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@@ -84,3 +84,23 @@ preprints/A-counterexample-to-integer-degree-harmonic-dimension-comparison-Septe
 preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
 preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Odd-Characteristic-September-26-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
 preprints/A-counterexample-to-Kaplanskys-quasitrace-conjecture-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/naimark-counterexample-zfc.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/wall-d2-counterexample.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/paper.pdf filter=lfs diff=lfs merge=lfs -text
+preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/directional-zero-one-finite-range.pdf filter=lfs diff=lfs merge=lfs -text
diff --git a/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/README.md b/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..98419fb36cb846d1f7a8dd9737367d5a6118a40c
--- /dev/null
+++ b/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/README.md
@@ -0,0 +1,18 @@
+# [A Counterexample to Kurosh’s Division-Ring Problem](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 23, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026,
+  author = {{OpenAI}},
+  title = {{A Counterexample to Kurosh's Division-Ring Problem}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/paper.pdf}{OAI:A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/application.tex b/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/application.tex
new file mode 100644
index 0000000000000000000000000000000000000000..f9c86fbe7fb90574c99c1a5b070cd4e81ed790be
--- /dev/null
+++ b/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/application.tex
@@ -0,0 +1,204 @@
+\section{The countable construction and the actual center}
+\label{sec:countable-construction}
+
+We now apply Theorem~\ref{thm:lifting} repeatedly.  Recall its
+full conclusion: from a finite-dimensional central division algebra $A/k$
+and a tuple of free series $X$ whose constant term generates $A$ and whose
+two linear coefficients are independent modulo inner directions, it
+produces such a tuple over a countable extension $k'$ and a central
+division algebra of strictly larger degree.  It preserves every old
+polynomial identity and gives a monic algebraic relation for any chosen
+series $w$.  The countability, generation, and two-direction conclusions
+are all needed to iterate this theorem.
+
+The resulting division algebras need not embed in one another.  We use
+an ultraproduct to retain their polynomial relations, and then prove
+separately that the resulting algebra has infinite dimension over its
+actual center.
+
+\subsection{Scheduling all polynomial expressions}
+
+Start with
+\[
+ k_0=\overline{\mathbb Q},\qquad A_0=k_0,\qquad
+ X^{(0)}=(z_1,z_2).
+\]
+Its constant pair $(0,0)$ generates $k_0$ as a unital $k_0$-algebra.
+The inner-direction space is zero, and its two linear coefficients
+$(1,0)$ and $(0,1)$ are independent.  Thus it satisfies the hypotheses
+of Theorem~\ref{thm:lifting}.
+
+Inductively we obtain countable fields
+\[
+ k_0\subseteq k_1\subseteq k_2\subseteq\cdots,
+\]
+finite-dimensional central division algebras $A_i/k_i$ of strictly
+increasing degrees, and pairs
+\[
+ X^{(i)}\in A_i\langle\!\langle z_1,z_2\rangle\!\rangle^2
+\]
+satisfying the same generation and transverse-independence hypotheses.
+Write $(x_i,y_i)=X^{(i)}_0$ for the constant pair; in particular,
+\begin{equation}
+ A_i=k_i[x_i,y_i].
+ \label{eq:stage-generation}
+\end{equation}
+
+Here is a schedule that ensures algebraicity of every expression
+needed in the limit.  When $k_i$ has been constructed, enumerate the
+countable set $k_i\langle x,y\rangle$ as
+$(p_{i,j})_{j\geq0}$.  Enumerate the pairs $(i,j)\in\mathbb N^2$ by
+increasing $i+j$, in a fixed order within each diagonal.  If $(i_n,j_n)$
+is the $n$th pair, beginning at $n=0$, then $i_n\leq n$.
+At stage $n$, apply the lifting theorem to
+\[
+ w=p_{i_n,j_n}(X^{(n)}).
+\]
+This expression is defined over $k_n$, since $k_{i_n}\subseteq k_n$.
+The identity $w-p_{i_n,j_n}(X^{(n)})=0$ is preserved, so the new relation
+for the lifted $w$ gives a nonzero monic polynomial
+\[
+ q_{i_n,j_n}(t)\in k_{n+1}[t]
+\]
+such that
+\begin{equation}
+ q_{i_n,j_n}\bigl(p_{i_n,j_n}(X^{(n+1)})\bigr)=0.
+ \label{eq:retained-formal-relation}
+\end{equation}
+Every later lifting preserves this polynomial identity, now regarded
+over its coefficient field $k_{n+1}$.  Taking constant terms shows that
+the corresponding identity holds at $(x_h,y_h)$ for every $h\geq n+1$.
+Each polynomial over each $k_i$ is treated at a finite stage.
+
+\subsection{A two-generated algebraic division ring}
+
+Fix a nonprincipal ultrafilter $\mathcal U$ on $\mathbb N$ and form
+(see \cite{Keisler2010} for the general construction)
+\[
+ \mathcal D=\prod_{i\geq0} A_i\big/\mathcal U.
+\]
+Here two sequences represent the same element when their equality set
+belongs to $\mathcal U$.  This is a division ring: a nonzero class has
+nonzero coordinates on a set in $\mathcal U$, and taking their
+coordinatewise inverses on that set gives its inverse.
+
+Put $k_\infty=\bigcup_i k_i$.  An element $c\in k_j$ determines a class
+in $\mathcal D$ by taking its image in the central field $k_i\subset A_i$
+for $i\geq j$ and choosing arbitrary coordinates at the finitely many
+earlier indices.  These classes are independent of the early choices
+and of $j$.  They define a central field embedding
+\[
+ k_\infty\hookrightarrow\mathcal D:
+\]
+the field operations agree at every sufficiently large index, and a
+nonzero field element stays nonzero at all those indices.  This
+construction uses only the inclusions of coefficient fields; it does
+not require embeddings between the algebras $A_i$.
+
+Let
+\[
+ x=[(x_i)_i],\qquad y=[(y_i)_i],\qquad
+ R=k_\infty[x,y]\subseteq\mathcal D.
+\]
+Every $r\in R$ has the form $p(x,y)$ for some
+$p\in k_j\langle x,y\rangle$.  The schedule and
+\eqref{eq:retained-formal-relation} provide a nonzero monic polynomial
+$q\in k_h[t]$, for a suitable $h$, such that
+$q(p(x_i,y_i))=0$ at every sufficiently large index.  Therefore
+$q(r)=0$ in $\mathcal D$.  Every element of $R$ is algebraic over
+$k_\infty$.
+
+This already makes $R$ a division ring.  For $0\ne r\in R$, choose a
+nonzero relation of least degree
+\[
+ a_0+a_1r+\cdots+a_nr^n=0,
+ \qquad a_j\in k_\infty.
+\]
+Its constant coefficient is nonzero: otherwise multiplication by
+$r^{-1}$ in the ambient division ring $\mathcal D$ would give a shorter
+relation.  Consequently
+\[
+ r^{-1}=-a_0^{-1}(a_1+a_2r+\cdots+a_nr^{n-1})\in R.
+\]
+Since $k_\infty$ is countable and $R$ is generated by two elements as an
+algebra, $R$ is countable.  Its characteristic is zero.
+
+\subsection{Infinite dimension and passage to the center}
+
+We first prove
+\begin{equation}
+ [R:k_\infty]=\infty.
+ \label{eq:infinite-auxiliary-dimension}
+\end{equation}
+For each $i$ and $\ell\geq0$, let $V_{i,\ell}$ be the $k_i$-span of the
+words in $x_i,y_i$ of length at most $\ell$, including the empty word
+$1$.  If $V_{i,\ell}=V_{i,\ell+1}$, then $V_{i,\ell}$ is stable under
+right multiplication by $x_i$ and $y_i$ and contains every word.
+By \eqref{eq:stage-generation}, it equals $A_i$.  Until this happens,
+the dimensions increase strictly, starting from $1$.
+
+Fix $\ell$.  The dimensions $[A_i:k_i]$ tend to infinity, so
+\[
+ \dim_{k_i}V_{i,\ell}\geq\ell+1
+\]
+for every sufficiently large $i$.  There are only finitely many
+subsets of size $\ell+1$ in the finite list of words of length at most
+$\ell$.  The ultrafilter therefore selects one fixed subset
+$w_0,\ldots,w_\ell$ that is linearly independent at a set of stages
+belonging to $\mathcal U$.
+
+These same words are independent over the embedded $k_\infty$ in $R$.
+Indeed, the coefficients of a putative relation
+\[
+ \sum_{j=0}^{\ell} c_j w_j(x,y)=0
+\]
+all belong to one field $k_h$.  Equality in the ultraproduct holds on
+a set in $\mathcal U$.  Intersecting it with the independence set and
+the tail $i\geq h$ gives a nonempty set where stage independence
+forces every $c_j$ to vanish.  Thus $R$ has arbitrarily large finite
+linearly independent sets, proving
+\eqref{eq:infinite-auxiliary-dimension}.
+
+Finally let $F=Z(R)$ be the actual center.  It contains $k_\infty$ and
+is algebraic over $k_\infty$, since every element of $R$ is.  In
+particular, every element of $R$ is algebraic over $F$.  Also
+\[
+ R=k_\infty[x,y]=F[x,y]=F(x,y),
+\]
+because $F\subseteq R$ and $R$ is already a division ring.
+
+We use finite descent of the multiplication table and generator
+coordinates; compare the proof of the local-finiteness criterion in
+\cite[Section~4]{DeoBienHai2019}. Suppose, for a contradiction, that
+$[R:F]=n<\infty$. Choose an
+$F$-basis $b_1=1,b_2,\ldots,b_n$ of $R$, and write
+\[
+ b_i b_j=\sum_{h=1}^n c_{ijh}b_h,
+ \qquad
+ x=\sum_{h=1}^n u_hb_h,
+ \qquad
+ y=\sum_{h=1}^n v_hb_h,
+\]
+with all coefficients in $F$.  Let $E$ be the subfield of $F$
+generated over $k_\infty$ by these finitely many coefficients.
+Algebraicity gives $[E:k_\infty]<\infty$.  The space
+\[
+ B=\sum_{h=1}^n E b_h
+\]
+contains $1,x,y$ and is closed under multiplication by the displayed
+structure constants.  It is therefore a $k_\infty$-subalgebra of $R$
+containing $k_\infty[x,y]=R$, so $B=R$.  This implies
+\[
+ [R:k_\infty]\leq n[E:k_\infty]<\infty,
+\]
+contrary to \eqref{eq:infinite-auxiliary-dimension}.
+
+Thus the countable characteristic-zero division ring $D=R$ is
+algebraic over its actual center $F$, while
+\[
+ D=F[x,y]=F(x,y),\qquad [D:F]=\infty.
+\]
+The finite set $\{x,y\}$ witnesses the failure of local finiteness.
+
+This proves Theorem~\ref{thm:main}, subject only to Theorem~\ref{thm:lifting},
+which we now prove.
diff --git a/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/correction.tex b/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/correction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..eacdb6156bba043dd98837aceeb8e858fc886a02
--- /dev/null
+++ b/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/correction.tex
@@ -0,0 +1,295 @@
+\section{Solving the free-series equations}
+\label{sec:free-correction}
+
+We now use the response and cyclic-sum results to keep the output of a
+substitution conjugate to its constant term.  The input retains two prescribed
+linear coefficients.  This is the point at which free words, rather than
+commuting parameters, enter the construction essentially.
+
+Work over a common split scalar field $E$ in which the preceding evaluations
+and spectral projections are defined.  All series substitutions below have
+small constant terms and are interpreted by the coefficientwise valuation
+convergence established earlier.  In particular, the letters of a free series
+commute with its matrix coefficients; the letters do not commute with each
+other.
+
+Put $m=11$ and index the blocks by $i\in\mathbb Z/m\mathbb Z$.
+Let $P_i$ be jointly independent small generic pairs of $b\times b$
+matrices.  Set $R=\operatorname{diag}(P_i)$, a pair of $mb\times mb$
+matrices.  The old coefficient algebra acts on a separate factor of dimension
+$d$.  Write $\lambda_{i,h}$ and $\pi_{i,h}$ for the distinct eigenvalues
+and spectral projections of $w(P_i)$.  Eigenvalues belonging to different
+blocks are distinct.  For a substitution $Z$ with constant term $R$, let
+$S_{i,h}(Z)$ denote the fixed-root Schur expression at $\lambda_{i,h}$,
+using the constant projection $\pi_{i,h}$ in the $i$-th block.
+
+We recall precisely the response information needed here.  In each block
+choose one eigenvalue label from every part of the active partition, giving
+$a$ selected labels.  The selected loop-response map is
+\[
+ D_i:M_b(E)^2\longrightarrow E^a.
+\]
+Put $L_i=\operatorname{im}D_i$ and $H_i=E^a/L_i$.
+Proposition~\ref{prop:compatible-relations} supplies a vector space
+$G=\mathcal G$ of dimension $g$ and coordinate isomorphisms
+\[
+ p_i:G\longrightarrow H_i^*.
+\]
+For a generic fixed input arrow $U$ on $i\leftarrow j$, varying its
+opposite arrow $V$ gives two selected responses
+\[
+ m_i(U,V)\in E^a,
+ \qquad m_j(V,U)\in E^a.
+\]
+The annihilator of these paired responses in $H_i^*\oplus H_j^*$ is
+exactly
+\begin{equation}
+ \{(p_i(c),p_j(c)):c\in G\}.
+ \label{eq:correction-paired-annihilator}
+\end{equation}
+The identifications $p_i$ are compatible for the whole collection of blocks
+and for all the generic arrows used below.
+
+The transfer result supplies the following additional facts.  If all Schur
+expressions vanish in degrees less than $\ell$, their degree-$\ell$
+coefficients are scalar on the projected eigenspaces.  The resulting tuples
+belong to their active spaces and are determined by their selected entries.
+Denote the selected error for a word $\omega$ of length $\ell$ in block
+$i$ by $e_{i,\omega}\in E^a$.  The cyclic-sum result says that, for every
+orbit $\mathcal O$ of words under cyclic rotation,
+\begin{equation}
+ \sum_{\omega\in\mathcal O}\sum_i
+ p_i(c)\bigl(e_{i,\omega}+L_i\bigr)=0
+ \qquad(c\in G).
+ \label{eq:correction-cyclic-input}
+\end{equation}
+These statements hold for every partial substitution under consideration,
+including after an extension of its scalar field.  The preceding results
+therefore supply all the hypotheses used in the next construction.
+
+\begin{proposition}
+\label{prop:eleven-block-correction}
+Prescribe generic input-arrow pairs on the edges
+\[
+ \begin{array}{c|c}
+ \text{letter}&\text{offsets }i-j\text{ of edges }i\leftarrow j\\ \hline
+ s_1&1,2\\
+ s_2&3,4.
+ \end{array}
+\]
+Under the response and cyclic-sum hypotheses above, there is a pair
+$Z(s_1,s_2)\in M_{mb}(E)\langle\!\langle s_1,s_2\rangle\!\rangle^2$
+with constant term $R$ and exactly these prescribed linear coefficients,
+for which
+\[
+ S_{i,h}(Z)=0\qquad\text{for every }i,h.
+\]
+It can be constructed by finite linear systems.  At step $\ell\ge3$ the
+new unknowns are coefficients of degree $\ell-1$ on the reverse edges
+and coefficients of degree $\ell$ on the diagonal blocks.
+\end{proposition}
+
+\begin{proof}
+None of the four forward offsets is zero.  The sum of two forward offsets
+lies between $2$ and $8$, and is therefore nonzero modulo $11$.
+Thus a path of one or two forward steps cannot return to its initial block.
+Constant input steps preserve each block.  The path expansions for the
+output and for its Schur expressions consequently show that the prescribed
+constant and linear terms, with zero higher coefficients, solve all the
+Schur equations through degree two.
+
+Suppose a partial substitution solves them in degrees less than
+$\ell\ge3$.  Add a correction $V B$ on a reverse edge $j\leftarrow i$,
+where $B$ is a word of length $\ell-1$, and add arbitrary degree-$\ell$
+loop corrections.  At the block-diagonal constant term, an off-diagonal
+input variation gives an output variation in the same off-diagonal block.
+Its compression on any fixed root block is zero.  Thus the reverse
+correction changes no Schur equation through degree $\ell-1$.
+
+Its contribution in degree $\ell$ must pair it with a single prescribed
+linear arrow.  There is a unique forward edge that returns along this
+reverse edge: the reverse offsets $10,9,8,7$ correspond respectively to
+$1,2,3,4$.  If that edge has coefficient $U$ and color $s_c$, the
+contributions are
+\begin{equation}
+ \begin{array}{c|c|c}
+ \text{block}&\text{word}&\text{selected response}\\ \hline
+ i&s_cB&m_i(U,V)\\
+ j&Bs_c&m_j(V,U).
+ \end{array}
+ \label{eq:correction-leading-pair}
+\end{equation}
+This is exactly the two-arrow Schur calculation defining the paired
+responses.  A previously present coefficient of degree at least two
+cannot interact with the new degree-$\ell-1$ coefficient in degree
+$\ell$.  Two new corrections have total degree at least
+$2(\ell-1)>\ell$.  The degree-$\ell$ variation is therefore linear.
+Loop corrections give $D_i$ independently at every vertex and every
+word of length $\ell$.
+
+We compute the annihilator of this finite correction map.  Let
+$f_{i,\omega}\in(E^a)^*$ be a collection of functionals annihilating its
+image.  The loop columns first imply that $f_{i,\omega}$ annihilates
+$L_i$.  Write its induced functional on $H_i$ uniquely as
+\[
+ f_{i,\omega}=p_i(c_{i,\omega}),\qquad c_{i,\omega}\in G.
+\]
+Here and below we use the same notation for the induced functional and
+its pullback to $E^a$.  By
+\eqref{eq:correction-paired-annihilator} and
+\eqref{eq:correction-leading-pair}, annihilation of the reverse columns
+is equivalent to
+\begin{equation}
+ c_{i,s_cB}=c_{j,Bs_c}
+ \quad\text{on every edge }i\leftarrow j\text{ of color }c.
+ \label{eq:correction-edge-equality}
+\end{equation}
+The two offsets of either color are consecutive integers, say $a,a+1$.
+Applying \eqref{eq:correction-edge-equality} to both edges out of $j$
+gives
+\[
+ c_{j+a,s_cB}=c_{j+a+1,s_cB}.
+\]
+As $j$ ranges over $\mathbb Z/11\mathbb Z$, this makes the value
+independent of the vertex.  Equation~\eqref{eq:correction-edge-equality}
+then identifies its value with that on the rotated word $Bs_c$.
+Iterating rotations shows that the annihilator consists exactly of one
+element of $G$ per cyclic orbit of words.  The converse follows from
+the same paired-annihilator identity.
+
+Equation~\eqref{eq:correction-cyclic-input} says that every such
+annihilator kills the degree-$\ell$ error.  Finite-dimensional linear
+duality therefore puts the negative error in the image of the correction
+map.  Choose a correction solving all selected equations.  Lower degrees
+remain solved, so transfer and the active partition imply that every
+nonselected equation vanishes in degree $\ell$ as well.
+
+Continue inductively.  A coefficient of degree $n$ can change only at
+steps $n$ and $n+1$, so the resulting free series is well-defined.
+Every fixed Schur coefficient is eventually zero and remains zero.
+The construction also covers $g=0$, when the annihilator is zero.
+For a periodic word the orbit consists of its distinct rotations; no
+multiplicity is inserted into the annihilator calculation or into
+\eqref{eq:correction-cyclic-input}.
+\end{proof}
+
+\begin{figure}[tbp]
+\centering
+\input{figures/eleven-block-offsets}
+\caption{The eleven-block input pattern. Indices lie in $\mathbb Z/11\mathbb Z$.
+The left panel shows the offset-one cycle; the right panel specifies all
+four forward offsets, translated at every vertex. Each forward arrow carries a
+generic pair of input matrices. At order $\ell\ge3$, a reverse correction
+has degree $\ell-1$; degree-$\ell$ loop corrections are not drawn.}
+\label{fig:eleven-blocks}
+\end{figure}
+
+Figure~\ref{fig:eleven-blocks} separates the generating cycle from the
+full local arrow pattern. The role of the four offsets is now explicit.  Avoiding returns of length
+one or two starts the induction without changing the linear data.
+Consecutive offsets within each color make its only obstructions
+independent of the vertex.  The remaining obstructions are exactly the
+cyclic sums already proved to vanish.
+
+\subsection{Fixed roots and Schur complements}
+
+The following elementary fact identifies the spectral equations with a
+single polynomial identity. Its coefficient ring need not be commutative.
+Let $E$ be a field, let $\mathfrak a=(s_1,s_2)$ be the augmentation ideal
+of the free algebra $E\langle s_1,s_2\rangle$, and put
+\[
+  T_N=E\langle s_1,s_2\rangle/\mathfrak a^{N+1}
+  \qquad (N\geq 0).
+\]
+Elements of $E$ are central in $T_N$. Reduction modulo $\mathfrak a$
+is called the constant term.
+
+\begin{lemma}\label{lem:schur-polynomial}
+Let $W\in M_d(T_N)$ have constant term
+\[
+  W_0=\sum_{h=1}^{e}\lambda_h\pi_h,
+  \qquad
+  \pi_h\pi_j=\delta_{hj}\pi_h,
+  \qquad
+  \sum_{h=1}^{e}\pi_h=I_d,
+\]
+where the $\lambda_h\in E$ are distinct and the $\pi_h\in M_d(E)$
+are nonzero projections. Set
+\[
+  q(t)=\prod_{h=1}^{e}(t-\lambda_h),
+  \qquad \bar\pi_h=I_d-\pi_h.
+\]
+On the complementary summand $\bar\pi_h T_N^d$, the operator
+\[
+  C_h=\bar\pi_h(W-\lambda_h I_d)\bar\pi_h
+\]
+is invertible. Define the Schur expression on $\pi_h T_N^d$ by
+\[
+  S_h=\pi_h(W-\lambda_h I_d)\pi_h
+      -\pi_hW\bar\pi_h C_h^{-1}\bar\pi_hW\pi_h.
+\]
+Then $q(W)=0$ if and only if $S_h=0$ for every $h$.
+\end{lemma}
+
+\begin{proof}
+The constant term of $C_h$ is
+$\sum_{j\ne h}(\lambda_j-\lambda_h)\pi_j$, which is invertible on
+$\bar\pi_h E^d$. A matrix over $T_N$ with invertible constant term is
+invertible: after factoring out its constant term, the inverse is a
+finite geometric series in a matrix with entries in $\mathfrak a$.
+This proves the assertion about $C_h$. All modules below are right
+$T_N$-modules, with matrices acting on column vectors from the left.
+
+Suppose first that every $S_h$ vanishes. Define
+$K_h:\pi_hT_N^d\longrightarrow T_N^d$ by
+\[
+  K_hu=u-C_h^{-1}\bar\pi_hW\pi_hu,
+\]
+where the second term lies in the complementary summand. Then
+$\pi_hK_hu=u$, and direct block multiplication gives
+\[
+  (W-\lambda_h I_d)K_hu=S_hu=0.
+\]
+The map $K=\sum_hK_h\pi_h$ has constant term $I_d$, so it is
+invertible. The displayed kernel equations imply
+$WK=KW_0$. Consequently
+\[
+  q(W)=Kq(W_0)K^{-1}=0.
+\]
+
+Conversely, suppose $q(W)=0$. Define
+\[
+  P_h=\prod_{j\ne h}
+       \frac{W-\lambda_j I_d}{\lambda_h-\lambda_j}.
+\]
+These products are unambiguous because their factors are polynomials
+in the same matrix with central scalar coefficients. Polynomial
+interpolation modulo $q$ shows that the $P_h$ are spectral
+idempotents; in particular,
+\[
+  (W-\lambda_h I_d)P_h=0,
+  \qquad P_h\equiv\pi_h\pmod{\mathfrak a}.
+\]
+Let $\iota_h:\pi_hT_N^d\longrightarrow T_N^d$ be inclusion. The
+endomorphism $A_h=\pi_hP_h\iota_h$ of $\pi_hT_N^d$ has constant
+term the identity, and is therefore invertible. Thus
+\[
+  \widetilde K_h=P_h\iota_hA_h^{-1}
+\]
+satisfies
+$\pi_h\widetilde K_h=\operatorname{id}_{\pi_hT_N^d}$ and
+$(W-\lambda_h I_d)\widetilde K_h=0$. The complementary component
+of the latter equation determines its remaining component uniquely:
+\[
+  \bar\pi_h\widetilde K_h
+    =-C_h^{-1}\bar\pi_hW\pi_h.
+\]
+Its $\pi_h$ component now gives $S_h=0$.
+\end{proof}
+
+For a matrix over $E\langle\!\langle s_1,s_2\rangle\!\rangle$,
+apply Lemma~\ref{lem:schur-polynomial} after reduction to $T_N$.
+It follows that the fixed-root Schur equations hold through degree
+$N$ exactly when the identity $q(W)=0$ holds through degree $N$.
+In particular, this equivalence applies at every finite stage of a
+formal correction, with the same polynomial $q$ throughout.
diff --git a/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/cyclic-obstruction.tex b/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/cyclic-obstruction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..423de574122da88a2622956c2a9546f8acd4e75f
--- /dev/null
+++ b/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/cyclic-obstruction.tex
@@ -0,0 +1,406 @@
+\section{Cyclic relations on the leading free-word errors}
+\label{sec:cyclic-obstructions}
+
+The space \(\mathcal G\) controls the linear corrections available
+at several blocks. We must show that the errors to be corrected
+satisfy its relations for individual cyclic orbits of free words.
+Commuting scalar substitutions would combine too many words. We
+instead substitute matrices, repeat each block, and attach hubs
+whose derivative columns identify the possible left nullspace.
+
+Traces of words and their cyclic symmetry are classical in matrix
+invariant theory; see \cite[\S4]{Procesi1976}. Here the response
+relations must first be proved to annihilate every matrix trace test.
+An explicit directed-cycle calculation then isolates each cyclic orbit.
+
+Fix \(m\geq1\) independent generic size-\(b\) points \(P_i\).
+Write \(I_i\) for their selected root labels, so \(|I_i|=a\), and
+let \(\mathcal G\subseteq\bigoplus_i H_{P_i}^*\) be the space
+of Proposition~\ref{prop:compatible-relations}. As before, all
+scalars may be extended to a common field containing the required
+splitting data.
+
+\begin{theorem}[Cyclic compatibility of the error]
+\label{thm:cyclic-compatibility}
+Let \(Z(s_1,s_2)\) be a pair of formal series in two free letters,
+with constant term \(\operatorname{diag}(P_1,\ldots,P_m)\).
+Suppose that the fixed-root Schur expressions \(S_{i,h}\), for
+every block \(i\) and every root label \(h\), vanish in every
+degree below \(\ell\), where \(\ell\geq1\).
+By transfer, their degree-\(\ell\) coefficients are scalar on
+their root spaces; write
+\[
+ [s^\omega]S_{i,h}=e_{i,h,\omega}\pi_{i,h},
+ \qquad |\omega|=\ell.
+\]
+For each word the scalar tuple at block \(i\) belongs to its
+active space. For every cyclic orbit \(\mathcal O\) of words of
+length \(\ell\) and every \(c\in\mathcal G\),
+\begin{equation}
+ \sum_{\omega\in\mathcal O}
+ \sum_{i=1}^m\sum_{h\in I_i}
+ c_{i,h}e_{i,h,\omega}=0.
+ \label{eq:cyclic-error-relation}
+\end{equation}
+The orbit sum runs over distinct words, including when a word has
+a nontrivial rotational stabilizer.
+\end{theorem}
+
+We prove the theorem in three steps. First, arbitrary matrix
+substitutions turn the free-word errors into matrices on a repeat
+factor. Second, a derivative test shows that every relation in
+\(\mathcal G\) annihilates their traces. Finally, directed-cycle
+matrices isolate each cyclic orbit.
+
+\begin{lemma}[Splitting repeated blocks]
+\label{lem:repeated-splitting}
+Fix \(L\geq1\) and arbitrary scalar matrices
+\(S_1,S_2\in M_L(E)\). Substitute \(s_c=\delta S_c\) into
+\(Z\), with a commuting Taylor parameter \(\delta\), and add
+an order-\(\delta^\ell\) input loop \(Y_i\) in each old block
+\(i\). Here \(Y_i\) is an arbitrary pair in
+\(M_b(E)\otimes M_L(E)\). Put
+\[
+ B_{i,h}=\sum_{|\omega|=\ell}
+ e_{i,h,\omega}S^\omega+D_{i,h}(Y_i),
+\]
+where \(D_{i,h}\) is applied entrywise to the repeat-factor
+coefficients. One can choose the \(Y_i\) so that every \(B_{i,h}\),
+for all root labels \(h\), has simple spectrum. For such a choice,
+after a finite splitting extension the leaf output has branches
+\[
+ \lambda_{i,h,j}(\delta)
+ =\lambda_{i,h}+\delta^\ell\theta_{i,h,j}
+    +O(\delta^{\ell+1}),\qquad 1\leq j\leq L,
+\]
+each scalar of multiplicity \(t\), where \(\theta_{i,h,j}\)
+are the eigenvalues of \(B_{i,h}\). Their spectral projections
+are integral in \(\delta\) and reduce to
+\(\pi_{i,h}\otimes\Pi_{i,h,j}\), where
+\(\Pi_{i,h,j}\) are the rank-one spectral projections of
+\(B_{i,h}\).
+\end{lemma}
+
+\begin{proof}
+Matrix substitution is legitimate coefficientwise in \(\delta\).
+The leading Schur expression on the repeated \((i,h)\)-space is
+\(I_t\otimes B_{i,h}\): the original errors give the first sum,
+and an order-\(\delta^\ell\) input loop contributes its first
+response. All lower orders are zero.
+
+For each label \(h\), \(D_{i,h}\) is a nonzero linear functional
+on the input directions. Its matrix-valued extension is therefore
+onto \(M_L(E)\). The discriminant of \(B_{i,h}\), as a polynomial
+in the loop coordinates, is consequently not identically zero.
+There are finitely many labels, so the product of these discriminants
+is nonzero. Over the infinite field in use, or a scalar extension,
+choose loops at which their product is nonzero. This choice is made
+for all labels, not merely the selected ones. For \(L=1\) the
+simple-spectrum condition is automatic.
+
+Separate the old distinct roots by a formal conjugation equal to
+the identity at \(\delta=0\). The separated \((i,h)\)-block is
+\begin{equation}
+ \lambda_{i,h}I+\delta^\ell(I_t\otimes B_{i,h})
+ +O(\delta^{\ell+1}).
+ \label{eq:old-cluster-expansion}
+\end{equation}
+Here is an exact comparison with the fixed-root Schur expression.
+Write \(M=W-\lambda_{i,h}I\) in the fixed root/complement
+decomposition, with blocks \(M_{uv}\). Its complementary block
+\(M_{22}\) is invertible. If \(F\) is the separated root block
+minus \(\lambda_{i,h}I\), its invariant columns can be written
+\(\binom QR\), with \(Q=I+O(\delta)\), \(R=O(\delta)\), and
+\(M\binom QR=\binom QR F\). Eliminating the lower row gives
+\[
+ \bigl(M_{11}-M_{12}M_{22}^{-1}M_{21}\bigr)Q
+   =\bigl(Q-M_{12}M_{22}^{-1}R\bigr)F.
+\]
+The left parenthesis is the fixed-root Schur expression. Both
+outer factors relating it to \(F\) reduce to identity at zero.
+Consequently it and \(F\) have the same first nonzero coefficient.
+Its vanishing below degree \(\ell\) forces the same vanishing
+for \(F\), and its degree-\(\ell\) coefficient is
+\(I_t\otimes B_{i,h}\). This proves
+\eqref{eq:old-cluster-expansion}.
+
+Within this block divide the change from \(\lambda_{i,h}I\) by
+\(\delta^\ell\). Its constant term is \(I_t\otimes B_{i,h}\),
+with \(L\) distinct eigenvalues. A second formal conjugation,
+integral in \(\delta\), separates these slopes into blocks of
+size \(t\). Across all old labels there are exactly
+\[
+ mLe_b=e_{bmL}
+\]
+such blocks. Their characteristic polynomials are pairwise coprime
+over the Laurent fraction field: different old roots are already
+distinct at order zero, and distinct slopes separate the blocks
+inside one old root. The minimal-degree bound at size \(bmL\)
+is \(e_{bmL}\). Each separated block contributes at least one
+to that degree, so each contributes exactly one and is scalar.
+The two integral conjugations also give integral spectral
+projections with the stated limits.
+\end{proof}
+
+Keep a choice as in Lemma~\ref{lem:repeated-splitting}. Adjoin
+\(L+1\) independent generic size-\(b\) hub blocks, indexed by
+\(p=1,\ldots,L\) and by \(*\), with spectra disjoint from the
+old blocks. The individual hub \(p\) sends an input arrow to
+repeat position \(p\) of every old block \(i\), with fixed generic
+value \(U_{i,p}\). The common hub \(*\) sends the same generic
+value \(U_{i,*}\) to every repeat position in old block \(i\).
+There are no reverse arrows at this test input. Arrow values for
+different pairs of old blocks and hubs can be chosen independently
+and to satisfy the surjectivity conditions of
+Lemma~\ref{lem:pair-quotient-surjectivity}.
+
+Figure~\ref{fig:hub-test} shows the distinction between the individual
+hubs and the common hub.
+
+\begin{figure}[tbp]
+\centering
+\input{figures/repeated-leaves-hubs}
+\caption{The hub test at $\delta=0$, for one old block $i$ and $L=3$.
+Hub $p$ sends a generic matrix pair only to position $p$; the common hub
+sends the same pair to all positions. Every old block uses these same
+$L+1$ hubs. At $\delta\ne0$ the leaf input may couple repeat positions.
+The individual hubs eliminate off-diagonal dual entries, and the common
+hub identifies the diagonal functionals.}
+\label{fig:hub-test}
+\end{figure}
+
+Treat the entire deformed leaf array as one block and order it
+before the hubs. Over the \(\delta\)-Laurent field this is a star
+with maximal root count. Put
+\[
+ N=mL+L+1.
+\]
+Select the \(L\) branches over each old selected label and the
+selected labels of all hubs, for a total of \(Na\) output rows.
+The rank of their derivative matrix is at most
+\(\rho_{bN}\leq Na-g\). This bound follows from the original
+trace-minor identities, evaluated coefficientwise at the present
+Taylor input; it does not require the specialized input to be
+generic.
+
+\newpage
+\begin{lemma}[The limiting derivative columns]
+\label{lem:limiting-columns}
+The following derivative columns at the preceding star have finite
+limits at \(\delta=0\).
+\begin{enumerate}
+\item A loop \(Y\otimes T\) inside old block \(i\) has limit
+\[
+ D_{i,h}(Y)\operatorname{tr}(\Pi_{i,h,j}T)
+\]
+on row \((i,h,j)\), and zero on all other rows. Hub loops give
+their own loop responses.
+\item For individual hub \(p\), a reverse arrow of value \(V\)
+from repeat position \(r\) in old block \(i\) has limit
+\begin{align*}
+ &m_{i,h}(P_i,Q_p;U_{i,p},V)
+      \operatorname{tr}(\Pi_{i,h,j}E_{pr})
+       &&\text{on row }(i,h,j),\\
+ &\mathbf 1_{p=r}\,
+   m_{p,k}(Q_p,P_i;V,U_{i,p})
+       &&\text{on hub row }(p,k),
+\end{align*}
+and zero elsewhere. Here \(E_{pr}\) is the repeat-factor matrix
+unit with row \(p\) and column \(r\).
+\item For the common hub, use arrow \(U_{i,*}\) and replace the
+repeat-factor matrix by
+\[
+ \sum_{p=1}^L E_{pr}.
+\]
+The hub contribution is the single response
+\(m_{*,k}(Q_*,P_i;V,U_{i,*})\).
+\item The derivative along the whole input \(\delta\)-curve,
+multiplied by \(\delta^{1-\ell}\), has limit
+\(\ell\theta_{i,h,j}\) on leaf rows and zero on hub rows.
+\end{enumerate}
+The matrix consisting of a finite spanning collection of these
+limiting columns has rank at most \(Na-g\).
+\end{lemma}
+
+\begin{proof}
+Over the Laurent field, apply Lemma~\ref{lem:star-derivatives} with
+the whole leaf array as one block. For loop derivatives, the
+bounded leaf projection tends to
+\(\pi_{i,h}\otimes\Pi_{i,h,j}\). Compression of the ordinary
+first response gives \(D_{i,h}(Y)\) on the multiplicity space,
+and taking trace divided by \(t\) gives the first formula.
+
+For paired derivatives on a leaf, the Schur inverse is on the hub
+at a root \(\lambda_{i,h,j}(\delta)\); that root stays away from
+the hub spectrum at \(\delta=0\). For paired derivatives on a
+hub, the inverse is on the leaf array at a hub root, which stays
+away from every old leaf root. These inverses are regular at
+\(\delta=0\). No inverse at the difference of two approaching
+leaf roots occurs in this calculation.
+
+At the reduction, the leaf input is diagonal between old blocks
+and repeated inside each old block. A forward arrow at position
+\(p\) followed by a reverse arrow from position \(r\) inserts
+\(E_{pr}\) on the leaf repeat factor. At the hub end, the repeat
+indices must match, giving \(\mathbf 1_{p=r}\). The ordinary
+paired-response formula and trace divided by \(t\) now give the
+second formula. Summing forward positions gives the common-hub
+formula. The bounded projections and the distant-root inverses
+justify all these limits coefficientwise.
+
+The curve column lies in the span of input derivative columns over
+the Laurent field by the chain rule. The expansion in
+Lemma~\ref{lem:repeated-splitting} gives
+\[
+ \delta^{1-\ell}\frac{d}{d\delta}
+       \lambda_{i,h,j}(\delta)
+ =\ell\theta_{i,h,j}+O(\delta).
+\]
+Hub roots are constant on the curve. Thus this column also has a
+finite limit. Choose bases for the finitely many loop and reverse
+direction spaces and include the curve column. Every
+\((Na-g+1)\)-minor of the resulting matrix vanishes over the
+Laurent field; since its entries have finite limits, these minors
+remain zero after specialization. This proves the last assertion.
+\end{proof}
+
+We now use the individual hubs to remove off-diagonal repeat-factor
+weights and the common hub to make all diagonal weights agree.
+The rank bound will then show that no element of \(\mathcal G\)
+has been lost.
+
+\begin{lemma}[The left nullspace]
+\label{lem:hub-nullspace}
+The left nullspace of the limiting matrix in
+Lemma~\ref{lem:limiting-columns}, including its curve column,
+has dimension \(g\). Its projection to the old blocks identifies
+it with \(\mathcal G\): for each \(c\in\mathcal G\), its leaf
+weights are \(c_{i,h}\) on every row \((i,h,j)\), independently
+of \(j\), and its hub weights are the uniquely determined
+compatible functionals. Consequently
+\begin{equation}
+ \sum_i\sum_{h\in I_i}c_{i,h}\operatorname{tr}(B_{i,h})=0
+ \qquad(c\in\mathcal G).
+ \label{eq:error-matrix-trace}
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Let \(\mu_{i,h,j}\) be leaf row weights in the left nullspace
+and let \(\nu_p,\nu_*\) be the hub weights. Define
+\[
+ T_{i,h}=\sum_{j=1}^L\mu_{i,h,j}\Pi_{i,h,j}\in M_L(E).
+\]
+The loop columns imply
+\[
+ \sum_{h\in I_i}D_{i,h}(Y)
+          \operatorname{tr}(T_{i,h}T)=0
+ \qquad\text{for every }Y,T.
+\]
+Thus, for every matrix entry \((r,p)\), the coordinate vector
+\( ((T_{i,h})_{rp})_{h\in I_i}\) defines a functional in
+\(H_{P_i}^*\). The hub weights similarly belong to their own
+loop annihilators.
+
+Fix an individual hub \(p\) and a reverse position \(r\ne p\).
+There is no hub response. Since
+\[
+ \operatorname{tr}(T_{i,h}E_{pr})=(T_{i,h})_{rp},
+\]
+the reverse columns say that the functional with these
+\((r,p)\)-entries kills the leaf endpoint projection of the paired
+map. That projection is onto \(H_{P_i}\), so all these entries
+vanish. As \(p,r\) vary, every \(T_{i,h}\) is diagonal. When
+\(r=p\), the corresponding diagonal-entry functional and the
+hub functional \(\nu_p\) are exactly a pair in the relation of
+Proposition~\ref{prop:compatible-relations}.
+
+For the common hub, the leaf expression is
+\[
+ \operatorname{tr}\left(T_{i,h}\sum_pE_{pr}\right)
+   =\sum_p(T_{i,h})_{rp}=(T_{i,h})_{rr}.
+\]
+For every \(r\), this diagonal-entry functional is therefore
+paired with the same \(\nu_*\). Projection of the pair relation
+onto the hub dual space is an isomorphism, so the diagonal-entry
+functionals are independent of \(r\). Compatibility for the
+collection of old blocks and hubs identifies them across all old
+blocks. Hence
+\[
+ T_{i,h}=c_{i,h}I_L,
+ \qquad c\in\mathcal G,
+\]
+and all hub weights are uniquely determined by \(c\). Since the
+rank-one projections \(\Pi_{i,h,j}\) are mutually orthogonal
+and sum to identity, this also gives
+\(\mu_{i,h,j}=c_{i,h}\). The map from the left nullspace to
+\(\mathcal G\) is consequently injective.
+
+There are \(Na\) rows, and the limiting rank is at most \(Na-g\)
+even after the curve column is included. The nullspace therefore
+has dimension at least \(g\). Its injection into the
+\(g\)-dimensional space \(\mathcal G\) is an isomorphism.
+In particular every \(c\in\mathcal G\), not just a possible
+subspace of relations, occurs. Pairing its null vector with the
+curve column and cancelling the nonzero integer \(\ell\) yields
+\[
+ 0=\sum_i\sum_{h\in I_i}\sum_{j=1}^L
+       c_{i,h}\theta_{i,h,j}
+  =\sum_i\sum_{h\in I_i}c_{i,h}\operatorname{tr}(B_{i,h}).
+\]
+If \(g=0\), the same proof gives zero nullspace and the assertion
+about \(c\) is vacuous. If \(L=1\), the off-diagonal step has
+no entries to remove; the common-hub step still gives precisely
+the stated conclusion.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{thm:cyclic-compatibility}]
+Apply Lemma~\ref{lem:hub-nullspace} to arbitrary \(S_1,S_2\)
+and to the auxiliary loops chosen in
+Lemma~\ref{lem:repeated-splitting}. The added-loop contribution
+to \eqref{eq:error-matrix-trace} is zero. Indeed, write
+\(Y_i=\sum_\alpha Y_{i,\alpha}\otimes T_{i,\alpha}\). Then
+\[
+ \sum_{h\in I_i}c_{i,h}\operatorname{tr}(D_{i,h}(Y_i))
+ =\sum_\alpha\operatorname{tr}(T_{i,\alpha})
+      \sum_{h\in I_i}c_{i,h}D_{i,h}(Y_{i,\alpha})=0,
+\]
+because \(c_i\) annihilates the loop image. We conclude that,
+for arbitrary matrix substitutions,
+\begin{equation}
+ \sum_{|\omega|=\ell}
+ \left(\sum_i\sum_{h\in I_i}c_{i,h}e_{i,h,\omega}\right)
+ \operatorname{tr}(S^\omega)=0.
+ \label{eq:all-matrix-trace-tests}
+\end{equation}
+Auxiliary field extensions impose no restriction: this equality
+has its coefficients in the original field, and extension is
+injective.
+
+We use a matrix-cycle test of the kind appearing in the constructive
+trace argument of \cite[Proposition~4.1]{KlepSpenko2014}, recording the
+rotational multiplicity explicitly. Fix a word
+\(\omega_0=c_0c_1\cdots c_{\ell-1}\), with letters
+\(c_j\in\{1,2\}\), and take \(L=\ell\). On vertices indexed
+modulo \(\ell\), set
+\[
+ S_c=\sum_{j:\,c_j=c}E_{j,j+1}\qquad(c=1,2).
+\]
+For a word \(\omega\) of length \(\ell\), its matrix trace
+counts the starting vertices at which one reads \(\omega\)
+around this directed cycle. It is zero unless \(\omega\) lies
+in the cyclic orbit \(\mathcal O\) of \(\omega_0\); for every
+distinct word in that orbit its value is the same positive integer
+\[
+ \frac{\ell}{|\mathcal O|}.
+\]
+This is the rotational stabilizer size and includes periodic and
+constant-letter words. Substitution into
+\eqref{eq:all-matrix-trace-tests}, followed by cancellation of that
+integer in characteristic zero, gives
+\eqref{eq:cyclic-error-relation}.
+\end{proof}
+
+The leading errors now satisfy every cyclic relation imposed by
+\(\mathcal G\). The colored correction graph will make these
+exactly the annihilators of its order-by-order correction map.
diff --git a/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/division-realization.tex b/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/division-realization.tex
new file mode 100644
index 0000000000000000000000000000000000000000..ccde7e09e5c778dd6b1e733044dd0860ffd561ee
--- /dev/null
+++ b/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/division-realization.tex
@@ -0,0 +1,240 @@
+\section{Realization in a central division algebra}
+\label{sec:division-realization}
+
+The preceding construction solved equations in split matrices.  We now
+place its constant blocks and prescribed arrows in one central division
+algebra.  Each subsequent correction is the solution of a finite linear
+system over its center.  This permits descent without any assertion that
+solutions of nonlinear equations descend.
+
+Throughout this section $k$ has characteristic zero and contains all roots
+of unity, and $A$ is a central division algebra of degree $d$ over $k$.
+The integer $b$ is the block size chosen earlier in the spectral case.
+In the case where $q_0(w)=0$ already holds as a free-series identity, any
+positive integer $b$ may be used.  Put $m=11$.
+
+\subsection{The two division-algebra factors}
+
+Choose independent central indeterminates $\alpha,\beta$ and put
+$k_0=k(\alpha,\beta)$.  Let $B_b$ be the symbol algebra over $k_0$
+generated by $u,v$ with
+\[
+ u^b=\alpha,\qquad v^b=\beta,\qquad vu=\zeta_buv,
+\]
+where $\zeta_b$ is a primitive $b$-th root of unity.  For $b=1$ this
+means $B_1=k_0$.  By the construction in Section~\ref{subsec:generic-division}, $B_b$
+is central simple of degree $b$.  Lemma~\ref{lem:symbol-division}
+shows that $A\otimes_k B_b$ is division and remains division under
+pure rational and central Laurent scalar extensions.  We will use
+these facts twice in adjoining the cyclic block factor.
+
+Fix a $k_0$-basis $(e_a)_{a=1}^{b^2}$ of $B_b$.  Over $k_0$ introduce
+the following jointly independent variables:
+\[
+ x_{c,a,j},\qquad y_{r,c,a,j},
+ \qquad
+ \begin{matrix}
+ c\in\{1,2\},\quad 1\le a\le b^2,\\
+ j\in\mathbb Z/m\mathbb Z,\quad r\in\{1,2,3,4\}.
+ \end{matrix}
+\]
+Let $L_0$ be their rational function field and let $\sigma$ increase
+the index $j$ by one, fixing $k_0$.  This automorphism has order exactly
+$m$.  Put $H_0=L_0^{\langle\sigma\rangle}$.  Artin's fixed-field theorem
+for a finite group of automorphisms gives a cyclic Galois extension
+$L_0/H_0$ of degree $m$; see \cite[Corollary~3.11, p.~38]{MilneFT2022}.
+
+Choose a further independent central indeterminate $a_*$ and form the
+cyclic algebra $C$ over $F_0=H_0(a_*)$ with coefficient field
+$L_0(a_*)$ and generator $\tau$, subject to
+\begin{equation}
+ \tau^m=a_*,\qquad \tau c\tau^{-1}=\sigma(c)
+ \quad(c\in L_0).
+ \label{eq:cyclic-realization-relations}
+\end{equation}
+It has vector-space decomposition
+$C=\bigoplus_{r=0}^{m-1}L_0(a_*)\tau^r$.
+After extending scalars to a field containing $L_0(a_*)$, represent
+$c\in L_0$ by the diagonal with entries $\sigma^{-i}(c)$ and
+$\tau$ by the cyclic forward shift, whose wrap entry is $a_*$.
+These matrices satisfy \eqref{eq:cyclic-realization-relations} and
+span all matrix units.  This gives the usual splitting of $C$ and
+shows that it is central simple of degree $m$; see
+\cite[Construction~2.5.1 and Proposition~2.5.2]{GilleSzamuely2006}.
+Our conjugation convention uses the inverse of their Galois generator.
+
+Define
+\[
+ \widetilde B_0=(B_b\otimes_{k_0}F_0)\otimes_{F_0}C,
+ \qquad D_0=A\otimes_k\widetilde B_0.
+\]
+Then $D_0$ is central simple of degree $dbm$ over $F_0$ by the tensor-product and scalar-extension properties of
+central simple algebras
+\cite[Chapter~IV, Corollary~2.8 and Proposition~2.15]{MilneCFT}.
+It is also division. Indeed
+\[
+ D_L=(A\otimes_k B_b)\otimes_{k_0}L_0
+\]
+is division by the rational-extension argument.  Extend $\sigma$ to
+$D_L$ by fixing the old algebra factors.  The skew Laurent division
+ring $D_L((t;\sigma))$ receives a unital homomorphism from $D_0$
+by $\tau\mapsto t$ and $a_*\mapsto t^m$.  The central element
+$t^m$ is transcendental over $H_0$.  Simplicity again makes this map
+injective, and finite dimension makes $D_0$ division.
+
+Finally put
+\[
+ F=F_0((\varepsilon)),\qquad
+ \widetilde B=\widetilde B_0\otimes_{F_0}F,\qquad
+ D=D_0\otimes_{F_0}F.
+\]
+The central Laurent argument shows that $D$ is a central division
+algebra over $F$ of degree $dbm$.  Input coefficients will lie in
+$\widetilde B$, so that they commute with the old factor $A$.
+
+\subsection{One realization of all generic data}
+
+For $r\in\mathbb Z/m\mathbb Z$, let
+\[
+ \mathcal V_r=
+ \bigl(B_b\otimes_{k_0}L_0\bigr)\tau^r
+ \otimes_{H_0}F
+ \ \subseteq\ \widetilde B.
+\]
+This notation uses the evident coefficient embedding and scalar
+extension; equivalently it is the $F$-span of that graded summand in
+$\widetilde B_0$.  Upon splitting, $\mathcal V_r$ consists exactly
+of all $b\times b$ entries at offset $r$ in the $m$-block matrix.
+In fact $L_0\otimes_{H_0}E\simeq E^m$ over a splitting field $E$,
+and $B_b$ becomes $M_b(E)$.  Multiplication by $\tau^r$ moves these
+independent diagonal entries to offset $r$, with nonzero wrap factors.
+In particular $\mathcal V_0$ gives all diagonal blocks, and
+$\mathcal V_{-r}$ gives all reverse entries for offset $r$.
+
+Define the two constant inputs and four arrow pairs by
+\[
+ P_c^0=\varepsilon\sum_a x_{c,a,0}e_a,
+ \qquad
+ U_{r,c}=\left(\sum_a y_{r,c,a,0}e_a\right)\tau^r
+ \quad(c=1,2).
+\]
+The prescribed initial substitution is the pair
+\begin{equation}
+ Z_c^{\mathrm{init}}(s_1,s_2)
+ =P_c^0+(U_{1,c}+U_{2,c})s_1
+             +(U_{3,c}+U_{4,c})s_2.
+ \label{eq:division-initial-input}
+\end{equation}
+After splitting, the diagonal blocks of $(P_1^0,P_2^0)$ are exactly
+the required independent small generic pairs.  The two input-component
+entries at every forward edge are independent generic entries, jointly
+with all the diagonal data.
+
+Here independence is over the old split coefficient field, which is the
+field used in the generic rank tests.  To check it, first split $A$ and
+$B_b$ using constants independent of the variables $x,y$.  The $m$
+conjugates of each coefficient use the disjoint sets of variables with
+different index $j$.  The split symbol basis is a matrix basis.
+Permuting those variables, changing that basis, and multiplying entries
+by powers of $a_*$ preserve their algebraic independence.  This verifies
+all generic determinant conditions simultaneously.  No independence
+over the larger invariant field $H_0$ is asserted or needed.
+
+We may perform the spectral calculations after extending $F$ to a
+split Laurent field and then to the finite extensions needed for
+eigenvalues.  Its constant coefficient field contains the old algebraic
+closure and the rational variables just chosen.  The matrix constant
+terms remain $\varepsilon$-small.  Arrow coefficients need not be
+small, since they occur only with positive free-word degree.
+
+\subsection{Linear descent of the correction}
+
+Assume first that $q_0(w)\ne0$.  Let $W_0=w(P_1^0,P_2^0)\in D$
+and choose its monic minimal polynomial
+\begin{equation}
+ q(t)\in F[t].
+ \label{eq:fixed-central-polynomial}
+\end{equation}
+We keep this polynomial fixed throughout the correction.  It is
+irreducible, because $F[W_0]$ is a finite-dimensional commutative
+domain, and separable because the characteristic is zero.  In a
+splitting field its roots are exactly the distinct eigenvalues of the
+constant output.  For completeness, left multiplication by $W_0$ on
+$D$ has characteristic polynomial a power of $q$, since $D$ is a
+vector space over $F[W_0]$.  After splitting, that left multiplication
+is a direct sum of copies of the action of the split matrix $W_0$.
+Every root of $q$ therefore occurs in that matrix, and no other
+eigenvalue occurs.  Separability makes its action semisimple.
+
+We solve $q(w(Z))=0$ in free series over $D$.  Initially take
+\eqref{eq:division-initial-input}.  Its split Schur equations vanish
+through degree two, so Lemma~\ref{lem:schur-polynomial} gives the same
+assertion for the polynomial relation.  Suppose inductively that a
+partial substitution solves this relation through degree $\ell-1$,
+where $\ell\ge3$.  Allow a new coefficient in
+$\mathcal V_{-r}^2$ at every word of length $\ell-1$, for
+$r=1,2,3,4$, and a new coefficient in $\mathcal V_0^2$ at every word
+of length $\ell$.  These are finite-dimensional $F$-spaces.
+
+Impose every coefficient equation of
+\begin{equation}
+ q(w(Z))=0\pmod{(s_1,s_2)^{\ell+1}}.
+ \label{eq:finite-central-correction}
+\end{equation}
+This is a finite affine linear system over $F$ in the allowed new
+coordinates.  Indeed two new insertions have degree at least
+$2(\ell-1)>\ell$.  At any fixed free-word degree, only boundedly
+many positive-degree insertions occur in the evaluation of the old
+series; the remaining insertions are from the $\varepsilon$-small
+constant input.  The finitely many previously fixed coefficients have
+a finite lower valuation bound.  Thus the sums defining each
+coefficient of \eqref{eq:finite-central-correction}, and its linear
+part in the new coordinates, converge in $D$ and are defined over $F$.
+Choosing an $F$-basis of $D$ turns them into finitely many ordinary
+scalar linear equations.
+
+After scalar extension, each permitted input space becomes exactly
+the collection of reverse edges or loops used in
+Proposition~\ref{prop:eleven-block-correction}.  The current partial
+substitution is a valid partial split solution by
+Lemma~\ref{lem:schur-polynomial}.  The proposition supplies a split
+correction, and that lemma converts its Schur vanishings back to all
+the equations in \eqref{eq:finite-central-correction}.  In particular
+the new degree-$\ell-1$ reverse coefficients preserve the already
+solved equations: their diagonal Schur variation at that degree is
+zero.  We have therefore solved the whole finite linear system after
+a field extension.  A matrix and its augmented matrix have the same
+ranks before and after extension, so the system already has a solution
+over $F$.
+
+Choose such a solution at every step.  The coefficient stabilization
+in Proposition~\ref{prop:eleven-block-correction} produces
+\[
+ Z\in\widetilde B\langle\!\langle s_1,s_2\rangle\!\rangle^2,
+ \qquad q(w(Z))=0,
+\]
+with the prescribed constant and linear terms.  Only finite linear
+systems have been descended.  Neither the roots of $q$ nor the split
+correction coefficients need belong to $F$.
+
+Since the input coefficients commute with $A$, evaluation at $Z$
+defines a substitution homomorphism
+\[
+ A\langle\!\langle z_1,z_2\rangle\!\rangle
+ \longrightarrow D\langle\!\langle s_1,s_2\rangle\!\rangle.
+\]
+It is well-defined by the same coefficientwise valuation bound: at
+fixed output degree, only boundedly many nonconstant input factors
+occur.  The bound also justifies multiplication and regrouping of the
+convergent coefficient sums.  Hence every old polynomial identity on
+the tuple and the chosen series is preserved by this substitution.
+
+If $q_0(w)=0$ already holds, use the very same division algebra and
+the initial substitution \eqref{eq:division-initial-input}, with any
+$b\ge1$.  No spectral correction is required; the old relation
+$q_0(w)$ remains zero under the substitution homomorphism.  Both cases
+produce a central division algebra of degree $11bd>d$, with identical
+generic forms for the constant and the two linear input coefficients.
+The next step restores generation and two transverse directions using
+these coefficients.  That step is needed in the unconstrained case
+as well as in the spectral case.
diff --git a/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/formal-generic.tex b/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/formal-generic.tex
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index 0000000000000000000000000000000000000000..7b647f71774408cf537aac6943d5cfb91ce7406a
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+++ b/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/formal-generic.tex
@@ -0,0 +1,406 @@
+\section{Formal evaluations and generic spectra}\label{sec:formal-generic}
+
+We begin the proof of Theorem~\ref{thm:lifting} by constructing a
+substitution $Z=(Z_1,Z_2)$ for the two free letters. Its coefficients will commute
+with the old algebra $A$, so applying it to $(X,w)$ preserves their
+polynomial identities. The matrix calculation focuses on $w$;
+Sections~\ref{sec:division-realization} and~\ref{sec:generation}
+realize the substitution in a division algebra of larger degree and
+restore generation and both transverse directions for $X(Z)$.
+
+Throughout the lifting argument, $k$ is a field of characteristic zero
+containing all roots of unity, $A$ is a central division algebra of degree
+$d$ over $k$, and
+\[
+ w\in A\langle\!\langle z_1,z_2\rangle\!\rangle.
+\]
+The letters commute with $A$, but not with each other.  Thus an element
+is a family $(a_\omega)_\omega$ indexed by finite words, written
+$\sum_\omega a_\omega z^\omega$, and multiplication uses concatenation
+of words.  Fix an algebraic closure $K$ of $k$ and a splitting
+$A\otimes_k K\simeq M_d(K)$; this uses the splitting and
+scalar-extension properties of central simple algebras
+\cite[Chapter~IV, Example~2.14(a) and Proposition~2.15]{MilneCFT}.
+
+If $P=(P_1,P_2)$ is a pair of $n\times n$ indeterminate matrices, put
+\[
+ W_n(P)=\sum_\omega a_\omega\otimes P^\omega
+       \in M_{dn}(K[[P]]).
+\]
+Here $K[[P]]$ is the ordinary commutative power series ring in the
+$2n^2$ entries.  We omit the subscript $n$ when the size is clear.
+The coefficient matrices act on the first tensor factor.  In
+particular, every input arrow between blocks is a pair of scalar
+matrices acting on the second factor.  It is not an arbitrary pair of
+matrices on both factors.
+
+In the spectral case we seek a substitution $Z=(Z_1,Z_2)$, with
+constant term $R$, for which $W(Z)$ is conjugate to a split semisimple
+constant output $W(R)$. If $\Lambda$ is the set of distinct eigenvalues
+of $W(R)$, this gives
+\[
+ q(W(Z))=0,\qquad q(t)=\prod_{\lambda\in\Lambda}(t-\lambda).
+\]
+The division realization will identify $q$ with the minimal polynomial
+over its central coefficient field. If $w$ already satisfies the
+minimal polynomial of $w(0)$, substitution retains that relation and
+the spectral correction is unnecessary.
+
+For the spectral construction, we will find matrix sizes with a common
+eigenvalue multiplicity, disjoint spectra at independent inputs, and
+scalar spectral blocks under commuting Taylor perturbations. These
+properties will make the eigenvalue responses of
+Section~\ref{sec:responses} well-defined. We first specify the
+evaluations and specializations used to establish them and to evaluate
+triangular auxiliary inputs later.
+
+\subsection{Convergence and specialization}\label{subsec:formal-conventions}
+
+Let $E$ be a complete discretely valued field containing $K$, with
+valuation $v$, such that $v(c)=0$ for $c\in K^\times$.  We use
+$E=C((\varepsilon))$, or a finite extension of such a field, and
+normalize $v(\varepsilon)=1$.  Matrix valuations mean the minimum of
+the entry valuations.  A matrix is \emph{small} if this minimum is
+positive.  A small generic point is
+$P=\varepsilon P^\circ$, where all entries of $P^\circ$ are
+algebraically independent over the previously specified constant
+field.  Several independent points can use the same $\varepsilon$.
+
+The finite extensions used to split spectra are separable in
+characteristic zero. The valuation extends to each such extension,
+which remains complete and discretely valued
+\cite[Theorem~7.38 and Remark~7.43]{MilneANT}; with the normalization
+$v(\varepsilon)=1$, its value group may be $\frac1e\mathbb Z$ for a
+positive integer $e$.
+
+\begin{lemma}[Admissible evaluations]\label{lem:admissible-evaluation}
+Suppose a constant input pair $R$ is block upper triangular with $b$
+blocks and small diagonal blocks.  Its strictly upper blocks may have
+arbitrary entries in $E$.  Let $H$ be a matrix pair of formal series
+of positive order in finitely many commuting or free variables.
+Then $W(R+H)$ is defined coefficientwise by convergent sums in $E$.
+The same assertion holds for products, finite determinants, and
+derivatives of fixed order of finitely many such entry series.
+
+These evaluations preserve formal entry identities.  They also
+remain valid coefficientwise in an additional parameter $\eta$ if
+the diagonal blocks have entries in $E[[\eta]]$ with small constant
+terms, while the strictly upper blocks and the coefficients of $H$
+have entries in $E((\eta))$.
+\end{lemma}
+
+\begin{proof}
+Fix a coefficient of degree $r$ in the variables of $H$.  An input
+path contributing to this coefficient uses at most $r$ positive-order
+terms of $H$.  Between these insertions it follows the constant upper
+triangular pair.  In each of the at most $r+1$ resulting intervals,
+there are at most $b-1$ strictly upper steps.  Thus at most
+$(r+1)(b-1)$ constant factors are strictly upper.  All remaining
+constant factors come from small diagonal blocks.
+
+Only finitely many coefficients of $H$ can occur at the chosen degree.
+Their valuations, and those of the finitely many strictly upper
+blocks, have a common lower bound.  If the original word has length
+$L$, its contribution consequently has valuation at least
+$cL-C_r$, where $c>0$ and $C_r$ is independent of the word.  There
+are finitely many words of each length, so the sum converges.  The
+same estimate applies after inserting the finitely many extra factors
+needed for a product or a derivative of fixed order.  Completeness
+allows regrouping of these sums, proving compatibility with
+multiplication and with identities.
+
+For the last assertion, first fix both the degree $r$ in $H$ and an
+$\eta$-coefficient.  The bounded number of strictly upper steps and
+of $H$-insertions bounds their total negative $\eta$-order.  Hence
+only a bounded number of positive-order diagonal $\eta$-terms can
+contribute, and only finitely many of their coefficients are used.
+Every other factor is a small constant diagonal term.  The preceding
+valuation estimate therefore applies again.  The resulting assertion
+is coefficientwise; no interchange of the two Laurent topologies is
+being asserted.
+\end{proof}
+
+We will use inverses of finite matrices as rational operations after
+evaluation.  The following elementary distinction is important.
+Although $(\varepsilon-Q)^{-1}$ has a formal expansion at $Q=0$ over
+$E$, that expansion need not converge after $Q=\varepsilon Q^\circ$.
+Accordingly, when transferring an identity involving an inverse, we
+first multiply by its determinant.  We evaluate the resulting
+numerator by Lemma~\ref{lem:admissible-evaluation}, and divide only
+when the evaluated determinant is nonzero.  Formal inverse expansions
+are used to define coefficients; they are not presumed to converge at
+every subsequent specialization.
+
+\begin{lemma}[Detection by independent residues]\label{lem:generic-detection}
+Let
+\[
+ F(Q)=\sum_\alpha c_\alpha Q^\alpha\in E[[Q_1,\ldots,Q_N]],
+ \qquad \inf_\alpha v(c_\alpha)>-\infty.
+\]
+Adjoin elements $Q_1^\circ,\ldots,Q_N^\circ$ whose residues are
+algebraically independent over the residue field of $E$, and complete
+with respect to the extended valuation.  Then
+$F(\varepsilon Q^\circ)$ is defined, and is nonzero whenever $F$ is
+nonzero.
+\end{lemma}
+
+\begin{proof}
+The valuations $v(c_\alpha)+|\alpha|$ tend uniformly to infinity
+with $|\alpha|$.  Their least value is therefore attained by a
+nonempty finite set of monomials.  After scaling to this valuation,
+the residue of their sum is a nonzero polynomial in the independent
+residues of the $Q_j^\circ$.  This proves both assertions.
+\end{proof}
+
+In applications, the coefficients of a cleared numerator are finite
+sums of products of base-point scalars and matrix-entry series.
+The path estimate gives a uniform valuation bound in the auxiliary
+coordinates.  Lemma~\ref{lem:generic-detection} therefore identifies
+formal identities in these coordinates with identities at independent
+small generic points.  The base-point valued field is constructed
+first; the auxiliary residues are adjoined afterward.  This order
+keeps their independence valid even after the base spectrum has been
+split.  The argument can be applied separately at each Taylor degree.
+
+We record two further field conventions.  Evaluation
+\[
+ K[[P]]\longrightarrow C(P^\circ)((\varepsilon)),
+ \qquad P\longmapsto\varepsilon P^\circ,
+\]
+is injective, by homogeneous degree or by
+Lemma~\ref{lem:generic-detection}.  It extends to the fraction field,
+and a finite splitting field embeds into a finite algebraic extension
+of the target.  Also, if $R$ is a small constant pair, then
+\begin{equation}
+K[[P]]\longrightarrow E[[\Delta]],\qquad P\longmapsto R+\Delta
+ \label{eq:generic-taylor-embedding}
+\end{equation}
+is injective: reduction modulo the valuation ideal sends the image of
+$f$ to $f(\Delta)$.  We use this observation for small block diagonal
+$R$.  Denominators in the fraction field remain nonzero, so the map
+extends to fraction fields.
+
+Finally, the Taylor substitution $P\mapsto P+\eta v$ maps
+$\operatorname{Frac}K[[P]]$ into
+$\operatorname{Frac}K[[P]][[\eta]]$, with reduction the identity.
+It extends to a finite separable extension $\Omega$ with values in
+$\Omega[[\eta]]$: choose a primitive element and lift its defining
+equation using its nonzero derivative.  The lift reducing to the
+chosen primitive element is unique, by coefficient recursion.  This
+convention lets us move a base point while retaining its labeled
+eigenvalue branches.
+
+\subsection{Generic division evaluations}\label{subsec:generic-division}
+
+Generic matrix evaluations have a useful divisibility property for
+their eigenvalue multiplicities.  We obtain it by first evaluating
+inside a division algebra.
+
+For $n\ge1$, choose independent central variables $\alpha,\beta$
+and a primitive $n$th root of unity $\zeta_n\in k$.  Over
+$k_0=k(\alpha,\beta)$ let $B_n$ be generated by $x,y$ with
+\[
+ x^n=\alpha,\qquad y^n=\beta,\qquad yx=\zeta_nxy.
+\]
+For $n=1$ this means $B_1=k_0$.  The reduced monomials
+$x^iy^j$, $0\le i,j<n$, form a basis.  Indeed they span by the
+relations, and after adjoining $n$th roots of $\alpha$ and $\beta$
+their clock-and-shift matrices are linearly independent.  These
+matrices span $M_n$, so $B_n$ is central simple of degree $n$.
+This is the standard symbol construction
+\cite[Corollary~2.5.5(1)]{GilleSzamuely2006}, with the inverse
+root of unity for our convention.
+
+\begin{lemma}\label{lem:symbol-division}
+The central simple algebra $A\otimes_k B_n$ is division.  Its pure
+rational scalar extensions and its central Laurent series extensions
+are division as well.
+\end{lemma}
+
+\begin{proof}
+Start with $A((x))$, where $x$ commutes with $A$, and let $\sigma$
+fix $A$ and send $x$ to $\zeta_nx$.  The skew Laurent series ring
+$A((x))((y;\sigma))$, with $ya=\sigma(a)y$, is a division ring by
+the leading-coefficient construction of inverses.  The elements
+$x^n,y^n$ are central and algebraically independent.  The defining
+relations give a unital map of $A\otimes_k B_n$ into this division
+ring.  Central simplicity makes the map injective.  A finite
+dimensional domain over a field is division, since multiplication by
+a nonzero element is an injective and hence surjective linear map.
+
+For a central division algebra $D/F$, the rational extension $D(t)$
+embeds in the division ring $D((t))$.  Iterating proves the rational
+assertion.  Because $D$ has a finite $F$-basis,
+$D\otimes_F F((\varepsilon))=D((\varepsilon))$, which is again
+division by leading coefficients.
+\end{proof}
+
+Choose a basis $b_1,\ldots,b_{n^2}$ of $B_n$, and let
+$c_{ij}$, $i=1,2$, $j=1,\ldots,n^2$, be independent variables.
+Evaluate $w$ at
+\[
+ P_i=\varepsilon\sum_j c_{ij}b_j.
+\]
+This gives an element of a central division algebra of degree $dn$
+over
+\[
+ F_n=k(\alpha,\beta,(c_{ij}))((\varepsilon)).
+\]
+After splitting $A$ and $B_n$ by constants independent of the $c_{ij}$,
+the $b_j$ become a basis of $M_n$.  The change from the $c_{ij}$ to
+the entries of $P_i/\varepsilon$ is therefore an invertible linear
+change of independent coordinates.  The resulting evaluation is a
+small generic scalar matrix evaluation in the sense fixed above.
+
+Let $e_n$ denote the minimal-polynomial degree of this element.  Its
+minimal polynomial is irreducible and separable over $F_n$.
+Consequently $W(P)$ is semisimple over a splitting field, and every
+distinct root has the same multiplicity
+\begin{equation}
+t_n=\frac{dn}{e_n}.
+ \label{eq:generic-multiplicity}
+\end{equation}
+For completeness, if $D/F$ has degree $N$ and $a\in D$ has minimal
+polynomial $q$ of degree $e$, then left multiplication by $a$ has
+characteristic polynomial $q^{N^2/e}$: view $D$ as a vector space
+over $F[a]$.  After splitting $D=M_N$, left multiplication acts by
+the matrix $a$ on each of its $N$ columns.  Its characteristic
+polynomial is therefore the $N$th power of the matrix characteristic
+polynomial.  Comparing root multiplicities gives $N/e$ in the
+matrix representation.  This also proves that $N/e$ is an integer.
+
+Linear independence of $1,a,\ldots,a^{e-1}$ is preserved by field
+extension.  Thus the same $e_n,t_n$ describe $W(P)$ over a splitting
+field of $\operatorname{Frac}K[[P]]$.
+
+\subsection{Nonconstant roots and stable multiplicity}
+\label{subsec:stable-multiplicity}
+
+Let $q_0$ be the minimal polynomial of $w(0)$ over $k$.  If
+$q_0(w)=0$ as a free series, the subsequent spectral construction is
+unnecessary; the unconstrained case of the lifting argument will use
+this existing relation.  In the rest of this section assume
+$q_0(w)\ne0$.
+
+A nonzero homogeneous part of $q_0(w)$ is detected by scalar matrices
+of every sufficiently large size.  To see this, choose a word of
+length $L$ with nonzero coefficient and use a path of $L+1$ vertices,
+placing its successive letters on the consecutive matrix-unit
+arrows.  The $(1,L+1)$ entry detects precisely that coefficient.
+Padding by zero blocks works for all larger sizes.  It follows that
+$q_0(W(P))\ne0$ at every generic size $n\ge n_0$, for some $n_0$.
+
+\begin{lemma}\label{lem:nonconstant-roots}
+For $n\ge n_0$, every distinct generic eigenvalue of $W(P)$ has a
+nonzero differential in the input entries.  Independent small generic
+points of sizes at least $n_0$ have disjoint spectra.  Their spectra
+also avoid the constant spectrum of $w(0)$.
+\end{lemma}
+
+\begin{proof}
+Use the division evaluation over $F_n$.  Each derivation
+$\partial/\partial c_{ij}$ acts coefficientwise on Laurent series
+and extends uniquely to algebraic extensions in characteristic zero.
+Uniqueness implies that it commutes with automorphisms of a splitting
+field over $F_n$.  If all these derivations killed one root, they
+would kill all its Galois conjugates.  Irreducibility makes these all
+the distinct roots.  Thus every coefficient of the reduced
+characteristic polynomial would have all its input derivatives zero.
+
+The coefficient of $\varepsilon^r$ in such a characteristic
+coefficient is a homogeneous polynomial of degree $r$ in the
+$c_{ij}$.  In characteristic zero a polynomial whose partial
+derivatives all vanish is constant.  All terms of positive
+$\varepsilon$-degree therefore vanish.  The characteristic polynomial
+is the characteristic polynomial of $w(0)$.  Semisimplicity then
+implies $q_0(W(P))=0$, a contradiction.
+
+Splitting $A,B_n$ uses constants independent of the inputs.  The
+invertible basis change described above, together with the nonzero
+factor $\varepsilon$, transports the differential assertion to
+ordinary matrix-entry coordinates.  For independent points $P,Q$,
+derivations in the $P$-coordinates kill the roots belonging to $Q$.
+They cannot kill a root belonging to $P$.  No root can therefore
+belong to both spectra.  The same argument excludes a constant root.
+\end{proof}
+
+Choose $p\ge n_0$ for which the positive integer $t_p$ is minimal
+among all $t_n$, $n\ge n_0$, and put
+\[
+ t=t_p,\qquad \mathcal S=\{pN:N\ge1\}.
+\]
+
+\begin{lemma}\label{lem:stable-multiplicity}
+For every $N\ge1$,
+\[
+ t_{pN}=t,\qquad e_{pN}=Ne_p.
+\]
+\end{lemma}
+
+\begin{proof}
+Specialize a size-$pN$ point to a diagonal sum of $N$ independent
+size-$p$ points.  Lemma~\ref{lem:nonconstant-roots} gives $Ne_p$
+distinct eigenvalues there.  The generic bound on minimal-polynomial
+degree survives this specialization: it is expressed by the vanishing
+of all $(e_{pN}+1)$-minors of the matrix whose columns are the
+vectorizations of $1,W,\ldots,W^{e_{pN}}$.  These are formal entry
+identities covered by Lemma~\ref{lem:admissible-evaluation}.
+Consequently $e_{pN}\ge Ne_p$.  Minimality of $t$ gives the opposite
+inequality,
+$e_{pN}=dpN/t_{pN}\le dpN/t=Ne_p$.
+\end{proof}
+
+We henceforth use sizes in $\mathcal S$.  At size $n$, the minimal
+polynomial always has degree at most $e_n=dn/t$ at every admissible
+specialization.  The following consequence controls all commuting
+Taylor deformations used below.
+
+\begin{lemma}[Scalar branches at maximal root count]
+\label{lem:scalar-branches}
+Let $V(T)$ be a matrix over a commuting formal Taylor ring $E[[T]]$.
+Suppose $V(0)$ is split semisimple with $e$ distinct eigenvalues and
+the minimal-polynomial degree of $V(T)$ over
+$\operatorname{Frac}E[[T]]$ is at most $e$.  Then a formal change of
+basis reducing to the identity separates $V(T)$ into $e$ scalar
+blocks.  The sizes of these blocks are their multiplicities at
+$T=0$.
+\end{lemma}
+
+\begin{proof}
+Use the eigenspace decomposition at zero.  Off-diagonal blocks can be
+removed recursively in total Taylor degree: the equation for the
+block between eigenvalues $\lambda_i,\lambda_j$ is multiplication
+by $\lambda_i-\lambda_j$, a nonzero scalar.  We obtain blocks
+$V_i(T)$ reducing to $\lambda_i I$.  Their characteristic polynomials
+have pairwise unit resultants, because their reductions have different
+roots.  Their minimal polynomials over the Taylor fraction field are
+therefore pairwise coprime.  The minimal-polynomial degree of their
+direct sum is the sum of their degrees.  There are $e$ nonempty
+blocks and the total degree is at most $e$, so every block has
+minimal-polynomial degree one.  Each $V_i(T)$ is scalar.  Its scalar
+entry belongs to $E[[T]]$, since the block entries do.
+\end{proof}
+
+In particular, the roots at a small semisimple point with $e_n$
+distinct roots have labeled scalar Taylor branches.  The splitting
+field of the generic point embeds in their Taylor fraction field by
+\eqref{eq:generic-taylor-embedding}; a labeling is chosen by reduction
+to the roots at the base.
+
+Let $\rho_n$ be the generic rank of the input Jacobian of
+\[
+ \operatorname{tr}W,\operatorname{tr}W^2,\ldots,
+ \operatorname{tr}W^{dn}.
+\]
+Its minors are formal entry identities, so $\rho_n$ bounds the
+corresponding specialized ranks.  At a maximal-root-count point the
+scalar branches satisfy
+\[
+ d\operatorname{tr}W^j
+   =jt\sum_{h=1}^{e_n}\lambda_h^{j-1}\,d\lambda_h.
+\]
+The first $e_n$ rows give an invertible Vandermonde matrix.  Hence
+the trace Jacobian and the eigenvalue Jacobian have the same rank
+there, including over a commuting Taylor fraction field.  We have
+therefore obtained both ingredients needed for the response
+calculations: scalar branches and specialization-stable rank bounds.
diff --git a/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/generation.tex b/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/generation.tex
new file mode 100644
index 0000000000000000000000000000000000000000..ed54e89a0405d91068449af83d7234bc6ee38ada
--- /dev/null
+++ b/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/generation.tex
@@ -0,0 +1,310 @@
+\section{Restoring generation and two transverse directions}
+\label{sec:generation}
+
+The preceding construction gives a substitution $Z(s_1,s_2)$ inside
+the centralizer of $A$ in a larger central division algebra.  It
+preserves the old polynomial identities and algebraizes the chosen
+series.  To complete Theorem~\ref{thm:lifting}, we must
+make the new constant tuple generate the larger algebra and retain
+two independent directions modulo inner derivations. We first use the
+original transverse directions to prove generation of the generic
+diagonal product and to control the arrows between its blocks.
+Translating the first free letter by a new central parameter then joins
+the blocks into a full matrix algebra. A single rank calculation shows
+that both directions remain independent modulo inner derivations.
+
+Throughout this section, brackets with a tuple are taken componentwise.
+Thus, for $C=(C_1,\ldots,C_u)$ in an algebra $B$, write
+\[
+ J(C)=\{([T,C_1],\ldots,[T,C_u]):T\in B\}.
+\]
+Polynomial relations on $X_0$ detect the two input directions to first
+order.
+
+\begin{lemma}[Derivatives of relations]
+\label{lem:derivatives-of-relations}
+Let $A$ be a finite-dimensional central simple algebra over a field
+$k$ of characteristic zero.  Let
+$X\in A\langle\!\langle z_1,z_2\rangle\!\rangle^u$ have constant
+tuple $X_0$ generating $A$, and suppose its two linear coefficients
+$X_1,X_2$ are independent modulo $J(X_0)$.  Define
+\[
+ \theta:k\langle x_1,\ldots,x_u\rangle\longrightarrow A,
+ \qquad \theta(f)=f(X_0),\qquad I=\ker\theta.
+\]
+Then the map taking $f\in I$ to the coefficients of $z_1,z_2$ in
+$f(X)$ is onto $A^2$.
+\end{lemma}
+
+\begin{proof}
+Denote these two coefficient maps by $\delta_1,\delta_2$.  Each
+satisfies
+\[
+ \delta_j(fg)=\theta(f)\delta_j(g)+\delta_j(f)\theta(g).
+\]
+Since $\theta$ is onto and $I$ is a two-sided ideal, the image
+$M=(\delta_1,\delta_2)(I)$ is an $A$-bimodule in $A^2$.
+
+Left and right multiplication generate $\operatorname{End}_k(A)$;
+see \cite[Chapter~IV, Corollary~2.9]{MilneCFT}.
+Indeed, after splitting $A$, multiplication by matrix units on the
+left and right gives every matrix unit on the vector space $A$;
+the resulting isomorphism
+$A\otimes_k A^{\mathrm{op}}\simeq\operatorname{End}_k(A)$ descends
+by finite-dimensional linear algebra.  Applying these endomorphism
+matrix units to $M\subseteq A\otimes_k k^2$ shows that
+$M=A\otimes_k V$ for a subspace $V\subseteq k^2$.
+If $V\ne k^2$, some nonzero $(a,b)\in k^2$ annihilates $V$.
+The map $a\delta_1+b\delta_2$ then vanishes on $I$ and descends to
+a derivation of $A$.
+
+Every $k$-linear derivation of $A$ is inner
+\cite[Theorem~8]{Jacobson1937Derivations}. For completeness, after
+splitting $A=M_d(E)$, a derivation $\partial$ is implemented by
+\[
+ T=\sum_i\partial(E_{i1})E_{1i}.
+\]
+Differentiating $E_{kl}E_{i1}=\delta_{li}E_{k1}$, multiplying by
+$E_{1i}$, and summing gives $\partial(E_{kl})=[T,E_{kl}]$.
+The equations for an implementing element on a fixed $k$-basis of
+$A$ form a linear system over $k$.  Solvability after scalar
+extension implies solvability over $k$.
+It follows that $aX_1+bX_2\in J(X_0)$, a contradiction.
+\end{proof}
+
+In particular, there are $f_1,f_2\in I$ whose derivative pairs are
+$(1,0)$ and $(0,1)$.  These relations recover the two generic input
+matrices from small evaluations of the tuple.
+
+\begin{lemma}[Generic blocks]
+\label{lem:generic-generating-blocks}
+Under the hypotheses of Lemma~\ref{lem:derivatives-of-relations},
+write $d=\deg A$ and split $A$ into $d\times d$ matrices.  Fix a
+positive integer $b$ and finitely many independent small generic input pairs
+\[
+ P_i=\varepsilon P_i^{\circ},\qquad 1\le i\le r,
+\]
+of $b\times b$ matrices, with jointly independent entries in the
+$P_i^{\circ}$.  Work over a common splitting Laurent field.
+Then each tuple $C_i=X(P_i)$ generates $M_{db}$, and the diagonal
+tuple $C=\operatorname{diag}(C_i)$ generates the full product of
+these matrix algebras.
+
+For distinct $i,j$, the output derivative of a generic input arrow
+$U=(U_1,U_2)$ from block $j$ to block $i$ is nonzero modulo
+\[
+ \{C_iT-TC_j:T\in M_{db}\}.
+\]
+More precisely, the image of the arrow derivatives in this quotient
+has dimension at least $b^2$.
+\end{lemma}
+
+\begin{proof}
+Let $E$ contain the splitting data and independent input coordinates.
+Consider integral elements of the $E((\varepsilon))$-algebra
+generated by $X(P_i)$.  Their reductions modulo $\varepsilon$
+contain $M_d(E)\otimes1$, because $X_0$ generates the old algebra.
+Lemma~\ref{lem:derivatives-of-relations} also gives
+\[
+ \varepsilon^{-1}f_j(X(P_i))
+ \equiv I_d\otimes P_{i,j}^{\circ}\pmod\varepsilon.
+\]
+A generic pair generates $M_b(E)$.  To see the relevant determinant
+is nonzero, specialize one matrix to a diagonal matrix with distinct
+entries and the other to a matrix with every entry nonzero.
+Interpolation and sandwiching give all matrix units.  Thus the
+reductions generate $M_d(E)\otimes M_b(E)=M_{db}(E)$.
+Choose finitely many integral algebra elements whose reductions
+form a matrix basis.  Their determinant is a unit, so they remain
+independent over $E((\varepsilon))$, proving full generation.
+
+The representations defined by different $C_i$ have distinct kernels.
+Indeed,
+\[
+ \operatorname{tr}f_1(C_i)
+  =\varepsilon d\operatorname{tr}P_{i,1}^{\circ}
+    +O(\varepsilon^2),
+\]
+and the leading coefficients differ.  Equal kernels would induce an
+isomorphism of the full matrix images carrying every polynomial
+evaluation in one to the other.  Such an isomorphism preserves trace:
+the trace of left multiplication is matrix size times matrix trace.
+This contradicts the display.
+The kernels are therefore distinct maximal two-sided ideals.
+Evaluation is onto their finite product.  Explicitly, for each pair
+$i\ne j$ choose a polynomial vanishing in block $j$ and equal to
+the identity in block $i$.  The product over $j\ne i$ vanishes in
+all other blocks and is the identity in block $i$; multiplying by
+a lift of any matrix gives that product coordinate.
+
+For the last assertion, compare the inner columns and arrow columns
+at $P_i=P_j=0$.  The inner image is
+$J(X_0)\otimes M_b$, of dimension $(d^2-1)b^2$.  The arrow derivative
+is
+\[
+ X_1\otimes U_1+X_2\otimes U_2.
+\]
+The original transverse independence implies that the combined map
+has rank $(d^2+1)b^2$ at this reduction.  Its generic rank is at
+least this number.  At a generic pair, the inner map has rank at
+most $(db)^2=d^2b^2$; in fact it is injective, since a nonzero
+intertwiner between the two simple representations would identify
+their kernels.  Subtracting this generic inner rank leaves at least
+$b^2$ independent arrow directions in the quotient.  A generic
+fixed arrow consequently has nonzero quotient class.
+\end{proof}
+
+All determinant conditions in Lemma~\ref{lem:generic-generating-blocks}
+hold on the same independent block and arrow coordinates used in
+the division realization.  Only finitely many blocks and arrows are
+involved, so their nonzero minors can be imposed simultaneously.
+In particular, the argument does not require replacing that realization
+by a different choice of data.
+
+We next isolate the translation argument.  Its hypotheses concern
+only the constant and linear coefficients; higher-order corrections
+may be arbitrary.
+
+\begin{lemma}[A translation preserving two directions]
+\label{lem:translation-two-directions}
+Let $E$ be a field of characteristic zero, let $n\ge1$ be an integer,
+let $m=11$, and index
+the blocks of $M_{mn}(E)$ by $\mathbb Z/m\mathbb Z$.
+Suppose
+\[
+ Y(s_1,s_2)\in M_{mn}(E)
+       \langle\!\langle s_1,s_2\rangle\!\rangle^u
+\]
+has constant tuple $C=\operatorname{diag}(C_i)$ generating
+$\prod_iM_n(E)$.  Suppose its two linear coefficients have the
+following supports:
+\[
+ (Y_1)_{ij}=0\ \text{unless }i-j\in\{1,2\},
+ \qquad
+ (Y_2)_{ij}=0\ \text{unless }i-j\in\{3,4\}.
+\]
+Assume every displayed edge coefficient is nonzero modulo the
+cross-inner image $\{C_iT-TC_j:T\in M_n(E)\}$.
+For a new central Laurent parameter $\gamma$, define
+\[
+ Y'(s'_1,s'_2)=Y(\gamma+s'_1,s'_2).
+\]
+Then $Y'_0=Y(\gamma,0)$ generates $M_{mn}(E((\gamma)))$, and
+$Y'_1,Y'_2$ are independent modulo $J(Y'_0)$.
+\end{lemma}
+
+\begin{proof}
+The substitution is well-defined.  In the coefficient of a fixed
+new word, a contribution from an old word of length $r$ contains
+$r$ minus the new word length factors of $\gamma$.  Thus only
+finitely many terms contribute at each $\gamma$ degree.  Coefficients
+converge in $E[[\gamma]]$, and the substitution respects products.
+
+Let $B=\prod_iM_n(E)$ and let $I_C$ be the kernel of evaluation of
+the free algebra at $C$.  First-letter differentiation on $I_C$
+has image a $B$-bimodule in $M_{mn}(E)$.  Its projection onto every
+first-color edge is nonzero.  Otherwise the derivative into that
+cross bimodule would descend to a derivation $\delta:B\to M_n(E)$,
+where the left and right actions are through the distinct endpoint
+factors $i,j$.  Such a derivation is inner.  In fact, for the central
+idempotent $e_i$ of factor $i$, differentiating $e_i a=a e_i$ gives
+\[
+ \delta(a)=a_iT-Ta_j,\qquad T=\delta(e_i).
+\]
+Applied to the generators, this contradicts the transverse edge
+hypothesis.  Since each full rectangular block is a simple bimodule
+for its two matrix factors, the derivative image contains the full
+block on every first-color edge.
+
+Divide the corresponding evaluations of elements of $I_C$ at
+$Y(\gamma,0)$ by $\gamma$.  Their reductions supply all these edge
+blocks.  Ordinary evaluations already supply the full diagonal
+product.  The offset-one edges form a directed cycle through all
+vertices, so products supply every matrix block.  A finite set of
+such elements with independent reductions therefore proves that
+$Y(\gamma,0)$ generates the full matrix algebra.
+
+It remains to check the two directions.  Put $N=mn$ and
+$J_0=J(C)$.  Since $C$ generates $B$, its commutant consists of the
+$m$ block scalars.  Hence $\dim J_0=N^2-m$.  Choose $N^2-m$ inner
+columns with independent reductions spanning $J_0$.  For a block
+scalar $c=\operatorname{diag}(c_i I_n)$, the additional column
+\[
+ \gamma^{-1}[c,Y(\gamma,0)]
+\]
+is integral and reduces to $[c,Y_1]$.  These columns have rank
+$m-1$ modulo $J_0$.  Indeed, $[c,Y_1]\in J_0$ forces $c_i=c_j$
+on each offset-one edge, by its transverse hypothesis, and the
+cycle forces all $c_i$ to agree.
+
+The two shifted linear coefficients reduce to $Y_1,Y_2$.  Suppose
+\[
+ aY_1+bY_2\in J_0+\{[c,Y_1]:c\text{ is block scalar}\}.
+\]
+At a second-color edge, only $bY_2$ contributes outside $J_0$,
+so $b=0$.  On the directed offset-one cycle, the relation then
+forces $a=c_i-c_j$ on each edge.  Summing gives $ma=0$, whence
+$a=0$ in characteristic zero.  Thus these two reductions are
+independent modulo the enlarged inner-column space.
+
+We have found $N^2-m+(m-1)+2=N^2+1$ independent reductions among
+inner columns and the two tangents.  They lift to independent
+columns over $E((\gamma))$.  Full generation of $Y'_0$ implies
+$\dim J(Y'_0)=N^2-1$, so the two tangent classes are independent.
+\end{proof}
+
+Apply the lemma to $Y=X(Z)$.  Lemma~\ref{lem:generic-generating-blocks}
+supplies its diagonal-product and edge hypotheses, while the
+eleven-block construction supplies precisely the two color supports.
+Consequently the translated tuple generates after splitting and has
+two independent tangent classes.  These are finite rank conditions,
+so they also hold over the division algebra's own center.
+The central Laurent extension by $\gamma$ preserves division.
+Both the old polynomial identities and the new algebraic relation
+are preserved by the convergent substitution.
+
+\subsection{Returning to a countable center}
+
+The complete Laurent fields used so far need not be countable.
+We finish by retaining only the scalar data required by the result.
+Use the notation of Section~\ref{sec:division-realization}:
+$F_0=H_0(a_*)$ is the construction field, $D_0$ is the central
+division algebra over it, and
+\[
+ L=F_0((\varepsilon))((\gamma))
+\]
+is the full shifted Laurent center. The field $F_0$ is countable:
+$L_0$ is generated by finitely many indeterminates over the countable
+field $k(\alpha,\beta)$, and $H_0$ is a subfield of $L_0$.
+The algebra $D_0$ has degree $dbm$, and $D_0\otimes_{F_0}L$
+is the central division algebra just constructed.
+
+Choose an $F_0$-basis of $D_0$. Let $k'\subseteq L$ be the field
+generated by $F_0$, the coefficients of the monic polynomial $q$
+(taken to be $q_0$ when $q_0(w)=0$),
+and every scalar coordinate in this basis of every coefficient
+of the translated series $X'$ and $w'$. There are countably many
+free words and finitely many coordinates per coefficient, so $k'$
+is countable.  Put
+\[
+ A'=D_0\otimes_{F_0} k'.
+\]
+This is central simple of degree $dbm$.  Its scalar extension to
+$L$ is division, so $A'$ embeds in a division ring and has no zero
+divisors.  A finite-dimensional algebra without zero divisors is
+a division algebra.  Thus $A'/k'$ has the required central division
+property and strictly larger degree.
+
+All coefficients of $X',w'$ and $q$ lie in this smaller model.
+The identities descend by its injective scalar extension to $L$.
+Generation and transverse independence descend as well, since
+the ranks of the finite coordinate matrices defining them do not
+change under field extension.  This proves every conclusion of
+Theorem~\ref{thm:lifting}.
+
+In the unconstrained case $q_0(w)=0$, the same argument applies
+without the spectral corrections: use the same eleven-block
+generic constant and colored linear terms, retain the relation
+$q_0$, and carry out the translation and countable restriction
+above.  Thus that case also increases degree while preserving
+generation and both directions.
diff --git a/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/global-defect.tex b/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/global-defect.tex
new file mode 100644
index 0000000000000000000000000000000000000000..469dbd6d99acd4b90a8874b3a5d3629a968dd03c
--- /dev/null
+++ b/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/global-defect.tex
@@ -0,0 +1,298 @@
+\section{The common space of spectral relations}
+\label{sec:global-defect}
+
+The active space describes the possible responses at one block. We now
+identify the relations that remain when many independent blocks are used
+together. The essential point is that the derivatives at different
+triangular inputs belong to one common space. A bound on their ranks
+separately would not give the relations needed for the free-series
+construction.
+
+We retain the preceding notation. The allowed sizes are
+\(\mathcal S=\{pN:N\geq1\}\); at size \(n\in\mathcal S\), there are
+\(e_n=dn/t\) distinct generic roots, each of multiplicity \(t\).
+The active space has dimension \(a_n\), and \(\rho_n\) is the generic
+rank of the eigenvalue differential, equivalently of the trace
+differential. In particular \(\rho_n\leq a_n\).
+We use the replication identity \(a_{pN}=Na_p\) and the following
+consequence of the preceding attainability argument. There are a size
+\(r\in\mathcal S\) and an integer \(K_0\geq1\) such that the loop
+responses at a generic size-\(r\) point, together with paired responses
+through \(K_0\) independent generic size-\(r\) partners, span its
+active space. In each pairing, either arrow may be fixed generically
+and the opposite arrow varied. All spaces below are extended to a
+common splitting field when several independent points are involved.
+
+\begin{lemma}[Derivatives at a star]
+\label{lem:star-derivatives}
+Let allowed-size blocks, evaluated by the preceding formal
+specialization conventions, have split semisimple outputs with
+disjoint spectra and the maximal total number of roots. Add input
+arrows only from designated
+hub blocks to designated leaf blocks, with no arrows between hubs or
+between leaves. The resulting input has the same roots and
+multiplicities as its diagonal part. A loop derivative at a block is
+its loop response \(D\), with zero entries at all other blocks. If
+\(U\) is a forward arrow between a leaf \(P\) and a hub \(Q\), a
+derivative in its reverse arrow \(V\) has entries
+\[
+ \bigl(m_h(P,Q;U,V)\bigr)_h,
+ \qquad
+ \bigl(m_k(Q,P;V,U)\bigr)_k
+\]
+at these two blocks and zero entries at the others.
+\end{lemma}
+
+\begin{proof}
+Order the leaves before the hubs. The input and its evaluated output
+are block upper triangular. The diagonal output blocks are the
+original ones; since their spectra are disjoint, the output is similar
+to their direct sum. In particular it is semisimple with the stated
+roots and multiplicities.
+
+Differentiate \(\operatorname{tr}(W^j)\). A closed block path
+contributing to a reverse-arrow derivative must use that reverse
+arrow once and its matching forward arrow once. There is no other
+way to return to the starting block in this star. Its contribution
+is therefore exactly the mixed-arrow contribution at the diagonal
+input. Loop derivatives use no off-diagonal return. The previously
+proved mixed-response formula gives the displayed pairs after
+converting trace derivatives to root derivatives. This conversion is
+invertible on all distinct roots: its entries are
+\(jt\lambda_h^{j-1}\), and the corresponding Vandermonde determinant
+is nonzero. The same reasoning applies to specialized blocks whenever
+the stated maximal-count and disjointness hypotheses hold.
+\end{proof}
+
+\begin{lemma}[A stable upper bound for the differential rank]
+\label{lem:tail-defect}
+One can choose an arbitrarily large allowed size \(b\) and integers
+\(a=a_b\) and \(g\geq0\), such that
+\[
+ \rho_b=a-g,
+ \qquad
+ \rho_{bM}\leq Ma-g \quad(M\geq1).
+\]
+At a generic size-\(b\) point choose one root from each part of the
+active partition. Projection onto these \(a\) selected roots
+identifies the active space with the full coordinate space. The
+quotient of that coordinate space by the loop-response image has
+dimension \(g\).
+\end{lemma}
+
+\begin{proof}
+For \(N\geq1\), put
+\[
+ g_N=Na_r-\rho_{rN}.
+\]
+Replication and \(\rho_{rN}\leq a_{rN}\) show that \(g_N\geq0\).
+For \(N\geq K_0\), take \(K_0\) of \(N\) independent size-\(r\)
+blocks as hubs and the remainder as leaves. By
+Lemma~\ref{lem:star-derivatives} and the attainability property, the
+projection of the derivative image onto the leaf active spaces is
+surjective. Reverse-arrow variables belonging to different leaves
+are independent. Thus
+\[
+ \rho_{rN}\geq(N-K_0)a_r,
+ \qquad g_N\leq K_0a_r.
+\]
+Here the specialized rank is bounded by the generic rank by the
+formal minor identities. The finitely many smaller values of \(N\)
+do not affect boundedness.
+
+A bounded sequence of nonnegative integers has an eventual minimum
+that is attained arbitrarily late. Choose \(N_0\geq K_0\) at such
+an occurrence, and write
+\[
+ g=g_{N_0},\qquad
+ g_N\geq g\ (N\geq N_0),\qquad
+ b=rN_0,\qquad a=N_0a_r.
+\]
+Then \(a=a_b\), \(\rho_b=a-g\), and
+\[
+ \rho_{bM}=Ma-g_{N_0M}\leq Ma-g.
+\]
+No eventual constancy of \(g_N\) is required. Finally, the active
+partition says that projection to one representative of each part
+is an isomorphism on the active space. The loop image has rank
+\(\rho_b\), so its quotient has dimension \(g\).
+\end{proof}
+
+Fix \(b,a,g\) as in Lemma~\ref{lem:tail-defect}. For each generic
+size-\(b\) point \(P\), let \(E_P\) be the \(a\)-dimensional
+space of selected-root coordinates, and let
+\[
+ L_P=\operatorname{im}D_P\subseteq E_P,
+ \qquad H_P=E_P/L_P.
+\]
+Thus \(\dim H_P=g\). We regard \(H_P^*\) as the space of
+covectors on \(E_P\) annihilating all loop responses.
+
+\begin{lemma}[Surjectivity at both ends of a pair]
+\label{lem:pair-quotient-surjectivity}
+Let \(P,Q\) be independent generic size-\(b\) points. For a fixed
+generic arrow \(U\) between them, varying the reverse arrow \(V\)
+gives a joint paired-response map into \(H_P\oplus H_Q\) whose
+projection to either summand is surjective. This holds whichever
+orientation is fixed.
+\end{lemma}
+
+\begin{proof}
+Specialize \(P\) and \(Q\) to diagonal sums of \(N_0\)
+independent size-\(r\) points. For each little block in \(P\), the
+\(N_0\geq K_0\) blocks in \(Q\), together with their components
+of the fixed generic arrow, provide enough partners to span its
+active space after loop columns are included. The reverse-arrow
+entries touching different little blocks of \(P\) vary independently.
+Compression at a root of one little block sees only the reverse
+entries at that block. Consequently loops and paired columns span
+all \(N_0a_r=a\) active coordinates at the \(P\)-end.
+
+The resulting nonzero rank minor persists at generic \(P,Q,U\).
+The labeled root data and the pair inverses are regular at the
+disjoint diagonal specialization, so this is precisely the
+determinant specialization justified earlier. Passing to the loop
+quotient gives surjectivity onto \(H_P\). The same proof works at
+the \(Q\)-end and with the other orientation fixed. The finitely
+many nonzero minor conditions hold simultaneously on a nonempty
+generic set.
+\end{proof}
+
+The next lemma supplies the common subspace promised at the start
+of this section. Its formal-coordinate argument is the reason we
+obtain compatible relations between pairs of blocks.
+
+\begin{lemma}[A common tangent space]
+\label{lem:common-tangent}
+Fix \(M\geq1\) independent generic size-\(b\) points and their
+selected roots. At inputs with this diagonal and arbitrary strictly
+upper block arrows, the collective span of all selected-root
+derivative columns has dimension at most \(Ma-g\). The span of
+all loop columns and all two-block paired columns has dimension
+exactly \(Ma-g\).
+\end{lemma}
+
+\begin{proof}
+Put \(R=Ma-g\). For \(M=1\) the assertion is the definition of
+the loop rank, so suppose \(M\geq2\). Use one block as a hub,
+ordered last. The loops span \(\bigoplus_i L_{P_i}\), of dimension
+\(M(a-g)\). Modulo that space, reverse derivatives project
+surjectively onto each of the \(M-1\) leaf quotients by
+Lemma~\ref{lem:pair-quotient-surjectivity}. These projections vary
+independently. The derivative rank at this star is therefore at
+least
+\[
+ M(a-g)+(M-1)g=R.
+\]
+The generic upper bound in Lemma~\ref{lem:tail-defect} is \(R\).
+Thus at a generic strictly upper-arrow array \(\mathcal U\) the
+selected-root derivative rank equals \(R\).
+
+Work over the field containing \(\mathcal U\) and the diagonal
+splitting data. Let \(F(H)\) be the formal Taylor map of the
+selected root shifts under unrestricted input-entry perturbations
+\(H\). The base has the maximal root count, so the scalar branch
+construction applies. Every \((R+1)\)-minor of \(dF\) is zero,
+and some \(R\)-minor is a unit at \(H=0\). Choose the corresponding
+\(R\) output coordinates and use them, together with the remaining
+input coordinates, as new formal source coordinates \((y,z)\).
+The formal inverse theorem gives
+\[
+ F(y,z)=(y,G(y,z)).
+\]
+The vanishing \((R+1)\)-minors now imply
+\(\partial G/\partial z=0\). In characteristic zero, coefficientwise
+differentiation gives \(G(y,z)=G(y)\). At any formal source point
+where \(F=0\), all derivative columns therefore belong to the fixed
+graph
+\[
+ \{(v,dG(0)v):v\in E^R\}.
+\]
+
+Shifting only the strictly upper arrows leaves every root unchanged.
+Apply the preceding conclusion to these shifts: their Jacobian
+columns all belong to this one graph, not merely to spaces of the
+same dimension.
+
+It remains to pass from formal shifts at \(\mathcal U\) to all
+upper-arrow arrays. Let \(J(U)\) be the selected Jacobian at such
+an array. It is polynomial in \(U\). Indeed a trace derivative
+has just one differentiated input step, and every other
+off-diagonal step is strictly upper; a contributing block path
+therefore uses a bounded number of upper arrows. The remaining
+diagonal sums converge by smallness. The conversion from trace
+derivatives to root derivatives has coefficients depending only
+on the fixed diagonal roots, so it preserves polynomial dependence.
+
+Every covector annihilating the preceding graph annihilates the
+polynomial matrix \(J(\mathcal U+S)\) as a formal identity in
+the upper-arrow shift \(S\). Substitution \(S=V-\mathcal U\)
+is valid in this polynomial identity. Hence it annihilates every
+coefficient of \(J(V)\). The span of those coefficients, formed
+over the original diagonal field, has dimension at most \(R\),
+since scalar extension preserves rank. It contains every specialized
+column and proves the collective-span assertion.
+
+Loop columns occur at the diagonal input. A paired column supported
+on two blocks occurs at the base with just the corresponding single
+upper arrow, by Lemma~\ref{lem:star-derivatives}. Allowing both
+arrows to vary makes the collection independent of which orientation
+is called forward. All these columns therefore lie in the common
+space. The one-hub star already gives \(R\) independent columns
+among them, proving equality.
+\end{proof}
+
+\begin{proposition}[Compatible dual relations]
+\label{prop:compatible-relations}
+For any finite nonempty collection of independent generic size-\(b\)
+points \(P_i\), there is a \(g\)-dimensional space
+\[
+ \mathcal G_{\{P_i\}}\subseteq\bigoplus_i H_{P_i}^*
+\]
+with the following properties.
+\begin{enumerate}
+\item Projection to each \(H_{P_i}^*\) is an isomorphism.
+\item It annihilates all joint paired responses, with both arrows
+arbitrary.
+\item For a fixed generic arrow between \(P_i,P_j\), the
+annihilator in \(H_{P_i}^*\oplus H_{P_j}^*\) of the variable
+reverse-arrow image is exactly the restriction of
+\(\mathcal G_{\{P_i\}}\).
+\item On adjoining independent blocks, restriction of the larger
+space is the original space after scalar extension.
+\end{enumerate}
+For one block this space is \(H_P^*\). These statements include
+\(g=0\).
+\end{proposition}
+
+\begin{proof}
+Define \(\mathcal G\) as the annihilator of all loop and joint
+paired columns in the selected coordinate space. By
+Lemma~\ref{lem:common-tangent}, it has dimension \(g\); annihilation
+of loops identifies it with a subspace of \(\bigoplus_i H_{P_i}^*\).
+If an element has zero component at one vertex \(i\), its pairing
+with columns on \((i,j)\) and the surjectivity onto \(H_{P_j}\)
+force its component at every other vertex \(j\) to vanish.
+Projection onto any one component is therefore injective, and hence
+an isomorphism by dimension.
+
+For a fixed generic arrow on \((i,j)\), the reverse-image in
+\(H_{P_i}\oplus H_{P_j}\) has dimension at least \(g\), since
+its projection onto either summand is surjective. It is killed by
+the \(g\)-dimensional restriction of \(\mathcal G\). The latter's
+annihilator has dimension \(2g-g=g\), so these two spaces coincide.
+This proves the exact pairwise assertion.
+
+Finally, restriction from a larger collection annihilates every old
+loop and pair column. It is injective, because projection to any
+retained vertex is injective, and both spaces have dimension \(g\).
+It is therefore onto. For a single block the same statement follows
+from its quotient dimension. All dimension arguments remain valid
+when \(g=0\).
+\end{proof}
+
+We henceforth write \(\mathcal G\) for these compatible spaces,
+specifying the blocks when needed. We have identified all linear
+relations shared by the loop and paired corrections. The next
+section proves that the leading free-word errors satisfy the
+corresponding cyclic relations.
diff --git a/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/introduction.tex b/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/introduction.tex
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index 0000000000000000000000000000000000000000..dd6445211fdde0d51776a05061604a5cd2f600b3
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+++ b/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/introduction.tex
@@ -0,0 +1,139 @@
+\section{Introduction}\label{sec:introduction}
+
+A division ring $D$ is \emph{algebraic over its center} $F=Z(D)$ if every
+$a\in D$ satisfies a nonzero polynomial in $F[t]$. For a subset $S\subseteq D$,
+write $F(S)$ for the division subring generated by $F\cup S$.
+The ring is \emph{locally finite} if
+\[
+ [F(S):F]<\infty\qquad\text{for every finite subset }S\subseteq D.
+\]
+The Kurosh problem for division rings asks whether algebraicity over the
+center implies local finiteness. We give a negative answer.
+
+All rings and subrings in this paper have the same identity. If $k$ is a
+central subfield and $x,y$ belong to a ring, $k[x,y]$ denotes the unital
+$k$-algebra generated by $x,y$; the generators need not commute.
+The free polynomial algebra on two formal generators is denoted
+$k\langle x,y\rangle$.
+
+\begin{theorem}\label{thm:main}
+There is a countable division ring $D$ of characteristic zero, with
+center $F=Z(D)$, and elements $x,y\in D$ such that
+\[
+ D=F[x,y]=F(x,y),\qquad [D:F]=\infty,
+\]
+while every element of $D$ is algebraic over $F$.
+\end{theorem}
+
+The same pair $\{x,y\}$ therefore witnesses the failure of local finiteness.
+The construction does not prescribe the center. It first produces an
+algebraic division ring over a countable central subfield, and a separate
+finite-descent argument proves infinite dimension over the actual center.
+The annihilating polynomial is allowed to depend on the element.
+
+\subsection{Historical context}
+
+Kurosh formulated his question in 1941 in connection with Burnside's
+finiteness problem for groups. He asked whether a division ring that is
+algebraic and finitely generated as an algebra over a central subfield
+must be finite-dimensional over that subfield
+\cite[Problem~(K), p.~240]{Kurosh1941}. The subfield in this formulation
+need not be the full center, and finite generation means algebra
+generation, not merely division generation.
+
+The broader problem for algebraic algebras has a different history.
+Golod's construction, based on the Golod--Shafarevich estimates, gives
+finitely generated infinite-dimensional nil algebras
+\cite[Example~1]{Golod1964}. Their unitizations are algebraic algebras
+with nonzero nilpotents, so they do not answer the division-ring question.
+
+There are positive results under additional hypotheses. Jacobson's
+bounded-degree theorem gives a polynomial identity, and Kaplansky's
+theorem makes algebraic algebras satisfying a polynomial identity
+locally finite; see \cite[Theorems~3 and~5]{Jacobson1952}.
+Bell, Drensky, and Sharifi later proved that a division ring whose
+elements satisfy nonzero polynomial relations of uniformly bounded degree $d$,
+with coefficients on the left in a possibly noncentral subfield, has
+dimension at most $d^2$ over its center
+\cite[Theorem~1.3]{BellDrenskySharifi2013}. Our construction imposes
+element-dependent relations without a uniform degree bound.
+
+Amitsur proved that a division algebra finitely generated as an algebra
+over an uncountable central field is automatically algebraic
+\cite[Corollary~$2'$]{Amitsur1956}. This conclusion is algebraicity,
+not local finiteness. Countability plays a different role here: it
+allows us to schedule all polynomial expressions in the construction.
+Goodearl and Letzter's recent work \cite{GoodearlLetzter2026} treats
+nonfinite generation as an algebra for classes of transcendental skew
+fields and explicitly distinguishes that question from the algebraic
+Kurosh problem. Theorem~\ref{thm:main} concerns algebraic division rings
+and their actual centers.
+
+The proof combines standard division-algebra constructions with matrix
+and formal-series calculations. Symbol and cyclic algebras provide
+division-ring models for the generic block data
+\cite[Section~2.5]{GilleSzamuely2006}. Traces organize cyclic word
+relations \cite[Section~4]{Procesi1976}, while inner derivations describe
+infinitesimal conjugation \cite[Theorem~8]{Jacobson1937Derivations}.
+The lifting argument supplies the additional response, compatibility,
+and generation statements needed to preserve two independent deformation
+directions while imposing successive algebraic relations.
+
+\subsection{The construction}
+
+The main ingredient is Theorem~\ref{thm:lifting}. Its input is a tuple of
+formal series in two noncommuting variables over a finite-dimensional
+central division algebra. The constant tuple generates the coefficient
+algebra, and the two linear coefficients are independent modulo changes
+coming from conjugation. The theorem makes any prescribed series
+algebraic over a larger countable central field. At the same time it
+increases the degree of the division algebra, preserves all previous
+polynomial identities, and restores both generation and the two independent
+directions. These last requirements allow the construction to continue.
+
+For the prescribed series $w$, the immediate goal is to find a formal
+substitution $Z=(Z_1,Z_2)$ for which $w(Z)$ satisfies the minimal
+polynomial of the chosen constant output $w(Z(0))$. The input $Z(0)$
+is a nonzero generic input, small with respect to an auxiliary valuation,
+not the original zero input.
+After splitting
+the coefficient algebra into matrices, we take the constant input to be
+a diagonal sum of independent generic matrix pairs. Varying a diagonal
+block gives a first eigenvalue response; adding an arrow to another block
+and an arrow back gives a paired response. Sections~\ref{sec:formal-generic}
+and~\ref{sec:responses} establish the formal evaluation rules and show that
+these responses partition the eigenvalue labels into linked groups.
+
+A transfer theorem puts each first unsolved noncommutative word coefficient
+in the same response space. It also turns proportional responses into
+functional relations between eigenvalue branches
+(Sections~\ref{sec:word-transfer} and~\ref{sec:linkage-replication}).
+The relations within each block then separate from a fixed-dimensional
+space of relations among blocks. To test the latter, we repeat each block
+and attach auxiliary blocks, called hubs. Individual hubs distinguish
+repeat positions; a common hub compares them. This matrix test proves
+that every leading error satisfies the required cyclic sums of word
+coefficients (Sections~\ref{sec:global-defect} and~\ref{sec:cyclic-obstructions}).
+
+An eleven-block arrangement of two colors supplies corrections whose
+only obstructions are those cyclic sums. We solve the equations one word
+length at a time in Section~\ref{sec:free-correction}. A cyclic division
+algebra realizes these matrix calculations over a division ring, and a
+Taylor translation restores generation and the two independent directions
+(Sections~\ref{sec:division-realization} and~\ref{sec:generation}).
+The corrections are solutions of finite affine linear systems; this is
+what permits their descent from a splitting field to the division
+algebra's center. If $w$ already satisfies its constant term's minimal
+polynomial, the spectral correction is unnecessary, but realization
+and translation are still needed to increase degree and restore the
+iteration hypotheses.
+
+Section~\ref{sec:countable-construction} first explains how the lifting
+theorem yields Theorem~\ref{thm:main}. The fields introduced during the
+iteration are countable, so every polynomial expression can be scheduled.
+A division-ring ultraproduct retains the resulting relations. Its
+two-generated subalgebra is already a division ring, and finite word
+sets prove that its dimension over the central coefficient field is
+infinite. The actual-center argument concludes this deduction. The
+remaining sections prove the lifting theorem and all its required
+compatibilities.
diff --git a/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/lifting.tex b/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/lifting.tex
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+++ b/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/lifting.tex
@@ -0,0 +1,81 @@
+\section{The lifting theorem}\label{sec:lifting}
+
+The construction proceeds through finite-dimensional central division
+algebras. At each step we impose an algebraic relation on one prescribed
+element while preserving the freedom to enlarge the algebra at the next step.
+Formal power series in two noncommuting variables record that freedom.
+
+For a $k$-algebra $A$, write
+\[
+ A\langle\!\langle z_1,z_2\rangle\!\rangle
+\]
+for the ring of formal series in words in $z_1,z_2$, with coefficients
+in $A$ commuting with the variables. Multiplication concatenates words
+and multiplies their coefficients in their original order. The ring is
+complete for the filtration by word length. For a tuple
+$X=(X^{(1)},\ldots,X^{(u)})$ of such series, let $X_0\in A^u$ be its
+constant tuple and let $X_1,X_2\in A^u$ be its coefficients of the two
+individual letters. Thus the subscripts $1,2$ denote letters, not degrees.
+Set
+\begin{equation}\label{eq:inner-tangent}
+ J_A(X_0)=\{([T,X_0^{(1)}],\ldots,[T,X_0^{(u)}]):T\in A\},
+ \qquad [T,a]=Ta-aT.
+\end{equation}
+The classes of $X_1,X_2$ modulo this subspace describe changes of the tuple
+that cannot be obtained by infinitesimal conjugation.
+
+\begin{theorem}[Algebraic lifting]\label{thm:lifting}
+Let $k$ be a countable field of characteristic zero containing every root
+of unity, and let $A$ be a finite-dimensional central division algebra
+over $k$. Suppose
+$X\in A\langle\!\langle z_1,z_2\rangle\!\rangle^u$ satisfies:
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item $X_0$ generates $A$ as a $k$-algebra;
+\item $X_1,X_2$ have linearly independent classes in $A^u/J_A(X_0)$.
+\end{enumerate}
+For every $w\in A\langle\!\langle z_1,z_2\rangle\!\rangle$, there exist
+a countable field extension $k'/k$, a central division algebra $A'/k'$ of
+strictly larger degree, and series
+\[
+ X'\in A'\langle\!\langle s_1,s_2\rangle\!\rangle^u,
+ \qquad w'\in A'\langle\!\langle s_1,s_2\rangle\!\rangle
+\]
+with the following properties:
+\begin{enumerate}[label=\textup{(\alph*)}]
+\item Every noncommutative polynomial identity over $k$ satisfied by the
+tuple $(X,w)$ is satisfied by $(X',w')$ as an identity of formal series.
+\item The constant tuple $X'_0$ generates $A'$ over $k'$, and the classes
+of $X'_1,X'_2$ in $(A')^u/J_{A'}(X'_0)$ are linearly independent.
+\item There is a monic nonzero polynomial $q\in k'[t]$ such that $q(w')=0$.
+\end{enumerate}
+Here the degree of a central division algebra $B/F$ is $\sqrt{[B:F]}$.
+\end{theorem}
+
+Theorem~\ref{thm:lifting} is the central construction. Its application to
+the Kurosh problem uses $u=2$ and addresses one polynomial in the two
+generators at a time. The fields are allowed to grow; they need not be
+algebraic extensions of their predecessors. The final step will therefore
+identify the actual center separately and prove infinite dimension over it.
+
+Figure~\ref{fig:response-interfaces} separates the two tasks within a
+lifting step. The response calculation makes the prescribed series
+algebraic; the final translation restores the hypotheses needed to repeat
+the step. The two transverse directions must survive together, not merely
+one at a time.
+
+\begin{figure}[tbp]
+\centering
+\input{figures/response-interfaces}
+\caption{The spectral case of Theorem~\ref{thm:lifting}.
+The two displayed classes at each end are linearly independent in the
+indicated quotient. The middle boxes describe the matrix calculation:
+allowable eigenvalue changes control the first unsolved word errors,
+and the cyclic relations are precisely the obstructions to correction.
+Realization and translation return a generating tuple over a division
+algebra of strictly larger degree. If $w$ already satisfies the minimal
+polynomial of $w(0)$, the middle spectral correction is omitted, but
+realization and translation remain necessary. No embedding tower of
+the successive division algebras is asserted.}
+\label{fig:response-interfaces}
+\end{figure}
+\clearpage
diff --git a/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/linkage.tex b/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/linkage.tex
new file mode 100644
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+++ b/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/linkage.tex
@@ -0,0 +1,317 @@
+\section{Functional linkage and replication}
+\label{sec:linkage-replication}
+
+We now show that the dimension of the active response space is additive
+under generic direct sums.  The main point is to turn the linear relations
+between linked responses into identities between eigenvalue branches.
+These identities will continue to control responses through partners of
+every allowed size.
+
+Retain the progression $\mathcal S=\{pN:N\geq 1\}$ and the common
+eigenvalue multiplicity $t$.  Thus, at size $n\in\mathcal S$, the number
+of distinct generic eigenvalues is $e_n=dn/t$.  We use the following
+previously established facts.  At a direct sum of independent small
+generic inputs of sizes in $\mathcal S$, the spectra are disjoint and
+have the maximal number of distinct eigenvalues.  Their commuting
+Taylor deformations therefore have scalar spectral blocks, by
+Lemma~\ref{lem:scalar-branches}.  The response identities admit the
+specializations of Lemma~\ref{lem:active-specialization}, and their
+leading terms satisfy Proposition~\ref{prop:word-transfer}.
+
+For a generic input $P$, let $\Omega_P$ be its spectral splitting
+field, and choose a constant input direction $v$ such that
+$D_h(P;v)\neq 0$ for every eigenvalue label $h$.  Recall the active-space
+partition $\mathcal P_P$, characterized by
+\begin{equation}
+ \mathcal T_P
+ =\left\{c\in\Omega_P^{e_n}:
+       \frac{c_h}{D_h(P;v)}
+       \text{ is constant on each member of }\mathcal P_P\right\}.
+ \label{eq:linkage-partition}
+\end{equation}
+Labels in one member of this partition are called linked.  In particular,
+$a_n=\dim_{\Omega_P}\mathcal T_P$ is the number of its members.
+The argument below uses the full coefficientwise definition of
+$\mathcal T_P$: it contains the loop responses $D(P;Y)$ and the
+coefficient vectors of $m(P,Q;U,V)$ for every partner size in
+$\mathcal S$.
+
+\subsection{A formal identity for linked branches}
+
+\begin{proposition}[Functional linkage]
+\label{prop:functional-linkage}
+Let $P$ be a small generic input of size $p$, and let $h,h'$ be linked
+labels at $P$.  Let $C_1,\ldots,C_s$ be small generic inputs of sizes in
+$\mathcal S$, jointly independent with $P$, where $s=0$ is allowed.
+Put $R=\operatorname{diag}(P,C_1,\ldots,C_s)$.
+Over a common spectral splitting field $E$, denote by
+$\Lambda_j^R(\Delta)$ the scalar eigenvalue branch of $W(R+\Delta)$
+whose value at zero is $\lambda_j(P)$, for $j=h,h'$.
+
+Define
+\[
+ g_j(\eta)=\lambda_j(P+\eta v)-\lambda_j(P)
+ \in\Omega_P[[\eta]],\qquad j=h,h'.
+\]
+Then $g_{h'}'(0)\neq 0$, and
+\begin{equation}
+ \Lambda_h^R(\Delta)-\lambda_h(P)
+ =\bigl(g_h\circ g_{h'}^{-1}\bigr)
+       \bigl(\Lambda_{h'}^R(\Delta)-\lambda_{h'}(P)\bigr).
+ \label{eq:functional-linkage}
+\end{equation}
+Here $g_{h'}^{-1}$ denotes the inverse for composition.  In particular,
+the univariate series in this identity depends only on $P,h,h'$ and $v$,
+and is unchanged when auxiliary blocks are adjoined.
+\end{proposition}
+
+\begin{proof}
+We first explain why the response relations persist while $P$ moves.
+By the Taylor convention in Section~\ref{subsec:formal-conventions},
+substitution $P\mapsto P+\eta v$ gives homomorphisms
+\[
+ \operatorname{Frac}K[[P]]
+ \longrightarrow \bigl(\operatorname{Frac}K[[P]]\bigr)[[\eta]],
+ \qquad
+ \Omega_P\longrightarrow\Omega_P[[\eta]],
+\]
+both reducing to the identity at $\eta=0$. The second uniquely
+extends the first through the finite separable extension
+$\Omega_P/\operatorname{Frac}K[[P]]$. Thus every nonzero element
+has image with invertible constant term, and the eigenvalues and
+projections map to their uniquely labeled Taylor deformations.
+
+For linked labels, Equation~\eqref{eq:linkage-partition} gives
+\begin{equation}
+ D_{h'}(P;v)c_h-D_h(P;v)c_{h'}=0
+ \label{eq:linked-response-relation}
+\end{equation}
+for every loop response and every coefficient vector of every paired
+response.  Apply the preceding homomorphism coefficientwise in the
+partner coordinates.  The resulting identities are exactly
+Equation~\eqref{eq:linked-response-relation} at the moved input
+$P+\eta v$, with the moved eigenvalues, projections, and $D$-factors.
+They remain valid in the triangular specializations used in
+Proposition~\ref{prop:word-transfer}.  More explicitly, first clear the
+finitely many auxiliary resolvent determinants.  At each Taylor degree,
+the number of strictly upper steps is bounded.  This permits their
+entries to have Laurent coefficients in $\eta$, while the diagonal
+entries are expanded at the original small inputs.  Coefficientwise
+convergence and specialization are then supplied by
+Lemma~\ref{lem:active-specialization}.  Thus the transfer argument
+applies over $E((\eta))$ with the transported response relation.
+
+Write $v_R$ for $v$ supported in the $P$-block and put
+$R(\eta)=R+\eta v_R$.  At $R(\eta)+H$, impose the equation
+\begin{equation}
+ \Lambda_{h'}^{R(\eta)}(H)=\lambda_{h'}(P+\eta v).
+ \label{eq:fixed-linked-branch}
+\end{equation}
+Choose linear coordinates $H=u v_R+H_{\mathrm{rest}}$, with
+$H_{\mathrm{rest}}$ in a fixed complementary space of input directions.
+The derivative of the left side in $u$ at $H=0$ is
+$D_{h'}(P+\eta v;v)$, a unit in $E[[\eta]]$.  Formal implicit
+substitution therefore eliminates $u$.  In particular, the quotient
+of $E[[\eta,H]]$ by Equation~\eqref{eq:fixed-linked-branch} is
+\begin{equation}
+ E[[\eta,H_{\mathrm{rest}}]],
+ \qquad
+ E[[\eta,H_{\mathrm{rest}}]]
+ \lhook\joinrel\longrightarrow
+ E((\eta))[[H_{\mathrm{rest}}]].
+ \label{eq:linkage-quotient-injection}
+\end{equation}
+The eliminated series has zero constant term in $H_{\mathrm{rest}}$.
+
+On this quotient, scalar-block separation implies that the fixed-root
+Schur expression $S_{h'}$ at $R(\eta)$ vanishes.  We claim that $S_h$
+also vanishes over the right-hand ring in
+Equation~\eqref{eq:linkage-quotient-injection}.  Induct on total degree
+in $H_{\mathrm{rest}}$.  Both Schur expressions vanish in degree zero.
+If they vanish below degree $l$, the commuting-parameter conclusion of
+Proposition~\ref{prop:word-transfer}, together with the transported
+relation~\eqref{eq:linked-response-relation}, gives scalar leading
+coefficients $[S_j]_l=c_{j,l}\pi_j$ for $j=h,h'$, with
+\[
+ D_{h'}(P+\eta v;v)c_{h,l}
+   =D_h(P+\eta v;v)c_{h',l}=0.
+\]
+Here the $c_{j,l}$ are scalar homogeneous polynomials in the remaining
+parameters over $E((\eta))$; the projections are the moved target
+projections. The first factor is nonzero, so $c_{h,l}=0$ and
+$[S_h]_l=0$.
+
+It follows that the $h$-branch is fixed as well:
+\[
+ \Lambda_h^{R(\eta)}(H)=\lambda_h(P+\eta v)
+\]
+on the quotient.  For clarity, the implication from $S_h=0$ uses the
+scalar blocks of Lemma~\ref{lem:scalar-branches}: the complementary
+block at the fixed root is invertible, and the determinant of the full
+matrix at that root is a unit times the $t$-th power of the shift of
+the $h$-branch.  The quotient is a domain, so the shift is zero.
+Both branch series and the eliminated coordinate are regular in
+$\eta$.  The injection in
+Equation~\eqref{eq:linkage-quotient-injection} therefore descends this
+equality to $E[[\eta,H_{\mathrm{rest}}]]$.
+
+Finally make the invertible linear coordinate change
+$\Delta=H+\eta v_R$.  Equation~\eqref{eq:fixed-linked-branch} becomes
+\[
+ \Lambda_{h'}^R(\Delta)-\lambda_{h'}(P)=g_{h'}(\eta).
+\]
+Since $g_{h'}'(0)=D_{h'}(P;v)\neq0$, it eliminates $\eta$ as
+$g_{h'}^{-1}(\Lambda_{h'}^R(\Delta)-\lambda_{h'}(P))$.
+Substitution in the equality for the $h$-branch gives
+Equation~\eqref{eq:functional-linkage}.
+\end{proof}
+
+\subsection{All partner sizes and the dimension formula}
+
+We next pass from an identity near a direct sum to an identity for
+every coefficient defining the generic active space.  This step is
+necessary because a bound proved using a fixed collection of partners
+would not bound $a_n$.
+
+\begin{proposition}[Replication of active dimensions]
+\label{prop:active-replication}
+For every positive integer $N$,
+\[
+ a_{pN}=N a_p.
+\]
+\end{proposition}
+
+\begin{proof}
+Put $n=pN$ and choose independent small generic inputs
+$P_1,\ldots,P_N$ of size $p$.  Write
+$R_1=\operatorname{diag}(P_1,\ldots,P_N)$ and use
+$x=R_1+\Delta_1$ to label the generic size-$n$ spectrum by its branches
+at $R_1$.
+
+This substitution embeds $K[[x]]$ in $E[[\Delta_1]]$.  Indeed, for
+an image of a series over $K$, the Taylor coefficients are uniformly
+integral in the small parameter $\varepsilon$.  Substituting
+$\Delta_1=\varepsilon X-R_1$, with independent coordinates $X$,
+is therefore convergent and recovers the original series at
+$\varepsilon X$.  Homogeneous expansion at that point detects every
+nonzero series.  The embedding extends to the fraction field.  All
+generic spectral roots are supplied by the scalar branches at $R_1$,
+so it also embeds a spectral splitting field $\Omega_x$ into
+$\operatorname{Frac}E[[\Delta_1]]$, with these labels.  The branches
+and spectral projections themselves are regular at $\Delta_1=0$.
+
+Fix a block $i$ and labels $h,h'$ in one member of
+$\mathcal P_{P_i}$.  Choose a constant direction $v_i$ supported in
+that block, nonzero under every $D_h(P_i;\cdot)$.  Apply
+Proposition~\ref{prop:functional-linkage} first at $R_1$.
+Differentiating its identity gives proportional loop responses at
+$x$.  Now adjoin an independent generic partner $Q$ of any fixed
+size in $\mathcal S$, and apply the same proposition to
+$\operatorname{diag}(R_1,Q)$.  Its univariate series is the same
+one.  At $\operatorname{diag}(x,Q)$, perturb the opposite input
+blocks by $uU$ and $zV$.  Either arrow alone preserves the spectrum,
+so the first derivatives in $u$ and $z$ vanish.  The coefficient of
+$uz$ in the branch is the paired response $m_h(x,Q;U,V)$.
+The mixed chain rule and the separate first derivative in $v_i$
+give
+\begin{equation}
+ D_{h'}(x;v_i)m_h(x,Q;U,V)
+   -D_h(x;v_i)m_{h'}(x,Q;U,V)=0.
+ \label{eq:replicated-pair-relation}
+\end{equation}
+The two $D$-factors have nonzero reductions at $\Delta_1=0$.
+
+We must verify this relation coefficientwise in $Q$, rather than only
+at its generic value.  Cross multiplication has already removed the
+response ratio.  Multiply further by the finitely many determinants
+of $W(Q)-\lambda_h(x)$ and $W(Q)-\lambda_{h'}(x)$ needed to clear
+the auxiliary inverses.  At each fixed $\Delta_1$-degree, the resulting
+series in the independent coordinates of $Q$ has coefficients
+uniformly bounded below in $\varepsilon$-valuation.  The branch and
+projection coefficients do not involve $Q$; the other factors are
+ordinary matrix series with two prescribed arrow steps and a fixed
+number of Taylor insertions.  Their path expansions give precisely
+this bound, as in Lemma~\ref{lem:admissible-evaluation}.
+Evaluation at small generic $Q$ therefore detects every coefficient.
+Equation~\eqref{eq:replicated-pair-relation} is consequently a
+formal identity in those coordinates.  Injectivity of the embedding
+of $\Omega_x$ descends it to the generic size-$n$ field.  Finally,
+the cleared determinants have nonzero constant terms in $Q$, so
+they can be canceled in $\Omega_x[[Q]]$.
+
+This argument applies separately to every partner size with the same
+labeling of $x$ and the same $D$-factors.  It therefore gives a
+relation on every coefficient vector used to define $\mathcal T_x$.
+Together with the corresponding loop relations, it says that
+$c_h/D_h(x;v_i)$ is constant on each old partition member within
+block $i$, for every $c\in\mathcal T_x$.  These partitions have
+$Na_p$ members in total, and all their normalizing factors are
+nonzero.  Thus $a_n\leq Na_p$.
+
+For the opposite inequality, use
+Lemma~\ref{lem:active-specialization} for each $P_i$, choosing all
+partners jointly independent.  At $R_1$, support each loop direction,
+and both arrows of each paired response, in its chosen block $i$.
+The responses then coincide with the original responses on that
+block and vanish on every other block.  They have rank $Na_p$.
+Evaluate the same finite collection first at $R_1+\Delta_1$.
+All branch data and partner resolvents are regular at
+$\Delta_1=0$, since the partner spectra are disjoint from that of
+$R_1$.  A generic rank bound smaller than $Na_p$ would make every
+minor of that order vanish over the embedded generic field, and hence
+also at the reduction.  The displayed specialization contradicts this.
+Therefore $a_n\geq Na_p$.
+\end{proof}
+
+\subsection{Partners of one common size}
+
+The dimension formula permits us to use a fixed matrix size on both
+ends of every paired response.  We will also need to fix one arrow
+and vary only its reverse.
+
+\begin{corollary}[Equal-size partners]
+\label{cor:equal-size-partners}
+There are $r\in\mathcal S$ and a positive integer $K_0$ with the
+following property.  Let $P,Q_1,\ldots,Q_{K_0}$ be independent small
+generic inputs of size $r$.  For generic fixed input arrows
+$U_j:P\leftarrow Q_j$, the loop columns and the images of
+\[
+ V\longmapsto m(P,Q_j;U_j,V),
+ \qquad V:Q_j\leftarrow P,\quad 1\leq j\leq K_0,
+\]
+span $\mathcal T_P$ after extension to the joint field of the data.
+The analogous assertion holds with the other arrow fixed.  The same
+$r$ and $K_0$ may be used for both assertions.
+\end{corollary}
+
+\begin{proof}
+Choose a finite witness list at size $p$ from
+Lemma~\ref{lem:active-specialization}, and let its partner sizes be
+$q_1,\ldots,q_M\in\mathcal S$.  Choose $r=pN$ with
+$r\geq q_j$ for every $j$.  Specialize the size-$r$ base to a direct
+sum of $N$ independent size-$p$ inputs.  For each block, use a fresh
+copy of the witness list.  Pad a size-$q_j$ partner to size $r$ by
+adjoining independent generic diagonal data of size $r-q_j$, omitting
+the padding if this difference is zero.  Give both arrows zero entries
+on the padding.  The response remains the original size-$p$ witness
+on its chosen base block and is zero on the other base blocks.
+
+The resulting columns have rank $Na_p=a_r$ by
+Proposition~\ref{prop:active-replication}.  All inputs at this
+specialization have maximal spectral count, and the partner spectra
+are disjoint from the base spectrum.  The regular-minor argument in
+the preceding proof therefore transfers this rank to generic
+size-$r$ base and partner inputs.
+
+Assign a separate partner to every paired column in this finite
+list.  At the witness specialization, choose its forward arrow to
+be the specified $U_j$ and test the variable reverse arrow on the
+specified $V_j$.  A minor of order $a_r$ is nonzero there.  Since
+paired responses are bilinear in their arrows, this minor remains
+nonzero for generic fixed forward arrows and suitable reverse-arrow
+columns.  This proves the first assertion with finitely many partners.
+The identical argument fixes the reverse arrows instead.  Increasing
+the number of independent partners to the larger of the two finite
+numbers proves both assertions with one $K_0$.  If no paired column
+is needed, one unused generic partner may be added so that $K_0>0$.
+\end{proof}
diff --git a/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/responses.tex b/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/responses.tex
new file mode 100644
index 0000000000000000000000000000000000000000..19fcb7eaaf7a1b227fb4101b85ac79f5cf14c36e
--- /dev/null
+++ b/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/responses.tex
@@ -0,0 +1,312 @@
+\section{The space of eigenvalue responses}\label{sec:responses}
+
+Fix a size $n\in\mathcal S$, a generic input pair $P$, and a splitting
+field $\Omega_P$ of $W(P)$ over $\operatorname{Frac}K[[P]]$.
+Write its distinct roots as $\lambda_h$, with spectral projections
+$\pi_h$, $1\le h\le e_n$.  Each projection has rank $t$.
+We also use these data after embedding into the valued fields of
+Section~\ref{sec:formal-generic}.
+
+We seek the possible leading changes of the eigenvalues at $P$.
+Besides varying $P$ itself, we allow an excursion through another
+input block and a return to $P$.  Their linear span will determine
+which eigenvalues can be varied independently.
+
+\subsection{Loop and paired responses}\label{subsec:response-formulas}
+
+Upper triangular block evaluations are also the basic mechanism of
+noncommutative difference-differential calculus; see
+\cite[Proposition~2.2 and Theorem~3.11]{KaliuzhnyiVerbovetskyiVinnikov2014}.
+Here the admissible evaluations of Section~\ref{sec:formal-generic}
+let us extract scalar eigenvalue responses from those block calculations.
+
+For a perturbation $\mathcal W$ of $W(P)$, put $\pi=\pi_h$,
+$\bar\pi=1-\pi$, and
+\[
+ C_h(\mathcal W)=\bar\pi(\mathcal W-\lambda_h)\bar\pi.
+\]
+This operator acts on the complementary space. Its constant term is
+invertible, so its inverse has a unique expansion in either commuting
+or free formal parameters. Define the fixed-root Schur expression by
+\begin{equation}
+S_h(\mathcal W)=
+ \pi(\mathcal W-\lambda_h)\pi
+ -\pi\mathcal W\bar\pi
+  C_h(\mathcal W)^{-1}
+  \bar\pi\mathcal W\pi.
+ \label{eq:fixed-schur}
+\end{equation}
+This expression records the obstruction to keeping the root $\lambda_h$
+fixed. For $u$ in the image of $\pi$, the vector
+\[
+ u-C_h(\mathcal W)^{-1}\bar\pi\mathcal W\pi u
+\]
+is the unique vector projecting to $u$ whose complementary component
+under $\mathcal W-\lambda_h$ vanishes. Its remaining component is
+$S_h(\mathcal W)u$. Thus $S_h(\mathcal W)=0$ means that the fixed-root
+kernel is a graph over the original eigenspace.
+
+When the constant output is split semisimple and these equations are
+imposed for every one of its distinct roots, the kernel graphs combine
+into a formal change of basis. The output remains conjugate to its
+constant term and satisfies that constant term's minimal polynomial.
+Lemma~\ref{lem:schur-polynomial} proves the equivalence with the
+polynomial identity at every finite word order.
+
+The same Schur expression is defined near $\operatorname{diag}(P,C_0)$
+if $\lambda_h$ is absent from $W(C_0)$, embedding $\pi_h$ at the $P$
+block. In this use, equations indexed by labels at $P$ record those
+fixed roots only.
+
+For a loop direction $Y=(Y_1,Y_2)$ at $P$, define the scalar
+$D_h(P;Y)$ by
+\begin{equation}
+\pi_h W'_P(Y)\pi_h=D_h(P;Y)\pi_h.
+ \label{eq:loop-response}
+\end{equation}
+This compression is scalar: apply Lemma~\ref{lem:scalar-branches}
+to the commuting deformation $P+xY$ and take its first coefficient.
+Equivalently, $D_h(P;Y)$ is the differential of the branch
+$\lambda_h$ in direction $Y$.  By
+Lemma~\ref{lem:nonconstant-roots}, each of these linear functionals is
+nonzero.  Because $K$ is infinite, we can choose a constant direction
+$v$ over $K$ with
+\begin{equation}
+D_h(P;v)\ne0\qquad(1\le h\le e_n).
+ \label{eq:nonzero-direction}
+\end{equation}
+Indeed the forbidden directions form a finite union of proper
+$K$-linear subspaces, even though their defining coefficients belong
+to $\Omega_P$.
+
+Let $Q$ be an independent generic point of size $n'\in\mathcal S$.
+Let $U$ be an input arrow $P\leftarrow Q$ and $V$ an input arrow
+$Q\leftarrow P$; each is a pair of rectangular scalar matrices.
+Form the triangular input pair
+\[
+ T=\begin{pmatrix}P&U&0\\0&Q&V\\0&0&P\end{pmatrix}.
+\]
+Set
+\begin{equation}
+M_h(P,Q;U,V)=\pi_h\left(
+ W(T)_{13}
+ -W(T)_{12}(W(Q)-\lambda_h)^{-1}W(T)_{23}
+ \right)\pi_h.
+ \label{eq:paired-response}
+\end{equation}
+Block paths show that this is bilinear in $U,V$.  The inverse exists
+because the spectra at independent points are disjoint.
+
+\begin{lemma}\label{lem:paired-scalar}
+There is a scalar $m_h(P,Q;U,V)$ such that
+\[
+ M_h(P,Q;U,V)=m_h(P,Q;U,V)\pi_h.
+\]
+It is the coefficient of $xy$ in the branch reducing to $\lambda_h$
+for the commuting input deformation
+\[
+ \begin{pmatrix}P&xU\\yV&Q\end{pmatrix}.
+\]
+The same assertion holds at an admissible specialized pair whose
+spectra are disjoint and whose combined output is split semisimple
+with the maximal number $e_{n+n'}$ of distinct roots.
+\end{lemma}
+
+\begin{proof}
+At $x=y=0$, disjointness and Lemma~\ref{lem:stable-multiplicity}
+give $e_n+e_{n'}=e_{n+n'}$ roots.  Thus
+Lemma~\ref{lem:scalar-branches} applies.  Either arrow alone leaves
+the spectrum unchanged, so the chosen branch has no terms depending
+on $x$ alone or on $y$ alone.
+
+Compute the $xy$ coefficient of its fixed-root Schur expression.
+The direct return from $P$ through $Q$ gives $W(T)_{13}$.  Eliminating
+the $Q$ block gives the second term in
+\eqref{eq:paired-response}.  The other eigenspaces in the $P$ block
+give no further term of this order: their couplings to $\pi_h$ have
+zero constant term and no first-order term under these off-diagonal
+input perturbations.  Replacing the fixed root by the scalar branch
+subtracts its $xy$ coefficient times $\pi_h$.  The resulting Schur
+expression vanishes, because the branch eigenspace projects
+isomorphically to the base eigenspace.  This proves the formula.
+The specialized assertion uses exactly the same argument and the
+specialization-stable degree bound.
+\end{proof}
+
+\subsection{Coefficient spans and their specializations}
+\label{subsec:active-space}
+
+At $Q=0$ the matrices $W(Q)-\lambda_h$ are invertible, since no
+$\lambda_h$ belongs to the spectrum of $w(0)$.  Hence
+\eqref{eq:paired-response} has a formal expansion in the entries of
+$Q$ over $\Omega_P$.  Define
+\[
+ m_h=\frac1t\operatorname{tr}M_h,
+ \qquad m=(m_h)_{h=1}^{e_n},\qquad D=(D_h)_{h=1}^{e_n}.
+\]
+Here $D=D(P;\cdot):M_n(\Omega_P)^2\to\Omega_P^{e_n}$ is the linear
+map of loop responses. Its columns are its values on the standard
+coordinate directions in the two input matrices.
+The scalar identity of Lemma~\ref{lem:paired-scalar} holds also for
+these formal expansions, by clearing their determinants and applying
+Lemma~\ref{lem:generic-detection}.
+
+Define $\mathcal T_P\subseteq\Omega_P^{e_n}$ to be the linear span
+of the columns of $D$ and of every coefficient vector of $m$, for
+every allowed partner size $n'\in\mathcal S$ and every pair of
+coordinate arrows $U,V$.  Write
+\[
+ a_n=\dim_{\Omega_P}\mathcal T_P.
+\]
+Although the definition permits infinitely many partner sizes and
+coefficients, the ambient space is finite dimensional.
+
+\begin{lemma}[Specialization and finite witnesses]
+\label{lem:active-specialization}
+The scalar identities for $M_h$ and every linear identity expressing
+$m\in\mathcal T_P$ hold after any admissible specialization of the
+partner input for which the required inverses exist.  For identities
+on a subset of labels, only the inverses for those labels are needed.
+
+After a field extension, the columns of $D$ and finitely many paired
+responses span $\mathcal T_P$.  These responses can be taken at
+independent small generic partner points, whose sizes need not be
+equal.
+\end{lemma}
+
+\begin{proof}
+For the first assertion, apply any $\Omega_P$-linear functional
+annihilating $\mathcal T_P$ to the formal tuple $m$.  The result is
+zero.  Multiply by the finitely many determinants of
+$W(Q)-\lambda_h$ that occur.  The cleared expression is a finite
+sum of products of base-point scalars and matrix-entry series; its
+auxiliary coefficients have a uniform valuation bound.  Its formal
+identity is therefore equivalent to its identity at a small generic
+partner, by Lemma~\ref{lem:generic-detection}.  It persists under the
+admissible specializations of Lemma~\ref{lem:admissible-evaluation}.
+Divide by the specialized determinants when they are nonzero.  The
+same argument, entry by entry, proves the scalar identities for
+$M_h$.  It also proves the projected assertion, without introducing
+any spectral labels for the partner input.
+
+Choose finitely many coefficient vectors, together with columns of
+$D$, that span $\mathcal T_P$, and select a nonzero minor witnessing
+their rank.  For each column requiring a partner, use a separate set
+of partner variables.  By multilinearity, the corresponding
+determinant of the series columns has that nonzero minor as the
+coefficient of a specified product of monomials.  It is therefore a
+nonzero formal series.  Clear the column denominators.  The numerator
+is still nonzero, and its coefficients are uniformly bounded below
+in valuation.  Independent small generic evaluations keep this
+numerator and its denominators nonzero, by
+Lemma~\ref{lem:generic-detection}.  The resulting values attain the
+full dimension.  The first assertion places all of them in the
+extended active space.
+
+This argument evaluates cleared numerators, not the possibly
+divergent geometric expansions of their inverses.  The distinction
+is precisely the one made in Section~\ref{subsec:formal-conventions}.
+\end{proof}
+
+The lemma also applies after adjoining Taylor or Laurent parameters.
+One first clears determinants and then works coefficientwise as in
+Lemma~\ref{lem:admissible-evaluation}.  In particular, it permits
+triangular partners with repeated base blocks, provided the inverses
+required for the selected target roots exist in the chosen Laurent
+field.
+
+\subsection{A partition of the eigenvalue labels}
+\label{subsec:active-partition}
+
+The active space has a stronger structure than a general linear
+subspace.  Its coordinates form groups whose responses are always in
+a fixed ratio.
+
+\begin{proposition}\label{prop:active-partition}
+For the direction $v$ chosen in \eqref{eq:nonzero-direction}, there
+is a partition of $\{1,\ldots,e_n\}$ such that
+\[
+ \mathcal T_P=
+ \left\{c\in\Omega_P^{e_n}:
+ \frac{c_h}{D_h(P;v)}\text{ is constant on each part}\right\}.
+\]
+\end{proposition}
+
+\begin{proof}
+We first prove that $\mathcal T_P$ is closed under the operation
+\begin{equation}
+(a,b)\longmapsto
+ \left(\frac{a_hb_h}{D_h(P;v)}\right)_h.
+ \label{eq:weighted-product}
+\end{equation}
+Choose two finite spanning sets supplied by
+Lemma~\ref{lem:active-specialization}, using independent partner
+points for the two sets.  It suffices to prove the assertion for a
+column $a$ in the first set and a column $b$ in the second, over their
+joint coefficient field $E$.
+
+Put $P'=P+\xi v$.  Use a triangular chain with endpoints $P,P$,
+middle point $P'$, an optional point $Q_1$ between the first endpoint
+and $P'$, and an optional point $Q_2$ between $P'$ and the last
+endpoint.  A direct step represents a loop column; an excursion
+through $Q_i$ represents a paired column.  Put the arrows defining
+$a$ and $b$ on the successive steps, with no other input arrows.
+Group the internal positions as a single partner $C(\xi)$.  Its
+size belongs to $\mathcal S$.
+
+For every target label, $W(C(\xi))-\lambda_h$ is invertible over
+$E((\xi))$.  The optional blocks have spectra disjoint from $P$.
+At the middle block, the branch reducing to $\lambda_h$ moves with
+nonzero derivative $D_h(P;v)$; all other branches are already away
+from $\lambda_h$.  Thus
+\begin{equation}
+\left.\xi\bigl(W(P')-\lambda_h\bigr)^{-1}\right|_{\xi=0}
+   =\frac{\pi_h}{D_h(P;v)}.
+ \label{eq:middle-residue}
+\end{equation}
+This follows directly by scalar branch separation.
+
+By Lemma~\ref{lem:active-specialization}, the paired response through
+$C(\xi)$ belongs to $\mathcal T_P\otimes E((\xi))$.  Eliminate
+the optional blocks first in its Schur formula.  Triangularity means
+that their elimination does not change the middle diagonal block.
+All terms are regular at $\xi=0$ except the middle inverse in
+\eqref{eq:middle-residue}.  The effective first-to-middle coupling,
+compressed by $\pi_h$ at both ends and evaluated at zero, is
+$a_h\pi_h$.  For a direct step this is
+\eqref{eq:loop-response}; for an excursion it is
+\eqref{eq:paired-response}.  The corresponding middle-to-last
+coupling is $b_h\pi_h$.
+
+The singular term occurs with the minus sign in the Schur
+complement.  Multiplying by $\xi$ and reducing at zero therefore
+gives
+\[
+ \left(-\frac{a_hb_h}{D_h(P;v)}\right)_h
+       \in\mathcal T_P\otimes E.
+\]
+Taking a Laurent coefficient preserves membership in this fixed
+finite dimensional space: apply its annihilating linear functionals.
+Bilinearity and the two spanning sets prove closure under
+\eqref{eq:weighted-product} after extension to $E$, and faithful
+scalar extension proves it over $\Omega_P$.
+
+Divide coordinates by the nowhere-zero vector $D(P;v)$.  The resulting
+space $\mathcal U\subseteq\Omega_P^{e_n}$ contains $1$, since
+$D(P;v)\in\mathcal T_P$, and is closed under coordinatewise
+multiplication.  Declare two coordinates equivalent if every element
+of $\mathcal U$ has the same value on them.  By avoiding finitely
+many proper linear subspaces, choose an element of $\mathcal U$
+whose values distinguish all the equivalence classes.  Lagrange
+interpolation in this element gives the indicator function of each
+class.  These indicators span all class-constant functions.  Thus
+$\mathcal U$ is exactly that algebra of functions, proving the
+proposition.
+\end{proof}
+
+We call two labels in the same part \emph{linked}.  The number of
+parts is $a_n$.  Selecting one representative from each part
+identifies $\mathcal T_P$ with the full coordinate space on those
+representatives.  The next stage must show that these response
+relations control higher-order free-letter errors, rather than only
+the two elementary perturbations used to define them.
diff --git a/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/transfer.tex b/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/transfer.tex
new file mode 100644
index 0000000000000000000000000000000000000000..e2d8005d1b33226340abb571f9ca5a41c89f605c
--- /dev/null
+++ b/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/build/sections/transfer.tex
@@ -0,0 +1,250 @@
+\section{Transfer of the first unsolved word coefficient}
+\label{sec:word-transfer}
+
+Our next objective is to control equations in free letters.  The first and
+paired responses describe matrix perturbations with one loop or two opposite
+arrows.  We show that they also contain the first unsolved coefficient of an
+arbitrary free-series perturbation.  The argument unfolds a word along a
+triangular chain and then replaces its internal states by a single auxiliary
+partner.  The only difficulty is that this partner initially has the target
+eigenvalue.  The following linear-algebra observation explains how to remove
+that singularity without changing the coefficient being computed.
+
+\begin{lemma}[An inverse between two vanishing couplings]
+\label{lem:killed-pole}
+Let $E$ be a field and let $M(\xi)\in\operatorname{End}_E(V)[[\xi]]$, where
+$V$ is finite dimensional.  Suppose that zero is a semisimple eigenvalue of
+$M_0=M(0)$, and let $\Pi$ be its spectral projection.  Suppose that
+\[
+ G=\Pi M'(0)\Pi\big|_{\ker M_0}
+\]
+is invertible.  Then $M(\xi)^{-1}$ exists over $E((\xi))$ and has a pole of
+order at most one.  Let $J$ be zero on $\ker M_0$ and equal to $M_0^{-1}$ on
+$\operatorname{im}M_0$.  If $L(\xi)$ and $B(\xi)$ are compatible matrix
+series satisfying
+\[
+ L(0)\Pi=0,\qquad \Pi B(0)=0,
+\]
+then
+\begin{equation}
+\label{eq:killed-pole-limit}
+ \lim_{\xi\to0}L(\xi)M(\xi)^{-1}B(\xi)=L(0)JB(0).
+\end{equation}
+
+Suppose additionally that $e$ is an idempotent, $f=1-e$,
+$fM_0f$ is invertible on $fV$, and $e:\ker M_0\to eV$ is an isomorphism.
+Under the same two coupling conditions,
+\begin{equation}
+\label{eq:fixed-complement-inverse}
+ L(0)JB(0)=L(0)f(fM_0f)^{-1}fB(0).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Use $V=\ker M_0\oplus\operatorname{im}M_0$.  In this decomposition
+\[
+ M(\xi)=
+ \begin{pmatrix}
+  \xi G+O(\xi^2)&O(\xi)\\
+  O(\xi)&M_0|_{\operatorname{im}M_0}+O(\xi)
+ \end{pmatrix}.
+\]
+The lower-right block is invertible and the Schur complement of that block
+is $\xi G+O(\xi^2)$.  The upper-left block of $M(\xi)^{-1}$ is consequently
+$\xi^{-1}G^{-1}+O(1)$; its off-diagonal blocks are bounded; and its
+lower-right block tends to $M_0^{-1}|_{\operatorname{im}M_0}$.
+The kernel components of both couplings are $O(\xi)$.  Multiplying these
+four blocks proves \eqref{eq:killed-pole-limit}.
+
+For the second assertion, write $b=B(0)u$.  The condition $\Pi b=0$ places
+$b$ in $\operatorname{im}M_0$.  Solutions of $M_0x=b$ differ by
+$\ker M_0$.  Since projection to $eV$ is an isomorphism on that kernel,
+there is a unique solution with $ex=0$.  Its $f$ coordinate is
+$(fM_0f)^{-1}fb$.  All solutions have the same image under $L(0)$ because
+$L(0)$ kills $\ker M_0$.  Comparing this solution with $Jb$ proves
+\eqref{eq:fixed-complement-inverse}.
+\end{proof}
+
+We now recall exactly which response identities enter the transfer
+argument.  At a small generic point $P$ of size $n\in\mathcal S$, the
+eigenvalues $\lambda_h$ have projections $\pi_h$ of rank $t$.  The first
+response is $D_h(P;Y)\pi_h$.  A pair of arrows through an independent
+partner $Q$ gives $m_h(P,Q;U,V)\pi_h$.  The columns of the first responses
+and the coefficient vectors of all these paired responses span
+$\mathcal T_P$.  By Lemma~\ref{lem:active-specialization}, their scalarity
+and all linear containment identities persist for a block-triangular
+partner after its finitely many resolvent determinants are cleared.  For a
+set of target labels $I$, only the determinants belonging to labels in $I$
+are needed.  We use the notation
+\[
+ \mathcal T_P|_I=\{(c_h)_{h\in I}:(c_h)_h\in\mathcal T_P\}.
+\]
+These spaces and identities are extended to the coefficient field in use.
+We also fix a direction $v$ with $D_h(P;v)\ne0$ for every target label.
+
+\begin{proposition}[Transfer of a leading coefficient]
+\label{prop:word-transfer}
+Let $R=\operatorname{diag}(P,C_0)$ be a small matrix input of total size $N\in
+\mathcal S$;
+the block $C_0$ may be absent.  Let $I$ be a set of labels at
+$P$ such that $W(C_0)-\lambda_h$ is invertible for every $h\in I$.
+Work over a field $E$ containing the labeled spectral data at $P$ and the
+entries of $C_0$, with the formal evaluations allowed by
+Lemma~\ref{lem:admissible-evaluation}.  Let
+\[
+ Z(s_1,s_2)\in M_N(E)\langle\!\langle s_1,s_2\rangle\!\rangle^2
+\]
+have constant term $R$.  These are input matrices: their scalar entries
+act on the input factor and commute with the old coefficient algebra.
+
+For $h\in I$, embed $\pi_h$ at the $P$ block, put $\bar\pi_h=1-\pi_h$,
+and form
+\[
+ S_h(Z)=\pi_h(W(Z)-\lambda_h)\pi_h
+ -\pi_hW(Z)\bar\pi_h
+ \bigl(\bar\pi_h(W(Z)-\lambda_h)\bar\pi_h\bigr)^{-1}
+ \bar\pi_hW(Z)\pi_h.
+\]
+Suppose that every $S_h(Z)$ vanishes in degrees less than $\ell$, where
+$\ell\ge1$.  For each word $\omega$ of length $\ell$, there are scalars
+$c_{h,\omega}\in E$ such that
+\[
+ [S_h(Z)]_\omega=c_{h,\omega}\pi_h,
+ \qquad (c_{h,\omega})_{h\in I}\in
+        (\mathcal T_P|_I)\otimes E.
+\]
+The same conclusion holds for homogeneous degree-$\ell$ coefficients of
+perturbations in finitely many ordinary commuting parameters.
+\end{proposition}
+
+\begin{proof}
+For $\ell=1$, only the $P$-to-$P$ input coefficient contributes to the
+compressed derivative, so the assertion is the first-response identity.
+Assume $\ell\ge2$ and fix a word $\omega$ of length $\ell$.
+
+\paragraph{Unfolding the word.}
+Use $\ell+1$ ordered states.  On consecutive steps put the letters of
+$\omega$, and evaluate a word by following that path.  Thus a coefficient
+of $Z$ for a substring becomes the corresponding strictly upper block of
+the input, while every diagonal block is $R$.  The $(1,\ell+1)$ block
+extracts precisely the coefficient of $\omega$.  This substitution is a
+homomorphism on the free-series ring modulo degree $\ell+1$, and it
+therefore respects every inverse with invertible constant term.
+
+Keep only $P$ at the two endpoint states, leaving all $R$ blocks at the
+$\ell-1$ internal states.  This operation does not change the required
+compressed endpoint coefficient.  At an endpoint, constant input steps
+preserve the blocks $P,C_0$, and the constant output commutes with
+$\pi_h$.  A contributing Schur path can therefore neither enter the
+first endpoint complement from its target projection nor leave the last
+endpoint complement for its target projection.  All complement
+eliminations relevant to the coefficient occur at internal states.
+
+Denote the internal input by $Q_0$.  Its size is $(\ell-1)N$ and hence
+belongs to $\mathcal S$.  Grouping the internal
+states, write the cropped input as
+\[
+ T_0=\begin{pmatrix}
+ P&G_1&Y\\0&Q_0&G_2\\0&0&P
+ \end{pmatrix}.
+\]
+Here each entry denotes a pair of input matrices.  In the output write
+$E_1=W(T_0)_{12}$, $E_2=W(T_0)_{23}$, and $F=W(T_0)_{13}$.
+
+\paragraph{The internal target eigenspace.}
+Fix $h\in I$, set $M_0=W(Q_0)-\lambda_h$, and let $e$ be the sum of the
+fixed copies of $\pi_h$ on its internal states.  Put $f=1-e$.
+The matrix $fM_0f$ is upper triangular with invertible diagonal blocks.
+Moreover its Schur complement on $eV$ is zero: every strictly upper
+coefficient is a coefficient for a proper substring of $\omega$, and
+these vanish by hypothesis.  Its kernel is consequently the graph
+\[
+ u\longmapsto u-f(fM_0f)^{-1}fM_0e u,
+ \qquad u\in eV.
+\]
+Projection onto $eV$ is an isomorphism on this kernel, whose dimension is
+$(\ell-1)t$.  Triangularity shows that this is also the full algebraic
+multiplicity of zero.  Thus zero is semisimple.  Let $\Pi$ be its
+spectral projection.
+
+The same argument applies to the chain consisting of the first endpoint
+and the internal states, and to the chain consisting of the internal
+states and the last endpoint.  They involve only proper substrings.
+Solving their triangular kernel equations shows that the couplings
+between their zero eigenspaces vanish:
+\begin{equation}
+\label{eq:transfer-coupling-annihilation}
+ \pi_hE_1\Pi=0,\qquad \Pi E_2\pi_h=0.
+\end{equation}
+Indeed any nonzero compressed coupling would reduce the kernel dimension
+below the sum of the diagonal zero multiplicities.  This argument uses
+semisimplicity only at the target eigenvalue.
+
+\paragraph{A regularized auxiliary partner.}
+In every internal copy of $R$, replace its $P$ block by $P+\xi v$, keeping
+all arrows fixed; call the resulting input $Q(\xi)$.  Put
+$M(\xi)=W(Q(\xi))-\lambda_h$.
+The spectral projection $\Pi$ is upper triangular in the state order,
+with diagonal equal to the fixed target projections.  The graph
+identification of its range with $eV$ is upper triangular with identity
+diagonal.  It follows that
+\[
+ \Pi M'(0)\Pi\big|_{\ker M_0}
+\]
+is upper triangular in this identification, with every diagonal block
+equal to $D_h(P;v)I_t$.  It is therefore invertible.  This verifies the
+simple-pole hypothesis of Lemma~\ref{lem:killed-pole}, including the
+effect of all internal triangular arrows.
+
+Let $T(\xi)$ be the corresponding full cropped input and use $E_1(\xi)$,
+$E_2(\xi)$, $F(\xi)$ for its output blocks.  For nonzero formal $\xi$, the
+compressed endpoint expression is
+\begin{align}
+ \mathcal E_h(\xi)
+ &=\pi_h\bigl(F(\xi)-E_1(\xi)M(\xi)^{-1}E_2(\xi)\bigr)\pi_h
+ \label{eq:transfer-regularized-expression}\\
+ &=D_h(P;Y)\pi_h+M_h(P,Q(\xi);G_1,G_2).
+ \nonumber
+\end{align}
+Its tuple is scalar and belongs to the projected active space over
+$E((\xi))$.  This is exactly the specialized paired-response identity:
+the auxiliary block is upper triangular, its diagonal entries are small
+blocks with a Taylor shift, and its strictly upper entries are fixed
+finite matrices.  Clear the target resolvent determinants first and
+apply Lemma~\ref{lem:active-specialization}; restore the inverses over
+$E((\xi))$, where the preceding argument proves they exist.  No
+specialization of an undefined inverse is being used.
+
+\paragraph{Identifying the limit.}
+By \eqref{eq:transfer-coupling-annihilation} and
+Lemma~\ref{lem:killed-pole}, expression
+\eqref{eq:transfer-regularized-expression} has the finite limit
+\[
+ \pi_h\bigl(F-E_1JE_2\bigr)\pi_h
+ =\pi_h\bigl(F-E_1f(fM_0f)^{-1}fE_2\bigr)\pi_h.
+\]
+The last expression eliminates precisely the fixed complementary
+spaces in the internal states.  By the unfolding and endpoint-cropping
+argument it is $[S_h(Z)]_\omega$.  In particular the limiting inverse has
+not changed the coefficient to one computed with a different choice of
+complement.  Taking the limit in the scalar-response identities and in
+the finitely many linear equations defining $\mathcal T_P|_I$ proves
+the assertion simultaneously for all target labels.
+
+Finally consider a perturbation in commuting variables $h_1,\ldots,h_s$.
+Substitute $h_j=a_jz$ for arbitrary scalar directions $a_j$ and one free
+letter $z$.  The lower homogeneous terms still vanish, so the result
+just proved applies to the degree-$\ell$ coefficient on every such
+line.  Scalarity and containment in a fixed finite-dimensional linear
+space are linear conditions.  Each of their annihilators therefore
+gives a homogeneous polynomial in the $a_j$ that vanishes identically.
+Over the infinite characteristic-zero field all its coefficients vanish.
+This proves the commuting-parameter assertion.
+\end{proof}
+
+Proposition~\ref{prop:word-transfer} is the bridge from responses to
+free-series equations.  It controls a first unsolved coefficient without
+assuming that the perturbation is commutative or that the auxiliary
+matrix has a separated spectrum before regularization.  We next use its
+commuting-parameter consequence to show that linked responses arise from
+actual formal relations between eigenvalues.
diff --git a/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/paper.pdf b/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..7789e74dabbca601ec9933f18eb67bfde361ac62
--- /dev/null
+++ b/preprints/A-Counterexample-to-Kuroshs-Division-Ring-Problem-September-23-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:9195b0a83fc633704a735973a068257d424a46196a74cad21176e49348147aec
+size 541378
diff --git a/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/README.md b/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..c8b7db18ced02ba4ce3051700fb998f7270a6b54
--- /dev/null
+++ b/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/README.md
@@ -0,0 +1,18 @@
+# [A counterexample to Ryser's covering conjecture](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 23, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026,
+  author = {{OpenAI}},
+  title = {{A counterexample to Ryser's covering conjecture}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/paper.pdf}{OAI:A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/build/construction.tex b/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/build/construction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..77fc3b53f831d469fe004b814e26d097d0e83f2c
--- /dev/null
+++ b/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/build/construction.tex
@@ -0,0 +1,857 @@
+\Needspace{14\baselineskip}
+\section{A configuration with two coordinatizations}
+\label{sec:gadget}
+
+We now use the deletion lemma to construct the exceptional configuration.
+We need two properties of the same set: many points on each undeleted
+line, and rigidity of its small line covers.  A change of coordinates
+lets us obtain both.  In the original coordinates we delete one line
+in every finite nonzero direction.  After the change of coordinates,
+the deleted sets become curves, so the line-survival conclusion of
+Lemma~\ref{lem:linecover} applies to every line.  It will force every
+small mixed cover by lines and individual points to be a parallel class.
+
+\begin{lemma}
+\label{lem:gadget}
+For every sufficiently large prime \(s\equiv2\pmod3\), there is
+a set \(Y\subseteq\F_s^2\) with the following properties.  For each
+\(d\in\F_s^*\), there is an affine line \(\ell_d\) of slope \(d\),
+and
+\[
+ Y=\F_s^2\setminus\bigcup_{d\in\F_s^*}\ell_d.
+\]
+For each \(d\in\F_s^*\), every line of slope \(d\) other than
+\(\ell_d\) contains at least five points of \(Y\).  The map
+\[
+ f:\F_s^2\longrightarrow\F_s^2,
+ \qquad f(x,y)=(x^3,y),
+\]
+is a bijection, and every affine line contains at least \(s/8\) points
+of \(f(Y)\).  Moreover, if \(a\) affine lines and \(b\) individual
+points cover \(f(Y)\), and
+\begin{equation}
+\label{con-weighted-budget}
+ a+\frac b2\le s,
+\end{equation}
+then \(a=s\), \(b=0\), and the lines form a full parallel class.
+Consequently every such mixed cover has weighted cost at least \(s\),
+and no cover of \(f(Y)\) by at most \(s\) lines contains two
+nonparallel lines.
+\end{lemma}
+
+\begin{proof}
+Let \(s\) tend to infinity through primes congruent to \(2\pmod3\).
+Since
+\(\gcd(3,s-1)=1\), cubing permutes \(\F_s^*\) and fixes zero.
+Thus \(f\) is bijective.  It maps horizontal lines to horizontal
+lines and vertical lines to vertical lines.  We first prove the two
+survival properties simultaneously with high probability, then use
+them to classify mixed covers satisfying~\eqref{con-weighted-budget}.
+
+\paragraph{Survivors in the original coordinates.}
+Choose the lines \(\ell_d\), independently for the \(s-1\) values
+of \(d\), uniformly within their parallel classes.  Fix a line
+\(\ell\) of finite nonzero slope \(d\), and temporarily ignore the
+choice \(\ell_d\).  Each of the other \(s-2\) choices deletes a
+uniformly distributed point of \(\ell\).  Let \(Z_\ell\) count
+the surviving points on \(\ell\).  Its expectation is
+\[
+ \mu:=\E Z_\ell=s(1-1/s)^{s-2}.
+\]
+Changing one of these choices changes the survivor count by at most
+two.  Apply the independent-trial jump bound and bounded-drift
+concentration estimate of
+\cite[Lemmas~A.2--A.3]{OpenAI2026Balanced} to the exposure martingale,
+with \(T=s-2\), \(L=2\), and \(u=\mu/2\).  The martingale has
+zero drift, and hence
+\[
+ \Pr(Z_\ell<\mu/2)
+ \le 2\exp\left(-\frac{\mu^2}{128(s-2)}\right)
+ =\exp(-\Omega(s)).
+\]
+There are fewer than \(s^2\) lines under consideration.  A union
+bound, together with \(\mu/2\ge5\) for sufficiently large \(s\),
+shows that all these counts are at least five with probability tending
+to one.  If \(\ell\ne\ell_d\), reinstating the deletion by
+\(\ell_d\) changes no point on \(\ell\), since the two lines are
+parallel.  Thus every required line in the original coordinates has
+at least five points of \(Y\).
+
+\paragraph{Survivors and almost-pencil covers in the new coordinates.}
+In the new coordinates, the possible deletion shapes are
+\[
+ F_{d,z}=\{(x^3,dx+z):x\in\F_s\},
+ \qquad d\in\F_s^*,\quad z\in\F_s.
+\]
+There are \(s(s-1)\) such shapes, each of size \(s\), and each of
+the \(s-1\) independent trials chooses one of them.  Every point
+is deleted by a specified trial with probability \(1/s\), so its
+overall survival probability is the uniform value
+\[
+ \delta_s=(1-1/s)^{s-1}\ge\frac14
+\]
+for all sufficiently large \(s\).
+
+Each shape meets every affine line in at most three points.  For a
+vertical line this follows from bijectivity of cubing; for a
+horizontal line it follows from \(d\ne0\).  For a line of finite
+nonzero slope \(m\), the intersection is determined by
+\[
+ dx+z=mx^3+u,
+\]
+a nonzero polynomial equation of degree three.  In particular, no
+shape is itself a line when \(s>5\).  All the hypotheses of
+Lemma~\ref{lem:linecover} hold with field size \(s\), a fixed
+integer constant, for example \(D=3\), survival lower bound
+\(\delta=1/4\), and \(\rho_0=1/100\).  The exact-line
+occurrence probabilities are zero, and the subfield restriction is
+automatic for a prime field.  With probability tending to one, every
+line consequently contains at least \(s/8\) points of \(f(Y)\),
+and its almost-cover conclusion holds for \(f(Y)\) as well.  We now
+combine that conclusion with the lower bound on every line to replace
+an almost pencil by a complete one.
+
+\paragraph{From an almost pencil to a full parallel class.}
+Assume these events hold, and consider a cover satisfying
+\eqref{con-weighted-budget}.  Repeated resources can be removed;
+alternatively, one may keep their original counts, since removal
+only reduces the budget.  The line portion has at most \(s\)
+distinct lines and leaves at most \(b\le2s\) points of \(f(Y)\)
+uncovered.  Lemma~\ref{lem:linecover} supplies at least
+\((1-\rho_0)s\) distinct lines in this portion through a common
+projective point \(p\).  There are at most \(\rho_0s\) other
+lines, and
+\[
+ b\le2(s-a)\le2\rho_0s.
+\]
+If an affine line through \(p\) were absent from the cover, all the
+chosen lines through \(p\) would cover at most one of its affine
+points.  The other lines would cover at most \(\rho_0s\) further
+points, and the individual point resources at most \(2\rho_0s\).
+Thus at most \(1+3\rho_0s\) points of this missing line would be
+covered.  For sufficiently large \(s\), this contradicts
+\[
+ \frac s8>1+3\rho_0s.
+\]
+Every affine line through \(p\) must therefore be present.  A
+finite \(p\) requires \(s+1\) lines, which is impossible.  An
+infinite \(p\) requires all \(s\) lines in its parallel class,
+and then \eqref{con-weighted-budget} forces \(a=s\) and \(b=0\).
+This also rules out repetitions in a cover attaining the budget.
+
+The events used in the two coordinate systems hold simultaneously
+with probability tending to one.  Consequently every sufficiently
+large prime \(s\equiv2\pmod3\) admits a successful configuration
+\(Y\).  A mixed cover of cost less than
+\(s\) would satisfy \eqref{con-weighted-budget} but not its
+necessary equality conclusion.  The two stated consequences follow.
+\end{proof}
+
+There are arbitrarily large primes congruent to \(2\pmod3\).  Indeed,
+if the list were finite and its product were \(P\), the integer
+\(3P-1\equiv2\pmod3\) would have a prime factor congruent to
+\(2\pmod3\), and none of the primes dividing \(P\) could divide it.
+Fix any sufficiently large prime \(s\equiv2\pmod3\), and choose
+\(Y\) and the omitted lines from Lemma~\ref{lem:gadget}.
+These data remain fixed for the rest of the proof.
+
+\section{Ambient placement}
+\label{sec:placement}
+
+We will place copies of \(Y\) in a larger affine plane, assign each
+copy the directions in which its labels will change, and delete the
+ambient lines of those directions that meet it.  The placement must
+meet two requirements.  Joins between different copies must avoid
+their changed directions, while the undeleted points must retain the
+pencil and line-survival properties of Lemma~\ref{lem:linecover}.
+We first specify the copies and direction sets, then prove the
+deletion estimates for independent pools of translates, and finally
+select one compatible translate from each pool.
+
+\subsection{Fields, copies, and direction sets}
+
+Let \(n\ge3\) be odd, and put \(q=s^n\). We identify \(\F_s\)
+with the prime subfield of \(\F_q\). If \(E\) is a subfield, then
+\(|E|=s^a\), where \(a=[E:\F_s]\), and multiplication of vector-space
+bases gives
+\[
+ n=a[\F_q:E].
+\]
+For a proper subfield, the index \([\F_q:E]\) is an odd integer
+larger than one, so it is at least three. Consequently
+\[
+ |E|=s^a\le s^{n/3}=q^{1/3}
+ \qquad\text{for every proper subfield }E.
+\]
+Thus Theorem~\ref{thm:incidence} and
+Lemma~\ref{lem:linecover} apply along this sequence of ambient
+fields.  In this section all constants may depend on the already
+fixed \(s\) and \(Y\), but not on \(n\).
+
+The odd-degree condition has now supplied the required subfield bound.
+It remains to assign every ambient direction to at least one copy,
+with bounded overlap between assignments.  Identify the ambient
+directions with
+\(\PP^1(\F_q)=\F_q\cup\{\infty\}\), using slope.  Choose a
+set \(\Lambda\) of representatives of
+\(\F_q^*/\F_s^*\), with \(1\in\Lambda\).  There will be one
+configuration group for each \(\lambda\in\Lambda\), together
+with one further group denoted by \(*\).  Write
+\[
+ \mathcal I=\Lambda\sqcup\{*\},
+ \qquad N_q=|\mathcal I|=\frac{q-1}{s-1}+1.
+\]
+Each index \(i\in\mathcal I\) will specify a pool of translates
+of one configuration.  We call this pool a group and will choose one
+translate from each group.
+
+For \(\lambda\in\Lambda\), define
+\[
+ M_\lambda(x,y)=(x,\lambda y),
+ \qquad S_\lambda=\lambda\F_s^*,
+ \qquad A_\lambda=\{0,\infty\}.
+\]
+Choose distinct \(u,v\in\F_s^*\) and define
+\[
+ M_*=
+ \begin{pmatrix}1&1\\u&v\end{pmatrix},
+ \qquad
+ S_*=(\F_s\cup\{\infty\})\setminus\{u,v\},
+ \qquad A_*=\{u,v\}.
+\]
+All these linear maps are invertible.  For \(M_*\), the image of
+a finite slope \(t\) is
+\[
+ \frac{u+vt}{1+t},
+\]
+where a zero denominator gives the infinite direction.  Its images
+of the horizontal and vertical directions are \(u\) and \(v\).
+The induced projective bijection therefore maps the \(s-1\)
+finite nonzero subfield slopes precisely to \(S_*\).  The same
+statement for \(M_\lambda\) follows from the slope map
+\(t\mapsto\lambda t\).
+
+For each \(i\in\mathcal I\), put
+\[
+ W_i=M_i(\F_s^2).
+\]
+The directions of joins of distinct points of \(W_i\) form exactly
+\(S_i\cup A_i\).  Each set \(S_i\) has size \(s-1\), and
+\(A_i\) consists of the two images of the coordinate axes.  The
+cosets \(S_\lambda\) partition all nonzero finite ambient slopes.
+The extra set \(S_*\) contains \(0\) and \(\infty\).  Hence
+every ambient direction belongs to at least one and at most two
+sets \(S_i\).  More precisely, the directions belonging to two
+sets are
+\begin{equation}
+\label{con-double-directions}
+ S_1\cap S_* = \F_s^*\setminus\{u,v\},
+\end{equation}
+and the two groups involved are always \(1\) and \(*\).
+
+The sets \(S_i\) are the directions whose labels will change, and
+\(A_i\) are the images of the two axis directions preserved by \(f\).
+For compatibility between copies we use the
+slightly larger direction sets
+\[
+ D_i=
+ \begin{cases}
+ S_i\cup A_i,&i\in\{1,*\},\\
+ S_i,&i\notin\{1,*\}.
+ \end{cases}
+\]
+In particular,
+\begin{equation}
+\label{con-conflict-directions}
+ D_1=D_*=\PP^1(\F_s),
+\end{equation}
+whereas the remaining \(D_i\) are the nonidentity multiplicative
+cosets of \(\F_s^*\).  Every direction occurs in at most two
+sets \(D_i\).  Since \(|W_i|=s^2\), \(|D_i|\le s+1\), and
+\(N_q=O_s(q)\), a fixed constant depending only on \(s\)
+satisfies all size and multiplicity assumptions of
+Theorem~\ref{thm:selection}.  In a conflict-free selection, including
+the axes in both \(D_1\) and \(D_*\) ensures that an axis line
+meeting one of these two selected copies cannot also meet the other.
+This will enter the weighted cover
+argument for the two groups sharing a direction
+in~\eqref{con-double-directions}.
+
+\subsection{Independent candidate pools}
+
+Let \(K\) be a fixed pool size supplied by
+Theorem~\ref{thm:selection}.  For each
+\(i\in\mathcal I\), choose independent uniform shifts
+\(b_{i,1},\ldots,b_{i,K}\in\F_q^2\), independently between
+groups, and write
+\[
+ P_{i,k}=b_{i,k}+W_i,
+ \qquad Y_{i,k}=b_{i,k}+M_iY.
+\]
+Associate to this candidate the following set of ambient deletion
+lines:
+\[
+ \mathcal B_{i,k}
+ =\{\ell:\operatorname{dir}(\ell)\in S_i,
+                  \ \ell\cap Y_{i,k}\ne\varnothing\}.
+\]
+
+\begin{lemma}
+\label{con-deletion-count}
+For every candidate and every \(d\in S_i\), exactly \(s-1\)
+lines of \(\mathcal B_{i,k}\) have direction \(d\).  Each of
+these lines contains at least five points of \(Y_{i,k}\).
+Consequently \(|\mathcal B_{i,k}|=(s-1)^2\).
+\end{lemma}
+
+\begin{proof}
+Undo the translation and apply \(M_i^{-1}\).  An ambient line in
+the given direction meeting \(P_{i,k}\) becomes a line through a
+point of \(\F_s^2\) in a finite nonzero subfield direction.  Its
+intersection with \(\F_s^2\) is the corresponding subfield
+line.  The \(s\) lines in that subfield parallel class extend to
+distinct parallel ambient lines.  Exactly one is the omitted line
+in the definition of \(Y\); the other \(s-1\) each contain at
+least five points of \(Y\), by Lemma~\ref{lem:gadget}.  These are
+precisely the lines meeting the candidate's retained configuration.
+\end{proof}
+
+Each candidate therefore prescribes a bounded list of deletion lines.
+We next apply the deletion lemma to all these independent lists at
+once.  Besides the almost-pencil and line-survival conclusions, we
+need one bound on how many groups a line can meet along joins from an
+external point.  All three properties will survive the subsequent
+selection of one candidate per group.
+
+\begin{lemma}
+\label{con-pool-properties}
+There are constants \(\delta\in(0,1)\) and
+\(\rho=\delta/10\), independent of \(n\), with the following
+properties with probability tending to one as \(n\) tends to
+infinity through odd integers. Let
+\[
+ X_{\rm pool}
+ =\F_q^2\setminus
+   \bigcup_{i\in\mathcal I}\bigcup_{k=1}^K
+          \bigcup_{\ell\in\mathcal B_{i,k}}\ell.
+\]
+\begin{enumerate}
+\item Every cover of \(X_{\rm pool}\) by at most \(q\) affine
+lines contains at least \((1-\rho)q\) lines through a common
+projective point.
+\item For every affine line \(\ell\), if only exact occurrences
+of \(\ell\) are ignored in the pool deletions, at least
+\(\delta q/2\) points of \(\ell\) remain.
+\item For every finite point \(p\) and affine line \(\ell\)
+not through \(p\), at most five groups \(i\) have a candidate
+\(P_{i,k}\) containing a point \(y\in\ell\) whose join to
+\(p\) has direction in \(S_i\).
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+We verify the hypotheses of Lemma~\ref{lem:linecover} for the full
+pool, which gives the first two assertions.  The third follows from
+a separate union bound over finite points and lines.
+
+Treat each candidate shift as one independent deletion trial, listing
+the lines of \(\mathcal B_{i,k}\).  There are \(KN_q=O_{s,K}(q)\)
+trials, each listing \(m=(s-1)^2\) lines.  We may take the shape
+family to be all \(q(q+1)\) ambient affine lines.  Each shape has
+size \(q\), and two distinct line shapes intersect in at most one
+point.  Thus the shape and trial bounds in
+Lemma~\ref{lem:linecover} hold with a sufficiently large fixed
+integer \(D\) depending only on \(s,K\).
+
+A candidate's deleting set is a uniform translate of a fixed union
+of \(m\) lines.  Its deletion probability at a fixed point is
+independent of the point and is at most \(m/q\).  Independence of
+the shifts makes the overall survival probability \(\delta_q\)
+uniform on \(\F_q^2\).  For sufficiently large \(q\),
+\(m/q\le1/2\) and \(N_q\le2q/(s-1)\).  Using
+\(\log(1-t)\ge-2t\) for \(0\le t\le1/2\), we obtain
+\[
+ \delta_q\ge (1-m/q)^{KN_q}
+ \ge \exp\bigl(-2KN_qm/q\bigr)
+ \ge \exp\bigl(-4K(s-1)\bigr).
+\]
+Fix
+\(\delta=\exp(-4K(s-1))\) and \(\rho=\delta/10\).
+
+For completeness, the exact-line hypothesis also has a uniform
+bound.  A fixed line \(\ell\) of direction \(d\) can occur only
+in candidates of groups with \(d\in S_i\), of which there are at
+most two.  In each such candidate, there are \(s-1\) translated
+lines in this direction.  Each fixed line has probability \(1/q\)
+of becoming \(\ell\) under the uniform shift.  Therefore the sum,
+over all trials, of the probabilities of listing \(\ell\) is at
+most
+\begin{equation}
+\label{con-exact-line-probability}
+ \frac{2K(s-1)}q.
+\end{equation}
+Increasing the fixed \(D\) if necessary verifies every hypothesis
+of Lemma~\ref{lem:linecover}.  Its conclusions give the first two
+properties.
+
+To prove the third, fix \(p\notin\ell\).  For one candidate in
+group \(i\), there are at most \(s-1\) possible points
+\(y\in\ell\) whose join to \(p\) has a direction in \(S_i\):
+each such direction contributes at most one intersection with
+\(\ell\).  A fixed point belongs to a uniform translate of
+\(W_i\) with probability \(s^2/q^2\).  The probability for this
+candidate is thus at most
+\[
+ \frac{(s-1)s^2}{q^2}.
+\]
+If six distinct groups meet the condition, choose one witnessing
+candidate from each.  There are at most \((KN_q)^6=O_{s,K}(q^6)\)
+choices, and their shifts are independent.  For the fixed pair
+\((p,\ell)\), the probability of six such groups is therefore
+\(O_{s,K}(q^{-6})\).  There are at most
+\(q^2q(q+1)=O(q^4)\) pairs \((p,\ell)\).  The union bound gives
+a total failure probability \(O_{s,K}(q^{-2})\), proving the claim.
+\end{proof}
+
+\subsection{Selection and transfer of the deletion estimates}
+
+The estimates just proved concern independent candidates.  We can now
+select compatible copies, using only inclusions to transfer the
+estimates to the selected deletion set.
+By Theorem~\ref{thm:selection}, with probability tending to one the
+same pool admits one candidate per group with no conflicts.
+Intersect this event with those of Lemma~\ref{con-pool-properties}.
+No independence between these events is needed.  For every
+sufficiently large odd integer \(n\), their intersection has positive
+probability, so fix a pool in the intersection and a conflict-free
+selection.  Denote the selected sets and shifts by
+\[
+ P_i=b_i+M_i(\F_s^2),\qquad Y_i=b_i+M_iY.
+\]
+Conflict avoidance gives the useful implication
+\begin{equation}
+\label{con-separation}
+ i\ne j,\quad \operatorname{dir}(\ell)\in D_i,\quad
+ \ell\cap P_i\ne\varnothing
+ \quad\Longrightarrow\quad \ell\cap P_j=\varnothing.
+\end{equation}
+In particular the \(P_i\) are pairwise disjoint: a common point
+would give a zero difference, which belongs to every conflict
+direction, and each \(D_i\) is nonempty.
+
+Let \(B\) be the union, as a set of lines, of the deletion lines
+of the selected candidates, and put
+\begin{equation}
+\label{con-selected-survivors}
+ X=\F_q^2\setminus\bigcup_{\ell\in B}\ell.
+\end{equation}
+Then \(X\supseteq X_{\rm pool}\).  Moreover, if \(\ell\notin B\),
+every selected deletion line belongs to the pool list after all
+exact occurrences of \(\ell\) have been omitted.  Hence \(X\)
+contains the survivor set of that modified pool process. This
+inclusion is deterministic: it holds even if \(\ell\) occurs in an
+unselected candidate and even though the selected shifts depend on the
+whole pool. In particular,
+\begin{equation}
+\label{con-selected-line-bound}
+ \ell\notin B
+ \quad\Longrightarrow\quad |X\cap\ell|\ge\frac{\delta q}{2}.
+\end{equation}
+The third property of Lemma~\ref{con-pool-properties} also remains
+available for the selection, since it was established for all
+candidates in the pool.
+
+This completes the placement: the copies are separated, covers of the
+ordinary survivors by at most \(q\) lines contain an almost pencil,
+every undeleted line contains at least \(\delta q/2\) survivors,
+and the bound of five groups remains valid.  The order of choices is
+\(s,Y\), then the selection constant and \(K\), then the deletion
+constants \(D,\delta,\rho\), and finally \(q=s^n\).
+Only after fixing these constants do we require \(q=s^n\) to be
+sufficiently large.  All further numerical inequalities below involve
+these fixed constants
+and can be imposed by enlarging this final threshold.  Thus the
+construction is available for every sufficiently large odd integer
+\(n\), with the same \(s,Y\); its pools and selected shifts may
+depend on \(n\). The constant used for the ambient deletion lemma
+may depend on \(K\), but it changes no hypothesis of the already
+applied selection theorem. A small value of \(\delta\) only increases
+the final lower threshold on \(q\).
+
+\section{The hypergraph and its covering number}
+\label{sec:hypergraph}
+
+The placement supplies the ordinary points \(X\) and the exceptional
+copies \(Y_i\).  We now turn them into edges.  Ordinary edges retain
+all their affine-line labels.  In an exceptional copy, the directions
+\(S_i\) instead use line labels read from \(f(Y)\); all remaining
+directions retain their ordinary labels.  We will first check that
+every pair of edges still meets, then use the rigidity of \(f(Y)\)
+to rule out covers of size \(q\).
+
+\subsection{Vertices, edges, and intersection}
+
+Set \(r=q+1\).  There is one vertex part \(V_d\) for each
+\(d\in\PP^1(\F_q)\).  For every ambient affine line \(\ell\)
+of direction \(d\), place an ordinary vertex \(o_\ell\) in
+\(V_d\).  If \(d\in S_i\), let \(t_i(d)\in\F_s^*\) be
+the unique subfield direction whose image under \(M_i\) is
+\(d\).  For each of the \(s\) subfield affine lines \(L\) of
+direction \(t_i(d)\), place a special vertex \(v_{i,d,L}\) in
+\(V_d\).  These subfield lines are read in the new coordinates
+used for \(f(Y)\).  All vertices are formal labels, tagged to be
+distinct between parts, between ordinary and special types, and
+between different groups.  In particular each \(V_d\) is finite,
+and the parts are pairwise disjoint.
+
+Write \(\ell_d(z)\) for the ambient line of direction \(d\)
+through \(z\), and \(L_t(w)\) for the subfield line of direction
+\(t\) through \(w\in\F_s^2\).  Each \(x\in X\) defines the
+ordinary edge
+\[
+ e_x=\{o_{\ell_d(x)}:d\in\PP^1(\F_q)\}.
+\]
+For each \(i\in\mathcal I\) and \(w\in Y\), put
+\(y_{i,w}=b_i+M_iw\) and define an exceptional edge by selecting,
+in part \(V_d\), the vertex
+\begin{equation}
+\label{con-exceptional-vertex}
+ \begin{cases}
+ v_{i,d,L_{t_i(d)}(f(w))},& d\in S_i,\\
+ o_{\ell_d(y_{i,w})},&d\notin S_i.
+ \end{cases}
+\end{equation}
+Denote this edge by \(e_{i,w}\), and let
+\[
+ H=\{e_x:x\in X\}\cup
+   \{e_{i,w}:i\in\mathcal I,\ w\in Y\}.
+\]
+Thus an exceptional edge changes labels in the assigned directions,
+without introducing any new part.  Every edge contains exactly one
+vertex from every part, so \(H\)
+is \((q+1)\)-partite and \((q+1)\)-uniform.  It is nonempty,
+because \(Y\ne\varnothing\).
+
+Figure~\ref{fig:two-coordinates} shows how to read an exceptional
+edge from its parameter \(w\).  The deletion lines were defined from
+its ordinary base point \(y_{i,w}\), while its special labels use
+\(f(w)\).  The two axis parallel classes are preserved by \(f\),
+which joins these descriptions in the intersection proof.
+
+\begin{figure}[htbp]
+\centering
+\input{figures/two-coordinates}
+\caption{The two label rules for the vertex of \(e_{i,w}\) in part
+\(V_d\).  The left branch uses the embedded point \(b_i+M_iw\);
+the right branch uses \(f(w)=(w_1^3,w_2)\) in the small plane.
+For each \(d\in A_i\), two parameters give the same ordinary label
+exactly when their images under \(f\) lie on the same corresponding
+horizontal or vertical line.
+The diagram records maps and label choices, rather than geometric
+positions in the finite plane.}
+\label{fig:two-coordinates}
+\end{figure}
+
+\begin{proposition}
+\label{con-intersection}
+The hypergraph \(H\) is intersecting.  Furthermore, all the edges
+in its displayed parametrization are distinct.
+\end{proposition}
+
+\begin{proof}
+There are four types of pairs, according to whether their edges are
+ordinary or exceptional and, in the latter case, whether their group
+indices agree.
+
+Two ordinary edges with distinct base points share the ordinary
+label of their join.  Next, an exceptional base point \(y\in Y_i\)
+lies on a deletion line in every direction of the nonempty set
+\(S_i\), so \(y\notin X\).  Its join with \(x\in X\) cannot
+have direction in \(S_i\): otherwise that join would be a selected
+deletion line containing \(x\).  Its ordinary label therefore
+belongs to both the exceptional edge and \(e_x\).
+
+For exceptional edges from distinct groups \(i,j\), the base
+points are distinct by \eqref{con-separation}.  That same separation
+condition ensures that their join has direction outside both
+\(D_i\) and \(D_j\), and hence outside \(S_i\cup S_j\).
+The two edges share its ordinary label.
+
+Finally, consider \(e_{i,w}\) and \(e_{i,w'}\) with \(w\ne w'\).
+The transformed points \(f(w),f(w')\) are distinct.  If their
+join has finite nonzero slope \(t\), the exceptional edges share
+the special label of this subfield line in the part of direction
+\(M_i(t)\).  If their join is horizontal or vertical, the relevant
+original coordinates are equal as well, because \(f\) preserves
+these two parallel classes bijectively.  The original base points
+then share an ordinary axis-direction label, in a direction of
+\(A_i\).  This completes the intersection proof.
+
+We also check that the parameters count distinct edges, as required
+when we count exceptional edges below.  Two ordinary labels in
+nonparallel directions determine the ordinary base point, so ordinary
+edges are distinct.
+An exceptional edge has special labels and cannot equal an ordinary
+edge.  Within one group, the special labels in any two distinct
+directions of \(S_i\) determine \(f(w)\), since the corresponding
+subfield lines are nonparallel.  As \(|S_i|=s-1\ge2\), equal
+exceptional edges within a group would imply \(w=w'\).  Finally,
+any special label tagged by group \(i\) is absent from every edge
+of a different group.  Exceptional edges in distinct groups are
+therefore distinct too.
+\end{proof}
+
+\Needspace{14\baselineskip}
+\subsection{Covering vertices in the small-plane coordinates}
+
+We identify the positions of the exceptional edges of group \(i\)
+with \(f(Y)\), via \(e_{i,w}\leftrightarrow f(w)\).  The action
+of a covering vertex means the subset of these positions whose edges
+contain that vertex.  The next lemma translates every such action
+into one of the two resources in Lemma~\ref{lem:gadget}: a subfield
+line or an individual point.
+
+\begin{lemma}
+\label{con-label-action}
+Fix a group \(i\).
+\begin{enumerate}
+\item A special vertex of group \(i\) covers the positions on one
+subfield line.  A special vertex of another group covers none.
+\item An ordinary vertex of direction in \(S_i\) covers none.
+An ordinary vertex of direction in \(A_i\), if it covers any
+positions, covers them on a single horizontal or vertical line in
+the \(f\) coordinates.
+\item An ordinary vertex of direction outside \(S_i\cup A_i\)
+covers at most one position.
+\end{enumerate}
+In particular, the positions covered by any one vertex are contained
+in a subfield affine line.
+\end{lemma}
+
+\begin{proof}
+The assertions for special vertices and altered ordinary directions
+follow directly from \eqref{con-exceptional-vertex}.  For an axis
+direction, undoing \(b_i,M_i\) fixes one original coordinate, so
+applying \(f\) gives the corresponding horizontal or vertical
+line.  A line in a direction outside \(S_i\cup A_i\) cannot
+contain two points of \(P_i\), since the join directions of that
+embedded subplane are exactly \(S_i\cup A_i\).  It therefore
+covers at most one exceptional position.  A singleton can be
+included in an arbitrary subfield line through it, and an empty
+set requires no covering resource.
+\end{proof}
+
+\subsection{The final cover argument}
+
+\begin{proof}[Completion of the proof of Theorem~\ref{thm:main}]
+We prove that \(H\) has no vertex cover of size at most \(q\).
+Suppose to the contrary that \(C\) is such a cover.  Its ordinary
+vertices, regarded as affine lines, must cover \(X\): special
+vertices belong to no ordinary edge.  These lines also cover
+\(X_{\rm pool}\).  Lemma~\ref{con-pool-properties} therefore
+gives a projective point \(p\) such that at least
+\((1-\rho)q\) ordinary vertices of \(C\) are lines through
+\(p\).  The line-survival bound will force every undeleted member
+of this pencil into \(C\).  We then count the vertices left for
+exceptional edges, separating centers at infinity, centers inside one
+of the sets \(P_i\), and finite centers outside them all.
+
+\paragraph{Every undeleted line in this pencil is forced.}
+We claim that
+\begin{equation}
+\label{con-forced-pencil}
+ p\in\ell,\quad\ell\notin B
+ \quad\Longrightarrow\quad o_\ell\in C,
+\end{equation}
+where incidence with an infinite \(p\) means that \(\ell\) has
+the corresponding direction.  If \(o_\ell\) were missing, all
+other lines of \(C\) through \(p\) would cover at most one
+affine point of \(\ell\).  There are at most \(\rho q\)
+vertices outside the large pencil subcollection.  Each of their
+ordinary lines covers at most one point of \(\ell\), and their
+special vertices cover no ordinary edge.  Consequently
+\[
+ |X\cap\ell|\le1+\rho q.
+\]
+This contradicts \eqref{con-selected-line-bound}, since
+\(\delta q/2>1+\rho q\) for sufficiently large \(q\) and
+\(\rho=\delta/10\).  At infinity the pencil lines are parallel,
+so the possible contribution of one central point is absent; the
+same upper bound still applies.
+
+Let \(F_p\) be the set of ordinary vertices forced by
+\eqref{con-forced-pencil}, and call the vertices of \(C\setminus F_p\)
+the additional vertices.  The deleted pencil lines determine how
+many additional vertices the assumed budget \(|C|\le q\) permits.
+
+\paragraph{An infinite center.}
+Suppose \(p\) is the point at infinity in direction \(d\).  Put
+\[
+ I_d=\{i\in\mathcal I:d\in S_i\},
+ \qquad z=|I_d|\in\{1,2\}.
+\]
+Only these groups contribute deletion lines in direction \(d\),
+and each contributes \(s-1\), by
+Lemma~\ref{con-deletion-count}.  At most \(z(s-1)\) of the
+\(q\) lines in this parallel class belong to \(B\).  Thus
+\[
+ |C\setminus F_p|\le z(s-1).
+\]
+None of the forced vertices covers any exceptional edge of a group
+in \(I_d\), because those groups use special vertices in
+direction \(d\).
+
+If \(z=1\), all positions of this group's \(f(Y)\) must therefore
+be covered by at most \(s-1\) additional vertices.  By
+Lemma~\ref{con-label-action}, their effects can be included in at
+most \(s-1\) subfield lines.  This contradicts
+Lemma~\ref{lem:gadget}.
+
+If \(z=2\), the two groups are \(1,*\), by
+\eqref{con-double-directions}.  An additional vertex can now meet
+both groups, so we use the weighted line-and-point budget instead of
+counting one line per vertex.  For each of these groups let \(a_i\)
+count the additional vertices that cover at least one of its
+positions and are either special vertices of this group or ordinary
+vertices in its axis directions \(A_i\).  Let \(b_i'\) count the
+other additional vertices covering a position of the group.  By
+Lemma~\ref{con-label-action}, the latter vertices each cover only
+a singleton.  The former supply subfield lines.  The weighted
+conclusion of Lemma~\ref{lem:gadget} gives
+\begin{equation}
+\label{con-two-group-lower}
+ a_i+\frac{b_i'}2\ge s
+ \qquad (i=1,*).
+\end{equation}
+
+Each additional vertex contributes at most one to the sum of the
+two left sides.  A special vertex is confined to its tagged group.
+An ordinary axis-direction vertex counted in \(a_i\) is a line
+meeting \(P_i\) with direction in \(A_i\subseteq D_i\); by
+\eqref{con-separation} it cannot meet the other group's \(P_j\).
+It therefore contributes nothing to that other group, even as a
+singleton.  Any vertex of neither exclusive type contributes at
+most \(1/2\) to each group.  This accounts for every type of
+additional vertex, including those covering neither group.
+Summing \eqref{con-two-group-lower}, we obtain
+\[
+ 2s\le a_1+\frac{b_1'}2+a_*+\frac{b_*'}2
+ \le |C\setminus F_p|\le2(s-1),
+\]
+a contradiction.
+
+\paragraph{A finite center inside an embedded configuration.}
+Now let \(p\) be finite, and write
+\[
+ h_p=|\{\ell\in B:p\in\ell\}|.
+\]
+There are \(q+1\) affine lines through \(p\), so
+\begin{equation}
+\label{con-finite-budget}
+ |F_p|=q+1-h_p,
+ \qquad |C\setminus F_p|\le h_p-1.
+\end{equation}
+The loss of one in the second inequality comes from the \(q+1\)
+members of a finite pencil and the assumed cover size \(q\).
+If \(h_p=0\), the forced vertices already exceed the permitted
+size of \(C\).  We may therefore assume \(h_p\ge1\) in both
+finite-center cases.
+
+Suppose \(p\in P_i\), for the necessarily unique such group.
+Every deletion line through \(p\) comes from this group: a line
+from another group would contradict \eqref{con-separation}, since
+its direction belongs to that other group's conflict set.  There
+is at most one such line in each direction of \(S_i\), and hence
+\begin{equation}
+\label{con-internal-pencil-size}
+ h_p\le s-1.
+\end{equation}
+Let \(w_p=M_i^{-1}(p-b_i)\in\F_s^2\), whether or not
+\(w_p\in Y\).  We account for the exceptional positions of this
+group covered by \(F_p\): they will all lie on two nonparallel
+lines in the new coordinates.  For a position with base point
+\(y\ne p\), its join with \(p\) has a direction in
+\(S_i\cup A_i\).  A direction in \(S_i\) cannot cover this
+edge by an ordinary vertex.  Any actual forced coverage must
+therefore lie along one of the two axis-direction lines through
+\(p\).  A position based at \(p\) itself, if present, lies on
+both these axes as well.
+
+Since \(f\) preserves the two coordinate parallel classes, all
+these covered positions are contained, in the new coordinates, in
+the horizontal and vertical lines through \(f(w_p)\).  The
+remaining positions are covered by the additional vertices; each
+can be included in one subfield line by
+Lemma~\ref{con-label-action}.  Equations
+\eqref{con-finite-budget} and \eqref{con-internal-pencil-size}
+would thus give a cover of \(f(Y)\) by at most
+\[
+ 2+(h_p-1)\le s
+\]
+lines containing the two nonparallel coordinate lines. These
+are legitimate covering resources whether or not their ordinary labels
+belong to \(F_p\): their union contains every position covered by
+\(F_p\). Removing any duplicate replacement lines preserves these
+two distinct lines and can only decrease the count.  This contradicts
+Lemma~\ref{lem:gadget}.
+
+\paragraph{A finite center outside all embedded configurations.}
+It remains to consider \(p\notin\bigcup_{i\in\mathcal I}P_i\).
+Here we use the five points on each deletion line, rather than the
+small-plane cover obstruction.  We will find \(5h_p\) exceptional
+edges missed by \(F_p\), while each additional vertex covers at
+most five of them.
+
+Each group contributes at most one deletion line through \(p\).
+Indeed, two distinct such lines cannot be parallel.  After undoing
+\(b_i,M_i\), their subfield affine equations have distinct
+subfield directions.  Solving these two equations uses a
+nonsingular linear system over \(\F_s\), so their intersection
+lies in \(\F_s^2\).  This would place \(p\) in \(P_i\), a
+contradiction.  Also, deletion lines from different groups cannot
+coincide, by \eqref{con-separation}.
+
+The \(h_p\) deletion lines through \(p\) therefore come from
+\(h_p\) distinct groups.  From each such group \(i\), choose five
+distinct points of \(Y_i\) on its contributing line, possible by
+Lemma~\ref{con-deletion-count}.  These give \(5h_p\) distinct
+exceptional edges, by Proposition~\ref{con-intersection}.  None
+is covered by a forced ordinary vertex: its unique join line to
+\(p\) is precisely its contributing deletion line, whose
+direction is altered for that group.
+
+Any one additional vertex covers at most five of these selected
+edges.  For a special vertex this follows because it belongs to
+only one group.  An ordinary line through \(p\) covers none,
+by the same unique-join and altered-direction argument.  Finally,
+let an ordinary vertex \(o_\ell\), with \(\ell\) not through
+\(p\), cover some selected edges.  The line \(\ell\) meets each
+contributing line in at most one point, so \(o_\ell\) covers at most
+one selected edge from each contributing group.  Every such group
+has a selected point \(y\in Y_i\cap\ell\) whose join to \(p\)
+has direction in \(S_i\).  The third part of
+Lemma~\ref{con-pool-properties}, which applies to all candidates
+and hence to the selection, bounds the number of these groups by
+five.  This proves the asserted bound for every additional vertex.
+
+By \eqref{con-finite-budget}, at most \(h_p-1\) additional
+vertices are available.  They can cover at most \(5(h_p-1)\) of
+the \(5h_p\) selected edges, a final contradiction.
+
+We have ruled out every possible projective center \(p\), and
+hence every vertex cover of size at most \(q\).  Therefore
+\(\tau(H)\ge q+1\).  Conversely, any edge of the nonempty
+intersecting hypergraph \(H\) is itself a vertex cover and has
+size \(q+1\).  It follows that
+\[
+ \tau(H)=q+1.
+\]
+The parameter choices in Section~\ref{sec:placement} give this
+hypergraph for every sufficiently large odd integer \(n\), with
+\(q=s^n\), after fixing any sufficiently large prime
+\(s\equiv2\pmod3\) and a successful \(Y\). Choose one such \(Y\)
+for each eligible \(s\), and let \(n_0(s)\ge3\) exceed all the
+resulting degree thresholds. This proves Theorem~\ref{thm:main}.
+Since there are arbitrarily large odd integers, it also gives infinitely
+many ranks \(r=q+1\).
+\end{proof}
diff --git a/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/build/figures/two-coordinates.tex b/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/build/figures/two-coordinates.tex
new file mode 100644
index 0000000000000000000000000000000000000000..e75b5b6380da9027c5bfb40a3ad525f6c95daccd
--- /dev/null
+++ b/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/build/figures/two-coordinates.tex
@@ -0,0 +1,19 @@
+\begin{tikzpicture}[
+  >=Latex,
+  point/.style={draw,rounded corners=2pt,align=center,inner sep=7pt},
+  labelbox/.style={draw,align=center,text width=5.1cm,inner sep=7pt},
+  every node/.style={font=\small}]
+\node[point] (w) at (0,0) {$w=(w_1,w_2)\in Y$};
+\node[point] (old) at (-3.5,-1.55)
+  {Ambient coordinates\\[3pt]$y_{i,w}=b_i+M_iw\in Y_i$};
+\node[point] (new) at (3.5,-1.55)
+  {Small-plane coordinates\\[3pt]$f(w)=(w_1^3,w_2)\in f(Y)$};
+\node[labelbox] (ordinary) at (-3.5,-3.65)
+  {Ordinary vertex in $V_d$\\[3pt]$o_{\ell_d(y_{i,w})}$};
+\node[labelbox] (special) at (3.5,-3.65)
+  {Special vertex in $V_d$\\[3pt]$v_{i,d,L_{t_i(d)}(f(w))}$};
+\draw[->] (w) -- node[above left,inner sep=2pt,xshift=-4pt,yshift=-2pt] {embed} (old);
+\draw[->] (w) -- node[above right,inner sep=2pt,xshift=4pt,yshift=-2pt] {apply $f$} (new);
+\draw[->] (old) -- node[left] {$d\notin S_i$} (ordinary);
+\draw[->] (new) -- node[right] {$d\in S_i$} (special);
+\end{tikzpicture}
diff --git a/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/build/incidence.tex b/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/build/incidence.tex
new file mode 100644
index 0000000000000000000000000000000000000000..085724f311bcda6ddd4c110201fb61f2499b8b9b
--- /dev/null
+++ b/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/build/incidence.tex
@@ -0,0 +1,707 @@
+\section{A uniform incidence bound}\label{sec:incidence}
+
+For a set $P$ of points and a set $\mathcal L$ of lines in
+$\PP^2(\F_q)$, write
+\begin{equation}\label{eq:inc-definition}
+ I(P,\mathcal L)
+ =\bigl|\{(p,\ell)\in P\times\mathcal L:p\in\ell\}\bigr|.
+\end{equation}
+Our aim is a power saving with constants independent of the field and
+its characteristic. In Section~\ref{sec:linecover}, this estimate will
+control intersections of a proposed line cover when no projective point
+lies on many of its lines. This is the incidence input to the conclusion
+that a small cover must contain almost a full pencil.
+
+\begin{theorem}[Uniform incidence saving]\label{thm:incidence}
+There exist absolute constants $\kappa\in(0,1/10)$ and $Q$ with the following
+property. Suppose that $q\ge Q$ is a prime power and every proper subfield
+of $\F_q$ has order at most $q^{1/3}$. If $P$ and $\mathcal L$ are sets of
+projective points and projective lines over $\F_q$ satisfying
+\begin{equation}\label{eq:inc-theorem-sizes}
+ |P|,|\mathcal L|\le q^{1+\kappa},
+\end{equation}
+then
+\begin{equation}\label{eq:inc-theorem-bound}
+ I(P,\mathcal L)\le q^{3/2-\kappa}.
+\end{equation}
+\end{theorem}
+
+The proof follows the incidence and sum--product strategy of
+Bourgain, Katz, and Tao~\cite[Sections 3--4 and 6]{BKT2004}. Its geometric
+step turns an incidence configuration with too many incidences into a
+dense set of pairs whose differences and quotients range over small
+sets. A graph argument then extracts one subset with both a small
+difference set and a small quotient set. Finally, scalar coverings show
+that such a subset cannot have size near the square root of the field
+order under the proper-subfield hypothesis.
+
+We develop the two algebraic ingredients first, then return to the
+incidence configuration. The proofs include the simultaneous extraction
+of additive and multiplicative control and the bounds on scalar
+expressions that make the argument uniform over fields of nonprime
+order.
+
+\subsection{Finite-set estimates}
+
+The first task is to pass from control of differences and quotients
+along the edges of a dense bipartite graph to control on all pairs in
+one subset. We begin with the sumset estimates needed to use the
+resulting additive control.
+
+For subsets of an abelian group, $X+Y$ and $X-Y$ denote the usual sumset
+and difference set. For a nonnegative integer $j$, the notation $jY$
+in this subsection means a sum of $j$ copies of $Y$, with $0Y=\{0\}$.
+Later, a field element multiplying a set will always mean field scaling;
+we will specify explicitly when an integer denotes an iterated sumset.
+
+The first two estimates below are Ruzsa's triangle inequality and
+covering lemma; the maximal-translates proof of the latter appears in
+\cite[p.~324]{Ruzsa1999}. The multiple-sum bound is a Pl\"unnecke--Ruzsa
+estimate, for which we use Petridis's minimizing-subset
+proof~\cite[Proposition 2.1 and Section 3]{Petridis2012}.
+
+\begin{lemma}[Triangle, covering, and multiple-sum estimates]
+\label{inc-sumset-estimates}
+Let the sets below be finite subsets of an abelian group.
+\begin{enumerate}
+\item If $Z$ is nonempty, then
+\begin{equation}\label{eq:inc-triangle}
+ |X-Y|\,|Z|\le |X-Z|\,|Z-Y|.
+\end{equation}
+\item If $Y$ is nonempty and $|X-Y|\le M|Y|$, then $X$ is contained in
+the union of at most $M$ translates of $Y-Y$.
+\item If $Y$ is nonempty and $|Y-Y|\le M|Y|$, then, for all nonnegative
+integers $u,v$,
+\begin{equation}\label{eq:inc-multiple-sums}
+ |uY-vY|\le M^{u+v}|Y|.
+\end{equation}
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+For the triangle inequality, choose one representation $d=x_d-y_d$
+for each $d\in X-Y$. The map
+\[
+ (d,z)\longmapsto (x_d-z,z-y_d)
+\]
+from $(X-Y)\times Z$ into $(X-Z)\times(Z-Y)$ is injective: summing the
+two coordinates recovers $d$, after which the chosen representation
+recovers $z$. This proves~\eqref{eq:inc-triangle}.
+
+For the covering statement, take a maximal set $T\subseteq X$ for
+which the translates $t-Y$, $t\in T$, are pairwise disjoint. Their union
+is contained in $X-Y$, so $|T|\le M$. Maximality implies that for every
+$x\in X$ some $x-Y$ intersects $t-Y$ with $t\in T$, whence
+$x\in t+(Y-Y)$.
+
+For the multiple-sum bound, the minimizing-subset argument of
+Petridis~\cite{Petridis2012} lets us iterate a one-step growth estimate.
+Put $Q_0=-Y$ and choose a nonempty
+$W\subseteq Y$ minimizing
+\[
+ \mu=\frac{|W+Q_0|}{|W|}.
+\]
+Then $\mu\le M$, and every $W'\subseteq W$, including the empty set,
+satisfies $|W'+Q_0|\ge\mu|W'|$. We claim that for every finite set $E_0$,
+\begin{equation}\label{eq:inc-petridis-growth}
+ |W+Q_0+E_0|\le\mu|W+E_0|.
+\end{equation}
+The assertion is immediate if $E_0$ is empty. Otherwise order its
+elements as $e_1,\ldots,e_k$. When the translate $W+e_i$ is added to
+the preceding translates, let
+\[
+ W_i'=\{w\in W:w+e_i\in\bigcup_{j<i}(W+e_j)\}.
+\]
+It contributes exactly $|W|-|W_i'|$ new elements. Moreover,
+$W_i'+Q_0+e_i$ is contained in the union of the preceding enlarged
+translates $W+Q_0+e_j$. Thus the contribution of $W+Q_0+e_i$ is at most
+\[
+ |W+Q_0|-|W_i'+Q_0|
+ \le\mu\bigl(|W|-|W_i'|\bigr).
+\]
+Summing proves~\eqref{eq:inc-petridis-growth}. Iterating this estimate
+gives $|W+jQ_0|\le\mu^j|W|$ for every nonnegative integer $j$.
+We now remove the auxiliary set $W$ by applying
+\eqref{eq:inc-triangle} to $uQ_0,vQ_0$ with intermediate set $-W$:
+\[
+ |uQ_0-vQ_0|\,|W|
+ \le |W+uQ_0|\,|W+vQ_0|
+ \le\mu^{u+v}|W|^2.
+\]
+Negation preserves cardinality, so $|uQ_0-vQ_0|=|uY-vY|$. Since
+$|W|\le|Y|$ and $\mu\le M$, this is~\eqref{eq:inc-multiple-sums}.
+\end{proof}
+
+We next carry out the extraction from a dense graph. The argument uses
+the common-neighborhood method in Gowers's quantitative refinement
+\cite[Section 4, Lemma 11 and Proposition 12]{Gowers1998} of the
+Balog--Szemer\'edi theorem~\cite{BalogSzemeredi1994}; see also
+\cite[Section 5, Lemmas 5.1--5.2]{FoxSudakov2011}. The point is to use the same walks
+for both additive and multiplicative edge labels. We can therefore
+obtain both bounds on one subset, which is what the subsequent
+subfield argument requires. In the geometric application, the edge
+labels will come from the four pencils through four chosen points.
+
+\begin{lemma}[Simultaneous difference and quotient extraction]
+\label{inc-simultaneous-extraction}
+Let $F$ be a field, let $A,B\subseteq F^*$ be finite, and let
+$G\subseteq A\times B$. Define
+\[
+ D=\{a-b:(a,b)\in G\},\qquad
+ J=\{a/b:(a,b)\in G\}.
+\]
+Suppose that, for real numbers $n>0$ and $V\ge2$,
+\begin{equation}\label{eq:inc-graph-hypotheses}
+ |A|,|B|,|D|,|J|\le Vn,
+ \qquad |G|\ge n^2/V.
+\end{equation}
+Then there is $S\subseteq A$ such that
+\begin{equation}\label{eq:inc-extracted-size}
+ \frac{3n}{8V^2}\le |S|\le Vn
+\end{equation}
+and
+\begin{equation}\label{eq:inc-extracted-expansion}
+ |S-S|,|S/S|\le 2^{20}V^{22}n.
+\end{equation}
+Here $S/S=\{a/a':a,a'\in S\}$.
+\end{lemma}
+
+\begin{proof}
+We first find a large set of vertices such that every ordered pair
+is joined by many walks of length four. Counting the possible edge
+labels of these walks will then bound both differences and quotients
+of the endpoints.
+
+Regard $G$ as a bipartite graph, and put
+$\tau=(512V^8)^{-1}$. Call an ordered pair $(a,a_2)\in A^2$ bad if
+it has fewer than $\tau n$ common neighbors in $B$. To find a
+neighborhood with few bad pairs, first note that vertices of $B$
+with fewer than $n/(2V^2)$ neighbors account for at most
+$|B|n/(2V^2)\le n^2/(2V)$ edges. Since every vertex of $B$ has at
+most $Vn$ neighbors, the set of vertices with at least $n/(2V^2)$
+neighbors has size at least $n/(2V^2)$.
+
+Summed over all $b\in B$, the number of bad ordered pairs in the
+neighborhood of $b$ is at most
+\[
+ |A|^2\tau n\le V^2\tau n^3.
+\]
+Averaging over the large neighborhoods just identified, one obtains
+a neighborhood $A_0$ of size at least $n/(2V^2)$ in which the number
+of bad ordered pairs is at most
+\[
+ 2V^4\tau n^2=\frac{n^2}{256V^4}
+ \le \frac{|A_0|^2}{64}
+ \le \frac{|A_0|^2}{16}.
+\]
+Let $S$ consist of those $a\in A_0$ having at most $|A_0|/4$ bad
+partners in $A_0$. At most $|A_0|/4$ elements are discarded, so
+$|S|\ge3|A_0|/4$, proving~\eqref{eq:inc-extracted-size}.
+
+The choice of $S$ ensures that, for every $a,a'\in S$, at least
+$|A_0|/2$ elements $a_2\in A_0$ form nonbad pairs with both endpoints.
+Each such $a_2$ gives at least $(\tau n)^2$
+ordered walks
+\[
+ a,b_1,a_2,b_2,a'
+\]
+whose four edges belong to $G$. Repeated vertices are allowed.
+Consequently each ordered endpoint pair in $S^2$ has at least
+\begin{equation}\label{eq:inc-walk-count}
+ \frac{|A_0|}{2}(\tau n)^2
+ \ge\frac{n^3}{2^{20}V^{18}}
+\end{equation}
+such walks. This lower bound now converts the limited number of edge
+labels into a bound on endpoint differences.
+
+For one of these walks, write its edge differences, always taking
+the $A$ coordinate minus the $B$ coordinate, as
+\[
+ d_1=a-b_1,\quad d_2=a_2-b_1,\quad
+ d_3=a_2-b_2,\quad d_4=a'-b_2.
+\]
+They satisfy $a-a'=d_1-d_2+d_3-d_4$. For a fixed endpoint pair,
+the four differences determine the walk uniquely: starting from $a$,
+one recovers successively $b_1,a_2,b_2,a'$. Choose one ordered
+endpoint pair for each element of $S-S$. The resulting collections
+of difference quadruples are disjoint for different differences,
+because their alternating sums differ. Counting in $D^4$ and using
+\eqref{eq:inc-walk-count} gives
+\[
+ |S-S|\le
+ \frac{|D|^4}{n^3/(2^{20}V^{18})}
+ \le 2^{20}V^{22}n.
+\]
+
+The same walks give the quotient bound on the same set $S$. Put
+\[
+ j_1=a/b_1,\quad j_2=a_2/b_1,\quad
+ j_3=a_2/b_2,\quad j_4=a'/b_2.
+\]
+All are nonzero, and $a/a'=j_1j_3/(j_2j_4)$. From a fixed initial
+$a$ and these four quotients one recovers
+$b_1=a/j_1$, $a_2=j_2b_1$, $b_2=a_2/j_3$, and $a'=j_4b_2$.
+Choosing one endpoint pair for each element of $S/S$ and counting
+in $J^4$ therefore proves the other bound in
+\eqref{eq:inc-extracted-expansion}.
+\end{proof}
+
+\subsection{Bounded scalar expressions and subfields}
+
+We now turn to the algebraic obstruction: a set near the square-root
+field scale cannot have both the small difference set and the small
+quotient set supplied by the extraction lemma. The first step is to
+bound sums involving scalars formed using a bounded number of field
+operations. We use the scalar-covering method of Bourgain, Katz, and
+Tao~\cite[Lemma 3.2 and Proposition 3.3]{BKT2004}, keeping the dependence
+on the number of operations explicit.
+
+An expression in a set $E\subseteq F$ is built from members of
+$E\cup\{0,1\}$ using binary addition, subtraction, and multiplication,
+and unary negation and inversion. Each of these operations counts
+as one operation. Inversion is allowed only when its argument is
+nonzero.
+
+\begin{lemma}[Uniform scalar coverings]\label{inc-scalar-coverings}
+Let $B$ be a nonempty finite subset of a field $F$, let $E\subseteq B$,
+and let $K\ge2$. Suppose that
+\begin{equation}\label{eq:inc-scalar-hypotheses}
+ |B-B|\le K|B|,
+ \qquad |B-eB|\le 2K^3|B|\quad(e\in E).
+\end{equation}
+Define
+\begin{equation}\label{eq:inc-scalar-exponents}
+ C_0=4,\qquad C_{m+1}=3C_m+5,
+ \qquad\text{so that }C_m=\frac{13\cdot3^m-5}{2}.
+\end{equation}
+If $\lambda\in F$ has an expression in $E$ using at most $m$
+operations, then $\lambda B$ is contained in at most $K^{C_m}$
+translates of $B-B$. In particular,
+\begin{equation}\label{eq:inc-scalar-sum-bound}
+ |E+\lambda E|\le |B+\lambda B|
+ \le K^{C_m+3}|B|.
+\end{equation}
+These bounds are uniform over all such expressions and all choices
+of their entries from $E$.
+\end{lemma}
+
+\begin{proof}
+Put $H=B-B$, so $H=-H$ and $0\in H$. In the remainder of this proof,
+integer multiples of $B$ or $H$ inside sumsets denote iterated sums.
+We prove the covering assertion by induction on expression length.
+To combine coverings at an induction step, we will need a fixed
+covering of $2H$ by translates of $H$. By
+Lemma~\ref{inc-sumset-estimates},
+\[
+ |2H-B|=|2B-3B|\le K^5|B|.
+\]
+The covering statement of that lemma consequently supplies a set
+$T_*\subseteq F$ with $|T_*|\le K^5$ such that
+\begin{equation}\label{eq:inc-two-differences-cover}
+ 2H\subseteq T_*+H.
+\end{equation}
+
+Every $e\in E$ has a covering $eB\subseteq T_e+H$ with
+$|T_e|\le2K^3\le K^4$, by
+\eqref{eq:inc-scalar-hypotheses} and the covering lemma.
+The scalars $0$ and $1$ also have coverings by one translate of $H$:
+use $\{0\}\subseteq H$ and $B\subseteq b_0+H$ for any $b_0\in B$.
+This proves the case $m=0$.
+
+We now check how the number of translates changes under each allowed
+operation. Suppose that
+\[
+ \lambda B\subseteq T+H,\qquad
+ \mu B\subseteq U+H,\qquad
+ |T|\le K^a,\quad |U|\le K^b.
+\]
+Negation replaces $T$ by $-T$. For addition and subtraction,
+\[
+ (\lambda\pm\mu)B
+ \subseteq \lambda B\pm\mu B
+ \subseteq T\pm U+2H
+ \subseteq T\pm U+T_*+H,
+\]
+so the number of translates is at most $K^{a+b+5}$. For a product,
+\[
+ \lambda\mu B
+ \subseteq \lambda U+\lambda H
+ =\lambda U+(\lambda B-\lambda B)
+ \subseteq\lambda U+T-T+T_*+H.
+\]
+This uses at most $K^{b+2a+5}$ translates and is valid even if
+$\lambda=0$. Finally, if $\lambda\ne0$, then
+\[
+ |\lambda B-B|
+ \le |T|\,|H-B|
+ =|T|\,|B-2B|
+ \le K^{a+3}|B|.
+\]
+Scaling by $\lambda^{-1}$ and then negating gives
+$|\lambda^{-1}B-B|\le K^{a+3}|B|$. The covering lemma now covers
+$\lambda^{-1}B$ by at most $K^{a+3}$ translates of $H$.
+
+These bounds give the claimed uniform induction: in an expression
+with at most $m+1$ operations, each immediate subexpression has at
+most $m$ operations, so every resulting covering uses at most
+$K^{3C_m+5}$ translates. This proves the covering assertion
+with~\eqref{eq:inc-scalar-exponents}.
+
+To deduce the sumset bound from the covering, let
+$\lambda B\subseteq T+H$ be the covering just obtained. Then
+\[
+ |B+\lambda B|
+ \le |T|\,|B+H|
+ =|T|\,|2B-B|
+ \le K^{C_m+3}|B|,
+\]
+again by Lemma~\ref{inc-sumset-estimates}. Since $E\subseteq B$,
+this also bounds $|E+\lambda E|$.
+\end{proof}
+
+When working along a sequence $q\to\infty$, a positive quantity
+$K=q^{o(1)}$ means that $\log K/\log q\to0$. Likewise,
+$|S|=q^{1/2+o(1)}$ means that $\log|S|/\log q\to1/2$.
+In particular, every fixed power of a subpower factor is still a
+subpower factor. This is why a uniform bound for expressions of
+fixed length suffices for the next lemma.
+
+\begin{lemma}[Exclusion of an intermediate approximate subfield]
+\label{inc-no-approximate-subfield}
+There is no sequence of prime powers $q\to\infty$, each satisfying
+the proper-subfield hypothesis of Theorem~\ref{thm:incidence}, together
+with sets $S\subseteq\F_q^*$ and numbers $K\ge2$ such that
+\begin{equation}\label{eq:inc-approximate-subfield-hypotheses}
+ |S|=q^{1/2+o(1)},\qquad K=q^{o(1)},\qquad
+ |S-S|,|S/S|\le K|S|.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Suppose that such a sequence exists. We first use multiplicative
+overlaps to obtain sets satisfying the scalar-covering hypotheses.
+Ratios of differences in a large subset will then form a subfield.
+The proper-subfield hypothesis will force that subfield to be the
+whole field, where averaging will contradict the scalar-covering
+bound.
+
+\paragraph{Extracting scalar-covering hypotheses.}
+Put $s=|S|$. The $s$ sets
+$S/a$, $a\in S$, each have size $s$, and their union is contained
+in $S/S$. Cauchy--Schwarz gives
+\begin{equation}\label{eq:inc-multiplicative-overlaps}
+ \sum_{a,b\in S}|S/a\cap S/b|
+ \ge\frac{s^4}{|S/S|}\ge\frac{s^3}{K}.
+\end{equation}
+Thus some $b\in S$ satisfies
+$\sum_{a\in S}|S/a\cap S/b|\ge s^2/K$.
+Since every summand is at most $s$, at least $s/(2K)$ values
+$a\in S$ have $|S/a\cap S/b|\ge s/(2K)$: otherwise the sum of
+the large summands would be less than $s^2/(2K)$ and that of all
+the other summands at most $s^2/(2K)$.
+
+Let $E$ be the set of $e=a/b$ for these values of $a$, and put
+$B=S/b$. Then
+\begin{equation}\label{eq:inc-overlap-subset}
+ E\subseteq B,\qquad |B|=s,\qquad
+ |E|\ge\frac{s}{2K},\qquad
+ |S\cap eS|\ge\frac{s}{2K}\quad(e\in E).
+\end{equation}
+The last inequality follows by multiplying $S/a\cap S/b$ by $a$.
+For $Z=S\cap eS$, the triangle inequality and the containments
+$Z\subseteq S,eS$ imply
+\[
+ |S-eS|
+ \le\frac{|S-Z|\,|Z-eS|}{|Z|}
+ \le\frac{(Ks)^2}{s/(2K)}=2K^3s.
+\]
+After scaling by $1/b$, we have
+\[
+ |B-B|\le Ks,\qquad |B-eB|\le2K^3s\quad(e\in E).
+\]
+Thus Lemma~\ref{inc-scalar-coverings} applies to $B$ and $E$.
+We have obtained uniform control for every scalar expressible using
+a bounded number of operations on elements of $E$.
+
+\paragraph{Ratios of differences form a subfield.}
+Write $t=|E|$. By~\eqref{eq:inc-overlap-subset} and
+\eqref{eq:inc-approximate-subfield-hypotheses},
+$t=q^{1/2+o(1)}$. In particular, $t\ge2$ for all sufficiently
+large fields in the sequence. Define
+\begin{equation}\label{eq:inc-ratio-set}
+ R_0=\left\{\frac{x-x'}{y-y'}:
+ x,x',y,y'\in E,\ y\ne y'\right\}.
+\end{equation}
+To prove closure under addition and multiplication, we apply the
+scalar-covering bound to the result of either operation.
+Every member of $R_0$ has an expression in $E$ with at most four
+operations. Consequently, for $r,r'\in R_0$, each of $r+r'$ and
+$rr'$ has an expression with at most nine operations. For either
+candidate $\lambda$, Lemma~\ref{inc-scalar-coverings} gives
+\[
+ |E+\lambda E|\le K^{C_9+3}s<t^2
+\]
+for all sufficiently large $q$, uniformly over $r,r'$.
+Indeed, the ratio of this upper bound to $t^2$ is at most
+$4K^{C_9+5}/s$, which tends to zero. Therefore the map
+$(x,y)\mapsto x+\lambda y$ from $E^2$ is not injective. A collision
+between distinct pairs satisfies $y\ne y'$, since otherwise also
+$x=x'$, and gives
+\[
+ \lambda=\frac{x'-x}{y-y'}\in R_0.
+\]
+It follows that $R_0$ is closed under addition and multiplication.
+It contains $0$ and $1$ and, directly from its definition, is closed
+under negation and reciprocals of nonzero elements. Thus $R_0$ is a
+subfield of $\F_q$.
+
+The size of $E$ now prevents this subfield from being proper.
+Choose distinct $e_0,e_1\in E$. The map
+\[
+ e\longmapsto\frac{e-e_0}{e_1-e_0}
+\]
+injects $E$ into $R_0$, so $|R_0|\ge t>q^{1/3}$ for large $q$.
+The proper-subfield hypothesis forces $R_0=\F_q$.
+
+\paragraph{A scalar with a large sumset.}
+Every field element consequently has a four-operation representation
+using elements of $E$. The scalar-covering estimate therefore bounds
+$|E+\lambda E|$ for every scalar. To reach a contradiction, we average
+over the field and find a scalar for which this sumset is too large.
+
+For $\lambda\in\F_q$, let $N_\lambda$ be the number of ordered
+quadruples $(x,y,x',y')\in E^4$ satisfying
+$x+\lambda y=x'+\lambda y'$. There are $t^2$ identical pairs,
+which give a collision for every $\lambda$. If $y\ne y'$, exactly
+one value of $\lambda$ gives a collision; if $y=y'$ but $x\ne x'$,
+none does. Hence
+\begin{equation}\label{eq:inc-average-collisions}
+ \frac1q\sum_{\lambda\in\F_q}N_\lambda
+ =t^2+\frac{t^4-t^3}{q}
+ \le t^2+\frac{t^4}{q}.
+\end{equation}
+For some $\lambda$, Cauchy--Schwarz applied to the fibers of
+$(x,y)\mapsto x+\lambda y$ therefore yields
+\begin{equation}\label{eq:inc-large-dilate-sum}
+ |E+\lambda E|
+ \ge\frac{t^4}{t^2+t^4/q}
+ \ge\frac12\min(t^2,q)
+ =q^{1-o(1)}.
+\end{equation}
+But $R_0=\F_q$, so this particular $\lambda$ itself has a
+four-operation representation in~\eqref{eq:inc-ratio-set}.
+Lemma~\ref{inc-scalar-coverings} bounds its sumset by
+$K^{C_4+3}s=q^{1/2+o(1)}$, contradicting
+\eqref{eq:inc-large-dilate-sum}. The expression length here remains
+bounded; no iteration of field generation is being used.
+\end{proof}
+
+\Needspace{10\baselineskip}
+\subsection{Proof of the incidence theorem}
+
+The algebraic ingredients are now in place. It remains to obtain the
+hypotheses of Lemma~\ref{inc-simultaneous-extraction} from a configuration
+with too many incidences. We use the projective-coordinate reduction
+in the incidence strategy of Bourgain, Katz, and
+Tao~\cite[Section 6]{BKT2004}: after controlling degrees, we choose four
+points whose pencils encode the two coordinates, their difference,
+and their quotient.
+
+\begin{proof}[Proof of Theorem~\ref{thm:incidence}]
+We argue by contradiction, allowing all quantitative losses to be
+subpower factors along a sequence of fields. Suppose the theorem is
+false. For a sequence of positive exponents
+$\kappa_j\to0$ in $(0,1/10)$, choose admissible counterexamples with
+$q_j\to\infty$. Such choices are possible because the failure of
+the theorem says that for each fixed exponent there are arbitrarily
+large counterexamples. Along this sequence, put
+$U=\max(2,q_j^{\kappa_j})$ and suppress the index $j$. We have
+\begin{equation}\label{eq:inc-counterexample-sequence}
+ |P|,|\mathcal L|\le Uq,\qquad
+ I(P,\mathcal L)\ge\frac{q^{3/2}}{U},\qquad
+ U\ge2,\qquad U=q^{o(1)}.
+\end{equation}
+All bounds below use absolute constants. Every assertion made for
+sufficiently large $q$ is along this sequence. Our goal is to construct
+from these counterexamples sets of the size and small-expansion form
+excluded by Lemma~\ref{inc-no-approximate-subfield}.
+
+\paragraph{Pruning the incidence graph.}
+We first retain many incidences while putting both an upper and a
+lower bound on every remaining degree. Consider the bipartite graph
+on $P$ and $\mathcal L$ whose edges are incidences. If $d(v)$ denotes a degree on either side, then counting
+ordered pairs on the opposite side and using the unique join of two
+distinct projective points, or the unique intersection of two
+distinct projective lines, gives
+\begin{equation}\label{eq:inc-degree-squares}
+ \sum_v d(v)^2\le2U^2q^2
+ \quad\text{on each side}.
+\end{equation}
+Here $\sum_vd(v)(d(v)-1)\le U^2q^2$, and
+$\sum_vd(v)=I(P,\mathcal L)\le U^2q^2$.
+
+Set
+\begin{equation}\label{eq:inc-degree-cutoffs}
+ H=16U^3q^{1/2},\qquad d=\frac{q^{1/2}}{8U^2}.
+\end{equation}
+Delete every vertex whose initial degree exceeds $H$. By
+\eqref{eq:inc-degree-squares}, the total number of incidences lost,
+counting both sides, is at most
+\[
+ \frac{4U^2q^2}{H}=\frac{q^{3/2}}{4U}.
+\]
+Next, iteratively delete vertices whose current degree is less than
+$d$. Each of the at most $2Uq$ original vertices is deleted at most
+once, so this stage loses at most
+$2Uq\,d=q^{3/2}/(4U)$ incidences. Denote the surviving sets again by
+$P,\mathcal L$. They satisfy
+\begin{equation}\label{eq:inc-pruned-bounds}
+ I(P,\mathcal L)\ge\frac{q^{3/2}}{2U},\qquad
+ \frac{q}{32U^4}\le |P|,|\mathcal L|\le Uq,
+ \qquad d\le d(v)\le H.
+\end{equation}
+The size lower bounds follow by dividing the surviving incidence
+count by the maximum degree $H$.
+
+The pruning also controls the intersection of $P$ with
+\emph{every} projective line, including lines not in $\mathcal L$.
+We will need this stronger form of control when discarding points
+on a few lines chosen later. Fix a projective line $\ell$. Each point
+of $P\cap\ell$ is incident with at least $d-1$ lines of $\mathcal L$ different from $\ell$.
+These lines are distinct for different points of $P\cap\ell$.
+Since $d\to\infty$, for large $q$ we obtain
+\begin{equation}\label{eq:inc-all-line-cap}
+ |P\cap\ell|
+ \le\frac{|\mathcal L|}{d-1}
+ \le16U^3q^{1/2}=H
+ \qquad\text{for every projective line }\ell.
+\end{equation}
+
+\paragraph{Four anchors with many common witnesses.}
+We seek four points, three collinear and the fourth off their line,
+together with many points joined to each of them by lines of
+$\mathcal L$. We obtain the four points by first counting such
+configurations around each possible common point and then averaging.
+
+For $x\in P$, let $N(x)$ consist of the points in $P\setminus\{x\}$
+joined to $x$ by a line of $\mathcal L$. Distinct lines through $x$
+have disjoint point sets away from $x$, so
+\begin{equation}\label{eq:inc-neighborhood-size}
+ |N(x)|\ge d(d-1)\ge\frac{q}{128U^4}
+\end{equation}
+for large $q$. Put $t_\ell=|N(x)\cap\ell|$. Summing degrees of the
+points of $N(x)$ gives
+\[
+ \sum_{\ell\in\mathcal L}t_\ell
+ \ge d|N(x)|\ge\frac{q^{3/2}}{1024U^6}.
+\]
+Lines with $t_\ell\le2$ contribute at most $2Uq$. Because $U=q^{o(1)}$,
+for sufficiently large $q$ this is at most half of the displayed
+lower bound. Consequently
+\[
+ \sum_{\ell:t_\ell\ge3}t_\ell
+ \ge\frac{q^{3/2}}{2048U^6}.
+\]
+For every integer $t\ge3$, one has
+$t(t-1)(t-2)\ge(2/9)t^3$. H\"older's inequality therefore shows
+that the number of ordered triples of distinct points of $N(x)$
+lying on a line of $\mathcal L$ is at least
+\begin{equation}\label{eq:inc-collinear-triples}
+ \begin{split}
+ \sum_{\ell\in\mathcal L}t_\ell(t_\ell-1)(t_\ell-2)
+ &\ge\frac{2}{9|\mathcal L|^2}
+       \left(\sum_{\ell:t_\ell\ge3}t_\ell\right)^3\\
+ &\ge\frac{q^{5/2}}{9\cdot2^{32}U^{20}}.
+ \end{split}
+\end{equation}
+By~\eqref{eq:inc-all-line-cap} and
+\eqref{eq:inc-neighborhood-size}, each such triple admits at least
+\[
+ \frac{q}{128U^4}-H\ge\frac{q}{256U^4}
+\]
+choices of a fourth point in $N(x)$ outside its line, once $q$ is
+large enough. Summing over $x$ and using
+\eqref{eq:inc-pruned-bounds}, we obtain at least
+\begin{equation}\label{eq:inc-witness-quadruple-count}
+ \frac{q^{9/2}}{9\cdot2^{45}U^{28}}
+\end{equation}
+ordered choices $(x,p_1,p_2,p_3,p_4)$ with $p_1,p_2,p_3$ distinct
+and collinear on a line of $\mathcal L$, with $p_4$ outside that
+line, and with all four $p_i$ in $N(x)$.
+
+We now reverse the count to fix the four anchor points while retaining
+many choices of the common point $x$. The number of possible ordered
+quadruples $(p_1,p_2,p_3,p_4)$ in this count is at most
+\begin{equation}\label{eq:inc-quadruple-upper-bound}
+ |\mathcal L|H^3|P|
+ \le2^{12}U^{11}q^{7/2}.
+\end{equation}
+Some quadruple thus has at least
+\begin{equation}\label{eq:inc-common-witnesses}
+ c_0\frac{q}{U^{39}}
+ \quad\text{witnesses }x,\qquad
+ c_0=\frac1{9\cdot2^{57}}.
+\end{equation}
+Discard the witnesses on the common line of $p_1,p_2,p_3$ and on
+the joins of $p_1,p_4$ and of $p_2,p_4$. The cap
+\eqref{eq:inc-all-line-cap} discards at most $3H$ witnesses.
+Since $U=q^{o(1)}$, this is eventually less than
+$c_0q/(2U^{39})$. At least that many witnesses remain. Their joins
+to the four anchors will supply the graph and the four small sets
+required by Lemma~\ref{inc-simultaneous-extraction}.
+
+\paragraph{The four pencils as coordinates.}
+Choose projective coordinates in which
+\[
+ p_1=(1:0:0),\qquad p_2=(0:1:0),\qquad
+ p_3=(1:1:0),\qquad p_4=(0:0:1).
+\]
+This is possible over $\F_q$: representatives of $p_1,p_2$ span
+their line, the representative of the distinct third point has
+both coefficients nonzero and permits rescaling of the first two,
+and a representative of $p_4$ completes a basis. The discarded
+lines are precisely the coordinate line at infinity and the two
+coordinate axes through $p_4$. Hence every remaining witness has
+coordinates $(a:b:1)$ with $a,b\ne0$.
+
+Let $G$ be the set of these pairs $(a,b)$, and let $A,B$ be its
+coordinate projections. Define $D$ and $J$ from $G$ as in
+Lemma~\ref{inc-simultaneous-extraction}. The chosen coordinates make
+the roles of the four pencils explicit: for a witness $(a:b:1)$,
+the joins to $p_2,p_1,p_3,p_4$, respectively, have equations
+\[
+ X=aZ,\qquad Y=bZ,\qquad
+ X-Y=(a-b)Z,\qquad X=(a/b)Y.
+\]
+Each join is a member of $\mathcal L$, by the witness property.
+Each anchor belongs to $P$ and has degree at most $H$.
+Distinct values of the corresponding parameter give distinct
+lines in its pencil. Thus
+\[
+ |A|,|B|,|D|,|J|\le H,
+ \qquad |G|\ge\frac{c_0q}{2U^{39}}.
+\]
+With
+\begin{equation}\label{eq:inc-graph-parameters}
+ n=q^{1/2},\qquad V=\frac{2}{c_0}U^{39},
+\end{equation}
+we have $V\ge2$, $V=q^{o(1)}$, and all the hypotheses
+\eqref{eq:inc-graph-hypotheses} hold.
+
+The incidence configuration now meets the extraction hypotheses.
+Lemma~\ref{inc-simultaneous-extraction} gives
+$S\subseteq\F_q^*$ satisfying
+\[
+ \frac{3q^{1/2}}{8V^2}\le |S|\le Vq^{1/2},\qquad
+ |S-S|,|S/S|\le2^{20}V^{22}q^{1/2}.
+\]
+In particular $|S|=q^{1/2+o(1)}$, and, on setting
+$K=2^{23}V^{24}$, we have $K\ge2$, $K=q^{o(1)}$, and
+$|S-S|,|S/S|\le K|S|$. This contradicts
+Lemma~\ref{inc-no-approximate-subfield}.
+
+We have excluded every sequence of counterexamples of the form
+\eqref{eq:inc-counterexample-sequence}. The negation of the theorem
+would have produced such a sequence, so some absolute
+$\kappa\in(0,1/10)$ and $Q$ satisfy the stated conclusion.
+\end{proof}
diff --git a/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/build/linecover.tex b/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/build/linecover.tex
new file mode 100644
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+++ b/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/build/linecover.tex
@@ -0,0 +1,447 @@
+\section{Line covers after random deletions}\label{sec:linecover}
+
+We regard the affine plane $\F_q^2$ as a subset of its projective
+completion $\PP^2(\F_q)$. Thus a pencil with centre at infinity is a
+parallel class of affine lines. The purpose of this section is to show
+that a set of at most $q$ lines covering almost all survivors of a
+suitable random deletion process must contain almost a full pencil.
+The deletions within a single trial may be dependent; independence is
+required only between trials.
+
+Classical covering results for finite planes are closely related by
+duality to blocking sets. Jamison and Brouwer--Schrijver established the
+affine blocking-set lower bound $2q-1$~\cite{Jamison1977,BrouwerSchrijver1978},
+and Blokhuis, Brouwer, and Sz\H{o}nyi studied partial projective covers
+and bounds depending on their uncovered points
+\cite[Section 1.1, Proposition 1.5]{BBS2010}. Here the covered set instead consists of the
+positive-density survivors of structured random deletions. We prove the
+pencil statement needed in this setting directly.
+
+There are two different concentration tasks. The total number of
+surviving points and the number on a fixed line have bounded changes
+under one trial; for these we use the elementary estimates
+in~\cite[Lemmas~A.2--A.3]{OpenAI2026Balanced}. The more delicate task
+is to retain many intersections of a short list of lines, simultaneously
+for every such list. A deletion trial rarely hits this small set, and
+we must use that rarity in the variance estimate. We prove the needed
+Bernstein-type martingale bound here, following the method of
+Freedman~\cite[Theorem (1.6) and Proposition (2.1)]{Freedman1975}.
+
+\begin{lemma}[Martingale concentration]\label{lem:martingale}
+Let $Z_i$, $1\leq i\leq n$, be martingale differences for a filtration
+$(\mathcal A_i)_{i=0}^n$. Suppose that $M,V>0$ and deterministic numbers
+$v_i\geq0$ satisfy, almost surely,
+\[
+ |Z_i|\leq M,\qquad
+ \E(Z_i^2\mid\mathcal A_{i-1})\leq v_i,\qquad
+ \sum_{i=1}^n v_i\leq V.
+\]
+Then, for every $u>0$,
+\begin{equation}\label{eq:lc-martingale}
+ \Pr\left(\left|\sum_{i=1}^n Z_i\right|\geq u\right)
+ \leq 2\exp\left[-\frac14
+   \min\left\{\frac{u^2}{V},\frac{u}{M}\right\}\right].
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+For $|z|\leq1$ we have $e^z\leq1+z+z^2$. Consequently, if
+$|\theta|\leq M^{-1}$, the conditional mean-zero property gives
+\[
+ \E(e^{\theta Z_i}\mid\mathcal A_{i-1})
+ \leq 1+\theta^2v_i\leq e^{\theta^2v_i}.
+\]
+Iteration and Markov's inequality bound the upper tail by
+$\exp(\theta^2V-\theta u)$ for $0<\theta\leq M^{-1}$.
+Take $\theta=\min\{u/(2V),1/M\}$. If $u\leq2V/M$, the exponent is
+$-u^2/(4V)$; otherwise it is at most $-u/(2M)$.
+Applying the same argument to $-Z_i$ proves
+\eqref{eq:lc-martingale}.
+
+\end{proof}
+
+\begin{lemma}[Line covers after random deletions]\label{lem:linecover}
+Let $q\to\infty$ through prime powers for which every proper subfield
+of $\F_q$ has order at most $q^{1/3}$. Fix a positive integer $D$ and
+constants $\delta>0$ and $0<\rho<1$. For each $q$, let $\mathcal F$ be a
+family of at most $Dq^D$ subsets of $\F_q^2$, called shapes, such that
+\begin{enumerate}
+ \item every $F\in\mathcal F$ has at most $Dq$ points;
+ \item for every $F\in\mathcal F$ and affine line $\ell$, either
+       $F=\ell$ or $|F\cap\ell|\leq D$.
+\end{enumerate}
+There are at most $Dq$ independent trials, each producing a random list
+of at most $D$ shapes from $\mathcal F$. Delete every point on any listed
+shape, and denote the surviving set by $X$. Assume that
+\begin{enumerate}
+ \item for each point $x$ and each trial, the probability that this
+       trial deletes $x$ is at most $D/q$;
+ \item $\Pr(x\in X)=\delta_q\geq\delta$ for every $x\in\F_q^2$;
+ \item for each affine line $\ell$, the sum over trials of the
+       probabilities that the trial lists the exact shape $\ell$ is
+       at most $D/q$.
+\end{enumerate}
+Write $X^\ell$ for the surviving set when all exact occurrences of the
+shape $\ell$ are ignored, with every other listed shape still deleted.
+With probability tending to one, both of the following statements
+hold:
+\begin{enumerate}
+ \item $|X^\ell\cap\ell|\geq\delta q/2$ for every affine line $\ell$;
+ \item every set of at most $q$ affine lines covering all but at most
+       $2q$ points of $X$ contains at least $(1-\rho)q$ lines through a
+       common projective point.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+We first establish a simultaneous lower bound on surviving
+cross-intersections for every admissible pair of short line lists,
+ignoring any deletion shape equal to one of those lines. After fixing
+such a realization, we show that any cover lacking an almost full
+pencil contains an admissible pair with too few surviving
+cross-intersections. This second step samples only within the fixed
+cover, with separate arguments according to its maximum concurrency.
+
+Fix the absolute constant $\kappa\in(0,1/10)$ of
+Theorem~\ref{thm:incidence}, and put $\epsilon=\kappa/10$.
+All constants below may depend on the fixed parameters, but not on
+$q$, the particular deletion distributions, or a subsequently chosen
+cover.
+
+\paragraph{Uniform point and line counts.}
+Expose the independent trials one at a time. Changing one trial
+changes $|X|$ by at most $2D^2q$, because the old and new lists contain
+at most $2D$ shapes in total and each has at most $Dq$ points.
+The exposure observation in~\cite[Lemma~A.3]{OpenAI2026Balanced}
+therefore bounds the absolute martingale increment by $L=2D^2q$.
+There are $T\le Dq$ trials, and $\E|X|=\delta_q q^2$. The fixed-time
+concentration estimate in~\cite[Lemma~A.2]{OpenAI2026Balanced}, with
+$u=q^{3/2+\epsilon}$, gives
+\[
+ \Pr\bigl(\bigl||X|-\delta_q q^2\bigr|\ge q^{3/2+\epsilon}\bigr)
+ \le2\exp\left(-\frac{q^{2\epsilon}}{32D^5}\right).
+\]
+If there are no trials, the count is deterministic and this bound
+holds as well. Equivalently, we may add a dummy trial of zero change.
+
+For the line count $|X^\ell\cap\ell|$, the rule ignoring exact copies
+of $\ell$ is applied separately to each trial. Changing one trial
+then changes the count by at most $2D^2$: each of its at most $2D$
+old or new shapes meets $\ell$ in at most $D$ points. With
+$L=2D^2$ and $u=q^{1/2+\epsilon}$, the same two elementary estimates
+and $T\le Dq$ give
+\[
+ \Pr\bigl(\bigl||X^\ell\cap\ell|-\E|X^\ell\cap\ell|\bigr|
+              \ge q^{1/2+\epsilon}\bigr)
+ \le2\exp\left(-\frac{q^{2\epsilon}}{32D^5}\right).
+\]
+
+Moreover, a point restored by ignoring $\ell$ can be restored only if
+some trial lists $\ell$ exactly. The probability of this event is at
+most $D/q$, by the third deletion hypothesis. Therefore
+\[
+ \delta_q q\leq \E|X^\ell\cap\ell|\leq\delta_q q+D.
+\]
+There are $q(q+1)$ affine lines. The superpolynomial tails and a union
+bound consequently show that, with probability tending to one,
+\begin{equation}\label{eq:lc-counts}
+ \begin{split}
+ |X|&=\delta_q q^2+O_D(q^{3/2+\epsilon}),\\
+ |X^\ell\cap\ell|&=\delta_q q+O_D(q^{1/2+\epsilon})
+       \qquad\text{for every affine line }\ell.
+ \end{split}
+\end{equation}
+The lower line estimate proves the first conclusion of the lemma.
+For the pencil conclusion we will also need to bound overlaps in an
+alleged cover. The same estimates give the uniform upper bound
+\begin{equation}\label{eq:lc-line-upper}
+ |X\cap\ell|\leq\delta_q q+O_D(q^{1/2+\epsilon}),
+\end{equation}
+because $X\subseteq X^\ell$.
+
+\paragraph{A simultaneous event for small line samples.}
+The counts just proved control a single line but do not yet restrict
+how a cover's lines meet. We now obtain a stronger test on short lists
+of lines: provided their intersections are distinct and not concentrated
+on a deletion shape, many of those intersections survive. The list
+length will be logarithmic in $q$, so that the variance-sensitive tail
+can pay for a union bound over all lists.
+
+Set
+\[
+ h=10(D+1),\qquad H_0=2Dh,\qquad
+ t=\lceil A\log q\rceil,
+\]
+where the fixed constant $A$ will be chosen below. Consider any ordered
+tuple of $2t$ distinct affine lines, separated into two lists of length
+$t$. Call the tuple admissible if its cross-list intersections form a
+set $P_0$ of $t^2$ distinct finite points, and
+\begin{equation}\label{eq:lc-occupancy}
+ |F\cap P_0|\leq H_0
+ \quad\text{for every }F\in\mathcal F
+       \text{ which is not one of the tuple lines}.
+\end{equation}
+For this tuple, ignore deletions by every shape equal to a tuple line,
+and let $Y_0$ count the survivors in $P_0$ after the remaining deletions.
+We claim that, with probability tending to one,
+\begin{equation}\label{eq:lc-tuple-event}
+ Y_0\geq\frac{\delta t^2}{2}
+ \qquad\text{for every admissible tuple}.
+\end{equation}
+
+To prove the claim, fix a tuple in advance. Omitting shapes is a
+deterministic operation on each trial, so the trials remain
+independent. Omission can only restore points, and hence
+$\E Y_0\geq\delta t^2$. By \eqref{eq:lc-occupancy}, each trial's
+contribution to the deletion set within $P_0$ has size at most
+$C_0=DH_0$. The probability that this contribution is nonempty is at
+most $Dt^2/q$, by a union bound over $P_0$ and the per-point deletion
+hypothesis.
+
+For completeness, the rarity of a nonempty contribution also controls
+the conditional variance of the exposure martingale. Fix the past,
+and let $g_i(w)$ be the conditional expected final value of $Y_0$ when
+the current trial has outcome $w$. Let $g_i(\varnothing)$ denote the
+hypothetical value obtained by making this trial delete nothing on
+$P_0$. Independence leaves the law of the future trials unchanged, so
+\[
+ |g_i(w)-g_i(\varnothing)|\leq C_0,
+\]
+with equality to zero whenever the trial's contribution is empty.
+The current trial also retains its original law conditional on the
+past. Thus the martingale increment has absolute value at most $2C_0$, and
+its conditional variance is at most
+\[
+ \E\bigl((g_i(w)-g_i(\varnothing))^2\mid\text{past}\bigr)
+ \leq C_0^2\frac{Dt^2}{q}.
+\]
+The sum of conditional variances is at most $D^2C_0^2t^2$, since there
+are at most $Dq$ trials. Applying Lemma~\ref{lem:martingale} at deviation
+$\delta t^2/2$ yields
+\begin{equation}\label{eq:lc-tuple-tail}
+ \Pr\left(Y_0<\frac{\delta t^2}{2}\right)
+ \leq2e^{-c_0t^2},\qquad
+ c_0=\min\left\{\frac{\delta^2}{16D^2C_0^2},
+                       \frac{\delta}{16C_0}\right\}>0.
+\end{equation}
+In particular, $c_0$ is independent of $A$.
+
+There are at most $(q(q+1))^{2t}$ ordered tuples. The logarithm of the
+union bound in \eqref{eq:lc-tuple-tail} is
+\[
+ -c_0t^2+4t\log q+O(t/q)+O(1)
+ =\bigl(-c_0A^2+4A+o(1)\bigr)(\log q)^2.
+\]
+Choose, for example, any fixed $A>5/c_0$. This proves
+\eqref{eq:lc-tuple-event} simultaneously for all admissible tuples.
+
+\paragraph{Consequences of an alleged cover.}
+The random part of the argument is complete. We now keep the deletion
+outcome fixed and show deterministically that the simultaneous events
+exclude any cover without an almost full pencil. Random sampling below
+is only a way to find a short witness inside a fixed cover.
+
+Fix a realization satisfying both \eqref{eq:lc-counts} and
+\eqref{eq:lc-tuple-event}. Suppose that the second conclusion of the
+lemma fails. From an offending cover remove every line which actually
+occurred as a listed deletion shape. Such a line contains no point of
+$X$, so the remaining line set $L$ still covers all but at most $2q$
+points of $X$. Put $m=|L|$, and for any projective point $p$ write
+$n_p=|\{\ell\in L:p\in\ell\}|$. Define
+\[
+ E_L=\sum_{x\in X}(n_x-1)_+,
+ \qquad (a)_+=\max\{a,0\}.
+\]
+The count of covered survivors and \eqref{eq:lc-line-upper} give
+\[
+ m\bigl(\delta_q q+O_D(q^{1/2+\epsilon})\bigr)
+ \geq\delta_q q^2-O_D(q^{3/2+\epsilon})-2q.
+\]
+Since $\delta_q\geq\delta$ and $m\leq q$, this implies
+\begin{equation}\label{eq:lc-cover-counts}
+ m=q-O_{D,\delta}(q^{1/2+\epsilon}),\qquad
+ E_L=O_D(q^{3/2+\epsilon}).
+\end{equation}
+For the second estimate, subtract the number of covered survivors
+from $\sum_{\ell\in L}|X\cap\ell|$ and use $m\leq q$.
+Finally, put
+\[
+ R=\max_{p\in\PP^2(\F_q)}n_p.
+\]
+The failure of the pencil conclusion implies $R\leq(1-\rho)q$.
+We will find an admissible tuple of lines from $L$ with fewer than
+$\delta t^2/2$ survivors in its cross-intersection set.
+
+\paragraph{Low concurrency: $R\leq q^{1/2+\kappa/4}$.}
+Let $T=q^{1/2-\kappa/3}$. The number of ordered pairs of distinct lines
+of $L$ meeting at a point $x\in X$ is $\sum_{x\in X}n_x(n_x-1)$.
+Points with $n_x<T$ contribute at most
+\begin{equation}\label{eq:lc-low-light}
+ T E_L=O(q^{2-7\kappa/30}).
+\end{equation}
+The set $P=\{x\in X:n_x\geq T\}$ has size
+\[
+ |P|\leq\frac{E_L}{T-1}
+      =O(q^{1+13\kappa/30})\leq q^{1+\kappa}
+\]
+for sufficiently large $q$. Theorem~\ref{thm:incidence}, applied to
+$P$ and $L$, therefore gives $\sum_{x\in P}n_x\leq q^{3/2-\kappa}$.
+The contribution of these points to the ordered-pair count is at most
+\begin{equation}\label{eq:lc-low-heavy}
+ R\sum_{x\in P}n_x\leq q^{2-3\kappa/4}.
+\end{equation}
+Together, \eqref{eq:lc-low-light} and \eqref{eq:lc-low-heavy} show that
+the fraction of ordered distinct line pairs meeting in $X$ is
+\begin{equation}\label{eq:lc-low-pair-fraction}
+ O\bigl(q^{-7\kappa/30}+q^{-3\kappa/4}\bigr)=o(1).
+\end{equation}
+
+Sample an ordered tuple of $2t$ distinct lines uniformly from $L$.
+The expected number of its cross pairs meeting in $X$ is $t^2$ times
+the fraction in \eqref{eq:lc-low-pair-fraction}. Markov's inequality
+shows that this number is less than $\delta t^2/2$ with probability
+tending to one.
+
+We next check the two admissibility conditions. A given sampled pair
+is parallel with probability at most $R/(m-1)$. Once two distinct
+lines have been sampled, at most $R$ population lines pass through
+their projective intersection, so the probability of concurrency for
+any specified three sampled positions is at most $R/(m-2)$. The
+probability of any parallel pair or concurrent triple in the tuple
+is consequently
+\begin{equation}\label{eq:lc-low-degeneracy}
+ O\left(\frac{(t^2+t^3)R}{m}\right)
+ =O\bigl((\log q)^3q^{-1/2+\kappa/4}\bigr)=o(1).
+\end{equation}
+If neither event occurs, every cross intersection is finite and the
+cross intersections are distinct: equality of two different cross
+intersections would make at least three of the tuple lines concurrent.
+
+To verify \eqref{eq:lc-occupancy}, fix $F\in\mathcal F$. Consider the
+event that $F$ is not a tuple line but contains more than $H_0$ distinct
+finite cross intersections. Each tuple line then contains at most
+$D$ points of $F$. Greedily choose a cross intersection on $F$ and its
+two representing positions, and discard all cross intersections on
+either of those lines. Each step discards at most $2D$ points. Since
+$H_0=2Dh$, this produces $h$ cross pairs whose $2h$ positions are
+distinct.
+
+Among the population $L$, there are at most $mDR$ ordered distinct
+line pairs, neither line equal to $F$, whose intersection lies in
+$F$. Indeed, the first line meets $F$ in at most $D$ points, and at
+each such point there are at most $R$ choices of the second line.
+There are at most $t^{2h}$ ordered choices of the $h$ cross pairs of
+positions. For each choice, the sampled lines at those $2h$ distinct
+positions are uniform among $(m)_{2h}$ assignments, where
+$(z)_a=z(z-1)\cdots(z-a+1)$. The probability of the event for this
+fixed shape is therefore at most
+\begin{equation}\label{eq:lc-low-shape}
+ \frac{t^{2h}(mDR)^h}{(m)_{2h}}
+ =O_{D,h}\left(\left(\frac{t^2R}{m}\right)^h\right).
+\end{equation}
+The numerator may count assignments with repeated population lines;
+this only enlarges the upper bound. Summing
+\eqref{eq:lc-low-shape} over all shapes gives
+\[
+ O\bigl((\log q)^{2h}q^{D-h(1/2-\kappa/4)}\bigr)=o(1),
+\]
+because $\kappa<1/10$ and $h=10(D+1)$ imply
+\[
+ D-h(1/2-\kappa/4)
+ <D-\frac{19}{4}(D+1)<0.
+\]
+Thus, with probability tending to one, the sampled tuple is admissible
+and has fewer than $\delta t^2/2$ cross pairs meeting in $X$.
+
+\paragraph{High concurrency: $R>q^{1/2+\kappa/4}$.}
+Choose a projective point $p$ with $n_p=R$, and take all $R$ lines of
+$L$ through $p$ as the pencil. Its complement has size
+\[
+ N=m-R\geq\rho q/2
+\]
+for sufficiently large $q$, by \eqref{eq:lc-cover-counts}; no
+complement line passes through $p$. Sample the first list of $t$ lines
+uniformly without replacement from the pencil, and independently
+sample the second list in the same way from the complement.
+
+Any cross intersection lies off $p$. At a cross intersection
+$x\in X$, exactly one pencil line passes through $x$, and the number
+of cross pairs meeting there is the number of complement lines
+through $x$, namely $n_x-1$. Thus at most $E_L$ of all $RN$ population
+cross pairs meet in $X$. Their fraction is at most
+\begin{equation}\label{eq:lc-high-pair-fraction}
+ \frac{E_L}{RN}
+ =O\bigl(q^{\epsilon-\kappa/4}\bigr)
+ =O(q^{-3\kappa/20})=o(1).
+\end{equation}
+As before, expectation and Markov's inequality show that fewer than
+$\delta t^2/2$ sampled cross pairs meet in $X$, with probability
+tending to one.
+
+For a fixed complement line, at most one pencil line meets it at
+infinity when $p$ is finite. When $p$ is at infinity, the pencil lines
+have direction $p$ and every complement line has a different
+direction, so all cross intersections are finite. The probability of
+any infinite cross intersection is therefore at most $t^2/R$.
+Two equal cross intersections must use the same pencil line, since
+distinct pencil lines meet only at $p$, which lies on no complement
+line. Given two complement lines, their projective intersection
+determines at most one affine pencil line, namely its join with $p$.
+If this join is the line at infinity, there is no affine candidate.
+Conditioning on the second list and taking a union bound over its
+pairs shows that the probability of a collision is $O(t^3/R)$.
+The total probability of a failure of finiteness or distinctness is
+therefore
+\begin{equation}\label{eq:lc-high-degeneracy}
+ O\left(\frac{t^2+t^3}{R}\right)
+ =O\bigl((\log q)^3q^{-1/2-\kappa/4}\bigr)=o(1).
+\end{equation}
+
+Fix $F\in\mathcal F$, and consider the event that $F$ is not a tuple
+line but contains more than $H_0$ distinct finite cross intersections.
+The same greedy argument gives $h$ cross pairs with disjoint
+positions. There are at most
+$DN$ population cross pairs, with neither line equal to $F$, whose
+intersection lies in $F$: a complement line distinct from $F$ meets
+$F$ in at most $D$ points, all off $p$, and each such point determines
+at most one pencil line. The two lists are sampled independently, so
+the probability of this occupancy violation for a fixed shape is at
+most
+\begin{equation}\label{eq:lc-high-shape}
+ \frac{t^{2h}(DN)^h}{(R)_h(N)_h}
+ =O_{D,h}\left(\left(\frac{t^2}{R}\right)^h\right).
+\end{equation}
+After summing over $\mathcal F$, the bound becomes
+\[
+ O\bigl((\log q)^{2h}q^{D-h(1/2+\kappa/4)}\bigr)=o(1),
+\]
+since
+\[
+ D-h(1/2+\kappa/4)<D-5(D+1)<0.
+\]
+The sampled tuple is consequently admissible and has fewer than
+$\delta t^2/2$ cross pairs meeting in $X$, with probability tending
+to one in this regime as well.
+
+\paragraph{The contradiction and the order of choices.}
+All sampling estimates above are uniform over line sets satisfying
+\eqref{eq:lc-cover-counts} and the appropriate bound on $R$. In either
+regime, for every sufficiently large $q$ there is therefore an
+admissible tuple from the fixed line set $L$ with
+\[
+ |X\cap P_0|<\delta t^2/2.
+\]
+No line of $L$ actually occurred as a listed deletion shape. Hence
+ignoring exact occurrences of its tuple lines changes none of the
+deletions in the fixed realization, and the tuple's count $Y_0$ is
+exactly $|X\cap P_0|$. This contradicts
+\eqref{eq:lc-tuple-event}.
+
+In particular, the cover and the tuple are allowed to depend on the
+realized deletions: the event \eqref{eq:lc-tuple-event} was established
+simultaneously for every admissible tuple before fixing the
+realization. Sampling from $L$ is used only to prove the existence of
+a tuple inside that fixed realization. No concentration estimate is
+conditioned on the choice of the cover. The contradiction proves the
+second conclusion, and finishes the proof.
+\end{proof}
diff --git a/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/build/paper.tex b/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/build/paper.tex
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@@ -0,0 +1,301 @@
+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage{lmodern}
+\input{glyphtounicode}
+\input{glyphtounicode-cmex}
+\pdfgentounicode=1
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta,positioning}
+\usepackage[margin=1in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{microtype}
+\usepackage{enumitem}
+\usepackage{booktabs,array}
+\usepackage{needspace}
+\usepackage{xcolor}
+\usepackage[hyphens]{url}
+\usepackage{hyperref}
+\hypersetup{colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,
+  urlcolor=blue!45!black,
+  pdftitle={A counterexample to Ryser's covering conjecture},pdfauthor={OpenAI}}
+\urlstyle{same}
+\setlength{\emergencystretch}{2em}
+\setlist{itemsep=3pt,topsep=5pt}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newtheorem{definition}[theorem]{Definition}
+\newcommand{\F}{\mathbb F}
+\newcommand{\PP}{\mathbb P}
+\newcommand{\R}{\mathbb R}
+\newcommand{\N}{\mathbb N}
+\newcommand{\E}{\mathbb E}
+\DeclareMathOperator{\Var}{Var}
+\DeclareMathOperator{\Cov}{Cov}
+\DeclareMathOperator{\supp}{supp}
+\title{A counterexample to Ryser's covering conjecture}
+\author{OpenAI}
+\date{September 23, 2026}
+
+\begin{document}
+\maketitle
+\begin{abstract}
+For every sufficiently large prime $s\equiv2\pmod3$ and every sufficiently
+large odd integer $n$, with the threshold depending on $s$, we construct
+an intersecting $(s^n+1)$-partite $(s^n+1)$-uniform hypergraph with covering
+number $s^n+1$. This disproves Ryser's covering conjecture in its
+intersecting case.
+\end{abstract}
+
+\section{Introduction}
+
+A vertex cover of a finite hypergraph $H$ is a set of vertices meeting
+every edge. Its covering number $\tau(H)$ is the minimum size of such a
+set, and its matching number $\nu(H)$ is the maximum number of pairwise
+disjoint edges. For an integer $r\ge2$, an $r$-partite $r$-uniform
+hypergraph has disjoint vertex parts $V_1,\ldots,V_r$, with every edge
+containing exactly one vertex from each part. Ryser's covering
+conjecture asserts that every such hypergraph satisfies
+\begin{equation}\label{eq:ryser}
+ \tau(H)\le(r-1)\nu(H).
+\end{equation}
+A nonempty hypergraph is intersecting when every two distinct edges
+meet; equivalently, its matching number is one. In this case the
+conjecture asks for a cover of size at most $r-1$. The vertices of a
+cover may lie in several different parts, which is the central
+obstacle in constructing a counterexample.
+
+For $r=2$, inequality~\eqref{eq:ryser} is K\H{o}nig's matching theorem.
+An equivalent formulation appears in Henderson's 1971
+thesis~\cite{Henderson1971}; Best and Wanless discuss the history of its
+attribution to Ryser~\cite{BestWanless2018}. Aharoni proved the full
+tripartite case~\cite{Aharoni2001}. In the intersecting case, the results
+through rank four are due to Gy\'arf\'as, and the rank-five result to
+Tuza, as recorded by Kir\'aly and T\'othm\'er\'esz~\cite{KiralyTothmeresz2017};
+see also Tuza's early study of the covering conjecture~\cite{Tuza1983}.
+Haxell and Scott proved $\tau(H)\le(r-\epsilon)\nu(H)$ for some
+$\epsilon>0$ when $r=4$ or $5$, without assuming
+intersection~\cite{HaxellScott2012}. White proved the particular
+rank-four case $\nu(H)=2$, obtaining $\tau(H)\le6$~\cite{White2026}.\footnote{White
+reports assistance from GPT-5.6 Sol and Claude in the Methods section
+of his preprint.}
+Under the additional assumption that any two distinct edges meet in
+exactly one vertex, Franceti\'{c}, Herke, McKay, and Wanless established
+the intersecting case through rank nine~\cite{FranceticHerkeMcKayWanless2017}.
+Bishnoi, Das, Morris, and Szab\'o obtained bounds when every pair of
+edges has a larger prescribed intersection~\cite{BDMS2021}.
+
+Finite planes supply the classical equality examples. Delete a point
+and all lines through it from a projective plane of order $q$. Partition
+the remaining points according to the deleted line through them, and
+take the remaining lines as hyperedges. The result is intersecting,
+has $q+1$ parts and covering number $q$. Thus equality in the proposed
+bound occurs whenever $r-1$ is a prime power. Work of Mansour, Song, and
+Yuster~\cite{MansourSongYuster2009} and Aharoni, Bar\'{a}t, and
+Wanless~\cite{AharoniBaratWanless2016} studies the structure and sparsity
+of equality examples. Abu-Khazneh, Bar\'{a}t, Pokrovskiy, and
+Szab\'{o} exploit the rigidity of minimum covers in a truncated plane
+and add exceptional edges to obtain further equality
+examples~\cite{AbuKhaznehBaratPokrovskiySzabo2019}. In that construction,
+the number of parts and the covering number both increase by one.
+Haxell and Scott combine affine planes with intersecting configurations
+to obtain covering number at least $r-4$ for every sufficiently large
+$r$, improved to $r-3$ when $r$ is even~\cite{HaxellScott2017}.
+
+Our construction also starts from a finite plane and inserts exceptional
+edges, but keeps one part for each ambient direction. It changes the
+line labels within these parts to raise the covering number beyond
+$q$. The following theorem gives the precise field orders for which
+this can be done.
+
+\begin{theorem}\label{thm:main}
+There exists $s_0$ such that, for every prime $s\ge s_0$ with
+$s\equiv2\pmod3$, there is an integer $n_0(s)\ge3$ with the following
+property. For every odd integer $n\ge n_0(s)$, put $q=s^n$.
+There is a finite intersecting $(q+1)$-partite $(q+1)$-uniform
+hypergraph $H$ satisfying
+\[
+ \tau(H)=q+1.
+\]
+In particular, the intersecting case of~\eqref{eq:ryser} is false.
+\end{theorem}
+
+Every edge of a nonempty intersecting hypergraph is itself a cover,
+so the assertion $\tau(H)\le q+1$ is automatic. The proof must construct
+an intersecting hypergraph and exclude every cover of size at most $q$.
+We first fix any sufficiently large eligible prime $s$, construct a
+configuration in $\F_s^2$, and then take $n$ sufficiently large among
+the odd integers. All subsequent thresholds may depend on that fixed
+configuration. The result is asymptotic; the proof does not give a
+numerical threshold.
+
+A companion paper~\cite{OpenAI2026Balanced} proves a different theorem:
+for every sufficiently large prime $q$, it constructs such a hypergraph
+with exactly $q+1$ vertices in every part and with every vertex
+nonisolated. Its construction modifies line partitions directly in the
+prime plane. Here the field order is a proper odd power, and the proof
+uses a small-field configuration with a nonlinear change of
+coordinates. The two constructions give distinct ranges of ranks.
+The only results imported from the companion are its three elementary
+estimates in Appendix~A: the affine second-moment identity
+(Lemma~A.1), bounded-drift concentration (Lemma~A.2), and the exposure
+bound for independent trials (Lemma~A.3).
+
+Lov\'asz proposed the stronger assertion that deleting $r-1$ vertices
+can always reduce the matching number. Clow, Haxell, and Mohar
+disproved that assertion at $r=3$~\cite{CHM2025}; Abiad, Garbe, Povill,
+and Spiegel subsequently gave an infinite family at $r=3$ and
+examples at $r=4$~\cite{AbiadGarbePovillSpiegel2025}. These examples
+are consistent with~\eqref{eq:ryser}. They exclude a prescribed
+matching-reduction procedure, whereas Theorem~\ref{thm:main}
+violates the covering inequality itself.
+
+\subsection*{Monochromatic tree covers}
+
+An $r$-edge-coloring here assigns exactly one of $r$ colors to each edge.
+A monochromatic tree cover is a family of monochromatic trees whose vertex
+sets cover the graph; the trees may overlap, and singleton trees are allowed.
+Gy\'arf\'as's conjecture asks whether $r-1$ such trees suffice for every
+$r$-edge-colored complete graph; see \cite[Conjecture~1]{Milicevic2017}.
+This is the covering formulation recorded after Theorem~2 in
+\cite[Section~3]{ErdosGyarfasPyber1991}, not the distinct vertex-disjoint
+tree-partition conjecture numbered Conjecture~2 there.
+
+\begin{corollary}\label{cor:tree-cover}
+Let $s_0$ and $n_0(s)$ be as in Theorem~\ref{thm:main}. For every prime
+$s\ge s_0$ with $s\equiv2\pmod3$ and every odd integer $n\ge n_0(s)$,
+put $r=s^n+1$. There is a finite simple complete graph with an
+$r$-edge-coloring for which the minimum number of monochromatic trees
+needed to cover its vertices is exactly $r$. The same minimum holds with
+connected monochromatic subgraphs or maximal monochromatic components in
+place of trees.
+\end{corollary}
+
+\begin{proof}
+Take the hypergraph $H$ from Theorem~\ref{thm:main}, with parts
+$V_1,\ldots,V_r$. For $e\in E(H)$, let $x_i(e)$ be the unique vertex in
+$e\cap V_i$. Form the complete graph $K$ with vertex set $E(H)$. For each
+pair of distinct hyperedges $e,f$, choose one index $i$ with
+$x_i(e)=x_i(f)$ and color the edge $ef$ by $i$. Such an index exists
+because $H$ is intersecting. Only one color is chosen even when the two
+hyperedges meet in several parts.
+
+Along a color-$i$ path, the value of $x_i$ is constant. Thus every connected
+color-$i$ subgraph determines a vertex of $V_i$ belonging to every
+hyperedge indexed by that subgraph. For a singleton, choose any color and
+the corresponding vertex of its hyperedge. If $t$ connected monochromatic
+subgraphs cover the vertices of $K$, the at most $t$ hypergraph vertices so
+chosen meet every edge of $H$. Hence $t\ge\tau(H)=r$.
+
+Conversely, fix a vertex of $K$. The $r$ monochromatic stars centered there,
+one for each color and allowing singleton stars, cover all vertices of $K$.
+They give a tree cover of size at most $r$. The lower bound applies to trees
+and to maximal monochromatic components as well, and the stars can be
+enlarged to maximal components. This proves all three stated minima.
+\end{proof}
+
+Thus Gy\'arf\'as's cover conjecture fails at these extension-field ranks.
+The examples admit no cover by $r-1$ connected monochromatic subgraphs even
+without a diameter restriction. They therefore also refute Mili\'cevi\'c's
+bounded-diameter strengthening \cite[Conjecture~6]{Milicevic2017}, whose
+diameter bound may depend on $r$, and the $\alpha=1$ case of the conjecture
+of DeBiasio, Kamel, McCourt, and Sheats
+\cite[Conjecture~2.8]{DeBiasioKamelMcCourtSheats2021}, whose diameter bound
+depends only on the independence number $\alpha$. Indeed, $\alpha(K)=1$,
+so the latter conjecture also demands a cover by at most $r-1$ such
+subgraphs.
+
+\subsection*{Proof strategy}
+
+In the affine-plane model, the parts are indexed by the $q+1$ directions
+in $\F_q^2$, and the vertices of a part are the $q$ lines of that
+direction. A point $x$ defines an edge consisting of its incident line
+in each direction. The edges of two points intersect at their join,
+but a full parallel class always gives a cover of size $q$. We retain
+many of these point edges and add exceptional edges whose labels are
+changed in some directions. The construction has three tasks: preserve
+intersection, restrict small covers of the retained point edges, and
+make the exceptional edges too costly for those covers.
+
+\paragraph{Restricting the ordinary cover.}
+Delete a bounded number of translated shapes in each of $O(q)$
+independent trials. The shapes are lines or curves that meet every
+line in boundedly many points. Thus a shape has bounded intersection
+with a line unless it is that line itself. Section~\ref{sec:linecover} proves that, under explicit
+uniform survival assumptions, every cover of almost all surviving
+points by at most $q$ lines contains almost a pencil. Here a pencil is
+a family of lines through one projective point, including a parallel
+class when that point is at infinity. Surviving points on each
+nondeleted line then force every nondeleted member of this pencil
+into the cover. To obtain the pencil statement over the required
+extension fields, Section~\ref{sec:incidence} first proves an incidence
+saving by the sum--product method of Bourgain, Katz, and
+Tao~\cite{BKT2004}. Its constants are uniform over fields whose proper
+subfields all have order at most $q^{1/3}$.
+
+\paragraph{One configuration in two coordinates.}
+In a fixed prime plane $\F_s^2$, remove one line in each finite nonzero
+direction and call the remaining set $Y$. The change of coordinates
+$f(x,y)=(x^3,y)$ is bijective when $s\equiv2\pmod3$. Original line
+labels determine which ambient point edges must be deleted to
+intersect the exceptional edges. The labels for altered directions
+are instead read from the lines through $f(w)$, for $w\in Y$.
+The map $f$ preserves the horizontal and vertical parallel classes;
+this ensures intersection within the exceptional configuration.
+Its nonlinear action on the other lines makes their images curves
+with at most three intersections with any line. Applying the deletion
+lemma in these new coordinates yields the crucial obstruction:
+covering $f(Y)$ by $a$ lines and $b$ individual points with
+$a+b/2\le s$ requires exactly one full parallel class and no point
+resources. Section~\ref{sec:gadget} proves this statement together
+with the ordinary-coordinate occupancy needed later.
+
+\paragraph{Placing the configurations and spending the cover budget.}
+Embed linearly transformed copies of $Y$ in $\F_q^2$ so that every
+ambient direction is altered in one or two copies. Arbitrary translates
+could create disjoint exceptional edges. We therefore sample a fixed
+number of independent candidate translates for each copy.
+Section~\ref{sec:selection} proves that one can choose compatible
+representatives from these pools, using an independent-transversal
+argument in the tradition of Haxell~\cite{Haxell1995,Haxell2001}.
+Section~\ref{sec:placement} proves the deletion estimates for the
+entire independent pool before making this selection. Keeping only
+the selected deletions restores ordinary points, so the needed
+estimates transfer by deterministic inclusion.
+
+Section~\ref{sec:hypergraph} gives the labels of every edge, proves
+intersection, and excludes a cover of size at most $q$. After the
+ordinary edges force a pencil, its center determines the remaining
+argument. At infinity, the uncovered exceptional configurations
+require more vertices than the missing parallel lines release.
+Inside an embedded configuration, the remaining budget would cover
+$f(Y)$ by at most $s$ lines including two nonparallel axis lines.
+Outside every configuration, each deleted pencil line supplies five
+uncovered exceptional edges, while every additional vertex covers at
+most five. In all three cases the available budget is too small.
+
+\paragraph{Conventions.}
+A direction is a one-dimensional linear subspace of $\F_q^2$; as a subspace
+it includes zero. The projective completion adds one point at infinity per
+direction. A pencil consists of the affine lines through a projective point:
+it has $q+1$ members for a finite center and $q$ members for a center at
+infinity. Thus concurrence always includes parallel lines.
+For a positive integer $j$, write $(a)_j=a(a-1)\cdots(a-j+1)$.
+All asymptotic constants may depend on the parameters explicitly held fixed.
+An event holds with high probability if its probability tends to one along
+the specified sequence of fields. The small field is fixed before the
+ambient field tends to infinity.
+
+\input{incidence}
+\input{linecover}
+\input{selection}
+\input{construction}
+
+\Needspace{32\baselineskip}
+\bibliographystyle{plain}
+\bibliography{references}
+\end{document}
diff --git a/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/build/references.bib b/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/build/references.bib
new file mode 100644
index 0000000000000000000000000000000000000000..cc21c5de07146f4cce60ea282086d2c2a833bfd3
--- /dev/null
+++ b/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/build/references.bib
@@ -0,0 +1,390 @@
+@phdthesis{Henderson1971,
+  author = {Henderson, John Robert},
+  title = {Permutation Decompositions of {(0,1)}-Matrices and Decomposition Transversals},
+  school = {California Institute of Technology},
+  year = {1971},
+  doi = {10.7907/J1Z1-SK19},
+  note = {\href{https://doi.org/10.7907/J1Z1-SK19}{doi:10.7907/J1Z1-SK19}},
+  url = {https://thesis.caltech.edu/5726/}
+}
+
+@article{Aharoni2001,
+  author = {Aharoni, Ron},
+  title = {{Ryser}'s conjecture for tripartite 3-graphs},
+  journal = {Combinatorica},
+  volume = {21},
+  number = {1},
+  pages = {1--4},
+  year = {2001},
+  doi = {10.1007/s004930170001},
+  note = {\href{https://doi.org/10.1007/s004930170001}{doi:10.1007/s004930170001}},
+  url = {https://doi.org/10.1007/s004930170001}
+}
+
+@article{Tuza1983,
+  author = {Tuza, Zsolt},
+  title = {{Ryser}'s conjecture on transversals of {$r$}-partite hypergraphs},
+  journal = {Ars Combinatoria},
+  volume = {16B},
+  pages = {201--209},
+  year = {1983},
+  note = {\href{https://combinatorialpress.com/ars/vol16b/}{Publisher record}},
+  url = {https://combinatorialpress.com/ars/vol16b/}
+}
+
+@article{BDMS2021,
+  author = {Bishnoi, Anurag and Das, Shagnik and Morris, Patrick and Szab{\'o}, Tibor},
+  title = {{Ryser}'s conjecture for {$t$}-intersecting hypergraphs},
+  journal = {Journal of Combinatorial Theory, Series A},
+  volume = {179},
+  pages = {105366},
+  year = {2021},
+  doi = {10.1016/j.jcta.2020.105366},
+  note = {\href{https://doi.org/10.1016/j.jcta.2020.105366}{doi:10.1016/j.jcta.2020.105366}},
+  eprint = {2001.04132},
+  archivePrefix = {arXiv},
+  primaryClass = {math.CO},
+  url = {https://arxiv.org/abs/2001.04132}
+}
+
+@article{BKT2004,
+  author = {Bourgain, Jean and Katz, Nets and Tao, Terence},
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+  journal = {Geometric and Functional Analysis},
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+  eprint = {math/0301343},
+  archivePrefix = {arXiv},
+  primaryClass = {math.CO},
+  url = {https://arxiv.org/abs/math/0301343}
+}
+
+@article{Petridis2012,
+  author = {Petridis, Giorgis},
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+  year = {2012},
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+  note = {\href{https://doi.org/10.1007/s00493-012-2818-5}{doi:10.1007/s00493-012-2818-5}},
+  eprint = {1101.3507},
+  archivePrefix = {arXiv},
+  primaryClass = {math.CO},
+  url = {https://arxiv.org/abs/1101.3507}
+}
+
+@article{Haxell1995,
+  author = {Haxell, P. E.},
+  title = {A condition for matchability in hypergraphs},
+  journal = {Graphs and Combinatorics},
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+  year = {1995},
+  doi = {10.1007/BF01793010},
+  note = {\href{https://doi.org/10.1007/BF01793010}{doi:10.1007/BF01793010}},
+  url = {https://doi.org/10.1007/BF01793010}
+}
+
+@article{CHM2025,
+  author = {Clow, Alexander and Haxell, Penny and Mohar, Bojan},
+  title = {A Counterexample to a Conjecture of {Lov{\'a}sz}},
+  journal = {Combinatorica},
+  volume = {46},
+  pages = {26},
+  year = {2026},
+  doi = {10.1007/s00493-026-00220-3},
+  note = {Article 26. \href{https://doi.org/10.1007/s00493-026-00220-3}{doi:10.1007/s00493-026-00220-3}},
+  eprint = {2505.05339},
+  archivePrefix = {arXiv},
+  primaryClass = {math.CO},
+  url = {https://link.springer.com/article/10.1007/s00493-026-00220-3}
+}
+
+
+@misc{BestWanless2018,
+  author = {Best, Darcy and Wanless, Ian M.},
+  title = {What did {Ryser} Conjecture?},
+  year = {2018},
+  eprint = {1801.02893},
+  archivePrefix = {arXiv},
+  primaryClass = {math.CO},
+  url = {https://arxiv.org/abs/1801.02893},
+  note = {\href{https://arxiv.org/abs/1801.02893}{arXiv:1801.02893}}
+}
+
+@article{HaxellScott2012,
+  author = {Haxell, P. E. and Scott, A. D.},
+  title = {On {Ryser}'s conjecture},
+  journal = {The Electronic Journal of Combinatorics},
+  volume = {19},
+  number = {1},
+  pages = {P23},
+  year = {2012},
+  doi = {10.37236/1175},
+  note = {\href{https://doi.org/10.37236/1175}{doi:10.37236/1175}}
+}
+
+@article{MansourSongYuster2009,
+  author = {Mansour, Toufik and Song, Chunwei and Yuster, Raphael},
+  title = {A comment on {Ryser}'s conjecture for intersecting hypergraphs},
+  journal = {Graphs and Combinatorics},
+  volume = {25},
+  number = {1},
+  pages = {101--109},
+  year = {2009},
+  doi = {10.1007/s00373-008-0821-9},
+  eprint = {0709.3138},
+  archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/0709.3138},
+  note = {\href{https://arxiv.org/abs/0709.3138}{arXiv:0709.3138}}
+}
+
+@article{AharoniBaratWanless2016,
+  author = {Aharoni, Ron and Bar{\'a}t, J{\'a}nos and Wanless, Ian M.},
+  title = {Multipartite hypergraphs achieving equality in {Ryser}'s conjecture},
+  journal = {Graphs and Combinatorics},
+  volume = {32},
+  number = {1},
+  pages = {1--15},
+  year = {2016},
+  doi = {10.1007/s00373-015-1575-9},
+  eprint = {1409.4833},
+  archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/1409.4833},
+  note = {\href{https://arxiv.org/abs/1409.4833}{arXiv:1409.4833}}
+}
+
+@article{FranceticHerkeMcKayWanless2017,
+  author = {Franceti{\'c}, Nevena and Herke, Sarada and McKay, Brendan D. and Wanless, Ian M.},
+  title = {On {Ryser}'s conjecture for linear intersecting multipartite hypergraphs},
+  journal = {European Journal of Combinatorics},
+  volume = {61},
+  pages = {91--105},
+  year = {2017},
+  doi = {10.1016/j.ejc.2016.10.004},
+  eprint = {1508.00951},
+  archivePrefix = {arXiv},
+  note = {\href{https://doi.org/10.1016/j.ejc.2016.10.004}{doi:10.1016/j.ejc.2016.10.004}}
+}
+
+@article{AbuKhaznehBaratPokrovskiySzabo2019,
+  author = {Abu-Khazneh, Ahmad and Bar{\'a}t, J{\'a}nos and Pokrovskiy, Alexey and Szab{\'o}, Tibor},
+  title = {A family of extremal hypergraphs for {Ryser}'s conjecture},
+  journal = {Journal of Combinatorial Theory, Series A},
+  volume = {161},
+  pages = {164--177},
+  year = {2019},
+  doi = {10.1016/j.jcta.2018.07.011},
+  eprint = {1605.06361},
+  archivePrefix = {arXiv},
+  note = {\href{https://doi.org/10.1016/j.jcta.2018.07.011}{doi:10.1016/j.jcta.2018.07.011}}
+}
+
+@article{HaxellScott2017,
+  author = {Haxell, P. E. and Scott, A. D.},
+  title = {A note on intersecting hypergraphs with large cover number},
+  journal = {The Electronic Journal of Combinatorics},
+  volume = {24},
+  number = {3},
+  pages = {P3.26},
+  year = {2017},
+  doi = {10.37236/6460},
+  eprint = {1609.05458},
+  archivePrefix = {arXiv},
+  note = {\href{https://doi.org/10.37236/6460}{doi:10.37236/6460}}
+}
+
+@incollection{Ruzsa1999,
+  author = {Ruzsa, Imre Z.},
+  title = {An analog of {Freiman}'s theorem in groups},
+  booktitle = {Structure theory of set addition},
+  series = {Ast{\'e}risque},
+  publisher = {Soci{\'e}t{\'e} Math{\'e}matique de France},
+  volume = {258},
+  pages = {323--326},
+  year = {1999},
+  url = {https://www.numdam.org/item/AST_1999__258__323_0/},
+  note = {\href{https://www.numdam.org/item/AST_1999__258__323_0/}{Primary source}}
+}
+
+@article{Gowers1998,
+  author = {Gowers, W. T.},
+  title = {A new proof of {Szemer{\'e}di}'s theorem for arithmetic progressions of length four},
+  journal = {Geometric and Functional Analysis},
+  volume = {8},
+  number = {3},
+  pages = {529--551},
+  year = {1998},
+  doi = {10.1007/s000390050065},
+  url = {https://doi.org/10.1007/s000390050065},
+  note = {\href{https://doi.org/10.1007/s000390050065}{doi:10.1007/s000390050065}}
+}
+
+@article{BalogSzemeredi1994,
+  author = {Balog, Antal and Szemer{\'e}di, Endre},
+  title = {A statistical theorem of set addition},
+  journal = {Combinatorica},
+  volume = {14},
+  number = {3},
+  pages = {263--268},
+  year = {1994},
+  doi = {10.1007/BF01212974},
+  url = {https://doi.org/10.1007/BF01212974},
+  note = {\href{https://doi.org/10.1007/BF01212974}{doi:10.1007/BF01212974}}
+}
+
+@article{FoxSudakov2011,
+  author = {Fox, Jacob and Sudakov, Benny},
+  title = {Dependent random choice},
+  journal = {Random Structures \& Algorithms},
+  volume = {38},
+  number = {1--2},
+  pages = {68--99},
+  year = {2011},
+  doi = {10.1002/rsa.20344},
+  eprint = {0909.3271},
+  archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/0909.3271},
+  note = {\href{https://arxiv.org/abs/0909.3271}{arXiv:0909.3271}}
+}
+
+@article{Haxell2001,
+  author = {Haxell, P. E.},
+  title = {A note on vertex list colouring},
+  journal = {Combinatorics, Probability and Computing},
+  volume = {10},
+  number = {4},
+  pages = {345--347},
+  year = {2001},
+  doi = {10.1017/S0963548301004758},
+  url = {https://doi.org/10.1017/S0963548301004758},
+  note = {\href{https://doi.org/10.1017/S0963548301004758}{doi:10.1017/S0963548301004758}}
+}
+
+@article{Freedman1975,
+  author={Freedman, David A.},
+  title={On tail probabilities for martingales},
+  journal={The Annals of Probability},
+  volume={3}, number={1}, pages={100--118}, year={1975},
+  doi={10.1214/aop/1176996452},
+  note = {\href{https://doi.org/10.1214/aop/1176996452}{doi:10.1214/aop/1176996452}}
+}
+
+@article{Jamison1977,
+  author={Jamison, Robert E.},
+  title={Covering finite fields with cosets of subspaces},
+  journal={Journal of Combinatorial Theory, Series A},
+  volume={22}, number={3}, pages={253--266}, year={1977},
+  doi={10.1016/0097-3165(77)90001-2},
+  note = {\href{https://doi.org/10.1016/0097-3165(77)90001-2}{doi:10.1016/0097-3165(77)90001-2}}
+}
+
+@article{BrouwerSchrijver1978,
+  author={Brouwer, A. E. and Schrijver, A.},
+  title={The blocking number of an affine space},
+  journal={Journal of Combinatorial Theory, Series A},
+  volume={24}, number={2}, pages={251--253}, year={1978},
+  doi={10.1016/0097-3165(78)90013-4},
+  note = {\href{https://doi.org/10.1016/0097-3165(78)90013-4}{doi:10.1016/0097-3165(78)90013-4}}
+}
+
+@article{BBS2010,
+  author={Blokhuis, A. and Brouwer, A. E. and Sz{\H o}nyi, T.},
+  title={Covering all points except one},
+  journal={Journal of Algebraic Combinatorics},
+  volume={32}, pages={59--66}, year={2010},
+  doi={10.1007/s10801-009-0204-1},
+  note = {\href{https://doi.org/10.1007/s10801-009-0204-1}{doi:10.1007/s10801-009-0204-1}}
+}
+
+@misc{White2026,
+  author = {White, Patrick},
+  title = {{Tuza}'s {Ryser}-conjecture claim for four-partite hypergraphs with matching number two},
+  year = {2026},
+  eprint = {2609.14281},
+  archivePrefix = {arXiv},
+  primaryClass = {math.CO},
+  note = {Version 1, 13 September 2026},
+  howpublished = {\href{https://arxiv.org/abs/2609.14281v1}{arXiv:2609.14281v1}},
+  url = {https://arxiv.org/abs/2609.14281v1}
+}
+
+
+@misc{OpenAI2026Balanced,
+  author = {{OpenAI}},
+  title = {{Balanced counterexamples to Ryser's conjecture at prime orders}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/Balanced-Counterexamples-to-Rysers-Conjecture-at-Prime-Orders-September-27-2026/paper.pdf}{OAI:Balanced-Counterexamples-to-Rysers-Conjecture-at-Prime-Orders-September-27-2026}},
+  year = {2026}
+}
+
+@misc{AbiadGarbePovillSpiegel2025,
+  author = {Abiad, Aida and Garbe, Frederik and Povill, Xavier and Spiegel, Christoph},
+  title = {Infinitely many counterexamples to a conjecture of {Lov{\'a}sz}},
+  year = {2025},
+  eprint = {2506.21286},
+  archivePrefix = {arXiv},
+  primaryClass = {math.CO},
+  note = {Version 2, 14 July 2025},
+  howpublished = {\href{https://arxiv.org/abs/2506.21286v2}{arXiv:2506.21286v2}},
+  url = {https://arxiv.org/abs/2506.21286v2}
+}
+
+@article{KiralyTothmeresz2017,
+  author = {Kir{\'a}ly, Zolt{\'a}n and T{\'o}thm{\'e}r{\'e}sz, Lilla},
+  title = {On {Ryser}'s Conjecture for $t$-Intersecting and Degree-Bounded Hypergraphs},
+  journal = {The Electronic Journal of Combinatorics},
+  volume = {24},
+  number = {4},
+  pages = {P4.40},
+  year = {2017},
+  doi = {10.37236/6448},
+  eprint = {1705.10024},
+  archivePrefix = {arXiv},
+  primaryClass = {math.CO},
+  note = {\href{https://doi.org/10.37236/6448}{doi:10.37236/6448}}
+}
+
+@article{ErdosGyarfasPyber1991,
+  author = {Erd{\H{o}}s, P. and Gy{\'a}rf{\'a}s, A. and Pyber, L.},
+  title = {Vertex coverings by monochromatic cycles and trees},
+  journal = {Journal of Combinatorial Theory, Series B},
+  volume = {51},
+  number = {1},
+  pages = {90--95},
+  year = {1991},
+  doi = {10.1016/0095-8956(91)90007-7},
+  note = {\href{https://doi.org/10.1016/0095-8956(91)90007-7}{doi:10.1016/0095-8956(91)90007-7}},
+  url = {https://doi.org/10.1016/0095-8956(91)90007-7}
+}
+
+@misc{Milicevic2017,
+  author = {Mili{\'c}evi{\'c}, Luka},
+  title = {Covering complete graphs by monochromatically bounded sets},
+  year = {2017},
+  eprint = {1705.09370},
+  archivePrefix = {arXiv},
+  primaryClass = {math.CO},
+  note = {Version 1, 25 May 2017},
+  howpublished = {\href{https://arxiv.org/abs/1705.09370v1}{arXiv:\allowbreak1705.09370v1}},
+  url = {https://arxiv.org/abs/1705.09370v1}
+}
+
+@misc{DeBiasioKamelMcCourtSheats2021,
+  author = {DeBiasio, Louis and Kamel, Yigal and McCourt, Grace and Sheats, Hannah},
+  title = {Generalizations and strengthenings of {Ryser}'s conjecture},
+  year = {2021},
+  eprint = {2009.07239},
+  archivePrefix = {arXiv},
+  primaryClass = {math.CO},
+  note = {Version 3, 3 November 2021},
+  howpublished = {\href{https://arxiv.org/abs/2009.07239v3}{arXiv:2009.07239v3}},
+  url = {https://arxiv.org/abs/2009.07239v3}
+}
diff --git a/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/build/selection.tex b/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/build/selection.tex
new file mode 100644
index 0000000000000000000000000000000000000000..118c3fca60c5a5443e57f046c5dc3f5ff44ca6df
--- /dev/null
+++ b/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/build/selection.tex
@@ -0,0 +1,538 @@
+\section{Selecting compatible translates}\label{sec:selection}
+
+The deletion estimates require independent random choices, while
+intersection of the exceptional edges requires geometric compatibility.
+We separate these tasks by sampling several independent translates of
+each bounded configuration and then selecting one from each pool.
+The theorem below supplies the selection step. Its input consists of
+finite point sets $W_i$ and sets $D_i$ of forbidden join directions;
+its output avoids every forbidden join between different selected sets.
+Because a direction is a linear subspace containing zero, the condition
+also rules out a common point between two selected configurations
+whenever at least one of their direction sets is nonempty.
+
+\begin{theorem}[Compatible translates from independent pools]\label{thm:selection}
+For every positive integer $D$ there is a positive integer $K=K(D)$
+with the following property.  Let $q$ tend to infinity through prime
+powers.  For each $q$, let $n\le Dq$, and for $1\le i\le n$ let
+$W_i\subseteq\F_q^2$ and let $D_i$ be a set of directions, with
+\[
+ |W_i|\le D,\qquad |D_i|\le D.
+\]
+Assume that each direction belongs to $D_i$ for at most $D$ indices $i$.
+For distinct indices $i,j$, declare the translates $b+W_i$ and
+$b'+W_j$ to conflict if there are $u\in W_i$, $v\in W_j$, and
+$d\in D_i\cup D_j$ such that
+\[
+ (b+u)-(b'+v)\in d.
+\]
+For each $i$, draw $K$ independent uniform shifts
+$b_{i,1},\ldots,b_{i,K}\in\F_q^2$, independently for all indices.
+With probability tending to one, one can choose $k_i\in\{1,\ldots,K\}$
+for every $i$ so that the translates $b_{i,k_i}+W_i$ have no conflicts.
+The convergence is uniform over all data satisfying the stated bounds.
+\end{theorem}
+
+To prove the theorem, make each candidate a vertex of a graph and
+join candidates from different pools when they conflict. We seek an
+independent set containing one vertex from each pool, called an
+independent transversal. Its nonexistence has a small domination
+certificate. We first prove the form of that assertion we need, then
+show that the independent shifts make every such certificate unlikely.
+The certificate argument is a weaker form of Haxell's
+independent-transversal domination theorem~\cite{Haxell2001},
+whose method originates in~\cite{Haxell1995}. We include its proof.
+Haxell's theorem uses total domination, which also requires neighbors
+for the vertices of the dominating set. Here we use the following
+weaker convention.
+For a graph $G$ and a set $S$ of its vertices, we say that $T\subseteq S$
+\emph{dominates $S$} if every vertex of $S\setminus T$ has a neighbor in
+$T$.  No condition is imposed on the vertices of $T$ themselves.
+
+\begin{lemma}[A domination certificate]\label{sel-domination}
+Let a finite simple graph have a vertex partition into nonempty parts
+$A_1,\ldots,A_n$.  If there is no independent set containing one
+vertex from each part, then there are a nonempty index set $I$ and a set
+\[
+ T\subseteq\bigcup_{i\in I}A_i,
+ \qquad |T|\le 2(|I|-1),
+\]
+such that $T$ dominates $\bigcup_{i\in I}A_i$.
+\end{lemma}
+
+\begin{proof}
+We grow a set of vertices with small size compared with the number
+of parts it reaches. Whenever it does not yet dominate those parts,
+we use an undominated vertex to reach another part. A minimality
+condition on the transversals prevents this growth from stopping
+without the required domination certificate.
+
+Choose an inclusion-minimal nonempty subfamily of parts with no
+independent transversal, and distinguish one part $A_*$ of this
+subfamily.  Let $\mathcal U$ be the family of independent transversals of
+all its other parts.  Minimality makes $\mathcal U$ nonempty.  The
+deficient subfamily has at least two parts, since every part is nonempty.
+
+Choose any $x_1\in A_*$.  Among $U\in\mathcal U$, minimize
+$|N(x_1)\cap U|$, where $N(x)$ denotes the neighborhood of $x$ in $G$.
+Fix a minimizer and write $Y_1=N(x_1)\cap U$.  This set is nonempty:
+otherwise adjoining $x_1$ would give an independent transversal of the
+deficient subfamily.
+
+Inductively, suppose that vertices $x_1,\ldots,x_j$ and sets
+$Y_1,\ldots,Y_j$ have been fixed with the following properties:
+\begin{enumerate}
+ \item The sets $Y_a$ are nonempty and pairwise disjoint, and some
+ $U\in\mathcal U$ jointly realizes all the equalities
+ \[
+   N(x_a)\cap U=Y_a\qquad(1\le a\le j).
+ \]
+ \item At stage $a$, the size $|Y_a|$ was minimized over all
+ $U\in\mathcal U$ satisfying the already fixed equalities for indices
+ smaller than $a$.
+ \item Each $x_a$ lies either in $A_*$ or in a part represented by a
+ vertex of some earlier $Y_b$, with $b<a$.
+\end{enumerate}
+Let $Y=\bigcup_{a=1}^jY_a$, let $I$ consist of the distinguished part
+and all parts represented in $Y$, and put
+\[
+ T=\{x_1,\ldots,x_j\}\cup Y.
+\]
+By joint realization, the vertices of $Y$ occupy distinct parts, none
+of them $A_*$.  Thus $|I|=|Y|+1$, while nonemptiness of the $Y_a$ gives
+$j\le |Y|$.  The third invariant shows that $T$ lies in the union of
+the parts indexed by $I$, and
+\[
+ |T|\le j+|Y|\le 2|Y|=2(|I|-1).
+\]
+If $T$ dominates this union, we have the required certificate.
+
+Otherwise, choose a vertex $x_{j+1}$ in that union, outside $T$, with
+no neighbor in $T$.  Among the transversals realizing all the current
+equalities, choose $U$ minimizing $|N(x_{j+1})\cap U|$, and set
+\[
+ Y_{j+1}=N(x_{j+1})\cap U.
+\]
+This set avoids all the previous $Y_a$, because $x_{j+1}$ has no
+neighbor in $T$.  We claim it is nonempty.
+
+If it were empty and $x_{j+1}\in A_*$, then $U\cup\{x_{j+1}\}$
+would be an independent transversal of the deficient subfamily.
+Otherwise, the part of $x_{j+1}$ contains a unique vertex $y$ of $Y$,
+say $y\in Y_b$.  The transversal $U$ uses $y$ in that part, so
+$x_{j+1}\notin U$.  Replace $y$ by $x_{j+1}$ to form
+\[
+ U'=(U\setminus\{y\})\cup\{x_{j+1}\}.
+\]
+Since $N(x_{j+1})\cap U=\varnothing$, this is again an independent
+transversal of the nondistinguished parts.  For every $a<b$, the vertex
+$y$ does not belong to $Y_a$, and the new vertex is not adjacent to
+$x_a$.  Consequently the exact earlier equalities are preserved:
+\[
+ N(x_a)\cap U'=Y_a\qquad(a<b).
+\]
+For $a=b$, the replacement removes the neighbor $y$ and adds no neighbor,
+so
+\[
+ |N(x_b)\cap U'|=|Y_b|-1.
+\]
+This contradicts the minimum imposed at stage $b$.  Notice that the
+contradiction uses only the equalities preceding stage $b$; preservation
+of later equalities is not required.
+
+Thus $Y_{j+1}$ is nonempty and all the induction invariants continue to
+hold.  Each continuation adds vertices in previously unrepresented
+parts to $Y$, whereas all of $Y$ lies in one transversal of the fixed
+nondistinguished subfamily.  There can be only finitely many
+continuations.  The process must therefore stop with a dominating set
+$T$, proving the lemma.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{thm:selection}]
+The case $n=0$ is immediate, so suppose $n\ge1$.  Fix $D$ and, for the
+moment, a constant integer $K\ge6$.  We will choose $K$ sufficiently
+large in terms of $D$ at the end.
+
+Form a graph whose vertices are the candidate positions $(i,k)$,
+partitioned into the $n$ pools.  Put an edge between candidates in
+different pools precisely when their translates conflict.  Candidate
+positions are distinct vertices even when their sampled shifts happen
+to coincide.  An independent transversal in this graph is exactly a
+selection as in the theorem.
+
+By Lemma~\ref{sel-domination}, it suffices to show that, with probability
+tending to one, there are no nonempty index sets $I$, with $|I|=m$, and
+sets $T$ of at most $2m$ candidate positions in those pools that dominate
+their union.  We use the slightly weaker bound $2m$ to simplify the
+counting.  Throughout the probability estimates, $I$ and $T$ are fixed
+sets of indices before any shifts are exposed.
+
+The proof splits according to $m$, the number of pools in the
+certificate. When $m$ is small, the forbidden lines associated with
+$T$ occupy only a small part of each candidate's possible shifts.
+When $m$ is proportional to $q$, that direct bound is insufficient.
+Instead, we expose $T$ in a bounded number of chunks and prove that
+a positive average proportion of shifts remains feasible. Crucially,
+this proportion will depend only on $D$, so we may subsequently choose
+$K$ large enough that the unused candidates defeat all certificates.
+
+\paragraph{Forbidden equations and counting possible certificates.}
+For every direction $d$ in use, fix a nonzero linear functional
+$\pi_d:\F_q^2\to\F_q$ whose kernel is $d$.  After exposing the shifts
+of the candidates $e\in T$, call a shift $b\in\F_q^2$ \emph{feasible
+for target $i$} if, for every such $e$, with source-pool index $j$, it
+avoids every equation
+\begin{equation}\label{sel-forbidden}
+ \pi_d(b)=\pi_d(b_e)+\pi_d(v-u),
+ \qquad
+ u\in W_i,\quad v\in W_j,\quad d\in D_i\cup D_j.
+\end{equation}
+We impose these prohibitions even when $i=j$.  This only makes
+feasibility more restrictive.  A feasible candidate outside $T$ has no
+neighbor in $T$, so its existence prevents domination.
+
+Set
+\begin{equation}\label{sel-constants}
+ C=2D^3,\qquad c_1=\frac1{4C},\qquad \alpha=\frac1{2C}.
+\end{equation}
+For one target $i$ and one exposed candidate $e$, there are at most $C$
+equations in \eqref{sel-forbidden}.  This remains true if the two lists
+$D_i$ and $D_j$ are counted separately, retaining duplicate directions.
+Each equation excludes one affine line, containing exactly $q$ shifts.
+
+The number of choices of $I$ is at most
+\[
+ \binom{n}{m}\le\left(\frac{eDq}{m}\right)^m.
+\]
+There is an absolute constant $C_*>0$ such that, for $K\ge6$, the number
+of candidate-index sets $T$ in these pools with $|T|\le2m$ is at most
+\begin{equation}\label{sel-certificate-count}
+ \sum_{t=0}^{2m}\binom{Km}{t}\le (C_*K^2)^m.
+\end{equation}
+For example, the usual bound
+$\sum_{t=0}^{r}\binom Nt\le(eN/r)^r$ for $1\le r\le N/2$
+applies with $N=Km$ and $r=2m$.
+
+\paragraph{The range $1\le m\le c_1q$.}
+Condition on all the exposed shifts in $T$.  A fresh uniform shift for
+any target $i\in I$ is infeasible with probability at most
+\[
+ \frac{C|T|}{q}\le\frac{2Cm}{q}\le\frac12.
+\]
+At least $(K-2)m$ candidate positions in the target pools remain
+unexposed.  Their shifts are conditionally independent and uniform.
+Thus the probability that $T$ dominates, for this fixed $I,T$, is at most
+\begin{equation}\label{sel-small-probability}
+ \left(\frac{2Cm}{q}\right)^{(K-2)m}.
+\end{equation}
+Combining this with the counts of $I$ and $T$, the probability of a
+dominating certificate of size parameter $m$ is at most
+\begin{equation}\label{sel-small-sum}
+ \left[
+  2CeD C_*K^2
+  \left(\frac{2Cm}{q}\right)^{K-3}
+ \right]^m.
+\end{equation}
+Choose $K$ large enough that
+\begin{equation}\label{sel-small-K}
+ 2CeD C_*K^2\,2^{-(K-3)}\le\frac12.
+\end{equation}
+Then the bracket in \eqref{sel-small-sum} is at most $1/2$ throughout
+this range, and tends to zero for every fixed positive $m$.  Summing
+over $m$ gives $o(1)$: for a fixed cutoff, the finite initial sum tends
+to zero, while the remaining terms are bounded by a geometric tail.
+
+\paragraph{A feasible-density estimate for $m>c_1q$.}
+We prove that there is a constant $\delta_1>0$, depending only on $D$
+and not on $K$, such that for every fixed $I,T$ in this range,
+\begin{equation}\label{sel-density-target}
+ \Pr\!\left[
+   \sum_{i\in I}|U_i^{\mathrm{fin}}|<\delta_1mq^2
+ \right]
+ \le \exp\bigl(-\Omega_D(q\log q)\bigr).
+\end{equation}
+Here $U_i^{\mathrm{fin}}$ is the set of shifts feasible for target $i$
+after all candidates in $T$ have been exposed.  The estimate is uniform
+in $I,T$ and the configuration data.  The lower cutoff on $q$ may
+depend on the fixed value of $K$.
+
+Expose the candidate positions of $T$ in a fixed order, divided into
+consecutive chunks of size at most $\lfloor\alpha q\rfloor$, with all
+but the last having that size.  For sufficiently large $q$,
+$\lfloor\alpha q\rfloor\ge\alpha q/2$.  Since $|T|\le2m\le2Dq$,
+the number of chunks is at most
+\begin{equation}\label{sel-chunk-count}
+ J_0=\left\lceil\frac{4D}{\alpha}+1\right\rceil.
+\end{equation}
+In particular, this bound does not involve $K$.  If $T$ is empty, the
+desired feasible-density estimate holds deterministically; otherwise
+there is at least one chunk.
+
+At the start of a chunk, condition on all shifts previously exposed.
+Let $U_i$ be the set of shifts still feasible for target $i$, and write
+\[
+ M=\sum_{i\in I}|U_i|.
+\]
+For the analysis of this chunk we keep all these sets frozen at their
+start values.  They are now deterministic, while the shifts belonging
+to the new chunk are independent uniform vectors.
+
+For an entry $e$ of the chunk, with source-pool index $j$, define $Z_e$
+by summing, over all equations \eqref{sel-forbidden} associated with
+$i\in I$ and $e$, the number of shifts $b\in U_i$ satisfying that
+equation.  Count the $D_i$ and $D_j$ lists separately, including
+duplicates.  A shift deleted more than once is therefore counted more
+than once, which is harmless for an upper bound on loss.  Each line
+contains at most $q$ shifts, and each of its right-hand sides is uniform
+in $\F_q$.  Consequently,
+\begin{equation}\label{sel-Z-basic}
+ 0\le Z_e\le Cmq,
+ \qquad \E Z_e\le \frac{CM}{q}.
+\end{equation}
+All expectations and variances in the following chunk calculation are
+conditional on the fixed past.
+
+\paragraph{The terms whose directions belong to the target.}
+The expectation bound alone does not control the loss in a chunk.
+We estimate its fluctuations by separating the two direction lists
+in each forbidden equation. Bounded reuse of target directions handles
+one list; the affine second moment will handle the source list.
+Write
+\[
+ Z_e=Z_e^{(1)}+Z_e^{(2)},
+\]
+where $Z_e^{(1)}$ uses the list $D_i$ in each target, and $Z_e^{(2)}$
+uses the list $D_j$ of the source.  Group the terms of $Z_e^{(1)}$ by
+direction.  The group for $d$ is a function only of $\pi_d(b_e)$.
+For distinct directions $d,d'$, the functionals $\pi_d,\pi_{d'}$ are
+linearly independent, so
+\[
+ b_e\longmapsto\bigl(\pi_d(b_e),\pi_{d'}(b_e)\bigr)
+\]
+is a bijection from $\F_q^2$ to itself.  The two coordinates are
+therefore independent and uniform.  The direction groups are pairwise
+independent, which suffices for adding their variances.
+
+There are at most $Dm$ such groups.  Each direction belongs to at most
+$D$ target direction sets; for each associated target-source pair
+there are at most $D^2$ choices of $(u,v)$.  Thus every group is bounded
+by $D^3q$, and
+\begin{equation}\label{sel-first-variance}
+ \operatorname{Var}(Z_e^{(1)})\le D^7mq^2.
+\end{equation}
+
+\paragraph{The terms whose directions belong to the source.}
+For each target $i$ and each direction $d$, define the variance of its
+parallel line slices by
+\begin{equation}\label{sel-slice-variance}
+ V_{i,d}=\frac1q\sum_{\ell\parallel d}
+ \left(|U_i\cap\ell|-\frac{|U_i|}{q}\right)^2,
+\end{equation}
+where the sum is over the $q$ affine lines of direction $d$.
+These variances obey the exact identity
+\begin{equation}\label{sel-energy}
+ \sum_d V_{i,d}
+   =|U_i|-\frac{|U_i|^2}{q^2}
+   \le |U_i|\le q^2,
+\end{equation}
+where the sum runs over all $q+1$ directions. This is precisely the
+normalized affine second-moment identity
+in~\cite[Lemma~A.1]{OpenAI2026Balanced}, applied to $U_i\subseteq\F_q^2$.
+That identity holds for every subset of every finite affine plane;
+it has no randomness, characteristic, or subfield hypothesis. Thus it
+applies after conditioning on any history of earlier chunks. This is
+the reason we can control all source directions even when the feasible
+sets $U_i$ have complicated shapes.
+
+Every individual summand of $Z_e^{(2)}$ associated with $i,d,u,v$ is a
+uniformly sampled line-slice count, with its slice index translated by
+a fixed offset.  Its variance is therefore $V_{i,d}$.  There are at
+most $D^3m$ summands, and each pair $(i,d)$ is repeated at most $D^2$
+times.  Cauchy--Schwarz applied to their centered sum, without any
+independence assumption, gives
+\begin{equation}\label{sel-second-variance}
+ \operatorname{Var}(Z_e^{(2)})
+ \le D^5m A_e,
+ \qquad
+ A_e:=\sum_{i\in I}\sum_{d\in D_j}V_{i,d}.
+\end{equation}
+
+Each source pool contributes at most $K$ entries to the chunk, and
+each direction belongs to at most $D$ source direction sets.
+Consequently, by \eqref{sel-energy},
+\begin{equation}\label{sel-exception-energy}
+ \sum_{e\text{ in the chunk}} A_e
+ \le KD\sum_{i\in I}\sum_d V_{i,d}
+ \le KDmq^2.
+\end{equation}
+Call an entry \emph{regular} if $A_e\le mq^{3/2}$.  There are at most
+$KD\sqrt q$ irregular entries.  For a regular entry,
+\eqref{sel-first-variance}, \eqref{sel-second-variance}, and
+$\operatorname{Var}(X+Y)\le2\operatorname{Var}(X)+2\operatorname{Var}(Y)$
+give
+\begin{equation}\label{sel-regular-variance}
+ \operatorname{Var}(Z_e)
+ \le 2D^7mq^2+2D^5m^2q^{3/2}.
+\end{equation}
+For a fixed constant $\gamma>0$, Chebyshev's inequality and
+\eqref{sel-Z-basic} therefore imply
+\begin{align}
+ \Pr\!\left[Z_e>\frac{CM}{q}+\gamma mq\right]
+ &\le \frac{2D^7}{\gamma^2m}
+       +\frac{2D^5}{\gamma^2\sqrt q}
+ \notag\\
+ &\le B_{D,\gamma}q^{-1/2}
+ \qquad\text{for regular }e,
+ \label{sel-regular-tail}
+\end{align}
+where $B_{D,\gamma}$ is independent of $K$.  In the last step we used
+$m>c_1q$.
+
+\paragraph{Conditional independence and loss over one chunk.}
+The preceding variance bounds show that each regular entry has a
+small chance of removing far more than its expected share. We now
+use independence within the chunk to show that few entries do so,
+and convert that statement into a deterministic bound on the total
+remaining feasible set.
+
+The regularity of an entry is determined by the frozen sets $U_i$ and
+its fixed source-pool index.  In particular, it does not depend on its
+fresh shift $b_e$.  After conditioning on the past, the collection of
+regular entries is fixed, and their variables $Z_e$ are independent
+functions of independent shifts.  Their common threshold in
+\eqref{sel-regular-tail} is also fixed under this conditioning.
+
+A union bound over subsets of regular entries, using the chunk size
+bound and \eqref{sel-regular-tail}, shows that the probability of at
+least $\gamma q$ threshold exceedances is at most
+\begin{equation}\label{sel-chunk-tail}
+ 2^{\lceil\alpha q\rceil}
+ \left(B_{D,\gamma}q^{-1/2}\right)^{\lceil\gamma q\rceil}
+ \le \exp(-c_{D,\gamma}q\log q)
+\end{equation}
+for all sufficiently large $q$, with $c_{D,\gamma}>0$ independent of
+$K$.  If there are fewer than $\lceil\gamma q\rceil$ regular entries,
+the event in question is empty and the bound remains valid.
+
+For each fixed $K$, we may also take $q$ sufficiently large that
+\begin{equation}\label{sel-K-cutoff}
+ KD\sqrt q\le\gamma q.
+\end{equation}
+Except on the event in \eqref{sel-chunk-tail}, all but at most
+$2\gamma q$ entries are regular and below the threshold: the exceptions
+include all irregular entries.  The actual loss of feasible shifts
+within the chunk is at most $\sum_eZ_e$, because these counts used the
+larger feasible sets at the start of the chunk.  Bounding each
+exceptional entry by $Cmq$ and all other entries by the threshold gives,
+for the end-of-chunk value $M'$, the deterministic inequality
+\begin{align}
+ M'
+ &\ge M-C\alpha M-\alpha\gamma mq^2-2C\gamma mq^2
+ \notag\\
+ &=(1-C\alpha)M-(\alpha+2C)\gamma mq^2
+ \notag\\
+ &=\frac12M-(\alpha+2C)\gamma mq^2.
+ \label{sel-chunk-recurrence}
+\end{align}
+
+Choose $\gamma>0$, depending only on $D$, so small that
+\begin{equation}\label{sel-gamma-choice}
+ 2(\alpha+2C)\gamma\le 2^{-J_0-1},
+ \qquad
+ \delta_1:=2^{-J_0-1}.
+\end{equation}
+At the beginning, before exposing $T$, every shift is feasible, so
+$M=mq^2$.  If all $J\le J_0$ chunks satisfy
+\eqref{sel-chunk-recurrence}, iteration gives
+\begin{align*}
+ \frac{M_{\mathrm{fin}}}{mq^2}
+ &\ge 2^{-J}-(\alpha+2C)\gamma
+               \sum_{a=0}^{J-1}2^{-a}\\
+ &\ge 2^{-J_0}-2(\alpha+2C)\gamma
+ \ge\delta_1.
+\end{align*}
+The conditional probability bound \eqref{sel-chunk-tail} holds for
+every possible past.  Averaging over the past and taking a union bound
+over the at most $J_0$ chunks proves \eqref{sel-density-target}.
+The constants $J_0$, $\gamma$, and $\delta_1$ depend only on $D$.
+The dependence on $K$ in \eqref{sel-K-cutoff} affects only how large
+$q$ must be, not these constants.
+
+\paragraph{Unused candidates and the large-range union bound.}
+We have established the required positive average feasible density
+with a failure probability of order $\exp(-\Omega_D(q\log q))$.
+There are only exponentially many candidate certificates, so this
+error can be summed over all of them. On the successful event, the
+unused candidates now supply the further exponential saving needed
+to rule out domination.
+
+Condition now on all the shifts belonging to $T$, and suppose the
+feasible-density event in \eqref{sel-density-target} succeeds.  Let
+$t_i$ be the number of exposed candidate positions from pool $i$.
+Every unused candidate from that pool remains independent and uniform,
+and is feasible with probability $|U_i^{\mathrm{fin}}|/q^2$.
+The probability that none of the unused candidates in the target pools
+is feasible is therefore at most
+\begin{align}
+ \prod_{i\in I}
+ \left(1-\frac{|U_i^{\mathrm{fin}}|}{q^2}\right)^{K-t_i}
+ &\le
+ \exp\!\left(-\sum_{i\in I}
+       (K-t_i)\frac{|U_i^{\mathrm{fin}}|}{q^2}\right)
+ \notag\\
+ &\le \exp\bigl(-(K\delta_1-2)m\bigr).
+ \label{sel-unused-candidates}
+\end{align}
+To obtain the last inequality, use
+\[
+ \sum_{i\in I}\frac{|U_i^{\mathrm{fin}}|}{q^2}\ge\delta_1m,
+ \qquad
+ 0\le\frac{|U_i^{\mathrm{fin}}|}{q^2}\le1,
+ \qquad
+ \sum_{i\in I}t_i=|T|\le2m.
+\]
+This argument also covers pools for which every candidate belongs to
+$T$; it does not assume $K-t_i$ is positive in every target pool.
+
+For $m>c_1q$, the number of pairs $I,T$ is at most $A_K^m$, where
+\begin{equation}\label{sel-large-count}
+ A_K=\frac{eD}{c_1}C_*K^2.
+\end{equation}
+For each fixed pair, domination can occur only if the feasible-density
+estimate fails or all unused candidates are infeasible despite its
+success.  The sum of the first failure probabilities over all
+$m,I,T$ in the large range is bounded by
+\[
+ \exp\bigl(O_{D,K}(q)\bigr)
+ \exp\bigl(-\Omega_D(q\log q)\bigr)=o(1).
+\]
+Here the number of choices is at most
+$\sum_{m\le Dq}A_K^m=\exp(O_{D,K}(q))$, with $K$ fixed.
+
+Finally choose $K$ sufficiently large, still depending only on $D$, so
+that both \eqref{sel-small-K} and
+\begin{equation}\label{sel-large-K}
+ A_K\exp(2-K\delta_1)<\frac12
+\end{equation}
+hold.  Such a choice exists because $\delta_1>0$ does not depend on $K$,
+whereas $A_K$ grows only quadratically in $K$.  By
+\eqref{sel-unused-candidates}, the sum of the remaining failure
+probabilities is then at most
+\[
+ \sum_{m>c_1q}
+ \left(A_K\exp(2-K\delta_1)\right)^m
+ \le\sum_{m>c_1q}2^{-m}=o(1).
+\]
+Together with the small-range estimate, this excludes all dominating
+certificates with probability tending to one.  Lemma~\ref{sel-domination}
+therefore supplies an independent transversal, proving the theorem.
+
+The choices have been made in the order
+\[
+ D\ ;\quad C,c_1,\alpha,J_0,\gamma,\delta_1\ ;\quad K\ ;\quad q.
+\]
+In particular, $K$ is fixed before taking $q$ to infinity, and all
+sufficiently-large-$q$ conditions may depend on that fixed $K$.
+Every estimate above is uniform over the configurations and direction
+sets subject to the bounds in the theorem.
+\end{proof}
diff --git a/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/paper.pdf b/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..254d97823a75b094640fad6dd2abf610b1d4e6de
--- /dev/null
+++ b/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:ba76ca1f7af545c68090e0cfa1defba423af058cb4c0ac6087e563b5123b1eb7
+size 544159
diff --git a/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/README.md b/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..fd70c8e1efeb2a7b202e0453db9943919b1471f1
--- /dev/null
+++ b/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/README.md
@@ -0,0 +1,18 @@
+# [A Counterexample to Wall's D(2) Problem](wall-d2-counterexample.pdf)
+
+**Author:** OpenAI
+
+**Date:** October 6, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-Counterexample-to-Walls-D2-Problem-October-6-2026,
+  author = {{OpenAI}},
+  title = {{A Counterexample to Wall's $D(2)$ Problem}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/wall-d2-counterexample.pdf}{OAI:A-Counterexample-to-Walls-D2-Problem-October-6-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/build/figures/tower.tex b/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/build/figures/tower.tex
new file mode 100644
index 0000000000000000000000000000000000000000..be8f198d4ca8a7f0f6e8178988b0c0da159591dc
--- /dev/null
+++ b/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/build/figures/tower.tex
@@ -0,0 +1,20 @@
+\begin{tikzpicture}[
+  >=Latex,
+  every node/.style={font=\normalsize},
+  every edge/.style={draw,->,semithick},
+  baseline=(current bounding box.center)
+]
+  \node (F) at (0,0) {$F$};
+  \node (next) at (3,1.1) {$A_{i+1}$};
+  \node (cover) at (6.2,1.1) {$\widehat A_i$};
+  \node (base) at (6.2,-1.1) {$A_i$};
+  \path (F) edge node[above,sloped] {$f_{i+1}$} (next);
+  \draw[semithick,{Hooks[right]}-Latex]
+    (next) -- node[above] {inclusion} (cover);
+  \path (cover) edge node[right] {$p_i$} (base);
+  \node[anchor=west,font=\small,align=left] at (6.85,0)
+    {infinite cyclic\\cover};
+  \path (F) edge node[below,sloped] {$f_i$} (base);
+  \node[font=\small] at (2.85,-1.85)
+    {$s(A_i)<s(A_{i+1})\leq s(F)$};
+\end{tikzpicture}
diff --git a/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/build/main.tex b/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/build/main.tex
new file mode 100644
index 0000000000000000000000000000000000000000..894ec6a054b27d51c91afcbe08ff7a117ac4d388
--- /dev/null
+++ b/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/build/main.tex
@@ -0,0 +1,20 @@
+\documentclass[11pt]{article}
+\input{preamble}
+\title{A Counterexample to Wall's \texorpdfstring{$D(2)$}{D(2)} Problem}
+\author{OpenAI}
+\date{October 6, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We give a negative answer to Wall's finite $D(2)$ problem. We construct
+a finite connected three-dimensional CW complex satisfying the $D(2)$
+finiteness condition but not homotopy equivalent to any finite CW complex
+of dimension at most two. The example has infinite fundamental group.
