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+preprints/A-Fixed-Particle-Test-for-Computation-in-a-Forced-Viscous-Flow-September-27-2026/manuscript.pdf filter=lfs diff=lfs merge=lfs -text
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preprints/A-Fixed-Particle-Test-for-Computation-in-a-Forced-Viscous-Flow-September-27-2026/README.md

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+# [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}
+}
+```
+

preprints/A-Fixed-Particle-Test-for-Computation-in-a-Forced-Viscous-Flow-September-27-2026/build/analytic-appendix.tex

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+\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.

preprints/A-Fixed-Particle-Test-for-Computation-in-a-Forced-Viscous-Flow-September-27-2026/build/analytic-tools.tex

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+\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}

preprints/A-Fixed-Particle-Test-for-Computation-in-a-Forced-Viscous-Flow-September-27-2026/build/main.tex

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+\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.
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+\end{thebibliography}
+\end{document}

preprints/A-Fixed-Particle-Test-for-Computation-in-a-Forced-Viscous-Flow-September-27-2026/manuscript.pdf

@@ -0,0 +1,3 @@
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preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/.gitignore

@@ -0,0 +1,4 @@
+/.build/
+/.paper.pdf.*
+/__pycache__/
+/verification/__pycache__/

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

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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}
+}
+```

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}

preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/main.tex

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+\pdfglyphtounicode{logicalandtext}{22C0}
+\pdfglyphtounicode{logicalanddisplay}{22C0}
+\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}

preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/build/references.bib

@@ -0,0 +1,329 @@
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+}
+
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+
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+
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+}
+
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+}
+
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+
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+}
+
+@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},
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+}
+
+@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},
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+  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}}
+}
+
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+  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},
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+  url = {https://arxiv.org/abs/2601.12165}
+}
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+  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},
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+  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},
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+  note = {\href{https://doi.org/10.1016/j.tcs.2008.09.025}{doi:10.1016/j.tcs.2008.09.025}}
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+}
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+  author = {Gao, Qiang and Lanzetta, Ryan A. and Ledwith, Patrick and Wang, Jie and Khalaf, Eslam},
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+  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},
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+  note = {\href{https://doi.org/10.1103/PhysRevB.29.5617}{doi:10.1103/PhysRevB.29.5617}}
+}

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}.

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}

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}

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.

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}

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}

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}

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}

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

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}
+}
+```
+

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

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}
+}
+```

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}

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}

preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/build/references.bib

@@ -0,0 +1,212 @@
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+  url = {https://annals.math.princeton.edu/2007/166-2/p01}
+}
+
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+}
+
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+}
+
+@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},
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+  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},
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+  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},
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+  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},
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+  journal = {Algebras and Representation Theory},
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+  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.},
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+}
+
+@misc{KontsevichSoibelman2008,
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+}
+
+@article{BallardFaveroKatzarkov2012,
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+}
+
+@misc{Aspinwall2004,
+  author = {Aspinwall, Paul S.},
+  title = {{D}-branes on {Calabi--Yau} manifolds},
+  year = {2004},
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+}
+
+@article{BorelSerre1958,
+  author = {Borel, Armand and Serre, Jean-Pierre},
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+@misc{Catanese2014,
+  author = {Catanese, Fabrizio},
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+}
+

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}

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}.

preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/build/sections/introduction.tex

@@ -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.

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}

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}

preprints/A-Gepner-stability-condition-on-every-smooth-quintic-threefold-September-24-2026/paper.pdf

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+version https://git-lfs.github.com/spec/v1
+oid sha256:f70b73596fffb0689874c768d3975cb8e16cbdede231c21baef0707ab1898f49
+size 482371

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}
+}
+```
+

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

preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/build/source/paper.tex

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+\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}

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

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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+
+\bibitem{AP}
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+{doi:10.1090/S0894-0347-04-00464-3}.
+
+\bibitem{BGT}
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+
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+{doi:10.1214/009117904000000856};
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+(electronic reprint).
+
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+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}
+V.~Chv\'atal and B.~Reed,
+\emph{Mick Gets Some (the Odds Are on His Side)},
+Proc. 33rd Annual Symposium on Foundations of Computer Science (1992),
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+\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};
+\href{https://arxiv.org/abs/1310.2728v6}{arXiv:1310.2728v6}.
+
+\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.
+\href{https://doi.org/10.1007/s00220-002-0699-y}
+{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

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}

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}

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.

