Add preprints files (part 9)

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@@ -327,3 +327,80 @@ preprints/Endpoint-pointwise-convergence-for-the-Schrodinger-equation-in-higher-
 preprints/Entanglement-with-zero-distillable-secret-key-in-local-dimension-ten-September-27-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
 preprints/Entropy-and-Face-Dimension-of-the-Perfect-Matching-Polytope-September-23-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
 preprints/Entropy-equality-and-local-symmetry-in-negative-curvature-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/sharp-one-dimensional-lieb-thirring-inequalities-matrix-potentials.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/quartic-torus-packets.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/primitive-sextic-torus-packets.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case-October-5-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Essential-Singularity-of-the-Correlation-Length-in-the-Planar-XY-Model-October-5-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Eventual-Categoricity-for-Abstract-Elementary-Classes-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Every-complex-K3-surface-is-Oka-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Every-four-dimensional-Kakeya-set-has-full-Hausdorff-dimension-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Exact-asymptotic-moduli-in-a-Daugavet-subspace-of-L1-September-27-2026/manuscript.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-3-2026/exact-bsd-low-selmer-corank.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Exact-Derandomization-of-Logarithmic-Space-L-equals-RL-equals-BPL-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Exact-mass-asymptotics-for-the-two-dimensional-O4-lattice-model-October-5-2026/exact-mass-o4.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Exact-orders-and-invariant-anticanonical-linear-systems-September-26-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Exact-quantum-factoring-over-a-fixed-finite-gate-set-September-25-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Existential-universal-real-sentences-in-the-counting-hierarchy-October-4-2026/etr-counting-hierarchy.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Expected-amenable-subalgebras-preserving-core-commutants-September-23-2026/Expected-amenable-subalgebras-preserving-core-commutants-September-23-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Exponential-decay-in-two-dimensional-classical-On-models-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Exponential-PSD-rank-of-positively-shifted-matching-matrices-October-5-2026/shifted-matching-psd.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Failure-of-Bounded-Approximation-in-a-Lipschitz-Free-Space-over-a-Uniformly-Discrete-Metric-Space-September-26-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Failure-of-finite-assembly-for-a-chromatic-overlap-at-the-prime-three-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Failure-of-rational-injectivity-for-maximal-coarse-assembly-October-5-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Fiber-functors-for-finite-symmetric-tensor-categories-in-positive-characteristic-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Filtered-chromatic-splitting-at-generic-primes-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Filtered-products-and-boundary-preserving-compression-in-complex-cobordism-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Finite-algebraic-envelopes-and-the-Boone-Higman-conjecture-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Finite-angular-cylinder-covers-below-the-half-area-bound-September-27-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Finite-Circle-Obstructions-Binary-Codes-and-Histogram-Embeddings-for-Edit-Distance-September-27-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Finite-Congruence-Lattices-Characterization-and-Undecidability-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Finite-cylinder-approximation-of-ruled-sets-September-27-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Finite-generation-for-the-Kn-local-sphere-September-24-2026/Finite-generation-for-the-Kn-local-sphere-September-24-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Finite-Instructions-and-Solenoidal-Shear-Flows-September-27-2026/manuscript.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Finite-Monoid-Computations-and-a-Uniform-Generalized-Star-Height-Bound-September-25-2026/Finite-Monoid-Computations-and-a-Uniform-Generalized-Star-Height-Bound-September-25-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds-October-5-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Finite-Smith-Toda-Complexes-at-Varying-Primes-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Finite-tensor-savings-and-exact-Fourier-circuits-September-25-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Finite-triangular-approximation-of-radial-sweeps-September-27-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Fixed-energy-universality-for-weak-Anderson-disorder-on-random-regular-graphs-October-5-2026/fixed-energy-universality-weak-anderson-disorder-random-regular-graphs.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Fixed-Points-of-Nonexpansive-Maps-in-Reflexive-Banach-Spaces-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Fontaine-Mazur-modularity-at-the-prime-2-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Foulkes-Conjecture-for-the-Sixth-Symmetric-Power-September-25-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Fourfold-nonvanishing-by-minimal-metrics-and-moving-jets-September-27-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Frobenius-Structures-on-Tame-Hecke-Eigensheaves-October-5-2026/tame-hecke-frobenius.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/From-critical-crossings-to-quenched-near-critical-universality-in-Voronoi-percolation-October-5-2026/critical-crossings-quenched-near-critical-universality-voronoi-percolation.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/From-partial-permutation-information-to-Fourier-bounds-September-26-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Fujitas-freeness-conjecture-September-23-2026/Fujitas-freeness-conjecture-September-23-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Full-support-of-the-zero-temperature-Sherrington-Kirkpatrick-order-parameter-September-27-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Functional-universality-of-critical-SK-autocorrelations-October-5-2026/critical-sk-autocorrelations.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Gaussian-free-field-limits-of-weighted-integer-Lipschitz-heights-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Generalized-Cartan-Hadamard-isoperimetry-and-Euclidean-equality-rigidity-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Generalized-ionization-energies-for-full-Coulomb-atoms-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Generalized-outer-electron-radii-of-neutral-Coulomb-atoms-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Generalized-Star-Height-at-Most-Four-September-25-2026/Generalized-Star-Height-at-Most-Four-September-25-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Generalized-Star-Height-at-Most-Three-September-25-2026/article.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Generic-C1-Future-Inextendibility-Near-Rotating-Subextremal-Kerr-Spacetimes-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Generic-Future-Inextendibility-with-Square-Integrable-Connection-Near-a-Fixed-Kerr-Spacetime-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Geometric-Programs-for-Solenoidal-Forcing-September-27-2026/manuscript.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Giglis-distributional-curvature-characterization-of-Alexandrov-spaces-September-24-2026/main.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Global-Arthur-Enhancements-of-Cuspidal-Excursion-Parameters-October-5-2026/manuscript.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Global-classical-solutions-of-the-three-dimensional-relativistic-Vlasov-Maxwell-system-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Global-quantum-geometric-Langlands-at-irrational-level-October-4-2026/quantum-langlands.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Global-Spherical-Shells-on-Minimal-Surfaces-of-Class-VII-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026/article.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/GOE-bulk-universality-for-fixed-degree-random-regular-graphs-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Graph-Decompositions-at-the-Integral-Expectation-Threshold-October-5-2026/graph-threshold-decompositions.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Ground-state-condensation-in-the-dilute-hard-sphere-gas-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two-October-5-2026/hamiltonian-fixed-points-below-stable-morse-number.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-October-5-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/Hard-sphere-fluctuations-on-the-regular-Boltzmann-lifespan-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text

preprints/Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026/README.md

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+# [Equality and rigidity in the spacetime Penrose inequality](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 27, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026,
+  author = {{OpenAI}},
+  title = {{Equality and rigidity in the spacetime Penrose inequality}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026/paper.pdf}{OAI:Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026}},
+  year = {2026}
+}
+```
+

preprints/Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026/build/three/06-global-classification.tex

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+\section{Global classification and recovery through the horizon}
+\label{q3:global:section}
+
+We continue with the finite-component class of
+Theorem~\ref{q3:intro:rigidity}, with no prescribed boundary genera.
+Starting from the complete static base and end estimates just proved,
+we compactify one end of the double at its actual regularity and retain
+all possible other ends while proving zero-mass rigidity. We then exclude the
+interior null set.  This order preserves the information needed to
+recover both the original metric and second fundamental form.
+
+\begin{lemma}[The two conformal metrics]
+\label{q3:static:conformal-completion}
+Define
+\[
+h_\pm=\left(\frac{1\pm\lambda}{2}\right)^4h
+\quad\text{on }\mathcal E.
+\]
+Both are scalar flat. The plus end satisfies
+\(h_+=\delta+O_k(r^{-1-\epsilon})\), hence has zero ADM mass.
+The minus end compactifies by one point to a nondegenerate
+\(C^{1,\epsilon}\cap W^{2,p}_{\mathrm{loc}}\) metric for some
+\(p>3\), with scalar curvature zero as a distribution at that point.
+\end{lemma}
+
+\begin{proof}
+Scalar flatness follows from
+Equation~\eqref{q3:static:static-equations} and the three-dimensional
+conformal scalar-curvature formula, since \(1\pm\lambda\) are harmonic.
+In terms of the preceding variables,
+\begin{equation}
+\label{q3:static:conformal-gamma}
+h_+=\cosh^4(U/2)\gamma,\qquad
+h_-=\sinh^4(U/2)\gamma.
+\end{equation}
+The first factor is \(1+O_k(r^{-2})\), proving the assertion for
+\(h_+\). Its ADM flux is \(O(r^{-\epsilon})\), so its mass is zero.
+
+For the minus end invert \(x=y/|y|^2\), write \(\rho=|y|\), and
+put \(R(y)=I-2y\otimes y/\rho^2\).
+The inversion differential is \(\rho^{-2}R(y)\), with
+\(R(y)^TR(y)=I\).
+Equation~\eqref{q3:static:gamma-expansion} gives
+\[
+I^*\gamma=\rho^{-4}
+\bigl(I+O_k(\rho^{1+\epsilon})\bigr),
+\qquad
+U(I(y))=-M\rho+(b_1\cdot y)\rho
+ +O_k(\rho^{2+\epsilon}).
+\]
+Derivatives of the angular matrix \(R(y)\) cost the corresponding
+powers of \(\rho^{-1}\), so the displayed metric estimate retains
+the symbol bounds. Dividing the expansion of \(\sinh(U/2)\) by
+\(\rho\), and using \(\epsilon<1\), gives
+\begin{equation}
+\label{q3:static:inverted-minus}
+I^*h_-=\left(\frac M2\right)^4
+\left[
+\left(1-\frac{4b_1\cdot y}{M}\right)I
+ +O_k(\rho^{1+\epsilon})
+\right].
+\end{equation}
+In particular, its only first-order term is linear and scalar.
+
+For the remainder \(E(y)\), the bounds
+\[
+|E|\leq C\rho^{1+\epsilon},\quad
+|DE|\leq C\rho^\epsilon,\quad
+|D^2E|\leq C\rho^{\epsilon-1}
+\]
+show that \(E(0)=DE(0)=0\) extends it as \(C^{1,\epsilon}\).
+To see the Hölder estimate between two points, use the gradient
+bound when their separation is comparable to their radii; otherwise
+integrate the Hessian along the short segment in a common annulus.
+Moreover choose
+\[
+3<p<\frac3{1-\epsilon}.
+\]
+Then \(D^2E\in L^p\). Integration by parts across small coordinate
+spheres has an error bounded by a constant times their area,
+because the first derivatives are bounded.
+It follows that the punctured classical derivatives are the weak
+derivatives of the extension.
+The Christoffel symbols are continuous with weak derivatives in
+\(L^p\), so scalar curvature is its usual \(L^p\) expression
+in \(\partial\Gamma\) and \(\Gamma\Gamma\).
+It vanishes off the point and therefore also as a distribution on
+the whole ball. There is no point curvature term.
+\end{proof}
+
+\subsection{Zero-mass rigidity with arbitrary remaining ends}
+\label{q3:static:spin-subsection}
+
+The static equations and end expansion are now established.  We have
+not yet proved that the stationary field is timelike throughout the
+original exterior.  The next step closes precisely that gap.
+The completed conformal manifold can still have ends corresponding
+to an interior zero set. We need a rigidity statement that permits
+these ends, and also the regularity in
+Lemma~\ref{q3:static:conformal-completion}.
+The smooth arbitrary-end positive mass theorem is established in
+\cite[Theorem B]{CecchiniZeidler2024}.
+That formulation also credits the earlier spinor rigidity theorem of
+Bartnik and Chru\'sciel \cite[Theorem~11.2]{BartnikChrusciel2005}.
+In our scalar-flat situation the following direct spinor proof also
+handles the compactified point. It uses Witten's
+identity~\cite{Witten1981}; no asymptotic condition is imposed on any
+other end.
+
+\begin{lemma}[Scalar-flat rigidity at a faster-decaying end]
+\label{q3:static:spin-rigidity}
+Let \((W,\mathfrak g)\) be a connected orientable complete
+three-dimensional Riemannian manifold without boundary.
+Assume that its metric is locally \(C^{1,\alpha}\cap W^{2,p}\),
+for some \(\alpha>0,p>3\), is scalar flat as a distribution,
+and has an end on which it is smooth with
+\[
+\mathfrak g=\delta+O_2(r^{-1-\epsilon}),\qquad \epsilon>0.
+\]
+The other ends can be arbitrary.
+Then \((W,\mathfrak g)\) is isometric to Euclidean space.
+The asserted regularity is sufficient at a compactified point and
+at a reflected hypersurface.
+\end{lemma}
+
+\begin{proof}
+An orientable three-manifold is spin.
+Choose a spin structure and use a smooth background metric to
+identify its spinor bundle with that for \(\mathfrak g\).
+The positive square-root change of orthonormal frames has the
+stated metric regularity.
+The spin connection consequently has continuous locally bounded
+coefficients, with weak first derivatives in \(L^p_{\mathrm{loc}}\).
+Write \(\nabla^S\) for that connection, \(D\) for the Dirac operator,
+and \(c\) for Clifford multiplication.
+
+The weak Lichnerowicz formula has the following integrated form for
+every compactly supported \(H^1\) spinor \(\phi\):
+\begin{equation}
+\label{q3:static:lichnerowicz}
+\|D\phi\|_2^2=\|\nabla^S\phi\|_2^2.
+\end{equation}
+Here and below norms and integrals are for \(\mathfrak g\).
+To check the formula at the indicated regularity, first use a smooth
+compactly supported spinor. Approximate the metric on its support
+in \(C^1\) and \(W^{2,p}\) by smooth metrics, using the same
+background-bundle identification.
+The smooth Lichnerowicz formula passes to the limit: connection
+coefficients converge uniformly and scalar curvatures converge
+in \(L^p\).
+The limiting scalar term is zero by the assumed distributional
+scalar flatness and the \(L^p\) curvature expression.
+Density then proves Equation~\eqref{q3:static:lichnerowicz} for compact
+\(H^1\) spinors.
+Thus no integration surface is retained around the compactified point.
+
+We next establish the coercivity needed to complete compact spinors.
+For a compactly supported scalar function \(f\) on the AE end,
+one-dimensional integration by parts along coordinate rays gives
+\begin{equation}
+\label{q3:static:radial-hardy}
+\int_{r_0}^\infty f^2\dd r
+\leq4\int_{r_0}^\infty r^2|\partial_rf|^2\dd r.
+\end{equation}
+The inner boundary term has the favorable sign.
+Apply this to \(f=|\phi|\) and integrate over angles.
+Metric comparability and
+\(|\dd|\phi||\leq|\nabla^S\phi|\) give
+\begin{equation}
+\label{q3:static:spin-hardy}
+\int_{\mathrm{AE}}r^{-2}|\phi|^2\dd V_{\mathfrak g}
+\leq C\|\nabla^S\phi\|_2^2.
+\end{equation}
+For any fixed compact \(K\subset W\), connect it to a fixed end
+annulus by a relatively compact connected domain.
+Poincaré's inequality with the norm on that annulus retained gives
+\[
+\||\phi|\|_{L^2(K)}
+\leq C_K\left(
+\|\dd|\phi|\|_{L^2(\text{domain})}
+ +\|\phi\|_{L^2(\text{annulus})}\right).
+\]
+For example, this version follows from the ordinary Poincaré
+inequality after subtracting the mean; the observed annulus
+controls that mean because it has positive measure.
+Equation~\eqref{q3:static:spin-hardy} controls its last term.
+We have proved
+\begin{equation}
+\label{q3:static:local-coercivity}
+\|\phi\|_{L^2(K)}\leq C_K\|\nabla^S\phi\|_2.
+\end{equation}
+Connectedness is the only global input in this propagation.
+
+Let \(\mathcal H\) be the completion of compactly supported smooth
+spinors in the norm \(\|\nabla^S\phi\|_2\).
+Equation~\eqref{q3:static:local-coercivity} embeds \(\mathcal H\)
+in \(L^2_{\mathrm{loc}}\), and the locally bounded connection
+then embeds it in \(H^1_{\mathrm{loc}}\).
+Both \(\nabla^S:\mathcal H\to L^2\) and
+\(D:\mathcal H\to L^2\) are well-defined, and
+Equation~\eqref{q3:static:lichnerowicz} makes the latter an isometry.
+Its range is therefore closed.
+Equation~\eqref{q3:static:spin-hardy} also passes to this completion.
+The local constants may depend on the chosen compact set; no uniform
+estimate over the remaining ends is required.
+
+Fix the ordinary orthonormalized coordinate frame on the distinguished
+end and a smooth cutoff \(\theta\), equal to one near infinity and
+zero before a slightly smaller end neighborhood.
+For each constant spinor \(z\in\mathbb C^2\), set
+\(\psi_0=\theta z\).
+Because the connection there is \(O(r^{-2-\epsilon})\),
+\(\nabla^S\psi_0,D\psi_0\in L^2\).
+Orthogonally project \(D\psi_0\) onto the closed subspace
+\(D\mathcal H\) and choose \(\eta\in\mathcal H\) such that
+\[
+D\eta=-\operatorname{proj}_{D\mathcal H}(D\psi_0).
+\]
+Set \(\psi=\psi_0+\eta\), \(w=D\psi\).
+Then \(w\in L^2\) and
+\[
+\int\langle w,D\phi\rangle\dd V_{\mathfrak g}=0
+\quad\text{for every compact smooth }\phi.
+\]
+Thus \(Dw=0\) weakly.
+Local elliptic regularity for a first-order elliptic operator with
+locally Lipschitz principal coefficients and bounded lower-order
+coefficients gives \(w\in H^1_{\mathrm{loc}}\).
+This regularity can also be seen by freezing the principal
+coefficients on small coordinate balls: the constant-coefficient
+Dirac estimate controls one derivative in \(L^2\), the oscillation
+is absorbed, and commutators of mollification are bounded by the
+local Lipschitz norm. Apply the estimate to regularized \(w\)
+and pass to a weak limit.
+The metric in the lemma has more than this coefficient regularity.
+
+Completeness supplies compactly supported Lipschitz distance cutoffs
+\(\chi_R\), equal to one on the radius-\(R\) ball and with
+\(|\dd\chi_R|\leq C/R\).
+Indeed a complete locally compact Riemannian length space is proper,
+and a piecewise linear cutoff of its distance has these properties.
+Equation~\eqref{q3:static:lichnerowicz} applies to \(\chi_Rw\), and
+the weak equation gives
+\[
+\|\nabla^S(\chi_Rw)\|_2^2
+=\|D(\chi_Rw)\|_2^2
+=\|c(\dd\chi_R)w\|_2^2
+\leq \frac C{R^2}\|w\|_2^2.
+\]
+On every fixed compact set the cutoff eventually equals one.
+It follows that \(\nabla^S w=0\).
+A parallel spinor has constant length on the connected manifold;
+the distinguished end has infinite volume, so \(w\in L^2\)
+forces \(w=0\).
+No condition at another end entered this argument.
+
+It remains to show that \(\psi\), rather than just \(D\psi\), is
+parallel. Define the continuous sesquilinear form
+\[
+\mathcal B(a,b)=
+\int\left(\langle\nabla^Sa,\nabla^Sb\rangle
+-\langle Da,Db\rangle\right)\dd V_{\mathfrak g}
+\]
+on fields for which the indicated derivatives are in \(L^2\).
+For compact \(\phi\), the polarized Lichnerowicz identity,
+with one slot compactly supported, gives
+\(\mathcal B(\psi_0,\phi)=0\).
+Approximate \(\eta\) in \(\mathcal H\) by compact spinors.
+These limits use the global derivative norms; no boundary flux of
+\(\eta\) at another end is invoked.
+Continuity and Equation~\eqref{q3:static:lichnerowicz} imply
+\[
+\mathcal B(\psi_0,\eta)=0,\qquad
+\mathcal B(\eta,\eta)=0,\qquad
+\mathcal B(\psi,\psi)=\mathcal B(\psi_0,\psi_0).
+\]
+The last expression is zero. To see this directly, integrate the
+Lichnerowicz identity for \(\psi_0\) up to a large end sphere.
+There is no contribution at another end because \(\psi_0\)
+vanishes there. On that sphere \(\psi_0=O(1)\) and
+\(\nabla^S\psi_0,D\psi_0=O(r^{-2-\epsilon})\);
+the boundary integral is \(O(r^{-\epsilon})\) and tends to zero.
+This is also the zero ADM boundary term of the spin proof.
+Since \(D\psi=w=0\), we conclude that
+\(\|\nabla^S\psi\|_2^2=0\).
+
+The parallel equation gives each resulting spinor a
+\(C^{1,\beta}_{\mathrm{loc}}\) representative for some \(\beta>0\).
+Indeed, \(H^1\subset L^6\) and the locally bounded connection first
+give \(W^{1,6}\), hence local H\"older continuity; the locally H\"older
+connection coefficients then make its first derivatives H\"older.
+Write \(\psi_z\) for this parallel spinor. The fixed cutoff,
+orthogonal projection, and inverse of \(D:\mathcal H\to D\mathcal H\)
+make \(z\mapsto\psi_z\) complex linear.
+For any point \(p\in W\), consider
+\[
+ E_p:\mathbb C^2\longrightarrow\mathbb S_p,
+ \qquad E_p(z)=\psi_z(p),
+\]
+where \(\mathbb S_p\) is the complex spinor fiber.
+If \(E_p(z)=0\), parallelism and connectedness give \(\psi_z=0\)
+everywhere. Then \(\eta=-\theta z\), which for \(z\ne0\) has infinite
+\(r^{-2}\)-weighted \(L^2\) norm on the distinguished end,
+contrary to Equation~\eqref{q3:static:spin-hardy}.
+Thus \(E_p\) is injective for every \(p\), and it is an isomorphism
+because the complex spinor rank is two. The images of a basis of
+\(\mathbb C^2\) form a global parallel spinor frame.
+
+Its Hermitian Gram matrix is constant and positive definite;
+a constant change of basis makes the frame orthonormal.
+The weak curvature equations annihilate this frame, so the spin
+curvature vanishes. Faithfulness of the spin representation on
+\(\mathfrak{spin}(3)\) gives zero Riemann curvature.
+Moreover, Clifford multiplication identifies the real tangent bundle
+with the traceless skew-Hermitian endomorphisms of the spinor bundle.
+Three fixed orthonormal matrices in this parallel spinor frame,
+with matrix inner product \(-\tfrac12\operatorname{tr}(AB)\),
+therefore give a global parallel orthonormal tangent frame.
+The dual coframe is closed by torsion-freeness and integrates
+locally to coordinates in which the metric is exactly Euclidean.
+This also removes any apparent metric singularity at the
+compactified point.
+
+Finally, completeness makes the universal flat cover Euclidean
+\(\R^3\). The global parallel frame makes its deck transformations
+translations. A nontrivial discrete translation group has rank
+at least one, and a quotient by it has volume growth at most
+quadratic. The distinguished AE end gives a cubic lower bound:
+radial paths in that end put a Euclidean annulus of radius
+comparable to \(R\) inside a metric ball of radius \(CR\).
+The two volume bounds are incompatible.
+The deck group is therefore trivial, proving the lemma.
+\end{proof}
+
+\begin{lemma}[The infinity component has no interior missing boundary]
+\label{q3:static:no-internal-null}
+The component \(\mathcal E\) is all of \(\Omega\).
+In particular \(N>0\) everywhere in the open exterior and the
+stationary metric is vacuum there.
+\end{lemma}
+
+\begin{proof}
+The complete base of Lemma~\ref{q3:static:base-completion} reaches every
+component of $S$. Take its complete two-copy double, use $h_+$ on
+the plus copy and $h_-$ on the minus copy, and identify corresponding
+boundary points. The signed reflected lapse expresses the joined
+metric as
+\[
+q=\left(\frac{1+\lambda_{\mathrm{signed}}}{2}\right)^4h_{\mathrm d}.
+\]
+It is smooth at each positive-lapse join and $C^2$ at each zero-lapse
+join. The matched second jets make its scalar curvature zero across
+every join, as well as on the two open sides. Compactify the unique
+AF end of the minus copy by
+Lemma~\ref{q3:static:conformal-completion}.
+
+We verify completeness of this conformal manifold without placing any
+restriction on the number or regularity of the possible internal-null
+ends. On the plus copy the factor $(1+\lambda)/2$ is at least $1/2$.
+Fix a sufficiently large original AF tail. On its complement the
+continuous function $\sqrt N$ is defined on a compact subset of the
+original closed exterior, equals zero on $S$ and the internal zero
+set, and is strictly less than one elsewhere by
+Lemma~\ref{q3:static:base-completion}. Its maximum on this compact set
+is therefore some $a_0<1$. Thus on the entire corresponding minus
+remainder $(1-\lambda)/2\geq(1-a_0)/2>0$ uniformly. On the double
+away from the minus AF tail the conformal metric is consequently
+bounded below by a fixed positive multiple of the complete doubled
+base metric. All joins are already attached. A finite-length escape
+can therefore only run down the minus AF tail, and its proved
+nondegenerate one-point compactification supplies its limit. More
+explicitly, truncate that tail at a large sphere. The other side of
+the sphere is complete with compact boundary by the uniform lower
+bound, and the compactified tail is a compact metric region with
+that same boundary; their gluing is complete.
+
+The conformal manifold is connected, orientable, and without boundary.
+It has local $C^{1,\alpha}\cap W^{2,p}$ regularity for some
+$\alpha>0$ and $p>3$: the finitely many joins are at least $C^2$,
+and the sole compactification point has the regularity established
+in Lemma~\ref{q3:static:conformal-completion}. Its scalar curvature is
+zero distributionally, including at that point, and its distinguished
+plus end has the fast decay required by
+Lemma~\ref{q3:static:spin-rigidity}. That lemma makes this entire
+manifold Euclidean space.
+
+Choose a sufficiently large sphere in the distinguished plus end.
+Under the Euclidean isometry it is a compact embedded sphere. Its
+end tail is the unbounded component of the complement: paths from
+large end radii to that sphere have lengths tending to infinity,
+and the tail has precisely that sphere as frontier. The other
+component is bounded and has compact closure. A sequence in
+$\mathcal E$ approaching an interior point with $N=0$ lies outside
+the chosen end tail and escapes every compact set of the complete
+conformal manifold. Indeed it cannot converge at an ordinary plus
+point by continuity of $N$, cannot converge on any joined component
+because every original boundary collar has no interior zero, and
+cannot converge at a point in the open minus copy or its compactified
+end. Such a sequence is impossible in the compact Euclidean
+complement. Thus $\mathcal E$ has no interior boundary in $\Omega$.
+
+It is a nonempty open and closed subset of connected $\Omega$, and
+hence equals $\Omega$. Vacuum now follows from
+Proposition~\ref{q3:variation:adjoint}, since $N>0$ everywhere in
+$\Omega$.
+\end{proof}
+
+\begin{lemma}[Equality forces a single spherical boundary component]
+\label{q3:static:one-boundary-component}
+In the Euclidean conformal double, every original component $S_i$
+is a round sphere of radius $a=(8\kappa)^{-1}$ and has original area
+$|S_i|_g=16\pi m^2=|S|_g$. Consequently $\ell=1$.
+\end{lemma}
+
+\begin{proof}
+The completed conformal double in the proof of
+Lemma~\ref{q3:static:no-internal-null} is Euclidean. In particular its
+plus metric $q=h_+$ is flat up to every original boundary component.
+Put $\phi=(1+\lambda)/2$. On each $S_i$,
+Lemma~\ref{q3:static:base-completion} gives
+\[
+\phi=\tfrac12,\qquad
+\partial_{\nu_h}\log\phi=\kappa,\qquad
+\mathrm{II}_h=0,\qquad h|_{TS_i}=g|_{TS_i}.
+\]
+The conformal second-fundamental-form formula, with normal pointing
+into the plus side, is therefore
+\[
+\mathrm{II}_q
+ =\phi^2\bigl(\mathrm{II}_h+
+ 2\partial_{\nu_h}\log\phi\,h|_{TS_i}\bigr)
+ =8\kappa\,q|_{TS_i},
+ \qquad q|_{TS_i}=\tfrac1{16}g|_{TS_i}.
+\]
+Let $F_i$ be the Euclidean position vector on $S_i$ and $\nu_i$
+its unit normal into the plus side. Its shape operator is
+$8\kappa I$, so the tangential differential of $F_i-a\nu_i$ is
+zero for $a=(8\kappa)^{-1}$. Connectedness makes $F_i-a\nu_i$
+a constant point $c_i$. Thus the image lies on the sphere of radius
+$a$ centered at $c_i$, with $\nu_i$ its outward radial normal.
+The image is open in that sphere by local invertibility and closed
+by compactness, hence is the whole sphere. The Euclidean embedding
+therefore identifies $S_i$ diffeomorphically with it.
+Since $q|_{TS_i}=g|_{TS_i}/16$, its area satisfies
+\begin{equation}
+\label{q3:static:individual-boundary-area}
+|S_i|_g=16|S_i|_q=64\pi a^2
+ =\frac{\pi}{\kappa^2}=16\pi m^2=|S|_g.
+\end{equation}
+Summing over the finite family yields
+$|S|_g=\sum_{i=1}^{\ell}|S_i|_g=\ell|S|_g$. The boundary is nonempty
+and $|S|_g>0$, hence $\ell=1$. In particular no connectedness or spherical
+topology assumption was needed before this step.
+\end{proof}
+
+\begin{lemma}[Identification of the base and its mass]
+\label{q3:static:schwarzschild-base}
+The regular completion of \((\Omega,h,\lambda)\) is the
+Schwarzschild static exterior
+\begin{equation}
+\label{q3:static:isotropic-base}
+h=\left(1+\frac a\rho\right)^4\delta,\qquad
+\lambda=\frac{1-a/\rho}{1+a/\rho},\qquad
+\rho\geq a,
+\end{equation}
+where \(a=m/2\).
+The completion is homeomorphic to the original \(\Mext\),
+and \(S\) is identified diffeomorphically with its horizon sphere
+in the regular base boundary structure.
+\end{lemma}
+
+\begin{proof}
+Lemma~\ref{q3:static:one-boundary-component} identifies the boundary
+in the Euclidean conformal double as one round sphere of radius
+$a=(8\kappa)^{-1}$. The plus side is its outside: its normal is the
+outward radial normal, its entire boundary is this sphere, and it
+contains the distinguished unbounded end. Center Euclidean
+coordinates at this sphere and write $\rho$ for their radius.
+
+In the resulting Euclidean coordinates, write
+\(\psi=2/(1+\lambda)\), so \(h=\psi^4\delta\).
+The scalar-flat conformal equation gives \(\Delta_\delta\psi=0\)
+outside the sphere, with \(\psi=2\) on it and \(\psi\to1\)
+at infinity. The function \(1+a/\rho\) has the same data.
+Their difference vanishes by the exterior maximum principle.
+This proves Equation~\eqref{q3:static:isotropic-base}.
+Its Schwarzschild mass is $2a=(4\kappa)^{-1}=m$, since
+$\kappa=1/(4m)$. In particular $a=m/2$.
+
+The regular base collar has the same underlying boundary topology
+as the original collar, even when its coordinate is
+\(\lambda=\sqrt N\) instead of the original defining function \(N\).
+Together with the unchanged interior this identifies its completion
+homeomorphically with \(\Mext\).
+An interior base isometry extends uniquely to the metric completions.
+On the regular boundary structures it is smooth: its boundary
+restriction preserves induced lengths, hence is a smooth boundary
+isometry, and normal geodesic coordinates extend it smoothly to
+a collar.
+Thus the completion has the claimed global topology, and
+\(\Omega\cong(a,\infty)\times S^2\) is simply connected.
+\end{proof}
+
+\subsection{The time graph and the Kruskal attachment}
+\label{q3:static:embedding-subsection}
+
+\begin{proof}[Proof of Proposition~\ref{q3:static:classification}]
+Lemma~\ref{q3:static:one-boundary-component} proves $\ell=1$ and the
+spherical topology of $S$. Lemma~\ref{q3:static:no-internal-null} proves
+\(N>0\) on \(\Omega\), and
+Lemma~\ref{q3:static:schwarzschild-base} gives the global static
+base with mass \(m\).
+For the reconstruction write \(M=m\), so \(\kappa=1/(4M)\).
+Simple connectivity and Lemma~\ref{q3:static:base-completion}
+give a global smooth function \(H\) with
+\begin{equation}
+\label{q3:static:time-primitive}
+\dd H=-A.
+\end{equation}
+In static Schwarzschild coordinates use time \(T=t+H(x)\).
+Equations~\eqref{q3:static:base-definitions} and
+\eqref{q3:static:time-primitive} give the exact identity
+\begin{equation}
+\label{q3:static:reconstructed-metric}
+-N\dd T^2+h
+=-u^2\dd t^2+g_{ij}(\dd x^i+X^i\dd t)
+                         (\dd x^j+X^j\dd t).
+\end{equation}
+The graph \(T=H(x)\), corresponding to \(t=0\), therefore induces
+the original metric \(g\).
+The stationary future unit normal is \(n=(\partial_t-X)/u\).
+With the second-fundamental-form convention of the theorem,
+\begin{equation}
+\label{q3:static:reconstructed-k}
+K_{\mathrm{graph}}
+=\frac1{2u}\bigl(\partial_tg-\mathcal L_Xg\bigr)
+=-\frac1{2u}\mathcal L_Xg
+=K.
+\end{equation}
+This calculation recovers the original \(K\), including a
+non-time-symmetric slice.
+
+We next establish smoothness in the original boundary structure.
+If \(u|_S=0\), Equation~\eqref{q3:static:double-norm} gives
+\[
+A=\frac{Y^\flat}{a^2-s^2|Y|_g^2}.
+\]
+This is smooth in the original collar.
+The primitive \(H\) therefore extends smoothly and finitely to \(S\).
+For instance, fix its values on one interior collar section and
+integrate its smooth normal derivative to the boundary.
+Tangential derivatives of this extension agree with those prescribed
+by \(\dd H=-A\), by continuity from the interior.
+
+If \(u|_S>0\), the numerator
+\(X^\flat-(2\kappa)^{-1}\dd N\) vanishes as a one-form on \(S\),
+by Equation~\eqref{q3:static:simple-norm}.
+Smooth division by the defining function \(N\) gives
+\begin{equation}
+\label{q3:static:log-time}
+A=\frac1{2\kappa}\dd\log N+\beta,\qquad
+H=-\frac1{2\kappa}\log N+B,
+\end{equation}
+where \(\beta\) and \(B\) are smooth in the original collar.
+The smoothness of \(B\) follows by applying the same normal-integration
+argument to its differential \(-\beta\).
+In this case \(H\to+\infty\) at the boundary, so static time itself
+is not a regular attachment coordinate.
+
+Let \(r\) be the Schwarzschild area radius. On the whole exterior
+\[
+N=1-\frac{2M}{r},\qquad
+r=\frac{2M}{1-N}.
+\]
+Normalize the tortoise coordinate and Kruskal coordinates by
+\begin{equation}
+\label{q3:static:kruskal-definitions}
+\begin{aligned}
+r_*&=r+2M\log\left(\frac r{2M}-1\right),\\
+\mathsf U&=-e^{-\kappa(T-r_*)},&
+\mathsf V&= e^{\kappa(T+r_*)}.
+\end{aligned}
+\end{equation}
+On the right exterior \(\mathsf U<0,\mathsf V>0\), and
+the Schwarzschild metric extends as
+\[
+\mathbf g_M
+=-\frac{32M^3}{r}e^{-r/(2M)}
+   \dd\mathsf U\,\dd\mathsf V+r^2\dd\omega^2.
+\]
+An exact useful form of Equation~\eqref{q3:static:kruskal-definitions}
+is
+\begin{equation}
+\label{q3:static:kruskal-factor}
+\mathsf U=-Q(N)\sqrt N\,e^{-\kappa T},\qquad
+\mathsf V= Q(N)\sqrt N\,e^{\kappa T},\qquad
+Q(N)=\frac{\exp\!\bigl(1/[2(1-N)]\bigr)}{\sqrt{1-N}}.
+\end{equation}
+The function \(Q\) is smooth and positive near zero, with
+\(Q(0)=\sqrt e\).
+
+If \(u|_S=0\), both \(H\) and
+\(\sqrt N=s\sqrt{a^2-s^2|Y|^2}\) are smooth.
+Substitution into Equation~\eqref{q3:static:kruskal-factor} gives
+\(\mathsf U=\mathsf V=0\) on \(S\) and
+\[
+\partial_s\mathsf U|_S
+=-\sqrt e\,\kappa e^{-\kappa H|_S}<0,\qquad
+\partial_s\mathsf V|_S
+=\sqrt e\,\kappa e^{\kappa H|_S}>0.
+\]
+This is a smooth attachment at the bifurcation sphere.
+If \(u|_S>0\), Equation~\eqref{q3:static:log-time} instead gives
+\begin{equation}
+\label{q3:static:future-attachment}
+\mathsf U=-Q(N)N e^{-\kappa B},\qquad
+\mathsf V=Q(N)e^{\kappa B}.
+\end{equation}
+These are smooth in the original collar;
+\(\mathsf U\) vanishes simply and
+\(\mathsf V|_S=\sqrt e\,e^{\kappa B|_S}>0\).
+Thus the attachment is on the future horizon, with each generator
+met once after the angular identification.
+
+The angular regularity requires a further check when
+\(u|_S>0\), because the regular base coordinate was \(\lambda\),
+not the original \(N\).
+On the smooth reflected collar in
+Equation~\eqref{q3:static:even-base}, normal projection onto its fixed
+boundary is invariant under \(\lambda\mapsto-\lambda\).
+This follows from uniqueness of the normal geodesics and reflection
+invariance of the metric.
+Schwarzschild angular coordinates are constant along these normal
+geodesics, and their boundary values are the smooth round-sphere
+identification from Lemma~\ref{q3:static:schwarzschild-base}.
+They are therefore smooth even functions of \(\lambda\).
+A smooth even function, with smooth tangential parameters, is a
+smooth function of \(N=\lambda^2\) for \(N\geq0\):
+all odd Taylor coefficients vanish, and repeated application of
+\((2\lambda)^{-1}\partial_\lambda\) to the Taylor remainder gives
+continuous derivatives of every order at zero.
+This proves the required angular smoothness in the original collar.
+In the zero boundary-lapse case the original collar is already a
+smooth base collar, so its completed base isometry supplies the
+angular extension directly.
+
+We have constructed a smooth map \(\iota:\Mext\to\) Schwarzschild.
+Equation~\eqref{q3:static:reconstructed-metric} extends to \(S\)
+by continuity, hence \(\iota^*\mathbf g_M=g\) there as well.
+The positive definiteness of \(g\) proves that \(\dd\iota\) is
+injective at every boundary point.
+Interior injectivity follows from the global base isometry;
+boundary injectivity follows from the sphere identification.
+The boundary image, at \(r=2M\), is disjoint from the open image,
+where \(r>2M\).
+The map is proper: a compact subset of Schwarzschild has bounded
+area radius, and its inverse image is closed in the compact base
+region corresponding to
+\([2M,R]\times S^2\).
+Thus the injective immersion is an embedding.
+
+The future normal extends smoothly even where \(u=0\).
+To avoid taking a limit of the expression involving \(u^{-1}\),
+choose the smooth future timelike Kruskal vector
+\(Z=\partial_{\mathsf U}+\partial_{\mathsf V}\) near the boundary.
+Let \(Z^\top\) be its tangential projection along the embedded
+spacelike hypersurface and set
+\[
+\widehat n=
+\frac{Z-Z^\top}
+{\sqrt{-\mathbf g_M(Z-Z^\top,Z-Z^\top)}}.
+\]
+The denominator is positive, and this is a smooth future unit
+normal. On the open exterior it agrees with the normal used in
+Equation~\eqref{q3:static:reconstructed-k}.
+Its second fundamental form is smooth and equals \(K\) in the
+interior, so equality extends to \(S\).
+
+There is also a smooth spacelike extension of the image through
+its boundary. In the positive boundary-lapse case, use
+\((\mathsf U,\omega)\) as hypersurface coordinates; its normal
+derivative is nonzero by
+Equation~\eqref{q3:static:future-attachment}.
+The remaining null coordinate is a smooth graph function and can
+be extended across \(\mathsf U=0\).
+In the zero boundary-lapse case use
+\((\chi,\omega)\), where
+\(\chi=(\mathsf V-\mathsf U)/2\); the displayed normal derivatives
+give \(\partial_s\chi>0\).
+Again extend the remaining graph function across zero.
+Smooth one-sided graph functions extend across a smooth boundary,
+and compactness of \(S\), together with openness of the spacelike
+condition, preserves spacelikeness on a sufficiently short collar.
+This extension makes no assertion about the original data behind
+\(S\).
+
+Finally, the exact inverse-metric identity gives
+\begin{equation}
+\label{q3:static:end-slope}
+|\dd H|_h^2
+=\left|\frac{X^\flat}{N}\right|_C^2
+=\frac{|X|_g^2}{u^2N}
+\longrightarrow\frac{|b|^2}{(b^0)^2}<1.
+\end{equation}
+Choose a constant \(a_0<1\) larger than the limiting slope norm.
+Along radial rays in the Schwarzschild base, uniformly in angle,
+\[
+|H(r,\omega)|
+\leq C+a_0\int_R^r
+\frac{\dd\rho}{\sqrt{1-2M/\rho}}
+=a_0r+O(\log r).
+\]
+Since \(r_*=r+O(\log r)\), it follows that
+\[
+H-r_*\longrightarrow-\infty,\qquad
+H+r_*\longrightarrow+\infty.
+\]
+This is approach to the chosen Schwarzschild spatial infinity.
+It allows a nonzero asymptotic tilt and does not impose \(P=0\)
+or \(K=0\).
+All conclusions of Proposition~\ref{q3:static:classification} follow.
+\end{proof}
+
+Propositions~\ref{q3:variation:adjoint} and~\ref{q3:static:classification}
+prove Theorem~\ref{q3:intro:rigidity}. They also prove
+Theorem~\ref{q3:intro:connected-rigidity}: the connected-case argument
+in Lemma~\ref{q3:variation:area-support} supplies the same localized
+multiplier identity, and all subsequent adjoint, static, and
+reconstruction steps use that identity rather than
+partial-coincidence outermostness.

preprints/Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026/paper.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:4f1aab59ba484eb8a903daed582b30ffe17bbd42392e233213a30abb3aadc710
+size 1274586

preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/README.md

@@ -0,0 +1,18 @@
+# [Equality cases in the sharp one-dimensional matrix Lieb–Thirring inequality](sharp-one-dimensional-lieb-thirring-inequalities-matrix-potentials.pdf)
+
+**Author:** OpenAI
+
+**Date:** October 5, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026,
+  author = {{OpenAI}},
+  title = {{Equality cases in the sharp one-dimensional matrix Lieb--Thirring inequality}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/sharp-one-dimensional-lieb-thirring-inequalities-matrix-potentials.pdf}{OAI:Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026}},
+  year = {2026}
+}
+```
+

preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/main.tex

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+\documentclass[11pt]{article}
+\input{preamble}
+\title{Equality cases in the sharp one-dimensional\newline matrix Lieb--Thirring inequality}
+\author{OpenAI}
+\date{October 5, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We classify all equality cases in the sharp one-dimensional Lieb--Thirring
+inequality for every finite matrix size and $1/2<\gamma<3/2$. For measurable
+Hermitian positive semidefinite potentials $W$ with
+$\int_\R\tr(W^{\gamma+1/2})<\infty$, equality holds precisely for direct sums,
+in one constant unitary basis, of scalar one-bound-state solitons and zero
+channels. The nonzero solitons may have independent scales and centers.
+\end{abstract}
+\setcounter{tocdepth}{1}
+\tableofcontents
+\clearpage
+\input{sections/introduction}
+\input{sections/preliminaries}
+\input{sections/action}
+\input{sections/spectral-bound}
+\input{sections/compactness}
+\input{sections/rigidity}
+\appendix
+\input{sections/appendix-trace}
+\input{sections/appendix-paths}
+\input{references}
+\end{document}

preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/preamble.tex

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+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage[margin=1.05in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs}
+\usepackage{graphicx,microtype,xcolor,tikz}
+\usetikzlibrary{arrows.meta,positioning,fit,calc}
+\usepackage{hyperref}
+\hypersetup{colorlinks=true,linkcolor=blue!45!black,citecolor=green!35!black,urlcolor=blue!55!black,pdftitle={Equality cases in the sharp one-dimensional matrix Lieb--Thirring inequality},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}
+\newcommand{\R}{\mathbb R}
+\newcommand{\C}{\mathbb C}
+\newcommand{\HS}{\mathrm{HS}}
+\DeclareMathOperator{\tr}{tr}
+\DeclareMathOperator{\Tr}{Tr}
+\DeclareMathOperator{\Sym}{Sym}
+\DeclareMathOperator{\sech}{sech}
+\DeclareMathOperator{\diag}{diag}
+\DeclareMathOperator{\op}{op}
+\numberwithin{equation}{section}
+\allowdisplaybreaks[2]
+\setlength{\emergencystretch}{1.5em}

preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/references.tex

@@ -0,0 +1,90 @@
+\begin{thebibliography}{99}
+\raggedright
+\bibitem{AizenmanLieb1978}
+Michael Aizenman and Elliott H. Lieb,
+\emph{On semi-classical bounds for eigenvalues of Schr\"odinger operators},
+Physics Letters A \textbf{66} (1978), no.~6, 427--429.
+\href{https://doi.org/10.1016/0375-9601(78)90385-7}{doi:10.1016/0375-9601(78)90385-7}.
+
+\bibitem{BenguriaLoss2000}
+Rafael Benguria and Michael Loss,
+\emph{A simple proof of a theorem of Laptev and Weidl},
+Mathematical Research Letters \textbf{7} (2000), no.~2--3, 195--203.
+\href{https://doi.org/10.4310/MRL.2000.v7.n2.a5}{doi:10.4310/MRL.2000.v7.n2.a5}.
+
+\bibitem{FrankGontierLewin2025}
+Rupert L. Frank, David Gontier and Mathieu Lewin,
+\emph{Optimizers for the finite-rank Lieb--Thirring inequality},
+American Journal of Mathematics \textbf{147} (2025), no.~2, 503--560.
+\href{https://doi.org/10.1353/ajm.2025.a954649}{doi:10.1353/ajm.2025.a954649}.
+
+\bibitem{HLW2000}
+Dirk Hundertmark, Ari Laptev and Timo Weidl,
+\emph{New bounds on the Lieb--Thirring constants},
+Inventiones Mathematicae \textbf{140} (2000), no.~3, 693--704.
+\href{https://doi.org/10.1007/s002220000077}{doi:10.1007/s002220000077}.
+
+\bibitem{HLT1998}
+Dirk Hundertmark, Elliott H. Lieb and Lawrence E. Thomas,
+\emph{A sharp bound for an eigenvalue moment of the one-dimensional
+Schr\"odinger operator},
+Advances in Theoretical and Mathematical Physics \textbf{2} (1998), no.~4,
+719--731.
+\href{https://doi.org/10.4310/ATMP.1998.v2.n4.a2}{doi:10.4310/ATMP.1998.v2.n4.a2}.
+
+\bibitem{Keller1961}
+Joseph B. Keller,
+\emph{Lower bounds and isoperimetric inequalities for eigenvalues of the
+Schr\"odinger equation},
+Journal of Mathematical Physics \textbf{2} (1961), no.~2, 262--266.
+\href{https://doi.org/10.1063/1.1703708}{doi:10.1063/1.1703708}.
+
+\bibitem{LaptevWeidl2000}
+Ari Laptev and Timo Weidl,
+\emph{Sharp Lieb--Thirring inequalities in high dimensions},
+Acta Mathematica \textbf{184} (2000), no.~1, 87--111.
+\href{https://doi.org/10.1007/BF02392782}{doi:10.1007/BF02392782}.
+
+\bibitem{LiebThirring1975}
+Elliott H. Lieb and Walter E. Thirring,
+\emph{Bound for the kinetic energy of fermions which proves the stability
+of matter},
+Physical Review Letters \textbf{35} (1975), no.~11, 687--689;
+erratum, \textbf{35} (1975), 1116.
+\href{https://doi.org/10.1103/PhysRevLett.35.687}{doi:10.1103/PhysRevLett.35.687}.
+
+\bibitem{LiebThirring1976}
+Elliott H. Lieb and Walter E. Thirring,
+\emph{Inequalities for the moments of the eigenvalues of the Schr\"odinger
+Hamiltonian and their relation to Sobolev inequalities},
+in \emph{Studies in Mathematical Physics: Essays in Honor of Valentine
+Bargmann}, Elliott H. Lieb, Barry Simon and Arthur S. Wightman (eds.),
+Princeton University Press, Princeton, NJ, 1976, 269--303.
+
+\bibitem{Milnor1965}
+John W. Milnor,
+\emph{Topology from the Differentiable Viewpoint},
+University Press of Virginia, Charlottesville, VA, 1965.
+
+\bibitem{MLT}
+OpenAI,
+\emph{Sharp one-dimensional Lieb--Thirring inequalities for matrix potentials},
+OpenAI Math Release preprint
+\href{https://github.com/openai/math/blob/main/preprints/Sharp-one-dimensional-Lieb-Thirring-inequalities-for-matrix-potentials-October-5-2026/sharp-matrix-lieb-thirring.pdf}{OAI:Sharp-one-dimensional-Lieb-Thirring-inequalities-for-matrix-potentials-October-5-2026},
+2026.
+
+\bibitem{ReadSchulz2026}
+Larry Read and Marvin R. Schulz,
+\emph{The sharp one-dimensional Lieb--Thirring inequality for the sum
+of eigenvalues},
+\href{https://arxiv.org/abs/2609.10478v2}{arXiv:2609.10478v2},
+September 14, 2026.
+
+\bibitem{Sard1942}
+Arthur Sard,
+\emph{The measure of the critical values of differentiable maps},
+Bulletin of the American Mathematical Society \textbf{48} (1942),
+883--890.
+\href{https://doi.org/10.1090/S0002-9904-1942-07811-6}{doi:10.1090/S0002-9904-1942-07811-6}.
+
+\end{thebibliography}

preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/sections/action.tex

@@ -0,0 +1,328 @@
+\section{An action estimate that retains the equality defects}
+\label{act:section}
+
+The finite-interval argument for the sharp inequality can also measure
+failure of its equality relations.  We retain the first-order square
+and a quantitative part of the matrix trace deficit.  The coefficient
+of this second defect will be independent of the number of rows.  This
+uniformity allows us to exhaust an eigenvalue list before knowing that
+it is finite.
+
+The construction adapts the field, connecting paths, and action
+identity of \cite[Sections~3--6]{MLT}.  Their analytic foundations are
+proved in Appendices~\ref{app:trace} and~\ref{app:paths}; the additional
+estimate below extracts operator-norm rigidity from the trace deficit.
+
+Fix positive integers $n,d$, a bounded interval
+$\Omega=(x_-,x_+)$, and
+$K=\operatorname{diag}(\ell_1,\ldots,\ell_n)$ with
+$0<\ell_i\leq\kappa$.  For
+$A\in H^1_0(\Omega;\R^{n\times d})$, set $\rho=AA^{\mathsf T}$
+and define
+\begin{equation}\label{act:energy}
+ \mathcal E_K(A)=\int_\Omega
+ \bigl(\|A'\|_{\HS}^2+\tr(K^2\rho)-\tr(\rho^q)\bigr)\,dx.
+\end{equation}
+For pointwise matrix values $A\in\R^{n\times d}$ and
+$B\in\Sym_n$, put
+\begin{equation}\label{act:defect}
+ \Phi_K(A,B)=\min\left\{1,
+       \|[B,K^2]\|_{\op}^2
+       +\|K^2-B^2-(AA^{\mathsf T})^r\|_{\op}^2\right\}.
+\end{equation}
+Thus $\Phi_K=0$ expresses two algebraic relations, while
+$A'-BA=0$ is a first-order differential relation.
+
+\begin{theorem}[Quantitative finite-interval action estimate]
+\label{act:quantitative-action}
+Let $0<\sigma<1$ and $\kappa>0$.  There is a constant
+$c_*=c_*(\sigma,\kappa)>0$ with the following property.
+For $n,d,\Omega,K$ as above, suppose
+$U\in H^1_0(\Omega;\R^{n\times d})$ satisfies
+\[
+                  \int_\Omega UU^{\mathsf T}\,dx=I_n.
+\]
+For every $\eta>0$ there exist a real lower triangular matrix
+$C\in\R^{n\times n}$ and a path
+$B\in C^1(\overline\Omega;\Sym_n)$ such that
+\[
+ B(x_-)=K,\qquad B(x_+)=-K,\qquad
+ \sup_{x\in\overline\Omega}\|B(x)\|_{\op}\leq\kappa,
+\]
+and, for $A=CU$,
+\begin{equation}\label{act:bound}
+ \mathcal E_K(A)\geq b_\sigma\sum_{i=1}^n\ell_i^{2\gamma}
+   +\int_\Omega\left(\|A'-BA\|_{\HS}^2
+                        +c_*\Phi_K(A,B)\right)\,dx-\eta.
+\end{equation}
+In particular, $c_*$ is independent of $n,d,\Omega,U$, and the
+smallest diagonal entry of $K$.
+\end{theorem}
+
+\subsection{Coercivity at the matrix-field constraint}
+
+The algebraic relations $\Phi_K(A,B)=0$ require $[B,K^2]=0$
+and $\rho=(K^2-B^2)^\sigma$.  For $0<\delta\leq1$, we use a
+locally Lipschitz field $M_\delta:\Sym_n\to\Sym_n$ extending
+the commuting rule
+\[
+ M_\delta(B)=(K^2-B^2+\delta I)^\sigma-\delta^\sigma I
+ \quad\text{if }[B,K^2]=0\text{ and }K^2-B^2\geq0.
+\]
+Thus the constraint $M_\delta(B)=\rho$ is a regularized form of
+the algebraic equality relation in this case.  The extension to
+arbitrary symmetric $B$ has a $C^1$ primitive
+$J_\delta:\Sym_n\to\R$ satisfying
+\begin{equation}\label{act:primitive}
+ dJ_\delta(B)[H]=-\tr(M_\delta(B)H).
+\end{equation}
+Proposition~\ref{field:properties} constructs these maps and proves
+their required properties.  Along a path $B$, the derivative identity
+converts $-\tr(M_\delta(B)B')$ into the total derivative of
+$J_\delta(B)$, which will supply the boundary term in the action
+identity.
+Its endpoint difference is explicit:
+\begin{align}
+ J_\delta(-K)-J_\delta(K)
+ &=\sum_{i=1}^n\int_{-\ell_i}^{\ell_i}
+     \bigl((\ell_i^2-s^2+\delta)^\sigma-\delta^\sigma\bigr)\,ds
+       \notag\\
+ &\longrightarrow b_\sigma\sum_{i=1}^n\ell_i^{2\gamma}
+       \qquad(\delta\downarrow0).
+       \label{act:endpoints}
+\end{align}
+The limit follows by dominated convergence and
+$2\sigma+1=2\gamma$.
+
+Define the scalar function
+\begin{equation}\label{act:F}
+ F(A,B)=\tr(\rho^q)-\tr\bigl((K^2-B^2)\rho\bigr),
+ \qquad \rho=AA^{\mathsf T}.
+\end{equation}
+The following lemma bounds this nonlinear term while keeping the
+algebraic defect.  The constraint in its hypothesis will later be
+enforced by a penalty.
+
+\begin{lemma}\label{act:constraint}
+The constant $c_*(\sigma,\kappa)>0$ can be chosen so that, whenever
+$0<\delta\leq1$, $\|B\|_{\op}\leq\kappa$, and
+$M_\delta(B)=AA^{\mathsf T}$, one has
+\begin{equation}\label{act:constraint-bound}
+ F(A,B)+c_*\Phi_K(A,B)\leq E_{n,\delta},
+ \qquad
+ E_{n,\delta}
+ =n\bigl(\delta\kappa^{2\sigma}
+           +\delta^\sigma(\kappa^2+\delta)\bigr)
+      +\delta^{2\sigma}.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Write $\rho=M_\delta(B)\geq0$.  Proposition~\ref{field:positive}
+gives
+\begin{equation}\label{act:positive-symbol}
+ Q_s:=s^2I-2sB+K^2\geq0\quad(s\in\R),
+ \qquad 0\leq\rho\leq\kappa^{2\sigma}I.
+\end{equation}
+Put $V=K^2-B^2+\delta I$ and $\beta=\sigma-\tfrac12$.
+With the positive normalization constant $c_\sigma$ specified in
+\eqref{trace:normalization}, define
+\begin{equation}\label{act:RN}
+ \begin{split}
+ R(t)&=\frac1\pi\int_\R
+           \bigl(Q_s+(\delta+t)I\bigr)^{-1}\,ds,\qquad t>0,\\
+ N&=c_\sigma\int_0^\infty
+           t^\beta\bigl(t^{-1/2}I-R(t)\bigr)\,dt
+       =\rho+\delta^\sigma I.
+ \end{split}
+\end{equation}
+The last identity is also part of Proposition~\ref{field:positive}.
+Since $(sI-B)^2+V=Q_s+\delta I\geq\delta I$ and $N\geq0$,
+Theorem~\ref{trace:deficit} applies.  It supplies
+$L(t)=R(t)^{-1}>0$ and Hermitian matrices $Z(t)$ satisfying
+\begin{equation}\label{act:LZ}
+ V+tI=L^2+Z^2+i[B,Z],\qquad
+ LZ+ZL+i[B,L]=0,
+\end{equation}
+and the exact identity
+\begin{align}
+ \Delta&:=\tr(NV)-\tr(N^q)\notag\\
+ &=c_\sigma\int_0^\infty t^\beta
+    \left(\tr(R(t)Z(t)^2)
+      +q\bigl\|R(t)^{-1/2}
+           -R(t)^{1/2}(tI+N^r)^{1/2}\bigr\|_{\HS}^2\right)\,dt
+       \geq0.
+       \label{act:trace-deficit}
+\end{align}
+The matrices in \eqref{act:LZ} may be complex Hermitian even though
+$A$ and $B$ are real; $i$ denotes the imaginary unit.
+
+We extract a dimension-independent bound from the part of the
+integral with $1\leq t\leq2$.  On this interval there are constants
+$a_\kappa,A_\kappa>0$ such that
+\begin{equation}\label{act:R-bounds}
+                       a_\kappa I\leq R(t)\leq A_\kappa I.
+\end{equation}
+For the upper bound, use $Q_s\geq0$ on a bounded interval in $s$,
+and $Q_s\geq\tfrac12s^2I$ for $|s|\geq\max\{1,4\kappa\}$.
+For the lower bound, restrict the integral to $|s|\leq1$, where
+$Q_s+(\delta+t)I\leq(4+2\kappa+\kappa^2)I$.
+These estimates depend on $\kappa$ alone.  Moreover
+$\|N\|_{\op}\leq\kappa^{2\sigma}+1$, so
+$H_t=(tI+N^r)^{1/2}$ and $L(t)$ are bounded in operator norm by
+constants depending only on $\sigma,\kappa$.
+
+The two integrands in \eqref{act:trace-deficit} control
+$\|Z(t)\|_{\HS}^2$ and $\|L(t)-H_t\|_{\HS}^2$:
+indeed $\tr(RZ^2)=\tr(ZRZ)$ and
+\[
+ R^{-1/2}-R^{1/2}H_t=R^{1/2}(L-H_t).
+\]
+It follows by averaging over $[1,2]$ that at some $t$ in this
+interval,
+\begin{equation}\label{act:small-squares}
+ \|Z(t)\|_{\op}+\|L(t)-H_t\|_{\op}
+        \leq C_1\sqrt\Delta,
+\end{equation}
+where $C_1$ depends only on $\sigma,\kappa$.
+When $\Delta=0$, the same conclusion follows by taking a point
+where both nonnegative integrands vanish.
+
+Suppose first that $\Delta\leq1$, and use this value of $t$.
+The first identity in \eqref{act:LZ} gives
+\[
+ V-N^r=(L^2-H_t^2)+Z^2+i[B,Z].
+\]
+The boundedness of $L,H_t,B$, together with
+\eqref{act:small-squares}, bounds the norm of this expression by
+$C_2\sqrt\Delta$.  To control the commutator, the second identity
+in \eqref{act:LZ} first yields
+$\|[B,L]\|_{\op}\leq2\|L\|_{\op}\|Z\|_{\op}$.
+Then
+\[
+ [B,V]=[B,L^2]+[B,Z^2]+i[B,[B,Z]]
+\]
+has the same bound.  Thus
+\begin{equation}\label{act:V-coercivity}
+ \|V-N^r\|_{\op}+\|[B,V]\|_{\op}
+       \leq C_3\sqrt\Delta.
+\end{equation}
+Here and below the constants in this proof depend only on
+$\sigma,\kappa$.
+
+The matrices $N=\rho+\delta^\sigma I$ and $\rho$ commute.
+Because $r>1$, the scalar function $y\mapsto y^r$ is Lipschitz on
+$[0,\kappa^{2\sigma}+1]$.  Consequently
+\[
+ \|N^r-\rho^r\|_{\op}\leq C_4\delta^\sigma.
+\]
+Since $[B,V]=[B,K^2]$ and $\delta\leq\delta^\sigma$,
+\eqref{act:V-coercivity} implies
+\[
+ \|[B,K^2]\|_{\op}^2
+   +\|K^2-B^2-\rho^r\|_{\op}^2
+       \leq C_5(\Delta+\delta^{2\sigma}).
+\]
+Choose $c_*>0$ sufficiently small, also with $c_*\leq1$.
+For $\Delta\leq1$ the preceding estimate gives
+\begin{equation}\label{act:coercivity}
+                   c_*\Phi_K(A,B)\leq\Delta+\delta^{2\sigma}.
+\end{equation}
+For $\Delta>1$ this follows directly from $\Phi_K\leq1$.
+This proves the required uniform control of the algebraic defect.
+
+It remains to relate $\Delta$ to $F$.  Expanding $N$ and $V$ gives
+\[
+ F(A,B)=-\Delta+\tr(\rho^q)-\tr(N^q)
+                  +\delta\tr\rho+\delta^\sigma\tr V.
+\]
+The first two powers are ordered because $N=\rho+\delta^\sigma I$.
+Also $\tr\rho\leq n\kappa^{2\sigma}$ and
+$V\leq(\kappa^2+\delta)I$.  Adding \eqref{act:coercivity} now
+proves \eqref{act:constraint-bound}.
+\end{proof}
+
+\subsection{The action identity and the penalty limit}
+
+\begin{proof}[Proof of Theorem~\ref{act:quantitative-action}]
+Fix $0<\delta\leq1$.  Proposition~\ref{path:connecting} supplies,
+for every $0<\varepsilon\leq1$, a lower triangular matrix
+$C_\varepsilon$ and a path $B_\varepsilon$ with the prescribed
+endpoints and operator-norm bound such that, for
+$A_\varepsilon=C_\varepsilon U$,
+\begin{equation}\label{act:path}
+ \varepsilon B_\varepsilon'
+       =M_\delta(B_\varepsilon)
+                    -A_\varepsilon A_\varepsilon^{\mathsf T}.
+\end{equation}
+For fixed $\delta,K,\Omega,U$, the matrices $C_\varepsilon$ are
+bounded uniformly in $0<\varepsilon\leq1$.
+
+Temporarily suppress the subscript $\varepsilon$ and write
+\[
+                 u_\delta(A,B)
+                      =\|M_\delta(B)-AA^{\mathsf T}\|_{\HS}^2.
+\]
+Since $A\in H^1_0$ and $B\in C^1$, the product and chain rules
+for absolutely continuous functions, \eqref{act:primitive}, and
+\eqref{act:path} yield almost everywhere
+\begin{align*}
+ \bigl(J_\delta(B)+\tr(A^{\mathsf T}BA)\bigr)'
+ &=2\tr((A')^{\mathsf T}BA)
+          -\tr\bigl((M_\delta(B)-AA^{\mathsf T})B'\bigr)\\
+ &=2\tr((A')^{\mathsf T}BA)-\varepsilon^{-1}u_\delta(A,B).
+\end{align*}
+Complete the square in the action and integrate this identity.
+Because $A$ vanishes at both endpoints, the result is the exact
+formula
+\begin{align}
+ \mathcal E_K(A)
+ &=J_\delta(-K)-J_\delta(K)\notag\\
+ &\quad+\int_\Omega\left(
+       \|A'-BA\|_{\HS}^2-F(A,B)
+                          +\varepsilon^{-1}u_\delta(A,B)\right)\,dx.
+       \label{act:identity}
+\end{align}
+
+We now use the penalty to pass from the constraint estimate in
+Lemma~\ref{act:constraint} to all matrix values on these paths.
+The continuous representative of $U$ is bounded on
+$\overline\Omega$.  The uniform bound on $C_\varepsilon$ and the
+bound $\|B_\varepsilon\|_{\op}\leq\kappa$ therefore place all pairs
+$(A_\varepsilon(x),B_\varepsilon(x))$ in one compact set
+$\mathcal K_\delta\subset\R^{n\times d}\times\Sym_n$ on which
+$\|B\|_{\op}\leq\kappa$.  On this set $F$, $\Phi_K$, and
+$u_\delta$ are continuous.  Lemma~\ref{act:constraint} implies
+\begin{equation}\label{act:penalty}
+ \limsup_{\varepsilon\downarrow0}
+ \sup_{(A,B)\in\mathcal K_\delta}
+    \bigl(F(A,B)+c_*\Phi_K(A,B)-\varepsilon^{-1}u_\delta(A,B)\bigr)
+       \leq E_{n,\delta}.
+\end{equation}
+Indeed, if the left side exceeded $E_{n,\delta}$, some sequence
+$\varepsilon_j\downarrow0$ and corresponding pairs in the compact
+set would make the expression at least $E_{n,\delta}+\tau$ for
+some $\tau>0$.  Boundedness of $F+c_*\Phi_K$ would force
+$u_\delta=O(\varepsilon_j)$ along that sequence.  A convergent
+subsequence would then give a limiting pair with $u_\delta=0$ and
+$F+c_*\Phi_K\geq E_{n,\delta}+\tau$, contradicting
+Lemma~\ref{act:constraint}.
+
+For any $\tau>0$, choose $\varepsilon$ small enough that the
+supremum in \eqref{act:penalty} is at most $E_{n,\delta}+\tau$.
+Substitution into \eqref{act:identity} gives
+\begin{align*}
+ \mathcal E_K(A_\varepsilon)
+ \geq{}&J_\delta(-K)-J_\delta(K)
+        -|\Omega|(E_{n,\delta}+\tau)\\
+ &+\int_\Omega\left(\|A_\varepsilon'-B_\varepsilon A_\varepsilon\|_{\HS}^2
+                +c_*\Phi_K(A_\varepsilon,B_\varepsilon)\right)\,dx.
+\end{align*}
+Finally, first choose $\delta>0$ so small that the endpoint error
+in \eqref{act:endpoints} and $|\Omega|E_{n,\delta}$ together are
+less than $\eta/2$.  Next choose $\tau>0$ with
+$|\Omega|\tau<\eta/2$, and then choose the required $\varepsilon$.
+These choices prove \eqref{act:bound}.  All compactness arguments
+were made after fixing $\delta$; no bound uniform in $\delta$ is
+needed.
+\end{proof}

preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/sections/appendix-paths.tex

@@ -0,0 +1,465 @@
+\section{The regularized field and connecting paths}
+\label{app:paths}
+
+The finite-interval action argument requires a symmetric matrix path
+from $K$ to $-K$ whose derivative measures the difference between a
+matrix field and $AA^{\mathsf T}$.  We construct both the field and the
+path here.  These are the constructions of \cite[Sections~4--5]{MLT};
+we include their proofs to make the analytic input to the equality
+argument self-contained.  The positivity statement below also
+identifies the resolvent to which Theorem~\ref{trace:deficit} applies.
+
+Throughout this appendix,
+$K=\operatorname{diag}(k_1,\ldots,k_n)$ with $k_i>0$,
+$\kappa=\max_i k_i$, and $\Sym_n$ denotes the real symmetric matrices.
+We use $\beta=\sigma-\tfrac12$ and the constant $c_\sigma$ from
+\eqref{trace:normalization}.
+
+\subsection{A field with a scalar boundary integral}
+
+For $\delta>0$, define on $y\geq0$
+\begin{equation}\label{field:f}
+ f_\delta(y)=\frac{c_\sigma}{\pi}\int_0^\infty t^\beta
+       \left(\frac1{t+\delta}-\frac1{t+\delta+y}\right)\,dt,
+\end{equation}
+and extend to $y<0$ by the tangent line $f_\delta'(0)y$.
+The resulting function is $C^1$, strictly increasing, and concave.
+Indeed, for $y\geq0$,
+\begin{equation}\label{field:f-derivative}
+ f_\delta'(y)=a_\sigma(y+\delta)^{\beta-1},
+ \qquad
+ a_\sigma=\frac{c_\sigma}{\pi}
+       \int_0^\infty\frac{u^\beta}{(1+u)^2}\,du>0.
+\end{equation}
+Both integrals converge since $-\tfrac12<\beta<\tfrac12$.
+The extension allows the functional calculus below even when its
+matrix argument has negative eigenvalues.
+
+Set
+\begin{equation}\label{field:definition}
+ \begin{split}
+ Q_s(B)&=s^2I-2sB+K^2,\\
+ M_\delta(B)&=\lim_{R\to\infty}\int_{-R}^{R}
+       \bigl(f_\delta(Q_s(B))-f_\delta(s^2)I\bigr)\,ds,
+       \qquad B\in\Sym_n,
+ \end{split}
+\end{equation}
+where the symmetric cutoff cancels the leading term
+$-2s f_\delta'(s^2)B$, which is odd in $s$.
+The scalar counterpart is
+\begin{equation}\label{field:mu}
+ \mu_\delta(z)=\int_\R
+     \bigl(f_\delta(s^2+z)-f_\delta(s^2)\bigr)\,ds,
+       \qquad z\in\R.
+\end{equation}
+
+\begin{proposition}[Regularized field]\label{field:properties}
+The limit in \eqref{field:definition} exists locally uniformly, and
+$M_\delta:\Sym_n\to\Sym_n$ is locally Lipschitz.  There is a
+$C^1$ function $J_\delta:\Sym_n\to\R$, normalized by
+$J_\delta(0)=0$, such that
+\begin{equation}\label{field:primitive}
+ dJ_\delta(B)[H]=-\tr(M_\delta(B)H).
+\end{equation}
+The scalar function $\mu_\delta$ is locally Lipschitz and strictly
+increasing, with $\mu_\delta(0)=0$, and
+\begin{equation}\label{field:mu-positive}
+       \mu_\delta(z)=(z+\delta)^\sigma-\delta^\sigma
+       \qquad(z\geq0).
+\end{equation}
+For every real unit vector $e$, and for every standard basis vector
+$e_i$, respectively,
+\begin{align}
+ e^{\mathsf T}M_\delta(B)e
+  &\leq\mu_\delta\bigl(e^{\mathsf T}K^2e
+                         -(e^{\mathsf T}Be)^2\bigr),
+       \label{field:direction}\\
+ Be_i=ye_i\quad&\Longrightarrow\quad
+ M_\delta(B)e_i=\mu_\delta(k_i^2-y^2)e_i.
+       \label{field:coordinate}
+\end{align}
+For each diagonal sign matrix $S$,
+\begin{equation}\label{field:equivariance}
+                 M_\delta(SBS)=SM_\delta(B)S.
+\end{equation}
+Finally,
+\begin{equation}\label{field:endpoints}
+ \begin{split}
+ J_\delta(-K)-J_\delta(K)
+ &=\sum_{i=1}^n\int_{-k_i}^{k_i}\mu_\delta(k_i^2-s^2)\,ds\\
+ &\longrightarrow b_\sigma\sum_{i=1}^n k_i^{2\sigma+1}
+       \qquad(\delta\downarrow0).
+ \end{split}
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+We first justify the matrix integral, including its local Lipschitz
+dependence.  If a scalar function $h$ is Lipschitz on an interval
+containing the spectra of Hermitian matrices $X,Y$, then
+\begin{equation}\label{field:HS-calculus}
+ \|h(X)-h(Y)\|_{\HS}\leq\operatorname{Lip}(h)\|X-Y\|_{\HS}.
+\end{equation}
+To see this, use eigenbases $e_i$ of $X$ and $v_j$ of $Y$.
+The mixed matrix entries of the two differences are
+$(h(\lambda_i)-h(\nu_j))\langle e_i,v_j\rangle$ and
+$(\lambda_i-\nu_j)\langle e_i,v_j\rangle$; square the scalar
+Lipschitz inequality and sum over $i,j$.
+
+On a bounded set of $B$'s the spectrum of
+$E_s(B)=-2sB+K^2$ is contained in $[-a|s|,a|s|]$ for all sufficiently
+large $|s|$.  For
+\[
+ h_s(y)=f_\delta(s^2+y)-f_\delta(s^2)-f_\delta'(s^2)y,
+\]
+Equation~\eqref{field:f-derivative} gives
+$\sup_{|y|\leq a|s|}|h_s'(y)|=O(|s|^{2\beta-3})$.
+It follows from \eqref{field:HS-calculus} that
+\[
+ f_\delta(Q_s(B))-f_\delta(s^2)I
+      =-2s f_\delta'(s^2)B+T_s(B),
+\]
+where both $\|T_s(B)\|_{\HS}$ and the local Lipschitz constant of
+$T_s$ are $O(|s|^{2\beta-2})$.  For the first bound, include
+$f_\delta'(s^2)K^2$ in $T_s$; for the second, apply
+\eqref{field:HS-calculus} to $h_s(E_s(B))$ and use
+$E_s(B_1)-E_s(B_2)=-2s(B_1-B_2)$.
+Because $2\beta-2<-1$, these remainders are integrable at infinity.
+The odd term cancels at each symmetric cutoff.  On bounded
+$s$-intervals, \eqref{field:HS-calculus} applies directly to
+$f_\delta$.  This proves the asserted convergence and regularity.
+
+For the primitive, choose a scalar antiderivative
+$g_\delta'=f_\delta$.  Trace differentiation gives
+$d\tr(g_\delta(X))[H]=\tr(f_\delta(X)H)$.
+This formula follows for polynomials by cyclicity and for
+$g_\delta$ by approximation in $C^1$ on compact spectral intervals.
+For $s\ne0$, the function
+\[
+ B\longmapsto
+ \frac{\tr(g_\delta(Q_s(B))-g_\delta(K^2))}{2s}
+                  +f_\delta(s^2)\tr B
+\]
+has differential
+$-\tr((f_\delta(Q_s(B))-f_\delta(s^2)I)H)$.
+At $s=0$ that differential is constant in $B$ and also has a
+primitive.  Thus the negative of every finite-cutoff field has
+zero integral around each closed piecewise smooth curve in
+$\Sym_n$.  Local uniform convergence passes this property to
+$-M_\delta$.  Path integration, starting at zero, defines
+$J_\delta$ and proves \eqref{field:primitive}.
+
+For bounded $z$, the integrand in \eqref{field:mu} and its
+Lipschitz constant in $z$ are $O(|s|^{2\beta-2})$ at infinity.
+This proves absolute convergence and local Lipschitz continuity
+of $\mu_\delta$.  Strict increase and $\mu_\delta(0)=0$ follow
+from the same properties of $f_\delta$.  For $z\geq0$,
+Tonelli's theorem applied to \eqref{field:f} gives
+\[
+ \mu_\delta(z)=c_\sigma\int_0^\infty t^\beta
+       \bigl((t+\delta)^{-1/2}-(t+\delta+z)^{-1/2}\bigr)\,dt.
+\]
+The normalization \eqref{trace:normalization}, after scaling,
+evaluates this as \eqref{field:mu-positive}.
+
+Translations in the variable $s$ remain legitimate with a
+symmetric cutoff.  More precisely, for fixed $a,z\in\R$,
+\begin{equation}\label{field:shift}
+ \lim_{R\to\infty}\int_{-R}^{R}
+   \bigl(f_\delta((s-a)^2+z)-f_\delta(s^2)\bigr)\,ds
+          =\mu_\delta(z).
+\end{equation}
+Indeed $F_z(s)=f_\delta(s^2+z)$ is even and
+$F_z'(s)=O(|s|^{2\beta-1})\to0$ at infinity.  Pairing the two
+short intervals produced by a translation bounds the difference
+between the shifted and unshifted cutoff integrals by
+\[
+ |a|^2\sup_{R-|a|\leq u\leq R+|a|}|F_z'(u)|,
+\]
+which tends to zero.
+
+Scalar concavity in an eigenbasis of $Q_s(B)$ now gives
+\[
+ e^{\mathsf T}f_\delta(Q_s(B))e
+     \leq f_\delta(e^{\mathsf T}Q_s(B)e).
+\]
+The scalar argument on the right is
+$(s-e^{\mathsf T}Be)^2+e^{\mathsf T}K^2e-(e^{\mathsf T}Be)^2$.
+Integration and \eqref{field:shift} prove \eqref{field:direction}.
+If $Be_i=ye_i$, functional calculus gives equality in the
+corresponding coordinate, proving \eqref{field:coordinate}.
+Since $SK^2S=K^2$, conjugation inside the defining integral proves
+\eqref{field:equivariance}.
+
+On diagonal matrices, \eqref{field:primitive} and
+\eqref{field:coordinate} read
+\[
+ dJ_\delta(\operatorname{diag}(b_1,\ldots,b_n))
+       =-\sum_i\mu_\delta(k_i^2-b_i^2)\,db_i.
+\]
+Integrating from $K$ to $-K$ gives the first equality in
+\eqref{field:endpoints}.  For $z\geq0$,
+$0\leq\mu_\delta(z)\leq z^\sigma$ and
+$\mu_\delta(z)\to z^\sigma$.  Dominated convergence and the
+substitution $s=k_i u$ give its stated limit.
+\end{proof}
+
+The field is defined on all symmetric matrices, but the action
+argument compares it with a positive matrix $AA^{\mathsf T}$.
+At such values its directional bound forces positivity of the
+entire quadratic symbol $Q_s(B)$.  This is the condition needed
+for the trace deficit formula.
+
+\begin{proposition}[Positive values of the field]\label{field:positive}
+If $M=M_\delta(B)\geq0$, then
+\begin{equation}\label{field:positive-bounds}
+ Q_s(B)\geq0\quad(s\in\R),\qquad
+             0\leq M\leq\kappa^{2\sigma}I.
+\end{equation}
+Set $V=K^2-B^2+\delta I$ and use $B,V$ in
+\eqref{trace:resolvents} and \eqref{trace:N}.  Then the symbol
+$(sI-B)^2+V=Q_s(B)+\delta I$ is bounded below by $\delta I$, and
+\begin{equation}\label{field:N}
+                         N=M+\delta^\sigma I.
+\end{equation}
+In particular, the hypotheses of Theorem~\ref{trace:deficit} hold.
+\end{proposition}
+
+\begin{proof}
+By \eqref{field:direction}, positivity of $M$ and strict increase
+of $\mu_\delta$ imply
+$e^{\mathsf T}K^2e\geq(e^{\mathsf T}Be)^2$ for every real unit
+vector $e$.  Completing the scalar square gives
+\[
+ e^{\mathsf T}Q_s(B)e=(s-e^{\mathsf T}Be)^2
+                +e^{\mathsf T}K^2e-(e^{\mathsf T}Be)^2\geq0.
+\]
+Thus $Q_s(B)\geq0$, also as a Hermitian matrix on $\C^n$.
+Moreover, \eqref{field:direction} gives
+$M\leq\mu_\delta(\kappa^2)I\leq\kappa^{2\sigma}I$, using
+$(a+b)^\sigma\leq a^\sigma+b^\sigma$.
+
+Write $Y_s=(sI-B)^2$.  Diagonalization of $B$ and
+\eqref{field:shift} allow replacement of the scalar subtraction
+$f_\delta(s^2)I$ in \eqref{field:definition} by $f_\delta(Y_s)$.
+Since both $Q_s(B)$ and $Y_s$ are positive, their difference under
+$f_\delta$ is, by \eqref{field:f},
+\[
+ \frac{c_\sigma}{\pi}\int_0^\infty t^\beta
+ \left((Y_s+(t+\delta)I)^{-1}
+          -(Q_s(B)+(t+\delta)I)^{-1}\right)\,dt.
+\]
+For fixed $B,\delta$, both matrices being inverted are bounded
+below by $c(1+t+s^2)I$.  Their difference is $K^2-B^2$, so the
+resolvent identity bounds the norm of the integrand by
+$C t^\beta(1+t+s^2)^{-2}$.  Integrating first in $s$ yields
+$C't^\beta(1+t)^{-3/2}$, which is integrable on $(0,\infty)$.
+Fubini's theorem is therefore applicable and gives
+\[
+ M=c_\sigma\int_0^\infty t^\beta
+        \bigl((t+\delta)^{-1/2}I-R(t)\bigr)\,dt.
+\]
+Subtracting this expression from \eqref{trace:N} and scaling
+\eqref{trace:normalization} gives \eqref{field:N}.
+\end{proof}
+
+\subsection{Choosing the lower triangular coefficient matrix}
+
+Let $\mathcal L_n$ be the real lower triangular $n\times n$
+matrices.  Their dimension equals that of $\Sym_n$.  This dimension
+match allows us to prescribe a symmetric terminal value by choosing
+$C\in\mathcal L_n$.  At a simple initial problem there is one choice
+up to signs of the rows.  Continuation preserves the parity of this
+number, and hence produces the required terminal value.
+
+\begin{proposition}[Connecting path]\label{path:connecting}
+Let $\Omega=(x_-,x_+)$ be bounded and let
+$U\in H^1_0(\Omega;\R^{n\times d})$, where $n,d\geq1$, satisfy
+\[
+                         \int_\Omega UU^{\mathsf T}\,dx=I_n.
+\]
+For every $\delta,\varepsilon>0$ there are
+$C_\varepsilon\in\mathcal L_n$ and
+$B_\varepsilon\in C^1(\overline\Omega;\Sym_n)$ such that, with
+$A_\varepsilon=C_\varepsilon U$,
+\begin{equation}\label{path:ode}
+ \varepsilon B_\varepsilon'
+       =M_\delta(B_\varepsilon)-A_\varepsilon A_\varepsilon^{\mathsf T},
+ \qquad B_\varepsilon(x_-)=K,\quad B_\varepsilon(x_+)=-K.
+\end{equation}
+They satisfy
+\begin{equation}\label{path:bounds}
+ \sup_{x\in\overline\Omega}\|B_\varepsilon(x)\|_{\op}\leq\kappa,
+ \qquad
+ \sup_{0<\varepsilon\leq1}\|C_\varepsilon\|_{\HS}<\infty,
+\end{equation}
+where the second bound is for fixed $\delta,K,\Omega$.
+\end{proposition}
+
+\begin{proof}
+Use the continuous representative of $U$.  Choose a smooth,
+nonnegative, compactly supported cutoff $\chi$ on $\Sym_n$, equal
+to one on $\{B:\|B\|_{\op}\leq\kappa\}$ and invariant under
+conjugation by diagonal sign matrices.  Averaging a cutoff over
+the finite sign group produces such a function.  The truncated
+field $\widehat M=\chi M_\delta$ is bounded and globally Lipschitz.
+
+Fix $\varepsilon>0$.  For $C\in\mathcal L_n$ and
+$\theta\in[0,1]$, the equation
+\begin{equation}\label{path:homotopy}
+ \varepsilon B'=\theta\widehat M(B)-(CU)(CU)^{\mathsf T},
+                         \qquad B(x_-)=K
+\end{equation}
+has a unique $C^1$ solution on $\overline\Omega$.  Define the
+continuous endpoint map
+\[
+                  \Phi(C,\theta)=B(x_+)+K\in\Sym_n.
+\]
+Let $\mathcal G$ be the group of diagonal sign matrices, acting
+by $C\mapsto SC$ on $\mathcal L_n$ and by conjugation on
+$\Sym_n$.  Equivariance of the field and uniqueness of the ODE give
+\begin{equation}\label{path:equivariance}
+                       \Phi(SC,\theta)=S\Phi(C,\theta)S.
+\end{equation}
+
+We record the three facts needed for continuation: compactness
+of the zero set, freeness of the sign action there, and one regular
+orbit of zeros at $\theta=0$.
+At a zero, integration of \eqref{path:homotopy} gives
+\begin{equation}\label{path:C-identity}
+       CC^{\mathsf T}=2\varepsilon K
+                       +\theta\int_\Omega\widehat M(B(x))\,dx.
+\end{equation}
+Consequently all zeros lie in a bounded set, and their joint set
+in $\mathcal L_n\times[0,1]$ is compact.  For
+$0<\varepsilon\leq1$, this calculation gives the uniform bound
+\begin{equation}\label{path:C-bound}
+ \|C\|_{\HS}^2\leq2\tr K
+       +n|\Omega|\sup_{B\in\Sym_n}\|\widehat M(B)\|_{\op}.
+\end{equation}
+
+No row of $C$ can vanish at a zero.  Otherwise the reflection
+changing only row $i$ would fix $C$; ODE uniqueness and
+\eqref{path:equivariance} would force $Be_i=y e_i$ throughout the
+interval.  Equation~\eqref{field:coordinate} would then give
+\[
+ \varepsilon y'=\theta\chi(B(x))\mu_\delta(k_i^2-y^2),
+                         \qquad y(x_-)=k_i.
+\]
+The coefficient $\chi(B(x))$ is continuous and the right side
+is locally Lipschitz in $y$.  Uniqueness forces $y\equiv k_i$,
+contradicting $y(x_+)=-k_i$.  Thus the action of $\mathcal G$
+on the zero set is free.
+
+At $\theta=0$ one has
+\begin{equation}\label{path:initial-map}
+                     \Phi(C,0)=2K-\varepsilon^{-1}CC^{\mathsf T}.
+\end{equation}
+Its lower triangular zeros are exactly
+$C=\operatorname{diag}(d_1,\ldots,d_n)$ with
+$d_i=\pm\sqrt{2\varepsilon k_i}$.  Indeed, the first row of
+$CC^{\mathsf T}=2\varepsilon K$ fixes $|d_1|$ and forces all
+other entries of the first column to vanish, and induction does
+the same for subsequent columns.  These $2^n$ zeros form one
+sign orbit.  At each, the derivative in a lower triangular
+direction $H$ has diagonal entries
+$-2\varepsilon^{-1}d_i H_{ii}$ and lower off-diagonal entries
+$-\varepsilon^{-1}d_j H_{ij}$ for $i>j$.  It is therefore an
+isomorphism $\mathcal L_n\to\Sym_n$.
+
+We next justify the continuation using only smooth approximation
+and the regular-value argument for parity
+\cite[Sections~2--4]{Milnor1965}.  Suppose there were no zero at
+$\theta=1$.  Compactness and freeness give a bounded invariant
+open set $O\subset\mathcal L_n$ containing the projection of
+every zero, whose closure lies where all rows are nonzero.
+There is a positive lower bound for $\|\Phi\|_{\HS}$ on
+\begin{equation}\label{path:gap-set}
+       (\partial O\times[0,1])\ \cup\ (\overline O\times\{1\}).
+\end{equation}
+Reparametrize $\theta$ by a smooth map equal to zero near zero
+and equal to one at one.  The resulting map $F$ agrees with
+\eqref{path:initial-map} on an initial collar and retains the gap
+on \eqref{path:gap-set}.
+
+Extend $F$ constantly beyond the two ends of the parameter
+interval, smooth by convolution, and splice to
+\eqref{path:initial-map} in a smaller initial collar.
+Averaging the approximation $\Psi_0$ in the form
+\[
+       \Psi(C,\theta)=\frac1{|\mathcal G|}
+                  \sum_{S\in\mathcal G}S\Psi_0(SC,\theta)S
+\]
+gives a smooth equivariant approximation $\Psi$ that still equals
+the initial map on that collar.  It can be made uniformly close
+enough on $\overline O\times[0,1]$ to preserve the positive gap.
+
+Freeness permits an equivariant perturbation with arbitrary target
+directions.  At each $C_0\in\overline O$, choose a ball whose
+distinct sign translates are disjoint, and a smooth bump $b$
+supported in this ball with $b(C_0)=1$.  For $H\in\Sym_n$ the map
+\[
+                        V_H(C)=\sum_{S\in\mathcal G}b(SC)SHS
+\]
+is equivariant and satisfies $V_H(C_0)=H$.
+Choose a basis of target matrices and then a finite collection of
+such neighborhoods covering $\overline O$.  This gives finitely
+many smooth equivariant maps $V_1,\ldots,V_N$ whose values span
+$\Sym_n$ at every point of $\overline O$.
+
+Let $h\geq0$ be smooth, vanish near zero, and be positive outside
+the collar where $\Psi$ is the initial map.  Choose it so that
+every zero of $h$ lies within that collar, and set
+\[
+       \Psi_\alpha(C,\theta)
+           =\Psi(C,\theta)+h(\theta)\sum_{j=1}^{N}\alpha_jV_j(C).
+\]
+For $\alpha$ in a sufficiently small open ball, the gap on
+\eqref{path:gap-set} persists.  At a zero of this family with
+$0<\theta<1$, its full differential in $(C,\theta,\alpha)$
+is onto: when $h(\theta)>0$ the parameter directions span the
+target, and when $h(\theta)=0$ the derivative in $C$ is the
+isomorphism computed for \eqref{path:initial-map}.
+Its interior zero set is consequently a smooth manifold.
+Sard's theorem \cite{Sard1942}, applied to projection of this
+manifold onto the parameter ball, gives arbitrarily small
+$\alpha$ for which zero is a regular value of
+$(C,\theta)\mapsto\Psi_\alpha(C,\theta)$.
+For completeness, if the full derivative is $(L_1,L_2)$ in
+the $(C,\theta)$ and $\alpha$ directions, regularity of that
+projection means that each $w$ admits $v$ with
+$L_1v+L_2w=0$.  Hence $\operatorname{ran}L_2\subset
+\operatorname{ran}L_1$, and surjectivity of $(L_1,L_2)$ implies
+surjectivity of $L_1$.
+
+For such an $\alpha$, the zero set in $O\times[0,1]$ is a
+compact smooth one-dimensional manifold with boundary.  The gap
+excludes boundary zeros except at $\theta=0$, where the initial
+collar gives exactly the $2^n$ regular zeros above.  The free
+sign action has a quotient that is again a compact
+one-dimensional manifold with boundary: a neighborhood disjoint
+from its nontrivial translates supplies the local interval or
+half-interval chart.  All $2^n$ boundary points form one orbit,
+so the quotient has exactly one boundary point.  This is
+impossible, since a compact one-dimensional manifold is a finite
+union of circles and closed intervals.  Therefore the original
+map $\Phi(\cdot,1)$ has a zero.
+
+It remains to remove the cutoff.  For the path at this zero and
+any fixed real unit vector $e$, put $y=e^{\mathsf T}Be$.
+Whenever $|y|>\kappa$, Equation~\eqref{field:direction} yields
+\[
+ e^{\mathsf T}M_\delta(B)e
+       \leq\mu_\delta(e^{\mathsf T}K^2e-y^2)<0.
+\]
+Since $\chi\geq0$ and $(CU)(CU)^{\mathsf T}\geq0$, the ODE
+implies $y'\leq0$ there.  Both endpoint values lie in
+$[-\kappa,\kappa]$.  An excursion above $\kappa$ cannot start
+from $\kappa$ while $y$ is nonincreasing; an excursion below
+$-\kappa$ cannot return to $-\kappa$ while $y$ is nonincreasing.
+The two endpoint conditions therefore rule out both excursions.
+This holds for every $e$, so $\|B(x)\|_{\op}\leq\kappa$.
+Consequently $\chi(B)=1$, giving \eqref{path:ode}, and
+\eqref{path:C-bound} supplies the remaining assertion in
+\eqref{path:bounds}.
+\end{proof}

preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/sections/appendix-trace.tex