+\end{abstract}
+\input{sections/01-introduction}
+\input{sections/02-tower}
+\input{sections/03-plus}
+\input{sections/04-construction}
+\bibliographystyle{alpha}
+\bibliography{references}
+\end{document}
diff --git a/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/build/preamble.tex b/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/build/preamble.tex
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+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage[margin=1in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{microtype,enumitem,needspace,placeins,xcolor,tikz,booktabs}
+\usetikzlibrary{arrows.meta,positioning,calc}
+\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref}
+\hypersetup{pdftitle={A Counterexample to Wall's D(2) Problem},pdfauthor={OpenAI}}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\newtheorem{remark}[theorem]{Remark}
+\newcommand{\C}{\mathbb C}
+\newcommand{\Z}{\mathbb Z}
+\newcommand{\Q}{\mathbb Q}
+\newcommand{\RP}{\mathbb{RP}}
+\newcommand{\Hom}{\operatorname{Hom}}
+\newcommand{\id}{\operatorname{id}}
+\newcommand{\im}{\operatorname{im}}
+\newcommand{\ab}{\mathrm{ab}}
+\newcommand{\rk}{\operatorname{rank}}
+\setlist[enumerate]{label=\textup{(\roman*)},leftmargin=*}
+\setlength{\emergencystretch}{1.5em}
+\allowdisplaybreaks[1]
diff --git a/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/build/references.bib b/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/build/references.bib
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@@ -0,0 +1,148 @@
+@article{Wall65,
+  author = {Wall, C. T. C.},
+  title = {Finiteness conditions for {CW}-complexes},
+  journal = {Annals of Mathematics. Second Series},
+  volume = {81},
+  number = {1},
+  year = {1965},
+  pages = {56--69},
+  doi = {10.2307/1970382}
+}
+
+@incollection{Wall79,
+  author = {Wall, C. T. C.},
+  title = {List of problems},
+  booktitle = {Homological Group Theory},
+  editor = {Wall, C. T. C.},
+  series = {London Mathematical Society Lecture Note Series},
+  volume = {36},
+  publisher = {Cambridge University Press},
+  address = {Cambridge},
+  year = {1979},
+  pages = {369--394}
+}
+
+@book{Johnson03,
+  author = {Johnson, F. E. A.},
+  title = {Stable Modules and the {D(2)}-Problem},
+  series = {London Mathematical Society Lecture Note Series},
+  volume = {301},
+  publisher = {Cambridge University Press},
+  address = {Cambridge},
+  year = {2003},
+  doi = {10.1017/CBO9780511550256}
+}
+
+@incollection{Cohen78,
+  author = {Cohen, J. M.},
+  title = {Complexes of cohomological dimension two},
+  booktitle = {Algebraic and Geometric Topology},
+  series = {Proceedings of Symposia in Pure Mathematics},
+  volume = {32},
+  publisher = {American Mathematical Society},
+  address = {Providence, RI},
+  year = {1978},
+  pages = {221--223},
+  note = {Part 2}
+}
+
+@article{Hambleton18,
+  author = {Hambleton, Ian},
+  title = {Two remarks on {Wall's D2} problem},
+  journal = {Mathematical Proceedings of the Cambridge Philosophical Society},
+  volume = {167},
+  year = {2019},
+  pages = {361--368}
+}
+
+@article{HofmannNicholson27,
+  author = {Hofmann, Tommy and Nicholson, John},
+  title = {Exotic presentations of quaternion groups and {Wall's D2} problem},
+  journal = {Journal of Algebra},
+  volume = {712},
+  year = {2027},
+  pages = {253--310},
+  doi = {10.1016/j.jalgebra.2026.06.040}
+}
+
+@article{Mannan09,
+  author = {Mannan, W. H.},
+  title = {{Quillen's} plus construction and the {D(2)} problem},
+  journal = {Algebraic \& Geometric Topology},
+  volume = {9},
+  number = {3},
+  year = {2009},
+  pages = {1399--1411},
+  doi = {10.2140/agt.2009.9.1399}
+}
+
+@article{Fox53,
+  author = {Fox, Ralph H.},
+  title = {Free differential calculus. {I}. {Derivation} in the free group ring},
+  journal = {Annals of Mathematics. Second Series},
+  volume = {57},
+  number = {3},
+  year = {1953},
+  pages = {547--560},
+  doi = {10.2307/1969736}
+}
+
+@article{Howie81Pairs,
+  author = {Howie, James},
+  title = {On pairs of 2-complexes and systems of equations over groups},
+  journal = {Journal f\"ur die reine und angewandte Mathematik},
+  volume = {324},
+  year = {1981},
+  pages = {165--174},
+  doi = {10.1515/crll.1981.324.165}
+}
+
+@article{LouderWilton17,
+  author = {Louder, Larsen and Wilton, Henry},
+  title = {Stackings and the {W}-cycles conjecture},
+  journal = {Canadian Mathematical Bulletin},
+  volume = {60},
+  number = {3},
+  year = {2017},
+  pages = {604--612}
+}
+
+@incollection{Quillen71,
+  author = {Quillen, Daniel},
+  title = {Cohomology of groups},
+  booktitle = {Actes du Congr\`es International des Math\'ematiciens (Nice, 1970)},
+  volume = {2},
+  publisher = {Gauthier-Villars},
+  address = {Paris},
+  year = {1971},
+  pages = {47--51}
+}
+
+@article{BridsonTweedale07,
+  author = {Bridson, Martin R. and Tweedale, Michael},
+  title = {Deficiency and abelianized deficiency of some virtually free groups},
+  journal = {Mathematical Proceedings of the Cambridge Philosophical Society},
+  volume = {143},
+  year = {2007},
+  pages = {257--264}
+}
+
+@article{MannanPopiel21,
+  author = {Mannan, W. H. and Popiel, Tomasz},
+  title = {An exotic presentation of {$Q_{28}$}},
+  journal = {Algebraic \& Geometric Topology},
+  volume = {21},
+  number = {4},
+  year = {2021},
+  pages = {2065--2084}
+}
+
+@article{Nicholson21,
+  author = {Nicholson, John},
+  title = {On {CW}-complexes over groups with periodic cohomology},
+  journal = {Transactions of the American Mathematical Society},
+  volume = {374},
+  number = {9},
+  year = {2021},
+  pages = {6531--6557}
+}
diff --git a/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/build/sections/01-introduction.tex b/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/build/sections/01-introduction.tex
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+++ b/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/build/sections/01-introduction.tex
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+\section{Introduction}\label{sec:introduction}
+
+Wall's finiteness conditions ask when algebraic restrictions on a CW
+complex force it to have a homotopy model of a prescribed dimension.
+Let $X$ be a finite connected CW complex, let $G=\pi_1(X)$, and let
+$\widetilde X$ be its universal cover. The $D(2)$ condition is
+\begin{equation}\label{eq:d2-condition}
+ H_i(\widetilde X;\Z)=0\quad(i>2),\qquad
+ H^3(X;M)=0\quad\text{for every finitely generated }\Z G\text{-module }M,
+\end{equation}
+where cohomology uses the corresponding local coefficients.
+The finite $D(2)$ problem asks whether every such $X$ is homotopy
+equivalent to a finite CW complex of dimension at most two
+\cite[Section~2]{Wall65}, \cite[Problem~D3]{Wall79}.
+The finiteness of both complexes is part of the question, and no
+asphericity hypothesis is imposed. This is also the formulation in
+Hofmann--Nicholson~\cite[p.~254]{HofmannNicholson27}.
+
+\begin{theorem}\label{thm:main}
+There is a finite connected three-dimensional CW complex $X$ such that
+\[
+ H_i(\widetilde X;\Z)=0\quad(i>2),\qquad
+ H^3(X;M)=0\quad\text{for every }\Z[\pi_1(X)]\text{-module }M,
+\]
+but $X$ is not homotopy equivalent to any finite CW complex of
+dimension at most two.
+\end{theorem}
+
+The example has infinite fundamental group; the finiteness assertion
+concerns its cell structure.
+
+\subsection{Background and approach}
+
+Wall introduced the dimension conditions in his work on finiteness
+for CW complexes~\cite{Wall65}. His reduction produces a finite
+three-dimensional model in the $D(2)$ case; the remaining question is
+whether its three-dimensional cells can also be removed up to homotopy.
+The problem reappeared as Problem~D3 in his 1979 list~\cite{Wall79}.
+It lies at the interface between two-dimensional topology and the
+realization of chain complexes over group rings. Johnson's work
+\cite{Johnson03} develops this algebraic viewpoint and the role of
+cancellation in passing from a stable realization to an actual
+two-dimensional complex.
+
+A stable form of the problem has a positive answer: adjoining
+sufficiently many two-spheres makes a finite $D(2)$ complex homotopy
+equivalent to a finite two-dimensional complex. This goes back to
+Cohen~\cite{Cohen78}; Hambleton~\cite[Lemma~2.1]{Hambleton18} gives
+a proof with the number of added spheres bounded by the number of
+three-dimensional cells. Such stabilization does not answer the
+unstabilized question. There are also substantial positive results
+for particular fundamental groups. Hambleton~\cite[Theorem~B]{Hambleton18}
+proves the $D(2)$ property for finite subgroups of $\operatorname{SO}(3)$.
+More recently, Hofmann and Nicholson~\cite[Theorem~A]{HofmannNicholson27}
+establish the $D(2)$ property for the generalized quaternion groups of
+orders $24$ and $32$ and recover the order-$28$ case previously proved
+using work of Mannan--Popiel and Nicholson~\cite{MannanPopiel21,Nicholson21}.
+In particular,
+they show that the longstanding candidate
+with fundamental group of order $32$, proposed by Cohen and Dyer,
+does have a finite two-dimensional model.
+
+Candidate counterexamples have also come from relation modules.
+A relation gap occurs when the relator subgroup of a free group needs
+more normal generators than its abelianization needs as a module over
+the quotient group ring.
+Bridson and Tweedale~\cite[Sections~3--4]{BridsonTweedale07} use
+coprime annihilation relations to reduce the number of module generators
+and construct finite $D(2)$ complexes with virtually free fundamental
+groups. Their failure to have finite two-dimensional models is conditional
+on a relation-gap assertion.
+
+We apply Quillen's plus construction~\cite{Quillen71}. It kills a
+perfect normal subgroup of the fundamental
+group while preserving homology with coefficients from the quotient.
+Mannan~\cite[Section~2]{Mannan09} shows directly that applying it to a
+finite two-dimensional complex along a finitely normally generated
+perfect subgroup produces a finite $D(2)$ complex. He also proves
+that every finite cohomologically two-dimensional three-dimensional
+complex arises in this way up to homotopy
+\cite[Theorem~3.4]{Mannan09}. Thus this construction supplies the
+dimension condition; the task is to find an obstruction to a
+two-dimensional model of its result.
+
+Our obstruction uses a character. For a connected space $Y$, a
+homomorphism $\rho:\pi_1(Y)\to\C^\times$ determines a
+one-dimensional complex local system $L_\rho$. We prove that, when
+$Y$ is a finite two-dimensional complex,
+\begin{equation}\label{eq:obstruction-overview}
+ H_2(Y;L_\rho)=0
+ \quad\Longrightarrow\quad
+ \rho(g)=1\ \text{for every finite-order }g\in\pi_1(Y).
+\end{equation}
+The proof gives the stronger statement that every map $F\to Y$ from
+a finite connected simplicial complex with $b_1(F;\Q)=0$ pulls $L_\rho$
+back to a system with trivial monodromy. The method is a tower of
+infinite cyclic covers and image subcomplexes, following the cyclic-tower
+method of Howie~\cite{Howie81Pairs}; see also the formulation of
+Louder--Wilton~\cite[Section~3]{LouderWilton17} and its account of
+the earlier work of Papakyriakopoulos and Stallings. The additional input
+is that vanishing of rank-one twisted second homology persists under
+these cyclic covers, by a Laurent-polynomial rank argument.
+
+\subsection{Proof structure}
+
+Section~\ref{sec:tower} establishes this obstruction independently
+of the construction. If a map $F\to Y$ violated it, its finite image
+would have nonpositive Euler characteristic and hence admit an
+infinite cyclic cover. The map lifts because $b_1(F;\Q)=0$.
+Twisted second homology still vanishes upstairs, so the argument
+can be repeated. Each lifted image contains strictly more simplices,
+although all are images of the same fixed triangulated $F$. This
+forces a contradiction.
+
+Section~\ref{sec:plus} presents the finite plus construction with
+its explicit cellular splitting. Section~\ref{sec:construction}
+then gives a five-generator, four-relator presentation and a perfect
+normal subgroup generated normally by two powers. The quotient has
+an element of order two detected by a complex character. An exact
+four-row boundary calculation shows that the presentation complex
+has zero second homology with that character. The plus construction
+preserves this vanishing and gives the required finite $D(2)$
+complex. Equation~\eqref{eq:obstruction-overview} excludes its
+two-dimensional realization.
+
+The geometric obstruction and the finite plus construction play separate
+roles: the former applies to any rank-one local system on a finite
+two-dimensional complex, while the latter is standard once a suitable
+perfect normal subgroup is available. Their combination makes the
+failure of two-dimensional realization visible in a small twisted
+cellular boundary matrix.
diff --git a/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/build/sections/02-tower.tex b/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/build/sections/02-tower.tex
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+\section{A tower obstruction in dimension two}
+\label{sec:tower}
+
+We first isolate the obstruction to a finite two-dimensional model.
+Its input is the vanishing of second homology with a rank-one local
+system.  Its conclusion restricts the maps into the complex from finite
+complexes with vanishing first rational Betti number.  The case of a
+Moore complex will then show that the corresponding character cannot
+detect a finite-order element.
+
+A \emph{rank-one complex local system} $L$ on a connected CW complex $T$
+assigns a one-dimensional complex vector space to each point, with
+parallel transport along paths that depends only on homotopy relative to
+the endpoints.  After choosing a basepoint and a basis of its fiber,
+its monodromy is a character
+\[
+  \rho:\pi_1(T)\longrightarrow\C^\times.
+\]
+We use ordinary homology with these coefficients.  In module notation,
+write $\C_\rho$ for the right $\Z[\pi_1(T)]$-module with
+$v\cdot g=\rho(g)v$.  If the universal-cover cellular chains carry the
+left deck action, then
+\[
+  C_*(T;L)
+  =\C_\rho\otimes_{\Z[\pi_1(T)]}C_*(\widetilde T;\Z).
+\]
+Thus cellular chains have finite support, even when $T$ is infinite.
+Equivalently, $C_j(T;L)$ is the direct sum of one coefficient fiber for
+each oriented $j$-cell, with boundary maps defined by parallel transport.
+
+Two elementary consequences will be used throughout the proof.  First,
+if $A\subseteq T$ is a subcomplex and $\dim T\leq2$, then
+\begin{equation}
+  H_2(A;L|_A)\longrightarrow H_2(T;L)
+  \quad\text{is injective}.
+  \label{eq:subcomplex-injection}
+\end{equation}
+Indeed, the cellular chain inclusion is injective, and in dimension at
+most two the second homology is the kernel of the second boundary.
+This argument requires no injectivity of $\pi_1(A)\to\pi_1(T)$.
+Second, on a connected complex the zeroth homology is the space of
+coinvariants,
+\[
+  H_0(T;L)
+  =\C/\langle (\rho(g)-1)v:g\in\pi_1(T),\ v\in\C\rangle.
+\]
+Consequently, nontrivial monodromy implies $H_0(T;L)=0$.
+
+The next lemma ensures that vanishing of second homology also survives
+the covering operation needed in the tower.
+
+\begin{lemma}[Infinite cyclic covers]
+\label{lem:cyclic-cover}
+Let $A$ be a finite connected CW complex of dimension at most two, let
+$L$ be a rank-one complex local system on $A$, and let
+$p:\widehat A\to A$ be a connected infinite cyclic cover.  If
+$H_2(A;L)=0$, then
+\[
+  H_2(\widehat A;p^*L)=0.
+\]
+\end{lemma}
+
+\begin{proof}
+Choose a generator $\tau$ of the deck group and put
+$R=\C[z,z^{-1}]$.  There is a canonical deck action on the pulled-back
+local system: in the fiber description of $p^*L$, it is
+\begin{equation}
+  (\widehat x,v)\longmapsto(\tau\widehat x,v),
+  \qquad v\in L_{p(\widehat x)}.
+  \label{eq:canonical-deck-action}
+\end{equation}
+The equality $p\tau=p$ makes this action compatible with parallel
+transport.  Let $z$ act by this deck transformation.  Choosing one lift
+of each cell of $A$ and a basis in its coefficient fiber identifies
+$C_j(\widehat A;p^*L)$ with a finite free $R$-module, of rank equal to
+the number of $j$-cells of $A$.
+
+Taking coinvariants of this deck action identifies translated cell
+lifts and their coefficient fibers.  It therefore gives an isomorphism
+of chain complexes
+\begin{equation}
+  C_*(\widehat A;p^*L)\otimes_R R/(z-1)
+  \cong C_*(A;L).
+  \label{eq:cyclic-specialization}
+\end{equation}
+In the chosen bases, let $D(z)$ be the matrix of the second boundary
+upstairs, and let $m$ be its number of columns.  The matrix $D(1)$ is
+the second boundary downstairs, so it is injective by the hypothesis.
+If $m=0$, there is nothing to prove.  Otherwise, some $m\times m$ minor
+of $D(1)$ is nonzero.  The corresponding minor of $D(z)$ is consequently
+a nonzero Laurent polynomial.  Thus $D(z)$ has full column rank over
+$\C(z)$ and is injective over $R$, since $R^m$ embeds in $\C(z)^m$.
+There are no cells above dimension two, and hence
+$H_2(\widehat A;p^*L)=\ker D(z)=0$.
+\end{proof}
+
+The following proof adapts the cyclic-tower method of
+Howie~\cite{Howie81Pairs}, in which one alternates subcomplex inclusions
+and infinite cyclic covers; see Louder--Wilton~\cite[Section~3]{LouderWilton17}.
+Lemma~\ref{lem:cyclic-cover} supplies the property that persists along
+the tower here: vanishing of twisted second homology.
+
+\begin{theorem}[Tower obstruction]
+\label{thm:tower}
+Let $Y$ be a finite connected CW complex of dimension at most two, and
+let $L$ be a rank-one complex local system on $Y$ with $H_2(Y;L)=0$.
+There is no map $f:F\to Y$ from a finite connected simplicial complex
+satisfying $b_1(F;\Q)=0$ for which $f^*L$ has nontrivial monodromy.
+\end{theorem}
+
+\begin{proof}
+Suppose that such a map exists.  We will repeatedly lift it to an
+infinite cyclic cover and replace that cover by the image of the lift.
+At every step the image will contain strictly more simplices, although
+the domain retains one fixed finite triangulation.  For a finite
+simplicial complex $A$, write $s(A)$ for its number of nonempty simplices.
+
+We first justify working simplicially without increasing the dimension
+of $Y$.  Its one-skeleton is a finite graph.  Subdivide that graph to
+make it simplicial and homotope each two-cell attaching map to an edge
+path.  After further subdivision if necessary, each nonconstant
+attaching path can be represented by a map from a polygonal circle
+with at least three edges, with every edge mapping homeomorphically
+onto a graph edge.  Replace the attachment by the mapping cylinder of this
+map, capped by a disk along its free boundary circle.  This merely
+inserts a collar into the attached disk.  The resulting cylinder has
+one rectangle for each traversal of a graph edge.  Each rectangle is
+embedded, with its two upper vertices in the separate domain circle
+and its two lower vertices at distinct endpoints of the graph edge.
+After compatible subdivisions of boundary edges, triangulate each
+rectangle from its own interior vertex and each capping disk from a
+new interior vertex.  This yields a finite simplicial complex even
+when different rectangles traverse the same graph edge: their
+interiors and upper edges remain distinct.  A constant attaching map
+contributes a triangulated two-sphere joined to the graph at one
+vertex.  Homotoping the attaching maps preserves the homotopy type,
+so this construction gives a finite simplicial model of dimension at
+most two.
+
+Transfer $L$ and $f$ across this homotopy equivalence.  Homotopy
+invariance preserves both $H_2(Y;L)=0$ and the nontrivial pullback
+monodromy.  After a sufficiently fine subdivision of $F$, simplicial
+approximation makes $f$ simplicial.  Fix this triangulation of $F$
+for the rest of the proof.  Let $A_0=f(F)$, let $f_0:F\to A_0$ be the
+corestriction, and let $L_0=L|_{A_0}$.  The simplicial image $A_0$ is
+a finite connected subcomplex, and
+\eqref{eq:subcomplex-injection} gives $H_2(A_0;L_0)=0$.
+
+Here is the inductive step.  Suppose that $A_i$ is a finite connected
+simplicial complex of dimension at most two, $f_i:F\to A_i$ is a
+simplicial map onto its image $A_i$, and $L_i$ satisfies
+\begin{equation}
+  H_2(A_i;L_i)=0,
+  \qquad f_i^*L_i\text{ has nontrivial monodromy}.
+  \label{eq:tower-invariants}
+\end{equation}
+The second condition implies that $L_i$ itself has nontrivial
+monodromy, so $H_0(A_i;L_i)=0$.  Since rank-one cellular chains have
+the same dimensions as ordinary complex cellular chains, the Euler
+characteristic gives
+\[
+  \chi(A_i)=-\dim_\C H_1(A_i;L_i)\leq0.
+\]
+On the other hand,
+\[
+  \chi(A_i)=1-b_1(A_i;\Q)+b_2(A_i;\Q).
+\]
+It follows that $b_1(A_i;\Q)\geq1$.  The finitely generated
+abelianization of $\pi_1(A_i)$ therefore has a nonzero free part,
+and there is an epimorphism
+$\phi_i:\pi_1(A_i)\twoheadrightarrow\Z$.
+
+Let $p_i:\widehat A_i\to A_i$ be the connected cover corresponding to
+$\ker\phi_i$.  The composite $\phi_i(f_i)_*$ is zero: every
+homomorphism $\pi_1(F)\to\Z$ factors through $H_1(F;\Z)$, which is
+finite because $F$ is finite and $b_1(F;\Q)=0$.
+Thus $f_i$ lifts to $\widetilde f_i:F\to\widehat A_i$.
+Give the cover the lifted triangulation.  The lift is simplicial on
+the already fixed triangulation of $F$, since on each domain simplex
+it is the lift of a simplicial map into a single target simplex.
+No new subdivision of $F$ is needed.
+
+Set
+\[
+  A_{i+1}=\widetilde f_i(F),
+  \qquad L_{i+1}=(p_i^*L_i)|_{A_{i+1}},
+\]
+and corestrict the lift to $f_{i+1}:F\to A_{i+1}$.
+Lemma~\ref{lem:cyclic-cover}, followed by
+\eqref{eq:subcomplex-injection}, gives
+$H_2(A_{i+1};L_{i+1})=0$.  Moreover,
+$f_{i+1}^*L_{i+1}\cong f_i^*L_i$, so the nontrivial monodromy
+persists.  These are precisely the conditions
+\eqref{eq:tower-invariants} needed to continue.  Figure~\ref{fig:tower}
+shows one step.
+
+\begin{figure}[tb]
+  \centering
+  \input{figures/tower}
+  \caption{One step of the tower.  The lifted map factors through its
+  finite image $A_{i+1}\subseteq\widehat A_i$, and the diagram commutes.
+  The domain $F$ keeps the same triangulation at every step; only the
+  indicated subcomplex arrow is an inclusion.}
+  \label{fig:tower}
+\end{figure}
+
+It remains to show that $s(A_i)$ increases strictly.  The restriction
+$p_i|_{A_{i+1}}$ maps onto $A_i$ and maps each
+simplex isomorphically onto a simplex.  Hence
+$s(A_{i+1})\geq s(A_i)$.  If equality held, there would be exactly
+one simplex above each simplex of $A_i$, compatibly with all faces.
+The restriction would then be a simplicial isomorphism.  Its inverse,
+followed by the inclusion into $\widehat A_i$, would give a section
+of $p_i$.  Such a section would make $(p_i)_*$ surjective, contrary
+to
+\[
+  \im(p_i)_*=\ker\phi_i\ne\pi_1(A_i).
+\]
+Thus $s(A_{i+1})>s(A_i)$.  But every simplex of $A_i$ is the image
+of a simplex in the fixed triangulation of $F$, so
+\[
+  s(A_0)<s(A_1)<s(A_2)<\cdots\leq s(F).
+\]
+This is impossible, and the theorem follows.
+\end{proof}
+
+\Needspace{5\baselineskip}
+The following consequence is the form used in the construction.
+
+\begin{corollary}[Characters and torsion]
+\label{cor:torsion}
+Let $Y$ be a finite connected CW complex of dimension at most two.
+If a character $\rho:\pi_1(Y)\to\C^\times$ satisfies
+$H_2(Y;\C_\rho)=0$, then $\rho(g)=1$ for every finite-order
+element $g\in\pi_1(Y)$.
+\end{corollary}
+
+\begin{proof}
+Suppose that $g$ has finite order $n$ and $\rho(g)\ne1$; in
+particular, $n\geq2$.  Let
+\[
+  M_n=S^1\cup_{\deg n}D^2
+\]
+be the Moore complex obtained by attaching a disk along a map of
+degree $n$.  A loop in $Y$ representing $g$ extends to a map
+$M_n\to Y$, because its $n$th power is nullhomotopic.  The cellular
+boundary $C_2(M_n;\Z)\to C_1(M_n;\Z)$ is multiplication by $n$,
+so $H_1(M_n;\Z)=\Z/n$ and $b_1(M_n;\Q)=0$.  The pullback
+character sends the circle generator to $\rho(g)\ne1$.
+
+Use the finite simplicial model for $M_n$ constructed in the proof of
+Theorem~\ref{thm:tower}, and compose its homotopy equivalence to
+$M_n$ with the map to $Y$.  This gives a finite connected simplicial
+domain with vanishing first rational Betti number and nontrivial
+pullback monodromy, contradicting that theorem.  For $n=2$ the
+standard model $M_2$ is $\RP^2$, the case needed below.
+\end{proof}
diff --git a/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/build/sections/03-plus.tex b/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/build/sections/03-plus.tex
new file mode 100644
index 0000000000000000000000000000000000000000..2d82278ba92126487d27748825e9ec201c11d808
--- /dev/null
+++ b/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/build/sections/03-plus.tex
@@ -0,0 +1,117 @@
+\section{A finite plus construction}\label{sec:plus}
+
+We next recall how a finitely normally generated perfect subgroup
+produces a finite $D(2)$ complex. This is the finite form of Quillen's
+plus construction~\cite{Quillen71} used by Mannan~\cite[Section~2]{Mannan09}.
+We give the construction because its cellular splitting proves both
+the required cohomological vanishing and the preservation of the twisted
+homology used in Section~\ref{sec:tower}.
+
+Recall that a group $K$ is \emph{perfect} if $K=[K,K]$, or equivalently
+if its abelianization $K_{\ab}$ is zero. A normal subgroup of a group
+$P$ is \emph{finitely normally generated} if it is the normal closure
+in $P$ of finitely many elements. This condition does not require
+finite generation as an abstract group.
+
+\begin{proposition}[Finite plus construction]\label{prop:plus}
+Let $B$ be a finite connected CW complex of dimension at most two,
+with $\pi_1(B)=P$. Suppose that the normal closure $K$ of elements
+$t_1,\ldots,t_r\in P$ is perfect, and put $G=P/K$.
+There is a finite complex $X\supseteq B$, obtained by attaching
+$r$ cells of dimension two and $r$ cells of dimension three, such that:
+\begin{enumerate}
+\item the inclusion induces the quotient $P\to G$ on fundamental groups;
+\item $H_i(\widetilde X;\Z)=0$ for every $i>2$, and
+      $H^3(X;M)=0$ for every left $\Z G$-module $M$;
+\item for every right $\Z G$-module $N$, the inclusion induces
+      $H_*(B;N)\cong H_*(X;N)$, with the coefficients on $B$
+      pulled back along $P\to G$.
+\end{enumerate}
+\end{proposition}
+
+\begin{proof}
+Attach two-dimensional cells along loops representing $t_1,\ldots,t_r$,
+and call the resulting complex $B'$. The van Kampen theorem gives
+$\pi_1(B')=G$. Let $\overline B$ be the full preimage of $B$ in the
+universal cover $\widetilde{B'}$. Since $P\to G$ is onto,
+$\overline B$ is connected; it is the covering of $B$ corresponding
+to $K$. Hence
+\[
+ H_1(\overline B;\Z)=K_{\ab}=0.
+\]
+
+Write $R=\Z G$ and regard cellular chains of these covers as left
+$R$-modules. The relative cellular complex
+$C_*(\widetilde{B'},\overline B)$ consists of $R^r$ in degree two
+and zero in every other degree. The homology exact sequence of the
+pair therefore gives a surjection
+\begin{equation}\label{eq:relative-surjection}
+ H_2(\widetilde{B'};\Z)
+ \longrightarrow H_2(\widetilde{B'},\overline B;\Z)=R^r.
+\end{equation}
+Choose classes $\alpha_1,\ldots,\alpha_r$ mapping to the standard
+basis vectors, where the basis records chosen lifts and orientations
+of the new cells. The space $\widetilde{B'}$ is simply connected, so
+the degree-two Hurewicz theorem represents each $\alpha_j$ by a map
+$S^2\to\widetilde{B'}$. Project these maps to $B'$ and attach one
+three-dimensional cell along each projected map. This defines $X$.
+Only finitely many cells have been added, and attaching the last cells
+does not change the fundamental group.
+
+The preimage of $B'$ in $\widetilde X$ is its universal cover, so we
+may retain the notation $\widetilde{B'}$ and $\overline B$. In the
+chosen cellular bases,
+\[
+ C_3(\widetilde X)=R^r,
+ \qquad C_2(\widetilde X)=C_2(\overline B)\oplus R^r.
+\]
+Let $q:C_2(\widetilde X)\to R^r$ be projection onto the second
+summand. The choice in \eqref{eq:relative-surjection} says exactly that
+\begin{equation}\label{eq:split-boundary}
+ q\,\partial_3=\id_{R^r}.
+\end{equation}
+Indeed, the relative cellular boundary of the $j$th three-dimensional
+cell is the $j$th basis vector; in the two-dimensional complex
+$\widetilde{B'}$, a homology class is represented by its cellular
+two-cycle without any ambiguity from three-boundaries.
+
+Equation~\eqref{eq:split-boundary} proves that $\partial_3$ is
+injective. There are no cells above dimension three, so
+$H_i(\widetilde X;\Z)=0$ for all $i>2$. It also gives the
+cohomological assertion directly. For any $R$-linear map
+$\varphi:C_3(\widetilde X)\to M$, its extension
+$\varphi q:C_2(\widetilde X)\to M$ satisfies
+\[
+ (\varphi q)\partial_3=\varphi.
+\]
+Thus the cochain boundary
+$\Hom_R(C_2(\widetilde X),M)\to\Hom_R(C_3(\widetilde X),M)$
+is onto, and $H^3(X;M)=0$.
+
+Finally, the relative cellular complex of $(\widetilde X,\overline B)$
+is
+\begin{equation}\label{eq:relative-complex}
+ 0\longrightarrow R^r\xrightarrow{\,\id\,}R^r
+   \longrightarrow0,
+\end{equation}
+in degrees three and two. It is contractible. The short exact sequence
+of absolute and relative cellular complexes is split in each degree,
+since the cells of $\overline B$ form subsets of the chosen cellular
+bases. Tensoring with any right $R$-module $N$ therefore leaves a short
+exact sequence of complexes whose relative term is still contractible.
+Its homology exact sequence gives
+\[
+ H_*(N\otimes_R C_*(\overline B))
+   \cong H_*(N\otimes_R C_*(\widetilde X)).
+\]
+The left side computes $H_*(B;N)$: the coefficient action of $P$
+factors through the deck group $G$ of $\overline B\to B$.
+The right side computes $H_*(X;N)$, proving the last assertion.
+\end{proof}
+
+The splitting in \eqref{eq:split-boundary} is stronger than ordinary
+third-homology vanishing: it makes the cohomology calculation work for
+every coefficient module. In the application, we will choose $B$ and
+$K$ so that a character of $G$ has vanishing second homology on $B$,
+yet detects a torsion element of $G$. Proposition~\ref{prop:plus}
+preserves that homology vanishing when passing to $X$.
diff --git a/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/build/sections/04-construction.tex b/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/build/sections/04-construction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..41e22e130ebcfbda7d0846aadfcc333f4312dc5a
--- /dev/null
+++ b/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/build/sections/04-construction.tex
@@ -0,0 +1,189 @@
+\section{The counterexample}\label{sec:construction}
+
+We now construct the presentation complex and perfect normal subgroup
+needed in Proposition~\ref{prop:plus}. The quotient must contain torsion
+detected by a character, and the corresponding local system on the
+presentation complex must have vanishing second homology.
+Perfectness will follow from
+annihilation relations with coprime coefficients in the subgroup's
+abelianization.
+Related uses of coprime annihilators for relation modules appear in
+Bridson--Tweedale~\cite[Proposition~3.3]{BridsonTweedale07}.
+
+Let $B$ be the presentation complex of
+\begin{equation}\label{eq:presentation}
+ P=\left\langle x_1,a_1,x_2,a_2,s\ \middle|\
+ \begin{aligned}
+ x_1a_1x_1^{-1}&=a_1^4,&
+ x_2a_2x_2^{-1}&=a_2^3,\\
+ x_1&=a_2^{13},&
+ s x_2 s^{-1}&=a_1^5
+ \end{aligned}\right\rangle.
+\end{equation}
+Thus $B$ has one vertex, five edges, and four two-dimensional cells.
+Define
+\[
+ t_1=x_1^2,\qquad t_2=x_2^3,\qquad
+ K=\langle\!\langle t_1,t_2\rangle\!\rangle_P,
+ \qquad G=P/K,
+\]
+where the double brackets denote normal closure in $P$.
+Sending $s$ to $1\in\Z$ and the other four generators to zero
+respects all these relations and descends to an epimorphism
+$G\to\Z$. In particular, $G$ is infinite.
+We use the commutator convention $[g,h]=ghg^{-1}h^{-1}$.
+
+\begin{lemma}\label{lem:perfect}
+The normal subgroup $K$ is perfect.
+\end{lemma}
+
+\begin{proof}
+The exponents are matched by $4^2-1=15=3\cdot5$ and
+$3^3-1=26=2\cdot13$, with $\gcd(15,26)=1$.
+Conjugation by $t_1=x_1^2$ sends $a_1$ to $a_1^{16}$, while
+conjugation by $t_2=x_2^3$ sends $a_2$ to $a_2^{27}$.
+Squaring $x_1=a_2^{13}$ and cubing $s x_2s^{-1}=a_1^5$ now give
+\begin{equation}\label{eq:commutators}
+ t_1=a_2^{26}=[t_2,a_2],
+ \qquad
+ s t_2 s^{-1}=a_1^{15}=[t_1,a_1].
+\end{equation}
+Write $K_{\ab}$ additively and let $u_j$ be the image of $t_j$.
+Conjugation makes $K_{\ab}$ a left $\Z G$-module: every element
+of $K$ acts trivially on its own abelianization. For $t\in K$,
+the image of $[t,a]$ is $(1-a)[t]$. Consequently
+\begin{equation}\label{eq:module-relations}
+ u_1=(1-a_2)u_2,
+ \qquad s u_2=(1-a_1)u_1.
+\end{equation}
+
+Since $t_1=a_2^{26}$, the element $a_2$ fixes $u_1$ and
+$a_2^{26}$ acts trivially on all of $K_{\ab}$. Applying
+$1+a_2+\cdots+a_2^{25}$ to the first equality gives
+\[
+ 26u_1=(1-a_2^{26})u_2=0.
+\]
+Likewise, $a_1$ fixes $s u_2$, because $s t_2s^{-1}=a_1^{15}$;
+and $a_1^{15}$ acts trivially on $K_{\ab}$. Applying
+$1+a_1+\cdots+a_1^{14}$ to the second equality gives
+\[
+ 15s u_2=(1-a_1^{15})u_1=0.
+\]
+The action of $s$ is invertible, so $15u_2=0$, and the first
+equality of \eqref{eq:module-relations} then gives $15u_1=0$.
+Since $15$ and $26$ are relatively prime, $u_1=0$. The second
+equality gives $s u_2=0$, hence $u_2=0$.
+
+Every element of $K$ is a product of $P$-conjugates of $t_1^{\pm1}$
+and $t_2^{\pm1}$. Thus $u_1,u_2$ generate $K_{\ab}$ under
+addition and the $P$-action. Their vanishing proves $K_{\ab}=0$.
+\end{proof}
+
+By Proposition~\ref{prop:plus}, there is a finite complex $X\supseteq B$
+with fundamental group $G$, obtained by adding two two-dimensional
+and two three-dimensional cells, such that
+\begin{equation}\label{eq:d2-for-x}
+ H_i(\widetilde X;\Z)=0\quad(i>2),\qquad
+ H^3(X;M)=0\quad\text{for every left }\Z G\text{-module }M.
+\end{equation}
+It remains to find the character that obstructs a two-dimensional model.
+
+Choose a primitive third root of unity $\zeta\in\C$. The assignments
+\begin{equation}\label{eq:character}
+ \rho(x_1)=-1,\quad \rho(a_1)=\zeta,\quad
+ \rho(x_2)=\zeta^2,\quad \rho(a_2)=-1,\quad \rho(s)=1
+\end{equation}
+satisfy the four relations of \eqref{eq:presentation}, since
+\[
+ \zeta=\zeta^4,\qquad -1=(-1)^3,\qquad
+ -1=(-1)^{13},\qquad \zeta^2=\zeta^5.
+\]
+They also send $t_1$ and $t_2$ to $1$. Hence they define a character
+$\rho:G\to\C^\times$. Denote its local system on $X$ and its
+pullback to $B$ by $L_\rho$.
+
+\Needspace{6\baselineskip}
+\begin{lemma}\label{lem:twisted-vanishing}
+The local system $L_\rho$ satisfies $H_2(X;L_\rho)=0$.
+\end{lemma}
+
+\begin{proof}
+Proposition~\ref{prop:plus} reduces the computation to $B$.
+Its twisted cellular groups in degrees two and one are $\C^4$ and
+$\C^5$, respectively. We compute the second boundary by the lifted
+edge-path sum, equivalently by Fox differentiation~\cite{Fox53}.
+In this convention the derivative of a word obeys
+\[
+ \frac{\partial(uv)}{\partial g}
+   =\frac{\partial u}{\partial g}
+      +u\frac{\partial v}{\partial g},\qquad
+ \frac{\partial g}{\partial g}=1,\qquad
+ \frac{\partial g^{-1}}{\partial g}=-g^{-1},
+\]
+and the derivatives with respect to other generators are zero.
+Evaluate their group-ring coefficients at $\rho$.
+
+Use the relators
+\[
+ \begin{split}
+ r_1&=x_1a_1x_1^{-1}a_1^{-4},\qquad
+ r_2=x_2a_2x_2^{-1}a_2^{-3},\\
+ r_3&=x_1a_2^{-13},\qquad
+ r_4=sx_2s^{-1}a_1^{-5}.
+ \end{split}
+\]
+With the edges ordered as $x_1,a_1,x_2,a_2,s$, the four boundary
+vectors, written as rows, are
+\begin{equation}\label{eq:fox-matrix}
+ D=\begin{pmatrix}
+ 1-\zeta & -2 & 0 & 0 & 0\\
+ 0&0&2&\zeta^2-1&0\\
+ 1&0&0&-1&0\\
+ 0&-(1+\zeta)&1&0&1-\zeta^2
+ \end{pmatrix}.
+\end{equation}
+Here is a direct check of the entries. For a relator
+$xax^{-1}a^{-n}$, the evaluated entries in the $x$ and $a$
+coordinates are
+\[
+ 1-\rho(a)^n,
+ \qquad \rho(x)-\sum_{j=0}^{n-1}\rho(a)^j.
+\]
+For $r_1$, the latter sum is
+$1+\zeta+\zeta^2+\zeta^3=1$; for $r_2$, it is $1-1+1=1$.
+For $r_3$, the alternating sum of thirteen terms is $1$.
+For $r_4$, the $a_1$ entry is
+$-\sum_{j=0}^4\zeta^j=-(1+\zeta)$, the $x_2$ entry is
+$\rho(s)=1$, and the $s$ entry is $1-\rho(x_2)=1-\zeta^2$.
+These give \eqref{eq:fox-matrix}.
+
+The minor using columns $x_1,a_1,x_2,s$ has determinant
+\begin{equation}\label{eq:nonzero-minor}
+ -4(1-\zeta^2)\ne0.
+\end{equation}
+Thus the four rows are independent, so the boundary
+$C_2(B;L_\rho)\to C_1(B;L_\rho)$ is injective.
+Since $B$ has no three-dimensional cells, $H_2(B;L_\rho)=0$,
+as required.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+The complex $X$ constructed above is finite and satisfies
+\eqref{eq:d2-for-x}. In $G$, the element $x_1$ has square one,
+and it is nonidentity because $\rho(x_1)=-1$. Thus its order is
+exactly two.
+
+Suppose that $X$ were homotopy equivalent to a finite connected
+complex $Y$ of dimension at most two. Transfer $\rho$ along the
+induced isomorphism of fundamental groups, giving a local system
+$L$ on $Y$. Homotopy invariance and Lemma~\ref{lem:twisted-vanishing}
+give $H_2(Y;L)=0$. But the transferred character takes the value
+$-1$ on an element of order two, contrary to
+Corollary~\ref{cor:torsion}.
+
+Concretely, a loop representing that element extends to a map
+$\RP^2\to Y$: the attaching loop of the two-dimensional cell of
+$\RP^2$ is its generator traversed twice. The pulled-back local
+system has monodromy $-1$, whereas $b_1(\RP^2;\Q)=0$.
+This is precisely the map excluded by Theorem~\ref{thm:tower}.
+\end{proof}
diff --git a/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/wall-d2-counterexample.pdf b/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/wall-d2-counterexample.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..44fd369996c1d7d3c60913f607ec793a6bc95690
--- /dev/null
+++ b/preprints/A-Counterexample-to-Walls-D2-Problem-October-6-2026/wall-d2-counterexample.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:e34de3716e64bba49880fff61808d5065477785596703bc0debf2379ad246c0c
+size 338287
diff --git a/preprints/A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026/README.md b/preprints/A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..d52a31aaeda31c98c5adc7e3ad9fb5c6bdd5b233
--- /dev/null
+++ b/preprints/A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026/README.md
@@ -0,0 +1,18 @@
+# [A Counterexample to the Group-Ring Determinant Conjecture](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 23, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026,
+  author = {{OpenAI}},
+  title = {{A Counterexample to the Group-Ring Determinant Conjecture}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026/paper.pdf}{OAI:A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026/build/main.tex b/preprints/A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026/build/main.tex
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+++ b/preprints/A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026/build/main.tex
@@ -0,0 +1,615 @@
+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage[margin=1in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{microtype}
+\usepackage{xcolor}
+\PassOptionsToPackage{hyphens}{url}
+\usepackage{hyperref}
+\usepackage{bookmark}
+\hypersetup{colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,
+  urlcolor=blue!45!black,pdfauthor={OpenAI},
+  pdftitle={A Counterexample to the Group-Ring Determinant Conjecture}}
+\input{glyphtounicode-cmex.tex}
+\pdfgentounicode=1
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\numberwithin{equation}{section}
+\newcommand{\F}{\mathbb F}
+\newcommand{\Z}{\mathbb Z}
+\newcommand{\Q}{\mathbb Q}
+\newcommand{\R}{\mathbb R}
+\newcommand{\C}{\mathbb C}
+\newcommand{\Mat}{\operatorname{M}}
+\newcommand{\diag}{\operatorname{diag}}
+\newcommand{\N}{\mathcal N}
+\setlength{\emergencystretch}{2em}
+\title{A Counterexample to the Group-Ring Determinant Conjecture}
+\author{OpenAI}
+\date{September 23, 2026}
+
+\begin{document}
+\maketitle
+\begin{abstract}
+We disprove the unrestricted group-ring Determinant Conjecture. We construct a finitely generated group $G$ and a square matrix over $\mathbb Z[G]$ that is invertible over $\mathbb Q[G]$ and has Fuglede--Kadison determinant strictly between zero and one. The logarithmic integral defining the determinant is finite.
+\end{abstract}
+
+\section{Introduction and proof overview}
+\label{sec:introduction}
+
+A nonsingular integer matrix has determinant of absolute
+value at least one. The group-ring Determinant Conjecture asks for an
+analogue when the entries are finite integer combinations of elements
+of a discrete group. The relevant determinant, introduced by Fuglede
+and Kadison~\cite{FK1952}, uses the trace of a logarithm rather than
+a finite-dimensional determinant polynomial.
+
+Here are the conventions that fix this question. Let $G$ be a discrete
+group and let $\Z[G]$ denote its ordinary group ring of finite sums.
+Use the left regular representation
+\[
+ (L_g\xi)(h)=\xi(g^{-1}h),\qquad \xi\in\ell^2(G).
+\]
+For $A\in\Mat_n(\Z[G])$, write $T_A$ for its operator on
+$\ell^2(G)^n$ and $e_i$ for the vector with $\delta_1$ in component
+$i$. The probability spectral measure of the positive square is
+\[
+ \mu_A(E)=\frac1n\sum_{i=1}^n
+ \langle \mathbf1_E(T_A^*T_A)e_i,e_i\rangle
+ \quad(E\subseteq[0,\infty)\text{ Borel}).
+\]
+The square-matrix instance of the conjecture is
+\begin{equation}\label{eq:conjecture}
+ \int_{(0,\infty)}\log t\,d\mu_A(t)\ge0.
+\end{equation}
+The zero atom is omitted and the general integral may be $-\infty$.
+The usual conjecture also permits rectangular matrices
+\cite[Conjecture 13.2]{Luck2002}; a square counterexample already
+contradicts it. Our example will have a bounded inverse, so its
+spectrum stays away from zero.
+
+Write $\N(G)$ for the group von Neumann algebra and use the
+\emph{unnormalized} matrix trace
+$\tau_n(T)=\sum_i\langle Te_i,e_i\rangle$, so $\tau_n(I_n)=n$.
+For invertible $T_A$ the determinant convention is
+\begin{equation}\label{eq:normalization}
+ \log\det_{\N(G)}(T_A)
+ =\frac12\tau_n\bigl(\log(T_A^*T_A)\bigr)
+ =\frac n2\int_{(0,\infty)}\log t\,d\mu_A(t).
+\end{equation}
+This extends the absolute value of the ordinary determinant.
+The inversion unitary $\delta_g\mapsto\delta_{g^{-1}}$ identifies
+our integral-matrix convention with right multiplication by $A^*$
+on row vectors. It fixes the identity vectors, and therefore
+preserves the corresponding positive-square spectral measures.
+
+\begin{theorem}\label{thm:main}
+There are a finitely generated discrete group $G_D$, an integer
+$n\ge1$, and $A\in\Mat_n(\Z[G_D])$ invertible over $\Q[G_D]$ such
+that
+\[
+ 0<\det_{\N(G_D)}(T_A)<1.
+\]
+The integral in Equation~\eqref{eq:conjecture} is finite and strictly
+negative. Thus the unrestricted group-ring Determinant Conjecture
+has a negative resolution.
+\end{theorem}
+
+\paragraph{The exact existence input.}
+A unital ring is directly finite if $ab=1$ implies $ba=1$.
+We use the following scalar failure of direct finiteness from the
+complete characteristic-two companion.
+
+\begin{theorem}[{\cite[Theorem 1.1]{CharacteristicTwo}}]
+\label{thm:input}
+There exist a finite field $K_2$ of characteristic two, a finitely
+generated group $H_2$, and finite sums $a_2,b_2\in K_2[H_2]$ with
+$a_2b_2=1$ and $b_2a_2\ne1$.
+\end{theorem}
+
+The companion's Section~6 specifies its field, group presentation
+and scalar sums by a terminating finite prescription. We use the
+existence theorem, not an executed numerical instance or a
+group-word decision procedure. The coefficient field is finite but
+is not asserted to be $\F_2$. There is no torsion-free hypothesis;
+the companion construction uses torsion. All steps after this exact
+input are proved below. In particular, a result only in odd
+characteristic would not supply the binary premise used here.
+
+\paragraph{Context and mathematical ancestry.}
+Determinant bounds are part of the theory of $L^2$-invariants
+\cite{Luck2002}. Approximation and closure arguments established
+such bounds for large classes of groups, including amenable groups
+\cite[Theorem 1.21, with its erratum]{Schick2001}; Elek and Szab\'o proved the
+determinant bound for sofic groups
+\cite[Theorem 5]{ElekSzabo2005}. Their proof uses finite graph
+approximations and the integrality of products of nonzero eigenvalues
+of positive integral matrices. Our construction instead starts from
+failure of direct finiteness in characteristic two. The cited sofic theorem implies, as a
+separate consequence, that its resulting group is not sofic.
+
+The lower-bound question is distinct from determinant
+\emph{approximation}. Kammeyer's counterexample to a finite-quotient
+limiting formula has determinant $2$ over the discrete Heisenberg
+group, while its principal congruence quotients of levels $3^i$, $i\ge1$,
+have determinants at most $5^{1/3}$
+\cite[Theorem 2 and Section 2]{Kammeyer2026}. That discrepancy does not violate
+the lower bound one.
+
+The finite-coordinate projection construction below is related to
+the commuting wreath-product projections of Dicks and Schick
+\cite[Section 3]{DicksSchick2002} and of Lehner, Neuhauser and
+Woess~\cite[Section 2]{LehnerNeuhauserWoess2008}. Here the
+projections separate one coefficient of a binary defect; we do not
+import a spectral-distribution theorem from those works. The
+analytic step is the classical tracial logarithmic derivative
+\cite[Lemma 2]{FK1952}, also present in the path-integral determinant
+viewpoint of de la Harpe and Skandalis
+\cite[Lemma 1 and Proposition 2(d)]{deLaHarpeSkandalis1984}. We give its proof and its
+application to block shears locally.
+
+\paragraph{How a binary defect produces a real trace gap.}
+Set $d=[K_2:\F_2]$. Representing multiplication by field elements
+on the $d$-dimensional binary vector space $K_2$ gives matrices
+$X,Y\in\Mat_d(\F_2[H_2])$ with $XY=I_d$ and a nonzero idempotent
+$C=I_d-YX$. One entry of $C$ has a nonzero coefficient. We enlarge
+the group to
+\[
+ G_D=\left(\bigoplus_{h\in H_2}C_3\right)\rtimes H_2,
+\]
+where $H_2$ acts by left translation on the coordinate labels. Averaging over the
+cyclic group at one coordinate and taking complementary averages
+at finitely many other coordinates isolates that coefficient.
+The result is a rational orthogonal projection $P$ of size $d+1$
+with trace $d+\theta$, for a positive rational number $\theta$.
+Its reduction modulo two nevertheless factors through $d$ columns.
+Section~\ref{sec:projection} proves the coefficient separation and
+this factorization in full.
+
+\paragraph{How the gap becomes an integral determinant.}
+For a general rational projection $P$ of size $m$, suppose an odd
+integer $D$ clears its denominators and $\bar P=uv$ factors through
+$r$ columns over $\F_2[G]$. Choose a power $q$ of two congruent to
+one modulo $D$. Section~\ref{sec:lifting} lifts the factorization modulo $q$
+by a finite geometric series in a corner ring. Arbitrary integral
+lifts then give an integral block matrix with an explicit rational
+inverse. Section~\ref{sec:analysis} removes its two block shears
+without changing the trace of the logarithm of its positive square.
+The remaining diagonal matrix expands the first $r$ coordinate
+summands by $q$ and contracts the range of
+$P$ by $q^{-1}$, giving
+\[
+ \log\det(T_A)=(r-\tau_m(P))\log q.
+\]
+Finally Section~\ref{sec:assembly} uses $r=d$ and
+$\tau_{d+1}(P)=d+\theta$ to obtain the exact value $q^{-\theta}$.
+The projection construction, modular lifting and trace calculation
+have distinct roles; none replaces the scalar existence input.
+
+\section{From the scalar defect to a projection}
+\label{sec:projection}
+
+We first produce the trace gap. All group-ring sums in this section
+are finite, and matrix multiplication keeps the indicated order.
+For a group-ring sum $c$, write $[1]c$ for its identity coefficient.
+The involution $*$ inverts group elements and conjugates coefficients;
+on matrices it also transposes the indices. Thus an orthogonal
+projection satisfies $P=P^*=P^2$ in the regular representation.
+
+\begin{lemma}\label{lem:binary}
+Let $K_2$ be a finite field of characteristic two and $H_2$ a group.
+For $d=[K_2:\F_2]$, there is an injective unital ring homomorphism
+\[
+ \rho:K_2[H_2]\longrightarrow\Mat_d(\F_2[H_2]).
+\]
+For $a_2,b_2\in K_2[H_2]$ with $a_2b_2=1\ne b_2a_2$, their images $X,Y$ satisfy
+$XY=I_d$ and $C:=I_d-YX\ne0$, with
+\begin{equation}\label{eq:defect}
+ C^2=C,\qquad XC=0,\qquad CY=0.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Choose a binary basis of $K_2$. The matrix of multiplication by
+$\lambda\in K_2$ is a unital ring representation; it is injective
+because its value on $1\in K_2$ recovers $\lambda$. Apply this map
+coefficientwise to a group-ring sum. The coefficient matrices
+commute with the formal group symbols, so products are preserved.
+Comparing the coefficient matrix at each group element proves
+injectivity. Thus $XY=I_d$ and $I_d-YX\ne0$. Now
+$(YX)^2=Y(XY)X=YX$, $X(I_d-YX)=X-(XY)X=0$ and
+$(I_d-YX)Y=Y-Y(XY)=0$, proving all the identities.
+\end{proof}
+
+\begin{proposition}\label{prop:separation}
+Let $X,Y\in\Mat_d(\F_2[H_2])$, $d\ge1$, satisfy
+$XY=I_d\ne YX$. In the group
+$G_D=(\bigoplus_{h\in H_2}C_3)\rtimes H_2$, where $H_2$ acts
+by left translation on coordinate labels, there is an orthogonal
+projection $P\in\Mat_{d+1}(\Z[1/3][G_D])$ and a finite set
+$S\subseteq H_2\setminus\{1\}$ such that
+\[
+ \tau_{d+1}(P)=d+\theta,\qquad
+ \theta=\frac13\left(\frac23\right)^{|S|}>0,
+ \qquad \bar P=uv,
+\]
+where $u$ has size $(d+1)\times d$ and $v$ has size $d\times(d+1)$
+over $\F_2[G_D]$. The integer $D=3^{1+|S|}$ clears all coefficients
+of $P$.
+\end{proposition}
+
+\begin{proof}
+Put $C=I_d-YX$. Choose an entry $C_{ij}$ and a group element
+$h_0$ whose coefficient in that entry is nonzero. The coefficient
+is $1$ because the field is $\F_2$. Left multiplication by
+$h_0^{-1}$ therefore gives
+\begin{equation}\label{eq:support}
+ h_0^{-1}C_{ij}=1+\sum_{s\in S}s
+\end{equation}
+for a finite $S\subseteq H_2\setminus\{1\}$, possibly empty.
+
+To isolate the identity term in Equation~\eqref{eq:support}, we
+seek a nonzero rational projection $p$ with $psp=0$ for every
+$s\in S$. The cyclic coordinates provide complementary projections
+that enforce these vanishing products. Let $z_h$ generate the cyclic
+coordinate labelled by $h$ and put
+\[
+ p_h=\frac{1+z_h+z_h^2}{3},\qquad
+ p=p_1\prod_{s\in S}(1-p_s).
+\]
+The $p_h$ are self-adjoint idempotents: averaging the nine products
+in $(1+z_h+z_h^2)^2$ gives three copies of each element of $C_3$.
+They commute because distinct coordinates commute. Consequently
+$p=p^*=p^2$, and the displayed product shows that $3^{1+|S|}p$
+is integral. This is the elementary commuting-coordinate method
+in the works cited above; compare
+\cite[Lemmas 3.1 and 3.4]{DicksSchick2002}.
+
+Conjugation obeys $s p_hs^{-1}=p_{sh}$. For $s\in S$, the
+product $sps^{-1}$ contains $p_s$, while $p$ contains $1-p_s$.
+All these coordinate factors commute, so
+\begin{equation}\label{eq:separation}
+ psp=p(sps^{-1})s=0.
+\end{equation}
+This argument does not need disjointness of the two collections of
+coordinate labels. Inverse pairs or finite-order elements in $S$
+cause no change.
+
+The original coordinates $\{1\}\cup S$ are distinct. The coefficient
+of the identity in their product is therefore the product of the
+individual identity coefficients. Hence
+\begin{equation}\label{eq:theta}
+ \tau_1(p)=[1]p=\frac13\prod_{s\in S}\frac23=\theta>0.
+\end{equation}
+For empty $S$ this means $p=p_1$ and $\theta=1/3$.
+
+The map $\Z[1/3]\to\F_2$ is defined because $3$ is odd. Reduce
+the projection modulo two and combine
+Equations~\eqref{eq:support} and~\eqref{eq:separation} to obtain
+\[
+ \bar p h_0^{-1} C_{ij}\bar p=\bar p.
+\]
+Let $f$ be the row of length $d$ with entry $\bar p h_0^{-1}$
+at position $i$ and zero elsewhere. Let $w$ be the column with
+entry $\bar p$ at position $j$ and zero elsewhere. Thus
+$fCw=\bar p$. Define
+\[
+ P=\diag(I_d,p),\qquad
+ u=\begin{pmatrix}X\\fC\end{pmatrix},\qquad
+ v=\begin{pmatrix}Y&Cw\end{pmatrix}.
+\]
+Using Equation~\eqref{eq:defect} in this order gives
+\[
+ uv=\begin{pmatrix}XY&XCw\\fCY&fC^2w\end{pmatrix}
+   =\diag(I_d,\bar p)=\bar P.
+\]
+The rational trace is $d+\theta$ by Equation~\eqref{eq:theta}.
+It is the rational orthogonal projection, not its binary reduction,
+that has this strictly positive extra trace.
+\end{proof}
+
+We have obtained a projection of real trace greater than $d$ but a
+binary factorization through $d$ columns. The next step produces
+a factorization through the same number of columns modulo a power of two, so
+that it can be inserted into an integral matrix.
+
+\section{Lifting the factorization and preserving integrality}
+\label{sec:lifting}
+
+We now work with any group $G$, an orthogonal rational projection
+$P\in\Mat_m(\Q[G])$, and a factorization of $\bar P$ through $r$
+columns over $\F_2[G]$. Assume that an odd integer $D$ clears the
+denominators of $P$. Here orthogonality means $P^2=P=P^*$, where
+$*$ transposes the matrix and inverts group elements.
+
+For a power $q>1$ of two, the trace gap suggests the absolute value
+of the operator we want: expand $r$ coordinate summands by $q$ and
+contract the range of $P$ by $q^{-1}$. Its
+logarithmic determinant should be $(r-\tau_m(P))\log q$, which is
+negative when $\tau_m(P)>r$. The difficulty is to realize this
+determinant by an integral matrix. We will add block shears whose
+off-diagonal entries come from a factorization of $P$ modulo $q$.
+The analytic justification that these shears preserve the determinant
+is given in Section~\ref{sec:analysis}.
+
+The first lemma lifts the binary factorization to every
+power $q$ of two; only idempotence is needed for this step.
+
+\begin{lemma}\label{lem:lift}
+Let $G$ be a group, $m,r\ge1$, $D$ an odd positive integer, and
+$P\in\Mat_m(\Q[G])$ satisfy $P^2=P$ and
+$DP\in\Mat_m(\Z[G])$. Suppose $\bar P=uv$ over $\F_2[G]$, where $u$ and $v$
+have respective sizes $m\times r$ and $r\times m$. For $q=2^k$,
+$k\ge1$, the reduction $e=P_q$ has a factorization
+$e=u_qv_q$ of the same sizes over $(\Z/q\Z)[G]$.
+\end{lemma}
+
+\begin{proof}
+Since $D$ is odd, every coefficient of $P$ has a well-defined
+reduction modulo $q$. Set $\mathcal R=(\Z/q\Z)[G]$ and choose
+arbitrary finite lifts $u_0,v_0$ of $u,v$ over $\mathcal R$.
+In the corner ring $e\Mat_m(\mathcal R)e$, whose identity is $e$,
+put
+\[
+ Z=eu_0v_0e,\qquad W=e-Z.
+\]
+Modulo two, $Z$ is $\bar Puv\bar P=\bar P^3=\bar P$.
+Every coefficient of $W$ is therefore even. A product of $k$
+such coefficients is divisible by $2^k$, giving $W^k=0$.
+This is nilpotence in the even ideal of the corner ring; the
+corner ring itself need not be nilpotent.
+
+Because $Z=e-W$, the finite geometric sum
+\[
+ Q=e+W+\cdots+W^{k-1}
+\]
+satisfies $QZ=ZQ=e-W^k=e$. When $k=1$, this sum is just $e$.
+Thus $Q$ is a corner inverse of $Z$, not a claimed inverse in
+the entire matrix ring. Set
+\[
+ u_q=Qeu_0,\qquad v_q=v_0e.
+\]
+Then $u_qv_q=Qeu_0v_0e=QZ=e$. Only finite sums and products
+were used, so the resulting supports remain finite.
+\end{proof}
+
+For the integral construction, choose $q=2^k>1$ with
+$q\equiv1\pmod D$. Such a power exists because two is a unit
+modulo the odd integer $D$; for $D=1$, any $2^k>1$ works.
+Choose integral lifts $U,V$ of the factors from Lemma~\ref{lem:lift}.
+The diagonal operator with the desired expansion and contraction is
+\[
+ J=-q^{-1}P-(I_m-P),\qquad B=\diag(qI_r,J).
+\]
+The minus signs leave its positive square unchanged and will produce
+the subtraction $UV-P$ needed for integrality. Prescribe the
+upper-left block $qI_r$ and off-diagonal blocks $V,U$. Eliminating
+these off-diagonal blocks subtracts $U(qI_r)^{-1}V=UV/q$ from the
+lower-right block. To leave $J$ as the Schur complement, that block
+must therefore be $UV/q+J$. Thus the factorization modulo $q$ is exactly what is
+needed to cancel the denominators in this block.
+
+\begin{lemma}\label{lem:integral}
+Suppose $P=P^*=P^2\in\Mat_m(\Q[G])$, an odd $D>0$ clears its
+denominators, and $\bar P=uv$ factors through $r\ge1$ columns.
+Let $q=2^k>1$ satisfy $q\equiv1\pmod D$. Choose arbitrary
+integral lifts $U,V$ of the factors supplied by Lemma~\ref{lem:lift}.
+Then
+\begin{equation}\label{eq:integral-matrix}
+ A=\begin{pmatrix}
+ qI_r&V\\
+ U&q^{-1}\bigl(UV-P-q(I_m-P)\bigr)
+ \end{pmatrix}
+ \in\Mat_{r+m}(\Z[G])
+\end{equation}
+has an inverse in $\Mat_{r+m}(\Q[G])$.
+\end{lemma}
+
+\begin{proof}
+The matrix $P+q(I_m-P)=qI_m+(1-q)P$ is integral because
+$D\mid(q-1)$. Its reduction modulo $q$ is $e=P_q$, which is
+also the reduction of $UV$. Thus every coefficient of
+$UV-P-q(I_m-P)$ is an integer divisible by $q$; the bottom-right
+block of Equation~\eqref{eq:integral-matrix} is integral.
+
+For $J$ and $B$ defined above, the complementary projection identities give
+$J^{-1}=-qP-(I_m-P)$. For $x\in[0,1]$ put
+\[
+ L_x=\begin{pmatrix}I_r&0\\xU/q&I_m\end{pmatrix},\qquad
+ R_x=\begin{pmatrix}I_r&xV/q\\0&I_m\end{pmatrix}.
+\]
+Their inverses negate the off-diagonal blocks, and multiplication
+without interchanging any factors gives
+\begin{equation}\label{eq:path}
+ M(x):=L_xBR_x=
+ \begin{pmatrix}qI_r&xV\\xU&x^2UV/q+J\end{pmatrix}.
+\end{equation}
+In particular $A=M(1)$, with the rational group-ring inverse
+$R_1^{-1}B^{-1}L_1^{-1}$. For general real $x$, both $M(x)$ and
+its inverse lie in the real group ring and depend continuously
+differentiably on $x$. No identities $PU=U$, $VP=V$, or
+$V=U^*$ are required of the chosen integer lifts.
+\end{proof}
+
+The construction is integral at its endpoint, though its simplifying
+path need not be integral. We
+now show that this path preserves the logarithmic determinant.
+
+\section{The trace calculation}
+\label{sec:analysis}
+
+Let $\mathcal A_n$ be the operator-norm closure of
+$\Mat_n(\C[G])$ in its left regular representation. For finite
+group-ring matrices define
+\[
+ \tau_n(C)=\sum_{i=1}^n[1]C_{ii}.
+\]
+The identity-coefficient formula gives
+$\tau_n(CD)=\tau_n(DC)$ by summing
+$(C_{ij})_g(D_{ji})_{g^{-1}}$ over $i,j,g$. Moreover
+$|\tau_n(C)|\le n\|C\|$ and
+$\tau_n(C^*)=\overline{\tau_n(C)}$. Thus it extends to a
+continuous trace on the unital $C^*$-algebra $\mathcal A_n$.
+
+\begin{lemma}\label{lem:logderivative}
+For a continuously differentiable path
+$M:[0,1]\to\mathcal A_n$ of invertible elements,
+\[
+ \frac{d}{dx}\tau_n\bigl(\log(M(x)^*M(x))\bigr)
+ =2\operatorname{Re}\tau_n\bigl(M'(x)M(x)^{-1}\bigr).
+\]
+\end{lemma}
+
+\begin{proof}
+Put $N=M^*M$. Continuity of $M$ and its inverse on the compact
+interval gives constants $0<\delta<c$ with
+$\delta I\le N(x)\le cI$ for all $x$. Write
+$W=I-N/c$, so $\|W\|\le\rho=1-\delta/c<1$. Functional calculus
+gives the uniformly norm-convergent series
+\[
+ \log N=(\log c)I-\sum_{j\ge1}\frac{W^j}{j}.
+\]
+Its derivative series is also uniformly norm-convergent, since
+\[
+ \left\|\frac{d}{dx}\frac{W^j}{j}\right\|
+ \le\sup_{y\in[0,1]}\|W'(y)\|\rho^{j-1}.
+\]
+In the derivative of $W^j$, cyclicity makes the traces of its
+$j$ terms equal; commutation of $N$ and $N'$ is not assumed.
+Using $W'=-N'/c$ therefore gives
+\[
+ \frac{d}{dx}\tau_n(\log N)
+ =\frac1c\sum_{j\ge1}\tau_n(W^{j-1}N')
+ =\tau_n(N^{-1}N').
+\]
+Since $N'=M'^*M+M^*M'$, the second summand contributes
+$\tau_n(M'M^{-1})$. Cyclicity and
+$MN^{-1}=(M^*)^{-1}$ turn the first into
+\[
+ \tau_n((M^*)^{-1}M'^*)
+ =\overline{\tau_n(M'M^{-1})}.
+\]
+Their sum is the asserted real part. This is the tracial
+logarithmic derivative of~\cite[Lemma 2]{FK1952}, written with
+the unnormalized trace used here.
+\end{proof}
+
+\begin{lemma}\label{lem:shears}
+Let $r,m\ge1$ and let $B\in\mathcal A_{r+m}$ be invertible.
+Let $S,T$ have respective sizes $m\times r$ and $r\times m$ over
+the operator-norm closure of $\C[G]$. For $x\in[0,1]$, let
+$L_x,R_x$ be lower and upper block shears with diagonal blocks
+$I_r,I_m$ and off-diagonal blocks $xS,xT$, respectively.
+Then $\tau_{r+m}(\log((L_xBR_x)^*(L_xBR_x)))$ is constant.
+\end{lemma}
+
+\begin{proof}
+Write $M=L_xBR_x$. Both shears have inverses obtained by negating
+their off-diagonal blocks. The ordered product rule gives
+\[
+ M'M^{-1}=L_x'L_x^{-1}
+ +(L_xB)(R_x'R_x^{-1})(L_xB)^{-1}.
+\]
+The matrices $L_x'L_x^{-1}$ and $R_x'R_x^{-1}$ have zero diagonal
+blocks, so each has trace zero. Cyclicity gives trace zero also for
+the conjugate $(L_xB)(R_x'R_x^{-1})(L_xB)^{-1}$.
+Lemma~\ref{lem:logderivative} now gives zero derivative.
+\end{proof}
+
+\begin{proposition}\label{prop:criterion}
+Let $G$ be a discrete group, $m,r\ge1$, and let
+$P=P^*=P^2\in\Mat_m(\Q[G])$. Suppose an odd $D>0$ clears its
+denominators and $\bar P=uv$ with
+$u\in\Mat_{m,r}(\F_2[G])$ and
+$v\in\Mat_{r,m}(\F_2[G])$. For every $q=2^k>1$ congruent to one
+modulo $D$, the matrix $A$ of Lemma~\ref{lem:integral} satisfies
+\begin{align}
+ \tau_{r+m}(\log(T_A^*T_A))
+   &=2(r-\tau_m(P))\log q,\label{eq:criterion-trace}\\
+ \log\det_{\N(G)}(T_A)&=(r-\tau_m(P))\log q.
+ \label{eq:criterion-det}
+\end{align}
+It is integral and rationally invertible, and its positive-square
+spectral measure has support in a compact subinterval of $(0,\infty)$.
+\end{proposition}
+
+\begin{proof}
+Lemma~\ref{lem:integral} has already established integrality and
+the finite-support rational inverse. Apply Lemma~\ref{lem:shears}
+to Equation~\eqref{eq:path}. At $x=0$ the matrix is
+$B=\diag(qI_r,-q^{-1}P-(I_m-P))$. Orthogonality gives
+\[
+ B^*B=\diag(q^2I_r,q^{-2}P+I_m-P),\qquad
+ \log(B^*B)=2\log q\,\diag(I_r,-P).
+\]
+Taking the trace and using shear invariance proves
+Equation~\eqref{eq:criterion-trace}; half this trace is
+Equation~\eqref{eq:criterion-det}. The regular representation
+of the rational group-ring inverse is bounded, so
+\[
+ \|T_A^{-1}\|^{-2}I\le T_A^*T_A\le\|T_A\|^2I.
+\]
+Its logarithm is therefore bounded. In the probability measure
+convention from the introduction this also gives
+\begin{equation}\label{eq:criterion-integral}
+ \int_{(0,\infty)}\log t\,d\mu_A(t)
+ =\frac{2(r-\tau_m(P))\log q}{r+m}.
+\end{equation}
+\end{proof}
+
+\section{Assembly and exact determinant}
+\label{sec:assembly}
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+Take $K_2,H_2,a_2,b_2$ from Theorem~\ref{thm:input}. Apply
+Lemma~\ref{lem:binary} with $d=[K_2:\F_2]\ge1$ and then
+Proposition~\ref{prop:separation}. It gives the group
+$G_D=(\bigoplus_{H_2}C_3)\rtimes H_2$ and a projection $P$
+of size $d+1$ whose binary reduction factors through $d$ columns.
+Its trace is $d+\theta$, where
+\[
+ \theta=\frac13\left(\frac23\right)^{|S|}>0,
+ \qquad D=3^{1+|S|}.
+\]
+The odd integer $D$ clears the coefficients of $P$. Choose
+$k\ge1$ with $q=2^k\equiv1\pmod D$.
+Proposition~\ref{prop:criterion}, with $r=d$ and $m=d+1$,
+produces $A\in\Mat_{2d+1}(\Z[G_D])$ with a rational group-ring
+inverse and
+\[
+ \log\det_{\N(G_D)}(T_A)=-\theta\log q.
+\]
+Since $q>1$ and $\theta>0$, this proves
+\begin{equation}\label{eq:final-det}
+ 0<\det_{\N(G_D)}(T_A)=q^{-\theta}<1.
+\end{equation}
+With $n=2d+1$, Equation~\eqref{eq:criterion-integral} reads
+\begin{equation}\label{eq:final-integral}
+ \int_{(0,\infty)}\log t\,d\mu_A(t)
+ =-\frac{2\theta\log q}{2d+1}<0.
+\end{equation}
+It is finite because of the spectral bounds in
+Proposition~\ref{prop:criterion}. Finally, a finite generating
+set of $H_2$ together with $z_1$ generates $G_D$, since
+$z_h=hz_1h^{-1}$. Thus $G_D$ is finitely generated.
+\end{proof}
+
+The target group differs from the companion's group and contains
+the cyclic coordinates of order three. The construction has a
+finite negative logarithmic determinant and no zero spectral mass;
+it is a failure of the asserted lower bound, not of determinant
+class. Applying the sofic determinant bound of
+\cite[Theorem 5]{ElekSzabo2005} to the positive integral matrix
+$A^*A$ proves that $G_D$ is not sofic. This last implication is an
+external consequence, not an ingredient in constructing $A$.
+
+\begingroup
+\raggedright
+\bibliographystyle{plain}
+\bibliography{references}
+\endgroup
+\end{document}
diff --git a/preprints/A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026/build/references.bib b/preprints/A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026/build/references.bib
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@@ -0,0 +1,108 @@
+@book{Luck2002,
+  author = {Wolfgang L{\"u}ck},
+  title = {{$L^2$-Invariants: Theory and Applications to Geometry and $K$-Theory}},
+  series = {Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge},
+  volume = {44},
+  publisher = {Springer},
+  year = {2002},
+  doi = {10.1007/978-3-662-04687-6}
+}
+
+@article{ElekSzabo2005,
+  author = {G{\'a}bor Elek and Endre Szab{\'o}},
+  title = {Hyperlinearity, essentially free actions and {$L^2$}-invariants.
+           {The} sofic property},
+  journal = {Mathematische Annalen},
+  volume = {332},
+  pages = {421--441},
+  year = {2005},
+  doi = {10.1007/s00208-005-0640-8}
+}
+
+@article{FK1952,
+  author = {Bent Fuglede and Richard V. Kadison},
+  title = {Determinant theory in finite factors},
+  journal = {Annals of Mathematics},
+  series = {2},
+  volume = {55},
+  number = {3},
+  pages = {520--530},
+  year = {1952},
+  doi = {10.2307/1969645}
+}
+
+@misc{CharacteristicTwo,
+  author = {{OpenAI}},
+  title = {{A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Characteristic Two}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026/paper.pdf}{OAI:A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026}},
+  year = {2026},
+  note = {Theorem 1.1 and Section 6}
+}
+
+@misc{Kammeyer2026,
+  author = {Holger Kammeyer},
+  title = {A counterexample to the determinant approximation conjecture},
+  year = {2026},
+  eprint = {2609.15567},
+  howpublished = {\href{https://arxiv.org/abs/2609.15567v1}{arXiv:2609.15567v1}},
+  archivePrefix = {arXiv},
+  primaryClass = {math.GR},
+  note = {Version 1, September 14, 2026},
+  url = {https://arxiv.org/abs/2609.15567v1}
+}
+
+@article{DicksSchick2002,
+  author = {Warren Dicks and Thomas Schick},
+  title = {The spectral measure of certain elements of the complex group ring of a wreath product},
+  journal = {Geometriae Dedicata},
+  volume = {93},
+  pages = {121--137},
+  year = {2002},
+  doi = {10.1023/A:1020381532489},
+  eprint = {math/0107145},
+  archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/math/0107145}
+}
+
+@article{LehnerNeuhauserWoess2008,
+  author = {Franz Lehner and Markus Neuhauser and Wolfgang Woess},
+  title = {On the spectrum of lamplighter groups and percolation clusters},
+  journal = {Mathematische Annalen},
+  volume = {342},
+  number = {1},
+  pages = {69--89},
+  year = {2008},
+  doi = {10.1007/s00208-008-0222-7},
+  eprint = {0712.3135},
+  archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/0712.3135}
+}
+
+@article{deLaHarpeSkandalis1984,
+  author = {de la Harpe, Pierre and Skandalis, Georges},
+  title = {D\'eterminant associ\'e \`a une trace sur une alg\`ebre de {Banach}},
+  journal = {Annales de l'Institut Fourier},
+  volume = {34},
+  number = {1},
+  pages = {241--260},
+  year = {1984},
+  doi = {10.5802/aif.958},
+  url = {https://www.numdam.org/articles/10.5802/aif.958/}
+}
+
+@article{Schick2001,
+  author = {Thomas Schick},
+  title = {{$L^2$}-determinant class and approximation of {$L^2$}-{Betti} numbers},
+  journal = {Transactions of the American Mathematical Society},
+  volume = {353},
+  number = {8},
+  pages = {3247--3265},
+  year = {2001},
+  doi = {10.1090/S0002-9947-01-02699-X},
+  eprint = {math/9807032},
+  archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/math/9807032v2},
+  note = {With the appended erratum in
+          \href{https://arxiv.org/abs/math/9807032v2}{arXiv:math/9807032v2}}
+}
diff --git a/preprints/A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026/paper.pdf b/preprints/A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026/paper.pdf
new file mode 100644
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--- /dev/null
+++ b/preprints/A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:3524b6f2390f17d5af1f06d76bfa0a6560cbdc3cb6d1d316fcdf17a715d0de40
+size 311036
diff --git a/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/README.md b/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..6eb3c1e279bc858aabd08a8790f91a072b81118b
--- /dev/null
+++ b/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/README.md
@@ -0,0 +1,20 @@
+# [A Counterexample to the Infinite Matroid Packing/Covering Conjecture](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 24, 2026
+
+## Citation
+
+To cite this manuscript:
+
+```bibtex
+@misc{OAI:A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026,
+  author = {{OpenAI}},
+  title = {{A Counterexample to the Infinite Matroid Packing/Covering Conjecture}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/paper.pdf}{OAI:A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/build/figures/double-ray.tex b/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/build/figures/double-ray.tex
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diff --git a/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/build/paper.tex b/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/build/paper.tex
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+\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref}
+\hypersetup{pdftitle={A Counterexample to the Infinite Matroid Packing/Covering Conjecture},pdfauthor={OpenAI}}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\numberwithin{equation}{section}
+\DeclareMathOperator{\cl}{cl}
+\setlist{itemsep=3pt,topsep=5pt}
+\title{A Counterexample to the Infinite Matroid\newline Packing/Covering Conjecture}
+\author{OpenAI}
+\date{September 24, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We construct in ZFC two self-dual partitional matroids on a countably infinite
+common ground set that admit neither a packing/covering partition nor an
+intersection witness. This disproves the unrestricted infinite matroid
+packing/covering and intersection conjectures and answers Jo\'o's question
+for two partitional matroids negatively. The examples are neither finitary
+nor cofinitary.
+\end{abstract}
+\begingroup
+\small
+\tableofcontents
+\endgroup
+\clearpage
+\input{sections/01-introduction}
+\input{sections/02-matroids}
+\input{sections/03-filters}
+\input{sections/04-local-bases}
+\input{sections/05-assembly}
+\input{sections/06-counterexample}
+\input{sections/07-consequences}
+\begingroup\small
+\bibliographystyle{plain}
+\bibliography{references}
+\endgroup
+\end{document}
diff --git a/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/build/references.bib b/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/build/references.bib
new file mode 100644
index 0000000000000000000000000000000000000000..36846bc5d881e63f43de69ad13f708febf8a1ff9
--- /dev/null
+++ b/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/build/references.bib
@@ -0,0 +1,83 @@
+@article{BDKPW2013,
+  author = {Bruhn, Henning and Diestel, Reinhard and Kriesell, Matthias and Pendavingh, Rudi and Wollan, Paul},
+  title = {Axioms for infinite matroids},
+  journal = {Advances in Mathematics},
+  volume = {239},
+  year = {2013},
+  pages = {18--46},
+  doi = {10.1016/j.aim.2013.01.011}
+}
+
+@article{BowlerCarmesin2015,
+  author = {Bowler, Nathan and Carmesin, Johannes},
+  title = {Matroid intersection, base packing and base covering for infinite matroids},
+  journal = {Combinatorica},
+  volume = {35},
+  number = {2},
+  year = {2015},
+  pages = {153--180},
+  doi = {10.1007/s00493-014-2953-2},
+  note = {Numbered locators refer to the \href{https://www.math.uni-hamburg.de/spag/dm/papers/PackingCovering.pdf}{author version dated 25 June 2014}}
+}
+
+@article{BowlerGeschke2016,
+  author = {Bowler, Nathan and Geschke, Stefan},
+  title = {Self-dual uniform matroids on infinite sets},
+  journal = {Proceedings of the American Mathematical Society},
+  volume = {144},
+  number = {2},
+  year = {2016},
+  pages = {459--471},
+  doi = {10.1090/proc/12667},
+  note = {Numbered locators refer to the \href{https://www.math.uni-hamburg.de/home/geschke/papers/UniformMatroid7.pdf}{author version dated 6 October 2014}}
+}
+
+@misc{GollinJoo2025,
+  author = {Gollin, J. Pascal and Jo{\'o}, Attila},
+  title = {Wild generalised truncation of infinite matroids},
+  year = {2025},
+  note = {\href{https://arxiv.org/abs/2504.05064v1}{arXiv:2504.05064v1}, 7 April 2025}
+}
+
+@article{Joo2024Packing,
+  author = {Jo{\'o}, Attila},
+  title = {On the packing/covering conjecture of infinite matroids},
+  journal = {Israel Journal of Mathematics},
+  volume = {261},
+  number = {2},
+  year = {2024},
+  pages = {765--790},
+  doi = {10.1007/s11856-023-2595-4}
+}
+
+@article{Joo2021Partitional,
+  author = {Jo{\'o}, Attila},
+  title = {Intersection of a partitional and a general infinite matroid},
+  journal = {Discrete Mathematics},
+  volume = {344},
+  number = {9},
+  year = {2021},
+  pages = {112514},
+  doi = {10.1016/j.disc.2021.112514}
+}
+
+@article{Joo2021Countable,
+  author = {Jo{\'o}, Attila},
+  title = {Proof of {Nash-Williams}' intersection conjecture for countable matroids},
+  journal = {Advances in Mathematics},
+  volume = {380},
+  year = {2021},
+  pages = {107608},
+  doi = {10.1016/j.aim.2021.107608}
+}
+
+@article{AharoniZiv1998,
+  author = {Aharoni, Ron and Ziv, Ran},
+  title = {The intersection of two infinite matroids},
+  journal = {Journal of the London Mathematical Society},
+  volume = {58},
+  number = {3},
+  year = {1998},
+  pages = {513--525},
+  doi = {10.1112/S0024610798006723}
+}
diff --git a/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/build/sections/01-introduction.tex b/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/build/sections/01-introduction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..42ba6f10f407fa296a9472ccaef1ea6e36f53fd4
--- /dev/null
+++ b/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/build/sections/01-introduction.tex
@@ -0,0 +1,202 @@
+\section{Introduction}\label{sec:introduction}
+
+A \emph{packing} of a family of matroids on a common ground set is a choice
+of pairwise disjoint spanning sets, one for each matroid. A \emph{covering}
+is a choice of independent sets, one for each matroid, whose union is the
+ground set. The infinite matroid packing/covering conjecture asks whether
+every family admits a partition of the ground set on which these two
+requirements can be satisfied in the following complementary senses.
+
+Throughout, matroids are standard infinite matroids, with the maximal
+extension axiom recalled in Section~\ref{sec:matroids}. For a matroid $M$
+on $E$ and $X\subseteq E$, write $M\!\upharpoonright X$ for restriction
+and
+\[
+ M.X:=M/(E\setminus X)=(M^*\!\upharpoonright X)^*
+\]
+for contraction onto $X$. A set is spanning if its closure is the whole
+ground set. A \emph{packing/covering partition} for $(M_i:i\in\Theta)$ is
+a partition $E=P\dot\cup C$ together with sets $S_i\subseteq P$ and
+$I_i\subseteq C$ such that
+\begin{align*}
+ S_i\cap S_j&=\varnothing &&(i\ne j),\\
+ \cl_{M_i\upharpoonright P}(S_i)&=P &&(i\in\Theta),\\
+ I_i&\text{ is independent in }M_i.C &&(i\in\Theta),\\
+ \bigcup_{i\in\Theta}I_i&=C.
+\end{align*}
+Bowler and Carmesin formulated the assertion that every set-indexed family
+has such a partition and established its relationship with infinite
+matroid intersection~\cite[Conjecture~1.3 and Corollary~3.9]{BowlerCarmesin2015}. We give a counterexample
+with two matroids.
+
+\begin{theorem}\label{thm:main}
+In ZFC, there exist a countably infinite set $E$ and two infinite matroids
+$M_0,M_1$ on $E$ such that $M_i^*=M_i$ on the same labelled ground set for
+$i=0,1$, and
+\[
+ I_0\cup I_1\ne E
+ \quad\text{for every independent }I_0\text{ in }M_0
+ \text{ and independent }I_1\text{ in }M_1.
+\]
+The pair $(M_0,M_1)$ has no packing/covering partition. Consequently, the
+infinite matroid packing/covering conjecture is false.
+\end{theorem}
+
+The last conclusion needs a separate argument: failure of a covering of
+$E$ alone does not rule out a mixed partition. For a self-dual pair,
+however, the spanning sets on $P$ and the independent sets in the
+contractions onto $C$ give disjoint spanning sets on $E$. Their complements
+are an independent covering. We prove this implication in
+Section~\ref{sec:matroids} before constructing the counterexample.
+
+\subsection*{Context and the local construction}
+
+The axioms of Bruhn, Diestel, Kriesell, Pendavingh and
+Wollan~\cite{BDKPW2013} provide an infinite matroid theory with restriction,
+contraction and duality. In particular, independent sets must extend to
+maximal independent sets inside \emph{every} prescribed subset of the
+ground set. This condition is a central part of our construction.
+
+A matroid is \emph{uniform} if replacing any element of any basis by any
+element outside that basis again gives a basis. Bowler and
+Geschke~\cite[Theorem~6 and the proof of Theorem~15]{BowlerGeschke2016} developed a basis criterion for infinite
+uniform matroids and constructed countable self-dual examples under
+Martin's axiom for countable partially ordered sets, and hence under the
+continuum hypothesis. Their criterion combines an
+antichain of bases, invariance under balanced finite changes, and a basis
+existence condition for intervals of subsets. Our assembly also uses the
+complementary interval-recursion pattern of their Lemma~12, while
+verifying the full independent-set axioms directly.
+
+Bowler and Geschke left open whether an infinite self-dual uniform matroid
+can be constructed in ZFC. Even without self-duality, ZFC existence of a
+uniform matroid of infinite rank and corank is recorded as open by Gollin
+and Jo\'o~\cite[Section~1]{GollinJoo2025}.
+
+Our local construction gives a self-dual uniform matroid $Q$ on a
+countable set in ZFC, with every basis and its complement infinite.
+It therefore answers both of these existence questions affirmatively.
+Its bases satisfy an additional ordinal inequality that obstructs
+independent coverings when copies of $Q$ are placed along a doubly
+infinite path. The two matroids in Theorem~\ref{thm:main} are countably
+infinite direct sums of these copies. In Jo\'o's terminology, a direct sum
+of uniform matroids is \emph{partitional}. His definition of uniformity is
+equivalent to requiring every subset to be independent or spanning
+\cite[Definition~1.2 and Proposition~2.2]{Joo2021Partitional}, the property
+proved for $Q$ in Remark~\ref{rem:uniform-local}. Thus both matroids in our
+pair are partitional. This distinction matters: Jo\'o proved an intersection
+theorem for an arbitrary infinite matroid paired with a direct sum of
+\emph{finitely many} uniform matroids, and asked whether intersection holds
+for two partitional matroids on a common countable ground set
+\cite[Theorem~1.4 and Question~1.5]{Joo2021Partitional}.
+
+For matroids $M,N$ on a common ground set $E$, the pair \emph{satisfies
+intersection} if there is a set $J$ independent in both matroids and a
+partition $J=J_M\dot\cup J_N$ such that
+\[
+ \cl_M(J_M)\cup\cl_N(J_N)=E.
+\]
+The unrestricted infinite Matroid Intersection Conjecture asserts that
+every pair of matroids on a common, possibly infinite, ground set satisfies this
+property~\cite[Definition~3.5 and Section~7]{BowlerCarmesin2015}.
+
+\begin{corollary}\label{cor:intersection}
+The same pair $(M_0,M_1)$ constructed in Theorem~\ref{thm:main} consists of
+partitional matroids and does not satisfy intersection. More explicitly,
+for every set $J$ independent in both $M_0$ and $M_1$ and every partition
+$J=J_0\dot\cup J_1$,
+\[
+ \cl_{M_0}(J_0)\cup\cl_{M_1}(J_1)\ne E.
+\]
+Consequently, the unrestricted infinite Matroid Intersection Conjecture is
+false in ZFC. The pair also gives a negative ZFC answer to Jo\'o's
+Question~1.5 for two partitional matroids on a common countable ground set.
+\end{corollary}
+
+\begin{proof}
+Each $M_i$ is a direct sum of countably infinitely many copies of the uniform
+matroid $Q$, so it is partitional. Bowler and Carmesin
+\cite[Proposition~3.6]{BowlerCarmesin2015} prove that, for any matroids $M,N$
+on the same ground set,
+\[
+ (M,N)\text{ satisfies intersection}
+ \quad\Longleftrightarrow\quad
+ (M,N^*)\text{ has a packing/covering partition}.
+\]
+Apply this equivalence with $M=M_0$ and $N=M_1$. Since $M_1^*=M_1$ on the same labelled ground set,
+the right-hand side concerns the identical pair $(M_0,M_1)$ and is false by
+Theorem~\ref{thm:main}. Its common ground set is countable, so it also
+satisfies the hypotheses of Jo\'o's question.
+\end{proof}
+
+Bowler and Carmesin also formulated separate Covering and Packing
+conjectures. Their universal equivalences imply that both conjectures are
+false in ZFC; Section~\ref{sec:separate-consequences} gives the precise
+formulations and deduction. This consequence concerns universal statements
+about matroid families and does not identify their witnessing families
+with the pair $(M_0,M_1)$.
+
+The examples are neither finitary nor cofinitary. Here a matroid is
+\emph{finitary} if a set is independent whenever all its finite subsets
+are independent, and \emph{cofinitary} if its dual is finitary. Nash-Williams' original intersection conjecture concerned finitary
+matroids; Aharoni and Ziv~\cite{AharoniZiv1998} gave an early published
+formulation. Jo\'o proved intersection for two finitary matroids on a countable
+ground set~\cite[Theorem~1.3]{Joo2021Countable}.
+His subsequent packing/covering theorem allows every set-indexed family
+on a countable ground set whose members are direct sums of finitary and
+cofinitary matroids~\cite[Theorem~1.2]{Joo2024Packing}.
+The examples here have no nonempty finitary or cofinitary direct summand,
+as we show in Section~\ref{sec:counterexample}, so they lie outside this
+mixed direct-sum class. Countability of the ground set is distinct from
+these structural hypotheses; the present result concerns unrestricted
+infinite matroids and does not refute Nash-Williams' original finitary
+conjecture.
+
+\subsection*{Proof overview}
+
+We first construct an ultrafilter $\mathcal U$ on a countable set $D$ and
+an ordinal-valued function $\rho$ on its members. Call a subset of $D$
+\emph{large} if it belongs to $\mathcal U$. This auxiliary ordinal rank, distinct from matroid rank, satisfies
+$\rho(X)\ge\rho(Y)$ whenever $X\subseteq Y$ are large. On two labelled copies $L,R$ of $D$, we construct
+a self-dual matroid $Q$ with the following property: if $B$ is a basis
+and its slice $B_L\subseteq D$ is large, then $D\setminus B_R$ is large
+and
+\[
+ \rho(B_L)>\rho(D\setminus B_R).
+\]
+This is the property used at the end of the proof. To build the pair,
+join consecutive integer vertices by \emph{bundles}, each a labelled
+copy of $D$, and place $Q$ on the two incident bundles at each vertex.
+Its $L$-coordinate points toward the central bundle between $-1$ and $0$,
+and its $R$-coordinate points outward. The copies at even and odd vertices
+form $M_0$ and $M_1$, respectively. In a putative independent covering, one starts with the large side of a
+central bundle and follows the path outward. Each successive copy of
+$Q$ strictly decreases the ordinal rank, an impossibility.
+
+Most of the proof constructs $Q$. Partition $D$ into rapidly growing
+finite blocks. The first ideal consists of subsets whose proportions
+in these blocks tend to zero. Within each equivalence class modulo this
+ideal, we define real-valued probes of a subset; a half-open interval
+condition on their lower or upper limits selects the bases. Finite
+changes shift these limits by their net cardinality, making basis
+exchange immediate. A separate interpolation argument supplies bases
+between suitable endpoints, including endpoints with infinite limits.
+This argument is the key to maximal extension.
+
+A second, larger ideal permits the insertion of sets that preserve the
+ordinal inequalities while separating previously chosen classes.
+Independent binary columns supply these separating sets simultaneously
+for fewer than continuum many requirements. We then address every
+interval of subsets in a transfinite recursion, inserting complementary
+classes of bases when necessary. At each stage there are fewer than
+continuum many earlier requirements; no regularity assumption on the
+continuum is used.
+
+Section~\ref{sec:filters} constructs the two ideals, the ultrafilter, the
+rank and the simultaneous reservations. Section~\ref{sec:local-bases}
+proves the interpolation properties within a single class.
+Section~\ref{sec:assembly} assembles the classes and proves all the
+matroid axioms, including maximal extension.
+Section~\ref{sec:counterexample} places $Q$ along the path and completes
+the ordinal descent argument. Section~\ref{sec:separate-consequences}
+derives the separate universal Covering and Packing consequences.
diff --git a/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/build/sections/02-matroids.tex b/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/build/sections/02-matroids.tex
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@@ -0,0 +1,119 @@
+\section{Matroids and the self-dual reduction}\label{sec:matroids}
+
+We first record the infinite-matroid conventions and explain why, for a
+self-dual pair, it suffices to rule out an independent covering of the
+whole ground set. This reduction is essential: a mixed packing/covering
+partition need not be a packing or a covering on the whole set.
+
+\subsection{Independence, closure, and duality}
+
+An infinite matroid on a set $F$ is a family $\mathcal I\subseteq 2^F$
+satisfying the following axioms
+\cite[Section~1.1]{BDKPW2013}:
+\begin{description}
+\item[(I1)] $\varnothing\in\mathcal I$.
+\item[(I2)] Every subset of a member of $\mathcal I$ belongs to $\mathcal I$.
+\item[(I3)] If $I\in\mathcal I$ is not inclusion-maximal and
+  $B\in\mathcal I$ is inclusion-maximal, there is an element
+  $e\in B\setminus I$ such that $I\cup\{e\}\in\mathcal I$.
+\item[(IM)] For every $I\in\mathcal I$ and $I\subseteq Y\subseteq F$,
+  the family $\{J\in\mathcal I:I\subseteq J\subseteq Y\}$ has an
+  inclusion-maximal member.
+\end{description}
+The inclusion-maximal independent sets are the \emph{bases}. In
+particular, (IM) requires maximal extensions inside every subset, not
+only inside $F$.
+
+For a matroid $N=(F,\mathcal I)$ and $X\subseteq F$, its closure is
+\[
+ \operatorname{cl}_N(X)=X\cup
+ \{e\in F:\text{there is an independent }I\subseteq X
+                 \text{ with }I\cup\{e\}\text{ dependent}\}.
+\]
+This operator is extensive, monotone, and idempotent
+\cite[Theorem~4.2]{BDKPW2013}. A set $S$ is \emph{spanning} if
+$\operatorname{cl}_N(S)=F$.
+Restrictions are matroids, and the complements of the bases of $N$
+are the bases of its dual $N^*$
+\cite[Theorems~3.1 and~3.4]{BDKPW2013}. We use
+\[
+ N.X=N/(F\setminus X)=(N^*\!\upharpoonright X)^*
+\]
+for contraction onto $X$. Self-duality will always mean the equality
+$N=N^*$ on the same labeled ground set.
+
+\begin{lemma}\label{lem:closure-duality}
+Let $N$ be an infinite matroid on $F$.
+\begin{enumerate}
+\item If $S\subseteq X\subseteq F$, then
+\[
+ \operatorname{cl}_{N\upharpoonright X}(S)
+   =\operatorname{cl}_N(S)\cap X.
+\]
+\item A subset $S\subseteq F$ is spanning if and only if it contains
+  a basis of $N$.
+\item A subset $S\subseteq F$ is spanning in $N$ if and only if
+  $F\setminus S$ is independent in $N^*$.
+\end{enumerate}
+\end{lemma}
+\begin{proof}
+The first assertion follows directly from the closure definition:
+for $e\in X$, independence of sets contained in $S\cup\{e\}$ is
+the same in $N$ and in its restriction to $X$.
+
+For the second, use (IM) to take a maximal independent subset $B$ of
+$S$. For every $e\in S\setminus B$, the set $B\cup\{e\}$ is
+dependent, so $S\subseteq\operatorname{cl}_N(B)$. Monotonicity and
+idempotence therefore give
+\[
+ \operatorname{cl}_N(B)=\operatorname{cl}_N(S).
+\]
+If $S$ is spanning, so is $B$. An independent spanning set is a basis:
+an independent extension by an element of its closure would contradict
+the defining independent witness for closure. Conversely, a basis spans,
+because adjoining any element outside it gives a dependent set; any set
+containing a basis consequently spans.
+
+If $S$ contains a basis $B$, then $F\setminus S$ is contained in
+the dual basis $F\setminus B$. Conversely, extend an independent
+$F\setminus S$ to a basis $B^*$ of $N^*$ using (IM).
+The primal basis $F\setminus B^*$ is contained in $S$.
+\end{proof}
+
+\subsection{From a mixed partition to an independent covering}
+
+\begin{lemma}\label{lem:selfdual-reduction}
+Let $N_0,N_1$ be matroids on the same set $F$, with $N_i=N_i^*$
+on the same labels. Suppose there is a partition
+$F=P\mathbin{\dot\cup}C$, disjoint sets $S_0,S_1\subseteq P$
+spanning $N_0\!\upharpoonright P,N_1\!\upharpoonright P$,
+respectively, and independent sets $I_i$ of $N_i.C$ such that
+$I_0\cup I_1=C$. Then $F$ can be covered by one independent set
+of each of $N_0,N_1$.
+\end{lemma}
+\begin{proof}
+Self-duality gives
+\[
+ N_i.C=(N_i\!\upharpoonright C)^*.
+\]
+Thus $T_i=C\setminus I_i$ spans $N_i\!\upharpoonright C$ by
+Lemma~\ref{lem:closure-duality}. Also
+$T_0\cap T_1=C\setminus(I_0\cup I_1)=\varnothing$.
+Consequently the two sets
+\[
+ A_i=S_i\cup T_i\qquad(i=0,1)
+\]
+are disjoint. The restriction-closure identity shows that
+$P\subseteq\operatorname{cl}_{N_i}(S_i)$ and
+$C\subseteq\operatorname{cl}_{N_i}(T_i)$, so $A_i$ spans the
+full matroid $N_i$. Its complement $F\setminus A_i$ is independent
+in $N_i^*=N_i$. Finally,
+\[
+ (F\setminus A_0)\cup(F\setminus A_1)=F
+\]
+because $A_0\cap A_1=\varnothing$.
+\end{proof}
+
+We will construct a self-dual pair with no independent covering.
+Lemma~\ref{lem:selfdual-reduction} will then exclude every mixed
+packing/covering partition for that pair.
diff --git a/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/build/sections/03-filters.tex b/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/build/sections/03-filters.tex
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@@ -0,0 +1,278 @@
+\section{An ultrafilter rank and simultaneous reservations}
+\label{sec:filters}
+
+We construct two tools on a countable set. The first is an ordinal-valued
+function on an ultrafilter whose value can be increased while preserving
+a prescribed lower bound on the set. The second supplies two disjoint
+sets that meet each of fewer than continuum many prescribed sets in
+positive upper block density. These two tools will allow us to choose bases
+with an ordinal inequality while also separating them from all earlier
+choices.
+
+\subsection{Blocks, columns, and two ideals}
+
+Let $\mathfrak c=2^{\aleph_0}$, also regarded as its initial ordinal.
+For each integer $m\ge0$, let
+\[
+ D_m=\{m\}\times\{0,1\}^{\{0,1\}^m},\qquad
+ D=\coprod_{m\ge0}D_m,\qquad w_m=|D_m|=2^{2^m}.
+\]
+Thus $D$ is countably infinite. The tags $m$ make its finite blocks
+disjoint. Their sizes satisfy
+\begin{equation}\label{eq:block-growth}
+ w_m\ge2\sum_{j<m}w_j\qquad(m\ge1).
+\end{equation}
+Indeed, equality holds for $m=1$. If the inequality holds for $m$, then
+$2\sum_{j<m+1}w_j\le3w_m\le w_m^2=w_{m+1}$, since $w_m\ge4$.
+
+The density-zero ideal on $D$ is
+\[
+ \mathcal J=\left\{X\subseteq D:
+       \frac{|X\cap D_m|}{w_m}\longrightarrow0\right\}.
+\]
+An ideal is a family closed under taking subsets and finite unions;
+we call a set \emph{positive} for the ideal if it does not belong to it.
+The displayed family is a proper ideal containing all finite sets.
+
+Let $L$ and $R$ be labelled copies of $D$, and write
+$E_0=L\mathbin{\dot\cup}R$. For $X\subseteq E_0$, its two slices,
+viewed as subsets of $D$, are denoted by $X_L$ and $X_R$.
+When writing lifts, we identify $E_0$ with $D\times\{L,R\}$.
+Write $W_m$ for the union of the two copies of $D_m$.
+The ideal $\mathcal J_0$ on $E_0$ consists of the sets whose two slices
+belong to $\mathcal J$. Equivalently,
+\[
+ X\in\mathcal J_0
+ \quad\Longleftrightarrow\quad
+ \frac{|X\cap W_m|}{|W_m|}\longrightarrow0.
+\]
+This equivalence follows because the doubled-block density is the
+average of the two nonnegative slice densities. Complements are always
+taken in the relevant ground set, $D$ or $E_0$.
+
+\begin{lemma}[Independent columns]\label{lem:columns}
+There are subsets $A_t,B_t,G_t$ of $D$, indexed by $t<\mathfrak c$,
+with the following property. For any $r$ distinct columns from these
+three families and any prescribed membership pattern on them, the
+proportion of that pattern in $D_m$ is exactly $2^{-r}$ for all
+sufficiently large $m$.
+\end{lemma}
+
+\begin{proof}
+Assign distinct infinite binary strings to all the columns; this is
+possible because $3\mathfrak c=\mathfrak c$. For a column with string
+$h$, put $(m,f)$ in that column precisely when
+$f(h\upharpoonright m)=1$.
+The prefixes of any fixed finite list of distinct strings are distinct
+for all sufficiently large $m$. Prescribing the values of $f$ at these
+$r$ arguments leaves $2^{2^m-r}$ functions among the $2^{2^m}$ choices
+in $D_m$, proving the assertion.
+\end{proof}
+
+We need a larger ideal in addition to $\mathcal J$. Define
+\begin{equation}\label{eq:larger-ideal}
+ \mathcal K=\left\{X\subseteq D:
+ X\cap\bigcap_{t\in F}A_t\in\mathcal J
+ \text{ for some finite }F\subseteq\mathfrak c\right\},
+\end{equation}
+where an empty intersection is $D$. This is the ideal generated by
+$\mathcal J$ and the complements $A_t^c$. To see the equivalence, split
+$X$ into its intersection with the positive $A$-columns in
+\eqref{eq:larger-ideal} and its remaining part, which is covered by
+their complements. In particular $\mathcal J\subseteq\mathcal K$.
+Lemma~\ref{lem:columns} shows that $D\notin\mathcal K$, so this ideal
+is proper.
+
+We will repeatedly use the following consequence. A consistent finite
+pattern prescribing positive membership in $A$-columns and arbitrary
+signs in $G$-columns is $\mathcal K$-positive. Intersecting such a pattern
+with any further finitely many positive $A$-columns still gives a
+pattern with a fixed positive eventual density, so
+\eqref{eq:larger-ideal} excludes membership in $\mathcal K$.
+
+Write $X\subseteq_{\mathcal K}Y$ if $X\setminus Y\in\mathcal K$.
+On $E_0$, let $\mathcal K_0$ be the ideal of sets whose two slices
+belong to $\mathcal K$. Thus $\mathcal J_0\subseteq\mathcal K_0$.
+The distinction between these ideals matters: changes in
+$\mathcal K_0$ will preserve an ordinal condition, whereas differences
+positive for $\mathcal J_0$ will keep the resulting bases incomparable.
+
+\subsection{The ultrafilter and its ordinal rank}
+
+An ultrafilter $\mathcal U$ on $D$ is a proper filter that contains
+exactly one of $X$ and $D\setminus X$ for every $X\subseteq D$.
+
+\begin{lemma}\label{lem:ultrafilter}
+There is an ultrafilter $\mathcal U$ on $D$ disjoint from $\mathcal K$,
+containing every $G_t$, and satisfying
+\begin{equation}\label{eq:finitely-many-columns}
+ X\in\mathcal U\quad\Longrightarrow\quad
+ \{t<\mathfrak c:X\subseteq_{\mathcal K}G_t\}
+ \text{ is finite}.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Call $Z\subseteq D$ forbidden if it is contained modulo $\mathcal K$
+in infinitely many of the $G_t$. Require a filter to contain all
+complements of members of $\mathcal K$, all $G_t$, and all complements
+of forbidden sets. We verify the finite-intersection property.
+
+Combine the finitely many $\mathcal K$-small exclusions into one set
+$N\in\mathcal K$. Let $F$ be the finite set of required positive
+$G$-indices, and let $Z_1,\ldots,Z_q$ be the forbidden sets to exclude.
+For each $j$, choose an index $t_j$ such that
+$Z_j\setminus G_{t_j}\in\mathcal K$, with the $t_j$ distinct and
+outside $F$. Each $Z_j$ has infinitely many witnessing indices, so
+these choices are possible. The pattern
+\[
+ P=\bigcap_{t\in F}G_t\cap\bigcap_{j=1}^qG_{t_j}^c
+\]
+is $\mathcal K$-positive. Moreover
+$P\cap Z_j\subseteq Z_j\setminus G_{t_j}\in\mathcal K$.
+It follows that
+\[
+ P\setminus\left(N\cup\bigcup_{j=1}^qZ_j\right)
+\]
+is still $\mathcal K$-positive: otherwise $P$ would be a finite union
+of members of $\mathcal K$. This nonempty set lies in the required
+finite intersection. Extend the resulting proper filter to an
+ultrafilter by the ultrafilter lemma. It contains no member of
+$\mathcal K$ and no forbidden set, as required.
+\end{proof}
+
+Fix such an ultrafilter. Call its members \emph{large} and its
+nonmembers \emph{small}. Small in this sense does not mean membership
+in either ideal. Every large set is $\mathcal J$-positive, and every
+complement of a $\mathcal K$-set is large.
+
+List all large subsets of $D$ as $(F_\xi)_{\xi<\mathfrak c}$,
+allowing repetitions if needed. For a large set $X$, define
+\begin{equation}\label{eq:ordinal-rank}
+ \rho(X)=\min\{\xi<\mathfrak c:
+                     F_\xi\subseteq_{\mathcal K}X\}.
+\end{equation}
+The set of indices is nonempty because $X$ occurs in the list.
+This is an auxiliary ordinal rank, distinct from matroid rank.
+
+\begin{lemma}[Rank invariance and monotonicity]\label{lem:rank}
+If $X\triangle Y\in\mathcal K$, then $X$ is large if and only if $Y$
+is large, and in that case $\rho(X)=\rho(Y)$.
+If $X,Y$ are large and $X\subseteq_{\mathcal K}Y$, then
+$\rho(X)\ge\rho(Y)$.
+\end{lemma}
+
+\begin{proof}
+The complement of $X\triangle Y$ belongs to $\mathcal U$.
+Intersecting it with either large set gives a large subset of the
+other, proving invariance of largeness. The two sets have exactly the
+same candidate indices in \eqref{eq:ordinal-rank}, proving rank
+invariance. Finally, $F_\xi\subseteq_{\mathcal K}X$ and
+$X\subseteq_{\mathcal K}Y$ imply
+$F_\xi\subseteq_{\mathcal K}Y$ by closure of the ideal under finite
+unions. Taking minima proves the last assertion.
+\end{proof}
+
+The next property supplies large sets of arbitrarily high rank inside
+any interval whose lower endpoint is small and upper endpoint is
+large.
+
+\begin{lemma}[Rank-raising thinning]\label{lem:thinning}
+Let $s\subseteq X\subseteq D$, with $s$ small and $X$ large, and let
+$\gamma<\mathfrak c$. There is a large set $u$ such that
+\[
+ s\subseteq u\subseteq X,\qquad \rho(u)>\gamma.
+\]
+\end{lemma}
+
+\begin{proof}
+For every $\xi\le\gamma$, the set
+$Y_\xi=F_\xi\cap(X\setminus s)$ is large. By
+\eqref{eq:finitely-many-columns}, it is contained modulo
+$\mathcal K$ in only finitely many $G_t$.
+The union of these finite sets of indices has cardinal at most
+$\max(\aleph_0,|\gamma+1|)<\mathfrak c$. Choose $t$ outside that union
+and put
+\[
+ u=s\cup(X\cap G_t).
+\]
+Then $u$ is large and lies between $s$ and $X$.
+If $F_\xi\subseteq_{\mathcal K}u$ for some $\xi\le\gamma$, then
+\[
+ Y_\xi\setminus G_t
+ =F_\xi\cap X\cap s^c\cap G_t^c
+ \subseteq F_\xi\setminus u\in\mathcal K,
+\]
+contrary to the choice of $t$. Thus $\rho(u)>\gamma$.
+\end{proof}
+
+This counting argument does not require regularity of
+$\mathfrak c$. For any fixed cardinal $\lambda<\mathfrak c$, a union
+of $\lambda$ finite sets has cardinal at most
+$\max(\lambda,\aleph_0)<\mathfrak c$. We will use this same estimate
+at each stage of a recursion of length $\mathfrak c$.
+
+\subsection{Two simultaneous reservations}
+
+The ordinal rank is unchanged by $\mathcal K$-small changes. We now
+show that such changes can nevertheless meet many prescribed sets
+in a way that remains positive for the smaller ideal $\mathcal J$.
+For $t<\mathfrak c$, define the following subsets of $E_0$, using the
+same labels in both copies:
+\[
+ H_{t,0}=(A_t^c\cap B_t^c)\times\{L,R\},\qquad
+ H_{t,1}=(A_t^c\cap B_t)\times\{L,R\}.
+\]
+They are disjoint, and both belong to $\mathcal K_0$ because their
+slices lie in $A_t^c$.
+
+\begin{lemma}[Simultaneous reservations]\label{lem:reservations}
+Let $\mathscr S$ be a family of fewer than $\mathfrak c$ subsets of
+$E_0$, each positive for $\mathcal J_0$. There is an index
+$t<\mathfrak c$ such that
+\[
+ S\cap H_{t,0}\notin\mathcal J_0
+ \quad\text{and}\quad
+ S\cap H_{t,1}\notin\mathcal J_0
+ \qquad(S\in\mathscr S).
+\]
+\end{lemma}
+
+\begin{proof}
+Fix first one $\mathcal J_0$-positive set $S$. Choose $\epsilon>0$
+such that the proportion of $S$ in $W_m$ is at least $\epsilon$ for
+infinitely many $m$. Call an index $t$ bad for $S$ if at least one
+of $S\cap H_{t,0}$ and $S\cap H_{t,1}$ belongs to $\mathcal J_0$.
+We prove that only finitely many indices are bad.
+
+Choose any finite list of distinct bad indices $t_1,\ldots,t_q$, and
+for each $i$ choose $k_i\in\{0,1\}$ with
+$S\cap H_{t_i,k_i}\in\mathcal J_0$. Put $H_i=H_{t_i,k_i}$.
+These cells use $2q$ distinct $A$- and $B$-columns. Avoiding $H_i$
+allows three of the four membership patterns on its two columns.
+Lemma~\ref{lem:columns} therefore gives $3^q$ avoiding patterns out of
+$4^q$ equally frequent patterns in every sufficiently late block.
+Duplicating the cells on the two copies does not change these proportions.
+In particular,
+\[
+ \frac{|W_m\setminus\bigcup_{i=1}^qH_i|}{|W_m|}
+ =\left(\frac34\right)^q
+\]
+for all sufficiently large $m$. Every point of $S\cap W_m$ either
+avoids all these cells or lies in one of the failing intersections, so
+\[
+ \frac{|S\cap W_m|}{|W_m|}
+ \le \left(\frac34\right)^q+
+       \sum_{i=1}^q\frac{|S\cap H_i\cap W_m|}{|W_m|}.
+\]
+Each summand tends to zero. Along the infinite subsequence on which
+$S$ has density at least $\epsilon$, this gives
+$\epsilon\le(3/4)^q$. Since $(3/4)^q\to0$, the size of any finite
+list of bad indices is bounded. Hence only finitely many indices are bad.
+
+For the family $\mathscr S$, take the union of these finite sets of
+bad indices. Its cardinal is less than $\mathfrak c$ by the estimate
+following Lemma~\ref{lem:thinning}. Any index outside that union
+works simultaneously for every $S\in\mathscr S$. If the family is
+empty, any index works.
+\end{proof}
diff --git a/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/build/sections/04-local-bases.tex b/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/build/sections/04-local-bases.tex
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+++ b/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/build/sections/04-local-bases.tex
@@ -0,0 +1,332 @@
+\section{Bases within a density-zero class}\label{sec:local-bases}
+
+We now construct families of candidate bases within one equivalence class
+modulo $\mathcal J_0$. The purpose is twofold: balanced finite changes
+must preserve the family, and intervals of subsets must contain or lie
+on one side of a member of the family. The latter property will supply
+maximal independent extensions in the eventual matroid.
+
+Fix $T\subseteq E_0$ such that $T$ and $T^c$ are
+$\mathcal J_0$-positive, and put
+\[
+ \mathcal C(T)=\{X\subseteq E_0:X\triangle T\in\mathcal J_0\}.
+\]
+We use $T$ as the chosen representative, or \emph{prototype}, of this class.
+Every member of $\mathcal C(T)$ and its complement are
+$\mathcal J_0$-positive. Indeed, an ideal-small member, together with its
+ideal-small symmetric difference from $T$, would make $T$ ideal-small;
+the same argument applies to complements.
+
+For $n\ge1$ and $m\ge0$, define
+\begin{align}
+ V_n&=\bigcup_{m<n}W_m,\qquad c_n=|V_n|,\notag\\
+ p_n^T(X)&=|X\cap V_n|-|T\cap V_n|,\notag\\
+ d_m^T(X)&=\frac{|X\cap W_m|-|T\cap W_m|}{|W_m|},\notag\\
+ f_n^T(X)&=p_n^T(X)+c_n\left(
+       \sup_{m\ge n}d_m^T(X)+\inf_{m\ge n}d_m^T(X)\right).
+       \label{eq:probes}
+\end{align}
+These quantities are defined for $X\in\mathcal C(T)$. The prefix count
+records a finite modification exactly once its affected blocks lie in
+$V_n$. The tail extrema allow later-block changes to influence earlier
+probes, and their sum changes sign when both $X$ and $T$ are complemented.
+Write
+\[
+ \ell_T(X)=\liminf_{n\to\infty}f_n^T(X),\qquad
+ h_T(X)=\limsup_{n\to\infty}f_n^T(X),
+\]
+allowing infinite values, and define two families
+\begin{align}
+ \mathcal B^-(T)&=\{X\in\mathcal C(T):-1<\ell_T(X)\le0\},\notag\\
+ \mathcal B^+(T)&=\{X\in\mathcal C(T):0\le h_T(X)<1\}.
+ \label{eq:local-bases}
+\end{align}
+We call these the lower-limit and upper-limit rules, respectively.
+In this section, a basis for a rule means a member of the indicated
+family; no matroid is asserted yet.
+
+\subsection{Probe estimates and continuity}
+
+\begin{lemma}[Probe properties]\label{lem:probe-properties}
+For $X,Y\in\mathcal C(T)$ the following statements hold.
+\begin{enumerate}
+\item $d_m^T(X)\to0$ and $p_n^T(X)/c_n\to0$.
+\item If $X\subseteq Y$, then
+\begin{equation}\label{eq:probe-monotone}
+ f_n^T(Y)-f_n^T(X)\ge |(Y\setminus X)\cap V_n|.
+\end{equation}
+\item Adding or deleting one element changes every probe by at most
+one. A finite modification with net change $b$ in cardinality satisfies
+$f_n^T(Y)=f_n^T(X)+b$ for all sufficiently large $n$.
+\item Complementation gives
+\[
+ f_n^{T^c}(X^c)=-f_n^T(X),\qquad
+ \mathcal B^+(T^c)=\{B^c:B\in\mathcal B^-(T)\}.
+\]
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+The first density limit follows from $X\triangle T\in\mathcal J_0$.
+Also $p_n^T(X)=\sum_{m<n}|W_m|d_m^T(X)$. For any $\varepsilon>0$,
+the terms after a fixed index have absolute density at most
+$\varepsilon$; their contribution divided by $c_n$ is at most
+$\varepsilon$, and the finite initial contribution divided by $c_n$
+tends to zero.
+
+Inclusion increases each block count, hence both tail extrema; the
+prefix increases by $|(Y\setminus X)\cap V_n|$. This proves the second
+assertion. If an element changes in a block $W_k$ with $k<n$, only the
+prefix changes. If $k\ge n$, each tail extremum changes by at most
+$1/|W_k|$. By \eqref{eq:block-growth},
+\[
+ |W_k|\ge 2\sum_{m<k}|W_m|\ge 2c_n,
+\]
+so the total change is at most one. After every block affected by a
+finite modification is in the prefix, the tails agree and the prefix
+changes by exactly $b$.
+
+Finally, complementing both $X$ and $T$ negates every prefix and block
+density. The supremum of the negatives is the negative infimum, and
+vice versa. This proves the last assertion, including its half-open
+endpoint conventions.
+\end{proof}
+
+The next continuity statement is needed both to stop finite deletion
+procedures and to pass to their decreasing limit. Its common
+density-zero bound is essential to controlling the tail extrema.
+
+\begin{lemma}[Continuity within a fixed interval]\label{lem:block-continuity}
+Suppose $I\subseteq X_k\subseteq Y$ for $k\ge1$, where
+$I,Y\in\mathcal C(T)$, and membership of every element in $X_k$
+eventually agrees with its membership in a set $X$. Then
+$X\in\mathcal C(T)$ and, for every fixed $n\ge1$,
+\[
+ f_n^T(X_k)\longrightarrow f_n^T(X).
+\]
+\end{lemma}
+
+\begin{proof}
+We have $I\subseteq X\subseteq Y$ and $Y\setminus I\in\mathcal J_0$,
+so $X\in\mathcal C(T)$. On every fixed finite union of blocks, $X_k$
+eventually agrees with $X$. On the remaining blocks,
+\[
+ |d_m^T(X_k)-d_m^T(X)|
+ \le \frac{|(Y\setminus I)\cap W_m|}{|W_m|},
+\]
+uniformly in $k$, and the right-hand side tends to zero. Thus the
+sequences of block densities converge uniformly in $m$. Supremum and
+infimum over a fixed tail change by at most the uniform difference,
+while the fixed prefix eventually stabilizes. Formula
+\eqref{eq:probes} gives the result.
+\end{proof}
+
+\begin{lemma}[Antichains and finite changes]\label{lem:local-antichain}
+Each family $\mathcal B^-(T)$ and $\mathcal B^+(T)$ contains $T$, is
+an antichain under inclusion, and is preserved by finite modifications
+that add and delete equally many elements.
+\end{lemma}
+
+\begin{proof}
+The probes of $T$ vanish. Balanced finite changes preserve the class
+and eventually preserve every probe, so they preserve either rule.
+
+Suppose $X\subsetneq Y$ both satisfy the lower-limit rule. A finite
+difference of cardinality $k\ge1$ gives
+$\ell_T(Y)=\ell_T(X)+k>0$, a contradiction. For an infinite difference,
+$|(Y\setminus X)\cap V_n|\to\infty$. The probes of $X$ are eventually
+bounded below because their lower limit is finite, so
+Lemma~\ref{lem:probe-properties} implies $f_n^T(Y)\to\infty$, again
+a contradiction. The upper-limit case follows by complementing and
+using the lower-limit rule for $T^c$.
+\end{proof}
+
+\subsection{Modifications from a positive pool}
+
+To interpolate between subsets, we need to move the limiting probes
+in either direction while remaining in $\mathcal C(T)$. A positive
+pool of available elements suffices, even though we use only a
+density-zero subset of it.
+
+\begin{lemma}[Small modifications with divergent probes]\label{lem:positive-pool}
+Let $X\in\mathcal C(T)$.
+If $G\subseteq X^c$ is $\mathcal J_0$-positive, there is
+$P\subseteq G$ in $\mathcal J_0$ such that
+$f_n^T(X\cup P)\to+\infty$.
+If $G\subseteq X$ is $\mathcal J_0$-positive, there is
+$P\subseteq G$ in $\mathcal J_0$ such that
+$f_n^T(X\setminus P)\to-\infty$.
+\end{lemma}
+
+\begin{proof}
+For additions, choose $\delta>0$ such that $G$ has density at least
+$\delta$ in arbitrarily late blocks. Suppress the superscript $T$,
+put $a_n=\sup_{m\ge n}|d_m(X)|$, and choose strictly increasing
+positive integers $N_j$ such that, for $n\ge N_j$,
+\begin{equation}\label{eq:spike-cutoff}
+ \frac{|p_n(X)|}{c_n}+2a_n+\frac1{\sqrt{c_n}}\le 2^{-j}.
+\end{equation}
+This is possible by Lemma~\ref{lem:probe-properties}.
+
+For each sufficiently large $j$, put $b_j=2^{1-j}<\delta$ and choose
+an increasing sequence of blocks $m_j\ge N_{j+1}$ in which $G$ has
+density at least $\delta$, also requiring
+$1/|W_{m_j}|\le\delta-b_j$. From $G\cap W_{m_j}$ select exactly
+$\lceil b_j|W_{m_j}|\rceil$ elements. Rounding is possible because
+this number is at most $\delta|W_{m_j}|$. Let $P$ be the union of these
+selections. Its nonzero block densities lie between $b_j$ and
+$b_j+1/|W_{m_j}|$, and hence tend to zero. Thus $P\in\mathcal J_0$.
+
+If $N_j\le n<N_{j+1}$, the selected block $m_j$ belongs to the tail
+in \eqref{eq:probes}. The tail supremum for $X\cup P$ is at least
+$b_j-a_n$, and its tail infimum is at least $-a_n$. The prefix can only
+increase. Therefore \eqref{eq:spike-cutoff} gives
+\[
+ \frac{f_n(X\cup P)}{c_n}
+ \ge -\frac{|p_n(X)|}{c_n}+b_j-2a_n
+ \ge \frac1{\sqrt{c_n}}.
+\]
+These intervals contain every sufficiently large $n$, proving
+divergence to $+\infty$. Apply the addition statement to $X^c$ and
+$T^c$, and use complement symmetry, to obtain the deletion statement.
+\end{proof}
+
+\subsection{Interpolation and interval alternatives}
+
+The finite-change rule rounds any finite limiting value into the
+interval $(-1,0]$. The remaining case has opposite infinite limits at
+the two endpoints. We handle it by finite deletions whose effect on
+earlier probes is summable.
+
+\begin{lemma}[Interpolation]\label{lem:local-interpolation}
+If $I\subseteq Y$ are in $\mathcal C(T)$,
+$\ell_T(I)\le-1$, and $\ell_T(Y)>0$, then there is
+$B\in\mathcal B^-(T)$ with $I\subseteq B\subseteq Y$.
+\end{lemma}
+
+\begin{proof}
+Suppress $T$ from the notation. If $\ell(I)=a$ is finite, add
+$\lfloor-a\rfloor$ elements from $Y\setminus I$. There are enough:
+a smaller finite gap of size $h$ would give
+$\ell(Y)=a+h\le-1$. The resulting lower limit belongs to $(-1,0]$.
+If $\ell(Y)=b$ is finite, delete $\lceil b\rceil$ gap elements from
+$Y$. A smaller finite gap of size $h$ would instead give
+$\ell(I)=b-h>0$. Again the resulting lower limit belongs to $(-1,0]$.
+
+It remains to consider $\ell(I)=-\infty$ and $\ell(Y)=+\infty$.
+Set $X_0=Y$ and construct decreasing sets $X_j$ by finite deletions.
+At stage $j\ge1$, do nothing if some $n\ge j$ already satisfies
+$f_n(X_{j-1})\le0$. Otherwise choose $M_j\ge j$ so large that
+\begin{equation}\label{eq:stage-loss}
+ 2c_j\sup_{m\ge M_j}
+       \frac{|(Y\setminus I)\cap W_m|}{|W_m|}<2^{-j}.
+\end{equation}
+This is possible because $Y\setminus I\in\mathcal J_0$.
+Delete elements of $X_{j-1}\setminus I$ in blocks $m\ge M_j$, one
+at a time in increasing block order, stopping as soon as
+$f_n\le0$ for some $n\ge j$.
+
+This stopping time is finite. If every eligible element were deleted,
+the resulting set would differ from $I$ by finitely many elements,
+so it would have lower limit $-\infty$. Fix a strictly negative probe
+with index $n\ge j$ for that set. By
+Lemma~\ref{lem:block-continuity}, the same probe is negative after
+finitely many of the deletions. Just before the stopping deletion all
+probes with index at least $j$ were positive. Each drops by at most
+one, so
+\begin{equation}\label{eq:stage-reset}
+ f_n(X_j)\ge-1\qquad(n\ge j)
+\end{equation}
+at every stage that makes deletions. For $n<j$ the prefix is unchanged
+during the whole stage, and each tail extremum decreases by at most
+the density supremum in \eqref{eq:stage-loss}. Since $c_n\le c_j$,
+the entire stage decreases $f_n$ by less than $2^{-j}$.
+
+Let $X_\infty=\bigcap_{j\ge0}X_j$. It lies between $I$ and $Y$ and
+belongs to $\mathcal C(T)$; Lemma~\ref{lem:block-continuity} applies
+to this decreasing sequence. Every stage leaves a witness
+$f_n\le0$ with $n\ge j$. Such a witness persists under later
+deletions and in the limit, so $\ell(X_\infty)\le0$.
+
+There is a first stage $j_0$ that makes deletions. Otherwise the
+unchanged set $Y$ would have a nonpositive probe in every tail,
+contrary to $\ell(Y)=+\infty$. Fix $n\ge j_0$. After stage $j_0$ its
+probe is at least $-1$. Through stage $n$, every later stage either
+leaves it unchanged or restores this bound by \eqref{eq:stage-reset}.
+Stages $j>n$ then cost less than $2^{-j}$ each. Passing to the limit,
+\[
+ f_n(X_\infty)\ge-1-\sum_{j>n}2^{-j}=-1-2^{-n}.
+\]
+Consequently $-1\le\ell(X_\infty)\le0$. If the lower limit exceeds
+$-1$, take $B=X_\infty$. If it equals $-1$, add one point of
+$Y\setminus X_\infty$ to obtain lower limit zero. There are infinitely
+many such points: a finite difference from $Y$ would contradict
+$\ell(Y)=+\infty$ by the finite-change rule. This completes the
+interpolation.
+\end{proof}
+
+The local rules have the following interval property. Unless a basis
+lies below the lower endpoint, between the endpoints, or above the upper
+endpoint, the prototype has a positive part outside the lower endpoint
+and the upper endpoint has a positive part outside the prototype.
+These are the two differences needed to separate a new prototype class
+from this one in Section~\ref{sec:assembly}.
+
+\begin{proposition}[Interval alternatives for one class]
+\label{prop:local-intervals}
+Fix $\sigma\in\{-,+\}$ and let $I\subseteq Y\subseteq E_0$.
+If
+\[
+ T\setminus I\in\mathcal J_0
+ \quad\text{or}\quad
+ Y\setminus T\in\mathcal J_0,
+\]
+there is $B\in\mathcal B^\sigma(T)$ satisfying at least one of
+\begin{equation}\label{eq:local-alternatives}
+ B\subseteq I,\qquad I\subseteq B\subseteq Y,
+ \qquad Y\subseteq B.
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+We first establish the interval alternatives when one or both endpoints
+belong to the class. For the lower-limit rule, if
+$X\in\mathcal C(T)$ has $\ell_T(X)\le-1$,
+Lemmas~\ref{lem:positive-pool} and~\ref{lem:local-interpolation}
+extend it to a basis by additions from any prescribed positive
+subset of $X^c$. If $\ell_T(X)>0$, their deletion versions give a
+basis contained in $X$, using any prescribed positive subset of $X$.
+Both unrestricted corrections are available because $X$ and $X^c$
+are positive.
+
+If both $I$ and $Y$ belong to $\mathcal C(T)$, a lower endpoint with
+positive lower limit supplies a basis below $I$, and an upper
+endpoint with lower limit at most $-1$ supplies a basis above $Y$.
+An endpoint that is itself a basis also gives an outcome in
+\eqref{eq:local-alternatives}. The only remaining case is
+$\ell_T(I)\le-1$ and $\ell_T(Y)>0$, when
+Lemma~\ref{lem:local-interpolation} gives a basis between them.
+
+If $I$ belongs to the class and $Y\setminus I$ is positive,
+correcting $I$ downwards, leaving it fixed when it is a basis, or
+correcting it upwards from the pool $Y\setminus I$ gives a basis
+below $I$ or between $I$ and $Y$.
+If $Y$ belongs to the class and the gap is positive, the
+corresponding corrections give a basis between $I$ and $Y$ or
+above $Y$.
+Complementation proves these same three conclusions for the
+upper-limit rule, exchanging the two one-endpoint conclusions.
+
+Now suppose $T\setminus I\in\mathcal J_0$.
+If $I\in\mathcal C(T)$, a $\mathcal J_0$-small gap puts $Y$ in the same class,
+whereas a positive gap permits the lower-endpoint conclusion just
+proved. In either case we obtain \eqref{eq:local-alternatives}.
+If $I\notin\mathcal C(T)$, then $T\cap I\in\mathcal C(T)$ and
+$I\setminus T$ is positive. Applying the lower-endpoint conclusion
+to $[T\cap I,I]$ gives a basis contained in $I$.
+
+Finally, if $Y\setminus T\in\mathcal J_0$, apply the result just
+proved to the prototype $T^c$ and the interval $[Y^c,I^c]$, with
+the opposite rule. Complementing its basis gives
+\eqref{eq:local-alternatives} for $T$ and the original rule.
+\end{proof}
diff --git a/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/build/sections/05-assembly.tex b/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/build/sections/05-assembly.tex
new file mode 100644
index 0000000000000000000000000000000000000000..468d695d8eb2e140a32bfb44be906b2a90589745
--- /dev/null
+++ b/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/build/sections/05-assembly.tex
@@ -0,0 +1,342 @@
+\section{Assembling a self-dual matroid}\label{sec:assembly}
+
+We now choose the prototype classes to which the local rules will be applied.
+There are two requirements: every resulting basis must impose the desired
+strict inequality on ordinal ranks, and enough classes must be present to
+allow maximal independent extensions inside every subset of $E_0$.
+The interval criterion used for the second requirement is due to
+Bowler and Geschke~\cite[Theorem~6]{BowlerGeschke2016}; we include the full
+verification of the matroid axioms after constructing the bases.
+
+\subsection{Admissible sets and two monotone zones}
+
+For $X\subseteq E_0=L\dot\cup R$, put
+\[
+ u(X)=X_L,\qquad y(X)=D\setminus X_R.
+\]
+Call $X$ \emph{admissible} if either
+\begin{equation}\label{eq:admissible}
+ \begin{aligned}
+ &u(X),y(X)\in\mathcal U
+       &&\text{and }\rho(u(X))>\rho(y(X)),\\
+ \text{or }\quad
+ &u(X),y(X)\notin\mathcal U
+       &&\text{and }\rho(D\setminus u(X))>
+                         \rho(D\setminus y(X)).
+ \end{aligned}
+\end{equation}
+All ranks appearing here are evaluated on members of $\mathcal U$.
+Define the \emph{Upper zone} to consist of those $X$ for which $u(X)$ is
+large and either $y(X)$ is small, or $y(X)$ is large and
+$\rho(u(X))\le\rho(y(X))$.
+Define the \emph{Lower zone} by requiring $X^c$ to lie in the Upper zone.
+Complements of subsets of $E_0$ are taken in $E_0$.
+
+\begin{lemma}\label{lem:zones}
+The Upper zone, Lower zone, and admissible sets partition $2^{E_0}$.
+The Upper zone is upward closed, the Lower zone is downward closed,
+and all three statuses are invariant under symmetric differences in
+$\mathcal K_0$.
+Complementation exchanges the two zones and preserves admissibility.
+Moreover, $\varnothing$ is Lower, $E_0$ is Upper, and every admissible
+set and its complement are $\mathcal J_0$-positive.
+\end{lemma}
+
+\begin{proof}
+Write $u=u(X)$ and $y=y(X)$.
+When $u$ is large and $y$ small, $X$ is Upper; when $u$ is small and $y$
+large, it is Lower.
+If both are large, comparison of their ranks partitions the possibilities
+into Upper and admissible.
+If both are small, comparison of the ranks of their complements partitions
+the possibilities into Lower and admissible.
+Complementing $X$ replaces both $u$ and $y$ by their complements, proving
+the asserted symmetries.
+
+Suppose that $X\subseteq X'$ and $X$ is Upper.
+Then $u\subseteq u'$ and $y'\subseteq y$, with $u'$ large.
+If $y'$ is small, $X'$ is Upper by definition.
+Otherwise $y$ is large as well, and Lemma~\ref{lem:rank} gives
+\[
+ \rho(u')\le\rho(u)\le\rho(y)\le\rho(y'),
+\]
+so again $X'$ is Upper.
+Downward closure of the Lower zone follows by complementation.
+The invariance assertion follows from Lemma~\ref{lem:rank} in the two
+coordinates.
+The statuses of $\varnothing$ and $E_0$ follow directly from the definitions.
+
+If the first line of~\eqref{eq:admissible} holds, $X_L=u$ and
+$(X^c)_R=y$ are large, hence $\mathcal J$-positive.
+If the second line holds, $X_R=D\setminus y$ and
+$(X^c)_L=D\setminus u$ are large instead.
+In either case $X$ and $X^c$ are $\mathcal J_0$-positive.
+\end{proof}
+
+\begin{lemma}[Crossing the zones]\label{lem:zone-crossing}
+If $A\subseteq Z\subseteq E_0$, with $A$ Lower and $Z$ Upper, then
+there is an admissible $B$ satisfying $A\subseteq B\subseteq Z$.
+\end{lemma}
+
+\begin{proof}
+The sets $s=A_L$ and $X=Z_L$ are respectively small and large, and
+$s\subseteq X$.
+We keep $B_R=A_R$, so write $y=D\setminus A_R$.
+If $y$ is large, Lemma~\ref{lem:thinning} gives a large $u$ between
+$s$ and $X$ with $\rho(u)>\rho(y)$; take $B_L=u$.
+If $y$ is small, apply the same Lemma between the small set
+$D\setminus X$ and the large set $D\setminus s$ to obtain a large $v$
+with $\rho(v)>\rho(D\setminus y)$.
+Taking $B_L=D\setminus v$ gives the second line
+of~\eqref{eq:admissible}.
+Both constructions respect the two endpoint inclusions.
+\end{proof}
+
+\subsection{Reserving separation from earlier classes}
+
+Different prototypes must have positive differences in both directions.
+This will ensure that their local bases cannot contain one another.
+The next Lemma arranges these two inequalities while preserving the
+endpoint statuses needed in Lemma~\ref{lem:zone-crossing}.
+
+\begin{lemma}\label{lem:insert-prototype}
+Let $\mathcal T$ be a family of fewer than $\mathfrak c$ admissible subsets
+of $E_0$.
+Suppose $A\subseteq Z\subseteq E_0$, that $A$ is not Upper and $Z$ is
+not Lower, and that, for every $T\in\mathcal T$,
+\begin{equation}\label{eq:available-separation}
+ T\setminus A\notin\mathcal J_0,
+ \qquad Z\setminus T\notin\mathcal J_0.
+\end{equation}
+Then there is an admissible $B$ between $A$ and $Z$ such that
+$B\setminus T$ and $T\setminus B$ are $\mathcal J_0$-positive for every
+$T\in\mathcal T$.
+\end{lemma}
+
+\begin{proof}
+Put $G=Z\setminus A$.
+For every $T\in\mathcal T$ with $A\setminus T\in\mathcal J_0$,
+include the trace $G\setminus T$ in a list.
+It is positive by~\eqref{eq:available-separation}, since
+$Z\setminus T=(A\setminus T)\cup(G\setminus T)$.
+For every $T$ with $T\setminus Z\in\mathcal J_0$, also include the
+positive trace $T\cap G$, using
+$T\setminus A=(T\setminus Z)\cup(T\cap G)$.
+There are fewer than $\mathfrak c$ traces.
+Lemma~\ref{lem:reservations} supplies disjoint $\mathcal K_0$-small sets
+$H_0,H_1$ that meet every listed trace $\mathcal J_0$-positively.
+An empty list presents no restriction on their choice.
+
+Modify only the gap:
+\begin{equation}\label{eq:gap-reservations}
+ A'=A\cup(H_0\cap G),\qquad
+ Z'=Z\setminus(H_1\cap G).
+\end{equation}
+Because $H_0$ and $H_1$ are disjoint and $G$ is disjoint from $A$, we have
+$A'\subseteq Z'$.
+The endpoint changes are $\mathcal K_0$-small, so $A'$ is not Upper and
+$Z'$ is not Lower by Lemma~\ref{lem:zones}.
+If an endpoint is admissible, take it for $B$.
+Otherwise $A'$ is Lower and $Z'$ Upper, and
+Lemma~\ref{lem:zone-crossing} gives an admissible $B$ between them.
+
+For any $T\in\mathcal T$, if $A\setminus T$ is positive then so is
+$B\setminus T$.
+Otherwise the reserved positive set $H_0\cap(G\setminus T)$ lies in
+$B\setminus T$.
+Likewise, if $T\setminus Z$ is positive it is already contained in
+$T\setminus B$; otherwise the reserved positive set $H_1\cap T\cap G$
+is contained in $T\setminus B$.
+This proves both inequalities.
+\end{proof}
+
+\subsection{Meeting every interval}
+
+We choose admissible prototypes in complementary pairs.
+For each first prototype $T$ we retain the local bases
+$\mathcal B^-(T)$, and for $T^c$ we retain $\mathcal B^+(T^c)$.
+These two families are complements of one another by the local rules.
+Once a pair has been inserted, its rule and all its bases remain fixed.
+The aim is that, for every interval $I\subseteq Y\subseteq E_0$ with infinite gap,
+some basis $B$ satisfies the following alternative:
+\begin{equation}\label{eq:interval-alternative}
+ B\subseteq I,\qquad I\subseteq B\subseteq Y,
+ \qquad\text{or}\qquad Y\subseteq B.
+\end{equation}
+
+\begin{proposition}\label{prop:basis-family}
+There is a nonempty family $\mathcal B$ of admissible subsets of $E_0$
+with the following properties:
+\begin{enumerate}
+ \item $\mathcal B$ is an antichain and is closed under complements.
+ \item If $B\in\mathcal B$, $F\subseteq B$ and $H\subseteq E_0\setminus B$
+ are finite with $|F|=|H|$, then $(B\setminus F)\cup H\in\mathcal B$.
+ \item For every $I\subseteq Y\subseteq E_0$ with $Y\setminus I$ infinite,
+ the alternative~\eqref{eq:interval-alternative} holds.
+\end{enumerate}
+\end{proposition}
+
+\begin{proof}
+As $E_0$ is countable, list all intervals with infinite gap in a sequence
+of length $\mathfrak c$.
+Recursively retain complementary pairs of admissible prototypes, requiring
+the two directed differences of any distinct prototypes to be
+$\mathcal J_0$-positive.
+At any stage, call bases in the already retained local families
+\emph{old bases}.
+They are closed under complements.
+
+Consider the current interval $I\subseteq Y$.
+If~\eqref{eq:interval-alternative} already holds for an old basis,
+skip this stage. Otherwise Proposition~\ref{prop:local-intervals}
+implies that every old prototype $T$ satisfies
+\begin{equation}\label{eq:unmet-interval}
+ T\setminus I\notin\mathcal J_0,
+ \qquad Y\setminus T\notin\mathcal J_0.
+\end{equation}
+Thus an unmet interval has the positive differences needed to reserve
+separation from every old class.
+
+An admissible set cannot contain an Upper set or be contained in a Lower
+set, by Lemma~\ref{lem:zones}. Thus if $I$ is Upper, we must seek the
+allowed basis below $I$; if $Y$ is Lower, we must seek it above $Y$.
+Otherwise we seek a basis between the endpoints. Accordingly choose
+\[
+ [A,Z]=
+ \begin{cases}
+ [\varnothing,I],& I\text{ is Upper},\\
+ [Y,E_0],&Y\text{ is Lower},\\
+ [I,Y],&\text{otherwise}.
+ \end{cases}
+\]
+The first two cases cannot coincide, by monotonicity of the Upper zone.
+In every case $A$ is not Upper and $Z$ is not Lower.
+We verify~\eqref{eq:available-separation} for all old prototypes.
+In the third case it is~\eqref{eq:unmet-interval}.
+In the first case $T\setminus A=T$ is positive.
+If $I\setminus T$ were $\mathcal J_0$-small, $I\cap T$ would remain Upper and be
+contained in $T$, forcing the admissible $T$ to be Upper.
+Thus $Z\setminus T=I\setminus T$ is positive.
+In the second case $Z\setminus T=T^c$ is positive.
+If $T\setminus Y$ were $\mathcal J_0$-small, $T\cap Y$ would remain admissible and be
+contained in the Lower set $Y$, a contradiction.
+Thus $T\setminus A=T\setminus Y$ is positive as well.
+
+At a stage $\alpha<\mathfrak c$ there are at most two prototypes per
+earlier stage, hence fewer than $\mathfrak c$ old prototypes.
+Lemma~\ref{lem:insert-prototype} therefore supplies an admissible $B$
+between $A$ and $Z$ with both positive differences from every old
+prototype.
+Insert $B$ with the minus rule and $B^c$ with the plus rule.
+The complement has both positive differences from every old $T$, because
+$T^c$ is also old and
+\[
+ B^c\setminus T=T^c\setminus B,\qquad
+ T\setminus B^c=B\setminus T^c.
+\]
+The two new prototypes have positive differences from one another by
+Lemma~\ref{lem:zones}.
+They therefore define new classes.
+All probes of $B$ relative to itself vanish, so $B\in\mathcal B^-(B)$.
+Its placement in $[A,Z]$ supplies one of the three outcomes
+in~\eqref{eq:interval-alternative} for the original interval.
+
+This recursion uses no regularity assumption on $\mathfrak c$.
+For each individual $\alpha$ below its initial ordinal, the numbers of
+predecessors, prototypes, and requested traces are all bounded by a fixed
+cardinal less than $\mathfrak c$.
+The reservation Lemma applies with that bound.
+Choices can be made using a fixed well-order of $2^{E_0}$.
+
+Let $\mathcal B$ be the union of all retained local basis families.
+It is nonempty, since the interval $[\varnothing,E_0]$ receives an outcome.
+Every member remains admissible under its $\mathcal J_0$-small change
+from its prototype, and the family is closed under complements.
+Lemma~\ref{lem:local-antichain} gives the antichain and finite-change
+properties within each class.
+For different classes, the positive directed differences of their
+prototypes remain positive after $\mathcal J_0$-small changes.
+Thus their bases are incomparable as well.
+Finally every interval outcome persists, because no local family is
+subsequently altered.
+\end{proof}
+
+\subsection{The maximal-extension axiom}
+
+The interval construction is complete.
+We now check that it produces a matroid, including maximal extension
+inside arbitrary subsets; finite exchange alone does not supply this
+last property.
+
+\begin{theorem}\label{thm:local-matroid}
+There is a matroid $Q$ on $E_0=L\dot\cup R$ whose bases are precisely
+the family $\mathcal B$ in Proposition~\ref{prop:basis-family}.
+It is equal to its dual on the same labels, and all its bases are admissible.
+In particular, for every basis $B$,
+\begin{equation}\label{eq:basis-rank-decrease}
+ B_L\in\mathcal U
+ \quad\Longrightarrow\quad
+ D\setminus B_R\in\mathcal U
+ \quad\text{and}\quad
+ \rho(B_L)>\rho(D\setminus B_R).
+\end{equation}
+Every independent set of $Q$ that is not a basis can be extended by any
+element outside it.
+\end{theorem}
+
+\begin{proof}
+Define the independent sets to be all subsets of members of $\mathcal B$.
+Since $\mathcal B$ is nonempty, this gives (I1), and heredity (I2) is
+immediate.
+The maximal independent sets are exactly $\mathcal B$:
+each independent set is contained in a member, and the antichain property
+prevents any such member from having a proper independent extension.
+
+Suppose $I$ is independent but not maximal, and choose $B_0\in\mathcal B$
+properly containing it.
+Let $e\in E_0\setminus I$.
+If $e\in B_0$, then $I\cup\{e\}$ is independent.
+Otherwise choose $f\in B_0\setminus I$; the balanced swap
+$(B_0\setminus\{f\})\cup\{e\}$ is a basis containing $I\cup\{e\}$.
+This proves the final assertion of the theorem.
+For any maximal independent $B_1$, its difference $B_1\setminus I$ is
+nonempty, since $B_1\subseteq I$ would contradict its maximality.
+Taking $e$ from this difference proves (I3).
+
+For (IM), fix an independent $I\subseteq Y\subseteq E_0$.
+If $Y\setminus I$ is finite, choose an independent extension of $I$
+containing a maximum number of elements of that finite gap.
+It is maximal within $Y$.
+If the gap is infinite, use~\eqref{eq:interval-alternative}.
+A basis $B\subseteq I$ forces $B=I$: put $I$ inside a basis $B'$ and
+apply the antichain property to $B\subseteq I\subseteq B'$.
+Thus in this case $I$ itself is maximal within $Y$.
+A basis between $I$ and $Y$ is the required maximal extension.
+A basis containing $Y$ makes $Y$ independent, so $Y$ itself is the
+required extension.
+This proves all four independent-set axioms.
+
+The bases of the dual matroid are the complements of the bases of $Q$.
+Since $\mathcal B$ is complement-closed, $Q^*=Q$ on $E_0$.
+Admissibility was established in Proposition~\ref{prop:basis-family},
+and~\eqref{eq:basis-rank-decrease} is its first case.
+\end{proof}
+
+\begin{remark}\label{rem:uniform-local}
+The matroid $Q$ is uniform in the sense that every subset is independent
+or spanning~\cite{BowlerGeschke2016}.
+Indeed, if $X$ is dependent, maximize an independent subset $I$ inside
+$X$ using (IM).
+If $I$ were not a basis of $Q$, the last assertion of
+Theorem~\ref{thm:local-matroid} would allow another element of $X$ to be
+added.
+Thus $X$ contains a basis and is spanning.
+Every basis and its complement are infinite by admissibility, so $Q$
+has infinite rank and corank.
+Every finite $F\subseteq E_0$ is independent: starting from any basis,
+insert the elements of $F$ that are missing and delete equally many
+elements outside $F$, using its infinitude and balanced finite changes.
+Yet $E_0$ is dependent, because every basis has a nonempty complement.
+Thus $Q$ is not finitary, and self-duality shows that it is not cofinitary
+either.
+\end{remark}
diff --git a/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/build/sections/06-counterexample.tex b/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/build/sections/06-counterexample.tex
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@@ -0,0 +1,153 @@
+\section{Two matroids on a double ray}\label{sec:counterexample}
+
+We now use the local matroid $Q$ from
+Theorem~\ref{thm:local-matroid} to construct the required pair.
+Recall that $Q=Q^*$ on $E_0=L\mathbin{\dot\cup}R$, where each
+of $L,R$ is a labeled copy of the countable set $D$. The ultrafilter
+$\mathcal U$ and ordinal rank $\rho$ have the following consequence
+for every basis $B$ of $Q$:
+\begin{equation}\label{eq:outward-rank}
+ B_L\in\mathcal U
+ \quad\Longrightarrow\quad
+ D\setminus B_R\in\mathcal U
+ \quad\text{and}\quad
+ \rho(B_L)>\rho(D\setminus B_R).
+\end{equation}
+The rank is antitone under inclusion of members of $\mathcal U$.
+An independent covering will propagate a member of $\mathcal U$
+along a ray, strictly decreasing this rank at every step.
+
+\subsection{Direct sums and the common ground set}
+
+For completeness, the direct sums we use satisfy all the infinite-matroid
+axioms. Let $(Q_j:j\in J)$ be matroids on pairwise disjoint sets $F_j$,
+where $J$ is countable. Declare $I\subseteq\bigcup_{j\in J}F_j$
+independent if $I\cap F_j$ is independent in every $Q_j$.
+Axioms (I1) and (I2) follow componentwise. The maximal independent
+sets are exactly the unions of one basis from each component: a
+nonmaximal component permits an extension of the union, and if all
+components are maximal no element can be adjoined.
+
+For (I3), if $I$ is not maximal, some $I\cap F_j$ is not maximal
+in $Q_j$. Given a global basis $B$, its part $B\cap F_j$ is a
+component basis. Apply (I3) there and adjoin the resulting element to
+$I$. For (IM), given independent $I\subseteq Y$, choose in each
+component a maximal independent extension of $I\cap F_j$ inside
+$Y\cap F_j$. Their union is a maximal independent extension of $I$
+inside $Y$. These choices are available in ZFC. This proves that the
+direct sum is a matroid, and also proves its stated description of
+bases. If every component is self-dual on its own labels, then the
+complement of each global basis is again a global basis, so the direct
+sum is self-dual on the same labels.
+
+Take vertices $v_n$ indexed by $n\in\mathbb Z$. Between $v_n$ and
+$v_{n+1}$ put the bundle
+\[
+ e_n=\{n\}\times D,
+ \qquad E=\mathop{\dot\bigcup}_{n\in\mathbb Z}e_n.
+\]
+Thus every bundle has the fixed label map $(n,d)\mapsto d$ onto $D$.
+The ground set $E$ is countably infinite: $D$ is a countably infinite
+union of nonempty finite blocks, and $E=\mathbb Z\times D$.
+Designate $e_{-1}$, joining $v_{-1}$ and $v_0$, as the central bundle.
+At every vertex put a copy $Q_n$ of $Q$ on $e_{n-1}\cup e_n$.
+The inward bundle is its $L$-coordinate, and the outward bundle its
+$R$-coordinate; explicitly,
+\[
+\begin{array}{c|cc}
+ &L&R\\\hline
+ n\le -1&e_n&e_{n-1}\\
+ n\ge 0&e_{n-1}&e_n.
+\end{array}
+\]
+Both middle vertices therefore use the central bundle as $L$.
+All copies use the same label maps, ultrafilter, and rank.
+
+
+
+Define
+\[
+ M_0=\bigoplus_{n\in 2\mathbb Z}Q_n,
+ \qquad
+ M_1=\bigoplus_{n\in 2\mathbb Z+1}Q_n.
+\]
+For either parity, the component supports partition $E$, since each
+bundle has exactly one endpoint of that parity. Hence these are
+matroids on the same countably infinite set $E$. The preceding
+direct-sum verification and $Q=Q^*$ give $M_i=M_i^*$ on the same
+labels for $i=0,1$.
+
+The parity placement and inward/outward coordinates are shown in
+Figure~\ref{fig:double-ray}.
+
+\begin{figure}[htbp]
+\centering
+\input{figures/double-ray.tex}
+\caption{Each segment represents a copy of $D$. A copy of $Q$ uses
+the two bundles incident with its vertex. Filled vertices contribute
+to $M_0$, open vertices to $M_1$. The central bundle is $e_{-1}$;
+from either endpoint, the $L$-coordinate points toward the center and
+the $R$-coordinate points outward.}
+\label{fig:double-ray}
+\end{figure}
+
+Every finite subset of $E$ is independent in each $M_i$, because
+all its component parts are finite and every finite subset of $Q$ is
+independent by Remark~\ref{rem:uniform-local}. But $E$ is dependent
+in each $M_i$: its intersection with any component support is the
+dependent whole ground set of that copy of $Q$. Thus neither $M_i$
+is finitary. Since $M_i=M_i^*$, neither is cofinitary either.
+
+In fact, neither matroid has a nonempty finitary or cofinitary direct
+summand. Suppose a nonempty set $F$ supports a finitary summand.
+Every subset of $F$ is independent, since all its finite subsets are
+independent, so $F$ is the summand's only basis. Thus every basis of
+$M_i$ contains $F$. But complements of bases are also bases of $M_i$,
+a contradiction. Dualizing the direct-sum basis description gives the
+same contradiction for a cofinitary summand.
+
+\subsection{The descending ranks}
+
+Suppose that independent sets $J_i$ of $M_i$ cover $E$.
+Replacing $J_1$ by $E\setminus J_0\subseteq J_1$, we obtain an
+independent partition $E=J_0\mathbin{\dot\cup}J_1$.
+Assign each element of a bundle to the endpoint whose parity agrees
+with its part in this partition. At every vertex, the assigned elements
+on its two incident bundles form an independent set of its copy of $Q$.
+
+The central bundle is partitioned between $v_{-1}$ and $v_0$.
+Exactly one of the two label sets belongs to $\mathcal U$.
+Start at that endpoint and move outward forever. Write $x_k\subseteq D$
+for the labels received by the $k$th vertex on its inward bundle.
+Thus $x_0\in\mathcal U$. If the current vertex receives $z_k$
+on its outward bundle, the next vertex receives exactly
+\[
+ x_{k+1}=D\setminus z_k
+\]
+there, because the assignment is a partition and the label maps agree.
+
+Assume inductively that $x_k\in\mathcal U$. The independent
+assignment at this vertex is contained in a basis $B_k$ of its copy
+of $Q$. Put
+\[
+ u_k=(B_k)_L,
+ \qquad y_k=D\setminus(B_k)_R.
+\]
+We have $x_k\subseteq u_k$, so $u_k\in\mathcal U$.
+Equation~\eqref{eq:outward-rank} gives $y_k\in\mathcal U$ and
+$\rho(u_k)>\rho(y_k)$. Since $z_k\subseteq(B_k)_R$,
+we have $y_k\subseteq x_{k+1}$; hence $x_{k+1}\in\mathcal U$.
+Antitonicity now yields
+\begin{equation}\label{eq:ordinal-descent}
+ \rho(x_k)\ \ge\ \rho(u_k)\ >\ \rho(y_k)\ \ge\
+ \rho(x_{k+1}).
+\end{equation}
+This produces an infinite strictly descending sequence of ordinals,
+which is impossible: its set of values has a least element, and the
+next value would be smaller.
+
+Thus $M_0,M_1$ have no independent covering of $E$.
+By Lemma~\ref{lem:selfdual-reduction}, they have no packing/covering
+partition either. Their common ground set is countably infinite and
+both matroids are self-dual on the same labels, completing the proof
+of Theorem~\ref{thm:main}.
diff --git a/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/build/sections/07-consequences.tex b/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/build/sections/07-consequences.tex
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+++ b/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/build/sections/07-consequences.tex
@@ -0,0 +1,48 @@
+\section{The separate Covering and Packing conjectures}
+\label{sec:separate-consequences}
+
+Bowler and Carmesin also formulated separate Covering and Packing
+conjectures. For a set-indexed family $\mathcal M=(M_i:i\in\Theta)$ on $E$,
+write $\mathcal M\!\upharpoonright Y=(M_i\!\upharpoonright Y:i\in\Theta)$
+and $\mathcal M.Y=(M_i.Y:i\in\Theta)$. Their Covering Conjecture asserts,
+for every such family, the equivalence
+\begin{equation}\label{eq:covering-conjecture}
+\begin{aligned}
+ &\mathcal M\text{ has a covering}\\
+ &\quad\Longleftrightarrow\quad
+ \text{for every }Y\subseteq E,\\
+ &\qquad\bigl[\mathcal M\!\upharpoonright Y\text{ has a packing}
+ \Longrightarrow
+ \mathcal M\!\upharpoonright Y\text{ has a covering}\bigr].
+\end{aligned}
+\end{equation}
+Their Packing Conjecture asserts, again for every such family,
+\begin{equation}\label{eq:packing-conjecture}
+\begin{aligned}
+ &\mathcal M\text{ has a packing}\\
+ &\quad\Longleftrightarrow\quad
+ \text{for every }Y\subseteq E,\\
+ &\qquad\bigl[\mathcal M.Y\text{ has a covering}
+ \Longrightarrow
+ \mathcal M.Y\text{ has a packing}\bigr].
+\end{aligned}
+\end{equation}
+These are the conjectures of~\cite[Conjectures~5.2 and~6.3]{BowlerCarmesin2015}.
+
+\begin{corollary}\label{cor:separate-covering-packing}
+The unrestricted separate Covering and Packing Conjectures of Bowler and
+Carmesin are both false in ZFC.
+\end{corollary}
+
+\begin{proof}
+Bowler and Carmesin's universal equivalences identify intersection for
+every pair, packing/covering for every pair, packing/covering for every
+set-indexed family, \eqref{eq:covering-conjecture} for every set-indexed
+family, and \eqref{eq:packing-conjecture} for every set-indexed family
+\cite[Section~7]{BowlerCarmesin2015}. Theorem~\ref{thm:main} falsifies the
+pairwise packing/covering assertion, so both separate universal conjectures
+are false. These last two failures are consequences of equivalence between
+universal statements; this argument does not identify their witnessing
+families with the pair $(M_0,M_1)$.
+\end{proof}
+
diff --git a/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/paper.pdf b/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/paper.pdf
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index 0000000000000000000000000000000000000000..72ddfd66c1ccd4dbdd86ea6a4e73129512900f6a
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+++ b/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:320eb5d308aabe2d7ab5d206329aed63091745888f021a77a9ad97d625477b9e
+size 404856
diff --git a/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/README.md b/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/README.md
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+++ b/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/README.md
@@ -0,0 +1,18 @@
+# [A Cyclic Polytabloid Proof of Saxl's Conjecture](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 24, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026,
+  author = {{OpenAI}},
+  title = {{A Cyclic Polytabloid Proof of Saxl's Conjecture}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/paper.pdf}{OAI:A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/build/paper.tex b/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/build/paper.tex
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+++ b/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/build/paper.tex
@@ -0,0 +1,68 @@
+\documentclass[11pt]{article}
+\usepackage[utf8]{inputenc}
+\usepackage{amsmath,amssymb,amsthm}
+\input{glyphtounicode-cmex}
+\pdfgentounicode=1
+\usepackage[margin=1in]{geometry}
+\usepackage{microtype,enumitem,booktabs}
+\usepackage{color}
+\definecolor{linkblue}{rgb}{0.05,0.15,0.35}
+\PassOptionsToPackage{hyphens}{url}
+\usepackage[colorlinks=true,linkcolor=linkblue,citecolor=linkblue,urlcolor=linkblue]{hyperref}
+\usepackage[nameinlink,noabbrev,capitalise]{cleveref}
+\hypersetup{pdftitle={A Cyclic Polytabloid Proof of Saxl's Conjecture},pdfauthor={OpenAI}}
+\numberwithin{equation}{section}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\newtheorem{example}[theorem]{Example}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\AddToHook{env/theorem/begin}{\crefalias{theorem}{theorem}}
+\AddToHook{env/lemma/begin}{\crefalias{theorem}{lemma}}
+\AddToHook{env/proposition/begin}{\crefalias{theorem}{proposition}}
+\AddToHook{env/corollary/begin}{\crefalias{theorem}{corollary}}
+\AddToHook{env/definition/begin}{\crefalias{theorem}{definition}}
+\AddToHook{env/example/begin}{\crefalias{theorem}{example}}
+\AddToHook{env/remark/begin}{\crefalias{theorem}{remark}}
+\newcommand{\C}{\mathbb C}
+\DeclareMathOperator{\Span}{span}
+\DeclareMathOperator{\Ind}{Ind}
+\DeclareMathOperator{\Res}{Res}
+\DeclareMathOperator{\Hom}{Hom}
+\DeclareMathOperator{\sgn}{sgn}
+\DeclareMathOperator{\id}{id}
+\DeclareMathOperator{\GL}{GL}
+\DeclareMathOperator{\Mat}{Mat}
+\DeclareMathOperator{\adj}{adj}
+\DeclareMathOperator{\tr}{tr}
+\DeclareMathOperator{\Sym}{Sym}
+\DeclareMathOperator{\Alt}{Alt}
+\DeclareMathOperator{\supp}{supp}
+\DeclareMathOperator{\diag}{diag}
+\DeclareMathOperator{\len}{len}
+\setlist[enumerate]{itemsep=3pt,topsep=5pt}
+\setlist[itemize]{itemsep=3pt,topsep=5pt}
+\setcounter{tocdepth}{2}
+\title{A Cyclic Polytabloid Proof of Saxl's Conjecture}
+\author{OpenAI}
+\date{September 24, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+For every staircase partition, we prove that the tensor square of the
+corresponding irreducible complex representation of the symmetric group
+contains every irreducible representation of that group. This proves
+Saxl's conjecture.
+\end{abstract}
+\input{sections/introduction.tex}
+\input{sections/specht-tools.tex}
+\input{sections/dominance.tex}
+\input{sections/band-quotient.tex}
+\input{sections/path-band.tex}
+\input{sections/completion.tex}
+\input{references.tex}
+\end{document}
diff --git a/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/build/references.tex b/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/build/references.tex
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+++ b/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/build/references.tex
@@ -0,0 +1,112 @@
+\begin{thebibliography}{HSTZ13}
+
+\bibitem[Bes18]{bessenrodt2018}
+Christine Bessenrodt,
+\emph{Critical classes, Kronecker products of spin characters, and the
+Saxl conjecture},
+Algebraic Combinatorics \textbf{1} (2018), no.~3, 353--369.
+\href{https://doi.org/10.5802/alco.18}{doi:10.5802/alco.18}.
+
+\bibitem[BBS21]{bessenrodtbowmansutton2021}
+C.~Bessenrodt, C.~Bowman, and L.~Sutton,
+\emph{Kronecker positivity and 2-modular representation theory},
+Transactions of the American Mathematical Society, Series B
+\textbf{8} (2021), 1024--1055.
+\href{https://doi.org/10.1090/btran/70}{doi:10.1090/btran/70}.
+
+\bibitem[BWZ10]{brownwilligenburgzabrocki2010}
+Andrew A.~H. Brown, Stephanie van Willigenburg, and Mike Zabrocki,
+\emph{Expressions for Catalan Kronecker products},
+Pacific Journal of Mathematics \textbf{248} (2010), no.~1, 31--48.
+\href{https://doi.org/10.2140/pjm.2010.248.31}{doi:10.2140/pjm.2010.248.31}.
+
+\bibitem[DG94]{drenskygiambruno1994}
+Vesselin Drensky and Antonio Giambruno,
+\emph{Cocharacters, codimensions and Hilbert series of the polynomial
+identities for $2\times2$ matrices with involution},
+Canadian Journal of Mathematics \textbf{46} (1994), no.~4, 718--733.
+\href{https://doi.org/10.4153/CJM-1994-040-6}{doi:10.4153/CJM-1994-040-6}.
+
+\bibitem[Ebr25]{ebrahimi2025}
+Mahdi Ebrahimi,
+\emph{The problem of deciding the positivity of Kronecker coefficients
+and Saxl conjecture}, preprint, 2025.
+\href{https://arxiv.org/abs/2511.03484v3}{arXiv:2511.03484v3}.
+
+\bibitem[FH91]{fultonharris1991}
+William Fulton and Joe Harris,
+\emph{Representation Theory: A First Course},
+Graduate Texts in Mathematics, vol.~129,
+Springer-Verlag, New York, 1991.
+\href{https://doi.org/10.1007/978-1-4612-0979-9}{doi:10.1007/978-1-4612-0979-9}.
+
+\bibitem[HR23]{harmanryba2023}
+Nate Harman and Christopher Ryba,
+\emph{A tensor-cube version of the Saxl conjecture},
+Algebraic Combinatorics \textbf{6} (2023), no.~2, 507--511.
+\href{https://doi.org/10.5802/alco.267}{doi:10.5802/alco.267}.
+
+\bibitem[HSTZ13]{heidestz2013}
+Gerhard Heide, Jan Saxl, Pham Huu Tiep, and Alexandre E.~Zalesski,
+\emph{Conjugacy action, induced representations and the Steinberg square
+for simple groups of Lie type},
+Proceedings of the London Mathematical Society (3)
+\textbf{106} (2013), no.~4, 908--930.
+\href{https://doi.org/10.1112/plms/pds062}{doi:10.1112/plms/pds062}.
+
+\bibitem[Ike15]{ikenmeyer2015}
+Christian Ikenmeyer,
+\emph{The Saxl conjecture and the dominance order},
+Discrete Mathematics \textbf{338} (2015), no.~11, 1970--1975.
+\href{https://doi.org/10.1016/j.disc.2015.04.027}{doi:10.1016/j.disc.2015.04.027}.
+\href{https://arxiv.org/abs/1410.6549v2}{arXiv:1410.6549v2}.
+
+\bibitem[Jam78]{james1978}
+G.~D. James,
+\emph{The Representation Theory of the Symmetric Groups},
+Lecture Notes in Mathematics, vol.~682,
+Springer-Verlag, Berlin--Heidelberg, 1978.
+\href{https://doi.org/10.1007/BFb0067708}{doi:10.1007/BFb0067708}.
+
+\bibitem[LN25]{letelliernam2025}
+Emmanuel Letellier and GyeongHyeon Nam,
+\emph{The Saxl conjecture and the tensor square of unipotent characters
+of $\GL_n(q)$},
+Algebraic Combinatorics \textbf{8} (2025), no.~4, 1119--1140.
+\href{https://doi.org/10.5802/alco.434}{doi:10.5802/alco.434}.
+
+\bibitem[Li21]{li2021}
+Xin Li,
+\emph{Saxl Conjecture for triple hooks},
+Discrete Mathematics \textbf{344} (2021), no.~6, article 112340.
+\href{https://doi.org/10.1016/j.disc.2021.112340}{doi:10.1016/j.disc.2021.112340}.
+
+\bibitem[LS17]{luosellke2017}
+Sammy Luo and Mark Sellke,
+\emph{The Saxl conjecture for fourth powers via the semigroup property},
+Journal of Algebraic Combinatorics \textbf{45} (2017), no.~1, 33--80.
+\href{https://arxiv.org/abs/1511.02387v2}{arXiv:1511.02387v2}.
+
+\bibitem[Ope26]{universaltensorsquares2026}
+OpenAI,
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+OpenAI Math Release preprint
+\href{https://github.com/openai/math/blob/main/preprints/Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026/main.pdf}{OAI:Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026},
+2026.
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+conjectures},
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+
+\end{thebibliography}
diff --git a/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/build/sections/band-quotient.tex b/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/build/sections/band-quotient.tex
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+++ b/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/build/sections/band-quotient.tex
@@ -0,0 +1,205 @@
+\section{A quotient obtained by cutting off a band}
+\label{sec:band-quotient}
+
+The next step transfers support from a smaller cyclic staircase module
+and a band into $W_m$. The projection must retain their specified
+generators and realize the full induced module, not just a submodule
+of it. We first define the two factors and then verify both requirements.
+
+Fix $s\in\{1,2\}$ and $h=m-s\ge1$, and put
+\begin{equation}\label{band:sets}
+ D=B_h,\qquad F=B_m\setminus D,\qquad
+ G=G_m=S_{B_m},\qquad H=S_D\times S_F.
+\end{equation}
+Thus $|F|=N_m-N_h=s(2m-s+1)/2$.
+Let $R_F$ and $C_F$ be the partitions of $F$ into its nonempty row
+and column intersections, with the orders inherited from $R_m$ and $C_m$.
+Each block has size at most $s$.  In two layers with alphabet
+$1,\ldots,s$, define
+\begin{equation}\label{band:generator}
+ u_{m,s}=v_{R_F}\otimes v_{C_F},\qquad
+ U_{m,s}=\C[S_F]u_{m,s}.
+\end{equation}
+The dependence on $m$ records the length of the band.
+Our goal is a quotient of $W_m$ induced from $W_h\boxtimes U_{m,s}$.
+
+In the disjoint-coset model of induction
+\cite[Section~3.3]{fultonharris1991}, translates of the inducing module
+form a direct sum. We record why disjoint coordinate sectors realize
+this model for the cyclic span of a specified vector, including the
+inverse map.
+
+\begin{lemma}[Induction from coordinate sectors]\label{lem:sector-induction}
+Let a finite group $G$ act transitively on a finite set $\Omega$.
+Suppose a $G$-module has a direct-sum decomposition
+$T=\bigoplus_{A\in\Omega}T_A$ with $gT_A=T_{gA}$.
+Fix $D\in\Omega$, let $H$ be its stabilizer, and take $z\in T_D$.
+If $A_0=\C[H]z$, then the map
+\begin{equation}\label{band:induced-map}
+ \Phi:\C[G]\otimes_{\C[H]}A_0\longrightarrow\C[G]z,
+ \qquad g\otimes a\longmapsto ga
+\end{equation}
+is an isomorphism of $G$-modules.
+\end{lemma}
+
+\begin{proof}
+The relation $(gh)\otimes a=g\otimes ha$, for $h\in H$, has the
+same image on both sides, so $\Phi$ is well-defined.  It is equivariant
+and surjective.  Choose $g_A\in G$ with $g_AD=A$ for every $A\in\Omega$,
+and let $\pi_A:T\to T_A$ be the coordinate projection.  Since
+\[
+ \C[G]z=\bigoplus_{A\in\Omega}g_AA_0,
+ \qquad g_AA_0\subseteq T_A,
+\]
+we have $g_A^{-1}\pi_A(y)\in A_0$ for $y\in\C[G]z$.  Define
+\begin{equation}\label{band:induced-inverse}
+ \Psi(y)=\sum_{A\in\Omega}
+          g_A\otimes g_A^{-1}\pi_A(y).
+\end{equation}
+Replacing $g_A$ by $g_Ah$ does not change this expression, because
+$g_Ah\otimes h^{-1}a=g_A\otimes a$ in the balanced tensor product.
+Clearly $\Phi\Psi(y)=\sum_A\pi_A(y)=y$.
+For $g\otimes a$, set $A=gD$ and write $g=g_Ah$ with $h\in H$.
+Only its $A$-sector is nonzero, and hence
+\[
+ \Psi\Phi(g\otimes a)=g_A\otimes ha=g\otimes a.
+\]
+Thus the displayed maps are inverse isomorphisms.
+\end{proof}
+
+\begin{proposition}[Band quotient]\label{prop:band-quotient}
+For the parameters in \Cref{band:sets}, there is a surjective
+$G_m$-equivariant map
+\begin{equation}\label{band:quotient}
+ W_m\twoheadrightarrow
+ \Ind_{S_D\times S_F}^{S_{B_m}}(W_h\boxtimes U_{m,s}).
+\end{equation}
+In particular, every irreducible constituent of the induced module on
+the right is a constituent of $W_m$.
+\end{proposition}
+
+\begin{proof}
+\emph{Projection and the high positions.}
+Use the ordinary tensor identification
+\[
+ E_m^{\otimes B_m}\otimes E_m^{\otimes B_m}
+ \cong (E_m\otimes E_m)^{\otimes B_m}.
+\]
+Put $E_{\mathrm{lo}}=\Span(e_1,\ldots,e_s)$ and
+$E_{\mathrm{hi}}=\Span(e_{s+1},\ldots,e_m)$.
+On a pair-letter basis vector define
+\[
+ p_s(e_a\otimes e_b)=
+ \begin{cases}
+ e_a\otimes e_b,& a,b\le s\text{ or }a,b>s,\\
+ 0,&\text{otherwise},
+ \end{cases}
+ \qquad P_s=p_s^{\otimes B_m}.
+\]
+The same local projection is used at every position, so $P_s$
+commutes with $G$.  Call a retained position \emph{high} when both
+letters are greater than $s$.
+
+In a retained word of $w_m$, let $A$ be the set of high positions.
+Alternation in the first layer gives row counts
+$r_i=\max(h+1-i,0)$, and alternation in the second gives column
+counts $d_j=\max(h+1-j,0)$, for $1\le i,j\le m$.
+For each $1\le k\le m$, these counts satisfy
+\begin{equation}\label{band:margin-equality}
+ \sum_{i=1}^k r_i
+ =\#\{(i,j)\in B_h:i\le k\}
+ =\sum_{j=1}^m\min(k,d_j).
+\end{equation}
+Column $j$ can contribute at most $\min(k,d_j)$ high positions to
+the first $k$ rows.  Equality of the sums in
+\Cref{band:margin-equality} therefore forces equality in every column.
+For $d_j>0$, take $k=d_j$: all its $d_j$ high positions occupy the
+first $d_j$ rows.  Columns with $d_j=0$ have no high positions.
+Consequently
+\begin{equation}\label{band:unique-high-set}
+ A=\{(i,j)\in B_m:i\le h+1-j\}=B_h=D.
+\end{equation}
+For the zero-one matrix recording membership in $A$, this is the
+equality case of the row and column margin inequalities
+\cite[Section~2]{ryser1957}. The specific staircase margins in
+\Cref{band:margin-equality} make every prefix bound an equality.
+
+\emph{Factorization with signs.}
+In a row or column block of length $r$, its intersection with $D$
+is its initial segment of length $a=\max(r-s,0)$; its intersection
+with $F$ has length $b=\min(r,s)$.
+The retained assignments put the high letters $s+1,\ldots,r$ on
+the first segment and the low letters $1,\ldots,b$ on the second.
+Let $\Alt$ denote the unnormalized alternation of a displayed list,
+placed in the indicated segment, with $\Alt(\varnothing)=1$.
+The part of the block tensor with this prescribed high set is exactly
+\begin{equation}\label{band:block-factorization}
+ (-1)^{ab}\,
+ \Alt(e_{s+1},\ldots,e_r)\otimes\Alt(e_1,\ldots,e_b).
+\end{equation}
+Here the high list is empty when $r\le s$.
+Indeed, every high letter precedes every low letter in the position
+order, giving $ab$ inversions; all other inversions are internal to
+the two independently permuted lists.  Every such pair of permutations
+occurs once.  In particular, when $r<s$ the formula is just the
+original alternation on $e_1,\ldots,e_r$.
+
+Rows and columns both have the length list $m,m-1,\ldots,1$.
+Thus their cross signs in \Cref{band:block-factorization} multiply to
+\[
+ \prod_{r=1}^m(-1)^{2\max(r-s,0)\min(r,s)}=1.
+\]
+Identify $E_h$ with $E_{\mathrm{hi}}$ by
+$e_a\mapsto e_{s+a}$, and write $\widetilde w_h$ and
+$\widetilde W_h$ for the images of $w_h$ and $W_h$ in the two
+high-letter layers on $D$.  The low portions are precisely the blocks
+of $R_F,C_F$.  Regrouping the factors over the disjoint sets $D,F$
+introduces no signs, and hence
+\begin{equation}\label{band:generator-factorization}
+ z:=P_s(w_m)=\widetilde w_h\otimes u_{m,s}\ne0.
+\end{equation}
+The nonvanishing follows also directly from the nonzero alternating
+tensors on each disjoint block.
+
+The two factors of $H=S_D\times S_F$ act independently on this
+tensor, while each acts diagonally on the two layers over its own
+position set.  Therefore
+\begin{align}
+ A_0:=\C[H]z
+ &=\Span\{(\sigma\widetilde w_h)\otimes(\tau u_{m,s}):
+             \sigma\in S_D,\ \tau\in S_F\}\notag\\
+ &=\widetilde W_h\boxtimes U_{m,s}
+ \cong W_h\boxtimes U_{m,s}.
+ \label{band:subgroup-orbit}
+\end{align}
+The second equality follows by expanding tensors of linear
+combinations of the two separate orbit sets.
+
+\emph{The coordinate sectors.}
+Let $\Omega$ be the set of subsets $A\subset B_m$ of size $|D|$.
+Within the pair-letter space define
+\[
+ T_A=(E_{\mathrm{hi}}\otimes E_{\mathrm{hi}})^{\otimes A}
+       \otimes
+       (E_{\mathrm{lo}}\otimes E_{\mathrm{lo}})^{\otimes(B_m\setminus A)}.
+\]
+These are the coordinate spans of words high exactly on $A$.
+They have disjoint word bases, so their sum is direct, and
+$gT_A=T_{gA}$.  By \Cref{band:generator-factorization}, $z\in T_D$;
+the stabilizer of $D$ in $G$ is exactly $H$.
+Apply \Cref{lem:sector-induction} and
+\Cref{band:subgroup-orbit} to obtain
+\[
+ \C[G]z\cong\Ind_H^G A_0
+       \cong\Ind_H^G(W_h\boxtimes U_{m,s}).
+\]
+Since $P_s$ is equivariant, $P_s(W_m)=\C[G]P_s(w_m)=\C[G]z$.
+Its restriction to $W_m$, followed by the inverse map $\Psi$ in
+\Cref{band:induced-inverse} and the identification of $A_0$ in
+\Cref{band:subgroup-orbit}, gives \Cref{band:quotient}.
+The image of this ambient projection need not lie in the two original
+Specht factors; the composition is still a surjective $G$-equivariant
+linear map onto
+the displayed module.  Semisimplicity over $\C$ proves the last
+assertion.
+\end{proof}
diff --git a/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/build/sections/completion.tex b/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/build/sections/completion.tex
new file mode 100644
index 0000000000000000000000000000000000000000..7399b740982de06ba2ca92160c49598d2af58c05
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+++ b/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/build/sections/completion.tex
@@ -0,0 +1,156 @@
+\section{Completion of the induction}
+\label{sec:completion}
+
+We now combine the two band quotients with the dominance case. The
+combinatorial point is that a partition outside the dominance range either
+has a long enough first row for a width-one cut, or has enough boxes in its
+first four rows for a width-two cut.
+
+\begin{samepage}
+\begin{lemma}[Horizontal-strip reduction]
+\label{comb:strip-reduction}
+Let $m\ge2$ and $\lambda\vdash N_m$. If $\lambda$ is not dominated by
+$\rho_m$, then at least one of the following holds.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item There is a partition $\nu\vdash N_{m-1}$ such that
+      $\lambda/\nu$ is a horizontal strip of size $m$.
+\item One has $m\ge5$, and there are partitions
+      \[
+      \nu=\lambda^{(0)}\subset\lambda^{(1)}\subset\cdots
+      \subset\lambda^{(q)}=\lambda,
+      \qquad q\le4,
+      \]
+      with $\nu\vdash N_{m-2}$ and every
+      $\lambda^{(i)}/\lambda^{(i-1)}$ a nonempty horizontal strip.
+      Their total size is $2m-1$.
+\end{enumerate}
+\end{lemma}
+\end{samepage}
+
+\begin{proof}
+Unlike the deletion in \Cref{lem:young-rule}, this reduction need not
+preserve a prescribed dominance inequality; a rightmost suffix suffices
+to preserve the partition.
+If $\lambda_1\ge m$, remove the bottom box in each of the rightmost $m$
+nonempty columns. Subtracting one from this suffix of the column-height
+sequence preserves weak decrease and nonnegativity. The remaining boxes
+therefore form a partition $\nu$, and the deleted boxes form a horizontal
+strip. Since $N_m-m=N_{m-1}$, this proves \textup{(i)}.
+
+Suppose instead that $\lambda_1\le m-1$. Failure of domination gives an
+index $k<m$ with
+\begin{equation}
+\label{comb:dominance-failure}
+\sum_{i=1}^k\lambda_i>km-\frac{k(k-1)}2.
+\end{equation}
+The index is less than $m$ because the first $m$ parts of the staircase
+already have sum $N_m$. For $k\le3$, however,
+\[
+\sum_{i=1}^k\lambda_i\le k(m-1)
+\le km-\frac{k(k-1)}2.
+\]
+Thus $4\le k<m$, and in particular $m\ge5$. Because the parts decrease,
+the mean of the first four is at least the mean of the first $k$. Hence
+\begin{equation}
+\label{comb:four-row-capacity}
+\sum_{i=1}^4\lambda_i
+\ge\frac4k\sum_{i=1}^k\lambda_i
+>4m-2(k-1)\ge2m+4>2m-1.
+\end{equation}
+
+A full sweep removes the bottom box of every nonempty column. If the
+current row lengths are $(\alpha_1,\alpha_2,\ldots)$, subtracting one from
+all its positive column heights gives row lengths
+$(\alpha_2,\alpha_3,\ldots)$. The remaining diagram is nested in the
+original one, and the removed cells form a horizontal strip of size
+$\alpha_1$. Four full sweeps starting from $\lambda$ would therefore
+remove $\lambda_1+\cdots+\lambda_4$ boxes.
+
+Perform full sweeps until the remaining number of boxes to be removed is
+at most the number of nonempty columns, then remove that number of bottom
+boxes from the rightmost columns. The suffix argument in the first
+paragraph shows that this last partial sweep also leaves a partition.
+By \Cref{comb:four-row-capacity}, at most four nonempty sweeps remove
+exactly $2m-1$ boxes. Their remainder has size
+$N_m-(2m-1)=N_{m-2}$. Reversing the sequence gives \textup{(ii)}.
+\end{proof}
+
+Four strips can be necessary. Take $m=13$ and $\lambda=(8^{11},3)$,
+where the exponent denotes eleven repeated parts equal to $8$.
+This partition has size $91=N_{13}$ and fails domination at $k=12$,
+since its first twelve parts sum to $91>90$.
+Every horizontal strip in its eight-column diagram has at most eight
+boxes, so three strips cannot remove the required $25=2m-1$ boxes.
+The sweeps remove $8+8+8+1=25$ boxes and leave
+$\nu=(8^7,7,3)$ of size $66=N_{11}$.
+
+\begin{proof}[Proof of \Cref{thm:cyclic-saxl}]
+We induct on $m$. The module $W_1$ is the one-dimensional representation
+of the one-element group, so the assertion holds at $m=1$.
+
+Let $m\ge2$, suppose the assertion holds for all smaller positive indices,
+and fix $\lambda\vdash N_m$. If $\rho_m\unrhd\lambda$, then
+\Cref{prop:dominated} supplies $S^\lambda$ in $W_m$. Otherwise apply
+\Cref{comb:strip-reduction}.
+
+In case \textup{(i)}, use the band cut with $s=1$, so
+$|D|=N_{m-1}$ and $|F|=m$. The width-one band module $U_{m,1}$ is
+trivial. By the induction hypothesis, $S_D^\nu$ is a constituent of
+$W_{m-1}$. By \Cref{lem:pieri}, $S_{B_m}^\lambda$ occurs in
+\[
+\Ind_{S_D\times S_F}^{S_{B_m}}
+\bigl(S_D^\nu\boxtimes\mathbf1\bigr).
+\]
+Consequently it occurs in the quotient from \Cref{prop:band-quotient}
+and hence in $W_m$.
+
+In case \textup{(ii)}, put $K=2m-1$ and use the cut with $s=2$.
+Write $b_i=|\lambda^{(i)}|-|\lambda^{(i-1)}|$, so that
+$b_1+\cdots+b_q=K$ and $q\le4$. Choose disjoint subsets of $F$ of
+these sizes and form the permutation module
+\[
+A=\Ind_{S_{b_1}\times\cdots\times S_{b_q}}^{S_F}\mathbf1
+  \cong\bigoplus_{\eta\vdash K}(S_F^\eta)^{\oplus a_\eta}.
+\]
+Repeated application of \Cref{lem:pieri}, followed by transitivity of
+induction, shows that $S_{B_m}^\lambda$ is a constituent of
+$\Ind_{S_D\times S_F}^{S_{B_m}}(S_D^\nu\boxtimes A)$. Equivalently,
+\begin{equation}
+\label{comb:positive-multiplicity}
+\begin{gathered}
+\sum_{\eta\vdash K}a_\eta c_{\nu,\eta}^{\lambda}>0,\\[3pt]
+c_{\nu,\eta}^{\lambda}
+=\dim\Hom_{S_{B_m}}\!\left(
+ S_{B_m}^\lambda,
+ \Ind_{S_D\times S_F}^{S_{B_m}}
+ (S_D^\nu\boxtimes S_F^\eta)\right).
+\end{gathered}
+\end{equation}
+Here the coefficients are nonnegative integers. Moreover, every
+constituent of $A$ has at most $q$ rows: starting from the empty diagram,
+each of the $q$ horizontal-strip additions can introduce at most one new
+row, because two new rows would both require a new box in the first
+column. Thus \Cref{comb:positive-multiplicity} provides some $\eta$ with
+at most four rows, $a_\eta>0$, and $c_{\nu,\eta}^{\lambda}>0$.
+
+\begin{samepage}
+By \Cref{prop:two-band}, this $S_F^\eta$ occurs in $U_{m,2}$.
+By induction, $S_D^\nu$ occurs in the specified cyclic module
+$W_{m-2}$. One copy of each suffices to give $S_{B_m}^\lambda$ in
+\[
+\Ind_{S_D\times S_F}^{S_{B_m}}(W_{m-2}\boxtimes U_{m,2}),
+\]
+which is a quotient of $W_m$ by \Cref{prop:band-quotient}.
+Complete reducibility again gives the constituent in $W_m$ itself.
+This completes the induction. Notice that case \textup{(ii)} has
+$m\ge5$, so every smaller module invoked has a positive index.
+\end{samepage}
+\end{proof}
+
+\begin{proof}[Proof of \Cref{thm:saxl}]
+The two factors generating $W_m$ have Specht type $\rho_m$.
+Thus $W_m$ is a submodule of
+$S^{\rho_m}\otimes_{\C}S^{\rho_m}$ under the diagonal action.
+\Cref{thm:cyclic-saxl} supplies every $S^\lambda$ in that submodule,
+so $g(\rho_m,\rho_m,\lambda)>0$ for every permitted $m$ and $\lambda$.
+\end{proof}
diff --git a/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/build/sections/dominance.tex b/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/build/sections/dominance.tex
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+++ b/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/build/sections/dominance.tex
@@ -0,0 +1,198 @@
+\section{The triangular cyclic module and dominance}
+\label{sec:dominance}
+
+We now fix the generator from the introduction exactly and state the
+cyclic support theorem used in the induction. The second task of this
+section is to detect every constituent dominated by the staircase,
+using a sign twist and an explicit permutation-module quotient.
+
+Following the triangular arrangement in \cite[Section~4]{ikenmeyer2015},
+for $m\ge1$ use the position set
+\begin{equation}
+ B_m=\{(i,j)\in\mathbb Z_{\ge1}^2:i+j\le m+1\},
+ \qquad G_m=S_{B_m}.
+ \label{dom:triangle}
+\end{equation}
+Its row and column set partitions are
+\[
+ R_m=\{R_{m,i}:1\le i\le m\},\qquad
+ R_{m,i}=\{(i,j):1\le j\le m+1-i\},
+\]
+\[
+ C_m=\{C_{m,j}:1\le j\le m\},\qquad
+ C_{m,j}=\{(i,j):1\le i\le m+1-j\}.
+\]
+Order each row by increasing $j$ and each column by increasing $i$.
+With $E_m=\C^m$ and its ordered basis $e_1,\ldots,e_m$, form the
+alternating block tensors of \Cref{lem:polytabloid} and set
+\begin{equation}
+ w_m=v_{R_m}\otimes v_{C_m},\qquad
+ W_m=\C[G_m]w_m
+ \ \subseteq\ E_m^{\otimes B_m}\otimes E_m^{\otimes B_m}.
+ \label{dom:cyclic-module}
+\end{equation}
+Both set partitions have block sizes $m,m-1,\ldots,1$.
+Since $\rho_m^t=\rho_m$, \Cref{lem:polytabloid} gives
+\begin{equation}
+ V_R:=\C[G_m]v_{R_m}\cong S^{\rho_m},\qquad
+ V_C:=\C[G_m]v_{C_m}\cong S^{\rho_m},\qquad
+ W_m\subseteq V_R\otimes V_C.
+ \label{dom:square-inclusion}
+\end{equation}
+Here and throughout, the action on a tensor product of two $G_m$-modules
+is diagonal.  Thus it suffices to prove the following stronger statement.
+
+\begin{theorem}[Cyclic staircase support]
+\label{thm:cyclic-saxl}
+For every $m\ge1$ and every partition $\lambda\vdash N_m$, the irreducible
+$S^\lambda$ occurs in $W_m$.
+\end{theorem}
+
+We prove \Cref{thm:cyclic-saxl} by induction in \Cref{sec:completion}.
+The present section establishes the case supplied by dominance.
+
+\begin{proposition}
+\label{prop:dominated}
+If $m\ge1$, $\lambda\vdash N_m$, and $\rho_m\unrhd\lambda$, then
+$S^\lambda$ occurs in $W_m$.
+\end{proposition}
+
+\begin{proof}
+Fix $m$, abbreviate $G=G_m$, and let $\epsilon$ be the one-dimensional
+sign representation, also denoting its character by $\epsilon$.
+By Young's rule, conjugation of dominance, and the sign-twist identity
+in \Cref{tools:dual-sign}, it is enough to show that
+$W_m\otimes\epsilon$ contains every constituent of $M^{\rho_m}$.
+We will find the cyclic module generated by
+$v_{R_m}\otimes v_{R_m}$ inside an isomorphic image of
+$W_m\otimes\epsilon$, then map that module onto $M^{\rho_m}$.
+The row subgroup
+\[
+ H_R=\prod_{i=1}^m S_{R_{m,i}}
+\]
+acts on $v_{R_m}$ by $\epsilon|_{H_R}$.  Consequently the unnormalized signed
+diagonal average is exactly
+\begin{equation}
+ \sum_{h\in H_R}\epsilon(h)\,h w_m
+ =v_{R_m}\otimes u,\qquad
+ u=\sum_{h\in H_R}h v_{C_m}.
+ \label{dom:signed-average}
+\end{equation}
+To see that $u\ne0$, let $a_0(i,j)=i$ and consider the corresponding word
+$e_{a_0}$.  It is fixed by $H_R$.  Its coefficient in $v_{C_m}$ is $1$:
+within each increasingly ordered column it is the identity assignment
+$1,2,\ldots,m+1-j$.  The coefficient of $e_{a_0}$ in every $h v_{C_m}$ is
+therefore also $1$, since $h^{-1}e_{a_0}=e_{a_0}$.  Its coefficient in $u$ is
+$|H_R|=\prod_{r=1}^m r!$, which is nonzero.
+
+We next identify the alternating line in $V_R$.  Frobenius reciprocity,
+the induction identity for a sign twist, and \Cref{lem:young-rule} give
+\begin{align}
+ \dim\Hom_{H_R}(\epsilon|_{H_R},\Res_{H_R}^{G}V_R)
+ &=\dim\Hom_G(\Ind_{H_R}^{G}\epsilon|_{H_R},V_R)\notag\\
+ &=\dim\Hom_G(M^{\rho_m}\otimes\epsilon,V_R)\notag\\
+ &=\dim\Hom_G(M^{\rho_m},V_R\otimes\epsilon)=1.
+ \label{dom:alternating-line}
+\end{align}
+For the last equality, $V_R\otimes\epsilon\cong S^{\rho_m}$ by
+self-conjugacy of the staircase, and $S^{\rho_m}$ has multiplicity one in
+$M^{\rho_m}$.  As $v_{R_m}$ is nonzero and alternating under $H_R$, it spans
+the line in \Cref{dom:alternating-line}.
+
+Twisting the diagonal tensor product can be done in its second factor:
+the reassociation
+\begin{equation}
+ (V_R\otimes V_C)\otimes\epsilon
+ \longrightarrow V_R\otimes(V_C\otimes\epsilon),\qquad
+ (x\otimes y)\otimes z\longmapsto x\otimes(y\otimes z)
+ \label{dom:twist-one-factor}
+\end{equation}
+is a $G$-isomorphism.  Choose a $G$-isomorphism
+$\phi:V_C\otimes\epsilon\longrightarrow V_R$, which exists by
+\Cref{dom:square-inclusion} and $\rho_m^t=\rho_m$, and choose $0\ne z\in\epsilon$.
+Since $u$ is $H_R$-fixed, $u\otimes z$ is alternating under $H_R$.
+It follows from \Cref{dom:alternating-line} that
+$\phi(u\otimes z)=c\,v_{R_m}$ for some $c\ne0$.
+Tensor \Cref{dom:signed-average} with $z$ and apply the reassociation in
+\Cref{dom:twist-one-factor}, followed by $\id\otimes\phi$.
+The right-hand side becomes $c(v_{R_m}\otimes v_{R_m})$.
+Thus the image of $W_m\otimes\epsilon$ under this isomorphism contains
+\begin{equation}
+ Z_m=\C[G](v_{R_m}\otimes v_{R_m})
+ \ \subseteq\ V_R\otimes V_R.
+ \label{dom:repeated-row-cyclic}
+\end{equation}
+
+We construct a surjection from $Z_m$ onto $M^{\rho_m}$.
+Let $E_{\mathrm{out}}$ have basis $f_1,\ldots,f_{2m-1}$ and define the linear map
+\[
+ q:E_m\otimes E_m\longrightarrow E_{\mathrm{out}},\qquad
+ q(e_a\otimes e_b)=f_{a+b-1}.
+\]
+Regroup the two input layers by position, using ordinary tensor
+reassociation, and apply $q$ at every position.  Then project onto the
+span $\mathcal M_\alpha$ of words in which letter $r$ occurs exactly $r$
+times for $1\le r\le m$, and no letter greater than $m$ occurs.
+Thus the content is $\alpha=(1,2,\ldots,m,0,\ldots,0)$ in the output
+alphabet.  Denote the resulting linear map by
+\begin{equation}
+ Q:E_m^{\otimes B_m}\otimes E_m^{\otimes B_m}
+ \longrightarrow\mathcal M_\alpha.
+ \label{dom:pair-letter-map}
+\end{equation}
+Both operations commute with position permutations: the same local map
+$q$ is used everywhere, and content is unchanged by $G$.  Hence $Q$ is
+$G$-equivariant.
+
+Consider any summand in the expansion of $v_{R_m}\otimes v_{R_m}$.
+On a row of length $r$, write its two assignments as
+$(a_1,\ldots,a_r)$ and $(b_1,\ldots,b_r)$, each a permutation of
+$1,\ldots,r$, and put $c_j=a_j+b_j-1$.  Then
+\begin{equation}
+ \sum_{j=1}^r c_j=r^2,
+ \qquad
+ \sum_{j=1}^r c_j^2-r^3
+ =\sum_{j=1}^r(c_j-r)^2\ge0.
+ \label{dom:square-minimum}
+\end{equation}
+For a term retained by the content projection, the sum of squared output
+letters over the whole triangle is $\sum_{r=1}^m r^3$.
+Summing \Cref{dom:square-minimum} over the rows forces equality on every
+row.  Thus $c_j=r$ at every position in the row of length $r$.
+There is only one possible retained output word, namely
+\[
+ t_m=\bigotimes_{(i,j)\in B_m}f_{m+1-i}.
+\]
+It remains to compute its coefficient.  On a row of length $r$, each of
+the $r!$ assignments $a_j=\pi(j)$ has the unique compatible assignment
+$b_j=r+1-\pi(j)$.  The latter is the permutation $\omega_r\pi$, where
+$\omega_r(j)=r+1-j$, so the product of the two alternating signs is
+\[
+ \sgn(\pi)\sgn(\omega_r\pi)
+ =\sgn(\omega_r)=(-1)^{\binom r2}.
+\]
+The row contributions are independent.  In particular, there is no
+cancellation, and the exact image is
+\begin{equation}
+ Q(v_{R_m}\otimes v_{R_m})
+ =\left((-1)^{\sum_{r=1}^m\binom r2}\prod_{r=1}^m r!\right)t_m\ne0.
+ \label{dom:noncancellation}
+\end{equation}
+The stabilizer of $t_m$ is exactly $H_R$, since its equal-letter sets are
+the rows of $B_m$.  Its orbit is a basis of $\mathcal M_\alpha$, and hence
+\begin{equation}
+ Q(Z_m)=\C[G]t_m=\mathcal M_\alpha
+ \cong\Ind_{H_R}^{G}\mathbf1\cong M^{\rho_m}.
+ \label{dom:permutation-quotient}
+\end{equation}
+This also explains why the increasing content $(1,2,\ldots,m)$ gives the
+usual module indexed by the decreasing partition $\rho_m$: reordering
+the content parts conjugates the standard Young subgroup.
+
+By \Cref{lem:young-rule}, every $S^\tau$ with $\tau\unrhd\rho_m$ occurs in
+the quotient in \Cref{dom:permutation-quotient}.  Semisimplicity and
+\Cref{dom:repeated-row-cyclic} imply that it occurs in $W_m\otimes\epsilon$.
+If $\rho_m\unrhd\lambda$, reversal of dominance under conjugation gives
+$\lambda^t\unrhd\rho_m^t=\rho_m$.  Taking $\tau=\lambda^t$ and twisting
+back by $\epsilon$ proves that $S^\lambda$ occurs in $W_m$.
+\end{proof}
diff --git a/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/build/sections/introduction.tex b/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/build/sections/introduction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..655a3448134287e8a60a79190e71161cea62d1da
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+++ b/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/build/sections/introduction.tex
@@ -0,0 +1,190 @@
+\section{Introduction}
+\label{sec:introduction}
+
+For a partition $\alpha\vdash n$, let $S^\alpha$ denote the corresponding
+irreducible complex representation of the symmetric group $S_n$.
+The Kronecker coefficient
+\[
+g(\alpha,\beta,\lambda)
+=\dim\Hom_{S_n}(S^\lambda,S^\alpha\otimes S^\beta)
+\]
+is the multiplicity of $S^\lambda$ in the tensor product, with $S_n$
+acting diagonally. Put
+\begin{equation}\label{intro:staircase}
+N_m=\frac{m(m+1)}2,\qquad \rho_m=(m,m-1,\ldots,1).
+\end{equation}
+We prove the following statement.
+
+\begin{theorem}[Saxl's conjecture]\label{thm:saxl}
+For every integer $m\ge1$ and every partition $\lambda\vdash N_m$,
+\[
+g(\rho_m,\rho_m,\lambda)>0.
+\]
+Equivalently, $S^{\rho_m}\otimes S^{\rho_m}$ contains every irreducible
+complex representation of $S_{N_m}$.
+\end{theorem}
+
+The assertion is uniform in both the staircase size and the constituent.
+The proof establishes it inside one explicitly generated submodule of
+the tensor square, a stronger conclusion that retains the vector needed
+at each induction step.
+
+The companion \cite[Theorem~1.1]{universaltensorsquares2026} uses the
+stronger cyclic statement of \Cref{thm:cyclic-saxl}, together with
+additional arguments, to construct, for every positive integer
+$n\notin\{2,4,9\}$, an irreducible complex representation of $S_n$ whose
+tensor square contains every irreducible representation of $S_n$.
+
+\subsection{History and significance}
+
+Saxl's conjecture belongs to a broader tensor-square covering problem:
+can a single irreducible representation have every irreducible as a
+constituent of its square? A motivating precedent comes from finite
+simple groups of Lie type. Heide, Saxl, Tiep, and Zalesski proved that
+the Steinberg square has this property, except for the groups
+$\mathrm{PSU}_d(q)$ with $d\ge3$ and $\gcd(d,2(q+1))=1$
+\cite[Theorem~1.2]{heidestz2013}.
+Saxl proposed the staircase assertion in a UCLA Combinatorics Seminar on
+20 March 2012, as recorded by Pak, Panova, and Vallejo
+\cite[Conjecture~1.2 and Footnote~2]{pakpanovavallejo2016}.
+Their work places the staircase square among tensor-square covering
+questions for symmetric groups and proves several families of positive
+Kronecker coefficients; in particular, their Theorem~1.4 covers hooks
+and two-row partitions for sufficiently large staircase size.
+Ikenmeyer proved positivity when $\lambda$ and $\rho_m$ are comparable
+in the dominance order \cite[Theorem~2.1]{ikenmeyer2015}.
+We use his triangular arrangement of two alternating tensor layers
+\cite[Section~4]{ikenmeyer2015}; his argument also yields all hook constituents
+\cite[Corollary~6.1]{ikenmeyer2015}.
+
+Character-theoretic and modular methods have supplied further substantial
+families. The character-value criterion of Pak, Panova, and Vallejo
+\cite{pakpanovavallejo2016} detects constituents using the principal-hook
+cycle type of a self-conjugate partition. Bessenrodt developed related
+critical-class and spin-character methods, including the double-hook
+case \cite{bessenrodt2018}. Li proved the triple-hook case using
+computer-assisted finite reductions \cite{li2021}.
+Here double and triple hooks are diagrams whose largest square has side
+two and three, respectively.
+Bessenrodt, Bowman, and Sutton connected Kronecker positivity to
+2-modular decomposition numbers; their results include all
+2-height-zero constituents, hence all odd-degree constituents, and
+framed staircases \cite[Theorems~5.2 and~5.5]{bessenrodtbowmansutton2021}.
+
+Another line of work addresses the tensor exponent and asymptotic
+coverage. Luo and Sellke used the semigroup property to prove that a
+random partition occurs in the staircase square with probability tending
+to one, for both the uniform and Plancherel measures
+\cite[Theorems~1.5 and~1.6]{luosellke2017}. They also found, in every
+sufficiently large degree, an irreducible representation whose fourth
+tensor power contains every irreducible
+\cite[Theorem~1.4]{luosellke2017}. Harman and Ryba proved full coverage
+for the tensor cube of every staircase, with combinatorial and modular
+proofs \cite[Theorem~1.1]{harmanryba2023}.
+Asymptotic coverage does not exclude exceptional constituents, while a
+cube or fourth power uses additional tensor factors; neither gives the square
+assertion in \Cref{thm:saxl}.
+
+Recent developments include semigroup constructions of further
+constituents and generalized-block constraints on support
+\cite{ebrahimi2025}, and a full
+staircase-square theorem for unipotent characters of $\GL_n(q)$
+\cite[Theorem~1.1]{letelliernam2025}. The latter is an analogue in a
+different character theory: ordinary Kronecker positivity implies the
+corresponding unipotent positivity, but the converse need not hold.
+
+The band calculation below also has a precise predecessor at the level
+of ambient representations. Brown, van Willigenburg, and Zabrocki proved
+that
+$S^{(m,m-1)}\otimes S^{(m,m-1)}$ is the multiplicity-free sum of all
+irreducibles of $S_{2m-1}$ indexed by partitions with at most four rows
+\cite[Corollary~4.1]{brownwilligenburgzabrocki2010}.
+Our induction needs each of these constituents in the cyclic module of
+one prescribed band vector, rather than merely somewhere in that
+ambient square. Explicit nonzero pairings supply this additional
+generator-specific statement. Throughout, the objective is support,
+or positivity, rather than a formula for Kronecker multiplicities.
+
+\subsection{The construction and its reusable steps}
+
+Place $N_m$ tensor positions in the triangle
+$B_m=\{(i,j):i,j\ge1,\ i+j\le m+1\}$.
+Let $R_m$ and $C_m$ be its row and column set partitions, ordered within
+each row by increasing $j$ and within each column by increasing $i$.
+Use the ordered basis $e_1,\ldots,e_m$ of $\C^m$. On a block of length
+$r$, alternate the initial letters $e_1,\ldots,e_r$, taking the signed
+sum over all permutations. Taking the product over row blocks gives
+$v_{R_m}$; doing the same over column blocks gives $v_{C_m}$.
+These are polytabloids of staircase type. Instead of beginning with an
+arbitrary vector in the tensor square, we work throughout with
+\[
+W_m=\C[S_{B_m}](v_{R_m}\otimes v_{C_m}),
+\]
+where $S_{B_m}$ permutes the positions in both layers simultaneously.
+\Cref{thm:cyclic-saxl} asserts that this particular cyclic module already
+contains every irreducible. We first prove that it contains every
+$S^\lambda$ for which $\lambda$ is dominated by $\rho_m$: each initial
+sum of the parts of $\lambda$ is at most the corresponding staircase
+sum. A sign twist and a permutation-module quotient give this starting
+case (\Cref{prop:dominated}). The following three steps reach the
+remaining partitions.
+
+\begin{enumerate}
+\item \textbf{A full induced quotient from a coordinate projection.}
+Separate a smaller triangle $B_{m-s}$ from a band of width $s=1$ or $2$.
+Call letters $1,\ldots,s$ low and letters $s+1,\ldots,m$ high.
+Each position carries one letter from each tensor layer.
+Project onto words whose two letters at each position are either both
+high or both low. The staircase row and column counts force the high
+positions of the projected generator to be exactly the smaller triangle.
+The generator then factors into the smaller cyclic generator and a band
+generator, whose cyclic module is denoted by $U_{m,s}$.
+Translates of the high-position set occupy disjoint
+coordinate sectors, so the quotient is the full induced module
+\[
+\Ind_{S_{B_{m-s}}\times S_{B_m\setminus B_{m-s}}}^{S_{B_m}}
+       (W_{m-s}\boxtimes U_{m,s}).
+\]
+\Cref{lem:sector-induction} isolates the general reason that sector
+separation gives an induced-module isomorphism, including an explicit
+inverse. \Cref{prop:band-quotient} gives the resulting quotient.
+
+\item \textbf{Constituents detected in the specified band generator.}
+A width-two band is an odd path of alternating pairs with fixed endpoint
+letters. At each position, a covector on the pair-letter space is
+represented by a two-by-two matrix. Pairing along the path evaluates a word of
+matrices and adjugates. To test a partition with at most four rows,
+place its tableau columns on consecutive path segments and alternate
+the covectors within each column. Each segment has length at most four.
+A four-element basis of $\Mat_2(\C)$ makes every such alternated segment
+matrix invertible.
+A simultaneous conjugation of the test basis makes the endpoint entry
+of the product nonzero. The resulting dual-polytabloid pairing proves
+that every shape with at most four rows occurs in $U_{m,2}$ itself
+(\Cref{prop:two-band}).
+
+\item \textbf{A horizontal-strip reduction with four steps.}
+If a shape outside this dominance range has a first row of length at
+least $m$, one horizontal $m$-strip suffices. Otherwise its first four
+rows contain more than $2m-1$ boxes. Successive sweeps of the bottom
+boxes of columns then remove exactly $2m-1$ boxes in at most four
+horizontal strips (\Cref{comb:strip-reduction}). In this second case,
+let $\nu$ be the remaining partition. Iterated Pieri supplies a shape
+$\eta\vdash2m-1$ with at most four rows such that $S^\lambda$ occurs
+in the representation induced from $S^\nu\boxtimes S^\eta$.
+The band calculation puts $S^\eta$ in $U_{m,2}$, while induction puts
+$S^\nu$ in $W_{m-2}$. The full induced quotient then gives the target
+constituent in $W_m$.
+\end{enumerate}
+
+The smaller factor in the quotient is the actual cyclic module $W_{m-s}$,
+and the band factor is generated by the actual projected vector.
+This is why both parts of the induction track generators instead of
+using ambient tensor-product support alone. Only one copy of a suitable
+intermediate irreducible is needed.
+
+\Cref{sec:specht-tools} fixes the tensor conventions and proves the
+needed form of Young's rule from Pieri. \Cref{sec:dominance} defines
+$W_m$ and proves the dominated case. The quotient and band calculations
+occupy \Cref{sec:band-quotient,sec:path-band};
+\Cref{sec:completion} proves the strip reduction and both main theorems.
diff --git a/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/build/sections/path-band.tex b/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/build/sections/path-band.tex
new file mode 100644
index 0000000000000000000000000000000000000000..d91d2325fd36e49d98453cc1f9682c70909e3a40
--- /dev/null
+++ b/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/build/sections/path-band.tex
@@ -0,0 +1,321 @@
+\section{Cyclic support of the bands}\label{sec:path-band}
+
+For the band step of the induction, it remains to determine the support
+needed in $U_{m,1}$ and $U_{m,2}$ for \Cref{prop:band-quotient}.
+The width-one factor is trivial.
+For width two we will detect every shape with at most four rows directly
+in the specified generator, using the invariant evaluation pairing.
+
+For width one, the band consists of the $m$ positions with $i+j=m+1$.
+Every row and column block is a singleton, and both alphabets consist only
+of the letter $1$. Thus $u_{m,1}$ is a nonzero constant word tensor, fixed
+by every position permutation, and
+\begin{equation}\label{path:one-band}
+ U_{m,1}\cong S^{(m)} \qquad (m\geq2).
+\end{equation}
+We now determine the support needed from the particular width-two
+generator.
+
+\subsection{The indexed path contraction}
+
+Let $m\geq3$, $K=2m-1$, and $F=B_m\setminus B_{m-2}$.
+Number its positions along the path by
+\begin{equation}\label{path:positions}
+ p_{2j-1}=(m+1-j,j)\quad(1\leq j\leq m),\qquad
+ p_{2j}=(m-j,j)\quad(1\leq j<m).
+\end{equation}
+In this numbering, the row blocks are the singleton $1$ and the pairs
+$(2,3),(4,5),\ldots,(K-1,K)$; the column blocks are
+$(1,2),(3,4),\ldots,(K-2,K-1)$ and the singleton $K$.
+Order each pair by increasing path index and let $u$ denote the resulting
+alternating band tensor. The row orders agree with increasing $j$, while
+each of the $m-1$ column pairs reverses increasing $i$. Consequently
+\begin{equation}\label{path:generator-sign}
+ u=(-1)^{m-1}u_{m,2},\qquad U_{m,2}=\C[S_F]u.
+\end{equation}
+\Cref{fig:band-path} illustrates the numbering and the consecutive
+segments used in \Cref{path:conjugation-example}.
+
+\begin{figure}[ht]
+\centering
+\setlength{\unitlength}{1pt}
+\begin{picture}(430,155)
+\small
+\put(90,148){\makebox(0,0){Band in $B_3$}}
+\put(10,139){\vector(1,0){25}}
+\put(39,139){\makebox(0,0){$j$}}
+\put(10,139){\vector(0,-1){25}}
+\put(10,108){\makebox(0,0){$i$}}
+\put(45,110){\color[gray]{0.65}\circle*{6}}
+\put(45,123){\makebox(0,0){$D=B_1$}}
+\put(45,30){\circle*{4}}
+\put(45,70){\circle*{4}}
+\put(90,70){\circle*{4}}
+\put(90,110){\circle*{4}}
+\put(135,110){\circle*{4}}
+\put(45,34){\vector(0,1){32}}
+\put(90,74){\vector(0,1){32}}
+\multiput(49,70)(8,0){4}{\line(1,0){4}}
+\put(81,70){\vector(1,0){5}}
+\multiput(94,110)(8,0){4}{\line(1,0){4}}
+\put(126,110){\vector(1,0){5}}
+\put(45,16){\makebox(0,0){$p_1$}}
+\put(30,70){\makebox(0,0){$p_2$}}
+\put(106,70){\makebox(0,0){$p_3$}}
+\put(90,124){\makebox(0,0){$p_4$}}
+\put(135,124){\makebox(0,0){$p_5$}}
+\put(305,148){\makebox(0,0){Dual-test columns for $\eta=(2,2,1)$}}
+\multiput(220,90)(40,0){3}{\circle*{4}}
+\put(350,90){\circle*{4}}
+\put(390,90){\circle*{4}}
+\put(224,90){\vector(1,0){32}}
+\multiput(264,90)(8,0){3}{\line(1,0){4}}
+\put(288,90){\vector(1,0){8}}
+\put(304,90){\vector(1,0){42}}
+\multiput(354,90)(8,0){3}{\line(1,0){4}}
+\put(378,90){\vector(1,0){8}}
+\put(220,76){\makebox(0,0){$p_1$}}
+\put(260,76){\makebox(0,0){$p_2$}}
+\put(300,76){\makebox(0,0){$p_3$}}
+\put(350,76){\makebox(0,0){$p_4$}}
+\put(390,76){\makebox(0,0){$p_5$}}
+\put(325,70){\color[gray]{0.5}\line(0,1){38}}
+\put(215,55){\line(1,0){90}}
+\put(215,55){\line(0,1){6}}
+\put(305,55){\line(0,1){6}}
+\put(260,43){\makebox(0,0){$r_1=3$}}
+\put(335,55){\line(1,0){60}}
+\put(335,55){\line(0,1){6}}
+\put(395,55){\line(0,1){6}}
+\put(365,43){\makebox(0,0){$r_2=2$}}
+\end{picture}
+\caption{The width-two band for $m=3$. Solid arrows are column pairs;
+dashed arrows are row pairs, all directed by increasing path index.
+Column arrows reverse increasing $i$, as in
+\Cref{path:generator-sign}. The row singleton is $p_1$ and the column
+singleton is $p_5$, each carrying the fixed letter $1$ in its layer.
+On the right, the dual-test tableau of shape $\eta=(2,2,1)$ has column
+lengths $r_1=3$ and $r_2=2$, giving the two bracketed consecutive segments.
+The gray vertical line separates these segments, which differ from the
+original pair blocks.}
+\label{fig:band-path}
+\end{figure}
+
+Put $E=\C^2$ with basis $e_1,e_2$ and $V=E\otimes E$.
+Ordinary tensor reassociation identifies the two band layers with
+$V^{\otimes K}$, using the position order in \Cref{path:positions}.
+Identify $V^*$ with $\Mat_2(\C)$ by letting a matrix $X$ take the value
+$X_{ab}$ on $e_a\otimes e_b$. All pairings below are bilinear evaluation
+pairings. Write
+\[
+ J=\begin{pmatrix}0&1\\-1&0\end{pmatrix}.
+\]
+
+\begin{lemma}[Path contraction]\label{path:contraction}
+For arbitrary $X_1,\ldots,X_K\in\Mat_2(\C)$,
+\begin{equation}\label{path:contraction-formula}
+ \left\langle u,X_1\otimes\cdots\otimes X_K\right\rangle
+ =(-1)^{m-1}
+ \left[X_1\adj(X_2)X_3\adj(X_4)\cdots
+                 X_{K-2}\adj(X_{K-1})X_K\right]_{11}.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+The two singleton blocks require the first row-layer index and last
+column-layer index to be $1$. The coefficient of an alternating pair in
+path order is $J_{ab}$. Hence the left side of
+\Cref{path:contraction-formula} is exactly
+\begin{equation}\label{path:index-sum}
+ \sum_{\substack{a_1,\ldots,a_K,b_1,\ldots,b_K\in\{1,2\}\\
+                  a_1=b_K=1}}
+ \left(\prod_{\ell=1}^{m-1}
+   J_{b_{2\ell-1},b_{2\ell}}J_{a_{2\ell},a_{2\ell+1}}\right)
+ \prod_{q=1}^K(X_q)_{a_qb_q}.
+\end{equation}
+Reading the contracted indices successively from $a_1$ to $b_K$ gives
+\[
+ \left[X_1JX_2^{\mathsf T}JX_3JX_4^{\mathsf T}J
+       \cdots X_{K-2}JX_{K-1}^{\mathsf T}JX_K\right]_{11}.
+\]
+The transpose at an even position records that its column index is
+encountered before its row index. For
+$X=\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)$,
+direct multiplication gives
+\begin{equation}\label{path:adjugate}
+ \adj(X)=\begin{pmatrix}d&-b\\-c&a\end{pmatrix}
+        =\tr(X)I-X,\qquad JX^{\mathsf T}J=-\adj(X).
+\end{equation}
+There are $m-1$ even positions, yielding the stated sign.
+In particular, adjugation is a linear operation on $\Mat_2(\C)$.
+\end{proof}
+
+\subsection{Alternating consecutive segments}
+
+For a shape with at most four rows, place its tableau columns on
+successive disjoint segments of the path. On a segment of length $r$,
+the dual polytabloid alternates the first $r$ elements of one common
+basis of $V^*$. The contraction formula therefore assigns an alternating
+matrix sum to each segment; these sums multiply in path order.
+We now choose the basis so that every such segment matrix, for
+$1\le r\le4$ and either starting parity, is invertible.
+
+Use the following ordered basis of $V^*$:
+\begin{equation}\label{path:matrix-basis}
+ M_1=I,\qquad
+ M_2=Z=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\qquad
+ M_3=T=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad
+ M_4=J=ZT.
+\end{equation}
+These matrices are linearly independent: $I,Z$ span the diagonal
+matrices and $T,J$ span the off-diagonal matrices. The last three are
+traceless and anticommute in pairs; moreover
+\[
+ Z^2=T^2=I,\qquad J^2=-I.
+\]
+Thus all four are invertible. Alternating evaluations on $Z,T,J$ are also
+used to test polynomial identities for two-by-two matrices with adjugation
+\cite[Section~3]{drenskygiambruno1994}; here the adjugations are determined
+by the path positions.
+
+For a global path position $q$, define the linear operation
+\[
+ D_q(X)=
+ \begin{cases}
+ X,&q\text{ odd},\\
+ \adj(X),&q\text{ even}.
+ \end{cases}
+\]
+For a segment of length $r\leq4$ beginning at position $p$, define its
+alternating matrix by the following sum of ordered products:
+\begin{equation}\label{path:segment-definition}
+ A_{r,p}=\sum_{\pi\in S_r}\sgn(\pi)
+      D_p(M_{\pi(1)})D_{p+1}(M_{\pi(2)})\cdots
+      D_{p+r-1}(M_{\pi(r)}).
+\end{equation}
+
+\begin{lemma}[Segment alternation]\label{path:segment}
+Let $e$ be the number of even positions among $p,\ldots,p+r-1$, and put
+$\delta=1$ if $p$ is even and $\delta=0$ otherwise. Then
+\begin{equation}\label{path:segment-formula}
+ A_{r,p}=r!(-1)^{e-\delta}M_1\cdots M_r.
+\end{equation}
+In particular, all these matrices are invertible. Their values are
+\begin{equation}\label{path:segment-table}
+ \begin{array}{c|cc}
+ r & p\text{ odd} & p\text{ even}\\ \hline
+ 1&I&I\\
+ 2&-2Z&2Z\\
+ 3&-6J&-6J\\
+ 4&-24I&24I
+ \end{array}
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Fix a permutation $\pi$ and let $j$ be the local slot occupied by $M_1=I$.
+Deleting $I$ leaves a permutation of $M_2,\ldots,M_r$.
+Its sign is exactly the sign acquired when those pairwise anticommuting
+matrices are reordered. Moving $I$ from its first canonical slot to slot
+$j$ contributes $(-1)^{j-1}$ to $\sgn(\pi)$ and changes no matrix product.
+It follows that
+\[
+ \sgn(\pi)M_{\pi(1)}\cdots M_{\pi(r)}
+       =(-1)^{j-1}M_1\cdots M_r.
+\]
+By \Cref{path:adjugate}, adjugation fixes $I$ and negates each of the
+other basis matrices. Write $\epsilon_j=1$ if $p+j-1$ is even, and $0$
+otherwise. Adjugation therefore contributes the additional sign
+$(-1)^{e-\epsilon_j}$. Since
+$\epsilon_j\equiv\delta+j-1\pmod2$, the total sign is
+$(-1)^{e-\delta}$, independently of $\pi$. All $r!$ summands in
+\Cref{path:segment-definition} consequently agree, proving
+\Cref{path:segment-formula}, including $r=1$.
+Finally, the canonical products for $r=1,2,3,4$ are respectively
+$I,Z,J,-I$, which gives \Cref{path:segment-table}.
+\end{proof}
+
+\subsection{Detection by a dual polytabloid}
+
+The segment calculation gives an invertible ordered product. Its fixed
+endpoint entry can nevertheless vanish. We now choose the dual test
+basis so that this entry is nonzero, while keeping the band generator
+and its endpoint letters unchanged.
+
+\begin{proposition}[Four-row support of the width-two band]\label{prop:two-band}
+For every $m\geq3$ and every partition $\eta\vdash 2m-1$ with
+$\len(\eta)\leq4$, the irreducible module $S^\eta$ occurs in $U_{m,2}$.
+\end{proposition}
+
+\begin{proof}
+Let $r_1,\ldots,r_c$ be the column lengths of $\eta$ in their usual order,
+so $1\leq r_j\leq4$ and $\sum_jr_j=K$. Partition the path into consecutive
+segments of these lengths, beginning at
+$s_j=1+\sum_{i<j}r_i$.
+For any ordered basis $N_1,\ldots,N_4$ of $V^*$, let $t_N$ be the dual
+tensor whose factor on segment $j$ is
+\[
+ \sum_{\pi\in S_{r_j}}\sgn(\pi)
+       N_{\pi(1)}\otimes\cdots\otimes N_{\pi(r_j)}.
+\]
+The segments are the columns of a tableau of shape $\eta$ and the initial
+basis entries label its rows. By \Cref{lem:polytabloid}, their product
+$t_N$ is a polytabloid, and
+\[
+ \mathcal T_N=\C[S_F]t_N\ \subseteq\ (V^*)^{\otimes K}
+ \quad\text{is irreducible of type }\eta.
+\]
+
+First take $N_i=M_i$. Multilinearity of
+\Cref{path:contraction-formula} and the definition of $t_M$ give
+\begin{equation}\label{path:ordered-factorization}
+ \langle u,t_M\rangle=(-1)^{m-1}[B]_{11},\qquad
+ B=A_{r_1,s_1}A_{r_2,s_2}\cdots A_{r_c,s_c}.
+\end{equation}
+Indeed, summing independently over the permutations within each segment
+is exactly the distributive expansion of the displayed ordered product.
+The segments are consecutive, so no matrix factors are interchanged in
+this factorization. By \Cref{path:segment}, $B$ is invertible.
+
+The identity $\adj(X)=\tr(X)I-X$ also shows that, for every
+$Q\in\GL_2(\C)$,
+\[
+ D_q(QXQ^{-1})=QD_q(X)Q^{-1}.
+\]
+Choose an eigenvector $v$ of $B$ with eigenvalue $\beta$; invertibility
+gives $\beta\ne0$. Choose $Q$ taking $v$ to the first coordinate vector
+and set $N_i=QM_iQ^{-1}$. Conjugation is an invertible linear operation
+on $V^*$, so these $N_i$ are still an ordered basis. Each alternating
+segment matrix is now $QA_{r_j,s_j}Q^{-1}$, and their ordered product is
+$QBQ^{-1}$. Since $(QBQ^{-1})e_1=\beta e_1$,
+\begin{equation}\label{path:nonzero-pairing}
+ \langle u,t_N\rangle=(-1)^{m-1}[QBQ^{-1}]_{11}
+                    =(-1)^{m-1}\beta\ne0.
+\end{equation}
+Only the test covectors have changed; $u$ and its endpoint letters remain
+those fixed in \Cref{path:generator-sign,path:index-sum}.
+
+The invariant evaluation pairing defines
+\begin{equation}\label{path:quotient-map}
+ \Phi:U_{m,2}\longrightarrow\mathcal T_N^*,\qquad
+ \Phi(x)(t)=\langle x,t\rangle.
+\end{equation}
+For $g\in S_F$, its equivariance follows explicitly from
+$\Phi(gx)(t)=\langle x,g^{-1}t\rangle=(g\Phi(x))(t)$.
+By \Cref{path:nonzero-pairing} this map is nonzero on the particular
+generator $u$. Its target is irreducible and, by the self-duality in
+\Cref{tools:dual-sign}, has type $\eta$. Thus $\Phi$ is surjective, and
+semisimplicity proves the claimed occurrence in $U_{m,2}$.
+\end{proof}
+
+\begin{example}\label{path:conjugation-example}
+The change of test basis can be essential. For $m=3$ and
+$\eta=(2,2,1)$, the column segments have lengths $3,2$, beginning at
+positions $1,4$. Their product in \Cref{path:ordered-factorization} is
+$(-6J)(2Z)=12T$, whose $(1,1)$ entry vanishes. With
+\[
+ Q=\begin{pmatrix}1&1\\1&2\end{pmatrix},\qquad
+ QTQ^{-1}=\begin{pmatrix}1&0\\3&-1\end{pmatrix},
+\]
+the resulting pairing is $12$, since $(-1)^{m-1}=1$.
+\end{example}
diff --git a/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/build/sections/specht-tools.tex b/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/build/sections/specht-tools.tex
new file mode 100644
index 0000000000000000000000000000000000000000..c30074d46d2224b8b19cbdbb852d0800c01d24fa
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+++ b/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/build/sections/specht-tools.tex
@@ -0,0 +1,222 @@
+\section{Specht modules and horizontal strips}
+\label{sec:specht-tools}
+
+We fix the tensor conventions used to identify alternating block vectors
+with Specht modules. The other input needed by the induction is the
+Pieri rule: it transfers constituent support across the removal of
+horizontal strips. We record its dominance consequences with proofs.
+
+All representations and tensor products are over $\C$ and are
+finite-dimensional.  A partition $\lambda\vdash n$ is padded with zero
+parts when necessary; its length is $\len(\lambda)$ and its conjugate is
+$\lambda^t$, where $\lambda^t_j=\#\{i:\lambda_i\ge j\}$.  We identify
+$\lambda$ with its diagram $\{(i,j):1\le j\le\lambda_i\}$.
+For partitions of the same size, write $\lambda\unrhd\mu$ if
+$\sum_{i=1}^k\lambda_i\ge\sum_{i=1}^k\mu_i$ for every $k\ge1$.
+Conjugation reverses this order:
+\begin{equation}\label{tools:conjugate-dominance}
+ \lambda\unrhd\mu\quad\Longleftrightarrow\quad
+ \mu^t\unrhd\lambda^t.
+\end{equation}
+Indeed, for every integer $q\ge1$,
+\[
+ \sum_i\max(\lambda_i-q,0)
+ =\max_{k\ge0}\left(\sum_{i=1}^k\lambda_i-kq\right),
+ \qquad
+ \sum_{j=1}^q\lambda^t_j
+ =n-\sum_i\max(\lambda_i-q,0).
+\]
+Dominance increases the first expression and therefore decreases the
+second.  Applying the resulting implication to the conjugate partitions
+proves the converse.
+
+\subsection{Position tensors and polytabloids}
+
+For a finite set $B$, write $S_B$ for its symmetric group. For
+$\lambda\vdash |B|$, write $S_B^\lambda$ for the irreducible $S_B$-module
+of type $\lambda$; omit the subscript when the position set is understood.
+If $E$ has
+ordered basis $e_1,\ldots,e_d$, the tensor space $E^{\otimes B}$ has word
+basis $e_a=\bigotimes_{b\in B}e_{a(b)}$, indexed by maps
+$a:B\to\{1,\ldots,d\}$.  The position action is
+$g e_a=e_{a\circ g^{-1}}$.  All regroupings of tensor factors are ordinary
+tensor identifications and introduce no signs.
+
+Let $L$ be a set partition of $B$ whose blocks have size at most $d$.
+For each block $A=(b_1,\ldots,b_r)$ choose an ordering and set
+\begin{equation}\label{tools:block-tensor}
+ v_A=\sum_{\pi\in S_r}\sgn(\pi)
+       \bigotimes_{i=1}^r e_{\pi(i)},\qquad
+ v_L=\bigotimes_{A\in L}v_A,
+\end{equation}
+where the $i$th factor of $v_A$ occupies position $b_i$.
+Changing a block ordering changes $v_L$ only by a sign.  Its nonzero
+word coefficients are $\pm1$: on each block of size $r$, the letters are
+a bijection with $\{1,\ldots,r\}$.
+
+\begin{lemma}[Alternating blocks as polytabloids]\label{lem:polytabloid}
+Define $\theta_i=\#\{A\in L:|A|\ge i\}$, so that the block sizes are
+the column lengths of $\theta$.  Then
+$\C[S_B]v_L\cong S^\theta$.
+The same assertion holds for any ordered basis of a dual vector space.
+\end{lemma}
+
+\begin{proof}
+Arrange the blocks as columns of a tableau $t$ of shape $\theta$, in
+nonincreasing order of their sizes, and place each ordered block from top
+to bottom.  Tableau entries are the distinct elements of $B$.
+A row tabloid $\{t'\}$ records the set of entries in each row.
+Let $M^\theta$ be the permutation module spanned by these row tabloids.
+Send $\{t'\}$ to
+the word assigning letter $i$ to every entry of row $i$.
+This is an $S_B$-equivariant bijection from the tabloid basis of
+$M^\theta$ to the word basis of content $\theta$.
+If $C_t$ is the subgroup permuting entries within each column of $t$,
+the polytabloid
+\[
+ e_t=\sum_{c\in C_t}\sgn(c)c\{t\}
+\]
+maps to $v_L$: in a column, the position action replaces a permutation
+of the letters by its inverse, which has the same sign.
+In particular this vector is nonzero, since the original row word has
+coefficient $1$.  The span of all translates of $e_t$ is the Specht
+module $S^\theta$, irreducible over $\C$ by the standard polytabloid
+construction \cite[Definitions~4.1 and~4.3, Theorem~4.12]{james1978}.
+This proves the assertion.
+The argument applies to a dual space with any chosen basis; alternatively,
+an invertible basis change on that space induces an invertible map on
+its position tensor power commuting with $S_B$.
+\end{proof}
+
+We use the usual complex representation-theoretic facts
+\cite[Theorems~4.12 and~6.7]{james1978}: the $S^\lambda$, for
+$\lambda\vdash n$, form the complete list of irreducible $S_n$-modules,
+and
+\begin{equation}\label{tools:dual-sign}
+ (S^\lambda)^*\cong S^\lambda,
+ \qquad S^\lambda\otimes\sgn\cong S^{\lambda^t}.
+\end{equation}
+Every complex representation of a finite group is semisimple
+\cite[Proposition~1.5 and Corollary~1.6]{fultonharris1991}, so any
+irreducible quotient is also a constituent.  For $H\le G$,
+Frobenius reciprocity \cite[Proposition~3.17]{fultonharris1991}
+takes the form
+\begin{equation}\label{tools:frobenius}
+ \Hom_G(\Ind_H^G A,V)
+ \cong\Hom_H(A,\Res_H^G V).
+\end{equation}
+Induction, modeled by $\C[G]\otimes_{\C[H]}A$, is transitive and
+distributes over direct sums.  The symbol
+$\boxtimes$ denotes the outer tensor product for separate group actions;
+ordinary tensor products of two $S_B$-modules carry the diagonal action.
+The natural pairing of $E^{\otimes B}$ and $(E^*)^{\otimes B}$ satisfies
+$\langle gu,gt\rangle=\langle u,t\rangle$.
+Thus pairing with an invariant subspace $T$ of the dual tensor space
+defines an equivariant map $u\mapsto(t\mapsto\langle u,t\rangle)$
+into $T^*$.
+
+\subsection{Pieri and the needed part of Young's rule}
+
+For diagrams $\nu\subseteq\lambda$, the skew diagram $\lambda/\nu$
+is a \emph{horizontal strip} if it has at most one box in each column.
+We use the empty partition and the trivial group $S_0$ in the following
+standard identity.
+
+\begin{lemma}[Pieri rule]\label{lem:pieri}
+For $\nu\vdash a$ and $b\ge0$,
+\begin{equation}\label{tools:pieri}
+ \Ind_{S_a\times S_b}^{S_{a+b}}
+      (S^\nu\boxtimes\mathbf1)
+ \cong
+ \bigoplus_{\substack{\lambda\vdash a+b,\ \lambda\supseteq\nu\\
+                      \lambda/\nu\text{ a horizontal strip}}}
+       S^\lambda.
+\end{equation}
+Every summand occurs once.
+\end{lemma}
+
+This is the one-row specialization of the Littlewood--Richardson rule
+\cite[Section~16, especially Rule~16.4]{james1978}: all $b$ entries of
+its skew filling are $1$, so column strictness requires a horizontal
+strip; the unique such filling satisfies the lattice-word condition.
+We give the dominance and multiplicity
+consequences needed here, including the constructive existence argument.
+
+\begin{lemma}[Young's rule: support and diagonal multiplicity]
+\label{lem:young-rule}
+Let $\theta=(\theta_1,\ldots,\theta_\ell)\vdash n$. The row-tabloid
+permutation module has the equivalent induced realization
+\[
+ M^\theta=
+ \Ind_{S_{\theta_1}\times\cdots\times S_{\theta_\ell}}^{S_n}
+ \mathbf1.
+\]
+Then $S^\tau$ occurs in $M^\theta$ if and only if
+$\tau\unrhd\theta$, and $S^\theta$ occurs once.
+The word module of any content obtained by reordering the parts of
+$\theta$ is also isomorphic to $M^\theta$.
+\end{lemma}
+
+\begin{proof}
+The case $n=0$ is immediate from the trivial representation of $S_0$.
+Assume henceforth that $n>0$.
+Transitivity of induction and \Cref{lem:pieri} show that the multiplicity
+of $S^\tau$ counts chains
+\begin{equation}\label{tools:strip-chain}
+ \varnothing=\nu^{(0)}\subseteq\nu^{(1)}\subseteq\cdots
+ \subseteq\nu^{(\ell)}=\tau,
+ \qquad |\nu^{(k)}/\nu^{(k-1)}|=\theta_k,
+\end{equation}
+whose successive differences are horizontal strips.
+Each strip raises the first column's height by at most one, so
+$\nu^{(k)}$ has at most $k$ rows.  Since it is contained in $\tau$,
+\[
+ \sum_{i=1}^k\theta_i=|\nu^{(k)}|
+ \le\sum_{i=1}^k\tau_i.
+\]
+For $k>\ell$, both initial sums equal $n$, since $\tau$ has at most
+$\ell$ rows.  This proves necessity of dominance.
+If $\tau=\theta$, equality forces
+$\nu^{(k)}$ to be exactly the first $k$ rows of $\theta$.
+These diagrams do form a chain of the required kind, proving
+multiplicity one.
+
+For sufficiency, suppose $\tau\unrhd\theta$ and induct on $\ell$.
+For $\ell=1$, necessarily $\tau=(n)$.
+For $\ell>1$, put $t=\theta_\ell$.
+There are at least $t$ columns of $\tau$, since
+$\tau_1\ge\theta_1\ge t$.  Delete a bottom box from each of the
+$t$ tallest columns, choosing the rightmost columns first among equal
+heights, and call the remaining diagram $\nu$.
+Its column heights are still nonincreasing: unequal adjacent heights
+differ by at least one, and within any group of equal heights precisely
+a suffix is lowered.  Thus $\nu$ is a partition and $\tau/\nu$ is a
+horizontal $t$-strip.  For every $k\ge1$ its loss in the first $k$ rows is
+\begin{equation}\label{tools:prefix-loss}
+ \sum_{i=1}^k\tau_i-\sum_{i=1}^k\nu_i
+       =\max(0,t-\tau_{k+1}):
+\end{equation}
+there are $\tau_{k+1}$ columns taller than $k$, and the rule selects these
+before any column whose bottom box belongs to the first $k$ rows.
+If the loss is zero, the desired dominance inequality for $\nu$ follows
+immediately.  If it is positive and $k<\ell$, then
+\[
+ \sum_{i=1}^k\nu_i
+ =\sum_{i=1}^{k+1}\tau_i-t
+ \ge\sum_{i=1}^{k+1}\theta_i-t
+ \ge\sum_{i=1}^k\theta_i.
+\]
+For completeness, dominance at $\ell$ gives $\len(\tau)\le\ell$,
+and dominance at $\ell-1$ gives $\tau_\ell\le t$.
+All $\tau_\ell$ columns of height $\ell$ are therefore selected, so
+the last row disappears and $\len(\nu)\le\ell-1$.
+Hence all remaining initial sums equal the total $n-t$, and
+$\nu\unrhd(\theta_1,\ldots,\theta_{\ell-1})$.
+Induction gives a chain ending at $\nu$; appending the deleted strip
+gives \Cref{tools:strip-chain} ending at $\tau$.
+
+Finally, the stabilizer of a word with prescribed letter multiplicities
+is the product of the symmetric groups on its equal-letter position
+sets.  Reordering those multiplicities conjugates this subgroup, giving
+the asserted isomorphism of permutation modules.
+\end{proof}
diff --git a/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/paper.pdf b/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/paper.pdf
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index 0000000000000000000000000000000000000000..4b88ba1fe782564c3330cd762e4ec4bb3de1859e
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+++ b/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:42c2824b06a1114993095d44e1226e7f15f17caaa09358d68fa5dfddedbd8f44
+size 360873
diff --git a/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/README.md b/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..f9bf72074ec79d4842c0caa58a2e62e297bcedde
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+++ b/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/README.md
@@ -0,0 +1,18 @@
+# [A Direct Proof of Optimal Max-Cut Hardness](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 23, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026,
+  author = {{OpenAI}},
+  title = {{A Direct Proof of Optimal Max-Cut Hardness}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/paper.pdf}{OAI:A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/figures/complementary-traversals.tex b/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/figures/complementary-traversals.tex
new file mode 100644
index 0000000000000000000000000000000000000000..11017874e3387a3d5b444540717f066a7ea20f37
--- /dev/null
+++ b/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/figures/complementary-traversals.tex
@@ -0,0 +1,20 @@
+\begin{tikzpicture}[x=1cm,y=1cm,every node/.style={font=\small},
+  mono/.style={draw=orange!80!black,line width=1.4pt},
+  slice/.style={draw=blue!65!black,line width=1.4pt,dashed}]
+\node[circle,draw,minimum size=7mm] (v) at (0,1.7) {$v$};
+\foreach \x/\name/\lab in {-3/a/{(1,0)},-2/b/{(1,1)},-.5/c/{(2,0)},.5/d/{(2,1)},2/e/{(3,0)},3/f/{(3,1)}} {
+ \node[circle,draw,fill=white,minimum size=3mm,inner sep=1pt] (\name) at (\x,0) {};
+ \node[below] at (\x,-.12) {$\lab$};
+ \draw[gray!40] (v) -- (\name);
+}
+\draw[mono] (v) -- (a);
+\draw[mono] (v) -- (d);
+\draw[mono] (v) -- (e);
+\draw[slice] (v) -- (c);
+\draw[slice] (v) -- (d);
+\node[circle,draw,fill=violet!65,minimum size=4mm,inner sep=1pt] at (.5,0) {};
+\draw[mono] (-3,-1) -- (-2.4,-1);
+\node[anchor=west] at (-2.3,-1) {one child per pair};
+\draw[slice] (-3,-1.5) -- (-2.4,-1.5);
+\node[anchor=west] at (-2.3,-1.5) {both in one pair};
+\end{tikzpicture}
diff --git a/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/main.tex b/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/main.tex
new file mode 100644
index 0000000000000000000000000000000000000000..d551e94cdd005d9a9d5e28f21af89dc7dcb01875
--- /dev/null
+++ b/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/main.tex
@@ -0,0 +1,72 @@
+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage{lmodern}
+\usepackage[margin=1.12in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs}
+\usepackage{microtype}
+\usepackage{enumitem}
+\usepackage{xcolor}
+\usepackage{tikz}
+\usepackage{hyperref}
+\hypersetup{colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black,pdftitle={A Direct Proof of Optimal Max-Cut Hardness},pdfauthor={OpenAI}}
+\usepackage[nameinlink,capitalise,noabbrev]{cleveref}
+\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{\F}{\mathbb F_2}
+\newcommand{\E}{\mathbb E}
+\newcommand{\one}{\mathbf 1}
+\newcommand{\GW}{\alpha_{\mathrm{GW}}}
+\DeclareMathOperator{\Aff}{Aff}
+\DeclareMathOperator{\Stab}{Stab}
+\DeclareMathOperator{\Inf}{Inf}
+\DeclareMathOperator{\Val}{Val}
+\DeclareMathOperator{\TV}{TV}
+\DeclareMathOperator{\supp}{supp}
+\DeclareMathOperator{\codim}{codim}
+\DeclarePairedDelimiter{\abs}{\lvert}{\rvert}
+\DeclarePairedDelimiter{\norm}{\lVert}{\rVert}
+\newcommand{\ip}[2]{\langle #1,#2\rangle}
+\newcommand{\bits}{\{-1,1\}}
+\newcommand{\Bfun}{B}
+\newcommand{\cut}{\operatorname{cut}}
+\newcommand{\eps}{\varepsilon}
+\newcommand{\ind}{\mathbf 1}
+\setlist{topsep=4pt,itemsep=2pt}
+\allowdisplaybreaks[2]
+
+\title{A Direct Proof of Optimal Max-Cut Hardness}
+\author{OpenAI}
+\date{September 23, 2026}
+
+\begin{document}
+\maketitle
+\begin{abstract}
+We prove that approximating Max-Cut on simple unweighted graphs within
+any fixed factor greater than the Goemans--Williamson constant is NP-hard.
+\end{abstract}
+
+\input{sections/introduction}
+\input{sections/preliminaries}
+\input{sections/tree}
+\input{sections/reduction}
+\input{sections/tree_fourier}
+\input{sections/hashing}
+\input{sections/extraction}
+\input{sections/decoding}
+\input{sections/conclusion}
+
+\appendix
+\input{sections/affine_games}
+\input{sections/unweighted}
+
+\bibliographystyle{plain}
+\bibliography{references}
+\end{document}
diff --git a/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/references.bib b/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/references.bib
new file mode 100644
index 0000000000000000000000000000000000000000..fd031e2ecfaa247cd24a9ca92f0af243f48fc41f
--- /dev/null
+++ b/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/references.bib
@@ -0,0 +1,214 @@
+@incollection{Karp72,
+  author = {Karp, Richard M.},
+  title = {Reducibility among Combinatorial Problems},
+  booktitle = {Complexity of Computer Computations},
+  editor = {Miller, Raymond E. and Thatcher, James W.},
+  publisher = {Plenum Press},
+  address = {New York},
+  pages = {85--103},
+  year = {1972},
+  doi = {10.1007/978-1-4684-2001-2_9},
+  url = {https://doi.org/10.1007/978-1-4684-2001-2_9}
+}
+
+@article{GW95,
+  author = {Goemans, Michel X. and Williamson, David P.},
+  title = {Improved Approximation Algorithms for Maximum Cut and
+           Satisfiability Problems Using Semidefinite Programming},
+  journal = {Journal of the ACM},
+  volume = {42},
+  number = {6},
+  pages = {1115--1145},
+  year = {1995},
+  doi = {10.1145/227683.227684},
+  url = {https://math.mit.edu/~goemans/PAPERS/maxcut-jacm.pdf}
+}
+
+@article{Hastad01,
+  author = {H{\aa}stad, Johan},
+  title = {Some Optimal Inapproximability Results},
+  journal = {Journal of the ACM},
+  volume = {48},
+  number = {4},
+  pages = {798--859},
+  year = {2001},
+  doi = {10.1145/502090.502098},
+  url = {https://www.csc.kth.se/~johanh/optimalinap.pdf}
+}
+
+@article{KKMO07,
+  author = {Khot, Subhash and Kindler, Guy and Mossel, Elchanan
+            and O'Donnell, Ryan},
+  title = {Optimal Inapproximability Results for {MAX-CUT}
+           and Other 2-Variable {CSPs}?},
+  journal = {SIAM Journal on Computing},
+  volume = {37},
+  number = {1},
+  pages = {319--357},
+  year = {2007},
+  doi = {10.1137/S0097539705447372},
+  url = {https://www.stat.berkeley.edu/~mossel/publications/max_cut_final.pdf}
+}
+
+@article{MOO10,
+  author = {Mossel, Elchanan and O'Donnell, Ryan and Oleszkiewicz, Krzysztof},
+  title = {Noise Stability of Functions with Low Influences:
+           Invariance and Optimality},
+  journal = {Annals of Mathematics},
+  volume = {171},
+  number = {1},
+  pages = {295--341},
+  year = {2010},
+  doi = {10.4007/annals.2010.171.295},
+  url = {https://annals.math.princeton.edu/2010/171-1/p05}
+}
+
+@inproceedings{OW08,
+  author = {O'Donnell, Ryan and Wu, Yi},
+  title = {An Optimal {SDP} Algorithm for {Max-Cut}, and Equally Optimal
+           Long Code Tests},
+  booktitle = {Proceedings of the Fortieth Annual ACM Symposium
+               on Theory of Computing},
+  pages = {335--344},
+  publisher = {ACM},
+  year = {2008},
+  doi = {10.1145/1374376.1374425},
+  url = {https://dl.acm.org/doi/10.1145/1374376.1374425}
+}
+
+@inproceedings{Raghavendra08,
+  author = {Raghavendra, Prasad},
+  title = {Optimal Algorithms and Inapproximability Results for Every {CSP}?},
+  booktitle = {Proceedings of the Fortieth Annual ACM Symposium
+               on Theory of Computing},
+  pages = {245--254},
+  publisher = {ACM},
+  year = {2008},
+  doi = {10.1145/1374376.1374414},
+  url = {https://dl.acm.org/doi/10.1145/1374376.1374414}
+}
+
+@article{KMS25,
+  author = {Khot, Subhash and Minzer, Dor and Safra, Muli},
+  title = {On Independent Sets, 2-to-2 Games and {Grassmann} Graphs},
+  journal = {Theory of Computing},
+  volume = {21},
+  number = {10},
+  pages = {1--55},
+  year = {2025},
+  doi = {10.4086/toc.2025.v021a010},
+  url = {https://theoryofcomputing.org/articles/v021a010/v021a010.pdf},
+  note = {Preliminary version in STOC 2017}
+}
+
+@article{DKKMS25,
+  author = {Dinur, Irit and Khot, Subhash and Kindler, Guy and Minzer, Dor and Safra, Muli},
+  title = {Towards a Proof of the 2-to-1 Games Conjecture?},
+  journal = {Theory of Computing},
+  volume = {21}, number = {11}, pages = {1--50}, year = {2025},
+  doi = {10.4086/toc.2025.v021a011},
+  url = {https://theoryofcomputing.org/articles/v021a011/v021a011.pdf},
+  note = {Preliminary version in STOC 2018}
+}
+
+@inproceedings{BKS19,
+  author = {Barak, Boaz and Kothari, Pravesh K. and Steurer, David},
+  title = {Small-Set Expansion in Shortcode Graph and the {2-to-2 Conjecture}},
+  booktitle = {10th Innovations in Theoretical Computer Science Conference
+               (ITCS 2019)},
+  editor = {Blum, Avrim},
+  series = {Leibniz International Proceedings in Informatics},
+  volume = {124},
+  pages = {9:1--9:12},
+  publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
+  year = {2019},
+  doi = {10.4230/LIPIcs.ITCS.2019.9},
+  url = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.9}
+}
+
+@article{KMS23,
+  author = {Khot, Subhash and Minzer, Dor and Safra, Muli},
+  title = {Pseudorandom Sets in {Grassmann} Graph Have Near-Perfect Expansion},
+  journal = {Annals of Mathematics},
+  volume = {198},
+  number = {1},
+  pages = {1--92},
+  year = {2023},
+  doi = {10.4007/annals.2023.198.1.1},
+  url = {https://annals.math.princeton.edu/2023/198-1/p01}
+}
+
+@inproceedings{Khot02,
+  author = {Khot, Subhash},
+  title = {On the Power of Unique 2-Prover 1-Round Games},
+  booktitle = {Proceedings of the Thirty-Fourth Annual ACM Symposium
+               on Theory of Computing},
+  pages = {767--775},
+  publisher = {ACM},
+  year = {2002},
+  doi = {10.1145/509907.510017},
+  url = {https://dl.acm.org/doi/10.1145/509907.510017}
+}
+
+@article{TSSW00,
+  author = {Trevisan, Luca and Sorkin, Gregory B. and Sudan, Madhu and Williamson, David P.},
+  title = {Gadgets, Approximation, and Linear Programming},
+  journal = {SIAM Journal on Computing}, volume = {29}, number = {6},
+  pages = {2074--2097}, year = {2000},
+  doi = {10.1137/S0097539797328847},
+  url = {https://doi.org/10.1137/S0097539797328847}
+}
+
+@article{FS02,
+  author = {Feige, Uriel and Schechtman, Gideon},
+  title = {On the Optimality of the Random Hyperplane Rounding Technique for {MAX CUT}},
+  journal = {Random Structures \& Algorithms}, volume = {20}, number = {3},
+  pages = {403--440}, year = {2002},
+  doi = {10.1002/rsa.10036}, url = {https://doi.org/10.1002/rsa.10036}
+}
+
+@article{Borell85,
+  author = {Borell, Christer},
+  title = {Geometric Bounds on the {Ornstein--Uhlenbeck} Velocity Process},
+  journal = {Zeitschrift f{\"u}r Wahrscheinlichkeitstheorie und Verwandte Gebiete},
+  volume = {70}, number = {1}, pages = {1--13}, year = {1985},
+  doi = {10.1007/BF00532234}, url = {https://doi.org/10.1007/BF00532234}
+}
+
+@article{FGLSS96,
+  author = {Feige, Uriel and Goldwasser, Shafi and Lov{\'a}sz, L{\'a}szl{\'o} and Safra, Shmuel and Szegedy, Mario},
+  title = {Interactive Proofs and the Hardness of Approximating Cliques},
+  journal = {Journal of the ACM},
+  volume = {43}, number = {2}, pages = {268--292}, year = {1996},
+  doi = {10.1145/226643.226652},
+  url = {https://www.cs.tau.ac.il/~safra/PapersAndTalks/FGLSS.pdf}
+}
+
+@article{AroraSafra98,
+  author = {Arora, Sanjeev and Safra, Shmuel},
+  title = {Probabilistic Checking of Proofs: A New Characterization of {NP}},
+  journal = {Journal of the ACM},
+  volume = {45}, number = {1}, pages = {70--122}, year = {1998},
+  doi = {10.1145/273865.273901},
+  url = {https://www.cs.umd.edu/~gasarch/TOPICS/pcp/AS.pdf},
+  note = {Preliminary version in FOCS 1992}
+}
+
+@article{ALMSS98,
+  author = {Arora, Sanjeev and Lund, Carsten and Motwani, Rajeev and Sudan, Madhu and Szegedy, Mario},
+  title = {Proof Verification and the Hardness of Approximation Problems},
+  journal = {Journal of the ACM},
+  volume = {45}, number = {3}, pages = {501--555}, year = {1998},
+  doi = {10.1145/278298.278306},
+  url = {https://people.csail.mit.edu/madhu/papers/1992/almss-journ.pdf},
+  note = {Preliminary version in FOCS 1992}
+}
+
+@article{BGS98,
+  author = {Bellare, Mihir and Goldreich, Oded and Sudan, Madhu},
+  title = {Free Bits, {PCPs}, and Nonapproximability---Towards Tight Results},
+  journal = {SIAM Journal on Computing}, volume = {27}, number = {3},
+  pages = {804--915}, year = {1998},
+  doi = {10.1137/S0097539796302531},
+  url = {https://doi.org/10.1137/S0097539796302531}
+}
diff --git a/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/affine_games.tex b/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/affine_games.tex
new file mode 100644
index 0000000000000000000000000000000000000000..24d1a6f8f4f87d6f5c32378bdc47ffdcbaa36742
--- /dev/null
+++ b/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/affine_games.tex
@@ -0,0 +1,572 @@
+\section{The affine game supplied by the 2-to-1 Games Theorem}
+\label{sec:affine-games}
+
+The Max-Cut reduction needs two features beyond the usual numerical
+2-to-1 game gap: the projections must be affine over $\F$, and the
+alphabet must be fixed before the completeness error is chosen.
+We derive these features from the construction and soundness analysis in
+\cite[Sections~4--5]{DKKMS25}.  Its Grassmann agreement hypothesis is
+supplied by the expansion theorem of Khot, Minzer, and Safra~\cite{KMS23}
+and the expansion-to-agreement implications of Barak, Kothari, and
+Steurer~\cite[Theorems~9--11]{BKS19}; the introduction of the published
+version of~\cite{DKKMS25} records this completion.
+
+We first describe the folded game and its affine projections.  The
+source's vertex definition excludes certain subspaces that its displayed
+sampler can produce.  We specify the resulting conditioning, complete the
+tables used in its soundness proof, and give the invariant-list argument
+with the needed exceptional-subspace estimates.  For the second player
+we use bounded lists on quotient spaces, keeping each restriction to
+the advice space separate.  Completeness then follows by choosing the
+initial 3Lin error last.
+
+\subsection{The folded game and its edge law}
+
+\paragraph{Starting instance.}
+Theorem~4.1 of~\cite{DKKMS25}, obtained from H\aa stad's
+theorem~\cite{Hastad01}, supplies an absolute constant $s_*<1$ such that
+for every sufficiently small $\varepsilon>0$ it is NP-hard to distinguish regular binary
+3Lin instances of value at least $1-\varepsilon$ from those of value at
+most $s_*$.  Regularity includes bounded variable occurrence, three
+distinct variables per equation, and at most one variable shared by two
+distinct equations.
+
+For now let $\ell\ge2$, $k_0\ge\ell$, and $\beta\in(0,1)$ be fixed
+parameters, with $\beta$ rational.  The two alphabet dimensions will be
+$\ell$ and $\ell-1$, so the dimension in \Cref{prop:affine-games} is
+$n=\ell-1$.  The soundness argument below chooses $\ell,k_0,\beta$
+using only $b$ and the absolute constant $s_*$.
+
+\paragraph{Spaces, labels, and folding.}
+Let $\mathcal Z$ be the variables of the 3Lin instance, let
+$V_0=\F^{\mathcal Z}$, and view each equation $e$ as
+\[
+  \langle x_e,\alpha\rangle=b_e,
+  \qquad x_e\in V_0,\quad b_e\in\F,
+\]
+where $x_e$ has its three variable coordinates equal to one.  A
+\emph{legitimate} tuple $U$ consists of $k_0$ disjoint equations with
+the additional separation condition imposed in
+\cite[Section~4.2]{DKKMS25}: variables from different equations of $U$
+do not occur together in any input equation.  Put
+\[
+  V_U=\F^{\operatorname{var}(U)},
+  \qquad H_U=\operatorname{span}\{x_e:e\in U\}.
+\]
+The vectors $x_e$, $e\in U$, are independent, so $\dim V_U=3k_0$,
+$\dim H_U=k_0$, and there is a unique linear function
+$h_U:H_U\to\F$ with $h_U(x_e)=b_e$.
+
+Before folding, a left question is $(U,L)$, where
+\[
+  L\leq V_U,\qquad \dim L=\ell,\qquad L\cap H_U=\{0\}.
+\]
+For a variable subset $V\subseteq\mathcal Z$, write $V_V=\F^V\le V_0$;
+the rule for sampling $V$ is given below. A right question is $(V,L')$,
+with $L'\leq V_V$ and
+$\dim L'=\ell-1$.  On an incident pair, $L'\leq L$.
+The respective answers are linear functions on $L$ and $L'$, and the
+constraint is restriction.  Choosing bases therefore identifies the
+unfolded alphabets with $\F^\ell$ and $\F^{\ell-1}$.
+
+The folding of~\cite[Section~4.2]{DKKMS25} puts $(U,L)$ and $(U_1,L_1)$
+in the same class when
+\[
+ L+H_U+H_{U_1}=L_1+H_U+H_{U_1}.
+\]
+Lemma~4.2 there proves that this is an equivalence relation and supplies,
+for every class $\mathcal C$,
+a fixed $\ell$-dimensional space $R_{\mathcal C}$ such that
+\begin{equation}
+  R_{\mathcal C}+H_U=L+H_U
+  \quad\text{for every }(U,L)\in\mathcal C.
+  \label{eq:affine-complements}
+\end{equation}
+Both sides have dimension $\ell+k_0$; consequently
+$R_{\mathcal C}\cap H_U=\{0\}$ as well.  A folded answer is a linear
+function $\sigma:R_{\mathcal C}\to\F$.  Its unfolding on $(U,L)$ is
+the restriction to $L$ of the unique extension
+\begin{equation}
+  \widetilde\sigma_U(r+h)=\sigma(r)+h_U(h),
+  \qquad r\in R_{\mathcal C},\quad h\in H_U.
+  \label{eq:affine-unfolding}
+\end{equation}
+
+For clarity, the affineness here is a property of the map between
+\emph{answer spaces}.  For $z\in L$, write uniquely
+$z=A_Uz+B_Uz$ with $A_Uz\in R_{\mathcal C}$ and $B_Uz\in H_U$.
+The map $A_U:L\to R_{\mathcal C}$ is a linear isomorphism by
+\eqref{eq:affine-complements}.  Thus unfolding is
+\[
+  \sigma\longmapsto\sigma\circ A_U+h_U\circ B_U,
+\]
+an affine bijection from $R_{\mathcal C}^*$ to $L^*$.
+Composing it with the surjective restriction $L^*\to(L')^*$ gives a
+surjective affine map with a one-dimensional kernel in its linear
+part.  Every fiber has exactly two elements.  This proves the required
+affine promise after bases are fixed.  Claim~4.4 of~\cite{DKKMS25}
+shows that the projections agree when several unfolded edges are
+aggregated into the same folded edge.  Equivalently, these edges may
+be kept separately with their individual weights.
+
+\paragraph{Validity of the edge sampling.}
+We specify the treatment of ill-formed samples omitted from the displayed
+sampler.  Sampling $k_0$ equations independently produces a
+nonlegitimate tuple with probability
+$O(k_0^2/|\mathcal Z|)$, by bounded occurrence.  For fixed $k_0$ this
+can be made arbitrarily small.  If necessary, take sufficiently many
+disjoint copies of the input instance; this preserves its value and
+regularity.
+
+Given a legitimate $U$, independently for each equation, the right
+variable set $V$ contains all three variables with probability $1-\beta$
+and otherwise one uniformly chosen variable.
+If $f_0$ equations contribute all three, then
+\[
+  \dim V_V=k_0+2f_0,
+  \qquad \dim(V_V\cap H_U)=f_0.
+\]
+For a uniformly chosen $(\ell-1)$-space $L'\leq V_V$, counting
+nonzero vectors gives
+\begin{align*}
+  \Pr[L'\cap H_U\ne\{0\}]
+  &\leq
+  \frac{(2^{f_0}-1)(2^{\ell-1}-1)}{2^{k_0+2f_0}-1}
+  \leq 2^{\ell-k_0}.
+\end{align*}
+If $L'\cap H_U=\{0\}$, a uniformly chosen $\ell$-space $L\leq V_U$
+containing $L'$ is obtained by choosing a nonzero vector of $V_U/L'$.
+It intersects $H_U$ nontrivially precisely when this vector belongs
+to $(H_U+L')/L'$.  Hence
+\[
+  \Pr[L\cap H_U\ne\{0\}\mid L']
+  =\frac{2^{k_0}-1}{2^{3k_0-\ell+1}-1}
+  \leq 2^{\ell-2k_0}.
+\]
+We condition the sampling on legitimacy and $L\cap H_U=\{0\}$.
+The discarded probability is at most
+\begin{equation}
+  \kappa=O(k_0^2/|\mathcal Z|)
+          +2^{\ell-k_0}+2^{\ell-2k_0}.
+  \label{eq:affine-invalid-probability}
+\end{equation}
+The two geometric terms can be made arbitrarily small by increasing
+$k_0$ after $\ell$ is fixed.  We take their sum at most $1/4$ and,
+by using disjoint copies of the input if needed, take the probability of
+nonlegitimacy at most $1/4$.  Thus the probability $q_{\rm all}$ of
+retaining a sample from the original independent-equation sampler is
+at least $1/2$.  We distinguish this total retained probability from
+the normalization after legitimacy has already been imposed.
+
+Let $P$ be that sampler after conditioning on legitimate $U$, and put
+$E=\{L\cap H_U=\{0\}\}$ and
+$\eta=2^{\ell-k_0}+2^{\ell-2k_0}$.  Our edge law is $R=P(\,\cdot\mid E)$,
+so $\TV(P,R)=P(E^c)\leq\eta$.  On the source's valid edge set define
+the subprobability weights
+\[
+ w(e)=\sum_{\substack{\omega\in E\\\omega\mapsto e}}P(\omega),
+ \qquad q_{\rm edge}=\sum_e w(e)=P(E)\geq1-\eta.
+\]
+Thus $R(e)=w(e)/q_{\rm edge}$. For any folded labeling, its acceptance
+under $R$ is its raw valid acceptance divided by $q_{\rm edge}$. The raw
+valid acceptance uses the law $P$ above, conditioned on legitimate $U$
+but not on $E$, with payoff zero on invalid outcomes. We next bound this
+quantity using completed tables in the published soundness analysis.
+
+\subsection{Completed tables and the equations encoded by their lists}
+
+The soundness proof uses tables on every $\ell$-dimensional subspace,
+although a folded labeling specifies answers only at valid vertices.
+We first extend those answers, preserving the consistency between the
+tables on $V_U$ and $V_V$.  We then show that the linear functions
+decoded from these tables still satisfy the equations in $U$.
+
+\paragraph{Completing the tables.}
+For a right variable set $V$ occurring with a legitimate $U$, define
+\[
+  H(V)=\operatorname{span}\{x_e:e\text{ is an input equation and }
+                                     \operatorname{var}(e)\subseteq V\}.
+\]
+Every input equation supported in $\operatorname{var}(U)$ belongs to
+$U$: an equation using two tuple blocks violates legitimacy, and a
+distinct equation inside one block would share three variables with
+that block's equation.  Since the equations of $U$ have disjoint
+supports, this proves
+\begin{equation}
+  H_U\cap V_V=H(V).
+  \label{eq:affine-validity-common-side-space}
+\end{equation}
+In particular, for $L\subseteq V_V$, validity of $(U,L)$ is independent
+of the containing legitimate tuple $U$.
+
+Fix a folded labeling.  Let $F_U[L]$ be its unfolded answer at each
+valid $(U,L)$, and the zero linear function whenever
+$L\cap H_U\ne\{0\}$.  For each $V$, define $F_V[L]$ using any
+containing $U$ when $L\cap H(V)=\{0\}$, and by zero otherwise.
+For valid $L$, all vertices $(U,L)$ with that same $L$ lie in one
+folding class; their unfolded answers agree by
+\cite[Claim~4.4]{DKKMS25}.  Equation~\eqref{eq:affine-validity-common-side-space}
+gives the same independence for invalid $L$.  Hence the completed tables
+satisfy
+\begin{equation}
+  F_U[L]=F_V[L]\qquad(L\subseteq V_V,\ \dim L=\ell).
+  \label{eq:affine-validity-completed-consistency}
+\end{equation}
+The auxiliary table $F_V$ is on $\ell$-spaces; it is distinct from
+the game's right-answer table on $(\ell-1)$-spaces.
+
+The completed $F_U$ has a second property.  Fix
+$K\supseteq H_U$ of dimension $\ell+k_0$.  All valid $L$ with
+$L+H_U=K$ belong to one folding class.  The common folded label and
+its $(H_U,h_U)$-extension give one linear function $f_K:K\to\F$
+with $f_K|_{H_U}=h_U$ and $F_U[L]=f_K|_L$ for every such $L$.
+This condition imposes no restriction on invalid entries.
+
+\paragraph{Agreement and list parameters.}
+The Grassmann agreement test for a table $F$ first chooses a uniform
+$(\ell-1)$-space, then two independent uniform $\ell$-spaces containing
+it, and accepts when their table entries agree on the common
+$(\ell-1)$-space.  Write $\operatorname{agreement}(F)$ for its
+acceptance probability.
+
+Fix a soundness parameter $\delta\in(0,1)$.  The agreement lemma
+\cite[Lemma~3.7]{DKKMS25} supplies constants
+$q,r,C$ and a positive function $\alpha(\ell)$ at agreement threshold
+$\delta^2/8$, as in the proof of its Lemma~5.3.  Set the thresholds
+\[
+  \tau_r=\min\{C/2,1/2\},\qquad
+  \tau_i=10^{-9}\tau_{i+1}^{12}\quad(0\le i<r).
+\]
+Thus $0<\tau_i\le1/2$, and the pairs supplied by the agreement lemma
+have agreement strictly above the relevant thresholds.
+We also fix the second list thresholds
+\[
+ \zeta_r=\tau_0/2,\qquad
+ \zeta_i=10^{-9}\zeta_{i+1}^{12}\quad(0\le i<r).
+\]
+Both vectors are fixed before choosing $\ell$;
+the function $\alpha(\ell)$ is positive once $\ell$ is fixed.
+We choose $\ell$ large enough for that agreement lemma and for the
+ordinary list bound~\cite[Lemma~3.12]{DKKMS25} on
+$(\ell-q)$-spaces with thresholds $(\zeta_i)$.  We also require
+$q\le\ell-1$ and
+\begin{equation}
+  2^{q-\ell}\le\tau_0/4.
+  \label{eq:affine-validity-ell-margin}
+\end{equation}
+The repetition parameter $k_0$ is chosen subsequently, large enough that
+the cited ambient-dimension requirements hold for every dimension
+between $k_0$ and $3k_0$ and for their quotients by a $q$-space.
+
+For a fixed $q$-space $Q$, a list records linear functions on spaces
+of codimension at most $r$ whose restrictions agree with the table on
+more than the designated fraction of $\ell$-spaces containing $Q$.
+Only pairs with no occurring extension are kept.  The next lemma
+shows that, for the completed tables above, these maximal pairs
+respect the original equations whenever $Q$ avoids their span.
+
+\begin{lemma}[The invariant-list assertion on admissible advice spaces]
+\label{lem:affine-validity-invariance}
+Let $X$ have dimension $3k_0$, let $H\le X$ have dimension $k_0$,
+and let $h:H\to\F$ be linear.  Suppose a table $F$ assigns a linear
+function to every $\ell$-space, and that for every
+$(\ell+k_0)$-space $K\supseteq H$ there is a linear function
+$f_K:K\to\F$ such that $f_K|_H=h$ and
+$F[L]=f_K|_L$ whenever $L+H=K$.
+Fix a $q$-space $Q$ with $Q\cap H=\{0\}$, and set
+\[
+  e_{k_0}=2^{\ell+r-2k_0+1}.
+\]
+Assume $q\le\ell$, $3k_0-r>\ell$, \eqref{eq:affine-validity-ell-margin},
+and $e_{k_0}\le\tau_0/4$.
+For $Q\subseteq W\subseteq X$ of codimension $i\le r$, say that
+$(g,W)$ occurs if $g:W\to\F$ is linear and
+\[
+  \Pr_{Q\subseteq L\subseteq W,\ \dim L=\ell}
+       [F[L]=g|_L]>\tau_i.
+\]
+Call it maximal if it has no occurring extension to a strictly larger
+space.  Every maximal occurring pair satisfies $H\subseteq W$ and
+$g|_H=h$.
+\end{lemma}
+
+\begin{proof}
+Put $H'=H\cap W$.  In $W/Q$, the image $(H'+Q)/Q$ has dimension
+$\dim H'$ because $Q\cap H=\{0\}$.  Counting nonzero vectors in a
+uniform $(\ell-q)$-space gives
+\begin{align*}
+ d_W&:=\Pr_{Q\subseteq L\subseteq W}[L\cap H\ne\{0\}]\\
+ &\le
+ \frac{(2^{\dim H'}-1)(2^{\ell-q}-1)}{2^{\dim W-q}-1}
+ \le 2^{\ell+\dim H'-\dim W+1}
+ \le e_{k_0}.
+\end{align*}
+The same bound holds with $W$ replaced by any containing space of
+codimension at most $r$.
+
+Partition the valid spaces $L$ with $Q\subseteq L\subseteq W$ by
+$Y=L+H'$.  Within a fixed class, the table is the restriction of one
+linear function on $Y$ respecting $h|_{H'}$; this follows by restricting
+$f_{L+H}$.  If $g|_{H'}\ne h|_{H'}$, the difference of these two
+functions is nonzero on $H'$.  A complement to $H'$ in $Y$ containing
+$Q$ is a graph of a linear map on an $(\ell-q)$-dimensional quotient.
+Each of its $\ell-q$ independent lift choices must satisfy one
+nontrivial linear equation for the two functions to agree.
+Thus agreement within this class is at most $2^{q-\ell}$ (and is zero
+if the functions already disagree on $Q$).  Total agreement is at most
+\[
+  d_W+(1-d_W)2^{q-\ell}
+  \le e_{k_0}+2^{q-\ell}\le\tau_0/2<\tau_i,
+\]
+a contradiction.  Hence $g|_{H'}=h|_{H'}$.
+
+There is consequently a unique extension $\widetilde g$ to
+$\widetilde W=W+H$ respecting $h$.  Under either of the two uniform
+distributions on valid $\ell$-spaces containing $Q$ in $W$ and in
+$\widetilde W$, the image of $L$ in
+\[
+  W/(W\cap H)\simeq\widetilde W/H
+\]
+is uniform among the $\ell$-spaces containing the image of $Q$.
+Indeed, every such image has the same number of complements containing
+$Q$, namely $2^{(\ell-q)\dim(H\cap W)}$ in $W$ and
+$2^{(\ell-q)\dim H}$ in $\widetilde W$.
+Moreover, agreement depends only on this image: $f_{L+H}$ and
+$\widetilde g$ agree on $H$, so their difference vanishes on one
+complement precisely when it vanishes on every complement.
+The conditional agreement on valid spaces is therefore identical in
+the two ambient spaces.  Since agreement in $W$ is at least $\tau_i$,
+agreement in $\widetilde W$ is at least
+\[
+  (1-d_{\widetilde W})\frac{\tau_i-d_W}{1-d_W}
+  \ge(1-e_{k_0})(\tau_i-e_{k_0})
+  \ge\tau_i-2e_{k_0}\ge\tau_i/2.
+\]
+If $H\not\subseteq W$, the codimension $j$ of $\widetilde W$ is
+strictly smaller than $i$, and $\tau_j\le\tau_{i-1}<\tau_i/2$.
+This gives an occurring extension, contradicting maximality.
+\end{proof}
+
+Applying this lemma with $X=V_U$, $H=H_U$, and $h=h_U$ shows that
+every maximal pair decoded from $F_U$ respects all equations in $U$,
+provided $Q\cap H_U=\{0\}$.  We next estimate the probability of
+that condition under the distribution used by the source's soundness
+proof.
+
+\subsection{Soundness, completeness, and the parameter order}
+
+\paragraph{The advice distribution.}
+The outer game in~\cite[Section~5.2]{DKKMS25} first samples $(U,V)$,
+then takes $Q$ uniformly from the $q$-spaces of $V_V$.  It sends
+$(U,Q)$ and $(V,Q)$ to its two players and asks for assignments on
+$\operatorname{var}(U)$ and $V$, respectively.  It accepts when those
+assignments agree on $V$ and the first satisfies every equation of $U$.
+If $f_0$ equations contributed all three variables, then
+$\dim V_V=k_0+2f_0$ and $\dim H(V)=f_0$.  Consequently, under this
+actual distribution, uniformly in $U,V$,
+\begin{equation}
+ \Pr[Q\cap H_U\ne\{0\}\mid U,V]
+ \le\frac{(2^{f_0}-1)(2^q-1)}{2^{k_0+2f_0}-1}
+ \le2^{q-k_0}.
+ \label{eq:affine-validity-advice-loss}
+\end{equation}
+We further increase $k_0$ until
+\begin{equation}
+ e_{k_0}\le\tau_0/4,
+ \qquad 2^{q-k_0}\le\delta\alpha(\ell)/8.
+ \label{eq:affine-validity-k-margins}
+\end{equation}
+
+\paragraph{From game acceptance to nonempty lists.}
+Suppose a folded labeling has raw valid acceptance greater than
+$\delta$ under $P$, the sampler conditioned on legitimate $U$ but not
+on validity of $L$.  Testing the completed $F_U[L]$ against the
+original right answer on $(V,L')$ can only increase this acceptance:
+valid entries are unchanged, and invalid outcomes previously had payoff
+zero.  We may therefore
+apply the agreement/list and covering arguments in
+\cite[Lemma~5.3, Claim~5.6, and Equation~(5.1)]{DKKMS25} to the
+completed tables. Before invoking invariance, these arguments use total
+linear tables, their agreement, and
+\eqref{eq:affine-validity-completed-consistency}; all are now defined
+on every sampled subspace.
+
+For fixed $U,Q$, let $\mathcal L_{U,Q}$ be the uniform law on
+$\ell$-spaces of $V_U$ containing $Q$.  Let $\mathcal L'_{U,Q}$
+first sample $V$ conditional on the outer question $(U,Q)$ and then
+a uniform $\ell$-space of $V_V$ containing $Q$.
+These are the conditional laws in~\cite[Lemma~4.11]{DKKMS25}.
+Indeed, if $N_q(d)$ denotes the number of $q$-spaces in $\F^d$, then
+for $Q\subseteq V_V$ a uniform $\ell$-space $L\le V_V$ satisfies
+\[
+ \Pr[Q\subseteq L\mid V]=\frac{N_q(\ell)}{N_q(\dim V_V)}.
+\]
+Thus conditioning the source's $V,L$ sampler on $Q\subseteq L$
+weights each $V$ proportionally to
+$\Pr[V\mid U]/N_q(\dim V_V)$, exactly as in the outer game.
+
+Let $E_1$ be the event
+$\operatorname{agreement}(F_U)>\delta^2/8$, let $E_2$ be
+\[
+ \TV(\mathcal L_{U,Q},\mathcal L'_{U,Q})
+ \le 2^{\ell+5}\sqrt\beta\,k_0^{1/4},
+\]
+and let $E_3$ be nonemptiness of the $Q$-list of $F_U$ at thresholds
+$(\tau_i)$.  Event $E_2$ is precisely the smoothness condition in
+the source's Equation~(4.3).
+For $E_3$, Lemma~3.7 of the source supplies an occurring pair for an
+$\alpha(\ell)$ fraction of uniform $Q$ whenever $E_1$ holds; any such
+pair has a maximal occurring extension.  This uses only existence of
+the first list, not a bound on its size.
+The cited arguments give these events joint probability at least
+$\delta\alpha(\ell)/4$.
+Intersecting with $E_4=\{Q\cap H_U=\{0\}\}$ leaves probability at
+least $\delta\alpha(\ell)/8$, by
+\eqref{eq:affine-validity-advice-loss}--\eqref{eq:affine-validity-k-margins}.
+The event $E_4$ depends only on $U,Q$.  Consequently it does not alter
+the subsequent conditional law of $V$ after $U,Q$ have been fixed.
+
+\paragraph{The second player's quotient lists.}
+For each question $(V,Q)$ and each linear form $\gamma:Q\to\F$,
+fix a linear extension $p_\gamma:V_V\to\F$.
+Let $\nu:V_V\to V_V/Q$ be the quotient map.  Define a total table
+$T_\gamma$ on the $(\ell-q)$-spaces of $V_V/Q$ by
+\[
+ T_\gamma[L/Q]=
+ \begin{cases}
+  \overline{F_V[L]-p_\gamma|_L},&F_V[L]|_Q=\gamma,\\
+  0,&F_V[L]|_Q\ne\gamma.
+ \end{cases}
+\]
+Here $Q\subseteq L\subseteq V_V$, $\dim L=\ell$, and the overline
+denotes the induced linear form on $L/Q$; it is defined in the first
+case because the difference vanishes on $Q$.
+
+For this total table, use the ordinary maximal list with thresholds
+$(\zeta_i)$: a pair $(g,S)$ occurs if $S\le V_V/Q$ has codimension
+$i\le r$ and $T_\gamma$ agrees with $g$ on more than a $\zeta_i$
+fraction of the $(\ell-q)$-spaces in $S$.
+By~\cite[Lemma~3.12]{DKKMS25}, each such list has size at most
+$M_0=M_0(\ell-q,r,\zeta_r)$, independent of $k_0$.
+The second player chooses a uniform $\gamma\in Q^*$, then a uniform
+pair $(g,S)$ from this list.  It lifts $g$ to the linear form
+$g\circ\nu+p_\gamma$ on $\nu^{-1}(S)$, extends it uniformly to
+$V_V$, and outputs the corresponding assignment.  If its chosen list
+is empty, it uses an arbitrary assignment.  The entire procedure uses
+only $(V,Q)$ and the fixed tables.
+
+The zero entries may create additional occurring extensions in an
+auxiliary table.  This causes no difficulty: the proof below needs
+an extension of a specified linear form, rather than an identification
+of these auxiliary maximal pairs with the maximal pairs of $F_V$.
+
+\paragraph{Success of the two strategies.}
+The first player chooses a uniform maximal pair $(f,W)$ from the
+$\tau$-list of $F_U$ and extends $f$ uniformly to $V_U$, using an
+arbitrary assignment if its list is empty.  This finite list need not
+have a size bound independent of $k_0$: the success estimate will hold
+for every one of its elements.  The second player uses the quotient
+lists just constructed.
+
+Fix $U,Q$ in $E_1\cap E_2\cap E_3\cap E_4$, and any first-player
+choice $(f,W)$, with $i=\operatorname{codim}_{V_U}W\le r$.
+By \Cref{lem:affine-validity-invariance}, $H_U\subseteq W$ and
+$f|_{H_U}=h_U$.  Every extension chosen by the first player therefore
+satisfies all equations of $U$.  This is the only use of the
+invariant-list assertion in the source's decoding argument.
+
+The covering step of~\cite[Lemmas~4.11 and~5.5 and the proof of
+Lemma~5.3]{DKKMS25}, using
+\eqref{eq:affine-validity-completed-consistency}, gives probability at
+least $0.3\tau_i\ge0.3\tau_0$ over $V$ that
+\[
+ \operatorname{codim}_{V_V}(W\cap V_V)=i,
+ \qquad
+ \Pr_{Q\subseteq L\subseteq W\cap V_V}
+       [F_V[L]=f|_L]>\tau_i/2.
+\]
+Here $L$ has dimension $\ell$.  Set $S_0=W\cap V_V$ and
+$\gamma=f|_Q$.  Every equality $F_V[L]=f|_L$ in this display is
+retained in $T_\gamma$.  Since
+$\zeta_i\le\tau_0/2\le\tau_i/2$, the quotient pair
+\[
+ \bigl(\overline{f|_{S_0}-p_\gamma|_{S_0}},\ S_0/Q\bigr)
+\]
+occurs for $T_\gamma$ and hence has a maximal occurring extension
+in that auxiliary table.  Lifting any such extension by the same
+$p_\gamma$ gives a function agreeing with $f$ on all of $S_0$.
+This conclusion concerns function extension and is unaffected by
+the extra agreements that zero completion may introduce.
+
+The second player selects this tag and one such maximal extension
+with probability at least $1/M$, where
+$M:=2^qM_0$ is independent of $k_0$.
+Its eventual answer then agrees with $f$ on $W\cap V_V$.
+The first player's uniform extension agrees with this
+answer on all of $V_V$ with probability $2^{-i}\ge2^{-r}$:
+the quotient $V_V/(W\cap V_V)$ has dimension $i$.
+Thus, conditionally on $E_1\cap E_2\cap E_3\cap E_4$, the strategy
+wins with probability at least
+\[
+ c_*:=2^{-r}\frac{0.3\tau_0}{M}>0.
+\]
+The overall success probability is at least
+$\delta\alpha(\ell)c_*/8$, a positive constant independent of $k_0$.
+The upper bound in~\cite[Lemma~5.4]{DKKMS25} tends to zero as
+$\beta k_0\to\infty$.  Choosing $k_0$ sufficiently large gives the
+same contradiction as in the source and proves that raw valid
+acceptance is at most $\delta$.
+
+\paragraph{Fixing soundness before completeness.}
+For the game in \Cref{prop:affine-games}, take $\delta=b/4$ and retain
+the valid raw weights.  Their total mass $q_{\rm edge}$ is at least
+$1/2$ by the preceding rejection estimates.  Normalization therefore
+gives
+\[
+  \Val\le\delta/q_{\rm edge}\le b/2<b.
+\]
+Every restriction above concerns only $b$ and parameters fixed from it.
+After choosing $\ell$, take $k_0=j^4$ and $\beta=j^{-3}$ with $j$
+sufficiently large.  Then $\beta k_0\to\infty$,
+$\beta\sqrt{k_0}\to0$, and $\sqrt\beta\,k_0^{1/4}\to0$.
+These limits satisfy the outer-game soundness and covering requirements,
+including $2^\ell\beta\le1/8$, as well as all the displayed margins.
+In particular, the alphabet dimension $n=\ell-1$ has been fixed using
+only $b$.
+
+\paragraph{Completeness with the alphabet fixed.}
+Let $\alpha:\mathcal Z\to\F$ satisfy all but an $\varepsilon$ fraction of
+the input equations, and also write $\alpha:V_0\to\F$ for its linear
+extension.  Give the folded left question $\mathcal C$ the answer
+$\alpha|_{R_{\mathcal C}}$ and give $(V,L')$ the answer $\alpha|_{L'}$.
+Whenever every equation of sampled $U$ is satisfied,
+$h_U=\alpha|_{H_U}$, so~\eqref{eq:affine-unfolding} is exactly
+$\alpha|_{R_{\mathcal C}+H_U}$.  The edge is therefore satisfied.
+This argument concerns the sampled edge; it does not require every
+member of its folding class to involve satisfied equations.
+
+Before either conditioning step, a union bound gives probability at most
+$k_0\varepsilon$ of sampling an unsatisfied equation.  Since the
+total retained probability $q_{\rm all}$ is at least $1/2$, the
+conditioned game's value is at least
+$1-k_0\varepsilon/q_{\rm all}\ge1-2k_0\varepsilon$.
+After $\ell,k_0$ have been fixed for
+soundness, choose $\varepsilon\leq a/(2k_0)$ in the initial 3Lin
+gap.  This gives the asserted completeness without altering $n$.
+
+\paragraph{Effectivity and size.}
+All spaces in an individual question have bounded dimension.  There
+are polynomially many $k_0$-tuples of input equations, and each tuple
+has only constantly many subspaces, bases, and local sampling
+outcomes.  The folding equivalence relation is explicitly testable
+by linear algebra.  To find $R_{\mathcal C}$ effectively, start from
+any $(U,L)\in\mathcal C$ and enumerate the $\ell$-spaces inside
+$L+H_U$, whose dimension is the fixed constant $\ell+k_0$; test each
+candidate against~\eqref{eq:affine-complements} for every class
+member.  Lemma~4.2 guarantees a successful candidate.  Gaussian
+elimination then computes bases and the affine edge maps.
+The finite sampling distributions, their conditioning, and the
+aggregation of parallel edges have rational weights of polynomial
+bit length.  These observations give the deterministic polynomial
+reduction claimed in the proposition.
+This completes the proof of \Cref{prop:affine-games}.
diff --git a/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/conclusion.tex b/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/conclusion.tex
new file mode 100644
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+++ b/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/conclusion.tex
@@ -0,0 +1,89 @@
+\section{Completing the hardness proof}\label{sec:completion}
+
+The parameter order is
+\[
+ (t,\eps,\mu)\longrightarrow(s,m,p_*,\theta)
+ \longrightarrow b\longrightarrow n\longrightarrow T\longrightarrow a.
+\]
+The extraction constants fix the required game soundness $b$ before the
+alphabet dimension $n$ is known. Repetition is then chosen to hide the
+planted coordinate, and the game's completeness error $a$ is chosen last.
+
+\begin{proof}[Proof of \Cref{thm:gap}]
+Fix $t$ and $\eps$ as in the theorem. Take $\delta=\eps$. This satisfies $0<\delta<1-B(t)$ because
+$\eps<(t-B(t))/4<(1-B(t))/4$. Choose a
+rational $0<\mu<1/2$ with $t\mu\le\eps/4$.
+Apply \Cref{prop:extraction} with $t,\delta,\mu$, fixing
+$s,m,p_*,\theta$, and hence $k,d,r$. Choose a rational
+\[
+ 0<b<2^{-2s}\frac{p_*\theta}{2}.
+\]
+By \Cref{prop:affine-games}, this fixes an alphabet dimension $n$ for
+which the game has arbitrarily small completeness error. Choose an
+integer $T$ satisfying
+\[
+ T\ge\frac{2^{n(2k-2s)}}{(p_*\theta)^2}.
+\]
+Finally choose a positive rational $a<1-b$ small enough that
+$2Ta\le\eps/4$, and start with the NP-hard game gap
+$\Val(\mathcal G)\ge1-a$ versus $\Val(\mathcal G)\le b$.
+
+In the YES case, \Cref{lem:completeness} gives cut value at least
+$(1+t)/2-\eps/2$, which is stronger than required.
+In the NO case, suppose a cut had value at least
+$(1+B(t))/2+\eps/2$. By \Cref{lem:cut-stability}, its odd average has
+expected slice stability at least $B(t)+\eps$.
+\Cref{prop:extraction,prop:decoding} would then imply
+$\Val(\mathcal G)\ge2^{-2s}p_*\theta/2>b$, a contradiction.
+Thus every cut has value less than $(1+B(t))/2+\eps/2$.
+
+For completeness, the test is a deterministic polynomial-size graph
+construction. The vertex set has
+\[
+ |X|^T\,2^{|\mathcal B|}
+ =|X|^T\,2^{2^{(n+1)T}}
+\]
+vertices. All factors except $|X|$ are constants for fixed $t,\eps$.
+There are polynomially many endpoint and edge tuples, and all
+word-generation sample spaces have constant size. Enumerate every test
+outcome and add its probability to the corresponding edge weight.
+The initial game weights, $t$, and $r$ are rational, so these weights
+have polynomial bit complexity. Conditional edge probabilities also
+have polynomial bit complexity. All alphabets and words can be enumerated
+in constant time relative to the input size. The resulting graph is
+therefore computable in deterministic polynomial time. It has a rational
+edge distribution and satisfies the asserted, weaker $\eps$-loss bounds.
+\end{proof}
+
+\begin{proof}[Proof of \Cref{thm:main}]
+Substituting $\rho=-t$ into \eqref{eq:gw-constant} gives
+\begin{equation}\label{eq:gw-ratio}
+ \GW=\min_{-1<t\le1}\frac{1+B(t)}{1+t},
+\end{equation}
+where the same formula defines $B(t)=2\arcsin(t)/\pi$ for negative $t$.
+For $-1<t\le0$ the ratio is at least one, whereas for $0<t<1$ it is
+less than one. At $t=1$ it equals one, and as $t\downarrow-1$ it
+diverges. Thus its minimum occurs in $(0,1)$.
+
+Fix $\GW<\alpha\le1$. By continuity and the density of the rationals,
+choose a rational $t\in(0,1)$ such that
+\[
+ \alpha\frac{1+t}{2}>\frac{1+B(t)}{2}.
+\]
+Choose positive rational $\eps$ and $\epsilon'$ small enough that
+$\eps<(t-B(t))/4$ and
+\begin{equation}\label{eq:strict-final-gap}
+ \alpha\left(\frac{1+t}{2}-\eps-\epsilon'\right)
+ >\frac{1+B(t)}{2}+\eps+\epsilon'.
+\end{equation}
+Apply \Cref{thm:gap}. Delete loops while retaining the original weight
+scale, so that the remaining weights sum to at most one. Apply
+\Cref{lem:unweighted} with error $\epsilon'$, producing a simple
+unweighted graph whose maximum cut, on the common scale $Q$, differs
+from the original maximum cut weight by at most $\epsilon'$.
+An $\alpha$-approximation algorithm on that graph returns, in the YES
+case, a cut of scaled value at least the left side of
+\eqref{eq:strict-final-gap}. In the NO case, no cut has scaled value
+greater than its right side. A fixed rational threshold strictly between them therefore
+distinguishes the NP-hard promise cases. This proves the theorem.
+\end{proof}
diff --git a/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/decoding.tex b/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/decoding.tex
new file mode 100644
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+++ b/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/decoding.tex
@@ -0,0 +1,374 @@
+\section{Decoding with a hidden coordinate}
+\label{sec:decoding}
+
+We now convert the Fourier mass supplied by Proposition~\ref{prop:extraction}
+into a labeling of the original affine game.  The difficulty is that the
+mass is measured after fixing a row context.  A labeler at an $X$-vertex
+does not know the projection on its incident edge, and therefore cannot
+usually lift that context to its own label space.  We resolve this by
+repeating the game coordinates and planting one coordinate on which the
+context and the latent resampling rows have zero linear coefficients.
+Hiding the planted coordinate makes the resulting
+distribution close to the original one.
+The comparison across the unknown projection then involves only the $2s$
+free shifts, so its loss is independent of $n$ and $T$.
+
+Throughout this section, the cut tables are fixed.  We use the tree
+parameters and the row coordinates of the preceding sections.  For each
+choice of a slice $\mathcal D$ and a node $v\in\mathcal D$, fix the
+coordinate change
+\[
+ R\longleftrightarrow (L,\Gamma),\qquad
+ L=(L_{gb})_{g\in[s],\,b\in\{0,1\}}\in H^{2s},
+ \quad \Gamma\in H^{k-2s},
+\]
+by choosing its representative leaves in a deterministic order.  This
+change uses only additions of row coordinates and depends only on the
+tree and $v$.  In particular, it commutes with pullback of affine forms.
+The masses $m_q$ always refer to these coordinates and to the uniform
+average over the slice tables in Equation~\eqref{eq:mass}.
+
+\subsection{Removing the row smoothing}
+
+Let $R^{\mathrm{new}}_1,\ldots,R^{\mathrm{new}}_k$ be independent uniform
+elements of $H$, and independently let $B_1,\ldots,B_k$ be Bernoulli
+variables with $\Pr(B_i=1)=1-r$.  Set
+\[
+ E_i=B_iR^{\mathrm{new}}_i,
+ \qquad E=(E_1,\ldots,E_k).
+\]
+For any fixed row array $R$, the array $R+E$ has the distribution of
+the one-copy row resampling used by the test.  Write $(E_L,E_\Gamma)$
+for the coordinates of $E$ under the above linear change of variables.
+
+\begin{lemma}[Removing smoothing]\label{lem:unsmooth}
+For every fixed $\underline y$, $\mathcal D$, $v$, $\Gamma$,
+$F_{\mathrm{out}}$, and label $q\in\mathcal A$,
+\begin{equation}\label{eq:unsmooth}
+ m_q(\bar G_{\underline y};\Gamma,F_{\mathrm{out}})
+ \leq
+ \E_{E,\,\underline e\mid\underline y}
+ m_q(G_{\underline e};\Gamma+E_\Gamma,F_{\mathrm{out}}).
+\end{equation}
+Here the noise $E$ is independent of the conditionally sampled edge
+tuple $\underline e$.
+\end{lemma}
+
+\begin{proof}
+For the fixed context, regard a function of $(L,F_{\mathcal D})$ as an
+element of the real Hilbert space with the uniform probability measure.
+Let $\Pi_q$ be the orthogonal projection onto the shift Fourier indices
+$Q$ satisfying $Q_{g0}+Q_{g1}=q$ for every $g$.  Then the corresponding
+mass is the squared norm of this projection.  The identity
+\[
+ \bar G_{\underline y}(L,\Gamma,F)
+ =\E_{E,\,\underline e\mid\underline y}
+ G_{\underline e}(L+E_L,\Gamma+E_\Gamma,F)
+\]
+and convexity of the squared Hilbert-space norm give the desired
+inequality.  Indeed, translation by $E_L$ commutes with $\Pi_q$ and
+preserves its norm: on each shift Fourier coefficient it only
+multiplies by a sign.
+\end{proof}
+
+Choose once and for all an ordering of each label space, and define
+\begin{equation}\label{eq:decoder-choice}
+ q^*=q^*(\underline y,\mathcal D,v,\Gamma,F_{\mathrm{out}})
+ \in\operatorname*{arg\,max}_{q\in\mathcal A}
+ m_q(\bar G_{\underline y};\Gamma,F_{\mathrm{out}})
+\end{equation}
+using that ordering to break ties.  This definition uses the original
+conditional edge average and the original row-smoothing operator.
+It will remain unchanged when we modify the distribution of the row
+data below.
+
+Sample $\underline y$, the slice, its uniform node, the context, and
+the outside tables as in Proposition~\ref{prop:extraction}, and then
+sample $R^{\mathrm{new}}$, $B$, and $\underline e\mid\underline y$.
+Combining that Proposition with Lemma~\ref{lem:unsmooth} gives
+\begin{equation}\label{eq:unplanted-score}
+ \E\,m_{q^*}(G_{\underline e};
+              \Gamma+E_\Gamma,F_{\mathrm{out}})
+ \geq p_*\theta.
+\end{equation}
+The mass in this expectation lies in $[0,1]$, since $G_{\underline e}$
+is bounded in absolute value by one.
+
+\subsection{Planting a coordinate}
+
+Write each affine row in coordinates on
+$\mathcal A=(\F^n)^T$.  Apart from its single constant coefficient, it
+has one block of $n$ linear coefficients for each of the $T$ slots.
+For a slot $a\in[T]$, let $Z_a$ be the event that the slot-$a$ blocks
+vanish in all $k-2s$ context rows and all $k$ latent rows
+$R^{\mathrm{new}}_i$.  No condition is imposed on any constant
+coefficient.  Under the original, unplanted law,
+\begin{equation}\label{eq:zero-slot-probability}
+ p_0:=\Pr(Z_a)=2^{-n(2k-2s)}.
+\end{equation}
+Conditional on the depth, slice, and node, the events $Z_a$ are
+independent: they concern disjoint blocks of independent uniform
+coefficients.
+
+The \emph{planted experiment} chooses a uniform mark $a_*\in[T]$ and
+sets these blocks to zero at $a_*$.  All other row coefficients, the
+noise masks, the outside tables, and the game edges retain their
+original distributions.  Equivalently, conditional on $a_*=a$, this
+is the unplanted law conditioned on $Z_a$.
+
+\begin{lemma}[Hiding the planted coordinate]\label{lem:planting}
+Let $\mathsf U$ be the unplanted law and $\mathsf P$ the planted law
+after forgetting the mark.  Both laws are taken on the full data
+\[
+ \bigl(\underline e,J_0,\mathcal D,v,F_{\mathrm{out}},
+        \Gamma,R^{\mathrm{new}},B\bigr).
+\]
+Then
+\begin{equation}\label{eq:planting-tv}
+ \TV(\mathsf P,\mathsf U)
+ \leq \frac12\sqrt{\frac{1-p_0}{Tp_0}}
+ \leq \frac{1}{2\sqrt{Tp_0}}.
+\end{equation}
+Consequently, if
+\begin{equation}\label{eq:decoder-repetition}
+ T\geq\frac{1}{p_0(p_*\theta)^2},
+\end{equation}
+then under the planted law,
+\begin{equation}\label{eq:planted-score}
+ \E_{\mathsf P}\,m_{q^*}(G_{\underline e};
+               \Gamma+E_\Gamma,F_{\mathrm{out}})
+ \geq \frac{p_*\theta}{2}.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Let $Z=\sum_{a=1}^T\one_{Z_a}$.  Averaging the conditional densities
+over the uniform mark gives
+\[
+ \frac{d\mathsf P}{d\mathsf U}=\frac{Z}{Tp_0}.
+\]
+Conditional on the tree data, $Z$ has the binomial distribution with
+parameters $T,p_0$; these parameters do not depend on the tree data.
+Thus Cauchy--Schwarz gives
+\[
+ \TV(\mathsf P,\mathsf U)
+ =\frac12\E_{\mathsf U}\abs{\frac{Z}{Tp_0}-1}
+ \leq\frac12\sqrt{\frac{1-p_0}{Tp_0}}.
+\]
+The score in Equation~\eqref{eq:unplanted-score} is a $[0,1]$-valued
+function of the displayed data without the mark.  In particular,
+$q^*$ is the fixed function defined in Equation~\eqref{eq:decoder-choice};
+it does not use the mark, the sampled incident edges beyond their
+$Y$-endpoints, or the sampled noise.  Its statistical dependence on
+zero slots causes no difficulty.  Expectations of this score under
+the two laws differ by at most their total variation distance.
+Equations~\eqref{eq:unplanted-score} and
+\eqref{eq:decoder-repetition} therefore imply
+Equation~\eqref{eq:planted-score}.
+\end{proof}
+
+\subsection{Fourier mass under an affine restriction}
+
+The next elementary observation accounts for the loss in decoding.
+Let $\rho:\widetilde{\mathcal A}\to\mathcal A$ be a surjective affine
+map with two-element fibers, and write
+\[
+ \widetilde H=\Aff(\widetilde{\mathcal A},\F),\qquad
+ \iota=\rho^*:H\hookrightarrow\widetilde H.
+\]
+Pullback is linear on the vector spaces of affine forms, even when
+$\rho$ has a nonzero translation part.  The image of $\iota$ has
+codimension one, and $\iota(\one)=\one$.
+Its dual restriction map
+$\iota^*:\widetilde H^*\to H^*$ has two-element fibers and, on
+evaluation functionals, agrees with $\rho$.
+
+\begin{lemma}[Restriction of shift mass]\label{lem:restriction}
+Fix a context and the outside tables.  Let
+$\widetilde f(\widetilde L,F_{\mathcal D})$ be a function on
+$\widetilde H^{2s}$ and the slice tables.  Suppose its shift Fourier
+support has the common-pair-sum property
+\[
+ \widetilde Q_{g0}+\widetilde Q_{g1}
+ =\widetilde Q_{h0}+\widetilde Q_{h1}
+ \quad\text{for all }g,h\in[s].
+\]
+Set $f(L,F_{\mathcal D})=
+\widetilde f(\iota L,F_{\mathcal D})$, where $\iota$ acts on each
+shift.  For every label $q\in\mathcal A$,
+\begin{equation}\label{eq:restriction-mass}
+ m_q(f)\leq 2^{2s}
+       \sum_{\substack{\widetilde q\in\widetilde{\mathcal A}\\
+                       \rho(\widetilde q)=q}}
+          \widetilde m_{\widetilde q}(\widetilde f).
+\end{equation}
+The masses on both sides use the common-pair-sum definition of
+Equation~\eqref{eq:mass}, with the respective affine row spaces.
+\end{lemma}
+
+\begin{proof}
+For every fixed choice of slice tables, Fourier inversion and
+character orthogonality give
+\[
+ \widehat f(Q)=
+ \sum_{\substack{\widetilde Q\\
+                  \iota^*\widetilde Q_{gb}=Q_{gb}\ \forall g,b}}
+       \widehat{\widetilde f}(\widetilde Q).
+\]
+The sum has $2^{2s}$ indices.  Hence its squared absolute value is at
+most $2^{2s}$ times the sum of the squared absolute values of its
+terms.  Now sum over all $Q$ with pair sums $q$.  Every nonzero
+coefficient on the right has a common pair sum $\widetilde q$ whose
+restriction is $q$.  Moreover,
+\[
+ \widetilde q(\one)=\widetilde q(\iota(\one))=q(\one)=1,
+\]
+so $\widetilde q$ is an evaluation functional, hence a label in
+$\widetilde{\mathcal A}$, and $\rho(\widetilde q)=q$.
+Each lifted index is counted only once.  Averaging over the slice
+tables proves Equation~\eqref{eq:restriction-mass}.
+\end{proof}
+
+\subsection{The endpoint labelers}
+
+Equation~\eqref{eq:planted-score} is an average over full edge tuples.
+To turn it into a labeling of the original game, we place the input edge
+at the planted coordinate and make all other edges public.  The zero
+coefficients ensure that the needed context lift does not require the
+projection on the input edge.
+
+\begin{proposition}[Decoding]\label{prop:decoding}
+Suppose the extraction conclusion of Proposition~\ref{prop:extraction}
+holds with constants $p_*,\theta>0$.  If $T$ satisfies
+Equation~\eqref{eq:decoder-repetition}, then the original affine
+game has value at least
+\begin{equation}\label{eq:decoded-value}
+ \Val(\mathcal G)\geq 2^{-2s}\frac{p_*\theta}{2}.
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+We construct randomized labelers that use only their own endpoint
+and shared randomness.  The entire game and the fixed cut tables
+are available as background data.  No efficiency claim about the
+labelers is needed to lower-bound game value.
+
+\paragraph{Shared data and the $Y$-labeler.}
+Given a random input edge $e=(x,y)$ from the original edge law,
+choose a uniform public mark $a_*\in[T]$.  At every other slot,
+publicly sample an independent edge, including its endpoints and
+projection.  Insert the input edge at the marked slot.  Its
+projection is used only in the analysis; it is not revealed to
+either labeler.  Conditional on any mark, the resulting full edge
+tuple has the product edge law; hence the tuple is independent of
+the mark.  Each labeler knows its own endpoint tuple, and both know
+all edges outside the marked slot.
+
+Publicly sample the depth, slice, node, outside tables, context,
+latent rows, and masks according to the planted experiment above.
+The row sampling uses the fixed alphabet dimensions, the public
+tree coordinates, and the mark, and is independent of the input edge.
+Together with the edge tuple, these data have exactly the planted
+law used in Equation~\eqref{eq:planted-score}.  The $Y$-labeler
+forms its endpoint tuple $\underline y$, computes the label $q^*$
+from Equation~\eqref{eq:decoder-choice}, and outputs its marked
+component $q^*_{a_*}$.  All quantities used in that computation are
+available to this labeler: in particular the conditional average
+defining $\bar G_{\underline y}$ can be computed from the game and
+cut as fixed background data.  It does not use the actual marked
+edge or its projection.
+
+\paragraph{The $X$-labeler.}
+To define the $X$-labeler, replace the marked factor of $\mathcal A$
+by the $X$-alphabet:
+\[
+ \widetilde{\mathcal A}
+ =\prod_{a\ne a_*}\F^n\ \times\ \F^{n+1},
+\]
+with factors in their original slot order.  Let
+$\mathcal B=(\F^{n+1})^T$.  The full projection factors as
+\[
+ \mathcal B\xrightarrow{\ \kappa\ }
+ \widetilde{\mathcal A}\xrightarrow{\ \rho\ }\mathcal A.
+\]
+Here $\kappa$ is the public edge projection at every unmarked slot
+and the identity at the marked slot; $\rho$ is the identity at
+every unmarked slot and the unknown map $\pi_e$ at the marked
+slot.  Thus $\kappa$ is known to the $X$-labeler, while $\rho$ is
+used only in the analysis.
+
+The planted zeros imply that every row in $\Gamma$ and
+$R^{\mathrm{new}}$ has zero linear coefficient block in the marked slot.
+The same holds for $E$, $E_\Gamma$, and
+$\Gamma'=\Gamma+E_\Gamma$, since the noise masks and context change
+only multiply by scalars and add rows.  Consequently
+$\widetilde\Gamma'=\rho^*\Gamma'$ is known to the $X$-labeler:
+it is obtained by leaving every unmarked coefficient and constant
+coefficient unchanged and setting all marked coefficients to zero.
+This description does not require $\pi_e$, including its affine
+translation part.
+
+The $X$-labeler forms the bounded raw function
+\[
+ \widetilde G(\widetilde R,F)
+ =\sigma^o_{\underline x}
+   \bigl(W_{\widetilde{\mathcal A}}(\widetilde R,F)
+                \circ\kappa\bigr),
+ \qquad \widetilde R\in\widetilde H^k.
+\]
+Fix its context to $\widetilde\Gamma'$ and its outside tables to
+the public $F_{\mathrm{out}}$, leaving all $2s$ shifts free in
+$\widetilde H$.  Let
+$\widetilde m_{\widetilde q}$ be its masses.  They are nonnegative
+and their sum over labels is at most one, by Parseval and
+$\abs{\widetilde G}\leq1$.  The labeler samples a label
+$\widetilde q$ with probability at least
+$\widetilde m_{\widetilde q}$ for each label, assigning any leftover
+probability to a fixed label.  It uses an additional independent
+public uniform random number to sample from this distribution and
+outputs the marked component of $\widetilde q$.  This sampling does
+not use $q^*$.
+
+\paragraph{Comparison on the sampled edge.}
+We verify the restriction hypothesis needed to compare these
+probabilities with the score.  For any $h\in\widetilde H$, shifting
+both child generators in one pair by $h$ multiplies the output
+of gate $v$ by $(-1)^{h(\widetilde l)}$ at every label
+$\widetilde l$.  Applying this shift in two distinct pairs leaves
+that output, and hence the whole word, unchanged.  These operations
+only translate the free shifts and preserve the context.
+Character orthogonality therefore shows that every shift Fourier
+index of $\widetilde G$ has a common pair sum.  This argument holds
+for each fixed choice of all gate tables.
+
+Word generation commutes with pullback of rows, pointwise in the
+label.  Since $\pi_{\underline e}=\rho\circ\kappa$, restricting
+each free shift of $\widetilde G$ to $\rho^*H$ yields exactly
+$G_{\underline e}$ with context $\Gamma'$ and the same tables.
+The map $\rho$ has two-element fibers, so Lemma~\ref{lem:restriction}
+applies and gives
+\[
+ \sum_{\rho(\widetilde q)=q^*}\widetilde m_{\widetilde q}
+ \geq 2^{-2s}
+ m_{q^*}(G_{\underline e};\Gamma',F_{\mathrm{out}}).
+\]
+Conditional on all the planted data, the left side lower-bounds
+the probability that the sampled full label projects to $q^*$.
+That event in particular implies satisfaction of the marked
+input edge.  Averaging and using Equation~\eqref{eq:planted-score}
+gives success probability at least $2^{-2s}p_*\theta/2$.
+
+Finally, fix all shared randomness, including the uniform number
+used to sample the $X$-label.  The same fixed data define the two
+strategies at every endpoint, so they give a deterministic labeling
+of the original game.  Their average value over the shared
+randomness has the lower bound just proved, so some fixed choice
+attains it.  This proves Equation~\eqref{eq:decoded-value}.
+\end{proof}
+
+The decoding loss depends on $s,p_*,\theta$, and hence on the target
+cut gap, but not on the alphabet dimension or the repetition
+length.  After choosing a game soundness smaller than the right
+side of Equation~\eqref{eq:decoded-value}, one may therefore fix
+the alphabet dimension and then choose $T$ by
+Equation~\eqref{eq:decoder-repetition}.
diff --git a/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/extraction.tex b/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/extraction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..27bf71cd2de73cc01667dc24a80022f171b89b7f
--- /dev/null
+++ b/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/extraction.tex
@@ -0,0 +1,298 @@
+\section{From excess stability to a label}\label{sec:extraction}
+
+This section contains the analytic statement used in the reduction. Its
+constants do not depend on the dimension of the affine label space.
+The functions may depend on an additional random parameter $\omega$, whose
+law is independent of all rows, gate tables, and the random slice. In the application,
+$\omega$ will be a tuple of right endpoints of the game.
+
+Let $\phi_\omega:\bits^{\mathcal A}\to[-1,1]$ be odd, meaning
+$\phi_\omega(-w)=-\phi_\omega(w)$, and set
+\[
+ f_\omega(R,F)=\phi_\omega(W(R,F)),\qquad
+ \bar f_\omega=T_r f_\omega,\qquad r=1-\mu/d.
+\]
+The word, tree, and random slice $\mathcal D$ are those of
+\Cref{sec:tree}. Write $F_{\mathrm{out}}$ for the tables outside
+$\mathcal D$. Once $\omega,\mathcal D,R,F_{\mathrm{out}}$ are fixed,
+$\bar f_\omega$ is a function of the independent slice-table bits.
+
+For a node $v\in\mathcal D$, use the free shift coordinates
+$(L_{gb})_{g\in[s],b\in\{0,1\}}$ and the remaining context
+$\Gamma\in H^{k-2s}$ from \Cref{lem:shift-coordinates}. A hat marked
+``shifts'' expands only these $2s$ free rows. For a label $q\in\mathcal A$
+(equivalently, $q\in H^*$ with $q(\one)=1$),
+define
+\begin{equation}\label{eq:mass}
+ m_q(f;\Gamma,F_{\mathrm{out}})
+ =\E_{F_{\mathcal D}}
+ \sum_{\substack{Q\in(H^*)^{2s}\\
+                  Q_{g0}+Q_{g1}=q\ (g\in[s])}}
+ \abs{\widehat f^{\,\mathrm{shifts}}(Q)}^2.
+\end{equation}
+The dependence on $v,\mathcal D$, and $\omega$ is suppressed. The sets of
+indices in this definition are disjoint as $q$ varies. For a fixed $q$,
+the mass includes every way to distribute that common sum between the
+two children of each pair. We seek such mass for the original smoothed
+function $\bar f_\omega$. The unsplit projection and a table-degree
+cutoff will serve only to find a label carrying this mass; neither
+projection is part of the conclusion. Keeping all the indices with a
+given common sum will let us compare masses under affine pullback in
+\Cref{sec:decoding}.
+
+In particular, if
+$|f|\le1$, then at every fixed context
+\begin{equation}\label{eq:mass-bound}
+ 0\le\sum_{q\in\mathcal A}m_q(f;\Gamma,F_{\mathrm{out}})\le1.
+\end{equation}
+
+\begin{proposition}[Dimension-independent extraction]\label{prop:extraction}
+Fix $t\in(0,1)$, $0<\delta<1-B(t)$, and $0<\mu<1/2$.
+There are tree parameters $s\ge2,m\ge1$ and constants $p_*,\theta>0$,
+depending only on $t,\delta,\mu$, with the following property.
+For any affine label space $\mathcal A$ and any family $\phi_\omega$ as above,
+if
+\begin{equation}\label{eq:excess-stability}
+ \E_{\omega,\mathcal D,R,F_{\mathrm{out}}}
+ \Stab_t^{F_{\mathcal D}}(\bar f_\omega)
+ \ge B(t)+\delta,
+\end{equation}
+then
+\begin{equation}\label{eq:extraction-event}
+ \Pr\left[
+   \max_{q\in\mathcal A}
+   m_q(\bar f_\omega;\Gamma,F_{\mathrm{out}})\ge\theta
+ \right]\ge p_*.
+\end{equation}
+Here we sample $\omega$, the random slice $\mathcal D$, and a uniform
+node $v\in\mathcal D$; conditional on these, we sample
+$F_{\mathrm{out}}$ and $\Gamma$ independently and uniformly.
+\end{proposition}
+
+\begin{proof}
+Excess stability first produces an influential slice entry. We retain
+that influence after the unsplit projection, discard the part even in
+its gate table, and express the remaining influence as an affine hash
+of vectors indexed by labels. The hashing lemma then forces one label
+to carry substantial conditional mass.
+
+All constants chosen below are independent of $\mathcal A$ and of the
+law of $\omega$. The degree cutoff $K$ and influence threshold $\tau$
+are chosen first. We leave $s,m$ unspecified during the estimates:
+Step~4 will choose $s$ to control the even and hashed contributions, and
+then $m$ to control the cost of the unsplit projection.
+
+\paragraph{Step 1: find an influential slice entry.}
+Simultaneously negating the slice tables negates the word. Since
+$\phi_\omega$ is odd and smoothing preserves this identity,
+$\bar f_\omega$ has mean zero in the slice bits for every fixed
+$\omega,\mathcal D,R,F_{\mathrm{out}}$. It also lies in $[-1,1]$.
+Apply \Cref{lem:mis} with $\xi=\delta/2$, obtaining $K\ge1$ and $\tau>0$.
+We may decrease $\tau$ to ensure $\tau\le1$.
+Since stability is at most one, \eqref{eq:excess-stability} implies that,
+with probability at least
+\[
+ \eta=\delta/2,
+\]
+some entry $z\in\F^s$ of some $v\in\mathcal D$ has
+$\Inf_{v,z}^{\le K}(\bar f_\omega)\ge\tau$.
+Indeed, if the probability of this event is $p$, the expected stability is
+at most $B(t)+\delta/2+p$.
+
+Let $S_v$ be the orthogonal row projection onto indices unsplit at $v$.
+Put
+\[
+ H_v=P_{\le K}^{F_{\mathcal D}}S_v\bar f_\omega.
+\]
+By \Cref{lem:unsplit}, the required bound on the projection loss is
+\begin{equation}\label{eq:choose-m}
+ \E\sum_{v\in\mathcal D}
+ \norm{(1-S_v)\bar f_\omega}_{2,R,F}^2
+ \le\frac{s}{2em\mu}\le\frac{\eta\tau}{8}.
+\end{equation}
+For any fixed $s$, the last inequality can be ensured by increasing $m$.
+The same expectation is the expectation over
+$\omega,\mathcal D,R,F_{\mathrm{out}}$ of the sum of the corresponding
+conditional squared norms in $F_{\mathcal D}$. Markov's inequality shows
+that this sum is at most $\tau/4$ except on an event of probability
+$\eta/2$. On the intersection with the influential-entry event, the
+triangle inequality for the projection defining that influence gives
+\[
+ \sqrt{\Inf_{v,z}(H_v)}
+ \ge\sqrt{\Inf_{v,z}^{\le K}(\bar f_\omega)}
+      -\norm{(1-S_v)\bar f_\omega}_{2,F_{\mathcal D}}
+ \ge\frac{\sqrt\tau}{2}.
+\]
+Consequently,
+\begin{equation}\label{eq:large-max-sum}
+ \E\sum_{v\in\mathcal D}\max_z\Inf_{v,z}(H_v)
+ \ge\frac{\eta\tau}{8}=:\Lambda.
+\end{equation}
+
+We will also use
+\begin{equation}\label{eq:total-influence}
+ \E\sum_{v\in\mathcal D}\sum_z\Inf_{v,z}(H_v)\le K.
+\end{equation}
+To verify it, integrate over all rows. Each $S_v$ is an orthogonal
+projection in the rows and commutes with the slice-bit Fourier projections.
+It can therefore only decrease the integrated influence at its entries.
+After dropping each $S_v$ separately, the left side is bounded by the
+total influence of $P_{\le K}^{F_{\mathcal D}}\bar f_\omega$, averaged over
+the remaining variables. Equation~\eqref{eq:degree-influence} and
+$|\bar f_\omega|\le1$ give \eqref{eq:total-influence}.
+No pointwise boundedness of $H_v$ is asserted or needed.
+
+\paragraph{Step 2: discard even dependence on a gate.}
+Fix $v\in\mathcal D$, and split $H_v$ into its even and odd parts under
+negation of $F_v$ alone. These parts have disjoint Fourier supports in the
+slice bits, also after retaining the monomials containing any specified
+entry. Thus
+\[
+ \Inf_{v,z}(H_v)=
+ \Inf_{v,z}(H_v^{\mathrm{even}})
+ +\Inf_{v,z}(H_v^{\mathrm{odd}}).
+\]
+By the sign action in \Cref{lem:row-symmetry}, the even part has
+$q_v(\one)=0$ on row support. Since the index is unsplit, all its child
+indices are either $0$ or $q_v$, so they too take value zero on $\one$.
+The domain-translation action in that lemma shows that the even part is
+invariant under translations of the domain of $F_v$, with the rows and
+outside tables fixed. These translations act transitively on its entries.
+All entry influences at $v$ are therefore equal. By
+\eqref{eq:total-influence},
+\begin{equation}\label{eq:even-influence}
+ \E\sum_{v\in\mathcal D}
+ \max_z\Inf_{v,z}(H_v^{\mathrm{even}})\le\frac{K}{2^s}.
+\end{equation}
+
+\paragraph{Step 3: express odd influences as affine hashes.}
+Fix $\omega,\mathcal D,v,\Gamma,F_{\mathrm{out}}$. The remaining row
+coordinates $L_{gb}$ are independent uniform elements of $H$. On the odd
+part, the common pair sum $q$ is a label, by
+\eqref{eq:labels-dual}, and hence is nonzero. An unsplit shift index with
+common sum $q$ puts $q$
+in the second child on a subset $I\subseteq[s]$ and in the first child
+elsewhere. Let $a_{q,I}(F_{\mathcal D})$ be its coefficient in
+$H_v^{\mathrm{odd}}$.
+
+Set
+\[
+ L_\Sigma=\sum_{g=1}^sL_{g0},\qquad J_g=L_{g0}+L_{g1},\qquad
+ \mathbf J(q)=(q(J_1),\ldots,q(J_s)).
+\]
+The $s+1$ forms $L_\Sigma,J_1,\ldots,J_s$ are independent and uniform: the
+displayed linear map from $H^{2s}$ onto $H^{s+1}$ is surjective. The
+remaining $s-1$ coordinates do not occur in the unsplit expansion.
+The character of the shift index $(q,I)$ is
+\[
+ \chi_q(L_\Sigma)(-1)^{\sum_{g\in I}q(J_g)}.
+\]
+On the other hand, the domain-translation identity gives
+\[
+ a_{q,I}(F_v(\cdot+z),F_{\mathcal D\setminus\{v\}})
+ =(-1)^{\sum_{g\in I}z_g}a_{q,I}(F_{\mathcal D}).
+\]
+With $\Psi_q=\sum_{I\subseteq[s]}a_{q,I}$ we obtain
+\begin{equation}\label{eq:odd-hash-form}
+ H_v^{\mathrm{odd}}
+ =\sum_{q\in\mathcal A}\chi_q(L_\Sigma)
+ \Psi_q\bigl(F_v(\cdot+\mathbf J(q)),
+                   F_{\mathcal D\setminus\{v\}}\bigr).
+\end{equation}
+The functions $a_{q,I}$ for distinct $I$ are orthogonal in the uniform
+slice bits: a domain translation whose two characters differ preserves
+their inner product and changes its sign. Consequently,
+\begin{equation}\label{eq:psi-mass}
+ \norm{\Psi_q}_{2,F_{\mathcal D}}^2
+ =\sum_I\norm{a_{q,I}}_{2,F_{\mathcal D}}^2
+ \le m_q(\bar f_\omega;\Gamma,F_{\mathrm{out}}).
+\end{equation}
+The inequality follows because the unsplit row projection is determined
+by the free shift index, and the degree and parity projections are
+orthogonal projections in the slice bits at each fixed context.
+Every $\Psi_q$ has degree at most $K$.
+
+Equation~\eqref{eq:odd-hash-form} has converted the free-shift characters
+into translations of a gate table. We next collect its Boolean Fourier
+coefficients into vectors whose squared norms are entry influences.
+These Fourier coefficients are indexed by sets of table entries; the
+earlier index $I\subseteq[s]$ instead specified a character of the
+table's domain $\F^s$.
+
+Denote by
+$\widehat\Psi_q(B,D)$ the slice Fourier coefficient with entry set
+$B\subseteq\F^s$ at gate $v$ and entry set $D$ on the other slice gates.
+For $z\in\F^s$ define a real vector $c(q,z)$, with coordinates indexed by
+pairs $(A,D)$ where $0\in A\subseteq\F^s$, by
+\[
+ c(q,z)_{A,D}=\widehat\Psi_q(z+A,D).
+\]
+Using entry sets relative to $z$ makes this vector space the same for
+every $z$ and every label $q$.
+Set $c(q,z)=0$ for functionals $q$ which are not labels. For the influence
+at $z$, a slice Fourier set $B$ containing $z$ has the unique form
+$B=z+A$ with $0\in A$. Translation by $\mathbf J(q)$ sends its coefficient
+to $\widehat\Psi_q(z+A+\mathbf J(q),D)$. Hence, exactly,
+\begin{equation}\label{eq:influence-hash}
+ \Inf_{v,z}(H_v^{\mathrm{odd}})
+ =\norm*{\sum_q\chi_q(L_\Sigma)c(q,z+\mathbf J(q))}^2.
+\end{equation}
+Writing $w_q=\sum_z\norm{c(q,z)}^2$ and $M_v=\sum_qw_q$, we have
+\begin{equation}\label{eq:wq-mq}
+ w_q=\sum_{B,D}|B|\abs{\widehat\Psi_q(B,D)}^2
+ \le K\norm{\Psi_q}_2^2
+ \le K m_q(\bar f_\omega;\Gamma,F_{\mathrm{out}}).
+\end{equation}
+The factor $|B|$ counts the entries whose influences contain that
+Fourier monomial; it is at most the monomial's total degree, which is
+at most $K$.
+Thus $M_v\le K$ at every context. The total-mass identity in
+\Cref{lem:hashing}, together with \eqref{eq:influence-hash}, also gives
+\begin{equation}\label{eq:sum-M}
+ \E\sum_{v\in\mathcal D}M_v
+ =\E\sum_{v\in\mathcal D}\sum_z
+     \Inf_{v,z}(H_v^{\mathrm{odd}})\le K.
+\end{equation}
+For this identity the free coordinates are integrated separately for each
+node; their different coordinate systems do not affect the uniform row law.
+
+\paragraph{Step 4: force a heavy label.}
+Choose $\zeta>0$ with $\zeta K\le\Lambda/4$. Apply
+\Cref{lem:hashing}, obtaining $\gamma>0$ and a lower bound on $s$.
+Choose $s\ge2$ large enough for that lemma and for
+$K2^{-s}\le\Lambda/4$. Then choose $m$ as in \eqref{eq:choose-m}.
+Finally put
+\begin{equation}\label{eq:extraction-constants}
+ M_0=\frac{\Lambda}{4\cdot2^m},\qquad
+ \theta=\frac{\gamma M_0}{K},\qquad
+ p_* =\frac{\Lambda}{4K\,2^m}.
+\end{equation}
+
+For each node, condition on $\omega,\mathcal D,v,\Gamma,F_{\mathrm{out}}$
+and integrate the free shifts. Partition these contexts into three classes:
+\begin{enumerate}[label=(\roman*)]
+\item $w_q\le\gamma M_v$ for every $q$;
+\item the first condition fails and $M_v<M_0$;
+\item $M_v\ge M_0$ and $w_q>\gamma M_v$ for some $q$.
+\end{enumerate}
+On class (i), the hashing lemma bounds the expected maximum odd influence
+by $\zeta M_v$. Summing and using \eqref{eq:sum-M} costs at most
+$\Lambda/4$. On class (ii), the maximum is bounded in expectation by the
+total influence, namely $M_v<M_0$. Since $|\mathcal D|\le2^m$, these
+contexts cost at most $\Lambda/4$. On class (iii), the same total-mass
+bound is at most $K$ per node.
+Equation~\eqref{eq:even-influence} bounds all even contributions by
+$\Lambda/4$. Let $N_{\mathrm{iii}}$ count the nodes whose induced
+contexts lie in class~(iii). Comparing all four contributions with
+\eqref{eq:large-max-sum} gives
+\[
+ \Lambda
+ \le K2^{-s}+\zeta K+2^m M_0+K\E N_{\mathrm{iii}}
+ \le\frac{3\Lambda}{4}+K\E N_{\mathrm{iii}}.
+\]
+Thus $\E N_{\mathrm{iii}}\ge\Lambda/(4K)$.
+At each such node \eqref{eq:wq-mq} supplies a label with
+$m_q>\gamma M_0/K=\theta$.
+Choosing a uniform node from the slice loses at most a factor $2^m$ and
+gives \eqref{eq:extraction-event}. This proves the proposition.
+\end{proof}
diff --git a/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/hashing.tex b/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/hashing.tex
new file mode 100644
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+++ b/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/hashing.tex
@@ -0,0 +1,251 @@
+\section{A vector-valued hashing lemma}\label{sec:hashing}
+
+The following lemma turns dispersion among characters into dispersion among
+table entries.  Its constants are independent of the dimension of the character
+space and of the dimension of the vectors being hashed. In the application,
+$q$ will index candidate labels and $z$ will index entries of a gate table.
+The resulting hashed vector will collect the Fourier coefficients
+contributing to one entry's influence, so its squared norm will be that influence. This is why
+we need a vector-valued statement rather than a scalar estimate.
+
+\begin{lemma}[Vector-valued hashing]\label{lem:hashing}
+Let $H$ be a finite-dimensional vector space over $\F$, write
+$\chi_q(u)=(-1)^{q(u)}$ for $q\in H^*$ and $u\in H$, let $\mathcal V$ be a
+real Hilbert space, let $s\ge1$ be an integer, and let
+\[
+ c(q,z)\in\mathcal V,
+ \qquad q\in H^*,\quad z\in\F^s.
+\]
+Set
+\[
+ w_q=\sum_{z\in\F^s}\norm{c(q,z)}^2,
+ \qquad M=\sum_{q\in H^*}w_q.
+\]
+Choose $L,J_1,\ldots,J_s$ independently and uniformly in $H$, and define
+\[
+ \mathbf J(q)=(q(J_1),\ldots,q(J_s)),
+ \qquad
+ U(z)=\sum_{q\in H^*}\chi_q(L)c(q,z+\mathbf J(q)).
+\]
+Then
+\begin{equation}\label{eq:hashing-total-mass}
+ \E\sum_{z\in\F^s}\norm{U(z)}^2=M.
+\end{equation}
+For every $\zeta>0$, there are $\gamma>0$ and $s_0\in\mathbb N$, depending
+only on $\zeta$, such that for every $s\ge s_0$,
+\begin{equation}\label{eq:hashing-dispersion}
+ \max_{q\in H^*}w_q\le\gamma M
+ \quad\Longrightarrow\quad
+ \E\max_{z\in\F^s}\norm{U(z)}^2\le\zeta M.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+All expectations over $H$ and $\F^s$ below use uniform probability measure;
+the displayed sums defining $w_q$ and $M$ are unnormalized.  Conditional on
+the $J_g$, orthogonality of the characters of $L$ gives
+\[
+ \E_L\norm{U(z)}^2
+ =\sum_q\norm{c(q,z+\mathbf J(q))}^2.
+\]
+Summing over $z$ proves \eqref{eq:hashing-total-mass}.  If $M=0$, every vector
+$c(q,z)$ is zero and \eqref{eq:hashing-dispersion} is immediate.  Henceforth
+assume $M>0$.
+
+Set $h=2^s$ and $\rho=\max_q w_q/M$. We will prove the quantitative
+bound below for every $C,D>0$, and then choose its parameters.
+The main estimate bounds the maximum squared norm by the square root of
+the sum of fourth powers. After a Fourier change of variables, most terms
+in that fourth moment force the same character $q$ in all four factors;
+their total is controlled by the dispersion of the character masses.
+Two truncations will control the remaining terms and the cost of this
+change to the array:
+\begin{equation}\label{eq:hashing-quantitative}
+ \E\max_z\norm{U(z)}^2
+ \le 2M\left[
+ \sqrt{2\rho+\frac{2C^2}{D^2}+\frac{3D^4}{h}}
+ +\left(\frac1{\sqrt C}+\frac1D\right)^2+\frac1h
+ \right].
+\end{equation}
+
+
+\paragraph{Fourier representation.}
+For $p\in\F^s$ and $u\in H$, define
+\[
+ \alpha_q(p)=\sum_{z\in\F^s}(-1)^{p\cdot z}c(q,z),
+ \qquad
+ A_p(u)=\sum_{q\in H^*}\chi_q(u)\alpha_q(p),
+ \qquad
+ S_p=L+\sum_{g=1}^s p_gJ_g.
+\]
+Fourier inversion in $z$ gives
+\begin{equation}\label{eq:hashing-inversion}
+ U(z)=\E_p(-1)^{p\cdot z}A_p(S_p).
+\end{equation}
+Writing $m_p=\E_u\norm{A_p(u)}^2$, Parseval's identity in the two groups gives
+\begin{equation}\label{eq:hashing-parseval}
+ m_p=\sum_q\norm{\alpha_q(p)}^2,
+ \qquad \E_p m_p=M,
+ \qquad \E_p\norm{\alpha_q(p)}^2=w_q.
+\end{equation}
+For distinct $p,p'$, the variables $S_p,S_{p'}$ are independent and uniform
+in $H$: the coefficient vectors $(1,p),(1,p')$ are linearly independent
+over $\F$.  Similarly, any three distinct $p_1,p_2,p_3$ give three
+independent uniform variables $S_{p_1},S_{p_2},S_{p_3}$.  Indeed, any
+nonempty subcollection of their coefficient vectors whose sum is zero
+must have even cardinality, and hence would consist of two equal vectors.
+
+\paragraph{Two truncations.}
+We first discard indices $p$ with unusually large mean squared norm
+$m_p$, and then discard unusually large values of $A_p(u)$ on the
+remaining indices. The first cutoff makes the change to each individual
+character coefficient uniformly small under the second cutoff. The
+second gives the pointwise bound needed when the fourth-moment indices
+coincide. We will control the discarded part using a first-moment bound
+and a variance bound from the pairwise independence of the samples $S_p$.
+Define
+\[
+ A'_p(u)=
+ \begin{cases}
+ A_p(u),&m_p\le CM\text{ and }\norm{A_p(u)}\le D\sqrt M,\\
+ 0,&\text{otherwise},
+ \end{cases}
+\]
+and put
+\[
+ \alpha'_q(p)=\E_u\chi_q(u)A'_p(u),
+ \qquad w'_q=\E_p\norm{\alpha'_q(p)}^2,
+ \qquad U'(z)=\E_p(-1)^{p\cdot z}A'_p(S_p).
+\]
+Pointwise truncation does not increase the total squared norm, so Parseval
+gives $\sum_qw'_q\le M$.  If $m_p\le CM$, then
+\[
+ \norm{\alpha_q(p)-\alpha'_q(p)}
+ \le\E_u\norm{A_p(u)}
+       \one_{\{\norm{A_p(u)}>D\sqrt M\}}
+ \le\frac{m_p}{D\sqrt M}
+ \le\frac{C\sqrt M}{D}.
+\]
+If $m_p>CM$, then $\alpha'_q(p)=0$.  Consequently,
+\begin{equation}\label{eq:hashing-truncated-mass}
+ \sum_qw'_q\le M,
+ \qquad
+ w'_q\le 2w_q+\frac{2C^2M}{D^2}.
+\end{equation}
+
+Let
+\[
+ X_p=\norm{(A_p-A'_p)(S_p)},\qquad X=\E_pX_p.
+\]
+Equation~\eqref{eq:hashing-inversion} and the triangle inequality imply
+$\max_z\norm{U(z)-U'(z)}\le X$.  The contribution to $\E X$ from the
+indices with $m_p>CM$ is at most
+\[
+ \E_p\one_{\{m_p>CM\}}\sqrt{m_p}
+ \le\frac{\E_p m_p}{\sqrt{CM}}=\sqrt{M/C}.
+\]
+The contribution from the remaining indices is at most
+$\E_p m_p/(D\sqrt M)=\sqrt M/D$.  Moreover, the $X_p$ are pairwise
+independent, by the pairwise independence of the $S_p$.  Since
+$A_p-A'_p$ is either $A_p$ or zero at each point,
+\[
+ \operatorname{Var}(X)
+ =h^{-2}\sum_p\operatorname{Var}(X_p)
+ \le h^{-2}\sum_p\E_u\norm{A_p(u)-A'_p(u)}^2
+ \le\frac Mh.
+\]
+We have proved
+\begin{equation}\label{eq:hashing-truncation-error}
+ \E\max_z\norm{U(z)-U'(z)}^2
+ \le M\left[\left(\frac1{\sqrt C}+\frac1D\right)^2+\frac1h\right].