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}

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}

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.

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.

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}
+}
+```
+

preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/build/bibliography.tex

@@ -0,0 +1,152 @@
+\begin{thebibliography}{99}
+
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+
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+Ann. Inst. Henri Poincar\'e Probab. Stat. \textbf{51} (2015), no.~3,
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+
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+Comm. Math. Phys. \textbf{395} (2022), no.~2, 773--857.
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+
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+
+\bibitem{DMS}
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+\emph{Liouville quantum gravity as a mating of trees},
+Ast\'erisque \textbf{427} (2021), viii+258 pp.
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+\href{https://arxiv.org/abs/1409.7055v4}{arXiv:1409.7055v4}.
+
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+A. M. Dykhne,
+\emph{Conductivity of a two-dimensional two-phase system},
+Soviet Phys. JETP \textbf{32} (1971), no.~1, 63--65;
+translated from Zh. Eksp. Teor. Fiz. \textbf{59} (1970), 110--115.
+
+\bibitem{FOT}
+M. Fukushima, Y. Oshima, and M. Takeda,
+\emph{Dirichlet Forms and Symmetric Markov Processes},
+second revised and extended edition,
+De Gruyter Studies in Mathematics, vol.~19,
+Walter de Gruyter, Berlin, 2011.
+\href{https://doi.org/10.1515/9783110218091}{doi:10.1515/9783110218091}.
+
+\bibitem{GRV}
+C. Garban, R. Rhodes, and V. Vargas,
+\emph{Liouville Brownian motion},
+Ann. Probab. \textbf{44} (2016), no.~4, 3076--3110.
+\href{https://doi.org/10.1214/15-AOP1042}{doi:10.1214/15-AOP1042}.
+
+\bibitem{GRVForm}
+C. Garban, R. Rhodes, and V. Vargas,
+\emph{On the heat kernel and the Dirichlet form of Liouville Brownian motion},
+Electron. J. Probab. \textbf{19} (2014), Paper No.~96, 1--25.
+\href{https://doi.org/10.1214/EJP.v19-2950}{doi:10.1214/EJP.v19-2950}.
+
+\bibitem{GHS}
+E. Gwynne, N. Holden, and X. Sun,
+\emph{A mating-of-trees approach for graph distances in random planar maps},
+Probab. Theory Related Fields \textbf{177} (2020), nos.~3--4, 1043--1102.
+\href{https://doi.org/10.1007/s00440-020-00969-8}{doi:10.1007/s00440-020-00969-8}.
+
+\bibitem{GMMetric}
+E. Gwynne and J. Miller,
+\emph{Existence and uniqueness of the Liouville quantum gravity metric
+for $\gamma\in(0,2)$},
+Invent. Math. \textbf{223} (2021), no.~1, 213--333.
+\href{https://doi.org/10.1007/s00222-020-00991-6}{doi:10.1007/s00222-020-00991-6}.
+
+\bibitem{GMConformal}
+E. Gwynne and J. Miller,
+\emph{Conformal covariance of the Liouville quantum gravity metric
+for $\gamma\in(0,2)$},
+Ann. Inst. Henri Poincar\'e Probab. Stat. \textbf{57} (2021), no.~2,
+1016--1031.
+\href{https://doi.org/10.1214/20-AIHP1105}{doi:10.1214/20-AIHP1105}.
+
+\bibitem{GM}
+E. Gwynne and J. Miller,
+\emph{Random walk on random planar maps: spectral dimension, resistance
+and displacement},