@@ -0,0 +1,287 @@
+\section{The exact integrated trace deficit}\label{app:trace}
+
+The matrix comparison needed in the action argument has an exact
+remainder.  We give the proof from \cite[Section~3]{MLT}, retaining
+both nonnegative terms.  Together they control the commutator and
+the algebraic equality relation used in the action argument.
+Throughout this appendix, $0<\sigma<1$, $\beta=\sigma-\tfrac12$, and
+$r=1/\sigma$, $p=1+\sigma$, $q=1+r$.
+Define $c_\sigma>0$ by
+\begin{equation}\label{trace:normalization}
+ c_\sigma^{-1}
+ =\int_0^\infty t^\beta
+       \bigl(t^{-1/2}-(t+1)^{-1/2}\bigr)\,dt.
+\end{equation}
+The integrand is $O(t^{\sigma-1})$ at zero and
+$O(t^{\sigma-2})$ at infinity.  Scaling therefore gives
+\begin{equation}\label{trace:scaling}
+ c_\sigma\int_0^\infty t^\beta
+       \bigl(t^{-1/2}-(t+y)^{-1/2}\bigr)\,dt=y^\sigma,
+ \qquad y\geq0.
+\end{equation}
+
+\begin{theorem}[Exact trace deficit]\label{trace:deficit}
+Let $n\geq1$ and let $B,V\in\C^{n\times n}$ be Hermitian.
+Suppose that
+\begin{equation}\label{trace:symbol}
+ Y_s=(sI-B)^2,\qquad X_s=Y_s+V\geq bI
+ \quad(s\in\R)
+\end{equation}
+for some $b>0$.  Set
+\begin{equation}\label{trace:resolvents}
+ R(t)=\frac1\pi\int_\R(X_s+tI)^{-1}\,ds,
+ \qquad P(t)=t^{-1/2}I-R(t),\qquad t>0,
+\end{equation}
+and
+\begin{equation}\label{trace:N}
+ N=c_\sigma\int_0^\infty t^\beta P(t)\,dt.
+\end{equation}
+These integrals converge absolutely in matrix norm, and $R(t)>0$.
+There are continuous Hermitian matrices $L(t)=R(t)^{-1}>0$ and
+$Z(t)$ satisfying
+\begin{equation}\label{trace:LZ}
+ V+tI=L^2+Z^2+i[B,Z],\qquad
+ LZ+ZL+i[B,L]=0.
+\end{equation}
+If $N\geq0$, then
+\begin{equation}\label{trace:deficit-identity}
+ \begin{split}
+ \tr(NV)-\tr(N^q)
+ =c_\sigma\int_0^\infty t^\beta\Bigl(
+    \tr(RZ^2)
+    +q\bigl\|R^{-1/2}-R^{1/2}(tI+N^r)^{1/2}\bigr\|_{\HS}^2
+    \Bigr)\,dt.
+ \end{split}
+\end{equation}
+In particular, $\tr(NV)\geq\tr(N^q)$.
+\end{theorem}
+
+The hypothesis concerns the full symbol $X_s$; the perturbation
+$V$ itself need not be positive.  The proof first recovers $L$ and
+$Z$ from solutions on the two half-lines, then evaluates a logarithmic
+integral, and finally completes a matrix square.
+
+\subsection{Half-line boundary maps}
+
+Quadratic growth in $s$, together with \eqref{trace:symbol}, gives
+an integrable bound for $X_s^{-1}$.  Hence $R(t)>0$ and
+$R(t)=O(1)$ as $t\downarrow0$.  Diagonalization of $B$ gives
+\[
+ \frac1\pi\int_\R(Y_s+tI)^{-1}\,ds=t^{-1/2}I.
+\]
+For $t\geq1$, both $X_s+tI$ and $Y_s+tI$ dominate
+$c(s^2+t)I$, with $c>0$ independent of $s,t$.
+The resolvent identity bounds their inverse difference by
+$C(s^2+t)^{-2}$, and therefore
+\begin{equation}\label{trace:P-bounds}
+ \|P(t)\|_{\op}=
+ \begin{cases}
+  O(t^{-1/2}),&t\downarrow0,\\
+  O(t^{-3/2}),&t\to\infty.
+ \end{cases}
+\end{equation}
+These bounds prove the asserted convergence of $N$.
+
+Fix $t>0$ and put $\nabla=\partial_x-iB$.  We shall use the
+equation
+\begin{equation}\label{trace:ode}
+ (-\nabla^2+V+tI)\psi=0.
+\end{equation}
+Matrix Riccati equations also appear in the commutation method for
+Schr\"odinger inequalities \cite{BenguriaLoss2000}.  Here their role
+is to compute the integrated resolvent through boundary data.
+
+For each $v\in\C^n$ there is a unique $H^1$ solution of
+\eqref{trace:ode} on either half-line with $\psi(0)=v$.
+Indeed, the sesquilinear form
+\[
+ a_t(\phi,\psi)=\int
+  \bigl(\langle\nabla\phi,\nabla\psi\rangle
+       +\langle\phi,(V+tI)\psi\rangle\bigr)\,dx,
+\]
+with inner products linear in the second variable, is coercive on
+$H^1(\R;\C^n)$: its Fourier symbol $X_s+tI$ dominates
+$c_t(1+s^2)I$.  Zero extension gives coercivity on the trace-zero
+subspace of either half-line.  Correcting any $H^1$ extension of
+$v$ by the Riesz representation theorem gives the unique weak
+solution.  The equation implies $\psi''\in L^2$, so this solution
+lies in $H^2$; both $\psi$ and $\psi'$ vanish at the infinite end.
+
+Define the right and left boundary matrices $S,T$ by
+\begin{equation}\label{trace:boundary}
+ \nabla\psi(0+)=-Sv,\qquad \nabla\psi(0-)=Tv.
+\end{equation}
+Integration by parts gives
+$a_t(\psi_v,\psi_w)=\langle v,Sw\rangle$ on the right and
+$a_t(\psi_v,\psi_w)=\langle v,Tw\rangle$ on the left.  Thus
+$S,T$ are Hermitian.  Translation and uniqueness show that
+$\nabla\psi=-S\psi$ throughout the right half-line and
+$\nabla\psi=T\psi$ throughout the left half-line.
+Substituting these relations into \eqref{trace:ode}, and allowing
+arbitrary boundary data, gives
+\begin{equation}\label{trace:riccati}
+ V+tI=S^2+i[B,S]=T^2-i[B,T].
+\end{equation}
+
+Glue the two solutions with boundary value $v$ at zero.
+The resulting continuous function has derivative jump
+$-(S+T)v$, and hence satisfies
+\[
+ (-\nabla^2+V+tI)\psi=\delta_0(S+T)v.
+\]
+Fourier inversion, with forward kernel $e^{-isx}$, gives
+\[
+ v=\frac1{2\pi}\int_\R(X_s+tI)^{-1}\,ds\,(S+T)v
+   =\tfrac12 R(t)(S+T)v.
+\]
+The inverse symbol is $O(s^{-2})$, so its Fourier inverse is
+continuous and this evaluation at zero is justified.  Therefore
+\[
+ L:=\tfrac12(S+T)=R^{-1}>0,\qquad Z:=\tfrac12(S-T)
+\]
+satisfy \eqref{trace:LZ}, by adding and subtracting
+\eqref{trace:riccati}.  The second equation in \eqref{trace:LZ}
+determines $Z$ uniquely from $L$, since $L>0$; in an eigenbasis
+of $L$, the map $Z\mapsto LZ+ZL$ multiplies its $(j,k)$ entry
+by the positive number $L_{jj}+L_{kk}$.  This also proves the
+continuity of $Z(t)$, because $R(t)$, and hence $L(t)$, is continuous.
+
+The diagonal entries of that same equation give $Z_{jj}=0$
+in an eigenbasis of $L$, so $\tr Z=0$.  Multiplication on both
+sides by $R=L^{-1}$ gives
+$i[R,B]=-(ZR+RZ)$.  Cyclicity of the trace now yields
+\begin{equation}\label{trace:RV}
+ \tr\bigl(R(V+tI)\bigr)=\tr L-\tr(RZ^2).
+\end{equation}
+In particular, the correction $\tr(RZ^2)=\tr(ZRZ)$ is nonnegative.
+We will also need the location of the spectrum of $B+iS$.
+The right solutions obey $\psi'=i(B+iS)\psi$.
+For an eigenvector with eigenvalue $z$, the solution is $e^{izx}v$;
+its square integrability forces $\operatorname{Im}z>0$.
+
+\subsection{A logarithmic integral}
+
+The next step converts the boundary matrices into a scalar identity
+that can be integrated in $t$.  Define
+\begin{equation}\label{trace:F}
+ F(t)=\frac{\tr V}{\sqrt t}
+       -\frac1\pi\int_\R
+          \log\frac{\det(X_s+tI)}{\det(Y_s+tI)}\,ds.
+\end{equation}
+The determinants are positive, and the logarithmic ratio is
+$O(|s|^{-2})$ at infinity.  Its derivative in $t$ is the trace
+of the resolvent difference, with an integrable bound locally
+uniform for $t>0$.  Thus the integral converges absolutely and
+$F$ is continuously differentiable.
+
+The first identity in \eqref{trace:riccati} factors the symbol as
+\[
+ X_s+tI=(sI-B+iS)(sI-B-iS).
+\]
+If $u_j+iv_j$ are the eigenvalues of $B+iS$, counted with
+algebraic multiplicity, then $v_j>0$ and
+\[
+ \det(X_s+tI)=\prod_{j=1}^n\bigl((s-u_j)^2+v_j^2\bigr).
+\]
+The denominator in \eqref{trace:F} has the same form, with centers
+the eigenvalues of $B$ and widths $\sqrt t$.
+Translations of the individual logarithms change their
+symmetric-cutoff integrals by $o(1)$: for
+$h(s)=\log(s^2+v^2)$,
+\[
+ \frac{d}{da}\int_{-M}^M h(s-a)\,ds
+       =h(M+a)-h(M-a)\longrightarrow0
+\]
+uniformly for bounded $a$.  After centering the factors, the identity
+\[
+ \int_\R\log\frac{s^2+v^2}{s^2+w^2}\,ds=2\pi(v-w),
+ \qquad v,w>0,
+\]
+follows by differentiation in $v$ and integration from $w$ to $v$.
+Since $\sum_jv_j=\tr S=\tr L$, we obtain
+\begin{equation}\label{trace:F-evaluation}
+ F(t)=\frac{\tr V}{\sqrt t}-2\tr(L-\sqrt t I).
+\end{equation}
+
+Set
+\begin{equation}\label{trace:G}
+ G(t)=\tr(L-2\sqrt t I+tR).
+\end{equation}
+Both functions are nonnegative.  For $G$, apply
+$\lambda-2\sqrt t+t/\lambda\geq0$ to the positive eigenvalues
+$\lambda$ of $L$.  For $F$, take the trace of \eqref{trace:LZ}
+and use \eqref{trace:F-evaluation} to obtain
+\[
+ F(t)=t^{-1/2}\tr\bigl((L-\sqrt t I)^2+Z^2\bigr)\geq0.
+\]
+Equations \eqref{trace:RV} and \eqref{trace:F-evaluation} give
+the identity
+\begin{equation}\label{trace:FG}
+ \tr(PV)=F+G+\tr(RZ^2).
+\end{equation}
+Every term on the right is nonnegative.  The bounds
+\eqref{trace:P-bounds} therefore show that $F$, $G$, and
+$\tr(RZ^2)$ are integrable against $t^\beta\,dt$, and that
+$t^{\beta+1}F(t)$ vanishes at both endpoints.
+
+Differentiating \eqref{trace:F} and using
+\eqref{trace:F-evaluation}--\eqref{trace:G} gives
+\[
+ F'(t)=-\frac{\tr V}{2t^{3/2}}+\tr P(t),
+ \qquad tF'(t)=-\tfrac12F(t)-G(t).
+\]
+Integration by parts, with the endpoint limits just proved, yields
+$\int_0^\infty t^\beta G\,dt
+ =\sigma\int_0^\infty t^\beta F\,dt$.
+Consequently, integration of \eqref{trace:FG} gives
+\begin{equation}\label{trace:weighted}
+ \tr(NV)=q c_\sigma\int_0^\infty t^\beta G(t)\,dt
+       +c_\sigma\int_0^\infty t^\beta\tr(RZ^2)\,dt.
+\end{equation}
+It remains to express the first integral as $\tr(N^q)/q$ plus
+a nonnegative remainder.
+
+\subsection{The noncommutative square}
+
+For a positive semidefinite matrix $D$, let
+\[
+ F_0(t,D)=\frac{\tr D}{\sqrt t}
+       -2\tr\bigl((tI+D)^{1/2}-\sqrt t I\bigr).
+\]
+For a scalar eigenvalue $d\geq0$, the corresponding summand is
+$\int_0^d(t^{-1/2}-(t+y)^{-1/2})\,dy$.
+Tonelli's theorem and \eqref{trace:scaling} therefore imply
+\begin{equation}\label{trace:F0}
+ c_\sigma\int_0^\infty t^\beta F_0(t,D)\,dt
+     =\frac1p\tr(D^p).
+\end{equation}
+Now expand the Hilbert--Schmidt square
+\begin{align}
+ \mathcal D_D(t)
+  &:=\bigl\|R^{-1/2}-R^{1/2}(tI+D)^{1/2}\bigr\|_{\HS}^2
+       \notag\\
+  &=\tr\bigl(R^{-1}+R(tI+D)-2(tI+D)^{1/2}\bigr)
+       \label{trace:square}\\
+  &=G(t)-\tr(P(t)D)+F_0(t,D).\notag
+\end{align}
+No commutation is used: the cross terms cancel adjacent factors
+$R^{-1/2}R^{1/2}$, and cyclicity changes the remaining quadratic
+term into $\tr(R(tI+D))$.
+All three terms in the last line are integrable by
+\eqref{trace:P-bounds}, \eqref{trace:FG}, and \eqref{trace:F0}.
+Thus
+\begin{equation}\label{trace:duality}
+ c_\sigma\int_0^\infty t^\beta G(t)\,dt
+  =\tr(ND)-\frac1p\tr(D^p)
+       +c_\sigma\int_0^\infty t^\beta\mathcal D_D(t)\,dt.
+\end{equation}
+When $N\geq0$, choose $D=N^r$.  Since $rp=q=1+r$,
+the first two terms on the right equal $\tr(N^q)/q$.
+Substitution into \eqref{trace:weighted} proves
+\eqref{trace:deficit-identity} and completes the proof of
+Theorem~\ref{trace:deficit}.
+
+If $B$ and $V$ commute, the symbol hypothesis implies $V\geq bI$.
+Simultaneous diagonalization then gives
+$R(t)=(tI+V)^{-1/2}$, $N=V^\sigma$, and $Z=0$.
+Both remainders vanish, as required by scalar equality.

preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/sections/compactness.tex

@@ -0,0 +1,379 @@
+\section{From equality to pointwise matrix relations}
+\label{comp:section}
+
+We now apply Theorem~\ref{act:quantitative-action} to finite collections of
+negative eigenfunctions.  Equality forces each of its nonnegative errors to
+vanish.  The main issue is that the number of eigenfunctions may grow without
+bound, whereas the potential has only $m$ channels.  We will use this fixed
+number of columns to control the tails of the resulting matrices.
+
+Throughout this section, $W$ is real symmetric, $W\geq0$ almost everywhere,
+and $0<\int_{\R}\tr(W^p)<\infty$.  Suppose that equality holds, in the form
+\begin{equation}
+ b_\sigma\sum_{i\in\mathcal I}k_i^{2\gamma}
+   =D_\sigma\int_{\R}\tr(W^p)\,dx,
+ \label{comp:equality}
+\end{equation}
+where $-k_i^2$ are the negative eigenvalues of $H_W$, repeated according to
+multiplicity, and
+\[
+ k_1\geq k_2\geq\cdots>0.
+\]
+By Lemma~\ref{pre:spectral}, the index set $\mathcal I$ is a nonempty finite
+initial segment of $\mathbb N$ or all of $\mathbb N$, each eigenvalue has
+finite multiplicity, and $k_i\to0$ in the infinite case.  Fix real
+orthonormal eigenfunctions $\varphi_i\in H^1(\R;\R^m)$ and regard them as
+row vectors.  Write
+\[
+ \mathscr H=\ell^2(\mathcal I;\R),\qquad
+ \Lambda=\operatorname{diag}(k_i:i\in\mathcal I),\qquad \kappa=k_1.
+\]
+For operators between real Hilbert spaces, the superscript $T$ will also
+denote the adjoint.
+
+\begin{proposition}[Pointwise relations at equality]
+\label{comp:relations}
+Under these assumptions, there are rows $a_i\in H^1(\R;\R^m)$,
+$i\in\mathcal I$, each a linear combination of eigenfunctions with
+eigenvalue $-k_i^2$, such that
+\begin{equation}
+ -a_i''-a_iW=-k_i^2a_i
+ \quad\text{in distributions.}
+ \label{comp:row-eigenfunction}
+\end{equation}
+The rows $a_i$ are allowed to vanish.  On a common set of full measure, they
+define a Hilbert--Schmidt operator $A(x):\R^m\to\mathscr H$, and there exists
+a bounded self-adjoint operator $B$ on $\mathscr H$ with
+$\|B\|_{\op}\leq\kappa$ such that
+\begin{equation}
+ [B,\Lambda^2]=0,\qquad
+ \Lambda^2-B^2=(AA^T)^r,\qquad
+ a_i'(x)=(BA)_{i,\cdot}\quad(i\in\mathcal I).
+ \label{comp:limit-relations}
+\end{equation}
+Moreover,
+\begin{equation}
+ W(x)=q\bigl(A(x)^TA(x)\bigr)^r
+ \quad\text{almost everywhere.}
+ \label{comp:potential-limit}
+\end{equation}
+The operator $B$ is asserted to exist separately at each such point $x$;
+no measurable choice is required.
+\end{proposition}
+
+We divide the proof into the construction of a sequence whose errors
+vanish, convergence of every fixed row, and control of the remaining rows.
+
+\subsection{Finite spectral approximations and vanishing errors}
+\label{comp:approximation}
+
+Let $n_j=j$ when $\mathcal I=\mathbb N$, and let $n_j=|\mathcal I|$ when
+$\mathcal I$ is finite.  Choose $0<\xi_j<1/j$ so small that
+\begin{equation}
+ \ell_{j,i}:=(k_i^2-\xi_j)^{1/2}>0\quad(1\leq i\leq n_j),
+ \qquad
+ \sum_{i\leq n_j}\bigl(k_i^{2\gamma}-\ell_{j,i}^{2\gamma}\bigr)<1/j.
+ \label{comp:common-shift}
+\end{equation}
+Set $K_j=\operatorname{diag}(\ell_{j,1},\ldots,\ell_{j,n_j})$.
+The same shift $\xi_j$ is used for every eigenvalue: this leaves the
+differences of their squares unchanged.
+
+The approximation leading to \eqref{bound:trial} can be made arbitrarily
+accurate in $H^1$.  Apply it to the first $n_j$ eigenfunctions, choosing
+the $H^1$ errors below $1/j$ and the form-matrix error below $\xi_j$.
+This gives real, smooth, compactly supported functions
+$u_{j,1},\ldots,u_{j,n_j}$, orthonormal in $L^2(\R;\R^m)$, such that
+\begin{equation}
+ \|u_{j,i}-\varphi_i\|_{H^1}<1/j,
+ \qquad
+ \bigl(h_W[u_{j,i},u_{j,l}]\bigr)_{i,l=1}^{n_j}\leq-K_j^2.
+ \label{comp:form-approximation}
+\end{equation}
+Here $h_W[\cdot,\cdot]$ is the polarized quadratic form, and the second
+inequality is the ordering of real symmetric matrices.
+
+Let $U_j$ be the matrix with rows $u_{j,i}$, and choose a bounded interval
+$\Omega_j$ containing their supports in its interior.  Apply
+Theorem~\ref{act:quantitative-action} with $K=K_j$, $d=m$, $\kappa=k_1$,
+and $\eta=1/j$.  It gives a lower triangular matrix $C_j$, a symmetric
+path $B_j$ with $\|B_j\|_{\op}\leq\kappa$, and $A_j=C_jU_j$.
+Extend $A_j$ by zero outside $\Omega_j$ and extend $B_j$ by $K_j$ to the
+left and by $-K_j$ to the right.  All the errors in that theorem vanish
+outside $\Omega_j$, so their integrals may be taken over $\R$.
+
+The lower triangular form of $C_j$ gives a useful additional error.  Put
+$Z_j=A_j^TA_j$, an $m\times m$ positive semidefinite matrix.  By
+\eqref{comp:form-approximation},
+\begin{equation}
+ \mathcal E_{K_j}(A_j)
+ \leq\int_{\R}\bigl(\tr(WZ_j)-\tr(Z_j^q)\bigr)\,dx-T_j,
+ \qquad
+ T_j=\sum_{l\leq i\leq n_j}(C_j)_{il}^{\,2}(k_l^2-k_i^2)\geq0.
+ \label{comp:triangular-gap}
+\end{equation}
+Indeed, the $i$th row of $A_j$ is $\sum_{l\leq i}(C_j)_{il}u_{j,l}$.
+Its quadratic form, plus $\ell_{j,i}^2$ times its squared $L^2$ norm, is
+at most
+\[
+ \sum_{l\leq i}(C_j)_{il}^{\,2}
+      (\ell_{j,i}^2-\ell_{j,l}^2)
+ =-\sum_{l\leq i}(C_j)_{il}^{\,2}(k_l^2-k_i^2).
+\]
+Summing over $i$ proves \eqref{comp:triangular-gap}, since $A_jA_j^T$ and
+$Z_j$ have the same nonzero eigenvalues, with multiplicity.  The ordering
+of the $k_i$ makes every summand of $T_j$ nonnegative.
+
+Recall the nonnegative Young deficit from Lemma~\ref{pre:young}:
+\begin{equation}
+ Y(W,Z)=D_\sigma\tr(W^p)-\tr(WZ)+\tr(Z^q)\geq0,
+ \qquad
+ Y(W,Z)=0\ \Longleftrightarrow\ W=qZ^r.
+ \label{comp:young-deficit}
+\end{equation}
+Combining Theorem~\ref{act:quantitative-action},
+\eqref{comp:equality}, and \eqref{comp:triangular-gap} yields
+\begin{align}
+ 0\leq T_j+\int_{\R}\Bigl(
+   Y(W,Z_j)+\|A_j'-B_jA_j\|_{\HS}^2
+       +c_*\Phi_{K_j}(A_j,B_j)\Bigr)\,dx
+ &\leq d_j,\label{comp:vanishing-errors}\\
+ d_j:=b_\sigma\left(
+    \sum_{i\in\mathcal I}k_i^{2\gamma}
+       -\sum_{i\leq n_j}\ell_{j,i}^{2\gamma}\right)+1/j
+ &\longrightarrow0.\notag
+\end{align}
+Here $c_*>0$ is fixed: its independence of $n_j$ is essential.  Thus all
+three integral errors tend to zero, as does $T_j$.
+
+We will also need a uniform bound on $A_j$.  The coercive estimate
+\eqref{pre:young-coercive} gives
+\begin{equation}
+ Y(W,Z_j)\geq\frac12\tr(Z_j^q)
+       -(2^{p-1}-1)D_\sigma\tr(W^p).
+ \label{comp:coercivity}
+\end{equation}
+Since $Z_j$ has size $m$, scalar H\"older's inequality gives
+\[
+ \|A_j\|_{\HS}^{2q}=(\tr Z_j)^q
+       \leq m^{q-1}\tr(Z_j^q).
+\]
+Equations~\eqref{comp:vanishing-errors} and \eqref{comp:coercivity}
+therefore bound $A_j$ uniformly in $L^{2q}(\R)$ with the
+Hilbert--Schmidt norm, independently of its number of rows.
+
+\subsection{Convergence of each fixed row}
+\label{comp:rows}
+
+For fixed $i\in\mathcal I$, the coefficients $(C_j)_{il}$, $l\leq i$,
+are bounded as $j\to\infty$.  To see this, choose a bounded interval $J_i$
+so large that the restricted Gram matrix
+\[
+ \left(\int_{J_i}\varphi_l(x)\varphi_s(x)^T\,dx
+       \right)_{l,s=1}^{i}\geq\frac34 I_i,
+\]
+where $I_i$ denotes the $i\times i$ identity matrix.
+Such an interval exists because the full Gram matrix is the identity.
+By \eqref{comp:form-approximation}, the corresponding restricted Gram
+matrix of the $u_{j,l}$ is at least one half of the identity for all
+sufficiently large $j$.  Hence
+\[
+ \frac12\sum_{l\leq i}(C_j)_{il}^{\,2}
+ \leq\int_{J_i}|(A_j)_{i,\cdot}|^2\,dx
+ \leq |J_i|^{1-1/q}
+        \left(\int_{J_i}\|A_j\|_{\HS}^{2q}\,dx\right)^{1/q}.
+\]
+The last expression is uniformly bounded.  Lower triangularity is used
+here to keep the number of coefficients in a fixed row finite.
+
+A diagonal subsequence now makes every coefficient $(C_j)_{il}$ converge
+to a limit $c_{il}$, for $l\leq i$.  From
+$T_j\to0$ and \eqref{comp:triangular-gap},
+\begin{equation}
+ c_{il}=0\quad\text{whenever }k_l>k_i.
+ \label{comp:gap-suppression}
+\end{equation}
+Define
+\[
+ a_i=\sum_{l\leq i}c_{il}\varphi_l.
+\]
+The sum is finite, and \eqref{comp:form-approximation} implies
+\begin{equation}
+ (A_j)_{i,\cdot}\longrightarrow a_i
+       \quad\text{in }H^1(\R;\R^m)
+       \quad\text{for every fixed }i.
+ \label{comp:row-convergence}
+\end{equation}
+The ordering of the eigenvalues and \eqref{comp:gap-suppression} show
+that only eigenfunctions with eigenvalue $-k_i^2$ occur in $a_i$.
+This proves \eqref{comp:row-eigenfunction}, including the possibility
+$a_i=0$.  The eigenfunction equations hold distributionally by testing
+the form equations against smooth compactly supported functions; the
+potential is locally integrable and the rows are locally bounded.
+
+Choose the continuous, locally absolutely continuous representatives
+of these rows.  The Sobolev embedding $H^1(\R)\subset C_b(\R)$ shows
+that \eqref{comp:row-convergence} also gives pointwise convergence at
+every $x$.  Passing to a further subsequence, we may arrange that the
+integrals of the nonnegative integrand in
+\eqref{comp:vanishing-errors} are summable, and that, for every fixed
+$i$, the squared $L^2$ errors of the derivatives in
+\eqref{comp:row-convergence} are summable.  A diagonal choice achieves
+both conditions, by making the first $j$ available derivative errors and the
+integral error less than $2^{-j}$ along the chosen subsequence.
+Consequently, on a common set of full measure,
+\begin{equation}
+ \begin{gathered}
+ Y(W,Z_j)\to0,\qquad
+ \|A_j'-B_jA_j\|_{\HS}\to0,\qquad
+ \Phi_{K_j}(A_j,B_j)\to0,\\
+ (A_j')_{i,\cdot}(x)\to a_i'(x)
+       \quad\text{for every }i\in\mathcal I.
+ \end{gathered}
+ \label{comp:pointwise-errors}
+\end{equation}
+We also exclude the null set where $W$ is not a finite positive
+semidefinite matrix.  It remains to show that convergence of the
+individual rows is strong enough to pass to the products in these
+relations.
+
+\subsection{Tail control in the row-index space}
+\label{comp:tails}
+
+Pad $A_j$, $B_j$, and $K_j$ with zeros in the unused coordinates of
+$\mathscr H$.  Thus $A_j:\R^m\to\mathscr H$, and $B_j,K_j$ are
+self-adjoint finite-rank operators on $\mathscr H$.  Since the omitted
+$k_i$ tend to zero,
+\begin{equation}
+ \|K_j^2-\Lambda^2\|_{\op}
+    \leq\max\{\xi_j,\sup_{i>n_j}k_i^2\}\longrightarrow0,
+ \label{comp:diagonal-limit}
+\end{equation}
+where a supremum over no indices is zero.
+
+Fix a point $x$ in the full-measure set from
+\eqref{comp:pointwise-errors} and suppress $x$ for the moment.  Write
+$\rho_j=A_jA_j^T$.  The definition of $\Phi$ and its convergence to zero
+give
+\begin{equation}
+ \|[B_j,K_j^2]\|_{\op}\to0,
+ \qquad e_j:=\|K_j^2-B_j^2-\rho_j^r\|_{\op}\to0.
+ \label{comp:pointwise-algebra-error}
+\end{equation}
+For $h\geq1$, let $P_h$ be the orthogonal projection onto the first $h$
+coordinates of $\mathscr H$, with all coordinates included if $h$ exceeds
+$|\mathcal I|$.  Set $Q_h=I-P_h$ and
+$s_h=\sup_{i>h}k_i^2$, again with the empty supremum equal to zero.
+Then $s_h\to0$.  For every unit vector $v$ in the range of $Q_h$,
+\begin{equation}
+ \|B_jv\|^2+\langle v,\rho_j^rv\rangle\leq s_h+e_j.
+ \label{comp:tail-quadratic}
+\end{equation}
+Both terms on the left are nonnegative.  Jensen's inequality for the
+spectral measure of $\rho_j$, using $r>1$, implies
+\[
+ \langle v,\rho_jv\rangle^r\leq\langle v,\rho_j^rv\rangle.
+\]
+It follows that
+$\|Q_h\rho_jQ_h\|_{\op}\leq(s_h+e_j)^{1/r}$.
+Although $\mathscr H$ may be infinite dimensional, $Q_h\rho_jQ_h$
+has rank at most $m$, because $A_j$ has exactly $m$ columns.  Its trace
+is therefore at most $m$ times its operator norm.  We obtain the two
+tail bounds
+\begin{equation}
+ \boxed{\quad
+ \|Q_hA_j\|_{\HS}^2\leq m(s_h+e_j)^{1/r},
+ \qquad
+ \|B_jQ_h\|_{\op}\leq(s_h+e_j)^{1/2}.
+ \quad}
+ \label{comp:tail-bounds}
+\end{equation}
+The first estimate converts decay of the eigenvalue parameters into
+decay of the total mass of all omitted rows.  Its factor is the fixed
+physical dimension $m$, rather than the number of retained
+eigenfunctions; Figure~\ref{comp:tail-figure} depicts this distinction.
+
+\begin{figure}[tb]
+ \centering
+ \begin{tikzpicture}[font=\small,>=Latex]
+  \fill[black!5] (0,2.05) rectangle (2.7,3.65);
+  \fill[blue!7] (0,0) rectangle (2.7,2.05);
+  \draw[thick] (0,0) rectangle (2.7,3.65);
+  \draw[thick] (0,2.05)--(2.7,2.05);
+  \draw[<->] (0,4.05)--node[above]{$m$ columns, fixed}(2.7,4.05);
+  \node at (1.35,2.83) {$P_hA_j$};
+  \node at (1.35,1.28) {$Q_hA_j$};
+  \node at (1.35,0.55) {$\vdots$};
+  \node[anchor=east,align=right] at (-0.2,2.83)
+       {first $h$\\rows};
+  \node[anchor=east,align=right] at (-0.2,1.08)
+       {all remaining\\rows};
+  \draw[->] (2.9,2.83)--(3.6,2.83);
+  \node[anchor=west,align=left] at (3.75,2.83)
+       {Finite-dimensional convergence\\for each fixed $h$};
+  \draw[->] (2.9,1.08)--(3.6,1.08);
+  \node[anchor=west,align=left] at (3.75,1.08)
+       {$\operatorname{rank}(Q_hA_j)\leq m$\\[4pt]
+        $\|Q_hA_j\|_{\HS}^2\leq m(s_h+e_j)^{1/r}$};
+ \end{tikzpicture}
+ \caption{The number of rows can grow with the number of negative
+ eigenfunctions, but the number of physical channels remains $m$.
+ After fixing the first $h$ rows, the identity
+ $K_j^2\approx B_j^2+(A_jA_j^T)^r$ bounds the operator norm in the
+ remaining rows.  The rank bound converts it into a Hilbert--Schmidt
+ bound, which tends to zero as first $j\to\infty$ and then $h\to\infty$.
+ The drawing is schematic; all matrices are padded by zero rows in
+ $\mathscr H$.}
+ \label{comp:tail-figure}
+\end{figure}
+
+For each fixed $h$, the first $h$ rows of $A_j$ converge by
+\eqref{comp:row-convergence}.  The first bound in
+\eqref{comp:tail-bounds} makes the remaining rows uniformly small by
+choosing $h$ large and then $j$ sufficiently large.  Thus $A_j$ is Cauchy in
+Hilbert--Schmidt norm and converges to an operator $A:\R^m\to\mathscr H$
+whose rows are precisely $a_i(x)$:
+\begin{equation}
+ A_j\longrightarrow A\quad\text{in Hilbert--Schmidt norm.}
+ \label{comp:HS-limit}
+\end{equation}
+This argument applies to the whole subsequence already selected and
+does not require a further choice depending on $x$.
+
+For $B_j$ we need only pointwise compactness.  Its finite compressions
+$P_hB_jP_h$ are bounded in finite-dimensional spaces, so a diagonal
+subsequence makes them converge for every $h$.  Self-adjointness and
+the second estimate in \eqref{comp:tail-bounds} give
+\[
+ \|B_j-P_hB_jP_h\|_{\op}
+   \leq2\|B_jQ_h\|_{\op}
+   \leq2(s_h+e_j)^{1/2}.
+\]
+The chosen subsequence consequently converges in operator norm to a
+self-adjoint $B$ with $\|B\|_{\op}\leq\kappa$.  This subsequence, and
+hence $B$, may depend on the point $x$.
+
+We can now pass to all the required products.  Hilbert--Schmidt
+convergence of $A_j$ implies operator norm convergence of
+$A_jA_j^T$ and of $A_j^TA_j$.  Continuous functional calculus for
+nonnegative operators then gives convergence of their $r$th powers.
+Together with \eqref{comp:diagonal-limit} and
+\eqref{comp:pointwise-algebra-error}, this proves
+\[
+ [B,\Lambda^2]=0,\qquad
+ \Lambda^2-B^2=(AA^T)^r.
+\]
+Also $B_jA_j\to BA$ in Hilbert--Schmidt norm.  Taking its $i$th row
+and using both the derivative convergence and the first-order error
+in \eqref{comp:pointwise-errors} gives
+$a_i'(x)=(BA)_{i,\cdot}$ for every $i$.  Finally, continuity of the
+finite-dimensional Young deficit and
+$Y(W,A_j^TA_j)\to0$ yield
+$Y(W,A^TA)=0$.  Lemma~\ref{pre:young} gives
+$W=q(A^TA)^r$.
+
+We have proved these statements at every point of one common set of
+full measure.  The construction selects no function $x\mapsto B(x)$:
+only the existence of a suitable operator at each such point enters
+the next section.  This completes the proof of
+Proposition~\ref{comp:relations}.

preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/sections/introduction.tex

@@ -0,0 +1,161 @@
+\section{Introduction}\label{intro:section}
+
+Lieb--Thirring inequalities compare the binding energy of a Schr\"odinger
+operator with an integral of its potential. Equality asks how the potential
+must arrange that binding. In one dimension, a scalar one-bound-state
+optimizer has a hyperbolic-secant profile. For a matrix potential, several
+such profiles can occupy orthogonal internal channels. The issue is whether
+equality also permits channels that rotate with position, or several bound
+states that interact within the same channel. We prove that throughout
+$1/2<\gamma<3/2$ every equality case consists of independent scalar profiles
+in one fixed basis.
+
+Let $m\ge1$ be finite, let $1/2<\gamma<3/2$, and set
+$p=\gamma+1/2$. For a measurable Hermitian positive semidefinite function
+$W:\R\to\C^{m\times m}$ satisfying
+\begin{equation}\label{intro:class}
+ \int_\R\tr(W(x)^p)\,dx<\infty,
+\end{equation}
+let $H_W=-d^2/dx^2-W$ be the operator associated with
+\begin{equation}\label{intro:form}
+ h_W[\psi]=\int_\R\|\psi'\|^2
+       -\int_\R\langle\psi,W\psi\rangle,
+ \qquad \psi\in H^1(\R;\C^m).
+\end{equation}
+Lemma~\ref{pre:spectral} proves that the form is closed and bounded below
+and that the negative spectrum is discrete, with possible accumulation
+only at zero. We write
+\[
+ \Tr(H_W)_-^\gamma=\sum_{\lambda_j(H_W)<0}|\lambda_j(H_W)|^\gamma,
+\]
+counting multiplicities. The finite-dimensional trace $\tr$ is unnormalized;
+matrix powers are defined by spectral functional calculus.
+
+\begin{theorem}[Equality classification]\label{thm:main}
+For every $W$ satisfying~\eqref{intro:class},
+\begin{equation}\label{intro:sharp}
+ \Tr(H_W)_-^\gamma\le C_\gamma\int_\R\tr(W^p),
+ \qquad
+ C_\gamma=\left(\frac{\gamma-1/2}{\gamma+1/2}\right)^{\gamma-1/2}
+       \frac{\Gamma(\gamma+1)}{\sqrt\pi\,\Gamma(\gamma+3/2)}.
+\end{equation}
+Put $r=(\gamma-1/2)^{-1}$. Equality holds if and only if there are
+$k\in\{0,\ldots,m\}$, a constant unitary matrix $U$, positive numbers
+$a_1,\ldots,a_k$, and real numbers $x_1,\ldots,x_k$ such that almost
+everywhere
+\begin{equation}\label{intro:classification}
+ W(x)=U\diag\bigl(w_1(x),\ldots,w_k(x),0,\ldots,0\bigr)U^*,
+ \quad
+ w_j(x)=(r+1)a_j^2\sech^2\!\bigl(ra_j(x-x_j)\bigr).
+\end{equation}
+The case $k=0$ means $W=0$. Each nonzero channel has exactly one negative
+eigenvalue, $-a_j^2$, and therefore every extremal potential has at most
+$m$ negative eigenvalues, counted with multiplicity.
+\end{theorem}
+
+The inequality in~\eqref{intro:sharp} is the sharp matrix inequality
+of~\cite[Theorem~1.1]{MLT}. Our contribution is its equality classification.
+We retain the analytic constructions in the proof, so that the inequality
+and its equality conditions are established here under the same measurable
+potential hypothesis. The scales and centers in~\eqref{intro:classification}
+are independent; repeated scales and identical profiles are allowed.
+
+\subsection{History and related results}
+Lieb and Thirring introduced spectral moment inequalities in their work
+on the kinetic energy of fermions and stability of matter
+\cite{LiebThirring1975,LiebThirring1976}. Their sharp constants distinguish
+the regimes in which binding is best concentrated in a single state or
+distributed among many states. Keller's optimization of the lowest
+Schr\"odinger eigenvalue~\cite{Keller1961} supplies the one-state problem
+underlying the scalar profiles in~\eqref{intro:classification}; their
+explicit form appears in~\cite[Section~4(B), Equations~(4.19)--(4.20)]{LiebThirring1976}.
+The present question concerns equality in the sum of all negative
+eigenvalue moments, with a matrix potential that need not commute with
+itself at different points.
+
+The sharp scalar endpoint $\gamma=1/2$ was proved by
+Hundertmark, Lieb and Thomas~\cite{HLT1998}; Hundertmark, Laptev and
+Weidl~\cite[Theorem~3.1]{HLW2000} extended it to operator-valued
+potentials. The semiclassical constant
+is sharp for $\gamma\ge3/2$; the lifting of moments of Aizenman and
+Lieb~\cite{AizenmanLieb1978} and the operator-valued inequalities of
+Laptev and Weidl~\cite{LaptevWeidl2000} are central to this development.
+Benguria and Loss~\cite{BenguriaLoss2000} gave a commutation proof of the
+matrix result at $\gamma=3/2$. More recently, Read and
+Schulz~\cite{ReadSchulz2026} proved the sharp scalar inequality at
+$\gamma=1$, with constant $4/(3\sqrt3\pi)$, and its operator-valued
+extension. The matrix companion~\cite{MLT} proves the sharp one-state
+constant for the whole open interval considered here.
+
+The endpoint $\gamma=3/2$ has a different equality structure. In the
+scalar finite-rank problem, Frank, Gontier and Lewin
+\cite[Theorem~5]{FrankGontierLewin2025} identify the optimizers as KdV
+multisolitons. In particular, several bound states can occur in a single
+scalar channel at that endpoint. The conclusion of
+Theorem~\ref{thm:main}, one bound state per nonzero channel, uses the
+strict inequality $r>1$ and is asserted only for the open interval.
+
+\subsection{Quantitative defects and fixed channels}
+The proof begins with the finite-interval matrix action method
+of~\cite{MLT}. For orthonormal real trial functions approximating finitely
+many negative eigenfunctions, place the functions in the rows of $U(x)$
+and choose a lower triangular matrix $C$. The rows of $A=CU$ remain in the
+successive trial spans.
+If $K$ is the diagonal matrix of positive square roots of the trial
+binding energies, the action involves $A'$, $K^2AA^T$, and a power of
+the density $AA^T$. A symmetric matrix path $B(x)$ connects $K$ to
+$-K$ and gives a lower bound for that action.
+
+For equality, the lower bound must retain more information. Besides the
+square $\|A'-BA\|_{\HS}^2$, we retain a term controlling
+\[
+ [B,K^2]\quad\hbox{and}\quad K^2-B^2-(AA^T)^r.
+\]
+Its coefficient depends on the largest binding energy and the exponent,
+but not on the number of trial functions. This uniformity is the first
+essential point: an extremal potential is not known in advance to have
+finitely many negative eigenvalues. The integrated trace identity of the
+matrix companion supplies the two nonnegative squares from which this
+estimate follows.
+
+The second point is compactness with a fixed number of columns. As the
+trial list grows, $A$ has more rows but still only $m$ columns. Its
+density therefore has rank at most $m$. The retained defect bounds the
+tail rows by the small binding energies in that tail. This yields strong
+pointwise convergence even for an infinite eigenvalue list and gives
+exact algebraic relations in the limit. Those relations separate the
+rows with different energies. A differential inequality then makes
+their row spaces independent of position; their mutual orthogonality
+leaves at most $m$ nonzero directions. On each resulting finite block,
+a matrix Riccati equation has a fixed diagonalizing basis and determines
+the profiles in~\eqref{intro:classification}.
+
+Section~\ref{pre:section} gives the spectral and convexity preliminaries.
+Section~\ref{act:section} proves the quantitative action estimate, and
+Section~\ref{bound:section} recovers the sharp spectral bound.
+Section~\ref{comp:section} passes to exact relations for an equality
+case. Section~\ref{rig:section} proves that the channels are fixed,
+solves the Riccati equation, and treats complex potentials and the
+converse. Appendices~\ref{app:trace} and~\ref{app:paths} prove the trace
+identity, regularized field construction, and connecting-path theorem
+used by the action estimate.
+
+\subsection{Notation}
+Throughout the proof,
+\begin{equation}\label{intro:parameters}
+ \sigma=\gamma-\tfrac12\in(0,1),\qquad
+ p=1+\sigma,\qquad r=\sigma^{-1},\qquad q=r+1,
+\end{equation}
+and
+\begin{equation}\label{intro:constants}
+ D_\sigma=\frac{\sigma^\sigma}{(1+\sigma)^{1+\sigma}},\qquad
+ b_\sigma=\int_{-1}^1(1-s^2)^\sigma\,ds,\qquad
+ C_\gamma=\frac{D_\sigma}{b_\sigma}.
+\end{equation}
+The last identity follows from the beta integral. We use the operator
+norm $\|\cdot\|_{\op}$ and the Hilbert--Schmidt norm
+$\|A\|_{\HS}^2=\tr(A^*A)$. The symbol $T$ denotes transpose of a real
+matrix, and $\Sym_n$ is the space of real symmetric $n\times n$ matrices
+with inner product $\tr(BH)$. The commutator is $[B,H]=BH-HB$.
+The identity matrix, or identity operator, has the size determined by
+its context.

preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/sections/preliminaries.tex

@@ -0,0 +1,83 @@
+\section{Spectral and convexity preliminaries}\label{pre:section}
+
+Two elementary facts let us work with measurable potentials and identify
+the equality condition in the potential estimate. The first is the form
+argument used in the matrix inequality~\cite[Lemma~7.1]{MLT}; we give its
+proof to fix the precise regularity available below.
+
+\begin{lemma}\label{pre:spectral}
+Let $p>1$ and let $W:\R\to\C^{m\times m}$ be measurable, Hermitian and
+positive semidefinite, with $\int_\R\tr(W^p)<\infty$.
+The form $h_W$ on $H^1(\R;\C^m)$ is closed and bounded below.
+Multiplication by $W^{1/2}$ is compact from $H^1$ to $L^2$.
+The negative spectrum of $H_W$ consists of isolated eigenvalues of finite
+multiplicity, with possible accumulation only at zero. Its eigenfunctions
+belong to $H^1$ and satisfy $-\psi''-W\psi=\lambda\psi$ distributionally.
+\end{lemma}
+
+\begin{proof}
+Set $w=\|W\|_{\op}$. Then $w^p\le\tr(W^p)$, so $w\in L^p$.
+The one-dimensional Sobolev estimate, applied to the norm of a vector
+function, gives $\|\psi\|_\infty^2\le2\|\psi\|_2\|\psi'\|_2$.
+H\"older's inequality, interpolation and Young's inequality imply
+\begin{equation}\label{pre:form-bound}
+ \int_\R\langle\psi,W\psi\rangle
+ \le 2^{1/p}\|w\|_p\|\psi'\|_2^{1/p}\|\psi\|_2^{2-1/p}
+ \le \varepsilon\|\psi'\|_2^2+C_\varepsilon\|\psi\|_2^2.
+\end{equation}
+A sufficiently shifted form norm is therefore equivalent to the $H^1$
+norm. This proves closedness and semiboundedness, and polarization gives
+continuity of the potential form on $H^1$.
+
+Write $T\psi=W^{1/2}\psi$. The truncated multiplier
+$T_{R,L}=\mathbf1_{\{|x|\le R,\,w\le L\}}T$ is compact by the compact
+embedding of $H^1([-R,R])$ into $L^2([-R,R])$. Moreover,
+\[
+ \|(T-T_{R,L})\psi\|_2^2
+ \le C\|w\mathbf1_{\{|x|>R\}\cup\{w>L\}}\|_p\|\psi\|_{H^1}^2.
+\]
+The coefficient tends to zero, so $T$ is compact.
+For each $a>0$, an infinite-dimensional spectral subspace in
+$(-\infty,-a]$ would contain an $L^2$-orthonormal sequence bounded in
+$H^1$ by~\eqref{pre:form-bound}. A subsequence converges weakly to zero
+in $H^1$, hence its images under $T$ converge strongly to zero. This
+contradicts
+$-a\ge h_W[\psi]\ge-\|T\psi\|_2^2$ for unit vectors in that subspace.
+Thus every such subspace is finite-dimensional. Finally, the weak form
+equation tested against smooth compactly supported functions is the stated
+distributional equation; its potential term is locally integrable because
+$W\in L^p_{\mathrm{loc}}$ and $\psi$ is locally bounded.
+\end{proof}
+
+\begin{lemma}[Trace Young inequality and its equality condition]\label{pre:young}
+For positive semidefinite matrices $W,Z$ of the same finite size, set
+\begin{equation}\label{pre:young-defect}
+ Y(W,Z)=D_\sigma\tr(W^p)-\tr(WZ)+\tr(Z^q).
+\end{equation}
+Then $Y(W,Z)\ge0$, with equality if and only if $W=qZ^r$.
+Also,
+\begin{equation}\label{pre:young-coercive}
+ Y(W,Z)\ge\tfrac12\tr(Z^q)
+       -(2^{p-1}-1)D_\sigma\tr(W^p).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Scalar maximization over $z\ge0$ gives
+$yz-z^q\le D_\sigma y^p$ for $y\ge0$, with equality precisely when
+$y=qz^r$. In an orthonormal eigenbasis $(v_j)$ of $Z$, let $z_j$ be its
+eigenvalues and put $y_j=\langle v_j,Wv_j\rangle$. Strict convexity of
+$t^p$ on $[0,\infty)$ gives
+\[
+ \tr(WZ)-\tr(Z^q)
+ \le D_\sigma\sum_j y_j^p
+ \le D_\sigma\sum_j\langle v_j,W^pv_j\rangle.
+\]
+Equality in the second inequality holds exactly when each $v_j$ belongs
+to an eigenspace of $W$. Equality in the first then requires
+$Wv_j=qz_j^rv_j$ for every $j$. This proves both directions of the
+equality assertion, including zero eigenvalues. Applying the same
+inequality to $2W,Z$ gives
+$\tr(WZ)\le\tfrac12\tr(Z^q)+2^{p-1}D_\sigma\tr(W^p)$,
+which is~\eqref{pre:young-coercive}.
+\end{proof}

preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/sections/rigidity.tex

@@ -0,0 +1,293 @@
+\section{Fixed channels and the equality profiles}
+\label{rig:section}
+
+The pointwise relations of Proposition~\ref{comp:relations} still involve
+an auxiliary operator $B$ that may depend on the point without a prescribed
+measurable choice.  We now extract constant subspaces from those relations.
+On each nonzero subspace, $B$ becomes a uniquely determined matrix and
+satisfies an ordinary differential equation.  This will identify every
+real equality potential; realification and a scalar calculation will then
+complete the proof of Theorem~\ref{thm:main}.
+
+\subsection{The channel subspaces are constant}
+
+Assume first that $W$ is real and is a nonzero equality potential.  Use the
+rows $a_i$, numbers $k_i$, and operator $\Lambda=\operatorname{diag}(k_i)$
+from Proposition~\ref{comp:relations}.  For each distinct value $\alpha>0$
+in the list $(k_i)$, let $n_\alpha$ be its finite multiplicity, and let
+$G_\alpha$ be the $n_\alpha\times m$ matrix whose rows are the $a_i$ with
+$k_i=\alpha$.  Each $G_\alpha$ has entries in $H^1(\R)$, and we use their
+continuous representatives.
+
+At every point in the common full-measure set of that proposition, choose
+one of the operators $B$ it supplies.  Since $B$ commutes with $\Lambda^2$,
+it preserves each of its eigenspaces.  Moreover,
+\[
+ AA^T=(\Lambda^2-B^2)^{1/r}
+\]
+commutes with both $B$ and $\Lambda^2$.  Writing $B_\alpha$ for the block
+on the $\alpha^2$-eigenspace therefore gives
+\begin{align}
+ G_\alpha G_\beta^T&=0 &&(\alpha\ne\beta),
+ \label{rig:orthogonal-blocks}\\
+ G_\alpha'&=B_\alpha G_\alpha,
+ & [B_\alpha,G_\alpha G_\alpha^T]&=0,
+ \label{rig:block-first-order}\\
+ \alpha^2I-B_\alpha^2&=(G_\alpha G_\alpha^T)^r,
+ &\|B_\alpha\|_{\op}&\le\kappa.
+ \label{rig:block-square}
+\end{align}
+These statements hold almost everywhere and require no compatibility
+between the choices of $B$ at different points.
+
+\begin{lemma}\label{rig:fixed-right-spaces}
+For every $\alpha$, the subspace
+\[
+ \mathcal R_\alpha=(\ker G_\alpha(x))^\perp\subset\R^m
+\]
+is independent of $x\in\R$.  The nonzero spaces $\mathcal R_\alpha$ are
+mutually orthogonal, and
+\begin{equation}\label{rig:dimension-bound}
+ \sum_\alpha\dim\mathcal R_\alpha\le m.
+\end{equation}
+In particular, only finitely many $G_\alpha$ are nonzero.  The potential
+$W$ has a continuous representative, equal to zero on the orthogonal
+complement of these spaces and equal to
+$q(G_\alpha^TG_\alpha)^r$ on $\mathcal R_\alpha$.  With this
+representative, every $G_\alpha$ is $C^2$.
+\end{lemma}
+
+\begin{proof}
+Fix $v\in\R^m$.  Equation~\eqref{rig:block-first-order} implies
+\[
+ \|(G_\alpha v)'(x)\|\le\kappa\|G_\alpha(x)v\|
+ \quad\text{for almost every }x.
+\]
+The function $G_\alpha v$ is locally absolutely continuous.  If it
+vanishes at $x_0$, integration and Gronwall's inequality, first to the
+right and then to the left of $x_0$, show that it vanishes everywhere.
+Consequently $\ker G_\alpha(x)$ is independent of $x$, including points
+outside the original full-measure set.  This proves the first assertion.
+
+The products in \eqref{rig:orthogonal-blocks} are continuous, so their
+almost-everywhere vanishing implies their vanishing everywhere.  Thus the
+constant row spaces $\mathcal R_\alpha$ are mutually orthogonal.  Any
+finite collection has total dimension at most $m$, proving
+\eqref{rig:dimension-bound} and finiteness of the nonzero blocks.
+
+On the full-measure set from Proposition~\ref{comp:relations},
+$A^TA=\sum_\alpha G_\alpha^TG_\alpha$.  Each summand acts on its own
+orthogonal space $\mathcal R_\alpha$, and only finitely many summands
+are nonzero.  Equation~\eqref{comp:potential-limit} therefore yields
+\begin{equation}\label{rig:continuous-potential}
+ W=q\sum_\alpha(G_\alpha^TG_\alpha)^r
+ \quad\text{almost everywhere}.
+\end{equation}
+The right-hand side is continuous and has the asserted restrictions;
+we henceforth use it as the representative of $W$.  The row eigenfunction
+equation \eqref{comp:row-eigenfunction} now reads
+\begin{equation}\label{rig:block-second-order}
+ G_\alpha''=\alpha^2G_\alpha-G_\alpha W
+ \quad\text{in distributions}.
+\end{equation}
+Its right-hand side is continuous.  Integrating this identity twice shows
+that the continuous representative of $G_\alpha$ is $C^2$.
+\end{proof}
+
+The row spaces already split the physical space $\R^m$ into finitely
+many constant channels.  To determine the potential within one of them,
+we also need a constant range for $G_\alpha$ in the eigenfunction index
+space $\R^{n_\alpha}$.
+
+\begin{lemma}\label{rig:fixed-left-spaces}
+For each $\alpha$ with $\ell:=\dim\mathcal R_\alpha>0$, the range of
+$G_\alpha(x)$ in $\R^{n_\alpha}$ is independent of $x$.  In constant
+orthonormal bases of this range and of $\mathcal R_\alpha$, the
+restriction of $G_\alpha$ is an invertible $C^2$ matrix
+$D:\R\to\R^{\ell\times\ell}$.  The matrix
+$B_0=D'D^{-1}$ is $C^1$ and satisfies, everywhere on $\R$,
+\begin{equation}\label{rig:active-identities}
+ B_0=B_0^T,\qquad
+ \alpha^2I-B_0^2=(DD^T)^r>0,\qquad
+ B_0'=r(B_0^2-\alpha^2I).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Choose a constant orthonormal basis of $\mathcal R_\alpha$, and let
+$E(x)$ be the matrix of the restriction of $G_\alpha(x)$ to this space.
+It has full column rank $\ell$ at every point, since the kernel of
+$G_\alpha$ is constant.  The projection onto its range is
+\[
+ P(x)=E(x)(E(x)^TE(x))^{-1}E(x)^T.
+\]
+The commutator in \eqref{rig:block-first-order} shows that $B_\alpha$
+preserves the range of $G_\alpha G_\alpha^T$, which is precisely the
+range of $E$.  Hence $E'=B_\alpha E$ has its columns in this range
+almost everywhere.  Differentiating the projection gives
+\begin{align*}
+ P'={}&(I-P)E'(E^TE)^{-1}E^T\\
+     &+E(E^TE)^{-1}(E')^T(I-P)=0
+ \quad\text{almost everywhere}.
+\end{align*}
+Since $P$ is $C^1$, it is constant.
+
+Choose a constant orthonormal basis of this range as well.  The resulting
+square matrix $D$ is invertible everywhere and $C^2$.  Almost everywhere,
+$D'D^{-1}$ is the restriction of $B_\alpha$ to its invariant range in
+these coordinates.  Equations~\eqref{rig:block-first-order} and
+\eqref{rig:block-square} give the symmetry and square identity in
+\eqref{rig:active-identities} almost everywhere; continuity extends
+them to every point.  Positivity is strict because $D$ is invertible.
+
+On $\mathcal R_\alpha$, Equation~\eqref{rig:continuous-potential}
+identifies the potential with $q(D^TD)^r$.  Equation~
+\eqref{rig:block-second-order} consequently becomes
+\[
+ D''=\alpha^2D-qD(D^TD)^r
+     =\alpha^2D-q(DD^T)^rD.
+\]
+The second equality follows, for example, by a singular-value
+decomposition.  Differentiating $B_0=D'D^{-1}$ and using $q=r+1$
+now gives
+\[
+ B_0'=D''D^{-1}-B_0^2
+      =\alpha^2I-q(\alpha^2I-B_0^2)-B_0^2
+      =r(B_0^2-\alpha^2I),
+\]
+as claimed.
+\end{proof}
+
+\subsection{The matrix Riccati equation separates the profiles}
+
+Fix a nonzero block and the matrices of Lemma~\ref{rig:fixed-left-spaces}.
+Diagonalize the symmetric matrix $B_0(x_*)$ at one point by a constant
+orthogonal change of range coordinates.  The right-hand side of its
+Riccati equation is a polynomial in $B_0$.  Uniqueness for this matrix
+ordinary differential equation therefore preserves the diagonal form.
+More explicitly, the strict inequality in \eqref{rig:active-identities}
+places each initial diagonal entry in $(-\alpha,\alpha)$.  There is a
+unique $y_j\in\R$ such that the scalar solution with that initial value is
+\begin{equation}\label{rig:tanh}
+ b_j(x)=-\alpha\tanh\bigl(r\alpha(x-y_j)\bigr),
+ \qquad 1\le j\le\ell.
+\end{equation}
+The diagonal matrix with these entries solves the same initial-value
+problem as $B_0$ on all of $\R$, so uniqueness gives equality everywhere.
+
+Equation~\eqref{rig:active-identities} implies that $DD^T$ is diagonal
+in these coordinates.  Since $D'=\operatorname{diag}(b_j)D$, each row
+of $D$ is a positive scalar multiple of its value at $x_*$.  None of
+these rows is zero, and their directions are mutually orthogonal because
+$DD^T$ is diagonal.  Taking their normalized directions as a constant
+orthonormal basis of $\mathcal R_\alpha$ makes $D$ diagonal with positive
+entries.  The potential in this fixed basis is therefore diagonal, with
+entries
+\begin{equation}\label{rig:profiles}
+ q\bigl(\alpha^2-b_j(x)^2\bigr)
+ =(r+1)\alpha^2\operatorname{sech}^2\bigl(r\alpha(x-y_j)\bigr).
+\end{equation}
+Combining these bases over the finitely many nonzero spaces
+$\mathcal R_\alpha$, and adding a basis of their orthogonal complement,
+proves the required necessity for real $W$.  There are at most $m$
+nonzero profiles by \eqref{rig:dimension-bound}.  The zero potential
+has the same description with no nonzero profile.
+
+\subsection{Complex Hermitian potentials}
+
+For a complex Hermitian potential $W$, take the realification
+$\widetilde W$ from \eqref{bound:realification}.  The proof of
+Proposition~\ref{bound:sharp} shows that it is equivalent to
+$W\oplus\overline W$ by a constant unitary matrix and that both the trace-power
+integral and the negative spectral moment are doubled.  Thus equality
+for $W$ implies equality for the real potential $\widetilde W$.
+
+The real result provides a constant unitary matrix $V$ such that
+\[
+ W(x)\oplus\overline W(x)
+   =V\operatorname{diag}(f_1(x),\ldots,f_{2m}(x))V^*
+ \quad\text{almost everywhere},
+\]
+where each $f_j$ is either a profile of the form \eqref{rig:profiles} or
+zero.  Let $P$ be the projection onto the first summand $\C^m$ and put
+$\widehat P=V^*PV$.  Since $P$ commutes with $W\oplus\overline W$,
+\[
+ (f_i(x)-f_j(x))\widehat P_{ij}=0
+ \quad\text{almost everywhere}.
+\]
+Consequently $\widehat P_{ij}=0$ whenever the functions $f_i,f_j$ are
+not equal almost everywhere.  Partition the indices into groups of
+identical functions, including the possible zero function.  The range
+of $\widehat P$ splits as the orthogonal sum of its intersections with
+these coordinate groups.  On each group the diagonal potential is a
+scalar multiple of the identity, so any constant orthonormal basis of
+that intersection diagonalizes its restriction.  Transporting these
+bases by $V$ to the first summand gives a constant unitary basis of
+$\C^m$ in which $W$ has the asserted profiles.  This proves necessity
+for complex Hermitian potentials.
+
+\subsection{Each listed potential attains equality}
+
+It remains to verify attainment, including the exact spectral moment,
+without appealing to a classification of scalar optimizers.  Fix $a>0$
+and $x_0\in\R$, and set
+\[
+ V(x)=(r+1)a^2\operatorname{sech}^2\bigl(ra(x-x_0)\bigr),
+ \qquad H=-\partial_x^2-V.
+\]
+The first-order factorization used in the commutation argument
+of~\cite[Section~II]{BenguriaLoss2000} also explains why this profile has
+only one negative eigenvalue.  Put
+\[
+ w(x)=a\tanh\bigl(ra(x-x_0)\bigr),
+ \qquad Q=\partial_x+w,\qquad \operatorname{Dom}Q=H^1(\R).
+\]
+Then $Q^*=-\partial_x+w$ on $H^1(\R)$.  Since $w$ and $w'$ are bounded,
+both product domains are $H^2(\R)$, and direct differentiation yields
+\begin{align}
+ Q^*Q&=H+a^2,
+ \label{rig:factorization}\\
+ QQ^*&=-\partial_x^2+a^2
+       +(r-1)a^2\operatorname{sech}^2\bigl(ra(x-x_0)\bigr)
+       \ge a^2.
+ \label{rig:partner}
+\end{align}
+Here the last inequality uses $r>1$.  The equation $Q\psi=0$ has the
+one-dimensional $L^2$ solution space
+\[
+ \ker Q=
+ \operatorname{span}\left\{
+    \operatorname{sech}^{1/r}\bigl(ra(x-x_0)\bigr)
+ \right\}.
+\]
+Thus $-a^2$ is a simple eigenvalue of $H$, and
+\eqref{rig:factorization} excludes eigenvalues below it.  If
+$H\phi=\lambda\phi$ with $-a^2<\lambda<0$, put
+$\mu=\lambda+a^2$ and $g=Q\phi$.  Then
+$\|g\|_2^2=\mu\|\phi\|_2^2>0$.  Moreover,
+$Q^*Q\phi=\mu\phi\in\operatorname{Dom}Q$, so
+$g\in\operatorname{Dom}(QQ^*)$ and $QQ^*g=\mu g$.  This contradicts
+\eqref{rig:partner}, since $\mu<a^2$.  By Lemma~\ref{pre:spectral},
+all negative spectrum is discrete.  Hence $-a^2$ is the sole negative
+eigenvalue.
+
+Finally, $2p-1=2\gamma$ and the substitution $s=\tanh y$ give
+\begin{align*}
+ \int_\R V(x)^p\,dx
+ &=\frac{(r+1)^p}{r}a^{2\gamma}
+     \int_\R\operatorname{sech}^{2p}y\,dy\\
+ &=\frac{(r+1)^p}{r}a^{2\gamma}
+     \int_{-1}^{1}(1-s^2)^{p-1}\,ds
+  =\frac{b_\sigma}{D_\sigma}a^{2\gamma},
+\end{align*}
+where $(r+1)^p/r=D_\sigma^{-1}$ and $p-1=\sigma$.  Consequently
+\[
+ \operatorname{Tr}(H)_-^\gamma
+   =a^{2\gamma}
+   =C_\gamma\int_\R V^p.
+\]
+The zero potential contributes zero to both sides.  Finite orthogonal
+direct sums add the spectral moments and trace-power integrals, and a
+constant unitary conjugation leaves both unchanged.  Every potential
+listed in Theorem~\ref{thm:main} therefore attains equality, completing
+its proof.

preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/build/sections/spectral-bound.tex