+\end{equation}
+
+\paragraph{The fourth moment.}
+For brevity write $B_p=A'_p(S_p)$.  Expanding the fourth power of the norm
+and summing the characters in $z$ gives the exact identity
+\begin{align}
+ \E\sum_z\norm{U'(z)}^4
+ &=h^{-3}\sum_{p_1,p_2,p_3}
+   \E\bigl[\ip{B_{p_1}}{B_{p_2}}
+                  \ip{B_{p_3}}{B_{p_1+p_2+p_3}}\bigr]\notag\\
+ &=\E_{p_1,p_2,p_3}\E
+   \bigl[\ip{B_{p_1}}{B_{p_2}}\ip{B_{p_3}}{B_{p_4}}\bigr],
+ \qquad p_4=p_1+p_2+p_3.
+ \label{eq:hashing-fourth-expansion}
+\end{align}
+Here $p_1,p_2,p_3$ are independent uniform indices in the second line.
+The factor $h^{-3}$ in the first line comes from four factors $h^{-1}$
+in Fourier inversion and the sum of $h$ characters over $z$.
+
+If any two of $p_1,p_2,p_3,p_4$ coincide, then some two of
+$p_1,p_2,p_3$ coincide.  This event has probability at most $3/h$.
+Since $\norm{B_p}\le D\sqrt M$, its contribution to
+\eqref{eq:hashing-fourth-expansion} is at most $3D^4M^2/h$ in absolute value.
+
+Now fix four distinct indices with $p_4=p_1+p_2+p_3$.  The first three
+$S_{p_i}$ are independent uniform elements of $H$, and
+$S_{p_4}=S_{p_1}+S_{p_2}+S_{p_3}$.  Expanding the four $A'_{p_i}$ in
+characters and averaging over those three independent elements yields
+\begin{equation}\label{eq:hashing-common-character}
+ \E\bigl[\ip{B_{p_1}}{B_{p_2}}\ip{B_{p_3}}{B_{p_4}}\bigr]
+ =\sum_q
+   \ip{\alpha'_q(p_1)}{\alpha'_q(p_2)}
+   \ip{\alpha'_q(p_3)}{\alpha'_q(p_4)}.
+\end{equation}
+In detail, characters indexed by $q_1,q_2,q_3,q_4$ produce the factor
+\[
+ \prod_{i=1}^3\chi_{q_i+q_4}(S_{p_i}),
+\]
+whose expectation is zero unless $q_1=q_2=q_3=q_4$.
+
+The right side of \eqref{eq:hashing-common-character} need not be
+nonnegative.  For an upper bound, replace each summand by
+$\prod_{i=1}^4\norm{\alpha'_q(p_i)}$, which is nonnegative, and then
+drop the restriction that the four indices be distinct.  For every fixed
+$q$, Cauchy--Schwarz gives
+\begin{align*}
+ \E_{p_1,p_2,p_3}\prod_{i=1}^4\norm{\alpha'_q(p_i)}
+ &\le
+ \left(\E\norm{\alpha'_q(p_1)}^2
+             \norm{\alpha'_q(p_2)}^2\right)^{1/2}
+ \left(\E\norm{\alpha'_q(p_3)}^2
+             \norm{\alpha'_q(p_4)}^2\right)^{1/2}\\
+ &=(w'_q)^2.
+\end{align*}
+The last equality uses the unconditioned independent uniform law of each
+pair $(p_1,p_2)$ and $(p_3,p_4)$; it does not assert independence after
+conditioning on distinctness. Thus the nondegenerate fourth moment is
+bounded by $\sum_q(w'_q)^2$: dispersion among characters now controls
+concentration among table entries. Combining these estimates with
+\eqref{eq:hashing-truncated-mass} proves
+\begin{align*}
+ \E\sum_z\norm{U'(z)}^4
+ &\le \sum_q(w'_q)^2+\frac{3D^4M^2}{h}\\
+ &\le M^2\left(2\rho+\frac{2C^2}{D^2}+\frac{3D^4}{h}\right).
+\end{align*}
+Therefore, using Cauchy--Schwarz once more,
+\begin{equation}\label{eq:hashing-truncated-maximum}
+ \E\max_z\norm{U'(z)}^2
+ \le\left(\E\sum_z\norm{U'(z)}^4\right)^{1/2}
+ \le M\sqrt{2\rho+\frac{2C^2}{D^2}+\frac{3D^4}{h}}.
+\end{equation}
+The inequality $(a+b)^2\le2a^2+2b^2$, together with
+\eqref{eq:hashing-truncation-error} and
+\eqref{eq:hashing-truncated-maximum}, proves
+\eqref{eq:hashing-quantitative}.
+
+Finally, set $\varepsilon=\min\{\zeta,1\}$ and choose
+\[
+ C=\frac{64}{\varepsilon},\qquad
+ D=\frac{8C}{\varepsilon},\qquad
+ \gamma=\frac{\varepsilon^2}{128},\qquad
+ 2^{s_0}\ge
+ \max\left\{\frac{192D^4}{\varepsilon^2},\frac{16}{\varepsilon}\right\}.
+\]
+If $s\ge s_0$ and $\rho\le\gamma$, the square-root term in
+\eqref{eq:hashing-quantitative} is at most $\varepsilon/4$.
+Also $D\ge\sqrt C$, so the other two terms in its brackets are at most
+$\varepsilon/16$ each.  The resulting bound is
+$3\varepsilon M/4\le\zeta M$, establishing
+\eqref{eq:hashing-dispersion} with constants depending only on $\zeta$.
+\end{proof}
diff --git a/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/introduction.tex b/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/introduction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..912186f714f2465079c48e9d1555bd904afa97d2
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+++ b/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/introduction.tex
@@ -0,0 +1,207 @@
+\section{Introduction}
+
+For a graph with nonnegative edge weights, Max-Cut asks for a partition of the
+vertices maximizing the weight of edges crossing the partition. A cut is
+encoded by signs on vertices, and an edge contributes exactly when its signs
+differ. For a maximization problem, an $\alpha$-approximation returns a
+feasible solution worth at least $\alpha$ times the optimum. Unless stated
+otherwise, the graph and its rational weights are given explicitly.
+We write $\operatorname{MaxCut}(G)$ for the maximum crossing weight and,
+when the edge weights sum to one, also denote this value by $\Val(G)$.
+The decision problem of whether a weighted graph has a cut of at least a
+given weight appears among Karp's original NP-complete
+problems~\cite{Karp72}. Approximation asks how much of this optimum can
+still be guaranteed in polynomial time.
+Goemans and Williamson~\cite{GW95} gave a semidefinite programming algorithm with random-hyperplane rounding
+for Max-Cut, with expected approximation ratio approaching
+\begin{equation}\label{eq:gw-constant}
+ \GW=\min_{-1\le \rho<1}
+ \frac{2\arccos\rho}{\pi(1-\rho)}
+ =0.878567\ldots.
+\end{equation}
+The expression compares two contributions of an edge whose endpoint
+vectors have inner product $\rho$: its semidefinite objective value is
+$(1-\rho)/2$, whereas a random hyperplane separates the vectors with
+probability $\arccos(\rho)/\pi$. The minimum ratio over $\rho$ is the
+uniform guarantee of this rounding rule.
+Determining whether this ratio is optimal under ordinary NP-hardness has
+been a central question in approximation algorithms. H\aa stad's
+hardness theorem~\cite{Hastad01}, combined with the gadgets of
+Trevisan, Sorkin, Sudan, and Williamson~\cite{TSSW00}, rules out ratios
+greater than $16/17$.
+Khot, Kindler, Mossel, and O'Donnell~\cite{KKMO07} established optimality of
+$\GW$ assuming the Unique Games Conjecture~\cite{Khot02}; the analytic input
+is the Majority Is Stablest theorem of Mossel, O'Donnell, and
+Oleszkiewicz~\cite{MOO10}.
+
+We resolve the unconditional optimality question for Max-Cut.
+
+\begin{theorem}\label{thm:main}
+For every fixed $\alpha$ with $\GW<\alpha\le1$, it is NP-hard to approximate
+Max-Cut within a factor of $\alpha$. This holds for simple unweighted graphs.
+\end{theorem}
+
+The statement means that a polynomial-time algorithm which always returns a
+cut of weight at least $\alpha$ times the maximum would imply
+$\mathrm P=\mathrm{NP}$. We prove the following more precise gap theorem.
+Write
+\begin{equation}\label{eq:B}
+ B(t)=\frac{2}{\pi}\arcsin t,\qquad 0<t<1.
+\end{equation}
+
+\begin{theorem}\label{thm:gap}
+Fix a rational $t\in(0,1)$ and a constant
+$0<\eps<(t-B(t))/4$. For graphs equipped with a rational probability
+distribution on edges, it is NP-hard to distinguish
+\begin{align*}
+ \textnormal{YES:}\quad&\Val\ge\frac{1+t}{2}-\eps,\\
+ \textnormal{NO:}\quad&\Val\le\frac{1+B(t)}{2}+\eps,
+\end{align*}
+where $\Val$ is the maximum cut weight.
+\end{theorem}
+
+\subsection*{Prior work and the direct route}
+
+The reduction of proof verification to approximation gaps has its roots in
+Feige, Goldwasser, Lov\'asz, Safra, and Szegedy~\cite{FGLSS96} and in the
+PCP theorems of Arora--Safra and Arora, Lund, Motwani, Sudan, and
+Szegedy~\cite{AroraSafra98,ALMSS98}. H\aa stad's Fourier analysis of
+proof-verification tests~\cite{Hastad01} made many of these gaps sharp.
+For Max-Cut, the Gaussian geometry of the semidefinite relaxation gives
+a particularly precise target. Feige and Schechtman~\cite{FS02}
+constructed integrality gaps approaching $\GW$ for this relaxation;
+a limitation of that relaxation alone does not establish hardness for
+arbitrary algorithms. The Unique Games reduction of
+Khot, Kindler, Mossel, and O'Donnell~\cite{KKMO07} connected the same
+geometric threshold to computational hardness.
+
+The original analysis of Khot, Kindler, Mossel, and O'Donnell proposed
+Majority Is Stablest as the analytic statement needed to obtain this
+threshold. Mossel, O'Donnell, and Oleszkiewicz proved it through a
+comparison between low-influence Boolean functions and Gaussian
+functions~\cite{MOO10}, transferring Borell's halfspace noise-stability
+bound~\cite{Borell85} to the Boolean setting. Their theorem leaves the
+Unique Games Conjecture as the hypothesis in the classical hardness
+reduction~\cite[Corollary~2.12]{MOO10}. We use Majority Is Stablest to
+identify an influential coordinate of our test, then prove the additional
+extraction needed to turn that coordinate into a source-game label.
+The algorithmic benchmark above is the ideal rounding ratio;
+finite-precision SDP optimization and random-hyperplane rounding give
+expected ratio $\GW-\eta$ for any fixed $\eta>0$~\cite[Section~2]{GW95}.
+
+The comparison between semidefinite approximation and hardness extends
+beyond a single worst-case ratio. O'Donnell and Wu~\cite{OW08}
+determined the full Max-Cut approximation curve under the Unique Games
+Conjecture, relating the attainable cut value to the instance's optimum.
+Raghavendra~\cite{Raghavendra08} established a general correspondence
+between semidefinite relaxations and Unique-Games-based hardness for
+finite constraint satisfaction problems. Our theorem concerns the
+unconditional worst-case ratio for Max-Cut; the proof below establishes
+the specific family of completeness and soundness gaps in
+\Cref{thm:gap}.
+
+In a projection game, each vertex of a bipartite graph receives a label
+from a specified alphabet, and the edge weights form a probability
+distribution. An edge accepts when its prescribed map sends the left label to the
+right label. Its value is the maximum accepted edge weight. A two-to-one
+constraint has exactly two preimages for every right label. Completeness
+and soundness are, respectively, the promised lower bound on the YES-case
+value and upper bound on the NO-case value.
+
+Our starting point is the published 2-to-1 game construction with imperfect
+completeness~\cite{DKKMS25}. The Grassmann-graph approach was introduced
+by Khot, Minzer, and Safra~\cite{KMS25}. Its soundness was completed
+through the connection between expansion and agreement established by
+Barak, Kothari, and Steurer~\cite{BKS19} and the expansion theorem of
+Khot, Minzer, and Safra~\cite{KMS23}. We need a precise affine form of the
+construction: its projections are surjective affine maps between binary
+spaces of dimensions differing by one, and its alphabet can be fixed for a
+desired soundness before the completeness error is made arbitrarily small.
+\Cref{prop:affine-games} states this input. Appendix~\ref{sec:affine-games}
+derives it from the published construction, specifying both the
+valid-subspace conditioning and the auxiliary candidate lists used in
+the soundness proof.
+Together with Majority Is Stablest, that construction supplies the
+external foundation of the argument. The Unique Games Conjecture enters
+only as historical context. The tree Fourier analysis, hashing bound,
+and hidden-coordinate decoding are proved here.
+
+\subsection*{The proof and its main ingredients}
+
+As in long-code reductions~\cite{BGS98,Hastad01}, a label $a$ is represented
+by the dictator $w\mapsto w(a)$, where $w$ ranges over Boolean words
+indexed by labels. The graph has one such word domain for each tuple of
+left vertices of the source game. Thus an arbitrary cut specifies a
+Boolean function on each domain. To sample an edge, the test chooses two
+left tuples through a common right tuple, generates two words on the
+right label space, and pulls them back through the sampled affine
+projections. It negates the second word. When the sampled constraints
+are satisfied, dictator labels give a cut probability of one half of one
+plus the correlation between the two words evaluated at a common label.
+
+The word generator is a finite rooted tree. Every internal node has $s$
+pairs of children and a random table from $\F^s$ to $\bits$; the leaves
+carry affine forms on the label space. At a label $a$, these forms give
+leaf signs, and the tables recursively combine the child signs into the
+root sign $W(a)$. The gate rule is chosen so that each monomial of its
+Walsh expansion keeps one child from every pair. A second exploration
+keeps both children of one randomly selected pair at each reached node.
+At any fixed depth, the two explorations have exactly one node in
+common. The test couples the tables reached by this second exploration
+at a random depth, using entrywise correlation $t$, and leaves all other
+tables unchanged. Every monomial therefore acquires precisely one factor
+of $t$. A small independent resampling of the affine rows gives the
+smoothing needed for soundness and only a small completeness loss.
+
+For an arbitrary cut, we first average its odd part over the source-game
+edges. The resulting function is balanced in the selected table entries,
+and a cut above the desired NO threshold makes its expected noise
+stability exceed $B(t)$. Majority Is Stablest then detects an influential
+entry. That entry is indexed by a gate input, not by a source-game label.
+The remaining task is to connect its influence to labels that the two
+endpoints of a source edge can choose consistently. Three steps address
+this task.
+
+\begin{enumerate}
+\item \emph{Local simplification of Fourier support.}
+The gate symmetries force the two child indices in each pair to have the
+same sum. After row smoothing, averaging the depth of the selected tables
+makes it inexpensive to retain indices which put this sum in one child of
+each pair and zero in the other. This simplification is imposed only at
+the influential gate. It is the structure that lets affine shifts act as
+translations of that gate's table.
+
+\item \emph{A vector-valued affine hashing lemma.}
+For independent affine shifts, a diffuse distribution of Fourier mass
+cannot create a large entry influence on average. The bound is uniform in
+the dimension of the row space. Its proof uses two truncations and a
+fourth-moment identity; it may be useful whenever translated coefficient
+arrays must be decoded without an alphabet-size loss.
+
+\item \emph{Decoding with a hidden coordinate.}
+The mass concentration is obtained after fixing many row coordinates.
+A left endpoint cannot usually construct the pullback of this context
+without knowing its incident projection. We use a product of the game
+and plant zero linear coefficients in one uniformly chosen factor of
+the context and of the latent resampling rows. After that factor is
+forgotten, the planted law is close to the original law. At the planted
+factor, the context can be pulled back without knowing the projection.
+Only the $2s$ free rows at the selected gate still have to be compared
+across it, at a cost of $2^{2s}$ in Fourier mass.
+\end{enumerate}
+
+The resulting decoding loss is independent of the source alphabet.
+This is essential: the alphabet dimension depends on the soundness we
+request, so a dimension-dependent loss could make the soundness choice
+circular. We first fix that loss and the source soundness, then the
+alphabet and the number of product factors, and finally the source
+completeness error. The quantifier order in
+\Cref{prop:affine-games} permits exactly this sequence.
+
+\Cref{sec:preliminaries} fixes the Fourier conventions.
+\Cref{sec:tree,sec:reduction} define the code and the Max-Cut test.
+\Cref{sec:tree-fourier,sec:hashing,sec:extraction} prove the analytic ingredients.
+\Cref{sec:decoding,sec:completion} give the decoding and finish the proof.
+Appendix~\ref{sec:affine-games} derives the precise starting game from the
+published construction, and Appendix~\ref{sec:unweighted} gives a deterministic
+conversion to simple unweighted graphs.
diff --git a/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/preliminaries.tex b/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/preliminaries.tex
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+++ b/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/preliminaries.tex
@@ -0,0 +1,171 @@
+\section{Preliminaries}\label{sec:preliminaries}
+
+The proof uses two Fourier expansions. Characters of affine rows encode
+candidate labels, whereas characters of Boolean table entries detect the
+influences to which Majority Is Stablest applies. We introduce both
+conventions, then state the exact affine-game input.
+
+All vector spaces over $\F$ are finite dimensional. All expectations on a
+finite vector space or Boolean cube use the uniform probability measure,
+unless another distribution is specified. All Fourier coefficients use
+this probability normalization. An $L^2$ norm always integrates over the
+variables displayed in its subscript; variables declared fixed are not
+integrated.
+
+\subsection{Affine rows and labels}
+
+Let $\mathcal A$ be a nonempty affine space over $\F$, with a fixed affine
+coordinate system, and set
+\[
+ H=\Aff(\mathcal A,\F).
+\]
+This is a vector space, containing the constant form $\one$.
+For $q\in H^*$ define $\chi_q(h)=(-1)^{q(h)}$.
+Evaluation identifies $\mathcal A$ with
+\begin{equation}\label{eq:labels-dual}
+ \{q\in H^*:q(\one)=1\}.
+\end{equation}
+Indeed, if $h(a)=h_0+\sum_i h_i a_i$, a functional taking value one on
+$\one$ has the unique form $q(h)=h_0+\sum_i h_i a_i$.
+We call precisely these functionals \emph{labels}; a general nonzero
+functional in $H^*$ need not be a label.
+
+For $f:H^k\to\mathbb R$ and $C=(q_1,\ldots,q_k)\in(H^*)^k$, put
+\[
+ \widehat f(C)=\E_R f(R)\prod_{i=1}^k\chi_{q_i}(R_i),\qquad
+ |C|=\#\{i:q_i\ne0\}.
+\]
+Parseval gives $\sum_C\widehat f(C)^2=\E_R f(R)^2$.
+The same notation is used for functions with additional variables, which
+remain fixed during the indicated Fourier expansion.
+
+For $0\le r\le1$, let $T_r$ be the row-noise operator: independently in each
+coordinate, retain the row with probability $r$ and otherwise replace it
+by an independent uniform row. Then
+\begin{equation}\label{eq:row-noise}
+ \widehat{T_rf}(C)=r^{|C|}\widehat f(C).
+\end{equation}
+This operator preserves the range $[-1,1]$. It is also convolution by an
+independent increment $E\in H^k$, whose coordinates are zero with
+probability $r$ and independent uniform rows with probability $1-r$.
+Convolution commutes with every fixed translation of the rows.
+
+\subsection{Boolean Fourier analysis}
+
+For a finite set $\Omega$ and $h:\bits^\Omega\to\mathbb R$, write
+\[
+ h(x)=\sum_{S\subseteq\Omega}\widehat h(S)\prod_{a\in S}x_a.
+\]
+Its influence and degree-$K$ influence at $a\in\Omega$ are
+\[
+ \Inf_a(h)=\sum_{S\ni a}\widehat h(S)^2,\qquad
+ \Inf_a^{\le K}(h)=\sum_{\substack{S\ni a\\|S|\le K}}
+ \widehat h(S)^2.
+\]
+Let $P_{\le K}$ be the orthogonal projection onto Fourier degree at most
+$K$. In particular,
+\begin{equation}\label{eq:degree-influence}
+ \sum_{a\in\Omega}\Inf_a(P_{\le K}h)
+ \le K\norm{h}_2^2.
+\end{equation}
+For $t$-correlated uniform bits $x,y$, independently coupled in each
+coordinate,
+\begin{equation}\label{eq:stability}
+ \Stab_t(h)=\E[h(x)h(y)]
+ =\sum_{S\subseteq\Omega}t^{|S|}\widehat h(S)^2.
+\end{equation}
+Thus stability is nonnegative for $t\ge0$. If $h\in[-1,1]$, it is at most
+one.
+
+\begin{lemma}[Majority Is Stablest with a degree cutoff]\label{lem:mis}
+Fix $0<t<1$ and $\xi>0$. There are an integer $K\ge1$ and $\tau>0$,
+depending only on $t,\xi$, such that every
+$h:\bits^\Omega\to[-1,1]$ with $\E h=0$ satisfies
+\[
+ \bigl(\Inf_a^{\le K}(h)<\tau\text{ for all }a\in\Omega\bigr)
+ \quad\Longrightarrow\quad
+ \Stab_t(h)\le B(t)+\xi.
+\]
+\end{lemma}
+
+\begin{proof}
+The untruncated assertion follows from Majority Is Stablest
+\cite[Theorem~4.4]{MOO10}. To match its $[0,1]$ convention, put
+$g=(h+1)/2$: then $\E g=1/2$, each nonconstant Fourier coefficient is
+halved, and $\Stab_t(h)=4\Stab_t(g)-1$. The Gaussian halfspace value
+for $g$ is $1/4+\arcsin(t)/(2\pi)$, which becomes $B(t)$ in our
+normalization; see \cite[Theorem~4.1 and Corollary~4.3]{MOO10} and
+\cite{Borell85}. Here is the degree-cutoff reduction to the untruncated
+assertion. For a cube-noise
+parameter $u<1$, put $h_u=T_u h$, using ordinary bit noise in this
+paragraph. Range and mean are preserved. Uniformly over $h$ of norm at
+most one,
+\[
+ 0\le\Stab_t(h)-\Stab_t(h_u)
+ \le\sup_{j\ge0}t^j(1-u^{2j})\longrightarrow0
+ \quad(u\uparrow1).
+\]
+The convergence follows by first choosing a finite degree cutoff to bound
+the tail $t^j$, and then taking $u$ close to one on the remaining degrees.
+Choose $u$ so that this error is at most $\xi/2$. Let $\delta>0$ be the
+untruncated influence threshold for error $\xi/2$. Choose $K$ with
+$u^{2(K+1)}\le\delta/2$, and take $\tau=\delta/2$. Then
+\[
+ \Inf_a(h_u)
+ \le\Inf_a^{\le K}(h)+u^{2(K+1)}\norm{h}_2^2
+ <\delta.
+\]
+Apply the untruncated theorem to $h_u$ and add the stability error.
+\end{proof}
+
+\subsection{Affine projection games}
+
+A weighted bipartite projection game consists of finite sets $X,Y$, a
+rational probability distribution $\lambda$ on a finite edge set
+$E$ with endpoint maps to $X$ and $Y$ (parallel edges are allowed), finite alphabets,
+and a projection $\pi_e$ for each edge. Its value is
+\[
+ \Val(\mathcal G)=
+ \max_{\ell_X,\ell_Y}\Pr_{e=(x,y)\sim\lambda}
+ [\pi_e(\ell_X(x))=\ell_Y(y)].
+\]
+In this paper the alphabets are $\F^{n+1}$ and $\F^n$, and every
+$\pi_e:\F^{n+1}\to\F^n$ is surjective affine.
+\Cref{prop:affine-games} supplies NP-hard gaps with this structure.
+We discard endpoints with zero marginal probability whenever conditional
+edge distributions are used.
+
+The starting theorem has the quantifier order
+\[
+ \forall b\in(0,1)\quad \exists n=n(b)\ge1\quad
+ \forall a\in(0,1-b).
+\]
+The precise promise is given below; its derivation from
+the published construction is in \Cref{sec:affine-games}.
+
+\begin{proposition}[Affine 2-to-1 Games Theorem]
+\label{prop:affine-games}
+For every constant $b\in(0,1)$ there is an integer $n\geq 1$ such that,
+for every constant $a\in(0,1-b)$, the following promise problem is NP-hard.
+Its input is a
+weighted bipartite game with vertex sets $X,Y$, respective label sets
+$\F^{n+1},\F^n$, and a surjective affine map
+\[
+  \pi_e:\F^{n+1}\longrightarrow\F^n
+\]
+on each edge $e=(x,y)$.  An edge is satisfied when
+$\pi_e(\lambda_X(x))=\lambda_Y(y)$.  The promise cases are
+\[
+  \mathrm{YES}:\quad \Val(\mathcal G)\geq 1-a,
+  \qquad
+  \mathrm{NO}:\quad \Val(\mathcal G)\leq b.
+\]
+The integer $n$ depends on $b$ alone.  For fixed $a,b$, the reduction is
+deterministic and has polynomial running time and output size.
+\end{proposition}
+
+A weighted graph is likewise specified by a rational distribution on
+unordered edges, allowing loops. A cut assigns signs to its vertices and
+satisfies an edge when its signs differ. Its value is its satisfaction
+probability. Loops have zero cut weight. The conversion in
+\Cref{lem:unweighted} keeps a common scale even when loops are deleted.
diff --git a/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/reduction.tex b/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/reduction.tex
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+\section{The Max-Cut reduction}\label{sec:reduction}
+
+Fix a rational $t\in(0,1)$, tree parameters $s,m$, a rational
+$0<\mu<1/2$, and $r=1-\mu/d$. The parameters will be chosen in
+\Cref{sec:completion}. Let $\mathcal G$ be an affine game as in
+\Cref{prop:affine-games}, with edge law $\lambda$ and alphabet dimensions
+$n+1,n$. Let $T\ge1$ be an integer. We use the product label spaces
+\[
+ \mathcal B=(\F^{n+1})^T,\qquad
+ \mathcal A=(\F^n)^T,
+\]
+and the row space $H=\Aff(\mathcal A,\F)$.
+
+\subsection{Vertices and edge distribution}
+
+For each $\underline x\in X^T$, make one graph vertex
+$(\underline x,w)$ for every word $w\in\bits^{\mathcal B}$.
+This is the standard long-code word domain~\cite{BGS98,Hastad01};
+the tree construction specifies how the test samples its queries.
+Thus an arbitrary cut is exactly a family of functions
+\[
+ \sigma_{\underline x}:\bits^{\mathcal B}\longrightarrow\bits.
+\]
+The functions receive only the resulting words; they do not receive the
+rows, gate tables, slice, or sampled game edges that produced them.
+
+The graph's edge distribution is the following test.
+\begin{enumerate}
+\item Draw $\underline y\in Y^T$ from the product right-endpoint marginal
+of $\lambda$. Independently draw two edge tuples
+$\underline e,\underline e'$ from the product conditional edge law given
+$\underline y$. Write $\underline x,\underline x'$ for their left
+endpoints and
+$\pi_{\underline e},\pi_{\underline e'}:\mathcal B\to\mathcal A$
+for their coordinatewise projections.
+
+\item Independently of all game data, draw a random slice $\mathcal D$.
+Draw $F,F'$ as in \Cref{lem:tree-correlation}: corresponding entries
+are independent $t$-correlated uniform pairs on $\mathcal D$, and
+identical uniform pairs off $\mathcal D$.
+
+\item Draw a uniform common row array $R\in H^k$. In each of two copies,
+independently retain each row with probability $r$ and otherwise resample
+it uniformly, obtaining $R^1,R^2$.
+
+\item Output the unordered edge whose endpoints are
+\begin{equation}\label{eq:test-queries}
+ \bigl(\underline x,W(R^1,F)\circ\pi_{\underline e}\bigr),
+ \qquad
+ \bigl(\underline x',-W(R^2,F')\circ\pi_{\underline e'}\bigr).
+\end{equation}
+\end{enumerate}
+All edge weights are the probabilities of these outcomes, aggregated over
+outcomes with the same unordered endpoints.
+
+\begin{lemma}[Completeness]\label{lem:completeness}
+If $\Val(\mathcal G)\ge1-a$, the graph has a cut of value at least
+\begin{equation}\label{eq:completeness}
+ (1-2Ta)\frac{1+t r^{2d}}2
+ \ge\frac{1+t}{2}-t\mu-2Ta.
+\end{equation}
+The first lower bound is useful when $2Ta\le1$, which is the regime used
+below.
+\end{lemma}
+
+\begin{proof}
+Choose game labels $\ell_X,\ell_Y$ satisfying an edge with probability at
+least $1-a$. Label each word vertex by evaluation at the tuple of left
+labels:
+\[
+ \sigma_{\underline x}(w)=
+ w\bigl(\ell_X(x_1),\ldots,\ell_X(x_T)\bigr).
+\]
+Each coordinate of either sampled edge tuple has the original edge law.
+A union bound shows that all $2T$ sampled constraints are satisfied except
+with probability at most $2Ta$. When all these constraints are satisfied,
+the two queried tuples
+project to the same label
+$l=(\ell_Y(y_1),\ldots,\ell_Y(y_T))\in\mathcal A$.
+The cut signs then have product
+$-W(R^1,F)(l)W(R^2,F')(l)$. The word-generation data are independent of
+the game edges, so \Cref{lem:tree-correlation} gives conditional cut
+probability $(1+t r^{2d})/2$. On the remaining event the cut probability
+is nonnegative. This proves the first bound when $2Ta\le1$; if $2Ta>1$
+that bound is negative and is automatic. Finally,
+$r^{2d}\ge1-2\mu$, and the loss due to the prefactor is at most $2Ta$,
+giving the second bound.
+\end{proof}
+
+\subsection{A large cut produces excess stability}
+
+An arbitrary cut need not change sign when its queried word is negated.
+We separate its two parities because the negative second query makes
+odd dependence increase the cut value and even dependence decrease it.
+For each cut table define its even and odd parts by
+\[
+ \sigma^{\mathrm{even}}(w)=\frac{\sigma(w)+\sigma(-w)}2,
+ \qquad
+ \sigma^o(w)=\frac{\sigma(w)-\sigma(-w)}2.
+\]
+For an edge tuple and a right-endpoint tuple, set
+\begin{align}
+ G_{\underline e}(R,F)
+ &=\sigma^o_{\underline x(\underline e)}
+       (W(R,F)\circ\pi_{\underline e}),\label{eq:edge-function}\\
+ G_{\underline y}(R,F)
+ &=\E_{\underline e\mid\underline y}G_{\underline e}(R,F),
+ \qquad \bar G_{\underline y}=T_rG_{\underline y}.
+ \label{eq:average-function}
+\end{align}
+Write $A_{\underline y}$ and $\bar A_{\underline y}=T_rA_{\underline y}$
+for the analogous averages using $\sigma^{\mathrm{even}}$.
+These functions are bounded in absolute value by one.
+Moreover,
+\[
+ G_{\underline y}(R,F)=\phi_{\underline y}(W(R,F)),\qquad
+ \phi_{\underline y}(w)=\E_{\underline e\mid\underline y}
+ \sigma^o_{\underline x(\underline e)}(w\circ\pi_{\underline e}),
+\]
+where $\phi_{\underline y}$ is odd and depends on no slice or row data.
+Thus $\bar G_{\underline y}$ has mean zero in the slice entries for
+every fixed $\underline y,\mathcal D,R,F_{\mathrm{out}}$, by the
+observation following \Cref{lem:tree-correlation}.
+The later extraction argument, \Cref{prop:extraction}, applies to precisely
+such bounded odd functions of the generated word.
+
+\begin{lemma}[Stability identity]\label{lem:cut-stability}
+For every cut of the constructed graph,
+\begin{equation}\label{eq:cut-stability}
+ \cut(\sigma)=\frac12+
+ \frac12\E\Stab_t^{F_{\mathcal D}}(\bar G_{\underline y})
+ -\frac12\E\Stab_t^{F_{\mathcal D}}(\bar A_{\underline y}),
+\end{equation}
+where both expectations are over
+$\underline y,\mathcal D,R,F_{\mathrm{out}}$.
+In particular,
+\begin{equation}\label{eq:cut-upper-stability}
+ \cut(\sigma)\le\frac12+
+ \frac12\E\Stab_t^{F_{\mathcal D}}(\bar G_{\underline y}).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Condition on $\underline y,\mathcal D,R,F_{\mathrm{out}}$.
+The sampled edge tuples and the two row resamplings are independent
+between copies. After averaging over them, the expected first sign as a
+function of $F_{\mathcal D}$ is $\bar A_{\underline y}+\bar G_{\underline y}$.
+The minus sign in the second queried word makes the expected second sign
+$\bar A_{\underline y}-\bar G_{\underline y}$ at $F'_{\mathcal D}$.
+The two slice inputs are ordinary $t$-correlated uniform cube inputs.
+The rows in these functions are the common latent array $R$; the two
+independent row resamplings have already been absorbed into $T_r$.
+Negating every slice entry leaves $\bar A_{\underline y}$ unchanged and
+negates $\bar G_{\underline y}$. Their Fourier supports therefore have
+opposite total parities, so the cross terms vanish. The expected sign
+product is
+$\Stab_t(\bar A_{\underline y})-\Stab_t(\bar G_{\underline y})$.
+An edge is cut with probability one half of one minus its sign product,
+giving \eqref{eq:cut-stability}. Nonnegativity of stability for $t>0$
+gives \eqref{eq:cut-upper-stability}.
+\end{proof}
+
+If a cut has value at least $(1+B(t))/2+\delta/2$, then its odd average
+has expected slice stability at least $B(t)+\delta$.
+The next three sections show that this excess stability yields a label
+carrying substantial Fourier mass, with constants independent of the game
+dimension and the number of repetitions; the formal statement is
+\Cref{prop:extraction}. \Cref{sec:decoding} then turns that mass into a
+labeling of a single copy of the original game.
diff --git a/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/tree.tex b/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/tree.tex
new file mode 100644
index 0000000000000000000000000000000000000000..953231fb9500d562a87808ae644a811d1f5d760d
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+++ b/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/tree.tex
@@ -0,0 +1,162 @@
+\section{A tree code with complementary traversals}
+\label{sec:tree}
+
+We use the affine label space $\mathcal A$ and row space
+$H=\Aff(\mathcal A,\F)$ of \Cref{sec:preliminaries}.
+The code below turns affine rows and Boolean gate tables into a word
+indexed by labels. Its two complementary traversals control the
+correlation seen by a cut that evaluates the word at a single label.
+
+\begin{definition}[Tree code]
+\label{def:tree-code}
+Fix integers $s\ge2$ and $m\ge1$, and put
+\[
+ k=(2s)^m,\qquad d=s^m.
+\]
+Let $\mathcal T$ be the complete rooted $2s$-ary tree of depth $m$.
+The children of each internal node $v$ are indexed by
+$(g,b)\in[s]\times\{0,1\}$.  Its leaves are indexed by $[k]$.
+To each internal node assign a table
+$F_v:\F^s\to\{-1,1\}$, and to each leaf assign a row $R_i\in H$.
+The word $W(R,F)\in\{-1,1\}^{\mathcal A}$ is defined pointwise as
+follows.  At $a\in\mathcal A$, leaf $i$ has sign $(-1)^{R_i(a)}$.
+An internal node with child signs $(Z_{g,b})$ outputs
+\begin{equation}
+ \left(\prod_{g=1}^s Z_{g,0}\right)F_v(z),
+ \qquad (-1)^{z_g}=Z_{g,0}Z_{g,1}.
+ \label{eq:gate-rule}
+\end{equation}
+The root output is $W(R,F)(a)$.
+\end{definition}
+
+We choose the tables to be noised by a second traversal of the tree.
+A \emph{random slice} is obtained by first choosing
+$J_0$ uniformly from $\{0,\ldots,m-1\}$.  Starting at the root,
+at each reached node of depth less than $J_0$ choose one of its $s$
+child pairs uniformly and independently, and continue through both
+children of that pair.  The set $\mathcal D$ of reached nodes at
+depth $J_0$ is the slice.
+
+Conditionally on $J_0=j$, the slice has $2^j$ nodes, and every fixed
+node of depth $j$ belongs to it with probability $s^{-j}$.
+We write $F_{\rm out}=(F_u)_{u\notin\mathcal D}$ for the tables
+outside the slice.
+
+The two traversals of the tree used below complement one another.
+The slice construction chooses one pair and keeps both children.
+The Walsh expansion of a gate keeps one child from every pair.
+Their intersection therefore follows a single path. Figure~\ref{fig:traversals}
+shows the local rule; iterating it gives one common node at every slice depth.
+
+\begin{figure}[ht]
+\centering
+\input{figures/complementary-traversals}
+\caption{Complementary choices at one gate, illustrated for $s=3$.
+A monomial traversal (solid) selects one child from each pair; the slice
+exploration (dashed) selects both children of one pair. Exactly one child
+is shared, shown in purple. Only this local selection is depicted;
+the same rule is repeated above the chosen slice.}
+\label{fig:traversals}
+\end{figure}
+
+\begin{lemma}[Complementary traversal and correlation]
+\label{lem:tree-correlation}
+For every fixed collection of tables $F$, the root output as a
+multilinear polynomial in the $k$ leaf signs has only degree-$d$
+monomials.  Each such monomial chooses one child from every pair
+at every node that it reaches.  Every resulting traversal meets
+every possible slice $\mathcal D$ at exactly one node.
+Consequently, negating all the tables on a slice negates $W$.
+
+Let $0<t<1$ and $0<r<1$.  Draw $F$ uniformly, and construct $F'$
+by leaving the tables outside $\mathcal D$ unchanged and coupling
+each pair of corresponding table entries on $\mathcal D$ as
+uniform signs of correlation $t$, independently across entries.
+Independently draw
+$R\in H^k$ uniformly, and form $R^1,R^2$ by retaining each row of
+$R$ with probability $r$ and otherwise replacing it by an
+independent uniform row, independently in the two copies.
+Then, for every $a\in\mathcal A$ and every fixed slice,
+\begin{equation}
+ \E\bigl[W(R^1,F)(a)W(R^2,F')(a)\mid\mathcal D\bigr]
+       =t r^{2d}.
+ \label{eq:tree-correlation}
+\end{equation}
+In particular, if $r=1-\mu/d$ with $0<\mu<1/2$, then
+$r^{2d}\ge1-2\mu$.
+\end{lemma}
+
+\begin{proof}
+For a fixed gate table, expand in its argument $z$:
+$F_v(z)=\sum_{I\subseteq[s]}\widehat F_v(I)(-1)^{\sum_{g\in I}z_g}$.
+Substituting into Equation~\eqref{eq:gate-rule}, the term indexed
+by $I$ is
+\[
+ \widehat F_v(I)
+ \prod_{g\notin I}Z_{g,0}\prod_{g\in I}Z_{g,1}.
+\]
+Iterating this expansion gives
+\begin{equation}
+ W_F(x)=\sum_{P\in\mathscr P}a_P(F)\prod_{i\in P}x_i,
+ \qquad |P|=d,
+ \label{eq:tree-polynomial}
+\end{equation}
+where $\mathscr P$ is the family of leaf sets produced by choosing
+one child per pair at each reached internal node.  The leaf set
+determines all these choices: a selected child has a nonempty
+intersection with that leaf set, whereas an unselected child has
+none.  Thus different traversals give different monomials, and
+$a_P(F)$ is the product of one Walsh coefficient of $F_v$ for each
+node reached by the traversal.
+
+At the root both a monomial traversal and the slice exploration
+are present.  At any common node above the slice depth, exactly
+one child is selected by both explorations.  Induction gives
+exactly one common node at the slice depth.  Hence each $a_P$
+contains exactly one factor from a table on $\mathcal D$.
+Negating every such table negates each coefficient in
+Equation~\eqref{eq:tree-polynomial} and therefore negates the word.
+Moreover, each Walsh coefficient of a slice table satisfies
+\[
+ \E[\widehat F'_v(I)\mid F_v]=t\widehat F_v(I).
+\]
+Conditional independence between tables therefore gives
+$\E[a_P(F')\mid F,\mathcal D]=t a_P(F)$.
+
+For fixed $a$, the signs
+$x_i^b=(-1)^{R_i^b(a)}$ are uniform, independent across $i$, and
+$\E[x_i^1x_i^2]=r^2$.  Thus the product expectation of two
+monomials in Equation~\eqref{eq:tree-polynomial} is zero unless
+their leaf sets agree, and is $r^{2d}$ when they agree.
+Since $W_F$ is Boolean, Parseval gives $\sum_P a_P(F)^2=1$.
+Averaging first over rows and then over $F'$ proves
+Equation~\eqref{eq:tree-correlation}.  The last claim is
+Bernoulli's inequality:
+$(1-\mu/d)^{2d}\ge1-2\mu$.
+\end{proof}
+
+The correlation identity describes a cut that evaluates the word at
+one label. To analyze arbitrary cuts, we also need functions that can
+depend on the entire word but do not see its generating rows or tables.
+We call $f$ a \emph{function of the code} if
+\begin{equation}
+ f(R,F)=\phi(W(R,F))
+ \label{eq:function-of-code}
+\end{equation}
+for a fixed function $\phi$ on words.  This class is closed under
+averaging over fixed choices of $\phi$, including choices that
+first pull a word back through a fixed affine map.
+For $r\in[0,1]$, let $T_r$ denote independent row resampling with
+retention probability $r$, and write $\bar f=T_rf$.
+The operator preserves bounds on the range of $f$.
+If $\phi$ is odd, Lemma~\ref{lem:tree-correlation} shows that both
+$f$ and $\bar f$ are odd under simultaneous negation of the slice
+tables.  In particular, they have mean zero in those table entries
+when all rows and outside tables are fixed.
+
+At these fixed data, the remaining random input is the cube of
+$|\mathcal D|2^s$ independent entries, indexed by
+$(v,z)\in\mathcal D\times\F^s$. Fourier degree and influence on this
+cube refer to subsets of \emph{entries}. This differs from the Walsh
+expansion of a fixed gate in its $s$-bit argument $z$ used in the
+correlation proof.
diff --git a/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/tree_fourier.tex b/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/tree_fourier.tex
new file mode 100644
index 0000000000000000000000000000000000000000..2c5dffec582858770004c4e85973a7ed1dace00e
--- /dev/null
+++ b/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/tree_fourier.tex
@@ -0,0 +1,239 @@
+\section{Fourier structure of the tree code}\label{sec:tree-fourier}
+
+The cut depends on the generated word, so every transformation that
+preserves that word also constrains the cut's row Fourier coefficients.
+At each gate these constraints give a common Fourier index shared by
+all child pairs. Negating the gate table distinguishes whether this
+index is an evaluation label. We then show that, at a random slice,
+little smoothed Fourier mass is lost by keeping the entire common index
+in just one child of each pair. Finally we express these retained
+indices in row coordinates that remain free after the other rows are
+fixed. This is the local form used in \Cref{sec:extraction}.
+Throughout the section, $f$ is a function of the code in the sense of
+\eqref{eq:function-of-code}.
+
+\subsection{Row Fourier support}
+
+For each node $v$, define its \emph{generator} $P(v)$, a set of leaves,
+recursively: $P(v)=\{v\}$ at a leaf, and
+$P(v)=P(v_{1,0})\cup P(v_{1,1})$ at an internal node.
+Thus a generator always follows both children of the first pair;
+unlike the slice, it involves no random choices.
+Adding the same $h\in H$ to all rows in $P(v)$ multiplies the
+subtree output at $a$ by $(-1)^{h(a)}$.  This follows by induction
+from Equation~\eqref{eq:gate-rule}: multiplying both children of
+one pair by the same sign leaves all gate inputs $z_g$ unchanged
+and multiplies the prefactor by that sign.
+
+For a row Fourier index $C=(q_i)_{i=1}^k\in(H^*)^k$, put
+\[
+ \chi_C(R)=\prod_i\chi_{q_i}(R_i),\qquad
+ |C|=|\{i:q_i\ne0\}|,\qquad
+ q_v=\sum_{i\in P(v)}q_i.
+\]
+Write $q_{v,gb}$ for the index of child $(g,b)$ of $v$.
+Thus, by definition, $q_v=q_{v,10}+q_{v,11}$.
+At fixed tables, the row expansion and its smoothing are
+\begin{equation}
+ f(R,F)=\sum_C\widehat f_C(F)\chi_C(R),\qquad
+ \widehat{\bar f}_C(F)=r^{|C|}\widehat f_C(F).
+ \label{eq:row-smoothing}
+\end{equation}
+
+\begin{lemma}[Row symmetries]
+\label{lem:row-symmetry}
+Let $f$ be a function of the code.  Every nonzero row Fourier
+coefficient of $f$ satisfies, at each internal node $v$,
+\begin{equation}
+ q_{v,g0}+q_{v,g1}=q_v\qquad(g\in[s]).
+ \label{eq:row-relations}
+\end{equation}
+For a table collection $F$, let $F^{v,-}$ negate $F_v$ alone,
+and let $F^{v,z}$ replace $F_v(u)$ by $F_v(u+z)$, leaving all
+other tables unchanged.  Then
+\begin{align}
+ \widehat f_C(F^{v,-})
+   &=(-1)^{q_v(\one)}\widehat f_C(F),
+   \label{eq:table-flip-action}\\
+ \widehat f_C(F^{v,z})
+   &=(-1)^{\sum_g z_g q_{v,g1}(\one)}\widehat f_C(F).
+   \label{eq:table-translation-action}
+\end{align}
+All these statements remain true after row smoothing.
+They are also preserved by row Fourier projections and by
+projection to any range of total Fourier degrees in the entries
+of a specified collection of whole gate tables.
+\end{lemma}
+
+\begin{proof}
+For each $g$, shifting the generators of both children in pair
+$g$ by $h$ has exactly the same effect on the subtree at $v$ as
+shifting $P(v)$ by $h$.  Consequently the two row translations
+have the same effect on the whole word.  Taking row Fourier
+coefficients yields
+\[
+ \chi_{q_{v,g0}+q_{v,g1}}(h)\widehat f_C(F)
+   =\chi_{q_v}(h)\widehat f_C(F)\qquad(h\in H).
+\]
+Distinct characters are distinct functions, proving
+Equation~\eqref{eq:row-relations}.
+
+Negating $F_v$ has the same effect on the word as adding
+$\one$ to all rows in $P(v)$.  This proves
+Equation~\eqref{eq:table-flip-action}.  To translate the input of
+$F_v$ by $z$, add $\one$ to the generator of its second child in
+each pair with $z_g=1$.  This changes precisely those gate input
+bits and leaves its prefactor unchanged.  Fourier transformation
+gives Equation~\eqref{eq:table-translation-action}.
+
+Row smoothing and row Fourier projections act diagonally on
+$C$, so preserve all three identities.  On the cube of
+table-entry signs, a domain translation permutes coordinates,
+and negating a gate table multiplies each monomial by its
+parity sign.  Both operations preserve total degree in any
+specified collection of whole gate tables and therefore
+commute with the stated degree projections.
+\end{proof}
+
+\subsection{An unsplit projection on a random slice}
+
+Equation~\eqref{eq:row-relations} assigns one common sum $q_v$ to every
+child pair. Equation~\eqref{eq:table-flip-action} then has a useful
+interpretation: coefficients odd under negating $F_v$ have
+$q_v(\one)=1$, so their common index is a label by
+\eqref{eq:labels-dual}. The individual child indices, however, can
+still be arbitrary functionals whose sum is $q_v$.
+
+We simplify this last freedom at one gate at a time. Keeping $q_v$ in
+one child of each pair leaves only the choice of that child, rather
+than two arbitrary functionals with prescribed sum.
+Call $C$ \emph{unsplit at $v$} if, for every $g$, the ordered pair
+$(q_{v,g0},q_{v,g1})$ is either $(q_v,0)$ or $(0,q_v)$.
+Let $S_v$ be the orthogonal row Fourier projection onto such
+indices.  When $q_v=0$, this requires every child index to be zero.
+The next bound averages the cost over the slice depth. It does not
+require an index to be unsplit at every gate of the tree.
+
+\begin{lemma}[Average cost of the unsplit projection]
+\label{lem:unsplit}
+Let $f$ be a function of the code with $|f|\le1$, let
+$r=1-\mu/d$ with $0<\mu<1/2$, and let $\bar f=T_rf$.
+Choose the random slice independently of $f$.  Then
+\begin{equation}
+ \E_{J_0,\mathcal D}\sum_{v\in\mathcal D}
+       \norm{(I-S_v)\bar f}_{2,R,F}^{2}
+       \le \frac{s}{2e m\mu}.
+ \label{eq:unsplit-bound}
+\end{equation}
+The same bound holds after averaging over additional background
+data determining $f$, provided the slice is chosen independently
+of those data.
+\end{lemma}
+
+\begin{proof}
+Fix an index $C$ satisfying Equation~\eqref{eq:row-relations},
+and let $N_j$ be the number of nodes at depth $j$ with nonzero
+index $q_v$.  If $q_v\ne0$, each child pair contains at least
+one nonzero index.  A failure of the unsplit condition at such a
+node supplies at least one additional nonzero child.  If $q_v=0$,
+a failure supplies at least two nonzero children, whereas its
+minimum contribution is zero.  Therefore the number $B_j$ of
+failures at depth $j$ satisfies
+\[
+ B_j\le N_{j+1}-sN_j.
+\]
+Using the slice inclusion probability and telescoping gives
+\begin{align*}
+ \E_{J_0,\mathcal D}\sum_{v\in\mathcal D}
+          \one_{\{C\text{ is split at }v\}}
+ &\le \frac1m\sum_{j=0}^{m-1}
+          \frac{N_{j+1}-sN_j}{s^j}\\
+ &=\frac{s}{m}\left(\frac{N_m}{s^m}-N_0\right)
+ \le\frac{s|C|}{md}.
+\end{align*}
+Here $N_m=|C|$.  By Equation~\eqref{eq:row-smoothing} and
+Parseval, the left side of Equation~\eqref{eq:unsplit-bound}
+is at most
+\[
+ \frac{s}{m}\sum_C
+   \left(r^{2|C|}\frac{|C|}{d}\right)
+           \E_F|\widehat f_C(F)|^2.
+\]
+For $x\ge0$, $xe^{-2\mu x}\le(2e\mu)^{-1}$, and
+$\log r\le-\mu/d$.  Hence every parenthesized factor is at
+most $(2e\mu)^{-1}$, while
+$\sum_C\E_F|\widehat f_C(F)|^2=\norm f_2^2\le1$.
+This proves the bound.  The proof is pointwise in the background
+data, so its average satisfies the same inequality.
+\end{proof}
+
+\subsection{Free shifts and the remaining row coordinates}
+
+The extraction argument must use the common index after fixing most
+of the rows. We therefore choose coordinates in which translating a
+child generator changes one free row and leaves all other coordinates
+fixed. The unsplit projection will then act separately at each fixed
+context, and its retained characters will have an explicit form.
+
+Fix a slice $\mathcal D$ and a node $v\in\mathcal D$.
+The $2s$ sets $P(v_{g,b})$ are nonempty and disjoint.
+Choose one representative leaf $i_{g,b}$ from each of them,
+using a rule depending only on the tree, and set
+$L_{g,b}=R_{i_{g,b}}$.  For every other leaf $i$ in that
+generator, replace $R_i$ by $R_i+L_{g,b}$.  Keep all rows outside
+these generators as they are.  Denote the resulting $k-2s$
+remaining coordinates by $\Gamma\in H^{k-2s}$.
+
+\begin{lemma}[Free-shift coordinates]
+\label{lem:shift-coordinates}
+The change of coordinates
+\[
+ H^k\longrightarrow H^{2s}\times H^{k-2s},\qquad
+ R\longmapsto((L_{g,b}),\Gamma)
+\]
+is a linear bijection.  In particular, under uniform rows the
+free shifts are independent uniforms conditional on $\Gamma$.
+The Fourier index of $L_{g,b}$ corresponding to a full row
+index $C$ is $Q_{g,b}=q_{v,g b}$.
+
+Consequently, at every fixed $\Gamma$, the operator $S_v$ acts
+as the Fourier projection in the $2s$ shifts onto precisely the
+unsplit shift indices.  For a function of the code or its row
+smoothing, the shift coefficients vanish unless the sums
+$Q_{g,0}+Q_{g,1}$ are the same for every $g$.  The table actions
+in Equations~\eqref{eq:table-flip-action} and
+\eqref{eq:table-translation-action} hold for each shift
+coefficient at this fixed context, with
+$q_v=Q_{1,0}+Q_{1,1}$ and $q_{v,g1}=Q_{g,1}$.
+\end{lemma}
+
+\begin{proof}
+The inverse map sets the representative row equal to $L_{g,b}$
+and recovers every other row in its generator by adding
+$L_{g,b}$ to its recorded difference.  This proves bijectivity.
+In a row character, the coefficient of $L_{g,b}$ is exactly
+$\sum_{i\in P(v_{g,b})}q_i=q_{v,g b}$.
+Thus the condition defining $S_v$ depends only on the shift
+index $Q$, not on any Fourier index of $\Gamma$.  Grouping the
+full Fourier expansion by $Q$ proves its asserted action on
+each fixed fiber.  The same grouping proves the support and
+table-action identities: all full row coefficients with a
+given $Q$ satisfy the same identities and carry the same signs.
+\end{proof}
+
+Thus, for a nonzero common index $q$, the retained shift indices are
+specified by the subset $I\subseteq[s]$ of pairs in which $q$ occurs
+in the second child. Their characters are
+\[
+ \prod_{g\notin I}\chi_q(L_{g0})
+ \prod_{g\in I}\chi_q(L_{g1}).
+\]
+If $q=0$, there is only the all-zero unsplit index, with character $1$.
+Negating $F_v$ multiplies the corresponding coefficient by
+$(-1)^{q(\one)}$. On the part odd in $F_v$, $q$ is therefore a label
+and hence nonzero, so the description by $q$ and $I$ is unique.
+For such a label, translating the argument of $F_v$ by $z\in\F^s$
+multiplies its coefficient by $(-1)^{\sum_{g\in I}z_g}$.
+Thus the same subset $I$ records both the row character and the table
+translation character. This is the local Fourier description used
+in \Cref{sec:extraction}, valid at each fixed context.
diff --git a/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/unweighted.tex b/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/unweighted.tex
new file mode 100644
index 0000000000000000000000000000000000000000..0ebb36f5c68940903ad2fa4104b105362ca3b521
--- /dev/null
+++ b/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/build/sections/unweighted.tex
@@ -0,0 +1,90 @@
+\section{Conversion to simple unweighted graphs}\label{sec:unweighted}
+
+We include an explicit deterministic conversion so that rational edge
+weights and loops introduce no additional complexity assumption. The
+comparison uses the original, unnormalized weight scale.
+We replace each vertex by a cluster and approximate each weighted edge
+by a bipartite graph whose edge count on every rectangle approximates the
+weighted-density prediction with a uniform absolute error.
+An arbitrary cut may split each cluster. Its cluster proportions then
+describe a random cut of the original graph, so the rectangle estimates
+control all cuts of the enlarged graph, not just cuts constant on clusters.
+
+\begin{lemma}\label{lem:unweighted}
+Let $G$ be a graph on $N$ vertices with nonnegative rational weights
+$w_{uv}$ on unordered distinct vertex pairs, and
+$\sum_{u<v}w_{uv}\le1$. For every fixed $\epsilon'>0$, one can construct
+in deterministic polynomial time a simple unweighted graph $G'$ and an
+integer scale $Q$ such that
+\[
+ \left|\frac{\operatorname{MaxCut}(G')}{Q}
+             -\operatorname{MaxCut}(G)\right|\le\epsilon'.
+\]
+The same conclusion holds when the input graph has loops, after deleting
+them without renormalizing its other weights.
+\end{lemma}
+
+\begin{proof}
+Choose a prime $p>\max\{2,2N^2/\epsilon'\}$. A prime of polynomial size can be found deterministically
+by testing successive integers, using Bertrand's postulate to bound the
+search. Replace every vertex by a cluster indexed by $\mathbb F_p^6$.
+For each $u<v$ let $I_{uv}\subseteq\mathbb F_p$ be any fixed set of
+$\lfloor p w_{uv}\rfloor$ residues, for example the first such residues
+in the usual ordering. Join $x$ in cluster $u$ to $y$ in cluster $v$
+exactly when
+\[
+ x\cdot y\in I_{uv}.
+\]
+There are no edges within clusters. This defines a simple graph on
+$Np^6$ vertices, constructible by enumerating all possible pairs.
+Take $Q=p^{12}$.
+
+We first estimate its edge count on arbitrary rectangles between two
+clusters. Fix a residue set $I$ and subsets $S,T\subseteq\mathbb F_p^6$.
+Writing $\psi(a)=\exp(2\pi i a/p)$, Fourier inversion gives
+\[
+ \one_I(a)=\sum_{h\in\mathbb F_p}\beta_h\psi(ha),\qquad
+ \beta_h=\frac1p\sum_{a\in I}\psi(-ha),\qquad
+ |\beta_h|\le1.
+\]
+For $h\ne0$, the matrix $M_h(x,y)=\psi(hx\cdot y)$ satisfies
+\[
+ M_hM_h^*=p^6 I,
+\]
+by character orthogonality, so its operator norm is $p^3$. Consequently,
+\begin{align*}
+ \left|\#\{(x,y)\in S\times T:x\cdot y\in I\}
+                -\frac{|I|}{p}|S||T|\right|
+ &\le\sum_{h\ne0}|\beta_h|p^3\sqrt{|S||T|}\\
+ &\le p^{10}.
+\end{align*}
+For $I=I_{uv}$, division by $p^{12}$ and
+$\bigl||I|/p-w_{uv}\bigr|\le1/p$ show that
+\begin{equation}\label{eq:rectangle-discrepancy}
+ \left|\frac{e_{uv}(S,T)}{p^{12}}
+       -w_{uv}\frac{|S|}{p^6}\frac{|T|}{p^6}\right|
+ \le\frac1p+\frac1{p^2}.
+\end{equation}
+
+Now fix an arbitrary cut of $G'$, and let $a_u\in[0,1]$ be the
+fraction of cluster $u$ on its positive side. Its crossing edges between
+clusters $u,v$ form two rectangles. Equation~\eqref{eq:rectangle-discrepancy}
+therefore compares their scaled count with
+\[
+ w_{uv}\bigl(a_u(1-a_v)+(1-a_u)a_v\bigr)
+\]
+to error at most $2/p+2/p^2$.
+Summing over the at most $N(N-1)/2$ pairs costs at most
+$N(N-1)(1/p+1/p^2)$. The displayed weighted expression, summed over
+pairs, is the expected cut weight obtained by independently assigning
+vertex $u$ a positive sign with probability $a_u$. It is at most
+$\operatorname{MaxCut}(G)$. This proves the desired upper bound for
+every cut of $G'$.
+
+Conversely, give all vertices in each cluster the sign of a maximum cut
+of $G$. The same rectangle bound compares its scaled weight with
+$\operatorname{MaxCut}(G)$ to at most the same error, giving the reverse
+inequality. Our choice of $p$ ensures
+$N(N-1)(1/p+1/p^2)<\epsilon'$, which proves the assertion.
+Loops contribute zero to every cut and can be deleted at the outset.
+\end{proof}
diff --git a/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/paper.pdf b/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..b56f78d3aae8c7a71309ec43b61e5ee28a0092ef
--- /dev/null
+++ b/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:32a11e793f6e379c8d750f8f3560a6b0648bee56441efd25bceea9c2b7ffb849
+size 540064
diff --git a/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/README.md b/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..230e5c26a9f59a1ae5bc0c273e890b1226d881a9
--- /dev/null
+++ b/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/README.md
@@ -0,0 +1,18 @@
+# [A counterexample to Naimark's problem in ZFC](naimark-counterexample-zfc.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 24, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026,
+  author = {{OpenAI}},
+  title = {{A counterexample to Naimark's problem in ZFC}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/naimark-counterexample-zfc.pdf}{OAI:A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/build/main.tex b/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/build/main.tex
new file mode 100644
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+++ b/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/build/main.tex
@@ -0,0 +1,49 @@
+\documentclass[11pt]{article}
+\pdfinfoomitdate=1
+\pdftrailerid{}
+\pdfsuppressptexinfo=15
+\usepackage[expansion=false]{microtype}
+\usepackage[margin=1.05in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{enumitem,booktabs,array,needspace}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta,positioning,calc}
+\usepackage[hyphens]{url}
+\usepackage[hidelinks,hypertexnames=false]{hyperref}
+\usepackage[nameinlink,capitalize,noabbrev]{cleveref}
+\hypersetup{pdftitle={A counterexample to Naimark's problem in ZFC},pdfauthor={OpenAI}}
+\allowdisplaybreaks[1]
+\numberwithin{equation}{section}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\newtheorem{remark}[theorem]{Remark}
+\DeclareMathOperator{\Ad}{Ad}
+\DeclareMathOperator{\id}{id}
+\title{A counterexample to Naimark's problem in ZFC}
+\author{OpenAI}
+\date{September 24, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We construct in ZFC a unital infinite-dimensional simple complex
+$C^*$-algebra with a faithful tracial state whose nonzero irreducible
+representations are all unitarily equivalent. This gives a negative answer
+to Naimark's problem without additional set-theoretic assumptions.
+\end{abstract}
+\tableofcontents
+\clearpage
+\input{sections/introduction}
+\input{sections/states}
+\input{sections/hnn}
+\input{sections/bridge}
+\input{sections/iteration}
+\input{sections/construction}
+{\small
+\bibliographystyle{plain}
+\bibliography{references}
+}
+\end{document}
diff --git a/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/build/references.bib b/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/build/references.bib
new file mode 100644
index 0000000000000000000000000000000000000000..c53d6cba25ee769ddd7cf08d9ce2fecd54d0b229
--- /dev/null
+++ b/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/build/references.bib
@@ -0,0 +1,170 @@
+@article{AkemannWeaver2004,
+  author = {Akemann, Charles and Weaver, Nik},
+  title = {Consistency of a counterexample to {Naimark}'s problem},
+  journal = {Proceedings of the National Academy of Sciences of the United States of America},
+  volume = {101},
+  number = {20},
+  pages = {7522--7525},
+  year = {2004},
+  doi = {10.1073/pnas.0401489101},
+  note = {\href{https://doi.org/10.1073/pnas.0401489101}{doi:\nolinkurl{10.1073/pnas.0401489101}}},
+  url = {https://pmc.ncbi.nlm.nih.gov/articles/PMC419638/}
+}
+
+@article{CalderonFarah2022,
+  author = {Calder{\'o}n, Daniel and Farah, Ilijas},
+  title = {Can you take {Akemann--Weaver}'s {$\diamondsuit_{\aleph_1}$} away?},
+  journal = {Journal of Functional Analysis},
+  volume = {285},
+  number = {5},
+  pages = {110017},
+  year = {2023},
+  doi = {10.1016/j.jfa.2023.110017},
+  note = {\href{https://doi.org/10.1016/j.jfa.2023.110017}{doi:\nolinkurl{10.1016/j.jfa.2023.110017}}. Numbered locators refer to \href{https://arxiv.org/abs/2006.06886v3}{arXiv:2006.06886v3}},
+  url = {https://arxiv.org/abs/2006.06886v3}
+}
+
+@article{KishimotoOzawaSakai2003,
+  author = {Kishimoto, Akitaka and Ozawa, Narutaka and Sakai, Sh{\^o}ichir{\^o}},
+  title = {Homogeneity of the pure state space of a separable {$C^*$}-algebra},
+  journal = {Canadian Mathematical Bulletin},
+  volume = {46},
+  number = {3},
+  pages = {365--372},
+  year = {2003},
+  doi = {10.4153/CMB-2003-038-3},
+  note = {\href{https://doi.org/10.4153/CMB-2003-038-3}{doi:\nolinkurl{10.4153/CMB-2003-038-3}}},
+  url = {https://arxiv.org/abs/math/0110152}
+}
+
+@article{AkemannAndersonPedersen1986,
+  author = {Akemann, Charles A. and Anderson, Joel and Pedersen, Gert K.},
+  title = {Excising states of {$C^*$}-algebras},
+  journal = {Canadian Journal of Mathematics},
+  volume = {38},
+  number = {5},
+  pages = {1239--1260},
+  year = {1986},
+  note = {\href{https://doi.org/10.4153/CJM-1986-063-7}{doi:\nolinkurl{10.4153/CJM-1986-063-7}}},
+  doi = {10.4153/CJM-1986-063-7}
+}
+
+@article{Kadison1957,
+  author = {Kadison, Richard V.},
+  title = {Irreducible operator algebras},
+  journal = {Proceedings of the National Academy of Sciences of the United States of America},
+  volume = {43},
+  number = {3},
+  pages = {273--276},
+  year = {1957},
+  doi = {10.1073/pnas.43.3.273},
+  note = {\href{https://doi.org/10.1073/pnas.43.3.273}{doi:\nolinkurl{10.1073/pnas.43.3.273}}},
+  url = {https://pmc.ncbi.nlm.nih.gov/articles/PMC528430/}
+}
+
+@article{BlanchardDykema2001,
+  author = {Blanchard, {\'E}tienne F. and Dykema, Kenneth J.},
+  title = {Embeddings of reduced free products of operator algebras},
+  journal = {Pacific Journal of Mathematics},
+  volume = {199},
+  number = {1},
+  pages = {1--19},
+  year = {2001},
+  doi = {10.2140/pjm.2001.199.1},
+  note = {\href{https://doi.org/10.2140/pjm.2001.199.1}{doi:\nolinkurl{10.2140/pjm.2001.199.1}}},
+  url = {https://msp.org/pjm/2001/199-1/pjm-v199-n1-p01-s.pdf}
+}
+
+@article{Ueda2008,
+  author = {Ueda, Yoshimichi},
+  title = {Remarks on {HNN} extensions in operator algebras},
+  journal = {Illinois Journal of Mathematics},
+  volume = {52},
+  number = {3},
+  pages = {705--725},
+  year = {2008},
+  note = {\href{https://doi.org/10.1215/ijm/1254143997}{doi:\nolinkurl{10.1215/ijm/1254143997}}},
+  doi = {10.1215/ijm/1254143997}
+}
+
+@misc{UedaPreprint2008,
+  author = {Ueda, Yoshimichi},
+  title = {A relationship between {HNN} extensions and amalgamated free products in operator algebras},
+  year = {2008},
+  note = {Corrected earlier version, arXiv:math/0601706v4},
+  howpublished = {\url{https://arxiv.org/abs/math/0601706v4}},
+  url = {https://arxiv.org/abs/math/0601706v4}
+}
+
+@article{AkemannWassermannWeaver2010,
+  author = {Akemann, Charles and Wassermann, Simon and Weaver, Nik},
+  title = {Pure states on free group {$C^*$}-algebras},
+  journal = {Glasgow Mathematical Journal},
+  volume = {52},
+  number = {1},
+  pages = {151--154},
+  year = {2010},
+  doi = {10.1017/S0017089509990383},
+  note = {\href{https://doi.org/10.1017/S0017089509990383}{doi:\nolinkurl{10.1017/S0017089509990383}}},
+  url = {https://doi.org/10.1017/S0017089509990383}
+}
+
+@article{Vaccaro2018,
+  author = {Vaccaro, Andrea},
+  title = {Trace spaces of counterexamples to {Naimark}'s problem},
+  journal = {Journal of Functional Analysis},
+  volume = {275},
+  number = {10},
+  pages = {2794--2816},
+  year = {2018},
+  doi = {10.1016/j.jfa.2018.06.018},
+  note = {\href{https://doi.org/10.1016/j.jfa.2018.06.018}{doi:\nolinkurl{10.1016/j.jfa.2018.06.018}}. See the corrigendum and the corrected \href{https://arxiv.org/abs/1711.05845v4}{arXiv:1711.05845v4}},
+  url = {https://arxiv.org/abs/1711.05845v4}
+}
+
+@article{Vaccaro2020,
+  author = {Vaccaro, Andrea},
+  title = {Corrigendum to ``Trace spaces of counterexamples to {Naimark}'s problem''},
+  journal = {Journal of Functional Analysis},
+  volume = {278},
+  number = {12},
+  pages = {108458},
+  year = {2020},
+  note = {\href{https://doi.org/10.1016/j.jfa.2020.108458}{doi:\nolinkurl{10.1016/j.jfa.2020.108458}}},
+  doi = {10.1016/j.jfa.2020.108458}
+}
+
+@misc{Tanaka2026,
+  author = {Tanaka, Ryotaro},
+  title = {A separably representable counterexample to {Naimark}'s problem in {ZFC}},
+  year = {2026},
+  note = {arXiv:2609.26930v1, September 22},
+  howpublished = {\url{https://arxiv.org/abs/2609.26930v1}},
+  eprint = {2609.26930},
+  archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/2609.26930v1}
+}
+
+@article{FarahHirshberg2017,
+  author = {Farah, Ilijas and Hirshberg, Ilan},
+  title = {Simple nuclear {$C^*$}-algebras not isomorphic to their opposites},
+  journal = {Proceedings of the National Academy of Sciences of the United States of America},
+  volume = {114},
+  number = {24},
+  pages = {6244--6249},
+  year = {2017},
+  doi = {10.1073/pnas.1619936114},
+  note = {\href{https://doi.org/10.1073/pnas.1619936114}{doi:\nolinkurl{10.1073/pnas.1619936114}}. Earlier version: \emph{A simple AF algebra not isomorphic to its opposite}, \href{https://arxiv.org/abs/1612.01170v2}{arXiv:1612.01170v2}},
+  url = {https://arxiv.org/abs/1612.01170v2}
+}
+
+@article{NaimpallyPeters2013,
+  author = {Naimpally, Som A. and Peters, James F.},
+  title = {Preservation of continuity},
+  journal = {Scientiae Mathematicae Japonicae},
+  volume = {76},
+  number = {2},
+  pages = {305--311},
+  year = {2013},
+  url = {https://www.jams.jp/scm/contents/e-2013-3/2013-26.pdf}
+}
diff --git a/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/build/sections/bridge.tex b/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/build/sections/bridge.tex
new file mode 100644
index 0000000000000000000000000000000000000000..7cee04dcde1ae1e81561a9925ac78b6a1903d574
--- /dev/null
+++ b/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/build/sections/bridge.tex
@@ -0,0 +1,357 @@
+\section{Joining two classes while preserving unique extensions}
+\label{sec:bridge}
+
+We now choose the abelian subalgebras in the reduced HNN construction.
+The resulting inclusion has two features needed in the iteration: every
+old pure state extends uniquely, and precisely the prescribed two old
+representation classes merge. The proof must control arbitrary old
+pure states, even though the defining data consist of only two peaking
+sequences.
+
+\begin{theorem}[The bridge]\label{na:bridge}
+Let $A$ be a unital $C^*$-algebra with a faithful tracial state $\tau$
+and no finite-dimensional irreducible representations. For $i=1,2$,
+let $(p_n^i)_{n\geq0}$ be a decreasing sequence of projections, with
+$p_0^i=1$, peaking to a pure state $f_i$. Suppose that $f_1\not\sim f_2$
+and
+\[
+ \tau(p_n^1)=\tau(p_n^2)=t_n,
+ \qquad 1=t_0>t_1>\cdots>0,
+ \qquad t_n\longrightarrow0.
+\]
+Put $D_i=C^*(p_n^i:n\geq0)$ and let
+$\theta:D_1\to D_2$ be determined by $\theta(p_n^1)=p_n^2$.
+Form the reduced HNN extension $H(A)$ of Proposition~\ref{na:hnn}
+using these data and their trace-preserving conditional expectations.
+Then every pure state $h$ of $A$ has a unique state extension
+$\widetilde h$ to $H(A)$. These extensions are pure, and for all pure
+states $h,k$ of $A$,
+\[
+ \widetilde h\sim\widetilde k
+ \quad\Longleftrightarrow\quad
+ h\sim k
+ \quad\text{or}\quad
+ \{[h],[k]\}=\{[f_1],[f_2]\}.
+\]
+The extension has a faithful tracial state extending $\tau$ and a
+faithful trace-preserving conditional expectation onto $A$. If $A$ is
+simple, then $H(A)$ is simple.
+\end{theorem}
+
+\begin{proof}
+Lemma~\ref{na:peaking-expectation} supplies the expectations onto $D_i$.
+The differences $p_n^i-p_{n+1}^i$ are nonzero, mutually orthogonal
+projections, so each $D_i$ is the algebra of continuous functions on
+$\mathbb N_0\cup\{\infty\}$. Matching these differences defines
+$\theta$, and the equality of trace values makes it trace preserving.
+Thus the reduced HNN construction applies. Its trace and expectation
+already give the corresponding assertions in the theorem. The work
+is to prove the assertions about pure states.
+
+\medskip\noindent
+\emph{Pass to the two-by-two construction.}
+Use the notation of Lemma~\ref{na:free-product} and
+Proposition~\ref{na:hnn}:
+\[
+ F=M_2(A),\qquad G=M_2(D_1),\qquad D=D_1\oplus D_2,
+ \qquad Z=F*_D G.
+\]
+Write $E_D^F:F\to D$ for the expectation, $e_i=e_{ii}$ for the
+common diagonal projections, and $s=e_{21}^G$. Recall that
+$H(A)=e_1Ze_1$, that $Z=M_2(H(A))$ using the matrix units of $F$,
+and that $v=e_{12}^F s$ satisfies $vp_n^1v^*=p_n^2$ in the corner.
+Put
+\[
+ P_n^i=p_n^i e_i\in F,\qquad \phi_i(a)=f_i(a_{ii})\quad(a\in F).
+\]
+Here $p_n^i e_i$ denotes the matrix with entry $p_n^i$ in position
+$(i,i)$ and zero elsewhere. The projections $P_n^i$ peak to $\phi_i$:
+their zeroth term is $e_i$, so a state taking value one on them is
+supported on that corner and restricts there to $f_i$. The states
+$\phi_1,\phi_2$ are pure and inequivalent, by matrix amplification of
+the GNS representations of $f_1,f_2$.
+Also $F$ has no finite-dimensional irreducible representations. Indeed,
+any such representation would restrict along $a\mapsto\operatorname{diag}(a,a)$
+to a nonzero finite-dimensional representation of $A$, whose image
+would supply a finite-dimensional irreducible representation of $A$.
+
+We shall prove unique extension and the exclusion of unintended joins
+for $F\subset Z$, and then return to the corner. By
+Lemma~\ref{na:unitary-states}, it suffices to use $\phi_i$ as the
+representative of its distinguished equivalence class. More explicitly,
+conjugating by a unitary of $F$ bijects the sets of state extensions.
+We first establish uniqueness for these representatives and for every
+pure state outside their two classes; only afterward will we use the
+resulting unique lifts.
+
+Consider pure states $h,k$ of $F$ satisfying one of the two tests
+\begin{equation}\label{eq:bridge-tests}
+ h=k,
+ \qquad\text{or}\qquad
+ h\not\sim k\ \text{ and }\quad
+ \{[h],[k]\}\ne\{[\phi_1],[\phi_2]\}.
+\end{equation}
+Whenever either state belongs to $[\phi_i]$, take it to be $\phi_i$
+itself. Choose decreasing positive contractive excision nets
+$x=x_\lambda$ for $h$ and $y=y_\nu$ for $k$ as in
+Lemma~\ref{na:excision}; at $\phi_i$, choose the sequence $P_n^i$
+by Lemma~\ref{na:projection-excision}.
+All limits involving both $x$ and $y$ below are on the product of the
+two index sets.
+
+We will show that $x_\lambda W y_\nu\to0$ in norm for every
+nonempty word $W$ in \eqref{na:matrix-word-span}.
+When $h=k$, this will force all state extensions to agree; for the
+inequivalent pairs in \eqref{eq:bridge-tests}, it will keep the lifted
+representations inequivalent. The stable unitary will then join the
+designated pair.
+
+\medskip\noindent
+\emph{A norm estimate for reduced words.}
+We claim that
+\begin{equation}\label{eq:bridge-word-decay}
+ \|x a_0b_1a_1\cdots b_m a_m y\|\longrightarrow0
+\end{equation}
+for every $m\geq1$, every $b_j\in\{s,s^*\}$, and every $a_j\in F$
+with
+\begin{equation}\label{eq:bridge-centering}
+ E_D^F(a_j)=0\qquad(0<j<m).
+\end{equation}
+By the spanning description \eqref{na:matrix-word-span}, these words,
+together with $F$, span a dense subspace of $Z$.
+
+Let $P_G$ be the orthogonal projection of the reduced free-product
+Hilbert $D$-module $K$ onto the summands starting with a factor in
+$L_G^\circ$. For $d\in F$, the decomposition
+$K=L_F\otimes_D K^{(F)}$ gives
+\begin{equation}\label{eq:bridge-module-bound}
+ \|dP_G\|\leq\|E_D^F(d^*d)\|^{1/2}.
+\end{equation}
+Indeed, a vector $\eta\in P_GK$ has the form $1\otimes\eta$
+in this decomposition, and
+\[
+ \|d\otimes\eta\|^2
+ =\|\langle\eta,E_D^F(d^*d)\eta\rangle\|
+ \leq\|E_D^F(d^*d)\|\,\|\eta\|^2.
+\]
+The expectation norm here is the norm in $D$.
+
+For $b\in\{s,s^*\}$ write $b=e_{\ell(b)}be_{r(b)}$.
+Thus $(\ell(s),r(s))=(2,1)$ and
+$(\ell(s^*),r(s^*))=(1,2)$. For a word in
+\eqref{eq:bridge-word-decay}, put
+\[
+ \ell=\ell(b_1),\qquad r=r(b_m),\qquad
+ w=b_1a_1\cdots a_{m-1}b_m.
+\]
+Each letter of this centered alternating core creates a new leg when
+applied, from right to left, to a word not starting with $G$. Therefore
+\[
+ (1-P_G)w(1-P_G)=0,
+ \qquad w=P_Gw+(1-P_G)wP_G.
+\]
+Set $d=xa_0e_\ell$ and $d'=e_ra_my$. We obtain
+\begin{equation}\label{eq:bridge-two-ends}
+ \|dwd'\|
+ \leq \|dP_G\|\,\|w\|\,\|d'\|
+       +\|d\|\,\|w\|\,\|P_Gd'\|.
+\end{equation}
+Using \eqref{eq:bridge-module-bound} and its adjoint version, it follows
+that \eqref{eq:bridge-word-decay} holds whenever both
+\begin{equation}\label{eq:bridge-endpoints}
+ \begin{split}
+ \|E_D^F(e_\ell a_0^*x^2a_0e_\ell)\|&\longrightarrow0,\\
+ \|E_D^F(e_r a_my^2a_m^*e_r)\|&\longrightarrow0.
+ \end{split}
+\end{equation}
+This sufficient criterion does not require the pair of endpoint
+states to satisfy \eqref{eq:bridge-tests}.
+
+\medskip\noindent
+\emph{When an endpoint decays.}
+In the first line of \eqref{eq:bridge-endpoints}, increase $x^2$ to $x$.
+The resulting elements are a decreasing net of positive continuous
+functions on the $\ell$th convergent sequence in the spectrum of $D$;
+they vanish on the other component. At a finite atom
+$Q=P_n^\ell-P_{n+1}^\ell$, their pointwise limit is
+\[
+ \frac{\tau_F(Qa_0^*r_ha_0Q)}{\tau_F(Q)}=0.
+\]
+Here $r_h$ is the support projection in $F^{**}$ and $\tau_F$ is the
+normalized matrix trace, extended normally to $F^{**}$. The numerator
+vanishes by traciality and Lemma~\ref{na:trace-null}; for example
+\[
+ 0\leq\tau_F(r_h a_0Qa_0^*)
+ \leq\|a_0Qa_0^*\|\tau_F(r_h)=0.
+\]
+At the point $\infty$, the limit is
+$\phi_\ell(a_0^*r_ha_0)$. It is zero if $h\not\sim\phi_\ell$,
+because the normal extension of the $\phi_\ell$ representation
+annihilates $r_h$, as in Lemma~\ref{na:cross-excision}.
+If $h=\phi_\ell$, the support identity instead gives
+\[
+ \phi_\ell(a_0^*r_ha_0)=|\phi_\ell(a_0)|^2.
+\]
+Thus the first endpoint decays unless $h=\phi_\ell$, and it still
+decays in that case when $\phi_\ell(a_0)=0$.
+
+The uniform convergence used here is Dini's theorem for monotone nets
+\cite[Theorem~1.5, p.~306]{NaimpallyPeters2013}. We recall its compactness
+argument. For a fixed
+$\varepsilon>0$, the open sets where a chosen function is below
+$\varepsilon$ cover the compact spectrum. Choose a finite subcover,
+and then an index dominating its finitely many indices. Monotonicity
+makes every later function uniformly smaller than $\varepsilon$.
+The second line of \eqref{eq:bridge-endpoints} is symmetric: it
+decays unless $k=\phi_r$, and also decays in that case when
+$\phi_r(a_m)=0$.
+
+\medskip\noindent
+\emph{Remove the matched endpoints.}
+Call the left endpoint matched when $h=\phi_\ell$, and the right
+endpoint matched when $k=\phi_r$. At each matched endpoint write
+\[
+ a_0=\phi_\ell(a_0)1+a_0^0,
+ \qquad\text{or}\qquad
+ a_m=\phi_r(a_m)1+a_m^0.
+\]
+Expand the word using these decompositions. In each resulting term,
+an unmatched endpoint already satisfies its estimate in
+\eqref{eq:bridge-endpoints}, and a matched endpoint with its
+zero-value coefficient does as well. We treat the scalar terms by
+moving the corresponding peak through its adjacent letter:
+\begin{equation}\label{eq:bridge-transfer}
+ P_n^{\ell(b)}b=bP_n^{r(b)},
+ \qquad bP_n^{r(b)}=P_n^{\ell(b)}b.
+\end{equation}
+These identities hold in $G$: the diagonal embedding of $D_2$ uses
+$\theta^{-1}$, so the two projections have the same matrix entry
+$p_n^1$ there.
+
+In the following displays, an untransferred endpoint coefficient
+means the coefficient chosen in the expansion. For $m\ge2$, after
+suppressing its scalar coefficient, a left scalar term becomes
+\[
+ P_n^{\ell(b_1)}b_1a_1\cdots b_ma_my
+ =b_1\bigl(P_n^{r(b_1)}a_1b_2\cdots b_ma_my\bigr).
+\]
+The peak now belongs to the other distinguished state, and $b_1$
+is a bounded outside factor. A right scalar choice similarly moves
+$P_t^{r(b_m)}$ to $P_t^{\ell(b_m)}$ and leaves $b_m$ as a bounded
+factor on the right. When both ends are transferred and $m\ge3$,
+the resulting expression,
+again up to its scalar coefficient, is
+\[
+ b_1\bigl(P_n^{r(b_1)}a_1b_2\cdots
+ b_{m-1}a_{m-1}P_t^{\ell(b_m)}\bigr)b_m.
+\]
+Suppose at least one core letter remains after the transfers. Each
+newly exposed endpoint coefficient is one of the original internal
+$a_j$. By \eqref{eq:bridge-centering},
+$\phi_1(a_j)=\phi_2(a_j)=0$: these states evaluate the two components
+of $E_D^F(a_j)$ at $\infty$. Thus every new endpoint satisfies its
+estimate, whether or not its transferred state matches the adjacent
+letter. Endpoints that were not transferred satisfy their estimates
+by the preceding expansion. Applying \eqref{eq:bridge-endpoints}
+to the remaining word therefore proves decay of each such term.
+This covers no transfers, one transfer when $m\ge2$, and two
+transfers when $m\ge3$, including a remaining core of one letter.
+The sufficient endpoint criterion requires no assumption that the
+transferred pair of states satisfies \eqref{eq:bridge-tests}.
+
+It remains to check the cases in which the core disappears.
+If $m=1$, both ends cannot match: the range and source of $s$ or
+$s^*$ are different, so double matching would be the designated pair
+excluded in \eqref{eq:bridge-tests}. For a single left transfer,
+say $h=\phi_i$, the remaining compression, up to a scalar and a
+bounded outside letter, is
+\[
+ P_n^j a_1y_\nu,\qquad j\ne i.
+\]
+The state $k$ is inequivalent to $\phi_j$. In the equal-state test it
+equals $\phi_i$; in the other test, equivalence to $\phi_j$ would
+again be the excluded designated pair. Hence this expression tends
+to zero by Lemma~\ref{na:cross-excision}. A single right transfer
+is symmetric.
+
+Finally let $m=2$ and transfer at both ends. Both original endpoints
+matched, so \eqref{eq:bridge-tests} forces $h=k=\phi_i$ for some
+$i$. Since each letter switches the two matrix indices,
+\[
+ \ell(b_1)=r(b_2)=i,
+ \qquad r(b_1)=\ell(b_2)=j\ne i.
+\]
+The remaining compression is $P_n^j a_1P_t^j$, with
+$\phi_j(a_1)=0$. If $q=\min(n,t)$, nestedness and excision give
+\[
+ \|P_n^j a_1P_t^j\|
+ \leq\|P_q^j a_1P_q^j\|\longrightarrow0.
+\]
+This is convergence on the product of the two sequence indices.
+Together with cross-excision, whose conclusion also holds on the
+product net, these cases prove \eqref{eq:bridge-word-decay} in full.
+
+\medskip\noindent
+\emph{Unique extensions and the exclusion of other joins.}
+In the equal-state test, let $\psi$ be any state of $Z$ extending $h$.
+Its GNS vector is fixed by every $x_\lambda$ and $y_\nu$: a positive
+contraction $x$ of state value one satisfies
+$\|(1-\pi_\psi(x))\xi_\psi\|^2\leq\psi(1-x)=0$.
+Consequently, for every tested word $W$,
+\[
+ \psi(W)=\psi(x_\lambda Wy_\nu)\longrightarrow0.
+\]
+Its values on $F$ are already prescribed. The dense span above
+therefore proves uniqueness; existence also follows directly by
+composing $h$ with the expectation onto $F$. Conjugacy by unitaries
+in $F$ now establishes uniqueness for every old pure state, including
+all states in the two distinguished classes. Their unique extensions
+are pure by Lemma~\ref{na:unique-lift}.
+
+Next take an inequivalent pair allowed in
+\eqref{eq:bridge-tests}. If its unique lifts were equivalent, realize
+them by unit vectors $\xi_h,\xi_k$ in one irreducible representation
+$\pi$ of $Z$. The represented excision elements fix the corresponding
+vectors, so
+\[
+ \langle\pi(z)\xi_k,\xi_h\rangle
+ =\langle\pi(x_\lambda zy_\nu)\xi_k,\xi_h\rangle.
+\]
+For $z\in F$, the right side tends to zero by cross-excision; for a
+tested nonempty word it tends to zero by
+\eqref{eq:bridge-word-decay}. Thus the matrix coefficient vanishes on
+all of $Z$. This contradicts cyclicity of $\xi_k$ in the irreducible
+representation. Unitary conjugacy within the old classes extends this
+exclusion to all representatives. Equivalent old states, on the other
+hand, have equivalent unique lifts by
+Lemmas~\ref{na:unitary-states} and~\ref{na:unique-lift}.
+
+\medskip\noindent
+\emph{Return to the corner and join the designated pair.}
+States of $A=e_1Fe_1$ identify with states of $F$ taking value one on
+$e_1$, by composition with compression. The same correspondence holds
+between $H(A)=e_1Ze_1$ and $Z$, and it commutes with restriction.
+Every extension to $Z$ of an $e_1$-supported state of $F$ is still
+$e_1$-supported. Hence this correspondence bijects extension sets and
+transfers unique extension to $A\subset H(A)$.
+
+Under $F=M_2(A)$ and $Z=M_2(H(A))$, these supported states have the
+matrix amplifications of their corner GNS representations, with the
+representing vector in the first summand. Purity and equivalence of
+classes are therefore preserved. The matrix-flip unitary of $F$
+moves the representative $\phi_2$ from the second summand to the
+first, so its class corresponds to $[f_2]$, just as the class of
+$\phi_1$ corresponds to $[f_1]$. The exclusion of all other joins
+thus descends to the corner.
+
+To see that the designated join actually occurs, use the stable
+unitary relation. The state $\widetilde f_2\circ\Ad v$ takes value
+one on every $p_n^1$, so its restriction to $A$ is $f_1$. Unique
+extension, now proved, gives
+\[
+ \widetilde f_1=\widetilde f_2\circ\Ad v.
+\]
+This proves the asserted equivalence criterion. If $A$ is simple,
+Lemma~\ref{na:simple-extension} applies to the faithful expectation
+and the unique-extension property, proving that $H(A)$ is simple.
+\end{proof}
diff --git a/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/build/sections/construction.tex b/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/build/sections/construction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..efb3c6cd5adc3fbe645a82f1003dcff979eda56e
--- /dev/null
+++ b/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/build/sections/construction.tex
@@ -0,0 +1,139 @@
+\section{The construction in ZFC}
+\label{sec:construction}
+
+We now choose the bridge steps so that every pure state eventually
+joins the class of the state peaked by $(p_n)$. Countable
+determination ensures that every pure state of the final algebra
+will have appeared early enough to be treated.
+
+\begin{theorem}[Kishimoto--Ozawa--Sakai]
+\label{na:homogeneity}
+Let $B$ be a separable $C^*$-algebra. If pure states $h,k$ of $B$
+have GNS representations with the same kernel, there is an
+automorphism $\gamma$ of $B$ such that $h\circ\gamma=k$.
+In particular, this holds for every pair of pure states of a
+separable simple $C^*$-algebra.
+\end{theorem}
+
+This is \cite[Theorem~1.1]{KishimotoOzawaSakai2003}, which in fact
+provides an asymptotically inner automorphism. We apply it only to
+the separable algebras $A[S]$ of \cref{na:inherited}.
+
+\begin{proof}[Proof of \cref{na:main}]
+Set $\kappa=\mathfrak c^+$. Every ordinal below $\kappa$ has
+cardinality at most $\mathfrak c$, and every countable subset of
+$\kappa$ is bounded: otherwise $\kappa$ would be a countable union
+of sets of cardinality at most $\mathfrak c$, a contradiction.
+
+\smallskip\noindent
+\emph{Fix the schedule first.}
+Before constructing any algebras, choose an injection
+\begin{equation}\label{eq:schedule}
+ \sigma:\kappa\times\kappa\longrightarrow\kappa,
+ \qquad \sigma(\beta,\eta)\ge\beta.
+\end{equation}
+To do so, enumerate $\kappa\times\kappa$ in order type $\kappa$.
+When a pair $(\beta,\eta)$ is encountered, choose an unused ordinal
+at least $\beta$. Fewer than $\kappa$ ordinals have been used,
+whereas the tail above $\beta$ has cardinality $\kappa$.
+This defines the entire schedule without referring to any states.
+
+\smallskip\noindent
+\emph{Construct the algebras in ordinal order.}
+Build a continuous bridge iteration $(X_\beta)_{\beta\le\kappa}$
+starting from the algebra of \cref{na:base-peak}. For each
+$\beta<\kappa$, once $X_\beta$ has been constructed, choose a list
+\[
+ (h_{\beta,\eta})_{\eta<\kappa}
+\]
+containing every pure state of $X_\beta$, with repetitions allowed.
+This is possible by \cref{na:state-count}, since
+$|\beta|\le\mathfrak c$ for $\beta<\kappa$.
+
+At stage $j<\kappa$, let $f$ be the state of $X_j$ peaked by
+$(p_n)$. If $j$ is outside the range of $\sigma$, make the identity
+step. Otherwise write $j=\sigma(\beta,\eta)$ and let $g$ be the
+unique lift of $h_{\beta,\eta}$ to $X_j$. The list for $\beta$ is
+already available because $\beta\le j$; when $\beta=j$, it is
+chosen before the successor step. If $g\sim f$, again make the
+identity step.
+
+Suppose $g\not\sim f$. By \cref{na:countable-determination}, choose
+a countable dependency-closed $S\subset j$ such that $g|_{A[S]}$
+is pure and uniquely determines $g$ on $X_j$. The restriction
+$f|_{A[S]}$ is pure by \cref{na:peaking-restriction}. The algebra
+$A[S]$ is separable, simple and unital by \cref{na:inherited}.
+Applying \cref{na:homogeneity}, choose an automorphism $\gamma$
+of $A[S]$ with
+\[
+ (g|_{A[S]})\circ\gamma=f|_{A[S]},
+ \qquad q_n^j=\gamma(p_n)\quad(n\ge0).
+\]
+If a state on $A[S]$ takes value one on every $q_n^j$, composing
+it with $\gamma$ gives the unique state taking value one on every
+$p_n$. Thus $(q_n^j)$ peaks to $g|_{A[S]}$. By
+\cref{na:closed-extension}, it peaks to $g$ on $X_j$ as well.
+
+The trace requirement deserves a separate check. For each $n$,
+$\gamma$ carries the unital matrix stage $M_{2^n}\subset A_0$
+onto a unital matrix subalgebra of $A[S]$, and carries its minimal
+projection $p_n$ to $q_n^j$. The ambient trace restricts to the
+unique normalized trace on this matrix algebra. Consequently,
+\[
+ \tau(q_n^j)=2^{-n}=\tau(p_n).
+\]
+This argument does not require an automorphism to preserve the
+specified trace on all of $A[S]$.
+
+We can therefore perform the bridge of \cref{na:bridge} with
+$(p_n),(q_n^j)$, and set $I_j=S$. At an identity step set
+$I_j=\varnothing$. At limit stages take the norm completion of
+the union. Each successor satisfies \textup{(U)} and has a faithful
+trace-preserving expectation onto its base. The results of
+\cref{sec:iteration} therefore apply throughout the recursion.
+
+All choices are set-sized. At every stage the algebra is generated
+by at most $\kappa$ elements, so its density is at most $\kappa$
+and its underlying cardinality at most $\lambda=\kappa^{\aleph_0}$.
+Fix a set of cardinality $\lambda$. Algebra structures on its
+subsets, together with their norms, traces, embeddings, states,
+automorphisms and lists, form a set of codes. Well-order that set
+to make the required choices; recode successor algebras with their
+embeddings and use norm inductive limits. Ordinary choice and
+transfinite recursion in ZFC suffice.
+
+\smallskip\noindent
+\emph{Every final pure state is treated.}
+Let $A=X_\kappa$ and let $h$ be any pure state of $A$.
+By \cref{na:countable-determination}, there is a countable
+dependency-closed $S\subset\kappa$ such that $h|_{A[S]}$ is pure
+and has unique extension $h$ to $A$. Boundedness of $S$ gives
+$\beta<\kappa$ with $A[S]\subset X_\beta$.
+The state $h|_{X_\beta}$ is the unique extension of $h|_{A[S]}$
+to $X_\beta$: any other extension could itself be extended to
+$A$, contradicting uniqueness there. Hence $h|_{X_\beta}$ is pure.
+
+Choose $\eta$ with $h_{\beta,\eta}=h|_{X_\beta}$ and put
+$j=\sigma(\beta,\eta)$. Its lift to $X_j$ is $h|_{X_j}$.
+At stage $j$ this lift either already belongs to the $p_n$-class
+or is joined to it by the bridge. By \cref{na:unique-chain}, the
+equivalence persists to $A$. Thus all pure states of $A$ have
+unitarily equivalent GNS representations.
+
+Every nonzero irreducible representation $\pi$ of the unital
+algebra $A$ is unital: the range of $\pi(1)$ is a nonzero closed
+invariant subspace. Every nonzero vector is cyclic by irreducibility,
+so each unit vector defines a pure state whose GNS representation
+is equivalent to $\pi$. The $p_n$-state supplies one such
+representation. Therefore $A$ has exactly one nonzero irreducible
+representation up to unitary equivalence.
+
+Finally, $A$ is simple and has a faithful tracial state by
+\cref{na:inherited} and the limit permanence facts. It is unital
+and contains the infinite-dimensional algebra $A_0$. An algebra
+$K(H)$ is unital only when $H$ is finite-dimensional:
+the identity on an infinite-dimensional Hilbert space is not
+compact, as an infinite orthonormal sequence has no norm-convergent
+subsequence. The finite-dimensional case, including $H=0$, cannot
+be isomorphic to $A$. This proves the theorem.
+\end{proof}
diff --git a/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/build/sections/hnn.tex b/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/build/sections/hnn.tex
new file mode 100644
index 0000000000000000000000000000000000000000..99a30bc8f4641e7bf48af7ce2f8a622af886b6bf
--- /dev/null
+++ b/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/build/sections/hnn.tex
@@ -0,0 +1,376 @@
+\section{Reduced HNN adjunctions}\label{sec:hnn}
+
+We now construct the extension that will join two pure-state classes.
+The construction itself has a more general input: two trace-preserving
+isomorphic subalgebras of a faithfully tracial algebra.  It adjoins a
+unitary implementing the isomorphism, while prescribing the expectation
+of every word in that unitary.  These prescriptions will also let us
+restrict the construction to smaller bases and interchange two
+adjunctions with data in the same base.
+
+The matrix-corner construction below is the reduced HNN construction of
+Ueda; see \cite[Proposition~2.2 and Remark~2.3]{UedaPreprint2008} and
+\cite{Ueda2008}.  We give the details, including the faithful
+representations and expectations needed later.  The corresponding
+expectation and embedding results for reduced amalgamated free products
+appear in \cite[Lemma~1.1 and Theorem~1.3]{BlanchardDykema2001}.
+
+\subsection{The reduced adjunction}
+
+Let $(A,\tau)$ be a unital $C^*$-algebra with faithful tracial state.
+Fix unital subalgebras $D_1,D_2\subset A$, trace-preserving conditional
+expectations $E_i:A\to D_i$, and a trace-preserving isomorphism
+$\theta:D_1\to D_2$.  A word in $A$ and a unitary $v$ has the form
+\begin{equation}\label{na:hnn-word}
+ a_0v^{\epsilon_1}a_1\cdots v^{\epsilon_m}a_m,
+ \qquad a_t\in A,\quad\epsilon_t\in\{+1,-1\}.
+\end{equation}
+For $m\geq1$, call it \emph{reduced} if every internal coefficient
+satisfies
+\begin{equation}\label{na:hnn-turns}
+ \begin{cases}
+ E_1(a_t)=0,&(\epsilon_t,\epsilon_{t+1})=(+1,-1),\\
+ E_2(a_t)=0,&(\epsilon_t,\epsilon_{t+1})=(-1,+1).
+ \end{cases}
+\end{equation}
+There is no condition at equal consecutive signs.
+
+\begin{proposition}\label{na:hnn}
+Let $(A,\tau)$ be a unital $C^*$-algebra with faithful tracial state,
+let $D_1,D_2\subset A$ be unital subalgebras with trace-preserving
+conditional expectations $E_i:A\to D_i$, and let
+$\theta:D_1\to D_2$ be a trace-preserving isomorphism.  There is a
+unital extension with a faithful tracial state extending $\tau$,
+\[
+ H(A)=C^*(A,v),\qquad vdv^*=\theta(d)\quad(d\in D_1),
+\]
+where $v$ is called the \emph{stable unitary}, and a trace-preserving
+conditional expectation $E_A:H(A)\to A$
+annihilating every nonempty reduced word
+\eqref{na:hnn-word}.  Among faithfully tracial extensions generated by
+$A$ and such a unitary, the relation and vanishing of the trace on
+nonempty reduced words determine $H(A)$ uniquely, by an isomorphism
+fixing $A$ and $v$.
+\end{proposition}
+
+We construct this adjunction as a matrix corner of an amalgamated free product.
+The proof below records the module and its two compression expectations;
+these will also be used in the pure-state bridge of Section~\ref{sec:bridge}.
+
+\subsection{Construction from an amalgamated free product}
+
+An element is \emph{centered} for a conditional expectation $E$ if it
+belongs to $\ker E$.
+
+\begin{lemma}\label{na:free-product}
+Let $F$ and $G$ be unital $C^*$-algebras with faithful tracial states,
+containing a common unital subalgebra $D$ on which the traces agree.
+Suppose that $E_D^F:F\to D$ and $E_D^G:G\to D$ are trace-preserving
+conditional expectations.  There is a unital $C^*$-algebra $Z$ generated
+by faithful copies of $F$ and $G$ agreeing on $D$, with a faithful
+tracial state extending both traces and faithful trace-preserving conditional
+expectations
+\[
+ E_D:Z\longrightarrow D,
+ \qquad E_F:Z\longrightarrow F,
+ \qquad E_D=E_D^F\circ E_F.
+\]
+The map $E_D$ vanishes on every nonempty alternating product of elements
+of $\ker E_D^F$ and $\ker E_D^G$.  The map $E_F$ vanishes on every product
+\begin{equation}\label{na:free-core}
+ a_0 b_1 a_1\cdots b_m a_m,
+ \qquad m\geq1,
+\end{equation}
+where $a_0,a_m\in F$, $b_t\in\ker E_D^G$, and
+$a_t\in\ker E_D^F$ for $0<t<m$.  The linear span of $F$ and these
+products is dense in $Z$.
+\end{lemma}
+
+\begin{proof}
+For $i\in\{F,G\}$, complete $i$ as a right Hilbert $D$-module $L_i$
+with inner product
+\[
+ \langle x,y\rangle_D=E_D^i(x^*y).
+\]
+Left multiplication by $a\in i$ is bounded, since
+\[
+ E_D^i(x^*a^*ax)\leq\|a\|^2E_D^i(x^*x),
+\]
+and its adjoint is left multiplication by $a^*$.  The expectation gives
+an orthogonal decomposition
+\[
+ L_i=D\oplus L_i^\circ,
+ \qquad L_i^\circ=\overline{\ker E_D^i}.
+\]
+Both summands are invariant under left and right multiplication by $D$.
+Form the Hilbert $D$-module
+\begin{equation}\label{na:free-module}
+ K=D\oplus
+ \bigoplus_{\substack{m\geq1,\ i_t\in\{F,G\}\\i_t\ne i_{t+1}}}
+ L_{i_1}^\circ\otimes_D\cdots\otimes_D L_{i_m}^\circ.
+\end{equation}
+Here $\otimes_D$ is the interior tensor product of Hilbert modules; the
+left $D$-action on a word acts on its first factor.  The first summand
+$D$ is called the vacuum summand, with distinguished vector $1$.
+
+Let $K^{(i)}$ be the vacuum summand together with all words not starting
+with $i$.  The decomposition of $L_i$ identifies
+\[
+ K=L_i\otimes_D K^{(i)}.
+\]
+Represent $i$ by left multiplication on the first tensor factor.  The
+two representations agree on $D$.  They are faithful: an element $a$
+annihilating the vacuum satisfies $E_D^i(a^*a)=0$, so its faithful
+factor trace gives $a=0$.  Let $Z$ be the $C^*$-algebra they generate in
+the adjointable operators on $K$.
+
+A centered element of $i$ acting on $K^{(i)}$ adds its own tensor
+factor at the front.  Thus a nonempty alternating centered product has
+zero vacuum compression.  Centering arbitrary factor elements shows
+that $D$ and these products span a dense subspace of $Z$.  Vacuum
+compression consequently takes values in the left copy of $D$ and
+defines the conditional expectation $E_D$.
+
+Write $\tau_D$ for the common trace on $D$.  We next prove that
+$\tau_Z=\tau_D\circ E_D$ is tracial and faithful.  For traciality,
+compare $\tau_Z(aw)$ and $\tau_Z(wa)$ when $a$ belongs to one factor
+and $w$ is an alternating centered word.  If $w$ has length at least
+two, both values vanish.  Indeed, multiplication at a same-factor end,
+followed by centering that end product, leaves either an alternating
+centered word or a $D$-coefficient times a nonempty such word.  At a
+different-factor end, first center $a$ and use $D$-bimodularity.  For a
+one-letter word, use the factor trace when the factors agree; when they
+differ, both values are zero.  For $w\in D$, use the factor trace.
+Density and repeated cyclic movement of a factor now prove traciality
+on all of $Z$.
+
+Suppose $\tau_Z(z^*z)=0$.  For every $t\in Z$, traciality gives
+\[
+ 0\leq\tau_Z(t^*z^*zt)
+ =\tau_Z(ztt^*z^*)
+ \leq\|t\|^2\tau_Z(zz^*)=0.
+\]
+The trace on $D$ is faithful, so $E_D((zt)^*(zt))=0$.  This is the
+$D$-valued inner product of $zt1$ with itself, and hence $zt1=0$.  Products of
+centered letters create all elementary tensors in
+\eqref{na:free-module}; thus $Z1$ is dense in $K$.  It follows that
+$z=0$, proving faithfulness of $\tau_Z$.
+
+Finally, let $P_F$ be the orthogonal projection onto
+$D\oplus L_F^\circ=L_F$.  This submodule reduces the $F$-action.
+A core $w=b_1a_1\cdots a_{m-1}b_m$ as in
+\eqref{na:free-core} sends $L_F$ into words starting with $G$, so
+$P_FwP_F=0$.  Since $P_F$ commutes with $F$, arbitrary endpoint
+coefficients satisfy
+\[
+ P_Fa_0wa_mP_F=a_0P_FwP_Fa_m=0.
+\]
+Successive centering shows that $F$ and the products
+\eqref{na:free-core} span a dense subspace.  Their compressions belong
+to the faithful left copy of $F$ on $L_F$, which is norm closed.
+Compression therefore defines a conditional expectation $E_F:Z\to F$.
+The vacuum lies in $L_F$, giving $E_D=E_D^F\circ E_F$ and trace
+preservation.  Faithfulness follows from faithfulness of $\tau_Z$.
+\end{proof}
+
+\begin{proof}[Proof of Proposition~\ref{na:hnn}]
+Take
+\[
+ F=M_2(A),\qquad G=M_2(D_1),\qquad D=D_1\oplus D_2.
+\]
+Embed $D$ in $F$ by $(d_1,d_2)\mapsto\operatorname{diag}(d_1,d_2)$
+and in $G$ by
+$(d_1,d_2)\mapsto\operatorname{diag}(d_1,\theta^{-1}(d_2))$.
+The normalized matrix traces agree on $D$, and the maps
+\[
+ E_D^F(a)=(E_1(a_{11}),E_2(a_{22})),\qquad
+ E_D^G(b)=(b_{11},\theta(b_{22}))
+\]
+are trace-preserving conditional expectations.  Apply
+Lemma~\ref{na:free-product} to obtain $Z$, and retain the module
+notation used in its proof.
+The diagonal matrix projections $e_i=e_{ii}$ are common to the two
+factors.  Set
+\begin{equation}\label{na:hnn-corner}
+ s=e_{21}^G,\qquad
+ H(A)=e_1Ze_1,\qquad
+ v=e_{12}^Fs,
+\end{equation}
+and identify $A$ with $e_1Fe_1$.  Then $v^*v=vv^*=e_1$.
+For $d\in D_1$, the element $sds^*$ is the second-corner copy of
+$\theta(d)$ in $F$, so $vdv^*=\theta(d)$ in the first corner.
+
+The matrix units of $F$ identify $Z$ with $M_2(e_1Ze_1)$.  Under this
+identification, the matrix algebra over $C^*(A,v)$ contains $F$ and
+$s=e_{21}^Fv$.  It contains $G$ as well, because $G$ is generated by
+$D$ and $s,s^*$.  Hence $H(A)=C^*(A,v)$.  The trace on the corner is
+$2\tau_Z$, and its restriction to $A$ is $\tau$.  The restriction of
+$E_F$ gives the required faithful trace-preserving expectation onto
+$A$.
+
+To check reduced words, write $q=e_{12}^F$, so $v=qs$ and
+$v^*=s^*q^*$.  Between consecutive $G$-letters the internal
+$F$-coefficients are
+\[
+\begin{array}{c|cccc}
+ (\epsilon_t,\epsilon_{t+1}) &(+,+)&(+,-)&(-,+)&(-,-)\\ \hline
+ \text{coefficient} &a_tq&a_t&q^*a_tq&q^*a_t.
+\end{array}
+\]
+The first and last are off diagonal.  The middle two have zero
+$D$-expectation precisely under the respective conditions in
+\eqref{na:hnn-turns}.  Since $s,s^*$ are centered, the expanded word
+is a product of the form \eqref{na:free-core}, and its expectation
+onto $F$, hence onto $A$, is zero.
+
+For uniqueness, split a coefficient at an opposite-sign turn as
+$(a-E_i(a))+E_i(a)$.  The first term is centered at that turn, while
+the second cancels two stable letters using the unitary relation.
+Repeating this operation expresses every polynomial as an $A$-term
+plus nonempty reduced words.  Thus the stipulated data determine the
+trace of every polynomial, and in particular all moments of $x^*x$
+for each polynomial $x$.
+
+For a faithful tracial state $t$ on any $C^*$-algebra,
+\begin{equation}\label{na:moment-norm}
+ \|x\|=\lim_{r\to\infty}t((x^*x)^r)^{1/(2r)}.
+\end{equation}
+To see this, restrict $t$ to the commutative algebra generated by
+$x^*x$ and the unit.  Its spectral measure has full support, by
+faithfulness, and its $r$th-root moments tend to the spectral maximum.
+Consequently identical polynomial trace moments give identical
+polynomial norms.  The generator correspondence is therefore
+isometric and extends to the asserted isomorphism.  This argument
+requires no expectation in the comparison extension beyond the stated
+trace-vanishing rule.
+\end{proof}
+
+We will also use the explicit spanning description of the matrix
+algebra $Z$.  In the model \eqref{na:hnn-corner}, every centered
+$G$-element is a sum of $D$-multiples of $s$ and $s^*$.  Absorb these
+$D$-coefficients into neighboring $F$-coefficients; their internal
+centering is preserved by bimodularity.  Thus $F$ and the products
+\begin{equation}\label{na:matrix-word-span}
+ a_0b_1a_1\cdots b_ma_m,
+ \quad m\geq1,\quad b_t\in\{s,s^*\},\quad a_t\in F,
+ \quad E_D^F(a_t)=0\ (0<t<m)
+\end{equation}
+have dense linear span in $Z$.
+
+\begin{corollary}\label{na:hnn-restriction}
+With the hypotheses of Proposition~\ref{na:hnn}, let $A'\subset A$
+be a unital subalgebra containing $D_1$ and $D_2$.  Then
+$C^*(A',v)\subset H(A)$ is the reduced adjunction $H(A')$ formed with
+the restricted trace and expectations $E_i|_{A'}$.  The expectation
+$E_A$ restricts to the expectation $H(A')\to A'$.
+\end{corollary}
+
+\begin{proof}
+Because $D_i\subset A'$, the restrictions $E_i|_{A'}$ remain
+conditional expectations onto $D_i$.  The subalgebra $C^*(A',v)$
+inherits a faithful trace, the unitary relation and the reduced-word
+trace vanishing.  Proposition~\ref{na:hnn} identifies it with $H(A')$.
+The reduction of polynomials in the preceding proof shows that $E_A$
+maps this subalgebra into $A'$ and has the stated restriction.
+\end{proof}
+
+\subsection{Interchanging adjunctions and taking limits}
+
+The construction now has its two required analytic features: a
+faithful expectation onto the base and a complete prescription of
+reduced-word moments.  The latter makes it possible to change the
+order of two adjunctions whose data were already present in a common
+base.
+
+\begin{proposition}\label{na:reorder}
+Let $O$ be a unital faithfully tracial $C^*$-algebra.  For each
+$t\in\{j,k\}$, let $D_1^t,D_2^t\subset O$, $E_i^t:O\to D_i^t$
+and $\theta_t:D_1^t\to D_2^t$ satisfy the hypotheses of
+Proposition~\ref{na:hnn}.  When forming an adjunction over $H_j(O)$
+or $H_k(O)$, extend the other data expectations by composition with
+the expectation onto $O$.  There is a trace-preserving isomorphism
+\begin{equation}\label{na:reorder-equation}
+ H_k(H_j(O))\ \cong\ H_j(H_k(O))
+\end{equation}
+fixing $O$ and the named stable unitaries $v_j,v_k$.
+\end{proposition}
+
+\begin{proof}
+Consider $C=H_k(H_j(O))$.  Corollary~\ref{na:hnn-restriction} places
+both $H_j(O)$ and $H_k(O)$ inside $C$.  Compose the expectations
+$C\to H_j(O)\to O$, and call the result $E_O$.  Its restriction to
+each smaller adjunction is that adjunction's expectation onto $O$,
+as can be checked on its reduced words.
+
+We prove that the two smaller adjunctions are free over $O$: an
+alternating product of elements from their respective kernels of
+$E_O$ has zero $E_O$-expectation.  Approximate each centered
+$H_k(O)$-element by finite sums of nonempty reduced $k$-words over
+$O$.  Such approximations follow by reducing a polynomial
+approximation and subtracting its $O$-expectation.  Keep the centered
+$H_j(O)$-elements as coefficients.
+
+When two $k$-words are joined across a centered element
+$b\in H_j(O)$, the new internal coefficient has the form $aba'$,
+where $a,a'\in O$.  Its expectation onto $O$ is
+$aE_O(b)a'=0$, so its expectation onto either $D_i^k$ is also zero.
+The resulting word is therefore a nonempty reduced $k$-word over
+$H_j(O)$, including at every new opposite-sign turn.  Initial and
+final $H_j(O)$-coefficients are allowed.  Its expectation onto
+$H_j(O)$ vanishes.  Passing to norm limits proves the freeness
+assertion.  An alternating product with no $k$-block is a single
+centered $j$-element and is already killed by $E_O$.
+
+The individual expectations and this freeness determine all joint
+$O$-valued moments.  Indeed, center the factors of a joint word; the
+fully alternating centered term vanishes, and every other term has
+fewer blocks after its $O$-coefficients are absorbed into neighbors.
+The resulting recursion is symmetric in $j$ and $k$.  Applying the
+same argument in the reverse construction gives equal scalar trace
+moments for every polynomial in $O,v_j,v_k$.  Faithfulness and
+\eqref{na:moment-norm} give the desired isomorphism.
+\end{proof}
+
+By Lemma~\ref{na:trace-expectation}, restrictions and compositions of
+these expectations agree whenever they have the same range and
+preserve the specified trace.
+
+\begin{lemma}\label{na:hnn-limit}
+Let $\lambda$ be a nonzero limit ordinal, and let
+$(O_\beta)_{\beta<\lambda}$ be an increasing family of unital
+$C^*$-algebras with compatible faithful tracial states.  Suppose that
+fixed subalgebras $D_1,D_2$ lie in every $O_\beta$, with a fixed
+trace-preserving isomorphism $\theta:D_1\to D_2$ and compatible
+trace-preserving conditional expectations $E_i^\beta:O_\beta\to D_i$.
+Use the embeddings of Corollary~\ref{na:hnn-restriction} to identify
+the reduced adjunctions $H(O_\beta)$ with the same stable unitary $v$.
+Then, with
+\[
+ O=\overline{\bigcup_{\beta<\lambda}O_\beta},\qquad
+ R=\overline{\bigcup_{\beta<\lambda}H(O_\beta)},
+\]
+the expectations extend to $O\to D_i$, and $R=H(O)$ with its
+specified generators and trace.
+\end{lemma}
+
+\begin{proof}
+The compatible norm-one expectations extend continuously to $O$ and
+retain complete positivity, bimodularity and trace preservation.
+Denote these extensions by $E_i$.
+The compatible traces extend faithfully to both completions.  For
+example, the quotient of $R$ by its trace-null ideal is injective,
+hence isometric, on every earlier faithfully tracial algebra.  It is
+therefore isometric on their dense union and on $R$, so that ideal
+is zero.  The same argument applies to $O$.
+
+We have $R=C^*(O,v)$, and the unitary relation persists.  Consider a
+nonempty reduced word with coefficients in $O$.  Approximate its
+finitely many coefficients in one common earlier $O_\beta$.  At a
+turn requiring $E_i(a)=0$, replace an approximation $b$ by
+$b-E_i^\beta(b)$.  This remains in $O_\beta$, is exactly centered,
+and differs from $a$ by at most twice the original error.  Thus the
+given word is a norm limit of reduced words at earlier stages, all
+with trace zero.  Proposition~\ref{na:hnn} identifies $R$ with $H(O)$.
+This finite-coefficient argument requires no assumption on the
+cofinality of $\lambda$.
+\end{proof}
diff --git a/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/build/sections/introduction.tex b/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/build/sections/introduction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..2e39ab37502985409779071bd4ed17daffa88c0b
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+++ b/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/build/sections/introduction.tex
@@ -0,0 +1,136 @@
+\section{Introduction}
+\label{sec:introduction}
+
+For a complex Hilbert space $H$, let $K(H)$ denote the $C^*$-algebra of
+compact operators on $H$. Every nonzero irreducible representation of
+$K(H)$ is unitarily equivalent to its defining representation.
+Naimark's problem asks whether this property characterizes the algebras
+$K(H)$. We give a negative answer in ZFC.
+
+\begin{theorem}\label{na:main}
+There exists a unital infinite-dimensional simple complex $C^*$-algebra
+$A$ with a faithful tracial state such that all its nonzero irreducible
+representations are unitarily equivalent. In particular, $A$ is not
+isomorphic to $K(H)$ for any complex Hilbert space $H$.
+\end{theorem}
+
+Here a representation is irreducible if it is nonzero and has no
+nontrivial closed invariant subspace. A tracial state $\tau$ is faithful if
+$\tau(a^*a)>0$ whenever $a\ne0$. The last assertion of the theorem follows
+from the first: $K(H)$ is unital only when $H$ is finite-dimensional.
+All algebras and Hilbert spaces in the proof are complex, and no
+separability assumption is placed on $A$.
+
+\subsection*{History and scope}
+Naimark's question dates to 1951; see the historical account of the
+question and Rosenberg's affirmative work in \cite{AkemannWeaver2004}.
+When the unique nonzero irreducible representation acts on a separable
+Hilbert space, the algebra is isomorphic to the compact operators;
+see \cite[Proposition~7.2]{CalderonFarah2022}. This condition concerns
+an irreducible representation, not merely a faithful representation
+on a separable Hilbert space.
+Akemann and Weaver constructed a counterexample
+assuming Jensen's diamond principle
+\cite[Theorem~5]{AkemannWeaver2004}. They also established independence
+for the existence of a counterexample generated by $\aleph_1$ elements
+\cite[Corollary~7]{AkemannWeaver2004}. The unrestricted existence question
+does not impose this bound on the number of generators.
+
+Subsequent work refined the set-theoretic hypotheses and the possible
+structure of counterexamples. Calder\'on and Farah obtain a
+counterexample from $\diamondsuit_{\mathrm{Cohen}}$ together with CH
+\cite[Theorem~5.4]{CalderonFarah2022}. Under diamond, Vaccaro constructs
+counterexamples whose tracial state spaces are affinely homeomorphic
+to any prescribed metrizable Choquet simplex
+\cite{Vaccaro2018,Vaccaro2020}. Thus the presence of a trace is already
+compatible with the earlier constructions.
+
+Tanaka gives another ZFC construction with a faithful tracial state
+\cite[Theorems~1.1, 4.1 and~5.3]{Tanaka2026}, so the existence conclusion
+of Theorem~\ref{na:main} is already present there. His construction
+starts from the CAR algebra and uses strong shell sums and rank-one
+defects. It also gives a faithful representation on a separable Hilbert
+space; that representation is not irreducible. Here we give an
+alternative proof organized around a selective bridge for faithfully
+tracial bases without finite-dimensional irreducible representations,
+reduced-HNN reordering, and dependency-closed
+unique extension and countable determination. The precise hypotheses
+and conclusions of these intermediate results are stated below.
+Our construction has length $\mathfrak c^+$, where
+$\mathfrak c=2^{\aleph_0}$; we make no assertion that this length is the
+density of the resulting algebra.
+
+\subsection*{The proof in outline}
+We work with pure states, whose Gelfand--Naimark--Segal (GNS)
+representations are irreducible.
+Two pure states are called equivalent when their GNS representations
+are unitarily equivalent. The construction repeatedly joins a chosen
+pair of equivalence classes. For this procedure to be controllable,
+each old pure state must have a unique state extension, and no other
+pair of old classes may become equivalent.
+
+The main analytic step starts with a faithfully tracial algebra having
+no finite-dimensional irreducible representations. It adjoins a unitary
+carrying one decreasing projection sequence to another. Each sequence specifies its state as
+the only state taking value one on every projection; we call it a
+\emph{peaking sequence}. When the two states are inequivalent and the
+sequences have matching trace
+values strictly decreasing to zero, the resulting reduced HNN extension
+has exactly the required pure-state properties
+(Theorem~\ref{na:bridge}). The extension is built from a reduced
+amalgamated free product using Ueda's matrix-corner construction
+\cite{UedaPreprint2008,Ueda2008}. Pure-state excision
+\cite{AkemannAndersonPedersen1986} and the cross-excision estimate of
+Akemann and Weaver \cite[Lemma~1]{AkemannWeaver2004} turn uniqueness
+of extension and the exclusion of unintended joins into norm-decay
+estimates for reduced words. The estimates apply to every old pure state.
+
+The second step makes every pure state accessible to the recursion.
+We start with a separable simple algebra, adjoin one unitary at each
+active successor stage, and take norm closures of unions at limits.
+Each adjunction depends on only countably many earlier generators.
+If a set of generators contains all these dependencies, we prove that
+every pure state on the algebra it generates has a unique extension
+to the whole algebra (Theorem~\ref{na:closed-extension}). To prove this,
+we postpone an omitted adjunction past the retained ones. The
+no-other-join assertion of the bridge keeps its two states inequivalent
+until the postponed adjunction is performed. For every pure state of
+the full algebra, excision provides a separable subalgebra generated
+by a countable dependency-closed set of steps on which the restriction
+is pure. The closed-subiteration theorem then makes the original state
+its unique extension to the full algebra
+(Lemma~\ref{na:countable-determination}).
+
+This last conclusion replaces the state-guessing role of a
+set-theoretic prediction principle. At every stage below
+$\mathfrak c^+$ there are at most $\mathfrak c$ pure states, so a fixed
+schedule can eventually treat them all. On each separable simple
+determining subalgebra, the homogeneity theorem of Kishimoto, Ozawa and
+Sakai \cite{KishimotoOzawaSakai2003} supplies the next peaking sequence.
+A final countable-determination argument catches every pure state of
+the limit algebra.
+
+The ingredients have several useful precedents. Akemann and Weaver
+combine pure-state homogeneity with extensions joining prescribed
+states \cite[Corollary~3]{AkemannWeaver2004}. Farah and Hirshberg realize prescribed countable equivalence patterns
+with unique pure-state extensions
+\cite[Lemma~2.3]{FarahHirshberg2017}; Vaccaro's corrected construction
+combines this method with control of the trace space
+\cite{Vaccaro2018,Vaccaro2020}.
+Akemann, Wassermann and
+Weaver prove pure-state restriction and unique-extension results for
+reduced free group algebras
+\cite[Proposition~2.3 and Theorem~2.6]{AkemannWassermannWeaver2010}.
+Calder\'on and Farah propose using separable subalgebras that uniquely
+determine the relevant states, followed by an amalgamation preserving
+unique extension \cite[Section~8, especially Question~8.1]{CalderonFarah2022}.
+The two features established here are the selective bridge for all
+old pure states and its compatibility with dependency-closed subiterations.
+Together they give the countable determination needed for the
+unrestricted recursion.
+
+Section~\ref{sec:states} proves the state-theoretic facts used below.
+Section~\ref{sec:hnn} constructs the reduced adjunction, and
+Section~\ref{sec:bridge} proves its pure-state properties.
+Section~\ref{sec:iteration} establishes countable determination;
+Section~\ref{sec:construction} carries out the final recursion.
diff --git a/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/build/sections/iteration.tex b/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/build/sections/iteration.tex
new file mode 100644
index 0000000000000000000000000000000000000000..196c59060932cecbd36c705637518f287b6c1322
--- /dev/null
+++ b/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/build/sections/iteration.tex
@@ -0,0 +1,337 @@
+\section{Closed subiterations and countable determination}
+\label{sec:iteration}
+
+We now iterate the bridge. The main point is that a pure state on the
+resulting algebra is uniquely determined on a suitable separable
+subalgebra. To prove this, we first show that one may retain any collection
+of bridge steps containing the dependencies of each of its members,
+and then perform all omitted steps afterward.
+
+\subsection{States and traces along a chain}
+
+A chain $(B_\alpha)_{\alpha\le\lambda}$ of unital inclusions is
+\emph{continuous} if
+$B_\gamma=\overline{\bigcup_{\alpha<\gamma}B_\alpha}$ at every nonzero
+limit ordinal $\gamma\le\lambda$.
+
+\begin{lemma}\label{na:unique-chain}
+Let $(B_\alpha)_{\alpha\le\lambda}$ be a continuous chain whose
+successor inclusions have property \textup{(U)}. Then every inclusion
+$B_\alpha\subset B_\beta$, for $\alpha\le\beta\le\lambda$, has
+\textup{(U)}. The unique lifts of equivalent pure states are equivalent.
+\end{lemma}
+\begin{proof}
+At a successor, unique extensions compose, and the intermediate lift
+is pure by \cref{na:unique-lift}. At a limit, two extensions agree on
+each earlier algebra, hence on their dense union. Existence follows
+from state extension. Transfinite induction gives the first assertion.
+For the second, a unitary implementing equivalence in the original
+algebra also conjugates the unique lifts; use
+\cref{na:unitary-states}.
+\end{proof}
+
+\begin{lemma}\label{na:inequivalent-limit}
+Let $(B_\alpha)_{\alpha\le\lambda}$ be continuous, with $\lambda$ a
+nonzero limit ordinal. Suppose pure states $h,k$ of $B_\lambda$ restrict
+to inequivalent pure states of $B_\alpha$ for all $\alpha$ in some tail
+below $\lambda$. Then $h\not\sim k$.
+\end{lemma}
+\begin{proof}
+Otherwise realize $h,k$ by unit vectors $\xi,\eta$ in one irreducible
+representation $\pi$ of $B_\lambda$. For each earlier $\alpha$ in the
+given tail, the cyclic subspaces
+\[
+ M_\alpha=\overline{\pi(B_\alpha)\xi},\qquad
+ N_\alpha=\overline{\pi(B_\alpha)\eta}
+\]
+are reducing and carry inequivalent irreducible representations of
+$B_\alpha$. The orthogonal projection onto $N_\alpha$, restricted to
+$M_\alpha$, is an intertwiner, so it is zero. Thus
+$\langle\pi(a)\xi,\eta\rangle=0$ for every $a\in B_\alpha$.
+Continuity gives the same equality for every $a\in B_\lambda$,
+contradicting cyclicity of $\xi$.
+\end{proof}
+
+We will also use two elementary permanence facts. A continuous union
+of simple unital algebras is simple: a proper quotient is injective,
+and hence isometric, on every earlier algebra, so it is isometric on
+their dense union and its completion. Compatible faithful tracial
+states remain faithful at the limit. Indeed, the extended trace's
+null ideal has zero intersection with every earlier algebra; the
+quotient by this ideal is again isometric on a dense union, so the
+ideal is zero. At successors, simplicity follows from
+\cref{na:simple-extension} whenever \textup{(U)} and a faithful
+conditional expectation are available.
+
+\subsection{The initial algebra and its subiterations}
+
+Let
+\[
+ A_0=\bigotimes_{m=1}^{\infty}M_2,\qquad
+ p_n=e_{11}^{\otimes n}\otimes1\quad(n\ge1),\qquad p_0=1.
+\]
+Here $A_0$ is the norm closure of its unital matrix subalgebras
+$M_{2^n}$, with the compatible normalized matrix traces.
+
+\begin{lemma}\label{na:base-peak}
+The algebra $A_0$ is separable, simple and infinite-dimensional, its
+trace is faithful, and $(p_n)$ peaks to a pure state $f_0$.
+Moreover, $\tau(p_n)=2^{-n}$. Every unital $C^*$-algebra containing $A_0$
+has no finite-dimensional irreducible representations.
+\end{lemma}
+\begin{proof}
+Simplicity and faithfulness follow from the permanence facts above
+applied to the matrix stages. Separability and infinite dimension are
+immediate from those stages. A state taking value one on $p_n$ must
+restrict to the vector state of the first basis vector on $M_{2^n}$.
+These vector states are compatible, so they determine exactly one
+state on the dense union and its completion. This proves peaking and
+purity. The trace formula is the normalized matrix trace formula.
+
+A nonzero finite-dimensional irreducible representation of an
+algebra containing $A_0$ restricts to a unital representation of
+$A_0$. Its kernel is zero by simplicity, which would embed the
+infinite-dimensional algebra $A_0$ in a finite-dimensional algebra.
+\end{proof}
+
+Consider a continuous chain $(X_\beta)_{\beta\le\delta}$ starting
+from $X_0=A_0$. At an \emph{active} index $j<\delta$, perform the
+bridge of \cref{na:bridge} using $(p_n)$ and a decreasing sequence
+of projections $(q_n^j)$ that peaks in $X_j$ to a state inequivalent
+to the $p_n$-state, with
+\[
+ q_0^j=1,\qquad \tau(q_n^j)=2^{-n}.
+\]
+Write $v_j$ for its stable unitary, so
+$X_{j+1}=C^*(X_j,v_j)$. At every other index set $X_{j+1}=X_j$.
+The traces are the compatible traces supplied by the bridges.
+By \cref{na:unique-chain}, $(p_n)$ peaks at every stage. For each
+active $j$, $(q_n^j)$ peaks at every stage from $j$ onward, and its
+state is equivalent to the $p_n$-state from stage $j+1$ onward.
+
+For $T\subset\delta$, define the unital subalgebra
+\[
+ A[T]=C^*\bigl(A_0,\{v_j:j\in T,\ j\text{ active}\}\bigr)
+ \subset X_\delta.
+\]
+Thus $A[\beta]=X_\beta$ when the ordinal $\beta$ is regarded as the
+set of smaller ordinals. Every element of $X_\delta$ lies in $A[T]$
+for some countable $T$: approximate it by a sequence of polynomials,
+each involving finitely many stable unitaries.
+
+Choose countable sets $I_j\subset j$ such that
+$q_n^j\in A[I_j]$ for every $n$ at each active $j$, and set
+$I_j=\varnothing$ otherwise. Call $T$ \emph{dependency-closed} if
+\begin{equation}\label{eq:dependency-closed}
+ j\in T\quad\Longrightarrow\quad I_j\subset T.
+\end{equation}
+Every countable set is contained in a countable dependency-closed
+set: repeatedly adjoin the dependencies of its members and take the
+union over the natural numbers.
+
+\begin{lemma}\label{na:inherited}
+For every dependency-closed $T\subset\delta$, the algebras
+$A[T\cap\beta]$, $\beta\le\delta$, form a continuous bridge
+iteration in the inherited order, with identity steps at indices
+outside $T$. In particular, $A[T]$ is simple and has a faithful
+tracial state. If $T$ is countable, $A[T]$ is separable.
+\end{lemma}
+\begin{proof}
+Let $k\in T$ be active. Both sequences for its bridge lie in
+$A[T\cap k]$, by \eqref{eq:dependency-closed}. The original
+expectations onto their generated abelian algebras restrict to this
+smaller base, since their ranges already lie in it.
+The restriction property \cref{na:hnn-restriction} identifies the
+algebra obtained by adjoining $v_k$ with that reduced HNN extension.
+
+The sequences peak to pure states of the smaller base by
+\cref{na:peaking-restriction}, applied to its inclusion in $X_k$.
+That same lemma gives each of these particular states its unique
+extension to $X_k$. If the two smaller states were equivalent,
+\cref{na:unitary-states} and uniqueness would make their extensions
+equivalent, contrary to the choice of the original active step.
+The smaller base contains $A_0$, so \cref{na:base-peak} excludes
+finite-dimensional irreducibles. All hypotheses of
+\cref{na:bridge} therefore hold in the inherited iteration.
+
+Continuity follows because a polynomial uses only finitely many
+indices. Simplicity follows by induction from
+\cref{na:simple-extension} and the limit observation above.
+Faithfulness follows either from the restricted trace on $X_\delta$
+or from the same permanence argument. Finally, a countable set of
+stable unitaries together with the separable algebra $A_0$ generates
+a separable algebra.
+\end{proof}
+
+\subsection{Moving omitted steps to the end}
+
+The preceding lemma proves unique extension along the inherited
+iteration. We need the stronger assertion that it holds from the
+entire inherited algebra into the original result.
+
+\begin{theorem}\label{na:closed-extension}
+For every dependency-closed $S\subset\delta$, the inclusion
+$A[S]\subset X_\delta$ has property \textup{(U)}.
+\end{theorem}
+\begin{proof}
+We will insert the omitted indices, in their original order, after
+$A[S]$. Recall that the ordinal $j$ is the set of indices below $j$,
+whereas $j+1$ also contains the index $j$ itself.
+Fix an active $j\notin S$, and for
+$j+1\le\beta\le\delta$ set
+\begin{equation}\label{eq:reordered-bases}
+ O_\beta=A[j\cup(S\cap\beta)],\qquad
+ R_\beta=A[(j+1)\cup(S\cap\beta)].
+\end{equation}
+These index sets are dependency-closed. Initially,
+$O_{j+1}=X_j$ and $R_{j+1}=X_{j+1}$; afterward both chains
+perform precisely the inherited steps with indices in $S$.
+
+\smallskip\noindent
+\emph{The omitted pair stays inequivalent on $O_\beta$.}
+Both $(p_n)$ and $(q_n^j)$ peak on every $O_\beta$ by
+\cref{na:peaking-restriction}, since they peak in $X_\delta$.
+Their states are initially inequivalent. Suppose $k>j$ is an active
+member of $S$, and they are still inequivalent on $O_k$.
+The $q_n^k$-sequence also lies in $O_k$ and peaks there.
+Each of these three particular states has a unique extension to
+$X_k$, by \cref{na:peaking-restriction}; this does not use the
+theorem being proved.
+
+In $X_k$, the $q_n^j$-state is equivalent to the $p_n$-state,
+because step $j$ has already been performed. The $q_n^k$-state is
+inequivalent to that state, because step $k$ is active. Equivalence
+on $O_k$ would pass to these particular unique extensions.
+Consequently, on $O_k$ the $q_n^k$-state is equivalent to neither
+the $p_n$-state nor the $q_n^j$-state. The $k$-bridge joins only
+the classes of its designated pair, by \cref{na:bridge}. It therefore
+does not join the two states of the omitted $j$-bridge.
+At limits their inequivalence persists by
+\cref{na:inequivalent-limit}. Transfinite induction proves the claim.
+
+\smallskip\noindent
+\emph{The omitted adjunction commutes past the inherited steps.}
+Write $H_j$ for the reduced HNN adjunction with the fixed data
+$(p_n),(q_n^j)$. We claim, respecting the trace and named generators,
+\begin{equation}\label{eq:omitted-adjunction}
+ R_\beta\cong H_j(O_\beta)
+ \qquad(j+1\le\beta\le\delta).
+\end{equation}
+This is true initially. At an active $k\in S$ above $j$, the data
+for both adjunctions lie in $O_k$. The expectation from
+$R_k=H_j(O_k)$ onto $O_k$, followed by either $k$-data expectation,
+is a trace-preserving expectation onto that data algebra. By
+\cref{na:trace-expectation}, it agrees with the expectation supplied
+there by the corresponding peaking sequence. The same observation
+applies with $j$ and $k$ interchanged. Thus the reordering
+isomorphism \cref{na:reorder} applies and gives
+\[
+ H_k\bigl(H_j(O_k)\bigr)
+ \cong H_j\bigl(H_k(O_k)\bigr),
+\]
+proving the next instance of \eqref{eq:omitted-adjunction}.
+Other successor steps are identities. Figure~\ref{fig:reordering}
+displays the two orders at an active index $k\in S$.
+
+\begin{figure}[htbp]
+\centering
+\begin{tikzpicture}[>=Stealth,every node/.style={font=\small}]
+ \node (o) at (0,0) {$O_k$};
+ \node (j) at (6.4,0) {$H_j(O_k)=R_k$};
+ \node (k) at (0,-2) {$H_k(O_k)=O_{k+1}$};
+ \node[align=center] (jk) at (6.4,-2)
+ {$R_{k+1}\cong H_k(H_j(O_k))$\\[3pt]
+  $\phantom{R_{k+1}}\cong H_j(H_k(O_k))$};
+ \draw[->] (o) -- node[above] {adjoin $v_j$} (j);
+ \draw[->] (o) -- node[left] {adjoin $v_k$} (k);
+ \draw[->] (j) -- node[right] {adjoin $v_k$} (jk);
+ \draw[->] (k) -- node[below] {adjoin $v_j$} (jk);
+\end{tikzpicture}
+\caption{Reordering two adjunctions whose data all lie in $O_k$.
+The arrows are inclusions. Both paths give the same faithfully
+tracial algebra, under an isomorphism fixing $O_k,v_j,v_k$.}
+\label{fig:reordering}
+\end{figure}
+
+At a limit, the isomorphisms already constructed agree on overlaps
+because they fix the named generators. The expectations from the
+$O_\beta$ onto the two fixed abelian algebras generated by $(p_n)$
+and $(q_n^j)$ also agree on overlaps, by
+\cref{na:trace-expectation}. Applying \cref{na:hnn-limit} to these
+fixed data identifies the closure of the preceding $H_j(O_\beta)$
+with $H_j$ of the limit base, proving
+\eqref{eq:omitted-adjunction} there.
+
+At $\beta=\delta$ we have
+$O_\delta=A[S\cup j]$ and
+$R_\delta=A[S\cup(j+1)]$.
+The two sequences still peak to inequivalent states on $O_\delta$,
+by the first part of the proof. Thus \cref{na:bridge} applies to
+\eqref{eq:omitted-adjunction}, giving \textup{(U)} for
+\[
+ A[S\cup j]\subset A[S\cup(j+1)].
+\]
+If $j\in S$ or $j$ is inactive, this inclusion is the identity.
+The continuous chain $\bigl(A[S\cup j]\bigr)_{j\le\delta}$ starts
+at $A[S]$ and ends at $X_\delta$; continuity follows because a
+polynomial involves only finitely many indices outside $S$.
+Now \cref{na:unique-chain} proves the theorem.
+\end{proof}
+
+\subsection{A separable algebra determines each pure state}
+
+The separable pure-restriction argument below is closely related to
+\cite[Proposition~2.3 and concluding Remark~1]{AkemannWassermannWeaver2010}.
+Here we arrange that the separable algebra is dependency-closed;
+\cref{na:closed-extension} then adds uniqueness of the full extension.
+
+\begin{lemma}\label{na:countable-determination}
+For every pure state $h$ of $X_\delta$, there is a countable
+dependency-closed $S\subset\delta$ such that $h|_{A[S]}$ is pure
+and $h$ is its unique state extension to $X_\delta$.
+\end{lemma}
+\begin{proof}
+Construct increasing countable dependency-closed sets $S_n$, starting
+with $S_0=\varnothing$. Choose a countable dense subset $D_n$ of
+$A[S_n]$. For each $a\in D_n$ and each integer $m\ge1$,
+\cref{na:excision} supplies a positive contraction $x_{a,m}$ in
+$X_\delta$ satisfying
+\[
+ h(x_{a,m})=1,\qquad
+ \bigl\|x_{a,m}(a-h(a)1)x_{a,m}\bigr\|<1/m.
+\]
+Each witness has countable support. Adjoin all these supports to
+$S_n$ and close under dependencies to obtain $S_{n+1}$.
+
+Set $S=\bigcup_n S_n$ and $B=A[S]$. Suppose
+$h|_B=t h_1+(1-t)h_2$ for states $h_1,h_2$ and $0<t<1$.
+Each $h_i$ takes value one on every witness $x_{a,m}$.
+A positive contraction of state value one fixes the state's GNS
+vector, so $h_i(a)=h_i(x_{a,m}a x_{a,m})$ and
+$h_i(x_{a,m}^2)=1$. Hence
+\[
+ |h_i(a)-h(a)|
+ =\bigl|h_i\bigl(x_{a,m}(a-h(a)1)x_{a,m}\bigr)\bigr|<1/m.
+\]
+Letting $m$ tend to infinity shows that both states agree with $h$
+on every $D_n$. Their union is dense in $B$, so $h|_B$ is pure.
+Its unique extension is $h$ by \cref{na:closed-extension}.
+\end{proof}
+
+\begin{corollary}\label{na:state-count}
+Put $\mathfrak c=2^{\aleph_0}$. If $|\delta|\le\mathfrak c$, then
+$X_\delta$ has at most $\mathfrak c$ pure states.
+\end{corollary}
+\begin{proof}
+There are at most
+\[
+ \max(\aleph_0,|\delta|)^{\aleph_0}
+ \le\mathfrak c^{\aleph_0}=\mathfrak c
+\]
+countable subsets of $\delta$. For each one, the corresponding
+algebra $A[S]$ is separable and has at most $\mathfrak c$ states:
+values on a countable dense subset determine a state. By
+\cref{na:countable-determination}, every pure state of $X_\delta$
+is the unique extension of one such restriction. There are at most
+$\mathfrak c\cdot\mathfrak c=\mathfrak c$ possibilities.
+\end{proof}
diff --git a/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/build/sections/states.tex b/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/build/sections/states.tex
new file mode 100644
index 0000000000000000000000000000000000000000..3c85f98e0443f5ffd9ae90c500655d4a34db2d75
--- /dev/null
+++ b/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/build/sections/states.tex
@@ -0,0 +1,307 @@
+\section{Pure states, excision, and peaking projections}\label{sec:states}
+
+The construction must preserve unique state extensions while changing selected
+equivalence classes.  This section records the state-theoretic tools for doing
+so.  Its main analytic fact is excision: near the support of a pure state,
+compression of an element approaches its scalar state value.
+
+All algebras and Hilbert spaces are complex.  Unless stated otherwise, algebras
+are unital and inclusions preserve the unit.  A \emph{state} on an algebra $B$
+is a positive linear functional of norm one, and it is \emph{pure} if it is an
+extreme point of the state space.  Write $(\pi_h,H_h,\xi_h)$ for the GNS
+representation of a state $h$.  Thus $\xi_h$ is a cyclic unit vector and
+$h(a)=\langle\pi_h(a)\xi_h,\xi_h\rangle$; purity is equivalent to
+irreducibility of $\pi_h$.  For pure states, write $h\sim k$ when $\pi_h$ and
+$\pi_k$ are unitarily equivalent.  We use
+$\Ad(u)(a)=uau^*$ for conjugation by a unitary.
+
+A \emph{tracial algebra} is a unital $C^*$-algebra equipped with a faithful
+tracial state $\tau$: one has $\tau(ab)=\tau(ba)$, and $\tau(a^*a)=0$ implies
+$a=0$.  Traces and other bounded functionals are extended normally to the
+enveloping von Neumann algebra $B^{**}$, with the same notation.  The normal
+extension of a trace is tracial, by separate ultraweak continuity.
+For a representation $\pi$ of $B$, its normal extension to $B^{**}$ is
+denoted by $\pi^{**}$.
+
+For an inclusion $B\subset C$, we shall use the property
+\begin{equation}\tag{U}\label{eq:U}
+ \text{every pure state of $B$ has exactly one state extension to $C$.}
+\end{equation}
+Here uniqueness is among all states.  States always extend across a unital
+inclusion, by the positive form of the Hahn--Banach theorem.
+
+\subsection{Unique extensions and peaking}
+
+\begin{lemma}\label{na:unique-lift}
+If a pure state $h$ of $B$ has a unique state extension $\widetilde h$ to
+$C\supset B$, then $\widetilde h$ is pure.
+\end{lemma}
+\begin{proof}
+In a nontrivial convex decomposition of $\widetilde h$, both restrictions to
+$B$ equal $h$ by purity.  Uniqueness then makes both states equal to
+$\widetilde h$.
+\end{proof}
+
+\begin{lemma}\label{na:unitary-states}
+Equivalent pure states of a unital $C^*$-algebra $B$ are conjugate by a unitary
+of $B$.  Consequently, if they each have a unique state extension to an
+algebra $C\supset B$, their extensions are equivalent.
+\end{lemma}
+\begin{proof}
+Realize the two states by unit vectors $\xi,\eta$ in a common irreducible
+representation $\pi$.  On their finite-dimensional span choose a unitary
+carrying $\xi$ to $\eta$ and a self-adjoint logarithm of that unitary.
+Kadison transitivity, including its self-adjoint form
+\cite[Theorem~1]{Kadison1957}, supplies a self-adjoint $a\in B$ whose action
+on this span is the chosen logarithm.  The span reduces $\pi(a)$, so
+$u=e^{ia}$ satisfies $\pi(u)\xi=\eta$.  If $h$ is represented by $\xi$
+and $k$ by $\eta$, then $k=h\circ\Ad(u^*)$.
+Conjugating their unique extensions by the same unitary proves the last
+assertion.
+\end{proof}
+
+A decreasing sequence of projections $(P_n)_{n\ge0}$ in $B$ \emph{peaks to}
+a state $h$ if $h$ is the only state satisfying $h(P_n)=1$ for every $n$.
+Such a state is pure: both states in any convex decomposition satisfy the
+same constraints.  The following elementary observation will let us pass to
+smaller algebras without assuming \eqref{eq:U} for the inclusion.
+
+\begin{lemma}[Peaking under restriction]\label{na:peaking-restriction}
+Let $B\subset C$ be a unital inclusion and let $(P_n)$ be a decreasing
+sequence of projections in $B$ that peaks to a state $H$ of $C$.
+Then $(P_n)$ peaks to $h=H|_B$ in $B$, and $H$ is the unique state extension
+of $h$ to $C$.
+\end{lemma}
+\begin{proof}
+Every state of $B$ satisfying the projection constraints extends to a state
+of $C$ with the same constraints, hence to $H$.  It must therefore equal
+$h$.  Every extension of $h$ also satisfies the constraints and must equal
+$H$.
+\end{proof}
+
+\subsection{Excision and inequivalent states}
+
+For a state $h$, its \emph{support projection} $r_h\in B^{**}$ is the least
+projection on which its normal extension takes value one.  We shall prove
+that this projection is minimal when $h$ is pure, together with the norm
+approximation that we need. The decreasing-net form of pure-state
+excision used here is proved by Akemann, Anderson, and Pedersen
+\cite[Proposition~2.2]{AkemannAndersonPedersen1986}. We record a proof
+in the net form used below.
+
+\begin{lemma}[Decreasing excision]\label{na:excision}
+Let $h$ be a pure state of a unital $C^*$-algebra $B$.  There is a decreasing
+net of positive contractions $x_\lambda\in B$ such that $h(x_\lambda)=1$,
+$x_\lambda$ decreases strongly to the minimal projection $r_h$ in $B^{**}$,
+and, for every $a\in B$,
+\begin{equation}\label{eq:excision}
+ \|x_\lambda(a-h(a)1)x_\lambda\|\longrightarrow0.
+\end{equation}
+Moreover, $r_h a r_h=h(a)r_h$ for every $a\in B$.
+\end{lemma}
+\begin{proof}
+Put
+\[
+ L=\{a\in B:h(a^*a)=0\},\qquad N=L\cap L^*.
+\]
+Here $L$ is a closed left ideal and $N$ is a $C^*$-subalgebra.  If $a\in L$,
+then $a^*a\in N$, since
+$h((a^*a)^2)\le\|a\|^2h(a^*a)=0$.
+
+First, $h$ is the only state annihilating $N$.  If another state $k$
+annihilates $N$, then $L$ is contained in the kernel of
+$a\mapsto\pi_k(a)\xi_k$.  The map
+$Q_h:B\to H_h$, $Q_h(a)=\pi_h(a)\xi_h$, is onto by one-vector Kadison
+transitivity.  This assertion concerns the whole Hilbert space, even when
+it is nonseparable.  Since $\ker Q_h=L$, the open mapping theorem for
+$B/L\to H_h$ shows that
+\[
+ T\bigl(\pi_h(a)\xi_h\bigr)=\pi_k(a)\xi_k
+\]
+defines a bounded operator $T:H_h\to H_k$.  It intertwines the
+representations.  Thus $T^*T$ is scalar by irreducibility of $\pi_h$;
+its value on $\xi_h$ shows that $T^*T=1$.  Taking vector coefficients gives
+$k=h$.
+
+We next construct the decreasing net.  Order $N_+$ by the usual operator
+order, a directed order, and set
+\[
+ e_s=1-(1+s)^{-1},\qquad x_s=1-e_s \qquad(s\in N_+).
+\]
+Functional calculus gives $e_s\in N$, and operator monotonicity of
+inversion makes $(e_s)$ increasing.  For $b\in N$ and
+$s\ge m(bb^*+b^*b)$,
+\[
+ \|(1-e_s)b\|^2
+ \le \frac1m\|(1+s)^{-1}s(1+s)^{-1}\|
+ \le\frac1{4m};
+\]
+the same estimate holds for $\|b(1-e_s)\|^2$.  Thus $(e_s)$ is an
+approximate identity of $N$, and $h(x_s)=1$.
+
+Let $q$ be the strong limit of $(e_s)$, the support projection of $N$.
+Every normal state supported on $1-q$ annihilates $N$, and hence is $h$;
+conversely, $h(1-q)=1$.  The corner $(1-q)B^{**}(1-q)$ therefore has
+exactly one normal state.  Normal states separate its self-adjoint
+elements, so this corner is scalar.  It follows that $1-q=r_h$ is
+minimal and $x_s\downarrow r_h$.  The identity $r_har_h=h(a)r_h$ follows
+by applying $h$ to this scalar corner.
+
+Finally, let $\rho_s$ be arbitrary states of $B$.  Every weak-star cluster
+point of the positive functionals
+$a\mapsto\rho_s(x_sax_s)$ annihilates $N$, by the approximate-identity
+estimates.  A nonzero cluster point, after normalization, is therefore
+$h$.  For self-adjoint $a$, a failure of \eqref{eq:excision} would give a
+subnet and norming states $\rho_s$ with a cluster point nonzero on
+$a-h(a)1$, a contradiction.  Real and imaginary parts give the conclusion
+for arbitrary $a$.
+\end{proof}
+
+\begin{lemma}[Projection excision]\label{na:projection-excision}
+If decreasing projections $(P_n)$ in $B$ peak to $h$, then
+$P_n\downarrow r_h$ strongly and
+\[
+ \|P_n(a-h(a)1)P_n\|\longrightarrow0\qquad(a\in B).
+\]
+In particular, if $h(a)=0$, then $\|P_n a P_t\|\to0$ on the product
+index set.
+\end{lemma}
+\begin{proof}
+The infimum of the $P_n$ in $B^{**}$ supports exactly one normal state,
+namely $h$.  As in the preceding proof, it is the minimal projection
+$r_h$.  For arbitrary states $\rho_n$, every cluster point $\psi$ of
+$\rho_n(P_n\,\cdot\,P_n)$ vanishes on $1-P_m$ for each fixed $m$.
+If $\psi\ne0$, its normalization satisfies every peaking constraint and
+therefore equals $h$.  The same norming-state argument proves excision.
+For the last assertion use nesting:
+\[
+ \|P_n a P_t\|\le \|P_{\min(n,t)}aP_{\min(n,t)}\|.
+\qedhere
+\]
+\end{proof}
+
+Excision also separates inequivalent states in norm.  We use the following
+form of the cross-excision estimate of Akemann and Weaver
+\cite[Lemma~1]{AkemannWeaver2004}.
+
+\begin{lemma}[Cross-excision]\label{na:cross-excision}
+Let $h\not\sim k$ be pure states of $B$, and let $(x_\lambda)$ and
+$(y_\nu)$ be decreasing nets as in Lemma~\ref{na:excision}, for $h$ and
+$k$ respectively.  Then, for every $a\in B$,
+\begin{equation}\label{eq:cross-excision}
+ \|x_\lambda a y_\nu\|\longrightarrow0
+\end{equation}
+on the product directed set.  Either net may be a peaking sequence of
+projections.
+\end{lemma}
+\begin{proof}
+The normal extension of $\pi_h$ annihilates $r_k$.  Otherwise a unit vector
+in the range of $\pi_h^{**}(r_k)$ would represent $k$, by
+$r_kar_k=k(a)r_k$.  Every nonzero vector in an irreducible representation
+is cyclic, so this would imply $h\sim k$.
+
+Consequently $h(ay_{\nu_0}a^*)\to0$ as $\nu_0$ increases.  For
+$\nu\ge\nu_0$, the inequalities $y_\nu^2\le y_\nu\le y_{\nu_0}$ give
+\begin{align*}
+ \|x_\lambda a y_\nu\|^2
+ &\le \|x_\lambda a y_{\nu_0}a^*x_\lambda\|\\
+ &\le h(ay_{\nu_0}a^*)
+   +\|x_\lambda(ay_{\nu_0}a^*-h(ay_{\nu_0}a^*)1)x_\lambda\|.
+\end{align*}
+Choose $\nu_0$ first, then use excision to choose $\lambda_0$.
+This proves smallness on a rectangular tail and hence the product-net
+limit.  Lemma~\ref{na:projection-excision} gives the stated projection
+case.
+\end{proof}
+
+\begin{lemma}[Trace-null supports]\label{na:trace-null}
+If $B$ has no finite-dimensional irreducible representations and $\tau$ is
+a tracial state on $B$, then $\tau(r_h)=0$ for every pure state $h$.
+\end{lemma}
+\begin{proof}
+The space $H_h$ is infinite-dimensional.  Choose orthonormal unit vectors
+$\eta_j$ and unitaries $u_j\in B$ with $\pi_h(u_j)\xi_h=\eta_j$, as in
+Lemma~\ref{na:unitary-states}.  The projections $u_jr_hu_j^*$ are
+pairwise orthogonal, since
+\[
+ r_hu_i^*u_jr_h=h(u_i^*u_j)r_h=0\qquad(i\ne j).
+\]
+They all have trace $\tau(r_h)$, so $m\tau(r_h)\le1$ for every positive
+integer $m$.
+\end{proof}
+
+\subsection{Expectations from peaking projections}
+
+We now use projection excision to construct the conditional expectations
+needed for the tracial extensions.  A \emph{conditional expectation}
+$E:C\to B$ is a unital completely positive map fixing $B$ and satisfying
+$E(b_1ab_2)=b_1E(a)b_2$ for $b_1,b_2\in B$.  It is \emph{faithful} if
+$E(a^*a)=0$ implies $a=0$.
+
+\begin{lemma}\label{na:trace-expectation}
+For a unital subalgebra $B$ of a faithfully tracial algebra $(C,\tau)$,
+there is at most one trace-preserving conditional expectation $C\to B$.
+If one exists, it is faithful.
+\end{lemma}
+\begin{proof}
+For such an expectation, bimodularity gives
+$\tau(b^*E(a))=\tau(b^*a)$ for every $b\in B$.  Two expectations therefore
+have difference $d\in B$ satisfying $\tau(d^*d)=0$, so they agree.
+Faithfulness follows from $\tau(E(a^*a))=\tau(a^*a)$.
+\end{proof}
+
+\begin{lemma}\label{na:peaking-expectation}
+Let $(B,\tau)$ be faithfully tracial.  Suppose decreasing projections
+$(p_n)_{n\ge0}$ peak to $h$, with $p_0=1$ and trace values strictly
+decreasing to zero.  Then
+$D=C^*(p_n:n\ge0)$ is isomorphic to
+$C(\mathbb N_0\cup\{\infty\})$, and there is a trace-preserving conditional
+expectation $E:B\to D$.  Writing $d_n=p_n-p_{n+1}$ for the finite-point
+atoms, it is given by
+\begin{equation}\label{eq:peaking-expectation}
+ E(a)(n)=\frac{\tau(d_nad_n)}{\tau(d_n)},\qquad E(a)(\infty)=h(a).
+\end{equation}
+\end{lemma}
+\begin{proof}
+The $d_n$ are nonzero orthogonal projections.  Their closed linear span is
+isomorphic to $c_0(\mathbb N_0)$, and adjoining the unit gives the claimed
+description of $D$.  Indeed $p_n=1-\sum_{j<n}d_j$ and $\|p_n\|=1$,
+since $\tau(p_n)>0$.  Multiplying by $p_n$ shows that $1$ has distance
+at least one from every linear combination of $d_0,\ldots,d_{n-1}$.
+
+The values in \eqref{eq:peaking-expectation} define a continuous function:
+by $d_n\le p_n$ and Lemma~\ref{na:projection-excision},
+\[
+ \left|\frac{\tau(d_nad_n)}{\tau(d_n)}-h(a)\right|
+ \le \|p_n(a-h(a)1)p_n\|\longrightarrow0.
+\]
+Every coordinate of $E$ is a state, so $E$ is unital and completely
+positive: positivity of each matrix amplification is tested pointwise
+in the commutative range.  The map fixes $D$.  To check bimodularity,
+an element of $D$ acts scalarly on each $d_n$; at infinity, $h|_D$ is the
+evaluation character.  Cauchy--Schwarz for $h$ then gives
+$h(ba)=h(b)h(a)$ and $h(ab)=h(a)h(b)$ for $b\in D$.
+These coordinate identities prove bimodularity.
+
+Finally, $\sum_n\tau(d_n)=1$ and the trace on $D$ has no mass at infinity.
+Summing the finite coordinates gives $\tau(E(a))=\tau(a)$, since the
+remaining trace satisfies
+$|\tau(p_Na)|\le\|a\|\tau(p_N)\to0$.
+\end{proof}
+
+Unique state extensions and a faithful expectation will also preserve
+simplicity, without a separate analysis of ideals in each extension.
+
+\begin{lemma}\label{na:simple-extension}
+Suppose $B\subset C$ satisfies \eqref{eq:U}, $B$ is simple, and there is a
+faithful conditional expectation $E:C\to B$.  Then $C$ is simple.
+\end{lemma}
+\begin{proof}
+Let $J$ be a proper closed two-sided ideal of $C$.  The quotient map
+$q:C\to C/J$ is injective on $B$, since $B$ is simple and $q(1)\ne0$.
+Every pure state $h$ of $B$ therefore extends through this quotient to a
+state of $C$ annihilating $J$.  By \eqref{eq:U}, this extension is $h\circ E$.
+Thus $h(E(a))=0$ for every positive $a\in J$ and every pure $h$ on $B$.
+Pure states separate positive elements, so $E(a)=0$; faithfulness gives
+$a=0$.  Hence $J=0$.
+\end{proof}
diff --git a/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/naimark-counterexample-zfc.pdf b/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/naimark-counterexample-zfc.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..452c9e55a5ebe92eb62ad0fc385e949baabbe4e4
--- /dev/null
+++ b/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026/naimark-counterexample-zfc.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:5f8f878435bb94400a8f7c0c556bfc48d4bfc00782aa68066b6d4dac2d1560a8
+size 371791
diff --git a/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/README.md b/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..a7568f88f8769ec9aab356b4a1b9e6d38439a3d4
--- /dev/null
+++ b/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/README.md
@@ -0,0 +1,17 @@
+# [A counterexample to Sidorenko's conjecture](paper.pdf)
+
+**Author:** OpenAI  
+**Date:** September 23, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-counterexample-to-Sidorenkos-conjecture-September-23-2026,
+  author = {{OpenAI}},
+  title = {{A counterexample to Sidorenko's conjecture}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/paper.pdf}{OAI:A-counterexample-to-Sidorenkos-conjecture-September-23-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/figures/corner-activation.tex b/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/figures/corner-activation.tex
new file mode 100644
index 0000000000000000000000000000000000000000..cbb649af8c567da43a85e4f0f2d92694d783d19a
--- /dev/null
+++ b/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/figures/corner-activation.tex
@@ -0,0 +1,27 @@
+\begin{figure}[htbp]
+\centering
+\begin{tikzpicture}[font=\small]
+  \fill[blue!7] (-1.25,0) ellipse (2.05 and 1.45);
+  \fill[orange!9] (1.25,0) ellipse (2.05 and 1.45);
+  \draw[blue!60!black,thick] (-1.25,0) ellipse (2.05 and 1.45);
+  \draw[orange!70!black,thick] (1.25,0) ellipse (2.05 and 1.45);
+  \node[blue!60!black] at (-2.4,1.35) {$U_e$};
+  \node[orange!70!black] at (2.4,1.35) {$U_f$};
+  \node at (-1.7,.75) {$(w,j)$};
+  \node at (-2.0,-.1) {$(v,l)$};
+  \node at (-1.45,-.85) {$(w,l)$};
+  \node at (1.7,.75) {$(z,j)$};
+  \node at (2.0,-.1) {$(v,m)$};
+  \node at (1.45,-.85) {$(z,m)$};
+  \fill[linkblue] (0,.55) circle (1.7pt);
+  \node[align=center] at (0,-.05) {$\kappa$\\$=(v,j)$};
+  \node at (0,-1.9) {$M_\kappa=U_e\cup U_f,\qquad
+                    \Gamma_\kappa=M_\kappa\setminus\{\kappa\}$};
+\end{tikzpicture}
+\caption{The seven coordinates available at the corner
+$\kappa=(v,j)$, where $j=\{v,w,z\}$, $e=\{v,w\}$ and
+$f=\{v,z\}$. The opposite faces on $e$ and $f$ are $l$ and $m$.
+The two activation branches keep the same six outer coordinates;
+one also keeps the common corner marked by the dot.}
+\label{fig:kernel-corner-activation}
+\end{figure}
diff --git a/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/figures/proof-overview.tex b/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/figures/proof-overview.tex
new file mode 100644
index 0000000000000000000000000000000000000000..d7d477b123ccb10d53d0e787e5d089e47e6feaa5
--- /dev/null
+++ b/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/figures/proof-overview.tex
@@ -0,0 +1,43 @@
+\begin{figure}[htbp]
+\centering
+\begin{tikzpicture}[
+  font=\small,
+  proofbox/.style={draw=linkblue!70,rounded corners=2pt,
+    fill=linkblue!4,align=center,inner sep=5pt,text width=3.4cm},
+  proofstep/.style={-{Latex[length=2mm]},thick,draw=linkblue!80}]
+  \node[proofbox,text width=6.0cm] (complex) at (0,0)
+    {Fixed triple system\\Section~\ref{sec:complex}};
+  \node[proofbox,text width=3.8cm] (activation) at (-3.7,-1.5)
+    {Activation supports\\and type scores\\Section~\ref{sec:kernel}};
+  \node[proofbox] (transverse) at (0.9,-1.5)
+    {Transverse contribution\\Section~\ref{sec:transverse}};
+  \node[proofbox] (tail) at (4.9,-1.5)
+    {Weighted singular\\bound\\Sections~\ref{sec:orbits}--\ref{sec:tail}};
+  \node[proofbox,text width=5.8cm] (rank) at (2.9,-2.9)
+    {Rank-layer limit\\Lemma~\ref{lem:rank-layer}};
+  \node[proofbox,text width=6.5cm] (negative) at (0,-4.3)
+    {Negative averaged coefficient\\in one limiting oriented density};
+  \node[proofbox,text width=6.5cm] (kernel) at (0,-5.6)
+    {Dilution and crossed product\\symmetric kernel with a strict deficit};
+  \node[proofbox,text width=6.5cm] (host) at (0,-6.9)
+    {Sampling a finite simple host $G$\\Lemma~\ref{lem:finite-host}};
+  \coordinate (estimatefork) at (2.9,-0.7);
+  \draw[proofstep] (complex.south west) -- (activation.north);
+  \draw[proofstep] (complex.south east) -- (estimatefork) -- (transverse.north);
+  \draw[proofstep] (estimatefork) -- (tail.north);
+  \draw[proofstep] (transverse.south) -- (rank.north west);
+  \draw[proofstep] (tail.south) -- (rank.north east);
+  \draw[proofstep] (activation.south) -- (negative.north west);
+  \draw[proofstep] (rank.south) -- (negative.north east);
+  \draw[proofstep] (negative) -- (kernel);
+  \draw[proofstep] (kernel) -- (host);
+\end{tikzpicture}
+\caption{Logical dependencies of the counterexample. The fixed triple
+system supplies the combinatorial properties used in both branches.
+The rank-layer limit realizes the averaged coefficient made negative
+by the finite activation construction. Section~\ref{sec:kernel}
+constructs the kernel and transfers its strict deficit to a finite
+simple host assuming Lemma~\ref{lem:rank-layer};
+Sections~\ref{sec:transverse}--\ref{sec:tail} then prove that input.}
+\label{fig:proof-overview}
+\end{figure}
diff --git a/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/macros.tex b/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/macros.tex
new file mode 100644
index 0000000000000000000000000000000000000000..23c43de820e4bfee4a0596f1d7deb820252ffa49
--- /dev/null
+++ b/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/macros.tex
@@ -0,0 +1,48 @@
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage[mathscr]{eucal}
+\usepackage[margin=1in]{geometry}
+\usepackage{microtype}
+\usepackage{booktabs,longtable,array}
+\usepackage{enumitem}
+\usepackage{graphicx}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta,positioning,calc,fit,backgrounds}
+\usepackage{xcolor}
+\definecolor{linkblue}{RGB}{31,67,102}
+\usepackage[colorlinks=true,linkcolor=linkblue,citecolor=linkblue,urlcolor=linkblue,
+  bookmarksnumbered=true,pdfusetitle]{hyperref}
+\usepackage[nameinlink,capitalise,noabbrev]{cleveref}
+
+\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}
+\numberwithin{equation}{section}
+
+\newcommand{\F}{\mathbb F}
+\newcommand{\R}{\mathbb R}
+\newcommand{\E}{\mathbb E}
+\renewcommand{\P}{\mathbb P}
+\newcommand{\1}{\mathbf 1}
+\DeclareMathOperator{\Sym}{Sym}
+\DeclareMathOperator{\Lag}{Lag}
+\DeclareMathOperator{\rank}{rank}
+\DeclareMathOperator{\diag}{diag}
+\DeclareMathOperator{\Hom}{hom}
+\DeclareMathOperator{\Var}{Var}
+\DeclareMathOperator{\Cov}{Cov}
+\DeclareMathOperator{\TV}{TV}
+
+\setlist{topsep=4pt,itemsep=2pt,parsep=0pt}
+\setlength{\emergencystretch}{2em}
+\setcounter{tocdepth}{1}
+\allowdisplaybreaks[1]
+\ifdefined\pdfinfoomitdate\pdfinfoomitdate=1\fi
+\ifdefined\pdftrailerid\pdftrailerid{}\fi
+\ifdefined\pdfsuppressptexinfo\pdfsuppressptexinfo=15\fi
diff --git a/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/main.tex b/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/main.tex
new file mode 100644
index 0000000000000000000000000000000000000000..7b55ebcef6643c64bff9c86b569f3692272a65d4
--- /dev/null
+++ b/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/main.tex
@@ -0,0 +1,29 @@
+\documentclass[11pt]{article}
+\input{macros}
+\title{A counterexample to Sidorenko's conjecture}
+\author{OpenAI}
+\date{September 23, 2026}
+
+\begin{document}
+\maketitle
+\begin{abstract}
+We disprove Sidorenko's conjecture with a bipartite graph on \(35\) vertices
+and \(66\) edges: its homomorphism density in some finite simple graph is
+smaller than the conjectured lower bound. The same connected graph also
+disproves the forcing conjecture: at one fixed density, asymptotically
+matching the edge and pattern densities does not imply quasirandomness.
+\end{abstract}
+\tableofcontents
+
+\input{sections/introduction}
+\input{sections/complex}
+\input{sections/kernel}
+\input{sections/transverse}
+\input{sections/orbits}
+\input{sections/planted}
+\input{sections/tail}
+\input{sections/consequences}
+
+\bibliographystyle{plain}
+\bibliography{references}
+\end{document}
diff --git a/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/references.bib b/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/references.bib
new file mode 100644
index 0000000000000000000000000000000000000000..e2db4f3963996a839bc52e04ea9b675af683cd88
--- /dev/null
+++ b/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/references.bib
@@ -0,0 +1,262 @@
+@article{chunggrahamwilson1989,
+  author = {Chung, F. R. K. and Graham, R. L. and Wilson, R. M.},
+  title = {Quasi-random graphs},
+  journal = {Combinatorica},
+  volume = {9}, number = {4}, pages = {345--362}, year = {1989},
+  doi = {10.1007/BF02125347},
+  url = {https://doi.org/10.1007/BF02125347}
+}
+
+@article{conlonfoxsudakov2010,
+  author = {Conlon, David and Fox, Jacob and Sudakov, Benny},
+  title = {An approximate version of {Sidorenko}'s conjecture},
+  journal = {Geometric and Functional Analysis},
+  volume = {20}, number = {6}, pages = {1354--1366}, year = {2010},
+  doi = {10.1007/s00039-010-0097-0},
+  url = {https://doi.org/10.1007/s00039-010-0097-0}
+}
+
+@article{conlonkimleelee2018,
+  author = {Conlon, David and Kim, Jeong Han and Lee, Choongbum and Lee, Joonkyung},
+  title = {Some advances on {Sidorenko}'s conjecture},
+  journal = {Journal of the London Mathematical Society},
+  series = {2}, volume = {98}, number = {3}, pages = {593--608}, year = {2018},
+  doi = {10.1112/jlms.12142}, url = {https://doi.org/10.1112/jlms.12142}
+}
+
+@article{conlonlee2017,
+  author = {Conlon, David and Lee, Joonkyung},
+  title = {Finite reflection groups and graph norms},
+  journal = {Advances in Mathematics}, volume = {315}, pages = {130--165}, year = {2017},
+  doi = {10.1016/j.aim.2017.05.009}, url = {https://doi.org/10.1016/j.aim.2017.05.009}
+}
+
+@article{conlonlee2021,
+  author = {Conlon, David and Lee, Joonkyung},
+  title = {{Sidorenko}'s conjecture for blow-ups},
+  journal = {Discrete Analysis}, year = {2021}, note = {Paper No.~2, 13 pp.},
+  doi = {10.19086/da.21472}, url = {https://doi.org/10.19086/da.21472}
+}
+
+@article{coregliano2024,
+  author = {Coregliano, Leonardo N.},
+  title = {Left-cut-percolation and induced-{Sidorenko} bigraphs},
+  journal = {SIAM Journal on Discrete Mathematics},
+  volume = {38}, number = {2}, pages = {1586--1629}, year = {2024},
+  doi = {10.1137/22M1526794},
+  url = {https://doi.org/10.1137/22M1526794}
+}
+
+@misc{coreglianorazborov2021,
+  author = {Coregliano, Leonardo N. and Razborov, Alexander A.},
+  title = {Biregularity in {Sidorenko}'s conjecture},
+  year = {2021},
+  eprint = {2108.06599},
+  archivePrefix = {arXiv},
+  note = {arXiv:2108.06599v2},
+  url = {https://arxiv.org/abs/2108.06599}
+}
+
+@incollection{erdoslovaszspencer1979,
+  author = {Erd{\H{o}}s, Paul and Lov{\'a}sz, L{\'a}szl{\'o} and Spencer, Joel},
+  title = {Strong Independence of Graphcopy Functions},
+  booktitle = {Graph Theory and Related Topics},
+  publisher = {Academic Press},
+  pages = {165--172},
+  year = {1979},
+  url = {https://users.renyi.hu/~p_erdos/1979-24.pdf}
+}
+
+@article{gowerswolf2011,
+  author = {W. T. Gowers and Julia Wolf},
+  title = {Linear forms and quadratic uniformity for functions on {$\mathbb{F}_p^n$}},
+  journal = {Mathematika}, volume = {57}, number = {2},
+  pages = {215--237}, year = {2011},
+  doi = {10.1112/S0025579311001264},
+  url = {https://arxiv.org/abs/1002.2209}
+}
+
+@article{hatami2010,
+  author = {Hatami, Hamed},
+  title = {Graph norms and {Sidorenko}'s conjecture},
+  journal = {Israel Journal of Mathematics},
+  volume = {175},
+  pages = {125--150},
+  year = {2010},
+  doi = {10.1007/s11856-010-0005-1},
+  url = {https://arxiv.org/abs/0806.0047}
+}
+
+@article{huowan1993,
+  author = {Huo, Yuanji and Wan, Zhexian},
+  title = {Non-symmetric association schemes of symmetric matrices},
+  journal = {Acta Mathematicae Applicatae Sinica, English Series},
+  volume = {9},
+  pages = {236--255},
+  year = {1993},
+  doi = {10.1007/BF02032918},
+  url = {https://link.springer.com/article/10.1007/BF02032918}
+}
+
+@article{imliliu2026,
+  author = {Im, Seonghyuk and Li, Ruonan and Liu, Hong},
+  title = {{Sidorenko}'s conjecture for subdivisions and theta substitutions},
+  journal = {Combinatorics, Probability and Computing},
+  volume = {35}, number = {2}, pages = {269--279}, year = {2026},
+  doi = {10.1017/S0963548325100242}, url = {https://doi.org/10.1017/S0963548325100242}
+}
+
+@article{kimleelee2016,
+  author = {Kim, Jeong Han and Lee, Choongbum and Lee, Joonkyung},
+  title = {Two approaches to {Sidorenko}'s conjecture},
+  journal = {Transactions of the American Mathematical Society},
+  volume = {368}, number = {7}, pages = {5057--5074}, year = {2016},
+  doi = {10.1090/tran/6487}, url = {https://doi.org/10.1090/tran/6487}
+}
+
+@article{kramertent2010,
+  author = {Kramer, Linus and Tent, Katrin},
+  title = {A {Maslov} cocycle for unitary groups},
+  journal = {Proceedings of the London Mathematical Society},
+  series = {3},
+  volume = {100},
+  pages = {91--115},
+  year = {2010},
+  doi = {10.1112/plms/pdp023},
+  url = {https://webhomes.maths.ed.ac.uk/~v1ranick/papers/kramertent.pdf}
+}
+
+@article{liszegedy2025,
+  author = {Li, X. and Szegedy, B.},
+  title = {On the logarithmic calculus and {Sidorenko}'s conjecture},
+  journal = {Acta Mathematica Hungarica}, volume = {177}, pages = {450--463}, year = {2025},
+  doi = {10.1007/s10474-026-01582-2}, url = {https://doi.org/10.1007/s10474-026-01582-2},
+  note = {First circulated as arXiv:1107.1153 in 2011; published online March 1, 2026}
+}
+
+@article{lovasz2011local,
+  author = {Lov{\'a}sz, L{\'a}szl{\'o}},
+  title = {Subgraph densities in signed graphons and the local {Simonovits--Sidorenko} conjecture},
+  journal = {Electronic Journal of Combinatorics},
+  volume = {18},
+  number = {1},
+  pages = {P127},
+  year = {2011},
+  doi = {10.37236/614},
+  url = {https://www.combinatorics.org/ojs/index.php/eljc/article/view/v18i1p127}
+}
+
+@article{lovaszszegedy2006,
+  author = {L{\'a}szl{\'o} Lov{\'a}sz and Bal{\'a}zs Szegedy},
+  title = {Limits of dense graph sequences},
+  journal = {Journal of Combinatorial Theory, Series B},
+  volume = {96}, number = {6}, pages = {933--957}, year = {2006},
+  doi = {10.1016/j.jctb.2006.05.002},
+  url = {https://doi.org/10.1016/j.jctb.2006.05.002}
+}
+
+@article{mckay1981,
+  author = {McKay, Brendan D.},
+  title = {Practical graph isomorphism},
+  journal = {Congressus Numerantium},
+  volume = {30},
+  year = {1981},
+  pages = {45--87},
+  url = {https://users.cecs.anu.edu.au/~bdm/nauty/pgi.pdf}
+}
+
+@article{schmidt2015,
+  author = {Kai-Uwe Schmidt},
+  title = {Symmetric bilinear forms over finite fields with applications to coding theory},
+  journal = {Journal of Algebraic Combinatorics},
+  volume = {42}, number = {2}, pages = {635--670}, year = {2015},
+  doi = {10.1007/s10801-015-0595-0},
+  url = {https://arxiv.org/abs/1410.7184}
+}
+
+@article{sidorenko1991,
+  author = {Sidorenko, A. F.},
+  title = {Inequalities for functionals generated by bipartite graphs},
+  journal = {Diskretnaya Matematika},
+  volume = {3},
+  number = {3},
+  pages = {50--65},
+  year = {1991},
+  url = {https://www.mathnet.ru/eng/dm804},
+  note = {In Russian. English translation in Discrete Mathematics and
+    Applications 2 (1992), no.~5, 489--504,
+    \href{https://doi.org/10.1515/dma.1992.2.5.489}{doi:10.1515/dma.1992.2.5.489}}
+}
+
+@article{sidorenko1993,
+  author = {Sidorenko, Alexander},
+  title = {A correlation inequality for bipartite graphs},
+  journal = {Graphs and Combinatorics},
+  volume = {9},
+  number = {2--4},
+  pages = {201--204},
+  year = {1993},
+  doi = {10.1007/BF02988307},
+  url = {https://doi.org/10.1007/BF02988307}
+}
+
+@article{skokanthoma2004,
+  author = {Skokan, Jozef and Thoma, Lubos},
+  title = {Bipartite subgraphs and quasi-randomness},
+  journal = {Graphs and Combinatorics},
+  volume = {20}, number = {2}, pages = {255--262}, year = {2004},
+  doi = {10.1007/s00373-004-0556-1},
+  url = {https://doi.org/10.1007/s00373-004-0556-1}
+}
+
+@incollection{simonovits1984,
+  author = {Simonovits, Mikl{\'o}s},
+  title = {Extremal graph problems, degenerate extremal problems, and
+    supersaturated graphs},
+  booktitle = {Progress in Graph Theory},
+  editor = {Bondy, J. A. and Murty, U. S. R.},
+  publisher = {Academic Press},
+  address = {Toronto},
+  pages = {419--437},
+  year = {1984},
+  url = {https://users.renyi.hu/~miki/waterloo.pdf}
+}
+
+@misc{szegedy2015entropy,
+  author = {Szegedy, Bal{\'a}zs},
+  title = {An information theoretic approach to {Sidorenko}'s conjecture},
+  year = {2015},
+  eprint = {1406.6738},
+  archivePrefix = {arXiv},
+  note = {arXiv:1406.6738v3; first posted in 2014},
+  url = {https://arxiv.org/abs/1406.6738}
+}
+
+@book{taylor1992,
+  author = {Donald E. Taylor},
+  title = {The Geometry of the Classical Groups},
+  series = {Sigma Series in Pure Mathematics}, volume = {9},
+  publisher = {Heldermann Verlag}, address = {Berlin}, year = {1992},
+  isbn = {3-88538-009-9},
+  url = {https://www.heldermann.de/SSPM/SSPM09/sspm09.htm}
+}
+
+@article{weisfeilerleman1968,
+  author = {Weisfeiler, B. Yu. and Leman, A. A.},
+  title = {A reduction of a graph to canonical form and an algebra arising during this reduction},
+  journal = {Nauchno-Technicheskaya Informatsiya, Seriya 2},
+  year = {1968},
+  pages = {12--16},
+  note = {No.~9. In Russian; English translation by Grigory Ryabov},
+  url = {https://www.iti.zcu.cz/wl2018/pdf/wl_paper_translation.pdf}
+}
+
+@misc{zhao2026conjugacy,
+  author = {Zhao, Yuqi},
+  title = {Conjugacy class averages and {Sidorenko}'s conjecture},
+  year = {2026},
+  eprint = {2606.15368},
+  archivePrefix = {arXiv},
+  howpublished = {arXiv:2606.15368v1},
+  url = {https://arxiv.org/abs/2606.15368}
+}
diff --git a/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/sections/complex.tex b/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/sections/complex.tex
new file mode 100644
index 0000000000000000000000000000000000000000..178b0e9da21867a727ea847dae8e537b9d719a70
--- /dev/null
+++ b/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/sections/complex.tex
@@ -0,0 +1,192 @@
+\section{The finite complex}
+\label{sec:complex}
+
+We specify the fixed triple system and certify the properties used in
+both the kernel construction and the singular-tail bound.
+
+Let $I=\{0,\ldots,12\}$ and $J=\{0,\ldots,21\}$. Associate to
+each $j\in J$ the three-element subset of $I$ in Table~\ref{tab:complex};
+we call these subsets \emph{faces} and also denote them by $j$.
+Let $E$ be the set of unordered point pairs contained in a face.
+The graph used throughout the paper is the incidence graph
+\[
+ H=(I\sqcup J,\mathscr C),\qquad
+ \mathscr C=\{(i,j):i\in j\}.
+\]
+The copies of $I$ and $J$ in this disjoint union are distinct even when
+their numerical labels agree. We call an incidence $(i,j)$ a
+\emph{corner}. Thus $H$ has $35$ vertices and $66$ edges.
+
+\begin{table}[htbp]
+\centering
+\small
+\setlength{\tabcolsep}{7pt}
+\begin{tabular}{c@{\qquad}ccc@{\qquad}ccc@{\qquad}c}
+\toprule
+Face&\multicolumn{3}{c}{Points}&\multicolumn{3}{c}{Opposite face}&Bits\\
+ $j$&$i_1$&$i_2$&$i_3$&$j_{12}$&$j_{13}$&$j_{23}$&\\
+\midrule
+0 &0&1&3   &1&2&4   &000\\
+1 &0&1&9   &0&3&7   &010\\
+2 &0&2&3   &3&0&8   &001\\
+3 &0&2&9   &2&1&9   &110\\
+4 &1&3&10  &0&5&11  &001\\
+5 &1&7&10  &6&4&16  &000\\
+6 &1&7&12  &5&7&18  &100\\
+7 &1&9&12  &1&6&21  &110\\
+8 &2&3&4   &2&9&10  &101\\
+9 &2&4&9   &8&3&12  &100\\
+10&3&4&11  &8&11&13 &111\\
+11&3&10&11 &4&10&17 &101\\
+12&4&8&9   &13&9&21 &101\\
+13&4&8&11  &12&10&19&011\\
+14&5&6&7   &15&16&18&000\\
+15&5&6&11  &14&17&19&110\\
+16&5&7&10  &14&17&5 &010\\
+17&5&10&11 &16&15&11&111\\
+18&6&7&12  &14&20&6 &001\\
+19&6&8&11  &20&15&13&010\\
+20&6&8&12  &19&18&21&011\\
+21&8&9&12  &12&20&7 &111\\
+\bottomrule
+\end{tabular}
+\caption{The faces of the complex. The entry $j_{ab}$ is the other
+face containing $\{i_a,i_b\}$. Bit positions are read from left to right.}
+\label{tab:complex}
+\end{table}
+
+Join two faces when they share a pair. This defines the
+\emph{face-neighbor graph}. For a pair $e\in E$, define
+\begin{equation}
+ a_e=\frac{1}{3}\#\{\text{bit positions at which the two faces on $e$
+ differ}\}.
+ \label{eq:complex-separation}
+\end{equation}
+
+Color refinement starts with a separate color for each part of $H$.
+A refinement step replaces a vertex's color by the ordered pair
+consisting of its old color and the histogram of its neighbors' old
+colors. This is the classical color-refinement procedure; see
+Weisfeiler and Leman~\cite{weisfeilerleman1968} and
+McKay~\cite[Sections~2.3--2.6]{mckay1981} for its equitable-partition interpretation.
+
+\begin{proposition}
+\label{prop:complex}
+The complex has the following properties.
+\begin{enumerate}
+\item Every pair in $E$ lies in exactly two faces, $|E|=33$, and the
+face-neighbor graph is connected.
+\item For each bit position, each of its two classes consists of
+$11$ faces covering all $13$ points. Each class has an ordering in
+which every face after the first shares a pair with an earlier face
+and introduces exactly one new point.
+\item Every $a_e$ is at least $1/3$, and some $a_e$ is greater than
+$1/3$.
+\item Color refinement of $H$, starting with a separate color for each
+part, eventually gives distinct colors to all vertices within either
+part.
+\end{enumerate}
+\end{proposition}
+
+\begin{proof}
+The neighbor entries in Table~\ref{tab:complex} pair the $66$
+face--pair incidences into $33$ reciprocal pairs. Comparing the
+triples in each indicated pair shows that their common point pair is
+exactly the one specified. No other face contains that pair, as is
+also checked by the same table.
+
+Table~\ref{tab:complex-orders} gives all six orders and their
+new-point certificates for the second assertion.
+
+\begin{table}[htbp]
+\centering
+\small
+\setlength{\tabcolsep}{5pt}
+\begin{tabular}{cc@{\qquad}ll}
+\toprule
+\multicolumn{2}{c}{Bit class}&Face order&New points after the first face\\
+Position&Value&&\\
+\midrule
+1&0&$0,1,2,4,5,16,14,18,20,19,13$
+ &$9,2,10,7,5,6,12,8,11,4$\\
+1&1&$3,9,8,10,11,17,15,12,21,7,6$
+ &$4,3,11,10,5,6,8,12,1,7$\\
+\addlinespace
+2&0&$0,2,8,9,12,4,11,5,6,18,14$
+ &$2,4,9,8,10,11,7,12,6,5$\\
+2&1&$1,3,7,21,20,19,13,10,15,17,16$
+ &$2,12,8,6,11,4,3,5,10,7$\\
+\addlinespace
+3&0&$0,1,3,9,7,6,5,16,14,15,19$
+ &$9,2,4,12,7,10,5,6,11,8$\\
+3&1&$2,8,10,11,4,17,13,12,21,20,18$
+ &$4,11,10,1,5,8,9,12,6,7$\\
+\bottomrule
+\end{tabular}
+\caption{An exposure order for each bit class. Every face after the
+first shares a pair with an earlier face and introduces the
+corresponding point in the last column. Bit positions are read from
+left to right in Table~\ref{tab:complex}.}
+\label{tab:complex-orders}
+\end{table}
+
+The root face contributes its three points, and these ten new points
+give all of $I$ in each case. The shared-pair condition follows from
+the neighbor columns of Table~\ref{tab:complex}. Thus every bit class
+is connected. The first two classes are joined, for example, by
+faces $2$ and $3$, so the whole face-neighbor graph is connected.
+Inspection of the bit labels gives separation in at least one
+position on every pair. The pair $\{0,2\}$ has adjacent labels
+$001$ and $110$, so in fact $a_{\{0,2\}}=1$.
+
+It remains to verify the color-refinement assertion. First consider
+the point graph $(I,E)$. Its point degrees $d_i$ and the numbers
+$n_4(i),n_5(i),n_6(i)$ of neighbors of degrees $4,5,6$ are as follows:
+\begin{equation}
+\begin{array}{c|c|ccc@{\qquad}c|c|ccc}
+i&d_i&n_4&n_5&n_6&i&d_i&n_4&n_5&n_6\\\hline
+0&4&1&0&3&7&5&1&3&1\\
+1&6&1&3&2&8&5&0&3&2\\
+2&4&1&1&2&9&6&2&3&1\\
+3&6&2&2&2&10&5&1&1&3\\
+4&5&1&1&3&11&6&1&4&1\\
+5&4&0&3&1&12&5&0&3&2\\
+6&5&1&3&1&&&&&
+\end{array}
+\label{eq:complex-colors}
+\end{equation}
+Only the pairs $\{4,10\}$, $\{6,7\}$, and $\{8,12\}$ have
+equal signatures in Equation~\eqref{eq:complex-colors}. The first is
+separated by adjacency to the already distinguished point $2$:
+$4$ is adjacent to $2$, whereas $10$ is not. The other two are
+separated by adjacency to the already distinguished point $11$:
+$6$ and $8$ are adjacent to $11$, whereas $7$ and $12$ are not.
+Hence refinement on the point graph is discrete.
+
+This point refinement can be recovered from incidence refinement.
+The degree of $i$ in $H$ equals $d_i$: counting other points in its
+incident faces gives twice each neighbor in $(I,E)$. More generally,
+suppose the current incidence color of every point determines a point
+coloring \(\ell\) through a fixed projection. One further incidence
+step determines, at each face \(j\), the color-indexed histogram
+\[
+ m_j(\gamma)=\#\{k\in j:\ell(k)=\gamma\}.
+\]
+The next step determines the multiset of these histograms over the
+faces incident to each point. For every color
+\(\gamma\), this gives
+\[
+ \sum_{j\ni i}
+ \left(m_j(\gamma)-\1_{\{\gamma=\ell(i)\}}\right)
+ =2\,\#\{k:\{i,k\}\in E,\ \ell(k)=\gamma\}.
+\]
+The subtraction removes only the occurrence of \(i\) in each face,
+even when other points have the same color. Each neighbor is counted
+twice because its pair with \(i\) lies in two faces. Dividing by two
+therefore recovers the point neighbor-color histogram.
+Thus two incidence steps recover each further point refinement step.
+After the point colors are discrete, one more step distinguishes all
+faces, since their three-element point sets are distinct. This proves
+the last assertion without any assumption about a chosen refinement
+implementation.
+\end{proof}
diff --git a/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/sections/consequences.tex b/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/sections/consequences.tex
new file mode 100644
index 0000000000000000000000000000000000000000..b8fe63813bd09d9de2257f3a017fe5b23d7910dd
--- /dev/null
+++ b/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/sections/consequences.tex
@@ -0,0 +1,146 @@
+\section{Amplification and the forcing consequence}
+\label{sec:consequences}
+
+Theorem~\ref{thm:main} is now proved. We derive the two consequences
+stated in the introduction directly from its finite host. The first
+uses tensor powers to amplify the homomorphism deficit and blow-ups
+to compare injective copies at a positive limiting density. The second
+keeps the edge density fixed while interpolating to an exact graphon
+equality, and then samples a nonquasirandom sequence of finite graphs.
+
+\subsection{Amplification and injective copies}
+
+\begin{proof}[Proof of Corollary~\ref{cor:amplification}]
+Take \(G\) from Theorem~\ref{thm:main} and put
+\(N=|V(G)|\), \(p_0=p(G)\), \(t_0=t(H,G)\), and
+\(\rho=t_0/p_0^{66}\). Simplicity and an edge give \(0<p_0<1\),
+and mapping the two parts of \(H\) to the endpoints of an edge gives
+\(t_0>0\). Thus \(0<\rho<1\).
+
+Let \(G^{\times r}\) be the categorical power, whose adjacency is
+coordinatewise. It is finite, simple and undirected, and
+homomorphisms into it factor coordinatewise. Hence
+\[
+p(G^{\times r})=p_0^r,\qquad
+t(H,G^{\times r})=t_0^r,\qquad
+\frac{t(H,G^{\times r})}{p(G^{\times r})^{66}}=\rho^r\longrightarrow0,
+\]
+which proves (i). This is the standard tensor-power principle
+\cite[Section~2]{conlonfoxsudakov2010}.
+
+For (ii), use instead the balanced independent blow-ups \(F_s=G[s]\):
+replace each vertex by an independent set of size \(s\), and each edge
+by the complete bipartite graph between its two sets. Then \(n_s=Ns\),
+\(p(F_s)=p_0\), and \(t(H,F_s)=t_0\), since each homomorphism into
+\(G\) has exactly \(s^{35}\) lifts. For \(n_s\ge35\), set
+\(q_s=\operatorname{Inj}(H,F_s)/(n_s)_{35}\). This is the homomorphism
+probability for a uniform vertex map conditioned on injectivity, so
+the collision union bound gives
+\[
+|q_s-t_0|\le1-\frac{(n_s)_{35}}{n_s^{35}}
+\le\frac{\binom{35}{2}}{n_s}.
+\]
+Meanwhile \(d_s=p_0n_s/(n_s-1)\to p_0\). Therefore
+\(q_s/d_s^{66}\to\rho<1\), proving (ii) with
+\(p_*=p_0\) and \(\delta_*=(1-\rho)/2\).
+\end{proof}
+
+\subsection{Failure of the forcing conjecture}
+
+\begin{proof}[Proof of Corollary~\ref{cor:forcing}]
+Let \(G\) be the host from Theorem~\ref{thm:main}, put
+\(N=|V(G)|\), and set \(p=p(G)\). Since \(G\) has an edge and is
+simple,
+\[
+0<p\le\frac{N-1}{N}<1.
+\]
+Let \(W_0\) be its adjacency kernel on the uniform probability space
+\(V(G)\). Independent vertex samples count all homomorphisms, so
+\[
+t(K_2,W_0)=p,\qquad t(H,W_0)=t(H,G)<p^{66}.
+\]
+Moreover, \(W_0\) is \(\{0,1\}\)-valued, and hence
+\(\|W_0-p\|_2^2=p(1-p)>0\).
+
+On the balanced sign space \(\{-1,1\}\), choose
+\(0<\delta\le\min(p,1-p)\) and define the rank-one perturbation
+\[
+W_1(a,b)=p+\delta ab.
+\]
+It takes values in \([0,1]\) and has mean \(p\). For
+\(A\subseteq E(H)\), let \(d_A(v)\) be the degree of \(v\) in the
+edge set \(A\). Expanding the product defining \(t(H,W_1)\) and
+averaging the independent signs gives
+\[
+t(H,W_1)=
+\sum_{\substack{A\subseteq E(H)\\ d_A(v)\text{ even for every }v}}
+p^{66-|A|}\delta^{|A|}>p^{66}.
+\]
+Indeed, the empty edge set contributes \(p^{66}\), every surviving
+term is nonnegative, and the edge set of a cycle contributes a
+strictly positive term. Such a cycle exists because a forest on
+\(35\) vertices has at most \(34\) edges, whereas \(H\) has \(66\).
+
+Lift the two kernels to the independent coordinates of
+\(\Omega=V(G)\times\{-1,1\}\), with product probability, and set
+\[
+W_s\bigl((x,a),(y,b)\bigr)
+=(1-s)W_0(x,y)+sW_1(a,b),\qquad 0\le s\le1.
+\]
+Each \(W_s\) is symmetric, takes values in \([0,1]\), and has mean
+\(p\). The endpoint lifts preserve their \(H\)-densities. The two
+centered summands depend on independent coordinate pairs and have
+mean zero, so they are orthogonal in \(L^2(\Omega^2)\). Thus
+\[
+\|W_s-p\|_2^2=(1-s)^2p(1-p)+s^2\delta^2>0
+\qquad(0\le s\le1).
+\]
+The function \(s\mapsto t(H,W_s)\) is a polynomial. Its endpoint
+values lie strictly below and above \(p^{66}\), respectively, so
+some \(s_*\in(0,1)\) satisfies \(t(H,W_{s_*})=p^{66}\). The norm
+identity shows that \(W=W_{s_*}\) is nonconstant.
+
+For completeness, this gives failure of asymptotic quasirandomness,
+not just a graphon equality. On the finite atom space of \(W\), put
+\[
+r(x)=\E_z W(x,z),\qquad c(x,y)=\E_z W(x,z)W(y,z).
+\]
+Two applications of Cauchy--Schwarz give
+\begin{align*}
+t(C_4,W)&=\E_{x,y}c(x,y)^2\\
+&\ge\bigl(\E_{x,y}c(x,y)\bigr)^2
+=\bigl(\E_z r(z)^2\bigr)^2\ge p^4.
+\end{align*}
+Equality would force \(r(x)=p\) for every \(x\) and
+\(c(x,y)=p^2\) for every pair \((x,y)\): every atom pair has positive
+mass, including the diagonal pairs.
+Then, for every \(x\),
+\[
+\E_z\bigl(W(x,z)-p\bigr)^2=c(x,x)-2pr(x)+p^2=0,
+\]
+contrary to nonconstancy. Hence \(t(C_4,W)>p^4\).
+
+Use the independent sampling construction for this finite kernel,
+as in Lemma~\ref{lem:finite-host} and
+\cite[Section~2.6, Lemma~2.4]{lovaszszegedy2006}. For each fixed finite graph \(F\),
+the collision count gives
+\(\E t(F,G_n)=t(F,W)+O_F(n^{-1})\). The covariance of the indicators
+for two vertex maps vanishes when their images are disjoint, and the
+proportion of pairs with overlapping images is \(O_F(n^{-1})\).
+Therefore \(\operatorname{Var}(t(F,G_n))=O_F(n^{-1})\). With tolerance
+\(n^{-1/4}\), Chebyshev's inequality and a union bound for
+\(F=K_2,H,C_4\) give total failure probability
+\(O(n^{-1/2})<1\) for all sufficiently large \(n\). Choose one
+successful realization for each such \(n\). The expectation estimate
+then gives a deterministic sequence of finite simple graphs with
+\[
+p(G_n)\to p,\qquad t(H,G_n)\to p^{66},\qquad
+t(C_4,G_n)\to t(C_4,W)>p^4.
+\]
+This sequence is not quasirandom of density \(p\), proving the claim.
+\end{proof}
+
+The graphon equality is exact, whereas the finite sequence matches the
+edge and $H$-densities only in the limit. The argument supplies one
+density $p\in(0,1)$ and does not assert failure of $p$-forcing at every
+density.
diff --git a/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/sections/introduction.tex b/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/sections/introduction.tex
new file mode 100644
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+++ b/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/sections/introduction.tex
@@ -0,0 +1,375 @@
+\section{Introduction}\label{sec:introduction}
+
+For finite simple undirected graphs \(H\) and \(G\) with
+\(|V(G)|\ge1\), write
+\[
+t(H,G)=\frac{\Hom(H,G)}{|V(G)|^{|V(H)|}},
+\qquad
+p(G)=\frac{2|E(G)|}{|V(G)|^2},
+\]
+where \(\Hom(H,G)\) counts every edge-preserving map, including
+noninjective maps. Sidorenko's conjecture asserts that
+\begin{equation}\label{eq:sidorenko}
+t(H,G)\ge p(G)^{|E(H)|}
+\end{equation}
+whenever \(H\) is bipartite and has at least one edge.
+We disprove this conjecture.
+
+\begin{theorem}\label{thm:main}
+Let \(H\) be the incidence graph of the \(22\) triples on \(13\) points
+in Table~\ref{tab:complex}. This bipartite graph has \(35\) vertices
+and \(66\) edges. There is a finite simple undirected graph \(G\)
+with at least one edge such that
+\[
+t(H,G)<p(G)^{66}.
+\]
+\end{theorem}
+
+We first construct a strict density gap for a finite nonnegative
+kernel and then retain it when sampling a finite simple host. This
+route specifies \(H\) explicitly and proves existence of \(G\), while
+the final density continues to count all homomorphisms.
+
+A finite kernel is a nonnegative function \(W(x,y)\) on two finite
+probability spaces. Its mean is the analogue of edge density, and its
+\(H\)-density is the expectation of the product of \(W\) over the
+incidences of \(H\), with independent samples from the first space at
+the point vertices and from the second space at the face vertices.
+For a symmetric kernel taking values in \([0,1]\), a simple graph
+can be sampled by choosing a label for each host vertex and using the
+kernel value as the probability of each edge.
+
+The same pattern has further consequences with distinct density
+scopes. Tensor powers make \(t(H,F)/p(F)^{66}\) arbitrarily small,
+with the host density allowed to vary. Balanced blow-ups retain a
+fixed deficit in injective labeled copies at one positive limiting
+edge density. At one fixed \(p\in(0,1)\), a nonconstant symmetric
+\([0,1]\)-valued kernel has mean \(p\) and \(H\)-density \(p^{66}\);
+sampling gives finite graphs whose edge and \(H\)-densities converge
+to these values while their four-cycle density has a limit strictly
+greater than \(p^4\).
+Corollaries~\ref{cor:amplification} and~\ref{cor:forcing} state these
+consequences precisely below.
+
+\subsection*{The negative correlation}
+
+The triple system has \(13\) points and \(22\) faces. Each of its
+\(33\) point pairs belongs to exactly two faces. This double
+occurrence converts a local sign constraint into a correlation
+involving four incidences.
+
+To see the algebra, attach a sign \(\xi_{ij}\) to each chosen active
+incidence and an independent uniform sign \(\eta_e\) to each point
+pair. The finite-field lemma compares the averaged matrix product with
+a sign model having a factor
+\(1+c_0\eta_e\xi_{ij}\xi_{kj}\) whenever both incidences of
+\(e=\{i,k\}\) on a face \(j\) are active; here
+\(c_0\in\{-1,1\}\). If the two faces on \(e\) are \(j,l\), then
+\[
+ \E_{\eta_e}
+ (1+c_0\eta_e\xi_{ij}\xi_{kj})
+ (1+c_0\eta_e\xi_{il}\xi_{kl})
+ =
+ 1+\xi_{ij}\xi_{kj}\xi_{il}\xi_{kl}.
+\]
+A factor used on only one face averages to \(1\). Thus the model
+retains the parity of the four incidence signs exactly when both
+face occurrences are selected. This is the finite correlation that
+the kernel construction turns negative.
+
+The matrix model uses a large fixed even dimension \(D=2r\).
+For a symmetric matrix difference over \(\F_q\), with \(q\) an odd
+prime, prescribe rank \(r\)
+and the determinant sign of the induced nondegenerate quotient form.
+Half rank is the key geometric constraint: when the difference
+between two point matrices is nonsingular, the two rank-\(r\) images
+through a common center are complementary, which relates their
+determinant signs. On configurations with nonsingular point
+differences, the normalized center integrals approximate the sign
+model in averaged \(L^1\), with a uniform bound on the local factors.
+A joint estimate shows that the determinant signs on the \(33\) point
+pairs are asymptotically independent, supplying the independent
+\(\eta_e\) in the limiting sign model.
+
+The kernel uses one independent matrix coordinate for each incidence.
+A type names the point or face that a sampled vertex is intended to
+represent. At an incident type pair, two choices of active coordinate
+set have opposite sign biases. They cancel every nonconstant first
+sign moment while retaining a signed overlap moment. Expanding the
+parity model assigns a set of selected coordinates to each point
+pair. The chosen supports permit exactly one collection of nonempty
+pair labels at the intended type assignment, and its product of
+overlap moments is negative. Scores obtained from color refinement
+make this assignment dominate all other part-preserving type maps,
+including maps with repetitions.
+
+The limiting density in one orientation now has a negative correction.
+Adding an inactive point type makes that correction occur with a factor
+\(\varepsilon^{13}\), while the uncontrolled correction in the
+opposite orientation has a factor \(\varepsilon^{22}\). Choosing
+\(\varepsilon>0\) sufficiently small and then taking \(q\) sufficiently
+large makes the product of the two finite oriented densities less than
+one. A crossed product of the rectangular kernel
+with its transpose then gives a symmetric kernel with the same strict
+deficit.
+
+The remaining issue is the contribution of singular point
+differences. Their probability is \(O(q^{-1})\), but the normalized
+rank indicators can grow as a power of \(q\), so the weighted
+contribution requires a separate bound. The graph
+\(\{(v,Xv):v\in\F_q^D\}\) of a symmetric matrix is a Lagrangian
+subspace of \(\F_q^D\oplus\F_q^D\) for
+\[
+ \omega((v,w),(v',w'))=v^{\mathsf T}w'-w^{\mathsf T}v'.
+\]
+The intersection of two such graphs projects isomorphically onto
+the kernel of their difference. Rank defects thereby become
+intersection dimensions.
+
+We compare the probability of each point-intersection profile with
+the gain from conditioning on its intersections with a face center.
+Three partitions of the face set give a global profile bound. If this
+first bound does not give decay, a quadratic estimate forces any large
+pair intersection to propagate through the connected complex until
+the extra separation of one pair makes the profile too costly. The
+remaining bounded dimensions reduce, after choosing \(D\), to a
+short list of integer equality cases. A final span count gives strict
+decay for those cases. The dimension and all finite construction
+parameters are fixed before \(q\) tends to infinity; constants in
+the \(q\)-estimates may depend on that fixed dimension.
+
+Figure~\ref{fig:proof-overview} summarizes these dependencies.
+
+\input{figures/proof-overview}
+
+\subsection*{History and relation to positive results}
+
+The conjecture grew out of questions about supersaturation: how many
+copies of a fixed bipartite graph must occur once the host has many
+edges? In a related study of graphcopy functions, Erd\H{o}s,
+Lov\'asz, and Spencer distinguished normalized homomorphism and
+injective-copy counts and proved their limiting agreement under
+increasing host orders~\cite{erdoslovaszspencer1979}.
+Simonovits's discussion of degenerate extremal problems placed
+supersaturation lower bounds in a general program~\cite{simonovits1984}.
+Sidorenko formulated the associated inequalities for functionals of
+nonnegative kernels~\cite{sidorenko1991,sidorenko1993}.  In the
+homomorphism formulation, the proposed lower bound is the density in
+a constant kernel of the same mean.  The insistence on all
+homomorphisms is part of this formulation; repeated images of vertices
+are allowed.  Theorem~\ref{thm:main} concerns this ordinary graph
+inequality.
+
+Trees, even cycles, and complete bipartite graphs are basic positive
+examples~\cite{sidorenko1993}. Several methods established broader
+classes. Conlon,
+Fox, and Sudakov proved the conjecture when one vertex is adjacent to
+the entire opposite part, and obtained more general comparisons with
+complete bipartite graphs~\cite{conlonfoxsudakov2010}.  A different
+approach studies norms derived from homomorphism densities.
+Hatami developed the graph-norm approach, recording Szegedy's observation
+that weakly norming graphs satisfy Sidorenko's inequality, and proved
+stronger subgraph inequalities and the hypercube case~\cite{hatami2010}.
+Conlon and Lee constructed weakly norming graphs from finite reflection
+groups~\cite{conlonlee2017}.  These results provide stronger inequalities
+than the comparison with a single edge and explain why many highly
+structured bipartite graphs satisfy the conjecture.
+
+Other proofs assemble a graph from pieces while keeping control of
+the associated homomorphism distribution.  Li and Szegedy's logarithmic
+calculus, first circulated in 2011, gives gluing results and a class of
+reflection trees~\cite{liszegedy2025}.  Szegedy developed an information
+theoretic formulation using entropy and conditional
+gluing~\cite{szegedy2015entropy}.  Kim, Lee, and Lee proved the conjecture
+for tree-arrangeable graphs and for Cartesian products of a Sidorenko
+graph with a tree~\cite{kimleelee2016}.  Conlon, Kim, Lee, and Lee
+introduced strong tree decompositions and established further
+subdivision results~\cite{conlonkimleelee2018}.  More recently, Im,
+Li, and Liu treated subdivisions and substitutions by even theta
+graphs under a local-density counting hypothesis for the base or suitable
+global path-multiplicity conditions~\cite{imliliu2026}.
+The gluing and substitution statements impose different structural
+conditions on their constituent graphs.
+
+Positive results also depend on how often neighborhoods of a given
+size occur. Conlon and Lee proved a divisibility criterion and a
+degree-count criterion when one part is regular, and showed that every
+bipartite graph has a sufficiently large one-sided blow-up satisfying
+the conjecture~\cite{conlonlee2021}. Coregliano subsequently replaced
+the divisibility hypothesis with weaker degree-count bounds and
+extended the reflection approach through left-cut-percolation and
+induced-Sidorenko bigraphs~\cite{coregliano2024}.  A positive result
+for a blow-up does not imply the corresponding result for the
+original graph.
+
+There are useful reductions of the host space as well. Coregliano and
+Razborov proved that, for a fixed pattern, a lower bound by one fixed
+positive multiple of the Sidorenko expression on every biregular
+bigraphon implies the exact Sidorenko inequality on every bigraphon.
+Here biregular means that the
+kernel integrals in either variable are constant. They also proved a
+reflective tree-decomposition theorem~\cite{coreglianorazborov2021}.
+In another direction, Zhao gave a conditional reduction through
+conjugacy-class averaging on finite groups and proved an inequality
+for one-subdivisions on the resulting conjugacy-averaged bipartite
+Cayley kernels~\cite[Theorems~1.3--1.4]{zhao2026conjugacy}.  The averaging monotonicity in the
+general reduction is a hypothesis.
+
+The incidence graph used here has parts of sizes $13$ and $22$,
+with degrees $4,5,6$ on the first part and degree $3$ on the second.
+It therefore has no vertex adjacent to the whole opposite part.
+Since it is connected and its first part is not regular, it is not
+weakly norming: Hatami's necessary condition requires regularity on
+each part~\cite{hatami2010}. Coregliano's sufficient degree-count
+hypothesis is not met in either orientation.  On the degree-$3$ side the relevant comparison is
+$22<\binom{13}{3}=286$; in the reverse orientation there are only
+$3,6,4$ vertices of degrees $4,5,6$, respectively, below the
+corresponding thresholds
+$\binom{22}{4},\binom{22}{5},\binom{22}{6}$~\cite[Theorem~3.6]{coregliano2024}.
+Its minimum degree is $3$, so it is not directly a graph obtained by
+the even-path substitutions above, which introduce vertices of
+degree $2$.  These observations locate the example relative to
+specific positive theorems; the negative inequality is established by
+the construction and estimates below.
+
+Finally, the conclusion is compatible with local versions of the
+conjecture.  For a bipartite graph with $m$ edges, Lov\'asz proved the
+inequality for a mean-one symmetric kernel $W$ satisfying
+$\lVert W-1\rVert_\infty\leq 1/(4m)$~\cite[Theorem~5.1]{lovasz2011local}.
+He also proved a cut-norm
+version under the additional bound $0\leq W\leq2$, with
+$\lVert W-1\rVert_\square\leq2^{-8m-2}$~\cite[Theorem~4.1]{lovasz2011local}.
+Here the kernels are on $[0,1]^2$, with Lebesgue measure, and
+\[
+ \lVert F\rVert_\square
+ =\sup_{S,T\subseteq[0,1]}
+   \left|\int_{S\times T}F(x,y)\,dx\,dy\right|,
+\]
+where $S,T$ are measurable. Thus the constant kernel has a neighborhood
+on which the inequality holds. A finite probability kernel can be
+represented on $[0,1]^2$ by a step function whose interval lengths
+are its atom probabilities. The mean-one normalization of our final
+symmetric kernel, having a strict deficit, cannot satisfy either set
+of local hypotheses above. Scaling the kernel into $[0,1]$ preserves
+the normalized density comparison.
+
+The construction also uses established geometric and algebraic
+frameworks in specific roles. Conditions on the rank and
+discriminant sign of a symmetric matrix difference are the relations
+in the association scheme of symmetric matrices studied by Huo and
+Wan~\cite{huowan1993}; Schmidt records this framework and the exact
+counts by rank and sign~\cite[Section~2]{schmidt2015}. Symplectic geometry
+provides the elementary Lagrangian and pair-intersection counting
+formulas~\cite{taylor1992}.
+The joint-sign argument uses the standard quadratic Gauss-sum rank
+bound~\cite[Lemma~3.3]{gowerswolf2011}, and the type scores adapt
+color refinement~\cite{weisfeilerleman1968,mckay1981}. We rederive
+the estimates used here and combine the signed rank-layer correlations
+with bounds for their weighted contribution on singular configurations.
+
+\subsection*{Amplification and injective copies}
+
+For a finite simple graph \(F\) of order \(n\ge35\), let
+\(\operatorname{Inj}(H,F)\)
+count injective edge-preserving maps, with no condition on nonedges,
+and write \((n)_{35}=n(n-1)\cdots(n-34)\). At ordinary edge density
+\(d=|E(F)|/\binom n2\), the labeled random-graph benchmark is
+\((n)_{35}d^{66}\), the expected count in \(G(n,d)\).
+
+\begin{corollary}[Amplification and injective supersaturation]
+\label{cor:amplification}
+For the same graph \(H\) as in Theorem~\ref{thm:main}, the following
+hold.
+\begin{enumerate}[label=(\roman*)]
+\item For every \(c>0\), there is a finite simple undirected graph
+\(F\) with \(p(F)>0\) such that
+\[
+t(H,F)<c\,p(F)^{66}.
+\]
+\item There are \(p_*\in(0,1)\), \(\delta_*>0\), and finite simple
+undirected graphs \(F_s\) of orders \(n_s\to\infty\) for which
+\(d_s=|E(F_s)|/\binom{n_s}{2}\to p_*\) and, for all sufficiently
+large \(s\),
+\[
+\operatorname{Inj}(H,F_s)
+\le(1-\delta_*)(n_s)_{35}d_s^{66}.
+\]
+\end{enumerate}
+\end{corollary}
+
+The proof in Section~\ref{sec:consequences} uses tensor powers and
+balanced blow-ups of a single host from Theorem~\ref{thm:main}.
+
+Part (ii) contradicts the injective Erd\H{o}s--Simonovits
+random-graph benchmark described in
+\cite[Section~1]{conlonfoxsudakov2010}: the sequence has one positive
+limiting density, so it eventually satisfies \(d_s>n_s^{-\gamma}\)
+for every fixed \(\gamma>0\), while retaining a fixed deficit.
+
+\subsection*{A consequence for the forcing conjecture}
+
+Fix \(0<p<1\). A graph \(F\) with \(m\) edges is \emph{\(p\)-forcing}
+if every sequence of finite simple graphs \(G_n\) with
+\(|V(G_n)|\to\infty\), \(p(G_n)\to p\), and
+\(t(F,G_n)\to p^m\) is quasirandom of density \(p\): for every fixed
+finite simple graph \(J\), one has
+\(t(J,G_n)\to p^{|E(J)|}\). It is \emph{forcing} if it is
+\(p\)-forcing for every \(p\in(0,1)\). The classical edge-and-four-cycle
+characterization of quasirandomness is due to Chung, Graham, and
+Wilson~\cite{chunggrahamwilson1989}. Skokan and Thoma studied which
+bipartite patterns can replace the four-cycle~\cite{skokanthoma2004}.
+Using this asymptotic formulation and following their question,
+Conlon, Fox, and Sudakov formulate the forcing conjecture: a graph is forcing if and
+only if it is bipartite and contains a cycle
+\cite[Section~1, equation~(2) and Conjecture~2]{conlonfoxsudakov2010}.
+
+For a symmetric \([0,1]\)-valued kernel \(W\) on a probability space,
+write
+\[
+t(F,W)=\E\prod_{uv\in E(F)}W(X_u,X_v),
+\]
+where the \(X_v\) are independent samples. In particular,
+\(t(K_2,W)=\E W\). We identify each finite probability kernel below
+with its step graphon on \([0,1]\).
+
+\begin{corollary}[Failure of the forcing conjecture]
+\label{cor:forcing}
+For the same connected bipartite graph \(H\) of
+Theorem~\ref{thm:main}, there are \(p\in(0,1)\) and a nonconstant
+graphon \(W\) such that
+\[
+t(K_2,W)=p,\qquad t(H,W)=p^{66}.
+\]
+Consequently, \(H\) is not \(p\)-forcing at this density and hence is
+not forcing, despite being bipartite and containing a cycle.
+\end{corollary}
+
+Section~\ref{sec:consequences} interpolates between finite kernels of
+the same mean to obtain the exact graphon equality, then uses independent
+sampling to obtain a sequence of finite graphs with the stated limiting
+densities.
+Conlon, Fox, and Sudakov note that the forcing conjecture is stronger
+than Sidorenko's conjecture~\cite[Section~1]{conlonfoxsudakov2010};
+the interpolation gives a concrete witness for this graph at one density.
+
+\subsection*{Organization and conventions}
+
+Section~\ref{sec:complex} gives the triple system and verifies its
+finite properties. Section~\ref{sec:kernel} states
+Lemma~\ref{lem:rank-layer} and uses it to prove
+Theorem~\ref{thm:main}. The next four sections prove that lemma:
+Section~\ref{sec:transverse} treats transverse matrices;
+Section~\ref{sec:orbits} gives the Lagrangian orbit counts;
+Section~\ref{sec:planted} bounds the local gains; and
+Section~\ref{sec:tail} proves strict decay of the singular
+contribution. Section~\ref{sec:consequences} then proves the amplification,
+injective-copy and forcing consequences stated above.
+
+All probability spaces used in our construction are finite. Expectations
+refer to the indicated independent samples, except where a
+conditional law is specified. Empty products equal \(1\).
+The field size \(q\) tends to infinity through odd primes. Every
+such limit is taken after fixing the dimension and all finite
+construction parameters. Dependence on \(D\) in the constants of
+these \(q\)-asymptotic estimates is allowed; bounds used to choose
+\(D\) will explicitly be uniform in \(D\).
diff --git a/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/sections/kernel.tex b/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/sections/kernel.tex
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+\section{Constructing a negative kernel correlation}
+\label{sec:kernel}
+
+This section reduces Theorem~\ref{thm:main} to a finite-field counting
+lemma. We first state that lemma, then construct a nonnegative kernel
+whose homomorphism density is too small. All probability spaces in
+the construction are finite; their atom probabilities need not be
+rational.
+
+\subsection{The rank-layer model}
+
+Let $q$ be an odd prime, let $\chi:\F_q\to\{0,1,-1\}$ be the
+quadratic character with $\chi(0)=0$, and let $\mathcal S_D$ be the
+space of symmetric $D$-by-$D$ matrices over $\F_q$, equipped with
+uniform probability. For a symmetric matrix of positive rank, its
+\emph{sign} is the quadratic character of the determinant of its
+induced nondegenerate symmetric form on the quotient by the radical.
+A change of basis multiplies this determinant by a square, so the
+sign is well defined. Fix a positive even dimension $D=2r$ and set
+\begin{equation}
+ K_\xi(Z)=
+ \frac{\1_{\{\rank Z=r,\ \operatorname{sign}(Z)=\xi\}}}
+ {\P_{Z'\in\mathcal S_D}(\rank Z'=r,\ \operatorname{sign}(Z')=\xi)},
+ \qquad \xi\in\{1,-1\},\qquad c_0=\chi((-1)^r).
+ \label{eq:kernel-rank-definition}
+\end{equation}
+Both denominators are positive: diagonal matrices realize either
+nonzero determinant square class in rank $r$. In particular,
+$K_\xi\ge0$ and $\E K_\xi=1$.
+
+\begin{lemma}[Rank-layer limit]
+\label{lem:rank-layer}
+There is an integer $D_0$ such that, for every fixed even $D\ge D_0$,
+the following assertion holds as $q$ tends to infinity through odd
+primes. Let $X_i$, $i\in I$, and $Y_j$, $j\in J$, be independent
+uniform elements of $\mathcal S_D$. For every set
+$B\subseteq\mathscr C$ and every choice of signs
+$\xi_{ij}\in\{1,-1\}$ on $B$,
+\begin{equation}
+ \E\prod_{(i,j)\in B}K_{\xi_{ij}}(X_i-Y_j)
+ =\E_\eta\prod_{j\in J}
+ \prod_{\substack{e=\{i,k\}\subset j\\(i,j),(k,j)\in B}}
+ \bigl(1+c_0\eta_e\xi_{ij}\xi_{kj}\bigr)+o(1),
+ \label{eq:rank-layer}
+\end{equation}
+where the $\eta_e$, $e\in E$, are independent uniform signs. The error
+is uniform over all such $B$ and sign choices. The same assertion
+holds with $Y_j-X_i$ in place of $X_i-Y_j$.
+\end{lemma}
+
+The averaging on the right has a useful exact form. For
+\(e=\{v,w\}\in E\), let \(j,l\) be its two faces and put
+\begin{equation}
+ U_e=\{(v,j),(w,j),(v,l),(w,l)\}\subseteq\mathscr C.
+ \label{eq:kernel-four-corners}
+\end{equation}
+Write \(\xi_\kappa=\xi_{ij}\) for \(\kappa=(i,j)\in B\).
+If only one of the two face factors on \(e\) is present, its average
+over \(\eta_e\) is \(1\). If both are present, their average is
+\(1+\prod_{\kappa\in U_e}\xi_\kappa\), since \(c_0^2=1\).
+Independence of the \(\eta_e\) therefore gives
+\begin{equation}
+ \E_\eta\prod_{j\in J}
+ \prod_{\substack{e=\{i,k\}\subset j\\(i,j),(k,j)\in B}}
+ (1+c_0\eta_e\xi_{ij}\xi_{kj})
+ =
+ \prod_{\substack{e\in E\\U_e\subseteq B}}
+ \left(1+\prod_{\kappa\in U_e}\xi_\kappa\right).
+ \label{eq:kernel-parity-model}
+\end{equation}
+Thus every fully active pair tests the parity of its four incidence
+signs, contributing \(0\) or \(2\). This is an identity for the
+integrated sign model in Equation~\eqref{eq:rank-layer}. The
+quantification over \(B\) is needed because each coordinate in the
+activation construction below selects its own active incidences.
+
+The proof of Lemma~\ref{lem:rank-layer} occupies
+Sections~\ref{sec:transverse}--\ref{sec:tail}.
+The dimension is fixed before the prime tends to infinity; uniformity
+in $B$ and the signs then concerns a fixed finite family. For the
+assertion with reversed matrix differences, negation multiplies the sign of a rank-$r$ matrix by
+$c_0$ and bijects the corresponding layers, whence
+$K_\xi(-Z)=K_{c_0\xi}(Z)$. Replacing every incidence sign by
+$c_0\xi_{ij}$ leaves each sign product on the right of
+Equation~\eqref{eq:rank-layer} unchanged.
+
+For a kernel $W:\mathcal X\times\mathcal Y\to[0,\infty)$ on two
+finite probability spaces, write
+\begin{equation}
+ t_{I,J}(W)=\E\prod_{(i,j)\in\mathscr C}W(x_i,y_j),
+ \qquad W^\top(y,x)=W(x,y),
+ \label{eq:kernel-bipartite-density}
+\end{equation}
+where all $x_i\in\mathcal X$ and $y_j\in\mathcal Y$ are sampled
+independently. Thus $t_{I,J}(W^\top)$ samples the $I$-vertices from
+$\mathcal Y$ and the $J$-vertices from $\mathcal X$.
+
+For a symmetric kernel $U:\Omega\times\Omega\to[0,\infty)$ on one
+finite probability space, write $t(H,U)=t_{I,J}(U)$, with every vertex
+sampled independently from $\Omega$; sampled labels may coincide.
+
+\subsection{Activation laws and the coordinate expansion}
+
+A type records which point or face of the fixed complex a sampled
+vertex is intended to represent. Independent matrix coordinates carry
+separate copies of the rank-layer model, and an activation law chooses
+which copies occur at a pair of types. We first leave these finite
+choices arbitrary and derive their contribution to the density.
+
+Let $\mathcal A,\mathcal B$ be finite type sets, with probability
+distributions $\pi,\nu$, and let $\mathcal K$ be a fixed finite
+coordinate set. Use the spaces
+\[
+ \mathcal X=\mathcal A\times\mathcal S_D^{\mathcal K},\qquad
+ \mathcal Y=\mathcal B\times\mathcal S_D^{\mathcal K},
+\]
+with the type distributions just specified and all matrix coordinates
+independent and uniform. For each $(a,b)\in\mathcal A\times\mathcal B$,
+choose a probability law on a subset $S\subseteq\mathcal K$ and signs
+$(\xi_\kappa)_{\kappa\in S}$. Define
+\begin{equation}
+ W_q\bigl((a,x),(b,y)\bigr)
+ =\E_{S,\xi\mid a,b}\prod_{\kappa\in S}
+ K_{\xi_\kappa}(x^\kappa-y^\kappa).
+ \label{eq:kernel-activation}
+\end{equation}
+An empty product is one. For fixed types, averaging over the matrices
+gives one, since each difference is uniform and distinct coordinates
+are independent. Therefore $\E W_q=1$, for every $q$.
+
+For $\alpha\subseteq\mathcal K$ put
+\[
+ \phi_\alpha(S,\xi)=
+ \1_{\{\alpha\subseteq S\}}\prod_{\kappa\in\alpha}\xi_\kappa,
+ \qquad
+ A_{ab}(\alpha,\beta)=\E_{S,\xi\mid a,b}\phi_\alpha\phi_\beta.
+\]
+Here $\phi_\alpha$ is defined to be zero when $\alpha\not\subseteq S$;
+in particular $\phi_\varnothing=1$. We will impose the moment condition
+\begin{equation}
+ \E_{S,\xi\mid a,b}\phi_\alpha=0
+ \quad\text{for every $a,b$ and every nonempty $\alpha$.}
+ \label{eq:kernel-zero-moments}
+\end{equation}
+If $\alpha\cap\beta=\varnothing$, then
+$\phi_\alpha\phi_\beta=\phi_{\alpha\cup\beta}$, so
+\begin{equation}
+ A_{ab}(\alpha,\beta)=0
+ \quad\text{if $\alpha\cap\beta=\varnothing$ and
+ $\alpha\cup\beta\ne\varnothing$.}
+ \label{eq:kernel-disjoint-zero}
+\end{equation}
+
+At a corner $(i,j)$, let $e(i,j),f(i,j)$ denote the two point pairs
+of $j$ containing $i$; their order is immaterial. Expanding
+Equation~\eqref{eq:kernel-parity-model} at every coordinate selects,
+for each pair \(e\), a subset \(\alpha_e\subseteq\mathcal K\) of
+coordinates at which its four-sign term is used. At a corner, the
+two labels meeting there contribute their overlap moment
+\(A_{ab}(\alpha_{e(i,j)},\alpha_{f(i,j)})\).
+For type assignments
+$a:I\to\mathcal A$, $b:J\to\mathcal B$, define
+\begin{equation}
+ C_A(a,b)=
+ \sum_{\substack{\alpha_e\subseteq\mathcal K\ (e\in E)\\
+                   \alpha_e\ne\varnothing\ (e\in E)}}
+ \prod_{(i,j)\in\mathscr C}
+ A_{a(i)b(j)}\bigl(\alpha_{e(i,j)},\alpha_{f(i,j)}\bigr).
+ \label{eq:kernel-coefficient}
+\end{equation}
+These are finite real numbers, with no restriction on either type
+map. In particular, repetitions of types are included.
+The next lemma identifies this all-nonempty label sum as the
+limiting nonconstant contribution at the fixed type assignment.
+
+\begin{lemma}[Coordinate expansion]
+\label{lem:kernel-expansion}
+Assume Equation~\eqref{eq:kernel-zero-moments} and fix an even dimension
+as in Lemma~\ref{lem:rank-layer}. For fixed type laws and activation
+laws,
+\begin{align}
+ t_{I,J}(W_q)&=1+
+ \sum_{a:I\to\mathcal A\,,\ b:J\to\mathcal B}
+ \left(\prod_{i\in I}\pi_{a(i)}\right)
+ \left(\prod_{j\in J}\nu_{b(j)}\right)C_A(a,b)+o(1).
+ \label{eq:kernel-expansion}
+\end{align}
+The corresponding formula for $W_q^\top$ has type maps
+$b:I\to\mathcal B$, $a:J\to\mathcal A$ and the corner factors
+$A_{a(j)b(i)}$.
+\end{lemma}
+
+\begin{proof}
+Conditional on all types, sample an activation law independently at
+each incidence when expanding the product in
+Equation~\eqref{eq:kernel-bipartite-density}. Conditional on these
+samples, the matrix integral factors across coordinates. At coordinate
+$\kappa$ apply Lemma~\ref{lem:rank-layer} with the active set of
+incidences for which $\kappa\in S$, and with their sampled signs.
+Use Equation~\eqref{eq:kernel-parity-model} at each coordinate.
+There are only finitely many activation and sign configurations.
+The model expectations are bounded, and their approximating matrix
+expectations are consequently bounded for large $q$. A finite product
+and then the finite average therefore preserve the $o(1)$ error.
+
+Expand these finite parity products, and let \(\alpha_e\) be the set
+of coordinates in which the four-sign term for \(e\) is selected. A
+coordinate selected for \(e\) requires activation at all four
+corners in \(U_e\) and contributes the product of their signs.
+At $(i,j)$ the two subsets on its incident pairs give exactly
+$\phi_{\alpha_{e(i,j)}}\phi_{\alpha_{f(i,j)}}$; averaging that
+incidence's activation and signs gives its factor \(A\). Thus the
+limiting contribution at the fixed type assignment is
+\begin{equation}
+ \sum_{\alpha_e\subseteq\mathcal K\ (e\in E)}
+ \prod_{(i,j)\in\mathscr C}
+ A_{a(i)b(j)}
+   \bigl(\alpha_{e(i,j)},\alpha_{f(i,j)}\bigr).
+ \label{eq:kernel-unrestricted-labels}
+\end{equation}
+Repeated vertex types leave the matrix draws at distinct source vertices
+independent. Repeated type pairs use independent samples from the same
+activation law at their distinct source incidences.
+
+The all-empty labeling contributes one. In any other nonzero term,
+no face can have both an empty and a nonempty pair label: the two
+meet at a corner, where Equation~\eqref{eq:kernel-disjoint-zero} makes
+the factor zero. Hence any face with one nonempty label has all three
+labels nonempty. The shared-pair labels and connectedness of the
+face-neighbor graph propagate this conclusion to every face. Thus
+the remaining labels are exactly those summed in
+Equation~\eqref{eq:kernel-coefficient}. This proves
+Equation~\eqref{eq:kernel-expansion}. For the transpose, the same
+argument uses the last assertion of Lemma~\ref{lem:rank-layer}.
+\end{proof}
+
+\subsection{A single negative term}
+
+The problem is now finite: choose activation laws so that the weighted
+coefficient sum in Equation~\eqref{eq:kernel-expansion} is negative.
+We first make the coefficient at the identity type assignment negative,
+and then choose the priors and thinning factors to make it dominate
+the other assignments.
+
+Take $\mathcal A=I$, $\mathcal B=J$, and one coordinate for each
+corner: $\mathcal K=\mathscr C$. The four-incidence set \(U_e\) from
+Equation~\eqref{eq:kernel-four-corners} is now also a subset of the
+coordinate set.
+At a corner $\kappa=(v,j)$, write $e,f$ for the two pairs of $j$
+through $v$ and put
+\begin{equation}
+ M_\kappa=U_e\cup U_f,
+ \qquad \Gamma_\kappa=U_e\mathbin\triangle U_f.
+ \label{eq:kernel-seven-six}
+\end{equation}
+Distinct faces cannot share two distinct pairs. Hence the opposite
+faces on $e,f$ are distinct and
+$U_e\cap U_f=\{\kappa\}$. In particular $M_\kappa$ has seven
+coordinates and $\Gamma_\kappa=M_\kappa\setminus\{\kappa\}$ has six.
+Figure~\ref{fig:kernel-corner-activation} displays these sets.
+
+\input{figures/corner-activation}
+
+Fix $0<\theta<1$. Give each corner a sign $\sigma_\kappa$, choosing
+exactly one negative sign, for example $\sigma_{(0,0)}=-1$ and all
+others $+1$. At the type pair $\kappa=(v,j)$ with $v\in j$ use this
+\emph{base law}:
+\begin{itemize}
+\item with probability $1/2$, set $S=M_\kappa$ and give each sign
+vector on $S$ probability
+$2^{-7}(1+\sigma_\kappa\theta\prod_{\gamma\in\Gamma_\kappa}\xi_\gamma)$;
+\item with probability $1/2$, set $S=\Gamma_\kappa$ and give each
+sign vector on $S$ probability
+$2^{-6}(1-\sigma_\kappa\theta\prod_{\gamma\in\Gamma_\kappa}\xi_\gamma)$.
+\end{itemize}
+These conditional masses are positive and sum to one because a
+nonempty sign monomial has zero average under uniform signs. At a
+nonincident type pair $(v,j)$ use $S=\varnothing$ deterministically.
+Write $A^{\mathrm{base}}$ for the resulting moments.
+
+Under either branch at a corner, the only nonempty sign monomial with
+possibly nonzero expectation has index $\Gamma_\kappa$. Its conditional
+expectations are respectively $\sigma_\kappa\theta$ and
+$-\sigma_\kappa\theta$. The two effects of the branch choice are
+\begin{equation}
+ \E\phi_{\Gamma_\kappa}
+ =\tfrac12\sigma_\kappa\theta-\tfrac12\sigma_\kappa\theta=0,
+ \qquad
+ A^{\mathrm{base}}_\kappa(U_e,U_f)
+ =\tfrac12\sigma_\kappa\theta.
+ \label{eq:kernel-branch-biases}
+\end{equation}
+The second identity holds because only the \(M_\kappa\) branch
+contains \(U_e\cup U_f\), while their sign product is the
+\(\Gamma_\kappa\) monomial. All base laws satisfy
+Equation~\eqref{eq:kernel-zero-moments}, but this overlap moment can
+have either sign.
+More explicitly, for $Q=\alpha\cup\beta$ and
+$R=\alpha\mathbin\triangle\beta$,
+\begin{align}
+ A^{\mathrm{base}}_\kappa(\alpha,\beta)
+ &=\tfrac12\1_{\{Q\subseteq M_\kappa\}}
+       \bigl(\1_{\{R=\varnothing\}}
+                +\sigma_\kappa\theta\1_{\{R=\Gamma_\kappa\}}\bigr)
+ \notag\\
+ &\quad+\tfrac12\1_{\{Q\subseteq\Gamma_\kappa\}}
+       \bigl(\1_{\{R=\varnothing\}}
+                -\sigma_\kappa\theta\1_{\{R=\Gamma_\kappa\}}\bigr).
+ \label{eq:kernel-base-moment}
+\end{align}
+
+\begin{lemma}[Support forcing]
+\label{lem:kernel-support}
+At the identity type assignments $a(i)=i$, $b(j)=j$, the sum in
+Equation~\eqref{eq:kernel-coefficient} for the base laws has exactly one
+nonzero term. Its labels and value are
+\begin{equation}
+ \alpha_e=U_e\quad(e\in E),\qquad
+ C_{A^{\mathrm{base}}}(\mathrm{id},\mathrm{id})
+ =-(\theta/2)^{66}.
+ \label{eq:kernel-negative-term}
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+We first prove the support identity
+\begin{equation}
+ \bigcap_{\kappa\in U_g}M_\kappa=U_g
+ \qquad(g\in E).
+ \label{eq:kernel-support-intersection}
+\end{equation}
+The same set \(\mathscr C\) indexes coordinates \(\gamma\) and
+activation corners \(\kappa\). Availability is symmetric:
+\(\gamma\in M_\kappa\) if and only if \(\kappa\in M_\gamma\).
+Both conditions mean that the two corners lie in a common set \(U_e\).
+Fix a coordinate $\gamma=(v,j)$, write $j=\{v,w,z\}$, and let
+$e=\{v,w\}$, $f=\{v,z\}$ have opposite faces $l,m$, respectively.
+This coordinate belongs to precisely $U_e$ and $U_f$. Consequently it
+is available at exactly the following activation corners: all three
+corners of $j$, the two $e$-endpoint corners of $l$, and the two
+$f$-endpoint corners of $m$. To be available at all four corners of
+$g$, its two faces must therefore be among $j,l,m$. The choice $j,l$
+forces $g=e$, and $j,m$ forces $g=f$, since distinct triples share at
+most one pair. The choice $l,m$ cannot work: its two endpoints would
+both have to lie in $\{v,w\}\cap\{v,z\}=\{v\}$. Thus the coordinate
+$\gamma$ is available at every corner of $g$ if and only if
+$g\in\{e,f\}$, equivalently if and only if $\gamma\in U_g$.
+This proves Equation~\eqref{eq:kernel-support-intersection}.
+
+In a nonzero identity-type term, $\alpha_g$ must lie in the maximal
+activation support at each of the four corners of $g$. Hence
+$\alpha_g\subseteq U_g$. At a corner $\kappa$ with pairs $e,f$, the
+two nonempty labels cannot be disjoint by
+Equation~\eqref{eq:kernel-disjoint-zero}. Since
+$U_e\cap U_f=\{\kappa\}$, both labels contain $\kappa$. Applying
+this at all four corners of $g$ gives $\alpha_g=U_g$.
+
+For these labels, Equation~\eqref{eq:kernel-branch-biases} gives
+the local factor \(\sigma_\kappa\theta/2\). Multiplying the $66$ factors,
+with exactly one negative $\sigma_\kappa$, proves the assertion.
+\end{proof}
+
+Each local matrix $A^{\mathrm{base}}_\kappa$ is positive semidefinite,
+being a matrix of second moments of the random variables $\phi_\alpha$.
+For example its principal block on $U_e,U_f$ is
+\[
+ \begin{pmatrix}1/2&\sigma_\kappa\theta/2\\
+                 \sigma_\kappa\theta/2&1/2\end{pmatrix},
+\]
+with eigenvalues $(1+\theta)/2$ and $(1-\theta)/2$. Thus the negative
+global term uses an admissible negative off-diagonal moment. The
+support argument proves that local diagonal alternatives cannot form
+any additional nonempty global labeling at the identity types.
+
+\subsection{Pinning the types by color refinement}
+
+The identity calculation alone does not control the kernel density:
+the expansion averages over every type assignment, including maps with
+repetitions. We now give the identity strictly larger score than every
+competitor. The following lemma applies to any bipartite graph whose
+part colors refine to distinct vertex colors.
+
+\begin{lemma}[Pinning by vertex and pair scores]
+\label{lem:pinning}
+Let $F=(P\sqcup Q,L)$ be a finite bipartite graph for which color
+refinement, starting with the two part colors, becomes discrete within
+each part. There are real vertex scores $s_t$, $t\in P\sqcup Q$,
+and pair scores $s_{ab}$, $(a,b)\in P\times Q$, such that the identity
+is the unique maximizer of
+\begin{equation}
+ \mathcal Q(f)=\sum_{u\in P\sqcup Q}s_{f(u)}
+              +\sum_{\{u,v\}\in L,\ u\in P}s_{f(u)f(v)}
+ \label{eq:kernel-pinning-score}
+\end{equation}
+over \emph{all} maps $f$ preserving the two parts. In particular the
+maximum has a positive gap from every other such map; no injectivity
+or adjacency-preservation condition is imposed.
+\end{lemma}
+
+\begin{proof}
+Let $\ell_0$ be the two part colors and, at round $t$, let $h_t(u)$
+be the histogram of $\ell_{t-1}$-colors among the neighbors of $u$.
+The new color is $\ell_t(u)=(\ell_{t-1}(u),h_t(u))$. Choose $R\ge1$
+rounds giving a discrete coloring. At round $t$ assign the vertex score
+$-\|h_t(a)\|^2$ to type $a$ and the pair score
+\[
+ 2h_t(a)(\ell_{t-1}(b))+2h_t(b)(\ell_{t-1}(a))
+\]
+to every $(a,b)\in P\times Q$, including nonedges. The total round
+score of \(f\) is
+\[
+ F_t(f)=-\sum_u\|h_t(f(u))\|^2
+ +2\sum_u\sum_{v\in N_F(u)}
+     h_t(f(u))\bigl(\ell_{t-1}(f(v))\bigr).
+\]
+Here \(N_F(u)\) is the neighbor set of \(u\). Both sums are indexed
+by the vertices and directed incidences of the source graph \(F\),
+even when \(f\) repeats a type.
+
+If $\ell_{t-1}(f(u))=\ell_{t-1}(u)$ for every $u$, the inner
+neighbor sum equals $\langle h_t(f(u)),h_t(u)\rangle$. Thus
+\begin{align}
+ F_t(f)
+ &=\sum_u\bigl(2\langle h_t(f(u)),h_t(u)\rangle
+                   -\|h_t(f(u))\|^2\bigr)\notag\\
+ &=\sum_u\|h_t(u)\|^2
+       -\sum_u\|h_t(f(u))-h_t(u)\|^2.
+ \label{eq:kernel-pinning-square}
+\end{align}
+The first equality uses only preservation of previous colors. It
+therefore applies to noninjective maps and to maps taking edges to
+nonedges. In this situation $F_t(\mathrm{id})-F_t(f)$ is a nonnegative
+integer, zero precisely when $f$ also preserves all new colors.
+
+To combine the rounds into a single score, set
+\[
+ M=\max_{f,\,1\le t\le R}|F_t(f)-F_t(\mathrm{id})|,
+ \qquad L_0=M+2,
+ \qquad \mathcal Q(f)=\sum_{t=1}^R L_0^{R-t}F_t(f).
+\]
+The maximum defining $M$ is finite. For a nonidentity map, let $t$
+be the first round at which a color fails to be preserved. All
+earlier scores tie the identity; the loss at round $t$ is at least
+$L_0^{R-t}$. Its possible later advantage is at most
+\[
+ M\sum_{j=0}^{R-t-1}L_0^j<L_0^{R-t}.
+\]
+Hence its total score is strictly smaller. A nonidentity map must
+have such a first round because the final colors are discrete.
+The weighted sums of the round vertex scores and pair scores are the
+required $s_t,s_{ab}$. Finiteness of the set of maps gives the positive
+gap.
+\end{proof}
+
+Apply Lemma~\ref{lem:pinning} to $H$, using
+Proposition~\ref{prop:complex}. Let its scores be $s_a$ on $I$, $s_b$
+on $J$, and $s_{ab}$ on $I\times J$. For a positive parameter $\rho$,
+choose priors and thinning probabilities
+\begin{equation}
+ \pi_a=\frac{e^{\rho s_a}}{Z_I},\qquad
+ \nu_b=\frac{e^{\rho s_b}}{Z_J},\qquad
+ \lambda_{ab}=e^{\rho(s_{ab}-s_{\max})},\qquad
+ s_{\max}=\max_{a,b}s_{ab},
+ \label{eq:kernel-priors}
+\end{equation}
+where $Z_I=\sum_{a\in I}e^{\rho s_a}$ and
+$Z_J=\sum_{b\in J}e^{\rho s_b}$. Since $0<\lambda_{ab}\le1$, the
+following is a probability law: use the entire base law with
+probability $\lambda_{ab}$, and use $S=\varnothing$ otherwise.
+Its moments satisfy the exact identity
+\begin{equation}
+ A^\lambda_{ab}(\alpha,\beta)
+ =\lambda_{ab}A^{\mathrm{base}}_{ab}(\alpha,\beta)
+ +(1-\lambda_{ab})\1_{\{\alpha=\beta=\varnothing\}}.
+ \label{eq:kernel-thinning}
+\end{equation}
+This preserves Equation~\eqref{eq:kernel-zero-moments}. In every
+all-nonempty term it introduces one factor $\lambda_{ab}$ per
+incidence, independently of the sizes of the coordinate subsets.
+
+Let $C(a,b)=C_{A^{\mathrm{base}}}(a,b)$ and let $t_0$ be the
+coefficient of the nonempty terms in
+Equation~\eqref{eq:kernel-expansion} for the thinned laws and these
+priors. By Equation~\eqref{eq:kernel-thinning},
+\begin{align}
+ t_0
+ &=Z_I^{-13}Z_J^{-22}e^{-66\rho s_{\max}}
+     \sum_{a:I\to I\,,\ b:J\to J}
+           C(a,b)e^{\rho\mathcal Q(a,b)}.
+ \label{eq:kernel-pinned-coefficient}
+\end{align}
+All normalizing factors outside the sum are common to every map.
+Here $\mathcal Q(a,b)$ means the score of the part-preserving map
+specified by $a,b$.
+
+Write $Q_*=\mathcal Q(\mathrm{id},\mathrm{id})$, and let
+\[
+ \delta_*=\min_{(a,b)\ne(\mathrm{id},\mathrm{id})}
+          (Q_*-\mathcal Q(a,b))>0,
+ \qquad
+ B_*=\sum_{(a,b)\ne(\mathrm{id},\mathrm{id})}|C(a,b)|<\infty.
+\]
+After division by $e^{\rho Q_*}$, the absolute value of the
+competitor sum is at most $B_*e^{-\rho\delta_*}$. Choose a fixed
+finite $\rho>0$ large enough that
+\begin{equation}
+ B_*e^{-\rho\delta_*}<(\theta/2)^{66};
+ \label{eq:kernel-finite-gap}
+\end{equation}
+if $B_*=0$, any positive $\rho$ suffices. By
+Equation~\eqref{eq:kernel-negative-term} this makes $t_0<0$.
+
+\subsection{Dilution, symmetrization, and finite hosts}
+
+We have made the limiting nonempty coefficient negative in one
+orientation. The symmetric kernel below multiplies the densities in
+both orientations. Dilution makes the negative term dominate the
+unknown term from the other orientation.
+
+Keep $\theta,\rho$, the priors, and all the thinned laws fixed. Add
+an inactive type $*$ to $\mathcal A=I$, giving it probability
+$1-\varepsilon$ and giving each old type $a$ probability
+$\varepsilon\pi_a$, where $0<\varepsilon<1$. Every pair involving
+$*$ uses $S=\varnothing$. Keep the $J$-type prior $\nu$ unchanged.
+Let $W_{q,\varepsilon}$ be the resulting kernel.
+
+For precision, define also the finite coefficient for the opposite
+orientation before dilution:
+\begin{align}
+ t_1={}&\sum_{b:I\to J\,,\ a:J\to I}
+       \left(\prod_{i\in I}\nu_{b(i)}\right)
+       \left(\prod_{j\in J}\pi_{a(j)}\right)
+       \sum_{\substack{\alpha_e\subseteq\mathscr C\ (e\in E)\\
+                       \alpha_e\ne\varnothing\ (e\in E)}}
+       \prod_{(i,j)\in\mathscr C}
+       A^\lambda_{a(j)b(i)}
+         \bigl(\alpha_{e(i,j)},\alpha_{f(i,j)}\bigr).
+ \label{eq:kernel-transpose-coefficient}
+\end{align}
+Both $t_0<0$ and $t_1\in\R$ are independent of $q$ and
+$\varepsilon$.
+
+In a nonempty-label term, a vertex with inactive type gives a zero
+factor on each of its incidences. Every vertex of $H$ has positive
+degree, so every original $I$-type draw must be active. There are
+$13$ such draws for $W_{q,\varepsilon}$ and $22$ for its transpose.
+Lemma~\ref{lem:kernel-expansion} therefore gives the exact limiting
+coefficients
+\begin{align}
+ t_{I,J}(W_{q,\varepsilon})
+   &=1+\varepsilon^{13}t_0+o(1),\notag\\
+ t_{I,J}(W_{q,\varepsilon}^\top)
+   &=1+\varepsilon^{22}t_1+o(1).
+ \label{eq:kernel-dilution}
+\end{align}
+These errors tend to zero with $q$ after $\varepsilon$ is fixed.
+In particular, the limiting product is
+\begin{equation}
+ (1+\varepsilon^{13}t_0)(1+\varepsilon^{22}t_1)
+ =1+\varepsilon^{13}
+      \bigl(t_0+\varepsilon^9t_1+\varepsilon^{22}t_0t_1\bigr)<1
+ \label{eq:kernel-product-gap}
+\end{equation}
+for all sufficiently small positive $\varepsilon$, because the
+bracket tends to $t_0<0$. Choose one such $\varepsilon$ and then a
+sufficiently large odd prime $q$. The cancellation of $c_0$ in the
+coordinate expansion makes the limiting coefficients independent of
+the congruence class of $q$. The resulting fixed kernel $W$ is
+nonnegative, finite, has mean one, and satisfies
+\begin{equation}
+ t_{I,J}(W)t_{I,J}(W^\top)<1.
+ \label{eq:kernel-oriented-gap}
+\end{equation}
+
+\begin{proposition}
+\label{prop:negative-kernel}
+Assuming Lemma~\ref{lem:rank-layer}, there is a symmetric kernel
+$U:\Omega\times\Omega\to[0,1]$ on a finite probability space with
+mean $\mu>0$ such that
+\begin{equation}
+ t(H,U)<\mu^{66}.
+ \label{eq:kernel-symmetric-gap}
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+Use the kernel $W:\mathcal X\times\mathcal Y\to[0,\infty)$ just
+constructed and set $\Omega=\mathcal X\times\mathcal Y$ with product
+probability. Define
+\begin{equation}
+ V\bigl((x,y),(x',y')\bigr)=W(x,y')W(x',y).
+ \label{eq:kernel-crossed-product}
+\end{equation}
+This kernel is symmetric, nonnegative, and has mean one: the two
+factors on the right use independent variables and each has mean
+one. In its product over $H$, the factors using $x_i$ for $i\in I$
+and $y_j$ for $j\in J$ are independent of those using $y_i$ for
+$i\in I$ and $x_j$ for $j\in J$. Therefore
+\[
+ t(H,V)=t_{I,J}(W)t_{I,J}(W^\top)<1.
+\]
+Let $L=\max V$, which is finite and at least one, and set $U=V/L$.
+Then $0\le U\le1$, $\mu=\E U=1/L>0$, and homogeneity gives
+$t(H,U)=L^{-66}t(H,V)<L^{-66}=\mu^{66}$.
+\end{proof}
+
+We use the standard independent sampling construction for graphons
+\cite[Section~2.6, Lemma~2.4]{lovaszszegedy2006}. The following argument keeps the collision
+error explicit and counts all homomorphisms.
+
+\begin{lemma}[Transfer to a finite simple host]
+\label{lem:finite-host}
+Suppose $U$ is a symmetric $[0,1]$-valued kernel on a finite
+probability space, $\mu=\E U>0$, and
+\[
+ \tau=t(H,U),\qquad \Delta=\mu^{66}-\tau>0.
+\]
+For any integer
+\begin{equation}
+ n>\frac{595+66\mu^{66}}{\Delta},
+ \label{eq:kernel-host-size}
+\end{equation}
+there is a finite simple graph $G$ on $n$ vertices with
+$t(H,G)<p(G)^{66}$.
+\end{lemma}
+
+\begin{proof}
+Sample labels $z_1,\ldots,z_n$ independently from the probability
+space of $U$. Conditional on these labels, put an edge on each
+unordered pair of distinct vertices $\{a,b\}$ independently with
+probability $U(z_a,z_b)$. No loops are drawn. This defines a random
+finite simple undirected graph $G_n$, and
+\begin{equation}
+ \E p(G_n)=(1-1/n)\mu.
+ \label{eq:kernel-host-mean}
+\end{equation}
+
+For every injective map $f:V(H)\to\{1,\ldots,n\}$, its image
+vertices have independent labels, and its $66$ source edges use
+distinct unordered host pairs. Its probability of being a
+homomorphism is exactly $\tau$. Among all $n^{35}$ maps, the
+proportion that identify any two source vertices is at most
+$\binom{35}{2}/n=595/n$, by the union bound. Each such map has a
+homomorphism indicator between zero and one; a collision on a
+source edge gives zero, while any repeated host edges are counted
+by their actual single Bernoulli trials. No independence claim for
+these maps is needed. Comparing their contribution with $\tau$
+times their proportion gives
+\begin{equation}
+ |\E t(H,G_n)-\tau|\le\frac{595}{n}.
+ \label{eq:kernel-collision-bound}
+\end{equation}
+Thus the density still counts all homomorphisms, including every
+noninjective one.
+
+Using $(1-1/n)^{66}\ge1-66/n$, Equations~\eqref{eq:kernel-host-size}
+and~\eqref{eq:kernel-collision-bound} give
+\begin{align*}
+ \E t(H,G_n)
+ &\le\mu^{66}-\Delta+595/n\\
+ &<\mu^{66}-66\mu^{66}/n\\
+ &\le\bigl((1-1/n)\mu\bigr)^{66}
+   =\bigl(\E p(G_n)\bigr)^{66}
+   \le\E\bigl[p(G_n)^{66}\bigr].
+\end{align*}
+The last inequality is convexity. Hence some realization has
+$t(H,G_n)<p(G_n)^{66}$. Such a realization cannot be edgeless.
+\end{proof}
+
+Proposition~\ref{prop:negative-kernel} and Lemma~\ref{lem:finite-host}
+prove Theorem~\ref{thm:main} conditional only on
+Lemma~\ref{lem:rank-layer}. To summarize the order of choices, fix a
+sufficiently large even $D$, the $66$ coordinates and $\theta$, then
+the scores and a finite $\rho$ satisfying Equation~\eqref{eq:kernel-finite-gap},
+then a positive $\varepsilon$ satisfying
+Equation~\eqref{eq:kernel-product-gap}, and then a sufficiently large
+odd prime $q$. Only after fixing and scaling this finite kernel are
+$\mu,\Delta$ and the host size $n$ chosen.
+Sections~\ref{sec:transverse}--\ref{sec:tail} establish the required
+rank-layer limit.
diff --git a/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/sections/orbits.tex b/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/sections/orbits.tex
new file mode 100644
index 0000000000000000000000000000000000000000..7da7ee247b4be60c3baf545a2e0f78309a667ee0
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+++ b/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/sections/orbits.tex
@@ -0,0 +1,527 @@
+\section{Lagrangian counts and ordered triple orbits}
+\label{sec:orbits}
+
+The contribution of singular differences is nonnegative, so for its upper
+bound we may discard the discriminant-sign restrictions after comparing
+the signed and unsigned normalizing masses. Their ratio stays bounded
+at fixed \(D\). Symmetric matrices then admit a
+useful interpretation as Lagrangian subspaces: nullity of a difference becomes
+intersection dimension.  We first give the exact counting formulas needed
+for this change of model.  We then prove a lower bound for every individual
+ordered triple orbit.  This stronger statement is needed for the change of
+measure in Section~\ref{sec:planted}.
+
+Throughout this section, dimensions are fixed and $q$ is an odd prime.
+The notation $A\asymp_D B$ means that the ratio is bounded above and below
+by positive constants depending only on $D$.  In particular, these constants
+will be uniform over all the subspaces and form classes that occur.
+As before, $b(n)=n(n+1)/2$.
+
+\subsection{Symplectic reduction and elementary counts}
+
+Symplectic bases, quotients, and Lagrangian counts are standard finite
+classical geometry; see Taylor~\cite[Theorem~7.4, Lemma~7.5, and Exercises~8.1--8.2]{taylor1992}. We give the arguments
+explicitly and then establish the individual ordered-orbit estimate
+needed here.
+
+Let $(\mathcal V,\omega)$ be a nondegenerate symplectic space of dimension
+$2N$ over $\F_q$.  A subspace is \emph{isotropic} if $\omega$ vanishes on it;
+a \emph{Lagrangian} is an isotropic subspace of dimension $N$.
+Write $\Lag(\mathcal V)$ for the set of Lagrangians, always equipped with
+uniform probability.  Orthogonal complements in this section refer to
+$\omega$.  Thus every Lagrangian $L$ satisfies $L^\perp=L$.
+
+We record explicitly the reduction facts used below.  An isotropic basis
+$e_1,\ldots,e_t$ extends to a symplectic basis.  To see this, choose vectors
+$f_1,\ldots,f_t$ with $\omega(e_i,f_j)=\delta_{ij}$, using nondegeneracy.
+If $a_{ij}=\omega(f_i,f_j)$, replace $f_i$ by
+$f_i+\sum_k(a_{ki}/2)e_k$.  The new $f_i$ pair to zero with one another,
+while their pairings with the $e_i$ are unchanged.  Their joint span is
+nondegenerate; repeat in its orthogonal complement.  In particular,
+symplectic automorphisms act transitively on isotropic subspaces of any
+specified dimension.
+
+For an isotropic $t$-space $S$, the quotient
+$S^\perp/S$ is symplectic of dimension $2(N-t)$.  The map
+\begin{equation}
+ L\longmapsto L/S
+ \quad\text{is a bijection from }
+ \{L\in\Lag(\mathcal V):S\subseteq L\}
+ \text{ to }\Lag(S^\perp/S).
+ \label{eq:orbits-containing-reduction}
+\end{equation}
+More generally, every Lagrangian $L$ has a Lagrangian reduction
+\begin{equation}
+ \overline L^{\,S}=\big((L\cap S^\perp)+S\big)/S.
+ \label{eq:orbits-general-reduction}
+\end{equation}
+Indeed, if $u=\dim(L\cap S)$, the pairing map $S\to L^*$ has kernel
+$S\cap L^\perp=S\cap L$ and rank $t-u$.  Its transpose therefore gives
+$\dim(L\cap S^\perp)=N-t+u$.  The space in
+Equation~\eqref{eq:orbits-general-reduction} is isotropic and has dimension
+$N-t$.  Choosing symplectic coordinates for $S$ and its dual also gives
+\begin{equation}
+ \mathcal V=(S\oplus S^*)\mathbin{\perp}\mathcal U,
+ \qquad \mathcal U\cong S^\perp/S.
+ \label{eq:orbits-quotient-extension}
+\end{equation}
+Here $S^*$ denotes a chosen isotropic dual subspace.  Every symplectic
+automorphism of the quotient extends by the identity on $S\oplus S^*$.
+In particular, it extends fixing $S$ pointwise.
+
+For $0\le t\le n$, put
+\begin{equation}
+ \genfrac{[}{]}{0pt}{}{n}{t}_q
+ =\prod_{i=0}^{t-1}\frac{q^{n-i}-1}{q^{t-i}-1},
+ \qquad
+ L_N=\prod_{j=1}^N(q^j+1),\qquad L_0=1.
+ \label{eq:orbits-basic-notation}
+\end{equation}
+An empty product is one.  Counting ordered independent $t$-tuples
+in $\F_q^n$ and dividing by the number of ordered bases of a
+$t$-space shows that $\genfrac{[}{]}{0pt}{}{n}{t}_q$ counts the
+$t$-dimensional subspaces of $\F_q^n$.  We shall also use
+\begin{equation}
+ |\operatorname{GL}_n(\F_q)|
+ =q^{n^2}\prod_{j=1}^n(1-q^{-j}),
+ \qquad |\operatorname{GL}_0(\F_q)|=1.
+ \label{eq:orbits-gl-order}
+\end{equation}
+
+\begin{lemma}[Lagrangian counting formulas]
+\label{lem:lagrangian-counts}
+In a symplectic $2N$-space, the following formulas hold.
+\begin{enumerate}
+ \item The number of Lagrangians is $L_N$.
+ \item A fixed isotropic $t$-space is contained in exactly $L_{N-t}$
+ Lagrangians.  Consequently its containment probability and the number of
+ isotropic $t$-spaces are, respectively,
+ \begin{equation}
+ \vartheta_N(t)=\frac{L_{N-t}}{L_N},\qquad
+ I_N(t)=\frac{L_N}{L_{N-t}}\genfrac{[}{]}{0pt}{}{N}{t}_q.
+ \label{eq:orbits-containment}
+ \end{equation}
+ \item For a fixed Lagrangian $A$, the number of Lagrangians $B$ with
+ $\dim(A\cap B)=h$ is $\genfrac{[}{]}{0pt}{}{N}{h}_q q^{b(N-h)}$.  Thus
+ \begin{equation}
+ \pi_{N,h}:=\P\{\dim(A\cap B)=h\}
+ =\frac{\genfrac{[}{]}{0pt}{}{N}{h}_q q^{b(N-h)}}{L_N}.
+ \label{eq:orbits-pair-probability}
+ \end{equation}
+ All ordered pairs with the same intersection dimension belong to one
+ symplectic orbit.
+\end{enumerate}
+For fixed $N$, these formulas give
+\begin{align}
+ L_N&=(1+O_N(q^{-1}))q^{b(N)},
+ &\genfrac{[}{]}{0pt}{}{n}{t}_q&=(1+O_n(q^{-1}))q^{t(n-t)},\\
+ \vartheta_N(t)&\asymp_N q^{-Nt+\binom t2},
+ &I_N(t)&\asymp_N q^{2Nt-(3t^2-t)/2},\\
+ \pi_{N,h}&\asymp_N q^{-b(h)}.
+ \label{eq:orbits-count-exponents}
+\end{align}
+\end{lemma}
+
+\begin{proof}
+Every line is isotropic.  Count pairs consisting of a line and a Lagrangian
+containing it.  Each Lagrangian has $(q^N-1)/(q-1)$ lines, and there are
+$(q^{2N}-1)/(q-1)$ lines in $\mathcal V$.  Reduction by a line gives
+$L_N(q^N-1)=(q^{2N}-1)L_{N-1}$.  Starting at $L_0=1$ yields the product
+in Equation~\eqref{eq:orbits-basic-notation}.
+Equation~\eqref{eq:orbits-containing-reduction} gives the containment
+count.  Counting pairs $(S,L)$ with $\dim S=t$ and $S\subseteq L$ gives
+$I_N(t)L_{N-t}=L_N\genfrac{[}{]}{0pt}{}{N}{t}_q$.
+
+For the pair count, first choose $H=A\cap B$ inside $A$.
+In $H^\perp/H$, the image of $B$ must be transverse to the image of $A$,
+where \emph{transverse} means having zero intersection.  There are exactly
+$q^{b(N-h)}$ such Lagrangians: after choosing complementary coordinate
+Lagrangians, they are graphs of symmetric linear maps from one coordinate
+space to the other.  Symmetry follows by evaluating the alternating form
+on two graph vectors.  This proves Equation~\eqref{eq:orbits-pair-probability}.
+To prove pair transitivity, first identify the common subspaces by
+symplectic bases.  In their common-space quotient, identify the transverse
+pairs by choosing a basis of one member and its dual basis in the other.
+The resulting quotient map extends by
+Equation~\eqref{eq:orbits-quotient-extension}; containment reduction is a
+bijection, so the extension identifies the original pairs.
+Finally, factoring the leading powers of $q$ out of the finite products
+proves all the displayed estimates, including their endpoint cases.
+\end{proof}
+
+\subsection{Comparison with the matrix chart}
+
+Take $N=D=2r$, identify $\mathcal V=\F_q^D\oplus(\F_q^D)^*$, and use
+\begin{equation}
+ \omega((x,f),(y,g))=g(x)-f(y).
+ \label{eq:orbits-coordinate-form}
+\end{equation}
+For $X\in\mathcal S_D$, write $\Gamma_X=\{(x,Xx):x\in\F_q^D\}$,
+where coordinates identify a symmetric matrix with a map to the dual.
+These are exactly the Lagrangians transverse to the fixed second
+coordinate space.  Denote this chart by $\mathcal A$.  Then
+\begin{equation}
+ |\mathcal A|=q^{b(D)},\qquad
+ \beta_D:=\frac{|\mathcal A|}{L_D}=\prod_{j=1}^D(1+q^{-j})^{-1},
+ \qquad
+ \dim(\Gamma_X\cap\Gamma_Y)=D-\rank(X-Y).
+ \label{eq:orbits-chart}
+\end{equation}
+Define the unsigned normalized incidence kernel by
+\begin{equation}
+ J(L,M)=\frac{\1_{\{\dim(L\cap M)=r\}}}{\pi_{D,r}}.
+ \label{eq:orbits-unsigned-kernel}
+\end{equation}
+It has mean one in either variable for every fixed value of the other.
+Let
+$p_\xi=\P\{\rank X=r,\ \operatorname{sign}(X)=\xi\}$,
+so that $K_\xi$ has normalization $p_\xi^{-1}$. This is the quantity
+denoted \(\pi_\xi\) in Equation~\eqref{eq:transverse-signed-mass};
+the letter \(p\) here distinguishes it from the Lagrangian pair
+probability \(\pi_{D,r}\).
+Lemma~\ref{lem:symmetric-counts} and
+Equation~\eqref{eq:orbits-pair-probability} imply
+\begin{equation}
+ p_\xi=(1/2+O_D(q^{-1}))q^{-b(r)},\qquad
+ \pi_{D,r}=(1+O_D(q^{-1}))q^{-b(r)}.
+ \label{eq:orbits-normalizers}
+\end{equation}
+
+\begin{lemma}[Chart comparison]
+\label{lem:chart-comparison}
+Let $B\subseteq\mathscr C$ be a set of active incidences, with prescribed
+signs $\xi_{ij}$.  Let $F$ be any nonnegative function of the $35$
+Lagrangians indexed by $I\sqcup J$.  For independent uniform matrices
+$X_i,Y_j\in\mathcal S_D$ and independent uniform Lagrangians $L_i,M_j$,
+\begin{align}
+ &\E\left[
+ F((\Gamma_{X_i})_i,(\Gamma_{Y_j})_j)
+ \prod_{(i,j)\in B}K_{\xi_{ij}}(X_i-Y_j)\right]\notag\\
+ &\qquad\le
+ \beta_D^{-35}\prod_{(i,j)\in B}\frac{\pi_{D,r}}{p_{\xi_{ij}}}
+ \ \E\left[F((L_i)_i,(M_j)_j)
+ \prod_{(i,j)\in B}J(L_i,M_j)\right].
+ \label{eq:orbits-chart-comparison}
+\end{align}
+The multiplier is bounded in terms of $D$ alone for all sufficiently large
+$q$, uniformly in the active set and its signs.  In particular, this
+comparison applies with $F$ the indicator that some pair in $E$ has
+positive intersection dimension, or the indicator of any specified
+intersection profile.
+\end{lemma}
+
+\begin{proof}
+On the chart, dropping the sign condition gives the pointwise inequality
+\begin{equation}
+ K_\xi(X-Y)\le
+ \frac{\pi_{D,r}}{p_\xi}J(\Gamma_X,\Gamma_Y).
+ \label{eq:orbits-pointwise-comparison}
+\end{equation}
+The graph variables have exactly the law of the uniform Lagrangian
+variables conditioned on all $35$ variables belonging to $\mathcal A$.
+That event has probability $\beta_D^{35}$.  Apply
+Equation~\eqref{eq:orbits-pointwise-comparison}, express the conditional
+expectation by inserting the event's indicator and dividing by its
+probability, and then remove the indicator by nonnegativity.  This gives
+Equation~\eqref{eq:orbits-chart-comparison} without any boundedness
+assumption on $F$ or on the normalized kernels.  Finally
+$\beta_D\to1$ and $\pi_{D,r}/p_\xi\to2$.
+\end{proof}
+
+\subsection{The cost of an ordered triple orbit}
+
+Let $L_1,L_2,L_3$ be Lagrangians in a symplectic $2D$-space.  Use opposite
+indices for their pair intersections and put
+\begin{equation}
+ \begin{gathered}
+ c=\dim(L_1\cap L_2\cap L_3),\qquad
+ h_i=\dim(L_j\cap L_k),\\
+ s_i=h_i-c,\qquad d=D-c,
+ \qquad \{i,j,k\}=\{1,2,3\}.
+ \end{gathered}
+ \label{eq:orbits-profile}
+\end{equation}
+These integers form the \emph{dimensional profile}.  Set
+\begin{equation}
+ C=(D+1)c+\sum_{i=1}^3b(s_i).
+ \label{eq:orbits-profile-cost}
+\end{equation}
+We call a dimensional profile \emph{feasible} if it is realized by an
+ordered triple of Lagrangians over the field under consideration.
+
+Here is why the lower bound below concerns individual orbits. Let
+\(\mu\) be the independent triple law, and let \(\nu\) be the law
+obtained by choosing a uniform center \(Y\) and then, independently
+given \(Y\), three uniform Lagrangians meeting \(Y\) in dimension \(r\).
+Section~\ref{sec:planted} will verify that the density of \(\nu\)
+relative to \(\mu\) is \(\E_Y\prod_i J(L_i,Y)\). This density is
+constant on an ordered symplectic orbit \(\mathcal O\), with value
+\(\nu(\mathcal O)/\mu(\mathcal O)\). The analogous ratio for a whole
+dimensional profile is only the \(\mu\)-weighted average of its orbit
+values. A pointwise bound for the center integral therefore needs a
+lower bound for \(\mu(\mathcal O)\) for every individual orbit.
+
+\begin{proposition}[Mass of ordered triple orbits]
+\label{prop:triple-orbits}
+For three independent uniform Lagrangians, every feasible dimensional
+profile has probability $O_D(q^{-C})$.  More strongly, there is a constant
+$\kappa_D>0$, independent of $q$ and of the particular orbit, such that
+for every occurring ordered symplectic orbit $\mathcal O$ of that profile,
+\begin{equation}
+ \kappa_D q^{-C}\le
+ \P\{(L_1,L_2,L_3)\in\mathcal O\}
+ \le \kappa_D^{-1}q^{-C}.
+ \label{eq:orbits-orbit-mass}
+\end{equation}
+The constant can be chosen for all odd primes.  Moreover, conditional on
+any fixed ordered pair having intersection dimension $h$, the probability
+that an independent third Lagrangian completes a specified dimensional
+profile is
+\begin{equation}
+ O_D(q^{-C+b(h)}).
+ \label{eq:orbits-conditional-profile}
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+We divide the argument into the common-space count, the classification
+above the pair intersections, and the orbit lower bound.
+
+\paragraph{Counting the forced subspaces.}
+Choose an isotropic common $c$-space $C_0$ and impose its containment in
+all three Lagrangians.  The number of choices times the containment
+probability is exactly
+\begin{equation}
+ I_D(c)\vartheta_D(c)^3\asymp_D q^{-(D+1)c}.
+ \label{eq:orbits-common-cost}
+\end{equation}
+Conditional on these three separate containments, their images in
+$C_0^\perp/C_0$ are independent uniform Lagrangians of a symplectic
+$2d$-space.  In that quotient we require common intersection zero.
+For the next part of the argument denote the reduced Lagrangians again
+by $L_i$, and write $H_i=L_j\cap L_k$, of dimension $s_i$.
+
+The $H_i$ are pairwise orthogonal, since any two lie in a common
+Lagrangian.  They are also in direct sum.  Indeed, a relation
+$x_1+x_2+x_3=0$ with $x_i\in H_i$ puts $x_i$ in $L_i$, since the other
+two summands lie there.  Hence each $x_i$ is in the zero common
+intersection.  Thus
+\begin{equation}
+ H_*:=H_1\oplus H_2\oplus H_3\text{ is isotropic},
+ \qquad a:=s_1+s_2+s_3\le d,
+ \qquad z:=d-a\ge0.
+ \label{eq:orbits-pair-sum}
+\end{equation}
+Choose $H_*$ and then its ordered decomposition into the $H_i$.
+The number of such choices is
+\begin{equation}
+ I_d(a)\frac{|\operatorname{GL}_a(\F_q)|}
+ {\prod_i|\operatorname{GL}_{s_i}(\F_q)|}
+ \asymp_D
+ q^{2da-(3a^2-a)/2+a^2-\sum_i s_i^2}.
+ \label{eq:orbits-pair-space-count}
+\end{equation}
+The stabilizer of an ordered direct-sum decomposition in
+$\operatorname{GL}_a$ consists exactly of the block diagonal maps,
+which proves the quotient in this formula.
+For these choices each $L_i$ must contain
+$A_i:=\bigoplus_{j\ne i}H_j$, of dimension $a-s_i$.
+Their independent containment probabilities give the additional factor
+$\prod_i\vartheta_d(a-s_i)$.  The total exponent is
+\begin{align}
+ &2da-\frac{3a^2-a}{2}+a^2-\sum_i s_i^2
+ +\sum_i\left[-d(a-s_i)+\binom{a-s_i}{2}\right]
+ =-\sum_i b(s_i).
+ \label{eq:orbits-pair-cost-cancellation}
+\end{align}
+Multiplying with Equation~\eqref{eq:orbits-common-cost}, and ignoring
+all conditions beyond these containments, proves the asserted profile
+upper bound.
+
+The profile upper bound is now established. The center-factor comparison
+also needs a lower bound for each individual ordered orbit; the remaining
+argument supplies that bound while keeping the intersections exact.
+
+\paragraph{Splitting off the pair intersections.}
+We next identify all invariants of an actual common-zero triple.  Fix
+$i$.  The pairing $H_i\to L_i^*$ is injective, since its kernel is
+$H_i\cap L_i^\perp=H_i\cap L_i=0$.  Its transpose is therefore
+surjective.  Choose an $s_i$-space $Q_i\subseteq L_i$ paired perfectly
+with $H_i$.  Both spaces are isotropic, so
+$B_i=H_i\oplus Q_i$ is a nondegenerate symplectic $2s_i$-space.
+Each $L_\ell$ contains a Lagrangian of $B_i$: it contains $Q_i$ when
+$\ell=i$ and $H_i$ otherwise.  Call that subspace $K_\ell$.
+Under the orthogonal decomposition $\mathcal V=B_i\mathbin\perp B_i^\perp$,
+the projection of a vector $x\in L_\ell$ to $B_i$ is orthogonal to
+$K_\ell$, hence belongs to $K_\ell$.  As $K_\ell\subseteq L_\ell$,
+subtracting the projection gives the actual splitting
+\begin{equation}
+ L_\ell=K_\ell\oplus(L_\ell\cap B_i^\perp),
+ \qquad \ell=1,2,3.
+ \label{eq:orbits-forced-splitting}
+\end{equation}
+For $j\ne i$, the space $H_j$ lies in $B_i^\perp$: it is orthogonal to
+$H_i$, and is orthogonal to $Q_i$ because $H_j,Q_i\subseteq L_i$.
+Thus the other pair intersections survive in the orthogonal complement,
+and the construction can be repeated there.  The zero-dimensional
+blocks require no choice.
+
+After splitting all three blocks, the remaining ordered triple is
+transverse in a symplectic space of dimension $2z$.  The space $H_*$
+is Lagrangian in the sum of the three blocks, so this remaining triple
+is precisely the reduction of the original triple to $H_*^\perp/H_*$
+as in Equation~\eqref{eq:orbits-general-reduction}.
+Symplectic bases identify any two corresponding pair blocks, respecting
+all three members of the ordered triple.  A symplectic map identifying
+the residual ordered triples extends over these block maps.  Consequently,
+the dimensions $s_i$ and the residual ordered orbit determine the entire
+common-zero ordered orbit.  For $c>0$, first identify the common spaces;
+Equation~\eqref{eq:orbits-containing-reduction} and the extension in
+Equation~\eqref{eq:orbits-quotient-extension} show that the same assertion
+lifts to the original space.  In particular there is no further invariant
+arising from extensions between the blocks.
+
+\paragraph{The mass of a residual orbit.}
+An ordered transverse pair of Lagrangians in dimension $2z$ can be
+identified with the two coordinate spaces.  A third Lagrangian
+transverse to both is then the graph of a nonsingular symmetric form.
+The stabilizer of the ordered pair consists of the maps induced by
+$g\in\operatorname{GL}_z(\F_q)$ on one coordinate space and its inverse
+dual on the other; its action on forms is congruence.  Thus residual
+ordered orbits correspond exactly to congruence orbits
+$\mathcal O_{\mathrm{form}}$ of nonsingular symmetric forms.
+For this classification of ordered transverse triples, see also
+Kramer and Tent~\cite[Theorem~6 and Section~2.3]{kramertent2010}.
+For three independent uniform residual Lagrangians, a prescribed
+residual orbit has probability exactly
+\begin{equation}
+ \frac{q^{b(z)}|\mathcal O_{\mathrm{form}}|}{L_z^2}.
+ \label{eq:orbits-residual-mass}
+\end{equation}
+Indeed, after the first Lagrangian is chosen, the second is transverse
+with probability $q^{b(z)}/L_z$; for each such ordered pair there are
+$|\mathcal O_{\mathrm{form}}|$ choices of the third among $L_z$
+Lagrangians.
+
+This probability has a positive lower bound depending only on $z$,
+uniformly in the form class.  Diagonalize a nonsingular symmetric form $F$
+and choose an orthogonal basis with specified nonzero norms.  An
+isometry is determined by the images of this basis.  At the stage when
+the remaining nondegenerate orthogonal complement has dimension $k$,
+there are at most $2q^{k-1}$ vectors of a specified nonzero norm: after
+diagonalizing, fix $k-1$ coordinates and solve a quadratic equation in
+the last one.  Each chosen vector has nonzero norm, so the next
+orthogonal complement is again nondegenerate.  Hence
+\begin{equation}
+ |\operatorname{Isom}(F)|\le
+ 2^z q^{z(z-1)/2},\qquad
+ |\mathcal O_{\mathrm{form}}|
+ \ge 2^{-z}q^{b(z)}\prod_{j=1}^z(1-q^{-j}).
+ \label{eq:orbits-isometry-bound}
+\end{equation}
+Orbit--stabilizer and Equation~\eqref{eq:orbits-gl-order} give the second
+inequality.  For example, for every odd prime the lower bound in
+Equation~\eqref{eq:orbits-residual-mass} is at least
+\begin{equation}
+ \gamma_z:=2^{-z}
+ \frac{\prod_{j=1}^z(1-3^{-j})}
+ {\prod_{j=1}^z(1+3^{-j})^2}>0.
+ \label{eq:orbits-residual-constant}
+\end{equation}
+For $z=0$, there is one residual Lagrangian and one empty form;
+$L_0=1$, $|\mathcal O_{\mathrm{form}}|=1$, and
+Equation~\eqref{eq:orbits-residual-mass} is exactly one.  We take
+$\gamma_0=1$.
+
+We have identified the residual form class and bounded its probability
+in its own symplectic space. We must now lift that estimate through
+the chosen pair-intersection spaces while keeping their intersections
+exact. We condition each Lagrangian separately before imposing the
+residual orbit.
+
+\paragraph{Independent residual laws and exact intersections.}
+Fix candidate spaces $H_i$ as in
+Equation~\eqref{eq:orbits-pair-space-count}, and sample the three
+Lagrangians independently subject to $A_i\subseteq L_i$.
+Require, separately for each sample, the avoidance event
+\begin{equation}
+ \mathcal E_i=\{L_i\cap H_*=A_i\}.
+ \label{eq:orbits-avoidance}
+\end{equation}
+In the containment quotient $A_i^\perp/A_i$, of half-dimension
+$d-(a-s_i)=z+s_i$, failure means meeting the fixed isotropic
+$s_i$-space $H_*/A_i$ in a line.  If $s_i>0$, a union bound and
+Equation~\eqref{eq:orbits-containment} give
+\begin{equation}
+ \P(\mathcal E_i^c\mid A_i\subseteq L_i)
+ \le \frac{q^{s_i}-1}{q-1}\frac1{q^{z+s_i}+1}
+ <\frac{q^{-z}}{q-1}\le\frac12.
+ \label{eq:orbits-avoidance-probability}
+\end{equation}
+If $s_i=0$, the failure probability is zero.  In particular, each
+avoidance probability is at least $1/2$ and is $1-O_D(q^{-1})$.
+The containments and avoidance events concern each sample separately,
+so after conditioning on all of them the three samples remain
+independent.
+
+Let $\widehat L_i$ be the reduction of $L_i$ to $H_*^\perp/H_*$.
+Each conditional $\widehat L_i$ is uniform on the Lagrangians of this
+quotient.  To verify this, extend any symplectic automorphism of the
+quotient by the identity on $H_*$ and a chosen dual, as in
+Equation~\eqref{eq:orbits-quotient-extension}.  The resulting map fixes
+every $H_i$ and $A_i$, preserves containment and avoidance, and takes
+reductions to their images under the prescribed quotient map.  The
+conditional law is therefore invariant under the full symplectic
+group of the quotient.  That group is transitive on its Lagrangians,
+so the law is uniform.  Since the reduction is a function of each
+sample individually, the three reductions are also independent.
+
+Require their ordered residual orbit to be the one of the fixed target
+orbit $\mathcal O$.  By Equation~\eqref{eq:orbits-residual-mass}, this
+has conditional probability at least $\gamma_z$.  These conditions
+force the selected pair intersections to be exact.  For example, if
+$x\in L_i\cap L_j$, then $x$ is orthogonal to every $H_\ell$:
+for each $\ell$, at least one of $L_i,L_j$ contains $H_\ell$.
+Thus $x\in H_*^\perp$, and its image lies in
+$\widehat L_i\cap\widehat L_j=0$.  It follows that $x\in H_*$.
+Avoidance now gives
+\begin{equation}
+ L_i\cap L_j\subseteq A_i\cap A_j=H_k,
+ \qquad \{i,j,k\}=\{1,2,3\}.
+ \label{eq:orbits-exact-pairs}
+\end{equation}
+The reverse inclusion was imposed from the start.  Hence the pair
+intersections are exactly the $H_k$, and the common intersection is
+zero.  The splitting argument proves that every resulting triple
+belongs to the prescribed ordered orbit in the common-space quotient,
+and then to $\mathcal O$ in the original space.
+
+For each choice of $C_0,H_1,H_2,H_3$, the required additional probability
+is at least $2^{-3}\gamma_z$.  Successful choices cannot count a triple
+twice: its exact common space is $C_0$ and its exact pair spaces in the
+quotient are the $H_i$.  We have proved the explicit lower bound
+\begin{align}
+ \P(\mathcal O)\ge{}&2^{-3}\gamma_z\,
+ I_D(c)\vartheta_D(c)^3
+ I_d(a)\frac{|\operatorname{GL}_a(\F_q)|}
+ {\prod_i|\operatorname{GL}_{s_i}(\F_q)|}
+ \prod_i\vartheta_d(a-s_i).
+ \label{eq:orbits-explicit-lower-bound}
+\end{align}
+Equations~\eqref{eq:orbits-common-cost} and
+\eqref{eq:orbits-pair-cost-cancellation} now give the lower bound in
+Equation~\eqref{eq:orbits-orbit-mass}.  The upper bound follows from
+the profile upper bound already proved.  All product bounds are
+uniform for $q\ge3$ and involve dimensions at most $D$; the explicit
+$\gamma_z$ is uniform over actual form classes.  Taking minima and
+maxima over these finitely many dimension tuples gives a single
+constant $\kappa_D$.
+
+Finally, pair transitivity in Lemma~\ref{lem:lagrangian-counts} makes
+the probability of completing a given profile the same for every
+fixed ordered pair of dimension $h$.  It equals the unconditional
+profile probability divided by $\pi_{D,h}$ when the prescribed pair
+dimension is $h$, and is zero otherwise.  The profile upper bound
+and Equation~\eqref{eq:orbits-pair-probability} give
+Equation~\eqref{eq:orbits-conditional-profile}.
+\end{proof}
diff --git a/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/sections/planted.tex b/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/sections/planted.tex
new file mode 100644
index 0000000000000000000000000000000000000000..e1a4636e66361dae0f5918e1245950060535a83c
--- /dev/null
+++ b/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/sections/planted.tex
@@ -0,0 +1,484 @@
+\section{Local bounds from the planted law}\label{sec:planted}
+
+We now bound the center integral for fixed point Lagrangians, including
+triples whose pair intersections are nonzero.  Throughout this section the
+ambient symplectic space $\mathcal V$ has dimension $2D$, where $D=2r$.
+We use the pair probability $\pi_{N,h}$ and the unsigned normalized
+incidence kernel $J$ from Section~\ref{sec:orbits}.  Thus
+\[
+ J(L,Y)=\frac{\1_{\{\dim(L\cap Y)=r\}}}{\pi_{D,r}},
+ \qquad \pi_{N,h}\asymp q^{-b(h)}.
+\]
+For a set $A\subseteq\{1,2,3\}$ of active incidences, put
+\begin{equation}\label{eq:planted-face-factor}
+ \mathcal F_A(L_1,L_2,L_3)
+   =\E_Y\prod_{i\in A}J(L_i,Y),
+\end{equation}
+where $Y$ is a uniform Lagrangian.  Constants in this section may depend
+on the fixed dimension $D$, but are uniform over the subspaces, their
+orbits, and the odd primes $q$.  Every exponent displayed below is an
+explicit expression in the dimensions; there is no unspecified error
+term in an exponent.
+
+\subsection{The density ratio and the common intersection}
+
+Let $\mu$ be the law of three independent uniform Lagrangians.  Define
+the \emph{planted law} $\nu$ by first choosing $Y$ uniformly and then,
+conditional on $Y$, choosing the three $L_i$ independently and uniformly
+subject to $\dim(L_i\cap Y)=r$.  Pair transitivity makes the conditioning
+probability equal to $\pi_{D,r}$ for every $Y$.  Hence, exactly,
+\begin{equation}\label{eq:planted-density}
+ \frac{d\nu}{d\mu}(L_1,L_2,L_3)
+ =\pi_{D,r}^{-3}\E_Y
+       \prod_{i=1}^3\1_{\{\dim(L_i\cap Y)=r\}}
+ =\mathcal F_{\{1,2,3\}}(L_1,L_2,L_3).
+\end{equation}
+The right side is invariant under simultaneous symplectic maps.
+Therefore, on any actual ordered symplectic orbit $\mathcal O$,
+\begin{equation}\label{eq:planted-orbit-ratio}
+ \mathcal F_{\{1,2,3\}}\big|_{\mathcal O}
+     =\frac{\nu(\mathcal O)}{\mu(\mathcal O)}.
+\end{equation}
+It is essential here to use the mass of this orbit in the denominator.
+If its common intersection has dimension $c$, its opposite pair
+intersection dimensions are $h_i$, and
+\begin{equation}\label{eq:planted-local-dimensions}
+ s_i=h_i-c,\qquad d=D-c,\qquad
+ C=(D+1)c+\sum_{i=1}^3 b(s_i),
+\end{equation}
+then Proposition~\ref{prop:triple-orbits} gives
+$\mu(\mathcal O)\asymp q^{-C}$, uniformly also when the residual
+transverse space has dimension zero.
+
+We estimate the numerator through four geometric choices. First record
+the actual common space \(C_0=L_1\cap L_2\cap L_3\), then count how
+the center \(Y\) meets it. In the quotient by \(C_0\), count the
+intersections of the three point Lagrangians with the reduced center.
+Finally count the remaining lift forms, which supply any additional
+pair intersections. These counts use the independent laws before the
+mutual point profile is imposed.
+
+The common space \(C_0\) is unique. Thus summing over
+isotropic $c$-spaces $C_0$, while retaining that equality, counts each
+triple of common dimension $c$ exactly once.  For a deterministic
+summand $C_0$, we evaluate probabilities under the original law of
+four independent uniform Lagrangians.  Under this law the
+three containments $C_0\subseteq L_i$ have total probability
+$\asymp q^{-3Dc+3\binom c2}$.  There are
+$\asymp q^{2Dc-(3c^2-c)/2}$ choices of $C_0$, so their combined
+contribution is
+\begin{equation}\label{eq:planted-common-count}
+ q^{2Dc-(3c^2-c)/2-3Dc+3\binom c2}
+      =q^{-(D+1)c}
+\end{equation}
+up to a constant factor.  These are the counts of
+Lemma~\ref{lem:lagrangian-counts}.
+
+For the center let $u=\dim(C_0\cap Y)$.  Choosing a prospective
+$u$-space in $C_0$ and requiring its containment in $Y$ gives
+\begin{align}
+ \P\{\dim(C_0\cap Y)=u\}
+ &\le O\bigl(q^{u(c-u)-Du+\binom u2}\bigr) \notag\\
+ &=O\bigl(q^{-u(D-c)-b(u)}\bigr).
+ \label{eq:planted-center-common-cost}
+\end{align}
+This bound includes $u=0$.  We now work in the symplectic quotient
+$C_0^\perp/C_0$ and write
+\[
+ L_i'=L_i/C_0,\qquad
+ Y'=\bigl((Y\cap C_0^\perp)+C_0\bigr)/C_0.
+\]
+Both are Lagrangians of dimension $d$.  There is an exact quotient
+identity
+\begin{equation}\label{eq:planted-quotient-intersection}
+ L_i'\cap Y'=\bigl((L_i\cap Y)+C_0\bigr)/C_0,
+ \qquad \dim(L_i'\cap Y')=\dim(L_i\cap Y)-u.
+\end{equation}
+Indeed, a class in the left side has a representative
+$y\in Y\cap C_0^\perp$.  Membership in $L_i/C_0$ means that
+$y+z\in L_i$ for some $z\in C_0$.  Since $C_0\subseteq L_i$,
+this implies $y\in L_i\cap Y$.  The converse is immediate, and the
+map from $L_i\cap Y$ to the quotient has kernel $C_0\cap Y$.
+On the required incidence event the last dimension is $r-u$.
+This also proves that no choice of lifts is missing from the quotient
+count.
+
+Put
+\begin{equation}\label{eq:planted-reduced-parameters}
+ r'=r-u,\qquad \delta=d-2r'=2u-c.
+\end{equation}
+For this counting step fix deterministic $C_0,Y$ with
+$\dim(C_0\cap Y)=u$.  Under the original independent product law,
+condition only on the three containments $C_0\subseteq L_i$.
+The quotient Lagrangians $L_i'$ are then independent and uniform,
+because containment reduction is a bijection.  The equality
+$C_0=\bigcap_i L_i$ is still an event to count.  Their three
+individual conditions $\dim(L_i'\cap Y')=r'$ consequently have
+probability $\pi_{d,r'}^3\asymp q^{-3b(r')}$.  Including the planted
+normalizer $\pi_{D,r}^{-3}$, the gain relative to
+Equation~\eqref{eq:planted-common-count} is
+\begin{align}
+ g_C
+ &=3\bigl[b(r)-b(r-u)\bigr]-u(D-c)-b(u) \notag\\
+ &=\frac32(2ru-u^2+u)-u(2r-c)-\frac12(u^2+u)
+      \notag\\
+ &=(r+c+1)u-2u^2.
+ \label{eq:planted-common-gain}
+\end{align}
+
+The preceding independent law is the law \emph{before} imposing the
+mutual intersection profile of the $L_i'$.  In particular, their
+common-zero condition is still an event to be counted.  Under their
+three individual center conditions the spaces
+\[
+ R_i=L_i'\cap Y'
+\]
+are independent uniform $r'$-subspaces of $Y'$.  Conditional on these
+individual $R_i$, the remaining choices of $L_i'$ are independent as
+well.  We use these product laws below, and never assert independence
+after conditioning on the mutual profile.
+
+\subsection{The subspace cost and the lift forms}
+
+The common-space calculation accounts for how the center meets $C_0$.
+It remains to count the prescribed pair intersections in the quotient:
+first inside $Y'$, and then through the remaining choices of the $L_i'$.
+
+Because the required quotient triple has common intersection zero,
+the $R_i$ must have common intersection zero.  Let
+\begin{equation}\label{eq:planted-subspace-parameters}
+ \begin{split}
+ p_i&=\dim(R_j\cap R_k),\qquad t_i=s_i-p_i,\qquad
+ P=\sum_i p_i,\qquad T=\sum_i t_i,\\
+ v&=d-\dim(R_1+R_2+R_3),\qquad
+ w=3r'-d-P+v=\frac{d-3\delta}{2}-P+v,
+ \end{split}
+\end{equation}
+where $\{i,j,k\}=\{1,2,3\}$.
+The pair spaces $R_j\cap R_k$ form a direct sum.  To see this, in a
+relation $a_1+a_2+a_3=0$ among the three pair spaces, membership of
+two terms in $R_i$ forces the remaining term into $R_i$ as well;
+each term therefore lies in the common intersection and vanishes.
+The related inequality $w\ge0$ follows in the abstract direct sum
+$R_1\oplus R_2\oplus R_3$.  The summation map has kernel of
+dimension $3r'-d+v$.  The three pair-relation spaces consist of
+vectors of the forms
+\[
+ (0,a,-a),\qquad (-b,0,b),\qquad (z,-z,0).
+\]
+Their sum can vanish only if $a=b=z$, hence only if all three vanish.
+They therefore contribute $P$ independent dimensions to that kernel.
+
+Here $p_i$ records the pair-intersection dimension already inside $Y'$,
+and $t_i=s_i-p_i$ is the target additional dimension from the remaining
+Lagrangian choices. The parameter $v$ is the span codimension in $Y'$;
+$w$ is the dimension of the summation-map kernel after quotienting by
+the subspace generated by the pair relations.
+
+\begin{lemma}\label{lem:planted-subspace-cost}
+For three independent uniform $r'$-subspaces of a $d$-space, the
+event of common intersection zero, opposite pair dimensions $p_i$,
+and span codimension $v$ has probability
+\begin{equation}\label{eq:planted-subspace-cost}
+ O\left(q^{-\delta P-\sum_i p_i^2-vw}\right).
+\end{equation}
+\end{lemma}
+\begin{proof}
+Write $n=d-v$.  First choose a prospective span $Z$ of dimension
+$n$.  There are $\asymp q^{nv}$ choices, and the probability that
+all three independent $r'$-subspaces lie in it is
+$\asymp q^{-3r'v}$.  Conditional on these separate containments,
+the three subspaces are independent uniform $r'$-subspaces of $Z$.
+
+The number of choices of three pair spaces in direct sum inside
+$Z$ is $O(q^{nP-\sum_i p_i^2})$.  Indeed, choose their $P$-dimensional
+sum and then an ordered direct-sum decomposition; the two exponents
+are $P(n-P)$ and $P^2-\sum_i p_i^2$.  Each $R_i$ must contain the
+sum of the other two pair spaces, of dimension $P-p_i$.  In $Z$
+this costs exponent $(n-r')(P-p_i)$, for a total of
+$2(n-r')P$.  We may drop the requirements that $Z$ be exactly the
+span and that these containments give exactly the pair intersections.
+Thus the total cost exponent is
+\begin{align*}
+ (3r'-d+v)v-nP+\sum_i p_i^2+2(n-r')P
+ &= (3r'-d+v)v+(d-v-2r')P+\sum_i p_i^2\\
+ &=\delta P+\sum_i p_i^2+v(3r'-d-P+v),
+\end{align*}
+which proves the claim.  Infeasible containments have probability zero.
+\end{proof}
+
+We next describe the remaining Lagrangian choices without any
+generic-position assumption.  Choose symplectic coordinates
+$Y'\oplus(Y')^*$ with pairing
+\[
+ \omega((y,\alpha),(z,\beta))=\beta(y)-\alpha(z).
+\]
+For an $r'$-subspace $R\subseteq Y'$, let $A=R^\circ\subseteq(Y')^*$
+be its annihilator.  A Lagrangian $L'$ meeting $Y'$ exactly in $R$
+projects into $A$ by orthogonality to $R$; its kernel is $R$,
+so the dimensions show that its image is all of $A$.  Its induced map
+$A\longrightarrow Y'/R\cong A^*$ is a symmetric bilinear form
+$F$ on $A$.  Conversely, every such form gives exactly one lift,
+namely
+\begin{equation}\label{eq:planted-lift-parametrization}
+ L'(R,F)=\{(y,\alpha):\alpha\in A,\quad
+              \beta(y)=F(\alpha,\beta)\text{ for every }\beta\in A\}.
+\end{equation}
+The set in Equation~\eqref{eq:planted-lift-parametrization} has dimension
+$d$, intersects $Y'$ in $R$, and is isotropic precisely because $F$
+is symmetric.  This proves the bijection, including when $A=0$.
+There are exactly $q^{b(d-r')}$ lifts for each $R$.  Under the
+product law already specified, conditional on the $R_i$, the lift
+forms $F_i$ on $A_i=R_i^\circ$ are therefore independent uniform
+symmetric forms.
+
+For the opposite pair put
+\begin{equation}\label{eq:planted-pair-annihilators}
+ Q_i=A_j\cap A_k=(R_j+R_k)^\circ,
+ \qquad \dim Q_i=d-2r'+p_i=\delta+p_i.
+\end{equation}
+Use cyclic differences
+\[
+ D_1=(F_2-F_3)|_{Q_1},\qquad
+ D_2=(F_3-F_1)|_{Q_2},\qquad
+ D_3=(F_1-F_2)|_{Q_3}.
+\]
+Projection of $L_j'\cap L_k'$ to $(Y')^*$ has kernel
+$R_j\cap R_k$ and image $\operatorname{rad}D_i$.  In fact, a vector
+$\alpha\in Q_i$ has simultaneous lifts exactly when the two
+prescribed classes in $Y'/R_j$ and $Y'/R_k$ agree in
+$Y'/(R_j+R_k)$.  Pairing this agreement against every
+$\beta\in Q_i$ is exactly the condition
+$(F_j-F_k)(\alpha,\beta)=0$.  Thus there is an exact sequence
+\begin{equation}\label{eq:planted-pair-exact-sequence}
+ \begin{gathered}
+ 0\longrightarrow R_j\cap R_k
+ \longrightarrow L_j'\cap L_k'
+ \longrightarrow\operatorname{rad}D_i\longrightarrow0,\\
+ \dim(L_j'\cap L_k')=p_i+\operatorname{nullity}D_i.
+ \end{gathered}
+\end{equation}
+Thus the required reduced pair profile is exactly the event
+$\operatorname{nullity}D_i=t_i$ for all $i$.
+
+The remaining task is therefore to bound the simultaneous nullities of
+the three difference forms. We first determine their exact joint law,
+rather than assuming independence.
+
+\begin{lemma}\label{lem:planted-difference-image}
+Let $A_1,A_2,A_3$ be arbitrary subspaces of a vector space, put
+$Q_i=A_j\cap A_k$, and put $V=A_1\cap A_2\cap A_3$.
+The image of the cyclic difference map on symmetric forms is exactly
+\begin{equation}\label{eq:planted-difference-image}
+ \left\{(D_1,D_2,D_3):
+       D_1|_V+D_2|_V+D_3|_V=0\right\}.
+\end{equation}
+Its codimension in the product of the three form spaces on the
+$Q_i$ is $b(\dim V)$.  Independent uniform $F_i$ give the uniform
+law on this image.
+\end{lemma}
+\begin{proof}
+The displayed constraint is necessary by cancellation.  Conversely,
+take any three target forms satisfying it, and prescribe the values
+of the three original forms on $V$ to be
+\[
+ H_1=0,\qquad H_2=-D_3|_V,\qquad H_3=D_2|_V.
+\]
+They have the required three cyclic differences on $V$.  On each
+$Q_i$ choose any symmetric extension of the prescribed $H_j$ for
+the positive term of its cyclic difference, and prescribe the
+negative term there by subtracting $D_i$.
+
+Each $F_i$ has now received prescriptions on exactly two subspaces
+of $A_i$.  Their intersection is $V$, and the prescriptions agree
+there.  Two symmetric forms on subspaces $B,C$ agreeing on
+$B\cap C$ always extend simultaneously: write
+$B+C=(B\cap C)\oplus B_0\oplus C_0$, retain the prescribed blocks,
+choose the mixed $B_0$--$C_0$ block arbitrarily, and then extend to
+the ambient space.  Apply this construction separately for each
+$F_i$.  It proves sufficiency even if the three $A_i$ or $Q_i$
+have further linear relations.
+
+Finally, the map taking a target triple to the sum of its restrictions
+on $V$ is onto the form space on $V$: extend any prescribed form
+from $V$ to one $Q_i$ and take the other two forms to be zero.
+Its kernel therefore has codimension $b(\dim V)$.  A linear map
+between finite vector spaces sends uniform measure to uniform measure
+on its image, proving the last assertion.
+\end{proof}
+
+In our application $V=(R_1+R_2+R_3)^\circ$ has dimension $v$.
+Consequently the difference law is exactly the law of three independent
+uniform target forms conditioned on the constraint in
+Equation~\eqref{eq:planted-difference-image}, an event of probability
+$q^{-b(v)}$.  A uniform symmetric form on an $m$-space has nullity
+at least $t$ with probability $O(q^{-b(t)})$.  For completeness,
+there are $O(q^{t(m-t)})$ prospective radical $t$-spaces, and
+vanishing on one imposes $mt-\binom t2$ independent coefficients;
+subtracting gives $-b(t)$.  This also covers $t=m=0$.
+Under the independent target law we may multiply the three bounds;
+conditioning costs at most $q^{b(v)}$.  We obtain, uniformly for
+each actual tuple $(R_1,R_2,R_3)$,
+\begin{equation}\label{eq:planted-nullity-cost}
+ \P\{\operatorname{nullity}D_i=t_i\ (1\le i\le3)\mid R_1,R_2,R_3\}
+   \le O\left(q^{-\sum_i b(t_i)+b(v)}\right).
+\end{equation}
+An upper bound larger than one is harmless.  The common-zero
+condition on the full lifts, or their membership in a particular
+ordered orbit, can only decrease this probability.  In particular,
+we do not replace those extra conditions by an independence claim.
+
+\subsection{The resulting gain}
+
+For each fixed parameter tuple, the preceding counts bound its
+contribution to the numerator in Equation~\eqref{eq:planted-orbit-ratio}
+by the product of the following factors, up to a constant depending
+on the fixed dimension:
+\begin{center}
+\begin{tabular}{@{}ll@{}}
+\toprule
+Choice & Upper-bound factor\\
+\midrule
+Common space and its three containments
+  & \(q^{-(D+1)c}\)\\
+Center intersection and normalized incidence conditions
+  & \(q^{g_C}\)\\
+Subspaces \(R_i\) inside the reduced center
+  & \(q^{-\delta P-\sum_i p_i^2-vw}\)\\
+Nullities of the restricted difference forms
+  & \(q^{-\sum_i b(t_i)+b(v)}\)\\
+\bottomrule
+\end{tabular}
+\end{center}
+The last factor includes the possible loss \(q^{b(v)}\) from
+conditioning on the shared restriction constraint. We sum over the
+finitely many tuples and divide by an individual orbit mass bounded
+below by a constant times \(q^{-(D+1)c-\sum_i b(s_i)}\). This cancels
+the common-space cost and gives the following pointwise gain.
+
+\begin{proposition}[Three active incidences]\label{prop:planted-gain}
+For a local profile $(h_1,h_2,h_3,c)$, we choose one pointwise gain
+exponent independent of \(q\) as follows. Let \(\mathcal U\) be the
+union, over all odd prime fields and all ordered orbits of this profile,
+of the parameter tuples arising from actual quadruples
+$(L_1,L_2,L_3,Y)$ with the stated point profile and
+$\dim(L_i\cap Y)=r$, using
+Equations~\eqref{eq:planted-reduced-parameters} and
+\eqref{eq:planted-subspace-parameters}. Their integer entries are bounded
+at fixed \(D\), so \(\mathcal U\) is finite. If \(\mathcal U\) is empty,
+the face factor is zero over every odd prime field. Otherwise the face
+factor with three active incidences is
+$O(q^g)$ uniformly over all triples in the profile, where
+\begin{equation}\label{eq:planted-gain}
+ g=\max_{\text{tuple}\in\mathcal U}\left\{
+ g_C+\sum_i b(s_i)-\delta P-\sum_i p_i^2-vw
+                   -\sum_i b(t_i)+b(v)\right\},
+\end{equation}
+so the selected exponent is independent of \(q\).
+
+Every tuple in \(\mathcal U\) satisfies the following necessary
+constraints:
+\begin{equation}\label{eq:planted-feasibility}
+ \begin{gathered}
+ 0\le u\le\min(c,r),\quad 0\le r'=r-u\le d,
+ \quad |\delta|\le c,\quad d-\delta=2r',\\
+ P+T=\sum_i s_i\le d,\qquad
+ 0\le t_i\le p_i+\delta,\qquad
+ 0\le v\le p_i+\delta\quad(1\le i\le3),\qquad w\ge0.
+ \end{gathered}
+\end{equation}
+In particular $T\le P+3\delta$ and $d+3\delta\ge0$.
+These conditions are not claimed sufficient for realizability.
+\end{proposition}
+\begin{proof}
+Fix an actual ordered orbit $\mathcal O$.  Start with four independent
+ambient Lagrangians and the normalizer $\pi_{D,r}^{-3}$ in
+Equation~\eqref{eq:planted-density}.  Partition its contribution
+according to the unique common space $C_0$, and then according to
+$u,p_i,t_i,v$.  Equations~\eqref{eq:planted-common-count}--\eqref{eq:planted-common-gain},
+Lemma~\ref{lem:planted-subspace-cost}, and
+Equation~\eqref{eq:planted-nullity-cost} give the numerator bound
+\[
+ \nu(\mathcal O)\le
+ O\left(\sum_{\text{actual parameter tuples}}
+ q^{-(D+1)c+g_C-\delta P-\sum_i p_i^2-vw
+                      -\sum_i b(t_i)+b(v)}\right).
+\]
+At intermediate counting steps we have discarded exactness or orbit
+conditions only to enlarge the count.  The parameter tuples indexing
+the original partition remain actual ones.  Dividing by the lower
+bound $\mu(\mathcal O)\gg q^{-(D+1)c-\sum_i b(s_i)}$ from
+Proposition~\ref{prop:triple-orbits} proves
+Equation~\eqref{eq:planted-gain}.  There are only boundedly many
+dimension tuples for fixed \(D\). The tuples from the current field
+belong to \(\mathcal U\), so its maximum gives a
+bound uniform in \(q\).
+
+The bounds on $u,r',\delta$ follow from their definitions and
+Equation~\eqref{eq:planted-quotient-intersection}.  The reduced pair
+intersections are in isotropic direct sum, giving
+$\sum_i s_i\le d$ as in Proposition~\ref{prop:triple-orbits}.
+Equation~\eqref{eq:planted-pair-exact-sequence} gives the bounds on
+$t_i$, and $V\subseteq Q_i$ gives those on $v$.  We already proved
+$w\ge0$.  Summing $t_i\le p_i+\delta$ gives
+$T\le P+3\delta$, and summing $v\le p_i+\delta$ gives
+$v\le P/3+\delta$.  Thus $P+3\delta\ge0$ and
+\[
+ d+3\delta\ge 6\delta+2P-2v
+       \ge\frac43(P+3\delta)\ge0,
+\]
+where the first inequality uses $w\ge0$.  There are further
+necessary conditions, such as $p_i\le r'$ and $P-p_i\le r'$.
+The later estimates need only Equation~\eqref{eq:planted-feasibility};
+maximizing over actual tuples preserves all the others automatically.
+\end{proof}
+
+The gain bound separates three sources of intersection: $u$ records
+how the center meets the common space, $p_i$ the pair intersections
+inside the reduced center, and $t_i$ the additional intersections
+supplied by lift forms. Their costs give one pointwise exponent for
+the center factor. The global comparison also needs the cases with
+fewer active incidences.
+
+\begin{lemma}[At most two active incidences]\label{lem:two-active-gain}
+For two active incidences whose point Lagrangians meet in dimension
+$h$, a valid gain exponent is
+\begin{equation}\label{eq:planted-two-active-gain}
+ g_2(h)
+ =\max_{\substack{p+t=h\,;\ 0\le t\le p\le r}}
+      \{b(h)-p^2-b(t)\}
+ =b(\lfloor h/2\rfloor).
+\end{equation}
+For zero or one active incidence, the face factor is exactly one.
+\end{lemma}
+\begin{proof}
+Use the two-point version of the planted density identity.  Under
+this law the two $R_i=L_i\cap Y$ are independent uniform
+$r$-subspaces of the $2r$-space $Y$.  Their intersection dimension
+$p$ has probability $O(q^{-p^2})$: for fixed $R_1$, choose a
+$p$-subspace of it, in $O(q^{p(r-p)})$ ways, and require its
+containment in $R_2$, at cost $q^{-pr}$.  Conditional on the two
+individual $R_i$, the two lift forms are independent uniform.  Their
+difference on $(R_1+R_2)^\circ$, of dimension $p$, is a uniform
+symmetric form, since restriction from either original form space
+is surjective.  Equation~\eqref{eq:planted-pair-exact-sequence}
+now gives $h=p+t$ with $0\le t\le p$, and the nullity event costs
+at most $O(q^{-b(t)})$.  The ambient pair orbit has mass
+$\asymp q^{-b(h)}$ by Lemma~\ref{lem:lagrangian-counts}.  Dividing
+gives the maximand in Equation~\eqref{eq:planted-two-active-gain}.
+
+For fixed $h$, let $f(p)=b(h)-p^2-b(h-p)$.  On its integer domain
+$\lceil h/2\rceil\le p\le\min(h,r)$,
+\[
+ f(p+1)-f(p)=h-3p-1<0.
+\]
+Its maximum is therefore at $p=\lceil h/2\rceil$, which is
+allowed since $h\le2r$.  If $h=2a$ the value is
+$b(2a)-a^2-b(a)=b(a)$; if $h=2a+1$ it is
+$b(2a+1)-(a+1)^2-b(a)=b(a)$.  The maximizing parameters are
+realizable: take two $r$-subspaces with intersection $p$, then a
+symmetric difference form of nullity $t=h-p$.  Pair transitivity
+makes this applicable to every pair of intersection dimension $h$.
+Finally, an empty product is one, and for one incidence the mean-one
+normalization of $J(L,Y)$ gives exactly one for every $L$.
+\end{proof}
diff --git a/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/sections/tail.tex b/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/sections/tail.tex
new file mode 100644
index 0000000000000000000000000000000000000000..dc11e07a592e70d3a13e3a005c72378c88edc351
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+++ b/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/sections/tail.tex
@@ -0,0 +1,724 @@
+\section{The singular contribution}\label{sec:tail}
+
+We now prove that configurations with a positive point-pair intersection
+make a vanishing contribution.  Throughout this section the ambient
+symplectic space has dimension $2D$, where $D=2r$, and the point
+Lagrangians $(L_i)_{i\in I}$ are independent and uniform.  We use the
+unsigned normalized kernel $J$ from Section~\ref{sec:orbits}.  For an
+activity set $B\subseteq\mathscr C$, put
+\[
+ F_j((L_i)_{i\in j})=
+ \E_Y\prod_{i\in j:\,(i,j)\in B}J(L_i,Y).
+\]
+Center integration gives the product $\prod_jF_j$.  The bounds of
+Section~\ref{sec:planted} are pointwise bounds on these factors, including
+on exceptional configurations of the point variables.
+
+For $e=\{i,k\}\in E$, set $h_e=\dim(L_i\cap L_k)$, and for $j\in J$ set
+$c_j=\dim\bigcap_{i\in j}L_i$.  These dimensions form a \emph{profile}.
+On a face we use opposite-pair indices $\ell=1,2,3$, so that
+$h_\ell=c+s_\ell$, $d=D-c$, and
+\[
+ C_j=(D+1)c_j+\sum_{\ell=1}^3b(s_\ell),
+ \qquad b(n)=\frac{n(n+1)}2.
+\]
+All constants in bounds of the form $O_D(q^x)$ may depend on $D$ and on
+the fixed complex.  Bounds on an exponent that are stated to be uniform
+in $D$ have a different role: they will allow us to choose $D$ before
+taking any limit in $q$.
+
+We first compare the probability cost of a point profile with the sum
+of its local gains.  For sufficiently large $D$,
+Lemma~\ref{lem:tail-residue} reduces the remaining cases to zero common
+dimensions and pair dimensions $0$ or $2$.
+Lemma~\ref{lem:tail-residual-counts} then gives stronger counts for these
+cases, separating pair spaces that form a direct sum from those that do not.
+
+\subsection{Exposing a profile}
+
+Recall that $a_e$ is the fraction of the three bit positions at which the
+two faces on $e$ differ.  Proposition~\ref{prop:complex} gives
+$a_e\ge 1/3$ for every $e$, and $a_e>1/3$ for at least one $e$.
+
+\begin{proposition}[Global profile cost]\label{prop:global-cost}
+For every prescribed profile, its probability is $O_D(q^{-\mathcal C})$,
+where
+\begin{equation}\label{eq:global-cost}
+ \mathcal C=\frac12\left(
+ \sum_{j\in J}C_j-\sum_{e\in E}(1-a_e)b(h_e)\right).
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+Fix one of the six classes obtained by prescribing one bit position and
+its value.  It consists of eleven faces, is connected in the face
+adjacency graph, and uses all thirteen points.  Choose a spanning tree
+and a parent-first order.  The first face uses three points.  Every
+subsequent face shares a pair with its parent and can add at most one
+point.  Since all thirteen points occur, the ten subsequent faces must
+each add exactly one fresh point.
+
+Expose all the point Lagrangians occurring in the first face.  Its
+prescribed profile has probability $O_D(q^{-C_j})$ by
+Proposition~\ref{prop:triple-orbits}.  At a subsequent face $j$, let $e$
+be its pair shared with its parent, and condition on the sigma-field
+generated by \emph{all} previously exposed point Lagrangians.  Earlier
+face-profile events are measurable in this sigma-field.  The fresh
+point remains independent and uniform.  If the already exposed pair
+has the prescribed intersection dimension $h_e$, pair transitivity and
+Proposition~\ref{prop:triple-orbits} give the conditional probability
+\[
+ \frac{\P(\text{the prescribed profile on }j)}
+      {\P(\dim(L_i\cap L_k)=h_e)}
+ =O_D(q^{-C_j+b(h_e)}).
+\]
+Here pair transitivity makes the ratio the same for every \emph{specific}
+pair of that dimension.  Thus the bound holds after conditioning on
+the full earlier configuration, however its older points are related.
+If the pair has the wrong dimension, the conditional probability is
+zero.  Iterated conditional expectation proves a class probability
+bound with cost $\sum_{j\text{ in class}}C_j$ minus $b(h_e)$ for its tree
+pairs.  Subtracting $b(h_e)$ for every internal pair of the class only
+weakens this bound.
+
+Let $A$ be the complete profile event and let $A_1,\ldots,A_6$ be the six
+class profile events.  Because $A\subseteq A_t$ for each $t$,
+\[
+ \P(A)^6\le\prod_{t=1}^6\P(A_t).
+\]
+This inequality does not require independence among the classes, all
+of which use all thirteen points.  Each face appears in three class
+bounds.  A pair $e$ is internal in exactly $3(1-a_e)$ classes, one for
+each position where its two faces have the same bit.  Taking sixth
+roots gives Equation~\eqref{eq:global-cost}.
+\end{proof}
+
+If the union in Proposition~\ref{prop:planted-gain} is empty for any
+three-active face, its factor is zero and the entire profile contribution
+vanishes; omit that profile. For each remaining profile and each face,
+choose an exponent $g_j$ with $F_j=O_D(q^{g_j})$ as in
+Proposition~\ref{prop:planted-gain} and Lemma~\ref{lem:two-active-gain}.
+For three active incidences choose a maximizing parameter tuple in
+Equation~\eqref{eq:planted-gain}. The choices range over finite sets
+at fixed $D$. The pointwise gain bounds
+and Proposition~\ref{prop:global-cost} imply
+\begin{equation}\label{eq:tail-profile-bound}
+ \E\left[\1_{\text{profile}}\prod_jF_j\right]
+ =O_D\left(q^{\sum_jg_j-\mathcal C}\right).
+\end{equation}
+In particular, center weighting introduces no new conditional
+independence assertion in the preceding exposure argument.
+
+\subsection{Local slack and its quadratic part}
+
+The profile bound gives decay when its probability cost exceeds the
+sum of its local gains. We separate this difference into face
+contributions and a nonnegative term from bit separation. A quadratic
+bound will control large intersection dimensions; separate span
+counts will handle the equality profiles.
+
+Define the baseline and slack on a face by
+\[
+ \beta_j=\frac{C_j}{2}-\frac16\sum_{e\subset j}b(h_e),
+ \qquad S_j=\beta_j-g_j,
+\]
+and define the nonnegative excess separation term
+\begin{equation}\label{eq:tail-excess}
+ \mathcal E=\frac12\sum_{e\in E}(a_e-1/3)b(h_e).
+\end{equation}
+Since each pair occurs twice,
+\begin{equation}\label{eq:tail-total-slack}
+\mathcal C-\sum_jg_j=\sum_j S_j+\mathcal E.
+\end{equation}
+
+Local slack can be negative. On a fully active face with
+\(L_1=L_2=L_3\), all common and pair dimensions equal \(D\), and
+\(F_j=\pi_{D,r}^{-2}\). Its gain exponent is \(2b(r)\), whereas
+\(\beta_j=b(D)/2\), so \(S_j=-D/4\). The global sum and separation
+below control this common-intersection mode.
+
+\begin{lemma}[Faces with at most two active incidences]\label{lem:tail-lower-active}
+Let $h=\max_{e\subset j}h_e$.  If $j$ has at most two active incidences,
+then
+\begin{equation}\label{eq:tail-lower-active}
+ S_j\ge0,
+ \qquad S_j\ge\frac{h^2}{24}-\frac h{12}.
+\end{equation}
+Equality $S_j=0$ requires $c_j=0$ and either all three pair dimensions
+zero or exactly one pair dimension equal to $2$.  The latter case
+requires exactly two active incidences, at that pair's endpoints.
+\end{lemma}
+
+\begin{proof}
+Put $m=\max_\ell s_\ell$ and choose $\ell_0$ with $s_{\ell_0}=m$.
+Using $h=c+m$ and $\sum s_\ell\le D-c$, expand
+\begin{align*}
+ 3C-\sum_\ell b(c+s_\ell)-2b(c+m)
+ &=3(D+1)c-5b(c)
+   -c\left(\sum_\ell s_\ell+2m\right)
+   +2\sum_{\ell\ne\ell_0}b(s_\ell)\\
+ &\ge b(c)+2\sum_{\ell\ne\ell_0}b(s_\ell).
+\end{align*}
+The last step uses $\sum s_\ell+2m\le3(D-c)$; substituting this upper
+bound leaves exactly $b(c)$ in the terms involving $c$.  Consequently
+\begin{equation}\label{eq:tail-baseline-lower}
+ \beta_j\ge\frac{b(h)}3+\frac{b(c)}6
+             +\frac13\sum_{\ell\ne\ell_0}b(s_\ell).
+\end{equation}
+The gain is zero with zero or one active incidence.  With two, it is
+$b(\lfloor h'/2\rfloor)$ for their pair dimension $h'\le h$.  For an
+integer $h\ge0$,
+\[
+ b(\lfloor h/2\rfloor)\le\frac{b(h)}3,
+ \qquad b(\lfloor h/2\rfloor)\le\frac{h(h+2)}8.
+\]
+The first follows separately for $h=2k$ and $h=2k+1$; its differences
+are $k(k-1)/6$ and $(k+1)(k+2)/6$, respectively, so equality holds
+exactly for $h=0,2$.  Combining these inequalities with
+Equation~\eqref{eq:tail-baseline-lower} proves nonnegativity and
+$S_j\ge b(h)/3-h(h+2)/8=h^2/24-h/12$.
+Equality forces $c=0$, $s_\ell=0$ for $\ell\ne\ell_0$, and $h=0$ or
+$2$.  When $h=2$, the gain must equal $1$, which requires the stated
+two active endpoints.  With at most one active incidence the positive
+baseline for $h>0$ excludes equality.
+\end{proof}
+
+For a face with three active incidences, retain the notation of
+Section~\ref{sec:planted}:
+\[
+ \begin{gathered}
+ u,\quad r'=r-u,\quad\delta=2u-c=d-2r',\quad
+ s_\ell=p_\ell+t_\ell,\\
+ P=\sum_\ell p_\ell,\quad T=\sum_\ell t_\ell,
+ \quad w=\frac{d-3\delta}{2}-P+v.
+ \end{gathered}
+\]
+The necessary constraints, which will suffice below, are
+\begin{equation}\label{eq:tail-feasible}
+ \begin{gathered}
+ c\ge|\delta|,\qquad P+T\le d,\qquad
+ p_\ell,t_\ell,v\ge0,\qquad
+ t_\ell,v\le p_\ell+\delta,\qquad w\ge0.
+ \end{gathered}
+\end{equation}
+In particular $T\le P+3\delta$, and hence
+$d+3\delta\ge P+T+3\delta\ge2T\ge0$.
+All the dimensions are integers, and $\delta=2u-c$ with $0\le u\le c$.
+
+Substituting the gain in Equation~\eqref{eq:planted-gain} into
+$S=\beta-g$ gives the exact identity
+\begin{equation}\label{eq:slack}
+ \begin{split}
+ S={}&\frac{c(d+\delta-1)-\delta(d+2)+2\delta^2}{4}\\
+ &+\sum_{\ell=1}^3\left[
+ \frac{(2p_\ell-t_\ell)(2p_\ell-t_\ell-1)}6
+ +\delta p_\ell-\frac c6(p_\ell+t_\ell)\right]
+ +vw-b(v).
+ \end{split}
+\end{equation}
+For clarity, the algebra separates into two identities.  The
+common-space part uses $D=d+c$ and $u=(c+\delta)/2$:
+\[
+ \frac{(D+1)c}{2}-\frac{b(c)}2
+ -\big((r+c+1)u-2u^2\big)
+ =\frac{c(d+\delta-1)-\delta(d+2)+2\delta^2}{4}.
+\]
+For each reduced pair, use $b(c+s)=b(c)+b(s)+cs$ and
+\[
+ p^2+b(t)-\frac23b(p+t)
+ =\frac{(2p-t)(2p-t-1)}6.
+\]
+These expansions yield every term of Equation~\eqref{eq:slack}.
+
+Let \(Q\) be the homogeneous quadratic part of its right-hand side:
+\begin{equation}\label{eq:tail-quadratic}
+\begin{split}
+ Q={}&\frac{c(d+\delta)-\delta d+2\delta^2}{4}\\
+ &+\sum_{\ell=1}^3\left[
+ \frac{(2p_\ell-t_\ell)^2}{6}
+ +\delta p_\ell-\frac c6(p_\ell+t_\ell)\right]
+ +v\left(\frac{d-3\delta}{2}-P\right)+\frac{v^2}{2}.
+\end{split}
+\end{equation}
+The remaining terms are linear:
+\begin{equation}\label{eq:tail-linear-loss}
+ S=Q-\frac c4-\frac\delta2-\frac{2P-T}{6}-\frac v2.
+\end{equation}
+
+The next estimate will turn the total slack into bounds on
+\(P,T,\delta\). When a pair dimension grows, these bounds force its
+common part to spread through neighboring faces until the excess
+separation is too large. Once all pair dimensions are bounded, we
+return to the exact slack and use its coefficient of \(d\).
+
+\begin{lemma}[Coercivity]\label{lem:coercivity}
+Every tuple satisfying Equation~\eqref{eq:tail-feasible} satisfies
+\begin{equation}\label{eq:coercivity}
+ Q\ge\frac1{342}(P^2+T^2+\delta^2).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Set
+\[
+ a=\left(P-\frac{d-3\delta}{2}\right)_+,
+ \qquad v_0=\min\left(\frac P3+\delta,
+                         \frac{P-T+3\delta}{2}\right).
+\]
+Both arguments of the minimum are nonnegative, by
+Equation~\eqref{eq:tail-feasible}.  Completing the square gives
+\[
+ vw-\frac{v^2}{2}
+ =\left(\frac{d-3\delta}{2}-P\right)v+\frac{v^2}{2}
+ \ge-\frac{a^2}{2}.
+\]
+The condition $w\ge0$ gives $a\le v$; furthermore
+$v\le\min_\ell(p_\ell+\delta)\le P/3+\delta$.  From $d\ge P+T$ we
+also obtain $a\le(P-T+3\delta)/2$.  Therefore $a\le v_0$, and
+\begin{equation}\label{eq:tail-scalar-quadratic}
+ \begin{split}
+ Q\ge{}&\frac{c(d+\delta)-\delta d+2\delta^2}{4}
+ +\frac{(2P-T)^2}{18}+\delta P
+ -\frac{c(P+T)}6-\frac{v_0^2}{2}.
+ \end{split}
+\end{equation}
+Here we also used $\sum_\ell(2p_\ell-t_\ell)^2\ge(2P-T)^2/3$.
+The coefficient of $c$ on the right is
+\[
+ \frac{d+\delta}{4}-\frac{P+T}{6}
+ \ge\frac{d+3\delta}{12}\ge0.
+\]
+Because \(c\ge|\delta|\), replacing \(c\) by \(|\delta|\) can only
+decrease the right side. We do so and treat the two signs of \(\delta\).
+
+First suppose $\delta\ge0$.  The remainder in
+Equation~\eqref{eq:tail-scalar-quadratic} becomes
+\[
+ F(T)=\frac{3\delta^2}{4}+\frac{(2P-T)^2}{18}
+            +\frac{(5P-T)\delta}{6}-\frac12
+            \min\left(\frac P3+\delta,
+                        \frac{P-T+3\delta}{2}\right)^2,
+ \qquad 0\le T\le P+3\delta.
+\]
+On $0\le T\le P/3+\delta$, the first argument attains the minimum,
+and $F'(T)=(T-2P)/9-\delta/6\le0$; at the right endpoint this
+derivative is $-5P/27-\delta/18$.  On the remaining interval the
+coefficient of $T^2$ is $1/18-1/8=-5/72$, so $F$ is concave.  Its
+minimum is therefore at one of the following two endpoints:
+\begin{align*}
+ F(P/3+\delta)&=\frac{32P^2+84P\delta+45\delta^2}{324},\\
+ F(P+3\delta)&=\frac{2P^2+12P\delta+27\delta^2}{36}.
+\end{align*}
+Both are at least $(P^2+\delta^2)/18$.  Moreover
+\[
+ P^2+T^2+\delta^2
+ \le P^2+(P+3\delta)^2+\delta^2
+ \le3P^2+19\delta^2\le19(P^2+\delta^2).
+\]
+This proves the desired coefficient $1/(18\cdot19)$ in this case,
+including $P=\delta=0$.
+
+Next suppose $\delta=-y<0$.  Put $x=P-3y\ge0$; then $0\le T\le x$.
+After setting $c=y$, the coefficient of $d$ is $y/2$.
+Using $d\ge P+T=x+3y+T$ in
+Equation~\eqref{eq:tail-scalar-quadratic} leaves
+\[
+ G(T)=\frac{y^2}{4}+\frac{(2x-T)y}{3}
+ +\frac{(2x-T)^2}{18}
+ -\frac12\min\left(\frac x3,\frac{x-T}{2}\right)^2.
+\]
+Its derivative up to $T=x/3$ is $(T-2x)/9-y/3\le0$; after that
+breakpoint its quadratic coefficient is again $-5/72$.  Thus its
+minimum is at least the smaller of its breakpoint and last-endpoint values,
+\begin{align*}
+ G(x/3)&=\frac{32x^2+180xy+81y^2}{324},\\
+ G(x)&=\frac{2x^2+12xy+9y^2}{36}.
+\end{align*}
+Each is at least $(x^2+y^2)/18$, including the degenerate interval
+$x=0$.  Finally,
+\[
+ P^2+T^2+y^2\le(x+3y)^2+x^2+y^2
+                 \le3x^2+19y^2\le19(x^2+y^2).
+\]
+The same constant $1/342$ follows, completing the proof.
+\end{proof}
+
+\subsection{Uniform control and equality profiles}
+
+Put $M=\max_{e\in E}h_e$.  On every face with three active incidences,
+$c\le M$, $P\le3(M-c)$, $\delta\le c$, and
+$v\le p_\ell+\delta\le s_\ell+c=h_\ell\le M$.
+Equation~\eqref{eq:tail-linear-loss} consequently gives the uniform bound
+\begin{equation}\label{eq:tail-uniform-linear}
+ S\ge Q-\frac{3M}{2}.
+\end{equation}
+Indeed the subtracted expression is at most
+$c/4+c/2+(M-c)+M/2=3M/2-c/4$.
+Suppose that the cost in Equation~\eqref{eq:global-cost} does not yet
+give a strict saving, so $\sum_jS_j+\mathcal E\le0$.
+Let $J_3$ denote the faces with three active incidences and let
+$h_j=\max_{e\subset j}h_e$.  Summing
+Lemma~\ref{lem:coercivity}, Equation~\eqref{eq:tail-uniform-linear},
+and Lemma~\ref{lem:tail-lower-active} gives
+\begin{equation}\label{eq:tail-budget}
+ \begin{split}
+ &\sum_{j\in J_3}
+ \frac{P_j^2+T_j^2+\delta_j^2}{342}
+ +\sum_{j\notin J_3}\frac{h_j^2}{24}+\mathcal E
+ \le33M.
+ \end{split}
+\end{equation}
+To see the constant explicitly, each three-active face can lose at
+most $3M/2$, and each lower-activity face loses at
+most $h_j/12\le M/12\le3M/2$; there are twenty-two faces in total.
+Every term on the left of
+Equation~\eqref{eq:tail-budget} is nonnegative.  Thus it bounds each
+term separately, despite the possible negative individual slacks.
+
+\begin{lemma}[Profiles not separated by the first cost]\label{lem:tail-residue}
+There is an even-dimensional threshold such that, for every even $D$
+above it, every feasible profile and every activity set for which
+$\sum_jg_j\ge\mathcal C$ has the following properties:
+\begin{equation}\label{eq:tail-residual-profile}
+ c_j=0\quad(j\in J),\qquad h_e\in\{0,2\}\quad(e\in E),
+ \qquad\sum_jg_j\le\frac23\sum_{e\in E}b(h_e).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Assume \(\sum_j S_j+\mathcal E\le0\). We first bound all pair
+dimensions independently of \(D\), the activity set, and the choices
+of maximizing tuples.
+
+\paragraph{A uniform bound for pair dimensions.}
+Equation~\eqref{eq:tail-budget} gives a constant \(K\ge1\), depending
+only on the fixed numerical coefficients, such that
+\[
+ P_j+T_j\le K\sqrt M\quad(j\in J_3),\qquad
+ h_j\le K\sqrt M\quad(j\notin J_3),\qquad
+ \mathcal E\le33M.
+\]
+The case \(M=0\) already has the required bound. Suppose \(M>0\),
+and choose a face \(j_0\) containing a pair of dimension \(M\).
+For sufficiently large \(M\), this face cannot have lower activity,
+because \(M>K\sqrt M\). On a three-active face every reduced pair
+dimension satisfies \(s_\ell\le P+T\le K\sqrt M\). Thus a pair of
+dimension \(h\) on that face forces its common dimension, and hence
+every pair dimension on the face, to be at least \(h-K\sqrt M\).
+
+Let \(\ell_*\) be the diameter of the fixed connected face-neighbor
+graph. We claim by induction on \(t\) that every face at distance
+\(t\) from \(j_0\) is three-active and every pair on it has dimension
+at least \(M-(t+1)K\sqrt M\). The preceding paragraph gives the case
+\(t=0\). When \(M\) is large enough that
+\[
+ M-(\ell_*+1)K\sqrt M\ge M/2>K\sqrt M,
+\]
+choose for a face at distance \(t+1\) its predecessor on a shortest
+path from \(j_0\). Their shared pair has dimension greater than
+\(K\sqrt M\), so the new face cannot have lower activity. Applying
+the three-active bound proves the claimed lower bound at distance
+\(t+1\).
+Induction therefore gives \(h_e\ge M/2\) for every pair \(e\).
