:global-child}
+In the valid construction of \Cref{prop:construction-valid}, let
+\(W'=(a',b')\) be a child of \(W=(a,b)\), with parent center \(v\),
+edge cutoff \(S\), new pruned list \(\K'\), and cutoff predicate
+\(\widetilde S\) satisfying \eqref{eq:cutoff-agreement}.
+In the edge's working frame, its high and low parts are the
+following global sets:
+\begin{align}
+ W'\setminus S
+ &=\Qual(\K')\cap P_{b'}\cap(J\setminus\wP{a'}),
+ \label{eq:global-high}\\
+ W'\cap S
+ &=D_{a'}\cap P_b\cap(J\setminus\wD{b'})
+                  \cap(J\setminus\wD v)\cap\widetilde S .
+ \label{eq:global-low}
+\end{align}
+\end{lemma}
+\begin{proof}
+Every high child job qualifies, precedes \(b'\), and, because
+it is later than \(a'\), is outside \(\wP{a'}\).
+Conversely, a job selected by the right side of
+\eqref{eq:global-high} is strictly before \(b'\) by precedence.
+If it were at or before \(a'\), the child's global
+Invariant~\ref{inv:past} would put it in \(\wP{a'}\).
+It is therefore strictly in \(W'\), and the child list selects
+exactly its high jobs.
+
+For \eqref{eq:global-low}, consider first a job selected by its
+right side. The two positive tests \(D_{a'}\cap P_b\) locate
+it strictly inside the parent \(W\): its time is after
+\(a'\ge a\) and before \(b\). This remains true with sentinel
+boundaries under our conventions.
+We can consequently use \eqref{eq:cutoff-agreement} to infer
+that the job is in \(S\).
+This cutoff is contained in the center-low set \(S_*\).
+A selected job at or after \(v\) would therefore lie in
+\(\wD v\), by the center separator or by membership in \(v\)
+itself. The corresponding exclusion places it in the open side
+\(E=(a,v)\).
+Within \(E\), a low job at or after \(b'\) is in
+\(\wD{b'}\), using the local separator at \(b'\).
+The remaining exclusion puts the selected job strictly between
+\(a'\) and \(b'\).
+
+Conversely, a low job of this gap lies in \(D_{a'}\cap P_b\)
+by \Cref{lem:edge-labels}, and it satisfies
+\(\widetilde S\) by \eqref{eq:cutoff-agreement}.
+Its time is strictly before both \(b'\) and \(v\), so it is
+in neither of their weak successor cones. It passes every test
+in \eqref{eq:global-low}.
+\end{proof}
+
+\begin{figure}[t]
+\centering
+\input{figures/global-gap-filters}
+\caption{The two global tests for a child.
+For the low part, positive cones locate a candidate inside the parent
+before its cutoff predicate is used; the two weak successor exclusions
+then restrict that candidate to the gap.
+For the high part, the global past-boundary invariant excludes
+qualifiers at or before \(a'\).
+The intersection factors are arranged in the order used by the proof.}
+\label{fig:gap}
+\end{figure}
+
+The localization order matters. Neither the cutoff extension
+\(\widetilde S\) nor the flattened historical extension needs to
+agree with its intended local set everywhere in \(J\).
+Positive cone tests first place the candidate in the domain where
+agreement is known. The construction never appends a membership
+predicate for all ancestor intervals.
+
+\subsection{Relative predicates for the marked splits}
+
+\begin{lemma}\label{lem:relative-splits}
+Every marked slot \(z\) in a node \(W\) has a bounded relative
+description of its earlier jobs in either the working direction
+that produced it or the original direction. The formulas use the stated
+atomic language and have fewer than \(500\) leaves, as counted in
+\Cref{lem:final-count}.
+\end{lemma}
+\begin{proof}
+Use the working frame for a side, and put
+\[
+ H_*=
+ \begin{cases}
+ P_b,&\same,\\
+ J\setminus D_a,&\switch.
+ \end{cases}
+\]
+For the center \(v\), the predicate \(B_v(H_*)\), restricted to
+\(W\), is its earlier set, by
+\Cref{lem:separator-description} and \eqref{eq:center-cutoff}.
+
+For an internal mark \(z\) in the open side \(E\), choose a walk
+cutoff \(S\) for which it is a local separator.
+The predicate
+\begin{equation}
+ F_z=B_v(H_*)\cap B_z(\widetilde S)
+ \label{eq:relative-split}
+\end{equation}
+gives its earlier set upon restriction to \(W\).
+The first factor selects \(E\) on \(W\); on that domain the
+second is the separator description from
+\Cref{lem:separator-description}.
+
+If the working frame is opposite to original time, \(F_z\cap W\)
+describes the original \emph{later} jobs. Replace \(F_z\) by
+\begin{equation}
+ (J\setminus F_z)\setminus z
+ \label{eq:reverse-split}
+\end{equation}
+to obtain the original earlier jobs after restriction to \(W\).
+The exclusion of the marked triple is necessary: complementation
+alone includes it. Apply the same conversion to the center
+predicate when appropriate.
+\end{proof}
+
+\begin{lemma}\label{lem:final-count}
+The predicates in \Cref{lem:global-child,lem:relative-splits}
+have fewer than \(500\) leaves.
+\end{lemma}
+\begin{proof}
+A pruned list or a carry contains at most five entries with fewer
+than \(40\) leaves each, so its union has at most \(195\) leaves.
+An empty union uses the false constant.
+The same-direction cutoff \eqref{eq:direct-same-cutoff} therefore
+needs at most \(196\) leaves; the switched one needs two.
+Use \(205\) as a common loose allowance.
+
+The high predicate \eqref{eq:global-high} uses at most
+\(195+2=197\) leaves. The low predicate
+\eqref{eq:global-low} uses at most \(205+4=209\).
+Their union is a global description of \(W'\) with at most
+\(406\) leaves.
+
+The center predicate has three leaves. The internal split
+\eqref{eq:relative-split} has at most \(3+(205+2)=210\),
+and conversion \eqref{eq:reverse-split} adds one.
+Complementation does not increase leaf counts.
+Every cone uses an actual triple or a sentinel constant; all ranks
+and thresholds are from the two allowed frames. Thus these are
+formulas in the stated atomic language.
+\end{proof}
+
+\begin{proof}[Proof of \Cref{thm:structure}]
+Use \Cref{prop:construction-valid} to construct marks and lists
+from a feasible full schedule.
+The root set \(J\) is represented by true.
+Every nonempty child has the global description supplied by
+\Cref{lem:global-child}. Every mark has the relative split
+description supplied by \Cref{lem:relative-splits}.
+\Cref{lem:final-count} places both kinds of description in
+\(\F\), independently of their depth.
+All branches terminate because the node sizes decrease strictly.
+These are exactly the conditions of \Cref{def:hierarchy}.
+\end{proof}