+Ann. Probab. \textbf{49} (2021), no.~3, 1097--1128.
+\href{https://doi.org/10.1214/20-AOP1471}{doi:10.1214/20-AOP1471}.
+
+\bibitem{GMSHarmonic}
+E. Gwynne, J. Miller, and S. Sheffield,
+\emph{Harmonic functions on mated-CRT maps},
+Electron. J. Probab. \textbf{24} (2019), Paper No.~58, 1--55.
+\href{https://doi.org/10.1214/19-EJP325}{doi:10.1214/19-EJP325}.
+
+\bibitem{GMS}
+E. Gwynne, J. Miller, and S. Sheffield,
+\emph{The Tutte embedding of the mated-CRT map converges to Liouville
+quantum gravity},
+Ann. Probab. \textbf{49} (2021), no.~4, 1677--1717.
+\href{https://doi.org/10.1214/20-AOP1487}{doi:10.1214/20-AOP1487}.
+
+\bibitem{GMS2}
+E. Gwynne, J. Miller, and S. Sheffield,
+\emph{An invariance principle for ergodic scale-free random environments},
+Acta Math. \textbf{228} (2022), no.~2, 303--384.
+\href{https://doi.org/10.4310/ACTA.2022.v228.n2.a2}{doi:10.4310/ACTA.2022.v228.n2.a2}.
+
+\bibitem{LP}
+R. Lyons and Y. Peres,
+\emph{Probability on Trees and Networks},
+Cambridge Series in Statistical and Probabilistic Mathematics, vol.~42,
+Cambridge University Press, New York, 2016.
+\href{https://doi.org/10.1017/9781316672815}{doi:10.1017/9781316672815}.
+
+\bibitem{IG4}
+J. Miller and S. Sheffield,
+\emph{Imaginary geometry IV: interior rays, whole-plane reversibility,
+and space-filling trees},
+Probab. Theory Related Fields \textbf{169} (2017), nos.~3--4, 729--869.
+\href{https://doi.org/10.1007/s00440-017-0780-2}{doi:10.1007/s00440-017-0780-2}.
+\href{https://arxiv.org/abs/1302.4738v2}{arXiv:1302.4738v2}.
+
+\bibitem{MSSphere}
+J. Miller and S. Sheffield,
+\emph{Liouville quantum gravity spheres as matings of finite-diameter trees},
+Ann. Inst. Henri Poincar\'e Probab. Stat. \textbf{55} (2019), no.~3,
+1712--1750.
+\href{https://doi.org/10.1214/18-AIHP932}{doi:10.1214/18-AIHP932}.
+\href{https://arxiv.org/abs/1506.03804v4}{arXiv:1506.03804v4}.
+
+\bibitem{Mullin}
+R. C. Mullin,
+\emph{On the enumeration of tree-rooted maps},
+Canad. J. Math. \textbf{19} (1967), 174--183.
+\href{https://doi.org/10.4153/CJM-1967-010-x}{doi:10.4153/CJM-1967-010-x}.
+
+\bibitem{ConformalMaps}
+OpenAI,
+\emph{Canonical conformal limits of subcritical FK planar maps},
+OpenAI Math Release preprint
+\href{https://github.com/openai/math/blob/main/preprints/Canonical-conformal-limits-of-subcritical-FK-planar-maps-September-24-2026/main.pdf}{OAI:Canonical-conformal-limits-of-subcritical-FK-planar-maps-September-24-2026},
+2026.
+
+\bibitem{MetricMaps}
+OpenAI,
+\emph{Metric-measure limits of subcritical FK and spanning-tree planar maps},
+OpenAI Math Release preprint
+\href{https://github.com/openai/math/blob/main/preprints/Metric-measure-limits-of-subcritical-FK-and-spanning-tree-planar-maps-September-24-2026/main.pdf}{OAI:Metric-measure-limits-of-subcritical-FK-and-spanning-tree-planar-maps-September-24-2026},
+2026.
+
+\end{thebibliography}

preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/build/paper.tex

@@ -0,0 +1,80 @@
+\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}

preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/build/sections/01-history.tex

@@ -0,0 +1,55 @@
+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.