+
+Fix a pair \(e_*\) with \(a_{e_*}>1/3\). Its contribution to
+\(\mathcal E\) is at least
+\[
+ \frac12(a_{e_*}-1/3)b(h_{e_*})
+ \ge \frac{a_{e_*}-1/3}{16}M^2.
+\]
+For large \(M\) this contradicts \(\mathcal E\le33M\). Hence there is
+an integer \(M_*\), independent of \(D\), the activity set, and the
+maximizing tuples, such that every profile under consideration
+satisfies \(M\le M_*\).
+
+\paragraph{Choose the dimension after this bound.}
+On a three-active face with \(M\le M_*\), the integers
+\(c,p_\ell,t_\ell,v\) lie between \(0\) and \(M_*\), and
+\(|\delta|\le M_*\). Put \(n_\ell=2p_\ell-t_\ell\). Substituting the
+definition of \(w\) into Equation~\eqref{eq:slack} gives
+\begin{equation}\label{eq:tail-leading-coefficient}
+ S=Ad+R,\qquad
+ A=\frac{c-\delta+2v}{4}=\frac{c-u+v}{2},
+\end{equation}
+where
+\[
+\begin{split}
+ R={}&\frac{c(\delta-1)-2\delta+2\delta^2}{4}
+ +\frac16\sum_{\ell=1}^3
+ \left[n_\ell(n_\ell-1)+6\delta p_\ell-c(p_\ell+t_\ell)\right]\\
+ &-\left(\frac{3\delta}{2}+P\right)v+\frac{v(v-1)}2.
+\end{split}
+\]
+Let \(C_*\) be the maximum of \(|R|\) over the finite integer box
+\(0\le c,p_\ell,t_\ell,v\le M_*\), \(|\delta|\le M_*\), with
+\(P=\sum_\ell p_\ell\). This bounds every actual remainder and is
+independent of \(D\).
+
+For an actual tuple, \(0\le u\le c\) and \(v\ge0\), so \(A\ge0\).
+Moreover \(c,u,v\) are integers: either \(A=0\) or \(A\ge1/2\).
+Choose an even \(D\) with
+\[
+ D>M_*+44C_*.
+\]
+If a three-active face had \(A>0\), it would have
+\(S\ge(D-M_*)/2-C_*\). Every other three-active face has
+\(S\ge-C_*\), every lower-activity face has nonnegative slack, and
+\(\mathcal E\ge0\). Since there are \(22\) faces, the total slack
+would be at least
+\[
+ \frac{D-M_*}{2}-22C_*>0,
+\]
+contrary to the hypothesis. Thus every three-active face has
+\(u=c\), \(\delta=c\), and \(v=0\).
+
+\paragraph{Integer equality profiles.}
+For these parameters, Equation~\eqref{eq:slack} becomes
+\begin{equation}\label{eq:tail-integer-slack}
+ S=\frac{3c^2-3c}{4}
+    +\frac16\sum_{\ell=1}^3
+       \left[n_\ell^2+(c-1)n_\ell+3cp_\ell\right].
+\end{equation}
+If \(c\ge2\), completing the square in \(n_\ell\) gives
+\[
+ S\ge\frac{3c(c-1)}4-\frac{(c-1)^2}{8}
+       =\frac{(c-1)(5c+1)}8>0.
+\]
+If \(c=1\), then \(S=\sum_\ell(n_\ell^2+3p_\ell)/6\ge0\),
+and equality forces \(p_\ell=t_\ell=0\) for every \(\ell\).
+If \(c=0\), the constraint \(t_\ell\le p_\ell\) gives the integer
+\(n_\ell\ge p_\ell\ge0\). Each \(n_\ell(n_\ell-1)\) is nonnegative,
+and its zeros give precisely
+\[
+ (p_\ell,t_\ell)=(0,0)\quad\text{or}\quad(1,1).
+\]
+Thus every slack is nonnegative at the chosen dimension. The assumed
+inequality forces every slack, and \(\mathcal E\), to be zero.
+
+A zero-slack face with \(c=1\) has all pair dimensions equal to \(1\).
+Any face across one of these pairs cannot be a lower-activity equality
+case from Lemma~\ref{lem:tail-lower-active}, and cannot be a
+three-active equality case with \(c=0\). It must have \(c=1\) as well.
+Connectivity would give \(h_e=1\) for every \(e\), contradicting
+\(\mathcal E=0\) at a pair with \(a_e>1/3\). The remaining equality
+cases have \(c_j=0\) and \(h_e\in\{0,2\}\). As \(S_j=0\), their gains
+equal their baselines, and
+\[
+ \sum_jg_j=\sum_j\frac13\sum_{e\subset j}b(h_e)
+          =\frac23\sum_e b(h_e).
+\]
+The choices of \(M_*\) and \(C_*\) were uniform in the activity set
+and maximizing tuples, so the dimension threshold works for all of
+them simultaneously.
+\end{proof}
+
+\subsection{Two counts for the remaining pair spaces}
+
+It remains to improve the cost for the profiles in
+Equation~\eqref{eq:tail-residual-profile}. Their nonzero pair
+intersections are two-dimensional, and each pair space must lie in
+the Lagrangians at both endpoints. A dependence among the pair
+spaces forces at least one extra dimension in the sum of the local
+endpoint spans, producing a factor \(q^{-D}\). For a direct family,
+the independent orthogonality constraints instead give the required
+cost after the endpoint containment factors cancel. The total
+ambient span need not be isotropic.
+
+\begin{lemma}[A strict span inequality]\label{lem:tail-span}
+Let $H_e$ be subspaces assigned to distinct pairs $e$ of a simple
+graph.  Set
+\[
+ k=\dim\sum_e H_e,\qquad k_i=\dim\sum_{e\ni i}H_e.
+\]
+Then $\sum_i k_i\ge2k$.  If the nonzero spaces do not form a direct
+sum, then $\sum_i k_i\ge2k+1$.
+\end{lemma}
+
+\begin{proof}
+For any ordering of the spaces, let $\Delta_e$ be the increment in
+the dimension of the global span when $H_e$ is added.  At each
+endpoint $i$ of $e$, let $\Delta_{e,i}$ be its increment in the local
+span.  The earlier local span is contained in the earlier global
+span, so $\Delta_{e,i}\ge\Delta_e$.  Summing gives
+$\sum_i k_i\ge2\sum_e\Delta_e=2k$.
+
+If the family is not direct, choose an inclusion-minimal subfamily
+$\mathcal T$ which is not direct.  Order its members first and let
+$e$ be its last member.  Put
+\[
+ \delta_*=\dim\left(H_e\cap
+          \sum_{f\in\mathcal T\setminus\{e\}}H_f\right)>0.
+\]
+At least one endpoint $i$ of $e$ fails to belong to some other member
+of $\mathcal T$: otherwise every other pair in $\mathcal T$ would
+contain both endpoints of $e$ and would equal $e$.  The members of
+$\mathcal T$ visible at this $i$, including $e$, are therefore a
+proper subfamily and form a direct sum by minimality.  Consequently
+\[
+ \Delta_{e,i}=\dim H_e=\Delta_e+\delta_*.
+\]
+All other increment comparisons remain nonnegative, and hence
+$\sum_i k_i\ge2k+\delta_*\ge2k+1$.  The argument uses the actual
+dimension $\delta_*$ and in particular includes a partial intersection
+of $H_e$ with the earlier span.
+\end{proof}
+
+\begin{lemma}[Costs for the remaining profiles]\label{lem:tail-residual-counts}
+For a profile satisfying Equation~\eqref{eq:tail-residual-profile},
+let $m$ be the number of pairs with $h_e=2$.  The contribution of
+configurations whose pair spaces are not direct is
+$O_D(q^{-D+1641})$.  The contribution of configurations whose pair
+spaces are direct is
+\[
+ O_D\left(q^{-\frac13\sum_e b(h_e)}\right).
+\]
+Here ``contribution'' includes the product of the center factors.
+The numerical remainder $1641$ is independent of $D$.
+\end{lemma}
+
+\begin{proof}
+Write $H_e=L_i\cap L_k$ for $e=\{i,k\}$, and use $k,k_i$ as in
+Lemma~\ref{lem:tail-span}.  We count possible $H_e$ and impose their
+containment in the independent $L_i$.  Discarding exactness of the
+intersections can only enlarge the count.  A nonisotropic local span
+cannot be contained in a Lagrangian and contributes zero.  For every
+isotropic local span of dimension $k_i$, its containment probability
+is $O_D(q^{-Dk_i+\binom{k_i}{2}})$ by
+Lemma~\ref{lem:lagrangian-counts}.
+
+For the nondirect case, first choose an arbitrary $k$-space in the
+$2D$-space, in $O_D(q^{k(2D-k)})$ ways, and then choose the $m$
+two-spaces inside it, in $O_D(q^{2m(k-2)})$ ways.  Requiring their
+span to be the chosen space would only reduce this upper bound.  On
+the stratum with fixed $k$ and $k_i$, multiplying the containment
+probabilities gives exponent
+\[
+ -D\left(\sum_i k_i-2k\right)
+ -k^2+2mk-4m+\sum_i\binom{k_i}{2}.
+\]
+The point degrees in the pair graph, given by
+Proposition~\ref{prop:complex}, are
+$(4,6,4,6,5,4,5,5,5,6,5,6,5)$.  Therefore
+\[
+ \sum_i\binom{k_i}{2}
+ \le\sum_i\binom{2\deg(i)}2=618.
+\]
+Since $m\le33$ and $-k^2+2mk\le m^2$, the part of the exponent
+independent of $D$ is at most
+\[
+ m^2-4m+618\le1575.
+\]
+Lemma~\ref{lem:tail-span} makes the coefficient of $-D$ at least
+one.  There are only boundedly many $k,k_i$, independently of $D$,
+because $k\le2m\le66$ and $k_i\le2\deg(i)$.  Thus the probability
+of this part of the profile is $O_D(q^{-D+1575})$.
+Its total gain is at most $\tfrac23\sum_e b(h_e)=2m\le66$, proving
+the first bound.  The ambient span in this argument was arbitrary;
+only the local spans were required to be isotropic.
+
+For the direct case, the following count applies to arbitrary fixed
+dimensions $h_e$.  Order the pairs and choose an ordered basis of
+each $H_e$.  Each new basis vector must be orthogonal to the previous
+vectors of its own space and to all vectors of earlier spaces sharing
+an endpoint.  In a direct family these previous vectors are linearly
+independent.  Nondegeneracy of the symplectic form identifies ambient
+vectors with linear functionals, so their orthogonality equations are
+independent, even if the previous vectors are mutually orthogonal.
+If there are $a$ such equations, there are exactly $q^{2D-a}$ vectors
+satisfying them.  Excluding choices which would destroy directness
+only decreases this count.
+
+Let $e\sim f$ mean distinct pairs sharing an endpoint, and count each
+unordered pair $\{e,f\}$ once.  Dividing the sequential basis count
+by the number of ordered bases in each space gives at most
+\[
+ O_D\left(q^{\,2D\sum_eh_e-\sum_eh_e^2
+ -\sum_e\binom{h_e}{2}-\sum_{e\sim f}h_eh_f}\right)
+\]
+possible direct families satisfying all local orthogonality conditions.
+Here $k_i=\sum_{e\ni i}h_e$, so
+\[
+ \sum_i k_i=2\sum_eh_e,
+ \qquad
+ \sum_i\binom{k_i}{2}
+ =2\sum_e\binom{h_e}{2}+\sum_{e\sim f}h_eh_f.
+\]
+Distinct pairs in a simple graph share at most one endpoint, which
+explains the coefficient of the last sum.  Multiplying by the
+containment probabilities cancels both the $D$ terms and the cross
+terms.  The remaining exponent is exactly
+\[
+ -\sum_eh_e^2+\sum_e\binom{h_e}{2}=-\sum_e b(h_e).
+\]
+The gain bound in Equation~\eqref{eq:tail-residual-profile} now leaves
+$-\tfrac13\sum_e b(h_e)$, as asserted.
+\end{proof}
+
+\subsection{Completion of the rank-layer estimate}
+
+\begin{proposition}[Vanishing singular contribution]\label{prop:singular-tail}
+For every sufficiently large even $D=2r$ and every activity set
+$B\subseteq\mathscr C$,
+\begin{equation}\label{eq:tail-vanishing}
+ \E_{L,Y}\left[
+ \1_{\{\exists e=\{i,k\}\in E:\,\dim(L_i\cap L_k)>0\}}
+ \prod_{(i,j)\in B}J(L_i,Y_j)\right]=o(1)
+\end{equation}
+as $q\to\infty$ through odd primes, with $D$ fixed.  The dimension
+threshold can be chosen for all activity sets simultaneously.
+\end{proposition}
+
+\begin{proof}
+Choose an even $D$ above the threshold in
+Lemma~\ref{lem:tail-residue} and also with $D>1641$.  These thresholds
+refer only to dimensions and the fixed complex.  For any profile with
+some $h_e>0$, either $\mathcal C>\sum_jg_j$, or it satisfies
+Equation~\eqref{eq:tail-residual-profile}.  In the first case,
+Equation~\eqref{eq:tail-profile-bound} gives a strictly negative
+$q$-exponent.  In the second, the two bounds in
+Lemma~\ref{lem:tail-residual-counts} are strictly negative because
+$D>1641$ and $\sum_e b(h_e)>0$.
+
+At this fixed \(D\), every selected gain is an integer and
+\(6\mathcal C\) is an integer, because \(3a_e\) is an integer.
+A strict first-cost saving is therefore at least \(1/6\). A singular
+direct residual profile saves at least \(1\), and the nondirect case
+also saves at least \(1\) because even \(D>1641\) implies \(D\ge1642\).
+There are finitely many profiles and activity sets, and the gain
+exponents were selected from the fixed finite unions in
+Proposition~\ref{prop:planted-gain}. Summing their vanishing
+contributions proves Equation~\eqref{eq:tail-vanishing}.
+The local maximizing tuples need not be realizable simultaneously:
+the residue argument uses only their necessary dimension inequalities
+and the shared profile dimensions.
+In particular, no simultaneous limit in $D$ and $q$ has been used.
+The bound $D>1641$ is needed here only for the nondirect residual
+case and is not asserted to be a threshold for the whole proof.
+\end{proof}
+
+\begin{proof}[Proof of Lemma~\ref{lem:rank-layer}]
+Choose an even $D$ large enough for both
+Proposition~\ref{prop:transverse-limit} and
+Proposition~\ref{prop:singular-tail}.  For the matrix variables let
+\[
+ \mathcal T=\{\det(X_i-X_k)\ne0\text{ for every }\{i,k\}\in E\}.
+\]
+On $\mathcal T$, Proposition~\ref{prop:transverse-limit} gives the
+right-hand side of Equation~\eqref{eq:rank-layer}, up to $o(1)$,
+for every fixed activity set and every choice of incidence signs.
+The integrand is nonnegative.  Lemma~\ref{lem:chart-comparison}
+bounds its contribution on $\mathcal T^c$ by a constant depending
+on $D$ times the unsigned Lagrangian contribution in
+Equation~\eqref{eq:tail-vanishing}: graph intersections project isomorphically
+onto the kernels of matrix differences, and dropping the prescribed signs
+only enlarges the integrand after the comparable normalizations.
+This contribution is $o(1)$ by
+Proposition~\ref{prop:singular-tail}.  Adding the transverse and
+singular parts proves Equation~\eqref{eq:rank-layer}.
+There are only finitely many activity sets and sign assignments, so
+the convergence is uniform over the choices required in
+Section~\ref{sec:kernel}.  This proves Lemma~\ref{lem:rank-layer}.
+The construction in Section~\ref{sec:kernel} therefore yields the strict
+symmetric-kernel inequality, and Lemma~\ref{lem:finite-host} gives a finite
+simple host $G$ with $t(H,G)<p(G)^{66}$, proving Theorem~\ref{thm:main}.
+\end{proof}
diff --git a/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/sections/transverse.tex b/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/build/sections/transverse.tex
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+\section{Transverse configurations and determinant signs}
+\label{sec:transverse}
+
+We begin the proof of Lemma~\ref{lem:rank-layer} with the configurations
+whose point differences are nonsingular. Throughout this section $D=2r$
+is fixed, $q$ tends to infinity through odd primes, and constants in
+$O(\cdot)$ may depend on $D$. We use $b(n)=n(n+1)/2$, including $b(0)=0$.
+The estimates concerning growing $D$ will be made separately in
+Section~\ref{sec:tail}.
+
+On triples with nonsingular point differences, for two or three active
+incidences on a face, the sign model has value
+\(2\) or \(8\) when every active pair satisfies
+\(\chi(\det(X_i-X_k))=c_0\xi_{ij}\xi_{kj}\), and value \(0\) when a
+match fails. For \(r>1\), we will approximate the normalized center
+factor by this model in \(L^1\) on these triples, and bound that
+factor uniformly there. These estimates
+permit multiplication over faces sharing point matrices. The
+preliminary counts fix the layer normalizations and enumerate the
+candidate subspaces; restriction-sign balance supplies the final
+factor \(1/2\). We then prove that the determinant signs on all
+\(33\) point pairs have an asymptotically independent joint law.
+
+\subsection{Symmetric forms over a finite field}
+
+The discriminant sign of a symmetric form is the quadratic character of
+the determinant of its induced nondegenerate form on the quotient by
+its radical. A change of basis multiplies this determinant by a square,
+so its character is well defined. In dimension zero the determinant is
+understood to be $1$.
+
+Rank and discriminant sign are the standard invariants of symmetric
+bilinear forms over a finite field of odd order. Exact counts by these
+invariants appear, for example, in
+Schmidt~\cite[Section~2, Proposition~2.1]{schmidt2015}.
+We give the elementary estimates needed here, with their proofs.
+
+\begin{lemma}[Subspaces and symmetric matrices]
+\label{lem:symmetric-counts}
+For $0\leq k\leq n$, the number of $k$-dimensional subspaces of
+$\F_q^n$ is
+\begin{equation}
+\label{eq:transverse-grassmann}
+ \genfrac{[}{]}{0pt}{}{n}{k}_q
+ =\prod_{a=0}^{k-1}\frac{q^n-q^a}{q^k-q^a}
+ =q^{k(n-k)}(1+O(q^{-1})).
+\end{equation}
+For a uniform matrix $M\in\mathcal S_n$, with $n\geq 1$,
+\[
+ \P(M\text{ is nonsingular})=1-O(q^{-1}),\qquad
+ \P(\chi(\det M)=\xi)=\frac12+O(q^{-1})
+ \quad(\xi\in\{+1,-1\}).
+\]
+For $1\leq k\leq n$, the probability of rank $k$ and prescribed
+discriminant sign is
+\begin{equation}
+\label{eq:transverse-signed-rank}
+ \left(\frac12+O(q^{-1})\right)q^{-b(n-k)}.
+\end{equation}
+Finally, for $0\leq u\leq n$,
+\begin{equation}
+\label{eq:transverse-nullity}
+ \P(\dim\ker M\geq u)=O(q^{-b(u)}).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Count ordered linearly independent $k$-tuples in $\F_q^n$ and divide
+by the number of ordered bases of $\F_q^k$ to obtain
+Equation~\eqref{eq:transverse-grassmann}. Its asymptotic follows by factoring
+out the displayed powers of $q$; all products have fixed length.
+
+For the nonsingularity and sign assertions, use induction on $n$.
+On the event that the leading $(n-1)$-dimensional principal minor $C$
+is nonsingular, write
+\[
+ M=\begin{pmatrix}C&z\\z^{\mathsf T}&a\end{pmatrix},\qquad
+ \det M=\det C\,(a-z^{\mathsf T}C^{-1}z).
+\]
+Given $C,z$, the final scalar factor is uniform on $\F_q$. It is zero
+with probability $q^{-1}$ and has either nonzero character with
+probability $(q-1)/(2q)$. The exceptional principal minor has probability
+$O(q^{-1})$ by induction. The same argument starts at $n=1$ with the
+empty principal minor.
+
+For the rank assertion, fix a $k$-subspace $U$ with basis-column matrix
+$P$. Every symmetric matrix with image $U$ is uniquely
+$PRP^{\mathsf T}$ for a nonsingular symmetric $k$-by-$k$ matrix $R$.
+Indeed, if $Q P$ is the identity, symmetry and the image condition give
+$M=P(QMQ^{\mathsf T})P^{\mathsf T}$, and multiplication by $Q,Q^{\mathsf T}$
+also proves uniqueness. The surjection $P^{\mathsf T}:\F_q^n\to\F_q^k$
+identifies the nondegenerate quotient form with $R$, so the discriminant
+sign is $\chi(\det R)$. Consequently the desired count is
+\[
+ \genfrac{[}{]}{0pt}{}{n}{k}_q\,q^{b(k)}
+ \left(\frac12+O(q^{-1})\right).
+\]
+Divide by $q^{b(n)}$ and use
+$k(n-k)+b(k)-b(n)=-b(n-k)$.
+
+For Equation~\eqref{eq:transverse-nullity}, sum over $u$-subspaces $U$
+contained in the radical. In a basis beginning with $U$, this containment
+sets to zero the first $u$ rows and columns, imposing
+$nu-\binom u2$ independent linear conditions. A uniform symmetric form
+therefore has $U$ in its radical with probability
+$q^{-nu+\binom u2}$. Multiplication by
+$\genfrac{[}{]}{0pt}{}{n}{u}_q=O(q^{u(n-u)})$ gives the assertion.
+\end{proof}
+
+In particular, writing $\pi_\xi$ for the mass of the signed rank-$r$
+layer in $\mathcal S_D$, we have
+\begin{equation}
+\label{eq:transverse-signed-mass}
+ \pi_\xi=\left(\frac12+O(q^{-1})\right)q^{-b(r)},
+ \qquad \pi_\xi^{-1}=2q^{b(r)}(1+O(q^{-1})).
+\end{equation}
+These are the normalizations in $K_\xi$.
+
+A subspace is \emph{totally isotropic} for a symmetric bilinear form
+if the form vanishes on every pair of vectors in that subspace.
+The next count is the split orthogonal counterpart of the symplectic
+counts in Section~\ref{sec:orbits}; see also Taylor~\cite[Exercise~11.3]{taylor1992}.
+
+\begin{lemma}[Split forms and their maximal isotropic subspaces]
+\label{lem:transverse-split}
+Let $r\geq1$. A nondegenerate symmetric form on a $2r$-dimensional
+space over $\F_q$
+has an $r$-dimensional totally isotropic subspace if and only if its
+determinant character is $c_0=\chi((-1)^r)$. Such a form is a direct
+orthogonal sum of $r$ hyperbolic planes. The total number of its
+$r$-dimensional totally isotropic subspaces is
+\begin{equation}
+\label{eq:transverse-isotropic-count}
+ N_r=\prod_{a=0}^{r-1}(1+q^a)
+ =2q^{r(r-1)/2}(1+O(q^{-1})).
+\end{equation}
+The count includes both families of maximal isotropic subspaces.
+\end{lemma}
+
+\begin{proof}
+First, a nondegenerate symmetric form has an orthogonal basis. A
+nonzero form has a vector of nonzero norm, since otherwise polarization
+would make the form zero. Split off this nondegenerate line and continue
+on its orthogonal complement.
+
+For any $a,b\in\F_q^\times$ and $t\in\F_q$, the two sets
+$\{ax^2:x\in\F_q\}$ and $\{t-by^2:y\in\F_q\}$ each have size
+$(q+1)/2$ and therefore meet. Thus $ax^2+by^2=t$ is solvable. A diagonal
+nondegenerate form in at least three variables consequently has a
+nonzero isotropic vector: solve $ax^2+by^2=-c$ and set the third
+coordinate to $1$. Given an isotropic vector $e\ne0$, choose $f$ with
+$\langle e,f\rangle=1$ and replace $f$ by
+$f-\langle f,f\rangle e/2$. The resulting plane has Gram matrix
+$\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)$ and splits
+off orthogonally. Repetition leaves an anisotropic space of dimension
+at most two. In dimension two, $ax^2+by^2$ is isotropic exactly when
+$\chi(-ab)=1$, or equivalently when its determinant character is
+$\chi(-1)$. Each hyperbolic plane also has determinant character
+$\chi(-1)$. This proves that the determinant condition is precisely
+the condition that the entire $2r$-space split into hyperbolic planes.
+
+A split form has the required isotropic subspace, spanned by one vector
+from each hyperbolic pair. Conversely, if such a subspace $U$ is given,
+split off a hyperbolic plane using $e\in U$. Its orthogonal complement
+has an isotropic $(r-1)$-subspace: for $u\in U$, replace $u$ by
+$u-\langle u,f\rangle e$. This linear map on $U$ has kernel
+$\langle e\rangle$, so its image has dimension $r-1$.
+Induction shows that the entire form is split.
+
+In split coordinates $(x,y)\in\F_q^r\oplus\F_q^r$, isotropy is the
+equation $x^{\mathsf T}y=0$. There are
+\[
+ (q^r-1)q^{r-1}+(q^r-1)
+ =(q^r-1)(q^{r-1}+1)
+\]
+nonzero isotropic vectors. The quotient $\ell^\perp/\ell$ by an
+isotropic line has dimension $2r-2$. Splitting off a hyperbolic plane
+through $\ell$ identifies this quotient with its nondegenerate
+orthogonal complement, whose determinant character is
+$\chi((-1)^{r-1})$. The criterion just proved makes the quotient split.
+Count pairs $(\ell,U)$ with
+$\ell\subset U$ and $\dim U=r$ to get
+\[
+ N_r\frac{q^r-1}{q-1}
+ =\frac{(q^r-1)(q^{r-1}+1)}{q-1}N_{r-1}.
+\]
+With $N_0=1$, this gives the product in
+Equation~\eqref{eq:transverse-isotropic-count}. In particular $N_1=2$;
+no choice of one isotropic family has been imposed in this count.
+\end{proof}
+
+The isotropic-subspace count does not yet determine the sign of a
+second form on those subspaces. The following averaged estimate
+supplies that sign information for any sufficiently large family.
+
+\begin{lemma}[Concentration of restriction signs]
+\label{lem:transverse-restrictions}
+Let $\mathcal U$ be any nonempty family of distinct $r$-subspaces of
+$\F_q^D$, with $r\geq1$, and let $Q$ be a uniform symmetric form.
+For $\xi\in\{+1,-1\}$ put
+\[
+ f_\xi(Q)=\frac1{|\mathcal U|}\sum_{U\in\mathcal U}
+ \1_{\{Q|_U\text{ nonsingular of sign }\xi\}}.
+\]
+Uniformly over the family $\mathcal U$,
+\begin{equation}
+\label{eq:transverse-restriction-error}
+ \E\left|f_\xi(Q)-\frac12\right|
+ =O\left(q^{-1/2}+|\mathcal U|^{-1/2}\right).
+\end{equation}
+The same bound holds after conditioning $Q$ to be nonsingular, or
+conditioning both $Q$ and $Q-L$ to be nonsingular, uniformly over every
+fixed symmetric form $L$.
+\end{lemma}
+
+\begin{proof}
+Restriction onto a fixed subspace is a surjective linear map on spaces
+of symmetric forms, so each summand has expectation
+$1/2+O(q^{-1})$ by Lemma~\ref{lem:symmetric-counts}.
+For distinct $U,V\in\mathcal U$ write $T=U\cap V$. Extend a basis of
+$T$ separately to bases of $U$ and $V$, and then to an ambient basis.
+In this basis, the coefficients defining the two restricted forms
+overlap exactly in the symmetric block on $T$. Thus any two forms on
+$U,V$ agreeing on $T$ extend simultaneously to the ambient space;
+given $Q|_T$, their remaining coefficients are independent and uniform.
+
+If $T$ has positive dimension, its restriction is nonsingular outside
+an event of probability $O(q^{-1})$; for $T=0$ this holds with probability
+one. Given a nonsingular restriction on $T$, the determinant on $U$
+is the determinant on $T$ times that of a uniform symmetric Schur
+complement of size $r-\dim T>0$. The corresponding Schur complement
+on $V$ is independent. Both signs are balanced up to $O(q^{-1})$,
+uniformly over the fixed common block and mixed coefficients. The
+covariance of the two indicator summands is therefore $O(q^{-1})$.
+The diagonal covariance is at most $1$, giving
+\[
+ \Var(f_\xi)=O(q^{-1}+|\mathcal U|^{-1}).
+\]
+Cauchy--Schwarz and the expectation estimate prove
+Equation~\eqref{eq:transverse-restriction-error}.
+
+For the last assertion, each of $Q$ and $Q-L$ is marginally uniform.
+The event $\mathcal A$ that the specified forms are nonsingular has
+probability $1-O(q^{-1})$, uniformly in $L$. Since the error being
+averaged is nonnegative,
+\[
+ \E\left[\left|f_\xi-\frac12\right|\,\middle|\,\mathcal A\right]
+ \leq \frac{\E|f_\xi-1/2|}{\P(\mathcal A)}.
+\]
+This proves the same bound under either conditioning.
+\end{proof}
+
+\subsection{Counting centers on one face}
+
+The following projection identity explains the special role of half
+rank.
+
+\begin{lemma}[Complementary images]
+\label{lem:transverse-projection}
+Suppose $B\in\mathcal S_{2r}$ is nonsingular and $A\in\mathcal S_{2r}$
+has rank $r$. Then
+\begin{equation}
+\label{eq:transverse-projection}
+ \rank(B-A)=r\quad\Longleftrightarrow\quad AB^{-1}A=A.
+\end{equation}
+If $A$ and $A-B$ have discriminant signs $\xi_1,\xi_2$, respectively,
+under these conditions, then
+\begin{equation}
+\label{eq:transverse-pair-sign}
+ \chi(\det B)=c_0\xi_1\xi_2.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+If $A$ and $B-A$ both have rank $r$, their images span the image of
+$B$, so these images are complementary. Since
+$AB^{-1}+(B-A)B^{-1}$ is the identity, uniqueness of the corresponding
+direct-sum decomposition makes $AB^{-1}$ the projection onto the first
+image. In particular $AB^{-1}A=A$. Conversely, this identity makes
+$AB^{-1}$ an idempotent of rank $r$. Its complementary idempotent has
+rank $r$ and, after multiplication by $B$, equals $B-A$.
+
+Choose basis-column matrices $P_1,P_2$ of the two complementary images
+and write $A=P_1R_1P_1^{\mathsf T}$ and
+$A-B=P_2R_2P_2^{\mathsf T}$. Then
+\[
+ B=(P_1\ P_2)\diag(R_1,-R_2)(P_1\ P_2)^{\mathsf T}.
+\]
+Taking determinant characters gives
+Equation~\eqref{eq:transverse-pair-sign}.
+\end{proof}
+
+For independent point matrices $(X_i)_{i\in I}$ define
+\begin{equation}
+\label{eq:transverse-event}
+ \eta_e^X=\chi(\det(X_i-X_k))\quad(e=\{i,k\}),\qquad
+ \mathcal T=\{X_i-X_k\text{ nonsingular for every }e\in E\}.
+\end{equation}
+The choice of orientation of $e$ is immaterial because $D$ is even.
+Each difference is uniform in $\mathcal S_D$, so
+$\P(\mathcal T^c)=O(q^{-1})$ by Lemma~\ref{lem:symmetric-counts}.
+For an activity pattern $B\subseteq\mathscr C$, let
+$A_j=\{i\in j:(i,j)\in B\}$ and define the local center factor and its
+sign model by
+\begin{align}
+ F_j(X)&=\E_{Y_j}\prod_{i\in A_j}K_{\xi_{ij}}(X_i-Y_j),
+ \label{eq:transverse-face-factor}\\
+ G_j(X)&=\prod_{\{i,k\}\subseteq A_j}
+       (1+c_0\eta^X_{\{i,k\}}\xi_{ij}\xi_{kj}).
+ \label{eq:transverse-model}
+\end{align}
+On transverse local configurations, $G_j$ equals $2$ for two active
+incidences whose pair signs match, and $8$ for three active incidences
+whose three pair signs match; it is zero when any required match fails.
+With at most one active incidence, both $F_j$ and $G_j$ equal $1$.
+
+\begin{lemma}[Local transverse approximation]
+\label{lem:local-transverse}
+Suppose $r>1$. For each face $j$, let $\mathcal T_j$ be the event that
+all three differences between its point matrices are nonsingular.
+Then, uniformly over the activity and sign choices,
+\begin{equation}
+\label{eq:transverse-local-l1}
+ \E\bigl[\1_{\mathcal T_j}|F_j-G_j|\bigr]=O(q^{-1/2}).
+\end{equation}
+Moreover $0\leq F_j\leq C_D$ on $\mathcal T_j$, for a constant $C_D$
+independent of $q$, and $0\leq G_j\leq8$ everywhere.
+\end{lemma}
+
+\begin{proof}
+First suppose all three incidences are active, and locally write their
+point matrices as $X_1,X_2,X_3$ and their signs as $\xi_1,\xi_2,\xi_3$.
+Set
+\[
+ B_0=X_1-X_2,\quad B_1=X_1-X_3,\quad A=X_1-Y,
+ \qquad Q=B_0^{-1},\quad L=B_0^{-1}-B_1^{-1}.
+\]
+On $\mathcal T_j$, the form
+$L=B_0^{-1}(B_1-B_0)B_1^{-1}$ is symmetric and nonsingular. Write a
+rank-$r$ matrix $A$ as $PRP^{\mathsf T}$. By
+Lemma~\ref{lem:transverse-projection}, its other two rank conditions are
+equivalent to
+\[
+ R(P^{\mathsf T}B_\nu^{-1}P)R=R\quad(\nu=0,1),
+\]
+or to
+\begin{equation}
+\label{eq:transverse-center-parametrization}
+ P^{\mathsf T}LP=0,\qquad
+ P^{\mathsf T}QP\text{ nonsingular},\qquad
+ R=(P^{\mathsf T}QP)^{-1}.
+\end{equation}
+Thus the centers satisfying the three rank conditions are in bijection
+with the maximal $L$-isotropic subspaces $U$ on which $Q$ restricts
+nonsingularly. Changing the basis of $U$ leaves the resulting $A$
+unchanged, and every $A$ recovers $U$ as its image, proving the stated
+bijection.
+
+Let $\mathcal M$ be the event that all three required pair matches
+$\eta^X_{\{a,b\}}=c_0\xi_a\xi_b$ hold. If any match fails, the signed
+count is zero by Equation~\eqref{eq:transverse-pair-sign}. If they all
+hold, then
+\[
+ \chi(\det L)
+ =\eta^X_{\{1,2\}}\eta^X_{\{1,3\}}\eta^X_{\{2,3\}}
+ =c_0,
+\]
+so $L$ is split by Lemma~\ref{lem:transverse-split}. In
+Equation~\eqref{eq:transverse-center-parametrization}, the sign of $A$
+is the sign of $Q|_U$, since inversion preserves a nonzero determinant
+character. Once it equals $\xi_1$, the pair matches with point $1$
+force the other two signs to equal $\xi_2,\xi_3$.
+
+For split $L$, let $f_{\xi_1}(Q;L)$ denote the fraction of its $N_r$
+maximal isotropic subspaces having a nonsingular restriction of sign
+$\xi_1$. On a matching transverse triple the exact normalized count is
+\begin{equation}
+\label{eq:transverse-three-normalization}
+ F_j=\frac{q^{-b(D)}N_r}{\pi_{\xi_1}\pi_{\xi_2}\pi_{\xi_3}}
+       f_{\xi_1}(Q;L)
+     =(16+O(q^{-1}))f_{\xi_1}(Q;L).
+\end{equation}
+Here the exponent cancels because
+$r(r-1)/2-b(2r)+3b(r)=0$. The factor $16$ combines the leading
+factor $2$ in $N_r$ with the three signed-layer normalization factors
+$2$; the restriction-sign fraction will supply a further factor $1/2$.
+
+We specify the probability law used for that fraction. Before any
+invertibility conditioning, $B_0,B_1$ are independent uniform symmetric
+matrices: this follows by conditioning on $X_1$ and translating the
+independent matrices $X_2,X_3$. After requiring $B_0,B_1$ to be
+nonsingular, inversion preserves their independent uniform laws on
+nonsingular symmetric matrices. Conditional also on a fixed value of
+$L=B_0^{-1}-B_1^{-1}$, the matrix $Q$ is therefore uniform on
+\[
+ \{Q\in\mathcal S_D:Q\text{ and }Q-L\text{ are nonsingular}\}.
+\]
+Each admissible $Q$ corresponds to exactly one ordered inverse pair
+$(Q,Q-L)$, which verifies the conditional uniformity directly.
+Requiring $L$ to be nonsingular is exactly the remaining transverse
+condition. Lemma~\ref{lem:transverse-restrictions}, uniformly for every
+fixed nonsingular split $L$, now gives
+\begin{equation}
+\label{eq:transverse-matching-error}
+ \E\left[\1_{\mathcal M}
+       \left|f_{\xi_1}(Q;L)-\frac12\right|
+       \,\middle|\,\mathcal T_j,L\right]
+ \leq \E\left[\left|f_{\xi_1}(Q;L)-\frac12\right|
+       \,\middle|\,\mathcal T_j,L\right]
+ =O(q^{-1/2}).
+\end{equation}
+We used $r>1$, for which $N_r^{-1/2}=O(q^{-1/2})$.
+The matching indicator is kept inside the expectation; no additional
+conditioning on the matches is used. For nonsplit $L$, the matching
+event is empty and both the signed count and its model vanish.
+Equations~\eqref{eq:transverse-three-normalization}
+and~\eqref{eq:transverse-matching-error} prove the local error estimate
+with limiting value $8$ on matches. The same exact formula, with
+$0\leq f_{\xi_1}\leq1$, gives a uniform bound on every transverse
+triple, including the exceptional restrictions.
+
+For two active incidences, relabel them $1,2$ and put $B=X_1-X_2$ and
+$Q=B^{-1}$. The same projection argument now parametrizes centers by
+all $r$-subspaces $U$ for which $Q|_U$ is nonsingular. On the pair match
+$\chi(\det B)=c_0\xi_1\xi_2$, only the sign $\xi_1$ of this restriction
+needs to be imposed; off the match the count is zero. If $f_{\xi_1}(Q)$
+is the fraction among all $\genfrac{[}{]}{0pt}{}{2r}{r}_q$ subspaces,
+the exact count on the match is
+\begin{equation}
+\label{eq:transverse-two-normalization}
+ F_j=\frac{q^{-b(D)}\genfrac{[}{]}{0pt}{}{2r}{r}_q}
+              {\pi_{\xi_1}\pi_{\xi_2}}
+       f_{\xi_1}(Q)
+     =(4+O(q^{-1}))f_{\xi_1}(Q).
+\end{equation}
+The exponent identity here is $r^2-b(2r)+2b(r)=0$. Conditional on $B$
+being nonsingular, $Q$ is uniform nonsingular symmetric. Apply
+Lemma~\ref{lem:transverse-restrictions} with this single conditioning
+and keep the matching indicator inside the expectation as before.
+This gives limiting value $2$, error $O(q^{-1/2})$, and a uniform
+bound on the nonsingular-pair event. Restricting further to
+$\mathcal T_j$ can only decrease the nonnegative error expectation.
+The remaining cases follow from the mean-one normalization of each
+$K_\xi$ and translation invariance.
+\end{proof}
+
+\subsection{Joint distribution of the pair signs}
+
+The pairs share point matrices, so marginal balance of their signs does
+not give their joint distribution. For each nonempty subset moment,
+we first establish a high-probability event for the difference minors
+at a selected incident vertex, then condition on all entries except
+its off-diagonal row vector and transpose and average over that vector.
+Table~\ref{tab:complex} shows that the degree of each
+vertex in $(I,E)$ is at most $6$:
+its degree there equals its number of incident faces, since each face
+contains two pairs through that point and each pair occurs twice.
+
+\begin{lemma}[Joint determinant signs]
+\label{lem:joint-signs}
+For fixed even $D\geq12$, the law of $(\eta_e^X)_{e\in E}$ converges
+in total variation to that of independent uniform signs on
+$\{+1,-1\}^E$.
+\end{lemma}
+
+\begin{proof}
+Fix a nonempty subset $F\subseteq E$, and choose a vertex $i$ incident
+with $s\geq1$ of its pairs. Write $N=D-1$ and, for every point $v$,
+write its matrix in block form
+\[
+ X_v=\begin{pmatrix}a_v&z_v^{\mathsf T}\\z_v&W_v\end{pmatrix},
+ \qquad W_v\in\mathcal S_N,\quad z_v\in\F_q^N.
+\]
+Let $k_1,\ldots,k_s$ be the selected neighbors of $i$, and put
+$M_\ell=W_i-W_{k_\ell}$. We first establish a high-probability event
+using these lower-right blocks, before conditioning on the other
+entries. Given $W_i$, the $M_\ell$ are independent uniform symmetric
+matrices. All are nonsingular except with probability $O(q^{-1})$.
+Conditional on their nonsingularity, their inverses are independent
+uniform nonsingular symmetric matrices.
+
+Let $Q_1,\ldots,Q_s$ instead be independent uniform matrices in
+$\mathcal S_N$. For any $t\in\F_q^s\setminus\{0\}$, the matrix
+$\sum_\ell t_\ell Q_\ell$ is uniform in $\mathcal S_N$: condition on
+all but a coordinate with nonzero coefficient. By
+Equation~\eqref{eq:transverse-nullity}, its probability of rank at most
+$s$ is $O(q^{-b(N-s)})$. The union bound over all nonzero $t$ gives
+\[
+ \P\left(\exists t\ne0:
+       \rank\left(\sum_\ell t_\ell Q_\ell\right)\leq s\right)
+ =O(q^{s-b(N-s)}).
+\]
+Conditioning all the $Q_\ell$ to be nonsingular divides this upper bound
+by a probability $1-O(q^{-1})$. It follows that the event
+\begin{equation}
+\label{eq:transverse-good-minors}
+ \mathcal G=\left\{\begin{array}{l}
+ M_\ell\text{ is nonsingular for every }\ell,\\
+ \rank(\sum_\ell t_\ell M_\ell^{-1})\geq s+1
+       \text{ for every }t\in\F_q^s\setminus\{0\}
+ \end{array}\right\}
+\end{equation}
+satisfies
+\begin{equation}
+\label{eq:transverse-good-probability}
+ \P(\mathcal G^c)=O(q^{-1}+q^{s-b(N-s)})=O(q^{-1}).
+\end{equation}
+Indeed $s\leq6$ and $D\geq12$ imply $N-s\geq5$ and
+$b(N-s)\geq15>s$. These probability bounds hold given every $W_i$
+and hence hold without conditioning as well.
+
+Only now condition on all entries of all point matrices except $z_i$.
+The event $\mathcal G$ is determined by this conditioning, whereas
+$z_i$ remains uniform on $\F_q^N$. On every such fixed configuration
+in $\mathcal G$, the determinant formula gives
+\begin{equation}
+\label{eq:transverse-schur-evaluations}
+ \det(X_i-X_{k_\ell})=\det(M_\ell)h_\ell(z_i),\qquad
+ h_\ell(z)=a_i-a_{k_\ell}
+ -(z-z_{k_\ell})^{\mathsf T}M_\ell^{-1}(z-z_{k_\ell}).
+\end{equation}
+Let $\mu$ be the law of $(h_1(z),\ldots,h_s(z))$ for uniform $z$.
+Fix a nontrivial additive character $\psi$ of $\F_q$ and define
+$\widehat\mu(t)=\E_z\psi(\sum_\ell t_\ell h_\ell(z))$.
+The quadratic part at every frequency $t\ne0$ has rank at least $s+1$
+by Equation~\eqref{eq:transverse-good-minors}.
+
+We use the standard rank bound for a quadratic Gauss sum; see
+Gowers and Wolf~\cite[Lemma~3.3]{gowerswolf2011}. The short differencing argument
+also fixes our normalization. A quadratic polynomial
+$p(z)=z^{\mathsf T}Hz+\lambda^{\mathsf T}z+c$ with symmetric quadratic
+matrix $H$ obeys
+\begin{align*}
+ \left|\E_z\psi(p(z))\right|^2
+ &=\E_{z,u}\psi(p(z+u)-p(z))\\
+ &=\E_u\psi(u^{\mathsf T}Hu+\lambda^{\mathsf T}u)
+               \E_z\psi(2z^{\mathsf T}Hu),
+\end{align*}
+whose absolute value is at most
+$\P_u(Hu=0)=q^{-\rank H}$. Here the inner average vanishes unless
+$Hu=0$, by additive-character orthogonality and the fact that $q$ is
+odd. Therefore
+$|\widehat\mu(t)|\leq q^{-(s+1)/2}$ for $t\ne0$, uniformly over all
+the fixed linear and constant terms in
+Equation~\eqref{eq:transverse-schur-evaluations}.
+
+Writing $\nu$ for uniform measure on $\F_q^s$, finite Fourier
+orthogonality and Cauchy--Schwarz yield the explicit bound
+\begin{align}
+\label{eq:transverse-fourier-tv}
+ \TV(\mu,\nu)
+ &=\frac12\sum_x|\mu(x)-q^{-s}|\\
+ &\leq\frac12\left(q^s\sum_x|\mu(x)-q^{-s}|^2\right)^{1/2}
+ =\frac12\left(\sum_{t\ne0}|\widehat\mu(t)|^2\right)^{1/2}
+ \leq\frac12q^{-1/2}.\nonumber
+\end{align}
+The identity in the middle follows by expanding the square and using
+$\sum_t\psi(t\cdot(x-y))=q^s\1_{\{x=y\}}$.
+
+Under the same fixed conditioning, all factors
+$\eta_e^X$ for $e\in F$ not incident to $i$ are constant. They and the
+characters of the minor determinants have absolute value at most one.
+By Equation~\eqref{eq:transverse-schur-evaluations}, the remaining part
+of the product is $\prod_{\ell=1}^s\chi(h_\ell(z_i))$. Its expectation
+under $\nu$ is zero, because $\E_{x\in\F_q}\chi(x)=0$ and $s\geq1$.
+The difference of expectations of a function bounded by one is at
+most twice total variation, so on $\mathcal G$ the absolute conditional
+expectation of the entire selected product is at most $q^{-1/2}$.
+On $\mathcal G^c$ it is at most one. Averaging and using
+Equation~\eqref{eq:transverse-good-probability} proves
+\begin{equation}
+\label{eq:transverse-subset-moments}
+ \E\prod_{e\in F}\eta_e^X=O(q^{-1/2})\qquad(F\ne\varnothing).
+\end{equation}
+
+Finally, replace any zero coordinate of $\eta^X$ by $+1$, obtaining a
+sign vector $\widetilde\eta^X$. A union bound and
+Lemma~\ref{lem:symmetric-counts} show that this replacement changes
+the vector with probability $O(q^{-1})$ and changes each subset moment
+by $O(q^{-1})$. For every $\varepsilon\in\{+1,-1\}^E$, the identity
+\[
+ \1_{\{\widetilde\eta^X=\varepsilon\}}
+ =2^{-|E|}\prod_{e\in E}(1+\varepsilon_e\widetilde\eta_e^X)
+\]
+expresses its probability through the subset moments. The empty
+subset contributes $2^{-|E|}$ and every other subset is $O(q^{-1/2})$
+by Equation~\eqref{eq:transverse-subset-moments}. Since $E$ is fixed,
+summing over all sign vectors proves convergence in total variation.
+\end{proof}
+
+\subsection{Assembling the transverse contribution}
+
+\begin{proposition}
+\label{prop:transverse-limit}
+For every fixed even $D\geq12$, every activity pattern
+$B\subseteq\mathscr C$, and every prescribed set of incidence signs,
+\begin{equation}
+\label{eq:transverse-limit}
+ \E\left[\1_{\mathcal T}
+       \prod_{(i,j)\in B}K_{\xi_{ij}}(X_i-Y_j)\right]
+ =\E_\eta\prod_{j\in J}\prod_{\{i,k\}\subseteq A_j}
+       (1+c_0\eta_{\{i,k\}}\xi_{ij}\xi_{kj})+o(1),
+\end{equation}
+where the $\eta_e$ are independent uniform signs. At each fixed $D$,
+the error is uniform over all activity patterns and incidence signs.
+\end{proposition}
+
+\begin{proof}
+Given the point matrices, the center variables are independent, so the
+left side is $\E[\1_{\mathcal T}\prod_jF_j]$. Increase $C_D$ from
+Lemma~\ref{lem:local-transverse} to at least $8$. The deterministic
+telescoping identity for two products gives, on $\mathcal T$,
+\[
+ \left|\prod_jF_j-\prod_jG_j\right|
+ \leq C_D^{|J|-1}\sum_j|F_j-G_j|.
+\]
+Since $\mathcal T\subseteq\mathcal T_j$, taking expectations and
+applying Equation~\eqref{eq:transverse-local-l1} bounds this difference
+by $O(q^{-1/2})$. This argument applies to faces sharing point matrices;
+it uses no independence of their local errors.
+
+The bound $\prod_jG_j\leq8^{|J|}$ and
+$\P(\mathcal T^c)=O(q^{-1})$ allow removal of
+$\1_{\mathcal T}$ from the model expectation. Lemma~\ref{lem:joint-signs}
+then replaces $\eta^X$ by independent uniform signs. Total variation
+controls every bounded function, including the displayed model even
+when $c_0$ varies with the prime $q$. Finally, there are only finitely
+many activity and sign choices, so the largest of their errors also
+tends to zero.
+\end{proof}
+
+Proposition~\ref{prop:transverse-limit} reduces
+Lemma~\ref{lem:rank-layer} to the weighted singular contribution
+\begin{equation}
+\label{eq:transverse-tail-target}
+ \E\left[\1_{\mathcal T^c}
+       \prod_{(i,j)\in B}K_{\xi_{ij}}(X_i-Y_j)\right]=o(1).
+\end{equation}
+The unweighted estimate $\P(\mathcal T^c)=O(q^{-1})$ does not establish
+Equation~\eqref{eq:transverse-tail-target}, because the normalized
+rank-layer indicators grow with $q$. Proposition~\ref{prop:singular-tail}
+will prove this estimate for all sufficiently large even $D$, after
+the geometric and exponent estimates in the next sections. The
+threshold $D\geq12$ above applies only to the transverse calculation
+and the joint-sign argument.
diff --git a/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/paper.pdf b/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..d95c94dc6b53a5f3fa92dbf3706dcfe826a870cb
--- /dev/null
+++ b/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:0c0d7b5eb1ed43c00ed2f60d2347b7d5511bad2218efe12c28fd9968ee549b0f
+size 639874
diff --git a/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/README.md b/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..094decc441c606c21a8ec24ca5bb53d71f990eb5
--- /dev/null
+++ b/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/README.md
@@ -0,0 +1,17 @@
+# [A counterexample to Tachikawa's second conjecture](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 23, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-counterexample-to-Tachikawas-second-conjecture-September-23-2026,
+  author = {{OpenAI}},
+  title = {{A counterexample to Tachikawa's second conjecture}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/paper.pdf}{OAI:A-counterexample-to-Tachikawas-second-conjecture-September-23-2026}},
+  year = {2026}
+}
+```
diff --git a/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/paper.tex b/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/paper.tex
new file mode 100644
index 0000000000000000000000000000000000000000..a935a9ac83184036900863a5abc246f852cac531
--- /dev/null
+++ b/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/paper.tex
@@ -0,0 +1,74 @@
+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage{lmodern}
+\usepackage[margin=1in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{booktabs,array}
+\usepackage{tikz-cd}
+\usetikzlibrary{arrows.meta,positioning}
+\usepackage[numbers,sort&compress]{natbib}
+\usepackage{microtype}
+\usepackage{xurl}
+\usepackage{xcolor}
+\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref}
+\usepackage[capitalise,nameinlink,noabbrev]{cleveref}
+\hypersetup{pdftitle={A counterexample to Tachikawa's second conjecture},pdfauthor={OpenAI}}
+\DeclareMathOperator{\Hom}{Hom}
+\DeclareMathOperator{\End}{End}
+\DeclareMathOperator{\Ext}{Ext}
+\DeclareMathOperator{\Tor}{Tor}
+\DeclareMathOperator{\im}{im}
+\DeclareMathOperator{\coker}{coker}
+\DeclareMathOperator{\rad}{rad}
+\DeclareMathOperator{\soc}{soc}
+\DeclareMathOperator{\add}{add}
+\DeclareMathOperator{\pd}{pd}
+\DeclareMathOperator{\idim}{id}
+\DeclareMathOperator{\domdim}{domdim}
+\DeclareMathOperator{\tr}{tr}
+\DeclareMathOperator{\Res}{Res}
+\DeclareMathOperator{\Cone}{Cone}
+\newcommand{\sHom}{\underline{\Hom}}
+\newcommand{\sEnd}{\underline{\End}}
+\newcommand{\cH}{\mathcal H}
+\newcommand{\cI}{\mathcal I}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\newtheorem{notation}[theorem]{Notation}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newtheorem{example}[theorem]{Example}
+\numberwithin{equation}{section}
+\setlength{\emergencystretch}{1.5em}
+\title{A counterexample to Tachikawa's second conjecture}
+\author{OpenAI}
+\date{September 23, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We disprove Tachikawa's second conjecture by constructing a
+finite-dimensional symmetric algebra over $k=\mathbb F_2(q,H_1,H_2)$
+with a finite-dimensional nonprojective module whose self-extension
+groups vanish in every positive degree. The associated endomorphism
+algebra also gives
+counterexamples to the classical, generalized, and strong Nakayama
+conjectures, the Auslander--Gorenstein conjecture, and the Wakamatsu
+tilting conjecture. These conclusions persist after every extension of $k$.
+\end{abstract}
+\input{sections/introduction}
+\input{sections/homological}
+\input{sections/transfer}
+\input{sections/algebra}
+\input{sections/stable}
+\input{sections/bimodule}
+\input{sections/triangular}
+\input{sections/consequences}
+\clearpage
+\bibliographystyle{plainnat}
+\bibliography{refs}
+\end{document}
diff --git a/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/refs.bib b/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/refs.bib
new file mode 100644
index 0000000000000000000000000000000000000000..e055f4201fe02bfd2f2c0b6ebffb84252c5b6043
--- /dev/null
+++ b/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/refs.bib
@@ -0,0 +1,398 @@
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+}
+
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+}
+
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+}
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+  title = {On the homological conjectures for {Artin} algebras},
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+  archivePrefix = {arXiv},
+  primaryClass = {math.RT},
+  note = {Version 2, September 24, 2026},
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+}
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+  primaryClass = {math.RT},
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+  archivePrefix = {arXiv},
+  primaryClass = {math.RT},
+  note = {Version 1, February 19, 2014},
+  url = {https://arxiv.org/abs/1402.4595v1}
+}
+
+@article{XiongZhang2012,
+  author = {Xiong, Bao-Lin and Zhang, Pu},
+  title = {{Gorenstein}-projective modules over triangular matrix {Artin} algebras},
+  journal = {Journal of Algebra and Its Applications},
+  volume = {11},
+  number = {4},
+  pages = {1250066},
+  year = {2012},
+  doi = {10.1142/S0219498812500661}
+}
+
+@article{BessenrodtHolmZimmermann2007,
+  author = {Bessenrodt, Christine and Holm, Thorsten and Zimmermann, Alexander},
+  title = {Generalized {Reynolds} ideals for non-symmetric algebras},
+  journal = {Journal of Algebra},
+  volume = {312},
+  number = {2},
+  pages = {985--994},
+  year = {2007},
+  doi = {10.1016/j.jalgebra.2007.02.028}
+}
+
+@article{Hattori1965,
+  author = {Hattori, Akira},
+  title = {Rank element of a projective module},
+  journal = {Nagoya Mathematical Journal},
+  volume = {25},
+  pages = {113--120},
+  year = {1965},
+  doi = {10.1017/S002776300001148X}
+}
+
+@article{LiuZhouZimmermann2012,
+  author = {Liu, Yuming and Zhou, Guodong and Zimmermann, Alexander},
+  title = {{Higman} ideal, stable {Hochschild} homology and
+           {Auslander-Reiten} conjecture},
+  journal = {Mathematische Zeitschrift},
+  volume = {270},
+  number = {3--4},
+  pages = {759--781},
+  year = {2012},
+  month = apr,
+  doi = {10.1007/s00209-010-0825-z},
+  url = {https://www.lamfa.u-picardie.fr/alex/LiuZhouZimmermannRevised2.pdf},
+  note = {Numbered results refer to the author version of November 19, 2010}
+}
+
+@misc{ErdmannEtAl2003,
+  author = {Erdmann, Karin and Holloway, Miles and Snashall, Nicole and
+            Solberg, {\O}yvind and Taillefer, Rachel},
+  title = {Support varieties for selfinjective algebras},
+  year = {2003},
+  eprint = {math/0311311},
+  archivePrefix = {arXiv},
+  note = {arXiv:math/0311311v1},
+  url = {https://arxiv.org/abs/math/0311311v1}
+}
+
+@book{Tachikawa1973,
+  author = {Tachikawa, Hiroyuki},
+  title = {Quasi-{Frobenius} Rings and Generalizations: {QF-3} and {QF-1} Rings},
+  series = {Lecture Notes in Mathematics},
+  volume = {351},
+  publisher = {Springer},
+  address = {Berlin, Heidelberg},
+  year = {1973},
+  note = {Notes by Claus Michael Ringel},
+  isbn = {978-3-540-06501-2},
+  doi = {10.1007/BFb0059997},
+  url = {https://link.springer.com/book/10.1007/BFb0059997}
+}
+
+@article{ChenXi2025TAMS,
+  author = {Chen, Hongxing and Xi, Changchang},
+  title = {Homological theory of self-orthogonal modules},
+  journal = {Transactions of the American Mathematical Society},
+  volume = {378},
+  number = {10},
+  pages = {7287--7335},
+  year = {2025},
+  doi = {10.1090/tran/9426},
+  url = {https://www.wemath.cn/~ccxi/Papers/Articles/cx-2025tams.pdf}
+}
+
+@article{ChenXi2025VG,
+  author = {Chen, Hongxing and Xi, Changchang},
+  title = {Virtually {Gorenstein} algebras of infinite dominant dimension},
+  journal = {Journal of Pure and Applied Algebra},
+  volume = {230},
+  number = {4},
+  pages = {108224},
+  year = {2026},
+  eprint = {2509.04990},
+  archivePrefix = {arXiv},
+  doi = {10.1016/j.jpaa.2026.108224},
+  url = {https://www.wemath.cn/~ccxi/Papers/Articles/VGADM.pdf}
+}
+
+@misc{BohmlerMarczinzik2026,
+  author = {B{\"o}hmler, Bernhard and Marczinzik, Ren{\'e}},
+  title = {{Tor} and {Ext} vanishing results for commutative {Artinian} rings},
+  year = {2026},
+  month = aug,
+  eprint = {2608.09701},
+  archivePrefix = {arXiv},
+  primaryClass = {math.AC},
+  note = {arXiv:2608.09701v1},
+  doi = {10.48550/arXiv.2608.09701},
+  url = {https://arxiv.org/abs/2608.09701v1}
+}
+
+@misc{ZhangZhou2026,
+  author = {Zhang, Xiaojin and Zhou, Panyue},
+  title = {The {Auslander--Reiten} conjecture for algebras with radical cube zero},
+  year = {2026},
+  month = sep,
+  eprint = {2609.08679},
+  archivePrefix = {arXiv},
+  primaryClass = {math.RT},
+  note = {arXiv:2609.08679v1},
+  url = {https://arxiv.org/abs/2609.08679v1}
+}
+
+@misc{OpenAIAR2026,
+  author = {{OpenAI}},
+  title = {{An explicit counterexample to the Auslander--Reiten conjecture}},
+  howpublished = {OpenAI Math Release preprint
+    \href{https://github.com/openai/math/blob/main/preprints/An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026/paper.pdf}{OAI:An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026}},
+  year = {2026}
+}
+
+@misc{OpenAIFinitistic2026,
+  author = {{OpenAI}},
+  title = {{An algebra of infinite little finitistic dimension}},
+  howpublished = {OpenAI Math Release preprint
+    \href{https://github.com/openai/math/blob/main/preprints/An-algebra-of-infinite-little-finitistic-dimension-September-23-2026/paper.pdf}{OAI:An-algebra-of-infinite-little-finitistic-dimension-September-23-2026}},
+  year = {2026}
+}
+
+@incollection{BuanSolberg1998,
+  author = {Buan, Aslak Bakke and Solberg, {\O}yvind},
+  title = {Relative cotilting theory and almost complete cotilting modules},
+  booktitle = {Algebras and modules {II}},
+  editor = {Reiten, Idun and Smal{\o}, Sverre O. and Solberg, {\O}yvind},
+  series = {CMS Conference Proceedings},
+  volume = {24},
+  pages = {77--92},
+  publisher = {American Mathematical Society},
+  address = {Providence, RI},
+  year = {1998},
+  note = {Numbered theorems refer to the author-hosted version.
+    \href{https://oyvinso.folk.ntnu.no/Papers/almostcompnakayama.pdf}{Manuscript PDF}}
+}
diff --git a/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/sections/algebra.tex b/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/sections/algebra.tex
new file mode 100644
index 0000000000000000000000000000000000000000..3a7b197ff4f2ead4e6f676d64131dcc4da6e1c89
--- /dev/null
+++ b/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/sections/algebra.tex
@@ -0,0 +1,339 @@
+\section{The ten-dimensional algebra}
+\label{a:section}
+
+We now construct an algebra \(C\) and a simple module \(s\) whose
+resolution supplies the degree-three extension classes used below.
+Besides this resolution, we need
+finite projective-dimension bounds for the dual regular modules;
+these bounds make derived induction admit a finite bimodule
+representative. The multiplication of $C$ is the one in
+\cite[Section~4.1]{OpenAIAR2026}; all properties needed here are proved
+directly.  We work over the fixed field
+\(k=\mathbb F_2(q,H_1,H_2)\), where the three parameters are
+algebraically independent.
+
+Let \(C\) have the following basis, arranged by its left and right
+idempotents:
+\begin{equation}
+\begin{array}{c|c@{\qquad}c|c}
+\text{corner}&\text{basis}&\text{corner}&\text{basis}\\ \hline
+eCe&e,x,y,z&eCf&u,v\\
+fCe&t,j&fCf&f,n.
+\end{array}
+\label{a:corners}
+\end{equation}
+The unit is \(1=e+f\).  The complete multiplication table, with the
+left factor indexing the row, is
+\begin{equation}
+\begingroup
+\setlength{\arraycolsep}{4pt}
+\begin{array}{c|cccccccccc}
+ &e&f&x&y&z&u&v&t&j&n\\ \hline
+e&e&0&x&y&z&u&v&0&0&0\\
+f&0&f&0&0&0&0&0&t&j&n\\
+x&x&0&0&qz&0&v&0&0&0&0\\
+y&y&0&z&0&0&v&0&0&0&0\\
+z&z&0&0&0&0&0&0&0&0&0\\
+u&0&u&0&0&0&0&0&y+qx&z&v\\
+v&0&v&0&0&0&0&0&qz&0&0\\
+t&t&0&j&q^2j&0&(1+q)n&0&0&0&0\\
+j&j&0&0&0&0&0&0&0&0&0\\
+n&0&n&0&0&0&0&0&qj&0&0
+\end{array}
+\endgroup
+\label{a:multiplication}
+\end{equation}
+
+\begin{lemma}\label{a:table}
+The table \eqref{a:multiplication} defines an associative unital algebra
+of dimension ten.  Its radical is the span of the eight basis vectors
+other than \(e,f\), and \((\rad C)^5=0\).
+\end{lemma}
+
+\begin{proof}
+Give the basis vectors the degrees
+\[
+\begin{gathered}
+\deg e=\deg f=0,\qquad \deg u=\deg t=1,\\
+\deg x=\deg y=\deg n=2,\qquad
+\deg v=\deg j=3,\qquad \deg z=4.
+\end{gathered}
+\]
+The table is homogeneous and respects the corners.  Thus associativity
+with an idempotent, or with incompatible adjacent corners, follows
+from \eqref{a:corners}.  For positive-degree triples of total degree
+greater than four, both bracketings vanish.  The remaining composable
+triples and the value of either bracketing are precisely
+\[
+\begin{array}{c|ccccc}
+abc&utu&tut&xut&yut&ntu\\ \hline
+(ab)c=a(bc)&(1+q)v&q(1+q)j&qz&qz&0
+\end{array}
+\]
+\[
+\begin{array}{c|cccccc}
+abc&unt&txu&tyu&utx&uty&tun\\ \hline
+(ab)c=a(bc)&qz&0&0&z&q^2z&0.
+\end{array}
+\]
+Every entry is obtained by two applications of
+\eqref{a:multiplication}; this exhausts the associativity check by
+trilinearity.  The positive-degree span is a nilpotent ideal with
+fifth power zero and quotient \(ke\oplus kf\cong k^2\).  It is therefore
+the radical.
+\end{proof}
+
+Let \(s\) be the one-dimensional left \(C\)-module on which \(f\)
+acts as the identity and every other basis vector acts as zero.
+Its vector space dual \(Ds\) is the right simple with the same character.
+
+The corner \(eCe\) gives a concrete connection with Schulz's
+example~\cite[Example~7]{Schulz1995}. The table identifies it with
+the quantum exterior algebra
+\[
+ eCe\cong k\langle x,y\rangle/(x^2,y^2,yx-q^{-1}xy).
+\]
+Moreover, the assignments
+\(t\mapsto x+q^{-1}y\) and \(j\mapsto q^{-1}z\) give an
+isomorphism of right \(eCe\)-modules
+\[
+ fCe\cong(x+q^{-1}y)eCe.
+\]
+The identities \(tx=j\), \(ty=q^2j\), and \(jx=jy=0\) verify
+right-linearity, and the two images form a basis of the right ideal.
+With Schulz's
+parameter \(p=q^{-1}\), this right ideal is the first syzygy of his
+module \((x+y)eCe\). The later left-module resolutions will use the
+same type of parameter shift within the ten-dimensional algebra.
+
+Write \(a^*\) for the basis vector of \(DC\) dual to a basis vector
+\(a\) of \(C\).  Form the trivial extension
+\begin{equation}
+T=C\ltimes DC,\qquad
+(a,\phi)(b,\psi)=(ab,a\psi+\phi b),\qquad
+(a\phi b)(c)=\phi(bca).
+\label{a:trivial-extension}
+\end{equation}
+By Lemma~\ref{tr:trivial-symmetric}, this is a symmetric algebra of
+dimension twenty, with symmetrizing functional
+\(t_T(a,\phi)=\phi(1)\).  In particular,
+\[
+t_T\bigl((a,\phi)(b,\psi)\bigr)=\phi(b)+\psi(a).
+\]
+We regard \(s\) as a \(T\)-module through the quotient \(T\to C\).
+Set
+\begin{equation}
+B=C\otimes C\subseteq E=T\otimes T.
+\label{a:tensor-algebras}
+\end{equation}
+Then \(\dim_k B=100\) and \(\dim_k E=400\).  The functional
+\(t_E=t_T\otimes t_T\) gives a symmetric nondegenerate product pairing,
+so \(E\) is symmetric.
+
+We record the radical and corner information that will enter the trace
+argument.  One has
+\begin{align}
+\rad T&=(\rad C)\oplus DC,\nonumber\\
+\rad B&=(\rad C)\otimes C+C\otimes(\rad C),\nonumber\\
+\rad E&=(\rad T)\otimes T+T\otimes(\rad T).
+\label{a:radicals}
+\end{align}
+For the first equality, the displayed ideal is a square-zero extension
+of the nilpotent ideal \(\rad C\), and its quotient is \(k^2\).
+For the tensor products the displayed ideals are sums of commuting
+nilpotent ideals, and their quotients are \(k^4\).  This proves all
+three equalities.  The four vertex idempotents of \(B\), also used in
+\(E\), are \(e\otimes e,e\otimes f,f\otimes e,f\otimes f\).
+For these vertices the corner dimensions are
+\begin{equation}
+\bigl(\dim_k(pB p')\bigr)_{p,p'}=
+\begin{pmatrix}
+16&8&8&4\\
+8&8&4&4\\
+8&4&8&4\\
+4&4&4&4
+\end{pmatrix}.
+\label{a:corner-parity}
+\end{equation}
+Indeed these dimensions are the products of the corresponding corner
+dimensions \(4,2,2,2\) of \(C\).  In particular, every entry is even.
+
+For \(i\in\mathbb Z\), put
+\[
+\ell_i=x+q^i y.
+\]
+We denote right multiplication by \(a\) by \(\rho_a\) and left
+multiplication by \(a\) by \(\lambda_a\).  Thus \(\rho_a\) is a map
+of left modules and \(\lambda_a\) is a map of right modules whenever
+the indicated source and target corners allow it.
+
+\begin{proposition}\label{a:dual-dimensions}
+The dual regular module \(DC\) has projective dimension at most two
+on both sides.  More precisely,
+\[
+D(eC)\cong Ce,\qquad D(Ce)\cong eC,
+\]
+and there are exact sequences
+\begin{align}
+0\longrightarrow Cf\xrightarrow{\rho_t}Ce
+ \xrightarrow{\rho_{\ell_{-2}}}Ce
+ \xrightarrow{e\mapsto j^*}D(fC)\longrightarrow0,
+\label{a:left-dual-resolution}\\
+0\longrightarrow fC\xrightarrow{\lambda_u}eC
+ \xrightarrow{\lambda_{\ell_0}}eC
+ \xrightarrow{e\mapsto v^*}D(Cf)\longrightarrow0.
+\label{a:right-dual-resolution}
+\end{align}
+Consequently
+\begin{equation}
+\pd_B({}_B E)\le4,\qquad
+\pd_{B^{\mathrm{op}}}(E_B)\le4,\qquad
+\idim_B({}_B B)\le4.
+\label{a:dimension-bounds}
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+The pairing \(eC\times Ce\to k\) that takes the coefficient of \(z\)
+in a product is nondegenerate.  In the ordered bases
+\[
+\mathcal B_{eC}=(e,x,y,z,u,v),\qquad
+\mathcal B_{Ce}=(e,x,y,z,t,j),
+\]
+its only nonzero entries are the products
+\(ez,xy,yx,ze,uj,vt\), with coefficients \(1,q,1,1,1,q\).
+Thus it has determinant \(q^2\ne0\).  The maps sending \(e\) to
+\(z^*\) give the stated left- and right-module isomorphisms.
+
+Here is the full calculation for the other dual summands.  The two
+surjections, on the ordered bases just specified, have the images
+\begin{align*}
+Ce\longrightarrow D(fC):\quad
+(e,x,y,z,t,j)&\longmapsto
+(j^*,t^*,q^2t^*,0,qn^*,f^*),\\
+eC\longrightarrow D(Cf):\quad
+(e,x,y,z,u,v)&\longmapsto
+(v^*,u^*,u^*,0,n^*,f^*).
+\end{align*}
+Each has rank four; their kernels are respectively
+\(\langle\ell_{-2},z\rangle_k\) and
+\(\langle\ell_0,z\rangle_k\).
+The intermediate maps have images and kernels
+\[
+\begin{array}{c|c|c}
+\text{map}&\text{image}&\text{kernel}\\ \hline
+\rho_{\ell_{-2}}:Ce\to Ce
+ &\langle\ell_{-2},z\rangle_k
+ &\langle y+qx,z,t,j\rangle_k\\
+\lambda_{\ell_0}:eC\to eC
+ &\langle\ell_0,z\rangle_k
+ &\langle y+qx,z,u,v\rangle_k.
+\end{array}
+\]
+For completeness, the initial injections are given by
+\begin{align*}
+\rho_t:(f,u,v,n)&\longmapsto(t,y+qx,qz,qj),\\
+\lambda_u:(f,t,j,n)&\longmapsto(u,y+qx,z,v).
+\end{align*}
+Their four images are independent and span the respective kernels in
+the table.  This proves exactness of
+\eqref{a:left-dual-resolution} and
+\eqref{a:right-dual-resolution}, including their orientations and
+the exceptional index \(-2\) in the first sequence.
+
+On either \(C\)-side, \(T=C\oplus DC\) has a projective resolution of
+length at most two.  Tensor two such resolutions over the field and
+totalize.  The resulting resolutions of the corresponding
+\(B\)-modules \(E\) have length at most four, proving the first two
+bounds in \eqref{a:dimension-bounds}.  Likewise
+\(DB\cong DC\otimes DC\) has right \(B\)-projective dimension at most
+four.  Dualizing a finite projective resolution gives an injective
+coresolution of \({}_B B\), and proves the last bound.
+\end{proof}
+
+We now resolve \(s\). The same multiplication maps will also determine
+the dual complex, including its right-module cohomology.
+
+\begin{proposition}\label{a:resolution}
+The augmentation \(Cf\to s\) extends to a projective resolution
+\(R\to s\) with
+\begin{equation}
+R_0=Cf,\qquad R_i=Ce\ (i\ge1),\qquad
+d_1=\rho_u,\qquad d_{i+2}=\rho_{\ell_i}\ (i\ge0).
+\label{a:simple-resolution}
+\end{equation}
+It satisfies
+\begin{equation}
+H^a\Hom_C(R,s)=
+\begin{cases}k&a=0,\\0&a\ne0,\end{cases}
+\qquad
+H^a\Hom_C(R,C)=
+\begin{cases}Ds&a=2,\\0&a\ne2,\end{cases}
+\label{a:dual-cohomology}
+\end{equation}
+where the second identification is an isomorphism of right
+\(C\)-modules.
+\end{proposition}
+
+\begin{proof}
+On the basis \((e,x,y,z,t,j)\) of \(Ce\), right multiplication gives
+\begin{align*}
+\rho_u(e,x,y,z,t,j)
+  &=(u,v,v,0,(1+q)n,0),\\
+\rho_{\ell_i}(e,x,y,z,t,j)
+  &=(\ell_i,q^{i+1}z,z,0,(1+q^{i+2})j,0).
+\end{align*}
+For every \(i\ge0\), the coefficient \(1+q^{i+2}\) is nonzero.
+Consequently
+\[
+\begin{array}{c|c|c}
+\text{map}&\text{image}&\text{kernel}\\ \hline
+\rho_u
+ &\langle u,v,n\rangle_k
+ &\langle\ell_0,z,j\rangle_k\\
+\rho_{\ell_i},\ i\ge0
+ &\langle\ell_i,z,j\rangle_k
+ &\langle\ell_{i+1},z,j\rangle_k.
+\end{array}
+\]
+The first image is \(\rad(Cf)\), and each later image is the preceding
+kernel.  This proves exactness at every index.  Since
+\(\Hom_C(Ce,s)=es=0\) and \(\Hom_C(Cf,s)=fs=k\), the first equality
+in \eqref{a:dual-cohomology} follows.
+
+Under \(\Hom_C(Cf,C)=fC\) and \(\Hom_C(Ce,C)=eC\), the dual cochain
+complex is
+\begin{equation}
+fC\xrightarrow{\lambda_u}eC
+ \xrightarrow{\lambda_{\ell_0}}eC
+ \xrightarrow{\lambda_{\ell_1}}eC
+ \xrightarrow{\lambda_{\ell_2}}\cdots,
+\label{a:dual-complex}
+\end{equation}
+starting in degree zero.  The first arrow is injective, with image
+\(\langle u,y+qx,z,v\rangle_k\).  For \(i\ge0\),
+\[
+\lambda_{\ell_i}(e,x,y,z,u,v)
+  =(\ell_i,q^iz,qz,0,(1+q^i)v,0).
+\]
+The image and kernel are therefore
+\[
+\begin{array}{c|c|c}
+ &\text{image}&\text{kernel}\\ \hline
+i=0&\langle\ell_0,z\rangle_k
+ &\langle\ell_{-1},z,u,v\rangle_k\\
+i\ge1&\langle\ell_i,z,v\rangle_k
+ &\langle\ell_{i-1},z,v\rangle_k.
+\end{array}
+\]
+Since \(q\ell_{-1}=qx+y\), the first kernel equals the image of
+\(\lambda_u\).  The only remaining cohomology is in degree two:
+\[
+\frac{\langle\ell_0,z,v\rangle_k}
+     {\langle\ell_0,z\rangle_k}=k\overline v.
+\]
+Finally \(vf=v\), \(ve=0\), and the only nonzero product of \(v\)
+with a radical basis vector is \(vt=qz\), which is a boundary.
+Thus the radical acts trivially on \(k\overline v\), and its right
+module structure is exactly \(Ds\).
+\end{proof}
diff --git a/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/sections/bimodule.tex b/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/sections/bimodule.tex
new file mode 100644
index 0000000000000000000000000000000000000000..22d686a1c53d3e765f55e90596bd1c558c3f1686
--- /dev/null
+++ b/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/sections/bimodule.tex
@@ -0,0 +1,696 @@
+\section{Derived induction and the trace obstruction}
+\label{b:section}
+
+We now construct the bimodule that will join the two twists in the
+triangular algebra.  The two-factor filtration follows the organization
+of \cite[Section~6]{OpenAIAR2026}. A related use of tensor products of
+bimodules attached to Hochschild classes occurs in support-variety
+theory~\cite[Definition~2.1 and Theorem~2.3]{ErdmannEtAl2003}.
+Here the bimodule comes from derived
+induction, and its evaluated lifting property is proved by a trace
+obstruction.
+
+\begin{proposition}\label{b:bimodule}
+There is a finite \(E\)-bimodule \(Y\), projective separately as a left and
+as a right \(E\)-module, such that
+\begin{equation}\label{b:W}
+ W^a:=\sHom_E(X,(YX)[a])
+ \cong
+ \begin{cases}
+ k,&a=-3,0,\\
+ 0,&a\notin\{-3,0\}.
+ \end{cases}
+\end{equation}
+For every \(H\in k^\times\), there is an ordinary bimodule map
+\(g_H:U_H\to Y\) whose evaluation, under \(U_HX=X\), represents a nonzero
+element of \(W^0\).  The same \(Y\) works for all \(H\), and all modules and
+maps are defined over \(k\).
+\end{proposition}
+
+The two assertions require different arguments. A filtration of derived
+induction will compute the groups \(W^a\). In particular, \(W^0=k\)
+provides nonzero stable \(E\)-module maps \(X\to YX\), but does not show
+that one is the evaluation of a bimodule map \(U_H\to Y\). The trace
+obstruction, combined with stable duality, will prove that this evaluation
+has nonzero image for every \(H\).
+
+\subsection{A finite representative of derived induction}
+
+Let \(K\) be the two-sided bar complex for derived induction:
+\[
+ K_i=E\otimes B^{\otimes i}\otimes E\quad(i\geq0),\qquad K_i=0\quad(i<0).
+\]
+This is the ordinary-algebra instance of the bar model for derived
+tensor functors; see~\cite[Sections~6.1 and~6.6]{Keller1994}.
+The finite representative needed here will follow from the separate
+projective-dimension bounds on the two sides.
+For \(i\geq1\), its differential is
+\begin{align*}
+ d_i(a\otimes b_1\otimes\cdots\otimes b_i\otimes c)
+ ={}&
+ ab_1\otimes b_2\otimes\cdots\otimes b_i\otimes c\\
+ &+\sum_{r=1}^{i-1}
+ a\otimes b_1\otimes\cdots\otimes b_rb_{r+1}
+       \otimes\cdots\otimes b_i\otimes c\\
+ &+a\otimes b_1\otimes\cdots\otimes b_{i-1}\otimes b_ic .
+\end{align*}
+As throughout, signs disappear in characteristic two.  Associativity
+gives \(d_{i-1}d_i=0\), since each double multiplication occurs twice.
+Multiplication in degree zero defines a chain map \(m:K\to E\).
+
+\begin{lemma}\label{b:bar-model}
+The complex \(K\) represents \(E\otimes_B^{\mathbf L}E\), has finite
+projective \(E^e\)-terms and has homology only in degrees \(0,\ldots,4\).
+The bimodule \(c_5K\) is projective on both sides.  Consequently its
+stable image
+\[
+ \cI=\langle K\rangle=(c_5K)[5]
+\]
+has finite side-projective representatives, as does \(\cI[-1]\).
+One such representative \(Y\) of \(\cI[-1]\) is obtained by taking four
+successive finite bimodule cosyzygies of \(c_5K\).
+\end{lemma}
+
+\begin{proof}
+The usual bar resolution of \(B\) has terms
+\(B\otimes B^{\otimes i}\otimes B\).  Its augmented complex is contracted
+as a complex of vector spaces by inserting a unit.  Inserting a unit at
+one end also proves that tensoring it with a module at the other end
+gives a projective resolution of that module on the remaining side.
+Inducing the two outer actions gives \(K\), with each term a finite free
+\(E^e\)-module.
+
+On the left, \(B\otimes B^{\otimes i}\otimes E\) is a projective
+\(B\)-resolution of \({}_BE\).  By Proposition~\ref{a:dual-dimensions}, this
+module has a projective resolution supported in degrees \(0,\ldots,4\).
+Comparison of projective resolutions gives a homotopy equivalence
+between these two resolutions.  Applying \(E\otimes_B-\) preserves the
+homotopy equivalence, so \(K\), as a left \(E\)-complex, is homotopy
+equivalent to a complex of projectives supported in those degrees.
+The same argument using \(E_B\) works on the right.  In particular
+\(H_i(K)=0\) for \(i>4\).
+
+For completeness, this equivalence proves projectivity of the specific
+cokernel \(c_5K\).  On either side, choose a homotopy from the identity
+of \(K\) to a map factoring through the complex ending in degree \(4\).
+In degree \(5\) it gives
+\[
+ 1_{K_5}=d_6h_5+h_4d_5.
+\]
+Modulo \(\im d_6\), the identity of \(c_5K\) therefore factors as
+\[
+ c_5K\xrightarrow{\ \overline d_5\ }K_4
+       \xrightarrow{\ q_5h_4\ }c_5K,
+\]
+where \(q_5:K_5\to c_5K\) is the quotient map.  Thus \(c_5K\) is a
+summand of a projective on each side.  This assertion concerns the two
+side-module structures; the two splittings need not be bimodule
+splittings.  The high exact sequences
+\(0\to c_{n+1}K\to K_n\to c_nK\to0\), for \(n\geq5\), now split
+on each side.  Induction shows that all higher cokernels are
+side-projective as well.
+
+To make the finite choice explicit, put \(N_0=c_5K\), and for
+\(r=0,1,2,3\) use
+\[
+ 0\longrightarrow N_r\longrightarrow
+ \Hom_k(E^e,N_r)\longrightarrow N_{r+1}\longrightarrow0,\qquad
+ n\longmapsto(a\longmapsto an).
+\]
+Here the \(E^e\)-action on the middle term is by right multiplication
+on its input.  The middle term is finite injective, hence projective
+because \(E^e\) is symmetric.  Since \(N_r\) is projective and therefore
+injective on each \(E\)-side, this sequence splits on both sides; it
+follows inductively that every \(N_r\) is side-projective.
+Take \(Y=N_4\).  Its stable class is
+\((c_5K)[4]=\cI[-1]\), as required.  Lemmas~\ref{h:stable-image}
+and~\ref{h:evaluation} give the stated stable images and their
+evaluation compatibilities.  We denote the stable map induced by
+\(m\) by \(\overline m:\cI\to E\).
+\end{proof}
+
+\subsection{The stable Hom profile in every degree}
+
+\begin{lemma}\label{b:profile}
+The bimodule \(Y\) of Lemma~\ref{b:bar-model} satisfies
+\eqref{b:W}.
+\end{lemma}
+
+\begin{proof}
+The evaluated complex is
+\[
+ K\otimes_E X=E\otimes B^{\otimes\bullet}\otimes X.
+\]
+Before inducing to \(E\), its bar model and \(R\otimes R\) are
+projective \(B\)-resolutions of \(X=s\otimes s\).  They are homotopy
+equivalent.  Induction preserves this equivalence and gives
+\begin{equation}\label{b:evaluated-model}
+ K\otimes_E X\simeq
+ (T\otimes_C R)\otimes(T\otimes_C R)
+ =\mathcal J\otimes\mathcal J .
+\end{equation}
+In particular, Lemma~\ref{h:evaluation} identifies its stable image
+with \(\cI X\).
+
+We retain the actual attaching maps in the triangle
+\[
+ s[2]\longrightarrow\mathcal J\longrightarrow s
+       \xrightarrow{\tau}s[3]
+\]
+of \eqref{e:triangle}.  Choose bounded-below projective
+models \(A_\bullet\) for \(s[2]\) and \(Q_\bullet\) for \(s\), and a
+chain map \(t:Q_\bullet\to A_\bullet[1]\) representing \(\tau\).
+The cone of \(t[-1]\) has underlying graded complex
+\(A_\bullet\oplus Q_\bullet\), differential
+\[
+ \begin{pmatrix}d_A&t\\0&d_Q\end{pmatrix},
+\]
+and represents \(\mathcal J\), by Lemma~\ref{h:derived-models}.
+Indeed the cone construction for the rotated triangle gives this
+identification; comparisons of projective
+models lift the connecting map and give homotopy equivalences of the
+cones.  Thus this is a homotopy equivalent projective model, with its
+specified subcomplex and quotient, rather than only an equality of
+homology groups.
+
+Tensor two copies of this model and shift by \([-4]\).
+Filter first by \(A_\bullet\otimes A_\bullet\), then add
+\(Q_\bullet\otimes A_\bullet\) and \(A_\bullet\otimes Q_\bullet\),
+and finally add \(Q_\bullet\otimes Q_\bullet\).
+Every inclusion is split in each graded degree, and the successive
+quotients resolve
+\begin{equation}\label{b:three-levels}
+ X,\qquad X[-2]\oplus X[-2],\qquad X[-4].
+\end{equation}
+The stable image of the full complex is
+\(\mathcal L=(\cI X)[-4]\).
+With the middle terms in the displayed order, the adjacent
+connecting maps are the shifted row and column
+\[
+  (\tau_1,\tau_2),\qquad
+  \binom{\tau_2}{\tau_1}.
+\]
+These are the cross differentials \(t\) in the relevant factor and
+the identity in the other factor.  Lemma~\ref{h:stable-image}
+identifies them with the connecting maps after taking stable images.
+
+We first compute the kernels and cokernels of these adjacent maps in
+every degree. On the polynomial groups the maps form the two-variable
+Koszul sequence; on the negative groups they form its dual. We then use the
+two long exact sequences of the filtration to control the full top
+attaching map, including its possible component into the bottom level.
+
+Write
+\(\mathcal P_m=k[\tau_1,\tau_2]_m\) for polynomials of total exponent
+\(m\), and put \(\mathcal P_m=0\) for \(m<0\).
+By Proposition~\ref{e:all-degrees},
+\[
+ \cH^{3m}=\mathcal P_m,\qquad
+ \cH^{-3m-1}=D\mathcal P_m\quad(m\geq0),
+\]
+and all other \(\cH^a\) vanish.  Multiplication by a degree-three
+class on the negative groups is transpose to polynomial
+multiplication.  For each integer \(l\), denote the adjacent maps by
+\[
+ \begin{aligned}
+ d_l:(\cH^{l-2})^2&\longrightarrow\cH^{l+1},&
+ d_l(u,v)&=\tau_1u+\tau_2v,\\
+ d'_l:\cH^{l-4}&\longrightarrow(\cH^{l-1})^2,&
+ d'_l(w)&=(\tau_2w,\tau_1w).
+ \end{aligned}
+\]
+Products in these formulas mean shifted postcomposition.  We claim
+\begin{align}
+ \ker d_l&=\im d'_{l-1},\label{b:koszul-middle}\\
+ \coker d_{l-1}&=
+ \begin{cases}k,&l=0,\\0,&l\ne0,\end{cases}
+ \qquad
+ \ker d'_l=
+ \begin{cases}k,&l=3,\\0,&l\ne3.\end{cases}
+ \label{b:koszul-ends}
+\end{align}
+
+To check the middle identity, the polynomial calculation for \(m\geq0\)
+is the exact sequence
+\begin{equation}\label{b:polynomial-koszul}
+ 0\longrightarrow\mathcal P_{m-1}
+ \xrightarrow{\,(\tau_2,\tau_1)^{\mathsf t}\,}
+ \mathcal P_m^2
+ \xrightarrow{\,(\tau_1,\tau_2)\,}
+ \mathcal P_{m+1}\longrightarrow0 .
+\end{equation}
+The row is onto since every positive-degree monomial is divisible
+by a variable.  If \(\tau_1u+\tau_2v=0\), coprimality implies
+\(u=\tau_2w\) and \(v=\tau_1w\), uniquely; the signs agree in
+characteristic two.  When the middle group is
+\((D\mathcal P_m)^2\), use the dual of
+\eqref{b:polynomial-koszul}, interchanging the two middle slots:
+\[
+ D\mathcal P_{m+1}\longrightarrow
+ (D\mathcal P_m)^2\longrightarrow D\mathcal P_{m-1}.
+\]
+At \(m=0\), the last term is the actual group \(\cH^2=0\), and the
+first map \(D\mathcal P_1\to(D\mathcal P_0)^2\) is an isomorphism.
+This is the seam \(l=1\).  If the middle group is unsupported, both
+its kernel and the image into it are zero.  This proves
+\eqref{b:koszul-middle} for every \(l\).
+
+For the row cokernel, a target \(\cH^l=\mathcal P_m\) gives
+\(\mathcal P_{m-1}^2\to\mathcal P_m\), which is onto for \(m\geq1\)
+and has cokernel \(k\) for \(m=0\).  A negative target
+\(\cH^l=D\mathcal P_m\) gives the dual of the injective column
+\(\mathcal P_m\to\mathcal P_{m+1}^2\), and is onto for every
+\(m\geq0\).  All other targets are zero.
+For the column kernel, a nonnegative source \(\mathcal P_m\) maps
+injectively into \(\mathcal P_{m+1}^2\).
+A source \(D\mathcal P_m\) for \(m\geq1\) is the dual of the
+surjective row \(\mathcal P_{m-1}^2\to\mathcal P_m\), so again the
+column is injective.  The remaining source is
+\(\cH^{-1}=D\mathcal P_0=k\), at \(l=3\), with zero target
+\((\cH^2)^2\).  These cases prove \eqref{b:koszul-ends}.
+They include the only possible multiplication from a supported
+negative degree across zero, namely \(\cH^{-1}\to\cH^2=0\).
+
+It remains to pass from adjacent maps to the whole filtered
+object.  Let \(\mathcal L_1\) be the stable image of the lower two
+levels of \eqref{b:three-levels}, and put
+\(N^l=\sHom_E(X,\mathcal L_1[l])\).
+The first two levels give
+\begin{equation}\label{b:first-two-levels}
+ 0\longrightarrow\coker d_{l-1}\longrightarrow N^l
+ \xrightarrow{q_l}\ker d_l\longrightarrow0.
+\end{equation}
+Let
+\[
+ r_l:\cH^{l-4}\longrightarrow N^{l+1}
+\]
+be the \emph{actual} connecting map obtained by attaching the top
+level.  Naturality of the quotient to the middle level gives
+\(q_{l+1}r_l=d'_l\).
+Thus \(\ker r_l\subseteq\ker d'_l\), which is zero unless \(l=3\).
+At that index \(N^4=0\): its left term in
+\eqref{b:first-two-levels} is \(\coker d_3=0\), and its right term
+is a kernel in \((\cH^2)^2=0\).  Hence
+\(\ker r_3=\cH^{-1}=k\).
+
+Moreover,
+\[
+ q_l(\im r_{l-1})=\im d'_{l-1}=\ker d_l.
+\]
+For \(l\ne0\), \(q_l\) is an isomorphism, so \(r_{l-1}\) is onto.
+At \(l=0\), the vanishing \(\cH^{-2}=0\) gives \(N^0=k\), while
+the whole incoming source of \(r_{-1}\) is \(\cH^{-5}=0\).
+Consequently \(\coker r_{l-1}\) is \(k\) at \(l=0\) and zero
+elsewhere.
+
+The final long exact sequence gives
+\[
+ 0\longrightarrow\coker r_{l-1}
+ \longrightarrow\sHom_E(X,\mathcal L[l])
+ \longrightarrow\ker r_l\longrightarrow0.
+\]
+The two scalar groups therefore survive precisely at \(l=0,3\).
+This argument permits an additional component of the top attaching
+map into the bottom level: \(N^4=0\) and \(\cH^{-5}=0\) already
+force the asserted kernel and cokernel for the full \(r_l\).
+Finally \(YX=(\cI X)[-1]=\mathcal L[3]\), so \(l=a+3\).
+This gives exactly \eqref{b:W}.
+\end{proof}
+
+\subsection{From perfect factorizations to an ordinary trace}
+
+The required Hom profile is now established. It remains to construct
+bimodule maps from the twists whose evaluations generate its degree-zero
+line. We first show that a particular character map cannot factor
+through a perfect complex; stable duality will then turn this
+obstruction into the required maps.
+
+Let \(\chi:E\to k\) be the character of \(X\), write
+\(\chi_B=\chi|_B\), and retain \(p_f=f\otimes f\).
+The \(B\)-bimodule structure on \(DB\) is
+\((a\lambda b)(c)=\lambda(bca)\).
+
+We use the bimodule \(DB\otimes B\), with its outer actions, to connect
+perfect factorizations with ordinary traces. Using the bound
+\(\idim_B B\leq4\), left coinduction will replace a perfect complex by
+a bounded complex with terms in \(\add(DB\otimes B)\). For an ordinary
+composite \(v:B\to DB\) through \(DB\otimes B\), the identification
+\(DB\otimes B\simeq\Hom_k(B,B)\) will express the scalar \(v(1)(p_f)\)
+as the trace of a linear operator.
+
+\begin{lemma}\label{b:ordinary-factorization}
+If a bimodule map \(v:B\to DB\) factors in the derived category
+through a perfect \(B^e\)-complex, then it factors by ordinary
+bimodule maps through an object of \(\add(DB\otimes B)\), where the
+tensor product has its outer bimodule actions.
+\end{lemma}
+
+\begin{proof}
+Represent the perfect intermediate object by a bounded complex
+\(P\) of finite projective bimodules.  We first replace it by a
+bounded complex \(P'\) whose terms belong to \(\add(DB\otimes B)\).
+
+For a finite bimodule \(L\), define
+\[
+ J_B(L)=\Hom_k(B,L),\qquad
+ (a\phi b)(c)=\phi(ca)b,\qquad
+ j_L(z)(c)=cz.
+\]
+The map \(j_L:L\to J_B(L)\) is a natural bimodule monomorphism.
+Set \(C^0(L)=L\) and
+\(C^{r+1}(L)=\coker j_{C^r(L)}\).  These are additive functors
+and preserve split retracts.  For a fixed \(d=4\) we have the
+functorial exact sequence
+\begin{equation}\label{b:coresolution}
+ 0\longrightarrow L\longrightarrow J_B(C^0(L))
+ \longrightarrow\cdots\longrightarrow J_B(C^{d-1}(L))
+ \longrightarrow C^d(L)\longrightarrow0.
+\end{equation}
+
+For the free bimodule \(L=B\otimes B\), this construction is the
+left-module coinduced coresolution of \(B\), tensored over \(k\)
+with the right regular module \(B\).  This follows from
+\(\Hom_k(B,B\otimes B)=\Hom_k(B,B)\otimes B\) and from exactness
+of tensoring vector spaces.  Each coinduced left-module term is
+a finite injective, hence belongs to \(\add(DB)\).
+The last left-module cokernel is also injective: by
+Proposition~\ref{a:dual-dimensions}, \(\idim({}_BB)\leq4\), and
+dimension shifting gives, for every finite left module \(V\),
+\[
+ \Ext_B^1(V,C^d(B))
+ \cong\Ext_B^{d+1}(V,B)=0.
+\]
+Here \(C^d(B)\) denotes the same coresolution construction on the
+left regular module.  Thus every term on the right of \(L\) in
+\eqref{b:coresolution}, including the last cokernel, lies in
+\(\add(DB\otimes B)\) when \(L=B\otimes B\).
+The assertion extends to every finite projective bimodule
+\(L\), since the functors in \eqref{b:coresolution} preserve finite
+sums and split retracts.
+
+Apply this one fixed finite coresolution to each term of \(P\).
+Naturality provides the horizontal maps, giving a bounded double
+complex.  Put the first coinduced term in homological degree \(0\)
+relative to \(L\), and the last cokernel in degree \(-d\).
+Its totalization \(P'\) is bounded and has all terms in
+\(\add(DB\otimes B)\).  The augmentation \(P\to P'\) is a
+quasi-isomorphism: filtering its cone by the finitely many original
+degrees has exact associated complexes
+\eqref{b:coresolution}.  Thus the given derived factorization can
+be written \(B\to P'\to DB\).
+
+For \(V\in\add(DB\otimes B)\) and \(i>0\), we next prove
+\begin{equation}\label{b:two-orthogonalities}
+ \Ext_{B^e}^i(B,V)=0,\qquad
+ \Ext_{B^e}^i(V,DB)=0.
+\end{equation}
+It suffices to take \(V=DB\otimes B\).
+Coinduction identifies, for a bimodule \(Q\),
+\[
+ \Hom_{B^e}(Q,DB\otimes B)
+ \cong\Hom_{B^{\mathrm{op}}}(\Res Q,B).
+\]
+A projective bimodule resolution of \(B\) restricts to a
+right-projective resolution of the free right module \(B\).
+This proves the first vanishing.
+For the second, tensor a left-projective resolution \(Q_\bullet\)
+of \(DB\) over \(k\) with the right regular module \(B\).
+It is a projective bimodule resolution of \(DB\otimes B\), and
+\[
+ \Hom_{B^e}(Q_\bullet\otimes B,DB)
+ \cong\Hom_B(Q_\bullet,DB).
+\]
+Its positive cohomology vanishes since \({}_BDB\) is injective.
+Both assertions pass to finite sums and summands.
+
+Finally choose a projective resolution \(Q_B\to B\) and an
+injective coresolution \(DB\to I_{DB}\), regarded as homological
+complexes.  There are natural quasi-isomorphisms of Hom complexes
+\begin{equation}\label{b:Hom-comparisons}
+ \begin{split}
+ \Hom^\bullet_{B^e}(B,P')&\longrightarrow
+       \Hom^\bullet_{B^e}(Q_B,P'),\\
+ \Hom^\bullet_{B^e}(P',DB)&\longrightarrow
+       \Hom^\bullet_{B^e}(P',I_{DB}).
+ \end{split}
+\end{equation}
+To verify them, filter by the finitely many terms of \(P'\).
+For each successive quotient the claim is one of
+\eqref{b:two-orthogonalities}, together with equality in degree
+zero and vanishing of negative Ext between modules.  A shift of
+a term merely shifts this comparison.  Hence terms in negative
+degrees cause no exception, and induction on this finite
+filtration proves \eqref{b:Hom-comparisons}.
+
+Taking degree-zero cohomology shows that both derived arrows
+have actual chain-map representatives
+\(\alpha:B[0]\to P'\) and \(\beta:P'\to DB[0]\).
+Their composite is the ordinary map \(\beta_0\alpha_0\), so it
+factors through \(P'_0\in\add(DB\otimes B)\).
+It equals \(v\) as an ordinary bimodule map, because
+\(\Hom_{\mathrm D(B^e)}(B,DB)=\Hom_{B^e}(B,DB)\).
+\end{proof}
+
+The scalar trace calculation below is related to the trace and
+corner-dimension calculation in
+\cite[Example~3.2(1) and proof of Theorem~3.8]{LiuZhouZimmermann2012}.
+Here the preceding lemma supplies the reduction from perfect
+factorizations, and the calculation uses the split corner table
+\eqref{a:corner-parity}.
+
+\begin{lemma}\label{b:trace-obstruction}
+The bimodule map
+\begin{equation}\label{b:character-map}
+ \eta:B\longrightarrow DB,\qquad
+ \eta(a)=\chi_B(a)\chi_B,
+\end{equation}
+does not factor in the derived category through a perfect
+\(B^e\)-complex.
+\end{lemma}
+
+\begin{proof}
+The character identities make \(\eta\) a bimodule map.
+By Lemma~\ref{b:ordinary-factorization}, a perfect factorization
+would give an ordinary factorization through
+\(\add(DB\otimes B)\).  We show that every such factorization
+\(B\to DB\) sends \(1\) to a functional vanishing on \(p_f\).
+
+First consider maps
+\[
+ B\xrightarrow{\alpha}DB\otimes B\xrightarrow{\beta}DB.
+\]
+Identify \(DB\otimes B\) with \(\Hom_k(B,B)\) by
+\(\lambda\otimes c\mapsto[x\mapsto\lambda(x)c]\).
+The element \(\alpha(1)\) satisfies
+\(a\alpha(1)=\alpha(1)a\).
+Under this identification, its corresponding linear map \(f_0\)
+therefore satisfies
+\[
+ f_0(xa)=f_0(x)a\quad(x,a\in B).
+\]
+Thus \(f_0(x)=d_0x\), where \(d_0=f_0(1)\).
+If \((c_i)\) is any vector-space basis of \(B\), with dual basis
+\((c_i^*)\), then
+\[
+ \alpha(1)=\sum_i c_i^*\otimes d_0c_i.
+\]
+Right linearity of \(\beta\) reduces it to the left-module
+endomorphism \(\lambda\mapsto\beta(\lambda\otimes1)\) of \(DB\).
+Duality identifies these endomorphisms with endomorphisms of the
+right regular module \(B\), which are left multiplications.
+Thus the endomorphism of \(DB\) is right multiplication by some
+\(b'\in B\).  Consequently
+\[
+ \beta(\lambda\otimes c)=\lambda b'c,\qquad
+ (\beta\alpha(1))(p_f)
+   =\sum_i c_i^*(b'd_0c_ip_f)
+   =\tr_k\bigl(L_{b'd_0}R_{p_f}\bigr).
+\]
+Here \(L_a\) and \(R_a\) are left and right multiplication on
+the vector space \(B\).  The order \(b'd_0\) follows from the
+dual action \((\lambda b'c)(p_f)=\lambda(b'cp_f)\).
+
+Write
+\(b'd_0=r+\sum_v a_vp_v\), where \(r\in\rad B\) and the
+\(p_v\) are its four vertex idempotents.
+Left and right multiplications commute, and \(p_f^2=p_f\), so
+\[
+ (L_rR_{p_f})^n=L_{r^n}R_{p_f}=0
+\]
+for sufficiently large \(n\).
+The radical part has trace zero.
+For each \(v\), the operator \(L_{p_v}R_{p_f}\) is a projection
+onto \(p_vBp_f\).  The corner dimensions of \(B\), obtained from
+the corner table \eqref{a:corner-parity}, are even; in particular
+\(\dim_k(p_vBp_f)=4\) for every \(v\).
+Each projection therefore has trace zero in characteristic two.
+Linearity gives \((\beta\alpha(1))(p_f)=0\).
+
+A factorization through a finite sum of \(DB\otimes B\) is a sum
+of such composites.  A factorization through a summand extends
+to that finite sum by its split inclusion and projection.
+The same vanishing thus holds throughout \(\add(DB\otimes B)\).
+But \(\eta(1)(p_f)=\chi_B(p_f)=1\), a contradiction.
+\end{proof}
+
+\subsection{Lifts with nonzero evaluation}
+
+We now apply the trace obstruction to obtain the maps required by
+Proposition~\ref{b:bimodule}.
+
+Put \(S=\Hom_k(X,X)\), with bimodule action
+\[
+ (atb)(x)=a\,t(bx).
+\]
+The representation gives a bimodule map \(\pi:E\to S\).
+Since \(X\) is one-dimensional, \(S=k\) with both actions given
+by \(\chi\).
+
+\begin{lemma}\label{b:stable-adjunction}
+For every finite side-projective bimodule \(L\), ordinary
+tensor--Hom adjunction induces a natural isomorphism
+\[
+ \sHom_{E^e}(L,S)\ \cong\ \sHom_E(LX,X).
+\]
+This assertion does not require \(S\) to be side-projective.
+\end{lemma}
+
+\begin{proof}
+The ordinary adjoint of \(f:L\to S\) is
+\(\widehat f(l\otimes x)=f(l)(x)\).
+A factorization of \(f\) through a finite \(E^e\)-projective
+evaluates to a factorization through an \(E\)-projective.
+
+Conversely embed \(L\) in a finite projective-injective
+\(E^e\)-module \(J\).
+The embedding splits as a right \(E\)-module map because
+\(L_E\) is projective, hence injective over symmetric \(E\).
+Thus \(LX\to JX\) is injective.
+Suppose \(\widehat f\) factors as \(LX\to P\to X\) with \(P\)
+finite \(E\)-projective.  Injectivity of \(P\) extends its first
+factor to \(JX\to P\).  Composing with \(P\to X\) and applying
+ordinary adjunction gives \(J\to S\) whose restriction to \(L\)
+is \(f\).  This is a bimodule projective factorization.
+The two factorization ideals therefore correspond exactly.
+Naturality is inherited from ordinary adjunction.
+\end{proof}
+
+\begin{lemma}\label{b:evaluated-lift}
+For every \(H\in k^\times\), there is a stable bimodule map
+\(U_H[1]\to\cI\) with nonzero evaluated stable map
+\(X[1]\to\cI X\).
+\end{lemma}
+
+\begin{proof}
+Write \(U=U_H\) and
+\(\phi=\pi\overline m:\cI\to S\).
+All shifts of \(U\) and \(\cI\) have finite side-projective
+representatives by Lemma~\ref{h:evaluation}.
+Consider the finite-dimensional linear map
+\[
+ \phi_*:\sHom_{E^e}(U[1],\cI)
+ \longrightarrow\sHom_{E^e}(U[1],S).
+\]
+Symmetric duality, Lemma~\ref{h:duality} applied over \(E^e\),
+identifies its transpose with
+\begin{equation}\label{b:transpose}
+ \sHom_{E^e}(S,U)\longrightarrow\sHom_{E^e}(\cI,U),
+ \qquad b\longmapsto b\phi.
+\end{equation}
+Indeed the fixed first argument is \(U[1]\), and its duality
+shift is \(U[1][-1]=U\); naturality in the second argument gives
+the displayed precomposition.
+
+Let \(\zeta=f^*\otimes f^*\in E\), as in
+Lemma~\ref{e:socle}.  Both its left and right actions
+are the character \(\chi\).  The map
+\[
+ b_0:S\longrightarrow U,\qquad b_0(1)=\zeta
+\]
+is a bimodule map even for the twisted right action, because
+\[
+ \zeta\cdot b=\zeta\sigma_H(b)
+       =\chi(\sigma_H(b))\zeta=\chi(b)\zeta.
+\]
+We will prove that \(b_0\phi\ne0\) stably.
+
+Suppose otherwise.  The derived map \(b_0\pi m:K\to U\)
+then has zero stable image and factors through a perfect
+\(E^e\)-complex by Lemma~\ref{h:perfect-factor}.
+To apply that lemma with projective models, one may take a
+bounded-below projective resolution \(Q\to U\) with finite terms and lift the
+global chain map \(K\to U\) to \(K\to Q\); its stable image is
+the same map \(b_0\phi\).
+
+Restriction of this factorization to \(B^e\) still has a perfect
+intermediate object.  Indeed an \(E^e\)-free module restricts to
+the outer bimodule \(E\otimes E\).  Tensoring the left and
+right \(B\)-projective resolutions of \(E\) from
+Proposition~\ref{a:dual-dimensions} gives a \(B^e\)-projective
+resolution of length at most \(8\).  Finite sums and summands
+have the same bound.  A bounded complex of these restricted
+modules is perfect by its finite term filtration.
+
+Let \(\operatorname{Bar}(B)\to B\) denote the bar resolution.
+Inclusion of both outer \(B\)-factors in \(E\) defines a
+\(B^e\)-chain map
+\[
+ \iota:\operatorname{Bar}(B)\longrightarrow\Res_{B^e}K,
+\]
+and hence a derived map \(B\to\Res_{B^e}K\).
+Since \(\sigma_H\) fixes \(B\), the decomposition
+\[
+ \Res_{B^e}U
+   =B\oplus(DC\otimes C)\oplus(C\otimes DC)\oplus DB
+\]
+is a decomposition of \(B\)-bimodules.  Let \(\rho\) be projection
+to the last summand.  The composite
+\[
+ \operatorname{Bar}(B)\xrightarrow{\iota}\Res K
+ \xrightarrow{\,b_0\pi m\,}\Res U\xrightarrow{\rho}DB
+\]
+is exactly the character map \(\eta\) of
+\eqref{b:character-map} composed with the bar augmentation:
+in degree zero it sends \(a\otimes c\) to
+\(\chi_B(ac)\chi_B\), since \(\rho(\zeta)=\chi_B\), and in
+higher degrees it is zero.
+Thus our supposed factorization would make \(\eta\) factor
+through a perfect \(B^e\)-complex.
+Lemma~\ref{b:trace-obstruction} excludes this possibility.
+We have proved \(b_0\phi\ne0\).
+
+It follows from \eqref{b:transpose} that \(\phi_*\) is nonzero.
+Choose \(\beta:U[1]\to\cI\) with \(\phi\beta\ne0\).
+Lemma~\ref{b:stable-adjunction}, applied only to the
+side-projective source \(U[1]\), identifies this nonzero class
+with the composite
+\[
+ U[1]X\xrightarrow{\beta X}\cI X
+       \xrightarrow{\overline m X}EX\cong X .
+\]
+This identity follows from ordinary adjunction and from the
+evaluation of the representation \(\pi\).
+The evaluated maps involving \(U[1],\cI,E\) are defined stably
+by Lemma~\ref{h:evaluation}; no stable evaluation functor on
+the nonside-projective module \(S\) is needed.
+The composite is nonzero, so \(\beta X\) itself is nonzero.
+The canonical shift comparison identifies \(U[1]X\) with
+\((UX)[1]=X[1]\), proving the assertion.
+\end{proof}
+
+\begin{proof}[Proof of Proposition~\ref{b:bimodule}]
+Take the fixed finite side-projective \(Y\) constructed in
+Lemma~\ref{b:bar-model}.
+Lemma~\ref{b:profile} proves its required Hom profile.
+For any \(H\in k^\times\), shift the map of
+Lemma~\ref{b:evaluated-lift} by \([-1]\) and identify
+\(\cI[-1]\) with \(Y\) to obtain a stable map \(U_H\to Y\).
+By the shift and evaluation compatibilities of
+Lemma~\ref{h:evaluation}, its evaluation is a nonzero class
+\(X\to YX\).
+A stable map between the fixed finite modules is represented
+by an ordinary bimodule map; choose one and call it \(g_H\).
+All Hom spaces, dualities, and selections used above are over
+the original field \(k\), so this requires no scalar extension.
+The construction of \(Y\) did not depend on \(H\).
+\end{proof}
diff --git a/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/sections/consequences.tex b/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/sections/consequences.tex
new file mode 100644
index 0000000000000000000000000000000000000000..e53a2a697b2ee612a74e5ff9bc3a6ecb377c45ce
--- /dev/null
+++ b/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/sections/consequences.tex
@@ -0,0 +1,305 @@
+\section{The endomorphism algebra and homological consequences}
+\label{sec:consequences}
+
+The symmetric pair constructed in Theorem~\ref{thm:main} has a
+self-orthogonal generator $A\oplus M$. The classical relation between
+such generators and dominant dimension turns the module counterexample
+into an algebra whose entire regular injective resolution is projective.
+We first isolate this step, then extract the modules in
+Corollaries~\ref{cor:homological} and~\ref{cor:finitistic}.
+
+\begin{proposition}\label{prop:endomorphism}
+Let $B$ be a finite-dimensional symmetric algebra over a field and let
+$N$ be a finite-dimensional nonprojective left $B$-module satisfying
+$\Ext_B^i(N,N)=0$ for all $i>0$. Put
+\[
+ G=B\oplus N,\qquad \Gamma=\End_B(G)^{\mathrm{op}},
+\]
+where endomorphisms compose in the ordinary order. Then
+\[
+ \domdim({}_\Gamma\Gamma)=\infty,
+ \qquad \idim_\Gamma\Gamma=\infty.
+\]
+In particular, $\Gamma$ is not self-injective.
+\end{proposition}
+
+\begin{proof}
+The module $G$ is a generator-cogenerator: it contains the regular
+module, and symmetry identifies $B$ with its dual, an injective
+cogenerator. Moreover
+\[
+ \Ext_B^i(G,G)=0\qquad(i>0).
+\]
+The $N,N$ summand vanishes by hypothesis. The summands with first
+argument $B$ vanish by projectivity, and the remaining summand with
+second argument $B$ vanishes by injectivity.
+
+Mueller's criterion~\cite[Lemma~3]{Mueller1968} says that, for a
+generator-cogenerator $G$, vanishing of these groups for
+$1\leq i\leq n$ is equivalent to dominant dimension at least $n+2$
+for its endomorphism algebra. His Theorem~4 identifies the left and
+right dominant dimensions, so the opposite convention in the
+definition of $\Gamma$ makes no difference to this assertion.
+Applying the criterion for every $n$ proves
+$\domdim({}_\Gamma\Gamma)=\infty$.
+
+We verify non-self-injectivity directly. The functor
+\[
+ H=\Hom_B(G,-):\add G\longrightarrow \Gamma\text{-}\mathrm{proj}
+\]
+is fully faithful and sends $G$ to the left regular module $\Gamma$.
+Explicitly, if $e\in\End_B(G)$ and $h:G\to V$, then
+$e^{\mathrm{op}}h=h\circ e$; full faithfulness follows first on
+finite sums of $G$ and then on their summands.
+Suppose that the left regular module $\Gamma$ were injective. Its direct
+summand $H(N)$ would be injective as well. Choose a finite injective envelope
+$N\hookrightarrow J$ over $B$. Since $B$ is symmetric,
+$J\in\add B\subseteq\add G$. Left exactness of $H$ gives a
+monomorphism $H(N)\hookrightarrow H(J)$, which splits by injectivity
+of $H(N)$. Full faithfulness gives a retraction $J\to N$ of the
+original inclusion. Thus $N$ is injective, hence projective over the
+symmetric algebra $B$, a contradiction.
+
+Finally, every term in the minimal injective resolution of $\Gamma$
+is projective by infinite dominant dimension. If its injective
+dimension were finite, this resolution would end in a finite exact
+sequence
+\[
+ 0\longrightarrow\Gamma\longrightarrow I^0\longrightarrow\cdots
+ \longrightarrow I^d\longrightarrow0
+\]
+of projective-injective terms. For $d>0$, the last surjection splits
+because $I^d$ is projective. Its kernel is then projective, and
+repeating backwards shows that the first inclusion splits. The case
+$d=0$ is immediate. In either case $\Gamma$ is a summand of $I^0$
+and is injective, contrary to the preceding paragraph. Hence
+$\idim_\Gamma\Gamma=\infty$.
+\end{proof}
+
+\begin{proof}[Proof of Corollary~\ref{cor:homological}]
+Fix a field extension $K/k$. The symmetric structure survives
+extension of scalars: since $A$ is finite-dimensional, the natural
+bimodule identification
+\[
+ K\otimes_k DA\simeq D_K(K\otimes_k A),
+ \qquad D_K=\Hom_K(-,K),
+\]
+turns $A\simeq DA$ into a symmetrizing isomorphism for $A_K$.
+
+We also record why the module hypotheses survive. For finite
+$A$-modules $U,V$, resolve $U$ by finite projective modules.
+Finite projectivity gives, in every degree,
+\[
+ K\otimes_k\Hom_A(P_i,V)
+ \simeq
+ \Hom_{A_K}(K\otimes_kP_i,K\otimes_kV).
+\]
+Flatness commutes with kernels and images, so
+\begin{equation}\label{c:basechange}
+ K\otimes_k\Ext_A^i(U,V)
+ \simeq \Ext_{A_K}^i(K\otimes_kU,K\otimes_kV)
+ \qquad(i\geq0).
+\end{equation}
+Thus $M_K$ is self-orthogonal. Choose a finite projective surjection
+$P\twoheadrightarrow M$ with kernel $L$. Its extension class in
+$\Ext_A^1(M,L)$ is nonzero, since $M$ is not projective.
+It remains nonzero under \eqref{c:basechange}, because extension of
+fields is faithful. The extended surjection therefore does not split,
+and $M_K$ is nonprojective. The degree-zero case of
+\eqref{c:basechange} is compatible with composition and also identifies
+\[
+ \Gamma_K\simeq K\otimes_k\End_A(A\oplus M)^{\mathrm{op}}.
+\]
+These are the same finite Ext and nonprojectivity base-change
+arguments used in the companion~\cite[Proposition~9.2]{OpenAIAR2026}.
+
+Apply Proposition~\ref{prop:endomorphism} to $A_K,M_K$, and abbreviate
+$\Gamma_K$ to $\Gamma$. Write
+\[
+ 0\longrightarrow\Gamma\longrightarrow I^0
+ \longrightarrow I^1\longrightarrow\cdots
+\]
+for its minimal injective resolution. Every $I^j$ is
+projective-injective and $\idim_\Gamma\Gamma=\infty$.
+This proves the first assertion and refutes the classical Nakayama
+conjecture. It also refutes the Auslander--Gorenstein conjecture:
+$\pd_\Gamma I^j=0\leq j$ for every $j\geq0$, whereas the left
+self-injective dimension is already infinite.
+
+There is a nonprojective indecomposable injective left $\Gamma$-module
+$J$. Indeed, finite-dimensional algebras have the same finite number
+of isomorphism classes of indecomposable projectives and indecomposable
+injectives, both indexed by the simple modules. If every indecomposable
+injective were projective, these classes would exhaust the indecomposable
+projectives, making the regular module injective. This contradicts
+Proposition~\ref{prop:endomorphism}. Since every $I^j$ is projective,
+none contains $J$ as a direct summand, giving the omitted-injective
+failure of the generalized Nakayama conjecture.
+
+Let $S=\soc J$. The socle of an indecomposable injective over a
+finite-dimensional algebra is simple, and $J$ is its injective
+envelope. If there were a nonzero map $S\to I^j$, it would be
+injective. Injectivity of $I^j$ would extend it to a map $J\to I^j$.
+The extension is injective because $S$ is essential in $J$, and it
+splits because $J$ is injective. This would make $J$ a summand of
+$I^j$, a contradiction. Hence $\Hom_\Gamma(S,I^j)=0$ for every
+$j\geq0$. Applying $\Hom_\Gamma(S,-)$ to the resolution gives
+\[
+ \Ext_\Gamma^i(S,\Gamma)=0\qquad(i\geq0).
+\]
+The nonzero module $S$ therefore refutes the strong Nakayama conjecture
+and also gives the simple-Ext formulation of generalized Nakayama
+failure. In particular, the degree-zero vanishing is part of the
+conclusion, not only the positive-degree vanishings.
+
+Finally let $W$ be the direct sum of one representative from each
+isomorphism class of indecomposable projective-injective left
+$\Gamma$-modules. There are finitely many such classes, so $W$ is
+finite, projective, and injective. In particular
+$\Ext_\Gamma^i(W,W)=0$ for $i>0$, and every $I^j$ belongs to
+$\add W$. The displayed injective resolution is therefore an
+infinite $\add W$-coresolution of $\Gamma$. Applying
+$\Hom_\Gamma(-,W)$ preserves its exactness because $W$ is injective.
+Thus $W$ is Wakamatsu tilting and has projective dimension zero.
+
+If $W$ were tilting, there would be a finite exact coresolution
+\[
+ 0\longrightarrow\Gamma\longrightarrow W^0\longrightarrow\cdots
+ \longrightarrow W^r\longrightarrow0,\qquad W^j\in\add W.
+\]
+Every $W^j$ is projective-injective. The backward splitting argument
+in Proposition~\ref{prop:endomorphism} would make $\Gamma$ injective,
+a contradiction. Hence $W$ is not tilting, which refutes the Wakamatsu
+tilting conjecture.
+\end{proof}
+
+The proof works separately over each extension $K/k$. It does not
+require a minimal injective resolution to remain minimal under scalar
+extension, or an indecomposable injective or simple module to remain
+indecomposable or simple. The construction of $\Gamma_K$ is compatible
+with scalar extension, and the required $J,S,W$ are chosen over $K$.
+
+The successive cokernels of the same injective resolution give the
+finitistic-dimension consequence.
+
+\begin{proof}[Proof of Corollary~\ref{cor:finitistic}]
+Fix a field extension $K/k$ and abbreviate $\Gamma_K$ to $\Gamma$.
+Use the minimal injective resolution of the left regular module over
+this field,
+\[
+ 0\longrightarrow {}_\Gamma\Gamma\longrightarrow I^0
+ \longrightarrow I^1\longrightarrow\cdots.
+\]
+Every $I^n$ is finite-dimensional and projective-injective by
+Corollary~\ref{cor:homological}. Set $C_0={}_\Gamma\Gamma$ and
+$C_n=\im(I^{n-1}\to I^n)$ for $n\geq1$. Exactness gives
+\[
+ 0\longrightarrow C_n\longrightarrow I^n\longrightarrow C_{n+1}
+ \longrightarrow0\qquad(n\geq0).
+\]
+Thus the $C_{n+1}$ are successive cokernels, and all $C_n$ are finite.
+
+Since $C_0$ and $I^0$ are projective, the first sequence gives
+$\pd_\Gamma C_1\leq1$. If $C_1$ were projective, its surjection from
+$I^0$ would split, making $\Gamma$ a direct summand of the injective
+module $I^0$. This would make $\Gamma$ injective, contrary to
+$\idim_\Gamma\Gamma=\infty$. Hence $\pd_\Gamma C_1=1$.
+
+Suppose $n\geq1$ and $\pd_\Gamma C_n=n$. Splicing a length-$n$
+projective resolution of $C_n$ with the displayed short exact sequence
+gives $\pd_\Gamma C_{n+1}\leq n+1$. For every left module $V$ and
+$j\geq1$, projectivity of $I^n$ and the long exact Ext sequence give
+\[
+ \Ext_\Gamma^j(C_n,V)\simeq\Ext_\Gamma^{j+1}(C_{n+1},V).
+\]
+There is a left module $V$ with $\Ext_\Gamma^n(C_n,V)\neq0$, since
+otherwise the Ext criterion for projective dimension would give
+$\pd_\Gamma C_n\leq n-1$. Thus
+$\Ext_\Gamma^{n+1}(C_{n+1},V)\neq0$, proving the reverse bound and
+$\pd_\Gamma C_{n+1}=n+1$. Induction gives the asserted exact dimensions.
+The finite left modules $C_n$ therefore make the little left finitistic
+dimension infinite. The big left finitistic dimension is at least the
+little one, so it is infinite as well.
+\end{proof}
+
+The remaining applications use the established failures of the homological
+conjectures together with results on tilting modules and stratifying ideals.
+
+A finite left module $P$ over a finite-dimensional algebra $\Lambda$
+is \emph{almost complete tilting} if it is a direct summand of a tilting
+module and has one fewer isomorphism class of indecomposable summands
+than there are isomorphism classes of simple left $\Lambda$-modules.
+Here tilting has the ordinary finite-projective-dimension meaning
+recalled in Section~\ref{sec:introduction}. An \emph{indecomposable
+complement} is a finite indecomposable module $Y$ such that $P\oplus Y$
+is tilting.
+
+\begin{corollary}[Infinitely many tilting complements]\label{cor:tilting-complements}
+For every field extension $K/k$, let $\Gamma_K$ be the algebra in
+Corollary~\ref{cor:homological}. There is a finite-dimensional projective
+almost complete tilting left $\Gamma_K^{\mathrm{op}}$-module $P_K$ with
+infinitely many pairwise nonisomorphic indecomposable complements.
+\end{corollary}
+
+\begin{proof}
+Fix $K$ and abbreviate $\Gamma_K$ to $\Gamma$. By
+Corollary~\ref{cor:homological}, $\Gamma$ fails the generalized Nakayama
+conjecture. Buan--Solberg~\cite[Theorems~3.1 and~3.4]{BuanSolberg1998}
+therefore give an injective almost complete cotilting left $\Gamma$-module
+$U$ with infinitely many pairwise nonisomorphic indecomposable complements
+$X$. The exact duality $D_K=\Hom_K(-,K)$ sends finite left
+$\Gamma$-modules to finite left $\Gamma^{\mathrm{op}}$-modules.
+It sends finite injective dimension to finite projective dimension,
+preserves self-orthogonality, and dualizes the finite cotilting resolution
+of the injective cogenerator into a finite tilting coresolution of the
+regular module. Thus $P_K=D_KU$ is projective and almost complete tilting,
+and $D_KX$ are its complements. Duality preserves indecomposability and
+distinct isomorphism classes, so there are infinitely many such
+complements. The witnesses are chosen separately over each extension
+$K/k$.
+\end{proof}
+
+The final application concerns decompositions of derived categories.
+For an idempotent $e$ in a finite-dimensional algebra $R$, the ideal
+$ReR$ is \emph{stratifying} if multiplication induces an isomorphism
+\[
+ Re\otimes_{eRe}eR\xrightarrow{\sim}ReR
+\]
+and $\Tor_i^{eRe}(Re,eR)=0$ for every $i>0$.
+It then induces the standard recollement relating the unbounded
+derived categories of $R/ReR$, $R$, and $eRe$; see
+\cite[Introduction]{ChenFangXi2024}.
+An \emph{idempotent stratification of length $n$}, in the convention of
+\cite[Definition~4.7]{ChenFangXi2024}, consists of nonzero mutually
+orthogonal idempotents $e_0,\ldots,e_n$ with sum $1$ such that, on putting
+$f_i=e_0+\cdots+e_i$, the ideal generated by $f_i$ in
+$f_{i+1}Rf_{i+1}$ is stratifying and $e_{i+1}\notin Rf_iR$ for
+$0\leq i<n$. Thus its length counts the successive nontrivial
+recollements obtained by passing through these corner algebras.
+
+\begin{corollary}[Idempotent stratifications]\label{cor:stratifying}
+For every field extension $K/k$ and every integer $L\geq1$, there is an
+auxiliary finite-dimensional indecomposable symmetric $K$-algebra $T_L$
+admitting an idempotent stratification of length at least $L$ in the sense
+of Chen, Fang and Xi. In particular, $T_L$ has a nonzero proper
+stratifying ideal, and $\mathcal D(T_L\text{-}\mathrm{Mod})$ admits a
+chain of at least $L$ nontrivial standard recollements through successive
+corner algebras. Here $\mathrm{Mod}$ denotes all left modules and
+$\mathcal D$ is the unbounded derived category.
+\end{corollary}
+
+\begin{proof}
+The base-change argument above gives a symmetric $K$-algebra $A_K$ and a
+nonprojective self-orthogonal module $M_K$. Choose a nonprojective
+indecomposable summand $N$ in one block $B$ of $A_K$. The block $B$ is
+indecomposable and symmetric, and $\Ext_B^i(N,N)=0$ for every $i>0$,
+since these groups are direct summands of $\Ext_{A_K}^i(M_K,M_K)$.
+
+Chen, Fang and Xi~\cite[proof of Theorem~5.13]{ChenFangXi2024} construct
+from such a pair, over any field, reduced mirror-reflective algebras $T_n$
+that are indecomposable and symmetric. Their Corollary~5.12(1), with
+Definition~4.7, gives an idempotent stratification of length at least
+$n$. Theorem~1.2(2) supplies the
+stated successive nontrivial standard recollements of unbounded derived
+categories. Taking $n=L$ proves the claim.
+\end{proof}
diff --git a/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/sections/homological.tex b/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/sections/homological.tex
new file mode 100644
index 0000000000000000000000000000000000000000..86b7bfafaf7bfc1cddca4b06248118540d545ff8
--- /dev/null
+++ b/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/sections/homological.tex
@@ -0,0 +1,466 @@
+\section{Homological preliminaries}\label{sec:homological}
+
+The tools in this section serve two passages in the construction.
+Complete-resolution comparison will turn the triangular cone's
+endomorphism calculation into the vanishing needed for transfer to a
+symmetric algebra. Stable duality will supply the negative-degree maps
+used in that calculation. Separately, projective models and their stable
+images will turn the derived-induction object into a finite bimodule,
+preserving its maps and connecting morphisms under evaluation.
+
+Throughout the construction, $k$ has characteristic two. Unless explicitly
+stated otherwise, algebras and modules are finite dimensional over $k$.
+Resolution and model complexes have finite-dimensional terms; the full Hom
+complexes defined below may have infinite-dimensional product terms.
+Modules are left modules unless a side is specified. Write
+$D=\Hom_k(-,k)$ and $G^e=G\otimes_kG^{\mathrm{op}}$; an unsubscripted tensor
+product is over $k$. Differentials lower homological degree, and
+\[
+ P[a]_j=P_{j-a},\qquad c_jP=\coker(d^P_{j+1}:P_{j+1}\longrightarrow P_j).
+\]
+Characteristic two removes the signs in shifts, tensor differentials and cones.
+For two complexes we use the full Hom complex
+\begin{equation}\label{h:full-hom}
+ \Hom_G^a(P,Q)=\prod_{j\in\mathbb Z}\Hom_G(P_j,Q_{j-a}),\qquad
+ (\delta f)_j=d^Q_{j-a}f_j+f_{j-1}d^P_j.
+\end{equation}
+For a module $N$, viewed in degree zero, $\Hom_G(P,N)^a=\Hom_G(P_a,N)$.
+The space $\sHom_G(M,N)$ is the quotient of $\Hom_G(M,N)$ by maps factoring
+through finite projectives.
+
+\subsection{Complete resolutions and stable duality}
+
+An exact complex $P$ of projectives is \emph{totally acyclic} if
+$\Hom_G(P,G)$ is exact. If $c_0P=M$, we call $P$ a complete resolution of $M$.
+The comparison below also applies when $G$ is not self-injective.
+Complete resolutions and passage from derived objects to stable modules
+belong to the framework developed by Buchweitz~\cite{Buchweitz2021}.
+We give the comparison and truncation arguments in our conventions.
+
+\begin{lemma}[Complete-resolution comparison]\label{h:comparison}
+Let $P$ be totally acyclic and let $Q$ be an exact complex of projectives.
+For every integer $a$ there are natural isomorphisms
+\begin{equation}\label{h:comparison-formula}
+ H^a\Hom_G^\bullet(P,Q)
+ \simeq\sHom_G(c_aP,c_0Q)
+ \simeq H^a\Hom_G(P,c_0Q).
+\end{equation}
+The first isomorphism is induced by cokernels of shifted chain maps and
+respects composition. For $a>0$ these spaces are
+$\Ext_G^a(c_0P,c_0Q)$.
+\end{lemma}
+
+\begin{proof}
+We first record the extension property supplied by total acyclicity. If $J$
+is projective, then $\Hom_G(P,J)$ is exact, since $J$ is a summand of a finite
+sum of copies of $G$. Identify $c_jP$ with $\im d_j^P\subset P_{j-1}$.
+For $u:c_jP\to J$, the map $u d_j^P:P_j\to J$ is a cocycle.
+Exactness gives $h:P_{j-1}\to J$ with $h d_j^P=u d_j^P$, so $h$ extends $u$.
+It follows at once that the cycles of $\Hom_G(P,N)^a$ are maps
+$c_aP\to N$ and its boundaries are exactly the maps factoring through
+projectives. This proves the second isomorphism, with $N=c_0Q$.
+
+For the first, begin in degree zero. A map $c_0P\to c_0Q$ lifts to
+$f_0:P_0\to Q_0$, and then successively to $f_j$ for $j>0$, using
+projectivity of $P_j$ and exactness of $Q$. To proceed downwards,
+$d_j^Qf_j$ vanishes on $\ker d_j^P=\im d_{j+1}^P$ by the chain equation
+already obtained in degree $j+1$. It therefore defines a map
+$c_jP=\im d_j^P\to Q_{j-1}$. The extension property gives $f_{j-1}$.
+Thus every cokernel map lifts to a full chain map.
+
+A homotopy induces a projective factorization on $c_0$. Conversely, suppose
+the cokernel map of $f$ factors as $c_0P\to J\to c_0Q$ with $J$ projective.
+Extend its first factor to $P_{-1}$ and lift its second factor to $Q_0$.
+Their composite is a homotopy component $h_{-1}:P_{-1}\to Q_0$ whose
+homotopy differential removes the induced map on cokernels. It remains to
+nullhomotope a chain map inducing zero there. For this remaining map set
+$h_{-1}=0$. For $j\geq0$, lift
+\[
+ f_j+h_{j-1}d_j^P:P_j\longrightarrow\im d_{j+1}^Q
+\]
+to $h_j:P_j\to Q_{j+1}$; the initial image assertion is the vanishing on
+$c_0$, and each later one follows from the preceding chain equation.
+For $j<0$, the map $f_j+d_{j+1}^Qh_j$ kills $\im d_{j+1}^P$ by the equation
+in degree $j+1$. It descends to $\im d_j^P$ and extends to
+$h_{j-1}:P_{j-1}\to Q_j$ by the same extension property. These equations
+give $f=dh+hd$ in every degree. The product in \eqref{h:full-hom} permits all
+these components; each equation uses only two of them.
+
+Replace $P$ by $P[-a]$ to obtain degree $a$. The construction by cokernels
+preserves compositions and shifts. Finally, the nonnegative tail of $P$
+resolves $c_0P$. In every positive cohomological degree its Hom complex has
+the same two adjacent differentials as $\Hom_G(P,c_0Q)$, proving the Ext
+assertion, including degree one.
+\end{proof}
+
+Recall that $G$ is \emph{symmetric} if there is a bimodule isomorphism
+$G\simeq DG$, or equivalently a linear form $t:G\to k$ such that
+$t(ab)=t(ba)$ and $(a,b)\mapsto t(ab)$ is nondegenerate. In this case finite
+projectives are injective, and every finite module embeds in a finite
+projective: dualize a finite projective surjection onto its dual.
+Splicing a projective resolution and a projective-injective coresolution
+therefore gives a complete resolution of every finite module; applying
+$\Hom_G(-,G)$ preserves its exactness because $G$ is injective.
+
+For such a resolution put $M[a]=c_{-a}P$ in the stable category. The exact
+sequences
+\begin{equation}\label{h:syzygy-sequence}
+ 0\longrightarrow c_{j+1}P\longrightarrow P_j
+ \longrightarrow c_jP\longrightarrow0
+\end{equation}
+show that $[1]$ is inverse syzygy. Lemma~\ref{h:comparison} makes these
+shifts independent of choices, functorial on stable maps, and compatible
+with composition. In particular,
+\begin{equation}\label{h:stable-ext}
+ H^a\Hom_G(P,N)=\sHom_G(M,N[a])\qquad(a\in\mathbb Z).
+\end{equation}
+For positive degrees the shifted composition is the Yoneda product, since
+both are computed by lifting maps to projective resolutions.
+
+A short exact sequence $0\to L\to V\to N\to0$ gives a long exact sequence
+of the groups in \eqref{h:stable-ext}: apply $\Hom_G(P,-)$ termwise.
+Its connecting map is composition with $\partial:N\to L[1]$. Concretely,
+lift a projective surjection $J\to N$ to $J\to V$ and restrict to its kernel.
+The resulting map $\Omega N\to L$, shifted once, is $\partial$.
+This agrees with the cochain connecting map, which lifts a cocycle and
+then applies the differential. We use this description to keep track of
+connecting maps, as well as the groups in the long exact sequences.
+
+The next lemma is the symmetric form of Auslander--Reiten duality,
+also called Tate duality; see
+\cite[Section~2, (2.3) and (2.7)--(2.8)]{Linckelmann2013}.
+Its composition pairing will determine the action on negative-degree
+self-maps.
+
+\begin{lemma}[Symmetric stable duality]\label{h:duality}
+For a symmetric algebra $G$ there are natural isomorphisms
+\begin{equation}\label{h:duality-formula}
+ D\sHom_G(M,N)\simeq\sHom_G(N,M[-1]).
+\end{equation}
+There is a linear functional $\lambda_M:\sHom_G(M,M[-1])\to k$ for which
+the perfect pairing is $(f,g)\mapsto\lambda_M(gf)$. Consequently, for every
+integer $a$ the pairing
+\begin{equation}\label{h:shifted-pairing}
+ \sHom_G(M,N[a])\times\sHom_G(N,M[-a-1])\longrightarrow k,
+ \qquad (\alpha,\beta)\longmapsto\lambda_M\bigl(\beta[a]\alpha\bigr)
+\end{equation}
+is perfect and natural.
+\end{lemma}
+
+\begin{proof}
+For every finite projective $J$ the finite-dimensional adjunction gives
+\[
+ D\Hom_G(J,N)\simeq\Hom_G(N,DG\otimes_GJ)\simeq\Hom_G(N,J).
+\]
+The first identification is immediate for $J=G$, hence for finite sums
+and summands; the second uses a fixed symmetrizing form. Apply this
+termwise to a complete resolution $P$ of $M$. Dualizing degree-zero
+cohomology of $\Hom_G(P,N)$ yields
+\[
+ \frac{\Hom_G\bigl(N,\ker(P_0\to P_{-1})\bigr)}
+ {\im\bigl(\Hom_G(N,P_1)\to
+                \Hom_G(N,\ker(P_0\to P_{-1}))\bigr)}.
+\]
+Here $\ker(P_0\to P_{-1})=c_1P=M[-1]$. A factorization through any
+projective lifts through $P_1\twoheadrightarrow c_1P$, so the displayed
+quotient is precisely $\sHom_G(N,M[-1])$. All identifications are natural;
+comparison of complete resolutions also gives naturality in $M$.
+
+Define $\lambda_M(g)$ by pairing $1_M$ with $g$. Naturality in $N$, applied
+to $f:M\to N$, identifies the pairing of $f$ and $g$ with
+$\lambda_M(gf)$. Replacing $N$ by $N[a]$ and using the shift equivalences
+gives \eqref{h:shifted-pairing}.
+\end{proof}
+
+\subsection{Projective models and their stable images}
+
+In Section~\ref{b:section}, derived induction first supplies a bounded
+derived bimodule. We will obtain the finite bimodule $Y$ from high
+cokernels of a projective model. To compute its evaluated stable Hom
+groups, we must also retain the actual maps attaching the model's
+successive pieces. The next two lemmas establish this passage from
+derived objects and connecting morphisms to finite stable modules.
+
+We write $\mathrm D^b(G)$ for the derived category of bounded complexes of
+finite modules. A complex is \emph{perfect} if it is derived isomorphic to
+a bounded complex of finite projectives. We will represent bounded derived
+objects by bounded-below complexes of finite projectives with bounded
+homology. The following elementary facts specify the models and maps we use.
+For a chain map $f:L\to N$, our cone convention is
+\begin{equation}\label{h:cone}
+ \Cone(f)_j=N_j\oplus L_{j-1},\qquad
+ d_j^{\Cone(f)}=
+ \begin{pmatrix}d_j^N&f_{j-1}\\0&d_{j-1}^L\end{pmatrix}.
+\end{equation}
+
+The projective-model statement is the bounded case of the standard
+$K$-projective formalism; see
+\cite[Proposition~1.4 and Example~3.2(a)]{Spaltenstein1988}, with the
+opposite indexing convention for complexes.
+
+\begin{lemma}[Derived maps and two-cell models]\label{h:derived-models}
+A bounded-below complex $P$ of projectives computes derived maps out of it
+by chain maps modulo homotopy. Every bounded complex of finite modules has
+such a projective model with finite terms. In particular, two such models
+of the same derived object are homotopy equivalent.
+
+Suppose a derived triangle is given by
+\[
+ L\longrightarrow V\longrightarrow R\xrightarrow{\partial}L[1].
+\]
+For projective models $P_L,P_R$ and a chain map
+$b:P_R\to P_L[1]$ representing $\partial$, a model of $V$ is the two-cell
+complex
+\begin{equation}\label{h:two-cell}
+ P_{V,j}=P_{L,j}\oplus P_{R,j},\qquad
+ d_j^{P_V}=\begin{pmatrix}d_j^{P_L}&b_j\\0&d_j^{P_R}\end{pmatrix}.
+\end{equation}
+Its subcomplex and quotient represent $L$ and $R$, and its connecting
+map is the specified $\partial$.
+\end{lemma}
+
+\begin{proof}
+Every chain map from $P$ to an acyclic complex is nullhomotopic: start at
+the lowest nonzero degree of $P$ and successively lift the residual map
+through the next differential of the target. The same argument applies
+to every shift. Thus the full Hom from $P$ sends quasi-isomorphisms to
+quasi-isomorphisms, which proves the assertion about derived maps. This is
+the $K$-projectivity of a bounded-below projective source; it does not assert
+$K$-projectivity of a complete resolution.
+
+For completeness, a bounded-below finite complex $V$ admits a projective
+model $\theta:Q\to V$ by the following induction. Having constructed lower
+terms, choose $Q_j$ finite projective surjecting onto
+\[
+ \{(x,y)\in V_j\oplus Q_{j-1}:d_Vx=\theta(y),\ d_Qy=0\}.
+\]
+Use its two components for $\theta_j$ and $d_j^Q$. Every cycle of the
+comparison cone is then a boundary, so $\theta$ is a quasi-isomorphism.
+A derived isomorphism between projective models and its inverse are
+represented by chain maps, and their composites are homotopic to the
+identities by the first assertion.
+
+To verify the last assertion, represent the first arrow of the triangle
+by $f:P_L\to P_V$ on any projective models. Its cone represents $R$, and
+the projection $\Cone(f)\to P_L[1]$ represents $\partial$.
+The cone of the deshifted projection has $P_V$ as a subcomplex with
+contractible quotient $\Cone(1_{P_L})$. It therefore represents $V$.
+Replacing the projective models by homotopy equivalent ones and the map
+by its prescribed representative preserves this conclusion: a homotopy
+commuting square induces a cone map with that homotopy as its off-diagonal
+component, and homotopic maps have isomorphic cones. The resulting cone
+$\Cone(b[-1])$ is exactly \eqref{h:two-cell}.
+\end{proof}
+
+A bounded-below complex with bounded homology represents a bounded derived
+object by good truncation. Also, a bounded complex whose terms have finite
+projective dimension is perfect: its finite filtration by terms expresses
+it as successive cones of shifts of modules with bounded projective
+resolutions. These observations allow us to use the preceding models
+without introducing infinite-dimensional terms.
+
+\begin{lemma}[Stable images]\label{h:stable-image}
+Let $G$ be symmetric and let $P$ be bounded below with projective terms and
+bounded homology. For $n$ sufficiently large the formula
+\begin{equation}\label{h:stable-image-formula}
+ \langle P\rangle=(c_nP)[n]
+\end{equation}
+defines a stable module, independent of $n$ up to natural stable
+isomorphism. It is functorial on derived maps represented by such models,
+commutes with shifts, sends a module to its stable class, and sends an Ext
+class to its corresponding stable map.
+
+A degreewise split short exact sequence $0\to P_L\to P_V\to P_R\to0$
+of these complexes gives a stable connecting map
+$\langle P_R\rangle\to\langle P_L\rangle[1]$ and the associated long exact
+stable Hom sequences. With respect to a graded splitting, this connecting
+map is the stable image of the cross differential
+$b:P_R\to P_L[1]$ in \eqref{h:two-cell}.
+\end{lemma}
+
+\begin{proof}
+Above the homology support, \eqref{h:syzygy-sequence} is exact for $P$.
+It identifies $(c_{n+1}P)[n+1]$ with $(c_nP)[n]$, naturally on chain maps.
+A homotopy changes $c_n(f)$ by a map factoring through $P_{n-1}$, so the
+construction descends to homotopy classes. Lemma~\ref{h:derived-models}
+then makes it functorial on derived maps. The identity
+$c_n(P[a])=c_{n-a}P$ gives shift compatibility. A projective resolution of
+a module recovers its successive syzygies, proving the assertion for
+modules. Restricting a lifted representative of an Ext class to high
+degrees gives the same successive syzygies of that map, proving the last
+assertion about classes.
+
+Choose $n$ so that all three complexes are exact in degrees at least $n$.
+Their cokernels give a short exact sequence
+\[
+ 0\longrightarrow c_nP_L\longrightarrow c_nP_V
+ \longrightarrow c_nP_R\longrightarrow0.
+\]
+For injectivity on the left, suppose
+$(x,0)=d_{n+1}^{P_V}(u,v)$. Then $d_{n+1}^{P_R}v=0$, so
+$v=d_{n+2}^{P_R}w$ for some $w$. The equation $d_Lb+bd_R=0$ gives
+$x=d_{n+1}^{P_L}(u+b_{n+2}w)$, as required. Surjectivity and exactness in
+the middle follow by lifting components in the graded splitting.
+
+There is a commutative diagram with exact rows
+\[
+\begin{tikzcd}[column sep=large]
+0\arrow[r]&c_{n+1}P_R\arrow[r]\arrow[d,"\overline b_{n+1}"']
+ &P_{R,n}\arrow[r]\arrow[d]
+ &c_nP_R\arrow[r]\arrow[d,equal]&0\\
+0\arrow[r]&c_nP_L\arrow[r]&c_nP_V\arrow[r]&c_nP_R\arrow[r]&0.
+\end{tikzcd}
+\]
+The middle vertical map sends $z$ to the class of $(0,z)$. On an element
+$d_{n+1}^{P_R}z$, its value is the class of $(b_{n+1}z,0)$, since
+$(b_{n+1}z,d_{n+1}^{P_R}z)$ is a boundary in $P_V$. Thus the left vertical
+map is induced by $b_{n+1}:P_{R,n+1}\to P_{L,n}$.
+Shifting the left vertical arrow by $[n+1]$ gives
+$\langle P_R\rangle\to\langle P_L\rangle[1]$. By the description after
+\eqref{h:stable-ext}, this is the connecting map of the bottom row shifted
+by $[n]$, and it is exactly $\langle b\rangle$. This also gives the
+asserted long exact sequences.
+\end{proof}
+
+The following criterion supplies the other use of stable images in
+Section~\ref{b:section}. There, an ordinary trace will rule out a perfect
+factorization of a derived map. The criterion then shows that its stable
+image is nonzero.
+
+\begin{lemma}[Vanishing stable image and perfect factorization]
+\label{h:perfect-factor}
+Over a symmetric algebra, a derived map between bounded derived objects
+whose stable image is zero factors in the derived category through a
+perfect complex.
+\end{lemma}
+
+\begin{proof}
+Choose bounded-below projective models $P,Q$ and represent the map by a
+global chain map $f:P\to Q$, using Lemma~\ref{h:derived-models}. For a
+sufficiently high $n$, both complexes are exact in degrees at least $n$
+and $c_n(f)$ factors through a finite projective $J$:
+\[
+ c_nP\xrightarrow{u}J\xrightarrow{v}c_nQ.
+\]
+Since $J$ is injective, extend $u$ across
+$c_nP=\im d_n^P\subset P_{n-1}$. Since $J$ is projective, lift $v$ to $Q_n$.
+The composite gives $h_{n-1}:P_{n-1}\to Q_n$. It makes
+$f_n+h_{n-1}d_n^P$ zero after projection to $c_nQ$, so this residual map
+lifts through $d_{n+1}^Q$ to $h_n:P_n\to Q_{n+1}$. Inductively, for $j>n$
+the chain equation and the preceding homotopy equation show that
+$f_j+h_{j-1}d_j^P$ lands in $\ker d_j^Q=\im d_{j+1}^Q$. Lift it to $h_j$.
+Set $h_j=0$ for $j<n-1$.
+
+The homotopic map $f'=f+dh+hd$ now vanishes in every degree $j\geq n$.
+Its chain equation in degree $n$ reads
+\begin{equation}\label{h:truncation-boundary}
+ f'_{n-1}d_n^P=0.
+\end{equation}
+Consequently its remaining components define a chain map into the actual
+homological subcomplex $Q_{\leq n-1}$, obtained by retaining $Q_j$ for
+$j\leq n-1$ and setting the higher terms to zero. Equation
+\eqref{h:truncation-boundary} verifies the boundary condition at the cut.
+Thus $f'$ factors through $Q_{\leq n-1}$, which is bounded and projective
+because $Q$ was bounded below. Composing with the chosen model maps gives
+the desired derived factorization. This also covers a map with a module
+as target: first lift it globally to a projective resolution of that
+module, and then apply this argument.
+\end{proof}
+
+\subsection{Bimodule evaluation}
+
+For a symmetric algebra $E$, the tensor product of symmetrizing forms makes
+$E^e$ symmetric. We call an $E$-bimodule \emph{side-projective} if it is
+projective as a left and as a right $E$-module, and write $LX=L\otimes_EX$.
+Side-projectivity does not assert projectivity over $E^e$.
+For the derived-induction construction, the passage to a finite stable
+bimodule must commute with evaluation at $X$, including the attaching
+maps just discussed. The next lemma establishes these compatibilities.
+
+\begin{lemma}[Finite shifts and evaluation]\label{h:evaluation}
+Every stable shift of a finite side-projective bimodule has a finite
+side-projective representative. Evaluation at a finite $E$-module $X$
+sends stable maps between these bimodules to stable $E$-module maps and
+has natural identifications
+\begin{equation}\label{h:evaluation-shifts}
+ (L[a])X\simeq(LX)[a],\qquad L(X[a])\simeq(LX)[a].
+\end{equation}
+It preserves the stable connecting maps of short exact sequences of
+side-projective bimodules, and these identifications are natural in both
+$L$ and $X$.
+
+If $P$ is a bounded-below complex of projective bimodules with bounded
+homology and all sufficiently high $c_nP$ are side-projective, then
+$P\otimes_EX$ has bounded homology and
+\begin{equation}\label{h:evaluation-image}
+ \langle P\rangle X\simeq\langle P\otimes_EX\rangle,
+\end{equation}
+naturally on maps and on connecting maps from degreewise split sequences
+of such complexes. The hypothesis on the high cokernels holds whenever
+$P$, considered separately on either side, is homotopy equivalent to a
+bounded complex of projectives on that side.
+\end{lemma}
+
+\begin{proof}
+Take a projective bimodule surjection onto $L$. Its kernel is
+side-projective because the sequence splits on each side. For a positive
+shift, use the finite projective-injective embedding
+\[
+ L\longrightarrow\Hom_k(E^e,L),\qquad
+ z\longmapsto(a\longmapsto az),
+ \qquad (r\varphi)(a)=\varphi(ar).
+\]
+Its target is injective by coinduction and projective by symmetry of
+$E^e$. The embedding splits on each side because $L$ is injective on
+that side, so its cokernel is again side-projective. Iteration supplies
+finite representatives for every shift.
+
+An $E^e$-projective evaluates to an $E$-projective: this is immediate for
+the free bimodule $E\otimes E$, which evaluates to $E\otimes X$, and
+passes to finite sums and summands. Hence evaluation respects projective
+factorizations. Every short exact sequence of side-projective bimodules
+splits on the right, and therefore remains exact after evaluation. In
+particular, evaluating the sequences just used to define shifts gives
+the first identification in \eqref{h:evaluation-shifts}. Applying the
+projective-presentation description of connecting maps to these same
+sequences shows that their actual connecting maps are preserved.
+
+For fixed $L$, right projectivity makes $L\otimes_E-$ exact, and left
+projectivity makes it preserve projectives. Applying it to projective
+resolutions and projective-injective coresolutions gives the second
+identification in \eqref{h:evaluation-shifts}. A bimodule map $L\to L'$
+gives a natural transformation on these complexes; thus evaluating it
+on a shifted module agrees stably with shifting its evaluation. The
+same reasoning proves naturality in $X$ and compatibility with
+composition.
+
+For the complex assertion, the high exact sequences
+$0\to c_{n+1}P\to P_n\to c_nP\to0$ split on the right. Their evaluations
+remain exact, proving bounded homology of $P\otimes_EX$; its terms are
+projective by the preceding argument. Right exactness gives the
+canonical identifications
+\[
+ c_n(P\otimes_EX)=(c_nP)\otimes_EX.
+\]
+Together with \eqref{h:evaluation-shifts}, these identify the two sides
+of \eqref{h:evaluation-image}. They commute with chain maps. For a
+filtered complex, the cross differential in
+Lemma~\ref{h:stable-image} evaluates to its tensor with $X$. That lemma's
+explicit diagram therefore identifies the evaluated connecting map with
+the connecting map after evaluation.
+
+Finally, suppose a homotopy equivalence on one side compares $P$ with a
+complex supported in degrees at most $d$. At any $n>d$, the homotopy
+between the identity and the composite through that complex gives
+\[
+ 1_{P_n}=d_{n+1}h_n+h_{n-1}d_n.
+\]
+Writing $\pi_n:P_n\to c_nP$, passage to the cokernel yields
+\[
+ 1_{c_nP}=(\pi_nh_{n-1})\overline d_n,
+ \qquad \overline d_n:c_nP\longrightarrow P_{n-1}.
+\]
+Thus $c_nP$ is a summand of a projective on that side. The separate
+homotopy equivalence on the other side proves the other projectivity.
+The homotopies need not be bimodule homotopies.
+\end{proof}
diff --git a/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/sections/introduction.tex b/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/sections/introduction.tex
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+++ b/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/sections/introduction.tex
@@ -0,0 +1,357 @@
+\section{Introduction}\label{sec:introduction}
+
+Let $A$ be a finite-dimensional algebra over a field $k$.
+A finite-dimensional $A$-module $M$ is \emph{self-orthogonal} if
+$\Ext_A^i(M,M)=0$ for every integer $i>0$.
+Tachikawa's second conjecture asserts that every finite-dimensional
+self-orthogonal module over a finite-dimensional self-injective
+algebra is projective.  This question belongs to the homological
+conjectures surrounding quasi-Frobenius algebras and dominant
+dimension; see Tachikawa's 1973 monograph~\cite[pp.~VI--VII]{Tachikawa1973} and the account
+of the conjecture in~\cite[p.~7288]{ChenXi2025TAMS}.
+The Auslander--Reiten conjecture, originating in
+\cite{AuslanderReiten1975}, asks for finite-dimensional algebras
+whether the condition $\Ext_A^i(M,M\oplus A)=0$ for every $i>0$
+forces $M$ to be projective. For a self-injective algebra, the groups
+$\Ext_A^i(M,A)$ vanish automatically, so Tachikawa's second conjecture
+is its self-injective case.
+
+Write $D=\Hom_k(-,k)$. An algebra $A$ is \emph{symmetric} if
+$A\simeq DA$ as $A$-bimodules; in particular, a symmetric algebra
+is self-injective. We prove the following result in this class.
+
+\begin{theorem}\label{thm:main}
+Let
+\[
+ k=\mathbb F_2(q,H_1,H_2),
+\]
+where $q,H_1,H_2$ are algebraically independent.
+There exist a finite-dimensional associative unital symmetric
+$k$-algebra $A$ and a finite-dimensional unital left $A$-module $M$
+such that $M$ is nonprojective and
+\[
+ \Ext_A^i(M,M)=0\qquad\text{for every integer }i>0.
+\]
+In particular, Tachikawa's second conjecture is false.
+\end{theorem}
+
+Since $A$ is self-injective, the same module satisfies
+$\Ext_A^i(M,M\oplus A)=0$ for every $i>0$.
+Thus Theorem~\ref{thm:main} also gives a symmetric counterexample to
+the Auslander--Reiten conjecture.
+
+\paragraph{Related results.}
+Schulz's earlier example of a nonprojective module without positive
+self-extensions uses an algebra over a skew field, outside the
+finite-dimensional-over-a-field setting considered here; see
+\cite{Schulz1994} and the distinction recalled in
+\cite[Introduction]{DiverisPurin2012}.
+His quantum-exterior example has nonzero first self-extension but no
+higher self-extensions
+\cite[Example~7]{Schulz1995}; the parameter-shifting syzygies in that
+example are an antecedent for our starting algebra.
+For symmetric Artin algebras, Diveris and Purin show that a component
+containing a nonprojective module with eventually vanishing
+self-extensions has stable part of type $\mathbb Z A_\infty$
+\cite[Theorem~A]{DiverisPurin2014}.
+Over an algebraically closed field, Erdmann classifies, up to Morita
+equivalence, the indecomposable
+weakly symmetric algebras with radical cube zero and nonzero radical
+square that admit such modules. The classification singles out quantum
+exterior and double Nakayama algebras with parameters that are not roots
+of unity~\cite[Theorem~1.1]{Erdmann2017}.
+
+Chen, Fang and Xi characterize the symmetric case through stratifying
+ideals and give conditional mirror-reflective constructions from
+gendo-symmetric counterexamples to the Nakayama conjecture
+\cite[Theorems~1.1 and~1.2]{ChenFangXi2024}.
+Chen and Xi relate self-orthogonal Nakayama-stable generators over
+self-injective algebras to additional categorical and
+Gorenstein--Morita conditions~\cite[Theorem~1.2 and Corollary~1.4]{ChenXi2025TAMS}.
+Their further criterion assumes virtual Gorensteinness of the relevant
+endomorphism algebra~\cite[Theorem~1.3]{ChenXi2025VG}.
+Zhang and Zhou prove the self-injective case for split algebras with
+radical cube zero~\cite[Corollary~4.2]{ZhangZhou2026} and subsequently
+treat Igusa--Todorov algebras~\cite[Theorem~3.2]{ZhangZhouIgusaTodorov2026}.
+Xia proves the Auslander--Reiten conjecture for quantum complete
+intersections~\cite[Corollary~4.3]{Xia2026}.
+In a different direction, B\"ohmler and Marczinzik construct a
+nonprojective module over a commutative local self-injective algebra
+whose first two self-extension groups vanish, while the third does
+not~\cite[Theorem~3.1 and the end of Section~3]{BohmlerMarczinzik2026}.
+Theorem~\ref{thm:main} establishes vanishing in every positive degree.
+
+Enomoto and Marczinzik prove that Tachikawa's second conjecture for the
+two-fold trivial extension of an Artin algebra implies the Auslander--Reiten
+conjecture for that algebra~\cite[Theorem~3.5]{Enomoto2026}.
+This reduction has a self-injective target, not a symmetric target in
+general. Our construction obtains a symmetric algebra by establishing
+the additional complete-resolution vanishing needed for transfer to
+the ordinary trivial extension.
+Enomoto and Marczinzik also prove an equivalence among the universal
+versions of the Auslander--Reiten, generalized Nakayama, Nakayama,
+and both Tachikawa conjectures
+over a fixed commutative Artinian coefficient
+ring~\cite[Theorem~1.1]{Enomoto2026}.
+
+\paragraph{The construction.}
+The companion article~\cite{OpenAIAR2026} constructs an explicit
+eight-simple triangular counterexample. We use the same ten-dimensional
+starting algebra and related cone and triangular-module constructions,
+and reproduce the algebra, resolutions, and arguments needed here.
+The passage to a symmetric algebra requires vanishing in negative
+degrees of a complete self-extension complex as well as positive Ext
+vanishing.
+
+Two steps organize the proof.  First, Proposition~\ref{tr:transfer}
+gives a general transfer principle. For a finite-dimensional algebra
+$\Lambda$, a doubly infinite exact complex $P_Z$ of finite projective
+$\Lambda$-modules is \emph{totally acyclic}
+if $\Hom_\Lambda(P_Z,\Lambda)$ is also exact. Suppose
+$P_Z$ is totally acyclic, $Z=\coker((P_Z)_1\to(P_Z)_0)$, and
+\begin{equation}\label{intro:profile}
+ H^a\Hom_\Lambda(P_Z,Z)=0
+ \qquad(a>0\ \text{or}\ a\leq-2).
+\end{equation}
+Then the trivial extension and induced module
+\[
+ A=\Lambda\ltimes D\Lambda,
+ \qquad M=A\otimes_\Lambda Z
+\]
+give the required symmetric example whenever $Z$ is nonprojective.
+The negative tail computes the positive extensions into
+$D\Lambda\otimes_\Lambda Z$, with the precise index $-a-1$.
+This transfer principle applies to any finite-dimensional algebra
+and module satisfying its hypotheses.
+
+Second, we construct the data required by this transfer. A
+ten-dimensional algebra $C$ has a one-dimensional simple module $s$.
+Put
+\[
+ T=C\ltimes DC,\qquad B=C\otimes_k C,
+ \qquad E=T\otimes_k T,\qquad X=s\otimes_k s.
+\]
+We regard $s$ as a $T$-module through $T\to C$, and $X$ as an
+$E$-module. Write $s[r]$ for $s$ placed in homological degree $r$.
+Derived induction of $s$ from $C$ to $T$ gives an object
+$\mathcal J=T\otimes_C^{\mathbf L}s$ in a triangle
+\[
+ s[2]\longrightarrow\mathcal J\longrightarrow s
+ \xrightarrow{\tau}s[3].
+\]
+The class $\tau$ generates $\Ext_T^*(s,s)=k[\tau]$.
+The tensor product $E$ is symmetric, and external products give the
+self-extension algebra $\Ext_E^*(X,X)=k[\tau_1,\tau_2]$, where
+$\tau_1=\tau\otimes1$ and $\tau_2=1\otimes\tau$ both have degree three.
+A stable map is a
+module map modulo maps factoring through projectives; over a symmetric
+algebra the shift $[1]$ is inverse syzygy. Symmetric duality determines
+the negative stable self-maps of $X$ from this polynomial algebra.
+
+Tensoring two copies of this triangle describes
+$E\otimes_B^{\mathbf L}X\simeq\mathcal J\otimes\mathcal J$.
+Its attaching maps act by $\tau_1$ and $\tau_2$. On nonnegative
+stable self-maps these give the two-variable Koszul sequence, and on
+negative self-maps they give its dual. This Koszul calculation, together
+with the full attaching maps, leaves only two scalar groups. The bar
+complex for $E\otimes_B^{\mathbf L}E$ provides a finite $E$-bimodule
+$Y$, projective separately on the left and right, whose evaluation
+represents $(\mathcal J\otimes\mathcal J)[-1]$ in the stable category.
+Its stable Hom groups are
+\[
+ W^a:=\sHom_E(X,(Y\otimes_E X)[a])\simeq
+ \begin{cases}k,&a=-3,0,\\0,&a\notin\{-3,0\}.\end{cases}
+\]
+For $H\in k^\times$, let $h_H$ fix $C$ and multiply $DC$ by $H$,
+and let $U_H$ be $E$ with right action twisted by $h_H\otimes h_H$.
+There is a natural identification $U_H\otimes_E X=X$.
+To combine two twists, we need the nonzero line $W^0$ to be reached
+by evaluating genuine bimodule maps $U_H\to Y$; a module map
+$X\to Y\otimes_E X$ alone does not supply a bimodule fiber.
+A trace obstruction for a character map $B\to DB$, combined with
+symmetric duality, produces precisely these maps $g_H$, with nonzero
+evaluations in degree zero. The sparse Hom profile and these bimodule
+lifts are the two conclusions of Proposition~\ref{b:bimodule}.
+
+After normalizing the maps for $H_1,H_2$, their fiber has a finite
+bimodule representative $F$, obtained as an ordinary kernel after
+adjoining a projective summand. It comes with a stable map
+$v:X\to F\otimes_E X$. The two twists act by distinct scalars on each
+nonzero positive or negative homogeneous self-map space of $X$.
+These actions make the self-extension comparison invertible in the
+required degree ranges. Forming a cone over
+\[
+ \Lambda=\begin{pmatrix}E&0\\ F&E\end{pmatrix}
+\]
+produces the totally acyclic resolution of $Z$ with
+\eqref{intro:profile}; the boundary degrees $1$ and $-2$ are checked
+separately. Figure~\ref{fig:proof-architecture} locates the bimodule
+construction and the final transfer in this argument.
+All bimodule representatives, kernels, and cokernels used to define
+the final algebra and module are finite-dimensional over the stated
+field.  Infinite complexes serve to prove their properties.
+
+\begin{figure}[ht]
+\centering
+\begin{tikzpicture}[>=Stealth,
+ box/.style={draw,rounded corners=2pt,align=center,text width=4.8cm,
+             inner sep=6pt,font=\small},
+ lab/.style={font=\footnotesize,align=center}]
+\node[box] (ex) at (0,3.7)
+ {$E=T\otimes T$, $X=s\otimes s$\\$\Ext_E^*(X,X)=k[\tau_1,\tau_2]$};
+\node[box] (y) at (6.6,3.7)
+ {$Y$ and $g_H:U_H\to Y$\\$W^a=0$ for $a\notin\{-3,0\}$};
+\node[box] (fv) at (6.6,1)
+ {$F$ and $v:X\to F\otimes_E X$};
+\node[box] (lz) at (0,1)
+ {$(\Lambda,Z)$ and $P_Z$\\$H^a\Hom_\Lambda(P_Z,Z)=0$\\for $a>0$ or $a\leq-2$};
+\node[box] (am) at (0,-1.6)
+ {$A=\Lambda\ltimes D\Lambda$\\$M=A\otimes_\Lambda Z$\\symmetric counterexample};
+\draw[->] (ex)--node[above=9mm,lab]{derived induction\\and trace obstruction}(y);
+\draw[->] (y)--node[right,lab]{two twists\\and finite fiber}(fv);
+\draw[->] (fv)--node[above=9mm,lab]{triangular cone}(lz);
+\draw[->] (lz)--node[right,lab]{trivial extension\\and induction}(am);
+\end{tikzpicture}
+\caption{Constructions and implications in the proof. The complete
+self-Hom vanishing for $(\Lambda,Z)$ supplies the hypotheses for
+transfer to $(A,M)$. Arrows indicate proof stages, not algebra maps.}
+\label{fig:proof-architecture}
+\end{figure}
+
+\paragraph{Homological consequences.}
+The pair $(A,M)$ gives counterexamples to several conjectures formulated
+through injective resolutions and tilting modules. Corollary~\ref{cor:homological}
+realizes five such failures using one endomorphism algebra and two
+associated modules. We state the relevant conjectures in their
+finite-dimensional left-module forms before giving the corollary.
+For a finite-dimensional algebra $\Gamma$, write
+\[
+ 0\longrightarrow {}_\Gamma\Gamma\longrightarrow I^0
+ \longrightarrow I^1\longrightarrow\cdots
+\]
+for its minimal injective resolution. Its \emph{dominant dimension} is
+the supremum of the integers $d\geq0$ for which $I^j$ is projective
+whenever $0\leq j<d$, and is infinite when every $I^j$
+is projective. The classical \emph{Nakayama conjecture} asserts that
+infinite dominant dimension forces self-injectivity.
+The \emph{generalized Nakayama conjecture} asserts that every
+indecomposable injective occurs as a summand of some $I^j$; equivalently,
+for every simple module $S$, some $\Ext_\Gamma^i(S,\Gamma)$ with
+$i\geq0$ is nonzero. The \emph{strong Nakayama conjecture} asserts
+that a finite module $V$ with $\Ext_\Gamma^i(V,\Gamma)=0$ for all
+$i\geq0$ is zero. The \emph{Auslander--Gorenstein conjecture} asserts
+that the bounds $\pd_\Gamma I^j\leq j$ for every $j\geq0$ force
+finite self-injective dimension on both sides. These formulations are
+recalled in~\cite[Introduction]{Zhang2018HomologicalConjectures}.
+
+Write $\add W$ for the direct summands of finite direct sums of a
+module $W$. A finite left $\Gamma$-module $W$ is \emph{tilting} if it
+has finite projective dimension, is self-orthogonal, and admits a
+finite exact coresolution
+\[
+ 0\longrightarrow\Gamma\longrightarrow W^0\longrightarrow\cdots
+ \longrightarrow W^r\longrightarrow0,\qquad W^j\in\add W.
+\]
+It is \emph{Wakamatsu tilting} if it is self-orthogonal and admits an
+exact coresolution
+\[
+ 0\longrightarrow\Gamma\longrightarrow W^0\longrightarrow W^1
+ \longrightarrow\cdots,\qquad W^j\in\add W,
+\]
+which remains exact after applying $\Hom_\Gamma(-,W)$. This is the
+finite left-module definition in~\cite[Definition~2.1]{Huang2006}.
+The \emph{Wakamatsu tilting conjecture} asserts that a Wakamatsu tilting
+module of finite projective dimension is tilting; see
+\cite{DivaaniAazarMahinFallahTousi2025} for a recent discussion.
+
+\begin{corollary}[Homological conjectures]\label{cor:homological}
+Let $A,M$ be as in Theorem~\ref{thm:main}. For every field extension
+$K/k$, put $A_K=K\otimes_k A$, $M_K=K\otimes_k M$, and
+\[
+ \Gamma_K=\End_{A_K}(A_K\oplus M_K)^{\mathrm{op}}.
+\]
+Then $\Gamma_K$ is a finite-dimensional $K$-algebra with the following
+properties, in the category of finite left $\Gamma_K$-modules.
+\begin{enumerate}
+\item Every term of the minimal injective resolution of $\Gamma_K$
+is projective, but $\idim_{\Gamma_K}\Gamma_K=\infty$.
+\item There is a nonzero simple module $S$ such that
+$\Ext_{\Gamma_K}^i(S,\Gamma_K)=0$ for every $i\geq0$.
+\item There is a projective-injective Wakamatsu tilting module $W$
+which is not tilting.
+\end{enumerate}
+Consequently the classical, generalized, and strong Nakayama conjectures,
+the Auslander--Gorenstein conjecture, and the Wakamatsu tilting conjecture
+are false for finite-dimensional algebras over $K$.
+\end{corollary}
+
+The opposite in this definition keeps our left-module convention. The
+endomorphism-algebra connection with dominant dimension is classical,
+going back to Mueller~\cite[Lemma~3 and Theorem~4]{Mueller1968}; the precise
+Tachikawa--Nakayama correspondence for a pair $(A,M)$ is also stated
+in~\cite[p.~7288]{ChenXi2025TAMS}. Section~\ref{sec:consequences}
+uses this connection and constructs the modules $S$ and $W$ directly.
+Taking $K$ to be an algebraic closure of $k$ gives examples over an
+algebraically closed field of characteristic two. This field contains
+the displayed transcendental parameters; the assertion does not concern
+the algebraic closure of a finite field.
+
+The same resolution also gives a finitistic-dimension consequence.
+For a finite-dimensional algebra, the \emph{little left finitistic
+dimension} is the supremum of the finite projective dimensions of
+finite left modules; the \emph{big left finitistic dimension} allows
+all left modules instead.
+
+\begin{corollary}[Finitistic dimensions]\label{cor:finitistic}
+Let $k$ be the field in Theorem~\ref{thm:main}, let $K/k$ be any field
+extension, and let $\Gamma_K$ be the algebra in
+Corollary~\ref{cor:homological}. For every integer $n\geq1$, there is a
+finite-dimensional left $\Gamma_K$-module $C_n$ such that
+\[
+ \pd_{\Gamma_K}C_n=n.
+\]
+Consequently $\Gamma_K$ has infinite little left finitistic dimension
+and infinite big left finitistic dimension.
+\end{corollary}
+
+This gives an independent characteristic-two example alongside the
+complex algebra of~\cite[Theorem~1.1]{OpenAIFinitistic2026}.
+
+Two further applications are developed in
+Section~\ref{sec:consequences}. The connection between generalized
+Nakayama failure and cotilting complements established by Buan and Solberg
+\cite[Theorems~3.1 and~3.4]{BuanSolberg1998} gives a projective module
+over $\Gamma_K^{\mathrm{op}}$ that can be completed to a tilting module
+in infinitely many ways, each by adjoining one indecomposable summand;
+these summands are pairwise nonisomorphic
+(Corollary~\ref{cor:tilting-complements}). Their introduction credits
+an independent proof of this connection to Happel and Unger
+\cite[Introduction]{BuanSolberg1998}. The mirror-reflective
+constructions of Chen, Fang and Xi~\cite{ChenFangXi2024} give auxiliary
+indecomposable symmetric algebras with arbitrarily long idempotent
+stratifications and the resulting chains of successive derived-category
+recollements specified in Corollary~\ref{cor:stratifying}.
+
+\paragraph{Organization.}
+Section~\ref{sec:homological} establishes the complete-resolution,
+duality, and finite-representative tools.
+Section~\ref{sec:transfer} proves the transfer to a symmetric algebra.
+Sections~\ref{a:section} and~\ref{e:section} construct the starting
+algebra and compute the stable self-maps and twist actions.
+Section~\ref{b:section} constructs the finite bimodule and proves the
+trace obstruction. Section~\ref{sec:triangular} forms the triangular
+module, establishes the complete vanishing profile, and applies the
+transfer to prove Theorem~\ref{thm:main}.
+Section~\ref{sec:consequences} studies the associated endomorphism
+algebra and proves Corollaries~\ref{cor:homological} and~\ref{cor:finitistic}.
+
+\paragraph{Conventions.}
+All algebras are associative and unital, all modules are unital left
+modules unless indicated otherwise, and all modules occurring in the
+construction are finite-dimensional.
+Unspecified tensor products are over $k$, and
+$G^e=G\otimes_k G^{\mathrm{op}}$ acts on $G$-bimodules by the usual
+outer actions.  Complexes are indexed homologically, with
+$P[a]_j=P_{j-a}$.  Since the field has characteristic two, signs in
+their differentials and comparison maps disappear.  The following
+section fixes the stable and derived conventions used throughout.
diff --git a/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/sections/stable.tex b/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/sections/stable.tex
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+\section{Polynomial stable endomorphisms and twists}
+\label{e:section}
+
+The dual cohomology in Proposition~\ref{a:resolution} produces a
+degree-three extension class over \(T\).  We first determine its
+products, then tensor two copies and use symmetric duality to determine
+all negative degrees as well.
+
+\begin{proposition}\label{e:polynomial}
+Put \(\mathcal J=T\otimes_C R\), where \(R\) is the resolution in
+\eqref{a:simple-resolution}.  There is a triangle in the derived
+category of \(T\)-modules
+\begin{equation}
+s[2]\longrightarrow\mathcal J\longrightarrow s
+ \xrightarrow{\tau}s[3],
+\label{e:triangle}
+\end{equation}
+and the connecting class generates the Yoneda algebra:
+\begin{equation}
+\Ext_T^*(s,s)=k[\tau],\qquad |\tau|=3.
+\label{e:one-polynomial}
+\end{equation}
+The complex \(\mathcal J\) is bounded below, has finite projective
+\(T\)-terms, and has bounded homology.
+\end{proposition}
+
+\begin{proof}
+Tensor the sequence of \((T,C)\)-bimodules
+\(0\to DC\to T\to C\to0\) with \(R\).  Projectivity of each
+\(R_i\) gives a short exact sequence of complexes of left
+\(T\)-modules
+\begin{equation}
+0\longrightarrow DC\otimes_C R
+ \longrightarrow\mathcal J\longrightarrow R\longrightarrow0.
+\label{e:coefficient-sequence}
+\end{equation}
+The two outside complexes have action inflated from \(C\).  For a
+finite projective left \(C\)-module \(V\), the natural evaluation map
+\[
+DC\otimes_C V\longrightarrow D\Hom_C(V,C),\qquad
+\phi\otimes v\longmapsto\bigl(f\mapsto\phi(f(v))\bigr)
+\]
+is an isomorphism, first for a finite free module and then for its
+summands.  Applying this termwise identifies the left complex in
+\eqref{e:coefficient-sequence} with \(D\Hom_C(R,C)\).  By
+\eqref{a:dual-cohomology}, its only homology is \(s\) in homological
+degree two, with the asserted \(C\)-module structure.  Good truncation
+therefore identifies it with \(s[2]\) in the derived category, also
+after inflating the action to \(T\).  Fix this identification.
+The quotient complex resolves \(s\), so
+\eqref{e:coefficient-sequence} gives \eqref{e:triangle}.
+Its middle complex has projective \(T\)-terms, and the triangle gives
+its bounded homology.
+
+Because \(\mathcal J\) is bounded below and projective, ordinary
+induction--restriction adjunction computes
+\[
+Q_i:=\Hom_{\mathrm D^b(T)}(\mathcal J,s[i])
+ =H^i\Hom_T(\mathcal J,s)
+ =H^i\Hom_C(R,s)
+ =\begin{cases}k&i=0,\\0&i\ne0.\end{cases}
+\]
+Apply derived Hom into shifts of \(s\) to \eqref{e:triangle}.
+With ordinary negative Ext interpreted as zero, the resulting long
+exact sequence contains
+\begin{equation}
+Q_{i-1}\longrightarrow\Ext_T^{i-3}(s,s)
+ \xrightarrow{\alpha\mapsto\alpha[3]\tau}
+ \Ext_T^i(s,s)\longrightarrow Q_i.
+\label{e:polynomial-sequence}
+\end{equation}
+At \(i=1\) this is \(k\to0\to\Ext_T^1(s,s)\to0\), so the first
+Ext group vanishes.  For every \(i\ge2\), the middle map in
+\eqref{e:polynomial-sequence} is an isomorphism.  In particular
+\(\Ext_T^2(s,s)=0\), and at \(i=3\) it sends the identity of \(s\)
+to the nonzero class \(\tau\).  Induction now shows that \(\tau^m\)
+is a basis of \(\Ext_T^{3m}(s,s)\) for every \(m\ge0\), and all
+other positive degrees vanish.  Since the isomorphisms are the
+displayed Yoneda compositions, this proves the algebra assertion
+\eqref{e:one-polynomial}.
+\end{proof}
+
+We now tensor two copies of this self-extension algebra. Set
+\begin{equation}
+X=s\otimes s,\qquad
+\cH^a=\sHom_E(X,X[a])\quad(a\in\mathbb Z).
+\label{e:stable-groups}
+\end{equation}
+Let \(\tau_1=\tau\otimes1\) and \(\tau_2=1\otimes\tau\) be the
+two external extension classes.  We write \(\mathcal P_m\) for the
+space of polynomials of total exponent \(m\) in \(\tau_1,\tau_2\);
+their cohomological degree is \(3m\).  This notation distinguishes
+these polynomial spaces from the projective resolutions used later.
+
+\begin{proposition}\label{e:all-degrees}
+The module \(X\) is nonprojective and
+\begin{equation}
+\cH^{\ge0}=k[\tau_1,\tau_2],\qquad |\tau_1|=|\tau_2|=3.
+\label{e:two-polynomial}
+\end{equation}
+In every integer degree, the only nonzero groups are
+\begin{equation}
+\cH^{3m}=\mathcal P_m,\qquad
+\cH^{-3m-1}=D\mathcal P_m\qquad(m\ge0).
+\label{e:profile}
+\end{equation}
+Under these identifications, postcomposition by a shifted
+\(\tau_i\) between negative groups is the transpose of multiplication
+by \(\tau_i\) on the polynomial spaces.  The maps
+\(\cH^{-1}\to\cH^2\) given by either class are zero.
+\end{proposition}
+
+\begin{proof}
+Choose a projective \(T\)-resolution \(Q\to s\).  The total complex
+of \(Q\otimes Q\) is a projective \(E\)-resolution of \(X\).  There
+are finitely many summands in each total degree, and termwise finite
+projectivity gives an isomorphism of cochain complexes
+\[
+\Hom_E(Q\otimes Q,s\otimes s)
+ \cong\Hom_T(Q,s)\otimes\Hom_T(Q,s).
+\]
+The cohomology of this tensor product is the tensor product of the
+cohomologies: over a field each complex splits into its homology and
+contractible summands.  Tensoring comparison maps of resolutions
+computes the external Yoneda products under this isomorphism.
+In particular the maps acting on the first and second tensor factors
+commute, since all interchange signs are \(+1\) in characteristic
+two.  Proposition~\ref{e:polynomial} therefore gives
+\(\Ext_E^*(X,X)=k[\tau_1,\tau_2]\) as algebras.
+
+Its degree-three part is nonzero, so \(X\) is not projective.  Also
+\(\End_E(X)=k\), and the identity cannot factor through a projective:
+such a factorization would exhibit \(X\) as a projective summand.
+Thus \(\sEnd_E(X)=k\).  Positive Ext agrees with positive stable Hom
+over the symmetric algebra \(E\), proving \eqref{e:two-polynomial}.
+Lemma~\ref{h:duality} gives
+\[
+\cH^{-a-1}\cong D\cH^a\qquad(a\ge0),
+\]
+which proves the support and dimensions in \eqref{e:profile}.
+
+To retain the actual multiplication maps, choose a nonzero functional
+\(\lambda:\cH^{-1}\to k\).  The composition pairing from
+Lemma~\ref{h:duality} identifies \(\cH^{-3m-1}\) with
+\(D\mathcal P_m\) by
+\begin{equation}
+(\beta,p)_m
+ =\lambda\bigl(p[-3m-1]\beta\bigr),
+\qquad
+\beta\in\cH^{-3m-1},\quad p\in\mathcal P_m.
+\label{e:pairing}
+\end{equation}
+It is perfect for every \(m\).  For \(m\ge1\), associativity of
+composition gives, for \(p\in\mathcal P_{m-1}\),
+\begin{equation}
+\bigl(\tau_i[-3m-1]\beta,p\bigr)_{m-1}
+ =(\beta,p[3]\tau_i)_m.
+\label{e:transpose}
+\end{equation}
+The right-hand side is pairing with polynomial multiplication by
+\(\tau_i\).  This proves the transpose assertion with its precise
+shift.  For \(m=0\), the target is \(\cH^2=0\), which proves the
+boundary assertion as well.
+\end{proof}
+
+One can make the negative action especially concrete.  Let
+\(\eta_{a,b}\in\cH^{-3(a+b)-1}\) be dual to
+\(\tau_1^a\tau_2^b\).  With multiplication here meaning
+postcomposition by the appropriately shifted class,
+\begin{equation}
+\tau_1\eta_{a,b}=
+ \begin{cases}\eta_{a-1,b}&a>0,\\0&a=0,\end{cases}
+\qquad
+\tau_2\eta_{a,b}=
+ \begin{cases}\eta_{a,b-1}&b>0,\\0&b=0.\end{cases}
+\label{e:negative-action}
+\end{equation}
+These formulas include the zero maps out of \(\eta_{0,0}\).
+Products of two negative-degree classes vanish by the support
+\eqref{e:profile}.
+
+For \(H\in k^\times\), define
+\begin{equation}
+h_H:T\longrightarrow T,\quad (a,\phi)\longmapsto(a,H\phi),
+\qquad \sigma_H=h_H\otimes h_H,
+\qquad U_H=E_{\sigma_H}.
+\label{e:twist-definition}
+\end{equation}
+Here \(E_{\sigma_H}\) is the regular left module with right action
+\(a\cdot b=a\sigma_H(b)\).  These are automorphisms, and \(U_H\)
+is projective on both sides.  The tensor functor \(U_H(-)\) is
+restriction by \(\sigma_H^{-1}\), as shown by the natural balanced
+isomorphism
+\begin{equation}
+U_H\otimes_E V\xrightarrow{\ \sim\ }
+\Res_{\sigma_H^{-1}}V,\qquad
+a\otimes v\longmapsto\sigma_H^{-1}(a)v.
+\label{e:inverse-restriction}
+\end{equation}
+The same convention applies to \(T_{h_H}\otimes_T-\).
+Let \(\chi:E\to k\) be the character of \(X\).  Since
+\(\chi\sigma_H=\chi\), we fix the resulting identification
+\(U_H X=X\).  The exact functor \(U_H(-)\) preserves projectives
+and acts on all groups \(\cH^a\), with the shift identifications of
+Lemma~\ref{h:evaluation}.
+
+To determine this action in negative degrees, we first choose an
+explicit generator in degree \(-1\). Once the action on this generator
+is known, the composition pairing determines the action on every
+negative group.
+
+\begin{lemma}\label{e:socle}
+Put
+\[
+p_f=f\otimes f,\qquad \zeta=f^*\otimes f^*\in E p_f.
+\]
+For every \(a,b\in E\),
+\(a\zeta b=\chi(a)\chi(b)\zeta\).
+The map \(i_X:X\to Ep_f\), \(i_X(1)=\zeta\), is an embedding with
+image in \((\rad E)p_f=\Omega X\).  Regarded as a map
+\(X\to\Omega X\), it is nonzero stably and spans \(\cH^{-1}\).
+\end{lemma}
+
+\begin{proof}
+The dual basis vector \(f^*\in DC\) is the character of the
+\(f\)-simple on \(C\).  The dual bimodule action in
+\eqref{a:trivial-extension} shows that multiplication on either side
+by an element of \(C\) scales \(f^*\) by that character; the
+square-zero ideal \(DC\) kills it on either side.  Tensoring proves
+the asserted action on \(\zeta\).  The projective cover
+\(Ep_f\to X\), \(ap_f\mapsto\chi(a)\), has kernel
+\((\rad E)p_f\) by \eqref{a:radicals}.  The nonzero vector \(\zeta\)
+lies in this kernel, so the displayed map is an embedding into
+\(\Omega X\).
+
+We verify nonfactorization through every finite projective.  Write
+\(J_E=\rad E\).  If \(b\in E\) satisfies \(J_Eb=0\), symmetry gives
+\[
+t_E(abj)=t_E(jab)=0\qquad(a\in E,\ j\in J_E),
+\]
+since \(ja\in J_E\).  Nondegeneracy therefore implies \(bJ_E=0\).
+For any map \(X\to E\), every vector in its image satisfies this
+condition.  Any left-module map \(E\to J_Ep_f\) has the form
+\(b\mapsto br\) with \(r\in J_Ep_f\), so its composite with
+\(X\to E\) is zero.  The same conclusion holds for finite sums and
+summands of \(E\).  Thus all maps \(X\to\Omega X\) factoring
+through a finite projective are zero, whereas \(i_X\ne0\).
+Finally \(\cH^{-1}=D\mathcal P_0\) is one-dimensional by
+Proposition~\ref{e:all-degrees}.
+\end{proof}
+
+\begin{proposition}\label{e:twists}
+For every \(H\in k^\times\) and every \(m\ge0\), the action of
+\(U_H(-)\) on the groups in \eqref{e:profile} is
+\begin{equation}
+\left.U_H\right|_{\cH^{3m}}=H^{-m}\,1_{\mathcal P_m},
+\qquad
+\left.U_H\right|_{\cH^{-3m-1}}=H^{m+2}\,1_{D\mathcal P_m}.
+\label{e:twist-weights}
+\end{equation}
+In particular, the negative action is the stated scalar on the entire
+\((m+1)\)-dimensional space.
+\end{proposition}
+
+\begin{proof}
+First twist \eqref{e:coefficient-sequence} by restriction along
+\(h_H^{-1}\).  Because \(h_H\) fixes \(C\), the map
+\(h_H\otimes1\) identifies its middle complex with the original
+one.  It induces the identity on the quotient \(R\) and multiplication
+by \(H\) on \(DC\otimes_C R\).  Write \(\tau_H\) for the twist of
+\(\tau\).  Passing to the fixed derived identifications yields a
+commutative diagram of triangles
+\[
+\begin{tikzcd}[column sep=large]
+s[2]\arrow[r]\arrow[d,"H"']&
+\Res_{h_H^{-1}}\mathcal J\arrow[r]
+ \arrow[d,"h_H\otimes1"]&
+s\arrow[r,"\tau_H"]\arrow[d,"1"]&
+s[3]\arrow[d,"H"]\\
+s[2]\arrow[r]&\mathcal J\arrow[r]&s\arrow[r,"\tau"]&s[3].
+\end{tikzcd}
+\]
+Naturality of the connecting morphism gives \(H\tau_H=\tau\).
+Hence each external class \(\tau_i\) is sent to
+\(H^{-1}\tau_i\), and preservation of products proves the first
+formula in \eqref{e:twist-weights}.
+
+For the negative part we first compute the action on \(\cH^{-1}\).
+Twist the projective presentation
+\[
+0\longrightarrow\Omega X\longrightarrow Ep_f
+ \longrightarrow X\longrightarrow0.
+\]
+Under \eqref{e:inverse-restriction}, its comparison with the original
+presentation is
+\[
+\sigma_H:\Res_{\sigma_H^{-1}}(Ep_f)\longrightarrow Ep_f.
+\]
+It is \(E\)-linear, since
+\(\sigma_H(\sigma_H^{-1}(a)b)=a\sigma_H(b)\), and induces the
+identity on \(X\).  On the kernels it sends
+\(\zeta\) to \(H^2\zeta\).  Thus the generator in
+Lemma~\ref{e:socle} is sent to \(H^2\) times itself, proving that
+\(U_H\) acts by \(H^2\) on \(\cH^{-1}\).
+
+Finally the exact twist functor preserves shifted composition.
+Applying it to the product in \eqref{e:pairing}, and using its actions
+on \(\mathcal P_m\) and \(\cH^{-1}\), gives
+\[
+\bigl(U_H(\beta),H^{-m}p\bigr)_m
+ =H^2(\beta,p)_m
+\qquad
+(\beta\in\cH^{-3m-1},\ p\in\mathcal P_m).
+\]
+Equivalently,
+\((U_H(\beta),p)_m=H^{m+2}(\beta,p)_m\) for every \(p\).
+Nondegeneracy of the pairing forces
+\(U_H(\beta)=H^{m+2}\beta\) for every \(\beta\), as asserted.
+\end{proof}
diff --git a/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/sections/transfer.tex b/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/sections/transfer.tex
new file mode 100644
index 0000000000000000000000000000000000000000..0ffb43fdacb838920d5e407effd34301e6291ab9
--- /dev/null
+++ b/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/sections/transfer.tex
@@ -0,0 +1,132 @@
+\section{Passing to a symmetric algebra}\label{sec:transfer}
+
+The final step of the construction has a general form.  It explains why
+we will need vanishing in negative degrees as well as positive ones.
+Throughout this section, $R$ is a finite-dimensional algebra over our
+field $k$ and $D=\Hom_k(-,k)$.
+
+We recall the standard symmetry of a trivial extension; see
+\cite[Proposition~3.1]{BessenrodtHolmZimmermann2007}.
+
+\begin{lemma}\label{tr:trivial-symmetric}
+The trivial extension $R\ltimes DR$ is a finite-dimensional symmetric
+algebra.  In particular, its left and right regular modules are
+injective in the categories of all unital modules.
+\end{lemma}
+
+\begin{proof}
+The dual bimodule structure is
+\[
+ (a\lambda b)(c)=\lambda(bca)
+ \qquad(a,b,c\in R,\ \lambda\in DR),
+\]
+and the multiplication is
+\[
+ (a,\lambda)(b,\mu)=(ab,a\mu+\lambda b).
+\]
+The bimodule identities give associativity, and $(1,0)$ is a unit.
+The functional $t(a,\lambda)=\lambda(1)$ gives the pairing
+\[
+ t\big((a,\lambda)(b,\mu)\big)=\mu(a)+\lambda(b).
+\]
+This pairing is symmetric and nondegenerate, and associativity of the
+algebra makes it an associative form.  Thus $R\ltimes DR$ is isomorphic
+to its dual as a bimodule.
+
+For any algebra $G$ over $k$, the natural isomorphism
+$\Hom_G(L,DG)\simeq\Hom_k(L,k)$ shows that $DG$ is an injective left
+$G$-module; the same argument applies on the right.  This statement
+holds for arbitrary unital $L$, since vector-space duality is exact.
+Applying it to the symmetric algebra $G=R\ltimes DR$ proves the final
+assertion.
+\end{proof}
+
+\begin{proposition}[Transfer through the trivial extension]
+\label{tr:transfer}
+Let $P$ be a totally acyclic complex of finite projective left
+$R$-modules, and put $Z=c_0P$.  Suppose that $Z$ is nonprojective and
+\begin{equation}\label{tr:two-sided-vanishing}
+ H^a\Hom_R(P,Z)=0
+ \qquad\text{for }a>0\text{ and for }a\leq -2.
+\end{equation}
+Then
+\[
+ A=R\ltimes DR,\qquad M=A\otimes_R Z
+\]
+are a finite-dimensional symmetric algebra and a finite-dimensional
+nonprojective left $A$-module satisfying
+\[
+ \Ext_A^a(M,M)=0\qquad(a>0).
+\]
+\end{proposition}
+
+\begin{proof}
+Write $\nu=DR\otimes_R-$ and set $I=\nu P$.  Finite projectivity gives
+natural isomorphisms
+\begin{equation}\label{tr:nakayama-evaluation}
+ DR\otimes_R P_j\simeq D\Hom_R(P_j,R),
+ \qquad
+ \Hom_R(Z,DR\otimes_R P_j)\simeq D\Hom_R(P_j,Z).
+\end{equation}
+Both follow first for finite free modules and then for their direct
+summands.  The module $\Hom_R(P_j,R)$ is finite projective on the right,
+so $I_j$ is injective on the left.  Total acyclicity of $P$ and exactness
+of $D$ show that the entire complex $I$ is exact.
+
+As a right $R$-module, $A=R\oplus DR$.  Consequently the restriction of
+$A\otimes_RP$ is the exact complex $P\oplus I$.  Every term
+$A\otimes_RP_j$ is finite projective over $A$, since induction preserves
+a split direct-summand presentation.  Right exactness identifies
+$c_0(A\otimes_RP)$ with $A\otimes_RZ=M$.  Thus the nonnegative tail of
+$A\otimes_RP$ is a projective resolution of $M$.  In particular, the
+tensor exactness needed here has been proved for this resolution;
+flatness of $A_R$ is not an additional hypothesis.
+
+Put $N=\nu Z$.  Again by right exactness, $N=c_0I$.  Exactness of $I$
+identifies this cokernel with $\im(I_0\to I_{-1})$.  Hence
+\begin{equation}\label{tr:injective-tail}
+ 0\longrightarrow N\longrightarrow I_{-1}
+ \longrightarrow I_{-2}\longrightarrow I_{-3}\longrightarrow\cdots
+\end{equation}
+is an injective coresolution.  If $C^j=\Hom_R(P_j,Z)$ with differential
+$\delta^j:C^j\to C^{j+1}$, applying $\Hom_R(Z,-)$ to
+\eqref{tr:injective-tail} gives, in cohomological degrees $0,1,2$,
+\[
+ DC^{-1}\xrightarrow{D\delta^{-2}}DC^{-2}
+ \xrightarrow{D\delta^{-3}}DC^{-3}.
+\]
+Both arrows surrounding degree one are present.  More generally, the
+term in degree $a\geq0$ is $DC^{-a-1}$.  Thus, for every $a>0$,
+\begin{equation}\label{tr:negative-duality}
+ \Ext_R^a(Z,N)\simeq D H^{-a-1}\Hom_R(P,Z)=0.
+\end{equation}
+The first of these equalities uses degree $-2$, which accounts for the
+endpoint in \eqref{tr:two-sided-vanishing}.
+
+Induction--restriction adjunction on the projective resolution already
+constructed now yields
+\begin{align*}
+ \Ext_A^a(M,M)
+ &\simeq \Ext_R^a\bigl(Z,\Res_R^A M\bigr)\\
+ &\simeq \Ext_R^a(Z,Z)\oplus\Ext_R^a(Z,N)=0
+ \qquad(a>0).
+\end{align*}
+Here the first summand vanishes by
+\eqref{tr:two-sided-vanishing} and Lemma~\ref{h:comparison}, and the
+second by \eqref{tr:negative-duality}.
+
+Lemma~\ref{tr:trivial-symmetric} proves that $A$ is symmetric.  The
+module $M$ is finite and unital.  Finally, if $M$ were a direct summand
+of $A^r$, tensoring the splitting with the quotient $A\to R$ would make
+\[
+ R\otimes_A M\simeq R\otimes_A(A\otimes_RZ)\simeq Z
+\]
+a direct summand of $R^r$.  A tensor functor preserves a split identity
+composite without any flatness assumption.  This contradicts
+nonprojectivity of $Z$.
+\end{proof}
+
+We will apply Proposition~\ref{tr:transfer} to a triangular algebra
+$R=\Lambda$.  The rest of the proof constructs its module $Z$ together
+with the complete resolution and the two vanishing ranges in
+\eqref{tr:two-sided-vanishing}.
diff --git a/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/sections/triangular.tex b/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/sections/triangular.tex
new file mode 100644
index 0000000000000000000000000000000000000000..04bef4ebcee21dfd9aae285f56e1a43704eeef19
--- /dev/null
+++ b/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/build/sections/triangular.tex
@@ -0,0 +1,320 @@
+\section{The triangular module and the counterexample}
+\label{sec:triangular}
+
+We now combine the two twists and apply the transfer principle of
+Proposition~\ref{tr:transfer}.  Retain $E,X$ and the finite
+side-projective bimodule $Y$ from Proposition~\ref{b:bimodule}.
+For brevity, write $LX=L\otimes_E X$ for an $E$-bimodule $L$, and set
+\[
+ W^a=\sHom_E(X,YX[a]).
+\]
+Thus $W^a=k$ for $a=-3,0$ and is zero otherwise.
+
+\subsection{The fiber of the two evaluated maps}
+
+For $i=1,2$, put $U_i=U_{H_i}$.  Choose the maps
+$g_i:U_i\to Y$ supplied by Proposition~\ref{b:bimodule}.
+Using the fixed identifications $U_iX\simeq X$, their evaluated stable
+classes are nonzero elements of the same one-dimensional space
+$W^0$.  Rescale them so that both classes equal a fixed $w\ne0$.
+Choose a finite free $E$-bimodule $Q$ with a surjection
+$\pi:Q\to Y$, and define $F$ by
+\begin{equation}\label{t:fiber}
+ 0\longrightarrow F\longrightarrow U_1\oplus U_2\oplus Q
+ \xrightarrow{(g_1,g_2,\pi)}Y\longrightarrow0.
+\end{equation}
+For example, the action gives such a surjection from $E^e\otimes_kY$.
+Since $Y$ is projective on each side, this sequence splits separately
+on the left and on the right.  In particular, $F$ is finite and
+side-projective.
+
+Let
+\[
+ \mathcal V^a=\sHom_E(X,FX[a]).
+\]
+Evaluating \eqref{t:fiber} on $X$ preserves exactness, and $QX$ is
+$E$-projective.  The stable long exact sequence is therefore
+\begin{equation}\label{t:fiber-hom}
+ (\cH^{a-1})^2\longrightarrow W^{a-1}
+ \longrightarrow\mathcal V^a
+ \xrightarrow{p^a}(\cH^a)^2\longrightarrow W^a.
+\end{equation}
+In degree zero the last map is
+$(c,d)\mapsto(c+d)w$.  It follows that $\mathcal V^0$ is a line with a
+unique element $v_0$ projecting to $(1,1)$.
+
+For a stable map $g:X\to X[a]$, let $F(g)$ denote its image under
+the exact functor $F\otimes_E-$, using the shift comparisons of
+Lemma~\ref{h:evaluation}.  Define
+\begin{equation}\label{t:delta}
+ \Delta^a:(\cH^a)^2\longrightarrow\mathcal V^a,
+ \qquad (g,h)\longmapsto F(g)v_0+v_0[a]h.
+\end{equation}
+
+\begin{lemma}\label{t:comparison-maps}
+The map $\Delta^0$ is surjective.  For every $a>0$ and every
+$a\leq-3$, $\Delta^a$ is an isomorphism, and $\Delta^{-2}$ is
+injective.
+\end{lemma}
+
+\begin{proof}
+The projection $p^a$ in \eqref{t:fiber-hom} is an isomorphism for
+$a>0$.  For $a\geq2$ this follows from the support of $W$; at $a=1$
+it also uses the surjectivity of $(\cH^0)^2\to W^0$.
+The same support argument applies for $a\leq-4$.
+At $a=-3$, both $W^{-4}$ and $\cH^{-3}$ are zero, so
+$\mathcal V^{-3}=0$ and $p^{-3}$ is again an isomorphism.
+The remaining relevant low degrees can be read directly from
+\eqref{t:fiber-hom}:
+\begin{equation}\label{t:boundary-fiber}
+ \mathcal V^{-2}=k,
+ \qquad \mathcal V^0=kv_0,
+ \qquad \mathcal V^1=0.
+\end{equation}
+
+Naturality of the bimodule projections $F\to U_i$ gives
+\begin{equation}\label{t:projected-delta}
+ p^a\Delta^a(g,h)=
+ \bigl(U_1(g)+h,\ U_2(g)+h\bigr).
+\end{equation}
+Indeed each projection sends $v_0$ to the identity of $X$, and
+evaluation of a bimodule map commutes with the specified shift
+comparisons.  By Proposition~\ref{e:twists}, at $a=3m>0$
+the map in \eqref{t:projected-delta} is the scalar matrix
+\[
+ \begin{pmatrix}H_1^{-m}&1\\H_2^{-m}&1\end{pmatrix}
+ \otimes 1_{\mathcal P_m},
+\]
+and at $a=-3m-1\leq-4$ it is
+\[
+ \begin{pmatrix}H_1^{m+2}&1\\H_2^{m+2}&1\end{pmatrix}
+ \otimes 1_{D\mathcal P_m}.
+\]
+In both cases $m\geq1$.  Algebraic independence of $H_1,H_2$
+implies that each determinant is nonzero in $k$, for every such $m$.
+In all other degrees in the asserted ranges, the source and target
+are zero by Proposition~\ref{e:all-degrees} and the already proved
+isomorphism $p^a$.
+
+Finally, $\Delta^0(c,d)=(c+d)v_0$ is surjective.  Since
+$\cH^{-2}=0$, the map $\Delta^{-2}:0\to\mathcal V^{-2}=k$ is
+injective.  Its surjectivity is neither claimed nor needed.
+\end{proof}
+
+\subsection{A cone over the triangular algebra}
+
+Form the finite-dimensional algebra
+\begin{equation}\label{t:triangular-algebra}
+ \Lambda=\begin{pmatrix}E&0\\F&E\end{pmatrix}.
+\end{equation}
+A left $\Lambda$-module is a triple
+$(L,N,\alpha:F\otimes_E L\to N)$ of left $E$-modules and an
+$E$-linear structure map, with action
+\[
+ \begin{pmatrix}a&0\\b&d\end{pmatrix}(x,y)
+ =\bigl(ax,\alpha(b\otimes x)+dy\bigr).
+\]
+The column constructions are standard in the theory of totally acyclic
+complexes over triangular matrix rings; see
+\cite[Lemma~2.2 and Proposition~2.3]{EshraghiEtAl2014v1}.
+Here $F$ is projective on each side, so the required tensor exactness
+and orthogonality conditions hold. We retain the complete resolution
+to compute its self-Hom groups. Define the two column functors
+\[
+ \mathsf D_0(L)=(L,FL,1),
+ \qquad \mathsf D_1(L)=(0,L,0).
+\]
+They are exact and preserve finite projectives: exactness uses right
+projectivity of $F$, and their values on $E$ are the two columns of
+the regular module $\Lambda$.  The descriptions of module maps give
+\begin{align}
+ \Hom_\Lambda(\mathsf D_i(L),\mathsf D_i(L'))
+   &\simeq\Hom_E(L,L')\quad(i=0,1),\label{t:diagonal-hom}\\
+ \Hom_\Lambda(\mathsf D_0(L),\mathsf D_1(L'))&=0,
+ \qquad
+ \Hom_\Lambda(\mathsf D_1(L),\mathsf D_0(L'))
+   \simeq\Hom_E(L,FL').\label{t:cross-hom}
+\end{align}
+In the first diagonal slot, the map on the lower component is $F(g)$
+when the map on the upper component is $g$.
+
+Let $p_f=f\otimes f$.  Use the embedding
+$i_X:X\hookrightarrow Ep_f$, $i_X(1)=\zeta$, established in
+Lemma~\ref{e:socle}.  Choose a complete resolution
+$P$ of $X$ over $E$ whose first negative term is
+$P_{-1}=Ep_f$, and whose degree-zero differential is
+$d_0=i_X\epsilon$, where $\epsilon:P_0\to X$ is the augmentation.
+Such a choice is possible because $E$ is symmetric.  The complex
+$FP$ is exact with projective terms, and $c_0(FP)=FX$.
+
+Choose an ordinary map $\widetilde v:X\to FX$ representing $v_0$.
+Lemma~\ref{h:comparison} lifts it to a chain map $P\to FP$.
+By \eqref{t:cross-hom} this is a chain map
+\[
+ V:\mathsf D_1(P)\longrightarrow\mathsf D_0(P).
+\]
+Set
+\begin{equation}\label{t:cone}
+ P_Z=\Cone(V),\qquad Z=c_0P_Z.
+\end{equation}
+Thus
+\[
+ (P_Z)_j=\mathsf D_0(P_j)\oplus\mathsf D_1(P_{j-1}),
+\]
+with cross differential $V_{j-1}$ from the second summand to the
+first summand in degree $j-1$.
+
+\begin{lemma}\label{t:finite-module}
+The complex $P_Z$ is totally acyclic.  Its degree-zero cokernel $Z$
+is finite-dimensional and nonprojective.
+\end{lemma}
+
+\begin{proof}
+Both column complexes have projective terms and are exact.  Since
+$\Lambda=\mathsf D_0(E)\oplus\mathsf D_1(E)$,
+\eqref{t:diagonal-hom}--\eqref{t:cross-hom} identify their Hom
+complexes into $\Lambda$ with sums of
+\[
+ \Hom_E(P,E),\qquad\Hom_E(P,F),\qquad0.
+\]
+These are exact: $P$ is totally acyclic and $F$ is finite projective
+on the left.  Hence both column complexes are totally acyclic, and
+so is their cone.
+
+For an explicit finite description, $Z$ is the triple with upper
+component $X$ and lower component
+\begin{equation}\label{t:explicit-module}
+ N_Z=(FX\oplus Ep_f)
+ \big/\{(\widetilde v(x),i_X(x)):x\in X\},
+\end{equation}
+whose structure map $FX\to N_Z$ is induced by the first summand.
+To verify this formula, first quotient the degree-zero lower
+component by the image of the differential in $\mathsf D_0(P)$.
+This gives $FX$ in its first summand.  The remaining relations arise
+from $P_0$ in the shifted second column.  Its two components induce
+$\widetilde v\epsilon$ and $i_X\epsilon$, respectively, yielding
+\eqref{t:explicit-module}.
+
+The upper-component functor sends the projective generators
+$\mathsf D_0(E)$ and $\mathsf D_1(E)$ to $E$ and $0$, and therefore
+sends every finite projective $\Lambda$-module to an $E$-projective.
+But $X$ is nonprojective, since its degree-three self-extension group
+is nonzero by Proposition~\ref{e:all-degrees}.  Thus $Z$ is
+nonprojective.
+\end{proof}
+
+\subsection{The full endomorphism complex}
+
+The complete resolution makes it possible to check both vanishing
+ranges with the same endomorphism calculation.
+
+\begin{proposition}\label{t:vanishing}
+The module and resolution in \eqref{t:cone} satisfy
+\[
+ H^a\Hom_\Lambda(P_Z,Z)=0
+ \qquad(a>0\ \text{or}\ a\leq-2).
+\]
+\end{proposition}
+
+\begin{proof}
+Write $A_0=\mathsf D_0(P)$ and $B_0=\mathsf D_1(P)$, so that the
+underlying graded complex of $P_Z$ is $A_0\oplus B_0[1]$.
+By \eqref{t:cross-hom}, the block
+$\Hom_\Lambda^\bullet(A_0,B_0[1])$ is zero.  Projection onto the two
+diagonal blocks therefore gives a short exact sequence of full Hom
+complexes
+\begin{equation}\label{t:end-sequence}
+\begin{split}
+ 0\longrightarrow\mathcal C^\bullet
+ \longrightarrow\End_\Lambda^\bullet(P_Z)
+ \longrightarrow
+ \End_\Lambda^\bullet(A_0)\oplus
+ \End_\Lambda^\bullet(B_0[1])
+ \longrightarrow0,\\[-2pt]
+ \mathcal C^\bullet=\Hom_\Lambda^\bullet(B_0[1],A_0).
+\end{split}
+\end{equation}
+The underlying degreewise Hom spaces are products as in
+Lemma~\ref{h:comparison}; the block decomposition has only four
+blocks.  In particular, it does not replace any product by a direct
+sum.
+
+The diagonal cohomology is $(\cH^a)^2$.  To fix the cross-block
+index explicitly, in degree $a$ we have
+\begin{align*}
+ \mathcal C^a
+ &=\prod_j\Hom_E(P_{j-1},FP_{j-a})\\
+ &=\prod_i\Hom_E(P_i,FP_{i-(a-1)})
+   =\Hom_E^{a-1}(P,FP).
+\end{align*}
+The differentials agree under this reindexing in characteristic two.
+Lemma~\ref{h:comparison} consequently gives
+$H^a\mathcal C^\bullet=\mathcal V^{a-1}$.
+
+The connecting map in \eqref{t:end-sequence}, from diagonal
+cohomology in degree $a$ to $H^{a+1}\mathcal C^\bullet=\mathcal V^a$,
+is precisely $\Delta^a$.  Indeed, lift two diagonal cocycles to the
+diagonal blocks of an endomorphism of the cone and take its
+cochain differential.  The cross component is the sum of its two
+compositions with $V$.  The first diagonal acts downstairs by $F$
+of the corresponding map, while the second acts by precomposition.
+Under the composition-compatible comparison in
+Lemma~\ref{h:comparison}, this gives
+$F(g)v_0+v_0[a]h$, as in \eqref{t:delta}.
+Thus the long exact sequence gives
+\begin{equation}\label{t:end-cohomology}
+ 0\longrightarrow\coker\Delta^{a-1}
+ \longrightarrow H^a\End_\Lambda^\bullet(P_Z)
+ \longrightarrow\ker\Delta^a\longrightarrow0.
+\end{equation}
+
+For $a>1$ or $a\leq-3$, both outer terms vanish by
+Lemma~\ref{t:comparison-maps}.  At $a=1$, they vanish by surjectivity
+of $\Delta^0$ and injectivity of $\Delta^1$.  At $a=-2$, they vanish
+by the isomorphism $\Delta^{-3}$ and injectivity of $\Delta^{-2}$.
+Finally, apply Lemma~\ref{h:comparison} to the totally acyclic
+complex $P_Z$ to identify this cohomology with
+$H^a\Hom_\Lambda(P_Z,Z)$.
+\end{proof}
+
+\begin{remark}\label{t:neighboring-degrees}
+The boundary degrees can also be located completely.
+Equation~\eqref{t:fiber-hom} gives
+$\mathcal V^{-1}=(\cH^{-1})^2=k^2$.
+By Proposition~\ref{e:twists}, the two rows of $\Delta^{-1}$ are
+$(H_i^2,1)$, so this map is invertible.  Since $\Delta^{-2}:0\to k$
+and $\ker\Delta^0=k$, \eqref{t:end-cohomology} yields
+\[
+ H^a\End_\Lambda^\bullet(P_Z)=
+ \begin{cases}
+ k,&a=-1,0,\\
+ 0,&a\notin\{-1,0\}.
+ \end{cases}
+\]
+In particular the nonzero cokernel of $\Delta^{-2}$ contributes in
+degree $-1$, while degree $-2$ vanishes as required by
+Proposition~\ref{tr:transfer}.
+\end{remark}
+
+\subsection{Completion of the proof}
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+Fix the finite bimodule maps in \eqref{t:fiber}, the complete
+resolution $P$, and the lifted map used in \eqref{t:cone}.
+These choices exist over $k$ by the preceding propositions.
+The algebra $\Lambda$ in \eqref{t:triangular-algebra} and its module
+$Z$ in \eqref{t:explicit-module} are finite-dimensional.
+By Lemma~\ref{t:finite-module}, $Z$ is nonprojective and has the
+totally acyclic resolution $P_Z$.
+Proposition~\ref{t:vanishing} supplies both hypotheses on its
+complete self-Hom groups in Proposition~\ref{tr:transfer}.
+Apply that proposition with $R=\Lambda$ to obtain
+\[
+ A=\Lambda\ltimes D\Lambda,
+ \qquad M=A\otimes_\Lambda Z.
+\]
+It proves that $A$ is finite-dimensional and symmetric, that $M$
+is finite-dimensional and nonprojective, and that
+$\Ext_A^i(M,M)=0$ for every $i>0$.  These are all the assertions of
+Theorem~\ref{thm:main}.
+\end{proof}
diff --git a/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/paper.pdf b/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..6f8f9b84ac8e79bbd0c3a594d2efe182dd5a14f2
--- /dev/null
+++ b/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:332c2535f60d4ad620a86b3c4d547c51510b183f881445d9a0807986b0544f67
+size 619479
diff --git a/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/README.md b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..0d9923a39acad5b0ab20abf44f672a3e965a60a5
--- /dev/null
+++ b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/README.md
@@ -0,0 +1,18 @@
+# [A counterexample to the Colin de Verdière chromatic conjecture](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 23, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026,
+  author = {{OpenAI}},
+  title = {{A counterexample to the Colin de Verdi{\`e}re chromatic conjecture}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/paper.pdf}{OAI:A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/main.tex b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/main.tex
new file mode 100644
index 0000000000000000000000000000000000000000..7bb6c4cc8163f7d5095adbff94a3f86748e77ce1
--- /dev/null
+++ b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/main.tex
@@ -0,0 +1,38 @@
+\documentclass[11pt]{article}
+\input{preamble}
+\title{A counterexample to the Colin de Verdière chromatic conjecture}
+\author{OpenAI}
+\date{September 23, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We disprove the Colin de Verdière chromatic conjecture by constructing
+graphs whose chromatic number exceeds their Colin de Verdière invariant
+by more than one. The examples have independence number at most two.
+In fact, their ordinary fractional chromatic number also exceeds their
+Colin de Verdière invariant by more than one.
+\end{abstract}
+\tableofcontents
+\input{sections/01-introduction}
+\input{sections/matrix-trees}
+\input{sections/matrix-rank}
+\input{sections/02-geometry}
+\input{sections/03-distributions}
+\input{sections/04-moments}
+\input{sections/05-realization}
+\input{sections/06-collision}
+\input{sections/07-phases}
+\input{sections/08-preparation}
+\input{sections/09-histograms}
+\input{sections/10-product-tests}
+\input{sections/11-common-limit}
+\input{sections/12-status-span}
+\input{sections/12b-asymmetric}
+\input{sections/13-graph-transfer}
+\appendix
+\input{sections/14-parameters}
+\input{sections/15-measures}
+\clearpage
+\bibliographystyle{plainnat}
+\bibliography{sources/references}
+\end{document}
diff --git a/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/preamble.tex b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/preamble.tex
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index 0000000000000000000000000000000000000000..ea88e1490f2d6decd4303d61ef51e91c196781bc
--- /dev/null
+++ b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/preamble.tex
@@ -0,0 +1,76 @@
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage{lmodern}
+\usepackage[margin=1.05in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{microtype,graphicx,booktabs,array,longtable}
+\usepackage{enumitem,etoolbox}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta,positioning,calc,fit}
+\usepackage[numbers,sort&compress]{natbib}
+\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref}
+\usepackage[capitalise,nameinlink,noabbrev]{cleveref}
+\hypersetup{pdftitle={A counterexample to the Colin de Verdière chromatic conjecture},pdfauthor={OpenAI}}
+\setlength{\parskip}{0.18em}
+\setlist{itemsep=0.18em,topsep=0.35em}
+\makeatletter
+\patchcmd{\l@section}{\setlength\@tempdima{1.5em}}
+  {\setlength\@tempdima{2.1em}}{}{\PackageError{contents}{Cannot adjust section number width}{}}
+\renewcommand*\l@subsection{\@dottedtocline{2}{2.1em}{3.1em}}
+\makeatother
+\numberwithin{equation}{section}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{corollary}[theorem]{Corollary}
+\newtheorem{conjecture}[theorem]{Conjecture}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\newtheorem{construction}[theorem]{Construction}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newtheorem{example}[theorem]{Example}
+\newcommand{\F}{\mathbb F_2}
+\newcommand{\E}{\mathbb E}
+\newcommand{\PP}{\mathbb P}
+\newcommand{\R}{\mathbb R}
+\newcommand{\N}{\mathbb N}
+\newcommand{\cB}{\mathcal B}
+\newcommand{\cX}{\mathcal X}
+\newcommand{\cV}{\mathcal V}
+\newcommand{\cE}{\mathcal E}
+\newcommand{\cI}{\mathcal I}
+\newcommand{\cK}{\mathcal K}
+\newcommand{\cL}{\mathcal L}
+\newcommand{\cR}{\mathcal R}
+\newcommand{\cS}{\mathcal S}
+\newcommand{\cD}{\mathcal D}
+\newcommand{\cP}{\mathcal P}
+\newcommand{\cQ}{\mathcal Q}
+\newcommand{\cW}{\mathcal W}
+\newcommand{\1}{\mathbf 1}
+\DeclareMathOperator{\rank}{rank}
+\DeclareMathOperator{\im}{im}
+\DeclareMathOperator{\Span}{span}
+\DeclareMathOperator{\GL}{GL}
+\DeclareMathOperator{\Hom}{Hom}
+\DeclareMathOperator{\End}{End}
+\DeclareMathOperator{\Tr}{Tr}
+\DeclareMathOperator{\supp}{supp}
+\DeclareMathOperator{\diag}{diag}
+\DeclareMathOperator{\cm}{cm}
+\DeclareMathOperator{\had}{h}
+\DeclareMathOperator{\Var}{Var}
+\DeclareMathOperator{\TV}{TV}
+\DeclareMathOperator{\codim}{codim}
+\DeclareMathOperator{\id}{id}
+\DeclareMathOperator{\Sym}{Sym}
+\newcommand{\pair}[2]{\langle #1,#2\rangle}
+\newcommand{\norm}[1]{\lVert #1\rVert}
+\newcommand{\abs}[1]{\lvert #1\rvert}
+\newcommand{\eps}{\varepsilon}
+\newcommand{\trans}{^{\mathsf T}}
+\ifdefined\pdfinfoomitdate\pdfinfoomitdate=1\fi
+\ifdefined\pdftrailerid\pdftrailerid{}\fi
+
+\newcommand{\rawlaw}{{\mathsf P_0}}
diff --git a/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/01-introduction.tex b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/01-introduction.tex
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--- /dev/null
+++ b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/01-introduction.tex
@@ -0,0 +1,320 @@
+\section{Introduction}\label{sec:introduction}
+
+The Colin de Verdière invariant connects graph colouring with the
+inertia and nullity of real symmetric matrices. We prove that the
+chromatic number need not be at most one more than this invariant.
+
+\subsection{The invariant and the main result}
+
+A real symmetric matrix \(M\), indexed by the vertices of a finite
+nonempty simple graph \(G=(V,E)\), is \emph{well-signed for \(G\)} if, for
+distinct vertices \(u,v\),
+\[
+ M_{uv}<0\quad\text{when }\{u,v\}\in E,\qquad
+ M_{uv}=0\quad\text{when }\{u,v\}\notin E.
+\]
+Its diagonal entries are unrestricted. Such a matrix has the
+\emph{Strong Arnold Property} if the only real symmetric matrix \(X\)
+satisfying
+\[
+ MX=0,\qquad X_{uu}=0\ (u\in V),\qquad
+ X_{uv}=0\ (\{u,v\}\in E)
+\]
+is \(X=0\). The Colin de Verdière invariant is
+\[
+ \mu(G)=\max\left\{\dim\ker M:\ 
+ \begin{array}{l}
+ M\text{ is well-signed for }G,\quad n_-(M)=1,\\
+ M\text{ has the Strong Arnold Property}
+ \end{array}\right\},
+\]
+where \(n_-(M)\) counts negative eigenvalues with multiplicity.
+This class is nonempty: perturb an invertible diagonal matrix with
+one negative entry by sufficiently small negative entries on the
+edges. Inertia and invertibility persist, and an invertible matrix
+has the Strong Arnold Property.
+
+Write \(\alpha(G)\) and \(\chi(G)\) for the independence number and
+chromatic number. The chromatic conjecture for this invariant asserts
+that \(\chi(G)\le\mu(G)+1\) for every finite nonempty simple graph.
+Our main result gives a negative resolution.
+
+\begin{theorem}\label{thm:main}
+There are finite graphs \(G\) of arbitrarily large order for which
+\[
+ \alpha(G)\le2,\qquad
+ \mu(G)+1<\frac{|V(G)|}{2}\le\chi(G).
+\]
+In particular, the Colin de Verdière chromatic conjecture is false.
+\end{theorem}
+
+We prove a stronger matrix obstruction than is necessary for the
+upper bound on \(\mu\): every well-signed matrix on the constructed
+graph with one negative eigenvalue has rank greater than
+\(|V(G)|/2+1\). The Strong Arnold Property is not used in this bound.
+The maximum corank in this larger class is the related parameter
+\(\kappa\) considered by Lovász and Schrijver
+\citep[pp.~572--574]{lovaszschrijver2017}. Thus the construction also
+satisfies \(\kappa(G)+1<|V(G)|/2\). Unlike \(\mu\), this larger
+parameter is not minor-monotone in general; our argument uses it only
+through the stated matrix class.
+
+\subsection{History, significance, and the construction}
+
+Colin de Verdière introduced \(\mu\) through the spectral study of
+Schrödinger operators on graphs \citep{cdv1990,hls1999}.
+The Strong Arnold Property expresses a transversality condition from
+that spectral framework \citep{cdv1988arnold}. The small values of the
+invariant have concrete geometric meanings: \(\mu\le2\) characterizes
+outerplanar graphs, \(\mu\le3\) planar graphs, and \(\mu\le4\)
+linklessly embeddable graphs \citep{cdv1990,lovaszschrijver1998,hls1999}.
+The proposed chromatic bound holds throughout the range
+\(\mu(G)\le4\) \citep[p.~35]{hls1999}.
+Gram representations and nullspace embeddings have provided geometric
+ways to study these matrices \citep{kotlovlovaszvempala1997,lovaszschrijver1999}.
+These connections make the proposed chromatic bound a natural test of
+how much colouring information the spectral invariant retains.
+
+Write \(h(G)\) for the largest \(t\) such that \(K_t\) is a minor of
+\(G\). Minor monotonicity of \(\mu\), together with
+\(\mu(K_t)=t-1\), gives \(h(G)\le\mu(G)+1\)
+\citep[\S1.3 and Theorem~2.4]{hls1999}. Thus Hadwiger's conjecture
+\(\chi(G)\le h(G)\) \citep{hadwiger1943} would imply the chromatic
+bound proposed by Colin de Verdière \citep{cdv1990,cdv1998,hls1999}.
+For completeness, the complete-graph identity follows directly from
+\(-J_t\): it has one negative eigenvalue and nullity \(t-1\), and the
+Strong Arnold Property is automatic on a complete graph. No admissible
+matrix can have rank zero. The examples in \Cref{thm:main} therefore
+also violate Hadwiger's inequality. The matrix obstruction below is
+proved directly; its conclusion does not follow from a clique-minor
+obstruction.
+
+A distinct connected-matching route to the failure of Hadwiger's inequality
+is given in the companion manuscript~\citep[Theorem~1.1 and Corollary~1.2]{OpenAIHadwigerCounterexample2026}.
+
+The construction starts with a finite-field relation of \emph{holes}.
+Its vertices are injective frames. A hole is witnessed by a tensor
+shared by two frames, two compatible affine-gradient equations, and a
+parity condition. Summing those equations around a triangle gives a
+contradiction. The complement of the hole relation consequently has
+independence number at most two, and so needs at least half its order
+in colours. Our remaining task is to force matrix rank strictly above
+half the order. For this purpose, ordinary control of connected
+matchings is insufficient: the matrix argument needs conflicts between
+specified classes of matching edges and between a matching and a
+separately chosen vertex set.
+
+The new probabilistic ingredient is a thinning of the allowed
+channel images that preserves the primal-frame distribution. It
+penalizes prescribed nonzero translates of channel combinations,
+turning the difficult phase alternative into one with a fixed
+positive mass of compatible pairs. This gives both a version of
+conflict supersaturation that distinguishes colours and an
+asymmetric edge--point version. We prove these extensions together
+with their common-limit and finite-sampling steps. The low-rank Boolean
+moment argument uses commuting multiplication operators, as in the
+flat-extension method for moments \citep{laurentmourrain2009}; here the
+characteristic-two, partly truncated version is proved directly. The
+probabilistic comparison combines entropy control with the
+energy-increment method of weak regularity \citep{cover2006,frieze1999}.
+For finite sampling we adapt the deterministic fingerprint procedure of
+Kleitman and Winston \citep{kleitmanwinston1982,samotij2015}, using a
+constrained entropy minimum as its potential \citep{csiszar1975}.
+
+The second ingredient is a matrix criterion for clique blow-ups.
+A finite-tree representation of hyperbolic flats locates large
+families of blocks that can be normalized simultaneously. A first
+spectral normalization produces an almost perfect matching of
+coordinates. A second normalization extracts additional positive
+directions from selected pair blocks. The resulting rank gain
+contradicts a kernel of approximately half the total dimension.
+The criterion needs only the graph properties stated next and
+may be used independently of the tensor construction.
+
+\subsection{The base graphs and the proof structure}
+
+A \emph{hole} of a graph is a pair of distinct nonadjacent vertices.
+Two edges \emph{conflict} if they are disjoint and all four cross
+pairs between their endpoints are holes. An edge and a vertex
+conflict if the vertex is distinct from both endpoints and both
+cross pairs are holes. Edges \emph{touch} if they have a common
+endpoint or an edge joins their endpoint sets. Thus two disjoint
+edges fail to touch exactly when they conflict.
+
+
+
+\noindent\begin{minipage}{\linewidth}
+\begin{theorem}[Base graphs]\label{thm:base-graphs}
+There is a constant \(a>0\) such that, for every \(\beta>0\),
+there are arbitrarily large integers \(m\) and graphs \(H\) on
+\(m\) vertices with \(\alpha(H)\le2\) and the following properties.
+\begin{enumerate}[label=\textup{(G\arabic*)},ref=G\arabic*]
+\item\label{item:base-colours}
+Every matching of size at least \(m/20\), partitioned into classes
+of size at most \(am\), contains two conflicting edges in different
+classes.
+\item\label{item:base-edge-point}
+Every matching of size at least \(\beta m\) and every vertex set
+of size at least \(\beta m\) contain an edge and a vertex in conflict.
+\item\label{item:base-holes}
+Every two vertex sets of size at least \(\beta m\) contain a cross
+hole.
+\end{enumerate}
+The two vertex sets in \ref{item:base-holes} may overlap.
+\end{theorem}
+\end{minipage}\par
+
+\begin{figure}[htbp]
+\centering
+\begin{tikzpicture}[scale=1.0,every node/.style={font=\small}]
+\node[circle,fill=black,inner sep=2pt,label=left:\(A_1\)] (a1) at (0,1) {};
+\node[circle,fill=black,inner sep=2pt,label=left:\(A_2\)] (a2) at (0,-1) {};
+\node[circle,fill=black,inner sep=2pt,label=right:\(B_1\)] (b1) at (3.8,1) {};
+\node[circle,fill=black,inner sep=2pt,label=right:\(B_2\)] (b2) at (3.8,-1) {};
+\draw[very thick] (a1)--(a2);
+\draw[very thick] (b1)--(b2);
+\foreach \x in {a1,a2} \foreach \y in {b1,b2}
+  \draw[dashed,blue!65!black] (\x)--(\y);
+\node at (0,1.65) {edge \(A\)};
+\node at (3.8,1.65) {edge \(B\)};
+\end{tikzpicture}
+\caption{A conflict: solid segments are graph edges, and dashed
+segments are holes. The construction forces such a configuration
+between different classes of a large matching.}
+\label{fig:four-holes}
+\end{figure}
+
+For a positive integer \(s\), the \emph{clique blow-up} \(H[s]\)
+replaces each vertex of \(H\) by a clique of size \(s\), and each
+edge by all edges between the two corresponding cliques.
+We also write \(H[K_s]\) for \(H[s]\). An independent set in \(H[s]\) uses at most one vertex from each clique
+and projects to an independent set in \(H\). Hence
+\(\alpha(H[s])\le2\), and \(\chi(H[s])\ge\lceil ms/2\rceil\).
+
+\Cref{thm:matrix-obstruction} proves that, when \(a\) is fixed,
+\(\beta\) is sufficiently small, and \(m\) is sufficiently large,
+properties \ref{item:base-colours}--\ref{item:base-holes} force every
+well-signed one-negative matrix on \(H[s]\) to have rank greater than
+\(ms/2+1\), for every sufficiently large \(s\).
+Choosing one base graph from \Cref{thm:base-graphs} and applying
+rank--nullity therefore proves \Cref{thm:main}. Notice the order:
+the base graph is fixed before the blow-up parameter tends to
+infinity.
+
+The independence-number estimate also yields a stronger
+fractional-colouring consequence. For a finite nonempty simple graph
+\(G\), let \(\mathcal I(G)\) be the family of its independent vertex
+sets. A fractional colouring assigns nonnegative real weights \(w_I\)
+to these sets with \(\sum_{I\ni v}w_I\ge1\) at every vertex \(v\).
+The ordinary fractional chromatic number \(\chi_f(G)\) is the minimum
+of \(\sum_Iw_I\) over all such assignments. An ordinary colouring is
+a feasible assignment with unit weights on its colour classes, so
+\(\chi_f(G)\le\chi(G)\). On the other hand, summing the vertex
+constraints gives
+\[
+ |V(G)|\le \sum_{v\in V(G)}\sum_{I\ni v}w_I
+ =\sum_{I\in\mathcal I(G)}|I|w_I
+ \le \alpha(G)\sum_{I\in\mathcal I(G)}w_I,
+\]
+and therefore \(\chi_f(G)\ge |V(G)|/\alpha(G)\).
+
+\begin{corollary}[Fractional-colouring obstruction]\label{cor:fractional-spectral}
+There are finite nonempty simple graphs \(G\) of arbitrarily large
+order for which
+\[
+ \chi_f(G)\ge \frac{|V(G)|}{\alpha(G)}
+ \ge \frac{|V(G)|}{2}
+ >\kappa(G)+1\ge\mu(G)+1\ge h(G).
+\]
+\end{corollary}
+\begin{proof}
+Take the constructed examples in Theorem~\ref{thm:main} and put
+\(m=|V(G)|\). They have \(\alpha(G)\le2\). Every well-signed
+one-negative matrix on these graphs has rank greater than \(m/2+1\),
+so rank--nullity gives corank less than \(m/2-1\). Maximizing in the
+larger matrix class yields \(\kappa(G)+1<m/2\).
+Every matrix admitted in the definition of
+\(\mu(G)\) is admitted for \(\kappa(G)\), so \(\mu(G)\le\kappa(G)\).
+The inequality \(h(G)\le\mu(G)+1\) was established above. Combining
+these bounds with the fractional-colouring estimate proves the chain.
+\end{proof}
+
+The same graphs also violate the fractional-colouring weakening
+of Hadwiger's conjecture discussed by
+\citet[p.~148]{reedseymour1998}: for every positive integer \(p\),
+excluding a \(K_{p+1}\) minor was predicted to give a fractional
+\(p\)-colouring. Their definition uses nonnegative rational weights
+with coverage exactly one at each vertex and total weight at most
+\(p\), which are admissible above; their result (1.3) proves the
+corresponding \(2p\)-bound.
+Taking \(p=h(G)\ge1\) in the corollary rules out the coefficient-one
+prediction. The fractional relaxation here concerns colouring,
+while \(h(G)\) remains the ordinary clique-minor number. The
+comparisons with \(\kappa\) and \(\mu\) show that the obstruction
+survives both fractional colouring and omission of the Strong Arnold
+Property.
+
+The list chromatic number \(\chi_{\mathrm{list}}(G)\) is the least
+integer \(k\) such that every assignment of finite colour lists of
+size at least \(k\) to the vertices admits a proper colouring from
+those lists.
+
+\begin{corollary}[A linear list-colouring bound]\label{cor:list-spectral}
+Let \(C\) be the absolute integer in
+\citet[Theorem~1.1]{OpenAILinearListHadwiger2026}. Every finite nonempty
+simple graph \(G\) satisfies
+\[
+ \chi_{\mathrm{list}}(G)\le C\bigl(\mu(G)+1\bigr).
+\]
+\end{corollary}
+\begin{proof}
+That theorem gives \(\chi_{\mathrm{list}}(G)\le Ch(G)\). Combining this
+with \(h(G)\le\mu(G)+1\), proved above, gives the result.
+\end{proof}
+
+\subsection{From distribution tests to finite graphs}
+
+To prove \Cref{thm:base-graphs}, we first work with probability laws
+on the finite set of frames. An ordered pair with no hole is called
+a \emph{unit}. The two frames within a unit may be strongly dependent.
+The tests in \Cref{thm:distribution-tests} give a positive lower bound
+for the probability that two independently sampled units conflict
+across distinct colours, and that a unit conflicts with an independently
+sampled point, under specified density bounds and, in the coloured
+test, an upper bound on each colour's mass.
+
+The algebraic part gives sufficient certificates for four cross holes,
+using scalar conditions and equalities between ordered tuples of frame images, called
+\emph{keys}. Once these equalities and scalar conditions hold,
+\Cref{thm:gradient-realization} prescribes the remaining inner products
+between primal and channel frames, and \Cref{prop:collision} bounds
+the probability of agreement with that prescription. The main
+probabilistic task is therefore to force overlap between the accepted
+key distributions despite dependence within each unit. Channel thinning
+supplies compatible scalar data. The histogram and product comparisons
+pass the actual query laws to a common limit, where finite-dimensional
+duality produces overlapping choices. Finally, the entropy fingerprint
+argument transfers the two distribution tests to one finite sample
+satisfying all three base-graph properties.
+
+The paper first proves the matrix criterion. It then constructs
+the base graphs. \Cref{sec:geometry,sec:distributions} define the
+frame model and thinned law and formulate the distribution tests.
+\Cref{sec:moments,sec:realization,sec:collision} supply the finite
+algebraic certificates and the collision criterion.
+\Cref{sec:phases,sec:preparation} establish compatibility on positive
+mass. \Cref{sec:histograms,sec:product-tests,sec:common-limit,sec:status-span}
+prove the comparison and overlap statements that establish the
+coloured distribution test. \Cref{sec:asymmetric} proves the
+independent-point distribution test. \Cref{sec:graph-transfer}
+transfers both tests to finite graphs, proving \Cref{thm:base-graphs},
+and \Cref{sec:parameters} records the parameter order and quantitative
+margins. \Cref{sec:measure-constructions} gives the compact measure
+constructions used in the limiting steps.
+
+In the probabilistic construction, \(n\) is its unbounded input
+dimension, \(N=M_0n\) is the ambient frame dimension, and
+\(m=2^{1000gN}\) is the sampled graph order. The graph threshold
+\(\beta\) is distinct from the selector length \(b\).
+The original raw probability law is denoted by \(\rawlaw\), and its
+thinned version by \(\pi\); \(\mu\) always denotes the graph invariant.
diff --git a/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/02-geometry.tex b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/02-geometry.tex
new file mode 100644
index 0000000000000000000000000000000000000000..cb35dd5c5fd3cce7a85bc4635bd9291ebb7a4063
--- /dev/null
+++ b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/02-geometry.tex
@@ -0,0 +1,317 @@
+\section{The tensor geometry and the hole relation}\label{sec:geometry}
+
+We construct a triangle-free relation of holes from tensor equations.
+The complement of this relation will be our base graph; the moment
+arguments in the next sections force its conflict properties.
+All vector spaces, linear maps, tensor products and ranks in the construction are over \(\F=\mathbb F_2\).
+We use the entrywise pairing of matrices:
+\[
+ \pair{A}{B}=\sum_{r,c}A_{rc}B_{rc}=\Tr(A\trans B).
+\]
+Probabilities and entropy take real values.
+
+\subsection{The order of parameters}
+
+We first fix the parameter hierarchy, then define the coefficient
+space, its tensor-valued frame embeddings, and the hole equations.
+The fixed constants below bound the information retained about a unit
+and the number of vectors whose equality will certify a conflict.
+Their order ensures that the colour-class scale precedes the graph
+threshold, and that the thinning rate precedes the ambient dimension
+multiplier.
+
+The construction has one asymptotic parameter \(n\).  Every numerical parameter below, including the dimension multiplier \(M_0\), is fixed before \(n\) tends to infinity.
+Set
+\begin{equation}\label{eq:early-constants}
+ C_0=1000,\qquad g=10^9+1,\qquad
+ D=4C_0g,\qquad M=2^{1000},\qquad
+ k_{\max}=56(g-1).
+\end{equation}
+In particular, \(g\) is odd.  The first pin budget and the entropy slack are
+\begin{equation}\label{eq:pin-budgets}
+ \zeta=\frac{1}{1000k_{\max}},
+ \qquad K_1=\left\lceil\frac{4(D+1)}{\zeta}\right\rceil.
+\end{equation}
+For the phase estimates and a common upper bound on all pin dimensions, define
+\begin{equation}\label{eq:law-budgets}
+ \begin{gathered}
+ r=2K_1+2,\qquad L_0=\lceil100(D+10)\rceil,\qquad
+ d_0=4rL_0,\\
+ u_0=10(K_1+d_0+1),\qquad
+ K=\left\lceil\frac{10(D+K_1+u_0+10)}{\zeta}\right\rceil .
+ \end{gathered}
+\end{equation}
+The budget \(K\) is a conservative upper bound; the construction uses only
+the first peeling, whose pin dimension is at most \(K_1\).
+The additional probability thresholds in \Cref{sec:phases} use only these early parameters and fixed numerical tolerances.  They do not depend on the selector or channel dimensions.
+
+The early probability margins determine a colour cap \(\eta_{\rm col}>0\)
+and then the graph class scale \(a>0\).  Given the graph threshold
+\(\beta>0\), choose the endpoint-density bound \(L\) required in
+\Cref{sec:graph-transfer}.  Next choose the selector parameters
+\(j_*,b\) and the mixer parameters through \(J\) as in
+\Cref{sec:moments}.  Choose \(r_0\) sufficiently large after those choices
+and all fixed density bounds, and put \(h=1000r_0\).  After \(h\), choose
+a sufficiently small thinning rate \(\rho>0\) as in
+\Cref{sec:distributions}.  Finally choose a sufficiently large integer
+\(M_0\), also relative to \(1/\rho\), and set
+\begin{equation}\label{eq:parameter-order}
+ \begin{gathered}
+ (C_0,g,D,M,k_{\max},\zeta,K_1;\ r,L_0,d_0,u_0,K)\\
+ \longrightarrow\ (\eta_{\rm col},a)
+ \ \longrightarrow\ (\beta,L)
+ \ \longrightarrow\ (j_*,b,\text{mixer parameters},J)\\
+ \longrightarrow\ (r_0,h=1000r_0)
+ \ \longrightarrow\ \rho
+ \ \longrightarrow\ M_0,\qquad
+ N=M_0n,\qquad m=2^{C_0gN}.
+ \end{gathered}
+\end{equation}
+We always take \(n\ge 2r_0\) and subsequently increase its lower bound as needed.
+The matrices used below have dimensions depending on \(n\); for each \(n\) they are chosen to satisfy \Cref{lem:generic-mixers}, before any law of units is considered.
+The complete ledger of inequalities is in \Cref{sec:parameters}.
+
+An \(O(n)\) bit count below has a constant depending only on parameters chosen before \(M_0\).
+It can therefore be made a prescribed small fraction of \(N\).
+This is different from an \(o(N)\) bound: after \(M_0\) is fixed, \(n/N=1/M_0\) is constant.
+Later fixed tests and tolerances may change how large \(n\) must be; they do not change any earlier construction parameter.
+
+\subsection{Moment matrices and cut profiles}
+
+Let
+\[
+ \cI=\mathbb Z/g\mathbb Z,\qquad
+ \cE=\binom{\cI}{2},\qquad
+ I_d=\{d+1,\ldots,d+(g-1)/2\}\subset\cI .
+\]
+Elements of \(\cI\) are \emph{tags}; elements of \(\cE\) are \emph{components}.
+The ordinary, or base, variables comprise \(g^2+3\) blocks of \(n\) bits:
+\[
+ O_{d,t}\quad(d,t\in\cI),\qquad S,\quad \#,\quad Z.
+\]
+A separate selector has \(b\) bits.  Its values \(s\in\F^b\) are called \emph{labels}, to distinguish them from tags.
+
+The \(O\)-blocks carry testers whose availability depends on the tag,
+while \(S\) carries a tester available at every tag.
+The additional \(\#\)-block lets the self-Gram form \(E\), defined
+below, distinguish different selector labels.
+The \(Z\)-block is absent from \(E\); its free coordinates and
+change-of-basis symmetry are used in the later parameter tests.
+
+Let
+\[
+ p_*=\sum_{j=0}^{j_*}\binom bj,\qquad
+ \cB=\F^{p_*}\otimes\F^{\,1+(g^2+3)n}.
+\]
+The first factor is indexed by squarefree selector monomials of degree at most \(j_*\); the second by the constant and the individual base variables.
+If \(p_s\) is the vector of selector-monomial evaluations and \(z\) is a base value, write
+\begin{equation}\label{eq:point-evaluation}
+ v(s,z)=p_s\otimes(1,z)\in\cB.
+\end{equation}
+Thus the row and column coordinates both represent monomials of base degree at most one and selector degree at most \(j_*\).
+
+Let \(W\subset\cB\otimes\cB\) be the span of the point moments \(v(s,z)v(s,z)\trans\).
+For \(l\in\cI\), let \(W_l\) be the span of those point moments for which
+\[
+ O_{d,t}=0\quad\text{whenever }l\notin I_d.
+\]
+Let \(W_*\) be defined by requiring every \(O\)-block to be zero.
+The cut-profile space is
+\begin{equation}\label{eq:cut-profile-space}
+ \cX=
+ \left\{x=(x_{\{l,l'\}})_{\{l,l'\}\in\cE}:
+ x_{\{l,l'\}}=w_l+w_{l'},\quad w_l\in W_l\right\}.
+\end{equation}
+A tuple \(w=(w_l)_{l\in\cI}\) as in this display is a \emph{representation} of \(x\).
+
+\begin{lemma}\label{lem:cut-kernel}
+The intersection of \(W_l\) over any strict majority of tags is \(W_*\).
+Two representations of the same cut profile differ by a constant tuple with value in \(W_*\).
+\end{lemma}
+\begin{proof}
+Fix a base coordinate in \(O_{d,t}\).  It is allowed at exactly \((g-1)/2\) tags.
+A strict majority therefore contains a tag where that coordinate is disallowed.
+Every row and column involving it is zero in \(W_l\) for that tag.
+A matrix in the intersection consequently has zero rows and columns at all \(O\)-coordinates.
+
+The coordinate projection that sets all \(O\)-variables to zero sends each point moment to a point moment defining \(W_*\).
+Applied on both matrix factors, it fixes the matrix just described.
+That matrix is therefore in \(W_*\), proving the nontrivial inclusion.
+
+If \(w,w'\) give the same differences, then
+\(w_l+w'_l=w_{l'}+w'_{l'}\) for every pair of tags.
+Their difference is a constant tuple.  Its value belongs to every \(W_l\), hence to \(W_*\).
+Conversely every such constant tuple lies in the kernel of the cut map.
+\end{proof}
+
+In each \(O_{d,t}\)-block and in \(S\), choose the first \(2r_0\) coordinates in disjoint ordered pairs.
+The corresponding linear functionals on moments are
+\[
+ \eta_{d,t}\bigl(v(s,z)v(s,z)\trans\bigr)
+   =\sum_{\rho=1}^{r_0}z_{O_{d,t},2\rho-1}z_{O_{d,t},2\rho},
+ \qquad
+ \eta_S\bigl(v(s,z)v(s,z)\trans\bigr)
+   =\sum_{\rho=1}^{r_0}z_{S,2\rho-1}z_{S,2\rho}.
+\]
+They are well defined by selecting the indicated ordered matrix entries.
+Put
+\[
+ \eta_O=\sum_{d,t}\eta_{d,t},\qquad \eta=\eta_O+\eta_S.
+\]
+For a cut profile represented by \(w\), define
+\begin{equation}\label{eq:cut-functionals}
+ \begin{aligned}
+ a_t(x)&=\sum_{l,d}\eta_{d,t}(w_l),&
+ a(x)&=\sum_t a_t(x)=\sum_l\eta_O(w_l),\\
+ b_t(x)&=\sum_{l\ne t}\eta_S(w_l),&
+ \chi_*(w)&=\sum_l\eta(w_l).
+ \end{aligned}
+\end{equation}
+The functions \(a_t,a,b_t\) are well defined on \(\cX\).
+Indeed, a constant kernel value in \(W_*\) contributes zero to every \(O\)-tester, while it contributes
+\((g-1)\eta_S(w_*)=0\) to \(b_t\).
+The quantity \(\chi_*(w)\) is used only when a representation has been specified.
+
+Define a symmetric bilinear form \(T^0\) on \(\cX\) by
+\begin{equation}\label{eq:T-zero}
+ T^0=a\otimes a+\sum_{t\in\cI}(a_t\otimes b_t+b_t\otimes a_t).
+\end{equation}
+For \(1\le j\le J\) and ordered components \(e,f\), choose endomorphisms \(L_j^{ef},R_j^{ef}\) of \(\cB\).
+Their required properties will be proved in \Cref{lem:generic-mixers}.
+Set
+\begin{equation}\label{eq:T-one}
+ \begin{split}
+ B^1(x,y)&=\sum_{j=1}^J\sum_{e,f\in\cE}
+   \pair{x_e}{L_j^{ef}y_f(R_j^{ef})\trans},\\
+ T^1(x,y)&=B^1(x,y)+B^1(y,x),\qquad
+ T=T^0+T^1,\qquad
+ r(x)=a+T(x,\cdot)\in\cX^*.
+ \end{split}
+\end{equation}
+Here and later the letter \(r\) in \(r(x)\) denotes an affine-gradient map; the early integer \(r\) in \Cref{eq:law-budgets} is its separately specified rank budget.
+Since the mixed terms in \(T^0(x,x)\) cancel, \(T^1\) is alternating, and \(a(x)^2=a(x)\), we have
+\begin{equation}\label{eq:T-diagonal}
+ T(x,x)=a(x)\qquad(x\in\cX).
+\end{equation}
+
+The prescribed self-frame Gram matrix is denoted by \(E\).
+It is the sum of the ordered tester matrices for \(\eta\) and a matrix \(E^\#\) characterized on evaluation vectors by
+\begin{equation}\label{eq:E-sharp}
+ v(s,z)\trans E^\#v(s',z')
+   =\sum_{\rho=1}^b(s_\rho+s'_\rho)\,z_\#\trans M_\rho z'_\# ,
+\end{equation}
+where \(M_\rho\) are \(n\)-by-\(n\) matrices chosen in \Cref{lem:generic-mixers}.
+The expression is a bilinear form on \(\cB\) because \(j_*\ge1\).
+It vanishes when the two point inputs coincide.
+Consequently
+\begin{equation}\label{eq:E-moments}
+ \pair{E}{w}=\eta(w)\qquad(w\in W).
+\end{equation}
+The \(Z\)-block does not occur in \(E\).
+In particular, any invertible linear change of the \(Z\)-variables preserves \(E\); this symmetry will be used in unary probability estimates.
+
+\subsection{Raw vertices}
+
+For each \(e\in\cE\), take copies \(V_e^+,V_e^-\) of \(\F^N\), paired by the usual dot product.
+We regard \(V_e^-\) as the dual of \(V_e^+\) through that pairing.
+A raw vertex, also called an \emph{orientation}, consists at each component of four maps
+\[
+ P_e,Q_e:\cB\longrightarrow V_e^+,V_e^-,
+ \qquad X_e,Y_e:\F^h\longrightarrow V_e^+,V_e^-,
+\]
+where the target is the first or second one as appropriate.
+They satisfy
+\begin{equation}\label{eq:raw-frame-law}
+ [P_e,X_e]\text{ and }[Q_e,Y_e]\text{ are injective},\qquad
+ P_e\trans Q_e=E,\quad X_e\trans Q_e=0,\quad P_e\trans Y_e=0.
+\end{equation}
+No value is prescribed for \(X_e\trans Y_e\).
+
+Write \(d=\dim\cB\).
+For all large \(n\), \(N\ge2(d+h)\), by the final choice of \(M_0\).
+The frame set is then nonempty.
+For example, take the plus columns to be the first \(d+h\) coordinate vectors.
+Prescribe the full \((d+h)\)-square Gram matrix to be
+\(\begin{psmallmatrix}E&0\\0&0\end{psmallmatrix}\).
+Lift its columns to minus vectors using the first \(d+h\) coordinates, and append distinct coordinate vectors in a complementary space of dimension \(d+h\).
+These minus columns are independent and have the desired Gram values.
+
+Take the uniform distribution on this finite frame set, independently over components.
+Let \(\Omega_n\) be the raw vertex space and \(\rawlaw=\rawlaw_n\) this distribution.
+We call \(P_e,Q_e\) the primal maps and \(X_e,Y_e\) the channel maps.
+The sampling law \(\pi=\pi_n\), defined in \Cref{sec:distributions},
+will preserve the \(\rawlaw\)-marginal of all primal maps and restrict
+the conditional channel choices.  All identities in this section hold
+for every raw vertex, and therefore under either law.
+There are at most \(2^{2|\cE|N(d+h)}\) raw vertices, so
+\begin{equation}\label{eq:raw-space-size}
+ \log_2|\Omega_n|=O(N^2).
+\end{equation}
+
+Use the common ambient tensor space
+\[
+ \cV=\bigoplus_{e\in\cE}V_e^+\otimes V_e^-.
+\]
+For a raw vertex \(o\), define
+\begin{equation}\label{eq:raw-embedding}
+ U_o x=(P_{o,e}x_eQ_{o,e}\trans)_{e\in\cE},
+ \qquad
+ u_o(\xi)=\sum_{e\in\cE}\pair{Y_{o,e}X_{o,e}\trans}{\xi_e}.
+\end{equation}
+On a simple tensor \(pq\trans\) in component \(e\), the latter pairing is
+\[
+ u_o(pq\trans)=(Y_{o,e}\trans p)\cdot(X_{o,e}\trans q).
+\]
+Thus the minus channel is paired with the first tensor factor and the plus channel with the second.
+All copies of coefficient spaces attached to different vertices are distinct, although the ambient \(V_e^\pm\) are common.
+
+Each \(U_o\) is injective: a left inverse of \(P_{o,e}\) and the transpose of a left inverse of \(Q_{o,e}\) recover \(x_e\) from \(P_{o,e}x_eQ_{o,e}\trans\).
+The two annihilator conditions in \Cref{eq:raw-frame-law} also give
+\begin{equation}\label{eq:self-contraction-zero}
+ u_oU_o=0.
+\end{equation}
+
+\begin{definition}\label{def:hole}
+Two raw vertices \(i,j\) have a \emph{hole} between them if there exist \(\lambda_i,\lambda_j\in\cX\) such that
+\begin{equation}\label{eq:hole-equations}
+ \begin{gathered}
+ U_i\lambda_i=U_j\lambda_j,\qquad
+ u_jU_i=r(\lambda_i),\qquad u_iU_j=r(\lambda_j),\\
+ a(\lambda_i)+a(\lambda_j)=1.
+ \end{gathered}
+\end{equation}
+The middle two identities are identities of linear functionals on the whole cut-profile space.
+\end{definition}
+
+\begin{lemma}\label{lem:hole-triangle-free}
+The hole relation is symmetric, has no loops, and is triangle-free.
+\end{lemma}
+\begin{proof}
+Symmetry follows by interchanging the endpoints.
+A loop would have \(U_i\lambda_i=U_i\lambda_j\), hence \(\lambda_i=\lambda_j\) by injectivity, contradicting the last equation of \Cref{eq:hole-equations}.
+
+Suppose that three distinct raw vertices have holes on all three pairs.
+For a directed incidence \(ij\), let \(x_{ij}\) be its witness at \(i\).
+For distinct \(i,j,k\), evaluate the gradient identity for \(ij\) on \(x_{ik}\):
+\[
+ u_j(U_ix_{ik})=a(x_{ik})+T(x_{ij},x_{ik}).
+\]
+Sum over the six ordered choices of \((i,j,k)\).
+For fixed \(j\), the two arguments on the left are equal by the sharing equation for the remaining edge \(ik\), so they cancel.
+For fixed \(i\), the two bilinear terms cancel by symmetry of \(T\).
+The remaining sum is
+\[
+ \sum_{\{i,j\}}\bigl(a(x_{ij})+a(x_{ji})\bigr)=1+1+1=1,
+\]
+a contradiction.
+\end{proof}
+
+Sample \(m\) raw vertices independently from the law \(\pi\) defined in
+\Cref{sec:distributions}, and form \(G\) on their \emph{positions}: two distinct positions are adjacent when their raw types have no hole.
+This gives a finite simple graph even when types repeat.
+Equal raw types are adjacent because holes have no loops.
+An independent triple of positions would therefore give three distinct raw types forming a hole triangle.
+By \Cref{lem:hole-triangle-free},
+\begin{equation}\label{eq:sample-independence}
+ \alpha(G)\le2,\qquad \chi(G)\ge\lceil m/2\rceil.
+\end{equation}
diff --git a/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/03-distributions.tex b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/03-distributions.tex
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+\section{The distribution problem and its preliminary tools}\label{sec:distributions}
+
+A \emph{unit} is an ordered pair of raw vertices with no hole between its endpoints.
+The endpoints of a unit may be dependent.
+When two units are sampled independently, their four endpoints need not be independent within either unit.
+The event of interest is that all four cross pairs have holes, as in \Cref{fig:four-holes}.
+
+The raw reference law \(\rawlaw\) is the uniform law of the exact frames in
+\Cref{eq:raw-frame-law}.  We will replace its conditional channel law by a
+thinned law \(\pi\), while preserving its primal marginal exactly.
+The reference image and peeling arguments below use the absolute joint cap
+\begin{equation}\label{eq:raw-law-caps}
+ \sigma\le 2^{DN}\rawlaw^2.
+\end{equation}
+This condition alone imposes no constant bound on either endpoint marginal.
+The stronger marginal hypotheses required for the distribution tests will
+always be stated separately, relative to \(\pi\).
+
+\subsection{Uniform frame laws}
+
+We record the linear algebra behind all reference image laws.
+For a subspace of a nominal coefficient space, its \emph{image} is its image under the relevant frame map.
+Nominal independence is independence before applying these maps.
+
+\begin{lemma}\label{lem:paired-frame-orbit}
+Let \(A,A':U\to V\) and \(B,B':W\to V^*\) be injective maps between finite-dimensional spaces over \(\F\).
+If
+\[
+ B\trans A=(B')\trans A',
+\]
+there is \(L\in\GL(V)\) with \(LA=A'\) and \(L^{-\mathsf T}B=B'\).
+Thus based individually injective plus and minus frames with a prescribed Gram matrix form one orbit under the simultaneous primal and dual action.
+\end{lemma}
+\begin{proof}
+Write \(\Phi=B\trans:V\to W^*\) and \(\Phi'=(B')\trans:V\to W^*\).
+Both are surjective.
+The based map \(f:A(U)\to A'(U)\) given by \(f(Au)=A'u\) satisfies \(\Phi'f=\Phi\).
+It therefore maps \(A(U)\cap\ker\Phi\) isomorphically onto \(A'(U)\cap\ker\Phi'\).
+Extend this restriction to an isomorphism \(L_0:\ker\Phi\to\ker\Phi'\).
+
+Choose a complement \(U_0\) of \(A(U)\cap\ker\Phi\) in \(A(U)\).
+Its images under \(\Phi\) are independent.
+Extend a basis of \(\Phi(U_0)\) to a basis of \(W^*\), and choose lifts of the added basis vectors under \(\Phi\) and \(\Phi'\), respectively.
+For the basis vectors in \(\Phi(U_0)\), choose the lifts in \(U_0\) and their images under \(f\).
+Define \(L\) to equal \(L_0\) on \(\ker\Phi\) and to carry each chosen lift to its primed lift.
+These are direct-sum decompositions of \(V\), so \(L\) is an isomorphism.
+It extends \(f\) and satisfies \(\Phi'L=\Phi\), which is equivalent to the two required frame identities.
+\end{proof}
+
+\begin{lemma}\label{lem:frame-images}
+The following statements hold for all sufficiently large \(n\).
+\begin{enumerate}[label=\textup{(\alph*)}]
+\item
+At independently sampled raw vertices and components, images of prescribed primal coefficient tuples that are independent separately at each vertex and sign are uniform subject to their individual injectivities and their prescribed same-vertex Gram matrices.
+For tuples using channels as well, the same assertion holds conditionally on the full self-channel Gram matrices \(X_e\trans Y_e\); the relevant full Gram matrix is
+\(\begin{psmallmatrix}E&0\\0&X_e\trans Y_e\end{psmallmatrix}\).
+
+\item
+Let a fixed tuple of nominal directions use arbitrary combinations of the two endpoints' primal and channel coefficients.
+Compute its rank separately in each component and sign in the \emph{direct sum over both endpoints}, and let the total be \(t\).
+For every preassigned \(\epsilon>0\), choosing \(M_0\) sufficiently large gives, under \(\rawlaw^2\),
+\begin{equation}\label{eq:reference-image-cap}
+ \PP\{\text{all those images equal prescribed vectors}\}
+ \le 2^{-(1-\epsilon)tN}\qquad(t\ge1).
+\end{equation}
+This is valid for all possible ranks \(t\), including \(t=O(n)\), not only for bounded tuples.
+
+\item
+The plus frame \([P,X]\) is uniform among injective maps.
+Conditional on it, the minus columns are independent uniforms on the affine spaces prescribed in \Cref{eq:raw-frame-law}, followed only by the conditioning that \([Q,Y]\) is injective.
+The failure probability of this last condition, before conditioning, is at most \(2^{2(d+h)-N}\) per component.
+
+Given \(P,Q\), the channels \(X,Y\) are independent uniform-column draws on \(\ker Q\trans\) and \(\ker P\trans\), respectively, conditioned on \([P,X]\) and \([Q,Y]\) being injective.
+The two rank-failure probabilities are exponentially small in \(N\).
+Each channel frame alone is uniform among injective maps \(\F^h\to\F^N\).
+For any fixed \(q\) independent vectors in its paired ambient space, the probability that transpose evaluation on their span is not injective is at most
+\begin{equation}\label{eq:channel-transpose-failure}
+ 2^{q-h+1}.
+\end{equation}
+The unconditional joint channel law is bounded above by a constant, depending on \(g,h\) but not on \(n\), times the completely independent uniform-column law.
+
+\item
+For \(p\) independent uniform plus vectors and \(q\) independent uniform minus vectors in paired copies of \(\F^N\), and a prescribed \(p\)-by-\(q\) Gram matrix \(G\), the probability of that Gram matrix together with individual injectivity of both frames is at most \(2^{-pq}\) and at least
+\begin{equation}\label{eq:bounded-gram-normalization}
+ 2^{-pq}(1-2^{p-N})(1-2^{p+q-N}).
+\end{equation}
+In particular, any consistent Gram specification on a bounded number of vectors, together with injectivity on each side, has probability bounded below by a positive constant for large \(N\).
+\end{enumerate}
+\end{lemma}
+\begin{proof}
+The raw law is invariant under \(L\) on each \(V_e^+\) and \(L^{-\mathsf T}\) on \(V_e^-\), independently at vertices and components.
+For primal tuples, their restricted Gram is fixed by \(E\).
+By \Cref{lem:paired-frame-orbit}, all based realizations with that Gram and individual injectivity lie in one orbit.
+Projection of an invariant uniform raw law is uniform on this orbit: an element carrying one partial realization to another bijects their full-frame completions.
+There is at least one completion because these tuples are restrictions of a raw frame.
+After fixing \(X_e\trans Y_e\), the same argument applies to arbitrary full-frame coefficient tuples.
+Independence over raw draws and components proves (a).
+
+For (c), the action is transitive on injective plus frames, so their marginal is uniform.
+Given a plus frame, each column of \(Q\) has specified evaluations against \(P,X\), while each column of \(Y\) has zero evaluations against \(P\).
+There is no other constraint until full minus injectivity is imposed.
+The remaining allowed set is therefore the product of those affine column spaces restricted to full rank.
+
+Each column space contains the common translation space
+\(\operatorname{ann}(\im[P,X])\), of dimension \(N-d-h\).
+For a nonzero coefficient vector \(c\in\F^{d+h}\), the corresponding linear combination of minus columns is consequently uniform on an affine space containing that translation space.
+Its chance of being zero is at most \(2^{-(N-d-h)}\).
+Union over the fewer than \(2^{d+h}\) nonzero coefficient vectors gives the asserted bound \(2^{2(d+h)-N}\).
+
+Alternatively, condition first on \(P,Q\).
+The allowed \(X\)-columns lie in \(\ker Q\trans\), the allowed \(Y\)-columns in \(\ker P\trans\), and their two full-rank restrictions are separate.
+This proves the stated conditional independence.
+For example, before imposing injectivity of \([P,X]\), any dependence with a nonzero \(X\)-coefficient has probability at most \(2^{-(N-d)}\); union over its at most \(2^{d+h}\) coefficient choices gives a bound \(2^{2d+h-N}\).
+The estimate for \(Y\) is identical.
+
+The marginal of \(X\) alone is invariant under \(\GL(\F^N)\) and supported on injective maps; transitivity gives its uniformity, and likewise for \(Y\).
+Before conditioning independent channel columns to be injective, their evaluations on a fixed independent \(q\)-tuple form a uniform \(h\)-by-\(q\) matrix.
+For each nonzero coefficient vector in \(\F^q\), the chance that its evaluation is zero is \(2^{-h}\).
+Union over those coefficient vectors gives at most \(2^{q-h}\), and channel injectivity has probability at least \(1/2\) for large \(N\).
+This proves \Cref{eq:channel-transpose-failure}.
+To compare their joint marginal with completely independent columns, condition on \(G=X\trans Y\).
+The conditional marginal is uniform on the orbit of two individually injective \(h\)-frames having Gram matrix \(G\).
+Under independent uniform columns that orbit has probability at least \(2^{-h^2-2}\) for large \(N\), by the argument for (d) below.
+Since the raw probability of any \(G\) is at most one, the density ratio on every such orbit is at most \(2^{h^2+2}\).
+Multiply over the fixed number of components to obtain the joint channel bound.
+Its possible dependence on \(h\) is harmless: \(h\) is fixed before \(n\) grows.
+
+We next prove (b), keeping explicit the dependence on the rank.
+Replace each tested tuple by a basis of its nominal span; inconsistent dependent prescriptions have probability zero.
+For the plus directions at the two endpoints, start with completely independent uniform plus columns and then impose individual full-frame injectivity.
+The image of a rank-\(t_+\) tuple in the joint nominal direct sum is exactly uniform on \(t_+\) independent ambient vectors before this conditioning.
+Indeed, at each ambient coordinate the coefficient map is a surjective linear map onto \(\F^{t_+}\), and different ambient coordinates use independent input bits.
+Full injectivity has probability \(1-2^{-\Omega(N)}\), uniformly over the given construction, so the plus prescription costs at most
+\((1-2^{-\Omega(N)})^{-1}2^{-t_+N}\).
+
+Now condition on all plus frames.
+At a component let \(S\) be the span of both endpoints' plus columns; then \(\dim S\le2(d+h)\).
+Before minus injectivity, every minus column has an affine distribution whose translation space contains \(S^\perp\).
+Split off an independent uniform \(S^\perp\)-part of each column and condition on the remaining parts.
+For a rank-\(t_-\) tuple of minus coefficient combinations, the \(S^\perp\)-parts of its images are independent uniform vectors in \(S^\perp\), because the coefficient map has rank \(t_-\).
+Thus prescribed minus images have probability at most
+\[
+ 2^{-(N-2(d+h))t_-}.
+\]
+The additional conditioning on the two minus frames being injective changes this by a factor \(1+2^{-\Omega(N)}\), by (c).
+Perform this argument componentwise.
+There is a fixed total conditioning factor bounded, for example, by \(4^{|\cE|}\), and the resulting bound is
+\[
+ 4^{|\cE|}\,2^{-t_+N-(N-2(d+h))t_-}.
+\]
+Choose \(M_0\) so that \(2(d+h)/N<\epsilon/2\) for large \(n\), and then increase \(n\) so that the fixed prefactor is at most \(2^{\epsilon N/2}\).
+Since \(t=t_++t_-\ge1\), this proves \Cref{eq:reference-image-cap} uniformly for every allowable \(t\).
+In particular, mixing the two endpoints in a direction causes no loss of an entire \(N\)-bit image constraint.
+
+Finally, to prove (d), the plus columns are independent with probability at least \(1-2^{p-N}\): each nonzero coefficient dependence has probability \(2^{-N}\).
+Given their independence, each minus column takes its required evaluations with probability \(2^{-p}\), independently, so the Gram costs \(2^{-pq}\).
+Under this Gram conditioning, every nonzero combination of minus columns is uniform on an affine space of dimension at least \(N-p\).
+Its probability of vanishing is at most \(2^{-(N-p)}\).
+Union over fewer than \(2^q\) combinations gives failure probability at most \(2^{p+q-N}\).
+This proves the lower bound.
+For the upper bound with plus injectivity required, condition on the plus columns and use the exact \(2^{-pq}\) Gram probability whenever they are independent.
+A partially specified consistent Gram contains some full specification; for bounded \(p,q\), the same positive lower bound for that full specification proves the last assertion.
+\end{proof}
+
+\subsection{Thinning the channel images}
+
+Write \(e_* = |\cE|\), \(d=\dim\cB\), and
+\[
+ q_h=2^h-1,\qquad k_h=2e_*q_h,\qquad
+ \varepsilon_{\rm ch}=10e_*2^h\rho.
+\]
+After the channel dimension \(h\) has been fixed, choose \(\rho>0\) so small
+that \(10^4e_*2^{2h}\rho<\zeta/100\) and that this quantity is smaller than
+every fixed rate slack used below.  We then choose \(M_0\) large enough
+also in terms of \(1/\rho\).  The channel frame in sign \(+\) is \(X_e\),
+and that in sign \(-\) is \(Y_e\).
+
+\begin{lemma}[Channel thinning]\label{lem:channel-thinning}
+Fix any bound \(q_{\rm test}\) on the lengths of the ambient tuples to be tested;
+this bound is fixed before \(\rho,M_0,n\).  For all sufficiently large \(n\)
+there are subsets \(A_e^\pm\subseteq V_e^\pm\), each of density
+\((1+o(1))2^{-\rho N}\), with the following properties.
+Keep the \(\rawlaw\)-marginal on all primal frames \((P_e,Q_e)_e\).
+Given those frames, condition the original channel law on
+\begin{equation}\label{eq:channel-allowed}
+ X_e c\in A_e^+,\qquad Y_e c\in A_e^-
+ \quad(e\in\cE, 0\ne c\in\F^h).
+\end{equation}
+This conditional event has positive probability for every primal frame.
+The resulting probability law \(\pi\) satisfies:
+\begin{enumerate}[label=\textup{(\alph*)}]
+\item Its primal marginal is exactly that of \(\rawlaw\), and
+\begin{equation}\label{eq:thinning-density}
+ \pi^2\le 2^{\varepsilon_{\rm ch}N}\rawlaw^2.
+\end{equation}
+\item For every fixed independent ambient \(q\)-tuple, \(q\le q_{\rm test}\),
+transpose evaluation by any specified channel frame has failure probability
+at most \(2^{q-h+1}+o(1)\) under \(\pi\), uniformly in the tuple.
+More generally every fixed finite union of these rank-failure tests has its
+\(\rawlaw\)-probability plus \(o(1)\) as an upper bound.
+\item Given any primal frames, any component and sign, and any nonzero
+\(v\in V_e^\pm\),
+\begin{equation}\label{eq:channel-translate}
+ \pi\bigl\{\exists c\ne0:\ X_ec\in A_e^++v
+              \mid(P,Q)\bigr\}\le2^{-\rho N/2},
+\end{equation}
+with the analogous statement for \(Y_e,A_e^-\).
+\end{enumerate}
+The error in \textup{(b)} is exponentially small in \(N\).
+The ambient tuple there is fixed before sampling the vertex; no uniform
+conditional rank bound for a tuple selected from the sampled primal frame
+is asserted.
+\end{lemma}
+\begin{proof}
+Include every ambient vector in each \(A_e^\pm\) independently with
+probability \(p=2^{-\rho N}\); subsets for different components and signs
+are independent.  Fix a complete primal frame \(f\), and let \(\nu_f\)
+be the original conditional channel law.  Define its survival fraction
+\(Z_f=\nu_f(\text{\Cref{eq:channel-allowed}})\), a random variable in the
+random subsets.  Channel injectivity makes the \(q_h\) nonzero values in
+each component and sign distinct.  Consequently
+\[
+ \mathbb E_A Z_f=p^{k_h}.
+\]
+For any transpose-rank test \(T\), or any union of at most \(N\)
+such tests, the numerator
+\(Z_{f,T}=\nu_f(T\cap\text{\Cref{eq:channel-allowed}})\) similarly has
+expectation \(p^{k_h}\nu_f(T)\).
+
+We establish simultaneous concentration of these fractions by the
+bounded-differences method \citep{azuma1967,mcdiarmid1989}, with the
+needed exponential-moment estimate included below.  By
+\Cref{lem:frame-images}(c), a prescribed nonzero channel combination,
+conditional on \(f\), has probability at most \(2^{-(N-d)+1}\) of taking
+any prescribed value for all sufficiently large \(N\).  Indeed before
+full-frame injectivity it is uniform on the appropriate annihilator,
+and injectivity has probability \(1-2^{-\Omega(N)}\), uniformly in \(f\).
+Changing one membership bit of a subset can therefore change any such
+fraction by at most
+\[
+ c_N=C_h2^{-N+d},
+\]
+where \(C_h\) is fixed in \(n\).  There are \(2e_*2^N\) membership bits,
+so the sum of their squared influence bounds is at most
+\[
+ S_N=C_{g,h}2^{-N+2d}.
+\]
+For clarity, expose these independent bits and use the Doob martingale
+of any one fraction.  A centered increment whose conditional range has
+length at most \(c\) has exponential moment at most \(\exp(t^2c^2/8)\).
+To see this, the second derivative of its log moment generating function
+is the variance under an exponential tilt, at most \(c^2/4\); integrate
+twice using value and derivative zero at the origin.  Multiplication of
+the conditional bounds and Markov's inequality give
+\[
+ \Pr_A(|Z-\mathbb E_AZ|>u)\le2\exp(-2u^2/S_N).
+\]
+Take \(u=\epsilon_N=2^{-N/10}\).  Since \(d/N\) can be made arbitrarily
+small by increasing \(M_0\), these tails are at most
+\(2\exp(-c_{g,h}2^{N/2})\).  There are \(2^{O(N^2)}\) primal frames and \(2^{O(N)}\) primitive
+transpose tests, since their tuple lengths are bounded by the fixed
+\(q_{\rm test}\).  The unions of at most \(N\) primitive tests therefore
+number at most \(2^{O(N^2)}\) as well.  A union bound gives simultaneous
+error at most \(\epsilon_N\) for all these denominators and test numerators.
+The same argument for \(2^{-N}|A_e^\pm|\), whose bit influences are
+\(2^{-N}\), gives \(|A_e^\pm|/2^N=(1+o(1))p\).
+
+We include the translated numerators in this simultaneous estimate.
+Fix \(f,e,\pm,v\ne0\), and let \(Z_{f,v}^{\pm,e}\) be the original
+conditional probability of survival and of the event in
+\Cref{eq:channel-translate}.  Consider, for example, sign \(+\).
+If \(X_ec+v\) coincides with another already required nonzero channel
+value \(X_ec'\), then \(v=X_e(c+c')\) is a nonzero channel value.
+The original probability of this possibility is at most
+\(C_h2^{-N+d}\), by the point bound and a union over the nonzero
+coefficients.  On that exceptional event we bound the extra test by one;
+the survival probability over the random subsets is still \(p^{k_h}\).
+In every other case \(X_ec+v\) is a new membership bit and supplies an
+additional factor \(p\).  This includes the case \(X_ec+v=0\), since
+zero is not among the required nonzero channel values.  Union over \(c\)
+yields
+\[
+ \mathbb E_AZ_{f,v}^{+,e}
+ \le p^{k_h}\bigl(C_h2^{-N+d}+q_hp\bigr).
+\]
+A changed membership bit can affect this numerator only by being an
+original channel value or a translated channel value.  Both have point
+probability at most \(C_h2^{-N+d}\); hence the same influence and
+concentration estimates apply.  The additional \(2^N\) choices of
+translate do not affect the union bound.  We may therefore fix one
+realization of all subsets for which every displayed concentration
+statement holds simultaneously.
+
+Our choice of \(\rho\) gives \(k_h\rho<1/100\).  Thus
+\[
+ Z_f=p^{k_h}(1+O(\epsilon_N/p^{k_h}))>0
+\]
+uniformly, and the relative error tends to zero exponentially.
+The pointwise density of \(\pi\) with respect to \(\rawlaw\) is the
+survival indicator divided by \(Z_f\).  Consequently
+\[
+ \frac{d\pi^2}{d\rawlaw^2}
+ \le4p^{-2k_h}\le2^{\varepsilon_{\rm ch}N}
+\]
+for large \(N\); the primal law was left unchanged by definition.
+Division of \(Z_{f,T}\) by \(Z_f\) gives
+\(\pi(T\mid f)=\nu_f(T)+O(\epsilon_N/p^{k_h})\), uniformly in every
+specified fixed test.  Average over the original primal marginal and
+apply \Cref{eq:channel-transpose-failure} to prove \textup{(b)}.  Every
+fixed finite union is among our tested unions once \(N\) is large enough;
+applying the same numerator estimate directly to that union gives its
+raw probability plus the stated error, without replacing the union
+probability by the sum of its constituent probabilities.  Finally,
+dividing the translated
+numerator by \(Z_f\) bounds it by
+\[
+ C_h2^{-N+d}+q_h2^{-\rho N}
+       +O(\epsilon_N/p^{k_h})\le2^{-\rho N/2}
+\]
+for all sufficiently large \(N\).  This proves \textup{(c)} and the lemma.
+\end{proof}
+
+\subsection{The two distribution tests}
+
+For independent units \(A=(A_1,A_2)\), \(B=(B_1,B_2)\), write
+\(A\bowtie B\) when all four pairs \((A_i,B_j)\) are holes.  For a point
+\(z\), write \(A\bowtie z\) when both \((A_i,z)\) are holes.
+These definitions also cover repeated raw types; the absence of hole
+loops automatically forbids a conflict with a shared endpoint.
+
+\begin{theorem}[Distribution tests]\label{thm:distribution-tests}
+There is a constant \(\eta_{\rm col}>0\), depending only on the early
+parameters and \(M\), with the following property.  Fix any \(L<\infty\)
+after \(\eta_{\rm col}\) and before the selector and channel parameters;
+choose all remaining parameters in the order of
+\Cref{eq:parameter-order,sec:parameters}.  For every sufficiently large
+\(n\), the law \(\pi\) constructed above satisfies both conclusions.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item Let \((A,c)\) have a probability law on units marked by a colour in
+any fixed finite palette.  The colour may be randomized conditional on
+\(A\).  Write \(\sigma_1,\sigma_2\) for the endpoint
+marginals of its projected unit law \(\sigma\), and suppose
+\begin{equation}\label{eq:coloured-law-caps}
+ \sigma_1\le M\pi,\qquad \sigma_2\le M\pi,\qquad
+ \sigma\le2^{3000gN}\pi^2,
+ \qquad \Pr(c=j)\le\eta_{\rm col}\quad\text{for every }j.
+\end{equation}
+For two independent draws \((A,c),(B,c')\),
+\[
+ \Pr(A\bowtie B, c\ne c')\ge2^{-100gN}.
+\]
+\item Let \(A\) have a unit law \(\sigma\) and let \(z\) be an independent
+point with law \(\theta\), where
+\begin{equation}\label{eq:asymmetric-law-caps}
+ \sigma_1\le L\pi,\qquad \sigma_2\le L\pi,\qquad
+ \sigma\le2^{3000gN}\pi^2,
+ \qquad \theta\le L\pi.
+\end{equation}
+Then \(\Pr(A\bowtie z)\ge2^{-100gN}\).
+\end{enumerate}
+The threshold for \(n\) may depend on the fixed palette and on \(L\).
+All inequalities between measures are pointwise on their finite spaces.
+\end{theorem}
+
+The two conclusions are proved after the collision and overlap arguments,
+in \Cref{prop:coloured-distribution,sec:asymmetric-test}, respectively.
+The finite graph transfer is given in \Cref{sec:graph-transfer}.
+Here we establish their leaf and image hypotheses.  In particular,
+\Cref{eq:thinning-density} places every projected test law below
+\(2^{(3000g+\varepsilon_{\rm ch})N}\rawlaw^2\), leaving strict slack
+below the absolute budget \(D=4000g\).
+
+\subsection{Sparse profiles arising from true intersections}
+
+For a matrix profile, its \emph{total component rank} means \(\sum_{e\in\cE}\rank x_e\).
+
+\begin{lemma}\label{lem:raw-intersections}
+Let \(\sigma\) satisfy the joint cap in \Cref{eq:raw-law-caps}.
+Outside a set of \(\sigma\)-probability \(2^{-\Omega(N)}\),
+\[
+ \sum_{e\in\cE}
+  \dim\bigl(\im P_{1,e}\cap\im P_{2,e}\bigr)\le2D.
+\]
+For any unit outside that exceptional set and any
+\(U_1\lambda_1=U_2\lambda_2\), both \(\lambda_i\) have total component rank at most \(2D\).
+Each has a unique representation \((w_{i,l})_l\) vanishing at a strict majority of tags, and
+\[
+ |\{l:w_{i,l}\ne0\}|\le4D/g,\qquad
+ \chi_*(w_1)=\chi_*(w_2).
+\]
+\end{lemma}
+\begin{proof}
+If the summed intersection dimension exceeds \(2D\), choose \(2D\) independent common-image vectors, componentwise.
+Write each as \(P_{1,e}a=P_{2,e}b\).
+Their nominal relation directions \((a,b)\) are independent in the direct sum of the two primal coefficient spaces at that component.
+Indeed a dependence between the pairs would, after applying \(P_{1,e}\), be the corresponding dependence between the selected common-image vectors.
+Across components their ranks add.
+
+There are at most \(2^{O(n)}\) ways to choose this bounded number of coefficient pairs and their component indices.
+For each choice, \Cref{eq:reference-image-cap} bounds the probability of their zero images by \(2^{-(1-\epsilon)2DN}\) under \(\rawlaw^2\).
+After paying \(2^{DN}\) for \(\sigma\), and making the coefficient count a sufficiently small fraction of \(N\) by the choice of \(M_0\), the union bound is \(2^{-\Omega(N)}\).
+
+Suppose \(U_1\lambda_1=U_2\lambda_2=\xi\).
+The column space of \(\xi_e\) lies in both primal plus spans.
+Since multiplication by the two injective frame maps preserves matrix rank,
+\[
+ \rank(\lambda_{i,e})=\rank(\xi_e)
+ \le\dim(\im P_{1,e}\cap\im P_{2,e}).
+\]
+This proves the total-rank assertion.
+
+Take any representation \((w_l)_l\) of one of the profiles.
+A component \(w_l+w_{l'}\) is nonzero whenever \(w_l\ne w_{l'}\), so there are at most \(2D\) unequal unordered pairs.
+If no value occurred at a strict majority of tags, write \(c_v\) for its multiplicities.
+Then
+\[
+ \#\{\text{unequal pairs}\}
+ =\frac12\left(g^2-\sum_vc_v^2\right)
+ \ge\frac12\left(g^2-\frac g2\sum_vc_v\right)
+ =\frac{g^2}{4}>2D,
+\]
+a contradiction.
+Let the majority have size \(g-z\).
+There are at least \(z(g-z)\ge zg/2\) unequal pairs, hence \(z\le4D/g\).
+Its common value belongs to \(W_*\) by \Cref{lem:cut-kernel}, so subtract it from every coordinate.
+The resulting representation vanishes at a strict majority.
+It is unique: two such representations differ by a constant, and their two majority zero sets intersect.
+
+Finally, equality of the component tensors gives equality of their ambient traces.
+By cyclicity of trace and \Cref{eq:E-moments},
+\[
+ \Tr(P_{i,e}\lambda_{i,e}Q_{i,e}\trans)
+ =\pair{P_{i,e}\trans Q_{i,e}}{\lambda_{i,e}}
+ =\eta(\lambda_{i,e}).
+\]
+For \(e=\{l,l'\}\), it follows that
+\[
+ \eta(w_{1,l})+\eta(w_{2,l})
+ =\eta(w_{1,l'})+\eta(w_{2,l'}).
+\]
+This value is independent of the tag.
+The two sparse representations together have at most \(8D/g=32000<g\) nonzero coordinates, so some tag has both entries zero.
+The common value is therefore zero.
+Summing over tags proves equality of the two specified \(\chi_*\)-values.
+\end{proof}
+
+\subsection{Peeling into leaves with exact pins}
+
+For a unit \(O=(1,2)\), introduce nominal coefficient spaces at each component
+\[
+ \cL_{O,e}
+   =\bigoplus_{i\in O}(\cB_i\oplus H_i^+),
+ \qquad
+ \cR_{O,e}
+   =\bigoplus_{i\in O}(\cB_i\oplus H_i^-),
+ \qquad H_i^\pm=\F^h.
+\]
+Their frame-image maps add the endpoint images in the common ambient \(V_e^\pm\).
+A \emph{pin space} is a specified subspace \(D_{O,e}^\pm\) of this nominal direct sum together with specified exact images of every vector in a chosen basis.
+It may mix both endpoints, and it may mix primal and channel coordinates.
+Once the basis images are known, every image on the pin space is fixed.
+
+For a tested tuple of directions, its \emph{rank modulo the pins} is the sum, over components and signs, of the dimensions of the spans of its images in the quotients by \(D_{O,e}^\pm\).
+The rank is nominal; it is not computed after applying the raw frames.
+
+\begin{lemma}[First peeling and restriction]\label{lem:peeling}
+Take the constants in \Cref{eq:pin-budgets,eq:law-budgets}, and choose the reference accuracy in \Cref{eq:reference-image-cap} with \(\epsilon<\zeta/4\).
+For every law \(\sigma\) satisfying the absolute joint cap \Cref{eq:raw-law-caps}, the following decompositions are available.
+\begin{enumerate}[label=\textup{(\alph*)}]
+\item
+After discarding the exception in \Cref{lem:raw-intersections} and at most \(2^{-\zeta N}\) further mass, the law is partitioned into \emph{leaves}.
+Each leaf has fixed pin spaces with total dimension at most \(K_1\), and its normalized law \(\sigma_\lambda\) satisfies
+\begin{equation}\label{eq:leaf-minentropy}
+ \sigma_\lambda\{\text{a specified exact-image constraint of rank }t
+                 \text{ modulo the pins}\}
+ \le 2^{-(1-\zeta)tN}
+ \qquad(t\ge1).
+\end{equation}
+The normalized leaf laws satisfy a common bound \(2^{O(N)}\rawlaw^2\), with the implied constant determined by the early parameters.
+
+\item
+Suppose a restriction of the mixed law has relative mass \(2^{-o(N)}\).
+One may discard an exponentially small fraction of that restricted law so that every remaining normalized leaf restriction satisfies
+\begin{equation}\label{eq:final-leaf-minentropy}
+ \sigma_\lambda\{\text{a specified exact-image constraint of rank }t
+                 \text{ modulo the pins}\}
+ \le 2^{-(1-2\zeta)tN}
+ \qquad(t\ge1),
+\end{equation}
+and still has density \(2^{O(N)}\) relative to \(\rawlaw^2\).
+For several successive restrictions, the same statement applies to their combined relative cost.
+
+\end{enumerate}
+In all parts the mixed law uses each leaf's actual mass, normalized only after the indicated discards.
+No equal reweighting of leaves is used.
+\end{lemma}
+\begin{proof}
+We give the finite procedure and its quantitative bounds.
+Work first with the original, unnormalized restriction of \(\sigma\) obtained by removing the intersection exception.
+In particular its density is still at most \(2^{DN}\) relative to \(\rawlaw^2\).
+Let \(R\) be its residual support.
+As long as its residual mass exceeds \(2^{-\zeta N}\), begin with its normalized restriction and no pins.
+If \Cref{eq:leaf-minentropy} fails, condition on a violating image constraint and add its nominal directions to the pins.
+Given the existing pins, a consistent tuple constraint can be reduced to a basis of its span modulo the pins.
+The other prescriptions are either automatic or inconsistent; an inconsistent event cannot violate a positive probability bound.
+Thus every step adds positive rank.
+
+If the cumulative added rank is \(u\), the unnormalized mass of the current part is greater than
+\[
+ 2^{-\zeta N-(1-\zeta)uN}.
+\]
+The cumulative directions are independent after taking a basis, so the absolute density cap and \Cref{eq:reference-image-cap} give the upper bound
+\[
+ 2^{DN-(1-\epsilon)uN}.
+\]
+Comparing the bounds yields
+\begin{equation}\label{eq:peeling-rank-comparison}
+ (\zeta-\epsilon)u<D+\zeta.
+\end{equation}
+This is strictly below the rank budget \(K_1\).
+The same comparison applies to a whole violating tuple at once, so a large tuple cannot jump beyond that budget.
+The rank therefore cannot increase indefinitely.
+The process reaches a part with no violating constraint; remove that entire part as a leaf and restart on the residual.
+The underlying raw unit space is finite, and every leaf is nonempty, so the procedure terminates with residual mass at most \(2^{-\zeta N}\).
+
+A leaf of rank \(u\le K_1\) has mass greater than
+\(2^{-\zeta N-(1-\zeta)uN}\).
+Dividing the original density cap by this lower bound shows, for example,
+\[
+ \sigma_\lambda\le
+ 2^{(D+K_1+1)N}\rawlaw^2.
+\]
+This proves (a), including uniformity of its density constant.
+
+For (b), write the current normalized mixture as
+\(\sum_\lambda w_\lambda\sigma_\lambda\).
+Let \(q_\lambda\) be the relative mass retained by the combined restriction in leaf \(\lambda\), and let
+\(q=\sum_\lambda w_\lambda q_\lambda=2^{-o(N)}\).
+Discard leaves for which \(q_\lambda<2^{-\zeta N}\).
+Their mass in the restricted law is at most
+\[
+ q^{-1}\sum_{\lambda:q_\lambda<2^{-\zeta N}}
+       w_\lambda q_\lambda
+ \le 2^{-\zeta N+o(N)}.
+\]
+On every other leaf, division by \(q_\lambda\) multiplies probabilities by at most \(2^{\zeta N}\).
+For \(t\ge1\),
+\[
+ 2^{\zeta N}2^{-(1-\zeta)tN}
+ \le2^{-(1-2\zeta)tN}.
+\]
+This proves the claimed min-entropy and density bounds.
+It also explains why the small mass bound is needed for the mixed law, rather than separately asserted for every old leaf.
+
+Throughout these operations, disintegration over the finite parts is exact.
+Multiplying each normalized part by its actual mass recovers the retained law.
+This proves the last assertion as well.
+\end{proof}
+
+The per-leaf density bounds in \Cref{lem:peeling} are joint bounds relative to \(\rawlaw^2\).
+They do not assert that each leaf has the original one-vertex marginal cap.
+If the initial mixed law has endpoint caps \(C\pi\), restricting it by
+mass \(q\) gives endpoint caps \((C/q)\pi\).  Its primal marginal caps
+are therefore \((C/q)\operatorname{pr}_*\rawlaw\), where
+\(\operatorname{pr}\) forgets all channel columns.  For \(q=2^{-o(N)}\)
+these are \(C2^{o(N)}\) times the original \emph{primal} law.
+The corresponding full-frame bound is \(C2^{o(N)}\pi\), not
+\(C2^{o(N)}\rawlaw\).  The absolute joint bound is
+\(2^{(D+o(1))N}\rawlaw^2\).
+\Cref{sec:probability-scales} records where these distinct controls are used.
+
+\begin{corollary}[Colours and actual leaf weights]\label{cor:coloured-peeling}
+For a law satisfying \Cref{eq:coloured-law-caps}, discard colours whose
+mass is below \(2^{-\zeta N}\), and apply \Cref{lem:peeling}(a) to each
+remaining conditional colour law.  The total discarded mass tends to
+zero exponentially, the leaves have at most \(K_1\) pins and satisfy
+\Cref{eq:leaf-minentropy}, and each leaf has an absolute density bound
+\(2^{(D+K_1+1)N}\rawlaw^2\).  With their actual retained masses these
+leaves form a mixed law whose endpoint marginals are bounded by
+\((1+o(1))M\pi\).  Any subsequent combined restriction of mass
+\(2^{-o(N)}\) has the first-pin freshness in
+\Cref{eq:final-leaf-minentropy}, after the trimming in
+\Cref{lem:peeling}(b).
+\end{corollary}
+\begin{proof}
+The palette size is fixed, so deleting light colours costs at most a
+fixed multiple of \(2^{-\zeta N}\).  Every retained conditional colour
+law has absolute density at most
+\(2^{(3000g+\varepsilon_{\rm ch}+\zeta)N}\rawlaw^2\), which is below
+\(2^{DN}\rawlaw^2\).  The intersection exception and the residual
+mass in \Cref{lem:peeling}(a) are exponentially small uniformly over
+these laws.  Multiply each normalized leaf by its actual colour mass
+and its mass within that colour.  Summing recovers exactly the retained
+restriction of the original law, so its endpoint marginals are bounded
+by the original ones divided by the total retained mass.  This proves
+the asserted pooled bounds.  The final assertion is precisely
+\Cref{lem:peeling}(b), applied to this mixture.  In particular no inverse
+colour mass enters the pooled marginal constant.
+\end{proof}
+
+\begin{corollary}[An independent pair as an unpinned leaf]
+\label{cor:independent-point-leaf}
+Let \(\theta\le L\pi\), and draw two independent \(\theta\)-points.
+The probability that they form an internal hole is \(2^{-\Omega(N)}\).
+Their law conditioned on being a unit is a single leaf with no pins,
+with endpoint marginals at most \((1+o(1))L\pi\), and it satisfies
+\Cref{eq:leaf-minentropy} for every rank \(t\ge1\).  Its effective spaces
+\(C_1,C_2\), defined in \Cref{sec:moments}, are consequently zero.
+\end{corollary}
+\begin{proof}
+Under the raw primal law, each \(P_e\) is uniform injective.  Two
+independent such images have nonzero intersection with probability at
+most \((2^d-1)^2/(2^N-1)\), by a union bound over nonzero coefficient
+pairs.  The primal marginals under \(\theta\) have density at most \(L\)
+relative to those laws.  Union over components therefore bounds the
+probability of any plus-span intersection by
+\(O_g(L^22^{2d-N})=2^{-\Omega(N)}\).
+If the plus spans are disjoint in every component, a tensor common to
+the two endpoint primal tensor spaces has every component zero: each
+of its column spaces lies in the corresponding plus-span intersection.
+Injectivity of the primal frame maps then makes the sharing witnesses
+zero, contrary to the parity sum one in
+\Cref{eq:hole-equations}.  A hole is thus included in the exceptional
+event just bounded.
+
+Before deleting internal holes the pair law is at most \(L^2\pi^2\).
+For a rank-\(t\) exact-image event, \Cref{eq:reference-image-cap} and
+\Cref{eq:thinning-density} bound its probability by
+\[
+ L^2 2^{\varepsilon_{\rm ch}N-(1-\epsilon)tN}.
+\]
+Take \(\epsilon<\zeta/4\).  Since
+\(\varepsilon_{\rm ch}<\zeta/4\) and \(t\ge1\), division by the
+nonhole probability \(1-2^{-\Omega(N)}\) leaves this at most
+\(2^{-(1-\zeta)tN}\) for large \(N\), uniformly for all ranks.
+The same normalization gives the endpoint caps.  Empty pin spaces have
+zero primal projections, which gives the assertion about \(C_i\).
+\end{proof}
+
+\subsection{Bilinear sign estimates}
+
+\begin{lemma}[Walsh estimates]\label{lem:walsh}
+Let all weights below have absolute value at most one.
+\begin{enumerate}[label=\textup{(\alph*)}]
+\item
+If \(x,y\) are independent uniform vectors and \(B(x,y)\) is a bilinear form of rank \(d\), then
+\[
+ \left|\E f(x)g(y)(-1)^{B(x,y)}\right|\le2^{-d/2}.
+\]
+Replacing either uniform law by a law of density at most \(C_i\) multiplies this bound by at most \(C_1C_2\).
+
+\item
+If \(\alpha,\beta\) are subprobability measures on \(\F^d\), with point masses at most \(p_1,p_2\), respectively, then
+\[
+ \left|\sum_{x,y}\alpha(x)\beta(y)f(x)g(y)(-1)^{x\cdot y}\right|
+ \le2^{d/2}(p_1p_2)^{1/2}.
+\]
+
+\item
+For independent uniform global \(N\)-bit vector slots in two groups, a cross-pairing character with coefficient matrix \(C\) has bit rank \(N\rank C\).
+In particular, if its cross coefficient pattern is nonzero, its correlation against two bounded weights, one on each group, is at most \(2^{-N/2}\).
+Arbitrary phases depending separately on the groups may be included in these weights.
+\end{enumerate}
+\end{lemma}
+\begin{proof}
+Let \(H\) be the \(2^d\)-square matrix with entries \(H_{xy}=(-1)^{x\cdot y}\).
+For \(x,x'\in\F^d\),
+\[
+ (HH\trans)_{xx'}=\sum_y(-1)^{(x+x')\cdot y}
+ =\begin{cases}2^d,&x=x',\\0,&x\ne x'.\end{cases}
+\]
+The second value is zero by pairing \(y\) with \(y+e_j\) at any coordinate where \(x+x'\) is nonzero.
+Thus the Euclidean operator norm of \(H\) is \(2^{d/2}\).
+
+For (b), set \(a_x=\alpha(x)f(x)\) and \(b_y=\beta(y)g(y)\).
+Then
+\[
+ |a\trans Hb|\le2^{d/2}\|a\|_2\|b\|_2,\qquad
+ \|a\|_2^2\le p_1\sum_x\alpha(x)\le p_1,
+\]
+and similarly for \(b\), proving the result.
+
+For (a), invertible changes of coordinates reduce the form to the dot product on its \(d\) active coordinates.
+Average each bounded weight over its inactive coordinates.
+Apply (b) to the uniform probabilities \(p_1=p_2=2^{-d}\).
+A bounded-density change can be absorbed into the weights after dividing their bounds by \(C_1,C_2\).
+
+Finally, the matrix of the full bit form in (c) is \(C\otimes I_N\).
+Choose bases reducing \(C\) to \(\operatorname{diag}(I_{\rank C},0)\); the tensor matrix then has rank \(N\rank C\).
+Apply (a), absorbing any within-group phases into the corresponding bounded weight.
+\end{proof}
diff --git a/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/04-moments.tex b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/04-moments.tex
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+\section{Low-rank moments and mixing forms}\label{sec:moments}
+
+This section supplies the finite algebra used in the queries.
+Low-rank moment representations bound the number of selector labels,
+protected labels keep attack atoms independent of the pins, and fixed
+mixing forms provide the rank estimates used in later parameter and
+status tests.
+
+The pin budget $K$ is fixed before the parameters chosen in this
+Section.  The arguments in this section concern the coefficient spaces and
+exact frame constraints; they apply to the thinned law $\pi$ as well as
+to $\rawlaw$.  Set
+\begin{equation}\label{eq:moment-parameters}
+ R_*=2(K+20),\qquad
+ j_*\ge 10(K+1)(R_*+20).
+\end{equation}
+All inequalities concerning ranks in this section are over $\F$.
+The selector size $b$ will be chosen below.  Put
+$p_*=\sum_{j=0}^{j_*}\binom bj$ and write $d_{\mathrm b}=(g^2+3)n$ for the
+number of ordinary base coordinates.  Thus
+$\cB=\F^{p_*}\otimes\F^{1+d_{\mathrm b}}$ and
+$v(s,z)=p_s\otimes(1,z)$, as in \Cref{sec:geometry}.
+The constant base coordinate has index $0$.
+
+The quotient by the radical and the commuting coordinate operators
+used below belong to the flat-extension approach to moment matrices;
+see \citet{laurentmourrain2009}. We prove the required Boolean
+version with separate selector and base-degree bounds.
+
+\subsection{Boolean moment matrices}
+
+\begin{lemma}\label{lem:base-moments}
+The span of $(1,z)(1,z)\trans$, $z\in\F^{d_{\mathrm b}}$, consists precisely of
+symmetric $(1+d_{\mathrm b})$-by-$(1+d_{\mathrm b})$ matrices $Z$ satisfying
+$Z_{ii}=Z_{0i}$ for every $i$.
+For any $u$ distinct selectors, their evaluation vectors $p_s$ are
+linearly independent if $j_*\ge u-1$.
+\end{lemma}
+\begin{proof}
+Every point matrix has the stated identities.  Conversely the space
+specified by these identities has a basis consisting of the matrix
+supported at $(0,0)$, the matrices supported at
+$(0,i),(i,0),(i,i)$, and the matrices supported at $(i,j),(j,i)$ for
+$0<i<j$.  The first is the point matrix at zero.  The second is the
+sum of the point matrices at zero and at the $i$th unit vector.  The
+third is the sum of the point matrices at zero, at the $i$th and $j$th
+unit vectors, and at their sum.  This proves the first assertion.
+
+To isolate a selector $s$ among $u$ distinct selectors, for each other
+selector choose a coordinate on which it differs from $s$, and take
+the product of the corresponding affine bit functions that equal one
+at $s$ and zero at the chosen other selector.  This polynomial has
+degree at most $u-1$ and evaluates to the indicator of $s$ on the
+specified set.  Applying these interpolating linear functionals to a
+relation among the evaluation vectors proves independence.
+\end{proof}
+
+\begin{lemma}[Label support]\label{lem:label-support}
+Every $w\in W$ of rank at most $R_*$ has a representation
+\begin{equation}\label{eq:label-decomposition}
+ w=\sum_{s\in L}(p_s\otimes I)Z_s(p_s\otimes I)\trans,
+ \qquad |L|\le R_*,
+\end{equation}
+where each $Z_s$ is symmetric, $(Z_s)_{ii}=(Z_s)_{0i}$, and
+$\sum_{s\in L}\rank Z_s\le R_*$.
+\end{lemma}
+\begin{proof}
+Choose a linear combination of point matrices representing $w$.
+It defines a linear functional $\Lambda$ on the Boolean functions
+of selector degree at most $2j_*$ and base degree at most two.
+The matrix $w$ is the matrix of the pairing
+$B(f,g)=\Lambda(fg)$ on functions of base degree at most one and
+selector degree at most $j_*$.  Products and degrees here are in
+the Boolean algebra, so each variable satisfies $x^2=x$.
+
+Let $V_j$ denote the subspace with selector degree at most $j$,
+and let $r_j$ be the rank of $B|_{V_j\times V_j}$.  The sequence
+$r_j$ is nondecreasing and bounded by $R_*$.  Because
+$j_*\ge 6R_*+4$, there is $j\ge4R_*+2$ such that
+\begin{equation}\label{eq:flat-levels}
+ r_j=r_{j+1}=r_{j+2}.
+\end{equation}
+Indeed, if no such three successive values occurred between
+levels $4R_*+2$ and $j_*$, every two successive increments in
+that interval would contain a strict increase, giving more than
+$R_*$ increases.
+
+Let $Q$ be the quotient of $V_{j+2}$ by the radical of its
+restricted pairing.  The induced pairing on $Q$ is nondegenerate.
+Equality of the first and last ranks in \eqref{eq:flat-levels}
+means that the image of $V_j$ is all of $Q$: its dimension is at
+least $r_j=\dim Q$.  Since the quotient pairing is nondegenerate,
+a vector of $V_j$ pairing to zero with $V_j$ consequently pairs
+to zero with $V_{j+2}$.
+For a selector coordinate $s_a$, define $M_a[f]=[s_a f]$ using
+$f\in V_j$.  This is well defined: if $[f]=0$, then for every
+$g\in V_j$,
+\[
+ B(s_a f,g)=B(f,s_a g)=0,
+\]
+and $V_j$ spans $Q$.  The same identity proves that $M_a$ is
+self-adjoint.  Replacing $s_bf$ by a representative in $V_j$ and
+transferring $s_a$ to the other argument gives
+\[
+ B(M_aM_b[f],[g])=\Lambda(s_as_bfg).
+\]
+All representatives needed in this comparison lie in $V_{j+2}$.
+Thus $M_aM_b=M_bM_a$ and, since $s_a^2=s_a$, $M_a^2=M_a$.
+
+Commuting idempotents over $\F$ have a simultaneous eigenspace
+splitting
+\[
+ Q=\bigoplus_{s\in L}Q_s,
+ \qquad M_a|_{Q_s}=s_a I.
+\]
+For completeness, a single idempotent splits its space into kernel
+and image; all the other commuting maps preserve both summands,
+so iteration proves the assertion.  Distinct joint eigenspaces
+are orthogonal, by self-adjointness of a coordinate on which their
+labels differ.  Each nonzero summand has positive dimension, so
+$|L|\le\dim Q\le R_*$.  Define $Z_s$ by pairing the projections
+onto $Q_s$ of the affine base coordinates.  Then $Z_s$ is symmetric
+and $\rank Z_s\le\dim Q_s$.
+
+Repeated application of the $M_a$ gives the class of a base
+coordinate times any selector monomial of degree at most $j$.
+It follows that \eqref{eq:label-decomposition} reproduces all
+moments of selector degree at most $2j$.  To verify the diagonal
+identity separately on a label, use an interpolating selector
+polynomial $e_s$ of degree at most $|L|-1$.  Its operator on $Q$
+is the projection onto $Q_s$.  The Boolean identities give
+\[
+ B(z_i e_s,z_i e_s)
+ =\Lambda(z_i e_s^2)
+ =B(e_s,z_i e_s),
+\]
+which is $(Z_s)_{ii}=(Z_s)_{0i}$.  The interpolation degrees are
+within $j$.
+
+Extend the right side of \eqref{eq:label-decomposition} to the
+full row and column indexing.  Its rank is at most
+$\sum_s\dim Q_s\le R_*$.  By \Cref{lem:base-moments} it belongs
+to $W$.  If its difference from $w$ were nonzero, choose a
+nonzero difference moment of minimal selector degree $d'>2j$.
+Write its selector monomial as the product on a set $U$ of size
+$d'$, and keep the two affine base coordinates in that moment
+fixed.  Index rows by the subsets $A\subset U$ of size
+$\lfloor d'/2\rfloor$ and columns by the complementary monomials
+$U\setminus A'$ with the same indexing.  The union
+$A\cup(U\setminus A')$ is $U$ exactly when $A=A'$; otherwise
+its degree is smaller than $d'$.  The difference on this square
+submatrix is therefore the identity.  Both its row and column
+degrees are at most $j_*$, whereas its rank is
+$\binom{d'}{\lfloor d'/2\rfloor}>2R_*$.  This contradicts the
+rank bound $2R_*$ for the difference and completes the proof.
+\end{proof}
+
+\subsection{Labels protected from the pins}
+
+For an endpoint $i$ of a leaf, let $S_{i,e}^{\pm}\subset\cB$
+be the projection of $D_{O,e}^{\pm}$ onto its individual primal
+coordinate block.  These are projections, not intersections;
+a stored pin may mix endpoints and channel coordinates.
+
+\begin{lemma}\label{lem:sparse-pin-labels}
+Put $L_0^{\rm lab}=R_*+14$.  At an endpoint, the union of the
+nonzero label supports of all vectors in all $S_{i,e}^{\pm}$
+that can be written as
+$\sum_{s\in L}p_s\otimes z_s$ with $|L|\le L_0^{\rm lab}$
+has size at most
+\begin{equation}\label{eq:label-budget}
+ B_0=2|\cE|K(R_*+14).
+\end{equation}
+\end{lemma}
+\begin{proof}
+For one projected space, choose a basis from its members admitting
+such sparse representations, for the span of all these members.
+There are at most $K$ chosen members.  Comparing any further sparse
+member with its expression in this basis uses at most
+$(K+1)(R_*+14)$ labels.  Their evaluation vectors are independent
+by \Cref{lem:base-moments} and \eqref{eq:moment-parameters}.
+Consequently every label of the further member occurs in one of
+the chosen basis representations.  There are at most
+$K(R_*+14)$ such labels per projected space.  Summing over signs
+and components proves the stated, deliberately generous, bound.
+\end{proof}
+
+Exclude these labels whenever an attack atom is selected at that
+endpoint.  For later finite exclusions set
+\[
+ B_*=10(B_0+10000g+1)
+\]
+and choose $b$ so large that
+\begin{equation}\label{eq:selector-size}
+ 2^{b-2j_*}>100g(B_*+1).
+\end{equation}
+The effective test space at endpoint $i$ is
+\begin{equation}\label{eq:effective-space}
+ C_i=\{x\in\cX:x_e\in S_{i,e}^+\otimes S_{i,e}^-
+                  \text{ for every }e\in\cE\}.
+\end{equation}
+Fix a basis of this space on each leaf.  Since the total stored
+pin dimension is at most $K$, the sum over components of the
+individual projection dimensions in either sign is at most $K$.
+Therefore
+\begin{equation}\label{eq:effective-bounds}
+ \dim C_i\le K^2,\qquad
+ \sum_e\rank x_e\le K\quad(x\in C_i).
+\end{equation}
+If pins are enlarged, retain the old effective spaces and their
+chosen basis data as well; they are subspaces of the new ones.
+
+\begin{definition}[Atoms and flavors]\label{def:atom}
+For a tag $l$, selector $s$, and base point $z$ allowed at $l$,
+let $\mathfrak a_l(s,z)\in\cX$ be the cut profile represented
+by $w_l=v(s,z)v(s,z)\trans$ and $w_t=0$ for $t\ne l$.
+Its diagonal bit is $q=\eta(w_l)$ and its role bit is
+$a(\mathfrak a_l(s,z))$.
+A generic flavor retains all base coordinates allowed at $l$.
+A shared-only flavor sets all $O$ blocks to zero.  A pure flavor
+sets $S$ to zero and retains a specified subset of the allowed
+$O$ blocks, setting the others to zero.  Every flavor retains
+both $\#$ and $Z$ in their entirety.
+At a fixed label, a parameter test samples all retained bits
+independently and uniformly.  Its tester summary is the list of
+all individual quadratic tester values.  In addition to this
+summary we may record the basis values of
+$T(\mathfrak a_l(s,z),\cdot)|_{C_i}$.
+\end{definition}
+
+\subsection{Choice of the mixers}
+
+For a quadratic function $f$ on a binary vector space, its polar form is
+$(u,v)\mapsto f(u+v)+f(u)+f(v)+f(0)$; its polar rank is the rank
+of this bilinear form.
+
+Fix, for example,
+\begin{equation}\label{eq:mixer-parameters}
+ A=10(K+1)p_*(g^2+5),\qquad s_0=4\lceil A\rceil,
+ \qquad J>100(s_0+K^2+1).
+\end{equation}
+These constants precede $r_0,h$ and $M_0$.
+
+\begin{lemma}[Uniform mixing forms]\label{lem:generic-mixers}
+For all sufficiently large $n$ one can choose the matrices
+$L_j^{ef},R_j^{ef}$ and $M_a$ in \Cref{sec:geometry} so that:
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item\label{item:mixer-unary}
+For every nonzero $x\in\cX$ of total component rank at most $K$,
+every tag, label and flavor, and every fixing of the non-$Z$
+base coordinates, the quadratic function
+$T(x,\mathfrak a_l(s,z))$ of the $n$ remaining $Z$ bits has polar
+rank at least $2(J-s_0)$.  It uses a space of linear forms on $Z$
+of bounded dimension, independent of $n$.  This space can be
+chosen before the non-$Z$ coordinates are fixed.
+\item\label{item:mixer-binary}
+For two distinct labels and any two flavors, every nonzero parity
+of the indexed $L_j^{ef},R_j^{ef}$ evaluations between the two
+point inputs, allowing each matrix in either order or both,
+and optionally either or both orders of $E$, has bilinear-part
+rank at least $n/5$.  Here a nonzero parity involving an $L$ or
+$R$ evaluation means that at least one such indexed ordered
+evaluation occurs.  The same rank bound holds for the three
+parities consisting of $E$ alone, its reverse alone, or their sum.
+\end{enumerate}
+The choices are made before any unit law is selected.
+\end{lemma}
+\begin{proof}
+A symmetric rank-$r$ matrix $x$ on a $d_B$-dimensional coordinate
+space can be written $ZHZ\trans$, where $Z$ has $r$ independent
+columns and $H$ is a nonsingular symmetric $r$-by-$r$ matrix.
+To verify the factorization, take a left inverse $L$ of the
+chosen column basis $Z$ and put $H=LxL\trans$.  The projection
+$ZL$ fixes $x$ on the left and, by symmetry, on the right;
+hence $ZHZ\trans=x$.  Its rank forces $H$ to be nonsingular.
+Neither $H$ nor the original matrix is required to have nonzero
+diagonal.  For a fixed component rank tuple of total at most
+$K$, counting $Z,H$ gives at most $2^{d_BK+K^2+K}$ choices.
+There are at most $(K+1)^{|\cE|}$ such rank tuples.  Therefore
+the number of profiles under consideration satisfies
+\begin{equation}\label{eq:low-rank-profile-count}
+ \#\{x\in\cX:\textstyle\sum_e\rank x_e\le K\}\le2^{An}
+\end{equation}
+for all sufficiently large $n$, since
+$d_B=p_*(1+(g^2+3)n)$.
+
+Choose all $L,R$ matrices independently and uniformly.  Fix a
+nonzero profile and write $x_e=Z_eH_eZ_e\trans$, with
+$r_e=\rank x_e$.  For a tag-$l$ atom the contribution from a
+single indexed matrix uses the forward forms $Z_e\trans L v$
+when $l\in f$, and the reverse forms $v\trans L Z_f$ when
+$l\in e$, and the corresponding forms for $R$.
+Regard these initially as separate formal variables.  Across
+all indices their number is at most
+\[
+ m_0=4J(g-1)K.
+\]
+Each forward or reverse pair contributes a bilinear product
+with nonsingular middle matrix $H_e$ or $H_f$.  These products
+use disjoint formal variables.  Since $x\ne0$, for each $j$
+there is at least one such nonzero product, so the formal
+quadratic has polar rank at least $2J$.
+
+We justify the distribution of their linear parts on $Z$ even
+when forward and reverse restrictions overlap.  For a fixed
+matrix $L$, specifying $Z_e\trans L$ and $LZ_f$ imposes at
+most $r_er_f$ compatibility bits, namely their common
+$Z_e\trans LZ_f$ block.  After restriction to the free $Z$
+directions, the resulting list remains uniform on a linear
+subspace of codimension at most $r_er_f$ in the space of all
+lists.  Equivalently its law is dominated by $2^{r_er_f}$ times
+the completely uniform law.  Over all matrices the domination
+factor is at most $2^{2JK^2}$; only matrices with both relevant
+incidences contribute a compatibility term.
+
+For a uniform $m$-by-$n$ matrix, the probability of corank at
+least $s_0+1$ is at most
+$2^{m(s_0+1)-(s_0+1)n}$: choose $s_0+1$ independent row
+relations and require all their $n$ evaluations to vanish.
+If $m<s_0+1$ the probability is zero.  The preceding domination
+therefore bounds the bad probability for a fixed profile, tag
+and label by
+\[
+ 2^{2JK^2+(s_0+1)m_0}\,2^{-(s_0+1)n}.
+\]
+There is no union over non-$Z$ values or flavors in this estimate:
+they affect affine shifts, not these linear parts on the always
+retained $Z$ block.  A union over at most $2^{An}g2^b$ choices
+tends to zero.  On the complementary event the image of the
+$Z$ inputs in the formal variable space has codimension at most
+$s_0$.  Restricting a bilinear form to a subspace of codimension
+$s_0$ reduces rank by at most $2s_0$: in a basis extending
+that subspace one deletes $s_0$ rows and $s_0$ columns, each
+deletion decreasing rank by at most one.  This proves
+\ref{item:mixer-unary}.  The formal list itself supplies the
+claimed bounded space of linear forms.  The term $T^0$ has no
+$Z$ variables and does not affect this argument.
+
+For \ref{item:mixer-binary}, distinct labels have independent
+selector evaluation vectors.  Their two $Z$ direction spaces
+are therefore disjoint.  Restricting a random matrix to these
+two spaces in the two orders reads independent off-diagonal
+blocks.  Any nonempty parity involving ordered $L,R$ evaluations
+is consequently a uniform $n$-by-$n$ matrix on these directions,
+even after all other matrices and $E$ have been fixed.
+
+For the parities using only $E$, choose its matrices $M_a$
+independently and uniformly.  On the two $\#$ direction spaces,
+$E$ restricts to
+\[
+ A_{s,s'}=\sum_{a=1}^b(s_a+s'_a)M_a,
+\]
+which is uniform because $s\ne s'$.  The reverse gives
+$A_{s,s'}\trans$, and their sum is alternating.  An
+$\lfloor n/2\rfloor$ square block with disjoint row and column
+indices is uniform also in this last case.  A uniform
+$d_1$-by-$d_2$ matrix has probability at most
+$2^{(d_1+d_2)r-d_1d_2}$ of rank at most $r$, by representing
+such a matrix as a product of a $d_1$-by-$r$ and an
+$r$-by-$d_2$ matrix.  Taking $r< n/5$ gives probability
+$2^{-\Omega(n^2)}$ in all the preceding cases, including the
+square block of side $\lfloor n/2\rfloor$.
+
+There are at most $2^{2b}$ ordered label pairs and
+$2^{4J|\cE|^2+2}$ parities to check.  These counts are fixed
+in $n$.  The $Z$ and $\#$ restrictions also show that the
+tester part of $E$ and the choice of flavor cannot reduce the
+ranks just proved.  A union bound establishes both properties
+simultaneously.  Fix one successful choice for each sufficiently
+large $n$.
+\end{proof}
+
+\begin{remark}\label{rem:Z-reparametrization}
+Both $\rawlaw$ and $\pi$ have an exact symmetry under a common
+$G\in\GL_n(2)$ change of the $Z$ input coordinates at one vertex,
+using that change at every component and in both primal modes.
+To see this, let $R_G$ be the induced automorphism of $\cB$: it
+acts as $G$ on the $Z$ block, as the identity on the constant and
+other base blocks, and as the identity on the selector factor.
+Replace
+\[
+ (P_e,Q_e,X_e,Y_e)
+ \quad\hbox{by}\quad
+ (P_eR_G,Q_eR_G,X_e,Y_e).
+\]
+Since $E$ has no $Z$ rows or columns, $R_G\trans E R_G=E$.
+All prescribed self-Grams, annihilator equations and full-frame
+injectivities are therefore preserved.  The inverse transformation
+uses $G^{-1}$, so this is a bijection preserving $\rawlaw$ and its
+primal marginal.
+
+Moreover $\im(P_eR_G)=\im P_e$ and
+$\im(Q_eR_G)=\im Q_e$.  Conditional on the primal maps, the
+admissible channel maps are consequently exactly the same before
+and after this transformation: their annihilator and injectivity
+conditions depend only on these primal spans.  The ambient channel
+maps themselves and all prescribed ambient thinning sets are fixed.
+Thus the allowed channel sets and their conditional normalizing
+counts are unchanged.  Explicitly, if $\mathcal S$ denotes the
+channel-thinning event and $c(P,Q)=\PP_{\rawlaw}(\mathcal S\mid P,Q)$,
+then $c(PR_G,QR_G)=c(P,Q)$.  Both the retained primal marginal
+and the conditional density $\1_{\mathcal S}/c(P,Q)$ defining
+$\pi$ in \Cref{sec:distributions} are therefore preserved.
+
+Tester summaries are unchanged.  The assertion concerns the two
+reference laws; $T^1$ need not be invariant.  Under the inverse
+change of variables in a parameter test, each resulting key and
+its tester data are preserved, while the bounded space of linear
+forms on $Z$ in \Cref{lem:generic-mixers},
+part~\ref{item:mixer-unary}, is reparametrized.  In particular the
+unfiltered empirical mass of every test depending only on that key
+and the unchanged tester data is preserved.  The same equality holds
+after conditioning on any feasible tester summary: this event is
+independent of $Z$, and its input normalizing count is unchanged.
+\end{remark}
+
+Choose $r_0$ only after the preceding parameters and the bounded
+rank requirements in \Cref{sec:realization,sec:phases}, with all
+fixed marginal-density bounds already specified; retain
+$h=1000r_0$.  Choose the thinning rate $\rho>0$ after $h$ as in
+\Cref{sec:distributions}.  Finally choose $M_0$, also relative to
+$1/\rho$, large enough for every stated nominal-coefficient and
+scalar-record cost to be the required small fraction of $N=M_0n$.  None of the finite-test accuracies
+used later changes these choices.
diff --git a/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/05-realization.tex b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/05-realization.tex
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+\section{Recipes and realization of the gradients}\label{sec:realization}
+
+This section is deterministic.  We construct cross Gram entries
+that turn a finite collection of matched vector keys into four
+hole witnesses.  The construction uses the exact frame identities,
+the stored nominal pin spaces and finite-dimensional linear
+algebra.  It therefore applies to every frame in the support of
+the thinned law $\pi$, and to every restricted leaf law, without
+any conditional uniformity assumption on its channels.
+
+We use the nominal coefficient spaces of \Cref{sec:distributions}.
+For a unit $O$ and component $e$, these are
+\[
+ \cL_{O,e}=\bigoplus_{i\in O}(\cB_i\oplus H_i^+),\qquad
+ \cR_{O,e}=\bigoplus_{i\in O}(\cB_i\oplus H_i^-),
+ \qquad H_i^\pm=\F^h.
+\]
+Their images are obtained by summing the corresponding frame
+columns in the common ambient space.  If $i'$ is the other
+endpoint of $O$, put $p_i=u_{i'}U_i\in\cX^*$.
+
+\subsection{Small tables and scalar recipes}
+
+\begin{definition}\label{def:small-table}
+For a leaf and component, put
+\[
+ \mathcal J_{O,e}^{\pm}
+ =\bigoplus_{i\in O}(S_{i,e}^{\pm}\oplus H_i^{\pm}).
+\]
+For two leaves belonging to units $A,B$, a small table specifies
+the two cross bilinear forms on
+$\mathcal J_{A,e}^+\times\mathcal J_{B,e}^-$ and
+$\mathcal J_{B,e}^+\times\mathcal J_{A,e}^-$, for every $e$.
+It is admissible for two orientations if it agrees with their
+actual cross Grams whenever at least one of the two table
+arguments is a stored pin direction.
+
+Let $H_{i,0}^{\pm}$ be the projection of the appropriate pin
+space onto $H_i^{\pm}$, and fix a complement
+$H_i^{\pm}=H_{i,0}^{\pm}\oplus H_{i,1}^{\pm}$.
+The complement is called private.  We identify the two channel
+coordinate copies with $\F^h$ in their common column order.
+The table is injecting if, for each component, each orientation
+of the roles $O,O'$, and $i\in O,z\in O'$, the following maps
+are injective:
+\begin{align}\label{eq:table-injection}
+ D_{O,e}^+\cap\cB_i&\longrightarrow\F^h/H_{z,0}^+,
+ &d&\longmapsto\text{table evaluation of $d$ against $Y_z$},\\
+ D_{O,e}^-\cap\cB_i&\longrightarrow\F^h/H_{z,0}^-,
+ &d&\longmapsto\text{table evaluation of $d$ against $X_z$}.
+ \notag
+\end{align}
+These quotient conditions use the standard coordinate dot
+pairing on the channels.  For instance, the first image is
+zero in the quotient precisely when it pairs to zero with
+all vectors annihilating $H_{z,0}^+$.
+\end{definition}
+
+Contractions on $C_i$ computed from the table carry a star.
+Thus $(u_zU_i)^*(x)$, for $x\in C_i$, is computed by writing
+$x_e$ in $S_{i,e}^+\otimes S_{i,e}^-$ and taking the dot product
+of its two table evaluations against $Y_z,X_z$.
+This is linear in $x$.  All coordinates involved have bounded
+dimension, although that bound may depend on $h$.
+
+For every cross pair $i,z$, choose between one and seven atoms
+at $i$ and the same number at $z$, with a bijection preserving
+the tag and the diagonal bit $q$ of each position.  At any
+one endpoint all labels in its two lists must be distinct and
+must avoid the exclusions of \Cref{lem:sparse-pin-labels}.
+Write $v_{iz}$ for the sum of its list toward $z$.  A scalar
+recipe requires, in both orientations of the two unit roles,
+\begin{equation}\label{eq:recipe-scalars}
+ \begin{gathered}
+ \sum_{\text{list toward }z}q=1,
+ \qquad a(v_{iz})+a(v_{zi})=1,\\
+ T(v_{iz},\cdot)|_{C_i}
+       =\bigl(a+(u_zU_i)^*\bigr)|_{C_i},\\
+ (p_i+a)(v_{iz})=(p_{i'}+a)(v_{i'z}).
+ \end{gathered}
+\end{equation}
+An atom $\mathfrak a_l(s,y)$ at $i$ has vector key
+\[
+ \bigl(P_{ie}v(s,y),\ Q_{ie}v(s,y)\bigr)_{e\ni l},
+\]
+with a fixed order of its $2(g-1)$ entries.  Matching keys means
+equality in this order at every matched atom position.
+
+\begin{lemma}[Table flags and binary prescriptions]\label{lem:small-tables}
+The following statements hold.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item For a fixed pair of leaves and a numerical small table,
+admissibility is the intersection of two unary orientation
+filters, one on each unit.  Its numerical coordinate data,
+including injection and the bases of the effective spaces,
+have bounded description length apart from the nominal
+coefficient bases, which require $O(n)$ bits.
+\item The key directions of the lists at either unit are jointly
+independent modulo its table-coordinate spaces, component by
+component and sign by sign.
+\item Given \eqref{eq:recipe-scalars}, there is a prescription of
+the binary $L,R$ evaluations between distinct atoms at each
+endpoint, depending only on their tags, tester summaries and
+$p$ bits, which makes
+\begin{equation}\label{eq:recipe-gradients}
+ r(v_{iz})(x)=
+ \begin{cases}
+  0,&x\text{ is an atom of the list toward }z,\\
+  p_{z'}(\widetilde x),&x\text{ is an atom toward }z',
+ \end{cases}
+\end{equation}
+where $z'$ is the other endpoint of the opposite unit and
+$\widetilde x$ is the matched atom there.
+\end{enumerate}
+\end{lemma}
+\begin{proof}
+A pin image is fixed on its leaf.  A cross entry with a pinned
+first argument therefore depends only on the other unit's
+orientation, and conversely for a pinned second argument.
+Entries with both arguments pinned are numerical tests on the
+leaf pair.  This proves the first assertion about filters.
+Use bases of the projected pin spaces to encode tensors of
+$C_i$: their coordinates lie in spaces of dimension at most
+$K^2$, and there are at most $K^2$ basis tensors.  The pin,
+projection and channel relationships and the table are
+bounded-dimensional matrices.  The nominal bases themselves
+have $O(K\dim\cB)=O(n)$ coefficients.
+
+For the second assertion, project a purported dependence onto
+an individual primal coordinate block.  It expresses a sum of
+at most fourteen attack rays as an element of $S_{i,e}^{\pm}$.
+Any nonzero such vector would have a sparse representation on
+attack labels and is forbidden by \Cref{lem:sparse-pin-labels}.
+Independence of those label factors and the nonzero constant
+coordinate of each point then makes all coefficients zero.
+The channels introduce no relation because the table space
+is the direct sum of its individual primal and channel blocks.
+
+For the last assertion, at one endpoint let $I_1,I_2$ be its
+two disjoint nonempty lists.  The matrix of $T$ on its atoms
+must be symmetric, with diagonal $a$ on each atom.
+Prescribing the two list sums of its rows is possible exactly
+when their self-pairings equal the corresponding diagonal
+sums and their mutual pairings agree.  To see sufficiency,
+subtract any symmetric matrix with the required diagonal.
+The remaining task prescribes an alternating form on the
+span of the two independent list indicators and its pairings
+with the whole coordinate space.  Extend those indicators to
+a basis, retain the prescribed entries in their rows and
+columns, and assign the remaining alternating entries
+arbitrarily.
+
+For \eqref{eq:recipe-gradients}, the self-pairing requirements
+hold because $T(v,v)=a(v)$.  Mutual consistency is
+\[
+ p_z(v_{zi})+p_{z'}(v_{z'i})
+       =a(v_{iz})+a(v_{iz'}).
+\]
+The coupled equation at the opposite unit gives the left side
+as $a(v_{zi})+a(v_{z'i})$, which equals the displayed right side
+by the two role-sum equations in \eqref{eq:recipe-scalars}.
+Thus the symmetric matrix exists.
+
+Every off-diagonal entry can be implemented by formal $L,R$
+bit prescriptions on that pair of atoms.  The value of $T^0$
+is already determined by their tester summaries.  Choose one
+indexed product term in $T^1$ linking a component of the first
+star to a component of the second star; set its two factors to
+give the desired correction and set all other product factors
+to zero.  Forward and reverse evaluations are distinguished.
+The two labels at this endpoint differ, so these are the
+indexed ordered evaluations covered by
+\Cref{lem:generic-mixers}.  This argument specifies the bits;
+their occurrence in parameter tests will be supplied by
+\Cref{lem:binary-mixing}.  It makes no assumption about
+independence of arbitrary actual evaluations at fixed points.
+\end{proof}
+
+\subsection{The deterministic realization theorem}
+
+\begin{theorem}[Gradient realization]\label{thm:gradient-realization}
+Suppose two units in fixed leaves have lists satisfying
+\eqref{eq:recipe-scalars}, the binary prescriptions yielding
+\eqref{eq:recipe-gradients}, equal matched keys, and an
+admissible injecting small table.  Freeze their actual cross
+Gram entries whenever either argument is a pin or a key
+direction.  Then there are abstract cross bilinear forms on
+the full nominal spaces, extending all these frozen entries,
+whose primal-versus-channel entries satisfy
+\begin{equation}\label{eq:realized-gradients}
+ u_zU_i=r(v_{iz})\quad\text{on }\cX
+ \qquad(i\text{ and }z\text{ in opposite units}).
+\end{equation}
+Only the primal-versus-channel entries need be prescribed in
+the final test.  Their number is at most
+$16|\cE|h\dim\cB=O(n)$, with its coefficient fixed before
+$M_0$.  Agreement of the actual cross Grams with these entries
+produces all four cross holes, with witnesses $(v_{iz},v_{zi})$.
+\end{theorem}
+\begin{proof}
+Equal keys imply
+$U_iv_{iz}=U_zv_{zi}$, and the role sums in
+\eqref{eq:recipe-scalars} supply the witness parity.
+It remains to construct the gradients.  We distinguish
+abstract Gram entries from the random event that the actual
+entries agree with them.
+
+\paragraph{A baseline of bounded primal rank.}
+At each component and sign choose a splitting of the table
+space as $\mathcal J=D\oplus\mathcal J'$, append the key
+space, and append an additional primal complement.
+The independence in \Cref{lem:small-tables} makes these
+splittings possible.  Retain the small table on the product
+of the two table spaces and the actual frozen entries in any
+pin or key row or column.  These prescriptions agree on their
+overlaps by admissibility.  Set the free blocks from an
+additional primal complement to the opposite $\mathcal J'$
+to zero, in both orientations, and set the additional-primal
+by additional-primal block to zero.  This defines baseline
+cross bilinear forms.
+
+Fix an opposite endpoint $z$.  Let
+\[
+ F^0_{z,e}:\cL_{O,e}\longrightarrow\F^h,\qquad
+ G^0_{z,e}:\cR_{O,e}\longrightarrow\F^h
+\]
+be evaluation against its $Y_z,X_z$ channels.  On additional
+primal inputs, evaluation of an opposite channel factors
+through its projection from $\mathcal J$ onto $D$; hence the
+rank there is at most $\dim D$.  The remaining primal inputs
+are in bounded-dimensional projected pin and key spaces.
+It follows that the ranks of both baseline maps on all primal
+inputs are bounded independently of $r_0,h,n$.  Arbitrary
+channel-channel entries of the small table do not affect this
+bound.  A uniform bound for either map on the combined primal
+inputs of the two endpoints is
+\begin{equation}\label{eq:baseline-rank-bound}
+ B_{\mathrm{lin}}=3K+28:
+\end{equation}
+the opposite pin projection costs at most $K$ dimensions,
+the two individual projected primal spaces cost at most $2K$,
+and the two endpoints have at most twenty-eight key rays.
+
+Let $u_z^0U_i$ denote the resulting contractions.  They give
+the table values on $C_i$.  On an attack atom toward $z$ they
+are zero, since its tensor is the equal-key tensor at $z$ and
+$u_zU_z=0$.  On an attack toward $z'$ they equal
+$p_{z'}$ of its matched atom, since that tensor is the
+corresponding tensor at $z'$.  These are precisely the values
+in \eqref{eq:recipe-gradients}.
+
+\paragraph{Allowed changes.}
+Let $\mathcal A_{O,e}^{\pm}$ be the span of all key directions
+on the indicated side of the unit, and put
+\[
+ \mathcal F_{O,e}^{\pm}=D_{O,e}^{\pm}+\mathcal A_{O,e}^{\pm}.