@@ -0,0 +1,65 @@
+\section{Recovering the sharp spectral bound}\label{bound:section}
+
+The action estimate implies the sharp inequality by the spectral passage
+of~\cite[Section~7]{MLT}. We record it before studying equality, so that
+the finite spectral moment used below has already been established.
+
+\begin{proposition}\label{bound:sharp}
+Every potential in Theorem~\ref{thm:main} satisfies
+\[
+ \Tr(H_W)_-^\gamma\le\frac{D_\sigma}{b_\sigma}
+                         \int_\R\tr(W^p).
+\]
+\end{proposition}
+
+\begin{proof}
+First suppose that $W$ is real symmetric. Choose any finite initial list
+of negative eigenvalues $-k_1^2,\ldots,-k_n^2$, with
+$k_1\ge\cdots\ge k_n>0$, and real orthonormal eigenfunctions $\varphi_i$.
+For $0<\xi<k_n^2$ put $\ell_i=(k_i^2-\xi)^{1/2}$ and
+$K=\diag(\ell_1,\ldots,\ell_n)$. There are real smooth compactly
+supported orthonormal functions $u_i$ with form matrix
+\begin{equation}\label{bound:trial}
+ (h_W[u_i,u_j])_{i,j=1}^n\le-K^2.
+\end{equation}
+Indeed, approximate the $\varphi_i$ in $H^1$, then multiply the family
+by the inverse square root of its Gram matrix. The Gram matrix tends to
+$I_n$, and the form matrix tends to $-\diag(k_i^2)$ by
+Lemma~\ref{pre:spectral}; an operator norm error below $\xi$ gives
+\eqref{bound:trial}.
+
+Let $U$ have rows $u_i$, and choose a bounded interval containing their
+supports. For every lower triangular $C$, the $i$th row $a_i$ of $A=CU$
+satisfies
+\[
+ h_W[a_i]+\ell_i^2\|a_i\|_2^2
+ \le\sum_{l\le i}C_{il}^2(\ell_i^2-\ell_l^2)\le0.
+\]
+Set $Z=A^TA$. The matrices $AA^T$ and $Z$ have the same nonzero
+eigenvalues, with multiplicity. Summing the preceding bound and applying
+Lemma~\ref{pre:young} yields
+\[
+ \mathcal E_K(A)
+ \le\int_\R\{\tr(WZ)-\tr(Z^q)\}
+ \le D_\sigma\int_\R\tr(W^p).
+\]
+Theorem~\ref{act:quantitative-action}, with its nonnegative defects
+discarded, gives
+$b_\sigma\sum_i\ell_i^{2\gamma}\le D_\sigma\int\tr(W^p)+\eta$
+for every $\eta>0$. Let $\eta\downarrow0$, then $\xi\downarrow0$,
+and finally exhaust the negative eigenvalues. The case of an empty
+negative spectrum is immediate.
+
+For complex Hermitian $W=E+iF$, with $E,F$ real, its realification is
+\begin{equation}\label{bound:realification}
+ \widetilde W=\begin{pmatrix}E&-F\\F&E\end{pmatrix}.
+\end{equation}
+The constant unitary matrix
+$2^{-1/2}\left(\begin{smallmatrix}I&iI\\I&-iI\end{smallmatrix}\right)$
+conjugates $\widetilde W$ to $W\oplus\overline W$. Hence $\widetilde W$
+is real symmetric and positive semidefinite,
+$\tr(\widetilde W^p)=2\tr(W^p)$, and
+$\Tr(H_{\widetilde W})_-^\gamma=2\Tr(H_W)_-^\gamma$.
+Here conjugation identifies the spectra of $H_W$ and $H_{\overline W}$.
+The real inequality, divided by two, proves the result.
+\end{proof}

preprints/Equality-cases-in-the-sharp-one-dimensional-matrix-Lieb-Thirring-inequality-October-5-2026/sharp-one-dimensional-lieb-thirring-inequalities-matrix-potentials.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:e88f3118d513e3133e672b809b58c6c7be43fcf09e802302ac0b569eeccd3298
+size 467633

preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/README.md

@@ -0,0 +1,18 @@
+# [Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 24, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026,
+  author = {{OpenAI}},
+  title = {{Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/paper.pdf}{OAI:Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026}},
+  year = {2026}
+}
+```
+

preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/adelic.tex

@@ -0,0 +1,139 @@
+\section{Adelic formulation and choices}\label{subsec:adelic-packet}
+
+The model of Lemma~\ref{lem:packet-model} also identifies the measure obtained from adelic
+torus Haar probability. We give the identification for arbitrary local
+type, so the lattice formulation includes all finite lattice and real
+diagonalization choices in the usual adelic formulation.
+
+Let $\mathbb A_f$ be the finite adeles of $\Q$, let
+$\widehat\Z=\prod_\ell\Z_\ell$, and put
+$\mathbb A_\Q=\R\times\mathbb A_f$ and
+$\mathbb A_K=K\otimes_\Q\mathbb A_\Q$.
+Write $\widehat R=R\otimes\widehat\Z$.
+Choose a rational basis $b_1,\ldots,b_n$ of $K$, and let
+$\rho(\alpha)$ denote multiplication by $\alpha$ on row coefficient
+vectors. It embeds the torus
+\[
+ \mathbf T_K=(\operatorname{Res}_{K/\Q}\mathbb G_m)/\mathbb G_m
+ \quad\hbox{in}\quad G=\operatorname{PGL}_{n/\Q}.
+\]
+The embedding matrix $B_\sigma=(\sigma_j(b_i))_{i,j}$ satisfies
+\begin{equation}\label{eq:regular-diagonalization}
+ \rho(\alpha)B_\sigma
+   =B_\sigma\operatorname{diag}(\sigma_1(\alpha),\ldots,\sigma_n(\alpha)).
+\end{equation}
+For $g_f\in G(\mathbb A_f)$, choose a lift to
+$\operatorname{GL}_n(\mathbb A_f)$ and set
+\[
+ \widehat M(g_f)=\widehat\Z^{\,n}g_f^{-1},\qquad
+ M(g_f)=K\cap\widehat M(g_f),
+\]
+using the chosen rational basis to identify coefficient vectors with
+elements of $K$. This is a full lattice, and its completion is
+$\widehat M(g_f)$: choose an integer $d\ge1$ with
+$d\widehat\Z^{\,n}\subset\widehat M(g_f)
+\subset d^{-1}\widehat\Z^{\,n}$, and patch in the finite quotient
+$d^{-1}\Z^n/d\Z^n$. Changing the projective lift multiplies it by a
+scalar finite idele. Every such idele is a rational scalar times an
+element of $\widehat\Z^\times$; the latter preserves every completed
+full lattice. Consequently this change only rescales $M(g_f)$ by
+a rational number.
+
+Choose $g_\infty\in G(\R)$ with
+$g_\infty^{-1}\mathbf T_K(\R)^\circ g_\infty=A_n$, using the
+odd-degree identification of $G(\R)$ with $\SL_n(\R)$.
+For $g=(g_\infty,g_f)$, the adelic packet is the projection of
+$\mathbf T_K(\Q)\backslash\mathbf T_K(\mathbb A_\Q)g$ to
+$G(\Q)\backslash G(\mathbb A_\Q)/G(\widehat\Z)$, equipped with
+the pushforward of torus Haar probability.
+
+\begin{proposition}\label{prop:adelic-packet}
+The adelic torus quotient is compact. Its packet and canonical
+probability agree with $\mathcal P_{K,M(g_f),\sigma'}$ and
+$\mu_{K,M(g_f),\sigma'}$ for an ordering $\sigma'$ of the real
+embeddings. Every prescribed local homothety type occurs this way.
+Changing the rational basis or the projective lifts does not change
+the packet; changing the allowed real diagonalizer only reorders the
+embeddings. Thus Theorem~\ref{thm:main} includes every such adelic choice.
+\end{proposition}
+
+\begin{proof}
+Write $M=M(g_f)$ and $\widehat M=\widehat M(g_f)$.
+First identify the ambient quotient with $X_n$. Given a finite matrix
+lift $g_f$, choose a basis matrix $c\in\operatorname{GL}_n(\Q)$
+for $\Q^n\cap\widehat\Z^{\,n}g_f^{-1}$. Then
+$cg_f\in\operatorname{GL}_n(\widehat\Z)$, and the corresponding
+real lattice is $\Z^ncg_\infty$, normalized to covolume one.
+Changing this basis or either matrix lift has no effect on that
+lattice. This constructs the usual quotient identification directly.
+For odd $n$, the signed real $n$th root gives
+$[h]\mapsto(\det h)^{-1/n}h$, identifying
+$G(\R)$ with $\SL_n(\R)$ and $G(\Z)$ with $\SL_n(\Z)$.
+
+Multiplication matrices modulo scalars give
+\begin{equation}\label{eq:idelic-torus-quotient}
+ \mathbf T_K(\Q)\backslash\mathbf T_K(\mathbb A_\Q)
+ \simeq K^\times\backslash\mathbb A_K^\times/\mathbb A_\Q^\times.
+\end{equation}
+Here the pointwise lifts can also be seen by elementary linear algebra:
+over any ground field $F_0$ containing $\Q$, a matrix whose projective
+class lies in $\mathbf T_K$ becomes a scalar multiple of a multiplication
+operator over an algebraic closure. It therefore commutes exactly with
+the regular representation already over $F_0$. The centralizer of that
+representation is the algebra of multiplication operators, so the matrix
+belongs to $(K\otimes_\Q F_0)^\times$, with scalar ambiguity $F_0^\times$.
+Finite lifts may be chosen integral at almost all primes.
+
+A finite idele $t_f$ acts on the completed lattice by
+$\widehat M\mapsto t_f^{-1}\widehat M$. Hence its translates are
+exactly the prescribed local homothety type. The stabilizer of
+$\widehat M$ among finite ideles is
+\[
+ \{z:z\widehat M=\widehat M\}
+       =(\cO(M)\otimes\widehat\Z)^\times:
+\]
+equality holds precisely when both $z$ and $z^{-1}$ preserve
+$\widehat M$. In the projective torus the stabilizer is the image
+of this group, since a representative that preserves the lattice
+up to a scalar idele can be rescaled to preserve it exactly.
+The unit subgroup is compact open in the finite ideles and has finite
+index in $\widehat R^\times$; at each prime not dividing $q$ it equals
+$R_\ell^\times$. Its projective image is compact open as well.
+No invertibility of $M$ over $\cO(M)$ is used.
+
+To describe the quotient and its measure, first divide
+\eqref{eq:idelic-torus-quotient} by the image of
+$\widehat R^\times$. Fractional-ideal factorization and
+$\mathbb A_\Q^\times
+=\Q^\times\R^\times\widehat\Z^\times$
+give one real fiber $E\backslash C$ for each ordinary ideal class.
+These fibers are compact by the unit theorem, and the omitted
+finite-unit group is compact. Thus the torus quotient is compact.
+At the stabilizer of $\widehat M$, the finite fibers instead retain
+the translates $\mathcal N(I)$. The same decomposition becomes
+\[
+ \coprod_{[I]}(\mathcal N(I)\times C)/E,
+\]
+with exactly the action in Lemma~\ref{lem:packet-model}.
+Haar measure pushes to uniform counting measure on each finite
+unit orbit and to the stated logarithmic Haar measure on $C$.
+After normalization it is therefore precisely the probability in
+that lemma. For $g_\infty=[B_\sigma]$,
+\eqref{eq:regular-diagonalization} identifies the resulting real
+lattices with its displayed map. This proves both the packet
+identification and the equality of measures, including the orbit
+volume weights.
+
+Finally, every full $M$ is obtained by choosing a basis matrix for
+its completed lattice and taking its inverse as $g_f$.
+Any other allowed real diagonalizer has the form $[B_\sigma dP]$,
+where $d$ is diagonal and $P$ is a permutation matrix, because it
+must permute the joint eigenspaces. The diagonal factor translates
+the real torus and permutes its sign components, preserving its
+Haar probability; the permutation reorders the embeddings.
+If a rational change of basis has matrix $c$, then
+$B'_\sigma=cB_\sigma$ and $\rho'=c\rho c^{-1}$, while the same
+packet data have representatives $g'_\infty=cg_\infty$ and
+$g'_f=cg_f$. Thus the torus is conjugated and the translated packet
+is left-multiplied by $c$, which disappears in the ambient rational quotient.
+\end{proof}

preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/analytic.tex

@@ -0,0 +1,189 @@
+\section{Residue-relative weighted ideal counts}\label{sec:analytic}
+
+Write
+\[
+ a_K(m)=\#\{\mathfrak a\subseteq R:\N\mathfrak a=m\},
+ \qquad \kappaK=\operatorname*{Res}_{s=1}\zeta_K(s)>0.
+\]
+We first retain the residue in the analytic estimates. All constants
+subscripted by $n$ are uniform over the fields under consideration.
+
+\begin{lemma}\label{lem:residue}
+For $D$ sufficiently large in terms of $n$,
+\begin{equation}\label{eq:residue}
+ h\zeta_K(1+h)\asymp_n\kappaK
+ \quad\left(0<h\le\frac4{\log D}\right),
+ \qquad \kappaK^{-1}\ll_n\log D.
+\end{equation}
+For the remaining fields, the first comparison holds uniformly for
+sufficiently small positive $h$, and $\kappaK^{-1}\ll_n1$.
+\end{lemma}
+
+\begin{proof}
+Put $L=\log D$. Stark's Lemma~3 gives at most one zero in
+$\operatorname{Re}s\ge1-(4L)^{-1}$,
+$|\operatorname{Im}s|\le(4L)^{-1}$; such a zero is real and simple.
+His Lemma~8, which also applies to nonnormal fields, says that a real
+zero $\beta\ge1-(4n!L)^{-1}$ forces a quadratic subfield of $K$
+\cite[Lemmas~3 and~8]{Stark1974}. Since $n$ is odd, there is no such
+subfield. Consequently, for a constant $c_n>0$, every nontrivial zero
+$\rho$ satisfies $|1-\rho|\ge c_n/L$.
+
+The real logarithmic derivative of the completed zeta function gives
+\[
+ S_K(\sigma):=\sum_\rho\operatorname{Re}\frac1{\sigma-\rho}
+ =\frac1\sigma+\frac1{\sigma-1}+\frac L2
+   +\frac{\zeta_K'}{\zeta_K}(\sigma)+G_n(\sigma),\qquad \sigma>1,
+\]
+where zeros are counted with multiplicity and
+\[
+ G_n(\sigma)=\frac n2
+ \left(\frac{\Gamma'(\sigma/2)}{\Gamma(\sigma/2)}-\log\pi\right).
+\]
+This is the Hadamard identity in
+\cite[Equation~(9)]{Stark1974}. The zero terms are nonnegative,
+$G_n$ is bounded on $[1,2]$, and the Euler product gives
+$\zeta_K'/\zeta_K\le0$ there. Thus, at $\sigma_0=1+L^{-1}$,
+$S_K(\sigma_0)\ll_n L$.
+
+This bound persists uniformly for $1<\sigma\le1+4/L$.
+Indeed, for $\rho=\beta+i\gamma$, set
+$X=L(1-\beta)$, $Y=L\gamma$, and $U=L(\sigma-1)$.
+Then $X\ge0$, $0<U\le4$, and $|X+iY|\ge c_n$. Hence
+\[
+ \frac{\operatorname{Re}(\sigma-\rho)^{-1}}
+      {\operatorname{Re}(\sigma_0-\rho)^{-1}}
+ =\frac{X+U}{X+1}
+   \frac{(X+1)^2+Y^2}{(X+U)^2+Y^2}
+ \le4(1+3/c_n)^2.
+\]
+Summing gives $0\le S_K(\sigma)\ll_n L$.
+For $F_K(\sigma)=(\sigma-1)\zeta_K(\sigma)$, it follows that
+\[
+ \frac{F_K'}{F_K}(\sigma)
+ =S_K(\sigma)-\frac1\sigma-\frac L2-G_n(\sigma)
+ =O_n(L).
+\]
+Since $F_K$ extends analytically to $1$ with $F_K(1)=\kappaK>0$,
+integration over an interval of length at most $4/L$ proves the
+comparison in \eqref{eq:residue}. At $\sigma_0$, the inequality
+$\zeta_K(\sigma_0)\ge1$ also gives $\kappaK\gg_n L^{-1}$.
+
+For completeness, only finitely many fields of this fixed degree
+have discriminant below the cutoff. One can see this directly here:
+nonzero vectors of $D^{-1/(2n)}\sigma(R)$ have absolute coordinate
+product at least $D^{-1/2}$, so Mahler's criterion puts these lattices
+in a fixed compact set. Local basis lifts and a finite cover give
+bases of $R$ with uniformly bounded conjugates. A basis element
+outside $\Q$ generates $K$, since $n$ is prime; its monic integral
+minimal polynomial therefore has bounded coefficients. There are
+only finitely many such polynomials. The simple pole of each of
+these finitely many zeta functions now gives the remaining claims
+uniformly for sufficiently small $h>0$.
+\end{proof}
+
+We also use the following uniform form of Shiu's theorem
+\cite{Shiu1980,Pollack2020}: if $b$ is nonnegative and multiplicative
+and $b(m)\le d_n(m)$, where $d_n$ is the $n$-fold divisor function,
+then
+\begin{equation}\label{eq:shiu}
+ \sum_{u-v<m\le u}b(m)
+ \ll_n\frac v{\log u}
+       \exp\left(\sum_{\ell\le u}\frac{b(\ell)}\ell\right)
+ \qquad(u^{1/2}\le v\le u).
+\end{equation}
+Here $\ell$ denotes a rational prime, and the threshold for $u$
+depends only on $n$. To check uniformity, the growth conditions in
+\cite[Theorem~1.1]{Pollack2020}, with all excluded residue sets
+empty, are supplied by
+$d_n(\ell^j)=\binom{j+n-1}{n-1}\le n^j$ and
+$d_n(m)\ll_{n,\eps}m^\eps$. Its strict interval condition
+$u^\beta<v\le u$ is satisfied by taking $\beta=1/4$; thus
+\eqref{eq:shiu} includes both endpoints of the displayed range.
+
+\begin{proposition}\label{prop:weighted-count}
+Fix $A_0\ge1$ and $0<\eta\le1$. For $Q$ sufficiently large in terms
+of $n,A_0,\eta$,
+\begin{equation}\label{eq:weighted-count}
+ \sum_{x-y<\N\mathfrak a\le x}P(\mathfrak a)
+ \ll_n \frac{\kappaK y}{q}
+ \qquad
+ (Q^{3/4}\le x\le A_0Q,\quad \eta x\le y\le x).
+\end{equation}
+The implied constant is independent of $A_0$ and $\eta$.
+\end{proposition}
+
+\begin{proof}
+Set $b(m)=a_K(m)\one_{(m,q)=1}$. This function is nonnegative,
+multiplicative, and bounded by $d_n$. Indeed, at every prime $\ell$,
+including the ramified primes, the local generating series for
+$a_K$ is $\prod_{\mathfrak p\mid\ell}(1-T^{f_{\mathfrak p}})^{-1}$;
+it is coefficientwise bounded by $(1-T)^{-n}$.
+Decomposing an ideal into its part over $S$ and its coprime part,
+the sum in \eqref{eq:weighted-count} becomes
+\begin{equation}\label{eq:weighted-split}
+ \sum_{\mathbf k}P(\mathbf k)
+       \sum_{(x-y)/d(\mathbf k)<m\le x/d(\mathbf k)}b(m).
+\end{equation}
+
+First suppose $d=d(\mathbf k)\le Q^{1/8}$, and put $u=x/d$.
+Then $u\ge Q^{5/8}\ge D^{1/4}$. For $h=1/\log u$,
+Lemma~\ref{lem:residue} gives
+$\zeta_K(1+h)\ll_n\kappaK\log u$.
+There is no upper restriction on $u$: increasing $u$ only decreases
+$h$, so the lemma applies even when $D$ stays bounded and $q$ grows.
+Positivity of the Euler logarithm yields
+\[
+ \sum_{\ell\le u}a_K(\ell)\ell^{-1-h}\le\log\zeta_K(1+h).
+\]
+Since $a_K(\ell)\le n$ and
+$\sum_{\ell\le u}(\log\ell)/\ell\ll\log u$, replacing
+$\ell^{-1-h}$ by $\ell^{-1}$ costs only $O_n(1)$.
+The latter prime-sum bound follows, for example, by comparing
+$\sum_{\ell\le u}\lfloor u/\ell\rfloor\log\ell$ with
+$\log(\lfloor u\rfloor!)$. Therefore
+\[
+ \exp\left(\sum_{\ell\le u}\frac{a_K(\ell)}\ell\right)
+ \ll_n\kappaK\log u.
+\]
+Furthermore,
+\[
+ \log Z_S=\sum_{\ell\in S}\frac{a_K(\ell)}\ell+O_n(1),
+ \qquad
+ \sum_{\substack{\ell\in S\\\ell>u}}
+       \frac{a_K(\ell)}\ell
+ \ll_n\frac{\log(2q)}{Q^{5/8}}\ll_n1.
+\]
+The first error is bounded by a constant times
+$\sum_\ell\ell^{-2}$. Removing the primes of $S$ thus gives
+\begin{equation}\label{eq:coprime-euler}
+ \exp\left(\sum_{\ell\le u}\frac{b(\ell)}\ell\right)
+ \ll_n\frac{\kappaK\log u}{Z_S}.
+\end{equation}
+For $Q$ sufficiently large at fixed $\eta$, the length
+$v=y/d\ge\eta u$ is at least $u^{1/2}$.
+Applying \eqref{eq:shiu}, then \eqref{eq:coprime-euler} and
+\eqref{eq:shell-sum}, bounds the contribution of these $d$ by
+\[
+ C_n\frac{\kappaK y}{Z_S}
+       \sum_{d(\mathbf k)\le Q^{1/8}}\frac{P(\mathbf k)}{d(\mathbf k)}
+ \le C_n\frac{\kappaK y}{q}.
+\]
+
+For $d>Q^{1/8}$, terms with $d>x$ vanish because the inner sum
+in \eqref{eq:weighted-split} then contains no positive integer.
+For the remaining terms, $x/d\ge1$, and the elementary bound
+$\sum_{m\le z}d_n(m)\le z(1+\log z)^{n-1}$ for $z\ge1$
+follows by summing reciprocals in the first $n-1$ factors.
+Using \eqref{eq:weight-tail}, with $s=1/(2n)$, the remaining
+contribution to \eqref{eq:weighted-split} is at most
+\[
+ C_n\frac{x}{q}Q^{-s/16}(1+\log x)^{n-1}
+ =o\left(\frac{\kappaK x}{q}\right).
+\]
+This is uniform in the stated range: Lemma~\ref{lem:residue}
+gives $\kappaK^{-1}\ll_n\log(3D)$, and $D\le Q^2$.
+At fixed $A_0,\eta$, increase the threshold until this term is at
+most $\kappaK\eta x/q\le\kappaK y/q$. The implied constant has
+not changed with $\eta$, proving the proposition.
+\end{proof}

preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/entropy.tex

@@ -0,0 +1,167 @@
+\section{Entropy and identification of the limit}\label{sec:entropy}
+
+The ball estimate has exponent $n$, whereas $A_n$ has dimension $n-1$.
+We follow the ordinary-ball strategy of
+\cite[Section~2.7.1]{ELMV2011}. We retain the estimate for every invariant
+probability dominated by a finite multiple of the limit, so it forces
+positive entropy after restriction to any positive-weight collection of
+ergodic components. Measure classification then identifies the limit.
+
+For an invariant probability $\nu$, write $h_\nu(a)$ for the
+measure-theoretic entropy of the map $x\mapsto xa$.
+Fix distinct real numbers $\lambda_1,\ldots,\lambda_n$ with sum zero and put
+\[
+ a(t)=\operatorname{diag}(e^{\lambda_1t},\ldots,e^{\lambda_nt}),
+ \qquad \lambda=\min_{i\ne j}|\lambda_i-\lambda_j|>0.
+\]
+
+\begin{proposition}[Entropy from ordinary balls]\label{prop:entropy-from-balls}
+Let $n\ge3$ and let $\mu$ be an $A_n$-invariant probability on $X_n$.
+Suppose that for every compact $\Omega\subset X_n$ there are
+$C_\Omega,r_\Omega>0$ such that
+\[
+ \mu(xB(r))\le C_\Omega r^n
+ \qquad(x\in\Omega,\ 0<r<r_\Omega).
+\]
+Then every $A_n$-invariant probability $\xi$ satisfying $\xi\le c\mu$
+for some finite $c$ has $h_\xi(a(1))\ge\lambda/3$.
+\end{proposition}
+
+\begin{proof}
+Fix $0<b<1/4$ and consider the two-sided neighborhoods
+\[
+ T_t=a(t)B(b)a(-t)\cap a(-t)B(b)a(t),\qquad t\ge0.
+\]
+If $h\in T_t$, then
+\begin{equation}\label{eq:tube-entries}
+ |h_{ii}-1|<b,\qquad
+ |h_{ij}|<b e^{-|\lambda_i-\lambda_j|t}\quad(i\ne j).
+\end{equation}
+Set $r=e^{-\lambda t}$.  Every term in the determinant expansion other
+than the product of the diagonal entries contains at least two
+off-diagonal entries.  Since $\det h=1$, it follows that
+$\prod_i h_{ii}=1+O_{n,b}(r^2)$.  Thus
+\[
+ d(h)=\operatorname{diag}\left(h_{11},\ldots,h_{n-1,n-1},
+                   \Bigl(\prod_{i<n}h_{ii}\Bigr)^{-1}\right)
+\]
+belongs to a fixed compact subset $\mathcal D\subset A_n$, and
+$d(h)^{-1}h\in B(Cr)$ for a constant depending only on $n,b$.
+
+In logarithmic coordinates, $\mathcal D$ is a bounded subset of $\R^{n-1}$.
+Covering it by a mesh of size comparable to $r$, and enlarging $C$ to
+allow for matrix multiplication, gives
+\begin{equation}\label{eq:tube-cover}
+ T_t\subset\bigcup_{j=1}^{N_t}d_jB(Cr),
+ \qquad d_j\in\mathcal D,\qquad N_t\le Cr^{-(n-1)}.
+\end{equation}
+For $x$ in a compact set $\Omega$, all the centers $xd_j$ lie in the
+fixed compact set $\Omega\mathcal D$.  The assumed ball estimate and domination
+therefore imply, for all sufficiently large $t$,
+\begin{equation}\label{eq:tube-mass}
+ \xi(xT_t)\le C_{\Omega,b,c}\,r^{-(n-1)}r^n
+             =C_{\Omega,b,c}e^{-\lambda t}
+ \qquad(x\in\Omega).
+\end{equation}
+Since $\xi$ is a probability, the constant can be enlarged to cover
+the finitely many integer times before this estimate holds.
+
+We use the entropy criterion of
+Einsiedler--Lindenstrauss--Michel--Venkatesh
+\cite[Corollary~3.3]{ELMV2009}.  In the notation
+\[
+ B^{(s,t)}=a(-s)Ba(s)\cap a(-t)Ba(t),
+\]
+the criterion states the following: if flow-invariant probabilities
+$\xi_i$ converge weakly to a probability $\xi$, and $t_i\to\infty$,
+then $h_\xi(a(1))\ge\eta$ provided that, for every compact set $\Omega$,
+there are an identity neighborhood $B$ and a constant $C_\Omega$ such that
+\[
+ \xi_i\bigl(xB^{(-t_i,t_i)}\bigr)
+       \le C_\Omega e^{-2\eta t_i}\qquad(x\in\Omega).
+\]
+This criterion applies on the noncompact lattice space; its requirement
+that the limit be a probability is essential.  Here take the constant
+sequence $\xi_i=\xi$, the neighborhood $B(b)$, $t_i=i$, and
+$\eta=\lambda/3$.  Since $B(b)^{(-t,t)}=T_t$,
+Equation~\eqref{eq:tube-mass} proves the required estimate.
+\end{proof}
+
+\begin{lemma}[Entropy in the ergodic components]\label{lem:entropy-components}
+Under the hypotheses of Proposition~\ref{prop:entropy-from-balls},
+almost every $A_n$-ergodic component of $\mu$ has positive entropy
+for $a(1)$.
+\end{lemma}
+
+\begin{proof}
+The group $A_n\simeq\R^{n-1}$ is locally compact and second countable,
+and its action on the standard Borel space $X_n$ is continuous.
+The ergodic decomposition theorem
+\cite[Theorem~5.2]{GreschonigSchmidt2000}, specialized to invariant
+probabilities, therefore supplies a measurable decomposition
+\[
+ \mu=\int_Z\nu_z\,\dd\tau(z)
+\]
+into $A_n$-invariant, $A_n$-ergodic probabilities over a factor fixed
+by $A_n$. Set $T(x)=xa(1)$.
+The function $z\mapsto h_{\nu_z}(T)$ is measurable.  Indeed, choose
+increasing finite Borel partitions $\mathcal P_k$ generating the Borel
+$\sigma$-algebra of $X_n$.  For every $T$-invariant probability $\nu$,
+\[
+ h_\nu(T)=\sup_k\inf_{N\ge1}\frac1N
+ H_\nu\left(\bigvee_{j=0}^{N-1}T^{-j}\mathcal P_k\right),
+\]
+where $H_\nu$ denotes the Shannon entropy of a finite partition.
+The right side is a countable combination of Borel functions of $\nu$.
+
+The component partition is fixed by $T$, since $a(1)\in A_n$.
+The entropy integral formula for a fixed invariant partition
+\cite[Theorem~9.8]{Rokhlin1967} consequently gives
+\begin{equation}\label{eq:entropy-decomposition}
+ h_\mu(T)=\int_Z h_{\nu_z}(T)\,\dd\tau(z).
+\end{equation}
+The same formula holds after restricting to any measurable collection
+of components.  It does not require the $\nu_z$ to be $T$-ergodic:
+the induced transformation on the component space is the identity
+and contributes zero entropy.
+
+If $Z_0=\{z:h_{\nu_z}(T)=0\}$ had weight $w=\tau(Z_0)>0$, then
+\[
+ \xi=\frac1w\int_{Z_0}\nu_z\,\dd\tau(z)
+\]
+would be an $A_n$-invariant probability with $\xi\le w^{-1}\mu$.
+Equation~\eqref{eq:entropy-decomposition}, applied to this restricted
+decomposition, would give $h_\xi(T)=0$, contradicting
+Proposition~\ref{prop:entropy-from-balls}.
+\end{proof}
+
+\paragraph{Measure classification.}
+Einsiedler--Katok--Lindenstrauss
+\cite[Theorem~1.3 and Corollary~1.4]{EKL2006} prove that, for prime
+$n\ge3$, an $A_n$-invariant and $A_n$-ergodic probability on
+$\operatorname{SL}_n(\R)/\operatorname{SL}_n(\Z)$ is Haar whenever
+some diagonal one-parameter subgroup has positive entropy.
+Their action is on the left.  The map
+$\Gamma g\mapsto g^{-1}\Gamma$, where $\Gamma=\operatorname{SL}_n(\Z)$,
+changes our right action by $a(t)$ into the left action by $a(-t)$.
+Since entropy is unchanged by inversion of an invertible transformation,
+their result applies to our convention without any additional hypothesis.
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+Let $\mu_i$ be the packet probabilities for a sequence in the theorem.
+Equation~\eqref{eq:discriminant-scale} gives $Q_i\to\infty$.
+By Proposition~\ref{prop:tightness}, this sequence is tight and every
+weak subsequential limit $\mu$ is an $A_n$-invariant probability.
+Proposition~\ref{prop:ball-bound} supplies the hypotheses of
+Proposition~\ref{prop:entropy-from-balls}.
+Lemma~\ref{lem:entropy-components} shows that almost every
+$A_n$-ergodic component of $\mu$ has positive entropy for $a(1)$.
+As $n$ is prime, the classification just stated makes every such
+component equal to $m_n$.  Integrating the components gives $\mu=m_n$.
+
+Thus every convergent subsequence has the same probability limit.
+Together with tightness, this proves $\mu_i\to m_n$ and excludes
+loss of mass.  All preceding estimates were for each prescribed local
+homothety packet separately, so no averaging over different local
+module types is used.
+\end{proof}

preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/geometry.tex

@@ -0,0 +1,150 @@
+\section{Nonescape and ordinary ball bounds}\label{sec:geometry}
+
+We apply the weighted count in two ways. Cubes about zero control
+the cusp; small boxes away from the coordinate hyperplanes control
+ordinary neighborhoods in the lattice space.
+
+\begin{proposition}\label{prop:tightness}
+Every sequence of the packet probabilities with $Q\to\infty$
+is tight. More precisely, for $0<\theta<1$,
+\begin{equation}\label{eq:cusp}
+ \limsup_{Q\to\infty}\mu_*
+ \{\Lambda:\exists\,0\ne v\in\Lambda,\ \|v\|_\infty<\theta\}
+       \le C_n\theta^n.
+\end{equation}
+Every subsequential probability limit is $A_n$-invariant.
+\end{proposition}
+
+\begin{proof}
+For $f=\one_{[-\theta,\theta]^n}$, the first $n-1$ logarithms
+in \eqref{eq:product-fiber} are at most $\log\theta$.
+The last coordinate bounds the sum of their gaps from
+$\log\theta$ by $\log(\theta^n/z)$. Thus
+\[
+ \cV_f(z)=\frac{2^n}{(n-1)!}
+           \bigl(\max\{0,\log(\theta^n/z)\}\bigr)^{n-1}.
+\]
+Put $X=\theta^nQ$ and $U=Q^{3/4}$. Up to an $n$-dependent
+constant, Proposition~\ref{prop:unfolding} reduces the first
+moment to
+\[
+ \frac1{\sqrt D\,\kappaK}
+ \sum_{\N\mathfrak a\le X}
+     P(\mathfrak a)\left(\log\frac X{\N\mathfrak a}\right)^{n-1}.
+\]
+If $X<1$, the sum is empty. Otherwise its logarithmic factor is
+at most $(\log Q)^{n-1}$, since $\theta<1$. The terms with
+$\N\mathfrak a\le\min(U,X)$ contribute at most
+$C_nQ^{-1/4}(\log Q)^{n-1}$, by applying
+Proposition~\ref{prop:weighted-count} at $x=y=U$.
+When $X>U$, define the truncated cumulative sum
+\[
+ A_U(t)=\sum_{U<\N\mathfrak a\le t}P(\mathfrak a),
+ \qquad U\le t\le X.
+\]
+Writing each logarithmic power as an integral and applying Tonelli gives
+\begin{align*}
+ \sum_{U<\N\mathfrak a\le X}
+   P(\mathfrak a)\left(\log\frac X{\N\mathfrak a}\right)^{n-1}
+ &= (n-1)\int_U^X A_U(t)
+      \left(\log\frac Xt\right)^{n-2}\frac{dt}{t} \\
+ &\ll_n \frac{\kappaK}{q}
+      \int_U^X\left(\log\frac Xt\right)^{n-2}\dd t
+ \ll_n \frac{\kappaK}{q}X.
+\end{align*}
+Here the same proposition at $x=y=t$ bounds $A_U(t)$, and the
+last inequality follows from
+$\int_0^1(\log(1/v))^{n-2}\dd v<\infty$.
+There is no lower boundary term, since $A_U(U)=0$, and the
+logarithmic weight vanishes at $X$. If $X\le U$, this part is absent.
+Dividing by $\sqrt D\,\kappaK$ gives $O_n(\theta^n)$.
+Since $E_f\ge1$ whenever the displayed cusp event occurs,
+this proves \eqref{eq:cusp}.
+
+Mahler's compactness criterion \cite[Section~3, Theorem~2]{Mahler1946} says that the
+complement of this cusp event is compact. Given a desired
+mass error, choose $\theta$ first and then $Q$ sufficiently large.
+The finitely many earlier packet supports are compact by
+Lemma~\ref{lem:packet-model}, so the whole sequence is tight.
+The lattice space is locally compact and second countable;
+weak compactness for tight probabilities gives probability
+subsequential limits. Invariance passes to them by testing
+against continuous compactly supported functions.
+\end{proof}
+
+For $r>0$, let
+\[
+ B(r)=\{h\in\SL_n(\R):
+             \max_{i,j}|h_{ij}-\delta_{ij}|<r\}.
+\]
+
+\begin{proposition}\label{prop:ball-bound}
+Let $\mu$ be a weak limit of packet probabilities along
+$Q\to\infty$. For every compact $\Omega\subset X_n$ there are
+$C_\Omega,r_\Omega>0$ such that
+\begin{equation}\label{eq:ball-bound}
+ \mu(xB(r))\le C_\Omega r^n
+       \qquad(x\in\Omega,\ 0<r<r_\Omega).
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+First fix a compact $V\subset(\R^\times)^n$ and $b_0\ge1$.
+For $w\in V$ and small $r$, take
+$f=\one_{w+[-b_0r,b_0r]^n}$. Each coordinate has a fixed sign,
+is uniformly separated from zero, and has logarithmic width
+$O_{V,b_0}(r)$. Its absolute product lies within $Cr$ of
+$z_w=\prod_j|w_j|$, so
+\[
+ \cV_f(z)\le Cr^{n-1}
+       \one_{[z_w-Cr,z_w+Cr]}(z)
+\]
+for some $C\ge1$ depending only on $V,b_0,n$.
+Choose $r_0$ small enough that $z_w>4Cr_0$ on $V$.
+Use Proposition~\ref{prop:weighted-count} with
+\[
+ x=(z_w+2Cr)Q,\qquad y=4CrQ.
+\]
+This slightly enlarged interval contains every contributing
+ideal norm. Uniformly in $w$, for sufficiently large $Q$ it has
+$Q^{3/4}\le x\le A_0Q$ with a fixed $A_0\ge1$, and
+$\eta x\le y\le x$ with $\eta=4Cr/A_0\le1$, after decreasing
+$r_0$ if necessary. For every fixed $r\in(0,r_0)$,
+Proposition~\ref{prop:unfolding} therefore gives, eventually,
+\begin{equation}\label{eq:vector-box}
+ \int E_{\one_{w+[-b_0r,b_0r]^n}}\dd\mu_*
+       \ll_{V,b_0,n}
+       \frac{r^{n-1}}{\sqrt D\,\kappaK}
+       \frac{\kappaK rQ}{q}=r^n .
+\end{equation}
+Here $Q=q\sqrt D$. The implicit constant is independent of $r$. The threshold for $Q$
+may depend on $r$, which is all that will be needed.
+
+We next choose a suitable lattice vector at each $x\in\Omega$.
+Every full lattice has a vector with no zero coordinate:
+for any basis, its combination with coefficients
+$(1,N,\ldots,N^{n-1})$ has coordinates that are nonzero
+polynomials in $N$, so an integer $N$ can avoid their finitely
+many roots. Near each center choose a continuous local lift to basis
+matrices, and continue the chosen vector using its fixed integer
+basis coefficients. On a sufficiently small neighborhood with compact
+closure in that chart, its coordinates remain bounded and bounded away
+from zero. A finite cover of $\Omega$ by such neighborhoods therefore
+gives choices $w_x$ lying in one compact
+$V\subset(\R^\times)^n$. This pointwise selection need not be
+continuous or measurable.
+
+For $h$ in $B(r)$, each coordinate of $w_xh-w_x$ has absolute
+value at most $r\sum_i|(w_x)_i|$. Thus one fixed $b_0$ makes
+every lattice in $xB(r)$ contain a nonzero vector in
+$w_x+[-b_0r,b_0r]^n$. Equation~\eqref{eq:vector-box} bounds
+the packet mass of $xB(r)$ by $C_\Omega r^n$.
+This set is open, so Portmanteau gives
+\[
+ \mu(xB(r))\le\liminf_i\mu_i(xB(r))\le C_\Omega r^n.
+\]
+The argument holds for every fixed $x$ and $r$ with the same
+constant on $\Omega$. It proves \eqref{eq:ball-bound} for
+all $0<r<r_\Omega$, without any exchange of shrinking-radius
+and packet limits.
+\end{proof}

preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/local.tex

@@ -0,0 +1,97 @@
+\section{Uniform control of the local weights}\label{sec:local}
+
+The ideal sum in Proposition~\ref{prop:unfolding} involves arbitrary
+local modules. We now isolate two estimates that are uniform in
+those modules and in their indices. The first gives the total
+normalization; the second makes large local norm contributions
+negligible.
+
+Let $\mathbf k=(\mathbf k_\ell)_{\ell\in S}$ range over nonnegative
+valuation tuples, and write
+\[
+ d_\ell(\mathbf k_\ell)
+   =\ell^{\sum_{\mathfrak p\mid\ell}f_{\mathfrak p}k_{\mathfrak p}},
+ \qquad d(\mathbf k)=\prod_{\ell\in S}d_\ell(\mathbf k_\ell),
+ \qquad P(\mathbf k)=\prod_{\ell\in S}P_\ell(\mathbf k_\ell).
+\]
+Thus $d$ is the norm of the $S$-supported part of an ideal. Also set
+\[
+ Z_\ell=\prod_{\mathfrak p\mid\ell}
+          (1-\ell^{-f_{\mathfrak p}})^{-1},\qquad
+ Z_S=\prod_{\ell\in S}Z_\ell.
+\]
+Empty products are one.
+
+\begin{lemma}\label{lem:local-moment}
+For the weights just defined,
+\begin{equation}\label{eq:shell-sum}
+ \sum_{\mathbf k}\frac{P(\mathbf k)}{d(\mathbf k)}=\frac{Z_S}{q}.
+\end{equation}
+If $s=1/(2n)$, then, for every $\eps>0$,
+\begin{equation}\label{eq:weight-moment}
+ \sum_{\mathbf k}P(\mathbf k)d(\mathbf k)^{s-1}
+       \ll_{n,\eps}\frac{q^\eps}{q}.
+\end{equation}
+In particular,
+\begin{equation}\label{eq:weight-tail}
+ \sum_{d(\mathbf k)>Q^{1/8}}\frac{P(\mathbf k)}{d(\mathbf k)}
+       \ll_n \frac1q Q^{-s/16}.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Normalize additive Haar measure on $R_\ell$ to have mass one.
+The valuation shell $\mathbf k_\ell$ has mass
+$Z_\ell^{-1}/d_\ell(\mathbf k_\ell)$. Its conditional law is
+invariant under the transitive unit action, so $P_\ell$ is
+exactly the fraction of this shell lying in $L_\ell$.
+Since $L_\ell$ has mass $1/q_\ell$, summing over shells and
+then multiplying over $S$ proves \eqref{eq:shell-sum}.
+Elements with a zero component have additive measure zero.
+
+Choose an additive-Haar uniform element of $L_\ell$, and denote
+its component valuations by $v_{\mathfrak p}$. The condition
+$R_\ell L_\ell=R_\ell$ implies that the projection to every
+component contains a unit. Modulo the $k$th power of its maximal
+ideal, the projection therefore contains all $\Z_\ell$ multiples
+of that unit, at least $\ell^{\lceil k/e_{\mathfrak p}\rceil}$
+elements. Uniform measure on $L_\ell$ induces uniform measure
+on this finite additive image. Consequently
+\begin{equation}\label{eq:valuation-tail}
+ \Pr(v_{\mathfrak p}\ge k)
+       \le\ell^{-\lceil k/e_{\mathfrak p}\rceil}\qquad(k\ge1).
+\end{equation}
+This argument does not require the component projections to be
+independent, or even to equal their full integer rings.
+
+Put $j=\max_{\mathfrak p\mid\ell}
+\lceil v_{\mathfrak p}/e_{\mathfrak p}\rceil$. Then
+$d_\ell(\mathbf v_\ell)\le\ell^{nj}$. If $j\ge a\ge1$, some
+$v_{\mathfrak p}$ is at least $e_{\mathfrak p}(a-1)+1$.
+Equation~\eqref{eq:valuation-tail} and a union bound give
+$\Pr(j\ge a)\le n\ell^{-a}$. Hence
+\begin{equation}\label{eq:local-moment}
+ \mathbb E_{L_\ell}d_\ell(\mathbf v_\ell)^s
+ \le1+n\sum_{a\ge1}\ell^{(ns-1)a}
+ \le1+C_n\ell^{-1/2}.
+\end{equation}
+Using the shell masses once more yields the exact factorization
+\[
+ \sum_{\mathbf k}P(\mathbf k)d(\mathbf k)^{s-1}
+   =\frac1q\prod_{\ell\in S}
+       \left(Z_\ell\,\mathbb E_{L_\ell}d_\ell(\mathbf v_\ell)^s\right).
+\]
+Each factor is at most
+$(1-\ell^{-1})^{-n}(1+C_n\ell^{-1/2})$. For fixed $\eps>0$
+this is at most $\ell^\eps$ at all sufficiently large primes;
+the finitely many remaining primes contribute a constant
+depending only on $n,\eps$. As $\prod_{\ell\in S}\ell\le q$,
+this proves \eqref{eq:weight-moment}. Finally, take
+$\eps=s/16$ and use $q\le Q$ to obtain
+\[
+ \sum_{d>Q^{1/8}}\frac{P(\mathbf k)}d
+ \le Q^{-s/8}\sum_{\mathbf k}P(\mathbf k)d^{s-1}
+ \ll_n q^{-1}Q^{-s/16}.
+ \qedhere
+\]
+\end{proof}

preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/packets.tex

@@ -0,0 +1,215 @@
+\section{The packet as a weighted ideal sum}\label{sec:packets}
+
+We first identify the packet measure exactly, including its signs and
+orbit volumes. Throughout the proof, $K$ is totally real of the fixed
+prime degree $n$, and
+\[
+ R=\cO_K,\quad E=R^\times,\quad D=|\Disc(K)|,\quad
+ \kappaK=\operatorname*{Res}_{s=1}\zeta_K(s).
+\]
+The absolute norm $\N I$ of a nonzero fractional $R$-ideal is extended
+multiplicatively from integral ideals. Thus
+$\operatorname{covol}(\sigma(I))=\sqrt D\,\N I$.
+We use standard ideal theory and the unit theorem
+\cite[Chapters~3--5]{Milne2020}, together with the analytic
+class-number formula \cite[Theorem~19.12]{Sutherland2021}.
+The classical packet interpretation is discussed in
+\cite[Sections~2 and~5]{ELMV2011}.
+
+\subsection{Saturation and local choices}
+
+Let $J_0=RM$, $q=[J_0:M]$, and $Q=q\sqrt D$. For a rational prime
+$\ell$, write $(J_0)_\ell=j_\ell R_\ell$, and put
+$L_\ell=j_\ell^{-1}M_\ell$. Then
+\begin{equation}\label{eq:local-normalization}
+ L_\ell\subset R_\ell,\qquad R_\ell L_\ell=R_\ell,\qquad
+ [R_\ell:L_\ell]=q_\ell,\qquad q=\prod_\ell q_\ell.
+\end{equation}
+Set $S=\{\ell:\ell\mid q\}$; off $S$, $L_\ell=R_\ell$.
+Since $qJ_0\subset M$, one has $qR\subset\cO(M)$.
+Conversely, a multiplier of $M$ has an integral endomorphism matrix
+on $M$ and hence is an algebraic integer. It follows that
+\begin{equation}\label{eq:discriminant-scale}
+ qR\subset\cO(M)\subset R,\qquad
+ D(M)\le Dq^{2n}\le Q^{2n}.
+\end{equation}
+Only the last implication, $D(M)\to\infty\Rightarrow Q\to\infty$,
+will be needed.
+
+These quantities are constant within the prescribed local type.
+Indeed, if $M'_\ell=c_\ell M_\ell$, multiplication by $c_\ell$
+identifies $(RM)_\ell/M_\ell$ with $(RM')_\ell/M'_\ell$.
+It also identifies the local multiplier orders, since multiplication
+in $K\otimes_\Q\Q_\ell$ is commutative. The multiplier order commutes
+with completion: in a $\Z$-basis of $M$, its defining condition is
+that the matrix of multiplication have integral entries, and the
+same condition over $\Z_\ell$ defines the local order. Thus $q$,
+$Q$, $\cO(M)$, and $D(M)$ are unchanged by local homothety.
+Different local types can nevertheless have the same multiplier order.
+
+Choose one fractional ideal $I$ in each ordinary ideal class of $R$.
+Let $\mathcal N(I)$ be the set of lattices $N$ locally homothetic to $M$
+with $RN=I$. If $I_\ell=i_\ell R_\ell$, its possible completions are
+exactly
+\begin{equation}\label{eq:local-choices}
+ N_\ell=i_\ell z_\ell L_\ell,\qquad z_\ell\in R_\ell^\times.
+\end{equation}
+Indeed, taking maximal-order spans in a local homothety forces the
+remaining multiplier to be a unit. Conversely, these local lattices
+lie between $qI_\ell$ and $I_\ell$ and patch uniquely by the primary
+decomposition of $I/qI$. Thus $\mathcal N(I)$ is finite,
+$[I:N]=q$ for all its members, and its cardinality
+$T=|\mathcal N(I)|$ is independent of $I$. The label count $T$ need
+not equal the saturation index $q$. Local unit Haar probability induces
+uniform probability on each finite set of local choices, so the choices at different
+primes are independent. Every global homothety class in the
+packet occurs in these sets, by moving its maximal-order span
+to the chosen ideal-class representative.
+
+\subsection{Real parameters and measure}
+
+Let
+\[
+ C=\{t\in(\R^\times)^n:\textstyle\prod_j|t_j|=1\}.
+\]
+Give $C$ counting measure on its $2^n$ sign components and measure
+$du_1\cdots du_{n-1}$ on the coordinates $u_j=\log|t_j|$.
+The embedding of $E$ is a discrete cocompact subgroup of $C$.
+If $F$ is a measurable fundamental domain, the unit theorem and
+the totally real case of the analytic class-number formula
+\cite[Theorem~19.12]{Sutherland2021} give
+\begin{equation}\label{eq:unit-volume}
+ \vol(F)=2^{n-1}R_K,\qquad
+ h_K\vol(F)=\sqrt D\,\kappaK,
+\end{equation}
+where $h_K$ is the ordinary class number and $R_K$ the regulator
+in deleted logarithmic coordinates. The factor $2^{n-1}$ comes
+from the $2^n$ signs and the logarithmic kernel $\{1,-1\}$.
+In the class-number formula this is the factor $2^n/w_K$, since
+the roots of unity in a totally real field are exactly $\{1,-1\}$.
+
+\begin{lemma}\label{lem:packet-model}
+The maps
+\[
+ (N,t)\longmapsto (Q\N I)^{-1/n}\sigma(N)t
+ \qquad (N\in\mathcal N(I),\ t\in C)
+\]
+induce a bijection from the disjoint union of the quotients by
+\[
+ \epsilon:(N,t)\longmapsto
+       (\epsilon N,\sigma(\epsilon)^{-1}t),\qquad \epsilon\in E,
+\]
+onto the union of the packet orbits.
+Counting measure on the $N$ labels times the stated measure on $C$
+descends to the sum of the orbit Haar measures. In particular,
+the packet probability is obtained by averaging uniformly over $I$
+and $\mathcal N(I)$ and integrating over $F$ with measure
+$dt/\vol(F)$.
+\end{lemma}
+
+\begin{proof}
+Surjectivity follows from \eqref{eq:local-choices}. A global
+homothety changes a normalized embedded lattice by an element
+of $C$. Since $n$ is odd, each pair of opposite sign patterns has
+exactly one representative in $W_n$; a common negative sign
+does not change a lattice.
+
+For injectivity, suppose two displayed normalized lattices coincide.
+Absorb the ratio of the normalizing scalars into the real diagonal
+map carrying $\sigma(N)$ onto $\sigma(N')$. Applying this map
+to one nonzero $\alpha\in N$ gives $\sigma(\beta)$ for some
+$\beta\in N'$. None of the coordinates of $\sigma(\alpha)$
+vanishes, so the map is multiplication by $\sigma(\beta/\alpha)$.
+Thus $N'=bN$ for $b\in K^\times$. Their maximal-order spans
+represent the same ideal class and therefore equal the same
+chosen $I$. Hence $bI=I$, so $b\in E$, and the parameters differ
+by exactly the stated action.
+
+Combine an $N$ label with a sign component of $C$. Its stabilizer in $E$
+is the totally positive subgroup of
+$\{\epsilon\in E:\epsilon N=N\}$, and has finite index in $E$.
+It acts faithfully by logarithmic translations. By injectivity,
+these are all periods of the corresponding $A_n$-orbit.
+Each orbit of labels therefore contributes exactly one compact
+$A_n$-orbit, with precisely its prescribed Haar volume.
+This proves both finiteness and the measure assertion.
+One may equally use $\mathcal N(I)\times F$ as a fundamental
+domain, since the $E$-action preserves counting measure times $dt$.
+The total measure of these fundamental domains is
+$h_KT\vol(F)$. Dividing by this quantity gives precisely the
+uniform averages over ideal classes and finite labels, followed
+by integration against $dt/\vol(F)$.
+\end{proof}
+
+\subsection{Containment probabilities and unfolding}
+
+Write $\mathfrak p\mid\ell$ for the primes of $R$ over $\ell$,
+with ramification indices $e_{\mathfrak p}$ and residue degrees
+$f_{\mathfrak p}$. Normalize $v_{\mathfrak p}$ by a uniformizer,
+so $v_{\mathfrak p}(\ell)=e_{\mathfrak p}$ and
+$\sum_{\mathfrak p\mid\ell}e_{\mathfrak p}f_{\mathfrak p}=n$.
+For a tuple $\mathbf k_\ell=(k_{\mathfrak p})_{\mathfrak p\mid\ell}$
+of nonnegative integers, define
+\[
+ P_\ell(\mathbf k_\ell)
+ =\Pr_{z\in R_\ell^\times}\{x\in zL_\ell\},
+\]
+where $x$ has those component valuations and the probability is
+normalized unit Haar measure. The definition is independent of $x$
+because the units act transitively on each valuation shell.
+For an integral nonzero $R$-ideal $\mathfrak a$, set
+\[
+ P(\mathfrak a)=\prod_{\ell\in S}
+       P_\ell((v_{\mathfrak p}(\mathfrak a))_{\mathfrak p\mid\ell}).
+\]
+For $\alpha\in I\setminus\{0\}$, the probability that a uniform
+$N\in\mathcal N(I)$ contains $\alpha$ is
+$P((\alpha)I^{-1})$.
+
+For a nonnegative Borel function $f$ on $\R^n$, put
+$E_f(\Lambda)=\sum_{0\ne v\in\Lambda}f(v)$.
+The product-fiber integral below integrates $f$ over the $C$-orbit
+with absolute coordinate product $z$, using logarithmic Haar measure
+and summing over all sign components:
+\begin{equation}\label{eq:product-fiber}
+ \cV_f(z)=\sum_{\omega\in\{\pm1\}^n}
+ \int_{\R^{n-1}}
+ f\left(\omega_1e^{u_1},\ldots,\omega_{n-1}e^{u_{n-1}},
+      \omega_n z e^{-\sum_{j<n}u_j}\right)
+ \dd u_1\cdots\dd u_{n-1},\qquad z>0.
+\end{equation}
+
+\begin{proposition}\label{prop:unfolding}
+Writing $\mu_*=\mu_{K,M,\sigma}$, one has
+\begin{equation}\label{eq:unfolding}
+ \int_{X_n}E_f\dd\mu_*
+ =\frac1{\sqrt D\,\kappaK}
+       \sum_{\mathfrak a\subset R,\ \mathfrak a\ne0}
+           P(\mathfrak a)\cV_f(\N\mathfrak a/Q).
+\end{equation}
+The identity holds for nonnegative extended integrals.
+\end{proposition}
+
+\begin{proof}
+Lemma~\ref{lem:packet-model} expresses the left side as
+\[
+ \frac1{h_K\vol(F)}
+ \sum_I\int_F\sum_{\alpha\in I\setminus\{0\}}
+   P((\alpha)I^{-1})
+   f((Q\N I)^{-1/n}\sigma(\alpha)t)\dd t.
+\]
+The map $\alpha\mapsto(\alpha)I^{-1}$ runs over the integral
+ideals in the inverse class of $I$, each fiber being one free
+$E$-orbit. Summing that orbit and integrating over $F$ unfolds
+to an integral over $C$, by Tonelli. The absolute coordinate
+product of the normalized vector is
+$\N((\alpha)I^{-1})/Q$. Translating the first $n-1$ logarithms
+and relabeling the signs identifies the $C$-integral with
+\eqref{eq:product-fiber}, with Jacobian one.
+Every integral ideal occurs in exactly one inverse class.
+Equation~\eqref{eq:unit-volume} supplies the coefficient in
+\eqref{eq:unfolding}.
+\end{proof}
+
+When $M$ is an $R$-ideal, $q=1$ and $P(\mathfrak a)=1$.
+Thus \eqref{eq:unfolding} recovers the maximal-order formula.

preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/paper.tex

@@ -0,0 +1,247 @@
+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage{amsmath,amssymb,amsthm}
+\usepackage[margin=1in]{geometry}
+\usepackage{microtype}
+\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
+\hypersetup{
+ pdftitle={Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type},
+ pdfauthor={OpenAI},
+ pdfsubject={Full-packet equidistribution for arbitrary local module types},
+ pdfkeywords={torus packets, equidistribution, prime degree, nonmaximal orders, entropy}
+}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newcommand{\R}{\mathbb R}
+\newcommand{\Z}{\mathbb Z}
+\newcommand{\Q}{\mathbb Q}
+\DeclareMathOperator{\N}{N}
+\DeclareMathOperator{\vol}{vol}
+\DeclareMathOperator{\Disc}{Disc}
+\DeclareMathOperator{\SL}{SL}
+\DeclareMathOperator{\diag}{diag}
+\newcommand{\kappaK}{\kappa_K}
+\newcommand{\dd}{\,d}
+\newcommand{\one}{\mathbf 1}
+\newcommand{\eps}{\varepsilon}
+\newcommand{\cO}{\mathcal O}
+\newcommand{\cV}{\mathcal V}
+\numberwithin{equation}{section}
+\title{Equidistribution of Prime-Degree Torus Packets\\
+with Arbitrary Local Type}
+\author{OpenAI}
+\date{September 24, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We prove the packet form of the higher-dimensional Duke equidistribution
+problem for totally real fields of any fixed prime degree at least five,
+allowing arbitrary local homothety types of full lattices. As the
+multiplier-order discriminant tends to infinity, the volume-weighted packet
+measures converge to Haar probability measure, with no escape of mass.
+\end{abstract}
+
+\section{Introduction}\label{sec:intro}
+
+Compact diagonal orbits in the space of unimodular lattices encode
+ideal classes and units of totally real number fields. The higher-dimensional
+Duke equidistribution problem asks whether arithmetically complete families
+of these orbits become uniformly distributed as their discriminants grow.
+The grouping matters: one local homothety type specifies a complete packet,
+whereas a multiplier order alone can combine several packets. We prove the
+prime-degree packet statement for every such local type.
+
+Fix a prime $n\ge5$, and write
+\[
+ X_n=\SL_n(\Z)\backslash\SL_n(\R),\qquad
+ A_n=\{\diag(e^{u_1},\ldots,e^{u_n}):\textstyle\sum_j u_j=0\}.
+\]
+We regard $X_n$ as the space of covolume-one row lattices, with the
+right action of $A_n$, and let $m_n$ be its $\SL_n(\R)$-invariant
+probability measure. Haar measure on $A_n$ is
+$du_1\cdots du_{n-1}$ in the displayed logarithmic coordinates.
+
+Let $K$ be a totally real field of degree $n$, let
+$\sigma=(\sigma_1,\ldots,\sigma_n)$ order its real embeddings, and let
+$M\subset K$ be a free rank-$n$ $\Z$-submodule spanning $K$ over $\Q$.
+We call such an $M$ a full lattice. Its multiplier order and discriminant
+are
+\[
+ \cO(M)=\{\alpha\in K:\alpha M\subset M\},\qquad
+ D(M)=|\Disc(K)|[\cO_K:\cO(M)]^2.
+\]
+Two such lattices are \emph{locally homothetic} if
+$M'_\ell=c_\ell M_\ell$ for every rational prime $\ell$, where
+$c_\ell\in(K\otimes_\Q\Q_\ell)^\times$ and subscripts denote completion.
+Choose representatives $M'$ for their $K^\times$-homothety classes
+within the local type of $M$, and normalize
+\[
+ \Lambda_{M',\sigma}
+ =\operatorname{covol}(\sigma(M'))^{-1/n}\sigma(M').
+\]
+The packet $\mathcal P_{K,M,\sigma}$ consists of the distinct orbits
+$\Lambda_{M',\sigma}wA_n$, with
+\[
+ w\in W_n=\{\diag(\epsilon_1,\ldots,\epsilon_n):
+       \epsilon_j\in\{\pm1\},\ \textstyle\prod_j\epsilon_j=1\}.
+\]
+These orbits are compact and their number is finite; the
+parametrization in Section~\ref{sec:packets} also proves these facts.
+For an orbit $O$, let $\nu_O$ be its invariant probability and
+$\vol(O)$ its volume for the specified Haar measure. Set
+\begin{equation}\label{eq:packet-probability}
+ \mu_{K,M,\sigma}
+ =\frac{\sum_{O\in\mathcal P_{K,M,\sigma}}\vol(O)\nu_O}
+        {\sum_{O\in\mathcal P_{K,M,\sigma}}\vol(O)}.
+\end{equation}
+
+\begin{theorem}\label{thm:main}
+For every fixed prime $n\ge5$ and every sequence
+$(K_i,M_i,\sigma_i)$ as above with \mbox{$D(M_i)\to\infty$},
+\[
+ \mu_{K_i,M_i,\sigma_i}\longrightarrow m_n.
+\]
+The convergence is weak convergence of probability measures. In
+particular, it holds against every continuous compactly supported
+function, and no mass escapes into the cusp.
+\end{theorem}
+
+Theorem~\ref{thm:main} permits nonmaximal orders of unbounded index
+and arbitrary local module types. The field discriminants may remain
+bounded or grow independently of those indices. The statement is
+separate for each local homothety type; it makes no assertion about an
+individual orbit, a convergence rate, or composite degrees.
+
+\subsection*{The maximal-order specialization}
+
+Take $M=\cO_K$. Every lattice in its local type is a fractional
+$\cO_K$-ideal, and its homothety class is an ordinary ideal class.
+Thus the packet is obtained by embedding one representative of each
+ideal class, normalizing its covolume to one, and taking the distinct
+sign translates of its positive diagonal orbit. For every sequence of
+totally real fields of degree $n$ with $|\Disc(K)|\to\infty$, the
+volume-weighted measures of these packets converge to $m_n$ by
+Theorem~\ref{thm:main}. This construction also explains why an individual
+ideal class is not the object being averaged.
+
+For general $M$, the same maximal-order ideal classes remain the global
+parameters, but each carries a finite set of local module choices.
+Proposition~\ref{prop:adelic-packet} in Appendix~\ref{subsec:adelic-packet}
+identifies this construction with the
+pushforward of adelic torus Haar probability, independently of the
+rational basis and finite-lattice presentation. The theorem therefore
+includes all these presentations of the maximal-order specialization,
+as well as arbitrary noninvertible modules over nonmaximal orders.
+
+\subsection*{Context and prior work}
+
+Linnik's ergodic approach connected arithmetic distribution problems
+with dynamics \cite{Linnik1968}. Duke proved equidistribution of the
+length-weighted closed geodesics associated with fundamental
+real-quadratic discriminants on the modular surface
+\cite[Theorem~1]{Duke1988}. In higher rank, Einsiedler, Lindenstrauss,
+Michel, and Venkatesh developed the relationship between arithmetic
+separation and entropy for periodic torus orbits \cite{ELMV2009}.
+Their cubic equidistribution theorem combines subconvexity, local
+analysis, and measure classification
+\cite[Theorem~1.4]{ELMV2011}. Their packet construction already
+distinguishes local homothety from the coarser condition of having the
+same multiplier order \cite[Section~5.5]{ELMV2011}.
+
+The higher-prime-degree extension described in
+\cite[Section~1.6.3]{ELMV2011} is conditional on subconvexity.
+Khayutin subsequently obtained entropy bounds using arithmetic
+separation and Galois invariants, under maximal-order and
+two-transitivity hypotheses \cite[Theorem~1.1]{Khayutin2019}.
+Lemke Oliver, Thorner, and Zaman proved equidistribution for sequences
+of orders in totally real fields of each fixed prime degree at least five,
+outside a small
+exceptional family of fields \cite[Theorem~2.3]{LOTZ2024}.
+Their result allows nonmaximal orders; the exceptional family has at
+most $O_{n,\eps}(X^\eps)$ fields of discriminant at most $X$.
+Theorem~\ref{thm:main} treats every field and every full-lattice local
+type in the stated prime degrees. It gives an unconditional affirmative
+answer to this packet form of the higher-dimensional Duke problem,
+including sequences in a fixed field with increasing order index.
+
+\subsection*{The arithmetic ingredient}
+
+The proof follows the counting-to-rigidity strategy of
+Einsiedler--Lindenstrauss--Michel--Venkatesh
+\cite[Section~2.7.1]{ELMV2011}: packet averages of lattice-vector
+counts control the cusp and small ordinary balls, and these bounds
+feed into positive-entropy rigidity. Their cubic argument obtains
+asymptotics for these averages using global and local subconvexity
+\cite[Sections~2.7.2--2.7.3]{ELMV2011}. For the same rigidity step,
+we obtain the needed upper bounds by retaining the Dedekind-zeta
+residue in the unfolded ideal count. Stark's exceptional-zero descent
+\cite{Stark1974}, together with Shiu's theorem in Pollack's uniform
+formulation \cite{Shiu1980,Pollack2020}, supplies the required bounds
+without a subconvexity hypothesis. Odd degree excludes the quadratic
+subfields that occur in Stark's exceptional-zero alternative.
+
+For an arbitrary prescribed local type, we keep maximal-order ideal
+counting but insert local containment probabilities: at each prime,
+the weight records the probability that a vector belongs to a randomly
+chosen unit translate of the prescribed module.
+The essential estimate is a small positive moment of the local norm,
+uniform over full local modules whose maximal-order span is the full
+ring. Its proof uses a unit in each component projection, not
+independence of the components or invertibility of the module.
+The unit-mass argument in \cite[Lemma~9.7]{ELMV2011} is an antecedent;
+here we prove the exact shell normalization and the positive moment
+for arbitrary local modules with this full-span condition.
+
+The exact unfolding belongs to the classical Hecke method, presented
+in the packet setting in \cite[Section~10]{ELMV2011}. The dynamical
+conclusion uses the two-sided tube criterion of
+\cite[Corollary~3.3]{ELMV2009} and the prime-degree measure
+classification of Einsiedler, Katok, and Lindenstrauss
+\cite[Corollary~1.4]{EKL2006}. We give the residue estimate,
+local-module bounds, and the passage to entropy in almost every
+ergodic component below.
+
+\subsection*{Proof outline}
+
+Put $R=\cO_K$, $D=|\Disc(K)|$, and
+$q=[RM:M]$. The natural counting scale is $Q=q\sqrt D$, and
+$D(M)\le Q^{2n}$. Section~\ref{sec:packets} expresses the packet
+mean of a lattice-vector count as a maximal-order ideal sum with
+local weights. Section~\ref{sec:local} proves an exact total-mass
+identity and a power-saving tail for these weights.
+Section~\ref{sec:analytic} combines them with a residue-relative
+zeta estimate and Shiu's theorem. Removing the Euler factors at
+primes dividing $q$ produces precisely the factor needed to cancel
+the local normalization.
+
+The resulting count controls both short lattice vectors and small
+boxes about vectors with nonzero coordinates. Section~\ref{sec:geometry}
+deduces tightness and an $O(r^n)$ bound for the mass of ordinary balls
+in any limit measure. Since $A_n$ has dimension $n-1$, this bound
+forces positive entropy in almost every $A_n$-ergodic component.
+Section~\ref{sec:entropy} gives this implication and applies the
+prime-degree measure-classification theorem. In that step it is
+essential to prove positive entropy for every invariant probability
+dominated by a finite multiple of the limit, not only for the limit
+itself: this rules out a positive-weight family of zero-entropy
+components.
+
+\input{packets}
+\input{local}
+\input{analytic}
+\input{geometry}
+\input{entropy}
+
+\appendix
+\input{adelic}
+
+{\small
+\bibliographystyle{plain}
+\bibliography{references}
+}
+\end{document}

preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/build/references.bib

@@ -0,0 +1,123 @@
+@book{Linnik1968,
+  author = {Linnik, Yurij V.},
+  title = {Ergodic Properties of Algebraic Fields},
+  series = {Ergebnisse der Mathematik und ihrer Grenzgebiete},
+  volume = {45}, publisher = {Springer}, address = {Berlin--Heidelberg},
+  year = {1968},
+  url = {https://link.springer.com/book/9783642866333}
+}
+
+@article{Duke1988,
+  author = {Duke, William},
+  title = {Hyperbolic distribution problems and half-integral weight {Maass} forms},
+  journal = {Inventiones Mathematicae},
+  volume = {92}, number = {1}, year = {1988}, pages = {73--90},
+  doi = {10.1007/BF01393993}
+}
+
+@article{ELMV2011,
+  author = {Einsiedler, Manfred and Lindenstrauss, Elon and Michel, Philippe and Venkatesh, Akshay},
+  title = {Distribution of periodic torus orbits and {Duke}'s theorem for cubic fields},
+  journal = {Annals of Mathematics},
+  volume = {173}, number = {2}, year = {2011}, pages = {815--885},
+  doi = {10.4007/annals.2011.173.2.5}
+}
+
+@article{ELMV2009,
+  author = {Einsiedler, Manfred and Lindenstrauss, Elon and Michel, Philippe and Venkatesh, Akshay},
+  title = {Distribution of periodic torus orbits on homogeneous spaces},
+  journal = {Duke Mathematical Journal},
+  volume = {148}, number = {1}, year = {2009}, pages = {119--174},
+  doi = {10.1215/00127094-2009-023}
+}
+
+@article{EKL2006,
+  author = {Einsiedler, Manfred and Katok, Anatole and Lindenstrauss, Elon},
+  title = {Invariant measures and the set of exceptions to {Littlewood}'s conjecture},
+  journal = {Annals of Mathematics},
+  volume = {164}, number = {2}, year = {2006}, pages = {513--560},
+  doi = {10.4007/annals.2006.164.513}
+}
+
+@article{LOTZ2024,
+  author = {Lemke Oliver, Robert J. and Thorner, Jesse and Zaman, Asif},
+  title = {An approximate form of {Artin}'s holomorphy conjecture and non-vanishing of {Artin} {$L$}-functions},
+  journal = {Inventiones Mathematicae},
+  volume = {235}, number = {3}, year = {2024}, pages = {893--971},
+  doi = {10.1007/s00222-023-01232-2},
+  eprint = {2012.14422}, archivePrefix = {arXiv}, primaryClass = {math.NT}
+}
+
+@article{Khayutin2019,
+  author = {Khayutin, Ilya},
+  title = {Arithmetic of double torus quotients and the distribution of periodic torus orbits},
+  journal = {Duke Mathematical Journal},
+  volume = {168}, number = {12}, year = {2019}, pages = {2365--2432},
+  doi = {10.1215/00127094-2019-0016},
+  eprint = {1510.08481}, archivePrefix = {arXiv}, primaryClass = {math.NT}
+}
+
+@article{Stark1974,
+  author = {Stark, H. M.},
+  title = {Some effective cases of the {Brauer--Siegel} theorem},
+  journal = {Inventiones Mathematicae},
+  volume = {23}, number = {2}, year = {1974}, pages = {135--152},
+  doi = {10.1007/BF01405166}
+}
+
+@article{Shiu1980,
+  author = {Shiu, Peter},
+  title = {A {Brun--Titchmarsh} theorem for multiplicative functions},
+  journal = {Journal f\"ur die reine und angewandte Mathematik},
+  volume = {313}, year = {1980}, pages = {161--170},
+  doi = {10.1515/crll.1980.313.161}
+}
+
+@article{Pollack2020,
+  author = {Pollack, Paul},
+  title = {Nonnegative multiplicative functions on sifted sets, and the square roots of {$-1$} modulo shifted primes},
+  journal = {Glasgow Mathematical Journal},
+  volume = {62}, number = {1}, year = {2020}, pages = {187--199},
+  doi = {10.1017/S0017089519000041},
+  note = {\href{https://www.pollack-math.net/brunmultiplicative4.pdf}{Corrected author version}},
+  url = {https://www.pollack-math.net/brunmultiplicative4.pdf}
+}
+
+@misc{Milne2020,
+  author = {Milne, James S.},
+  title = {Algebraic Number Theory},
+  year = {2020}, note = {Version 3.08},
+  url = {https://www.jmilne.org/math/CourseNotes/ANT.pdf}
+}
+
+@misc{Sutherland2021,
+  author = {Sutherland, Andrew V.},
+  title = {The Analytic Class Number Formula},
+  year = {2021},
+  note = {MIT 18.785, Fall 2021, Lecture 19, November 15},
+  url = {https://ocw.mit.edu/courses/18-785-number-theory-i-fall-2021/mit18_785f21_lec19.pdf}
+}
+
+@article{Mahler1946,
+  author = {Mahler, Kurt},
+  title = {On lattice points in {$n$}-dimensional star bodies. {I}. {Existence} theorems},
+  journal = {Proceedings of the Royal Society of London. Series A},
+  volume = {187}, number = {1009}, year = {1946}, pages = {151--187},
+  doi = {10.1098/rspa.1946.0072}
+}
+
+@article{Rokhlin1967,
+  author = {Rokhlin, V. A.},
+  title = {Lectures on the entropy theory of measure-preserving transformations},
+  journal = {Russian Mathematical Surveys},
+  volume = {22}, number = {5}, year = {1967}, pages = {1--52},
+  doi = {10.1070/RM1967v022n05ABEH001224}
+}
+
+@article{GreschonigSchmidt2000,
+  author = {Greschonig, Gernot and Schmidt, Klaus},
+  title = {Ergodic decomposition of quasi-invariant probability measures},
+  journal = {Colloquium Mathematicum},
+  volume = {84/85}, number = {2}, year = {2000}, pages = {495--514},
+  doi = {10.4064/cm-84/85-2-495-514}
+}

preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/paper.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:7e6a0188abd45ae4520d4592c773d264e029751e3ef6fb2623388aee8b0f56c0
+size 404189

preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/README.md

@@ -0,0 +1,18 @@
+# [Equidistribution of primitive quartic torus packets for arbitrary orders](quartic-torus-packets.pdf)
+
+**Author:** OpenAI
+
+**Date:** October 5, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026,
+  author = {{OpenAI}},
+  title = {{Equidistribution of primitive quartic torus packets for arbitrary orders}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/quartic-torus-packets.pdf}{OAI:Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026}},
+  year = {2026}
+}
+```
+

preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/additive.tex

@@ -0,0 +1,861 @@
+\section{Additive distribution of the conductor weights}
+\label{sec:additive}
+
+The factors \(f_p^*\) introduced in Section~\ref{sec:exterior} retain the
+local lattice condition of an order at the primes in \(S\).  We now average their
+product on the real trace plane.  The estimate will also permit a residue
+class at a small modulus prime to \(S\); this is the form needed to average
+the factors at the remaining primes in Section~\ref{sec:sieve}.
+
+Fix a nonzero integer \(m\).  In the three real embeddings of \(F\), write
+\(z=(z_1,z_2,z_3)\), so that \(\Tr z=z_1+z_2+z_3\).  Choose two distinct
+indices \(i,j\), a fixed nonnegative function
+\(\Phi\in C_c^\infty(\R^2)\) which is at least one on \([-1,1]^2\), and a
+fixed nonnegative \(\chi\in C_c^\infty(\R)\) with
+\[
+ \chi(m)=1,\qquad
+ \supp\chi\subset(m-\tfrac13,m+\tfrac13).
+\]
+For \(c=(c_i,c_j)\in\R^2\) and positive \(r_i,r_j\), put
+\begin{equation}
+ \phi(z)=
+ \Phi\left(\frac{z_i-c_i}{r_i},\frac{z_j-c_j}{r_j}\right)\chi(\Tr z),
+ \qquad
+ A_\phi=r_i r_j\int_{\R^2}\Phi(u,v)\,du\,dv .
+ \label{eq:add-box}
+\end{equation}
+Thus \(A_\phi\) is the integral of \(\phi\) on \(\Tr z=m\), using \(z_i,z_j\)
+as coordinates on that plane.  Fix \(r_*>0\), keep \(r_i,r_j\le r_*\), and
+require the supports of their planar factors to lie in a fixed
+bounded region.  The supports of all the resulting \(\phi\)'s then lie in a
+fixed compact subset \(\Omega_\infty\) of \(\R^3\) on which \(\Tr z\) stays
+away from zero.  All constants below may depend on \(m,\Phi,\chi\), the
+upper bound for the scales, and this bounded region.  They do not depend on
+the locations \(c\) or on the scales within their specified range.
+
+For \(p\notin S\), the algebra \(F\otimes\Q_p\) is unramified and its
+integral trace map \(R_{F,p}\to\Z_p\) is onto.  The affine set
+\[
+ \mathcal T_{m,p}=\{z\in R_{F,p}:\Tr z=m\}
+\]
+is consequently a translate of a free rank-two direct summand of \(R_{F,p}\).
+A residue class on this set modulo \(p^a\) means a translate of
+\(p^a\ker(\Tr:R_{F,p}\to\Z_p)\).  Equivalently, if \(z_p^0\in\mathcal T_{m,p}\)
+represents the class, its members satisfy \(z-z_p^0\in p^aR_{F,p}\).
+There are \(p^{2a}\) such classes.
+
+\begin{proposition}[Additive distribution]
+\label{prop:additive-equidistribution}
+There are absolute constants \(\tau,\theta>0\) with the following property.
+Let \(K,\OO,F\) be as in Section~\ref{sec:exterior}, put
+\(X=[\OO_K:\OO]\sqrt{|\Disc(K)|}\), and take \(S\) as in that section for
+the fixed nonzero integer \(m\).  In Equation~\eqref{eq:add-box}, choose any
+two real partition coordinates and scales
+\[
+ X^{-\tau}\le r_i,r_j\le r_*
+\]
+whose planar supports stay in the fixed bounded region above.  Let
+\(1\le l\le X^\theta\) be prime to every prime in \(S\).  For each \(p\mid l\)
+choose any \(z_p^0\in\mathcal T_{m,p}\), specifying a class modulo
+\(p^{v_p(l)}\).  Then
+\begin{equation}
+ \sum_{\substack{z\in F,\ \Tr z=m\\
+                  z\in R_{F,p}\ (p\notin S)\\
+                  z-z_p^0\in p^{v_p(l)}R_{F,p}\ (p\mid l)}}
+       \phi(z)\prod_{p\in S}f_p^*(z)
+ =
+ A_\phi\,l^{-2}\prod_{p\in S}d_p(m)
+       \bigl(1+O(X^{-\theta})\bigr).
+ \label{eq:add-equidistribution}
+\end{equation}
+Here \(d_p\) is the trace density of Lemma~\ref{lem:fourier-scalar-density}.
+The error is uniform over all the fields, orders, ordered real embeddings,
+permitted box locations and scales, moduli \(l\), and chosen residue classes.
+The implied constant and the lower threshold for \(X\) depend only on the
+fixed real data listed above and on \(m\).
+\end{proposition}
+
+\subsection{Finite expansion and Poisson summation}
+
+We first assemble the local inverse transforms into a finite global
+expansion.  This is the point at which the fact that the cubic algebra is a
+field is essential.  Let \(\mathfrak d_F\) be the different of \(F\), and
+write \(\mathfrak d_p=\mathfrak d_F R_{F,p}\) for its localization.
+For a choice of denominator shells at the primes of \(S\), write
+\[
+ \mathfrak h=\prod_{p\in S}\mathfrak h_p,\qquad
+ H=\prod_{p\in S}H_p=\Nm\mathfrak h,\qquad
+ G_0=\prod_{p\in S}p^{g_p}.
+\]
+Local ideals here and below are identified with global ideals supported at
+the corresponding primes.  The functions \(K_{p,\mathfrak h_p}\) and their
+boundary sets \(\partial_{\mathfrak h_p}(z)\) are those of
+Lemma~\ref{lem:fourier-shell-estimates}.  Set
+\[
+ \mathscr L_{\mathfrak h}=\{p\in S:g_p>0\text{ or }p\le5\}.
+\]
+
+\begin{lemma}[Global bounds for denominator profiles]
+\label{lem:add-profile-bounds}
+For every profile one has
+\begin{equation}
+ G_0\le [R_F:C_F]\le H G_0.
+ \label{eq:add-profile-indices}
+\end{equation}
+Suppose that \(z\in F\) is integral outside \(S\), its real image lies in
+\(\Omega_\infty\), and
+\(\prod_{p\in S}K_{p,\mathfrak h_p}(z)\ne0\).  Define
+\[
+ T_\partial=T_\partial(\mathfrak h,z)
+   =\prod_{\substack{p\in S\setminus\mathscr L_{\mathfrak h}\\
+                     \mathfrak q\in\partial_{\mathfrak h_p}(z)}}
+       \Nm\mathfrak q ,
+\]
+where an empty product is one.  There is a constant depending only on
+\(\Omega_\infty\) such that
+\begin{align}
+ H&\ll D_FG_0^{24}T_\partial,
+ &T_\partial&\le\left(\prod_{p\in S}p\right)^3,
+ \label{eq:add-profile-norm}\\
+ \left|\prod_{p\in S}K_{p,\mathfrak h_p}(z)\right|
+   &\ll 8^{|S|}T_\partial^{-1/2}.
+ \label{eq:add-profile-inverse}
+\end{align}
+All such \(z\) lie in a fractional \(R_F\)-ideal
+\(\mathfrak a_{\mathfrak h}\), independent of \(z\), with
+\begin{equation}
+ \Nm(\mathfrak a_{\mathfrak h})^{-1}
+       \le 30^{12}D_FG_0^{24}.
+ \label{eq:add-profile-support}
+\end{equation}
+
+Set \(C_3=371\).  For all sufficiently large \(X\), depending only on the
+fixed real data and \(m\), a profile contributing at any such \(z\) has
+\(H\le X^{C_3}\).  For these \(X\) and every \(z\in F\) integral outside
+\(S\) with real image in \(\Omega_\infty\),
+\begin{equation}
+ \prod_{p\in S}f_p^*(z)
+  =\sum_{\substack{\mathfrak h=(\mathfrak h_p)_{p\in S}\\H\le X^{C_3}}}
+        \prod_{p\in S}K_{p,\mathfrak h_p}(z).
+ \label{eq:add-profile-expansion}
+\end{equation}
+The number of profiles in this sum is \(O_\eta(X^\eta)\) for every fixed
+\(\eta>0\), uniformly in the arithmetic data.
+\end{lemma}
+
+\begin{proof}
+Let \(M_p=\mathfrak h_p\cap C_{F,p}\).  The additive lattice identity
+\[
+ [R_{F,p}:C_{F,p}]
+   =[R_{F,p}:C_{F,p}+\mathfrak h_p]\,[\mathfrak h_p:M_p]
+\]
+has first factor between \(1\) and
+\([R_{F,p}:\mathfrak h_p]=H_p\).  The index of \(C_F\) is supported on \(S\)
+by Lemma~\ref{lem:ext-order-scale}; multiplying gives
+Equation~\eqref{eq:add-profile-indices}.
+
+At \(p\in\mathscr L_{\mathfrak h}\), the local support is contained in
+\(p^{-4(g_p+1)}\mathfrak h_p\mathfrak d_p^{-1}\).
+At every other \(p\in S\), the support loses at most one power of each
+boundary prime \(\mathfrak q\), and loses no power in the other components.
+It follows that a supporting \(z\) belongs to
+\[
+ \mathfrak b_{\mathfrak h,z}
+  =\mathfrak h\mathfrak d_F^{-1}
+     \prod_{p\in\mathscr L_{\mathfrak h}}
+           (pR_F)^{-4(g_p+1)}
+     \prod_{\substack{p\in S\setminus\mathscr L_{\mathfrak h}\\
+                      \mathfrak q\in\partial_{\mathfrak h_p}(z)}}
+           \mathfrak q^{-1}.
+\]
+The real trace condition excludes \(z=0\).  Since \(F\) is a field, its norm
+is nonzero, and containment of the principal ideal \((z)\) in
+\(\mathfrak b_{\mathfrak h,z}\) gives
+\[
+ |\Nm_{F/\Q}z|
+   =\Nm(\mathfrak b_{\mathfrak h,z})
+        [\mathfrak b_{\mathfrak h,z}:(z)]
+   \ge \frac{H}
+      {D_FT_\partial\prod_{p\in\mathscr L_{\mathfrak h}}p^{12(g_p+1)}}.
+\]
+The left side is bounded above on \(\Omega_\infty\).  For \(g_p>0\) we have
+\(g_p+1\le2g_p\); the only additional factors, for \(g_p=0\), are at
+\(2,3,5\).  Hence
+\[
+ \prod_{p\in\mathscr L_{\mathfrak h}}p^{12(g_p+1)}
+       \le30^{12}G_0^{24}.
+\]
+This proves the first inequality of Equation~\eqref{eq:add-profile-norm}.
+Each boundary prime occurs once, and
+\(\sum_{\mathfrak q\mid p}f_{\mathfrak q}\le
+\sum_{\mathfrak q\mid p}e_{\mathfrak q}f_{\mathfrak q}=3\);
+this proves the bound for \(T_\partial\).  Equation~\eqref{eq:add-profile-inverse}
+is the product of the boundary bounds of
+Lemma~\ref{lem:fourier-shell-estimates}; its absolute implied constant
+accounts for the three small primes.
+
+Dropping \(\mathfrak h_p\) in the first support condition enlarges that
+support.  In the second, the exponent \(v_{\mathfrak q}(\mathfrak h_p)-1\)
+at a boundary is nonnegative.  Thus every supporting
+point belongs to the ideal
+\[
+ \mathfrak a_{\mathfrak h}
+   =\mathfrak d_F^{-1}
+      \prod_{p\in\mathscr L_{\mathfrak h}}
+             (pR_F)^{-4(g_p+1)} ,
+\]
+which gives Equation~\eqref{eq:add-profile-support}.
+
+Lemma~\ref{lem:ext-order-scale} gives
+\(\Delta_F\ll X^{28}\),
+\(D_F\le\Delta_F\), and
+\([R_F:C_F]\le\Delta_F^{1/2}\).  The definition of \(S\) gives
+\(\prod_{p\in S}p\ll_m X^2\).  Equations
+\(\eqref{eq:add-profile-indices}\)--\(\eqref{eq:add-profile-norm}\) therefore
+give \(H\ll X^{28+24\cdot14+6}=X^{370}\).  The fixed implied constant is
+absorbed by \(X^{371}\) for sufficiently large \(X\).
+
+A nonzero field element has nonzero image in every component of
+\(F\otimes\Q_p\).  Lemma~\ref{lem:fourier-inverse-expansion} therefore
+identifies the density \(f_p^*(z)\) with its pointwise finite sum of inverse
+shells at all the present \(z\).  The norm bound just proved excludes every
+product with \(H>X^{371}\), uniformly in \(z\), and proves
+Equation~\eqref{eq:add-profile-expansion}.  Finally, for any fixed \(u>0\),
+Rankin's inequality bounds the number of profiles with \(H\le X^{C_3}\) by
+\[
+ X^{C_3u}\prod_{p\in S}(1-p^{-u})^{-3}.
+\]
+There are at most three prime-ideal components over each \(p\).  The radical
+bound on \(S\) implies \(|S|=O(\log X/\log\log X)\), since the product of
+\(k\) distinct primes is at least \(k!\).  The displayed Euler product is
+then \(X^{o(1)}\) for fixed \(u\).  Taking \(u\) arbitrarily small proves
+the asserted bound for every \(\eta>0\).
+\end{proof}
+
+We apply this expansion before imposing Poisson summation.  For \(p\notin S\)
+let
+\[
+ \eta_{p,l}(z)=
+ \begin{cases}
+  \mathbf 1_{z_p^0+p^{v_p(l)}R_{F,p}}(z),&p\mid l,\\
+  \mathbf 1_{R_{F,p}}(z),&p\nmid l .
+ \end{cases}
+\]
+Let \(\eta_{\mathfrak h,l}\) be the finite-adelic test function whose local
+factor is \(K_{p,\mathfrak h_p}\) at \(p\in S\) and \(\eta_{p,l}\) elsewhere.
+By Lemma~\ref{lem:fourier-shell-estimates}, every \(K_{p,\mathfrak h_p}\)
+is supported on elements of integral trace.  The factors outside \(S\) are
+supported in \(R_{F,p}\) and also have integral trace.  Thus any \(z\in F\)
+with \(\eta_{\mathfrak h,l}(z)\ne0\) has rational trace integral at every
+finite prime, hence an integer.  Only \(m\) meets \(\supp\chi\).  The finite
+profile expansion in Lemma~\ref{lem:add-profile-bounds} consequently shows
+that the left side of
+Equation~\eqref{eq:add-equidistribution}, denoted by \(\Sigma\), equals
+\begin{equation}
+ \Sigma
+   =\sum_{\substack{\mathfrak h\\H\le X^{C_3}}}
+       S_{\mathfrak h},\qquad
+ S_{\mathfrak h}
+   =\sum_{z\in F}\phi(z)\eta_{\mathfrak h,l}(z).
+ \label{eq:add-original-profiles}
+\end{equation}
+Here \(C_3=371\) is fixed.  The preceding lemma makes the profile interchange
+finite.  Each inner sum also has finitely many terms, since its finite
+conditions place \(z\) in a fractional ideal and its real support is compact.
+
+In the local results of Section~\ref{sec:local-fourier}, choose the finite
+components of the standard character of \(\mathbb A_\Q/\Q\); they all have
+conductor \(\Z_p\).  Compose this global character with the field trace.
+At a finite place the self-dual measure satisfies
+\(\vol(R_{F,p})=|D_F|_p^{1/2}\), while at infinity it is Lebesgue measure
+in the three embeddings.  The real covolume of \(R_F\) is \(\sqrt{D_F}\)
+and \(\prod_p\vol(R_{F,p})=D_F^{-1/2}\).  Thus the diagonal copy of \(F\)
+has covolume one for these measures.  Write
+\[
+ \widehat\phi(A)=\int_{\R^3}\phi(z)\psi_\infty(\Tr(Az))\,dz
+\]
+for the real transform.  Field Poisson summation in this
+normalization \cite[Section~4.2, Lemma~4.2.4]{Tate1967} gives
+\begin{equation}
+ S_{\mathfrak h}
+   =\sum_{A\in F}\widehat\phi(A)
+             \widehat\eta_{\mathfrak h,l}(A),\qquad
+ \widehat\eta_{\mathfrak h,l}(A)
+   =\prod_{p\in S}\gamma_{p,\mathfrak h_p}(A)
+       \prod_{p\notin S}\widehat\eta_{p,l}(A).
+ \label{eq:add-poisson}
+\end{equation}
+Here \(\gamma_{p,\mathfrak h_p}\) is the shell restriction of \(\gamma_p\),
+extended by zero, and hats use the chosen character.  Each finite test is
+Schwartz--Bruhat, so the formula is also ordinary lattice Poisson summation
+with congruence classes.  At \(p\notin S\), trace duality is unimodular and
+\[
+ \widehat\eta_{p,l}(A)
+   =p^{-3v_p(l)}
+       \psi_p(\Tr(Az_p^0))
+       \mathbf 1_{p^{-v_p(l)}R_{F,p}}(A)
+       \quad (p\mid l);
+\]
+for \(p\nmid l\) it is \(\mathbf 1_{R_{F,p}}\).
+In particular every such transform is bounded in absolute value by one.
+The finite Fourier factors in Equation~\eqref{eq:add-poisson} are supported
+on the fractional ideal
+\begin{equation}
+ \Gamma_{l,\mathfrak h}=l^{-1}\mathfrak h^{-1},\qquad P=l^3H,
+ \qquad
+ \Nm(\Gamma_{l,\mathfrak h})=P^{-1}.
+ \label{eq:add-frequency-lattice}
+\end{equation}
+
+The following elementary geometry of this ideal will be used both to count
+frequencies and to express the local congruences in global integer
+coordinates.
+
+\begin{lemma}[Lattice bounds for a cubic ideal]
+\label{lem:add-ideal-lattice}
+Let \(\mathfrak a\) be a fractional ideal of a totally real cubic field of
+discriminant \(D_F\), and write \(\Nm(\mathfrak a)=P^{-1}\) with \(P>0\).
+Let \(B_Y=\{x\in\R^3:\|x\|\le Y\}\).
+In embedding coordinates it has covolume \(\sqrt{D_F}/P\), first Euclidean
+minimum \(\gg P^{-1/3}\), and, for \(Y>0\),
+\begin{equation}
+ \#\{A\in\mathfrak a:\|A\|\le Y\}
+  \ll 1+YP^{1/3}+Y^2P^{2/3}+\frac{Y^3P}{\sqrt{D_F}} .
+ \label{eq:add-lattice-count}
+\end{equation}
+There is a \(\Z\)-basis \(b_1,b_2,b_3\) of \(\mathfrak a\) in which every
+\(A=\sum n_i b_i\) with \(\|A\|\le Y\) satisfies
+\[
+ |n_i|\ll YP^{1/3}\quad(1\le i\le3).
+\]
+All constants here are absolute.
+\end{lemma}
+
+\begin{proof}
+The covolume formula follows from the discriminant.  For \(0\ne A\in
+\mathfrak a\), containment \((A)\subset\mathfrak a\) implies
+\(|\Nm_{F/\Q}A|\ge \Nm\mathfrak a=P^{-1}\).  The product inequality in
+three real coordinates gives
+\(|\Nm A|\le(\|A\|/\sqrt3)^3\), proving the first-minimum bound.
+
+Let \(\lambda_1\le\lambda_2\le\lambda_3\) be the Euclidean successive
+minima.  A lattice in fixed dimension has a basis of bounded orthogonality
+defect, by Mahler's basis theorem \cite[Section~2, Theorem~1]{Mahler1946}.
+For such a basis, order its lengths increasingly and write its defect as
+\(\omega=\prod_i\|b_i\|/\covol(\mathfrak a)\ll1\).
+Cramer's rule and Hadamard's inequality bound the norm of the \(i\)-th
+coordinate functional by \(\omega/\|b_i\|\).  Since
+\(\|b_i\|\ge\lambda_i\), counting its possible integer coordinates gives
+\[
+ \#(\mathfrak a\cap B_Y)\ll
+       \prod_{i=1}^3(1+Y/\lambda_i).
+\]
+Minkowski's second theorem for the Euclidean ball, in the formulation of
+\cite[Theorem~4.3]{Widmer2010}, gives
+\(\lambda_1\lambda_2\lambda_3\asymp\covol(\mathfrak a)\).
+Expanding the last product, using \(\lambda_2\ge\lambda_1\), and inserting
+the first-minimum and covolume bounds proves
+Equation~\eqref{eq:add-lattice-count}.  The same coordinate functional
+bound with \(\|b_i\|\ge\lambda_1\) proves the final assertion.
+\end{proof}
+
+\subsection{The nonscalar contribution}
+
+Scalar frequencies \(A\in\Q\subset F\) in
+Equation~\eqref{eq:add-poisson} record only the trace.  The main work is to
+show that the remaining frequencies contribute an absolute power saving.
+We prove that estimate before evaluating the scalar terms.
+
+\begin{lemma}[Nonscalar contribution]
+\label{lem:add-nonscalar}
+There are absolute \(\tau,\theta,\epsilon_F>0\), with
+\(\epsilon_F>2\tau+3\theta\), such that for \(C_3=371\) and the data of
+Proposition~\ref{prop:additive-equidistribution} with these
+\(\tau,\theta\),
+\begin{equation}
+ \sum_{\substack{\mathfrak h\\H\le X^{C_3}}}
+   \left|
+    \sum_{A\in F\setminus\Q}
+       \widehat\phi(A)\widehat\eta_{\mathfrak h,l}(A)
+   \right|
+     \ll X^{-\epsilon_F}.
+ \label{eq:add-nonscalar}
+\end{equation}
+The constants have the same uniformity as in that proposition.
+\end{lemma}
+
+\begin{proof}
+We use \(\lambda=1/3\).  Lemma~\ref{lem:ext-order-scale}, which gives
+\(\Delta_F\ge X^{1/2}/2\), implies
+\begin{equation}
+ \Delta_F\ge X^\lambda
+ \label{eq:add-order-lower}
+\end{equation}
+for all sufficiently large \(X\).
+
+After bounding the real Fourier tail, we split according to the field
+discriminant.  If \(D_F\) is large, comparing it with the discriminant of
+the order generated by a retained nonscalar frequency after clearing its
+denominators forces \(H\) large.  Otherwise the lower bound for
+\(\Delta_F\) makes the index \([R_F:C_F]\) large.  For profiles with small
+\(G_0\), this forces \(H\) large; inverse support and boundary decay then
+bound the possibly signed original sum \(S_{\mathfrak h}\).  Poisson
+expresses its full nonscalar Fourier sum as \(S_{\mathfrak h}\) minus the
+separately estimated scalar series.  For profiles with large \(G_0\), the
+inverse depths attached to each frequency and its chosen shifts provide
+either sufficient pointwise decay or congruences; in the latter range a
+further discriminant comparison supplies the required denominator size.
+
+The positive exponents
+\(d,e,e_1,e_2,w,\theta,\tau\) below will be fixed at the end of the proof;
+put \(Y=X^w\) and \(s=w+\theta\).
+
+First the frequencies outside \(\|A\|\le Y\) can be discarded.  For every
+fixed integer \(r\ge0\), integration by parts in the fixed real coordinates
+gives, uniformly over the box family,
+\begin{equation}
+ |\widehat\phi(A)|\ll_r
+       X^{r\tau}(1+\|A\|)^{-r}.
+ \label{eq:add-real-decay}
+\end{equation}
+Indeed each derivative of order \(r\) costs at most a constant times
+\(X^{r\tau}\), and all supports are in one compact set.
+Lemma~\ref{lem:fourier-shell-estimates} and the bounds for \(p\notin S\) give
+\(|\widehat\eta_{\mathfrak h,l}(A)|\ll H^{-1}\).
+The first-minimum bound also gives the rough count
+\(\#(\Gamma_{l,\mathfrak h}\cap B_R)\ll(1+RP^{1/3})^3\).
+For \(R\ge Y\ge1\) this is \(O(R^3P)\), since \(P=l^3H\ge1\).
+Dyadic summation of Equation~\eqref{eq:add-real-decay}, for \(r>3\), yields
+\begin{equation}
+ \sum_{\substack{A\in F\\\|A\|>Y}}
+   |\widehat\phi(A)\widehat\eta_{\mathfrak h,l}(A)|
+  \ll_r X^{r\tau}l^3Y^{3-r}
+  \le X^{r\tau+3\theta+w(3-r)}\,O_r(1).
+ \label{eq:add-real-tail}
+\end{equation}
+This also proves absolute convergence in Equation~\eqref{eq:add-poisson}.
+The profile count costs only \(X^{o(1)}\).  When bounding Fourier terms
+directly, we may therefore restrict to the ball and use the uniform bound
+\(|\widehat\phi|\ll1\) there.
+
+Suppose first that \(D_F\ge X^d\).  A nonscalar \(A\) generates \(F\), since
+the field has degree three.  Moreover \(HR_F\subset\mathfrak h\): the
+integer \(H=|R_F/\mathfrak h|\) annihilates this finite group.  Thus
+\(lHA\in R_F\) for \(A\in\Gamma_{l,\mathfrak h}\).  The discriminant of the
+order \(\Z[lHA]\) is a positive integer multiple of \(D_F\), whereas the
+product of the squared differences of its three real conjugates is
+\(O((lHY)^6)\).  Existence of a nonscalar frequency in the ball therefore
+implies
+\[
+ D_F\ll(lHY)^6,\qquad H\gg X^{d/6-s}.
+\]
+Apply Equation~\eqref{eq:add-lattice-count} to
+\(\Gamma_{l,\mathfrak h}\) and multiply it by \(H^{-1}\).  The four resulting
+terms are bounded by constants times
+\begin{equation}
+ X^{-d/6+s}
+ +X^{-d/9+5s/3}
+ +X^{-d/18+7s/3}
+ +X^{-d/2+3s}.
+ \label{eq:add-large-field}
+\end{equation}
+All have a fixed power saving once \(s<d/100\).  If the ball contains no
+nonscalar frequency there is nothing to count for that profile.
+
+We now assume \(D_F<X^d\).  When \(d\le\lambda/3\),
+Equation~\eqref{eq:add-order-lower} gives
+\[
+ [R_F:C_F]=(\Delta_F/D_F)^{1/2}\ge X^{\lambda/3}.
+\]
+Profiles with \(G_0\le X^e\) are best estimated in their original form
+\(S_{\mathfrak h}\).  Equation~\eqref{eq:add-profile-indices} implies
+\(H\ge X^{\lambda/3-e}\).  All supporting points lie in
+\(\mathfrak a_{\mathfrak h}\), whose reciprocal norm is
+\(O(D_FG_0^{24})\) by Equation~\eqref{eq:add-profile-support}.
+The first-minimum bound of Lemma~\ref{lem:add-ideal-lattice}, applied in the
+fixed real region, gives \(O(D_FG_0^{24})\) possible points.
+At each such point, Equations
+\(\eqref{eq:add-profile-norm}\)--\(\eqref{eq:add-profile-inverse}\) give
+\[
+ \left|\prod_{p\in S}K_{p,\mathfrak h_p}(z)\right|
+   \ll X^{o(1)}
+        \left(\frac{D_FG_0^{24}}{H}\right)^{1/2}.
+\]
+Here and below \(X^{o(1)}\) denotes \(O_\eta(X^\eta)\) for every fixed
+\(\eta>0\), uniformly in the arithmetic data; it includes factors
+\(8^{|S|}\).  It follows that
+\begin{equation}
+ |S_{\mathfrak h}|
+   \ll X^{-\lambda/6+(3/2)d+(73/2)e+o(1)}.
+ \label{eq:add-small-index-profile}
+\end{equation}
+The use of the original sum has gained the boundary factor from the inverse
+transform.  It remains to check that its scalar portion is small as well,
+so that this bounds the nonscalar portion.
+
+If a rational scalar has denominator \(p^k\) at \(p\), its denominator
+ideal in \(R_{F,p}\) is \(p^kR_{F,p}\); its component exponents are
+\(e_{\mathfrak q}k\), so its norm is \(p^{3k}\), including at ramified
+primes.  Thus a scalar in this profile has its \(S\)-part denominator
+\(t=H^{1/3}\), or the profile contains no scalar.  Its denominator away
+from \(S\) divides \(l\), so there are
+\(O(1+YlH^{1/3})\) such rationals in the ball.  Their absolute contribution
+is at most
+\begin{equation}
+ O\bigl(H^{-1}+YlH^{-2/3}\bigr).
+ \label{eq:add-small-index-scalars}
+\end{equation}
+By \(H\ge X^{\lambda/3-e}\) this also saves a power for the choices below.
+Equation~\eqref{eq:add-poisson} expresses the full dual sum as
+\(S_{\mathfrak h}\).  Subtracting its scalar frequencies in the ball and
+using Equation~\eqref{eq:add-real-tail} for the scalar tail therefore bounds
+the full nonscalar sum for these profiles by
+Equations~\eqref{eq:add-small-index-profile}--\eqref{eq:add-small-index-scalars}
+and the tail.  No positivity of an individual inverse shell is needed.
+
+It remains to treat \(D_F<X^d\) and \(G_0>X^e\).  For each frequency in
+the denominator shell of the profile choose at each \(p\in S\) one of the shifts \(s_p\) of
+Lemma~\ref{lem:fourier-shift}, and write
+\[
+ \alpha_p=A+s_p,\qquad \beta_p=\alpha_p^{-1},\qquad
+ J=\prod_{p\in S}p^{j_p},
+\]
+where \(j_p\) is the least nonnegative integer with
+\(p^{j_p}\beta_p\in C_{F,p}\).  All sums involving \(J\) below are over
+frequencies in this profile.  The pointwise local estimate now reads
+\[
+ |\widehat\eta_{\mathfrak h,l}(A)|
+       \ll H^{-1}J^{-1/2}.
+\]
+For frequencies with \(J\ge X^{e_2}\), the rough lattice count immediately
+gives
+\begin{equation}
+ \sum_{\substack{\|A\|\le Y\\J\ge X^{e_2}}}
+  |\widehat\phi(A)\widehat\eta_{\mathfrak h,l}(A)|
+  \ll \frac{(1+YP^{1/3})^3}{H}\,X^{-e_2/2}
+  \ll X^{3s-e_2/2}.
+ \label{eq:add-large-j}
+\end{equation}
+We used \(H,l,Y\ge1\).  This leaves \(J<X^{e_2}\), where the extra factor in
+the Fourier bound may be too small.  The order discriminant then forces a
+large denominator ideal, and the definition of \(j_p\) supplies congruences
+among its possible frequencies.
+
+For a nonscalar \(A\) let
+\(\operatorname{disc}(A)=\Disc(1,A,A^2)\in\Q^\times\).
+The equality \(F=\Q[A]\) remains
+\(F\otimes\Q_p=\Q_p[A]\) after tensoring.  Shifting by a scalar and
+inverting an element with nonzero components preserve the generated
+algebra, so \(p^{j_p}\beta_p\) generates \(F\otimes\Q_p\).
+The full-rank order \(\Z_p[p^{j_p}\beta_p]\) is contained in \(C_{F,p}\).
+The identities
+\[
+ \operatorname{disc}(\beta_p)
+   =\frac{\operatorname{disc}(A)}
+          {\Nm_{F_p/\Q_p}(\alpha_p)^4},
+ \qquad
+ \operatorname{disc}(p^{j_p}\beta_p)
+   =p^{6j_p}\operatorname{disc}(\beta_p)
+\]
+follow by taking the three differences of conjugates and by scaling.
+Discriminant divisibility therefore gives
+\[
+ v_p(\operatorname{disc}(A))
+  -4v_p(\Nm\alpha_p)+6j_p\ge v_p(\Delta_F).
+\]
+The shift lemma says \(v_p(\Nm\alpha_p)=-\log_pH_p\) for \(p>5\), and
+differs from this by at most three at \(2,3,5\).
+For \(p\notin S\), \(lA\) is integral and \(C_{F,p}=R_{F,p}\) is unramified,
+so \(v_p(\operatorname{disc}(A))\ge-6v_p(l)\).
+Combining these valuation bounds for the nonzero rational
+\(\operatorname{disc}(A)\), with only an absolute factor from \(2,3,5\),
+gives
+\begin{equation}
+ Y^6\gg |\operatorname{disc}(A)|
+   \gg \Delta_F\,J^{-6}H^{-4}l^{-6}.
+ \label{eq:add-frequency-discriminant}
+\end{equation}
+The first inequality uses \(\|A\|\le Y\).
+Thus for \(J<X^{e_2}\), Equations
+\(\eqref{eq:add-order-lower}\) and \(\eqref{eq:add-frequency-discriminant}\)
+imply
+\begin{equation}
+ H\gg X^{\lambda/4-(3/2)(s+e_2)}.
+ \label{eq:add-denominator-lower}
+\end{equation}
+In particular, once \(s+e_2<\lambda/12\) with a margin, every nonscalar
+frequency under consideration has \(H\ge X^{\lambda/8}\) for large \(X\).
+
+For the remaining count, group frequencies according to their \(j_p\)'s and
+the chosen \(s_p\)'s.  The number of exponent lists with \(J<X^{e_2}\) is
+\(X^{o(1)}\), by the same Rankin argument as for profiles; the bounded lists
+of shifts cost \(4^{|S|}=X^{o(1)}\).  Groups containing no nonscalar
+frequency contribute zero and are omitted.  Fix one of the other groups.
+Apply the basis
+part of Lemma~\ref{lem:add-ideal-lattice} to \(\Gamma_{l,\mathfrak h}\).
+All frequencies in the ball have integer coordinates in a cube
+\([-L,L]^3\), where we choose \(L\) comparable to the upper bound
+\(YP^{1/3}\).  This choice, together with
+Equation~\eqref{eq:add-denominator-lower}, gives
+\begin{equation}
+ L\asymp YP^{1/3},\qquad
+ L\gg X^{\lambda/24},\qquad
+ L^3/H\ll X^{3s}.
+ \label{eq:add-coordinate-cube}
+\end{equation}
+
+At \(p\in S\) choose a generator \(i_p\) of the maximal-order ideal
+\(\mathfrak h_p\), and set \(x=i_p(A+s_p)\).  Localizing the global basis
+of \(\Gamma_{l,\mathfrak h}\) at \(p\), then multiplying it by \(i_p\), gives
+a \(\Z_p\)-basis of \(R_{F,p}\), because \(p\nmid l\).  Accordingly the
+map from the three integer coordinates to \(x\) is an affine
+\(\operatorname{GL}_3(\Z_p)\) change of coordinates.  This statement holds
+also for a product of ramified local fields.
+
+Smith normal form for \(\mathfrak h_p\cap C_{F,p}\) inside \(\mathfrak h_p\)
+gives a primitive linear form \(\ell_p:R_{F,p}\to\Z_p\), meaning that its
+image is \(\Z_p\), and an integer
+\(g_p'\ge g_p/3\) such that
+\[
+ \ell_p\bigl(i_p^{-1}(\mathfrak h_p\cap C_{F,p})\bigr)
+       \subset p^{g_p'}\Z_p.
+\]
+Indeed the three elementary-divisor exponents sum to \(g_p\), and we use
+the coordinate with largest exponent.  If \(p>5\), the shift lemma makes
+\(x\) a unit and \(\beta_p=i_px^{-1}\in\mathfrak h_p\).  The condition
+\(p^{j_p}\beta_p\in C_{F,p}\) implies
+\begin{equation}
+ \ell_p(x^{-1})\equiv0\pmod {p^k}
+       \quad\text{for every }1\le k\le(g_p'-j_p)_+.
+ \label{eq:add-inverse-congruence}
+\end{equation}
+Inversion permutes the units in the finite algebra
+\(R_{F,p}/p^kR_{F,p}\).  A primitive linear form has \(p^{2k}\) zeros among
+the \(p^{3k}\) additive residues.  Thus the unit condition and
+Equation~\eqref{eq:add-inverse-congruence} retain at most the fraction
+\(p^{-k}\) of all residues modulo \(p^k\).  Nilpotents in the finite algebra
+do not affect this argument.
+
+For \(p=2,3,5\), the shift lemma gives \(x\in R_{F,p}\) and
+\(x^{-1}\in p^{-1}R_{F,p}\).  Put \(y=p x^{-1}\).  Then
+\(p^{j_p}i_py=p(p^{j_p}\beta_p)\) still lies in
+\(\mathfrak h_p\cap C_{F,p}\), and the congruence in
+Equation~\eqref{eq:add-inverse-congruence} holds for \(y\) in place of
+\(x^{-1}\).  On the permitted locus the inverse identities
+\[
+ y(x)-y(x')=p x^{-1}(x'-x)(x')^{-1},\qquad
+ x-x'=p^{-1}x\bigl(y(x')-y(x)\bigr)x'
+\]
+show respectively that \(x\bmod p^{k+1}\) determines \(y\bmod p^k\), and
+that a fixed residue of \(y\bmod p^k\) confines \(x\) to one residue modulo
+\(p^{k-1}\).  The permitted locus has \(v_{\mathfrak q}(x)=0\) on the pole
+components and \(0\le v_{\mathfrak q}(x)\le e_{\mathfrak q}\) on the other
+components.  These conditions are determined modulo \(p^2\), since a
+change in \(p^2R_{F,p}\) has component valuation at least \(2e_{\mathfrak q}\).
+At modulus \(p^{k+1}\), each of the at most \(p^{2k}\) residues in
+the linear kernel therefore allows at most \(p^6\) residues of \(x\).
+The retained fraction is at most \(p^{3-k}\), hence \(O(p^{-k})\) with an
+absolute constant at these three primes.  This proves the required
+congruence fraction with only one extra power of a small prime in its
+modulus.
+
+Write \(a_p=(g_p'-j_p)_+\).  The available congruence powers satisfy
+\begin{equation}
+ \prod_{p\in S}p^{a_p}
+   \ge G_0^{1/3}/J
+   > X^{e/3-e_2}.
+ \label{eq:add-available-congruences}
+\end{equation}
+We will choose \(e/3-e_2>2e_1\).  If every prime with \(a_p>0\) is at most
+\(X^{e_1}\), multiply available factors \(p\), one at a time, until their
+product \(M_1\) first reaches \(X^{e_1/2}\).  Equation
+\(\eqref{eq:add-available-congruences}\) ensures this is possible, and
+\[
+ X^{e_1/2}\le M_1\le X^{3e_1/2}<X^{2e_1}.
+\]
+Impose just the corresponding congruences.  The Chinese remainder theorem
+combines their retained fractions into \(O(M_1^{-1})\).  Their joint modulus
+is \(M_1\) times at most \(2\cdot3\cdot5\), because only the small-prime
+tests need one extra power.  The affine integral coordinate changes above
+preserve these fractions in the global coordinates.  If
+\(2e_1<\lambda/24\), this joint modulus is \(o(L)\) by
+Equation~\eqref{eq:add-coordinate-cube}.  Counting each admissible residue
+class in the cube then gives
+\begin{equation}
+ O(L^3/M_1)=O(L^3X^{-e_1/2})
+ \label{eq:add-composite-congruence-count}
+\end{equation}
+possible frequencies.
+
+Otherwise some available prime satisfies \(p>X^{e_1}\); for large \(X\)
+it exceeds \(5\).  Use its congruence modulo \(p\).  Set
+\(B=R_{F,p}/pR_{F,p}\), a three-dimensional commutative \(\mathbf F_p\)-algebra,
+possibly with nilpotents, and reduce \(\ell_p\) modulo \(p\).  Choose an
+\(\mathbf F_p\)-basis of \(B\) and let \(x\) be the universal element of
+\(B\otimes_{\mathbf F_p}\mathbf F_p[X_1,X_2,X_3]\) in this basis.
+Let \(M_x\) be its multiplication matrix and put
+\(\mathcal N_B(x)=\det M_x\).  This is a nonzero polynomial of degree three,
+since \(\mathcal N_B(1)=1\).  On the unit open set \(\mathcal N_B\ne0\),
+\[
+ \ell_p(x^{-1})=\frac{P_\ell(x)}{\mathcal N_B(x)},\qquad
+ P_\ell(x)=\ell_p\bigl(\operatorname{adj}(M_x)1_B\bigr),
+\]
+where \(P_\ell\) has degree at most two.  This numerator is nonzero.
+Indeed the affine space parametrizing elements of the vector space \(B\)
+has coordinate ring \(\mathbf F_p[X_1,X_2,X_3]\), which is a domain even
+when multiplication in \(B\) has nilpotents.  The unit open set is dense,
+and inversion is an automorphism of that open set.  The nonzero linear
+polynomial \(\ell_p\) stays nonzero on the open set and after composition
+with inversion.  The displayed formula, or injectivity of localization at
+\(\mathcal N_B\), then forces \(P_\ell\ne0\).
+The affine \(\operatorname{GL}_3(\mathbf F_p)\) change from global lattice
+coordinates preserves nonvanishing.
+
+Let \(N_L=2\lfloor L\rfloor+1\asymp L\) be the number of possible integers in
+one coordinate of the cube.  A nonzero polynomial of bounded degree has
+\(O(N_L^2)\) zeros on a product of three residue intervals of length \(N_L<p\);
+this follows by induction on the number of variables, viewing it as a
+polynomial in the last variable and bounding the slices on which all its
+coefficients vanish.  If \(N_L\ge p\), partition the cube into boxes of side
+at most \(p\) and use the same bound there.  The number of global integer
+coordinates satisfying \(P_\ell=0\bmod p\) is consequently
+\begin{equation}
+ O\bigl(L^3/p+L^2\bigr)
+   =O\bigl(L^3(X^{-e_1}+X^{-\lambda/24})\bigr).
+ \label{eq:add-prime-congruence-count}
+\end{equation}
+This bounds the subset for which \(x\) is a unit as well.
+Multiplying Equations
+\(\eqref{eq:add-composite-congruence-count}\)--\(\eqref{eq:add-prime-congruence-count}\)
+by \(H^{-1}\), and using Equation~\eqref{eq:add-coordinate-cube}, gives
+respectively the bounds
+\[
+ O(X^{3s-e_1/2}),\qquad
+ O(X^{3s-e_1}+X^{3s-\lambda/24})
+\]
+for the fixed group of \(j_p\)'s and shifts.  Their \(X^{o(1)}\) number is
+harmless once these exponents are negative with a margin.  This completes
+the count of nonscalar frequencies in the ball.
+
+We finish by making the parameter choices and all margins explicit.  Put
+\(\lambda_0=\min(\lambda,1)\) and take
+\begin{equation}
+ \begin{aligned}
+ d=e&=\frac{\lambda_0}{1000},&
+ e_1&=\frac e{100},&
+ e_2&=\frac{e_1}{100},\\
+ w&=\frac{e_2}{100},&
+ \theta=\tau&=\frac w{100},&
+ \epsilon_F&=\frac{e_2}{4}.
+ \end{aligned}
+ \label{eq:add-parameters}
+\end{equation}
+Then \(s=101e_2/10000\),
+\[
+ s<d/100,\quad d<\lambda/3,\quad
+ e/3-e_2>2e_1,\quad 2e_1<\lambda/24,\quad
+ s+e_2<\lambda/12.
+\]
+The saving in Equation~\eqref{eq:add-small-index-profile} is at least
+\[
+ \lambda/6-(3/2)d-(73/2)e
+    \ge (1/6-0.038)\lambda_0,
+\]
+and Equation~\eqref{eq:add-small-index-scalars} has a still larger margin
+than \(\epsilon_F\), since \(e\le\lambda/1000\).
+All four exponents in Equation~\eqref{eq:add-large-field} also have margins
+at least \(29d/900>\epsilon_F\), by \(s<d/100\).
+Among the remaining ball estimates the smallest
+margin is in Equation~\eqref{eq:add-large-j}, namely
+\[
+ e_2/2-3s=\frac{4697}{10000}e_2>\epsilon_F;
+\]
+the congruence margins \(e_1/2-3s\), \(e_1-3s\), and
+\(\lambda/24-3s\) are larger.  With \(r=100\),
+the exponent in Equation~\eqref{eq:add-real-tail} is
+\[
+ r\tau+3\theta+w(3-r)=-\frac{9597}{10000}e_2.
+\]
+All \(X^{o(1)}\) costs from profiles, shifts, and exponent lists can thus be
+absorbed while retaining \(\epsilon_F=e_2/4\).  This proves
+Equation~\eqref{eq:add-nonscalar}.  Finally
+\(2\tau+3\theta=e_2/2000<\epsilon_F\), as asserted.
+\end{proof}
+
+\subsection{The scalar term and completion}
+
+\begin{proof}[Proof of Proposition~\ref{prop:additive-equidistribution}]
+Use the constants in Equation~\eqref{eq:add-parameters}.
+Equations~\eqref{eq:add-original-profiles} and \(\eqref{eq:add-poisson}\),
+together with Lemma~\ref{lem:add-nonscalar}, leave the scalar frequencies
+with absolute error \(O(X^{-\epsilon_F})\).  We first remove the profile
+cutoff from those frequencies.
+
+For \(a\in\Q\), let \(t=\prod_{p\in S}p^{\max(0,-v_p(a))}\) be its
+\(S\)-part denominator, with \(t=1\) for \(a=0\).  As noted above, its
+unique profile has \(H=t^3\).
+Lemma~\ref{lem:fourier-scalar-density} gives the exact product
+\(\prod_{p\in S}\gamma_p(a)=t^{-3}\), including at \(2,3,5\).
+The other denominator divides \(l\), so all such scalars with a given \(t\)
+lie in \((tl)^{-1}\Z\).  The change of real variables
+\((z_i,z_j,z_k)\mapsto(z_i,z_j,\Tr z)\) has absolute Jacobian one.
+Writing \(\mathbf 1=(1,1,1)\), it gives the exact identity
+\[
+ \widehat\phi(a\mathbf 1)=A_\phi\,\widehat\chi(a).
+\]
+For any fixed \(r>1\), rapid decay of the fixed function \(\chi\) implies
+\(\sum_{n\in\Z}|\widehat\chi(n/(tl))|\ll_r tl\).
+The scalars omitted by \(H\le X^{C_3}\) therefore contribute at most
+\begin{equation}
+ \sum_{t>X^{C_3/3}}t^{-3}
+       \sum_{n\in\Z}|\widehat\phi((n/(tl))\mathbf 1)|
+    \ll A_\phi\,l\,X^{-C_3/3}.
+ \label{eq:add-scalar-tail}
+\end{equation}
+We have enlarged the set of \(t\)'s to all positive integers, which is an
+upper bound.  With \(C_3=371\), this is \(O(X^{-\epsilon_F})\), since
+\(A_\phi\ll1\), \(l\le X^\theta\), and \(371/3-\theta>\epsilon_F\).
+The same calculation without the cutoff proves
+absolute convergence of the unrestricted scalar series.
+
+That series is one-dimensional Poisson summation for the trace marginals.
+At infinity the marginal is \(A_\phi\chi(t)\).  At \(p\in S\) it is
+\(d_p(t)\mathbf 1_{\Z_p}(t)\).  At \(p\mid l\), the onto integral trace map
+and two kernel coordinates show that the indicator
+\(z_p^0+p^{v_p(l)}R_{F,p}\) has trace density
+\[
+ p^{-2v_p(l)}\mathbf 1_{m+p^{v_p(l)}\Z_p}(t)
+\]
+with respect to Haar measure of mass one on \(\Z_p\).  At every other
+finite prime the marginal is \(\mathbf 1_{\Z_p}\).
+Their Fourier transforms at a rational \(a\) are exactly the finite factors
+in Equation~\eqref{eq:add-poisson} evaluated at the scalar \(a\), because
+\(\Tr(az)=a\Tr z\).
+
+For completeness, this one-dimensional application of Poisson does not
+require a smoothness assumption on \(d_p\).  Its Fourier coefficients on
+\(\Q_p/\Z_p\) of denominator \(p^k\) are \(p^{-3k}\), and there are
+\(p^k-p^{k-1}\) such characters.  Their absolute sum is finite.
+Truncating each expansion gives a compactly supported locally constant
+density on \(\Z_p\), to which Poisson applies.  The densities converge
+uniformly by Lemma~\ref{lem:fourier-scalar-density}, and the preceding
+absolute scalar bound permits passage to the limit on the Fourier side.
+On the original side, the finite marginals force the rational trace to be
+an integer.  Only \(m\) meets the real cutoff.  Hence the unrestricted
+scalar contribution is precisely
+\[
+ A_\phi\,l^{-2}\prod_{p\in S}d_p(m).
+\]
+
+Lemma~\ref{lem:fourier-scalar-density} bounds the product of \(d_p(m)\)
+above and below by positive absolute constants.  Also
+\(A_\phi\gg X^{-2\tau}\) by Equation~\eqref{eq:add-box}, and
+\(l^{-2}\ge X^{-2\theta}\).  Dividing the combined absolute error
+\(O(X^{-\epsilon_F})\) by the main term therefore costs at most
+\(O(X^{2\tau+2\theta})\).  Since
+\(\epsilon_F>2\tau+3\theta\), the relative error is \(O(X^{-\theta})\).
+Every estimate used fixed real derivative and support bounds and constants
+independent of the fields, orders, residues, and box locations and scales.
+This proves Equation~\eqref{eq:add-equidistribution} with its stated
+uniformity.
+\end{proof}

preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/conclusion.tex

@@ -0,0 +1,287 @@
+\section{Exclusion of the obstruction}\label{sec:conclusion}
+
+We apply Theorem~\ref{thm:exterior-count} to the exterior planes in
+Proposition~\ref{prop:rig-obstruction}.  Four small exterior coordinates
+give two small signed products, whose range in the trace plane has area
+of order $r^4$.  The integral along a diagonal orbit contributes
+logarithmic factors; we will show that they are summable at this scale.
+
+\subsection{The exterior vector integral}
+
+Fix a partition $b_1=ij|kl$ of $\{1,2,3,4\}$, and denote the other two
+partitions by $b_2,b_3$.  For $m\in\Z\setminus\{0\}$, $L>1$, and
+$0<r<1$, define $N_{m,L,r}(\Lambda)$ to be the number of
+$v\in\bigwedge^2\Lambda$ satisfying
+\begin{equation}\label{eq:end-vector-box}
+ Q(v)=m,\qquad |v_{ij}|,|v_{kl}|<L,\qquad
+ |v_{ab}|<r\quad(ab\ne ij,kl).
+\end{equation}
+This count is finite because $\bigwedge^2\Lambda$ is a lattice in
+$\bigwedge^2\R^4$ and the specified box is bounded.  To see that it is
+Borel, choose a local continuous determinant-one basis around a lattice
+with $k$ witnesses and keep their distinct integer exterior labels fixed.
+They remain distinct, and their $Q$-values remain $m$ because
+$Q((\bigwedge^2g)w)=\det(g)Q(w)$.  Their coordinates vary continuously,
+so the strict inequalities persist in a neighborhood.  Thus
+$\{N_{m,L,r}\ge k\}$ is open for every integer $k\ge1$.
+Recall that the three real embeddings of $F$ are indexed by
+$\mathcal B$.  For a nonzero $z\in F$, set
+\begin{equation}\label{eq:end-logarithms}
+ \mathcal L_{L,r}(z)
+  =\log_+\frac{L^2}{|z_{b_1}|}\,
+   \log_+\frac{r^2}{|z_{b_2}|}\,
+   \log_+\frac{r^2}{|z_{b_3}|}.
+\end{equation}
+All three conjugates of a nonzero element of the field $F$ are nonzero,
+so the expression is defined.
+
+In Equation~\eqref{eq:ext-multiplicity}, $M(z)$ sums over
+the fixed maximal-order ideal representatives $I$ and the $U^+$-orbits
+of nonzero raw $w\in\bigwedge^2_{\Z}I$ with $\Psi(w)/\Nm I=z$.  Each
+orbit has weight $\prod_pP_p(I,w)$, the average membership of $w$ in
+$\bigwedge^2_{\Z}L'$ over the local labels
+$L'\in\mathscr L_{\OO}(I)$.
+
+\begin{lemma}[Exterior unfolding]\label{lem:end-unfold}
+For every primitive totally real quartic field $K$, order $\OO$, and
+ordering of its real embeddings, let $d_U=[U:U^+]$.  With the preceding
+choices of $b_1,m,L,r$, use the same fixed system of maximal-order ideal
+representatives in the packet model and in the definition of $M$.  Then
+\begin{equation}\label{eq:end-unfold}
+ \int_{X_4}N_{m,L,r}(\Lambda)\,d\mu_{\OO,\sigma}(\Lambda)
+ =\frac{4}{d_U\sqrt D\,\kappa}
+       \sum_{t\in\{m,-m\}}
+       \sum_{\substack{z\in F\\\Tr z=t}}
+          M(z)\mathcal L_{L,r}(z).
+\end{equation}
+The index satisfies $2\le d_U\le16$.
+\end{lemma}
+
+\begin{proof}
+We use the packet integral in Equation~\eqref{eq:pkt-integration}.
+For its maximal-order ideal representative $I$ and local label
+$L'\in\mathscr L_{\OO}(I)$, the normalized lattice at the sign vector
+$s=(s_1,\ldots,s_4)\in\{\pm1\}^4$ and the logarithmic parameter
+$(u_1,\ldots,u_4)\in\R^4$ with $u_1+\cdots+u_4=0$ is
+\[
+ (X\Nm I)^{-1/4}\sigma(L')
+       \diag(s_1e^{u_1},\ldots,s_4e^{u_4}).
+\]
+For a raw vector $w\in\bigwedge^2 I$, its normalized exterior coordinates
+are
+\[
+ y_{ab}=(X\Nm I)^{-1/2}s_as_b e^{u_a+u_b}w_{ab}.
+\]
+If $z=\Psi(w)/\Nm I$ and $\epsilon(s)=\prod_a s_a$, then
+Equation~\eqref{eq:ext-signed-products} gives
+\begin{equation}\label{eq:end-signed-products}
+ (y_{12}y_{34},-y_{13}y_{24},y_{14}y_{23})
+       =\epsilon(s)z,\qquad
+ Q(y)=\epsilon(s)\Tr z.
+\end{equation}
+There are eight sign vectors for each value of $\epsilon(s)$.
+
+The kernel of the signature map on $U$ is $U^+$, so
+$d_U=|\operatorname{sgn}U|\le16$; the signature of $-1$ gives
+$d_U\ge2$.  After averaging over $L'$, the vector-count integrand in
+Equation~\eqref{eq:pkt-integration} is invariant under
+$v\mapsto\sigma(e)v$ for $e\in U$, since $L'\mapsto eL'$ permutes the
+labels and transports their exterior lattices.  Replacing
+the fundamental domain for $U$ in $T_\infty$ by one for $U^+$ therefore
+multiplies its integral by $d_U$, which we compensate by division by
+$d_U$.  The latter domain consists of all sign components, with a
+fundamental domain for the positive unit logarithms in each component.
+
+For a given raw $w$, its average membership over the local labels is
+\[
+ \frac1{N_{\OO}}\sum_{L'\in\mathscr L_{\OO}(I)}
+       \mathbf 1_{\{w\in\bigwedge^2L'\}}
+     =\prod_p P_p(I,w),
+\]
+the weight in Equation~\eqref{eq:ext-multiplicity}.  This weight is constant on its
+$U^+$-orbit.  That action is free: every embedding coordinate of the
+nonzero raw vector $w\in\bigwedge^2_{\Z}I\subset\bigwedge^2_{\Q}K$ is
+nonzero by Lemma~\ref{lem:ext-products}.  A stabilizing positive unit
+$e$ would therefore satisfy $\sigma_a(e)\sigma_b(e)=1$ for every pair
+$ab$, forcing $e=1$.  Summing the translates of each raw vector thus unfolds the
+positive-unit fundamental domain to all of the three-dimensional space
+$u_4=-u_1-u_2-u_3$, with multiplicity one.  The packet normalization
+remaining after the uniform label average is
+$h_K\vol(\mathscr F_U)=\sqrt D\,\kappa$ by
+Equation~\eqref{eq:pkt-unit-volume}.  Hence this unfolding has coefficient
+$1/(d_U\sqrt D\,\kappa)$ before the sign sum and the integral in $u$.
+All the sums and integrals are nonnegative, so Tonelli's theorem applies.
+
+Choose one coordinate from each opposite pair.  The logarithms of the
+absolute values of these coordinates are affine functions of
+$(u_1,u_2,u_3)$.  For the choices $12,13,14$, their linear parts are
+\[
+ u_1+u_2,\qquad u_1+u_3,\qquad -u_2-u_3.
+\]
+Their matrix has determinant of absolute value two.  Choosing the
+opposite member of a pair negates its row, and ordering the partitions
+only permutes rows, so the same absolute determinant holds for every
+choice.  If the two entries in an opposite pair have product of absolute
+value $|z_b|$ and both must be less than $h>0$ in absolute value, the
+logarithm of the chosen entry lies in an interval of length
+$\log_+(h^2/|z_b|)$.  Therefore, for a raw vector with the required
+trace after the sign change, its integral over $(u_1,u_2,u_3)$ is
+$\frac12\mathcal L_{L,r}(z)$.
+
+The $U^+$-orbit sum of the weights, including the sum over $I$, is exactly
+$M(z)$ by Equation~\eqref{eq:ext-multiplicity}.  Equation
+\eqref{eq:end-signed-products} allows raw trace $m$ for the eight positive
+determinant signs and raw trace $-m$ for the other eight.  Multiplying
+eight by the Jacobian factor $1/2$ gives the factor $4$ in
+Equation~\eqref{eq:end-unfold}.  The unit $-1$, although it acts trivially
+on exterior vectors, has already been accounted for in the index $d_U$.
+\end{proof}
+
+\begin{proposition}[Small exterior boxes]\label{prop:end-tube}
+For every fixed $m\in\Z\setminus\{0\}$ and $L>1$, there are constants
+$r_0>0$ and $C_{m,L}>0$ with the following property.  For every sequence
+of the fields, orders, and embedding orderings in Theorem~\ref{thm:main}
+with $X_i\to\infty$, and every fixed $0<r<r_0$,
+\begin{equation}\label{eq:end-tube}
+ \limsup_{i\to\infty}
+   \int_{X_4}N_{m,L,r}(\Lambda)\,
+           d\mu_{\OO_i,\sigma_i}(\Lambda)
+       \le C_{m,L}r^4.
+\end{equation}
+The constants are independent of the sequence, of $r$, and of the choice
+of partition $b_1$.
+\end{proposition}
+
+\begin{proof}
+Take $r_0<1$ so small that $2r_0^2\le |m|/2$.  For each
+$t\in\{m,-m\}$, fix the coordinate rectangle
+\[
+ \Omega_t=\{z\in\mathcal H_t:|z_{b_2}|,|z_{b_3}|\le1\}.
+\]
+Now let $0<r<r_0$.  In the trace-$t$ sum in
+Equation~\eqref{eq:end-unfold}, only points with
+$|z_{b_2}|,|z_{b_3}|<r^2$ contribute, and these points lie in $\Omega_t$.
+Since their three coordinates have sum $t$,
+\[
+ |z_{b_1}|\ge |m|-2r^2\ge |m|/2.
+\]
+Thus the first logarithm in Equation~\eqref{eq:end-logarithms} is bounded
+in terms of $m,L$ on the contributing points, independently of $r$.
+For every $z\in\Omega_t$ with $M(z)>0$, Lemma~\ref{lem:ext-support} gives
+$|z_b|\gg_{\Omega_t}X^{-84}$ for each $b$.  Hence the last two logarithms
+in Equation~\eqref{eq:end-logarithms} are $O(\log X)$, uniformly for
+$0<r<1$.
+
+We first remove the points in the two strips
+\begin{equation}\label{eq:end-strips}
+ |z_{b_2}|\le4X^{-\tau}
+       \quad\text{or}\quad
+ |z_{b_3}|\le4X^{-\tau}.
+\end{equation}
+For large $X$, each strip within $\Omega_t$ is covered by a rectangle
+of width $8X^{-\tau}$ and length $2$ in the coordinates
+$z_{b_2},z_{b_3}$, and these rectangles lie in $\Omega_t$.  Both sides
+meet the minimum length in
+Theorem~\ref{thm:exterior-count} for large $X$.  That theorem bounds the
+sum of $M(z)$ over their union, divided by $\sqrt D\,\kappa$, by
+$O(X^{-\tau})$.  The logarithmic bounds just proved show that their
+contribution to Equation~\eqref{eq:end-unfold} is
+\[
+ O_{m,L}\bigl(X^{-\tau}(\log X)^2\bigr)=o(1).
+\]
+The use of the rectangle estimate here controls the weighted mass
+directly; it does not require a pointwise bound for $M(z)$.
+
+For the remaining points, separate the signs of $z_{b_2},z_{b_3}$ and
+use the dyadic intervals
+\[
+ r^2 2^{-a-1}<|z_{b_2}|\le r^2 2^{-a},\qquad
+ r^2 2^{-b-1}<|z_{b_3}|\le r^2 2^{-b},
+ \qquad a,b\ge0.
+\]
+Each signed dyadic rectangle lies in $\Omega_t$.
+Each interval has width half its upper endpoint.  If it meets
+$|z_{b_j}|>4X^{-\tau}$, that width exceeds $2X^{-\tau}$.  Hence every
+dyadic rectangle meeting the remaining region satisfies the minimum
+side lengths of Theorem~\ref{thm:exterior-count}; any boundary rectangle
+may be included in full.  Its area is $O(r^4 2^{-a-b})$, while the last
+two factors of Equation~\eqref{eq:end-logarithms} are at most
+$(a+1)\log2$ and $(b+1)\log2$.  Theorem~\ref{thm:exterior-count} gives
+\[
+ \frac1{\sqrt D\,\kappa}
+   \sum_{z\text{ in that rectangle}} M(z)
+       \ll_{m,L} r^4 2^{-a-b}.
+\]
+The sum over the four choices of signs and over all $a,b$ is therefore
+at most
+\[
+ C_{m,L}r^4
+       \sum_{a,b\ge0}(a+1)(b+1)2^{-a-b}
+ \ll_{m,L}r^4.
+\]
+All constants are independent of $r$: the rectangles $\Omega_t$ and the
+uniform rectangle constant were fixed before choosing $r$.  The two
+strips tend to zero for each fixed $r$, and $4/d_U\le2$ in
+Equation~\eqref{eq:end-unfold}.  These observations prove
+Equation~\eqref{eq:end-tube} for both traces and all three partitions.
+\end{proof}
+
+\subsection{Identification of every probability limit}
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+For the sequence in that theorem,
+$X_i=\sqrt{|\Disc(\OO_i)|}$ tends to infinity.  By
+Proposition~\ref{prop:ord-tightness}, the packet probabilities are tight,
+and every subsequence has a further weakly convergent subsequence with
+an $A_4$-invariant probability limit $\mu$.
+Proposition~\ref{prop:ord-balls} gives the local ball bound required by
+Corollary~\ref{cor:rig-positive-components}.  Hence almost every component in
+the $A_4$-ergodic decomposition of $\mu$ has positive entropy for a
+diagonal flow.  Proposition~\ref{prop:rig-obstruction} reduces the
+identification of these components to the exclusion of its three
+exterior-plane events.
+
+Fix a partition $b_1$, a nonzero integer $m$, and $L>1$.  The event
+\[
+ E_{m,L,r}=\{\Lambda\in X_4:N_{m,L,r}(\Lambda)\ge1\}
+\]
+is open by the observation following Equation~\eqref{eq:end-vector-box}.
+
+The open-set inequality for weak convergence and
+Proposition~\ref{prop:end-tube} now give, along the convergent subsequence,
+\[
+ \mu(E_{m,L,r})
+  \le\liminf_i\mu_{\OO_i,\sigma_i}(E_{m,L,r})
+  \le\limsup_i\int N_{m,L,r}\,d\mu_{\OO_i,\sigma_i}
+  \le C_{m,L}r^4
+\]
+for every sufficiently small fixed $r>0$.  In particular,
+\[
+ \mu\left(\bigcap_{n\ge2}E_{m,L,1/n}\right)
+ \le \inf_{\substack{n\ge2\\1/n<r_0}} C_{m,L}n^{-4}=0.
+\]
+Every vector in one of the obstruction planes with $Q\ne0$ has a
+nonzero integer $Q$-value by Equation~\eqref{eq:rig-Q-integral}, and its
+two entries are less than some integer $L\ge2$ in absolute value.
+An exact witness belongs to $E_{m,L,1/n}$ for every $n\ge2$.  Thus the
+Borel event in Equation~\eqref{eq:rig-events} satisfies
+\[
+ \mathcal E_{b_1}
+ \subset
+ \bigcup_{m\in\Z\setminus\{0\}}\ \bigcup_{\substack{L\in\Z\\L\ge2}}
+       \bigcap_{n\ge2}E_{m,L,1/n}.
+\]
+The right side is a countable union of null sets, so
+$\mu(\mathcal E_{b_1})=0$.  The same argument applies to
+all three partitions.  Integrating these null events over the ergodic
+decomposition shows that the proper alternative in
+Proposition~\ref{prop:rig-obstruction} has zero total component weight.
+Almost every component is therefore $m_4$, and hence $\mu=m_4$.
+
+Every subsequential probability limit is now identified as $m_4$.
+Tightness implies that the entire sequence converges weakly to $m_4$,
+which gives the asserted convergence against every $f\in C_c(X_4)$.
+The same tightness statement supplies, for each $\epsilon>0$, one
+compact set carrying at least $1-\epsilon$ of every sufficiently late
+packet probability.  This is the required no-escape assertion.
+\end{proof}

preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/exterior.tex

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+\section{The cubic resolvent and exterior multiplicities}
+\label{sec:exterior}
+
+The obstruction in Section~\ref{sec:rigidity} is detected by an exterior
+vector concentrated in one pair of opposite coordinates.  We shall count
+such vectors through their three signed opposite products.  After packet
+normalization these products form a triple in a cubic field, and
+concentration near one exterior plane confines
+two of them to a small rectangle.  Our goal is a uniform count in that
+rectangle.  We first define the count and reduce its multiplicities to local
+densities.  We then construct the integral quadratic structure that will
+control those densities in the next two sections.
+
+\subsection{The three products and the counting problem}
+
+Let \(\widetilde K\subset\R\) be the normal closure of \(K\), and let
+\(G=\Gal(\widetilde K/\Q)\) act on the four ordered embeddings of \(K\).
+This permutation action is primitive: a nontrivial block would give a field
+strictly between \(\Q\) and \(K\).  A transitive subgroup of \(S_4\) has
+order \(4,8,12\), or \(24\).  A subgroup of order \(4\) or \(8\) is contained
+in a Sylow \(2\)-subgroup, which preserves a partition into two pairs.
+It follows that
+\[
+G=\mathfrak A_4\quad\hbox{or}\quad G=S_4.
+\]
+Both groups act transitively on the six two-element subsets of
+\(\{1,2,3,4\}\) and on the set of pair partitions
+\[
+\mathcal B=\{12|34,\ 13|24,\ 14|23\}.
+\]
+Define
+\begin{equation}
+ F=\left\{(z_b)_{b\in\mathcal B}\in\widetilde K^{\mathcal B}:
+       g(z_b)=z_{gb}\ \text{for all }g\in G,\ b\in\mathcal B\right\}.
+ \label{eq:ext-resolvent}
+\end{equation}
+Evaluation at any partition identifies this algebra with the fixed field
+of its stabilizer.  Thus \(F\) is a totally real cubic field, and its three
+real embeddings are the coordinates in Equation~\eqref{eq:ext-resolvent}.
+In particular, \(\Tr_{F/\Q}z=\sum_b z_b\).  This is the classical cubic
+resolvent from the action on pair partitions; for its relation to cubic
+resolvent rings, see \cite[Section~2.3]{Bhargava2004III}.  We write
+\(R_F=\OO_F\) and \(D_F=|\Disc(F)|\).
+
+Put
+\[
+ V=\bigwedge\nolimits^2_{\Q}K,\qquad
+ L_V=\bigwedge\nolimits^2_{\Z}\OO.
+\]
+For \(w=\sum_r x_r\wedge y_r\in V\), its embedding coordinates are
+\[
+ w_{ij}=\sum_r
+ \bigl(\sigma_i(x_r)\sigma_j(y_r)-\sigma_j(x_r)\sigma_i(y_r)\bigr)
+ \quad(i<j),\qquad w_{ji}=-w_{ij}.
+\]
+Choose a basis \(\beta_1,\ldots,\beta_4\) of \(\OO\) whose real embedding
+matrix \(A_{\OO}=(\sigma_i(\beta_j))\) has positive determinant.  Write
+\(\delta_{\OO}=\det A_{\OO}\in\widetilde K\).  Its selected real value is
+\(X=q\sqrt D\), and
+\begin{equation}
+ \delta_{\OO}^{\,2}=\Disc(\OO)=X^2.
+ \label{eq:ext-embedding-determinant}
+\end{equation}
+Define the quadratic expression
+\begin{equation}
+ \Psi(w)=\delta_{\OO}^{-1}
+       (w_{12}w_{34},-w_{13}w_{24},w_{14}w_{23}).
+ \label{eq:ext-product-map}
+\end{equation}
+
+\begin{lemma}[The product map]
+\label{lem:ext-products}
+The expression in Equation~\eqref{eq:ext-product-map} defines a rational
+quadratic map \(\Psi:V\to F\).  In the exterior basis of \(L_V\) induced
+by \((\beta_i)\), write \(w=\sum_{i<j}t_{ij}\beta_i\wedge\beta_j\).
+Then
+\begin{equation}
+ Q_V(w):=\Tr_{F/\Q}\Psi(w)
+       =t_{12}t_{34}-t_{13}t_{24}+t_{14}t_{23}.
+ \label{eq:ext-integral-form}
+\end{equation}
+If \(w\in V\) is nonzero, all six \(w_{ij}\), and hence all three
+components of \(\Psi(w)\), are nonzero.  For every extension \(k/\Q\),
+\(a\in(K\otimes_\Q k)^\times\), \(c\in k\), and \(w\in V\otimes_\Q k\),
+\begin{equation}
+ \Psi\bigl(c(\bigwedge\nolimits^2 a)w\bigr)
+       =c^2\Nm_{K\otimes k/k}(a)\Psi(w),
+ \label{eq:ext-covariance}
+\end{equation}
+where \(\bigwedge^2a\) is the exterior square of multiplication by \(a\).
+\end{lemma}
+
+\begin{proof}
+An element \(g\in G\) permutes the rows of \(A_{\OO}\), so it multiplies
+\(\delta_{\OO}\) by the sign of that permutation.  Permuting the four
+indices in the three signed products in
+Equation~\eqref{eq:ext-product-map} permutes the partitions and introduces
+the same common sign.  Division by \(\delta_{\OO}\) gives precisely the
+equivariance in Equation~\eqref{eq:ext-resolvent}.  This is an identity of
+quadratic polynomials in the rational coordinates of \(w\), and therefore
+defines a rational quadratic map to \(F\).
+
+The Pfaffian identity for the exterior square of a four-dimensional
+linear map is
+\[
+ Q\bigl((\bigwedge\nolimits^2 A_{\OO})t\bigr)
+       =\det(A_{\OO})Q(t),
+ \qquad
+ Q(t)=t_{12}t_{34}-t_{13}t_{24}+t_{14}t_{23}.
+\]
+It follows either by expanding the six minors or by evaluating
+\(\frac12 t\wedge t\) in the top exterior power.  This proves
+Equation~\eqref{eq:ext-integral-form}.  If a coordinate of a rational
+\(w\) vanishes, Galois transitivity on the unordered pairs makes all six
+coordinates vanish.  The exterior embedding map is injective, so \(w=0\).
+Finally, multiplication by \(a\) multiplies \(w_{ij}\) by \(a_i a_j\)
+after splitting, where \(a_i\) are its four embeddings.  Each opposite
+product is multiplied by \(\prod_i a_i=\Nm(a)\), and the scalar \(c\)
+contributes \(c^2\).  This proves Equation~\eqref{eq:ext-covariance}
+after splitting and hence over \(k\).
+\end{proof}
+
+We relate these products to the packet model of Section~\ref{sec:packets}.
+Let \(I\) be one of its maximal-order ideal-class representatives, let
+\(L\subset I\) be one of its local labels, and let
+\(v=(v_i)\in T_\infty\).  If \(w\in\bigwedge^2_{\Z}L\), the corresponding
+exterior vector in the normalized lattice has coordinates
+\[
+ \widehat w_{ij}=(X\Nm I)^{-1/2}w_{ij}v_i v_j .
+\]
+Since \(\prod_i|v_i|=1\), the number
+\(\epsilon(v)=\prod_i v_i\) is \(1\) or \(-1\).  Thus
+\begin{equation}
+ (\widehat w_{12}\widehat w_{34},
+  -\widehat w_{13}\widehat w_{24},
+  \widehat w_{14}\widehat w_{23})
+   =\epsilon(v)\,z(I,w),\qquad
+ z(I,w):=\frac{\Psi(w)}{\Nm I}.
+ \label{eq:ext-signed-products}
+\end{equation}
+In particular \(Q(\widehat w)=\epsilon(v)\Tr z(I,w)\).  This records
+exactly how the diagonal signs act on the products.
+
+Lemma~\ref{lem:ext-products} and the normalization above show that every
+lattice in a packet lies outside each \(\mathcal E_b\).  To prove that weak
+limits give these events zero mass, we need quantitative control of exterior
+vectors near \(V_b\); the count below supplies the arithmetic input for that
+control.
+
+For a rational prime \(p\), fix a generator \(I_p=\xi_pR_p\) as in the
+packet model, and give \(R_p^\times\) its Haar probability.  For
+\(0\ne w\in\bigwedge^2_{\Z}I\), define
+\begin{equation}
+ P_p(I,w)=\int_{R_p^\times}
+ \mathbf1_{\bigwedge^2_{\Z_p}(\xi_p t\OO_p)}(w)\,d^\times t .
+ \label{eq:ext-label-probability}
+\end{equation}
+Changing \(\xi_p\) by an \(R_p\)-unit leaves this probability unchanged.
+For \(p\nmid q\), it is \(1\).  Hence \(\prod_p P_p(I,w)\) is a finite
+product.  Multiplication by \(U^+\) preserves these probabilities by
+translation of the local unit variables.  For \(e\in U^+\), the algebraic
+norm \(\Nm_{K/\Q}(e)=1\), so Equation~\eqref{eq:ext-covariance} also
+shows that \(z(I,w)\) is unchanged.  We use \(U^+\) to unfold the positive
+real parameters on each choice of signs.
+Its action on nonzero exterior vectors is free: if a totally positive
+unit \(e\) fixes \(w\), then \(e_i e_j=1\) for every pair by
+Lemma~\ref{lem:ext-products}; the four positive \(e_i\) must all be \(1\).
+
+For \(z\in F\), define the nonnegative weighted multiplicity
+\begin{equation}
+ M(z)=
+ \sum_{\substack{I,\;
+    w\in(\bigwedge^2_{\Z}I\setminus\{0\})/U^+\\
+    z(I,w)=z}}
+       \prod_p P_p(I,w).
+ \label{eq:ext-multiplicity}
+\end{equation}
+The sum is initially allowed to have value \(+\infty\); the local bounds
+below prove its finiteness on the nonzero trace slices used here.
+It depends on \(K,\OO\), the ordering of
+the embeddings, and the chosen system of maximal-order ideal
+representatives, which we suppress from the notation.  In fact replacing
+\(I\) by \(aI\) and transporting \(w\) by \(\bigwedge^2a\) multiplies
+that class's products by \(\operatorname{sgn}(\Nm_{K/\Q}a)\), since
+\(\Nm(aI)=|\Nm_{K/\Q}a|\Nm I\).  We keep the representative system fixed.
+A term of positive
+weight admits a local label at every prime.  These labels patch to a
+global \(L\), and local membership implies \(w\in\bigwedge^2_{\Z}L\).
+Equation~\eqref{eq:ext-signed-products}, with positive signs, then shows
+that \(\Tr z(I,w)\) is the integral value of \(Q\) on a unimodular
+exterior lattice.  Thus
+\begin{equation}
+ M(z)>0\quad\Longrightarrow\quad
+ z\in F^\times,\qquad \Tr_{F/\Q}z\in\Z.
+ \label{eq:ext-trace-support}
+\end{equation}
+
+For \(m\in\Z\setminus\{0\}\), put
+\[
+ \mathcal H_m=\{(y_b)_{b\in\mathcal B}\in\R^{\mathcal B}:
+                      \textstyle\sum_b y_b=m\}.
+\]
+A rectangle in \(\mathcal H_m\) means the inverse image of a product of
+two intervals under any two distinct partition coordinates.  Its area
+is the product of those interval lengths.  The following estimate is
+the arithmetic input needed to exclude the obstruction.
+
+\begin{theorem}[Planar exterior count]
+\label{thm:exterior-count}
+There is an absolute \(\tau>0\) with the following property.  Fix
+\(m\in\Z\setminus\{0\}\) and a bounded set
+\(\Omega\subset\mathcal H_m\).  For every primitive totally real quartic
+field \(K\), every order \(\OO\subset K\), every ordering of its real
+embeddings, and every chosen system of fractional \(R\)-ideal
+representatives for the ordinary classes, let \(M\) be defined by
+Equation~\eqref{eq:ext-multiplicity}.
+For all sufficiently large \(X\), every rectangle
+\(\mathcal R\subset\Omega\) whose two side lengths are at least
+\(X^{-\tau}\) satisfies
+\begin{equation}
+ \sum_{\substack{z\in F,\ \Tr z=m\\z\in\mathcal R}}M(z)
+ \ll_{m,\Omega}\kappa\sqrt D\,\operatorname{area}(\mathcal R).
+ \label{eq:ext-planar-count}
+\end{equation}
+The implied constant and the threshold for \(X\) are uniform in the
+field, the order, the embedding ordering, the ideal representative system,
+the position of the rectangle, and its side lengths and aspect ratio.
+\end{theorem}
+
+Here and below we identify \(z\in F\) with its real triple.  To see the
+scale of this statement, suppose \(Q(\widehat w)=m\) and the four
+coordinates outside \(12|34\) have absolute value less than \(r\).
+The signed products of \(\widehat w\) then have the form
+\((m-y_2-y_3,y_2,y_3)\) with \(|y_2|,|y_3|<r^2\).  Their region has area
+\(O(r^4)\).  Equation~\eqref{eq:ext-signed-products} allows either trace
+\(m\) or \(-m\) for \(z(I,w)\), so the theorem is used at both fixed
+traces.  Sections~\ref{sec:local-fourier}--\ref{sec:sieve} prove
+Theorem~\ref{thm:exterior-count}; the remainder of this section supplies
+its local and algebraic inputs.
+
+\subsection{The local product fibers}
+
+Fix \(m\ne0\) for the rest of the section.  Choose once a sufficiently
+large absolute \(p_0\), and put
+\begin{equation}
+ S=\{p:p\mid Dqm\}\ \cup\ \{p:p\le p_0\}.
+ \label{eq:ext-prime-set}
+\end{equation}
+For fixed \(m\),
+\(\prod_{p\in S}p\ll_m Dq\le X^2\).  At primes in \(S\) we shall keep
+the exact additive density of the products.  Outside \(S\), integral
+valuations give a simpler bound.
+
+The following comparison explains what a fiber remembers about an exterior
+vector.  For \(k=\Q\) or \(k=\Q_p\), put \(K_k=K\otimes_\Q k\) and
+\(V_k=V\otimes_\Q k\), and take embedding coordinates over a splitting
+field of the etale algebra \(K_k\).
+
+\begin{lemma}[Comparison of two product fibers]
+\label{lem:ext-comparison}
+Let \(w,w'\in V_k\) have all six embedding coordinates nonzero.  Suppose
+that their coordinate ratios \(r_{ij}=w'_{ij}/w_{ij}\) satisfy
+\[
+ r_{12}r_{34}=r_{13}r_{24}=r_{14}r_{23}=s\in k^\times.
+\]
+There exist \(a\in K_k^\times\) and \(c\in k^\times\) such that
+\[
+ w'=c(\bigwedge\nolimits^2 a)w,\qquad c^2\Nm_{K_k/k}(a)=s.
+\]
+The pair \((a,c)\) is unique up to
+\((a,c)\mapsto(da,d^{-2}c)\), \(d\in k^\times\).
+\end{lemma}
+
+\begin{proof}
+Over the splitting field set
+\[
+ c_0=\frac{r_{12}r_{13}}{r_{23}},\qquad
+ (a_1,a_2,a_3,a_4)=
+ \left(1,\frac{r_{23}}{r_{13}},\frac{r_{23}}{r_{12}},
+                 \frac{r_{14}r_{23}}{r_{12}r_{13}}\right).
+\]
+The three equal opposite products give \(r_{ij}=c_0a_i a_j\) for every
+pair.  If two such factorizations are divided, all products of two of
+the four coordinatewise quotients are equal.  Comparing pairs that
+share one index shows that the four quotients are equal.  Hence the
+projective tuple \([a_1:a_2:a_3:a_4]\) is uniquely determined.
+
+The ratios \(r_{ij}\) are Galois-equivariant without alternating signs.
+Uniqueness makes the projective tuple equivariant for the permutation
+action on the four embeddings.  Express it in a \(k\)-basis of \(K_k\).
+The resulting projective coefficient vector is fixed by the usual
+Galois action.  Dividing by one nonzero coefficient makes every
+coefficient fixed, hence in \(k\).  It gives \(a\in K_k^\times\), since
+all four embedding components are nonzero.  The remaining coefficient
+\(c\) in \(r_{ij}=c a_i a_j\) is Galois-fixed as well.  This proves
+existence over \(k\), including when \(K_k\) is a product of nonsplit
+fields.  The projective uniqueness proves the stated ambiguity.
+Multiplying two opposite equalities gives \(c^2\Nm(a)=s\).
+\end{proof}
+
+For a prime \(p\), \(F_p^\times\) denotes the elements nonzero in every
+field factor of \(F_p\).  In the local formulas below, \(\Nm(a)\) denotes
+the algebraic norm \(\Nm_{K_p/\Q_p}(a)\).  For a global pair in
+Equation~\eqref{eq:ext-multiplicity}, choose a representative
+\(0\ne w\in\bigwedge^2_\Z I\) of its \(U^+\)-orbit and put
+\[
+ x=(\bigwedge\nolimits^2\xi_p)^{-1}w,\qquad
+ u=\frac{\Nm_{K_p/\Q_p}(\xi_p)}{\Nm I}.
+\]
+The valuations of the numerator and denominator in \(u\) are equal:
+both are the norm valuation of \(I_p\).  Thus \(u\in\Z_p^\times\), and
+Equation~\eqref{eq:ext-covariance} gives \(u\Psi(x)=z(I,w)\).
+Inverting the unit variable in Equation~\eqref{eq:ext-label-probability}
+gives
+\begin{equation}
+ P_p(I,w)=
+ \Pr_{a\in R_p^\times}\bigl[(\bigwedge\nolimits^2a)x\in L_{V,p}\bigr].
+ \label{eq:ext-unit-membership}
+\end{equation}
+
+These representatives lead us to consider, for arbitrary \(z\in F_p^\times\),
+the full local fiber
+\[
+ \mathscr F_p(z)=
+ \{(x,u)\in V_p\times\Z_p^\times:u\Psi(x)=z\}.
+\]
+Define the abelian local group and its unit subgroup by
+\begin{align}
+ \mathscr G_p
+  &=\{(a,c)\in K_p^\times\times\Q_p^\times:
+                 c^2\Nm(a)\in\Z_p^\times\}/
+        \{(d,d^{-2}):d\in\Q_p^\times\}, \notag\\
+ \mathscr U_p
+  &=\operatorname{image}(R_p^\times\times\Z_p^\times)
+       \ \subset\mathscr G_p.
+ \label{eq:ext-local-group}
+\end{align}
+The quotient in the first line means that \(d\) acts by
+\((a,c)\mapsto(da,d^{-2}c)\).  The action on the fiber is
+\begin{equation}
+ (a,c)\cdot(x,u)=
+ \left(c(\bigwedge\nolimits^2a)x,\,
+                    \frac{u}{c^2\Nm(a)}\right).
+ \label{eq:ext-fiber-action}
+\end{equation}
+Lemma~\ref{lem:ext-comparison}, with \(s=u/u'\), shows that this action
+is free and transitive when the fiber is nonempty.  The explicit ratios
+in its proof also give analytic local inverses, so this is an analytic
+identification of the fiber with \(\mathscr G_p\).  The group
+\(\mathscr U_p\) is compact and open.  The kernel of its unit
+parametrization is exactly \(d\in\Z_p^\times\): if both \(a\) and \(da\)
+are \(R_p\)-units, then \(v_p(d)=0\).  In particular this assertion
+holds at ramified primes.
+
+Choose a reference in a nonempty fiber and use it to identify the fiber
+with \(\mathscr G_p\).  For a coset
+\(\mathfrak c\in\mathscr G_p/\mathscr U_p\), let
+\(\alpha_p(\mathfrak c;z)\) be the fraction of that coset for which
+\(x\in L_{V,p}\), using Haar probability on \(\mathscr U_p\).
+We call the coset accepted when \(\alpha_p(\mathfrak c;z)>0\).
+The fraction is independent of the representative of the coset.  The sum of
+these fractions is independent of the reference.  For an empty fiber
+we declare this sum to be zero.
+
+A scalar \(c\in\Z_p^\times\) preserves \(L_{V,p}\).  The pushforward of
+the product Haar probability on \(R_p^\times\times\Z_p^\times\) is
+Haar probability on \(\mathscr U_p\).  Hence the representative constructed
+from \((I,w)\) occupies a coset \(\mathfrak c\) for which
+\(P_p(I,w)=\alpha_p(\mathfrak c;z(I,w))\).
+
+We now measure the sum over all local cosets.  Fix an additive character
+\(\psi_p\) of \(\Q_p\) with conductor \(\Z_p\), and let \(dz\) be the
+self-dual Haar measure on \(F_p\) for the trace pairing.  Thus
+\begin{equation}
+ \vol_{dz}(R_{F,p})=|D_F|_p^{1/2}.
+ \label{eq:ext-self-dual-measure}
+\end{equation}
+
+For every \(p\), consider the law of the random variable
+\begin{equation}
+ u\Psi(x),\qquad x\ \text{uniform additive in }L_{V,p},\qquad
+ u\ \text{uniform in }\Z_p^\times,
+ \label{eq:ext-density-law}
+\end{equation}
+with the two choices independent.  The next lemma proves that this law
+has a density with respect to \(dz\); we denote its locally constant
+version on \(F_p^\times\) by \(f_p^*(z)\).  Only the densities for
+\(p\in S\) will enter the multiplicity bound.
+
+\begin{lemma}[Local density and disintegration]
+\label{lem:ext-density}
+The law in Equation~\eqref{eq:ext-density-law} has a density, with a
+locally constant version on \(F_p^\times\).  For every \(z\in F_p^\times\),
+only finitely many cosets are accepted, and their fractions satisfy the
+exact identity
+\begin{equation}
+ \sum_{\mathfrak c\in\mathscr G_p/\mathscr U_p}
+        \alpha_p(\mathfrak c;z)
+ =
+ |D|_p^{-1/2}
+ \frac{1-p^{-1}}
+      {\displaystyle\prod_{\mathfrak p\mid p,\ \mathfrak p\subset R}
+                      (1-\Nm(\mathfrak p)^{-1})}
+ f_p^*(z).
+ \label{eq:ext-density-disintegration}
+\end{equation}
+The sum and the density are both zero if \(\mathscr F_p(z)\) is empty.
+The identity holds at every prime, including primes dividing \(q\) and
+the small ramified primes.
+\end{lemma}
+
+\begin{proof}
+Let \(\Phi_p(x,u)=u\Psi(x)\).  If \(\Phi_p(x,u)\in F_p^\times\), all six
+split coordinates of \(x\) are nonzero.  After splitting, the derivative
+in \(x\) is the derivative of three products on three disjoint
+two-dimensional planes.  It has rank three, and therefore has rank
+three over \(\Q_p\).  The map is a submersion there.  The remaining
+input set is a finite union of proper algebraic zero loci and has
+additive measure zero.  Local change of variables gives an absolutely
+continuous pushforward on the submersion locus, hence a density for
+the full law.  To obtain a locally constant version at a fixed
+\(z\in F_p^\times\), cover its inverse image in the compact set
+\(L_{V,p}\times\Z_p^\times\) by finitely many submersion charts,
+refined to disjoint compact open pieces.  The compact complement has
+image avoiding a sufficiently small neighborhood of \(z\).
+On that neighborhood each chart contributes a locally constant
+density.  This also treats an empty inverse image.
+
+Give \(V_p\) its additive measure for which \(L_{V,p}\) has mass one,
+before restricting to that lattice.  We first compute the fiber form for
+the product input measure after disintegration against \(dz\), and then
+the mass of \(\mathscr U_p\) for that form.  In a \(\Q_p\) submersion chart,
+the relative absolute four-form is obtained by evaluating the input top
+form on four fiber basis vectors and three lifts of a target basis, then
+dividing by the \(dz\)-form on that target basis.  We may compute this
+identity after extension to a splitting field, still evaluating on \(\Q_p\) tangent
+vectors and using the fixed extension of \(|p|_p=p^{-1}\).  We use the
+same interpretation for the scalar quotient below.
+
+Over a splitting field, the exterior embedding matrix has determinant
+\(\delta_{\OO}^{\,3}\).  Consequently its measure form in the six
+embedding coordinates is
+\[
+ |\delta_{\OO}|_p^{-3}\left|\bigwedge_{i<j}dx_{ij}\right|.
+\]
+Likewise \(dz\) is the absolute embedding-coordinate form
+\(\left|\bigwedge_{b\in\mathcal B}dz_b\right|\).  To verify the latter
+normalization, evaluate this form on an \(R_{F,p}\)-basis.  The square
+of its embedding determinant is the discriminant, so its absolute
+value is \(|D_F|_p^{1/2}\), as in
+Equation~\eqref{eq:ext-self-dual-measure}.
+
+After splitting, compute the relative form by retaining
+\(x_{12},x_{13},x_{14},u\) and substituting the three product equations
+for \(x_{34},x_{24},x_{23}\).  The three change-of-variable factors are
+\(|\delta_{\OO}/(u x_{12})|_p\),
+\(|\delta_{\OO}/(u x_{13})|_p\), and
+\(|\delta_{\OO}/(u x_{14})|_p\).  They cancel
+\(|\delta_{\OO}|_p^{-3}\), since \(u\) is a unit.  Haar probability on
+\(\Z_p^\times\) is \((1-p^{-1})^{-1}|du/u|\).  The resulting fiber form is
+\begin{equation}
+ \frac{1}{1-p^{-1}}
+ \left|\frac{dx_{12}}{x_{12}}\wedge
+       \frac{dx_{13}}{x_{13}}\wedge
+       \frac{dx_{14}}{x_{14}}\wedge\frac{du}{u}\right|.
+ \label{eq:ext-fiber-form}
+\end{equation}
+The fiber action preserves this measure.  Indeed its determinant on
+\(V_p\) is \(c^6\Nm(a)^3=(c^2\Nm(a))^3\), of absolute value one,
+and it multiplies \(u\) by a unit while fixing \(z\).
+
+It remains to compute the mass of \(\mathscr U_p\) in
+Equation~\eqref{eq:ext-fiber-form}.  Put
+\(\ell_i=da_i/a_i\) and \(s=dc/c\) over the splitting field.  The
+pullbacks of its four logarithmic differentials under
+Equation~\eqref{eq:ext-fiber-action} are
+\[
+ \ell_1+\ell_2+s,\quad \ell_1+\ell_3+s,\quad
+ \ell_1+\ell_4+s,\quad -\ell_1-\ell_2-\ell_3-\ell_4-2s.
+\]
+The vertical scalar direction \(d\) has vector
+\((1,1,1,1,-2)\).  Taking the first \(\ell_i\) as its coordinate and
+using the slice \(\ell_1=0\), the matrix on the four remaining
+differentials is
+\[
+ \begin{pmatrix}
+  1&0&0&1\\
+  0&1&0&1\\
+  0&0&1&1\\
+  -1&-1&-1&-2
+ \end{pmatrix},
+ \qquad \det=1.
+\]
+The vertical coordinate has coefficient \(1\).  Equivalently, the
+wedge of the four displayed pullbacks is the contraction of
+\((\bigwedge_i\ell_i)\wedge s\) by the vertical vector, up to sign.
+Evaluating this identity on four \(\Q_p\) tangent lifts of a quotient basis
+gives the same absolute quotient form for nonsplit \(K_p\).  On a patch
+where a \(\Q_p\)-linear coefficient \(\lambda(a)\) is nonzero, normalizing
+it to its value at a reference point gives a genuine \(\Q_p\) section;
+after shrinking a unit-group patch, the normalizing scalar is a unit.
+Since \(\lambda(da)=d\lambda(a)\), the vertical logarithmic derivative is
+\(1\).  Thus the contraction is the quotient of the parameter form by
+\(|dd/d|\), with quotient Jacobian \(1\), including at \(p=2\).
+
+For an \(R_p\)-basis, the multiplicative embedding form on \(K_p^\times\)
+is the additive basis form multiplied by its embedding determinant
+and divided by \(\Nm(a)\).  On \(R_p^\times\) the latter is a unit.
+The mass of this unit set is consequently
+\[
+ |D|_p^{1/2}
+ \prod_{\mathfrak p\mid p,\ \mathfrak p\subset R}
+               (1-\Nm(\mathfrak p)^{-1}).
+\]
+The \(c\)-unit mass and the scalar-kernel \(d\)-unit mass are both
+\(1-p^{-1}\), and cancel when taking the quotient.  Haar integration
+on the compact unit groups assembles the local sections.  Including the
+factor \((1-p^{-1})^{-1}\) in Equation~\eqref{eq:ext-fiber-form}, the
+mass of each \(\mathscr U_p\)-coset is
+\begin{equation}
+ \frac{|D|_p^{1/2}}{1-p^{-1}}
+ \prod_{\mathfrak p\mid p,\ \mathfrak p\subset R}
+               (1-\Nm(\mathfrak p)^{-1}).
+ \label{eq:ext-unit-coset-mass}
+\end{equation}
+This calculation uses the actual image of the unit group, so it
+requires no surjectivity assertion about the norm on units.
+
+The accepted part of the fiber is the inverse image of \(z\) in the
+compact set \(L_{V,p}\times\Z_p^\times\).  Since \(\mathscr U_p\) is
+open, this compact set meets only finitely many of its cosets.
+Integrating its indicator in the fiber multiplies the mass in
+Equation~\eqref{eq:ext-unit-coset-mass} by
+\(\sum_{\mathfrak c}\alpha_p(\mathfrak c;z)\).  The result is the
+density \(f_p^*(z)\).  Solving for the sum proves
+Equation~\eqref{eq:ext-density-disintegration}.
+\end{proof}
+
+In particular the order index has disappeared from the prefactor in
+Equation~\eqref{eq:ext-density-disintegration}: its contribution through
+\(\delta_{\OO}\) cancels in the three products.  Its effect remains
+inside the density of the lattice \(L_{V,p}\).  The discriminant in the
+prefactor is \(D\) because the translating parameters are the
+maximal-order units \(R_p^\times\).
+
+\subsection{Global multiplicity and the unramified factors}
+
+The local identity counts every local coset.  We next bound how many
+global pairs can occupy one assignment of those cosets.
+
+\begin{lemma}[Global pairs in one assignment of cosets]
+\label{lem:ext-global-cosets}
+Fix \(z\in F^\times\).  Choose a reference in each nonempty local fiber
+\(\mathscr F_p(z)\).  For any assignment
+\(\mathfrak c_p\in\mathscr G_p/\mathscr U_p\), at most \(32\) pairs
+\[
+ I,\qquad w\in(\bigwedge^2_{\Z}I\setminus\{0\})/U^+,\qquad z(I,w)=z
+\]
+occupy the assigned coset at every prime.
+\end{lemma}
+
+\begin{proof}
+For two such pairs, Lemma~\ref{lem:ext-comparison} over \(\Q\) gives
+\(w'=c(\bigwedge^2a)w\) with \(a\in K^\times\), \(c\in\Q^\times\), since
+the common opposite ratio is \(\Nm(I')/\Nm(I)\).  Relative to the local
+generators \(\xi_p,\xi'_p\), their local comparison is represented by
+\((a\xi_p/\xi'_p,c)\).  Equality of the local cosets means that some
+\(d_p\in\Q_p^\times\) satisfies
+\[
+ d_pa\xi_p/\xi'_p\in R_p^\times,\qquad d_p^{-2}c\in\Z_p^\times.
+\]
+In particular \(v_p(c)=2v_p(d_p)\) is even at every finite prime.
+Therefore \(c=\epsilon s^2\) with \(s\in\Q^\times\) and
+\(\epsilon\in\{1,-1\}\).  Absorbing \(s\) into \(a\) makes
+\(a\xi_p/\xi'_p\) a unit for every \(p\).  Hence \(I'=aI\) as fractional
+maximal-order ideals.  The chosen representatives of the ordinary ideal
+classes then give \(I'=I\) and \(a\in R^\times=U\).
+
+There are at most \([U:U^+]\le16\) possibilities for the unit action
+modulo \(U^+\), distinguished by the four real signs, and two
+possibilities for \(\epsilon\).  This proves the bound \(32\).
+\end{proof}
+
+At \(p\notin S\), both local algebras are unramified,
+\(\OO_p=R_p\), and \(\delta_{\OO}\) is a unit in a splitting field.
+Let \(\varphi_p\) be Frobenius acting on the four embeddings and on
+\(\mathcal B\).  Its orbits on \(\mathcal B\) correspond to primes
+\(\mathfrak q\mid p\) of \(F\), of residue degrees
+\(f_{\mathfrak q}\).  The map from two-element subsets to partitions
+sends \(ij\) to \(ij|kl\).  Above a Frobenius orbit of partitions its
+pair labels form either two orbits or one.  Define
+\[
+ \epsilon_{\mathfrak q}=
+ \begin{cases}
+  1,&\text{if the pair labels form two orbits},\\
+ -1,&\text{if they form one orbit},
+ \end{cases}
+ \qquad
+ b_{\mathfrak q}(j)=
+ \begin{cases}
+  j+1,&\epsilon_{\mathfrak q}=1,\\
+  \mathbf1_{\{2\mid j\}},&\epsilon_{\mathfrak q}=-1
+ \end{cases}
+ =\sum_{r=0}^j\epsilon_{\mathfrak q}^{\,r}.
+\]
+For \(z\in R_{F,p}\cap F_p^\times\), put
+\begin{equation}
+ j_{\mathfrak q}=v_{\mathfrak q}(z),\qquad
+ b_p(z)=\prod_{\mathfrak q\mid p}b_{\mathfrak q}(j_{\mathfrak q}),
+ \qquad n_p(z)=\max_{\mathfrak q\mid p}j_{\mathfrak q}.
+ \label{eq:ext-unramified-factor}
+\end{equation}
+Here \(v_{\mathfrak q}\) is the integral uniformizer valuation; it has
+\(v_{\mathfrak q}(p)=1\) at these primes.  For probability statements
+one may set \(n_p=\infty\), \(b_p=0\) when a component vanishes, a set
+of Haar measure zero.
+
+\begin{lemma}[Unramified local data]
+\label{lem:ext-unramified-data}
+Let \(p\notin S\).  A local fiber with an accepted coset has
+\(z\in R_{F,p}\), and for \(z\in R_{F,p}\cap F_p^\times\),
+\begin{equation}
+ \sum_{\mathfrak c\in\mathscr G_p/\mathscr U_p}
+       \alpha_p(\mathfrak c;z)\le b_p(z).
+ \label{eq:ext-unramified-cosets}
+\end{equation}
+One has \(b_p=1\) when \(n_p=0\), and \(0\le b_p\le(1+n_p)^3\).
+
+Give the affine lattice
+\[
+ \{z\in R_{F,p}:\Tr_{F_p/\Q_p}z=m\}
+\]
+its Haar probability.  With \(t=p^{-1}\) and
+\(f_{\mathfrak p}\) the residue degrees of the primes of \(K\) above \(p\),
+\begin{align}
+ \E_p b_p
+   &=(1-t)(1-t^3)
+       \prod_{\mathfrak p\mid p}(1-t^{f_{\mathfrak p}})^{-1}
+     =1+\frac{a_K(p)-1}{p}+O(p^{-2}), \label{eq:ext-unramified-mean}\\
+ \P_p(n_p\ge k)&\le3p^{-k}\qquad(k\ge1). \label{eq:ext-unramified-tail}
+\end{align}
+The error constant is absolute.  In any integral affine coordinates
+on the trace lattice, \(\{n_p\ge k\}\) is determined modulo \(p^k\),
+and \(b_p\mathbf1_{\{n_p=k\}}\) is determined modulo \(p^{k+1}\)
+for every finite \(k\ge0\).
+\end{lemma}
+
+\begin{proof}
+An accepted coset has a representative \(x\in L_{V,p}=\bigwedge^2 R_p\).
+Use the extended valuation with \(v_p(p)=1\) on a splitting field.
+Its six valuations \(k_{ij}=v_p(x_{ij})\) are nonnegative integers,
+invariant under Frobenius.  Since \(u\) and \(\delta_{\OO}\) are units,
+\[
+ k_{ij}+k_{kl}=j_{\mathfrak q}
+ \quad\text{for the partition }ij|kl\text{ over }\mathfrak q .
+\]
+This proves integrality of \(z\).  If the pair labels over
+\(\mathfrak q\) have two Frobenius orbits, one of their two valuations
+can be chosen from \(0,\ldots,j_{\mathfrak q}\).  If there is one orbit,
+the two valuations are equal, which requires even \(j_{\mathfrak q}\).
+The number of records is therefore at most \(b_p(z)\).
+
+Two representatives with the same six valuations have comparison
+\((a,c)\) satisfying
+\[
+ v_p(c)+v_p(a_i)+v_p(a_j)=0\qquad(i<j).
+\]
+Subtraction makes all four \(v_p(a_i)\) equal to a common \(r\), and
+\(v_p(c)=-2r\).  Because \(K_p\) is unramified, \(r\in\Z\).
+Replacing \(a\) by \(p^{-r}a\) and \(c\) by \(p^{2r}c\) puts both in
+the unit subgroup.  Thus a record specifies at most one coset, also
+for nonsplit unramified factors.  The subgroup \(\mathscr U_p\)
+preserves \(\bigwedge^2R_p\), so its coset fraction is either \(0\)
+or \(1\).  This proves Equation~\eqref{eq:ext-unramified-cosets}.
+The bounds at depth zero and at general depth follow immediately
+from the definition and the at most three factors.
+
+The trace from an unramified integer ring to \(\Z_p\) is onto, as is
+seen after reduction from the nonzero finite-field trace.  If
+\(F_p\) has more than one factor, the trace-\(m\) coset therefore
+projects uniformly onto each chosen factor: the other factors can
+supply any required trace.  More generally, for integers
+\(r_{\mathfrak q}\ge0\), the probability that
+\(v_{\mathfrak q}(z)\ge r_{\mathfrak q}\) for all \(\mathfrak q\) is
+\[
+ \begin{cases}
+  t^{\sum_{\mathfrak q}f_{\mathfrak q}r_{\mathfrak q}},
+       &\min_{\mathfrak q}r_{\mathfrak q}=0,\\
+  0,&\min_{\mathfrak q}r_{\mathfrak q}>0.
+ \end{cases}
+\]
+The first line follows by projecting onto the constrained factors
+while a factor with \(r_{\mathfrak q}=0\) supplies the trace.
+The second uses \(p\nmid m\).  It also covers the case that \(F_p\)
+is a field.
+
+Expanding \(b_{\mathfrak q}(j)=\sum_{r=0}^j\epsilon_{\mathfrak q}^r\)
+and summing the absolutely convergent geometric series gives
+\begin{equation}
+ \E_p b_p=
+ \frac{1-(\prod_{\mathfrak q}\epsilon_{\mathfrak q})t^3}
+      {\prod_{\mathfrak q}(1-\epsilon_{\mathfrak q}t^{f_{\mathfrak q}})}.
+ \label{eq:ext-geometric-mean}
+\end{equation}
+The five Frobenius types give the following data.  A pair
+\((f,\epsilon)\) in the second column is attached to one prime of \(F\).
+\[
+\begin{array}{c|c|c|c}
+ \text{cycle type on four labels}
+ & (f_{\mathfrak q},\epsilon_{\mathfrak q})
+ & \prod_{\mathfrak p\mid p}(1-t^{f_{\mathfrak p}})
+ & a_K(p)-1\\ \hline
+ 1^4 &(1,+),(1,+),(1,+)&(1-t)^4&3\\
+ 2\,1^2 &(1,+),(2,+)&(1-t)^2(1-t^2)&1\\
+ 2^2 &(1,+),(1,-),(1,-)&(1-t^2)^2&-1\\
+ 3\,1 &(3,+)&(1-t)(1-t^3)&0\\
+ 4 &(1,-),(2,-)&1-t^4&-1
+\end{array}
+\]
+In every row \(\prod_{\mathfrak q}\epsilon_{\mathfrak q}=1\) and
+\[
+ \prod_{\mathfrak q}(1-\epsilon_{\mathfrak q}t^{f_{\mathfrak q}})
+   =\frac{\prod_{\mathfrak p\mid p}(1-t^{f_{\mathfrak p}})}{1-t}.
+\]
+Substitution in Equation~\eqref{eq:ext-geometric-mean} proves the exact
+mean.  Its linear term is the number of degree-one primes of \(K\)
+minus one, which is \(a_K(p)-1\); the five fixed rows give the uniform
+error term.
+
+If \(F_p\) is a field, \(n_p\ge1\) would make its trace divisible by
+\(p\), and is impossible.  Otherwise the uniform projection above
+gives
+\(\P_p(v_{\mathfrak q}(z)\ge k)=p^{-f_{\mathfrak q}k}\).
+Taking the union over at most three factors proves
+Equation~\eqref{eq:ext-unramified-tail}.  Finally, the trace map is
+onto, so the affine trace lattice has two integral coordinates.
+Projection to each unramified factor modulo \(p^{k+1}\) is determined
+by these coordinates modulo \(p^{k+1}\).  The valuations truncated
+at \(k+1\), and hence the asserted threshold and finite-depth
+functions, have the stated congruence dependence.  This remains true
+on the component-vanishing set with the convention above.
+\end{proof}
+
+\begin{proposition}[Multiplicities from local densities]
+\label{prop:ext-multiplicity-majorant}
+For \(z\in F\) with \(\Tr z=m\ne0\), a positive value of \(M(z)\)
+requires \(z\) to be integral outside \(S\).  For such integral \(z\),
+\begin{equation}
+ M(z)\le32
+ \prod_{p\in S}
+ \left(
+ |D|_p^{-1/2}
+ \frac{1-p^{-1}}
+      {\displaystyle\prod_{\mathfrak p\mid p,\ \mathfrak p\subset R}
+                         (1-\Nm(\mathfrak p)^{-1})}
+ f_p^*(z)\right)
+ \prod_{p\notin S}b_p(z).
+ \label{eq:ext-local-majorant}
+\end{equation}
+All products in this assertion are finite at every contributing \(z\).
+\end{proposition}
+
+\begin{proof}
+The field \(F\) and the condition \(\Tr z=m\ne0\) imply
+\(z\in F^\times\).  At every prime the set of accepted cosets is
+finite by Lemma~\ref{lem:ext-density}.  At \(p\notin S\),
+Lemma~\ref{lem:ext-unramified-data} proves the integrality requirement
+and bounds the local sum by \(b_p(z)\).  The set
+\[
+ E_z=S\cup\{p:z\notin R_{F,p}^\times\}
+\]
+is finite, since a nonzero global \(z\) is a unit at all but finitely
+many primes.  At \(p\notin E_z\), \(b_p=1\), and each coset fraction is
+\(0\) or \(1\), so there is at most one accepted coset, of fraction
+\(1\).  If some prime has no accepted coset, then \(M(z)=0\) and the
+claimed inequality is immediate.  Otherwise the accepted coset is
+unique with fraction \(1\) outside \(E_z\), and each prime inside the
+finite set \(E_z\) has finitely many accepted cosets.  Thus there are
+only finitely many assignments of accepted cosets, and all local sums
+outside \(E_z\) equal \(1\).
+
+Choose local references.  Equation~\eqref{eq:ext-unit-membership}
+identifies the weight of a global pair at each prime with its coset
+fraction \(\alpha_p\).  By Lemma~\ref{lem:ext-global-cosets}, at most
+\(32\) global pairs occupy any assignment of accepted cosets.
+Summing over the finite set of assignments, including ones not
+realized globally, bounds \(M(z)\) by \(32\) times the product of the
+local sums of \(\alpha_p\).  Use
+Equation~\eqref{eq:ext-density-disintegration} at \(p\in S\) and
+Equation~\eqref{eq:ext-unramified-cosets} outside \(S\).
+This proves Equation~\eqref{eq:ext-local-majorant} and finiteness.
+\end{proof}
+
+The unramified mean in Equation~\eqref{eq:ext-unramified-mean} has the
+Euler factor needed for the residue \(\kappa\).  The remaining problem
+in Theorem~\ref{thm:exterior-count} is to average the densities
+\(f_p^*\) for \(p\in S\) before inserting these unramified factors.
+For a frequency \(\alpha\in F_p\), testing the law \(u\Psi(x)\) against
+the character \(z\mapsto\psi_p(\Tr_{F_p/\Q_p}(\alpha z))\) produces the
+quadratic phase \(u\Tr_{F_p/\Q_p}(\alpha\Psi(x))\).
+We express these forms through a single unimodular bilinear form and an
+action of \(F\).  The elements of \(F\) whose operators preserve \(L_V\)
+will then form the order that controls the local Fourier analysis.
+
+\subsection{The integral quadratic structure and its order}
+
+Let \(B_V\) be the polarization of \(Q_V\) without division by two:
+\[
+ B_V(x,y)=Q_V(x+y)-Q_V(x)-Q_V(y).
+\]
+Over a splitting field, write \(V_b\) for the plane spanned by the
+two coordinate axes belonging to the partition \(b\), the same
+coordinate planes used over \(\R\) in Section~\ref{sec:rigidity}.
+For an equivariant triple \(\alpha=(\alpha_b)\in F\), let \(\alpha_b\)
+scale both coordinate axes of \(V_b\).
+
+\begin{lemma}[Self-adjoint quadratic structure]
+\label{lem:ext-quadratic-structure}
+The plane-scalar action descends to a rational action
+\(\rho:F\to\End_{\Q}(V)\) for which \(V\) has rank two over \(F\).
+The form \(Q_V\) in an exterior \(\Z\)-basis of \(L_V\) is the three
+split pairs in Equation~\eqref{eq:ext-integral-form}; in particular
+\(B_V\) is unimodular on \(L_V\).  The action \(\rho\) is self-adjoint
+for \(B_V\), and for \(\alpha\in F\),
+\begin{align}
+ Q_\alpha(x):=\Tr_{F/\Q}(\alpha\Psi(x))
+    &=\tfrac12 B_V(\rho(\alpha)x,x), \label{eq:ext-polar-operator}\\
+ B_\alpha(x,y)&=B_V(\rho(\alpha)x,y), \notag\\
+ \det_{\Q}\rho(\alpha)&=\Nm_{F/\Q}(\alpha)^2,\qquad
+ \Psi(\rho(\alpha)x)=\alpha^2\Psi(x). \label{eq:ext-rank-two}
+\end{align}
+Here \(B_\alpha\) is the polarization of \(Q_\alpha\).
+The same assertions hold after base change to every \(\Q_p\).
+\end{lemma}
+
+\begin{proof}
+Galois permutes the planes \(V_b\), including possible alternating
+signs on their coordinate axes, and permutes the scalars \(\alpha_b\)
+in the same way.  The plane-scalar action therefore commutes with
+Galois and descends.  After splitting there are three planes of
+dimension two, proving the rank assertion.  Equation~\eqref{eq:ext-integral-form}
+shows that the matrix of \(B_V\) pairs the three opposite pairs by
+unit off-diagonal entries, with the displayed signs, so it is
+unimodular.  On each plane, multiplying both coordinates by
+\(\alpha_b\) is self-adjoint for its split pairing.  The contribution
+of that plane to \(\frac12B_V(\rho(\alpha)x,x)\) is
+\(\alpha_b\Psi_b(x)\), which proves
+Equation~\eqref{eq:ext-polar-operator}.  Polarization gives the next
+identity.  The determinant and the last quadratic identity follow
+by multiplying, respectively, the two equal eigenvalues and the
+two coordinate scalars on each plane.  All these are rational
+identities, hence persist under base change.
+\end{proof}
+
+Define the stabilizer order and its discriminant by
+\begin{equation}
+ C_F=\{\alpha\in F:\rho(\alpha)L_V\subset L_V\},\qquad
+ \Delta_F=\Disc(C_F).
+ \label{eq:ext-stabilizer-order}
+\end{equation}
+It is a ring containing \(1\) and a full \(\Z\)-lattice: this follows
+from the rationality of \(\rho\) by clearing denominators on a basis
+of \(F\).  It is contained in \(R_F\), since an \(\alpha\) whose
+matrix on \(L_V\) is integral is a root of that integral monic
+characteristic polynomial.  Thus it is an order.
+The stabilizer condition is a finite list of matrix-entry integrality
+conditions in a lattice basis, so it commutes with completion:
+\begin{equation}
+ C_{F,p}=C_F\otimes_\Z\Z_p
+   =\{\alpha\in F_p:\rho(\alpha)L_{V,p}\subset L_{V,p}\}.
+ \label{eq:ext-local-stabilizer}
+\end{equation}
+The following comparison ensures that its discriminant tends to infinity
+with the discriminant of \(\OO\), even when the field \(K\) stays fixed.
+
+\begin{lemma}[Scale of the stabilizer order]
+\label{lem:ext-order-scale}
+There is an absolute constant \(C\) such that
+\begin{equation}
+ \frac12 X^{1/2}\le
+ \Delta_F=D_F[R_F:C_F]^2\le C X^{28}.
+ \label{eq:ext-order-scale}
+\end{equation}
+More precisely, with \(d=\Disc(\OO)=X^2\),
+\[
+ d^2R_F\subset C_F\subset R_F,\qquad D_F\ll D^2.
+\]
+In particular \(C_{F,p}=R_{F,p}\) for \(p\nmid Dq\).
+\end{lemma}
+
+\begin{proof}
+Let \(E=\bigwedge^2 A_{\OO}\) be the exterior embedding matrix.
+Its entries are integral in every local splitting ring, and
+\(\det E=\delta_{\OO}^{\,3}\).  Since \(d=\delta_{\OO}^{\,2}\),
+\[
+ d^2 E^{-1}=\delta_{\OO}\operatorname{adj}(E)
+\]
+also has integral entries.  For \(\alpha\in R_{F,p}\), its
+plane-scalar matrix over the splitting ring is integral.
+Consequently
+\(d^2\rho(\alpha)=d^2E^{-1}\diag(\alpha_b,\alpha_b)E\)
+is integral over the splitting ring.  Its entries
+are in \(\Q_p\), so they lie in \(\Z_p\).  This proves
+\(d^2R_F\subset C_F\) and \([R_F:C_F]\le d^6\).
+
+If \(p\nmid D\), the etale algebra \(R_p\) splits over an unramified
+extension of \(\Z_p\); its splitting field and cubic resolvent are
+unramified.  Hence every prime dividing \(D_F\) divides \(D\).
+For \(p>3\), a cubic algebra is tamely ramified and its discriminant
+exponent is at most \(2\).  At \(2\) and \(3\), the finitely many
+local extensions of degree at most three have bounded discriminant
+exponents \cite[Proposition~7.64 and Remark~7.65]{MilneANT}.
+It follows that \(D_F\ll D^2\).  The discriminant-index formula now
+gives
+\[
+ \Delta_F=D_F[R_F:C_F]^2\ll D^2d^{12}\le X^{28}.
+\]
+The containment \(d^2R_F\subset C_F\) also proves equality of the
+local orders away from \(Dq\).
+
+For the reverse inequality, we first explain why a projection
+denominator controls \(\Disc(\OO)\).  Write \(\operatorname{tr}\) for
+matrix trace, put \(\mathscr E=\End_\Z(\OO)\), and let
+\(\mathscr W_K\subset\End_\Q(K)\) be the rational subspace of
+multiplication matrices.  Its intersection
+\(\mathscr L=\mathscr W_K\cap\mathscr E\) is saturated.  It is
+multiplication by \(\OO\): if \(\alpha\OO\subset\OO\), then
+\(\alpha=\alpha\cdot1\in\OO\), and the converse follows because
+\(\OO\) is a ring.  The pairing
+\((T,S)\mapsto\operatorname{tr}(TS)\) is unimodular on
+\(\mathscr E\), since its matrix units are dual.  Its restriction
+to \(\mathscr W_K\) is the nondegenerate field trace pairing.
+Let \(\mathcal E_K\) be the orthogonal projection onto
+\(\mathscr W_K\).
+
+For the restricted pairing one has
+\begin{equation}
+ \mathcal E_K(\mathscr E)=\mathscr L^\vee.
+ \label{eq:ext-projected-dual}
+\end{equation}
+One inclusion follows by pairing a projected integral matrix with
+\(\mathscr L\).  Conversely, an integer functional on the primitive
+sublattice \(\mathscr L\) extends to one on \(\mathscr E\).
+Unimodularity represents the extension by an element of
+\(\mathscr E\); its orthogonal projection represents the original
+functional on \(\mathscr L\).  Nondegeneracy identifies that
+projection with the prescribed element of \(\mathscr L^\vee\).
+Consequently, if a positive integer \(N\) satisfies
+\(N\mathcal E_K(\mathscr E)\subset\mathscr E\), then
+\[
+ \mathscr L^\vee\subset N^{-1}\mathscr L,\qquad
+ X^2=\Disc(\OO)=[\mathscr L^\vee:\mathscr L]\le N^4.
+\]
+It remains to produce the denominator \(N=2\Delta_F\).
+
+Choose trace-dual bases \(a_i\) of \(C_F\) and \(b_i\) of \(C_F^\vee\),
+where
+\[
+ C_F^\vee=\{\beta\in F:\Tr_{F/\Q}(\beta C_F)\subset\Z\}.
+\]
+The adjugate of the trace Gram matrix gives
+\(\Delta_F b_i\in C_F\).  Define on \(\End_\Q(V)\)
+\begin{equation}
+ \mathcal E_F(T)=\sum_{i=1}^3\rho(a_i)T\rho(b_i).
+ \label{eq:ext-centralizer-projection}
+\end{equation}
+For two embeddings \(\sigma,\tau\) of \(F\), the dual-basis identity is
+\[
+ \sum_i\sigma(a_i)\tau(b_i)=
+ \begin{cases}1,&\sigma=\tau,\\0,&\sigma\ne\tau.\end{cases}
+\]
+Thus, after splitting, \(\mathcal E_F\) extracts exactly the three
+diagonal blocks \(\End(V_b)\).  It is the orthogonal projection
+for the matrix trace pairing, and
+\begin{equation}
+ \Delta_F\mathcal E_F(\End_\Z(L_V))\subset\End_\Z(L_V).
+ \label{eq:ext-centralizer-denominator}
+\end{equation}
+This denominator conclusion uses the trace-dual lattice only; it
+does not require that lattice to be an invertible \(C_F\)-ideal.
+
+We transfer this projection to matrices on \(K\) explicitly.
+Let
+\[
+ j(T)(v\wedge w)=Tv\wedge w+v\wedge Tw
+ \qquad(T\in\End_\Q(K)).
+\]
+There is a natural contraction
+\(\mathfrak c:\End_\Q(\bigwedge^2K)\to\End_\Q(K)\).
+For clarity, in a basis \(e_1,\ldots,e_4\) write
+\[
+ S(e_j\wedge e_l)=\sum_{i<k}S^{ik}_{jl}e_i\wedge e_k\qquad(j<l),
+\]
+and extend these coefficients antisymmetrically in \(i,k\) and in
+\(j,l\), with zero coefficients when either pair repeats.  Then
+\[
+ \mathfrak c(S)^i_j=\sum_{k=1}^4 S^{ik}_{jk}.
+\]
+This is the contraction of
+\(\bigwedge^2K\otimes\bigwedge^2K^\vee\) in one covariant and one
+contravariant index, so it is independent of the basis.  Both \(j\)
+and \(\mathfrak c\) are integral in the \(\OO\) and exterior
+\(\OO\)-bases.  Expanding their coefficients gives
+\begin{equation}
+ \mathfrak c(j(T))=2T+\operatorname{tr}(T)\id_K.
+ \label{eq:ext-contraction}
+\end{equation}
+Indeed an off-diagonal coefficient of \(T\) occurs for the two
+remaining indices \(k\); a diagonal coefficient gives
+\(\sum_{k\ne i}(T^i_i+T^k_k)=2T^i_i+\operatorname{tr}(T)\).
+
+In embedding coordinates, an off-diagonal entry of \(j(T)\) changes
+one index of a two-element subset.  Such subsets belong to distinct
+opposite-pair planes.  It follows that
+\(\mathcal E_F(j(T))=j(\diag T)\) over the splitting field.
+Equation~\eqref{eq:ext-contraction} now proves the formula
+\begin{equation}
+ \mathcal E_K(T)=
+ \tfrac12\bigl(\mathfrak c(\mathcal E_F(j(T)))-\operatorname{tr}(T)\id_K\bigr)
+ \label{eq:ext-multiplication-projection}
+\end{equation}
+for the projection defined above: its right side is \(\diag T\)
+over the splitting field.  A Galois-equivariant diagonal tuple is
+the tuple of embeddings of an element of \(K\), so these diagonal
+matrices are precisely \(\mathscr W_K\).  Diagonal extraction is
+orthogonal for matrix trace.  Finally,
+Equation~\eqref{eq:ext-centralizer-denominator} and integrality of
+\(j\) and \(\mathfrak c\) give
+\[
+ 2\Delta_F\mathcal E_K(\End_\Z(\OO))\subset\End_\Z(\OO).
+\]
+Taking \(N=2\Delta_F\) in the preceding lattice bound gives
+\[
+ X^2\le(2\Delta_F)^4.
+\]
+This is the lower inequality in Equation~\eqref{eq:ext-order-scale}.
+\end{proof}
+
+For \(x\in L_{V,p}\) and \(\alpha\in C_{F,p}\),
+Equation~\eqref{eq:ext-polar-operator} gives two integral pairings:
+\[
+ \Tr\Psi(x)=Q_V(x)\in\Z_p,\qquad
+ 2\Tr(\alpha\Psi(x))=B_V(\rho(\alpha)x,x)\in\Z_p.
+\]
+They remain integral after multiplying \(\Psi(x)\) by a \(\Z_p\)-unit.
+Thus the products pair integrally with the scalar \(1\) and with
+\(2C_{F,p}\).  Define the order and its trace dual
+\[
+ C_F^\sharp=\Z+2C_F,\qquad
+ (C_F^\sharp)^\vee
+   =\{z\in F:\Tr_{F/\Q}(zC_F^\sharp)\subset\Z\}.
+\]
+This trace dual gives the integral support for the additive count.
+The same support will bound the total depth of the unramified factors
+and prevent a nonzero real product coordinate from being too small.
+
+\begin{lemma}[Trace-dual support and product size]
+\label{lem:ext-support}
+One has
+\begin{equation}
+ [C_F:C_F^\sharp]=4,\qquad
+ \Delta_F^\sharp:=\Disc(C_F^\sharp)=16\Delta_F\ll X^{28},
+ \label{eq:ext-sharp-discriminant}
+\end{equation}
+and \(\Tr:(C_F^\sharp)^\vee\to\Z\) is onto.
+For \(p\notin S\), \(C_{F,p}^\sharp=(C_{F,p}^\sharp)^\vee=R_{F,p}\).
+For every prime \(p\), the density \(f_p^*\) on \(F_p^\times\) is
+supported on \((C_{F,p}^\sharp)^\vee\).  Consequently
+\[
+ M(z)>0\ \Longrightarrow\ z\in(C_F^\sharp)^\vee.
+\]
+The following stronger containment is used to average the majorant in
+Proposition~\ref{prop:ext-multiplicity-majorant}: if \(z\in F^\times\) is
+integral outside \(S\) and \(\prod_{p\in S}f_p^*(z)>0\), then
+\(z\in(C_F^\sharp)^\vee\).
+
+Fix \(m\ne0\) and a bounded
+\(\Omega\subset\mathcal H_m\).  Every
+\(z\in(C_F^\sharp)^\vee\cap\Omega\) satisfies
+\begin{align}
+ &\Delta_F^\sharp z\in R_F,\qquad
+ 1\le|\Nm_{F/\Q}(\Delta_F^\sharp z)|\ll_\Omega X^{84},
+       \label{eq:ext-norm-bound}\\
+ &\prod_{p\notin S}p^{n_p(z)}
+       \le|\Nm_{F/\Q}(\Delta_F^\sharp z)|,\qquad
+ |z_b|\gg_\Omega X^{-84}\quad(b\in\mathcal B).
+ \label{eq:ext-depth-coordinate-bound}
+\end{align}
+The constants are uniform in all arithmetic data and in the point
+of \(\Omega\).
+\end{lemma}
+
+\begin{proof}
+Since \(C_F\) is an order, \(C_F\cap\Q=\Z\), so \(1\) is primitive
+in its lattice.  A basis \(1,a_2,a_3\) of \(C_F\) gives the basis
+\(1,2a_2,2a_3\) of \(C_F^\sharp\).  This proves the index and
+discriminant assertions in Equation~\eqref{eq:ext-sharp-discriminant}.
+The functional taking \(1\) to \(1\) and the other two basis vectors
+to \(0\) is represented in the trace dual.  Hence trace from that
+dual is onto \(\Z\).  Lemma~\ref{lem:ext-order-scale}, and \(2\in S\),
+give \(C_{F,p}^\sharp=R_{F,p}\) outside \(S\).
+
+The two integrality identities preceding the lemma show that the
+law in Equation~\eqref{eq:ext-density-law} is supported on
+\((C_{F,p}^\sharp)^\vee\).  For a contributing global pair, use its local
+representation \(z=u\Psi(x)\) and Equation~\eqref{eq:ext-fiber-action}
+to translate \(x\) into \(L_{V,p}\), adjusting \(u\) by the inverse
+norm of that unit.  This proves local trace-dual membership
+at every prime.  Intersecting the local lattices proves the stated
+global support.  For the second support assertion, the density factors
+give this membership at \(p\in S\).  At \(p\notin S\), the order is
+\(R_{F,p}\), whose trace pairing is unimodular because \(F_p\) is
+unramified; thus \(R_{F,p}^\vee=R_{F,p}\), and the assumed integrality
+gives membership there as well.
+
+The adjugate of the trace Gram matrix of \(C_F^\sharp\) gives
+\(\Delta_F^\sharp(C_F^\sharp)^\vee\subset C_F^\sharp\subset R_F\).
+If \(z\in\Omega\), then \(\Tr z=m\ne0\), so \(z\ne0\); since \(F\)
+is a field, all its embeddings are nonzero.  The norm
+\(\Nm(\Delta_F^\sharp z)\) is consequently a nonzero integer.  The
+boundedness of \(\Omega\) and \(\Delta_F^\sharp\ll X^{28}\) give
+\[
+ 1\le|\Nm(\Delta_F^\sharp z)|
+       =(\Delta_F^\sharp)^3\prod_{b\in\mathcal B}|z_b|
+       \ll_\Omega X^{84}.
+\]
+At \(p\notin S\), \(z\) is integral and the local algebra is
+unramified.  Hence
+\[
+ n_p(z)\le\sum_{\mathfrak q\mid p}
+               f_{\mathfrak q}v_{\mathfrak q}(z)
+       \le v_p\bigl(\Nm(\Delta_F^\sharp z)\bigr).
+\]
+The product of these inequalities gives the depth bound because
+\(\Nm(\Delta_F^\sharp z)\) is an integer.  Finally, if the coordinates of
+\(\Omega\) have absolute value at most \(B\), with \(B\ge1\), then
+for each \(b\)
+\[
+ |z_b|=\frac{|\Nm(\Delta_F^\sharp z)|}
+              {(\Delta_F^\sharp)^3\prod_{b'\ne b}|z_{b'}|}
+       \ge\frac{1}{(\Delta_F^\sharp)^3 B^2}\gg_\Omega X^{-84}.
+\]
+This proves Equation~\eqref{eq:ext-depth-coordinate-bound}.
+\end{proof}

preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/introduction.tex

@@ -0,0 +1,253 @@
+\section{Introduction}\label{sec:introduction}
+
+Ideal classes in a totally real number field give compact orbits of a
+diagonal group on a space of lattices.  A natural arithmetic question is
+whether the finite collections of orbits obtained in this way become
+uniformly distributed as their discriminants grow.  We prove this for the
+complete packets attached to invertible ideals of every order in a primitive
+totally real quartic field.  The order may have arbitrarily large index in
+the ring of integers.
+
+Let
+\[
+ X_4=\mathrm{SL}_4(\Z)\backslash\mathrm{SL}_4(\R)
+\]
+be the space of covolume-one row lattices in $\R^4$, and let $m_4$ be its
+invariant probability measure.  Put
+\[
+ H_4=\{u\in\R^4:u_1+u_2+u_3+u_4=0\},\qquad
+ A_4=\{\diag(e^{u_1},e^{u_2},e^{u_3},e^{u_4}):u\in H_4\}.
+\]
+The group $A_4$ acts on $X_4$ by right multiplication.  Its Haar measure
+is $du_1\,du_2\,du_3$, with $u_4=-u_1-u_2-u_3$.
+
+A quartic field $K$ is \emph{primitive} if it has no field strictly between
+$\Q$ and $K$.  Let $K$ be primitive and totally real, let $\OO\subseteq\OO_K$
+be an order, and fix an ordering $\sigma=(\sigma_1,\ldots,\sigma_4)$ of its
+real embeddings.  Here an order is a full-rank subring containing $1$.
+A fractional $\OO$-ideal $I$ is \emph{proper} if its multiplier order
+$\{a\in K:aI\subseteq I\}$ is $\OO$, and it is \emph{invertible} if
+$IJ=\OO$ for some fractional $\OO$-ideal $J$.  The ordinary ideal class
+group $\Pic(\OO)$ consists of proper invertible fractional $\OO$-ideals
+modulo $I\sim aI$ for $a\in K^\times$.
+
+For such an ideal put
+\begin{equation}\label{eq:intro-lattice}
+ \Lambda_{I,\sigma}
+   =\covol(\sigma(I))^{-1/4}\sigma(I)\in X_4.
+\end{equation}
+A covolume-one lattice is represented here by any positively oriented
+basis; two such bases differ by an element of $\mathrm{SL}_4(\Z)$.
+Let $W_{\mathrm{all}}$ be the sixteen diagonal matrices with entries in
+$\{1,-1\}$.  Choose one ideal in each ordinary class and form the finite
+set of distinct orbits
+\[
+ \mathcal P_{\OO,\sigma}
+ =\{\Lambda_{I,\sigma}wA_4:I\text{ a class representative},\
+                                      w\in W_{\mathrm{all}}\}.
+\]
+A matrix in $W_{\mathrm{all}}$ acts on the underlying lattice, so a sign
+matrix of determinant $-1$ also gives a point of $X_4$.  Using all signs
+makes the set independent of the chosen ordinary ideal representatives.
+Every orbit is compact by the unit theorem; Section~\ref{sec:packets}
+proves these assertions and describes the associated measure.
+
+For an orbit $\mathcal Y\in\mathcal P_{\OO,\sigma}$, let $\nu_{\mathcal Y}$
+be its invariant probability and let $\vol(\mathcal Y)$ be its volume for
+the specified Haar measure.  Define
+\begin{equation}\label{eq:intro-measure}
+ \mu_{\OO,\sigma}
+ =\frac{\displaystyle\sum_{\mathcal Y\in\mathcal P_{\OO,\sigma}}
+              \vol(\mathcal Y)\nu_{\mathcal Y}}
+        {\displaystyle\sum_{\mathcal Y\in\mathcal P_{\OO,\sigma}}
+              \vol(\mathcal Y)}.
+\end{equation}
+Each distinct orbit occurs once in these sums.
+
+\begin{theorem}\label{thm:main}
+Let $(K_i,\OO_i,\sigma_i)$ be any sequence in which $K_i$ is a primitive
+totally real quartic field, $\OO_i$ is an order in $K_i$, and $\sigma_i$ is
+an ordering of its real embeddings.  If $|\Disc(\OO_i)|\to\infty$, then
+\[
+ \int_{X_4} f\,d\mu_{\OO_i,\sigma_i}
+       \longrightarrow \int_{X_4} f\,dm_4
+       \qquad\text{for every }f\in C_c(X_4).
+\]
+There is no escape of mass: for every $\varepsilon>0$ there is a compact
+$\mathcal K\subset X_4$ such that
+$\mu_{\OO_i,\sigma_i}(\mathcal K)\ge1-\varepsilon$ for all sufficiently
+large $i$.
+\end{theorem}
+
+In particular, the field may remain fixed while the order index tends to
+infinity.  The packet in Theorem~\ref{thm:main} uses every class in
+$\Pic(\OO)$, so its ideals are locally free of rank one over the specified
+order.  The proof uses this complete class average in both of its counting
+arguments.
+
+\subsection{History and context}
+
+For real quadratic fields, compact diagonal orbits are closed geodesics on
+the modular surface.  Linnik's ergodic method and Skubenko's theorem for
+positive discriminants established arithmetic distribution under the
+condition $\left(\frac d p\right)=1$ for the discriminant $d$ at a fixed
+odd prime $p$ \cite{Linnik1968,Skubenko1962}.
+Using Iwaniec's estimates for half-integral weight forms \cite{Iwaniec1987},
+Duke removed that restriction for the geodesic collections associated with
+positive fundamental discriminants \cite[Theorem~1]{Duke1988}.
+Chelluri extended the corresponding statement to the unit tangent bundle,
+as recorded in \cite[Section~1.2, p.~253]{ELMV2012}.
+Einsiedler, Lindenstrauss, Michel, and Venkatesh later gave a proof using
+entropy for all positive nonsquare order discriminants
+\cite[Theorem~2.3]{ELMV2012}.
+
+In higher degree the diagonal group has higher rank, and the arithmetic
+packets lie in spaces of higher-dimensional lattices.  Einsiedler,
+Lindenstrauss, Michel, and Venkatesh developed the discriminant and volume
+framework connecting separation of periodic torus orbits to entropy
+\cite{ELMV2009}.  Their cubic theorem proves equidistribution of packets
+that include those arising from nonmaximal totally real cubic orders
+\cite[Theorem~1.4]{ELMV2011}.  The local homothety class of a lattice is
+part of their packet data; a multiplier order alone need not specify it
+\cite[Section~5.5]{ELMV2011}.  Their higher-dimensional discussion also
+identifies the possibility of proper intermediate homogeneous limits
+\cite[Section~1.6.3]{ELMV2011}.  The theorem of Einsiedler, Katok, and
+Lindenstrauss makes a diagonal-ergodic probability with positive entropy
+homogeneous \cite[Theorem~1.3]{EKL2006}; in prime dimension at least three
+its homogeneous conclusion is already Haar \cite[Corollary~1.4]{EKL2006}.
+
+For each fixed prime $p\ge5$, Lemke Oliver, Thorner, and Zaman proved
+equidistribution for the orbit collections attached to orders in totally
+real degree-$p$ fields as the order discriminant grows, outside an
+exceptional set containing $O_{p,\varepsilon}(Q^\varepsilon)$ fields of
+field discriminant at most $Q$ \cite[Theorem~2.3]{LOTZ2024}.
+The separate preprint \cite[Theorem~1.1]{PrimeDegreePackets} treats every
+packet in each fixed prime degree at least five and permits arbitrary local module types.
+We adapt its ordinary-vector method in the argument below.
+For quartic packets, Khayutin obtained entropy bounds for probability
+weak-$*$ limits of maximal-order packet measures with two-transitive Galois
+groups, including the primitive quartic types $\mathfrak A_4$ and $S_4$
+\cite[Theorem~1.1]{Khayutin2019Tori}.  Wieser and Yang obtain entropy bounds
+for certain maximal-order quartic packets whose fields have a quadratic
+subfield, under further hypotheses \cite{WieserYang2026}.  The present
+argument must establish tightness and positive entropy in almost every
+diagonal-ergodic component for arbitrary orders, and then exclude the
+proper homogeneous limits allowed in degree four.
+
+The arithmetic use of a sieve to rule out intermediate algebraic limits
+has a precedent in Khayutin's work on joint distributions of CM points
+\cite{Khayutin2019CM}.  Local Fourier analysis for nonmaximal orders already
+appears in the cubic work of Einsiedler, Lindenstrauss, Michel, and Venkatesh
+\cite[Proposition~9.9]{ELMV2011}.  Here the object being transformed is a
+specific quadratic product map on an exterior lattice.  The resulting
+estimate controls arbitrary conductor depth in a cubic-resolvent count,
+before the remaining multiplicative weights are inserted by a sieve.
+
+\subsection{The two counts and the conductor estimate}
+
+Write
+\[
+ \begin{gathered}
+ R=\OO_K,\qquad D=|\Disc(K)|,\qquad q=[R:\OO],\\
+ X=q\sqrt D=\sqrt{|\Disc(\OO)|},\qquad
+ \kappa=\Res_{s=1}\zeta_K(s).
+ \end{gathered}
+\]
+We also write $a_K(n)$ for the number of nonzero integral $R$-ideals of
+norm $n$.  Both counting arguments preserve the factor $\sqrt D\,\kappa$
+that normalizes the packet after its finite local labels are averaged.
+
+Sections~\ref{sec:packets} and~\ref{sec:ordinary} treat ordinary lattice
+vectors.  An invertible $\OO$-ideal is locally a unit translate of $\OO$
+inside a maximal-order ideal.  Averaging those translates gives an exact
+unfolding into a weighted sum over integral $R$-ideals.  A local moment
+bound controls the order index, while Stark's exceptional-zero result and
+the absence of a quadratic subfield retain the residue $\kappa$ in the
+ideal count.  Counting short vectors proves tightness.  Counting vectors
+in boxes away from the coordinate hyperplanes gives an $O(r^4)$ bound for
+radius-$r$ neighborhoods in $X_4$, uniformly on each compact set of centers
+in every probability limit.  For a diagonal flow with distinct weights,
+a two-sided tube is covered by $O(r^{-3})$ such neighborhoods with centers
+in a fixed compact enlargement, where $r$ decreases exponentially with the
+tube length.  The resulting exponential decay also holds for normalized
+averages over sets of $A_4$-ergodic components with positive measure,
+because they are dominated by a multiple of the limit measure.  This forces
+positive entropy in almost every component.
+
+Section~\ref{sec:rigidity} applies measure classification and identifies
+the remaining obstruction.  A proper positive-entropy component has two
+coordinate blocks of dimension two.  Each lattice in its supporting orbit
+then has a vector in its exterior square supported on the two opposite
+coordinates $ij$ and $kl$, with both coefficients nonzero.  There are three
+choices of the partition $ij|kl$.
+
+For an exterior vector $v=(v_{ij})_{i<j}$, the positive diagonal action
+preserves the three signed products
+\[
+ (v_{12}v_{34},\,-v_{13}v_{24},\,v_{14}v_{23}).
+\]
+Their sum is an integer on the exterior square of a unimodular lattice.
+For the arithmetic lattices in our packets, Section~\ref{sec:exterior}
+places the products in the three embeddings of a cubic resolvent field
+$F$.  Fixing their sum at a nonzero integer $m$ puts them on a trace-$m$
+plane.  If the four exterior coordinates outside $ij,kl$ have size at most
+$r$, two product coordinates have size at most $r^2$.  Thus the geometric
+obstruction is tested by a rectangle of area $O(r^4)$ on that plane.
+Theorem~\ref{thm:exterior-count} supplies the corresponding arithmetic
+rectangle bound, with the multiplicity of every product and the packet
+normalization included.
+
+The main new work lies in proving that bound uniformly in $\OO$.  At primes
+dividing $Dq$, the difficult local factors in the multiplicity are given by
+the density of the pushforward of Haar measure on the exterior square of the
+order lattice, with a unit scalar, under the quadratic product map.
+Its exact fiber measure accounts for ramification and for the order index.
+An element of $F$ acts on each opposite-coordinate plane by the value of the
+corresponding real embedding; the resulting action descends to the rational
+exterior space $\bigwedge^2_{\Q}K$.  The elements preserving
+$\bigwedge^2_{\Z}\OO$ form an order $C_F$.  Its discriminant is bounded
+below by a fixed positive power of $X$ and above by a fixed power of $X$,
+even if the field $F$ remains fixed.
+
+Section~\ref{sec:local-fourier} analyzes these densities through their
+quadratic Fourier transforms.  Denominator ideals organize the frequencies;
+the stabilizer order controls the decay of the Fourier coefficients and the
+support and size of the corresponding inverse transforms.  The local proof
+is expressed in terms of the quadratic lattice, its action by a cubic
+\'etale algebra, and its stabilizer order, a formulation suitable for
+analogous cubic-resolvent counts.  In Section~\ref{sec:additive}, Poisson
+summation gives the trace density from scalar frequencies.  Complementary
+estimates control the total nonscalar contribution using discriminant
+separation, bounds for the Fourier and inverse transforms, and congruences
+supplied by inverse quadratic forms.  The estimate is uniform for small
+residue moduli and for shrinking rectangles.
+
+At the remaining primes, elementary valuation records give a local factor
+with mean $1+(a_K(p)-1)/p+O(p^{-2})$.  Section~\ref{sec:sieve} averages
+these factors by a weighted Selberg sieve.  The subtraction of $1$ in the
+local mean removes the logarithm in the prime-sum bound, leaving $\kappa$.
+Finally, Section~\ref{sec:conclusion} unfolds exterior vectors along the
+diagonal group.  The fiber lengths are logarithmic and summable over the
+rectangles above.  The resulting $O(r^4)$ bound excludes every proper
+homogeneous component and proves Theorem~\ref{thm:main}.
+
+\subsection{Conventions}
+
+The notation $\Nm_{E/k}$ denotes the algebraic field norm.  Ideal norms
+$\Nm(\mathfrak a)$ are positive absolute norms, extended multiplicatively
+to fractional ideals; absolute values are written explicitly for field
+norms when needed.  For a rational prime $p$,
+a subscript $p$ denotes tensoring a field with $\Q_p$ or a lattice with
+$\Z_p$.  For a prime $\mathfrak p$ or $\mathfrak q$ of a local field we use
+the integer valuation with value $1$ on a uniformizer, and write $e$ and $f$
+for its ramification index and residue degree.  A normalized extension of
+$v_p$ has $v_p(p)=1$, and $|p|_p=p^{-1}$.  Ideal norms include the residue
+degrees.  We use $\mathfrak p$ for primes of $K$ and $\mathfrak q$ for primes
+of $F$ when the two occur together.
+
+All logarithms are natural, and $\log_+t=\max(0,\log t)$ for $t>0$.
+An upper bound $X^{o(1)}$ means $O_\eta(X^\eta)$ for every fixed $\eta>0$.
+Unless stated otherwise, constants are uniform in $K$, $\OO$, and the
+ordering of the embeddings.  In the exterior estimates a nonzero trace $m$
+and a bounded real region are fixed before $X$ tends to infinity; permitted
+dependence on them is stated in the relevant results.

preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/local-fourier.tex

@@ -0,0 +1,680 @@
+\section{Local Fourier estimates}\label{sec:local-fourier}
+
+The exterior multiplicity bound in Section~\ref{sec:exterior} uses an
+additive density at each prime in $S$.  We now estimate its Fourier transform.
+Scalar frequencies detect only the trace and have an exact, order-independent
+value.  For the remaining frequencies we record both decay and the support of
+the inverse transform on a fixed denominator shell.  The support is needed
+because the denominator at a prime dividing the order index can be arbitrarily
+large.
+
+\subsection{Quadratic data and the trace marginal}
+
+Fix a rational prime $p$.  We first specify the local hypotheses used in the
+argument.  Let
+\[
+ E=\prod_{\nu=1}^r E_\nu,\qquad R_E=\prod_{\nu=1}^r R_\nu
+\]
+be a cubic \'etale $\Q_p$-algebra and its maximal order.  Here $E_\nu$ is a
+field, $R_\nu$ is its integer ring, and $r\le3$.  Write
+$\mathfrak q_\nu=(\pi_\nu)$ for the maximal ideal of $R_\nu$,
+$v_\nu(\pi_\nu)=1$, $e_\nu=v_\nu(p)$, and
+$q_\nu=|R_\nu/\mathfrak q_\nu|=p^{f_\nu}$.  Thus
+\[
+ \sum_\nu e_\nu f_\nu=3,
+ \qquad |\Nm_{E_\nu/\Q_p}a|_p=q_\nu^{-v_\nu(a)}.
+\]
+Let $\mathfrak d_\nu=\mathfrak q_\nu^{d_\nu}$ be the different of
+$E_\nu/\Q_p$, and put $\mathfrak d_E=\prod_\nu\mathfrak d_\nu$.
+
+Let $W$ be a free $E$-module of rank two, with its unital action denoted by
+$\rho:E\longrightarrow\End_{\Q_p}(W)$.  Suppose that $B$ is a nondegenerate
+symmetric $\Q_p$-bilinear form satisfying
+\[
+ B(\rho(a)x,y)=B(x,\rho(a)y)\qquad(a\in E,\ x,y\in W).
+\]
+Let $L$ be a full $\Z_p$-lattice in $W$ which is self-dual for $B$:
+\[
+ L=\{y\in W:B(y,L)\subset\Z_p\}.
+\]
+Assume also that $Q_1(x)=B(x,x)/2$ is integral on $L$.  For the scalar
+calculation we shall use the stronger property that in some integral
+coordinates on $L$,
+\begin{equation}\label{eq:fourier-split-form}
+ Q_1(x_1,y_1,x_2,y_2,x_3,y_3)
+       =x_1y_1-x_2y_2+x_3y_3.
+\end{equation}
+The division by $2$ takes place in $\Q_p$.  Equation~\eqref{eq:fourier-split-form}
+asserts the required integrality also when $p=2$.  The shell estimates below
+use only $Q_1(L)\subset\Z_p$; the exact split expression is used for the
+scalar marginal.
+
+For $a\in E$, set $Q_a(x)=B(\rho(a)x,x)/2$.  The nondegenerate trace pairing
+on $E$ determines a unique quadratic map $\Psi_E:W\longrightarrow E$ by
+\begin{equation}\label{eq:fourier-polar-identity}
+ \Tr_{E/\Q_p}(a\Psi_E(x))=Q_a(x)
+       =\frac12 B(\rho(a)x,x)\qquad(a\in E).
+\end{equation}
+Define the stabilizer order and the auxiliary order
+\[
+ \mathcal C_L=\{a\in E:\rho(a)L\subset L\},
+ \qquad \mathcal C_L^\sharp=\Z_p+2\mathcal C_L.
+\]
+Indeed $\mathcal C_L$ is an order in $R_E$.  It is a subring containing $1$,
+and a sufficiently large power of $p$ times $R_E$ preserves $L$.  Conversely,
+an element preserving $L$ satisfies the monic characteristic polynomial of
+its integral matrix on $L$.  The action is faithful, so the element is
+integral over $\Z_p$ and belongs to $R_E$.
+
+In the application, $E=F_p$, $W=V_p$, $L=L_{V,p}$, $B=B_V$, and
+$\mathcal C_L=C_{F,p}$.  Lemma~\ref{lem:ext-quadratic-structure} supplies
+all the stated hypotheses and identifies $\Psi_E$ with the base change of
+$\Psi$.  We use the abstract notation temporarily to include every product
+algebra and every ramification type in one argument.
+
+Choose an additive character $\psi_p$ of $\Q_p$ with conductor $\Z_p$.
+We give $W$ the self-dual Haar measure for $(x,y)\mapsto\psi_p(B(x,y))$;
+it gives $L$ mass $1$.  On $E$ use the self-dual measure for
+$(A,z)\mapsto\psi_p(\Tr_{E/\Q_p}(Az))$, denoted by $dz$ or $dA$ according
+to the variable.  If $\mathfrak a$ is a fractional $R_E$-ideal, its
+annihilator for this pairing is
+$\mathfrak a^\perp=\mathfrak a^{-1}\mathfrak d_E^{-1}$.  In particular
+\begin{equation}\label{eq:fourier-self-dual-volume}
+ \vol(R_E)=\prod_\nu q_\nu^{-d_\nu/2}\le1.
+\end{equation}
+To see the normalization, $R_E^\perp=\mathfrak d_E^{-1}$ and
+$\vol(R_E)\vol(R_E^\perp)=1$, whereas
+$[\mathfrak d_E^{-1}:R_E]=\prod_\nu q_\nu^{d_\nu}$.
+For $E=F_p$, Equation~\eqref{eq:fourier-self-dual-volume} reads
+$\vol(R_{F,p})=|D_F|_p^{1/2}$.
+
+Let $\mu_p$ be the law of
+\begin{equation}\label{eq:fourier-density-law}
+ u\Psi_E(x),\qquad x\in L\text{ with additive probability},
+ \qquad u\in\Z_p^\times\text{ with Haar probability},
+\end{equation}
+with $x$ and $u$ independent.  This is the law defining $f_p^*$ in
+Lemma~\ref{lem:ext-density}.  The following regularity fact specifies the
+version of that density which we shall invert pointwise.
+
+\begin{lemma}\label{lem:fourier-density}
+The measure $\mu_p$ has a density $f_p^*$ with respect to $dz$.  It has a
+finite locally constant version at every $z=(z_\nu)\in E$ with
+$z_\nu\ne0$ for all $\nu$.  This version vanishes outside
+\[
+ (\mathcal C_L^\sharp)^\vee
+ =\{z\in E:\Tr_{E/\Q_p}(z\mathcal C_L^\sharp)\subset\Z_p\}.
+\]
+In particular $\Tr_{E/\Q_p}(u\Psi_E(x))\in\Z_p$ on the domain of
+Equation~\eqref{eq:fourier-density-law}.
+\end{lemma}
+
+\begin{proof}
+Define an $E$-valued bilinear form $b_E$ by
+\[
+ \Tr_{E/\Q_p}(a b_E(x,y))=B(\rho(a)x,y)\qquad(a\in E).
+\]
+Self-adjointness of the action makes $b_E$ symmetric and $E$-bilinear.
+It is nondegenerate because $B$ is nondegenerate.  Equation~\eqref{eq:fourier-polar-identity}
+then gives $\Psi_E(x)=b_E(x,x)/2$ and
+$d\Psi_{E,x}(h)=b_E(x,h)$.  On the $\nu$-th field factor, a nonzero
+component of $\Psi_E(x)$ implies that the component $x_\nu$ of $x$ is
+nonzero.  The nondegenerate form on the rank-two $E_\nu$-space makes
+$h_\nu\mapsto b_E(x_\nu,h_\nu)$ a nonzero, hence onto, $E_\nu$-linear
+functional.  It follows that $\Psi_E$ is a $\Q_p$-submersion at every
+preimage of an output with nonzero components.  Multiplication by a unit
+$u$ is invertible on $E$, so the map $(x,u)\mapsto u\Psi_E(x)$ is a
+submersion at the same points.
+
+For each $\nu$ the polynomial $\Psi_{E,\nu}$ is nonzero: a nondegenerate
+symmetric form in characteristic zero has a nonzero quadratic form.  Choose
+a nonzero $\Q_p$-coordinate polynomial of this component.  Its zero set in
+the open lattice $L$ has Haar measure zero, by the usual one-variable root
+bound followed by induction and Fubini.  Off the union of
+these null sets, the local submersion theorem gives absolute continuity of
+the pushforward.  For a fixed output with nonzero components, its preimage
+in the compact set $L\times\Z_p^\times$ is compact.  Cover this preimage
+by finitely many compact-open submersion charts, refined to disjoint
+compact-open pieces within those charts.  The image of the remaining compact
+set avoids a sufficiently small neighborhood of the output.  After shrinking
+that neighborhood, integration in the fiber coordinates of the pieces gives
+a finite constant density on it.  This proves the asserted
+pointwise version, including value zero when the neighborhood misses the
+image.
+
+Finally, for $c\in\mathcal C_L$ and $r\in\Z_p$,
+\[
+ \Tr_{E/\Q_p}((r+2c)\Psi_E(x))
+       =rQ_1(x)+B(\rho(c)x,x)\in\Z_p\qquad(x\in L).
+\]
+Multiplication by $u\in\Z_p^\times$ preserves this inclusion.  The image
+of the compact domain is therefore contained in the displayed trace dual,
+which proves the support and trace assertions.
+\end{proof}
+
+We use the Fourier convention
+\[
+ \widehat f(A)=\int_E f(z)\psi_p(\Tr_{E/\Q_p}(Az))\,dz,
+ \qquad
+ \check g(z)=\int_E g(A)\psi_p(-\Tr_{E/\Q_p}(Az))\,dA.
+\]
+The transform of the measure in Equation~\eqref{eq:fourier-density-law} is
+\begin{equation}\label{eq:fourier-gamma}
+ \gamma_p(A)=\widehat{f_p^*}(A)
+ =\E_{u\in\Z_p^\times}\int_L\psi_p(uQ_A(x))\,dx.
+\end{equation}
+
+\begin{lemma}[The scalar trace density]\label{lem:fourier-scalar-density}
+Suppose that the integral scalar form is exactly
+$Q_1=x_1y_1-x_2y_2+x_3y_3$ as in
+Equation~\eqref{eq:fourier-split-form}.  Regard $a\in\Q_p$ as a scalar in
+$E$.  In the quartic application this scalar form is $Q_1=Q_V$.  Then, for
+every prime $p$, including $p=2$,
+\begin{equation}\label{eq:fourier-scalar-coefficients}
+ \gamma_p(a)=
+ \begin{cases}
+ 1,&a\in\Z_p,\\
+ p^{-3k},&v_p(a)=-k<0.
+ \end{cases}
+\end{equation}
+The trace of the law in Equation~\eqref{eq:fourier-density-law} has density
+$d_p(t)\mathbf1_{\Z_p}(t)$ for additive measure giving $\Z_p$ mass $1$.
+This density is continuous on $\Z_p$ and satisfies
+\begin{equation}\label{eq:fourier-scalar-density-bound}
+ |d_p(t)-1|\le\frac1{p(p+1)}\qquad(t\in\Z_p).
+\end{equation}
+In particular $5/6\le d_p(t)\le7/6$, and for every finite set of primes
+$S$ and every choice $t_p\in\Z_p$,
+\[
+ \prod_{p\in S}d_p(t_p)\asymp1
+\]
+with absolute upper and lower constants.
+\end{lemma}
+
+\begin{proof}
+For fixed $u\in\Z_p^\times$ and $v_p(a)=-k<0$, integration in the second
+coordinate gives
+\[
+ \int_{\Z_p^2}\psi_p(ua xy)\,dx\,dy
+ =\int_{\Z_p}\mathbf1_{\{ua x\in\Z_p\}}\,dx=p^{-k}.
+\]
+For $a\in\Z_p$ the integral is $1$.  The sign of $a$ has no effect, so the
+three independent pairs in Equation~\eqref{eq:fourier-split-form} give
+Equation~\eqref{eq:fourier-scalar-coefficients}, even before averaging $u$.
+This calculation uses the integral split expression and does not divide by
+$2$ in $\Z_p$.
+
+The scalar coefficient is the Fourier coefficient of the trace law, since
+$\Tr(a z)=a\Tr(z)$.  That law is supported on $\Z_p$ by
+Lemma~\ref{lem:fourier-density}.  The character group of $\Z_p$ is
+$\Q_p/\Z_p$, and it has $p^k-p^{k-1}$ characters of exact denominator
+$p^k$.  Hence its nonconstant Fourier coefficients have total absolute sum
+\[
+ \sum_{k\ge1}(p^k-p^{k-1})p^{-3k}
+       =\frac{1-p^{-1}}{p^2-1}=\frac1{p(p+1)}.
+\]
+Fourier inversion on the compact group $\Z_p$ is therefore uniformly
+convergent and gives the continuous density and
+Equation~\eqref{eq:fourier-scalar-density-bound}.  The lower bound is
+positive at every prime, and $\sum_p1/(p(p+1))<\infty$.  Taking logarithms
+of the upper and lower bounds proves the last assertion.
+\end{proof}
+
+\subsection{Denominator shells and quadratic transformation}
+
+For $A=(A_\nu)\in E$, its denominator ideal is
+\[
+ \mathfrak h_p(A)=\prod_\nu\mathfrak q_\nu^{h_\nu(A)},\qquad
+ h_\nu(A)=\max\{0,-v_\nu(A_\nu)\},
+\]
+where $v_\nu(0)=+\infty$.  For an integral $R_E$-ideal
+$\mathfrak h_p=\prod_\nu\mathfrak q_\nu^{h_\nu}$, its shell is thus
+\[
+ \mathscr S_{\mathfrak h_p}
+ =\prod_{h_\nu=0}R_\nu\ \times\!
+  \prod_{h_\nu>0}\pi_\nu^{-h_\nu}R_\nu^\times.
+\]
+The integral components of a shell may be zero.  Define
+\begin{equation}\label{eq:fourier-shell-indices}
+ H_p=\Nm(\mathfrak h_p)=\prod_\nu q_\nu^{h_\nu},\qquad
+ p^{g_p}=[\mathfrak h_p:\mathfrak h_p\cap\mathcal C_L].
+\end{equation}
+The shell is a compact open subset of $\mathfrak h_p^{-1}$.  We shall shift
+its frequencies by integral scalars so that the quadratic form is
+nondegenerate, without changing the integral in
+Equation~\eqref{eq:fourier-gamma}.
+
+\begin{lemma}[A scalar shift on a shell]\label{lem:fourier-shift}
+Put $\iota_p=1$ for $p\le5$ and $\iota_p=0$ for $p>5$.  For every
+$A\in\mathscr S_{\mathfrak h_p}$ there is an $s\in\{0,1,2,3\}$ such that
+$\alpha=A+s\in E^\times$ and
+\[
+ v_\nu(\alpha_\nu)=-h_\nu\quad(h_\nu>0),\qquad
+ 0\le v_\nu(\alpha_\nu)\le e_\nu\iota_p\quad(h_\nu=0).
+\]
+Writing $\beta=\alpha^{-1}$, one has
+\begin{equation}\label{eq:fourier-shift-bounds}
+ \beta\in p^{-\iota_p}\mathfrak h_p,
+ \qquad p^{-3\iota_p}H_p
+       \le |\Nm_{E/\Q_p}\alpha|_p\le H_p.
+\end{equation}
+In particular the norm is exactly $H_p$ when $p>5$.  If
+\begin{equation}\label{eq:fourier-j-definition}
+ j_p=j_p(A,s)=\min\{j\ge0:p^j\beta\in\mathcal C_L\},
+\end{equation}
+then $j_p$ is finite and $j_p\le g_p+\iota_p$.  Finally, for any generator
+$i\in E^\times$ of the ideal $\mathfrak h_p$, the element $x=i(A+s)$ is in
+$R_E$, is a unit of $R_E$ if $p>5$, and satisfies
+$x^{-1}\in p^{-\iota_p}R_E$.
+\end{lemma}
+
+\begin{proof}
+An integral scalar shift leaves every pole valuation $-h_\nu<0$ unchanged.
+For an integral component and $p>5$, the condition that $A_\nu+s$ fail to be
+a unit forbids at most one residue $s\in\mathbf F_p$: it forbids none if
+the residue of $-A_\nu$ does not belong to the scalar subfield.  There are
+at most three components, and $0,1,2,3$ are distinct modulo $p$.  Some choice
+therefore makes all integral components units.
+
+For $p\le5$, normalize the extension of $v_p$ by $v_p(p)=1$, so that on
+$E_\nu$ it is $v_\nu/e_\nu$.  Distinct elements $s,s'$ of the list satisfy
+$v_p(s-s')\le1$.  The ultrametric inequality shows that for a fixed integral
+component at most one candidate can satisfy $v_p(A_\nu+s)>1$.  Again there
+are at most three components, so a candidate remains for which all their
+valuations are at most $1$.  Thus $A_\nu+s$ is nonzero in every integral
+component.
+
+The bounds on $\beta$ follow componentwise.  In the norm, the possible
+loss from the integral components is at most
+$p^{-\iota_p\sum_\nu e_\nu f_\nu}=p^{-3\iota_p}$, proving
+Equation~\eqref{eq:fourier-shift-bounds}.  The group
+$\mathfrak h_p/(\mathfrak h_p\cap\mathcal C_L)$ has order $p^{g_p}$ and is
+killed by $p^{g_p}$.  Thus
+\begin{equation}\label{eq:fourier-order-containment}
+ p^{g_p}\mathfrak h_p\subset\mathcal C_L.
+\end{equation}
+Multiplying the containment for $\beta$ by $p^{g_p+\iota_p}$ proves the
+claim about $j_p$.  Finally $v_\nu(i_\nu)=h_\nu$.  The components of
+$i\alpha$ consequently have valuation zero on the pole factors and valuation
+between zero and $e_\nu\iota_p$ on the others.  This proves the assertions
+about $x$ and $x^{-1}$.
+\end{proof}
+
+For the shift in Lemma~\ref{lem:fourier-shift},
+$Q_{A+s}=Q_A+sQ_1$ and $Q_1(L)\subset\Z_p$.  Thus for every fixed unit $u$
+\begin{equation}\label{eq:fourier-scalar-shift-invariance}
+ \int_L\psi_p(uQ_{A+s}(x))\,dx
+       =\int_L\psi_p(uQ_A(x))\,dx.
+\end{equation}
+The following quadratic transformation turns the norm of $A+s$ into decay,
+and leaves an integral involving its inverse.
+
+\begin{lemma}[Quadratic Fourier transformation]\label{lem:fourier-quadratic-transform}
+Let $W_0$ be a finite-dimensional $\Q_p$-space with a nondegenerate symmetric
+form $B_0$, and let $L_0\subset W_0$ be a full $\Z_p$-lattice self-dual for
+$B_0$.  Use the self-dual Haar
+measure, which gives $L_0$ mass $1$.  For an invertible $B_0$-self-adjoint
+operator $P$, put $q_P(x)=B_0(Px,x)/2$.  There is a phase $\omega(P)$ with
+$|\omega(P)|=1$ such that
+\begin{equation}\label{eq:fourier-general-quadratic}
+ \int_{L_0}\psi_p(q_P(x))\,dx
+ =\omega(P)|\det P|_p^{-1/2}
+       \int_{L_0}\psi_p(-q_{P^{-1}}(x))\,dx.
+\end{equation}
+The normalized phase is invariant under linear isometry of the quadratic
+form: $\omega(T^*PT)=\omega(P)$ for every invertible $T$, where $T^*$ is
+the $B_0$-adjoint.
+
+For the local data above, $\alpha\in E^\times$, $\beta=\alpha^{-1}$ and
+$u\in\Z_p^\times$, this gives
+\begin{equation}\label{eq:fourier-quadratic-transform}
+ \int_L\psi_p(uQ_\alpha(x))\,dx
+ =\omega_p(u,\alpha)|\Nm_{E/\Q_p}\alpha|_p^{-1}
+       \int_L\psi_p(-u^{-1}Q_\beta(x))\,dx,
+ \qquad |\omega_p(u,\alpha)|=1.
+\end{equation}
+For every $r\in E^\times$ one has
+$\omega_p(u,\alpha r^2)=\omega_p(u,\alpha)$.
+All assertions hold at $p=2$.
+\end{lemma}
+
+\begin{proof}
+We use Weil's Fourier formula for a quadratic character
+\cite[Chapter~I, \S14, Theorem~2, p.~161]{Weil1964}.
+We record its normalization for the transform
+\[
+ \mathcal F_{B_0}h(y)=\int h(x)\psi_p(B_0(x,y))\,dx.
+\]
+Completing the square gives
+\[
+ q_P(x)+B_0(x,y)=q_P(x+P^{-1}y)-q_{P^{-1}}(y).
+\]
+Let $B_N=p^{-N}L_0$ and $c_N=\int_{B_N}\psi_p(q_P(x))\,dx$.  On any fixed
+compact set of $y$, translation by $P^{-1}y$ preserves $B_N$ for all
+sufficiently large $N$.  The transform of
+$\mathbf1_{B_N}\psi_p(q_P)$ on that set is therefore
+$c_N\psi_p(-q_{P^{-1}})$.  If this identity is paired with a
+Schwartz--Bruhat function, its left side stabilizes with $N$, since the
+Fourier transform of the test function has compact support.  Choosing a
+test function whose integral against $\psi_p(-q_{P^{-1}})$ is nonzero shows
+that $c_N$ stabilizes as well.  This proves the distributional shape of the
+formula.
+
+Differencing the absolute square of $c_N$ and integrating in one variable
+gives
+\[
+ |c_N|^2=\vol(B_N)
+   \int_{B_N\cap P^{-1}B_N^\perp}\psi_p(q_P(h))\,dh.
+\]
+For sufficiently large $N$, the intersection is $P^{-1}p^NL_0$ and
+$q_P$ is integral on it.  The right side then equals
+$\vol(p^{-N}L_0)\vol(P^{-1}p^NL_0)=|\det P|_p^{-1}$.
+Hence the stabilized constant is $\omega(P)|\det P|_p^{-1/2}$ with unit
+phase.  The transform of $\mathbf1_{L_0}$ is $\mathbf1_{L_0}$, so pairing
+the distribution identity with this function gives
+Equation~\eqref{eq:fourier-general-quadratic}.
+
+Under $q_{T^*PT}=q_P\circ T$, changing variables multiplies the Gaussian
+constant by $|\det T|_p^{-1}$ and its normalizing determinant square root by
+$|\det T|_p$.  Their product, the phase, is unchanged; this is also the
+isometry invariance in Weil~\cite[Chapter~II, \S25, p.~173]{Weil1964}.
+For $P=u\rho(\alpha)$ its inverse is $u^{-1}\rho(\beta)$.  Rank two over
+$E$ gives $\det\rho(\alpha)=\Nm_{E/\Q_p}(\alpha)^2$, and $|u|_p=1$.
+This proves Equation~\eqref{eq:fourier-quadratic-transform}.  Taking
+$T=\rho(r)$ proves the last phase identity.  The polar operator of $q_P$ is
+$P$, because its polarization is $B_0(Px,y)$.  In particular this proof
+never assumes that $2$ is a unit of $\Z_p$.
+\end{proof}
+
+\subsection{The inverse transform on a shell}
+
+Extend the restriction of $\gamma_p$ to a shell by zero, and denote its
+inverse transform by
+\begin{equation}\label{eq:fourier-shell-transform}
+ \gamma_{p,\mathfrak h_p}(A)
+   =\mathbf1_{\mathscr S_{\mathfrak h_p}}(A)\gamma_p(A),
+ \qquad
+ K_{p,\mathfrak h_p}(z)
+   =\int_E\gamma_{p,\mathfrak h_p}(A)
+             \psi_p(-\Tr_{E/\Q_p}(Az))\,dA.
+\end{equation}
+The next lemma gives the estimates for these functions which will be used
+in Poisson summation.  Its quantitative support assertions separate the case where
+the denominator ideal is contained in the stabilizer order.
+
+\begin{lemma}[Intrinsic shell estimates]\label{lem:fourier-shell-estimates}
+Let $E$ be the cubic algebra above, let $W$ have rank two over $E$ with
+self-adjoint action for a nondegenerate symmetric form $B$, and let $L$ be
+self-dual for $B$, with $Q_1(L)\subset\Z_p$.  Use the stabilizer order
+$\mathcal C_L=\{a\in E:\rho(a)L\subset L\}$ and $\Psi_E,\gamma_p$ from
+Equations~\eqref{eq:fourier-polar-identity} and \eqref{eq:fourier-gamma}.
+Fix an integral ideal
+$\mathfrak h_p=\prod_\nu\mathfrak q_\nu^{h_\nu}$, and use
+$H_p,g_p$ from Equation~\eqref{eq:fourier-shell-indices}.
+The zero-extended function $\gamma_{p,\mathfrak h_p}$ and its inverse
+$K_{p,\mathfrak h_p}$ are Schwartz--Bruhat functions.  Moreover,
+\[
+ \supp K_{p,\mathfrak h_p}\subset(\mathcal C_L^\sharp)^\vee
+ \subset\{z\in E:\Tr_{E/\Q_p}z\in\Z_p\}.
+\]
+For
+$A\in\mathscr S_{\mathfrak h_p}$ choose $s$ and $j_p=j_p(A,s)$ as in
+Lemma~\ref{lem:fourier-shift}.  Then
+\begin{equation}\label{eq:fourier-shell-decay}
+ |\gamma_p(A)|\le p^{3\iota_p}H_p^{-1}p^{-j_p/2},
+ \qquad
+ \|K_{p,\mathfrak h_p}\|_\infty
+       \le p^{3\iota_p}\vol(R_E)\le p^{3\iota_p}.
+\end{equation}
+Here $\iota_p=\mathbf1_{\{p\le5\}}$, so the first constant is $1$ at
+every $p>5$ and is at most $125$ at the three remaining primes.
+
+If $g_p>0$ or $p\le5$, then
+\begin{equation}\label{eq:fourier-inverse-support}
+ \supp K_{p,\mathfrak h_p}
+       \subset p^{-4(g_p+1)}\mathfrak h_p\mathfrak d_E^{-1}.
+\end{equation}
+If $g_p=0$ and $p>5$, then support instead requires, for each $\nu$,
+\begin{equation}\label{eq:fourier-boundary-support}
+ z_\nu\in
+ \begin{cases}
+ \mathfrak d_\nu^{-1},&h_\nu=0,\\
+ \mathfrak q_\nu^{h_\nu-1}\mathfrak d_\nu^{-1},&h_\nu>0.
+ \end{cases}
+\end{equation}
+For a $z$ satisfying these support conditions, define its set of boundary
+primes by
+\[
+ \partial_{\mathfrak h_p}(z)
+ =\{\mathfrak q_\nu:h_\nu>0,
+       \ z_\nu\notin\mathfrak q_\nu^{h_\nu}\mathfrak d_\nu^{-1}\}.
+\]
+In this case
+\begin{equation}\label{eq:fourier-boundary-decay}
+ |K_{p,\mathfrak h_p}(z)|
+ \le 8\vol(R_E)
+       \prod_{\mathfrak q_\nu\in\partial_{\mathfrak h_p}(z)}q_\nu^{-1/2}.
+\end{equation}
+Every constant is uniform in the lattice, the order $\mathcal C_L$, the
+shell, and the field factors and their ramification.
+\end{lemma}
+
+\begin{proof}
+For $c\in\mathcal C_L^\sharp$, Equation~\eqref{eq:fourier-polar-identity}
+shows that $Q_c(L)\subset\Z_p$.  Consequently $\gamma_p(A+c)=\gamma_p(A)$.
+Each denominator shell is invariant under translation by $R_E$: adding an
+integral element preserves every negative component valuation and preserves
+the set of integral components.  Since $\mathcal C_L^\sharp\subset R_E$,
+the zero-extended shell restriction retains this period.  Its inverse is
+therefore supported in the annihilator $(\mathcal C_L^\sharp)^\vee$, whose
+elements have integral trace because $1\in\mathcal C_L^\sharp$.
+The order $\mathcal C_L^\sharp$ is an open lattice, and the shell is compact
+open.  Thus the zero extension in Equation~\eqref{eq:fourier-shell-transform}
+is Schwartz--Bruhat, as is its Fourier inverse.
+
+For a fixed unit $u$ put
+$I_\beta(u)=\int_L\psi_p(-u^{-1}Q_\beta(x))\,dx$.
+Polarization and additive orthogonality on the self-dual lattice give
+\[
+ |I_\beta(u)|^2
+ =\int_{\substack{h\in L\\\rho(\beta)h\in L}}
+       \psi_p(-u^{-1}Q_\beta(h))\,dh.
+\]
+Indeed, in the difference $Q_\beta(y+h)-Q_\beta(y)$ the term linear in $y$
+is $B(\rho(\beta)h,y)$, whose integral is the indicator of
+$\rho(\beta)h\in L$.  The absolute value of the last display is at most the
+relative mass of $L\cap\rho(\beta)^{-1}L$.  If
+$p^{a_1},\ldots,p^{a_6}$ are the Smith elementary divisors of
+$\rho(\beta)$ on $L$, then
+\[
+ j_p=\max\{0,-\min_i a_i\},\qquad
+ [L:L\cap\rho(\beta)^{-1}L]
+       =p^{\sum_i\max(0,-a_i)}\ge p^{j_p}.
+\]
+The equality for $j_p$ is exactly the definition of $\mathcal C_L$.
+It follows that $|I_\beta(u)|\le p^{-j_p/2}$.  Apply
+Equations~\eqref{eq:fourier-scalar-shift-invariance} and
+\eqref{eq:fourier-quadratic-transform}, then use
+Equation~\eqref{eq:fourier-shift-bounds} and average over $u$.  This proves
+the first bound in Equation~\eqref{eq:fourier-shell-decay}.  The shell is
+contained in $\mathfrak h_p^{-1}$, which has volume
+$H_p\vol(R_E)$.  Integration of the first bound, with $p^{-j_p/2}\le1$,
+proves the second.
+
+We next prove Equation~\eqref{eq:fourier-inverse-support}.  Put
+$N=4(g_p+1)$ and take $t\in p^N\mathfrak h_p^{-1}$.  For $A$ in the shell
+choose $s$ as in Lemma~\ref{lem:fourier-shift}, and write
+$\alpha=A+s$, $\beta=\alpha^{-1}$.  The component valuations of $t$ are
+at least $-h_\nu+e_\nu N$.  Translation by $t$ therefore preserves the
+shell, and the same $s$ still has the valuation properties of that lemma.
+Set $\zeta=t\beta$.  Equation~\eqref{eq:fourier-shift-bounds} gives
+\[
+ \zeta\in p^{N-\iota_p}R_E,\qquad
+ \alpha+t=\alpha(1+\zeta),\qquad
+ (\alpha+t)^{-1}-\beta
+       =-\frac{t\beta^2}{1+\zeta}
+       \in p^{N-2\iota_p}\mathfrak h_p.
+\]
+Here $1+\zeta$ is a unit.  It is a square unit in each factor when $p$ is
+odd, by Hensel's lemma on principal units.  When $p=2$, one has
+$\zeta\in8R_E$.  In the $\nu$-th field the Hensel inequality for
+$Y^2-(1+\zeta_\nu)$ at $Y=1$ is
+$v_\nu(\zeta_\nu)\ge3e_\nu>2e_\nu=2v_\nu(2)$, so it is a square unit
+there as well, including for ramified dyadic factors.
+
+Moreover $N-2\iota_p\ge g_p+v_p(2)$.  Equation~\eqref{eq:fourier-order-containment}
+therefore places the inverse difference in $2\mathcal C_L$.  Its $Q$-value
+on $L$ is integral by Equation~\eqref{eq:fourier-polar-identity}.
+In Equation~\eqref{eq:fourier-quadratic-transform}, the inverse integral is
+unchanged by this difference.  The norm is unchanged by the unit $1+\zeta$,
+and the phase is unchanged because that unit is a square.  The original
+integral, for each fixed $u$, is consequently unchanged under $A\mapsto A+t$.
+So is its unit average.  Translation by $p^N\mathfrak h_p^{-1}$ preserves
+the shell as a set, hence preserves its zero-extended function.  The inverse
+transform is supported on the annihilator of this group, namely
+$p^{-N}\mathfrak h_p\mathfrak d_E^{-1}$.  This proves
+Equation~\eqref{eq:fourier-inverse-support} with the absolute exponent $4$.
+
+It remains to prove the sharper assertions for $g_p=0$ and $p>5$.  The
+equality $g_p=0$ says $\mathfrak h_p\subset\mathcal C_L$.  For each factor
+with $h_\nu=0$, its component idempotent $1_\nu$ and its whole component
+ring $1_\nu R_\nu$ lie in $\mathfrak h_p$.  The self-adjoint projections
+$\rho(1_\nu)$ preserve $L$ and split off their lattices orthogonally.
+The complementary projection $1-\sum_{h_\nu=0}\rho(1_\nu)$ also preserves
+$L$.  Each resulting summand is self-dual, because the splitting is
+orthogonal and $L$ is self-dual.
+
+On an $h_\nu=0$ summand, an integral $A_\nu$ preserves its lattice, so
+$Q_{A_\nu}$ is integral there since $2$ is a unit.  That factor of the
+integral in Equation~\eqref{eq:fourier-gamma} is $1$.  On the combined
+summand of all $h_\nu>0$, the frequency $A_+$ is invertible.  Its inverse,
+extended by zero on the other factors, lies in
+$\mathfrak h_p\subset\mathcal C_L$.  Its inverse quadratic integral in
+Equation~\eqref{eq:fourier-general-quadratic} is therefore also $1$.
+On that summand the square root of the polar determinant has absolute value
+$H_p$, because each positive factor still has rank two.  Applying the formula
+therefore shows that the remaining frequency function is $H_p^{-1}$ times a
+unit-average of Gaussian phases.  Its absolute value is at most $H_p^{-1}$.
+If there are no positive
+$h_\nu$, this conclusion has $H_p=1$ and the empty summand contributes $1$.
+
+Let $\chi_\nu$ be the quadratic character of
+$(R_\nu/\mathfrak q_\nu)^\times$ for a factor with $h_\nu>0$.  At odd
+residue characteristic every principal unit is a square, so unit square
+classes are exactly the two residue square classes.  Phase invariance in
+Lemma~\ref{lem:fourier-quadratic-transform} shows that the bounded unit-average
+just obtained depends only on the tuple of these classes.  This use of phase
+invariance concerns the ambient quadratic form; a unit need not preserve
+the lattice.  Fourier expansion on the product of the two-element
+square-class groups expresses the shell function as a sum of at most
+$2^{|\{\nu:h_\nu>0\}|}\le8$ terms of the form
+\[
+ c_{\boldsymbol\varepsilon}H_p^{-1}
+ \prod_{h_\nu=0}\mathbf1_{R_\nu}(A_\nu)
+ \prod_{h_\nu>0}
+   \mathbf1_{\pi_\nu^{-h_\nu}R_\nu^\times}(A_\nu)
+   \chi_\nu(\overline{\pi_\nu^{h_\nu}A_\nu})^{\varepsilon_\nu},
+ \qquad |c_{\boldsymbol\varepsilon}|\le1,
+\]
+where $\varepsilon_\nu\in\{0,1\}$.  The bound for the coefficients follows
+from the normalized finite Fourier expansion of a function of absolute
+value at most $1$.  It also covers any coupling of components caused by
+the common average over $u$.
+
+We compute the inverse of each factor.  Put
+$\psi_\nu=\psi_p\circ\Tr_{E_\nu/\Q_p}$ and use the component self-dual
+measure.  For $h_\nu=0$, the integral over $R_\nu$ is
+$\vol(R_\nu)\mathbf1_{\mathfrak d_\nu^{-1}}(z_\nu)$.
+For $h=h_\nu>0$, the normalized inverse of the trivial unit character is
+\[
+ q_\nu^{-h}\int_{\pi_\nu^{-h}R_\nu^\times}
+       \psi_\nu(-Az)\,dA
+ =\vol(R_\nu)\left(
+    \mathbf1_{\mathfrak q_\nu^h\mathfrak d_\nu^{-1}}(z)
+   -q_\nu^{-1}\mathbf1_{\mathfrak q_\nu^{h-1}\mathfrak d_\nu^{-1}}(z)
+                     \right).
+\]
+This follows by subtracting the integral over $\pi_\nu^{-h+1}R_\nu$ from
+that over $\pi_\nu^{-h}R_\nu$ and using their annihilators.  For the quadratic
+unit character, its function of $a=\pi_\nu^hA$ is invariant under addition
+by $\mathfrak q_\nu$.  This forces the normalized inverse to vanish outside
+$\mathfrak q_\nu^{h-1}\mathfrak d_\nu^{-1}$.  It also vanishes on
+$\mathfrak q_\nu^h\mathfrak d_\nu^{-1}$, where the additive character on
+$R_\nu$ is trivial and the sum of $\chi_\nu$ is zero.  At the intervening
+boundary the additive character descends to a nontrivial character $\eta$
+of the residue field $k_\nu$, and the normalized integral is
+$\vol(R_\nu)q_\nu^{-1}$ times its quadratic Gauss sum.
+
+For completeness, if $\chi$ is the quadratic character of a finite field
+$k$ of odd order $q$ and $\eta$ is a nontrivial additive character, then
+\[
+ \left|\sum_{a\in k^\times}\chi(a)\eta(a)\right|^2
+ =\sum_{t\in k^\times}\chi(t)
+       \sum_{b\in k^\times}\eta((t-1)b)
+ =(q-1)-\sum_{\substack{t\in k^\times\\t\ne1}}\chi(t)=q.
+\]
+Thus the quadratic factor has magnitude
+$\vol(R_\nu)q_\nu^{-1/2}$ at the boundary.  The trivial factor there has
+the smaller magnitude $\vol(R_\nu)q_\nu^{-1}$.  Multiplying these
+component estimates, using $H_p=\prod_{h_\nu>0}q_\nu^{h_\nu}$ and
+$\prod_\nu\vol(R_\nu)=\vol(R_E)$, and summing at most eight terms proves
+Equations~\eqref{eq:fourier-boundary-support} and
+\eqref{eq:fourier-boundary-decay}.  The annihilator computation retains the
+different in every ramified factor.
+\end{proof}
+
+The density need not be locally constant on the component-vanishing locus.
+For the nonvanishing outputs that occur in the global field, the shell
+inverses nevertheless give a finite pointwise expansion.
+
+\begin{lemma}[Pointwise inverse expansion]\label{lem:fourier-inverse-expansion}
+If $z=(z_\nu)\in E$ has $z_\nu\ne0$ for every $\nu$, then
+\begin{equation}\label{eq:fourier-inverse-expansion}
+ f_p^*(z)=\sum_{\mathfrak h_p}K_{p,\mathfrak h_p}(z),
+\end{equation}
+where the sum is over integral $R_E$-ideals and only finitely many summands
+are nonzero.
+\end{lemma}
+
+\begin{proof}
+Let $B_n=p^{-n}R_E$.  Fubini and additive orthogonality give
+\[
+ \int_{B_n}\gamma_p(A)\psi_p(-\Tr_{E/\Q_p}(Az))\,dA
+       =\vol(B_n)\,\mu_p(z+p^n\mathfrak d_E^{-1}).
+\]
+The self-dual normalizations give
+\[
+ B_n^\perp=p^n\mathfrak d_E^{-1},\qquad
+ \vol(B_n)\vol(B_n^\perp)=1.
+\]
+Thus the right side is the average density on that neighborhood.  By
+Lemma~\ref{lem:fourier-density} it equals $f_p^*(z)$ for sufficiently large
+$n$.  The ball $B_n$ is the disjoint union of the finitely many shells with
+$h_\nu\le ne_\nu$.
+
+There can in fact be only finitely many nonzero shell inverses at this $z$.
+The injection
+$\mathfrak h_p/(\mathfrak h_p\cap\mathcal C_L)\hookrightarrow R_E/\mathcal C_L$
+gives $g_p\le v_p([R_E:\mathcal C_L])$.  In
+Equation~\eqref{eq:fourier-inverse-support}, a nonzero value requires
+\[
+ h_\nu\le v_\nu(z_\nu)+d_\nu+4e_\nu(g_p+1).
+\]
+In Equation~\eqref{eq:fourier-boundary-support}, it requires
+$h_\nu\le v_\nu(z_\nu)+d_\nu+1$ whenever $h_\nu>0$.
+All right sides are bounded for this $z$, because its component valuations
+are finite.  The shell sum therefore stabilizes, and the preceding ball
+inversion proves Equation~\eqref{eq:fourier-inverse-expansion}.
+\end{proof}
+
+The exact scalar coefficients in Lemma~\ref{lem:fourier-scalar-density}
+supply the trace marginal in the next section.  The nonscalar estimate uses
+the pointwise decay in Equation~\eqref{eq:fourier-shell-decay}, the inverse
+supports in Equations~\eqref{eq:fourier-inverse-support} and
+\eqref{eq:fourier-boundary-support}, and the boundary bound in
+Equation~\eqref{eq:fourier-boundary-decay}.  In the case $g_p=0$ and $p>5$,
+the last upper bound contains a factor $q_\nu^{-1/2}$ for each actual boundary
+component where the support permits one fewer power of $\mathfrak q_\nu$.
+Once the real boxes are specified, Section~\ref{sec:additive} combines the
+supports with the nonzero field norm to restrict the denominator profiles in
+Poisson summation.

preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/macros.tex

@@ -0,0 +1,30 @@
+\newcommand{\Q}{\mathbb Q}
+\newcommand{\R}{\mathbb R}
+\newcommand{\Z}{\mathbb Z}
+\newcommand{\C}{\mathbb C}
+\newcommand{\E}{\mathbb E}
+\renewcommand{\P}{\mathbb P}
+\newcommand{\OO}{\mathcal O}
+\newcommand{\eps}{\varepsilon}
+\DeclareMathOperator{\Tr}{Tr}
+\DeclareMathOperator{\Nm}{N}
+\DeclareMathOperator{\Gal}{Gal}
+\DeclareMathOperator{\Disc}{Disc}
+\DeclareMathOperator{\Res}{Res}
+\DeclareMathOperator{\covol}{covol}
+\DeclareMathOperator{\vol}{vol}
+\DeclareMathOperator{\diag}{diag}
+\DeclareMathOperator{\rank}{rank}
+\DeclareMathOperator{\End}{End}
+\DeclareMathOperator{\Pic}{Pic}
+\DeclareMathOperator{\supp}{supp}
+\DeclareMathOperator{\id}{id}
+\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}

preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/main.tex

@@ -0,0 +1,30 @@
+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern,amsmath,amssymb,amsthm,mathtools}
+\usepackage[mathscr]{eucal}
+\usepackage[margin=1.05in]{geometry}
+\usepackage{microtype,graphicx,xcolor,booktabs}
+\usepackage{hyperref}
+\hypersetup{colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black,pdftitle={Equidistribution of primitive quartic torus packets for arbitrary orders},pdfauthor={OpenAI}}
+\input{macros}
+\title{Equidistribution of primitive quartic torus packets\\for arbitrary orders}
+\author{OpenAI}
+\date{October 5, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+Let $K$ range over totally real quartic fields with no proper intermediate field, and let $\OO$ be any order in $K$. We prove that the packets of periodic diagonal orbits attached to invertible $\OO$-ideal classes, weighted by orbit volume, equidistribute with no escape of mass as $|\Disc(\OO)|$ tends to infinity. The proof combines measure rigidity with a cubic-resolvent estimate that remains uniform at primes dividing the order index.
+\end{abstract}
+\tableofcontents
+\input{introduction}
+\input{packets}
+\input{ordinary}
+\input{rigidity}
+\input{exterior}
+\input{local-fourier}
+\input{additive}
+\input{sieve}
+\input{conclusion}
+\bibliographystyle{plain}
+\bibliography{references}
+\end{document}

preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/ordinary.tex

@@ -0,0 +1,771 @@
+\section{Ordinary vectors and probability limits}\label{sec:ordinary}
+
+We first control vectors in the lattices themselves.  A count near the
+origin will prevent escape into the cusp, and a count near a vector with
+no zero coordinate will bound the mass of small neighborhoods in \(X_4\).
+The latter bound is the input to the entropy argument in
+Section~\ref{sec:rigidity}.  We follow the method of
+\cite[Sections~2--5]{PrimeDegreePackets}, proving the estimates with
+constants uniform over primitive totally real quartic fields, their orders,
+and the orderings of their embeddings.  The absence of a quadratic
+subfield supplies the residue estimate needed in this degree.
+
+\subsection{Containment probabilities and unfolding}
+
+For a rational prime \(p\), write
+\[
+ R_p=\prod_{\mathfrak p\mid p}R_{\mathfrak p}.
+\]
+Here \(R_{\mathfrak p}\) is the integer ring of the completion
+\(K_{\mathfrak p}\).  Let \(v_{\mathfrak p}\) be its integer valuation,
+normalized to be \(1\) on a uniformizer, and put
+\[
+ e_{\mathfrak p}=v_{\mathfrak p}(p),\qquad
+ f_{\mathfrak p}=[R/\mathfrak p:\Z/p\Z].
+\]
+Thus \(\Nm\mathfrak p=p^{f_{\mathfrak p}}\) and
+\(\sum_{\mathfrak p\mid p}e_{\mathfrak p}f_{\mathfrak p}=4\);
+see \cite[Theorem~3.34]{MilneANT}.  For \(p\mid q\) and a tuple
+\(\boldsymbol l_p=(l_{\mathfrak p})_{\mathfrak p\mid p}\) of
+nonnegative integers, call the set of \(x\in R_p\) with
+\(v_{\mathfrak p}(x_{\mathfrak p})=l_{\mathfrak p}\) in every component
+the valuation shell \(\boldsymbol l_p\).  Choose \(x\) in this shell
+and define
+\begin{equation}\label{eq:ord-local-probability}
+ P_p(\boldsymbol l_p)
+   =\P_{t_p\in R_p^\times}\{x\in t_p\OO_p\},
+\end{equation}
+where the probability is normalized unit Haar measure.  The definition
+is independent of \(x\): the unit group acts transitively on each set
+with specified component valuations.
+
+For a nonzero integral \(R\)-ideal \(\mathfrak a\), set
+\begin{equation}\label{eq:ord-ideal-weight}
+ P(\mathfrak a)=
+   \prod_{p\mid q}
+   P_p\bigl((v_{\mathfrak p}(\mathfrak a))_{\mathfrak p\mid p}\bigr).
+\end{equation}
+An empty product, including the case \(q=1\), is \(1\).  If
+\(0\ne b\in I\), its image has no zero local component and
+\((b)I^{-1}\) is integral.  In the coordinates
+\(I_p=\xi_pR_p\), the valuations of \(\xi_p^{-1}b\) are those of
+\((b)I^{-1}\).  Lemma~\ref{lem:pkt-local-labels} and the independence
+of the local choices therefore give
+\begin{equation}\label{eq:ord-membership}
+ \P_{L\in\mathscr L_{\OO}(I)}\{b\in L\}
+       =P((b)I^{-1}).
+\end{equation}
+
+For a nonnegative Borel function \(f\) on \(\R^4\), write
+\[
+ E_f(\Lambda)=\sum_{0\ne x\in\Lambda}f(x).
+\]
+This is a Borel function on \(X_4\): in a local basis chart it is a
+countable sum of the Borel functions obtained by evaluating \(f\) at
+the integer linear combinations of that basis.
+For \(z>0\), define the integral over the surface with absolute
+coordinate product \(z\) by
+\begin{equation}\label{eq:ord-fiber}
+ \mathcal V_f(z)=
+ \sum_{\omega\in\{\pm1\}^4}
+ \int_{\R^3}
+ f\bigl(\omega_1e^{u_1},\omega_2e^{u_2},\omega_3e^{u_3},
+        \omega_4z e^{-u_1-u_2-u_3}\bigr)\,du_1\,du_2\,du_3 .
+\end{equation}
+The signs and logarithmic measure are exactly those of \(T_\infty\).
+
+\begin{proposition}[Ordinary vector unfolding]\label{prop:ord-unfolding}
+For every nonnegative Borel \(f\) on \(\R^4\),
+\begin{equation}\label{eq:ord-unfolding}
+ \int_{X_4}E_f\,d\mu_*
+ =\frac1{\sqrt D\,\kappa}
+   \sum_{\substack{\mathfrak a\subset R\\\mathfrak a\ne0}}
+       P(\mathfrak a)\,
+       \mathcal V_f\bigl(\Nm\mathfrak a/X\bigr).
+\end{equation}
+Both sides are allowed to be infinite.
+\end{proposition}
+
+\begin{proof}
+This is the unfolding of \cite[Proposition~2.2]{PrimeDegreePackets}
+with the packet normalization of Proposition~\ref{prop:pkt-model}.
+Average the vector sum over \(L\) in
+Equation~\eqref{eq:pkt-integration}, and use
+Equation~\eqref{eq:ord-membership}.  The factor \(N_{\OO}\) cancels,
+and the left side becomes
+\[
+ \frac1{h_K\vol(\mathscr F_U)}
+ \sum_I\int_{\mathscr F_U}
+   \sum_{0\ne b\in I}P((b)I^{-1})f(c_I\sigma(b)v)\,dv .
+\]
+For fixed \(I\), the map \(b\mapsto(b)I^{-1}\) takes \(I\setminus\{0\}\)
+onto the integral ideals in the inverse class of \(I\).  Each fiber
+is one free \(U\)-orbit.  Indeed, two generators of the same principal
+ideal differ by a unique unit.  The coefficient \(P((b)I^{-1})\) is
+constant on that orbit.  Summing its unit translates and integrating
+over \(\mathscr F_U\) gives
+\[
+ \int_{\mathscr F_U}\sum_{\epsilon\in U}
+       f(c_I\sigma(\epsilon b)v)\,dv
+     =\int_{T_\infty}f(c_I\sigma(b)v)\,dv .
+\]
+The translates tile \(T_\infty\), including its sign components.
+All changes in the order of integration and summation are justified
+by Tonelli's theorem.
+
+The absolute product of the coordinates of \(c_I\sigma(b)\) is
+\[
+ c_I^4|\Nm_{K/\Q}(b)|
+ =\frac{\Nm((b)I^{-1})}{X}.
+\]
+Translation of the first three logarithmic coordinates and relabeling
+of the signs therefore identify the last integral with
+\(\mathcal V_f(\Nm((b)I^{-1})/X)\), with Jacobian one.  Each integral
+ideal belongs to the inverse class of exactly one chosen \(I\).
+Equation~\eqref{eq:pkt-unit-volume} now gives the coefficient in
+Equation~\eqref{eq:ord-unfolding}.
+\end{proof}
+
+\subsection{The mass and tail of the local weights}
+
+The ideal sum in Equation~\eqref{eq:ord-unfolding} need not have simple
+coefficients at primes dividing \(q\).  We will use its exact total
+mass over valuation shells and a positive moment of their norms.
+Both statements are the order case of
+\cite[Lemma~3.1]{PrimeDegreePackets}; the argument below explains why
+they are uniform even when the local order has large index.
+
+For \(p\mid q\), put
+\[
+ q_p=[R_p:\OO_p],\qquad
+ N_p(\boldsymbol l_p)=p^{\sum_{\mathfrak p\mid p}
+                                  f_{\mathfrak p}l_{\mathfrak p}},
+ \qquad
+ Z_p=\prod_{\mathfrak p\mid p}(1-p^{-f_{\mathfrak p}})^{-1}.
+\]
+For a collection \(\boldsymbol l=(\boldsymbol l_p)_{p\mid q}\), write
+\[
+ d(\boldsymbol l)=\prod_{p\mid q}N_p(\boldsymbol l_p),\qquad
+ P(\boldsymbol l)=\prod_{p\mid q}P_p(\boldsymbol l_p),\qquad
+ Z_q=\prod_{p\mid q}Z_p .
+\]
+Thus \(d(\boldsymbol l)\) is the norm of the part of an integral ideal
+supported above primes dividing \(q\), and \(\prod_{p\mid q}q_p=q\).
+
+\begin{lemma}[Local mass and moment]\label{lem:ord-shells}
+With the preceding notation,
+\begin{equation}\label{eq:ord-shell-mass}
+ \sum_{\boldsymbol l}\frac{P(\boldsymbol l)}{d(\boldsymbol l)}
+       =\frac{Z_q}{q}.
+\end{equation}
+For every \(\eta>0\) there is a constant \(C_\eta\), independent of
+\(K,\OO\), such that
+\begin{equation}\label{eq:ord-shell-moment}
+ \sum_{\boldsymbol l}
+       P(\boldsymbol l)d(\boldsymbol l)^{-1+1/8}
+       \le C_\eta\,\frac{q^\eta}{q}.
+\end{equation}
+In particular there is an absolute constant \(C\) for which
+\begin{equation}\label{eq:ord-shell-tail}
+ \sum_{d(\boldsymbol l)>X^{1/8}}
+       \frac{P(\boldsymbol l)}{d(\boldsymbol l)}
+       \le \frac Cq X^{-1/128}.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Normalize additive Haar measure \(m_p\) on \(R_p\) by \(m_p(R_p)=1\).
+In a component \(R_{\mathfrak p}\), the set of elements of valuation
+\(l_{\mathfrak p}\) has measure
+\((1-p^{-f_{\mathfrak p}})p^{-f_{\mathfrak p}l_{\mathfrak p}}\).
+Consequently the shell \(\boldsymbol l_p\) in \(R_p\) has measure
+\begin{equation}\label{eq:ord-shell-measure}
+ m_p(\text{shell }\boldsymbol l_p)
+       =\frac1{Z_p\,N_p(\boldsymbol l_p)}.
+\end{equation}
+Its conditional additive measure is invariant under the transitive
+unit action.  It is therefore the image of unit Haar probability.
+By Equation~\eqref{eq:ord-local-probability},
+\(P_p(\boldsymbol l_p)\) is exactly the fraction of this shell lying
+in \(\OO_p\).  The elements with a zero component have measure zero,
+and \(m_p(\OO_p)=q_p^{-1}\).  Summing
+Equation~\eqref{eq:ord-shell-measure} over the intersections with
+\(\OO_p\) gives
+\[
+ \sum_{\boldsymbol l_p}
+       \frac{P_p(\boldsymbol l_p)}{N_p(\boldsymbol l_p)}
+       =\frac{Z_p}{q_p}.
+\]
+Multiplication over \(p\mid q\) proves
+Equation~\eqref{eq:ord-shell-mass}.
+
+For the moment, choose \(x\) with normalized additive Haar probability
+on \(\OO_p\), and let \(v_{\mathfrak p}=v_{\mathfrak p}(x_{\mathfrak p})\).
+The projection \(H_{\mathfrak p}\) of \(\OO_p\) into
+\(R_{\mathfrak p}\) contains the embedded copy of \(\Z_p\), because
+\(1\in\OO_p\).  Its induced probability is normalized Haar on
+\(H_{\mathfrak p}\).  For an integer \(k\ge1\), the image of this
+copy of \(\Z_p\) modulo the ideal of valuation \(k\) has
+\(p^{\lceil k/e_{\mathfrak p}\rceil}\) elements.  Hence the image of
+\(H_{\mathfrak p}\) has at least that many elements, and
+\begin{equation}\label{eq:ord-valuation-tail}
+ \P_{\OO_p}\{v_{\mathfrak p}\ge k\}
+       \le p^{-\lceil k/e_{\mathfrak p}\rceil}.
+\end{equation}
+This uses only the size of the additive image of the projection; it
+does not require the projections to be independent.
+
+Set
+\[
+ j=\max_{\mathfrak p\mid p}
+              \left\lceil\frac{v_{\mathfrak p}}{e_{\mathfrak p}}\right\rceil.
+\]
+If \(j\ge a\ge1\), then some \(v_{\mathfrak p}\) is at least
+\(e_{\mathfrak p}(a-1)+1\).  Equation~\eqref{eq:ord-valuation-tail}
+and a union bound over at most four components give
+\(\P_{\OO_p}\{j\ge a\}\le4p^{-a}\).  On the other hand,
+\[
+ N_p(x)=p^{\sum_{\mathfrak p\mid p}f_{\mathfrak p}v_{\mathfrak p}}
+       \le p^{j\sum_{\mathfrak p\mid p}e_{\mathfrak p}f_{\mathfrak p}}
+       =p^{4j}.
+\]
+These assertions hold outside the null sets where a component vanishes.
+The tail formula for the expectation of \(p^{j/2}\) now yields
+\begin{align}
+ \E_{\OO_p}N_p(x)^{1/8}
+ &\le \E_{\OO_p}p^{j/2} \notag\\
+ &=1+\sum_{a\ge1}
+       (p^{a/2}-p^{(a-1)/2})\P_{\OO_p}\{j\ge a\}
+ \le 1+4p^{-1/2}.
+ \label{eq:ord-local-moment}
+\end{align}
+
+Applying Equation~\eqref{eq:ord-shell-measure} once more gives the
+exact factorization
+\[
+ \sum_{\boldsymbol l}
+       P(\boldsymbol l)d(\boldsymbol l)^{-1+1/8}
+ =\frac1q\prod_{p\mid q}
+       \left(Z_p\,\E_{\OO_p}N_p(x)^{1/8}\right).
+\]
+Each factor is at most
+\((1-p^{-1})^{-4}(1+4p^{-1/2})\).  For any fixed \(\eta>0\), this is
+at most \(p^\eta\) at all sufficiently large primes.  The product of
+the factors at the finitely many remaining primes is bounded by a
+constant depending only on \(\eta\).  Since
+\(\prod_{p\mid q}p\le q\), this proves
+Equation~\eqref{eq:ord-shell-moment}.  Finally, take \(\eta=1/128\)
+there and use \(q\le X\):
+\[
+ \sum_{d(\boldsymbol l)>X^{1/8}}\frac{P(\boldsymbol l)}{d(\boldsymbol l)}
+ \le X^{-1/64}
+       \sum_{\boldsymbol l}P(\boldsymbol l)d(\boldsymbol l)^{-1+1/8}
+ \le \frac Cq X^{-1/128}.
+\]
+This proves Equation~\eqref{eq:ord-shell-tail}.
+\end{proof}
+
+\subsection{Ideal counts relative to the residue}
+
+The shell identity supplies the factor \(q^{-1}\).  To use it in
+Equation~\eqref{eq:ord-unfolding}, the count of the remaining ideal
+part must retain the factor \(\kappa\).  Recall
+\[
+ a_K(n)=\#\{\mathfrak a\subset R:\mathfrak a\ne0,\ \Nm\mathfrak a=n\},
+ \qquad n\ge1,
+\]
+and let \(d_4\) be the fourfold divisor function.  Unique factorization
+of ideals makes \(a_K\) multiplicative.  At every rational prime,
+including a ramified one, its local generating series is
+\[
+ \sum_{j\ge0}a_K(p^j)T^j
+       =\prod_{\mathfrak p\mid p}(1-T^{f_{\mathfrak p}})^{-1}.
+\]
+Each factor is coefficientwise at most \((1-T)^{-1}\), and there are
+at most four factors.  Hence \(a_K(n)\le d_4(n)\) for every \(n\);
+in particular \(a_K(p)\le4\).
+
+\begin{lemma}[Residue and prime sums]\label{lem:ord-residue}
+There are absolute constants \(D_0>1\), \(h_0>0\), \(Y_0>1\), and
+\(C\ge1\) with the following properties for every primitive totally
+real quartic field \(K\).
+\begin{enumerate}
+\item If \(D\ge D_0\), then
+\begin{equation}\label{eq:ord-residue-comparison}
+ C^{-1}\kappa\le h\zeta_K(1+h)\le C\kappa
+       \qquad(0<h\le4/\log D).
+\end{equation}
+If \(D<D_0\), the same inequalities hold for \(0<h\le h_0\).
+\item One has
+\begin{equation}\label{eq:ord-residue-lower}
+ \kappa^{-1}\le C\log(3D).
+\end{equation}
+\item Whenever \(Y\ge\max\{Y_0,D^{1/4}\}\),
+\begin{equation}\label{eq:ord-prime-sum}
+ \exp\left(\sum_{p\le Y}\frac{a_K(p)}p\right)
+       \le C\kappa\log Y .
+\end{equation}
+\end{enumerate}
+The sum in Equation~\eqref{eq:ord-prime-sum} is over rational primes.
+\end{lemma}
+
+\begin{proof}
+We first treat large \(D\), and put \(L=\log D\).
+Stark's zero theorem \cite[Lemma~3]{Stark1974} gives at most one
+nontrivial zero of \(\zeta_K\) in
+\[
+ \Re s\ge1-\frac1{4L},\qquad |\Im s|\le\frac1{4L};
+\]
+any such zero is real and simple.  His
+Lemma~8 \cite[Lemma~8]{Stark1974}, valid also for nonnormal fields,
+states in degree four that a real zero
+\(\beta\ge1-(96L)^{-1}\) would force a quadratic subfield of \(K\).
+Primitivity excludes that subfield.  It follows that every nontrivial
+zero \(\rho\) satisfies
+\begin{equation}\label{eq:ord-zero-disk}
+ |1-\rho|\ge \frac{c_0}{L},\qquad c_0=\frac1{192}.
+\end{equation}
+Indeed, a zero in the smaller open disk would lie in Stark's rectangle,
+would be real, and would be in the forbidden real interval.
+
+For real \(s>1\), the logarithmic derivative of the completed zeta
+function gives
+\begin{equation}\label{eq:ord-zero-sum}
+\begin{aligned}
+ S_K(s):=\sum_\rho\Re\frac1{s-\rho}
+ &=\frac1s+\frac1{s-1}+\frac L2+
+       \frac{\zeta_K'}{\zeta_K}(s)+G(s),\\
+ G(s)&=2\left(\frac{\Gamma'(s/2)}{\Gamma(s/2)}-\log\pi\right);
+\end{aligned}
+\end{equation}
+see \cite[Equation~(9)]{Stark1974}.  The real zero sum converges and
+has nonnegative terms, since \(\Re\rho\le1\).  At \(s_0=1+L^{-1}\),
+the Euler product gives \(\zeta_K'/\zeta_K(s_0)\le0\).  The function
+\(G\) is bounded on \([1,2]\), so for sufficiently large \(D\),
+\[
+ 0\le S_K(s_0)\ll L.
+\]
+This estimate persists for all \(1<s\le1+4/L\).  To see this term by
+term, write \(\rho=\beta+i\gamma\).  Since \(\beta\le1\),
+\[
+ s-\beta\le4(s_0-\beta),\qquad
+ |s-\rho|\ge|1-\rho|.
+\]
+Equation~\eqref{eq:ord-zero-disk} consequently gives
+\[
+ |s_0-\rho|\le |s-\rho|+\frac3L
+              \le (1+3/c_0)|s-\rho|.
+\]
+Using \(\Re(s-\rho)^{-1}=(s-\beta)/|s-\rho|^2\), we obtain
+\[
+ \Re\frac1{s-\rho}
+ \le4(1+3/c_0)^2\Re\frac1{s_0-\rho}.
+\]
+Thus \(S_K(s)\ll L\) on the entire interval.
+
+Let \(\mathcal Z_K(s)=(s-1)\zeta_K(s)\).  Rearranging
+Equation~\eqref{eq:ord-zero-sum} shows that
+\[
+ \frac{\mathcal Z_K'}{\mathcal Z_K}(s)
+       =S_K(s)-\frac1s-\frac L2-G(s)=O(L)
+       \qquad(1<s\le1+4/L).
+\]
+The function \(\mathcal Z_K\) extends to \(s=1\) with value
+\(\kappa>0\).  Integrating its logarithmic derivative across an
+interval of length at most \(4/L\) proves
+Equation~\eqref{eq:ord-residue-comparison}, with an absolute constant.
+At \(s_0\), the inequality \(\zeta_K(s_0)\ge1\) then gives
+\(\kappa\gg L^{-1}\).
+
+There are only finitely many fields with \(D<D_0\), by Hermite's
+finiteness theorem \cite[Theorem~8.43]{MilneANT}.  For each of them,
+\(h\zeta_K(1+h)\) tends to the positive residue \(\kappa\) as
+\(h\downarrow0\).  Taking a common sufficiently small \(h_0\) over
+this finite set proves the remaining comparison, and their positive
+residues have a common positive lower bound.  Enlarging \(C\) proves
+Equation~\eqref{eq:ord-residue-lower} for every field.
+
+Choose \(Y_0\) sufficiently large that \(1/\log Y_0\le h_0\), and
+let \(Y\ge\max\{Y_0,D^{1/4}\}\).  For \(h=1/\log Y\), the comparison
+just proved applies: in the large \(D\) case \(h\le4/\log D\), and
+in the bounded \(D\) case \(h\le h_0\).  Positivity in the Euler
+logarithm gives
+\[
+ \sum_{p\le Y}a_K(p)p^{-1-h}\le\log\zeta_K(1+h).
+\]
+Moreover, \(a_K(p)\le4\), \(1-p^{-h}\le h\log p\), and the elementary
+Chebyshev bound followed by partial summation imply
+\[
+ \sum_{p\le Y}a_K(p)(p^{-1}-p^{-1-h})
+ \le4h\sum_{p\le Y}\frac{\log p}{p}
+ \ll h\log Y=1.
+\]
+It follows that
+\[
+ \sum_{p\le Y}\frac{a_K(p)}p
+ \le \log\zeta_K(1+h)+O(1)
+ \le \log(\kappa\log Y)+O(1),
+\]
+which proves Equation~\eqref{eq:ord-prime-sum}.  There is no upper
+restriction on \(Y\).  This observation is what permits \(D\) to stay
+bounded while the order index, and hence \(X\), grows.
+\end{proof}
+
+We also need an estimate for ideals in an interval.  The following
+uniform form of Shiu's bound \cite{Shiu1980} is the one recorded in
+\cite[Equation~(4.2)]{PrimeDegreePackets}.
+
+\begin{lemma}\label{lem:ord-shiu}
+There are absolute constants \(C,u_0\) such that, for every
+nonnegative multiplicative function \(b\) with \(b(n)\le d_4(n)\),
+every real \(u\ge u_0\), and every \(u^{1/2}\le v\le u\),
+\begin{equation}\label{eq:ord-shiu}
+ \sum_{u-v<n\le u}b(n)
+ \le C\frac v{\log u}
+       \exp\left(\sum_{p\le u}\frac{b(p)}p\right).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Apply \cite[Theorem~1.1]{Pollack2020} with no excluded residue classes
+and with its parameter \(\beta=1/4\).  Its two growth conditions hold
+with bounds independent of \(b\):
+\[
+ b(p^j)\le d_4(p^j)=\binom{j+3}{3}\le4^j,\qquad
+ b(n)\le d_4(n)\ll_\epsilon n^\epsilon
+ \quad(\epsilon>0).
+\]
+The theorem's constant and threshold depend only on these growth
+bounds and on the fixed parameter.  Its strict length condition
+\(u^\beta<v\) is satisfied even at \(v=u^{1/2}\) once \(u>1\).
+This gives Equation~\eqref{eq:ord-shiu}, including both endpoints.
+\end{proof}
+
+\begin{proposition}[Weighted ideal intervals]\label{prop:ord-weighted-intervals}
+There is an absolute constant \(C\) such that, for each fixed
+\(A\ge1\) and \(c>0\), there is a threshold \(X_0(A,c)\) with the
+following property.  For every \(K,\OO,\sigma\) under consideration
+with \(X\ge X_0(A,c)\), and all real \(x,y\) satisfying
+\[
+ X^{3/4}\le x\le AX,\qquad cx\le y\le x,
+\]
+one has
+\begin{equation}\label{eq:ord-weighted-interval}
+ \sum_{\substack{\mathfrak a\subset R,\ \mathfrak a\ne0\\
+                  x-y<\Nm\mathfrak a\le x}}P(\mathfrak a)
+       \le C\,\frac{\kappa y}{q}.
+\end{equation}
+The constant \(C\) is independent of \(A,c\); the threshold may
+depend on them.
+\end{proposition}
+
+\begin{proof}
+The range is empty for \(c>1\), so suppose \(0<c\le1\).
+We adapt the separation of the local norm from the remaining ideal
+in \cite[Proposition~4.2]{PrimeDegreePackets}.  Put
+\[
+ b(n)=a_K(n)\mathbf 1_{(n,q)=1}.
+\]
+This is nonnegative and multiplicative, and it is at most \(d_4\).
+Every integral ideal splits uniquely into its part above the primes
+dividing \(q\), described by one tuple \(\boldsymbol l\), and a part
+coprime to \(qR\).  The left side of
+Equation~\eqref{eq:ord-weighted-interval} is therefore
+\begin{equation}\label{eq:ord-ideal-split}
+ \sum_{\boldsymbol l}P(\boldsymbol l)
+       \sum_{(x-y)/d(\boldsymbol l)<n\le x/d(\boldsymbol l)}b(n).
+\end{equation}
+
+First take \(d=d(\boldsymbol l)\le X^{1/8}\), and put \(u=x/d\),
+\(v=y/d\).  Then
+\[
+ u\ge X^{5/8}\ge D^{1/4},\qquad v\ge cu.
+\]
+The second inequality implies \(v\ge u^{1/2}\) when \(X\) is large
+in terms of \(c\).  The first inequality puts \(u\) in the range of
+Equation~\eqref{eq:ord-prime-sum}, once \(X\) is above an absolute
+threshold.  To remove the primes dividing \(q\) from that Euler bound,
+observe that
+\begin{equation}\label{eq:ord-euler-removal}
+ \log Z_q=\sum_{p\mid q}\frac{a_K(p)}p+O(1),\qquad
+ \sum_{\substack{p\mid q\\p>u}}\frac{a_K(p)}p
+       \ll\frac{\log(2q)}u\ll1 .
+\end{equation}
+For the first identity, expand
+\(\log Z_p=\sum_{\mathfrak p\mid p}\sum_{j\ge1}p^{-jf_{\mathfrak p}}/j\).
+The terms of order \(p^{-1}\) count exactly the primes of residue
+degree one, and the remaining terms are \(O(p^{-2})\), uniformly
+in the splitting type.  Their sum over rational primes is bounded.
+For the second estimate, use \(a_K(p)\le4\), at most
+\(\log(2q)/\log2\) primes dividing \(q\), and
+\(u\ge X^{5/8}\), \(q\le X\).
+Consequently Equations~\eqref{eq:ord-prime-sum} and
+\eqref{eq:ord-euler-removal} give
+\[
+ \exp\left(\sum_{p\le u}\frac{b(p)}p\right)
+       \ll \frac{\kappa\log u}{Z_q}.
+\]
+Lemma~\ref{lem:ord-shiu} and
+Equation~\eqref{eq:ord-shell-mass} now bound the contribution of
+these tuples to Equation~\eqref{eq:ord-ideal-split} by
+\[
+ C\frac{\kappa y}{Z_q}
+       \sum_{d(\boldsymbol l)\le X^{1/8}}
+             \frac{P(\boldsymbol l)}{d(\boldsymbol l)}
+ \le C\frac{\kappa y}{q}.
+\]
+
+It remains to control tuples with \(d>X^{1/8}\).  Those with \(d>x\)
+contribute nothing.  For \(z\ge1\), summing the first three factors
+of the divisor convolution gives
+\[
+ \sum_{n\le z}d_4(n)
+ =\sum_{a_1a_2a_3a_4\le z}1
+ \le z\left(\sum_{a\le z}\frac1a\right)^3
+ \le z(1+\log z)^3 .
+\]
+Use this with \(z=x/d\), and then
+Equation~\eqref{eq:ord-shell-tail}.  The remaining contribution is
+at most
+\[
+ Cx(1+\log x)^3
+       \sum_{d(\boldsymbol l)>X^{1/8}}
+             \frac{P(\boldsymbol l)}{d(\boldsymbol l)}
+ \le C\frac{x}{q}X^{-1/128}(1+\log x)^3 .
+\]
+For fixed \(A\), once \(X\ge A\) the condition \(x\le AX\) gives
+\(\log x\le2\log X\).  By
+Equation~\eqref{eq:ord-residue-lower} and \(D\le X^2\), the ratio of
+this last bound to \(\kappa x/q\) is at most
+\[
+ C X^{-1/128}(\log(3X))^4=o(1).
+\]
+This bound is independent of the field and the order.  Increase the
+threshold in terms of \(A,c\) until the ratio is at most \(c\).
+The tail is then at most \(\kappa cx/q\le\kappa y/q\).  The constant
+in the resulting bound has not changed with \(A\) or \(c\), proving
+the proposition.
+\end{proof}
+
+\subsection{Short vectors and small neighborhoods}
+
+We now apply the weighted interval bound to the two vector counts
+announced at the start of the section.  If \(f_r=\mathbf 1_{[-r,r]^4}\), direct integration
+in Equation~\eqref{eq:ord-fiber} gives
+\begin{equation}\label{eq:ord-cube-fiber}
+ \mathcal V_{f_r}(z)=\frac{16}{6}
+                       \bigl(\log_+(r^4/z)\bigr)^3 .
+\end{equation}
+Indeed, in a fixed sign component put \(s_i=\log r-u_i\) for
+\(1\le i\le3\).  The cube conditions become \(s_i\ge0\) and
+\(\sum_{i=1}^3s_i\le\log(r^4/z)\), a simplex of the displayed volume
+divided by sixteen.
+
+\begin{proposition}[Tightness]\label{prop:ord-tightness}
+There is an absolute constant \(C\) such that, for every fixed
+\(0<r<1\),
+\begin{equation}\label{eq:ord-cusp}
+ \limsup_{X\longrightarrow\infty}
+ \mu_*\{\Lambda:\text{some }0\ne v\in\Lambda
+                         \text{ satisfies }\|v\|_\infty<r\}
+       \le Cr^4 .
+\end{equation}
+The limit superior ranges over all primitive totally real quartic
+fields, all their orders, and all orderings of their real embeddings.
+Consequently, for every sequence of these packet measures
+\(\mu_i=\mu_{\OO_i,\sigma_i}\) with \(X_i\to\infty\) and every
+\(\epsilon>0\), there is a compact \(\mathcal K_\epsilon\subset X_4\) with
+\(\mu_i(\mathcal K_\epsilon)\ge1-\epsilon\) for all sufficiently large \(i\).
+Such a sequence is tight, and every weak subsequential limit is an
+\(A_4\)-invariant probability on \(X_4\).
+\end{proposition}
+
+\begin{proof}
+The count \(E_{f_r}\) dominates the indicator in
+Equation~\eqref{eq:ord-cusp}.  Put \(T=r^4X\) and \(T_0=X^{3/4}\).
+By Equations~\eqref{eq:ord-unfolding} and \eqref{eq:ord-cube-fiber},
+it suffices to bound
+\begin{equation}\label{eq:ord-cube-sum}
+ \frac1{\sqrt D\,\kappa}
+ \sum_{\Nm\mathfrak a\le T}
+       P(\mathfrak a)
+       \bigl(\log(T/\Nm\mathfrak a)\bigr)^3 .
+\end{equation}
+The sum is empty when \(T<1\).  Otherwise its logarithmic factor is
+at most \((\log X)^3\), since \(\Nm\mathfrak a\ge1\) and \(r<1\).
+Apply Proposition~\ref{prop:ord-weighted-intervals} at
+\(x=y=T_0\), with \(A=c=1\).  The part with
+\(\Nm\mathfrak a\le\min\{T,T_0\}\) is at most
+\[
+ C\frac{\kappa T_0}{q}(\log X)^3.
+\]
+After division by \(\sqrt D\,\kappa=X\kappa/q\), this contributes
+\(O(X^{-1/4}(\log X)^3)=o(1)\).
+
+If \(T>T_0\), put
+\[
+ A_{T_0}(t)=\sum_{T_0<\Nm\mathfrak a\le t}P(\mathfrak a)
+ \qquad(T_0\le t\le T).
+\]
+The same interval estimate at \(x=y=t\) gives
+\(A_{T_0}(t)\le C\kappa t/q\), since \(T\le X\).  Writing each
+logarithmic cube as an integral and using Tonelli yields
+\begin{align*}
+ \sum_{T_0<\Nm\mathfrak a\le T}
+       P(\mathfrak a)\bigl(\log(T/\Nm\mathfrak a)\bigr)^3
+ &=3\int_{T_0}^T A_{T_0}(t)
+                   \bigl(\log(T/t)\bigr)^2\frac{dt}{t}\\
+ &\le C\frac{\kappa}{q}
+              \int_0^T\bigl(\log(T/t)\bigr)^2\,dt
+ \le C\frac{\kappa T}{q}.
+\end{align*}
+The last integral is finite after the substitution \(t=Ts\).
+This part of Equation~\eqref{eq:ord-cube-sum} is therefore \(O(r^4)\).
+If \(T\le T_0\), it is absent.  The constants used at \(A=c=1\) are
+absolute, so the resulting estimate proves
+Equation~\eqref{eq:ord-cusp}.  This is the compactness argument of
+\cite[Proposition~5.1]{PrimeDegreePackets}, now using the quartic
+weighted interval bound.
+
+Mahler's compactness criterion \cite[Section~3, Theorem~2]{Mahler1946}
+says that
+\[
+ \mathcal K_r=\{\Lambda\in X_4:\|v\|_\infty\ge r
+                                      \text{ for all }0\ne v\in\Lambda\}
+\]
+is compact.  Choose \(r\) so that \(Cr^4<\epsilon/2\).
+Equation~\eqref{eq:ord-cusp} then gives
+\(\mu_i(\mathcal K_r)\ge1-\epsilon\) for all sufficiently large \(i\).
+Each of the finitely many preceding packet measures is supported on a
+finite union of compact orbits, by Proposition~\ref{prop:pkt-model}.
+Enlarging the compact set to include those supports proves tightness
+of the entire sequence.  The space \(X_4\) is locally compact,
+second countable, and metrizable; the weak compactness theorem for
+tight probabilities therefore gives probability subsequential limits.
+For \(a\in A_4\) and \(f\in C_c(X_4)\), the equality
+\[
+ \int f(xa)\,d\mu_i(x)=\int f(x)\,d\mu_i(x)
+\]
+passes to a weak limit, since both test functions are continuous with
+compact support.  This proves \(A_4\)-invariance.
+\end{proof}
+
+\begin{lemma}[Boxes away from the coordinate hyperplanes]\label{lem:ord-boxes}
+Let \(V_0\subset(\R^\times)^4\) be compact, and let \(b>0\).
+There are constants \(C,r_0>0\), depending only on \(V_0,b\), such
+that for every fixed \(0<r<r_0\) there is a threshold
+\(X_0(V_0,b,r)\) for which
+\begin{equation}\label{eq:ord-vector-box}
+ \int_{X_4}E_{\mathbf 1_{v+[-br,br]^4}}\,d\mu_*
+       \le Cr^4\qquad(v\in V_0,\ X\ge X_0(V_0,b,r)).
+\end{equation}
+The threshold and constants are uniform over \(K,\OO,\sigma\) and
+over the centers \(v\in V_0\).
+\end{lemma}
+
+\begin{proof}
+The assertion is immediate if \(V_0\) is empty.  Otherwise choose
+\(0<\epsilon\le M\) such that
+\(\epsilon\le|v_i|\le M\) for every \(v\in V_0\) and \(i\).
+For \(r_0<\epsilon/(2b)\), each box in the statement lies in one
+coordinate sign component.  Its first three logarithmic widths satisfy
+\[
+ \log\frac{|v_i|+br}{|v_i|-br}\ll_{V_0,b}r .
+\]
+The product map \(x\mapsto\prod_i|x_i|\) has uniformly bounded first
+derivatives on these boxes.  If \(t_v=\prod_i|v_i|\), it follows
+from Equation~\eqref{eq:ord-fiber} that, for one \(C_1\ge1\),
+\begin{equation}\label{eq:ord-box-fiber}
+ \mathcal V_{\mathbf 1_{v+[-br,br]^4}}(z)
+       \le C_1r^3\,\mathbf 1_{[t_v-C_1r,t_v+C_1r]}(z)
+       \qquad(v\in V_0,\ 0<r<r_0).
+\end{equation}
+Let \(t_->0\) and \(t_+<\infty\) be the minimum and maximum of \(t_v\)
+on \(V_0\), and shrink \(r_0\) so that \(6C_1r_0<t_-\).
+For a fixed \(r\) in this range set
+\[
+ x=(t_v+2C_1r)X,\qquad y=4C_1rX.
+\]
+The possible ideal norms in Equation~\eqref{eq:ord-box-fiber} lie
+strictly inside \((x-y,x)\).  The enlargement avoids any issue with
+integer endpoints.  There is a fixed \(A\ge1\), depending only on
+\(V_0,b\), such that \(x\le AX\) for all centers and radii under
+consideration.  For sufficiently large \(X\), uniformly in the
+centers, \(x\ge X^{3/4}\).  Also \(y\le x\) by the choice of \(r_0\),
+and \(y\ge cx\) with \(c=4C_1r/A>0\), after decreasing \(r_0\) if
+necessary so that \(c\le1\).
+
+Proposition~\ref{prop:ord-weighted-intervals} applies for every fixed
+\(r\), with this \(c\) and the fixed \(A\).  Its constant is independent
+of \(c\), so the sum of the relevant ideal weights is at most
+\(C\kappa rX/q\), with a constant independent of \(r\).  Combining
+this with Equations~\eqref{eq:ord-unfolding} and
+\eqref{eq:ord-box-fiber}, and using \(\sqrt D=X/q\), proves
+Equation~\eqref{eq:ord-vector-box}.  Only the threshold for \(X\)
+depends on the fixed radius.
+\end{proof}
+
+For \(r>0\), define the open identity neighborhood
+\begin{equation}\label{eq:ord-ball-neighborhood}
+ B(r)=\{h\in\mathrm{SL}_4(\R):
+              \max_{i,j}|h_{ij}-\delta_{ij}|<r\}.
+\end{equation}
+
+\begin{proposition}[Local bounds in probability limits]\label{prop:ord-balls}
+Let \(\mu\) be a weak probability limit of packet measures along
+\(X_i\to\infty\).  For every compact \(\Omega\subset X_4\), there
+are constants \(C_\Omega,r_\Omega>0\) such that
+\begin{equation}\label{eq:ord-balls}
+ \mu(xB(r))\le C_\Omega r^4
+       \qquad(x\in\Omega,\ 0<r<r_\Omega).
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+The assertion is immediate if \(\Omega\) is empty, so assume otherwise.
+We use the passage from vector boxes to ordinary neighborhoods in
+\cite[Proposition~5.2]{PrimeDegreePackets}.  First, every full lattice
+has a vector with no zero coordinate.  If \(\Lambda=\Z^4g\), each
+coordinate of \((1,N,N^2,N^3)g\) is a nonzero polynomial in \(N\):
+its coefficients are a column of the invertible matrix \(g\).
+An integer \(N\) outside their finitely many roots gives the required
+vector.
+
+For each \(x_0\in\Omega\), choose a continuous local lift \(g(x)\) of
+nearby lattices to \(\mathrm{SL}_4(\R)\), and an integer row \(z\)
+for which \(zg(x_0)\) has no zero coordinate.  On a sufficiently
+small neighborhood with compact closure in this chart, \(zg(x)\)
+stays bounded and its coordinates stay bounded away from zero.
+A finite collection of these neighborhoods covers \(\Omega\).
+Consequently we may select, for every \(x\in\Omega\), a vector
+\(v_x\in\Lambda_x\) so that all selected vectors lie in one compact
+set \(V_0\subset(\R^\times)^4\).  The selection is used pointwise and
+need not be measurable.
+
+Choose \(M\) with \(|(v_x)_i|\le M\) for all selected vectors, and
+fix \(b>4M\).  If \(y=xh\) with \(h\in B(r)\), then
+\(v_xh\in\Lambda_y\), and
+\[
+ |(v_xh-v_x)_j|
+ \le \sum_i |(v_x)_i|\,|h_{ij}-\delta_{ij}|<br.
+\]
+Thus every lattice in \(xB(r)\) has a nonzero vector in
+\(v_x+[-br,br]^4\).  Lemma~\ref{lem:ord-boxes} gives, for each fixed
+small \(r>0\) and all sufficiently large \(i\),
+\[
+ \mu_i(xB(r))
+ \le \int E_{\mathbf 1_{v_x+[-br,br]^4}}\,d\mu_i
+ \le C_\Omega r^4
+ \qquad(x\in\Omega).
+\]
+The constants are common to all \(x\) and \(r\) in the stated ranges;
+the index after which this estimate holds may depend on \(r\).
+The set \(xB(r)\) is open in \(X_4\).  For each fixed \(x,r\),
+Portmanteau's inequality therefore yields
+\[
+ \mu(xB(r))\le\liminf_{i\to\infty}\mu_i(xB(r))
+       \le C_\Omega r^4.
+\]
+This proves the bound for every \(r\) with the same constant.  It uses
+no interchange between a shrinking radius and the packet limit.
+\end{proof}

preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/packets.tex

@@ -0,0 +1,298 @@
+\section{Packets from arbitrary orders}\label{sec:packets}
+
+Fix a primitive totally real quartic field \(K\), an order \(\OO\subset K\),
+and an ordering \(\sigma=(\sigma_1,\ldots,\sigma_4)\) of its real embeddings,
+as in Theorem~\ref{thm:main}.  We describe the packet using finitely many
+local lattice choices above each ideal class of the maximal order.  This
+description lets us average a containment condition at each finite prime
+before performing an integral at the real place.  The construction is
+adapted from the packet model in
+\cite[Section~2, Lemma~2.1]{PrimeDegreePackets}; we give the argument with
+all sixteen sign patterns in degree four.
+
+Throughout the paper put
+\begin{equation}\label{eq:pkt-scale}
+ R=\OO_K,\qquad D=|\Disc(K)|,\qquad q=[R:\OO],\qquad
+ X=q\sqrt D,\qquad \kappa=\Res_{s=1}\zeta_K(s).
+\end{equation}
+In particular, \(|\Disc(\OO)|=q^2D=X^2\).  Absolute ideal norms are
+extended multiplicatively to fractional \(R\)-ideals.  Thus
+\begin{equation}\label{eq:pkt-covolume}
+ \covol(\sigma(I))=\sqrt D\,\Nm I
+\end{equation}
+for every nonzero fractional \(R\)-ideal \(I\): this is the defining
+discriminant identity for \(I=R\), follows by index for integral \(I\),
+and then follows for fractional \(I\) after multiplication by an integer.
+For a full \(\Z\)-lattice \(J\subset K\), write
+\(J_p=J\otimes_\Z\Z_p\), and put
+\[
+ K_p=K\otimes_\Q\Q_p,\qquad R_p=R\otimes_\Z\Z_p,\qquad
+ \OO_p=\OO\otimes_\Z\Z_p .
+\]
+Here \(K_p\) is a product of finite extensions of \(\Q_p\), and \(R_p\)
+is the product of their integer rings.  We use the usual arithmetic of
+fractional ideals in the Dedekind domain \(R\); see
+\cite[Theorems~3.7, 3.20, and~3.29]{MilneANT}.
+
+\subsection{The finite local choices}
+
+Choose one fractional \(R\)-ideal \(I\) in each ordinary ideal class of
+\(R\).  Write \(h_K\) for the number of these classes.  At every prime
+choose a generator
+\[
+ I_p=\xi_p R_p,\qquad \xi_p\in K_p^\times.
+\]
+Such a generator exists because each component of \(R_p\) is a discrete
+valuation ring.  Let \(\mathscr L_{\OO}(I)\) be the set of full lattices
+\(L\subset I\) whose completions have the form
+\begin{equation}\label{eq:pkt-local-choices}
+ L_p=\xi_p t_p\OO_p,\qquad t_p\in R_p^\times
+ \quad\text{for every }p.
+\end{equation}
+Replacing \(\xi_p\) by another generator merely changes \(t_p\) by a
+unit, so this set does not depend on the chosen generators.
+
+\begin{lemma}\label{lem:pkt-local-labels}
+The set \(\mathscr L_{\OO}(I)\) is finite, and its cardinality is
+\begin{equation}\label{eq:pkt-label-count}
+ N_{\OO}=\prod_{p\mid q}[R_p^\times:\OO_p^\times],
+\end{equation}
+independently of \(I\).  Its members satisfy
+\[
+ [I:L]=q,\qquad RL=I,
+\]
+and are proper invertible fractional \(\OO\)-ideals.  Every ordinary
+class of proper invertible fractional \(\OO\)-ideals is represented,
+up to multiplication by \(K^\times\), by a member of one of the sets
+\(\mathscr L_{\OO}(I)\).  Uniform probability on
+\(\mathscr L_{\OO}(I)\) is induced by independent normalized Haar
+probabilities for \(t_p\in R_p^\times\).
+\end{lemma}
+
+\begin{proof}
+The additive group \(R/\OO\) has order \(q\), so \(qR\subset\OO\).
+For every \(p\) and every unit \(t_p\), it follows that
+\[
+ qI_p=\xi_p t_p qR_p\subset \xi_p t_p\OO_p\subset I_p.
+\]
+At \(p\nmid q\) the middle lattice is \(I_p\).  At the remaining
+primes its image is a submodule of the finite module \(I_p/qI_p\).
+The primary decomposition of \(I/qI\) identifies its \(p\)-primary
+part with \(I_p/qI_p\).  Thus a choice of these local submodules
+patches to a unique subgroup of \(I/qI\), and its inverse image in
+\(I\) is the unique lattice with the prescribed completions.
+
+The stabilizer of \(\OO_p\) under multiplication by \(R_p^\times\)
+is exactly \(\OO_p^\times\).  Indeed, \(t_p\OO_p=\OO_p\) implies
+that both \(t_p\) and \(t_p^{-1}\) belong to \(\OO_p\), by applying
+the equality and its inverse to \(1\).  The converse is immediate.
+The local choices therefore form the finite homogeneous space
+\(R_p^\times/\OO_p^\times\).  Its invariant probability is uniform
+and is the image of normalized unit Haar measure.  Patching over the
+finitely many primes dividing \(q\) proves
+Equation~\eqref{eq:pkt-label-count} and the assertion about
+probabilities.  The index of the lattice in
+Equation~\eqref{eq:pkt-local-choices} is
+\([R_p:\OO_p]\), and its \(R_p\)-span is \(I_p\) because \(1\in\OO_p\).
+Taking products of the indices and comparing all completions gives
+\([I:L]=q\) and \(RL=I\).
+
+We record why the local condition describes exactly invertible ideals
+of the order.  A full fractional \(\OO\)-ideal \(J\) is invertible
+if and only if
+\begin{equation}\label{eq:pkt-local-principality}
+ J_p=\eta_p\OO_p\quad\text{for some }\eta_p\in K_p^\times
+ \quad\text{at every }p.
+\end{equation}
+For the forward implication, complete an equality
+\(JJ^{-1}=\OO\).  The resulting equality over \(\OO_p\) expresses
+\(1\) as a finite sum \(\sum_i a_i b_i\), with \(a_i\in J_p\) and
+\(b_i\in J_p^{-1}\).  At each maximal ideal \(\mathfrak m\) of the
+semilocal ring \(\OO_p\), some \(a_i b_i\) is a unit after localization,
+and that \(a_i\) generates \((J_p)_{\mathfrak m}\).  To obtain one
+generator for all the maximal ideals, choose such an \(a_{\mathfrak m}\)
+for each \(\mathfrak m\), and use the Chinese remainder theorem
+\cite[Theorem~1.14]{MilneANT} to
+choose \(c_{\mathfrak m}\in\OO_p\) congruent to \(1\) at
+\(\mathfrak m\) and to \(0\) at every other maximal ideal.  Then
+\(\eta_p=\sum_{\mathfrak m}c_{\mathfrak m}a_{\mathfrak m}\) generates
+\((J_p)_{\mathfrak m}\) modulo \(\mathfrak m\) at each \(\mathfrak m\);
+Nakayama's lemma \cite[Lemma~1.9]{MilneANT} shows that it generates each localization.  Hence it
+generates \(J_p\).  Since \(J_p\) spans \(K_p\) over \(\Q_p\),
+\(\eta_p\) has no zero component and lies in \(K_p^\times\).
+
+Conversely, define
+\[
+ J^{-1}=\{a\in K:aJ\subset\OO\}.
+\]
+This definition commutes with completion: after choosing \(\Z\)-bases
+of \(J\) and \(\OO\), the containment is a finite collection of
+integrality conditions on the coordinates of multiplication by \(a\),
+and the same conditions over \(\Z_p\) describe the completed lattice.
+Under Equation~\eqref{eq:pkt-local-principality} we therefore have
+\((J^{-1})_p=\eta_p^{-1}\OO_p\).  Consequently
+\((JJ^{-1})_p=\OO_p\) for every \(p\), which implies \(JJ^{-1}=\OO\).
+This proves the equivalence.  It also proves properness: if \(aJ\subset J\)
+for an invertible \(J\), multiplication by \(J^{-1}\) gives
+\(a\OO\subset\OO\), and hence \(a\in\OO\).  Thus its multiplier order
+is \(\OO\).
+
+The lattices in Equation~\eqref{eq:pkt-local-choices} are stable under
+\(\OO_p\) at every prime, so comparison of completions shows that
+\(\OO L\subset L\).  They are therefore fractional \(\OO\)-ideals
+satisfying Equation~\eqref{eq:pkt-local-principality}.  In the other
+direction, let \(J\) be an invertible fractional \(\OO\)-ideal.  Its \(R\)-span
+\(RJ\) is a fractional \(R\)-ideal.  Multiplying \(J\) by an element
+of \(K^\times\), we may arrange \(RJ=I\) for the chosen representative
+of this maximal-order class.  If \(J_p=\eta_p\OO_p\), then
+\(\eta_p R_p=I_p=\xi_pR_p\), so \(\eta_p=\xi_p t_p\) for a unit
+\(t_p\in R_p^\times\).  Thus the resulting \(J\) belongs to
+\(\mathscr L_{\OO}(I)\), as required.
+\end{proof}
+
+\subsection{Real parameters, identifications, and volume}
+
+Let \(U=R^\times\), and let \(U^+\) be its totally positive subgroup.
+Put
+\[
+ T_\infty=\{v=(v_1,\ldots,v_4)\in(\R^\times)^4:
+                         \textstyle\prod_i|v_i|=1\}.
+\]
+We give \(T_\infty\) counting measure on its sixteen sign components
+and measure \(du_1\,du_2\,du_3\) on the logarithms
+\(u_i=\log|v_i|\), with \(u_4=-u_1-u_2-u_3\).  Denote this measure
+by \(dv\).  The unit theorem
+\cite[Theorems~5.1 and~5.9]{MilneANT} says that
+\[
+ \ell(U)=\{(\log|\sigma_i(\epsilon)|)_{i=1}^4:\epsilon\in U\}
+\]
+is a full lattice in the hyperplane \(\sum_i u_i=0\).  Its
+covolume for the deleted-coordinate measure is the ordinary regulator
+\(R_K\).  Its kernel on \(U\) consists of \(1\) and \(-1\), since
+these are the roots of unity in a totally real field.
+
+Let \(\mathscr F_U\) be a measurable fundamental domain for the embedded
+group \(\sigma(U)\) in \(T_\infty\).  Reducing logarithms modulo
+\(\ell(U)\) leaves sixteen sign choices, on which the remaining kernel
+\(\{1,-1\}\) identifies pairs of opposite signs.  It follows that
+\(\vol(\mathscr F_U)=8R_K\).  The analytic class-number formula in
+signature \((4,0)\), with two roots of unity, gives
+\begin{equation}\label{eq:pkt-unit-volume}
+ h_K\vol(\mathscr F_U)=8h_KR_K=\sqrt D\,\kappa;
+\end{equation}
+see \cite[Chapter~V, Theorem~2.4(a)]{MilneCFT}.
+
+For \(L\in\mathscr L_{\OO}(I)\), set
+\[
+ c_I=(X\Nm I)^{-1/4}.
+\]
+Equations~\eqref{eq:pkt-covolume} and \([I:L]=q\) show that
+\(c_I\sigma(L)v\) has covolume one for every \(v\in T_\infty\).
+This notation denotes the corresponding row lattice, and thus a point
+of \(X_4\), after choosing a basis of determinant one.  This convention
+also applies when the diagonal \(v\) has negative determinant.
+
+\begin{proposition}[Packet model and volume]\label{prop:pkt-model}
+The maps
+\[
+ (L,v)\longmapsto c_I\sigma(L)v
+ \qquad
+ (L\in\mathscr L_{\OO}(I),\ v\in T_\infty)
+\]
+give a bijection from the disjoint union over the chosen \(I\) of the
+quotients by
+\begin{equation}\label{eq:pkt-unit-action}
+ \epsilon:(L,v)\longmapsto
+       (\epsilon L,\sigma(\epsilon)^{-1}v),\qquad \epsilon\in U,
+\end{equation}
+onto \(\bigcup_{\mathcal Y\in\mathcal P_{\OO,\sigma}}\mathcal Y\subset X_4\).
+Counting measure on the \(L\)'s times \(dv\) descends to the sum of
+the Haar measures of those orbits.  In particular,
+\begin{equation}\label{eq:pkt-total-volume}
+ \sum_{\mathcal Y\in\mathcal P_{\OO,\sigma}}\vol(\mathcal Y)
+       =N_{\OO}\sqrt D\,\kappa .
+\end{equation}
+Writing \(\mu_*=\mu_{\OO,\sigma}\), for every nonnegative Borel
+function \(\Phi\) on \(X_4\) one has
+\begin{equation}\label{eq:pkt-integration}
+ \int_{X_4}\Phi\,d\mu_*
+ =\frac{1}{h_KN_{\OO}\vol(\mathscr F_U)}
+   \sum_I\sum_{L\in\mathscr L_{\OO}(I)}
+       \int_{\mathscr F_U}\Phi(c_I\sigma(L)v)\,dv .
+\end{equation}
+The identity permits infinite values on both sides.
+\end{proposition}
+
+\begin{proof}
+The action in Equation~\eqref{eq:pkt-unit-action} preserves the
+parameter set, since \(\epsilon I=I\) for \(\epsilon\in U\), and
+preserves its image lattice.  Lemma~\ref{lem:pkt-local-labels} shows
+that every ordinary invertible \(\OO\)-ideal class occurs.  If two
+ideals differ by multiplication by \(\alpha\in K^\times\), their
+normalized embedded lattices differ by the diagonal with coordinates
+\(\sigma_i(\alpha)/|\Nm_{K/\Q}(\alpha)|^{1/4}\), an element of
+\(T_\infty\).  Finally \(T_\infty\) is the union of all sixteen
+sign translates of \(A_4\).  These facts prove surjectivity.
+
+Suppose two displayed normalized lattices coincide.  After absorbing
+the ratio of their positive scalars, this equality gives a diagonal
+map carrying \(\sigma(L)\) onto \(\sigma(L')\).  Choose \(0\ne b\in L\)
+and let \(b'\in L'\) be its image.  None of the four coordinates of
+\(\sigma(b)\) is zero.  The diagonal entries must therefore be
+\(\sigma_i(b')/\sigma_i(b)=\sigma_i(b'/b)\).  The map is multiplication
+by \(\alpha=b'/b\), and \(L'=\alpha L\).  Taking \(R\)-spans gives
+\(I'=\alpha I\).  The chosen representatives then force \(I'=I\)
+and \(\alpha I=I\), so \(\alpha\in U\).  The scalar normalizations
+are now equal, and the two parameters differ by exactly
+Equation~\eqref{eq:pkt-unit-action}.  This proves injectivity, including
+all identifications involving signs.  Equality of covolume-one row
+lattices is the same as equality in \(X_4\): any two positively oriented
+bases of the same lattice differ by an element of \(\mathrm{SL}_4(\Z)\).
+
+It remains to check that this identification gives the stipulated
+orbit weights.  Write
+\(\OO^{\times,+}=\OO^\times\cap U^+\).  The \(U\)-stabilizer of a
+label \(L\) is \(\OO^\times\), by its multiplier order.  The stabilizer
+of \(L\) together with a sign pattern is therefore
+\(\OO^{\times,+}\).  This subgroup has finite index in \(U\).
+For \(q=1\) this follows directly from \(\OO=R\).  For \(q>1\),
+the kernel of \(U\longrightarrow(R/qR)^\times\) lies in
+\(\OO^\times\), since both a unit congruent to \(1\) modulo \(qR\)
+and its inverse belong to \(\OO\), and imposing positive signs costs
+at most a further factor of sixteen.  The same diagonal-identification
+argument shows that all periods of the corresponding \(A_4\)-orbit
+are precisely \(\ell(\OO^{\times,+})\).  They form a full lattice,
+so the orbit is compact.
+
+Combining a label with a sign produces \(16N_{\OO}\) finite choices
+above each \(I\).  Each \(U\)-orbit of these choices has
+\([U:\OO^{\times,+}]\) elements and parametrizes one \(A_4\)-orbit.
+On that orbit the quotient of logarithmic Lebesgue measure is its
+Haar measure, with volume
+\[
+ \covol\bigl(\ell(\OO^{\times,+})\bigr)
+ =[\ell(U):\ell(\OO^{\times,+})]R_K
+ =\frac{[U:\OO^{\times,+}]}{2}R_K .
+\]
+The last equality uses \(\ker(\ell|_U)=\{1,-1\}\) and
+\(-1\notin\OO^{\times,+}\).  Consequently the number of distinct
+orbits times their common volume is
+\[
+ \frac{16h_KN_{\OO}}{[U:\OO^{\times,+}]}
+ \cdot \frac{[U:\OO^{\times,+}]}2 R_K
+ =8h_KN_{\OO}R_K.
+\]
+This proves both the assertion about the descended measure and
+Equation~\eqref{eq:pkt-total-volume}, by
+Equation~\eqref{eq:pkt-unit-volume}.
+
+One can also take \(\mathscr L_{\OO}(I)\times\mathscr F_U\) as a
+fundamental domain for the diagonal action in
+Equation~\eqref{eq:pkt-unit-action}: reducing \(v\) into
+\(\mathscr F_U\) transports \(L\) by the same unique unit, outside
+the usual null boundary set.  Counting measure times \(dv\) is
+invariant under this action.  These domains have total measure
+\(h_KN_{\OO}\vol(\mathscr F_U)\), equal to the total volume just
+computed.  Dividing their integral by that quantity proves
+Equation~\eqref{eq:pkt-integration}.
+\end{proof}

preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/references.bib

@@ -0,0 +1,311 @@
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+}
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+  year = {1947},
+  pages = {64--67}
+}
+
+@article{Rankin1938,
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+}
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+  author = {Koukoulopoulos, Dimitris},
+  title = {The Distribution of Prime Numbers},
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+  publisher = {American Mathematical Society},
+  address = {Providence, RI},
+  year = {2019},
+  isbn = {978-1-4704-4754-0},
+  url = {https://dms.umontreal.ca/~koukoulo/documents/publications/primes.pdf}
+}
+
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+  author = {Einsiedler, Manfred and Lindenstrauss, Elon and Michel, Philippe and Venkatesh, Akshay},
+  title = {Distribution of periodic torus orbits on homogeneous spaces},
+  journal = {Duke Mathematical Journal},
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+  number = {1},
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+  doi = {10.1215/00127094-2009-023},
+  url = {https://people.math.ethz.ch/~einsiedl/ELMV1-final-DMJ.pdf}
+}
+
+@article{EKL2006,
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+  title = {Invariant measures and the set of exceptions to {Littlewood}'s conjecture},
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+  url = {https://annals.math.princeton.edu/2006/164-2/p04}
+}
+
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+  author = {Rokhlin, V. A.},
+  title = {Lectures on the entropy theory of measure-preserving transformations},
+  journal = {Russian Mathematical Surveys},
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+  doi = {10.1070/RM1967v022n05ABEH001224}
+}
+
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+}
+
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+  author = {Milne, James S.},
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+  year = {2020},
+  note = {Version 3.08},
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+}
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+}
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+  howpublished = {OpenAI Math Release preprint
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+  year = {2026}
+}
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+}
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+}
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+}

preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/rigidity.tex

@@ -0,0 +1,498 @@
+\section{The homogeneous obstruction in degree four}\label{sec:rigidity}
+
+For a sequence with \(X\to\infty\), Proposition~\ref{prop:ord-tightness}
+shows that every weak subsequential limit of the packet measures is an
+\(A_4\)-invariant probability.  Proposition~\ref{prop:ord-balls}
+supplies the compact ball estimate in Equation~\eqref{eq:ord-balls}.
+We first turn that estimate
+into positive entropy in almost every \(A_4\)-ergodic component.
+Measure classification then makes those components homogeneous.
+In degree four, the proper homogeneous components that remain have two
+coordinate blocks of dimension two.  We will show that each of them
+carries an exterior-square lattice vector in one of three fixed planes.
+The arithmetic part of the proof will exclude these three events.
+
+\subsection{Entropy from the compact ball estimate}
+
+Fix distinct real numbers \(\lambda_1,\ldots,\lambda_4\) with sum zero,
+and set
+\[
+ a(t)=\diag(e^{\lambda_1t},\ldots,e^{\lambda_4t}),
+ \qquad
+ \lambda_a=\min_{i\ne j}|\lambda_i-\lambda_j|>0.
+\]
+For an \(a(\R)\)-invariant probability \(\xi\), let \(h_\xi(a(1))\)
+denote the measure-theoretic entropy of right multiplication by \(a(1)\).
+The argument below is the degree-four specialization of the
+ordinary-ball argument in
+\cite[Proposition~6.1 and Lemma~6.2]{PrimeDegreePackets}, which in turn
+follows the strategy of \cite[Section~2.7.1]{ELMV2011}.
+We include the argument because its statement for every dominated
+invariant probability is needed when passing to ergodic components.
+
+Here is the entropy input.  For an open identity neighborhood \(B\)
+in \(\mathrm{SL}_4(\R)\), write
+\[
+ B^{(s,t)}=a(-s)Ba(s)\cap a(-t)Ba(t).
+\]
+The following is the specialization to \(X_4\) of the
+Einsiedler--Lindenstrauss--Michel--Venkatesh tube criterion
+\cite[Corollary~3.3 and Equation~(3.4)]{ELMV2009}.  The flow \(a(t)\)
+is \(\R\)-diagonalizable, as required there.
+
+\begin{theorem}[Entropy criterion for two-sided tubes]
+\label{thm:rig-tube-criterion}
+Let \(\xi_i\) and \(\xi\) be \(a(\R)\)-invariant Borel probabilities
+on \(X_4\), with \(\xi_i\) converging weak-* to \(\xi\).
+Suppose that \(t_i>0\) tend to infinity and that \(\eta>0\).
+Assume that for every compact \(\Omega\subset X_4\) there are an open
+identity neighborhood \(B\subset\mathrm{SL}_4(\R)\) and a constant
+\(C_\Omega\) such that
+\[
+ \xi_i\bigl(xB^{(-t_i,t_i)}\bigr)
+ \le C_\Omega e^{-2\eta t_i}
+ \qquad(x\in\Omega,\ i\ge1).
+\]
+Then \(h_\xi(a(1))\ge\eta\).
+\end{theorem}
+
+In this criterion \(B\) may depend on \(\Omega\), and the
+probabilities need not be ergodic under the flow.  The requirement
+that the weak-* limit is a probability is part of the theorem.
+
+\begin{proposition}[Entropy for dominated probabilities]
+\label{prop:rig-dominated-entropy}
+Let \(\mu\) be an \(A_4\)-invariant probability on \(X_4\) satisfying
+the compact ball bound in Equation~\eqref{eq:ord-balls} for every compact set of
+centers.  If \(\xi\) is an \(A_4\)-invariant probability with
+\(\xi\le c\mu\) for some finite \(c\), then
+\[
+ h_\xi(a(1))\ge \lambda_a/3.
+\]
+\end{proposition}
+
+\begin{proof}
+Use the entrywise neighborhoods \(B(r)\) defined in
+Equation~\eqref{eq:ord-ball-neighborhood}.  Fix \(0<b<1/4\) and put
+\[
+ T_t=a(t)B(b)a(-t)\cap a(-t)B(b)a(t),
+ \qquad r=e^{-\lambda_a t}\quad(t\ge0).
+\]
+For \(h\in T_t\), the two conjugations give
+\begin{equation}\label{eq:rig-tube-entries}
+ |h_{ii}-1|<b,\qquad
+ |h_{ij}|<b e^{-|\lambda_i-\lambda_j|t}\le br
+ \quad(i\ne j).
+\end{equation}
+In the determinant expansion, every term except the product of the
+diagonal entries contains at least two off-diagonal entries.
+Since \(\det h=1\), Equation~\eqref{eq:rig-tube-entries} implies
+\(\prod_i h_{ii}=1+O(r^2)\).  The diagonal matrix
+\[
+ d(h)=\diag\bigl(h_{11},h_{22},h_{33},
+                         (h_{11}h_{22}h_{33})^{-1}\bigr)
+\]
+therefore belongs to a fixed compact subset
+\(\mathcal D_\infty\subset A_4\), and \(d(h)^{-1}h\in B(Cr)\).
+The constant \(C\) here and below may depend on \(b\) and the fixed flow.
+Indeed, the first three diagonal entries of \(d(h)^{-1}h\) are one,
+the fourth differs from one by \(O(r^2)\), and all its off-diagonal
+entries are \(O(r)\).
+
+The logarithmic coordinates on \(A_4\) identify
+\(\mathcal D_\infty\) with a bounded subset of \(\R^3\).
+A mesh of spacing comparable to \(r\), followed by a fixed enlargement
+to account for matrix multiplication, gives
+\begin{equation}\label{eq:rig-tube-cover}
+ T_t\subset\bigcup_{j=1}^{N_t}d_jB(Cr),
+ \qquad d_j\in\mathcal D_\infty,\qquad N_t\le Cr^{-3}.
+\end{equation}
+For a compact \(\Omega\subset X_4\), the centers \(xd_j\) lie in the
+compact set \(\Omega\mathcal D_\infty\).  Domination and
+Equation~\eqref{eq:ord-balls} now imply
+\begin{equation}\label{eq:rig-tube-mass}
+ \xi(xT_t)\le C_{\Omega,c}r^{-3}r^4
+             =C_{\Omega,c}e^{-\lambda_a t}
+ \qquad(x\in\Omega)
+\end{equation}
+for all sufficiently large \(t\).
+Apply Theorem~\ref{thm:rig-tube-criterion} to the constant sequence
+\(\xi_i=\xi\), with \(t_i=i\), \(B=B(b)\), and
+\(\eta=\lambda_a/3\).  The identity \(B(b)^{(-t,t)}=T_t\) and
+\(2\lambda_a/3<\lambda_a\) verify its estimate for large \(i\).
+For the finitely many earlier \(i\), enlarge \(C_{\Omega,c}\), using
+that \(\xi\) is a probability.  The criterion gives the assertion.
+\end{proof}
+
+\begin{corollary}[Positive entropy in the ergodic components]
+\label{cor:rig-positive-components}
+Let \(\mu\) be an \(A_4\)-invariant probability on \(X_4\) satisfying
+Equation~\eqref{eq:ord-balls} on every compact set of centers.  Almost every
+component in its \(A_4\)-ergodic decomposition has positive
+\(a(1)\)-entropy.  In particular, this holds for every probability
+limit in Proposition~\ref{prop:ord-balls}.
+\end{corollary}
+
+\begin{proof}
+The group \(A_4\simeq\R^3\) is locally compact and second countable,
+and acts continuously on the standard Borel space \(X_4\).
+The ergodic decomposition theorem
+\cite[Theorem~5.2]{GreschonigSchmidt2000}, specialized to invariant
+probabilities, gives a measurable decomposition
+\[
+ \mu=\int_Z\nu_z\,d\tau(z)
+\]
+into \(A_4\)-invariant, \(A_4\)-ergodic probabilities over a factor
+fixed by \(A_4\).  Put \(T(x)=xa(1)\).
+
+We record why a restriction to the zero-entropy components is
+measurable.  Choose increasing finite Borel partitions
+\(\mathcal P_k\) generating the Borel \(\sigma\)-algebra of \(X_4\).
+For any \(T\)-invariant probability \(\nu\), the
+Kolmogorov--Sinai approximation theorem and subadditivity give
+\[
+ h_\nu(T)=\sup_k\inf_{N\ge1}\frac1N
+ H_\nu\left(\bigvee_{j=0}^{N-1}T^{-j}\mathcal P_k\right),
+\]
+where \(H_\nu\) is the Shannon entropy of a finite partition.
+Each finite-partition entropy is a Borel function of \(\nu\), so
+\(z\mapsto h_{\nu_z}(T)\) is measurable.
+For each probability under consideration, complete the standard Borel
+probability space; it is then a normalized Lebesgue space.
+The entropy integral formula for an invariant component factor
+\cite[Theorem~9.8]{Rokhlin1967} gives
+\begin{equation}\label{eq:rig-entropy-decomposition}
+ h_\mu(T)=\int_Z h_{\nu_z}(T)\,d\tau(z),
+\end{equation}
+and the same formula after restricting and normalizing \(\tau\) on
+any measurable subset of positive measure.  This formula applies
+because the component factor is fixed by \(T\).  It does not require
+the individual \(\nu_z\) to be \(T\)-ergodic.
+
+If \(Z_0=\{z:h_{\nu_z}(T)=0\}\) had measure \(w>0\), then
+\[
+ \xi=\frac1w\int_{Z_0}\nu_z\,d\tau(z)
+\]
+would be an \(A_4\)-invariant probability with \(\xi\le w^{-1}\mu\).
+The restricted form of Equation~\eqref{eq:rig-entropy-decomposition}
+would give \(h_\xi(T)=0\), contrary to
+Proposition~\ref{prop:rig-dominated-entropy}.
+\end{proof}
+
+\subsection{Finite-volume subgroups containing the diagonal group}
+
+We next use the measure-classification theorem of
+Einsiedler--Katok--Lindenstrauss
+\cite[Theorem~1.3; definition in Conjecture~1.1]{EKL2006}.
+We state the exact specialization that supplies the homogeneous
+measures needed here.
+
+\begin{theorem}[Positive-entropy measure classification]
+\label{thm:rig-ekl-classification}
+For \(n\ge3\), let \(A_n\) be the group of positive diagonal matrices
+in \(\mathrm{SL}_n(\R)\).  Let \(\nu\) be an \(A_n\)-invariant and
+\(A_n\)-ergodic probability on
+\(\mathrm{SL}_n(\R)/\mathrm{SL}_n(\Z)\), with the left action.
+If \(h_\nu(a_0(1))>0\) for some one-parameter subgroup
+\(a_0:\R\to A_n\), then there are a
+closed connected subgroup \(L\subset\mathrm{SL}_n(\R)\) containing
+\(A_n\) and a point \(x\) such that \(Lx\) is closed and \(\nu\) is
+the \(L\)-invariant probability on \(Lx\).
+\end{theorem}
+
+No ergodicity for the chosen one-parameter subgroup is assumed.
+For \(n=4\), inversion
+\(\mathrm{SL}_4(\Z)g\mapsto g^{-1}\mathrm{SL}_4(\Z)\)
+turns right multiplication by \(a(t)\) into left multiplication by
+\(a(-t)\), and entropy is unchanged upon inverting an invertible
+transformation.  Thus, on our right quotient, the theorem says that
+a positive-entropy \(A_4\)-ergodic probability is invariant
+probability on an orbit \(\Lambda J\), where \(J\) is closed and
+connected, contains \(A_4\), and the stabilizer of \(\Lambda\) is a
+lattice in \(J\).
+
+For a partition \(\mathcal I=\{I_1,\ldots,I_s\}\) of
+\(\{1,2,3,4\}\) into nonempty subsets, define
+\begin{equation}\label{eq:rig-block-group}
+ J_{\mathcal I}
+ =\left\{g\in\mathrm{SL}_4(\R):
+ \begin{array}{l}
+ g_{ij}=0\text{ if \(i,j\) belong to different \(I_v\)},\\
+ \det(g|_{\R^{I_v}})>0\text{ for every }v
+ \end{array}\right\}.
+\end{equation}
+This is the identity component of the corresponding block-diagonal
+group.  Lindenstrauss and Weiss identified the equal-block phenomenon
+for orbit closures containing a compact diagonal orbit
+\cite[Theorem~1.3]{LindenstraussWeiss2001}; their result also enters the
+proof of \cite[Corollary~1.4]{EKL2006}.  The next lemma gives a direct
+finite-volume argument and records the rational structure of the blocks.
+
+\begin{lemma}[Equal blocks and a rational commutant]
+\label{lem:rig-equal-blocks}
+Let \(J\) be a closed connected subgroup of \(\mathrm{SL}_4(\R)\)
+containing \(A_4\), and let \(\Lambda\in X_4\).  Suppose that the
+stabilizer \(\Gamma_\Lambda=\{j\in J:\Lambda j=\Lambda\}\) is a
+lattice in \(J\).
+Then \(J=J_{\mathcal I}\) for a partition \(\mathcal I\) whose blocks
+all have the same size.
+
+More precisely, put \(W_\Q=\Q\Lambda\) and
+\(W_\R=W_\Q\otimes_\Q\R=\R^4\).  There is a totally real number field
+\(E\subset\End_\Q(W_\Q)\), of degree \(s=|\mathcal I|\), whose scalar
+extension to \(\R\) is exactly the commutant of \(J\) in
+\(\End_\R(W_\R)\).  The coordinate blocks \(\R^{I_v}\) are the real
+eigenspaces of \(E\) on \(W_\R\).
+\end{lemma}
+
+\begin{proof}
+Let \(\mathfrak j\) be the Lie algebra of \(J\).
+It contains the trace-zero diagonal algebra and is invariant under
+diagonal conjugation.  The other weight spaces in
+\(\mathfrak{sl}_4(\R)\) are the distinct one-dimensional spaces
+\(\R E_{ij}\), \(i\ne j\), where \(E_{ij}\) is a matrix unit.
+An invariant subspace is a sum of these weight spaces, so
+\(\mathfrak j\) consists of the diagonal algebra and a selection of
+the root spaces.
+
+The group \(J\) is unimodular.
+One way to see this standard fact is to use the quotient
+criterion for Haar measures: existence of an invariant quotient
+measure and triviality of the modular character of the discrete
+group \(\Gamma_\Lambda\) imply that the modular character of \(J\)
+is trivial on \(\Gamma_\Lambda\).  It therefore descends to
+\(\Gamma_\Lambda\backslash J\).  Its pushforward of invariant
+probability would be invariant under translations by its image.
+A nontrivial positive dilation admits no invariant probability on
+\(\R_{>0}\), as its disjoint dilates of a fundamental interval all
+have the same mass.  Hence the modular character is trivial on \(J\).
+
+Draw a directed edge \(i\to j\) when \(E_{ij}\in\mathfrak j\).
+The trace of the adjoint action of
+\(\diag(u_1,\ldots,u_4)\), with \(\sum_i u_i=0\), on
+\(\mathfrak j\) is the sum of the selected roots \(u_i-u_j\).
+Unimodularity makes this sum zero.  Its coefficient at \(u_i\) is
+the outgoing degree of \(i\) minus its incoming degree.  These
+coefficients sum to zero, and a linear form vanishing on
+\(\sum_i u_i=0\) has all coefficients equal.  Thus every vertex
+has equal outgoing and incoming degrees.
+
+Every edge now lies within a strongly connected component.
+Otherwise the finite directed acyclic graph of components would
+have a source with an outgoing edge; summing degree balance over
+that source would give a positive outgoing degree and zero incoming
+degree.  For distinct indices, the bracket identity
+\([E_{ij},E_{jk}]=E_{ik}\) makes the edge relation transitive.
+Each strongly connected component is consequently a complete
+directed graph.  These components form a partition \(\mathcal I\),
+and \(\mathfrak j\) is the Lie algebra of \(J_{\mathcal I}\).
+Connected Lie subgroups with the same Lie algebra agree, giving
+\(J=J_{\mathcal I}\).
+
+We use \(\Gamma_\Lambda\) to recover a rational structure on the
+block scalars.  Its elements preserve \(\Lambda\), hence act
+rationally on \(W_\Q\).  Let
+\[
+ \mathcal C_\Q
+ =\{M\in\End_\Q(W_\Q):M\gamma=\gamma M
+                         \text{ for every }\gamma\in\Gamma_\Lambda\}.
+\]
+In a lattice basis the commutation equations have integer
+coefficients.  A finite subcollection of these linear equations
+defines the same solution space, by finite dimensionality.
+It follows that \(\mathcal C_\Q\otimes_\Q\R\) is the real
+commutant of \(\Gamma_\Lambda\).
+
+We claim that this is also the real commutant of \(J\).
+Let \(M\) be a real endomorphism commuting with \(\Gamma_\Lambda\).
+The continuous map
+\[
+ \Gamma_\Lambda\backslash J\longrightarrow\End_\R(W_\R),
+ \qquad \Gamma_\Lambda j\longmapsto j^{-1}Mj
+\]
+is well-defined.  Its pushforward of invariant probability is
+invariant under conjugation by \(A_4\).  Each off-diagonal
+coordinate is multiplied by a nontrivial positive dilation under
+some element of \(A_4\).  A probability on \(\R\) invariant under
+such a dilation is supported at zero: the annuli
+\(\{x:r^k\le |x|<r^{k+1}\}\), \(k\in\Z\), for a fixed \(r>1\),
+are disjoint dilates covering \(\R\setminus\{0\}\).
+Thus every off-diagonal coordinate of \(j^{-1}Mj\) is zero almost
+everywhere.  Invariant probability on the transitive space
+\(\Gamma_\Lambda\backslash J\) has full support; continuity makes
+\(j^{-1}Mj\) diagonal for every \(j\in J\).
+In particular \(M\) is diagonal.  Conjugating by
+\(1+tE_{ij}\) for \(i,j\) in the same block forces its \(i\)-th
+and \(j\)-th diagonal entries to agree.  Therefore \(M\) is scalar
+on each block, which is exactly the condition that it commute with
+\(J\).  The reverse inclusion is immediate, proving the claim.
+
+It follows that \(\mathcal C_\Q\otimes_\Q\R\simeq\R^s\).
+In particular, \(\mathcal C_\Q\) is commutative and has no
+nonzero nilpotents.  The structure theorem for finite reduced
+commutative algebras over \(\Q\) expresses it as a product of
+number fields, all totally real because its real scalar extension
+has no complex factor.
+
+This product has only one factor.  If
+\(e\in\mathcal C_\Q\) were an idempotent other than \(0\) and \(1\),
+then \(W=e(W_\R)\) would be a nonempty proper union of coordinate
+blocks, rational relative to \(\Lambda\).  Every
+\(\gamma\in\Gamma_\Lambda\) preserves the full lattice
+\(\Lambda\cap W\) in \(W\), so
+\(\lvert\det(\gamma|_W)\rvert=1\).
+The continuous character
+\[
+ \chi:J\longrightarrow\R_{>0},\qquad
+ \chi(j)=\lvert\det(j|_W)\rvert
+\]
+is therefore trivial on \(\Gamma_\Lambda\).
+It is surjective: if \(k=\dim W\), a diagonal matrix may scale the
+coordinates in \(W\) by \(e^{(4-k)t}\) and those in its complement
+by \(e^{-kt}\).  Its total determinant is one and its determinant
+on \(W\) is \(e^{k(4-k)t}\).
+The character \(\chi\) would consequently push the invariant
+probability on \(\Gamma_\Lambda\backslash J\) to a probability on
+\(\R_{>0}\) invariant under all multiplicative translations.
+The disjoint-interval argument above rules this out.
+Thus \(\mathcal C_\Q\) is a field, which we denote by \(E\).
+
+Since \(E\otimes_\Q\R\simeq\R^s\), its degree is \(s\).
+The space \(W_\Q\) is a vector space over \(E\), and
+\[
+ W_\Q\otimes_\Q\R
+ \simeq\bigoplus_{\tau:E\hookrightarrow\R}
+                 W_\Q\otimes_{E,\tau}\R.
+\]
+Each summand has dimension \(\dim_E W_\Q=4/s\).
+These summands are the eigenspaces of the block-scalar algebra,
+namely the coordinate blocks.  The blocks therefore have equal size.
+\end{proof}
+
+For a positive-entropy homogeneous probability the lemma leaves
+\(J=\mathrm{SL}_4(\R)\) or two blocks of size two.
+Indeed, the only other equal-block possibility is four singleton
+blocks, giving \(J=A_4\).  A lattice in \(A_4\simeq\R^3\) has
+compact quotient, and translations on this torus have zero entropy:
+for a flat metric they are isometries, so the iterated Bowen metrics
+equal the original metric and the topological entropy is zero;
+the variational principle bounds measure entropy by topological entropy.
+
+For a possible two-block limit orbit \(\Lambda J\), the quadratic field
+\(E\subset\End_\Q(\Q\Lambda)\) has real scalar extension equal to
+the commutant of \(J\) in \(\End_\R(\R^4)\).
+The construction supplies no embedding of \(E\) into the packet fields
+\(K_i\), so their primitivity does not by itself exclude this orbit.
+We will exclude it by proving that packet limits give zero mass to the
+exterior obstruction below.
+
+\subsection{The three exterior obstruction events}
+
+Let \(e_1,\ldots,e_4\) be the coordinate basis of \(\R^4\).
+For \(w\in\bigwedge^2\R^4\), write \(w_{ij}\), \(i<j\), for its
+coefficients in the basis \(e_i\wedge e_j\), and define
+\begin{equation}\label{eq:rig-Q}
+ Q(w)=w_{12}w_{34}-w_{13}w_{24}+w_{14}w_{23}.
+\end{equation}
+The identity
+\[
+ w\wedge w=2Q(w)\,e_1\wedge e_2\wedge e_3\wedge e_4
+\]
+shows that
+\(Q((\bigwedge^2g)w)=\det(g)Q(w)\) for \(g\in\mathrm{GL}_4(\R)\).
+Expressing a unimodular lattice in a determinant-one basis gives
+\begin{equation}\label{eq:rig-Q-integral}
+ Q\bigl(\bigwedge\nolimits^2\Lambda\bigr)\subset\Z
+ \qquad(\Lambda\in X_4).
+\end{equation}
+This also covers the sign translates in the packet definition:
+a coordinate reflection of determinant \(-1\) changes only the
+sign of \(Q\).
+
+Let
+\[
+ \mathcal B=\{12|34,\ 13|24,\ 14|23\}.
+\]
+For \(b=ij|kl\in\mathcal B\), with the indices within each pair
+increasing, put
+\begin{equation}\label{eq:rig-planes}
+ V_b=\operatorname{span}_\R\{e_i\wedge e_j,e_k\wedge e_l\}.
+\end{equation}
+The restriction of \(Q\) to \(V_b\) is, up to sign, the product of
+its two displayed coordinates.  Define
+\begin{equation}\label{eq:rig-events}
+ \mathcal E_b
+ =\left\{\Lambda\in X_4:
+   \text{there exists }w\in\bigwedge\nolimits^2\Lambda\cap V_b
+   \text{ with }Q(w)\ne0\right\}.
+\end{equation}
+Each \(\mathcal E_b\) is \(A_4\)-invariant and Borel.
+For Borel measurability, take a countable cover of \(X_4\) by local
+charts admitting continuous lattice bases.  In a chart, the six
+integer coefficients of an exterior vector label it continuously.
+For each label, membership in \(V_b\) is a set of continuous
+coordinate equalities, and \(Q(w)\ne0\) is an open condition.
+Taking the countable union over labels and charts proves the claim.
+Invariance follows because \(A_4\) preserves \(V_b\) and \(Q\).
+
+\begin{proposition}[The quartic homogeneous obstruction]
+\label{prop:rig-obstruction}
+Let \(\nu\) be an \(A_4\)-invariant, \(A_4\)-ergodic probability
+on \(X_4\).  Suppose that \(h_\nu(a_0(1))>0\) for some
+one-parameter subgroup \(a_0:\R\to A_4\).
+Then either \(\nu=m_4\), or there exists
+\(b\in\mathcal B\) such that \(\nu(\mathcal E_b)=1\).
+In the latter case every lattice \(\Lambda'\) in the homogeneous orbit carrying
+\(\nu\) has a vector in \(\bigwedge^2\Lambda'\cap V_b\) with both
+coordinates nonzero and with nonzero integer \(Q\)-value.
+\end{proposition}
+
+\begin{proof}
+Theorem~\ref{thm:rig-ekl-classification}, in the right-quotient
+form explained above, represents \(\nu\) as invariant probability
+on \(\Lambda J\) for a connected closed subgroup \(J\) containing
+\(A_4\).  By Lemma~\ref{lem:rig-equal-blocks} and the
+positive-entropy assumption, either \(J=\mathrm{SL}_4(\R)\), which
+gives \(\nu=m_4\), or \(J\) has two coordinate blocks of size two.
+
+In the second case write the blocks as \(\{i,j\}\) and \(\{k,l\}\),
+and put \(b=ij|kl\).  The commutant field \(E\) in
+Lemma~\ref{lem:rig-equal-blocks} is quadratic.  Choose a nonzero
+\(\alpha\in E\) with \(\Tr_{E/\Q}(\alpha)=0\).
+Its two scalar values on the blocks are \(c\) and \(-c\), with
+\(c\ne0\).  The rational endomorphism \(\alpha\) of
+\(W_\Q=\Q\Lambda\) induces a rational endomorphism
+\[
+ D_\alpha:\bigwedge\nolimits^2 W_\Q\longrightarrow
+                    \bigwedge\nolimits^2 W_\Q,\qquad
+ D_\alpha(v\wedge w)=\alpha(v)\wedge w+v\wedge\alpha(w).
+\]
+On the exterior square of the first block it acts by \(2c\), and
+on that of the second by \(-2c\).  It vanishes on the cross-block
+summand.  Its real image is therefore exactly
+\[
+ \bigwedge\nolimits^2\R^{\{i,j\}}
+ \oplus \bigwedge\nolimits^2\R^{\{k,l\}}=V_b.
+\]
+Because \(D_\alpha\) is rational, \(V_b\) is a rational subspace
+relative to \(\bigwedge^2 W_\Q\).
+
+It follows that \(V_b\cap\bigwedge^2\Lambda\) is a rank-two lattice
+in \(V_b\).  It contains a vector off both coordinate axes: take
+two independent lattice vectors; if neither is already off the
+axes, their sum is.  For this vector \(Q(w)\ne0\), and
+Equation~\eqref{eq:rig-Q-integral} makes its value a nonzero integer.
+Every element of \(J\) preserves each of the two exterior lines,
+multiplying its coefficient by the nonzero determinant of the
+corresponding block.  The translated vector therefore has both
+coordinates nonzero at every lattice of \(\Lambda J\).
+This proves \(\nu(\mathcal E_b)=1\).
+\end{proof}
+
+Corollary~\ref{cor:rig-positive-components} and
+Proposition~\ref{prop:rig-obstruction} reduce identification of a
+packet probability limit \(\mu\) to proving
+\(\mu(\mathcal E_b)=0\) for all three \(b\in\mathcal B\).
+Equation~\eqref{eq:rig-Q-integral} lets us test those events one
+nonzero integer \(Q\)-value at a time.  The next sections establish
+the exterior-vector estimate used for this exclusion.

preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/build/manuscript/sieve.tex

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+\section{Averaging the unramified factors}\label{sec:sieve}
+
+Proposition~\ref{prop:additive-equidistribution} distributes the
+nonnegative mass $\phi(z)\prod_{p\in S}f_p^*(z)$ among residue classes at
+primes outside $S$.  To prove Theorem~\ref{thm:exterior-count}, we must
+insert the remaining
+factors $b_p(z)$ while retaining their local means.  Some of those means are
+less than one, so the argument must also keep track of the probability that
+a prime does not divide any component of $z$.  We first obtain a weighted
+sieve estimate with this feature and then combine its local means with the
+Euler factors in Proposition~\ref{prop:ext-multiplicity-majorant}.
+
+\subsection{Residue functions and the local bounds}
+
+Fix $m\in\Z\setminus\{0\}$ and a bounded region of the real trace-$m$
+plane.  Throughout the sieve argument, $\phi$ is one of the nonnegative
+smooth box functions in Proposition~\ref{prop:additive-equidistribution},
+with its planar support in this region.  Its two scales lie between
+$X^{-\tau}$ and
+fixed constants.  Define a measure on $F$ by
+\begin{equation}\label{eq:sieve-measure}
+ \nu_\phi
+ =\sum_{\substack{z\in F,\ \Tr z=m\\z\text{ integral outside }S}}
+       \phi(z)\prod_{p\in S}f_p^*(z)\,\delta_z,
+ \qquad
+ V_\phi=A_\phi\prod_{p\in S}d_p(m).
+\end{equation}
+Here $\delta_z$ is unit point mass at $z$.  All its coefficients are
+nonnegative, and Proposition~\ref{prop:additive-equidistribution} with
+modulus one gives a finite total mass.
+Lemma~\ref{lem:fourier-scalar-density} gives
+\begin{equation}\label{eq:sieve-volume}
+ V_\phi\asymp A_\phi
+\end{equation}
+uniformly in $S$: for large primes the logarithms of $d_p(m)=1+O(p^{-2})$
+have an absolutely convergent sum, and the finitely many small-prime
+densities have uniform positive upper and lower bounds.
+
+For a positive integer $l$ prime to every prime of $S$, the integral
+trace-$m$ coset has $l^2$ residue classes modulo $l$ in its two integral
+coordinates.  Write $\E_l W$ for the uniform mean of a function $W$ on
+these classes, and also write $W(z)$ for its pullback to points integral
+outside $S$.
+
+\begin{lemma}[Nonnegative residue functions]\label{lem:sieve-residue-function}
+With $\phi$ and $V_\phi$ as in Equation~\eqref{eq:sieve-measure}, let
+$1\le l\le X^\theta$ be an integer prime to every prime of $S$.
+For every nonnegative
+function $W$ on the residue classes of the integral trace-$m$ coset
+modulo $l$,
+\begin{equation}\label{eq:sieve-residue-function}
+ \int W\,d\nu_\phi
+   =V_\phi\E_l W+O\bigl(V_\phi X^{-\theta}\E_l W\bigr).
+\end{equation}
+The implied constant is uniform in the fields, orders, embedding
+orderings, permitted box locations and scales, $l$, and $W$.  It may
+depend on $m$, the fixed bounded region, and the fixed smooth box function.
+\end{lemma}
+
+\begin{proof}
+For each residue class $a$, Equation~\eqref{eq:add-equidistribution} gives
+\[
+ \nu_\phi(a)=V_\phi l^{-2}(1+\epsilon_a),
+ \qquad |\epsilon_a|\le C_{\mathrm{add}}X^{-\theta},
+\]
+with the same $C_{\mathrm{add}}$ for every $a$ and $l$ in the stated range.
+Multiply by
+$W(a)$ and sum.  Since $W(a)\ge0$, the absolute value of the summed error
+is at most
+$C_{\mathrm{add}}V_\phi X^{-\theta}l^{-2}\sum_a W(a)
+ =C_{\mathrm{add}}V_\phi X^{-\theta}\E_l W$.
+This also covers $\E_l W=0$.  In particular the error retains the local
+mean even when that mean is very small.
+\end{proof}
+
+We recall the information about the functions to be inserted.  For
+$p\notin S$, let $\E_p$ and $\P_p$ denote expectation and probability for
+normalized additive Haar measure on
+$\{z\in R_{F,p}:\Tr z=m\}$.  On $R_{F,p}\cap F_p^\times$, recall that
+$n_p(z)=\max_{\mathfrak q\mid p}v_{\mathfrak q}(z)$ is the largest
+component valuation, with $v_{\mathfrak q}(p)=1$; see
+Equation~\eqref{eq:ext-unramified-factor}.  On the null set where a
+component vanishes, keep the convention $n_p=\infty$, $b_p=0$.
+Lemma~\ref{lem:ext-unramified-data} permits one fixed integer $C\ge1$
+such that, on the nonvanishing locus,
+\begin{align}
+ &b_p\ge0,\qquad b_p=1\ \text{if }n_p=0,\qquad
+ b_p\le C(1+n_p)^C,\label{eq:sieve-local-weight}\\
+ &\P_p(n_p\ge j)\le Cp^{-j}\quad(j\ge1),\qquad
+ \vartheta_p:=\P_p(n_p\ge1)\le C/p,\quad
+ \vartheta_p\le\tfrac12. \label{eq:sieve-local-tail}
+\end{align}
+The event $\{n_p\ge j\}$ and the function
+$b_p\mathbf 1_{\{n_p=j\}}$ are determined by the two integral coordinates
+modulo $p^{C(j+1)}$ for every integer $j\ge1$.  They have extensions
+defined by congruence classes on the whole local coset, agreeing with their values at every
+global point.  A prime with $n_p(z)\ge1$ will be called active at $z$.
+In particular,
+\begin{equation}\label{eq:sieve-mean-lower}
+ \E_p b_p\ge1-\vartheta_p\ge\tfrac12.
+\end{equation}
+
+Lemma~\ref{lem:ext-support} places the support of $\nu_\phi$ in the
+trace-$m$ coset of $(C_F^\sharp)^\vee$.  On this support all $n_p(z)$,
+$p\notin S$, are finite.  The lemma's norm and depth conclusions bound
+the product of the depth factors by $O(X^{84})$ in our fixed bounded real region.  Thus,
+for a fixed $B>84$ and all sufficiently large $X$,
+\begin{equation}\label{eq:sieve-depth-product}
+ \prod_{p\notin S}p^{n_p(z)}\le X^B
+ \qquad\bigl(\nu_\phi(\{z\})>0\bigr).
+\end{equation}
+
+Two consequences will be used repeatedly.  There is a fixed $A_0>1$ such
+that $b_p(z)\le A_0^{n_p(z)}$.  This follows from
+Equation~\eqref{eq:sieve-local-weight} for $n_p\ge1$, since
+$\sup_{j\ge1}(C(1+j)^C)^{1/j}<\infty$, and from $b_p=1$ at depth zero.
+Also, with
+\[
+ \mathcal W(z)=\prod_{p\notin S}b_p(z),
+\]
+the product is finite on the support of $\nu_\phi$ and satisfies
+\begin{equation}\label{eq:sieve-pointwise-weight}
+ \mathcal W(z)\ll_\epsilon X^\epsilon
+ \qquad(\epsilon>0).
+\end{equation}
+Indeed, for any $t>0$, the inequality
+$C(1+j)^C\le p^{tj}$ holds for all $j\ge1$ once $p$ is sufficiently large.
+For each of the remaining finitely many primes it holds with a constant
+factor independent of $j$.  Multiplying these inequalities and using
+Equation~\eqref{eq:sieve-depth-product}, with $t=\epsilon/B$, proves
+Equation~\eqref{eq:sieve-pointwise-weight}.  All constants here are uniform
+in the arithmetic data and in the permitted boxes.
+
+\begin{proposition}[Weighted averaging]\label{prop:sieve-averaging}
+For the measure $\nu_\phi$ in Equation~\eqref{eq:sieve-measure}, there is
+a fixed $\delta>0$ such that
+\begin{equation}\label{eq:sieve-average}
+ \int\mathcal W(z)\,d\nu_\phi(z)
+   \ll V_\phi\prod_{\substack{p\le X^\delta\\p\notin S}}\E_p b_p.
+\end{equation}
+The choice of $\delta$ and the implied constant are independent of the
+field, order, embedding ordering, and the permitted box location and
+scales.  They may depend on $m$, the fixed bounded real region, and the
+fixed smooth box function.
+\end{proposition}
+
+The proof first counts bounded products of prescribed positive depths,
+while requiring inactivity at any other chosen small primes.  The
+inactivity condition preserves the factors in
+Equation~\eqref{eq:sieve-mean-lower}.  We then bound the contribution of two
+exceptional sets of points and truncate the remaining depth products within
+the modulus range of
+Lemma~\ref{lem:sieve-residue-function}.
+
+\subsection{Prescribed depths and inactive primes}
+
+Write
+$\mathcal P_y=\{p\le y:p\notin S\}$.  Chebyshev's estimate
+$\sum_{p\le y}(\log p)/p\ll\log y$ and
+Equation~\eqref{eq:sieve-local-tail} give a fixed $C_*\ge1$ such that
+\[
+ \sum_{p\in\mathcal P_y}\vartheta_p\log p\le C_*\log y
+ \qquad(y\ge2).
+\]
+Choose fixed positive parameters satisfying
+\begin{equation}\label{eq:sieve-parameters}
+ 20(C+1)\eta<\theta,\qquad
+ 0<\delta<\min\{\eta/4,\eta/(4C_*)\},
+ \qquad T_0=X^\eta,\quad Y=X^\delta.
+\end{equation}
+Here $T_0$ truncates the squarefree Selberg divisors and is the stopping
+threshold for depth products, whose crossing prefixes may reach $T_0^2$;
+$Y$ bounds the small primes outside $S$ handled by the sieve.
+For every $\mathcal I\subset\mathcal P_Y$, Mertens' estimate and
+$-\log(1-\vartheta_p)\le2\vartheta_p\le2C/p$ give, for a fixed $C_1$,
+\begin{equation}\label{eq:sieve-inactivity-lower}
+ \prod_{p\in\mathcal I}(1-\vartheta_p)
+       \gg(\log X)^{-C_1}.
+\end{equation}
+The bound is uniform in the subset $\mathcal I$ and in $S$.
+
+\begin{lemma}[Prescribed depths and inactive primes]\label{lem:sieve-assignment}
+Let $\mathcal A,\mathcal I$ be disjoint subsets of $\mathcal P_Y$, and
+assign an integer $j_p\ge1$ to each $p\in\mathcal A$.  Suppose that
+$\prod_{p\in\mathcal A}p^{j_p}\le T_0^2$.  Then
+\begin{equation}\label{eq:sieve-assignment}
+ \int
+   \prod_{p\in\mathcal A}
+      b_p\mathbf 1_{\{n_p=j_p\}}
+   \prod_{p\in\mathcal I}\mathbf 1_{\{n_p=0\}}\,
+        d\nu_\phi
+ \ll V_\phi
+   \prod_{p\in\mathcal A}\E_p
+      \bigl(b_p\mathbf 1_{\{n_p=j_p\}}\bigr)
+   \prod_{p\in\mathcal I}(1-\vartheta_p).
+\end{equation}
+The constant is uniform in the two subsets and all their assigned depths.
+\end{lemma}
+
+\begin{proof}
+We use the Selberg upper-bound square
+\cite{Selberg1947}; see also \cite[Chapter~21]{Koukoulopoulos2019}.
+Its calculation is included because the relative error in
+Lemma~\ref{lem:sieve-residue-function} must preserve the assigned means.
+If $\vartheta_p=0$, the activity event at $p$ is a union of residue classes
+of Haar measure zero, hence is empty on global points.  Such a prime can
+be removed from $\mathcal I$.  For the remaining primes put
+\[
+ \vartheta(d)=\prod_{p\mid d}\vartheta_p,\qquad
+ h(d)=\prod_{p\mid d}\frac{\vartheta_p}{1-\vartheta_p}
+\]
+for squarefree $d$ supported on $\mathcal I$.  Both functions take value
+one at $d=1$.  Let $\mathcal D$ be the set of those integers with
+$d\le T_0$, and put $\mathcal H=\sum_{d\in\mathcal D}h(d)$.
+
+For coefficients $\lambda_d$ supported on $\mathcal D$ with
+$\lambda_1=1$, the square
+\[
+ \left(\sum_{d\in\mathcal D}\lambda_d
+       \prod_{p\mid d}\mathbf 1_{\{n_p\ge1\}}\right)^2
+\]
+majorizes the indicator that every prime of $\mathcal I$ is inactive.
+Its expectation in the independent product of the local probability
+spaces is
+\begin{equation}\label{eq:sieve-quadratic}
+ \sum_{d,e\in\mathcal D}\lambda_d\lambda_e\vartheta([d,e])
+   =\sum_{l\in\mathcal D}\frac1{h(l)}
+       \left(\sum_{\substack{d\in\mathcal D\\l\mid d}}
+                      \lambda_d\vartheta(d)\right)^2.
+\end{equation}
+Here $[d,e]$ denotes the least common multiple.  The identity follows
+by expanding the right side and using
+$\vartheta([d,e])
+ =\vartheta(d)\vartheta(e)\sum_{l\mid(d,e)}h(l)^{-1}$.
+
+The set $\mathcal D$ is closed under divisors.  Finite M\"obius inversion
+therefore permits the choice
+\[
+ \sum_{\substack{d\in\mathcal D\\l\mid d}}\lambda_d\vartheta(d)
+       =\frac{\mu_{\mathrm{Mob}}(l)h(l)}{\mathcal H},
+ \qquad l\in\mathcal D,
+\]
+where $\mu_{\mathrm{Mob}}$ is the M\"obius function.  Explicitly,
+\[
+ \lambda_d=
+  \frac{\mu_{\mathrm{Mob}}(d)h(d)}
+       {\vartheta(d)\mathcal H}
+  \sum_{\substack{r\le T_0/d\\(r,d)=1\\dr\in\mathcal D}}h(r).
+\]
+Consequently $\lambda_1=1$, the value of
+Equation~\eqref{eq:sieve-quadratic} is $\mathcal H^{-1}$, and
+\begin{equation}\label{eq:sieve-coefficients}
+ |\lambda_d|
+ \le\frac{h(d)}{\vartheta(d)}
+ =\prod_{p\mid d}(1-\vartheta_p)^{-1}
+ \le2^{\omega(d)}\le T_0.
+\end{equation}
+
+To estimate $\mathcal H$, give every squarefree integer supported on
+$\mathcal I$ probability proportional to $h(d)$, without the bound
+$d\le T_0$.  In this finite product distribution, prime $p$ is selected
+with probability $\vartheta_p$.  Thus
+\[
+ \E\log d=\sum_{p\in\mathcal I}\vartheta_p\log p
+       \le C_*\log Y<\tfrac14\log T_0.
+\]
+Markov's inequality shows that at least half the mass lies at $d\le T_0$.
+It follows that
+\begin{equation}\label{eq:sieve-selberg-main}
+ \mathcal H\ge\tfrac12\prod_{p\in\mathcal I}(1+h(p)),
+ \qquad
+ \mathcal H^{-1}\le2\prod_{p\in\mathcal I}(1-\vartheta_p).
+\end{equation}
+
+It remains to average this square against $\nu_\phi$ together with the
+assigned weights.  Write
+\[
+ A_{\mathcal A}=
+   \prod_{p\in\mathcal A}\E_p
+        \bigl(b_p\mathbf 1_{\{n_p=j_p\}}\bigr).
+\]
+The assigned weight is defined modulo
+\[
+ \prod_{p\in\mathcal A}p^{C(j_p+1)}
+ \le\left(\prod_{p\in\mathcal A}p^{j_p}\right)^{2C}
+ \le T_0^{4C}.
+\]
+For a pair $d,e\in\mathcal D$, its activity indicators require only the
+additional modulus $[d,e]^{2C}\le T_0^{4C}$.  The two prime sets are
+disjoint, so the total modulus is at most
+\begin{equation}\label{eq:sieve-modulus-budget}
+ T_0^{8C}\le X^\theta
+\end{equation}
+and is prime to every prime of $S$.  The function before multiplication
+by $\lambda_d\lambda_e$ is nonnegative, and its residue mean is
+$A_{\mathcal A}\vartheta([d,e])$ by the Chinese remainder theorem.
+Lemma~\ref{lem:sieve-residue-function} therefore gives the corresponding
+main term $V_\phi A_{\mathcal A}\vartheta([d,e])$ and an error of absolute
+value at most
+$C V_\phi X^{-\theta}A_{\mathcal A}\vartheta([d,e])$.
+There are at most $T_0^2$ pairs and, by
+Equation~\eqref{eq:sieve-coefficients}, each coefficient product has
+absolute value at most $T_0^2$.  Summing the errors absolutely costs at
+most
+\begin{equation}\label{eq:sieve-square-error}
+ C V_\phi A_{\mathcal A}X^{-\theta}T_0^4.
+\end{equation}
+This estimate keeps $A_{\mathcal A}$ even if it is very small.
+The main term is $V_\phi A_{\mathcal A}\mathcal H^{-1}$ by
+Equation~\eqref{eq:sieve-quadratic}.  Finally,
+$X^{-\theta}T_0^4=o((\log X)^{-C_1})$ by
+Equation~\eqref{eq:sieve-parameters}.  Equations
+\eqref{eq:sieve-inactivity-lower} and
+\eqref{eq:sieve-selberg-main} absorb
+Equation~\eqref{eq:sieve-square-error} and prove the lemma.
+\end{proof}
+
+\subsection{Truncating and summing the depths}
+
+\begin{proof}[Proof of Proposition~\ref{prop:sieve-averaging}]
+Put
+$P_Y=\prod_{p\in\mathcal P_Y}\E_p b_p$.  Equations
+\eqref{eq:sieve-mean-lower} and
+\eqref{eq:sieve-inactivity-lower} imply
+\begin{equation}\label{eq:sieve-product-lower}
+ P_Y\gg(\log X)^{-C_1}.
+\end{equation}
+First observe that primes exceeding $Y$ cost only a fixed factor.
+Equations~\eqref{eq:sieve-depth-product} and
+\eqref{eq:sieve-pointwise-weight}, together with $b_p\le A_0^{n_p}$, give
+\begin{equation}\label{eq:sieve-large-primes}
+ \prod_{\substack{p>Y\\p\notin S}}b_p(z)
+ \le A_0^{\sum_{p>Y,\ p\notin S}n_p(z)}
+ \le \exp(B\log A_0/\delta).
+\end{equation}
+
+Consider the points for which
+$\prod_{p\in\mathcal P_Y}p^{n_p(z)}\le T_0$.
+Assign all their active primes in $\mathcal P_Y$ and require inactivity
+at the other primes in $\mathcal P_Y$.  These assignments satisfy
+Lemma~\ref{lem:sieve-assignment}.  After applying it, we may enlarge the
+sum of its nonnegative upper bounds by allowing every subset and every
+positive depth.  The resulting product is
+\[
+ V_\phi\prod_{p\in\mathcal P_Y}
+   \left(1-\vartheta_p+
+       \sum_{j\ge1}\E_p(b_p\mathbf 1_{\{n_p=j\}})\right)
+   =V_\phi P_Y.
+\]
+Together with Equation~\eqref{eq:sieve-large-primes}, this bounds the
+contribution from these points.
+
+For a remaining point, the product through $\mathcal P_Y$ exceeds $T_0$.
+Order its active primes in $\mathcal P_Y$ increasingly and stop at the
+first prefix whose product exceeds $T_0$.  If every individual factor
+$p^{n_p(z)}$ for $p\in\mathcal P_Y$ is at most $T_0$, the prefix has product in
+$(T_0,T_0^2]$, since its preceding product and its last factor are both
+at most $T_0$.  If also
+$\prod_{p\in\mathcal P_{y_0}}p^{n_p(z)}\le T_0$, for a fixed cutoff
+$y_0>4$ to be chosen below independently of $X$, its last prime exceeds
+$y_0$.  The next two estimates allow these restrictions at negligible
+weighted cost.
+
+First, if $p^{n_p(z)}>T_0$ for $p\in\mathcal P_Y$, then
+$n_p(z)\ge J_p$, where
+$J_p=\lfloor\log T_0/\log p\rfloor+1$.  The threshold is determined modulo
+\[
+ p^{C(J_p+1)}\le T_0^C Y^{2C}\le X^\theta.
+\]
+Lemma~\ref{lem:sieve-residue-function} and
+Equation~\eqref{eq:sieve-local-tail} bound its $\nu_\phi$-measure by
+$O(V_\phi p^{-J_p})=O(V_\phi/T_0)$.  Summing over $p\in\mathcal P_Y$
+gives $O(V_\phi Y/T_0)$.  By Equation~\eqref{eq:sieve-pointwise-weight},
+the contribution after multiplication by $\mathcal W$ is
+$O_\epsilon(V_\phi X^{\epsilon+\delta-\eta})$.
+Choose $\epsilon<\eta-\delta$.  This is smaller than $V_\phi P_Y$ by a
+power of $X$, in view of Equation~\eqref{eq:sieve-product-lower}.
+
+We also need the analogous removal for a fixed nonempty finite set
+$\mathcal Q$ of primes, even though some of them may belong to the varying
+set $S$.  Put $N=|\mathcal Q|$.  If
+$\prod_{p\in\mathcal Q\setminus S}p^{n_p(z)}>T_0$, some prime in
+$\mathcal Q\setminus S$ has $p^{n_p(z)}>T_0^{1/N}$.  Put
+$J_{p,\mathcal Q}=\lfloor\log T_0/(N\log p)\rfloor+1$.  The event
+$n_p\ge J_{p,\mathcal Q}$ is defined modulo at most
+$T_0^{C/N}p^{2C}\le X^\theta$ for large $X$, and its local probability
+is at most $Cp^{-J_{p,\mathcal Q}}\le C T_0^{-1/N}$.
+Lemma~\ref{lem:sieve-residue-function} therefore bounds the measure of
+the union by
+$O_{\mathcal Q}(V_\phi T_0^{-1/N})$.  After multiplication by
+Equation~\eqref{eq:sieve-pointwise-weight} with $\epsilon<\eta/N$, its
+weighted contribution again saves a power of $X$ relative to $V_\phi$.
+If $\mathcal Q\setminus S$ is empty the event is empty.  Thus this
+removal is uniform in the possible intersections of $\mathcal Q$ with $S$.
+
+Choose a fixed $M_0>0$ and then a fixed $y_0>4$ such that
+\begin{equation}\label{eq:sieve-crossing-parameters}
+ \eta M_0>2B\log A_0+1,\qquad M_0/\log y_0<\tfrac14.
+\end{equation}
+Apply the finite-set removal with $\mathcal Q=\{p:p\le y_0\}$.
+These choices precede the choice of the small $\epsilon$ used in that
+removal, and all are independent of $X$.
+
+The retained first-crossing prefixes therefore have product in
+$(T_0,T_0^2]$ and last prime greater than $y_0$.  Group these assignments
+by the interval containing their last prime:
+\begin{equation}\label{eq:sieve-buckets}
+ \sqrt y<p\le y,\qquad y=Y^{1/2^i},\quad i=0,1,2,\ldots.
+\end{equation}
+Only groups with $y>y_0$ occur.  Every unassigned active prime is greater
+than the last assigned prime and hence greater than $\sqrt y$.  The total
+weight of those primes, including primes beyond $Y$, is therefore at most
+\begin{equation}\label{eq:sieve-unassigned}
+ \exp\left(\frac{2B\log A_0}{\log y}\log X\right).
+\end{equation}
+For each assignment in the group, require inactivity at its unassigned
+primes in $\mathcal P_{\sqrt y}$ and apply
+Lemma~\ref{lem:sieve-assignment}.  We use Rankin's power majorant
+\cite{Rankin1938}: since the assigned product exceeds $T_0$, its upper
+bound can be multiplied by
+\[
+ T_0^{-s}\prod_{\text{assigned }p}p^{s j_p}\ge1,
+ \qquad s=M_0/\log y.
+\]
+Now enlarge the sum of these nonnegative upper bounds to all subsets and
+all positive depths through $y$.  This enlarges only a numerical sum after
+the lemma has been applied to the genuine prefixes; it invokes no residue
+estimate for the newly admitted products above $T_0^2$.
+
+For $p\in\mathcal P_{\sqrt y}$, the option of no assignment has weight
+$1-\vartheta_p$ and the assigned options have total weight
+$\sum_{j\ge1}\E_p(b_p\mathbf 1_{\{n_p=j\}})p^{sj}$.  Their sum is
+$\E_p(b_p p^{s n_p})$.  For $\sqrt y<p\le y$ outside $S$, inactivity was
+not imposed, so the option of no assignment has weight one.  The total
+there is $\E_p(b_p p^{s n_p})+\vartheta_p$.  Thus the group's contribution
+is at most
+\begin{equation}\label{eq:sieve-bucket-bound}
+ C V_\phi T_0^{-s}
+ \exp\left(\frac{2B\log A_0}{\log y}\log X\right)
+ \prod_{p\in\mathcal P_{\sqrt y}}\E_p(b_p p^{s n_p})
+ \prod_{\substack{\sqrt y<p\le y\\p\notin S}}
+       \bigl(\E_p(b_p p^{s n_p})+\vartheta_p\bigr).
+\end{equation}
+
+We estimate these local moments uniformly for $y>y_0$.  By
+Equation~\eqref{eq:sieve-crossing-parameters}, $s<1/4$.  The total
+contribution of depths $j\ge2$ to $\E_p(b_p p^{s n_p})$ is at most
+\[
+ C^2\sum_{j\ge2}(1+j)^C p^{-(1-s)j}\ll_C p^{-3/2}.
+\]
+At depth one, the increase over $\E_p b_p$ is bounded by
+\[
+ \frac{C'}p(p^s-1)
+   \ll_{M_0}\frac{\log p}{p\log y}
+ \qquad(p\le y),
+\]
+because $0\le s\log p\le M_0$.  Chebyshev's estimate shows that these
+increases have bounded sum over $p\le y$.  Mertens' estimate also gives
+$\sum_{\substack{\sqrt y<p\le y\\p\notin S}}\vartheta_p\ll1$.  Since
+$\E_p b_p\ge1/2$, comparison of the products in
+Equation~\eqref{eq:sieve-bucket-bound} with their base means gives
+\[
+ \prod_{p\in\mathcal P_{\sqrt y}}\E_p(b_p p^{s n_p})
+ \prod_{\substack{\sqrt y<p\le y\\p\notin S}}
+       \bigl(\E_p(b_p p^{s n_p})+\vartheta_p\bigr)
+ \ll_{M_0}\prod_{p\in\mathcal P_y}\E_p b_p.
+\]
+For a fixed $C_2$, a final use of
+Equation~\eqref{eq:sieve-mean-lower} and Mertens' estimate yields
+\[
+ \prod_{p\in\mathcal P_y}\E_p b_p
+ \le P_Y\prod_{\substack{y<p\le Y\\p\notin S}}
+                 (1-\vartheta_p)^{-1}
+ \ll P_Y\left(\frac{\log Y}{\log y}\right)^{C_2}.
+\]
+By Equation~\eqref{eq:sieve-crossing-parameters}, the bound for the group
+in Equation~\eqref{eq:sieve-bucket-bound} is consequently at most
+\[
+ C_{M_0}V_\phi P_Y
+ \exp\left(-\frac{\log X}{\log y}\right)
+ \left(\frac{\log Y}{\log y}\right)^{C_2}.
+\]
+For the values of $y$ in Equation~\eqref{eq:sieve-buckets}, its sum is
+bounded by
+\[
+ C_{M_0}V_\phi P_Y
+       \sum_{i\ge0}\exp(-2^i/\delta)\,2^{iC_2}
+ \ll V_\phi P_Y.
+\]
+Together with the small-product part and the two discarded sets, this
+proves Equation~\eqref{eq:sieve-average}.
+\end{proof}
+
+\subsection{The planar exterior count}
+
+\begin{proof}[Proof of Theorem~\ref{thm:exterior-count}]
+Fix the nonzero trace and bounded region in that theorem.  Let
+$\mathcal R$ be any of its rectangles, in the specified two partition
+coordinates.  Choose an allowed smooth function $\phi$ that majorizes
+$\mathbf 1_{\mathcal R}$, with its two scales equal to the side lengths
+of $\mathcal R$.  Its support lies in a fixed enlargement of the bounded
+region and
+\[
+ A_\phi\asymp\operatorname{area}(\mathcal R).
+\]
+The constants are uniform over the positions, side lengths, and aspect
+ratios of the rectangles.  Proposition~\ref{prop:ext-multiplicity-majorant}
+and Proposition~\ref{prop:sieve-averaging} give
+\begin{equation}\label{eq:sieve-before-euler}
+ \sum_{\substack{\Tr z=m\\z\in\mathcal R}}M(z)
+ \ll \sqrt D\, A_\phi
+       \prod_{p\in S}(1-p^{-1})Z_p
+       \prod_{p\in\mathcal P_Y}\E_p b_p,
+ \qquad
+ Z_p=\prod_{\substack{\mathfrak p\mid p\\\mathfrak p\subset R}}
+                (1-\Nm(\mathfrak p)^{-1})^{-1}.
+\end{equation}
+Here $\prod_{p\in S}|D|_p^{-1/2}=\sqrt D$, because $S$ contains every
+prime divisor of $D$, and Equation~\eqref{eq:sieve-volume} absorbed the
+trace densities.  The restriction to integral points outside $S$ in the
+multiplicity proposition agrees with the support of $\nu_\phi$.
+
+The exact unramified mean in Lemma~\ref{lem:ext-unramified-data} is
+\[
+ \E_p b_p=(1-p^{-1})(1-p^{-3})Z_p\qquad(p\notin S).
+\]
+Since $1-p^{-3}\le1$, the two Euler products on the right of
+Equation~\eqref{eq:sieve-before-euler} are at most the product of
+$(1-p^{-1})Z_p$ over $p\in S\cup\mathcal P_Y$.
+Ideal factorization in the degree-four maximal order gives, uniformly
+also at ramified primes,
+\[
+ \log Z_p=\frac{a_K(p)}p+O(p^{-2}).
+\]
+Thus
+\begin{equation}\label{eq:sieve-euler}
+ \prod_{p\in S}(1-p^{-1})Z_p
+       \prod_{p\in\mathcal P_Y}\E_p b_p
+ \ll
+ \exp\left(\sum_{p\in S\ \mathrm{or}\ p\le Y}
+                    \frac{a_K(p)-1}{p}\right).
+\end{equation}
+
+For fixed $m$, the radical of $S$ is polynomial in $X$.  Choose a fixed
+$c\ge1/2$ large enough that, for all sufficiently large $X$, every prime
+in $S$ and every prime at most $Y$ is at most $Z=X^c$.  Since
+$a_K(p)\ge0$, the exponent in Equation~\eqref{eq:sieve-euler} is at most
+\[
+ \sum_{p\le Z}\frac{a_K(p)}p-\sum_{p\le Y}\frac1p.
+\]
+We have $Z\ge D^{1/4}$ because $X=q\sqrt D\ge\sqrt D$.
+Equation~\eqref{eq:ord-prime-sum} and Mertens' estimate now give
+\[
+ \exp\left(\sum_{p\in S\ \mathrm{or}\ p\le Y}
+                    \frac{a_K(p)-1}{p}\right)
+ \ll \frac{\kappa\log Z}{\log Y}
+ =\frac c\delta\,\kappa.
+\]
+This also applies when $D$ remains bounded and $q$ tends to infinity,
+by the stated uniform range of Lemma~\ref{lem:ord-residue}.
+Substitution in Equation~\eqref{eq:sieve-before-euler} proves
+\[
+ \sum_{\substack{\Tr z=m\\z\in\mathcal R}}M(z)
+       \ll\kappa\sqrt D\,\operatorname{area}(\mathcal R).
+\]
+The only box-dependent quantity in the proof is $A_\phi$, which is
+comparable to the area with a fixed constant.  All other estimates are
+uniform over the box translations and the two scales between
+$X^{-\tau}$ and the fixed upper bounds.  The local multiplicity majorant
+has the same constant for every chosen ideal representative system, and
+its right side is independent of that choice.  Thus the bound is uniform
+in the representative system as well.  This proves the precise
+uniformity asserted in Theorem~\ref{thm:exterior-count}.
+\end{proof}

preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/quartic-torus-packets.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:95c2a937da74097afe5d155a412fa89a2c4560a8231f10c4230244a8c6bed0f5
+size 759557

preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/README.md

@@ -0,0 +1,18 @@
+# [Equidistribution of Primitive Sextic Torus Packets](primitive-sextic-torus-packets.pdf)
+
+**Author:** OpenAI
+
+**Date:** October 5, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026,
+  author = {{OpenAI}},
+  title = {{Equidistribution of Primitive Sextic Torus Packets}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/primitive-sextic-torus-packets.pdf}{OAI:Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026}},
+  year = {2026}
+}
+```
+

preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/figures/product-mechanism.tex

@@ -0,0 +1,102 @@
+\begin{figure}[tbp]
+\centering
+\begin{tikzpicture}[
+  x=1cm,y=1cm,
+  font=\small,
+  >={Stealth[length=2mm]},
+  patch/.style={draw=black!75,line width=.55pt,fill=black!4},
+  selected/.style={draw=blue!65!black,line width=.55pt,fill=blue!16},
+  coarse/.style={draw=black!70,dashed,line width=.65pt},
+  fine/.style={draw=black!35,line width=.25pt},
+  bins/.style={draw=black!70,densely dotted,line width=.5pt},
+  note/.style={align=left,font=\footnotesize},
+  leader/.style={draw=black!65,line width=.4pt,->}
+]
+\path[use as bounding box] (0,-5.15) rectangle (14.2,4.35);
+
+\node[anchor=west,font=\small\bfseries] at (0,4.1)
+  {(a) Contract $W$: identify the $U$ factor and prove independence};
+\node[align=center,font=\footnotesize] at (2.35,3.67)
+  {one $q$-cell; full $w,u$ ranges};
+\draw[patch] (.65,1.3) rectangle (4.05,3.35);
+\draw[selected] (2.15,1.3) rectangle (2.75,3.35);
+\node[font=\footnotesize,anchor=north] at (2.45,1.19) {$w$-cell};
+\node[anchor=east] at (.51,3.2) {$u$};
+\node[anchor=north] at (3.9,1.19) {$w$};
+
+\draw[->,line width=.8pt] (4.65,2.35) -- (7.1,2.35);
+\node[above] at (5.87,2.45) {$a_0^{t}$, $t\asymp T$};
+\node[below,font=\footnotesize,align=center] at (5.87,2.18)
+  {$W$ contracts; $U$ is fixed\\$T\prec F$};
+
+\draw[coarse] (7.95,1.15) rectangle (10.15,3.5);
+\draw[patch] (8.55,1.3) rectangle (9.55,3.35);
+\draw[selected] (8.991,1.3) rectangle (9.168,3.35);
+\foreach \x in {8.6,8.7,8.8,8.9,9.0,9.1,9.2,9.3,9.4,9.5}
+  \draw[fine] (\x,1.3) -- (\x,3.35);
+\foreach \y in {1.78,2.31,2.84}
+  \draw[bins] (7.95,\y) -- (10.15,\y);
+\node[font=\footnotesize] at (9.05,3.73) {output $U$ chart};
+\node[note,anchor=west] at (10.58,3.18)
+  {One coarse transverse cell\\contains the whole patch.};
+\draw[leader] (10.5,3.15) -- (10.17,3.15);
+\node[note,anchor=west] at (10.58,2.26)
+  {Fine transverse cells\\resolve the $w$-label.};
+\draw[leader] (10.5,2.24) -- (9.28,2.24);
+\node[note,anchor=west] at (10.58,1.42) {Common $U$ bins};
+\draw[leader] (10.5,1.48) -- (10.04,1.78);
+\node[align=center,font=\footnotesize] at (7.1,.5)
+  {Mesh comparison: revealing the $w$-cell changes the $U$-bin law by $o(1)$ in sampling mean.};
+
+\node[anchor=west,font=\small\bfseries] at (0,-.22)
+  {(b) Expand a $W$ tile: identify the remaining $W$ factor};
+\node[align=center,font=\footnotesize] at (2.35,-.86)
+  {one stage-$k$ tile\\shape $\Delta^k\mathcal F$};
+\draw[patch] (2.0,-3.7) rectangle (2.7,-1.55);
+\draw[selected] (2.0,-2.68) rectangle (2.7,-2.43);
+\foreach \x in {2.175,2.35,2.525}
+  \draw[bins] (\x,-3.7) -- (\x,-1.55);
+\node[anchor=east] at (1.86,-1.72) {$u$};
+\node[anchor=north] at (2.65,-3.81) {$w$};
+\node[note,anchor=east] at (1.72,-2.57) {$u$-cell\\depth $S$};
+\draw[leader] (1.8,-2.57) -- (1.98,-2.57);
+
+\draw[->,line width=.8pt] (3.65,-2.57) -- (6.5,-2.57);
+\node[above] at (5.08,-2.45) {$a_0^{-(k+m_0)}$};
+\node[below,font=\footnotesize,align=center] at (5.08,-2.73)
+  {$W$ expands; $U$ is fixed\\$S\prec T\prec F$, $k\asymp T$};
+
+\node[align=center,font=\footnotesize] at (8.63,-.88)
+  {fixed output shape\\$\Delta^{-m_0}\mathcal F$};
+\draw[patch] (7.05,-3.7) rectangle (10.2,-1.55);
+\draw[selected] (7.05,-2.68) rectangle (10.2,-2.43);
+\foreach \x in {7.8375,8.625,9.4125}
+  \draw[bins] (\x,-3.7) -- (\x,-1.55);
+\draw[coarse] (6.9,-2.92) rectangle (10.35,-2.2);
+\foreach \y in {-2.63,-2.58,-2.53,-2.48}
+  \draw[fine] (7.05,\y) -- (10.2,\y);
+\node[note,anchor=west] at (10.58,-1.94)
+  {Only the $u$-subpatch fits\\in one coarse transverse\\cell for the $W$ chart.};
+\draw[leader] (10.5,-2.02) -- (10.35,-2.2);
+\node[note,anchor=west] at (10.58,-3.18)
+  {Fine transverse cells\\resolve the $u$-label.};
+\draw[leader] (10.5,-3.05) -- (9.72,-2.6);
+\node[align=center,font=\footnotesize] at (8.63,-4.04)
+  {Stage-$(k+j)$ subtiles resolve $W$ tests.};
+\node[align=center,font=\footnotesize,text width=14cm] at (7.1,-4.78)
+  {Information comparison: an $O(S)$ budget over $T$ stages makes the $u$-label change\\
+   the descendant-tile law by $o(1)$ in sampling mean, at a suitable stage $k\asymp T$.};
+\end{tikzpicture}
+\caption{The two different comparisons in the product rule. Each input picture
+fixes a $q$-cell at depth $F$; the coordinate $q$ is suppressed. Dashed outlines
+denote coarse output transverse cells; thin solid lines denote fine transverse
+cells. Dotted lines denote fixed $U$ bins in (a) and stage-$(k+j)$ $W$ subtiles in (b).
+Shading marks a labelled subpatch, not a density. In (a), contraction permits
+one coarse transverse-cell comparison for the whole patch and its subpatches.
+In (b), expansion leaves the full $U$ interval broad. The full patch is compared
+with $V$ data and the thin $u$-subpatch with $W$ data; the information estimate
+compares their descendant-tile laws and hence their resolved $W$ tests. All widths and
+tile shapes are schematic: $W$ can have several dimensions and its tiles need
+not be rectangles. The resulting factor identifications are projective.}
+\label{fig:product-mechanism}
+\end{figure}

preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/main.tex

@@ -0,0 +1,88 @@
+\documentclass[11pt]{article}
+\ifdefined\pdfinfoomitdate\pdfinfoomitdate=1\fi
+\ifdefined\pdftrailerid\pdftrailerid{}\fi
+\ifdefined\pdfsuppressptexinfo\pdfsuppressptexinfo=15\fi
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage[margin=1in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{microtype}
+\usepackage{enumitem,booktabs}
+\usepackage{xcolor,tikz}
+\usetikzlibrary{arrows.meta,positioning,calc,patterns,decorations.pathreplacing}
+\usepackage[numbers,sort&compress]{natbib}
+\usepackage[hyphens]{url}
+\usepackage{hyperref}
+\hypersetup{colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,
+ urlcolor=blue!45!black,pdfauthor={OpenAI},
+ pdftitle={Equidistribution of Primitive Sextic Torus Packets}}
+\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{\R}{\mathbb R}
+\newcommand{\Z}{\mathbb Z}
+\newcommand{\N}{\mathbb N}
+\newcommand{\Q}{\mathbb Q}
+\newcommand{\C}{\mathbb C}
+\newcommand{\E}{\mathbb E}
+\newcommand{\Prb}{\mathbb P}
+\newcommand{\one}{\mathbf 1}
+\newcommand{\dd}{\mathop{}\!d}
+\newcommand{\idtext}[1]{\begingroup\urlstyle{same}\nolinkurl{#1}\endgroup}
+\DeclareMathOperator{\supp}{supp}
+\DeclareMathOperator{\SL}{SL}
+\DeclareMathOperator{\GL}{GL}
+\DeclareMathOperator{\Ad}{Ad}
+\DeclareMathOperator{\diag}{diag}
+\DeclareMathOperator{\Tr}{Tr}
+\DeclareMathOperator{\covol}{covol}
+\DeclareMathOperator{\Disc}{Disc}
+\DeclareMathOperator{\vol}{vol}
+\DeclareMathOperator{\Res}{Res}
+\DeclareMathOperator{\Fix}{Fix}
+\DeclareMathOperator{\Mat}{Mat}
+\DeclareMathOperator{\End}{End}
+\DeclareMathOperator{\Var}{Var}
+\DeclareMathOperator{\dist}{dist}
+\DeclareMathOperator{\diam}{diam}
+\DeclareMathOperator{\sgn}{sgn}
+\DeclareMathOperator{\rank}{rank}
+\setlist[enumerate]{leftmargin=*,itemsep=4pt}
+\setlength{\emergencystretch}{2em}
+\setcounter{tocdepth}{1}
+\title{Equidistribution of Primitive Sextic Torus Packets}
+\author{OpenAI}
+\date{October 5, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We prove that the complete ideal-class packets of maximal orders in
+totally real sextic fields with no proper intermediate fields become
+equidistributed, with no escape of mass, in the space of unimodular
+lattices as their field discriminants tend to infinity. The limit is
+Haar probability measure. Each packet includes all coordinate-sign
+translates and is weighted by diagonal-orbit volume.
+\end{abstract}
+\tableofcontents
+\input{sections/introduction}
+\input{sections/arithmetic}
+\input{sections/homogeneous}
+\input{sections/conditioning}
+\input{sections/product}
+\input{sections/activity}
+\input{sections/entropy}
+\input{sections/combination}
+\input{sections/assembly}
+\begingroup
+\small\raggedright
+\setlength{\bibsep}{3pt}
+\bibliographystyle{plainnat}
+\bibliography{references}
+\endgroup
+\end{document}

preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/references.bib

@@ -0,0 +1,198 @@
+@book{Linnik1968,
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+  volume = {45},
+  publisher = {Springer-Verlag},
+  address = {Berlin--Heidelberg--New York},
+  year = {1968},
+  note = {Translated from the Russian by M. S. Keane}
+}
+
+@article{Duke1988,
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+  title = {Distribution of periodic torus orbits on homogeneous spaces},
+  journal = {Duke Math. J.},
+  volume = {148},
+  number = {1},
+  year = {2009},
+  pages = {119--174},
+  doi = {10.1215/00127094-2009-023}
+}
+
+@article{ELMV2011,
+  author = {Einsiedler, Manfred and Lindenstrauss, Elon and Michel, Philippe and Venkatesh, Akshay},
+  title = {Distribution of periodic torus orbits and {Duke}'s theorem for cubic fields},
+  journal = {Ann. of Math. (2)},
+  volume = {173},
+  number = {2},
+  year = {2011},
+  pages = {815--885},
+  doi = {10.4007/annals.2011.173.2.5}
+}
+
+@article{ELMV2012,
+  author = {Einsiedler, Manfred and Lindenstrauss, Elon and Michel, Philippe and Venkatesh, Akshay},
+  title = {The distribution of closed geodesics on the modular surface, and {Duke}'s theorem},
+  journal = {Enseign. Math. (2)},
+  volume = {58},
+  number = {3--4},
+  year = {2012},
+  pages = {249--313},
+  doi = {10.4171/LEM/58-3-2}
+}
+
+@article{EKL2006,
+  author = {Einsiedler, Manfred and Katok, Anatole and Lindenstrauss, Elon},
+  title = {Invariant measures and the set of exceptions to {Littlewood}'s conjecture},
+  journal = {Ann. of Math. (2)},
+  volume = {164},
+  number = {2},
+  year = {2006},
+  pages = {513--560},
+  doi = {10.4007/annals.2006.164.513}
+}
+
+@article{Khayutin2019,
+  author = {Khayutin, Ilya},
+  title = {Arithmetic of double torus quotients and the distribution of periodic torus orbits},
+  journal = {Duke Math. J.},
+  volume = {168},
+  number = {12},
+  year = {2019},
+  pages = {2365--2432},
+  doi = {10.1215/00127094-2019-0016}
+}
+
+@article{WieserYang2026,
+  author = {Wieser, Andreas and Yang, Pengyu},
+  title = {A uniform {Linnik} basic lemma and entropy bounds},
+  journal = {Comment. Math. Helv.},
+  year = {2026},
+  note = {Published online January 22, 2026},
+  doi = {10.4171/CMH/613}
+}
+
+@article{Stark1974,
+  author = {Stark, Harold M.},
+  title = {Some effective cases of the {Brauer--Siegel} theorem},
+  journal = {Invent. Math.},
+  volume = {23},
+  number = {2},
+  year = {1974},
+  pages = {135--152},
+  doi = {10.1007/BF01405166}
+}
+
+@article{Shiu1980,
+  author = {Shiu, Peter},
+  title = {A {Brun--Titchmarsh} theorem for multiplicative functions},
+  journal = {J. Reine Angew. Math.},
+  volume = {313},
+  year = {1980},
+  pages = {161--170},
+  doi = {10.1515/crll.1980.313.161}
+}
+
+@article{Pollack2020,
+  author = {Pollack, Paul},
+  title = {Nonnegative multiplicative functions on sifted sets, and the square roots of {$-1$} modulo shifted primes},
+  journal = {Glasg. Math. J.},
+  volume = {62},
+  number = {1},
+  year = {2020},
+  pages = {187--199},
+  doi = {10.1017/S0017089519000041},
+  note = {Corrected author version available at \url{https://www.pollack-math.net/brunmultiplicative4.pdf}}
+}
+
+@book{Cassels1959,
+  author = {Cassels, J. W. S.},
+  title = {An Introduction to the Geometry of Numbers},
+  series = {Grundlehren der mathematischen Wissenschaften},
+  volume = {99},
+  publisher = {Springer-Verlag},
+  address = {Berlin--G{\"o}ttingen--Heidelberg},
+  year = {1959}
+}
+
+@article{EK2003,
+  author = {Einsiedler, Manfred and Katok, Anatole},
+  title = {Invariant measures on {$G/\Gamma$} for split simple {Lie} groups {$G$}},
+  journal = {Comm. Pure Appl. Math.},
+  volume = {56},
+  number = {8},
+  pages = {1184--1221},
+  year = {2003},
+  doi = {10.1002/cpa.10092}
+}
+
+@article{EK2005,
+  author = {Einsiedler, Manfred and Katok, Anatole},
+  title = {Rigidity of measures---the high entropy case and non-commuting foliations},
+  journal = {Israel J. Math.},
+  volume = {148},
+  pages = {169--238},
+  year = {2005},
+  doi = {10.1007/BF02775436}
+}
+
+@incollection{EL2010,
+  author = {Einsiedler, Manfred and Lindenstrauss, Elon},
+  title = {Diagonal actions on locally homogeneous spaces},
+  booktitle = {Homogeneous Flows, Moduli Spaces and Arithmetic},
+  series = {Clay Mathematics Proceedings},
+  volume = {10},
+  publisher = {American Mathematical Society},
+  address = {Providence, RI},
+  year = {2010},
+  pages = {155--241}
+}
+
+@article{Rokhlin1967,
+  author = {Rokhlin, V. A.},
+  title = {Lectures on the entropy theory of measure-preserving transformations},
+  journal = {Russian Math. Surveys},
+  volume = {22},
+  number = {5},
+  year = {1967},
+  pages = {1--52},
+  doi = {10.1070/RM1967v022n05ABEH001224}
+}
+
+@book{CoverThomas2006,
+  author = {Cover, Thomas M. and Thomas, Joy A.},
+  title = {Elements of Information Theory},
+  edition = {Second},
+  publisher = {Wiley-Interscience},
+  address = {Hoboken, NJ},
+  year = {2006},
+  doi = {10.1002/047174882X}
+}
+
+@misc{PrimeDegreePackets,
+  author = {{OpenAI}},
+  title = {{Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026/paper.pdf}{OAI:Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026}},
+  year = {2026}
+}
+
+@misc{QuarticPackets,
+  author = {{OpenAI}},
+  title = {{Equidistribution of primitive quartic torus packets for arbitrary orders}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026/quartic-torus-packets.pdf}{OAI:Equidistribution-of-Primitive-Quartic-Torus-Packets-for-Arbitrary-Orders-October-5-2026}},
+  year = {2026}
+}

preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/activity.tex

@@ -0,0 +1,289 @@
+\section{Active roots and their support}\label{sec:activity}
+
+The product rule turns the plaque measures into a directed graph at each
+augmented state.  We first show that its arrows compose and describe the
+support of the positive-root plaques.  We then realize this graph by
+measurable indicators on the original spaces, retaining the long-wall
+comparison needed for the entropy argument.
+
+Write $\mathsf P$ for the law of the limiting augmented state
+$\omega=(x,(P^B)_B)$.  For $j\ne k$, define
+\[
+ a_{jk}(\omega)
+   =\mathbf 1_{\{m^{U_{jk}}_\omega\text{ is not supported on }\{1\}\}}.
+\]
+We call $jk$ \emph{active} when $a_{jk}=1$.  This definition does not depend
+on the representative of the projective measure.  It is measurable:
+after normalizing by a positive integral against a bump near the identity,
+activity is detected by a countable family of nonnegative continuous
+tests supported away from the identity.  All assertions below are made
+on a common set of full $\mathsf P$-measure on which the comparison and
+product rules hold.
+
+\begin{proposition}[Composition and positive-root support]
+\label{prop:activity}
+Almost surely, activity is transitively closed:
+\[
+ a_{pq}(\omega)a_{qr}(\omega)
+       \bigl(1-a_{pr}(\omega)\bigr)=0
+       \qquad(p,q,r\text{ distinct}).
+\]
+Let $b\in L$ be regular, let $V$ be the pattern group of all
+$b$-positive roots, and let $V_\omega^+$ be the subgroup generated by
+the active $b$-positive root groups.  These active roots form a pattern,
+and
+\[
+                   \supp m^V_\omega\subset V_\omega^+.
+\]
+Both conclusions hold simultaneously for every regular $b$.
+\end{proposition}
+
+\begin{proof}
+First observe that Theorem~\ref{thm:product} factors the measure of any
+pattern group into its one-dimensional root measures, in a suitable
+order.  Indeed, the cone generated by its roots is pointed.  Choose a
+root $\alpha$ on an exposed extremal ray.  No other root in the pattern
+is proportional to $\alpha$, so a diagonal logarithm can vanish on
+$\alpha$ and be strictly negative on every remaining root.  Removing
+$\alpha$ leaves a normal pattern subgroup $W$: a sum of two roots in
+the original pattern cannot equal $\alpha$, since extremality would
+force both summands onto its ray.  Thus the hypotheses of
+Theorem~\ref{thm:product} hold for $V=WU_\alpha$.  Iterate, allowing
+$W=\{1\}$ at the last step.
+
+To prove transitivity, fix distinct $p,q,r$ and consider the
+three-root Heisenberg group generated by $U_{pq},U_{qr},U_{pr}$.
+Suppose that $a_{pr}=0$.  Factoring first with $U_{pq}$ as the final
+factor, and then with $U_{qr}$ as the final factor, gives two
+descriptions of the same projective measure.  Write
+$u_{jk}(t)=1+tE_{jk}$.  Since the $U_{pr}$ factor is a point mass, the
+first description is supported on the matrices
+\[
+ u_{qr}(t)u_{pq}(s)=1+sE_{pq}+tE_{qr},
+\]
+whereas the other is the pushforward of the product of the two root
+measures under
+\[
+ u_{pq}(s)u_{qr}(t)
+     =1+sE_{pq}+tE_{qr}+stE_{pr}.
+\]
+If both roots were active, each root measure would give positive mass
+to some compact parameter set disjoint from zero.  Their product would
+then give positive mass to matrices with nonzero $pr$ entry, contrary
+to the first description.  The projective identities apply on compact
+ranges containing these sets, so all masses used here are finite.
+This proves transitivity.
+
+For the support assertion, use the successive factorization already
+proved.  Every inactive root factor is a point mass at the identity;
+every other factor is supported on its own active root group.  The
+product is therefore supported on the subgroup generated by the active
+roots.  Transitivity says exactly that the active positive roots are
+closed under root addition, so this subgroup is a pattern group and is
+closed.  There are only finitely many positive-root patterns, which
+also gives the simultaneous assertion.
+\end{proof}
+
+The entropy argument will use plaques enlarged by the diagonal group.
+The following consequence retains the activity at the sampled state.
+
+\begin{lemma}[Support on plaques enlarged by $A$]\label{lem:va-support}
+With $V$ and $V_\omega^+$ as in Proposition~\ref{prop:activity}, the
+limiting $VA$-plaque probability, written in displacement coordinates
+from its sampled point, is supported on $V_\omega^+A$ within its chart.
+\end{lemma}
+
+\begin{proof}
+Use a product box with coordinates $\psi(q)va$, with product coordinate
+domains.  Local $A$-invariance gives, for each $i$, a product
+decomposition
+\[
+                d\mu_i(q,v,a)=d\nu_i(q,v)\,da
+\]
+inside the box, where $da$ is Haar measure restricted to its coordinate
+box and the normalization is absorbed into $\nu_i$.  For $VA$ plaques
+the transverse coordinate is $q$; for $V$ plaques it is $(q,a)$.
+Choose a common $q$ grid and supplement it by an $a$ grid for the latter
+conditioning.  Conditional on a $q$ cell, the $VA$ probability is the
+product of a probability on $v$ and normalized Haar measure on $a$.
+Conditioning also on an $a$ cell leaves the $v$ probability unchanged.
+These are exact identities before taking a limit.  The interior-depth
+comparisons permit these compatible grids in the defining data, and
+the identities pass to the joint limit by continuous tests and the
+null-boundary convention.
+
+At a sample $x=\psi(q_s)v_sa_s$, the $V$ displacements corresponding to
+varying $v$ at fixed $a_s$ are
+\[
+                         a_s^{-1}v_s^{-1}va_s.
+\]
+By Proposition~\ref{prop:activity}, their probability is supported on
+$V_\omega^+$.  A general $VA$ displacement is
+\[
+ a_s^{-1}v_s^{-1}va
+   =\bigl(a_s^{-1}v_s^{-1}va_s\bigr)\bigl(a_s^{-1}a\bigr),
+\]
+and hence belongs to $V_\omega^+A$ almost surely.  The argument applies
+on inner chart boxes; exhausting them gives the statement on the whole
+plaque box, whose boundary has zero probability.
+\end{proof}
+
+\begin{lemma}[Indicators along the sequence]\label{lem:activity-indicators}
+Fix an interior scale $R_i$ allowed in the long-wall comparison of
+Proposition~\ref{prop:plaque-comparison}, with ratio-separated depths on
+both sides of it inside $(l_{-,i},l_{+,i})$.  There exist measurable
+functions $g_{jk,i}:X\to\{0,1\}$ such that
+\[
+ \operatorname{Law}_{\mu_i}\bigl(Z_i(x),(g_{jk,i}(x))_{j\ne k}\bigr)
+ \ \Longrightarrow\
+ \operatorname{Law}_{\mathsf P}
+       \bigl(\omega,(a_{jk}(\omega))_{j\ne k}\bigr).
+\]
+They satisfy asymptotic transitivity:
+\[
+ \int_X g_{pq,i}(x)g_{qr,i}(x)
+                  \bigl(1-g_{pr,i}(x)\bigr)\,d\mu_i(x)=o(1)
+       \qquad(p,q,r\text{ distinct}),
+\]
+and, for every fixed $C<\infty$,
+\begin{equation}\label{eq:wall-invariance}
+ \sup_{\substack{b\in\ker\beta_{jk}\\ \|b\|\le CR_i}}
+ \int_X\bigl|g_{jk,i}(xa(b))-g_{jk,i}(x)\bigr|\,d\mu_i(x)
+                         \longrightarrow0.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Let $\mathcal S$ be the separable metric state space used for $Z_i$,
+and put $A_{jk}=\{\omega:a_{jk}(\omega)=1\}$.  Choose
+$\epsilon_r\downarrow0$.  For each root and each $r$, there is a
+$\mathsf P$-continuity set $D_{jk,r}\subset\mathcal S$ such that
+\[
+              \mathsf P(D_{jk,r}\mathbin\triangle A_{jk})
+                      <\epsilon_r.
+\]
+For completeness, use a countable family of continuous coordinate
+tests generating the Borel sets of $\mathcal S$.  For each test choose
+a countable dense family of thresholds outside the atoms of its
+distribution under $\mathsf P$.
+The finite Boolean combinations of the resulting threshold sets have
+null boundary and approximate every measurable set in
+$\mathsf P$-measure.  Thus the approximations may be made using
+finitely many data tests at each stage.
+
+For fixed $r$, set
+$g^{(r)}_{jk,i}(x)=\mathbf 1_{D_{jk,r}}(Z_i(x))$.
+Weak convergence of the states and the null boundaries give joint
+convergence of the states and these finitely many indicators.  Since
+the exact indicators are transitive, the limiting transitivity defect
+for any triple is at most $3\epsilon_r$.
+
+We next prove the uniform wall estimate at this fixed approximation
+level.  Fix a root, $C$, and any sequence
+$b_i\in\ker\beta_{jk}$ with $\|b_i\|\le CR_i$.  Both marginals of
+\[
+                    \bigl(Z_i(x),Z_i(xa(b_i))\bigr)
+\]
+have the ordinary state law, because $\mu_i$ is $A$-invariant.
+Consequently these paired laws are tight.  In every subsequential
+limit their two marginals are $\mathsf P$, and
+Proposition~\ref{prop:plaque-comparison} identifies the two projective
+$U_{jk}$ measures.  Their exact activity indicators are therefore
+equal.  Applying the continuity-set approximation separately in the
+two marginals shows that every such limit has
+\[
+ \mathbb E\bigl|
+       \mathbf 1_{D_{jk,r}}(\omega)
+        -\mathbf 1_{D_{jk,r}}(\omega')\bigr|
+                    \le 2\epsilon_r.
+\]
+The boundary of the tested event is contained in the union of the two
+marginal boundaries, so the corresponding prelimit expectations
+converge along this subsequence.  Since the sequence $b_i$ was
+arbitrary, a sequence chosen within a vanishing error of the supremum
+proves
+\[
+ \limsup_i\ \sup_{\substack{b\in\ker\beta_{jk}\\\|b\|\le CR_i}}
+ \int_X\bigl|g^{(r)}_{jk,i}(xa(b))-g^{(r)}_{jk,i}(x)\bigr|\,d\mu_i(x)
+                      \le2\epsilon_r.
+\]
+This argument uses limits of paired states even when $b_i$ diverges.
+
+Finally choose $r=r(i)\to\infty$ sufficiently slowly.  At stage $r$
+impose the preceding estimates for all roots, all triples, and all
+integers $1\le C\le r$, together with the first $r$ tests of a
+convergence-determining family for the joint state and indicator law.
+The limiting laws at fixed $r$ differ from the exact activity law by
+at most the sum of the finitely many approximation errors.  A
+diagonal choice therefore gives the asserted joint convergence,
+vanishing transitivity defects, and \eqref{eq:wall-invariance} for
+$g_{jk,i}=g^{(r(i))}_{jk,i}$.  Bounds for integral $C$ imply the bounds
+for every fixed finite $C$.
+\end{proof}
+
+We conclude with the relation between activity and pieces having
+countable intersections with root plaques.
+It is this relation that will exclude proper homogeneous components
+once all roots have been shown to be active.
+
+\begin{lemma}[Inactivity on pieces with countable plaque sections]\label{lem:transverse-inactive}
+Let $S\subset X$ be a Borel set whose intersection with each local
+$U_{jk}$ plaque is countable.  Then
+\[
+            a_{jk}(\omega)=0
+         \quad\text{for $\mathsf P$-almost every state with }x\in S.
+\]
+In particular, this applies to the fixed-point pieces of
+Lemma~\ref{lem:fixed-pieces} for roots crossing distinct eigenvalues:
+each such piece meets a sufficiently small root plaque in at most one
+point.  It also applies to countable unions of these pieces.
+\end{lemma}
+
+\begin{proof}
+Work first in a $U_{jk}$ flow box with coordinates $(q,h)$ and plaque
+probability $P^B$.  For each $q$, the section
+\[
+                  S_q=\{h:\psi(q)h\in S\}
+\]
+is countable.  The sampling identity for this single chart states
+that the conditional law of the sampled $h$, given $(q,P^B)$, is
+$P^B$.  A probability measure assigns mass zero to those points of a
+countable set which are not its atoms.  Consequently
+\[
+ \mathsf P\{x\in S\cap B,\ P^B(\{h\})=0\}=0.
+\]
+No conditioning on the other plaque probabilities is used here.
+Projective agreement in the chart identifies an atom at the sampled
+$h$ with an atom at the identity of $m^{U_{jk}}_\omega$.  A countable
+flow-box cover therefore shows that this root measure has an atom at
+the identity almost surely over $S$.
+
+It remains to show that, under the full state law, an atom at the
+identity forces the entire root measure to be supported there.
+Identify $U_{jk}$ with $\R$ by $t\mapsto1+tE_{jk}$.  Choose a fixed
+$b\in L$ with $\lambda=e^{\beta_{jk}(b)}>1$.  Fixed-motion equivariance
+in Proposition~\ref{prop:plaque-comparison}, together with
+$A$-invariance of the prelimit sampling law, says that the projective
+law of $m^{U_{jk}}_\omega$ is invariant under the pushforward by
+$t\mapsto\lambda t$.
+
+The event that this measure has an atom at zero is itself invariant
+under dilation.  If that event has positive probability, condition
+the projective-measure law on it and normalize each measure $m$ by
+its atom: $\nu=m/m(\{0\})$.  This is a measurable normalization of a
+locally finite measure, with $\nu(\{0\})=1$, and its law remains
+dilation-invariant.  For every compact interval $K$ disjoint from
+zero,
+\[
+ (t\mapsto\lambda^n t)_*\nu(K)
+         =\nu(\lambda^{-n}K)\longrightarrow0.
+\]
+Indeed, $\lambda^{-n}K$ lies in a shrinking punctured neighborhood of
+zero, whose mass tends to zero by local finiteness.  Dilation
+invariance makes each quantity in this display have the same
+distribution as $\nu(K)$.  Almost-sure convergence to zero therefore
+forces $\nu(K)=0$ almost surely.  A countable family of such intervals
+covers $\R\setminus\{0\}$, and hence $\nu=\delta_0$ almost surely.
+The atom event of probability zero is harmless.  Combining this
+conclusion with the first part proves the lemma.
+\end{proof}

preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/arithmetic.tex

@@ -0,0 +1,677 @@
+\section{Arithmetic estimates for the packets}\label{sec:arithmetic}
+
+This section establishes the three estimates on which the dynamical
+argument rests: uniform control of the cusp, a dimension bound for
+probability limits, and exponential decay on two-sided diagonal tubes.
+The first two follow by counting ordinary lattice vectors.  For the
+third, we count an ordinary vector together with a dual vector; their
+coordinatewise product lies in the codifferent and can be counted in a
+thin slice.
+
+For the volume, unfolding, and ordinary-vector estimates, we adapt
+the method of \cite[Sections~2--6]{PrimeDegreePackets} and
+\cite[Section~3]{QuarticPackets}.  We reproduce the needed arguments
+in the packet and Haar normalizations of this paper.
+
+Throughout this section, $K$ is a totally real field of degree six with
+no proper intermediate field, and $\sigma$ is an ordering of its real
+embeddings.  Write
+\[
+ D=|\Disc(K)|,\qquad Q=D^{1/2},\qquad
+ \kappa=\mathop{\rm Res}_{s=1}\zeta_K(s),\qquad
+ d_K(l)=\#\{\mathfrak b\subset\mathcal O_K:N\mathfrak b=l\}.
+\]
+All ideals counted here are nonzero.  For a sequence of fields and
+embedding orderings, we use the corresponding notation
+\[
+ D_i=|\Disc(K_i)|,\qquad Q_i=D_i^{1/2},\qquad
+ \kappa_i=\mathop{\rm Res}_{s=1}\zeta_{K_i}(s),\qquad
+ \mu_i=\mu_{K_i,\sigma_i}.
+\]
+Unless another dependence is indicated, constants below are absolute
+for this fixed degree.  In particular, they are independent of the
+field and of the ordering of its embeddings.
+
+\subsection{Volume and unfolding}
+
+We first make the sign convention in the packet definition explicit.
+The space $X=\SL_6(\Z)\backslash\SL_6(\R)$ parametrizes ordinary
+unimodular lattices: a lattice has a positively oriented basis, and
+any two such bases differ by an element of $\SL_6(\Z)$.  Consequently
+a diagonal sign matrix acts on lattices even when its determinant is
+$-1$.  The resulting lattice again represents a point of $X$ by choosing
+a positively oriented basis.  There is no additional orientation
+coordinate.
+
+Let $h_K$ and $R_K$ denote the ordinary class number and regulator,
+respectively.  Our convention for $R_K$ is the covolume of the absolute
+logarithms of the units after deleting the sixth coordinate.  Put
+\[
+ U=\mathcal O_K^\times,\qquad
+ U^+=\{\epsilon\in U:\sigma_j(\epsilon)>0\ (1\le j\le6)\},
+ \qquad s=[U:U^+].
+\]
+For a nonnegative Borel function $f$ on $\R^6$, define its lattice sum by
+\[
+ \widehat f(\Lambda)=\sum_{v\in\Lambda\setminus\{0\}}f(v).
+\]
+For $p>0$, define a logarithmic fiber integral
+\begin{align}
+ V_f(p)
+  =\sum_{\epsilon\in\{\pm1\}^6}\int_{\R^5}
+  f\bigl(\epsilon_1e^{u_1},\ldots,\epsilon_5e^{u_5},
+     \epsilon_6p e^{-u_1-\cdots-u_5}\bigr)
+       \,\dd u_1\cdots\dd u_5 .                 \label{eq:fiber-integral}
+\end{align}
+Thus $V_f(p)$ integrates over the set on which the absolute product of
+the six coordinates is $p$, with the Haar normalization used for $A$.
+
+\begin{proposition}[Packet volume and vector unfolding]
+\label{prop:packet-volume}
+Every orbit in $\mathcal P_{K,\sigma}$ has $A$-volume $(s/2)R_K$,
+and there are $h_K2^6/s$ distinct orbits.  In particular, the total
+packet volume is $Q\kappa$.  For every nonnegative Borel function $f$,
+\begin{equation}
+ \int_X\widehat f\,\dd\mu_{K,\sigma}
+   =\frac1{Q\kappa}\sum_{l\ge1}d_K(l)V_f(l/Q),
+                                                        \label{eq:vector-unfolding}
+\end{equation}
+where both sides are allowed to be infinite.
+\end{proposition}
+
+\begin{proof}
+The only roots of unity in a totally real field are $1$ and $-1$.
+The kernel of the absolute logarithmic map on $U$ therefore has
+order two.  Its image on $U^+$ has index $s/2$ in its image on $U$.
+The unit theorem now gives covolume $(s/2)R_K$ for the positive-unit
+logarithms in the coordinates $u_1,\ldots,u_5$.
+
+We check that these are exactly the stabilizers and identifications
+in the packet.  Suppose a real diagonal map sends $\sigma(I)$ onto
+$\sigma(J)$.  Choose $0\ne d\in I$, and let its image be $\sigma(e)$
+with $e\in J$.  All embeddings of $d$ are nonzero, so the diagonal
+entries of the map are $\sigma_j(e/d)$.  It is multiplication by
+$e/d\in K^\times$, and hence $J=(e/d)I$.  This observation also
+applies to normalized lattices after their positive covolume factors
+have been undone.  It follows that different ordinary ideal classes
+give disjoint packet orbits.  For a fixed ideal, the diagonal
+automorphisms are its units, because $(I:I)=\mathcal O_K$.
+
+The stabilizer in $A$ is consequently the image of $U^+$.
+Two sign translates belong to the same $A$-orbit exactly when their
+signs differ by a unit signature.  The signature image has cardinality
+$s$, giving $2^6/s$ orbits for each ordinary class and the stated orbit
+volume.  The analytic class number formula, in this normalization, is
+\[
+ Q\kappa=2^5h_KR_K.
+\]
+It agrees with the product of the orbit number and orbit volume.
+
+For completeness, let $\mathcal F$ be a fundamental domain for the
+positive-unit logarithms in the traceless diagonal algebra.  Summing
+over ordinary ideal-class representatives and all sign matrices
+counts each orbit exactly $s$ times.  Thus for any nonnegative $F$ on
+$X$,
+\begin{equation}
+ \int_XF\,\dd\mu_{K,\sigma}
+ =\frac1{sQ\kappa}\sum_{[I]}\sum_w
+   \int_{\mathcal F}F(\Lambda_{I,\sigma}w a(u))\,\dd u.
+                                                        \label{eq:packet-integral}
+\end{equation}
+Here $\dd u=\dd u_1\cdots\dd u_5$ and the sixth coordinate is minus
+their sum.
+
+Apply this identity to $F=\widehat f$.  A nonzero $d\in I$ determines
+the integral ideal
+\[
+ \mathfrak b=(d)I^{-1},
+ \qquad
+ \prod_{j=1}^6
+ \left|\covol(\sigma(I))^{-1/6}\sigma_j(d)\right|
+ =\frac{N\mathfrak b}{Q},
+\]
+because $\covol(\sigma(I))=QNI$.  The ideal $\mathfrak b$
+determines the class $[I]=[\mathfrak b]^{-1}$, and every integral
+$\mathfrak b$ occurs.  For that ideal, the possible $d$ form a single
+$U$-orbit, or $s$ orbits under $U^+$.  Summing a positive-unit orbit
+and integrating over $\mathcal F$ unfolds to the entire traceless
+diagonal algebra.  Multiplication by a representative of $U/U^+$
+translates the absolute logarithms and permutes the signs.  After the
+sum over $w$, each of these $s$ orbits therefore contributes
+$V_f(N\mathfrak b/Q)$.  Their number cancels the factor $s$ in
+\eqref{eq:packet-integral}, proving \eqref{eq:vector-unfolding}.
+All rearrangements are justified by nonnegativity.
+\end{proof}
+
+\subsection{Uniform counts of ideals}
+
+The normalization by $\kappa$ in \eqref{eq:vector-unfolding} makes it
+important to keep this same factor in upper bounds for ideal counts.
+We obtain it from a zero-free neighborhood of $1$, followed by Shiu's
+estimate for nonnegative multiplicative functions.
+
+\begin{lemma}[Uniform ideal estimates]\label{lem:ideal-estimate}
+For all sufficiently large discriminants $D$ in the present family,
+\begin{align}
+ \kappa&\gg (\log D)^{-1},                         \label{eq:residue-lower}\\
+ \sum_{p\le D}\frac{d_K(p)}p
+   &\le \log(\kappa\log D)+O(1),                 \label{eq:prime-sum}\\
+ \sum_{x-y<l\le x}d_K(l)&\ll y\kappa
+ \quad\left(D^{1/4}\le x\le D,
+                    \ x^{1/2}\le y\le x\right).  \label{eq:ideal-interval}
+\end{align}
+In \eqref{eq:prime-sum}, $p$ ranges over rational primes.  The constants
+and the discriminant threshold are uniform in $K$.
+\end{lemma}
+
+\begin{proof}
+We first record the precise zero exclusion needed here.  Stark's
+Lemma~3 shows that the rectangle
+\[
+ \Re s\ge1-\frac1{4\log D},\qquad
+ |\Im s|\le\frac1{4\log D}
+\]
+contains at most one nontrivial zero of $\zeta_K$, and that such a
+zero is real and simple.  His Lemma~8 applies also to nonnormal
+fields: a real zero $\beta$ with
+\[
+ 1-\frac1{4\cdot6!\log D}\le\beta<1
+\]
+would be a zero of the zeta function of a quadratic subfield of $K$
+\cite[Lemmas~3 and~8]{Stark1974}.  Our field has no such subfield.
+Consequently, for an absolute $c_0>0$,
+\begin{equation}
+ |1-\rho|\ge \frac{c_0}{\log D}
+ \quad\text{for every nontrivial zero $\rho$ of $\zeta_K$}.
+                                                        \label{eq:zero-exclusion}
+\end{equation}
+One can take $c_0=1/2880$.  This use of Stark's result does not require
+$K/\Q$ to be normal.
+
+Set $s_0=1+1/\log D$.  The logarithmic derivative of the completed
+Dedekind zeta function gives, for real $1<s\le s_0$,
+\begin{equation}
+ S(s):=\sum_\rho\Re\frac1{s-\rho}
+   =\frac1{s-1}+\frac12\log D
+        +\frac{\zeta'_K(s)}{\zeta_K(s)}+O(1).
+                                                        \label{eq:zeta-derivative}
+\end{equation}
+The zeros are counted with multiplicity.  The real-part sum converges,
+and the gamma-factor contribution is bounded uniformly because the
+degree and signature are fixed.  Each summand on the left is
+nonnegative, and $\zeta'_K(s_0)/\zeta_K(s_0)\le0$ by its Dirichlet
+series.  Hence $S(s_0)\ll\log D$.
+
+The same estimate holds throughout $(1,s_0]$.  Indeed, writing
+$a=1-\Re\rho\ge0$, $v=\Im\rho$, and $u=s-1$, the relevant term is
+\[
+ \frac{a+u}{(a+u)^2+v^2}.
+\]
+For $0<u\le1/\log D$, its numerator does not exceed its numerator
+at $s_0$, while \eqref{eq:zero-exclusion} gives
+$a^2+v^2\ge c_0^2/(\log D)^2$.  The denominator at $s_0$ is
+therefore at most a constant multiple of the denominator at $s$.
+Termwise comparison proves the asserted uniform estimate for $S(s)$.
+Integrating \eqref{eq:zeta-derivative} after moving $1/(s-1)$ to the
+other side now gives
+\begin{equation}
+ \zeta_K(s_0)\asymp\kappa\log D.                 \label{eq:zeta-residue-comparison}
+\end{equation}
+More explicitly, the logarithmic derivative of
+$(s-1)\zeta_K(s)$ is $S(s)-\tfrac12\log D+O(1)$, whose integral
+over this interval is $O(1)$, and its limit at $s=1$ is $\kappa$.
+Since $\zeta_K(s_0)\ge1$, this proves \eqref{eq:residue-lower}.
+
+The Euler product gives
+\[
+ \sum_{p\le D}\frac{d_K(p)}{p^{s_0}}
+      \le\log\zeta_K(s_0).
+\]
+Also $d_K(p)\le6$ and
+\[
+ \sum_{p\le D}d_K(p)\left(\frac1p-\frac1{p^{s_0}}\right)
+ \le\frac6{\log D}\sum_{p\le D}\frac{\log p}{p}\ll1,
+\]
+where the final estimate follows from Chebyshev's bound by partial
+summation.  Equation~\eqref{eq:zeta-residue-comparison} now proves
+\eqref{eq:prime-sum}.
+
+To obtain the interval estimate, we use Shiu's theorem
+\cite{Shiu1980}, in the formulation of
+\cite[Theorem~1.1]{Pollack2020} with empty excluded residue sets.
+For a nonnegative multiplicative function $f$, the hypotheses are
+\[
+ f(p^a)\le A_1^a,
+ \qquad
+ f(m)\le A_2(\epsilon)m^\epsilon
+       \quad(\epsilon>0).
+\]
+For any fixed $0<\beta<1/2$, the conclusion is
+\begin{equation}
+ \sum_{x-y<m\le x}f(m)
+ \ll\frac y{\log x}
+     \exp\left(\sum_{p\le x}\frac{f(p)}p\right)
+ \qquad(x^\beta<y\le x).                         \label{eq:shiu-form}
+\end{equation}
+Both the constant and the sufficiently-large threshold for $x$
+depend only on $\beta,A_1$, and the function $A_2$.
+
+The coefficients $d_K$ are multiplicative and are bounded by the
+ordinary six-fold divisor function $d_6$.  In particular,
+\[
+ d_K(p^a)\le\binom{a+5}{5}\le6^a,
+ \qquad d_K(m)\le d_6(m)\ll_\epsilon m^\epsilon.
+\]
+The hypotheses of \eqref{eq:shiu-form} thus hold with constants
+independent of $K$.  We take, for example, $\beta=1/3$; the assumed
+length $y\ge x^{1/2}$ lies in the required range.  Since $x\le D$,
+\eqref{eq:prime-sum} bounds the exponential factor by
+$O(\kappa\log D)$.  Finally $x\ge D^{1/4}$ implies
+$\log D/\log x\le4$, proving \eqref{eq:ideal-interval}.
+\end{proof}
+
+\subsection{The cusp and balls in the space of lattices}
+
+For a lattice $x\in X$, let $m(x)$ be the length of its shortest
+nonzero vector in the sup norm on $\R^6$.  We use the sup norm only
+to simplify the fiber integral; replacing it by any fixed norm
+changes the constants.
+
+\begin{proposition}[Uniform cusp bound]\label{prop:cusp}
+There are $C>0$ and $c>0$ such that, for all sufficiently large $D$,
+\[
+ \mu_{K,\sigma}\{x:m(x)\le r\}\le Cr^c
+       \qquad(0<r<1).
+\]
+Consequently the packet probabilities form a tight family as
+$D\to\infty$, and every weak subsequential limit is an
+$A$-invariant probability measure on $X$.
+\end{proposition}
+
+\begin{proof}
+Take $f=\one_{[-r,r]^6}$ and put $X_r=Qr^6$.  For $0<p\le r^6$,
+the change of variables $t_j=\log r-u_j$ in
+\eqref{eq:fiber-integral} identifies the positive-sign integration
+region with
+\[
+ t_j\ge0\quad(1\le j\le5),\qquad
+ \sum_{j=1}^5t_j\le\log(r^6/p).
+\]
+Thus
+\[
+ V_f(p)=\frac{2^6}{5!}\bigl(\log(r^6/p)\bigr)^5
+       \quad(0<p\le r^6),
+ \qquad V_f(p)=0\quad(p>r^6).
+\]
+Since the occurrence of a short vector is bounded by its count,
+\eqref{eq:vector-unfolding} gives
+\begin{equation}
+ \mu_{K,\sigma}\{m\le r\}
+ \ll\frac1{Q\kappa}
+       \sum_{l\le X_r}d_K(l)
+                   \bigl(1+\log(X_r/l)\bigr)^5.          \label{eq:cusp-sum}
+\end{equation}
+The sum is empty if $X_r<1$.
+
+Suppose first that $r\ge D^{-1/48}$, so $X_r\ge D^{3/8}$.
+For $l\ge D^{1/4}$, divide the range of summation into intervals
+$(2^{-j-1}X_r,2^{-j}X_r]$.  Whenever such an interval contains one
+of these $l$, its upper endpoint is at least $D^{1/4}$ and at most
+$Q$.  Lemma~\ref{lem:ideal-estimate}, with interval length half the
+upper endpoint, bounds its ideal count by
+$O(\kappa 2^{-j}X_r)$.  Its weight in \eqref{eq:cusp-sum} is
+$O((j+1)^5)$.  The total contribution of these terms before division
+by $Q\kappa$ is therefore at most
+\[
+ C\kappa X_r\sum_{j\ge0}2^{-j}(j+1)^5\ll\kappa X_r.
+\]
+For the remaining terms, the elementary divisor estimate
+$\sum_{l\le Y}d_6(l)\ll Y(\log(2Y))^5$ gives an upper bound
+$CD^{1/4}(\log D)^{10}$ before normalization.  In view of
+\eqref{eq:residue-lower}, its normalized contribution is at most
+\[
+ CD^{-1/4}(\log D)^{11}\ll r^6;
+\]
+the last inequality follows from $r^6\ge D^{-1/8}$.
+This proves an $O(r^6)$ bound in the present range.
+
+Now let $r<D^{-1/48}$ and $X_r\ge1$.  Dyadic summation of the
+same elementary divisor estimate yields
+\[
+ \sum_{l\le X_r}d_6(l)\bigl(1+\log(X_r/l)\bigr)^5
+      \ll X_r(\log(2X_r))^5.
+\]
+Using $X_r\le Q$ and \eqref{eq:residue-lower},
+\eqref{eq:cusp-sum} is therefore at most
+$Cr^6(\log D)^6$.  But $\log D<48\log(1/r)$ in this range, and
+$r^6(\log(1/r))^6\ll r^3$.  The claimed estimate follows, for
+example with $c=3$, after increasing $C$.
+
+Mahler's compactness criterion says that
+$\{x:m(x)\ge r\}$ is compact for every $r>0$.  The bound just proved
+therefore gives tightness.  Any weak limit has mass one, and
+$A$-invariance passes to the limit because right translation by each
+fixed element of $A$ is continuous and preserves compact supports.
+\end{proof}
+
+Let $B(r)$ denote an open matrix ball about the identity in $G$, for
+$r$ small; all fixed smooth local metrics give equivalent estimates.
+
+\begin{proposition}[Ordinary balls in a probability limit]
+\label{prop:ball}
+Suppose $D_i\to\infty$ and $\mu_i$ converges weakly to a probability
+$\mu$.  For every compact $\Omega\subset X$, there are constants
+$C_\Omega,r_\Omega>0$ such that
+\[
+ \mu(xB(r))\le C_\Omega r^6
+       \qquad(x\in\Omega,\ 0<r<r_\Omega).
+\]
+\end{proposition}
+
+\begin{proof}
+At any lattice $x_0$, choose a nonzero lattice vector none of whose
+coordinates is zero.  Such a vector exists because a full-rank lattice
+cannot be contained in a finite union of proper linear subspaces.
+In an injective local chart at $x_0$, continue this marked vector as
+$v(x)$.  By reducing the chart, its coordinates remain bounded and
+bounded away from zero.  A finite collection of these charts covers
+$\Omega$, so these bounds and all local derivative bounds can be
+taken uniform.
+
+For $x$ in one of the charts and sufficiently small $r$, every lattice
+in $xB(r)$ has a nonzero vector in the ordinary Euclidean ball
+$B_{\R^6}(v(x),C_1r)$.  Let $f_{x,r}$ be the indicator of that ball.
+Its support has absolute coordinate product in an interval $J_{x,r}$ of length
+at most $C_2r$; these intervals all lie in a fixed compact subinterval
+of $(0,\infty)$ when $r<r_\Omega$.  On this support the first five
+logarithmic coordinates each vary by $O(r)$.  Consequently
+\[
+ V_{f_{x,r}}(p)\le C_3r^5\one_{J_{x,r}}(p).
+\]
+Fix $r>0$.  For all sufficiently large $i$, uniformly in the finitely
+many charts and in $x\in\Omega$, the ideal norms in $Q_iJ_{x,r}$
+can be covered by an interval with upper endpoint comparable to $Q_i$
+and length at most $C_4Q_ir$.  We may choose that length comparable
+to $Q_ir$, so it is at least the square root of its upper endpoint
+for large $i$.  The upper endpoint lies between $D_i^{1/4}$ and
+$D_i$.  Lemma~\ref{lem:ideal-estimate} and
+\eqref{eq:vector-unfolding} therefore give
+\[
+ \mu_i(xB(r))
+ \le\int_X\widehat f_{x,r}\,\dd\mu_i
+ \le\frac{C_3r^5}{Q_i\kappa_i}
+       \sum_{l\in Q_iJ_{x,r}}d_{K_i}(l)
+ \le C_5r^6.
+\]
+The constants are independent of $x$ and $r$ in the stated range;
+the threshold for $i$ may depend on $r$.  Since $xB(r)$ is open,
+the open-set inequality for weak convergence gives the same bound
+for $\mu(xB(r))$.  This proves the assertion for each $x$ and $r$
+with the uniform constants supplied by the finite cover.
+\end{proof}
+
+\subsection{Trace slices of the codifferent}
+
+The vector estimate has now supplied tightness and the ordinary-ball
+bound.  To control diagonal tubes, we need to count a quantity which
+is nearly preserved by block-diagonal matrices.  For an ordinary
+vector $v$ and a dual vector $y$, that quantity is a partial sum of
+the products $v_jy_j$.  The following lattice estimate makes such
+partial sums effective arithmetic coordinates.
+
+Let $\mathfrak D_K$ be the different of $K$, so
+\[
+ \mathfrak D_K^{-1}
+ =\{z\in K:\Tr_{K/\Q}(z\mathcal O_K)\subset\Z\}.
+\]
+Give $\R^6$ its Euclidean inner product, let
+$\mathbf e=(1,\ldots,1)$, and put
+\[
+ H=\mathbf e^\perp,
+ \qquad \mathcal L_K=\sigma(\mathfrak D_K^{-1})\cap H.
+\]
+
+\begin{lemma}[Small cells in a trace slice]\label{lem:trace-slice}
+The lattice $\mathcal L_K$ has Euclidean covolume $\sqrt6/Q$ in
+$H$ and admits a fundamental parallelotope of diameter
+$O(D^{-1/30})$.  Let $S$ be a nonempty proper subset of
+$\{1,\ldots,6\}$.  For fixed $M>0$, uniformly in $m\in\Z$ and
+intervals $J\subset\R$ of length $\delta>0$,
+\begin{equation}
+ \#\left\{z\in\mathfrak D_K^{-1}:
+     \Tr_{K/\Q}z=m,
+     \ |\sigma_j(z)|\le M\ (1\le j\le6),
+     \ \sum_{j\in S}\sigma_j(z)\in J\right\}
+ \ll_M Q\bigl(\delta+D^{-1/30}\bigr).
+                                                        \label{eq:trace-slice-count}
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Since $1$ is primitive in the free abelian group $\mathcal O_K$,
+choose an integral basis $\alpha_1=1,\alpha_2,\ldots,\alpha_6$.
+Let $\beta_1,\ldots,\beta_6$ be its trace-dual basis, so
+$\Tr_{K/\Q}(\alpha_j\beta_k)=\delta_{jk}$.  By the definition of
+the codifferent, these $\beta_k$ form a basis of
+$\mathfrak D_K^{-1}$.  In particular,
+$\Tr_{K/\Q}\beta_k=\delta_{1k}$, which proves that the trace map
+onto $\Z$ is surjective and that
+$\sigma(\beta_2),\ldots,\sigma(\beta_6)$ form a basis of
+$\mathcal L_K$.  For $2\le j,k\le6$, orthogonal projection gives
+\[
+ \bigl\langle\operatorname{proj}_H\sigma(\alpha_j),
+                    \sigma(\beta_k)\bigr\rangle=\delta_{jk}.
+\]
+These projected vectors are therefore the dual basis inside $H$.
+Since $\operatorname{proj}_H\sigma(\alpha_1)=0$, we obtain
+\begin{equation}
+ \mathcal L_K^*=\operatorname{proj}_H\sigma(\mathcal O_K).
+                                                        \label{eq:projected-dual}
+\end{equation}
+
+Take a nonzero vector in the right side of
+\eqref{eq:projected-dual}, arising from $\alpha\in\mathcal O_K$.
+The vector is nonzero precisely when $\alpha\notin\Q$.
+Primitivity of the field implies $\Q(\alpha)=K$, so the order
+$\Z[\alpha]$ has discriminant at least $D$.  Differences of
+embedding coordinates are unchanged by orthogonal projection to
+$H$.  It follows that
+\[
+ D\le |\Disc(\Z[\alpha])|
+   =\prod_{j<k}|\sigma_j(\alpha)-\sigma_k(\alpha)|^2
+   \le C\|\operatorname{proj}_H\sigma(\alpha)\|^{30}.
+\]
+Thus the first successive minimum of $\mathcal L_K^*$ is at least
+$cD^{1/30}$.  Transference of successive minima in dimension five
+gives
+\[
+ \lambda_5(\mathcal L_K)
+       \ll\lambda_1(\mathcal L_K^*)^{-1}
+       \ll D^{-1/30}.
+\]
+A fixed-dimensional lattice basis bound then provides a basis for
+$\mathcal L_K$ all of whose lengths are $O(D^{-1/30})$; these
+standard geometry-of-numbers facts may be found in \cite{Cassels1959}.
+The associated fundamental parallelotope has the asserted diameter.
+
+The full codifferent lattice has covolume $Q^{-1}$.  Its trace
+planes have spacing $1/\sqrt6$, since the trace takes every integer
+value and has normal vector $\mathbf e$.  Decomposing a basis into
+five vectors in $H$ and a vector of trace one therefore gives
+\[
+ Q^{-1}=\frac1{\sqrt6}\covol_H(\mathcal L_K).
+\]
+
+The trace-$m$ points, if any, form a translate of $\mathcal L_K$.
+Tile that affine hyperplane by translates of the fundamental
+parallelotope, assigning boundaries measurably.  A cell attached to
+a point counted in \eqref{eq:trace-slice-count} lies in the
+$O(D^{-1/30})$ enlargement of its bounded slice.  The linear
+functional $z\mapsto\sum_{j\in S}z_j$ is nonconstant on $H$ because
+$S$ is nonempty and proper.  The five-dimensional volume of the
+enlarged slice is consequently
+$O_M(\delta+D^{-1/30})$, uniformly in its location and in $m$.
+Dividing by the cell covolume $\sqrt6/Q$ proves the counting bound.
+\end{proof}
+
+\subsection{Exponential decay on diagonal tubes}
+
+For $D_i$ sufficiently large, define
+\begin{equation}
+ T_{-,i}=\frac{\log D_i}{(\log\log D_i)^{2/3}},\qquad
+ T_{+,i}=\frac{\log D_i}{(\log\log D_i)^{1/3}}.
+                                                        \label{eq:time-range}
+\end{equation}
+Both tend to infinity, their ratio tends to infinity, and
+$T_{+,i}=o(\log D_i)$.  The last property ensures that the cells in
+Lemma~\ref{lem:trace-slice} are much smaller than the slices required
+at these times.
+
+\begin{proposition}[Two-sided tube estimate]\label{prop:bowen}
+Let $S$ be a nonempty proper subset of $\{1,\ldots,6\}$, and let
+$b=(b_1,\ldots,b_6)\in L$ satisfy
+\[
+ |b_j-b_k|\ge1\qquad(j\in S,\ k\notin S).
+\]
+For every compact $\Omega\subset X$, there is an identity
+neighborhood $O\subset G$ such that, for all sufficiently large
+$i$, all $x\in\Omega$, and all $T_{-,i}\le t\le T_{+,i}$,
+\begin{equation}
+ \mu_i\bigl(x(a(tb)Oa(-tb)\cap a(-tb)Oa(tb))\bigr)
+       \le e^{-t/3}.                              \label{eq:bowen-bound}
+\end{equation}
+The neighborhood and discriminant threshold may be chosen
+simultaneously for any fixed finite collection of pairs $(S,b)$.
+\end{proposition}
+
+\begin{proof}
+For a lattice $\Lambda$, its Euclidean dual is
+$\Lambda^*=\{y:\langle v,y\rangle\in\Z\text{ for all }v\in\Lambda\}$.
+At each point of $\Omega$, choose nonzero marked vectors in the
+lattice and its dual, both avoiding all coordinate hyperplanes.
+Continue the pair in a local lattice chart.  Finitely many such
+charts cover $\Omega$, and the marked vectors in these charts lie
+in fixed bounded boxes separated from the coordinate hyperplanes.
+Their inner product is an integer, constant in each chart.  Indeed,
+under right multiplication by $h$, the pair transforms as
+$v\mapsto vh$, $y\mapsto yh^{-\mathsf T}$.
+
+Choose $O$ sufficiently small in matrix coordinates.  If
+\[
+ h\in a(tb)Oa(-tb)\cap a(-tb)Oa(tb),
+\]
+then $h$ is uniformly close to the identity, and each entry crossing
+the cut $S\sqcup S^c$ is $O(e^{-t})$.  To verify the latter bound,
+the two membership conditions respectively bound its $(j,k)$ entry
+by a constant times $e^{t(b_j-b_k)}$ and $e^{-t(b_j-b_k)}$;
+their minimum is $O(e^{-t|b_j-b_k|})$.  Reducing $O$ also makes
+$h^{-1}$ uniformly bounded.  Let $P_S$ be the diagonal projection
+onto the coordinates in $S$.  Then
+\[
+ hP_Sh^{-1}-P_S=[h,P_S]h^{-1}=O(e^{-t}).
+\]
+For the marked pair $v_0,y_0$ at the center $x$, put
+$v=v_0h$ and $y=y_0h^{-\mathsf T}$.  These vectors remain in fixed
+bounded boxes avoiding the coordinate hyperplanes, and
+\begin{align*}
+ \sum_{j=1}^6v_jy_j&=\langle v_0,y_0\rangle=:m\in\Z,\\
+ \sum_{j\in S}v_jy_j
+   &=v_0hP_Sh^{-1}y_0^{\mathsf T}
+     =v_0P_Sy_0^{\mathsf T}+O(e^{-t}).
+\end{align*}
+All constants are uniform over the finite cover of $\Omega$.
+Thus membership in the tube forces the existence of an ordinary
+and dual vector pair satisfying these boundedness, trace, and
+partial-sum conditions.  The short partial-sum interval may depend
+on $x$.
+
+We count such pairs in the packet.  For a representative ideal $I$
+and a packet lattice $c\sigma(I)wa$, where
+$c=\covol(\sigma(I))^{-1/6}$, its dual is
+\[
+ c^{-1}\sigma(\mathfrak D_K^{-1}I^{-1})wa^{-1}.
+\]
+Consequently a nonzero pair comes from
+\[
+ d\in I\setminus\{0\},\qquad
+ \ell\in\mathfrak D_K^{-1}I^{-1}\setminus\{0\}.
+\]
+The scalar, sign, and diagonal factors cancel in their
+coordinatewise product:
+\begin{equation}
+ v_jy_j=\sigma_j(z),\qquad z=d\ell\in\mathfrak D_K^{-1}.
+                                                        \label{eq:pair-product}
+\end{equation}
+The allowed $z$ therefore belong to a fixed bounded embedding box,
+have trace $m$, and have their $S$-sum in an interval of length
+$O(e^{-t})$.
+
+For a fixed nonzero allowed $z$, the ordinary vector's ideal
+$\mathfrak b=(d)I^{-1}$ must divide $(z)\mathfrak D_K$.  In fact
+\[
+ \mathfrak b\bigl((\ell)I\mathfrak D_K\bigr)
+       =(z)\mathfrak D_K,
+\]
+and both factors on the left are integral.  Conversely, this
+divisibility ensures $z/d\in\mathfrak D_K^{-1}I^{-1}$ whenever
+$(d)I^{-1}=\mathfrak b$.  For a fixed $d$ and $z$, there is exactly
+one possible $\ell$, namely $z/d$.
+
+Let $f$ be the indicator of a fixed bounded box, or of a fixed finite
+union of boxes, containing all possible ordinary vectors $v$ and
+separated from the coordinate hyperplanes.  Its logarithmic fiber
+integrals are bounded uniformly in $p>0$, because the first five
+absolute logarithms range over a fixed bounded set.  Repeating
+the unfolding of Proposition~\ref{prop:packet-volume}, with the
+additional restriction $\mathfrak b\mid(z)\mathfrak D_K$, gives
+\begin{equation}
+ \int_X\#\{\text{admissible pairs with product }z\}
+           \,\dd\mu_{K,\sigma}
+ \le\frac1{Q\kappa}
+       \sum_{\mathfrak b\mid(z)\mathfrak D_K}
+          V_f(N\mathfrak b/Q)
+ \ll\frac1{Q\kappa}
+          \#\{\mathfrak b:\mathfrak b\mid(z)\mathfrak D_K\}.
+                                                        \label{eq:pair-unfolding}
+\end{equation}
+Here we have dropped the restrictions on the dual vector after
+fixing $z$, which can only increase the count.  The divisibility
+restriction is unchanged by multiplication of $d$ by a unit, so
+the same unfolding and sign multiplicities apply.
+
+The embedding bound on $z$ implies
+$N((z)\mathfrak D_K)=D|N_{K/\Q}z|\ll D$.
+For an integral ideal of norm $N$, its number of ideal divisors is
+at most $d(N)^6$, where $d$ is the ordinary divisor function: over
+each rational prime there are at most six prime ideals, and every
+one of their exponents is at most the exponent of that prime in
+$N$.  The classical divisor bound consequently gives, uniformly
+for our allowed $z$,
+\begin{equation}
+ \#\{\mathfrak b:\mathfrak b\mid(z)\mathfrak D_K\}
+       \le\exp\left(C\frac{\log D}{\log\log D}\right).
+                                                        \label{eq:ideal-divisors}
+\end{equation}
+
+Lemma~\ref{lem:trace-slice} bounds the number of allowed products
+by $O(Q(e^{-t}+D^{-1/30}))$.  Uniformly for $t\le T_{+,i}$,
+\[
+ D_i^{-1/30}=o(e^{-t}),
+\]
+because $T_{+,i}/\log D_i\to0$.  The product count is therefore
+$O(Q_i e^{-t})$.  The event defining the tube is bounded by the
+number of admissible pairs, so summing \eqref{eq:pair-unfolding}
+over these products and using \eqref{eq:ideal-divisors} gives
+\[
+ \mu_i\bigl(x(a(tb)Oa(-tb)\cap a(-tb)Oa(tb))\bigr)
+ \ll\kappa_i^{-1}
+      \exp\left(-t+C\frac{\log D_i}{\log\log D_i}\right).
+\]
+Finally, \eqref{eq:residue-lower} and $t\ge T_{-,i}$ show that
+the logarithm of the factor multiplying $e^{-t}$ is $o(t)$,
+uniformly throughout the time range.  Enlarging the discriminant
+threshold makes the last display at most $e^{-t/3}$.
+The same choices work for finitely many cuts and diagonal elements
+by taking the smallest of their identity neighborhoods and the
+largest of their thresholds.
+\end{proof}

preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/assembly.tex

@@ -0,0 +1,36 @@
+\section{Proof of the equidistribution theorem}\label{sec:assembly}
+
+We now return to the packets of Theorem~\ref{thm:main}.
+Proposition~\ref{prop:cusp} and Mahler's compactness criterion give
+tightness. Take an arbitrary subsequence with a weak probability limit
+$\mu$. It is $A$-invariant, and Proposition~\ref{prop:ball} gives
+the ordinary-ball estimate required by
+Proposition~\ref{prop:positive-entropy}. Theorem~\ref{thm:ekl}
+therefore makes almost every $A$-ergodic component of $\mu$
+homogeneous.
+
+Apply the intermediate-conditioning construction to this subsequence,
+passing further to a subsequence wherever necessary. The augmented
+limiting state still projects to $\mu$ on $X$.
+Proposition~\ref{prop:bowen} supplies the tube estimate throughout
+the required time interval. Hence
+Theorem~\ref{thm:full-activity} makes every root active almost surely
+in this state.
+
+If proper homogeneous components had positive total weight,
+Lemma~\ref{lem:fixed-pieces} would place positive $\mu$-mass on the
+countable union of their fixed-point pieces. At least one piece would
+have positive mass. Its fixing diagonal is nonidentity, so it has
+two unequal diagonal entries. The corresponding root plaques meet
+that piece in at most countably many points. By
+Lemma~\ref{lem:transverse-inactive}, that root is inactive almost
+surely on states projecting to the piece, a contradiction.
+Thus almost every component is the $G$-invariant probability, and
+$\mu=m_6$.
+
+Every weakly convergent subsequence supplied by tightness has the same
+limit. It follows that the original sequence converges to $m_6$ on
+$C_c(X_6)$. Proposition~\ref{prop:cusp} already gives the compact sets
+required for the nonescape assertion. The estimates are uniform in the
+embedding ordering, so the conclusion holds for the arbitrary choices
+in Theorem~\ref{thm:main}.

preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/combination.tex

@@ -0,0 +1,462 @@
+\section{Combining the root orientations}\label{sec:combination}
+
+Proposition~\ref{prop:segment-activity} guarantees activity somewhere on
+every long segment across each oriented cut.  It does not yet assert that
+the roots are active at the same point.  The wall invariance of the root
+indicators supplies the missing relation: on the scale $R_i$, an indicator
+can be read as a function of the value of its root alone.  Transitivity
+then becomes an approximate addition rule for functions on the real line.
+We first obtain activity in every real interval and then use two-edge
+cycles to show that the inactive sets have measure zero.
+
+\begin{theorem}[Full activity]\label{thm:full-activity}
+For every ordered pair $j\ne k$ in $\{1,\ldots,6\}$,
+\begin{equation}\label{eq:full-activity}
+ \int_X g_{jk,i}\,\dd\mu_i\longrightarrow1.
+\end{equation}
+Consequently every root is active almost surely in the limiting state law.
+\end{theorem}
+
+\subsection{Functions of a root coordinate}
+
+For each root $\beta$, fix $b_\beta\in L$ such that
+$\beta(b_\beta)=1$.  If $x$ is sampled with law $\mu_i$, define the random
+measurable functions
+\begin{equation}\label{eq:root-line-functions}
+ G_{\beta,i}^{x}(s)=g_{\beta,i}\bigl(xa(R_i s b_\beta)\bigr),
+ \qquad s\in\R.
+\end{equation}
+They take values in $\{0,1\}$.  Regard them as elements of the compact
+metrizable space
+\[
+ \mathcal B=\{f\in L^\infty_{\mathrm{loc}}(\R):0\leq f\leq1\},
+\]
+with weak-* convergence on every bounded interval.  More explicitly,
+one may use a countable dense set of $L^1$ test functions on each
+$[-q,q]$, $q\in\N$, to metrize this topology.  The finite product
+$\mathcal B^{30}$ is also compact and metrizable.  After passing to a
+subsequence, the laws of the arrays in \eqref{eq:root-line-functions}
+converge to a law of functions $(F_\beta)_\beta$ in this space.
+We also write $G_{jk,i}^{x}=G_{\beta_{jk},i}^{x}$ and
+$F_{jk}=F_{\beta_{jk}}$.
+
+\begin{lemma}\label{lem:root-line-laws}
+For any such convergent subsequence, the limiting law and its prelimit
+arrays have the following properties.
+\begin{enumerate}
+ \item Almost surely, for every $v_0\in L$ with rational coordinates,
+ every oriented cut $J$, and every positive rational $\tau$,
+ \begin{equation}\label{eq:positive-cut-integral}
+  \sum_{\beta\in\mathcal E_J}\int_0^\tau
+  F_\beta\bigl(\beta(v_0+t b_J)\bigr)\,\dd t>0.
+ \end{equation}
+ \item For distinct $j,k,\ell$, and every bounded rectangle
+ $B\subset\R^2$, the nonnegative defects satisfy
+ \begin{equation}\label{eq:line-transitivity-defect}
+  \E_{x\sim\mu_i}\int_B
+  G_{jk,i}^{x}(s)G_{k\ell,i}^{x}(t)
+  \bigl(1-G_{j\ell,i}^{x}(s+t)\bigr)\,\dd s\,\dd t
+  \longrightarrow0.
+ \end{equation}
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+For $v$ in a fixed bounded subset of $L$, the vector
+$v-\beta(v)b_\beta$ belongs to $\ker\beta$ and remains bounded.
+Wall invariance \eqref{eq:wall-invariance}, followed by invariance of
+$\mu_i$ under the common translation, gives
+\begin{equation}\label{eq:common-translate-comparison}
+ \sup_{v\text{ in a fixed bounded set}}
+ \int_X\left|
+ G_{\beta,i}^{x}\bigl(\beta(v)\bigr)
+ -g_{\beta,i}\bigl(xa(R_i v)\bigr)
+ \right|\,\dd\mu_i(x)\longrightarrow0.
+\end{equation}
+
+Fix the parameters in assertion~1.  The left side of
+\eqref{eq:positive-cut-integral} is a continuous function of the
+weak-* array, because $\beta(b_J)\ne0$ for every crossing root.
+By \eqref{eq:common-translate-comparison}, its prelimit differs in
+$L^1(\mu_i)$ by $o(1)$ from
+\[
+ Y_i(x)=\sum_{\beta\in\mathcal E_J}\int_0^\tau
+ g_{\beta,i}\bigl(xa(R_i(v_0+t b_J))\bigr)\,\dd t.
+\]
+Suppose that the limiting observable were zero on a set of probability
+$p>0$.  Convergence in law and nonnegativity allow us to choose thresholds
+$\varepsilon_i\downarrow0$ and events $E_i$ with
+$\mu_i(E_i)\geq p/2$ such that the original prelimit observable is at
+most $\varepsilon_i$ on $E_i$.  To see this directly, first choose a
+fixed sufficiently small continuity threshold, use convergence in law,
+and then decrease the threshold along a diagonal subsequence.
+The normalized restrictions $\mu_i|_{E_i}/\mu_i(E_i)$ consequently have
+$Y_i$ expectation tending to zero.  Translate these probabilities by
+$a(R_i v_0)$ and call the results $\xi_i$.  They satisfy
+$\xi_i\leq(2/p)\mu_i$, whereas the change of variable $u=R_i t$
+gives
+\[
+ \frac1{R_i}\int_0^{\tau R_i}\int_X
+ A_{J,i}\bigl(xa(u b_J)\bigr)\,\dd\xi_i(x)\,\dd u
+ \longrightarrow0.
+\]
+This contradicts Proposition~\ref{prop:segment-activity}.  The parameters
+in assertion~1 range over a countable set, so the assertion holds
+simultaneously almost surely.
+
+For assertion~2, the roots $\beta_{jk}$ and $\beta_{k\ell}$ are linearly
+independent.  Choose $v(s,t)$ linearly in $(s,t)$ with
+\[
+ \beta_{jk}(v(s,t))=s,\qquad
+ \beta_{k\ell}(v(s,t))=t.
+\]
+It follows that $\beta_{j\ell}(v(s,t))=s+t$.  On a bounded rectangle,
+\eqref{eq:common-translate-comparison} permits replacing all three
+functions in the defect by their indicators at the common point
+$xa(R_i v(s,t))$, with total expected error $o(1)$.  The expected defect
+there is $o(1)$ by the approximate transitivity in
+Lemma~\ref{lem:activity-indicators}; its distribution is unchanged by
+the translation.  Integrating over the rectangle proves the claim.
+\end{proof}
+
+The transitivity defect is nonlinear in the functions, so we do not pass
+it directly through weak-* convergence.  Instead we select deterministic
+arrays for which the defects themselves tend to zero.
+
+\begin{lemma}\label{lem:deterministic-root-arrays}
+If \eqref{eq:full-activity} fails for some root, there is a subsequence
+and a deterministic choice of arrays $(G_{\beta,i})_\beta$, each taken
+from \eqref{eq:root-line-functions}, with the following properties:
+\begin{enumerate}
+ \item $G_{\beta,i}\to F_\beta$ weak-* on bounded intervals, and the
+ deterministic limiting functions satisfy
+ \eqref{eq:positive-cut-integral} for all its stated parameters;
+ \item every defect in \eqref{eq:line-transitivity-defect}, without its
+ expectation, tends to zero on every bounded rectangle;
+ \item for some root $\beta_0$ and some bounded interval $I$ of positive
+ length,
+ \begin{equation}\label{eq:limiting-deficiency}
+  \int_I(1-F_{\beta_0}(s))\,\dd s>0.
+ \end{equation}
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+Pass to a subsequence on which
+$\int_X(1-g_{\beta_0,i})\,\dd\mu_i\geq\varepsilon$ for a fixed
+$\varepsilon>0$.  Invariance of $\mu_i$ and Fubini's theorem imply, for
+any fixed interval $I$,
+\[
+ \E\int_I(1-G_{\beta_0,i}^{x}(s))\,\dd s
+ =|I|\int_X(1-g_{\beta_0,i})\,\dd\mu_i
+ \geq\varepsilon|I|.
+\]
+The integral is a bounded weak-* continuous observable, so the same
+inequality holds for the limiting law.  This law therefore gives positive
+probability to a positive deficiency.  Choose a point $F$ in its support
+that has a positive deficiency and satisfies assertion~1 of
+Lemma~\ref{lem:root-line-laws}.
+
+For each integer $q$, choose a neighborhood $U_q$ of $F$ of diameter
+less than $1/q$.  It has positive limiting probability, and thus the
+prelimit probabilities of $U_q$ are bounded below by a positive number
+for all sufficiently large indices.  There are finitely many root
+compositions.  By \eqref{eq:line-transitivity-defect}, the sum of their
+defects on $[-q,q]^2$ has expectation tending to zero.  Choose an index
+so large that, by Markov's inequality, the probability that this sum
+exceeds $1/q$ is less than half the probability of $U_q$.  There is
+therefore a sampled array in $U_q$ with total defect at most $1/q$.
+Make these choices with increasing indices.  They converge to $F$ and
+have vanishing defects on every bounded rectangle, by nonnegativity
+and containment in some $[-q,q]^2$.
+\end{proof}
+
+\subsection{From cut activity to every interval}
+
+For the rest of the argument, fix deterministic arrays as in
+Lemma~\ref{lem:deterministic-root-arrays}.  The labels are still
+$\{0,1\}$-valued; their weak-* limits may initially take any value in
+$[0,1]$.
+
+\begin{lemma}\label{lem:positive-root-intervals}
+For every root $\beta$ and every nonempty open interval $I\subset\R$,
+\[
+ \int_I F_\beta(s)\,\dd s>0.
+\]
+\end{lemma}
+
+\begin{proof}
+Fix a root $\beta_{jk}$ and a desired interval $I$; replacing it by a
+bounded nonempty open subinterval only strengthens what we must prove.
+Choose a rational
+$v_0\in L$ so that $\beta_{jk}(v_0)$ lies in $I$.  Choose $r>0$ small
+enough that
+\[
+ [\beta_{jk}(v_0)-6r,\ \beta_{jk}(v_0)+6r]\subset I.
+\]
+For every root $\alpha$, put
+$I_\alpha=(\alpha(v_0)-r,\alpha(v_0)+r)$.  Draw the directed edge
+$p\to q$ if $\int_{I_{\beta_{pq}}}F_{pq}>0$.
+
+This directed graph is strongly connected.  Indeed, for any nonempty
+proper subset $J$, choose a positive rational $\tau$ sufficiently small
+that $\alpha(v_0+t b_J)\in I_\alpha$ for every
+$\alpha\in\mathcal E_J$ and $0\leq t\leq\tau$.
+Equation~\eqref{eq:positive-cut-integral} then supplies an edge from
+$J$ to its complement.  A finite directed graph with an outgoing edge
+across every nontrivial cut is strongly connected: otherwise the set of
+vertices reachable from a suitable vertex is a nonempty proper set
+with no outgoing edge.
+
+Take a simple directed path
+$j=p_0,p_1,\ldots,p_m=k$, where $m\leq5$.  Write
+$\alpha_h=\beta_{p_{h-1}p_h}$ and $I_h=I_{\alpha_h}$.  Weak-*
+convergence gives positive lower bounds, for sufficiently large $i$, on
+\[
+ \int_{I_h}G_{\alpha_h,i}(s)\,\dd s,
+ \qquad 1\leq h\leq m.
+\]
+The product of these integrals is consequently bounded below by a
+positive constant.  Interpret it as the measure of tuples
+$(s_1,\ldots,s_m)\in\prod_h I_h$ on which all the path labels equal
+one.
+
+Except for a set of tuple measure tending to zero, successive composition
+of these labels gives
+\[
+ G_{p_0p_h,i}(s_1+\cdots+s_h)=1,
+ \qquad 1\leq h\leq m.
+\]
+Here is the integration detail.  At step $h$, the failure of this
+implication, if all earlier implications held, is bounded by the
+transitivity defect for $p_0,p_{h-1},p_h$.  The path is simple, so these
+are distinct vertices.  Integration in $s_1,\ldots,s_{h-1}$ pushes
+Lebesgue measure on bounded intervals to a measure of bounded density
+in their sum.  Multiplying by previous indicator restrictions can only
+decrease this measure.  The defect integral on a fixed bounded
+rectangle therefore bounds the failure measure by a fixed multiple of
+an $o(1)$ quantity.  Summing over the finitely many steps proves the
+assertion.  If $m=1$, no composition is necessary.
+
+The final sum $s_1+\cdots+s_m$ lies within $mr$ of
+$\alpha_1(v_0)+\cdots+\alpha_m(v_0)=\beta_{jk}(v_0)$, hence in $I$.
+Its pushforward density is bounded above by a fixed constant.  Since
+a positive measure of tuples has final label one, we obtain
+$\liminf_i\int_I G_{jk,i}>0$.  Weak-* convergence proves the lemma.
+\end{proof}
+
+We have now located positive activity in every interval for every root.
+To eliminate a positive inactive set, we will translate an active set
+by sums of parameters from two opposite auxiliary roots.  Composing
+successively with those two roots returns to the original root while
+adding their two parameters.  Interval positivity supplies parameter
+sets of positive measure on both sides of zero.  By mixing normalized
+Lebesgue measure on these sets, we will obtain bounded densities of
+mean zero, bounded support, and variance bounded away from zero.
+The sum of two independent parameters with these densities describes
+one cycle increment.  For a fixed bounded interval
+$I$, the argument needs a fixed number of independent cycle increments
+to have density bounded below on $I-I$: this makes every part of the
+active set interact with every part of the inactive set inside $I$.
+The next lemma gives that lower bound uniformly over the densities
+that arise.
+
+\subsection{A uniform convolution estimate}
+
+\begin{lemma}\label{lem:convolution}
+Fix $B,M,v,L>0$.  Consider the probability densities $p$ on $\R$ such
+that
+\[
+ \supp p\subset[-B,B],\qquad 0\leq p\leq M,
+ \qquad \int x p(x)\,\dd x=0,
+ \qquad \int x^2p(x)\,\dd x\geq v.
+\]
+There are an integer $m\geq2$ and $c>0$, depending only on
+$B,M,v,L$, such that
+\[
+ p^{*m}(x)\geq c\qquad(|x|\leq L)
+\]
+for every such density.  The inequality uses the continuous representative
+of the convolution density.
+\end{lemma}
+
+\begin{proof}
+If the family is empty there is nothing to prove.  Otherwise write
+$\sigma_p^2=\int x^2p(x)\,\dd x$, so
+$v\leq\sigma_p^2\leq B^2$, and let
+\[
+ \phi_p(t)=\int e^{itx}p(x)\,\dd x.
+\]
+Bounded support and the zero mean give the uniform expansion
+\begin{equation}\label{eq:uniform-characteristic-expansion}
+ \phi_p(t)=1-\frac12\sigma_p^2t^2+O(B^3|t|^3).
+\end{equation}
+Applying the same expansion to
+$|\phi_p(t)|^2$, the characteristic function of the difference of two
+independent variables with density $p$, shows that for a sufficiently
+small $\delta>0$, depending only on $B,v$,
+\begin{equation}\label{eq:small-frequency-domination}
+ |\phi_p(t)|\leq e^{-vt^2/4}\qquad(|t|\leq\delta).
+\end{equation}
+
+There is also a uniform gap below one outside this neighborhood.  For
+$|t|\geq\delta$, an angle $\theta$, and $0<a<\pi$, the set
+\[
+ \{x\in[-B,B]:\operatorname{dist}(tx-\theta,2\pi\Z)<a\}
+\]
+has Lebesgue measure at most
+$a(2B/\pi+4/\delta)$.  Indeed, each complete period contributes the
+fraction $a/\pi$ of its length, and at most two incomplete periods
+contribute at the ends.  Choose $a$ so small that $M$ times this bound
+is at most $1/2$.  With $\theta=\arg\phi_p(t)$, at least half of the
+probability lies outside this arc.  Taking the real part after rotation
+by $e^{-i\theta}$ gives
+\begin{equation}\label{eq:large-frequency-gap}
+ \sup_p\sup_{|t|\geq\delta}|\phi_p(t)|
+ \leq\frac{1+\cos a}{2}=:\rho<1.
+\end{equation}
+If $\phi_p(t)=0$, the inequality is immediate and the angle is
+irrelevant.  Notice that the estimate holds at arbitrarily large
+frequencies; no uniform version of the Riemann--Lebesgue lemma is needed.
+
+Plancherel's identity yields
+\[
+ \int_\R|\phi_p(t)|^2\,\dd t
+ =2\pi\int_\R p(x)^2\,\dd x\leq2\pi M.
+\]
+For $m\geq2$, $\phi_p^m$ is integrable.  Fourier inversion therefore
+defines a continuous representative of $p^{*m}$.  We claim the following
+uniform local central limit estimate:
+\begin{equation}\label{eq:uniform-local-clt}
+ \sup_p\sup_{x\in\R}
+ \left|
+ \sqrt m\,p^{*m}(x)
+ -\frac1{\sqrt{2\pi}\,\sigma_p}
+   \exp\left(-\frac{x^2}{2m\sigma_p^2}\right)
+ \right|\longrightarrow0.
+\end{equation}
+To verify it, change variables $u=t\sqrt m$ in the inversion formula.
+The difference in \eqref{eq:uniform-local-clt} is at most $1/(2\pi)$
+times
+\[
+ \int_\R\left|
+ \phi_p(u/\sqrt m)^m-e^{-\sigma_p^2u^2/2}
+ \right|\,\dd u.
+\]
+On each fixed bounded $u$ interval, the expansion
+\eqref{eq:uniform-characteristic-expansion} proves uniform convergence
+of the integrand to zero; the logarithm may be taken near one, and its
+error is $O(B^3|u|^3/\sqrt m)$.  On
+$|u|\leq\delta\sqrt m$, \eqref{eq:small-frequency-domination} bounds
+the first term by $e^{-vu^2/4}$, while the Gaussian term is bounded by
+$e^{-vu^2/2}$.  Thus the contribution outside a large fixed bounded
+interval but inside this region is uniformly small.  On
+$|u|>\delta\sqrt m$, \eqref{eq:large-frequency-gap} and Plancherel give
+\[
+ \int_{|u|>\delta\sqrt m}|\phi_p(u/\sqrt m)|^m\,\dd u
+ \leq2\pi M\sqrt m\,\rho^{m-2}\longrightarrow0.
+\]
+The Gaussian tail there also tends to zero uniformly.  These three
+estimates prove \eqref{eq:uniform-local-clt}.
+
+For $|x|\leq L$ and all sufficiently large $m$, the Gaussian term in
+\eqref{eq:uniform-local-clt} is at least
+$1/(2\sqrt{2\pi}B)$, uniformly in $p$.  Increase $m$ so that the
+error is at most $1/(4\sqrt{2\pi}B)$.  Then
+\[
+ p^{*m}(x)\geq\frac1{4\sqrt{2\pi}B\sqrt m}
+ \quad(|x|\leq L),
+\]
+which is the required conclusion.
+\end{proof}
+
+\subsection{Cycles remove the inactive sets}
+
+\begin{proof}[Proof of Theorem~\ref{thm:full-activity}]
+Suppose the theorem fails, and take the deterministic arrays and limiting
+functions from Lemma~\ref{lem:deterministic-root-arrays}.  Fix an edge
+$j\to k$, and choose $\ell$ distinct from $j,k$.
+Lemma~\ref{lem:positive-root-intervals} supplies, for each of the roots
+$\ell j$ and $j\ell$, positive limiting integral on each of the
+intervals $(-2,-1)$ and $(1,2)$.  Accordingly, for all sufficiently
+large $i$, each of the four sets where the relevant $G$ equals one
+has length bounded below by a positive constant.
+
+For each auxiliary root, normalize Lebesgue measure on its two sets
+to probability densities $p_i^-$ and $p_i^+$.  Their means belong to
+$[-2,-1]$ and $[1,2]$, respectively.  There is a convex combination
+\[
+ p_i=\lambda_i p_i^-+(1-\lambda_i)p_i^+
+\]
+with mean zero.  The coefficients can be chosen in $[1/3,2/3]$ by
+solving the mean-zero equation.  Thus these densities have support in
+$[-2,2]$, a common finite upper bound, and variance at least one.
+They are supported entirely where the corresponding auxiliary
+indicator is one.  Denote the density for $\ell j$ by $p_i$ and that
+for $j\ell$ by $q_i$.
+
+Starting from $G_{jk,i}(t)=1$, composition with a parameter $s$ in the
+support of $p_i$ gives the edge $\ell k$, and composition with a
+parameter $s'$ in the support of $q_i$ returns to $jk$:
+\[
+ \ell j+jk=\ell k,\qquad j\ell+\ell k=jk.
+\]
+Both are root compositions with three distinct vertices.  Their defect
+estimates imply, for every fixed bounded interval $I$,
+\begin{equation}\label{eq:one-cycle-error}
+ \int_I\int_\R\int_\R
+ G_{jk,i}(t)\bigl(1-G_{jk,i}(t+s+s')\bigr)
+ p_i(s)q_i(s')\,\dd s\,\dd s'\,\dd t\longrightarrow0.
+\end{equation}
+Indeed, except on null sets of parameters, both auxiliary labels are
+one.  The indicator in \eqref{eq:one-cycle-error} is then bounded by
+the sum of the two composition defects.  For the first defect,
+integrate out $s'$ and use the uniform density bound on $p_i$.
+For the second, the variable $t+s$ has a bounded density on a fixed
+bounded interval when $t\in I$ is integrated with Lebesgue measure
+and $s$ with $p_i$.  The bound on $q_i$ and the defect estimate for
+$(j\ell,\ell k)$ finish the verification.
+
+Put $r_i=p_i*q_i$.  These cycle-increment densities have support in
+$[-4,4]$, a common density bound, mean zero, and variance at least two.
+We can iterate \eqref{eq:one-cycle-error} any fixed number $m$ of times
+to get
+\begin{equation}\label{eq:repeated-cycle-error}
+ \int_I\int_\R
+ G_{jk,i}(t)\bigl(1-G_{jk,i}(t+u)\bigr)
+ r_i^{*m}(u)\,\dd u\,\dd t\longrightarrow0.
+\end{equation}
+For clarity, along a sequence of $m$ increments, a path starting with
+label one and ending with label zero has a first transition from one
+to zero.  Bound its indicator by the sum of the indicators of these
+transitions.  The intermediate positions lie in
+$I+[-4m,4m]$.  After integrating all preceding increments, their
+distributions have densities bounded by one relative to Lebesgue
+measure on that larger interval, because the initial $t$ is integrated
+over $I$ and all increments have probability laws.  Each term is
+therefore bounded by the one-cycle error on the larger interval.
+There are only $m$ terms, proving \eqref{eq:repeated-cycle-error}.
+
+Fix a bounded interval $I$ of positive length and apply
+Lemma~\ref{lem:convolution} to the $r_i$, with $L$ large enough that
+$I-I\subset[-L,L]$.  Choose its fixed power $m$ and lower bound
+$c>0$.  In \eqref{eq:repeated-cycle-error}, retain only the pairs
+$t,t+u\in I$.  We obtain
+\[
+ c\left(\int_I G_{jk,i}(t)\,\dd t\right)
+   \left(\int_I(1-G_{jk,i}(t))\,\dd t\right)
+ \longrightarrow0.
+\]
+The first factor has positive lower limit by
+Lemma~\ref{lem:positive-root-intervals}.  Hence
+$\int_I(1-G_{jk,i})\to0$ for every such interval.  Weak-* convergence
+shows that $F_{jk}=1$ almost everywhere.  The argument applies to every
+root, contradicting the deficiency
+\eqref{eq:limiting-deficiency}.  Thus \eqref{eq:full-activity} holds.
+
+By Lemma~\ref{lem:activity-indicators}, the sequence indicators converge
+jointly in law to the activity indicators of the limiting state.  Since
+each has expectation tending to one, every limiting root indicator
+equals one almost surely.  There are only thirty roots, so this holds
+for all of them simultaneously.
+\end{proof}

preprints/Equidistribution-of-Primitive-Sextic-Torus-Packets-October-5-2026/build/sections/conditioning.tex

@@ -0,0 +1,664 @@
+\section{Conditional probabilities at intermediate resolutions}
+\label{sec:conditioning}
+
+We now pass from the measures $\mu_i$ to conditional probabilities along
+unipotent plaques. The conditioning must be performed before taking the
+limit: exact conditional measures of a compact diagonal orbit along a
+unipotent group contain no information about its distribution at the
+scales of the arithmetic estimate. We instead condition on small
+\emph{transverse cells}. The freedom to choose their depth over a broad
+range will make the resulting data compatible with changes of charts and
+with certain diagonal motions whose length tends to infinity.
+
+Throughout this section, $G=\SL_n(\R)$, $X=\SL_n(\Z)\backslash G$, and
+$n=6$ in the application. We assume only that $\mu_i$ are $A$-invariant
+probability measures converging weakly to an $A$-invariant probability
+$\mu$, and that the available times satisfy
+\[
+ T_{-,i}\longrightarrow\infty,
+ \qquad T_{+,i}/T_{-,i}\longrightarrow\infty.
+\]
+The particular arithmetic values of these times will be used later.
+All subsequences taken below preserve these assumptions.
+
+\subsection{Plaque charts and their extensions}
+
+Write $L$ for the traceless diagonal algebra and $a(b)=\exp(b)$ for
+$b\in L$. For $j\ne k$, set
+\[
+ \beta_{jk}(b)=b_k-b_j,
+ \qquad U_{jk}=\{1+tE_{jk}:t\in\R\}.
+\]
+These signs correspond to the right action on $X$: the displacement $u$
+at $x$ becomes $a(-b)ua(b)$ at $xa(b)$, and its $jk$ parameter is
+multiplied by $e^{\beta_{jk}(b)}$.
+
+A \emph{pattern} is a set of roots positive on some element of $L$ and
+closed under root addition whenever the sum is a root. Its
+\emph{pattern group} $V$ consists of the unipotent matrices with arbitrary
+entries in those positions and zero entries elsewhere. After permuting
+coordinates it is a triangular unipotent group. All these groups are
+normalized by $A$, and there are only finitely many of them. We also use
+the groups $VA$, but only in local charts.
+
+For any such group $H$, a plaque chart is an injective smooth map
+\begin{equation}
+ \Phi:Q\times B\longrightarrow X,
+ \qquad \Phi(q,h)=\psi(q)h,
+ \label{eq:plaque-chart}
+\end{equation}
+where $Q$ is an open box in a complementary transverse manifold and
+$B$ is an open bounded coordinate box in $H$. The map extends smoothly
+and injectively to a slightly larger domain. The sets with $q$ fixed
+are its plaques. Smooth reparametrizations of $h$, allowed to depend
+smoothly on $q$, describe the same plaques and will also be used. A
+smaller chart with closure in the displayed chart supplies a fixed
+margin from its boundary.
+
+The comparison of probabilities on different bounded portions of a
+plaque requires charts extending beyond a prescribed displacement
+range. The following elementary recurrence observation supplies them.
+
+\begin{lemma}[No periods in a pattern group]\label{lem:no-periods}
+Let $\nu$ be an $A$-invariant probability on $X$ and let $V$ be a
+pattern group. For $\nu$-almost every $x$,
+\[
+ xv=x,\quad v\in V \quad\Longrightarrow\quad v=1.
+\]
+At every such $x$, the image $xK$ of any prescribed compact
+displacement range $K\subset V$ is contained in an embedded plaque
+with a transverse neighborhood. Consequently, for any compact displacement range in $V$
+and any $\epsilon>0$, finitely many inner plaque charts cover a set of
+$\nu$-measure at least $1-\epsilon$, and their outer charts contain that
+range about every point of the corresponding inner charts.
+\end{lemma}
+
+\begin{proof}
+Choose $b\in L$ such that $a(-tb)va(tb)\to1$ as $t\to\infty$ for every
+$v\in V$. At a point recurrent under $a(b)$, a relation $xv=x$ gives
+\[
+ xa(kb)\,a(-kb)va(kb)=xa(kb).
+\]
+A subsequence of these points returns to a compact neighborhood of
+$x$. Stabilizers of points in that neighborhood contain no nonidentity
+element in a fixed neighborhood of $1$: this follows from the
+discreteness of the lattice and local quotient charts. The displayed
+relation is therefore impossible for $v\ne1$. Poincar\'e recurrence
+applies at almost every point.
+
+The orbit map $v\mapsto xv$ is then an injective immersion. Its
+restriction to a compact set is an embedding. Take a slightly larger
+compact coordinate domain in $V$ and use a complementary transverse
+slice. Compactness permits shrinking this slice until its product
+with the prescribed plaque domain is injective. Indeed, a failure
+along slices shrinking to $x$ would give either two distinct points
+of the limiting embedded plaque with the same image, or contradict
+local injectivity near one of its points. Shrinking once more provides
+inner charts with margins. A countable subcover and continuity from
+below give the finite-cover assertion.
+\end{proof}
+
+Fix a countable stock of plaque charts and inner charts with the
+following properties. It is a basis wherever local charts are needed;
+for pattern groups it also supplies the extensions in
+Lemma~\ref{lem:no-periods}, for each compact displacement range in a
+countable exhaustion. It includes subboxes, fixed subdivisions of the
+plaque coordinates into arbitrarily fine bins, and the simultaneous
+coordinate usages described next. Chart and bin boundaries have
+$\mu$-measure zero. Such choices are possible by varying coordinate
+endpoints; for each coordinate only countably many levels can have
+positive measure. One may start with multiplication charts and choose
+countably many centers, transversals, and domain sizes dense among
+the choices with compact closure and an injective enlargement.
+
+Two simultaneous usages will matter. If $V=WU$, where $U$ is a root
+group, $W$ is a normal pattern subgroup, and the patterns are disjoint,
+include charts
+\[
+ (q,w,u)\longmapsto\psi(q)wu.
+\]
+Their $V$, $U$, and $W$ transverse coordinates are, respectively,
+$q$, $(q,w)$, and $(q,u)$. Normality gives the last assertion; the
+$w$ coordinate is a smoothly reparametrized $W$ coordinate when $u$
+is fixed. Also include charts $(q,v,a)\mapsto\psi(q)va$ for $VA$, with
+the simultaneous $V$ transverse coordinate $(q,a)$. All chart bounds,
+plaque extensions, bin sizes, and fixed margins in a comparison are
+chosen before $i$ tends to infinity.
+
+\subsection{A broad range of almost unchanged conditionals}
+
+In a chart $\mathcal B=\Phi(Q\times B)$, restrict $\mu_i$ to
+$\mathcal B$. Partition $Q$ by a shifted dyadic grid of side $2^{-l}$.
+For a point $x=\Phi(q,h)$ in the chart, let
+\[
+ P^{\mathcal B}_{i,l}(x)
+   =\text{the conditional distribution of $h$ given the grid cell of $q$}.
+\]
+The conditioning uses the restricted measure. Zero-mass columns may
+be assigned any probability. An expectation involving this chart
+means integration over $\mathcal B$ with the unnormalized measure
+$\mu_i|_{\mathcal B}$; equivalently, one can normalize that measure and
+then multiply the expectation by $\mu_i(\mathcal B)$.
+
+For positive sequences $r_i,s_i$, write $r\prec s$ if $r_i/s_i\to0$.
+An interior depth will mean a sequence $d_i$ with
+$l_-\prec d\prec l_+$, where the endpoints are supplied by the next
+lemma. Indices $i$ on scales will often be suppressed.
+
+\begin{lemma}[A common plateau]\label{lem:plateau}
+After passage to a subsequence there are integer depths $l_{-,i}$ and
+$l_{+,i}$ such that
+\begin{equation}\label{eq:plateau-room}
+ T_-\prec l_-\prec l_+\prec T_+.
+\end{equation}
+For every chart and every fixed finite plaque-bin partition in the
+countable stock, the expected $\ell^1$ distance between the
+conditional bin-probability vectors at depths $l_-$ and $l_+$ tends
+to zero, averaged also over a uniform shift of the nested dyadic
+grid. The same conclusion holds between any two intervening depths
+for that grid.
+
+The shifts can be selected so that these conclusions hold for every
+fixed chart and fixed bin partition and so that random-grid boundary
+estimates hold at the two endpoint depths and at any prescribed
+countable collection of interior depth sequences. When coordinate
+usages share transverse labels,
+their grids may share the corresponding shifts. At interior depths,
+conditional bin probabilities are unchanged in expected $\ell^1$
+distance by using auxiliary shifted grids selected with the same
+boundary convention.
+\end{lemma}
+
+\begin{proof}
+Fix a chart, a finite bin partition, and a grid shift. If $p_l$ is the
+conditional bin-probability vector, its coordinates form bounded
+martingales in the increasing transverse grid sigma-fields. Thus
+\[
+ \int_{\mathcal B}\|p_v-p_u\|_2^2\,\dd\mu_i
+ =\int_{\mathcal B}\|p_v\|_2^2\,\dd\mu_i
+  -\int_{\mathcal B}\|p_u\|_2^2\,\dd\mu_i,
+ \qquad u\le v.
+\]
+The sum of these differences over disjoint depth intervals is at most
+$1$. This remains true after averaging in the shift. Enumerate all
+fixed chart/bin choices and assign them positive summable weights.
+Their weighted sum has the same bounded total variation.
+
+Put $L_i=\log(T_{+,i}/T_{-,i})$. Divide the interval of logarithmic
+depths into $m_i$ consecutive intervals, where $m_i\to\infty$ and
+$m_i=o(L_i)$. Discard its first and last intervals. Some remaining
+interval has weighted martingale cost $O(m_i^{-1})$. Its endpoints,
+rounded to integers, give $l_-$ and $l_+$. The discarded end intervals
+and the length $L_i/m_i\to\infty$ give all three ratios in
+\eqref{eq:plateau-room}. Each fixed chart/bin choice has cost tending
+to zero; Cauchy--Schwarz gives the asserted $\ell^1$ conclusion.
+Orthogonality of martingale increments gives it for intervening
+depths as well.
+
+For the boundary assertion, fix a point $q$ in a Euclidean transverse
+coordinate domain. Under a uniform grid shift, the probability that
+$q$ is within $\rho 2^{-l}$ of a depth-$l$ grid wall is $O(\rho)$,
+uniformly in $q,l$, and the sampling measure. Integrating first in
+the shift gives the same estimate for any measure on that domain.
+Choose $\rho_i\downarrow0$ slowly. Markov's inequality, followed by
+a diagonal choice over the chart/bin and depth lists, selects shifts
+with both the conditional-probability estimates and boundary masses
+tending to zero. Common labels can use common shift variables:
+each marginal shift remains uniform, which is all the argument uses.
+
+Here is the comparison with an auxiliary grid at an interior depth
+$d$. Outside negligible boundary mass, one of its cells lies inside
+one depth-$l_-$ cell and is a union of depth-$l_+$ cells, apart from
+the latter cells meeting its walls. The relevant relative collar
+widths are $O(2^{l_--d})$ and $O(2^{d-l_+})$. Both tend to zero much
+faster than the chosen $\rho_i$. Summing the martingale bin errors
+over the contained fine cells compares the auxiliary conditional
+law to the coarse one. The discarded mass changes the weighted
+conditional $\ell^1$ error by at most twice that mass. This proves
+the last assertion. The same argument compares any two of the
+selected grids by sandwiching them between these endpoint grids.
+\end{proof}
+
+We make the selection convention precise for later use. All defining
+charts, simultaneous coordinate usages, fixed bin subdivisions, and
+depth sequences belong to countable lists. A diagonal selection
+enforces their estimates for each fixed member; it does not assert
+uniformity over the whole infinite list at a finite index $i$.
+Within $[l_-,l_+]$ any fixed finite number of scales can be chosen
+with successive ratios tending to infinity and with room at both
+ends. For instance, use distinct fixed-fraction interpolations on
+the logarithmic depth interval. Different arguments may reuse such
+scales. Auxiliary grids chosen for one comparison need not define
+the data, because Lemma~\ref{lem:plateau} compares their fixed-bin
+laws with those of the defining grids. Such a grid may also be
+chosen later at an interior depth not in the original list; it
+does not change the augmented states defined below. We always let $\rho_i$ decrease
+slowly enough to dominate the exponentially small relative collar
+widths arising from these scale separations. None of these conventions
+requires absolute continuity.
+
+\subsection{The augmented probability space}
+
+Choose a defining interior depth for every chart and selected grids
+as above. Record at $x$ its conditional plaque probability in each
+chart containing it, and an isolated dummy symbol for every other
+chart. Write $Z_i(x)$ for the resulting tuple together with $x$.
+Each plaque probability is regarded as a probability on the compact
+closure of its coordinate domain. The countable product of these
+compact spaces, together with the $X$ coordinate, is a separable
+metrizable space. Tightness of $\mu_i$ therefore permits passage to
+a subsequence for which the laws of $Z_i(x)$ converge. Denote the
+limiting state by
+\[
+ \omega=\bigl(x,(P^{\mathcal B})_{\mathcal B}\bigr).
+\]
+Its $x$ marginal is $\mu$. Fixed-bin probabilities pass to the limit:
+the averaged plaque probability in a chart is its plaque-coordinate
+marginal, so the null-boundary choices imply that the limiting
+probabilities assign zero mass to the chosen bin boundaries almost
+surely.
+
+\begin{lemma}[Sampling one plaque probability]\label{lem:sampling}
+In any fixed chart, conditional on its transverse coordinate $q$
+and its own limiting probability $P^{\mathcal B}$, the sampled
+plaque coordinate $h$ has law $P^{\mathcal B}$. In particular,
+$h\in\supp P^{\mathcal B}$ almost surely on that chart. This assertion
+conditions on this one probability, rather than on the entire
+augmented state.
+\end{lemma}
+
+\begin{proof}
+Let $q_i^0$ be a representative of the cell containing $q$. At the
+sequence level, $P^{\mathcal B}_{i}$ is a function of that cell.
+Consequently, for bounded continuous tests $f$ and $F$,
+\[
+ \int_{\mathcal B} f(h)F(q_i^0,P^{\mathcal B}_{i})\,\dd\mu_i
+ =\int_{\mathcal B}P^{\mathcal B}_{i}(f)
+                  F(q_i^0,P^{\mathcal B}_{i})\,\dd\mu_i.
+\]
+On compact inner boxes, replacing $q_i^0$ by $q$ incurs an error
+tending to zero. Let $i\to\infty$ and then exhaust the chart by
+inner boxes. Its boundary has zero limiting mass. This proves the
+conditional-law identity. The standard support assertion for a
+sample from a probability measure follows, for example, by using a
+countable basis of the plaque domain.
+\end{proof}
+
+The remaining task in this section is to show that probabilities
+recorded in different charts describe the same measure along a
+plaque, up to normalization. We first isolate the estimate that
+allows many small transverse columns to be regrouped without losing
+control of the error.
+
+\subsection{Aggregating fine columns into patches}
+
+In an output chart use the defining nested grid at depths $l_-$ and
+$l_+$. Write $E$ for a coarse cell and $e$ for a fine cell contained
+in $E$. Fix finitely many plaque bins $J_j$, and set
+\[
+ M_e=\mu_i(\text{$e$ column}),\qquad
+ M_{e,j}=\mu_i(\text{$e$ column with }h\in J_j),\qquad
+ P_E(j)=\Prb_i(h\in J_j\mid q\in E).
+\]
+Here and below the chart restriction is understood. The plateau
+estimate is precisely
+\begin{equation}\label{eq:mass-error}
+ \delta_i:=\sum_E\sum_{e\subset E}\sum_j
+          |M_{e,j}-M_eP_E(j)|\longrightarrow0.
+\end{equation}
+
+\begin{lemma}[Comparison on trimmed patches]\label{lem:patch-comparison}
+In the preceding chart, let $p$ range over pairwise disjoint
+measurable patches, each lying transversely in one coarse cell
+$E(p)$. Suppose a subset of each patch has the product form
+\[
+ p^{\mathrm{in}}=
+ \bigcup_{e\in I_p}\bigl(e\times B_p\bigr),
+ \qquad B_p=\bigcup_{j\in J(p)}J_j,
+\]
+and the sum of the discarded masses $\mu_i(p\setminus p^{\mathrm{in}})$
+is at most $\epsilon$. The collections $I_p,J(p)$ may depend on
+$i,p$, and the number of patches need not be bounded. If $L_p$ is
+the plaque-bin law conditional on $p$, then, with errors weighted by
+$\mu_i(p)$,
+\[
+ L_p\quad\hbox{agrees with}\quad
+ \frac{\one_{B_p}P_{E(p)}}{P_{E(p)}(B_p)}
+ \quad\text{to total expected }\ell^1\text{ error }
+ O(\epsilon)+o(1).
+\]
+An arbitrary law is used if the denominator vanishes. The
+unnormalized version is
+\begin{equation}\label{eq:patch-unnormalized}
+ \sum_p\mu_i(p)
+ \left\|P_{E(p)}(B_p)L_p-
+             \one_{B_p}P_{E(p)}\right\|_1
+ \le O(\epsilon)+o(1).
+\end{equation}
+If patches and a disjoint refinement into subpatches admit such trims with the same
+coarse cell and the same plaque-bin union, their conditional bin
+laws agree with one another with the corresponding weighted error.
+
+These assertions also hold for bounded continuous tests, up to their
+uniform continuity error on the fixed bins. A test may depend on
+the sampled point, provided its norm and continuity modulus in the
+tested plaque variable are uniform. At the output sample,
+$P_{E(p)}$ may be replaced in fixed-bin expressions by the recorded
+data probability, with an additional error tending to zero.
+\end{lemma}
+
+\begin{proof}
+For a trimmed patch, sum the errors in \eqref{eq:mass-error} over
+$e\in I_p$ and $j\in J(p)$. The resulting comparison is between
+its unnormalized bin-mass vector and
+\[
+ \left(\sum_{e\in I_p}M_e\right)
+             \one_{B_p}P_{E(p)}.
+\]
+Every pair $(e,j)$ occurs in at most one trimmed patch, since those
+sets are disjoint. Hence the total unnormalized error is at most
+$\delta_i$. Restoring discarded mass adds at most $\epsilon$.
+For two nonnegative finite vectors $v,w$, comparison after
+normalization, weighted by $\|v\|_1$, costs at most
+$2\|v-w\|_1$; this follows by adding and subtracting
+$w/\|v\|_1$. The same inequality covers zero mass by an arbitrary
+choice of normalized vector. It proves the first assertion and,
+on multiplying by $P_{E(p)}(B_p)\le1$, proves
+\eqref{eq:patch-unnormalized}.
+
+Apply the same argument separately to patches and subpatches and
+use their common restricted vector $P_E|_{B_p}$ to obtain their
+comparison. Approximating a continuous test by its values on bins
+gives the remaining test assertion. The bound is in $\ell^1$, so it
+is uniform over all choices of bounded bin values, including values
+depending on the sample. Finally, the plateau controls the expected
+fixed-bin distance between the coarse conditional vector and the
+recorded vector. Restricting to disjoint patches does not increase
+that expectation.
+\end{proof}
+
+In applications the bins and charts are fixed first. Their sizes
+may then be chosen sufficiently small to achieve any prescribed
+limsup error, after which $i$ tends to infinity. Thus ``arbitrarily
+small error'' below means an actual bound by a prescribed
+$\epsilon>0$ in this order of choices. It is not a claim about
+uniform convergence over all subdivisions.
+
+\subsection{Projective agreement and diagonal motions}
+
+Center the probability in a $V$ chart at its sampled point: if the
+sample has group coordinate $h_s$, the group displacement of $h$ is
+$h_s^{-1}h$. For a smoothly reparametrized chart, use the corresponding
+group elements in this formula. Two nonzero measures on a common
+domain agree \emph{projectively} if one is a positive scalar multiple
+of the other there.
+
+\begin{proposition}[Comparison of plaque data]\label{prop:plaque-comparison}
+For the augmented limit constructed above the following assertions
+hold.
+\begin{enumerate}
+\item For every pattern group $V$, its centered probabilities in
+different charts agree projectively on their common plaque sheet
+near the sampled point. They extend to a measurable nonzero
+projective locally finite measure $m^V_\omega$ on $V$, with
+$1\in\supp m^V_\omega$. The chart probabilities represent its
+restrictions on all bounded displacement ranges contained in their
+plaque domains.
+\item Fix $b\in L$. In every subsequential joint limit of
+\[
+ \bigl(Z_i(x),Z_i(xa(b))\bigr),\qquad x\sim\mu_i,
+\]
+the two $V$ measures are related projectively by the pushforward
+under $v\mapsto a(-b)va(b)$.
+\item Fix a root $\beta$ and an interior time scale $R$ with room
+for interior depths on both sides. For every fixed $C<\infty$ and
+every sequence
+\[
+ b_i\in\ker\beta,\qquad \|b_i\|\le CR_i,
+\]
+the measures $m^{U_\beta}$ in every subsequential joint limit of
+$\bigl(Z_i(x),Z_i(xa(b_i))\bigr)$ agree projectively. This assertion
+holds for every such sequence $b_i$; the defining grids and state
+construction are unchanged.
+\end{enumerate}
+\end{proposition}
+
+\begin{proof}
+We prove one local comparison that covers all three assertions. The
+motion is either the identity, a fixed $a(b)$, or $a(b_i)$ with
+$b_i\in\ker\beta$ and $V=U_\beta$. Denote its induced conjugation on
+$V$ by $c$. In the first two cases $c$ is fixed; in the third case
+$c$ is the identity.
+
+We display the geometry in group-element plaque coordinates. For
+a reparametrized chart, write its original coordinate as $t\in B_0$
+and its group element as $h=r_q(t)$, where the maps $r_q$ and their
+inverses are smooth on the fixed extended chart. Its group-coordinate
+domain is $B(q)=r_q(B_0)$; for a multiplication chart $B(q)$ is
+constant. The conditional probabilities and bins remain in the
+original fixed coordinates. When converting a probability to group
+coordinates at a sample, use $r_{q_s}$. Across a depth-$F$ cell this
+conversion differs from the actual $r_q$ by $O(2^{-F})$, which has
+vanishing effect on every continuous test considered below.
+
+In the long-motion case choose an input
+transverse depth $F$ with
+\begin{equation}\label{eq:comparison-scales}
+ l_-\prec R\prec F\prec l_+,
+\end{equation}
+For a fixed motion, it suffices to take $l_-\prec F\prec l_+$;
+all factors $\exp(O(R))$ below are then bounded constants.
+The depth-$F$ input grid may be an auxiliary grid selected for this
+comparison. Thus $R$ and $F$ need not belong to the countable list
+used to define the states; Lemma~\ref{lem:plateau} compares the
+auxiliary laws back to the recorded ones. Its boundary selection
+uses $\mu_i$ and the scale bound, and does not depend on the
+particular $b_i$ within that bound.
+The input patches are the depth-$F$ transverse cells times the full
+plaque-coordinate domain. Their conditional laws are the input
+probabilities at depth $F$.
+
+\smallskip\noindent
+\emph{Fitting the images into output charts.}
+Choose input inner charts and fixed displacement test ranges first.
+An input chart may have any fixed plaque extension needed for these
+ranges. By Lemma~\ref{lem:no-periods}, output inner charts can be
+chosen whose outer charts accommodate the entire image plaque
+range, with a margin. A finite such cover omits arbitrarily small
+limiting mass. The output distribution is again $\mu_i$, so the
+same is true for the sequence. If an image patch meets one of these
+inner charts, its whole image fits in the corresponding outer chart
+for all sufficiently large $i$. This last assertion follows from
+the transverse diameter estimate below. Assign such patches to
+output charts, or work with each applicable pair of charts; the
+finite number of pairs only multiplies error bounds by a fixed
+constant.
+
+More precisely, let $B(q)$ and $B'(q')$ be the full input and output
+domains in group coordinates, and let $(q_s,h_s),(q'_s,h'_s)$ be the
+two samples. The output extension is required to contain
+\begin{equation}\label{eq:full-plaque-margin}
+ h'_s\,c\bigl(h_s^{-1}\overline{B(q_s)}\bigr)\subset B'(q'_s)
+ \quad\text{with a positive boundary margin}.
+\end{equation}
+This condition concerns the entire input plaque domain. It is an
+open condition on the paired sample coordinates: the domains are
+smooth images of fixed compact closures. Their possible centered
+ranges lie in a fixed compact subset of $V$, so the extension lemma
+supplies output covers with this property.
+
+In the two charts the map has the form
+\begin{equation}\label{eq:plaque-transition}
+ (q,h)\longmapsto\bigl(\theta_i(q),\,d_i(q)c(h)\bigr).
+\end{equation}
+The first coordinate is independent of $h$, because the motion
+normalizes $V$. The factor $d_i(q)$ is a left translation in $V$.
+It remains in a fixed compact set on the comparisons in question:
+both plaque-coordinate ranges and $c$ are fixed. For the varying
+motion, right multiplication and its inverse have differential norm
+at most $\exp(O(R))$ in a left invariant metric on $G$ and the
+induced quotient metric. Fixed chart changes have bounded
+differentials. Consequently both $\theta_i(q)$ and $d_i(q)$ vary over an
+input cell by
+\begin{equation}\label{eq:transverse-image-width}
+O\bigl(2^{-F}\exp(O(R))\bigr)=o(2^{-l_-}).
+\end{equation}
+For reparametrized charts, the input domains $B(q)$ also vary by
+$O(2^{-F})$ over the cell, and their admitted output ranges vary by
+$O(2^{-F}\exp(O(R)))$. The fixed boundary margins absorb these
+variations. Thus the same estimate applies to those charts.
+The same bounds hold for pullbacks of paths in the extended charts.
+They justify following one plaque sheet throughout its prescribed
+range, rather than including a different sheet at a return of the
+orbit. They also justify the fitting assertion: vary $q$ along the
+short path in its input cell and then follow the fixed plaque range
+inside the output extension.
+
+On one such patch, $\theta_i$ is injective. To see this, the admitted
+output plaque ranges have a common open subset, by the margin and
+the small variation of $d_i(q)$. If two transverse inputs had the
+same image, one could choose the same output plaque point in this
+common subset, contradicting injectivity of the motion and of the
+input chart. This observation permits product trims in output
+coordinates, even if many distinct input patches return to the
+same output box.
+
+\smallskip\noindent
+\emph{Trimming to fine columns and fixed plaque bins.}
+Discard image patches meeting a coarse transverse wall. By
+\eqref{eq:transverse-image-width}, every point of such a patch is in
+a collar whose relative width tends to zero. The grid convention,
+applied to the output distribution $\mu_i$, makes the discarded
+mass $o(1)$. Each remaining patch lies transversely in one coarse
+cell $E$.
+
+Take fine transverse cells whose closures lie in its projected
+transverse range, and plaque bins admitted over the whole range.
+Their products form an inner set as in
+Lemma~\ref{lem:patch-comparison}. Here are the two loss estimates.
+First, a path varying the transverse output coordinate through one
+fine cell pulls back a distance
+\begin{equation}\label{eq:fine-pullback-width}
+ O\bigl(2^{-l_+}\exp(O(R))\bigr)=o(2^{-F}).
+\end{equation}
+Away from the input cell walls and a fixed small collar of the
+input chart boundary, it therefore stays in the input transverse
+range and in the chart. Input grid-wall loss is $o(1)$ by the
+selected shifts; chart-boundary loss can be made arbitrarily small
+by its null limiting mass. This verifies that almost all sampled
+fine transverse cells are admitted.
+
+Second, on each plaque the transformation in
+\eqref{eq:plaque-transition} has bounded distortion, uniformly in
+$i$, and its dependence on $q$ tends to zero over the patch. A
+sample away from a small collar of the input plaque boundary thus
+lies in an output plaque bin admitted over the whole transverse
+range, provided the fixed bins are chosen sufficiently fine. The
+collar has arbitrarily small sampling mass, again by the fixed
+null-boundary choice. For a reparametrized chart these collars are
+pulled back to its original fixed coordinate boundaries, which
+have zero limiting mass. Continuous tests expressed in the fixed
+coordinates pull back with uniform continuity moduli on the compact
+chart ranges.
+This gives arbitrarily small total trim loss.
+No bound on the number of patches is involved.
+
+\smallskip\noindent
+\emph{Comparing the conditional laws.}
+The image of $\mu_i$ restricted to an input patch is exactly
+$\mu_i$ restricted to its image, by $A$ invariance (or trivially for
+the identity motion). Apply Lemma~\ref{lem:patch-comparison} to its
+trim. The input patch law, converted by
+\eqref{eq:plaque-transition}, agrees with $P_E$ restricted to the
+admitted bins, up to arbitrarily small weighted error.
+
+To express this without dividing by a possibly small admitted
+mass, take two continuous displacement tests $f,\phi$ supported
+with margin inside the input plaque range. Their output bin
+approximations are wholly inside the admitted bins. The
+unnormalized comparison says that each input test integral,
+multiplied by $P_E(B_p)$, agrees with the corresponding output
+$P_E$ integral. Cross-multiplication therefore gives
+\begin{equation}\label{eq:projective-cross-products}
+ P_{\mathrm{in}}(f)P_{\mathrm{out}}(\phi\circ c^{-1})
+ -P_{\mathrm{in}}(\phi)P_{\mathrm{out}}(f\circ c^{-1})
+ \longrightarrow0
+\end{equation}
+in probability on the compared samples. All integrals here are in
+displacement coordinates centered at the respective samples.
+The small $q$ variation changes the tests by a uniformly vanishing
+amount; sample-dependent centering is permitted by the uniform
+test assertion in Lemma~\ref{lem:patch-comparison}. The input
+depth-$F$ law can be replaced by its recorded probability by
+Lemma~\ref{lem:plateau}; the output coarse law can be replaced by
+the output recorded probability by the same lemma. This proves
+\eqref{eq:projective-cross-products} for the data that actually
+define the states.
+
+For completeness, the errors just used are localized as follows.
+Fix a pair of charts and open conditions placing the samples in
+inner boxes, requiring the full-plaque margin
+\eqref{eq:full-plaque-margin}, and placing the displacement test
+supports inside the input plaque range with positive margins.
+The patch fits and trims hold for all sequence samples satisfying
+these conditions, uniformly for
+the chosen motion bound. Trim accuracy may be made arbitrarily
+good independently of the fixed cover or these margin conditions.
+Thus the cross-product discrepancy tends to zero in probability
+on each such event. Exhaust by stricter margins and use continuous
+tests on compact inner boxes. Every joint distributional limit
+satisfies the corresponding exact identity there, since its tests
+and coordinate changes are continuous. Countably many charts,
+margin conditions, and tests suffice.
+
+\smallskip\noindent
+\emph{Gluing along one plaque.}
+Apply this result first to the identity motion. Compare any chart
+at the sample with a chart extending far enough to contain its
+bounded plaque domain. By Lemma~\ref{lem:sampling}, each probability
+has positive mass on every neighborhood of the sampled point.
+Choose a nonnegative continuous test supported in a common such
+neighborhood and positive at that point. Its integral is positive
+for both probabilities. The cross-product identities then give
+projective agreement for every continuous test with compact support
+in the first plaque domain.
+
+To compare two bounded domains, use a single extension containing
+both, on the same plaque sheet. Exhaust $V$ by compact connected
+coordinate ranges containing $1$ in their interiors. Successive
+extensions agree after normalization near $1$, and so define a
+locally finite measure on the union. Local finiteness follows
+because each compact set lies in one extension carrying a finite
+chart probability with a finite positive normalization factor.
+The identity lies in its support by the sampling lemma. This
+construction does not require chaining through intermediate support
+points: each comparison uses a chart containing both the identity
+neighborhood and the entire displacement range in question.
+Using countably many tests and choosing normalizing bumps from a
+countable collection makes the resulting projective measure
+measurable.
+
+\smallskip\noindent
+\emph{Equivariance, including the long motion.}
+For a fixed diagonal motion, the same cross-product identities
+give pushforward by $c(v)=a(-b)va(b)$, first on bounded test ranges
+and then globally by the extensions just constructed. For the
+varying motion in $\ker\beta$, $c$ is the identity, and all estimates
+above are uniform for $\|b_i\|\le CR_i$: the only growing chart cost
+is $\exp(O(R_i))$, dominated by
+\eqref{eq:comparison-scales}. In a paired-state limit the second
+base point need not be a fixed function of the first. This causes
+no difficulty. The local identities hold on every paired open
+margin event, and these events exhaust probability one because
+both marginals admit the required extended charts. Their exact
+identities therefore hold in every such limit. Since the argument
+applies to an arbitrary sequence $b_i$ under the displayed bound,
+it proves the last assertion with the stated quantifiers.
+\end{proof}
+
+The measures $m^V_\omega$ retain information obtained by conditioning
+the original sequence, while having the chart consistency and
+diagonal comparison properties of ordinary plaque measures. The
+next step is to relate the data for different pattern groups;
+that requires a product argument, rather than a further change of
+coordinates.