+\]
+Write bars for the quotients
+\[
+ \overline{\cL}_{O,e}
+ =\cL_{O,e}/\bigl(\mathcal F_{O,e}^{+}
+                      +\textstyle\bigoplus_iH_{i,1}^{+}\bigr),
+ \quad
+ \overline{\cR}_{O,e}
+ =\cR_{O,e}/\bigl(\mathcal F_{O,e}^{-}
+                      +\textstyle\bigoplus_iH_{i,1}^{-}\bigr).
+\]
+We may change $F_z^0,G_z^0$ by any maps
+\begin{equation}\label{eq:allowed-channel-changes}
+ \delta F_{z,e}:\overline{\cL}_{O,e}
+                     \longrightarrow(H_{z,0}^-)^\perp,
+ \qquad
+ \delta G_{z,e}:\overline{\cR}_{O,e}
+                     \longrightarrow(H_{z,0}^+)^\perp.
+\end{equation}
+Here each map is extended to a bilinear change supported on
+its opposite channel block and zero on the opposite primal
+blocks.  It vanishes on all frozen entries: the domain
+quotient kills frozen directions on one side, while on the
+other side a pin's channel projection lies in $H_{z,0}$ and
+is annihilated by the allowed values.  A reciprocal-role
+change in the same Gram block contributes nothing on this
+map's primal inputs, because it is supported on a channel
+argument there.  On private-by-private blocks both changes
+are zero.  They may add further channel-channel values, which
+are irrelevant to the primal contractions.  Thus all choices
+below may be made independently for opposite endpoints and
+for the reciprocal unit roles as far as primal-input values
+are concerned.
+
+\paragraph{The unrestricted linear response.}
+Fix $z$ and solve simultaneously for the two $i\in O$.
+Consider the linear space of responses formed by
+\[
+ F_z^0\cdot\delta G_z+\delta F_z\cdot G_z^0
+\]
+and all bilinear forms on
+$\overline{\cL}_{O,e}\times\overline{\cR}_{O,e}$, with their
+contractions summed over components on each individual cut
+profile.  We claim that the pair of target forms
+\begin{equation}\label{eq:target-difference}
+ \bigl(r(v_{iz})-u_z^0U_i\bigr)_{i\in O}
+\end{equation}
+belongs to this response space.
+
+Let $(x_i)_{i\in O}\in\cX^O$ annihilate all responses.
+Annihilating the arbitrary pure bilinear forms says that at
+each component the sum of the two nominal tensors has zero
+image in the product of barred quotients.  Project further
+onto the individual primal spaces modulo
+$S_{i,e}^{\pm}$ and the individual key spans.  This projection
+is well defined on the barred spaces, since every pin
+projects into $S_{i,e}^{\pm}$.  It shows that $x_{i,e}$ vanishes
+after both such quotients.  A matrix killed after quotienting
+its two factors by spaces of dimensions $d_+,d_-$ has rank
+at most $d_++d_-$: in bases extending those spaces all entries
+outside their rows and columns vanish.  Consequently
+\[
+ \rank x_{i,e}\le2K+28\le R_*.
+\]
+Apply \Cref{lem:label-support} to every component.
+
+For an attack label at $i$, compare its label summand with
+all key directions in that component.  Their combined label
+set has at most $R_*+14$ elements, whose direction spaces
+$p_s\otimes\F^{1+d_{\mathrm b}}$ form a direct sum.  Any vector of
+$S_{i,e}^{\pm}$ in this sum has zero attack-label component,
+by \Cref{lem:sparse-pin-labels}.  Thus the attack summand
+projects to zero modulo just its individual key ray if the
+atom uses this component, and is zero if there is no such
+ray.  In the former case, writing its affine base vector as
+$v=(1,y)$, a symmetric matrix killed in the two quotients
+by $v$ has the form
+\[
+ Z=cvv\trans+vd\trans+dv\trans.
+\]
+Indeed choose $v$ as the first basis vector: the surviving
+matrix can be supported only on its first row and column.
+The moment identities in the original affine coordinates
+now give
+\[
+ Z_{ii}=cv_i,\qquad Z_{0i}=cv_i+d_i+d_0v_i,
+\]
+so $d_i=d_0v_i$ and $Z=cvv\trans$.
+
+The cut relation on any triangle of tags separates label by
+label: the union of the three supports has at most $3R_*$
+labels, within the interpolation bound.  At this attack
+label the nonzero components lie on its tag star; the
+triangle relation forces their scalar coefficients to be
+equal.  Subtract that scalar multiple of the attack atom
+from $x_i$.  Do this for every attack label.  Each subtracted
+atom annihilates the response space, because both its factor
+directions are keys.  The remaining profiles still
+annihilate the response and have only nonattack label
+support.  Removing these separated label blocks does not
+increase their component ranks or label support sizes.
+
+At one component contract a remaining $x_{i,e}$ on the minus
+side by a primal functional $\phi$ at $i$ annihilating
+$S_{i,e}^-$ and all individual key directions.  Extend $\phi$
+by zero on other endpoint and channel blocks; it is a
+functional on $\overline{\cR}_{O,e}$.  Pure annihilation shows
+that the resulting individual plus vector $d$ is a sum of a
+pin, key directions, and private channel directions.  In
+this representation all key coefficients vanish.  To see
+this, project onto each individual primal block.  At the
+other endpoint the projected pin would be a sparse sum of
+attack rays.  At endpoint $i$ it is the difference of the
+nonattack vector $d$, supported on at most $R_*$ labels, and
+a sum of at most fourteen attack rays.  In both cases the
+label exclusion prevents any attack coefficient.  The
+channel components of the pin lie in the protected spaces
+$H_0$, whereas the remaining channel term lies in their
+chosen complements $H_1$.  They must both vanish.  Therefore
+\[
+ d\in D_{O,e}^+\cap\cB_i.
+\]
+Test the derivative response using
+$\delta G_{z,e}=\phi\otimes c$ for arbitrary
+$c\in(H_{z,0}^+)^\perp$.  It gives
+$F_{z,e}^0(d)\cdot c=0$.  Hence the table evaluation of $d$
+is zero modulo $H_{z,0}^+$, and injection in
+\eqref{eq:table-injection} yields $d=0$.
+The reciprocal contraction proves the same assertion in the
+minus mode.
+
+It follows that the remaining component tensors lie in the
+products of the individual projected pin plus key spaces.
+Their nonattack support removes the key directions by the
+same sparse-label argument.  Thus the remaining profiles
+belong to $C_i$.  We have proved that every annihilator is a
+sum of effective-space profiles and attack atoms.  The target
+\eqref{eq:target-difference} vanishes on the former by
+\eqref{eq:recipe-scalars} and on the latter by
+\eqref{eq:recipe-gradients} and the baseline values.
+In a finite-dimensional vector space the image of a linear
+map is the annihilator of the kernel of its dual: this
+follows by extending a basis of the image to a basis of the
+target space.  It proves the claimed linear solvability.
+
+\paragraph{A low-rank representation of the target.}
+For one $i$, the form $r(v_{iz})$ has a matrix representation
+on the components of rank at most
+\begin{equation}\label{eq:unconstrained-gradient-rank}
+ 15r_0+14J|\cE|
+\end{equation}
+per component.  Here is an explicit routing for its tester
+part.  Let $w=(w_t)$ be the representation given by its atom
+list.  The list condition gives $\chi_*(w)=\sum q=1$.
+On an allowed $O_{d,t}$ block at any tag the coefficient in
+$a+T^0(v_{iz},\cdot)$ is
+\[
+ 1+a(v_{iz})+b_t(v_{iz})
+ =1+\chi_*(w)+\eta_S(w_t)=\eta_S(w_t).
+\]
+There are at most seven tags $t$ with nonzero $\eta_S(w_t)$.
+Route this tester from each allowed tag $l\in I_d$ to $d$
+through the component $\{l,d\}$.  It contributes nothing at
+$d$, where $O_{d,t}$ is disallowed.  At a component this uses
+at most fourteen ordinary testers, allowing either endpoint
+as center.  For the shared tester, route a star centered at
+$t$ with coefficient $a_t(v_{iz})$.  Its contribution at the
+center is $(g-1)a_t(v_{iz})=0$, and at any other tag it gives
+the desired term in $a_t(v_{iz})b_t$.  Consolidating gives at
+most one shared tester per component.  Each ordered tester
+matrix has rank $r_0$, proving the $15r_0$ bound.
+
+For $T^1(v_{iz},\cdot)$, each of the two orientations of an
+indexed term supplies a matrix of rank at most the component
+rank of $v_{iz}$.  This profile is a sum of at most seven
+atoms, so its total component rank is at most $7(g-1)$.
+The total contribution per component has rank at most
+$14J(g-1)\le14J|\cE|$.  This proves
+\eqref{eq:unconstrained-gradient-rank}.
+
+\paragraph{Removing the bounded-dimensional obstructions.}
+Let $M_{i,e}$ be the preceding representing matrices.  On each
+individual primal factor choose a projection $A_{i,e}^{\pm}$
+whose kernel is the sum of its projected pin and key spaces.
+The matrix
+$(A_{i,e}^+)\trans M_{i,e}A_{i,e}^-$, extended by zero outside
+that individual block, is a permitted pure quotient form.
+Its sum over $i$ has rank at most twice
+\eqref{eq:unconstrained-gradient-rank}.  Moreover
+\begin{align}\label{eq:bounded-rank-removal}
+ M_{i,e}-(A_{i,e}^+)\trans M_{i,e}A_{i,e}^-
+ &= (I-A_{i,e}^+)\trans M_{i,e}
+       +(A_{i,e}^+)\trans M_{i,e}(I-A_{i,e}^-),\\
+ \rank\bigl(M_{i,e}-(A_{i,e}^+)\trans M_{i,e}A_{i,e}^-\bigr)
+ &\le \rank(I-A_{i,e}^+)+\rank(I-A_{i,e}^-)
+ \le 2K+28.\notag
+\end{align}
+Subtract these pure forms from the linear task.  Its residual,
+including the baseline contraction, has a per-individual
+matrix representation of bounded rank independent of
+$r_0,h,n$, and remains linearly soluble.
+
+In any unrestricted solution, replace the derivative output
+maps by bounded-rank maps in the same allowed value spaces,
+preserving all their dot products with the opposite baseline
+values on primal inputs.  Such replacement is possible as
+follows.  The map from the allowed value space to the dual
+of the span of those baseline values has bounded rank.
+Choose a linear section of its image and compose with that
+map.  This has bounded-dimensional image and preserves every
+observed pairing.  Postcomposing the original derivative map
+with this value projection leaves its primal derivative
+response unchanged and preserves its domain quotient.
+The two new derivative maps have bounded rank.  Their extra
+product $\delta F_z\cdot\delta G_z$ is also a bounded-rank
+pure form, which we include with opposite sign in the
+remaining pure task.  That task still has an unrestricted
+pure solution.  Its per-individual target matrices have rank at
+most
+\begin{equation}\label{eq:residual-rank-bound}
+ R_{\mathrm{res}}=2K+28+4B_{\mathrm{lin}}=14K+140.
+\end{equation}
+Here the first term bounds the projection difference, one
+$B_{\mathrm{lin}}$ bounds the baseline contraction, two bound
+the derivative terms, and one bounds their extra product.
+
+\paragraph{Compression of the remaining pure task.}
+We show that the latter task has a pure solution of bounded
+rank.  At endpoint $i$, expand all projected pin and key
+vectors, and all row and column forms in its bounded-rank
+target representation, along selector coordinates and base
+blocks.  Their counts depend on earlier construction
+parameters but not on $r_0,h,n$.  On each ordinary base block
+let $U$ be the span of the vectors to fix and $K_0$ the common
+kernel of the forms to preserve.  Choose a complement in
+$K_0$ to $U\cap K_0$, and project along that complement onto
+a subspace containing $U$.  The resulting endomorphism fixes
+$U$, preserves all the forms, and has rank at most
+$\dim U+\codim K_0$.  There are at most
+$2|\cE|(K+14)$ vectors to fix and at most
+$2|\cE|R_{\mathrm{res}}$ forms to preserve at one endpoint
+before selector expansion.  Thus an explicit bound for the
+rank on an ordinary block is
+\begin{equation}\label{eq:block-compression-bound}
+ D_{\mathrm{blk}}
+ =2p_*|\cE|(K+14+R_{\mathrm{res}}).
+\end{equation}
+
+Use these endomorphisms blockwise, fix the constant base
+coordinate, and act trivially on the selector factor.  Use
+the same resulting map at every component and in both modes
+at this endpoint.  It sends each allowed point to an allowed
+point and therefore preserves all $W_l$ and the cut domain.
+It fixes the individual projected pin and key vectors.
+Taking identity on channels consequently fixes every mixed
+pin itself and preserves private channel subspaces.  The
+map descends to the barred spaces and has bounded rank there:
+only the bounded protected channel dimensions survive their
+quotients.  On either barred space a uniform bound is
+\begin{equation}\label{eq:quotient-compression-bound}
+ D_{\mathrm{quo}}
+ =2p_*\bigl(1+(g^2+3)D_{\mathrm{blk}}\bigr)+2K.
+\end{equation}
+The first term counts the two compressed primal blocks and
+the second bounds their surviving channel projections.
+
+Precompose an unrestricted pure solution on its two arguments
+with these maps.  Its rank per component becomes bounded.
+On individual cut profiles its value is unchanged, because
+the compressed profile remains in that cut domain and every
+form in the target representation is preserved.  It is
+therefore the required bounded-rank pure solution.
+
+\paragraph{Factoring the pure forms through channels.}
+Restore the earlier projected pure forms.  Their sum with
+the compressed correction has rank at most
+\begin{equation}\label{eq:final-pure-rank}
+ 30r_0+28J|\cE|+D_{\mathrm{quo}}
+\end{equation}
+per component for the fixed opposite endpoint.  In particular
+the additive constant is independent of $r_0,h,n$.
+In the two allowed channel value spaces impose, in addition,
+orthogonality to the opposite baseline and bounded derivative
+values on primal inputs.  These are only boundedly many
+linear restrictions.  Each remaining value space has
+codimension at most $K+2B_{\mathrm{lin}}$, so their dot pairing
+has rank at least $h-2K-4B_{\mathrm{lin}}$: restricting a
+nondegenerate pairing by codimensions $c_1,c_2$ loses at most
+$c_1+c_2$ in rank.  With $h=1000r_0$, the sufficient requirement
+is
+\begin{equation}\label{eq:realization-r0-requirement}
+ 970r_0>28J|\cE|+D_{\mathrm{quo}}+2K+4B_{\mathrm{lin}}.
+\end{equation}
+Every quantity on its right is fixed before $r_0$, so this is
+one of the permitted requirements on that choice.
+
+Factor a rank-$r$ desired pure bilinear form as
+$\sum_{a=1}^r\ell_a\otimes m_a$.  Choose vectors $f_a,g_a$
+in these remaining channel spaces with $f_a\cdot g_b$ equal
+to the Kronecker delta; they exist by the pairing rank.
+The changes $\sum_a\ell_af_a$ and $\sum_am_ag_a$ realize
+that form as their dot product.  Their cross terms with the
+baseline and the previous small derivative changes vanish
+on primal inputs by the additional orthogonality conditions.
+They obey \eqref{eq:allowed-channel-changes} and hence
+preserve every frozen entry.  This realizes the target
+exactly for both endpoints $i\in O$.
+
+Repeat for both opposite endpoints and both unit roles.
+The independence of primal-input changes already proved
+ensures that the resulting two cross Gram blocks realize
+all equations \eqref{eq:realized-gradients} simultaneously.
+Only primal-channel entries have been used.  In each of the
+two blocks there are at most $8h\dim\cB$ such entries per
+component, proving the asserted count.  Actual agreement
+with them gives all gradient equations; equal keys and the
+role sums give the other witness conditions.  All four
+cross pairs are therefore holes.
+\end{proof}
diff --git a/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/06-collision.tex b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/06-collision.tex
new file mode 100644
index 0000000000000000000000000000000000000000..5268b1ae337d6784a287f9818b15c74713788aac
--- /dev/null
+++ b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/06-collision.tex
@@ -0,0 +1,337 @@
+\section{From overlapping key distributions to four holes}\label{sec:collision}
+
+The deterministic construction becomes useful when its keys collide.
+The principal cost is equality of ambient vectors; the remaining
+primal-channel scalar entries have a smaller cost.  We give a
+criterion that retains this distinction, including for cells selected
+using the parameters on the opposite side.  The two unit laws may
+be different.  We also keep colour and type tests in the successful
+event throughout the argument.
+
+\subsection{Queries and their reference measure}
+
+A numerical template fixes the shape of all lists in a scalar
+recipe: positions, endpoints, tags, labels, flavors and desired
+$q$ bits, with matching tags and $q$ bits across the two sides.
+It also fixes each atom's \emph{numerical unary status}: its tester
+summary, role $a(\mathrm{atom})$, basis values of
+$T(\mathrm{atom},\cdot)|_{C_i}$, and $p_i$ bit.
+It fixes the within-endpoint $L,R$ bit prescriptions of
+\Cref{lem:small-tables}. Its labels are distinct at every endpoint.
+For a fixed pair of leaves it may specify a small table and two
+separate orientation filters, including its admissibility filters.
+Only choices for which equal accepted keys satisfy the hypotheses
+of \Cref{thm:gradient-realization} will be used.
+
+The query at a unit draws fresh independent point parameters at
+all its positions and evaluates their keys.  Parameters may be
+uniform on the retained flavor coordinates, or normalized
+conditional on the prescribed, possible $q$ bit at each position.
+This latter conditioning depends only on point parameters and
+has a fixed positive probability whenever it is possible.  In
+particular the two units' parameter draws remain independent.
+No normalization is made for the other status, binary, Gram or
+orientation acceptance conditions.  Every resulting accepted
+measure is a subprobability measure.
+
+Let $k$ be the number of ambient vector slots on one template
+side, counting components and signs.  There are at most four
+lists of seven atoms, each with $2(g-1)$ slots, so
+\begin{equation}\label{eq:key-rank-budget}
+ k\le 56(g-1)=k_{\max}.
+\end{equation}
+Let $\cK_n$ be the set of global tuples of these $k$ vectors
+satisfying the prescribed paired diagonal bit at every position
+and component, and having every off-position plus/minus inner
+product within a component equal to zero.  Write $\nu_n$ for
+the uniform probability measure on $\cK_n$.
+
+The set $\cK_n$ has density bounded below by a positive constant
+in the space of all $k$ independent uniform vectors, and
+\begin{equation}\label{eq:key-space-size}
+ |\cK_n|\le 2^{kN}.
+\end{equation}
+Indeed the number of slots is fixed.  With probability tending
+to one the plus vectors in each component are independent;
+given them, each minus vector fulfills a fixed list of
+independent linear equations with the prescribed probability
+$2^{-r}$, where $r$ is the number of equations on it.  Their
+total number is fixed.  This supplies a constant lower bound,
+including all diagonal and off-position requirements.
+Queries are accepted only when their keys belong to $\cK_n$.
+
+For a pair of leaves, let $d_A,d_B$ be the densities relative
+to $\nu_n$ of the separate accepted key measures, integrating
+over their normalized leaf orientation laws and parameter draws.
+The densities include their orientation filters and all query
+conditions but are not renormalized for success.  Thus
+$\int d_A\,d\nu_n,\int d_B\,d\nu_n\le1$.
+Both filters may depend on the ordered pair of leaves, while
+each is a predicate on its own orientation and query parameters.
+A test on the leaf pair, such as distinct colours or prescribed
+law types, is written as an external indicator
+$J_n(\ell_A,\ell_B)\in\{0,1\}$.  Equivalently, one can set
+both accepted measures to zero when that test fails.
+
+\begin{proposition}[Filtered collision criterion]\label{prop:collision}
+Let $\sigma_A,\sigma_B$ be two laws on units, possibly carrying
+finitely many colour or type marks.  After forgetting these
+marks, suppose each law has density at most $2^{3000gN}$
+relative to $\pi\otimes\pi$.
+For $O\in\{A,B\}$, let $\widetilde\sigma_O$ be a normalized
+restriction of $\sigma_O$ of relative mass at least $2^{-o(N)}$,
+decomposed into leaves satisfying
+\eqref{eq:final-leaf-minentropy}.  On each side mix the leaves
+with their actual masses in $\widetilde\sigma_O$.
+
+Fix a required predicate $\Psi(A,B)$ on the marked units and
+an allowed leaf-pair indicator $J_n$.
+Use only numerical query options whose joint acceptance on
+an allowed leaf pair, including their separate orientation
+filters, implies $\Psi(A,B)$.  Suppose these accepted queries satisfy
+\begin{equation}\label{eq:overlap-criterion}
+ \E_{\text{independent leaf pair}}
+       J_n(\ell_A,\ell_B)\int d_A d_B\,d\nu_n\ge 2^{-o(N)}.
+\end{equation}
+Then independent draws $A\sim\sigma_A$, $B\sim\sigma_B$ satisfy
+$\Psi(A,B)$ and have all four cross holes with probability at least
+\begin{equation}\label{eq:collision-conclusion}
+ 2^{-(k_{\max}+.03)N-o(N)}.
+\end{equation}
+It suffices to obtain \eqref{eq:overlap-criterion} after skipping
+some leaf pairs, or after summing over a bounded number of
+numerical options.  In particular, a positive subsequential
+overlap lower bound yields \eqref{eq:collision-conclusion} along
+that subsequence.  With $\sigma_A=\sigma_B$ and
+$\Psi$ the distinct-colour test, the conclusion retains that
+test.  The statement also applies when the two sides have
+different prescribed law types.
+\end{proposition}
+\begin{proof}
+First note where thinning enters the hypotheses.  By
+\Cref{lem:channel-thinning}, with
+$\varepsilon_{\rm ch}=10|\cE|2^h\rho$, the projected unit laws
+satisfy
+\[
+ \sigma_O\le 2^{3000gN}\pi^{\otimes2}
+ \le 2^{(3000g+\varepsilon_{\rm ch})N}
+          \rawlaw^{\otimes2}.
+\]
+The choice of $\rho$ leaves this exponent strictly below
+$D=4000g$, also after a restriction of mass $2^{-o(N)}$.
+Thus the reference image estimates and first peeling are
+charged against the original law $\rawlaw$; they supply the
+all-rank leaf hypothesis used here.  No uniform conditional
+channel law under $\sigma_O$ or $\pi$ is used in this proof.
+
+We first work with one shape of vector slots and one option per
+leaf pair.  A bounded number of shapes or options will be
+handled at the end.  We count only leaf pairs with $J_n=1$,
+and every accepted measure already contains its filters
+enforcing $\Psi$ on these pairs.  Leaves are drawn independently
+from their two actual mass distributions; none of the following
+estimates requires those distributions to be equal.
+
+\paragraph{The numerical records.}
+Fix two leaves, a key $q\in\cK_n$, and point parameter values
+$a,b$ on the two sides.  Freeze the images of all pin and key
+directions.  The pin images are known from the leaves and the
+key images are the entries of $q$.  At each unit record the
+pairings of all its nominal columns with these opposite frozen
+images.  Record any numerical statuses in
+\Cref{eq:recipe-scalars,eq:recipe-gradients} not already fixed
+by the query conditions.  For fixed parameters these records,
+together with acceptance, partition the unit's orientations
+into unary cells.  A partition is allowed to depend on the
+opposite parameters; it does not depend on the opposite
+orientation once its frozen images have been specified.
+
+There are at most
+\begin{equation}\label{eq:record-count}
+ L_n=2^{Cn}
+\end{equation}
+cells on either side, for a constant $C$ fixed before $M_0$.
+To verify the count, there are $O(\dim\cB+h)=O(n)$ columns
+and only boundedly many pin and key vectors to pair against.
+Nominal projected-pin bases use $O(n)$ coefficients.  The
+basis tensors of $C_i$ are encoded in their bounded projected
+pin bases, rather than as full matrices on $\cB$; this uses
+boundedly many further coordinates.  Parameter values require
+$O(n)$ bits and the remaining table and status information has
+bounded size.  The ambient pin and key images are already
+fixed and are not charged again as new records.
+
+The numerical records determine every frozen cross entry.
+Together with the fixed algebraic data, leaves, parameters,
+and table, they determine a choice of the abstract solution
+in \Cref{thm:gradient-realization}.  One can make this choice
+deterministic by taking the first solution in fixed coordinate
+orders on the finite spaces.  Its unknown entries need not
+be encoded as additional records.  In particular, a global
+signature cylinder or a complicated orientation filter is
+only an acceptance predicate, not an input requiring a new
+coordinate description for this solver.
+
+\paragraph{Pruning at a fixed parameter pair.}
+Put
+\begin{equation}\label{eq:cell-threshold}
+ \tau_n=2^{-(k+.01)N}.
+\end{equation}
+For the fixed parameter pair $(a,b)$ and key $q$, let
+$\alpha_a^b(c;q)$ be the mass of a first-side accepted
+orientation in record cell $c$, measured in its normalized
+leaf law.  Define $\beta_b^a(d;q)$ similarly.  Denote their
+sums over records by $\alpha_a(q),\beta_b(q)$; these sums
+do not depend on the opposite parameter, since it only
+refines the record partition.  Their sums over all keys
+are at most one.
+
+Discard for this fixed pair only cells of mass less than
+$\tau_n$.  The product mass lost from small first-side cells
+at key $q$ is at most
+\[
+ L_n\tau_n\,\beta_b(q).
+\]
+Sum over keys and average the independent parameter weights.
+The loss in collision probability is at most $L_n\tau_n$,
+since $\sum_q\beta_b(q)\le1$ for every $b$.
+Discarding small second-side cells loses at most the same amount.
+Because $\nu_n$ is uniform, the overlap integral is
+$|\cK_n|$ times the probability of an accepted key collision.
+Thus the total overlap loss, for any fixed leaf pair and
+also after averaging leaf pairs, is at most
+\begin{equation}\label{eq:fubini-pruning}
+ 2|\cK_n|L_n\tau_n
+ \le 2^{1+Cn-.01N}
+ \le 2^{-.005N}
+\end{equation}
+for the chosen sufficiently large $M_0$ and all sufficiently
+large $n$.
+
+Every retained cell is its original unary cell and has mass
+at least $\tau_n$.  We have not intersected the retained
+sets over different opposite parameter values; such an
+intersection could reduce a cell below its threshold.
+Instead all subsequent estimates are made for this fixed
+parameter pair and its unaltered cells, and then averaged.
+This preserves both their mass bounds and the product of
+the two conditional orientation laws.
+
+\paragraph{Image entropy inside a retained cell.}
+By \Cref{lem:small-tables}, the $k$ nominal key directions
+are jointly independent modulo the pins, with rank computed
+separately by component and sign and jointly across the two
+endpoints.  Suppose a further tuple has rank $t\ge1$ modulo
+pins and these key directions.  Apply
+\eqref{eq:final-leaf-minentropy} to the combined tuple of
+rank $k+t$, prescribe the known key images and any images
+of the further tuple, and divide by the cell mass at least
+$\tau_n$.  Its largest conditional point mass is at most
+\begin{align}\label{eq:cell-pmax}
+ 2^{-(1-2\zeta)(k+t)N+(k+.01)N}
+ &=2^{[-(1-2\zeta)t+.01+2\zeta k]N}
+ \le2^{-.95tN}.
+\end{align}
+Here $k\le k_{\max}$ and
+$\zeta=1/(1000k_{\max})$, so $.01+2\zeta k\le .012$.
+The inequality holds for every $t\ge1$, including tuples
+whose rank grows with $n$.  Further restrictions defining
+the cell cause no additional loss: the numerator was bounded
+by an event containing the entire cell and the prescribed
+images, and only the original cell mass has been divided out.
+
+\paragraph{Testing the remaining scalar Gram entries.}
+For a retained cell pair the abstract target is fixed.  Let
+$s_n$ be the number of its tested primal-channel entries in
+the two cross Gram blocks.  By
+\Cref{thm:gradient-realization} and the final choice of $M_0$,
+\[
+ s_n\le16|\cE|h\dim\cB<.01N.
+\]
+Expand the indicator that all these entries agree with the
+target as a sum of $2^{s_n}$ binary characters, with the
+factor $2^{-s_n}$.
+
+A character has a coefficient tensor in the direct sum of
+the two cross nominal tensor products, over all components.
+Take its image after quotienting every nominal factor by
+pins and keys.  If this image is zero, its value depends
+only on frozen cross entries.  More explicitly the kernel
+of a tensor-product quotient is the sum of tensors with a
+frozen factor on at least one side; all their pairings have
+been recorded.  The actual and target Grams agree there,
+so the disagreement character is identically one.
+
+Otherwise let $t\ge1$ be the sum of the ranks of the quotient
+coefficient matrices, over components and the two cross
+orientations.  A rank factorization expresses its phase,
+up to a constant and two separate phases involving frozen
+images, as
+\[
+ \sum_{a=1}^t X_a\cdot Y_a.
+\]
+On each unit the corresponding coefficient directions have
+joint rank $t$ modulo its frozen space.  The two different
+cross orientations use opposite nominal signs, so their
+ranks add.  By \eqref{eq:cell-pmax}, each tuple of actual
+images has largest point mass at most $2^{-.95tN}$ under
+its conditional cell law.  The two laws are independent.
+Absorb the separated phases into bounded weights.
+\Cref{lem:walsh}, in dimension $tN$, bounds the absolute
+character mean by
+\begin{equation}\label{eq:gram-character-bound}
+ 2^{tN/2}\bigl(2^{-.95tN}2^{-.95tN}\bigr)^{1/2}
+ =2^{-.45tN}\le2^{-.45N}.
+\end{equation}
+
+There is at least the trivial character with value one, and
+all other zero-quotient characters also have value one.
+Consequently the conditional agreement probability on every
+retained cell pair is at least
+\[
+ 2^{-s_n}\bigl(1-2^{s_n}2^{-.45N}\bigr)
+ \ge2^{-s_n-1}\ge2^{-.02N}
+\]
+for all sufficiently large $n$.  No independence of the
+individual tested Gram bits is asserted or required.
+
+\paragraph{Averaging and the exponent.}
+The assumed overlap is subexponentially large.  The error
+in \eqref{eq:fubini-pruning} is exponentially small, so the
+retained collision probability, after averaging leaves and
+independent parameters, is at least
+$2^{-o(N)}/|\cK_n|$.  Integrating the preceding scalar
+agreement bound gives four holes with probability at least
+\[
+ 2^{-(k+.02)N-o(N)}
+\]
+under $\widetilde\sigma_A\otimes\widetilde\sigma_B$, with
+$\Psi$ satisfied.  This implication is valid even
+if many successful queries certify the same orientation
+pair: the fresh query experiment is a probability space,
+and its successful event is contained in the event that
+the underlying orientations satisfy $\Psi$ and have all four
+holes.  Each original law dominates its restricted normalized
+law by its own restriction mass.  The product of those two
+masses is at least $2^{-o(N)}$, which gives the same additional
+cost even when the laws and restrictions are different.
+
+If there are boundedly many options, sum the corresponding
+overlap quantities and choose one whose expectation is at
+least the sum divided by their number, or partition into
+these fixed cases.  Shapes of $\cK_n$ and varying finite
+numerical formats can likewise be separated into boundedly
+many cases.  These constant factors are absorbed by the
+slack from $.02$ to $.03$ in
+\eqref{eq:collision-conclusion}.  Finally
+\[
+ k_{\max}+.03=56(g-1)+.03<100g,
+\]
+so \eqref{eq:collision-conclusion} contradicts a sequence
+whose probability of four holes satisfying the required
+predicate $\Psi$ is less than $2^{-100gN}$.  In particular,
+failure of a filtered distribution test forces vanishing
+overlap only for its permitted options; the argument makes
+no assertion about the overlaps of excluded colour or type
+pairs.
+\end{proof}
diff --git a/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/07-phases.tex b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/07-phases.tex
new file mode 100644
index 0000000000000000000000000000000000000000..55789fbc642d44970fbda6336f8ac2cfa26c3847
--- /dev/null
+++ b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/07-phases.tex
@@ -0,0 +1,745 @@
+\section{Phase estimates before key collisions}
+\label{sec:phases}
+
+The estimates in this section supply the compatible small tables
+used in \Cref{prop:status-preparation}.  Channel thinning makes the
+retained mass and the compatibility margin positive constants chosen
+before the channel size.  The original frame law is \(\rawlaw\),
+whereas all marginal bounds below are against the thinned law \(\pi\).
+
+Throughout this section, different units are independent, but the two
+endpoints of one unit may have arbitrary dependence.  A marked unit is
+a unit together with choices
+\(\lambda_i\in C_i^{\mathrm{old}}\), \(i=1,2\), where
+\(C_i^{\mathrm{old}}\) denotes its effective space after the first
+peeling.  Marks are permitted to depend on the entire unit.  Put
+\begin{equation}
+ \Delta_O=U_1\lambda_1+U_2\lambda_2,\qquad
+ u_{O,S}=\sum_{i\in S}u_i\quad(S\subseteq\{1,2\}).
+ \label{eq:delta-phase}
+\end{equation}
+The total component rank of \(\Delta_O\) is at most
+\(r=2K_1+2\).  We use \(L_0,d_0,u_0,K\) from
+\Cref{eq:law-budgets,eq:pin-budgets}; in particular,
+\[
+ L_0=\lceil100(D+10)\rceil,\qquad d_0=4rL_0,\qquad
+ u_0=10(K_1+d_0+1).
+\]
+Write \(\varepsilon_{\rm ch}=10|\cE|2^h\rho\), as in
+\Cref{lem:channel-thinning}.  A marked law with endpoint marginals
+at most \(\Lambda\pi\) has unchanged primal marginal bound
+\(\Lambda\operatorname{pr}_*\rawlaw\), but its full-frame bound is only
+\(\Lambda2^{\varepsilon_{\rm ch}N}\rawlaw\).
+Here \(\Lambda\) is a fixed constant specified before \(h\).
+In the coloured test it comes from the pooled law and the total mass
+retained, and does not depend on the probability of an individual
+colour.  The starting absolute pair cap is \(2^{3000gN}\pi^2\).
+After the first peeling and restrictions of fixed positive mass it
+is at most \(2^{(D+.01)N}\rawlaw^2\), by
+\Cref{lem:channel-thinning} and \(D=4000g\).
+Every use of a full-frame image bound below explicitly pays the
+loss \(\varepsilon_{\rm ch}N\).  We choose
+\(\varepsilon_{\rm ch}<\min\{\zeta/100,10^{-4}\}\), after \(h\),
+and choose \(M_0\) afterward to make each primal ratio loss as small
+as required.  A fixed multiplicative constant is absorbed by taking
+\(n\) sufficiently large.
+
+\begin{definition}
+A componentwise cover is a collection of subspaces
+\(T_e^\pm\subseteq V_e^\pm\).  A tensor \(\xi=(\xi_e)_e\) is covered
+by it if
+\[
+ \xi_e\in T_e^+\otimes V_e^-+V_e^+\otimes T_e^-
+ \quad\text{for every }e.
+\]
+The dimensions of the two modes of the cover are
+\(\sum_e\dim T_e^+\) and \(\sum_e\dim T_e^-\).
+\end{definition}
+
+\begin{lemma}[Rank growth without a cover]
+\label{lem:no-cover-rank}
+Let \(r\ge1\) be an integer, set \(m_r=100r+10\) and
+\(q_r=2^{-m_r}\), and let \(s\) be a power of two with
+\(s\ge2^{m_r}\).  Suppose \(T_1,\ldots,T_s\) are independent copies
+of a random tensor of total component rank at most \(r\).
+If every fixed componentwise cover with dimension at most \(rs\)
+in each mode has probability less than \(p\), then
+\begin{equation}
+ \PP\left\{\rank\left(\sum_{j=1}^sT_j\right)<q_rs/4\right\}
+ \le 4^s p^{q_rs/2}.
+ \label{eq:no-cover-rank}
+\end{equation}
+Here rank means the sum of component ranks.
+\end{lemma}
+
+\begin{proof}
+Regard all components as blocks in the direct sums of the two mode
+spaces.  Their column and row spaces are direct sums of
+componentwise spaces, so that every cover obtained in this proof is
+componentwise.  We first make a deterministic observation about any
+realized list of tensors.
+
+Expose a subset \(E\) of the indices and quotient the two modes by
+the column and row spans of the exposed tensors.  If \(b=s-|E|\)
+indices remain, let \(C_j,R_j\) be their projected original column
+and row spaces.  Define their span deficits by
+\[
+ \delta_C=\sum_{j\notin E}\dim C_j-\dim\sum_{j\notin E}C_j,\qquad
+ \delta_R=\sum_{j\notin E}\dim R_j-\dim\sum_{j\notin E}R_j.
+\]
+The projected sum has rank at least
+\begin{equation}
+ \#\{j\notin E:\overline T_j\ne0\}-\delta_C-\delta_R.
+ \label{eq:rank-deficits}
+\end{equation}
+Indeed, before adding the mode spaces, place the projected tensors
+on the separate formal summands \(C_j\otimes R_j\).  Their block
+diagonal sum has rank at least the displayed number of nonzero
+tensors.  Applying the two addition maps can decrease rank by at
+most their kernel dimensions, which are \(\delta_C,\delta_R\).
+
+There is an exposed set leaving \(b=s/2^j\) indices for some
+\(0\le j<m_r\) such that
+\(\delta_C+\delta_R\le b/4\).  To prove this, order the realized
+direction spaces by a uniform random permutation.  In either mode,
+let \(a_t\) be the expected increase of their span dimension at the
+\(t\)-th step.  Submodularity and exchangeability give
+\[
+ r\ge a_1\ge a_2\ge\cdots\ge a_s\ge0.
+\]
+For \(b_j=s/2^j\), set
+\[
+ f_j=a_{s-b_j+1},\qquad
+ v_j=f_j-\frac1{b_j}\sum_{t=s-b_j+1}^s a_t.
+\]
+Each remaining index has expected projected dimension \(f_j\)
+immediately after the first \(s-b_j\) exposures.  Consequently its
+mode's expected deficit divided by \(b_j\) is \(v_j\).
+The average increment on the first half of this tail is at least
+\(f_{j+1}\).  It follows that
+\[
+ v_j\le f_j-f_{j+1}+\tfrac12v_{j+1}.
+\]
+Summing for \(0\le j<m_r\) yields
+\[
+ \sum_{j=0}^{m_r-1}v_j
+ \le2(f_0-f_{m_r})-v_0+v_{m_r}\le2r.
+\]
+The two modes therefore have total expected deficit ratios at most
+\(4r\) across these \(m_r\) scales.  Some scale has expected ratio
+less than \(1/4\), and some permutation realizes a ratio at most
+\(1/4\), as claimed.
+
+If the original sum has rank less than \(q_rs/4\), its projection
+at this exposed set has no larger rank.  By
+\Cref{eq:rank-deficits}, at most
+\(q_rs/4+b/4\le b/2\) remaining tensors have nonzero projections.
+At least \(q_rs/2\) of them thus belong to the cover formed by the
+exposed column and row spans.  Both cover dimensions are at most
+\(rs\).
+
+For fixed disjoint index sets \(E,J\), with
+\(|J|=\lceil q_rs/2\rceil\), condition only on the tensors indexed
+by \(E\).  The cover is then fixed, and the tensors indexed by \(J\)
+are still independent.  The probability that all are covered is
+at most \(p^{|J|}\).  There are at most \(2^s\) choices for each
+index set.  Taking their union proves \Cref{eq:no-cover-rank}.
+\end{proof}
+
+\begin{theorem}[Phase alternative]
+\label{thm:phase-alternative}
+Suppose a marked-unit law \(\sigma\) satisfies
+\[
+ \sigma_i\le\Lambda\pi\quad(i=1,2),\qquad
+ \sigma\le2^{(D+.01)N}\rawlaw^2,
+\]
+where \(\Lambda\) is fixed before the channel size, and suppose
+\(\Delta_O\ne0\) throughout.  For all sufficiently large \(n\),
+one can retain a sublaw of relative mass at least
+\(\delta_{\rm ph}>0\), depending only on the early parameters,
+such that one of the following holds for independent draws \(A,B\):
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item every nontrivial phase mean
+\[
+ \E(-1)^{u_{B,S}(\Delta_A)+u_{A,R}(\Delta_B)},
+ \qquad(S,R)\ne(\varnothing,\varnothing),
+\]
+has absolute value less than \(.005\);
+\item writing \(F(A,B)=u_{B,\{1,2\}}(\Delta_A)\), one has
+\(F(A,B)=0\) for every pair of retained units, and every displayed
+mean for which \(|S|=1\) or \(|R|=1\) is \(2^{-\Omega(N)}\).
+\end{enumerate}
+All marks remain in their first effective spaces.  No additional
+image directions are pinned.  Neither the retained-mass bound nor
+the tiny-cover dimension used in the proof depends on selector or
+channel sizes; the lower bound required of \(h\) may depend on
+\(\Lambda\).
+\end{theorem}
+
+\begin{proof}
+We first choose the parameters and treat the case with no large cover.
+In the cover case we pass to offset records. A tiny cover either makes
+\(F\) identically zero or permits a gain from thinning; in the absence
+of a tiny cover, a fresh-direction count gives mixing.
+
+Write
+\[
+ c_s=10^{-12}(1+r)^{-4},\qquad q_0=2^{-100r-10},
+ \qquad \theta=10^{-4},\qquad \Gamma=D+2r+10,
+\]
+and let \(s\) be the largest power of two not exceeding \(c_sN\).
+For large \(n\), it satisfies the size requirement of
+\Cref{lem:no-cover-rank} and \(s\ge c_sN/2\).
+Choose \(p_0>0\), using only these early parameters, so small that
+\begin{equation}
+ \frac{q_0}{2}\log_2(1/p_0)
+ \ge 2+\log_2(1/\theta)+\frac{2\Gamma}{c_s}.
+ \label{eq:early-p0}
+\end{equation}
+When the channel size is chosen, require also
+\begin{equation}
+ \frac{hq_0}{4}
+ \ge \log_2(1/\theta)+\frac{2\Gamma}{c_s}.
+ \label{eq:early-h-moment}
+\end{equation}
+Further lower bounds for \(h\) arising below will again involve
+only early parameters.
+
+\medskip\noindent
+\emph{No large cover.}
+Suppose no fixed cover of dimension at most \(rs\) in each mode
+has mass at least \(p_0\).  Fix a candidate value \(\xi\) of
+\(\Delta_B\), a nonempty \(S\), and a set \(R\).  Define the
+unit-side sign \(\psi(A)=(-1)^{u_{A,R}(\xi)}\).  Against completely
+iid channel columns at the endpoints selected by \(S\), expansion
+of the even \(s\)-th moment gives
+\begin{align}
+ &\E_{\mathrm{iid\ channels}}
+ \left|\E_A\psi(A)(-1)^{u_{B,S}(\Delta_A)}\right|^s
+ \notag\\
+ &\hspace{25mm}\le
+ \E_{A_1,\ldots,A_s}
+       2^{-h\rank(\Delta_{A_1}+\cdots+\Delta_{A_s})}.
+ \label{eq:phase-moment}
+\end{align}
+For one channel pair the expectation of its bilinear character
+on a tensor \(\xi\) is \(2^{-\rank\xi}\).  Independence over
+channels and components proves this assertion; selecting both
+endpoints only squares the factor.  The signs \(\psi(A_j)\)
+have absolute value one and can be removed for an upper bound.
+By \Cref{lem:no-cover-rank}, the last expression is at most
+\[
+ 2^{2s}p_0^{q_0s/2}+2^{-hq_0s/4}.
+\]
+The channel marginal comparison in \Cref{lem:frame-images}
+multiplies this bound by a constant depending on \(h\) and the
+number of components, but independent of \(n\).
+Markov's inequality and
+\Cref{eq:early-p0,eq:early-h-moment} bound the probability that
+the inner mean has magnitude at least \(\theta\) by
+\(C_h2^{-\Gamma N}\).
+
+There are at most \(C_r2^{2rN}\) tensors of total component rank
+at most \(r\): choose their component ranks, then factor each
+matrix through that rank.  The component-rank choices and the
+overcounting factors are bounded independently of \(n\).
+Take a union over these candidate values and the bounded choices
+of \(S,R\), and then pay the joint-law density
+\(2^{(D+.01)N}\).  The exceptional \(B\)-mass is exponentially
+small.  On all other \(B\)'s the conditional \(A\)-mean is at most
+\(\theta\), even after inserting the actual value \(\Delta_B\).
+Every required phase mean is therefore at most
+\(\theta+2^{-\Omega(N)}<.005\).  Interchanging \(A,B\) deals
+with \(S=\varnothing\).  This proves the first alternative
+without a further restriction.
+
+\medskip\noindent
+\emph{A large cover and its offsets.}
+Otherwise retain mass at least \(p_0\) in a fixed cover
+\((T_e^+,T_e^-)\) of dimension at most \(rs\) in each mode.
+Each individual primal frame span avoids these fixed subspaces
+except on exponentially small mass.  Indeed its dimension is
+\(\dim\cB\), its location is uniform under \(\rawlaw\), and
+\[
+ \PP_{\rawlaw}\{\im P_{ie}\cap T_e^+\ne0\}
+ \le 2^{\dim\cB+\dim T_e^+-N+1};
+\]
+the same estimate holds for the minus mode.  Here
+\(rs\le rc_sN\) and \(\dim\cB/N\) is sufficiently small.
+The primal marginal bound \(\Lambda\operatorname{pr}_*\rawlaw\) permits
+deletion of these exceptions; this step does not use a full-frame
+bound against \(\rawlaw\).  Normalize the retained law, still denoted \(\sigma\).
+Its marginal bound \(\sigma_i\le L\pi\) has
+\(L\le2\Lambda/p_0\) for all sufficiently large \(n\).
+Thus this fixed bound may also depend on the given \(\Lambda\),
+whereas the retained-mass bound remains independent of \(\Lambda\).
+
+Project both modes of each component modulo \(T_e^\pm\).
+Because \(\Delta\) is covered, the two endpoint tensors have the
+same quotient.  The projections are injective on the individual
+primal spans, so they preserve each tensor rank.  Choose a shortest
+outer-product factorization of the common quotient tensor and
+lift each factor to the two endpoint primal spans.  Componentwise,
+the difference has the form
+\begin{equation}
+ \Delta=\sum_\alpha
+   \bigl(p_\alpha b_\alpha\trans+
+         a_\alpha q_\alpha\trans+
+         a_\alpha b_\alpha\trans\bigr),
+ \quad
+ a_\alpha\in T^+,\quad b_\alpha\in T^- .
+ \label{eq:cover-lifting}
+\end{equation}
+The first endpoint is the reference here; its factors are
+\(p_\alpha,q_\alpha\), and those at the second endpoint are
+\(p_\alpha+a_\alpha,q_\alpha+b_\alpha\).
+The factor lists at either endpoint are independent in their
+respective modes.
+
+Choose bases for the spans of the offsets \(a_\alpha,b_\alpha\)
+in each component.  Their combined ordered list is denoted
+\(\omega_O\), and its size is \(t_O\), with
+\begin{equation}
+ 1\le t_O\le2r.
+ \label{eq:offset-rank}
+\end{equation}
+The lower bound follows from \(\Delta\ne0\).
+Collect the companions of these offset basis vectors in a tuple
+\(Z_O\): minus companions pair with plus offsets and vice versa.
+The companions are independent within each component and mode.
+For example, writing \(b_\alpha=\sum_kc_{k\alpha}\omega_k^-\)
+gives a full-row-rank matrix \((c_{k\alpha})\), so the vectors
+\(\sum_\alpha c_{k\alpha}p_\alpha\) are independent.
+Using the second endpoint as reference changes each companion
+by a known combination of offsets, and preserves independence.
+
+An offset record consists of these basis vectors, their component
+and sign assignments, and the bounded coefficient arrays in
+\Cref{eq:cover-lifting}.  Its logarithmic count is at most
+\[
+ 2r^2s+O_{r,g}(1)<.002N
+\]
+for sufficiently large \(n\).  Choosing a nominal representation
+of a companion tuple in one endpoint's primal frame costs
+\(2^{O(r\dim\cB)}=2^{O(n)}\) possibilities.  This last cost has
+an arbitrarily small ratio exponent after choosing \(M_0\).
+Fix deterministic conventions for all these records and
+factorizations; their choices can be arbitrary functions of the
+marked unit.
+
+For a linear tensor functional \(u\), let \(\Phi_u(\omega)\)
+be the tuple obtained by contracting \(u\) with the offset basis
+vectors.  In particular, on a plus offset \(a\) its corresponding
+plus output is \(X(Y\trans a)\); on a minus offset \(b\) the
+minus output is \(Y(X\trans b)\).  At fixed two offset records
+the phase in the theorem is, up to a sum of two unary phases,
+\begin{equation}
+ Z_A\cdot\Phi_{u_{B,S}}(\omega_A)
+ +Z_B\cdot\Phi_{u_{A,R}}(\omega_B).
+ \label{eq:offset-walsh}
+\end{equation}
+The unary phases are evaluations of the last term of
+\Cref{eq:cover-lifting}; fixed offset records make them separate
+functions of \(A\) and \(B\).
+
+\medskip\noindent
+\emph{Singleton phase estimates.}
+All record laws in the following calculation are unnormalized
+subprobabilities of \(\sigma\).  For any record, the companion
+tuple alone has point masses at most
+\begin{equation}
+ p_{\max}(Z_O)\le2^{-t_ON+.01N}.
+ \label{eq:companion-pmax}
+\end{equation}
+To obtain this, enumerate its nominal primal representations,
+apply the individual primal frame-image estimate and the unchanged
+primal marginal bound \(L\operatorname{pr}_*\rawlaw\), and choose \(M_0\) large enough to absorb the
+coefficient count and lost primal codimensions.
+
+Suppose \(S=\{j\}\), and use endpoint \(j\) as the reference
+for the \(B\)-companions.  For fixed \(\omega_A\), discard
+pairs on which the coefficient lists
+\(Y_j\trans a\) or \(X_j\trans b\), for its independent offsets,
+fail full rank in any component.  Under the original law, comparison with iid columns bounds the
+transpose-rank failure on \(d\) independent ambient vectors by
+\(2^{d-h+1}\) for large \(n\).  The fixed-list assertion of
+\Cref{lem:channel-thinning} bounds the same event under \(\pi\)
+by \(2^{d-h+1}+o(1)\), uniformly over the fixed list.  For each
+fixed \(A\), the list being tested is fixed before drawing \(B\).
+Consequently the discarded pair mass is at most
+\[
+ L\,O(r)\,2^{2r-h}+o(1)<10^{-4},
+\]
+on making \(h\) sufficiently large in terms of the fixed marginal
+cap and the early rank bound.
+
+On the remaining pairs, the tuple
+\((Z_B,\Phi_{u_{B,j}}(\omega_A))\) has point masses at most
+\begin{equation}
+ 2^{-(t_B+t_A)N+(.02+\varepsilon_{\rm ch})N}.
+ \label{eq:singleton-joint-pmax}
+\end{equation}
+Here is a direct conditional count.  Enumerate a nominal
+representation of \(Z_B\), first test its values, and condition
+on the primal frames at endpoint \(j\).  The two channel maps
+are independent uniform-column draws in the appropriate
+annihilators, apart from a bounded injectivity conditioning.
+Freeze the values of \(Y\trans a\) and \(X\trans b\).  Their
+number of possibilities is bounded in \(n\), and their lists
+have full rank on the retained event.  Ignore the equations
+that originally defined these coefficient values.  Prescribing
+their images under \(X\) and \(Y\) then costs at least
+\((N-\dim\cB)t_A\) bits, up to a bounded factor.
+This is a count under \(\rawlaw\), conditional on its primal
+frames.  Replacing its channel law by that of \(\pi\) costs at
+most \(2^{\varepsilon_{\rm ch}N}\); applying the endpoint marginal
+bound costs the fixed factor \(L\).  Together with the primal
+companion test this gives \Cref{eq:singleton-joint-pmax}.  The ignored equations only
+enlarge the event, so no independence between a coefficient
+list and the map defining it has been assumed.
+
+Apply \Cref{lem:walsh} to \Cref{eq:offset-walsh}, on
+\((t_A+t_B)N\) bits.  On the \(A\)-side the point-mass bound for
+the entire tuple is no larger than
+\Cref{eq:companion-pmax}; on the \(B\)-side use
+\Cref{eq:singleton-joint-pmax}.  Separate unary phases have
+absolute value one.  The contribution for two fixed records is
+at most
+\[
+ 2^{(t_A+t_B)N/2}
+ \left(2^{-t_AN+.01N}
+       2^{-(t_A+t_B)N+(.02+\varepsilon_{\rm ch})N}\right)^{1/2}
+ =2^{(-t_A/2+.015+\varepsilon_{\rm ch}/2)N}.
+ \]
+There are at most \(2^{.004N}\) record pairs.  By
+\Cref{eq:offset-rank}, their total is at most \(2^{-.480N}\).
+Thus every singleton phase mean is bounded by
+\(10^{-4}+2^{-.480N}\).  The same proof applies when
+\(|R|=1\).
+
+\medskip\noindent
+\emph{A tiny cover or mixing of the full endpoint sum.}
+Put \(\delta=.0005\).  If a cover of total dimension at most
+\(d_0\) contains the whole offset list with probability at least
+\(\delta/4\), restrict to those units.  Intersect its spaces componentwise with the preceding large cover.
+This preserves every captured offset list and does not increase its
+dimension.  Each individual primal span therefore still avoids it.  Delete units whose channel transpose at either
+endpoint is not injective on a fixed basis of any component and
+sign of this tiny cover.  By \Cref{lem:channel-thinning} and the
+endpoint bound, the deleted relative mass is at most
+\[
+ L'\,O(d_0+|\cE|)2^{d_0-h}+o(1),
+ \qquad L'\le 4L/\delta.
+\]
+Take \(h\) so this is less than \(1/2\).  This leaves relative mass
+at least \(\delta/8\) in the large-cover law.  All future offset
+lists in this tiny cover now pass the singleton rank tests.
+
+Repeat \Cref{eq:cover-lifting} with the tiny cover as the quotient
+directions.  Fix an offset record: the actual offset basis, its
+component and sign assignments, and its bounded binary coefficient
+arrays.  These data have at most
+\[
+ R_0=2^{100(r+1)^2(d_0+|\cE|+1)}
+\]
+possibilities.  Indeed at most \(2r\) vectors are chosen from a
+fixed space of dimension \(d_0\), at most \(2r\) component/sign
+assignments are made, and all coefficient arrays have bounded
+sizes in terms of \(r,|\cE|\).  The deliberately generous exponent
+bounds the product of these finite counts.  Retain a record of
+mass at least \(1/R_0\).  This fixes a common ordered tuple
+\(\omega\), of length \(1\le t\le2r\), but fixes neither the
+companions nor any frame image.  In particular this restriction
+is not an additional pinning step.
+
+Split once more, retaining at least half the mass, according as
+\(\Phi_{u_{O,\{1,2\}}}(\omega)\) is zero or nonzero.  All
+normalizations in this paragraph are bounded by early constants.
+In the zero case each term in \Cref{eq:cover-lifting} has one
+factor from \(\omega\), so
+\[
+ u_{B,\{1,2\}}(\Delta_A)=0
+ \quad\hbox{for every retained } A,B.
+\]
+This includes the last, offset--offset term of the lift.  The
+singleton count already proved, with no rank discard because the
+whole tiny cover passes the transpose tests, gives exponentially
+small singleton means under this fixed restriction.  Thus
+alternative \textup{(ii)} holds.
+
+In the nonzero case we claim the stronger full-sum point bound
+\begin{equation}
+ \PP_\sigma\{Z_O=z,
+       \Phi_{u_{O,\{1,2\}}}(\omega)=\phi\}
+ \le 2^{-tN-\rho N/3}
+ \quad(\phi\ne0).
+ \label{eq:tp-fullsum-pmax}
+\end{equation}
+Here \(\sigma\) denotes the normalized retained law; its endpoint
+marginals are at most \(L''\pi\) for an early constant \(L''\).
+To verify the claim, choose endpoint one as reference in the lift.
+Enumerate all nominal primal descriptions of its \(t\) independent
+companions; their count is \(2^{O(n)}\).  For each description,
+prescribing the companion images \(z\) has probability at most
+\(2^{-tN+O(n)}\) under the primal marginal of \(\pi\), which is
+exactly the primal marginal of \(\rawlaw\).
+Choose a nonzero coordinate \(d\) of the prescribed output \(\phi\).
+The corresponding contraction of each endpoint's channel form is
+a nonzero channel combination, because its transpose is injective
+on the fixed cover basis.  Call these combinations \(v_1,v_2\).
+They lie in the same prescribed component/sign subset, and
+\(v_1+v_2=d\ne0\).  Necessarily, therefore, endpoint one has a
+nonzero channel combination in the translate by \(d\) of its
+allowed subset.  Conditional on all its primal frames, this
+endpoint-one event has \(\pi\)-probability at most
+\(2^{-\rho N/2}\), by \Cref{lem:channel-thinning}.
+
+This is a necessary event on endpoint one alone.  Consequently
+its probability under the dependent two-endpoint law is bounded
+using only the endpoint marginal \(L''\pi\): no independence of
+the endpoints, and no conditional density bound for endpoint two,
+is used.  Sum the nominal descriptions to obtain
+\[
+ \PP_\sigma\{Z_O=z,\Phi=\phi\}
+ \le L''2^{-tN+C n}2^{-\rho N/2}
+\]
+for a constant \(C\) fixed before \(M_0\).
+Choose \(M_0\) so \(C/M_0<\rho/12\), and then \(n\) so
+\(\log_2 L''/N<\rho/12\).  This proves
+\Cref{eq:tp-fullsum-pmax}.
+
+For the other side of the Walsh estimate the companion-only
+bound can now be sharpened to
+\begin{equation}
+ p_{\max}(Z_O)\le2^{-tN+\rho N/12}.
+ \label{eq:tp-fine-companion-pmax}
+\end{equation}
+Its errors are solely \(O(n)\) primal image and nominal-description
+costs plus a fixed normalization, so increasing \(M_0\) after
+\(\rho\) proves this bound.  There is no full-frame thinning loss
+in \Cref{eq:tp-fine-companion-pmax}.
+For a phase containing a full-sum term, apply \Cref{lem:walsh} to
+\Cref{eq:offset-walsh}, using
+\Cref{eq:tp-fullsum-pmax} on that term's side and
+\Cref{eq:tp-fine-companion-pmax} on the other.  Both records are
+already fixed, so there is no exponential record union.  The
+absolute value is at most
+\[
+ 2^{2tN/2}
+ \left(2^{-tN+\rho N/12}2^{-tN-\rho N/3}\right)^{1/2}
+ =2^{-\rho N/8}.
+\]
+When the other phase term is absent, its output coordinates are
+zero; the companion bound still bounds the full tuple.  Phases
+containing a singleton have the previous exponential estimate.
+Every nontrivial phase is of one of these types, so the nonzero
+case gives alternative \textup{(i)}.
+
+Assume instead that no such cover captures mass \(\delta/4\).
+Fix an \(A\)-offset list \(v=\omega_A\), and use endpoint \(1\)
+as reference throughout the next calculation.  Call \(B\)
+frequent for this list if its offset record \(m\), its companion
+value \(z\), and its output \(\phi=\Phi_{u_{B,\{1,2\}}}(v)\)
+have joint point probability greater than
+\(2^{-(t_B+.5)N}\).  For fixed \(m,z\), define the deterministic
+list
+\begin{equation}
+ {\cal F}_v(m,z)=
+ \left\{\phi:
+  \sigma\bigl(m_B=m,Z_B=z,
+       \Phi_{u_{B,\{1,2\}}}(v)=\phi\bigr)
+       >2^{-(t_B+.5)N}\right\}.
+ \label{eq:frequent-list}
+\end{equation}
+By \Cref{eq:companion-pmax},
+\(\abs{{\cal F}_v(m,z)}\le2^{.51N}\).
+
+We claim that the frequent-pair mass is at most \(\delta\).
+Otherwise a set of \(B\)'s of mass at least \(\delta/2\) has
+frequent \(A\)-mass at least \(\delta/2\).  Draw \(L_0\)
+independent \(A\)-probes.  At each stage the spans of all past
+offsets have total dimension at most \(2rL_0<d_0\).  The mass
+of a probe whose whole offset list lies in those spans is
+less than \(\delta/4\).  Thus each stage has conditional
+probability at least \(\delta/4\) of being frequent and having
+a fresh offset vector.  Select one fresh vector by a fixed
+rule using only the probe lists.  Averaging fixes a probe
+collection with independent selected witnesses in each
+component and mode, and with successful \(B\)-mass at least
+\begin{equation}
+ q_*=(\delta/2)(\delta/4)^{L_0}>0.
+ \label{eq:fresh-probe-mass}
+\end{equation}
+This lower bound is independent of selector and channel sizes.
+Require
+\begin{equation}
+ L\,O(L_0)\,2^{L_0-h}<q_*/2.
+ \label{eq:early-h-probes}
+\end{equation}
+By \Cref{lem:channel-thinning}, the successful \(B\)'s can
+then be trimmed so that the channel
+transpose at endpoint \(1\) has full rank on all these fixed
+witness vectors, while leaving mass at least \(q_*/2\), after absorbing the uniform
+\(o(1)\) error in a strict version of \Cref{eq:early-h-probes}.
+
+Now estimate this event under the raw law \(\rawlaw^2\).
+Condition on all of endpoint \(2\) and on the primal frames
+at endpoint \(1\).  Enumerate the offset record \(m\), and
+enumerate a nominal primal tuple representing \(Z_B\) at
+endpoint \(1\).  There are at most \(2^{.002N+O(n)}\) such
+choices.  For each choice the ambient value \(z\), all the
+lists in \Cref{eq:frequent-list}, and their endpoint-\(2\)
+translations are fixed.  The event defined by the adaptive
+marks is contained in the union of these raw events.  This
+uses no conditional density assertion about the marked law,
+and does not enumerate \(2^{t_BN}\) arbitrary ambient
+companion values.
+
+Project each of the \(L_0\) lists onto the coordinate of its
+chosen fresh witness.  Their total product size is at most
+\(2^{.51L_0N}\).  Freeze the bounded lists of channel
+coefficient values on the witnesses.  On their full-rank
+event, prescribing the corresponding endpoint-\(1\) output
+tuple costs at least \(.99L_0N\) bits, by the same annihilator
+count as in \Cref{eq:singleton-joint-pmax}.  The upper
+probability under \(\sigma\), after paying its joint density,
+is therefore at most
+\[
+ 2^{(D+.01)N+.002N+O(n)-.48L_0N+O(1)}.
+\]
+Choose \(M_0\) to make the \(O(n)/N\) term small.
+Because \(L_0\ge100(D+10)\), this is exponentially small,
+contradicting the mass \(q_*/2\).  The claim follows.
+
+Discard the frequent events in a full-sum phase test.
+For fixed records, this is a restriction on \(B\) alone once
+the \(A\)-offset record is fixed.  The \(B\)-tuple point bound
+is now \(2^{-(t_B+.5)N}\); the \(A\)-tuple bound is
+\(2^{-(t_A-.01)N}\).  Walsh on \((t_A+t_B)N\) bits gives
+\(2^{-.245N}\), before the \(2^{.004N}\) record count.
+Thus every phase with \(S=\{1,2\}\) has magnitude at most
+\(\delta+2^{-\Omega(N)}\).  Interchanging the units covers
+the cases with \(R=\{1,2\}\).  Together with the singleton
+estimates, all nontrivial means are less than \(.005\).
+
+All retained laws have relative mass at least
+\begin{equation}
+ \delta_{\rm ph}=\frac{p_0\delta}{64R_0}>0.
+ \label{eq:tp-phase-retained-mass}
+\end{equation}
+The large-cover span deletion leaves at least \(p_0/2\);
+the tiny-cover restriction, transpose deletion, record restriction
+and zero/nonzero split retain at least
+\(p_0\delta/(32R_0)\) in total.  This proves the stated, slightly
+smaller uniform bound.  The lower bounds on \(h\) depend on the
+fixed marginal caps after these early restrictions.  Only after
+\(h\) do we choose the thinning rate \(\rho\); the new full-sum
+estimate then requires \(M_0\) large relative to \(1/\rho\).
+No inverse mass depending on \(n\) or on a numerical table is paid.
+\end{proof}
+
+\begin{corollary}[A uniform phase compatibility margin]
+\label{tp:phase-compatibility}
+On either retained law of \Cref{thm:phase-alternative}, for each
+fixed \(c\in\F\) and all sufficiently large \(n\),
+\begin{equation}
+ \PP\{u_{B,1}(\Delta_A)=u_{B,2}(\Delta_A)
+       =u_{A,1}(\Delta_B)=u_{A,2}(\Delta_B)=c\}\ge1/32.
+ \label{eq:tp-four-equal-phases}
+\end{equation}
+Also \(\PP\{F(A,B)+F(B,A)=0\}\ge .49\), and each single
+prescribed phase bit has probability at least \(.49\).
+\end{corollary}
+\begin{proof}
+In alternative \textup{(i)}, Fourier inversion of the four bits
+gives every prescribed pattern probability at least
+\((1-15\cdot .005)/16>1/32\).  The even-sum condition and each
+single-bit condition have probability at least
+\((1-.005)/2>.49\).
+In alternative \textup{(ii)}, both full endpoint sums are zero
+identically.  Among the sixteen Fourier characters, the four
+with even support in each endpoint pair therefore equal one.
+Their coefficients for the common pattern \((c,c,c,c)\) are all
+one, for either value of \(c\).  Each of the other twelve
+characters contains a singleton and has mean \(2^{-\Omega(N)}\).
+Thus the probability in \Cref{eq:tp-four-equal-phases} is
+\(1/4+O(2^{-\Omega(N)})\).  The remaining assertions follow
+from the same zero sums and singleton means.
+\end{proof}
+
+\begin{lemma}[Injection on independent pairs]
+\label{lem:injection}
+Let \(\sigma_A,\sigma_B\) be two possibly different marked-unit
+laws, each with pins of total dimension at most \(K\).
+Suppose all four mixed endpoint marginals are at most
+\(\Lambda\pi\), where \(\Lambda\) is fixed before \(h\).
+Draw \(A\sim\sigma_A\) and \(B\sim\sigma_B\) independently;
+the endpoints within either unit may be dependent.  For every
+fixed \(\eps>0\), choosing \(h\) sufficiently large in terms
+of \(K,\eps,g\) makes the probability that the pair fails a
+small-table injection condition at most \(\eps\), for all
+sufficiently large \(n\).  This is an unconditional estimate,
+so its error can be subtracted from any independently
+established compatibility mass.
+\end{lemma}
+\begin{proof}
+It suffices to treat one component, endpoint pair and sign, and
+then sum over at most \(16|\cE|\) tests.
+All copies of \(A\) used below are drawn from \(\sigma_A\)
+independently of \(B\sim\sigma_B\).  The argument uses only
+this independence and the separate marginal bounds, and does
+not require the two unit laws to coincide.  Let \(V_A\) be the image
+of the individual pinned primal space \(D_{A,e}^+\cap\cB_i\),
+and let \(H_B\subseteq\F^h\) be the protected coefficient space
+at the opposite tested endpoint.  Both have dimension at most
+\(K\).  Failure means that some \(0\ne v\in V_A\) has
+\(Y_B\trans v\in H_B\).  Both the pins and \(H_B\) may depend
+on their entire units.
+
+Put \(\ell=\lfloor N/(10(K+1))\rfloor\).  The probability that
+an independent new individual primal span meets a fixed space
+of dimension at most \(K\ell\) is at most
+\[
+ \Lambda 2^{\dim\cB+K\ell-N+1}=2^{-\Omega(N)}.
+\]
+This uses only the unchanged primal marginal, and tests the whole
+primal span, so it is uniform over the adaptive pin choices.
+If a fixed \(B\) has failure probability greater than
+\(\eta>0\), successive independent draws
+\(A_1,\ldots,A_\ell\) can each witness failure while the spaces
+\(V_{A_j}\) remain mutually independent, with conditional
+probability at least \(c=\eta/2\) at every step.  The probability
+of this complete test is at least \(c^\ell\).
+
+Conversely fix any such independent spaces.  There are at most
+\(2^{K\ell}\) choices of one nonzero witness in each, and the
+chosen witnesses are independent.  For iid \(Y_B\) columns their
+transpose images are independent uniform elements of \(\F^h\).
+For a fixed \(H_B\) of dimension at most \(K\), the probability
+that all these images belong to it is at most
+\(2^{-(h-K)\ell}\).  There are at most
+\((K+1)2^{Kh}\) possible subspaces \(H_B\), independently of
+\(n\).  The original channel marginal comparison in
+\Cref{lem:frame-images} therefore bounds this event under
+\(\rawlaw\) by \(C_{K,h}2^{-(h-2K)\ell}\).
+Here the tested list has length proportional to \(N\); we use
+this absolute reference-law count, rather than the bounded-list
+rank assertion of \Cref{lem:channel-thinning}.
+
+The endpoint bound against \(\pi\) and the absolute thinning
+density loss multiply this upper bound by
+\(\Lambda2^{\varepsilon_{\rm ch}N}\).  Fubini and the lower
+bound \(c^\ell\) give
+\[
+ \PP_B\{\PP_A(\mathrm{failure}\mid B)>\eta\}
+ \le C_{K,h}\Lambda
+ 2^{-(h-2K-\log_2(1/c))\ell+\varepsilon_{\rm ch}N}.
+\]
+Take
+\(h>2K+\log_2(1/c)+30(K+1)\), before choosing
+\(\varepsilon_{\rm ch}<10^{-4}\).  The right side is
+exponentially small.  The unconditional failure probability
+for this test is at most \(\eta+2^{-\Omega(N)}\).
+Choose \(\eta<\eps/(32|\cE|)\), sum the tests, and then take
+\(n\) sufficiently large.  This proves the assertion.
+\end{proof}
diff --git a/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/08-preparation.tex b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/08-preparation.tex
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+\section{Preparation of the two-endpoint statuses}
+\label{sec:preparation}
+
+We work along a sequence of unit laws with endpoint marginals at
+most \(M\pi\) and projected pair density at most
+\(2^{3000gN}\pi^2\), as in the first part of
+\Cref{thm:distribution-tests}.  A colour is an additional mark and
+may be randomized conditional on the unit.  Passing to subsequences
+is allowed.  By \Cref{lem:channel-thinning}, the absolute joint
+density relative to \(\rawlaw^2\) has ample slack below
+\(2^{DN}\).  Apply \Cref{cor:coloured-peeling}, or
+\Cref{lem:peeling} when there is no colour, and discard the
+exception in \Cref{lem:raw-intersections}.  This removes only
+\(o(1)\) total mass.
+
+All restrictions in this section act on the resulting \emph{pooled
+law}, with the actual leaf weights.  Although a colour is fixed on
+each leaf, no prediction or phase argument is applied separately to
+its normalized colour class.  In particular the endpoint marginal
+bound depends on the total retained mass, not on the smallest
+colour mass or the number of colours.  All effective spaces in the
+following definition are the first spaces
+\(C_i^{\mathrm{old}}\), with total pin dimension at most \(K_1\).
+They will not be enlarged.
+
+Let \(\operatorname{pr}\) forget the channel columns and retain the
+primal frames.  The marginal fact used below is exactly
+\[
+ \operatorname{pr}_*\pi=\operatorname{pr}_*\rawlaw.
+\]
+Thus a restriction of fixed positive pooled mass has endpoint law
+at most a fixed multiple of \(\pi\), and its primal projection is
+at most that same multiple of \(\operatorname{pr}_*\rawlaw\).
+This does not assert a constant full-frame density relative to
+\(\rawlaw\).
+
+For a tag \(l\) and a diagonal bit \(q\), let
+\(\Omega_{l,q,n}\) be the probability space of global plus and minus
+vector slots on the star of \(l\), with the uniform law subject to
+paired diagonal \(q\) in every component. Requiring its individual
+vectors to be nonzero changes this law by exponentially small total
+variation, which will be harmless.
+
+\begin{definition}[Common prediction]
+\label{def:prediction}
+A prediction along a subsequence consists of common binary
+functions
+\[
+ f_{i,n}(l,q,\cdot):\Omega_{l,q,n}\longrightarrow\F,
+ \qquad i=1,2,
+\]
+a set of units of mass at least \(10^{-6}\), and, on every
+unit in this set, choices
+\[
+ \lambda_i\in C_i^{\mathrm{old}},\qquad
+ \alpha_i\in\F,\qquad
+ {\cal E}_i\subseteq\F^b,\quad |{\cal E}_i|\le B_*,
+\]
+such that the following uniform statement holds.  There is
+\(\varepsilon_n\to0\) for which, at both endpoints, at every
+label outside \({\cal E}_i\), at every tag, and in every
+permitted flavor, a uniform parameter atom \(x\) satisfies
+\begin{equation}
+ p_i(x)=T(\lambda_i,x)+\alpha_i a(x)
+          +f_{i,n}(l,q,\operatorname{key}(x))
+ \label{eq:prediction}
+\end{equation}
+except with probability at most \(\varepsilon_n\).
+Here \(p_i=u_{i'}U_i\), where \(i'\) is the other endpoint.
+The functions \(f_{i,n}\) are the same for all retained units.
+The marks \(\lambda_i,\alpha_i,{\cal E}_i\) may depend on
+the whole unit.
+\end{definition}
+
+The errors in \Cref{def:prediction} refer to uniform parameters
+in each flavor, before conditioning on a tester bit.  Conditioning
+on a possible fixed tester value only multiplies the error by a
+fixed constant.  There are finitely many labels, tags and flavors,
+with their numbers fixed independently of \(n\).
+
+If a prediction exists, retain its marked population and
+pigeonhole the values of the two \(\alpha_i\)'s, of
+\[
+ c=a(\lambda_1)+a(\lambda_2),
+\]
+and of the indicator \(\Delta=0\).
+This loses a factor at most \(16\), leaving mass at least
+\(6.25\cdot10^{-8}>10^{-12}\).
+Pass to a subsequence on which every reference error probability
+of every binary combination of \(f_1,f_2,q\) has a limit, for
+each tag and bit.  This requires only finitely many comparisons.
+
+For sequences of common functions, write \(f\simeq g\) if their
+disagreement probability tends to zero on every
+\(\Omega_{l,q,n}\), and identify sequences related in this way.
+Then quotient their binary vector space by the span of the common
+function with value \(q\).
+Denote the resulting class by \([f]\), and put
+\begin{equation}
+ d_i=1+\alpha_i,\qquad e_i=(d_i,[f_i]).
+ \label{eq:status-class}
+\end{equation}
+The pairs \(e_i\) are \emph{prediction classes}; a query's numerical
+unary status records the scalar outcomes specified in
+\Cref{sec:collision}.
+The support of this pair of statuses is
+\(\{i:e_i\ne0\}\).  These statuses are common to all retained
+units, although the marked \(\lambda_i\)'s can vary.
+
+\begin{lemma}[Exactification of empty status]
+\label{lem:empty-status}
+If \(e_1=e_2=0\), deleting \(o(1)\) unit mass makes
+\begin{equation}
+ p_i=T(\lambda_i,\cdot)+a
+ \quad\text{on all of }\cX,\qquad i=1,2,
+ \label{eq:exact-prediction}
+\end{equation}
+an exact identity on every retained unit.
+The same conclusion holds on any marked population of fixed
+positive mass satisfying \Cref{eq:prediction} with
+\(\alpha_i=1\) and \(f_i\simeq\gamma_iq\); the threshold
+\(10^{-6}\) in \Cref{def:prediction} is not needed here.
+\end{lemma}
+
+\begin{proof}
+Empty status gives \(\alpha_i=1\) and
+\(f_i\simeq\gamma_iq\) for a fixed \(\gamma_i\in\F\).
+Restriction to any fixed positive population only changes the
+primal marginal bound by a fixed factor, which is all the proof uses.
+At any fixed tag, label and parameter point, the key depends
+only on the primal frames.  Under \(\pi\) its law is therefore
+the same as under \(\rawlaw\): the single-key reference up to
+an exponentially small injectivity error, by
+\Cref{lem:frame-images}.  The bounded \emph{primal} marginal
+density transfers the reference disagreement of \(f_i\) and
+\(\gamma_iq\) to an average \(o(1)\) parameter error on the
+marked population.  No channel coordinate is tested in this
+comparison.
+There are only finitely many endpoint, tag and label choices.
+Markov's inequality therefore allows deletion of \(o(1)\)
+unit mass so that this error tends to zero uniformly on
+the remaining units and choices.
+Combining it with \Cref{eq:prediction} gives, at every
+nonexceptional label,
+\[
+ p_i(x)+T(\lambda_i,x)+a(x)+\gamma_iq(x)=0
+\]
+with parameter error tending to zero in the generic flavor.
+
+For fixed label and tag the left side is a Boolean polynomial
+of base degree at most two.  A nonzero Boolean polynomial of
+degree at most \(k\) has nonzero probability at least \(2^{-k}\)
+under uniform bits.  For completeness, this follows by induction:
+choose a coordinate with a nonzero derivative; the derivative
+has degree at most \(k-1\), and whenever it is nonzero at least
+one of the two values on that coordinate is nonzero.
+The constant nonzero polynomial is the initial case.
+Thus the displayed relation, once its error is below \(1/4\),
+is exact for every allowed base value.
+
+For any fixed allowed base value its dependence on the selector
+has degree at most \(2j_*\).  It vanishes at every selector
+outside a set of size at most \(B_*\).
+The choice of \(b\) gives
+\(B_*<2^{b-2j_*}\), so the same polynomial bound forces it
+to vanish at the excluded labels also.
+
+Use one shared-only base point with \(\eta_S=1\), the same at
+every tag.  The sum of its \(g\) star atoms is zero in the cut
+space.  Summing the exact relation over these atoms gives
+\(0=\gamma_i g=\gamma_i\), because \(g\) is odd and
+the first three terms are linear on \(\cX\).
+Consequently \(\gamma_i=0\).
+The point atoms span the cut space by its definition, proving
+\Cref{eq:exact-prediction}.
+\end{proof}
+
+\subsection{A comparison for affine parameter slices}
+
+The next elementary comparison is stated separately because the
+global predictors in \Cref{def:prediction} need not be polynomial.
+All tests in it are common functions of the displayed image
+columns; they are not arbitrary orientation-dependent tests.
+
+\begin{lemma}[Affine-slice comparison]
+\label{lem:synthetic-slice}
+Fix a selector, a flavor, and a fixed number \(d\) of affine
+parameter variables.  Write a random affine coefficient map as
+\[
+ C(t,u)=p_s\otimes(t,z_0t+Lu),
+ \qquad (t,u)\in\F\oplus\F^d.
+ \]
+Assume the flavor retains an \(n\)-bit \(Z\)-block absent from
+\(E\).  Suppose we condition the affine coefficients on
+\(C\trans EC=G\), where \(G\) is fixed, this event has probability
+at least a fixed \(\beta>0\), and it depends on no \(Z\)
+coefficient.
+Let \({\cal T}_n\) be any common test of absolute value at most
+one on the image column arrays \((P_eC,Q_eC)_e\), using a fixed
+set of components.  Its empirical average over the conditioned
+affine coefficients, with the primal frame sampled from
+\(\operatorname{pr}_*\rawlaw=\operatorname{pr}_*\pi\), has variance
+\[
+ O(2^{-N/2}+2^{2d+1-n})
+ \]
+around its mean under uniform injective image column arrays
+with internal Gram \(G\).
+The constants may depend on the fixed column counts and
+\(\beta\), but the estimate is uniform over the tests
+\({\cal T}_n\).
+\end{lemma}
+
+\begin{proof}
+The \(Z\)-coefficients remain independent uniform bits after
+the conditioning.  The columns of \(C\) are independent
+except with probability \(O(2^{d-n})\).
+For two independent conditioned maps \(C,C'\), a relation
+\[
+ C(a,u)+C'(a',u')=0
+ \]
+has \(a=a'\), by its constant coordinate.  Its \(Z\)-part is
+\[
+ a(z_0+z_0')+L_Zu+L'_Zu'=0.
+ \]
+The matrix \([z_0+z_0',L_Z,L'_Z]\) is a uniform
+\(n\times(2d+1)\) bit matrix.  A union over its nonzero
+coefficient vectors bounds the failure of combined independence
+by \((2^{2d+1}-1)2^{-n}\).
+In particular, the two translation columns do not share a
+forced nominal constant direction.
+
+For fixed independent nominal columns, the restricted-image
+transitivity in \Cref{lem:frame-images} gives the uniform
+injective law with its specified Gram.  One batch consequently
+has the claimed reference law, independently of the remaining
+nominal coefficients.
+For two batches, compared with two independent copies of this
+reference law, the actual images have finitely many additional
+mutual Gram entries prescribed.  Joint injectivity contributes
+only \(O(2^{-N+O(d)})\) to the comparison: under independent
+uniform vector slots, a dependence among a fixed number of
+columns has this probability, and the fixed Gram conditioning
+has bounded reciprocal probability.
+
+Expand the mutual Gram indicator in binary characters, collapsing
+any repeated identical entries.  A nonzero character has a
+nonzero coefficient matrix on at least one interaction between
+the two batches.  More explicitly the coefficient blocks pairing
+first-batch plus vectors with second-batch minus vectors and
+first-batch minus vectors with second-batch plus vectors give
+cross-batch bit rank
+\(N(\rank A+\rank B)\ge N\).
+The internal Gram constraints, individual injectivity indicators,
+and the two arbitrary tests are separate bounded weights on
+the two batches; their normalization factors are fixed.
+\Cref{lem:walsh} bounds every nontrivial character contribution
+by \(O(2^{-N/2})\).  The same computation without the tests
+gives the normalization of the mutual Gram conditioning.
+There are only a fixed number of characters.  Thus the
+conditional two-batch product expectation differs from the
+product of the one-batch expectations by \(O(2^{-N/2})\),
+uniformly over the nominal mutual Gram values.
+
+Averaging the coefficients, and including the probability of
+nominal dependence, proves the asserted second-moment estimate.
+The one-batch mean differs from the reference mean only by
+the already bounded nominal-dependence error.
+\end{proof}
+
+\begin{lemma}[Equal coefficients when the tensor difference vanishes]
+\label{lem:zero-delta}
+On a predicted branch with \(\Delta=0\) throughout,
+\(\alpha_1=\alpha_2\).
+\end{lemma}
+
+\begin{proof}
+Suppose otherwise.  By \Cref{lem:raw-intersections}, the two
+canonical sparse representations \(w^{(i)}=(w_l^{(i)})_l\)
+of the marked \(\lambda_i\)'s have the same value of \(\chi_*\).
+Each has at most \(4D/g\) nonzero tag entries.
+As \(\alpha_1\ne\alpha_2\), on every unit exactly one endpoint
+has \(\chi_*(w^{(i)})+\alpha_i=1\).
+Fix an endpoint \(i\) on a positive fraction of the units.
+Pigeonhole the set
+\[
+ T_0=\{t:\eta_S(w_t^{(i)})=1\},
+ \qquad |T_0|\le4D/g,
+ \]
+and choose a single selector outside the stored exceptions
+on a positive fraction of this remaining population.
+The latter is possible by averaging over selectors, since
+\(B_*<2^{b-1}\).
+All these restrictions have fixed positive mass in \(n\);
+their endpoint primal marginal is at most
+\(\Lambda_0\operatorname{pr}_*\rawlaw\) for a fixed \(\Lambda_0\).
+No restriction made solely for this contradiction will be
+used later to choose a channel-size threshold.
+
+At each tag \(l\), use the pure flavor retaining exactly the
+allowed \(O_{d,t}\) blocks with \(t\notin T_0\), setting all
+\(S\) bits to zero and retaining the \(\#,Z\) blocks.
+For a pure tag atom, \(a(x)=q(x)\) and every \(b_t(x)=0\).
+The coefficient of its tester \(\eta_{d,t}\) in
+\(T^0(\lambda_i,x)+\alpha_i a(x)\) is
+\[
+ a(\lambda_i)+b_t(\lambda_i)+\alpha_i
+ =\chi_*(w^{(i)})+\eta_S(w_t^{(i)})+\alpha_i=1
+ \]
+on the retained \(t\)'s.  Hence \Cref{eq:prediction} gives
+\begin{equation}
+ f_i(l,q,\operatorname{key}(x))+q
+ =p_i(x)+T^1(\lambda_i,x)
+ \label{eq:pure-prediction}
+\end{equation}
+with uniform \(o(1)\) parameter error.
+
+The right side of \Cref{eq:pure-prediction} is a sum of at
+most
+\begin{equation}
+ s_1=(g-1)(h+2JK_1)
+ \label{eq:synthetic-product-count}
+\end{equation}
+products of affine functions of the base variables.
+Indeed \(p_i\) contributes at most \(h\) products in each
+of the \(g-1\) star components.  Factoring the old
+\(\lambda_i\)'s, whose total component rank is at most
+\(K_1\), gives at most \(2J(g-1)K_1\) products for \(T^1\).
+The selected pure tester retains at least
+\begin{equation}
+ \frac{g-1}{2}(g-4D/g)r_0>2s_1+1
+ \label{eq:synthetic-tester-room}
+\end{equation}
+disjoint ordered bit pairs.  To check the last inequality,
+use \(D=4000g\) and \(h=1000r_0\).  After division by
+\((g-1)r_0\), the left side is \((g-16000)/2\) and the
+right side is at most \(2000+4JK_1/r_0+o(1)\).
+The fixed value \(g=10^9+1\) and the later choice of
+sufficiently large \(r_0\) make the inequality hold.
+
+Put \(m_1=2gs_1+1\), and introduce synthetic variables
+\(\xi,\upsilon\in\F^{m_1}\) and
+\[
+ q_*=\sum_{j=1}^{m_1}\xi_j\upsilon_j.
+ \]
+For every component independently, choose uniform injective
+maps of the affine coefficient columns
+\((1,\xi,\upsilon)\) into the plus and minus ambient spaces
+with full paired Gram
+\[
+ \pair{\text{plus}(1,\xi,\upsilon)}
+       {\text{minus}(1,\xi',\upsilon')}
+ =\sum_j\xi_j\upsilon'_j.
+ \]
+Such maps exist for large \(n\), since the number of columns
+is fixed.  Their point evaluations form one synthetic array
+of global component keys; all tag stars use the corresponding
+entries of this same array.
+
+We next prove two properties of this array with probability
+tending to one.
+
+\medskip\noindent
+\emph{Shared-only parity.}
+In a local shared-only test use the fixed selector and the
+same point at every tag.  The sum of the tag atoms is zero
+in \(\cX\).  Summing \Cref{eq:prediction} therefore gives
+\[
+ \sum_l f_i(l,q,\operatorname{key}_l)=0
+ \]
+except on \(o(1)\) parameter mass, uniformly on the retained
+units.  Both tester bits have probability at least \(1/4\):
+the bias of the sum of \(r_0\) independent bit-pair products
+is \(2^{-r_0}\).
+
+Apply \Cref{lem:synthetic-slice} with no affine variables
+(\(d=0\)), the shared-only flavor, and internal Gram \((q)\).
+The Gram event depends only on the fixed tester coordinates
+and has probability at least \(1/4\).  The common test is
+the displayed parity predicate of the global keys in all
+components.  Its empirical frequency under the primal reference
+law concentrates at its global reference probability.  The
+endpoint primal marginal bound
+\(\Lambda_0\operatorname{pr}_*\rawlaw\), and its frequency
+\(1-o(1)\) on the retained population, force that reference
+probability to tend to one.
+At every point of the synthetic array the point-key marginal
+is this reference, up to the exponentially small nonzero-vector
+conditioning.  There are a fixed finite number of points.
+Thus, with probability tending to one,
+\begin{equation}
+ \sum_l f_i(l,q_*,\operatorname{key}_l)=0
+ \quad\text{at every synthetic point}.
+ \label{eq:synthetic-parity}
+\end{equation}
+
+\medskip\noindent
+\emph{Affine-product representations on every small slice.}
+Fix sets \(I,J\subseteq\{1,\ldots,m_1\}\), each of size at
+most \(2s_1+1\), and set all \(\xi\) and \(\upsilon\)
+coordinates outside \(I,J\), respectively, to zero.
+There are \(d=|I|+|J|\le4s_1+2\) remaining variables.
+At a given tag, map this slice into its selected local pure
+flavor by independent uniform affine functions for every
+retained base bit, including independent translations.
+Condition their full oriented Gram to equal the synthetic
+slice Gram.
+
+This conditioning has a fixed positive probability.
+At the fixed selector \(E^\#\) vanishes on pairs.
+There are at most \(2s_1+1\) active terms
+\(\xi_j\upsilon'_j\) in the target Gram, so
+\Cref{eq:synthetic-tester-room} lets us route them into
+distinct retained ordered tester pairs, and set all other
+tester affine coefficients to zero.  If \(t_0\) scalar
+tester coordinates are prescribed, this assignment alone
+has probability \(2^{-t_0(d+1)}>0\), independent of \(n\).
+The \(Z\)-affine coefficients remain iid.
+At every fixed slice point the unconditioned base input is
+uniform; after conditioning, its density is bounded by the
+fixed reciprocal Gram probability.
+
+Consequently \Cref{eq:pure-prediction} holds simultaneously
+at all points of the slice except on \(o(1)\) of these
+parametrizations, uniformly on the marked population.
+Its right side pulls back to a sum of at most \(s_1\)
+affine products.  Consider only the event
+\begin{quote}
+the array of values of the common function \(f_i+q_*\) on
+this specified slice admits such an affine-product
+representation.
+\end{quote}
+This is a common test of the image column arrays.  It contains
+neither the mark \(\lambda_i\) nor any coefficients of the
+unit-specific representation.  Its empirical success
+probability is \(1-o(1)\) on the retained population.
+\Cref{lem:synthetic-slice} and the bounded primal marginal
+therefore force its probability under the synthetic slice
+law to tend to one.  That law is also the marginal of the
+corresponding slice in the full synthetic array, by restricted
+frame transitivity.
+
+There are only finitely many choices of tag and slice, with
+the number fixed independently of \(n\).  We may therefore
+require all these events and \Cref{eq:synthetic-parity}
+simultaneously.  Notice that we embedded only the small
+slices locally; no local realization of the entire synthetic
+tester was required.
+
+Choose one synthetic array on which all these properties hold,
+and define
+\[
+ F_l(\xi,\upsilon)
+   =f_i(l,q_*,\operatorname{key}_l)+q_*.
+ \]
+For each tag define its Boolean cross-coefficient matrix by
+\[
+ C_l[j,k]=F_l(e_j,e_k)+F_l(e_j,0)
+                 +F_l(0,e_k)+F_l(0,0).
+ \]
+On a coordinate slice where \(F_l\) is a sum of \(s_1\)
+affine products, this is the matrix of \(\xi_j\upsilon_k\)
+coefficients.  One affine product contributes a sum of two
+rank-one matrices to this cross matrix.  Every
+\((2s_1+1)\)-square minor of \(C_l\) is therefore singular,
+so \(\rank C_l\le2s_1\).
+On the other hand \Cref{eq:synthetic-parity}, together with
+odd \(g\), gives \(\sum_lF_l=q_*\) and hence
+\[
+ \sum_l C_l=I_{m_1}.
+ \]
+Rank subadditivity would imply
+\(m_1\le\sum_l\rank C_l\le2gs_1\), contradicting
+\(m_1=2gs_1+1\).  This proves \(\alpha_1=\alpha_2\).
+\end{proof}
+
+\subsection{Compatible tables and the prepared alternatives}
+
+For an admissible small table, a contraction on a marked
+\(\lambda_i\) means its table contraction.  This is well defined
+because \(\lambda_i\in C_i^{\mathrm{old}}\) and all first effective
+spaces are retained.  We use the following phase requirements:
+\begin{equation}
+\begin{array}{ll}
+\displaystyle
+u_{B,S^\circ}(\Delta_A)+u_{A,S^\circ}(\Delta_B)
+       =|S^\circ|d,
+&
+\begin{gathered}
+\text{if the nonzero }e_i\text{ all coincide},\\[-1mm]
+S^\circ=\{i:e_i\ne0\}\ne\varnothing,\quad
+ d=\text{their first bit};
+\end{gathered}
+\\[4mm]
+u_{B,j}(\Delta_A)=c,\quad u_{A,i}(\Delta_B)=c
+\quad(i,j=1,2),
+&\text{if }e_1=e_2=0.
+\end{array}
+\label{eq:phase-compatibility}
+\end{equation}
+In all other status cases there is no additional phase condition.
+The four equations in the second line all use the \emph{same}
+previously fixed bit \(c=a(\lambda_1)+a(\lambda_2)\).
+For a candidate table these contractions are computed from its
+entries and the numerical coordinates of the marks.
+
+\begin{proposition}[Status preparation]
+\label{prop:status-preparation}
+There are constants \(\delta_*,c_*>0\), fixed before the selector
+and channel sizes, with the following property.  Along a subsequence,
+the original pooled unit law has a restriction of mass at least
+\(\delta_*\) whose first pin dimension is at most \(K_1\) and whose
+normalized leaves satisfy \Cref{eq:final-leaf-minentropy}.  One of
+the following descriptions applies.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item There is no prediction as in \Cref{def:prediction} on the
+first pooled law before the final negligible leaf trimming.  For
+independent draws from the prepared law, the probability of an
+admissible injecting actual small table is greater than \(.99\).
+\item A prediction with fixed \(\alpha_i,c\) and common classes
+\(e_i\) is retained.  Its marks and exceptions are preserved, and
+\Cref{eq:prediction} has uniform \(o(1)\) error on every retained
+unit.  If both classes vanish, \Cref{eq:exact-prediction} is exact
+on every unit.  For independent draws from the prepared law,
+the probability of an admissible injecting actual table satisfying
+\Cref{eq:phase-compatibility} is at least \(1/64\).
+\end{enumerate}
+In particular the corresponding successful pair event, measured
+in two draws from the original pooled law, has mass at least
+\(c_*\).  One may take
+\begin{equation}
+ \delta_* = \frac{10^{-6}\delta_{\mathrm{ph}}}{64},
+ \qquad c_* = \frac{\delta_*^2}{64},
+ \qquad 0<\eta_{\mathrm{col}}<\frac{c_*}{4},
+ \label{eq:preparation-early-margins}
+\end{equation}
+where \(0<\delta_{\mathrm{ph}}\le1\) is the early retained-mass
+constant in \Cref{thm:phase-alternative}.  The colour bound in
+\Cref{eq:preparation-early-margins} can be chosen using the initial
+\(M\) and early geometric constants only; the marginal constant
+\(L\) of the second distribution test does not enter it.
+
+After forgetting colour and other marks, the prepared law
+\(\sigma'\) satisfies
+\begin{equation}
+ \sigma'_i\le\frac{M}{\delta_*}\pi\quad(i=1,2),
+ \qquad
+ \sigma'\le 2^{(D+o(1))N}\rawlaw^2.
+ \label{eq:prepared-marginals}
+\end{equation}
+The endpoint primal marginals consequently have the same constant
+bound relative to \(\operatorname{pr}_*\rawlaw\).  Every normalized
+leaf has joint density at most \(2^{O(N)}\rawlaw^2\).  The successful
+table event is detected by a finite list of numerical tables and
+separate unary orientation flags, together with leaf-pair tests.
+The size of this list is fixed in \(n\), but need not be early.
+\end{proposition}
+
+\begin{proof}
+Consider first the pooled law after the first peeling and
+intersection deletion.  If no prediction exists on any subsequence,
+apply \Cref{lem:injection} to independent draws with no additional
+acceptance condition.  Its failure tolerance can be chosen below
+\(1/128\), so the success probability is greater than \(.99\) for
+large \(n\).  The actual table is automatically admissible.  Later
+leaf trimming removes only \(o(1)\) mass.  A prediction on the
+trimmed law with mass bounded strictly above \(10^{-6}\) would give
+one on the preceding law, by multiplying its mass by the retained
+proportion and preserving its common functions and unitwise error.
+This is the form of the no-prediction conclusion used below.
+
+Otherwise retain a prediction population and pigeonhole
+\(\alpha_1,\alpha_2,c\) and the indicator \(\Delta=0\), as above.
+Its relative mass is at least \(10^{-6}/16\).  If the statuses
+are empty, perform the \(o(1)\) deletion in
+\Cref{lem:empty-status}.  All subsequent restrictions then preserve
+the exact identities \Cref{eq:exact-prediction}.
+
+Suppose \(\Delta=0\).  By \Cref{lem:zero-delta},
+\(\alpha_1=\alpha_2\).  If precisely one status is active, its
+first bit cannot be \(1\): that would give \(\alpha_i=0\) there
+and \(\alpha_{i'}=1\) at the inactive endpoint.  Thus the right
+side of the first line of \Cref{eq:phase-compatibility} is zero.
+With two equal active statuses it is also zero, because \(2d=0\)
+over \(\F\).  Every left side vanishes when \(\Delta=0\), so
+all these conditions hold automatically.
+
+If both statuses are empty and \(c=1\), then
+\(U_1\lambda_1=U_2\lambda_2\), while the two identities
+\Cref{eq:exact-prediction} give the affine-gradient equations
+for an internal hole and \(c=1\) gives its unequal \(a\)-values.
+This contradicts the assumption that the pair is a unit.
+Consequently \(c=0\), and all four required phase bits vanish.
+Thus compatibility is automatic also in this case.
+
+Suppose next that \(\Delta\ne0\).  Apply
+\Cref{thm:phase-alternative} to the pooled marked law.  It retains
+at least the early fraction \(\delta_{\mathrm{ph}}\), and gives
+either full mixing or the identity
+\[
+ F(A,B)=u_{B,\{1,2\}}(\Delta_A)=0
+ \quad\text{for every two retained units }A,B,
+\]
+together with exponentially small singleton character means.
+For clarity, write the four phase bits as
+\[
+ X_j=u_{B,j}(\Delta_A),\qquad
+ Y_i=u_{A,i}(\Delta_B),\qquad i,j\in\{1,2\}.
+\]
+Full mixing bounds the absolute mean of every nonconstant
+character of \((X_1,X_2,Y_1,Y_2)\) by \(.005\).  Binary Fourier
+inversion therefore gives every prescribed four-bit pattern
+probability at least
+\[
+ \frac{1-15(.005)}{16}>\frac1{32}.
+\]
+Each condition in \Cref{eq:phase-compatibility} admits such a
+pattern.  In particular the empty-status pattern is exactly
+\((c,c,c,c)\).
+
+In the other alternative, \(X_1=X_2\) and \(Y_1=Y_2\)
+identically.  If there is one active status, the required bit is
+\(X_i+Y_i=d\); its nontrivial character has singleton support in
+both units, so its probability is \(1/2+o(1)\).  If two active
+statuses coincide, the condition is
+\(F(A,B)+F(B,A)=0\), which holds identically.  With empty statuses
+put \(X=X_1=X_2\) and \(Y=Y_1=Y_2\).  The characters of
+\(X,Y,X+Y\) all have singleton representatives.  Hence
+\begin{align*}
+ \PP\{X_1=X_2=Y_1=Y_2=c\}
+ &=\frac14\left(1+(-1)^c\E(-1)^X
+                    +(-1)^c\E(-1)^Y+\E(-1)^{X+Y}\right)\\
+ &=\frac14+o(1)>\frac1{32}.
+\end{align*}
+This explicitly requires the same fixed \(c\) on both sides.
+The remaining status cases impose no condition.  Thus all branches
+have conditional compatibility probability at least \(1/32\) for
+large \(n\), in agreement with \Cref{tp:phase-compatibility}.
+
+Apply the unconditional form of \Cref{lem:injection} to this
+retained law, with failure below \(1/128\).  The probability of a
+compatible injecting actual table is then at least
+\[
+ \frac1{32}-\frac1{128}-o(1)>\frac1{64}.
+\]
+This is subtraction of exceptional mass; it makes no independence
+assumption about injection and compatibility.  The injection
+estimate uses the endpoint bound relative to \(\pi\), together
+with the unchanged primal marginal and the allowed thinning loss
+in the full-frame estimates.  Its tolerance is fixed before \(h\).
+
+We now justify the claimed leaf estimates without adding any pins.
+Let \(w_\lambda\) be the original actual weights of the first
+leaves, and let \(q_\lambda\) be the fraction of leaf \(\lambda\)
+retained by all restrictions just made.  Discard the retained parts
+with \(q_\lambda<2^{-\zeta N}\).  Their total unnormalized mass
+is at most
+\[
+ \sum_{\lambda:q_\lambda<2^{-\zeta N}}
+       w_\lambda q_\lambda\le 2^{-\zeta N}.
+\]
+On each remaining leaf, division by \(q_\lambda\) weakens
+\Cref{eq:leaf-minentropy} to
+\[
+ 2^{\zeta N}2^{-(1-\zeta)tN}
+ \le 2^{-(1-2\zeta)tN}\qquad(t\ge1),
+\]
+as required.  Its joint density gains at most \(2^{\zeta N}\)
+and remains \(2^{O(N)}\rawlaw^2\).  The pin spaces and all old
+marks are unchanged.  This exponentially small deletion changes
+the successful pair probability by \(o(1)\), already allowed above.
+
+The first peeling loses \(o(1)\) mass.  On a predicted branch,
+the finite initial selection costs at most \(16\cdot10^6\), and
+the phase restriction costs at most \(1/\delta_{\mathrm{ph}}\).
+All further losses tend to zero; for large \(n\) their combined
+retained factor, including first peeling, is at least \(1/4\).
+This proves the unit-mass lower bound \(\delta_*\) in
+\Cref{eq:preparation-early-margins}.  The no-prediction branch
+retains \(1-o(1)\) mass and satisfies the same lower bound.
+Multiplying the conditional pair-success bound by
+\(\delta_*^2\) proves the original-scale bound \(c_*\).
+The density statements follow by dividing the initial bounds by
+the retained unit mass and using \Cref{eq:thinning-density}.
+All marginal constants required by the phase and injection steps
+are fixed in advance, for example below \(2M/\delta_*\), so their
+channel-size thresholds can be imposed after these early margins
+have been selected.
+
+Finally, every numerical table has bounded coordinate dimensions.
+The marked tensors are recorded by their coordinates in the first
+effective bases, and the exception sets have bounded descriptions.
+For a fixed leaf pair and a fixed numerical option, the two
+admissibility conditions are separate unary orientation flags,
+since the pin images are fixed on each leaf.  Fixing numerical
+mark coordinates is also a unary flag.  The phase requirements
+then depend only on the table entries and those coordinates.
+An actual successful table supplies at least one successful
+option in this finite list.  This proves the asserted finite-data
+form without selecting one table during preparation or conditioning
+the two draws jointly on table success.
+\end{proof}
+
+\begin{corollary}[Removing equal colours]
+\label{tp:preparation-colours}
+Suppose the original colour masses are \(p_\gamma\le
+\eta_{\mathrm{col}}\).  The successful prepared pair event of
+\Cref{prop:status-preparation}, with the further requirement that
+its two colours differ, has mass at least
+\(c_*-\eta_{\mathrm{col}}>3c_*/4\) in the original pooled law.
+\end{corollary}
+
+\begin{proof}
+Two independent original draws have equal colours with probability
+\[
+ \sum_\gamma p_\gamma^2\le
+ \eta_{\mathrm{col}}\sum_\gamma p_\gamma=\eta_{\mathrm{col}}.
+\]
+Subtract this upper bound from the original-scale successful mass
+\(c_*\).  Equivalently, in a restriction of mass \(\delta\), the
+equal-colour probability is at most
+\(\eta_{\mathrm{col}}/\delta^2\).  This calculation applies also
+to randomized colours, which are simply part of the original
+marked probability space.  It uses actual colour probabilities
+and imposes no lower bound on them.
+\end{proof}
+
+\begin{remark}[No feedback from later accuracies]
+\label{rem:preparation-accuracies}
+The margins \(\delta_*,c_*\), the choice of
+\(\eta_{\mathrm{col}}\), and the injection tolerance are fixed
+before \(h\).  At this stage success means that some numerical
+table passes.  The later choice of one option from a finite list
+may have a much smaller mass depending on \(h\), which is
+irrelevant to the earlier injection estimate.  The errors in the
+affine-slice comparison and in subsequent fixed-test limits are
+handled by increasing \(n\); their slice counts, partitions and
+flag thresholds do not change the channel size or pin budget.
+Only the first peeling has been used throughout.
+\end{remark}
diff --git a/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/09-histograms.tex b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/09-histograms.tex
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+\section{Histogram and parameter estimates}\label{sec:histograms}
+
+We prepare the estimates needed to pass successful queries to a common
+limit. Histogram control gives uniform integrability and finite nets
+within each leaf. Typicality compares prescribed common tests with
+their raw reference averages, binary mixing controls the specified
+parameter constraints, and unary positivity supplies the base-coordinate
+choices used in \Cref{sec:status-span}.
+
+All construction parameters, including $M_0$, are fixed throughout this
+section, and $N=M_0n$. Constants in estimates may depend on those
+parameters. A bounded number always means a number bounded independently
+of $n$. In particular, the numbers of tags, labels, coordinates in a
+small table, and positions in a query are bounded in this sense.
+
+\subsection{Query measures and the distinction between tests and flags}
+
+A position specifies an endpoint of a unit, a tag $l$, a label $s$, and
+one of the flavors from \Cref{sec:moments}. Its parameter is a uniform
+allowed base input. We may condition this input on a possible value of
+$q$, or on a feasible tester summary. The probability of any such fixed
+conditioning is bounded below by a positive constant. The $Z$ block
+remains uniform and independent of the other input coordinates. The
+parameters of different positions are independent, even when some
+positions have the same label. At a fixed orientation $O$, the key at
+position $j$ is denoted by $K_j(O,z_j)$. A query consists of a fixed
+bounded list of positions and their fresh parameters.
+The key uses only the primal maps $P_e,Q_e$.  Write
+$\operatorname{pr}(O)$ for the tuple of those maps, with both endpoints
+included when $O$ is a unit.  Forgetting the channel maps and any
+auxiliary marks in a law will always mean this projection.
+
+For a position of tag $l$ and diagonal value $q$, let
+$\Omega_{l,q,n}$ be the uniform law on its global vector slots, subject
+to the prescribed plus--minus diagonal $q$ at each component of its star.
+For a query, write
+\[
+ \overline\Omega_n=\bigotimes_j\Omega_{l_j,q_j,n}.
+\]
+Let $\omega_n$ instead be uniform on all its vector slots without any
+Gram conditions. Finally, $\nu_n$ is uniform on $\cK_n$, the set obtained
+by imposing all the individual diagonals and all the off-position Gram
+zeros required in \Cref{sec:collision}. Each of these bounded lists of
+Gram conditions has probability bounded below under $\omega_n$.
+
+We will use three different kinds of assertions.
+\begin{enumerate}[label=(\roman*)]
+\item A \emph{common key test} is a prescribed function of global keys.
+It may depend on $n$ and on the law under consideration, but it is not
+chosen using the particular orientation being queried.
+\item A \emph{common input-and-key test} may also use the nominal
+parameters. The concentration result below applies to these more
+general tests. The positivity result for basis statuses applies only to
+common key tests.
+\item A histogram \emph{flag} is any one of a fixed bounded list of bits
+depending on an orientation and its query parameters. The histogram
+compactness result permits arbitrary filters of the orientation and
+these flags. In particular a flag may use all channel maps; the bits
+$p_i$ used later are such flags. Later successful queries use a narrower class: products of
+unary status requirements, the specified Gram and $L,R$ constraints,
+and an additional orientation-only filter. The product comparison in
+\Cref{sec:product-tests} is stated for that narrower class.
+\end{enumerate}
+All filtered measures in this section are subprobability measures. We do
+not divide them by the probabilities of their filters.
+Colours and law types may be retained as finite marks on the
+orientations and on the leaves.  The number of their possible values
+is fixed before $n$ grows.  Neither a mark nor a channel-dependent
+flag becomes a common key test merely by being recorded.
+
+\subsection{Finite Gram comparisons}
+
+We first record the counting and Fourier facts used several times below.
+
+\begin{lemma}[Gram normalization]\label{lem:histogram-gram-normalization}
+Let $a$ plus slots and $b$ minus slots in one ambient component be
+independent uniform vectors, and prescribe their entire $a$ by $b$ Gram
+matrix. If $a+b\le N/2$, the probability of that prescription together
+with injectivity in each mode is
+\[
+ 2^{-ab}\bigl(1+O(2^{-N+a+b})\bigr),
+\]
+uniformly in the prescribed matrix. The same statement holds for a
+fixed collection of independent endpoint/component groups, with the
+sum of their Gram-bit counts in place of $ab$ and a fixed factor in the
+error bound. In particular, for a bounded number of slots all relevant
+Gram normalizations and their ratios are bounded above and below by
+positive constants, uniformly in the nominal inputs.
+\end{lemma}
+
+\begin{proof}
+The plus slots are linearly independent with probability
+$\prod_{j=0}^{a-1}(1-2^{j-N})=1-O(2^{-N+a})$. Given such plus slots,
+each minus slot has probability $2^{-a}$ of satisfying its prescribed
+values. Conditional on those values the minus slots are independent
+uniform vectors in specified affine cosets of one common annihilator
+of dimension $N-a$. For any nonzero combination of the $b$ slots,
+the resulting vector is uniform in an affine coset of that annihilator.
+Its probability of being zero is at most $2^{-(N-a)}$. A union bound
+over fewer than $2^b$ combinations proves the claimed injectivity error.
+Multiplication over independent groups gives the last assertion.
+\end{proof}
+
+When the number of slots is bounded, dropping an injectivity condition
+in a normalized Gram law changes a bounded integral by
+$O(2^{-N+O(1)})$. For the growing batches used below we will instead
+retain injectivity in the denominator and drop it only from a
+nonnegative numerator; this avoids paying an inappropriate additive
+error before a density bound.
+
+\begin{lemma}[Comparison of two bounded batches]
+\label{lem:histogram-batch-comparison}
+Consider two bounded batches of prescribed nominal columns in one raw
+endpoint, or in two independent raw endpoints. Suppose their union is
+nominally independent in every endpoint, component, and mode. Let $F$
+and $G$ be bounded functions of the separate image batches, respectively.
+The difference between their joint raw expectation and the product of
+their separate raw expectations is
+\[
+ O\bigl(\norm{F}_\infty\norm{G}_\infty 2^{-N/2}\bigr).
+\]
+The bound is uniform in the independent nominal columns and all their
+prescribed same-endpoint Gram entries. The functions may also depend
+on the fixed nominal inputs determining their own batch.
+\end{lemma}
+
+\begin{proof}
+By \Cref{lem:frame-images}, the image laws are uniform laws on the
+indicated independent vectors with their same-endpoint Gram entries
+prescribed. Relative to the product of the two separate batch laws,
+the union law adds only the mutual Gram entries and joint injectivity.
+The latter may be omitted at an exponentially small error because the
+number of columns is bounded.
+
+Expand the indicators of the mutual Gram entries into binary
+characters. Their values, which are determined by the nominal inputs,
+enter this expansion only as signs. In a nontrivial character, after
+identical descriptions of an unordered slot pair have been collapsed,
+the coefficient matrix between the two batches is nonzero. Its bit
+rank is at least $N$. The individual batch Gram densities can be
+included in the two separate bounded weights. \Cref{lem:walsh} bounds
+this character by a constant times
+$\norm{F}_\infty\norm{G}_\infty2^{-N/2}$.
+There are only boundedly many characters. Applying the same expansion
+with $F=G=1$ controls the normalization, while
+\Cref{lem:histogram-gram-normalization} bounds all internal normalization
+factors. The zero character gives the product of separate expectations.
+\end{proof}
+
+\begin{lemma}[Product tests under the key reference]
+\label{lem:reference-product}
+For a fixed query shape and arbitrary global single-key functions
+$f_{j,n}$ with $\abs{f_{j,n}}\le1$,
+\[
+ \int\prod_jf_{j,n}(k_j)\,d\nu_n
+ =\prod_j\int f_{j,n}\,d\Omega_{l_j,q_j,n}
+   +O(2^{-N/2}).
+\]
+The bound is uniform over the functions.
+\end{lemma}
+
+\begin{proof}
+Start with $\overline\Omega_n$ and expand the off-position Gram
+conditions. For any nonzero character, choose an interacting pair of
+positions and fix all other positions. The remaining phase has rank
+at least $N$ between the two selected position groups. The prescribed
+single-position diagonal densities and the unary functions are
+separately bounded weights. \Cref{lem:walsh} gives the error above.
+The identical expansion with all functions one gives the normalization
+of the off-position Gram event. Its zero-character contribution is a
+fixed positive constant, and all other contributions are exponentially
+small. Dividing proves the assertion.
+\end{proof}
+
+\subsection{A many-query Fourier bound}
+
+Here and in the histogram lemma, a leaf is normalized to have mass one.
+Its law may be very far from having bounded individual marginals.
+
+\begin{lemma}[Many queries and prescribed cells]
+\label{lem:histogram-many-queries}
+Fix a bounded query shape. There are constants $c_1,c_2>0$, independent
+of any key-space partition, with the following property. Put
+$\ell=\lfloor c_1n\rfloor$ and repeat the query independently $\ell$
+times at a common orientation. Let $A_1,\ldots,A_\ell$ be cells of a
+partition of the full slot space, with weights
+$w_j=\omega_n(A_j)$. Suppose the partition has bounded size and its
+nonempty weights have a positive lower bound along the sequence under
+consideration. Conditional on any nominal inputs whose queried columns
+are jointly independent in every endpoint, component, and mode, the
+raw image probability that each query $j$ lies in $A_j$ is at most
+\begin{equation}\label{eq:histogram-many-query-bound}
+ 2^{c_2\ell}\prod_{j=1}^{\ell}w_j
+\end{equation}
+for all sufficiently large $n$. How large $n$ must be may depend on the
+partition and its weight bound. The constants $c_1,c_2$ do not.
+
+The probability, over the repeated fresh parameters, that the required
+nominal independence fails tends to zero exponentially in $n$, uniformly
+over the common orientation.
+\end{lemma}
+
+\begin{proof}
+Let $k$ bound the number of vector slots in one draw, enlarging it by a
+fixed factor when convenient. At any endpoint and component, a new
+point column contains the component $p_s\otimes Z$ with a fresh uniform
+$n$-bit vector $Z$. Its probability of lying in a previously exposed
+space of dimension $v$ is at most $2^{-n+v}$: the projection of that
+space onto its $p_s\otimes Z$ direction space has dimension at most
+$v$. This remains valid with repeated labels, and also after the input
+is conditioned on $q$ or a tester summary. A union bound, with
+$k\ell$ a small fraction of $n$, proves the independence assertion.
+
+Fix independent nominal inputs. The raw image law is an iid global
+slot law conditioned on all required same-endpoint Gram entries and
+on injectivity. Divide the Gram entries into those within a draw and
+those between draws, after collapsing duplicate entries. Write their
+numbers as $b_{\mathrm{in}}$ and $b_{\mathrm{cross}}$, respectively.
+We have $b_{\mathrm{in}}=O(\ell)$. The normalization in the denominator
+is at least a fixed constant times
+$2^{-b_{\mathrm{in}}-b_{\mathrm{cross}}}$ by
+\Cref{lem:histogram-gram-normalization}. In the nonnegative numerator,
+drop injectivity and the within-draw conditions, and expand only the
+cross-draw conditions. It remains to bound
+\[
+ 2^{O(\ell)}\sum_{\chi}
+ \abs{\E_{\omega_n^{\otimes\ell}}
+          \chi\prod_{j=1}^{\ell}\1_{A_j}},
+\]
+where the zero character has contribution $\prod_jw_j$.
+
+Represent a nonzero character by a symmetric binary matrix on the
+distinguished slot indices, grouped into $\ell$ draw blocks of size
+at most $k$. Its diagonal draw blocks are zero. A coefficient records
+one unordered interacting slot pair; in particular there is no
+nontrivial character represented by the zero matrix. Let $t\ge1$ be
+the maximum rank of a coefficient rectangle whose row draw-block set
+and column draw-block set are disjoint.
+
+We claim that the number of such patterns with this value of $t$ is at
+most $2^{Ck^2\ell t}$ for an absolute constant $C$. Choose a nonsingular
+$t$ by $t$ minor witnessing the rank, with row-block set $I$ and
+column-block set $J$. Each of $I,J$ has at most $t$ blocks. Specify all
+entries incident with those blocks. If $u,v$ lie in two distinct
+unselected blocks, border the chosen minor by row $u$ and column $v$.
+The enlarged row and column block sets are still disjoint. Its rank
+cannot exceed $t$, so its bottom-right entry is uniquely determined by
+the other entries and the invertible minor. Entries in a single
+unselected diagonal block are already zero. Thus the specified entries
+determine the whole pattern. Their number is $O(k^2\ell t)$; the choices
+of the minor add at most $2t\log_2(k\ell)$ bits, which is absorbed in
+the asserted bound.
+
+For one pattern, fix slots outside the draw blocks $I\cup J$. Across
+$I$ and $J$ the phase has bit rank at least $tN$. Phases internal to
+each group or involving the fixed slots are separate weights, and the
+cell indicators within the two groups are bounded by one. The Walsh
+estimate gives $2^{-tN/2}$. Integrating the other cell indicators gives
+\begin{equation}\label{eq:histogram-character-bound}
+ \abs{\E\chi\prod_j\1_{A_j}}
+ \le 2^{-tN/2}\prod_{j\notin I\cup J}w_j
+ \le 2^{-tN/2}w_{\min}^{-2t}\prod_jw_j,
+\end{equation}
+where $w_{\min}=\min_jw_j$. This argument treats components and modes
+as distinguished slots; their only effect is to constrain which
+coefficient entries may be nonzero.
+
+The sum of nonzero characters, divided by $\prod_jw_j$, is therefore
+at most
+\[
+ \sum_{t\ge1}
+ 2^{Ck^2\ell t-tN/2+2t\log_2(1/w_{\min})}.
+\]
+Choose $c_1$ small enough both for nominal independence and for
+$Ck^2c_1<M_0/8$. For any fixed positive weight bound, the displayed sum
+is at most $\sum_{t\ge1}2^{-tN/4}$ for all sufficiently large $n$.
+It can consequently be absorbed in the within-draw factor
+$2^{O(\ell)}$. This proves \eqref{eq:histogram-many-query-bound} with a
+constant independent of the partition and its positive weight bound.
+\end{proof}
+
+\subsection{Uniform integrability and per-leaf nets}
+
+For a finite partition $\cP$ and reference weights $w_C$, use relative
+entropy in bits,
+\[
+ D_2(p\Vert w)=\sum_{C\in\cP}p_C\log_2\frac{p_C}{w_C},
+ \qquad 0\log_2 0=0.
+\]
+Only nonempty reference cells are used.
+
+For the finite entropy chain rule and Pinsker inequality in these units,
+see \citet[Theorem~2.5.3 and Lemma~11.6.1]{cover2006}. We include the
+short derivation of the inequality and its constants in bits.
+For probability vectors $p,q$, put $A=\{j:p_j>q_j\}$. Convexity of
+$t\log t$ (the log-sum inequality) collapses their relative entropy
+to that of the two probabilities $p(A),q(A)$. Binary relative entropy
+in natural units has second derivative $1/[u(1-u)]\ge4$ in its first
+argument and has value and first derivative zero at $u=q(A)$.
+Since $p(A)-q(A)=\norm{p-q}_1/2$, Taylor's integral formula gives
+\begin{equation}\label{eq:histogram-pinsker}
+ D_2(p\Vert q)\ge \frac{\norm{p-q}_1^2}{2\ln2}
+ \ge\tfrac12\norm{p-q}_1^2.
+\end{equation}
+Zero probabilities follow by continuity, with infinite relative entropy
+when appropriate.
+
+\begin{lemma}[Uniform histogram control]\label{lem:histogram-control}
+Fix a query shape, a constant $C_L$, and an integer $F$. Consider any
+collection of normalized leaf laws $\nu_L$ satisfying
+\begin{equation}\label{eq:histogram-leaf-cap}
+ \nu_L\le 2^{C_LN}\rawlaw^2.
+\end{equation}
+The following assertions are uniform over those leaves and over the
+bounded numerical choices of a query.
+\begin{enumerate}[label=(\roman*)]
+\item If $A_n$ is a sequence of slot sets with
+$\omega_n(A_n)\to0$, then the probability that the unfiltered query
+key belongs to $A_n$, averaged on any choice of leaves, tends to zero.
+Equivalently, the averaged key laws are uniformly absolutely continuous
+in this asymptotic small-set sense.
+\item On each leaf, choose any $F$ binary flags of the orientation and
+query parameters. Let $\mathcal H_{L,n}$ be the family of
+unnormalized key densities obtained by filtering by any event of the
+orientation and the flag values. For every $\eps>0$, this family has an
+$L^1(\omega_n)$ $\eps$-net of cardinality bounded independently of $n$
+and of the leaf, for all sufficiently large $n$. The net itself may
+depend on the leaf.
+\item Both conclusions persist after restricting the query to $\cK_n$
+and expressing the subdensities relative to $\nu_n$. In particular these
+subdensities are uniformly integrable: for every $\eta>0$ there is
+$R<\infty$ such that, for all sufficiently large $n$, all their tails
+above $R$ have integral at most $\eta$.
+\end{enumerate}
+The flag functions may vary with the leaf and with $n$. The number of
+flags is fixed. No bound on the normalized marginal of an individual
+leaf is assumed here.
+The displayed cap concerns the orientation law after any marks are
+forgotten.  Marks can be fixed on a leaf or included among the
+bounded flag data.
+\end{lemma}
+
+\begin{proof}
+We first obtain a uniform bound for coarse conditional information.
+Fix a partition with a bounded number $r$ of cells and with every
+nonempty reference weight bounded below by a positive constant. At an
+orientation $O$, let $H_O(C)$ be its input probability of cell $C$.
+Repeated queries are independent with this cell law. Uniformly over
+probability vectors on the fixed $r$-simplex, the empirical histogram of
+$\ell=\lfloor c_1n\rfloor$ samples converges to $H_O$. Relative entropy
+against the fixed positive reference weights is uniformly continuous
+on that simplex: each cell frequency has variance at most $1/(4\ell)$,
+and $u\log u$ is continuous at zero. Consequently, on orientations with
+$D_2(H_O\Vert w)>T+1$, the empirical histogram has entropy greater
+than $T$ with probability $1-o(1)$, uniformly over those orientations.
+The nominal independence event of
+\Cref{lem:histogram-many-queries} can be imposed simultaneously at a
+further $o(1)$ loss, also uniformly over orientations.
+
+There are at most $(\ell+1)^r$ empirical types. For a type $v$, the
+multinomial bound gives
+\[
+ \frac{\ell!}{\prod_C(\ell v_C)!}\prod_Cw_C^{\ell v_C}
+ \le 2^{-\ell D_2(v\Vert w)}.
+\]
+Indeed the probability of that same type under cell probabilities $v$
+is at most one, which bounds its multinomial coefficient by
+$\prod_Cv_C^{-\ell v_C}$.
+Apply \eqref{eq:histogram-many-query-bound} for each cell sequence of a
+type, and then pay \eqref{eq:histogram-leaf-cap}. We obtain
+\begin{align}\label{eq:histogram-entropy-tail}
+ &(1-o(1))\nu_L\{D_2(H_O\Vert w)>T+1\}\notag\\
+ &\hspace{1cm}\le
+ 2^{C_LM_0n+(c_2-T)\ell+r\log_2(\ell+1)}.
+\end{align}
+Choose $T>c_2+C_LM_0/c_1+2$. This bound is exponentially small.
+The density cap has been applied only on the event of
+nominal independence. Its failure was removed in the uniform
+per-orientation lower bound, not multiplied by $2^{C_LN}$ as an
+additive raw exceptional probability.
+
+Since $D_2(H_O\Vert w)\le\log_2(1/w_{\min})$, it follows that
+\begin{equation}\label{eq:histogram-coarse-entropy}
+ \limsup_{n\to\infty}\E_{O\sim\nu_L}D_2(H_O\Vert w)\le C_H
+\end{equation}
+for a constant $C_H$ independent of the fixed partition size and its
+positive weight bound. The onset of the bound may depend on both.
+The constant may depend on $C_L,M_0$ and the query shape.
+
+Let $V$ denote the flag vector, let $C_{\cP}(K)$ denote the cell of
+$\cP$ containing the key $K$, and let $p_{O,V}$ be its cell law
+conditional on $O,V$. Average this law with the actual joint
+distribution of $(O,V)$; flag values of probability zero may be given
+any conditional law. The entropy chain rule gives
+\[
+ \E D_2(p_{O,V}\Vert w)
+ =\E D_2(H_O\Vert w)+I(C_{\cP}(K);V\mid O)
+ \le\E D_2(H_O\Vert w)+F.
+\]
+Indeed, split each logarithm $\log_2(p_{O,V}(C)/w_C)$ through
+$H_O(C)$. The first term averages to $I(C_{\cP}(K);V\mid O)$; the second
+averages to $\E D_2(H_O\Vert w)$ because averaging $p_{O,V}$ over
+the conditional flag law gives $H_O$. The information term is at
+most the entropy of the $F$-bit flag vector, hence at most $F$.
+Thus \eqref{eq:histogram-coarse-entropy} holds for these conditional
+histograms with $C_H+F$ in place of $C_H$.
+
+For the small-set assertion, suppose by contradiction that
+$\omega_n(A_n)\to0$ but the averaged query probability of $A_n$ is
+bounded below by $a>0$ on a sequence of leaves. Fix $0<\delta<1/2$ and
+enlarge $A_n$ to a set of reference weight tending to $\delta$; the
+uniform slot atoms tend to zero, so this is possible. On the resulting
+two-cell partition, the binary entropy inequality gives
+\[
+ D_2((p,1-p)\Vert(\delta,1-\delta))
+ \ge p\log_2(1/\delta)-1.
+\]
+Taking expectations and limits in
+\eqref{eq:histogram-coarse-entropy} yields
+$a\log_2(1/\delta)\le C_H+1$. Sending $\delta$ to zero is a
+contradiction. This proves (i), and the same assertion holds for every
+filtered submeasure by domination.
+
+We next prove the net assertion. For a partition $\cP$, write
+$\Pi_{\cP}d$ for its conditional-average density relative to
+$\omega_n$. If some filtered density $d$ satisfies
+$\norm{d-\Pi_{\cP}d}_1>\eps$, split every cell by the set where
+$d-\Pi_{\cP}d$ is positive. On the refined partition the filtered cell
+masses differ in $\ell^1$ by more than $\eps$ from their parent masses
+subdivided in reference proportions. Since the filter is a function
+of $(O,V)$ with values in $[0,1]$, the triangle inequality implies that
+the \emph{expected} corresponding discrepancy of the unfiltered
+conditional vectors $p_{O,V}$ is at least $\eps$.
+
+For any refinement, the relative entropy chain rule and
+\eqref{eq:histogram-pinsker} give
+\begin{align}\label{eq:histogram-refinement-gain}
+ &\E D_2(p_{\mathrm{child}}\Vert w_{\mathrm{child}})
+ -\E D_2(p_{\mathrm{parent}}\Vert w_{\mathrm{parent}})\notag\\
+ &\quad\ge \frac12
+ \left(\E\norm{p_{\mathrm{child}}-
+    \operatorname{split}_{w}(p_{\mathrm{parent}})}_1\right)^2.
+\end{align}
+Here $\operatorname{split}_{w}$ divides each parent mass among its
+children in their reference proportions. To see the last step, apply
+the usual conditional relative entropy identity inside each parent,
+then Pinsker, and then Jensen first with the parent masses and then
+over $(O,V)$.
+
+A bounded number of refinements must suffice uniformly asymptotically.
+For otherwise choose any large fixed depth $d$ and a sequence of
+leaves with $d$ successive witnessing refinements. Their final
+partitions have at most $2^d$ cells. Pass to a subsequence on which the
+reference weights of the entire finite tree converge and the laws of
+the conditional histogram vectors converge. A final cell of zero
+limiting reference weight has zero limiting expected conditional mass
+by (i), and hence zero conditional mass almost surely. Delete these
+zero-weight cells in the limit. For a positive-weight parent, reference
+subdivision ratios converge. For a zero-weight parent, both the actual
+and subdivided total masses tend to zero in expectation. The positive
+discrepancy at every refinement consequently persists in the limit,
+at least as $\eps/2$ if needed.
+
+The limiting final conditional entropy is at most $C_H+F$. Indeed,
+before taking limits merge all zero-limiting-weight final cells into
+one positive cell, and use the conditional version of
+\eqref{eq:histogram-coarse-entropy} on that partition. All its limiting
+weights are positive, so entropy is continuous. This merging is used
+only for the final upper bound. The remaining positive limit cells
+still form a genuine refinement tree, on which
+\eqref{eq:histogram-refinement-gain} telescopes. That identity forces
+the final entropy to be at least $d\eps^2/8$, a contradiction for large
+fixed $d$.
+
+Apply the preceding argument with $\eps/2$. On a partition of bounded
+size $r_0'$ that approximates every family member, quantize each cell
+mass downwards in increments $\eps/(2r_0')$. The $L^1$ error in the
+piecewise-constant density is the sum of the cell-mass errors, at most
+$\eps/2$. There are at most
+$(1+2r_0'/\eps)^{r_0'}$ possible quantized mass lists. They form the
+required net, proving (ii).
+
+For (iii), the restriction of a density $d$ relative to $\omega_n$ has
+density $\omega_n(\cK_n)d$ on $\cK_n$ relative to $\nu_n$. Restriction
+therefore cannot increase the $L^1$ distance. A $\nu_n$-small set is
+also $\omega_n$-small, so (i) transfers as well. Every resulting
+subdensity has integral at most one; its superlevel set above $R$ has
+reference measure at most $1/R$. Assertion (i), applied along any
+contrary sequence, makes the integrals on those superlevel sets tend
+to zero uniformly as $R\to\infty$. This is uniform integrability.
+\end{proof}
+
+\begin{remark}\label{rem:histogram-leaf-versus-mixed}
+The preceding lemma allows a leaf to have strongly nonuniform
+histograms. For example, forcing a whole primal frame map into a
+fixed ambient subspace of constant codimension costs $2^{O(n)}$ and
+can be consistent with \eqref{eq:histogram-leaf-cap}. It puts its keys
+on a fixed small reference set. The uniform entropy constant is
+therefore allowed to depend on the leaf density exponent and on $M_0$.
+The next results concern typical orientations under a mixed law and
+use a different, stronger marginal hypothesis.
+\end{remark}
+
+\subsection{Typical empirical integrals under a paired law}
+
+Write $\rawlaw^{\mathrm{pr}}=\operatorname{pr}_*\rawlaw$ for the raw
+single-endpoint primal law.  In the rest of the section let $\rho_n$
+be a law on units satisfying, after forgetting any finite marks,
+for a fixed $C$ and some $a_n\to0$,
+\begin{equation}\label{eq:histogram-mixed-caps}
+ \rho_n\le2^{CN}\rawlaw^2,
+ \qquad \operatorname{pr}_*(\rho_n)_i
+       \le M2^{a_nN}\rawlaw^{\mathrm{pr}}\quad(i=1,2).
+\end{equation}
+Changing the fixed factor $M$ does not matter. Since $M_0$ is fixed,
+$a_nN=o(n)$. The law may have arbitrary dependence between its two
+endpoints. A decomposition into leaves does not change
+\eqref{eq:histogram-mixed-caps} for its mixture.
+The second inequality is solely a primal marginal bound.  No
+$M2^{o(N)}\rawlaw$ bound on the full endpoint law is required or
+asserted.  In particular $\operatorname{pr}_*\pi=\rawlaw^{\mathrm{pr}}$
+allows endpoint caps relative to the thinned law $\pi$ to supply this
+hypothesis.  Projecting the first inequality also gives
+$\operatorname{pr}_*\rho_n\le2^{CN}(\rawlaw^{\mathrm{pr}})^2$.
+
+\begin{lemma}[Typicality of common input-and-key tests]
+\label{lem:typicality}
+Let $F_n$ be a uniformly bounded common function of the inputs and
+primal keys of a fixed bounded query, with no other orientation
+dependence. In particular it does not inspect channel maps or the
+orientation's marks.
+Put
+\[
+ A_n(O)=\E_z F_n(z,K(O,z)),\qquad
+ m_n=\E_{O\sim\rawlaw^2,z}F_n(z,K(O,z)).
+\]
+For every fixed $\eps>0$, under \eqref{eq:histogram-mixed-caps},
+\[
+ \rho_n\{\abs{A_n-m_n}>\eps\}=2^{-\Omega(n)}.
+\]
+Repeated labels in the query are allowed. The constants may depend on
+the query and $\eps$, but the estimate is uniform over tests with the
+given sup bound. The exceptional set may depend on the test.
+
+For a query at just one endpoint, its raw variance about its raw mean
+is $2^{-\Omega(n)}$. This variance bound is also uniform when an
+arbitrary external input-and-image datum is fixed as an argument of
+the test. This datum may include a channel frame from a separately
+fixed vertex; uniformity refers to each fixed datum.
+\end{lemma}
+
+\begin{proof}
+Every empirical integral and exceptional event used in this proof
+is measurable with respect to the primal maps.  We can therefore
+work throughout with the projected law, paying only the projected
+marginal bound in \eqref{eq:histogram-mixed-caps}.  Raw expectations
+are unchanged by this projection.  Rescale the test to have absolute
+value at most one. First consider
+one raw endpoint and duplicate its parameter batch. The two batches
+have jointly independent nominal columns except with
+$2^{-\Omega(n)}$ probability, because their $Z$ inputs are independent
+and their total number of columns is bounded. With the nominal inputs
+fixed on this event, \Cref{lem:histogram-batch-comparison} compares the
+product of the two tests with the product of their separate raw
+expectations, at error $O(2^{-N/2})$. Averaging the two independent
+input batches and bounding the dependent-input event trivially proves
+\begin{equation}\label{eq:histogram-one-endpoint-variance}
+ \Var_{O\sim\rawlaw}\bigl(\E_zF_n(z,K(O,z))\bigr)
+ \le C_1 2^{-c_1'n}.
+\end{equation}
+All these estimates are uniform in a fixed external datum, since it
+only changes the separate bounded test functions. Chebyshev's
+inequality proves the one-endpoint assertion at every fixed
+tolerance.
+
+For the paired assertion, assume that both endpoints are used; the
+other case follows at once from the marginal bound. Let $D_2$ denote
+the second endpoint's input-and-image data under its raw model, and
+for a first primal orientation $o$ define
+\[
+ h_o(D_2)=\E_{z_1}F_n(z_1,K_1(o,z_1),D_2),\qquad
+ \bar h(D_2)=\E_{o\sim\rawlaw}h_o(D_2).
+\]
+Fix a small constant $\eta>0$. By
+\eqref{eq:histogram-one-endpoint-variance}, uniformly in $D_2$,
+$\PP_{\rawlaw^{\mathrm{pr}}}\{\abs{h_o(D_2)-\bar h(D_2)}>\eta\}$ is exponentially
+small. Fubini and Markov show that, outside an exponentially small
+set of first primal orientations, the raw-model probability of the bad data
+set
+\[
+ B_o=\{D_2:\abs{h_o(D_2)-\bar h(D_2)}>\eta\}
+\]
+is at most $2^{-c n}$ for some $c>0$. The projected marginal cap in
+\eqref{eq:histogram-mixed-caps} still makes the discarded
+first-endpoint mass exponentially small.
+
+Fix any of the remaining first orientations. We claim that a raw
+second orientation $o'$ has empirical input probability at least
+$\eta$ of $B_o$ only with probability $2^{-\Omega(n^2)}$. Draw
+$\ell'=\lfloor c_2'n\rfloor$ independent second-endpoint input
+batches. For an orientation with empirical bad probability at least
+$\eta$, at each step success and nominal independence of the new
+columns modulo the previous ones have conditional probability at
+least
+\[
+ \eta-O(2^{-n+O(\ell')})\ge\eta/2.
+\]
+The failure bound depends only on fresh uniform $Z$ coordinates and
+holds for any past inputs. Thus all successes together with joint
+nominal independence have probability at least $(\eta/2)^{\ell'}$.
+
+For independent nominal columns, the joint raw image law of these
+batches has density at most $2^{C_2(\ell')^2}$ relative to the product
+of the individual raw batch laws. Indeed it adds only
+$O((\ell')^2)$ mutual Gram conditions and injectivity; count their
+normalizations by \Cref{lem:histogram-gram-normalization} and bound the
+indicator in the numerator by one. After averaging the inputs, the
+all-success probability on the independence event is consequently at
+most
+\[
+ 2^{C_2(\ell')^2}\PP(D_2\in B_o)^{\ell'}
+ \le2^{C_2(\ell')^2-cn\ell'}.
+\]
+Choose $c_2'>0$ small enough that $C_2(c_2')^2<cc_2'/2$ and that
+nominal independence is available. Dividing by
+$(\eta/2)^{\ell'}$ proves the claim.
+
+This quadratic-exponential estimate is uniform in the fixed good
+first primal orientation. Hence the set of paired primal orientations with a good
+first endpoint but a bad oversampling second endpoint has
+$(\rawlaw^{\mathrm{pr}})^2$-mass $2^{-\Omega(n^2)}$. Multiplying by the projected joint density
+$2^{CN}=2^{CM_0n}$ still leaves quadratic-exponentially small mass.
+On every remaining pair,
+\[
+ \abs{\E_{z_2}h_o(D_2)-\E_{z_2}\bar h(D_2)}\le3\eta.
+\]
+Finally, $\bar h$ is a common input-and-key test of the second
+endpoint. Apply the one-endpoint estimate again and its projected marginal cap
+to compare its empirical integral with its raw mean to within
+$\eta$. Taking $4\eta<\eps$ proves the lemma.
+\end{proof}
+
+\begin{remark}\label{rem:typicality-no-test-supremum}
+Uniformity in the test means that each prescribed test has a small
+exceptional set with the stated bound. It does not mean that one
+orientation is simultaneously good for every possible global test.
+The later arguments use only finitely many common tests at a time.
+For two distinct laws, apply these results on a mixture with fixed
+positive type weights and retain the type as a mark.  A test prescribed
+for each of the finitely many types is handled by taking the union
+of its exceptional sets.  The same argument permits colours to be
+recorded on leaves while using the original, mass-weighted mixture
+for every marginal estimate.  It does not require the normalized
+individual colour classes to satisfy a common marginal cap.
+Restriction to a sublaw of inverse-polynomial mass, or more generally
+of reciprocal mass $2^{o(n)}$, preserves
+\eqref{eq:histogram-mixed-caps}. Such restriction need not preserve a
+uniform cap on every individually normalized old leaf. The mixed-law
+lemmas do not require that additional conclusion.
+\end{remark}
+
+\subsection{Mixing of the binary parameter constraints}
+
+\begin{lemma}[Binary mixing]\label{lem:binary-mixing}
+Fix a query template whose labels are distinct at each used endpoint.
+Impose the off-position Gram zeros of $\cK_n$ and a consistent fixed
+specification of the within-endpoint ordered $L,R$ evaluations used
+in \Cref{sec:realization}. Let $\mathcal A_n(O,z)$ be their joint
+indicator. There is a template constant $\kappa>0$ such that, outside
+an exponentially small set of orientations under
+\eqref{eq:histogram-mixed-caps},
+\begin{equation}\label{eq:binary-mixing}
+ \E_z\mathcal A_n(O,z)\prod_jw_j(O,z_j)
+ =\kappa\prod_j\E_{z_j}w_j(O,z_j)+O(2^{-c n})
+\end{equation}
+uniformly over all unary weights with $\abs{w_j}\le1$. These weights
+may depend on the entire fixed orientation. The input laws may be
+conditioned separately on feasible diagonal values or tester summaries.
+The exceptional set is independent of the unary weights.
+\end{lemma}
+
+\begin{proof}
+Collapse all identical Gram entries, including the repeated
+same-endpoint entries that equal the same $E$ evaluation, and impose
+only one copy of each. Consistency means that their prescribed values
+respect these repetitions. Expand the resulting binary indicators
+into characters. The zero character has coefficient $\kappa=2^{-b}$,
+where $b$ is the number of the resulting specified bits. We show that
+every nonzero character has an exponentially small integral against
+the product of unary weights.
+
+If an interacting pair of positions belongs to the same endpoint,
+their labels are distinct. The second property of
+\Cref{lem:generic-mixers} says that every nonzero parity of its ordered
+$L,R$ bits, with its possible $E$ bits in either order, has bilinear
+rank at least $n/5$ on the retained free variables. This includes
+characters involving only $E$ bits.
+
+For positions from opposite endpoints, we establish the analogous
+claim for their global Gram bits except on a
+$2^{-\Omega(n^2)}$ raw exceptional set. Write $d=\dim\cB+h$ for the
+full plus-frame dimension. At one component, an intersection of
+dimension at least $v=\lfloor n/10\rfloor$ between the two full plus
+spans gives $v$ independent image relations. Their coefficient
+choices number at most $2^{2dv}$, and the frame image estimate bounds
+each event by $2^{-(1-\epsilon)vN}$, for a fixed small $\epsilon$.
+The choice of $M_0$, hence $d/N$ small, makes this
+$2^{-\Omega(n^2)}$. There are only boundedly many components.
+
+Condition on plus frames off this exception. Choose a nonzero pair
+pattern and a component where, after exchanging the endpoints if
+necessary, it uses a plus slot at endpoint 1 paired with a minus slot
+at endpoint 2. At least $n-v$ of the first position's $Z$ directions
+are independent modulo the second endpoint's full plus frame. The
+second position's nominal minus $Z$ directions are independent.
+Conditional on the plus frames, the latter minus columns have
+independent uniform values on the appropriate affine annihilator
+spaces, before their full-rank conditioning. Their evaluations on
+these $n-v$ directions therefore form a uniform rectangular matrix,
+up to a fixed translation. Terms in the reverse order and other
+components may be frozen first; they use separate minus-frame
+randomness. The final injectivity conditioning costs only a bounded
+multiplicative factor.
+
+A uniform $a$ by $b$ binary matrix has rank at most $r$ with
+probability at most $2^{(a+b)r-ab}$, by writing it as a product of an
+$a$ by $r$ and an $r$ by $b$ matrix and counting. With
+$a\ge n-v$ and $b=n$, this shows that the bilinear rank is greater
+than $n/5$ except with probability $2^{-\Omega(n^2)}$. Union over the
+boundedly many nonzero pair patterns. The joint density in
+\eqref{eq:histogram-mixed-caps} preserves this exceptional scale.
+
+For any nonzero full character, choose a pair with nonzero coefficient
+pattern and fix the other position inputs. The remaining terms become
+separate unary phases. The selected pair has rank at least $n/5$ by
+one of the preceding two arguments. \Cref{lem:walsh} gives an error
+$O(2^{-n/10})$ against the two remaining bounded unary weights. Input
+conditioning costs only the fixed reciprocal probabilities of the
+diagonal values or tester summaries; those events can be absorbed
+into the respective weights. Summing the bounded number of
+characters proves \eqref{eq:binary-mixing}. All estimates were uniform
+in the unary weights, so the exceptional set does not depend on them.
+\end{proof}
+
+\subsection{Positive basis fractions on common key cells}
+
+We give the small-cell argument explicitly, because its uniform lower
+fraction will be used on a countable collection of signature cells.
+
+\begin{lemma}[Quadratic signs on affine subspaces]
+\label{lem:histogram-quadratic-bias}
+Let $Q$ be a quadratic function on a binary vector space and let its
+polar form have rank $2r$. Its uniform sign bias has magnitude at
+most $2^{-r}$. On an affine subspace of codimension $c$, the restricted
+polar rank is at least $2r-2c$, and its sign bias is at most
+$2^{-r+c}$ whenever this bound is less than one.
+\end{lemma}
+
+\begin{proof}
+Squaring the mean sign and translating one variable gives
+\[
+ \abs{\E_x(-1)^{Q(x)}}^2
+ =\E_h(-1)^{Q(h)+Q(0)}\1_{\{B(h,\cdot)=0\}},
+\]
+where $B$ is the polar form. Its absolute value is at most the kernel
+proportion $2^{-2r}$. Restricting a bilinear form to a codimension-$c$
+subspace in each of its two factors can decrease rank by at most
+$2c$. Apply the first assertion on that affine subspace, where
+translation affects only the linear and constant terms.
+\end{proof}
+
+\begin{lemma}[Unary reference law and base-status positivity]
+\label{lem:unary-positivity}
+Fix an endpoint, tag, label, flavor, and feasible tester summary with
+diagonal $q$. For a bounded number of common global key tests, its
+empirical conditional key frequencies differ from
+$\Omega_{l,q,n}$ by any prescribed fixed tolerance except on
+exponentially small mixed mass under
+\eqref{eq:histogram-mixed-caps}.
+
+At each unit, let $C_i\subset\cX$ be any orientation-dependent space
+of dimension at most $K^2$ all of whose elements have total component
+rank at most $K$, and choose any basis. Fix a sequence of global
+single-key sets $A_n$ with
+$\Omega_{l,q,n}(A_n)\ge\delta>0$. Except on exponentially small mixed
+mass, every basis assignment has, conditional on the tester summary
+and on $K_i\in A_n$, fraction at least
+$(7/8)2^{-\dim C_i}$.
+For the fixed common key sets, the exceptional event depends only on
+the primal frames and is chosen uniformly over all nominal profiles
+of the indicated rank before $C_i$ or its basis is selected. Thus
+these selections may depend on the channels and all orientation
+marks. We use the first effective spaces from
+\eqref{eq:effective-space}; the larger generality in this statement
+requires no further peeling.
+
+For the full generic flavor and inputs conditional just on $q$, the
+following conservative constant is valid:
+\begin{equation}\label{eq:uniform-base-fraction}
+ \beta_{\mathrm{base}}=2^{-K^2-1}4^{-(g^2+1)}>0.
+\end{equation}
+For every feasible specified tester summary and basis assignment,
+outside an exponentially small exceptional set,
+\begin{equation}\label{eq:histogram-base-cell-lower}
+ \PP_z\{K_i\in A_n,\ \text{the specified summary and basis bits}\mid q\}
+ \ge\beta_{\mathrm{base}}\,\Omega_{l,q,n}(A_n).
+\end{equation}
+In particular each joint choice of the role $a(\operatorname{atom})$
+and the basis bits has this lower bound, after choosing a compatible
+summary. Both roles are available for either possible generic $q$.
+No prescribed value of the channel flag $p_i$ is included in these
+lower bounds. If that bit is recorded together with the role and
+basis bits, the bound holds only after summing over its two values.
+
+The lower constants do not depend on $\delta$, on the key tests, or on
+the orientation-dependent choice of $C_i$. The onset of the
+asymptotics and the exceptional-set constants may depend on a fixed
+positive lower bound $\delta$. The exceptional set is allowed to
+depend on the prescribed tests: there is no simultaneous assertion
+over all measurable key sets at one orientation.
+\end{lemma}
+
+\begin{proof}
+For each fixed point parameter, the raw key law is the single-key
+reference up to an exponentially small injectivity error, by
+\Cref{lem:frame-images}. The first conclusion therefore follows from
+the one-endpoint part of \Cref{lem:typicality}, also after conditioning
+on a feasible summary.
+
+Fix a nonzero profile $x\in\cX$ of total component rank at most $K$,
+and consider the sign
+\[
+ s_x(z)=(-1)^{T(x,\operatorname{atom}(z))}.
+\]
+By the first property of \Cref{lem:generic-mixers}, its dependence on
+$Z$ factors through a bounded-dimensional space of linear forms,
+independent of the other input values, and its $Z$-polar rank is at
+least $2(J-s_0)$. Put
+\[
+ \gamma=2^{-K^2-3},\qquad b_*=2^{-J+2s_0}<\gamma.
+\]
+The strict inequality follows from the chosen lower bound on $J$.
+
+Apply a uniform $G\in\GL_n(\F)$ simultaneously to the $Z$ input
+coordinates in all primal frames at the queried endpoint. This
+preserves the raw primal law because $E$ has no $Z$ rows or columns.
+We perform this averaging on $\rawlaw^{\mathrm{pr}}$, not on the
+thinned conditional channel law. It also
+preserves the empirical mass of a global key cell: change variables
+in the uniform $Z$ input and leave the other coordinates unchanged.
+After this change of variables the cell event is fixed and the sign's
+bounded row-bit space is moved by the inverse transformation.
+
+Fix a starting orientation for which the cell, conditional on the
+summary, has input mass at least $\delta/2$. We claim that at most
+$2^{-2An}$ of the transformations can make its conditional absolute
+sign bias exceed $\gamma$, for sufficiently large $n$. Suppose the
+contrary and retain one bias sign, losing at most a factor two. For
+any fixed integer $u$, choose $u$ such transformations successively so
+that each new row-bit space meets the span of the previous ones in
+dimension at most $s_0$. This is possible for large $n$. Indeed, for
+a uniformly moved space of bounded dimension and a fixed space of
+bounded dimension, the probability of an intersection of dimension
+greater than $s_0$ is $O_u(2^{-s_0n})$: choose $s_0$ independent
+vectors in the former and bound the probability that their uniformly
+independent images all lie in the latter. The number of such choices
+and target vectors is bounded independently of $n$. Since
+$s_0=4\lceil A\rceil>2A$, this excluded fraction is smaller than the
+retained fraction at each of the fixed number of choices.
+
+Let $E_0$ be the fixed cell event in the conditional input probability
+space and let $\delta'=\PP(E_0)\ge\delta/2$. Denote the chosen signs,
+with their common bias sign adjusted to be positive, by $s_1,\ldots,s_u$.
+Let $\mathcal F_{j-1}$ contain all non-$Z$ inputs and the row bits of
+the preceding chosen spaces, and put
+\[
+ m_j=\E(s_j\mid\mathcal F_{j-1}),\qquad D_j=s_j-m_j.
+\]
+Conditioning the current row bits on the past imposes at most $s_0$
+linear conditions on their distribution. The polar rank can therefore
+lose at most $2s_0$ further dimensions. By
+\Cref{lem:histogram-quadratic-bias},
+\[
+ \abs{m_j}\le b_*
+\]
+pointwise, including after the non-$Z$ inputs are fixed. The $D_j$ are
+martingale differences with second moments at most one, so
+$\norm{u^{-1}\sum_jD_j}_2\le u^{-1/2}$. On the other hand every
+selected sign has conditional mean greater than $\gamma$ on $E_0$.
+Consequently
+\begin{equation}\label{eq:histogram-tiny-cell-martingale}
+ (\gamma-b_*)\delta'
+ \le \abs{\E\1_{E_0}\,u^{-1}\sum_jD_j}
+ \le\sqrt{\delta'/u}.
+\end{equation}
+Here the intrinsic bias costs $b_*\delta'$, because its conditional
+mean is bounded pointwise; it does not cost an absolute $b_*$ before
+division by the cell mass. Taking
+$u>2/[\delta(\gamma-b_*)^2]$ contradicts
+\eqref{eq:histogram-tiny-cell-martingale}. This proves the transformation
+claim with $J$ fixed independently of the cell mass.
+
+Integrate that claim over the starting raw primal orientation, using frame-law
+invariance. The first part of the lemma makes the cell mass at least
+$\delta/2$ outside an exponentially small set. More precisely, the
+transformation estimate is used only on orientations with this mass;
+that property is itself invariant under the transformations. This
+cell-mass exceptional set is common to all profiles and is discarded
+only once. For a fixed $x$, the event that the cell mass is at least
+$\delta/2$ and the conditional sign bias is bad has raw probability
+at most $2^{-2An}$. There are at most $2^{An}$ nonzero
+profiles of the required rank for all sufficiently large $n$, by
+\Cref{lem:generic-mixers}. Union over them, and over the bounded
+numerical position and summary choices. The resulting event is still
+exponentially small, also after the projected marginal inflation in
+\eqref{eq:histogram-mixed-caps}. This union is taken before making
+any channel-dependent choice of effective space or basis: its
+exceptional event is a function of the primal frames alone.
+
+On its complement every nonzero linear combination of a basis of
+$C_i$ has conditional sign bias at most $\gamma$, regardless of how
+the space and basis were selected from the unit. Fourier inversion,
+for $d_i=\dim C_i$, gives for every assignment
+\[
+ \PP(\text{assignment}\mid E_0,\text{summary})
+ \ge2^{-d_i}\bigl(1-(2^{d_i}-1)\gamma\bigr)
+ \ge(7/8)2^{-d_i}.
+\]
+This also covers $d_i=0$.
+
+It remains to check the uniform generic constant. For one retained
+tester block, the sum of $r_0\ge1$ independent bit-pair products has
+probabilities $(1\pm2^{-r_0})/2$, both at least $1/4$. Distinct tester
+blocks use disjoint input bits. There are at most $g^2+1$ such blocks,
+so every feasible summary has probability at least
+$4^{-(g^2+1)}$. Conditional on its compatible value of $q$, its
+probability is no smaller. Its empirical key-cell mass, conditional
+on that summary, is at least $(3/4)\Omega_{l,q,n}(A_n)$ outside an
+exponentially small set by the first part. Combining this with the
+basis lower fraction gives \eqref{eq:histogram-base-cell-lower}, since
+$(3/4)(7/8)>1/2$. In the generic flavor there is at least one retained
+$O$ tester and the independent shared tester. Prescribing their
+parities realizes either $a$ role at either fixed $q$, and a feasible
+summary for each has the same lower probability bound.
+\end{proof}
+
+\begin{remark}\label{rem:unary-positivity-key-only}
+The key-only hypothesis cannot be omitted. For a fixed nonzero
+profile $x_0$, the input event
+$\{z:T(x_0,\operatorname{atom}(z))=0\}$ has zero conditional mass for
+the opposite basis bit. This is an allowed input-and-key test for
+concentration, but is not a common global key cell for positivity.
+The constants in \Cref{lem:unary-positivity} apply on every fixed
+positive limiting key cell, however small. Countably many resulting
+cell inequalities can be passed through a diagonal subsequence;
+they do not assert simultaneous finite-$n$ control of all possible
+cells. An orientation-only subfilter preserves an almost-sure
+limiting positivity statement, whereas an arbitrary additional input
+filter need not.
+\end{remark}
diff --git a/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/10-product-tests.tex b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/10-product-tests.tex
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+\section{Weak product testing}\label{sec:product-tests}
+
+The preceding histogram lemma gives compact families within each
+leaf. We now prove a different statement: a fixed global test on a
+tuple of keys can be replaced, for the integrations that occur here,
+by a bounded sum of products of global single-key functions. Both its
+reference and its actual-query conclusions are needed.
+
+Fix a query template with $r$ positions and with labels distinct at
+each used endpoint. Its input laws and reference measures are those
+of \Cref{sec:histograms}. In particular
+$\overline\Omega_n=\bigotimes_{j=1}^r\Omega_{l_j,q_j,n}$ and
+$\nu_n$ is its off-position-Gram-conditioned reference on $\cK_n$.
+The additional prescribed binary tests are the within-endpoint
+ordered $L,R$ evaluations from \Cref{sec:realization}. Let
+$\mathcal A_n(O,z)$ be the successful binary indicator from
+\Cref{lem:binary-mixing}, including the off-position Gram zeros.
+All common functions below inspect primal keys only. Channel maps
+may enter the unary input weights and the orientation filters.
+
+\begin{theorem}[Weak product comparison]\label{thm:weak-product}
+Let $W_n$ be any sequence of real functions on $\cK_n$ with
+$\norm{W_n}_\infty\le B$, where $B$ is fixed. Extend it to the product
+of the single-key spaces with the same bound. For every $\eps>0$
+there are constants $L,C<\infty$, independent of $n$, and functions
+\begin{equation}\label{eq:weak-product-approximant}
+ S_n(k_1,\ldots,k_r)
+   =\sum_{a=1}^{L}c_{a,n}\prod_{j=1}^{r}f_{a,j,n}(k_j),
+ \qquad
+ \abs{c_{a,n}}\le C,\quad \abs{f_{a,j,n}}\le1,
+\end{equation}
+with the following properties.
+\begin{enumerate}[label=(\roman*)]
+\item For arbitrary global unary functions $g_{j,n}$ of bound one,
+\begin{equation}\label{eq:weak-product-reference}
+ \abs{\int(W_n-S_n)\prod_jg_{j,n}(k_j)\,d\nu_n}\le\eps
+\end{equation}
+for all sufficiently large $n$.
+\item Under any mixed unit law satisfying
+\eqref{eq:histogram-mixed-caps}, outside an exponentially small
+exceptional set of orientations,
+\begin{equation}\label{eq:weak-product-actual}
+ \sup_{\abs{w_j}\le1}
+ \abs{\E_z\mathcal A_n(O,z)
+       (W_n-S_n)(K(O,z))\prod_jw_j(O,z_j)}\le\eps.
+\end{equation}
+The unary input weights may depend on the entire fixed orientation
+and on any additional marks fixed with it. The exceptional set is
+independent of these weights. Thus one may prescribe channel flags
+such as $p_i$ in a unary weight without making them common key tests
+or asserting positivity for their individual values.
+\end{enumerate}
+The functions $f_{a,j,n}$ are global key functions; they can be chosen
+to be indicators of sets, or finite linear combinations of such
+indicators with uniformly bounded complexity. They do not depend on
+the sampled orientation. The constants may depend on $B,\eps$ and
+the fixed template and construction parameters, but not on $n$.
+
+An additional orientation-only filter of bound one may multiply
+\eqref{eq:weak-product-actual} without changing it. In particular,
+after averaging normalized leaves, the same error control applies to
+unnormalized admissibility-filtered query measures, even when the
+filter is chosen using the opposite leaf. No assertion is made for
+an arbitrary additional joint input filter.
+For distinct marked laws, use their fixed-weight mixture, with caps
+checked after forgetting the marks. A retained ordered type pair or
+different-colour condition is an orientation/leaf-pair filter, so the
+same conclusion applies before normalization. All leaf averages use
+their actual weights; no marginal hypothesis on a separately
+normalized colour class is needed.
+\end{theorem}
+
+The proof uses a finite family of twists. If $I,J$ partition a set of
+positions into two nonempty groups, let $\mathcal T_{I,J}$ consist of
+all products of their available inter-position global Gram signs:
+the dot products of a plus slot at one position with a minus slot at
+another position in the same component. These signs are defined on
+the product reference whether or not that Gram is prescribed by a
+raw same-endpoint frame law. Every such twist equals one on
+$\cK_n$. The number of twists is a constant depending only on the
+query shape.
+
+For a real function $D$ on the keys of $I\cup J$, define its twisted
+cut norm by
+\begin{equation}\label{eq:twisted-cut-norm}
+ \norm{D}_{\square,\mathcal T}
+ =\max_{\tau\in\mathcal T_{I,J}}
+   \sup_{\abs{f},\abs{g}\le1}
+   \abs{\int D(k_I,k_J)\tau(k_I,k_J)f(k_I)g(k_J)
+                  \,d\overline\Omega_{I\cup J,n}}.
+\end{equation}
+Here $f,g$ may be arbitrary functions on their whole respective
+groups. Only after the recursion below will they become unary.
+
+We use the finite-partition energy-increment method underlying weak
+regularity; see \citet{frieze1999}. The additional Gram conditioning
+needed here is retained explicitly in the following self-contained proof.
+
+\begin{lemma}[Finite twisted regularity]
+\label{lem:product-twisted-regularity}
+Let $\abs{W}\le B$ on the product key space of $I\cup J$, and let
+$\delta>0$. There is a decomposition
+\begin{equation}\label{eq:twisted-regularity-decomposition}
+ W=\sum_{a=1}^{L'}c_a f_a(k_I)g_a(k_J)\tau_a(k_I,k_J)+D,
+\end{equation}
+where $\abs{f_a},\abs{g_a}\le1$, $\tau_a\in\mathcal T_{I,J}$,
+$\abs{c_a}\le B$, $\norm{D}_\infty\le2B$, and
+$\norm{D}_{\square,\mathcal T}\le\delta$. The number $L'$ is bounded
+in terms of $B,\delta$ and the twist count only. The functions $f_a,g_a$
+may be chosen as indicators of cells in finite group partitions.
+\end{lemma}
+
+\begin{proof}
+Let $\Gamma$ be the finite list of crossing Gram bits generating the
+twists. Begin with the trivial partitions of the two groups. For any
+current pair of finite partitions, condition $W$ on their cells and
+on $\Gamma$; call the resulting conditional expectation $S$. Then
+$\norm{S}_\infty\le B$ and $\norm{W-S}_\infty\le2B$.
+
+If $D=W-S$ has twisted cut norm greater than $\delta$, choose a
+twist and witnesses $f,g$ giving correlation greater than $\delta$.
+Quantize $f,g$ uniformly within their bounds finely enough, with a
+mesh depending only on $B,\delta$, that their quantized product times
+the twist still correlates with $D$ by more than $\delta/2$. Refine
+the group partitions by the values of these quantized functions.
+The witness product is measurable for the new conditioning algebra
+and has $L^2$ norm at most one. Thus the new conditional expectation
+of $D$ has $L^2$ norm at least $\delta/2$. Since $D$ is orthogonal
+to the old conditioning algebra, the squared $L^2$ norm of the new
+conditional expectation of $W$ increases by at least $\delta^2/4$.
+It is bounded above by $B^2$. At most $4B^2/\delta^2+1$
+refinements are possible. Each increases the partition sizes by a
+factor bounded in terms of $B,\delta$, so the final partitions have
+bounded size.
+
+On each product of group cells, the final conditional expectation is
+a bounded function of $\Gamma$. Assign value zero to any impossible
+combination of a cell pair and Gram bits. Its finite Fourier
+expansion on the Gram bits has coefficients of absolute value at
+most $B$. Multiplying by the two cell indicators gives
+\eqref{eq:twisted-regularity-decomposition}. The decomposition is
+exact everywhere on the product key space, regardless of which Gram
+bit values occur in a particular cell pair.
+\end{proof}
+
+We next prove that a small twisted-cut remainder is negligible on
+actual queries. This is the step that is not supplied by unary
+histograms alone.
+
+\begin{lemma}[A terminal remainder on actual inputs]
+\label{lem:product-terminal-remainder}
+Let $I,J$ be a nontrivial partition of some of the template positions.
+Let $D_n$ be a common function of their global keys with
+$\norm{D_n}_\infty\le B_1$ and
+$\norm{D_n}_{\square,\mathcal T}\le\delta$. Let $\psi_n(I,J)$
+be any product of the specified nominal $L,R$ signs and global Gram
+signs crossing from $I$ to $J$, evaluated on their actual inputs and
+keys. For each fixed $\varepsilon_{\mathrm{emp}}>0$, except on exponentially small mixed
+mass,
+\begin{equation}\label{eq:product-remainder-actual}
+ \sup_{\abs{f},\abs{g}\le1}
+ \abs{\E_{z_I,z_J}D_n(K_I,K_J)\psi_n(I,J)f(O,z_I)g(O,z_J)}
+ \le\bigl(C_3\delta+\varepsilon_{\mathrm{emp}}+o(1)\bigr)^{1/4}.
+\end{equation}
+The constant $C_3$ and the vanishing error depend only on $B_1$ and
+the fixed template. The weights may depend arbitrarily on the fixed
+orientation. The exceptional set does not depend on those weights.
+\end{lemma}
+
+\begin{proof}
+For one fixed orientation, put
+$U(I,J)=D_n(K_I,K_J)\psi_n(I,J)$ and use independent copies of the
+parameter groups $I,J$. Cauchy--Schwarz in $I$ gives
+\[
+ \abs{\E U(I,J)f(I)g(J)}^2
+ \le\E_I\bigl(\E_JU(I,J)g(J)\bigr)^2.
+\]
+Expand the square, and apply Cauchy--Schwarz in the two $J$ variables.
+The factors $g(J_0)g(J_1)$ have bound one. We obtain
+\begin{equation}\label{eq:product-four-corner}
+ \abs{\E U(I,J)f(I)g(J)}^4
+ \le Q_n(O):=
+ \E_{I_0,I_1,J_0,J_1}
+       \prod_{a,b\in\{0,1\}}U(I_a,J_b).
+\end{equation}
+In particular $Q_n(O)\ge0$: it is the expectation over $J_0,J_1$ of
+the square of $\E_IU(I,J_0)U(I,J_1)$. All separate weights, including
+their orientation dependence, have disappeared.
+
+The right-hand side is a bounded common input-and-key test, involving
+two fresh copies of each original position. It depends only on the
+primal maps: the channel-dependent weights and all marks have
+disappeared in \eqref{eq:product-four-corner}, and the nominal $L,R$
+signs involve only the displayed fresh parameters. Repeated labels across
+these copies are permitted by \Cref{lem:typicality}. It follows that,
+outside an exponentially small mixed exceptional set,
+\begin{equation}\label{eq:product-four-corner-typical}
+ Q_n(O)\le\E_{\rawlaw^2}Q_n+\varepsilon_{\mathrm{emp}}.
+\end{equation}
+We bound the raw mean by $C_3\delta+o(1)$.
+
+Fix the nominal inputs of the four groups. Their columns are jointly
+independent within each endpoint, component, and mode except with
+$2^{-\Omega(n)}$ probability, since all copies have fresh $Z$ inputs.
+The dependent-input contribution is $o(1)$ by the fixed sup bound.
+On the independence event, \Cref{lem:frame-images} gives the uniform
+image law with prescribed same-endpoint Grams and injectivity.
+Ignore injectivity at an exponentially small error and express this
+law relative to the product of all the individual single-key
+references. There are only boundedly many additional Gram bits, and
+their normalization is bounded uniformly in the nominal inputs by
+\Cref{lem:histogram-gram-normalization}. Expand their indicators into
+characters. The nominal $L,R$ phases are now constants. The global
+Gram phases already in the four copies of $\psi_n$ can be combined
+with the expansion characters.
+
+For completeness, distinguish a slot by its group $I$ or $J$, its
+copy, its position, component, and sign. Remove the diagonals already
+in the reference and collapse repeated descriptions of the same
+unordered slot dot. Every remaining Gram coefficient belongs to
+exactly one of the following classes:
+\begin{enumerate}[label=(\alph*)]
+\item both slots lie in one of $I_0,I_1,J_0,J_1$;
+\item the slots join $I_0$ to $I_1$, or $J_0$ to $J_1$;
+\item the slots lie in one of the four $I_a$--$J_b$ quadrants.
+\end{enumerate}
+Class (a) contributes bounded separate group phases. Suppose a
+nonzero class-(b) pattern between $I_0,I_1$ remains. Fix both $J$
+groups. The four $D_n$ factors split into two bounded functions of
+$I_0$ and $I_1$. The class-(c) phases become separate functions of
+those two groups, as do all class-(a) phases. Their only remaining
+interaction is the nonzero coefficient rectangle between $I_0$ and
+$I_1$, whose bit rank is at least $N$. The individual diagonal
+densities may also be absorbed into the respective bounded weights.
+\Cref{lem:walsh} bounds this term by $O(2^{-N/2})$. The same argument
+applies to a nonzero $J_0$--$J_1$ pattern.
+
+If neither class-(b) pattern remains, attach each class-(c) pattern
+as an available twist to its corresponding $D_n(K_{I_a},K_{J_b})$.
+Fix $I_1,J_1$. The remaining integral in $I_0,J_0$ has one factor
+$D_n$ times an available twist, tested against bounded separate
+functions. The two other unfixed $D_n$ factors have bound $B_1$ and
+the fixed factor has bound $B_1$. The twisted cut estimate therefore
+bounds this term by $B_1^3\delta$, with the harmless fixed
+normalization factors included in the constant.
+
+These cases exhaust the character expansion. The groups $I$ and $J$
+may each contain positions from both endpoints of the unit: this only removes some
+same-endpoint Gram bits from the expansion and does not create a new
+class. A nonzero coefficient rectangle has rank at least one in
+distinguished global slots, hence bit rank at least $N$.
+The two dots $P_a\cdot Q_b$ and $P_b\cdot Q_a$, for example, use
+different slots and are not duplicates. The prescribed nominal Gram
+values affect character coefficients only by signs.
+
+Summing the bounded number of terms and averaging the nominal inputs
+gives $\E_{\rawlaw^2}Q_n\le C_3\delta+o(1)$. Combine this with
+\eqref{eq:product-four-corner} and
+\eqref{eq:product-four-corner-typical}. The same $Q_n(O)$ bounds every
+choice of separate weights, so the exceptional set is independent of
+them.
+\end{proof}
+
+\begin{proof}[Proof of \Cref{thm:weak-product}]
+If $r=1$, quantize the values of $W_n$ in $[-B,B]$ with mesh at most
+$\eps/2$, choosing representatives in the same interval. The resulting
+$S_n$ satisfies $\norm{W_n-S_n}_\infty\le\eps/2$ and is a linear
+combination of at most $1+\lceil4B/\eps\rceil$ global level-set
+indicators, with coefficients bounded by $B$. Both required errors
+are at most $\eps/2$, since the reference measure is a probability
+measure and the actual-query indicator and weights have bound one.
+This includes $B=0$, when $S_n=0$. For $r\ge2$,
+apply \Cref{lem:product-twisted-regularity} to a nontrivial split of
+the full set of positions. Recursively apply it to group factors in
+the structured terms until every factor is unary. Whenever a
+remainder is produced, leave that whole term terminal; do not expand
+its other factors. We obtain the exact decomposition, on the
+unconditioned product reference,
+\begin{equation}\label{eq:product-recursive-decomposition}
+ W_n=S_n^{\mathrm{tw}}+\sum_{v\in\mathcal V_n}R_{v,n}.
+\end{equation}
+Each structured term is a product of unary global functions times
+available Gram twists. Each terminal term has the form
+\begin{equation}\label{eq:product-terminal-form}
+ R_{v,n}(k)=c_{v,n}D_{v,n}(k_I,k_J)
+       \prod_{H}f_{v,H,n}(k_H)\,\tau_{v,n}(k),
+\end{equation}
+where $I,J$ partition one group, the groups $H$ are disjoint outside
+$I\cup J$, all outside functions are bounded, $\tau_{v,n}$ is a
+product of available Gram signs, and $D_{v,n}$ has a specified small
+twisted cut norm on $I|J$. The sup bounds, numbers of terms, and
+partition complexities are all bounded independently of $n$.
+
+Here is an explicit way to organize the tolerance choices. Along
+any structured branch a split replaces one group by two, so at most
+$r-1$ splits are needed before all groups are unary. At any fixed
+depth the number of pending terms and all their coefficient and
+factor bounds have already been bounded using previous choices.
+Allocate at most $\eps/(2r)$ of each desired final error to that
+depth, dividing this budget among its pending terms and their known
+outside bounds. Choose the next cut tolerances small enough for the
+reference estimates below and for the fourth roots in
+\Cref{lem:product-terminal-remainder}. These choices determine a
+finite bound on the next level's number of structured terms.
+Continue for at most $r-1$ levels. The empirical tolerance $\varepsilon_{\mathrm{emp}}$ for
+each terminal test is chosen at the same stage, sufficiently small
+for its allotted fourth-root error. All choices are constants; none
+requires a change in the construction parameters or in $M_0$.
+
+On $\cK_n$, every Gram twist in $S_n^{\mathrm{tw}}$ equals one.
+Removing them gives the function $S_n$ in
+\eqref{eq:weak-product-approximant}. Its unary factors are indicators
+of the finite partitions used in the recursion; coefficients and
+term counts have the asserted bounds.
+
+To prove the reference assertion, multiply a terminal term
+\eqref{eq:product-terminal-form} by a product of arbitrary bounded
+unary tests and expand the off-position Gram event defining $\nu_n$.
+Fix all positions outside its remainder. Their group factors are
+bounded constants. Gram phases internal to $I$ or $J$ and phases
+involving a fixed outside position are separate bounded group
+weights. The only other phases are available crossing twists.
+Thus the integral is bounded by the twisted cut norm of $D_{v,n}$
+times a constant depending only on its known bounds and the fixed
+Gram normalization. The tolerance allocation makes the sum of these
+errors at most $\eps$, proving \eqref{eq:weak-product-reference}.
+This estimate is uniform over all unary reference test functions.
+
+For the actual-query assertion, expand $\mathcal A_n$ into its
+Gram and nominal $L,R$ characters. The number of terms is bounded.
+Consider a terminal term and fix the outside position parameters.
+All phases internal to $I$ or $J$, or involving an outside position,
+can be absorbed into separate group weights. The remaining crossing
+phases are a product $\psi_n(I,J)$ of precisely the form in
+\Cref{lem:product-terminal-remainder}. It is independent of the
+outside parameter values: a binary bit involving an outside
+position has already been put into a separate weight. All original
+unary weights, even when they depend on the orientation, can also be
+included in these two weights.
+
+Apply \Cref{lem:product-terminal-remainder}, scaling the known
+outside bounds as necessary. Its common four-copy test is
+independent of the fixed outside parameters, so one exceptional
+orientation set controls every such fixation. Integrate the outside
+parameters, sum the bounded binary-character expansion, and sum the
+terminal terms. The chosen cut and empirical tolerances make the
+total at most $\eps$ for sufficiently large $n$. The union of the
+finitely many exceptional sets is exponentially small. Since each
+four-copy estimate is uniform over its separate weights, the final
+exceptional set is independent of all unary input weights and of
+orientation-only filters. This proves
+\eqref{eq:weak-product-actual}.
+\end{proof}
+
+\begin{corollary}[Factorized integration of the approximant]
+\label{cor:weak-product-factorized-integration}
+For the structured sum in \eqref{eq:weak-product-approximant} and
+arbitrary bounded unary input weights, outside exponentially small
+mixed mass,
+\begin{align}\label{eq:weak-product-factorized-integration}
+ &\E_z\mathcal A_n(O,z)S_n(K(O,z))\prod_jw_j(O,z_j)\notag\\
+ &\qquad=
+ \kappa\sum_{a=1}^{L}c_{a,n}
+       \prod_j\E_{z_j}
+        \bigl[f_{a,j,n}(K_j(O,z_j))w_j(O,z_j)\bigr]+o(1).
+\end{align}
+The error is uniform over the unary weights. An orientation-only
+filter may be included before averaging this identity over leaves.
+In particular the same statement applies to marked mixtures and
+retained type/colour pairs as in \Cref{thm:weak-product}.
+\end{corollary}
+
+\begin{proof}
+Apply \Cref{lem:binary-mixing} to each structured term, with
+$f_{a,j,n}(K_j)w_j$ as its unary weights. The number of terms and
+their bounds are fixed at the chosen approximation accuracy, so the
+sum of the exponentially small errors tends to zero. The exceptional
+set in that lemma is independent of these weights.
+\end{proof}
+
+\begin{remark}[What the two comparisons do not assert]
+\label{rem:weak-product-filter-class}
+The reference error in \eqref{eq:weak-product-reference} is weak
+testing error, not $L^1$ error. The actual comparison allows arbitrary
+orientation-dependent \emph{unary} parameter weights, and the
+specified binary Gram/$L,R$ conditions. It does not allow an
+unprescribed joint input filter to be inserted without proof.
+For instance, two complementary bounded tuple densities determined
+by a high-rank global bilinear bit can have nearly uniform unary
+marginals and zero overlap. A two-element histogram net would not
+exclude that example. The additional actual-query conclusion of
+\Cref{thm:weak-product}, proved by the four-copy expansion, is what
+rules it out for the successful-query format used here. All factors
+added to the common signature lists by this theorem are global key
+functions. Nominal input status bits remain weights, not common key
+cells in \Cref{lem:unary-positivity}.
+\end{remark}
diff --git a/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/11-common-limit.tex b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/11-common-limit.tex
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+\section{A common limit for successful queries}
+\label{sec:common-limit}
+
+We work with two independently sampled unit laws, which may be different.
+Each has a family of first-peeling leaves satisfying the per-leaf image
+bounds of \Cref{lem:peeling}; their mixtures satisfy the absolute joint
+and projected primal marginal bounds of \Cref{eq:histogram-mixed-caps}.
+The restrictions supplied by \Cref{prop:status-preparation} are one
+application; an independent pair of points will be another.
+All arguments in this section concern a sequence with $n\to\infty$, with
+the construction parameters, including $M_0$, fixed. We freely pass to
+subsequences, using the compactness and measurable-density constructions
+in \Cref{sec:measure-constructions}. The leaves retain their actual mass weights. Conditional on
+a leaf, its orientation law is normalized; a restriction imposed on a
+query is never normalized away. More explicitly, if $w_{A,n}(L)$ and
+$w_{B,n}(L')$ are the respective leaf probabilities, every average below
+uses $w_{A,n}(L)w_{B,n}(L')$. Neither a uniform law on leaves nor a new
+normalization of an individual colour is permitted. The marginal
+hypotheses apply to these mixtures, not to their normalized leaves.
+
+We also fix a permitted-pair indicator
+$J_n(L_A,L_B)\in\{0,1\}$. It can test a finite colour mark or the
+ordered types of the two laws. Randomized colours are sampled as part
+of the original unit and then included in the leaf record. In
+particular, different-colour acceptance is a leaf-pair test. All
+statements about vanishing overlap and product zero below are restricted
+to $J_n=1$; no assertion about the complementary pairs is needed.
+
+The limit has two tasks. First, own-leaf nets and global projections
+pass truncated overlaps to a common space; uniform integrability then
+removes the truncation. Second, the actual-query and reference
+conclusions of \Cref{thm:weak-product} identify each limiting density
+for the successful-query format \eqref{eq:limit-success-format}
+from its unary status measures.
+\Cref{lem:limit-zero-product,lem:limit-density} combine these tasks
+in \Cref{prop:limit-overlap}.
+A limit of unary marginals alone would not justify this conversion.
+
+\subsection{Options and the permitted filters}
+
+\begin{definition}[Numerical option]
+\label{def:limit-option}
+A numerical option specifies a recipe shape with between one and seven
+positions on each cross edge, together with the following data:
+\begin{enumerate}
+\item the endpoint, tag, label, flavor, and diagonal bit at every position;
+\item the full numerical unary statuses required at those positions,
+including the tester summary, the role $a(\mathrm{atom})$, the basis
+coordinates of $T(\mathrm{atom},\cdot)|_{C_i}$, and the $p_i$ bit;
+\item the required within-endpoint binary evaluations, and the
+off-position Gram conditions defining the reference key space;
+\item the small tables and the finite coordinate formats in which they
+are written.
+\end{enumerate}
+The two sides have matching tags and diagonal bits position by position.
+All labels used at one endpoint are distinct. Coordinate formats of
+smaller dimension are padded by zeros.
+\end{definition}
+
+Once the construction parameters have been chosen, the collection of
+options is finite. In particular, a large selector length does not make
+this collection grow with $n$. A basis of each $C_i$ was chosen on its
+own leaf. Its tensor coordinates in the projected pin bases, its $a$
+values, the pin/projection relations, and the mandatory support
+exclusions all
+have bounded finite descriptions. No arbitrary ambient tensor is included
+in this metadata.
+
+Fix an option $\omega$ and a pair of leaves $(L_A,L_B)$. Call the option
+\emph{usable} on this pair if its finite metadata satisfy the label and
+injection requirements and its numerical statuses imply
+\eqref{eq:recipe-scalars}, with its binary prescriptions supplying
+\eqref{eq:recipe-gradients}. Denote this indicator by
+$U_\omega(L_A,L_B)$. Usability is determined by bounded discrete metadata.
+The phase conditions in \Cref{prop:status-preparation} will help us find
+usable options; they need not be added to the definition of usability.
+Likewise, prediction exceptions may remain orientation metadata. They
+restrict our later choice of an option, rather than its definition of
+usability on an entire leaf pair.
+
+For a specified table, admissibility against pinned images is checked
+separately on the two orientations. Write these flags as
+$F_{A,\omega}$ and $F_{B,\omega}$. They may depend on both leaves, but
+each is a function of only one orientation once the leaves are fixed.
+The successful query on one side therefore consists exactly of
+\begin{equation}
+\label{eq:limit-success-format}
+\begin{aligned}
+&\text{an orientation-only flag}\ \times\
+  \prod_{\ell}\text{a unary status indicator at $\ell$}\\
+&\hspace{3em}\times\ \text{the specified binary and Gram indicators}.
+\end{aligned}
+\end{equation}
+The parameter input at each position is independent and uniform in its
+specified flavor, conditional on the requested feasible diagonal bit.
+Only the diagonal conditioning is included in this input normalization.
+The other successful-query factors remain unnormalized.
+
+\begin{remark}[Two different classes of flags]
+\label{rem:limit-filter-boundary}
+The density nets in \Cref{lem:histogram-control} cover arbitrary
+orientation filters and a bounded family of additional numerical flags.
+The factorization argument below uses the narrower format
+\eqref{eq:limit-success-format}. It does not introduce an arbitrary
+additional joint predicate on the point parameters. In particular,
+membership in a nominal basis-bit cell is a status requirement, not a
+global key test on which all basis assignments are asserted to remain
+possible.
+\end{remark}
+
+For the key shape of $\omega$, let $(\cK_n,\nu_n)$ be the uniform
+reference space from \Cref{sec:collision}. Define $d_{A,n}^\omega$ and
+$d_{B,n}^\omega$ to be the densities of the two successful queries with
+respect to $\nu_n$, after averaging orientations in their normalized
+leaves. Their integrals are at most one. Set
+\begin{equation}
+\label{eq:limit-option-overlap}
+I_n(\omega)=
+\E_{L_A,L_B} J_n(L_A,L_B)U_\omega(L_A,L_B)
+\int_{\cK_n}d_{A,n}^\omega d_{B,n}^\omega\,d\nu_n.
+\end{equation}
+Here the leaves are independent before the permitted-pair test, usability,
+or flag passage is tested. All restrictions in the integrand have already been charged to
+the subdensities. By \Cref{prop:collision}, a positive lower bound for
+$I_n(\omega)$ along a subsequence suffices for the desired collision.
+We shall analyze the contrary possibility
+\begin{equation}
+\label{eq:limit-vanishing-overlap}
+I_n(\omega)\longrightarrow0\qquad\text{for every numerical option }\omega.
+\end{equation}
+
+\subsection{A global projection for independent leaf families}
+
+The nets supplied by \Cref{lem:histogram-control} are chosen separately
+for each leaf. The next elementary Hilbert-space argument produces the
+common partitions that will be needed in the limit.
+
+\begin{lemma}[Global projections with adaptive filter choices]
+\label{lem:limit-global-projection}
+Let $(K_n,\nu_n)$ be finite probability spaces. On each first-side leaf
+$L$ and each second-side leaf $L'$ let there be a family of subdensities.
+Assume that, for each fixed $R<\infty$ and $\varepsilon>0$, their
+truncations at height $R$ have $L^1(\nu_n)$ nets of uniformly bounded
+size, with the nets chosen using their own leaf only. The bounds may
+depend on $R,\varepsilon$ but not on $n$ or the leaf.
+
+For every $R<\infty$ and $\eta>0$, there are deterministic partitions
+$\mathcal P_n$ of uniformly bounded size such that the following holds.
+These partitions may depend on the leaf laws, but are chosen before
+the leaves are sampled.
+For independent leaves, choose a member $f$ and a member $g$ of the two
+truncated families, allowing either choice to depend on both leaves.
+If $P_n$ is conditional expectation onto $\mathcal P_n$, then
+\begin{equation}
+\label{eq:limit-projection-error}
+\E\bigl|\pair{f}{g}-\pair{P_nf}{P_ng}\bigr|\le\eta.
+\end{equation}
+The same conclusion holds for every deterministic refinement of
+$\mathcal P_n$ chosen independently of the sampled leaves.
+Multiplication of the discrepancy by any indicator determined by the
+leaf pair does not increase the bound.
+\end{lemma}
+
+\begin{proof}
+We may truncate net representatives to $[0,R]$. Since
+$\norm{f-g}_2^2\le R\norm{f-g}_1$ for functions in this interval,
+we obtain $L^2$ nets with uniformly bounded size, say at most $m$.
+Pad shorter nets by zero functions. Write their members as $f_{L,a}$
+and $g_{L',b}$, and form the positive operators on $L^2(\nu_n)$
+\[
+C_A=\E_L\sum_{a=1}^m f_{L,a}\otimes f_{L,a},\qquad
+C_B=\E_{L'}\sum_{b=1}^m g_{L',b}\otimes g_{L',b}.
+\]
+Here $(f\otimes f)v=f\pair{f}{v}$. Both traces are at most
+$T=mR^2$.
+
+Fix $\delta>0$. There are at most $T/\delta$ eigenvalues of $C_A$
+larger than $\delta$. A unit eigenfunction $e$ for such an eigenvalue
+$\lambda$ satisfies, pointwise,
+\[
+|e(x)|
+\le\lambda^{-1}\E_L\sum_a |f_{L,a}(x)|\,
+  |\pair{f_{L,a}}{e}|
+\le T/\lambda\le T/\delta.
+\]
+Quantize these finitely many bounded eigenfunctions with a common finite
+partition. Its conditional expectation $P$ can be made to satisfy
+$\norm{(1-P)e}_2\le\varepsilon$ for every one of them. With $Q=1-P$,
+the spectral decomposition gives
+\[
+\norm{Q C_A Q}_{\mathrm{op}}
+\le\delta+T\varepsilon^2.
+\]
+The number of partition cells is bounded in terms of
+$m,R,\delta,\varepsilon$. If $P'$ is a refinement and $Q'=1-P'$,
+then $Q'=Q'Q=QQ'$, so the same operator bound holds for
+$Q'C_AQ'$.
+
+For net representatives, independence of the leaves gives
+\begin{align*}
+\E_{L,L'}\sum_{a,b}
+ \bigl|\pair{f_{L,a}}{g_{L',b}}-
+       \pair{P'f_{L,a}}{P'g_{L',b}}\bigr|^2
+&=\Tr(Q'C_AQ'C_B)\\
+&\le(\delta+T\varepsilon^2)T.
+\end{align*}
+This estimate permits pairwise-adaptive choices: the squared error for
+any chosen pair of representatives is pointwise at most the displayed
+sum. Cauchy--Schwarz bounds its expected absolute error. Approximating
+arbitrary family members by their net representatives costs at most a
+constant times $R$ times the $L^2$ net error, since orthogonal
+projections are contractions. Choose the net accuracy, then $\delta$
+and $\varepsilon$, sufficiently small in terms of $\eta$. This proves
+\eqref{eq:limit-projection-error}, also for refinements. The final
+indicator assertion follows from the absolute-value bound before the
+indicator is applied.
+\end{proof}
+
+For a fixed numerical option, all parameter-dependent predicates needed
+by one side can be put in a bounded own-leaf flag family: its own basis
+bits, tester outcomes, and prescribed binary tests are already fixed by
+the option. All remaining dependence on the opposite leaf is an
+orientation filter. Consequently \Cref{lem:histogram-control} supplies
+the net hypothesis of \Cref{lem:limit-global-projection} for all such
+filters simultaneously. No net need be chosen anew using the opposite
+leaf.
+
+\subsection{Common signatures and stored conditional measures}
+
+\begin{lemma}[Common signature model]
+\label{lem:limit-model}
+There is a subsequence and a common limiting experiment with the
+following properties.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item For each tag and diagonal bit $(l,q)$ there is a compact
+bit-sequence space $\Xi_{l,q}$ with reference law $m_{l,q}$.
+For each key shape there is a compact tuple-test space $\Upsilon$
+with reference law $\rho$. Here $\rho$ denotes this measure,
+distinct from the scalar thinning parameter in \Cref{lem:channel-thinning}.
+Its single-position projections
+$\xi_\ell$ have the independent laws $m_{l_\ell,q_\ell}$.
+\item A base orientation array $\theta$ contains its bounded discrete
+metadata and its single-atom status measures. For a unary type
+$t=(i,l,s,\varphi,q)$ and a full numerical status $\upsilon$, these measures
+have jointly measurable densities
+\begin{equation}
+\label{eq:limit-unary-densities}
+h^\theta_{t,\upsilon}:\Xi_{l,q}\longrightarrow[0,1],\qquad
+\sum_\upsilon h^\theta_{t,\upsilon}=1\quad m_{l,q}\text{-almost everywhere}.
+\end{equation}
+For the full generic flavor, the sum of these densities over statuses
+with any specified role and basis vector in the recorded actual dimension
+is bounded below by a positive construction constant, almost everywhere.
+Compatible finer tester
+requirements have the lower fractions supplied by
+\Cref{lem:unary-positivity}.
+\item There is first a random limiting leaf-pair record $Z$, which
+includes two conditional laws $\Lambda_A,\Lambda_B$ of the arrays and
+their finite flag vectors. Conditional on $Z$, the two orientation
+records are drawn independently from these laws. After flags are
+dropped and $Z$ is averaged out, $\theta_A,\theta_B$ are independent
+with their respective base laws; they are identically distributed when
+the two original laws coincide. The record also includes the limiting
+permitted-pair bit $J(Z)$.
+\item Every discrete assertion of table usability, injection, and any
+imposed phase compatibility is retained, jointly with $J$ and with its
+probability. Thus the preparation guarantees from
+\Cref{prop:status-preparation}, when imposed, remain valid in this
+experiment: the no-prediction branch has a passed injecting table on
+mass at least $.99$, and the predicted branch has the stated positive
+mass of passed compatible injecting tables. Each imposed prediction
+whose error tends to zero is exact in the limiting status measures
+outside its stored exceptions. No prediction is required for the
+construction of the model itself.
+\end{enumerate}
+All single-key tests used to define $\Xi_{l,q}$ are functions of global
+keys only.
+\end{lemma}
+
+\begin{proof}
+Choose countable lists of global binary tests on each
+$\Omega_{l,q,n}$. Include every imposed global predictor bit, if there is one. A key's values under this list form its unary signature. For each
+key shape also choose countably many binary tests on $\cK_n$, including
+the pullbacks of all unary tests at its positions. Their values form a
+tuple signature.
+
+The lists are constructed by countable closure. For every numerical
+option, positive integer truncation height, and successively improving
+fixed accuracy, include the cells of a partition provided by
+\Cref{lem:limit-global-projection}. For every tuple cylinder test and
+successively improving accuracy, include all unary indicator factors
+needed in \Cref{thm:weak-product}; close under the finite Boolean
+operations needed to record them. Repeat these operations. Every finite
+stage has bounded complexity independent of $n$, so it defines sequences
+of tests for all sufficiently large $n$; finitely many missing initial
+values may be assigned arbitrarily. Bounded coefficients in the
+approximations are also included among the scalar coordinates that will
+converge. Crucially, regularity in \Cref{thm:weak-product} is performed
+on key spaces. Its new unary factors remain global key functions.
+Nominal input phases handled in the proof of that theorem are not added
+to the signature lists.
+
+At an orientation, and for every unary type $t$, record the empirical
+mass of each finite signature cylinder together with each numerical
+status outcome. Include the finite metadata needed for the chosen
+$C_i$ bases and all numerical options; in predicted cases include the
+old marked profiles in these bases, the fixed role coefficients, and
+their exception sets. Call this array $\theta_n$. The cylinder masses
+lie in $[0,1]$ and obey consistent finite additivity identities. Their
+closed consistency space, together with the finite metadata, is compact
+and metrizable.
+
+For each pair of leaves retain its permitted-pair bit $J_n$ and, as part
+of the same random record, the \emph{conditional measures} of $(\theta_n,\text{flag vector})$ on
+each side. Also retain the full and truncated successful-query measures
+on all tuple cylinders, for each option. These are measures of mass at
+most one on compact bit-sequence spaces; their spaces are compact for
+weak convergence. Take a common subsequence of the laws on the resulting
+countable product, and simultaneously of the deterministic reference
+laws and the bounded approximation coefficients. Consistent limiting
+cylinder masses define countably additive measures on the bit-sequence
+spaces, for example by first defining the measure on finite prefixes
+and taking the unique measure with those finite-dimensional laws.
+
+Cylinder evaluation is continuous because bit cylinders are clopen.
+For each fixed unary cylinder, \Cref{lem:typicality} says that the sum
+of its empirical status masses differs in probability from its reference
+mass by a quantity tending to zero. This test uses only primal images.
+The projected primal marginal assumption on the actual mixture is the
+one needed here; no original full-frame marginal bound and no assertion
+about every individually normalized leaf is used. The status bits may
+depend arbitrarily on channels: summing all statuses removes those bits. Countably many limiting cylinder equalities imply that
+the total status measure is $m_{l,q}$ almost surely. Each status measure
+is therefore dominated by $m_{l,q}$, giving
+\eqref{eq:limit-unary-densities}. Versions jointly measurable in
+$(\theta,\xi)$ are obtained by taking the limits of their conditional
+density averages on increasing finite cylinder partitions, as in
+\Cref{lem:cylinder-densities}. An exceptional null set for the base law
+also has $\Lambda_A$- and $\Lambda_B$-mass zero for almost every stored
+record on the respective side, because averaging these conditional laws
+recovers that side's base law. The same is true after any flag restriction.
+
+For a generic role/basis assignment, apply
+\Cref{lem:unary-positivity} to each fixed cylinder of positive limiting
+reference weight. The lower fraction is independent of the cell;
+the onset of the estimate may depend on its positive weight. All
+profiles of the relevant rank are covered by that lemma before the
+profile is selected, so the chosen $C_i$ may vary with the entire
+orientation, including its channels. The $p_i$ flag is summed over in
+this positivity statement; no value of $p_i$ is prescribed as a positive
+base coordinate. Its lower inequalities pass to
+the limiting cylinder masses. Zero-reference cylinders already have
+zero total status mass. The cylinder inequalities extend by finite
+unions and monotone approximation to all measurable sets, giving the
+claimed almost-everywhere lower densities. This argument does not
+condition on arbitrary nominal input cells.
+
+By \Cref{lem:reference-product}, every finite collection of unary
+signature tests has its product reference law in the tuple limit.
+Therefore the unary projections under $\rho$ are independent with the
+stated marginal laws.
+
+There is no need to interchange disintegration and weak convergence.
+The conditional laws themselves were retained as random coordinates.
+For bounded continuous tests, taking their product is continuous in
+these two laws. Consequently sampling from
+$\Lambda_A\otimes\Lambda_B$ conditional on the limiting record $Z$
+reproduces the limiting joint orientation tests. Before flags are
+applied, the two base orientations were independent under their
+respective mixture laws at every $n$, so independence with those
+respective laws remains true after averaging $Z$ and dropping flags.
+If the laws agree, their limits agree as well. Finite metadata are discrete coordinates, hence all their
+incidence and compatibility identities are retained. This proves the
+table assertions and their mass statements. Finally the prediction
+errors tend uniformly to zero on the retained orientations. Since the
+predictor bits are included as unary tests, each such error is a finite
+status-cylinder mass and becomes zero in the limit.
+\end{proof}
+
+\subsection{Truncation and the zero-product implication}
+
+Fix the common model from \Cref{lem:limit-model}. For an option and a
+side, let $\eta_R(Z)$ and $\eta(Z)$ denote the limiting successful-query
+measures obtained respectively from $\min(d_n,R)\nu_n$ and $d_n\nu_n$.
+All are measures on the tuple space for that shape.
+
+\begin{lemma}[Passage of vanishing overlaps]
+\label{lem:limit-zero-product}
+Assume \eqref{eq:limit-vanishing-overlap}. For almost every $Z$, the
+measures $\eta_A(Z),\eta_B(Z)$ are absolutely continuous with respect
+to $\rho$. For each usable option their densities have product zero
+$\rho$-almost everywhere on the event $J(Z)=1$.
+\end{lemma}
+
+\begin{proof}
+For each integer $R$, cylinder inequalities give
+$0\le\eta_R\le R\rho$, and the measures increase with $R$.
+The uniform small-set conclusion of \Cref{lem:histogram-control}
+also gives uniform integrability of the successful-query densities.
+Indeed, since their masses are at most one,
+$\nu_n\{d_n>R\}\le1/R$; uniform small-set control bounds the mass on
+this set by a quantity tending to zero as $R\to\infty$, uniformly
+asymptotically in the leaf and in its orientation filter. Thus there is
+a deterministic function $\varepsilon_R\to0$ such that, in the limit,
+\[
+0\le\eta(Z)(\Upsilon)-\eta_R(Z)(\Upsilon)\le\varepsilon_R
+\quad\text{almost surely}.
+\]
+It follows that $\eta=\sup_R\eta_R$. In particular it has a density
+which is the increasing supremum of the truncated limiting densities.
+
+Fix $R$. The prelimit inner product of the two truncated densities,
+multiplied by $J_n$ and usability, tends to zero because it is bounded above by
+\eqref{eq:limit-option-overlap}. Choose a global partition with projection
+error at most $\varepsilon$ in
+\Cref{lem:limit-global-projection}. For that finite partition the inner
+product of the projected densities is
+\[
+\sum_{C:\,\nu_n(C)>0}
+ \frac{\eta_{A,R,n}(C)\eta_{B,R,n}(C)}{\nu_n(C)}.
+\]
+This expression passes to its limiting counterpart in expectation,
+jointly with the discrete permitted-pair and usability indicators. A cell whose reference
+weight tends to zero contributes at most $R^2\nu_n(C)$ and may be
+omitted. On the other cells ordinary continuity applies.
+
+Take increasing cylinder partitions which generate all tuple tests and
+include global projection partitions at accuracies tending to zero.
+The refinement assertion in \Cref{lem:limit-global-projection} preserves
+the error estimate. Conditional expectations of the limiting bounded
+densities on these partitions converge in $L^2(\rho)$: this follows,
+for instance, by approximating an $L^2$ function by cylinder-simple
+functions and using that conditional expectation is an $L^2$
+contraction. Their inner products therefore converge in expectation,
+being bounded by $R^2$. We obtain
+\[
+\E_Z J(Z)U_\omega(Z)
+ \int \frac{d\eta_{A,R}}{d\rho}
+          \frac{d\eta_{B,R}}{d\rho}\,d\rho=0.
+\]
+All terms are nonnegative. Taking the countable intersection of the
+resulting almost-sure assertions, then increasing $R$, proves the same
+product-zero conclusion for the full densities on $J=1$. Multiplication
+by $J$ is legitimate at the projection step because the projection
+error bound is in absolute value and permits every leaf-pair indicator.
+There are only finitely many options.
+\end{proof}
+
+\subsection{Identification of the full limiting density}
+
+For a side of an option, let $\alpha_Z$ be the subprobability law of
+$\theta$ obtained from its conditional law $\Lambda$ by retaining the
+appropriate orientation flag. Let $t_\ell,\upsilon_\ell$ be its required unary
+type and status at position $\ell$. Write $\kappa>0$ for the binary
+mixing constant for this side and numerical prescription. Constants on
+the two sides need not be denoted by the same letter.
+
+\begin{lemma}[Successful-query density]
+\label{lem:limit-density}
+For almost every $Z$, the full limiting successful-query measure on
+this side has density
+\begin{equation}
+\label{eq:limit-density}
+G_Z(u)=\kappa\int
+  \prod_{\ell}h^\theta_{t_\ell,\upsilon_\ell}
+                    \bigl(\xi_\ell(u)\bigr)\,d\alpha_Z(\theta)
+\qquad (u\in\Upsilon)
+\end{equation}
+with respect to $\rho$.
+\end{lemma}
+
+\begin{proof}
+The displayed expression is jointly measurable and nonnegative, by
+\Cref{lem:limit-model}. We prove equality against every tuple cylinder
+indicator $W$.
+
+At a fixed accuracy $\varepsilon$, apply
+\Cref{thm:weak-product} to its prelimit global key test $W_n$.
+On the reference key space its structured approximation has the form
+\[
+W_n^{(\varepsilon)}(k)
+=\sum_{a=1}^{m_\varepsilon} c_{a,n}
+        \prod_{\ell}\phi_{a,\ell,n}(k_\ell),
+\]
+where the number of terms and all bounds are fixed at this accuracy.
+The unary factors may be taken to be finite linear combinations of
+indicator factors already in the common test lists. In the
+actual-query assertion of that theorem, take as unary weights the
+numerical status indicators of this option. These may depend on the
+orientation. The extra admissibility condition is orientation-only.
+Thus, after averaging over leaves, the successful mass tested by $W_n$
+differs from that tested by $W_n^{(\varepsilon)}$ by at most the
+prescribed accuracy and a vanishing error.
+
+For clarity, this last use does not require typicality separately on
+every leaf. At a typical orientation the product-testing error is
+uniform over the allowed bounded unary weights. Multiplying by any
+orientation flag only decreases its bound. After both leaves are
+averaged, the mass of exceptional orientations is bounded by the mixed
+orientation error in \Cref{lem:typicality}. An opposite-leaf-dependent
+choice of the flag cannot increase that unconditional error.
+
+Apply \Cref{lem:binary-mixing} to each structured term. Again its unary
+weights are the status indicators times bounded global unary factors.
+The resulting expression is
+\begin{equation}
+\label{eq:limit-structured-mass}
+\kappa\int\sum_a c_{a,n}
+ \prod_\ell\left(
+   \int\phi_{a,\ell,n}\,d\eta^{\theta_n}_{t_\ell,\upsilon_\ell,n}
+             \right)d\alpha_{Z_n}(\theta_n),
+\end{equation}
+up to a vanishing error at this fixed complexity. Here the measures
+inside the product are the empirical single-atom status measures at
+the given orientation. Every integral in parentheses is a finite
+linear combination of status-cylinder coordinates. The coefficients
+and conditional measures were stored in the compactification.
+Consequently \eqref{eq:limit-structured-mass} converges jointly with
+the successful cylinder mass to the analogous expression in the limit.
+The preceding errors are controlled in mean absolute value over $Z$.
+
+It remains to compare that limiting structured expression with the
+integral of $W G_Z$. This is where the \emph{reference} conclusion of
+\Cref{thm:weak-product} is also needed. Its error bound, uniform against
+products of bounded unary functions, passes to the limiting reference
+law first for cylinder-simple unary functions, since all the relevant
+finite test probabilities converge. It then holds for all bounded
+measurable unary functions on the signature spaces: approximate each
+one in $L^1(m_{l,q})$ by cylinder-simple functions, and telescope the
+product. At this fixed approximation accuracy the structured test has
+a fixed finite bound, so this passage costs a quantity tending to zero.
+
+We may therefore use the unary functions
+$h^\theta_{t_\ell,\upsilon_\ell}$ in the limiting reference comparison, for
+each $\theta$, and integrate it under the subprobability law
+$\alpha_Z$. Independence of the unary projections under $\rho$ gives
+\begin{align*}
+\int W^{(\varepsilon)}(u)G_Z(u)\,d\rho(u)
+&=\kappa\int\sum_a c_a
+ \prod_\ell\left(
+   \int\phi_{a,\ell}(\xi)
+                h^\theta_{t_\ell,\upsilon_\ell}(\xi)\,dm_{l_\ell,q_\ell}(\xi)
+              \right)d\alpha_Z(\theta).
+\end{align*}
+This is exactly the limit of \eqref{eq:limit-structured-mass}.
+Replacing $W^{(\varepsilon)}$ by $W$ in this last integral costs at most
+$\kappa$ times the reference approximation accuracy. Combining the
+empirical and reference estimates and sending $\varepsilon\to0$ yields
+\[
+\E_Z\left|\eta(Z)(W)-\int W G_Z\,d\rho\right|=0.
+\]
+There are countably many cylinder indicators. Equality on all of them
+determines the measures and proves the lemma.
+\end{proof}
+
+\begin{proposition}[The limiting overlap obstruction]
+\label{prop:limit-overlap}
+Assume \eqref{eq:limit-vanishing-overlap}. In the common experiment of
+\Cref{lem:limit-model}, almost every sampled pair of orientation records
+with $J(Z)=1$ has the following property: there is no usable numerical
+option whose two orientation flags pass and for which, at every matched position,
+\begin{equation}
+\label{eq:limit-positive-generator}
+\int_{\Xi_{l,q}}
+ h^{\theta_A}_{t_A,\upsilon_A}(\xi)
+ h^{\theta_B}_{t_B,\upsilon_B}(\xi)\,dm_{l,q}(\xi)>0.
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+For a usable option, \Cref{lem:limit-zero-product,lem:limit-density}
+give $\int G_{A,Z}G_{B,Z}\,d\rho=0$ for almost every $Z$ with $J(Z)=1$.
+Fubini's theorem and independence of the unary projections express
+this integral as
+\[
+\kappa_A\kappa_B
+\int\!\int
+ \prod_\ell\left(
+  \int h^{\theta_A}_{t_{A,\ell},\upsilon_{A,\ell}}(\xi)
+       h^{\theta_B}_{t_{B,\ell},\upsilon_{B,\ell}}(\xi)
+                       \,dm_{l_\ell,q_\ell}(\xi)\right)
+ \,d\alpha_{A,Z}(\theta_A)\,d\alpha_{B,Z}(\theta_B).
+\]
+All factors are nonnegative and both $\kappa$ constants are positive.
+Hence the product inside is zero for almost every passed pair from
+the conditional laws. Averaging $Z$ and taking the finite union over
+options proves the assertion.
+\end{proof}
diff --git a/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/12-status-span.tex b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/12-status-span.tex
new file mode 100644
index 0000000000000000000000000000000000000000..1915898ce1fea4ff14d234c5d0570f5beb55b400
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@@ -0,0 +1,642 @@
+\section{Solving the status constraints}
+\label{sec:status-span}
+
+We now construct the options forbidden by \Cref{prop:limit-overlap}.
+The argument takes place in the common signature model, but the
+no-prediction alternative will require a transfer back to global
+functions on the finite key spaces.
+We show that a positive mass of permitted pairs with soluble scalar
+constraints forces positive key overlap. Applied to the prepared
+coloured law, the collision criterion then supplies a four-hole event
+between different colours.
+
+Write the endpoints of the two units as $(A,1),(A,2)$ and
+$(B,1),(B,2)$. In this section $h_{A,i}$ or $h_{B,j}$ denotes an atom's
+role bit $a(\mathrm{atom})$, not the channel dimension. The basis status
+vector at an endpoint is denoted by $\mathbf x$, and its dimension is
+the recorded dimension of that endpoint's effective space $C_i$.
+
+\subsection{Generators and the coupled linear system}
+
+Fix two orientation arrays $\theta_A,\theta_B$ satisfying the
+almost-sure conclusions of \Cref{lem:limit-model}. Some labels at each
+endpoint may be excluded. In particular, the mandatory support
+exclusions from \Cref{eq:label-budget} are always excluded.
+
+\begin{definition}[Available generator]
+\label{def:status-generator}
+On the cross edge $(i,j)$, a vector
+\begin{equation}
+\label{eq:status-generator-vector}
+g=(q,h_{A,i},h_{B,j},\mathbf x_{A,i},\mathbf x_{B,j},p_{A,i},p_{B,j})
+\end{equation}
+is an available generator if some common tag, surviving label at each
+end, and allowed flavors give positive overlap as in
+\eqref{eq:limit-positive-generator} for these entries. Entries of the
+tester summary not displayed in \eqref{eq:status-generator-vector}
+are summed over. A generator is retained with one such certificate of
+its tags, labels, and flavors.
+
+Let $V_{ij}$ be the binary span of these generators. Its \emph{base
+projection} forgets the final two $p$ entries and has target
+\[
+B_{ij}=\F\oplus\F\oplus\F\oplus
+             \F^{\dim C_{A,i}}\oplus\F^{\dim C_{B,j}}.
+\]
+\end{definition}
+
+For any surviving pair of labels and any tag, the full generic flavor
+gives at least one available generator above every base vector. Indeed, at fixed $q$
+the sum of the status densities with any requested role/basis assignment
+is bounded below almost everywhere by
+\Cref{lem:limit-model}. The product of the two such sums has positive
+integral on the common signature space. Splitting its finite sum over
+the two $p$ bits yields an available generator. Both diagonal bits are
+feasible in the full generic flavor. Thus
+\begin{equation}
+\label{eq:status-full-projection}
+V_{ij}\longrightarrow B_{ij}\text{ is surjective},\qquad
+\dim\ker(V_{ij}\longrightarrow B_{ij})\le2.
+\end{equation}
+This remains true after any exclusions leaving surviving generic labels.
+
+Fix a small table. Its local target basis vectors are the evaluations of
+\[
+(a+(u_{B,j}U_{A,i})^*)|_{C_{A,i}},\qquad
+(a+(u_{A,i}U_{B,j})^*)|_{C_{B,j}},
+\]
+in the recorded bases. Denote these vectors by
+$\mathbf t_{A,i;j}$ and $\mathbf t_{B,j;i}$, respectively. We seek one
+sum vector $v_{ij}\in V_{ij}$ for each cross edge, subject to
+\begin{equation}
+\label{eq:status-local-targets}
+\begin{aligned}
+q_{ij}&=1,& h_{A,i;ij}+h_{B,j;ij}&=1,\\
+\mathbf x_{A,i;ij}&=\mathbf t_{A,i;j},&
+\mathbf x_{B,j;ij}&=\mathbf t_{B,j;i},
+\end{aligned}
+\end{equation}
+\begin{equation}
+\label{eq:status-coupled-targets}
+\begin{aligned}
+\sum_{i=1}^2(p_{A,i;ij}+h_{A,i;ij})&=0 &&(j=1,2),\\
+\sum_{j=1}^2(p_{B,j;ij}+h_{B,j;ij})&=0 &&(i=1,2).
+\end{aligned}
+\end{equation}
+These are precisely the scalar recipe equations
+\eqref{eq:recipe-scalars}. The occurrence of a given endpoint in two
+different sums is distinguished by its edge subscript.
+By \Cref{thm:gradient-realization}, these are the effective-space
+conditions needed for the full gradients; the attack-atom conditions
+are supplied by \eqref{eq:recipe-gradients}. This use does not require
+pins to be supported at individual endpoints. The realization theorem
+already treats the mixed pin spaces through their projections and the
+injection hypothesis.
+
+\begin{lemma}[Dual relations]
+\label{lem:status-duality}
+Any obstruction to the scalar system has a nonzero vector of weights
+$(s_1,s_2,r_1,r_2)\in\F^4$. On every edge span $V_{ij}$ these give
+a relation
+\begin{equation}
+\label{eq:status-dual-relation}
+s_j(p_{A,i}+h_{A,i})+r_i(p_{B,j}+h_{B,j})
+=\mathbf c_{ij}\cdot\mathbf x_{A,i}
+ +\mathbf c'_{ij}\cdot\mathbf x_{B,j}
+ +\beta_{ij}(h_{A,i}+h_{B,j})+\gamma_{ij}q
+\end{equation}
+for suitable coefficients. The scalar system is soluble if, for every
+such relation, the sum of the right sides evaluated at the local targets
+is zero.
+\end{lemma}
+
+\begin{proof}
+Apply finite-dimensional duality to the linear constraint map on
+$\bigoplus_{i,j}V_{ij}$. The coefficients of the four coupled equations
+are $s_j,r_i$, giving \eqref{eq:status-dual-relation} separately on each
+edge. If they were all zero, the local projection onto
+$(q,h_{A,i}+h_{B,j},\mathbf x_{A,i},\mathbf x_{B,j})$ would be
+surjective by \eqref{eq:status-full-projection}, forcing all other dual
+coefficients to vanish. Hence a nonzero obstruction uses some coupled
+equation. Conversely the constraint target lies in the image exactly
+when it annihilates every dual relation. The coupled targets are zero,
+so their evaluated contribution is the sum stated in the lemma.
+\end{proof}
+
+\subsection{A dual pattern on positive mass gives a global prediction}
+
+Call a pair of arrays \emph{obstructable} if there are total exclusion
+sets of size at most $B_*$ at each endpoint, containing the mandatory
+exclusions, and some nonzero weights for which
+\eqref{eq:status-dual-relation} holds throughout. We do not require the
+relation to take a nonzero value on a particular table target. This
+stronger event is useful because its complement works for every table.
+
+\begin{lemma}[Extraction of a global prediction]
+\label{lem:status-global-prediction}
+On the no-prediction alternative of \Cref{prop:status-preparation}, the
+probability that the iid base arrays are obstructable is less than $.1$.
+\end{lemma}
+
+\begin{proof}
+Suppose otherwise. Let $E_s$ be the event that a witnessing relation
+has some $s_j=1$, and define $E_r$ similarly. Interchanging the two iid
+units interchanges these events, so they have the same probability.
+Their union is the obstructable event. Therefore
+$\PP(E_s)\ge .05$.
+
+Fix a witness with $s_j=1$. For each first-side endpoint $i$, rewrite
+its edge relation as
+\begin{align}
+p_{A,i}+\mathbf c_{ij}\cdot\mathbf x_{A,i}
+                 +(1+\beta_{ij})h_{A,i}
+={}&r_i p_{B,j}+\mathbf c'_{ij}\cdot\mathbf x_{B,j}
+                 +(r_i+\beta_{ij})h_{B,j}+\gamma_{ij}q.
+\label{eq:status-deterministic-comparison}
+\end{align}
+For any surviving label/flavor choices and any common feasible $(l,q)$,
+this equality holds almost surely under the product of the two status
+distributions conditional on the same signature. Otherwise a violating
+pair of numerical statuses would have positive overlap and would be a
+generator violating the relation. The two bits in
+\eqref{eq:status-deterministic-comparison} are conditionally independent
+at a fixed signature. If two independent bits agree almost surely, both
+are constant almost surely with their common value: writing their
+success probabilities as $a,b$, the mismatch probability
+$a(1-b)+(1-a)b$ is zero only for $(a,b)=(0,0)$ or $(1,1)$.
+
+Compare all surviving first-side choices to one fixed generic surviving
+choice at $(B,j)$. Its total conditional status mass is one on every
+signature space, and its generic flavor admits either $q$. It follows
+that, simultaneously over all tags, feasible bits, labels and flavors
+remaining at $(A,i)$,
+\begin{equation}
+\label{eq:status-extracted-prediction}
+p_{A,i}=T(\lambda_{A,i},\mathrm{atom})+\alpha_{A,i}h_{A,i}
+                    +f_i(l,q,\xi),
+\end{equation}
+where $\lambda_{A,i}$ is the basis combination specified by
+$\mathbf c_{ij}$ and $\alpha_{A,i}=1+\beta_{ij}$. The coefficients
+are common to these choices. The offset is also common across them;
+one does not choose a new offset for every label or flavor.
+
+There are very few possible offsets after the second-side array is
+fixed. If $r_i=0$, generic positivity of the role and basis at $(B,j)$
+forces $\mathbf c'_{ij}=0$ and $\beta_{ij}=0$ in
+\eqref{eq:status-deterministic-comparison}. The offset is then either
+zero or $q$. If $r_i=1$, it is, up to a common multiple of $q$, the
+deterministic value of
+\begin{equation}
+\label{eq:status-opposite-offset}
+p_{B,j}+\mathbf c'\cdot\mathbf x_{B,j}+\alpha' h_{B,j}.
+\end{equation}
+For this fixed array and endpoint there is at most one possible
+deterministic expression \eqref{eq:status-opposite-offset}, including
+its coefficients, even if the permitted exclusions vary. Indeed two
+candidate exclusion sets of size at most $B_*$ have a common surviving
+generic label, since the label universe is larger than $2B_*$. At that
+label every role/basis assignment has positive density conditional on
+the signature. Subtracting the two claimed deterministic expressions
+therefore forces their coefficient differences to be zero, and then
+their offset functions agree almost everywhere on every $(l,q)$ space.
+The same generic label can be used at every tag and diagonal bit.
+
+Thus, for a fixed second-side array, each choice of $j$ supplies at most
+four offsets for each first-side endpoint: the two multiples of $q$,
+and the unique function \eqref{eq:status-opposite-offset} plus those
+two multiples, if that function exists. Across both choices of $j$
+there are at most $2\cdot4^2=32$ pairs $(f_1,f_2)$.
+
+All events used here are measurable. There are finitely many types,
+status vectors, coefficients, and exclusion sets, and generator
+availability is a positivity test on the integral of jointly
+measurable densities. By Fubini, choose a fixed second-side array for
+which the first-side mass of $E_s$ exceeds $.04$. One of the at most
+32 pairs of offset functions then gives
+\eqref{eq:status-extracted-prediction} on first-side mass at least
+$\delta_0=.04/32$, allowing its coefficients and exclusion sets to vary
+with the first-side array. This is much larger than $10^{-6}$.
+
+We now transfer these fixed signature functions to a prediction in the
+finite problem. This step is necessary because the definition of
+prediction asks for global functions of actual keys. For any
+$\varepsilon>0$, approximate each fixed binary function $f_i$ by a
+binary cylinder function $f_i^{(\varepsilon)}$, with disagreement
+measure at most $\varepsilon$ on each of the finitely many $(l,q)$
+reference spaces. Such approximation follows by approximating the
+measurable event where $f_i=1$ in measure by the algebra of finite
+cylinders. The total unary measure of every type is its reference
+measure, so replacing $f_i$ increases every relation error in
+\eqref{eq:status-extracted-prediction} by at most $\varepsilon$.
+
+For fixed cylinder functions and a fixed choice of coefficient bits
+and exclusions, all these error probabilities are finite linear
+combinations of status-cylinder coordinates. Their maximum over the
+remaining unary types is continuous. Minimizing over the finitely many
+permitted coefficient/exclusion choices is also continuous, separately
+on each discrete coordinate format. The strict event that this minimum
+is below $2\varepsilon$ consequently has limiting probability at least
+$\delta_0$. Weak convergence implies that its prelimit probability
+exceeds $\delta_0/2$ for all sufficiently large indices. One may see
+this directly by approximating the indicator of an open strict-error
+event from below by continuous functions.
+
+Choose decreasing $\varepsilon_m\to0$ and increasing prelimit indices
+at which the last mass bound holds. At index $m$ use the two global
+functions obtained by evaluating $f_i^{(\varepsilon_m)}$ on the actual
+global key signature. Keep the orientations in the strict-error event
+and choose one of their successful coefficient/exclusion records.
+The kept mass is at least $\delta_0/2>10^{-6}$, and the maximum
+relation error over all remaining unary types tends uniformly to zero.
+Conditional-on-$q$ errors give the same conclusion for the unconditioned
+flavor tests. These are precisely the requirements of
+\Cref{def:prediction}. The no-prediction branch has only the first
+effective spaces and only negligible trimming, so this mass margin
+also gives a prediction before that trimming. This contradiction
+proves the lemma.
+\end{proof}
+
+\subsection{The predicted dual relations}
+
+In the predicted alternatives, exclude the stored prediction exceptions
+as well as the mandatory support exclusions. Write the marked profiles
+as $\lambda_{A,i}$ and $\lambda_{B,i}$. The constants $\alpha_i$ and
+the functions $f_i$ are common by endpoint index across units. Put
+$d_i=1+\alpha_i$. The prepared class $e_i=(d_i,[f_i])$ is as in
+\eqref{eq:status-class}: functions are first identified almost
+everywhere on all tag/bit signature spaces, and then quotiented by
+the span of the one function $q$, with a common coefficient on those
+spaces.
+
+In a limiting table computation, use the notation
+\begin{equation}
+\label{eq:status-table-delta}
+\mathsf u_{B,j}(\Delta_A)
+ =\sum_{i=1}^2 (u_{B,j}U_{A,i})^*(\lambda_{A,i}),
+\qquad
+\mathsf u_{A,i}(\Delta_B)
+ =\sum_{j=1}^2 (u_{A,i}U_{B,j})^*(\lambda_{B,j}).
+\end{equation}
+These are finite contractions in the recorded table coordinates. The
+notation does not require retaining an ambient tensor $\Delta$ in the
+compactification.
+
+\begin{lemma}[Predicted scalar compatibility]
+\label{lem:status-predicted}
+For a predicted pair with a table satisfying the compatibility
+conditions of \Cref{prop:status-preparation}, the system
+\eqref{eq:status-local-targets}--\eqref{eq:status-coupled-targets}
+is soluble. The assertion survives any further bounded label
+exclusions leaving generic labels.
+\end{lemma}
+
+\begin{proof}
+The exact limiting prediction is
+\[
+p_{A,i}=\boldsymbol\lambda_{A,i}\cdot\mathbf x_{A,i}
+                 +\alpha_i h_{A,i}+f_i,
+\qquad
+p_{B,j}=\boldsymbol\lambda_{B,j}\cdot\mathbf x_{B,j}
+                 +\alpha_j h_{B,j}+f_j,
+\]
+where bold $\boldsymbol\lambda$ denotes coordinates in the effective
+basis. Substitute this into a dual relation
+\eqref{eq:status-dual-relation}. At a fixed signature the two generic
+status distributions have every pair of role/basis assignments
+available. Comparing their coefficients forces
+\begin{align}
+\mathbf c_{ij}&=s_j\boldsymbol\lambda_{A,i},&
+\mathbf c'_{ij}&=r_i\boldsymbol\lambda_{B,j},&
+\beta_{ij}&=s_jd_i=r_id_j,
+\label{eq:status-forced-coefficients}\\
+s_jf_i+r_if_j&=\gamma_{ij}q.
+\label{eq:status-forced-functions}
+\end{align}
+Thus
+\begin{equation}
+\label{eq:status-class-equations}
+s_je_i=r_ie_j\qquad(i,j\in\{1,2\}).
+\end{equation}
+Equality of functions here is exactly the almost-everywhere equality
+recorded in the prepared subsequential class.
+
+If both $e_i$ vanish, \Cref{lem:empty-status} supplies the exact
+identity $p_i=T(\lambda_i,\mathrm{atom})+h_i$ on all atoms. Hence on
+each span $p_i+h_i=\boldsymbol\lambda_i\cdot\mathbf x_i$.
+At the local targets, the first coupled sum for a fixed $j$ is
+\[
+\sum_i a(\lambda_{A,i})+
+       \mathsf u_{B,j}(\Delta_A)
+=c+\mathsf u_{B,j}(\Delta_A)=0
+\]
+by the empty-status compatibility condition. The reciprocal coupled
+sums vanish in the same way. Local surjectivity therefore solves all
+constraints in this case.
+
+Suppose that some $e_i$ is nonzero. A nonzero solution of
+\eqref{eq:status-class-equations} is possible only when all nonzero
+$e_i$ are equal, and then both weight vectors are the indicator of
+their support $S$. To check this without any assumption on the dimension
+of the function space, first consider a support with one index, say
+$e_1\ne0,e_2=0$. The off-diagonal equations force $r_2=s_2=0$,
+and the diagonal gives $s_1=r_1=1$ for a nonzero solution. If both
+$e_i$ are nonzero, the diagonal equations give $s_i=r_i$.
+The off-diagonal equation $s_2e_1=s_1e_2$ either makes both weights
+zero, or makes both one and $e_1=e_2$. These cases exhaust the
+possibilities.
+
+If the two nonzero classes are unequal, no dual obstruction exists.
+Otherwise let $d$ be the common first bit on $S$. With
+$s_i=r_i=\1_{i\in S}$, \eqref{eq:status-forced-functions} is symmetric
+in $i,j$. Its diagonal is zero. Since the function $q$ is not zero
+on the family of reference spaces, this gives
+$\gamma_{ij}=\gamma_{ji}$ and $\gamma_{ii}=0$. The sum of all
+four $q$-contributions to a target is therefore zero. The total role
+contribution is
+\[
+\sum_{i,j}\beta_{ij}=|S|d\quad\text{in }\F.
+\]
+
+Evaluate the basis terms in
+\eqref{eq:status-forced-coefficients} at their local targets.
+The $a$-terms sum to $|S|c+|S|c=0$, because
+\Cref{prop:status-preparation} fixed the same value
+$c=\sum_i a(\lambda_i)$ in both units. The remaining terms are
+\[
+\sum_{j\in S}\mathsf u_{B,j}(\Delta_A)
+ +\sum_{i\in S}\mathsf u_{A,i}(\Delta_B).
+\]
+By \eqref{eq:phase-compatibility}, this equals $|S|d$.
+Together with the role contribution it is zero. Thus every dual
+relation vanishes on the target, and \Cref{lem:status-duality} proves
+solvability. The argument used only surviving generic labels and the
+prediction on those labels, so additional exclusions of the indicated
+kind do not change it.
+\end{proof}
+
+\begin{lemma}[Scalar solvability on positive mass]
+\label{lem:status-span}
+In each positive-mass alternative of
+\Cref{prop:status-preparation}, the common experiment has positive mass
+of passed injecting tables for which the scalar system is soluble,
+robustly under the exclusions needed below. On the no-prediction branch
+this holds whenever the total exclusions stay within $B_*$ per endpoint.
+On a predicted branch it holds after the stored exceptions and any
+further bounded exclusions leaving generic labels.
+\end{lemma}
+
+\begin{proof}
+Without prediction, the probability of an obstructable base pair is
+less than $.1$ by \Cref{lem:status-global-prediction}. This statement
+uses the unflagged iid base law. In the full conditional-law experiment,
+a passed injecting table exists on mass at least $.99$ by
+\Cref{lem:limit-model}. Their intersection therefore has mass greater
+than $.89$. On it no nonzero dual pattern exists for any exclusions
+within the stated budget, so \Cref{lem:status-duality} solves the
+scalar system for every such table. No independence of the two events
+is being assumed.
+
+With prediction, a passed compatible injecting table exists on positive
+mass in the same experiment. The exact prediction and all almost-sure
+unary properties continue to hold on that event, and
+\Cref{lem:status-predicted} gives the assertion.
+\end{proof}
+
+\subsection{Stable spans and fresh short lists}
+
+It remains to replace a vector in an edge span by a short list of actual
+generators, using distinct labels across both lists at an endpoint.
+The small dimension in \eqref{eq:status-full-projection} is decisive.
+
+\begin{lemma}[Fresh lists of at most seven generators]
+\label{lem:fresh-lists}
+On any pair for which scalar solvability survives the bounded exclusions
+specified in \Cref{lem:status-span}, further bounded exclusions can be made so that every scalar solution on the resulting
+four edge spans can be realized by between one and seven available
+generators per edge. The two endpoint labels of each generator are
+retained with its certificate, and all labels used at an endpoint are
+distinct across its two incident lists. Every realized matched
+position has positive status overlap.
+\end{lemma}
+
+\begin{proof}
+Begin with the mandatory support exclusions and, in the predicted
+case, the stored exceptions. More generally, start with the initial
+exceptions allowed by the scalar-solvability hypothesis. Whenever deleting at most 100 additional
+labels at each endpoint can shrink any of the four edge spans, make
+such a deletion. Each span continues to project onto its full base
+space. Its dimension exceeds that base dimension by at most two, so
+there are at most eight dimension-reducing steps in total. At the end,
+every edge span is unchanged by deleting up to 100 further labels at
+each endpoint. There remain generic labels, since
+\eqref{eq:selector-size} makes the label universe much larger than all
+these bounded exclusions. On the no-prediction branch the total budget,
+including later fresh-label removals, is at most $B_0+900<B_*$.
+On a predicted branch it is at most the stored exceptions plus this
+quantity, still leaving many generic labels. Scalar solvability is
+retained by the scalar-solvability hypothesis. Fix one solution on these stabilized
+spans.
+
+We prove a statement for one edge after any labels used on earlier
+edges have also been reserved. Write its stabilized span as $V$, its
+base space as $B$, and its projection as $\operatorname{pr}_B$. Let $G_L$ be the
+generators avoiding the current reserved sets $L$. Their span is still
+$V$. Let $H_L$ be the span of sums of one, two, or three members of
+$G_L$ whose total base projection is zero and whose endpoint labels
+are mutually disjoint within that word. We claim
+\begin{equation}
+\label{eq:status-short-kernel}
+H_L=\ker\operatorname{pr}_B.
+\end{equation}
+
+To prove the claim, take two generator lifts of the same $b\in B$.
+There is a third generic lift of $b$ at labels avoiding both of them
+and $L$. Comparing each original lift to this third one gives an
+allowed two-generator kernel word. Hence all lifts of $b$ have the
+same class modulo $H_L$. Call that class $\ell(b)$. A generator
+above base zero is itself an allowed one-generator kernel word, so
+$\ell(0)=0$. Choose mutually label-disjoint generic lifts of
+$b,c,b+c$. Their sum is an allowed three-generator kernel word,
+and therefore $\ell(b)+\ell(c)=\ell(b+c)$. The generators span $V$,
+so this additive lift identifies $V/H_L$ with $B$: the induced
+projection is inverse to $\ell$. This proves
+\eqref{eq:status-short-kernel}, including the case of zero base
+projection.
+
+For the desired vector in $V$, first choose a generic singleton lift
+of its base projection. Reserve its labels. The remaining discrepancy
+belongs to $\ker\operatorname{pr}_B$, whose dimension is at most two. On the remaining
+labels \eqref{eq:status-short-kernel} still holds because stability has
+preserved $V$. If the kernel is zero no correction is needed. If it is
+one-dimensional, choose a nonzero short kernel word and use it if
+needed. If it is two-dimensional, choose a nonzero short word, reserve
+its at most three labels at each end, and apply
+\eqref{eq:status-short-kernel} again. The new short words still span
+the full kernel, so one lies outside the first word's direction.
+These two words form a basis and the discrepancy is the sum of an
+appropriate subset of them. This uses at most
+$1+3+3=7$ generators. The initial lift is always retained, so the list
+is nonempty; reserving labels before each choice ensures disjointness
+even when two generators have the same numerical bit vector.
+
+Carry out this procedure on the four edges in turn. Each endpoint is
+incident to only two lists and uses at most fourteen labels in total.
+All removals and auxiliary freshness comparisons in the procedure are
+within the 100-label stability allowance. The generators keep their
+certificates of positive overlap, proving the lemma.
+\end{proof}
+
+\subsection{Positive overlap on permitted pairs}
+
+\begin{proposition}[Positive-mass overlap]
+\label{prop:positive-overlap}
+Consider two independent unit laws with actual leaf weights, the mixed
+caps \eqref{eq:histogram-mixed-caps}, the final per-leaf image bounds,
+and a permitted-pair indicator $J_n$ as in \Cref{sec:common-limit}.
+Suppose their common model has positive mass of pairs with $J=1$
+admitting a passed injecting table whose scalar system is soluble after
+every bounded exclusion used by \Cref{lem:fresh-lists}. Then some fixed
+numerical option has expected unnormalized, permitted-pair overlap
+bounded below by a positive constant along a subsequence. Consequently
+it satisfies \eqref{eq:overlap-criterion}, with the same permitted-pair
+predicate. The two laws need not coincide. If the original laws
+additionally satisfy the hypotheses of \Cref{prop:collision}, its
+probability bound holds along the same subsequence, with any initial
+restrictions of reciprocal mass $2^{o(N)}$ charged to that probability.
+\end{proposition}
+
+\begin{proof}
+Work on the subsequence defining the common model in the hypothesis.
+If the conclusion failed, finiteness of the option collection would
+give \eqref{eq:limit-vanishing-overlap} along this same subsequence.
+Apply \Cref{prop:limit-overlap} in this model.
+
+By hypothesis, there is positive model mass of permitted pairs with a
+table whose flags pass and whose scalar solution survives the needed
+exclusions. Apply \Cref{lem:fresh-lists}. This gives between one and
+seven matched generators per edge with fresh allowed labels. Their
+sum vectors obey \eqref{eq:status-local-targets} and
+\eqref{eq:status-coupled-targets}; equivalently, the lists obey all
+scalar recipe equations. At each generator the unspecified tester
+bits were only aggregated. Expanding the positive overlap integral
+as a finite sum over the two full tester summaries yields at least
+one pair of summaries for which the overlap is still positive.
+Make that refinement independently at each position.
+
+The scalar consistency argument in
+\Cref{sec:realization}, in particular \eqref{eq:recipe-gradients},
+now supplies a consistent numerical prescription of the required
+within-endpoint binary bits from these summaries and $p$ bits. Together
+with the already passed table, these data define a usable numerical
+option. The only parameter filters added are precisely the unary
+statuses and the prescribed binary conditions in
+\eqref{eq:limit-success-format}. At every matched position the refined
+status overlap is positive. This contradicts
+\Cref{prop:limit-overlap} on a positive-mass event.
+
+No pointwise uniform lower bound for these individual positive
+integrals is needed. The options form a finite collection and every
+product has finitely many nonnegative factors. A positive-mass event
+on which some such product is positive gives positive expectation
+for at least one option. Its lower bound is a constant on the chosen
+subsequence, however small that constant may be. The preceding
+contradiction therefore proves the stated subsequential lower bound.
+
+For this probability consequence, assume additionally that the original
+laws satisfy the hypotheses of \Cref{prop:collision}. The successful
+densities are unnormalized with respect to the retained law, and
+\Cref{prop:collision} explicitly charges any law
+restriction of reciprocal mass $2^{o(N)}$. A positive constant, or
+its product with such restriction costs, is $2^{-o(N)}$. The resulting
+four-hole probability, including the permitted-pair predicate, has
+exponent at most $k_{\max}+.03+o(1)$, whereas
+\[
+100g-(k_{\max}+.03)=44g+55.97>0.
+\]
+Thus these overlaps are on the scale required for
+\Cref{thm:distribution-tests}.
+\end{proof}
+
+
+\subsection{The coloured unit distribution test}
+
+\begin{proposition}[Different-colour conflicts]
+\label{prop:coloured-distribution}
+Let the early constant $\eta_{\mathrm{col}}>0$ satisfy
+\[
+ \eta_{\mathrm{col}}<\frac14c_* ,\qquad
+ c_*:=\frac{\delta_*^2}{64},
+\]
+where $\delta_*$ is the retained-mass margin in
+\eqref{eq:preparation-early-margins}. This choice depends on the original
+endpoint bound $M$ and the early geometric constants, and precedes the
+selector and channel dimensions. For every fixed finite palette and all
+sufficiently large $n$, a unit law with both endpoint marginals bounded
+by $M\pi$, pair law after forgetting colour bounded by $2^{3000gN}\pi^2$, and each
+colour of mass at most $\eta_{\mathrm{col}}$ satisfies
+\[
+ \Pr\{A\text{ and }B\text{ have four cross holes and different colours}\}
+ \ge 2^{-100gN},
+\]
+where $A,B$ are independent draws from that law. In particular,
+\Cref{thm:distribution-tests}\textup{(i)} holds.
+\end{proposition}
+
+\begin{proof}
+Suppose that the assertion fails along an unbounded sequence of $n$.
+The laws and their randomized colour marks may vary with $n$; the
+palette is fixed. Refine by colour and discard the exponentially light
+colour classes as in \Cref{lem:peeling}. Peel each remaining conditional
+class, but retain the original mass of every leaf. Their mixture is a
+restriction of the original law with total mass $1-o(1)$ before the
+fixed preparation restrictions. Hence its endpoint marginals remain
+bounded by a fixed multiple of $M\pi$ and its primal marginals by the
+same multiple of the original primal law. The conditioning of a small
+colour may change a leaf's joint density bound; this is already included
+in its absolute density and image budgets. It never changes the mixed
+marginal constant used next.
+
+Apply \Cref{prop:status-preparation} to this pooled law, carrying its
+colour as a mark. In the predicted case this yields a retained law of
+original mass at least $\delta_*$, and an event of passed compatible
+injecting tables of conditional pair mass at least $1/64$. Its scalar
+systems are soluble by \Cref{lem:status-predicted}, robustly under the
+fresh-label exclusions. In the no-prediction case the retained mass is
+also at least $\delta_*$ and \Cref{lem:status-span} supplies conditional
+pair mass greater than $.89$. In both cases the successful scalar-table
+event has mass at least $c_*$ in two draws from the original law.
+
+If the original colour probabilities are $p_c$, their equal-colour
+pair mass is
+\[
+ \sum_c p_c^2\le (\max_c p_c)\sum_c p_c
+ \le\eta_{\mathrm{col}}.
+\]
+The equal-colour part of any subevent, including a prepared subevent,
+has no larger original mass. Subtracting it therefore leaves at least
+$c_*-\eta_{\mathrm{col}}>0$ of successful scalar-table pairs of different
+colours. Equivalently, if preparation retained mass $\delta$, its
+normalized equal-colour pair mass is at most
+$\eta_{\mathrm{col}}/\delta^2$. This subtraction takes place before any
+pigeonholing of a numerical table. Thus neither the number of tables nor
+the palette size enters the early choice of $\eta_{\mathrm{col}}$.
+
+Use the permitted-pair indicator
+$J_n(L_A,L_B)=\mathbf1_{\{\operatorname{col}(L_A)
+\ne\operatorname{col}(L_B)\}}$. Store this bit in the common model.
+The preceding original-mass lower bound, together with the retained
+prediction and injection assertions of \Cref{lem:limit-model}, gives
+positive limiting mass on $J=1$ satisfying the hypothesis of
+\Cref{prop:positive-overlap}. That proposition yields a fixed option
+with positive subsequential overlap on different-colour pairs.
+The filtered form of \Cref{prop:collision} now gives probability at
+least $2^{-(k_{\max}+.03)N-o(N)}$ for a four-hole event on those same
+pairs. Because $k_{\max}+.03<100g$, this is larger than
+$2^{-100gN}$ for all sufficiently large indices, contradicting the
+chosen violating sequence.
+
+The overlap constant is allowed to depend on the subsequential model.
+This causes no loss in the uniform eventual assertion: any sequence
+violating it has a further subsequence with one fixed positive constant,
+and the fixed exponential gap just used defeats that sequence.
+\end{proof}
diff --git a/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/12b-asymmetric.tex b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/12b-asymmetric.tex
new file mode 100644
index 0000000000000000000000000000000000000000..81d7ff8bde6021931d78ccb664851cb8fe89f379
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@@ -0,0 +1,569 @@
+\section{Conflicts between a unit and an independent point}
+\label{sec:asymmetric}
+\label{sec:asymmetric-test}
+
+We prove the second distribution test.  Its two endpoint bounds are
+separate assumptions: throughout this section, \(\sigma\) is a law
+on units, \(\nu\) is a law on points, and
+\begin{equation}
+\label{eq:asym-caps}
+ \sigma\le 2^{3000gN}\pi^2,
+ \qquad \sigma_1,\sigma_2\le L\pi,
+ \qquad \nu\le L\pi.
+\end{equation}
+Here \(\pi\) is the thinned law and \(\rawlaw\) is the original
+frame law.  In particular only their primal marginals agree.  We
+use \(\varepsilon_{\rm ch}=10|\cE|2^h\rho\) as an upper bound
+on the exponent lost in comparing \(\pi^2\) with \(\rawlaw^2\).
+All restrictions made below have fixed positive mass, so all their
+constant marginal factors can be specified before the channel size
+is chosen.
+
+Draw \(A\sim\sigma\) and, independently, two points
+\(B_1,B_2\sim\nu\).  We use this independent experiment for all
+calculations involving the two points.  By
+\Cref{cor:independent-point-leaf},
+\begin{equation}
+\label{eq:asym-internal-hole}
+ \PP\{B_1B_2\text{ is a hole}\}
+ \le O_g\bigl(L^2 2^{2\dim\cB-N}\bigr)=o(1).
+\end{equation}
+Indeed the two primal spans in each component are disjoint except
+on the displayed exceptional mass.  On their complement, the first
+hole equation forces both profiles to be zero, contradicting the
+last hole equation.  Only after the independent-point calculations
+will we delete these internal holes.  The resulting unit law has a
+single unpinned leaf and
+\begin{equation}
+\label{eq:asym-zero-space}
+ C_{B,1}=C_{B,2}=0.
+\end{equation}
+The full-frame image bounds needed for that leaf pay
+\(\varepsilon_{\rm ch}N\); they do not assert an unchanged
+full-frame marginal.  Conditioning on a nonhole costs \(1+o(1)\).
+
+\subsection{No common prediction for the point side}
+
+Write \(\Omega_{l,q,n}\) for the single-key reference space from
+\Cref{sec:preparation}.  For a channel frame \(v\) and a key
+\(k=((x_e,y_e))_{e\ni l}\) in this space, put
+\[
+ \chi_v(k)=(-1)^{u_v(k)},\qquad
+ u_v(k)=\sum_{e\ni l}
+       (Y_{v,e}\trans x_e)\cdot(X_{v,e}\trans y_e).
+\]
+Every component has \(x_e\cdot y_e=q\).  This notation agrees
+with evaluating \(u_v\) on the tensor represented by the key.
+
+\begin{lemma}[Channel correlations on a common key]
+\label{lem:asym-channel-orthogonality}
+Let \(v,w\) be two frames for which the concatenated channel
+columns \([X_{v,e},X_{w,e}]\) and
+\([Y_{v,e},Y_{w,e}]\) have rank \(2h\), in every component
+of the star of \(l\).  Uniformly in these frames and in \(q\),
+\begin{align}
+ \E_{\Omega_{l,q,n}}\chi_v
+   &=2^{-h(g-1)}+O_g(2^{-N+4h}),\label{eq:asym-channel-mean}\\
+ \E_{\Omega_{l,q,n}}\chi_v\chi_w
+   &=2^{-2h(g-1)}+O_g(2^{-N+4h}).
+   \label{eq:asym-channel-correlation}
+\end{align}
+For independent frames of law \(\nu\le L\pi\), these rank
+conditions fail with probability \(o(1)\).
+\end{lemma}
+
+\begin{proof}
+Before imposing \(x\cdot y=q\) in one component, the four
+channel evaluation vectors
+\[
+ (Y_v\trans x,Y_w\trans x,X_v\trans y,X_w\trans y)
+\]
+are uniform on \(\F^{4h}\).  Conditional on their values,
+\(x,y\) are independent uniforms on affine subspaces of
+codimension \(2h\).  The pairing between their translation
+spaces has rank at least \(N-4h\).  For a bilinear form of
+rank \(R\), its sign average on an affine product is either
+zero or has absolute value \(2^{-R}\): average first in one
+variable, leaving at most a \(2^{-R}\) fraction of the other
+variable.  Thus every fixed evaluation tuple has diagonal
+probability \(\frac12+O(2^{-N+4h})\).  Conditioning on the
+diagonal changes its uniform law by \(O(2^{-N+4h})\) in total
+variation.  Deleting zero key vectors incurs an additional
+\(O(2^{-N})\) error.
+
+For independent uniform bits \(a,b\),
+\(\E(-1)^{ab}=1/2\).  There are \(h\) such products for one
+frame and \(2h\) independent products for the pair of frames
+in each of the \(g-1\) components.  Independence across
+components proves the two formulas.
+
+Under \(\rawlaw^2\), two independent channel frames have
+concatenation rank less than \(2h\) with probability at most
+\(O_g(2^{2h-N})\), by exposing their uniform injective columns.
+The domination
+\(\nu^2\le L^2 2^{\varepsilon_{\rm ch}N}\rawlaw^2\)
+still makes this probability exponentially small, since
+\(\varepsilon_{\rm ch}<1/2\).  This proves the last assertion.
+\end{proof}
+
+\begin{lemma}[Absence of a point-side prediction]
+\label{lem:asym-no-point-prediction}
+In the common limit of the independent two-point law, there is
+no positive mass on which, at either endpoint, a prediction
+\begin{equation}
+\label{eq:asym-point-prediction}
+ p_{B,j}=\alpha h_{B,j}+f(l,q,\xi)
+\end{equation}
+holds for a fixed measurable common-signature function \(f\),
+a fixed bit \(\alpha\), and all surviving unary types after
+bounded label exclusions.  The assertion holds for every fixed
+positive mass, however small, and after deletion of internal holes.
+\end{lemma}
+
+\begin{proof}
+Suppose the limiting mass is positive.  Approximate \(f\), on
+each of the finitely many tag/diagonal reference spaces, by binary
+cylinder functions.  For a fixed approximation, the maximum
+prediction error over the surviving unary types is a continuous
+function of the finitely many stored status-cylinder coordinates.
+Minimizing over the finite allowed exclusion choices preserves
+continuity.  The strict-error transfer in the proof of
+\Cref{lem:status-global-prediction} therefore gives a subsequence
+and global key functions \(f_n\) for which the finite prediction
+error tends to zero on a set of pairs of mass at least some
+\(\delta>0\).  The functions are fixed before sampling these
+pairs.  We use no supremum over functions chosen by a sampled
+orientation.
+
+There is a fixed surviving generic label on positive pair mass:
+average over the finite label set and retain one of its positive
+terms.  Fix also a tag, a feasible diagonal, and a feasible generic
+tester summary \(\tau\).  This summary has a fixed positive
+nominal probability and fixes the role \(a(\tau)\).  Conditioning
+the vanishing error on it still leaves vanishing error.  Absorb
+the term \(\alpha a(\tau)\) into the global binary test
+\(\phi_n(k)\).  Write the two points in the relevant order as
+\(U,V\), so that \(p_{B,j}\) evaluates the channel form of
+\(V\) on the primal key supplied by \(U\).  Set
+\[
+ \begin{split}
+ D_n(U,V)&=\PP_z\{u_V(K_U(z))\ne\phi_n(K_U(z))
+                       \mid \tau\},\\
+ d_n(V)&=\PP_{k\sim\Omega_{l,q,n}}
+                         \{u_V(k)\ne\phi_n(k)\}.
+ \end{split}
+\]
+For each fixed \(V\), the one-endpoint assertion of
+\Cref{lem:typicality} applies with that entire frame as external
+data.  Its exceptional set depends on this prescribed test, and
+its bound is uniform in the external data.  The raw mean is
+\(d_n(V)+o(1)\) by the single-key frame law.  The marginal
+of the primal frame of \(U\) is at most \(L\) times its
+\(\rawlaw\)-marginal.  Consequently, for every fixed
+\(t>0\),
+\[
+ \PP_{U,V\sim\nu}\{|D_n(U,V)-d_n(V)|>t\}=o(1).
+\]
+For example the variance proof bounds the left side, apart from
+the negligible reference error, by \(LC2^{-cn}/t^2\).
+Choose \(t_n\downarrow0\) slowly enough for the exceptional
+probability still to vanish.  If \(D_n\le\epsilon_n\to0\)
+on the successful pair set, its projection to the \(V\)-coordinate
+therefore yields
+\[
+ G_n=\{v:d_n(v)\le\epsilon_n+t_n\},\qquad
+ \nu(G_n)\ge\delta-o(1).
+\]
+Any adaptive choice of the successful pair set is harmless here:
+the typicality exception was bounded on the independent product
+law before restricting it.
+
+Take two independent frames \(V,W\) from \(\nu\) conditioned
+on \(G_n\).  The probability of channel rank failure is at most
+its unconditioned \(o(1)\) probability divided by
+\(\nu(G_n)^2\), and hence still tends to zero.  On a pair
+with full ranks, agreement of both channel signs with the same
+test gives
+\[
+ \E_{\Omega_{l,q,n}}\chi_V\chi_W
+ \ge 1-4(\epsilon_n+t_n).
+\]
+This contradicts \eqref{eq:asym-channel-correlation}, whose
+right side tends to \(2^{-2h(g-1)}<1\).  This use of two
+successful frames is essential: it introduces no required lower
+bound on \(\delta\) besides positivity.  Finally,
+\eqref{eq:asym-internal-hole} changes all bounded error or event
+probabilities by \(o(1)\), so deleting internal holes cannot
+create the asserted limiting prediction.
+\end{proof}
+
+\subsection{A singleton phase estimate}
+
+The point side needs only one phase calculation.  In the following
+lemma the marks on \(A\) belong to its first effective spaces,
+and \(\Delta_A=U_{A,1}\lambda_{A,1}+U_{A,2}\lambda_{A,2}\).
+
+\begin{lemma}[One point, then two independent points]
+\label{lem:asym-singleton-phase}
+Let a marked unit law have a fixed marginal bound relative to
+\(\pi\) and an absolute joint bound \(2^{(D+o(1))N}\)
+relative to \(\rawlaw^2\).  Suppose \(\Delta_A\ne0\)
+throughout, with its component rank bounded by \(r\), and fix
+\(c\in\F\).  One can retain a fixed positive fraction of
+the unit law such that, for an independent point \(Z\sim\nu\),
+\[
+ \PP\{u_Z(\Delta_A)=c\text{ and }R(A,Z)\}>.49.
+\]
+Here \(R\) is either the whole space or the transpose-rank
+condition on the offset list used below.  The retained fraction
+can be fixed before \(h\); the sufficiently large choice of
+\(h\) may depend on the fixed marginal constants.  With two
+independent points, under this same retained unit law,
+\begin{equation}
+\label{eq:asym-two-phase-bits}
+ \PP\{u_{B_1}(\Delta_A)=u_{B_2}(\Delta_A)=c,
+                  \ R(A,B_1),R(A,B_2)\}>.49^2.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Use the cover threshold \(p_0\), even moment \(s\), and
+rank-growth constants in \Cref{lem:no-cover-rank} and the proof
+of \Cref{thm:phase-alternative}.  Explicitly, \(s\) is the
+largest power of two at most \(c_sN\), where
+\(c_s=10^{-12}(1+r)^{-4}\), so \(s\ge c_sN/2\) for large
+\(n\).  Put \(\theta=10^{-4}\) and
+\(\Gamma=D+2r+10\).  If no cover of the specified
+size has unit mass at least \(p_0\), expand the moment against
+independent uniform channel columns of the one point:
+\[
+ \E_{\rm iid\ channels}
+     \left|\E_A(-1)^{u_Z(\Delta_A)}\right|^s
+ \le \E_{A_1,\ldots,A_s}
+       2^{-h\rank(\Delta_{A_1}+\cdots+\Delta_{A_s})}
+ \le 4^s p_0^{q_0s/2}+2^{-hq_0s/4}.
+\]
+This is \eqref{eq:phase-moment} with no reciprocal phase and no
+candidate value of \(\Delta_Z\).  The inequalities
+\eqref{eq:early-p0} and \eqref{eq:early-h-moment}, followed
+by Markov's inequality at threshold \(\theta\), bound the
+raw channel exception by \(C_h2^{-\Gamma N}\).  The channel
+comparison in \Cref{lem:frame-images} is included in the fixed
+factor \(C_h\).  Paying \(L2^{\varepsilon_{\rm ch}N}\)
+once leaves an exponentially small point-law exception.  Thus
+\(\left|\E_{A,Z}(-1)^{u_Z(\Delta_A)}\right|<.01\)
+for large \(n\).  Take \(R\) to be the whole space; each
+value of the bit has probability greater than \(.495\).
+
+Otherwise retain the \(p_0\) mass in a fixed cover and delete
+the negligible set whose individual primal spans meet that
+cover.  The deletion uses the unchanged primal marginal.  The
+lifting identity \eqref{eq:cover-lifting} gives offset records
+with at most \(2^{.002N}\) possibilities and offset basis size
+\(1\le t\le2r\).  At one fixed record, let \(Z_A\) be its
+companion tuple.  Its unnormalized subprobability law satisfies
+\[
+ p_{\max}(Z_A)\le2^{-tN+.01N}
+\]
+by \eqref{eq:companion-pmax}.  For that record let \(R(A,Z)\)
+require the channel transposes of \(Z\) to have full rank on
+the plus and minus offset lists.  The transpose estimate in
+\Cref{lem:channel-thinning}, multiplied by the fixed point
+marginal factor, gives
+\[
+ \PP\{R(A,Z)\text{ fails}\}
+ \le L\,O(r)2^{2r-h}+o(1)<.01
+\]
+after increasing \(h\).  We may, and do, make this bound less
+than \(.005\).
+
+On the retained rank event the output tuple
+\(\Phi_{u_Z}(\omega_A)\) has point masses at most
+\(2^{-tN+.01N}\).  To check this directly, condition on the
+point's primal frame and freeze the bounded lists of transpose
+coefficients on the offsets.  Each list has full rank.  Ignoring
+the equations defining those coefficients, their prescribed
+images under the opposite channel maps cost
+\(t(N-\dim\cB)\) bits.  The number of coefficient lists is
+fixed in \(n\).  The density loss from thinning and the ratio
+\(\dim\cB/N\) are absorbed into \(.01N\).  This is the
+singleton image calculation of \eqref{eq:singleton-joint-pmax}
+with no point-side companion tuple.
+
+The phase is \(Z_A\cdot\Phi_{u_Z}(\omega_A)\), plus a
+function of the point alone at this fixed record.  The Walsh
+bound \Cref{lem:walsh} therefore bounds its unnormalized signed
+mass, including the rank filter, by
+\[
+ 2^{tN/2}
+ \left(2^{-tN+.01N}2^{-tN+.01N}\right)^{1/2}
+ =2^{(-t/2+.01)N}.
+\]
+Summing over records gives at most \(2^{-.488N}\).  Since
+\(\PP(R)> .995-o(1)\), the mass where \(R\) holds and
+the phase bit has either prescribed value exceeds \(.49\).
+This proves the singleton assertion.
+
+For the last assertion put
+\[
+ f(A)=\PP_{Z\sim\nu}\{u_Z(\Delta_A)=c, R(A,Z)\mid A\}.
+\]
+The two points are conditionally independent given \(A\), with
+the identical separate rank filter, so the left side of
+\eqref{eq:asym-two-phase-bits} is
+\[
+ \E_A f(A)^2\ \ge\ (\E_A f(A))^2\ >\ .49^2.
+\]
+This argument asserts an averaged singleton estimate; it does
+not require a pointwise estimate for each unit.  Internal holes
+have not yet been deleted.
+\end{proof}
+
+\subsection{The mixed dual relations and the conflict bound}
+
+\begin{proposition}[The asymmetric distribution test]
+\label{prop:asymmetric-distribution}
+For every fixed \(L\), with the late parameters chosen as above,
+all sufficiently large \(n\) and all laws satisfying
+\eqref{eq:asym-caps} obey
+\[
+ \PP_{A\sim\sigma,\ Z\sim\nu}
+       \{A\text{ and }Z\text{ conflict}\}\ge2^{-100gN}.
+\]
+Thus \Cref{thm:distribution-tests}\textup{(ii)} holds.
+\end{proposition}
+
+\begin{proof}
+Suppose a violating sequence exists.  Peel the \(A\) law once
+by \Cref{lem:peeling}\textup{(a)}; retain the actual leaf weights
+and the resulting first effective spaces.  After making all
+restrictions on this law, we will apply part \textup{(b)} once
+to restore the required normalized-leaf image bounds without
+adding pins.  The margins between \(3000g\) and
+\(D=4000g\) absorb thinning and fixed normalizations.
+Throughout, the mixed marginal bounds used in
+\Cref{sec:histograms,sec:common-limit} concern the primal frames.
+
+\paragraph{Preparing the unit and the point pair.}
+Make the following subsequential choice on the \(A\) law.
+If an empty prediction exists on mass at least \(10^{-6}\),
+retain it.  Here empty means \(\alpha_1=\alpha_2=1\) and
+the offsets are \(\gamma_1q,\gamma_2q\) with fixed bits
+\(\gamma_i\), modulo errors tending to zero in the reference
+spaces.  \Cref{lem:empty-status} exactifies it, after negligible
+deletion, to
+\begin{equation}
+\label{eq:asym-exact-A}
+ p_{A,i}=T(\lambda_{A,i},\cdot)+a
+                  \quad\hbox{on all of }\cX,\qquad i=1,2.
+\end{equation}
+In particular its proof eliminates the multiples of \(q\),
+using the odd number of star atoms.  Fix
+\(c=a(\lambda_{A,1})+a(\lambda_{A,2})\) and the indicator
+of \(\Delta_A=0\), losing a factor at most four.  If no
+such empty prediction exists, retain the first law without
+prediction preparation.  Only this specialized empty prediction
+is tested; arbitrary common predictions need not be excluded.
+
+In the predicted branch with \(\Delta_A\ne0\), apply
+\Cref{lem:asym-singleton-phase}.  In the branch with
+\(\Delta_A=0\), necessarily \(c=0\).  Indeed the identities
+\eqref{eq:asym-exact-A} give
+\((u_{A,1}+u_{A,2})U_{A,i}=T(\lambda_{A,i},\cdot)+a\),
+since \(u_{A,i}U_{A,i}=0\); if \(c=1\), these profiles and
+\(\Delta_A=0\) satisfy all hole equations for the unit \(A\),
+a contradiction.  Thus in either predicted branch the independent
+experiment has fixed positive mass on
+\begin{equation}
+\label{eq:asym-compatible}
+ u_{B,j}(\Delta_A)=c\qquad(j=1,2).
+\end{equation}
+In the nonzero case this mass exceeds \(.49^2\) under the
+normalized retained \(A\) law.  Before constructing either
+common model, apply \Cref{lem:peeling}\textup{(b)} to the
+combined restriction made on \(A\).  More explicitly, let
+\(w_\lambda\) be the original first-leaf weights and let
+\(q_\lambda\) be the fraction retained in leaf \(\lambda\)
+by all the \(A\)-only restrictions just made.  Their total
+mass \(q=\sum_\lambda w_\lambda q_\lambda\) is bounded below
+by a fixed positive constant.  Discard the leaves with
+\(q_\lambda<2^{-\zeta N}\).  They have mass at most
+\(2^{-\zeta N}/q=o(1)\) in the restricted law; every remaining
+normalized leaf has the image bound
+\eqref{eq:final-leaf-minentropy} and an absolute density
+\(2^{O(N)}\) relative to \(\rawlaw^2\).  The pins and first
+effective spaces have not changed, and the actual remaining
+leaf weights are retained.  This operation depends only on
+\(A\), so the two points remain independent conditional on
+\(A\).  On the predicted branch the two-bit compatibility mass
+is still at least \(.49^2-o(1)\).
+
+Now delete internal \(B\) holes, costing \(o(1)\) by
+\eqref{eq:asym-internal-hole}, and take the single unpinned
+\(B\) leaf from \Cref{cor:independent-point-leaf}.
+Apply the independent-law version of \Cref{lem:injection} to
+these two laws, with failure probability less than \(.01\).
+On the predicted branch compatibility and injection therefore
+coexist on mass at least \(.49^2-.01-o(1)>.22\), for large
+\(n\).  On the branch without prediction a passed injecting
+actual table has mass at least \(.99\).  These are subtractions
+of exceptional mass, and require no independence between the
+exceptional events and compatibility.
+
+\paragraph{Separate flags for candidate tables.}
+We record this mass in the permitted finite-table format.
+For each leaf pair enumerate all numerical candidate small
+tables \(T\), as in \Cref{def:limit-option}; there are finitely
+many, with a bound independent of \(n\).  A fixed \(T\) has
+separate admissibility flags \(F_{A,T}(A)\) and \(F_{B,T}(B)\)
+by \Cref{lem:small-tables}.  In the predicted branch define
+\begin{equation}
+\label{eq:asym-table-flag}
+ H_{A,T}(A)=\1\left\{
+   \sum_{i=1}^2(u_{B,j}U_{A,i})^*_T(\lambda_{A,i})=c
+       \quad(j=1,2)\right\}.
+\end{equation}
+The star denotes contraction in the numerical table \(T\),
+as in \eqref{eq:status-table-delta}.  It is calculated from
+the entries of \(T\), the recorded tensor coordinates of the
+\(C_{A,i}\) bases, and the bounded coordinate vectors of the
+marks \(\lambda_{A,i}\).  Thus, for a fixed leaf pair and
+\(T\), \(H_{A,T}\) is a flag of \(A\) alone.  Include the
+mark coordinates among the finite orientation metadata and
+replace \(F_{A,T}\) by \(F_{A,T}H_{A,T}\).  In the branch
+without prediction set \(H_{A,T}=1\).  Injection is a finite
+matrix-rank test on the candidate table and the leaf metadata.
+
+Each pair counted by the ambient compatibility and injection
+estimate has at least one passed candidate: take \(T\) to be
+its actual small table.  Therefore the union of these permitted
+candidate-table events has at least the mass just established.
+We store their entire finite flag vector in the common model.
+This argument does not impose equality of a candidate table
+with the actual cross entries between private directions;
+such an equality would be a joint condition.  The actual table
+is used only to witness the existence of a passed candidate.
+Nor do we condition the two unit laws on the ambient phase
+event \eqref{eq:asym-compatible}.  They remain independent
+laws with the separate flags just described.
+
+\paragraph{Solving the mixed scalar system.}
+Construct the common model for these two different unit laws,
+retaining the ordered \(A\)--\(B\) type filter.  The dual
+relations \eqref{eq:status-dual-relation} become
+\begin{equation}
+\label{eq:asym-dual}
+ s_j(p_{A,i}+h_{A,i})+r_i(p_{B,j}+h_{B,j})
+ =\mathbf c_{ij}\cdot\mathbf x_{A,i}
+       +\beta_{ij}(h_{A,i}+h_{B,j})+\gamma_{ij}q,
+\end{equation}
+because \(C_{B,j}=0\).  We first show that no relation with
+some \(r_i=1\) can occur on positive model mass, for any of
+the permitted bounded exclusions.  If it did, Fubini fixes an
+\(A\) array with a positive-mass set of such \(B\) arrays.
+Pigeonhole its finite coefficient and exclusion data and fix a
+surviving generic label on the \(A\) side.  At a fixed common
+signature, the two status distributions are independent.  The
+two expressions in \eqref{eq:asym-dual} must agree almost
+surely, because a mismatching positive-density pair would be an
+available generator violating the relation.  Two independent
+bits which agree almost surely are both deterministic.  The
+generic \(A\) type has total status density one on every
+signature space, so its deterministic value gives a fixed
+common function on all surviving \(B\) types.  Solving
+\eqref{eq:asym-dual} for \(p_{B,j}\) gives
+\[
+ p_{B,j}=(1+\beta_{ij})h_{B,j}+f(l,q,\xi).
+\]
+The strict-error cylinder transfer used in
+\Cref{lem:asym-no-point-prediction} applies, and that lemma
+rules this out.  The finite number of possible dual and
+exclusion records implies that, almost surely, every remaining
+dual relation has \(r_1=r_2=0\).
+
+In the branch without an empty prediction, suppose obstructable
+pairs had probability at least \(.1\).  A nonzero weight now
+has some \(s_j=1\).  Generic positivity of the two possible
+\(B\)-role values in \eqref{eq:asym-dual} forces
+\(\beta_{ij}=0\).  The same relation, for both \(i\), is
+then
+\begin{equation}
+\label{eq:asym-empty-extraction}
+ p_{A,i}=T(\lambda_{A,i},\mathrm{atom})
+                         +h_{A,i}+\gamma_{ij}q.
+\end{equation}
+It holds on all surviving unary types, by comparison to a fixed
+generic \(B\) type of total density one.  These are empty
+predictions.  There are only two choices of \(j\) and four
+offset pairs \((\gamma_{1j}q,\gamma_{2j}q)\); therefore
+one choice occurs on \(A\) mass at least \(.1/8\).
+The marked profiles and exclusions may still vary with \(A\),
+as allowed in \Cref{def:prediction}; their number is not a
+further pigeonhole cost.  Cylinder transfer with strict error
+events, as in \Cref{lem:status-global-prediction}, loses at
+most a factor two in this mass and supplies a finite empty
+prediction of mass much larger than \(10^{-6}\).  This is a
+contradiction.  Thus obstructable mass is less than \(.1\),
+whereas passed injecting tables have mass at least \(.99\).
+Their intersection with nonobstructable pairs has mass greater
+than \(.89\).  On it \Cref{lem:status-duality} solves the
+scalar system for every such table and every allowed exclusion.
+
+On the predicted branch, substitute \eqref{eq:asym-exact-A}
+into \eqref{eq:asym-dual}.  The full generic base projection
+forces
+\[
+ \mathbf c_{ij}=s_j\boldsymbol\lambda_{A,i},
+ \qquad \beta_{ij}=0,\qquad\gamma_{ij}=0.
+\]
+For \(s_j=0\) all these coefficients are zero.  Fix any
+passed injecting candidate table \(T\), including the flag
+\(H_{A,T}=1\).  At its local targets the dual evaluation is
+\[
+ \sum_{i,j}\mathbf c_{ij}\cdot\mathbf t_{A,i;j}
+ =\sum_j s_j\left(
+       \sum_i a(\lambda_{A,i})
+       +\sum_i(u_{B,j}U_{A,i})^*_T(\lambda_{A,i})\right)=0
+\]
+by \eqref{eq:asym-table-flag} and
+\(\sum_i a(\lambda_{A,i})=c\).  This calculation uses the
+selected candidate's starred contractions throughout; it does
+not require that candidate to equal the actual cross table of
+the sampled pair.  Every dual relation vanishes on its target.
+By \Cref{lem:status-duality}, the scalar system is soluble on
+the positive mass having a passed compatible injecting
+candidate.  The reasoning uses only surviving generic labels
+and hence persists through the further bounded exclusions for
+fresh lists.
+
+\paragraph{Positive overlap and collision.}
+In either branch we now have the hypothesis of the mixed,
+filtered \Cref{prop:positive-overlap}.  Equivalently,
+\Cref{lem:fresh-lists} supplies at most seven generators per
+cross edge, with distinct labels at each endpoint and positive
+status overlap at every matched position.  The scalar recipe
+and gradient realization produce a usable numerical option;
+the filtered zero-product conclusion of
+\Cref{prop:limit-overlap} cannot hold on this positive mass.
+Consequently some option has a positive subsequential expected
+unnormalized overlap.  This positive constant need not be
+uniform over all possible limit models.
+
+The collision criterion \Cref{prop:collision} therefore gives
+four cross holes between \(A\) and \(B\) with probability
+\[
+ 2^{-(k_{\max}+.03+o(1))N}.
+\]
+All retained-law normalizations have fixed positive mass and
+are included in the \(o(1)N\) loss.  In particular these four
+holes imply that \(A\) conflicts with \(B_1\).  Viewed in
+the original independent experiment, the latter has exactly
+the probability of a conflict with one point of law \(\nu\).
+Since
+\[
+ k_{\max}+.03=56(g-1)+.03<100g,
+\]
+this contradicts the assumed bound below \(2^{-100gN}\) for
+all sufficiently large indices of the violating sequence.
+The asserted uniform eventual distribution test follows.
+\end{proof}
diff --git a/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/13-graph-transfer.tex b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/13-graph-transfer.tex
new file mode 100644
index 0000000000000000000000000000000000000000..6e9ec80266aca2f26b351b989feeb9b7ddfc7b60
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@@ -0,0 +1,374 @@
+\section{From distribution tests to finite graphs}\label{sec:graph-transfer}
+
+We now prove \Cref{thm:base-graphs}.  The only probabilistic input to
+this section is \Cref{thm:distribution-tests}, whose two parts were
+proved in \Cref{prop:coloured-distribution,prop:asymmetric-distribution}.
+The sample uses the thinned law \(\pi\).  Write
+\begin{equation}\label{eq:transfer-scales}
+ Q_N=2^{3000gN},\qquad \epsilon_N=2^{-100gN},\qquad
+ m=2^{1000gN},\qquad \Omega=\supp\pi.
+\end{equation}
+All spaces in this section are finite.  In particular, \(\pi\) is
+strictly positive on \(\Omega\), and
+\(\log|\Omega|=O(N^2)\): specifying all frame entries uses
+\(O(N^2)\) bits, and restricting their support by thinning only
+decreases its size.  We regard the fixed construction parameters as
+constants in this estimate.
+
+\subsection{Entropy and terminal covers}
+
+For probabilities \(p,q\) on a finite space with \(q>0\), define
+\[
+ \mathsf D(p\Vert q)=\sum_xp(x)\log\frac{p(x)}{q(x)},
+ \qquad 0\log0=0.
+\]
+Logarithms in entropy expressions are natural logarithms.
+
+This is the finite convex information-projection inequality underlying
+the entropy method of \citet{csiszar1975}.
+
+\begin{lemma}[Entropy increase after a deletion]\label{gt:entropy}
+Let \(\mathcal P\) be a nonempty compact convex family of probabilities
+and let \(p\) minimize \(\mathsf D(\cdot\Vert q)\) on \(\mathcal P\).
+Then \(p\) is positive on the union of the feasible supports.  For
+every \(p'\in\mathcal P\) supported on \(S\),
+\[
+ \mathsf D(p'\Vert q)-\mathsf D(p\Vert q)
+ \ge \mathsf D(p'\Vert p)\ge-\log p(S).
+\]
+\end{lemma}
+\begin{proof}
+The minimum exists by continuity.  If \(p(x)=0<p'(x)\), the
+right derivative of entropy along \((1-t)p+tp'\) is \(-\infty\)
+at zero; all terms on the support of \(p\) have finite derivative.
+This contradicts minimality.  Differentiating now on the common
+feasible support gives
+\[
+ \sum_x(p'(x)-p(x))\log\frac{p(x)}{q(x)}\ge0.
+\]
+Subtracting the two entropies proves the first inequality.  If
+\(p'\) is supported on \(S\), then
+\[
+ \mathsf D(p'\Vert p)
+ =\mathsf D(p'\Vert p(\cdot\mid S))-\log p(S).
+\]
+Relative entropy is nonnegative: apply \(\log t\le t-1\) with
+\(t=q(x)/p'(x)\) and sum on the positive support of \(p'\).
+This proves the second inequality as well.
+\end{proof}
+
+We require a terminal cover for coloured units that accounts for
+four different capacities.  The following elementary separation
+argument supplies the finite linear programming duality needed here.
+
+\begin{lemma}[Resource prices]\label{gt:prices}
+Let \(\mathcal R\) be a nonempty finite collection of objects, each
+having a nonnegative load vector \(v_r\in\R^d\), and let
+\(c\in\R_{\ge0}^d\) be the vector of capacities.  If no probability
+\((p_r)_{r\in\mathcal R}\) has \(\sum_rp_rv_r\le c\)
+coordinatewise, there are nonnegative prices \(w\in\R^d\) with
+\[
+ w\cdot c<1,\qquad w\cdot v_r\ge1\quad(r\in\mathcal R).
+\]
+\end{lemma}
+\begin{proof}
+The compact convex sets
+\(C=\operatorname{conv}\{v_r:r\in\mathcal R\}\) and
+\(B=\prod_{j=1}^d[0,c_j]\) are disjoint.  Choose a closest pair
+\(z\in C,b\in B\), and put \(u=z-b\ne0\).  For fixed \(z\),
+the nearest point of \(B\) is its coordinatewise truncation, so
+\(b_j=\min(z_j,c_j)\), \(u\ge0\), and \(u\cdot b=u\cdot c\).
+For every \(v\in C\), comparison with \((1-t)z+tv\) at
+\(t=0\) gives \(u\cdot(v-z)\ge0\).  Consequently
+\[
+ u\cdot v\ge u\cdot z
+   =u\cdot c+\|u\|^2>u\cdot c\ge0.
+\]
+Take \(w=u/(u\cdot z)\).  This proves both assertions using
+only minimization on compact subsets of a Euclidean space.
+\end{proof}
+
+\begin{lemma}[The coloured terminal cover]\label{gt:coloured-cut}
+Fix a palette \([q]=\{1,\ldots,q\}\).  Suppose that a set
+\(R\subseteq\Omega^2\times[q]\) supports no probability \(\sigma\)
+whose two endpoint marginals are at most \(M\pi\), whose projection
+onto \(\Omega^2\) is at most \(Q_N\pi^2\), and whose mass on
+each colour is at most \(\eta_{\mathrm{col}}\).
+There are sets \(S\subseteq\Omega\), \(E\subseteq\Omega^2\)
+and \(C\subseteq[q]\) such that
+\begin{equation}\label{eq:coloured-terminal-cover}
+ \pi(S)<\frac8M,\qquad
+ \pi^2(E)<\frac4{Q_N},\qquad
+ |C|<\frac4{\eta_{\mathrm{col}}},
+\end{equation}
+and each \((x,y,c)\in R\) has \(x\in S\), \(y\in S\),
+\((x,y)\in E\), or \(c\in C\).
+\end{lemma}
+\begin{proof}
+The assertion is immediate for empty \(R\).  Otherwise introduce
+separate resources for each left vertex, right vertex, ordered pair,
+and colour, with capacities
+\[
+ M\pi(x),\quad M\pi(y),\quad Q_N\pi(x)\pi(y),\quad
+ \eta_{\mathrm{col}},
+\]
+respectively.  An object \((x,y,c)\) uses one unit of each of its
+four resources.  Apply \Cref{gt:prices}.  Write the resulting prices
+as \(\ell_x,r_y,e_{xy},d_c\); their capacity-weighted sum is less
+than one, and
+\(\ell_x+r_y+e_{xy}+d_c\ge1\) on \(R\).
+Take all resources of price at least \(1/4\).  They cover \(R\).
+Let \(S\) be the union of the high-price left and right vertex
+sets.  In fact
+\[
+ \pi(S)\le4\sum_x\pi(x)(\ell_x+r_x)<4/M<8/M.
+\]
+The corresponding pair and colour inequalities are
+\(\pi^2(E)<4/Q_N\) and \(|C|<4/\eta_{\mathrm{col}}\).
+This uses the capacity on the pair projection, so all colours of
+one pair share a single pair resource.
+\end{proof}
+
+The next cover is a capacitated max-flow/min-cut argument
+\citep{fordfulkerson1956}; we include the finite proof for the real
+capacities used here.
+
+\begin{lemma}[The uncoloured terminal cover]\label{gt:flow-cut}
+If \(R\subseteq\Omega^2\) supports no probability with both
+marginals at most \(L\pi\) and joint law at most \(Q_N\pi^2\),
+then there are \(S\subseteq\Omega\) and \(E\subseteq\Omega^2\)
+with
+\begin{equation}\label{eq:uncoloured-terminal-cover}
+ \pi(S)<1/L,\qquad \pi^2(E)<1/Q_N,
+\end{equation}
+such that each pair in \(R\) meets \(S\) or belongs to \(E\).
+A set \(T\subseteq\Omega\) supports no probability at most
+\(L\pi\) precisely when \(\pi(T)<1/L\).
+\end{lemma}
+\begin{proof}
+Make a directed network from a source to a left copy of \(\Omega\),
+then to a right copy, then to a sink.  The source and sink arcs
+have capacities \(L\pi(x)\) and \(L\pi(y)\).  The middle arc
+\(x\to y\), present for \((x,y)\in R\), has capacity
+\(Q_N\pi(x)\pi(y)\).  A flow of value one is exactly a feasible
+unit law, and a larger flow can be scaled to value one.
+
+A maximum flow exists by compactness.  Its value is less than one.
+In its residual network there is no source-to-sink path, since
+augmenting by the smallest residual capacity on such a path would
+increase the value.  Let \(Z\) be the vertices reachable from the
+source.  Every original arc leaving \(Z\) is saturated; every
+original arc entering \(Z\) carries zero flow.  Summing flow
+conservation over \(Z\) therefore shows that its cut capacity is
+the maximum flow value, hence is less than one.
+
+Let \(A\) be the left types outside \(Z\), and \(B\) the right
+types inside \(Z\).  The cut inequality reads
+\[
+ L\pi(A)+L\pi(B)
+ +Q_N\pi^2\bigl(R\cap((\Omega\setminus A)
+                         \times(\Omega\setminus B))\bigr)<1.
+\]
+Take \(S=A\cup B\) and take \(E\) to be the pair set in the
+last summand.  This proves the cover and both bounds.  For the
+point assertion, a law of total mass one on \(T\) bounded by
+\(L\pi\) exists exactly when \(L\pi(T)\ge1\): necessity follows
+by summing, and sufficiency follows by normalizing \(\pi|_T\).
+\end{proof}
+
+\subsection{Finite fingerprints}
+
+Call two coloured units adjacent when their colours differ and
+their four cross pairs are holes.  This is a finite simple graph:
+a unit cannot conflict with itself, because a vertex has no hole
+to itself.  For a fixed palette, consider its independent sets.
+Also consider the bipartite graph with raw units on one side and
+points of \(\Omega\) on the other, with adjacency meaning an
+edge--point conflict.  In this graph the relevant objects are pairs
+\((I,J)\) having no adjacency between \(I\) and \(J\).
+
+The recording-and-replay construction follows the graph fingerprint
+method of Kleitman and Winston \citep{kleitmanwinston1982}; see also
+\citet{samotij2015}. Its potential here is the constrained entropy
+minimum, whose increase makes the terminal count small.
+
+\begin{lemma}[Fingerprint bound]\label{gt:fingerprints}
+There are families of terminal sets, respectively terminal pairs
+of sets, containing every such independent set, respectively every
+such pair \((I,J)\), with the following properties.  Each coloured
+terminal has an infeasible coloured region as in
+\Cref{gt:coloured-cut}; in each bipartite terminal at least one of
+the two feasible regions in \Cref{gt:flow-cut} is empty.
+The number of terminals of either kind is
+\begin{equation}\label{eq:transfer-terminal-count}
+ \exp\bigl(O(N^3 2^{100gN})\bigr)=\exp(o(m)).
+\end{equation}
+The constants may depend on the fixed palette and on \(L\).
+\end{lemma}
+\begin{proof}
+Start with the entire relevant graph as residual set.  On a
+nonempty feasible region choose the entropy minimizer, breaking
+all choices deterministically.  For coloured units the reference
+probability is \(\pi^2\) times the uniform law on \([q]\).
+The projected pair cap gives density at most \(qQ_N\) against
+this reference.  The minimum entropy is therefore in
+\([0,\log(qQ_N)]\).
+By \Cref{thm:distribution-tests}(i), the average minimizer mass
+of a residual neighbourhood is at least \(\epsilon_N\).
+Select a vertex of maximal neighbourhood mass.  If it belongs
+to the independent set being contained, record it and delete its
+neighbours.  Otherwise delete only that vertex.  These operations
+preserve containment of the target independent set.
+
+For the bipartite problem, choose separate entropy minimizers on
+the two residual sides, against \(\pi^2\) and \(\pi\).
+Their sum lies in \([0,\log Q_N+\log L]\).
+By \Cref{thm:distribution-tests}(ii), some vertex on either side
+has opposite minimizer neighbourhood mass at least
+\(\epsilon_N\).  Choose deterministically a vertex of maximal
+such mass, including its side in the choice.  If it belongs to
+the corresponding target set, record it and delete its neighbours
+on the opposite side; otherwise delete only the chosen vertex.
+The absence of edges between \(I\) and \(J\) preserves both
+containments.
+
+In either procedure every step removes a vertex: a recorded
+vertex has a nonempty neighbourhood.  Stop when feasibility
+fails (on either side in the bipartite case).  Deleting vertices
+can only increase each minimum entropy.  If a recording leaves
+the relevant feasible region nonempty, \Cref{gt:entropy} increases
+its minimum by at least
+\(-\log(1-\epsilon_N)\ge\epsilon_N\).
+Consequently the number of recordings is at most
+\begin{equation}\label{eq:transfer-fingerprint-length}
+ 1+\left\lceil
+ \frac{\max\{\log(qQ_N),\log Q_N+\log L\}}
+      {\epsilon_N}\right\rceil
+ =O(N2^{100gN}).
+\end{equation}
+The extra one permits a last recording that destroys feasibility.
+
+After recording a vertex, its residual neighbourhood is empty
+forever.  Since a selected vertex always has positive neighbourhood
+mass, it cannot be selected again.  The set of recorded vertices,
+with side information in the bipartite case, thus determines the
+entire procedure: replay its deterministic choices, performing a
+recording exactly when the selected vertex is in that set, and
+otherwise deleting it.  In particular it determines the terminal.
+The total number of available vertices has logarithm \(O(N^2)\).
+Bounding the number of fingerprints of length at most
+\eqref{eq:transfer-fingerprint-length} by the corresponding power
+of one plus that number proves \eqref{eq:transfer-terminal-count}.
+\end{proof}
+
+\subsection{One sample satisfies all three properties}
+
+Choose the colour threshold \(\eta_{\mathrm{col}}>0\) from
+\Cref{thm:distribution-tests}(i), and fix
+\begin{equation}\label{eq:transfer-a-L}
+ 0<a<\min\{1/100,\eta_{\mathrm{col}}/800\},\qquad
+ L>10^6/\beta.
+\end{equation}
+Here we may assume \(0<\beta\le1\), since increasing \(\beta\)
+only weakens the required graph properties.  The order is
+\(\eta_{\mathrm{col}},a\), then \(\beta,L\), followed by the
+remaining construction parameters as in \Cref{sec:parameters}.
+Let \(X_1,\ldots,X_m\) be independent samples from \(\pi\).
+On positions \([m]\), put an edge between distinct positions
+exactly when their types have no hole.  Repeated types are allowed.
+By \Cref{lem:hole-triangle-free}, this graph has independence
+number at most two for every sample.
+
+First consider any matching whose edges have been partitioned
+into classes of size at most \(am\).  Merge successive classes
+until a merged class first has size at least \(am\), and repeat,
+leaving a possible final smaller class.  Every merged class has
+size at most \(2am\).  As a matching has at most \(m/2\) edges,
+there are at most
+\[
+ q=\left\lceil\frac1{2a}\right\rceil+1
+\]
+merged classes.  Assign these the fixed palette \([q]\).
+Different merged colours imply different original classes.
+
+For every coloured terminal, fix a cover \((S,E,C)\) from
+\Cref{gt:coloured-cut}.  Put \(k=\lceil m/100\rceil\).
+The probability that at least \(k\) sampled positions lie in
+\(S\) is at most
+\[
+ \binom mk(8/M)^k\le(800e/M)^k=\exp(-\Omega(m)).
+\]
+A fixed collection of \(k\) disjoint ordered position pairs
+gives independent \(\pi^2\)-draws, and is entirely in \(E\)
+with probability less than \((4/Q_N)^k\).  There are at most
+\(m^{2k}\) such collections, so its union bound is
+\begin{equation}\label{eq:transfer-pair-tail}
+ m^{2k}(4/Q_N)^k=2^{-(1000gN-2)k}.
+\end{equation}
+Union over \eqref{eq:transfer-terminal-count}.  With probability
+\(1-\exp(-\Omega(m))\), simultaneously every coloured terminal
+has fewer than \(m/100\) exceptional positions and fewer than
+\(m/100\) disjoint exceptional ordered pairs.  The strict bounds
+use \(k-1<m/100\).
+
+Suppose there were a matching of size at least \(m/20\) with no
+conflict between original classes.  Orient its edges arbitrarily,
+merge its classes as above, and take the set of resulting coloured
+raw types.  This is an independent set in the coloured conflict
+graph, so is contained in a terminal.  Every matching edge is
+covered by an exceptional position, an exceptional ordered pair,
+or an exceptional colour.  The first two covers each account for
+fewer than \(m/100\) edges.  The last accounts for fewer than
+\[
+ \frac4{\eta_{\mathrm{col}}}\,2am
+ <m/100
+\]
+edges.  Their sum is less than \(3m/100<m/20\), a contradiction.
+This proves the first graph property.  Multiple sampled edges
+may have the same raw type; the counting uses their disjoint
+sample positions and does not assume distinct types.
+
+For the bipartite terminals put \(k_\beta=\lceil\beta m/100\rceil\).
+At a terminal with infeasible unit region, fix \((S,E)\) from
+\Cref{gt:flow-cut}.  At a terminal with infeasible point region,
+its entire point region has \(\pi\)-mass less than \(1/L\).
+For every one of these vertex sets, the position tail is at most
+\[
+ \binom m{k_\beta}L^{-k_\beta}
+ \le\left(\frac{100e}{\beta L}\right)^{k_\beta}
+ =\exp(-\Omega_\beta(m)).
+\]
+The exceptional ordered-pair tail is at most
+\[
+ m^{2k_\beta}Q_N^{-k_\beta}=2^{-1000gNk_\beta}.
+\]
+Another union over \eqref{eq:transfer-terminal-count} shows that
+all these sets simultaneously contain fewer than \(\beta m/100\)
+sample positions, and all pair exceptions contain fewer than that
+many disjoint ordered pairs, with probability tending to one.
+
+If a matching and vertex set, each of size at least
+\(\beta m/10\), had no edge--point conflict, their raw type sets
+would give a pair \((I,J)\) contained in a bipartite terminal.
+An infeasible point region cannot contain that vertex set.  An
+infeasible unit region covers fewer than \(2\beta m/100\)
+matching edges, also a contradiction.  We have proved the second
+property even at the stronger threshold \(\beta m/10\).
+There is no disjointness requirement between the original vertex
+set and matching; a conflict itself ensures that its point differs
+from both edge endpoints.
+
+Finally let \(A,B\subseteq[m]\) each have size at least \(\beta m\).
+A maximal matching in the induced graph on \(A\) leaves at most
+two vertices unmatched: the unmatched vertices are independent,
+and the whole graph has independence number at most two.
+Its size is at least \((|A|-2)/2\ge\beta m/10\) for all
+sufficiently large \(m\).  Apply the second property to this
+matching and \(B\).  The resulting edge--point conflict supplies
+a hole from \(A\) to \(B\), proving the third property.
+
+All the required events hold together with probability tending
+to one.  Their intersection therefore contains a finite sample
+for every sufficiently large allowed \(n\); since
+\(m=2^{1000gM_0n}\) tends to infinity, this proves
+\Cref{thm:base-graphs}.
diff --git a/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/14-parameters.tex b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/14-parameters.tex
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+\section{Parameters, ranks, and probability scales}
+\label{sec:parameters}
+
+This Appendix collects the choices made throughout the proof and
+checks their dependencies. All quantities chosen before \(n\) are
+constants. They can be very large; no upper bound on their size is
+needed, and none of these choices depends on an adversarial unit law.
+The fixed mixers may be chosen separately at each sufficiently large
+\(n\), by \Cref{lem:generic-mixers}.
+
+\subsection{The acyclic order of construction}
+
+The early constants in
+\eqref{eq:early-constants}--\eqref{eq:parameter-order} are
+\begin{align}
+ C_0&=1000,& g&=10^9+1,&
+ D&=4C_0g,& M&=2^{1000},\nonumber\\
+ k_{\max}&=56(g-1),&
+ \zeta&=\frac{1}{1000k_{\max}},&
+ K_1&=\left\lceil\frac{4(D+1)}{\zeta}\right\rceil .
+ \label{eq:ledger-early}
+\end{align}
+In particular \(g\) is odd and \(D=4000g\).
+The early rank budgets are
+\begin{align}
+ r&=2K_1+2,&
+ L_0&=\lceil100(D+10)\rceil,&
+ d_0&=4rL_0,\nonumber\\
+ u_0&=10(K_1+d_0+1),&
+ K&\in\N,\qquad
+ K\ge\frac{10(D+K_1+u_0+10)}{\zeta}.
+ \label{eq:ledger-law}
+\end{align}
+These are the bounds in
+\eqref{eq:law-budgets} and \eqref{eq:pin-budgets};
+the construction takes \(K\) to be the least integer meeting
+the displayed lower bound.  Only the first peeling is used.
+The auxiliary number \(u_0\) is retained in the numerical definition
+of a conservative bound \(K\ge K_1\); it does not count another
+operation in the present construction.
+
+The remaining early tolerances in
+\Cref{thm:phase-alternative,lem:injection} include
+\begin{equation}
+ c_s=10^{-12}(1+r)^{-4},\qquad
+ q_0=2^{-100r-10},\qquad
+ \delta_{\mathrm{cov}}=.0005.
+ \label{eq:ledger-cover-tolerances}
+\end{equation}
+For each \(n\), the moment order is the largest power of two
+\(s\le c_sN\); it is even for sufficiently large \(n\), and
+\(s\ge c_sN/2\).
+Choose \(p_0>0\) sufficiently small in terms of \(r,D,c_s,q_0\)
+and the fixed phase tolerance. Fix the transpose-injection error
+thresholds as well, with their total over all component and endpoint
+roles below the required constant error, such as \(.001\).
+These thresholds precede selector and channel sizes.
+For the bounded-tuple transpose tests in
+\Cref{lem:channel-thinning}, we may fix
+\[
+ q_{\rm test}=d_0+L_0+2r+2K+10.
+\]
+This covers the tiny covers, fixed fresh probes, and bounded pin
+and marked-profile tests.  The growing lists of length proportional
+to \(N\) in \Cref{lem:injection} instead use an absolute
+reference-law count, as explained in its proof.
+
+The exponential demands of the no-cover argument are simultaneously
+satisfiable in this order. Fix a phase-test threshold \(\vartheta>0\)
+smaller than the required fixed phase accuracy, and choose a constant
+\(\Gamma>D+2r+1\) large enough to absorb the indicated density and
+bounded-rank tensor counts. With \(L_\vartheta=\log_2(1/\vartheta)\),
+it suffices to arrange
+\begin{align}
+ \frac{c_s}{2}
+ \left(\frac{q_0}{2}\log_2(1/p_0)-2-L_\vartheta\right)
+     &>\Gamma,\label{eq:ledger-p0-demand}\\
+ \frac{c_s}{2}
+ \left(\frac{hq_0}{4}-L_\vartheta\right)
+     &>\Gamma.\label{eq:ledger-h-moment-demand}
+\end{align}
+The first determines a sufficiently small \(p_0\) using only early
+quantities; the second is a later lower bound on \(h\).
+Indeed the two moment terms are
+\(2^{2s}p_0^{q_0s/2}\) and \(2^{-hq_0s/4}\), and Markov at threshold
+\(\vartheta\) costs \(2^{sL_\vartheta}\).
+The comparison between raw and independent channel laws has a factor
+depending on \(h\), but it is constant in \(n\) and does not alter the
+linear exponents.
+
+For the bounded tiny-cover record use
+\[
+ R_0=2^{100(r+1)^2(d_0+|\cE|+1)},\qquad
+ \delta_{\rm ph}=\frac{p_0\delta_{\mathrm{cov}}}{64R_0},\qquad
+ \delta_* =10^{-6}\delta_{\rm ph}/64.
+\]
+The preparation argument supplies these early retained-mass bounds
+and \(c_*=\delta_*^2/64>0\), as in
+\eqref{eq:preparation-early-margins}.  They measure retained unit
+mass and successful pair mass in the original pooled law with
+endpoint bound \(M\pi\).  Choose
+\begin{equation}\label{eq:ledger-colours}
+ 0<\eta_{\mathrm{col}}<c_*/4,\qquad
+ 0<a<\min\{1/100,\eta_{\mathrm{col}}/800\}.
+\end{equation}
+A colour partition is refined into leaves with their actual weights;
+preparation is performed on their pooled law.  Thus these margins
+are independent of both the number of colours and the smallest
+positive colour mass.  Indeed original equal-colour pair mass is
+at most \(\sum_c p_c^2\le\eta_{\mathrm{col}}\); after restriction
+of mass \(\delta_*\) its normalized mass is at most
+\(\eta_{\mathrm{col}}/\delta_*^2\).  Either calculation gives
+strictly positive successful mass after the colour exclusion.
+
+Now prescribe \(0<\beta\le1\), and choose \(L>10^6/\beta\), as
+in \eqref{eq:transfer-a-L}.  Larger values of \(\beta\) follow
+by weakening the graph assertions.  For the matrix obstruction
+one chooses \(\beta\) sufficiently small in terms of \(a\),
+which is permitted here.  Fix the additional bounded marginal
+factors and positive injection/phase tolerances required by the
+asymmetric test with this \(L\).  These later fixed tolerances
+may depend on \(L\); they do not change the preceding
+\(\delta_*,c_*,\eta_{\mathrm{col}}\).  In particular the two
+preparation problems need not have the same success margin.
+
+Next choose the row and selector parameters:
+\begin{align}
+ R_*&=2(K+20),&
+ j_*&\ge10(K+1)(R_*+20),\nonumber\\
+ B_0&=2|\cE|K(R_*+14),&
+ B_*&=10(B_0+10000g+1),\nonumber\\
+ 2^{b-2j_*}&>100g(B_*+1),&
+ p_*&=\sum_{j=0}^{j_*}\binom bj,\nonumber\\
+ A&=10(K+1)p_*(g^2+5),&
+ s_0&=4\lceil A\rceil,\qquad
+ J>100(s_0+K^2+1).
+ \label{eq:ledger-selectors}
+\end{align}
+Integer inequalities are met by rounding upward when needed.
+This agrees with \Cref{lem:label-support,lem:generic-mixers}.
+In particular \(j_*\) exceeds the interpolation degree required for
+\((K+1)(R_*+14)\) labels. The length \(b\) is chosen before \(A\),
+and hence before \(s_0,J\).
+
+Now choose a sufficiently large integer \(r_0\), always setting
+\begin{equation}
+                         h=1000r_0.
+ \label{eq:ledger-channels}
+\end{equation}
+The demands are finite in number and depend only on preceding
+constants:
+\begin{enumerate}
+\item the moment demand \eqref{eq:ledger-h-moment-demand};
+\item fixed-cover and fresh-probe transpose-injection errors in
+  \Cref{thm:phase-alternative};
+\item the unconditional transpose-injection estimate in \Cref{lem:injection};
+\item residual ranks and private-channel space in
+  \Cref{thm:gradient-realization};
+\item the tester rank inequality in \Cref{lem:zero-delta}.
+\end{enumerate}
+Each is eventually satisfied by increasing \(r_0\).
+Quantitative checks appear below.
+
+After fixing \(h\), choose the thinning rate \(\rho>0\).
+It is enough to require
+\begin{equation}\label{eq:ledger-thinning-rate}
+ 10^4|\cE|2^{2h}\rho
+   <\min\{\zeta/100,10^{-6},\epsilon_{\mathrm{img}},
+                  \epsilon_{\mathrm{inj}}\},
+\end{equation}
+where the positive constants \(\epsilon_{\mathrm{img}}\) and
+\(\epsilon_{\mathrm{inj}}\) are chosen below the reserved
+full-frame image and injection rate margins.  Those margins
+are fixed before this choice.  Put
+\[
+ \kappa_{\mathrm{thin}}=2|\cE|(2^h-1),\qquad
+ \tau_{\mathrm{thin}}=10|\cE|2^h\rho.
+\]
+The thinning lemma gives
+\begin{equation}\label{eq:ledger-thinned-density}
+ \pi^2\le2^{\tau_{\mathrm{thin}}N}\rawlaw^2,
+ \qquad 3000g+\tau_{\mathrm{thin}}<D-1.
+\end{equation}
+Thus the distribution-test pair cap leaves a fixed positive
+reserve below the absolute cap \(2^{DN}\rawlaw^2\), even after
+bounded normalizations and removal of exponentially light colours.
+The survival fraction is asymptotic to
+\(2^{-\kappa_{\mathrm{thin}}\rho N}\).
+Because \(\kappa_{\mathrm{thin}}\rho<.01\), the simultaneous
+additive concentration error \(2^{-.1N}\) in the construction
+of the thinning sets is exponentially smaller than this fraction.
+Its use in conditional probabilities is therefore justified.
+
+Finally choose a sufficiently large integer \(M_0\), also relative
+to \(1/\rho\), and put
+\begin{equation}
+                  N=M_0n,\qquad m=2^{C_0gN}.
+ \label{eq:ledger-final-dimensions}
+\end{equation}
+The coefficient of every \(O(n)\) count paid as a fraction of \(N\)
+has then been fixed. Choose \(M_0\) to make all those fractions
+smaller than their prescribed slacks. Only afterward does \(n\)
+tend to infinity through integers with \(n\ge2r_0\).
+The resulting dependency order is
+\[
+ \begin{gathered}
+ (C_0,g,D,M,k_{\max},\zeta,K_1)
+ \longrightarrow
+ (r,L_0,d_0,u_0,K,\text{early tolerances},p_0,\delta_*,c_*)\\
+ \longrightarrow(\eta_{\mathrm{col}},a)
+ \longrightarrow(\beta,L,\text{asymmetric tolerances})\\
+ \longrightarrow(R_*,j_*,B_0,B_*,b,p_*,A,s_0,J)
+ \longrightarrow(r_0,h)\longrightarrow\rho
+ \longrightarrow M_0\longrightarrow n .
+ \end{gathered}
+\]
+Here \(b\) is only the selector length; \(\beta\) is the graph
+threshold.  The fixed palette has size at most
+\(\lceil1/(2a)\rceil+1\), and changes only bounded complexities
+and the sufficiently-large-\(n\) onset.
+
+\subsection{The channel demands precede later tests}
+
+In the frequent-output argument, a fixed collection of \(L_0\)
+fresh probes succeeds on B-mass at least
+\begin{equation}
+       q_*=(\delta_{\mathrm{cov}}/2)
+              (\delta_{\mathrm{cov}}/4)^{L_0}>0.
+ \label{eq:ledger-probe-mass}
+\end{equation}
+The one-vertex density factor depends only on the early marked-law
+restrictions and \(p_0\).
+Transpose-injection failure on the fixed fresh vectors is at most
+that factor times \(O(L_0)2^{L_0-h}\).
+It can be made smaller than \(q_*/2\) by increasing \(h\), without
+consulting a selector choice or a future common test.
+The later raw-event exponent is at most
+\[
+       -(.99-.51)L_0N+DN+\text{small linear errors}.
+\]
+The specified \(L_0\) leaves a margin close to \((.48L_0-D)N\).
+The small coefficient-count errors are paid by \(M_0\).
+
+For \Cref{lem:injection} only the unconditional accepting
+family is needed.  The independent primal-span part uses the
+unchanged primal marginals.  For the channel part, the
+estimate with \(\ell=\lfloor N/(10(K+1))\rfloor\) compares
+\(c^\ell\) with \(2^{-(h-2K)\ell+o(N)}\), where \(c>0\)
+is fixed by the required failure tolerance.  A sufficient demand is
+\begin{equation}\label{eq:ledger-injection-demand}
+ h>2K+\log_2(1/c)+30(K+1).
+\end{equation}
+It leaves a positive linear exponent margin.  The thinning loss
+in \eqref{eq:ledger-thinning-rate} is chosen smaller than that
+margin; fixed marginal factors cost only constants.  In particular
+this estimate does not require a table-dependent tolerance or a
+new pinning step.
+
+For channel realization, the explicit bounds in
+\Cref{eq:baseline-rank-bound,eq:residual-rank-bound,eq:block-compression-bound,eq:quotient-compression-bound}
+are
+\[
+\begin{aligned}
+ B_{\mathrm{lin}}&=3K+28,&
+ R_{\mathrm{res}}&=14K+140,\\
+ D_{\mathrm{blk}}&=2p_*|\cE|(K+14+R_{\mathrm{res}}),&
+ D_{\mathrm{quo}}&=2p_*\bigl(1+(g^2+3)D_{\mathrm{blk}}\bigr)+2K.
+\end{aligned}
+\]
+The pure correction rank per component is at most
+\(30r_0+28J|\cE|+D_{\mathrm{quo}}\), and the remaining channel
+pairing has rank at least \(h-2K-4B_{\mathrm{lin}}\).
+Thus \Cref{eq:realization-r0-requirement} asks precisely for
+\begin{equation}
+       970r_0>28J|\cE|+D_{\mathrm{quo}}+2K+4B_{\mathrm{lin}}.
+ \label{eq:ledger-realization-demand}
+\end{equation}
+Every quantity on the right is independent of \(h,r_0,n\)
+and is fixed by parameters through \(J\).
+The baseline first factors through bounded pin/key projections on
+primal inputs, then derivative corrections of bounded rank are
+chosen, and the small residual is compressed before the
+\(r_0\)-sized pure terms are factored through private channels.
+
+The tester inequality in \Cref{lem:zero-delta} is
+\begin{equation}
+ \frac{g-1}{2}\left(g-\frac{4D}{g}\right)r_0
+          >2(g-1)(h+2JK_1)+1.
+ \label{eq:ledger-pure-tester-demand}
+\end{equation}
+Substituting \(D=4000g\) and \(h=1000r_0\) gives
+\[
+       \frac{g-20000}{2}\,r_0
+             >4JK_1+\frac1{g-1}.
+\]
+The coefficient on the left is positive for the specified \(g\).
+This is therefore another compatible lower bound on \(r_0\).
+
+\subsection{Pin, atom, and tensor-rank ledger}
+
+\begin{center}
+\begin{tabular}{@{}p{.39\textwidth}p{.53\textwidth}@{}}
+\toprule
+Quantity & Bound and role\\
+\midrule
+First pin dimension & \(K_1\), summed over components and signs.\\
+Pin dimension used throughout & At most \(K_1\), hence at most
+the conservative budget \(K\).\\
+Tiny-cover dimension & At most \(d_0=4rL_0\); its offset
+record has a bounded number of possibilities.\\
+Effective space & \(\dim C_i\le K^2\); every member has total component
+rank at most \(K\).\\
+Attack atoms at one endpoint & At most \(14\), across its two lists.\\
+Component rank in the annihilator argument &
+\(2(K+14)\le R_*=2(K+20)\).\\
+List length per cross edge & At most \(7\) on either side.\\
+Atom positions in one unit's full query & At most \(4\cdot7=28\).\\
+General key-vector slots &
+\(28\cdot2(g-1)=k_{\max}=56(g-1)\).\\
+\bottomrule
+\end{tabular}
+\end{center}
+
+Each tag atom occupies \(g-1\) star components and supplies one plus
+and one minus vector at each. These are vector counts, not scalar
+Gram counts.
+The label exclusions in \Cref{lem:sparse-pin-labels} give nominal
+independence modulo the pins, as proved in
+\Cref{lem:small-tables}, part~\textup{(ii)}.
+The effective-space dimension is counted in projected pin bases:
+for a fixed endpoint \(\sum_e\dim S_{i,e}^\pm\le K\), so
+\[
+ \sum_e\dim(S_{i,e}^+\otimes S_{i,e}^-)
+ \le
+ \left(\sum_e\dim S_{i,e}^+\right)
+ \left(\sum_e\dim S_{i,e}^-\right)
+ \le K^2.
+\]
+Thus an effective tensor basis is encoded in bounded projected
+coordinates, not as unrestricted matrices with \(O(n^2)\) entries.
+
+The unprojected representation of one gradient in
+\Cref{thm:gradient-realization} has per-component rank at most
+\[
+                         15r_0+14J|\cE|.
+\]
+Two endpoint contributions give \(30r_0+O(1)\).
+If \(A_+,A_-\) remove bounded pin/key subspaces, then
+\[
+ \rank(M-A_+\trans M A_-)
+       \le\rank(I-A_+)+\rank(I-A_-).
+\]
+Hence this projection contributes only bounded residual rank,
+without multiplying the tester rank by a pin-dependent constant.
+The later blockwise compression and derivative corrections preserve
+that independence from \(r_0\).
+
+\subsection{Scalar records and the final dimension multiplier}
+
+Write
+\begin{equation}
+                  d_n=\dim\cB=p_*\bigl(1+(g^2+3)n\bigr).
+ \label{eq:ledger-column-dimension}
+\end{equation}
+Then
+\[
+       \frac{d_n}{N}
+       =\frac{p_*(g^2+3)}{M_0}+\frac{p_*}{M_0n}.
+\]
+This ratio, and the ratio after adding the bounded channel dimension,
+can be made arbitrarily small by choosing \(M_0\) last.
+In particular the image estimate can use a fixed sufficiently small
+\(\varepsilon<\zeta/4\).  For the tiny-cover calculation impose,
+in addition, that the total primal coefficient-count and image
+loss be smaller than \(\rho/12\).  The same choice leaves room
+for the injective frames and prescribed self-Gram values.
+
+The collision records are linear in \(n\).
+There are at most \(56\) point inputs across two units, costing at most
+\[
+                         56(g^2+3)n+56b
+\]
+bits. A list of \(K\) nominal pin directions uses at most
+\(2K(d_n+h)\) coefficient bits, plus bounded component, sign and rank
+indices.
+Once pin projections are fixed, a basis of \(C_i\) uses at most
+\(K^4\) binary coefficients in its tensor-coordinate space;
+there are four endpoints.
+Tester summaries, numerical small tables, projection relationships
+and finite statuses have bounded lengths in \(n\).
+The construction matrices \(E,L,R\) are fixed data and are not
+encoded afresh in each query record.
+
+The total frozen dimension of a unit is at most \(K+k_{\max}\),
+over all components and signs.
+There are \(2(d_n+h)\) nominal columns of a given sign at each
+component. Recording all interactions of both units' columns with
+opposite frozen images therefore uses at most
+\[
+                         4(d_n+h)(K+k_{\max})
+\]
+scalar bits.
+The deterministic solution is selected from these data and the point
+parameters. No unrestricted \(O(n^2)\)-entry matrix is appended.
+The number of record cells, including bounded nominal choices when
+needed, is consequently at most
+\begin{equation}
+                         2^{C_{\mathrm{rec}}n+C_{\mathrm{rec},0}},
+ \label{eq:ledger-record-count}
+\end{equation}
+for constants fixed before \(M_0\).
+
+Only mixed primal-channel Gram entries are tested in the final target.
+For one cross Gram block and component, the two endpoint primal
+summands have total dimension \(2d_n\) and the channel summands have
+total dimension \(2h\) on either side. The two mixed products
+contain at most \(8d_nh\) entries.
+There are two cross Gram blocks; hence at most
+\begin{equation}
+                              16|\cE|\,h\,d_n
+ \label{eq:ledger-tested-gram-bits}
+\end{equation}
+bits are tested. Frozen or redundant entries only reduce this count.
+This is \(O(n)\), with coefficient already fixed.
+
+Choose \(M_0\) to meet every earlier coefficient/codimension slack
+and, in particular, so that for sufficiently large \(n\),
+\begin{equation}
+ \log_2(\text{record-cell count})<.005N,\qquad
+ \text{tested scalar Gram bits}<.01N.
+ \label{eq:ledger-collision-bit-slacks}
+\end{equation}
+The smaller image and pin-count slacks form a finite collection
+and can be met by the same final enlargement of \(M_0\).
+
+The conditional point-mass estimate on a surviving cell is
+\[
+ \begin{split}
+ \log_2 p_{\max}/N
+  &\le -(1-2\zeta)(k+t)+(k+.01)\\
+  &=-(1-2\zeta)t+.01+2\zeta k
+       \le-.95t
+       \qquad(t\ge1,\ k\le k_{\max}).
+ \end{split}
+\]
+Here \(2\zeta k\le.002\).
+A nonzero quotient character of rank \(t\) has mean at most
+\[
+       2^{tN/2}
+       \bigl(2^{-.95tN}2^{-.95tN}\bigr)^{1/2}
+             =2^{-.45tN}.
+\]
+This pays for fewer than \(.01N\) scalar bits, leaving agreement
+probability at least \(2^{-.02N}\).
+For each fixed parameter pair and key, pruning cells below
+\(2^{-(k+.01)N}\) loses overlap at most
+\[
+ 2|\cK_n|\,2^{C_{\mathrm{rec}}n+C_{\mathrm{rec},0}}
+       2^{-(k+.01)N},
+\]
+which is exponentially small since \(|\cK_n|\le2^{kN}\) and
+\eqref{eq:ledger-collision-bit-slacks} holds.
+This uses Fubini at a fixed parameter pair, without intersecting
+pruning decisions over all opposite parameters.
+
+\subsection{Probability scales and later analytic choices}
+\label{sec:probability-scales}
+
+The marginal distinctions used in the proof are summarized below.
+A bounded multiplier may depend on a fixed preparation restriction;
+it is independent of \(n\).
+
+\begin{center}
+\begin{tabular}{@{}p{.36\textwidth}p{.56\textwidth}@{}}
+\toprule
+Mechanism & Scale and permissible cost\\
+\midrule
+Thinned reference law & Primal marginal exactly that of
+\(\rawlaw\); full pair density at most
+\(2^{\tau_{\mathrm{thin}}N}\rawlaw^2\).\\
+Original tested unit law & Joint cap \(Q_N\pi^2\); both endpoint
+marginals at most \(M\pi\), or \(L\pi\) for the asymmetric test.\\
+Prepared mixed law & Absolute joint cap within the reserved
+\(2^{DN}\rawlaw^2\) budget; bounded marginals relative to
+\(\pi\), hence bounded primal marginals relative to the original
+primal law.\\
+Normalized leaves & A common \(2^{C_{\mathrm{leaf}}N}\rawlaw^2\)
+cap and fresh-image min-entropy with at most \(2\zeta\) slack;
+no bounded individual full-frame marginal is asserted.\\
+Fixed-cost preparation restrictions & Positive mass fixed before
+\(h\); bounded logarithmic costs are absorbed by rate reserves.\\
+Single-endpoint typicality & Error \(2^{-\Omega(n)}\), tested on
+primal frames; arbitrary channel-dependent flags remain flags.\\
+Paired oversampling and cross-rank failures & Error
+\(2^{-\Omega(n^2)}\), surviving \(2^{O(N)}=2^{O(n)}\)
+joint inflation.\\
+Tiny-cover compatible pairs & Positive mass fixed before \(h\),
+using either the identically zero full-sum phase or the additional
+thinning gain.\\
+Key equality & Factor \(|\cK_n|^{-1}\ge2^{-kN}\),
+\(k\le k_{\max}\).\\
+Final four-hole probability &
+\(2^{-(k_{\max}+.03)N-o(N)}\), exceeding
+\(2^{-100gN}\) for sufficiently large \(n\).\\
+\bottomrule
+\end{tabular}
+\end{center}
+
+For clarity, the extra gain from thinning is not obtained by
+subtracting the full-frame density loss from \(\rho\).
+On a fixed tiny-cover record, the companion tuple of length
+\(t\) has primal point probability at most
+\(2^{-tN+\rho N/12}\).  If the full output is nonzero, one
+endpoint must have a nonzero channel combination in a prescribed
+nonzero translate of its allowed set.  Conditional on its primal
+frame, \Cref{lem:channel-thinning} bounds that event by
+\(2^{-\rho N/2}\).  The point probability of companion plus
+nonzero full output is therefore at most
+\(2^{-tN-\rho N/3}\), after absorbing fixed normalizations.
+The Walsh estimate pairs one such bound with a companion-only
+bound, giving at most
+\[
+ 2^{(t_A+t_B)N/2}
+ \bigl(2^{-t_AN+\rho N/12}
+       2^{-t_BN-\rho N/3}\bigr)^{1/2}
+ \le2^{-\rho N/12}.
+\]
+There is no exponential union over these records: one of their
+boundedly many values was fixed at an early constant mass cost.
+If the full output is zero on the retained record, the lifted
+phase vanishes for every ordered retained pair.  Both cases
+supply constant compatibility mass.  No further peeling or
+inverse-polynomial compatibility branch is used.
+
+Fixed-cost restrictions are compatible with the leaf trimming in
+\Cref{lem:peeling}.  More generally, if a later analytical restriction
+has total mass \(q_n=2^{-o(N)}\), leaves on which its relative mass
+is below \(2^{-\zeta N}\) contribute at most
+\[
+ 2^{-\zeta N}/q_n=2^{-(\zeta-o(1))N}
+\]
+of the restricted law.  On the other leaves division costs at most
+\(2^{\zeta N}\), changing \((1-\zeta)tN\) to at worst
+\((1-2\zeta)tN\) for every positive rank \(t\).  This trimming
+preserves the old pins.  Marginal hypotheses for later empirical
+common tests concern the primal projection; none requires replacing
+\(\pi\) by \(\rawlaw\) in a full-frame marginal inequality.
+
+Partitions, entropy bounds, weak-product accuracies and finite
+transformation sample sizes are analytical choices made after the
+construction constants.  They may depend on \(M_0\), a fixed positive
+test-cell mass, or a fixed accuracy.  They do not change pin ranks,
+key-slot counts or the scalar record coefficient in
+\eqref{eq:ledger-record-count}.  For each finite common-test stage
+these choices change only constants and the required onset in
+\(n\); a diagonal subsequence passes each fixed finite stage in
+turn.  No rate uniform over all such stages is needed.
+
+For example, unary positivity uses
+\(\gamma=2^{-K^2-3}\) and \(b_{\mathrm{int}}=2^{-J+2s_0}<\gamma\).
+On a reference key cell of mass at least \(\delta>0\), the proof
+first obtains conditional input mass at least \(\delta/2\), so it
+can use
+\[
+ u>\frac{2}{\delta(\gamma-b_{\mathrm{int}})^2}
+\]
+transformations.  This changes the constant in
+\(O_u(2^{-s_0n})\), while \(s_0=4\lceil A\rceil\) and the profile
+count \(2^{An}\) remain fixed.  Thus a smaller cell does not require
+a larger construction parameter \(J\).
+
+The final collision margin is
+\[
+ 100g-[56(g-1)+.03]=44g+55.97>0.
+\]
+This positive exponent gap absorbs any fixed positive subsequential
+overlap constant.  A violating sequence for either distribution
+test would yield one limit model with positive successful overlap,
+which contradicts its asserted upper bound \(2^{-100gN}\).
+Uniformity over all limit models is unnecessary.
+
+Finally, \Cref{sec:graph-transfer} uses
+\(Q_N=2^{3000gN}\) and \(m=2^{1000gN}\).  Entropy fingerprints
+have logarithmic count
+\[
+ O(N^3 2^{100gN})=o(m).
+\]
+For either fixed positive linear choice of \(k\), the coloured
+exceptional-pair estimate is
+\[
+ m^{2k}(4/Q_N)^k=2^{-(1000gN-2)k}.
+\]
+The coloured position exception has mass less than \(8/M\);
+at \(k=\lceil m/100\rceil\) its tail is at most
+\((800e/M)^k\).  Fewer than \(4/\eta_{\mathrm{col}}\)
+exceptional colours, each of size at most \(2am\), account for
+fewer than \(m/100\) matching edges by
+\eqref{eq:ledger-colours}.  In the asymmetric transfer use
+\(k=\lceil\beta m/100\rceil\); vertex and point exceptional
+sets have mass below \(1/L\), with tail at most
+\((100e/(\beta L))^k\).  These bounds pay for the
+\(\exp(o(m))\) terminal union and prove the graph properties
+at \(m/20\) and \(\beta m/10\), respectively.
diff --git a/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/15-measures.tex b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/15-measures.tex
new file mode 100644
index 0000000000000000000000000000000000000000..6b225a952b5de00a2d02925497ce069285f78632
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+++ b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/15-measures.tex
@@ -0,0 +1,125 @@
+\section{Compact measure constructions used in the proof}
+\label{sec:measure-constructions}
+
+The spectral argument and the common signature model use compactness
+only for compact metric spaces, countable arrays, and bounded finite
+measures. We record constructions for these particular uses, including
+the jointly measurable densities in \Cref{lem:limit-model}.
+
+\begin{lemma}[Compact arrays and finite measures]
+\label{lem:compact-measures}
+Let $X$ be a compact metric space and $C<\infty$.
+The positive Borel measures on $X$ of mass at most $C$ form a compact
+metrizable space for convergence against continuous functions.
+Countable products of compact metric spaces are compact metrizable.
+If $\nu_j$ converges weakly to $\nu$ and $F\subseteq X$ is closed,
+then $\limsup_j\nu_j(F)\le\nu(F)$.
+\end{lemma}
+\begin{proof}
+Choose nested finite Borel partitions of $X$ whose cell diameters tend
+to zero. Such partitions result by taking a finite cover by balls of
+arbitrarily small radius, making it disjoint by successive differences,
+and refining the previous partition. Given a sequence of measures,
+successive subsequences make every cell mass converge, as well as the
+total mass. The limiting cell masses are nonnegative and consistent
+under refinement.
+
+Realize these masses by subdividing an interval of the limiting total
+length, first into intervals of the masses of the first partition,
+then recursively into intervals for its refinements. Outside the
+countable set of subdivision endpoints, each interval point specifies
+a nested sequence of cells. Representatives from those cells form a
+Cauchy sequence: all representatives after level $j$ belong to one
+cell of diameter tending to zero. Compactness gives a limiting point
+of $X$. The representative step maps are measurable, so their limit
+is measurable. Push forward interval length by this limit map.
+Uniform continuity shows that the integral of each continuous function
+is the limit of its finite representative sums. The same sums
+approximate the integrals of the original measures uniformly. This
+proves weak subsequential compactness, including zero limiting mass.
+
+Continuous functions on $X$ have a countable uniformly dense family.
+For instance, finite nets from a countable dense subset of $X$, together
+with continuous distance cutoffs and rational values, give uniformly
+accurate piecewise weighted approximations to each continuous function.
+If $(f_j)$ is such a family, convergence of the integrals of all $f_j$
+metrizes weak convergence on measures of bounded mass, by uniform
+approximation. The preceding subsequence argument therefore gives
+compactness in that metric. For a countable product use the metric
+$\sum_j2^{-j}\min\{1,d_j\}$; successive coordinate subsequences and a
+diagonal choice prove its compactness.
+
+Finally the continuous functions
+$f_k(x)=\max\{0,1-k\,d(x,F)\}$ decrease to $\mathbf1_F$.
+For every $k$, weak convergence gives
+$\limsup_j\nu_j(F)\le\int f_k\,d\nu$.
+Bounded monotone approximation as $k\to\infty$ gives the stated
+closed-set inequality.
+\end{proof}
+
+In particular, consistent masses of finite bit cylinders define a
+measure on the compact bit-sequence space: apply the interval
+subdivision construction directly to the successive bit prefixes.
+If the masses depend measurably on a stored record, all subdivision
+thresholds are measurable functions of that record. Using an additional
+uniform interval coordinate gives a jointly measurable sample. The same
+construction works for a compact metric target using the nested
+partitions above: evaluation of a measure on a Borel cell is measurable
+in the weak topology. For closed cells this follows from continuous
+distance cutoffs, and a monotone-class argument gives all Borel cells. Two
+independent interval coordinates sample independently from the two
+conditional measures stored in \Cref{lem:limit-model}. Thus the argument
+retains those conditional measures before taking limits; it does not
+interchange an unrecorded conditional distribution with a weak limit.
+
+\begin{lemma}[Densities of dominated cylinder measures]
+\label{lem:cylinder-densities}
+Let $X$ be a compact bit-sequence space with Borel probability $m$,
+let $\theta$ have a probability law, and suppose $\nu_\theta$ is a
+measurable family of measures satisfying $0\le\nu_\theta\le m$.
+There is a jointly measurable $h(\theta,x)\in[0,1]$ such that,
+for almost every $\theta$, $\nu_\theta(A)=\int_Ah(\theta,x)\,dm(x)$
+for every Borel $A$. Conditional averages of an $L^2(m)$ function
+on increasing finite-prefix partitions converge in $L^2(m)$.
+\end{lemma}
+\begin{proof}
+Let $\mathcal P_k$ be the partition by the first $k$ bits, and set
+$h_k(\theta,x)=\nu_\theta(C)/m(C)$ when $x\in C\in\mathcal P_k$
+and $m(C)>0$, with value zero on null cells. These functions are
+jointly measurable and lie in $[0,1]$. Their successive differences
+are orthogonal in $L^2(d\Pr(\theta)\,dm(x))$, because the average of
+$h_{k+1}-h_k$ over each parent cell is zero for every $\theta$.
+The squared norms of $h_k$ are bounded by one. Hence $(h_k)$ is
+Cauchy in $L^2$, and has an $L^2$ limit $h$. One can see completeness
+here directly: choose a subsequence with summable $L^2$ norms of its
+successive differences; their $L^1$ norms are also summable, giving
+an almost-everywhere convergent sum with the required $L^2$ tail bound.
+A further almost-everywhere convergent subsequence makes $h$ jointly
+measurable with values in $[0,1]$.
+
+For each cylinder $C$, all sufficiently late $h_k$ integrate over $C$
+to $\nu_\theta(C)$. Convergence in the product $L^2$ space gives
+the same identity for $h$, first in $L^1$ of $\theta$ and thus almost
+surely. There are countably many cylinders, so these identities hold
+simultaneously outside one null set. They then extend to all Borel
+sets: the class of sets approximable in measure by the finite cylinder
+algebra is a sigma-algebra, and both measures are dominated by $m$.
+
+Finite-cell averaging is an $L^2$ contraction by Cauchy--Schwarz.
+Cylinder-simple functions are dense in $L^2(m)$, by simple-function
+approximation and the same cylinder approximation of measurable sets.
+Approximating a function by such a simple function and using the
+contraction proves the final assertion.
+\end{proof}
+
+Rescaling gives the same density construction for $\nu_\theta\le Rm$.
+For countably many ordered measures, the versions can be chosen
+simultaneously with their almost-everywhere ordering; their increasing
+supremum represents the increasing supremum of the measures. This is
+the use in \Cref{lem:limit-zero-product}. The integration identity for
+the sampled kernels holds first for rectangle indicators. The class of
+sets satisfying it is closed under complements and disjoint countable
+unions; hence it holds on the product sigma-algebra and then for
+nonnegative or bounded measurable integrands by simple-function
+approximation. These constructions justify the limiting integrations
+and projections used in \Cref{sec:common-limit}.
diff --git a/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/matrix-rank.tex b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/matrix-rank.tex
new file mode 100644
index 0000000000000000000000000000000000000000..ec5026f1ea622e7f89ae47636f46005df80f3fb4
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+++ b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/matrix-rank.tex
@@ -0,0 +1,718 @@
+\section{The matrix obstruction}\label{mr:section}
+
+We now prove the conditional matrix statement.  Only the three
+properties of the base graph enter this section; the Strong Arnold
+Property is not needed.  Throughout, $v=ms$ denotes the order of the
+clique blow-up.  Matrix traces and squared Frobenius norms will be divided
+by $v$, even when the matrix under consideration has fewer than $v$
+rows.
+
+\begin{theorem}[Matrix obstruction]\label{thm:matrix-obstruction}
+For every $a\in(0,1]$ there are constants $\beta_0(a)>0$ and
+$m_0(a)\in\mathbb N$ with the following property.  Let
+$0<\beta\le\beta_0(a)$, let $m\ge m_0(a)$, and let $G$ be an
+$m$-vertex graph satisfying the following conditions:
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item Every matching of size at least $m/20$, partitioned into classes
+of size at most $am$, contains two conflicting edges in different
+classes.
+\item Every matching of size at least $\beta m$ and every vertex set of
+size at least $\beta m$ contain a conflicting edge--point pair.
+\item Every two vertex sets of size at least $\beta m$ have a cross hole.
+\end{enumerate}
+Then there is $s_0=s_0(G,a,\beta)$ such that, for every integer
+$s\ge s_0$, every real symmetric well-signed matrix $M$ for the clique
+blow-up $G[K_s]$ with exactly one negative eigenvalue satisfies
+\[
+  \rank M>\frac{ms}{2}+1.
+\]
+Consequently $\mu(G[K_s])+1<ms/2$ for all these $s$.
+\end{theorem}
+
+The proof uses two normalizations.  A first normalization, separately
+inside the vertex cliques, produces almost disjoint coordinate pairs.
+A second normalization, inside groups of pairs having the same two base
+labels, supplies additional positive directions.  The tree lemmas ensure
+that both normalizations can be used in one bounded Gram matrix.  We first
+give the linear-algebra estimates that make the errors independent of the
+bounds furnished by the tree.
+
+\subsection{Positive blocks and bounded spectral limits}
+
+For a real symmetric matrix $A$, write $n_+(A)$ and $n_-(A)$ for its
+positive and negative indices, counting multiplicity.  The notation
+$\|A\|$ means the Euclidean operator norm, and
+$\|A\|_{\mathrm F}^2=\Tr(A^{\mathsf T}A)$.  A symmetric matrix with
+at most one negative eigenvalue is a Gram matrix for the Lorentz form
+\[
+  \langle x,y\rangle=-x_0y_0+\sum_{i\ge1}x_i y_i:
+\]
+diagonalize the matrix and multiply each eigenvector coordinate by the
+square root of the absolute value of its eigenvalue.  Unused positive
+coordinates can be added.  Every restriction or coefficient pullback of
+this Gram matrix still has at most one negative eigenvalue.  For lists
+$X,Y$ of Lorentz vectors, $\langle X,Y\rangle$ denotes their rectangular
+Gram matrix; the same letter denotes the corresponding synthesis map.
+
+The next facts are standard consequences of symmetric $M$-matrix
+theory and the Perron--Frobenius theorem; see
+\citet[Theorem~5.2]{FiedlerSchneider1983Analytic} and
+\citet[Theorem~I]{DebreuHerstein1953Nonnegative}.
+Here an $M$-matrix has the form $cI-A$ with $A$ entrywise
+nonnegative and $c$ at least its spectral radius. We include the
+elementary proofs for the precise symmetric cases used below.
+
+\begin{lemma}[Normalization of a positive block]\label{mr:positive-block}
+Let $D$ be real symmetric with nonpositive off-diagonal entries.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item If $D$ is positive definite, its positive inverse square root
+$D^{-1/2}$ is entrywise nonnegative and has positive diagonal entries.
+\item Suppose $D$ is positive semidefinite and the graph of its strictly
+negative off-diagonal entries is connected.  If $D$ is singular, its
+kernel is one-dimensional and is spanned by a vector all of whose
+coordinates are positive.  Every proper principal submatrix of $D$ is
+positive definite.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+For (i), choose $c>\lambda_{\max}(D)$ and put $A=I-D/c$.
+All entries of $A$ are nonnegative, and its eigenvalues lie in $[0,1)$.
+The convergent binomial series gives
+\begin{equation}\label{mr:inverse-root}
+ D^{-1/2}=c^{-1/2}\sum_{k=0}^{\infty}
+       \frac{\binom{2k}{k}}{4^k}A^k.
+\end{equation}
+Every term is entrywise nonnegative, and the constant term has positive
+diagonal.
+
+For (ii), take $c>\lambda_{\max}(D)$ and consider $A=cI-D$.
+It is symmetric and entrywise nonnegative, with positive diagonal and
+connected off-diagonal support.  A unit vector maximizing its Rayleigh
+quotient may be replaced by its coordinatewise absolute value, so a
+nonnegative maximizing eigenvector exists.  Its eigenvalue is positive.
+The eigenvector equation and connectivity imply that it has no zero
+coordinate: a zero coordinate would force zeros at every adjacent
+coordinate and then everywhere.  Moreover, every vector for the largest
+eigenvalue has one sign.  Indeed, equality in the absolute-value
+Rayleigh inequality forces equal signs across each positive edge, while
+its absolute value is a nonnegative maximizing eigenvector and thus has
+full support.  If the largest eigenspace had dimension at least two, it
+would contain a nonzero vector orthogonal to the positive eigenvector;
+that vector could not have one sign.  The largest eigenvalue is therefore
+simple.
+
+When $D$ is singular and positive semidefinite, this largest eigenvalue
+of $A$ is $c$, so the asserted kernel description follows.  If a proper
+principal submatrix were singular, extend a nonzero vector of zero
+quadratic form by zero coordinates.  Its quadratic form under $D$ is
+zero, hence it belongs to $\ker D$ by the spectral theorem.  This
+contradicts full support.
+\end{proof}
+
+\begin{lemma}[Spectral support from signs and inertia]
+\label{mr:spectral-support}
+Let $v_j\to\infty$, and let $B_j$ be symmetric entrywise nonnegative
+matrices of orders $d_j\le v_j$, with zero diagonal and
+$\sup_j\|B_j\|<\infty$.  Suppose $n_-(I-B_j)\le1$.
+Every weak subsequential limit of
+\[
+ \nu_j=\frac1{v_j}\sum_{i=1}^{d_j}\delta_{\lambda_i(B_j)}
+\]
+is supported on $[-1,1]$ and has first moment zero.  In particular, for
+each $\varepsilon>0$, the number of eigenvalues of $I-B_j$ outside
+$[-\varepsilon,2+\varepsilon]$ is $o(v_j)$.
+
+If also
+\[
+ \liminf_j\frac{d_j}{v_j}\ge1-\delta,
+ \qquad \rank(I-B_j)\le v_j/2+1,
+\]
+then
+\begin{equation}\label{mr:spectral-error}
+ \limsup_j\frac{\|B_j^2-I\|_{\mathrm F}^2}{v_j}\le2\delta.
+\end{equation}
+The constant in this inequality is independent of the operator bound.
+\end{lemma}
+
+\begin{proof}
+By \Cref{lem:compact-measures}, all measures have bounded mass and support in a common compact interval,
+so every subsequence has a weakly convergent further subsequence.  Fix
+such a limit $\nu$, and write $\theta=\nu(\mathbb R)\le1$.
+At most one eigenvalue of $B_j$ exceeds $1$, hence $\nu$ is supported
+on $(-\infty,1]$.  For every integer $k\ge0$, entrywise nonnegativity
+gives $\Tr B_j^{2k+1}\ge0$, and therefore
+$\int x^{2k+1}\,d\nu(x)\ge0$.  If $\nu$ gave mass $h>0$ to
+$(-\infty,-1-\varepsilon]$, this integral would be at most
+$\theta-h(1+\varepsilon)^{2k+1}$, which is negative for large $k$.
+This proves the support assertion.  The zero diagonal gives
+$\int x\,d\nu(x)=0$.  The asserted eigenvalue count follows by applying
+these facts to any putative subsequence on which the normalized
+exceptional count stays positive.
+
+For the additional assertion, pass to a subsequence realizing the
+limsup in \eqref{mr:spectral-error} and then to a weak limit as above.
+The nullity assumption supplies at least $d_j-v_j/2-1$ eigenvalues
+equal to $1$.  Since singleton sets are closed, weak convergence gives
+$r:=\nu(\{1\})\ge\theta-1/2$.  Consequently
+\[
+ \int_{x\ne1}(1+x)\,d\nu(x)=\theta-2r
+       \le1-\theta\le\delta.
+\]
+For $-1\le x\le1$ we have
+$(1-x^2)^2\le1-x^2\le2(1+x)$, and the first expression vanishes at
+$x=1$.  Polynomial integrals converge on the common compact support,
+so
+\[
+ \lim_j\frac{\|B_j^2-I\|_{\mathrm F}^2}{v_j}
+  =\int(1-x^2)^2\,d\nu(x)\le2\delta.
+\]
+In this argument the bound on $\|B_j\|$ is fixed before $j$ tends to
+infinity.  It removes the contribution of any exceptional eigenvalue
+to each fixed moment; it does not enter the resulting bound.
+\end{proof}
+
+Row-maximum rounding also appears in the study of nonnegative
+orthogonality constraints \citep{JiangMengWenChen2023Nonnegative,ChenHeZhang2025Stiefel}.
+Here symmetry and zero diagonal produce a partial matching, with
+bounds independent of the matrix order after normalization by $v$.
+
+\begin{lemma}[Extraction of coordinate pairs]\label{mr:matching}
+Let $B$ be a symmetric entrywise nonnegative $d\times d$ matrix with
+zero diagonal, where $d\le v$.  Suppose
+\[
+ \|B^2-I\|_{\mathrm F}^2\le\varepsilon v,
+ \qquad 0<\varepsilon\le 8^{-4},
+ \qquad \tau=\varepsilon^{1/4}.
+\]
+There is a matching of coordinates with adjacency matrix $W$, extended
+by zero on the omitted coordinates, such that
+\begin{align}
+ \#\{\text{omitted coordinates}\}&\le5\tau^2v,
+       \label{mr:matching-count}\\
+ \|B-W\|_{\mathrm F}^2&\le8\tau^2v,
+       \label{mr:matching-error}\\
+ |B_{ij}-1|&\le\tau\quad\text{on every retained pair }\{i,j\}.
+       \label{mr:matching-entry}
+\end{align}
+Every retained pair has $B_{ij}>0$.
+\end{lemma}
+
+\begin{proof}
+Put
+\[
+ d_i=\sum_jB_{ij}^2,
+ \qquad r_i=d_i^2-\sum_j B_{ij}^4\ge0.
+\]
+The diagonal entries of $B^2-I$ show that
+$\sum_i(d_i-1)^2\le\varepsilon v$.  Nonnegativity also gives
+\begin{align*}
+ \sum_i r_i
+ &=\sum_{j\ne k}\sum_iB_{ij}^2B_{ik}^2\\
+ &\le\sum_{j\ne k}\left(\sum_iB_{ij}B_{ik}\right)^2
+ \le\varepsilon v.
+\end{align*}
+Call row $i$ good when $|d_i-1|\le\tau$ and $r_i\le\tau^2$.
+There are at most $2\tau^2v$ bad rows.  In a good row, let
+$a_i=\max_j B_{ij}^2$.  Then
+\[
+ a_i\ge\frac{\sum_jB_{ij}^4}{d_i}
+      =d_i-r_i/d_i,
+ \qquad d_i-a_i\le\frac{\tau^2}{1-\tau}.
+\]
+Because $\tau\le1/8$, it follows that
+\[
+ a_i\ge\frac{1-2\tau}{1-\tau}\ge\frac67,
+ \qquad d_i-a_i\le\frac87\tau^2,
+ \qquad |\sqrt{a_i}-1|\le\tau.
+\]
+For the last inequality, use
+$1-\tau-\tau^2/(1-\tau)\ge(1-\tau)^2$ and
+$a_i\le1+\tau\le(1+\tau)^2$.
+There is a unique entry attaining $a_i$.  If its other endpoint is
+good, symmetry forces it to be the dominant entry there as well, since
+its square is at least $6/7$ whereas every nondominant squared entry in
+a good row is at most $8\tau^2/7$.  These good--good pairs form the
+required matching.
+
+For any set $S$ of rows, Cauchy--Schwarz gives
+\begin{equation}\label{mr:row-mass}
+ \sum_{i\in S}d_i
+ \le |S|+\sqrt{|S|\varepsilon v}.
+\end{equation}
+Thus the total squared row mass at bad rows is at most
+$(2+\sqrt2\tau)\tau^2v$.  Every good row whose dominant endpoint is
+bad contributes at least $6/7$ to this mass, by symmetry.  The total
+number of omitted coordinates is consequently at most
+\[
+ \left(2+\frac76(2+\sqrt2\tau)\right)\tau^2v
+ \le5\tau^2v.
+\]
+For a retained row, its contribution to $\|B-W\|_{\mathrm F}^2$ is
+at most $\tau^2+8\tau^2/7=15\tau^2/7$.  By
+\eqref{mr:row-mass}, the omitted rows contribute at most
+$(5+\sqrt5\tau)\tau^2v$.  Since $d\le v$ and $\tau\le1/8$, the
+sum is less than $8\tau^2v$.  The remaining assertions have already
+been proved.
+\end{proof}
+
+\begin{lemma}[A trace bound across the second normalization]
+\label{mr:trace-cross}
+For each $j$, let $H_j,Q_j$ be two lists of Lorentz vectors whose joint
+Gram matrix has operator norm at most a fixed $K$.  Let $v_j\to\infty$.
+Suppose that for every $\varepsilon>0$ the Gram matrix
+$\langle Q_j,Q_j\rangle$ has $o(v_j)$ eigenvalues above
+$2+\varepsilon$.  If
+\[
+ \limsup_j\frac{\Tr\langle H_j,H_j\rangle}{v_j}\le\eta,
+\]
+then
+\begin{equation}\label{mr:trace-cross-bound}
+ \limsup_j\frac{\|\langle H_j,Q_j\rangle\|_{\mathrm F}^2}{v_j}
+ \le2\eta.
+\end{equation}
+The conclusion is independent of $K$.
+\end{lemma}
+
+\begin{proof}
+Let $J_j$ be the joint Gram matrix.  Its negative part has rank at most
+one and norm at most $K$.  Replacing $J_j$ by its positive part changes
+any normalized trace by $O(K/v_j)$ and changes a Frobenius norm divided
+by $\sqrt{v_j}$ by $O(K/\sqrt{v_j})$.  Write the resulting positive
+semidefinite matrix as
+\[
+ P_j=\begin{pmatrix}A_j&C_j\\ C_j^{\mathsf T}&D_j\end{pmatrix}.
+\]
+A rank-one perturbation changes the number of eigenvalues above any
+fixed threshold by at most one, by the variational characterization of
+eigenvalues.  Thus the spectral projector $E_{\mathrm{hi}}$ of $D_j$
+for $(2+\varepsilon,\infty)$ has rank $o(v_j)$.  Put
+$E_{\mathrm{lo}}=I-E_{\mathrm{hi}}$.  Factor $P_j$ as a Euclidean
+Gram matrix, say $A_j=U_j^{\mathsf T}U_j$,
+$D_j=V_j^{\mathsf T}V_j$ and $C_j=U_j^{\mathsf T}V_j$.
+Since $\|V_jE_{\mathrm{lo}}\|^2\le2+\varepsilon$,
+\[
+ \|C_jE_{\mathrm{lo}}\|_{\mathrm F}^2
+ \le(2+\varepsilon)\Tr A_j.
+\]
+On the other spectral subspace,
+$\|C_jE_{\mathrm{hi}}\|_{\mathrm F}^2\le K^2
+\rank E_{\mathrm{hi}}=o(v_j)$.  Divide by $v_j$ and take the
+limsup.  This first bounds $\|C_j\|_{\mathrm F}/\sqrt{v_j}$.
+The original cross block differs by Frobenius norm at most $K$,
+so it has the same squared normalized limsup bound, at most
+$(2+\varepsilon)\eta$.  Finally let
+$\varepsilon\downarrow0$.  In particular the hypotheses force
+$\eta\ge0$, since the normalized trace of the original $H_j$ block
+has nonnegative liminf.
+\end{proof}
+
+\subsection{The first normalization and the candidate groups}
+
+\begin{proof}[Proof of \Cref{thm:matrix-obstruction}]
+Use the constant $c=c(a)>0$ from \Cref{mt:weighted-bag}; explicitly we
+may take
+\begin{equation}\label{mr:constants}
+ \begin{aligned}
+ K(a)&=\left\lceil\frac5a\right\rceil
+             \left(\left\lceil\log_2\frac5a\right\rceil+1\right),
+ &c(a)&=\frac1{40K(a)},\\
+ \delta_0(a)&=\min\left\{\frac1{3\cdot8^4},\frac{c(a)}{12},
+                   \frac13\left(\frac{c(a)}{2340}\right)^4\right\},
+ &\beta_0(a)&=\frac{\delta_0(a)}{10},\\
+ m_0(a)&=\left\lceil\max\left\{\frac2a,
+                                  \frac{10}{\delta_0(a)},2\right\}
+                         \right\rceil.
+ \end{aligned}
+\end{equation}
+Fix $a,\beta,m,G$ as in the theorem with these choices, and put
+\begin{equation}\label{mr:error-parameters}
+ q=\lceil\beta m\rceil,
+ \qquad\delta=5(\beta+1/m),
+ \qquad\tau=(3\delta)^{1/4}.
+\end{equation}
+These choices imply
+\begin{equation}\label{mr:smallness}
+ \delta\le\delta_0,
+ \qquad\tau\le\frac18,
+ \qquad\tau\le\frac{c}{2340},
+ \qquad\frac qm\le\frac{\delta}{5}\le\frac c{60},
+ \qquad am\ge2.
+\end{equation}
+
+Suppose the conclusion fails.  There is then an unbounded sequence of
+integers $s$ and corresponding well-signed matrices $M_s$ with one
+negative eigenvalue such that, writing $v=ms$,
+\begin{equation}\label{mr:contradiction-rank}
+ \rank M_s\le v/2+1.
+\end{equation}
+All subsequences below keep $a,\beta,m$ and the graph $G$ fixed, and
+$s\to\infty$ is the only limiting operation.  An expression $o(v)$
+has ratio to $v$ tending to zero in this limit; its constants may depend
+on the fixed graph and subsequence.  The displayed constants in the
+errors involving $\delta$ or $\tau$ are absolute.
+
+Represent $M_s$ by Lorentz Gram vectors, one for each blow-up vertex.
+An original vertex label $u\in V(G)$ specifies its $s$-column clique
+block.  By passing to a subsequence, the set of labels whose clique
+blocks have a negative eigenvalue can be fixed.  Two such labels cannot
+be a hole: zero cross Gram would make their negative directions
+orthogonal and give two negative eigenvalues of $M_s$.  These labels
+therefore form a clique in $G$.  There are fewer than $2q$ of them,
+since two disjoint $q$-element subsets of a larger clique would violate
+condition (iii).  Discard these labels.
+
+The other clique blocks are positive semidefinite.  They have strictly
+negative off-diagonal entries; \Cref{mr:positive-block} shows that
+deleting at most one coordinate makes each positive definite.  Work
+with $s\ge2$, and apply \Cref{mt:finite-tree} to their positive spans.
+For every hole between two labels the blocks are orthogonal, so their
+tree subtrees intersect.  By \Cref{mt:single-bag}, there is a bag
+missing fewer than $3q$ of the supplied single labels.  Keep the labels
+in this bag and denote their set by $U$.  There are fewer than $5q$
+missing original labels.  The retained coordinate dimension $d_s$
+satisfies
+\begin{equation}\label{mr:first-dimension}
+ d_s\ge(m-5q)(s-1),
+ \qquad \liminf_{s\to\infty}\frac{d_s}{v}\ge1-\delta.
+\end{equation}
+The tree and the chosen bag are fixed on a further subsequence.
+
+Normalize each retained clique block by its positive inverse square
+root, and denote the resulting column lists by $X_u$, $u\in U$.
+Their diagonal Gram blocks are identities.  By
+\Cref{mr:positive-block}, the normalizing coefficient matrices are
+entrywise nonnegative with positive diagonal.  Thus cross blocks remain
+zero on holes and strictly negative on base edges.  The full Gram matrix
+of the $X_u$ is
+\begin{equation}\label{mr:first-gram}
+ A_s=I-B_s,
+ \qquad B_s\ge0\ \text{entrywise},
+ \qquad (B_s)_{uu}=0\ \text{as a whole diagonal block}.
+\end{equation}
+The bag bound from \Cref{mt:finite-tree,mt:flat-normalization} makes
+$\|B_s\|$ uniformly bounded in $s$.  The rank is at most that of
+$M_s$, and its negative index is at most one.  Consequently
+\Cref{mr:spectral-support} gives
+\[
+ \limsup_{s\to\infty}\frac{\|B_s^2-I\|_{\mathrm F}^2}{v}
+ \le2\delta.
+\]
+Since $\delta>0$ is fixed, the left-hand quantity before taking a
+limsup is at most $3\delta=\tau^4$ for all sufficiently large $s$.
+Apply \Cref{mr:matching} to obtain a coordinate matching with $p_s^0$
+pairs and adjacency matrix $W_s$.  Then
+\begin{equation}\label{mr:first-matching}
+ \begin{split}
+ \liminf_s\frac{p_s^0}{v}
+     &\ge\frac12-\frac\delta2-\frac52\tau^2,\\
+ \|B_s-W_s\|_{\mathrm F}^2&\le8\tau^2v,
+ \qquad |(B_s)_{ij}-1|\le\tau
+          \quad\text{on retained pairs}.
+ \end{split}
+\end{equation}
+Every pair connects two different adjacent base labels, because
+$B_s$ vanishes within each original block and on base holes.
+
+Group the pairs by their unordered label pair $e=\{u,w\}\in E(G)$.
+If group $e$ contains $k_e(s)$ pairs, pass to a subsequence on which
+every $k_e(s)/s$ converges, and put
+\begin{equation}\label{mr:weights}
+ w_e=\lim_s k_e(s)/s,
+ \qquad \sum_{e\ni u}w_e\le1,
+ \qquad \sum_e w_e\le m/2.
+\end{equation}
+There are only finitely many types, so we may also fix whether each
+group is empty or its endpoint Gram block is positive definite,
+singular positive semidefinite, or has a negative eigenvalue.
+
+Groups in the last category will not be candidates for the second
+normalization.  Their types are pairwise touching: a conflict between
+two of them would make their cross Gram zero and give two orthogonal
+negative directions.  Choose a maximal ordinary matching of these
+types in the base graph, of size $k$.  If $k\ge m/20$, putting each edge
+in its own class contradicts (i), since $am\ge1$.  Thus $k<m/20$.
+The $2k$ endpoints of this maximal matching cover all the types under
+consideration.  The load bound in \eqref{mr:weights} therefore gives
+\begin{equation}\label{mr:indefinite-weight}
+ \sum_{e:\,\text{group has a negative eigenvalue}}w_e
+       \le2k<m/10.
+\end{equation}
+Their coordinate pairs are retained in the first matching.
+
+For every group of positive weight with positive semidefinite endpoint
+Gram, that Gram has the form
+\[
+ \begin{pmatrix}I&-C\\-C^{\mathsf T}&I\end{pmatrix},
+ \qquad C\ \text{entrywise strictly positive}.
+\]
+The graph of negative entries is connected.  If the group is singular,
+delete one whole matching pair.  By \Cref{mr:positive-block} its
+remaining Gram block is positive definite.  Positive weight ensures
+that the group is eventually nonempty after this deletion.  There are
+at most $\binom m2$ groups, so these deletions cost $O(m^2)=o(v)$
+coordinates.  Call the resulting positive definite groups of positive
+weight the \emph{candidate groups}.  Groups of zero limiting weight
+need not be candidates and keep their pairs in the first matching.
+By \eqref{mr:first-matching}, \eqref{mr:indefinite-weight} and
+$\tau\le1/8$, the total candidate weight satisfies
+\begin{equation}\label{mr:candidate-weight}
+ \sum_{e\text{ candidate}}w_e
+ \ge m\left(\frac25-\frac\delta2-\frac52\tau^2\right)
+ >\frac{3m}{10}.
+\end{equation}
+
+\subsection{A common bag for both normalizations}
+
+Apply \Cref{mt:finite-tree} again, this time to the single-label
+positive spans for all $u\in U$ together with the positive spans of all
+candidate groups.  These spans are allowed to overlap.  The required
+subtree contacts hold in each of the following cases: two single labels
+form a hole; two group types conflict; or a group type and a single
+label conflict.  In each case all cross Gram entries vanish, so the
+positive spans are orthogonal and their flats intersect.
+
+The candidate weights satisfy \Cref{mt:weighted-bag}, so a bag has
+group weight at least $cm$.  Apply \Cref{mt:common-bag} to the supplied
+single labels $U$ and these groups, using $q\le cm/60$.  It yields a
+bag containing all but fewer than $5q$ labels of $U$, and containing
+candidate group weight at least $cm/2$.  The common-bag proof uses all
+root labels absent after its threshold edge, so it applies equally
+when the losses occur one label at a time.
+
+Discard from the remaining coordinate matching every pair incident to one of the
+missing labels of $U$.  This removes whole label-pair groups and at most
+$5qs$ pairs.  Denote the surviving total number of pairs by $p=p_s$.
+Let $\mathcal T$ be the candidate groups in the common bag whose
+endpoints both remain, and let $t=t_s$ be their total number of pairs.
+Since the load at a missing label is at most one, the limiting weight
+lost from the bag is at most $5q$.  Thus
+\begin{align}
+ \liminf_s\frac pv
+ &\ge\frac12-\frac\delta2-\frac52\tau^2-\frac{5q}{m}
+ \ge\frac12-3\tau^2,
+       \label{mr:final-pair-count}\\
+ \lim_s\frac tv
+ &\ge\frac c2-\frac{5q}{m}\ge\frac c3.
+       \label{mr:extra-pair-count}
+\end{align}
+For the last inequality in \eqref{mr:final-pair-count}, use
+$5q/m\le\delta=\tau^4/3$ and $\tau\le1$.  All group counts have
+limits after the earlier subselection, so $t/v$ does as well.
+
+Write $X$ for the list of the $2p$ surviving matched endpoint columns,
+still in the first normalization. For $e\in\mathcal T$, let $k_e$
+denote its remaining number of pairs and write $X_e$ for its
+$2k_e$-column sublist, $D_e=\langle X_e,X_e\rangle$, and
+\begin{equation}\label{mr:second-normalization}
+ Q_e=X_eD_e^{-1/2},
+ \qquad Q=(Q_e)_{e\in\mathcal T}.
+\end{equation}
+The common bag gives a uniform operator bound for the joint Gram of
+the first-normalized single blocks and all $Q_e$.  Passing to the
+sublist $X$ does not increase this bound.  This point controls $Q_e$
+even when some eigenvalues of $D_e$ tend to zero and the coefficients
+in $D_e^{-1/2}$ become large.
+
+\begin{figure}[tb]
+\centering
+\begin{tikzpicture}[
+  box/.style={draw,rounded corners,align=center,text width=4.2cm,
+              minimum height=1.85cm,inner sep=5pt},
+  >=Latex,font=\small]
+\node[box] (first) {First normalization\\
+  $\langle X_u,X_u\rangle=I$\\coordinate pairs};
+\node[box,right=1.1cm of first] (sumdiff) {Pair differences $R$\\
+  $\langle R,R\rangle\approx2I$\\pair sums $H$\\have small trace};
+\node[box,below=1.15cm of first] (second) {Second normalization\\
+  $\langle Q_e,Q_e\rangle=I$\\within candidate groups};
+\node[box,right=1.1cm of second] (extra) {Exact complements $Q_eL_e$\\
+  $k_e$ extra directions\\orthogonal to $R_e$};
+\draw[->] (first)--(sumdiff);
+\draw[->] (first)--node[left,align=right]{one common\\tree bag}(second);
+\draw[->] (second)--(extra);
+\draw[->] (sumdiff)--node[right,align=left]{small cross Gram\\via $H$}(extra);
+\end{tikzpicture}
+\caption{The two normalizations.  The common bag bounds all indicated
+Gram matrices.  The trace estimate for $H$, followed by a spectral cutoff
+at $2$, controls the interaction with the second normalization using an
+absolute constant.}
+\label{mr:two-scales}
+\end{figure}
+
+\subsection{The additional positive directions}
+
+For each surviving pair $j=\{i,i'\}$ choose an order of its endpoints
+and form
+\[
+ R_j=\frac{X_i-X_{i'}}{\sqrt2},
+ \qquad H_j=\frac{X_i+X_{i'}}{\sqrt2}.
+\]
+The coefficient maps producing either list from $X$ are Euclidean
+isometries.  The restriction of $I-W_s$ to retained pair endpoints has
+Gram $2I$ on the difference coefficients.  Hence
+\eqref{mr:matching-error} and contraction of Frobenius norm give
+\begin{equation}\label{mr:R-bound}
+ \|\langle R,R\rangle-2I_p\|_{\mathrm F}^2\le8\tau^2v.
+\end{equation}
+Also $\langle H_j,H_j\rangle=1-(B_s)_{ii'}$, so
+\begin{equation}\label{mr:H-trace}
+ |\Tr\langle H,H\rangle|\le p\tau\le\frac{\tau v}{2}.
+\end{equation}
+These statements persist after all the pair deletions: they are
+restrictions of the original matching estimates.
+
+The coefficients in each $D_e^{-1/2}$ are entrywise nonnegative.
+Distinct groups have disjoint endpoint coordinates, and all cross Gram
+entries between such coordinates are nonpositive.  Coordinates with
+the same base label have zero interaction, since the first diagonal
+blocks are identities.  Therefore
+\begin{equation}\label{mr:Q-sign}
+ \langle Q,Q\rangle=I-C_s,
+ \qquad C_s\ge0\ \text{entrywise},
+ \qquad (C_s)_{ee}=0\ \text{as a whole group block}.
+\end{equation}
+It is bounded in operator norm and has at most one negative eigenvalue.
+By \Cref{mr:spectral-support}, its limiting spectral support lies in
+$[0,2]$: more precisely, for every fixed $\varepsilon>0$ there are
+$o(v)$ eigenvalues outside $[-\varepsilon,2+\varepsilon]$.
+The joint Gram of $(H,Q)$ is bounded by the common bag.  Applying
+\Cref{mr:trace-cross} to \eqref{mr:H-trace} yields the absolute estimate
+\begin{equation}\label{mr:HQ-bound}
+ \limsup_s\frac{\|\langle H,Q\rangle\|_{\mathrm F}^2}{v}
+ \le\tau.
+\end{equation}
+In particular, no factor depending on the norm of the common bag
+appears here.
+
+Fix a candidate group $e\in\mathcal T$. Let $U_e$ be the
+$2k_e\times k_e$ Euclidean isometry whose columns are the coefficient
+vectors of its pair differences. Then
+\[
+ R_e=X_eU_e=Q_e T_e,
+ \qquad T_e=D_e^{1/2}U_e.
+\]
+Because $D_e$ is positive definite, $T_e$ has rank exactly $k_e$.
+Choose a $2k_e\times k_e$ matrix $L_e$ whose columns are an orthonormal
+basis of $(\im T_e)^\perp$.  Thus
+\begin{equation}\label{mr:exact-complement}
+ L_e^{\mathsf T}L_e=I_{k_e},
+ \qquad\langle R_e,Q_eL_e\rangle=T_e^{\mathsf T}L_e=0.
+\end{equation}
+This uses the full span of all the group's original difference
+directions; no small singular direction is discarded.  Write
+$L=\diag(L_e)_{e\in\mathcal T}$ and $Y=QL$.  The list $Y$ has
+exactly $t$ columns.
+
+For a pair $j$ outside group $e$, each of the two numbers
+$x=\langle X_i,(Q_e)_k\rangle$ and
+$y=\langle X_{i'},(Q_e)_k\rangle$ is nonpositive.  Consequently
+\begin{equation}\label{mr:sign-domination}
+ |\langle R_j,(Q_e)_k\rangle|
+    =\frac{|x-y|}{\sqrt2}
+    \le-\frac{x+y}{\sqrt2}
+    =-\langle H_j,(Q_e)_k\rangle.
+\end{equation}
+This includes pairs in noncandidate groups.  Let $Z$ be
+$\langle R,Q\rangle$ with the entries between each $R_e$ and its
+own $Q_e$ replaced by zero.  Then
+$\|Z\|_{\mathrm F}^2\le\|\langle H,Q\rangle\|_{\mathrm F}^2$,
+and \eqref{mr:exact-complement} gives
+$\langle R,Y\rangle=ZL$.  Since $L$ is an isometry, Frobenius
+contraction and \eqref{mr:HQ-bound} imply
+\begin{equation}\label{mr:RY-bound}
+ \limsup_s\frac{\|\langle R,Y\rangle\|_{\mathrm F}^2}{v}
+ \le\tau.
+\end{equation}
+
+Put $E_s=\langle Y,Y\rangle=L^{\mathsf T}\langle Q,Q\rangle L$.
+Every diagonal group block of $E_s$ is an identity, so
+\begin{equation}\label{mr:Y-trace}
+ \Tr E_s=t.
+\end{equation}
+This is a Euclidean coefficient compression.  The variational
+characterization of eigenvalues shows that the number of eigenvalues
+of $E_s$ above $2+\varepsilon$ is at most that for
+$\langle Q,Q\rangle$, hence is $o(v)$.  Its norm is bounded.
+If $k_s$ denotes its number of eigenvalues greater than $1/4$, use
+the cutoff $2+\varepsilon=5/2$ to obtain
+\[
+ t=\Tr E_s\le\frac{t-k_s}{4}+\frac52k_s+o(v).
+\]
+Possible negative eigenvalues only decrease the left-hand trace
+contribution in this upper estimate.  Rearranging gives
+\begin{equation}\label{mr:Y-positive-count}
+ k_s\ge t/3-o(v).
+\end{equation}
+
+Finally consider the joint Gram matrix of $(R,Y)$.  By
+\eqref{mr:R-bound}, the number of eigenvalues of
+$\langle R,R\rangle$ at most $1/4$ is at most
+\[
+ \frac{16}{49}\|\langle R,R\rangle-2I_p\|_{\mathrm F}^2
+ \le\frac{128}{49}\tau^2v.
+\]
+Restrict the two diagonal blocks to their spectral spaces above
+$1/4$.  This leaves at least
+$p-(128/49)\tau^2v$ dimensions on the $R$ side and
+$t/3-o(v)$ on the $Y$ side.  The restricted cross matrix has squared
+Frobenius norm at most $\|\langle R,Y\rangle\|_{\mathrm F}^2$.
+Delete from the first side its left singular directions with singular
+values greater than $1/8$.  The number deleted is at most
+$64\|\langle R,Y\rangle\|_{\mathrm F}^2$.  The remaining diagonal
+blocks are greater than $I/4$, and the cross operator norm is at most
+$1/8$.  For their coefficient vectors $x,y$, the joint quadratic form
+is bounded below by
+\[
+ \frac14(\|x\|^2+\|y\|^2)-\frac14\|x\|\|y\|
+ \ge\frac18(\|x\|^2+\|y\|^2),
+\]
+so this remaining space is positive definite.  Using
+\eqref{mr:RY-bound}, the positive index of the joint Gram is at least
+\begin{equation}\label{mr:rank-gain}
+ p+\frac t3-\frac{128}{49}\tau^2v-64\tau v-o(v).
+\end{equation}
+Every column used here is a linear combination of the original Lorentz
+Gram columns for $M_s$.  The joint Gram is therefore a coefficient
+pullback of $M_s$, and its positive index is at most $\rank M_s$.
+This argument does not require the combined list $(R,Y)$ to be
+linearly independent.
+
+Combine \eqref{mr:final-pair-count}, \eqref{mr:extra-pair-count} and
+\eqref{mr:rank-gain}.  Since $\tau\le1/8$,
+\begin{align*}
+ \liminf_s\frac{\rank M_s}{v}
+ &\ge\frac12+\frac c9-64\tau-\frac{275}{49}\tau^2\\
+ &\ge\frac12+\frac c9-65\tau
+ \ge\frac12+\frac c{12},
+\end{align*}
+where the last inequality uses $\tau\le c/2340$.
+This contradicts \eqref{mr:contradiction-rank}, whose normalized
+right-hand side tends to $1/2$.  Hence only finitely many $s$ can
+violate the stated rank bound.  The positive gap also shows directly
+why the literal additive $+1$ is harmless for all sufficiently large
+$s$.
+
+Every matrix in the definition of $\mu(G[K_s])$ is among the matrices
+just considered.  Rank--nullity gives
+$\dim\ker M<ms/2-1$, and taking the maximum yields
+$\mu(G[K_s])+1<ms/2$ as asserted.
+\end{proof}
+
+For completeness, the matrix class defining $\mu(H)$ is nonempty for
+every finite nonempty graph $H$.  Start from a diagonal matrix with one diagonal
+entry $-1$ and all others $1$, and put the same sufficiently small
+negative number in each edge position.  The perturbation has operator
+norm less than $1/2$ when its size is sufficiently small, so the matrix
+is nonsingular with exactly one negative eigenvalue.  Nonsingularity
+makes the Strong Arnold Property immediate.
+
+If the base graph additionally satisfies $\alpha(G)\le2$, then
+$\alpha(G[K_s])\le2$: an independent set uses at most one vertex of
+each clique, and its base labels are independent.  Thus
+$\chi(G[K_s])\ge\lceil ms/2\rceil>\mu(G[K_s])+1$.  Together with
+\Cref{thm:base-graphs}, the theorem above proves \Cref{thm:main}.
+One may also recover the chromatic number exactly as
+$\chi(G[K_s])=ms-\nu(\overline{G[K_s]})$, where $\nu$ is the maximum
+matching size: every color class has one or two vertices, and the
+two-vertex classes are precisely disjoint hole pairs.
diff --git a/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/matrix-trees.tex b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sections/matrix-trees.tex
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+\section{Finite trees for Lorentz blocks}\label{mt:section}
+
+The matrix argument needs a way to normalize several positive definite
+blocks at once, even when their dimensions tend to infinity.  We first
+give a geometric construction whose constants depend only on the number
+of blocks.  We then use the three conflict properties of
+\Cref{thm:base-graphs} to select the blocks that will be normalized
+together.  All vector spaces and bilinear forms in this section are real.
+Logarithms in the hyperbolic estimates are natural logarithms.
+
+\subsection{Hyperbolic flats and block normalization}
+
+A \emph{Lorentz space} is a finite-dimensional real vector space with
+bilinear form
+\[
+ [x,y]=-x_0y_0+\sum_{j=1}^d x_jy_j.
+\]
+Its future unit hyperboloid is
+\[
+ \mathbb H^d=\{x:[x,x]=-1,\ x_0>0\},
+ \qquad d_{\mathbb H}(x,y)=\operatorname{arcosh}(-[x,y]).
+\]
+Unused positive coordinates may always be added, so we may take
+\(d\ge1\).  The orthogonal complement of a future unit vector is positive
+definite.  Indeed, if \(x=(x_0,u)\) and \([x,(t,v)]=0\), then
+\(t=u\cdot v/x_0\), and
+\[
+ |v|^2-t^2\ge |v|^2/x_0^2.
+\]
+An orthonormal basis of this complement, together with \(x\), therefore
+gives Lorentz coordinates in which \(x=(1,0)\).  In these coordinates any
+other future unit vector has the form
+\((\cosh r,\sinh r\,u)\), with \(|u|=1\) and \(r\ge0\).
+In particular \(-[x,y]\ge1\).  If \(y,z\) have radii \(r,s\) about
+\(x\), the ordinary Cauchy--Schwarz inequality gives
+\[
+ -[y,z]=\cosh r\cosh s-\sinh r\sinh s\,(u\cdot v)
+ \le \cosh(r+s).
+\]
+This proves the triangle inequality for \(d_{\mathbb H}\); its other
+metric properties follow from the same coordinates.
+
+The segment from \(x\) to \(y\), of length \(\ell>0\), is
+\begin{equation}\label{mt:hyperbolic-segment}
+ \gamma(t)=\frac{\sinh(\ell-t)}{\sinh\ell}x
+            +\frac{\sinh t}{\sinh\ell}y,
+ \qquad 0\le t\le\ell.
+\end{equation}
+In coordinates about \(x\) it is \((\cosh t,\sinh t\,u)\), so
+\(d_{\mathbb H}(\gamma(t),\gamma(t'))=|t-t'|\).
+When \(x=y\) the segment is constant.  A subspace \(P\) is
+\emph{positive} if the Lorentz form restricted to it is positive
+definite.  Its associated \emph{flat} is
+\[
+ F_P=P^\perp\cap\mathbb H^d.
+\]
+Orthogonal decomposition by a positive subspace leaves exactly one
+negative square, so \(F_P\) is nonempty.  Formula
+\eqref{mt:hyperbolic-segment} shows that it contains the segment between
+any two of its points.
+
+\begin{lemma}[Normalization near a flat]\label{mt:flat-normalization}
+Let \(P\) be positive and write \(x=x_P+x_\perp\) for the orthogonal
+decomposition of \(x\in\mathbb H^d\) relative to \(P\).  Then
+\begin{equation}\label{mt:flat-distance}
+ \sinh^2 d_{\mathbb H}(x,F_P)=[x_P,x_P].
+\end{equation}
+Suppose \(P_1,\ldots,P_k\) are positive subspaces whose flats have
+distance at most \(R\) from one common point.  Choose any
+Lorentz-orthonormal basis in each \(P_i\), concatenate these bases into
+a list \(A\), and let \([A,A]\) be its Gram matrix.  Then
+\begin{equation}\label{mt:block-operator-bound}
+ \|[A,A]\|_{\mathrm{op}}
+ \le k(\cosh^2R+\sinh^2R).
+\end{equation}
+This bound does not depend on the dimensions of the subspaces.  The
+subspaces are allowed to overlap.  If \(P_i\perp P_j\), then
+\(F_{P_i}\cap F_{P_j}\ne\varnothing\).
+\end{lemma}
+\begin{proof}
+Put \(q=[x_P,x_P]\).  Then \([x_\perp,x_\perp]=-1-q\), and
+\(y=x_\perp/\sqrt{1+q}\) is future unit timelike.  For \(z\in F_P\),
+the hyperbolic coordinates above, now in \(P^\perp\), give
+\[
+ -[x,z]=-[x_\perp,z]\ge\sqrt{1+q},
+\]
+with equality at \(z=y\).  This proves \eqref{mt:flat-distance}.
+
+Use the common point as the time coordinate.  For the basis matrix
+\(A_i\), let \(t_i\) be its time-coordinate row and \(S_i\) its
+spatial-coordinate matrix.  The projection formula gives
+\(\|t_i\|\le\sinh R\), and orthonormality gives
+\[
+ S_i^{\mathsf T}S_i=I+t_i^{\mathsf T}t_i,
+ \qquad \|S_i\|_{\mathrm{op}}\le\cosh R.
+\]
+For the concatenated matrices \(S,t\), Cauchy--Schwarz over the \(k\)
+blocks gives \(\|S\|_{\mathrm{op}}^2\le k\cosh^2R\) and
+\(\|t\|^2\le k\sinh^2R\).  Since
+\([A,A]=S^{\mathsf T}S-t^{\mathsf T}t\), this proves
+\eqref{mt:block-operator-bound}.  Finally, if \(P_i\perp P_j\), their
+sum is positive, and its orthogonal complement contains a future unit
+timelike vector perpendicular to both.
+\end{proof}
+
+\subsection{A finite configuration and its segment points}
+
+A \emph{finite metric tree} is a finite tree whose edges are intervals
+of positive lengths, equipped with path distance; a single point is
+also allowed.  Its segment between \(a,b\) is denoted \([a,b]_T\).
+
+Finite hyperbolic configurations admit tree approximations with
+logarithmic additive error through Gromov's construction
+\citep{Gromov1987Hyperbolic}. Its maximum--minimum chain closure is
+recorded explicitly by \citet[Section~2, equation~(11)]{CornectMartinezPedroza2025Tree}.
+We give a direct hyperboloid argument with an error depending only on
+the number of marked points; the subsequent segment and flat lemmas
+supply the additional assertions needed for matrix normalization.
+
+\begin{lemma}[Finite tree approximation]\label{mt:finite-metric}
+For \(\ell\ge1\), put
+\[
+ K(\ell)=4+2\log\bigl(\max\{1,\ell-1\}\bigr).
+\]
+Given any \(\ell\) points \(z_1,\ldots,z_\ell\) in a hyperboloid, in any
+dimension, there is a finite metric tree with marked points
+\(z_1^T,\ldots,z_\ell^T\) such that
+\begin{equation}\label{mt:endpoint-error}
+ \bigl|d_T(z_i^T,z_j^T)-d_{\mathbb H}(z_i,z_j)\bigr|\le K(\ell).
+\end{equation}
+The tree may be taken to be the hull of its marked points.
+\end{lemma}
+\begin{proof}
+Take \(z_1\) as origin and write
+\(z_i=(\cosh r_i,\sinh r_i\,u_i)\).  A direction with \(r_i=0\)
+can be chosen arbitrarily.  For distinct indices set
+\[
+ q_{ij}=\frac{|u_i-u_j|}{2},\qquad
+ a_{ij}=\min\{r_i,r_j,-\log q_{ij}\},
+\]
+with \(-\log0=+\infty\), and set \(a_{ii}=r_i\).  The exact identity
+\begin{equation}\label{mt:radial-identity}
+ \cosh d_{\mathbb H}(z_i,z_j)
+ =\cosh(r_i-r_j)+2\sinh r_i\sinh r_j\,q_{ij}^2
+\end{equation}
+implies
+\begin{equation}\label{mt:raw-depth-error}
+ \left|d_{\mathbb H}(z_i,z_j)-(r_i+r_j-2a_{ij})\right|\le4.
+\end{equation}
+Here is a dimension-free verification.  If \(\min(r_i,r_j)<1\),
+the triangle inequality and its reverse put both
+\((r_i+r_j-d_{\mathbb H}(z_i,z_j))/2\) and \(a_{ij}\) between zero
+and \(\min(r_i,r_j)\), giving error at most two.  Otherwise, writing
+\[
+ D=\max\{|r_i-r_j|,\ r_i+r_j+2\log q_{ij}\}
+   =r_i+r_j-2a_{ij},
+\]
+the two terms in \eqref{mt:radial-identity} give
+\(e^D/4\le\cosh d_{\mathbb H}(z_i,z_j)\le2e^D\).
+Use \(\log u\le\operatorname{arcosh}u\le\log(2u)\) for \(u\ge1\)
+to obtain an error at most \(\log4<4\).
+
+For \(i\ne j\), define \(A_{ij}\) as the largest bottleneck value
+\(\min_{0\le t<k}a_{i_ti_{t+1}}\) over chains
+\(i=i_0,\ldots,i_k=j\), and put \(A_{ii}=r_i\).
+Deleting loops never decreases a bottleneck, so only simple chains
+of at most \(\ell-1\) edges are needed.  If a chain has bottleneck \(h\),
+its endpoint radii are at least \(h\), and the triangle inequality
+for the direction chords gives
+\(q_{ij}\le(\ell-1)e^{-h}\).  Consequently
+\begin{equation}\label{mt:bottleneck-bound}
+ a_{ij}\le A_{ij}
+ \le a_{ij}+\log\bigl(\max\{1,\ell-1\}\bigr),\qquad
+ A_{ij}\le\min(r_i,r_j).
+\end{equation}
+Concatenating chains also gives
+\(A_{ij}\ge\min(A_{it},A_{tj})\) for every \(t\).
+Take intervals \([0,r_i]\), identifying their points of equal height
+\(t\) whenever \(t\le A_{ij}\).  The last inequality makes this an
+equivalence relation at every height.  These equivalence classes are
+nested as height decreases; their finitely many splitting heights
+therefore form a rooted finite tree.  The distance between the two
+marked upper endpoints is
+\(r_i+r_j-2A_{ij}\).  Equations
+\eqref{mt:raw-depth-error}--\eqref{mt:bottleneck-bound} prove
+\eqref{mt:endpoint-error}.  Restricting to the hull of the marked
+points preserves their distances.
+\end{proof}
+
+Endpoint approximation alone does not justify lifting a point of a
+tree hull back into a flat.  The next lemma supplies this additional
+step, uniformly in the location of the point along its segment.
+
+\begin{lemma}[Comparison of segment points]\label{mt:segment-comparison}
+Let \(T\) and the points \(z_i^T\) satisfy
+\eqref{mt:endpoint-error}.  For \(\theta\in[0,1]\), associate the
+point at fraction \(\theta\) of \([z_i^T,z_j^T]_T\) with the point
+at the same fraction of the hyperbolic segment \([z_i,z_j]\).
+If \(p,q\) are obtained in this way, from any two pairs of marked
+points, and \(\widehat p,\widehat q\) are their associated hyperbolic
+points, then
+\begin{equation}\label{mt:segment-error}
+ \bigl|d_{\mathbb H}(\widehat p,\widehat q)-d_T(p,q)\bigr|
+ \le 11K(\ell+2).
+\end{equation}
+For a degenerate segment any choice of its fractional parameter gives
+the same conclusion.
+\end{lemma}
+\begin{proof}
+We first record an identity valid in any metric tree.  If
+\(p\in[a,b]_T\) and \(q\in[c,d]_T\), then
+\begin{equation}\label{mt:tree-segment-identity}
+ d_T(p,q)=\max_{e\in\{a,b\},\ f\in\{c,d\}}
+ \{d_T(e,f)-d_T(e,p)-d_T(f,q)\}.
+\end{equation}
+Every term is at most the left side by the triangle inequality.
+If \(p\ne q\), choose an endpoint \(e\) whose path from \(p\)
+does not enter the component toward \(q\), and an endpoint \(f\)
+whose path from \(q\) does not enter the component toward \(p\).
+Such choices exist because \(p,q\) lie on their respective segments;
+an endpoint equal to the segment point is allowed.  Then the path
+from \(e\) to \(f\) passes through \(p,q\), giving equality.  If
+\(p=q\), either choose an endpoint equal to this common point, or
+choose endpoints in different components of its complement; again
+equality holds.
+
+A second exact tree identity will locate the new marked points.
+If $P_\theta$ is at fraction $\theta$ of $[A,B]$ and $X$ is any
+point of the tree, then
+\begin{equation}\label{mt:tree-point-identity}
+ d(X,P_\theta)=\max\{d(A,X)-\theta d(A,B),\,
+                     d(B,X)-(1-\theta)d(A,B)\}.
+\end{equation}
+Indeed, if the projection of $X$ onto $[A,B]$ is at distance $u$
+from $A$ and at distance $h$ from $X$, both sides equal
+$h+|u-\theta d(A,B)|$.  This also holds for a degenerate segment.
+
+Apply \Cref{mt:finite-metric} to the original configuration together
+with $\widehat p,\widehat q$, obtaining a tree $T'$ with error
+$K'=K(\ell+2)$.  Denote their marked points by $p',q'$.  Suppose
+$\widehat p$ is at fraction $\theta$ of the hyperbolic segment
+$[a,b]$, and let $\widetilde p$ have that fraction on
+$[a',b']_{T'}$.  The two exact hyperbolic distances from $\widehat p$
+to its endpoints are $\theta d_{\mathbb H}(a,b)$ and
+$(1-\theta)d_{\mathbb H}(a,b)$.  In
+\eqref{mt:tree-point-identity}, each corresponding difference is
+therefore at most $2K'$.  Thus
+\[
+ d_{T'}(p',\widetilde p)\le2K',\qquad
+ d_{T'}(q',\widetilde q)\le2K'.
+\]
+Together with the $K'$ distance error for the pair
+$\widehat p,\widehat q$, this gives
+\[
+ |d_{\mathbb H}(\widehat p,\widehat q)
+       -d_{T'}(\widetilde p,\widetilde q)|\le5K'.
+\]
+Between the original endpoint sets, distances in $T$ and $T'$
+differ by at most $K(\ell)+K'\le2K'$.  Distances from an endpoint
+to its fractional segment point are that fraction, or its complement,
+times the segment length.  Each term in
+\eqref{mt:tree-segment-identity} therefore changes by at most
+$6K'$, and hence
+\[
+ |d_T(p,q)-d_{T'}(\widetilde p,\widetilde q)|\le6K'.
+\]
+The total error is at most $11K'$.  All identities remain valid when
+an endpoint segment has length zero.
+\end{proof}
+
+\begin{proposition}[Finite tree principle]\label{mt:finite-tree}
+For each index \(s\), let \(P_{1,s},\ldots,P_{k,s}\) be positive
+subspaces in a Lorentz space, whose dimension may depend on \(s\).
+Fix a set \(\mathcal D\) of pairs of distinct labels, and assume
+\[
+ \sup_s d_{\mathbb H}(F_{P_{i,s}},F_{P_{j,s}})\le D<\infty
+ \qquad (\{i,j\}\in\mathcal D).
+\]
+After passing to a subsequence, there are a finite combinatorial tree
+\(T\) and a nonempty connected vertex set \(S_i\subseteq V(T)\)
+for every label, with the following properties.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item \(S_i\cap S_j\ne\varnothing\) for every
+\(\{i,j\}\in\mathcal D\).
+\item For every vertex \(x\), called a \emph{bag}, choose any
+Lorentz-orthonormal basis in every \(P_{i,s}\) with \(x\in S_i\).
+The Gram matrix of the concatenated list has operator norm bounded
+uniformly in \(s\).  The bound depends only on \(k,D\), and not on
+the dimensions or on the bases.
+\end{enumerate}
+In particular, mutually orthogonal blocks can always be required to
+have intersecting subtrees.
+\end{proposition}
+\begin{proof}
+For every required pair, choose a point in each of its two flats at
+distance at most \(D+1\); an infimum need not be attained.  Choose
+also one point in each flat.  This gives
+\(\ell=k+2|\mathcal D|\le k^2\) marked representatives.  Apply
+\Cref{mt:finite-metric} and work in their tree hull \(K_s\).
+Let \(H_{i,s}\) be the hull of representatives chosen from the
+\(i\)-th flat, and set
+\[
+ R=\frac{D+1+K(\ell)}2,\qquad
+ U_{i,s}=\{x\in K_s:d_T(x,H_{i,s})\le R\}.
+\]
+The \(R\)-neighborhood of a subtree in a tree is a closed subtree: the
+segment between two of its points lies in the union of their paths
+to the original subtree and the subtree itself, and stays within
+distance \(R\) of it.
+For a required pair, the midpoint of its two witness representatives
+belongs to both neighborhoods.  Thus every required contact is
+preserved inside \(K_s\).
+
+Suppose \(x\in U_{i,s}\cap U_{j,s}\).  Choose
+\(h_i\in H_{i,s}\), \(h_j\in H_{j,s}\) at distance at most \(R\)
+from \(x\).  Every point of a finite tree hull lies on a segment
+between two of its defining representatives, allowing a repeated
+representative when the hull is a point.  For example, after fixing
+one representative the union of its segments to all the others is
+already their hull.  Lift \(h_i,h_j\) at their fractional parameters
+along the corresponding actual hyperbolic segments.  Their lifts
+\(z_i,z_j\) belong to their respective flats by convexity, and
+\Cref{mt:segment-comparison} gives
+\begin{equation}\label{mt:bag-radius}
+ d_{\mathbb H}(z_i,z_j)\le 2R+11K(\ell+2)=:R_*.
+\end{equation}
+For any nonempty bag, fix one of these lifts.  It has distance at
+most \(R_*\) from every flat in the bag.  By
+\Cref{mt:flat-normalization}, their joint Gram norm is at most
+\(k(\cosh^2R_*+\sinh^2R_*)\).
+
+It remains to obtain a single finite combinatorial pattern.  Suppress
+the degree-two vertices of \(K_s\) except marked representatives;
+there are at most \(2\ell\) vertices and at most \(2\ell\) edges.
+Each \(U_{i,s}\) meets an edge in an interval or the empty set.
+Subdivide every edge at all endpoints of these intervals.  This
+adds at most \(2k\) vertices per edge, so the resulting tree has at
+most \(2\ell(2k+1)\) vertices.  Each \(U_{i,s}\) is now a subcomplex.
+There are only finitely many trees of this bounded size with \(k\)
+specified nonempty connected vertex sets.  Along a subsequence the
+labelled pattern is constant.  Take this pattern for \(T,S_i\).
+Equation \eqref{mt:bag-radius} proves the uniform bag assertion.
+The final assertion follows from \Cref{mt:flat-normalization} with
+\(D=0\).
+\end{proof}
+
+For a finite list of flat sequences, a preliminary subsequence can
+also arrange that every pairwise distance is either uniformly bounded
+or tends to infinity.  For each pair in turn, take a bounded subsequence
+if one exists, and otherwise the distance tends to infinity; previous
+choices remain valid after further subselection.  Applying
+\Cref{mt:finite-tree} with all bounded pairs as required contacts
+preserves every such pair.  No assertion that the remaining subtrees
+are disjoint is needed.
+
+\subsection{Weighted extraction from the conflict properties}
+
+We use a finite graph \(G\) on \(m\) vertices with the three
+properties in \Cref{thm:base-graphs}, with parameters \(a,\beta>0\).
+For clarity, the properties used here are:
+\begin{enumerate}[label=\textup{(\alph*)}]
+\item\label{mt:property-colour}
+Every matching of size at least \(m/20\), partitioned into classes
+of size at most \(am\), contains conflicting edges from different
+classes.
+\item\label{mt:property-point}
+Every matching of size at least \(\beta m\) and every vertex set of
+size at least \(\beta m\) contain an edge and a vertex in conflict.
+\item\label{mt:property-hole}
+Every two vertex sets of size at least \(\beta m\) have a cross hole.
+\end{enumerate}
+A hole is a nonedge between distinct vertices.  Edges conflict when
+they are disjoint and all four cross pairs are holes.  An edge and a
+vertex conflict when the vertex is outside the edge and both cross
+pairs are holes.  Thus a shared endpoint prevents a conflict.
+We may decrease \(a\) and assume \(0<a\le1\).  Put
+\begin{equation}\label{mt:constants}
+ q=\lceil\beta m\rceil,\qquad
+ D_a=\lceil\log_2(5/a)\rceil+1,\qquad
+ K_a=\lceil5/a\rceil D_a,\qquad
+ c=c(a)=\frac1{40K_a}.
+\end{equation}
+
+An edge family \(\mathcal E\subseteq E(G)\) has \emph{loads at most
+one} if its nonnegative weights satisfy
+\begin{equation}\label{mt:loads}
+ \sum_{e\in\mathcal E:\,v\in e}w_e\le1
+ \qquad(v\in V(G)).
+\end{equation}
+Then \(\sum_e w_e\le m/2\).  We only need edges of positive weight.
+
+\begin{lemma}[Matching from bounded loads]\label{mt:load-matching}
+Every edge family of total weight \(W\) satisfying
+\eqref{mt:loads} has a matching of size at least \(W/2\).
+If it contains no conflicting pair and \(am\ge1\), then
+\(W<m/10\).
+\end{lemma}
+\begin{proof}
+Take a maximal matching of size \(k\) in the family.  Its \(2k\)
+endpoints cover every edge, so the load bound gives \(W\le2k\).
+In the second assertion, \(k\ge m/20\) would violate
+Property~\ref{mt:property-colour} with each matching edge given its
+own class.  Hence \(k<m/20\), proving the claim.
+\end{proof}
+
+\begin{lemma}[A bag with almost all vertex labels]\label{mt:single-bag}
+Let \(V_0\subseteq V(G)\), and assign a nonempty subtree
+\(S_v\) of a finite combinatorial tree to each \(v\in V_0\).
+Assume \(S_u\cap S_v\ne\varnothing\) whenever \(uv\) is a hole.
+Then some tree vertex belongs to all but fewer than \(3q\) of the
+subtrees.
+\end{lemma}
+\begin{proof}
+At a tree vertex \(x\), each subtree missing \(x\) is wholly
+contained in a unique component of \(T-x\).  No two components
+can each contain at least \(q\) whole subtrees: taking \(q\) labels
+from each would give two vertex sets with no cross hole, contrary
+to Property~\ref{mt:property-hole}.  If there is a component with
+at least \(q\) whole subtrees, orient the first edge from \(x\)
+toward that component.  Each vertex has at most one outgoing edge.
+An edge cannot be oriented both ways, since its two orientations
+would again supply two separated sets of \(q\) labels.  Following
+arrows in a finite tree must therefore end at a sink \(r\).
+
+Every component of \(T-r\) contains at most \(q-1\) whole
+subtrees.  If \(q=1\), no subtree misses \(r\).  If \(q\ge2\)
+and at least \(3q-2\) labels missed \(r\), greedily unite components
+until their total first reaches \(q\).  This total is at most
+\(2q-2\); the remaining components contain at least \(q\) labels.
+These two separated families contradict
+Property~\ref{mt:property-hole}.  The sink thus misses fewer than
+\(3q-2\), and in particular fewer than \(3q\), labels.
+\end{proof}
+
+\begin{lemma}[A bag with positive edge weight]\label{mt:weighted-bag}
+Suppose \(am\ge2\).  Let an edge family \(\mathcal E\) satisfy
+\eqref{mt:loads} and have total weight greater than \(3m/10\).
+Assign a nonempty subtree \(S_e\) of a finite tree to each edge,
+and suppose conflicting edges have intersecting subtrees.
+Then some vertex \(z\) satisfies
+\[
+ \sum_{e:\,z\in S_e}w_e\ge cm.
+\]
+\end{lemma}
+\begin{proof}
+Assign every edge's weight to one chosen vertex of its subtree.
+Every finite tree with an assigned nonnegative mass has a vertex
+whose removal leaves each component with at most half the total
+mass.  To see this, minimize the sum of mass times graph distance
+from the vertex.  Moving to a neighbor in a component of mass
+greater than half the total would strictly decrease that sum.
+Such a minimizing vertex is a weighted centroid.
+
+Starting with \(T\), cut at a weighted centroid in every current
+component whose assigned mass exceeds \(am/10\), and repeat.
+At depth \(j\) each component has mass at most \(m/2^{j+1}\),
+since the original mass is at most \(m/2\).  Consequently at most
+\(D_a\) rounds are needed.  At each depth there are at most
+\(\lceil5/a\rceil\) components requiring a cut, because their
+masses sum to at most \(m/2\).  Thus at most \(K_a\) cut vertices
+are used.  Mass located at a cut vertex is removed at that cut.
+
+Suppose every bag has edge weight less than \(cm\).  Delete every
+edge whose subtree meets a cut vertex.  The deleted weight is less
+than \(K_acm=m/40\), so the surviving weight \(W'\) satisfies
+\(W'>m/4\).  Each surviving subtree is wholly contained in one
+component of the cut tree; its assigned vertex lies there too.
+The total surviving weight in each such component is at most
+\(am/10\).  Use the components as classes of surviving edges.
+Subtrees of different classes are disjoint, so edges in different
+classes cannot conflict.
+
+Put \(Q=\lfloor am\rfloor\), so \(Q\ge am/2\).  Greedily choose
+a matching, with at most \(Q\) selected edges from any class,
+until no further edge can be added.  If \(k\) edges have been
+selected, every surviving edge either meets one of their \(2k\)
+endpoints or belongs to a class with \(Q\) selected edges.  The
+first family has weight at most \(2k\).  There are at most
+\(k/Q\) saturated classes, each of weight at most \(am/10\).
+Therefore
+\[
+ \frac m4<W'\le 2k+\frac{k}{Q}\frac{am}{10}
+ \le\frac{11}{5}k.
+\]
+It follows that \(k>5m/44>m/20\).  Its class sizes are at most
+\(Q\le am\), but it has no conflict between different classes.
+This contradicts Property~\ref{mt:property-colour}.
+\end{proof}
+
+\begin{proposition}[A common bag]\label{mt:common-bag}
+Suppose the hypotheses of \Cref{mt:single-bag,mt:weighted-bag}
+hold in the same tree.  Suppose additionally that
+\(S_e\cap S_v\ne\varnothing\) whenever the edge \(e\) and the
+vertex \(v\in V_0\) conflict.  If
+\(q\le cm/4\), there is a vertex \(x\) such that
+\begin{equation}\label{mt:common-counts}
+ |\{v\in V_0:x\notin S_v\}|<5q,
+ \qquad \sum_{e:\,x\in S_e}w_e\ge\frac{cm}{2}.
+\end{equation}
+If all edges of \(\mathcal E\) have both endpoints in \(V_0\) and
+\(q\le cm/30\), deleting every edge incident with a missing vertex
+leaves bag weight at least \(cm/3\).
+\end{proposition}
+\begin{proof}
+Let \(r\) be the vertex bag from \Cref{mt:single-bag}, let
+\(z\) be the edge bag from \Cref{mt:weighted-bag}, and let
+\(A=\{v\in V_0:r\in S_v\}\).  Along the path from \(r\) to
+\(z\), each \(S_v\) with \(v\in A\) is an initial interval.
+Thus the number of absent labels from \(A\) is nondecreasing.
+If fewer than \(2q\) are absent at \(z\), use \(x=z\).
+It misses fewer than \(3q+2q\) supplied vertex labels and has
+edge weight at least \(cm\).
+
+Otherwise let \(xy\) be the first edge of this path, in the
+direction from \(r\) to \(z\), for which fewer than \(2q\)
+labels of \(A\) are absent at \(x\), and at least \(2q\) are
+absent at \(y\).  Set
+\[
+ A_- =\{v\in A:y\notin S_v\},\qquad
+ \mathcal E_+=\{e:z\in S_e,\ x\notin S_e\}.
+\]
+Every subtree indexed by \(A_-\) is contained in the \(x\)-side
+of \(T-xy\): it contains \(r\) and avoids \(y\).  Every subtree
+indexed by \(\mathcal E_+\) is contained in the \(y\)-side: it
+contains \(z\) and avoids \(x\).  In particular these two
+families are disjoint.  This uses \emph{all} root labels absent at
+\(y\), including labels that ceased to occur earlier on the path;
+it does not require many labels to disappear on the single edge
+\(xy\).  See \Cref{mt:fig-path}.
+
+If the bag weight at \(x\) were less than \(cm/2\), then
+\(\sum_{e\in\mathcal E_+}w_e>cm/2\).
+By \Cref{mt:load-matching}, \(\mathcal E_+\) has a matching of
+size greater than \(cm/4\), hence at least \(q\) since
+\(q\le cm/4\).  Also \(|A_-|\ge2q\).  No selected edge conflicts
+with any label in \(A_-\), because such a conflict would force
+intersecting subtrees.  Taking any \(q\) of these labels violates
+Property~\ref{mt:property-point}.  Thus the bag at \(x\) has weight
+at least \(cm/2\); it misses fewer than \(5q\) supplied labels.
+
+Finally, the load bound shows that removing all edges incident
+with these missing vertices costs less than \(5q\) weight.
+If \(q\le cm/30\), the surviving bag weight is at least
+\(cm/2-5q\ge cm/3\), as asserted.
+\end{proof}
+
+\begin{figure}[htbp]
+\centering
+\begin{tikzpicture}[x=1cm,y=1cm,every node/.style={font=\small}]
+\draw[very thick] (0,0)--(8,0);
+\foreach \x in {0,1.5,3,4.3,5.4,6.7,8}
+  \fill (\x,0) circle (1.7pt);
+\draw (1.5,0)--(1.1,-.55);
+\draw (3,0)--(3.4,-.55);
+\draw (6.7,0)--(7.05,-.55);
+\node[below] at (0,0) {$r$};
+\node[below] at (4.3,0) {$x$};
+\node[below] at (5.4,0) {$y$};
+\node[below] at (8,0) {$z$};
+\draw[dashed,thick] (4.85,-.9)--(4.85,2.4);
+\node[below,align=center] at (4.85,-.92) {cut edge $xy$};
+\draw[blue!65!black,very thick] (0,.55)--(1.5,.55);
+\draw[blue!65!black,very thick] (0,.95)--(3,.95);
+\draw[blue!65!black,very thick] (0,1.35)--(4.3,1.35);
+\node[blue!65!black,align=center] at (2.05,2.0)
+ {root labels absent at $y$\\at least $2q$ in total};
+\draw[red!65!black,very thick] (5.4,.55)--(8,.55);
+\draw[red!65!black,very thick] (6.0,.95)--(8,.95);
+\draw[red!65!black,very thick] (6.7,1.35)--(8,1.35);
+\node[red!65!black,align=center] at (6.65,2.0)
+ {groups at $z$\\absent at $x$};
+\end{tikzpicture}
+\caption{The path argument uses cumulative absence.  The blue intervals
+show intersections with the path of root subtrees that avoid \(y\);
+some may end well before \(x\).  The red intervals come from group
+subtrees containing \(z\) and avoiding \(x\).  The two whole families
+lie on opposite sides of the cut, so no edge--vertex conflict can
+occur between them.}
+\label{mt:fig-path}
+\end{figure}
+
+\begin{remark}[Use with coordinate matchings]\label{mt:coordinate-interface}
+In the matrix application a base label indexes at most \(s\)
+coordinates, and the edge weight is the limit of the number of matched
+coordinate pairs of that type divided by \(s\).  A matching gives
+\eqref{mt:loads} automatically.  Deleting all pairs incident with the
+fewer than \(5q\) labels missing from the common bag removes fewer
+than \(5qs\) pairs.  It removes entire edge types, so any previously
+chosen normalization inside a retained type stays the same.
+The finite tree principle applies jointly to the original positive
+label blocks and the positive blocks of the candidate edge types.
+These subspaces may overlap.  A common bag consequently bounds all
+their normalized Gram interactions at once.  Passing to coordinate
+subsets is a compression by an isometry, so it preserves that bound.
+\end{remark}
diff --git a/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sources/references.bib b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sources/references.bib
new file mode 100644
index 0000000000000000000000000000000000000000..de71ac67ddf7b2c5ca86c51fd870f5daa52bb44d
--- /dev/null
+++ b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/build/sources/references.bib
@@ -0,0 +1,396 @@
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+}
+
+
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+  title = {Three observations on the {Colin de Verdi{\`e}re} spectral graph parameter},
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+}
+
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+  author = {Gromov, M.},
+  title = {Hyperbolic groups},
+  editor = {Gersten, S. M.},
+  booktitle = {Essays in Group Theory},
+  series = {Mathematical Sciences Research Institute Publications},
+  volume = {8},
+  pages = {75--263},
+  publisher = {Springer},
+  address = {New York},
+  year = {1987},
+  doi = {10.1007/978-1-4613-9586-7_3}
+}
+
+
+@article{CornectMartinezPedroza2025Tree,
+  author = {Cornect, Anders and Mart{\'i}nez-Pedroza, Eduardo},
+  title = {{Gromov}'s approximating tree and the all-pairs bottleneck paths problem},
+  journal = {Involve},
+  note = {To appear; arXiv:2408.05338v2},
+  year = {2025},
+  eprint = {2408.05338},
+  archivePrefix = {arXiv},
+  primaryClass = {cs.CG},
+  url = {https://arxiv.org/abs/2408.05338v2}
+}
+
+
+@article{FiedlerSchneider1983Analytic,
+  author = {Fiedler, Miroslav and Schneider, Hans},
+  title = {Analytic functions of {$M$}-matrices and generalizations},
+  journal = {Linear and Multilinear Algebra},
+  volume = {13},
+  number = {3},
+  pages = {185--201},
+  year = {1983},
+  doi = {10.1080/03081088308817519}
+}
+
+
+@article{DebreuHerstein1953Nonnegative,
+  author = {Debreu, Gerard and Herstein, I. N.},
+  title = {Nonnegative square matrices},
+  journal = {Econometrica},
+  volume = {21},
+  number = {4},
+  pages = {597--607},
+  year = {1953},
+  doi = {10.2307/1907925}
+}
+
+
+@article{ChenHeZhang2025Stiefel,
+  author = {Chen, Xiaojun and He, Yifan and Zhang, Zaikun},
+  title = {Tight error bounds for the sign-constrained {Stiefel} manifold},
+  journal = {SIAM Journal on Optimization},
+  volume = {35},
+  number = {1},
+  pages = {302--329},
+  year = {2025},
+  doi = {10.1137/24M1659030}
+}
+
+
+@article{JiangMengWenChen2023Nonnegative,
+  author = {Jiang, Bo and Meng, Xiao and Wen, Zaiwen and Chen, Xiaojun},
+  title = {An exact penalty approach for optimization with nonnegative orthogonality constraints},
+  journal = {Mathematical Programming},
+  volume = {198},
+  pages = {855--897},
+  year = {2023},
+  doi = {10.1007/s10107-022-01794-8}
+}
+
+@article{reedseymour1998,
+  author = {Reed, Bruce and Seymour, Paul},
+  title = {Fractional Colouring and {Hadwiger}'s Conjecture},
+  journal = {Journal of Combinatorial Theory, Series B},
+  volume = {74},
+  number = {2},
+  pages = {147--152},
+  year = {1998},
+  doi = {10.1006/jctb.1998.1835},
+  url = {https://cgm.cs.mcgill.ca/~breed/SummerNSERC04/frachad.pdf}
+}
+
+@misc{OpenAILinearListHadwiger2026,
+  author = {{OpenAI}},
+  title = {{A linear list-coloring bound in terms of the Hadwiger number}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026/paper.pdf}{OAI:A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026}},
+  year = {2026}
+}
+
+@misc{OpenAIHadwigerCounterexample2026,
+  author = {{OpenAI}},
+  title = {{A counterexample to Hadwiger's conjecture}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Hadwigers-conjecture-September-23-2026/paper.pdf}{OAI:A-counterexample-to-Hadwigers-conjecture-September-23-2026}},
+  year = {2026}
+}
diff --git a/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/paper.pdf b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..16eb3140b4d465dd54e0ebcc24014a02c94b45e6
--- /dev/null
+++ b/preprints/A-counterexample-to-the-Colin-de-Verdiere-chromatic-conjecture-September-23-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:97677c7a183884f099bff03396dacf857ae723f03aa9485819bfb1c52273d4be
+size 1094760
diff --git a/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/README.md b/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..b9ce49dc08fa2ba94006ea444438b3f88654d44f
--- /dev/null
+++ b/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/README.md
@@ -0,0 +1,18 @@
+# [A counterexample to the Monge ansatz for the three-marginal Coulomb cost](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 25, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026,
+  author = {{OpenAI}},
+  title = {{A counterexample to the Monge ansatz for the three-marginal Coulomb cost}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/paper.pdf}{OAI:A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/build/figures/components.tex b/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/build/figures/components.tex
new file mode 100644
index 0000000000000000000000000000000000000000..833f1dfd4b3bd42243c42bda81206525ef175cf8
--- /dev/null
+++ b/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/build/figures/components.tex
@@ -0,0 +1,26 @@
+\begin{figure}[!b]
+\centering
+\begin{tikzpicture}[>=Stealth, every node/.style={font=\small},
+ component/.style={draw,rounded corners,minimum width=2.25cm,minimum height=.85cm,align=center}]
+\node[component,fill=black!4] (B) at (0,0) {$B$ near $0$\\mass $1/3$};
+\node[component,draw=blue!60!black,fill=blue!4] (Y1) at (3.8,0)
+ {$Y_1(B)$ near $e_1$\\mass $1/6$};
+\node[component,draw=blue!60!black,fill=blue!4] (W1) at (-3.8,0)
+ {$W_1(B)$ near $-e_1$\\mass $1/6$};
+\node[component,draw=orange!60!black,fill=orange!5] (Y2) at (0,1.9)
+ {$Y_2(B)$ near $e_2$\\mass $1/6$};
+\node[component,draw=orange!60!black,fill=orange!5] (W2) at (0,-1.9)
+ {$W_2(B)$ near $-e_2$\\mass $1/6$};
+\draw[->,blue!60!black] (B) -- node[above] {$Y_1$} (Y1);
+\draw[->,blue!60!black] (B) -- node[above] {$W_1$} (W1);
+\draw[->,orange!60!black] (B) -- node[right] {$Y_2$} (Y2);
+\draw[->,orange!60!black] (B) -- node[right] {$W_2$} (W2);
+\end{tikzpicture}
+\caption{Schematic positions of the five disjoint components, with their
+marginal masses. Every optimal plan is concentrated on triples joining
+a central point to its two images with the same subscript. The arrows
+denote local diffeomorphisms, not straight transport trajectories.
+For each indicated map \(H\) and Borel set \(S\subset B\),
+\(\mu(H(S))=\mu(S)/2\).}
+\label{fig:components}
+\end{figure}
diff --git a/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/build/paper.tex b/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/build/paper.tex
new file mode 100644
index 0000000000000000000000000000000000000000..1356ac431e6c83ebd4598984579cedc820cbe93a
--- /dev/null
+++ b/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/build/paper.tex
@@ -0,0 +1,62 @@
+\documentclass[11pt]{article}
+\usepackage[margin=1in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{microtype}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta}
+\usepackage{xcolor}
+\definecolor{linkblue}{RGB}{20,55,100}
+\usepackage[colorlinks=true,linkcolor=linkblue,citecolor=linkblue,urlcolor=linkblue]{hyperref}
+\hypersetup{
+  pdftitle={A counterexample to the Monge ansatz for the three-marginal Coulomb cost},
+  pdfauthor={OpenAI},
+  pdfsubject={Smooth identical marginals with no optimal transport maps for Coulomb and inverse-power Riesz interactions},
+  pdfkeywords={multi-marginal optimal transport, Coulomb cost, Riesz cost, Monge ansatz, nonattainment}
+}
+\usepackage{url}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newcommand{\R}{\mathbb R}
+\newcommand{\Id}{\operatorname{Id}}
+\newcommand{\supp}{\operatorname{supp}}
+\newcommand{\Leb}{\mathcal L}
+\newcommand{\Sym}{\operatorname{Sym}}
+\newcommand{\E}{E_*}
+\newcommand{\dd}{\,d}
+\newcommand{\calH}{\mathcal H}
+\newcommand{\calU}{\mathcal U}
+\DeclareMathOperator{\diag}{diag}
+\setlength{\emergencystretch}{1.5em}
+\title{A counterexample to the Monge ansatz\\
+for the three-marginal Coulomb cost}
+\author{OpenAI}
+\date{September 25, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We construct a smooth compactly supported probability density on
+\(\R^3\), with a smooth compactly supported square root, for which the
+three-marginal Coulomb transport minimum is not attained by any pair of
+measure-preserving Borel maps. Nevertheless, the Monge and Kantorovich
+infima agree: we construct preserving maps whose costs approach the
+Kantorovich minimum. For every \(d\ge2\) and \(s>0\), we also construct a
+smooth identical marginal with the same nonattainment and equality-of-infima
+properties for the inverse-power interaction
+\(\sum_{i<j}|x_i-x_j|^{-s}\) on \(\R^d\).
+\end{abstract}
+
+\input{sections/introduction}
+\input{sections/splitting}
+\input{sections/local}
+\input{sections/certificate}
+\input{sections/proof}
+\input{sections/infima}
+\input{sections/riesz}
+
+\bibliographystyle{plain}
+\bibliography{references}
+\end{document}
diff --git a/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/build/references.bib b/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/build/references.bib
new file mode 100644
index 0000000000000000000000000000000000000000..a18b46615cd90c63e445708a371551ff2d4e167e
--- /dev/null
+++ b/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/build/references.bib
@@ -0,0 +1,133 @@
+@article{CDM2015,
+  author = {Colombo, Maria and Di Marino, Simone},
+  title = {Equality between {Monge} and {Kantorovich} multimarginal problems with {Coulomb} cost},
+  journal = {Annali di Matematica Pura ed Applicata},
+  volume = {194},
+  pages = {307--320},
+  year = {2015},
+  doi = {10.1007/s10231-013-0376-0},
+  note = {\href{https://doi.org/10.1007/s10231-013-0376-0}{doi:10.1007/s10231-013-0376-0}.
+          Author version, April 11, 2013: \url{https://cvgmt.sns.it/media/doc/paper/2127/MultiColDim.pdf}}
+}
+
+@article{CS2016,
+  author = {Colombo, Maria and Stra, Federico},
+  title = {Counterexamples in multimarginal optimal transport with {Coulomb} cost and spherically symmetric data},
+  journal = {Mathematical Models and Methods in Applied Sciences},
+  volume = {26},
+  number = {6},
+  pages = {1025--1049},
+  year = {2016},
+  doi = {10.1142/S021820251650024X},
+  note = {\href{https://doi.org/10.1142/S021820251650024X}{doi:10.1142/S021820251650024X}}
+}
+
+@incollection{FGG2023,
+  author = {Friesecke, Gero and Gerolin, Augusto and Gori-Giorgi, Paola},
+  title = {The strong-interaction limit of density functional theory},
+  editor = {Canc{\`e}s, Eric and Friesecke, Gero},
+  booktitle = {Density Functional Theory: Modeling, Mathematical Analysis, Computational Methods, and Applications},
+  series = {Mathematics and Molecular Modeling},
+  publisher = {Springer},
+  address = {Cham},
+  pages = {183--266},
+  year = {2023},
+  doi = {10.1007/978-3-031-22340-2_4},
+  note = {\href{https://doi.org/10.1007/978-3-031-22340-2_4}{doi:10.1007/978-3-031-22340-2\_4}.
+          Preprint version (2022): \href{https://arxiv.org/abs/2202.09760v1}{arXiv:2202.09760v1}}
+}
+
+@article{GKR2019,
+  author = {Gerolin, Augusto and Kausamo, Anna and Rajala, Tapio},
+  title = {Nonexistence of optimal transport maps for the multimarginal repulsive harmonic cost},
+  journal = {SIAM Journal on Mathematical Analysis},
+  volume = {51},
+  number = {3},
+  pages = {2359--2371},
+  year = {2019},
+  doi = {10.1137/18M1186514},
+  note = {\href{https://doi.org/10.1137/18M1186514}{doi:10.1137/18M1186514}}
+}
+
+@article{CFK2013,
+  author = {Cotar, Codina and Friesecke, Gero and Kl{\"u}ppelberg, Claudia},
+  title = {Density Functional Theory and Optimal Transportation with {Coulomb} Cost},
+  journal = {Communications on Pure and Applied Mathematics},
+  volume = {66},
+  number = {4},
+  pages = {548--599},
+  year = {2013},
+  doi = {10.1002/cpa.21437},
+  note = {\href{https://doi.org/10.1002/cpa.21437}{doi:10.1002/cpa.21437}.
+          Preprint: \href{https://arxiv.org/abs/1104.0603v1}{arXiv:1104.0603v1}}
+}
+
+@article{CDDM2015,
+  author = {Colombo, Maria and De Pascale, Luigi and Di Marino, Simone},
+  title = {Multimarginal Optimal Transport Maps for One-dimensional Repulsive Costs},
+  journal = {Canadian Journal of Mathematics},
+  volume = {67},
+  number = {2},
+  pages = {350--368},
+  year = {2015},
+  doi = {10.4153/CJM-2014-011-x},
+  note = {\href{https://doi.org/10.4153/CJM-2014-011-x}{doi:10.4153/CJM-2014-011-x}}
+}
+
+@article{Pass2013,
+  author = {Pass, Brendan},
+  title = {Remarks on the semi-classical {Hohenberg--Kohn} functional},
+  journal = {Nonlinearity},
+  volume = {26},
+  number = {9},
+  pages = {2731--2744},
+  year = {2013},
+  doi = {10.1088/0951-7715/26/9/2731},
+  note = {\href{https://doi.org/10.1088/0951-7715/26/9/2731}{doi:10.1088/0951-7715/26/9/2731}.
+          Preprint: \href{https://arxiv.org/abs/1211.2766}{arXiv:1211.2766}}
+}
+
+
+@article{BDK2022,
+  author = {Bindini, Ugo and De Pascale, Luigi and Kausamo, Anna},
+  title = {On deterministic solutions for multi-marginal optimal transport with {Coulomb} cost},
+  journal = {Communications on Pure and Applied Analysis},
+  volume = {21},
+  number = {4},
+  pages = {1189--1208},
+  year = {2022},
+  doi = {10.3934/cpaa.2022015},
+  note = {\href{https://doi.org/10.3934/cpaa.2022015}{doi:10.3934/cpaa.2022015}.
+          Preprint: \href{https://arxiv.org/abs/2011.05063v2}{arXiv:2011.05063v2}}
+}
+
+@misc{Friesecke2026,
+  author = {Friesecke, Gero},
+  title = {Mass splitting in the time-discrete generalized {Euler} equations and non-{Monge} solutions in multi-marginal optimal transport},
+  year = {2026},
+  note = {\href{https://arxiv.org/abs/2601.02616v1}{arXiv:2601.02616v1}}
+}
+
+@article{SGS2007,
+  author = {Seidl, Michael and Gori-Giorgi, Paola and Savin, Andreas},
+  title = {Strictly correlated electrons in density-functional theory: A general formulation with applications to spherical densities},
+  journal = {Physical Review A},
+  volume = {75},
+  number = {4},
+  pages = {042511},
+  year = {2007},
+  doi = {10.1103/PhysRevA.75.042511},
+  note = {\href{https://arxiv.org/abs/cond-mat/0701025v2}{arXiv:cond-mat/0701025v2}}
+}
+
+@article{Rosenblatt1952,
+  author = {Rosenblatt, Murray},
+  title = {Remarks on a Multivariate Transformation},
+  journal = {The Annals of Mathematical Statistics},
+  volume = {23},
+  number = {3},
+  pages = {470--472},
+  year = {1952},
+  doi = {10.1214/aoms/1177729394},
+  note = {\href{https://doi.org/10.1214/aoms/1177729394}{doi:10.1214/aoms/1177729394}}
+}
diff --git a/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/build/sections/certificate.tex b/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/build/sections/certificate.tex
new file mode 100644
index 0000000000000000000000000000000000000000..c2b500475f18e2d79d693ce132df54012a97a351
--- /dev/null
+++ b/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/build/sections/certificate.tex
@@ -0,0 +1,124 @@
+\section{A supporting inequality on five components}\label{sec:certificate}
+
+The two local inequalities use the same central potential. We can
+therefore combine their functions into one potential on five disjoint
+components. To identify its entire equality set, we must also show
+strict inequality for every component pattern other than permutations
+of the two opposite-pair triples \((0,a_k,-a_k)\).
+
+\begin{proposition}\label{prop:certificate}
+There are pairwise disjoint open balls \(U_0,U_k^+,U_k^-\) centered
+at \(0,a_k,-a_k\), respectively, and a bounded continuous function
+\(u:U\to\R\), where
+\[
+ U=U_0\cup U_1^+\cup U_1^-\cup U_2^+\cup U_2^-,
+\]
+such that
+\begin{equation}\label{eq:certificate}
+ c(x_1,x_2,x_3)\ge\sum_{i=1}^3u(x_i)
+ \quad\text{on }U^3.
+\end{equation}
+Equality holds precisely at the triples in \(U^3\) which, after a
+permutation, have the form
+\begin{equation}\label{eq:contact}
+ (X_k(y),y,Z_k(y))\in U_0\times U_k^+\times U_k^-,
+ \qquad y\in U_k^+,\quad k\in\{1,2\}.
+\end{equation}
+The maps \(X_k,Z_k\) are diffeomorphisms on \(U_k^+\), with
+\[
+ X_k(U_k^+)\subset U_0,\qquad Z_k(U_k^+)\subset U_k^-.
+\]
+\end{proposition}
+
+\begin{proof}
+Use the functions and neighborhoods of Lemma~\ref{lem:local}.
+Choose small balls contained in these neighborhoods, taking
+\(U_0\) inside both central neighborhoods. On the respective
+components define
+\[
+ u=u_0\ \text{on }U_0,\qquad
+ u=w_k\ \text{on }U_k^+,\qquad
+ u=v_k\ \text{on }U_k^-.
+\]
+The balls will be shrunk below. Since the functions are smooth near
+their centers, we can first restrict to fixed preliminary balls on
+which their absolute values are bounded by one constant \(M>0\).
+Take the final balls sufficiently small that their closures are
+pairwise disjoint.
+
+At the five centers the values of \(u\) are \(5/2,0,0,0,0\).
+For a triple of distinct centers other than a permutation of
+\((0,a_k,-a_k)\), there are two
+possibilities. If zero is absent, its potential sum is zero and
+its cost is positive. If zero is present, the other two centers
+come from different opposite pairs. Their cost is
+\[
+ 2+\frac1{\sqrt2}>\frac52,
+\]
+so again there is a strict gap. There are finitely many such
+triples, and the cost and local potentials are continuous near
+each one. All these strict inequalities therefore persist on
+the corresponding products after sufficiently small restrictions.
+
+For a repeated-center type, two coordinates belong to the same
+ball. If all radii are at most \(r\), the distance between those
+two coordinates is at most \(2r\), and their pair cost is at least
+\(1/(2r)\), with the usual interpretation at coincidence.
+Taking \(r\) small enough that \(1/(2r)>3M\) makes the cost
+strictly larger than the potential sum on every such product.
+
+The inequalities of Lemma~\ref{lem:local} remain valid when their
+domains are restricted, with the same equality characterization.
+Permuting a triple leaves both \(c\) and the sum of the three
+values of \(u\) unchanged. Thus the two prescribed component types and
+their permutations satisfy
+\eqref{eq:certificate}, with equality exactly as in
+\eqref{eq:contact}; every other type has strict inequality.
+Finally, keep \(U_0,U_k^-\) fixed and shrink each \(U_k^+\) once
+more. Since \(X_k(a_k)=0\) and \(Z_k(a_k)=-a_k\), continuity
+ensures
+\[
+ X_k(U_k^+)\subset U_0,\qquad Z_k(U_k^+)\subset U_k^-.
+\]
+All inequalities and diffeomorphism properties survive this
+restriction. The function \(u\) is bounded and continuous on
+the final disjoint union \(U\).
+\end{proof}
+
+We now use the central point as parameter for both equality families.
+This gives four diffeomorphisms on one common ball, as required by the
+selection obstruction and the construction of the smooth marginal.
+
+\begin{lemma}\label{lem:common-ball}
+There is an open ball \(B\) of positive radius centered at zero,
+contained in \(U_0\), on which the maps
+\[
+ Y_k=(X_k|_{U_k^+})^{-1}|_B,\qquad W_k=Z_k\circ Y_k
+ \quad(k=1,2)
+\]
+are well defined and are diffeomorphisms onto open images.
+They satisfy
+\[
+ Y_k(B)\subset U_k^+,\qquad W_k(B)\subset U_k^-,
+\]
+and every triple \((x,Y_k(x),W_k(x))\), \(x\in B\), is an
+equality triple in \eqref{eq:certificate}.
+In particular, the five sets
+\[
+ B,\quad Y_1(B),\quad W_1(B),\quad Y_2(B),\quad W_2(B)
+\]
+are pairwise disjoint.
+\end{lemma}
+
+\begin{proof}
+Both \(X_k(U_k^+)\) are open neighborhoods of zero. Choose
+a positive-radius ball \(B\) about zero inside their intersection
+and \(U_0\). The inverse maps then take \(B\) into \(U_k^+\),
+and Proposition~\ref{prop:certificate} gives
+\(Z_k(Y_k(B))\subset U_k^-\).
+The inverse and composition of the relevant diffeomorphisms are
+diffeomorphisms onto open images. The identities
+\(X_k(Y_k(x))=x\) and \(W_k=Z_k\circ Y_k\) give equality by
+Proposition~\ref{prop:certificate}. Disjointness follows from the
+five component balls.
+\end{proof}
diff --git a/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/build/sections/infima.tex b/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/build/sections/infima.tex
new file mode 100644
index 0000000000000000000000000000000000000000..e3a69d30f2ba82fa25e9173a318cf4555b944d43
--- /dev/null
+++ b/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/build/sections/infima.tex
@@ -0,0 +1,206 @@
+\section{Equality of the two infima}\label{sec:infima}
+
+The obstruction above concerns attainment, not the value of the Monge
+infimum. We prove this directly for the constructed marginal. The
+partition and cost-oscillation argument adapts the approximation method
+of Colombo and Di Marino~\cite[author version, Lemma~2.3 and proof of
+Theorem~2.1]{CDM2015}. Their general theorem treats the Coulomb singularity
+and cyclic maps for identical nonatomic marginals. Here the separated
+components allow an elementary proof, including the required Borel maps.
+
+\begin{proposition}\label{prop:equal-infima}
+For the probability measure \(\mu\) of Theorem~\ref{thm:main},
+\[
+ \inf_{\substack{T_2,T_3:\R^3\to\R^3\ \mathrm{Borel}\\
+                 (T_2)_\#\mu=(T_3)_\#\mu=\mu}}
+ \int c(x,T_2(x),T_3(x))\dd\mu(x)=\E(\mu).
+\]
+Every pair in this infimum has cost strictly greater than the common
+infimum, but pairs with finite cost approach it.
+\end{proposition}
+
+The approximation keeps the finite component structure of the optimizer
+\(\pi_0\). We partition the components into small cells and prescribe the
+mass of a graph plan in each product of three cells to equal that of
+\(\pi_0\). Within each product the two plans may differ, but the components
+remain separated and the cost oscillation tends to zero with the cell
+diameter. To realize the prescribed masses, we split each source cell
+into pieces and send each piece into two specified target cells, with
+the normalized restriction of \(\mu\) as each target law.
+The following elementary transport fact supplies those maps; its proof
+is given at the end of the section.
+
+\begin{lemma}\label{lem:borel-transport}
+Let \(d\ge2\) be an integer. If \(\alpha\) and \(\beta\) are absolutely
+continuous Borel probability measures on \(\R^d\), there is a Borel map
+\(Q:\R^d\to\R^d\) such that \(Q_\#\alpha=\beta\).
+\end{lemma}
+
+\begin{proof}[Proof of Proposition~\ref{prop:equal-infima}]
+Write \(K_0=\supp g\), and let \(K_1,\ldots,K_4\) be its images under
+the four maps in \(\calH\). These are nonempty compact sets in the five
+disjoint components, and \(\mu(K)=1\) for \(K=\bigcup_{a=0}^4K_a\).
+In particular,
+\[
+ \eta=\min_{a\ne b}\operatorname{dist}(K_a,K_b)>0.
+\]
+The optimal coupling \(\pi_0\) constructed in Section~\ref{sec:proof}
+is supported on products of three distinct component sets.
+
+Fix \(\delta>0\). Partition each \(K_a\) into finitely many Borel cells
+of diameter at most \(\delta\), for example by a finite grid of half-open
+cubes. Discard the cells of zero \(\mu\)-mass and denote the remaining
+cells by \(A_1,\ldots,A_N\). They partition a set of full \(\mu\)-measure
+and each belongs to a single \(K_a\). Put
+\[
+ q_{ijk}=\pi_0(A_i\times A_j\times A_k).
+\]
+The marginal constraints on \(\pi_0\) read
+\[
+ \sum_{j,k}q_{ijk}=\mu(A_i),\qquad
+ \sum_{i,k}q_{ijk}=\mu(A_j),\qquad
+ \sum_{i,j}q_{ijk}=\mu(A_k).
+\]
+For each fixed \(i\), split \(A_i\) into Borel sets \(B_{ijk}\) of masses
+\(q_{ijk}\), ignoring zero-mass pieces. Such a split can be made by
+successive first-coordinate slabs: the first-coordinate marginal of
+\(\mu|_{A_i}\) is absolutely continuous, so its cumulative distribution
+is continuous and takes every value between zero and \(\mu(A_i)\).
+Endpoints of the slabs have zero mass; any unused null set is included
+in a fixed piece.
+
+For each \(q_{ijk}>0\), Lemma~\ref{lem:borel-transport} supplies maps on
+\(B_{ijk}\) sending the probability \(q_{ijk}^{-1}\mu|_{B_{ijk}}\),
+respectively, to
+\[
+ \frac{\mu|_{A_j}}{\mu(A_j)}
+ \quad\hbox{and}\quad
+ \frac{\mu|_{A_k}}{\mu(A_k)}.
+\]
+Reset any exceptional values outside the indicated target cell to a
+fixed point in that cell. These changes are Borel and occur on null
+sets. Glue the maps over the finitely many pieces to obtain \(T_2,T_3\),
+and give them a fixed value on the remaining Borel null set in \(\R^3\).
+Different source pieces may use the same normalized target law. Their
+weights \(q_{ijk}\) add to the full mass of the target cell, so no target
+partition is needed. Indeed,
+for any Borel set \(D\),
+\[
+ \mu(T_2^{-1}(D))
+ =\sum_{i,j,k:q_{ijk}>0}q_{ijk}
+       \frac{\mu(D\cap A_j)}{\mu(A_j)}
+ =\sum_j\mu(D\cap A_j)=\mu(D),
+\]
+and the same calculation proves \((T_3)_\#\mu=\mu\).
+The graph coupling \(\pi_\delta=(\Id,T_2,T_3)_\#\mu\) gives each product
+\(A_i\times A_j\times A_k\) exactly the mass \(q_{ijk}\).
+
+Whenever \(q_{ijk}>0\), the three cells lie in three distinct component
+sets. Thus all pairwise distances in their product are at least
+\(\eta\), including after the redistribution within the cells. If
+\(x\) and \(y\) are two triples in the same such product, then
+\(|x_\ell-y_\ell|\le\delta\) for each coordinate. Consequently
+\[
+ \left|\frac1{|x_\ell-x_r|}-\frac1{|y_\ell-y_r|}\right|
+ \le\frac{2\delta}{\eta^2},
+ \qquad |c(x)-c(y)|\le\frac{6\delta}{\eta^2}.
+\]
+Both costs are finite, bounded by \(3/\eta\) on all active products.
+Since \(\pi_\delta\) and \(\pi_0\) have identical masses on these
+products, summing their oscillation bounds gives
+\[
+ \left|\int c\dd\pi_\delta-\int c\dd\pi_0\right|
+ \le\frac{6\delta}{\eta^2}.
+\]
+Letting \(\delta\downarrow0\) proves the required upper bound on the
+Monge infimum. The reverse bound follows because every admissible graph
+plan is a coupling. Strict nonattainment is Theorem~\ref{thm:main}.
+\end{proof}
+
+The argument uses a uniform separation estimate on the approximating
+plans themselves; it does not pass the singular Coulomb cost through
+weak convergence. No classification or uniqueness of optimal couplings
+is needed.
+
+\begin{proof}[Proof of Lemma~\ref{lem:borel-transport}]
+We construct a Borel map from each such probability to uniform measure
+on \((0,1)^d\), and a Borel map in the reverse direction. Let \(f\) be a
+nonnegative Borel probability density. The forward map is Rosenblatt's
+successive conditional-distribution transform~\cite{Rosenblatt1952}; we
+also give the inverse quantile construction and make the Borel choices
+explicit in both directions.
+
+\emph{The forward map.}
+We successively transform each coordinate by its conditional distribution
+function given the preceding coordinates. Write
+\(x_{1:j}=(x_1,\ldots,x_j)\), with \(x_{1:0}\) the empty prefix, and set
+\[
+ f_{1:j}(x_{1:j})=\int_{\R^{d-j}}f(x_{1:j},z)\dd z
+ \quad(1\le j<d),\qquad f_{1:d}=f,\qquad f_{1:0}=1.
+\]
+The prefix densities are Borel, have integral one, and are finite almost
+everywhere. Tonelli's theorem gives
+\(\int_\R f_{1:j}(a,r)\dd r=f_{1:j-1}(a)\) for every prefix \(a\),
+including the empty prefix when \(j=1\). For \(j=1,\ldots,d\), define
+\[
+ F_j(a,t)=\frac{\int_{-\infty}^t f_{1:j}(a,r)\dd r}
+                 {f_{1:j-1}(a)},\qquad a\in\R^{j-1}.
+\]
+This formula is used where its denominator belongs to \((0,\infty)\);
+on the remaining fibers use instead the fixed distribution function
+\(t\mapsto(1+e^{-t})^{-1}\). Thus every fiber is a continuous distribution
+function with limits zero and one, and all \(F_j\) are jointly Borel.
+For \(j\ge2\), the exceptional prefix fibers have zero probability under
+\(f_{1:j-1}(a)\dd a\); there are no exceptional fibers when \(j=1\).
+
+For a random variable with a continuous distribution function \(F\),
+the variable \(F(X)\) is uniform on \((0,1)\). Indeed, continuity and the
+limits of \(F\) allow a cut at each probability level \(u\in(0,1)\);
+any interval on which \(F=u\) has probability zero, and hence
+\(\mathbb P(F(X)\le u)=u\). At stage \(j\), this identity and Fubini show
+that, conditionally on the preceding original prefix, the new transformed
+coordinate is uniform.
+It is therefore independent of that prefix, and hence of all preceding
+transformed coordinates. A finite induction over \(j=1,\ldots,d\) shows that the Borel map
+\[
+ R_f(x)=\bigl(F_j(x_{1:j-1},x_j)\bigr)_{j=1}^d
+\]
+sends \(f\dd x\) to product uniform measure on \((0,1)^d\). Its values lie
+in \([0,1]^d\), including on exceptional fibers. This argument does not
+require a density to be positive everywhere or a distribution function to
+be strictly increasing.
+
+\emph{The reverse map.}
+Given independent uniform coordinates, we successively recover the
+original conditional laws by quantiles. For each \(j\)
+and \(0<u<1\), define
+\[
+ q_j(a,u)=\inf\{r\in\mathbb Q:F_j(a,r)\ge u\}.
+\]
+This value is finite, and
+\[
+ \{(a,u):q_j(a,u)<t\}
+ =\bigcup_{\substack{r\in\mathbb Q\\r<t}}
+      \{(a,u):F_j(a,r)\ge u\}
+\]
+proves joint Borel measurability on \(0<u<1\). At \(u=0,1\), define
+\(q_j(a,u)=0\). For \(0<u<1\), continuity of \(F_j(a,\cdot)\) gives
+\[
+ q_j(a,u)\le t\quad\Longleftrightarrow\quad u\le F_j(a,t).
+\]
+Consequently \(q_j(a,U)\) has distribution function \(F_j(a,\cdot)\)
+for uniform \(U\).
+For \(u\in[0,1]^d\), recursively set
+\[
+ z_j=q_j(z_{1:j-1},u_j)\quad(j=1,\ldots,d),
+ \qquad S_f(u)=(z_1,\ldots,z_d),
+\]
+starting with the empty prefix. This finite recursion defines a Borel map.
+For independent uniform inputs, induction shows that after step \(j\) the
+output prefix has density \(f_{1:j}\): the preceding prefix has density
+\(f_{1:j-1}\), its exceptional fibers have probability zero, and on every
+other fiber the quantile has the conditional distribution just defined.
+Thus \(S_f\) has output law \(f\dd x\).
+If \(f\) and \(g\) are densities of \(\alpha\) and \(\beta\), respectively,
+then \(S_g\circ R_f\) has the required property.
+\end{proof}
diff --git a/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/build/sections/introduction.tex b/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/build/sections/introduction.tex
new file mode 100644
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@@ -0,0 +1,149 @@
+\section{The problem and the result}\label{sec:introduction}
+
+For a Borel probability measure \(\mu\) on \(\R^3\), let
+\(\Pi(\mu,\mu,\mu)\) denote the probability measures on \((\R^3)^3\)
+with all three coordinate marginals equal to \(\mu\). We use the
+Coulomb cost
+\begin{equation}\label{eq:cost}
+ c(x_1,x_2,x_3)=\frac1{|x_1-x_2|}
+                 +\frac1{|x_1-x_3|}
+                 +\frac1{|x_2-x_3|},
+\end{equation}
+where \(c=+\infty\) if two coordinates coincide, and write
+\[
+ \E(\mu)=\inf_{\pi\in\Pi(\mu,\mu,\mu)}\int c\dd\pi.
+\]
+Here and below \(|\cdot|\) is the Euclidean norm.
+A \emph{Monge plan} is a coupling of the form
+\((\Id,T_2,T_3)_\#\mu\), where \(T_2,T_3:\R^3\to\R^3\) are
+Borel maps and \((T_2)_\#\mu=(T_3)_\#\mu=\mu\).
+For a Borel map \(T\), the pushforward is defined by
+\(T_\#\mu(A)=\mu(T^{-1}(A))\).
+For a map defined on a set of full measure, we use the same notation
+for the pushforward of the measure restricted to that set.
+
+The question is whether absolute continuity of \(\mu\), together
+with \(\E(\mu)<\infty\), guarantees an optimal Monge plan.
+We give a counterexample with additional regularity.
+
+\begin{theorem}\label{thm:main}
+There is a nonnegative density \(\rho\) on \(\R^3\) such that
+\[
+ \int_{\R^3}\rho(x)\dd x=1,\qquad
+ \rho\in C_c^\infty(\R^3),\qquad
+ \sqrt{\rho}\in C_c^\infty(\R^3),
+\]
+with the following properties. For \(\mu=\rho\dd x\), the value
+\(\E(\mu)\) is finite and is attained by a coupling in
+\(\Pi(\mu,\mu,\mu)\), but every pair of Borel maps
+\(T_2,T_3:\R^3\to\R^3\) satisfying
+\((T_2)_\#\mu=(T_3)_\#\mu=\mu\) obeys
+\[
+ \int_{\R^3}c\bigl(x,T_2(x),T_3(x)\bigr)\dd\mu(x)>\E(\mu).
+\]
+The inequality permits the left-hand side to be infinite.
+\end{theorem}
+
+The maps in the theorem are arbitrary Borel maps. In particular, the
+conclusion concerns existence of any Monge optimizer and includes
+maps with no cyclic relation or regularity.
+The theorem also excludes optimal symmetrizations of admissible
+Monge plans, since permutation averaging preserves their marginals
+and Coulomb cost.
+
+\paragraph{History and significance.}
+Coulomb multi-marginal transport arises in the strong-interaction
+limit of density functional theory. In this setting the description
+of optimal configurations by measure-preserving maps is usually
+called the Monge or co-motion ansatz. The co-motion formulation was
+developed in the strictly correlated electron literature; see Seidl,
+Gori-Giorgi and Savin~\cite{SGS2007} and the account by Friesecke,
+Gerolin and Gori-Giorgi~\cite{FGG2023}.
+The distinction between equality of optimal values and attainment
+by maps is essential. Colombo and Di Marino~\cite[author version, Theorem~1.1]{CDM2015} proved
+that the Coulomb Kantorovich minimum equals the infimum over
+measure-preserving cyclic maps for identical non-atomic marginals.
+This provides approximating maps without asserting that a
+minimizing map exists. Section~\ref{sec:infima} gives a direct proof of
+equality of the two infima for the particular marginal constructed here.
+
+For two particles, Cotar, Friesecke and
+Kl{\"u}ppelberg~\cite[arXiv version, Theorem~3.6]{CFK2013}
+proved existence and uniqueness of an optimal map for absolutely
+continuous marginals in every dimension. For arbitrary particle number
+on the line, Colombo, De Pascale and Di Marino~\cite[Theorem~1.1]{CDDM2015}
+constructed optimal cyclic maps for identical nonatomic marginals with
+finite transport value. Pass~\cite{Pass2013} constructed Coulomb
+optimizers that are not induced by maps for radial data; the existence
+of such an optimizer does not exclude a different optimizer of Monge
+form. Colombo and Stra~\cite{CS2016} gave counterexamples to the
+prescribed radial quantile-cyclic constructions. Bindini, De Pascale and
+Kausamo~\cite{BDK2022} subsequently constructed smooth densities for
+the reduced radial cost of the planar three-particle problem for which
+none of the four prescribed maps III, DDI, DID and IDD is optimal.
+This radial cost is obtained by minimizing the Coulomb interaction over
+angles; it is different from the Coulomb cost on the line.
+
+The general higher-dimensional map-attainment question with at least
+three particles is recorded in the 2022 preprint
+of~\cite[arXiv version, Theorem~2.4 and the following discussion]{FGG2023}.
+The three-particle Coulomb question in \(\R^3\) was still stated as open
+in January 2026 by Friesecke~\cite[Section~3.2]{Friesecke2026}.
+Theorem~\ref{thm:main} excludes every preserving Borel pair for a smooth
+identical marginal. With the particle-density normalization \(n=3\rho\),
+one has \(\int n=3\) and \(\sqrt n\in H^1(\R^3)\). Thus the example
+belongs to the physical density class \(D_3\)
+of~\cite[arXiv version, Eq.~(12)]{FGG2023}. This is a density-regularity
+statement; no external potential or ground-state realization is asserted.
+
+Our construction adapts the shared-component splitting mechanism
+of Gerolin, Kausamo and Rajala~\cite{GKR2019} for the repulsive
+harmonic cost \(-\sum_{i<j}|x_i-x_j|^2\).
+For the Coulomb interaction, the additional task is to build two
+families of triples with a common supporting potential. We obtain
+these families and their supporting inequalities by local calculus.
+
+\paragraph{Idea of the proof.}
+The construction has two parts: a geometric condition that forces every
+optimal plan onto two prescribed families of triples, and a mass imbalance that
+prevents a preserving map from choosing between them.
+Let \(B\) be a small ball about zero, and let \(e_1,e_2\) be the first
+two coordinate unit vectors. We construct smooth diffeomorphisms \(Y_k,W_k\), for \(k=1,2\), from \(B\) onto small
+components near \(e_k,-e_k\), respectively. The two families are
+\[
+ (x,Y_k(x),W_k(x)),\qquad x\in B,\quad k\in\{1,2\},
+\]
+together with their coordinate permutations; see
+Figure~\ref{fig:components}.
+
+The triples \((0,e_k,-e_k)\) have the same Coulomb cost and the same
+central gradient. This allows the two local families to share a
+supporting potential on \(B\). Completing this potential on the four
+outer components gives an inequality
+\(c(x_1,x_2,x_3)\ge u(x_1)+u(x_2)+u(x_3)\), with equality precisely
+on the two families. Strict cost gaps exclude the other triples of distinct centers;
+the Coulomb singularity excludes repeated components after the
+neighborhoods have been made small enough.
+
+For a smooth probability measure \(\nu\) on \(B\), we form the
+marginal \(\mu\) by giving the central component measure \(\nu/3\)
+and each outer component measure \((Y_k)_\#\nu/6\) or
+\((W_k)_\#\nu/6\). Randomly choosing a branch
+and permuting its three coordinates then gives an optimal coupling
+with this common marginal. Every optimizer must saturate the supporting
+inequality. In particular, on the central component, a deterministic
+map must choose one of the four branch images. If it chooses a branch
+\(H\) on a Borel set \(S\subset B\), it sends mass \(\nu(S)/3\)
+into \(H(S)\), which has marginal mass only \(\nu(S)/6\).
+This contradicts preservation on every selection set of positive mass.
+The exact image-measure relation, rather than the component totals
+alone, excludes arbitrarily irregular choices of branch.
+
+\input{figures/components}
+
+Sections~\ref{sec:splitting}--\ref{sec:proof} establish the mass obstruction,
+construct and certify the two local families, and form the smooth marginal
+and an explicit optimizer. Section~\ref{sec:infima} constructs preserving
+maps whose costs approach that optimum.
+Section~\ref{sec:riesz} extends the construction to every inverse-power
+Riesz cost in every Euclidean dimension at least two.
diff --git a/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/build/sections/local.tex b/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/build/sections/local.tex
new file mode 100644
index 0000000000000000000000000000000000000000..ee992762cfc912a0654a6af6e9bf4967ee2b1ba8
--- /dev/null
+++ b/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/build/sections/local.tex
@@ -0,0 +1,119 @@
+\section{Two local Coulomb branches}\label{sec:local}
+
+Put \(a_1=e_1=(1,0,0)\) and \(a_2=e_2=(0,1,0)\).
+At each of the two triples \((0,a_k,-a_k)\),
+\begin{equation}\label{eq:center-jets}
+ c(0,a_k,-a_k)=\frac52,\qquad
+ \nabla_xc(0,a_k,-a_k)=0,\qquad
+ \nabla_zc(0,a_k,-a_k)=\frac54a_k.
+\end{equation}
+The equal cost values and vanishing central gradients allow the two
+families to share one central potential.
+
+Near each center triple, we use the positive outer point \(y\) as the parameter and find
+the central and negative outer points by minimization. Subtracting
+suitable concave quadratic potentials in those two coordinates makes
+the minimum unique; its value defines the potential at \(y\).
+We also require both minimizing coordinates to depend diffeomorphically
+on \(y\). We can then invert the central coordinate and obtain both
+outer components as diffeomorphic images of a common central set.
+
+\begin{lemma}\label{lem:local}
+There are a number \(K>0\), open neighborhoods \(P_0^k,P_k^+,P_k^-\)
+of \(0,a_k,-a_k\), respectively, and smooth maps
+\(X_k,Z_k:P_k^+\to\R^3\), for \(k=1,2\), with the following
+properties:
+\begin{enumerate}
+ \item Each of \(X_k\) and \(Z_k\) is a diffeomorphism onto its
+ open image, and
+ \[
+  X_k(a_k)=0,\qquad Z_k(a_k)=-a_k.
+ \]
+ \item Define
+ \[
+ u_0(x)=\frac52-\frac K2|x|^2,\qquad
+ v_k(z)=\frac54a_k\cdot(z+a_k)-\frac K2|z+a_k|^2.
+ \]
+ There is a smooth function \(w_k:P_k^+\to\R\), with
+ \(w_k(a_k)=0\), such that on
+ \(P_0^k\times P_k^+\times P_k^-\),
+ \begin{equation}\label{eq:local-certificate}
+ c(x,y,z)\ge u_0(x)+w_k(y)+v_k(z).
+ \end{equation}
+ Equality holds if and only if
+ \((x,z)=(X_k(y),Z_k(y))\).
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+For a parameter \(K>0\), use the potentials in the statement and set
+\[
+ F_k(x,y,z)=c(x,y,z)-u_0(x)-v_k(z).
+\]
+By \eqref{eq:center-jets}, its \((x,z)\)-gradient vanishes at the
+\(k\)-th center triple. We choose \(K\) to obtain a unique minimizing
+pair on a small product neighborhood for each fixed \(y\), and to make each coordinate of the
+resulting minimizing pair an invertible function of \(y\).
+
+The Coulomb cost is smooth near both center triples. At the
+\(k\)-th center, write
+\[
+ M_k=D^2_{(x,z)(x,z)}c,\qquad
+ N_k=D_y\nabla_{(x,z)}c.
+\]
+The \((x,z)\)-Hessian of \(F_k\) there is \(M_k+KI_6\), while its
+mixed derivative in \(y\) is \(N_k\). For \(h(b)=|b|^{-1}\), \(b\ne0\),
+differentiation gives
+\begin{equation}\label{eq:pair-hessian}
+ D^2h(b)=|b|^{-3}
+ \left(3\frac{bb^{\mathsf T}}{|b|^2}-I_3\right).
+\end{equation}
+This matrix has eigenvalues \(2|b|^{-3},-|b|^{-3},-|b|^{-3}\)
+and is therefore invertible. The two \(3\times3\) blocks of \(N_k\) are
+\[
+ -D^2h(-a_k),\qquad -D^2h(-2a_k),
+\]
+and hence are both invertible. The matrices \(M_k\) are fixed
+symmetric \(6\times6\) matrices. For all sufficiently large \(K\),
+both \(M_k+KI_6\) are positive definite. Moreover,
+\[
+ K(M_k+KI_6)^{-1}N_k
+   =(I_6+K^{-1}M_k)^{-1}N_k\longrightarrow N_k.
+\]
+Invertibility is an open condition on \(3\times3\) matrices, so
+we may choose a single \(K\) for which both blocks of
+\((M_k+KI_6)^{-1}N_k\) are invertible for each \(k\).
+
+Fix such a \(K\). The implicit function theorem applied to
+\(\nabla_{(x,z)}F_k=0\) gives a smooth stationary branch
+\[
+ \nabla_{(x,z)}F_k(X_k(y),y,Z_k(y))=0,
+\]
+with the prescribed center values. Differentiating this equation
+at \(y=a_k\) gives
+\[
+ D(X_k,Z_k)(a_k)=-(M_k+KI_6)^{-1}N_k.
+\]
+The two block invertibilities allow the inverse function theorem
+to be applied separately to \(X_k\) and \(Z_k\).
+After restricting to one sufficiently small neighborhood of \(a_k\),
+both maps are diffeomorphisms onto their open images.
+
+By continuity, the \((x,z)\)-Hessian of \(F_k\) remains positive
+definite on a product of sufficiently small open balls around the
+three centers. Shrink the \(y\)-ball so that the stationary pair
+\((X_k(y),Z_k(y))\) belongs to the product of the other two balls.
+For fixed \(y\), this latter product is convex. The positive
+definite Hessian makes \(F_k(\cdot,y,\cdot)\) strictly convex on it,
+so its stationary pair is its unique minimum there.
+
+Define its minimum value by
+\[
+ w_k(y)=F_k(X_k(y),y,Z_k(y)).
+\]
+The preceding minimum property gives
+\eqref{eq:local-certificate} with exactly the stated equality set.
+The center values in \eqref{eq:center-jets} give \(w_k(a_k)=0\).
+All neighborhoods can be restricted further as needed to retain
+the separate diffeomorphism properties.
+\end{proof}
diff --git a/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/build/sections/proof.tex b/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/build/sections/proof.tex
new file mode 100644
index 0000000000000000000000000000000000000000..5ac0795bfedb7600aaf1f3669300a927e0486194
--- /dev/null
+++ b/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/build/sections/proof.tex
@@ -0,0 +1,105 @@
+\section{The marginal and exclusion of every Monge optimizer}
+\label{sec:proof}
+
+We now prove Theorem~\ref{thm:main}. Let the balls, maps, and
+potential be as in Proposition~\ref{prop:certificate} and
+Lemma~\ref{lem:common-ball}. Set
+\[
+ \calH=\{Y_1,W_1,Y_2,W_2\}.
+\]
+
+\paragraph{A smooth marginal.}
+Choose a nonnegative function \(g\in C_c^\infty(B)\) with
+\(\int_Bg^2\dd x=1\), and let
+\(\nu=g^2\dd x\), regarded as a probability measure on \(\R^3\).
+Define
+\begin{equation}\label{eq:mu}
+ \mu=\frac13\nu+\frac16\sum_{H\in\calH}H_\#\nu.
+\end{equation}
+These coefficients sum to one. For \(H\in\calH\), define
+\[
+ g_H(y)=
+ \begin{cases}
+  g(H^{-1}(y))\,|\det DH^{-1}(y)|^{1/2},&y\in H(B),\\
+  0,&y\notin H(B).
+ \end{cases}
+\]
+The formula is smooth on \(H(B)\). Its support is contained in
+the compact set \(H(\supp g)\subset H(B)\), so extension by zero
+is smooth on \(\R^3\). Thus \(g_H\in C_c^\infty(\R^3)\), and
+change of variables gives \(H_\#\nu=g_H^2\dd y\).
+
+It follows that \(\mu=\rho\dd x\), where
+\[
+ \rho=\frac13g^2+\frac16\sum_{H\in\calH}g_H^2.
+\]
+The functions here are nonnegative and have pairwise disjoint
+supports, since their supports lie in the five disjoint
+components. Consequently
+\[
+ \sqrt{\rho}=\frac1{\sqrt3}g
+                +\frac1{\sqrt6}\sum_{H\in\calH}g_H.
+\]
+This proves both asserted smoothness properties, compact support,
+and normalization. Also \(\mu(U)=1\) and
+\(\mu|_B=(1/3)\nu\).
+
+\paragraph{A finite optimal coupling.}
+Sample \(x\) with law \(\nu\), choose \(k\) uniformly from
+\(\{1,2\}\), independently of \(x\), and apply an independent
+uniform permutation to the three entries of
+\[
+ (x,Y_k(x),W_k(x)).
+\]
+Let \(\pi_0\) be the resulting law.
+In any fixed coordinate the central point has coefficient \(1/3\),
+and each specified outer map has coefficient
+\((1/2)(1/3)=1/6\). Hence every coordinate marginal is exactly
+\(\mu\).
+
+All sampled triples satisfy \eqref{eq:certificate} with equality.
+The boundedness of \(u\) therefore gives
+\begin{equation}\label{eq:optimal-value}
+ \int c\dd\pi_0=3\int_Uu\dd\mu<\infty.
+\end{equation}
+Conversely, every \(\pi\in\Pi(\mu,\mu,\mu)\) has
+\(\pi(U^3)=1\): each of the three events \(x_i\notin U\)
+has probability zero. Integrating \eqref{eq:certificate} yields
+\[
+ \int c\dd\pi\ge 3\int_Uu\dd\mu.
+\]
+Thus \(\pi_0\) attains \(\E(\mu)\), and the latter is finite.
+Only the inequality on \(U^3\) is used.
+
+For any optimal \(\pi\), the nonnegative slack
+\(c(x_1,x_2,x_3)-\sum_i u(x_i)\) on \(U^3\) has integral zero.
+Indeed the cost is finite and the potential sum is bounded, so
+subtraction and integration are legitimate.
+It follows that \(\pi\) is concentrated on the equality triples
+in \eqref{eq:contact}.
+
+\paragraph{No optimal maps.}
+Suppose that Borel maps \(T_2,T_3\), each preserving \(\mu\),
+induce an optimal plan. The preceding equality condition holds
+at \((x,T_2(x),T_3(x))\) for \(\mu\)-almost every \(x\), hence
+for \(\nu\)-almost every \(x\in B\).
+
+The first coordinate \(x\in B\subset U_0\) is the unique central
+coordinate of an equality triple. The remaining coordinates lie
+one each in \(U_k^+\) and \(U_k^-\) for some \(k\).
+If their values in those components are \(y,z\), then
+\(x=X_k(y)\) and \(z=Z_k(y)\).
+Injectivity of \(X_k\) forces
+\[
+ y=Y_k(x),\qquad z=W_k(x).
+\]
+Allowing either order of these coordinates and either value of
+\(k\), we obtain
+\[
+ T_2(x)\in\{Y_1(x),W_1(x),Y_2(x),W_2(x)\}
+ \quad\text{for \(\nu\)-almost every }x\in B.
+\]
+This contradicts Lemma~\ref{lem:splitting} applied to
+\eqref{eq:mu}. Thus no Monge plan attains the minimum, completing
+the proof of Theorem~\ref{thm:main}.\qed
+
diff --git a/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/build/sections/riesz.tex b/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/build/sections/riesz.tex
new file mode 100644
index 0000000000000000000000000000000000000000..1dd8b637ff41db5d9d8a17a10568995b7fcf97c6
--- /dev/null
+++ b/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/build/sections/riesz.tex
@@ -0,0 +1,160 @@
+\section{Inverse-power Riesz interactions}\label{sec:riesz}
+
+The construction is not specific to the Coulomb exponent or to three
+spatial dimensions. For an integer \(d\ge2\) and a real number \(s>0\),
+define on \((\R^d)^3\)
+\begin{equation}\label{eq:riesz-cost}
+ c_s(x_1,x_2,x_3)=\sum_{1\le i<j\le3}|x_i-x_j|^{-s},
+\end{equation}
+with \(c_s=+\infty\) if two coordinates coincide. For a Borel probability
+measure \(\mu\) on \(\R^d\), let \(\Pi(\mu,\mu,\mu)\) now denote the
+couplings on \((\R^d)^3\), and write
+\[
+ E_s(\mu)=\inf_{\pi\in\Pi(\mu,\mu,\mu)}\int c_s\dd\pi.
+\]
+
+\begin{theorem}\label{thm:riesz}
+For every integer \(d\ge2\) and every real \(s>0\), there is a
+nonnegative density \(\rho\) on \(\R^d\) such that
+\[
+ \int_{\R^d}\rho(x)\dd x=1,\qquad
+ \rho\in C_c^\infty(\R^d),\qquad
+ \sqrt{\rho}\in C_c^\infty(\R^d),
+\]
+with the following properties. For \(\mu=\rho\dd x\), the value
+\(E_s(\mu)\) is finite and attained by a coupling, but every pair of
+Borel maps \(T_2,T_3:\R^d\to\R^d\) with
+\((T_2)_\#\mu=(T_3)_\#\mu=\mu\) satisfies
+\[
+ \int_{\R^d}c_s\bigl(x,T_2(x),T_3(x)\bigr)\dd\mu(x)>E_s(\mu).
+\]
+The left-hand side may be infinite.
+Nevertheless,
+\[
+ \inf_{\substack{T_2,T_3:\R^d\to\R^d\ \mathrm{Borel}\\
+                 (T_2)_\#\mu=(T_3)_\#\mu=\mu}}
+ \int c_s(x,T_2(x),T_3(x))\dd\mu(x)=E_s(\mu),
+\]
+and pairs with finite cost approach this infimum.
+\end{theorem}
+
+The density and all local construction constants may depend on
+\((d,s)\). The physical Coulomb discussion in
+Section~\ref{sec:introduction} refers to the specialization
+\(d=3,s=1\); the interaction in \eqref{eq:riesz-cost} is the stated
+inverse power, not a logarithmic kernel.
+
+\begin{proof}
+Fix \(d\ge2\) and \(s>0\), and put \(a_1=e_1\) and \(a_2=e_2\)
+in \(\R^d\). The measure argument already applies in this dimension.
+We verify the three properties of the cost that it needs: local
+equality families given by diffeomorphisms, strict supporting
+inequalities on all other component products, and a uniform
+cost-oscillation bound on separated products.
+
+\paragraph{Local branches.}
+For \(h_s(b)=|b|^{-s}\), \(b\ne0\), the center jets are
+\[
+ c_s(0,a_k,-a_k)=2+2^{-s},\qquad
+ \nabla_xc_s(0,a_k,-a_k)=0,\qquad
+ \nabla_zc_s(0,a_k,-a_k)=s(1+2^{-s-1})a_k.
+\]
+As before, we keep the positive outer point \(y\) free and minimize
+in the other two coordinates. The potentials in Lemma~\ref{lem:local}
+are replaced by
+\[
+ u_0(x)=2+2^{-s}-\frac K2|x|^2,\qquad
+ v_k(z)=s(1+2^{-s-1})a_k\cdot(z+a_k)-\frac K2|z+a_k|^2.
+\]
+The central potential is common to the two branches, and
+\(F_k=c_s-u_0-v_k\) has vanishing \((x,z)\)-gradient at each center.
+Direct differentiation gives
+\[
+ D^2h_s(b)=s|b|^{-s-2}
+ \left((s+2)\frac{bb^{\mathsf T}}{|b|^2}-I_d\right).
+\]
+Its radial eigenvalue is \(s(s+1)|b|^{-s-2}\), and its tangential
+eigenvalues are \(-s|b|^{-s-2}\), so it is invertible.
+
+At the \(k\)-th center, define \(M_k\) and \(N_k\) as in
+Section~\ref{sec:local}, with \(c_s\) in place of \(c\).
+The two \(d\times d\) blocks of \(N_k\) are
+\(-D^2h_s(-a_k)\) and \(-D^2h_s(-2a_k)\), hence are invertible,
+while \(M_k\) is a symmetric \(2d\times2d\) matrix. For sufficiently
+large \(K\), both \(M_k+KI_{2d}\) are positive definite and both
+\(d\times d\) blocks of \((M_k+KI_{2d})^{-1}N_k\) are invertible, because
+\[
+ K(M_k+KI_{2d})^{-1}N_k
+ =(I_{2d}+K^{-1}M_k)^{-1}N_k\longrightarrow N_k.
+\]
+One \(K\) works for the two branches. The implicit and inverse function
+arguments of Lemma~\ref{lem:local} now give separate local
+diffeomorphisms \(X_k,Z_k\) on \(d\)-dimensional neighborhoods.
+The cost is smooth near the collision-free center triples; after
+shrinking the product balls, the \((x,z)\)-Hessian of \(F_k\) remains positive
+definite. Strict convexity on the product of the \(x\)- and \(z\)-balls
+therefore gives the same local supporting inequality and exact equality
+set, with \(w_k(a_k)=0\).
+
+\paragraph{The supporting inequality and the marginal.}
+The five center values of the potential are now
+\(2+2^{-s},0,0,0,0\). A distinct-center triple containing zero but
+not an opposite pair has cost
+\[
+ 2+2^{-s/2}>2+2^{-s}.
+\]
+A distinct-center triple without zero has positive cost and potential
+sum zero. These finitely many strict gaps persist on small products
+by continuity. If two points lie in one ball of radius at most \(r\),
+their pair cost is at least \((2r)^{-s}\). This exceeds \(3M\) for
+sufficiently small \(r\), where \(M\) bounds the absolute values of
+the preliminary potentials. Positivity of the other pair terms and
+permutation symmetry of \(c_s\) therefore give the full supporting
+inequality and exact contact set described in
+Proposition~\ref{prop:certificate}, now in \(\R^d\).
+The proof of Lemma~\ref{lem:common-ball} then also applies in \(\R^d\).
+
+On that common ball \(B\), choose a nonnegative
+\(g\in C_c^\infty(B)\) with \(\int_B g^2\dd x=1\), and form \(\nu\) and
+\(\mu\) by \eqref{eq:mu} using the four resulting diffeomorphisms.
+The change-of-variables construction in Section~\ref{sec:proof},
+now with \(d\)-dimensional Jacobians, gives the same smooth compactly
+supported density and square root: each image function is supported
+compactly inside its open component, and the five supports are
+disjoint. The same random choice of a branch and permutation gives
+an optimal coupling \(\pi_0\) with the three marginals equal to
+\(\mu\). The bounded supporting potential proves its cost is finite
+and forces every optimal coupling onto the contact set. Applying
+Lemma~\ref{lem:splitting} to the four branch maps then excludes
+every marginal-preserving Borel pair, exactly as in
+Section~\ref{sec:proof}.
+
+\paragraph{Approximation by preserving maps.}
+The five compact component sets have a positive mutual separation
+\(\eta\), and \(\pi_0\) uses only products of three distinct components.
+Repeat the cell partition and redistribution from the proof of
+Proposition~\ref{prop:equal-infima} in \(\R^d\), using
+Lemma~\ref{lem:borel-transport}. The first-coordinate marginals used
+to split the cells are absolutely continuous by Fubini in every such
+dimension. The redistribution gives a graph coupling
+\(\pi_\delta\) of a preserving Borel pair with the same mass as
+\(\pi_0\) on each product of cells of diameter at most \(\delta\).
+Every active product still has all pair distances at least \(\eta\).
+For two triples \(x,y\) in one such product, the scalar mean value
+theorem gives
+\[
+ \left||x_\ell-x_r|^{-s}-|y_\ell-y_r|^{-s}\right|
+ \le 2s\delta\,\eta^{-s-1},\qquad
+ |c_s(x)-c_s(y)|\le6s\delta\,\eta^{-s-1}.
+\]
+Both costs are bounded by \(3\eta^{-s}\) on the active products.
+Summing their oscillations against the common product masses yields
+\[
+ \left|\int c_s\dd\pi_\delta-\int c_s\dd\pi_0\right|
+ \le6s\delta\,\eta^{-s-1}\longrightarrow0.
+\]
+The reverse inequality for the infima holds because every graph plan
+is a coupling. This proves the final assertion. In particular, no
+restriction relating \(s\) to \(d\) is needed: the optimizer and
+these approximating plans stay away from all collisions.
+\end{proof}
diff --git a/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/build/sections/splitting.tex b/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/build/sections/splitting.tex
new file mode 100644
index 0000000000000000000000000000000000000000..0781a5281709933589118fb54938692081b3a294
--- /dev/null
+++ b/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/build/sections/splitting.tex
@@ -0,0 +1,52 @@
+\section{An obstruction to deterministic selection}\label{sec:splitting}
+
+We isolate the measure-theoretic step. A homeomorphism onto an open
+image sends Borel sets to Borel sets, since its inverse is
+continuous.
+
+\begin{lemma}\label{lem:splitting}
+Let \(d\ge2\) be an integer, let \(B\subset\R^d\) be open, let
+\(\nu\) be a Borel probability measure with \(\nu(B)=1\), and let
+\(H_1,\ldots,H_4:B\to\R^d\) be homeomorphisms onto open images.
+Suppose that \(B,H_1(B),\ldots,H_4(B)\) are pairwise disjoint.
+Define
+\begin{equation}\label{eq:split-measure}
+ \mu=\frac13\nu+\frac16\sum_{j=1}^4(H_j)_\#\nu.
+\end{equation}
+There is no Borel map \(T:\R^d\to\R^d\) such that
+\(T_\#\mu=\mu\) and
+\begin{equation}\label{eq:selection}
+ T(x)\in\{H_1(x),\ldots,H_4(x)\}
+ \quad\text{for \(\nu\)-almost every }x\in B.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+For each \(j\), define the Borel set
+\[
+ S_j=\{x\in B:T(x)=H_j(x)\}.
+\]
+Its image \(H_j(S_j)\) is Borel. Pairwise disjointness of the
+component images and injectivity of \(H_j\) give
+\[
+ \mu(S_j)=\frac13\nu(S_j),\qquad
+ \mu(H_j(S_j))=\frac16\nu(S_j).
+\]
+Since \(S_j\subset T^{-1}(H_j(S_j))\), preservation of \(\mu\)
+would imply
+\[
+ \frac13\nu(S_j)
+ \le \mu(T^{-1}(H_j(S_j)))
+ =\mu(H_j(S_j))
+ =\frac16\nu(S_j).
+\]
+Thus \(\nu(S_j)=0\) for all four indices, contradicting
+\eqref{eq:selection}.
+\end{proof}
+
+The argument does not require \(\nu\) to be uniform or even
+absolutely continuous. Additional points mapped into \(H_j(S_j)\)
+can only increase its preimage mass. Matching the total mass of
+each component by a partition therefore cannot remove the
+obstruction.
+
diff --git a/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/paper.pdf b/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..1dfc690c6ae2a3b800f4c542c106d219101cccf8
--- /dev/null
+++ b/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:46caabb46552913329e6a1a6226ad2dc196bd231d117562d0495f19a5832ae70
+size 322531
diff --git a/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/README.md b/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..63762d632bfc608bc47d266e59e375e40f995821
--- /dev/null
+++ b/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/README.md
@@ -0,0 +1,18 @@
+# [A counterexample to the coarse Novikov conjecture](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 23, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026,
+  author = {{OpenAI}},
+  title = {{A counterexample to the coarse Novikov conjecture}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/paper.pdf}{OAI:A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/build/figures/disc.tex b/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/build/figures/disc.tex
new file mode 100644
index 0000000000000000000000000000000000000000..02de6f7b9ae7caca6e2979dfa3141550e060c02e
--- /dev/null
+++ b/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/build/figures/disc.tex
@@ -0,0 +1,30 @@
+\begin{tikzpicture}[scale=.77,>=Stealth,font=\small]
+ \foreach \r in {1,2,3} {
+  \draw[gray!65] (0:\r) -- (60:\r) -- (120:\r) -- (180:\r) -- (240:\r) -- (300:\r) -- cycle;
+ }
+ \foreach \ang in {0,60,...,300} {\draw[gray!35] (0,0)--(\ang:3);}
+ \draw[very thick,blue!55!black] (0:3)--(60:3)--(120:3)--(180:3)--(240:3)--(300:3)--cycle;
+ \draw[dashed,thick,->] (0,0)--(19:3.65) node[right]{cut};
+ \fill (0,0) circle (2pt) node[below left] {$Q_3$};
+ \foreach \r/\q in {1/2,2/1,3/0} {
+  \fill (120:\r) circle (2pt);
+  \node[above left] at (120:\r) {$Q_{\q}$};
+ }
+ \node[blue!55!black] at (0,-3.05) {$C_j$: singleton fibers};
+ \node at (5.9,2.85) {equal depth $k$};
+ \foreach \y in {1.5,1.95,2.4} {
+  \fill (4.7,\y) circle (2pt);
+  \fill (7.1,\y) circle (2pt);
+  \draw[->,gray!75] (4.8,\y)--(7,\y);
+ }
+ \node at (5.9,1) {$q\longmapsto qg_k^a$};
+ \node at (4.7,-.25) {$Q_k$};
+ \node at (7.1,-.25) {$Q_{k+1}$};
+ \fill (4.7,-1.2) circle (2pt);
+ \foreach \y in {-.7,-1.2,-1.7} {
+  \fill (7.1,\y) circle (2pt);
+  \draw[gray!75] (4.8,-1.2)--(7,\y);
+ }
+ \node at (5.9,-2.1) {$\bar z=qg_k^a$};
+ \node at (5.9,-2.8) {within a fiber: $q\longleftrightarrow sq$};
+\end{tikzpicture}
diff --git a/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/build/main.tex b/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/build/main.tex
new file mode 100644
index 0000000000000000000000000000000000000000..aea37cb8ee923d2319f9e811f7788f8ec7d48e25
--- /dev/null
+++ b/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/build/main.tex
@@ -0,0 +1,56 @@
+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage[margin=1.05in]{geometry}
+\usepackage{microtype,amsmath,amssymb,amsthm,mathtools}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta}
+\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref}
+\hypersetup{pdftitle={A counterexample to the coarse Novikov conjecture},pdfauthor={OpenAI}}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\theoremstyle{remark}\newtheorem{remark}[theorem]{Remark}
+\newcommand{\Z}{\mathbb Z}
+\newcommand{\Q}{\mathbb Q}
+\newcommand{\C}{\mathbb C}
+\newcommand{\R}{\mathbb R}
+\newcommand{\N}{\mathbb N}
+\newcommand{\HH}{\mathcal H}
+\newcommand{\KK}{\operatorname{KK}}
+\newcommand{\SL}{\operatorname{SL}}
+\newcommand{\tr}{\operatorname{tr}}
+\newcommand{\prop}{\operatorname{prop}}
+\newcommand{\Ad}{\operatorname{Ad}}
+\newcommand{\supp}{\operatorname{supp}}
+\newcommand{\lf}{\mathrm{lf}}
+\newcommand{\id}{\mathrm{id}}
+\newcommand{\abs}[1]{\lvert #1\rvert}
+\newcommand{\norm}[1]{\lVert #1\rVert}
+\title{A counterexample to the coarse Novikov conjecture}
+\author{OpenAI}
+\date{September 23, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We disprove the coarse Novikov conjecture: ordinary coarse assembly need not
+be rationally injective for uniformly discrete spaces of bounded geometry.
+We construct a coarse disjoint union of finite graphs of uniformly bounded
+degree and an infinite-order class in its degree-one coarse K-homology whose
+image under ordinary coarse assembly in the K-theory of the reduced, locally
+compact Roe algebra vanishes.
+\end{abstract}
+\tableofcontents
+\clearpage
+\input{sections/01-introduction}
+\input{sections/02-conventions}
+\input{sections/03-geometry}
+\input{sections/04-persistence}
+\input{sections/05-transfer}
+\input{sections/06-vanishing}
+\clearpage
+\phantomsection
+\addcontentsline{toc}{section}{References}
+\bibliographystyle{plain}
+\bibliography{references}
+\end{document}
diff --git a/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/build/references.bib b/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/build/references.bib
new file mode 100644
index 0000000000000000000000000000000000000000..929b1efe7cf89a2fdda60c74d78ee3cb3557cc14
--- /dev/null
+++ b/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/build/references.bib
@@ -0,0 +1,216 @@
+@book{BHV,
+  author    = {Bekka, Bachir and de la Harpe, Pierre and Valette, Alain},
+  title     = {Kazhdan's Property {(T)}},
+  series    = {New Mathematical Monographs},
+  volume    = {11},
+  publisher = {Cambridge University Press},
+  address   = {Cambridge},
+  year      = {2008},
+  note      = {Numbered locators refer to the author manuscript dated February 23, 2007},
+  doi       = {10.1017/CBO9780511542749},
+  url       = {https://perso.univ-rennes1.fr/bachir.bekka/KazhdanTotal.pdf}
+}
+
+@book{Blackadar,
+  author    = {Blackadar, Bruce},
+  title     = {{K}-Theory for Operator Algebras},
+  edition   = {2},
+  series    = {Mathematical Sciences Research Institute Publications},
+  volume    = {5},
+  publisher = {Cambridge University Press},
+  address   = {Cambridge},
+  year      = {1998},
+  doi       = {10.1017/9781009701907},
+  url       = {https://bruceblackadar.com/Mathematics/book6.pdf}
+}
+
+@book{Khalkhali,
+  author    = {Khalkhali, Masoud},
+  title     = {Basic Noncommutative Geometry},
+  series    = {EMS Series of Lectures in Mathematics},
+  publisher = {European Mathematical Society},
+  address   = {Z{\"u}rich},
+  year      = {2009},
+  doi       = {10.4171/061},
+  url       = {https://www.math.uwo.ca/faculty/khalkhali/files/BNCG_1sted.pdf}
+}
+
+@book{WY,
+  author    = {Willett, Rufus and Yu, Guoliang},
+  title     = {Higher Index Theory},
+  series    = {Cambridge Studies in Advanced Mathematics},
+  volume    = {189},
+  publisher = {Cambridge University Press},
+  address   = {Cambridge},
+  year      = {2020},
+  note      = {Numbered locators refer to the author draft dated December 9, 2019, at \url{https://math.hawaii.edu/~rufus/higherindextheory}},
+  doi       = {10.1017/9781108867351},
+  url       = {https://doi.org/10.1017/9781108867351}
+}
+
+@article{Kazhdan,
+  author  = {Kazhdan, D. A.},
+  title   = {Connection of the dual space of a group with the structure of its close subgroups},
+  journal = {Functional Analysis and Its Applications},
+  volume  = {1},
+  number  = {1},
+  pages   = {63--65},
+  year    = {1967},
+  doi     = {10.1007/BF01075866},
+  url     = {https://www.mathnet.ru/eng/faa2807}
+}
+
+@misc{WYerrata,
+  author = {Willett, Rufus and Yu, Guoliang},
+  title  = {Errata to ``{Higher Index Theory}''},
+  year   = {2025},
+  note   = {December 8, 2025. \url{https://math.hawaii.edu/~rufus/Errata.pdf}}
+}
+
+@article{DFW,
+  author  = {Dranishnikov, A. N. and Ferry, Steven C. and Weinberger, Shmuel},
+  title   = {Large {Riemannian} manifolds which are flexible},
+  journal = {Annals of Mathematics},
+  series  = {2},
+  volume  = {157},
+  number  = {3},
+  pages   = {919--938},
+  year    = {2003},
+  doi     = {10.4007/annals.2003.157.919},
+  url     = {https://annals.math.princeton.edu/wp-content/uploads/annals-v157-n3-p06.pdf}
+}
+
+@article{HLS,
+  author  = {Higson, Nigel and Lafforgue, Vincent and Skandalis, Georges},
+  title   = {Counterexamples to the {Baum--Connes} conjecture},
+  journal = {Geometric and Functional Analysis},
+  volume  = {12},
+  number  = {2},
+  pages   = {330--354},
+  year    = {2002},
+  doi     = {10.1007/s00039-002-8249-5},
+  url     = {https://vlafforg.perso.math.cnrs.fr/files/counterex.pdf}
+}
+
+@book{Roe93,
+  author = {Roe, John},
+  title = {Coarse Cohomology and Index Theory on Complete Riemannian Manifolds},
+  series = {Memoirs of the American Mathematical Society},
+  volume = {104},
+  note = {Number 497},
+  publisher = {American Mathematical Society},
+  address = {Providence, RI},
+  year = {1993},
+  doi = {10.1090/memo/0497}
+}
+
+
+@incollection{HR95,
+  author = {Higson, Nigel and Roe, John},
+  title = {On the coarse {Baum--Connes} conjecture},
+  booktitle = {Novikov Conjectures, Index Theorems and Rigidity, Vol. 2},
+  editor = {Ferry, Steven C. and Ranicki, Andrew and Rosenberg, Jonathan},
+  series = {London Mathematical Society Lecture Note Series},
+  volume = {227},
+  publisher = {Cambridge University Press},
+  address = {Cambridge},
+  pages = {227--254},
+  year = {1995},
+  doi = {10.1017/CBO9780511629365.008}
+}
+
+
+@article{Yu98,
+  author = {Yu, Guoliang},
+  title = {The {Novikov} conjecture for groups with finite asymptotic dimension},
+  journal = {Annals of Mathematics},
+  series = {2},
+  volume = {147},
+  number = {2},
+  pages = {325--355},
+  year = {1998},
+  doi = {10.2307/121011}
+}
+
+
+@article{Yu00,
+  author = {Yu, Guoliang},
+  title = {The coarse {Baum--Connes} conjecture for spaces which admit a uniform embedding into {Hilbert} space},
+  journal = {Inventiones Mathematicae},
+  volume = {139},
+  number = {1},
+  pages = {201--240},
+  year = {2000},
+  doi = {10.1007/s002229900032}
+}
+
+
+@article{STY02,
+  author = {Skandalis, Georges and Tu, Jean-Louis and Yu, Guoliang},
+  title = {The coarse {Baum--Connes} conjecture and groupoids},
+  journal = {Topology},
+  volume = {41},
+  number = {4},
+  pages = {807--834},
+  year = {2002},
+  doi = {10.1016/S0040-9383(01)00004-0}
+}
+
+
+@article{WY12I,
+  author = {Willett, Rufus and Yu, Guoliang},
+  title = {Higher index theory for certain expanders and {Gromov} monster groups, {I}},
+  journal = {Advances in Mathematics},
+  volume = {229},
+  number = {3},
+  pages = {1380--1416},
+  year = {2012},
+  doi = {10.1016/j.aim.2011.10.024},
+  note = {Theorem numbering follows the author version at \url{https://math.hawaii.edu/~rufus/expgirth1.pdf}},
+  url = {https://math.hawaii.edu/~rufus/expgirth1.pdf}
+}
+
+
+@article{KY06,
+  author = {Kasparov, Gennadi and Yu, Guoliang},
+  title = {The coarse geometric {Novikov} conjecture and uniform convexity},
+  journal = {Advances in Mathematics},
+  volume = {206},
+  number = {1},
+  pages = {1--56},
+  year = {2006},
+  doi = {10.1016/j.aim.2005.08.004}
+}
+
+
+@article{FS14,
+  author = {Finn-Sell, Martin},
+  title = {Fibred coarse embeddings, a-{T}-menability and the coarse analogue of the {Novikov} conjecture},
+  journal = {Journal of Functional Analysis},
+  volume = {267},
+  number = {10},
+  pages = {3758--3782},
+  year = {2014},
+  doi = {10.1016/j.jfa.2014.09.012},
+  url = {https://arxiv.org/abs/1304.3348}
+}
+
+
+@article{DWY23,
+  author = {Deng, Jintao and Wang, Qin and Yu, Guoliang},
+  title = {The coarse {Baum--Connes} conjecture for certain relative expanders},
+  journal = {Advances in Mathematics},
+  volume = {424},
+  pages = {109047},
+  year = {2023},
+  doi = {10.1016/j.aim.2023.109047},
+  url = {https://arxiv.org/abs/2102.10617}
+}
+
+@misc{OpenAITorsionFreeBC2026,
+  author = {{OpenAI}},
+  title = {{A torsion-free counterexample to reduced Baum--Connes injectivity}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Counterexample-to-Reduced-Baum-Connes-Injectivity-September-23-2026/paper.pdf}{OAI:A-Torsion-Free-Counterexample-to-Reduced-Baum-Connes-Injectivity-September-23-2026}},
+  year = {2026}
+}
diff --git a/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/build/sections/01-introduction.tex b/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/build/sections/01-introduction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..39faa6d20b81e95f8f0cb66f7739226dd3e9b7c8
--- /dev/null
+++ b/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/build/sections/01-introduction.tex
@@ -0,0 +1,146 @@
+\section{Introduction}
+
+Coarse index theory relates the large-scale geometry of a space to the
+K-theory of an algebra of controlled operators. Its assembly map sends
+geometric K-homology classes to their analytic indices. The injectivity
+question asks whether that index retains the information in a class after
+one has passed to large scales. Roe developed this viewpoint and its
+Novikov-type formulation \cite{Roe93}; Higson and Roe clarified
+the role of bounded geometry and coarsened K-homology \cite{HR95}.
+Here we construct a bounded-geometry space for which ordinary reduced
+coarse assembly loses a class of infinite order.
+
+A metric space $X$ is uniformly discrete if distinct points have distance
+bounded below by a positive constant. It has bounded geometry if, for every
+$R<\infty$, the cardinalities of its radius-$R$ balls have a common finite
+bound. For such a space, let $P_r(X)$ be the Rips complex: its simplices are the
+finite subsets of $X$ of diameter at most $r$. We use locally finite analytic
+K-homology and its coarsening,
+\[
+ K_i^{\lf}(Y)=\KK_i(C_0(Y),\C),\qquad
+ KX_i(X)=\varinjlim_{r\to\infty}K_i^{\lf}(P_r(X)).
+\]
+Here and throughout, $i$ is taken modulo two. The reduced, locally compact
+Roe algebra $C^*(X)$ is the operator-norm closure of the locally compact
+finite-propagation operators on $\ell^2(X)\otimes\HH_0$, where $\HH_0$ is an
+infinite-dimensional separable Hilbert space. Section~\ref{sec:conventions}
+specifies these conventions and the ordinary coarse assembly map
+\[
+ \mu_X:KX_i(X)\longrightarrow K_i(C^*(X)).
+\]
+The coarse Baum--Connes conjecture asks whether this map is an isomorphism
+\cite[Conjecture~6.6]{HR95}. The rational coarse Novikov assertion is that
+$\mu_X\otimes\id_{\Q}$ is injective. Passing through all Rips scales is
+essential: a cycle at one scale can bound at a larger scale and therefore
+represent zero in $KX_*$. An obstruction to injectivity must survive every
+such coarsening while its analytic index vanishes.
+
+Yu proved coarse assembly isomorphism for bounded-geometry spaces of finite
+asymptotic dimension \cite[Theorem~7.1]{Yu98}, and subsequently for those admitting a
+coarse embedding into Hilbert space \cite{Yu00}. Skandalis, Tu and Yu placed
+the latter result in a groupoid framework \cite{STY02}. Further positive
+injectivity results include coarse embeddings into uniformly convex Banach
+spaces \cite[Theorem~1.1]{KY06} and fibred coarse embeddings into Hilbert space
+\cite[Theorem~34]{FS14}. These theorems impose geometric hypotheses beyond bounded
+geometry. Our result shows that bounded geometry alone does not suffice.
+
+We prove the following integral statement.
+
+\begin{theorem}\label{thm:main}
+There is a uniformly discrete bounded-geometry metric space $X$, which is a
+coarse disjoint union of finite connected graphs of uniformly bounded degree,
+and an infinite-order element $\alpha\in KX_1(X)$ such that
+\[
+ \mu_X(\alpha)=0\quad\text{in }K_1(C^*(X)).
+\]
+Consequently, ordinary coarse assembly for $X$ fails to be injective both
+integrally and after tensoring its domain and target with $\Q$.
+\end{theorem}
+
+\subsection*{Predecessors and the scope of the obstruction}
+Yu already constructed a non-bounded-geometry example with a nontrivial
+coarse K-homology class of zero index, using spheres of increasing dimension
+and size \cite[Section~8, Proposition~8.1]{Yu98}. Dranishnikov, Ferry and
+Weinberger constructed integral noninjectivity examples for a map from
+locally finite K-homology to Roe-algebra K-theory, also allowing spaces
+without bounded geometry \cite[Introduction, pp.~920--921]{DFW}.
+Their Remark~6.2 distinguishes those examples from the Rips-coarsened
+formulation: the relevant classes die under coarsening. Theorem~\ref{thm:main}
+retains bounded geometry and detects its class in the actual direct limit.
+
+Higson, Lafforgue and Skandalis constructed injectivity and surjectivity
+obstructions for several groupoid assembly maps \cite{HLS}. Their Hausdorff
+groupoid example in Section~5 combines a degree-one Bott class with a
+Kazhdan projection to produce a kernel class. This is a close methodological
+antecedent of the winding and constant-vector averaging used here.
+Their coarse-space construction in Section~6 instead establishes
+non-surjectivity for specific congruence expander unions
+\cite[Section~6, Proposition~11]{HLS}. These conclusions must be separated:
+a groupoid kernel example does not itself provide the bounded-geometry
+space and ordinary Rips-domain kernel in Theorem~\ref{thm:main}.
+
+Expander phenomena also coexist with injectivity. Willett and Yu proved
+ordinary coarse injectivity for coarse disjoint unions of finite graphs
+whose girth tends to infinity; with an additional weak-expansion hypothesis,
+surjectivity fails \cite[Theorem~1.5]{WY12I}. The fibers in our construction
+have short cycles, so persistent boundary winding does not imply that
+large-girth hypothesis. More recent isomorphism results for certain relative
+expanders likewise depend on specific embedding hypotheses for the kernel
+and quotient families \cite[Theorem~1.1]{DWY23}. The organization of our
+vertices into finite fibers does not assert such an embedding or a finite-group
+extension. Throughout, our target is the reduced, locally compact Roe algebra;
+maximal completions and coefficient-free group assembly are different
+questions.
+The independent companion article~\cite[Theorem~1.1]{OpenAITorsionFreeBC2026}
+concerns group assembly, which for a torsion-free group maps the compactly supported K-homology
+of its classifying space to the K-theory of its reduced group algebra. It
+constructs a finitely generated torsion-free group for which this
+coefficient-free map is not rationally injective.
+
+\subsection*{How the construction works}
+Begin with finite discs $D_j$ of radii $j$ in the triangular lattice, and let $C_j$ be their
+boundary cycles. Above each vertex of $D_j$ place a finite congruence quotient
+of $\SL_3(\Z)$. The quotient is trivial on $C_j$ and grows with depth inside
+the disc. Cayley edges connect vertices within a fiber; reduction maps and
+twisted matchings connect neighboring fibers. The resulting finite graph is
+denoted by $X_j$. Thus $C_j$ is also a cycle in $X_j$.
+
+Let $B$, $D$ and $X$ be the separated unions of the $C_j$, $D_j$ and $X_j$,
+respectively. The family of oriented boundary classes gives an element of
+$K_1^{\lf}(P_1(B))$, and its image defines $\alpha\in KX_1(X)$. Two features
+of the fibers have different roles.
+
+First, the twisting prevents short loops in $X_j$ from winding around the
+center once $j$ is sufficiently large. At any prescribed Rips scale this
+produces an integral cocycle whose restriction to $C_j$ has winding one.
+The associated K-theory pairing detects every nonzero integer multiple of
+$\alpha$ at that scale. Sections~\ref{sec:geometry} and
+\ref{sec:persistence} prove that $\alpha$ has infinite order in the direct
+limit.
+
+Second, property~(T) uniformly approximates the projections onto constant
+vectors in all the finite fibers. Averaging into those vectors defines an
+isometry $V:\ell^2(D)\otimes\HH_0\to\ell^2(X)\otimes\HH_0$.
+The right translations used for twisting fix the constant vectors, so they
+do not interfere with this averaging.
+Section~\ref{sec:transfer} proves that conjugation by $V$ maps $C^*(D)$ into
+$C^*(X)$, despite the unbounded sizes of the fibers. On the boundary fibers
+this isometry is the ordinary inclusion. Since the boundary classes bound
+the filled discs, naturality of assembly then gives $\mu_X(\alpha)=0$ in
+Section~\ref{sec:vanishing}.
+
+The transfer argument isolates a useful principle: uniform finite-propagation
+approximation of constant-fiber projections, together with controlled
+operators between adjacent fibers, can produce a Roe-algebra homomorphism
+without a corresponding coarse map. This principle accounts for the
+difference between the persistent winding in $X$ and the vanishing of its
+index through $D$.
+
+
+Section~\ref{sec:conventions} fixes the analytic K-homology, Rips metrics,
+and index naturality used by both branches. Section~\ref{sec:geometry}
+constructs the common graph family and its short-loop estimate.
+Section~\ref{sec:persistence} then proves persistence, independently of
+averaging. Section~\ref{sec:transfer} proves the operator transfer, and
+Section~\ref{sec:vanishing} combines its exact boundary restriction with
+the filled-disc argument to annihilate the same class.
diff --git a/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/build/sections/02-conventions.tex b/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/build/sections/02-conventions.tex
new file mode 100644
index 0000000000000000000000000000000000000000..1ec7cc45f973f01ef63b8f62ff54f4387515cf94
--- /dev/null
+++ b/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/build/sections/02-conventions.tex
@@ -0,0 +1,129 @@
+\section{Coarse assembly and locally finite classes}\label{sec:conventions}
+
+We collect the functorial facts needed to carry a family of boundary classes
+through the construction. All spaces below are proper metric spaces, and all
+maps used to push forward locally finite K-homology are continuous and proper.
+A map between discrete metric spaces is \emph{coarse} if it is proper and
+bornologous: for every $R$ there is $S$ such that points at distance at most
+$R$ have images at distance at most $S$.
+
+\subsection{Rips complexes and their coarse structure}
+For a uniformly discrete bounded-geometry space $Y$, the complex $P_r(Y)$ is
+locally finite and has uniformly bounded dimension at each fixed $r$.
+We use its usual simplicial topology. A concrete proper metric compatible
+with that topology and its coarse structure is convenient when the complex
+has disconnected components. Regard a point as a finitely supported
+probability measure on $Y$ whose support has diameter at most $r$, and use
+the transportation metric
+\begin{equation}\label{eq:wasserstein}
+ d_r(\nu,\eta)=\inf_{\pi}\sum_{x,y\in Y}d_Y(x,y)\pi(x,y),
+\end{equation}
+where $\pi$ ranges over the couplings of $\nu$ and $\eta$.
+On each finite subcomplex this metric gives the usual topology, equivalently
+the topology from spherical simplices. The inclusion $y\mapsto\delta_y$ is
+isometric, and every point is within distance $r$ of every vertex in its
+support. Moreover, a closed ball about $\delta_y$ of radius $R$ has all its supports
+in the finite set $B_Y(y,R+r)$. Indeed, for $x,z\in\supp(\nu)$,
+$d_Y(y,x)\le d_Y(y,z)+r$; averaging over $z$ gives
+$d_Y(y,x)\le d_r(\delta_y,\nu)+r$. The closed ball is a closed subset
+of a finite subcomplex and is compact. Local finiteness gives the asserted
+topology globally.
+
+In particular, choosing a support vertex gives a coarse inverse to the
+vertex inclusion. Distances between chosen support vertices and between
+the corresponding points differ by at most $2r$. Thus
+\eqref{eq:wasserstein} realizes the usual coarse structure determined by
+supports; it also supplies finite distances between disconnected simplicial
+components.
+
+\paragraph{Comparison with spherical metrics.}
+One can compare this metric directly with the standard semi-spherical extension:
+allow paths in spherical simplices, normalized to have edge length one,
+and jumps between vertices of cost $d_Y$. Write $d_{\mathrm{ss}}$ for the
+infimum of their lengths. At fixed $r$, simplex dimensions have a common
+bound. On each simplex, the transportation distance is bounded by a
+constant depending only on $r$ times its spherical distance; this follows
+by comparing barycentric coordinates in the bounded dimensions and using
+$d_r\le (r/2)\norm{\nu-\eta}_1$. The same inequality, with constant one,
+holds for vertex jumps. Conversely, one may travel to a support vertex,
+make one vertex jump, and travel to the endpoint. Thus, for a finite
+constant $c_r$,
+\[
+ d_r\le c_r d_{\mathrm{ss}},\qquad d_{\mathrm{ss}}\le d_r+2.
+\]
+The second inequality uses that the minimum distance between the two
+supports is at most $d_r$. The inequalities give the same coarse structure;
+the finite-subcomplex argument above gives the usual simplicial topology.
+These choices give the standard Rips coarsening and Roe-algebra
+identifications \cite[Definition~7.2.8, Proposition~7.2.11 and Theorem~7.2.16]{WY}.
+The inclusions for increasing
+$r$ are continuous proper coarse maps. A proper Lipschitz map of vertex
+spaces induces the corresponding proper simplicial maps of Rips complexes.
+The support-vertex selections are used for the Roe-algebra K-theory
+identifications; locally finite K-homology is pushed forward by the
+continuous proper simplicial maps. The selector compositions before and
+after an inclusion differ by a bounded distance, since their values lie
+in the same bounded support. This also applies to a proper Lipschitz map
+of vertex spaces, with the bound given by the diameter of its image support.
+Thus the Roe K-theory identifications respect
+these maps and the passage to larger scales.
+
+\subsection{Roe algebras and naturality}
+On an ample module over a proper metric space, an operator is locally compact
+if multiplication on either side by any function in $C_0$ makes it compact.
+Its propagation is at most $R$ if its matrix coefficients between sets at
+distance greater than $R$ vanish. For a discrete bounded-geometry space $Y$
+on $\ell^2(Y)\otimes\HH_0$, a finite-propagation operator is locally compact
+exactly when each of its matrix entries in $\mathcal B(\HH_0)$ is compact.
+Indeed, the propagation bound leaves only finitely many nonzero entries in
+each row and column; finite-support functions give compact products, and
+$C_0(Y)$ follows by uniform approximation.
+
+The locally finite analytic index at each Rips scale takes values in the
+Roe algebra of that complex. Coarse equivalence with the vertex space
+identifies its K-theory with $K_*(C^*(Y))$. The compatible indices define
+$\mu_Y$. We use the usual reduced Roe index and its naturality
+\cite[Sections~5.1 and~6.7, Lemma~7.1.3 and Theorem~7.2.16]{WY}, with the corrected localization
+comparison in the authors' errata \cite[items~30--31]{WYerrata}.
+All spaces used in this application are nonempty, and their ample modules
+may be chosen with infinite multiplicity. In particular, for a proper
+Lipschitz inclusion $f:Y\to Z$ and its covering isometry
+$F(\delta_y\otimes v)=\delta_{f(y)}\otimes v$,
+\begin{equation}\label{eq:naturality}
+ \mu_Z\circ f_*=(\Ad F)_*\circ\mu_Y.
+\end{equation}
+On vertex Roe algebras $\Ad F$ is an actual homomorphism; the induced map on
+K-theory is independent of the covering choices. All the maps to which we
+apply \eqref{eq:naturality} will be inclusions of this form.
+
+\subsection{Families and winding pairings}
+For a countable topological disjoint union of second-countable locally compact
+spaces $Y_j$, all the algebras $C_0(Y_j)$ and their countable direct sum are
+separable. Countable additivity in the first variable of $\KK$
+\cite[Theorem~19.7.1 and Section~19.8.10]{Blackadar} gives
+\begin{equation}\label{eq:additivity}
+ K_i^{\lf}\left(\coprod_jY_j\right)
+ =\KK_i\left(\bigoplus_j C_0(Y_j),\C\right)
+ \cong\prod_j K_i^{\lf}(Y_j).
+\end{equation}
+This is an integral product. Restriction to a clopen union of components is
+the corresponding coordinate projection, induced contravariantly by
+extension by zero on $C_0$.
+
+For an oriented circle $C$, choose its analytic fundamental class $[C]$ so
+that its index pairing with a unitary of winding one is $1$. The circle index
+formula then reads
+\begin{equation}\label{eq:winding}
+ \langle[f],[C]\rangle=\deg(f),\qquad f:C\to S^1.
+\end{equation}
+This fixes the otherwise harmless sign convention for the Dirac or Toeplitz
+description of the class \cite[Example~4.2.4]{Khalkhali}.
+An integral simplicial $1$-cocycle on a finite complex $Y$ determines a class
+in $H^1(Y;\Z)$, represented by a map $f:Y\to S^1$, and hence a unitary
+class $[f]\in K_1(C(Y))$. For an included oriented circle, naturality of the
+index pairing identifies its pairing with the sum of the cocycle around
+that circle \cite[Proposition~18.7.1 and Corollary~18.10.3]{Blackadar}.
+One can see the representing map directly: map vertices to
+$1$, traverse each edge with the assigned integer winding, and extend over
+triangles because their winding sums vanish. Higher extensions exist since
+$\pi_k(S^1)=0$ for $k\ge2$.
diff --git a/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/build/sections/03-geometry.tex b/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/build/sections/03-geometry.tex
new file mode 100644
index 0000000000000000000000000000000000000000..e073174a33ed4a47c24a228e566a351d0aac2252
--- /dev/null
+++ b/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/build/sections/03-geometry.tex
@@ -0,0 +1,243 @@
+\section{Finite discs with congruence fibers}\label{sec:geometry}
+
+We construct finite graphs $X_j$ containing distinguished boundary cycles
+$C_j$. The goal of this section is a uniform statement about short loops:
+at any fixed length bound, loops sufficiently far out in the family have
+zero signed crossing count for a fixed ray from the center of the underlying
+disc. This count agrees with winding whenever the projected loop avoids
+the center.
+
+\subsection{Congruence quotients and a separated holonomy}
+We need a quotient tower with uniformly bounded successive kernels, so that
+lifting base edges preserves bounded degree. We will also choose compatible
+twists whose trace separation rules out nonzero crossing count on short
+closed paths in sufficiently late graphs $X_j$. The choice of $\SL_3(\Z)$ supplies
+both this quotient tower and the uniform averaging estimates used in
+Section~\ref{sec:transfer}.
+
+Let $\Gamma=\SL_3(\Z)$ and let $S$ be the finite symmetric generating set of
+elementary matrices $I\pm E_{ab}$, $a\ne b$. To see that these generate,
+apply the Euclidean algorithm by determinant-one row operations to the
+primitive first column of an integral determinant-one matrix, then to its
+remaining $2$-by-$2$ block. Each integral elementary operation is a power
+of a member of $S$. Put
+\[
+ Q_0=\{1\},\qquad Q_k=\SL_3(\Z/2^k\Z)\quad(k\ge1),
+\]
+with their reduction maps. All reductions from $\Gamma$, and between
+successive $Q_k$, are surjective. Indeed, over $\Z/2^k\Z$ an invertible
+matrix has a unit in its first column. Elementary row operations move it to
+the first position and clear the remaining entries. Continuing reduces the
+matrix to a diagonal one. The identity
+\[
+ \begin{pmatrix}0&u\\-u^{-1}&0\end{pmatrix}
+ =\begin{pmatrix}1&u\\0&1\end{pmatrix}
+  \begin{pmatrix}1&0\\-u^{-1}&1\end{pmatrix}
+  \begin{pmatrix}1&u\\0&1\end{pmatrix}
+\]
+expresses the needed determinant-one row interchanges and diagonal
+matrices by elementary matrices. Every elementary matrix over the quotient
+lifts to an integral one, itself generated by $S$.
+There is a uniform bound
+\begin{equation}\label{eq:kernelbound}
+ h:=2^9\ \ge\ \abs{\ker(Q_{k+1}\to Q_k)}\quad(k\ge0).
+\end{equation}
+For $k\ge1$, a kernel element has the form $I+2^kA$ modulo $2^{k+1}$ and is
+specified by the nine binary entries of $A$; the same bound holds for $Q_1$.
+
+Choose a $2$-adic unit $t\in1+2\Z_2$ transcendental over $\Q$. Such a choice
+exists because $1+2\Z_2$ is uncountable and the algebraic elements over
+$\Q$ are countable. For every nonzero integer $m$,
+\begin{equation}\label{eq:transcendentaltrace}
+ t^m+t^{-m}+1\notin\Z.
+\end{equation}
+Otherwise multiplying the alleged equality by $t^{\abs m}$ gives a nonzero
+polynomial over $\Z$ satisfied by $t$.
+Define
+\[
+ g_k=\operatorname{diag}(t,t^{-1},1)\pmod{2^k}\in Q_k\quad(k\ge1),
+ \qquad g_0=1.
+\]
+These elements are compatible with reduction.
+
+\begin{lemma}[Trace separation]\label{lem:trace}
+For each integer $A\ge1$ there is an integer $N(A)\ge1$ such that, whenever
+$\ell\ge N(A)$, $0<\abs m\le A$, and $w\in\Gamma$ has word length at most
+$A$ with respect to $S$,
+\[
+ \tr(g_\ell^m)\ne\tr(w)\pmod{2^\ell}.
+\]
+\end{lemma}
+\begin{proof}
+There are only finitely many pairs $(m,w)$ in question. Each difference
+$t^m+t^{-m}+1-\tr(w)$ is a nonzero $2$-adic integer by
+\eqref{eq:transcendentaltrace}. Choose $N(A)$ greater than all their
+$2$-adic valuations.
+\end{proof}
+
+\subsection{The base discs and the lifted edges}
+Use the triangular lattice with coordinates $(a,b)\in\Z^2$ and neighbor
+differences
+\[
+ \pm(1,0),\quad\pm(0,1),\quad\pm(1,-1).
+\]
+Embed its two coordinate axes at angle $\pi/3$. Set
+\[
+ \rho(a,b)=\max\{\abs a,\abs b,\abs{a+b}\}.
+\]
+For $j\ge2$, let $D_j$ be the finite induced graph on $\rho\le j$,
+equipped with its graph metric. Its elementary triangles fill a closed
+topological disc. The boundary $C_j$, given by $\rho=j$, is a cycle of
+length $6j$; we equip its vertex set with its own cycle metric. When used as a
+topological circle, $C_j$ denotes the geometric realization of this cycle.
+The depth of a vertex
+$u\in D_j$ is
+\[
+ k_j(u)=j-\rho(u).
+\]
+Depths of neighboring vertices differ by at most one.
+
+Fix a ray from the origin meeting no other lattice vertex. For an oriented
+lattice edge $(u,v)$ not incident to the origin, let $a(u,v)\in\{-1,0,1\}$
+be its signed intersection with this ray. Set $a(u,v)=0$ for edges incident
+to the origin. Choose the sign so that the positively oriented boundary
+has total charge one. Then $a(v,u)=-a(u,v)$, and the sum of $a$ on any
+closed edge path avoiding the origin equals its winding number.
+
+The vertices of $X_j$ are
+\[
+ \{(u,q):u\in D_j,\ q\in Q_{k_j(u)}\}.
+\]
+Put undirected edges of length one as follows; repeated edges and loops
+may be discarded.
+\begin{enumerate}
+\item Within the fiber over $u$, join $q$ to $sq$ for every $s\in S$.
+\item If $(u,v)$ is an oriented base edge with equal depth $k$, join
+$(u,q)$ to $(v,qg_k^{a(u,v)})$.
+\item If $(u,v)$ goes from depth $k+1$ to depth $k$, join
+$(u,q)$ to $(v,\bar qg_k^{a(u,v)})$, where $\bar q$ denotes reduction.
+The reverse direction uses the inverse edge relation.
+\end{enumerate}
+Left Cayley multiplication and right holonomy commute. This choice of
+sides will allow a short closed path to yield a conjugacy relation in a
+single quotient.
+
+Each $X_j$ is connected: the fibers are connected Cayley graphs, and every
+base edge has nonempty relations to both endpoint fibers. Each vertex has
+degree at most
+\begin{equation}\label{eq:degree}
+ \Delta=\abs S+6h.
+\end{equation}
+Indeed, a base neighbor gives a bijection, a single parent, or at most $h$
+children. Boundary fibers are trivial, so the cycle $C_j$ includes in $X_j$
+by $u\mapsto(u,1)$. Figure~\ref{fig:disc} shows the depth labels and
+the two types of inter-fiber edge relation.
+
+These graphs do have uniformly short cycles. In a fiber of depth $k\ge1$,
+the commuting generators $A=I+E_{12}$ and $B=I+E_{13}$ give the four
+distinct vertices $q,Aq,BAq,Bq$ of a Cayley square. Thus the short-loop
+statement below concerns their charge, not the absence of short cycles.
+
+\begin{figure}[t]
+\centering
+\input{figures/disc}
+\caption{The base disc and its fibers, shown schematically for $j=3$.
+The shell at depth $k$ carries $Q_k$; the boundary carries $Q_0=\{1\}$.
+Crossing the dashed ray applies right multiplication by $g_k^{a(u,v)}$,
+with reduction to the shallower level when the depths differ. The center
+has depth $j$. The insets show selected inter-fiber edges: a twisted
+matching at equal depth and a shallow vertex with its deep children.
+In each inset, $a=a(u,v)$ for the base edge oriented from the left-hand
+fiber to the right-hand fiber.
+The number of drawn vertices is schematic, not the quotient cardinality.}
+\label{fig:disc}
+\end{figure}
+
+\subsection{Separated unions and their boundary maps}
+Choose $b_j\in C_j$. In each of the three unions
+\[
+ B=\coprod_{j\ge2}C_j,\qquad D=\coprod_{j\ge2}D_j,\qquad
+ X=\coprod_{j\ge2}X_j,
+\]
+join the chosen basepoints of successive components by an auxiliary edge
+of length $M_j=100+j^2$; in $X_j$ use $(b_j,1)$.
+Use the induced path distance on the original vertex sets, without adding
+vertices along these auxiliary edges. Inside each component this is its
+original graph or cycle metric: leaving a component cannot provide a
+shortcut back into it. The distances between distinct components tend to
+infinity outside a finite family, so these are coarse disjoint unions.
+
+The spaces are uniformly discrete and proper, and have bounded geometry.
+To check the last assertion directly, the weighted graph has degree at
+most $\Delta+2$ and every edge has length at least one. Hence a ball of
+radius $R$ has at most
+$\sum_{m=0}^{\lfloor R\rfloor}(\Delta+2)^m$ vertices. The same argument
+applies to $B$ and $D$ with their smaller degree bounds. There are proper
+$1$-Lipschitz inclusions
+\begin{equation}\label{eq:boundarymaps}
+ e:B\longrightarrow D,\quad u\longmapsto u,
+ \qquad i:B\longrightarrow X,\quad u\longmapsto(u,1).
+\end{equation}
+Every path in $B$ maps to a path of the same length, including the auxiliary
+links. Properness follows from injectivity and finiteness of bounded subsets
+in the target. The use of the intrinsic cycle metric on $B$ ensures these
+Lipschitz assertions.
+
+For later use, at every fixed distance bound $R$ only finitely many pairs
+$(u,v)\in D\times D$ from different components satisfy $d_D(u,v)\le R$.
+Such a path must cross a link of length at most $R$. There are only finitely
+many such links, and both endpoints of the pair lie within distance $R$
+of their endpoints. Those balls are finite.
+
+\subsection{Short loops have zero charge far out}
+For an edge path in $X_j$, define its charge to be the sum of $a(u,v)$ on
+its projected base steps; fiber steps contribute zero. Charges are additive
+under concatenation and change sign under reversal.
+
+\begin{lemma}[Small-loop lemma]\label{lem:smallloops}
+For each integer $A\ge1$ there is $J(A)$ such that, for all $j\ge J(A)$,
+every closed edge path in $X_j$ of length at most $A$ has charge zero.
+One may take $J(A)=\max\{2,N(A)+\lfloor A/2\rfloor\}$, with $N(A)$ from
+Lemma~\ref{lem:trace}.
+\end{lemma}
+\begin{proof}
+Consider a closed path of length $n\le A$ and projected vertices in $D_j$.
+Suppose its charge $m$ is nonzero. We first show that every projected vertex
+$u$ satisfies $\rho(u)\le\lfloor n/2\rfloor$.
+If not, choose a linear functional
+$f\in\{\pm a,\pm b,\pm(a+b)\}$ attaining $\rho(u)$.
+Every other vertex is reachable from $u$ along a shorter arc of the closed
+path of length at most $\lfloor n/2\rfloor$. Since $f$ changes by at most
+one in each step, including the stationary projections of fiber steps,
+the entire polygon lies in the open half-plane $f>0$. This argument applies
+to each occurrence of a vertex, so repeated vertices and edges cause no
+change. The polygon is contractible in that half-plane, avoids the origin,
+and has winding zero, contradicting its charge. In particular, the same
+argument covers paths whose projection initially might have met the origin:
+the supposed vertex outside the bound would force the whole projection to
+avoid it.
+
+Let $\ell$ be the minimum depth along the path. The preceding bound gives
+$\ell\ge j-\lfloor A/2\rfloor$. Reduce every fiber coordinate along the
+path to $Q_\ell$. A fiber step becomes left multiplication by an element of
+$S$. An oriented base step becomes right multiplication by
+$g_\ell^{a(u,v)}$. This is immediate in the equal-depth and deep-to-shallow
+cases. In the reverse case the defining edge relation says
+$q_{\mathrm{shallow}}=\bar q_{\mathrm{deep}}g_k^{-a}$, so that
+$\bar q_{\mathrm{deep}}=q_{\mathrm{shallow}}g_k^a$, as required.
+
+Commuting left and right multiplications and using closure of the path,
+we obtain
+\[
+ w z g_\ell^m=z
+\]
+for its starting coordinate $z\in Q_\ell$ and a word $w$ of length at most
+$A$ in $S$. Consequently $g_\ell^m=z^{-1}w^{-1}z$ and
+$\tr(g_\ell^m)=\tr(w^{-1})$ modulo $2^\ell$.
+Here $0<\abs m\le A$. If $j\ge J(A)$, then $\ell\ge N(A)$ and this
+contradicts Lemma~\ref{lem:trace} applied to $w^{-1}$.
+\end{proof}
+
+The qualification ``$j$ sufficiently large'' is part of the conclusion.
+It is precisely what is needed for a test at each fixed Rips scale; no
+single short-loop bound is asserted simultaneously for all components.
diff --git a/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/build/sections/04-persistence.tex b/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/build/sections/04-persistence.tex
new file mode 100644
index 0000000000000000000000000000000000000000..67d821eb428ae7b780ae05900b3ade1561b9e091
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+++ b/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/build/sections/04-persistence.tex
@@ -0,0 +1,78 @@
+\section{An infinite-order coarse K-homology class}\label{sec:persistence}
+
+The small-loop lemma converts winding on the base lattice into an integral
+cocycle on each sufficiently late Rips component. We use this cocycle to
+test the same boundary family at every scale.
+
+All links between components of $B$ have length greater than one, and each
+$C_j$ has length $6j\ge12$. Thus $P_1(B)$ is exactly the topological disjoint
+union of the circles $C_j$. By \eqref{eq:additivity}, their positively
+oriented analytic classes determine
+\[
+ \beta_1=([C_j])_{j\ge2}\in K_1^{\lf}(P_1(B)).
+\]
+Let $\beta\in KX_1(B)$ be its direct-limit image. For each integer $r\ge1$,
+let $i_r:P_1(B)\to P_r(X)$ be the map induced by the boundary inclusion,
+and put
+\[
+ \alpha_r=(i_r)_*\beta_1,\qquad \alpha=i_*\beta\in KX_1(X).
+\]
+The classes $\alpha_r$ represent one and the same element $\alpha$.
+
+\begin{proposition}\label{prop:persistence}
+For every integer $r\ge1$ there is a homomorphism
+\[
+ \lambda_r:K_1^{\lf}(P_r(X))\longrightarrow\Z
+ \quad\text{such that}\quad \lambda_r(\alpha_r)=1.
+\]
+Consequently, every nonzero integer multiple of $\alpha$ is nonzero in
+$KX_1(X)$.
+\end{proposition}
+\begin{proof}
+Choose $j$ sufficiently large that $j\ge J(3r)$ and that both auxiliary
+links adjacent to $X_j$ have length greater than $r$. Then $Y=P_r(X_j)$ is
+a finite clopen component of $P_r(X)$.
+
+For an oriented edge $(x,y)$ of $Y$, choose an edge path in $X_j$ from $x$
+to $y$ of length at most $r$, and let $b_r(x,y)$ be its charge. Two choices
+differ by a closed path of length at most $2r$, so
+Lemma~\ref{lem:smallloops} makes this integer independent of the choice.
+Reversing a path gives $b_r(y,x)=-b_r(x,y)$. Around any Rips triangle, the
+three chosen paths form a closed path of length at most $3r$, again with
+zero charge. Therefore $b_r$ is an integral simplicial $1$-cocycle on $Y$.
+
+On each unit boundary edge of $C_j$ we may choose that edge itself. Hence
+the sum of $b_r$ around the positively oriented $C_j$ is one. By the
+construction in Section~\ref{sec:conventions}, there is a map
+$f_r:Y\to S^1$ whose unitary class pairs with the included boundary as
+\[
+ \langle[f_r],(C_j\hookrightarrow Y)_*[C_j]\rangle=1.
+\]
+Define $\lambda_r$ by restriction of locally finite K-homology to the
+clopen component $Y$, followed by this index pairing. Restriction of
+$\alpha_r$ is exactly the included $j$th boundary class. Indeed,
+$i_r^{-1}(Y)=C_j$, and pullback along $i_r$ carries extension by zero from
+$C(Y)$ to extension by zero from $C(C_j)$, after restriction to $C_j$.
+Applying the contravariant first variable of $\KK$ gives the claimed
+coordinate projection in \eqref{eq:additivity}, regardless of connections
+among the other Rips components. This proves the displayed claim.
+Equivalently, one may extend $f_r$ by $1$ outside $Y$ to obtain a
+unitary in the unitization of $C_0(P_r(X))$; it is a compactly supported
+K-theory test for the locally finite class.
+
+For any integer $n\ne0$, we have $\lambda_r(n\alpha_r)=n\ne0$ at every
+integer scale $r\ge1$. If $n\alpha$ vanished in the direct limit, its
+representative $n\alpha_1$ would vanish at some later scale, and then at
+some integer scale. This contradicts the pairing. Integer scales are
+cofinal, so the conclusion holds in $KX_1(X)$.
+\end{proof}
+
+The component and the compactly supported test may change with $r$. The
+argument requires one test at each scale, and does not assert a single
+test persisting at all scales. We have proved that $\alpha$ has infinite
+order integrally. Since the kernel of $A\to A\otimes_{\Z}\Q$ is the
+torsion subgroup for any abelian group $A$, it follows that
+\begin{equation}\label{eq:rationalnonzero}
+ \alpha\otimes1\ne0\quad\text{in }KX_1(X)\otimes_{\Z}\Q.
+\end{equation}
+No product in \eqref{eq:additivity} has been rationalized separately.
diff --git a/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/build/sections/05-transfer.tex b/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/build/sections/05-transfer.tex
new file mode 100644
index 0000000000000000000000000000000000000000..f63378ff432605b120921b767a7a379e9623676b
--- /dev/null
+++ b/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/build/sections/05-transfer.tex
@@ -0,0 +1,168 @@
+\section{Averaging defines a Roe-algebra homomorphism}\label{sec:transfer}
+
+We now prove the analytic input for vanishing. Averaging over each fiber
+will embed the module of the filled base discs into the module of $X$.
+The main point is to show that the resulting spatial homomorphism takes
+values in the locally compact Roe algebra, uniformly over growing fibers.
+
+\subsection{A general transfer criterion}
+The following criterion separates that operator argument from the particular
+congruence construction. Propagation of an operator between two fiber
+subspaces is measured using the ambient metric of $X$.
+
+\begin{proposition}[Transfer through finite fibers]\label{prop:transfer}
+Let $D$ and $X$ be countable uniformly discrete bounded-geometry spaces.
+Suppose $X=\coprod_{u\in D}F_u$ is a partition into nonempty finite sets.
+Write $\xi_u=\abs{F_u}^{-1/2}\sum_{x\in F_u}\delta_x$ and let $P$ be the
+orthogonal projection on $\ell^2(X)$ onto the span of the $\xi_u$.
+Assume:
+\begin{enumerate}
+\item $P$ is an operator-norm limit of finite-propagation contractions $P_n$
+on $\ell^2(X)$;
+\item for every $R<\infty$ there is $s_R<\infty$ such that, whenever
+$d_D(u,v)\le R$, there is a contraction
+$O_{uv}:\ell^2(F_v)\to\ell^2(F_u)$ of propagation at most $s_R$ with
+$O_{uv}\xi_v=\xi_u$.
+\end{enumerate}
+For the isometry
+\[
+ V:\ell^2(D)\otimes\HH_0\longrightarrow\ell^2(X)\otimes\HH_0,
+ \qquad V(\delta_u\otimes h)=\xi_u\otimes h,
+\]
+conjugation defines a $*$-homomorphism
+$\Ad V:C^*(D)\to C^*(X)$ between the reduced, locally compact Roe algebras.
+\end{proposition}
+\begin{proof}
+The empty case is immediate, so assume $D$ is nonempty. We first lift the
+supported blocks of an operator by a finite decomposition into matchings.
+Sandwiching this lift between the constant-fiber projections gives the
+desired conjugate, and the approximants to those projections place it in
+the Roe algebra.
+
+Let $T\in\mathcal B(\ell^2(D)\otimes\HH_0)$ be locally compact and of
+propagation at most $R$. Its entries $T_{uv}$ are compact operators on
+$\HH_0$. Set $b_R=\sup_u\abs{B_D(u,R)}$. The bipartite support graph of
+these entries has degree at most $b_R$, so its edges partition into at most
+$N_R=2b_R-1$ matchings. For completeness, enumerate the edges and color
+them greedily with $2b_R-1$ colors: at most $2(b_R-1)$ already colored
+edges meet the endpoints of the next edge. A matching has no repeated row
+or column.
+
+For each matching form the block operator with entries
+$O_{uv}\otimes T_{uv}$. Orthogonality of both its source and target blocks
+gives norm at most $\sup_{u,v}\norm{T_{uv}}\le\norm T$. Summing the
+finitely many matching operators gives a bounded operator $A_T$ with
+\begin{equation}\label{eq:matchingnorm}
+ \norm{A_T}\le N_R\norm T,\qquad \prop(A_T)\le s_R.
+\end{equation}
+Its vertex-level entries are scalar multiples of compact operators, so
+finite propagation implies local compactness.
+
+Put $\widehat P=P\otimes1_{\HH_0}$ and
+$\widehat P_n=P_n\otimes1_{\HH_0}$. Compression of the $(u,v)$ block gives
+\[
+ p_u O_{uv}p_v=\lvert\xi_u\rangle\langle\xi_v\rvert,
+ \qquad \widehat P A_T\widehat P=VTV^*,
+\]
+where $p_u$ is the rank-one projection onto $\xi_u$. Put
+$d_n=\prop(P_n)$. For every $n$,
+$\widehat P_n A_T\widehat P_n$ has propagation at most $2d_n+s_R$.
+Its $(x,y)$ entry sums only over intermediate vertices
+$z\in B_X(x,d_n)$ and $w\in B_X(y,d_n)$. These sets are finite, and
+each summand is a scalar multiple of the compact entry $(A_T)_{zw}$.
+Thus the sandwich is locally compact and belongs to $C^*(X)$. Moreover,
+\begin{equation}\label{eq:sandwich}
+ \norm{\widehat P_n A_T\widehat P_n-VTV^*}
+ \le 2\norm{P_n-P}N_R\norm T\longrightarrow0.
+\end{equation}
+It follows that $VTV^*\in C^*(X)$.
+
+Such $T$ are dense in $C^*(D)$, and conjugation by an isometry is norm
+continuous. Therefore its range is contained in $C^*(X)$ on the whole
+algebra. Finally $V^*V=1$ proves multiplicativity and preservation of
+adjoints. The auxiliary choices in $A_T$ are used only to prove this range
+inclusion; they need not depend linearly on $T$.
+\end{proof}
+
+The amplified projection $\widehat P$ in this proof is a bounded operator,
+not a locally compact Roe-algebra element. Its diagonal compression at
+$x\in F_u$ is $\abs{F_u}^{-1}1_{\HH_0}$, which is not compact.
+The proof establishes local compactness for the sandwiches, where it is
+actually needed.
+
+\subsection{Uniform approximation from property (T)}
+Return to the finite fibers $F_u=\{u\}\times Q_{k_j(u)}$ of our graph $X$.
+Kazhdan introduced property~(T) and proved its higher-rank and lattice
+instances \cite{Kazhdan}. For $\SL_3(\Z)$ it provides
+a constant $\kappa>0$ for the fixed
+generating set $S$: in any unitary representation without invariant vectors,
+each unit vector is moved by at least $\kappa$ by some $s\in S$
+\cite[Proposition~1.3.2 and Example~1.7.4]{BHV}.
+
+Let $\lambda_k$ denote the left regular representation on $\ell^2(Q_k)$,
+and define
+\[
+ L_k=1-\frac1{\abs S}\sum_{s\in S}\lambda_k(s).
+\]
+The sum retains multiplicities when elements of $S$ have the same image
+in $Q_k$. Since $S$ is symmetric, $0\le L_k\le2$. Its kernel consists
+exactly of the constants, since $S$ generates $Q_k$. On the orthogonal
+complement of the constants,
+\[
+ \langle L_k\eta,\eta\rangle
+ =\frac1{2\abs S}\sum_{s\in S}\norm{\lambda_k(s)\eta-\eta}^2
+ \ge\delta\norm\eta^2,\qquad \delta=\frac{\kappa^2}{2\abs S}>0.
+\]
+If $p_k$ denotes projection onto the normalized constant vector, it follows
+that
+\begin{equation}\label{eq:gap}
+ \norm{(1-L_k/2)^n-p_k}\le(1-\delta/2)^n.
+\end{equation}
+For $k=0$ the approximation is exact. Take the direct sum of these lazy
+averaging powers over all fibers and call it $P_n$. They are contractions
+of propagation at most $n$, since each Cayley generator moves at most one
+edge in $X$. Equation~\eqref{eq:gap} proves uniform norm convergence to
+the constant-fiber projection $P$. This verifies the first hypothesis of
+Proposition~\ref{prop:transfer}.
+
+\subsection{Controlled operators between fibers}
+It remains to verify the second hypothesis. Equal-depth edges act by the
+permutation prescribed by their right holonomy. For an edge from a shallow
+vertex $u$ of depth $k$ to a deep vertex $v$ of depth $k+1$, put
+$a=a(u,v)$ and $h_k=\abs{\ker(Q_{k+1}\to Q_k)}$. The graph relation gives
+an isometry $J_a:\ell^2(Q_k)\to\ell^2(Q_{k+1})$ and its adjoint:
+\begin{equation}\label{eq:childmap}
+ J_a\delta_q=h_k^{-1/2}
+       \sum_{\bar z=qg_k^a}\delta_z,
+ \qquad
+ J_a^*\delta_z=h_k^{-1/2}\delta_{\bar z g_k^{-a}}.
+\end{equation}
+Every coefficient lies on a graph edge, so these maps have propagation at
+most one. The child sets have the common cardinality $h_k$ and are disjoint;
+this proves the isometry assertion. Since $\abs{Q_{k+1}}=h_k\abs{Q_k}$,
+both $J_a$ and its adjoint carry the normalized constant vectors to the
+corresponding normalized constants: each deep coefficient in the image
+under $J_a$ is $(h_k\abs{Q_k})^{-1/2}$, and each shallow coefficient
+in the image under $J_a^*$ is $h_k(h_k\abs{Q_{k+1}})^{-1/2}$.
+These formulas also apply to the singleton quotient $Q_0$.
+The equal-depth permutations have the
+same property.
+
+For $u,v$ in the same base disc with $d_D(u,v)\le R$, choose a base path
+from $v$ to $u$ of length at most $\lfloor R\rfloor$. Composing the edge
+operators just described gives a contraction $O_{uv}$ of propagation at
+most $\lfloor R\rfloor$, with $O_{uv}\xi_v=\xi_u$. This includes $u=v$
+by using the identity on its fiber.
+
+For the cross-component pairs at distance at most $R$, use
+$O_{uv}=\lvert\xi_u\rangle\langle\xi_v\rvert$. There are only finitely
+many such pairs by Section~\ref{sec:geometry}, and their fibers are finite.
+Consequently their propagations have a common finite bound. Combining this
+bound with $\lfloor R\rfloor$ supplies $s_R$ in
+Proposition~\ref{prop:transfer}. We conclude that
+\begin{equation}\label{eq:Phi}
+ \Phi=\Ad V:C^*(D)\longrightarrow C^*(X)
+\end{equation}
+is a $*$-homomorphism.
+This conclusion is an operator-norm statement in the reduced Roe algebra.
+It does not require $V$ to cover a coarse map from $D$ to $X$.
diff --git a/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/build/sections/06-vanishing.tex b/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/build/sections/06-vanishing.tex
new file mode 100644
index 0000000000000000000000000000000000000000..779a46c17a37a23ef30c4e21b005bb0d6d047a01
--- /dev/null
+++ b/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/build/sections/06-vanishing.tex
@@ -0,0 +1,79 @@
+\section{The boundary index vanishes}\label{sec:vanishing}
+
+We finish by applying the transfer homomorphism to the boundary family
+constructed in Section~\ref{sec:persistence}. Only the actual boundary
+inclusions $e:B\to D$ and $i:B\to X$ use assembly naturality.
+
+Let $W_j$ be the filled triangulated disc underlying the lattice graph $D_j$,
+and put $W=\coprod_{j\ge2}W_j$. Every elementary triangle has vertices at
+pairwise distance one, so there is a factorization by continuous proper maps
+\[
+ P_1(B)=\coprod_j C_j\longrightarrow W\longrightarrow P_1(D).
+\]
+The second map includes each elementary triangle as a Rips simplex.
+At scale one, no simplex of $P_1(D)$ joins different $D_j$.
+A compact subset of this topological disjoint union therefore meets only
+finitely many of its finite components. Its inverse image in $W$ is a closed
+subset of finitely many compact discs, proving properness of the second map;
+the first map is proper for the same reason. Each $W_j$ is contractible;
+homotopy invariance in the first variable of $\KK$ and
+$\KK_1(\C,\C)=K_1(\C)=0$
+\cite[Proposition~17.9.1 and Section~19.8.3]{Blackadar} therefore give
+\[
+ K_1^{\lf}(W)\cong\prod_j\KK_1(C(W_j),\C)
+ =\prod_j\KK_1(\C,\C)=0.
+\]
+Thus the whole family $\beta_1$ dies in $K_1^{\lf}(P_1(D))$, and
+\begin{equation}\label{eq:filledzero}
+ e_*\beta=0\quad\text{in }KX_1(D).
+\end{equation}
+
+Write $\HH_Y=\ell^2(Y)\otimes\HH_0$ for $Y=B,D,X$. The covering isometries
+$E:\HH_B\to\HH_D$ and $I:\HH_B\to\HH_X$ of $e$ and $i$ are
+\[
+ E(\delta_u\otimes h)=\delta_u\otimes h,
+ \qquad I(\delta_u\otimes h)=\delta_{(u,1)}\otimes h
+ \quad(u\in B).
+\]
+Put $e_{\mathrm{Roe}}=\Ad E$ and $i_{\mathrm{Roe}}=\Ad I$.
+Because the boundary fibers are the
+singleton $Q_0$, the averaging isometry of Section~\ref{sec:transfer}
+satisfies the exact identity
+\[
+ VE=I.
+\]
+It follows on the entire Roe algebra $C^*(B)$ that
+\begin{equation}\label{eq:operatorfactor}
+ \Phi\circ e_{\mathrm{Roe}}=i_{\mathrm{Roe}}.
+\end{equation}
+
+The following diagram displays the naturality square for $e$ and its
+operator-algebra continuation. The lower composite equals
+$(i_{\mathrm{Roe}})_*$ by \eqref{eq:operatorfactor}:
+\[
+\begin{tikzpicture}[baseline=(current bounding box.center),>=Stealth]
+ \node (a) at (0,1.3) {$KX_1(B)$};
+ \node (b) at (3.9,1.3) {$KX_1(D)$};
+ \node (c) at (0,0) {$K_1(C^*(B))$};
+ \node (d) at (3.9,0) {$K_1(C^*(D))$};
+ \node (f) at (7.4,0) {$K_1(C^*(X))$};
+ \draw[->] (a)--node[above]{$e_*$}(b);
+ \draw[->] (a)--node[left]{$\mu_B$}(c);
+ \draw[->] (b)--node[right]{$\mu_D$}(d);
+ \draw[->] (c)--node[below]{$(e_{\mathrm{Roe}})_*$}(d);
+ \draw[->] (d)--node[below]{$\Phi_*$}(f);
+\end{tikzpicture}
+\]
+Naturality for $i$, followed by \eqref{eq:operatorfactor}, naturality for
+$e$, and \eqref{eq:filledzero}, now gives
+\begin{align*}
+ \mu_X(\alpha)
+ &= (i_{\mathrm{Roe}})_*\mu_B(\beta)\\
+ &= \Phi_*(e_{\mathrm{Roe}})_*\mu_B(\beta)\\
+ &= \Phi_*\mu_D(e_*\beta)=0.
+\end{align*}
+The class here is exactly $\alpha=i_*\beta$ tested at every Rips scale in
+Proposition~\ref{prop:persistence}. That proposition makes it an
+infinite-order class, while \eqref{eq:rationalnonzero} makes
+$\alpha\otimes1$ nonzero in the rationalized domain. Its rational index
+is zero as well. This proves Theorem~\ref{thm:main}.
diff --git a/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/paper.pdf b/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..84884be5171fbee973dbdf213014519fcdfb458a
--- /dev/null
+++ b/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:785e448758d66154bb5c80341d8267bfa5340e3167bf4dfd90eab4881f94b578
+size 396022
diff --git a/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/README.md b/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..2938b9b994bb965a0433ae3006d69cd9b3e50483
--- /dev/null
+++ b/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/README.md
@@ -0,0 +1,18 @@
+# [A counterexample to the nearby Lagrangian conjecture](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 23, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026,
+  author = {{OpenAI}},
+  title = {{A counterexample to the nearby Lagrangian conjecture}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/paper.pdf}{OAI:A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/build/figures/momentum-separation.tex b/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/build/figures/momentum-separation.tex
new file mode 100644
index 0000000000000000000000000000000000000000..d0a61ede2d5be7d50dd5a810e633743f57436fdf
--- /dev/null
+++ b/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/build/figures/momentum-separation.tex
@@ -0,0 +1,33 @@
+\begin{figure}[tbp]
+\centering
+\begin{tikzpicture}[x=0.72cm,y=0.72cm,font=\small,
+  every node/.style={inner sep=2pt},line cap=round]
+  \draw[->,black!60] (0,0.5)--(5.3,0.5) node[right] {$v^\perp$};
+  \draw[->,black!60] (0,0.5)--(0,4.4) node[above] {$v$};
+  \draw[densely dashed,black!50] (3.4,0.5)--(3.4,4.03);
+  \draw[line width=1pt] (3.4,1.48)--(3.4,3.78);
+  \fill (3.4,1.48) circle (2pt) node[right] {$w'$};
+  \fill (3.4,3.78) circle (2pt) node[right] {$w$};
+  \draw[->] (2.97,1.6)--(2.97,3.66)
+    node[midway,left] {$tv$};
+  \fill[black!60] (3.4,0.5) circle (1.6pt);
+  \node[anchor=north,align=center,font=\footnotesize] at (3.4,0.24)
+    {same projection\\onto $v^\perp$};
+  \node[anchor=west] at (7,3.85) {Equal sphere momenta force};
+  \node[anchor=west] at (7,3.22) {$w-w'=tv$.};
+  \node[anchor=west] at (7,2.35) {For $b\notin U_0$, base momenta satisfy};
+  \node[anchor=west] at (7,1.55)
+    {$\|\Delta p_B\|\geq R(b)|t|(K\delta_0-C_1/c)$.};
+  \node[anchor=west] at (7,0.7)
+    {$K\delta_0>C_1/c,\quad \Delta p_B=0\ \Longrightarrow\ t=0$.};
+\end{tikzpicture}
+\caption{Schematic of the two momentum tests for two critical points at
+fixed base coordinates $(b,v)$, with $\Delta p_B$ the difference of the $B$-momenta. Sphere
+momenta detect the component of $w$ perpendicular to $v$; the varying
+scale separates the remaining pairs outside $U_0$.
+Within $U_0$, the fixed quadratic critical equation has a unique solution.
+The two-dimensional drawing represents an orthogonal decomposition only.
+The estimate also holds when the segment from $w'$ to $w$ passes through
+zero.}
+\label{fig:momentum-separation}
+\end{figure}
diff --git a/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/build/figures/regular-band.tex b/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/build/figures/regular-band.tex
new file mode 100644
index 0000000000000000000000000000000000000000..e7f3fae1a6c1420982c85ca81a73ada4776205ea
--- /dev/null
+++ b/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/build/figures/regular-band.tex
@@ -0,0 +1,68 @@
+\begin{figure}[tbp]
+\centering
+\begin{tikzpicture}[x=0.72cm,y=0.72cm,font=\small,
+  every node/.style={inner sep=2pt},line cap=round,line join=round]
+  \node at (2.8,5.25) {(a) One interval stabilization};
+  \fill[black!10] (0,1) rectangle (1.6,4);
+  \fill[black!22] (1.6,1) rectangle (5.6,4);
+  \draw[line width=1.1pt] (0,1)--(0,4);
+  \draw[densely dashed,line width=0.9pt] (0,1)--(1.6,1);
+  \draw[densely dashed,line width=0.9pt] (0,4)--(1.6,4);
+  \draw[black!55,dotted] (1.6,1)--(1.6,4);
+  \draw[line width=1.1pt] (1.6,1)--(5.6,1)--(5.6,4)--(1.6,4);
+  \node[rotate=90] at (-0.45,2.5) {incoming};
+  \node at (3.6,2.5) {$C_b\times J$};
+  \node[align=center,font=\footnotesize] at (0.8,4.53)
+    {fixed\\vertical};
+  \node[font=\footnotesize] at (3.65,4.48) {outgoing};
+  \node[align=center,font=\footnotesize] at (0.8,0.47)
+    {added collar};
+  \node[font=\footnotesize] at (3.65,0.47) {outgoing};
+
+  \node at (13.7,5.25) {(b) Extension into the fixed $Y$-cylinder};
+  \fill[black!4] (8,0.7) rectangle (19.4,4.3);
+  \fill[black!22] (8,0.7) rectangle (10.4,4.3);
+  \path[fill=black!22]
+    (10.4,1.92) .. controls (12.25,1.92) and (13.7,1.58) .. (15.8,1.58)
+    --(15.8,2.98) .. controls (13.7,2.98) and (12.25,3.08) .. (10.4,3.08)
+    --cycle;
+  \draw[black!45,densely dotted] (10.4,0.7)--(10.4,4.3);
+  \draw[black!45,densely dotted] (15.8,0.7)--(15.8,4.3);
+  \draw[black!55,densely dashed] (8,0.7)--(19.4,0.7);
+  \draw[black!55,densely dashed] (8,4.3)--(19.4,4.3);
+  \draw[line width=1.1pt] (8,0.7)--(8,4.3);
+  \draw[line width=0.7pt] (19.4,0.7)--(19.4,4.3);
+  \draw[line width=1.1pt]
+    (10.4,0.7)--(10.4,1.92)
+    .. controls (12.25,1.92) and (13.7,1.58) .. (15.8,1.58)
+    --(15.8,2.98)
+    .. controls (13.7,2.98) and (12.25,3.08) .. (10.4,3.08)
+    --(10.4,4.3);
+  \node[rotate=90] at (7.55,2.5) {incoming};
+  \node[rotate=90] at (9.15,2.5) {$Y\times[-1,0]$};
+  \node at (12.95,2.5) {$\mathcal R_b\cong C_b\times J$};
+  \node at (13.0,3.68) {$P_{+,b}$};
+  \node at (13.0,1.12) {$P_{-,b}$};
+  \node[align=center] at (17.58,2.5) {upper collar\\$Y\times[1,2]$};
+  \node[anchor=east,font=\footnotesize] at (19.4,4.65)
+    {complement $\cong E_b\times I$};
+  \node[font=\footnotesize] at (10.4,0.18) {$0$};
+  \node[font=\footnotesize] at (8,0.18) {$-1$};
+  \node[font=\footnotesize] at (15.8,0.18) {$1$};
+  \node[font=\footnotesize] at (19.4,0.18) {$2$};
+  \draw[->,thin] (16.7,-0.33)--(19.4,-0.33)
+    node[midway,below] {$t$};
+\end{tikzpicture}
+\caption{Face bookkeeping for one fiber, before corner rounding.
+In (a), the dashed segments are the fixed vertical faces on the added
+incoming collar $\Sigma\times[-1,0]\times J$; both sides beyond that
+collar are movable outgoing faces. In (b), the dark region is
+$W_b=(Y\times[-1,0])\cup\mathcal R_b$, and its heavy interface with
+the light complement is $E_b=\partial_+W_b$. The product identification
+of the complement is proved over the possibly varying bundle $E\to B$.
+The angular direction is schematic: the upper and lower dashed display
+edges are not boundary faces of the closed sphere $Y$. The drawing
+asserts neither radial monotonicity nor a product trivialization of
+$C\to B$.}
+\label{fig:regular-band}
+\end{figure}
diff --git a/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/build/main.tex b/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/build/main.tex
new file mode 100644
index 0000000000000000000000000000000000000000..fdd8ae79b0236fe763a6920329e75b1f7a7e6656
--- /dev/null
+++ b/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/build/main.tex
@@ -0,0 +1,26 @@
+\documentclass[11pt]{article}
+\input{preamble}
+\title{A counterexample to the\\nearby Lagrangian conjecture}
+\author{OpenAI}
+\date{September 23, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We disprove the unrestricted nearby Lagrangian conjecture. For some
+sufficiently large even integer $N$, we construct a closed exact smoothly
+embedded Lagrangian in $T^*(S^9\times S^{N-1})$ that is diffeomorphic to
+the base but is not Hamiltonian isotopic to the zero section.
+\end{abstract}
+\input{sections/introduction}
+\input{sections/tubes}
+\input{sections/morse}
+\input{sections/obstruction}
+\input{sections/embedding}
+\input{sections/transport}
+\begingroup
+\raggedright
+\setlength{\bibsep}{5pt}
+\bibliographystyle{plainurl}
+\bibliography{references}
+\endgroup
+\end{document}
diff --git a/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/build/preamble.tex b/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/build/preamble.tex
new file mode 100644
index 0000000000000000000000000000000000000000..e6e4c23f9db5f5e7e800cd9be27aab50a5bf08fe
--- /dev/null
+++ b/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/build/preamble.tex
@@ -0,0 +1,36 @@
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage{mathpazo}
+\usepackage[margin=1in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{tikz}
+\usepackage[numbers,sort&compress]{natbib}
+\usepackage{url}
+\urlstyle{same}
+\usepackage[unicode,colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
+\hypersetup{pdftitle={A counterexample to the nearby Lagrangian conjecture},pdfauthor={OpenAI},pdfsubject={Exact embedded Lagrangians and stable smooth tube spaces},pdflang={en}}
+\providecommand{\doi}[1]{doi: \href{https://doi.org/#1}{\nolinkurl{#1}}}
+\renewcommand{\bibsection}{\section*{\refname}\addcontentsline{toc}{section}{\refname}}
+\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{\T}{\mathbf T}
+\newcommand{\Hs}{H^s}
+\newcommand{\Wh}{\operatorname{Wh}}
+\newcommand{\Crit}{\operatorname{Crit}}
+\newcommand{\Diff}{\operatorname{Diff}}
+\newcommand{\id}{\operatorname{id}}
+\newcommand{\norm}[1]{\lVert#1\rVert}
+\newcommand{\abs}[1]{\lvert#1\rvert}
+\DeclareMathOperator{\im}{im}
+\DeclareMathOperator{\supp}{supp}
+\DeclareMathOperator{\Int}{int}
+\allowdisplaybreaks[1]
+\setlength{\emergencystretch}{1em}
diff --git a/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/build/references.bib b/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/build/references.bib
new file mode 100644
index 0000000000000000000000000000000000000000..2b5eb3c2ad82cd479cdd856ee031518f16b411c2
--- /dev/null
+++ b/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/build/references.bib
@@ -0,0 +1,344 @@
+@article{Arnold1986,
+  author = {Arnol'd, V. I.},
+  title = {First steps in symplectic topology},
+  journal = {Russian Mathematical Surveys},
+  volume = {41},
+  number = {6},
+  pages = {1--21},
+  year = {1986},
+  doi = {10.1070/RM1986v041n06ABEH004221},
+  url = {https://www.mathnet.ru/eng/rm2268}
+}
+
+@article{Abouzaid2012,
+  author = {Abouzaid, Mohammed},
+  title = {Nearby {Lagrangians} with vanishing {Maslov} class are homotopy equivalent},
+  journal = {Inventiones Mathematicae},
+  volume = {189},
+  pages = {251--313},
+  year = {2012},
+  doi = {10.1007/s00222-011-0365-0},
+}
+
+@article{Kragh2013,
+  author = {Kragh, Thomas},
+  title = {Parametrized ring-spectra and the nearby {Lagrangian} conjecture},
+  journal = {Geometry \& Topology},
+  volume = {17},
+  number = {2},
+  pages = {639--731},
+  year = {2013},
+  note = {With an appendix by Mohammed Abouzaid},
+  doi = {10.2140/gt.2013.17.639},
+  url = {https://msp.org/gt/2013/17-2/gt-v17-n2-p02-p.pdf}
+}
+
+@article{AK2018,
+  author = {Abouzaid, Mohammed and Kragh, Thomas},
+  title = {Simple homotopy equivalence of nearby {Lagrangians}},
+  journal = {Acta Mathematica},
+  volume = {220},
+  number = {2},
+  pages = {207--237},
+  year = {2018},
+  doi = {10.4310/ACTA.2018.v220.n2.a1},
+}
+
+@article{ACGK2025,
+  author = {Abouzaid, Mohammed and Courte, Sylvain and Guillermou, St\'{e}phane and Kragh, Thomas},
+  title = {Twisted generating functions and the nearby {Lagrangian} conjecture},
+  journal = {Duke Mathematical Journal},
+  volume = {174},
+  number = {5},
+  pages = {949--1011},
+  year = {2025},
+  doi = {10.1215/00127094-2024-0052},
+}
+
+@misc{AACK2025,
+  author = {Abouzaid, Mohammed and {\'{A}lvarez-Gavela}, Daniel and Courte, Sylvain and Kragh, Thomas},
+  title = {Normal invariant of nearby {Lagrangians} via twisted derivative},
+  year = {2025},
+  howpublished = {arXiv:2505.12515v2},
+  note = {Version 2, 22 October 2025},
+  url = {https://arxiv.org/abs/2505.12515v2}
+}
+
+@misc{AIS2026,
+  author = {{\'{A}lvarez-Gavela}, Daniel and Igusa, Kiyoshi and Sullivan, Michael G.},
+  title = {Legendrian and {Lagrangian} higher torsion},
+  year = {2026},
+  howpublished = {arXiv:2603.28007v2},
+  note = {Version 2, 19 August 2026},
+  url = {https://arxiv.org/abs/2603.28007v2}
+}
+
+@misc{CP2025,
+  author = {Courte, Sylvain and Porcelli, Noah},
+  title = {On the parametrised {Whitehead} torsion of families of nearby {Lagrangian} submanifolds},
+  year = {2025},
+  howpublished = {arXiv:2506.06110v1},
+  note = {Version 1, 6 June 2025},
+  url = {https://arxiv.org/abs/2506.06110v1}
+}
+
+@article{Rognes2002,
+  author = {Rognes, John},
+  title = {Two-primary algebraic {$K$}-theory of pointed spaces},
+  journal = {Topology},
+  volume = {41},
+  number = {5},
+  pages = {873--926},
+  year = {2002},
+  doi = {10.1016/S0040-9383(01)00005-2},
+}
+
+@inproceedings{Rognes2014,
+  author = {Rognes, John},
+  title = {Algebraic {$K$}-theory of strict ring spectra},
+  booktitle = {Proceedings of the International Congress of Mathematicians ({Seoul}, 2014)},
+  volume = {II},
+  pages = {1259--1283},
+  year = {2014},
+  url = {https://arxiv.org/abs/1403.5998v1}
+}
+
+@article{Kragh2026,
+  author = {Kragh, Thomas},
+  title = {Generating functions in {$\mathbb{R}^{2n}$} and the {Hatcher--Waldhausen} map},
+  journal = {Inventiones Mathematicae},
+  year = {2026},
+  volume = {245},
+  pages = {1377--1435},
+  note = {Published online 18 May 2026},
+  doi = {10.1007/s00222-026-01428-2},
+}
+
+@incollection{Waldhausen1982,
+  author = {Waldhausen, Friedhelm},
+  title = {Algebraic {$K$}-theory of spaces, a manifold approach},
+  booktitle = {Current trends in algebraic topology, Part 1 ({London}, {Ont.}, 1981)},
+  series = {CMS Conference Proceedings},
+  volume = {2},
+  pages = {141--184},
+  publisher = {American Mathematical Society},
+  address = {Providence, RI},
+  year = {1982},
+  url = {https://www.math.uni-bielefeld.de/~fw/algebraic_K_theory_of_spaces_a_manifold_approach.pdf}
+}
+
+@book{WJR2013,
+  author = {Waldhausen, Friedhelm and Jahren, Bj{\o}rn and Rognes, John},
+  title = {Spaces of {PL} Manifolds and Categories of Simple Maps},
+  series = {Annals of Mathematics Studies},
+  volume = {186},
+  publisher = {Princeton University Press},
+  address = {Princeton, NJ},
+  year = {2013},
+  doi = {10.1515/9781400846528},
+  isbn = {978-0-691-15775-7},
+}
+
+@article{Igusa1988,
+  author = {Igusa, Kiyoshi},
+  title = {The stability theorem for smooth pseudoisotopies},
+  journal = {K-Theory},
+  volume = {2},
+  number = {1--2},
+  pages = {1--355},
+  year = {1988},
+  url = {https://people.brandeis.edu/~igusa/Papers/StabilityBig.pdf}
+}
+
+@article{Bott1959,
+  author = {Bott, Raoul},
+  title = {The stable homotopy of the classical groups},
+  journal = {Annals of Mathematics. Second Series},
+  volume = {70},
+  number = {2},
+  pages = {313--337},
+  year = {1959},
+  doi = {10.2307/1970106},
+}
+
+@article{Adams1966,
+  author = {Adams, J. F.},
+  title = {On the groups {$J(X)$}. {IV}},
+  journal = {Topology},
+  volume = {5},
+  number = {1},
+  pages = {21--71},
+  year = {1966},
+  doi = {10.1016/0040-9383(66)90004-8},
+}
+
+@article{IWX2023,
+  author = {Isaksen, Daniel C. and Wang, Guozhen and Xu, Zhouli},
+  title = {Stable homotopy groups of spheres: from dimension 0 to 90},
+  journal = {Publications Math\'{e}matiques de l'IH\'{E}S},
+  volume = {137},
+  pages = {107--243},
+  year = {2023},
+  doi = {10.1007/s10240-023-00139-1},
+  url = {https://pmihes.centre-mersenne.org/articles/10.1007/s10240-023-00139-1/}
+}
+
+@incollection{Hatcher1978,
+  author = {Hatcher, A. E.},
+  title = {Concordance spaces, higher simple-homotopy theory, and applications},
+  booktitle = {Algebraic and geometric topology ({Proc. Sympos. Pure Math.}, {Stanford Univ.}, {Stanford, Calif.}, 1976), Part 1},
+  series = {Proceedings of Symposia in Pure Mathematics},
+  volume = {32},
+  pages = {3--21},
+  publisher = {American Mathematical Society},
+  address = {Providence, RI},
+  year = {1978},
+  url = {https://pi.math.cornell.edu/~hatcher/Papers/ConcordanceSpaces.pdf}
+}
+
+@article{Hofer1985,
+  author = {Hofer, Helmut},
+  title = {Lagrangian embeddings and critical point theory},
+  journal = {Annales de l'Institut Henri Poincar\'e C, Analyse non lin\'eaire},
+  volume = {2},
+  number = {6},
+  year = {1985},
+  pages = {407--462},
+  url = {https://www.numdam.org/article/AIHPC_1985__2_6_407_0.pdf}
+}
+
+@article{LaudenbachSikorav1985,
+  author = {Laudenbach, Fran{\c c}ois and Sikorav, Jean-Claude},
+  title = {Persistance d'intersection avec la section nulle au cours d'une isotopie hamiltonienne dans un fibr\'e cotangent},
+  journal = {Inventiones Mathematicae},
+  volume = {82},
+  year = {1985},
+  pages = {349--357},
+  doi = {10.1007/BF01388807}
+}
+
+@article{Hind2012,
+  author = {Hind, Richard},
+  title = {Lagrangian unknottedness in {Stein} surfaces},
+  journal = {Asian Journal of Mathematics},
+  volume = {16},
+  number = {1},
+  year = {2012},
+  pages = {1--36},
+  doi = {10.4310/AJM.2012.v16.n1.a1}
+}
+
+@article{DGI2016,
+  author = {Dimitroglou Rizell, Georgios and Goodman, Elizabeth and Ivrii, Alexander},
+  title = {Lagrangian isotopy of tori in {$S^2\times S^2$} and {$\mathbb{C}P^2$}},
+  journal = {Geometric and Functional Analysis},
+  volume = {26},
+  year = {2016},
+  pages = {1297--1358},
+  doi = {10.1007/s00039-016-0388-1}
+}
+
+@article{FSS2008,
+  author = {Fukaya, Kenji and Seidel, Paul and Smith, Ivan},
+  title = {Exact {Lagrangian} submanifolds in simply-connected cotangent bundles},
+  journal = {Inventiones Mathematicae},
+  volume = {172},
+  year = {2008},
+  pages = {1--27},
+  doi = {10.1007/s00222-007-0092-8}
+}
+
+@article{Nadler2009,
+  author = {Nadler, David},
+  title = {Microlocal branes are constructible sheaves},
+  journal = {Selecta Mathematica (N.S.)},
+  volume = {15},
+  number = {4},
+  year = {2009},
+  pages = {563--619},
+  doi = {10.1007/s00029-009-0008-0}
+}
+
+@book{Milnor1965,
+  author = {Milnor, John},
+  title = {Lectures on the {h}-Cobordism Theorem},
+  note = {Notes by L. Siebenmann and J. Sondow},
+  publisher = {Princeton University Press},
+  address = {Princeton, NJ},
+  year = {1965},
+  doi = {10.1515/9781400878055}
+}
+
+@book{Hatcher2002,
+  author = {Hatcher, Allen},
+  title = {Algebraic Topology},
+  publisher = {Cambridge University Press},
+  year = {2002},
+  url = {https://pi.math.cornell.edu/~hatcher/AT/AT.pdf}
+}
+
+@misc{BPT2025,
+  author        = {Borodzik, Maciej and Powell, Mark and Teichner, Peter},
+  title         = {Link concordance implies link homotopy},
+  year          = {2025},
+  eprint        = {2503.04348},
+  archivePrefix = {arXiv},
+  primaryClass  = {math.GT},
+  note          = {Version 1},
+  doi           = {10.48550/arXiv.2503.04348},
+  url           = {https://arxiv.org/abs/2503.04348v1}
+}
+
+@book{GolubitskyGuillemin1973,
+  author    = {Golubitsky, Martin and Guillemin, Victor},
+  title     = {Stable Mappings and Their Singularities},
+  series    = {Graduate Texts in Mathematics},
+  volume    = {14},
+  publisher = {Springer-Verlag},
+  address   = {New York--Heidelberg},
+  year      = {1973},
+  doi       = {10.1007/978-1-4615-7904-5}
+}
+
+@article{Chaperon1984,
+  author = {Chaperon, Marc},
+  title = {Une id\'ee du type g\'eod\'esiques bris\'ees pour les syst\`emes hamiltoniens},
+  journal = {Comptes Rendus de l'Acad\'emie des Sciences de Paris, S\'erie I},
+  volume = {298},
+  number = {13},
+  pages = {293--296},
+  year = {1984}
+}
+
+@article{Sikorav1987,
+  author = {Sikorav, Jean-Claude},
+  title = {Probl\`emes d'intersections et de points fixes en g\'eom\'etrie hamiltonienne},
+  journal = {Commentarii Mathematici Helvetici},
+  volume = {62},
+  number = {1},
+  pages = {62--73},
+  year = {1987},
+  doi = {10.1007/BF02564438}
+}
+
+@incollection{EG1998,
+  author = {Eliashberg, Yakov and Gromov, Mikhail},
+  title = {Lagrangian intersection theory: finite-dimensional approach},
+  booktitle = {Geometry of Differential Equations},
+  series = {American Mathematical Society Translations, Series 2},
+  volume = {186},
+  pages = {27--118},
+  publisher = {American Mathematical Society},
+  address = {Providence, RI},
+  year = {1998},
+  doi = {10.1090/trans2/186/02}
+}
+
+@misc{PorcelliSmith2026,
+  author = {Porcelli, Noah and Smith, Ivan},
+  title = {Bordism from quasi-isomorphism},
+  year = {2026},
+  howpublished = {arXiv:2509.21587v2},
+  note = {Version 2, 8 January 2026},
+  url = {https://arxiv.org/abs/2509.21587v2}
+}
diff --git a/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/build/sections/embedding.tex b/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/build/sections/embedding.tex
new file mode 100644
index 0000000000000000000000000000000000000000..a7b806a80a06430305a62bacf15a75a1e83eb87a
--- /dev/null
+++ b/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/build/sections/embedding.tex
@@ -0,0 +1,121 @@
+\section{An exact embedded realization}\label{sec:embedding}
+
+We now turn the nonzero tube family of Section~\ref{sec:tubes} into a
+closed exact embedded Lagrangian. A sphere factor records the direction
+of the vertical gradient. Its momenta separate branches except in one
+direction, and a scale varying over $S^9$ separates the remaining pairs.
+
+We first recall the elementary generating-family criterion, including the immersion statement needed below. If $M$ is a smooth manifold, a smooth function $F:M\times\R^r\to\R$ is a \emph{Morse family} when zero is a regular value of the full map $d_wF:M\times\R^r\to(\R^r)^*$. Individual fiber Hessians need not be invertible. Write
+\[
+ \Crit(F)=\{(q,w):d_wF(q,w)=0\},\qquad
+ \Gamma_F(q,w)=(q,d_qF(q,w)).
+\]
+
+\begin{lemma}\label{lem:generating-map}
+For a Morse family, $\Gamma_F:\Crit(F)\to T^*M$ is an exact Lagrangian immersion, with primitive $F|_{\Crit(F)}$.
+\end{lemma}
+
+\begin{proof}
+The critical locus is smooth of dimension $\dim M$. In local coordinates, its defining differential is the surjective matrix
+\[
+ A=(F_{wq},F_{ww}).
+\]
+A tangent vector killed by $d\Gamma_F$ has $\delta q=0$, $F_{ww}\delta w=0$, and $F_{qw}\delta w=0$. Since $F_{ww}$ is symmetric, these equations say $A^{\mathsf T}\delta w=0$. Surjectivity of $A$ forces $\delta w=0$. Thus $\Gamma_F$ is an immersion. On the critical locus $d_wF=0$, and hence
+\[
+ \Gamma_F^*\lambda=d(F|_{\Crit(F)}).
+\]
+Its pullback of $d\lambda$ is zero. The dimension just computed proves that the immersion is Lagrangian.
+\end{proof}
+
+Take the family $g_b$ from Lemma~\ref{lem:localized-family}, with $B=S^9$, $N=2m$, and a height function $\theta:B\to\R$. Fix a metric on $B$. Choose a union $U_0$ of neighborhoods of the two critical points of $\theta$ on which $g_b$ is the fixed quadratic form. Thus
+\[
+ \delta_0:=\min_{B\setminus U_0}\norm{d\theta}>0.
+\]
+Smooth angular dependence and homogeneity give a constant $C_1$ such that
+\begin{equation}\label{eq:mixed-derivative-bound}
+ \norm{d_wd_bg_b(w)}\leq C_1\norm{w}.
+\end{equation}
+Here $d_bg_b$ is a covector at $b$ and the norm is the corresponding operator norm. Its extension at $w=0$ is $C^1$, with derivative zero there.
+
+\begin{proposition}\label{prop:embedded-realization}
+On $Q=B\times S^{N-1}$ there is a smooth Morse family $F:Q\times\R^N\to\R$ which equals $g_b(w)$ outside a uniform vertical ball and whose generating map is a diffeomorphism from its critical locus onto a closed exact embedded Lagrangian $L\subset T^*Q$. Its critical locus, and hence $L$, is diffeomorphic to $Q$. The infinity class on each slice $B\times\{v_0\}$ is the chosen class $a$.
+\end{proposition}
+
+\begin{proof}
+Choose
+\begin{equation}\label{eq:scale-choice}
+ K\delta_0>C_1/c,
+\end{equation}
+where $c\norm{w}\leq\norm{\nabla g_b(w)}\leq C\norm{w}$. Set
+\[
+ R(b)=\Lambda e^{K\theta(b)}.
+\]
+Let $G_b$ be smooth on all of $\R^N$ and agree with $g_b$ for $\norm{w}\geq1$; multiplying by a radial cutoff which vanishes near zero gives such a family. Let $\beta:[0,\infty)\to[0,1]$ be smooth, equal to one on $[0,1]$ and zero on $[2,\infty)$. Define
+\begin{equation}\label{eq:embedded-family}
+ F(b,v,w)=G_b(w)-R(b)\beta(\norm{w}/T)\langle v,w\rangle,
+ \qquad v\in S^{N-1}.
+\end{equation}
+Choose $\Lambda$ so large that
+\[
+ R_{\min}>\max\left\{C,\ \sup_{b,\,\norm{w}\leq1}\norm{\nabla G_b(w)}\right\}.
+\]
+With
+\[
+ D_\beta=\sup_{s\geq0}\bigl(\abs{\beta(s)}+s\abs{\beta'(s)}\bigr),
+\]
+choose
+\begin{equation}\label{eq:T-choice}
+ T>\max\{1,R_{\max}/c,D_\beta R_{\max}/c\}.
+\end{equation}
+These choices are made in the order $K$, $\Lambda$, $T$.
+
+There are no vertical critical points with $\norm{w}\leq1$, by the choice of $R_{\min}$. For $\norm{w}\geq T$, the vertical derivative of the cutoff linear term has norm at most $D_\beta R(b)$, so
+\[
+ \norm{d_wF}\geq c\norm{w}-D_\beta R(b)>0.
+\]
+All critical points therefore satisfy precisely
+\begin{equation}\label{eq:embedded-critical}
+ \nabla g_b(w)=R(b)v.
+\end{equation}
+Conversely every solution of this equation has
+\[
+ 1<R(b)/C\leq\norm{w}\leq R(b)/c<T,
+\]
+so it is a critical point of~\eqref{eq:embedded-family}. By homogeneity, the critical locus is smoothly parameterized by
+\begin{equation}\label{eq:critical-parametrization}
+ (b,u)\longmapsto
+ \left(b,\frac{\nabla g_b(u)}{\norm{\nabla g_b(u)}},
+ \frac{R(b)u}{\norm{\nabla g_b(u)}}\right),
+ \qquad u\in S^{N-1}.
+\end{equation}
+Its inverse recovers $u=w/\norm{w}$.
+
+To check regularity of the critical equation, vary $v$ in $v^\perp$: its contribution to the differential is $-R\delta v$. Radial variation $\delta w=w$ contributes $\nabla g_b(w)=Rv$, because $\nabla g_b$ is homogeneous of degree one. These variations span $\R^N$. Hence $F$ is a Morse family, and Lemma~\ref{lem:generating-map} gives an exact Lagrangian immersion.
+
+It remains to prove injectivity. Two points with the same image must first have the same base coordinates $(b,v)$. At critical points the cutoffs are constant and $G_b=g_b$. Equality of the covectors in the sphere direction, which are
+\[
+ d_vF(\dot v)=-R(b)\langle w,\dot v\rangle,
+ \qquad \dot v\perp v,
+\]
+therefore implies $w-w'=tv$ for some $t\in\R$. Both endpoints, and their whole straight segment, have norm at most $R(b)/c$. The bound~\eqref{eq:mixed-derivative-bound} yields
+\begin{equation}\label{eq:collision-bound}
+ \norm{d_bg_b(w)-d_bg_b(w')}
+ \leq (C_1/c)R(b)\abs{t}.
+\end{equation}
+This use of the fundamental theorem of calculus remains valid if the segment passes through zero, since $d_bg_b$ is $C^1$ there.
+
+The difference in the $B$-momenta is
+\[
+ d_bg_b(w)-d_bg_b(w')-KR(b)t\,d\theta_b.
+\]
+For $b\notin U_0$, its norm is at least
+\[
+ R(b)\abs{t}\bigl(K\delta_0-C_1/c\bigr).
+\]
+By~\eqref{eq:scale-choice}, equality of those momenta forces $t=0$. For $b\in U_0$, the function $g_b$ is the fixed nondegenerate quadratic form, and~\eqref{eq:embedded-critical} already has a unique solution for each $v$. Thus the immersion is injective everywhere.
+
+Figure~\ref{fig:momentum-separation} summarizes the two momentum tests.
+\input{figures/momentum-separation}
+
+The parameter space in~\eqref{eq:critical-parametrization} is compact and has no boundary. A compact injective immersion is an embedding, so its image is a closed exact embedded $L\cong Q$. In fact Euler's identity gives the global primitive $F|_{\Crit(F)}=-g_b(w)$, although the exactness argument did not require this formula. Finally, for $\norm{w}\geq2T$, the family equals $g_b(w)$ independently of $v$. Its restriction at infinity to any fixed sphere slice therefore represents $a$.
+\end{proof}
diff --git a/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/build/sections/introduction.tex b/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/build/sections/introduction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..ad83cfd73490e10560a387e00ae49ec4e6ac9773
--- /dev/null
+++ b/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/build/sections/introduction.tex
@@ -0,0 +1,173 @@
+\section{Introduction}\label{sec:introduction}
+
+For a closed connected smooth manifold $Q$, write $\pi:T^*Q\to Q$ for
+the cotangent projection and
+$\lambda_{(q,p)}(v)=p(d\pi(v))$ for the canonical one-form. A smooth
+submanifold $L\subset T^*Q$ is Lagrangian if it has dimension $\dim Q$
+and $d\lambda$ vanishes on it; it is exact if $\lambda|_L=df$ for a
+globally defined function $f:L\to\R$. Here closed means compact without
+boundary. Arnold's nearby Lagrangian conjecture asks whether every closed
+exact embedded Lagrangian in $T^*Q$ is Hamiltonian isotopic to the zero
+section. This is the modern formulation used, for example, in
+\cite{Abouzaid2012,Kragh2013}; Arnold's discussion of Lagrangian knots
+provides the broader historical setting \cite{Arnold1986}.
+
+Throughout this paper our conventions are
+\[
+  \omega=d\lambda,\qquad \iota_{X_{H_t}}\omega=-dH_t.
+\]
+A Hamiltonian isotopy has a smooth time-dependent Hamiltonian supported
+in one fixed compact subset of $T^*Q$. For compact Lagrangians this
+entails no restriction: multiplying a Hamiltonian by a cutoff equal to
+one on a neighborhood of the compact isotopy trace preserves its flow
+along that trace.
+
+\begin{theorem}\label{thm:main}
+For some sufficiently large even integer $N$, the manifold
+$Q=S^9\times S^{N-1}$ admits a closed exact smoothly embedded Lagrangian
+$L\subset T^*Q$ such that $L$ is diffeomorphic to $Q$ and no compactly
+supported Hamiltonian isotopy carries the zero section to $L$.
+\end{theorem}
+
+The theorem resolves the unrestricted nearby Lagrangian conjecture
+negatively. Both the base and the Lagrangian in this example are
+connected and simply connected. The obstruction concerns how $L$ lies
+in the cotangent bundle, even though its abstract smooth manifold agrees
+with the base.
+
+\subsection{Context and predecessors}
+
+Classical intersection theorems of Hofer and Laudenbach--Sikorav constrain
+Hamiltonian deformations of the zero section
+\cite{Hofer1985,LaudenbachSikorav1985}. Generating functions provide one
+route to such results: their critical points encode intersections, so
+finite-dimensional critical-point theory can detect symplectic phenomena.
+The construction and transport of these functions under Hamiltonian
+isotopy were developed by Chaperon, Laudenbach and Sikorav
+\cite{Chaperon1984,LaudenbachSikorav1985,Sikorav1987}; Eliashberg and Gromov give a systematic
+account, including Chekanov's contact-isotopy lifting theorem
+\cite[Theorem~4.1.1]{EG1998}. Positive isotopy results in low
+dimension include Hind's theorem for Lagrangian spheres in $T^*S^2$
+\cite[Theorem~1]{Hind2012} and the cotangent-torus theorem of Dimitroglou
+Rizell, Goodman and Ivrii \cite[Theorem~B]{DGI2016}. The former, together
+with the projection theorem recalled next and the classification of
+closed surfaces, gives the closed exact case over $S^2$.
+
+The topology of the cotangent projection has become increasingly rigid.
+Fukaya, Seidel and Smith, and independently Nadler, established
+cohomological and categorical constraints under additional assumptions
+on the base, Maslov class and spin structures \cite{FSS2008,Nadler2009}.
+Abouzaid proved that the projection is a homotopy equivalence when the
+Maslov class vanishes \cite{Abouzaid2012}. Kragh's
+parametrized spectra, together with Abouzaid's appendix, removed that
+hypothesis \cite[Corollary~1.2 and Theorem~E.2]{Kragh2013}.
+Abouzaid and Kragh subsequently proved that the projection is a simple
+homotopy equivalence \cite[Theorem~1]{AK2018}. These results concern the
+topology of the projection and do not identify the ambient Hamiltonian
+isotopy class.
+
+More recent work reaches finer topological information. Abouzaid,
+Courte, Guillermou and Kragh constructed twisted generating functions
+and proved that the stable Lagrangian Gauss map, which records the
+stabilized tangent Lagrangian planes, induces zero on homotopy groups
+\cite{ACGK2025}. The normal invariant of a homotopy equivalence to $Q$
+lies in $[Q,G/O]$, where $G/O$ is the homotopy fiber of the map from
+stable real vector bundles to stable spherical fibrations. It measures
+part of the obstruction to making the equivalence a diffeomorphism.
+Abouzaid, \mbox{\'{A}lvarez-Gavela},
+Courte and Kragh used the twisted derivative on tube spaces to constrain
+the normal invariant of the projection, in particular proving that it
+is $2$-torsion \cite[Corollary~1.2]{AACK2025}. Porcelli and Smith's
+Floer-homotopy approach places this normal invariant in the image of the
+stable Hopf action $B(G/O)\to G/O$ \cite[Theorem~6]{PorcelliSmith2026}.
+These are further necessary conditions on a nearby
+Lagrangian, distinct from its ambient isotopy class.
+
+Smooth tube spaces connect generating functions with Waldhausen's
+algebraic $K$-theory of spaces \cite{Waldhausen1982,WJR2013}.
+\mbox{\'{A}lvarez-Gavela}, Igusa and Sullivan developed tube torsion for
+Legendrians and constructed Legendrians whose tube torsion is nontrivial
+although their base projections are homotopic to diffeomorphisms
+\cite[Theorem~1.15]{AIS2026}. They also identify stable tube homotopy
+classes as a finer invariant underlying the real cohomological torsion
+\cite[Remark~3.32]{AIS2026}. Their examples lie in the jet space
+$J^1Q=T^*Q\times\R$; forgetting the last coordinate can introduce double
+points. The present construction uses a particular nonzero stable tube
+class and proves that every possible pair of coincident cotangent
+branches is separated.
+
+Courte and Porcelli constrain the parametrized Whitehead torsion of
+families of nearby Lagrangians, proving weak vanishing when the base has
+Euler characteristic zero \cite[Theorems~1.2 and~1.5, Corollary~1.6]{CP2025}.
+Their parameters index whole Lagrangians in a fixed cotangent bundle.
+Below, $S^9$ instead indexes the functions at infinity of one generating
+family. These two families, and the invariants attached to them, have
+different roles. The one-critical-function and cobordism arguments are
+also related to Kragh's study of generating functions and the
+Hatcher--Waldhausen map \cite{Kragh2026}. We prove the required statement
+for our varying homogeneous data at infinity.
+
+\subsection{The construction and its obstruction}
+
+We describe the proof in the language of generating families. For a
+smooth function $F(q,w)$ on $Q\times\R^r$, the equations
+\[
+ d_wF(q,w)=0,\qquad (q,w)\longmapsto(q,d_qF(q,w))
+\]
+define an exact Lagrangian immersion when the full vertical-gradient
+equation is regular. This means regularity with respect to both $q$ and
+$w$, not invertibility of every fiber Hessian. An embedding requires the
+additional global injectivity argument supplied in
+Section~\ref{sec:embedding}.
+
+Our generating families agree outside a fixed vertical ball with
+degree-two homogeneous functions $g_b$, where $b\in S^9$. The negative
+domain of $g_b$ on the unit sphere is a smooth tube: a domain isotopic
+to the negative domain of a nondegenerate quadratic form. Adding positive and
+negative quadratic variables defines the stable smooth tube space $\T$.
+The homotopy type of each negative domain is a sphere, so such a family
+also determines a stable spherical fibration, classified by a map to
+$BG$. Section~\ref{sec:tubes} defines these models and their map
+$\gamma:\T\to BG$. Waldhausen's tube fibration, together with Rognes's
+low-degree smooth Whitehead group calculation
+\cite{Waldhausen1982,Rognes2002,Rognes2014}, yields
+\[
+ 0\ne a\in\pi_9\T,\qquad \gamma_*a=0.
+\]
+Thus the smooth tube family remains nontrivial after its associated
+stable spherical fibration has become trivial.
+
+Sections~\ref{sec:morse} and~\ref{sec:obstruction} prove that these
+infinity data cannot occur in a family with exactly one nondegenerate
+critical point in each fiber. Morse theory first gives a fiber homotopy
+equivalence between the sphere of negative eigendirections at that point
+and the negative domain at infinity. After the local quadratic model has been made constant, the
+regular zero level between an inner and an outer sphere forms a family
+of $h$-cobordisms. Thickening this family by one interval and extending
+it into a fixed cylinder produces a cobordism with a product complement.
+Stability and the classification of smooth families then give a product
+for the whole original family. This permits the negative domains to be
+deformed to a fixed tube, contradicting the choice of $a$.
+
+Section~\ref{sec:embedding} realizes $a$ using the extra sphere factor
+in $Q=S^9\times S^{N-1}$. A positive scale $R(b)$ and a linear coupling
+to $v\in S^{N-1}$ arrange that the entire critical locus satisfies
+\[
+ \nabla g_b(w)=R(b)v.
+\]
+Homogeneity parametrizes this locus by $S^9\times S^{N-1}$. The sphere
+and base momenta are the two components $d_vF$ and $d_bF$ of the generated
+covector. Equality of the sphere momenta confines the difference between two putative
+coincident branches to the line spanned by $v$. Varying $R$ over $S^9$
+makes the base momenta separate the remaining pairs. Near the critical
+points of that variation, the tube family has been chosen quadratic,
+and the critical equation already has a unique solution.
+
+Finally, Section~\ref{sec:transport} follows the generating family
+through a hypothetical Hamiltonian isotopy to the zero section. Its
+direct small-step construction adds fixed split quadratic forms and
+preserves the stable class of the homogeneous data. At the zero section,
+the critical-locus projection would be a diffeomorphism, giving a unique
+nondegenerate critical point over each base point. Restricting to a fixed
+$S^9$ slice then gives precisely the family excluded by
+Proposition~\ref{prop:one-critical}, completing the contradiction.
diff --git a/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/build/sections/morse.tex b/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/build/sections/morse.tex
new file mode 100644
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+++ b/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/build/sections/morse.tex
@@ -0,0 +1,148 @@
+\section{Families with one critical point}\label{sec:morse}
+
+We now establish the obstruction used in the proof. A family is \emph{homogeneous at infinity} if it agrees, outside one fixed ball in every fiber, with a family of degree-two homogeneous functions. Smoothness of a homogeneous function below means smoothness away from the origin, together with its $C^1$ extension at the origin.
+
+\begin{proposition}\label{prop:one-critical}
+Let $B=S^9$ and let $p\geq20$. Suppose that $f_b:\R^{2p}\to\R$, $b\in B$, is a smooth family satisfying the following conditions:
+\begin{enumerate}
+\item it is homogeneous at infinity, with smooth homogeneous data in $T_{p,p}$;
+\item each $f_b$ has exactly one critical point, which is nondegenerate and varies smoothly with $b$;
+\item the stable spherical class of the family at infinity is zero in $[B,BG]$.
+\end{enumerate}
+Then its stable tube class is zero in $[B,\T]$.
+\end{proposition}
+
+The proof occupies this section and Section~\ref{sec:obstruction}. We first fix the local quadratic model and explain the necessary Morse theory on the noncompact fibers. Comparing a one-critical function with a smooth $h$-cobordism by removing a normal neighborhood of its unstable disk is also central to Kragh's account of generating functions and the Hatcher--Waldhausen map \cite{Kragh2026}. Here the data at infinity vary with the parameter, and we keep their smooth tube class throughout the comparison.
+
+\subsection{Normalization and the negative sphere bundle}
+
+Translate the critical section to the origin by compactly supported fiber diffeomorphisms. Such a family is obtained by flowing a cutoff of the constant translation vector field: choose its plateau to contain all line segments from zero to the critical section. Subtract the critical value using a cutoff equal to one on the ball of radius $A$ and zero outside the ball of radius $2A$, with $A$ sufficiently large. On its transition annulus the derivative of this correction is $O(A^{-1})$, whereas the original gradient has norm at least $cA$. Thus no new critical points appear, and the data at infinity are unchanged.
+
+Let $Q_b$ be the quadratic term at the resulting critical point. Taylor expansion, uniformly over $B$, gives
+\[
+ f_b(w)-Q_b(w)=O(\norm{w}^3),\qquad
+ d_w(f_b-Q_b)=O(\norm{w}^2).
+\]
+A radial cutoff on a sufficiently small ball replaces $f_b$ by $Q_b$ near zero. The extra derivative is $O(\norm{w}^2)$ on the transition annulus, while $\norm{\nabla Q_b(w)}\geq c_0\norm{w}$ for a uniform $c_0>0$. We have therefore arranged
+\begin{equation}\label{eq:local-quadratic}
+ f_b(0)=0,\qquad f_b=Q_b\text{ near }0,
+\end{equation}
+without changing the unique critical point or the infinity data.
+
+\begin{lemma}\label{lem:morse-sublevels}
+Suppose a family satisfying~\eqref{eq:local-quadratic} has one nondegenerate critical point and smooth homogeneous data of type $T_{k,l}$ at infinity, where $k,l\geq3$. Its Morse index is $k$. Moreover, the unit sphere bundle of its negative eigenspaces is fiber homotopy equivalent to the family of negative domains at infinity.
+\end{lemma}
+
+\begin{proof}
+Choose a radius $a>0$ inside the common quadratic neighborhood and a radius $R_0$ with an inner collar on which the family is already homogeneous. Take a small uniform $\delta>0$ and write
+\[
+ M_-^b=\{f_b\leq-\delta\},\qquad M_+^b=\{f_b\leq\delta\}.
+\]
+We first show that $M_-^b$ is homotopy equivalent to the negative domain at infinity. On its compact part $\norm{w}\leq R_0$, choose a descending field, radial outward near $\norm{w}=R_0$. This boundary prescription is possible because homogeneity gives
+\[
+ df_b(\partial_r)=2f_b/r<0
+\]
+there. Interpolate with negative gradient away from that collar. The conditions are strict linear inequalities, so these choices can be patched smoothly, also over $b$.
+
+The field cannot leave through $f_b=-\delta$. It must reach the outer sphere in finite time: on the compact region its rate of descent has a positive lower bound, while $f_b$ has a finite lower bound. The exit is transverse. Stopping the flow at its first exit and fixing all points outside the ball gives a deformation retraction onto
+\[
+ M_-^b\cap\{r\geq R_0\}.
+\]
+The globally stopped homotopy is continuous, which is sufficient here. If $h_b$ denotes the angular function at infinity, angular projection of this outer portion has, over $u$ with $h_b(u)<0$, the radial fiber
+\[
+ \left[\max\left\{R_0,\sqrt{\frac{\delta}{-h_b(u)}}\right\},\infty\right).
+\]
+Contracting each such interval to its continuous lower endpoint proves the homotopy equivalence. The open and closed negative domains have the same homotopy type, since zero is a regular value. In particular,
+\begin{equation}\label{eq:negative-sublevel-type}
+ M_-^b\simeq S^{k-1}.
+\end{equation}
+
+On $f_b\geq\delta$, normalized negative gradient has $df_b$-derivative $-1$ and bounded speed. Indeed the gradient is bounded away from zero on the compact part and grows at least linearly at infinity. Extend this to a bounded smooth field by cutting off below $\delta$, away from zero. Each point with $f_b>\delta$ reaches the level $\delta$ after time $f_b-\delta$. Stopped flow gives a deformation retraction of $\R^{k+l}$ onto $M_+^b$, so
+\begin{equation}\label{eq:positive-sublevel-type}
+ M_+^b\simeq *.
+\end{equation}
+
+We give the local handle argument to justify its use for these noncompact sublevels. If the Morse index is $j$, choose local linear coordinates with \mbox{$Q_b=-\norm{x}^2+\norm{y}^2$}, where $x\in\R^j$. The coordinate changes and their inverses have uniformly bounded norms; a global frame is not needed. Choose a smooth function $\mu$ on $[0,\infty)$ with
+\[
+ -1<\mu'\leq0,\qquad \mu(0)>\delta,
+ \qquad \mu(s)=0\text{ for }s\geq2\delta.
+\]
+Modify the function in this neighborhood to
+\[
+ \widetilde f_b=f_b-\mu(\norm{x}^2+2\norm{y}^2).
+\]
+The modification is supported where $f_b<\delta$. Its derivatives in the chart are
+\[
+ -2(1+\mu')x,\qquad 2(1-2\mu')y.
+\]
+Thus zero is its only critical point and has value below $-\delta$. We have $\{\widetilde f_b\leq\delta\}=M_+^b$. Bounded normalized descent on the slab between $-\delta$ and $\delta$ retracts this set, relative to $M_-^b$, onto $\{\widetilde f_b\leq-\delta\}$.
+
+The latter set retracts, relative to $M_-^b$, onto
+\[
+ M_-^b\cup D,\qquad D=\{y=0,\ \norm{x}^2\leq\delta\}.
+\]
+For membership of $D$, note that $s+\mu(s)\geq\mu(0)>\delta$ for $0\leq s\leq\delta$. At an added point outside $M_-^b$, shrink $y$ radially to zero if $\norm{x}^2\leq\delta$, and otherwise to length $\sqrt{\norm{x}^2-\delta}$. The modified function decreases during this motion, and it agrees continuously with fixing $M_-^b$ at the boundary. This proves the claimed relative deformation.
+
+The relative homology of $(M_+^b,M_-^b)$ is therefore that of one $j$-disk modulo its boundary. Equations~\eqref{eq:negative-sublevel-type}--\eqref{eq:positive-sublevel-type} force $j=k$, and the boundary map makes $\partial D$ a generator of $H_{k-1}(M_-^b;\Z)$. Within the local negative cone, this sphere expands to $aS(E_b^-)$, where $E_b^-$ is the negative eigenspace in the original Euclidean metric.
+
+Flow this sphere to the outer sphere using the descending field above and project angularly. The construction varies smoothly with $b$. On each fiber it takes a homology generator to a generator in the negative domain. Both source and target have the homotopy type of the simply connected sphere $S^{k-1}$. They have CW type, so the integral homology isomorphism is a homotopy equivalence by the homological Whitehead theorem \cite[Corollary~4.33]{Hatcher2002}. The fiberwise criterion for maps of bundles over a CW base makes it a fiber homotopy equivalence. This identifies the claimed spherical fibration.
+\end{proof}
+
+Return to $k=l=p$ in Proposition~\ref{prop:one-critical}. By Lemma~\ref{lem:morse-sublevels}, its spherical hypothesis says that the stable class of $E^-$ has zero image under $BJ$. The injectivity of $\pi_9BO\to\pi_9BG$ established in Section~\ref{sec:tubes} implies that this bundle class is zero. The map $\operatorname{Gr}_p(\R^{2p})\to BO(p)$ has homotopy fiber the $(p-1)$-connected Stiefel manifold $V_p(\R^{2p})$, so this map is $p$-connected. The stabilization $BO(p)\to BO$ is also $p$-connected. For $p\geq20$ both induce isomorphisms on $\pi_9$. Hence the negative-eigenspace map is nullhomotopic in this finite Grassmannian.
+
+The space of nondegenerate forms of signature $(p,p)$ retracts onto the corresponding sign forms. We can consequently choose a smooth homotopy $Q_{b,s}$ of nondegenerate forms from $Q_b$ to the fixed $q_{p,p}$, with uniform lower gradient bound $c_1\norm{w}$. Insert this homotopy into a much smaller punctured quadratic ball, using a radial parameter $s(r)$ which is one near zero, zero near the outer boundary, and satisfies
+\[
+ r\abs{s'(r)}\leq\varepsilon.
+\]
+Such a cutoff exists on an arbitrarily long logarithmic radius interval. Differentiating $Q_{b,s(r)}(w)$ adds an error $O(\varepsilon r)$ to its quadratic gradient. A sufficiently small $\varepsilon$ excludes new critical points. The parameter is constant near zero, so the resulting function is smooth and equals the fixed $q_{p,p}$ there.
+
+\subsection{A smoothing lemma for stabilization}
+
+Direct sums of homogeneous functions are naturally $C^1$. The following explicit adjustment restores smoothness away from the total origin and will also be used in Hamiltonian transport.
+
+\begin{lemma}\label{lem:stabilization-smoothing}
+Let $g_b(w)$ be a smooth homogeneous family with $\norm{\nabla g_b(w)}\geq c\norm{w}$, and let $q'(z)$ be a fixed nondegenerate quadratic form. Put $W=(w,z)$ and $G_0(b,W)=g_b(w)+q'(z)$. Suppose that $F_b(W)$ is smooth and that
+\[
+ \abs{F_b-G_0}\leq M_0,\qquad
+ \norm{d_WF_b-d_WG_0}\leq M_1
+\]
+uniformly. Then, outside an arbitrarily large fixed ball, $F$ can be modified to agree exactly with a smooth homogeneous family $G_\eta$. The modification creates no new critical points, preserves a neighborhood of every original critical point, and the homogeneous family $G_\eta$ is homotopic through regular $C^1$ homogeneous functions to $G_0$.
+
+The construction is uniform over compact parameter spaces, or over any parameter domain on which the bounds used here are uniform.
+\end{lemma}
+
+\begin{proof}
+There is $c_0>0$ such that $\norm{\nabla_WG_0}\geq c_0\norm{W}$. For $W\ne0$ choose a smooth cutoff $\rho_\eta$ which vanishes on $[0,\eta]$, equals one on $[2\eta,\infty)$, and has derivative bounded by $C/\eta$, and set
+\[
+ G_\eta(b,W)=g_b(w)\rho_\eta\left(\frac{\norm{w}}{\norm{W}}\right)+q'(z).
+\]
+The potentially nonsmooth term vanishes in an open cone around $w=0$. Elsewhere it is smooth, so $G_\eta$ is smooth away from $W=0$. On the support of $1-\rho_\eta$, we have $\norm{w}\leq2\eta\norm{W}$. Homogeneity and the angular derivative bound give
+\begin{equation}\label{eq:angular-smoothing-estimates}
+ \abs{G_\eta-G_0}\leq C_2\eta^2\norm{W}^2,
+ \qquad
+ \norm{d_WG_\eta-d_WG_0}\leq C_2\eta\norm{W}.
+\end{equation}
+In the second estimate the derivative of $\norm{w}/\norm{W}$ costs at most $\norm{W}^{-1}$ where the cutoff changes. For small $\eta$, the straight homotopy from $G_0$ to $G_\eta$ retains a linear nonzero-gradient bound.
+
+Let $\alpha_A$ vanish for $\norm{W}\leq A$, equal one for $\norm{W}\geq2A$, and have derivative bounded by $C_3/A$. Replace $F$ by
+\[
+ F_A=(1-\alpha_A)F+\alpha_AG_\eta.
+\]
+On the transition annulus its derivative differs from $d_WG_0$ by at most
+\[
+ M_1+C_2\eta\norm{W}
+ +\frac{C_3}{A}\bigl(M_0+C_2\eta^2\norm{W}^2\bigr).
+\]
+Relative to $\norm{W}$ this is $O(\eta+\eta^2+A^{-1})$. Choose $\eta$ small and then $A$ large so that the bound is less than $c_0\norm{W}/2$. No critical points occur in the annulus or its exterior. The original critical points already lie in a uniform ball, since $d_WF$ differs from a linearly bounded-below gradient by at most $M_1$. Taking $A$ beyond that ball preserves their neighborhoods. The function is unchanged near zero and thus smooth everywhere.
+\end{proof}
+
+Adding positive squares increases the normal disk dimension of the negative
+tube $S^{k-1}\times D^l$ while leaving the dimension of its sphere core
+$S^{k-1}\times\{0\}$ fixed. We use this freedom to allow the final
+deformation of these cores through embeddings in Section~\ref{sec:obstruction}.
+
+Apply the lemma to the normalized $f_b(w)+\norm{z}^2$. Its difference from the homogeneous sum, and that of its vertical derivative, are bounded because $f_b-g_b$ is supported in a fixed $w$-ball. We obtain a family, still with one critical point and with a fixed local model, whose infinity data have type $T_{k,l}$ with
+\begin{equation}\label{eq:unbalanced-indices}
+ k=p,\qquad l>k+9.
+\end{equation}
+Both indices remain at least twenty. Its local model is $q_{k,l}$. If its tube family is null at this stage, adding the same number of negative squares as the positive squares just added gives nullity in the diagonal stable system. Fixed reordering of coordinate blocks does not affect nullity. It remains to prove that this enlarged tube family is null, which is the purpose of the next section.
diff --git a/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/build/sections/obstruction.tex b/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/build/sections/obstruction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..1328d9d80c1cee630987f188f47add24a8d1ecf6
--- /dev/null
+++ b/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/build/sections/obstruction.tex
@@ -0,0 +1,321 @@
+\section{The zero-level cobordism}\label{sec:obstruction}
+
+We continue with the normalized family from Section~\ref{sec:morse}: its unique critical point is zero, its local model is the fixed $q_{k,l}$, its data at infinity are smooth and homogeneous, and~\eqref{eq:unbalanced-indices} holds. Put
+\[
+ Y=S^{k+l-1},\qquad
+ \Sigma=\{q_{k,l}=0\}\cap Y\cong S^{k-1}\times S^{l-1}.
+\]
+Choose radii $a,R_0$ with collars in the local quadratic and outer homogeneous regions, respectively. Identify their annulus with $Y\times[0,1]$ by a smooth increasing radius $r(t)$, with $r'>0$. Write
+\[
+ s_b(u,t)=\frac{f_b(r(t)u)}{r(t)^2}.
+\]
+Near $t=0$ and $t=1$ this function is independent of $t$, and equals $q_{k,l}|_Y$ and the angular infinity data $h_b^1$, respectively. Differentials in this section are in the annulus variables, with $b$ fixed; we also write $f_b$ for its pullback to the annulus.
+
+\subsection{A regular band and its two complementary regions}
+
+Since $f_b$ has no critical point on the annulus and
+\begin{equation}\label{eq:zero-level-differential}
+ df_b=r^2ds_b\quad\text{on }s_b=0,
+\end{equation}
+zero is a regular value of $s_b$. Its zero set is transverse to both end faces. Compactness gives a small uniform $\epsilon>0$ and a band identification
+\begin{equation}\label{eq:zero-band}
+ C_b=\{s_b=0\},\qquad
+ \mathcal R_b=\{\abs{s_b}\leq\epsilon\}\cong C_b\times J,
+ \qquad J=[-\epsilon,\epsilon].
+\end{equation}
+To obtain it, choose a field with $ds_b$-derivative one in a slightly larger band, tangent to the end faces and purely angular and independent of $t$ on their collars. Such choices patch because the condition $ds_b(V)=1$ is affine. Integrating the field identifies its second coordinate with $s_b$. At the incoming end the choice can be fixed independently of $b$, so the resulting collar embedding $\Sigma\times J\hookrightarrow Y$ is fixed.
+
+\begin{lemma}\label{lem:signed-flows}
+On
+\[
+ P_{\pm,b}=\{(u,t):\ \pm s_b(u,t)\geq\epsilon\}
+\]
+there are smooth fields $V_{\pm,b}$, depending smoothly on $b$, such that
+\begin{align}
+ \pm df_b(V_{\pm,b})&>0 &&\text{on }P_{\pm,b},\label{eq:field-f}\\
+ dt(V_{\pm,b})&>0 &&\text{near }t=0,1,\label{eq:field-t}\\
+ \pm ds_b(V_{\pm,b})&>0 &&\text{on }s_b=\pm\epsilon.\label{eq:field-s}
+\end{align}
+The fields can be chosen with a common prescription extending across the whole angular sphere on each end collar.
+\end{lemma}
+
+\begin{proof}
+By~\eqref{eq:zero-level-differential}, near the zero set a field can make $df_b$ and $ds_b$ both strictly positive, or both strictly negative. The same holds at the two side levels for small uniform $\epsilon$. On an end collar write $s_b=h$, angular and independent of $t$, and use
+\[
+ V=\partial_t+h\nabla_Yh.
+\]
+Then
+\[
+ dt(V)=1,\quad ds_b(V)=h\norm{\nabla_Yh}^2,\quad
+ df_b(V)=h\bigl(2rr'+r^2\norm{\nabla_Yh}^2\bigr).
+\]
+These give the required signs, including strictness on the side levels, since $dh\ne0$ near $h=0$. Away from the prescribed regions, use strict signed gradient ascent or descent for $f_b$. Convex patching preserves every applicable strict linear inequality, and can keep the displayed prescription on smaller end collars. The construction is smooth over the compact parameter space. Fields on cornered regions are understood to extend locally across their faces.
+\end{proof}
+
+Let $T_-^0$ and $T_{-,b}^1$ be the truncated negative domains $s_b\leq-\epsilon$ on the inner and outer spheres, viewed in angular coordinates. They are isotopic to the respective negative tubes.
+
+\begin{lemma}\label{lem:residual-hcob}
+The negative field gives an embedding
+\[
+ j_b:T_-^0\hookrightarrow\Int T_{-,b}^1
+\]
+which is a homotopy equivalence. It also identifies $C_b$ with
+\[
+ T_{-,b}^1\setminus\Int j_b(T_-^0).
+\]
+In particular, the family $C_b$, with its fixed incoming identification with $\Sigma$, is a smooth family of h-cobordisms.
+\end{lemma}
+
+\begin{proof}
+For $V_{-,b}$ the inner face and side face are entry faces, and the outer face is the only exit. Every trajectory exits in finite forward time: on the compact region $P_{-,b}$, $-df_b(V_{-,b})$ has a positive lower bound and $f_b$ has bounded range. Backwards, the same argument forces a trajectory to reach an entry face.
+
+Flow to the first intersection with $t=1$ maps each entry face smoothly and injectively into $T_{-,b}^1$. Transversality to the exit face gives smooth dependence of the exit time, also at the side's outer edge where the time is zero. Extensions across the corner faces show that these maps have full rank up to the edges. Their images meet precisely along the shared inner edge and otherwise are disjoint, by uniqueness of flow; backwards flow gives their union as the entire outer tube. The image of the inner face lies in the interior of the outer tube, since the field points strictly inward from the side. The band identification~\eqref{eq:zero-band} identifies that side with $C_b$ and proves the stated decomposition.
+
+Choose $\delta>0$ as in Lemma~\ref{lem:morse-sublevels}, also small enough that the sphere $a(S^{k-1}\times\{0\})$ in the fixed negative eigenspace lies in $f_b\leq-\delta$. This sphere is the core of the inner tube. Its image under the negative flow remains in this sublevel. The core represents a generator there by that lemma, and therefore its image at $R_0$ represents a generator in the outer radial portion. Angular projection carries it to a generator in the open negative domain. Inclusion of $T_{-,b}^1$ into this open domain is a homotopy equivalence for our small $\epsilon$. Thus $j_b$ has degree $\pm1$ on the sphere homotopy types of its source and target. These types are simply connected, so $j_b$ is a homotopy equivalence.
+
+Excision with collars now gives
+\[
+ H_*(C_b,\partial_-C_b;\Z)
+ \cong H_*(T_{-,b}^1,j_b(T_-^0);\Z)=0.
+\]
+This includes connectedness of $C_b$. The two tubes and their common boundary are simply connected, so van Kampen applied to the decomposition implies $\pi_1C_b=0$. The cobordism is orientable, and Poincar\'e--Lefschetz duality gives vanishing of the relative homology at its outgoing end as well. Both boundary components are diffeomorphic to $S^{k-1}\times S^{l-1}$ and hence simply connected. All these smooth spaces have CW type, so the homological Whitehead theorem \cite[Corollary~4.33]{Hatcher2002} makes both boundary inclusions homotopy equivalences. Regularity and properness give a smooth bundle of these cobordisms over $B$, including their boundary collars, with fixed incoming identification.
+\end{proof}
+
+\subsection{Stabilizing the cobordism inside a fixed cylinder}
+
+The family $C_b$ gives a class $c_B\in[B,H(\Sigma)]$. We need a
+product identification relative to the incoming end that varies over all
+of $B$. We first thicken $C_b$ by one interval and
+extend it to $Y$. The resulting family has a product complement in a
+fixed cylinder. We will then use stability to recover the original
+family class from this enlarged one.
+
+In the space $H(X)$ recalled in Section~\ref{sec:tubes}, the incoming end is identified with $X$; when
+$\partial X\ne\varnothing$, the vertical boundary is the prescribed
+product $\partial X\times I$. There is no identification of the outgoing
+end with $X$. All these data have collars. Equivalently one may bend the
+vertical face into the outgoing face, so incoming and outgoing boundaries
+meet along $\partial X$. This is the collared model of
+\cite[Definitions~1.1.1 and~1.1.3]{WJR2013}. The product component is
+\[
+ H(X)_0\simeq BP(X),\qquad
+ P(X)=\Diff(X\times I\ \mathrm{rel}\
+ X\times\{0\}\cup\partial X\times I),
+\]
+with the equivalent convention of diffeomorphisms fixed near the indicated
+faces and product near the outgoing face. Thus a null classifying map
+into this component gives a product family relative to the incoming end.
+
+In our application, $\Sigma$ is simply connected and has dimension at least $38$.
+The smooth $h$-cobordism theorem
+\cite[Theorem~9.1]{Milnor1965} makes every $h$-cobordism on $\Sigma$ a
+product relative to its incoming end: its dimension is at least $39$,
+and its boundary inclusions are homotopy equivalences. If a product
+parametrization restricts to $a:\Sigma\to\Sigma$ at the incoming end,
+precomposing it with $a^{-1}\times\id_I$ fixes that end pointwise.
+Thus $H(\Sigma)$ is connected.
+
+The interval stabilization used here is
+$\sigma:H(X)\to H(X\times J)$, where $J$ is a closed interval. For $X$
+closed and $C$ an $h$-cobordism from $X$ to $N$, add a fixed incoming
+collar to form
+\[
+ C^-=(X\times[-1,0])\cup_X C.
+\]
+Use $C^-\times J$, with the following faces before rounding:
+\begin{equation}\label{eq:stabilization-faces}
+ \begin{aligned}
+ \text{incoming: }&X\times\{-1\}\times J,\\
+ \text{vertical: }&X\times[-1,0]\times\partial J,\\
+ \text{outgoing: }&(N\times J)\cup(C\times\partial J).
+ \end{aligned}
+\end{equation}
+The vertical face has its stated product identification; the remaining
+sides belong to the outgoing end. Corner bending turns these into the
+usual collared cobordism faces. For manifolds with boundary, retain the
+existing product vertical faces as well and round their product corners.
+This is the product construction of \cite[Definition~1.1.3]{WJR2013}.
+
+If $e:X\hookrightarrow Y$ is a codimension-zero embedding, extension
+$e_!:H(X)\to H(Y)$ glues the cobordism to the trivial cylinder on
+$Y\setminus\Int e(X)$ along the product vertical boundary. In the bent
+collared model, the same operation is
+\begin{equation}\label{eq:hcob-extension}
+ C\longmapsto(Y\times I)\cup_{X\times I}C,
+\end{equation}
+where $X\times I\subset C$ is the incoming collar. This expression shows
+that extension commutes with interval stabilization, using the product
+collars and their fixed corner adjustments.
+
+Add trivial collars $Y\times[-1,0]$ and $Y\times[1,2]$ to the annulus and set, with corners to be rounded,
+\begin{equation}\label{eq:band-cobordism}
+ W_b=(Y\times[-1,0])\cup\mathcal R_b.
+\end{equation}
+Its incoming boundary is the fixed $Y\times\{-1\}$.
+Figure~\ref{fig:regular-band} records these faces and the complementary
+regions used below.
+\input{figures/regular-band}
+
+\begin{lemma}\label{lem:band-is-stabilization}
+The family $W_b$ represents the extension to $Y$ of the interval stabilization of $C_b$.
+\end{lemma}
+
+\begin{proof}
+Adjoin a fixed incoming collar to $C_b$, obtaining
+\[
+ C_b^-=(\Sigma\times[-1,0])\cup C_b.
+\]
+Its interval stabilization is $C_b^-\times J$, with incoming $\Sigma\times J$. In the convention of~\eqref{eq:stabilization-faces}, the two faces $C_b\times\partial J$ and the outgoing end times $J$ form the movable boundary. The added incoming-collar portions supply the fixed vertical boundary when a boundary-relative model is used. Extending across $\Sigma\times J\hookrightarrow Y$ replaces that incoming collar by $Y\times[-1,0]$. The result is exactly~\eqref{eq:band-cobordism}, using~\eqref{eq:zero-band} for the attached piece.
+
+This construction works on transition maps as well. Local product trivializations of $C_b$ fix its incoming collar and can be made product on outgoing collars, without fixing their outgoing end. Extend them by identity over the added collar and take their product with $\id_J$. A fixed bending of the two interval coordinates gives the usual boundary-relative stabilized cylinder. Near the lower gluing seam the transitions are identity; near the upper seam they preserve both interval coordinates and act only in the remaining boundary variable. Thus fixed planar corner adjustments descend through the transitions. Extending by identity over the complement in $Y$ gives the same family as~\eqref{eq:band-cobordism}. This is a single interval stabilization with both side faces movable.
+\end{proof}
+
+\begin{lemma}\label{lem:product-complement}
+The complement of $W_b$ up to $Y\times\{2\}$ is, after compatible corner rounding, a product cylinder over the outgoing boundary of $W_b$, smoothly in $b$. Consequently $[W]=0$ in $[B,H(Y)]$.
+\end{lemma}
+
+\begin{proof}
+Before rounding, the outgoing boundary of $W_b$ comprises the two sign domains at $t=0$, the two band sides, and the band end at $t=1$. Its complement is the union of $P_{-,b}$ and $P_{+,b}$ with the upper collar $Y\times[1,2]$. Use Lemma~\ref{lem:signed-flows} on the sign regions. On and above the upper end, its common angular prescription extends across the band and has $dt>0$. It can be continued to the top with that property. The resulting field points from every outgoing face of $W_b$ into the complement, and its only exit face is $t=2$.
+
+We describe a rounding transverse to this field. At a lower corner put
+\[
+ v=t,\qquad w=\pm s_b-\epsilon;
+\]
+the complement is the quadrant $v,w\geq0$. At an upper corner put $v=t-1$ and use the same $w$; the complement is the union $v\geq0$ or $w\geq0$. At either corner the field has $dv(V)>0$ and $dw(V)>0$. Round the interface by a curve in the first quadrant at the bottom, and in the third quadrant at the top, smoothly joining the unchanged straight parts. Choose its conormal into the complement as a nonzero nonnegative combination of $dv,dw$. This makes the rounded interface transverse with the same orientation.
+
+The product end coordinates allow fixed small planar roundings over all parameters. If the stabilized piece and its exterior are kept as separate pieces at the bottom seam, move their dividing wall in the small corner neighborhood so it reaches the rounded interface transversely. All transition maps there are identity. Each piece is then a fixed planar adjustment of its original collared model, so Lemma~\ref{lem:band-is-stabilization} still describes the rounded family.
+
+Every forward trajectory reaches $t=2$ in finite time. In sufficiently small lower and upper collars, $dt(V)$ has a uniform positive lower bound; this bounds transit time there and prevents reentry across their inner thresholds in the wrong direction. In the remaining compact middle, a trajectory stays in one sign region and its signed $f_b$ increases at a rate bounded below, with bounded range. Its time there is therefore finite. It cannot exit across the entry boundary. The same argument backwards reaches the rounded outgoing boundary of $W_b$.
+
+Transversality gives smooth hitting times in the initial point and in $b$. Rescaling these times gives a product diffeomorphism of the complementary cobordism relative to its entry, say $E_b=\partial_+W_b$. This proves a product identification over the entire possibly varying bundle $E\to B$; it does not merely assert that the individual fibers are products.
+
+Finally, adding $E_b\times I$ to the outgoing end changes no incoming-relative h-cobordism bundle class. Extend the transverse field a small uniform distance into $W$ to obtain a collar $E\times[-\eta,0]$. A fixed increasing smooth diffeomorphism $[-\eta,1]\to[-\eta,0]$, equal to the identity near $-\eta$, compresses the added cylinder into this collar, and is identity on the rest of $W$. It is defined without choosing a trivialization of $E\to B$. The enlarged total is the literal product $B\times Y\times[-1,2]$, relative to its incoming boundary. Hence $[W]=0$.
+\end{proof}
+
+\subsection{Recovering the original cobordism family}
+
+We have shown that the extension to $Y$ of one interval stabilization
+of $C_b$ is a product family. We now show that this operation detects
+its class over $B=S^9$.
+
+The stable parametrized $h$-cobordism theorem
+\eqref{eq:stable-hcob-theorem} is natural for this extension
+\cite[Theorems~0.1 and~0.3]{WJR2013}.
+
+The quantitative statements needed below are as follows. The single
+stabilization $H(X)\to H(X\times J)$ is $r$-connected provided
+\begin{equation}\label{eq:igusa-range}
+ \dim X\geq\max(2r+8,3r+5).
+\end{equation}
+Every subsequent stage also satisfies this dimension bound, so the map
+$H(X)\to\Hs(X)$ is $r$-connected as well. Moreover, an
+$(r+1)$-connected map $X\to Y$ induces an $r$-connected map
+$\Hs(X)\to\Hs(Y)$. For the first statement we use Igusa's stability
+theorem \cite{Igusa1988} in the form of
+\cite[Theorem~2.16]{CP2025}; the second is
+\cite[Theorem~2.17]{CP2025}. Extension is the map of
+\cite[Definition~2.15]{CP2025}. As specified in
+\cite[Remark~2.13]{CP2025}, these statements use the WJR product
+convention \eqref{eq:stabilization-faces}. Connectivity here means isomorphisms on
+$\pi_i$ for $i<r$ and a surjection on $\pi_r$.
+
+\begin{proposition}\label{prop:hcob-injectivity}
+Let $k,l\geq20$, and put
+\[
+ Y=S^{k+l-1},\qquad
+ \Sigma=\{q_{k,l}=0\}\cap Y\cong S^{k-1}\times S^{l-1}.
+\]
+For a closed collar embedding $e:\Sigma\times J\hookrightarrow Y$, the
+composite
+\begin{equation}\label{eq:hcob-injectivity-map}
+ H(\Sigma)\xrightarrow{\sigma}H(\Sigma\times J)
+ \xrightarrow{e_!}H(Y)\longrightarrow\Hs(Y)
+\end{equation}
+induces an isomorphism on $\pi_9$ in the product component. Consequently,
+if a sphere family in $H(\Sigma)$ has null image under this composite,
+the family is itself null.
+\end{proposition}
+
+\begin{proof}
+Here $\dim\Sigma=k+l-2\geq38$, whereas the bound in
+\eqref{eq:igusa-range} for $r=10$ is $35$. Hence
+$H(\Sigma)\to\Hs(\Sigma)$ is $10$-connected. The composite
+$H(\Sigma)\to H(\Sigma\times J)\to\Hs(\Sigma\times J)$ has the same
+connectivity: its target is the tail of the defining stabilization
+telescope. It therefore induces an isomorphism on $\pi_9$.
+
+Set $d=\min(k,l)-1$. The source of $e$ is $(d-1)$-connected, and
+$Y$ has vanishing homotopy groups in degrees at most $d$. Thus $e$ is
+$d$-connected: it is an isomorphism below degree $d$ and a surjection in
+degree $d$. The stable extension
+$\Hs(\Sigma\times J)\to\Hs(Y)$ is consequently $(d-1)$-connected,
+with $d-1\geq18$, and also induces an isomorphism on $\pi_9$.
+Compatibility of extension and stabilization identifies the composite of
+these two isomorphisms with \eqref{eq:hcob-injectivity-map}.
+
+The space $H(\Sigma)$ is connected, as shown above.
+For a connected target, free homotopy classes of maps from $S^9$ are
+orbits of based classes under the fundamental group action. The orbit
+of zero is just zero. Thus the same detection of nullity applies to
+sphere families without a specified basepoint.
+\end{proof}
+
+By Proposition~\ref{prop:hcob-injectivity}, Lemmas~\ref{lem:band-is-stabilization} and~\ref{lem:product-complement} imply
+\begin{equation}\label{eq:zero-cobordism-class}
+ c_B=0\quad\text{in }[B,H(\Sigma)].
+\end{equation}
+The passage to free sphere classes is harmless: the fundamental-group action fixes zero. The classification of smooth families by this h-cobordism space now supplies incoming-relative product trivializations of the whole family $C_b$, not just of its individual fibers; see \cite[Proposition~2.11]{CP2025}.
+
+\subsection{Removing the cobordism and straightening the cores}
+
+\begin{proof}[Completion of the proof of Proposition~\ref{prop:one-critical}]
+First replace the outer negative domains by their truncated domains $T_{-,b}^1$; moving a regular boundary through a sufficiently small angular collar is an isotopy. By Lemma~\ref{lem:residual-hcob} and~\eqref{eq:zero-cobordism-class}, each of these domains is $j_b(T_-^0)$ with a smoothly varying product collar attached. Shrink that collar: if its product coordinate is $[0,1]$, retain only $[0,s]$ and let $s$ decrease to zero. The outer boundaries vary through embeddings, including at $s=0$, where they equal $\partial j_b(T_-^0)$. Thus the family is homotopic as tube domains to $j_b(T_-^0)$.
+
+Fix $T_-^0\cong S^{k-1}\times D^l$ and let
+\[
+ e_b:S^{k-1}\hookrightarrow Y,\qquad e_b(u)=j_b(u,0)
+\]
+be its cores. We may replace $j_b(T_-^0)$ by small round normal neighborhoods of $e_b$. Here is a precise version that removes framing choices. First shrink its disk factors to a uniformly small radius. In metric normal-bundle charts of the cores write
+\[
+ j_b(u,z)=\bigl(\psi_b(u,z),\xi_b(u,z)\bigr),\qquad
+ \psi_b(u,0)=u,\quad \xi_b(u,0)=0.
+\]
+On this smaller domain use, for $0<t\leq1$, the embeddings
+\begin{equation}\label{eq:normal-linearization}
+ (u,z)\longmapsto
+ \bigl(\psi_b(u,tz),\ t^{-1}\xi_b(u,tz)\bigr).
+\end{equation}
+These are compositions of the original embedding with dilation in the disk and normal directions. The estimates $\norm{\xi_b(u,tz)}\leq Ct\norm{z}$ keep them in the chosen chart. At $t=0$ the formula extends smoothly to $(u,D_z\xi_b(u,0)z)$. The normal derivative is an isomorphism, so the limit is also an embedding. Its fiber ellipsoids can be changed to round balls by interpolation of positive definite metrics, with the common disk radius chosen sufficiently small. At the level of domains this requires no fixed normal framing.
+
+The core family is homotopic through embeddings to the fixed standard core. Indeed, as maps into $Y=S^{k+l-1}$, both it and the constant standard family are nullhomotopic because
+\[
+ \dim(B\times S^{k-1})=k+8<k+l-1.
+\]
+Choose a smooth homotopy, constant on collars of its two time endpoints. Its parameter dimension, including time, is ten. Relative fiberwise multijet transversality avoids double points since
+\[
+ 10+2(k-1)<k+l-1.
+\]
+Vertical rank-defect strata have codimension at least $l+1>10+(k-1)$ and are avoided as well.
+Here we use the parametric multijet theorem in the form of
+\cite[Theorem~2.3.4 and Corollary~2.3.5(iii)]{BPT2025}, and explain
+its compact-parameter relative application. Cover the compact parameter
+manifold $B\times[0,1]$, away from smaller endpoint collars, by finitely
+many coordinate charts. In each chart multiply the local perturbations
+by a parameter cutoff supported away from the endpoint collars. Local
+source cutoffs allow independent variations at distinct source points,
+as required for the two-point jets. Apply transversality successively on
+compact subsets covering these charts; sufficiently small subsequent
+perturbations preserve avoidance already obtained on compact subsets.
+The rank estimate first gives fiberwise immersions. Compactness and
+$C^1$ openness then give a uniform neighborhood of the diagonal on
+which distinct sufficiently close source points remain distinct.
+Apply the two-point argument on the compact complement of this
+neighborhood. The strict dimension bounds make each transverse
+bad-stratum preimage empty. The endpoint families remain fixed, and
+their embedding property persists on smaller endpoint collars by
+$C^1$ openness. This proves that every member of the resulting homotopy
+is an embedding. Uniformly small round normal neighborhoods follow this
+family to a fixed tube.
+
+All the resulting homotopies of domains give homotopies of tube data. Use boundary collars for local defining-function choices; the fibers of these choices are convex, so partitions of unity in parameters give global choices. Continuous families in the smooth topology suffice for the homotopy class; when smooth parameter dependence is needed, relative approximation preserves embeddings and regularity on compact parameter sets. We have proved nullity of the enlarged family in $T_{k,l}$. Adding the missing negative squares returns to the diagonal system and proves the proposition.
+\end{proof}
diff --git a/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/build/sections/transport.tex b/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/build/sections/transport.tex
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@@ -0,0 +1,544 @@
+\section{Hamiltonian transport and the obstruction}
+\label{sec:transport}
+
+We now transport the generating family of
+Proposition~\ref{prop:embedded-realization} under a hypothetical Hamiltonian
+isotopy.  The construction below keeps track of its actual homogeneous
+data at infinity.  Each step adds a fixed quadratic form of split
+signature, followed by a homotopy through regular homogeneous data.
+In particular, the construction applies to the nonconstant tube family
+used here. The construction follows the classical strategy of
+composing with small Hamiltonian steps and adding split quadratic
+variables \cite[Sections~1.3--1.7]{Sikorav1987}; generating families and their invariance
+under isotopy are also developed in \cite{EG1998}.
+For a related generating-family lifting statement, see
+\mbox{\'{A}lvarez-Gavela} et al.~\cite[Lemma~3.23]{AIS2026}; the argument needed here is given directly.
+
+We use the Morse families and generating maps $\Gamma_F$ of
+Section~\ref{sec:embedding}. For any manifold $M$, write $M_0\subset T^*M$
+for its zero section. Every family constructed below has
+$\Gamma_F$ a diffeomorphism from $\Crit(F)$ onto the Lagrangian it
+generates.
+
+\subsection{A Euclidean cylinder and a compactly supported extension}
+
+Retain $N=2m$ and $Q=S^9\times S^{N-1}$ from the preceding construction,
+and put $d=10+N$.  View $Q$ as the product of unit spheres in
+$\R^{10}\times\R^N=\R^d$.  For $0<\nu<1$, the map
+\[
+ \iota:Q\times(-\nu,\nu)^2\longrightarrow\R^d,
+ \qquad
+ \iota((b,v),\alpha)=((1+\alpha_1)b,(1+\alpha_2)v)
+\]
+is a diffeomorphism onto a bounded open neighborhood $U$ of $Q$.
+Its inverse, the projection $r_Q:U\to Q$, and its two radial
+coordinates extend smoothly to a neighborhood of $\overline U$.
+The cotangent lift gives the canonical identification
+\[
+ T^*U\cong T^*Q\times T^*((-\nu,\nu)^2),
+ \qquad
+ \lambda_U=\lambda_Q+p_\alpha\cdot d\alpha.
+\]
+If $F$ is the family of Proposition~\ref{prop:embedded-realization},
+then
+\begin{equation}
+ F^{(0)}(x,w)=F(r_Q(x),w)
+ \label{eq:transport-initial-family}
+\end{equation}
+is a Morse family whose generated image is the cylinder
+\begin{equation}
+ \mathcal L_0=
+ \{(q,\alpha;p_q,0):(q,p_q)\in L,\ \alpha\in(-\nu,\nu)^2\}.
+ \label{eq:transport-cylinder}
+\end{equation}
+Its critical parametrization is a diffeomorphism.  All its vertical
+critical points lie in a uniform ball, and it has the stated smoothness
+and homogeneous estimates uniformly up to $\overline U$.
+
+\begin{lemma}[Extension of the reverse isotopy]
+\label{lem:transport-cylinder-extension}
+Suppose that a smooth compactly supported Hamiltonian isotopy of $T^*Q$
+sends $Q_0$ to $L$.  There is a smooth Hamiltonian on $T^*\R^d$,
+supported for all times in one compact subset of $T^*U$, whose flow
+$\widetilde\varphi_t$ has the following properties.  Every trajectory
+starting on $\mathcal L_0$ has constant radial position $\alpha$.
+For some $\nu_c>0$, the final image over
+$U_c=\iota(Q\times(-\nu_c,\nu_c)^2)$ is precisely the zero section:
+\begin{equation}
+ \widetilde\varphi_1(\mathcal L_0)\cap T^*U_c=(U_c)_0.
+ \label{eq:transport-central-zero}
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Let $\varphi_t$ be the assumed original flow, generated by $H_t$.
+The path $\varphi_{1-t}\circ\varphi_1^{-1}$ starts at the identity,
+ends at $\varphi_1^{-1}$, and is generated by
+\begin{equation}
+ H_t^Q=-H_{1-t}.
+ \label{eq:transport-reverse-sign}
+\end{equation}
+Indeed differentiation of this path gives $-X_{H_{1-t}}$ at its
+current point; the fixed composition on the right introduces no
+conjugation of the Hamiltonian.
+
+Choose $0\le\zeta\le1$ supported in $(-\nu,\nu)^2$ and identically
+$1$ on $(-\nu_c,\nu_c)^2$.  Choose $0\le\xi\le1$ of compact support
+in $\R^2$, with $\xi=1$ on a ball to be specified.  On $T^*U$ set
+\begin{equation}
+ \widetilde H_t(q,\alpha;p_q,p_\alpha)
+   =H_t^Q(q,p_q)\zeta(\alpha)\xi(p_\alpha),
+ \label{eq:transport-extended-hamiltonian}
+\end{equation}
+and extend by zero to $T^*\R^d$.  This is smooth with uniform compact
+support: $Q$ is compact, $H_t^Q$ has uniformly compact momentum
+support, and both additional coordinate pairs are cut off.
+
+The convention $\iota_{X_H}d\lambda=-dH$ gives the radial equations
+\begin{equation}
+ \dot\alpha=H_t^Q\zeta\,\nabla\xi(p_\alpha),
+ \qquad
+ \dot p_\alpha=-H_t^Q\nabla\zeta\,\xi(p_\alpha).
+ \label{eq:transport-radial-equations}
+\end{equation}
+Set
+\[
+ M=\sup_{t,q,p_q}\abs{H_t^Q(q,p_q)}\,\sup_\alpha\norm{\nabla\zeta(\alpha)}.
+\]
+This number is independent of the size of the plateau of $\xi$.
+Take $\xi=1$ on $\{\norm{p_\alpha}\le M+1\}$.  A trajectory starting on
+$\mathcal L_0$ starts with $p_\alpha=0$, and the second equation gives
+$\norm{p_\alpha(t)}\le Mt$ for $0\le t\le1$.  Hence it stays in this
+plateau, the first equation vanishes, and $\alpha$ is constant.
+When $\alpha\in(-\nu_c,\nu_c)^2$, also $\nabla\zeta=0$; then
+$p_\alpha=0$ and the $T^*Q$ motion is exactly the reverse isotopy.
+It sends $L$ to $Q_0$.  Since all radial positions on the entire
+cylinder are fixed, no other part of that cylinder enters the central
+slab.  This proves \eqref{eq:transport-central-zero}.
+\end{proof}
+
+\subsection{A compact mixed generating function for each small step}
+
+Fix a compact subset $K\Subset T^*U$ containing the support of
+$\widetilde H_t$ in its interior for every $t$.  Points outside $K$
+are stationary, and uniqueness of flow lines prevents a trajectory
+from crossing into the stationary exterior in finite time.  In
+particular, all the flow maps and their successive time increments
+are supported in this common compact region.  Smoothness in time and
+compactness allow a finite subdivision
+\[
+ 0=t_0<t_1<\cdots<t_s=1
+\]
+such that every map
+$\psi_j=\widetilde\varphi_{t_j}\widetilde\varphi_{t_{j-1}}^{-1}$
+is as close to the identity in $C^1$ as required below.
+
+\begin{lemma}[Mixed coordinates and the action primitive]
+\label{lem:transport-mixed-generator}
+For a sufficiently small increment, write
+$\psi(x,p)=(X,P)$ in the Euclidean cotangent coordinates.  The pair
+$(X,p)$ is a global coordinate system on its graph.  There is a smooth
+compactly supported function $\sigma(X,p)$ such that
+\begin{equation}
+ x=X+\sigma_p(X,p),\qquad P=p+\sigma_X(X,p).
+ \label{eq:transport-mixed-graph}
+\end{equation}
+One can choose a single compact $D\Subset U$ containing the spatial
+support projection of all these functions, independently of a
+subsequent refinement of the subdivision.  Also
+$I+\sigma_{pX}$ is invertible, and refinement makes
+$\norm{\sigma_p}_\infty$ arbitrarily small.
+\end{lemma}
+
+\begin{proof}
+Require $\norm{D_xX-I}<\varepsilon<1$ uniformly.  For each fixed $p$,
+the map $x\mapsto X(x,p)$ is a local diffeomorphism and satisfies
+\[
+ \norm{X(x_1,p)-X(x_2,p)}\ge(1-\varepsilon)\norm{x_1-x_2}.
+\]
+It is injective and proper, since it is the identity outside a compact
+set.  Its image is both open and closed in $\R^d$, so it is onto.
+The inverse $x=x(X,p)$ is smooth jointly in $(X,p)$.
+
+For a step from time $a$ to time $b$, let $\varphi_{s,a}$ denote
+its partial flow.  Cartan's formula and the sign convention give
+\[
+ \mathcal L_{X_{\widetilde H_s}}\lambda
+   =d\bigl(\lambda(X_{\widetilde H_s})-\widetilde H_s\bigr).
+\]
+Consequently the action integral
+\begin{equation}
+ A(x,p)=\int_a^b
+  \bigl(\lambda(X_{\widetilde H_s})-\widetilde H_s\bigr)
+          (\varphi_{s,a}(x,p))\,ds
+ \label{eq:transport-action}
+\end{equation}
+satisfies $\psi^*\lambda-\lambda=dA$.  It is compactly supported:
+outside $K$ the trajectory is stationary and its integrand is zero.
+On the graph, expressed in mixed coordinates, put
+\[
+ S(X,p)=A(x(X,p),p)+x(X,p)\cdot p.
+\]
+Then
+\[
+ dS=P\cdot dX+x\cdot dp.
+\]
+Writing $S=X\cdot p+\sigma$ gives
+\eqref{eq:transport-mixed-graph}.  Explicitly,
+\begin{equation}
+ \sigma(X,p)=A(x(X,p),p)+(x(X,p)-X)\cdot p.
+ \label{eq:transport-compact-sigma}
+\end{equation}
+This expression vanishes outside the compact mixed-coordinate image
+of $K$.  Its spatial projection lies in a fixed compact subset of
+$U$, because both the input and output trajectories remain in $K$.
+Enlarge that projection slightly to obtain $D\Subset U$.  Thus for
+every $X\in U\setminus D$, $\sigma$ and all its derivatives vanish
+on a neighborhood of $X$, for all momenta.  Finally
+$I+\sigma_{pX}=D_Xx$ is invertible, and
+$\sigma_p=x-X$ tends uniformly to zero with the length of the time
+increment.
+\end{proof}
+
+\subsection{Transport of one Morse family}
+
+The following step supplies the analytic induction.  The variables
+called $w$ include all the stabilizing variables introduced at earlier
+steps.
+
+\begin{proposition}[One transport step with controlled infinity data]
+\label{prop:transport-step}
+Suppose $F_o:U\times\R^r\to\R$ is a smooth Morse family, with
+$\Gamma_{F_o}$ a diffeomorphism onto its generated Lagrangian
+$\mathcal L_o\subset T^*U$.  Suppose that it is smooth up to
+$\overline U$, with uniform bounds on compact vertical sets, and
+equals a homogeneous function $g_o(x,w)$ for $\norm{w}\ge R_o$, where
+$g_o$ is smooth for $w\ne0$ and
+\begin{equation}
+ \norm{\nabla_wg_o(x,w)}\ge c_o\norm{w},
+ \qquad \norm{d_xg_o(x,w)}\le C_o\norm{w}^2,
+ \qquad c_o>0.
+ \label{eq:transport-old-bounds}
+\end{equation}
+For a small step $\psi$ as in
+Lemma~\ref{lem:transport-mixed-generator}, there is a smooth Morse
+family on $U\times\R^{r+2d}$ generating $\psi(\mathcal L_o)$ with
+the same properties.  Its critical locus corresponds
+diffeomorphically to $\Crit(F_o)$.  On every fixed compact parameter
+slice in $U$, its homogeneous data represent the split quadratic
+stabilization of the preceding tube class.
+\end{proposition}
+
+\begin{proof}
+Introduce $z,p\in\R^d$. Before choosing cutoffs, consider the local expression
+\[
+ F_o(X+z,w)+\sigma(X,p)-z\cdot p,
+ \qquad X+z\in U.
+\]
+Its $p$- and $z$-equations are $z=\sigma_p(X,p)$ and
+$p=\nabla_xF_o(X+z,w)$. Together with the old $w$-equation,
+these are precisely the mixed graph equations for composition with
+$\psi$: $z$ is the displacement from the new base point $X$ to the
+old one, and $p$ is the old momentum. The cutoffs below keep the
+shifted base point in $U$ and switch off the shift for large
+$\norm{w}$. We will then compare the result with
+$g_o(X,w)-z\cdot p$ to restore homogeneity at infinity.
+
+\emph{Choice of cutoffs.}
+Choose $\zeta_1:U\to[0,1]$ of compact support, identically $1$ on
+an open set $V$ containing $D$.  Fix $\delta_0>0$ so that
+\begin{equation}
+ 2\delta_0<\operatorname{dist}(\operatorname{supp}\zeta_1,\partial U),
+ \qquad \norm{\sigma_p}_\infty<\delta_0.
+ \label{eq:transport-buffer}
+\end{equation}
+The first choice is made before refining the subdivision to obtain
+the second.  Let $\beta_1:\R^d\to[0,1]$ be $1$ for
+$\norm{z}\le\delta_0$ and $0$ for $\norm{z}\ge2\delta_0$.
+Choose a smooth radial $\chi:\R^r\to[0,1]$, equal to $1$ on a ball
+strictly larger than $\{\norm{w}\le R_o\}$, of compact support, and such
+that
+\begin{equation}
+ \norm{\nabla\chi(w)}\le\delta_1/\norm{w}\quad(w\ne0).
+ \label{eq:transport-log-cutoff}
+\end{equation}
+Here $\delta_1>0$ may be arbitrarily small.  For example, after fixing
+the inner radius $R_1>R_o$, take
+$\chi(w)=\vartheta(\log(\norm{w}/R_1)/T)$ outside that radius, where
+$\vartheta$ is a fixed smooth cutoff equal to $1$ near $(-\infty,0]$
+and $0$ near $[1,\infty)$; choosing $T$ sufficiently large gives
+\eqref{eq:transport-log-cutoff}.  Extend it by $1$ at smaller radii.
+
+Define
+\begin{align}
+ \kappa(X,z,w)&=\zeta_1(X)\beta_1(z)\chi(w),
+ &x&=X+\kappa(X,z,w)z,\label{eq:transport-kappa}\\
+ \widehat F(X,w,z,p)
+   &=F_o(X+\kappa(X,z,w)z,w)+\sigma(X,p)-z\cdot p.
+ \label{eq:transport-shifted-family}
+\end{align}
+The shifted argument always lies in $U$.  Indeed, when the shift is
+nonzero, $X\in\operatorname{supp}\zeta_1$ and its length is less
+than $2\delta_0$; the buffer in \eqref{eq:transport-buffer} places
+that entire ball in $U$.  When the shift vanishes the assertion is
+immediate.
+
+\emph{The full critical equations.}
+Write $a=\nabla_xF_o(x,w)$, identifying Euclidean covectors and
+vectors.  Direct differentiation, before discarding any cutoff
+terms, gives
+\begin{align}
+ \widehat F_p&=\sigma_p(X,p)-z,\label{eq:transport-critical-p}\\
+ \widehat F_z&=\kappa a+(a\cdot z)\nabla_z\kappa-p,
+       \label{eq:transport-critical-z}\\
+ \widehat F_w&=\nabla_wF_o(x,w)+(a\cdot z)\nabla_w\kappa,
+       \label{eq:transport-critical-w}\\
+ \widehat F_X&=a+(a\cdot z)\nabla_X\kappa+\sigma_X(X,p).
+       \label{eq:transport-output}
+\end{align}
+At any vertical critical point, \eqref{eq:transport-critical-p}
+forces $z=\sigma_p(X,p)$, hence $\norm{z}<\delta_0$ and $\beta_1=1$
+on a neighborhood of that point.  When $\norm{w}\ge R_o$, the old
+function is homogeneous at every allowed shifted base point.
+Equations \eqref{eq:transport-old-bounds} and
+\eqref{eq:transport-log-cutoff} therefore imply
+\begin{equation}
+ \norm{\widehat F_w}
+ \ge(c_o-C_o\delta_0\delta_1)\norm{w}.
+ \label{eq:transport-outer-gradient}
+\end{equation}
+Choose $\delta_1$ so that $C_o\delta_0\delta_1<c_o/2$.
+Thus no vertical critical point has $\norm{w}\ge R_o$; at every one of
+them $\chi=1$ on a neighborhood as well.
+
+There are two open regions to consider, namely $V$ and $U\setminus D$;
+they cover $U$.
+For $X\in V$, the three cutoff factors are $1$ near a solution, and
+the critical equations become
+\begin{equation}
+ \nabla_wF_o(x,w)=0,
+ \qquad p=\nabla_xF_o(x,w),
+ \qquad x=X+\sigma_p(X,p),
+ \qquad x=X+z.
+ \label{eq:transport-supported-critical}
+\end{equation}
+They express exactly composition with the graph
+\eqref{eq:transport-mixed-graph}, and the generated covector is
+\begin{equation}
+ \widehat F_X=p+\sigma_X(X,p)=P.
+ \label{eq:transport-supported-output}
+\end{equation}
+The critical equation is regular here.  In local total coordinates
+$(x,w,p,X)$, first impose the regular old equation
+$\nabla_wF_o=0$.  With $(x,w)$ fixed, the equation
+$p-\nabla_xF_o=0$ has identity coefficient in $p$.  With $(x,w,p)$
+fixed, the last graph equation has invertible coefficient
+$I+\sigma_{pX}$ in $X$.  This triangular order proves surjectivity
+of the differential of the full new vertical-gradient map.
+
+For $X\in U\setminus D$, the function $\sigma$ vanishes on a spatial
+neighborhood for all $p$.  The equations instead reduce to
+\begin{equation}
+ z=0,\qquad x=X,\qquad
+ \nabla_wF_o(X,w)=0,\qquad
+ p=\zeta_1(X)\nabla_xF_o(X,w).
+ \label{eq:transport-outside-critical}
+\end{equation}
+The auxiliary variable $p$ may therefore differ from the old
+momentum.  Nevertheless, \eqref{eq:transport-output} gives exactly
+\begin{equation}
+ \widehat F_X=\nabla_xF_o(X,w),
+ \label{eq:transport-outside-output}
+\end{equation}
+because all the spatial cutoff terms are multiplied by $z=0$.
+This is the identity correspondence, which is the true graph of
+$\psi$ over this region.  For regularity, first use the equation
+$-z=0$ to eliminate $z$, then the equation
+$\zeta_1\nabla_xF_o-p=0$ to eliminate $p$; the remaining equation
+is the regular old vertical-gradient equation.  This also checks
+the entire transition region of $\zeta_1$.  The two descriptions
+agree on the overlap, where $\zeta_1=1$.
+
+\emph{The correspondence of critical loci.}
+For an old critical point $(x,w)$ put
+$p_o=\nabla_xF_o(x,w)$ and $(X,P)=\psi(x,p_o)$.  Define
+\begin{equation}
+ (x,w)\longmapsto
+ \bigl(X,w,z=x-X,\ p=\zeta_1(X)p_o\bigr).
+ \label{eq:transport-critical-correspondence}
+\end{equation}
+If $X\in V$, these are the mixed graph equations, with $p=p_o$.
+If $X\notin D$, the true graph is the identity, so $x=X$, $z=0$,
+and \eqref{eq:transport-outside-critical} applies.  Thus
+\eqref{eq:transport-critical-correspondence} is a smooth map into
+$\Crit(\widehat F)$ on all of $\Crit(F_o)$.  Its inverse is
+\begin{equation}
+ (X,w,z,p)\longmapsto(X+\kappa(X,z,w)z,w).
+ \label{eq:transport-critical-inverse}
+\end{equation}
+The preceding equations show that these maps are inverse and that
+the generated image is exactly $\psi(\mathcal L_o)$.
+The step preserves $T^*U$, since its support is compactly contained
+there.  Thus no old or new critical point is lost at the base
+boundary, and $\Gamma_{\widehat F}$ is a diffeomorphism onto the transported
+image.  All new solutions obey a uniform bound: $\norm{w}<R_o$,
+$\norm{z}<\delta_0$, and
+\[
+ \norm{p}\le
+ \sup\{ \norm{\nabla_xF_o(x,w)}:x\in\overline U, \norm{w}\le R_o\}<\infty.
+\]
+
+\emph{Restoring exact homogeneity at infinity.}
+Set $W=(w,z,p)$ and consider the homogeneous reference
+\begin{equation}
+ G_0(X,W)=g_o(X,w)-z\cdot p.
+ \label{eq:transport-homogeneous-reference}
+\end{equation}
+It is $C^1$ everywhere, homogeneous of degree two, and satisfies
+\begin{equation}
+ \norm{\nabla_WG_0}^2
+  =\norm{\nabla_wg_o}^2+\norm{z}^2+\norm{p}^2
+  \ge\min(c_o^2,1)\norm{W}^2.
+ \label{eq:transport-reference-gradient}
+\end{equation}
+Both the value error and its \emph{full} vertical first derivative
+are uniformly bounded.  To verify this explicitly, write
+\begin{align}
+ E(X,W)&=\widehat F(X,W)-G_0(X,W)\notag\\
+  &=\bigl[F_o(X+\kappa z,w)-F_o(X,w)\bigr]
+      +\bigl[F_o(X,w)-g_o(X,w)\bigr]+\sigma(X,p).
+ \label{eq:transport-error-decomposition}
+\end{align}
+The first bracket and every term in its vertical derivative can be
+nonzero only with $w,z$ in fixed bounded sets, where the required
+derivatives of $F_o$ are uniformly bounded.  The second bracket and
+its vertical derivative are bounded because it vanishes for
+$\norm{w}\ge R_o$.  The final term has compact support.  The expression
+$-z\cdot p$, although unbounded, has cancelled exactly.  Thus for
+some finite $M_0,M_1$,
+\begin{equation}
+ \abs{E(X,W)}\le M_0,
+ \qquad \norm{\nabla_WE(X,W)}\le M_1
+ \quad(X\in\overline U, W\in\R^{r+2d}).
+ \label{eq:transport-bounded-error}
+\end{equation}
+No compact-support assertion for $E$ in all its variables is needed.
+
+Apply Lemma~\ref{lem:stabilization-smoothing} with
+$q'(z,p)=-z\cdot p$, uniformly over $\overline U$. Its homogeneous
+smoothing is
+\begin{equation}
+ G_\eta(X,W)=
+   \rho_\eta(\norm{w}/\norm{W})g_o(X,w)-z\cdot p
+ \quad(W\ne0),
+ \label{eq:transport-angular-smoothing}
+\end{equation}
+where $\rho_\eta$ vanishes for arguments at most $\eta$ and is $1$
+for arguments at least $2\eta$. Choose $\eta$ small and then $A$
+sufficiently large as in the lemma, with $A$ beyond the uniform
+critical-locus bound. Let $\tau_A$ be the radial cutoff from that
+construction and set
+\begin{equation}
+ F_n=(1-\tau_A)\widehat F+\tau_A G_\eta.
+ \label{eq:transport-final-step-family}
+\end{equation}
+The lemma shows that this function is smooth, equals $G_\eta$ for
+$\norm{W}\ge2A$, and has no new critical points. It leaves a
+neighborhood of the entire critical locus unchanged, so $F_n$ has
+the same regular critical locus and generated parametrization as
+$\widehat F$. It also supplies a straight homotopy from $G_0$ to
+$G_\eta$ through regular $C^1$ homogeneous functions.
+
+The new homogeneous function $G_\eta$ is smooth off zero, smoothly
+dependent on $X$ up to $\overline U$, and has a uniform positive
+gradient margin on the unit sphere.  Homogeneity and compactness
+there give the next constants of the form
+\eqref{eq:transport-old-bounds}; the remaining uniform smooth bounds
+follow directly from the construction.  The quadratic form
+$-z\cdot p$ is nondegenerate with signature $(d,d)$: in fixed linear
+coordinates $z=a+b$, $p=a-b$ it equals $-\norm{a}^2+\norm{b}^2$.
+The space of nondegenerate forms of signature $(d,d)$ is connected.
+Choose a fixed path in this space from $-z\cdot p$ to the standard
+split form in the chosen added coordinates. Its direct sum with
+$g_o(X,w)$ has nonzero total gradient off the origin, since both block
+gradients can vanish only there. This path is independent of $X$, so
+direct sum defines a homotopy over the whole base. Together with the
+straight homotopy to $G_\eta$, it preserves the free stable tube class
+on every fixed compact parameter slice.
+This proves all the assertions.
+\end{proof}
+
+\subsection{The final critical point and the contradiction}
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+Take the class $a\ne0$ with $\gamma_*a=0$ supplied by
+Proposition~\ref{prop:kernel-class}, and the closed exact embedded
+Lagrangian $L\subset T^*(S^9\times S^{N-1})$ supplied by
+Proposition~\ref{prop:embedded-realization}.  Suppose that a
+Hamiltonian isotopy as in the theorem sent the zero section to $L$.
+Lemma~\ref{lem:transport-cylinder-extension} supplies a compactly
+supported Euclidean Hamiltonian whose final cylinder is the zero
+graph over $U_c$.  Start with
+\eqref{eq:transport-initial-family} and apply
+Proposition~\ref{prop:transport-step} to each of its $s$ increments.
+Only finitely many steps are required. The subdivision is fixed by the
+mixed-coordinate and spatial-buffer conditions, which depend on the
+Hamiltonian and its support. At each step the new family constants
+change only the logarithmic cutoff, smoothing parameter and outer
+radius; they impose no further restriction on the step length.
+Thus the estimates and cutoff radii may be chosen anew at each step.
+We obtain a smooth family
+\[
+ F^{(s)}:U\times\R^{2p}\longrightarrow\R,
+ \qquad p=m+ds,
+\]
+with exactly homogeneous smooth-off-zero data at infinity, whose
+regular critical locus maps diffeomorphically onto
+$\widetilde\varphi_1(\mathcal L_0)$.
+
+Over $U_c$, the full critical-locus projection
+\begin{equation}
+ \Crit(F^{(s)})\big|_{U_c}\longrightarrow U_c
+ \label{eq:transport-final-projection}
+\end{equation}
+is a diffeomorphism: it is the generating parametrization followed
+by projection of the zero graph.  In particular there is precisely
+one critical point over every $X\in U_c$, and these points form a
+smooth section.  Their vertical Hessians are invertible.  Indeed,
+if $v\in\R^{2p}$ were in the kernel of such a Hessian, then
+\[
+ D(\partial_WF^{(s)})_{(X,W)}(0,v)
+   =\partial_W^2F^{(s)}(X,W)v=0.
+\]
+Since the full critical equation is regular, its zero set has this
+kernel as its tangent space.  Thus $(0,v)$ would be tangent to
+$\Crit(F^{(s)})$ and in the kernel of the differential of
+\eqref{eq:transport-final-projection}.  That projection is a
+diffeomorphism, so $v=0$; the square Hessian is nonsingular.
+
+Fix $v_0\in S^{N-1}$ and restrict the base to
+\[
+ b\longmapsto\iota((b,v_0),0),\qquad b\in S^9.
+\]
+The resulting smooth family $f_b:\R^{2p}\to\R$ therefore has
+exactly one nondegenerate fiber critical point, varying smoothly
+with $b$.  Its homogeneous data belong to $T_{p,p}$, since each
+transport step added signature $(d,d)$.  On this fixed slice the
+initial homogeneous family was $g_b$, and every subsequent step
+preserved its class after split stabilization.  Hence, writing
+$g_\infty:S^9\to\mathbf T$ for the final stable tube family,
+\[
+ [g_\infty]_{\mathrm{free}}=[a]_{\mathrm{free}}\ne[*],
+ \qquad \gamma\circ g_\infty\simeq *.
+\]
+This formulation does not require the smoothing homotopies to fix a
+chosen basepoint.  Equivalently, the basepoint paths supplied by
+these homotopies identify the resulting based class with $a$.
+All hypotheses of Proposition~\ref{prop:one-critical} hold, including
+the uniform homogeneous behavior and the required large indices.
+That proposition makes the same tube family freely null in the
+stable tube space, contradicting the non-null class of
+Proposition~\ref{prop:kernel-class}.  The assumed compactly supported
+Hamiltonian isotopy does not exist.  This closed exact embedded $L$
+is therefore the required counterexample.
+\end{proof}
diff --git a/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/build/sections/tubes.tex b/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/build/sections/tubes.tex
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@@ -0,0 +1,258 @@
+\section{Stable tubes and a kernel class}\label{sec:tubes}
+
+The output of this section is a nonzero family of smooth tube data over
+$S^9$ whose associated stable spherical fibration is trivial. We first
+specify the function model and its stabilizations, then obtain the class
+from the stable smooth $h$-cobordism space.
+
+\subsection{Functions and domains}
+
+For positive integers $k,l$, write
+\[
+ q_{k,l}(x,y)=-\norm{x}^{2}+\norm{y}^{2},
+ \qquad (x,y)\in\R^k\oplus\R^l.
+\]
+Let $T_{k,l}$ be the path component of $q_{k,l}|_{S^{k+l-1}}$ in
+\[
+ \{h\in C^1(S^{k+l-1},\R):0\text{ is a regular value of }h\},
+\]
+with its $C^1$ topology. We also regard its elements as homogeneous functions
+\begin{equation}\label{eq:homogeneous-extension}
+ g(ru)=r^2h(u),\qquad r>0,\quad u\in S^{k+l-1},\qquad g(0)=0.
+\end{equation}
+Indeed $g$ is $C^1$ on the whole vector space, since $g(w)=O(\norm{w}^2)$
+and $dg(w)=O(\norm{w})$, and
+\begin{equation}\label{eq:angular-gradient}
+ \nabla g(ru)=r\bigl(2h(u)u+\nabla_Sh(u)\bigr).
+\end{equation}
+The radial and tangential terms are orthogonal. Consequently $0$ is a
+regular value of $h$ precisely when $g$ has no critical point away from $0$.
+When $h$ is smooth, $g$ is smooth off the origin, and its negative domain
+\[
+ D(h)=\{u\in S^{k+l-1}:h(u)\leq0\}
+\]
+is a smooth tube: a domain isotopic to
+$D(q_{k,l})\cong S^{k-1}\times D^l$.
+
+We will pass between functions and domains in families. Here are the
+details of that passage. For a fixed smooth domain $D$ with cooriented
+boundary, the smooth functions defining $D$ have negative sign in
+$\Int D$, positive sign outside $D$, and positive outward normal derivative
+on $\partial D$. Their convex combinations retain all three properties.
+Thus the space of defining functions for $D$ is convex. A collar of
+$\partial D$ supplies one such function. For nearby domains, express the
+boundaries as normal graphs and extend their graph isotopies to ambient
+isotopies supported in the collar. Pulling back the chosen function gives
+local continuous choices of defining functions. Over a compact parameter
+space a partition of unity combines these choices: at each parameter all
+the terms define the same domain, so convexity applies. This construction
+also works relative to a subspace on which defining functions are already
+chosen, by interpolation with that choice in a neighborhood. In particular,
+a homotopy of smooth domains lifts to a homotopy of defining functions,
+with prescribed endpoints.
+
+Smooth angular functions and $C^1$ angular functions give the same
+homotopy groups. To see this directly, for a compact family put
+\[
+ \mu=\min_{b,u}\bigl(4h_b(u)^2+\norm{\nabla_Sh_b(u)}^2\bigr)^{1/2}>0.
+\]
+Sufficiently close approximation in angular $C^1$ norm preserves this
+inequality, also throughout straight interpolation. Smooth finitely many
+angular functions and take a smooth partition of unity in the parameters
+to obtain such an approximation. Cutoffs in the parameters give the
+relative version. Applying the same argument to a homotopy proves both
+surjectivity and injectivity on homotopy groups. These descriptions agree
+with the smooth tube models of \cite[\S3.6]{AIS2026}; see also
+\cite[Definitions~3.10 and~3.36]{AACK2025} for homogeneous $C^1$
+functions and the tube-type space with its $C^1$ topology.
+
+Direct sum is well defined in this model:
+\[
+ (g\oplus g')(w,z)=g(w)+g'(z).
+\]
+Its gradient vanishes only when both block gradients vanish, hence only at
+$(w,z)=0$. Direct sums of paths from the standard forms also show that the
+indices add. Although $g\oplus g'$ can fail to be smooth on a coordinate
+axis, it is $C^1$ there; subsequent angular smoothing preserves its tube
+class. Adding positive squares has the usual geometric interpretation.
+In the sphere chart $u=w/\norm{w}$, $v=z/\norm{w}$, its negative domain is
+\[
+ h(u)+\norm{v}^2\leq0.
+\]
+This is the disk thickening of the old negative domain, with the boundary
+rounded. Adding negative squares gives the corresponding construction on
+the positive domain.
+
+Fix the coordinate identifications putting negative coordinates before
+positive ones. Additions of positive and negative squares then form a
+commuting two-parameter stabilization system. The diagonal is cofinal, and
+we use the based mapping telescope
+\begin{equation}\label{eq:stable-tubes}
+ \T=\operatorname*{hocolim}_{m}
+ \bigl(T_{m,m}\xrightarrow{\,\oplus q_{1,1}\,}T_{m+1,m+1}\bigr).
+\end{equation}
+In particular, a nullhomotopy after adding any fixed numbers of positive
+and negative squares implies stable nullity: adding the missing squares
+reaches a diagonal stage.
+
+\subsection{The stable kernel}
+
+Let $G$ be the stable space of homotopy self-equivalences of spheres,
+stabilized by suspension; $BG$ classifies stable spherical fibrations.
+The negative domains of a tube family have the homotopy type of spheres,
+and their stable fiber homotopy class defines a map $\gamma:\T\to BG$.
+To describe its fiber, let $H(X)$ be the smooth $h$-cobordism space
+of a compact smooth manifold $X$, with identified incoming end and
+prescribed product vertical boundary. Its stable version is
+\[
+ \Hs(X)=\operatorname*{hocolim}_{r}H(X\times J^r),
+\]
+where $J$ is a closed interval and the maps are interval product
+stabilizations. For $X=*$ this is equivalently the stabilization of
+disks of increasing dimension. We will specify the collared product
+model when applying it in Section~\ref{sec:obstruction}.
+
+We use the stable parametrized $h$-cobordism theorem
+\cite[Theorems~0.1 and~0.3]{WJR2013} in the form
+\begin{equation}\label{eq:stable-hcob-theorem}
+ \Hs(X)\simeq\Omega\Wh^{\mathrm{diff}}(X).
+\end{equation}
+Here
+$\Wh^{\mathrm{diff}}(X)$ denotes the infinite loop space of the cofiber
+of the unit $\Sigma^\infty X_+\to A(X)$, where $A(X)$ is Waldhausen's
+algebraic $K$-theory spectrum of $X$.
+
+Waldhausen's tube theorem identifies the fiber of $\gamma$ with
+$\Hs(*)$.
+
+\begin{theorem}[Waldhausen's tube fibration]\label{thm:tube-fibration}
+For the stable smooth tube space \eqref{eq:stable-tubes} there is a
+homotopy fiber sequence
+\begin{equation}\label{eq:tube-fibration}
+ \Hs(*)\longrightarrow\T\xrightarrow{\gamma}BG.
+\end{equation}
+The rigid tube map $\iota:BO\to\T$, defined by nondegenerate quadratic
+forms, satisfies $\gamma\circ\iota\simeq BJ$, where
+$BJ:BO\to BG$ is the stable spherical fibration map.
+\end{theorem}
+
+This is the smooth tube theorem of \cite{Waldhausen1982}, with the
+function and domain comparison and the rigid map described in
+\cite[\S3.6, Theorem~3.30 and Remark~3.34]{AIS2026}. The negative-domain
+frame model in \cite[\S4.1, Definition~4.3, Propositions~4.4, 4.6, 4.7 and Remark~4.5]{AACK2025}
+identifies this particular map $\gamma$ with the classifying map of the
+negative-domain spherical fibration. To identify the rigid composite,
+retract the negative domain of a quadratic form onto
+the unit sphere of its negative eigenspace. Stabilizing the corresponding
+Grassmannians gives $BO$, and taking those sphere bundles is exactly
+$BJ$. The comparison of the partition and spherical-domain models
+is a stable equivalence; we use no finite-stage equivalence between
+those models. All equivalences and fiber sequences here are understood
+up to weak homotopy equivalence.
+
+\begin{proposition}\label{prop:kernel-class}
+There exists
+\begin{equation}\label{eq:kernel-class}
+ 0\ne a\in\pi_9\T,\qquad \gamma_*a=0.
+\end{equation}
+Moreover, $(BJ)_*:\pi_9BO\to\pi_9BG$ is injective.
+\end{proposition}
+
+\begin{proof}
+Bott periodicity gives $\pi_8O\cong\pi_9O\cong\Z/2$
+\cite[Corollary to Theorem~II, equation~(1.5)]{Bott1959}.
+The stable $J$-homomorphisms in both degrees are injective
+\cite[Theorem~9.5]{Adams1966}. We also use
+\[
+ \pi_9^S\cong(\Z/2)^3,
+ \qquad
+ \bigl(\pi_{10}\Wh^{\mathrm{diff}}(*)\bigr)_{(2)}
+ \cong\Z/8\oplus(\Z/2)^2.
+\]
+The first value appears in \cite[Table~1, degree~9]{IWX2023}; the second
+is Rognes's computation \cite[Theorem~5.8]{Rognes2002}, also recorded in
+\cite[Theorem~3.3]{Rognes2014}. Only the two-primary assertion is used.
+
+In positive degrees $\pi_iG=\pi_i^S$. One may use either based or
+unbased sphere equivalences here, since the evaluation fibration has
+base a sphere whose connectivity tends to infinity under stabilization.
+The relevant part of the long exact sequence of
+\eqref{eq:tube-fibration} is
+\begin{equation}\label{eq:tube-long-exact}
+ \pi_{10}\T\longrightarrow\pi_{10}BG
+ \xrightarrow{\partial}\pi_9\Hs(*)
+ \longrightarrow\pi_9\T\xrightarrow{\gamma_*}\pi_9BG.
+\end{equation}
+The image of $\pi_{10}BO$ in $\pi_{10}BG=\pi_9^S$ is the
+order-two image of $J_9$. Theorem~\ref{thm:tube-fibration} puts it in
+$\ker\partial$. Since $\pi_9^S$ has order eight,
+$\abs{\im\partial}\leq4$. On the other hand,
+\eqref{eq:stable-hcob-theorem} identifies
+\[
+ \pi_9\Hs(*)=\pi_{10}\Wh^{\mathrm{diff}}(*),
+\]
+whose two-primary part has order $32$. Thus $\partial$ is not surjective.
+Exactness in \eqref{eq:tube-long-exact} implies that the image in
+$\pi_9\T$ is nonzero and is contained in $\ker\gamma_*$, proving
+\eqref{eq:kernel-class}. Finally, $\pi_9BO\to\pi_9BG$ is $J_8$, which
+is injective by the other degree of Adams's theorem cited above.
+\end{proof}
+
+\begin{lemma}\label{lem:localized-family}
+Fix $B=S^9$ and a standard height function $\theta:B\to\R$. The class
+$a$ in \eqref{eq:kernel-class} has, for arbitrarily large $m\geq20$,
+a representative $b\mapsto h_b$ in $T_{m,m}$ such that
+$(b,u)\mapsto h_b(u)$ is smooth and $h_b=q_{m,m}|_{S^{2m-1}}$ on
+neighborhoods of both critical points of $\theta$. Its homogeneous
+extension $g_b$ satisfies, for constants $0<c\leq C<\infty$,
+\begin{equation}\label{eq:uniform-homogeneous-gradient}
+ c\norm{w}\leq\norm{\nabla_wg_b(w)}\leq C\norm{w}
+ \qquad(b\in B,\ w\in\R^{2m}).
+\end{equation}
+For any fixed metric on $B$, there is also a constant $C_1$ such that
+\begin{equation}\label{eq:base-mixed-bound}
+ \norm{d_wd_bg_b(w)}\leq C_1\norm{w}.
+\end{equation}
+The derivatives in \eqref{eq:base-mixed-bound} extend continuously to
+$w=0$ with value zero.
+\end{lemma}
+
+\begin{proof}
+A map from the compact sphere to the mapping telescope
+\eqref{eq:stable-tubes} has bounded height. A finite initial portion of
+the telescope retracts onto its last stage. Hence $a$ is represented by
+a based map $S^9\to T_{m,m}$ at a finite stage. Further stabilization
+preserves its stable class and makes $m$ arbitrarily large.
+
+Choose an oriented disk in $B$ whose closure avoids the two critical
+points of $\theta$. Collapsing the complement of its interior gives a
+degree-one map $B\to S^9$, constant at the target basepoint off that
+disk. Equivalently one can take a smooth map of degree one supported
+there, obtained by a relative smooth approximation of this collapse
+map. Precomposition preserves the based homotopy class, since a
+degree-one self-map of $S^9$ is based homotopic to the identity. The
+resulting family is standard on neighborhoods of both critical points.
+
+Apply the relative angular approximation described above, retaining the
+standard function on smaller such neighborhoods. More explicitly,
+approximate the continuous map into $C^1(S^{2m-1})$ by a finite smooth
+partition of unity in $b$ multiplying smooth angular functions; then
+multiply its difference from $q_{m,m}$ by a parameter cutoff which is
+zero on the smaller neighborhoods and one outside the original
+standard neighborhoods. The approximation remains uniformly close in
+angular $C^1$ norm. The positive regularity margin and straight
+interpolation show that its stable class is still $a$ and its spherical
+class is zero.
+
+For this smooth family, the continuous positive function
+$\norm{2h_b(u)u+\nabla_Sh_b(u)}$ has a positive minimum and a finite
+maximum on $B\times S^{2m-1}$. Formula~\eqref{eq:angular-gradient}
+gives \eqref{eq:uniform-homogeneous-gradient}, including the origin.
+Finally $d_bg_b(ru)=r^2d_bh_b(u)$. Differentiating in $w$ gives a
+degree-one expression whose angular coefficients are bounded by
+compactness, proving \eqref{eq:base-mixed-bound}. The degree-two value
+bound and degree-one derivative bound also prove the asserted $C^1$
+extension at zero. As before, the nonzero based stable class cannot
+be freely nullhomotopic, because the fundamental group action fixes
+zero.
+\end{proof}
diff --git a/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/paper.pdf b/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..7473b5c975cae576a3c16102b0b36107400187df
--- /dev/null
+++ b/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:ce7a208ddd7ef666a814114a9c92bbb782eda8ea82e21502f7624709539a9a34
+size 303836
diff --git a/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026.pdf b/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..f01765aef44ca578ebf0c0ecc3878edeccb65fe9
--- /dev/null
+++ b/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:9eb71358fe8d9e08b450286ec6f40cb9f1452f3a49ed7801429485c2676526a6
+size 555598
diff --git a/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/README.md b/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..f8ee6ef9f69c28e6487e81087fe798fd387ea396
--- /dev/null
+++ b/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/README.md
@@ -0,0 +1,17 @@
+# [A cubic lower bound for border determinantal complexity of the permanent](A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 24, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026,
+  author = {{OpenAI}},
+  title = {{A cubic lower bound for border determinantal complexity of the permanent}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026.pdf}{OAI:A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026}},
+  year = {2026}
+}
+```
diff --git a/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/figures/proof-flow.tex b/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/figures/proof-flow.tex
new file mode 100644
index 0000000000000000000000000000000000000000..0c93080f36d563ea5bab8c2d8a67ca23e9ba9ce7
--- /dev/null
+++ b/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/figures/proof-flow.tex
@@ -0,0 +1,72 @@
+\begin{figure}[tbp]
+\centering
+\setlength{\fboxsep}{7pt}
+\fbox{\begin{minipage}{0.90\linewidth}
+\centering
+\textbf{One integral coefficient polynomial}\\[3pt]
+$\displaystyle G_D(X)=[s^qt^q]\per_k(X+sI+tD)
+\in\mathbb Z[\theta_1,\ldots,\theta_k][X_{ij}]$\\[3pt]
+$\displaystyle D=\diag(\theta_1,\ldots,\theta_k),\qquad
+r=k-2q,\qquad T=\binom{q+2}{2}$
+\end{minipage}}
+
+\smallskip
+\begin{tabular}{@{}c@{\hspace{0.025\linewidth}}c@{}}
+\begin{minipage}{0.43\linewidth}\centering
+$\swarrow$\quad over $\C$, every parameter choice
+\end{minipage}
+&
+\begin{minipage}{0.43\linewidth}\centering
+$\searrow$\quad over $K=\overline{\mathbb F}_2$, choose distinct parameters
+\end{minipage}
+\\[4pt]
+\fbox{\begin{minipage}{0.405\linewidth}\centering
+\textbf{Permanent projection}\\[3pt]
+$\displaystyle \per_m(A_m(X))=a_0G_D(X)$\\[3pt]
+$m\ge11k-2q,\qquad a_0\ne0$
+\end{minipage}}
+&
+\fbox{\begin{minipage}{0.405\linewidth}\centering
+\textbf{Plane-curve geometry}\\[3pt]
+$\displaystyle \nabla G_{\bar D}(X)=0
+\ \Longrightarrow\ \delta(C_X)\ge T$\\[3pt]
+$\displaystyle \codim V(\nabla G_{\bar D})\ge T-k$
+\end{minipage}}
+\\[5pt]
+\begin{minipage}{0.43\linewidth}\centering
+$\searrow$\quad apply at the complex parameters supplied by geometry
+\end{minipage}
+&
+\begin{minipage}{0.43\linewidth}\centering
+$\swarrow$\quad proper specialization and a transverse linear section
+\end{minipage}
+\end{tabular}
+
+\smallskip
+\fbox{\begin{minipage}{0.90\linewidth}
+\centering
+\textbf{A smooth affine restriction over $\C$}\\[3pt]
+A nonzero homogeneous form $G$ of degree $r$ in $d$ variables has a smooth
+projective zero set and satisfies $G=\per_m\circ\Phi_m$ for a fixed affine map $\Phi_m$.\\[3pt]
+$\displaystyle r-1\ge\frac{k}{4},\qquad
+d-1\ge\frac{k^2}{200}$
+\end{minipage}}
+
+\smallskip
+$\big\downarrow$\quad the polar count for smooth homogeneous forms
+\smallskip
+
+\fbox{\begin{minipage}{0.90\linewidth}
+\centering
+$\displaystyle
+\bdc(\per_m)\ge\bdc(G)
+\ge\frac{(r-1)(d-1)}{4e}
+\ge\frac{k^3}{3200e}$
+\end{minipage}}
+\caption{The two uses of the integral coefficient formula meet over
+$\C$. The graph projection applies to every complex parameter choice;
+the characteristic-two branch supplies a suitable complex choice through
+the proper parameter family. The displayed scale bounds use
+$q=\lfloor(k-1)/3\rfloor$, $k\ge128$, and $m\ge11k-2q$.}
+\label{fig:proof-flow}
+\end{figure}
diff --git a/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/figures/selector-wiring.tex b/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/figures/selector-wiring.tex
new file mode 100644
index 0000000000000000000000000000000000000000..b4f44b41fe8eebf214570b625baec12c5fc417ea
--- /dev/null
+++ b/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/figures/selector-wiring.tex
@@ -0,0 +1,63 @@
+\begin{figure}[tbp]
+\centering
+\begingroup
+\setlength{\unitlength}{1pt}
+\begin{picture}(460,202)
+\linethickness{0.45pt}
+\put(174,129){\framebox(112,55){}}
+\put(174,25){\framebox(112,55){}}
+\put(72,104){\line(1,0){316}}
+\qbezier(72,104)(123,122)(174,140)
+\qbezier(286,140)(337,122)(388,104)
+\put(72,171){\line(1,0){102}}
+\put(286,171){\line(1,0){102}}
+\qbezier(72,104)(123,86)(174,68)
+\qbezier(286,68)(337,86)(388,104)
+\put(72,37){\line(1,0){102}}
+\put(286,37){\line(1,0){102}}
+\put(72,104){\circle*{3.2}}
+\put(388,104){\circle*{3.2}}
+\put(72,171){\circle*{3.2}}
+\put(388,171){\circle*{3.2}}
+\put(72,37){\circle*{3.2}}
+\put(388,37){\circle*{3.2}}
+\put(174,171){\circle*{3.2}}
+\put(174,140){\circle*{3.2}}
+\put(286,171){\circle*{3.2}}
+\put(286,140){\circle*{3.2}}
+\put(174,68){\circle*{3.2}}
+\put(174,37){\circle*{3.2}}
+\put(286,68){\circle*{3.2}}
+\put(286,37){\circle*{3.2}}
+\put(230,162){\makebox(0,0){\small color 1: \(H\)}}
+\put(230,151){\makebox(0,0){\scriptsize 3 rows \(+\) 3 columns}}
+\put(181,171){\makebox(0,0)[l]{\scriptsize \(c_2\)}}
+\put(181,140){\makebox(0,0)[l]{\scriptsize \(c_1\)}}
+\put(279,171){\makebox(0,0)[r]{\scriptsize \(r_2\)}}
+\put(279,140){\makebox(0,0)[r]{\scriptsize \(r_1\)}}
+\put(230,57){\makebox(0,0){\small color 2: \(H\)}}
+\put(230,46){\makebox(0,0){\scriptsize 3 rows \(+\) 3 columns}}
+\put(181,68){\makebox(0,0)[l]{\scriptsize \(c_1\)}}
+\put(181,37){\makebox(0,0)[l]{\scriptsize \(c_2\)}}
+\put(279,68){\makebox(0,0)[r]{\scriptsize \(r_1\)}}
+\put(279,37){\makebox(0,0)[r]{\scriptsize \(r_2\)}}
+\put(64,104){\makebox(0,0)[r]{\shortstack[r]{\small shared main\\[-1pt]\small row \(i\)}}}
+\put(396,104){\makebox(0,0)[l]{\shortstack[l]{\small shared main\\[-1pt]\small column \(i\)}}}
+\put(72,192){\makebox(0,0){\small gate 1 row \(i\)}}
+\put(388,192){\makebox(0,0){\small gate 1 column \(i\)}}
+\put(72,16){\makebox(0,0){\small gate 2 row \(i\)}}
+\put(388,16){\makebox(0,0){\small gate 2 column \(i\)}}
+\put(230,113){\makebox(0,0){\small \(X_{ii}\)}}
+\put(122,132){\makebox(0,0){\small \(-2\)}}
+\put(121,76){\makebox(0,0){\small \(-2\theta_i\)}}
+\end{picture}
+\endgroup
+\caption{Wiring after the two replacements at a fixed diagonal position
+\(i\). Each box is a separate copy of \(H\) on three fresh rows and
+three fresh columns. Its incoming ports \(c_1,c_2\) are internal columns
+and its outgoing ports \(r_1,r_2\) are internal rows; their placement is
+schematic. The two copies share the main endpoints and have distinct
+gate and internal vertices. Unlabeled boundary edges have weight \(1\).
+The ordinary edge \(X_{ii}\) remains; other main and gate edges are omitted.}
+\label{fig:selector-wiring}
+\end{figure}
diff --git a/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/main.tex b/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/main.tex
new file mode 100644
index 0000000000000000000000000000000000000000..775834c75c94d514b1b85e522a76e3040fadf3f8
--- /dev/null
+++ b/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/main.tex
@@ -0,0 +1,98 @@
+\documentclass[11pt]{article}
+\def\DoNotLoadEpstopdf{}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage[margin=1in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{microtype}
+\usepackage{booktabs,array}
+\usepackage{placeins}
+\usepackage{etoolbox}
+\usepackage{xcolor}
+\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref}
+\usepackage[figure]{hypcap}
+\usepackage[nameinlink,capitalize,noabbrev]{cleveref}
+\makeatletter
+\newcommand\chResultRefstepcounter[2][]{%
+  \let\chSavedCrefRefstepcounter\cref@old@refstepcounter
+  \let\cref@old@refstepcounter\Hy@theorem@refstepcounter
+  \refstepcounter[#1]{#2}%
+  \let\cref@old@refstepcounter\chSavedCrefRefstepcounter
+}
+\AddToHook{begindocument/end}{%
+  \patchcmd{\cref@thmnoarg}
+    {\refstepcounter}{\Hy@theorem@refstepcounter}
+    {}{\PackageError{ch418}{Could not patch the plain theorem counter path}{%
+      Recheck the amsthm, hyperref, and cleveref versions.}}%
+  \patchcmd{\cref@thmoptarg}
+    {\refstepcounter}{\chResultRefstepcounter}
+    {}{\PackageError{ch418}{Could not patch the shared theorem counter path}{%
+      Recheck the amsthm, hyperref, and cleveref versions.}}%
+}
+\makeatother
+\AddToHook{cmd/thebibliography/after}{%
+  \addcontentsline{toc}{section}{\refname}%
+}
+\pdfinfoomitdate=1
+\pdftrailerid{}
+\pdfsuppressptexinfo=15
+\hypersetup{pdftitle={A cubic lower bound for border determinantal complexity of the permanent},pdfauthor={OpenAI}}
+\numberwithin{equation}{section}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newcommand{\C}{\mathbb C}
+\newcommand{\PP}{\mathbb P}
+\newcommand{\cO}{\mathcal O}
+\DeclareMathOperator{\Mat}{Mat}
+\DeclareMathOperator{\per}{per}
+\DeclareMathOperator{\dc}{dc}
+\DeclareMathOperator{\bdc}{\overline{dc}}
+\DeclareMathOperator{\adj}{adj}
+\DeclareMathOperator{\tr}{tr}
+\DeclareMathOperator{\length}{length}
+\DeclareMathOperator{\Hom}{Hom}
+\DeclareMathOperator{\Hilb}{Hilb}
+\DeclareMathOperator{\GL}{GL}
+\DeclareMathOperator{\End}{End}
+\DeclareMathOperator{\Spec}{Spec}
+\DeclareMathOperator{\Gr}{Gr}
+\DeclareMathOperator{\rank}{rank}
+\DeclareMathOperator{\im}{im}
+\DeclareMathOperator{\coker}{coker}
+\DeclareMathOperator{\diag}{diag}
+\DeclareMathOperator{\Sing}{Sing}
+\DeclareMathOperator{\Supp}{Supp}
+\DeclareMathOperator{\codim}{codim}
+\newcommand{\dd}{\mathrm d}
+\title{A cubic lower bound for border determinantal complexity of the permanent}
+\author{OpenAI}
+\date{September 24, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We prove that the complex border determinantal complexity of the $m\times m$
+permanent is $\Omega(m^3)$, allowing arbitrary affine-linear determinant
+representations and coefficientwise limits. It also gives cubic lower bounds
+for exact determinantal complexity and for the numbers of vertices and edges
+in affine-linear algebraic branching programs, including coefficientwise
+limits with a fixed vertex or edge budget.
+\end{abstract}
+\tableofcontents
+\input{sections/introduction}
+\input{sections/models}
+\input{sections/selector}
+\input{sections/adjugate}
+\input{sections/conductor}
+\input{sections/dimension}
+\input{sections/transfer}
+\input{sections/polar}
+\input{sections/consequences}
+\bibliographystyle{plain}
+\bibliography{references}
+\end{document}
diff --git a/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/references.bib b/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/references.bib
new file mode 100644
index 0000000000000000000000000000000000000000..e919901664a17e6cb77ccfa2fe982e314e1ce55d
--- /dev/null
+++ b/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/references.bib
@@ -0,0 +1,313 @@
+@article{LandsbergManivelRessayre2010,
+  author = {Landsberg, Joseph M. and Manivel, Laurent and Ressayre, Nicolas},
+  title = {Hypersurfaces with degenerate duals and the {Geometric Complexity Theory Program}},
+  journal = {Commentarii Mathematici Helvetici},
+  volume = {88},
+  number = {2},
+  pages = {469--484},
+  year = {2013},
+  doi = {10.4171/CMH/292},
+  url = {https://ems.press/journals/cmh/articles/11963}
+}
+
+@article{LandsbergRessayre2017,
+  author = {Landsberg, Joseph M. and Ressayre, Nicolas},
+  title = {Permanent v. determinant: An exponential lower bound assuming symmetry and a potential path towards {Valiant}'s conjecture},
+  journal = {Differential Geometry and its Applications},
+  volume = {55},
+  pages = {146--166},
+  year = {2017},
+  doi = {10.1016/j.difgeo.2017.03.017},
+  url = {https://arxiv.org/abs/1508.05788v2}
+}
+
+@misc{SheshadriBorder2026,
+  author = {Sheshadri, Karthik},
+  title = {A near-quadratic lower bound on the border determinantal complexity of {$\sum_i x_i^n$} via conormal specialization},
+  howpublished = {\href{https://arxiv.org/abs/2606.13628v1}{arXiv:2606.13628v1}},
+  year = {2026},
+  note = {11 June 2026},
+  doi = {10.48550/arXiv.2606.13628},
+  url = {https://arxiv.org/abs/2606.13628v1}
+}
+
+@misc{SheshadriNoGo2026,
+  author = {Sheshadri, Karthik},
+  title = {The Exact Reach of Conormal Invariants in Determinantal Complexity: a Quadratic No-Go Theorem},
+  howpublished = {\href{https://arxiv.org/abs/2606.15970v1}{arXiv:2606.15970v1}},
+  year = {2026},
+  note = {14 June 2026},
+  doi = {10.48550/arXiv.2606.15970},
+  url = {https://arxiv.org/abs/2606.15970v1}
+}
+
+@article{FGvS1999,
+  author = {Fantechi, Barbara and G{\"o}ttsche, Lothar and van Straten, Duco},
+  title = {Euler number of the compactified {Jacobian} and multiplicity of rational curves},
+  journal = {Journal of Algebraic Geometry},
+  volume = {8},
+  number = {1},
+  pages = {115--133},
+  year = {1999},
+  url = {https://arxiv.org/abs/alg-geom/9708012}
+}
+
+@article{CHT2023,
+  author = {Christ, Karl and He, Xiang and Tyomkin, Ilya},
+  title = {On the {Severi} problem in arbitrary characteristic},
+  journal = {Publications Math\'ematiques de l'IH\'ES},
+  volume = {137},
+  pages = {1--45},
+  year = {2023},
+  doi = {10.1007/s10240-022-00135-x},
+  url = {https://pmihes.centre-mersenne.org/articles/10.1007/s10240-022-00135-x/}
+}
+
+@misc{StacksProject,
+  author = {{The Stacks Project Authors}},
+  title = {The {Stacks} Project},
+  howpublished = {\url{https://stacks.math.columbia.edu}},
+  year = {2026},
+  note = {Tags 032L, 00PD, 0D4I, and 054K; accessed 24 September 2026},
+  url = {https://stacks.math.columbia.edu}
+}
+
+
+@article{MignonRessayre2004,
+  author = {Mignon, Thierry and Ressayre, Nicolas},
+  title = {A quadratic bound for the determinant and permanent problem},
+  journal = {International Mathematics Research Notices},
+  volume = {2004},
+  number = {79},
+  pages = {4241--4253},
+  year = {2004},
+  doi = {10.1155/S1073792804142566},
+  url = {https://academic.oup.com/imrn/article-abstract/2004/79/4241/793411}
+}
+
+
+@inproceedings{Valiant1979Algebra,
+  author = {Valiant, L. G.},
+  title = {Completeness Classes in Algebra},
+  booktitle = {Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing},
+  series = {STOC '79},
+  publisher = {Association for Computing Machinery},
+  year = {1979},
+  pages = {249--261},
+  doi = {10.1145/800135.804419},
+  url = {https://doi.org/10.1145/800135.804419}
+}
+
+
+@article{Valiant1979Permanent,
+  author = {Valiant, L. G.},
+  title = {The Complexity of Computing the Permanent},
+  journal = {Theoretical Computer Science},
+  volume = {8},
+  number = {2},
+  year = {1979},
+  pages = {189--201},
+  doi = {10.1016/0304-3975(79)90044-6},
+  url = {https://doi.org/10.1016/0304-3975(79)90044-6}
+}
+
+
+@article{Tyomkin2013,
+  author = {Tyomkin, Ilya},
+  title = {On {Zariski}'s theorem in positive characteristic},
+  journal = {Journal of the European Mathematical Society},
+  volume = {15},
+  number = {5},
+  pages = {1783--1803},
+  year = {2013},
+  doi = {10.4171/JEMS/403},
+  url = {https://ems.press/journals/jems/articles/11402}
+}
+
+
+@article{Kleiman1974,
+  author = {Kleiman, Steven L.},
+  title = {The transversality of a general translate},
+  journal = {Compositio Mathematica},
+  volume = {28},
+  number = {3},
+  pages = {287--297},
+  year = {1974},
+  url = {https://www.numdam.org/item/CM_1974__28_3_287_0/}
+}
+
+
+@article{AlperBogartVelasco2017,
+  author = {Alper, Jarod and Bogart, Tristram and Velasco, Mauricio},
+  title = {A Lower Bound for the Determinantal Complexity of a Hypersurface},
+  journal = {Foundations of Computational Mathematics},
+  volume = {17}, number = {3}, pages = {829--836}, year = {2017},
+  doi = {10.1007/s10208-015-9300-x},
+  url = {https://arxiv.org/abs/1505.02205v1}
+}
+
+
+@article{Piene1978,
+  author = {Piene, Ragni},
+  title = {Polar classes of singular varieties},
+  journal = {Annales scientifiques de l'{\'E}cole Normale Sup{\'e}rieure},
+  series = {4}, volume = {11}, number = {2}, pages = {247--276}, year = {1978},
+  doi = {10.24033/asens.1346},
+  url = {https://www.numdam.org/articles/10.24033/asens.1346/}
+}
+
+@article{DellHusfeldtMarxTaslamanWahlen2014,
+  author = {Dell, Holger and Husfeldt, Thore and Marx, D{\'a}niel and Taslaman, Nina and Wahl{\'e}n, Martin},
+  title = {Exponential Time Complexity of the Permanent and the {Tutte} Polynomial},
+  journal = {ACM Transactions on Algorithms},
+  volume = {10}, number = {4}, pages = {21:1--21:32}, year = {2014},
+  doi = {10.1145/2635812},
+  note = {Preprint: arXiv:1206.1775v1; numbered locators in the text refer to this version},
+  url = {https://arxiv.org/abs/1206.1775v1}
+}
+
+@article{vonZurGathen1987,
+  author = {von zur Gathen, Joachim},
+  title = {Permanent and determinant},
+  journal = {Linear Algebra and its Applications},
+  volume = {96},
+  pages = {87--100},
+  year = {1987},
+  doi = {10.1016/0024-3795(87)90337-5},
+  url = {https://www.sciencedirect.com/science/article/pii/0024379587903375}
+}
+
+@article{Meshulam1989,
+  author = {Meshulam, Roy},
+  title = {On two extremal matrix problems},
+  journal = {Linear Algebra and its Applications},
+  volume = {114/115},
+  pages = {261--271},
+  year = {1989},
+  doi = {10.1016/0024-3795(89)90465-5},
+  url = {https://meshulam.net.technion.ac.il/files/2021/06/4.pdf}
+}
+
+@article{Cai1990,
+  author = {Cai, Jin-yi},
+  title = {A note on the determinant and permanent problem},
+  journal = {Information and Computation},
+  volume = {84},
+  number = {1},
+  pages = {119--127},
+  year = {1990},
+  doi = {10.1016/0890-5401(90)90036-H},
+  url = {https://www.sciencedirect.com/science/article/pii/089054019090036H}
+}
+
+@article{CaiChenLi2010,
+  author = {Cai, Jin-Yi and Chen, Xi and Li, Dong},
+  title = {Quadratic lower bound for permanent vs. determinant in any characteristic},
+  journal = {Computational Complexity},
+  volume = {19},
+  number = {1},
+  pages = {37--56},
+  year = {2010},
+  doi = {10.1007/s00037-009-0284-2},
+  url = {https://pages.cs.wisc.edu/~jyc/papers/per-det.pdf}
+}
+
+@misc{Grenet2012,
+  author = {Grenet, Bruno},
+  title = {An Upper Bound for the Permanent versus Determinant Problem},
+  howpublished = {Author preprint},
+  year = {2012},
+  note = {25 May 2012},
+  url = {https://membres-ljk.imag.fr/Bruno.Grenet/publis/Gre11.pdf}
+}
+
+@article{MulmuleySohoni2001,
+  author = {Mulmuley, Ketan D. and Sohoni, Milind},
+  title = {Geometric Complexity Theory {I}: An Approach to the {P} vs. {NP} and Related Problems},
+  journal = {SIAM Journal on Computing},
+  volume = {31},
+  number = {2},
+  pages = {496--526},
+  year = {2001},
+  doi = {10.1137/S009753970038715X},
+  url = {https://epubs.siam.org/doi/10.1137/S009753970038715X}
+}
+
+@article{MulmuleySohoni2008,
+  author = {Mulmuley, Ketan D. and Sohoni, Milind},
+  title = {Geometric Complexity Theory {II}: Towards explicit obstructions for embeddings among class varieties},
+  journal = {SIAM Journal on Computing},
+  volume = {38},
+  number = {3},
+  pages = {1175--1206},
+  year = {2008},
+  doi = {10.1137/080718115},
+  url = {https://arxiv.org/abs/cs/0612134v1}
+}
+
+@misc{KumarVolk2026,
+  author = {Kumar, Mrinal and Volk, Ben Lee},
+  title = {A Quadratic Lower Bound on Determinantal Complexity},
+  howpublished = {\href{https://arxiv.org/abs/2609.34462v1}{arXiv:2609.34462v1}},
+  year = {2026},
+  note = {28 September 2026},
+  doi = {10.48550/arXiv.2609.34462},
+  url = {https://arxiv.org/abs/2609.34462v1}
+}
+
+@article{Galazka2017,
+  author = {Ga{\l}{\k a}zka, Maciej},
+  title = {Vector bundles give equations of cactus varieties},
+  journal = {Linear Algebra and its Applications},
+  volume = {521},
+  pages = {254--262},
+  year = {2017},
+  doi = {10.1016/j.laa.2016.12.005},
+  note = {Preprint: arXiv:1605.08005v1; numbered locators in the text refer to this version},
+  url = {https://arxiv.org/abs/1605.08005v1}
+}
+
+@misc{Baranovsky1998,
+  author = {Baranovsky, Vladimir},
+  title = {On Punctual {Quot} Schemes for Algebraic Surfaces},
+  howpublished = {\href{https://arxiv.org/abs/alg-geom/9703038v3}{arXiv:alg-geom/9703038v3}},
+  year = {1998},
+  note = {Version of 18 July 1998; first submitted 28 March 1997},
+  doi = {10.48550/arXiv.alg-geom/9703038},
+  url = {https://arxiv.org/abs/alg-geom/9703038v3}
+}
+
+@article{AdamsHarnadHurtubise1993,
+  author = {Adams, M. R. and Harnad, J. and Hurtubise, J.},
+  title = {Darboux Coordinates and {Liouville-Arnold} Integration in Loop Algebras},
+  journal = {Communications in Mathematical Physics},
+  volume = {155},
+  number = {2},
+  pages = {385--413},
+  year = {1993},
+  doi = {10.1007/BF02097398},
+  note = {Preprint: arXiv:hep-th/9210089v1; numbered locators in the text refer to this version},
+  url = {https://arxiv.org/abs/hep-th/9210089v1}
+}
+
+@inproceedings{Wampler1996,
+  author = {Wampler, Charles W.},
+  title = {Isotropic coordinates, circularity, and {Bezout} numbers: Planar kinematics from a new perspective},
+  booktitle = {ASME 1996 Design Engineering Technical Conferences},
+  year = {1996},
+  note = {Paper 96-DETC/MECH-1210},
+  doi = {10.1115/96-DETC/MECH-1210},
+  url = {https://academicweb.nd.edu/~cwample1/Preprints/DETC1996_MECH1210_IsotropicCircularity.pdf}
+}
+
+@article{Premet2003,
+  author = {Premet, Alexander},
+  title = {Nilpotent commuting varieties of reductive {Lie} algebras},
+  journal = {Inventiones Mathematicae},
+  volume = {154},
+  number = {3},
+  pages = {653--683},
+  year = {2003},
+  url = {https://eprints.maths.manchester.ac.uk/494/2/comm.pdf},
+  note = {\href{https://eprints.maths.manchester.ac.uk/494/2/comm.pdf}{Author manuscript}, Corollary~4.1}
+}
diff --git a/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/sections/adjugate.tex b/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/sections/adjugate.tex
new file mode 100644
index 0000000000000000000000000000000000000000..b21c75202144a3cfd15f6d83f814d8fc48b79b96
--- /dev/null
+++ b/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/sections/adjugate.tex
@@ -0,0 +1,309 @@
+\section{The adjugate scheme in characteristic two}
+\label{sec:adjugate}
+
+We now study the coefficient formula \eqref{eq:selector-coefficient} over
+\[
+ K=\overline{\mathbb F}_2.
+\]
+In this characteristic the permanent equals the determinant.  This lets
+us associate a plane curve and a finite adjugate scheme to every matrix
+\(X\).  Vanishing of the gradient will force this finite scheme to have
+large length.
+
+Fix an integer \(k\ge2\), pairwise distinct elements
+\(\theta_1,\ldots,\theta_k\in K\), and
+\(D=\operatorname{diag}(\theta_1,\ldots,\theta_k)\).  Set
+\begin{equation}
+ \label{eq:adjugate-parameters}
+ q=\left\lfloor\frac{k-1}{3}\right\rfloor,\qquad
+ r=k-2q,\qquad c=r-1,\qquad T=\binom{q+2}{2}.
+\end{equation}
+Thus
+\begin{equation}
+ \label{eq:adjugate-degrees}
+ 2q+c=k-1,\qquad 0\le q\le c.
+\end{equation}
+For a polynomial, brackets denote extraction of the indicated monomial
+coefficient.  We use the homogeneous degree-\(r\) polynomial
+\begin{equation}
+ \label{eq:adjugate-coefficient-polynomial}
+ G_D(X)=[s^qt^q]\per_k(X+sI_k+tD)
+       =[s^qt^q]\det(X+sI_k+tD).
+\end{equation}
+Statements about its gradient below refer to the common zero locus of
+all its \(k^2\) entrywise partial derivatives; no equation \(G_D=0\) is
+included in that definition.
+
+For each fixed \(X\in\Mat_k(K)\), put
+\begin{equation}
+ \label{eq:adjugate-curve}
+ A=tD+uX,\qquad M=sI_k+A,\qquad
+ F_X(s,t,u)=\det M,\qquad C_X=V(F_X)\subset\PP^2_K.
+\end{equation}
+Here \(F_X\) has degree \(k\), with coefficient \(1\) at \(s^k\).
+We abbreviate \(C_X\) to \(C\) when \(X\) is fixed.
+
+\begin{lemma}[Projection of the auxiliary curve]
+ \label{lem:adjugate-projection}
+For every \(X\in\Mat_k(K)\), the curve \(C_X\) is reduced and the map
+\[
+ \pi:C_X\longrightarrow\PP^1_K,\qquad [s:t:u]\longmapsto[t:u]
+\]
+is finite flat of degree \(k\) and is \'{e}tale over a nonempty open
+subset of the base.  Every irreducible component of \(C_X\) meets the
+line \(u=0\), and at each point of that intersection \(M\) has rank
+\(k-1\).  Moreover, \(\pi^*\cO_{\PP^1}(1)=\cO_C(1)\), and
+\begin{equation}
+ \label{eq:adjugate-splitting}
+ \pi_*\cO_C=\bigoplus_{j=0}^{k-1}\cO_{\PP^1}(-j),
+ \qquad
+ \mathcal E:=\pi_*\cO_C(k-1)
+       =\bigoplus_{j=0}^{k-1}\cO_{\PP^1}(k-1-j).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+The restriction
+\begin{equation}
+ \label{eq:adjugate-fixed-fiber}
+ F_X(s,t,0)=\prod_{i=1}^k(s+t\theta_i)
+\end{equation}
+is squarefree.  If a positive-degree irreducible homogeneous factor
+\(H\) occurred at least twice in \(F_X\), its restriction to \(u=0\)
+could not vanish identically: otherwise \(u\) would divide \(F_X\),
+contrary to \eqref{eq:adjugate-fixed-fiber}.  That restriction is a
+nonzero homogeneous polynomial of the same positive degree as \(H\).
+Its square would divide \eqref{eq:adjugate-fixed-fiber}, a contradiction.
+Thus \(C\) is reduced.  The restriction of each irreducible factor is
+likewise a positive-degree binary form, so it has a projective zero
+over \(K\).  Hence every component meets \(u=0\).
+
+The points on that line are
+\[
+ P_i=[-\theta_i:1:0]\qquad(1\le i\le k).
+\]
+At \(P_i\), the matrix \(M\) is diagonal with exactly one zero entry.
+It has rank \(k-1\), and its \(i\)-th diagonal adjugate entry is
+\(\prod_{j\ne i}(\theta_j-\theta_i)\ne0\).
+
+The only possible base point of the projection is \([1:0:0]\), which
+does not belong to \(C\) because \(F_X(1,0,0)=1\).
+On the base chart \(t\ne0\), use \(v=u/t\) and \(y=s/t\).  The inverse
+image has coordinate algebra
+\[
+ K[v,y]\big/\det(yI_k+D+vX).
+\]
+Its defining polynomial is monic of degree \(k\) in \(y\); consequently
+this algebra is free over \(K[v]\), with basis
+\(1,y,\ldots,y^{k-1}\).  The chart \(u\ne0\) gives the same conclusion,
+so \(\pi\) is finite flat of degree \(k\).
+At \(v=0\) the monic polynomial has the distinct roots
+\(-\theta_1,\ldots,-\theta_k\).
+Its resultant with its \(y\)-derivative is nonzero there, and hence
+nonzero on an open subset of the base.  Over that open subset the
+finite flat map is \'{e}tale.  In particular its generic algebra is
+a product of finite separable field extensions, also when \(k\) is even.
+
+The sections \(t,u\) have no common zero on \(C\), so they identify
+\(\pi^*\cO_{\PP^1}(1)\) with \(\cO_C(1)\).
+The two monic-polynomial bases satisfy
+\[
+ (s/u)^j=(t/u)^j(s/t)^j\qquad(0\le j<k)
+\]
+on their overlap.  These are exactly the transition functions of
+\(\cO_{\PP^1}(-j)\).  This proves the first splitting in
+\eqref{eq:adjugate-splitting}; tensoring by \(\cO_{\PP^1}(k-1)\)
+and using the projection formula proves the second.
+\end{proof}
+
+The fixed distinct-root fiber has established reducedness and a finite
+flat, generically separable projection.  We next use the adjugate to turn a vanished
+coefficient into a lower bound on the length of a finite scheme.
+
+Each entry of \(\adj M\) is a homogeneous form of degree \(k-1\).
+Define its ideal on \(C\) intrinsically by
+\begin{equation}
+ \label{eq:adjugate-ideal}
+ J=\operatorname{im}\bigl(\cO_C(1-k)^{k^2}
+                  \xrightarrow{\ \text{entries of }\adj M\ }\cO_C\bigr),
+ \qquad Z=V_C(J).
+\end{equation}
+Thus a local trivialization of \(\cO_C(k-1)\) turns the adjugate
+entries into generators of \(J\).  Changing the trivialization
+multiplies all generators by a unit and leaves the ideal unchanged.
+By Lemma~\ref{lem:adjugate-projection}, each component of \(C\)
+contains a point at which an adjugate entry is nonzero.  The support
+of \(\cO_C/J\) therefore contains no component of \(C\), so \(Z\)
+is a finite scheme.  It may have nonreduced structure.
+
+\begin{lemma}[Generation by adjugate entries]
+ \label{lem:adjugate-generation}
+Every homogeneous form of degree \(k-1\) that vanishes
+scheme-theoretically on \(Z\) is a constant \(K\)-linear combination
+of the entries of \(\adj M\).
+\end{lemma}
+
+\begin{proof}
+All sheaves on the base in this proof are on \(\PP^1_K\).
+The adjugate entries define a morphism
+\[
+ \psi:\Mat_k(\cO)\longrightarrow\mathcal E,\qquad
+       Y\longmapsto\tr(Y\adj M).
+\]
+The trace pairs matrix entries in transposed order, so its image is
+the \(\cO\)-span of all adjugate entries.  We first show that
+\begin{equation}
+ \label{eq:adjugate-image}
+ \operatorname{im}\psi=\pi_*(J(k-1)).
+\end{equation}
+Work on either standard affine chart of the base and trivialize
+\(\cO(1)\).  Write its coordinate ring as \(R\), the affine
+\(s\)-coordinate as \(y\), and the corresponding matrix as \(A_0\).
+The curve algebra is \(B=R[y]/(f)\), where
+\(f=\det(yI_k+A_0)\).  Let \(Q\) be the \(R\)-span in \(B\) of the
+entries of \(\adj(yI_k+A_0)\).
+The adjugate identity gives, in \(B\),
+\[
+ y\adj(yI_k+A_0)=-A_0\adj(yI_k+A_0).
+\]
+Thus \(Q\) is stable under multiplication by \(y\).  Since \(B\) is
+generated by \(y\) over \(R\), \(Q\) is already the \(B\)-ideal
+generated by these entries.  Restoring the local trivialization
+proves \eqref{eq:adjugate-image} on both charts.
+
+Put \(\mathcal K=\ker\psi\).  To lift every global section of
+\(\operatorname{im}\psi\) to a constant matrix, it is enough to prove
+\(H^1(\mathcal K)=0\).  We obtain this vanishing by comparing
+\(\mathcal K\) with the image of the commutator with \(A\).
+
+The matrix \(A=tD+uX\) has entries in \(\cO(1)\), so the commutator
+defines the morphism
+\[
+ \chi:\Mat_k(\cO(-1))\longrightarrow\Mat_k(\cO),\qquad
+       Y\longmapsto AY-YA.
+\]
+The adjugate commutes with \(A\), and cyclicity of trace gives
+\(\psi\chi=0\).  Put \(\mathcal I=\operatorname{im}\chi\); then
+\(\mathcal I\subset\mathcal K\).
+
+At the base point \([t:u]=[1:0]\), the commutator is \([D,-]\).
+On the matrix unit \(E_{ij}\) it acts by multiplication by
+\(\theta_i-\theta_j\).  These multipliers are nonzero for \(i\ne j\),
+including in characteristic two.  Hence the commutator has rank
+\(k^2-k\) at that point.
+At the same point, the images under \(\psi\) of the diagonal matrix
+units are the classes of
+\[
+ h_i(y)=\prod_{j\ne i}(y+\theta_j)
+ \quad\text{in}\quad K[y]\big/\prod_j(y+\theta_j).
+\]
+Evaluation at the \(k\) distinct roots gives a diagonal matrix with
+nonzero diagonal, so these classes are independent and \(\psi\)
+has rank \(k\), the rank of \(\mathcal E\).  Its generic rank is
+therefore \(k\), and its generic kernel has dimension \(k^2-k\).
+The rank of \(\chi\) at \([1:0]\) shows that its generic rank is at
+least \(k^2-k\), while \(\mathcal I\subset\mathcal K\) gives the
+opposite inequality.  Thus \(\mathcal I\) and \(\mathcal K\) have
+equal generic rank.
+The quotient \(\mathcal K/\mathcal I\) is therefore a coherent
+torsion sheaf on \(\PP^1\).
+
+We now use two exact sequences to compute \(H^1(\mathcal K)\).
+The first is
+\[
+ 0\longrightarrow\ker\chi\longrightarrow
+ \cO(-1)^{k^2}\longrightarrow\mathcal I\longrightarrow0.
+\]
+Since \(H^1(\PP^1,\cO(-1))=0\) and coherent \(H^2\) vanishes on
+\(\PP^1\), its cohomology sequence gives \(H^1(\mathcal I)=0\).
+These two facts follow, for example, from the usual computation
+with the two standard affine charts of \(\PP^1\).
+Next, a coherent torsion sheaf on \(\PP^1\) has finite support and
+vanishing \(H^1\).  The sequence
+\[
+ 0\longrightarrow\mathcal I\longrightarrow\mathcal K
+ \longrightarrow\mathcal K/\mathcal I\longrightarrow0
+\]
+therefore gives \(H^1(\mathcal K)=0\).
+Finally, the sequence
+\[
+ 0\longrightarrow\mathcal K\longrightarrow\cO^{k^2}
+ \longrightarrow\operatorname{im}\psi\longrightarrow0
+\]
+shows that constant matrices surject onto
+\[
+ H^0(\PP^1,\operatorname{im}\psi)=H^0(C,J(k-1)),
+\]
+where the equality uses \eqref{eq:adjugate-image} and finite
+pushforward.
+
+Let \(P\) be a degree-\(k-1\) homogeneous form vanishing on the
+scheme \(Z\).  Its restriction to \(C\) is a section of \(J(k-1)\),
+so the preceding surjectivity expresses it on \(C\) as a constant
+linear combination of the adjugate entries.
+Their difference is a degree-\(k-1\) form vanishing on \(C\).
+Since the homogeneous ideal of this plane hypersurface is generated
+by \(F_X\), of degree \(k\), the difference is zero.
+Thus the equality holds as an equality of homogeneous forms.
+\end{proof}
+
+\begin{proposition}[Length forced by the gradient]
+ \label{prop:adjugate-length}
+With the parameters in \eqref{eq:adjugate-parameters}, let
+\(X\in\Mat_k(K)\) satisfy \(\nabla G_D(X)=0\).
+Then its adjugate zero scheme \(Z\) satisfies
+\[
+ \length Z\ge T=\binom{q+2}{2}.
+\]
+\end{proposition}
+
+\begin{proof}
+Homogeneity and entrywise differentiation give
+\[
+ G_D(X)=[s^qt^qu^r]F_X,\qquad
+ \frac{\partial F_X}{\partial X_{ij}}=u(\adj M)_{ji}.
+\]
+Thus the gradient hypothesis is precisely the entrywise condition
+\begin{equation}
+ \label{eq:adjugate-coefficient-zero}
+ [s^qt^qu^c]\adj M=0,\qquad c=r-1.
+\end{equation}
+This calculation uses no division by the degree \(r\).
+By Lemma~\ref{lem:adjugate-generation}, the functional
+\(\Lambda=[s^qt^qu^c]\) consequently vanishes on every degree-\(k-1\)
+form that vanishes scheme-theoretically on \(Z\).
+
+The next comparison is the line-bundle case of Ga{\l}{\k a}zka's
+catalecticant bound under this annihilation condition.  It uses the
+multiplication map \(P\mapsto(Q\mapsto\Lambda(PQ))\), where \(P\) and
+\(Q\) have degrees \(q\) and \(k-1-q\), respectively
+\cite[Theorem~4 and Propositions~6, 7, and~9 in arXiv:1605.08005v1]{Galazka2017}.
+The complementary monomials below compute this map directly over \(K\).
+
+Suppose that a degree-\(q\) form
+\[
+ P=\sum_{i+j+\ell=q}a_{ij\ell}s^it^ju^\ell
+\]
+vanishes on \(Z\).  For each triple in this sum, the monomial
+\[
+ s^{q-i}t^{q-j}u^{c-\ell}
+\]
+has nonnegative exponents because \(i,j,\ell\le q\le c\).
+Its product with \(P\) still vanishes on \(Z\), and has degree
+\(2q+c=k-1\).  Exactly one term contributes to its selected
+coefficient, giving
+\[
+ 0=\Lambda\bigl(s^{q-i}t^{q-j}u^{c-\ell}P\bigr)=a_{ij\ell}.
+\]
+Hence \(P=0\).  Restriction therefore induces an injection
+\[
+ H^0(\PP^2,\cO(q))\lhook\joinrel\longrightarrow
+ H^0(Z,\cO_Z(q)).
+\]
+The space on the left has dimension \(\binom{q+2}{2}\).
+On a finite scheme over \(K\), an invertible sheaf is free of rank
+one on each Artin local component, so its space of global sections
+has dimension equal to the scheme's length.  The space on the right
+thus has dimension \(\length Z\), including all nonreduced
+multiplicities.  The stated inequality follows.
+\end{proof}
diff --git a/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/sections/conductor.tex b/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/sections/conductor.tex
new file mode 100644
index 0000000000000000000000000000000000000000..20875a60dfb282246de670aee81de73fca29d3ca
--- /dev/null
+++ b/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/sections/conductor.tex
@@ -0,0 +1,253 @@
+\section{The conductor and the normalization defect}
+\label{sec:conductor}
+
+We next bound the length of the adjugate subscheme by the singularities
+of the plane curve.  For a reduced projective curve \(C\), let
+\(\nu:\widetilde C\to C\) be its normalization and set
+\begin{equation}
+ \delta(C)=
+ \dim_K H^0\bigl(C,\nu_*\cO_{\widetilde C}/\cO_C\bigr).
+ \label{eq:conductor-delta}
+\end{equation}
+The normalization may be disconnected.  The comparison below follows the
+evaluation-ideal argument of Fantechi--G\"ottsche--van Straten
+\cite[Proposition C.2 and Corollary C.3]{FGvS1999}.
+Their setting is over \(\C\); we give the local proof needed here in
+arbitrary characteristic.
+
+\begin{lemma}
+\label{lem:conductor-local}
+Let \(k\ge1\) be an integer, let \(K\) be an algebraically closed field,
+and let \(S\) be a discrete
+valuation ring containing \(K\), with residue field \(K\) and fraction
+field \(Q\).  For \(A_0\in\Mat_k(S)\), put
+\[
+ M_0=yI+A_0,\qquad f=\det M_0,\qquad B=S[y]/(f).
+\]
+Suppose that \(f\) is separable over \(Q\).  Let \(R\) be the normalization
+of \(B\) in \(B_Q=B\otimes_S Q\), let
+\[
+ \mathfrak c=\{a\in B:aR\subset B\},
+\]
+and let \(J_0\subset B\) be the ideal generated by the entries of
+\(\adj M_0\).  Then
+\begin{equation}
+ \mathfrak c\subset J_0,\qquad
+ \length_S(B/J_0)\le
+ \length_S(B/\mathfrak c)=\length_S(R/B).
+ \label{eq:conductor-local-length}
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Monicity makes \(B\) free over \(S\), with basis
+\(1,y,\ldots,y^{k-1}\).  In particular, it embeds into \(B_Q\), a product
+of finite separable field extensions of \(Q\), and is reduced.  The integral closures of
+\(S\) and \(B\) in this algebra agree, since \(B\) is integral over \(S\).
+They are finite over \(S\).  One way to see the finiteness directly is to
+use the sum of the field trace pairings on the factors of \(B_Q\).
+This pairing is nondegenerate, so the lattice dual to \(B\) is finite
+free over \(S\).  If \(z\in B_Q\) is integral over \(S\), then each
+\(zy^j\), for \(0\le j<k\), is integral because \(y\) is integral.
+Its trace lies in the integrally closed ring \(S\), so \(z\) belongs
+to the dual lattice.
+The normalization is an \(S\)-submodule of that finite lattice and is
+therefore finite.  The same finiteness conclusion follows by applying
+the normalization theorem \cite[Tag 032L]{StacksProject} to each field
+factor.
+
+It follows that \(R\) is semilocal.  Its localizations at maximal ideals
+are one-dimensional Noetherian normal local domains, hence discrete
+valuation rings \cite[Tag 00PD]{StacksProject}.  All their residue fields
+are \(K\).  As a finite torsion-free \(S\)-module, \(R\) is free of
+rank \(k\), and \(R/B\) has finite length.
+
+\medskip
+\noindent
+\emph{The adjugate ideal as an evaluation ideal.}
+Consider the \(B\)-module
+\[
+ L=\operatorname{coker}(M_0:B^k\longrightarrow B^k).
+\]
+The same cokernel over \(S[y]\) is \(S^k\), with \(y\) acting as
+\(-A_0\): polynomial-vector division by the monic matrix \(yI+A_0\)
+reduces every class uniquely to a constant vector.  The adjugate identity
+shows that \(f\) already kills this cokernel.  Thus \(L\) is free of
+rank \(k\) over \(S\).
+
+Over a splitting field of \(f\), the action of \(-A_0\) has \(k\)
+distinct eigenvalues, each with a one-dimensional eigenspace.
+Consequently \(L_Q\) is rank one over every field factor of \(B_Q\),
+and \(L_Q\simeq B_Q\) as a \(B_Q\)-module.  Fixing such an isomorphism
+embeds the \(S\)-free module \(L\) as a lattice in \(B_Q\).
+
+Define its evaluation ideal by
+\[
+ \tau_B(L)=
+ \operatorname{im}\left(
+ \operatorname{Hom}_B(L,B)\otimes_B L\longrightarrow B
+ \right).
+\]
+A homomorphism \(L\to B\) is a row annihilating \(M_0\) modulo \(f\).
+Lift the row to \(b\in S[y]^{1\times k}\), so that
+\(bM_0=fb'\) for another polynomial row \(b'\).
+Multiplication by the adjugate gives
+\[
+ fb=fb'\adj M_0.
+\]
+We may cancel the nonzero \(f\) in the domain \(S[y]\), obtaining
+\(b=b'\adj M_0\).  Conversely, each adjugate row annihilates \(M_0\)
+modulo \(f\).  Evaluation on the standard generators of \(L\) therefore
+gives
+\begin{equation}
+ \tau_B(L)=J_0.
+ \label{eq:conductor-evaluation}
+\end{equation}
+
+\medskip
+\noindent
+\emph{The conductor is contained in the evaluation ideal.}
+The fractional ideal \(I=RL\subset B_Q\) is free of rank one at every
+maximal localization of \(R\).  We claim that some \(h\in L\) generates
+it at all these localizations at once.  Choose \(S\)-generators
+\(l_1,\ldots,l_k\) of \(L\).  At a maximal ideal \(\mathfrak m\) of \(R\),
+their images span the one-dimensional \(K\)-space
+\(I_{\mathfrak m}/\mathfrak m I_{\mathfrak m}\).  The coefficients
+\((a_1,\ldots,a_k)\in K^k\) for which \(\sum_i a_i l_i\) has zero
+image form a proper linear subspace.  There are only finitely many such
+maximal ideals.  Since \(K\) is infinite, we can avoid all these subspaces.
+The resulting \(h=\sum_i a_i l_i\in L\) generates every \(I_{\mathfrak m}\),
+and hence \(I=hR\).
+
+In particular, \(h\) is nonzero in every field factor of \(B_Q\).
+The rescaled module \(L'=h^{-1}L\) satisfies
+\[
+ B\subset L'\subset R:
+\]
+it contains \(1\), and \(L\subset RL=hR\).
+For \(a\in\mathfrak c\), multiplication by \(a\) defines a homomorphism
+\(L'\to B\) whose value at \(1\) is \(a\).
+Evaluation ideals are invariant under module isomorphism, so
+\[
+ \mathfrak c\subset\tau_B(L')
+ =\tau_B(L)=J_0.
+\]
+
+We have identified the adjugate ideal with an evaluation ideal and shown
+that it contains the conductor. It remains to compute the conductor
+colength in terms of the normalization.
+
+\medskip
+\noindent
+\emph{The conductor colength.}
+Let \(\lambda:B\to S\) extract the coefficient of \(y^{k-1}\) in
+the monic remainder, and consider
+\begin{equation}
+ \Phi:B\longrightarrow B^\vee:=\operatorname{Hom}_S(B,S),
+ \qquad
+ \Phi(a)(b)=\lambda(ab).
+ \label{eq:conductor-pairing}
+\end{equation}
+This is \(B\)-linear for the action
+\((a\cdot\ell)(b)=\ell(ab)\) on \(B^\vee\).
+In the basis \(1,y,\ldots,y^{k-1}\), its matrix has entry
+\(\lambda(y^{i+j})\) in position \((i,j)\).
+That entry is zero when \(i+j<k-1\), and is one when \(i+j=k-1\).
+Reversing the columns therefore gives a triangular matrix with diagonal
+entries one.  Thus \(\Phi\) is an isomorphism in every characteristic.
+The trace pairing above was used over \(Q\) to prove finiteness of \(R\).
+This coefficient pairing supplies the self-duality over \(S\) used below.
+
+The adjunction
+\[
+ \operatorname{Hom}_S(R,S)
+ \simeq
+ \operatorname{Hom}_B(R,\operatorname{Hom}_S(B,S))
+\]
+sends \(\ell\) to the map
+\(x\mapsto[b\mapsto\ell(bx)]\).
+Using \(\Phi\), it identifies \(\operatorname{Hom}_S(R,S)\) with
+\(\operatorname{Hom}_B(R,B)\).
+Every homomorphism in the latter module becomes a \(B_Q\)-linear
+endomorphism of \(B_Q\), hence multiplication by some \(a\in B_Q\).
+It takes \(R\) into \(B\) exactly when \(aR\subset B\).
+Since \(1\in R\), this forces \(a\in B\), and the possible multipliers
+are precisely \(\mathfrak c\).
+
+This identification respects restriction to \(B\): for a multiplier \(a\),
+the corresponding functional restricts to
+\(b\mapsto\lambda(ab)=\Phi(a)(b)\).
+Hence restriction
+\begin{equation}
+ R^\vee=\operatorname{Hom}_S(R,S)\longrightarrow B^\vee
+ \quad\hbox{has image }\Phi(\mathfrak c).
+ \label{eq:conductor-dual-image}
+\end{equation}
+To compute its colength, choose bases for the two free \(S\)-lattices
+in which the inclusion \(B\subset R\) has elementary divisors
+\(\pi^{n_1},\ldots,\pi^{n_k}\), where \(\pi\) is a uniformizer and
+the \(n_i\) are nonnegative.  The quotient \(R/B\) has length
+\(\sum_i n_i\), and the restriction on dual lattices has the same
+elementary divisors.  By \eqref{eq:conductor-pairing} and
+\eqref{eq:conductor-dual-image}, its cokernel is isomorphic to
+\(B/\mathfrak c\).  This proves
+\[
+ \length_S(B/\mathfrak c)=\length_S(R/B).
+\]
+The inclusion \(\mathfrak c\subset J_0\) now gives
+\eqref{eq:conductor-local-length}.
+\end{proof}
+
+\begin{proposition}
+\label{prop:conductor-length}
+For each curve \(C=C_X\) and adjugate subscheme \(Z\) constructed in
+Section~\ref{sec:adjugate},
+\begin{equation}
+ \length Z\le\delta(C).
+ \label{eq:conductor-global-length}
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+By Lemma~\ref{lem:adjugate-projection}, the projection
+\(\pi:C\to\PP^1_K\) is finite and flat, with a distinct-root fiber.
+It is therefore generically separable.  At a closed point \(b\) of the
+base, take \(S=\cO_{\PP^1,b}\) and trivialize \(\cO(1)\).
+If \(y\) is the resulting \(s\)-coordinate and \(A_0\) represents
+\(tD+uX\), the curve algebra is
+\[
+ B=S[y]/\bigl(\det(yI+A_0)\bigr).
+\]
+The stalk of \(\pi_*(\cO_C/J)\) is \(B/J_0\), with \(J_0\) as in
+Lemma~\ref{lem:conductor-local}.  Its normalization \(R\) gives the
+stalk \(R/B\) of
+\(\pi_*(\nu_*\cO_{\widetilde C}/\cO_C)\).
+Indeed, normalization commutes with this localization.
+Changes of trivialization multiply the adjugate generators by units,
+so these are the intrinsic ideals and quotient sheaves.
+
+Both quotient sheaves have finite support.  At every closed point of
+the base, their finite-length \(S\)-modules have length equal to their
+\(K\)-dimension, because the residue field is \(K\).
+Equivalently, each such length is the sum of the lengths at the curve
+points above that base point; no ramification factor enters.
+Summing \eqref{eq:conductor-local-length} over \(b\) therefore gives
+\[
+ \length Z
+ =\sum_b\length_{\cO_{\PP^1,b}}(B/J_0)
+ \le
+ \sum_b\length_{\cO_{\PP^1,b}}(R/B)
+ =\delta(C).
+\]
+Only generic separability was needed.  Nonreduced closed fibers and
+wild ramification are allowed by the coefficient pairing
+\eqref{eq:conductor-pairing}.
+\end{proof}
+
+In particular, Proposition~\ref{prop:adjugate-length} yields
+\begin{equation}
+ \nabla G_D(X)=0
+ \quad\Longrightarrow\quad
+ \delta(C_X)\ge T.
+ \label{eq:conductor-gradient-defect}
+\end{equation}
diff --git a/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/sections/consequences.tex b/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/sections/consequences.tex
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@@ -0,0 +1,134 @@
+\section{The permanent lower bound and its consequences}
+\label{sec:consequences}
+
+Proposition~\ref{prop:smooth-projection} supplies a smooth form with an
+exact affine permanent projection. Combining it with
+Theorem~\ref{thm:smooth-border} proves the main lower bound.
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+Let \(m\ge1408\) be an integer and put \(k=\lfloor m/11\rfloor\).
+Then \(k\ge128\).  With \(q=\lfloor(k-1)/3\rfloor\), we have
+\(q\ge0\) and
+\[
+ m\ge11k\ge11k-2q.
+\]
+Proposition~\ref{prop:smooth-projection} supplies a nonzero homogeneous
+form \(G\) of degree \(r\) in \(d\) variables, with smooth projective
+zero set, and a fixed affine map \(\Phi_m\) such that
+\(G=\per_m\circ\Phi_m\).  Lemma~\ref{lem:affine-substitution} and
+Theorem~\ref{thm:smooth-border} give
+\[
+ \bdc(\per_m)\ge\bdc(G)
+ \ge\frac{(r-1)(d-1)}{4e}
+ \ge\frac{k^3}{3200e},
+\]
+using the two scale bounds in Proposition~\ref{prop:smooth-projection}.
+The affine substitution is fixed before any determinant approximation
+is chosen, so this step applies to the unrestricted coefficientwise
+border model.
+
+Finally,
+\[
+ k\ge\frac{m}{11}-1\ge\frac{m}{12},
+\]
+where the last inequality holds for \(m\ge132\).  Therefore
+\[
+ \bdc(\per_m)\ge\frac{m^3}{5\,529\,600e},
+\]
+which proves the stated choices of \(c\) and \(m_0\).
+\end{proof}
+
+\subsection{Determinantal complexity and branching programs}
+
+The border lower bound gives the same estimate for exact determinant
+representations. A direct determinant construction also transfers it to
+algebraic branching programs and their fixed-budget limits.
+
+\begin{corollary}[Exact determinantal complexity]
+\label{cor:exact-complexity}
+For every integer $m\ge1408$,
+\[
+ \dc(\per_m)\ge \frac{m^3}{5\,529\,600e}.
+\]
+\end{corollary}
+\begin{proof}
+An exact determinant representation is also a constant sequence of
+approximating determinants. Hence $\bdc(f)\le\dc(f)$ for every $f$, and
+Theorem~\ref{thm:main} applies.
+\end{proof}
+
+\begin{corollary}[Affine-linear algebraic branching programs]
+\label{cor:abp-complexity}
+Over $\C$, an algebraic branching program is a finite directed acyclic
+graph with distinct designated source $s$ and sink $t$, whose edges carry
+affine-linear forms; its output is the sum of the commutative products of
+labels along all directed $s$-to-$t$ paths. Such a program computing
+$\per_m$ requires $\Omega(m^3)$ vertices and $\Omega(m^3)$ edges, even after
+trimming to the vertices and edges on source-to-sink paths. The same
+bounds hold if $\per_m$ is a coefficientwise (equivalently, Zariski)
+limit of outputs with a fixed vertex budget, respectively a fixed edge
+budget, along the approximation.
+\end{corollary}
+\begin{proof}
+Let a program with $N$ vertices have output $f$. Sum parallel-edge labels
+and topologically order the vertices. With adjacency convention
+$A_{v,u}=\text{label}(u\to v)$, the matrix $A$ is strictly triangular,
+$\det(I-A)=1$, and $(I-A)^{-1}=I+A+\cdots+A^{N-1}$. The path-sum formula
+and the block determinant identity give
+\[
+ f=e_t^{\mathsf T}(I-A)^{-1}e_s
+  =\det\begin{pmatrix}I-A&e_s\\-e_t^{\mathsf T}&0\end{pmatrix}.
+\]
+This is an affine-linear determinant of size $N+1$. Padding with isolated
+vertices covers all programs with at most $N$ vertices. Taking closures
+therefore gives $\bdc(f)\le N+1$ for every output in the fixed-budget
+border model, even when the graph varies. After summing parallel edges,
+there are finitely many graph types for a fixed vertex budget, and each
+output depends polynomially on the affine-label coefficients. Hence the
+output set is constructible, so its two closures agree by the argument
+in Section~\ref{sec:models}.
+
+For a nonzero output, trim away all vertices and edges outside
+source-to-sink paths. The remaining underlying graph is connected, so
+$N\le E+1$, where $E$ is its edge count. Thus $\bdc(f)\le E+2$.
+For a fixed-edge-budget approximation to the nonzero permanent, discard
+any zero outputs and trim each remaining program separately; the same
+uniform determinantal bound passes to the limit. The finite graph-type
+argument also applies to this trimmed edge-budget model.
+Theorem~\ref{thm:main} now yields both cubic lower bounds.
+\end{proof}
+
+\subsection{Smooth initial forms}
+
+Theorem~\ref{thm:smooth-border} also applies locally to a polynomial that
+need not be homogeneous. Its first nonzero homogeneous term can be obtained
+as a coefficientwise limit of translated and rescaled copies of that polynomial.
+
+\begin{corollary}[Smooth initial form]
+\label{cor:smooth-initial-form}
+Let $g\in\C[x_1,\ldots,x_d]$, where $d\ge2$, and let $a\in\C^d$.
+Suppose that the lowest nonzero homogeneous term $G_r(x)$ of $g(a+x)$
+has degree $r\ge2$ and defines a smooth hypersurface in $\PP^{d-1}$.
+Then
+\[
+ \bdc(g)\ge\frac{(r-1)(d-1)}{4e}.
+\]
+The same lower bound holds for $\dc(g)$.
+\end{corollary}
+\begin{proof}
+Fix a size $n$ for which $g$ belongs to the coefficientwise closure
+$\mathcal C_n$ of affine determinants of size $n$. Then $\deg g\le n$,
+and $\mathcal C_n$ is closed in the space of polynomials of degree at
+most $n$. For every fixed $t\in\C\setminus\{0\}$,
+Lemma~\ref{lem:affine-substitution} gives
+\[
+ t^{-r}g(a+tx)\in\mathcal C_n.
+\]
+Writing $g(a+x)=\sum_{j=r}^{\deg g}G_j(x)$ as a sum of homogeneous
+terms, the polynomial on the left is
+$\sum_{j=r}^{\deg g}t^{j-r}G_j(x)$. It converges coefficientwise to
+$G_r$ as $t\to0$. Closedness therefore gives $G_r\in\mathcal C_n$.
+Theorem~\ref{thm:smooth-border} now bounds this same size $n$ below by
+$(r-1)(d-1)/(4e)$. Taking the least admissible $n$ proves the border
+assertion, and $\dc(g)\ge\bdc(g)$ proves the exact assertion.
+\end{proof}
diff --git a/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/sections/dimension.tex b/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/sections/dimension.tex
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@@ -0,0 +1,367 @@
+\section{From curve defects to matrix codimension}
+\label{sec:dimension}
+
+The preceding sections show that a critical matrix of the coefficient
+polynomial produces a reduced plane curve with large normalization defect.
+We now count these matrices by separating the choice of the curve from the
+choice of a matrix that produces it.
+
+\begin{samepage}
+\begin{proposition}[Gradient codimension]\label{prop:gradient-codimension}
+Let $K=\overline{\mathbb F}_2$, let $k\geq2$, and set
+\[
+ q=\left\lfloor\frac{k-1}{3}\right\rfloor,
+ \qquad T=\binom{q+2}{2}.
+\]
+For pairwise distinct $\theta_1,\ldots,\theta_k\in K$, put
+$D=\operatorname{diag}(\theta_1,\ldots,\theta_k)$ and
+\[
+ G_D(X)=[s^qt^q]\per_k(X+sI+tD),
+ \qquad X\in\Mat_k(K).
+\]
+Then
+\begin{equation}
+ \dim\{X\in\Mat_k(K):\nabla G_D(X)=0\}
+ \leq k^2+k-T.
+ \label{eq:gradient-dimension}
+\end{equation}
+\end{proposition}
+\end{samepage}
+
+For the proof, fix these parameters and write
+\[
+ \Gamma_D=\{X\in\Mat_k(K):\nabla G_D(X)=0\}.
+\]
+Consider the morphism
+\begin{equation}
+ \label{eq:determinant-curve-map}
+ f_D:\Mat_k(K)\longrightarrow
+ \PP H^0(\PP^2_K,\cO(k)),\qquad
+ X\longmapsto[F_X],\qquad
+ F_X=\det(sI+tD+uX).
+\end{equation}
+The coefficient of $s^k$ is one, so a projective fiber fixes the
+polynomial $F_X$ itself and hence the embedded curve $C_X$.
+Every curve in the image is reduced by
+Lemma~\ref{lem:adjugate-projection}.  For $X\in\Gamma_D$,
+Propositions~\ref{prop:adjugate-length} and~\ref{prop:conductor-length}
+give
+\begin{equation}
+ \label{eq:dimension-gradient-defect}
+ T\leq\length Z\leq\delta(C_X).
+\end{equation}
+We will bound the dimension of $f_D(\Gamma_D)$ using this defect, then
+bound the fibers of $f_D$ by associating a finite subscheme of the fixed
+curve to each matrix in the fiber.
+
+\subsection{Curves with large normalization defect}
+
+Recall that, for a reduced curve $C$ with normalization
+$\nu:\widetilde C\to C$,
+\[
+ \delta(C)=\length(\nu_*\cO_{\widetilde C}/\cO_C).
+\]
+The normalization may be disconnected.  We will keep track of its
+components explicitly when applying the dimension theorem for integral
+curves.
+
+Over any algebraically closed field, let $b\geq1$ and
+$0\leq g\leq(b-1)(b-2)/2$ be integers.  The locus of integral plane
+curves of degree $b$ and normalization genus $g$ is locally closed and,
+when nonempty, has dimension $3b+g-1$
+\cite[Lemma~2.6, Proposition~2.7, and Remark~2.8]{CHT2023}.
+This is the cited theorem with surface $\PP^2$ and line bundle $\cO(b)$.
+The locus includes every integral curve of the stated genus: the surface
+has no singular points to exclude, and nodality is not a hypothesis.
+
+\begin{lemma}
+Let $K$ be an algebraically closed field\label{lem:severi-dimension}, and let $k\geq1$ and $T\geq0$
+be integers.  In the degree-$k$ plane linear system over $K$, the locus
+of reduced curves satisfying $\delta(C)\geq T$ is constructible.
+If nonempty, it has dimension at most
+\[
+ \frac{k(k+3)}2-T.
+\]
+\end{lemma}
+
+\begin{proof}
+Write $C=C_1\cup\cdots\cup C_a$, where the integral components have
+degrees $b_i$ and normalization genera $g_i$.  The Euler characteristics
+of the normalization and the curve are $a-\sum_i g_i$ and $1-p_a(C)$.
+The normalization exact sequence therefore gives
+\[
+ \delta(C)=p_a(C)-\sum_{i=1}^a g_i+a-1,
+ \qquad p_a(C)=\frac{(k-1)(k-2)}2.
+\]
+In particular, the term $a-1$ accounts for the disconnected normalization,
+and the identity includes the contribution of intersections between
+different components.
+
+For fixed degrees and genera, multiplying the equations of the
+components defines a morphism from the product of their integral curve
+loci to the degree-$k$ linear system.  Restricting to distinct factors
+gives precisely the reduced curves of this type.  The image is
+constructible, and its dimension is at most
+\begin{align*}
+ \sum_{i=1}^a(3b_i+g_i-1)
+ &=3k+\sum_{i=1}^a g_i-a\\
+ &=\frac{k(k+3)}2-\delta(C).
+\end{align*}
+There are only finitely many degree and genus types:
+$a\leq k$, $\sum b_i=k$, and
+$0\leq g_i\leq(b_i-1)(b_i-2)/2$.
+The normalization identity determines $\delta(C)$ from each type.
+The required locus is therefore a finite union of constructible images
+with the stated dimension bound.
+\end{proof}
+
+\subsection{A row subscheme determines a matrix}
+
+Fix a curve $C$ in the image of $f_D$, and let
+$\mathcal F_C=f_D^{-1}([C])$.  Our remaining task is to bound this
+matrix fiber.  For a critical matrix, its full-adjugate scheme $Z$
+supplied the defect bound \eqref{eq:dimension-gradient-defect} on its
+curve.  We now use a row obtained from the adjugate to distinguish all
+matrices in the fixed fiber $\mathcal F_C$.
+
+Put $N=k(k-1)/2$.  For $X\in\mathcal F_C$, define
+\[
+ R_X=(1,\ldots,1)\adj(sI+tD+uX),
+\]
+and define an ideal and its subscheme on the fixed curve by
+\begin{equation}
+ \label{eq:row-adjugate-ideal}
+ J'_X=\operatorname{im}\bigl(
+ \cO_C(1-k)^k\xrightarrow{\ R_X\ }\cO_C\bigr),
+ \qquad Z'_X=V_C(J'_X).
+\end{equation}
+Thus $J'_X$ is locally generated by the row entries after trivializing
+their common twist.  We use the prime to distinguish this row subscheme
+from the full-adjugate scheme $Z$.
+
+After transposition, the equations $R_X=0$ say that an adjugate matrix
+annihilates a fixed vector.  Adams, Harnad and Hurtubise use equations
+of this form for spectral divisors in a generic complex setting
+\cite[Section~1b, equation~(1.29) in arXiv:hep-th/9210089v1]{AdamsHarnadHurtubise1993}.
+Here we prove the uniform finite length of $Z'_X$ directly.  The common
+boundary frame then makes this subscheme determine the matrix uniquely
+on a fixed embedded curve.
+
+\begin{lemma}[The row subscheme]
+For each $X\in\mathcal F_C$\label{lem:row-adjugate-scheme}, the scheme $Z'_X$ is finite of length $N$.
+If $X,X'\in\mathcal F_C$ satisfy $Z'_X=Z'_{X'}$ as subschemes of $C$,
+then $X=X'$.
+\end{lemma}
+
+\begin{proof}
+The projection $\pi:C\to\PP^1_{t:u}$ and the bundle
+\[
+ \mathcal E=\pi_*\cO_C(k-1)
+ =\bigoplus_{j=0}^{k-1}\cO(k-1-j)
+\]
+are fixed throughout $\mathcal F_C$.
+The entries of $R_X$ give a map
+$\rho_X:\cO^k\to\mathcal E$ on $\PP^1$.
+At $b_0=[1:0]$, trivialize by $t$ and put $y=s/t$.  For every $X$,
+the map on this fiber sends the $i$-th standard basis vector to
+\begin{equation}
+ \label{eq:row-boundary-frame}
+ h_i(y)=\prod_{j\ne i}(y+\theta_j)
+ \quad\text{in}\quad
+ K[y]\big/\prod_j(y+\theta_j).
+\end{equation}
+Evaluation at the distinct roots $-\theta_i$ shows that these $k$
+classes form a basis.  In particular, all the maps $\rho_X$ restrict
+to the same isomorphism at $b_0$.  Each $\rho_X$ is therefore
+generically full rank and injective as a sheaf map.  Its cokernel is
+torsion of length
+\begin{equation}
+ \label{eq:row-adjugate-length}
+ \length(\coker\rho_X)
+ =\deg\mathcal E
+ =\sum_{j=0}^{k-1}(k-1-j)=N.
+\end{equation}
+
+On $C$, the adjugate identity gives
+\[
+ sR_X=-R_X(tD+uX).
+\]
+On either affine base chart, the span of the row entries over the base
+ring is consequently stable under the affine $s$-coordinate.  This
+coordinate generates the finite curve algebra over the base, so the
+span is already the ideal generated over the curve algebra, with its
+twist.  Hence
+\[
+ \operatorname{im}\rho_X=\pi_*(J'_X(k-1)).
+\]
+Finite pushforward is exact, so
+\[
+ \coker\rho_X=\pi_*(\cO_{Z'_X}(k-1)).
+\]
+This pushforward has finite support, and $\pi$ is finite, so
+$\cO_{Z'_X}$ has finite support as well.  Twisting preserves length,
+as does finite pushforward over the algebraically closed ground field.
+Equation \eqref{eq:row-adjugate-length} thus proves that $Z'_X$ is
+finite of length $N$.
+
+Suppose now that $Z'_X=Z'_{X'}$.  Their ideal sheaves on the fixed
+curve are equal, so $\rho_X$ and $\rho_{X'}$ have the same image inside
+the same bundle $\mathcal E$.  Since both maps are injective, there is
+an automorphism $U$ of $\cO^k$ with
+\[
+ \rho_{X'}=\rho_XU.
+\]
+Every such automorphism is an invertible constant matrix.  Restricting
+this equality to $b_0$, where both maps are the same isomorphism
+\eqref{eq:row-boundary-frame}, gives $U=I$ and hence
+$\rho_{X'}=\rho_X$.
+
+Multiplication by $s$ maps $\mathcal E$ to $\mathcal E(1)$, and the
+row identity says
+\[
+ s\rho_X=-\rho_X(1)(tD+uX).
+\]
+The analogous equation for $X'$ now gives
+$\rho_X(1)u(X-X')=0$.  The map $\rho_X(1)$ is injective, so
+$u(X-X')=0$ as a matrix of sections of $\cO(1)$.  Since $u$ is a
+nonzero section and $X-X'$ is constant, this forces $X=X'$.
+The same argument works after extending the ground field, so the
+uniqueness statement also holds for geometric points.
+\end{proof}
+
+The lemma reduces the fiber count to the possible length-$N$ subschemes
+of $C$.  We next bound the dimension of their Hilbert scheme in a form
+valid in characteristic two.
+
+\subsection{Finite subschemes of a plane curve}
+
+For a punctual quotient of $\cO$, Baranovsky's construction of punctual
+Quot schemes over $\C$ uses commuting nilpotent operators and a cyclic
+vector after choosing a basis \cite[Section~2, Lemma~2.2]{Baranovsky1998}.
+For the classical punctual-plane Hilbert dimension bound in arbitrary
+characteristic, see \cite[Corollary~4.1]{Premet2003}.
+The centralizer count below supplies the dimension bound needed here in
+every characteristic.
+
+\begin{lemma}\label{lem:hilbert-curve-dimension}
+Let $C\subset\PP^2_K$ be a reduced curve over an algebraically closed
+field $K$, and let $N\geq0$.  Then
+\[
+ \dim\operatorname{Hilb}^N(C)\leq N.
+\]
+\end{lemma}
+
+\begin{proof}
+Fix a point of the affine plane and an integer $\ell\geq1$, and
+translate the point to the origin.  Inside
+$\operatorname{Hilb}^{\ell}(\mathbb A^2_K)$, let $P_\ell$ be the closed
+locus of quotients on which $(x,y)^\ell$ acts by zero.
+Its geometric points are precisely the punctual subschemes supported at
+the origin: in a length-$\ell$ local algebra, the $\ell$-th power of
+the maximal ideal is zero.
+
+The frame bundle of the tautological quotient algebra over $P_\ell$
+has dimension $\dim P_\ell+\ell^2$.  In a frame, multiplication by
+the two coordinates gives commuting nilpotent matrices $A,B$ of size
+$\ell$, and the unit gives a cyclic vector $v\in K^\ell$, meaning
+$K[A,B]v=K^\ell$.  These data recover the ideal as
+\[
+ \{f\in K[x,y]:f(A,B)v=0\},
+\]
+and recover its frame.  Thus the resulting morphism from the frame
+bundle to the finite-type scheme of such triples is injective on geometric points.
+Its geometric fibers have dimension zero, which suffices for a
+dimension upper bound.
+
+There are finitely many nilpotent Jordan types for $A$.  If its linear
+centralizer has dimension $j$, then its group centralizer is the open
+set of invertible elements in that vector space.  The conjugacy orbit
+of $A$ therefore has dimension $\ell^2-j$.  Within its centralizer,
+the nilpotent choices of $B$ have dimension at most $j-1$: they satisfy
+$\det B=0$, whereas the determinant is one at the identity matrix in
+the same centralizer.  This argument works in every characteristic.
+Consequently the commuting nilpotent pairs have dimension at most
+$\ell^2-1$.  The choice of the cyclic vector adds at most $\ell$
+parameters, so
+\[
+ \dim P_\ell+\ell^2\leq\ell^2+\ell-1,
+ \qquad \dim P_\ell\leq\ell-1.
+\]
+
+Fix $p\in C$ and choose an affine plane chart around it, translated so
+that $p$ is the origin.  Inside the Hilbert scheme of this affine part
+of $C$, impose the same condition that the $\ell$-th power of the
+maximal ideal of $p$ acts by zero.  The resulting locus is a closed
+subscheme of $P_\ell$, and its geometric points are exactly the
+length-$\ell$ subschemes of $C$ supported at $p$.  Its dimension is
+therefore at most $\ell-1$, including when $p$ is singular.
+Fix a positive length type
+$\ell_1+\cdots+\ell_a=N$.  Consider ordered distinct points
+$(p_1,\ldots,p_a)$ of $C$ together with subschemes of lengths $\ell_i$
+supported at $p_i$.  This incidence is algebraic: containment in the
+$\ell_i$-th infinitesimal neighborhood of the moving point can be
+imposed using the $\ell_i$-th power of the diagonal ideal on $C\times C$.
+The support parameters have dimension at most $a$, and the fibers have
+dimension at most $\sum_i(\ell_i-1)=N-a$.  The incidence therefore
+has dimension at most $N$.  Taking disjoint unions of its subschemes
+maps it to $\operatorname{Hilb}^N(C)$.  Finitely many length types
+cover all geometric points of this Hilbert scheme, proving the claim.
+The case $N=0$ is immediate.
+\end{proof}
+
+\subsection{The matrix fibers and the gradient locus}
+
+We now use the universal family of length-$N$ subschemes to compare the
+matrix fiber with $\operatorname{Hilb}^N(C)$.
+
+\begin{proposition}[Dimension of a matrix fiber]\label{prop:matrix-fiber}
+For the morphism $f_D$ in \eqref{eq:determinant-curve-map}, every fiber
+has dimension at most $N=k(k-1)/2$.
+\end{proposition}
+
+\begin{proof}
+Fix a curve $C$ in the image and its fiber $\mathcal F_C$.  Form
+\[
+ \mathcal I_C=
+ \{(X,Z')\in\mathcal F_C\times\operatorname{Hilb}^N(C):
+ R_X|_{Z'}=0\}.
+\]
+This is a closed incidence.  Indeed, restrict the row entries to the
+universal finite flat subscheme over the Hilbert scheme and push forward
+their common twist.  This produces sections of a vector bundle on the
+product, whose zero locus is $\mathcal I_C$.
+
+Every matrix $X$ occurs in the incidence with its subscheme $Z'_X$.
+Conversely, row vanishing on $Z'$ means $J'_X\subset I_{Z'}$, so
+$Z'\subset Z'_X$.  The scheme $Z'$ has length $N$ by its membership
+in the Hilbert scheme, and $Z'_X$ has length $N$ by
+Lemma~\ref{lem:row-adjugate-scheme}.  Therefore they are equal.
+The same lemma shows that each fiber of $\mathcal I_C$ over
+$\operatorname{Hilb}^N(C)$ has at most one geometric point.  Such a
+finite-type fiber has dimension zero when nonempty, even if it is
+nonreduced.  Projection to $\mathcal F_C$ is surjective on geometric
+points, so
+\[
+ \dim\mathcal F_C
+ \leq\dim\mathcal I_C
+ \leq\dim\operatorname{Hilb}^N(C)
+ \leq N,
+\]
+where the last inequality is Lemma~\ref{lem:hilbert-curve-dimension}.
+\end{proof}
+
+\begin{proof}[Proof of Proposition~\ref{prop:gradient-codimension}]
+By \eqref{eq:dimension-gradient-defect}, every curve in
+$f_D(\Gamma_D)$ has normalization defect at least $T$.
+Lemma~\ref{lem:severi-dimension} bounds the dimension of this
+constructible image by $k(k+3)/2-T$.  Each fiber of the restriction
+of $f_D$ to $\Gamma_D$ is contained in a full matrix fiber and has
+dimension at most $N$ by Proposition~\ref{prop:matrix-fiber}.
+Consequently
+\[
+ \dim\Gamma_D
+ \leq\frac{k(k+3)}2-T+\frac{k(k-1)}2
+ =k^2+k-T,
+\]
+as asserted.
+\end{proof}
diff --git a/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/sections/introduction.tex b/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/sections/introduction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..1deec8449b3e26db097eab9cb031a16629d772e9
--- /dev/null
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@@ -0,0 +1,257 @@
+\section{Introduction}
+\label{sec:introduction}
+
+For a matrix $X=(x_{ij})$ of independent variables, the permanent is
+\[
+ \per_m(X)=\sum_{\sigma\in S_m}\prod_{i=1}^m x_{i,\sigma(i)}.
+\]
+The permanent--determinant problem asks how large a determinant of
+affine-linear forms is needed to express $\per_m$. Valiant connected
+determinant projections to arithmetic formulas: his construction represents
+a polynomial computed by a formula with $s$ addition or multiplication
+operations as a determinant of order $s+2$
+\cite[Theorem~1]{Valiant1979Algebra}. His algebraic completeness theorem
+shows that, in characteristic different from two, polynomial families
+defined by summing polynomial-size formulas over Boolean strings of
+polynomial length are projections of polynomial-size permanents
+\cite[Theorem~2]{Valiant1979Algebra}. Valiant's projections substitute
+individual variables or field constants. The exact model below allows
+every matrix entry to be an arbitrary affine-linear form.
+
+For $f\in\C[x_1,\ldots,x_N]$, its \emph{determinantal complexity}
+$\dc(f)$ is the least positive integer $n$ for which
+\[
+ f=\det\left(A_0+\sum_{j=1}^N x_jA_j\right),
+ \qquad A_0,\ldots,A_N\in\Mat_n(\C).
+\]
+Its \emph{border determinantal complexity} $\bdc(f)$ is the least positive
+integer $n$ such that $f$ lies in the coefficientwise closure of
+\begin{equation}
+ \left\{\det\left(A_0+\sum_{j=1}^N x_jA_j\right):
+ A_0,\ldots,A_N\in\Mat_n(\C)\right\}.
+ \label{eq:border-definition}
+\end{equation}
+The closure is taken in the finite-dimensional space of polynomials of
+degree at most $n$; a polynomial of larger degree cannot belong to it.
+We allow $f$ to be constant or zero. Coefficientwise closure agrees with
+Zariski closure here, as explained in Section~\ref{sec:models}. The size
+$n$ stays fixed along an approximating sequence, while the complex
+coefficients of its matrix entries are unrestricted.
+
+\begin{theorem}
+\label{thm:main}
+There exist absolute constants $c>0$ and $m_0$ such that, for every integer
+$m\ge m_0$,
+\[
+ \bdc(\per_m)\ge c m^3.
+\]
+One may take $c=1/(5\,529\,600e)$ and $m_0=1408$.
+\end{theorem}
+
+The constants are not optimized. Proposition~\ref{prop:model-equivalence}
+identifies this coefficientwise model with the projective determinant-orbit
+closure formulation, including equal permanent and determinant orders.
+
+The same numerical lower bound holds for exact determinantal complexity.
+There is also a consequence for algebraic branching programs. An
+affine-linear algebraic branching program is a finite directed acyclic
+graph with a source and a sink and an affine-linear form on each edge; it
+computes the sum of products of edge labels along all source-to-sink paths.
+Over $\C$, computing $\per_m$ requires $\Omega(m^3)$ vertices and
+$\Omega(m^3)$ edges. The same bounds hold for coefficientwise limits when
+the relevant vertex or edge budget remains fixed along the approximation,
+even if the graph varies. Section~\ref{sec:consequences} gives the precise
+statements and determinant reductions.
+
+\subsection{Earlier bounds and geometric methods}
+
+The early matrix-order lower bounds were linear in the permanent order $m$.
+For the variable-or-constant projection model, von zur Gathen published a
+bound due to Babai and Seress with leading term $\sqrt{2}\,m$ and a loss of
+$6\sqrt m$ \cite[Theorem~4.4]{vonZurGathen1987}. For arbitrary affine
+representations over $\C$, Meshulam's final theorem
+\cite[pp.~270--271]{Meshulam1989} and Cai's dimension argument
+\cite[Theorem~1.5 and Section~2.2]{Cai1990} give the scale
+$\sqrt{2}\,m-O(1)$. Their arguments compare the dimension of the affine
+matrix image with the minimum rank that an exact permanent representation
+must have.
+
+Mignon and Ressayre raised the exact lower bound to
+$\dc(\per_m)\ge m^2/2$ in characteristic zero
+\cite{MignonRessayre2004}. Their proof compares Hessians at a zero:
+affine pullback cannot increase Hessian rank, the Hessian of an order-$n$
+determinant has rank at most $2n$ at a noninvertible matrix, and, for
+$m\ge2$, the permanent has a zero where its Hessian has full rank $m^2$.
+Cai, Chen and Li extended the quadratic scale to positive characteristic
+different from two by constructing sparse zeros that avoid the large
+factorial factors in the earlier Hessian calculation
+\cite[Theorem~2.3]{CaiChenLi2010}.
+
+On the upper side, Grenet's construction realizes $\per_m$ as a determinant
+of order $2^m-1$, using a branching program indexed by the subsets of
+$\{1,\ldots,m\}$ \cite[unnumbered Theorem, p.~3]{Grenet2012}.
+Its determinant entries are individual variables and constants, so it is
+also an upper bound in the unrestricted exact model.
+
+Mulmuley and Sohoni's geometric complexity theory program studies
+permanent--determinant separation through orbit closures of determinants
+and padded permanents, seeking equations or representation-theoretic
+obstructions to containment
+\cite{MulmuleySohoni2001,MulmuleySohoni2008}.
+Landsberg, Manivel and Ressayre proved the unrestricted complex border
+bound $\bdc(\per_m)\ge m^2/2$ by studying hypersurfaces with degenerate
+duals and the effect of padding
+\cite[Theorem~1.1.1]{LandsbergManivelRessayre2010}.
+Theorem~\ref{thm:main} raises the lower-bound exponent from two to three
+in that same model. It remains a fixed polynomial bound. Whether the
+permanent has superpolynomial determinantal complexity, or superpolynomial
+border determinantal complexity, remains open.
+
+Landsberg and Ressayre later proved exponential optima for exact
+representations with symmetry requirements. Each specified projective
+symmetry of the input must lift to a projective symmetry of the determinant
+that fixes the pencil's constant matrix, preserves the image of its linear
+part, and induces the given input action there. Over $\C$, for $m\ge3$,
+requiring this for left multiplication by invertible diagonal and
+permutation matrices gives minimum order $2^m-1$. Lifting the full
+projective symmetry group of the permanent gives minimum order
+$\binom{2m}{m}-1$ in the same range
+\cite[Theorems~2.1 and~2.8; Propositions~2.9 and~2.10]{LandsbergRessayre2017}.
+These hypotheses specify exact representation classes; the border model
+of Theorem~\ref{thm:main} allows arbitrary affine determinant
+approximations of fixed order.
+
+Singularities also give geometric lower bounds for exact representations.
+For a homogeneous polynomial of degree greater than two whose affine
+singular locus has codimension greater than four, Alper, Bogart and Velasco
+prove
+$\dc(f)\ge\operatorname{codim}\operatorname{Sing}(f)+1$
+\cite[Theorem~1.2]{AlperBogartVelasco2017}.
+For the permanent, matrices with two zero columns show that this
+codimension is at most $2m$, so this obstruction alone has only a linear
+scale \cite[Remark~1.5]{AlperBogartVelasco2017}.
+Our construction instead obtains a coefficient polynomial from a permanent
+and finds a linear subspace on which its zero hypersurface is smooth. Its
+degree is linear and the subspace dimension is quadratic in an auxiliary
+matrix size.
+
+For a smooth complex form of degree $r\ge2$ in $d\ge2$ variables, the classical
+top polar degree is $r(r-1)^{d-2}$
+\cite[Corollary~3.7]{Piene1978}.
+The left and right kernel equations for the determinant matrix and the
+Schur-complement reduction used below also appear in Sheshadri's June 2026
+preprint \cite[Theorem~3 and Section~3.1]{SheshadriBorder2026}, as part of
+its proposed conormal approach to border bounds for diagonal power sums.
+Kumar and Volk's September 2026 preprint uses such kernel equations and a
+multihomogeneous count to obtain a quadratic \emph{exact} lower bound for
+$\sum_{i=1}^n x_i^n$
+\cite[Theorem~1.1 and Section~3]{KumarVolk2026}.
+Theorem~\ref{thm:smooth-border} proves the version needed here, including
+persistence of finitely many simple polar intersections in one sufficiently
+close determinant under an arbitrary coefficientwise approximation. Its
+lower bound is proportional to the product $(r-1)(d-1)$.
+
+\subsection{The coefficient construction and the proof}
+
+To use this product at cubic scale, we seek an affine restriction of a
+permanent of size proportional to $k$ that has degree proportional to $k$
+in quadratically many variables and has a smooth projective zero set.
+Let $k\ge1$ and $0\le q\le\lfloor k/2\rfloor$ be integers. With
+indeterminates $\theta_1,\ldots,\theta_k$, put
+$D=\diag(\theta_1,\ldots,\theta_k)$ and define
+\[
+ G_D(X)=[s^qt^q]\per_k(X+sI+tD)
+ \in\mathbb Z[\theta_1,\ldots,\theta_k][X_{ij}].
+\]
+This polynomial is homogeneous of degree $r=k-2q$ in the $k^2$ entries
+of $X$. We use the same notation after specializing the parameters in a
+field. For the main construction we assume $k\ge128$ and take
+$q=\lfloor(k-1)/3\rfloor$; then $r$ is linear in $k$ and
+$T=\binom{q+2}{2}$ is quadratic in $k$.
+
+The integral formula has two uses. Over $\C$, two count gates and a
+cancellation gadget realize $G_D$, up to a fixed nonzero scalar, as an
+affine restriction of a permanent of size $11k-2q$
+(Proposition~\ref{prop:selector}). This holds for every complex choice of
+the parameters. The scalar vanishes in characteristic two, so the geometric
+argument specializes the integral coefficient formula itself. We will use
+the complex projection at the complex parameter choice that geometry
+eventually supplies.
+
+Let $K=\overline{\mathbb F}_2$, choose pairwise distinct
+$\bar\theta_1,\ldots,\bar\theta_k\in K$, and write
+$\bar D=\diag(\bar\theta_1,\ldots,\bar\theta_k)$. Since permanent
+and determinant agree over $K$, every matrix $X$ defines a plane curve
+\[
+ C_X=V\bigl(\det(sI+t\bar D+uX)\bigr)\subset\PP^2_K.
+\]
+Its fiber on $u=0$ consists of the distinct points
+$[-\bar\theta_i:1:0]$. This fixed fiber makes the curve reduced and its
+projection to $[t:u]$ finite and generically separable, and it controls the adjugate
+of $sI+t\bar D+uX$ on the curve. The full adjugate defines a finite
+subscheme $Z\subset C_X$. If all entrywise derivatives of $G_{\bar D}$
+vanish at $X$, the coefficient $[s^qt^qu^{r-1}]$ of every adjugate entry
+vanishes. We prove that these
+entries span all degree-$(k-1)$ forms vanishing on $Z$, so the same
+coefficient vanishes on every such form. For a degree-$q$ monomial
+$s^it^ju^\ell$, multiplication by $s^{q-i}t^{q-j}u^{r-1-\ell}$ lets this
+functional recover its coefficient in any degree-$q$ form. These exponents
+are nonnegative because $r-1\ge q$. Thus restriction of degree-$q$ forms to $Z$ is
+injective, giving the length bound in Proposition~\ref{prop:adjugate-length}:
+\[
+ K[s,t,u]_q\hookrightarrow H^0(Z,\cO_Z(q)),
+ \qquad \length Z\ge\dim_K K[s,t,u]_q=T.
+\]
+
+The length of $Z$ is bounded above by the \emph{normalization defect}
+$\delta(C_X)$, the length of the quotient between the normalization's
+structure sheaf and the curve's structure sheaf. The comparison follows
+the evaluation-ideal argument of Fantechi, G\"ottsche and van Straten
+over $\C$ \cite[Proposition~C.2 and Corollary~C.3]{FGvS1999};
+Section~\ref{sec:conductor} proves the needed conductor inclusion and
+colength identity for our generically separable pencils in arbitrary
+characteristic.
+Thus every critical matrix lies over a curve with $\delta(C_X)\ge T$.
+
+Applying the arbitrary-characteristic Severi dimension theorem of Christ,
+He and Tyomkin \cite[Lemma~2.6, Proposition~2.7, and Remark~2.8]{CHT2023}
+to the integral components bounds the locus of curves with this defect.
+Their result builds on Tyomkin's dimension bound for toric pairs
+\cite[Theorem~1.2(1)]{Tyomkin2013}. The sum
+$R_X=(1,\ldots,1)\adj(sI+t\bar D+uX)$ of the adjugate rows controls
+the matrices over one fixed curve: its zero subscheme has length
+$k(k-1)/2$, and the common
+boundary frame makes that subscheme determine the matrix uniquely.
+A Hilbert-scheme estimate and an algebraic incidence then bound the
+dimension of each matrix fiber. Combining the curve and matrix bounds gives
+\[
+ \dim\{X:\nabla G_{\bar D}(X)=0\}\le k^2+k-T,
+ \qquad
+ \codim\{X:\nabla G_{\bar D}(X)=0\}\ge T-k.
+\]
+
+The projective zero scheme of the raw partial derivatives forms a proper
+family over the integral parameter space. Specialization transfers the
+dimension bound to a nonempty open set of complex parameters. Over $\C$,
+Euler's identity identifies these gradient zeros with the hypersurface's
+singular scheme, and a general linear section avoids that scheme and meets
+the smooth locus transversely. Proposition~\ref{prop:smooth-projection}
+then supplies, for $k\ge128$ and the stated choice of $q$, a nonzero form
+$G$ of degree $r$ in $d=\lfloor k^2/100\rfloor$ variables whose
+projective zero set is smooth. For each $m\ge11k-2q$, it also supplies
+a fixed affine map $\Phi_m$ satisfying $G=\per_m\circ\Phi_m$.
+
+Figure~\ref{fig:proof-flow} shows how the two uses of the coefficient
+formula meet and how the polar bound converts their output into the cubic
+estimate.
+
+\input{figures/proof-flow}
+
+Sections~\ref{sec:selector}--\ref{sec:transfer} establish this
+construction. Section~\ref{sec:polar} then proves the smooth-form bound
+stated in Section~\ref{sec:models}, using selected simple polar
+intersections and the left and right kernels of one nearby determinant.
+Section~\ref{sec:consequences} combines the two results and gives the
+exact, branching-program, and smooth-initial-form consequences.
+
+\FloatBarrier
diff --git a/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/sections/models.tex b/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/sections/models.tex
new file mode 100644
index 0000000000000000000000000000000000000000..9c037daf3baba1a3b9c178dc99a0af545cd219ac
--- /dev/null
+++ b/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/sections/models.tex
@@ -0,0 +1,142 @@
+\section{Border models and smooth homogeneous forms}
+\label{sec:models}
+
+We first identify the two border models precisely. All closures in this
+section are over $\C$. In a finite-dimensional polynomial space, coefficientwise
+convergence means ordinary convergence of the coefficients of every monomial.
+The image of a polynomial map is constructible
+\cite[Theorem~29.23.3, Tag~054K]{StacksProject}, and a complex constructible
+set has the same Euclidean and Zariski closures. For the latter fact, decompose
+the set into finitely many locally closed subsets: each contains a Zariski
+open dense subset of its closure, which is also Euclidean dense. These facts
+apply both to the polynomial map taking an affine matrix to its determinant
+and to the action map defining a linear-group orbit. Thus coefficientwise
+sequences can be used for either closure below.
+
+\begin{lemma}[Affine substitution]
+\label{lem:affine-substitution}
+Let $f$ be a complex polynomial, let $A$ be an affine map into its variable
+space, and let $a\ne0$ be a complex scalar. Then
+\[
+ \bdc(a f\circ A)\le \bdc(f).
+\]
+\end{lemma}
+\begin{proof}
+Fix a size $n$ and determinants $Q_j=\det M_j$ converging coefficientwise to
+$f$, with all entries of $M_j$ affine-linear. The entries of $M_j\circ A$
+are still affine-linear. Substitution by the fixed map $A$ is a continuous
+linear map on spaces of polynomials of degree at most $n$, so
+$Q_j\circ A\to f\circ A$. Multiplying one row of each substituted matrix
+by $a$ gives the assertion. This argument imposes no bound on the matrix
+entries in the approximating sequence.
+\end{proof}
+
+For $n\ge m$, let $W_n=\Mat_n(\C)$ and regard $\det_n$ as a degree-$n$
+form on $W_n$. Choose $m^2$ independent linear coordinates $x_{ij}$ on $W_n$.
+If $n>m$, choose one more independent coordinate $z$ and set
+\[
+ P_{n,m}=z^{n-m}\per_m(x).
+\]
+If $n=m$, set $P_{m,m}=\per_m(x)$; no padding coordinate is needed.
+Let $\mathcal D_n$ be the projective Zariski closure of the orbit of
+$[\det_n]$ under invertible linear substitutions on $W_n$.
+
+\begin{proposition}[Equivalence of the two border models]
+\label{prop:model-equivalence}
+For every $n\ge m\ge1$,
+\[
+ \bdc(\per_m)\le n
+ \quad\Longleftrightarrow\quad
+ [P_{n,m}]\in\mathcal D_n.
+\]
+\end{proposition}
+
+\begin{proof}
+Every linear map $L:W_n\to W_n$ is a limit of invertible ones. Consequently
+$\det_n\circ L$ belongs to the affine closure of the determinant orbit.
+This orbit is invariant under nonzero scalar multiplication: multiplying
+one row of a matrix variable by $a\ne0$ multiplies its determinant by $a$
+and is an invertible linear substitution. Its affine closure is therefore
+the cone over $\mathcal D_n$. In particular a nonzero form belongs to this
+affine closure exactly when its projective class belongs to $\mathcal D_n$.
+
+Suppose first that $\bdc(\per_m)\le n$. Identity blocks let us take every
+approximating matrix to have size exactly $n$. Write
+\[
+ Q_j(x)=\det\left(A_{j,0}+\sum_{a,b=1}^m x_{ab}A_{j,ab}\right)
+ \longrightarrow\per_m(x).
+\]
+For $n>m$, homogenize to degree $n$ using the independent coordinate $z$:
+\[
+ Q_j^{\mathrm{hom}}(x,z)=\det\left(zA_{j,0}+\sum_{a,b=1}^m x_{ab}A_{j,ab}\right)
+ \longrightarrow z^{n-m}\per_m(x).
+\]
+Homogenization here is a fixed linear map of coefficient spaces. Each
+displayed determinant is a linear substitution of $\det_n$ on $W_n$,
+so its limit belongs to the same closed cone. The coordinates fit because
+$m^2+1\le n^2$ when $n>m$.
+
+For $n=m$, instead take the degree-$m$ homogeneous component of each $Q_j$.
+It is precisely
+\[
+ \det\left(\sum_{a,b=1}^m x_{ab}A_{j,ab}\right),
+\]
+and these components converge to $\per_m$. They again are linear
+substitutions of $\det_m$. This proves the forward implication without
+introducing an extra variable in the case $n=m$.
+
+Conversely, the cone observation and equality of the two closures give
+determinant-orbit representatives converging coefficientwise to $P_{n,m}$.
+When $n>m$, set $z=1$ and all unused coordinates to zero; when $n=m$, just
+use the $m^2$ coordinates $x_{ab}$. Each resulting polynomial is an affine
+determinant of size $n$, and the limit is $\per_m$. This proves the converse.
+Finally, a determinant of size less than $m$ has degree less than $m$, as
+does every polynomial in its coefficientwise closure. Thus
+$\bdc(\per_m)\ge m$, and the stated range $n\ge m$ covers every size
+that can satisfy the affine condition.
+\end{proof}
+
+The quantitative obstruction used in the proof depends on the degree and
+the number of variables of a smooth homogeneous polynomial. Here smoothness
+means that the projective zero set has no point at which all first partial
+derivatives vanish.
+
+\begin{theorem}[Lower bound for smooth homogeneous forms]
+\label{thm:smooth-border}
+Let $G\in\C[x_1,\ldots,x_d]$ be a nonzero homogeneous polynomial of degree
+$r\ge2$, where $d\ge2$. If its zero set in $\PP^{d-1}$ is smooth, then
+\[
+ \bdc(G)\ge \frac{(r-1)(d-1)}{4e}.
+\]
+\end{theorem}
+
+The complete proof is in Section~\ref{sec:polar}. For a constant vector
+$w\in\C^d$, write
+\[
+ \partial_wG=\sum_{j=1}^d w_j\frac{\partial G}{\partial x_j}
+\]
+for the first directional derivative, or \emph{first polar}, of $G$.
+If $G$ is a coefficientwise limit of affine determinants of size $n$, the proof
+chooses $d-2$ first polars whose common intersection with the hypersurface
+consists of $r(r-1)^{d-2}$ simple points. It then shows that these selected
+points persist for one sufficiently close determinant and satisfy
+\begin{equation}
+ r(r-1)^{d-2}
+ \le 2^{d-2}\binom{2n-1}{d-1}
+ \le \left(\frac{4en}{d-1}\right)^{d-1}.
+ \label{eq:smooth-count-synopsis}
+\end{equation}
+The binomial coefficient is understood to be zero when $d-1>2n-1$;
+in that case the positive left side already rules out the approximation.
+The first inequality comes from lifting the selected points to the right
+and left kernel lines of the determinant matrix and counting the resulting
+bilinear equations. Taking the $(d-1)$st root gives the theorem. The proof
+establishes persistence and simplicity of those selected lifts using only
+coefficientwise closeness of the determinant polynomials, so it allows
+unbounded coefficients in the approximating matrices.
+
+This comparison specifies our construction's target. A nonzero homogeneous
+form of degree proportional to $k$ in quadratically many variables, with
+a smooth projective zero set, forces cubic determinant size. We now produce
+such a form as a fixed affine restriction of a permanent whose size is
+linear in $k$.
diff --git a/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/sections/polar.tex b/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/sections/polar.tex
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+\section{Counting polars of a smooth hypersurface}\label{sec:polar}
+
+We now prove Theorem~\ref{thm:smooth-border}. A smooth degree-$r$
+hypersurface in $\PP^{d-1}$ supplies $r(r-1)^{d-2}$ simple intersections
+with first polars. These points persist for one sufficiently close affine
+determinant. Right and left kernel variables then express each selected
+intersection by bilinear equations, whose multidegrees give the numerical
+comparison stated in Section~\ref{sec:models}. The determinant kernel-incidence
+count and its Schur-complement reduction
+build on \cite[Theorem~3 and Section~3.1]{SheshadriBorder2026}. We give the
+argument needed here for finitely many simple polar intersections,
+including unrestricted affine limits and possible excess components of
+the incidence.
+
+For a constant vector $w\in\C^N$, write
+\[
+ \partial_w F=\sum_{j=1}^N w_j\frac{\partial F}{\partial x_j}.
+\]
+We call this an ambient first polar: after restricting to an affine slice,
+the direction $w$ need not be tangent to the slice. A common zero of $s$
+equations in $s$ local coordinates is \emph{simple} if its full Jacobian
+is invertible.
+
+\subsection{Simple polar intersections}
+
+The numerical count below is the classical top polar degree of a smooth
+hypersurface \cite[Corollary~3.7]{Piene1978}. We give the transversality
+argument to retain the individual simple points needed in the limit.
+General-transversality principles of this kind are developed in
+\cite[Corollary~5]{Kleiman1974}; the argument below works directly over
+$\C$ with the incidence of polar directions.
+
+\begin{lemma}\label{polar:simple-points}
+Let $r,d\ge2$, and let $G\in\C[x_1,\ldots,x_d]$ be homogeneous of degree
+$r$. Suppose that $Y=V(G)\subset\PP^{d-1}$ is a smooth hypersurface.
+There are constant directions $w_1,\ldots,w_{d-2}\in\C^d$ for which
+\begin{equation}\label{polar:projective-certificate}
+ G=0,\qquad \partial_{w_i}G=0\quad(1\le i\le d-2)
+\end{equation}
+has exactly $\mu=r(r-1)^{d-2}$ distinct simple projective solutions.
+There is a linear form $L_0$ nonzero at every one of these points.
+\end{lemma}
+
+\begin{proof}
+The sections $\partial_wG$ of $\cO_Y(r-1)$ have no common zero on $Y$:
+otherwise all partial derivatives of $G$ would vanish at a point of the
+smooth hypersurface. We spell out why general choices give simple points.
+If $d=2$, the smooth degree-$r$ hypersurface in $\PP^1$ already consists
+of $r$ distinct points, and there are no polar equations. Assume $d\ge3$.
+
+Put $\mathcal P=(\C^d)^{d-2}$ and consider
+\[
+ \mathcal I=\{(x,w_1,\ldots,w_{d-2})\in Y\times\mathcal P:
+                  \partial_{w_i}G(x)=0\text{ for every }i\}.
+\]
+For each $x$, the equation in each separate $w_i$ block is a nonzero
+linear equation. Thus $\mathcal I$ is the total space of a vector bundle
+of rank $(d-1)(d-2)$ over $Y$. In particular it is smooth, and
+$\dim\mathcal I=d(d-2)=\dim\mathcal P$. The incidence is closed in
+$Y\times\mathcal P$, so its projection to $\mathcal P$ is proper because
+$Y$ is projective. It is surjective: $G$ and any
+$d-2$ chosen polars are $d-1$ positive-degree homogeneous equations in
+$d$ variables, so the projective dimension theorem gives a common
+projective zero.
+
+Characteristic-zero generic smoothness makes the generic fiber of each
+dominating component smooth. Remove the proper images of the
+non-dominating components and of the locus where the projection is not
+smooth. Properness makes these images closed, and generic smoothness
+ensures that they omit the generic point of $\mathcal P$. The resulting
+nonempty open subset has smooth zero-dimensional fibers. Each such fiber
+is a finite set of reduced points. At each point the differentials of the polar
+sections on $Y$ are independent. Since $dG\ne0$ in local projective
+coordinates, the full system~\eqref{polar:projective-certificate} is
+simple there.
+
+This is a proper intersection of hypersurfaces of degrees
+$r,r-1,\ldots,r-1$ in $\PP^{d-1}$. Projective B\'ezout therefore gives
+exactly $r(r-1)^{d-2}$ points, each of multiplicity one.
+Finally, the linear forms vanishing at at least one of these finitely
+many points form a finite union of proper hyperplanes in $(\C^d)^*$.
+Choose $L_0$ outside that union.
+\end{proof}
+
+\subsection{A count that permits excess components}
+
+The kernel equations below need not meet properly everywhere, so we count
+only their selected simple zeros. The coefficient in the next lemma is
+the multihomogeneous B\'ezout number. Its standard specialization bound
+controls nonsingular isolated zeros even when other components are present
+\cite[Sections~3--3.2]{Wampler1996}. We give a direct proof of the bilinear
+case needed here.
+
+\begin{lemma}\label{polar:product-count}
+Let $Y=\prod_{j=1}^s\PP^{a_j}$, where $a_j\ge0$ and
+$M=\sum_j a_j\ge1$. For each $1\le\ell\le M$, let $F_\ell$ be a
+multihomogeneous polynomial of degree one in two distinct coordinate
+groups, indexed by $u_\ell,v_\ell$, and degree zero in the other groups.
+Any finite collection of distinct simple common zeros of these equations
+has size at most
+\begin{equation}\label{polar:product-coefficient}
+ [H_1^{a_1}\cdots H_s^{a_s}]
+       \prod_{\ell=1}^M(H_{u_\ell}+H_{v_\ell}).
+\end{equation}
+Other zeros and components of any dimension are allowed.
+\end{lemma}
+
+\begin{proof}
+First construct a system with the displayed number of zeros, all simple.
+Give each occurrence of a coordinate group in an equation a different
+label. On $\PP^{a_j}$ choose the labeled linear forms so that every
+subfamily of at most $a_j+1$ coefficient vectors is independent.
+In coordinates $[z_0:\cdots:z_{a_j}]$, forms
+\[
+ z_0+t z_1+\cdots+t^{a_j}z_{a_j}
+\]
+with distinct values of $t$ have this property by the Vandermonde
+determinant. When $a_j=0$, use the nowhere-zero form $z_0$ on $\PP^0$.
+Replace equation $\ell$ by $Q_\ell=L_\ell R_\ell$, using its two
+labeled forms on factors $u_\ell$ and $v_\ell$.
+
+At a common zero of all $Q_\ell$, let $b_j$ count the labeled forms
+vanishing in factor $j$. General position implies $b_j\le a_j$.
+Each of the $M$ equations requires at least one vanishing form, and no
+label occurs in two equations. Hence
+\[
+ M\le\sum_j b_j\le\sum_j a_j=M.
+\]
+Exactly one labeled factor vanishes in each equation, and exactly $a_j$
+forms vanish in factor $j$. Conversely, choosing one factor of each
+equation, with exactly $a_j$ choices assigned to factor $j$, determines
+one point of $Y$: the selected $a_j$ hyperplanes in that factor meet in
+one point. No additional labeled form vanishes there, because $a_j+1$
+such hyperplanes have empty intersection. Different choices therefore
+give different points. Expanding the product in
+\eqref{polar:product-coefficient} counts exactly these choices; denote
+that coefficient by $B$.
+
+Every comparison zero is simple. In equation $\ell$, one factor is zero
+and the other is nonzero, so its differential is a nonzero multiple of
+the selected linear form's differential. In each projective factor the
+selected hyperplanes have independent differentials. The full Jacobian
+is therefore invertible. This construction also covers $B=0$, when the
+comparison system has no common zero.
+
+It remains to compare an arbitrary system with this one. Interpolate by
+\[
+ F_{\ell,t}=(1-t)Q_\ell+tF_\ell\qquad(t\in\C).
+\]
+The common-zero incidence is closed in $Y\times\C$. On it, rank
+deficiency of the differential in the $Y$ directions is a closed
+algebraic condition. Indeed, it is given by maximal Jacobian minors in
+each projective chart. Coordinate changes act invertibly on the
+differential, and changing a trivialization multiplies an equation by a
+unit. At a common zero the derivative of that unit contributes zero,
+so the rank condition is independent of the chart.
+
+Since $Y$ is projective, the parameter values having a rank-deficient
+common zero form a Zariski-closed subset $\mathcal B\subset\C$.
+The comparison system shows that $0\notin\mathcal B$, so $\mathcal B$
+is finite. For $t\notin\mathcal B$, all zeros are isolated by the
+implicit function theorem. Their set is closed in the compact space
+$Y$, hence finite. The number of zeros is locally constant on
+$\C\setminus\mathcal B$: the implicit function theorem preserves each
+zero in a disjoint neighborhood, and compactness prevents additional
+zeros from appearing outside those neighborhoods as the parameter varies.
+The same argument covers an empty fiber. Because the complement of a
+finite subset of $\C$ is connected, this number is always the number at
+$t=0$, namely $B$.
+
+Finally, choose any finite set of simple zeros at $t=1$. They persist in
+disjoint neighborhoods for parameters sufficiently close to $1$. Such
+parameters can be chosen outside $\mathcal B$, where the total count is
+$B$. The selected set has at most $B$ members. This conclusion does not
+assign multiplicities to, or impose conditions on, any excess components
+at $t=1$.
+\end{proof}
+
+\subsection{Ambient polars of affine determinant limits}
+
+We next apply the product count to an arbitrary affine determinant
+approximation. The directions in the statement remain ambient directions;
+this permits us to use a projective polar certificate on an affine chart.
+
+\begin{proposition}\label{polar:ambient-bound}
+Let $N,n,\mu\ge1$ and $h\ge0$ be integers. Suppose that
+$P\in\C[x_1,\ldots,x_N]$ is a coefficientwise limit of determinants of
+$n\times n$ matrices of affine-linear forms. Let
+$\Psi:\C^{h+1}\to\C^N$ be an affine map and
+$w_1,\ldots,w_h\in\C^N$ constant directions. If
+\begin{equation}\label{polar:affine-certificate}
+ P(\Psi(\xi))=0,\qquad
+ (\partial_{w_i}P)(\Psi(\xi))=0\quad(1\le i\le h)
+\end{equation}
+has at least $\mu$ distinct simple zeros, then
+\begin{align}
+ \mu&\le [H^{h+1}U^{n-1}V^{n-1}]
+            (H+U)^n(H+V)^{n-1}(U+V)^h\label{polar:ambient-coefficient}\\
+  &\le 2^h\binom{2n-1}{h+1},\notag\\
+ n&\ge\frac{h+1}{4e}\mu^{1/(h+1)}.\label{polar:ambient-size}
+\end{align}
+The binomial coefficient is zero if $h+1>2n-1$.
+\end{proposition}
+
+\begin{proof}
+\emph{Choose one sufficiently close determinant.}
+Fix $\mu$ distinct simple zeros of~\eqref{polar:affine-certificate}.
+All output polynomials lie in the finite-dimensional coefficient space
+of degree at most $n$. With $\Psi$ and the $w_i$ fixed, the map
+\[
+ (Q,\xi)\longmapsto
+   \bigl(Q(\Psi(\xi)),(\partial_{w_1}Q)(\Psi(\xi)),\ldots,
+                         (\partial_{w_h}Q)(\Psi(\xi))\bigr)
+\]
+is polynomial in the coefficients of $Q$ and the coordinates of $\xi$.
+At every selected zero for $P$, its $\xi$-Jacobian is invertible.
+The complex implicit function theorem preserves that simple zero for all
+sufficiently close coefficient vectors. Take disjoint neighborhoods of
+the $\mu$ zeros and intersect their finitely many coefficient neighborhoods.
+There is an actual determinant in the resulting neighborhood,
+\begin{equation}\label{polar:chosen-matrix}
+ Q(x)=\det A(x),\qquad
+ A(x)=A_0+\sum_{j=1}^N x_jA_j,\qquad A_j\in\Mat_n(\C),
+\end{equation}
+for which the same polar system has $\mu$ distinct simple zeros.
+This choice constrains only the coefficients of $Q$. The matrix entries
+may be arbitrary complex numbers, however large; no bound on an
+approximation order or on coefficients of a parametrized family is used.
+
+\emph{Choose a common chart for the right kernels.}
+At each selected zero, the first row of the polar Jacobian is nonzero,
+so $d(Q\circ\Psi)\ne0$. If $A(\Psi(\xi))$ had rank at most $n-2$,
+its adjugate would vanish and the determinant derivative in every direction
+would be zero. Since its determinant is zero, its rank is exactly $n-1$.
+It has unique right and left projective kernel lines.
+
+Choose a linear functional nonzero on every one of the finitely many
+right kernel lines. Such functionals are the complement of a finite union
+of proper hyperplanes. A fixed change of basis in the matrix domain makes
+this functional the first coordinate. Equivalently, replace $A$ by $AS$
+for a suitable invertible constant matrix $S$. This multiplies $Q$ and
+all its polars by the same nonzero constant and preserves simplicity.
+Continue to write $A,Q$ for the resulting matrix and polynomial. Every
+selected right kernel now has a representative with first coordinate one.
+Put
+\[
+ B_i=\partial_{w_i}A=\sum_{j=1}^N(w_i)_j A_j.
+\]
+These are constant matrices, and
+$\partial_{w_i}Q=\tr(\adj(A)B_i)$.
+
+\emph{Lift to a square system of bilinear equations.}
+Let $A^{\mathrm{hom}}(\xi_0,\xi)$ be the entrywise homogenization of the affine matrix
+$A\circ\Psi$, so $A^{\mathrm{hom}}(1,\xi)=A(\Psi(\xi))$. On
+\[
+ \PP^{h+1}_{[\xi_0:\xi]}\times\PP^{n-1}_p\times\PP^{n-1}_b
+\]
+use the equations
+\begin{equation}\label{polar:kernel-system}
+ A^{\mathrm{hom}} p=0,\qquad
+ (b^{\mathsf T}A^{\mathrm{hom}})_j=0\quad(2\le j\le n),\qquad
+ b^{\mathsf T}B_i p=0\quad(1\le i\le h).
+\end{equation}
+There are $n+(n-1)+h=h+2n-1$ equations, equal to the product dimension.
+In the chart $p_1\ne0$, the omitted first left-kernel equation follows
+from the others, since
+\[
+ 0=b^{\mathsf T}(A^{\mathrm{hom}} p)
+   =p_1(b^{\mathsf T}A^{\mathrm{hom}})_1+
+       \sum_{j=2}^n p_j(b^{\mathsf T}A^{\mathrm{hom}})_j.
+\]
+At a selected point, rank $n-1$ implies
+$\adj(A)=\rho p b^{\mathsf T}$ for normalized kernel representatives
+and some $\rho\ne0$: every column of the adjugate is in the right kernel,
+every row is in the left kernel, and some cofactor is nonzero.
+Consequently $\partial_{w_i}Q=\rho b^{\mathsf T}B_i p$ there.
+Each selected affine zero therefore has a unique lift satisfying
+\eqref{polar:kernel-system}.
+
+The lift is not yet known to be simple. We prove this locally before
+applying Lemma~\ref{polar:product-count}; other components of the kernel
+system will impose no further requirement.
+
+\begin{samepage}
+\emph{Eliminate the kernel coordinates.}
+Fix one selected lift and normalize $p_1=1$. Choose a nonzero coordinate
+of its left kernel vector, move that row to the first position, and
+normalize the corresponding left coordinate to one. These are local
+coordinate choices, which can differ between selected points. The
+cofactor obtained by deleting this first row and first column is nonzero,
+because it is the corresponding entry of $\rho p b^{\mathsf T}$.
+In these row orders write the ambient matrix near $\Psi(\xi)$ as
+\[
+ A(x)=\begin{pmatrix}a(x)&\beta(x)\\\gamma(x)&E(x)\end{pmatrix},
+ \qquad
+ p=\binom{1}{u},\qquad b=\binom{1}{v},
+\]
+where $E$ has size $(n-1)\times(n-1)$. The chosen cofactor makes $E$
+invertible at the selected point, so restrict to an ambient neighborhood
+on which it is invertible. When $n=1$, $E$ is the empty matrix with
+determinant one, and the auxiliary vectors $u,v$ are empty.
+
+\end{samepage}
+The same row permutation is applied to every $B_i$. The last $n-1$ right
+equations and all retained left equations give
+\[
+ \gamma+Eu=0,\qquad \beta+v^{\mathsf T}E=0.
+\]
+Their auxiliary Jacobian has invertible diagonal blocks $E,E^{\mathsf T}$,
+so they eliminate the kernel coordinates holomorphically as
+\begin{equation}\label{polar:solved-kernels}
+ u=-E^{-1}\gamma,\qquad v^{\mathsf T}=-\beta E^{-1}.
+\end{equation}
+The remaining right equation, evaluated on the slice, is
+\[
+ s(\xi)=S(\Psi(\xi))=0,
+ \qquad S(x)=a(x)-\beta(x)E(x)^{-1}\gamma(x).
+\]
+The Schur determinant identity gives
+\begin{equation}\label{polar:schur-determinant}
+ Q(x)=\sigma\det(E(x))S(x),
+\end{equation}
+where $\sigma\in\{1,-1\}$ is the sign of the row permutation. It follows
+by subtracting $\beta E^{-1}$ times the lower block rows from the first
+row. Here $S$ is defined on an ambient neighborhood where $E$ is
+invertible, so it can be differentiated in each ambient direction $w_i$.
+
+After substituting~\eqref{polar:solved-kernels}, let
+$g_i(\xi)$ denote the bilinear equation $b^{\mathsf T}B_i p$.
+For any constant ambient direction $w$, differentiation of $S$ gives
+\begin{align*}
+ \partial_w S
+ &=\partial_w a-(\partial_w\beta)E^{-1}\gamma
+       +\beta E^{-1}(\partial_w E)E^{-1}\gamma
+       -\beta E^{-1}(\partial_w\gamma)\\
+ &=(1,-\beta E^{-1})(\partial_w A)
+                 \binom{1}{-E^{-1}\gamma}.
+\end{align*}
+Thus $g_i=(\partial_{w_i}S)\circ\Psi$. In particular this computation
+does not identify an ambient derivative with a derivative along the slice.
+If $f=Q\circ\Psi$ and $q_i=(\partial_{w_i}Q)\circ\Psi$, differentiating
+\eqref{polar:schur-determinant} in the ambient space yields
+\begin{equation}\label{polar:reduced-system}
+ \begin{gathered}
+ f=c\,s,\qquad q_i=c\,g_i+c_i s,\\
+ c=\sigma(\det E)\circ\Psi\ne0,\qquad
+ c_i=\sigma(\partial_{w_i}\det E)\circ\Psi.
+ \end{gathered}
+\end{equation}
+The original equation vector $(f,q_1,\ldots,q_h)$ and the reduced vector
+$(s,g_1,\ldots,g_h)$ therefore differ by a triangular holomorphic matrix
+with diagonal entries $c$. At a common zero their Jacobians differ by
+the value of that invertible matrix. The original zero was simple, so
+the reduced Jacobian is invertible.
+
+Finally, a tangent vector annihilated by the full kernel-system Jacobian
+first satisfies the differentials of the eliminated equations. Their
+invertible auxiliary block determines its $(u,v)$ component from its
+$\xi$ component. The remaining equations and the invertible reduced
+Jacobian force that $\xi$ component to be zero, and hence the auxiliary
+component is zero as well. The full square Jacobian is invertible.
+All $\mu$ selected lifts are thus distinct simple zeros of
+\eqref{polar:kernel-system}.
+
+\emph{Count the selected lifts.}
+The three groups of equations in~\eqref{polar:kernel-system} have
+multidegrees and multiplicities
+\[
+ \begin{array}{c|c|c}
+ \text{equations}&\text{multidegree}&\text{number}\\ \hline
+ A^{\mathrm{hom}} p &(1,1,0)&n\\
+ (b^{\mathsf T}A^{\mathrm{hom}})_j, j\ge2 &(1,0,1)&n-1\\
+ b^{\mathsf T}B_i p &(0,1,1)&h.
+ \end{array}
+\]
+Lemma~\ref{polar:product-count} gives the first inequality
+in~\eqref{polar:ambient-coefficient}, regardless of other zeros or
+components. The product has nonnegative coefficients. Setting $U=V=1$
+sums the coefficients with the same $H$-exponent, and hence bounds the
+particular coefficient by
+\[
+ [H^{h+1}]\,2^h(H+1)^{2n-1}
+       =2^h\binom{2n-1}{h+1}.
+\]
+Since $\mu>0$, we must have $h+1\le2n-1$. For integers $1\le b\le a$,
+\[
+ \binom ab\le\frac{a^b}{b!}\le\left(\frac{ea}{b}\right)^b;
+\]
+the last bound follows from
+$\log(b!)\ge\int_1^b\log t\,dt\ge b\log b-b$.
+Taking $(h+1)$st roots gives
+\[
+ \mu^{1/(h+1)}
+ \le2^{h/(h+1)}\frac{e(2n-1)}{h+1}
+ \le\frac{4en}{h+1},
+\]
+which is~\eqref{polar:ambient-size}. No comparison between $n$ and
+the number $N$ of ambient variables, or between $n$ and $h+2$, was assumed.
+\end{proof}
+
+\subsection{The smooth-hypersurface lower bound}
+
+We have proved that a fixed finite set of simple polar intersections
+persists under arbitrary affine determinant approximation, and that its
+kernel lifts satisfy the multidegree bound. We now use the certificate
+provided by the smooth hypersurface.
+
+\begin{proof}[Proof of Theorem~\ref{thm:smooth-border}]
+Let $n$ be any size for which $G$ is a coefficientwise limit of affine
+$n\times n$ determinants. Choose the directions and $L_0$ from
+Lemma~\ref{polar:simple-points}. The affine hyperplane $L_0=1$ identifies
+with the projective chart $L_0\ne0$ and contains representatives of all
+$\mu=r(r-1)^{d-2}$ selected points. Parametrize it by an affine isomorphism
+\[
+ \Psi:\C^{d-1}\longrightarrow\{x\in\C^d:L_0(x)=1\}.
+\]
+Restricting the homogeneous equations to this hyperplane is precisely
+their dehomogenization in that projective chart. Thus the equations
+\[
+ G\circ\Psi=0,\qquad
+ (\partial_{w_i}G)\circ\Psi=0\quad(1\le i\le d-2)
+\]
+have $\mu$ distinct simple affine solutions. The directions remain constant
+vectors in $\C^d$; they need not be tangent to $L_0=1$.
+Apply Proposition~\ref{polar:ambient-bound} with $N=d$, $h=d-2$, and
+$P=G$. It gives
+\[
+ n\ge\frac{d-1}{4e}
+       \bigl(r(r-1)^{d-2}\bigr)^{1/(d-1)}
+   \ge\frac{(r-1)(d-1)}{4e},
+\]
+where the last inequality uses $r\ge r-1>0$. This holds for every
+admissible size $n$, so taking the minimum proves the theorem.
+\end{proof}
diff --git a/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/sections/selector.tex b/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/sections/selector.tex
new file mode 100644
index 0000000000000000000000000000000000000000..fc94d448cc5da8e0590cfae5dfad59f57112fb03
--- /dev/null
+++ b/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/sections/selector.tex
@@ -0,0 +1,272 @@
+\section{Selecting a coefficient by a permanent projection}
+\label{sec:selector}
+
+We realize a coefficient of a diagonally shifted permanent as an affine
+projection of a larger permanent.  The construction selects prescribed
+exponents in two auxiliary variables while keeping the larger permanent's
+size linear in the order of the shifted permanent.
+
+Fix integers $k\ge1$ and $0\le q\le\lfloor k/2\rfloor$, and put
+$r=k-2q$.  Let $\theta_1,\ldots,\theta_k$ be indeterminates and put
+$D=\operatorname{diag}(\theta_1,\ldots,\theta_k)$.  Let
+$X=(X_{ij})_{1\le i,j\le k}$ be a matrix of independent variables.
+Define the integral coefficient polynomial
+$G_D\in\mathbb Z[\theta_1,\ldots,\theta_k][X_{ij}:1\le i,j\le k]$ by
+\begin{equation}
+ G_D(X)=[s^qt^q]\per_k(X+sI+tD).
+ \label{eq:selector-coefficient}
+\end{equation}
+The brackets denote coefficient extraction in the auxiliary variables
+$s,t$.  For a tuple of values $\theta_i$ in any field, we use the same
+notation $G_D$ for the specialized polynomial.  Write $[k]=\{1,\ldots,k\}$.
+Expanding the diagonal entries gives
+\begin{equation}
+ G_D(X)=
+ \sum_{\substack{S,T\subseteq[k],\ |S|=|T|=q\\S\cap T=\varnothing}}
+ \left(\prod_{i\in T}\theta_i\right)\per(X_{C,C}),
+ \qquad C=[k]\setminus(S\cup T).
+ \label{eq:selector-subsets}
+\end{equation}
+Indeed, $S$ records the diagonal positions contributing $s$, and $T$
+those contributing $t\theta_i$.  The two sets are disjoint because each
+row and column is used once in a permanent term.  After these fixed
+diagonal positions are removed, the remaining permutation is an arbitrary
+permutation of $C$, and these data recover the original term uniquely.
+Thus no additional factorial occurs in
+\eqref{eq:selector-subsets}.  Every summand is homogeneous of degree $r$
+in $X$.
+We use the convention that the permanent of the empty matrix is $1$.
+
+We use the matching interpretation of a permanent.  A matrix's rows and
+columns form the two vertex classes of a bipartite graph, with each entry
+as the weight of the edge joining its row to its column.  The permanent is the
+sum, over perfect matchings, of the products of their edge weights.
+Negative weights are allowed.  Weighted matching gadgets and cancellation are
+classical tools of Valiant's projection framework \cite{Valiant1979Algebra}.
+The practice of fixing external choices before summing internal
+completions also appears in his permanent reduction
+\cite[Lemma~3.1 and its proof]{Valiant1979Permanent}.
+Signed gadgets enforcing equality of edge choices also appear in
+Dell, Husfeldt, Marx, Taslaman and Wahl\'en
+\cite[Section~3 and Figure~2]{DellHusfeldtMarxTaslamanWahlen2014}.
+Here we give the weights and completion counts needed for a construction
+of linear size.
+
+\begin{proposition}[Coefficient projection]\label{prop:selector}
+Let $k\ge1$ and $0\le q\le\lfloor k/2\rfloor$ be integers, put $r=k-2q$,
+and let $D=\operatorname{diag}(\theta_1,\ldots,\theta_k)$ with
+$\theta_i\in\C$.  For every integer $m\ge11k-2q$, there is an $m\times m$
+matrix $A(X)$ of complex affine-linear forms such that
+\begin{equation}
+ \per_m(A(X))=a_0G_D(X),
+ \qquad a_0=(-2)^{2k}\big((k-q)!\big)^4\ne0.
+ \label{eq:selector-scalar}
+\end{equation}
+Its entrywise homogenization $B(X,z)$ is a matrix of homogeneous linear
+forms whose permanent is the degree-$m$ homogenization of $a_0G_D$:
+\begin{equation}
+ \per_m(B(X,z))=a_0z^{m-r}G_D(X).
+ \label{eq:selector-homogeneous}
+\end{equation}
+The construction requires no distinctness or nonvanishing assumption on
+the entries of $D$.
+\end{proposition}
+
+\begin{proof}
+Start with $k$ main rows and columns carrying the edges $X_{ij}$.
+At each diagonal position add two labeled special edges, with weights
+\[
+ w_{i,1}=1,\qquad w_{i,2}=\theta_i.
+\]
+We call them the first and second color.  At this intermediate stage
+they are parallel to the ordinary edge $X_{ii}$.  The three labels record
+the choices $X_{ii}$, $s$, and $t\theta_i$ in the diagonal entry, with the
+factors $s,t$ omitted from the two special weights.  The coefficient of
+$s^qt^q$ in the shifted permanent is therefore the weighted sum of main
+matchings using exactly $q$ special edges of each color, as in
+\eqref{eq:selector-subsets}.
+We will count each color in a separate gate and couple each special
+main edge to the corresponding distinguished edge of its gate.
+
+\emph{A gate imposing one prescribed count.}
+Put $h=k-q$.  Take $k$ original rows and columns, with distinguished
+diagonal edges of weight $1$, and $h$ additional rows and columns.
+Join every original row to every additional column and every additional
+row to every original column, all with weight $1$.  There are no other
+edges.  Every additional row must occupy an original column, and every
+additional column must be occupied by an original row.  Consequently
+exactly $q$ distinguished edges occur in a perfect matching.
+
+For each prescribed distinguished set $S\subseteq[k]$ of size $q$, the
+remaining original rows match the additional columns in $h!$ ways, and
+the additional rows match the remaining original columns in $h!$ ways.
+These choices are independent.  Hence the multiplicity for this exact
+set is
+\begin{equation}
+ (h!)^2=((k-q)!)^2.
+ \label{eq:selector-gate-count}
+\end{equation}
+This also covers $q=0$: all original rows and columns then match across
+the two groups.
+
+\emph{A gadget equating two edge choices.}
+Call the main and gate vertices \emph{outer}.  For one special main edge
+of weight $w$ and its distinguished gate edge
+of weight $1$, write the outer row endpoints as $u_1,u_2$ and the outer
+column endpoints as $v_1,v_2$, respectively.  These two edges have distinct
+endpoints.  Delete them and add three fresh internal rows and three fresh
+internal columns, with weight matrix
+\begin{equation}
+ H=\begin{pmatrix}-1&-1&1\\-1&-1&1\\1&1&1\end{pmatrix}.
+ \label{eq:selector-gadget}
+\end{equation}
+For $i=1,2$, add an incoming edge from $u_i$ to internal column $i$ and
+an outgoing edge from internal row $i$ to $v_i$.  All four weights are
+$1$, except that the first incoming edge has weight $-2w$.
+
+Fix the subset $I\subseteq\{1,2\}$ of incoming edges used and the
+subset $O\subseteq\{1,2\}$ of outgoing edges used.  Let $\Phi_w(I,O)$
+be the total weight of their internal completions, including the weights
+of those boundary edges.  If $|I|\ne|O|$,
+the remaining internal row and column sets have different sizes, so
+there is no completion and $\Phi_w(I,O)=0$.  Otherwise
+\begin{equation}
+ \Phi_w(I,O)=(-2w)^{\mathbf1_{\{1\in I\}}}
+ \per H_{\{1,2,3\}\setminus O,\,\{1,2,3\}\setminus I}.
+ \label{eq:selector-boundary-weight}
+\end{equation}
+The complete table of $\Phi_w$ is
+\begin{equation}
+ \begin{array}[t]{c|rrrr}
+ I\backslash O&\varnothing&\{1\}&\{2\}&\{1,2\}\\\hline
+ \varnothing&-2&0&0&0\\
+ \{1\}&0&0&0&0\\
+ \{2\}&0&0&0&0\\
+ \{1,2\}&0&0&0&-2w.
+ \end{array}
+ \label{eq:selector-boundary-table}
+\end{equation}
+To verify the six balanced cases, first $\per H=-2$.  Deleting any one
+row and any one column indexed in $\{1,2\}$ leaves
+$\left(\begin{smallmatrix}-1&1\\1&1\end{smallmatrix}\right)$, whose
+permanent is $-1+1=0$.  This includes the crossed choices
+$I=\{1\},O=\{2\}$ and $I=\{2\},O=\{1\}$.  Deleting both interface
+rows and columns leaves the entry $H_{33}=1$; the four external edge
+weights then have product $-2w$.  The ten unbalanced cases have already
+been excluded.
+
+Thus $\Phi_w(I,O)$ vanishes unless $I=O=\varnothing$ or
+$I=O=\{1,2\}$.  In these two states the weights $-2$ and $-2w$
+reproduce neither outer edge being used or both being used, with the
+common factor $-2$.  The four balanced singleton states can have
+individual internal matchings; it is their sums that cancel.
+
+\emph{Combining the two counts.}
+Take two disjoint copies of the count gate, one for each color.
+For every $i\in[k]$ and $c\in\{1,2\}$, replace the special main edge
+of weight $w_{i,c}$ and its distinguished gate edge by the equality
+gadget.  Use fresh internal rows and columns for all $2k$ gadgets.
+At a fixed index $i$, the two replacements share the main row and column,
+while their internal vertices and their gate endpoints remain distinct.
+Figure~\ref{fig:selector-wiring} shows these attachments.
+
+\input{figures/selector-wiring}
+
+Group the full matching sum by the set $E$ of chosen edges outside the
+internal $H$-blocks.
+This set includes all chosen incoming and outgoing edges and must cover
+each outer vertex exactly once.  Let $w_{\mathrm{out}}(E)$ be the product
+of the weights of the chosen edges whose two endpoints are outer.
+For the gadget indexed by $(i,c)$, write $I_{i,c},O_{i,c}$ for its
+boundary state.  Since the internal vertex sets are disjoint, the total
+weight of all completions of $E$ is
+\[
+ w_{\mathrm{out}}(E)
+ \prod_{i=1}^k\prod_{c=1}^2
+ \Phi_{w_{i,c}}(I_{i,c},O_{i,c}).
+\]
+Each factor $\Phi_{w_{i,c}}$ already includes its boundary edge weights;
+$w_{\mathrm{out}}(E)$ contains only the remaining chosen weights.
+By \eqref{eq:selector-boundary-table}, this
+product vanishes unless every gadget has $I_{i,c}=O_{i,c}=\varnothing$
+or $I_{i,c}=O_{i,c}=\{1,2\}$.  Unbalanced states have no completion,
+and balanced singleton states cancel.
+
+Now restrict to choices $E$ for which every gadget has one of these two states.
+For each gadget with $I_{i,c}=O_{i,c}=\{1,2\}$, replace its four boundary
+edges by the original special main edge and distinguished gate edge.
+For an empty state, insert neither original edge.  Because $E$ covers every
+outer vertex exactly once, this contraction gives a perfect matching in the
+graph before replacement, with each special main edge used if and only if
+its paired gate edge is used.  Conversely, each such matching determines
+the set $E$ and its boundary states uniquely.  The table assigns $-2w_{i,c}$
+to a used pair of original weight $w_{i,c}$, and $-2$ to an unused pair of weight
+$1$.  Hence its contribution is its original matching weight multiplied
+by exactly one factor $-2$ for each of the $2k$ gadgets.  Shared main
+endpoints enter through the matching constraint enforced when $E$ is fixed;
+the internal factorization uses disjoint vertices.
+
+Each gate now forces exactly $q$ special edges of its color.  The two
+selected sets are disjoint by the main matching condition.  For these
+sets, the two gates contribute $((k-q)!)^4$ by
+\eqref{eq:selector-gate-count}, while the main matching sum is the
+corresponding summand of \eqref{eq:selector-subsets}.  The $2k$ gadgets
+contribute $(-2)^{2k}$.  This proves \eqref{eq:selector-scalar} for the
+unpadded construction.
+
+Its number of rows, and separately its number of columns, is
+\begin{equation}
+ k+2(2k-q)+3(2k)=11k-2q.
+ \label{eq:selector-size}
+\end{equation}
+These terms count the main graph, the two count gates, and the fresh
+internal blocks.  The final graph has no parallel special edges: their
+replacements lead to distinct internal columns.  Its weight matrix
+therefore has entries that are variables or constants.  Append an
+identity block to reach order $m$; its unique matching has weight $1$,
+so the permanent identity is preserved.
+
+Finally, write this padded affine matrix as
+$A(X)=A_0+\sum_{i,j}X_{ij}A_{ij}$ and set
+$B(X,z)=zA_0+\sum_{i,j}X_{ij}A_{ij}$.  For $z\ne0$, homogeneity gives
+\[
+ \per_m B(X,z)
+ =z^m\per_m A(X/z)
+ =a_0z^mG_D(X/z)
+ =a_0z^{m-r}G_D(X).
+\]
+Since $m\ge k\ge r$, both sides are polynomials.  Their equality for
+$z\ne0$ proves their identity also at $z=0$.  The identity block is
+homogenized along with every other constant entry, so it contributes
+the additional power of $z$ required in
+\eqref{eq:selector-homogeneous}.
+\end{proof}
+
+The affine identity yields a comparison for every fixed affine substitution.
+
+\begin{corollary}\label{cor:selector-border}
+Under the hypotheses of Proposition~\ref{prop:selector}, let $d\ge1$ be an
+integer and let
+$\Lambda:\C^d\to\Mat_k(\C)$ be any fixed affine map.  Then
+\[
+ \bdc(G_D\circ\Lambda)\le\bdc(\per_m).
+\]
+In particular, this holds for every fixed linear restriction of $G_D$.
+\end{corollary}
+
+\begin{proof}
+Proposition~\ref{prop:selector} gives
+$G_D\circ\Lambda=a_0^{-1}\per_m\circ A\circ\Lambda$.
+Apply Lemma~\ref{lem:affine-substitution} to the affine map
+$A\circ\Lambda$ and the nonzero scalar $a_0^{-1}$.
+Both are fixed before any determinant approximation is chosen.
+\end{proof}
+
+\begin{remark}\label{rem:selector-characteristic}
+The coefficient formula \eqref{eq:selector-coefficient} and the subset
+formula \eqref{eq:selector-subsets} are integral and can be specialized
+over any field.  The projection used for the complex border comparison
+requires the nonzero scalar $a_0$, whose factor $(-2)^{2k}$ vanishes in
+characteristic two.  The characteristic-two argument uses the integral
+coefficient expression specialized directly to $\overline{\mathbb F}_2$.
+\end{remark}
diff --git a/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/sections/transfer.tex b/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/sections/transfer.tex
new file mode 100644
index 0000000000000000000000000000000000000000..0b04628579d50f3dad75fdb00e22f3c0f56ff709
--- /dev/null
+++ b/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026/build/source/sections/transfer.tex
@@ -0,0 +1,309 @@
+\section{Characteristic transfer and smooth linear sections}
+\label{sec:transfer}
+
+Proposition~\ref{prop:gradient-codimension} bounds an affine gradient
+locus in characteristic two.  We now transfer that bound to complex
+parameters, then choose a linear section on which the coefficient
+polynomial defines a smooth projective hypersurface.  The final proof
+will combine the resulting affine permanent projection with
+Theorem~\ref{thm:smooth-border} to obtain the cubic lower bound.
+
+\subsection{Transferring the gradient bound}
+
+The proper family used for specialization is the projective zero scheme
+of the partial derivatives.  In characteristic two this scheme need not
+be the singular scheme of the hypersurface itself.  After passing to
+characteristic zero, Euler's identity will identify the two.
+
+\begin{lemma}[Specialization of a gradient locus]
+\label{lem:characteristic-transfer}
+Let
+\[
+ F(\theta,x)\in
+ \mathbb Z[\theta_1,\ldots,\theta_a][x_1,\ldots,x_v]
+\]
+be homogeneous of degree \(r\ge2\) in \(x\), where \(v\ge2\), and let
+\(1\le\delta\le v\) be an integer.  Suppose some
+\(\bar\theta\in\overline{\mathbb F}_2^{\,a}\) satisfies
+\[
+ \dim V_{\mathbb A^v_{\overline{\mathbb F}_2}}
+       \bigl(\partial_{x_1}F_{\bar\theta},\ldots,
+             \partial_{x_v}F_{\bar\theta}\bigr)
+ \le v-\delta.
+\]
+Then there is a nonempty Zariski open subset
+\(U\subset\mathbb A^a_{\C}\) such that, for every \(\theta\in U\),
+the polynomial \(F_\theta\) is nonzero and
+\begin{equation}
+ \dim\operatorname{Sing}V_{\PP^{v-1}_{\C}}(F_\theta)
+ \le v-1-\delta.
+ \label{eq:transfer-singular-bound}
+\end{equation}
+We use dimension \(-1\) for the empty projective scheme.
+\end{lemma}
+
+\begin{proof}
+Put \(R=\mathbb Z[\theta_1,\ldots,\theta_a]\) and
+\(S=\operatorname{Spec}R\).  The homogeneous ideal
+\[
+ I=(\partial_{x_1}F,\ldots,\partial_{x_v}F)\subset R[x_1,\ldots,x_v]
+\]
+defines a closed subscheme
+\begin{equation}
+ \mathcal Z=\operatorname{Proj}(R[x]/I)\subset\PP^{v-1}_S.
+ \label{eq:transfer-projective-family}
+\end{equation}
+Its structural morphism \(\pi:\mathcal Z\to S\) is projective, hence proper.
+Differentiation commutes with specialization, and the closed projective
+subscheme in \eqref{eq:transfer-projective-family} commutes with base
+change.  Thus its geometric fibers are exactly the projective
+gradient-zero schemes of the specialized polynomials.
+
+The derivatives have positive homogeneous degree \(r-1\).
+At \(\bar\theta\), their affine zero set is a cone.  If its
+projectivization is nonempty, its dimension is one less than that of the
+cone: on each projective coordinate chart the corresponding punctured
+cone is a product with \(\mathbb G_m\).  If the projectivization is
+empty, the same upper bound holds with our convention.  Hence
+\[
+ \dim\mathcal Z_{\bar\theta}\le v-1-\delta.
+\]
+Let \(s\in S\) be the image of the geometric point \(\bar\theta\).
+The dimension of a scheme of finite type over a field is unchanged by
+extending that field, so the same bound holds for \(\mathcal Z_s\).
+If \(v-1-\delta\ge0\), upper semicontinuity of fiber dimension for the
+proper morphism \(\pi\) gives an open neighborhood \(U_S\) of \(s\)
+on which this bound holds \cite[Tag~0D4I]{StacksProject}.
+If \(v-1-\delta=-1\), the fiber \(\mathcal Z_s\) is empty.
+Properness makes \(\pi(\mathcal Z)\) closed, so
+\(U_S=S\setminus\pi(\mathcal Z)\) is an open neighborhood of \(s\)
+whose fibers are all empty.  Thus the required neighborhood exists
+in every case, without a flatness assumption.
+
+Since \(S\) is integral, \(U_S\) contains its generic point, which has
+characteristic zero.  To see explicitly that the open has
+complex points, choose a nonempty principal open
+\(D(h)\subset U_S\), with \(0\ne h\in R\).  The polynomial \(h\)
+remains nonzero over \(\C\), so \(D(h)_\C\) is nonempty.  We may take
+\(U=(U_S)_\C\).  The same invariance under field extension shows that
+every complex fiber over \(U\) satisfies
+\begin{equation}
+ \dim V_{\PP^{v-1}_{\C}}(\nabla F_\theta)
+ \le v-1-\delta.
+ \label{eq:transfer-complex-gradient}
+\end{equation}
+
+Since \(\delta>0\), \(F_\theta\) cannot be the zero polynomial:
+otherwise the left side of \eqref{eq:transfer-complex-gradient}
+would be \(v-1\).  Over \(\C\), Euler's identity gives
+\[
+ rF_\theta=\sum_{i=1}^v x_i\partial_{x_i}F_\theta.
+\]
+Thus the ideals \((F_\theta,\nabla F_\theta)\) and
+\((\nabla F_\theta)\) agree.  These equations describe the actual
+projective hypersurface singular scheme.  Indeed, on the chart
+\(x_i\ne0\), put \(z_j=x_j/x_i\) for \(j\ne i\) and
+\(f_i=F_\theta/x_i^r\).  The hypersurface has equation \(f_i=0\)
+and cotangent presentation
+\[
+ \cO_{V(f_i)}
+ \xrightarrow{\,df_i\,}
+ \cO_{V(f_i)}^{\,v-1}
+ \longrightarrow\Omega_{V(f_i)/\C}\longrightarrow0.
+\]
+Its \((v-2)\)-nd Fitting ideal is generated by the local partial
+derivatives of \(f_i\).  On \(f_i=0\), these are the homogeneous
+gradient equations in this chart: Euler's identity supplies the
+derivative in the omitted \(x_i\) direction.  The Jacobian criterion
+therefore identifies this scheme with the singular scheme.
+Equation~\eqref{eq:transfer-complex-gradient} proves
+\eqref{eq:transfer-singular-bound}.
+\end{proof}
+
+All coefficients in \eqref{eq:transfer-projective-family} are integral;
+in particular, the construction does not invert \(2\) or \(r\).
+Division by \(r\) occurs only in characteristic zero.  In applying the
+lemma to \(G_D\), we use its coefficient formula
+\eqref{eq:selector-coefficient}, before multiplying by the projection
+scalar \(a_0\), which vanishes in characteristic two.
+
+\subsection{A smooth section of the required dimension}
+
+Avoiding the singular locus of a hypersurface is one condition on a
+linear section.  The section must also meet its smooth locus
+transversely.  The following incidence calculation establishes both
+conditions and keeps track of the projective dimension.
+
+\begin{proposition}[Smooth linear section]\label{prop:smooth-slice}
+Let \(F\) be a nonzero homogeneous complex polynomial of degree \(r\ge2\)
+on a vector space \(V\) of dimension \(v\).  Set
+\(Y=V_{\PP(V)}(F)\), and suppose
+\[
+ \dim\operatorname{Sing}Y\le v-1-\delta
+\]
+for an integer \(\delta\ge1\).  If \(2\le d<v\) and
+\(d-1<\delta\), then a nonempty Zariski open subset of
+\(\operatorname{Gr}(d,V)\) consists of subspaces \(H\) for which
+\(F|_H\) is nonzero and defines a smooth degree-\(r\) hypersurface
+in \(\PP(H)\).
+\end{proposition}
+
+\begin{proof}
+Write \(\mathcal G=\operatorname{Gr}(d,V)\), so
+\(\dim\mathcal G=d(v-d)\).  A point of \(\mathcal G\) is a vector
+subspace of dimension \(d\), whose projectivization has dimension
+\(d-1\).
+
+First consider pairs \((p,H)\) with
+\(p\in\operatorname{Sing}Y\) and \(p\in\PP(H)\).
+For fixed \(p\), the choices of \(H\) form
+\(\operatorname{Gr}(d-1,V/p)\), of dimension \((d-1)(v-d)\);
+here \(p\) also denotes its line in \(V\).  This incidence has
+dimension at most
+\[
+ (v-1-\delta)+(d-1)(v-d)
+ =\dim\mathcal G+d-1-\delta
+ <\dim\mathcal G.
+\]
+Its projection to \(\mathcal G\) is closed because
+\(\operatorname{Sing}Y\) is projective.  It is a proper subset, so
+outside it the section avoids every singular point of \(Y\).
+
+For \(p=[x]\in Y_{\rm reg}\), the nonzero differential \(dF_x\)
+has a kernel \(K_p\subset V\) of dimension \(v-1\).  Euler's
+identity at \(F(x)=0\) shows that \(p\subset K_p\).
+If \(p\subset H\), the restricted equation is singular at \(p\)
+exactly when
+\[
+ p\subset H\subset K_p.
+\]
+The homogeneous partial derivatives define the Gauss morphism
+\(Y_{\rm reg}\to\PP(V^*)\), \(p=[x]\mapsto[dF_x]\).
+The pairs above form a closed incidence in \(Y_{\rm reg}\times\mathcal G\):
+they are the inverse image of the closed flag incidence
+\(p\subset H\subset\ker\ell\) under \((p,H)\mapsto(p,H,[dF_x])\).
+For fixed \(p\), these subspaces form
+\(\operatorname{Gr}(d-1,K_p/p)\), of dimension
+\((d-1)(v-d-1)\).  Since \(\dim Y_{\rm reg}\le v-2\), this
+tangency incidence has dimension at most
+\[
+ (v-2)+(d-1)(v-d-1)=d(v-d)-1.
+\]
+Its image in \(\mathcal G\) is constructible, and the closure of
+that image has dimension at most \(\dim\mathcal G-1\).
+Removing this closure and the preceding singular-point incidence
+image leaves a nonempty open subset of the irreducible
+Grassmannian.
+
+For \(H\) in that open, every zero of \(F|_H\) has nonzero
+differential on \(H\).  The restriction cannot vanish identically:
+if it did, every point \(p\in\PP(H)\) would lie in \(Y\), singular
+avoidance would put it in \(Y_{\rm reg}\), and differentiation of
+the zero restriction would force \(H\subset K_p\), contrary to
+the tangency exclusion.
+
+Thus \(F|_H\) is a nonzero homogeneous polynomial of degree \(r\).
+Since \(d\ge2\), it defines a nonempty hypersurface of dimension
+\(d-2\) in \(\PP(H)\).  The nonzero differentials prove its
+smoothness.
+\end{proof}
+
+\subsection{A smooth form obtained from the permanent}
+
+We now apply the two geometric results to the coefficient polynomial.
+The resulting form has linear degree and quadratically many variables,
+and is an affine projection of a permanent of linear size.
+
+\begin{samepage}
+\begin{proposition}[A smooth form obtained from the permanent]
+Let \(k\ge128\) be an integer\label{prop:smooth-projection}, and put
+\[
+ q=\left\lfloor\frac{k-1}{3}\right\rfloor,\qquad
+ r=k-2q,\qquad d=\left\lfloor\frac{k^2}{100}\right\rfloor.
+\]
+There is a nonzero homogeneous polynomial
+\(G\in\C[x_1,\ldots,x_d]\) of degree \(r\) such that its zero set in
+\(\PP^{d-1}_{\C}\) is smooth and, for every integer \(m\ge11k-2q\),
+there is an affine map \(\Phi_m:\C^d\to\Mat_m(\C)\) satisfying
+\[
+ G=\per_m\circ\Phi_m.
+\]
+The parameters also satisfy
+\[
+ r-1\ge\frac{k}{4},\qquad d-1\ge\frac{k^2}{200}.
+\]
+\end{proposition}
+\end{samepage}
+
+\begin{proof}
+Set \(T=\binom{q+2}{2}\).  Since \(k\ge128\),
+\[
+ q\ge\frac{k-4}{3}\ge\frac{k}{4},\qquad
+ T\ge\frac{q^2}{2}\ge\frac{k^2}{32}.
+\]
+Consequently
+\begin{equation}
+ \Delta:=T-k
+ \ge\frac{k^2}{32}-k
+ \ge\frac{k^2}{64}>0.
+ \label{eq:transfer-codimension-size}
+\end{equation}
+Also \(T\le(k+2)(k+5)/18<k^2\) for these \(k\), so
+\(1\le\Delta\le k^2\).
+
+Choose pairwise distinct
+\(\bar\theta_1,\ldots,\bar\theta_k\in\overline{\mathbb F}_2\)
+and put \(\bar D=\operatorname{diag}(\bar\theta_1,\ldots,\bar\theta_k)\).
+Proposition~\ref{prop:gradient-codimension} gives
+\[
+ \dim V_{\mathbb A^{k^2}_{\overline{\mathbb F}_2}}(\nabla G_{\bar D})
+ \le k^2+k-T=k^2-\Delta.
+\]
+The coefficient formula \eqref{eq:selector-coefficient}, with the
+\(\theta_i\) treated as indeterminates, is integral in these parameters
+and homogeneous of degree \(r\) in the \(k^2\) matrix variables.
+Moreover,
+\[
+ r-1=k-2q-1
+ \ge\frac{k-1}{3}\ge\frac{k}{4},
+\]
+so \(r\ge2\).  Lemma~\ref{lem:characteristic-transfer} supplies a
+nonempty open subset \(U\subset\mathbb A^k_{\C}\).  Choose
+\(\theta=(\theta_1,\ldots,\theta_k)\in U(\C)\) and put
+\(D=\operatorname{diag}(\theta_1,\ldots,\theta_k)\).  Then \(G_D\ne0\)
+and
+\[
+ \dim\operatorname{Sing}V_{\PP^{k^2-1}_{\C}}(G_D)
+ \le k^2-1-\Delta.
+\]
+The complex coefficient projection permits arbitrary diagonal entries
+for this \(D\).
+
+For \(d=\lfloor k^2/100\rfloor\), we have \(2\le d<k^2\), and
+\eqref{eq:transfer-codimension-size} gives
+\[
+ d-1<\frac{k^2}{100}<\frac{k^2}{64}\le\Delta.
+\]
+Proposition~\ref{prop:smooth-slice} therefore supplies a subspace
+\(H\subset\Mat_k(\C)\) of dimension \(d\) such that \(G_D|_H\)
+is nonzero and defines a smooth degree-\(r\) hypersurface in \(\PP(H)\).
+Choose a linear isomorphism \(\Lambda:\C^d\to H\) and set
+\(G=G_D\circ\Lambda\).  This gives the asserted smooth form.  The
+remaining scale estimate is
+\[
+ d-1\ge\frac{k^2}{100}-2\ge\frac{k^2}{200},
+\]
+where the last inequality holds for \(k\ge20\).
+
+Finally, \(0\le3q\le k-1\), so \(0\le q\le\lfloor k/2\rfloor\).
+For each integer
+\(m\ge11k-2q\), Proposition~\ref{prop:selector} gives a matrix
+\(A_m(X)\) of affine-linear forms with
+\[
+ \per_m(A_m(X))=a_0G_D(X),\qquad a_0\ne0.
+\]
+Compose \(A_m\) with \(\Lambda\) and multiply its first row by
+\(a_0^{-1}\).  The resulting affine map
+\(\Phi_m:\C^d\to\Mat_m(\C)\) satisfies
+\(\per_m\circ\Phi_m=G\), as required.
+\end{proof}
diff --git a/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/README.md b/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..e9655145f785536f4972a906fd2df312f9d92758
--- /dev/null
+++ b/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/README.md
@@ -0,0 +1,18 @@
+# [Three fixed points on the symplectic quadric threefold](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 23, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026,
+  author = {{OpenAI}},
+  title = {{Three fixed points on the symplectic quadric threefold}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/paper.pdf}{OAI:A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/build/main.tex b/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/build/main.tex
new file mode 100644
index 0000000000000000000000000000000000000000..3dac156377782243b9ba6193f9bba8f0a76f5984
--- /dev/null
+++ b/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/build/main.tex
@@ -0,0 +1,46 @@
+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage[margin=1in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{microtype}
+\usepackage{xcolor}
+\usepackage{hyperref}
+\hypersetup{colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black,pdftitle={Three fixed points on the symplectic quadric threefold},pdfauthor={OpenAI}}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\numberwithin{equation}{section}
+\DeclareMathOperator{\Crit}{Crit}
+\DeclareMathOperator{\CritSet}{crit}
+\DeclareMathOperator{\Fix}{Fix}
+\DeclareMathOperator{\Ham}{Ham}
+\newcommand{\id}{\mathrm{id}}
+\newcommand{\R}{\mathbb R}
+\newcommand{\C}{\mathbb C}
+\newcommand{\CP}{\mathbb C\mathrm P}
+\allowdisplaybreaks[1]
+\setlength{\emergencystretch}{1.5em}
+\title{Three fixed points on the symplectic quadric threefold}
+\author{OpenAI}
+\date{September 23, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We construct a smooth Hamiltonian diffeomorphism of the complex quadric
+threefold with exactly three fixed points, at least one of which is
+degenerate. The critical number and the unit-inclusive rational cup length
+of this manifold are both four. Thus the example disproves the unrestricted
+critical-number and rational cup-length forms of the Arnold conjecture.
+\end{abstract}
+\input{sections/00-introduction}
+\input{sections/01-reduction}
+\input{sections/02-quadric}
+\input{sections/04-three-points}
+\input{sections/03-critical-bound}
+\input{sections/05-conclusion}
+\input{references}
+\end{document}
diff --git a/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/build/references.tex b/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/build/references.tex
new file mode 100644
index 0000000000000000000000000000000000000000..3ce00d8a8c3b86cf9a179e852ddce7c7e586fec7
--- /dev/null
+++ b/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/build/references.tex
@@ -0,0 +1,124 @@
+\begin{thebibliography}{99}
+\bibitem{Arnold1986}
+V.~I.~Arnol'd,
+\emph{First steps in symplectic topology},
+Russian Mathematical Surveys \textbf{41} (1986), no.~6, 1--21.
+\href{https://doi.org/10.1070/RM1986v041n06ABEH004221}{doi:10.1070/RM1986v041n06ABEH004221}.
+
+\bibitem{BHS2018}
+L.~Buhovsky, V.~Humili\`ere, and S.~Seyfaddini,
+\emph{A $C^0$ counterexample to the Arnold conjecture},
+Inventiones Mathematicae \textbf{213} (2018), no.~2, 759--809.
+\href{https://doi.org/10.1007/s00222-018-0797-x}{doi:10.1007/s00222-018-0797-x}.
+
+\bibitem{ConleyZehnder1983}
+C.~C.~Conley and E.~Zehnder,
+\emph{The Birkhoff--Lewis fixed point theorem and a conjecture of V.~I.~Arnold},
+Inventiones Mathematicae \textbf{73} (1983), 33--49.
+\href{https://doi.org/10.1007/BF01393824}{doi:10.1007/BF01393824}.
+
+\bibitem{Floer1989}
+A.~Floer,
+\emph{Symplectic fixed points and holomorphic spheres},
+Communications in Mathematical Physics \textbf{120} (1989), no.~4, 575--611.
+\href{https://doi.org/10.1007/BF01260388}{doi:10.1007/BF01260388}.
+
+\bibitem{Fortune1985}
+B.~Fortune,
+\emph{A symplectic fixed point theorem for $\CP^n$},
+Inventiones Mathematicae \textbf{81} (1985), no.~1, 29--46.
+\href{https://doi.org/10.1007/BF01388770}{doi:10.1007/BF01388770}.
+
+\bibitem{FukayaOno1999}
+K.~Fukaya and K.~Ono,
+\emph{Arnold conjecture and Gromov--Witten invariant},
+Topology \textbf{38} (1999), no.~5, 933--1048.
+\href{https://doi.org/10.1016/S0040-9383(98)00042-1}{doi:10.1016/S0040-9383(98)00042-1}.
+
+\bibitem{Golovko2020}
+R.~Golovko,
+\emph{On variants of Arnold conjecture},
+Archivum Mathematicum \textbf{56} (2020), no.~5, 277--286.
+\href{https://doi.org/10.5817/AM2020-5-277}{doi:10.5817/AM2020-5-277}.
+
+\bibitem{Gong2025}
+W.~Gong,
+\emph{Degenerate symplectic fixed points and Gromov--Witten invariants},
+\href{https://arxiv.org/abs/2507.04191v5}{arXiv:2507.04191v5}, December 21, 2025.
+
+\bibitem{Hofer1988}
+H.~Hofer,
+\emph{Lusternik--Schnirelman-theory for Lagrangian intersections},
+Annales de l'Institut Henri Poincar\'e C, Analyse non lin\'eaire
+\textbf{5} (1988), no.~5, 465--499.
+\href{https://doi.org/10.1016/S0294-1449(16)30339-0}{doi:10.1016/S0294-1449(16)30339-0}.
+
+\bibitem{Jiao2026}
+H.~Jiao,
+\emph{A smooth Hamiltonian diffeomorphism with two fixed points on
+$S^2\times S^2$},
+\href{https://arxiv.org/abs/2609.33626v1}{arXiv:2609.33626v1}, September 27, 2026.
+
+\bibitem{LiuTian1998}
+G.~Liu and G.~Tian,
+\emph{Floer homology and Arnold conjecture},
+Journal of Differential Geometry \textbf{49} (1998), no.~1, 1--74.
+\href{https://doi.org/10.4310/jdg/1214460936}{doi:10.4310/jdg/1214460936}.
+
+\bibitem{LusternikSchnirelmann1934}
+L.~Lusternik and L.~Schnirelmann,
+\emph{M\'ethodes topologiques dans les probl\`emes variationnels.
+I. Espaces \`a un nombre fini de dimensions},
+Actualit\'es scientifiques et industrielles 188, Hermann, Paris, 1934.
+
+\bibitem{Ma2013}
+R.~Ma,
+\emph{Proofs on Arnold conjectures},
+\href{https://arxiv.org/abs/0808.0613v7}{arXiv:0808.0613v7}, July 5, 2013.
+
+\bibitem{Mare1997}
+A.-L.~Mare,
+\emph{Topology of isotropy orbits},
+doctoral dissertation, Universit\"at Augsburg, 1997.
+\href{https://uregina.ca/~mareal/papers/Teza.pdf}{Author-hosted text}.
+
+\bibitem{Palais1979}
+R.~S.~Palais,
+\emph{The principle of symmetric criticality},
+Communications in Mathematical Physics \textbf{69} (1979), no.~1, 19--30.
+\href{https://doi.org/10.1007/BF01941322}{doi:10.1007/BF01941322}.
+
+\bibitem{Ruan1999}
+Y.~Ruan,
+\emph{Virtual neighborhoods and pseudo-holomorphic curves},
+Turkish Journal of Mathematics \textbf{23} (1999), no.~1, 161--232.
+\href{https://journals.tubitak.gov.tr/math/vol23/iss1/10/}{Publisher record};
+\href{https://arxiv.org/abs/alg-geom/9611021v2}{arXiv:alg-geom/9611021v2}.
+
+\bibitem{RudyakOprea1999}
+Y.~B.~Rudyak and J.~Oprea,
+\emph{On the Lusternik--Schnirelmann category of symplectic manifolds
+and the Arnold conjecture},
+Mathematische Zeitschrift \textbf{230} (1999), no.~4, 673--678.
+\href{https://doi.org/10.1007/PL00004709}{doi:10.1007/PL00004709}.
+
+\bibitem{Takens1968}
+F.~Takens,
+\emph{The minimal number of critical points of a function on a compact
+manifold and the Lusternik--Schnirelman category},
+Inventiones Mathematicae \textbf{6} (1968), no.~3, 197--244.
+\href{https://doi.org/10.1007/BF01404825}{doi:10.1007/BF01404825}.
+
+\bibitem{Weber2019}
+J.~Weber,
+\emph{Conley pairs in geometry---Lusternik--Schnirelmann theory and more},
+Expositiones Mathematicae \textbf{37} (2019), no.~1, 25--47.
+\href{https://doi.org/10.1016/j.exmath.2017.10.003}{doi:10.1016/j.exmath.2017.10.003}.
+
+\bibitem{Yorke1969}
+J.~A.~Yorke,
+\emph{Periods of periodic solutions and the Lipschitz constant},
+Proceedings of the American Mathematical Society \textbf{22} (1969),
+no.~2, 509--512.
+\href{https://doi.org/10.1090/S0002-9939-1969-0245916-7}{doi:10.1090/S0002-9939-1969-0245916-7}.
+\end{thebibliography}
diff --git a/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/build/sections/00-introduction.tex b/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/build/sections/00-introduction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..9af2072ff5fcb60efff3a587e9a95beac67391f0
--- /dev/null
+++ b/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/build/sections/00-introduction.tex
@@ -0,0 +1,159 @@
+\section{Introduction}\label{sec:introduction}
+
+Arnold's fixed-point conjecture compares Hamiltonian dynamics with the
+critical-point theory of smooth functions. For a sufficiently short
+autonomous Hamiltonian flow on a closed symplectic manifold, its fixed points are
+exactly the critical points of its Hamiltonian. The conjecture asks
+whether an arbitrary Hamiltonian diffeomorphism must retain the minimum
+number of critical points forced by the topology of the manifold.
+Arnold's formulation explicitly includes the count of geometrically
+distinct points~\cite[Section~2]{Arnold1986}. We construct a smooth
+counterexample on the complex quadric threefold, which also violates the
+weaker rational cup-length bound.
+
+\paragraph{The two degenerate bounds.}
+For a closed smooth manifold $M$, write
+\[
+ \CritSet(h)=\{q\in M:dh_q=0\},\qquad
+ \Crit(M)=\min_{h\in C^\infty(M,\R)}\#\CritSet(h).
+\]
+The minimum here permits degenerate critical points, whereas the
+\emph{Morse number} minimizes only over Morse functions. Let
+$\operatorname{cuplength}(M;\mathbb Q)$ be one plus the maximum number
+of positive-degree rational cohomology classes with nonzero cup product.
+Thus our cup-length convention includes the unit. The
+Lusternik--Schnirelmann inequality gives
+\[
+ \operatorname{cuplength}(M;\mathbb Q)\le\Crit(M);
+\]
+Section~\ref{sec:critical-bound} recalls the cup-product argument in a
+form that applies to arbitrary smooth functions.
+
+A Hamiltonian diffeomorphism of a symplectic manifold $(M,\omega)$ is
+the time-one map of a smooth time-dependent Hamiltonian flow. We write
+$\Ham(M,\omega)$ for these maps, use the convention
+$\iota_{X_{H_t}}\omega=dH_t$, and set
+$\Fix(\phi)=\{q\in M:\phi(q)=q\}$.
+The \emph{critical-number Arnold bound} asserts that
+\begin{equation}\label{eq:arnold-bound}
+ \#\Fix(\phi)\ge\Crit(M)
+\end{equation}
+for every $\phi\in\Ham(M,\omega)$ on every closed connected symplectic
+manifold. Its weaker \emph{rational cup-length} form replaces $\Crit(M)$
+by $\operatorname{cuplength}(M;\mathbb Q)$. Neither formulation assumes
+nondegeneracy; see Golovko~\cite[Conjectures~1.4 and~1.6]{Golovko2020}.
+A fixed point $q$ is nondegenerate when $1$ is not an eigenvalue of
+$d\phi_q$.
+
+Let
+\begin{equation}\label{eq:quadric}
+ Q^3=\bigl\{[z_0:z_1:z_2:z_3:z_4]\in\CP^4:
+                 z_0^2+z_1^2+z_2^2+z_3^2+z_4^2=0\bigr\},
+\end{equation}
+with the restricted Fubini--Study symplectic form, normalized in
+Section~\ref{sec:quadric}.
+
+\begin{theorem}\label{thm:counterexample}
+There is a smooth Hamiltonian diffeomorphism $\phi$ of the closed connected
+symplectic six-manifold $Q^3$ such that
+\begin{equation}\label{eq:counterexample}
+ \#\Fix(\phi)=3<4=\Crit(Q^3)
+                  =\operatorname{cuplength}(Q^3;\mathbb Q).
+\end{equation}
+At least one fixed point of $\phi$ is degenerate.
+\end{theorem}
+
+This disproves both unrestricted smooth bounds. The count includes all
+fixed points, so imposing a contractibility condition on their
+Hamiltonian trajectories cannot restore either inequality. The
+nondegenerate homological Arnold inequalities concern a different
+hypothesis and remain compatible with this example.
+
+The count of three is sharp. Gong~\cite[Section~1.1.1]{Gong2025}
+obtains at least $n$ fixed points for every Hamiltonian diffeomorphism
+of the standard complex quadric of complex dimension $n\ge2$.
+His normalization of the Fubini--Study form differs from ours by a
+positive scalar, which does not change the Hamiltonian group: rescaling
+the form and a generating Hamiltonian by the same scalar preserves its
+vector field. Thus our example attains his bound for $n=3$.
+
+\paragraph{Historical context and related work.}
+The positive cases explain why the unrestricted comparison is natural.
+Conley and Zehnder proved that every Hamiltonian diffeomorphism of the
+standard symplectic torus $T^{2n}$ has at least $2n+1$ fixed
+points~\cite{ConleyZehnder1983}, and Fortune obtained the bound $n+1$
+on standard $\CP^n$~\cite{Fortune1985}. Hofer obtained cup-length bounds without
+nondegeneracy when $\pi_2(M)=0$~\cite[Theorem~4]{Hofer1988}.
+Rudyak and Oprea proved the full critical-number bound when both the
+symplectic class and the first Chern class vanish on $\pi_2(M)$
+\cite[Corollary~4.3]{RudyakOprea1999}. The quadric has spheres of
+positive symplectic area and lies outside these latter hypotheses.
+
+Floer's pseudoholomorphic-curve methods established homological Morse
+inequalities for nondegenerate Hamiltonian fixed points, including the
+monotone setting~\cite{Floer1989}. The lower bound by the sum of rational
+Betti numbers was subsequently established for nondegenerate Hamiltonian
+diffeomorphisms on every closed symplectic manifold by Fukaya and Ono,
+Liu and Tian, and Ruan~\cite{FukayaOno1999,LiuTian1998,Ruan1999}.
+The distinction between these bounds
+and the degenerate formulations above is essential; see
+\cite[Theorem~1.3 and Conjectures~1.4 and~1.6]{Golovko2020}.
+Ma~\cite[Theorem~1.5]{Ma2013} asserted that the minimum number of fixed
+points over all Hamiltonian diffeomorphisms always equals $\Crit(M)$.
+Theorem~\ref{thm:counterexample} contradicts that assertion.
+
+Regularity also matters. Buhovsky, Humili\`ere, and Seyfaddini constructed
+Hamiltonian \emph{homeomorphisms} with a single fixed point on every closed
+connected symplectic manifold of real dimension at least four
+\cite{BHS2018}. Our construction stays in the smooth Hamiltonian group.
+In a preprint posted September 27, 2026, Jiao~\cite[Theorem~1.1]{Jiao2026}
+reports a smooth Hamiltonian diffeomorphism of the equal-area product
+$S^2\times S^2$ with two fixed points, using local saddle merging and
+mixed generating functions. The construction here instead uses a
+finite-order map on a six-dimensional quadric. None of these
+fixed-point results is an input to our construction.
+
+\paragraph{Proof mechanism.}
+The key is to transfer a critical-point count from a fixed submanifold
+to the fixed-point count of a nearby Hamiltonian map. Let $A$ be a
+finite-order Hamiltonian map with fixed submanifold $F$, whose fixed
+tangent vectors are exactly those tangent to $F$. Extend a function
+$f$ on $F$ to an $A$-invariant function $K$ on the ambient manifold,
+and let $B_s$ be its Hamiltonian flow. If $A^m=\id$, invariance gives
+\[
+ (A\circ B_\varepsilon)^m=B_{m\varepsilon}.
+\]
+For sufficiently small $\varepsilon>0$, the right side has only
+stationary fixed points. Any fixed point of $A\circ B_\varepsilon$
+is therefore a critical point of $K$ fixed by $A$. Finite-group
+averaging makes ambient criticality on $F$ equivalent to criticality
+of $f$, giving
+\[
+ \Fix(A\circ B_\varepsilon)=\CritSet(f).
+\]
+This criticality step is the finite-group case of Palais's principle
+of symmetric criticality~\cite[Section~5]{Palais1979}; the power identity
+excludes additional fixed points away from $F$.
+
+For the quadric, take the Hamiltonian involution
+\[
+ A[z_0:z_1:z_2:z_3:z_4]=[z_0:-z_1:-z_2:-z_3:-z_4].
+\]
+Its fixed locus is the quadric surface $Q^2\cong S^2\times S^2$.
+This product admits functions with only three critical points, as
+recorded by Mare~\cite[Proposition~3.2.5]{Mare1997} through an extension
+of Takens's sphere-product construction~\cite{Takens1968}. We give an
+explicit formula and compute its entire critical set. The isolated
+degenerate critical point becomes a degenerate Hamiltonian fixed point.
+In contrast, symplectic volume forces every smooth function on the
+ambient $Q^3$ to have at least four critical points, and a Hamiltonian
+rotation realizes that minimum.
+
+Section~\ref{sec:reduction} proves the finite-order transfer, including
+the short-period estimate and invariant extension.
+Section~\ref{sec:quadric} constructs the Hamiltonian involution and
+its tangent splitting. Section~\ref{sec:three-points} gives the
+three-critical-point function. Section~\ref{sec:critical-bound}
+proves the critical-number and rational cup-length equalities on $Q^3$,
+and Section~\ref{sec:conclusion} assembles the construction and
+verifies degeneracy.
diff --git a/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/build/sections/01-reduction.tex b/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/build/sections/01-reduction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..138017d9c8be4c27a2066f723ac68fb83dca6d61
--- /dev/null
+++ b/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/build/sections/01-reduction.tex
@@ -0,0 +1,163 @@
+\section{Perturbing a finite-order Hamiltonian map}
+\label{sec:reduction}
+
+A short Hamiltonian perturbation of a finite-order map can replace its
+fixed submanifold by the critical set of a function on that submanifold.
+The tangent-space hypothesis below is the usual \emph{clean fixed-set}
+condition; we state it explicitly to identify the geometric input.
+
+\begin{proposition}
+\label{prop:reduction}
+Let \((M,\omega)\) be a closed symplectic manifold, and let
+\(A\in\Ham(M,\omega)\) satisfy \(A^m=\id\) for some positive integer
+\(m\). Suppose that \(F=\Fix(A)\) is a closed embedded submanifold and
+that
+\[
+  T_qF=\ker(dA_q-\id)
+  \qquad\text{for every }q\in F.
+\]
+For every smooth function \(f:F\to\R\), there is a smooth
+\(A\)-invariant extension \(K:M\to\R\) such that, if \(B_s\) is its
+Hamiltonian flow, then for every sufficiently small \(\varepsilon>0\)
+the map
+\[
+  \phi_\varepsilon=A\circ B_\varepsilon
+\]
+is Hamiltonian and satisfies
+\[
+  \Fix(\phi_\varepsilon)=\CritSet(f),
+\]
+where the critical set on the right is viewed as a subset of \(M\).
+\end{proposition}
+
+The proof combines finite-group averaging with the exclusion of short
+nonconstant periodic trajectories. We establish the latter fact first.
+Lipschitz control of a vector field gives a lower bound for its
+nonconstant periods, a classical principle for which Yorke proved the
+sharp Euclidean estimate~\cite{Yorke1969}. The following elementary
+local argument suffices here.
+
+\begin{lemma}
+\label{lem:short-periods}
+Let \(X\) be a smooth vector field on a closed manifold \(M\), and let
+\(\Psi_t\) be its flow. There is a number \(\delta>0\) such that
+\[
+  \Fix(\Psi_t)=\{q\in M:X(q)=0\}
+  \qquad\text{for every }0<t<\delta.
+\]
+\end{lemma}
+
+\begin{proof}
+The flow is defined for all time because \(M\) is compact. Suppose that
+there were points \(q_j\) and times \(t_j>0\) with
+\[
+  t_j\longrightarrow0,\qquad
+  \Psi_{t_j}(q_j)=q_j,\qquad X(q_j)\ne0.
+\]
+After passing to a subsequence, assume that \(q_j\to q\). Choose
+coordinates near \(q\), and a Euclidean ball about its coordinate image
+whose closure lies inside the coordinate domain. By continuity of the
+flow, all sufficiently late trajectory segments
+\(\{\Psi_s(q_j):0\le s\le t_j\}\) lie in this ball. The coordinate vector
+field \(V\) is Lipschitz there with a constant \(L\ge0\), since its
+derivative is bounded on the closed ball.
+
+Write \(u_j(s)\) for the coordinate trajectory and set
+\[
+  C_j=\max_{0\le s\le t_j}|\dot u_j(s)|>0.
+\]
+For any \(s,r\in[0,t_j]\), convexity of the ball and the Lipschitz
+estimate give
+\[
+ \begin{split}
+  |\dot u_j(s)-\dot u_j(r)|
+   &=|V(u_j(s))-V(u_j(r))|\\
+   &\le L|u_j(s)-u_j(r)|
+    \le LC_j|s-r|
+    \le LC_jt_j.
+ \end{split}
+\]
+The closed trajectory has zero mean coordinate velocity:
+\(\int_0^{t_j}\dot u_j(r)\,dr=0\). At a time \(s_j\) of maximum speed,
+it follows that
+\[
+  C_j
+  =\left|
+     \frac1{t_j}\int_0^{t_j}
+        \bigl(\dot u_j(s_j)-\dot u_j(r)\bigr)\,dr
+    \right|
+  \le LC_jt_j.
+\]
+Dividing by \(C_j\) contradicts \(t_j\to0\). Thus all fixed points of a
+sufficiently short positive-time map are zeros of \(X\). Conversely,
+every zero of \(X\) is stationary and hence fixed by every \(\Psi_t\).
+\end{proof}
+
+\begin{proof}[Proof of Proposition~\ref{prop:reduction}]
+The first step is to extend $f$ so that criticality along $F$ is equivalent
+to ambient criticality at points of $F$. This is the finite-group case
+of symmetric criticality~\cite{Palais1979}, which we verify directly.
+Extend \(f\) smoothly to a function \(\widetilde f\) on \(M\): in
+submanifold charts, extend independently of the normal coordinates,
+and combine these local extensions using a partition of unity. Define
+\[
+  K=\frac1m\sum_{j=0}^{m-1}\widetilde f\circ A^j.
+\]
+Then \(K\circ A=K\) and \(K|_F=f\).
+
+Fix \(q\in F\), put \(D=dA_q\), and consider
+\[
+  P_q=\frac1m\sum_{j=0}^{m-1}D^j.
+\]
+Since \(D^m=\id\), we have \(DP_q=P_q\). Moreover, \(P_qv=v\) whenever
+\(Dv=v\). Thus \(P_q\) is a projection onto
+\(\ker(D-\id)=T_qF\). Differentiating the averaged extension gives,
+for every \(v\in T_qM\),
+\[
+  dK_q(v)=d\widetilde f_q(P_qv)=df_q(P_qv).
+\]
+Consequently,
+\begin{equation}
+\label{eq:critical-restriction}
+  \CritSet(K)\cap F=\CritSet(f).
+\end{equation}
+
+Because \(A\) is symplectic and \(K\circ A=K\), its differential carries
+the Hamiltonian vector field \(X_K\) to itself. Uniqueness of solutions
+therefore gives \(A\circ B_s=B_s\circ A\) for every \(s\). Apply
+Lemma~\ref{lem:short-periods} to \(X_K\), and choose
+\(0<\varepsilon<\delta/m\). We then have
+\[
+  \phi_\varepsilon^m=B_{m\varepsilon},
+  \qquad
+  \Fix(B_{m\varepsilon})=\CritSet(K).
+\]
+If \(\phi_\varepsilon(q)=q\), then \(q\) is fixed by
+\(B_{m\varepsilon}\), so \(dK_q=0\). It follows that
+\(B_\varepsilon(q)=q\), and hence \(A(q)=q\). Conversely, every point
+of \(\CritSet(K)\cap F\) is fixed by \(\phi_\varepsilon\).
+Equation~\ref{eq:critical-restriction} proves the asserted equality
+of fixed and critical sets.
+
+For completeness, the composition is generated by a smooth
+Hamiltonian on the unit time interval. Choose a Hamiltonian path
+\(A_t\) from \(\id\) to \(A\), generated by \(H_t\), and a smooth
+nondecreasing function \(a:[0,1]\to[0,1]\) equal to \(0\) near \(0\)
+and to \(1\) near \(1\). The time-dependent Hamiltonian
+\[
+  L_t(x)=
+  \begin{cases}
+    2\varepsilon a'(2t)K(x),&0\le t\le\tfrac12,\\[2pt]
+    2a'(2t-1)H_{a(2t-1)}(x),&\tfrac12\le t\le1
+  \end{cases}
+\]
+is smooth, since both pieces vanish near the joining time. Its flow
+first follows \(B_{\varepsilon a(2t)}\), and then
+\(A_{a(2t-1)}\circ B_\varepsilon\), so its time-one map is
+\(\phi_\varepsilon\).
+\end{proof}
+
+The proposition imposes no bound on the total number of critical
+points of \(K\), and requires invariance under $A$, not under an isotopy
+from $\id$ to $A$. A critical point \(q\notin F\) satisfies
+\(\phi_\varepsilon(q)=A(q)\ne q\), so it contributes no fixed point.
diff --git a/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/build/sections/02-quadric.tex b/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/build/sections/02-quadric.tex
new file mode 100644
index 0000000000000000000000000000000000000000..0673d90696fb6d376c07404a1127b3249aa57f5b
--- /dev/null
+++ b/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/build/sections/02-quadric.tex
@@ -0,0 +1,138 @@
+\section{A Hamiltonian involution of a quadric}
+\label{sec:quadric}
+
+We now construct the manifold and involution to which
+Proposition~\ref{prop:reduction} will apply.  Let $M\subset\CP^4$ be the
+quadric defined in~\eqref{eq:quadric}.
+
+\begin{proposition}\label{prop:quadric}
+The manifold $M$ is closed, connected, and of real dimension six. For
+the restricted Fubini--Study form $\omega$, normalized below, the map
+\[
+ A[z_0:z_1:z_2:z_3:z_4]
+   =[z_0:-z_1:-z_2:-z_3:-z_4]
+\]
+is a Hamiltonian involution.  Its fixed set is the smooth submanifold
+\[
+ F=M\cap\{z_0=0\}\cong S^2\times S^2.
+\]
+For every $q\in F$, the differential $dA_q$ acts as $+\id$ on $T_qF$
+and as $-\id$ on a complementary real two-dimensional subspace of
+$T_qM$.
+\end{proposition}
+
+\begin{proof}
+In any standard affine chart of $\CP^4$, the equation defining $M$
+takes the form $1+\sum_j w_j^2=0$.  Its differential cannot vanish on
+its zero set: that would force all the $w_j$ to be zero.  Thus $M$ is
+a smooth complex hypersurface, of complex dimension three.  As a
+closed subset of projective space it is compact, and it has no
+boundary.
+
+To see connectedness, write a nonzero isotropic vector as $z=x+iy$
+with $x,y\in\R^5$.  Its real and imaginary parts satisfy
+\[
+ |x|=|y|>0,\qquad x\cdot y=0.
+\]
+After rescaling, they form an ordered orthonormal pair.  Every such
+pair extends to a positively oriented orthonormal basis, so these
+pairs form one orbit of the connected group $\mathrm{SO}(5)$.  Their
+map $(x,y)\mapsto[x+iy]$ onto $M$ proves that $M$ is connected.
+
+We construct the symplectic form with its normalization explicit.
+On the unit sphere $S^9\subset\C^5$, put
+\[
+ \alpha=\sum_{j=0}^4(x_j\,dy_j-y_j\,dx_j).
+\]
+The form $d\alpha$ is invariant under scalar phase rotations.  If
+$v\in T_zS^9$, then
+\[
+ d\alpha(iz,v)
+   =-2\operatorname{Re}\sum_{j=0}^4\overline{z_j}v_j=0.
+\]
+Consequently it descends under the quotient map
+$p:S^9\to\CP^4$ to a two-form $\sigma$, characterized by
+$p^*\sigma=d\alpha$.  Local sections show that $\sigma$ is smooth
+and closed.  A projective tangent vector has a horizontal lift
+$v\in\C^5$ satisfying $\sum_j\overline{z_j}v_j=0$; multiplication by
+$i$ on these lifts induces the complex structure.  Since
+\[
+ d\alpha(v,iv)=2|v|^2,
+\]
+$\sigma$ is positive on every complex tangent line.  Its restriction
+$\omega=\sigma|_M$ is therefore closed and nondegenerate.
+
+We use a family of Hamiltonian rotations: equal speeds will give the
+involution, and unequal speeds will later give a function with four
+critical points on $M$. For $a,b,t\in\R$, let $\Phi_t^{a,b}$ keep $z_0$
+fixed and act on $(z_1,z_2)^{\mathsf T}$ and $(z_3,z_4)^{\mathsf T}$ by
+$R_{at}$ and $R_{bt}$, respectively, where
+\[
+ R_\theta=
+ \begin{pmatrix}
+  \cos\theta&-\sin\theta\\
+  \sin\theta&\cos\theta
+ \end{pmatrix}.
+\]
+These real orthogonal, complex unitary transformations preserve the
+quadric and $\alpha$. If $Z_{a,b}$ is their generating vector field
+on $S^9$, Cartan's formula
+gives
+\[
+ \iota_{Z_{a,b}}d\alpha=-d\bigl(\alpha(Z_{a,b})\bigr).
+\]
+The function $\alpha(Z_{a,b})$ is phase invariant, because the rotations
+commute with scalar multiplication.  It thus descends to a smooth
+function $g_{a,b}$ on projective space. The displayed identity descends
+as well, so $G_{a,b}=-g_{a,b}|_M$ generates the projected rotations on
+$M$ with our convention $\iota_{X_{G_{a,b}}}\omega=dG_{a,b}$.
+Explicitly,
+\begin{equation}\label{eq:rotation-hamiltonian}
+ G_{a,b}([z])=
+ \frac{2a\,\operatorname{Im}(\overline{z_1}z_2)
+       +2b\,\operatorname{Im}(\overline{z_3}z_4)}
+      {\sum_{j=0}^4|z_j|^2}.
+\end{equation}
+In particular, $A=\Phi_1^{\pi,\pi}$ is Hamiltonian and preserves $\omega$.
+
+A projective line fixed by $A$ lies in one of the two eigenspaces of
+$\operatorname{diag}(1,-1,-1,-1,-1)$.  The positive eigenspace gives
+the single point $[1:0:0:0:0]$, which does not lie in $M$.  The
+negative eigenspace gives precisely $F=M\cap\{z_0=0\}$.  To identify
+$F$, use the invertible complex-linear map
+\[
+ (z_1,z_2,z_3,z_4)\longmapsto
+ \begin{pmatrix}
+  z_1+iz_2&z_3+iz_4\\
+  -z_3+iz_4&z_1-iz_2
+ \end{pmatrix}.
+\]
+Its determinant is $z_1^2+z_2^2+z_3^2+z_4^2$.  It therefore
+identifies $F$ with the projectivized nonzero rank-one matrices.
+The map
+\[
+ \CP^1\times\CP^1\longrightarrow F,
+ \qquad ([u],[v])\longmapsto[uv^{\mathsf T}]
+\]
+is a diffeomorphism: a nonzero column and row recover the two
+projective factors, smoothly on the corresponding matrix charts.
+Finally, stereographic coordinates identify each $\CP^1$ with
+$S^2$.  Only this smooth identification is needed below.
+
+For the tangent-space assertion, fix $q\in F$ and choose an affine
+chart $z_j=1$ with $j>0$.  In its normalized coordinates, $A$ sends
+$w_0$ to $-w_0$ and leaves all other coordinates fixed.  The equation
+of $M$ is
+\[
+ w_0^2+1+\sum_{\substack{k>0\\k\ne j}}w_k^2=0.
+\]
+At $q$, where $w_0=0$, its linearization places no constraint on
+the $w_0$ direction.  The remaining differential is nonzero, so its
+kernel in the other coordinates is exactly $T_qF$.  Thus
+\[
+ T_qM=T_qF\oplus\C\frac{\partial}{\partial w_0},
+\]
+with the asserted $+1$ and $-1$ actions.  This also verifies directly
+that $F$ is a smooth complex surface. The positivity established above
+therefore makes $\omega|_F$ symplectic.
+\end{proof}
diff --git a/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/build/sections/03-critical-bound.tex b/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/build/sections/03-critical-bound.tex
new file mode 100644
index 0000000000000000000000000000000000000000..f07643f65651af2c6e2f5de3709809cbb8532624
--- /dev/null
+++ b/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/build/sections/03-critical-bound.tex
@@ -0,0 +1,146 @@
+\section{The critical number and cup length of the quadric}
+\label{sec:critical-bound}
+
+We now show that four is both the critical number and the rational cup
+length of the ambient quadric. The lower bound is the classical
+cup-product obstruction from Lusternik--Schnirelmann
+theory~\cite{LusternikSchnirelmann1934}; see
+Weber~\cite[Theorems~3.5 and 3.6]{Weber2019} for a modern treatment
+valid without a Morse assumption. We give the differential-form proof,
+then obtain the matching upper bound from the unequal-speed rotations
+of Section~\ref{sec:quadric}.
+
+\begin{lemma}\label{lem:critical-bound}
+Let $M$ be a closed smooth manifold and $r\ge1$ an integer. Suppose there are classes
+$a_j\in H^{d_j}(M;\R)$, with $d_j>0$ for $1\le j\le r$, such that
+$a_1\smile\cdots\smile a_r\ne0$. Then every smooth function on $M$
+has at least $r+1$ critical points.
+\end{lemma}
+
+\begin{proof}
+Suppose that $h:M\to\R$ has at most $r$ critical points.
+List its distinct critical values as $c_1<\cdots<c_k$, where $1\le k\le r$;
+the absolute minimum and maximum are among them.
+For a fixed Riemannian metric let $g_t$ be the negative-gradient flow
+of $h$, which is complete by compactness, and put $P_b=\{h\le b\}$.
+We first cover $M$ by $k$ open sets, each diffeomorphic to a finite
+disjoint union of coordinate balls. Every closed positive-degree form
+will be exact on each of these sets; the cover will therefore force
+every product of $k$ positive-degree cohomology classes to vanish.
+
+Around the critical points at $c_i$, choose an open set $U_i$ that is a
+union of disjoint coordinate balls, and a smaller neighborhood $V_i$
+with $\overline V_i\subset U_i$. A bound on the speed of the flow and
+the positive distance between $M\setminus U_i$ and $\overline V_i$
+give $T_i>0$ such that trajectories starting outside $U_i$ stay outside
+$V_i$ for $0\le t\le T_i$.
+
+Choose a closed level band $\{|h-c_i|\le\delta_i\}$ narrow enough that
+it contains no critical points outside $V_i$. On the part outside
+$V_i$, compactness gives a positive lower bound $b_i$ for
+$|\nabla h|^2$; if that set is empty, any positive $b_i$ will do.
+Choose $0<\epsilon_i<\delta_i$ so that the intervals
+$[c_i-\epsilon_i,c_i+\epsilon_i]$ are separated and
+$2\epsilon_i<T_i b_i$. Along the negative-gradient flow,
+\[
+ \frac{d}{dt}h(g_t(q))=-|\nabla h(g_t(q))|^2.
+\]
+A point of $P_{c_i+\epsilon_i}\setminus U_i$ either already lies below
+$c_i-\epsilon_i$ or decreases at rate at least $b_i$ until it reaches
+that level. It follows that
+\begin{equation}\label{eq:sublevel-drop}
+ g_{T_i}(P_{c_i+\epsilon_i}\setminus U_i)
+       \subset P_{c_i-\epsilon_i}.
+\end{equation}
+For $i>1$, a further fixed flow time $S_i>0$ carries $P_{c_i-\epsilon_i}$
+into $P_{c_{i-1}+\epsilon_{i-1}}$: the compact band between these levels
+contains no critical points and has a positive gradient-norm lower bound.
+
+Since $c_1$ is the absolute minimum, $P_{c_1-\epsilon_1}$ is empty,
+and \eqref{eq:sublevel-drop} shows that $P_{c_1+\epsilon_1}\subset U_1$.
+Inductively, cover $P_{c_i+\epsilon_i}$ by $U_i$ and the preimages under
+$g_{T_i+S_i}$ of the $i-1$ sets covering the preceding upper sublevel.
+Equation~\eqref{eq:sublevel-drop} proves that these sets cover.
+Each $g_t$ is a diffeomorphism of $M$, so every member remains
+diffeomorphic to a finite disjoint union of coordinate balls. The last
+upper sublevel is all of $M$, since $c_k$ is the absolute maximum.
+
+Write the resulting cover as $O_1,\ldots,O_k$. Choose closed forms
+$\alpha_j$ representing $a_j$ for $1\le j\le k$. Since these forms
+have positive degree, the Poincar\'e lemma on each ball gives
+$\alpha_j=d\lambda_j$ on $O_j$. Choose a smooth partition of unity
+$\psi_1,\ldots,\psi_k$ subordinate to the cover. Let
+$\chi:[0,1]\to[0,1]$ be smooth, zero on $[0,1/(4k)]$ and one on
+$[1/(2k),1]$, and put $\rho_j=\chi(\psi_j)$. At every point some
+$\psi_j\ge1/k$, so some $\rho_j$ is identically one nearby.
+Each $\rho_j\lambda_j$ extends smoothly by zero outside $O_j$.
+
+The closed forms
+\[
+ \alpha'_j=\alpha_j-d(\rho_j\lambda_j)
+\]
+represent the same classes as $\alpha_j$. Near every point, at least one
+of these forms vanishes, and hence $\alpha'_1\wedge\cdots\wedge\alpha'_k=0$.
+Thus $a_1\smile\cdots\smile a_k=0$, contradicting the nonvanishing
+of $a_1\smile\cdots\smile a_r$ because $k\le r$.
+\end{proof}
+
+Extension of coefficients $H^*(M;\mathbb Q)\to H^*(M;\R)$ is injective
+for a closed manifold and respects cup products. The lemma therefore
+also gives
+\begin{equation}\label{eq:cup-critical}
+ \operatorname{cuplength}(M;\mathbb Q)\le\Crit(M),
+\end{equation}
+with the unit-inclusive convention of Section~\ref{sec:introduction}.
+It also recovers the symplectic-volume obstruction used in our example.
+
+\begin{corollary}\label{cor:symplectic-critical}
+Every smooth function on a closed connected symplectic manifold of real
+dimension $2n$, with $n\ge1$, has at least $n+1$ critical points.
+\end{corollary}
+
+\begin{proof}
+The class $[\omega]^n$ is nonzero, since
+$\int_M\omega^n>0$ in the symplectic orientation. Apply
+Lemma~\ref{lem:critical-bound} with each $a_j=[\omega]$.
+\end{proof}
+
+\begin{corollary}\label{cor:quadric-critical}
+For the quadric threefold in \eqref{eq:quadric},
+\[
+ \Crit(Q^3)=\operatorname{cuplength}(Q^3;\mathbb Q)=4.
+\]
+\end{corollary}
+
+\begin{proof}
+Let $h=c_1(\mathcal O_{Q^3}(1))\in H^2(Q^3;\mathbb Z)$ be the
+hyperplane class. Since $Q^3$ has degree two in $\CP^4$,
+\[
+ h^3=2[\mathrm{pt}],
+\]
+where $[\mathrm{pt}]$ evaluates to one on the fundamental class.
+This product remains nonzero over $\mathbb Q$, giving
+$\operatorname{cuplength}(Q^3;\mathbb Q)\ge4$.
+
+For the matching upper bound, take the Hamiltonian $G_{1,2}$ of
+\eqref{eq:rotation-hamiltonian}. Its critical points are precisely the
+zeros of its Hamiltonian vector field. That field is induced on $Q^3$
+by the complex-linear generator
+\[
+ T=\operatorname{diag}(0,J,2J),
+ \qquad J=\begin{pmatrix}0&-1\\1&0\end{pmatrix}.
+\]
+The induced projective vector field vanishes at $[z]$ exactly when
+$Tz\in\C z$. The eigenvalues $0,\pm i,\pm2i$ of $T$ are distinct,
+so its projective zeros are its five eigenlines. The zero eigenline
+$[1:0:0:0:0]$ misses $Q^3$, whereas the other four lines
+\[
+ [0:1:\pm i:0:0],\qquad [0:0:0:1:\pm i]
+\]
+all satisfy the quadric equation. Thus $G_{1,2}$ has exactly four
+critical points on $Q^3$. Together with \eqref{eq:cup-critical}, this gives
+\[
+ 4\le\operatorname{cuplength}(Q^3;\mathbb Q)
+   \le\Crit(Q^3)\le4.
+\]
+\end{proof}
diff --git a/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/build/sections/04-three-points.tex b/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/build/sections/04-three-points.tex
new file mode 100644
index 0000000000000000000000000000000000000000..6eaa44003b44fea93d78c0842d6de14a95c804dc
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+++ b/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/build/sections/04-three-points.tex
@@ -0,0 +1,103 @@
+\section{Three critical points on the fixed submanifold}
+\label{sec:three-points}
+
+We next construct a function on the fixed submanifold with three
+critical points. The comparison with functions on the ambient quadric
+will follow in Section~\ref{sec:critical-bound}.
+Three-critical-point functions on products of spheres belong to the
+classical study of critical-point minima and category by
+Takens~\cite{Takens1968}; the $S^2\times S^2$ case is recorded by
+Mare~\cite[Proposition~3.2.5]{Mare1997}. We use the explicit formula below
+and determine its entire critical set directly, without importing an
+existence theorem or assuming that its critical points are Morse.
+
+\begin{proposition}
+\label{prop:three-points}
+Let $p=(0,0,1)$ and $e=(1,0,0)$ in $\R^3$.
+The smooth function
+\begin{equation}
+\label{eq:three-point-function}
+  f\colon S^2\times S^2\longrightarrow\R,
+  \qquad f(x,y)=e\cdot x+(x-p)\cdot y,
+\end{equation}
+has exactly three critical points.
+\end{proposition}
+
+\begin{proof}
+For a fixed $x\ne p$, the $y$-dependence is a height function on the
+second sphere, with critical points $y=\pm(x-p)/|x-p|$.
+At $x=p$ that height function is constant, but criticality of the full
+function still requires its derivative in the $x$ directions to vanish.
+We therefore separate this exceptional fiber from the two height branches.
+
+The constrained critical-point equations are
+\begin{equation}
+\label{eq:critical-multipliers}
+  e+y=\lambda x,\qquad x-p=\mu y,
+  \qquad |x|=|y|=1,
+\end{equation}
+for real multipliers $\lambda,\mu$.
+If $x=p$, the first equation gives $y=-e+\lambda p$.
+Its unit norm forces $\lambda=0$, yielding the unique point
+\[
+  q_0=(p,-e),\qquad f(q_0)=0.
+\]
+
+Suppose now that $x\ne p$ and set $t=|x-p|>0$.
+The second equation in \eqref{eq:critical-multipliers} gives
+$y=s(x-p)/t$ for a sign $s\in\{1,-1\}$.
+The first then becomes
+\[
+  t e-s p=(\lambda t-s)x.
+\]
+Since the left side is nonzero, $\lambda t-s\ne0$ and $x$ lies in
+the plane spanned by $e$ and $p$. Write $x=(u,0,w)$.
+The definition of $t$ gives $w=1-t^2/2$.
+Taking the first equation in \eqref{eq:critical-multipliers}
+against the tangent vector $(w,0,-u)$ gives
+\[
+  w+\frac{s u}{t}=0,
+  \qquad\text{so}\qquad u=-s t w.
+\]
+Consequently the unit-norm condition reduces to
+\begin{equation}
+\label{eq:critical-factorization}
+  0=(1-t^2/2)^2(1+t^2)-1
+   =\frac{t^4(t^2-3)}4.
+\end{equation}
+Because $t>0$, this forces $t=\sqrt3$, $w=-1/2$, and
+$u=s\sqrt3/2$. In particular, the other pole $x=-p$, for which $t=2$,
+is excluded. The two resulting candidates are
+\begin{equation}
+\label{eq:other-critical-points}
+  q_s=
+  \left(
+    \left(\frac{s\sqrt3}{2},0,-\frac12\right),
+    \left(\frac12,0,-\frac{s\sqrt3}{2}\right)
+  \right),
+  \qquad s\in\{1,-1\}.
+\end{equation}
+Both satisfy \eqref{eq:critical-multipliers} with
+$\lambda=\mu=s\sqrt3$, and
+\[
+  f(q_s)=s\,\frac{3\sqrt3}{2}.
+\]
+The cases $x=p$ and $x\ne p$ exhaust the domain, proving the count.
+\end{proof}
+
+\begin{remark}
+\label{rem:degenerate-critical-point}
+The critical point $q_0$ is degenerate. Indeed, in local coordinates
+\[
+  x=(u,v,\sqrt{1-u^2-v^2}),\qquad
+  y=(-\sqrt{1-a^2-b^2},a,b),
+\]
+the function has expansion
+\[
+  f=va+O\bigl(\lVert(u,v,a,b)\rVert^3\bigr).
+\]
+Its Hessian therefore has eigenvalues $1,-1,0,0$, so its rank is two.
+The complete critical-point calculation proves that this degenerate point
+is isolated. This is why a count restricted to Morse functions would not
+capture the example.
+\end{remark}
diff --git a/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/build/sections/05-conclusion.tex b/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/build/sections/05-conclusion.tex
new file mode 100644
index 0000000000000000000000000000000000000000..65e97dc4c1e5af190fd6cb10d48ccf213ed4bd6f
--- /dev/null
+++ b/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/build/sections/05-conclusion.tex
@@ -0,0 +1,53 @@
+\section{The Hamiltonian counterexample}
+\label{sec:conclusion}
+
+We now apply the finite-order reduction to the function on the fixed
+surface and compare the resulting fixed-point count with the ambient
+critical number and cup length.
+
+\begin{proof}[Proof of Theorem~\ref{thm:counterexample}]
+Take the symplectic quadric $(M,\omega)$ and Hamiltonian involution $A$
+from Proposition~\ref{prop:quadric}. Its fixed submanifold $F$ is
+diffeomorphic to $S^2\times S^2$.
+Transport the function of Proposition~\ref{prop:three-points} to $F$;
+it has exactly three critical points.
+
+Proposition~\ref{prop:reduction} provides a smooth $A$-invariant extension
+$K\colon M\to\R$ with $K|_F=f$.
+Let $B_s$ denote its Hamiltonian flow. For all sufficiently small
+$\varepsilon>0$, the map
+\[
+  \phi=A\circ B_\varepsilon
+\]
+is a Hamiltonian diffeomorphism and satisfies
+$\Fix(\phi)=\CritSet(f)$, with the critical points viewed as points of $F$.
+The exact comparison in Corollary~\ref{cor:quadric-critical} now gives
+\[
+  \#\Fix(\phi)=3<4=\Crit(M)
+     =\operatorname{cuplength}(M;\mathbb Q).
+\]
+It remains to verify the asserted degeneracy. Write
+$\omega_F=\omega|_F$, which is symplectic,
+and let $X_f$ be the Hamiltonian vector field of $f$ on $(F,\omega_F)$.
+Invariance of $K$ makes $X_K$ tangent to $F$, and restricting its defining
+equation to $TF$ gives $X_K|_F=X_f$. Indeed, at $q\in F$ invariance gives
+$dA_qX_K(q)=X_K(q)$, and the fixed tangent space is $T_qF$.
+Thus $B_s$ preserves $F$, and its restriction is the Hamiltonian flow
+of $f$ on $F$. At the critical point $q_0$ of
+Remark~\ref{rem:degenerate-critical-point}, differentiating
+$\iota_{X_f}\omega_F=df$ gives
+\[
+  \omega_F\bigl(DX_f(q_0)v,w\bigr)
+  =\operatorname{Hess}_{q_0}f(v,w).
+\]
+Thus any nonzero vector $v$ in the Hessian kernel satisfies
+$DX_f(q_0)v=0$. Since $A|_F=\id$ and $q_0$ is an equilibrium,
+\[
+  d\phi_{q_0}v
+   =\exp\bigl(\varepsilon DX_f(q_0)\bigr)v
+   =v.
+\]
+The fixed point $q_0$ therefore has eigenvalue $1$ in its linearization.
+This argument takes place entirely in the invariant subspace $T_{q_0}F$;
+no assumption on the normal Hessian of the extension $K$ is needed.
+\end{proof}
diff --git a/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/paper.pdf b/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..2476de04185fec18edcaac9760fdcb80450a8646
--- /dev/null
+++ b/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:bebf04613245ab3cd52f29724c9fe43e2943c07dfabd96fb21fd5394f175c474
+size 342652
diff --git a/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/README.md b/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..c7002c2a4f40b085a3a60e610b377fe1f3fca629
--- /dev/null
+++ b/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/README.md
@@ -0,0 +1,17 @@
+# [A density-uniform condensate bound for dilute Bose gases](paper.pdf)
+
+OpenAI  
+September 27, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026,
+  author = {{OpenAI}},
+  title = {{A density-uniform condensate bound for dilute Bose gases}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/paper.pdf}{OAI:A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/figures/endpoint-changes.tex b/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/figures/endpoint-changes.tex
new file mode 100644
index 0000000000000000000000000000000000000000..6a96f3ab9081c4bcf638ca551eec1eb3f8afa488
--- /dev/null
+++ b/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/figures/endpoint-changes.tex
@@ -0,0 +1,49 @@
+\begin{figure}[H]
+\centering
+\begin{tikzpicture}[x=1.35cm,y=1.15cm,>=Stealth,
+ every node/.style={font=\small}]
+ \begin{scope}
+  \node at (1,1.55) {Forward insertion $O$};
+  \foreach \j in {0,1,2} {
+   \node (s\j) at (\j,1) {$S^C_{i_\j}$};
+  }
+  \node (y1) at (1,0) {$Y_{i_1}$};
+  \node (y2) at (2,0) {$Y_{i_2}$};
+  \node (z) at (2.7,0) {$z$};
+  \node[gray] at (0,0) {$Y_{i_0}$};
+  \draw[->] (s0)--node[midway,above right] {$i_1$} (y1);
+  \draw[->] (s1)--node[midway,above right] {$i_2$} (y2);
+  \draw[->,thick] (s2)--node[midway,right] {$i_0$} (z);
+  \node at (0,-.5) {$a$};
+  \node at (1,-.5) {middle cell};
+  \node at (2.35,-.5) {$b$};
+ \end{scope}
+ \begin{scope}[xshift=6.0cm]
+  \node at (1,1.55) {Reverse insertion $B$};
+  \foreach \j in {0,1,2} {
+   \node (t\j) at (\j,1) {$S^C_{i_\j}$};
+  }
+  \node (x) at (-.7,0) {$x$};
+  \node (v0) at (0,0) {$Y_{i_0}$};
+  \node (v1) at (1,0) {$Y_{i_1}$};
+  \node[gray] at (2,0) {$Y_{i_2}$};
+  \draw[->,thick] (t0)--node[midway,left] {$i_2$} (x);
+  \draw[->] (t1)--node[midway,above left] {$i_0$} (v0);
+  \draw[->] (t2)--node[midway,above left] {$i_1$} (v1);
+  \node at (-.35,-.5) {$a$};
+  \node at (1,-.5) {middle cell};
+  \node at (2,-.5) {$b$};
+ \end{scope}
+\end{tikzpicture}
+\caption{Endpoint assignments relative to the same original closed-state starts
+$S^C$, for a tuple of three labels in successive
+cells. The forward change opens slot $i_0$ at $z\in b$; the reverse
+change opens slot $i_2$ at $x\in a$. Gray points remain outer endpoints
+but cease to be inner endpoints. Each arrow is labeled by its current
+slot and reassigns the original start at its tail to that slot. The two panels
+illustrate the single-tuple changes separately; the second-moment comparison
+may use different tuples for the forward and reverse changes. Arrows show
+only assignments, not sampled
+Brownian trajectories or a spatial straightness condition.}
+\label{fig:endpoints}
+\end{figure}
diff --git a/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/main.tex b/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/main.tex
new file mode 100644
index 0000000000000000000000000000000000000000..649da8f406709eeb39468bc7837f275f99fa1980
--- /dev/null
+++ b/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/main.tex
@@ -0,0 +1,61 @@
+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage{lmodern,microtype}
+\usepackage[a4paper,margin=28mm]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{booktabs,array,enumitem,tikz,float}
+\usetikzlibrary{arrows.meta,positioning}
+\usepackage[numbers,sort&compress]{natbib}
+\usepackage{xurl}
+\usepackage[hidelinks,hypertexnames=false]{hyperref}
+\hypersetup{pdftitle={A density-uniform condensate bound for dilute Bose gases},
+ 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{\Z}{\mathbb Z}
+\newcommand{\N}{\mathbb N}
+\newcommand{\dd}{\,\mathrm d}
+\DeclareMathOperator{\Tr}{Tr}
+\DeclareMathOperator{\supp}{supp}
+\DeclareMathOperator{\dist}{dist}
+\allowdisplaybreaks[2]
+\setlength{\emergencystretch}{2em}
+\pdfinfoomitdate=1
+\pdftrailerid{}
+\pdfsuppressptexinfo=15
+\title{A density-uniform condensate bound for dilute Bose gases}
+\author{OpenAI}
+\date{September 27, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+For each fixed bounded measurable, nonnegative, radial interaction of finite
+range in three dimensions that is not zero almost everywhere, we prove
+a positive lower bound on the constant-orbital
+condensate fraction that is uniform over all sufficiently small densities
+and all temperatures between zero and the square of the density.
+The interaction and density remain fixed in the thermodynamic limit.
+\end{abstract}
+\tableofcontents
+\input{sections/introduction}
+\input{sections/setup}
+\input{sections/energy}
+\input{sections/localization}
+\input{sections/slab}
+\input{sections/atypical}
+\input{sections/defects}
+\input{sections/geometry}
+\input{sections/opening}
+\input{sections/comparison}
+\input{sections/conclusion}
+\bibliographystyle{plainnat}
+\bibliography{references}
+\end{document}
diff --git a/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/references.bib b/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/references.bib
new file mode 100644
index 0000000000000000000000000000000000000000..5060e38ef6483145489aa2b33031cb2fc412de79
--- /dev/null
+++ b/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/references.bib
@@ -0,0 +1,405 @@
+@article{LiebSeiringer2002BEC,
+  author = {Lieb, Elliott H. and Seiringer, Robert},
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+  journal = {Physical Review Letters},
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+  pages = {170409},
+  year = {2002},
+  doi = {10.1103/PhysRevLett.88.170409},
+  eprint = {math-ph/0112032},
+  archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/math-ph/0112032}
+}
+@article{DeuchertSeiringer2020GP,
+  author = {Deuchert, Andreas and Seiringer, Robert},
+  title = {{Gross--Pitaevskii} Limit of a Homogeneous {Bose} Gas at Positive Temperature},
+  journal = {Archive for Rational Mechanics and Analysis},
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+  eprint = {1901.11363},
+  archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/1901.11363}
+}
+@article{BrenneckeBrooksCaraciOldenburg2025,
+  author = {Brennecke, Christian and Brooks, Morris and Caraci, Cristina and Oldenburg, Jakob},
+  title = {A Short Proof of {Bose--Einstein} Condensation in the {Gross--Pitaevskii} Regime and Beyond},
+  journal = {Annales Henri Poincar{\'e}},
+  volume = {26},
+  pages = {1353--1373},
+  year = {2025},
+  doi = {10.1007/s00023-024-01465-8},
+  eprint = {2401.00784},
+  archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/2401.00784}
+}
+@article{ChongLiangNam2026,
+  author = {Chong, Jacky J. and Liang, Hao and Nam, Phan Th{\`a}nh},
+  title = {Kinetic localization via {Poincar{\'e}}-type inequalities and applications to the condensation of {Bose} gases},
+  journal = {Journal of Functional Analysis},
+  volume = {291},
+  number = {5},
+  pages = {111542},
+  year = {2026},
+  doi = {10.1016/j.jfa.2026.111542},
+  eprint = {2510.20493},
+  archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/2510.20493}
+}
+@article{FournaisEtAl2026StrongFreeEnergy,
+  author = {Fournais, S. and Junge, L. and Girardot, T. and Morin, L. and Olivieri, M. and Triay, A.},
+  title = {The Free Energy of Dilute {Bose} Gases at Low Temperatures Interacting Via Strong Potentials},
+  journal = {Annales Henri Poincar{\'e}},
+  year = {2026},
+  doi = {10.1007/s00023-026-01676-1},
+  url = {https://doi.org/10.1007/s00023-026-01676-1}
+}
+@misc{Junge2026Propagation,
+  author = {Junge, Lukas},
+  title = {Propagation of Condensation via {Neumann} Localization in the Dilute {Bose} Gas},
+  year = {2026},
+  eprint = {2603.20776v3},
+  archivePrefix = {arXiv},
+  primaryClass = {math-ph},
+  url = {https://arxiv.org/abs/2603.20776v3},
+  note = {Version 3, 4 April 2026}
+}
+@article{Ueltschi2006FeynmanCycles,
+  author = {Ueltschi, Daniel},
+  title = {{Feynman} cycles in the {Bose} gas},
+  journal = {Journal of Mathematical Physics},
+  volume = {47},
+  pages = {123303},
+  year = {2006},
+  doi = {10.1063/1.2383008},
+  eprint = {math-ph/0605002},
+  archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/math-ph/0605002}
+}
+@article{Feynman1953AtomicLambda,
+  author = {Feynman, Richard P.},
+  title = {Atomic Theory of the {$\lambda$} Transition in Helium},
+  journal = {Physical Review},
+  volume = {91},
+  number = {6},
+  pages = {1291--1301},
+  year = {1953},
+  doi = {10.1103/PhysRev.91.1291},
+  url = {https://doi.org/10.1103/PhysRev.91.1291}
+}
+@article{LiebRuskai1973Entropy,
+  author = {Lieb, Elliott H. and Ruskai, Mary Beth},
+  title = {Proof of the Strong Subadditivity of Quantum-Mechanical Entropy},
+  journal = {Les rencontres physiciens-math{\'e}maticiens de Strasbourg--RCP25},
+  volume = {19},
+  pages = {36--55},
+  year = {1973},
+  note = {Expos{\'e} 5; with an appendix by B. Simon},
+  url = {https://numdam.org/item/RCP25_1973__19__A5_0/}
+}
+@article{BenjaminiPemantlePeres1998,
+  author = {Benjamini, Itai and Pemantle, Robin and Peres, Yuval},
+  title = {Unpredictable paths and percolation},
+  journal = {The Annals of Probability},
+  volume = {26},
+  number = {3},
+  pages = {1198--1211},
+  year = {1998},
+  doi = {10.1214/aop/1022855749},
+  eprint = {math/9701227},
+  archivePrefix = {arXiv},
+  note = {Preprint title: Paths with exponential intersection tails and oriented percolation}
+}
+
+@article{AbbeMassoulieMontanariSlySrivastava2018,
+  author = {Abbe, Emmanuel and Massouli{\'e}, Laurent and Montanari, Andrea and Sly, Allan and Srivastava, Nikhil},
+  title = {Group synchronization on grids},
+  journal = {Mathematical Statistics and Learning},
+  volume = {1},
+  number = {3/4},
+  pages = {227--256},
+  year = {2018},
+  doi = {10.4171/MSL/6},
+  eprint = {1706.08561},
+  archivePrefix = {arXiv}
+}
+
+@misc{GarbanSpencer2023,
+  author = {Garban, Christophe and Spencer, Thomas},
+  title = {Continuous symmetry breaking along the {Nishimori} line},
+  year = {2023},
+  eprint = {2109.01617v3},
+  archivePrefix = {arXiv},
+  primaryClass = {math.PR},
+  note = {Version 3, 27 March 2023; first submitted 3 September 2021},
+  doi = {10.1063/5.0087024},
+  url = {https://arxiv.org/abs/2109.01617v3}
+}
+@misc{Suto2023TwoStep,
+  author = {S{\"u}t{\H{o}}, Andr{\'a}s},
+  title = {{Bose--Einstein} Condensation of Interacting Bosons: A Two-Step Proof},
+  year = {2023},
+  eprint = {2305.18959},
+  archivePrefix = {arXiv},
+  primaryClass = {math-ph},
+  doi = {10.48550/arXiv.2305.18959},
+  url = {https://arxiv.org/abs/2305.18959},
+  note = {Version 1, 30 May 2023}
+}
+
+@article{PenroseOnsager1956,
+  author = {Penrose, Oliver and Onsager, Lars},
+  title = {{Bose--Einstein} Condensation and Liquid Helium},
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+}
+@article{Yang1962ODLRO,
+  author = {Yang, C. N.},
+  title = {Concept of Off-Diagonal Long-Range Order and the Quantum Phases of Liquid {He} and of Superconductors},
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+}
+
+@article{Bogolubov1947,
+  author = {Bogolubov, N.},
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+}
+
+@article{Bose1924,
+  author = {Bose, Satyendra Nath},
+  title = {{Plancks Gesetz und Lichtquantenhypothese}},
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+  year = {1924},
+  doi = {10.1007/BF01327326},
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+}
+
+@article{KennedyLiebShastry1988XY,
+  author = {Kennedy, Tom and Lieb, Elliott H. and Shastry, B. Sriram},
+  title = {The {XY} Model Has Long-Range Order for All Spins and All Dimensions Greater than One},
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+}
+
+@article{DysonLiebSimon1978,
+  author  = {Dyson, Freeman J. and Lieb, Elliott H. and Simon, Barry},
+  title   = {Phase Transitions in Quantum Spin Systems with Isotropic and Nonisotropic Interactions},
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+  number  = {4},
+  pages   = {335--383},
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+  doi     = {10.1007/BF01106729}
+}
+
+@article{Einstein1924IdealGas,
+  author  = {Einstein, Albert},
+  title   = {{Quantentheorie des einatomigen idealen Gases}},
+  journal = {Sitzungsberichte der Preussischen Akademie der Wissenschaften},
+  year    = {1924},
+  pages   = {261--267},
+  url     = {https://echo-old.mpiwg-berlin.mpg.de/ECHOdocuView?url=%2Fpermanent%2Fecho%2Feinstein%2Fsitzungsberichte%2FPG8B073X%2Findex.meta},
+  language = {German}
+}
+
+@article{Einstein1925IdealGas,
+  author  = {Einstein, Albert},
+  title   = {{Quantentheorie des einatomigen idealen Gases. Zweite Abhandlung}},
+  journal = {Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse},
+  year    = {1925},
+  pages   = {3--14},
+  url     = {https://echo-old.mpiwg-berlin.mpg.de/ECHOdocuView?url=%2Fpermanent%2Fecho%2Feinstein%2Fsitzungsberichte%2FR1Y4X6GP%2Findex.meta&viewMode=index},
+  language = {German}
+}
+
+@incollection{Fournais2021,
+  author = {Fournais, S{\o}ren},
+  title = {Length scales for {BEC} in the dilute {Bose} gas},
+  booktitle = {Partial Differential Equations, Spectral Theory, and Mathematical Physics: The Ari Laptev Anniversary Volume},
+  editor = {Exner, Pavel and Frank, Rupert L. and Gesztesy, Fritz and Holden, Helge and Weidl, Timo},
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+  pages = {115--133}, publisher = {EMS Press}, year = {2021},
+  doi = {10.4171/ECR/18-1/7}, eprint = {2011.00309}, archivePrefix = {arXiv},
+  url = {https://ems.press/books/ecr/187/3520}
+}
+
+@article{Dyson1957,
+  author = {Dyson, Freeman J.},
+  title = {Ground-State Energy of a Hard-Sphere Gas},
+  journal = {Physical Review}, volume = {106}, pages = {20--26},
+  year = {1957}, doi = {10.1103/PhysRev.106.20}
+}
+
+@article{LiebYngvason1998,
+  author = {Lieb, Elliott H. and Yngvason, Jakob},
+  title = {Ground State Energy of the Low Density {Bose} Gas},
+  journal = {Physical Review Letters}, volume = {80}, pages = {2504--2507},
+  year = {1998}, doi = {10.1103/PhysRevLett.80.2504},
+  eprint = {cond-mat/9712138}, archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/cond-mat/9712138}
+}
+
+@misc{BastiEtAl2026HardSpheres,
+  author = {Basti, Giulia and Brooks, Morris and Cenatiempo, Serena and Olgiati, Alessandro and Schlein, Benjamin},
+  title = {The {Lee--Huang--Yang} energy for a dilute gas of hard spheres: an upper bound},
+  year = {2026},
+  eprint = {2603.13084},
+  archivePrefix = {arXiv},
+  primaryClass = {math-ph},
+  doi = {10.48550/arXiv.2603.13084},
+  note = {Version 2, September 21, 2026},
+  url = {https://arxiv.org/abs/2603.13084v2}
+}
+
+@misc{BastiEtAl2026GeneralPotentials,
+  author = {Basti, Giulia and Brooks, Morris and Cenatiempo, Serena and Olgiati, Alessandro and Schlein, Benjamin},
+  title = {{Lee--Huang--Yang} Energy for Dilute {Bose} Gases: an Upper Bound for General Potentials},
+  year = {2026},
+  eprint = {2609.25314},
+  archivePrefix = {arXiv},
+  primaryClass = {math-ph},
+  note = {Version 1, September 21, 2026},
+  url = {https://arxiv.org/abs/2609.25314v1}
+}
+
+@article{Timar2013,
+  author = {Tim{\'a}r, {\'A}d{\'a}m},
+  title = {Boundary-connectivity via graph theory},
+  journal = {Proceedings of the American Mathematical Society},
+  volume = {141},
+  number = {2},
+  pages = {475--480},
+  year = {2013},
+  doi = {10.1090/S0002-9939-2012-11333-4},
+  eprint = {0711.1713},
+  archivePrefix = {arXiv},
+  primaryClass = {math.PR},
+  url = {https://arxiv.org/abs/0711.1713v2}
+}
+
+@article{AriasCastroCandesHelgasonZeitouni2008,
+  author = {Arias-Castro, Ery and Cand{\`e}s, Emmanuel J. and Helgason, Hannes and Zeitouni, Ofer},
+  title = {Searching for a trail of evidence in a maze},
+  journal = {The Annals of Statistics},
+  volume = {36}, number = {4}, pages = {1726--1757}, year = {2008},
+  doi = {10.1214/07-AOS526},
+  url = {https://doi.org/10.1214/07-AOS526}
+}
+
+@article{BergerPeres2013,
+  author = {Berger, Noam and Peres, Yuval},
+  title = {Detecting the trail of a random walker in a random scenery},
+  journal = {Electronic Journal of Probability},
+  volume = {18},
+  number = {87},
+  year = {2013},
+  doi = {10.1214/EJP.v18-2367},
+  eprint = {1210.0314},
+  archivePrefix = {arXiv},
+  primaryClass = {math.PR}
+}
+
+@misc{PeresSly2014,
+  author = {Peres, Yuval and Sly, Allan},
+  title = {Rigidity and tolerance for perturbed lattices},
+  year = {2014}, eprint = {1409.4490}, archivePrefix = {arXiv},
+  primaryClass = {math.PR}, url = {https://arxiv.org/abs/1409.4490}
+}
+
+@article{LeeHuangYang1957,
+  author = {Lee, T. D. and Huang, Kerson and Yang, C. N.},
+  title = {Eigenvalues and Eigenfunctions of a {Bose} System of Hard Spheres and Its Low-Temperature Properties},
+  journal = {Physical Review},
+  volume = {106},
+  number = {6},
+  pages = {1135--1145},
+  year = {1957},
+  doi = {10.1103/PhysRev.106.1135}
+}
+
+@article{FournaisSolovej2023,
+  author = {Fournais, S{\o}ren and Solovej, Jan Philip},
+  title = {The energy of dilute {Bose} gases {II}: the general case},
+  journal = {Inventiones Mathematicae},
+  volume = {232},
+  number = {2},
+  pages = {863--994},
+  year = {2023},
+  doi = {10.1007/s00222-022-01175-0},
+  eprint = {2108.12022},
+  archivePrefix = {arXiv},
+  url = {https://arxiv.org/abs/2108.12022v2}
+}
+
+@misc{GalandaPinamonti2025Equilibrium,
+  author = {Galanda, Stefano and Pinamonti, Nicola},
+  title = {Equilibrium states for non relativistic {Bose} gases with condensation},
+  year = {2025},
+  eprint = {2509.25101},
+  archivePrefix = {arXiv},
+  primaryClass = {math-ph},
+  note = {Version 1, 29 September 2025},
+  url = {https://arxiv.org/abs/2509.25101v1}
+}
+
+@misc{GalandaPinamonti2026GP,
+  author = {Galanda, Stefano and Pinamonti, Nicola},
+  title = {Equilibrium states for non relativistic {Bose} gases and the {Gross--Pitaevskii} limit},
+  year = {2026},
+  eprint = {2606.25775},
+  archivePrefix = {arXiv},
+  primaryClass = {math-ph},
+  note = {Version 2, 7 July 2026},
+  url = {https://arxiv.org/abs/2606.25775v2}
+}
+
+@article{HaggstromMossel1998,
+  author = {H{\"a}ggstr{\"o}m, Olle and Mossel, Elchanan},
+  title = {Nearest-neighbor walks with low predictability profile and percolation in {$2+\varepsilon$} dimensions},
+  journal = {The Annals of Probability},
+  volume = {26},
+  number = {3},
+  pages = {1212--1231},
+  year = {1998},
+  doi = {10.1214/aop/1022855750},
+  url = {https://faculty.wharton.upenn.edu/wp-content/uploads/2014/07/Nearest_Neighbor_Walks_with_Low_Predictability_Profile_and_Percolation_1.pdf}
+}
+
+@article{DeuchertSeiringerYngvason2019,
+  author = {Deuchert, Andreas and Seiringer, Robert and Yngvason, Jakob},
+  title = {{Bose--Einstein} Condensation in a Dilute, Trapped Gas at Positive Temperature},
+  journal = {Communications in Mathematical Physics},
+  volume = {368},
+  number = {2},
+  pages = {723--776},
+  year = {2019},
+  doi = {10.1007/s00220-018-3239-0},
+  url = {https://d-nb.info/1169529054/34}
+}
diff --git a/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/atypical.tex b/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/atypical.tex
new file mode 100644
index 0000000000000000000000000000000000000000..77f399be4253abe0fa77f99f06172588d2d4ec8f
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+++ b/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/atypical.tex
@@ -0,0 +1,459 @@
+\section{A joint bound for atypical trajectories}
+\label{sec:atypical}
+
+The bridge criteria require a bound for many failures at once.  A
+single-bridge estimate is insufficient here: the interacting closed measure
+does not make its bridges independent.  We obtain a localized exponential
+moment by assigning an auxiliary color to each tested bridge, retaining its
+spatial confinement, and estimating the resulting partition function.
+
+We use the free-energy increment bound \eqref{eq:6} and the density bound
+\eqref{eq:10}.  More explicitly, the two inputs needed in this section are
+\[
+ F_j-F_{j-1}\le C_v\rho\quad(1\le j\le N),
+ \qquad
+ \langle n_B\rangle_{J,-W}\le C_v\rho|B|
+ \quad(0\le J\le N, 0\le W\le\rho),
+\]
+where $B$ is any union of coarse cells and the second expectation is in the
+$J$-particle equilibrium state with external potential $-W$.  In particular,
+the constants in these inputs are fixed before the trajectory cutoff $D$
+and the isolation parameter $\zeta$ are chosen.
+
+Recall that $\delta=s^2$, $r=\zeta\rho^{-1/3}$, and the paths in the slab
+carry the lift convention specified in Section~\ref{sec:slab}.  For a
+nonempty union $U$ of coarse cells, let $J_U^{\mathrm{fail}}$ count the
+slots ending in $U$ for which at least one of the following holds: the
+lifted path leaves the ball of radius $Ds$ about its end; on the first
+time interval of length $r$ it leaves the ball of radius $r/10$ about
+its start; or on the last such interval it leaves that ball about its
+end.
+
+\begin{proposition}[Localized exponential moment for trajectory failures]
+\label{prop:atypical-moment}
+Fix $K$ as in the cell construction.  There exist $D_{\min}$, $C_v$, and
+$\kappa>0$, depending on the fixed potential and on $K$, such that the
+following holds.  Fix $D\ge D_{\min}$ and $0<\zeta\le1$.  For sufficiently
+small $\rho>0$ and sufficiently large volume at that density, uniformly
+for $\rho/2\le N/V\le2\rho$ and $0<T\le\rho^2$, every nonempty coarse
+cell union $U$ satisfies
+\begin{equation}
+ \log\mathbb E_{\mathrm{cl}}\exp(J_U^{\mathrm{fail}})
+ \le
+ \left[C_v\sum_{j\ge0}D_j^3e^{-\kappa D_j^2}
+       +o_\rho(1)\right]\rho|U|,
+ \qquad D_j=2^jD.
+ \tag{17}\label{eq:17}
+\end{equation}
+The constants $C_v,\kappa$ do not depend on $D$ or $\zeta$.  The term
+$o_\rho(1)$ is at fixed $v,K,D,\zeta$ and is uniform in the stated
+particle numbers, volumes, and temperatures.
+\end{proposition}
+
+\subsection{A localized kernel for one marked bridge}
+
+Partition the failure event into disjoint bins.  In bin $j\ge0$, the
+lifted maximum distance from the end belongs to $(D_js,2D_js]$.  Bin
+$*$ consists of a failure of one of the two short-interval tests with
+no macro failure.  Write
+\[
+ h_j=D_j,\quad b_j=D_j^2\quad(j\ge0),
+ \qquad h_*=D,\quad b_*=r.
+\]
+Here and below a sum over bins includes $*$.  Let
+\[
+ \Omega_j=\{x\in\Lambda_L:d(x,U)<(2h_j+4)s\}.
+\]
+If this set is the whole torus, its Laplacian has periodic boundary
+conditions; otherwise we use the Dirichlet Laplacian.  Choose a
+Lipschitz function $0\le p_j\le1$, equal to one when $d(x,U)\le2h_js$,
+vanishing when $d(x,U)\ge(2h_j+2)s$, and satisfying
+$|\nabla p_j|\le C/s$.  It has support compactly contained in $\Omega_j$
+unless $\Omega_j$ is the whole torus.  The elementary coarse-cell cover
+bound gives
+\[
+ |\Omega_j|\le C h_j^3|U|,
+ \qquad
+ \Omega_j\subset B_j,\qquad |B_j|\le C h_j^3|U|,
+\]
+for suitable coarse-cell unions $B_j$.  The constants also work when
+the enlarged set covers or winds around the torus.
+
+Let $K_j(E,S)$ be the unnormalized free single-bridge kernel with end
+$E\in U$ and failure in bin $j$.  There are absolute constants $c,C>0$
+and a fixed $c_1>1$ such that
+\begin{equation}
+ K_j(E,S)
+ \le C e^{-cb_j}p_j(E)p_j(S)
+          [e^{c_1\delta\Delta_{\Omega_j}}](E,S).
+ \tag{18}\label{eq:18}
+\end{equation}
+These constants are independent of $D$ and $\zeta$ after the
+small-density restrictions below are imposed.
+
+To prove this, fix a lift of $E$ and consider separately each lift of
+$S$, with displacement $d$ from that lift of $E$.  Put
+$k_t(d)=(4\pi t)^{-3/2}e^{-|d|^2/(4t)}$.  In a macro bin the event
+forces $|d|\le2h_js$.  If $|d|>h_js/2$, widening the Gaussian gives
+\[
+ k_\delta(d)\le C e^{-ch_j^2}k_{c_1\delta}(d).
+\]
+If $|d|\le h_js/2$, the zero-end bridge fluctuation must have maximum
+at least $h_js/2$.  The Brownian bridge maximum tail is at most
+$Ce^{-ch_j^2}$, while $k_\delta\le c_1^{3/2}k_{c_1\delta}$.
+This proves the same bound for this case.
+
+For bin $*$ one has $|d|\le Ds$.  Once $s\ge40D$, the linear drift
+on a time interval of length $r$ is at most $r/40$.  If $B$ is
+Brownian motion with generator $\Delta$, the fluctuation is
+$B_t-(t/\delta)B_\delta$.  The maximal inequality on $[0,r]$ bounds
+the chance that the first term exceeds a fixed positive multiple
+of $r$ by $Ce^{-cr}$.  The corresponding estimate for the second
+term is $Ce^{-c\delta}$, since exceeding that multiple of $r$
+requires $|B_\delta|\ge c\delta$.  We take $r\le\delta/2$.
+Time reversal gives the same bound at the other endpoint.  Thus
+the bin-$*$ kernel for each retained lift is at most
+$Ce^{-cr}k_{c_1\delta}(d)$.
+
+It remains to justify the Dirichlet kernel on the right of
+\eqref{eq:18}, rather than a free kernel without localization.
+For every retained lift, the straight segment from $E$ to $S$
+is within distance $2h_js$ of the chosen lift of $E$.  Every lifted
+path at distance less than $s$ from that segment projects into
+$\Omega_j$.  A bridge of diffusion coefficient $c_1$ over time
+$\delta=s^2$ has probability at least a fixed $q_*>0$ of remaining
+in this tube, by scaling its zero-end fluctuation.  Consequently
+the killed torus kernel is at least
+\[
+ q_*\sum_{\substack{\text{retained lifts of }S}}
+                  k_{c_1\delta}(d).
+\]
+The terms are disjoint winding classes of torus paths.  This remains
+valid for extents greater than $L$ and for $\Omega_j=\Lambda_L$.
+On the support of $K_j$, both $p_j(E)$ and $p_j(S)$ equal one.
+Summing the preceding bounds over lifts proves \eqref{eq:18}.
+
+\subsection{The colored operator and its large powers}
+
+Choose $\kappa>0$ sufficiently small, and set
+\[
+ \eta_j=e^{-\kappa b_j},\qquad M_j=\kappa b_j/\delta.
+\]
+By increasing the lower restrictions on $D$ and $r$, the prefactor
+in \eqref{eq:18}, including the marking factor $e-1$, is bounded by
+$\eta_j e^{-\delta M_j}=e^{-2\kappa b_j}$.  The choice of $\kappa$
+is independent of $D$ and $\zeta$.
+
+We will represent marked bridges by auxiliary particle colors. The small
+coefficient $\eta_j$ limits the enlargement of the map into the colored
+space, while the positive gap $M_j$ will pay the free-energy cost of
+removing a particle from the original interacting color. Normalizing this
+map will introduce a local attractive potential. The density estimate
+controls its cost on the original color, and the gap also absorbs it on
+the marked colors.
+
+The enlarged one-particle Hilbert space is
+\[
+ \mathfrak h_e=L^2(\Lambda_L)
+                    \oplus\bigoplus_j L^2(\Omega_j),
+\]
+with the first summand called the original color and the others
+indexed by the failure bins.  Define
+\[
+ v_e f=\bigl(f,(\sqrt{\eta_j}p_jf)_j\bigr),
+ \qquad V_e=v_e^{\otimes N}.
+\]
+The restrictions to the relevant domains in each marked component
+are understood.  On the labeled $N$-fold tensor product, define
+$\mathbb H$ by its closed quadratic form, separately in each
+color assignment.  Original-color particles have their original
+kinetic energy and the full pair potential between themselves.
+A particle of color $j$ has one-body operator
+$-c_1\Delta_{\Omega_j}+M_j$.  All interactions incident to marked
+particles are omitted.  The operator and the embedding commute
+with, or intertwine, coordinate permutations.  They therefore
+restrict to the symmetric $N$-particle spaces.  On those spaces put
+\[
+ \mathcal T=V_e^*e^{-\delta\mathbb H}V_e.
+\]
+
+Indeed, expanding
+\[
+ e^{J_U^{\mathrm{fail}}}
+ =\prod_{i=1}^N\left(1+(e-1)
+        \mathbf1_{\{E_i\in U,\ i\text{ fails}\}}\right)
+\]
+marks subsets of labels.  For each marked subset, decompose its
+failure events into bins and drop precisely the interactions
+incident to marked labels.  This increases the nonnegative path
+weight.  Applying \eqref{eq:18} to each marked path leaves the
+original interacting kernel on the other labels.  The sum over
+all color assignments is exactly the kernel of $\mathcal T$.
+Thus $\mathcal T$ majorizes the tested slab kernel pointwise on
+labeled coordinates.  The nonnegative outer kernel, including
+its symmetrization projector, gives
+\[
+ \mathbb E_{\mathrm{cl}}e^{J_U^{\mathrm{fail}}}
+ \le Z_N^{-1}\Tr_{\mathrm{sym}}
+                     e^{-(\beta-\delta)H}\mathcal T.
+\]
+For $\beta>\delta$, Schatten H\"older with conjugate exponents
+$\beta/(\beta-\delta)$ and $\beta/\delta$ yields
+\begin{equation}
+ \mathbb E_{\mathrm{cl}}e^{J_U^{\mathrm{fail}}}
+ \le
+ \left(\frac{\Tr_{\mathrm{sym}}\mathcal T^{\beta/\delta}}
+                   {Z_N}\right)^{\delta/\beta}.
+ \tag{19}\label{eq:19}
+\end{equation}
+
+The embedding has norm larger than one.  We next account for its
+effect on the high power in \eqref{eq:19} without losing the spatial
+localization.  Define the bounded Lipschitz function
+\[
+ a(x)=\frac12\log\left(1+\sum_j\eta_jp_j(x)^2\right),
+ \qquad A=\sum_{i=1}^N a(x_i).
+\]
+The same position multiplier acts in every color.  One has
+$V_e=e^A U_e$, where $U_e$ is an isometry: at the one-particle
+level the squared norm of the color vector is $e^{2a(x)}$.
+Consequently
+\[
+ \mathcal T=U_e^*XU_e,\qquad
+ X=e^Ae^{-\delta\mathbb H}e^A.
+\]
+Eigenvalue min--max for a positive compact compression gives
+$\lambda_k(\mathcal T)\le\lambda_k(X)$ for every $k$, including
+multiplicities.  Hence the trace of any positive power of
+$\mathcal T$ is bounded by that of $X$.
+
+\begin{lemma}[Trace bound for a Lipschitz sandwich]
+\label{lem:operator-sandwich}
+For the operators just defined, put
+\[
+ W_0=2a/\delta+c_1|\nabla a|^2,
+ \qquad
+ \widetilde{\mathbb H}=\mathbb H-\sum_{i=1}^N W_0(x_i).
+\]
+Then
+\begin{equation}
+ \Tr X^{\beta/\delta}
+       \le\Tr e^{-\beta\widetilde{\mathbb H}}.
+ \tag{20}\label{eq:20}
+\end{equation}
+In fact the analogous inequality holds with every positive power
+$q$ on the left and time $q\delta$ on the right.
+\end{lemma}
+
+\begin{proof}
+We interpolate from the identity to $X$ and bound the rate of change
+of the squared norm by the quadratic form of $\widetilde{\mathbb H}$.
+Applying the same estimate to exterior powers controls products of
+eigenvalues; an integral identity then gives the trace bound for every
+positive power.
+
+First, the heat traces below are finite and $X$ is positive compact.
+At fixed $N,L,\rho$, the gaps
+$M_j$ tend to infinity along the macro bins.  Domain monotonicity
+and the torus heat-kernel bound give, for each $t>0$,
+\[
+ \sum_j e^{-tM_j}\Tr e^{c_1t\Delta_{\Omega_j}}
+ \le C V(V^{-1}+t^{-3/2})\sum_j e^{-tM_j}<\infty.
+\]
+Including the original color, the free labeled $N$-particle trace
+is the $N$th power of a finite one-particle trace.  The nonnegative
+original-color pair potentials only increase the Hamiltonian in
+the form sense.  Min--max therefore gives compact resolvent and
+finite positive-time heat traces for $\mathbb H$.  Subtracting
+the bounded potential $\sum_iW_0(x_i)$ preserves these properties.
+The bounded multiplier $e^A$ also makes $X$ trace class.
+
+For $0\le t\le\delta$ define
+\[
+ U_t=e^{f(t)A}e^{-t\mathbb H}e^A,
+ \qquad f(t)=-1+2t/\delta.
+\]
+Thus $U_0=1$ and $U_\delta=X$.  Bounded Lipschitz multiplication
+preserves the closed form domain of $\mathbb H$.  This includes
+the Dirichlet conditions on marked colors and the weighted
+sum of color-gap terms.  For any initial vector $x$ and $t>0$,
+spectral calculus gives
+\[
+ e^{-f(t)A}U_tx=e^{-t\mathbb H}e^Ax\in D(\mathbb H).
+\]
+The vector $y(t)=U_tx$ is norm differentiable for $t>0$, belongs
+to the form domain, and satisfies
+\[
+ y'(t)=\frac{2A}{\delta}y(t)
+             -e^{f(t)A}\mathbb H e^{-f(t)A}y(t).
+\]
+The last expression is well-defined on this evolving vector.
+Its real inner product with $y$ can be evaluated using the
+closed form between $e^{fA}y$ and $e^{-fA}y$; no assertion that a
+Lipschitz multiplier preserves $D(\mathbb H)$ is needed.
+
+In a fixed color component, the diffusion coefficient $c_i$ of
+coordinate $i$ is either $1$ or $c_1$.  The product rule gives
+almost everywhere
+\[
+ \operatorname{Re}\left[
+  \overline{\nabla_i(e^{fA}y)}\cdot\nabla_i(e^{-fA}y)\right]
+ =|\nabla_i y|^2-f^2|\nabla a(x_i)|^2|y|^2.
+\]
+The cross terms are purely imaginary and all potentials commute
+with these multipliers.  Since $|f|\le1$ and $c_i\le c_1$, the
+preceding identities imply
+\[
+ \frac12\frac{\dd}{\dd t}\|U_tx\|^2
+ \le-\widetilde{\mathfrak q}[U_tx],
+\]
+where $\widetilde{\mathfrak q}$ is the form of
+$\widetilde{\mathbb H}$.
+
+Let $E_1\le E_2\le\cdots$ be the eigenvalues of
+$\widetilde{\mathbb H}$, counting multiplicity.  Apply the same
+calculation to the tensor power $U_t^{\otimes k}$, with the sums
+of $A$ and $\mathbb H$ over $k$ copies of the enlarged symmetric
+many-particle Hilbert space.  Restrict these copies to their
+antisymmetric subspace.  In the eigenbasis of
+$\widetilde{\mathbb H}$, the smallest eigenvalue of the copy sum
+there is $E_1+\cdots+E_k$.  It follows that
+\[
+ \frac12\frac{\dd}{\dd t}\|U_t^{\otimes k}x\|^2
+ \le-(E_1+\cdots+E_k)\|U_t^{\otimes k}x\|^2
+\]
+for every antisymmetric $x$ and $t>0$.  Integrate from a positive
+time and then let that time tend to zero, using $U_0=1$ and
+strong continuity.  The norm of the $k$th exterior power of
+$U_\delta$ is consequently at most
+$e^{-\delta(E_1+\cdots+E_k)}$.
+
+The exterior-power norm is the product of the largest $k$
+singular values.  Since $U_\delta=X$ is positive and injective,
+write its eigenvalues in decreasing order as $x_i>0$.  We obtain
+\[
+ \sum_{i=1}^k\log x_i
+       \le-\delta\sum_{i=1}^k E_i\qquad(k\ge1).
+\]
+These partial-sum inequalities imply, for every real $t$,
+\[
+ \sum_i(\log x_i-t)_+
+       \le\sum_i(-\delta E_i-t)_+.
+\]
+Indeed each side is the supremum over $k\ge0$ of the
+corresponding shifted partial sum; for fixed $t$ only finitely
+many positive terms occur.  For every $q>0$ the identity
+\[
+ e^{qu}=\int_{\mathbb R}(u-t)_+q^2e^{qt}\,\dd t
+\]
+and Tonelli's theorem now give
+$\sum_i x_i^q\le\sum_i e^{-q\delta E_i}$.  Taking
+$q=\beta/\delta$ proves \eqref{eq:20}.
+\end{proof}
+
+\subsection{The partition-function estimate with confined colors}
+
+The preceding lemma has converted the large power of the marked-slab
+operator into an ordinary heat trace. It remains to compare that trace
+with the original partition function. Spatial confinement is essential
+here: it makes the final cost proportional to $|U|$, the volume where
+failures are tested, rather than to the whole torus.
+
+Set
+\[
+ \Sigma=\sum_j\eta_jh_j^3
+       =\sum_{j\ge0}D_j^3e^{-\kappa D_j^2}
+                            +D^3e^{-\kappa r}.
+\]
+From the definition of $a$ and Cauchy--Schwarz,
+\[
+ 0\le W_0\le\frac C\delta\sum_j\eta_j\mathbf1_{\Omega_j}.
+\]
+For example,
+\[
+ |\nabla a|^2
+ =\frac{|\sum_j\eta_jp_j\nabla p_j|^2}
+                  {(1+\sum_j\eta_jp_j^2)^2}
+ \le\frac C\delta\sum_j\eta_j\mathbf1_{\Omega_j}.
+\]
+All constants are independent of the number or sizes of the
+domains.  Taking $D$ large and then $\rho$ small makes
+$\|W_0\|_\infty\le\rho$, because
+$\delta\ge K^2\rho^{-1}$ and $r\to\infty$.
+
+For $0\le J\le N$, differentiate the original-color partition
+function in the strength of $-W_0$.  The density input, applied
+to the covers $B_j$ and to every external potential $-tW_0$ with
+$0\le t\le1$, gives
+\[
+ \log Z_J^{(-W_0)}-\log Z_J
+ =\beta\int_0^1\langle n_{W_0}\rangle_{J,-tW_0}\,\dd t
+ \le\frac{\beta C_v\rho|U|}\delta\Sigma.
+\]
+The free-energy increment input also gives
+$Z_J/Z_N\le e^{\beta C_v\rho(N-J)}$.
+
+To use these inequalities, decompose the enlarged symmetric
+space according to the occupation of each color:
+\[
+ \operatorname{Sym}^N\mathfrak h_e
+ \cong
+ \bigoplus_{J+\sum_j k_j=N}
+  \operatorname{Sym}^J L^2(\Lambda_L)
+       \otimes\bigotimes_j\operatorname{Sym}^{k_j}L^2(\Omega_j).
+\]
+Each number sector is one tensor product.  There is no
+multinomial multiplicity in its trace: the normalized
+symmetrization over color assignments is the unitary realizing
+this decomposition.  In that sector, the cost
+$e^{\beta C_v\rho(N-J)}$ is one factor per marked particle.
+On color $j$, subtracting $W_0$ and this cost leaves an operator
+bounded below by
+\[
+ -c_1\Delta_{\Omega_j}
+                  +M_j-\|W_0\|_\infty-C_v\rho
+ \ge -c_1\Delta_{\Omega_j}+M_j/2.
+\]
+For this last inequality use
+$M_j\ge\kappa b_j\rho/(4K^2)$, choose $D$ sufficiently large
+depending on $v,K$, and then make $r$ sufficiently large.
+The same choices ensure $\beta M_j/2\ge1$ in every color.
+
+Sum the marked occupation numbers and drop their total-number
+constraint.  This bounds the sum by a product of ideal Bose
+grand partition functions with gaps $M_j/2$.  In each color its
+logarithm is at most
+\[
+ \sum_{\ell\ge1}\frac{e^{-\ell\beta M_j/2}}\ell
+                       \Tr e^{c_1\ell\beta\Delta_{\Omega_j}}
+ \le C|\Omega_j|(V^{-1}+T^{3/2})e^{-\beta M_j/2}.
+\]
+The estimate uses the killed heat-kernel bound by the periodic
+one and the geometric series with $\beta M_j/2\ge1$.
+The resulting sum over colors is finite, so the passage from
+finitely many colors to countably many is justified by
+monotone convergence of the nonnegative occupation sums.
+Combining this bound with \eqref{eq:19}, the compression
+inequality, and \eqref{eq:20}, we obtain the quantitative estimate
+\begin{equation*}
+ \log\mathbb E_{\mathrm{cl}}e^{J_U^{\mathrm{fail}}}
+ \le C_v\rho|U|\Sigma
+  +C\delta T|U|(V^{-1}+T^{3/2})
+                     \sum_jh_j^3e^{-\beta M_j/2}.
+\end{equation*}
+
+This also proves the required uniformity.  Since $\beta\ge\delta$,
+the color sum is bounded at fixed $D$ for all sufficiently small
+$\rho$, using the summable macro terms
+$D_j^3e^{-\kappa D_j^2/2}$ and the micro term
+$D^3e^{-\kappa r/2}$.  Moreover $V\ge s^3$,
+$\delta\le4K^2/\rho$, and $T\le\rho^2$, so
+\[
+ \frac{\delta T}{\rho}(V^{-1}+T^{3/2})
+ \le C_K(\rho^{3/2}+\rho^3).
+\]
+The micro contribution $D^3e^{-\kappa r}$ to $\Sigma$ tends
+to zero at fixed $D,\zeta$.  Substituting these estimates in the
+partition-function bound proves \eqref{eq:17}, with
+all errors uniform in the temperature and the thermodynamic
+volume, as asserted.
diff --git a/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/comparison.tex b/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/comparison.tex
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@@ -0,0 +1,554 @@
+\section{Cross comparison in the slab}\label{sec:comparison}
+
+
+The construction of Section~\ref{sec:opening} reduces the condensate
+bound to the following estimate.
+
+\begin{proposition}[Uniform second moment]\label{prop:second-moment}
+After fixing the cutoffs as above, there is a constant $C_v<\infty$
+such that, for all sufficiently small densities and sufficiently large
+volumes at each density,
+\[
+                    \int f^2\dd\mu\le C_v.
+\]
+The constant is uniform for $\rho/2\le N/V\le2\rho$ and
+$0<T\le\rho^2$.
+\end{proposition}
+
+We prove this by comparing the two changes in \eqref{eq:27} with a
+single reverse insertion. Interactions away from encounters must cancel:
+a fixed cost at every path cell would grow with the volume. The
+analytic comparison below leaves only a fixed cost for shared labels
+and a vanishing cost for nearby distinct bridges. To retain this gain
+after averaging, we must also preserve the configuration-dependent
+path and label choices. We establish that preservation before removing
+their constraints from any conditional integral.
+
+\subsection{The reverse insertion and the slot partition}
+
+Fix a term of \eqref{eq:27}. The tuple $I'$ is selected along
+$P'=P(C,\gamma')$ and produces the open state $O$; the density at $O$
+uses a tuple $I$ along $P=P(\widehat C,\gamma)$. Their first label
+$i_0$ is common. Write $e$ for the last label of $I$, and introduce
+the reverse state $B$ by the following endpoint assignments:
+\[
+\begin{array}{c|c|c}
+ \text{state}&\text{starting tuple}&\text{inner ending tuple}\\ \hline
+ C&S^C&Y\\
+ O&p_{I'}S^C&Y\text{ with }Y_{i_0}\text{ replaced by }z\\
+ \widehat C&p_I^{-1}p_{I'}S^C&Y\\
+ B&p_I^{-1}S^C&Y\text{ with }Y_e\text{ replaced by }x.
+\end{array}
+\]
+The outer tuple $Y$ is unchanged in all four states. The table specifies
+endpoints; the path variables retained or integrated in each state are
+described next. Figure~\ref{fig:endpoints} shows the two single-tuple
+assignments. In the cross term, the forward state uses $I'$ and the
+reverse state uses $I$.
+\input{figures/endpoint-changes}
+
+Identify a tuple with its slot set when taking intersections and
+differences, and partition the selected slots as
+\[
+ \mathsf h=I\cap I',\quad
+ \mathsf d=\mathsf h\cup\operatorname{pred}_I(\mathsf h),\quad
+ \mathsf j=I\setminus\mathsf d,\quad
+ \mathsf l=I'\setminus\mathsf h,\quad
+ \mathsf L=(I\cup I')^c .
+\]
+Thus $\mathsf h$ contains the shared labels, while $\mathsf d$ also
+contains their predecessors on $I$. In particular
+\(|\mathsf d|\le2|\mathsf h|\) and $e\in\mathsf d$, because the
+first label is shared. The two unshared portions that can cancel are
+$\mathsf j$ and $\mathsf l$; $\mathsf L$ contains the untouched slots.
+Let $J$ count cells of $P$ within distance $R_*$ of $P'$ and conversely,
+adding both counts. Call these occurrences near and all others far.
+
+All \(\mathsf L\) paths agree throughout. On \(\mathsf j\), \(C=O\) for endpoints and paths since the slots are off \(I'\), while \(B\) has the endpoints of \(\widehat C\): the \(I\)-successor is not shared, hence also off \(I'\), and \(S_i^B=S_{\operatorname{succ}_I(i)}^C=S_{\operatorname{succ}_I(i)}^O=S_i^{\widehat C}\) there. On \(\mathsf l\), \(B\) has the endpoints of \(C\), and \(\widehat C=O\). The separate path integrations off \(\mathsf L\) and our identification for the comparison are thus:
+\[
+\begin{array}{c|l|l}
+ \text{set} & \text{path variables in (27)} & \text{use in }B\\ \hline
+ \mathsf j & C_{\mathsf j}=O_{\mathsf j},\ \widehat C_{\mathsf j}
+            & B_{\mathsf j}=\widehat C_{\mathsf j}\ \text{initially}\\
+ \mathsf l & C_{\mathsf l},\ O_{\mathsf l}=\widehat C_{\mathsf l}
+            & B_{\mathsf l}=C_{\mathsf l}\\
+ \mathsf d & C_{\mathsf d},\ \widehat C_{\mathsf d},\ O_{\mathsf h}
+            & B_{\mathsf d}\ \text{fresh}.
+\end{array}
+\]
+We will also reintegrate some of \(\mathsf j\) below; \(O\) on \(\mathsf d\setminus\mathsf h\) equals \(C\) there.
+
+We work first on the domain of the original nonzero weights and cuts in (27). All paths there on \(I\cup I'\) in \(C,O,\widehat C\) satisfy \(\mathrm{cut}_R\), by the two cuts in (27) and the two eligibilities. Displacements even for all endpoints needed in conditional integrations are at most \((D+10)s\): in \(O\) use the forward displacement rule from \(C\) on \(I'\) and from \(\widehat C\) on \(I\) (indeed \(S^O=p_I S^{\widehat C}\)). This also covers the non-selected slots on the union in \(C,\widehat C\), which individually have the same endpoints as in \(O\). For \(B\) on \(\mathsf d\), if the successor is shared use its start from an eligible slot of \(C\), close to the successor cell (for \(e\) use that \(x\) is in the first cell); otherwise the start agrees with \(\widehat C\).
+
+For the beginning and end anchors of the integrated rows where (16) is needed the scores against \(\mathsf L\) are \(\le C_0\). In \(O\) the starts on \(I'\) come from eligible starts of \(C\), on \(I\) from eligible starts of \(\widehat C\); the respective end anchors are covered by eligibility or the dummy \(z\) test. Similarly \(B_{\mathsf j},B_{\mathsf l}\) use eligible endpoints from \(\widehat C,C\); on \(\mathsf d\) use the successor rule just given and end eligibility of \(\widehat C\) or the dummy \(x\) test. Paths in \(\mathsf L\) are the same in the scores. Also any pair between \(\mathsf j,\mathsf l\) in \(O\), and in \(B\), has starts separated by more than \(r\) and likewise ends: by the indicated eligible sources the starts used on both sets are isolated in the common start multiset, of which each of \(O,B\) assigns distinct elements (indices) to distinct slots. These ordinary end slots are isolated among \(Y\) by eligibility in \(\widehat C,C\) respectively (no separation from the exceptional inner end is needed).
+
+All confined bridge measures use nearest lifts. Write \(\pi_i^X\) for the probability bridge on that lift before imposing the cut, and factor out the kernels \(k\) of those individual lifts (not the periodic sums), with product denoted \(K_X(G)\) on a set \(G\). Factor the kernels from the cut normalizations as well (write \(\overline{\mathcal Z}\) for the remaining normalization). The proposed-path kernels cancel, and relative to using baseline kernels the ratio left in (27) off \(\mathsf L\) is
+\[
+ \frac{K_C(I\cup I') K_{\widehat C}(I)}{K_B(I\cup I') K_O(I)}
+      =\frac{K_C(\mathsf d)K_{\widehat C}(\mathsf d)}
+             {K_B(\mathsf d)K_O(\mathsf d)}\le e^{C|\mathsf h|}.      \tag{28}\label{eq:28}
+\]
+
+\subsection{Conditional laws and the two denominator bounds}
+
+For \(G=\mathsf j\) or \(\mathsf l\) write \(\Phi_G^X\) for the exponential of minus the interactions internal to \(G\) and with \(\mathsf L\) only, and \(Z[G,X]\) for its integral with the probability bridges and cuts \(R\), fixing \(\mathsf L\). Use empty-product conventions.
+
+\paragraph{A one-path marginal bound.}
+In these normalized group distributions an individual path marginal is bounded by \(C_4\pi_i^X\). The bound also holds if some other paths of the group are frozen on the cut, normalizing conditionally with their interactions. Indeed, conditional on all other paths, the individual normalization for its incident weight is at least \(p_*\) times the exponential of minus the expected interactions under its free bridge conditioned on the cut, by Jensen. Against \(\mathsf L\) use (16) divided by \(p_*\); other confined paths of the group can interact only from nearby end cells (distance at most \(2R+5\)), a bounded number of slots depending on \(R\) by simplicity, and the one-fixed-trajectory expectation bound applies (again divide by \(p_*\)). Thus this incident normalization is at least \(p_* e^{-C/p_*}\). Positivity \(v\ge0\) now gives a uniform conditional density upper bound \(C_4\) relative to \(\pi_i^X\), possibly large but independent of density after fixing the cut and score constants; average the bound for marginals. We use this observation for \(\mathsf j\) with the endpoints of \(C\) (equal to those of \(O\)) or \(B\), and \(\mathsf l\) with those of \(O\).
+
+The needed lower normalizations are
+\[
+ \begin{split}
+     \overline{\mathcal Z}_{I'}(O)&\ge
+              Z[\mathsf l,O]\exp[-C|\mathsf h|-o_\rho(1)J],\\
+     \overline{\mathcal Z}_{I}(O)&\ge
+              Z[\mathsf j,C]\exp[-C|\mathsf h|-o_\rho(1)J].
+ \end{split}                                                          \tag{29}\label{eq:29}
+\]
+Here the small coefficients tend to zero after all cut and eligibility
+constants have been fixed. For the first, sample \(\mathsf l\) with its normalized group law, and \(\mathsf h\) independently with product free bridges conditioned on the cuts. Writing \(\mathcal V_{G,G'}\) for the summed interactions between disjoint groups (internal pairs once for \(\mathcal V_{G,G}\)), the ratio to \(Z[\mathsf l,O]\) is the product of free cut probabilities on \(\mathsf h\) times
+\[
+ \mathbb E\exp[-\mathcal V_{\mathsf l,\mathsf j}
+               -\mathcal V_{\mathsf l,\mathsf d\setminus\mathsf h}
+               -\mathcal V_{\mathsf h,\mathsf L\cup\mathsf j\cup(\mathsf d\setminus\mathsf h)\cup\mathsf l}
+               -\mathcal V_{\mathsf h,\mathsf h}]
+\]
+with the complement frozen in \(O\). The interaction of \(\mathsf l\) with the frozen \(\mathsf j\) costs at most \(C/\sqrt r\) in expectation per nearby pair: it is zero in the boundary intervals by endpoint separation and both restraints (distance at least \(r-2r/10>R_v\)), and use marginal domination after those intervals. There are at most \(C(R)J\) possible pairs, since end cells must be within \(2R+5\) and are distinct within each set. Interaction with frozen \(\mathsf d\setminus\mathsf h\) costs at most \(C|\mathsf d|\), without needing the small bound. The independently sampled paths cost at most \(C|\mathsf h|\) with \(\mathsf L\) by (16) and with the other integrated or frozen paths by the fixed-trajectory bound and confinement (only a bounded number of slots in a fixed-radius end-cell neighborhood in these latter groups, allowing the special end). Jensen gives the first bound. 
+
+For the second denominator, sample $\mathsf j$ with its group law
+(the endpoints of $C$, which agree with those of $O$ on this set),
+and sample $\mathsf d$ with independent free cut bridges. The frozen
+selected complement is now $\mathsf l$. The missing interactions are
+\[
+ \mathcal V_{\mathsf j,\mathsf l},\qquad
+ \mathcal V_{\mathsf d,\mathsf L\cup\mathsf j\cup\mathsf l},\qquad
+ \mathcal V_{\mathsf d,\mathsf d}.
+\]
+The first sum costs $C r^{-1/2}J$, using the same endpoint separation
+in $O$ and the marginal bound for $\mathsf j$. The other sums and the
+free cut probabilities cost $C|\mathsf d|\le2C|\mathsf h|$. Jensen's
+inequality gives the second line of \eqref{eq:29}.
+
+\paragraph{The remaining normalized factors.}
+Using \eqref{eq:28} and \eqref{eq:29}, drop interaction costs from the numerator weights by positivity except for: internal \(\mathsf L\); \(\Phi_{\mathsf j}^C,\Phi_{\mathsf l}^C\) from the closed weight; \(\Phi_{\mathsf l}^O\) from the proposal; \(\Phi_{\mathsf j}^{\widehat C}\) from the hypothetical old weight. 
+
+To spell out the remaining integrations, let \(\pi_G^X\) denote probability-bridge products, \(c_G^X=\mathbf 1_{\mathrm{cut}_R(X_G)}\), and \(d\nu_G^X=c_G^X\Phi_G^X d\pi_G^X/Z[G,X]\) for the normalized groups. Besides the exponential comparison factors, the choice weights/constraints, \(K_B(I\cup I')\) and the original internal measure on \(\mathsf L\), the upper integrand on the nonzero domain now uses
+\[
+ (c_{\mathsf l}^C\Phi_{\mathsf l}^C\,d\pi_{\mathsf l}^C)
+ (c_{\mathsf j}^{\widehat C}\Phi_{\mathsf j}^{\widehat C}\,d\pi_{\mathsf j}^{\widehat C})
+ \ d\nu_{\mathsf j}^C\,d\nu_{\mathsf l}^O\,
+ d\pi_{\mathsf d}^C\,d\pi_{\mathsf d}^{\widehat C}\,d\pi_{\mathsf h}^O .
+\]
+
+
+In particular the extras \(C_{\mathsf j},O_{\mathsf l}\) occur as normalized factors (fixing just endpoints and \(\mathsf L\) in their laws); all cuts inserted in these measures hold on the original domain. We still account for constraints and far choice weights below. For a near occurrence in cell \(Q\) bound the value of the choice factor on the nonzero domain by \([(1-\varepsilon)n_Q]^{-1}\) for count \(n_Q\) of that outer cell. We can and do carry the indicator of the original constraints (including both nonzero choices) even when bounding such values; in particular the counts along both paths are in the stated occupancy interval there.
+
+\subsection{Restoring the baseline interaction weight}
+
+We describe the weight restoration in \(B\). Put in \(\mathsf j_n\subseteq\mathsf j\) those with end cell within \(2R+5\) of some \(\mathsf l\) cell; other \(\mathsf j\) cannot interact with \(\mathsf l\) on the cuts. 
+
+\paragraph{Fixed data and feasible fibers.}
+Throughout this step, all endpoints, tuples, cell paths and skeletons
+are fixed. The path integrations divide as follows:
+\[
+\begin{array}{c|l}
+ \text{role}&\text{path variables}\\ \hline
+ \text{fixed in both integrals}
+   &\mathsf L,\quad B_{\mathsf l}=C_{\mathsf l},\quad
+             B_{\mathsf j\setminus\mathsf j_n}
+                       =\widehat C_{\mathsf j\setminus\mathsf j_n}\\
+ \text{integrated in the original term}
+   &\widehat C_{\mathsf j_n},\quad C_{\mathsf j},\quad O_{\mathsf l},
+      \quad C_{\mathsf d},\quad\widehat C_{\mathsf d},\quad O_{\mathsf h}\\
+ \text{integrated in the baseline term}&B_{\mathsf j_n},\quad B_{\mathsf d}.
+\end{array}
+\]
+Call the fixed data \emph{feasible} if some choice of the remaining
+original paths satisfies both nonzero selection weights, their specified
+adaptations, and the cuts. Call such a choice a \emph{witness}. We only
+need an upper comparison on feasible data: otherwise the original inner
+integral vanishes. The probability laws of the extra original paths
+depend only on the fixed endpoints and $\mathsf L$, not on
+$\widehat C_{\mathsf j_n}$.
+
+\paragraph{Restoring the missing interactions.}
+First consider interaction weights alone. The bound needed for replacing
+the $\widehat C_{\mathsf j_n}$ integral by a baseline integral is
+\begin{equation}
+ \Phi_{\mathsf l}^B
+      \int c_{\mathsf j_n}^B\Phi_{\mathsf j}^B\,d\pi_{\mathsf j_n}^B
+ \ \le\ e^{C|\mathsf h|+o_\rho(1)J}
+      \int c_{\mathsf j_n}^B c_{\mathsf d}^B
+       \exp\!\Big[-\sum_{\{i,i'\}\not\subseteq\mathsf L}
+                          \mathcal V_{ii'}(B)\Big]\,
+                       d\pi_{\mathsf j_n}^B d\pi_{\mathsf d}^B ,
+\tag{29a}\label{eq:restoring-weights}
+\end{equation}
+where the sum is over unordered pairs of distinct slots indicated. 
+
+Indeed, normalize
+$c_{\mathsf j_n}^B\Phi_{\mathsf j}^B\,d\pi_{\mathsf j_n}^B$ with
+$B_{\mathsf j\setminus\mathsf j_n}$ frozen, and independently sample
+$\mathsf d$ by free bridges conditioned on their cuts. The earlier
+one-path marginal bound applies also to this partially frozen group:
+the score against $\mathsf L$ is still bounded, and only boundedly many
+frozen or integrated $\mathsf j$ paths can be nearby. The ratio of the
+full cut integral on the right to the product on the left is the free
+cut probability product on $\mathsf d$, multiplied by the expectation
+of the exponential of minus the missing interactions.
+
+The $\mathsf j_n$--$\mathsf l$ interactions cost $o_\rho(1)J$ by
+marginal domination, endpoint separation and the boundary restraints.
+The frozen remainder of $\mathsf j$ cannot interact with $\mathsf l$
+on the cuts. Interactions incident to $\mathsf d$ cost
+$C|\mathsf d|$ by the free cut bounds, including \eqref{eq:16}.
+Jensen and the cut probabilities prove \eqref{eq:restoring-weights}.
+Its endpoint and frozen-path hypotheses follow from original feasibility,
+independently of the reintegrated paths.
+
+This inequality concerns interaction weights alone. To apply it to the
+original integral, we must still account for the far selection factors
+and the path-adaptation constraints. The next step proves that their
+needed properties hold for every baseline cut replacement.
+
+\subsection{Tracking the constraints in this comparison}
+
+The original choices depend on which cells are usable and which slots
+are eligible. We now compare those tests between any original witness
+and every replacement of $B_{\mathsf j_n},B_{\mathsf d}$ on the cuts,
+with the other data fixed as above. This universal comparison will
+justify using \eqref{eq:restoring-weights} inside the constrained integral.
+
+In $B$, use the outer tuple $Y$, including $Y_e$, for cell counts
+and end isolation, and declare the exceptional slot $e$ ineligible.
+Other eligibility tests use the starts and paths as before; the
+exceptional path is included in all score sums. Let $S_0(B)$ be the
+resulting unusable cells. Define $S_1(B)$ by the stricter score threshold
+$C_0-\Delta_A$ and the stronger required eligible fraction
+$1-\varepsilon/2$, with the same occupancy interval. Thus
+$S_0(B)\subseteq S_1(B)$. Write $S(C),S(\widehat C)$ for the
+unusable cells in the two witness states. The table compares the tests;
+its last column gives the ineligible fraction causing failure when the
+count lies in the allowed interval.
+\[
+\begin{array}{c|c|c|c}
+ \text{bad-cell set}&\text{allowed count}&\text{score cutoff}
+                                      &\text{failure fraction}\\ \hline
+ S(C),S(\widehat C),S_0(B)&[b_-m,b_+m]&C_0&>\varepsilon\\
+ S_1(B)&[b_-m,b_+m]&C_0-\Delta_A&>\varepsilon/2\\
+ \text{defective cells}
+    &[2b_-m,b_+m/2]&C_0-\Delta_A&\ge\varepsilon/4.
+\end{array}
+\]
+Counts outside the displayed interval also cause failure. The symmetric
+defective test uses actual inner endpoints, whereas $S_0(B),S_1(B)$
+use the outer-end convention just specified. The witness states $C$
+and $\widehat C$ are closed, so their inner ends are $Y$.
+
+\begin{lemma}[Stability under every cut replacement]\label{lem:replacement-stability}
+Fix feasible endpoint and path data as in the preceding subsection.
+For every replacement of $B_{\mathsf j_n},B_{\mathsf d}$ on the cuts
+and every original witness $C,\widehat C$, one has
+\[
+ \begin{gathered}
+        S(C),S(\widehat C)\ \subseteq S_1(B),\\
+        S(C)=S_0(B)\ \text{pointwise outside distance }R_*\text{ of }P,\\
+        S(\widehat C)=S_0(B)\ \text{pointwise outside distance }R_*\text{ of }P'.
+\end{gathered}                                                       \tag{30}\label{eq:30}
+\]
+In the far cells specified by the last two lines, all individual
+eligibility indicators and eligible counts agree as well.
+\end{lemma}
+\begin{proof}
+The state $C$ differs from $B$ only on $I$, and $\widehat C$ differs
+from $B$ only on $\mathsf l,\mathsf d,\mathsf j_n$. All changed paths
+in either comparison are confined. Their slots lie on at most two simple
+cell paths, with the special inner end of $B$ in $a$.
+
+At an unchanged slot's anchor, the change in either score is at most
+$\Delta_A$. Indeed only changed paths with actual inner ends within
+$(R+R_A)s$ of the anchor can contribute. In either state their number
+is at most $2(2R+2R_A+4)^3+1$, by outer-cell counting and the one moved
+exceptional end; also $\int_0^{\delta/2}w\le3$. Start isolation is
+unchanged because the unordered start tuple is unchanged, and ordinary
+outer-end isolation is unchanged as well. Consequently the eligible
+count in either witness is at least the stricter eligible count of $B$
+minus two per outer cell, allowing for the changed slots. If a cell
+passes the $S_1(B)$ test, then its witness eligible count is at least
+$(1-\varepsilon/2)n_Q-2\ge(1-\varepsilon)n_Q$ for sufficiently large
+$m$. The occupancy counts are identical. This proves the inclusion
+in \eqref{eq:30} by contraposition.
+
+For exact equality, changed paths comparing with \(C\) lie near \(P\). Comparing with \(\widehat C\), their end cells lie within \(2R+5\) of \(P'\): note \(\mathsf d\) uses shared slots or their predecessors, including the last cell \(b\), and the special end is in \(a\). In a cell beyond the respective distance \(R_*\) there are no changed slots; an unchanged macro-conforming path has both anchors close enough to its cell that scores are exactly unchanged by the indicator cutoff in (15), and a macro failure stays ineligible anyway. The choice of \(R_*\) more than covers these distances. This proves equality of all actual eligibility flags and counts in the respective far cells.
+\end{proof}
+
+\paragraph{Lists containing every feasible pair.}
+Define an explicit path predicate given \(B,\gamma,\gamma'\). Partition \(S_1(B)\) into components with adjacency distance up to \(K_b=2R_*+10\). Call components bridges if within \(R_*+2\) of both skeletons (set distances), and put \(b_{\rm tot}\) equal to the sum of their sizes. For the list of possible \(P\), take arbitrary subsets of cells in the bridge components as bad; from every nonbridge component intersecting \(\gamma\) take its intersection with \(S_0(B)\) as bad, ignoring the others. Run the deterministic cell-set adaptation for these sets, listing paths when it accepts geometrically. Similarly list \(P'\) using \(\gamma'\). Let \(G\) be the predicate that both given paths are on the respective lists.
+
+The lists depend only on $B,\gamma,\gamma'$; $G$ tests membership of
+the given cell-path pair and has no further dependence on its labels.
+
+\begin{lemma}[Path lists and their encounter bound]\label{lem:path-lists}
+For every original witness and every baseline cut replacement in
+Lemma~\ref{lem:replacement-stability}, the witness paths $P,P'$ satisfy
+$G$. For arbitrary $B$, the lists contain at most $4^{b_{\rm tot}}$
+path pairs. Every listed pair satisfies
+\[
+                       J\le J_0+C(R_*)b_{\rm tot}.
+                       \tag{31}\label{eq:32}
+\]
+Here $J_0$ counts encounters of the original skeletons, as in
+Lemma~\ref{lem:skeleton}.
+\end{lemma}
+\begin{proof}
+First fix a witness and any cut replacement. A nonbridge component of
+$S_1(B)$ intersecting $\gamma$ stays outside distance $R_*$ of $P'$.
+If it approached a skeleton vertex of $P'$, it would be a bridge. If it
+approached a detour vertex, it would join, in the $K_b$ adjacency, the
+potential component containing that detoured obstacle, which is hit by
+$\gamma'$. It would again be a bridge. Therefore \eqref{eq:30} identifies
+the actual bad subset of $\widehat C$ on this whole nonbridge component
+as its intersection with $S_0(B)$.
+
+On bridge components, choose the guessed subsets to be the actual bad
+cells of $\widehat C$. Every actual nearest-neighbor bad component lies
+in a single $K_b$-component of $S_1(B)$, and every such component hit by
+$\gamma$ is retained. The reconstructed set therefore removes only whole
+actual bad components not hit by $\gamma$. Lemma~\ref{lem:detour}
+preserves acceptance and the adapted path under precisely these
+deletions. Hence $P$ is listed. The same argument with $C,\gamma'$
+gives $P'$. This reasoning holds for every replacement.
+
+There are at most $2^{b_{\rm tot}}$ guesses for each path, giving
+$4^{b_{\rm tot}}$ pairs. To bound their encounters, first count near
+occurrences that lie on their own skeleton and have an opposite-skeleton
+vertex within distance $R_*$. Since the adapted paths are simple, these
+occurrences inject into the corresponding counts of $J_0$.
+
+Every remaining near occurrence is a detour vertex or is close to an
+opposite detour vertex. Any detour participating in such an encounter
+belongs to a bridge component: it is adjacent to its obstacle, which
+is hit by its skeleton, and either lies close to the other skeleton or
+joins the component of an opposite detoured obstacle in the $K_b$
+adjacency. Across both paths, these detours use at most $12b_{\rm tot}$
+vertices by Lemma~\ref{lem:detour}. Their fixed-radius neighborhoods
+contain at most $C(R_*)b_{\rm tot}$ cells. Simplicity bounds the number
+of path occurrences in those cells by twice this number. This proves
+\eqref{eq:32} with coefficient one on $J_0$. The argument also applies
+to guessed subsets: their nearest-neighbor obstacles remain within
+components of $S_1(B)$.
+\end{proof}
+
+\paragraph{The common far-choice product.}
+For a slot, its choice factor in $B$ is its eligible indicator divided
+by the eligible count in its outer cell, using the $S_0(B)$ criteria
+and assigning zero when the count is zero. Let $H_F(B)$ be the product
+of these factors for the selected far occurrences of both paths.
+Unlike the path lists, this product depends on the selected labels.
+Its indicators and denominators are nevertheless all evaluated on the
+same fixed baseline configuration.
+
+Lemma~\ref{lem:replacement-stability} identifies these factors exactly
+with the original far factors: $P$ uses $\widehat C$ and $P'$ uses $C$.
+At feasible fixed data their product therefore has a common value
+$H_*$ for all original witnesses and all cut replacements. Indeed,
+the equality holds between \emph{any} witness and \emph{every}
+replacement, and the replacement cuts are nonempty. This is the
+selection property needed to apply \eqref{eq:restoring-weights}.
+
+
+
+Explicitly, at those fixed variables let \(\mathcal D\) impose the original remaining constraints, \(H_{\rm old}\) be the far product there, and use the product probability measure
+\[
+d\mathcal P=d\nu_{\mathsf j}^C\, d\nu_{\mathsf l}^O\,
+ d\pi_{\mathsf d}^C\,d\pi_{\mathsf d}^{\widehat C}\,d\pi_{\mathsf h}^O
+\]
+on the extras (its laws do not depend on the integrated \(\widehat C_{\mathsf j_n}\)). Having bounded the near factors, the needed inequalities on feasibility, omitting the cut indicators equal to one on the fixed paths in \(\mathsf j,\mathsf l\), are
+\[
+\begin{split}
+ &\Phi_{\mathsf l}^B
+ \int c_{\mathsf j_n}^{\widehat C}\Phi_{\mathsf j}^{\widehat C}
+       \mathbf 1_{\mathcal D} H_{\rm old}\,d\pi_{\mathsf j_n}^{\widehat C}\,d\mathcal P\\
+ &\quad\le H_*\,\Phi_{\mathsf l}^B
+      \int c_{\mathsf j_n}^B\Phi_{\mathsf j}^B\,d\pi_{\mathsf j_n}^B\\
+ &\quad\le e^{C|\mathsf h|+o_\rho(1)J}
+      \int c_{\mathsf j_n}^B c_{\mathsf d}^B
+       \exp\!\Big[-\sum_{\{i,i'\}\not\subseteq\mathsf L}
+                          \mathcal V_{ii'}(B)\Big]\mathbf 1_G H_F(B)\,
+                       d\pi_{\mathsf j_n}^B d\pi_{\mathsf d}^B .
+\end{split}
+\]
+The first inequality removes the original constraints only after
+factoring out their common far product. The second uses
+\eqref{eq:restoring-weights} and Lemma~\ref{lem:path-lists}:
+$\mathbf1_GH_F(B)=H_*$ everywhere on the cut replacements. At
+infeasible data the original inner integral is zero, so its comparison
+with the nonnegative baseline integral is automatic. Thus the argument
+compares inner integrals at fixed data, without choosing replacements
+according to the number of witnesses they admit.
+
+For precision, denote the original conditional path integral by
+\[
+\begin{split}
+ \mathcal I_{I,I'}={}&\int W(C)h_{\gamma'}(C,I')
+ \frac{\mathbf1_{\mathrm{cut}_R(O_{I'})}W_{I'}^O}
+      {\mathcal Z_{I'}(O)}\\
+ &\hspace{5mm}\cdot
+ \frac{\mathbf1_{\mathrm{cut}_R(O_I)}}{\mathcal Z_I(O)}
+                    W_I^{\widehat C}h_\gamma(\widehat C,I).
+\end{split}
+\]
+The integrations are over the original closed paths, proposed paths on
+$I'$, and hypothetical old paths on $I$, with the endpoints, tuples, and
+skeletons fixed. Let $\mathbf1_{\mathrm{occ}}$ require that the outer
+counts lie in $[b_-m,b_+m]$ in every cell of both paths.
+
+\begin{lemma}[Integrated cross comparison]\label{lem:cross}
+Fix two simple star cell paths from $a$ to $b$, their label tuples
+matching the outer $Y$-cells and having a common first label,
+two skeletons, and endpoint data $Y,S^C,x,z$ as in \eqref{eq:27}.
+With $B$, the near occurrences, $G$, and $H_F(B)$ defined above, one has
+\[
+\begin{split}
+ \mathcal I_{I,I'}\le{}&
+ e^{C|\mathsf h|+\sigma_\rho J}\mathbf1_{\mathrm{occ}}
+ \prod_{\substack{\text{near occurrences}\\\text{in cell }Q}}
+                              [(1-\varepsilon)n_Q(Y)]^{-1}\\
+ &\hspace{8mm}\cdot\int W(S^B,E^B;d\omega)\,\mathbf1_G H_F(B).
+\end{split}\tag{32}\label{eq:31}
+\]
+Here $C<\infty$ is fixed and $\sigma_\rho\ge0$ tends to zero as
+$\rho\downarrow0$ after the cutoffs are fixed, uniformly in all permitted
+volumes and temperatures. A cell on both paths is counted twice in the
+product. The right side is zero when the occupancy condition fails.
+\end{lemma}
+\begin{proof}
+The preceding inner-integral comparison proves the displayed bound on
+every feasible fiber. On an infeasible fiber the original contribution
+is zero. We explain the remaining bookkeeping to specify precisely how
+the full integral is recovered.
+Keep $\mathsf L,B_{\mathsf l},B_{\mathsf j\setminus\mathsf j_n}$ fixed
+and integrate the extensions in $\mathsf j_n,\mathsf d$ as in the
+displayed ledger. Whenever the original domain is nonempty, all the
+nearest-kernel factorizations apply. Restore the baseline kernels from
+\eqref{eq:28}, then integrate the fixed paths. Finally drop the remaining
+cut and nearest-lift restrictions from the baseline integral by
+positivity, obtaining \eqref{eq:31}. Its predicates are measurable
+functions of the baseline paths, specified by finite cell algorithms.
+This use of the conditional comparison is valid pointwise at the fixed
+data: a positive original inner integral has a witness, and otherwise
+the required upper bound is automatic.
+
+\end{proof}
+
+The conditional comparison is now established. To average its path-list
+cost, we need the defect bound under the reverse insertion. Under
+$\mu_{ab}$, with $e$ the opened slot in each term, every specified
+cell set $\mathcal U$ satisfies
+\[
+                  \mu_{ab}(\mathcal U\subseteq S_1(B))
+                              \le C M^{-2} q^{|\mathcal U|},
+                  \qquad q=q_{\rm cl}^{1/4},                          \tag{33}\label{eq:33}
+\]
+including the empty set bound. To compare to the symmetric defective test replace the exceptional outer endpoint by its actual inner end for counting and isolation. Counts change by at most one; only a bounded number of other end-isolation flags can change: those isolated before but not after are mutually separated by more than \(r\) and within \(r\) of the new point, and analogously in reverse (use packing). Their other tests are unchanged. Allow also the exceptional slot itself. For a potential bad cell with count \(n_Q(Y)\) in the occupancy interval, the ineligible count in the actual stricter-score test is thus at least \((\varepsilon/2)n_Q(Y)-C\), giving at least fraction \(\varepsilon/4\) for large \(m\); counts outside the interval imply the widened count defect after any change by one. This works pointwise in all the specified cells. Now apply \eqref{eq:22} and \eqref{eq:23}.
+
+\subsection{Summing the comparison}
+
+The conditional comparison is now complete, including its selection
+predicates. We sum first over labels at fixed baseline paths, then over
+the finite lists of possible cell paths. The remaining environment
+average will be controlled by the insertion defect bound and the two
+skeleton moments.
+
+\paragraph{Summing label choices.}
+Sum \eqref{eq:31} over candidate cell-path pairs and their tuples; this
+candidate range depends on $a,b,Y$, but not on the starts. For each
+tuple pair change variables from $S^C$ to $S^B$. The outer kernel is
+invariant. Group terms by the last label $e$ of $I$, whose outer end
+lies in $b$. At fixed $e$ and baseline configuration, the baseline
+measure, $S_0,S_1$, and the path lists are independent of every other
+label choice.
+
+Fix a listed pair satisfying the occupancy bound. A far cell occurs on
+only one path, and summing its choice factor from $H_F(B)$ costs at
+most one. Each near occurrence contributes a factor
+$(1-\varepsilon)^{-1}$. After removing these factors, the remaining
+cellwise sums are
+\[
+\begin{array}{c|c|c}
+ \text{cell}&\text{label choices}&\text{upper bound}\\ \hline
+ \text{ordinary common cell }Q&\text{two independent labels}
+       &1+(e^C-1)/n_Q\\
+ \text{first cell }a&\text{one shared label}&e^C/n_a\\
+ \text{last cell }b&\text{first tuple's label fixed to }e
+       &[1+(e^C-1)/n_b]/n_b.
+\end{array}
+\]
+For example, at an ordinary common cell, the $n_Q$ coincident choices
+carry $e^C$ and the $n_Q(n_Q-1)$ distinct choices carry one, all divided
+by $n_Q^2$. At $a$ the two tuples must share their first label; at $b$
+one of the two labels is fixed by the grouping. The endpoint cells are
+distinct. Simplicity makes these sums independent across cells, since
+labels in different outer cells cannot coincide. Using $n_Q\ge b_-m$
+and including $e^{\sigma_\rho J}$ gives the bound
+\[
+                           \frac{C}{m^2}e^{\kappa_0 J},
+                       \qquad \kappa_0=2\varepsilon+o_\rho(1).
+\]
+Lemma~\ref{lem:path-lists} now sums the cell-path pairs, giving
+$C m^{-2}\exp(\kappa_0J_0+C'b_{\rm tot})$ with fixed $C'$.
+
+\paragraph{Recovering the reverse insertion.}
+There is no remaining dependence on the dummy-for-\(B\) point \(z\) within \(b\); its base integral there gives \(1/M\). Together with the prefactor of (27) we have \(m/(m^2 M)=1/N\); summing \(e\) with \(Y_e\) in \(b\) and integrating the baseline with \(x\) in \(a\) using \(dx/V\) now gives exactly $\mu_{ab}$, with outer endpoint in $b$ and new
+inner endpoint in $a$, as defined in Section~\ref{sec:defects}. Thus
+\[
+                 \int f^2\dd\mu
+         \le C\sum_{a,b\ {\rm valid}} \mathbb E_{\gamma,\gamma'}\!
+               \left[e^{\kappa_0 J_0}
+                   \int e^{C' b_{\rm tot}}\,d\mu_{ab}\right].         \tag{34}\label{eq:34}
+\]
+
+\paragraph{The exponential moment of rare components.}
+For fixed skeletons, we claim
+\[
+ \int e^{C'b_{\rm tot}}\,d\mu_{ab}
+ \le \frac C{M^2}\exp\!\left[\vartheta(q)S_{K_b}\right],
+ \qquad \vartheta(q)\longrightarrow0\quad(q\downarrow0),
+\]
+where $S_{K_b}$ is the soft-distance sum of Lemma~\ref{lem:skeleton}.
+This is the estimate that allows the environment average in
+\eqref{eq:34} to use the second skeleton moment.
+
+Expand the product over realized bridge components of $S_1(B)$ with
+factors $1+(e^{C'|T|}-1)$. Bound it by the sum over unordered
+collections of disjoint connected sets $T$ in the $K_b$ adjacency,
+each within $R_*+2$ of both skeletons, requiring their cells to lie in
+$S_1(B)$. Apply \eqref{eq:33} to each union. Summing the products and
+then dropping disjointness gives
+\[
+                   C M^{-2}
+                     \exp\Big(\sum_T(e^{C'}q)^{|T|}\Big)
+\]
+(the sum in the exponent is over individual qualifying sets). 
+
+Root their counting near vertex occurrences \(z\in\gamma\); there are a bounded number of root cells per occurrence and at most \(C(K_b)^j\) connected sets of size \(j\) per root by the bounded degree. To reach the other skeleton requires \(d_\infty(z,\gamma')\le 2(R_*+2)+K_b j\). Consequently the exponent sum is at most
+\[
+                        o_q(1)\sum_{z\in\gamma}
+                                     e^{-d_\infty(z,\gamma')/K_b}
+\]
+by a geometric series: per root sum \((C(K_b)e^{C'}q)^j\) only at sizes with \(d_\infty(z,\gamma')/K_b\le j+2(R_*+2)/K_b\), extracting the displayed distance decay. 
+
+The geometric-series bound proves the claim with a coefficient
+$\vartheta(q)\to0$. Take density small enough that $2\kappa_0\le\alpha$
+and $2\vartheta(q)\le\xi(K_b)$. Cauchy--Schwarz and
+\eqref{eq:24}--\eqref{eq:25} then give, for each valid pair,
+\[
+ \mathbb E_{\gamma,\gamma'}
+ \left[e^{\kappa_0J_0}\int e^{C'b_{\rm tot}}\,d\mu_{ab}\right]
+ \le \frac C{M^2}
+       \bigl(\mathbb Ee^{2\kappa_0J_0}\bigr)^{1/2}
+       \bigl(\mathbb Ee^{2\vartheta(q)S_{K_b}}\bigr)^{1/2}
+ \le \frac C{M^2}.
+\]
+There are at most $M^2$ ordered pairs. Substitution into
+\eqref{eq:34} proves Proposition~\ref{prop:second-moment}.
diff --git a/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/conclusion.tex b/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/conclusion.tex
new file mode 100644
index 0000000000000000000000000000000000000000..127ea42a74504062042b418c1a39be6a7b1c27f6
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+++ b/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/conclusion.tex
@@ -0,0 +1,67 @@
+\section{Choice of constants and the limits}\label{sec:conclusion}
+
+We assemble the finite-volume estimate and make its uniformity explicit.
+The cutoffs are fixed from the potential before the density is reduced;
+the volume threshold is chosen afterward at each fixed density. This
+order also permits the zero-temperature limit at fixed volume.
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+The change of measure in Section~\ref{sec:opening} has mass at least
+$\upsilon/2$. Proposition~\ref{prop:second-moment} gives
+$\int f^2\dd\mu\le C_v$,
+where $C_v$ is independent of $\rho,L,N,T$ after the fixed cutoff choices.
+Since $\mu$ has total mass $q_0$, Cauchy--Schwarz yields
+\[
+ q_0\ge \frac{\upsilon^2}{4C_v}=:c_*(v)>0.
+\]
+We specify why all choices can be made without introducing density
+or temperature dependence into this constant.
+
+Choose the parameters in this order.
+\begin{enumerate}[label=\textup{(\arabic*)}]
+\item Fix the localization multipliers in the upper density estimates,
+reserving the positive energy fraction used for isolation in
+\eqref{eq:13}. Fix the small particle-addition fraction $\eta$, then
+$K$ large enough for the lower density estimate, and then the occupancy
+constants $b_-,b_+$.
+\item Choose the macro cutoff $D$ large, with
+$\varepsilon=\alpha(D)/20$ as in Lemma~\ref{lem:skeleton}.
+The quantity $\alpha(D)$ decreases only polynomially in $D$, whereas
+the macro failure bound in \eqref{eq:17} decreases as a Gaussian tail.
+Thus both the marked-color gap condition and the required macro
+smallness for \eqref{eq:22} hold.
+\item Choose $\zeta>0$ small enough for the isolation tail, and then
+$C_0$ large enough for the joint score tails and the two uniform dummy
+anchor tests. All these constants are now fixed, depending only on $v$.
+\item Decrease $\rho_*(v)$ to satisfy every small-density restriction.
+In particular $r=\zeta\rho^{-1/3}$, $m\asymp\rho^{-1/2}$, and
+$\lambda_0m=r\sqrt\rho\,m\asymp\rho^{-1/3}$ tend to infinity.
+The joint defect parameter tends to zero. Fixed conditional bridge
+bounds may be large, but the encounter terms they multiply contain
+$r^{-1/2}$ or $m^{-1}$ and hence tend to zero. Therefore
+$2\kappa_0\le\alpha$, and twice the coefficient in the soft-distance
+component bound is within the threshold of \eqref{eq:25}.
+\item At each $\rho<\rho_*(v)$ choose $L_0(\rho,v)$ large enough for
+all torus lifts, grids, and geometric rejection bounds. None of these
+conditions requires a further temperature restriction within
+$0<T\le\rho^2$.
+\end{enumerate}
+This gives the claimed finite-volume lower bound at positive temperature.
+
+At fixed such $N,L$, the labelled Schr\"odinger semigroup is positivity
+improving: the free heat kernel is strictly positive and the bounded
+interaction gives a strictly positive Feynman--Kac weight.
+Compact resolvent and positivity improvement give a unique positive
+normalized ground eigenfunction. Permutation invariance then makes it
+symmetric, so it is also the unique bosonic ground state. The normalized
+Gibbs state converges in trace norm to its projector as
+$\beta\to\infty$. The occupation observable
+$N^{-1}\sum_i(|u_0\rangle\langle u_0|)_i$ is bounded by one;
+its expectation therefore converges as well. The lower bound, already
+uniform over $\beta\ge\rho^{-2}$, survives at $T=0$.
+
+Finally fix $\rho$ and $T$ in the stated ranges. Every sequence with
+$L\to\infty$ and $N/L^3\to\rho$ eventually satisfies both the density
+band and the volume threshold. The same $c_*(v)$ applies to each late
+member, proving the asserted lower limit.
+\end{proof}
diff --git a/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/defects.tex b/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/defects.tex
new file mode 100644
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+++ b/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/defects.tex
@@ -0,0 +1,250 @@
+\section{Defective cells and an insertion comparison}\label{sec:defects}
+
+We combine occupation, isolation, and trajectory estimates to control
+an arbitrary specified set of unusable cells. The final comparison
+of this section transfers that control from a probability measure
+on closed paths to the finite measure obtained by opening one end.
+
+Fix
+$\Delta_A=1000(2R_A+2R+20)^3$.
+In this section all symmetric path tests use the actual inner
+endpoints. A cell is \emph{defective} if its end count is below
+$2b_-m$ or above $b_+m/2$, or if at least $\varepsilon/4$ of its
+slots fail eligibility with score threshold $C_0-\Delta_A$
+in place of $C_0$. Thus every unusable cell is defective.
+
+\begin{lemma}[Scores at separated anchors]\label{lem:anchor-scores}
+Let $U$ be a cell union and let $\mathcal D$ be any collection of
+anchors mutually separated by more than $r$, all within distance
+$Ds$ of $U$. There is a coarse-cell cover $S$ of the
+$(D+R_A)s$ neighborhood of $U$, with
+$|S|\leq C(D+R_A+2)^3|U|$, such that, pathwise,
+\begin{equation}
+ \sum_{a\in\mathcal D}\mathcal A^-(a;\omega)
+ \leq C\int_0^{\delta/2}(w(t)+r^{-3})n_S(\omega(t))\,dt.
+ \tag{21}\label{eq:21}
+\end{equation}
+The analogous inequality holds for the end scores after time
+reversal. The assertion holds also when the anchors are chosen
+from the paths by an arbitrary measurable rule, provided their
+separation and location hold for each realization.
+There is $B_0=C(1+K^2/\zeta^3)$ such that the exponential moment
+of the right side of (21), divided by $B_0$, is at most
+$\exp(C_v\rho|S|)$ under the closed measure.
+\end{lemma}
+
+\begin{proof}
+Lift the anchors periodically to $\mathbb R^3$. Their distinct
+lifts remain separated by more than $r$ once $L>2r$.
+For a point $y$ and $a_t=\sqrt{1+t}$, packing disjoint balls
+of radius $r/2$ bounds the number of anchors in a ball of radius
+$u$ by $C(1+u/r)^3$. Summing in shells of width
+$\max(r,a_t)$, with the Gaussian weight, gives
+\[
+ \sum_{a\in\mathcal D}
+ e^{-d(y,a)^2/[1000(1+t)]}
+ \leq C\bigl(1+(1+t)^{3/2}r^{-3}\bigr).
+\]
+For the torus expression, select a nearest lift of each anchor;
+it is bounded by the full lifted Gaussian sum just estimated.
+Only $y$ in $S$ can contribute after the distance indicator
+in (15), because the anchors are within $Ds$ of $U$.
+Multiplication by $w(t)$ now proves (21), since
+$w(t)(1+t)^{3/2}\leq C$. This proof is deterministic for every
+set of separated anchors and therefore permits adaptive choices.
+
+The integral of the scalar weight on the right of (21) is at
+most $C(1+\delta r^{-3})\leq B_0$.
+Divide by $B_0$ and regard the result as an average of the
+variables $n_S(\omega(t))$, padding with a zero variable if
+the total averaging mass is less than one. Jensen's inequality
+for the exponential, stationarity of the one-time closed law,
+and (11) at $\lambda=1/2$ give
+\[
+ \mathbb E_{\mathrm{cl}}\exp\left[
+ B_0^{-1}C\int_0^{\delta/2}(w(t)+r^{-3})n_S(\omega(t))\,dt
+ \right]
+ \leq\mathbb E_Pe^{n_S}
+ \leq\mathbb E_Pe^{n_{\phi_S}}
+ \leq e^{C_v\rho|S|}.
+\]
+Time reversal gives the other half of the assertion.
+\end{proof}
+
+\begin{proposition}[Specified defective cells]\label{prop:defective-cells}
+Choose $D$ and $\varepsilon$ so that
+$C_v\sum_{j\geq0}D_j^3e^{-\kappa D_j^2}
+\leq10^{-4}b_-\varepsilon$ in (17). Then choose $\zeta>0$
+sufficiently small in terms of these fixed constants, followed
+by $C_0>\Delta_A$ sufficiently large. For small enough density,
+there is $q_{\mathrm{cl}}=q_{\mathrm{cl}}(\rho)\to0$ as
+$\rho\downarrow0$ such that every collection $\mathcal U$
+of specified cells satisfies
+\begin{equation}
+ P_{\mathrm{cl}}(\text{every cell of }\mathcal U
+                         \text{ is defective})
+ \leq q_{\mathrm{cl}}^{|\mathcal U|}.
+ \tag{22}\label{eq:22}
+\end{equation}
+The bound and the convergence of $q_{\mathrm{cl}}$ are uniform
+in the allowed temperatures and sufficiently large volumes.
+Increasing $C_0$ preserves the statement.
+\end{proposition}
+
+\begin{proof}
+Let $u=|\mathcal U|\geq1$. On the event in question, some
+subcollection of at least $u/3$ cells shares one of three
+reasons: low count, high count, or at least
+$\varepsilon/4$ ineligible slots with counts in
+$[2b_-m,b_+m/2]$. For the first two reasons (12) applies
+to the union of that subcollection.
+
+Fix a subcollection of $u'$ cells of the third kind, and let
+$U'$ be its union. It contains at least
+$\varepsilon b_-mu'/2$ ineligible slots. Assign to each such
+slot the first failed test in the following ordered list:
+trajectory failure; end nonisolation; start nonisolation;
+beginning score failure; end score failure. For each test
+after the first, the slot is macro-conforming; for each score
+test, its respective anchor is isolated. At least one reason
+therefore has at least
+$k'=\varepsilon b_-mu'/20$ slots. Using this slightly smaller
+threshold avoids any integer-rounding convention.
+
+For trajectories, exponential Markov and (17) give probability
+at most
+\[
+ \exp\{[10^{-4}b_-\varepsilon+o_\rho(1)]\rho|U'|-k'\},
+\]
+which is at most $e^{-ck'}$ for sufficiently small density,
+since $\rho|U'|\leq2mu'$.
+For end nonisolation use (13) with $S=U'$.
+For start nonisolation, macro conformity places all those starts
+within distance $Ds$ of $U'$. A coarse cover of that neighborhood
+has volume at most $C(D+2)^3|U'|$, so (13) applies to it at
+time zero. Fix $\zeta$ so small that in both applications
+$C_v\zeta^3\rho|S|\leq k'/2$.
+Their probabilities are then at most $e^{-\lambda_0 k'/2}$.
+
+For either score reason the failed anchors are mutually separated
+by more than $r$. The end anchors lie in $U'$, and the start
+anchors lie within distance $Ds$ of $U'$. If at least $k'$
+of them have score greater than $C_0-\Delta_A$, their score
+sum exceeds $(C_0-\Delta_A)k'$. Apply
+Lemma~\ref{lem:anchor-scores} and exponential Markov. The
+probability is at most
+\[
+ \exp\{C_v\rho|S|-(C_0-\Delta_A)k'/B_0\}.
+\]
+The ratio $\rho|S|/k'$ is bounded in terms of the already
+fixed constants. Choose $C_0$ large enough that this last
+bound is at most $e^{-k'}$.
+
+For small density, $\lambda_0=\zeta\rho^{1/6}\leq1$.
+All the estimates for a fixed subcollection are thus bounded
+by a fixed constant times $e^{-c\lambda_0mu'}$, including
+the two count reasons. There are at most $2^u$ choices of
+a subcollection for each reason and five failed-test choices.
+Since $u'\geq u/3$, the resulting bound has the form
+$C_1^u e^{-c\lambda_0mu/3}$ after increasing $C_1$.
+Now
+\[
+ \lambda_0m\geq\tfrac12\zeta K^3\rho^{-1/3}\longrightarrow\infty.
+\]
+Absorbing $C_1^u$ proves (22), for example with
+$q_{\mathrm{cl}}=C_1e^{-c\lambda_0m/3}$ after replacing
+$m$ by its stated density-dependent lower bound. Each preceding
+bound was uniform in temperature and volume, as required.
+The empty collection has probability one and satisfies (22).
+\end{proof}
+
+We have obtained a joint defect bound under the closed probability
+law. The second-moment comparison will instead produce a finite
+measure with one endpoint opened. The following transfer retains both
+the small defect parameter and the cell-pair mass factor $M^{-2}$.
+
+For cells $a,b$, define the positive measure $\mu_{ab}$ as
+follows. In (14), sum over a slot $i$, restrict its outer endpoint
+$Y_i$ to cell $b$, and replace its inner endpoint by a point
+$z\in a$ integrated with Lebesgue measure. All other inner
+ends remain $Y_j$. Multiply by $1/(NV)$, retain normalization
+by $Z_N$, and integrate all the other variables as before.
+The slot index may be retained as part of the measure space.
+An event on the actual slab paths is \emph{symmetric} when
+it is invariant under simultaneous relabeling of all slots;
+it must not single out the opened slot.
+
+\begin{proposition}[Transfer to an open insertion]\label{prop:insertion-comparison}
+For any measurable symmetric slab event $\mathcal E$,
+\begin{equation}
+ \mu_{ab}(\mathcal E)
+ \leq\frac C{M^2}P_{\mathrm{cl}}(\mathcal E)^{1/4}.
+ \tag{23}\label{eq:23}
+\end{equation}
+In particular, the insertion mass is at most $C/M^2$ and
+the joint bound (22) transfers to this measure with exponent
+$q_{\mathrm{cl}}^{|\mathcal U|/4}$ and prefactor $C/M^2$.
+All tests in this application use the actual inner endpoints.
+\end{proposition}
+
+\begin{proof}
+Two applications of Cauchy--Schwarz give the fourth root: the first
+separates the two insertion factors, and the second bounds a local
+particle count on the event $\mathcal E$ by its second moment.
+Let $K_{\mathcal E}(E,S)$ be the tested slab kernel obtained
+by integrating $W(S,E;d\omega)\mathbf1_{\mathcal E}$.
+It need not be self-adjoint, but it is pointwise nonnegative,
+is bounded above by the untested slab kernel, and commutes
+with simultaneous coordinate permutations. Put
+$U_{\mathcal E}=K_{\mathcal E}e^{-(\beta/2-\delta)H}$.
+For a one-particle function $f$, let $a(f)$ denote annihilation,
+mapping the $N$-particle symmetric space to the $(N-1)$-particle
+one. The usual normalization gives
+$a^*(\mathbf1_b)a(\mathbf1_a)
+=\sum_i(|\mathbf1_b\rangle\langle\mathbf1_a|)_i$.
+The definition of $\mu_{ab}$ and semigroup cyclicity therefore give
+\[
+ \mu_{ab}(\mathcal E)=\frac1{NVZ_N}
+ \operatorname{Tr}_{\mathrm{sym}}
+ e^{-\beta H/2}a^*(\mathbf1_b)a(\mathbf1_a)U_{\mathcal E}.
+\]
+Apply Hilbert--Schmidt Cauchy--Schwarz to
+$a(\mathbf1_b)e^{-\beta H/2}$ and
+$a(\mathbf1_a)U_{\mathcal E}$. Since a cell has volume $s^3$,
+the one-body Cauchy--Schwarz inequality gives
+$a^*(\mathbf1_a)a(\mathbf1_a)\leq s^3n_a$, and similarly
+for $b$. Translation invariance yields
+$\operatorname{Tr}e^{-\beta H}n_b=Z_Nm$.
+Thus the first squared Hilbert--Schmidt norm is at most
+$s^3Z_Nm$.
+
+For the second norm use
+$\operatorname{Tr}U_{\mathcal E}^*n_aU_{\mathcal E}$.
+This is an integral of products of nonnegative kernels.
+Precisely, on labeled coordinates one may insert the projector
+and use the kernel of $U_{\mathcal E}P_{\mathrm{sym}}$; it
+has the same trace on its symmetric range and obeys
+\[
+ 0\leq U_{\mathcal E}P_{\mathrm{sym}}
+ \leq e^{-\beta H/2}P_{\mathrm{sym}}
+ \quad\hbox{pointwise in kernels}.
+\]
+Replace only the adjoint kernel in that trace by the untested
+one. This increases the integral and, by cyclicity, gives
+\[
+ \operatorname{Tr}_{\mathrm{sym}}U_{\mathcal E}^*n_aU_{\mathcal E}
+ \leq\operatorname{Tr}_{\mathrm{sym}}
+ e^{-(\beta-\delta)H}n_aK_{\mathcal E}
+ =Z_N\mathbb E_{\mathrm{cl}}[n_a\mathbf1_{\mathcal E}].
+\]
+Here $n_a$ counts the actual inner endpoints. Its second
+moment in the closed measure is at most $Cm^2$ by (11),
+so the last expression is at most
+$CZ_NmP_{\mathrm{cl}}(\mathcal E)^{1/2}$.
+The second squared Hilbert--Schmidt norm is consequently at
+most $Cs^3Z_NmP_{\mathrm{cl}}(\mathcal E)^{1/2}$.
+Taking square roots and dividing by $NVZ_N$ gives
+$Cs^3m/(NV)$ times the fourth root of the probability.
+Since $m=N/M$ and $s^3=V/M$, this factor is $C/M^2$,
+which proves (23).
+\end{proof}
diff --git a/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/energy.tex b/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/energy.tex
new file mode 100644
index 0000000000000000000000000000000000000000..6e0ded16c387925463adbe8412b2b2f738353d7d
--- /dev/null
+++ b/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/energy.tex
@@ -0,0 +1,179 @@
+\section{Energy and free-energy estimates}\label{sec:local-energy}\label{sec:energy}
+
+This section gives the particle-number costs needed to localize the gas.
+Throughout, $0<T\leq\rho^2$, $\beta=T^{-1}$, and
+$\rho/2\leq N/V\leq2\rho$. All statements hold after decreasing a
+potential-dependent upper bound on $\rho$ and increasing the volume
+threshold at each fixed density. The Hamiltonian with a bounded external
+potential $-W$ is denoted by $H_{j,L}^{(-W)}$; its symmetric partition
+function and free energy are $Z_j^{(-W)}$ and
+$F_j^{(-W)}=-T\log Z_j^{(-W)}$. We set $Z_0=1$ and $F_0=0$.
+For a trace-one state $\sigma$, write
+$\mathcal F^{(-W)}(\sigma)=\operatorname{Tr}H^{(-W)}\sigma-TS(\sigma)$.
+Here $S(\sigma)=-\operatorname{Tr}(\sigma\log\sigma)$ is the
+von Neumann entropy, with $0\log0=0$.
+Constants may depend on $v$, and later on already fixed dimensionless
+parameters, but never on $\rho,L,N,T$.
+
+\begin{lemma}[A Neumann cell bound]\label{lem:neumann-cell}
+There are $b_0,c_v>0$, depending only on the fixed potential, such that
+for a cube $Q$ of side $b\geq b_0$, every $y\in Q$, and every
+$f\in H^1(Q)$,
+\begin{equation}
+ \int_Q\bigl(|\nabla f|^2+v(x-y)|f|^2\bigr)\,dx
+ \geq c_v b^{-3}\int_Q|f|^2\,dx.\tag{1}\label{eq:1}
+\end{equation}
+The same conclusion holds for fixed positive coefficients in front of
+the two terms, with changed constants. Consequently, for disjoint
+cubes $Q$ of common side $b$, kinetic energy plus nonnegative pair
+energy bounds below the multiplication operator
+\begin{equation}
+ c_v b^{-3}\sum_Q n_Q\mathbf1_{\{n_Q\geq2\}}.\tag{2}\label{eq:2}
+\end{equation}
+Both statements apply to torus cubes smaller than the torus, and to
+Dirichlet states extended by zero.
+\end{lemma}
+
+\begin{proof}
+Choose $\nu>0$ for which the radial set
+$A=\{x:v(x)\geq\nu\}$ has positive measure. Increase $b_0$ so that
+$b_0/2>R_v$ and $b_0\geq1$. For each $y\in Q$ choose, independently
+in the three coordinate directions, an orientation with available
+length at least $b/2$. The resulting octant of $y+A$ lies in $Q$ and
+has measure $|A|/8=:a_v>0$ by radial symmetry. Call it $A_y$.
+For $f_0=|Q|^{-1}\int_Q f$, the scale-invariant Sobolev--Poincar\'e
+inequality gives $\|f-f_0\|_6\leq C\|\nabla f\|_2$. Thus
+\[
+ a_v|f_0|^2\leq2\int_{A_y}|f|^2+
+ 2\int_{A_y}|f-f_0|^2
+ \leq2\nu^{-1}\int_Qv(x-y)|f|^2+
+ C a_v^{2/3}\int_Q|\nabla f|^2.
+\]
+Moreover $\|f-f_0\|_2^2\leq Cb^2\|\nabla f\|_2^2$.
+Since $b^2\leq b^3$ for $b\geq1$, these estimates give (1).
+Positive fixed energy coefficients only change the constants.
+
+For (2), cut configuration space according to the cube containing
+each coordinate and discard cross-cube interactions. This is Neumann
+bracketing: the restrictions of an $H^1$ function to the pieces are
+admissible with no boundary condition at the cuts. Within a cube
+containing $j\geq2$ coordinates, pair them disjointly. For each pair
+apply (1) in the first coordinate with the other coordinate held
+fixed, and then integrate the other coordinates. No coordinate
+kinetic term or pair interaction is used twice. The number of pairs
+is $\lfloor j/2\rfloor\geq j/3$. Summing the resulting inequalities
+proves (2). The argument is unaffected by additional nonnegative
+periodic images of $v$. Zero extension gives the Dirichlet assertion.
+\end{proof}
+
+\begin{lemma}[An entropy bound]\label{lem:kinetic-entropy}
+If $\Omega$ is an open subset of the torus with Dirichlet kinetic
+energy, or the full torus with periodic kinetic energy, then for
+every $\alpha>0$ and every finite-energy symmetric trace-one $j$-particle state,
+\begin{equation}
+ \alpha\operatorname{Tr}\Bigl(\sum_{i=1}^j-\Delta_i\Bigr)\sigma
+ -TS(\sigma)
+ \geq-Tj-C_\alpha T|\Omega|(V^{-1}+T^{3/2}).\tag{3}\label{eq:3}
+\end{equation}
+In particular a state of finite kinetic energy has finite entropy.
+\end{lemma}
+
+\begin{proof}
+The periodic heat kernel satisfies
+$K_t(0)\leq C(V^{-1}+t^{-3/2})$ by its Gaussian image sum,
+or equivalently by its Fourier series. Killing at $\partial\Omega$
+decreases its diagonal, so
+$\operatorname{Tr}e^{t\alpha\Delta_\Omega}
+\leq C_\alpha|\Omega|(V^{-1}+t^{-3/2})$.
+The ideal Bose grand partition function with fugacity $e^{-1}$ has
+logarithm
+\[
+ \sum_{l\geq1}\frac{e^{-l}}l
+ \operatorname{Tr}e^{l\beta\alpha\Delta_\Omega}
+ \leq C_\alpha|\Omega|(V^{-1}+T^{3/2}).
+\]
+Its $j$-particle term is $e^{-j}$ times the canonical partition
+function. Applying the Gibbs variational inequality to that term
+gives (3). For an arbitrary finite-energy state, the same inequality
+follows first on finite spectral truncations and then by monotone
+entropy approximation, or directly from nonnegativity of relative
+entropy to the ideal Gibbs state. The right side is finite, proving
+the entropy assertion as well.
+\end{proof}
+
+\begin{proposition}[Particle-number free-energy bounds]\label{prop:increments}
+Uniformly in the temperature range under consideration,
+\begin{align}
+ F_N&\geq c_v\rho N,\tag{4}\label{eq:4}\\
+ F_{N+k}&\geq F_N+c_v\rho k\qquad(k\geq0).\tag{5}\label{eq:5}
+\end{align}
+For every bounded $W\geq0$, with no upper bound on its size required,
+\begin{equation}
+ F_j^{(-W)}-F_{j-1}^{(-W)}\leq C_v\rho
+ \qquad(1\leq j\leq4\rho V).\tag{6}\label{eq:6}
+\end{equation}
+The constant in (6) is independent of $W$.
+\end{proposition}
+
+\begin{proof}
+Partition the torus into equal cubes of side in $[b,2b]$, where
+$b=(4/\rho)^{1/3}$. The number of cubes is at most
+$V/b^3=\rho V/4\leq N/2$ when $L$ is sufficiently large.
+Reserve a fixed fraction of kinetic energy for (3), and apply (2)
+to the rest of the kinetic energy and the interaction. Since
+$\sum_Q n_Q\mathbf1_{\{n_Q\geq2\}}\geq N-\#\{Q\}\geq N/2$,
+the energy contribution is at least $c_v\rho N$.
+The entropy loss is bounded by
+$TN+C T(1+VT^{3/2})$. Divided by $\rho N$, these terms tend uniformly
+to zero as $\rho\downarrow0$ and the volume threshold increases:
+$T/\rho\leq\rho$, $T/(\rho N)\leq\rho/N$, and
+$VT^{5/2}/(\rho N)\leq2\rho^3$. This proves (4), after changing $c_v$.
+
+We next show that $F_j/j$ is nondecreasing in $j\geq1$.
+Regard a symmetric $j$-particle state as a state on the labeled tensor
+product and denote by $S_i$ the entropy of its trace-one
+$i$-particle marginal; put $S_0=0$. Strong subadditivity~\cite[Theorem~2 and Section~IV]{LiebRuskai1973Entropy} applied to
+two overlapping $i$-coordinate subsystems with overlap of size
+$i-1$ gives $2S_i\geq S_{i-1}+S_{i+1}$. The case $i=1$ uses
+ordinary subadditivity. Permutation symmetry identifies the
+marginals occurring here. Therefore $S_i$ is concave and
+$S_j/j\leq S_{j-1}/(j-1)$ for $j\geq2$.
+The one-body energy per particle is unchanged by passage to the
+$(j-1)$-particle marginal. The pair energy per particle decreases,
+because its coefficient is $(j-1)/2$ before passage and $(j-2)/2$
+afterwards, and the common two-body expectation is nonnegative;
+for $j=2$ the latter energy is zero. The marginal remains supported
+on the symmetric subspace. Applying these observations to the Gibbs
+state and then using its marginal as a variational state proves
+$F_j/j\geq F_{j-1}/(j-1)$. All entropies are finite by (3).
+Consequently
+$F_{N+k}\geq(N+k)F_N/N$, which with (4) proves (5).
+
+The lower increments now quantify the free-energy gain from adding
+particles. To obtain the complementary upper cost, uniformly even in
+an attractive external field, we insert one particle directly in the
+path representation.
+
+Retain in the symmetrized Feynman--Kac
+formula only those permutations in which the extra label is fixed.
+There are $(j-1)!$ such permutations against the factor $1/j!$.
+The added particle is a free torus loop whose start is integrated
+over the torus. After normalization by $VK_\beta(0)$ its position
+at every time is uniform, independently of all old trajectories.
+Thus its expected total interaction with the old paths is
+$\beta(j-1)\|v\|_1/V$. Its external weight is at least one because
+$W\geq0$. Jensen's inequality, conditional on all old paths, gives
+\[
+ Z_j^{(-W)}\geq\frac{VK_\beta(0)}j
+ e^{-\beta(j-1)\|v\|_1/V}Z_{j-1}^{(-W)}.
+\]
+Since $K_\beta(0)\geq(4\pi\beta)^{-3/2}$, the positive part of the
+prefactor's contribution to the increment is bounded by
+$T\log_+(C\rho T^{-3/2})$. For small $\rho$, the derivative of
+$t\mapsto t\log(C\rho t^{-3/2})$ on $0<t\leq\rho^2$ is
+$\log(C\rho t^{-3/2})-3/2\geq\log(C\rho^{-2})-3/2>0$.
+Its maximum on this interval is therefore at most
+$C\rho^2(1+|\log\rho|)\leq C\rho$.
+The interaction contribution is at most $4\|v\|_1\rho$.
+This proves (6), uniformly even as $T\downarrow0$.
+\end{proof}
diff --git a/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/geometry.tex b/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/geometry.tex
new file mode 100644
index 0000000000000000000000000000000000000000..3f8163613a2fd72d7bb187fba185325bb67c9f85
--- /dev/null
+++ b/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/geometry.tex
@@ -0,0 +1,357 @@
+\section{Cell paths}\label{sec:geometry}
+
+
+We now construct paths between distant cells. Their purpose is to keep the
+second moment of an averaged endpoint change bounded as the volume grows.
+The first step controls encounters of two random skeletons; the second
+step adapts each skeleton to the actual unusable cells.
+
+\subsection{Random skeletons and their encounter moments}\label{sec:skeletons}
+
+The use of unpredictable paths and intersection moments follows
+Benjamini--Pemantle--Peres~\cite[Theorem~1.3 and Lemma~3.1]{BenjaminiPemantlePeres1998}.
+The independent block velocities below are a continuous dyadic version
+of the aligned block-drift construction of H\"aggstr\"om and
+Mossel~\cite[Section~3, Proposition~3.1]{HaggstromMossel1998}.
+Finite-endpoint path averaging appears in the synchronization argument
+of Abbe, Massouli\'e, Montanari, Sly, and
+Srivastava~\cite[Section~6, Lemma~6.1]{AbbeMassoulieMontanariSlySrivastava2018}
+and in the symmetry-breaking argument of Garban and
+Spencer~\cite[Section~2, Lemma~2.5]{GarbanSpencer2023}.
+We give an explicit dyadic construction and prove the thickened and
+soft-distance estimates needed here.
+
+The construction in this subsection is purely geometric. Let
+$\mathcal G_n=(\Z/n\Z)^3$, and fix an ordered pair $a,b$ with lifted
+displacement $(w,d_2,d_3)$ satisfying
+\[
+ \lceil n/10\rceil\le w\le\lfloor n/8\rfloor,
+ \qquad |d_2|,|d_3|\le w.
+\]
+Call such a pair admissible. Admissible pairs form a fraction at least
+an absolute $\upsilon>0$ of all ordered pairs for sufficiently large $n$.
+All coordinates in the construction are first taken relative to a fixed
+lift of $a$, and then projected to $\mathcal G_n$.
+
+The first coordinate will advance from $0$ to $w$. In the other two
+coordinates we add a random displacement to the straight interpolation
+between the endpoints. We read that displacement forward up to the
+midpoint and backward afterward, so it vanishes at both endpoints.
+This fixes the endpoint without conditioning the random increments;
+the two halves deliberately use the same displacement variables.
+
+For $q=2,3$, let
+$(U^{(q)}_{\ell,j})_{\ell,j\ge0}$ be mutually independent random
+variables, uniform on $[-1,1]$. Fix
+$c_{\rm vel}=(1-2^{-1/4})/2$, and set
+\[
+ v_i^{(q)}=c_{\rm vel}\sum_{\ell\ge0}2^{-\ell/4}
+          U^{(q)}_{\ell,\lfloor(i-1)/2^\ell\rfloor},
+ \qquad Z_t^{(q)}=\sum_{i=1}^t v_i^{(q)},\qquad Z_0^{(q)}=0.
+\]
+The series converges absolutely, and $|v_i^{(q)}|\le1/2$.
+At integer times $t=0,\ldots,w$ use the base vertices
+\[
+ \left(t,\left\lfloor td_2/w+Z^{(2)}_{\min(t,w-t)}\right\rfloor,
+          \left\lfloor td_3/w+Z^{(3)}_{\min(t,w-t)}\right\rfloor\right).
+\]
+Starting from the base vertex at time $t$, first advance one step in
+the first coordinate, then take the signed second-coordinate steps,
+and finally the signed third-coordinate steps needed to reach the
+base vertex at time $t+1$. Omit zero-length moves. Begin the vertex
+list with $a$ and append each newly reached vertex once, without
+repeating a shared endpoint of successive pieces. Each transverse
+coordinate changes by at most two; thus the resulting
+nearest-neighbor skeleton $\gamma=(z_0,\ldots,z_l)$ has at most five
+steps per time interval and at most five listed occurrences at each
+first-coordinate level $t\ge1$, with one occurrence at level zero.
+The law depends only on the torus and the ordered endpoints.
+Independent skeletons use independent arrays of the $U$'s.
+The dyadic weights are chosen so that a previously unused block of
+$k$ increments supplies a transverse displacement of order $k^{3/4}$.
+In two transverse coordinates this gives the summable conditional
+density bound $Ck^{-3/2}$ proved below.
+
+For vertices on the torus write
+\[
+ d_\infty(x,y)=\min_{k\in\Z^3}\|\widetilde x-\widetilde y+nk\|_\infty,
+ \qquad d_\infty(x,\gamma')=\min_{z'\in\gamma'}d_\infty(x,z'),
+\]
+where the tildes denote arbitrary lifts. For two independent skeletons
+$\gamma,\gamma'$ with the same ordered endpoints and a fixed radius
+$R_*\ge1$, define
+\[
+ J_0=\sum_{z\in\gamma}\mathbf1_{\{d_\infty(z,\gamma')\le R_*\}}
+       +\sum_{z'\in\gamma'}\mathbf1_{\{d_\infty(z',\gamma)\le R_*\}},
+ \qquad
+ S_B=\sum_{z\in\gamma}e^{-d_\infty(z,\gamma')/B}\quad(B>0).
+\]
+Every sum uses the listed vertex occurrences, including the endpoints.
+An occurrence contributes once to $J_0$ when at least one occurrence
+of the other skeleton lies within the stated distance.
+
+\begin{lemma}[Skeleton encounters]\label{lem:skeleton}
+For the independent skeletons just constructed, there are absolute
+constants $c_{\mathrm{hit}}>0$ and $C<\infty$ such that, for every
+$R_*\ge1$,
+\[
+             \mathbb E e^{\alpha J_0}\le C,\qquad
+             \alpha=\frac{c_{\rm hit}}{(1+R_*)^2}.
+             \tag{24}\label{eq:24}
+\]
+For every fixed $B>0$ there are $\xi(B)>0$ and $C(B)<\infty$ such that
+\[
+             \mathbb E e^{\xi(B)S_B}\le C(B).
+             \tag{25}\label{eq:25}
+\]
+Both estimates are uniform in $n$ and in every admissible ordered pair.
+\end{lemma}
+\begin{proof}
+\emph{Conditional density bound.}
+In this proof write
+\(Z_t=(Z_t^{(2)},Z_t^{(3)})\) for the pair of transverse sums,
+and write \(Z'_t\) for the corresponding pair of the other skeleton.
+Put \(h=\lfloor w/2\rfloor\). Condition on all the variables
+generating \(\gamma'\). For the first skeleton define the dyadic blocks
+\[
+ I_{\ell,j}=\{j2^\ell+1,\ldots,(j+1)2^\ell\},
+\]
+and let \(\mathcal F_t\) be generated by the entire primed array and
+the variables \(U^{(q)}_{\ell,j}\) for which
+\(I_{\ell,j}\cap\{1,\ldots,t\}\ne\varnothing\), with \(q=2,3\).
+Thus it reveals exactly the variables of the first array used through time
+\(t\), including the large blocks that continue past that time.
+For \(t+k\le h\), \(k\ge1\), the future index interval contains
+a full aligned dyadic block of length at least \(k/4\): take a
+power of two between \(k/4\) and \(k/2\) and skip less than one
+such block to align, or use length one for \(k=1\). Its variable
+has not been used and contributes a uniform term of width at least
+\(c k^{3/4}\) to the sum, even after conditioning on the other
+variables. Indeed, a block at level \(\ell\) contributes
+\(c_{\rm vel}2^{3\ell/4}\) times its variable. Using one such
+variable in each independent coordinate gives a conditional
+two-coordinate density of \(Z_{t+k}\), given \(\mathcal F_t\),
+bounded by \(Ck^{-3/2}\).
+
+\emph{Same-time encounter tests.}
+Write \(D_t=|Z_t-Z'_t|_\infty\), for \(0\le t\le h\).
+Denote the two base vertices at time \(t\) by \(b_t,b'_t\).
+An occurrence \(z\) at first-coordinate level \(t\) is within
+sup-norm distance two of \(b_t\). If \(z'\) is an occurrence of
+the other skeleton at level \(t'\), then
+\( |t-t'|\le d_\infty(z,z')\): first-coordinate differences
+are at most \(w\le n/8\), so they do not wrap. Each base step
+has sup-norm size at most two. The triangle inequality therefore gives
+\[
+ d_\infty(b_t,b'_t)\le 3d_\infty(z,z')+4.
+\]
+The same-time lifted transverse differences also do not wrap.
+The linear drifts are common, each random deviation has absolute
+value at most \(w/4\), and rounding changes their difference by
+at most one. Thus these differences have magnitude at most
+\(w/2+1<n/2\). With \(u=\min(t,w-t)\), it follows that
+\[
+ D_u\le 3d_\infty(z,z')+5.
+\]
+
+Consequently
+\[
+ J_0\le C\sum_{t=0}^{h}
+             \mathbf1_{\{D_t\le C(R_*+1)\}}.
+\]
+The reflected times \(t\) and \(w-t\) have the same deviation
+\(Z_t\); their different linear drifts are common to both skeletons
+and cancel in the difference. Thus the second half contributes only
+another fixed multiplicity to this bound.
+
+For the distance-weighted sum, take \(z'\) to be a closest
+occurrence of \(\gamma'\). The same inequality gives
+\[
+ e^{-d_\infty(z,\gamma')/B}\le
+ e^{5/(3B)}e^{-D_u/(3B)},\qquad
+ S_B\le C(B)\sum_{t=0}^{h}e^{-D_t/(CB)}.
+\]
+These are the two deterministic reductions used in the moment estimate.
+
+\emph{Exponential moment.}
+Let \(X_t\in[0,1]\) denote either of these first-half tests.
+They are adapted to \(\mathcal F_t\), since all variables of
+\(\gamma'\) were conditioned on at the outset. Integrating the
+conditional density bound against the indicator, respectively
+the decaying exponential, and summing over the positive time
+increments gives the uniform estimate
+\[
+ \mathbb E\left[\sum_{u=t+1}^{h}X_u\,\middle|\,\mathcal F_t\right]
+ \le b_0,
+ \qquad
+ b_0=\begin{cases}
+ C(R_*+1)^2,&X_u=\mathbf1_{\{|Z_u-Z'_u|_\infty\le C(R_*+1)\}},\\
+ C(B),&X_u=e^{-|Z_u-Z'_u|_\infty/(CB)}.
+ \end{cases}
+\]
+For every \(j\ge1\), the expected sum of products on strictly
+ordered \(j\)-tuples is therefore at most \((b_0+1)^j\).
+To see this, condition the last-index sum on the revealment at
+each preceding index in the nested sum, and include \(X_0\le1\)
+when bounding the one-term sum. Now expand
+\[
+ e^{\theta\sum_{t=0}^{h}X_t}
+ =\prod_{t=0}^{h}\bigl(1+(e^{\theta X_t}-1)\bigr).
+\]
+The inequality \(e^{\theta x}-1\le(e^\theta-1)x\) for
+\(0\le x\le1\) bounds its expectation by the geometric series
+\(\sum_{j\ge0}[(e^\theta-1)(b_0+1)]^j\).
+Taking \(\theta\) to be a sufficiently small constant times
+\(1/(b_0+1)\), and accounting for the preceding sum-comparison
+constants, proves (24) and (25).
+\end{proof}
+
+\subsection{Detours around unusable components}
+
+We now apply the geometric construction to the coarse grid of
+Section~\ref{sec:setup}. This is the point at which the physical cutoffs
+enter: set
+\[
+ R_* = \lceil20(R+R_A+D+20)\rceil,\qquad
+ \varepsilon=\alpha/20,
+\]
+decreasing the absolute constant in $\alpha$ if necessary.
+Choose $D$ large enough for the earlier conditions with this choice.
+This is possible because \eqref{eq:17} has Gaussian decay in $D$,
+whereas $R_*$ grows only linearly. Choose $\zeta,C_0$ and then the
+small-density threshold as in Proposition~\ref{prop:defective-cells}.
+Increase $C_0$ also so that, for two uniform anchors, the probability
+of any end-score failure at threshold $C_0$ is less than $\upsilon/8$;
+the spatial average after \eqref{eq:16} gives this last requirement.
+
+The skeleton law does not see the unusable cells. We next adapt each
+sampled skeleton to those cells by a deterministic detour procedure.
+This step uses nearest-neighbor components of bad cells but allows
+sup-norm-one steps on the resulting path. Here and below, bad means
+unusable in the sense of Section~\ref{sec:slab}. Reject the configuration
+if any nearest-neighbor component of bad cells has size \(\ge n/4\).
+
+A smaller component \(T\) lifts consistently to a finite connected
+lattice set, up to translation. Indeed, simple cycles in it are too
+short to wrap, and closed walks decompose into simple cycles and
+backtracks. Its bounding box has coordinate diameters less than
+\(n/4\), since two of its vertices can be joined within \(T\) by
+fewer than \(n/4\) steps. Define its \emph{fill} by adjoining the
+finite nearest-neighbor components of its lattice complement, and
+project the result back to the torus. The fill lies in the same
+bounding box: a vertex outside that box connects straight outwards
+to infinity without meeting \(T\). Reject if either endpoint lies
+in any projected fill.
+
+The detour procedure scans an accepted skeleton. On entering an
+outermost fill, meaning one not contained in another fill, it replaces
+the portion through its last visit there by an exterior-boundary path.
+The next proof establishes the boundary
+connectivity that permits this replacement with sup-norm-one steps.
+Fix all tie-breaking rules for boundary paths in advance, and use
+chronological loop erasure for the resulting walk.
+
+\begin{lemma}[Deterministic detours]\label{lem:detour}
+Apply the rejection rules just specified to a nearest-neighbor skeleton
+from $a$ to $b$ and a set of bad cells in $(\Z/n\Z)^3$. If the skeleton
+is accepted, the procedure below produces a simple path with sup-norm
+steps of length one from $a$ to $b$, using only good cells. Every vertex
+is a skeleton vertex or lies on the exterior vertex boundary of a bad
+component hit by the skeleton. A component $T$ is detoured at most once
+and contributes at most $6|T|$ detour vertices. Removing any whole bad
+components not hit by an accepted skeleton preserves acceptance and its
+output path.
+\end{lemma}
+\begin{proof}
+\emph{Fills and projection.}
+We first establish the boundary facts needed by the procedure.
+In the lattice the fill is connected, since each hole neighbors
+the obstacle. Its exterior is connected, since a finite set has
+just one infinite complementary component. An edge from the fill
+to its exterior starts in the obstacle itself.
+
+For disjoint non-neighboring connected obstacles, their fills
+are nested or disjoint and non-neighboring. Indeed, each obstacle
+is in a single complementary component of the other. If it is in
+a hole, its fill stays there: outside that hole one connects to
+infinity avoiding the obstacle in it, either in the exterior of
+the enclosing obstacle or by using that connected enclosing
+obstacle, also to escape from any of its other holes. If the two
+obstacles are mutually in the exteriors, each connected fill
+avoids the other obstacle and thus stays in the other's exterior.
+Adjacency would imply adjacency of the obstacles themselves by
+the crossing-edge property.
+
+These facts survive projection to the torus. An intersection or
+adjacency can be lifted with corresponding translates of the two
+obstacles and their fills. The small box size avoids self-overlap
+and preserves cardinality, so proper nesting stays proper. We
+therefore use the outermost fills in the construction below.
+
+\emph{Connected exterior boundary.}
+The exterior vertex boundary consists of the exterior neighbors
+of the obstacle. It has at most \(6|T|\) vertices. We claim that
+it is connected in the star graph, whose steps have sup-norm one.
+
+We give a self-contained cycle-space proof of this boundary
+connectivity fact; see Tim\'ar~\cite[arXiv version~2,
+Lemmas~1--2]{Timar2013}. Consider the edge cut from the
+lattice fill to its exterior. Link two cut edges when they belong
+to a common elementary square. All cut edges belong to one link
+component. Otherwise choose one link component. Every square
+contains an even number of its edges: it contains an even number
+of edges of the full cut, and all those edges are linked. The
+same parity holds for every lattice cycle, since squares generate
+cycles modulo two. Explicitly, in a closed walk commute steps in
+different axes, each swap changing the walk by an elementary
+square modulo two, and then cancel backtracks. Counting the
+selected edges modulo two along paths from a fixed vertex now
+defines a vertex potential whose differing-neighbor edges are
+exactly the selected edges. This potential is constant on the
+connected fill and on its connected exterior. Its cut must
+therefore be either empty or the full cut, contradicting the
+choice of a proper nonempty link component. This proves the claim
+about cut edges. Their linked exterior endpoints are star
+neighbors or identical, proving boundary connectivity.
+
+Project this connected boundary back to the torus. Any edge
+leaving the projected fill lifts from within the fill to outside
+and hence goes from the projected obstacle to this projected
+boundary. The boundary vertices are good: the small box size
+places them outside the projected fill, and if one were bad it
+would belong to the adjacent bad component. Thus each outermost
+fill has a connected star boundary of at most \(6|T|\) usable
+vertices, and every nearest-neighbor crossing uses that boundary.
+This is the boundary information needed by the algorithm.
+
+\emph{Detour construction.}
+Scan the nearest-neighbor skeleton starting outside the outermost
+fills. If the next vertex enters one, replace the portion through
+the \emph{last} skeleton vertex in that same fill by a
+star-boundary path from the exterior predecessor to the exterior
+successor. These are boundary vertices by nearest-neighbor
+crossing, and the successor is outside all outermost fills by
+non-adjacency. Choose a deterministic simple path on the projected
+boundary, with fixed tie-breaking depending only on that component
+and the two vertices. Projection permits this connection regardless
+of the endpoint lifts used. Continue scanning from the successor.
+
+Finally apply chronological loop erasure to obtain a simple star
+path \(P\) from \(a\) to \(b\) in good cells. Each detoured
+component contributes at most \(6|T|\) detour vertices, all
+adjacent to it. The last-visit rule ensures that each component
+is detoured at most once; loop erasure can only remove vertices.
+
+\emph{Deletion invariance.}
+Suppose the original configuration is accepted. Any fill that
+intersects the skeleton must have its obstacle hit by the
+skeleton: the start is exterior and the skeleton has
+nearest-neighbor steps. Enclosing fills of such obstacles are
+hit as well. The construction is therefore determined by the
+actual bad components intersecting the skeleton. Deleting other
+\emph{whole} bad components in this accepted configuration
+preserves acceptance and leaves both the detours and the final
+loop-erased path unchanged.
+\end{proof}
+
+Rejection by large components has closed probability tending to zero at large \(n\) by (22): count connected sets of size \(\lceil n/4\rceil\) by at most \(M C^{\lceil n/4\rceil}\) (walk a rooted spanning tree), and a larger component contains one. For either given endpoint, containment in a small fill has probability at most \(\sum_{j\ge1} C(j+1)^3 C^j q_{\rm cl}^j\), since its bad component of size \(j\) must be within distance \(j\). This can be made arbitrarily small by the density restriction.
diff --git a/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/history.tex b/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/history.tex
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+The ideal-gas theory begins with Bose's counting of light quanta
+\cite{Bose1924} and Einstein's extension to a monatomic ideal gas,
+including the condensation prediction in his second paper
+\cite{Einstein1924IdealGas,Einstein1925IdealGas}.
+Bogoliubov's treatment of weakly interacting bosons introduced an
+approximate quasiparticle description based on macroscopic occupation of
+one mode \cite{Bogolubov1947}.
+
+For interacting particles, Penrose and Onsager formulated condensation
+through a macroscopic eigenvalue of the one-particle density matrix and
+applied this viewpoint to liquid helium \cite{PenroseOnsager1956}.
+Yang developed the broader framework of off-diagonal long-range order,
+covering Bose condensation and fermion pairing \cite{Yang1962ODLRO}.
+These criteria identify the order to be proved; establishing it for a
+specified interacting Hamiltonian remains a separate mathematical problem.
+
+Rigorous lattice antecedents include Dyson, Lieb, and Simon's
+low-temperature long-range order for the spin-$\tfrac12$ nearest-neighbor
+quantum XY model on the simple cubic lattice in dimensions $d\ge3$
+\cite{DysonLiebSimon1978}, and Kennedy, Lieb, and Shastry's ground-state
+long-range order for all spins on hypercubic lattices in dimensions $d\ge2$
+\cite{KennedyLiebShastry1988XY}.
+For spin-$\tfrac12$, the spin--hard-core-boson correspondence relates
+these results to condensation of half-filled lattice bosons.
+Passing to a continuum Hamiltonian with a fixed interaction
+and a prescribed density requires additional arguments.
+
+The distinction between the thermodynamic and Gross--Pitaevskii limits is
+essential. Lieb and Seiringer proved complete condensation in the
+Gross--Pitaevskii limit, with the interaction scaled so that $Na$ remains
+fixed, where $a$ is the scattering length
+\cite{LiebSeiringer2002BEC}. Deuchert and Seiringer subsequently treated
+positive-temperature homogeneous gases with scattering length
+$a_N=a_vL/N$, where $a_v$ is the scattering length of the
+unscaled potential, and controlled the one-particle density matrix of approximate
+Gibbs minimizers \cite{DeuchertSeiringer2020GP}. These results allow the
+interaction scale to vary with particle number. The problem considered
+here instead fixes the potential and a positive density before taking
+the volume to infinity.
+
+The limits between Gross--Pitaevskii scaling and a fixed-density
+thermodynamic limit have several forms. Brennecke, Brooks, Caraci, and
+Oldenburg proved complete ground-state condensation in the
+Gross--Pitaevskii regime and beyond
+\cite{BrenneckeBrooksCaraciOldenburg2025}. Fournais treated low-energy
+states on growing boxes in a joint dilute and increasing-volume limit
+\cite[Theorem~1.2]{Fournais2021}. Chong, Liang, and Nam gave simpler
+proofs of low-energy condensation using kinetic localization and
+Poincar\'e-type inequalities \cite{ChongLiangNam2026}.
+For the unit-box interaction $N^{2-2\kappa}v(N^{1-\kappa}x)$,
+rescaling to a fixed potential gives density $N^{3\kappa-2}$;
+the regimes $\kappa<2/3$ therefore couple dilution to increasing
+particle number. Junge's low-temperature canonical Gibbs-state
+result uses Neumann boxes whose size remains coupled to
+the dilute limit \cite[Corollary~6]{Junge2026Propagation}.
+
+The thermodynamic energy problem has a separate history. Dyson's
+hard-sphere bounds and the leading dilute asymptotics of Lieb and
+Yngvason established the first-order energy scale
+\cite{Dyson1957,LiebYngvason1998}. The next correction was predicted
+by Lee, Huang, and Yang's pseudopotential calculation
+\cite{LeeHuangYang1957}. Fournais and Solovej proved the corresponding
+lower bound for nonnegative radial finite-range potentials, including hard cores
+\cite[Theorems~1.1 and~1.3]{FournaisSolovej2023}; recent upper bounds
+attain the same coefficient for hard spheres and for general
+nonnegative even finite-range potentials
+\cite{BastiEtAl2026HardSpheres,BastiEtAl2026GeneralPotentials}.
+These energy results, and low-temperature free-energy estimates for
+strong potentials \cite{FournaisEtAl2026StrongFreeEnergy}, do not by
+themselves determine occupation of a single orbital: the one-particle
+kinetic gap tends to zero as the volume grows.
+
+A distinct thermodynamic-limit result appears in S\"ut\H{o}'s two-step preprint
+\cite{Suto2023TwoStep}. Its Theorem~1.2 assumes nonnegativity both of the
+potential and of its Fourier transform, together with integrability,
+a Fourier-moment condition, and spatial decay, and states condensation
+above a temperature-dependent density threshold. Fourier positivity is
+an additional restriction: for example, a nonnegative spherical square
+barrier has a Fourier transform that changes sign. That result therefore
+does not cover all bounded nonnegative radial finite-range potentials,
+and its fixed-temperature threshold does not alone provide the joint
+low-density and low-temperature uniformity sought here.
+
+Galanda and Pinamonti construct algebraic equilibrium states with a
+condensate background under quantitative convergence conditions for a
+spatially localized interaction \cite[Theorem~6.8]{GalandaPinamonti2025Equilibrium}.
+Their subsequent analysis treats renormalized correlation functions in
+Gross--Pitaevskii scaling, with a zero-temperature limit in its final
+convergence theorem \cite[Theorem~7.3]{GalandaPinamonti2026GP}.
+These constructions use a different state formulation and coupled
+interaction, regulator, and temperature limits from the fixed-potential
+canonical thermodynamic limit considered here.
+
+The trajectory formulation descends from Feynman's treatment of the
+Bose gas \cite{Feynman1953AtomicLambda}; for the canonical
+Feynman--Kac representation and its symmetrization and one-particle
+normalizations, see \cite[Appendix~A]{Ueltschi2006FeynmanCycles}.
+The random-path comparison has a different ancestry.
+Benjamini, Pemantle, and Peres constructed paths with exponentially
+decaying intersection tails \cite{BenjaminiPemantlePeres1998}.
+H\"aggstr\"om and Mossel developed a hierarchical random-drift
+construction with low predictability; its independent block variables
+are a close antecedent of the dyadic velocities used below
+\cite[Section~3, Proposition~3.1]{HaggstromMossel1998}.
+Abbe, Massouli\'e, Montanari, Sly, and Srivastava used finite-endpoint
+path averages and overlap estimates for group synchronization
+\cite{AbbeMassoulieMontanariSlySrivastava2018}; Garban and Spencer used
+this approach for classical disordered-spin symmetry breaking along the Nishimori line
+\cite{GarbanSpencer2023}. Here the finite cell paths, their thickened
+intersection estimates, and the continuum bridge comparison require
+separate proofs.
diff --git a/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/introduction.tex b/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/introduction.tex
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+\section{The model and the uniform bound}\label{sec:introduction}
+
+For a homogeneous gas, Bose--Einstein condensation means that one orbital
+contains a positive fraction of all particles in the equilibrium state.
+We consider the spatially constant orbital and keep both the physical
+interaction and a positive density fixed as the volume grows.
+The bound below uses the same positive fraction throughout a sufficiently
+dilute interval of densities.
+
+Let $v:\R^3\to[0,\infty)$ be bounded, measurable, radial, and supported
+in a ball of finite radius, with $v$ not zero almost everywhere.
+On the torus $\Lambda_L=(\R/L\Z)^3$, of volume $V=L^3$, let
+$v_L(x)=\sum_{k\in\Z^3}v(x+Lk)$ and define
+\[
+ H_{N,L}=-\sum_{i=1}^N\Delta_i+
+                \sum_{1\le i<j\le N}v_L(x_i-x_j)
+ \quad\hbox{on }L^2_{\mathrm{sym}}(\Lambda_L^N).
+\]
+The units are $\hbar^2/(2m_{\mathrm{particle}})=k_B=1$.
+At $T>0$, the state is the canonical Gibbs state
+$\Gamma_{N,L,T}=e^{-H_{N,L}/T}/\Tr e^{-H_{N,L}/T}$; at $T=0$, it is
+the ground state. Write $\gamma^{(1)}_{N,L,T}$ for its one-particle
+density matrix, normalized to trace $N$, and put $u_0=V^{-1/2}$.
+
+\begin{theorem}\label{thm:main}
+For every potential $v$ as above there are constants
+$\rho_*(v)>0$ and $c_*(v)>0$ with the following property.
+For every $0<\rho<\rho_*(v)$ there is $L_0(\rho,v)<\infty$ such that,
+for every $L\ge L_0(\rho,v)$, every integer $N\ge1$ satisfying
+$\rho/2\le N/L^3\le2\rho$, and every $0\le T\le\rho^2$,
+\[
+ \frac{\langle u_0,\gamma^{(1)}_{N,L,T}u_0\rangle}{N}
+ \ge c_*(v).
+\]
+Consequently, for each such fixed density $\rho$ and fixed
+$0\le T\le\rho^2$, every sequence with $L\to\infty$ and
+$N/L^3\to\rho$ has lower limiting condensate fraction at least $c_*(v)$.
+\end{theorem}
+
+The volume threshold may depend on the density; the condensate lower
+bound does not. The Hamiltonian is the exact interacting Hamiltonian,
+and its potential is not rescaled with the number of particles.
+The conclusion is macroscopic occupation of the constant orbital with
+a positive lower bound throughout the admitted dilute interval. The
+zero-temperature state is unambiguous: positivity of the finite-volume
+heat kernel gives a unique bosonic ground state, as shown at the end
+of the proof.
+
+\subsection{Context and antecedents}
+\input{sections/history}
+
+\subsection{How the comparison works}
+
+Let $q_0=\langle u_0,\gamma^{(1)}u_0\rangle/N$.
+The Feynman--Kac formula expresses the partition function by positive
+weights on closed collections of paths. Replacing one matching endpoint
+by an independently integrated spatial point gives a positive measure
+$\mu$ of total mass $q_0$. We construct a measure $\nu$ by changing a
+short portion of closed paths, with mass at least a fixed $a>0$, and
+show that its density $f=\dd\nu/\dd\mu$ has bounded second moment.
+Cauchy--Schwarz then gives
+\[
+                    a^2\le q_0\int f^2\dd\mu.
+\]
+Thus the main problem is a comparison of positive path measures.
+
+Partition space into cells of side $s$ comparable to $\rho^{-1/2}$,
+and change paths only during a time interval of length $\delta=s^2$.
+The mean count $m=Ns^3/V$ is comparable to $\rho^{-1/2}$, making
+many particle choices available in a typical cell as the density decreases,
+while the slab remains short compared with $1/T$ throughout the
+positive-temperature range.
+
+The local analysis must control simultaneous failures in specified
+families of cells. Energy and entropy estimates first give occupation
+tails and separation of most endpoints at the shorter scale
+$r=\zeta\rho^{-1/3}$, with $\zeta>0$ fixed and small. A particle is
+eligible when its endpoints are separated, its path stays near them
+during short initial and final time intervals, and it satisfies overall
+confinement and integrated interaction-score bounds for the path
+ensemble. The interacting
+bridges are not independent. A separate localized partition-function
+comparison bounds their joint trajectory failures, with a cost
+proportional to the volume of the tested cells. Together with the
+one-time estimates and score bounds, this makes specified sets of
+unusable cells unlikely.
+
+Between two distant cells, choose a random lattice path, called a
+skeleton, whose encounters with an independent copy have bounded
+exponential moments. Here an encounter records a vertex within a
+fixed cell distance of the other path; a second estimate weights all
+vertices by exponentially decaying distance. These two geometric
+bounds are established independently of the gas in
+Section~\ref{sec:skeletons}. Detour around unusable components. Select one eligible particle from
+each cell on the resulting simple path, and cyclically reassign their
+starting points. One end is opened and the selected bridges are sampled
+from their exactly normalized interacting law. This defines $\nu$.
+
+The second moment involves two independently chosen paths in related
+configurations. Estimating both changes against the original closed
+configuration would incur a cost proportional to their lengths.
+Instead, a reverse insertion supplies a baseline in which contributions
+far from encounters cancel. Two effects remain small as $\rho$ decreases:
+an encounter of distinct separated bridges has interaction cost tending
+to zero at rate $r^{-1/2}$, and coincident particle choices have probability
+of order $1/m$.
+
+Changing trajectories also changes which cells are usable and which
+particles are eligible, so the two routes cannot be treated as independent
+of the gas. A stricter cell test in the reverse configuration controls
+these changes: away from encounters the selection weights agree, while
+rare components give a controlled list of possible detours. Keeping
+those selection weights throughout the comparison yields the required
+second-moment bound after averaging over the environment and the two
+paths.
+
+Sections~\ref{sec:energy}--\ref{sec:defects} prove the local analytic
+bounds. Section~\ref{sec:geometry} constructs the paths and detours.
+Section~\ref{sec:opening} defines the change of measure, and
+Section~\ref{sec:comparison} proves its second-moment estimate with all
+normalizations and selection constraints retained.
+Section~\ref{sec:conclusion} fixes the parameters in order and passes
+the uniform positive-temperature bound to zero temperature.
diff --git a/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/localization.tex b/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/localization.tex
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+++ b/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/localization.tex
@@ -0,0 +1,350 @@
+\section{Localization, occupation, and isolation}\label{sec:localization}
+
+We now obtain simultaneous probability bounds for specified collections
+of cells. The important scale is $s\asymp K\rho^{-1/2}$: a cell contains
+$m=Ns^3/V\asymp\rho s^3$ particles on average, and $m$ tends to infinity
+as the density decreases. All cutoffs below are real and Lipschitz;
+their gradients and the resulting form identities are understood
+almost everywhere.
+
+The estimates use the same variational comparison. Spatial localization
+gives a lower bound for the free-energy cost of a tilted state, while
+diagonal tilting gives an upper bound for that cost in terms of a gradient
+error and the tilt's normalization. Comparing the two bounds controls
+the derivative of a logarithmic exponential moment. Positive density
+tilts control excess occupation, negative tilts control shortages, and
+a tilt of the close-particle count will give isolation.
+
+\begin{lemma}[Two-color localization and diagonal tilting]\label{lem:localization-tilt}
+Let $p^2+q^2=1$, and suppose multiplication by $p$ maps the one-particle
+form domain into $H^1_0(\Omega)$ for an open region $\Omega$.
+For a finite-energy symmetric trace-one $J$-particle state $\sigma$, let $w_j$ be
+the distribution of the number of particles in the $p$ color, and
+let $\sigma_{\mathrm{in},j},\sigma_{\mathrm{out},j}$ be the two
+conditional marginal states. Then
+\begin{equation}
+ \mathcal F^{(-W)}(\sigma)+\operatorname{Tr}
+ n_{|\nabla p|^2+|\nabla q|^2}\sigma
+ \geq\sum_{j=0}^Jw_j\bigl[
+ \mathcal F^{(-W)}(\sigma_{\mathrm{in},j})+
+ \mathcal F^{(-W)}(\sigma_{\mathrm{out},j})\bigr]-TH(w).
+ \tag{7}\label{eq:7}
+\end{equation}
+Here $H(w)=-\sum_jw_j\log w_j$, with $0\log0=0$, is the Shannon
+entropy, and $H(w)\leq C+\sum_jjw_j$ and
+$\sum_jjw_j=\operatorname{Tr}n_{p^2}\sigma$.
+
+Let $\Gamma=Z_N^{-1}e^{-\beta H_{N,L}}$, let $P$ be its position
+law, and let $G$ be a bounded symmetric Lipschitz real function of
+positions. Put $g=e^G$, $M_G=\mathbb E_Pe^{2G}$ and
+$\sigma=g\Gamma g/M_G$. For expectation under its position law
+write $\mathbb E_g$. Then
+\begin{equation}
+ \mathcal F(\sigma)-F_N\leq
+ \mathbb E_g\sum_i|\nabla_iG|^2+
+ T\bigl(2\mathbb E_gG-\log M_G\bigr).
+ \tag{8}\label{eq:8}
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+The geometric localization and entropy split behind (7) also appear in
+\cite[Section~3, Lemma~3.1]{DeuchertSeiringerYngvason2019}.
+We give the decomposition into fixed color-number sectors explicitly.
+The one-particle map $f\mapsto(pf,qf)$ is an isometry into the direct
+sum of the two one-particle spaces. Its symmetric tensor power is
+therefore an isometry. Dephase the resulting state with respect to
+the first-color number. The $j$-sector is unitarily identified with
+$\operatorname{Sym}^jL^2(\Omega)\otimes
+\operatorname{Sym}^{J-j}L^2(\Lambda_L)$: normalized symmetrization
+over color assignments gives this unitary, because the assignments
+are orthogonal. In particular there is no additional multiplicity
+or binomial entropy term. The product rule and
+$p\nabla p+q\nabla q=0$ give exactly the kinetic error in (7).
+Retain the same-color pair interactions and drop the other ones;
+nonnegativity makes this an energy decrease. One-body potentials
+are preserved in sum. Isometry preserves entropy, dephasing cannot
+decrease it, and subadditivity in each sector bounds it by
+$H(w)+\sum_jw_j[S(\sigma_{\mathrm{in},j})+
+S(\sigma_{\mathrm{out},j})]$. This proves (7).
+Comparison of $w$ with the geometric law
+$(1-e^{-1})e^{-j}$ gives $H(w)\leq-\log(1-e^{-1})+\mathbb E j$.
+Conditional on a labeled position configuration, the color choices
+are independent, with probabilities $p(x_i)^2$ and $q(x_i)^2$.
+This proves the number formula and also the diagonal interpretation
+of every color count used below. Finite-energy approximation and
+Lemma~\ref{lem:kinetic-entropy} justify all the entropy operations.
+
+For (8), define
+$\tau=\Gamma^{1/2}g^2\Gamma^{1/2}/M_G$.
+The nonzero spectra of $\tau$ and $\sigma$ agree: they are the
+spectra of $B^*B$ and $BB^*$ for $B=g\Gamma^{1/2}/\sqrt{M_G}$.
+If $H\psi=E\psi$, testing the eigenvalue equation with $g^2\psi$
+gives the form identity
+\[
+ \langle g\psi,Hg\psi\rangle
+ =E\langle\psi,g^2\psi\rangle+
+ \langle\psi,\sum_i|\nabla_i g|^2\psi\rangle.
+\]
+After summing in an eigenbasis,
+$\operatorname{Tr}H\sigma=\operatorname{Tr}H\tau+
+\mathbb E_g\sum_i|\nabla_iG|^2$. Both energies and entropies are
+finite, since $g$ and its gradients are bounded at fixed $N,L$.
+
+To compare relative entropies, approximate $g^2$ uniformly by positive
+simple functions on finite partitions into symmetric measurable
+sets $E$. Set $p_E=\operatorname{Tr}\Gamma\mathbf1_E$ and, for
+$p_E>0$, $\tau_E=\Gamma^{1/2}\mathbf1_E\Gamma^{1/2}/p_E$.
+The unmodified and tilted states are mixtures of the same $\tau_E$,
+with weights $p_E$ and $p_Eg_E^2/M_G$. Monotonicity of relative
+entropy under discarding the classical label (a partial trace;
+see \cite[Corollary~(3.2) and Section~IV]{LiebRuskai1973Entropy}) therefore bounds
+$D(\tau\Vert\Gamma)$ for this approximation by the classical
+relative entropy of these weights. Uniform approximation gives
+trace-norm convergence of the states and convergence of the
+classical entropies, since $g$ is bounded above and away from zero.
+Lower semicontinuity then gives
+$D(\tau\Vert\Gamma)\leq2\mathbb E_gG-\log M_G$.
+Finally $\mathcal F(\tau)-F_N=TD(\tau\Vert\Gamma)$ proves (8).
+\end{proof}
+
+For a nonempty union $U$ of coarse cells, set
+$\phi_U(x)=e^{-d(x,U)/s}$, where distance is the Euclidean torus
+distance. A neighborhood of radius $hs$ around $U$ is covered by
+at most $C(h+2)^3$ times as many cells as $U$.
+
+\begin{lemma}[Localized density bounds]\label{lem:density-tilts}
+Let $D_0=H_0\rho$, where $H_0$ is a sufficiently large fixed number.
+For $J\leq N$, $0\leq W\leq\rho$, and every finite-energy
+symmetric trace-one $J$-particle state,
+\begin{equation}
+ \mathcal F^{(-W)}(\sigma)-F_J^{(-W)}
+ \geq(c_vD_0-C_v\rho-Cs^{-2})\mathbb E_\sigma n_{\phi_U}
+       -C_vD_0^2|U|.\tag{9}\label{eq:9}
+\end{equation}
+In the corresponding equilibrium state,
+\begin{equation}
+ \mathbb E n_U\leq C_v\rho|U|.\tag{10}\label{eq:10}
+\end{equation}
+For the unmodified $N$-particle position law,
+\begin{equation}
+ \log\mathbb E_Pe^{2\lambda n_{\phi_U}}
+ \leq C_v\lambda\rho|U|\qquad(0\leq\lambda\leq1).
+ \tag{11}\label{eq:11}
+\end{equation}
+In particular $\mathbb E_P n_Q^2\leq C m^2$ for every cell $Q$.
+The proof of (9) can reserve a fixed positive fraction of the
+inside kinetic and pair energy for a further nonnegative lower bound.
+\end{lemma}
+
+\begin{proof}
+For $h\geq0$ take $p_h=1$ at distance at most $hs$ from $U$,
+and interpolate by a cosine to zero between distances $hs$ and
+$(h+1)s$. Set $q_h=(1-p_h^2)^{1/2}$ using the corresponding sine.
+Both gradients vanish off this transition and
+$|\nabla p_h|^2+|\nabla q_h|^2\leq Cs^{-2}$ there. Choose
+an open support region $\Omega_h$ inside the $(h+2)s$
+neighborhood; if that neighborhood covers the torus, use the torus.
+Its volume is at most $C(h+4)^3|U|$.
+
+Apply (7). Inside, partition the torus into cubes of side in
+$[D_0^{-1/3},2D_0^{-1/3}]$. For small density this side is between
+$b_0$ and $s$. The cubes meeting $\Omega_h$ number at most
+$CD_0(h+4)^3|U|$. Applying (2) to part of the inside energy of
+zero-extended Dirichlet states gives
+$c_vD_0j-CD_0^2(h+4)^3|U|$.
+Another fixed fraction of kinetic energy gives (3). An additional
+fixed fraction may remain unused. The external potential costs at
+most $\rho j$. Outside, (6) gives
+$F_{J-j}^{(-W)}\geq F_J^{(-W)}-C_v\rho j$.
+The classical entropy of the color number costs at most $T(C+j)$.
+Since $T\leq\rho^2$, all coefficients of $j$ other than
+$c_vD_0$ are bounded by $C_v\rho$. All volume errors, including
+$CT[1+|\Omega_h|(V^{-1}+T^{3/2})]$, are at most
+$C_vD_0^2(h+4)^3|U|$ after reducing the density: here
+$|U|\geq s^3$ and $V\geq s^3$.
+
+Average in $h$ with probability density $e^{-h}\,dh$. At each $x$,
+$\int_0^\infty p_h(x)^2e^{-h}\,dh\geq\phi_U(x)$.
+The averaged gradient cost is at most $Cs^{-2}\phi_U(x)$:
+the transition can occur only for
+$h\in[\max(0,d(x,U)/s-1),d(x,U)/s]$.
+The moment of $(h+4)^3$ is finite. Taking $H_0$ large makes
+$c_vD_0-C_v\rho$ positive and yields (9).
+In equilibrium the left side is zero; because
+$n_U\leq n_{\phi_U}$ this proves (10).
+
+For (11), take $G=\lambda n_{\phi_U}$ in (8).
+The distance function is one-Lipschitz, so
+$\sum_i|\nabla_iG|^2\leq\lambda^2s^{-2}n_{\phi_U}$.
+Also $G\geq0$, hence $\log M_G\geq0$ and the entropy term is
+at most $2T\lambda\mathbb E_g n_{\phi_U}$.
+Comparison with (9), increasing the fixed $H_0$ if necessary,
+absorbs both terms and gives
+$\mathbb E_g n_{\phi_U}\leq C_v\rho|U|$ uniformly in
+$0\leq\lambda\leq1$. Integrate
+$\partial_\lambda\log M_G=2\mathbb E_g n_{\phi_U}$
+from zero to obtain (11).
+For a single cell, exponential Markov gives
+$P(n_Q\geq t)\leq\min(1,e^{Cm-2t})$.
+Integration of $2tP(n_Q>t)$ gives $\mathbb E n_Q^2\leq C(m+1)^2$,
+which is at most $Cm^2$ since $m\to\infty$ uniformly with dilution.
+\end{proof}
+
+The mean bound alone does not guarantee that a cell supplies a particle
+to a spatial route. We next control simultaneous shortages and excesses
+in any specified family of cells. The lower tail uses the free-energy
+gain from placing extra particles in an underoccupied region; the upper
+tail follows from the exponential density estimate.
+
+\begin{lemma}[Simultaneous lower and upper occupation tails]\label{lem:occupation-tails}
+There are fixed $0<b_-<1/4$ and $b_+>4$ such that for every union
+$U$ of $u\geq1$ specified cells, each of the events
+$\{n_Q<2b_-m\text{ for every }Q\subset U\}$ and
+$\{n_Q>b_+m/2\text{ for every }Q\subset U\}$ has probability at most
+\begin{equation}
+ C e^{-cmu}.\tag{12}\label{eq:12}
+\end{equation}
+The constants and the required large fixed $K$ depend only on $v$.
+\end{lemma}
+
+\begin{proof}
+In each cell of $U$ choose $p=1$ on the central cube of half-side
+$s/4$, vanishing outside the central cube of half-side $s/3$.
+Cosine--sine interpolation gives squared localization cost at most
+$Cs^{-2}$ on these supports. Choose a squared smooth cutoff
+$0\leq\phi\leq1$, supported in the cell interiors, equal to one
+on these supports, with $|\nabla\phi|^2\leq C\phi/s^2$.
+The union of cell interiors is the Dirichlet region for $p$.
+Take one normalized smooth bump in each central cube of half-side
+$s/8$ and sum the bumps with equal amplitude to get a normalized
+orbital $\varphi$. Then
+$\int|\nabla\varphi|^2\leq Cs^{-2}$ and
+$\|\varphi\|_\infty^2\leq C/|U|$.
+For small density, its support has distance greater than $R_v$
+from $\operatorname{supp}q$.
+
+The conditional outside state is supported in
+$(\operatorname{supp}q)^{N-j}$, directly from the coloring map.
+In a color sector with $j$ inside particles, add $k$ particles in
+$\varphi$ to the outside state. The map on outside wave functions is
+\[
+ \psi\longmapsto
+ \binom{N-j+k}{k}^{1/2}P_{\mathrm{sym}}
+ (\psi\otimes\varphi^{\otimes k}).
+\]
+Distinct assignments of the new labels have disjoint supports, so
+this is an isometry. It preserves entropy and has additive kinetic
+energy; the separated supports also eliminate all old--new
+interactions. A new--new pair has expectation at most
+$\|v\|_1\|\varphi\|_\infty^2$. Therefore
+\[
+ \mathcal F(\sigma_{\mathrm{out},j})
+ \geq F_{N-j+k}-Ck/s^2-C_vk^2/|U|.
+\]
+Choose a sufficiently small fixed $\eta>0$ and put
+$k=\lfloor\eta\rho|U|\rfloor$.
+Since $\rho|U|\geq\rho s^3\to\infty$, small density ensures
+$k\geq\eta\rho|U|/2$, and $N+k\leq4\rho V$.
+By (5), followed by (6) for $j$ backward steps,
+$F_{N-j+k}\geq F_N+c_v\rho k-C_v\rho j$.
+Use (3) and positivity inside. Choose $\eta$ small enough to absorb
+$C_vk^2/|U|$ and then $K$ large enough to absorb $Ck/s^2$ into a
+fixed fraction of $c_v\rho k$. The remaining entropy errors are
+small compared with $\rho^2|U|$ for small density.
+Applying (7) and using $p^2\leq\phi$ gives, for every state to
+which the argument is applied,
+\[
+ \mathcal F(\sigma)-F_N
+ \geq c_v'\rho^2|U|-C_v\rho\mathbb E_\sigma n_\phi.
+\]
+Here the localization term has been absorbed in the latter error,
+since $s^{-2}\leq\rho/K^2$.
+
+Now set $G=-\lambda n_\phi$, $0\leq\lambda\leq1$, in (8).
+Its gradient contribution is at most
+$C\lambda^2s^{-2}\mathbb E_gn_\phi$; its entropy contribution is
+at most $-T\log M_G$, because $G\leq0$.
+Thus, after changing constants,
+$C_v\rho\mathbb E_g n_\phi-T\log M_G\geq c_v'\rho^2|U|$.
+Put $f(\lambda)=-\log M_G$. It is nondecreasing and
+$f'(\lambda)=2\mathbb E_gn_\phi$. While
+$f(\lambda)\leq a\rho|U|$, one has
+$Tf(\lambda)\leq a\rho^3|U|$, so for sufficiently small density
+$f'(\lambda)\geq c_v''\rho|U|$. Choose fixed
+$0<a<c_v''$. Starting at $f(0)=0$, this differential inequality
+forces $f(1)\geq a\rho|U|$. We have proved
+$\mathbb E e^{-2n_\phi}\leq e^{-a\rho|U|}$.
+
+On the lower-count event, $n_\phi\leq2b_-mu$, so its probability
+is at most $e^{4b_-mu-a\rho|U|}$. Since
+$mu/2\leq\rho|U|\leq2mu$, sufficiently small $b_-$ gives
+$e^{-cmu}$. On the upper-count event,
+$n_{\phi_U}\geq b_+mu/2$; (11) at $\lambda=1$ bounds its
+probability by $e^{C_v\rho|U|-b_+mu}$. A sufficiently large
+$b_+$ proves the other bound in (12).
+\end{proof}
+
+Occupation control supplies many particles per cell. The bridge
+comparison will additionally require well-separated endpoints. The
+next estimate shows that a prescribed region rarely contains many
+particles with a neighbor at the shorter scale $r$.
+
+\begin{lemma}[Isolation at a shorter scale]\label{lem:isolation}
+Fix $0<\zeta\leq1$ and set $r=\zeta\rho^{-1/3}$ and
+$\lambda_0=r\sqrt\rho$. A particle is nonisolated if another
+particle is within torus distance $r$. For any nonempty union $S$
+of coarse cells and any $k'\geq0$,
+\begin{equation}
+ P(\#\{\text{nonisolated particles in }S\}\geq k')
+ \leq\exp\{C_v\lambda_0\zeta^3\rho|S|-\lambda_0 k'\}.
+ \tag{13}\label{eq:13}
+\end{equation}
+The constant $C_v$ can be chosen independently of decreasing
+$\zeta$; the density threshold is allowed to depend on $\zeta$.
+\end{lemma}
+
+\begin{proof}
+Partition the torus into cubes of side in $[r,2r]$.
+For each cube meeting $S$ choose a cutoff $0\leq\chi\leq1$ that is one on
+the concentric triple cube, vanishes outside the concentric
+quintuple cube, and satisfies $|\nabla\chi|\leq C/r$.
+Let $Q=\sum_\chi(n_\chi-1)_+$. A nonisolated particle in a
+fine cube has a partner in that cube's triple cube. If there are
+$a$ nonisolated particles in the original cube, its cutoff has
+$n_\chi\geq\max(a,2)$, and hence
+$a\leq2(n_\chi-1)_+$. Summing gives the deterministic bound
+$\#\{\text{nonisolated in }S\}\leq2Q$.
+Bounded overlap of the quintuple cubes gives
+$\sum_i|\nabla_i Q|^2\leq Cr^{-2}n_{U_0}$,
+where $U_0$ is a coarse-cell cover of their union with
+$|U_0|\leq C|S|$. The cubes admit a coloring with a fixed number
+of colors, disjoint within each color. This can be obtained from
+the bounded-degree intersection graph and works also on the torus.
+
+Repeat the proof of (9) for $U_0$, reserving a fixed fraction of
+inside energy for (2) on these quintuple cubes, divided among the
+colors. In every localization mask $p_h=1$ on $U_0$, so those
+particles are certainly assigned to the inside color. Also
+$(n_\chi-1)_+\leq n_{\operatorname{supp}\chi}
+\mathbf1_{\{n_{\operatorname{supp}\chi}\geq2\}}$.
+The reserved energy therefore contributes
+$c_vr^{-3}\mathbb E_\sigma Q$. The small-density conditions ensure
+that the fine cubes exceed $b_0$; all overlap constants are
+independent of $\zeta$.
+
+Apply this augmented lower bound to $G=\lambda Q$ for
+$0\leq\lambda\leq\lambda_0$. Its gradient cost in (8) is
+at most $C\rho\mathbb E_g n_{U_0}$, absorbed by the density term
+of (9) after fixing $H_0$ sufficiently large. The entropy cost is
+at most $2T\lambda\mathbb E_gQ$, absorbed by
+$c_vr^{-3}\mathbb E_gQ$ for small density. It follows that
+$\mathbb E_gQ\leq C_vr^3\rho^2|S|
+=C_v\zeta^3\rho|S|$.
+Integration of $\partial_\lambda\log\mathbb E e^{2\lambda Q}
+=2\mathbb E_gQ$ yields
+$\log\mathbb E e^{2\lambda_0Q}
+\leq C_v\lambda_0\zeta^3\rho|S|$.
+Since $2Q\geq\#\{\text{nonisolated in }S\}$, exponential Markov
+proves (13).
+\end{proof}
diff --git a/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/opening.tex b/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/opening.tex
new file mode 100644
index 0000000000000000000000000000000000000000..7bd03d4d6f3004abce7742574e3c9992513d73bd
--- /dev/null
+++ b/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/opening.tex
@@ -0,0 +1,166 @@
+\section{Opening the slab}\label{sec:opening}
+
+The local estimates and cell paths now define a change from the closed
+Gibbs law to an open path measure. We specify the target measure, the
+accepted choices, and the interacting proposal in that order. Keeping
+its normalization exact will let the two copies of the change cancel
+in the next section.
+
+Let \(q_0=\langle u_0,\gamma^{(1)}u_0\rangle/N\). Starting from
+\eqref{eq:14}, average a slot \(i_0\) with factor \(1/N\) and replace
+its inner end by \(E_{i_0}=z\), integrated with \(dz/V\). All other
+inner ends remain \(Y_i\), and every outer endpoint \(Y_i\) is still
+integrated. The resulting finite measure has total mass \(q_0\): it
+inserts \(N^{-1}\sum_i(|u_0\rangle\langle u_0|)_i\) between the two
+kernels in the trace. Add a dummy point \(x\), uniform in the cell of
+\(Y_{i_0}\), and retain \(i_0\) as a coordinate. Denote this augmented
+measure by \(\mu\); the dummy coordinate does not change its mass.
+
+\subsection{The base law and acceptance}
+Our base law is the closed probability \eqref{eq:14}, denoted by the
+configuration $C$, together with independent uniform points $x,z$ of
+law $dx\,dz/V^2$. In the slab notation, configurations are tuples of
+paths rather than density operators; their starting tuples carry
+superscripts, such as $S^C$, and a subscript restricts their paths to
+a set of slots.
+
+Set $a=\operatorname{cell}(x)$ and $b=\operatorname{cell}(z)$.
+The first index of $\mu_{ab}$ in Section~\ref{sec:defects} specifies
+the new inner endpoint's cell, and the second specifies the outer
+endpoint's cell. Thus the two orientations used below are
+\[
+\begin{array}{c|c|c|c}
+ &\text{opened slot}&\text{outer endpoint}&\text{new inner endpoint}\\ \hline
+ \text{forward insertion }\mu_{ba}&i_0&Y_{i_0}\in a&z\in b\\
+ \text{reverse insertion }\mu_{ab}&e&Y_e\in b&x\in a.
+\end{array}
+\]
+The reverse slot $e$ will be the last label on the selected path.
+Only valid ordered pairs, meaning the admissible pairs of
+Section~\ref{sec:skeletons}, are used. For such a pair draw a skeleton
+and apply the deterministic adaptation to the unusable cells of $C$.
+Require geometric acceptance and the two tests
+$\mathcal A^+(x;C),\mathcal A^+(z;C)\le C_0$.
+Denote these combined conditions by $\operatorname{acc}(C,\gamma)$.
+For an invalid pair the acceptance indicator and all weights below
+are zero. The base average acceptance mass is at least $\upsilon/2$:
+the valid-pair fraction is at least $\upsilon$, and the geometric
+rejection and dummy-score probabilities can each be made smaller
+than a fixed fraction of it by the choices already specified.
+
+\subsection{Selecting one eligible slot in each path cell}
+On acceptance let $P(C,\gamma)$ be the ordered simple star path.
+For a cell $Q$, write
+\[
+ g_Q(C)=\sum_{i:Y_i\in Q}\mathbf1_{\{i\text{ eligible in }C\}}.
+\]
+Independently in each path cell choose a uniform eligible slot.
+This gives an ordered tuple $I=(i_0,\ldots,i_k)$, one label per
+cell. For any candidate tuple define, on acceptance,
+\[
+\begin{split}
+ h_\gamma(C,I)={}&
+ \mathbf1_{\{(\operatorname{cell}(Y_{i_0}),\ldots,
+                      \operatorname{cell}(Y_{i_k}))=P(C,\gamma)\}}\\
+ &\quad\cdot\prod_{j=0}^k
+       \frac{\mathbf1_{\{i_j\text{ eligible in }C\}}}
+            {g_{\operatorname{cell}(Y_{i_j})}(C)}.
+\end{split}
+\]
+Set the weight to zero off acceptance, and interpret a factor with
+zero eligible count as zero. The cells of $P$ are distinct and
+usable, so on acceptance the displayed factors are defined and
+$\sum_Ih_\gamma(C,I)=1$. Equivalently, the sum ranges over candidate
+simple star paths from $a$ to $b$ and label choices in their outer
+$Y$-cells. This range depends on $a,b,Y$, whereas the weight retains
+all dependence on the actual paths of $C$.
+
+\subsection{The conditional interacting proposal}
+For a tuple $I$, use cyclic predecessor and successor, acting as the
+identity off the tuple, and let \((p_I S)_i=S_{\operatorname{pred}_I(i)}\).
+Transform the starts to \(S^O=p_I S^C\). The inner ends \(E^O\) agree
+with $Y$ except at slot \(i_0\), whose end is replaced by \(z\);
+paths off $I$ are kept. Resample paths in \(I\) from their conditional
+interacting distribution restricted by \(\mathrm{cut}_R\).
+
+More precisely, for a state \(X\) with fixed endpoints and complement
+paths, let \(W_G^X\) use the free bridge measures on \(G\) and the
+exponential of minus the interactions incident to \(G\), each pair once.
+Write \(\mathcal Z_G(X)=\int\mathbf 1_{\mathrm{cut}_R(G)} W_G^X\).
+The proposal on \(I\) uses this cut measure in \(O\), divided by
+\(\mathcal Z_I(O)\). The new displacements are at most \((D+10)s\)
+by eligibility and star adjacency: the start from the last cell goes to
+the end \(z\). Thus the normalization is positive.
+
+\subsection{The density and its second moment}
+
+Averaging a path-dependent likelihood before estimating its second
+moment also appears in random-trail detection and perturbed-lattice
+reassignment~\cite{AriasCastroCandesHelgasonZeitouni2008,
+BergerPeres2013,PeresSly2014}. In the present setting the conditional
+interacting measures and both proposal normalizers must be retained
+explicitly.
+
+Write \(\nu\) for the resulting subprobability measure on the open
+variables. Its mass is the base acceptance probability, hence at least
+\(\upsilon/2\). Its density relative to \(\mu\) is
+\[
+ f = m\,\mathbb E_\gamma\sum_{I:\,\mathrm{first}(I)=i_0}
+       \frac{\mathbf 1_{\mathrm{cut}_R(O_I)}}{\mathcal Z_I(O)}
+              \int W_I^{\widehat C}\, h_\gamma(\widehat C,I),                             \tag{26}\label{eq:26}
+\]
+where the hypothetical closed state \(\widehat C\) has
+\(S^{\widehat C}=p_I^{-1}S^O\), \(E^{\widehat C}=Y\), and paths off
+\(I\) as in \(O\). A term with zero old weight is set to zero even
+if its normalizer vanishes; positive old weight gives the displacement
+condition and a positive normalizer.
+
+To verify the density, change start variables for each candidate tuple.
+The Jacobian is one and the outer kernel is invariant under this
+permutation. Write \(W_{\bar I}^{\rm int}\) for the complement bridge
+measures with only their internal interactions. This factor agrees
+before and after the change. Integrating the old paths on \(I\) leaves
+\[
+ W_{\bar I}^{\rm int}
+   \left[\int W_I^{\widehat C} h_\gamma(\widehat C,I)\right]
+     \mathbf 1_{\mathrm{cut}_R(O_I)}W_I^O/\mathcal Z_I(O)
+\]
+(all incident weights and normalizers use the displayed fixed complement).
+The corresponding factor in the open measure is
+\(W_{\bar I}^{\rm int}W_I^O\). The remaining normalization comes from
+the label and dummy coordinate:
+\[
+ \frac{\text{base density of }x}
+      {\text{target index factor}\times\text{target density of }x}
+ =\frac{V^{-1}}{N^{-1}s^{-3}}=\frac{N}{M}=m.
+\]
+Here both measures restrict $x$ to the cell containing $Y_{i_0}$,
+and both use $dz/V$. This proves \eqref{eq:26}, equivalently by
+integration against nonnegative test functions. Skeleton expectations
+are for the given cell pair, with zero weight for invalid pairs.
+
+Cauchy--Schwarz gives
+\((\upsilon/2)^2\le q_0\int f^2\dd\mu\). To compute the second
+moment, use \(\int f^2\dd\mu=\int f\dd\nu\): evaluate
+\eqref{eq:26} at a transformation from the base with independent
+skeleton \(\gamma'\) and tuple \(I'\). The resulting integral is
+\[
+ \begin{split}
+ &m\,\mathbb E_{x,z,\gamma',\gamma}\, Z_N^{-1}\int
+ F(Y,S^C)\,dY\,dS^C
+ \sum_{I',I:\,\mathrm{first}(I')=\mathrm{first}(I)}\int
+   W(C) h_{\gamma'}(C,I')\\
+ &\hspace{30mm}\cdot
+     \frac{\mathbf 1_{\mathrm{cut}_R(O_{I'})}W_{I'}^O}
+          {\mathcal Z_{I'}(O)}
+     \frac{\mathbf 1_{\mathrm{cut}_R(O_I)}}{\mathcal Z_I(O)}
+          W_I^{\widehat C} h_\gamma(\widehat C,I).
+ \end{split}                                                          \tag{27}\label{eq:27}
+\]
+The path integrations are over all closed paths, then proposed paths
+on \(I'\), and finally hypothetical old paths on \(I\). Here \(W(C)\)
+is the full slab weight, \(S^O=p_{I'}S^C\), and $x,z$ have their base
+uniform law. Both tuples must have the same first slot, which is summed
+over. All integrands are nonnegative, so Tonelli permits the changes in
+integration order used next. Section~\ref{sec:comparison} bounds
+\eqref{eq:27} uniformly.
diff --git a/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/setup.tex b/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/setup.tex
new file mode 100644
index 0000000000000000000000000000000000000000..d0cbc523bfa5d7c783a01175efe29f86bbde386d
--- /dev/null
+++ b/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/setup.tex
@@ -0,0 +1,37 @@
+\section{Scales and conventions}\label{sec:setup}
+
+We use the Hamiltonian of Section~\ref{sec:introduction} also on labelled
+coordinates, and suppress $L$ when it is fixed. We choose an everywhere
+nonnegative bounded Borel radial representative of $v$ vanishing outside
+its range ball. Changing this representative on a null set changes neither
+the operator nor the path integrals: at interior times the free relative
+positions have densities. Write $R_v$ for a range bound, and use
+$\|v\|_1,\|v\|_\infty$ on $\R^3$. In formulas on the torus,
+$v$ abbreviates its periodization $v_L$; infinite-space norms always
+refer to the original potential.
+
+The orthogonal symmetrizer is included exactly once in a trace kernel.
+If $k_t$ is the labelled $N$-particle kernel, then
+\[
+ k_t^{\mathrm{sym}}(X,Y)=\frac1{N!}\sum_{\pi\in\mathfrak S_N}
+                         k_t(X,\pi Y),\qquad
+ Z_N=\int k_\beta^{\mathrm{sym}}(X,X)\dd X.
+\]
+Thus integration over labelled coordinates carries no further factorial.
+We work with $\rho/2\le N/V\le2\rho$, $0<T\le\rho^2$, and
+$\beta=T^{-1}$. Every volume restriction below is imposed at fixed $\rho$,
+uniformly over this temperature interval. Constants $C,c>0$ may change
+between occurrences and depend on $v$ and previously fixed cutoffs,
+but not on $\rho,L,N,T$. The final order of those cutoff choices is
+recorded in Section~\ref{sec:conclusion}.
+
+We use a periodic grid \(\mathcal G=(\mathbb Z/n\mathbb Z)^3\) of equal cubes (fix the assignment at faces), with
+\[
+ n=\left\lfloor\frac{L}{K\rho^{-1/2}}\right\rfloor,\qquad
+ s=L/n\in[K\rho^{-1/2},2K\rho^{-1/2}],\qquad
+ M=n^3,\quad m=N/M .
+\]
+Here \(K\ge1\) is a fixed large constant chosen below. Thus \(\rho s^3/2\le m\le 2\rho s^3\). 
+
+A union \(U\) of \(u\ge1\) cells has volume \(|U|=u s^3\). A neighborhood out to distance \(h s\), \(h\ge0\), can be covered by at most \(C(h+2)^3 u\) cells, including on the torus. Physical distances \(d(\cdot,\cdot)\) are Euclidean torus distances unless indicated otherwise. Cell index distances will use the periodic sup norm. For a function \(p\) of position put \(n_p(X)=\sum_i p(x_i)\) on configurations (also the multiplication operator); similarly \(n_U\) for counting particles in \(U\).
+
diff --git a/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/slab.tex b/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/slab.tex
new file mode 100644
index 0000000000000000000000000000000000000000..d79847018b93b5388464bd5a01b446dab7a2b5d4
--- /dev/null
+++ b/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/build/sections/slab.tex
@@ -0,0 +1,167 @@
+\section{A short imaginary-time slab}\label{sec:slab}
+
+The next estimates describe the part of the Gibbs path measure that
+will be changed later. Set $\delta=s^2$. For sufficiently small
+density, $\beta>2\delta$, because
+$\delta\leq4K^2\rho^{-1}$ and $\beta\geq\rho^{-2}$.
+Let $P_{\mathrm{sym}}$ be the orthogonal symmetrization projector
+on labeled coordinates. Write $F(Y,S)$ for the kernel, from $Y$
+to $S$, of $e^{-(\beta-\delta)H}P_{\mathrm{sym}}$.
+It is nonnegative and invariant under separate permutations of
+each tuple. The inner slab has labeled paths $\omega_i$ from
+$S_i$ to $E_i$, with unnormalized free torus bridge measures
+$\mathsf K_{S_i,E_i}$. Put
+\[
+ \mathcal V_{ij}(\omega)=\int_0^\delta
+ v(\omega_i(t)-\omega_j(t))\,dt,\qquad
+ W(S,E;d\omega)=e^{-\sum_{i<j}\mathcal V_{ij}(\omega)}
+ \prod_{i=1}^N\mathsf K_{S_i,E_i}(d\omega_i).
+\]
+In the closed measure $E=Y$, and the joint probability measure is
+\begin{equation}
+ Z_N^{-1}F(Y,S)\,dY\,dS\,W(S,Y;d\omega).
+ \tag{14}\label{eq:14}
+\end{equation}
+Denote this closed probability law by $P_{\mathrm{cl}}$ and its
+expectation by $\mathbb E_{\mathrm{cl}}$.
+The normalization is the semigroup trace identity; the canonical
+permutation and one-particle conventions agree with
+\cite[Appendix~A, (A.1), (A.8)--(A.10), and (A.12)--(A.18)]{Ueltschi2006FeynmanCycles}. At each fixed
+time, permutation-invariant position tests have the equilibrium
+law $P$: insert the test as a multiplication operator, split the
+semigroup at that time, and use trace cyclicity.
+
+For clarity, a torus bridge includes its lift choice. Fix a
+representative of its end, lift its start in every possible way,
+and assign a displacement $d_\ell$ to that lift. Its weight is
+$k(d_\ell)=(4\pi\delta)^{-3/2}e^{-|d_\ell|^2/(4\delta)}$
+times the law of a Euclidean bridge with generator $\Delta$,
+projected to the torus. This gives the unnormalized torus bridge
+after summation over $\ell\in\mathbb Z^3$. All lifted tests below
+refer to this lift, not to a pointwise choice of nearest torus
+representatives along the trajectory.
+
+Fix a large number $D$, put $R=D+30$ and $R_A=2R+20$, and later
+choose $0<\varepsilon<1/2$. The short-distance scale is
+$r=\zeta\rho^{-1/3}$ from Lemma~\ref{lem:isolation}. The order of
+choices allows $\zeta$ to depend on $D,\varepsilon$, followed by
+a small-density restriction. In particular $r\to\infty$,
+$r/s\to0$, and $2r<\delta$.
+For an anchor $a$, define
+\begin{equation}
+ \begin{aligned}
+ \mathcal A^-(a;\omega)&=\int_0^{\delta/2}w(t)
+ \sum_i e^{-d(\omega_i(t),a)^2/[1000(1+t)]}
+ \mathbf1_{\{d(\omega_i(t),a)\leq R_As\}}\,dt,\\
+ &\hspace{15mm}w(t)=\min(1,t^{-3/2}),\qquad w(0)=1.
+ \end{aligned}
+ \tag{15}\label{eq:15}
+\end{equation}
+Define $\mathcal A^+$ by replacing $\omega_i(t)$ by
+$\omega_i(\delta-t)$. A subcollection in place of $\omega$ means
+that only its paths occur in the sum.
+
+A closed slot $i$ is \emph{eligible} if all of the following hold:
+its start $S_i$ and end $Y_i$ are each at distance greater than $r$
+from the other points of the respective tuple; its lifted path
+stays within $Ds$ of its end; in the first and last time intervals
+of length $r$ it stays within $r/10$ of its respective start and
+end; and both $\mathcal A^-(S_i;\omega)$ and
+$\mathcal A^+(Y_i;\omega)$ are at most a fixed threshold $C_0$.
+A cell is \emph{usable} if its end count belongs to
+$[b_-m,b_+m]$ and at least a proportion $1-\varepsilon$ of those
+slots are eligible.
+For an individual bridge, $\mathrm{cut}_R$ denotes just the lifted
+macro condition with radius $Rs$ and the same two interval
+conditions with radius $r/10$. In this definition every condition
+uses that bridge's own endpoints.
+
+\begin{lemma}[Single-bridge estimates]\label{lem:single-bridge}
+Suppose a bridge's endpoints have torus distance at most
+$(D+10)s$. When $L$ is sufficiently large compared with $Rs$,
+$\mathrm{cut}_R$ permits only the nearest lift. Under the probability
+bridge $\pi_{S,E}$ for that lift, its probability is at least a
+constant $p_*>0$. Under the same unconditioned probability bridge
+$\pi_{S,E}$, the expected interaction with any fixed trajectory
+$\omega_j$ is at most $C_v$, and the contribution
+of times $[r,\delta-r]$ is at most $C_v/\sqrt r$.
+For any fixed collection $\omega_{\mathrm{fix}}$,
+\begin{equation}
+ \mathbb E_{\pi_{S,E}}\left[
+ \mathbf1_{\mathrm{cut}_R}
+ \sum_{j\in\mathrm{fix}}\int_0^\delta
+ v(\omega(t)-\omega_j(t))\,dt\right]
+ \leq C(v,D)\bigl[
+ \mathcal A^-(S;\omega_{\mathrm{fix}})+
+ \mathcal A^+(E;\omega_{\mathrm{fix}})\bigr].
+ \tag{16}\label{eq:16}
+\end{equation}
+Finally, for any fixed collection of at most $N$ paths and a
+uniform torus point $x$,
+$\mathbb E_x\mathcal A^\pm(x;\omega)\leq CK^2$.
+These bounds are independent of temperature, density, and volume
+after the stated choices; $p_*$ may be fixed after $D$.
+\end{lemma}
+
+\begin{proof}
+Take $L>4Rs$. Any nonnearest lift has displacement greater than
+$Rs$, so it fails the cut already at time zero. In the nearest
+lift write the bridge as its straight interpolation plus
+$B_t-(t/\delta)B_\delta$, where $B$ has generator $\Delta$.
+By Brownian scaling the event that this fluctuation has supremum
+at most $s$ has a fixed positive probability; on it the macro
+distance is at most $(D+11)s<Rs$.
+The deterministic drift during the first interval is at most
+$(D+10)r/s$, which is at most $r/20$ once $s\geq20(D+10)$.
+The Brownian maximal bound and the Gaussian tail of $B_\delta$
+show that the remaining fluctuation exceeds $r/20$ on this
+interval with probability at most $Ce^{-cr}$: separately bound
+$\sup_{t\leq r}|B_t|$ by $r/40$ and
+$|B_\delta|$ by $\delta/40$. The same argument applies at the
+other end by reversal. Subtracting these two small failure
+probabilities from the macro event proves the lower bound $p_*$.
+
+At time $t$ the nearest-lift bridge is Gaussian with coordinate
+variance $2t(\delta-t)/\delta$. For
+$t'=\min(t,\delta-t)$ this variance lies in $[t',2t']$.
+Its periodic density is bounded by $C(t')^{-3/2}$ when
+$t'\leq\delta\ll L^2$. Thus the expectation of $v$ against any
+fixed point is at most $C_vw(t')$: use $\|v\|_\infty$ for
+$t'\leq1$ and the density bound with $\|v\|_1$ for $t'\geq1$.
+Integration gives a bound $C_v$ on the full interaction and
+$C_v\int_r^\infty t^{-3/2}\,dt\leq C_v/\sqrt r$ on the
+middle interval.
+
+To retain spatial decay, first take $t\leq\delta/2$ and use
+the start $S$ as anchor. The interpolation center is within
+$(D+10)t/s\leq(D+10)\sqrt t$ of $S$.
+For $t\geq1$, the Gaussian density integrated over the periodically
+lifted range balls around a fixed point $y$ is at most
+\[
+ C(v,D)t^{-3/2}\sum_{\ell\in\mathbb Z^3}
+ e^{-|y+\ell L-S|^2/(16t)}.
+\]
+Indeed $|u+b|^2\geq|u|^2/2-|b|^2$ absorbs the center displacement
+and the range radius, and $|b|\leq R_v+(D+10)\sqrt t$.
+Extracting $e^{-d(y,S)^2/(32t)}$ leaves a uniformly bounded
+periodic Gaussian sum, because $t\leq\delta\ll L^2$.
+For $0<t\leq1$, use probability at most one within a fixed
+distance $2(R_v+D+10)$ of $S$. Beyond this distance, a hit
+requires a Gaussian fluctuation at least $d(y,S)/2$; its tail
+has the same, or stronger, spatial decay after increasing
+$C(v,D)$. We have thus bounded the expected interaction density
+by
+$C(v,D)w(t)e^{-d(y,S)^2/[1000(1+t)]}$.
+
+On the cut, a possible hit also requires
+$d(y,S)\leq (R+D+10)s+R_v<R_As$ for sufficiently small density.
+This follows by bounding the bridge's distance from the end by
+$Rs$ and the endpoint displacement by $(D+10)s$.
+Consequently the indicator in (15) may be inserted in this upper
+bound. Reverse time for the other half, sum over the fixed
+trajectories, and integrate to obtain (16).
+Finally the integral over a uniform anchor of the Gaussian factor
+is at most $C(1+t)^{3/2}/V$. Since
+$w(t)(1+t)^{3/2}\leq C$, the mean score is at most
+$CN\delta/V\leq C\rho s^2\leq CK^2$.
+\end{proof}
diff --git a/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/paper.pdf b/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..450ee96ba84d91b61fba6418193beb471b957c0a
--- /dev/null
+++ b/preprints/A-density-uniform-condensate-bound-for-dilute-Bose-gases-September-27-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:5dceb5d5d293d6dcbc651560e65a2475219bf7ed9489ad55d9b5e49e75b1b0b0
+size 669895
diff --git a/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/README.md b/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..0af0bf6e5a2fcc92d9a1ec5a6678c3a66a787458
--- /dev/null
+++ b/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/README.md
@@ -0,0 +1,18 @@
+# [A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 23, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026,
+  author = {{OpenAI}},
+  title = {{A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf}{OAI:A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/build/figures/proof-dependencies.tex b/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/build/figures/proof-dependencies.tex
new file mode 100644
index 0000000000000000000000000000000000000000..14b0d38b8aaea83d30cc662967169b2eec983efc
--- /dev/null
+++ b/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/build/figures/proof-dependencies.tex
@@ -0,0 +1,63 @@
+\begin{figure}[tbp]
+\centering
+\begin{tikzpicture}[
+  font=\small,
+  stage/.style={draw, line width=.45pt, align=center,
+    inner xsep=5pt, inner ysep=5pt},
+  flow/.style={-{Stealth[length=2mm]}, line width=.55pt}
+]
+\node[stage, text width=6.4cm] (strict) at (-3.65,0)
+  {Strict entropy branch\\
+   Entropy optimizer\\
+   Lemma~\ref{lem:critical-two-alternatives}(i)};
+\node[stage, text width=6.4cm] (gap) at (3.65,0)
+  {Gap branch\\
+   Gap eigenfunction\\
+   Lemma~\ref{lem:critical-two-alternatives}(ii)};
+
+\node[stage, text width=5.8cm] (critical) at (0,-1.55)
+  {Common critical law and vector field\\
+   Proposition~\ref{prop:critical-sequence}};
+\draw[flow] (strict.south) -- (critical.north west);
+\draw[flow] (gap.south) -- (critical.north east);
+
+\node[stage, text width=4.0cm, minimum height=1.5cm]
+  (prediction) at (-4.85,-3.2)
+  {Prediction curve\\
+   and posterior estimates\\
+   Prop.~\ref{prop:prediction}, Lemma~\ref{lem:posterior-outliers}};
+\node[stage, text width=4.0cm, minimum height=1.5cm]
+  (hierarchy) at (0,-3.2)
+  {Symmetric\\tensor hierarchy\\
+   Proposition~\ref{prop:hierarchy}};
+\node[stage, text width=4.0cm, minimum height=1.5cm]
+  (matrix) at (4.85,-3.2)
+  {Matrix estimates\\and frozen law\\
+   Proposition~\ref{prop:frozen-law}};
+\draw[flow] (critical.west) -| (prediction.north);
+\draw[flow] (prediction.east) -- (hierarchy.west);
+\draw[flow] (hierarchy.east) -- (matrix.west);
+
+\node[stage, text width=9.3cm] (contradiction) at (0,-5.45)
+  {Integrated information contradiction\\
+   Propositions~\ref{prop:information-upper}
+   and~\ref{prop:information-lower},\\
+   Lemma~\ref{lem:scalar-divergence}};
+\draw[flow] (matrix.south) -- (4.85,-4.5)
+  -- node[above, align=center] {matrix bounds} (0,-4.5)
+  -- (contradiction.north);
+\draw[flow] (critical.east) -- (7.45,-1.55)
+  -- (7.45,-5.45)
+  -- node[above, align=center] {entropy bound} (contradiction.east);
+\end{tikzpicture}
+\caption{Main stages of the contradiction argument. At the regularized
+law, the strict entropy branch means that the logarithmic Sobolev
+constant exceeds the Poincar\'e constant; in the gap branch the two
+constants coincide. The logarithmic Sobolev inequality of the common
+critical law, together with the square-root energy estimates of
+Section~\ref{sec:matrices}, supplies the information upper bound.
+The frozen matrix law supplies the scalar lower bound. After the
+sequence limit at fixed parameters, its scale integral becomes
+arbitrarily large as the diffusion time and covariance cap grow together.}
+\label{fig:proof-dependencies}
+\end{figure}
diff --git a/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/build/glyph-mappings.tex b/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/build/glyph-mappings.tex
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diff --git a/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/build/main.tex b/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/build/main.tex
new file mode 100644
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--- /dev/null
+++ b/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/build/main.tex
@@ -0,0 +1,75 @@
+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage[margin=1.05in]{geometry}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{microtype}
+\usepackage{enumitem}
+\usepackage{graphicx,booktabs}
+\usepackage{tikz}
+\usetikzlibrary{arrows.meta,positioning}
+\usepackage[numbers,sort&compress]{natbib}
+\usepackage{url}
+\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
+\input{glyph-mappings}
+\pdfinfoomitdate=1
+\pdftrailerid{}
+\pdfsuppressptexinfo=15
+\hypersetup{pdftitle={A dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures},pdfauthor={OpenAI}}
+\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}
+\numberwithin{equation}{section}
+\newcommand{\R}{\mathbb R}
+\newcommand{\N}{\mathbb N}
+\newcommand{\E}{\mathbb E}
+\newcommand{\Prob}{\mathbb P}
+\newcommand{\1}{\mathbf 1}
+\newcommand{\Id}{\mathrm{Id}}
+\newcommand{\Ical}{\mathcal I}
+\newcommand{\MI}{\mathsf I}
+\newcommand{\HS}{\mathrm{HS}}
+\newcommand{\op}{\mathrm{op}}
+\DeclareMathOperator{\Ent}{Ent}
+\DeclareMathOperator{\Var}{Var}
+\DeclareMathOperator{\Cov}{Cov}
+\DeclareMathOperator{\tr}{tr}
+\DeclareMathOperator{\rank}{rank}
+\DeclareMathOperator{\supp}{supp}
+\DeclareMathOperator{\Dom}{Dom}
+\DeclareMathOperator{\Law}{Law}
+\setlist[enumerate]{itemsep=3pt,topsep=5pt}
+\setlist[itemize]{itemsep=3pt,topsep=5pt}
+\title{A dimension-free logarithmic Sobolev inequality\\
+for subgaussian log-concave measures}
+\author{OpenAI}
+\date{September 23, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We prove that every centered log-concave probability measure with a
+Lebesgue density on $\mathbb R^n$ and linear subgaussian parameter $a$
+satisfies $\operatorname{Ent}_\mu(f^2)\le Ca^2\int|Df|^2\,d\mu$
+for compactly supported smooth $f$, with one universal constant $C$.
+This resolves positively the dimension-free logarithmic Sobolev conjecture
+for subgaussian log-concave measures.
+\end{abstract}
+\input{sections/01-introduction}
+\newpage
+\input{sections/02-calculus}
+\input{sections/03-critical}
+\input{sections/04-prediction}
+\input{sections/05-hierarchy}
+\input{sections/06-matrices}
+\input{sections/07-observations}
+\bibliographystyle{plain}
+\begingroup
+\raggedright
+\bibliography{references}
+\endgroup
+\end{document}
diff --git a/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/build/references.bib b/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/build/references.bib
new file mode 100644
index 0000000000000000000000000000000000000000..723f432268caded43fd7b5dfcc169a4f579e4b46
--- /dev/null
+++ b/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/build/references.bib
@@ -0,0 +1,227 @@
+@article{Gross1975,
+  author = {Gross, Leonard},
+  title = {Logarithmic {Sobolev} inequalities},
+  journal = {American Journal of Mathematics},
+  volume = {97},
+  number = {4},
+  pages = {1061--1083},
+  year = {1975},
+  doi = {10.2307/2373688},
+  url = {https://www.jstor.org/stable/2373688}
+}
+
+@article{OttoVillani2000,
+  author = {Otto, Felix and Villani, C{\'e}dric},
+  title = {Generalization of an Inequality by {Talagrand} and Links with the Logarithmic {Sobolev} Inequality},
+  journal = {Journal of Functional Analysis},
+  volume = {173},
+  number = {2},
+  pages = {361--400},
+  year = {2000},
+  doi = {10.1006/jfan.1999.3557},
+  url = {https://doi.org/10.1006/jfan.1999.3557}
+}
+
+@incollection{BakryEmery1985,
+  author = {Bakry, Dominique and {\'E}mery, Michel},
+  title = {Diffusions hypercontractives},
+  booktitle = {S{\'e}minaire de probabilit{\'e}s, {XIX}, 1983/84},
+  series = {Lecture Notes in Mathematics},
+  volume = {1123},
+  publisher = {Springer},
+  address = {Berlin},
+  pages = {177--206},
+  year = {1985},
+  doi = {10.1007/BFb0075847},
+  url = {https://numdam.org/item/SPS_1985__19__177_0/}
+}
+
+@incollection{Ledoux1999,
+  author = {Ledoux, Michel},
+  title = {Concentration of measure and logarithmic {Sobolev} inequalities},
+  booktitle = {S{\'e}minaire de probabilit{\'e}s, {XXXIII}},
+  series = {Lecture Notes in Mathematics},
+  volume = {1709},
+  publisher = {Springer},
+  address = {Berlin},
+  pages = {120--216},
+  year = {1999},
+  doi = {10.1007/BFb0096511},
+  url = {https://numdam.org/item/SPS_1999__33__120_0/}
+}
+
+@article{Bizeul2026,
+  author = {Bizeul, Pierre},
+  title = {On the log-{Sobolev} constant of log-concave vectors},
+  journal = {Journal of Functional Analysis},
+  volume = {290},
+  number = {9},
+  pages = {111368},
+  year = {2026},
+  doi = {10.1016/j.jfa.2026.111368},
+  eprint = {2306.12997v3},
+  archivePrefix = {arXiv},
+  primaryClass = {math.FA},
+  note = {Cited theorem numbers refer to arXiv:2306.12997v3, 16 February 2026; first posted in 2023},
+  url = {https://arxiv.org/abs/2306.12997v3}
+}
+
+@article{KlartagLehec2024,
+  author = {Klartag, Bo'az and Lehec, Joseph},
+  title = {Isoperimetric inequalities in high-dimensional convex sets},
+  journal = {Bulletin of the American Mathematical Society},
+  volume = {62},
+  number = {4},
+  pages = {575--642},
+  year = {2025},
+  doi = {10.1090/bull/1869},
+  eprint = {2406.01324v2},
+  archivePrefix = {arXiv},
+  primaryClass = {math.FA},
+  note = {Cited theorem numbers refer to arXiv:2406.01324v2, 29 November 2024},
+  url = {https://arxiv.org/abs/2406.01324v2}
+}
+
+@article{GuoShamaiVerdu2005,
+  author = {Guo, Dongning and Shamai (Shitz), Shlomo and Verd{\'u}, Sergio},
+  title = {Mutual information and minimum mean-square error in {Gaussian} channels},
+  journal = {IEEE Transactions on Information Theory},
+  volume = {51},
+  number = {4},
+  pages = {1261--1282},
+  year = {2005},
+  month = apr,
+  doi = {10.1109/TIT.2005.844072},
+  eprint = {cs/0412108v1},
+  archivePrefix = {arXiv},
+  note = {Cited theorem and equation numbers refer to arXiv:cs/0412108v1},
+  url = {https://arxiv.org/abs/cs/0412108v1}
+}
+
+@misc{Viaclovsky2004,
+  author = {Viaclovsky, Jeff},
+  title = {Differential Analysis, {MIT} 18.156: Lecture Notes},
+  year = {2004},
+  howpublished = {MIT OpenCourseWare, Spring 2004},
+  note = {Lectures 3, 10, 16, 18, 19, and 24; notes prepared from the instructor's handwritten notes by Zuoqin Wang (Lectures 0--11) and Yanir Rubinstein (Lectures 12--24)},
+  url = {https://ocw.mit.edu/courses/18-156-differential-analysis-spring-2004/pages/lecture-notes/}
+}
+
+@article{Bobkov1999,
+  author = {Bobkov, Sergey G.},
+  title = {Isoperimetric and analytic inequalities for log-concave probability measures},
+  journal = {The Annals of Probability},
+  volume = {27},
+  number = {4},
+  pages = {1903--1921},
+  year = {1999},
+  doi = {10.1214/aop/1022874820},
+  url = {https://doi.org/10.1214/aop/1022874820}
+}
+
+@article{Ito1951,
+  author = {It{\^o}, Kiyosi},
+  title = {Multiple {Wiener} integral},
+  journal = {Journal of the Mathematical Society of Japan},
+  volume = {3}, number = {1}, pages = {157--169}, year = {1951},
+  doi = {10.2969/jmsj/00310157}
+}
+
+@article{ArakiYamagami1981,
+  author = {Araki, Huzihiro and Yamagami, Shigeru},
+  title = {An inequality for {Hilbert--Schmidt} norm},
+  journal = {Communications in Mathematical Physics},
+  volume = {81}, number = {1}, pages = {89--96}, year = {1981},
+  doi = {10.1007/BF01941801}
+}
+
+@article{Beigi2013,
+  author = {Beigi, Salman},
+  title = {Sandwiched {R{\'e}nyi} divergence satisfies data processing inequality},
+  journal = {Journal of Mathematical Physics},
+  volume = {54}, number = {12}, pages = {122202}, year = {2013},
+  doi = {10.1063/1.4838855},
+  eprint = {1306.5920v6}, archivePrefix = {arXiv}, primaryClass = {quant-ph},
+  note = {Cited theorem numbers refer to arXiv:1306.5920v6},
+  url = {https://arxiv.org/abs/1306.5920v6}
+}
+
+@article{Tropp2018,
+  author = {Tropp, Joel A.},
+  title = {Second-order matrix concentration inequalities},
+  journal = {Applied and Computational Harmonic Analysis},
+  volume = {44}, number = {3}, pages = {700--736}, year = {2018},
+  doi = {10.1016/j.acha.2016.07.005},
+  eprint = {1504.05919}, archivePrefix = {arXiv}, primaryClass = {math.PR},
+  note = {Cited section and lemma numbers refer to the author preprint revised 3 August 2016},
+  url = {https://tropp.caltech.edu/papers/Tro16-Second-Order-Matrix-preprint.pdf}
+}
+
+@article{Milman2010,
+  author = {Milman, Emanuel},
+  title = {Isoperimetric and concentration inequalities: Equivalence under curvature lower bound},
+  journal = {Duke Mathematical Journal},
+  volume = {154}, number = {2}, pages = {207--239}, year = {2010},
+  doi = {10.1215/00127094-2010-038},
+  eprint = {0902.1560v2}, archivePrefix = {arXiv}, primaryClass = {math.DG},
+  note = {Cited theorem numbers refer to arXiv:0902.1560v2, 1 December 2009},
+  url = {https://arxiv.org/abs/0902.1560v2}
+}
+
+@article{KlartagPutterman2023,
+  author = {Klartag, Bo'az and Putterman, Eli},
+  title = {Spectral monotonicity under {Gaussian} convolution},
+  journal = {Annales de la Facult\'e des Sciences de Toulouse. Math\'ematiques},
+  series = {6}, volume = {32}, number = {5}, pages = {939--967}, year = {2023},
+  doi = {10.5802/afst.1759},
+  url = {https://www.numdam.org/item/10.5802/afst.1759.pdf}
+}
+
+@article{KimMilman2012,
+  author = {Kim, Young-Heon and Milman, Emanuel},
+  title = {A generalization of {Caffarelli}'s contraction theorem via (reverse) heat flow},
+  journal = {Mathematische Annalen},
+  volume = {354}, number = {3}, pages = {827--862}, year = {2012},
+  doi = {10.1007/s00208-011-0749-x},
+  eprint = {1002.0373v3}, archivePrefix = {arXiv}, primaryClass = {math.FA},
+  url = {https://arxiv.org/abs/1002.0373v3}
+}
+
+@article{EldanKoehlerZeitouni2022,
+  author = {Eldan, Ronen and Koehler, Frederic and Zeitouni, Ofer},
+  title = {A spectral condition for spectral gap: fast mixing in high-temperature {Ising} models},
+  journal = {Probability Theory and Related Fields},
+  volume = {182}, number = {3-4}, pages = {1035--1051}, year = {2022},
+  doi = {10.1007/s00440-021-01085-x},
+  eprint = {2007.08200v2}, archivePrefix = {arXiv}, primaryClass = {math.PR},
+  note = {Cited theorem and equation numbers refer to arXiv:2007.08200v2, 6 August 2021},
+  url = {https://arxiv.org/abs/2007.08200v2}
+}
+
+@article{ChenEldan2022,
+  author = {Chen, Yuansi and Eldan, Ronen},
+  title = {Localization schemes: A framework for proving mixing bounds for {Markov} chains},
+  journal = {Duke Mathematical Journal},
+  volume = {174}, number = {8}, year = {2025},
+  doi = {10.1215/00127094-2024-0063},
+  eprint = {2203.04163v2}, archivePrefix = {arXiv}, primaryClass = {math.PR},
+  note = {Cited section and equation numbers refer to arXiv:2203.04163v2, 6 June 2022},
+  url = {https://arxiv.org/abs/2203.04163v2}
+}
+
+@article{Lieb1973,
+  author = {Lieb, Elliott H.},
+  title = {Convex trace functions and the {Wigner--Yanase--Dyson} conjecture},
+  journal = {Advances in Mathematics},
+  volume = {11}, number = {3}, pages = {267--288}, year = {1973},
+  doi = {10.1016/0001-8708(73)90011-X},
+  url = {https://www.numdam.org/item/RCP25_1973__19__A4_0/}
+}
+
+@article{Rothaus1986,
+  author = {Rothaus, O. S.},
+  title = {Hypercontractivity and the {Bakry--Emery} criterion for compact {Lie} groups},
+  journal = {Journal of Functional Analysis},
+  volume = {65}, number = {3}, pages = {358--367}, year = {1986},
+  doi = {10.1016/0022-1236(86)90025-X}
+}
diff --git a/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/build/sections/01-introduction.tex b/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/build/sections/01-introduction.tex
new file mode 100644
index 0000000000000000000000000000000000000000..17e26d7f6710d986241c1d65d4daae1b6caff506
--- /dev/null
+++ b/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/build/sections/01-introduction.tex
@@ -0,0 +1,254 @@
+\section{Introduction}\label{sec:introduction}
+
+A logarithmic Sobolev inequality controls the entropy of a function by
+its Dirichlet energy.  Besides implying a Poincar\'e inequality, it gives
+Gaussian concentration for Lipschitz functions.  For a log-concave
+measure, a natural question is whether Gaussian tails of all
+\emph{linear} functions already force this stronger conclusion.
+We prove that they do, with a constant independent of dimension.
+
+For a probability measure \(\nu\) and a nonnegative integrable function \(g\),
+write
+\[
+ \Ent_\nu(g)=\int g\log g\,d\nu
+       -\left(\int g\,d\nu\right)\log\left(\int g\,d\nu\right),
+\]
+with \(0\log0=0\).  A Lebesgue density \(\rho\) on \(\R^n\) is log-concave
+if its positive set is convex and \(\log\rho\) is concave there.
+We call a centered random vector \(X\) \emph{linearly \(a\)-subgaussian} if
+\begin{equation}\label{eq:intro-subgaussian}
+ \sup_{|\theta|=1}\E\exp\!\left(\frac{\langle X,\theta\rangle^2}{a^2}\right)
+ \le2.
+\end{equation}
+This definition concerns only linear functions of \(X\).  We write \(D\)
+for the Euclidean gradient.
+
+\begin{theorem}\label{thm:main}
+There is an absolute constant \(C<\infty\) with the following property.
+For every \(n\ge1\), every centered log-concave probability measure
+\(\mu\) with a Lebesgue density on \(\R^n\), and every \(a>0\)
+satisfying \eqref{eq:intro-subgaussian} for \(X\sim\mu\),
+\[
+ \Ent_\mu(f^2)\le C a^2\int_{\R^n}|Df|^2\,d\mu
+ \qquad\text{for every }f\in C_c^\infty(\R^n).
+\]
+The same constant works in every dimension.
+\end{theorem}
+
+No smoothness of the density or positive lower curvature bound is required.
+In particular, the theorem includes uniform measures on convex bodies.
+The proof will use smooth positive densities only after a reduction that
+preserves any hypothetical sequence of counterexamples.
+
+The Otto--Villani implication from logarithmic Sobolev to quadratic
+transport--entropy inequalities \cite[Theorem~1]{OttoVillani2000}
+gives the following standard consequence, with the same constant.
+Write $\mathcal P_2(\R^n)$ for the Borel probabilities with finite
+second moment. For $\alpha,\beta\in\mathcal P_2(\R^n)$, set
+\[
+ W_2(\alpha,\beta)^2
+ =\inf_{\pi\in\Pi(\alpha,\beta)}\int|x-y|^2\,d\pi(x,y),
+\]
+where $\Pi(\alpha,\beta)$ is the set of their couplings. Relative
+entropy is $H(\alpha\mid\beta)=\int f\log f\,d\beta$ if
+$\alpha=f\beta$, with its possible value $+\infty$, and is
+$+\infty$ if $\alpha\not\ll\beta$.
+
+\begin{corollary}[Quadratic transport--entropy inequality]
+\label{cor:intro-transport}
+Under the assumptions of Theorem~\ref{thm:main}, every
+$\nu\in\mathcal P_2(\R^n)$ satisfies
+\[
+ W_2(\nu,\mu)^2\le Ca^2 H(\nu\mid\mu),
+\]
+with the same absolute constant $C$ as in that theorem.
+\end{corollary}
+
+The proof in Section~\ref{sec:transport-consequence} gives the
+approximation argument for nonsmooth log-concave densities.
+
+\subsection{History and significance}
+
+Gross's work on logarithmic Sobolev inequalities connected the
+Gaussian inequality with hypercontractivity and established a
+framework in which the constants remain independent of dimension
+\cite{Gross1975}.  The diffusion approach of Bakry and \'Emery
+\cite{BakryEmery1985} gives the inequality for a density $e^{-V}$
+when $D^2V\ge\kappa\Id$ with $\kappa>0$: in the convention used
+below, its constant is at most $\kappa^{-1}$.
+Log-concavity alone asserts only $D^2V\ge0$ when the potential is
+smooth.  The question addressed here is whether a global assumption
+on the tails of linear functions can supply the logarithmic Sobolev
+bound without positive curvature.
+
+A logarithmic Sobolev inequality implies Gaussian concentration for
+all Lipschitz functions through Herbst's exponential-moment argument;
+see \cite[Section~2.3]{Ledoux1999}.  Under log-concavity, Gaussian
+concentration for all Lipschitz functions also gives Gaussian
+isoperimetry by Milman's theorem \cite[Theorem~1.1]{Milman2010},
+and hence a logarithmic Sobolev inequality
+\cite[Lemma~3.3]{Bobkov1999}, with dimension-independent bounds.
+The question here asks whether control of linear functions suffices.
+Uniformly subgaussian linear marginals are necessary, up to universal
+constants, by the same Herbst argument.
+Bizeul formulated their sufficiency within the log-concave class in 2023
+\cite[Conjecture~2]{Bizeul2026}.  Theorem~\ref{thm:main} resolves
+this conjecture for centered log-concave laws with a Lebesgue density,
+under the precise marginal condition
+\eqref{eq:intro-subgaussian}.
+
+Bobkov's norm-based criterion
+\cite[Theorem~1.3]{Bobkov1999} bounds the entropy coefficient by
+a universal multiple of the squared subgaussian norm of $|X|$.
+Under \eqref{eq:intro-subgaussian}, H\"older's inequality gives
+$\E\exp(|X|^2/(na^2))\le2$, so that criterion yields a coefficient
+of order $na^2$.  Bizeul's improvement
+\cite[Theorem~4]{Bizeul2026} has dimension dependence $n^{1/4}$
+in the parameter $\rho_{\mathrm{LS}}$ defined by
+$\Ent(f^2)\le2\rho_{\mathrm{LS}}^2\int|Df|^2$.
+Thus for linear subgaussian scale $a$ it gives an entropy coefficient
+of order $\sqrt n\,a^2$.
+Bizeul also obtained the dimension-free conclusion for rotationally
+invariant log-concave laws \cite[Theorem~8]{Bizeul2026}.
+Klartag and Lehec discuss the conjecture and these bounds in
+\cite[Section~8.3, Conjecture~76 and Theorem~77]{KlartagLehec2024}.
+The theorem's marginal hypothesis concerns more than covariance:
+it controls the tails of every linear function, while its conclusion
+controls the entropy of nonlinear tests.
+
+Our preliminary calculus uses the Bakry--\'Emery semigroup argument,
+including its entropy dissipation identity.  We give the anisotropic
+and approximation forms needed for Gaussian posteriors in
+Section~\ref{sec:calculus}.  The other classical input to that
+calculus comes from Gaussian channels.  Guo, Shamai, and Verd\'u
+relate the derivative of mutual information to minimum mean-square
+estimation error \cite[Theorem~2]{GuoShamaiVerdu2005}; their
+relative-entropy formulation \cite[Theorem~5, Equation~(66)]{GuoShamaiVerdu2005}
+gives the Gaussian prediction identity used here.  We prove its
+conditional integrated version with our variance and diffusion-time
+conventions.  The entropy-loss comparison in
+Lemma~\ref{lem:entropy-loss} requires an additional argument:
+repeated observations of independent copies make the cost of discarding
+conditional-mean directions negligible.  That comparison links
+Gaussian prediction to the tensor near-equalities described next.
+
+\subsection{Proof strategy}
+
+Write \(R(\nu)\) for the best constant in the convention
+\begin{equation}\label{eq:intro-lsi-convention}
+ \Ent_\nu(f^2)\le2R(\nu)\int|Df|^2\,d\nu.
+\end{equation}
+Failure of Theorem~\ref{thm:main} would give, after truncation,
+a weak quadratic tilt, and dilation, a sequence of compactly
+supported log-concave laws \(\mu\) such that
+\(R(\mu)=1\) while their linear subgaussian parameters tend to zero.
+The argument derives incompatible descriptions of fluctuations
+associated with these normalized laws.
+
+\paragraph{From an extremal function to a predictable vector field.}
+For a fixed \(s>0\), let \(G\) be a standard Gaussian independent of
+\(X\sim\mu\).  An extremizer for the entropy inequality of
+\(X+\sqrt sG\) yields a second probability law
+\(\eta\) and a vector field
+\[
+ w=D\log\frac{d\eta}{d\mu_s},
+ \qquad \mu_s=\Law(X+\sqrt s\,G).
+\]
+An exact change-of-reference identity shows that \(\eta\) itself
+satisfies a logarithmic Sobolev inequality.
+The Euler equation and translation tests make \(w\) an approximate
+first eigenfunction of the diffusion associated with \(\eta\).
+If the entropy constant instead coincides with the Poincar\'e
+constant, take \(\eta=\mu_s\) and \(w=D\varphi\), where
+\(\varphi\) is a gap eigenfunction.  Both cases are treated directly.
+
+The crucial extra information concerns prediction of \(w(Y)\) from
+a noisy observation of \(Y\sim\eta\).  With
+\(I=\E_\eta|w|^2\) and a fresh independent standard Gaussian \(G\),
+we prove the limiting identity
+\[
+ \frac{\E\big|\E[w(Y)\mid Y+\sqrt r\,G]\big|^2}{I}
+ \longrightarrow e^{-r}
+\]
+on a fixed positive interval of noise variances.
+It follows by matching entropy contraction with an entropy-loss
+comparison proved using repeated Gaussian observations.
+At each observation, we discard at most one direction: the difference
+between the conditional means under two competing priors.
+Independent copies make the accumulated cost of discarding these
+directions negligible.
+
+\paragraph{From prediction to tensor near-equalities.}
+Gaussian polynomial orthogonality turns the exact prediction curve
+into nearly sharp derivative estimates.
+Starting with \(T_0=w\), we construct symmetric tensor fields \(T_l\)
+whose derivatives approximate \(T_{l+1}\), all with asymptotically the
+same squared norm and all concentrated near the first nonzero spectral value
+of the diffusion.  The tensor estimates also control curvature and
+exclude concentration in the relevant small-dimensional subspaces.
+
+For each tensor \(T_l\), regard one index as a row and all remaining
+indices as a column, and form the positive matrix \(T_lT_l^*\).
+Averaging \(I^{-1}T_lT_l^*\) over a long band of levels gives a
+positive matrix field \(\rho\) of asymptotic mean trace one.  Its square root is nearly an equality case in
+Hilbert--Schmidt Poincar\'e.  Symmetry in the many tensor indices
+forces the infinitesimal matrix coefficients to commute
+asymptotically in a weighted Hilbert--Schmidt norm.  Gaussian linear combinations of these matrices, frozen at the initial
+diffusion point, consequently have scalar Gaussian moments under the
+weighted trace.  A Hermite expansion along the stationary diffusion
+then gives covariance fluctuations with the lognormal factor
+\[
+ \exp(\sqrt{8t}\,Z-4t),\qquad Z\sim N(0,1),
+\]
+whose mean is one.
+
+\paragraph{An information contradiction.}
+Conditionally on the stationary point, observe a centered Gaussian
+vector whose covariance is a bounded increasing function of
+\(T\rho\), where \(T>0\) is a scale and the covariance cap is
+adjustable.  Its marginal law is a mixture of these Gaussian laws.
+Together with the square-root energy estimates, the logarithmic Sobolev
+inequality of \(\eta\) gives a universal asymptotic upper bound on the
+mutual information, integrated against \(dT/T^2\).
+The matrix fluctuation law gives a lower bound by a one-dimensional
+Gaussian mixture with the lognormal variance above.
+After taking the sequence limit at fixed diffusion time and covariance
+cap, that lower bound becomes arbitrarily large as these two
+parameters increase together.
+
+The scale integral is essential.  In a spectral block where
+\(\rho\) has size \(\beta^2\), the substitution \(x=T\beta^2\) gives
+\[
+ \frac{dT}{T^2}=\beta^2\frac{dx}{x^2}.
+\]
+This is exactly the coefficient multiplying the unnormalized trace on
+that block.
+Consequently the information lower bound retains all of the matrix
+mass even when the spectrum is arbitrarily diffuse.
+
+Figure~\ref{fig:proof-dependencies} records the two constructions of the
+critical law and the main dependencies of the information comparison.
+\input{figures/proof-dependencies}
+
+\subsection{Components of the argument}
+
+Two ingredients have formulations beyond the final contradiction.
+The entropy-loss comparison of Lemma~\ref{lem:entropy-loss} relates
+relative entropy under common Gaussian convolution to conditional
+prediction of the original relative score.  Its proof separates
+the convexity input from the finite-copy observation argument.
+The matrix reduction in Section~\ref{sec:matrices} converts symmetric
+tensor near-equalities into a weighted scalar fluctuation law,
+without requiring simultaneous diagonalization in varying
+dimensions.  Section~\ref{sec:observations} then tests that law through
+bounded covariance observations and integrates over scale.
+
+For orientation, Section~\ref{sec:calculus} records the analytic
+conventions and approximation facts.
+Section~\ref{sec:critical} constructs the critical laws and vector
+fields.
+Section~\ref{sec:prediction} proves the prediction identity,
+and Section~\ref{sec:hierarchy} constructs the tensor hierarchy.
+Sections~\ref{sec:matrices} and \ref{sec:observations} turn that
+hierarchy into the contradiction.
+
diff --git a/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/build/sections/02-calculus.tex b/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/build/sections/02-calculus.tex
new file mode 100644
index 0000000000000000000000000000000000000000..48770caba4a889bc6fc5afbc72ab29dfea4bb52e
--- /dev/null
+++ b/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/build/sections/02-calculus.tex
@@ -0,0 +1,376 @@
+\section{Curvature, diffusion, and entropy calculus}\label{sec:calculus}
+
+We first establish the analytic facts used to construct and study the
+probability measures in the proof. Two time conventions will be kept
+distinct: convolution with a Gaussian of variance $r$ has generator
+$\Delta/2$, whereas a stationary diffusion has generator $-A$ and noise
+coefficient $\sqrt2$. These conventions determine every entropy
+dissipation factor below.
+
+For probabilities $p,\pi$ with $p\ll\pi$, define their relative entropy by
+\[
+ H(p\mid\pi)=\int\log\frac{dp}{d\pi}\,dp.
+\]
+Set $H(p\mid\pi)=+\infty$ when absolute continuity fails. Write
+$\gamma$ for standard Gaussian measure in the relevant dimension, and set
+\[
+ \Ical(p\mid\pi)=4\int\left|D\sqrt{\frac{dp}{d\pi}}\right|^2d\pi.
+\]
+The value is $+\infty$ unless the density exists and its square root
+belongs to the weighted Sobolev space. When the densities are positive
+and smooth this is $\int|D\log(dp/d\pi)|^2dp$. Our convention is
+\[
+ \Ent_\pi(f^2)\le 2R(\pi)\int|Df|^2d\pi,
+ \qquad 2H(p\mid\pi)\le R(\pi)\Ical(p\mid\pi).
+\]
+The Poincar\'e constant $R^P(\pi)$ uses the convention
+\[
+ \Var_\pi(f)=\int f^2d\pi-\left(\int f\,d\pi\right)^2
+ \le R^P(\pi)\int|Df|^2d\pi.
+\]
+Linearization of the entropy inequality gives $R^P(\pi)\le R(\pi)$.
+
+For smooth positive densities the Sobolev space is the completion of
+$C_c^\infty$ in the squared norm $\int(f^2+|Df|^2)d\pi$.
+Cutoffs, truncation of the values, and smoothing on compact sets show
+that it also consists of the functions with a weak gradient in this
+norm. The same approximation and lower semicontinuity of entropy
+extend a compact-test logarithmic Sobolev inequality to this space.
+One convenient proof of the latter lower semicontinuity is the formula
+\begin{equation}\label{eq:calculus-entropy-duality}
+ H(p\mid\pi)=\sup_{g\ \mathrm{bounded}}
+ \left\{\int g\,dp-\log\int e^g d\pi\right\}.
+\end{equation}
+Jensen's inequality proves the upper bound; bounded truncations of
+$\log(dp/d\pi)$ give equality. The same formula implies data
+processing under a measurable map or a Markov kernel, by conditional
+Jensen. Conditional entropy identities below are always averaged over
+the conditioning variable. In particular, disintegrating a joint
+density and splitting its logarithm gives the chain rule
+\[
+ H(P_{U,Z}\mid Q_U\otimes\nu)
+ =H(P_U\mid Q_U)+\E_{P_U}H(P_{Z\mid U}\mid\nu).
+\]
+It holds in the extended nonnegative sense by truncation. Mutual
+information is $\MI(U;Z)=H(P_{U,Z}\mid P_U\otimes P_Z)$.
+Taking $g(z)=v\cdot z$ in Equation~\eqref{eq:calculus-entropy-duality}
+and optimizing over $v$ also gives
+$H(p\mid\gamma)\ge|\E_pZ|^2/2$ when the mean exists.
+
+The following is the anisotropic Bakry--\'Emery curvature criterion
+\cite{BakryEmery1985}.  We include its semigroup proof and the convex
+approximation needed for Gaussian posteriors with nonsmooth support.
+
+\begin{lemma}[Positive curvature]\label{lem:curvature}
+Let $S$ be a positive definite matrix on $\R^n$, and let $U$ be a
+proper lower semicontinuous convex function, possibly taking the value
+$+\infty$. Suppose
+\[
+ d\pi(x)=Z^{-1}\exp\{-\tfrac12\langle x,Sx\rangle-U(x)\}\,dx
+\]
+is a probability with a full dimensional support. Then
+\[
+ \Ent_\pi(f^2)\le2\E_\pi\langle Df,S^{-1}Df\rangle,
+ \qquad
+ \Var_\pi(f)\le\E_\pi\langle Df,S^{-1}Df\rangle.
+\]
+The inequalities hold first for compact smooth tests, and then for
+functions obtained by finite-energy approximation.
+\end{lemma}
+
+\begin{proof}
+A linear change of variables reduces the assertion to $S=\Id$.
+First suppose $V(x)=|x|^2/2+U(x)$ is smooth with bounded Hessian.
+The diffusion with generator $L=\Delta-DV\cdot D$ has globally
+Lipschitz drift and invariant law $\pi$. If $J_t$ is the derivative
+of its trajectory with respect to the starting point, then
+$\dot J_t=-D^2V(X_t)J_t$, whence $\|J_t\|_{\op}\le e^{-t}$.
+For smooth bounded positive $q$, bounded away from zero,
+weighted Cauchy--Schwarz gives
+\[
+ \frac{|DP_tq|^2}{P_tq}
+ \le e^{-2t}P_t\left(\frac{|Dq|^2}{q}\right).
+\]
+Integration by parts yields
+\[
+ \frac{d}{dt}\int (P_tq)\log(P_tq)\,d\pi
+ =-\int\frac{|DP_tq|^2}{P_tq}\,d\pi.
+\]
+Couple a trajectory started at $x$ to an independent stationary
+initial point with the same driving Brownian motion. Their distance
+is at most $e^{-t}$ times their initial distance. Since $\pi$ has
+Gaussian tails, this proves convergence of $P_tq$ to $\E_\pi q$
+for bounded Lipschitz $q$, and hence convergence of its entropy to
+zero. Integrating the preceding two displays gives
+$\Ent_\pi(q)\le\frac12\int|Dq|^2/q\,d\pi$. Apply this to
+$q=f^2+\varepsilon$ and let $\varepsilon\downarrow0$.
+
+For general $U$, take its quadratic infimal convolutions
+\[
+ U_j(x)=\inf_z\{U(z)+j|x-z|^2\}.
+\]
+The minimizer is unique, and monotonicity of the subgradient of $U$
+shows that $DU_j$ is Lipschitz. The functions $U_j$ are convex and
+increase to $U$. A fixed affine minorant $U(z)\ge\ell\cdot z+c$
+gives $U_j(x)\ge\ell\cdot x+c-|\ell|^2/(4j)$.
+Mollify $U_j$ with a smooth compactly supported kernel at radii
+tending to zero. The resulting functions are smooth, convex, and
+have bounded Hessian; the affine lower bound persists with a
+uniform additive error. Consequently their normalized densities
+converge almost everywhere and in $L^1$, dominated by an integrable
+Gaussian times an exponential of a linear function. The compact-test
+inequality passes to the limit. Boundary values of $U$ do not matter,
+since the boundary of a full dimensional convex support has
+Lebesgue measure zero. Finally, insert $f=1+\varepsilon g$,
+using finite-energy cutoffs of the constant, and compare the
+coefficients of $\varepsilon^2$ to obtain the variance inequality.
+\end{proof}
+
+We shall also use two immediate consequences with their constants.
+If independent laws satisfy inequalities with quadratic energy
+forms $Q_1,Q_2$, their product satisfies the inequality with energy
+$Q_1+Q_2$. Indeed, decompose entropy into conditional entropy and
+the entropy of the fiber integral, and use
+\[
+ \left|D_x\left(\int f(x,y)^2d\pi_2(y)\right)^{1/2}\right|^2
+ \le\int|D_xf(x,y)|^2d\pi_2(y).
+\]
+Approximation removes zeros of the fiber norm. The variance version
+uses conditional variance and differentiation of the fiber mean.
+In particular, independent addition gives
+\begin{equation}\label{eq:calculus-convolution}
+ R(\Law(X+Z))\le R(\Law(X))+R(\Law(Z)).
+\end{equation}
+Second, Herbst's argument (see \cite[Section~2.3]{Ledoux1999})
+shows that, if $R(\pi)\le R$, every integrable $1$-Lipschitz function
+$u$ satisfies
+\begin{equation}\label{eq:calculus-herbst}
+ \log\E_\pi e^{t(u-\E_\pi u)}\le Rt^2/2.
+\end{equation}
+For a bounded smooth $u$, applying the inequality to $e^{tu/2}$,
+and writing $\psi(t)=\log\E e^{tu}$, gives
+$t\psi'(t)-\psi(t)\le Rt^2/2$. Integrate the derivative of
+$\psi(t)/t$ from zero. Bounded Lipschitz truncations and Fatou's
+lemma prove the assertion in general. In particular the conclusion
+applies to linear forms whenever their first moments are finite.
+
+\begin{lemma}[Dirichlet operator and stationary diffusion]
+\label{lem:dirichlet-core}
+Let $\pi$ be a smooth, strictly positive probability density on
+$\R^n$. On $C_c^\infty(\R^n)$ define
+\[
+ A_\pi=-\Delta-D\log\pi\cdot D.
+\]
+This nonnegative symmetric operator is essentially self-adjoint in
+$L^2(\pi)$, and $C_c^\infty$ is a core for its closure. In
+particular, if $f\in L^2(\pi)$ and the distribution $A_\pi f$
+belongs to $L^2(\pi)$, then $f\in\Dom(A_\pi)$. Its semigroup
+$P_u=e^{-uA_\pi}$ preserves constants and is realized by the
+stationary diffusion
+\begin{equation}\label{eq:calculus-diffusion}
+ dY_u=\sqrt2\,dB_u+D\log\pi(Y_u)\,du,
+ \qquad Y_0\sim\pi.
+\end{equation}
+For $f\in\Dom(A_\pi)$ the stationary It\^o formula holds in $L^2$,
+with drift $-A_\pi f$ and martingale integrand $\sqrt2\,Df$.
+\end{lemma}
+
+\begin{proof}
+Integration by parts gives
+$\langle f,A_\pi g\rangle=\int Df\cdot Dg\,d\pi$.
+If $v\in L^2(\pi)$ is orthogonal to
+$(1+A_\pi)C_c^\infty$, then $(1+A_\pi)v=0$
+distributionally.  Conjugating by $\sqrt\pi$ reduces this to a
+Poisson equation with a smooth local potential.  The local Newtonian
+potential estimates and bootstrap make $v$ smooth; see
+\cite[Lecture~24, p.~2; Lecture~16, p.~2]{Viaclovsky2004}. Take a compact cutoff $\chi_R$, equal to one on the ball
+of radius $R$, with $|D\chi_R|\le C/R$. Testing by
+$\chi_R^2v$ and using $2ab\le a^2/2+2b^2$ gives
+\[
+ \int\chi_R^2v^2d\pi+\tfrac12\int\chi_R^2|Dv|^2d\pi
+ \le2\int|D\chi_R|^2v^2d\pi.
+\]
+Thus $v=0$. The range of $1+A_\pi$ is dense. Nonnegativity
+gives $\|(1+A_\pi)f\|_2\ge\|f\|_2$, so the range of the
+closure is also closed and therefore equals $L^2$. The closure is
+self-adjoint: for $g$ in its adjoint domain, solve
+$(1+\overline A)h=(1+A^*)g$; then $g-h$ is in the orthogonal
+complement of the range of $1+A$, hence vanishes. This also proves
+the asserted maximal distributional domain characterization.
+
+The cutoffs $\chi_R$ converge to $1$ in the form norm with energy
+tending to zero. Hence $1$ belongs to the nullspace of the closed
+form and $P_u1=1$. To identify the diffusion, solve the stochastic
+equation locally and kill it on exiting each ball. Its kernels are
+symmetric and sub-Markov, as follows either from the Dirichlet
+heat equation or integration by parts. Increasing the balls gives
+the minimal sub-Markov semigroup. The stopped It\^o formula on
+compact tests identifies its generator as an extension of $-A_\pi$.
+Essential self-adjointness identifies it with $P_u$, which loses
+no mass. Started in $\pi$, the resulting diffusion is stationary
+and nonexplosive. Finally approximate $f$ in graph norm by compact
+smooth functions. Their gradients converge in $L^2(\pi)$ since
+$\|Dg\|_2^2=\langle g,Ag\rangle$. Stationarity and the
+It\^o isometry pass every term of their formulas to the limit.
+\end{proof}
+
+\begin{lemma}[Entropy dissipation]\label{lem:entropy-calculus}
+For the semigroup of Lemma~\ref{lem:dirichlet-core}, if
+$p_0=q_0\pi$ has finite relative entropy and
+$p_u=(P_uq_0)\pi$, then, for $0\le a\le b$,
+\begin{equation}\label{eq:calculus-stationary-entropy}
+ H(p_a\mid\pi)-H(p_b\mid\pi)
+ \ge\int_a^b\Ical(p_u\mid\pi)\,du.
+\end{equation}
+If instead both probability measures evolve by Gaussian convolution,
+$p_r=p_0*\gamma_r$ and $q_r=q_0*\gamma_r$, then
+\begin{equation}\label{eq:calculus-heat-entropy}
+ H(p_a\mid q_a)-H(p_b\mid q_b)
+ \ge\frac12\int_a^b\Ical(p_r\mid q_r)\,dr
+\end{equation}
+whenever the entropy on the left at time $a$ is finite. Here
+$\gamma_r$ is the centered Gaussian with covariance $r\Id$,
+and time zero is allowed. For the standard Gaussian
+Ornstein--Uhlenbeck semigroup, Equation~\eqref{eq:calculus-stationary-entropy}
+is an equality, including time zero when the initial entropy is
+finite. Each assertion also holds after integration over conditioning
+labels whose noises are independent conditional on those labels.
+\end{lemma}
+
+\begin{proof}
+For a bounded density $q$ bounded away from zero, the form chain
+rule, first for smooth functions and then by form approximation,
+gives
+\[
+ \frac{d}{du}\int(P_uq)\log(P_uq)\,d\pi
+ =-\int\frac{|DP_uq|^2}{P_uq}\,d\pi.
+\]
+Spectral smoothing puts $P_uq$ in the operator domain for $u>0$;
+the bounded range permits $1+\log(P_uq)$ as a form test.
+Approximate an arbitrary finite-entropy density by truncating it
+above and below and normalizing. The initial entropies converge.
+The semigroup contracts $L^1$, so densities converge at every
+later time. Lower semicontinuity of entropy and of
+$4\int|D\sqrt q|^2d\pi$, followed by Fatou's lemma in time,
+gives Equation~\eqref{eq:calculus-stationary-entropy} from time zero.
+Restarting the argument gives arbitrary $a$.
+
+For simultaneous heat evolution put $z=p_r/q_r$. Direct
+differentiation, using $\partial_r p_r=\Delta p_r/2$ and the
+same equation for $q_r$, gives
+\begin{equation}\label{eq:calculus-heat-chain}
+ (\partial_r-\tfrac12\Delta)\{q_r\Phi(z)\}
+ =-\tfrac12q_r\Phi''(z)|Dz|^2.
+\end{equation}
+Here use smooth convex $\Phi_k$ with
+$\Phi_k(1)=\Phi_k'(1)=0$, with second derivatives increasing to
+$1/z$, and with second derivatives supported in compact subsets
+of $(0,\infty)$. Such functions have at most linear growth and
+increase to $z\log z-z+1$. Integrate
+Equation~\eqref{eq:calculus-heat-chain} against $\chi_R$ with
+$|\Delta\chi_R|\le C/R^2$. At each fixed $k$ the cutoff error
+vanishes because $q_r\Phi_k(p_r/q_r)\le C_k(p_r+q_r)$.
+Let $R\to\infty$ and then $k\to\infty$. Monotone convergence
+gives the entropy and information terms, and in particular the
+stated inequality on positive time intervals. Data processing
+gives $H(p_\varepsilon\mid q_\varepsilon)\le H(p_0\mid q_0)$;
+let $\varepsilon\downarrow0$ to include the initial time.
+
+For Ornstein--Uhlenbeck evolution relative to $\gamma$, apply
+the same convex truncations to the ratio. Its generator is
+$\Delta-x\cdot D$. Expanding radial cutoffs have bounded
+generator, tending pointwise to zero, so the error vanishes by
+domination at fixed $k$. This gives equality on positive time
+intervals. At time zero, relative entropy is continuous: data
+processing gives its upper bound and lower semicontinuity under
+weak convergence gives its lower bound. Monotone convergence of
+the nonnegative time integral includes zero. Tonelli's theorem
+proves all conditional versions.
+\end{proof}
+
+We use the relative-entropy form of the Gaussian-channel identity of
+Guo, Shamai, and Verd\'u \cite[Theorem~5, Equation~(66)]{GuoShamaiVerdu2005}.
+The proof records explicitly its conditional integrated form.
+
+\begin{lemma}[Gaussian signals]\label{lem:gaussian-signals}
+Let $S$ be an $\R^d$-valued random vector with finite second
+moment, let $U$ be any side information, and let $G$ be standard
+Gaussian independent of $(S,U)$. Write $Z_t=\sqrt tS+G$ and
+\[
+ d(t)=\E_U H(\Law(Z_t\mid U)\mid\gamma).
+\]
+Then $d(0)=0$, $d(t)\le t\E|S|^2/2$, and
+\begin{equation}\label{eq:calculus-gaussian-signal}
+ d(t)=\frac12\int_0^t
+ \E\left|\E[S\mid Z_v,U]\right|^2dv.
+\end{equation}
+The integrand is nondecreasing in $v$. The formula without side
+information is obtained by taking $U$ constant.
+\end{lemma}
+
+\begin{proof}
+Conditional convexity of relative entropy and
+$H(\gamma(\cdot-\sqrt t s)\mid\gamma)=t|s|^2/2$
+give the upper bound. Gaussian kernel differentiation gives
+\[
+ D_z\log\frac{d\Law(Z_t\mid U)}{d\gamma}(z)
+ =\sqrt t\,\E[S\mid Z_t=z,U].
+\]
+The differentiation is justified at each positive $t$ by the
+Gaussian kernel and the finite second moment, or by truncating
+$S$ and passing locally and in the conditional squared norm.
+Applying Ornstein--Uhlenbeck evolution for time $u$ replaces $t$
+by $te^{-2u}$. Its entropy dissipation identity and
+$dt/du=-2t$ give $d'(t)=\frac12\E|\E[S\mid Z_t,U]|^2$
+almost everywhere on $(0,\infty)$. The upper bound implies
+$d(t)\to0$ at zero, proving the integrated identity. For
+$0<v<t$, the channel at $v$ is obtained from the channel at $t$
+by multiplying its output by $\sqrt{v/t}$ and adding fresh
+Gaussian noise of covariance $(1-v/t)\Id$. Conditional Jensen
+then proves monotonicity of prediction energy.
+\end{proof}
+
+\subsection{The transport--entropy consequence}
+\label{sec:transport-consequence}
+
+\begin{proof}[Proof of Corollary~\ref{cor:intro-transport}]
+Jensen's inequality in each coordinate gives
+$\int|x|^2d\mu\le na^2\log2$, so $\mu\in\mathcal P_2(\R^n)$.
+Only the case $H(\nu\mid\mu)<\infty$ needs proof. For $s>0$, put
+$\gamma_s=N(0,s\Id)$, $\mu_s=\mu*\gamma_s$, and
+$\nu_s=\nu*\gamma_s$. The density of $\mu_s$ is smooth and strictly
+positive. For $X\sim\mu$ and an independent standard Gaussian $G$,
+its potential $V_s$ satisfies
+\[
+ D^2V_s(y)=s^{-1}\Id-s^{-2}\Cov(X\mid X+\sqrt sG=y)\ge0.
+\]
+Indeed, the posterior has the original convex potential plus
+$|x-y|^2/(2s)$, so Lemma~\ref{lem:curvature} bounds its covariance
+by $s\Id$. Gaussian differentiation is valid without compact support.
+Thus $\mu_s$ is centered and log-concave, with a smooth convex potential.
+
+Set $\beta_s=\sqrt{8s/3}$ and $a_s=a+\beta_s$. For a scalar standard
+Gaussian $Z$, $\E\exp(sZ^2/\beta_s^2)=2$. The inequality
+\[
+ \frac{(u+t)^2}{(a+\beta_s)^2}
+ \le\frac{a}{a+\beta_s}\frac{u^2}{a^2}
+     +\frac{\beta_s}{a+\beta_s}\frac{t^2}{\beta_s^2}
+\]
+and H\"older's inequality show that $\mu_s$ satisfies
+\eqref{eq:intro-subgaussian} with parameter $a_s$.
+Theorem~\ref{thm:main}, with the Sobolev extension in
+Section~\ref{sec:calculus}, gives entropy coefficient $Ca_s^2$.
+In the conventions of \cite[Definitions~1--2]{OttoVillani2000},
+$\Ent(f^2)\le L\int|Df|^2$ is $\mathrm{LSI}(2/L)$, whose
+transport conclusion is $W_2^2\le LH$. Their Theorem~1 applies to
+the smooth convex potential of $\mu_s$, and hence
+\[
+ W_2(\nu_s,\mu_s)^2\le Ca_s^2H(\nu_s\mid\mu_s)
+ \le Ca_s^2H(\nu\mid\mu).
+\]
+The last inequality is data processing for the common Gaussian
+kernel, as in Section~\ref{sec:calculus}. Coupling a vector with
+itself plus independent Gaussian noise gives
+$W_2(\mu_s,\mu),W_2(\nu_s,\nu)\le\sqrt{ns}$.
+Letting $s\downarrow0$ at fixed $n$, with $a_s\to a$, proves the claim.
+\end{proof}
diff --git a/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/build/sections/03-critical.tex b/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/build/sections/03-critical.tex
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@@ -0,0 +1,815 @@
+\section{Critical laws after Gaussian regularization}\label{sec:critical}
+
+Suppose that the conclusion of Theorem~\ref{thm:main} is false.
+This section constructs, from a sequence of counterexamples, smooth
+probabilities $\eta$ satisfying a logarithmic Sobolev inequality and
+vector fields $w$ that nearly satisfy $A_\eta w=w$. There are two
+possible constructions. An attained nonlinear entropy quotient gives
+a score $w$; equality of the logarithmic Sobolev and Poincar\'e
+constants gives the gradient of a gap eigenfunction. We shall retain
+both alternatives throughout.
+
+\subsection{Normalization and Gaussian regularization}
+
+\begin{lemma}[Normalized counterexamples]\label{lem:critical-normalization}
+Failure of Theorem~\ref{thm:main} produces a sequence of centered,
+compactly supported, full dimensional log-concave probabilities $\mu$
+such that
+\begin{equation}\label{eq:critical-normalization}
+ R(\mu)=1,\qquad
+ \sup_{|\theta|=1}\E_\mu e^{\langle X,\theta\rangle^2/a^2}
+ \le2,\qquad a\longrightarrow0.
+\end{equation}
+For an absolute constant $C$, they also satisfy
+\begin{equation}\label{eq:critical-linear-mgf}
+ \log\E_\mu e^{v\cdot X}\le Ca^2|v|^2,
+ \qquad \Cov_\mu(X)\le Ca^2\Id.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Choose laws and compact smooth tests for which the entropy-to-energy
+ratio divided by the squared marginal parameter tends to infinity.
+The tests may be chosen with positive energy: a smooth test of zero
+energy is constant on the connected interior of the convex support,
+and its entropy vanishes. Restriction of the law to a sufficiently
+large ball preserves each test quotient arbitrarily closely. Multiply
+the restricted density by $e^{-\varepsilon|x|^2/2}$ and normalize,
+with $\varepsilon>0$ sufficiently small to preserve that quotient
+again. Both operations preserve log-concavity, their combined density
+relative to the original law can be bounded by $2$, and
+Lemma~\ref{lem:curvature} gives a finite logarithmic Sobolev
+constant for the resulting compact law.
+
+A bounded density increase changes the exponential-square expectation
+by at most that factor. If $\E e^{Z^2/A^2}\le4$, Jensen gives
+$\E e^{Z^2/(2A^2)}\le2$. Centering costs only another absolute
+factor: $\E Z^2\le A^2\log4$, while
+$(Z-\E Z)^2\le2Z^2+2(\E Z)^2$; increase the denominator first
+to make the resulting exponential expectation bounded by an absolute
+constant, and once more by Jensen to make it at most $2$.
+These arguments apply separately to every unit linear form with
+the same constants. Translation preserves the test quotient. Thus
+the centered compact laws still have $R/A^2\to\infty$ with
+$0<R<\infty$. Dilation by $R^{-1/2}$ gives
+Equation~\eqref{eq:critical-normalization}.
+
+For completeness, if a centered scalar $Z$ obeys
+$\E e^{Z^2/a^2}\le2$, then
+$\E|Z|^k\le2a^k\Gamma(k/2+1)$, by integrating its tail.
+Expansion of $e^{tZ}$, with the linear term absent, therefore gives
+$\log\E e^{tZ}\le Ca^2t^2$ for $a|t|\le c$.
+For $a|t|\ge c$, use
+$tZ\le Z^2/a^2+a^2t^2/4$ and absorb $\log2$ into
+$Ca^2t^2$. Apply this to $Z=X\cdot\theta$. The second moment
+bound follows either from the exponential-square bound or by
+differentiating the centered moment generating function at zero.
+\end{proof}
+
+We suppress the counterexample index, but all limits in this section
+refer to this sequence. Let $G$ be a fresh standard Gaussian and set
+\[
+ \mu_s=\Law(X+\sqrt sG),\quad R_s=R(\mu_s),\quad
+ b_s(y)=\E[X\mid X+\sqrt sG=y],\quad V_s=-\log\mu_s.
+\]
+Here and below $\mu_s$ also denotes its strictly positive density.
+Gaussian differentiation and posterior covariance give
+\begin{equation}\label{eq:critical-posterior}
+ DV_s(y)=\frac{y-b_s(y)}s,\qquad
+ M_s(y)=Db_s(y)=\frac{\Cov(X\mid y)}s,\qquad
+ K_s(y)=D^2V_s(y)=\frac{\Id-M_s(y)}s.
+\end{equation}
+The posterior of $X$ has a convex potential plus $|x-y|^2/(2s)$.
+Lemma~\ref{lem:curvature}, applied to its linear forms, therefore
+implies
+\begin{equation}\label{eq:critical-curvature}
+ 0\le M_s\le\Id,\qquad 0\le K_s\le s^{-1}\Id.
+\end{equation}
+All these functions are smooth. Compact support of $X$ also gives
+$|b_s|\le\sup_{\supp\mu}|x|$.
+
+The monotonicity below follows from the heat-flow contraction of
+Klartag and Putterman \cite[Theorem~1.2 and Section~3]{KlartagPutterman2023},
+which uses the transport construction of Kim and Milman
+\cite{KimMilman2012}. We recall the argument for our regularized laws.
+
+\begin{lemma}[Bounds for the regularized constants]
+\label{lem:critical-flow-bounds}
+The function $s\mapsto R_s$ is nondecreasing and $1$-Lipschitz on
+$(0,\infty)$, and
+\begin{equation}\label{eq:critical-basic-R}
+ 1\le R_s\le1+s.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+The heat equation for $\mu_s$ is the continuity equation with
+velocity $DV_s/2$. On a compact positive-time interval this velocity
+is globally Lipschitz and grows at most linearly, so its smooth flow
+is a diffeomorphism carrying $\mu_s$ to $\mu_t$. Its Jacobian
+satisfies $\dot J=K_tJ/2$. By positivity of $K_t$, the flow
+expands distances and its inverse is $1$-Lipschitz when $t\ge s$.
+Transport a test by that inverse. Its entropy is unchanged and
+its Dirichlet energy cannot increase, proving $R_t\ge R_s$.
+Equation~\eqref{eq:calculus-convolution} gives
+$R_{s+h}\le R_s+h$ and $R_s\le1+s$. For every compact smooth
+test, its entropy and energy converge to those under $\mu$ as
+$s\downarrow0$. Taking the supremum of its quotient gives
+$\liminf_{s\downarrow0}R_s\ge R(\mu)=1$. Monotonicity now
+proves the remaining bound.
+\end{proof}
+
+\subsection{Attainment above the Gaussian and Poincar\'e thresholds}
+
+On compact manifolds, Rothaus obtains nonconstant positive solutions
+of the logarithmic Euler equation when the sharp logarithmic Sobolev
+constant exceeds the Poincar\'e constant \cite[p.~358]{Rothaus1986}.
+The proof below must also control loss of entropy in the Gaussian tails.
+The next two lemmas establish this control and attainment. Their constants
+may depend on a single fixed law and its dimension. Uniform estimates
+along the counterexample sequence begin with the translation argument.
+
+\begin{lemma}[Compactness and the Gaussian entropy defect]
+\label{lem:critical-defect}
+Fix $s>0$ and a compactly supported probability $\mu$ on $\R^n$,
+and put $\nu=\mu*\gamma_s$. Its form domain embeds compactly
+into $L^2(\nu)$. For every $\varepsilon>0$ there is a finite
+$C_\varepsilon$, depending on this fixed law, such that
+\begin{equation}\label{eq:critical-defective}
+ \Ent_\nu(u^2)\le2(s+\varepsilon)\|Du\|_2^2
+                  +C_\varepsilon\|u\|_2^2.
+\end{equation}
+If $u_j\to u$ in $L^2(\nu)$, $\|u_j\|_2=1$, and
+$\|Du_j\|_2^2\to E_*<\infty$, then
+\begin{equation}\label{eq:critical-entropy-defect}
+ \limsup_j\Ent_\nu(u_j^2)
+ \le\Ent_\nu(u^2)+2s(E_*-\|Du\|_2^2).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Write $r=d\nu/d\gamma_s$. The Gaussian kernel formula gives
+$D\log r=b_s/s$, which is bounded, and hence
+$|\log r(y)|\le C(1+|y|)$. For $v=u\sqrt r$,
+\[
+ \Ent_\nu(u^2)=\Ent_{\gamma_s}(v^2)
+                         -\int v^2\log r\,d\gamma_s,
+ \qquad Dv=\sqrt r\,(Du+\tfrac12uD\log r).
+\]
+Gaussian integration by parts followed by Cauchy--Schwarz gives
+\[
+ \int |y|^2v^2d\gamma_s
+ =sn\|v\|_2^2+2s\int v\,y\cdot Dv\,d\gamma_s
+ \le2sn\|v\|_2^2+4s^2\|Dv\|_2^2.
+\]
+Initially this calculation is for compact tests, and cutoffs extend
+it by lower semicontinuity. The linear-growth entropy correction
+is consequently bounded by
+$\delta\|Dv\|_2^2+C_\delta\|v\|_2^2$ for every $\delta>0$.
+Gaussian logarithmic Sobolev and the displayed derivative formula,
+with another arbitrarily small Cauchy--Schwarz parameter, prove
+Equation~\eqref{eq:critical-defective}. The same moment estimate
+shows tightness of the $L^2(\nu)$ mass of any bounded form
+sequence. On a fixed ball, the smooth density is bounded above
+and away from zero, so the ordinary local Rellich theorem gives
+$L^2$ precompactness. A diagonal extraction and tightness prove
+compactness globally.
+
+We give the tail argument for the sharp coefficient $2s$ in
+Equation~\eqref{eq:critical-entropy-defect}. Put
+$v_{j,L}=(|u_j|-L)_+$ and $m_{j,L}=\|v_{j,L}\|_2^2$.
+Strong $L^2$ convergence implies
+$\lim_{L\to\infty}\sup_jm_{j,L}=0$. The energies of
+$v_{j,L}$ and $|u_j|\wedge L$ sum to $\|Du_j\|_2^2$.
+Lower semicontinuity for the clipped functions therefore gives
+\begin{equation}\label{eq:critical-tail-energy}
+ \lim_{L\to\infty}\limsup_j\|Dv_{j,L}\|_2^2
+ \le E_*-\|Du\|_2^2.
+\end{equation}
+For a nonnegative $v$ of squared norm $m\le1$, the negative
+part of $v^2\log v^2$ has integral at most $C\sqrt m$
+(use $-z\log z\le C\sqrt z$ on $0\le z\le1$).
+Since $m\log m\le0$, Equation~\eqref{eq:critical-defective}
+thus gives
+\[
+ \int v^2\log_+v^2d\nu
+ \le2(s+\varepsilon)\|Dv\|_2^2
+          +C_\varepsilon m+C\sqrt m.
+\]
+If $|u_j|>ML$, where $M>1$ is fixed, then
+$|u_j|\le c_M v_{j,L}$ with $c_M=M/(M-1)$. Consequently
+\[
+ u_j^2\log_+u_j^2
+ \le c_M^2v_{j,L}^2\log_+v_{j,L}^2
+                +c_M^2\log(c_M^2)v_{j,L}^2.
+\]
+The integrals of bounded truncations of $z^2\log z^2$ pass to
+the limit by strong $L^2$ convergence; its negative part does as
+well, being bounded and continuous. The last two bounds and
+Equation~\eqref{eq:critical-tail-energy}, with limits in the order
+$j$, $L$, $M$, $\varepsilon$, show that the lost positive
+entropy is at most $2s(E_*-\|Du\|_2^2)$. The normalization
+terms are zero because $\|u\|_2=\|u_j\|_2=1$.
+\end{proof}
+
+The threshold $s$ in the entropy-defect estimate controls escape into the
+Gaussian tails.  The Poincar\'e constant controls maximizing sequences
+that converge to constants.  The next lemma proves attainment when the
+logarithmic Sobolev constant exceeds both thresholds.
+
+\begin{lemma}[Existence and positivity of an optimizer]\label{lem:optimizer}
+Let $\nu=\mu*\gamma_s$, where $s>0$ and $\mu$ is compactly
+supported. Suppose its logarithmic Sobolev and Poincar\'e constants
+$R=R(\nu)$ and $P=R^P(\nu)$ satisfy
+$\infty>R>\max(P,s)$. Then there is a smooth strictly positive
+nonconstant $f$ with
+\begin{equation}\label{eq:critical-optimizer}
+ \int f^2d\nu=1,\qquad
+ \Ent_\nu(f^2)=2R\int|Df|^2d\nu,\qquad
+ A_\nu f=\lambda f\log f,\quad\lambda=R^{-1}.
+\end{equation}
+The last equation is initially distributional and then classical.
+\end{lemma}
+
+\begin{proof}
+Take nonnegative unit-norm maximizing tests $f_j$. Replacing a test
+by its absolute value preserves entropy and energy, so this choice
+loses nothing. Equation~\eqref{eq:critical-defective}, with
+$s+\varepsilon<R$, bounds their energies. We first exclude
+energies tending to zero; this is the only place the strict
+inequality $R>P$ is required.
+
+Suppose $\varepsilon_j^2=\|Df_j\|_2^2\to0$, and write
+$c_j=\E_\nu f_j$ and $h_j=(f_j-c_j)/\varepsilon_j$.
+Poincar\'e gives $\|h_j\|_2^2\le P$, while
+$\|Dh_j\|_2^2=1$ and
+$1-c_j^2=\varepsilon_j^2\|h_j\|_2^2$.
+Thus $c_j\to1$. By compactness, after extraction,
+$h_j\to h$ in $L^2$, where
+\[
+ \E h=0,\qquad \|Dh\|_2^2\le1,\qquad
+ \|h\|_2^2\le P\|Dh\|_2^2.
+\]
+For $c>0$, $\varepsilon>0$ and $z\ge-c/\varepsilon$ define
+\[
+ Q_{c,\varepsilon}(z)=\varepsilon^{-2}
+ \left[(c+\varepsilon z)^2\log\frac{(c+\varepsilon z)^2}{c^2}
+                  -(c+\varepsilon z)^2+c^2\right].
+\]
+Normalization and $-\log c_j^2-1+c_j^2\ge0$ give
+\begin{equation}\label{eq:critical-small-energy-start}
+ \varepsilon_j^{-2}\Ent(f_j^2)
+ \le\int Q_{c_j,\varepsilon_j}(h_j)\,d\nu.
+\end{equation}
+Here is the scalar estimate needed to control unbounded $h_j$.
+For $1/2\le c\le1$ and $0<\varepsilon\le1/2$,
+\begin{equation}\label{eq:critical-scalar-Q}
+ 0\le Q_{c,\varepsilon}(z)
+ \le(2+2\log2)z^2+z^2\log_+z^2,
+ \qquad Q_{c,\varepsilon}(z)\longrightarrow2z^2
+\end{equation}
+uniformly on bounded $z$-intervals as $c\to1$ and
+$\varepsilon\to0$. To verify the bound, put $t=\varepsilon z/c$.
+Then $Q=z^2B(t)$, where Taylor's integral formula gives
+\[
+ B(t)=2+4\int_0^1(1-r)\log(1+rt)\,dr,
+ \qquad -1\le t<\infty.
+\]
+For $t\le0$, $B(t)\le2$. For $t\ge0$, use
+$t\le z$ to obtain
+$B(t)\le2+2\log(1+z)\le2+2\log2+\log_+z^2$.
+Continuity at $t=0$ proves the compact convergence.
+
+The tail proof in Lemma~\ref{lem:critical-defect} did not use
+unit norm until its last sentence. In particular, for any bounded
+form sequence $u_j\to u$ in $L^2$ with energies tending to $E_*$,
+it proves
+\begin{equation}\label{eq:critical-general-tail}
+ \lim_{K\to\infty}\limsup_j
+ \int_{|u_j|>K}u_j^2\log_+u_j^2d\nu
+ \le2s(E_*-\|Du\|_2^2).
+\end{equation}
+Apply the compact convergence in Equation~\eqref{eq:critical-scalar-Q}
+on $|h_j|\le K$ and its upper bound on the complement. Strong
+$L^2$ convergence removes the quadratic tail as $K\to\infty$;
+Equation~\eqref{eq:critical-general-tail} controls the entropy tail.
+Equation~\eqref{eq:critical-small-energy-start} consequently gives
+\[
+ \limsup_j\varepsilon_j^{-2}\Ent(f_j^2)
+ \le2\|h\|_2^2+2s(1-\|Dh\|_2^2)
+ \le2\max(P,s)<2R.
+\]
+This contradicts the maximizing property.
+
+We may therefore extract $f_j\to f$ strongly in $L^2$ with
+$\|Df_j\|_2^2\to E_*>0$. Put $E=\|Df\|_2^2$. The entropy
+defect and the sharp inequality give
+\[
+ 2RE_*\le\Ent(f^2)+2s(E_*-E)
+         \le2RE+2s(E_*-E).
+\]
+Since $R>s$ and $E\le E_*$, equality forces $E=E_*>0$.
+Thus $f$ is a nonnegative, nonconstant optimizer.
+
+The homogeneous functional
+$\Ent(g^2)-2R\|Dg\|_2^2$ is nonpositive and is zero at $f$.
+Its derivative along every compact smooth variation $\zeta$ is
+\[
+ 4\int f\log f\,\zeta\,d\nu
+       -4R\int Df\cdot D\zeta\,d\nu=0.
+\]
+The entropy normalization term cancels the additional linear term.
+Differentiation is legitimate also at zeros because
+$z^2\log z^2$ has derivative zero at zero. This proves the Euler
+equation in Equation~\eqref{eq:critical-optimizer}.
+
+We have obtained a nonnegative optimizer with positive energy.  To use
+its logarithmic derivative, we still need local regularity and strict
+positivity.  We spell out the weak-solution regularity step, using the
+interior Newtonian-potential $L^p$ and Schauder estimates; see
+\cite[Lecture~24, p.~2; Lecture~10, Proposition~1]{Viaclovsky2004}.
+Write $\nu=e^{-V}$ and set $v=e^{-V/2}f\ge0$.  The function $V$ is smooth,
+and direct differentiation of the weak equation gives
+\[
+ -\Delta v=\lambda v\log v+
+ \left(\frac{\lambda V}{2}-\frac{|DV|^2}{4}
+                         +\frac{\Delta V}{2}\right)v.
+\]
+The conjugating factor and its reciprocal are bounded on each fixed
+ball.  Since $|z\log z|\le C_\delta(1+z^{1+\delta})$ for $z\ge0$,
+local $L^p$ integrability of $v$ puts the right side in
+$L^q_{\rm loc}$, where $q=p/(1+\delta)>1$.  Localize that right side
+on a larger ball and take its Newtonian potential with the sign solving
+$-\Delta u=F$.  The cited $L^q$ estimate gives $u\in W^{2,q}_{\rm loc}$;
+the difference $v-u$ is distributionally harmonic and hence smooth by
+Weyl's lemma \cite[Lecture~3, Theorem~2]{Viaclovsky2004}.
+This proves the required $W^{2,q}$ gain directly for the weak solution.
+
+For $n\ge2$, start with $p=2$ and choose
+$0<\delta<\min(1,2/n)$.  Before boundedness is reached, Sobolev
+embedding improves reciprocal integrability from $1/p$ to
+$(1+\delta)/p-2/n$, a decrease greater than $1/n$ because $p\ge2$.
+At a critical exponent use any sufficiently large finite integrability
+exponent and continue once more.  After finitely many steps $q>n/2$,
+so $v$, and thus $f$, is locally bounded and H\"older continuous.
+In dimension one the same conclusion follows by twice integrating the
+local Poisson equation.  The function $z\log z$ is locally H\"older
+of every exponent below one on $[0,\infty)$; reducing the exponent
+if necessary, the Schauder estimate therefore gives
+$f\in C^{2,\alpha}_{\rm loc}$ for some $\alpha>0$.
+All constants in this paragraph concern one fixed law and dimension;
+none enters a uniform estimate along the counterexample sequence.
+
+If $f(x_0)=0$, choose a ball $B$ about $x_0$ with
+$0\le f\le b<1$ on its closure. Start the $\nu$-diffusion at
+$x_0$ and stop at its exit time $\tau$ from $B$. For
+$Z_t=f(Y_{t\wedge\tau})$, $m(t)=\E Z_t$, and
+$g(z)=-\lambda z\log z$, the stopped It\^o formula gives, for almost
+every $t$,
+\[
+ m'(t)=\E[\1_{t<\tau}g(Z_t)]\le\E g(Z_t)\le g(m(t)),
+ \qquad m(0)=0.
+\]
+Indeed $g$ is nonnegative and concave on $[0,b]$; the first
+inequality adds its nonnegative stopped boundary contribution.
+Also $m'\ge0$. If $m(t_1)>0$, let $t_a$ be the first time at
+which $m=a$, for $0<a<m(t_1)$. Integration gives
+\[
+ \int_a^{m(t_1)}\frac{dz}{-\lambda z\log z}
+ \le t_1-t_a\le t_1,
+\]
+contradicting divergence as $a\downarrow0$. Hence $m=0$.
+The killed nondegenerate diffusion has positive probability of
+reaching each open subset of $B$ while staying in $B$: one can
+follow a smooth interior path by requiring Brownian motion to
+remain in a small tube about its corresponding driving path,
+and use local Lipschitz continuity of the drift. Brownian tube
+probabilities are positive by its finite increment densities and
+bridge continuity. Consequently $f$ vanishes throughout $B$.
+Its zero set is both open and closed, impossible in connected
+$\R^n$ for a unit-norm function. Thus $f>0$, and smooth elliptic
+bootstrap makes $f$ smooth everywhere.
+\end{proof}
+
+An optimizer supplies a second probability with its own logarithmic
+Sobolev inequality. The next identity explains why no convexity
+assumption on this second probability is needed.
+
+\begin{samepage}
+\begin{lemma}[Change of reference at an optimizer]
+\label{lem:critical-change-reference}
+Under Lemma~\ref{lem:optimizer}, suppose in addition that
+$\nu=e^{-V}$ is log-concave with bounded $K=D^2V$.
+Define
+\[
+ \eta=f^2\nu,\qquad h=\log f^2,\qquad w=Dh,\qquad
+ I=\E_\eta|w|^2.
+\]
+Then $I=2\lambda H(\eta\mid\nu)>0$, $R(\eta)\le1/\lambda$,
+each component of $w$ belongs to $\Dom(A_\eta)$, and
+\begin{equation}\label{eq:critical-score-equation}
+ A_\nu h=\lambda h+\tfrac12|w|^2,
+ \qquad A_\eta w=(\lambda\Id-K)w.
+\end{equation}
+Moreover,
+\begin{equation}\label{eq:critical-curvature-mean}
+ \E_\eta\langle w,Kw\rangle\le\lambda|\E_\eta w|^2.
+\end{equation}
+\end{lemma}
+\end{samepage}
+
+\begin{proof}
+Since $w=2Df/f$, $I=4\|Df\|_{L^2(\nu)}^2$; optimizer
+equality gives $I=2\lambda H(\eta\mid\nu)>0$.
+The chain rule applied to its Euler equation gives the first
+identity in Equation~\eqref{eq:critical-score-equation}. Commuting
+one derivative with $A_\nu$ gives
+\[
+ A_\nu w=D(A_\nu h)-Kw
+       =\lambda w+(Dw)w-Kw.
+\]
+Since $A_\eta=A_\nu-w\cdot D$ and $Dw$ is symmetric, the
+second identity follows distributionally. Its right side is in
+$L^2(\eta)$, since $K$ is bounded and $w\in L^2(\eta)$.
+Lemma~\ref{lem:dirichlet-core} therefore puts $w$ in the
+operator domain; no unproved integrability of higher derivatives
+has been used.
+
+Let $p=g^2\eta$ have unit mass, where $g$ is compactly supported
+and smooth. Direct expansion and compactly supported integration
+by parts give
+\begin{align*}
+ \Ical(p\mid\nu)
+ &=\Ical(p\mid\eta)+\int g^2|w|^2d\eta
+                    +2\int D(g^2)\cdot w\,d\eta\\
+ &=\Ical(p\mid\eta)+\int g^2\{|w|^2+2A_\eta h\}\,d\eta\\
+ &=\Ical(p\mid\eta)+2\lambda\int h\,dp.
+\end{align*}
+These formulas may be read in square-root form at zeros of $g$.
+On the other hand
+$H(p\mid\nu)=H(p\mid\eta)+\int h\,dp$. Thus
+\begin{equation}\label{eq:critical-exact-deficit}
+ \Ical(p\mid\nu)-2\lambda H(p\mid\nu)
+ =\Ical(p\mid\eta)-2\lambda H(p\mid\eta).
+\end{equation}
+Apply the inequality for $\nu$ to the compact smooth function
+$gf$. Its deficit is nonnegative, so the right side is
+nonnegative. Homogeneity and the Sobolev approximation of
+Section~\ref{sec:calculus} prove $R(\eta)\le1/\lambda$.
+Finally the domain identity gives
+\[
+ \E_\eta|Dw|^2=\lambda I-\E_\eta\langle w,Kw\rangle,
+\]
+while Poincar\'e for $\eta$ gives the lower bound
+$\lambda(I-|\E_\eta w|^2)$. Subtract to obtain
+Equation~\eqref{eq:critical-curvature-mean}.
+\end{proof}
+
+\subsection{Uniform estimates in the two alternatives}
+
+\begin{lemma}[Translation gain]\label{lem:translation-gain}
+Let $\mu$ satisfy Equation~\eqref{eq:critical-normalization}, with
+$a\le1$. Fix a compact interval $J\subset(0,1/2)$. There is a
+constant $C_J$, independent of the dimension and of the support
+radius of $\mu$, with the following property. For $s\in J$, put
+$\lambda=R_s^{-1}$. If $\sigma$ is any probability with
+\[
+ D_0=H(\sigma\mid\mu_s)<\infty,\qquad
+ F_0=\Ical(\sigma\mid\mu_s)<\infty,\qquad p=\E_\sigma Y,
+\]
+then
+\begin{equation}\label{eq:critical-translation-gain}
+ F_0-2\lambda D_0
+ \ge(s^{-2}-\lambda/s)|p|^2
+                   -C_J\sqrt a\,(F_0+D_0).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+The centered linear moment generating functions under $\mu_s$
+obey
+\[
+ \log\E_{\mu_s}e^{v\cdot Y}\le C_J|v|^2,
+ \qquad
+ \log\E_{\mu_s}e^{v\cdot b_s(Y)}\le Ca^2|v|^2.
+\]
+The first follows by independent Gaussian addition and
+Equation~\eqref{eq:critical-linear-mgf}; the second follows by
+conditional Jensen from that equation. Entropy duality, optimized
+over the scalar multiple of each unit vector, gives
+\begin{equation}\label{eq:critical-entropic-means}
+ |p|\le C_J\sqrt{D_0},\qquad
+ |\E_\sigma b_s(Y)|\le Ca\sqrt{D_0}.
+\end{equation}
+All expectations exist. More explicitly, compact support of $X$
+makes $\E_{\mu_s}e^{c|Y|^2}<\infty$ for some $c>0$;
+entropy duality then implies $\E_\sigma|Y|^2<\infty$.
+The latter fact is used only for integrability, not for a uniform
+bound in dimension.
+
+Let $\sigma_u$ be the translate of $\sigma$ by $-up$,
+$0\le u\le1$, and let $D_u=H(\sigma_u\mid\mu_s)$.
+Change of variables in entropy and the Hessian bound give
+\begin{align*}
+ D_u
+ &=D_0+\E_\sigma[V_s(Y-up)-V_s(Y)]\\
+ &\le D_0-u|p|^2/s+u\,p\cdot\E_\sigma b_s(Y)/s
+                                +u^2|p|^2/(2s)
+ \le C_JD_0.
+\end{align*}
+Thus Equation~\eqref{eq:critical-entropic-means}, applied to
+$\sigma_u$, implies
+$|\E_\sigma b_s(Y-up)|\le C_Ja\sqrt{D_0}$ uniformly in $u$.
+Differentiating the exact entropy expression, and then integrating,
+gives
+\begin{equation}\label{eq:critical-translated-entropy}
+ D_1=D_0-\frac{|p|^2}{2s}+O_J(aD_0).
+\end{equation}
+Indeed its derivative is
+$-(1-u)|p|^2/s+p\cdot\E_\sigma b_s(Y-up)/s$.
+
+Write $r(Y)=D\log(d\sigma/d\mu_s)(Y)$ in the weak
+square-root sense. Its mean is
+\begin{equation}\label{eq:critical-score-mean}
+ \E_\sigma r=\E_\sigma DV_s=(p-\E_\sigma b_s)/s.
+\end{equation}
+To justify this for finite Fisher information, the Lebesgue density
+$q$ of $\sigma$ has weak derivative
+$Dq=q(r-DV_s)\in L^1$: $r$ is square integrable and $DV_s$
+has linear growth. Integrating against expanding cutoffs gives
+$\int Dq=0$ and hence Equation~\eqref{eq:critical-score-mean}.
+
+In the moving coordinates $Y$, the score of $\sigma_1$ relative
+to $\mu_s$ is
+\[
+ r(Y)-p/s+d(Y)/s,\qquad d(Y)=b_s(Y)-b_s(Y-p).
+\]
+Since $0\le Db_s\le\Id$, integration along the segment gives
+\[
+ |d(Y)|^2
+ =\left|\int_0^1M_s(Y-up)p\,du\right|^2
+ \le\int_0^1|M_s(Y-up)p|^2du
+ \le p\cdot d(Y).
+\]
+The mean estimates at the two endpoints therefore imply
+\begin{equation}\label{eq:critical-posterior-translation-error}
+ \E_\sigma|d|^2\le C_JaD_0.
+\end{equation}
+Expanding the score square and using
+Equation~\eqref{eq:critical-score-mean},
+\[
+ \E_\sigma|r-p/s|^2
+ =F_0-|p|^2/s^2+2p\cdot\E_\sigma b_s/s^2.
+\]
+The remaining cross term is bounded by
+$C_J\sqrt{aD_0}\,(\sqrt{F_0}+\sqrt{D_0})$, and the
+$d$-square term by $C_JaD_0$. Consequently the translated
+information is finite and satisfies
+\begin{equation}\label{eq:critical-translated-information}
+ F_1=F_0-|p|^2/s^2+O_J(\sqrt a\,(F_0+D_0)).
+\end{equation}
+These calculations require only the weak derivatives already
+justified, so they apply also when the density of $\sigma$ vanishes.
+Apply logarithmic Sobolev to $\sigma_1$, namely
+$F_1\ge2\lambda D_1$, and combine
+Equations~\eqref{eq:critical-translated-entropy} and
+\eqref{eq:critical-translated-information}. Since
+$2/3\le\lambda\le1$ on the time interval in question, this
+proves Equation~\eqref{eq:critical-translation-gain}.
+\end{proof}
+
+We can now obtain estimates that are uniform along the counterexample
+sequence. Write $R_s^P$ for the optimal Poincar\'e constant of
+$\mu_s$. By linearization $R_s^P\le R_s$. Since $R_s\ge1>s$
+on $(0,1/2)$, the strict alternative $R_s>R_s^P$ meets all
+the hypotheses of Lemma~\ref{lem:optimizer}.
+
+\begin{lemma}[Critical fields in both alternatives]
+\label{lem:critical-two-alternatives}
+Fix $J\subset(0,1/2)$ compact and $s\in J$, and put
+$\lambda=1/R_s$. There are a smooth positive probability $\eta$
+and a smooth vector field $w$, with
+$0<I:=\E_\eta|w|^2<\infty$, such that
+\begin{equation}\label{eq:critical-common-identities}
+ R(\eta)\le\lambda^{-1},\qquad
+ w\in\Dom(A_\eta)^n,\qquad
+ A_\eta w=(\lambda\Id-K_s)w.
+\end{equation}
+They can be chosen as follows.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item If $R_s>R_s^P$, take the optimizer $f$ of
+Lemma~\ref{lem:optimizer} and set
+\[
+ \eta=f^2\mu_s,\qquad w=D\log f^2,
+ \qquad I=2\lambda H(\eta\mid\mu_s).
+\]
+\item If $R_s=R_s^P$, there is a centered nonconstant gap
+eigenfunction $\varphi$ satisfying
+$A_{\mu_s}\varphi=\lambda\varphi$. Take
+\[
+ \eta=\mu_s,\qquad w=D\varphi,
+ \qquad I=\lambda\E_{\mu_s}\varphi^2.
+\]
+\end{enumerate}
+In either alternative,
+\begin{equation}\label{eq:critical-uniform-smallness}
+ |\E_\eta w|^2+\E_\eta|K_sw|^2\le C_J\sqrt a\,I,
+ \qquad
+ \E_\eta\langle w,K_sw\rangle\le\lambda|\E_\eta w|^2.
+\end{equation}
+In alternative \textup{(ii)} the factor $\sqrt a$ can be
+replaced by $a^2$. Coordinates belong to $\Dom(A_\eta)$ in
+both alternatives.
+\end{lemma}
+
+\begin{proof}
+In the first alternative the identities and the last inequality
+are Lemma~\ref{lem:critical-change-reference}. Apply
+Lemma~\ref{lem:translation-gain} to $\sigma=\eta$, where
+$F_0=I$ and $D_0=I/(2\lambda)$, so the deficit vanishes.
+The coefficient $s^{-2}-\lambda/s$ is bounded below by a
+positive constant on $J$. Thus, with $p=\E_\eta Y$,
+\[
+ |p|^2\le C_J\sqrt a\,I,
+ \qquad |\E_\eta b_s|\le C_Ja\sqrt I.
+\]
+Equation~\eqref{eq:critical-score-mean} applied to $\eta$ now
+gives $|\E_\eta w|^2\le C_J\sqrt a\,I$.
+Since $K_s^2\le s^{-1}K_s$, the curvature-mean inequality
+gives the asserted estimate for $K_sw$.
+
+In the second alternative, compact form embedding supplies a
+minimizer of the Rayleigh quotient among mean-zero functions of
+unit norm. Its positive eigenvalue is $1/R_s^P=\lambda$.
+The nullspace consists of constants, since zero weak gradient on
+$\R^n$ implies constancy. Compactness therefore makes this a
+positive gap, and interior elliptic regularity makes its
+eigenfunction $\varphi$ smooth. Commuting a derivative gives
+$A_{\mu_s}D\varphi=(\lambda\Id-K_s)D\varphi$.
+Its right side is in $L^2$, so
+Lemma~\ref{lem:dirichlet-core} gives the operator-domain assertion.
+Taking the energy pairing and then Poincar\'e gives, exactly as
+before,
+\[
+ \E\langle w,K_sw\rangle\le\lambda|\E w|^2.
+\]
+Coordinates belong to the operator domain because both $Y$ and
+$A_{\mu_s}Y=(Y-b_s)/s$ belong to $L^2(\mu_s)$. Integration
+by parts, or self-adjointness with these coordinate tests, yields
+\[
+ \E w=\lambda\E[\varphi Y]
+      =\frac{\E[\varphi Y]-\E[\varphi b_s]}s,
+ \qquad
+ (1-\lambda s)\E[\varphi Y]=\E[\varphi b_s].
+\]
+Conditional Jensen gives
+\[
+ \E_{\mu_s}b_sb_s^*\le\E_\mu XX^*\le Ca^2\Id.
+\]
+For every unit vector $v$, Cauchy--Schwarz thus bounds
+$|\E[\varphi\,v\cdot b_s]|$ by $Ca\|\varphi\|_2$.
+Taking the supremum over $v$ introduces no dimension factor.
+Since $1-\lambda s\ge1/2$ and
+$I=\lambda\|\varphi\|_2^2$, it follows that
+$|\E w|\le C_Ja\sqrt I$. The bound for $K_sw$ follows
+from $K_s^2\le s^{-1}K_s$ once again.
+
+It remains to justify the coordinate assertion for the first
+alternative. Finite $H(\eta\mid\mu_s)$ and the Gaussian
+tails of $\mu_s$ imply $\E_\eta|Y|^2<\infty$, by the
+entropy argument in the proof of Lemma~\ref{lem:translation-gain}.
+The distributional identity
+\begin{equation}\label{eq:critical-coordinate-generator}
+ A_\eta Y=(Y-b_s)/s-w
+\end{equation}
+has an $L^2(\eta)$ right side, because $b_s$ is bounded for
+the fixed law and $w\in L^2(\eta)$. The maximal-domain
+assertion of Lemma~\ref{lem:dirichlet-core} applies.
+\end{proof}
+
+\begin{proposition}[Limit of the regularized constants]
+\label{prop:regularized-constants}
+For the sequence in Lemma~\ref{lem:critical-normalization},
+\begin{equation}\label{eq:critical-R-limit}
+ R_s\longrightarrow1\qquad\text{for every }0<s<1/2.
+\end{equation}
+At every differentiability point of $R_s$, the fields of
+Lemma~\ref{lem:critical-two-alternatives} satisfy the contact
+identity
+\begin{equation}\label{eq:critical-contact-derivative}
+ R_s'=R_s\,\frac{\E_\eta\langle w,K_sw\rangle}{I}.
+\end{equation}
+In particular $0\le R_s'\le C_J\sqrt a$ almost everywhere
+on each fixed compact $J\subset(0,1/2)$.
+\end{proposition}
+
+\begin{proof}
+Fix $s$, and let $\Phi_{s,t}$ be the heat continuity flow from
+Lemma~\ref{lem:critical-flow-bounds}, for $t$ on either side
+of $s$. Transport a form-domain test $g$ by
+$g_t=g\circ\Phi_{s,t}^{-1}$. Its distribution under $\mu_t$
+is fixed, so both its entropy and its variance are constant.
+The Jacobian equation $\partial_tD\Phi_{s,t}=K_tD\Phi_{s,t}/2$
+gives
+\begin{equation}\label{eq:critical-transport-energy}
+ \left.\frac{d}{dt}\right|_{t=s}
+ \int|Dg_t|^2d\mu_t
+ =-\int\langle Dg,K_sDg\rangle d\mu_s.
+\end{equation}
+For compact smooth $g$ this follows by differentiating the
+inverse-transpose Jacobian in the gradient. For a form-domain
+$g$, the change-of-variables formula still holds by Sobolev
+approximation. The flow derivatives and their time derivatives
+are uniformly bounded on a compact time interval, so the
+derivative of the integrand is dominated by $C|Dg|^2$.
+Dominated convergence proves
+Equation~\eqref{eq:critical-transport-energy} without a smoothness
+or compact-support restriction on $g$.
+
+In the strict alternative take $g=f$ and set
+$Q(t)=\Ent_{\mu_t}(f_t^2)/(2\int|Df_t|^2d\mu_t)$.
+The denominator is positive near $s$, and $Q(t)\le R_t$ for
+both signs of $t-s$, with $Q(s)=R_s$. At a differentiability
+point of $R_s$, this two-sided contact implies $Q'(s)=R_s'$.
+Equation~\eqref{eq:critical-transport-energy} and
+$w=2Df/f$ give Equation~\eqref{eq:critical-contact-derivative}.
+In the gap alternative take $g=\varphi$ and instead use
+$Q(t)=\Var_{\mu_t}(\varphi_t)/\int|D\varphi_t|^2d\mu_t$.
+Now $Q(t)\le R_t^P\le R_t$ and $Q(s)=R_s^P=R_s$.
+The same contact argument gives the identical derivative formula,
+with $w=D\varphi$.
+
+Equation~\eqref{eq:critical-uniform-smallness} bounds this
+derivative by $C_J\sqrt a$ in either alternative, with constants
+independent of which alternative holds at each time. Lipschitz
+continuity makes $R_s$ absolutely continuous. For fixed
+$0<\delta<s<1/2$, integration on $[\delta,s]$ gives
+\[
+ 1\le R_s\le R_\delta+C_{[\delta,s]}\sqrt a\,(s-\delta)
+             \le1+\delta+C_{[\delta,s]}\sqrt a\,(s-\delta).
+\]
+First take the counterexample-sequence limit and then let
+$\delta\downarrow0$. This proves
+Equation~\eqref{eq:critical-R-limit}.
+\end{proof}
+
+We finish by fixing the Gaussian time and collecting precisely the
+objects used in the following sections. The fixed time will not be
+sent to zero.
+
+\begin{proposition}[The critical sequence]\label{prop:critical-sequence}
+If Theorem~\ref{thm:main} fails, there is a sequence of dimensions
+and centered compactly supported log-concave laws $\mu$ satisfying
+Equation~\eqref{eq:critical-normalization}. Fix $s=1/8$, form
+$\mu_s,b_s,M_s,K_s$ as in Equation~\eqref{eq:critical-posterior},
+and put $\lambda=R_s^{-1}$. After passage to a subsequence there
+are smooth positive probabilities $\eta$ and smooth vector fields
+$w$ such that, with $A=A_\eta$ and $I=\E_\eta|w|^2$,
+\begin{gather}
+ \lambda\longrightarrow1,\qquad R(\eta)\le\lambda^{-1},
+ \qquad 0<I<\infty,\qquad w\in\Dom(A)^n,
+ \label{eq:critical-sequence-data}\\
+ A w=(\lambda\Id-K_s)w,\qquad
+ \|(A-1)w\|_{L^2(\eta)}+|\E_\eta w|=o(\sqrt I),
+ \label{eq:critical-near-eigenfunction}\\
+ \E_\eta|K_sw|^2=o(I),\qquad
+ \E_\eta|Dw|^2=(1+o(1))I.
+ \label{eq:critical-sequence-energy}
+\end{gather}
+The spectral gap of $A$ is at least $\lambda$. Each coordinate
+function belongs to $\Dom(A)$. Exactly one of the following
+constructions can be used throughout the subsequence:
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item $R_s>R_s^P$, $\eta=f^2\mu_s$ for the positive unit-norm
+optimizer, $w=D\log f^2$, and
+$I=2\lambda H(\eta\mid\mu_s)$. The exact deficit identity in
+Equation~\eqref{eq:critical-exact-deficit} holds for compact square-root
+tests relative to $\eta$, and
+$A Y=(Y-b_s)/s-w$.
+\item $R_s=R_s^P$, $\eta=\mu_s$,
+$A\varphi=\lambda\varphi$, $\E_\eta\varphi=0$,
+$w=D\varphi$, and $I=\lambda\E_\eta\varphi^2$.
+One may rescale $\varphi$ so that $I\to0$.
+\end{enumerate}
+All $o$-terms refer to the counterexample sequence with fixed
+finite parameters. In particular no relation between dimension,
+$I$, and the rate of convergence is asserted or needed here.
+\end{proposition}
+
+\begin{proof}
+Choose an infinite subsequence with the same alternative at
+$s=1/8$. Lemma~\ref{lem:critical-two-alternatives} gives all
+structural identities and the curvature and mean bounds.
+Proposition~\ref{prop:regularized-constants} gives $\lambda\to1$.
+Consequently
+\[
+ (A-1)w=(\lambda-1)w-K_sw=o_{L^2}(\sqrt I).
+\]
+The energy pairing yields
+$\E|Dw|^2=\lambda I-\E\langle w,K_sw\rangle=(1+o(1))I$.
+Poincar\'e with constant $\lambda^{-1}$ is the asserted spectral
+gap. The coordinate and branch-specific identities have already
+been proved. In the gap alternative all estimates are homogeneous
+in $\varphi$, so multiplying it by positive scalars can enforce
+$I\to0$ without changing any normalized assertion.
+\end{proof}
diff --git a/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/build/sections/04-prediction.tex b/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/build/sections/04-prediction.tex
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+\section{Efficient probes and the prediction curve}
+\label{sec:prediction}
+
+Fix $s=1/8$ and take one of the two subsequences supplied by
+Proposition~\ref{prop:critical-sequence}.  We continue to suppress the
+counterexample index.  Thus $\lambda\to1$, $\eta$ satisfies the logarithmic
+Sobolev inequality with constant at most $1/\lambda$, and, with
+$A=A_\eta$, $w$ belongs coefficientwise to $\Dom(A)$ and satisfies
+\begin{equation}
+ I=\E_\eta|w|^2>0,
+ \qquad \|(A-1)w\|_{L^2(\eta)}+|\E_\eta w|=o(\sqrt I).
+ \label{eq:prediction-input}
+\end{equation}
+In the strict case, $\eta=f^2\mu_s$, $w=D\log f^2$, and
+$2H(\eta\mid\mu_s)=I/\lambda$.  In the gap case,
+$\eta=\mu_s$, $w=D\varphi$, and
+$A_{\mu_s}\varphi=\lambda\varphi$, where
+$\E_{\mu_s}\varphi=0$ and $I=\lambda\E_{\mu_s}\varphi^2$.
+All limits in this section are along the selected subsequence, with the
+displayed auxiliary parameters fixed unless a different order is specified.
+
+Our objective is to determine how much of $w(Y_0)$ can be predicted from
+$Y_0+\sqrt rG$.  The strict case first requires an entropy estimate for an
+additional Gaussian observation of $w$.  We then prove an entropy-loss
+comparison for arbitrary smooth alternatives to a log-concave law.  These
+two estimates determine the prediction curve; linearization of the same
+comparison handles the gap case.
+
+\begin{lemma}[Efficient Gaussian probes]
+\label{lem:efficient-probes}
+In the strict case, let $Y_0\sim\eta$ and let $G'$ be an independent
+standard Gaussian vector in the same Euclidean space.  For $T\ge0$ define
+\[
+ L_T=\sqrt T\,w(Y_0)+G',\qquad
+ d(T)=H(\Law(L_T)\mid\gamma),
+\]
+where $\gamma$ is standard Gaussian measure.  For every finite $M$,
+\begin{equation}
+ \sup_{0\le T\le M}d(T)=o(I).
+ \label{eq:efficient-probes}
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Run the stationary diffusion with law $\eta$, using Brownian noise
+independent of $G'$, and write $P_u=e^{-uA}$.
+Proposition~\ref{prop:critical-sequence} puts each coordinate in
+$\Dom(A)$ and gives the identity
+\begin{equation}
+ -Ay=-y/s+w+b_s/s.
+ \label{eq:prediction-coordinate-generator}
+\end{equation}
+In particular, $\E_\eta|Y_0|^2<\infty$. The logarithmic Sobolev
+inequality for $\eta$ then gives
+\[
+ \E_\eta e^{v\cdot(Y_0-\E Y_0)}
+ \le e^{|v|^2/(2\lambda)}.
+\]
+Integrating this inequality over a Gaussian $v$ proves
+$\E_\eta e^{c'|Y_0|^2}<\infty$ for a sufficiently small $c'>0$.
+Here the choice of $c'$ may depend on the individual law and dimension.
+This exponential-square moment will also permit application of the
+entropy-loss comparison below.
+
+For a fixed $T\le M$, set
+\[
+ J_u=\MI(L_T;Y_u),\qquad
+ F_u=\E\,\Ical(\Law(Y_u\mid L_T)\mid\eta),\qquad
+ Q_u=\E|\E[Y_u\mid L_T]|^2.
+\]
+The Gaussian observation identity gives
+$J_0=TI/2-d(T)$.  Entropy dissipation, including its conditional version,
+gives $-dJ_u\ge F_u\,du$, and in particular
+$\int_0^1F_u\,du\le J_0$.  If $\sigma_{u,l}=\Law(Y_u\mid L_T=l)$,
+then the entropy chain rule and the strict-case normalization give
+\begin{equation}
+ \E H(\sigma_{u,L_T}\mid\mu_s)=J_u+\frac{I}{2\lambda},
+ \qquad
+ \E\Ical(\sigma_{u,L_T}\mid\mu_s)=F_u+I.
+ \label{eq:probe-reference-decomposition}
+\end{equation}
+For the second equality, write the conditional relative score with
+respect to $\mu_s$ as the sum of $w$ and the conditional relative score
+with respect to $\eta$.  The latter scores have conditional average zero
+at each $Y_u$, since the conditional likelihoods average to one.  This
+cancels the cross term.  At the almost every time where $F_u<\infty$,
+the assertion also follows weakly by differentiating that marginal
+identity: Cauchy--Schwarz gives locally integrable averaged derivatives.
+
+Apply Lemma~\ref{lem:translation-gain} to $\sigma_{u,l}$ and average
+over $l$.  Since $s^{-2}-\lambda/s$ stays positive, Equations
+\eqref{eq:probe-reference-decomposition} give, on $[0,1]$,
+\begin{equation}
+ -dJ_u\ge(2\lambda J_u+c_sQ_u)\,du-d\mathcal E_u,
+ \qquad \mathcal E([0,1])=o_M(I),
+ \label{eq:probe-decay}
+\end{equation}
+where $c_s>0$ is fixed and $\mathcal E$ is a nonnegative error measure.
+Indeed the error in that lemma is bounded by a number tending to zero
+times $F_u+I+J_u+I/(2\lambda)$, whose integral is $O_M(I)$.
+The measure formulation avoids imposing differentiability on $J_u$.
+
+The spectral theorem and Equation~\eqref{eq:prediction-input} imply
+\begin{equation}
+ \sup_{0\le u\le1}\|P_uw-e^{-u}w\|_2
+ \le \|(A-1)w\|_2=o(\sqrt I).
+ \label{eq:probe-semigroup}
+\end{equation}
+Reversibility gives
+$\E[L_T\mid Y_u]=\sqrt T P_uw(Y_u)$.  Relative entropy with respect
+to a standard Gaussian is at least half the square of the mean, by
+testing its variational formula with linear functions.  Consequently
+\[
+ J_u+d(T)=\E H(\Law(L_T\mid Y_u)\mid\gamma)
+ \ge\frac T2\|P_uw\|_2^2
+ =\frac{TI}{2}e^{-2u}+o_M(I).
+\]
+Integrating Equation~\eqref{eq:probe-decay} with factor $e^{2\lambda u}$
+and comparing at $u=1$ yields
+\begin{equation}
+ \int_0^1Q_u\,du\le C_s d(T)+o_M(I).
+ \label{eq:probe-position-control}
+\end{equation}
+Here the difference between $e^{-2\lambda}$ and $e^{-2}$ contributes
+only $o_M(I)$.
+
+It remains to turn control of the predicted position into control of the
+predicted score.  The pushforward of $\mu_s$ by $b_s$ has the centered
+linear moment bound
+\[
+ \log\E_{\mu_s}e^{v\cdot b_s}\le Ca^2|v|^2,
+\]
+by conditional Jensen and the corresponding bound for the original
+signal $X$.  The entropy variational formula, optimized over $v$, thus
+gives
+\[
+ \E|\E[b_s(Y_v)\mid L_T]|^2
+ \le Ca^2\bigl(J_v+I/(2\lambda)\bigr)=o_M(I).
+\]
+Condition Equation~\eqref{eq:prediction-coordinate-generator} on $L_T$
+and use variation of constants.  By
+Equation~\eqref{eq:probe-semigroup} and the contraction property of
+conditional expectation,
+\begin{equation}
+ \E[Y_u\mid L_T]
+ =e^{-u/s}\E[Y_0\mid L_T]
+ +\frac{e^{-u}-e^{-u/s}}{1/s-1}\E[w(Y_0)\mid L_T]
+ +o_{L^2,M}(\sqrt I),
+ \label{eq:probe-coordinate-solution}
+\end{equation}
+uniformly for $0\le u\le1$.  The two scalar functions multiplying the
+conditional means are linearly independent on $[0,1]$: the first is
+nonzero at zero, whereas the second vanishes there and is nonzero for
+$u>0$.  Their $2\times2$ Gram matrix is therefore positive definite.
+Applying its smallest-eigenvalue bound to each coordinate and then
+averaging, Equations~\eqref{eq:probe-position-control} and
+\eqref{eq:probe-coordinate-solution} show that
+\[
+ \E|\E[w(Y_0)\mid L_T]|^2\le C_s d(T)+o_M(I).
+\]
+By Lemma~\ref{lem:gaussian-signals}, the left side is $2d'(T)$ for
+almost every $T$.  The error is uniform for $T\in[0,M]$, and $d(0)=0$.
+Integrating this differential inequality proves
+Equation~\eqref{eq:efficient-probes}.  No lower bound on $I$ is used.
+\end{proof}
+
+The next comparison relates the entropy lost under heat flow to the
+conditional mean of the initial relative score.  The conditional mean
+need not equal the relative score at positive heat time, so entropy
+dissipation alone does not give this comparison.
+
+The choice of observation directions is related to function-preserving
+stochastic localization. Eldan--Koehler--Zeitouni
+\cite[Section~2, Lemma~2]{EldanKoehlerZeitouni2022} use controls
+approximating projection orthogonal to the covariance of the signal
+with a selected function. Chen and Eldan
+\cite[Sections~2.4.2 and~3.2.3, Equation~(27)]{ChenEldan2022}
+express the entropy drift through the difference of means under a
+measure and its normalized likelihood tilt, in a continuous-time
+setup with positive definite controls. Below we prove the cancellation
+and its error directly for finite Gaussian observations with singular
+projections, and use independent copies to control the discarded
+directions.
+
+\begin{lemma}[Entropy loss under a Gaussian observation]
+\label{lem:entropy-loss}
+Let $p$ and $q$ be smooth strictly positive probability densities on
+$\R^d$, with $q$ log-concave.  Suppose
+\[
+ H_0=H(p\mid q)<\infty,\qquad
+ w=D\log(p/q),\qquad I=\E_p|w|^2<\infty,
+ \qquad \E_p e^{c|Y|^2}<\infty
+\]
+for some $c>0$, where $Y\sim p$. Let $G$ be a standard Gaussian
+vector independent of $Y$. For $r>0$, let
+$U_r=Y+\sqrt rG$, let $p_r$ and $q_r$ be the Gaussian convolutions
+of $p$ and $q$, and put
+$\bar w_r(U_r)=\E_p[w(Y)\mid U_r]$ and $H_r=H(p_r\mid q_r)$.
+Then, for every $z>0$,
+\begin{equation}
+ 2(H_0-H_z)\le\int_0^z\E_p|\bar w_r(U_r)|^2\,dr.
+ \label{eq:prediction-entropy-loss}
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+The proof adds weak observations that increase the curvature of the
+base posterior while losing asymptotically no relative entropy.
+At each step we remove the direction in which the two posterior means
+differ. The retained output means then agree, and we will show that
+its relative entropy, averaged over the retained history, is
+$O(\delta^2)$ for observation strength $\delta$.
+Independent copies make the cost of removing these directions vanish
+after division by the number of copies. That last comparison concerns
+the score residuals after full observations, whose conditional
+independence will be essential.
+
+We keep the one-copy problem fixed. The three successive limits will
+be refinement of a finite observation mesh, increase of the number of
+independent copies, and increase of the total additional precision.
+Set $b=1/z$ and take $N$ independent
+copies.  Write $Y=(Y_1,\ldots,Y_N)$ and
+$W=(w(Y_1),\ldots,w(Y_N))$; the two possible priors are $p^{\otimes N}$
+and $q^{\otimes N}$.  Expectations in the proof use $p^{\otimes N}$.
+
+\emph{Additional observations and their entropy cost.}
+First observe $Z_0=\sqrt bY+G_0$.  The chain rule says that the
+remaining posterior relative entropy, averaged over this observation,
+is $N(H_0-H_z)$.  Fix $u>0$ and an integer $m$, and put $\delta=u/m$.
+For $i=0,\ldots,m-1$, let $\mathcal H_i$ be the retained history before
+step $i$.  Compute the difference of the two posterior means given
+this history and let $P_i$ be the orthogonal projection onto its span;
+set $P_i=0$ when the difference vanishes.  Retain only
+\[
+ (1-P_i)Z_{i+1},\qquad Z_{i+1}=\sqrt\delta Y+G_{i+1},
+\]
+where all Gaussian noises are independent.  The same measurable rule
+is used under both priors.  One may choose versions of the posterior
+means by their integral formulas; the positive densities make the rule
+well defined on all histories of positive probability.
+
+For each realized retained history the likelihood, as a function of
+$Y$, is a Gaussian quadratic factor of precision
+\begin{equation}
+ B=b\Id+\delta\sum_{i=0}^{m-1}(1-P_i)\succeq b\Id.
+ \label{eq:entropy-loss-precision}
+\end{equation}
+There is no extra likelihood factor from the choices of $P_i$: each
+choice was already determined by the retained past.  The factor is the
+same for both priors.  Thus the posterior base has curvature at least
+$B$, and the posterior relative score is still $W$.
+
+We next bound the relative entropy lost in the additional observations.
+Condition on a retained history at one step and put $Q=1-P_i$.
+The projected posterior means agree.  After subtracting their common
+mean, let $X_c$ denote either centered projected signal in
+$\operatorname{ran}Q$, whose dimension is at most $Nd$.  The base
+posterior has curvature at least $b\Id$.  Its centered linear moment
+bound and integration over a Gaussian vector give
+\[
+ \E_q[e^{\kappa|X_c|^2}\mid\mathcal H_i]
+ \le (1-2\kappa/b)^{-Nd/2},\qquad 0<2\kappa<b.
+\]
+For the alternative, write $a_i=\E_p[Y\mid\mathcal H_i]$.  For any
+sufficiently small $\kappa>0$, conditional Jensen gives
+\begin{align*}
+ \E_p[e^{\kappa|Q(Y-a_i)|^2}\mid\mathcal H_i]
+ &\le e^{2\kappa|a_i|^2}
+       \E_p[e^{2\kappa|Y|^2}\mid\mathcal H_i]\\
+ &\le\bigl(\E_p[e^{2\kappa|Y|^2}\mid\mathcal H_i]\bigr)^2.
+\end{align*}
+Squaring and applying conditional Jensen once more shows that its
+square, averaged over $\mathcal H_i$, is bounded by
+$\E_p e^{8\kappa|Y|^2}<\infty$.  These bounds hold for every history
+sigma-field, so their constants do not depend on the mesh or on the
+adaptive choices.
+
+Let $h_p,h_q$ be the output densities of
+$\sqrt\delta X_c+G$ in $\operatorname{ran}Q$ and let $\phi$ be standard
+Gaussian density there.  Centering and Jensen give the pointwise bound
+\[
+ \frac{h_q(x)}{\phi(x)}
+ =\E_q e^{\sqrt\delta x\cdot X_c-\delta|X_c|^2/2}
+ \ge e^{-\delta Nd/(2b)}.
+\]
+For either conditional signal, using a conditionally independent copy
+$X_c'$ and integrating the Gaussian kernel gives
+\[
+ \left\|\frac h\phi-1\right\|_{L^2(\phi)}^2
+ =\E[e^{\delta X_c\cdot X_c'}\mid\mathcal H_i]-1.
+\]
+The linear term vanishes because $X_c$ is centered.  The bound
+$|e^t-1-t|\le t^2e^{|t|}/2$, followed by
+$|X_c\cdot X_c'|\le(|X_c|^2+|X_c'|^2)/2$, bounds the last expression
+by $C\delta^2$ times the square of a small conditional
+exponential-square moment.  The preceding moment bounds therefore give
+\[
+ \E H(h_p\mid h_q)
+ \le\E\chi^2(h_p\mid h_q)
+ \le2e^{\delta Nd/(2b)}\E\left(
+ \left\|h_p/\phi-1\right\|_2^2+
+ \left\|h_q/\phi-1\right\|_2^2\right)
+ \le C_{N,b,p}\delta^2.
+\]
+This statement includes the zero-dimensional observed subspace, for
+which the loss is zero.  Summing the $m$ steps loses
+$O_{N,b,p}(u\delta)$ entropy.  Apply the anisotropic curvature
+inequality, Lemma~\ref{lem:curvature}, to the final posterior pair.
+For fixed $N$ and $u$ we obtain
+\begin{equation}
+ 2N(H_0-H_z)-o_{\delta\downarrow0}(1)
+ \le\E W^*B^{-1}W.
+ \label{eq:entropy-loss-posterior-lsi}
+\end{equation}
+
+\emph{Comparison with full observations.}
+We now estimate the effect of the rank-one exclusions. Work in the
+Hilbert space of square-integrable vector fields on the signal and all
+the full observations $Z_0,\ldots,Z_m$.  In particular,
+$\|W\|_2^2=NI$.  Completing the square pointwise gives
+\begin{equation}
+ b\|W\|_2^2-b^2\E W^*B^{-1}W
+ =\min_v\left\{b\|W-v\|_2^2+
+              \delta\sum_{i=0}^{m-1}\|(1-P_i)v\|_2^2\right\}.
+ \label{eq:entropy-loss-variational}
+\end{equation}
+Its minimizer is $v_*=bB^{-1}W$, so $\|v_*\|_2\le\|W\|_2$.
+Let $\mathcal F_i=\sigma(Z_0,\ldots,Z_i)$ and let $e_i$ be
+conditional expectation onto $\mathcal F_i$.  The sigma-fields are
+nested, and $P_i$ is $\mathcal F_i$-measurable because the retained
+history is a function of this full history. We compare the original
+minimum with one that penalizes the part of $v$ left unpredicted by
+the full observations: replace $1-P_i$ by $1-e_i$, and denote the
+resulting minimum by $M_0$. The nested projections make this new
+minimum explicitly computable in terms of Gaussian prediction.
+We first find its minimizer, then use its residuals in a dual test
+to bound the original minimum below in terms of $M_0$.
+Its minimizer is
+\[
+ v_0=b\left(b+\delta\sum_{i=0}^{m-1}(1-e_i)\right)^{-1}W.
+\]
+Here is its explicit form.  Set $\alpha_k=b/(b+k\delta)$ for
+$0\le k\le m$.  On $\operatorname{ran}e_0$ the multiplier is one;
+on $\operatorname{ran}(e_k-e_{k-1})$ it is $\alpha_k$; on
+$\operatorname{ran}(1-e_{m-1})$ it is $\alpha_m$.  Consequently
+\begin{equation}
+ v_0=\alpha_mW+
+       \sum_{k=0}^{m-1}(\alpha_k-\alpha_{k+1})e_kW.
+ \label{eq:entropy-loss-minimizer}
+\end{equation}
+All coefficients are nonnegative and sum to one.  Put
+$g_i=(1-e_i)v_0$.  Equation~\eqref{eq:entropy-loss-minimizer} gives
+\begin{equation}
+ g_i=\alpha_m(W-e_iW)+
+ \sum_{k=i+1}^{m-1}(\alpha_k-\alpha_{k+1})(e_kW-e_iW).
+ \label{eq:entropy-loss-residual}
+\end{equation}
+The weights in this formula sum to $\alpha_{i+1}\le1$.
+
+We need a bound uniform over every subdivision.  The retained histories need not factor across copies.  We use
+independence only for the full observation arrays: those belonging to
+different copies are independent under $p^{\otimes N}$.  Conditioning
+on $\mathcal F_i$ conditions each such array on its own past.  Thus
+$e_kW_j$ depends only on copy $j$'s observations
+through step $k$.  Conditional on $\mathcal F_i$, the blocks $g_{i,j}$
+are independent and centered, even though they involve future full
+observations.  To see this directly, factor the joint law into the
+independent copy arrays and condition each factor on its own past;
+the resulting conditional kernel remains a product.  By conditional
+Jensen applied to Equation~\eqref{eq:entropy-loss-residual},
+\begin{equation}
+ Z_{i,j}:=\tr\Cov(g_{i,j}\mid\mathcal F_i)
+ \le\E[|W_j-e_iW_j|^2\mid\mathcal F_i]
+ \le\E[|W_j|^2\mid\mathcal F_i].
+ \label{eq:entropy-loss-block-bound}
+\end{equation}
+For example, every term with $k>i$ is bounded in conditional squared
+mean by $\E[|W_j-e_iW_j|^2\mid\mathcal F_i]$ by the tower property;
+the nonnegative weights of total at most one preserve this bound.
+
+The right sides in Equation~\eqref{eq:entropy-loss-block-bound} are
+uniformly integrable over all these sigma-fields.  Explicitly, if
+$Z\ge0$ is integrable and $M=\E[Z\mid\mathcal F]$, then
+\begin{equation}
+ \E[M\1_{\{M>K\}}]\le2\E[Z\1_{\{Z>K/2\}}].
+ \label{eq:entropy-loss-ui}
+\end{equation}
+Indeed $M\le K/2+\E[Z\1_{\{Z>K/2\}}\mid\mathcal F]$, and on
+$\{M>K\}$ the last conditional expectation is at least $M/2$.
+The conditional covariance of $g_i$ is block diagonal.  Since $P_i$
+is measurable in $\mathcal F_i$ and has rank at most one,
+\[
+ \E|P_ig_i|^2\le\E\max_{1\le j\le N}Z_{i,j}.
+\]
+The maximum is at most $K+\sum_jZ_{i,j}\1_{\{Z_{i,j}>K\}}$.
+Equations~\eqref{eq:entropy-loss-block-bound} and
+\eqref{eq:entropy-loss-ui} show that, uniformly in $i,m$, and $\delta$,
+\begin{equation}
+ \E|P_ig_i|^2\le Nq_N,\qquad
+ q_N:=\inf_{K>0}\left\{\frac KN+
+           2\E_p[|w(Y)|^2\1_{\{|w(Y)|^2>K/2\}}]\right\}\longrightarrow0.
+ \label{eq:entropy-loss-rank-error}
+\end{equation}
+No moment of the score beyond its second moment is required.  In
+particular this bound permits $P_i$ to depend adaptively on all copies.
+
+For completeness, the comparison of the two minima follows from a
+dual test.  For every $v$,
+\[
+ \|(1-P_i)v\|_2^2
+ \ge2\langle v,(1-P_i)g_i\rangle-\|g_i\|_2^2.
+\]
+The normal equation for $v_0$ is
+$b(W-v_0)=\delta\sum_i g_i$.  Therefore the minimum over $v$ of
+\[
+ b\|W-v\|_2^2+2\delta\sum_i\langle v,g_i\rangle
+                  -\delta\sum_i\|g_i\|_2^2
+\]
+is attained at $v_0$ and equals $M_0$.  Use the displayed linear bound
+at $v_*$ in Equation~\eqref{eq:entropy-loss-variational}.  By
+Equation~\eqref{eq:entropy-loss-rank-error}, the true minimum is at least
+\begin{equation}
+ M_0-2\delta\sum_i\|v_*\|_2\|P_ig_i\|_2
+ \ge M_0-2uN\sqrt{Iq_N}.
+ \label{eq:entropy-loss-dual-error}
+\end{equation}
+
+\emph{The prediction integral.}
+It remains to compute $M_0$ and take the limits. The same orthogonal
+decomposition into nested projection increments gives
+\begin{equation}
+ \frac{M_0}{N}
+ =\sum_{i=0}^{m-1}
+ \frac{b^2\delta}{(b+i\delta)(b+(i+1)\delta)}
+       \left(I-\|e_iW_1\|_2^2\right).
+ \label{eq:entropy-loss-discrete-minimum}
+\end{equation}
+Indeed a component first seen after $j$ increments has penalty
+$bj\delta/(b+j\delta)$, and
+\[
+ \frac{bj\delta}{b+j\delta}
+ =\sum_{i=0}^{j-1}
+   \frac{b^2\delta}{(b+i\delta)(b+(i+1)\delta)}.
+\]
+Full observations through step $i$ have the sufficient Gaussian
+statistic of precision $b+i\delta$.  Completing the likelihood square
+proves sufficiency directly.  Hence
+\[
+ \|e_iW_1\|_2^2
+ =\E_p|\bar w_{1/(b+i\delta)}(U_{1/(b+i\delta)})|^2.
+\]
+This squared prediction norm is nondecreasing with precision, since a
+weaker observation can be obtained by adding independent Gaussian
+noise.  It is bounded by $I$, so Equation
+\eqref{eq:entropy-loss-discrete-minimum} converges as $m\to\infty$ to
+\[
+ b^2\int_0^u
+ \frac{I-\E_p|\bar w_{1/(b+t)}(U_{1/(b+t)})|^2}{(b+t)^2}\,dt.
+\]
+First take this mesh limit at fixed $N,u$.  It removes the entropy
+error in Equation~\eqref{eq:entropy-loss-posterior-lsi}, while
+Equation~\eqref{eq:entropy-loss-dual-error} remains uniform in the
+mesh.  Divide by $N$, then let $N\to\infty$, using $q_N\to0$.
+Equations~\eqref{eq:entropy-loss-posterior-lsi} and
+\eqref{eq:entropy-loss-variational} give
+\[
+ 2(H_0-H_z)
+ \le\frac Ib-\int_0^u
+ \frac{I-\E_p|\bar w_{1/(b+t)}(U_{1/(b+t)})|^2}{(b+t)^2}\,dt
+ =\frac{I}{b+u}
+   +\int_{1/(b+u)}^{1/b}\E_p|\bar w_r(U_r)|^2\,dr.
+\]
+Finally let $u\to\infty$.  Since $1/b=z$ and the integrand is bounded
+by $I$, this proves Equation~\eqref{eq:prediction-entropy-loss} in the
+claimed order of limits.
+\end{proof}
+
+We return to the critical sequence.  For $Y_0\sim\eta$ and independent
+Gaussian noise, define
+\[
+ U_r=Y_0+\sqrt rG,\qquad
+ \bar w_r(U_r)=\E[w(Y_0)\mid U_r],\qquad
+ m(r)=I^{-1}\E|\bar w_r(U_r)|^2.
+\]
+In the strict case we also use the label $L_T$ of
+Lemma~\ref{lem:efficient-probes}, constructed with fresh Gaussian noise
+independent of $(Y_0,G)$. Thus $L_T$ and $U_r$ are conditionally
+independent given $Y_0$. Set
+\[
+ p_T(r)=I^{-1}\E|\E[w(Y_0)\mid U_r,L_T]|^2,
+ \qquad p_0(r)=m(r).
+\]
+
+\begin{proposition}[The prediction curve]
+\label{prop:prediction}
+For the strict and gap critical sequences, for every fixed
+$0<r<1/4$,
+\begin{equation}
+ m(r)\longrightarrow e^{-r}.
+ \label{eq:prediction-curve}
+\end{equation}
+In the strict case, for every fixed finite $T\ge0$ one also has
+$p_T(r)\to e^{-r}$.  In the gap case the eigenfunction may be rescaled
+so that $I\to0$, without changing Equation~\eqref{eq:prediction-curve}
+or Equation~\eqref{eq:prediction-input}.
+\end{proposition}
+
+\begin{proof}
+For $0\le r<1/4$ put
+\[
+ \theta(r)=\exp\left(-\int_0^r\frac{dt}{R_{s+t}}\right).
+\]
+Proposition~\ref{prop:regularized-constants} gives
+$\theta(r)=e^{-r}+o(1)$, uniformly on compact intervals under
+consideration.  The common heat flow dissipates relative entropy at
+one half the relative Fisher information.  Thus its entropy relative
+to $\mu_{s+r}$ is at most $\theta(r)$ times its initial entropy,
+by the logarithmic Sobolev inequality at the contemporaneous base law.
+This assertion applies to conditional laws after averaging as well.
+
+In the strict case write
+$H_r=H(\Law(U_r)\mid\mu_{s+r})$ and
+$H_0=I/(2\lambda)$.  The entropy of $U_r$ given $L_T$, relative to
+$\mu_{s+r}$ and averaged over the label, is at most
+$\theta(r)(H_0+TI/2)$, since initially it is $H_0+\MI(Y_0;L_T)$.
+The chain rule gives
+\begin{align*}
+ \frac I2\int_0^T p_v(r)\,dv
+ &=\E H(\Law(L_T\mid U_r)\mid\gamma)\\
+ &=\E H(\Law(U_r\mid L_T)\mid\mu_{s+r})+d(T)-H_r\\
+ &\le\frac I2\theta(r)(1/\lambda+T)+d(T).
+\end{align*}
+The first equality is Lemma~\ref{lem:gaussian-signals}, conditionally
+on $U_r$.  Fix $v\ge0$ and then $T>v$.  Since $p_t(r)$ is
+nondecreasing in signal strength $t$, Lemma
+\ref{lem:efficient-probes} yields
+\[
+ (T-v)\limsup p_v(r)\le e^{-r}(1+T).
+\]
+The sequence limit is taken with $T$ fixed.  Allowing this fixed $T$
+to be arbitrarily large proves
+\begin{equation}
+ \limsup p_v(r)\le e^{-r}.
+ \label{eq:prediction-upper}
+\end{equation}
+
+Lemma~\ref{lem:entropy-loss} applies to $p=\eta$ and $q=\mu_s$;
+the exponential-square moment was proved in the proof of
+Lemma~\ref{lem:efficient-probes}.  Since $H_z\le\theta(z)H_0$, it gives
+\[
+ \int_0^z m(r)\,dr\ge\frac{1-\theta(z)}{\lambda}
+ \qquad(0<z<1/4).
+\]
+The reverse limiting inequality follows from
+Equation~\eqref{eq:prediction-upper}, $0\le m\le1$, and reverse
+Fatou.  Hence
+\begin{equation}
+ \int_0^z m(r)\,dr\longrightarrow1-e^{-z}.
+ \label{eq:prediction-integral}
+\end{equation}
+The function $m$ is nonincreasing in $r$.  For $0<h<r$ with
+$r+h<1/4$, its value at $r$ is bounded below and above by its
+averages on $[r,r+h]$ and $[r-h,r]$, respectively.  Equation
+\eqref{eq:prediction-integral} determines the limits of both averages;
+letting $h\downarrow0$ gives Equation~\eqref{eq:prediction-curve}.
+Finally $m(r)\le p_T(r)$ and Equation~\eqref{eq:prediction-upper}
+give the assertion for every fixed $T$.
+
+We now prove the gap assertion without using additional probes.
+First linearize Lemma~\ref{lem:entropy-loss}.  Let $q$ be a bounded
+smooth centered test under $\mu_s$, with bounded gradient, and let
+\[
+ d\eta_\varepsilon=(1+\varepsilon q)\,d\mu_s,
+ \qquad |\varepsilon|\|q\|_\infty<1.
+\]
+This alternative has a finite exponential-square moment.  Write
+$\bar q_r=\E_{\mu_s}[q(Y_0)\mid U_r]$ and
+$\overline{Dq}_r=\E_{\mu_s}[Dq(Y_0)\mid U_r]$.
+The output density ratio is $1+\varepsilon\bar q_r$, while the
+predicted initial relative score is exactly
+\[
+ \E_{\eta_\varepsilon}[D\log(1+\varepsilon q)(Y_0)\mid U_r]
+ =\frac{\varepsilon\overline{Dq}_r}{1+\varepsilon\bar q_r}.
+\]
+Consequently its squared norm averaged under the alternative output
+is
+$\varepsilon^2\E_{\mu_{s+r}}[|\overline{Dq}_r|^2/
+(1+\varepsilon\bar q_r)]$.  The entropy expansion
+$H((1+\varepsilon q)\mu_s\mid\mu_s)
+=\varepsilon^2\|q\|_2^2/2+o(\varepsilon^2)$, and its output version,
+now give
+\begin{equation}
+ \|q\|_2^2-\|\bar q_z\|_2^2
+ \le\int_0^z\|\overline{Dq}_r\|_2^2\,dr.
+ \label{eq:prediction-linearized-loss}
+\end{equation}
+Indeed the denominator is bounded away from zero uniformly in $r$,
+and conditional Jensen bounds the integrand by a constant times
+$\|Dq\|_2^2$, justifying the limit.  Approximation in the form norm
+extends Equation~\eqref{eq:prediction-linearized-loss} to $q=\varphi$:
+centered compact smooth approximants have the required boundedness,
+and all conditional expectation maps are $L^2$ contractions.  To retain
+compact support while centering, subtract the approximant's mean times
+one fixed compact smooth function of mean one; this correction tends
+to zero in the form norm.
+
+For clarity, we also give the heat variance identity used here.  For
+a centered scalar or finite-dimensional vector field $g\in L^2(\mu_s)$,
+let $g_r=\E[g(Y_0)\mid U_r]$ under the base law.  Then
+\begin{equation}
+ \frac d{dr}\E_{\mu_{s+r}}|g_r|^2
+ =-\E_{\mu_{s+r}}|Dg_r|^2.
+ \label{eq:prediction-heat-variance}
+\end{equation}
+For bounded $g$, both the numerator density $g_r\mu_{s+r}$ and the
+denominator density $\mu_{s+r}$ solve the heat equation, and the
+quotient calculation with the square function gives
+$(\partial_r-\Delta/2)(\mu_{s+r}|g_r|^2)
+=-\mu_{s+r}|Dg_r|^2$.  Integrate against expanding cutoffs, whose
+Laplacians tend uniformly to zero.  The bound
+$|g_r|\le\|g\|_\infty$ removes the cutoff terms and gives the
+integrated identity on positive time intervals.  Its initial endpoint
+follows from $\|g_r(U_r)-g(Y_0)\|_2\to0$: for a bounded continuous
+test use $g(U_r)$ as a predictor and dominated convergence under the
+coupling $U_r=Y_0+\sqrt rG$, and then approximate in $L^2(\mu_s)$
+and use conditional contraction.  Approximate a general $L^2$ field by bounded
+ones.  Applying the same identity to their differences shows that
+their gradients are Cauchy in the space-time energy norm, while
+conditional contraction gives convergence of both endpoint norms.
+This proves the integrated version of Equation
+\eqref{eq:prediction-heat-variance} in full generality.
+
+Poincar\'e at $\mu_{s+r}$ and preservation of the mean therefore imply
+\[
+ \E_{\mu_{s+r}}|g_r|^2\le\theta(r)\E_{\mu_s}|g|^2.
+\]
+Apply this first to $\varphi$, and then to $w-\E w$.
+The latter application and Equation~\eqref{eq:prediction-input} give
+$m(r)\le\theta(r)+o(1)$.  The former and
+$\|\varphi\|_2^2=I/\lambda$, inserted into Equation
+\eqref{eq:prediction-linearized-loss}, give
+$\int_0^z m(r)\,dr\ge(1-\theta(z))/\lambda$.
+The same integral squeeze and monotonicity argument prove
+Equation~\eqref{eq:prediction-curve}.  Multiplying $\varphi$ by any
+positive scalar multiplies every squared quantity by its square, so
+we may finally choose these scalars to arrange $I\to0$.
+\end{proof}
+
+The hierarchy construction will use the following consequences of the
+prediction curve.  The estimates distinguish conditional covariance
+tails, which are needed only in the strict case, from bounds on
+deterministic subspaces, which hold in both cases.
+
+\begin{lemma}[Posterior covariance and outlier subspaces]
+\label{lem:posterior-outliers}
+Use the critical sequence and notation of
+Proposition~\ref{prop:prediction}, with $I\to0$ in the gap case.
+The following assertions hold.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item For every fixed $c<\infty$, uniformly over deterministic
+orthogonal projections $\Pi$ with $\rank\Pi\le cI$,
+\begin{equation}
+ \E|\Pi w|^2=o(I).
+ \label{eq:prediction-small-subspaces}
+\end{equation}
+\item For every $\epsilon>0$, for all sufficiently far members of the
+sequence there is a deterministic subspace $V_\epsilon$ of dimension
+at most $C_\epsilon I$ such that, for every $v\perp V_\epsilon$,
+\begin{equation}
+ v^*(\E ww^*)v\le\epsilon|v|^2,\qquad
+ v^*(\E b_sb_s^*)v\le\epsilon|v|^2,\qquad
+ v^*(\E M_s)v\le\epsilon|v|^2.
+ \label{eq:prediction-outlier-subspaces}
+\end{equation}
+In the gap case $V_\epsilon$ may be taken to be zero, and
+$\|\E b_sb_s^*\|_{\op}+\|\E M_s\|_{\op}=o(1)$.
+\item In the strict case, for every fixed $r\in(0,1/4)$ and
+$\epsilon>0$, the conditional covariance
+$C_r=\Cov(w(Y_0)\mid U_r)$ satisfies
+\begin{equation}
+ \E\tr\bigl(C_r\1_{\{C_r>\epsilon\Id\}}\bigr)=o(I).
+ \label{eq:conditional-tail}
+\end{equation}
+The indicator denotes the spectral projection.  No assertion about
+additional Gaussian probes or Equation~\eqref{eq:conditional-tail}
+is needed for the gap case.
+\end{enumerate}
+\end{lemma}
+
+\begin{proof}
+We first prove the strict-case spectral estimates.  If a random vector
+$S$ has covariance $C$ and mean $a$, the affine estimator
+\[
+ a+C(\Id+C)^{-1}(S+G-a)
+\]
+of $S$ from $S+G$ is its least-squares affine predictor.  Its mean
+squared norm is $|a|^2+\tr(C^2(\Id+C)^{-1})$, by direct expansion of the
+covariances.  Conditional expectation has at least this squared
+norm, because it is the orthogonal projection onto all functions of
+the observation.  The same calculation holds conditionally on any
+side information, using fresh independent Gaussian noise.
+
+By the Gaussian signal derivative formula and its monotonicity,
+\[
+ \E|\E[w\mid L_1]|^2
+ \le2\bigl(d(2)-d(1)\bigr)\le2d(2)=o(I).
+\]
+Apply the affine prediction bound to $S=w$ and $C=\Cov_\eta w$.
+For every fixed $\epsilon>0$,
+\begin{equation}
+ \tr\bigl(C\1_{\{C>\epsilon\Id\}}\bigr)=o(I),
+ \label{eq:prediction-unconditional-tail}
+\end{equation}
+since $t^2/(1+t)\ge\epsilon t/(1+\epsilon)$ for $t>\epsilon$.
+For every projection of rank at most $cI$,
+\[
+ \E|\Pi w|^2
+ \le\epsilon cI+\tr\bigl(C\1_{\{C>\epsilon\Id\}}\bigr)
+            +|\E w|^2.
+\]
+First take the sequence limit, then $\epsilon\downarrow0$.
+Equation~\eqref{eq:prediction-input} proves
+Equation~\eqref{eq:prediction-small-subspaces} uniformly in $\Pi$.
+In the gap case a projection of rank at most $cI$ is eventually zero,
+which proves the same assertion.
+
+Conditionally on $U_r$, the additional observation $L_1$ has signal
+$w$ and covariance $C_r$.  Orthogonality of successive conditional
+expectations and the conditional affine prediction bound yield
+\[
+ I\bigl(p_1(r)-m(r)\bigr)
+ \ge\E\tr\bigl(C_r^2(\Id+C_r)^{-1}\bigr).
+\]
+Proposition~\ref{prop:prediction} makes the left side $o(I)$.
+The same scalar eigenvalue inequality proves
+Equation~\eqref{eq:conditional-tail}.
+
+We finally bound the dimensions of the deterministic outlier spaces.
+Let
+\[
+ D=\E_\eta\E[XX^*\mid X+\sqrt sG=Y_0],
+\]
+where the inner posterior kernel always refers to the original prior
+$\mu$.  Conditional Jensen and the definition of $M_s$ imply
+\begin{equation}
+ \E_\eta b_sb_s^*\preceq D,\qquad
+ \E_\eta M_s\preceq D/s.
+ \label{eq:prediction-posterior-domination}
+\end{equation}
+For each deterministic projection $Q$ of rank $l\ge1$, the original
+subgaussian assumption gives
+\begin{equation}
+ \log\E_\mu\exp\left(\frac{|QX|^2}{Ca^2}\right)\le Cl.
+ \label{eq:prediction-subspace-moment}
+\end{equation}
+Indeed choose a $1/2$-net of the unit sphere in
+$\operatorname{ran}Q$ with at most $5^l$ points.  The norm $|QX|$ is
+at most twice the largest absolute linear form over this net, and
+bounding the exponential of a maximum by the sum proves the claim.
+
+In the strict case, form the joint law by first drawing $Y_0$ from
+$\eta$ and then drawing $X$ from the original posterior kernel.
+Its entropy relative to the original joint law of
+$(X,X+\sqrt sG)$ is exactly $H_0=I/(2\lambda)$.  Data processing
+therefore bounds the entropy of its $X$-marginal relative to $\mu$ by
+$H_0$.  Applying the entropy variational inequality to
+Equation~\eqref{eq:prediction-subspace-moment} gives
+\begin{equation}
+ \tr(QD)\le Ca^2(H_0+l).
+ \label{eq:prediction-posterior-trace}
+\end{equation}
+If the spectral subspace of $D$ above $\tau>0$ has dimension $l$,
+then $\tau l\le Ca^2(H_0+l)$.  For sufficiently small $a$ this implies
+$l\le2Ca^2H_0/\tau=O_\tau(I)$; the assertion is trivial when $l=0$.
+The spectral subspace of $\E ww^*$ above $\epsilon$ has dimension
+at most $I/\epsilon$, since its trace is $I$.  Take $V_\epsilon$ to
+be the span of this subspace and the spectral subspace of $D$ above
+$s\epsilon$.  Its dimension is at most $C_\epsilon I$, and
+Equation~\eqref{eq:prediction-posterior-domination} gives all three
+bounds in Equation~\eqref{eq:prediction-outlier-subspaces}.
+
+In the gap case, averaging the original posterior under
+$\eta=\mu_s$ gives $D=\E_\mu XX^*\preceq Ca^2\Id$.  Thus
+Equation~\eqref{eq:prediction-posterior-domination} gives the two
+claimed operator bounds directly.  Also
+$\|\E ww^*\|_{\op}\le I\to0$, so $V_\epsilon=\{0\}$ suffices
+eventually.  This completes all assertions.
+\end{proof}
diff --git a/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/build/sections/05-hierarchy.tex b/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/build/sections/05-hierarchy.tex
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+\section{A symmetric hierarchy of nearly first eigenfunctions}
+\label{sec:hierarchy}
+
+The prediction curve from Section~\ref{sec:prediction} determines much more
+than the norm of a conditional mean.  Its Gaussian Bessel inequality is
+asymptotically saturated at every fixed derivative order.  We use this
+saturation to construct symmetric tensors whose derivatives remain nearly
+first eigenfunctions.  We then show that every tensor has negligible energy
+in the curvature directions of $\mu_s$.  Throughout this section, coefficient
+norms are unnormalized Euclidean norms, including for tensors of high order.
+
+For a tensor field with $d$ slots, differentiation appends a last slot.  A
+matrix subscript, such as $K_1$, specifies the slot on which it acts.  Write
+$\|F\|_2^2=\E_\eta|F|^2$ and
+$\langle F,G\rangle_2=\E_\eta\langle F,G\rangle$.  Every occurrence of $A$ on a
+tensor means its componentwise action.  The following formulation records
+precisely the inputs used in the construction; the dimension and $I$ may
+vary along the sequence.
+
+\begin{proposition}[Symmetric tensor hierarchy]\label{prop:hierarchy}
+Consider a sequence of smooth positive probability densities $\eta$ on
+finite-dimensional Euclidean spaces, with finite second moments.  Let
+$A=-\Delta-D\log\eta\cdot D$, and suppose that $\eta$ satisfies Poincar\'e
+with constant at most $\lambda^{-1}$, where $\lambda\to1$.  Fix $s=1/8$.
+Suppose smooth vector fields $b,w$ and a symmetric matrix field $K$ satisfy
+\[
+ D\log\eta=-\frac{y-b(y)}s+\omega(y),\qquad
+ Db=\Id-sK,\qquad 0\le K\le s^{-1}\Id,
+\]
+where either $\omega=w$ throughout the sequence or $\omega=0$ throughout
+the sequence.  Assume $b,w\in L^2(\eta)$, $w$ is a gradient, and, with
+$I=\|w\|_2^2>0$,
+\begin{equation}\label{eq:hierarchy-initial}
+ w\in\Dom A,\qquad
+ \|(A-1)w\|_2=o(\sqrt I),\qquad
+ |\E_\eta w|=o(\sqrt I),\qquad
+ \langle w,Kw\rangle_2=o(I).
+\end{equation}
+Assume also the following prediction and projection properties.
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item For $Y\sim\eta$ and an independent standard Gaussian $G$, set
+\[
+ U_r=Y+\sqrt r\,G,\quad h_r(u)=\E[w(Y)\mid U_r=u],\quad
+ m(r)=I^{-1}\E|h_r(U_r)|^2.
+\]
+For every fixed $0<r<1/4$, one has $m(r)\to e^{-r}$.
+\item For every fixed $c<\infty$,
+\begin{equation}\label{eq:hierarchy-input-projection}
+ \sup_{\substack{\Pi^2=\Pi=\Pi^*\ \mathrm{deterministic}\\
+                   \rank\Pi\le cI}}
+       \|\Pi w\|_2^2=o(I).
+\end{equation}
+For every $\epsilon>0$ there is a deterministic projection $\Pi_\epsilon$
+of rank at most $C_\epsilon I$ such that
+\begin{equation}\label{eq:hierarchy-input-outliers}
+ \left\|\E[(1-\Pi_\epsilon)bb^*(1-\Pi_\epsilon)]\right\|_{\op}
+ +\left\|\E[(1-\Pi_\epsilon)\omega\omega^*
+                  (1-\Pi_\epsilon)]\right\|_{\op}\le\epsilon
+\end{equation}
+eventually along the sequence.
+\item If $\omega=w$, the conditional covariance
+$C_r=\Cov(w(Y)\mid U_r)$ satisfies, for every fixed $r\in(0,1/4)$ and
+$a>0$,
+\begin{equation}\label{eq:hierarchy-input-tail}
+ \E\tr\bigl(C_r\1_{\{C_r>a\}}\bigr)=o(I).
+\end{equation}
+No conditional covariance-tail assumption is made when $\omega=0$.
+\end{enumerate}
+Then there are totally symmetric tensor fields $T_l$ of order $l+1$, for
+every fixed integer $l\ge0$, with $T_0=w$, such that
+\begin{equation}\label{eq:hierarchy-eigen}
+ \begin{gathered}
+ \|T_l\|_2^2=(1+o(1))I,\qquad
+ \|(A-1)T_l\|_2=o(\sqrt I),\\
+ \|DT_l-T_{l+1}\|_2=o(\sqrt I).
+ \end{gathered}
+\end{equation}
+\begin{equation}\label{eq:hierarchy-curvature}
+ \langle T_l,K_1T_l\rangle_2=o(I).
+\end{equation}
+\begin{equation}\label{eq:hierarchy-projection}
+ \sup_{\substack{\Pi^2=\Pi=\Pi^*\ \mathrm{deterministic}\\
+                   \rank\Pi\le cI}}
+      \|\Pi_1T_l\|_2^2=o(I)
+ \qquad\text{for every fixed }c<\infty.
+\end{equation}
+Every $T_l$ belongs componentwise to $\Dom A$.  For $l\ge1$ it can be
+chosen in $\bigcap_{p\ge1}\Dom A^p$.  By symmetry, the specified slot in
+Equations~\eqref{eq:hierarchy-curvature} and
+\eqref{eq:hierarchy-projection} may be any slot.  All assertions can be
+made simultaneous for any fixed finite collection of levels and parameters.
+\end{proposition}
+
+In the application, take $b=b_s$ and $K=K_s$.  Proposition~\ref{prop:critical-sequence}
+supplies the drift, Poincar\'e, and initial estimates;
+Proposition~\ref{prop:prediction} and Lemma~\ref{lem:posterior-outliers}
+supply the remaining inputs.  In the gap branch, $\omega=0$ and
+$\|\E bb^*\|_{\op}=o(1)$, so one may take $\Pi_\epsilon=0$ in
+Equation~\eqref{eq:hierarchy-input-outliers}.  The normalization $I\to0$
+makes Equation~\eqref{eq:hierarchy-input-projection} automatic: eventually
+the only projection of rank at most $cI$ is zero.  Thus no strict-branch
+probe estimate has been added to the gap-branch hypotheses.
+
+We prove the proposition in three steps.  First we construct the
+eigenfunction and derivative hierarchy.  Second we expand its evolution
+along stationary diffusion and, in the strict branch, deduce operator-norm
+tail estimates.  Finally those estimates justify a curvature integration
+by parts.
+
+\subsection{Gaussian Bessel saturation and the induction}
+
+We use the classical correspondence between Hermite polynomials and
+multiple Wiener integrals \cite[Sections~3--5]{Ito1951}, with the
+ordered-index normalization recorded below.  The saturation argument
+will come from the prediction curve, in addition to this orthogonality.
+
+For a standard Gaussian vector $G\in\R^n$, define its symmetric Hermite
+tensors $\mathfrak h_j(G)$ by
+\begin{equation}\label{eq:hierarchy-hermite-definition}
+ e^{\xi\cdot G-|\xi|^2/2}
+ =\sum_{j\ge0}\frac1{j!}\,\mathfrak h_j(G):\xi^{\otimes j}.
+\end{equation}
+Here a colon contracts all indicated slots and $\mathfrak h_0=1$.
+Comparing coefficients in the expectation of the product of two such
+exponentials gives, for symmetric order-$j$ and order-$k$ tensors $a,b$,
+\begin{equation}\label{eq:hierarchy-hermite-isometry}
+ \E[(a:\mathfrak h_j(G))(b:\mathfrak h_k(G))]
+ =\1_{\{j=k\}}j!\langle a,b\rangle.
+\end{equation}
+This identity uses ordered coefficient indices; it explains all factorials
+below.
+
+\begin{lemma}[Bessel inequality for the prediction]\label{lem:hierarchy-bessel}
+For the data of Proposition~\ref{prop:hierarchy}, define
+$g_r(y)=\E_G h_r(y+\sqrt r\,G)$.  For each $r>0$,
+\begin{equation}\label{eq:hierarchy-bessel}
+ \langle w,g_r\rangle_2=m(r)I,\qquad
+ \sum_{j\ge0}\frac{r^j}{j!}\|D^jg_r\|_2^2\le m(r)I.
+\end{equation}
+In particular, every fixed derivative appearing here is in the form
+domain of $A$.
+\end{lemma}
+
+\begin{proof}
+The first equality is conditional expectation:
+\[
+ \E\langle w(Y),h_r(U_r)\rangle=\E|h_r(U_r)|^2.
+\]
+For fixed $y$, Gaussian integration by parts identifies the order-$j$
+Hermite coefficient of $h_r(y+\sqrt r\,G)$ as
+$r^{j/2}D^jg_r(y)$.  Its orthogonal projection onto that chaos is
+\[
+ \frac{r^{j/2}}{j!}D^jg_r(y):\mathfrak h_j(G),
+\]
+where the vector output slot is not contracted.  By
+Equation~\eqref{eq:hierarchy-hermite-isometry}, the squared norm of this
+projection is $r^j|D^jg_r(y)|^2/j!$.  Bessel's inequality, integrated in
+$y\sim\eta$, proves the asserted sum.
+
+These calculations require only
+$\E|h_r(Y+\sqrt rG)|^2<\infty$.  To justify the derivatives, on a compact
+set of starting points use positivity of $\eta$ on a slightly larger
+compact set.  Integrating translated Gaussian kernels over that larger
+set bounds each fixed differentiated kernel by a constant times the
+mixture density of $U_r$.  The enlargement absorbs the polynomial factors
+from differentiation.  Thus the kernel integrals are classically smooth
+locally; alternatively all identities first hold distributionally.  The
+Bessel bounds for orders $j$ and $j+1$ put $D^jg_r$ and its weak derivative
+in $L^2(\eta)$, hence in the form domain by cutoff and local smoothing.
+The local constants need not be uniform in the dimension: only the exact
+integrated Bessel inequality is used quantitatively.
+\end{proof}
+
+Two elementary Hilbert-space facts will prevent dimension factors and
+unjustified differentiations in the induction.
+
+\begin{lemma}[Cross-covariance and spectral smoothing]
+\label{lem:hierarchy-smoothing}
+If $F$ is a square-integrable random variable with values in a
+finite-dimensional Hilbert space and $Z$ is a
+centered Euclidean random vector with $\Cov Z\le c\Id$, then
+\begin{equation}\label{eq:hierarchy-crosscov}
+ |\E(F\otimes Z)|_{\HS}^2\le c\E|F|^2.
+\end{equation}
+Next, let $A$ be a sequence of nonnegative Dirichlet operators with kernel
+the constants and spectral gap at least $\lambda\to1$.  If fields $S$ in
+their form domains satisfy
+\[
+ |\E S|^2=o(I),\qquad \|S\|_2^2=(1+o(1))I,\qquad
+ \|DS\|_2^2\le(1+o(1))I,
+\]
+then $\widetilde S=e^{1-A}S$ satisfies
+\begin{equation}\label{eq:hierarchy-smoothing}
+ \|\widetilde S-S\|_2+\|D(\widetilde S-S)\|_2=o(\sqrt I),
+ \qquad \|(A-1)\widetilde S\|_2=o(\sqrt I).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+The map $u\mapsto u\cdot Z$ from Euclidean space to scalar $L^2$ has
+operator norm at most $\sqrt c$.  Apply its adjoint to each scalar
+coefficient of $F$ and sum the squared bounds.  This proves
+Equation~\eqref{eq:hierarchy-crosscov} without a factor equal to the number
+of coordinates of either variable.
+
+For the second assertion, sum the coefficient spectral measures of
+$S-\E S$ and call the resulting measure $\nu$.  Then
+\[
+ \int(x-\lambda)\,d\nu(x)
+ =\|DS\|_2^2-\lambda\|S-\E S\|_2^2=o(I),
+\]
+where the left side is nonnegative.  For every fixed $\delta>0$, the
+spectral mass and the spectral energy outside
+$[1-\delta,1+\delta]$ are $o(I)$: the part below $1-\delta$ is absent
+eventually, and on $x\ge1+\delta$ both $1$ and $x$ are bounded by a
+$\delta$-dependent constant times $x-\lambda$.  The constant component
+has squared norm $o(I)$ as well.
+
+On the remaining interval the multipliers $e^{1-x}-1$ and
+$(x-1)e^{1-x}$ tend uniformly to zero as $\delta\downarrow0$.  Outside
+it, $(1+x)|e^{1-x}-1|^2\le C(1+x)$ and
+$(x-1)^2e^{2-2x}\le C$.  Integrate these inequalities against the
+spectral measure, first take the sequence limit and then let
+$\delta\downarrow0$.  This gives Equation~\eqref{eq:hierarchy-smoothing}.
+The same functional calculus shows that $e^{1-A}S$ belongs to every
+$\Dom A^p$.
+\end{proof}
+
+\begin{lemma}[Construction before the curvature estimate]
+\label{lem:hierarchy-construction}
+Under the hypotheses of Proposition~\ref{prop:hierarchy}, there exist
+totally symmetric fields $T_l$ satisfying
+Equations~\eqref{eq:hierarchy-eigen} and
+\eqref{eq:hierarchy-projection}, with the stated operator domains.
+Moreover $|\E T_l|=o(\sqrt I)$ for every fixed $l$.
+\end{lemma}
+
+\begin{proof}
+Start with $T_0=w$.  Suppose the assertions have been constructed through
+level $l$, and put $L=l+1$ and $V=DT_l$.  Self-adjointness gives
+\begin{equation}\label{eq:hierarchy-raw-derivative}
+ \|V\|_2^2=\langle T_l,AT_l\rangle_2=(1+o(1))I.
+\end{equation}
+If $\Pi$ is an admissible deterministic projection acting on an existing
+tensor slot, it commutes with $A$ and differentiation.  Consequently
+\[
+ \|\Pi_1V\|_2^2
+ =\langle\Pi_1T_l,A\Pi_1T_l\rangle_2
+ \le\|\Pi_1T_l\|_2^2+
+       \|\Pi_1T_l\|_2\|(A-1)T_l\|_2=o(I).
+\]
+The error is uniform over projections of rank at most $cI$ for fixed $c$.
+
+Fix $0<r<1/4$.  Set $V_j=T_j$ for $0\le j\le l$ and $V_L=V$.
+Successive form integration by parts gives
+\begin{equation}\label{eq:hierarchy-correlations}
+ I^{-1}\langle D^jg_r,V_j\rangle_2\longrightarrow e^{-r}
+ \qquad(0\le j\le L).
+\end{equation}
+Indeed the assertion at $j=0$ is
+Equation~\eqref{eq:hierarchy-bessel}.  For $j\le l$,
+\[
+ \langle D^{j+1}g_r,DT_j\rangle_2
+ =\langle D^jg_r,AT_j\rangle_2
+ =\langle D^jg_r,T_j\rangle_2+o(I).
+\]
+Replacing $DT_j$ by $T_{j+1}$ when $j<l$ costs $o(I)$.
+All test derivatives and their form energies are finite by
+Lemma~\ref{lem:hierarchy-bessel}; its bound controls every error for
+fixed $r$ and $j$.  No uniform estimate as $r\downarrow0$ has been used.
+
+Here is the precise Bessel budget.  Equation~\eqref{eq:hierarchy-correlations}
+and $\|V_j\|_2^2/I\to1$ imply
+\[
+ \liminf I^{-1}e^{2r}\|D^jg_r\|_2^2\ge1
+ \qquad(0\le j\le L).
+\]
+Multiply Equation~\eqref{eq:hierarchy-bessel} by $e^{2r}/I$ and subtract
+these minimum contributions.  The resulting bounds are
+\begin{align}
+ \limsup\frac{\|e^rD^Lg_r-V\|_2^2}{I}
+ &\le\frac{L!}{r^L}
+       \left(e^r-\sum_{j=0}^{L}\frac{r^j}{j!}\right)=O_L(r),
+ \label{eq:hierarchy-budget-error}\\
+ \limsup\frac{e^{2r}\|D^{L+1}g_r\|_2^2}{I}
+ &\le\frac{(L+1)!}{r^{L+1}}
+       \left(e^r-\sum_{j=0}^{L}\frac{r^j}{j!}\right)=1+O_L(r).
+ \label{eq:hierarchy-budget-energy}
+\end{align}
+For the first line, expand the squared difference and use the correlation
+at order $L$; its normalized limit subtracts exactly one from the
+normalized squared derivative norm.  Thus the exponential-series tail
+accounts for the entire excess, rather than merely bounding the norms.
+
+Choose $r\downarrow0$ sufficiently slowly along the sequence in these
+two estimates.  The field $S_0=e^rD^Lg_r$ then satisfies
+\[
+ \|S_0-V\|_2=o(\sqrt I),\qquad
+ \|DS_0\|_2^2\le(1+o(1))I.
+\]
+It is symmetric in its last $L$ slots; $V$ is symmetric in its first
+$L$ slots.  For $L\ge2$, the permutations of those two sets of slots
+generate every permutation of the $L+1$ slots.  Indeed their adjacent
+transpositions together are all the adjacent transpositions.  If a
+permutation fixes $S_0$, its action changes $V$ by at most
+$2\|S_0-V\|_2$; a permutation fixing $V$ costs zero.  Expressing each
+permutation as a fixed finite word therefore shows that $V$ is
+$o(\sqrt I)$ from its full symmetrization.  For $L=1$, $V=Dw$ is
+already symmetric because $w$ is a gradient.  Let $S$ be the full
+symmetrization of $S_0$.  Averaging coefficient permutations is an
+orthogonal contraction and commutes with differentiation, so
+\begin{equation}\label{eq:hierarchy-symmetric-approximant}
+ \|S-V\|_2=o(\sqrt I),\qquad
+ \|S\|_2^2=(1+o(1))I,\qquad
+ \|DS\|_2^2\le(1+o(1))I.
+\end{equation}
+The preceding projection estimate for $V$ now holds on its last slot
+too, since a coefficient-slot permutation moves it to an old slot with
+an $o(\sqrt I)$ error uniform over projections.
+
+To apply Poincar\'e sharply we still need to center $V$.  Write
+$C_y=\Cov(T_l,y)$, $C_b=\Cov(T_l,b)$, and
+$C_\omega=\Cov(T_l,\omega)$, with the vector variable occupying the
+last slot.  Testing the equation for $AT_l$ against centered coordinates
+gives
+\begin{equation}\label{eq:hierarchy-two-center-identities}
+ \E V=C_y+E_l,\qquad |E_l|=o(\sqrt I),\qquad
+ \E V=s^{-1}(C_y-C_b)-C_\omega.
+\end{equation}
+For clarity, the first exact identity before the error is
+$\E V=\E[(AT_l)\otimes(y-\E y)]$.  Poincar\'e gives
+$\Cov_\eta y\le\lambda^{-1}\Id$, and
+Equation~\eqref{eq:hierarchy-crosscov} bounds its error by
+$\lambda^{-1/2}\|(A-1)T_l\|_2$.  This is the needed dimension-free
+estimate.  The second identity follows by integration by parts using
+$Ay=(y-b)/s-\omega$.  These coordinate tests are legitimate in the
+form domain; their distributional $A$-images are in $L^2$ by the stated
+second-moment assumptions, hence they are also in $\Dom A$.
+
+Project the last slot onto $1-\Pi_\epsilon$.  The two covariance terms
+involving $b$ and $\omega$ have norm at most $C\sqrt{\epsilon I}$ by
+Equations~\eqref{eq:hierarchy-input-outliers} and
+\eqref{eq:hierarchy-crosscov}.  Subtracting the two identities in
+Equation~\eqref{eq:hierarchy-two-center-identities}, and using $s\ne1$,
+therefore bounds this part of $\E V$ by
+$C_s\sqrt{\epsilon I}+o(\sqrt I)$.  On the $\Pi_\epsilon$ part,
+\[
+ |(\Pi_\epsilon)_{L+1}\E V|
+ \le\|(\Pi_\epsilon)_{L+1}V\|_2=o(\sqrt I)
+\]
+by the projected-energy estimate and $\rank\Pi_\epsilon\le C_\epsilon I$.
+First take the sequence limit and then let $\epsilon\downarrow0$.
+This proves $|\E V|=o(\sqrt I)$, and thus $|\E S|=o(\sqrt I)$.
+
+Lemma~\ref{lem:hierarchy-smoothing} applies to $S$.  Define
+\[
+ T_{l+1}=e^{1-A}S.
+\]
+It is symmetric, lies in every operator-power domain, is
+$o(\sqrt I)$ from $V$, and satisfies the strong near-eigenfunction
+estimate.  Its mean is $e\E S=o(\sqrt I)$.  Finally, coefficient
+projections commute with the semigroup, and $\|e^{1-A}\|_{2\to2}\le e$.
+Together with $S-V=o_{L^2}(\sqrt I)$, this preserves the uniform
+projected-energy estimates.  The induction is complete.  Each step used
+only a finite collection of limits; diagonal choices give the assertion
+for all fixed levels.
+\end{proof}
+
+\subsection{Expansion along stationary diffusion}
+
+The fields just constructed retain the same norm under repeated
+differentiation.  This makes their stochastic Taylor expansion summable
+with a factorial remainder.  We use the ordered-integral form of the
+Wiener--It\^o correspondence \cite[Section~5]{Ito1951}.  Let $Y_t$ be
+the stationary diffusion with
+invariant density $\eta$,
+\[
+ dY_t=\sqrt2\,dB_t+D\log\eta(Y_t)\,dt,\qquad Y_0\sim\eta,
+\]
+where $Y_0$ is independent of the driving Brownian motion.
+
+\begin{lemma}[Hermite expansion of the stationary evolution]
+\label{lem:hermite-expansion}
+Let symmetric fields $T_l$ satisfy Equation~\eqref{eq:hierarchy-eigen}.
+For fixed integers $l,m\ge0$ and fixed $t>0$, set $N=B_t/\sqrt t$.  Then
+\begin{equation}\label{eq:hierarchy-diffusion-expansion}
+ T_l(Y_t)=e^{-t}\sum_{j=0}^m\frac{(2t)^{j/2}}{j!}
+       T_{l+j}(Y_0):\mathfrak h_j(N)+\mathcal E_{l,m,t},
+\end{equation}
+where the colon contracts the last $j$ tensor slots and
+\begin{equation}\label{eq:hierarchy-diffusion-remainder}
+ \limsup\frac{\|\mathcal E_{l,m,t}\|_2}{\sqrt I}
+ \le\frac{(2t)^{(m+1)/2}}{\sqrt{(m+1)!}}.
+\end{equation}
+For $0\le t\le T<\infty$ the same bound holds uniformly with $t$
+replaced on the right by $T$ and with the expansion at $t=0$ interpreted
+by continuity.  The limits are also simultaneous for any fixed finite
+set of starting levels.  In particular the normalized remainder tends
+to zero when the sequence limit is followed by $m\to\infty$.
+\end{lemma}
+
+\begin{proof}
+The operator-domain It\^o formula from Lemma~\ref{lem:dirichlet-core}
+gives, for each fixed $d$,
+\begin{align*}
+ e^vT_d(Y_v)
+ ={}&T_d(Y_0)+\sqrt2\int_0^v e^uT_{d+1}(Y_u):dB_u\\
+ &+\int_0^v e^u(1-A)T_d(Y_u)\,du
+   +\sqrt2\int_0^v e^u(DT_d-T_{d+1})(Y_u):dB_u.
+\end{align*}
+Stationarity, Cauchy--Schwarz for the time integral, and It\^o isometry
+show that the last two terms are $o(\sqrt I)$, uniformly for
+$v\in[0,T]$ in squared-mean norm.  Substitute this formula for the
+integrand at the next level, and repeat $m+1$ times.  For fixed $l,m,T$,
+every replacement error remains $o(\sqrt I)$; iterated isometry sums
+coefficient squares and introduces no dimension factor.
+
+After multiplication by $e^{-t}$, the remaining integral has coefficient
+$2^{(m+1)/2}$, is over the ordered time simplex of length $t$, and has
+innermost integrand $e^{u-t}T_{l+m+1}(Y_u)$.  Since $u\le t$,
+its squared norm is at most
+\[
+ 2^{m+1}\frac{t^{m+1}}{(m+1)!}
+       \|T_{l+m+1}\|_2^2
+ =(1+o(1))I\frac{(2t)^{m+1}}{(m+1)!}.
+\]
+This proves the remainder estimate and its uniform version.
+The other terms have coefficients evaluated at $Y_0$.  Symmetry in
+their Brownian slots turns their ordered iterated integrals into
+$t^{j/2}\mathfrak h_j(N)/j!$.  To verify the factor directly, expand
+the Brownian exponential martingale
+$\exp(\xi\cdot B_t-t|\xi|^2/2)$ by iterated It\^o integration and
+compare its coefficients with
+Equation~\eqref{eq:hierarchy-hermite-definition}.  The additional
+$\sqrt2$ at each stochastic integration gives exactly
+$(2t)^{j/2}/j!$ in
+Equation~\eqref{eq:hierarchy-diffusion-expansion}.
+\end{proof}
+
+\subsection{Operator tails in the strict branch}
+
+For a tensor $F$, a one-slot flattening is the matrix whose row index is
+one specified tensor slot and whose column index is the tuple of all
+remaining slots.  Its Hilbert--Schmidt norm is exactly $|F|$.
+
+In the strict branch, put $Q=Dw$. The drift decomposition gives
+$D^2(-\log\eta)=K-Q$. Viewing $DF$ as a matrix with its derivative
+slot as row index, commuting a derivative with $A$ formally gives
+\[
+ D(AF)=A(DF)+(K-Q)DF.
+\]
+For $F=T_l$, both $Q$ and $DF$ belong to $L^2$, but their product
+$QDF$ need not belong to $L^2$. We cannot yet use this formula to
+deduce the curvature estimate. The conditional covariance bound will
+instead let us approximate $DF$ by tests $P$ whose uniform operator
+norms tend to zero. Then
+$\|QP\|_2\le\|P\|_{\op,\infty}\|Q\|_2=o(\sqrt I)$,
+which is the product bound needed for the justified identity below.
+This step is needed only when $\omega=w$; in the gap branch $Q=0$.
+
+\begin{lemma}[Vanishing operator tails]\label{lem:hierarchy-operator-tails}
+In the strict branch of Proposition~\ref{prop:hierarchy}, $Q=Dw$ and
+the fields $T_l$ constructed in Lemma~\ref{lem:hierarchy-construction}
+satisfy the following property.  For every fixed $\delta>0$, each
+one-slot flattening $F$ of $T_l$, $l\ge1$, and also $F=Q$, obeys
+\begin{equation}\label{eq:hierarchy-operator-tail}
+ \E\tr\bigl(FF^*\1_{\{FF^*>\delta^2\}}\bigr)=o(I).
+\end{equation}
+Equivalently, clipping every singular value at any fixed positive cutoff
+changes the field by $o(\sqrt I)$ in $L^2$.
+\end{lemma}
+
+\begin{proof}
+We first prove the assertion for $Q$.  Couple the heat observations by
+$U_v=Y_0+W_v$, where $W$ is a standard Brownian motion independent of
+$Y_0\sim\eta$, and retain $h_v=\E[w(Y_0)\mid U_v]$.  Put
+$H_v=Dh_v(U_v)$ wherever this derivative is square integrable.  The
+conditional covariance has the exact representation
+\begin{equation}\label{eq:hierarchy-conditional-bracket}
+ C_r=\int_0^r\E[H_vH_v^*\mid U_r]\,dv,
+ \qquad
+ \int_0^r\E|H_v|^2\,dv=(1-m(r))I.
+\end{equation}
+To check the first identity and its normalization, let $p_v$ denote the
+density of $U_v$.  The entries of $p_vh_v$ solve the heat equation with
+generator $\Delta/2$.  Direct differentiation gives the matrix identity
+\[
+ (\partial_v-\tfrac12\Delta)(p_vh_vh_v^*)
+ =-p_vDh_v(Dh_v)^*.
+\]
+Integrate it against the later heat kernel and divide by $p_r$.  The
+initial term becomes $\E[ww^*\mid U_r]$, and the final term is
+$h_rh_r^*$, which proves the first formula.  Taking the trace and
+expectation proves the second formula.  For nonsmooth or unbounded
+initial $w$, perform the calculation on scalar projections with convex
+truncations increasing to the square, and then use polarization.  Spatial
+cutoffs and the square-integrability of $w$ justify the positive-time
+passage.  At time zero, approximation of $w$ in $L^2(\eta)$ by bounded
+continuous functions gives
+$\E|w-\E[w\mid U_v]|^2\to0$.  Thus there is no missing boundary term
+in Equation~\eqref{eq:hierarchy-conditional-bracket}.
+
+By Equation~\eqref{eq:hierarchy-initial},
+$\|Q\|_2^2=\langle w,Aw\rangle_2=(1+o(1))I$.
+For almost every $v>0$, convolution differentiation and form integration
+by parts give
+\begin{equation}\label{eq:hierarchy-derivative-correlation}
+ \E\langle Q(Y_0),H_v\rangle
+ =\langle Dw,Dg_v\rangle_2
+ =\langle Aw,g_v\rangle_2=m(v)I+o(I).
+\end{equation}
+The error is uniform in $v$: Jensen's inequality gives
+$\|g_v\|_2\le\sqrt I$, so it is bounded by
+$\|(A-1)w\|_2\sqrt I$.  The convolution derivative identity follows
+first with cutoffs and bounded functions, then by the square-integrable
+derivative bounds in Equation~\eqref{eq:hierarchy-conditional-bracket}.
+
+Fix a small $z>0$ with $2z<1/4$.  Combining the last two equations and
+using $0\le m(v)\le1$ yields the exact limiting discrepancy
+\begin{align}
+ \lim\frac1I\int_0^{2z}\E|Q(Y_0)-H_v|^2\,dv
+ &=2z+1-e^{-2z}-2\int_0^{2z}e^{-v}\,dv\\
+ &=2z-1+e^{-2z}=O(z^2).
+ \label{eq:hierarchy-integrated-discrepancy}
+\end{align}
+Dominated convergence is applicable to $m(v)$, so this calculation does
+not require uniform convergence of the prediction curve near zero.
+
+We select a time in $[z,2z]$ while preserving a covariance-tail bound.
+For a fixed $\kappa>0$, define
+\[
+ a(r)=I^{-1}\E\tr\left(\frac{C_r}{r}
+                   \1_{\{C_r/r>\kappa\}}\right).
+\]
+For fixed $r\in[z,2z]$, Equation~\eqref{eq:hierarchy-input-tail} gives
+$a(r)\to0$, and $a(r)\le1/z$ follows from $\E\tr C_r\le I$.
+Consequently $\int_z^{2z}a(r)\,dr\to0$.  On the other hand,
+Equation~\eqref{eq:hierarchy-integrated-discrepancy} and Markov's
+inequality show that the set of times satisfying
+\[
+ \E|Q(Y_0)-H_r|^2\le C zI
+\]
+has measure at least $z/2$, with an absolute $C$ and for sufficiently
+far terms of the sequence.  The set where $a(r)$ exceeds a quantity
+tending to zero has measure tending to zero.  Select $r\in[z,2z]$
+in the intersection.  Thus $r$ may vary with the sequence index, but
+\begin{equation}\label{eq:hierarchy-selected-time}
+ \E|Q(Y_0)-H_r|^2\le CzI,\qquad
+ \E\tr\left(\frac{C_r}{r}\1_{\{C_r/r>\kappa\}}\right)=o(I).
+\end{equation}
+All constants in this selection are for fixed $z,\kappa$.
+
+We compare $B=H_rH_r^*$ with $C=C_r/r$ in trace norm.  For matrices
+$U,V$,
+\begin{equation}\label{eq:hierarchy-product-trace}
+ \|UU^*-VV^*\|_1
+ \le |U-V|_{\HS}(|U|_{\HS}+|V|_{\HS}).
+\end{equation}
+Compare both $B$ and $C$ with $\E[QQ^*\mid U_r]$.  For $B$, conditional
+Jensen's inequality and the first bound in
+Equation~\eqref{eq:hierarchy-selected-time} give an $O(\sqrt z)I$
+bound on the expected trace-norm difference.  For $C$, use the first
+identity in Equation~\eqref{eq:hierarchy-conditional-bracket}, followed
+by Equation~\eqref{eq:hierarchy-product-trace} and Cauchy--Schwarz in
+both probability and time.  The integrated discrepancy is $O(z^2)I$,
+whereas
+\[
+ \int_0^r\E(|H_v|+|Q|)^2\,dv
+ \le 2(1-m(2z))I+4z\|Q\|_2^2.
+\]
+Here we used $r\le2z$ and the nonnegative integral in the second
+identity of Equation~\eqref{eq:hierarchy-conditional-bracket}.
+The normalized limsup of the right side is $O(z)$, by prediction at
+the fixed time $2z$. No prediction limit at the selected varying time
+is needed.
+Division by $r\ge z$ therefore also costs at most $O(\sqrt z)I$.
+We have proved, in normalized limsup,
+\begin{equation}\label{eq:hierarchy-bracket-comparison}
+ \E\|B-C\|_1\le O(\sqrt z)I.
+\end{equation}
+
+Fix $\delta>0$, take $\kappa=\delta^2/2$ above, and set
+$P=\1_{\{B>\delta^2\}}$.  Positivity of $C$ implies
+\[
+ \tr(BP)\le\frac{\delta^2}2\rank P
+      +\tr\bigl(C\1_{\{C>\delta^2/2\}}\bigr)+\|B-C\|_1.
+\]
+Since $\delta^2\rank P\le\tr(BP)$, the first term is absorbed into
+the left side.  Equations~\eqref{eq:hierarchy-selected-time} and
+\eqref{eq:hierarchy-bracket-comparison} bound the remaining high
+singular-value energy by $O(\sqrt z)I+o(I)$.  Clip $H_r$ at $\delta$;
+its squared distance from the operator-norm ball of radius $\delta$ is
+at most this energy.  The first estimate in
+Equation~\eqref{eq:hierarchy-selected-time} then gives the same bound,
+up to constants, for the distance of $Q$ from that ball.  Let $z\downarrow0$
+after the sequence limit.  The squared distance is $o(I)$ for every
+fixed $\delta>0$.  Applying this distance assertion at cutoff
+$\delta/2$ bounds the energy of singular values above $\delta$ by
+four times the squared distance, proving
+Equation~\eqref{eq:hierarchy-operator-tail} for $Q$.  Since
+$\|T_1-Q\|_2=o(\sqrt I)$, it also proves the assertion for $T_1$.
+
+It remains to propagate this estimate to higher levels.  Fix $l\ge1$
+and $t>0$.  Apply Lemma~\ref{lem:hermite-expansion} with starting level
+$1$.  Conditional on $Y_0$, the Hermite chaoses are orthogonal even
+after applying an arbitrary measurable projection $P(Y_0)$ to the
+first tensor slot.  The chaos containing $T_l(Y_0)$ has squared
+coefficient
+\[
+ c_{l,t}=e^{-2t}\frac{(2t)^{l-1}}{(l-1)!}>0.
+\]
+Use the expansion to order $m\ge l-1$, first take the sequence limit,
+and then let $m\to\infty$ in its remainder bound.  The resulting
+inequality is
+\begin{equation}\label{eq:hierarchy-random-projection}
+ \liminf\frac{
+   \E|P(Y_0)_1T_1(Y_t)|^2-c_{l,t}\E|P(Y_0)_1T_l(Y_0)|^2}{I}\ge0.
+\end{equation}
+The remainder estimates are uniform over such projections, since their
+operator norms are at most one.
+
+Choose $P(Y_0)$ to project onto the singular values greater than a fixed
+$\epsilon>0$ of the first-slot flattening of $T_l(Y_0)$.  Then
+\[
+ \E\rank P\le\epsilon^{-2}\|T_l\|_2^2
+ =(1+o(1))\epsilon^{-2}I.
+\]
+For every $\delta>0$, let $S_\delta$ be the singular-value clipping of
+$T_1$ at $\delta$.  Pointwise,
+\[
+ |P(Y_0)S_\delta(Y_t)|_{\HS}^2
+ \le\delta^2\rank P(Y_0).
+\]
+Stationarity controls the $o(I)$ clipping error for $T_1(Y_t)$.
+Hence the normalized limsup of the first expectation in
+Equation~\eqref{eq:hierarchy-random-projection} is at most
+$\delta^2/\epsilon^2$.  Send $\delta\downarrow0$ to obtain the tail
+claim at threshold $\epsilon$ for $T_l$.  No independence between
+$P(Y_0)$ and $T_1(Y_t)$ was required.  Tensor symmetry supplies every
+other one-slot flattening.
+\end{proof}
+
+\subsection{The curvature estimate and its domains}
+
+It remains to prove the curvature assertion of the hierarchy.
+We first justify the differentiated operator identity against a test
+whose product with the curvature is in $L^2$. The preceding tail
+estimate will supply such a test in the strict branch; the gap branch
+will use $T_{l+1}$ directly.
+
+\begin{samepage}
+\begin{lemma}[Weighted divergence and the differentiated operator]
+\label{lem:hierarchy-divergence}
+Let $\eta$ be a smooth positive probability density, let
+$A=-\Delta-D\log\eta\cdot D$, and put $H=D^2(-\log\eta)$.
+Let $P=(P_{i\alpha})$ have one vector slot $i$ and a finite-dimensional
+coefficient index $\alpha$.  Suppose every coefficient belongs to
+$\Dom A$ and $HP\in L^2(\eta)$.  Then the weighted adjoint divergence
+\[
+ (D^*P)_\alpha=-\sum_i\partial_iP_{i\alpha}
+                   -\sum_i(\partial_i\log\eta)P_{i\alpha}
+\]
+belongs to $L^2(\eta)$ and satisfies
+\begin{equation}\label{eq:hierarchy-divergence-bound}
+ \|D^*P\|_2^2\le\|DP\|_2^2+\langle P,HP\rangle_2.
+\end{equation}
+For every coefficient field $F\in\Dom A$,
+\begin{equation}\label{eq:hierarchy-differentiated-operator}
+ \langle AF,D^*P\rangle_2
+ =\langle DF,AP\rangle_2+\langle DF,HP\rangle_2.
+\end{equation}
+In particular this identity requires $HP\in L^2$, not $HDF\in L^2$.
+\end{lemma}
+\end{samepage}
+
+\begin{proof}
+For compactly supported smooth $P$, commuting two first derivatives in
+weighted integration by parts gives
+\begin{equation}\label{eq:hierarchy-divergence-identity}
+ \|D^*P\|_2^2
+ =\sum_{i,j,\alpha}
+       \langle\partial_jP_{i\alpha},\partial_iP_{j\alpha}\rangle_2
+   +\langle P,HP\rangle_2.
+\end{equation}
+Cauchy--Schwarz applied to the two arrays with $i,j$ interchanged bounds
+the sum by $\|DP\|_2^2$, without a dimension factor.
+
+For the stated $P$, membership in $\Dom A$ gives $P\in H^2_{\rm loc}$
+by local elliptic regularity and $DP\in L^2(\eta)$.  Choose smooth
+cutoffs $0\le\chi_R\le1$ tending to one with
+$\|D\chi_R\|_\infty\to0$.  Equation~\eqref{eq:hierarchy-divergence-identity}
+applies to $\chi_RP$ by local Sobolev approximation.  Since
+\[
+ \|D(\chi_RP)-DP\|_2\longrightarrow0,
+ \qquad
+ \langle\chi_RP,H\chi_RP\rangle_2
+   \longrightarrow\langle P,HP\rangle_2,
+\]
+its right side is bounded.  The second convergence uses
+$|\langle P,HP\rangle|\in L^1$, a consequence of $P,HP\in L^2$.
+Weak compactness in $L^2$ and distributional convergence of
+$D^*(\chi_RP)$ identify the limit as $D^*P$.  Lower semicontinuity
+proves Equation~\eqref{eq:hierarchy-divergence-bound}.  Only first
+derivatives of the cutoffs have entered the estimate.
+
+For compactly supported smooth $F$, differentiate the scalar operator:
+\[
+ D(AF)=A(DF)+HDF.
+\]
+Pair this identity with $P$, use symmetry of $H$, and integrate by
+parts to obtain Equation~\eqref{eq:hierarchy-differentiated-operator}.
+For general $F\in\Dom A$, use the operator core from
+Lemma~\ref{lem:dirichlet-core} to choose compact smooth $F_j$ with
+$F_j\to F$ and $AF_j\to AF$ in $L^2$.  This convergence also holds
+in the form norm, since
+\[
+ \|D(F_j-F)\|_2^2
+ =\langle F_j-F,A(F_j-F)\rangle_2\longrightarrow0.
+\]
+All terms in Equation~\eqref{eq:hierarchy-differentiated-operator}
+therefore converge: the three fixed factors are $D^*P$, $AP$, and
+$HP$, each in $L^2$.  This proves the claimed global identity.
+\end{proof}
+
+\begin{proof}[Completion of the proof of Proposition~\ref{prop:hierarchy}]
+Lemma~\ref{lem:hierarchy-construction} supplies the fields and all
+assertions except Equation~\eqref{eq:hierarchy-curvature}.  Fix $l\ge0$
+and write $F=T_l$.  View $DF$ as a matrix with its derivative slot as
+row index and all old tensor slots as column index.  We construct a
+matrix field $P$ satisfying
+\begin{equation}\label{eq:hierarchy-curvature-test}
+ \|P-DF\|_2+\|(A-1)P\|_2=o(\sqrt I),\qquad
+ \|DP\|_2=O(\sqrt I),\qquad \|QP\|_2=o(\sqrt I),
+\end{equation}
+where $Q=Dw$ in the strict branch and $Q=0$ in the gap branch.
+
+In the strict branch, apply Lemma~\ref{lem:hierarchy-operator-tails}
+to the corresponding flattening of $T_{l+1}$.  Choose a singular-value
+cutoff $\delta\downarrow0$ sufficiently slowly that the clipped field
+$C$ obeys
+\[
+ \|C-T_{l+1}\|_2=o(\sqrt I),\qquad
+ \|C(y)\|_{\op}\le\delta.
+\]
+Set $P=e^{1-A}C=eP_1C$.  Markov averaging and convexity of the
+operator-norm ball give $\|P(y)\|_{\op}\le e\delta$.  The spectral
+multipliers $e^{1-x}$, $(x-1)e^{1-x}$, and $\sqrt x\,e^{1-x}$ are
+bounded for $x\ge0$.  Moreover
+$|e^{1-x}-1|\le C|x-1|$ there.  These bounds, the approximation of
+$C$, and the near-eigenfunction assertion for $T_{l+1}$ give the first
+two estimates in Equation~\eqref{eq:hierarchy-curvature-test}.
+Finally $\|Q\|_2^2=(1+o(1))I$ and
+\[
+ \|QP\|_2\le\|P\|_{\op,\infty}\|Q\|_2=o(\sqrt I).
+\]
+The clipping is only used to construct $P$, so it need not preserve
+tensor symmetry.  In the gap branch take $P=T_{l+1}$ directly.
+The first two estimates follow from
+Equation~\eqref{eq:hierarchy-eigen}, and the last is immediate from
+$Q=0$.  No operator-tail argument is needed in this branch.
+
+The drift hypothesis gives $H=D^2(-\log\eta)=K-Q$ in either case.
+Both $KP$ and $QP$ are in $L^2$, and the operator-domain assertion for
+$P$ was retained by its construction.  Lemma~\ref{lem:hierarchy-divergence}
+therefore applies.  In particular,
+\[
+ \|D^*P\|_2^2
+ \le\|DP\|_2^2+s^{-1}\|P\|_2^2+\|P\|_2\|QP\|_2=O(I).
+\]
+Insert $AF=F+(A-1)F$ and $AP=P+(A-1)P$ into
+Equation~\eqref{eq:hierarchy-differentiated-operator}.  The terms
+$\langle F,D^*P\rangle_2$ and $\langle DF,P\rangle_2$ cancel.
+Consequently
+\begin{align*}
+ \langle DF,KP\rangle_2
+ ={}&\langle(A-1)F,D^*P\rangle_2
+     -\langle DF,(A-1)P\rangle_2
+     +\langle DF,QP\rangle_2
+ =o(I).
+\end{align*}
+Since $\|K\|_{\op,\infty}\le s^{-1}$ and
+$P-DF=o_{L^2}(\sqrt I)$, this gives
+\[
+ \langle DF,KDF\rangle_2=o(I).
+\]
+Replacing $DF$ by $T_{l+1}$ costs $o(I)$ by the derivative estimate.
+It proves Equation~\eqref{eq:hierarchy-curvature} at level $l+1$ on
+the derivative slot, and hence on every slot by symmetry.  Level zero
+was an initial assumption.  All claims of the proposition follow.
+\end{proof}
+
+To use many tensor degrees at once, first impose the estimates on a
+fixed finite list of levels, projection-rank constants, and times.
+Then increase that list sufficiently slowly along the original sequence.
+For example, one may choose $k\to\infty$ so that the normalized errors
+in Proposition~\ref{prop:hierarchy} tend to zero uniformly through
+level $3k$, while $\sup_{l\le3k}\|T_l\|_2^2/I$ stays bounded.
+The same diagonal applies to any fixed number of terms in
+Lemma~\ref{lem:hermite-expansion} and any fixed bounded time interval.
+Thus averaging over a band of degrees or shifting that band by a fixed
+integer does not change any of the preceding normalized estimates.
diff --git a/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/build/sections/06-matrices.tex b/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/build/sections/06-matrices.tex
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+\section{The reduced square root and a frozen matrix algebra}\label{sec:matrices}
+
+The hierarchy of Proposition~\ref{prop:hierarchy} contains tensors of
+arbitrarily high order whose derivatives have almost the same energy as the
+tensors themselves.  We first transfer this near equality to the positive
+square root of their one-slot reduction.  We then describe that square root
+along the stationary diffusion by matrices frozen at the initial point.
+The construction uses weighted traces throughout; in particular, no lower
+bound on a nonzero eigenvalue will be required.
+
+\subsection{Sharp energy for the reduced square root}
+
+All limits in this section are along the counterexample sequence.  We make
+the following diagonal choice explicit.  For each finite integer $K$, impose
+the conclusions of Proposition~\ref{prop:hierarchy} through level $3K$, the
+fixed-projection conclusions for rank coefficients at most $K$, and the
+estimates of Lemma~\ref{lem:hermite-expansion} for every starting level
+$0\le l<2K$ and every truncation order $0\le m\le K$, on time
+intervals $0\le t\le K$.  Only levels through $3K$ enter these
+expansions, including their remainders.  For each such finite list, the
+hierarchy defects and the excesses over the normalized Hermite remainder
+bounds can be made arbitrarily small.  Choose an integer $k\to\infty$
+slowly enough that these defects and excesses tend to zero, uniformly over
+the corresponding list.  Further auxiliary parameters below will be chosen
+by the same finite-list procedure.  In particular, normalized endpoint
+energies remain bounded when a band of $k$ levels is shifted by a fixed
+number of levels.
+
+Regard the direct sum
+\[
+ \Psi=\bigoplus_{k\le l<2k}\frac{T_l}{\sqrt{kI}}
+\]
+as a matrix by separating the first tensor index from the remaining
+indices and the direct-sum label.  Define
+\begin{equation}\label{eq:matrix-reduction}
+ \rho=\Psi\Psi^*,\qquad R=\rho^{1/2}.
+\end{equation}
+High tensor degree provides many interchangeable slots, while averaging
+adjacent degrees makes each fixed shift cost only endpoint terms of
+order $k^{-1}$.  These two features will force sharp energy estimates
+for the positive matrix field $R$.
+
+Here $R$ is a positive matrix field, distinct from the scalar log-Sobolev
+constant $R(\nu)$.  Write $\|\cdot\|_2$ for the field norm formed with
+ordinary Hilbert--Schmidt coefficient norms.  The hierarchy gives
+\begin{equation}\label{eq:matrix-normalization}
+ \E\tr R^2=\|\Psi\|_2^2\longrightarrow1,
+ \qquad \|D\Psi\|_2^2\longrightarrow1.
+\end{equation}
+
+\begin{lemma}[Centering and the square-root energy]\label{lem:matrix-centering}
+The field $R$ belongs to the Dirichlet form domain and satisfies
+\[
+ \|\E R\|_{\HS}=o(1),\qquad
+ \liminf\|DR\|_2^2\ge1,\qquad \|DR\|_2^2=O(1).
+\]
+\end{lemma}
+\begin{proof}
+For rectangular matrices $X,Y$, the Araki--Yamagami inequality
+\cite[Theorem~1]{ArakiYamagami1981} gives the following square-root bound.
+We give a finite-dimensional proof by self-adjoint dilation:
+\begin{equation}\label{eq:matrix-root-lipschitz}
+ \|\sqrt{XX^*}-\sqrt{YY^*}\|_{\HS}^2
+ \le 2\|X-Y\|_{\HS}^2.
+\end{equation}
+Indeed the absolute value of the dilation of $X$ has diagonal blocks
+$\sqrt{XX^*}$ and $\sqrt{X^*X}$.  If $A,B$ are self-adjoint and $f$ is
+scalar Lipschitz, expansion in an eigenbasis of $A$ and an eigenbasis of $B$
+shows that
+$\|f(A)-f(B)\|_{\HS}\le\operatorname{Lip}(f)\|A-B\|_{\HS}$.
+Apply this with $f(x)=|x|$.  Difference quotients, followed by form
+approximation, prove that $R$ is in the form domain and that
+$\|DR\|_2^2\le2\|D\Psi\|_2^2$.
+
+The curvature conclusion of the hierarchy yields
+\[
+ \E\tr((1-M_s)R^2)=o(1).
+\]
+We also have
+\begin{equation}\label{eq:matrix-independent-centering}
+ \tr((\E M_s)(\E R^2))=o(1).
+\end{equation}
+To see the latter assertion in the strict branch, fix $\delta>0$ and let
+$P_\delta$ be the spectral projection of $\E M_s$ onto $(\delta,\infty)$.
+Lemma~\ref{lem:posterior-outliers} gives
+$\rank P_\delta=O_\delta(I)$; the deterministic-projection conclusion of
+Proposition~\ref{prop:hierarchy} therefore gives
+$\tr(P_\delta\E R^2)=o(1)$.  Since $0\le M_s\le1$, the left side of
+Equation~\eqref{eq:matrix-independent-centering} is at most
+$o(1)+\delta\E\tr R^2$.  Send $\delta$ to zero after the sequence limit.
+In the gap branch use directly $\|\E M_s\|_{\op}=o(1)$.
+
+Let $R'$ be evaluated at an independent point.  Insert $1-M_s+M_s$
+between $R$ and $R'$ in
+$\|\E R\|_{\HS}^2=\E\tr(RR')$.
+Hilbert--Schmidt Cauchy--Schwarz bounds the term containing $1-M_s$ by
+\[
+ \bigl(\E\tr((1-M_s)R^2)\bigr)^{1/2}
+ \bigl(\E\tr R^2\bigr)^{1/2}=o(1).
+\]
+The other term is bounded by
+\[
+ \bigl(\E\tr(M_sR^2)\bigr)^{1/2}
+ \bigl(\tr((\E M_s)(\E R^2))\bigr)^{1/2}=o(1).
+\]
+Poincar\'e for $\eta$, whose constant tends to one, now gives the lower
+bound on $\|DR\|_2^2$ from Equation~\eqref{eq:matrix-normalization}.
+\end{proof}
+
+The next estimate is the reason for using tensors with many symmetric
+slots.  It compares averaging an action over those slots with the
+geometric, rather than arithmetic, weighting by the marginal matrix.
+
+\begin{lemma}[Averaging over symmetric tensor slots]\label{lem:matrix-copied-slots}
+Let $\psi$ be a finite direct sum of symmetric tensors, each with at least
+$k\ge2$ slots.  Let $\rho$ be its one-slot reduction and $R=\sqrt\rho$.
+For a self-adjoint matrix $H$, let $H_{\mathrm{av}}$ act on each component
+by averaging the action of $H$ over its slots.  Then
+\begin{equation}\label{eq:matrix-copied-slots}
+ \|H_{\mathrm{av}}\psi\|^2
+ \le\tr(RHRH)+\frac{2}{k}\tr(\rho H^2).
+\end{equation}
+\end{lemma}
+\begin{proof}
+For a component $\psi_d$ with $d$ slots, put
+\[
+ a_d=\langle\psi_d,H_{(1)}^2\psi_d\rangle,
+ \qquad c_d=\langle\psi_d,H_{(1)}H_{(2)}\psi_d\rangle.
+\]
+Symmetry and Cauchy--Schwarz give $|c_d|\le a_d$, and direct expansion gives
+\[
+ \|H_{\mathrm{av}}\psi\|^2
+ =\sum_d\left(\frac{a_d}{d}+\left(1-\frac1d\right)c_d\right)
+ \le\sum_d c_d+\frac2k\sum_d a_d.
+\]
+It remains to prove $|\sum_d c_d|\le\tr(RHRH)$.  This correlation bound
+can also be deduced from the Wigner--Yanase case of Lieb's trace-concavity
+theorem \cite{Lieb1973}; we give a proof by weighted interpolation.
+
+Define the linear map $C$ by
+\[
+ \tr(XC(Y))=\sum_d\langle\psi_d,X_{(1)}Y_{(2)}\psi_d\rangle.
+\]
+Flattening all remaining indices exhibits $C(Y)$ as a sum of terms
+$A_\alpha Y^{\mathsf T}A_\alpha^*$.  Thus $C$ is positive and
+$C(1)=C^*(1)=\rho$.  Restrict to the support of $\rho$, on which inverse
+powers are defined, and set
+\[
+ U(Y)=\rho^{-1/2}C(Y)\rho^{-1/2}.
+\]
+The map $U$ is unital and contractive in operator norm: its preceding
+representation is an isometric compression of copies of $Y^{\mathsf T}$.
+It also satisfies $\tr(\rho U(Y))=\tr(\rho Y)$.
+For $0\le\Re z\le1$ introduce
+\[
+ U_z(Y)=\rho^{z/2}
+ U(\rho^{-z/2}Y\rho^{-z/2})\rho^{z/2}.
+\]
+On $\Re z=0$, multiplication on either side by imaginary powers of $\rho$
+is isometric, so $U_z$ is operator-norm contractive.  At $z=1$,
+\[
+ U_1(Y)=C(\rho^{-1/2}Y\rho^{-1/2}),\qquad U_1^*(1)=1.
+\]
+Its adjoint is a unital positive map with the same compression and
+transpose representation, hence is operator-norm contractive.  Duality
+makes $U_1$ trace-norm contractive; imaginary powers give the same
+conclusion on $\Re z=1$.
+
+The interpolation used here is the finite-dimensional weighted Schatten
+argument; compare \cite[Section~III, Theorem~4]{Beigi2013}.  We give the details for
+our positive correlation map.
+Given matrices of Hilbert--Schmidt norm one, continue the singular values
+of the input by powers $2z$, and those of the matrix paired with its
+output by powers $2(1-z)$, keeping the unitary factors fixed.  The bilinear
+trace pairing is analytic in the strip, bounded by one on both boundary
+lines by the preceding endpoint bounds and trace duality.  The three-lines
+inequality bounds it by one at $z=1/2$.  Zero singular values follow by
+approximation.  This proves $\|U_{1/2}(Y)\|_{\HS}\le\|Y\|_{\HS}$.
+Take $Y=\rho^{1/4}H\rho^{1/4}$.  Then
+\[
+ \tr(YU_{1/2}(Y))=\tr(HC(H))=\sum_d c_d,
+ \qquad \|Y\|_{\HS}^2=\tr(RHRH).
+\]
+Cauchy--Schwarz proves the claim.  Compression to the support leaves the
+correlation unchanged: every tensor slot lies in that support.  Complexification, used
+only for interpolation, preserves the conclusion for real tensors.
+\end{proof}
+
+\begin{lemma}[Saturation of the square-root energy]\label{lem:matrix-saturation}
+There are measurable symmetric matrix fields $H_i$, $1\le i\le n$, solving
+\begin{equation}\label{eq:matrix-sylvester}
+ D_iR=\frac{H_iR+RH_i}{2},
+\end{equation}
+with zero kernel--kernel block, such that
+\begin{equation}\label{eq:matrix-saturation}
+ \|DR\|_2^2\longrightarrow1,\qquad
+ \sum_i\E\|[H_i,R]\|_{\HS}^2=o(1),\qquad
+ \sum_i\E\tr(R^2H_i^2)\longrightarrow1.
+\end{equation}
+Moreover,
+\begin{equation}\label{eq:matrix-action-errors}
+ \sum_i\left(\|D_iR-H_iR\|_2^2+
+ \|D_i\Psi-(H_i)_{(1)}\Psi\|_2^2\right)=o(1).
+\end{equation}
+The first slot can be replaced by any existing slot in each component.
+\end{lemma}
+\begin{proof}
+In an eigenbasis of $R$ with eigenvalues $r_a$, define
+$(H_i)_{ab}=2(D_iR)_{ab}/(r_a+r_b)$ when $r_a+r_b>0$, and zero otherwise.
+The kernel--kernel block of $D_iR$ vanishes almost everywhere.  One way to
+check this for a form-domain positive matrix field is to use absolutely
+continuous line representatives: at a differentiability point, the
+quadratic form on every vector in the kernel has a local minimum zero,
+and polarization gives the assertion.  The formula is measurable and gives
+\[
+ \sum_i\|H_iR\|_{\HS}^2\le4|DR|^2.
+\]
+Set
+\[
+ e=\|DR\|_2^2,\quad a=\sum_i\E\tr(R^2H_i^2),\quad
+ c=\sum_i\E\|[H_i,R]\|_{\HS}^2,\quad d=\|D\Psi\|_2^2.
+\]
+The Sylvester equation implies the exact identities
+\begin{equation}\label{eq:matrix-energy-identities}
+ e=a-\frac c4,\qquad
+ \sum_i\E\tr(RH_iRH_i)=e-\frac c4.
+\end{equation}
+Differentiating $R^2=\Psi\Psi^*$ in the weak form sense gives
+\[
+ e=\sum_i\E\langle D_i\Psi,(H_i)_{(1)}\Psi\rangle
+   =\sum_i\E\langle D_i\Psi,(H_i)_{\mathrm{av}}\Psi\rangle.
+\]
+For example, the first equality follows by pairing $D_i(R^2)$ with $H_i/2$;
+all products are integrable by the displayed bound.  The second uses the
+symmetry of each $D_i\Psi$ on its existing slots.
+Lemma~\ref{lem:matrix-copied-slots} and Cauchy--Schwarz yield
+\[
+ e^2\le d\left(e-\frac c4+\frac{2a}{k}\right).
+\]
+Since $a\le4e$, $d\to1$, and $\liminf e\ge1$, this first forces $e\to1$
+and then $c\to0$; Equation~\eqref{eq:matrix-energy-identities} gives $a\to1$.
+Finally,
+\[
+ \sum_i\|D_iR-H_iR\|_2^2=\frac c4,
+ \qquad
+ \sum_i\|D_i\Psi-(H_i)_{(1)}\Psi\|_2^2=d+a-2e.
+\]
+Both tend to zero.  Symmetry gives the assertion for the other slots.
+\end{proof}
+
+\subsection{Bounded columns and spectral blocks}
+
+We next make the column $(H_i)_i$ bounded in operator norm.  This requires
+a weighted tail estimate; an unweighted truncation would be inadequate
+when $R$ has many small eigenvalues.  Throughout the modifications below,
+$H_i$ denotes the current column; the exact Sylvester identity
+\eqref{eq:matrix-sylvester} refers to the original column.  The approximate
+derivative-action and balance estimates are preserved as stated in the
+following lemmas.
+
+\begin{lemma}[Balance and bounded columns]\label{lem:matrix-bounded-column}
+The matrices in Lemma~\ref{lem:matrix-saturation} satisfy
+\begin{equation}\label{eq:matrix-balance}
+ \E\left\|\sum_iH_i\rho H_i-\rho\right\|_1=o(1),\qquad
+ \E\left\|R\left(\sum_iH_i^2\right)R-\rho\right\|_1=o(1).
+\end{equation}
+They can be replaced by symmetric matrix fields, still denoted $H_i$, so
+that Equation~\eqref{eq:matrix-action-errors}, the commutator conclusion of
+Equation~\eqref{eq:matrix-saturation}, and
+Equation~\eqref{eq:matrix-balance} hold, and
+\begin{equation}\label{eq:matrix-column-bound}
+ S:=\sum_iH_i^2\le4\Id,\qquad
+ \E\|(S-1)R\|_{\HS}^2=o(1).
+\end{equation}
+The total change satisfies $\sum_i\E\|(H_i^{\mathrm{new}}-H_i^{\mathrm{old}})R\|_{\HS}^2=o(1)$.
+\end{lemma}
+\begin{proof}
+We repeatedly use the dimension-free inequality
+\begin{equation}\label{eq:matrix-reduction-perturbation}
+ \|XX^*-YY^*\|_1
+ \le(\|X\|_{\HS}+\|Y\|_{\HS})\|X-Y\|_{\HS}.
+\end{equation}
+By Equation~\eqref{eq:matrix-action-errors}, the first sum in
+Equation~\eqref{eq:matrix-balance} is close in expected trace norm to the
+one-slot reduction of $D\Psi$, with the derivative index included among
+the column indices.  The derivative coupling identifies this with
+\[
+ \frac1{kI}\sum_{k\le l<2k}T_{l+1}T_{l+1}^*.
+\]
+The trace-norm error from the coupling is $o(1)$ by
+Equation~\eqref{eq:matrix-reduction-perturbation} and
+Equation~\eqref{eq:matrix-normalization}.  Shifting the band changes its
+reduction only by its two endpoints, whose expected trace-norm cost is
+\[
+ \frac{\|T_{2k}\|_2^2+\|T_k\|_2^2}{kI}=O(k^{-1}).
+\]
+This proves the first balance equation.  Comparing the arrays of matrices
+$(H_iR)_i$ and $(RH_i)_i$ proves the second by the same inequality and the
+summed commutator estimate.
+
+For the original, possibly unbounded column write
+$\mathbf H v=(H_iv)_i$, $S=\mathbf H^*\mathbf H$, and $J=(1+S)^{-1}$.
+Let $\mathbf K=\mathbf HR-R^{\oplus n}\mathbf H$, so that
+$\E\|\mathbf K\|_{\HS}^2=o(1)$.  Algebra gives
+\[
+ [S,R]=\mathbf H^*\mathbf K-\mathbf K^*\mathbf H,
+ \qquad [R,J]=J[S,R]J.
+\]
+The bounds
+\[
+ \|\mathbf HJ\mathbf H^*\|_{\op}\le1,\qquad
+ \|\mathbf HJ\|_{\op}\le\tfrac12,\qquad \|J\|_{\op}\le1
+\]
+therefore imply $\|\mathbf H[R,J]\|_{\HS}\le2\|\mathbf K\|_{\HS}$.
+For every fixed $f(x)=a+b/(1+x)$, it follows that
+\[
+ \E\|\mathbf H[f(S),R]\|_{\HS}^2=o(1).
+\]
+The squared norms of $\mathbf Hf(S)R$ and $\mathbf HRf(S)$ differ by $o(1)$:
+the latter has bounded mean-square norm, and the preceding bound controls
+their difference.  Testing the second balance equation against the bounded
+matrix $f(S)^2$ consequently gives
+\[
+ \E\tr\bigl(R^2(S-1)f(S)^2\bigr)=o(1).
+\]
+Use $f=1,1+J,1-J$ and
+$(S-1)^2/(1+S)=(S-1)-2(S-1)J$ to obtain
+\begin{equation}\label{eq:matrix-column-tail}
+ \E\tr\left(R^2\frac{(S-1)^2}{1+S}\right)=o(1).
+\end{equation}
+All these expressions are integrable: their absolute values are bounded
+by a constant times $\tr(R^2(1+S))$.
+
+Let $Q=\1_{(4,\infty)}(S)$ and $P=1-Q$.  On this spectral region,
+$(S-1)^2/(1+S)$ dominates both a positive constant and a positive multiple
+of $S$.  Thus
+\[
+ \E\tr(R^2Q)+\E\tr(R^2SQ)=o(1).
+\]
+The input truncation costs
+$\sum_i\E\|H_iQR\|_{\HS}^2=\E\tr(R^2SQ)=o(1)$.
+The output truncation costs
+\[
+ \sum_i\E\|QH_iR\|_{\HS}^2
+ =\E\tr\left(Q\sum_iH_i\rho H_i\right)
+ \le\E\tr(Q\rho)+\E\left\|\sum_iH_i\rho H_i-\rho\right\|_1=o(1).
+\]
+Consequently replacing $H_i$ by $PH_iP$ has total right-$R$ squared cost
+$o(1)$.  The new column obeys
+$\sum_i(PH_iP)^2\le PSP\le4P$.
+
+Here is the perturbation accounting needed both now and below.  If
+symmetric $\widetilde H_i$ have
+\begin{equation}\label{eq:matrix-weighted-change}
+ \Delta=\sum_i\E\|(\widetilde H_i-H_i)R\|_{\HS}^2=o(1),
+\end{equation}
+then symmetry gives the same bound with $R$ on the left.  Each derivative
+error in Equation~\eqref{eq:matrix-action-errors} changes in norm by at most
+$\sqrt\Delta$, and the summed commutator norm changes by at most
+$2\sqrt\Delta$.  The two balance errors change in expected trace norm by
+at most
+\[
+ \left(\Bigl[\sum_i\E\|H_iR\|_{\HS}^2\Bigr]^{1/2}
+ +\Bigl[\sum_i\E\|\widetilde H_iR\|_{\HS}^2\Bigr]^{1/2}\right)\sqrt\Delta,
+\]
+by Equation~\eqref{eq:matrix-reduction-perturbation}, and this is $o(1)$.
+The new summed energy is bounded by the old one plus the perturbation in
+norm.  These statements apply to the compression just made.
+
+For the bounded column, $[S,R]=\mathbf H^*\mathbf K-\mathbf K^*\mathbf H$
+satisfies $\|[S,R]\|_2\le4\|\mathbf K\|_2=o(1)$.
+Put $T=S-1$, so $\|T\|_{\op}\le3$.
+The second balance equation says $\E\|RTR\|_1=o(1)$.  Finally,
+\[
+ \|TR\|_{\HS}^2
+ =\tr(TRTR)+\tr(T[T,R]R).
+\]
+The expectation of the first term is at most $3\E\|RTR\|_1$ in absolute
+value; the second is at most
+$3\|[T,R]\|_2\|R\|_2=o(1)$.  This proves the last assertion.
+\end{proof}
+
+\begin{lemma}[Commutation and exchange of indices]\label{lem:matrix-index-exchange}
+For the bounded columns just constructed,
+\begin{equation}\label{eq:matrix-index-exchange}
+ \sum_{i,j}\E\|[H_i,H_j]R\|_{\HS}^2=o(1),\qquad
+ \E\sum_{i,a,b}|(H_iR)_{ab}-(H_aR)_{ib}|^2=o(1).
+\end{equation}
+These conclusions also hold after a symmetric perturbation satisfying
+Equation~\eqref{eq:matrix-weighted-change}, provided the new column has a
+uniform operator bound.
+\end{lemma}
+\begin{proof}
+Equation~\eqref{eq:matrix-action-errors} implies that moving $H_i$ between
+any two existing slots costs $o(1)$ in summed mean square.  Three slots
+suffice to exchange two factors.  In the following chain, each $\simeq$
+means that the difference has squared norm $o(1)$ after expectation and
+summation over $i,j$:
+\begin{align*}
+ (H_i)_{(1)}(H_j)_{(1)}\Psi
+ &\simeq (H_i)_{(1)}(H_j)_{(2)}\Psi
+  = (H_j)_{(2)}(H_i)_{(1)}\Psi\\
+ &\simeq (H_j)_{(2)}(H_i)_{(3)}\Psi
+  = (H_i)_{(3)}(H_j)_{(2)}\Psi\\
+ &\simeq (H_i)_{(3)}(H_j)_{(1)}\Psi
+  = (H_j)_{(1)}(H_i)_{(3)}\Psi\\
+ &\simeq (H_j)_{(1)}(H_i)_{(1)}\Psi.
+\end{align*}
+The equalities use commutation on different slots.  Each move costs at
+most the one-factor error
+multiplied by the column bound: for any tensor $v$,
+\[
+ \sum_i\|(H_i)_{(r)}v\|^2\le4\|v\|^2.
+\]
+The squared norm of the sum of the four errors is at most four times their
+summed squared norms.  Thus the total commutator cost on $\Psi$ is $o(1)$.
+Its cost on $\Psi$ equals its cost on $R$, since both have reduction $R^2$.
+
+For the second conclusion, write $\Psi=RJ_0$, where $J_0J_0^*$ is the
+support projection of $R$.  The matrix-valued array
+$Z_{ia,b}=(H_iR)_{ab}-(H_aR)_{ib}$ has its column index in that support,
+so $\|ZJ_0\|_{\HS}=\|Z\|_{\HS}$.  The entries of $ZJ_0$ express the
+exchange of the derivative index and the first tensor index in
+$(H_i)_{(1)}\Psi$.  Equation~\eqref{eq:matrix-action-errors} and the
+approximation of $D\Psi$ by fully symmetric $T_{l+1}$ give the claim.
+For perturbed matrices, the derivative errors and the slot-moving errors
+have the bounds established after Equation~\eqref{eq:matrix-weighted-change};
+the same four moves use the new uniform column bound.  The index-exchange
+norm itself changes by at most $2\sqrt\Delta$.
+\end{proof}
+
+We have obtained a bounded column acting almost identically on the
+symmetric tensor slots.  To use Gaussian matrix polynomials, we now put its
+weighted estimates into a tracial form.  The following construction is
+performed only at the initial point of the diffusion, so a varying
+eigenbasis will never be differentiated.
+
+\begin{lemma}[Spectral blocks and a bounded input map]\label{lem:matrix-blocks}
+After enlarging the initial probability space by an independent uniform
+variable, there are a positive block matrix
+$R_*=\bigoplus_\beta\beta\Id_\beta$, with zero on $\ker R$, and symmetric
+matrices $H_i$ supported on entries whose row, column, and outer index
+belong to the same positive block, with the following properties.
+Their total change satisfies Equation~\eqref{eq:matrix-weighted-change},
+$R_*\le R\le(1+\epsilon)R_*$ on the positive support for
+$\epsilon\to0$, and the column bound $S\le4\Id$ holds.  With the positive
+weighted trace
+\begin{equation}\label{eq:matrix-weighted-trace}
+ \tau(U)=\E\sum_\beta\beta^2\tr_\beta U_\beta,
+\end{equation}
+where the expectation also includes any fresh noises present, we have
+$\tau(1)\to1$ and
+\begin{equation}\label{eq:matrix-tracial-defects}
+ \tau((S-1)^2)=o(1),\quad
+ \sum_{i,j}\tau(|[H_i,H_j]|^2)=o(1),\quad
+ \E\sum_\beta\beta^2\sum_{i,a,b\in\beta}
+ |(H_i)_{ab}-(H_a)_{ib}|^2=o(1).
+\end{equation}
+For a vector $v$ in a positive block define $H[v]=\sum_{i\in\beta}v_iH_i$.
+We can further impose, pointwise,
+\begin{equation}\label{eq:flattening-bound}
+ \|H[v]\|_{\HS}\le2|v|.
+\end{equation}
+The bound uses ordinary, unnormalized Hilbert--Schmidt norm.
+All the derivative-action and balance conclusions above continue to hold
+for these matrices in the initial coordinates.
+\end{lemma}
+\begin{proof}
+Choose an orthogonal eigenframe $O$ for $R$ at the initial point.  Transform
+all three indices: in that frame the matrices are
+\[
+ \widehat H_i=\sum_jO_{ji}O^*H_jO.
+\]
+This preserves the column bound, all the summed norms, and the
+index-exchange estimate, since these are Euclidean tensor contractions.
+Finite-dimensional spectral calculus and an ordered orthonormal basis
+completion give measurable choices, including at repeated eigenvalues.
+
+Let $h=\log(1+\epsilon)$ and take an independent $\theta$ uniform on
+$[0,h)$.  Group positive eigenvalues in
+$[e^{\theta+mh},e^{\theta+(m+1)h})$, $m\in\mathbb Z$, and round those in
+each nonempty block down to $\beta=e^{\theta+mh}$.  The zero eigenspace is
+kept separate.  Write
+\[
+ \kappa=\sum_i\E\|[H_i,R]\|_{\HS}^2=o(1),\qquad
+ a=\sum_i\E\|H_iR\|_{\HS}^2=O(1).
+\]
+For $0<\delta<h$, a pair of positive eigenvalues with
+$|\log(r_a/r_b)|\ge\delta$ satisfies
+$r_b^2\le C\delta^{-2}(r_a-r_b)^2$, when $\delta\le1$.
+If their logarithmic distance is smaller than $\delta$, the probability
+that a block boundary separates them is at most $\delta/h$.  Consequently
+the right-$R$ squared cost of deleting entries with row and column in
+different blocks is bounded by
+\begin{equation}\label{eq:matrix-block-error}
+ C\delta^{-2}\kappa+\frac{\delta}{h}a+\kappa.
+\end{equation}
+The final term covers a zero row and positive column directly by the
+commutator.  A zero column has no right-$R$ cost.  For example, choose
+$\delta\to0$ slowly enough that $\kappa/\delta^2\to0$, and then
+$h\to0$ slowly enough that $\delta/h\to0$.
+
+Deleting entries whose outer index belongs to another block has vanishing
+cost as well.  On the set where $i$ and $b$ have different block labels,
+\[
+ |(H_i)_{ab}r_b|^2
+ \le2|(H_i)_{ab}r_b-(H_a)_{ib}r_b|^2
+   +2|(H_a)_{ib}r_b|^2.
+\]
+Sum over the indices.  The first term is the second defect of
+Equation~\eqref{eq:matrix-index-exchange}; the second is a row--column block
+exclusion of the kind already bounded in
+Equation~\eqref{eq:matrix-block-error}.  This includes outer indices in the
+zero eigenspace.  Within each block the resulting matrices have the form
+$P_\beta\widehat H_iP_\beta$ for $i\in\beta$, and vanish otherwise.  Thus
+\[
+ \sum_{i\in\beta}(P_\beta\widehat H_iP_\beta)^2
+ \le P_\beta\left(\sum_i\widehat H_i^2\right)P_\beta\le4P_\beta.
+\]
+They are symmetric and their total change has the cost in
+Equation~\eqref{eq:matrix-weighted-change}.
+
+For every block matrix $X$ one has the precise comparison
+\begin{equation}\label{eq:matrix-weight-comparison}
+ \tau(X^*X)\le\E\|XR\|_{\HS}^2
+ \le(1+\epsilon)^2\tau(X^*X).
+\end{equation}
+In particular $\tau(1)\to1$.
+The perturbation bounds following Equation~\eqref{eq:matrix-weighted-change}
+prove the derivative, commutator, and balance conclusions for these
+matrices.  The bounded-column argument at the end of
+Lemma~\ref{lem:matrix-bounded-column} again gives
+$\E\|(S-1)R\|_{\HS}^2=o(1)$, so
+Equation~\eqref{eq:matrix-weight-comparison} proves the first tracial defect.
+Lemma~\ref{lem:matrix-index-exchange} gives the second.  For the third,
+all three indices now lie in one block, and the index-exchange defect
+is exactly weighted by $r_b^2$; replacing this by $\beta^2$ only decreases
+it.  Matrix symmetry and exchange of $i,a$ together give asymptotic
+symmetry under every permutation of the three indices.
+
+It remains to bound the map from the outer index to matrix coefficients.
+On each block let
+\[
+ Fv=\sum_i v_iH_i,\qquad (Gv)_{ab}=(H_bv)_a.
+\]
+The map $G$ has operator norm at most two because
+$\|Gv\|_{\HS}^2=\langle v,Sv\rangle\le4|v|^2$.
+Total symmetry just established says
+\[
+ \E\sum_\beta\beta^2\|F-G\|_{\HS(\R^{\dim\beta},\HS)}^2=o(1).
+\]
+Clip the singular values of $F$ to two, obtaining $F'=FB$ for a real
+positive contraction $B$ on the input space.  Metric projection onto the
+operator-norm ball in Hilbert--Schmidt norm gives
+$\|F-F'\|_{\HS}\le\|F-G\|_{\HS}$; this also follows immediately by
+singular-value decomposition and the variational characterization of
+singular values.  Thus the block-weighted squared cost is $o(1)$.
+Writing $H'_i=\sum_j B_{ji}H_j$ shows that every $H'_i$ remains symmetric.
+For each vector $v$,
+\[
+ \sum_i\|H'_iv\|^2\le\sum_j\|H_jv\|^2,
+\]
+so the column bound is preserved, as is the block support.  The weighted
+cost controls the right-$R$ cost by
+Equation~\eqref{eq:matrix-weight-comparison}.  Therefore the derivative and
+balance errors have exactly the bounds stated after
+Equation~\eqref{eq:matrix-weighted-change}; index exchange changes by at
+most twice the perturbation norm, and the four-move argument controls
+the commutators.  Repeating the bounded-column last step and then
+Equation~\eqref{eq:matrix-weight-comparison} proves all three tracial defects
+for $H'_i$.  The defining singular-value clipping proves
+Equation~\eqref{eq:flattening-bound}.
+\end{proof}
+
+\subsection{Gaussian moments and the frozen diffusion law}
+
+The matrices within a block need not commute exactly.  The next lemma
+shows that their Gaussian linear combinations nevertheless have the
+scalar Gaussian joint moments in the weighted limit.  We include the
+moment bounds because they also justify the unbounded exponential needed
+for the diffusion law.
+
+\begin{lemma}[Weighted Gaussian moments]\label{lem:matrix-gaussian-moments}
+Let the block matrices and $\tau$ be those of
+Lemma~\ref{lem:matrix-blocks}.  For independent standard Gaussian vectors
+$N_1,\ldots,N_q$, independent of the initial data, put $D_r=H[N_r]$.
+Then, for every fixed word $r_1,\ldots,r_m$,
+\begin{equation}\label{eq:matrix-gaussian-joint-law}
+ \tau(D_{r_1}\cdots D_{r_m})
+ \longrightarrow \E(Z_{r_1}\cdots Z_{r_m}),
+\end{equation}
+where $Z_1,\ldots,Z_q$ are independent scalar standard normals.
+Products of any fixed length can be reordered with error tending to zero
+in $L^2(\tau)$.  For every integer $p\ge1$ and real $u\ge0$,
+\begin{equation}\label{eq:matrix-gaussian-moment-bound}
+ \E_{N_r}\tr_\beta D_r^{2p}
+ \le4^p(2p-1)!!\tr_\beta1,
+ \qquad
+ \tau(e^{u|D_r|})\le2e^{2u^2}\tau(1).
+\end{equation}
+If $\mathfrak h_j$ is the tensor Hermite polynomial of
+Lemma~\ref{lem:hermite-expansion} and $\mathfrak h_j^{(1)}$ is its
+one-dimensional version, then for each fixed $j$,
+\begin{equation}\label{eq:matrix-wick-reduction}
+ \tau\left(\left|
+ \sum_{i_1,\ldots,i_j}H_{i_j}\cdots H_{i_1}
+      \mathfrak h_j(N_1)_{i_1\ldots i_j}
+ -\mathfrak h_j^{(1)}(D_1)\right|^2\right)=o(1).
+\end{equation}
+\end{lemma}
+\begin{proof}
+We first prove a conditional moment bound on one fixed block, allowing a
+general column bound $S\le C\Id$.  We use the Gaussian trace-moment
+integration-by-parts argument of \cite[Lemma~6.1 and Section~7]{Tropp2018};
+the weighted near-commutation limit will be proved afterward.
+Write $D=\sum_iN_iH_i$.
+Gaussian integration by parts gives
+\[
+ \E_N\tr D^{2p}
+ =\sum_{j=0}^{2p-2}\E_N\sum_i
+   \tr(H_iD^jH_iD^{2p-2-j}).
+\]
+In an eigenbasis of $D$ with eigenvalues $d_a$, the absolute value of a
+summand is bounded by
+\[
+ \sum_{i,a,b}|(H_i)_{ab}|^2|d_b|^j|d_a|^{2p-2-j}.
+\]
+For $p>1$, the scalar weighted arithmetic--geometric mean inequality
+bounds the product of powers by
+\[
+ \frac{j}{2p-2}|d_b|^{2p-2}
+ +\left(1-\frac{j}{2p-2}\right)|d_a|^{2p-2}.
+\]
+Since $\sum_{i,b}|(H_i)_{ab}|^2=S_{aa}\le C$ and similarly with $a,b$
+exchanged, each summand is at most $C\tr|D|^{2p-2}$.
+For $p=1$ the assertion is $\tr S\le C\tr1$.
+Induction proves
+$\E_N\tr D^{2p}\le C^p(2p-1)!!\tr1$.
+After multiplying by the block weights, all fixed moments are uniformly
+bounded.  Also $e^{u|x|}\le2\cosh(ux)$, so the even-moment series gives
+\[
+ \tau(e^{u|D|})
+ \le2\sum_{p\ge0}\frac{u^{2p}C^p(2p-1)!!}{(2p)!}\tau(1)
+ =2e^{Cu^2/2}\tau(1).
+\]
+This proves Equation~\eqref{eq:matrix-gaussian-moment-bound} with $C=4$.
+
+For clarity, the norm $\|X\|_{p,\tau}=\tau(|X|^p)^{1/p}$ is formed on
+the direct sum of the blocks and then averaged over the classical random
+variables.  Schatten H\"older followed by scalar H\"older gives
+\[
+ \|X_1\cdots X_m\|_{p,\tau}
+ \le\prod_{j=1}^m\|X_j\|_{p_j,\tau},
+ \qquad \sum_j\frac1{p_j}=\frac1p.
+\]
+These bounds use the actual weights $\beta^2$, with no factor depending
+on a block dimension.  The finite-matrix Schatten inequality can itself
+be obtained by the same singular-value three-lines argument as in
+Lemma~\ref{lem:matrix-copied-slots}, interpolating the endpoint bounds
+with one trace norm and the other norms operator norms, and then applying
+duality.
+
+For independent $N_r,N_s$,
+\[
+ \|[D_r,D_s]\|_{2,\tau}^2
+ =\sum_{i,j}\tau(|[H_i,H_j]|^2)=o(1).
+\]
+All its higher fixed moments are bounded by H\"older and the moment bound
+just proved.  Interpolation between $L^2(\tau)$ and an arbitrarily high
+fixed $L^p(\tau)$ therefore makes this commutator tend to zero in every
+fixed finite power norm.  Applying H\"older to the factors on either side
+of a commutator shows that an adjacent transposition in a word of fixed
+length costs $o(1)$ in $L^2(\tau)$.  A finite sequence of transpositions
+proves the reordering assertion.
+
+Now expand a mixed moment by Gaussian pairings, conditional on the
+matrices.  Pairings must join indices of the same Gaussian vector.  For
+each pair introduce its own independent standard Gaussian vector, shared
+only between the two factors in that pair.  This represents the
+contracted matrix word exactly as a conditional expectation.  Reorder the
+word until each pair is adjacent; the error is $o(1)$ even in
+$L^2(\tau)$.  Averaging the independent vectors then gives $S^{m/2}$.
+For each fixed integer $b$, boundedness of $S$ gives
+\[
+ \|S^b-1\|_{2,\tau}
+ \le\left(\sum_{a=0}^{b-1}4^a\right)\|S-1\|_{2,\tau}=o(1).
+\]
+Each pairing consequently contributes $\tau(1)+o(1)=1+o(1)$.
+Odd moments vanish.  The number of admissible pairings is precisely the
+scalar independent-normal moment, proving
+Equation~\eqref{eq:matrix-gaussian-joint-law}.
+
+Finally expand the tensor Hermite polynomial using its defining
+exponential
+$\exp(\langle v,N\rangle-|v|^2/2)$.
+A term consists of free $N$ indices and a number of signed Kronecker-delta
+pairs.  Represent each delta pair by its own independent Gaussian vector,
+as above, and move its two matrix factors together.  Conditional averaging
+is contractive in $L^2(\tau)$, so the reordering error remains $o(1)$.
+The resulting factors $S$ can be replaced by one, even between fixed
+powers of $D_1$, by H\"older and interpolation applied to the bounded
+matrix $S-1$.  The resulting polynomial is
+\[
+ \sum_{b=0}^{\lfloor j/2\rfloor}
+ \frac{(-1)^b j!}{2^bb!(j-2b)!}D_1^{j-2b}
+ =\mathfrak h_j^{(1)}(D_1).
+\]
+There are only finitely many terms for fixed $j$, proving
+Equation~\eqref{eq:matrix-wick-reduction}.
+\end{proof}
+
+\begin{proposition}[Frozen diffusion law]\label{prop:frozen-law}
+Let $Y_u$ be the stationary diffusion with invariant law $\eta$ and driving
+Brownian motion $B_u$, and fix $t>0$.  Make the matrix construction of
+Lemma~\ref{lem:matrix-blocks} at $Y_0$, using an independent random block
+offset.  In its orthonormal frame put $N=B_t/\sqrt t$; conditionally on the
+initial data and block offset, $N$ is a standard Gaussian vector.  Define,
+in the initial spatial coordinates,
+\begin{equation}\label{eq:matrix-frozen-covariance}
+ \sigma=\bigoplus_\beta\beta^2
+       \exp\bigl(\sqrt{8t}\,H[N]-4t\Id_\beta\bigr),
+\end{equation}
+and set $\sigma=0$ on the omitted kernel.  Then
+\begin{equation}\label{eq:matrix-frozen-law}
+ \E\|R(Y_t)-\sigma^{1/2}\|_{\HS}^2\longrightarrow0.
+\end{equation}
+The matrices defining $\sigma$ obey the bounded-column and input-map
+bounds \eqref{eq:matrix-column-bound} and \eqref{eq:flattening-bound},
+the weighted defects \eqref{eq:matrix-tracial-defects}, and the mixed
+moment law of Lemma~\ref{lem:matrix-gaussian-moments}.
+\end{proposition}
+\begin{proof}
+The proof has three steps: replace the added tensor indices in the
+hierarchy expansion by products of the initial matrices, identify their
+Wick contractions, and sum the resulting scalar Hermite series.
+All limits involving the degree of that series are taken after the
+counterexample-sequence limit, with $t$ fixed.
+
+The derivative-action estimate and the hierarchy coupling give
+\begin{equation}\label{eq:matrix-one-step}
+ \frac1{kI}\sum_{k\le l<2k}\sum_i
+ \|T_{l+1,i\,\cdot}-(H_i)_{(1)}T_l\|_2^2=o(1).
+\end{equation}
+It also holds when the band is shifted by any fixed integer $a\ge0$.
+Indeed the same matrices have been constructed using levels $k$ through
+$2k-1$.  Removing or adding at most $a$ levels in the error sum costs
+$O_C(a/k)$, since the unnormalized squared norm of each one-step
+error is at most
+\[
+ 2\|T_{l+1}\|_2^2+2C\|T_l\|_2^2=O_C(I),
+\]
+by the bounded column and the diagonal endpoint control.  This also
+explains why the matrices may already include the block and input
+clipping modifications: their cost on $\Psi$ is the right-$R$ cost in
+Equation~\eqref{eq:matrix-weighted-change}.
+
+Iterating Equation~\eqref{eq:matrix-one-step}, for each fixed $j$ we may
+replace the extra-index tensor $T_{l+j}$ by
+\[
+ (i_1,\ldots,i_j)\longmapsto
+ (H_{i_j})_{(1)}\cdots(H_{i_1})_{(1)}T_l
+\]
+in the normalized band mean-square norm, with error $o(1)$.
+More explicitly, if $e_a$ is the root mean-square error in the one-step
+estimate on the band shifted by $a$, the root error after $j$ steps is at
+most
+\[
+ \sum_{a=0}^{j-1}C^{a/2}e_{j-1-a}.
+\]
+Here multiplication by a column of $H_i$ costs at most $\sqrt C$ in the
+summed coefficient norm.  Each $e_a=o(1)$ at fixed $a$, so the displayed
+bound vanishes.  Contraction with $\mathfrak h_j(N)$ multiplies squared
+coefficient norm by at most $j!$: it first symmetrizes the contracted
+indices, an orthogonal projection, and Gaussian Hermite orthogonality
+then gives the factor $j!$.
+
+Apply Lemma~\ref{lem:hermite-expansion} to the band of levels in $\Psi$.
+For each fixed $m$, the preceding replacements and
+Equation~\eqref{eq:matrix-wick-reduction} give
+\[
+ \Psi(Y_t)=P_{m,t}(D)\Psi(Y_0)+\mathcal E_{m,t},\qquad D=H[N],
+\]
+where
+\[
+ P_{m,t}(x)=e^{-t}\sum_{j=0}^m
+          \frac{(2t)^{j/2}}{j!}\mathfrak h_j^{(1)}(x)
+\]
+and
+\begin{equation}\label{eq:matrix-hermite-error}
+ \limsup\|\mathcal E_{m,t}\|_2
+ \le \frac{(2t)^{(m+1)/2}}{\sqrt{(m+1)!}}.
+\end{equation}
+To justify the conversion of a tracial polynomial error to the norm of
+its action on $\Psi(Y_0)$, use
+$\|X\Psi(Y_0)\|_{\HS}=\|XR\|_{\HS}$ and
+Equation~\eqref{eq:matrix-weight-comparison}.  The band version of the
+remainder bound follows from the uniform finite-list diagonal fixed at
+the start of the section.  Its right side tends to zero as $m\to\infty$.
+
+The spectral measures of $D$ averaged with $\tau$ have total mass tending
+to one, converge in moments to the standard Gaussian by
+Equation~\eqref{eq:matrix-gaussian-joint-law}, and have uniformly bounded
+exponential moments at every fixed linear parameter by
+Equation~\eqref{eq:matrix-gaussian-moment-bound}.  These facts imply weak
+convergence to the standard Gaussian and convergence of the integrals of
+every continuous function bounded by a polynomial times a fixed linear
+exponential.  Here is a direct justification: the second moment gives
+tightness; the higher moment bounds give uniform integrability of each
+polynomial, so every subsequential limit has the Gaussian moments.  Its
+exponential moments determine its characteristic function by its power
+series, hence determine the law.  Using a larger exponential parameter
+makes the tails of any of the specified test functions uniformly small.
+
+The scalar Hermite generating identity gives, with $Z$ standard normal,
+\[
+ e^{-t}\sum_{j\ge0}\frac{(2t)^{j/2}}{j!}\mathfrak h_j^{(1)}(Z)
+ =\exp(\sqrt{2t}\,Z-2t)
+\]
+in squared mean.  In fact its squared tail norm is
+$e^{-2t}\sum_{j>m}(2t)^j/j!$.  Define the matrix multiplier
+\[
+ M=\exp(\sqrt{2t}\,D-2t\Id).
+\]
+The just-proved convergence of spectral integrals, applied with each
+fixed $m$, yields
+\begin{equation}\label{eq:matrix-exponential-limit}
+ \lim\tau(|M-P_{m,t}(D)|^2)
+ =e^{-2t}\sum_{j>m}\frac{(2t)^j}{j!},\qquad
+ \tau(M^2)\longrightarrow1.
+\end{equation}
+Together with Equations~\eqref{eq:matrix-hermite-error} and
+\eqref{eq:matrix-weight-comparison}, first sending the sequence index and
+then $m$ to infinity, this proves
+\[
+ \E\|\Psi(Y_t)-M\Psi(Y_0)\|_{\HS}^2=o(1).
+\]
+There is no operator-norm bound on $M$ in this argument; only its weighted
+trace moments are used.
+
+Write $\Psi(Y_0)=RJ_0$, with $J_0J_0^*$ equal to the support projection.
+On each block $(R-R_*)^2\le\epsilon^2R_*^2$, and $R_*$ is scalar there.
+By cyclicity and positivity of trace,
+\[
+ \E\|M(R-R_*)J_0\|_{\HS}^2
+ \le\epsilon^2\tau(M^2)=o(1).
+\]
+Thus $\Psi(Y_t)$ is close in squared mean to $MR_*J_0$.
+Its reduction is exactly
+\[
+ (MR_*J_0)(MR_*J_0)^*
+ =MR_*^2M=\bigoplus_\beta\beta^2
+       \exp(\sqrt{8t}\,H[N]-4t\Id_\beta)=\sigma.
+\]
+Finally Equation~\eqref{eq:matrix-root-lipschitz} transfers the approximation
+of these rectangular matrices to the approximation of their positive
+reduced square roots, proving Equation~\eqref{eq:matrix-frozen-law}.
+\end{proof}
+
+The resulting description retains the weights $\beta^2$ and the ordinary
+trace on every block.  It supplies both a bounded matrix column and the
+input bound of Equation~\eqref{eq:flattening-bound}, while allowing any
+number of positive blocks and eigenvalues arbitrarily close to zero.
+These are the properties needed for the covariance observations in
+Section~\ref{sec:observations}.
diff --git a/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/build/sections/07-observations.tex b/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/build/sections/07-observations.tex
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+\section{Covariance observations and the contradiction}
+\label{sec:observations}
+
+The matrices constructed in Section~\ref{sec:matrices} describe fluctuations
+whose information content is incompatible with the logarithmic Sobolev
+inequality of the stationary law.  We compare two bounds for the same
+Gaussian covariance observation.  The first follows from logarithmic
+Sobolev and is universal after integration over its scale.  The second
+uses the frozen matrices to recover a scalar lognormal covariance
+experiment.  Integration over scale is essential: it assigns to every
+spectral block exactly the weight used by the tracial limit, regardless
+of the sizes of its eigenvalues.
+
+Throughout this section, limits refer to the counterexample sequence.
+All observation times, caps, compact parameter intervals, and polynomial
+degrees are fixed before taking that limit.  We suppress its index.
+Recall that $Y_t$ is stationary with law $\eta$, that
+$R(\eta)\le\lambda^{-1}$ with $\lambda\to1$, and that $R$ without a
+measure argument denotes the positive matrix field.  In particular,
+Equations~\eqref{eq:matrix-normalization}
+and~\eqref{eq:matrix-saturation} give
+\begin{equation}\label{eq:obs-energy-input}
+ \E\tr R^2\longrightarrow1,
+ \qquad \E|DR|^2\longrightarrow1.
+\end{equation}
+To distinguish the original Sylvester matrices from the subsequently
+modified block matrices, denote the former by $\widehat H_i$.  Equations
+\eqref{eq:matrix-sylvester} and~\eqref{eq:matrix-saturation} assert
+\begin{equation}\label{eq:obs-sylvester-input}
+ D_iR=\tfrac12(\widehat H_iR+R\widehat H_i),
+ \qquad
+ \sum_i\E\|[\widehat H_i,R]\|_{\HS}^2=o(1).
+\end{equation}
+Only the upper bound below uses these original matrices.
+
+Fix $L\ge1$ and define
+\begin{equation}\label{eq:obs-covariance-definition}
+ f_L(q)=\frac{q}{1+q/L},\qquad
+ C_T(y)=\Id+f_L(TR(y)^2),\qquad
+ Z_T=C_T(Y_t)^{1/2}G,\quad T>0,
+\end{equation}
+where $G$ is an independent standard Gaussian vector.  Thus
+$\Id\le C_T\le(1+L)\Id$.  We write $H(P\mid Q)$ for relative entropy
+and $\MI(U;V)$ for mutual information, including the usual averaged
+conditional version when a conditioning variable is displayed.
+
+\subsection{The integrated information upper bound}
+
+\begin{lemma}[Integrated spectral multipliers]
+\label{lem:obs-multipliers}
+For $v,w\ge0$, not both zero, put
+\begin{equation}\label{eq:obs-multiplier}
+ M_L(v,w)=(v+w)^2\int_0^\infty
+ \frac{dT}{
+ (1+Tv^2/L)(1+Tw^2/L)
+ (1+(1+1/L)Tv^2)(1+(1+1/L)Tw^2)}.
+\end{equation}
+Set $M_L(0,0)=0$.  Then
+\begin{equation}\label{eq:obs-multiplier-bounds}
+ M_L(v,w)\le4\log(1+L)
+ \quad\hbox{for all }v,w\ge0,
+ \qquad
+ M_L(v,w)\le9
+ \quad\hbox{if }\tfrac12\le v/w\le2.
+\end{equation}
+The second bound concerns positive $v,w$ and is independent of $L$.
+\end{lemma}
+
+\begin{proof}
+Write $a=1/L$ and $b=1+1/L$, so $b-a=1$.  If
+$r=\max(v,w)>0$, discard the two denominator factors associated with
+the smaller eigenvalue and substitute $q=Tr^2$.  This gives
+\[
+ M_L(v,w)\le4\int_0^\infty
+ \frac{dq}{(1+aq)(1+bq)}
+ =4\log(b/a)=4\log(1+L).
+\]
+For the identity in the middle, integration up to $Q$ gives
+$\log(1+bQ)-\log(1+aQ)$, which tends to $\log(b/a)$.
+If $v,w$ are positive and comparable as in the statement, let
+$r_0=\min(v,w)$.  Discard the two factors with coefficient $a$, and
+use $b\ge1$.  The remaining integral is at most
+$\int_0^\infty(1+Tr_0^2)^{-2}\,dT=r_0^{-2}$.
+Since $v+w\le3r_0$, this proves the second bound.
+\end{proof}
+
+\begin{proposition}[Universal integrated upper bound]
+\label{prop:information-upper}
+There is an absolute constant $C_0$ such that, for every fixed $t>0$
+and $L\ge1$, the observations in
+Equation~\eqref{eq:obs-covariance-definition} satisfy
+\begin{equation}\label{eq:obs-information-upper}
+ \limsup\int_0^\infty\MI(Y_t;Z_T)\,\frac{dT}{T^2}\le C_0.
+\end{equation}
+In particular, one may take $C_0=9/4$.
+\end{proposition}
+
+\begin{proof}
+Let $p_T(z\mid y)$ denote the density of the centered Gaussian with
+covariance $C_T(y)$, and let $p_T(z)$ be its marginal density.
+For each output $z$, the posterior density with respect to $\eta$ is
+$q_z(y)=p_T(z\mid y)/p_T(z)$.  Applying $2\lambda H\le\Ical$ to
+this density and then averaging in $z$ gives
+\begin{equation}\label{eq:obs-likelihood-lsi}
+ 2\lambda\MI(Y_t;Z_T)
+ \le\E\big|D_y\log p_T(Z_T\mid Y_t)\big|^2.
+\end{equation}
+Here and below the expectation on the right includes the observation
+noise.  The posterior normalization does not depend on $y$ and therefore
+does not contribute to the derivative.
+
+We compute that derivative explicitly.  For any positive covariance
+$C$ and its derivative $D_iC$, differentiation of its determinant and
+inverse yields
+\[
+ D_i\log p(z\mid y)
+ =\tfrac12\left(
+ z^*C^{-1}(D_iC)C^{-1}z-\tr(C^{-1}D_iC)\right).
+\]
+Conditional on $y$, set $B_i=C^{-1/2}(D_iC)C^{-1/2}$ and
+$g=C^{-1/2}z$.  The vector $g$ is standard Gaussian.  Expanding its
+fourth moments gives
+$\E(g^*B_ig-\tr B_i)^2=2\tr B_i^2$; indeed the three Gaussian
+pairings give $(\tr B_i)^2+2\tr B_i^2$ before subtracting the mean.
+Consequently Equation~\eqref{eq:obs-likelihood-lsi} becomes
+\begin{equation}\label{eq:obs-gaussian-fisher}
+ \MI(Y_t;Z_T)\le\frac1{4\lambda}
+ \E\sum_i\tr\big(C_T^{-1}D_iC_T\,C_T^{-1}D_iC_T\big).
+\end{equation}
+
+These calculations also apply to the Sobolev matrix field $R$.  To see
+this without assuming additional regularity, fix $T,L$ and the finite
+dimension of one member of the sequence.  The map
+$R\mapsto C_T$ has bounded derivative on positive semidefinite
+matrices, and its eigenvalues lie in $[1,1+L]$.  For each fixed $z$,
+the map $C\mapsto p(z\mid C)^{1/2}$ has bounded derivative on this
+covariance interval.  The Sobolev chain rule therefore puts
+$q_z^{1/2}$ in the form domain; $p_T(z)>0$ and is a harmless fixed
+normalization.  The Sobolev approximation at the opening of
+Section~\ref{sec:calculus} extends the logarithmic Sobolev inequality
+to this form-domain function.  The squared derivatives
+integrate in $z$ to the nonnegative finite expression in
+Equation~\eqref{eq:obs-gaussian-fisher}.  Approximation in the form
+domain and Tonelli thus justify both that equation and the subsequent
+scale integration.  The finite-dimensional bounds used for form-domain
+membership do not enter its numerical coefficient.
+
+In an eigenbasis of $R$, write $v,w$ for the row and column
+eigenvalues.  The covariance function is
+\[
+ c_T(v)=1+f_L(Tv^2)
+       =\frac{1+bTv^2}{1+aTv^2},
+ \qquad a=1/L,\quad b=1+1/L.
+\]
+Its divided difference, including its derivative at $v=w$, is
+\[
+ \frac{c_T(v)-c_T(w)}{v-w}
+ =\frac{T(v+w)}{(1+aTv^2)(1+aTw^2)}.
+\]
+Multiplying its square by $c_T(v)^{-1}c_T(w)^{-1}$ and integrating
+against $dT/T^2$ gives exactly the multiplier in
+Equation~\eqref{eq:obs-multiplier} for the squared entry
+$|(D_iR)_{vw}|^2$.  When $v=w=0$, the covariance derivative is zero,
+which explains the value assigned to $M_L(0,0)$.
+
+It remains to obtain a bound independent of the fixed cap.  For
+noncomparable pairs with at least one positive eigenvalue,
+Equation~\eqref{eq:obs-sylvester-input} gives entrywise
+\[
+ |(D_iR)_{vw}|^2
+ =\frac{(v+w)^2}{4(v-w)^2}
+       |[\widehat H_i,R]_{vw}|^2
+ \le\frac94|[\widehat H_i,R]_{vw}|^2.
+\]
+This includes a pair with exactly one zero eigenvalue.  Their total
+expected derivative energy is therefore $o(1)$.  The kernel--kernel
+derivative is zero almost everywhere, as established in the Sylvester
+construction, and in any case has zero covariance multiplier.
+Lemma~\ref{lem:obs-multipliers} now yields
+\[
+ \int_0^\infty\MI(Y_t;Z_T)\,\frac{dT}{T^2}
+ \le\frac1{4\lambda}
+ \left(9\E|DR|^2+4\log(1+L)\,o(1)\right).
+\]
+Take the sequence limit at this fixed $L$, and use
+Equation~\eqref{eq:obs-energy-input}.  Stationarity makes the bound
+independent of $t$.  This proves the proposition.  In particular, the
+cap-dependent bound on noncomparable pairs is used only before its
+vanishing error has been removed.
+\end{proof}
+
+\subsection{Stability on the entire scale interval}
+
+We next transfer observations to the frozen covariance of
+Proposition~\ref{prop:frozen-law}.  A bound on a compact interval of
+$T$ would not suffice, because the useful scale of a block depends on
+its eigenvalue.  The following comparison holds on all positive
+scales.
+
+\begin{samepage}
+\begin{lemma}[Integrated covariance stability]
+\label{lem:obs-integrated-stability}
+For $T>0$ let $h_T(v)=\sqrt{1+f_L(Tv^2)}$, $v\ge0$, and put
+\[
+ K_L=\int_0^\infty
+ \frac{dq}{(1+q/L)^3(1+(1+1/L)q)}.
+\]
+Then $0<K_L\le\log(1+L)$, and for arbitrary positive semidefinite
+matrices $A,B$ of the same size,
+\begin{equation}\label{eq:obs-matrix-stability}
+ \int_0^\infty\|h_T(A)-h_T(B)\|_{\HS}^2\,\frac{dT}{T^2}
+ \le K_L\|A-B\|_{\HS}^2.
+\end{equation}
+Let $\sigma$ be the frozen covariance in
+Proposition~\ref{prop:frozen-law}, and couple
+\[
+ \widetilde Z_T=(\Id+f_L(T\sigma))^{1/2}G
+\]
+to $Z_T$ using the same independent Gaussian.  For every fixed $t,L$,
+\begin{equation}\label{eq:obs-observation-stability}
+ \int_0^\infty\E|Z_T-\widetilde Z_T|^2\,\frac{dT}{T^2}=o(1).
+\end{equation}
+\end{lemma}
+\end{samepage}
+
+\begin{proof}
+Direct differentiation gives
+\[
+ h_T'(v)=\frac{Tv}
+ {(1+Tv^2/L)^{3/2}(1+(1+1/L)Tv^2)^{1/2}}.
+\]
+For $v>0$, substitution $q=Tv^2$ therefore shows
+\[
+ \int_0^\infty|h_T'(v)|^2\,\frac{dT}{T^2}=K_L.
+\]
+Discarding two of the three factors $1+q/L$ bounds $K_L$ by the
+integral evaluated in Lemma~\ref{lem:obs-multipliers}, namely
+$\log(1+L)$.  If $v\ge w\ge0$, the fundamental theorem of calculus
+and Cauchy--Schwarz give
+\[
+ |h_T(v)-h_T(w)|^2
+ \le(v-w)\int_w^v|h_T'(r)|^2\,dr.
+\]
+Integrating in $T$ and using Tonelli proves the scalar bound
+$K_L(v-w)^2$.  The value of the derivative integral at the single
+endpoint $r=0$ does not affect this argument, so it covers $w=0$.
+
+Choose an orthonormal eigenbasis $(e_a)$ of $A$ with eigenvalues $v_a$
+and an orthonormal eigenbasis $(f_b)$ of $B$ with eigenvalues $w_b$.
+The entry of their functional-calculus difference between these two
+bases is
+\[
+ \langle e_a,(h_T(A)-h_T(B))f_b\rangle
+   =(h_T(v_a)-h_T(w_b))\langle e_a,f_b\rangle.
+\]
+Multiply the scalar bound by $|\langle e_a,f_b\rangle|^2$ and sum
+over $a,b$.  The sum on the right is
+$K_L\|A-B\|_{\HS}^2$, proving
+Equation~\eqref{eq:obs-matrix-stability}.  Finally, conditional on all
+data except $G$, Gaussian isometry gives
+\[
+ \E_G|Z_T-\widetilde Z_T|^2
+ =\|h_T(R(Y_t))-h_T(\sigma^{1/2})\|_{\HS}^2.
+\]
+Apply the matrix bound and the squared-mean convergence in
+Proposition~\ref{prop:frozen-law}.  The resulting factor $K_L$ is
+fixed while the sequence limit is taken.
+\end{proof}
+
+\subsection{Conditional information and blockwise tests}
+
+Let $\mathcal D$ be the initial position, random spectral partition,
+chosen orthonormal frame, and frozen block matrices from
+Proposition~\ref{prop:frozen-law}.  Block formulas below use components
+in this $\mathcal D$-measurable frame.  Every information term
+$\MI(Y_t;Z_T)$ continues to refer to the original observation in
+ambient coordinates; conditional on $\mathcal D$, conversion to
+the block coordinates is invertible.  Orthogonal invariance makes
+the components of the fresh Gaussian noise conditionally standard
+and independent of the diffusion data.  Conditional on $\mathcal D$,
+$N=B_t/\sqrt t$ is standard Gaussian.  On a positive block with
+rounded eigenvalue $\beta$, write $H[v]=\sum_{i\in\beta}v_iH_i$.
+The frozen covariance is
+\begin{equation}\label{eq:obs-frozen-covariance}
+ \sigma_\beta=\beta^2
+       \exp\big(\sqrt{8t}\,H[N]-4t\Id\big).
+\end{equation}
+All $H_i$ here are the modified matrices supported on their initial
+blocks.  Their bounds, from
+Equations~\eqref{eq:flattening-bound} and~\eqref{eq:matrix-tracial-defects},
+will be used in the form
+\begin{align}
+ &S=\sum_iH_i^2\le C\Id,
+ \qquad \|H[v]\|_{\HS}^2\le C_1|v|^2,
+ \label{eq:obs-block-bounds}\\
+ &\tau(\Id)\longrightarrow1,
+ \qquad\tau((S-\Id)^2)=o(1),
+ \qquad\sum_{i,j}\tau(|[H_i,H_j]|^2)=o(1).
+ \label{eq:obs-block-defects}
+\end{align}
+The constants $C,C_1\ge1$ are absolute.  The entries of the three-index
+array $(H_i)_{ab}$ are also asymptotically symmetric in all three
+indices in the weighted squared norm.  As before,
+\[
+ \tau(U)=\E\sum_\beta\beta^2\tr_\beta U_\beta,
+\]
+where expectation includes any Gaussian variables appearing in $U$.
+No trace in this definition or in
+Equation~\eqref{eq:obs-block-bounds} is normalized by block dimension.
+
+\begin{lemma}[Conditional Ornstein--Uhlenbeck score]
+\label{lem:obs-ou-score}
+Let $N$ be a standard Gaussian vector conditional on $\mathcal D$,
+and let $Z$ be any observation for which
+$\MI(N;Z\mid\mathcal D)<\infty$.  With independent standard
+Gaussian $G'$ set
+\[
+ N_u=\alpha N+\sqrt{1-\alpha^2}\,G',\qquad \alpha=e^{-u},\quad u>0.
+\]
+The relative score of its posterior, with respect to the standard
+Gaussian marginal, is
+\begin{equation}\label{eq:obs-ou-score}
+ D\log\frac{p_{N_u\mid Z,\mathcal D}}{\gamma}(N_u)
+ =\frac{\alpha}{1-\alpha^2}
+   \E[N-\alpha N_u\mid N_u,Z,\mathcal D].
+\end{equation}
+For every compact interval $\mathcal J\subset(0,\infty)$,
+\begin{equation}\label{eq:obs-ou-information}
+ \MI(N;Z\mid\mathcal D)
+ \ge\int_{\mathcal J}\frac{\alpha^2}{(1-\alpha^2)^2}
+ \E\left|\E[N-\alpha N_u\mid N_u,Z,\mathcal D]\right|^2\,du.
+\end{equation}
+For the covariance observation in
+Equation~\eqref{eq:obs-covariance-definition},
+\begin{equation}\label{eq:obs-data-processing}
+ \MI(Y_t;Z_T)\ge\MI(N;Z_T\mid\mathcal D).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Put $q=1-\alpha^2$.  Given a posterior law $\pi(dn)$ of $N$ at
+fixed $(Z,\mathcal D)$, the density at $y$ after adding the OU noise
+is the integral of
+$(2\pi q)^{-d/2}\exp(-|y-\alpha n|^2/(2q))$ against $\pi$.
+Differentiating this kernel and dividing by its integral gives
+\[
+ D\log p(y\mid Z,\mathcal D)
+ =-\frac{y-\alpha\E[N\mid N_u=y,Z,\mathcal D]}q.
+\]
+The score of the standard Gaussian is $-y$.  Subtracting it proves
+Equation~\eqref{eq:obs-ou-score}.  Kernel differentiation is justified
+first with $N$ truncated; on compact positive time intervals its
+derivatives are bounded by Gaussian kernels times polynomials, and
+truncation can then be removed by conditional second moments.
+
+The OU semigroup has generator $\Delta-y\cdot D$.  For a positive
+posterior density $r_u$ relative to $\gamma$, integration by parts
+gives
+\[
+ \frac{d}{du}\int r_u\log r_u\,d\gamma
+ =-\int\frac{|Dr_u|^2}{r_u}\,d\gamma.
+\]
+This is the entropy dissipation calculation of
+Lemma~\ref{lem:entropy-calculus}; positive-time smoothing, truncation
+of the density and spatial cutoffs justify it for finite-entropy
+posteriors.  Averaging over $(Z,\mathcal D)$ and integrating over
+$\mathcal J$ bounds its nonnegative right side by the initial
+conditional information.  Substitution of the score formula proves
+Equation~\eqref{eq:obs-ou-information}.
+
+For the last assertion, observation noise is independent of the
+initial data and diffusion path, and its law depends on them only
+through $Y_t$.  Hence
+$(\mathcal D,N)\longrightarrow Y_t\longrightarrow Z_T$ is a Markov
+chain.  The chain rule and data processing give
+\[
+ \MI(N;Z_T\mid\mathcal D)
+ \le\MI(\mathcal D,N;Z_T)\le\MI(Y_t;Z_T).
+\]
+These informations are finite for each member of the sequence:
+the covariance interval $[\Id,(1+L)\Id]$ bounds the output entropy
+from above by that of a Gaussian of covariance $(1+L)\Id$ and
+bounds its conditional entropy from below by that of a standard
+Gaussian.  Thus $\MI(Y_t;Z_T)\le(d/2)\log(1+L)$ in dimension $d$.
+This finiteness assertion is used only to justify the identities.
+\end{proof}
+
+\begin{lemma}[Block tests and the scale Jacobian]
+\label{lem:obs-block-tests}
+Fix compact intervals $\mathcal J,\mathcal X\subset(0,\infty)$, and
+write $\kappa(u)=\alpha^2/(1-\alpha^2)^2$, where $\alpha=e^{-u}$.
+Let $P(u,x,a,b)$ be a real polynomial, evaluated on matrices with any
+one fixed ordering of each monomial.  On each positive block define
+\[
+ A_\beta=H_\beta[N_\beta],\qquad
+ A_{u,\beta}=H_\beta[(N_u)_\beta],\qquad
+ B_{\beta,x}=H_\beta[(Z_{x/\beta^2})_\beta],
+\]
+and $P_{\beta,u,x}=P(u,x,A_{u,\beta},B_{\beta,x})$.  Then
+\begin{align}
+ \int_0^\infty\MI(Y_t;Z_T)\,\frac{dT}{T^2}
+ \ge{}&\int_{\mathcal J}\int_{\mathcal X}\kappa(u)
+ \E\sum_\beta\beta^2\tr_\beta
+ \bigl(2(A_\beta-\alpha A_{u,\beta})P_{\beta,u,x}
+ \notag\\[-2pt]
+ &\hspace{43mm}-C_1P_{\beta,u,x}^*P_{\beta,u,x}\bigr)
+ \,\frac{dx}{x^2}\,du.
+ \label{eq:obs-block-test-bound}
+\end{align}
+Here the constant $C_1$ is the absolute constant in
+Equation~\eqref{eq:obs-block-bounds}.
+\end{lemma}
+
+\begin{proof}
+For a block, consider the linear map
+$\mathcal H:v\mapsto H[v]$ between its Euclidean coordinate space
+and its matrix space with the ordinary Hilbert--Schmidt norm.
+Its adjoint is
+\[
+ (\mathcal H^*Q)_i=\tr(H_iQ),
+\]
+because $H_i$ is real symmetric.  Taking the adjoint in
+Equation~\eqref{eq:obs-block-bounds} gives
+\begin{equation}\label{eq:obs-adjoint-bound}
+ \sum_i|\tr(H_iQ)|^2\le C_1\tr(Q^*Q).
+\end{equation}
+This step has no factor involving the dimension of the block.
+
+At fixed $u,T$, select precisely those positive blocks for which
+$x=T\beta^2\in\mathcal X$.  On a selected block set
+\[
+ V_i=\tr_\beta
+ \bigl(H_iP(u,T\beta^2,H[(N_u)_\beta],H[(Z_T)_\beta])\bigr),
+ \qquad i\in\beta,
+\]
+and set all other components, including those in the kernel, equal
+to zero.  This vector is measurable with respect to
+$(N_u,Z_T,\mathcal D)$.  If
+$m=\E[N-\alpha N_u\mid N_u,Z_T,\mathcal D]$, completing the square
+gives
+\[
+ \E|m|^2\ge2\E(N-\alpha N_u)\cdot V-\E|V|^2.
+\]
+The coordinates of distinct selected blocks are orthogonal.  Their
+costs therefore add, and Equation~\eqref{eq:obs-adjoint-bound} bounds
+the cost by $C_1$ times the sum of their unnormalized squared matrix
+norms.  Their linear terms contract exactly to
+\[
+ (N-\alpha N_u)\cdot V
+ =\sum_{\beta\text{ selected}}\tr_\beta
+ \bigl((H[N_\beta]-\alpha H[(N_u)_\beta])P\bigr).
+\]
+Substitute these two statements in
+Lemma~\ref{lem:obs-ou-score} and integrate in $T$.
+
+For each realized positive block separately, the change of variable
+$x=T\beta^2$ has the exact Jacobian
+\begin{equation}\label{eq:obs-scale-jacobian}
+ T=\frac{x}{\beta^2},\qquad dT=\frac{dx}{\beta^2},\qquad
+ \boxed{\displaystyle\frac{dT}{T^2}=\beta^2\frac{dx}{x^2}}.
+\end{equation}
+Thus the block selection becomes $x\in\mathcal X$, and the
+coefficient of each block trace becomes precisely $\beta^2$.
+This proves Equation~\eqref{eq:obs-block-test-bound}.
+
+For completeness, all its polynomial terms are integrable on the
+compact parameter rectangle.  The moment bounds of
+Lemma~\ref{lem:matrix-gaussian-moments} control $A_\beta$ and
+$A_{u,\beta}$ under the weighted trace.  Conditional on the
+initial data, diffusion path, and OU noise, $B_{\beta,x}$ is a
+Gaussian matrix series whose vector covariance is at most
+$(1+L)\Id$.  If its coefficients are $K_j=\sum_i Q_{ij}H_i$
+for a square root $Q$ of that covariance, then for every vector $v$
+\[
+ \sum_j|K_jv|^2\le(1+L)\sum_i|H_iv|^2\le C(1+L)|v|^2.
+\]
+The same Gaussian even-moment proof therefore gives, for every fixed
+integer $p\ge1$,
+\begin{equation}\label{eq:obs-gaussian-moment-bound}
+ \E_G\tr_\beta|B_{\beta,x}|^{2p}
+ \le[C(1+L)]^p(2p-1)!!\tr_\beta\Id.
+\end{equation}
+Trace H\"older bounds each mixed word by its factors' moments.
+After multiplying by $\beta^2$ and summing, the right side is a
+constant times $\tau(\Id)$, which is finite and bounded along the
+sequence.  The measure $\kappa(u)\,du\,dx/x^2$ is finite on the
+chosen rectangle.  Tonelli applies to the square terms, while these
+moment bounds justify Fubini for the linear terms.  One can
+equivalently first truncate $V$ in the conditional-mean inequality
+and then remove that truncation in this integrated squared norm.
+\end{proof}
+
+\subsection{Transfer of the polynomial tests to scalar observations}
+
+The scale substitution has now produced the tracial weights of the
+matrix limit.  We next identify that limit for the matrices containing
+the observation.  Two replacements are needed: first replace $Z_T$
+by its frozen version on the whole scale interval, and then use the
+approximate symmetry of the three-index array to move a bounded
+matrix function through $H[\,\cdot\,]$.
+
+\begin{lemma}[Replacement in integrated polynomial tests]
+\label{lem:obs-polynomial-replacement}
+In the notation of Lemma~\ref{lem:obs-block-tests}, put
+$\widetilde B_{\beta,x}
+=H_\beta[(\widetilde Z_{x/\beta^2})_\beta]$.
+For every fixed polynomial in
+$u,x,A_\beta,A_{u,\beta},B_{\beta,x}$ and their adjoints, its
+weighted trace integral over $\mathcal J\times\mathcal X$ changes
+by $o(1)$ if $B_{\beta,x}$ is replaced by
+$\widetilde B_{\beta,x}$.
+\end{lemma}
+
+\begin{proof}
+The flattening bound and the inverse substitution in
+Equation~\eqref{eq:obs-scale-jacobian} give
+\begin{align*}
+ &\E\sum_\beta\beta^2\int_{\mathcal X}
+ \|B_{\beta,x}-\widetilde B_{\beta,x}\|_{\HS}^2\,\frac{dx}{x^2}
+ \\
+ &\quad=\int_0^\infty\E\sum_{\beta:T\beta^2\in\mathcal X}
+ \|H_\beta[(Z_T-\widetilde Z_T)_\beta]\|_{\HS}^2\,\frac{dT}{T^2}
+ \\
+ &\quad\le C_1\int_0^\infty
+ \E|Z_T-\widetilde Z_T|^2\,\frac{dT}{T^2}=o(1),
+\end{align*}
+by Lemma~\ref{lem:obs-integrated-stability}.  All projected errors
+are summed before this comparison; no small block weight is divided
+out.  Multiplication by the bounded weight $\kappa(u)$ and
+integration in the compact interval $\mathcal J$ preserve $o(1)$.
+
+To pass from this squared norm to arbitrary fixed products, include
+the parameter integrals in the finite tracial functional
+\[
+ \mathcal T(U)=\int_{\mathcal J}\int_{\mathcal X}
+ \kappa(u)\E\sum_\beta\beta^2\tr_\beta U_{\beta,u,x}
+ \,\frac{dx}{x^2}\,du.
+\]
+Every matrix in the statement has bounded $L^p(\mathcal T)$ norm
+for every fixed finite $p$.  For $B$ this was proved in
+Equation~\eqref{eq:obs-gaussian-moment-bound}; for $\widetilde B$
+the same proof applies because its covariance is also at most
+$(1+L)\Id$.  The remaining matrices are covered by the Gaussian
+moment lemma.  For any $q>p\ge2$, interpolation with
+$1/p=\theta/2+(1-\theta)/q$ gives
+\[
+ \|B-\widetilde B\|_{L^p(\mathcal T)}
+ \le\|B-\widetilde B\|_{L^2(\mathcal T)}^\theta
+       \|B-\widetilde B\|_{L^q(\mathcal T)}^{1-\theta}=o(1).
+\]
+For lower powers, H\"older and boundedness of
+$\mathcal T(\Id)$ give the same conclusion.  Expand the difference
+of two words of length $r$ by replacing one factor at a time.
+Trace H\"older with all exponents $r$ bounds each resulting trace
+by the product of its factors' $L^r(\mathcal T)$ norms.  The changed
+factor tends to zero and the others remain bounded.  There are
+only finitely many such terms.  This proves the claim, including
+polynomial squared norms and linear terms with $A-\alpha A_u$.
+\end{proof}
+
+For fixed $t,L$, define the scalar function
+\begin{equation}\label{eq:obs-g-function}
+ g_x(v)=\sqrt{1+f_L\big(xe^{\sqrt{8t}\,v-4t}\big)},
+ \qquad x>0,\ v\in\R.
+\end{equation}
+Then $1\le g_x\le\sqrt{1+L}$.  By
+Equation~\eqref{eq:obs-frozen-covariance}, the frozen observation on
+a block at $T=x/\beta^2$ is exactly
+$g_x(A_\beta)G_\beta$.  Thus
+$\widetilde B_{\beta,x}=H_\beta[g_x(A_\beta)G_\beta]$.
+
+\begin{lemma}[Polynomial density under exponential moments]
+\label{lem:obs-polynomial-density}
+Let $\nu$ be a finite positive measure on $\R^k$ with
+$\int e^{a|w|}\,d\nu(w)<\infty$ for every $a>0$.  Polynomials
+are dense in $L^p(\nu)$ for every $1\le p<\infty$.
+\end{lemma}
+
+\begin{proof}
+Suppose a continuous linear functional on $L^p(\nu)$ annihilates
+all polynomials.  Its representing density $F$ belongs to
+$L^{p'}(\nu)$, where $p'=p/(p-1)$ and $p'=\infty$ if $p=1$.
+H\"older and the hypothesis show that the finite signed measure
+$F\,d\nu$ has exponential moments of every linear magnitude.
+For each $\xi\in\R^k$, the function
+\[
+ z\longmapsto\int e^{z\xi\cdot w}F(w)\,d\nu(w)
+\]
+is entire: on every compact set of complex $z$, its derivatives
+are dominated by an integrable exponential.  All its derivatives
+at zero vanish because $(\xi\cdot w)^j$ is a polynomial.
+Its Taylor series and hence the entire function are zero.
+Evaluating at $z=\mathrm i$ proves that the Fourier transform of
+$F\,d\nu$ vanishes.  Fourier uniqueness for finite measures can
+here be seen by convolving with a centered Gaussian: the convolution
+has zero Fourier transform and an integrable Fourier inversion
+formula, so is zero for every positive Gaussian variance; letting
+that variance tend to zero shows the original measure is zero.
+Thus $F=0$ almost everywhere.  Duality, or equivalently separation
+of a proper closed subspace, proves the density assertion.
+\end{proof}
+
+\begin{lemma}[Nonlinear mixed trace law]
+\label{lem:obs-nonlinear-trace-law}
+Let $n,g',g$ be independent scalar standard Gaussians and let
+$n_u=\alpha n+\sqrt{1-\alpha^2}\,g'$.  For every fixed mixed
+polynomial, the weighted trace moments of
+\[
+ \big(A_\beta,A_{u,\beta},\widetilde B_{\beta,x}\big)
+\]
+converge to the corresponding scalar moments of
+\[
+ \big(n,n_u,g_x(n)g\big).
+\]
+The assertion holds after integration over compact positive $u,x$
+intervals against $\kappa(u)\,du\,dx/x^2$, and permits fixed
+polynomial dependence on $u,x$.  Adjoints in a matrix word become
+the identical real scalar factors in its limit.
+\end{lemma}
+
+\begin{proof}
+First we prove the matrix comparison
+\begin{equation}\label{eq:obs-index-swap-comparison}
+ H[g_x(A)G]-g_x(A)H[G]\longrightarrow0
+ \quad\hbox{in weighted squared Hilbert--Schmidt norm}.
+\end{equation}
+All notation in this calculation is on one block.  Define
+\[
+ J(v)_{ab}=(H_bv)_a,
+ \qquad D(v)=H[v]-J(v),
+ \qquad \Delta_{abi}=(H_i)_{ab}-(H_b)_{ai}.
+\]
+Put $q=g_x(A)$ and $B=H[G]$.  The exact columnwise identity is
+\begin{equation}\label{eq:obs-index-swap-identity}
+ H[qG]-qB=D(qG)+K-qD(G),
+ \qquad K_{\cdot b}=[H_b,q]G.
+\end{equation}
+Indeed the $b$th column of $J(qG)-qJ(G)$ is
+$H_bqG-qH_bG$.  Conditional Gaussian averaging therefore bounds
+the square of the left side by
+\begin{equation}\label{eq:obs-index-swap-error}
+ 6(1+L)\sum_{a,b,i}|\Delta_{abi}|^2
+       +3\sum_b\|[H_b,g_x(A)]\|_{\HS}^2.
+\end{equation}
+For the first and last terms in
+Equation~\eqref{eq:obs-index-swap-identity}, we used
+$\|q\|_{\op}\le\sqrt{1+L}$ and the Gaussian isometry for the
+linear map $D$.
+
+The function $g_x$ is Lipschitz with a bound depending only on
+$t,L$, uniformly for all $x>0$.  In fact, with
+$a=\sqrt{8t}$ and $r=xe^{av-4t}/L$, differentiation gives
+\[
+ |g_x'(v)|
+ =\frac{aLr}{2(1+r)^2g_x(v)}\le\frac{aL}{8}.
+\]
+In an eigenbasis of $A$ with eigenvalues $r_a$, the squared
+commutator norm is
+\[
+ \|[H_b,g_x(A)]\|_{\HS}^2
+ =\sum_{a,c}|(H_b)_{ac}|^2|g_x(r_c)-g_x(r_a)|^2
+ \le\operatorname{Lip}(g_x)^2\|[H_b,A]\|_{\HS}^2.
+\]
+This is a scalar Lipschitz estimate for matrix entries in the
+Hilbert--Schmidt norm.  Averaging $A=\sum_iN_iH_i$ over $N$ gives
+\[
+ \E_N\sum_b\|[H_b,A]\|_{\HS}^2
+ =\sum_{b,i}\|[H_b,H_i]\|_{\HS}^2.
+\]
+Multiply Equation~\eqref{eq:obs-index-swap-error} by the block
+weight, sum, and average.  The total symmetry defect and
+Equation~\eqref{eq:obs-block-defects} prove
+Equation~\eqref{eq:obs-index-swap-comparison}, uniformly in $x$ at
+fixed $t,L$.
+
+Every fixed higher moment of both sides of this comparison is
+bounded.  Conditional on $A$ and the initial data,
+$H[g_x(A)G]$ is a Gaussian matrix series with coefficient-square
+sum at most $(1+L)S$, by the quadratic-form argument used in
+Equation~\eqref{eq:obs-gaussian-moment-bound}.  Also the Schatten
+ideal inequality gives
+\[
+ \|g_x(A)H[G]\|_{p}\le\sqrt{1+L}\,\|H[G]\|_{p}
+\]
+on each block.  Interpolation and telescoping products, exactly as
+in Lemma~\ref{lem:obs-polynomial-replacement}, show that these
+two matrices may be interchanged in every fixed polynomial trace.
+
+It remains to identify traces containing $g_x(A)H[G]$.
+Lemma~\ref{lem:matrix-gaussian-moments} gives the joint scalar
+Gaussian moment law for $A=H[N]$, $H[G']$, and $H[G]$.
+In particular, $A_u=\alpha A+\sqrt{1-\alpha^2}H[G']$ has the
+required joint law with them.  To include the bounded continuous
+function $g_x(A)$, fix $x$ and a mixed word of length $r$.
+Choose a polynomial $p$ approximating $g_x$ in a sufficiently high
+finite $L^q(\gamma_1)$ norm, where $\gamma_1$ is the scalar
+standard Gaussian law.  Choose $q$ larger than all the H\"older
+exponents needed for this fixed word, and take the approximation
+error at most $1$; then $\|p\|_{L^q(\gamma_1)}
+\le\sqrt{1+L}+1$.  Such approximations exist by the
+polynomial-density argument in
+Lemma~\ref{lem:obs-polynomial-density}, applied to one Gaussian
+coordinate.  The scalar spectral measures of $A$ under $\tau$
+converge to $\gamma_1$ by the moment and exponential-moment bounds
+in Lemma~\ref{lem:matrix-gaussian-moments}.  Their higher moments
+are uniformly bounded.  Consequently, for even $q$,
+\begin{equation}\label{eq:obs-functional-approximation}
+ \tau\bigl(|g_x(A)-p(A)|^q\bigr)
+ \longrightarrow\E|g_x(n)-p(n)|^q.
+\end{equation}
+To justify this last passage directly, cut the scalar spectral
+variable off at $|v|\le M$, use weak convergence for the bounded
+continuous integrand there, and bound the complement by a higher
+uniform moment of $A$.  Since $g_x$ is bounded and $p$ has fixed
+degree, that tail tends to zero as $M\to\infty$.
+
+Replace each occurrence of $g_x(A)$ by $p(A)$.  Telescoping the
+word and applying trace H\"older bounds the replacement error
+by a constant times the $L^q$ error in
+Equation~\eqref{eq:obs-functional-approximation}, with all other
+factors controlled by fixed higher moments.  For factors $p(A)$,
+the preceding $L^q$ bound and moment convergence give a bound
+independent of how accurately $p$ approximates $g_x$.
+Thus the error constant stays bounded as that approximation is
+improved.  The remaining word
+is polynomial in the three independent Gaussian matrix series,
+so its trace has the scalar Gaussian limit.  Let the scalar
+approximation error tend to zero.  This proves the claimed limit
+at each fixed $u,x$; the scalar third coordinate is
+$g_x(n)g$.
+
+Finally, the moment bounds for the original, bounded function
+$g_x$ are uniform on the compact parameter rectangle, as are the
+bounds on polynomial coefficients in $u,x$ and on $\alpha$.
+They dominate each mixed trace by an integrable constant.
+Dominated convergence thus permits parameter integration.  This
+last argument uses the uniform bounds for $g_x$ itself, and does
+not require choosing one polynomial approximant uniformly in $x$.
+\end{proof}
+
+The preceding lemmas identify the polynomial tests of the actual
+observation with those of the scalar variance mixture.  We can now
+optimize the tests and integrate the scalar OU dissipation.  The
+constant in the resulting comparison is exactly the constant of
+the unnormalized flattening estimate.
+
+\begin{proposition}[Scalar information lower bound]
+\label{prop:information-lower}
+For every fixed $t>0$ and $L\ge1$, let $g_x$ be defined by
+Equation~\eqref{eq:obs-g-function}, and let $n,g$ be independent
+scalar standard Gaussians.  Then
+\begin{equation}\label{eq:obs-information-lower}
+ \liminf\int_0^\infty\MI(Y_t;Z_T)\,\frac{dT}{T^2}
+ \ge C_1^{-1}\int_0^\infty
+       \MI\big(n;g_x(n)g\big)\,\frac{dx}{x^2}.
+\end{equation}
+The constant $C_1$ is universal and is independent of $t,L$.
+\end{proposition}
+
+\begin{proof}
+Fix compact positive intervals $\mathcal J,\mathcal X$, and let
+$n,g',g$ be independent scalar standard Gaussians.  Write
+$n_u=\alpha n+\sqrt{1-\alpha^2}\,g'$, and fix a scalar polynomial
+$P(u,x,a,b)$.  Give its matrix version any fixed
+ordering of the factors of each monomial.  Lemmas
+\ref{lem:obs-block-tests}, \ref{lem:obs-polynomial-replacement},
+and~\ref{lem:obs-nonlinear-trace-law}, in that order, imply
+\begin{align}
+ &\liminf\int_0^\infty\MI(Y_t;Z_T)\,\frac{dT}{T^2}
+ \notag\\
+ &\quad\ge\int_{\mathcal J}\int_{\mathcal X}\kappa(u)
+ \E\big[2(n-\alpha n_u)P(u,x,n_u,g_x(n)g)
+ \notag\\[-2pt]
+ &\hspace{58mm}-C_1P(u,x,n_u,g_x(n)g)^2\big]
+       \,\frac{dx}{x^2}\,du.
+ \label{eq:obs-scalar-test-bound}
+\end{align}
+If the matrix polynomial is not symmetric, its prelimit cost is
+$\tr(P^*P)$ as in Equation~\eqref{eq:obs-block-test-bound}.
+The mixed trace law takes this cost to the real scalar $P^2$,
+so symmetry of the chosen ordering is unnecessary.
+
+The sequence limit in Equation~\eqref{eq:obs-scalar-test-bound}
+has been taken for each fixed polynomial.  We now optimize the
+tests in one squared-mean space containing the compact parameters.
+Give $(u,x)$ the finite measure $\kappa(u)\,du\,dx/x^2$ on
+$\mathcal J\times\mathcal X$, independently of the scalar
+Gaussians $n,g',g$, and put
+\begin{equation}\label{eq:obs-scalar-observed-vector}
+ W=(u,x,n_u,g_x(n)g),\qquad r=n-\alpha n_u.
+\end{equation}
+The first two coordinates of $W$ are bounded.  At each fixed
+$u,x$, the variable $n_u$ is standard Gaussian, and
+$|g_x(n)g|\le\sqrt{1+L}|g|$.
+Cauchy--Schwarz therefore gives all linear exponential moments
+for its finite law, uniformly on the compact parameter rectangle.
+Lemma~\ref{lem:obs-polynomial-density} shows that polynomials in
+the four coordinates of $W$ are dense in this $L^2$ space.
+
+Since $\E[r^2\mid u,x]=1-\alpha^2\le1$, the conditional mean
+$m(W)=\E[r\mid W]$ belongs to the same space.  For the finite
+joint measure just defined, completion of the square gives
+\[
+ \int\big(2rP(W)-C_1P(W)^2\big)
+ =C_1^{-1}\int m^2-C_1\int(P-m/C_1)^2.
+\]
+Density makes the infimum of the final squared error zero, using
+single four-variable polynomials in the integrated space.
+Taking their supremum and then disintegrating with respect to
+$(u,x)$ turns the right side of
+Equation~\eqref{eq:obs-scalar-test-bound} into
+\begin{equation}\label{eq:obs-scalar-score-lower}
+ C_1^{-1}\int_{\mathcal J}\int_{\mathcal X}
+ \frac{\alpha^2}{(1-\alpha^2)^2}
+ \E\left|\E[n-\alpha n_u\mid n_u,g_x(n)g]\right|^2
+ \,\frac{dx}{x^2}\,du.
+\end{equation}
+The conditional expectation here is taken at each fixed $u,x$.
+All approximation constants used before this optimization may
+depend on the fixed polynomial, $t,L$, and the compact intervals.
+The coefficient $C_1$ depends only on
+Equation~\eqref{eq:obs-adjoint-bound}.
+
+We verify the scalar information identity including its endpoints.
+Fix $x>0$ and write $z_x=g_x(n)g$.  Its information is finite:
+conditional variance lies in $[1,1+L]$, so Gaussian entropy
+comparison gives
+\begin{equation}\label{eq:obs-scalar-information-finite}
+ 0\le\MI(n;z_x)\le\tfrac12\log(1+L).
+\end{equation}
+Indeed $\E z_x^2\le1+L$ bounds its entropy above by
+$\frac12\log(2\pi e(1+L))$, while its conditional entropy is
+$\frac12\log(2\pi e)+\E\log g_x(n)$ and is at least
+$\frac12\log(2\pi e)$.
+Apply the scalar version of Lemma~\ref{lem:obs-ou-score} and its
+entropy dissipation identity between times $a$ and $b$, where
+$0<a<b<\infty$.  This gives
+\begin{align*}
+ \MI(n_a;z_x)-\MI(n_b;z_x)
+ =\int_a^b\frac{\alpha^2}{(1-\alpha^2)^2}
+ \E\left|\E[n-\alpha n_u\mid n_u,z_x]\right|^2\,du.
+\end{align*}
+As $a\downarrow0$, the joint pair $(n_a,z_x)$ converges in law
+to $(n,z_x)$.  Lower semicontinuity of relative entropy, applied
+to the joint law and the product of its marginals, gives
+$\MI(n;z_x)\le\liminf_{a\downarrow0}\MI(n_a;z_x)$.
+The reverse inequality follows from the Markov chain
+$z_x\longrightarrow n\longrightarrow n_a$ and data processing.
+Thus the initial endpoint is exactly $\MI(n;z_x)$.
+At the other endpoint, the same chain gives
+\[
+ 0\le\MI(n_u;z_x)\le\MI(n_u;n)
+       =-\tfrac12\log(1-\alpha^2)\longrightarrow0.
+\]
+The equality follows by subtracting the conditional Gaussian
+entropy of variance $1-\alpha^2$ from that of the standard
+Gaussian marginal.  We have proved
+\begin{equation}\label{eq:obs-scalar-ou-identity}
+ \MI(n;z_x)=\int_0^\infty
+ \frac{\alpha^2}{(1-\alpha^2)^2}
+ \E\left|\E[n-\alpha n_u\mid n_u,z_x]\right|^2\,du.
+\end{equation}
+
+Finally exhaust both positive parameter axes by compact intervals
+in Equation~\eqref{eq:obs-scalar-score-lower}.  Its integrand is
+nonnegative.  Monotone convergence and
+Equation~\eqref{eq:obs-scalar-ou-identity} prove
+Equation~\eqref{eq:obs-information-lower}.  In particular, the
+polynomial optimization and the exhaustion take place after the
+counterexample-sequence limit, so neither requires a uniform
+approximation over unbounded parameters.
+\end{proof}
+
+\subsection{Divergence of the scalar scale integral}
+
+The upper and lower comparisons are now in the same information
+units, with universal constants $C_0,C_1$.  To contradict them, it
+suffices to let the fixed diffusion time and the fixed covariance
+cap grow together.  The following elementary test detects the
+large values of the scalar lognormal variance.
+
+\begin{lemma}[Scalar information divergence]
+\label{lem:scalar-divergence}
+Let $n,g$ be independent scalar standard Gaussians and, for
+$t\ge1$, define
+\[
+ \ell=\exp(\sqrt{8t}\,n-4t),\qquad
+ z_x=\sqrt{1+f_t(x\ell)}\,g.
+\]
+There are absolute constants $c>0$ and $h_0\ge1$ such that, for
+all $t\ge h_0$,
+\begin{equation}\label{eq:obs-scalar-divergence}
+ \int_0^\infty\MI(n;z_x)\,\frac{dx}{x^2}
+ \ge c\E\big[\ell\1_{\{\ell\ge e^{2t}\}}\big]
+                  \log(t/h_0)
+ \ge\frac c2\log(t/h_0).
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+The Gaussian moment generating formula gives $\E\ell=1$.
+Let $P_{x,n}$ be the conditional law of $z_x$ given $n$, and
+let $Q_x$ be its marginal law.  Thus
+\[
+ \MI(n;z_x)=\E_n H(P_{x,n}\mid Q_x).
+\]
+Set $p_0=\Prob\{|g|\ge\sqrt2\}>0$.  Suppose first that
+$\ell\ge e^{2t}$, and choose a scale with
+$h=x\ell\in[h_0,t]$.  Since $h\le t$,
+\[
+ \Var(z_x\mid n)=1+\frac{h}{1+h/t}\ge h/2.
+\]
+It follows that the event $A_h=\{|z|\ge\sqrt h\}$ has
+conditional probability $p=P_{x,n}(A_h)\ge p_0$.
+
+To bound its marginal probability, use an independent variable
+$\ell'$ with the law of $\ell$.  On $x\ell'\le1$, the
+conditional variance of the mixture component is at most $2$,
+so its probability of $A_h$ is at most $2e^{-h/4}$ by the
+Gaussian exponential bound.  On the complementary event,
+Markov's inequality and $\E\ell'=1$ give probability at most
+$x$.  Hence
+\begin{equation}\label{eq:obs-binary-marginal-bound}
+ q=Q_x(A_h)\le2e^{-h/4}+x
+       \le2e^{-h/4}+te^{-2t}\le3e^{-h/4}.
+\end{equation}
+The last inequality holds for $t\ge1$ and $h\le t$, because
+$te^{-2t}\le e^{-t/4}\le e^{-h/4}$.
+
+Apply data processing to the event indicator.  The binary
+relative entropy is bounded below by
+\begin{align*}
+ H(P_{x,n}\mid Q_x)
+ &\ge p\log(p/q)+(1-p)\log\big((1-p)/(1-q)\big)\\
+ &\ge p\log(1/q)-\log2\\
+ &\ge p_0(h/4-\log3)-\log2.
+\end{align*}
+Choose once and for all
+\[
+ h_0\ge\max\{1,\,8\log3+8\log2/p_0\},
+ \qquad c=p_0/8.
+\]
+The preceding expression is then at least $ch$ for
+$h\in[h_0,t]$.  This entropy estimate is made separately for
+each conditional law $P_{x,n}$; the chosen event may therefore
+depend on $n$ through $h=x\ell$.
+
+Nonnegativity of relative entropy permits restricting the scale
+integral to these choices of $n$ and $x$.  Tonelli and the change
+of variable $x=h/\ell$ yield
+\begin{align*}
+ \int_0^\infty\MI(n;z_x)\,\frac{dx}{x^2}
+ &\ge\E\left[\1_{\{\ell\ge e^{2t}\}}
+       \int_{h_0/\ell}^{t/\ell}
+           H(P_{x,n}\mid Q_x)\,\frac{dx}{x^2}\right]\\
+ &\ge c\E\left[\ell\1_{\{\ell\ge e^{2t}\}}
+                         \int_{h_0}^t\frac{dh}{h}\right]\\
+ &=c\E\big[\ell\1_{\{\ell\ge e^{2t}\}}\big]\log(t/h_0).
+\end{align*}
+Under the probability change of measure with density $\ell$,
+completion of the square makes $n$ a Gaussian of mean
+$\sqrt{8t}$ and variance $1$.  Therefore $\log\ell$ under that
+measure is Gaussian of mean $4t$ and variance $8t$.  If $\Phi$
+is the standard Gaussian distribution function, then
+\[
+ \E\big[\ell\1_{\{\ell\ge e^{2t}\}}\big]
+   =\Phi(\sqrt{t/2})\ge\tfrac12.
+\]
+This proves both inequalities in
+Equation~\eqref{eq:obs-scalar-divergence}.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+Suppose that the asserted dimension-free bound fails.
+Lemma~\ref{lem:critical-normalization} gives compact centered
+counterexamples with normalized logarithmic Sobolev constant $1$
+and linear subgaussian parameter tending to zero.
+Proposition~\ref{prop:regularized-constants} shows that their
+positive-time regularized constants converge to $1$.
+Proposition~\ref{prop:critical-sequence} supplies the stationary
+laws and near-eigenfunction fields at the fixed regularization
+time $s=1/8$, in either the strict entropy branch or the spectral
+gap branch.  Proposition~\ref{prop:prediction} gives the exact
+prediction curve in both cases.  Proposition~\ref{prop:hierarchy}
+then gives the symmetric tensor hierarchy, and
+Proposition~\ref{prop:frozen-law} constructs its matrix square
+root and frozen block description.  These are precisely the
+inputs used in the present section.
+
+For each fixed $t>0$ and $L\ge1$, combine
+Propositions~\ref{prop:information-upper}
+and~\ref{prop:information-lower}.  They give
+\begin{equation}\label{eq:obs-final-scalar-bound}
+ \int_0^\infty
+ \MI\left(n;
+ \sqrt{1+f_L(xe^{\sqrt{8t}\,n-4t})}\,g\right)
+       \,\frac{dx}{x^2}\le C_0C_1,
+\end{equation}
+where $C_0C_1$ is independent of both fixed parameters.
+Choose an arbitrarily large but finite $t\ge h_0$ and the
+fixed cap $L=t$.  Lemma~\ref{lem:scalar-divergence} makes the
+left side at least $(c/2)\log(t/h_0)$, which exceeds $C_0C_1$
+for a sufficiently large choice of $t$.  This is a contradiction.
+
+All counterexample limits used to obtain
+Equation~\eqref{eq:obs-final-scalar-bound} were taken with this
+$t,L$ fixed.  Only afterward was their numerical size chosen.
+Thus no uniform estimate on the intermediate errors as
+$t,L\to\infty$ is needed.  The contradiction excludes an
+unbounded ratio $R(\mu)/a^2$ over the measures in the theorem.
+Finally the convention
+$\Ent_\mu(f^2)\le2R(\mu)\int|Df|^2\,d\mu$
+absorbs the factor $2$ into a universal constant and gives the
+asserted inequality for every smooth compactly supported real
+test function.
+\end{proof}
diff --git a/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf b/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..e70f1cafa6dc620ec257fa5cc9eacc7b1f06bd73
--- /dev/null
+++ b/preprints/A-dimension-free-logarithmic-Sobolev-inequality-for-subgaussian-log-concave-measures-September-23-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:767a5eb7cd819af7e33c8b867c41403aac36ec015c89c1d2beb186abca9ec89e
+size 712415
diff --git a/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/README.md b/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..7da62a0c8387d58169346c3a35a4aaa1ef8ce9ef
--- /dev/null
+++ b/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/README.md
@@ -0,0 +1,18 @@
+# [A direct proof of the complete Crouzeix inequality](paper.pdf)
+
+**Author:** OpenAI
+
+**Date:** September 26, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026,
+  author = {{OpenAI}},
+  title = {{A direct proof of the complete Crouzeix inequality}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/paper.pdf}{OAI:A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/build/figures/normal.tex b/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/build/figures/normal.tex
new file mode 100644
index 0000000000000000000000000000000000000000..c7e1553cc339dcb871a1b6a6974f7c1cbab7e564
--- /dev/null
+++ b/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/build/figures/normal.tex
@@ -0,0 +1,21 @@
+\begin{figure}[ht]
+\centering
+\begin{tikzpicture}[scale=.83,>=Latex]
+ \draw[thick] (0,0) circle (1.05);
+ \node at (0,0) {$\D$};
+ \fill (1.05,0) circle (2pt);
+ \node[below right] at (1.05,0) {$\lambda$};
+ \draw[->] (1.05,0) -- (1.8,0);
+ \draw[->] (2.1,.5) to[bend left=15] node[above] {$h$} (4,.5);
+ \draw[thick] (5.35,0) ellipse (1.1 and .85);
+ \node at (5.25,-.15) {$\Omega$};
+ \draw[dashed] (6.45,-1.3) -- (6.45,1.3);
+ \fill (6.45,0) circle (2pt);
+ \draw[->,thick] (6.45,0) -- (7.6,0) node[right] {$q(\lambda)$};
+ \node[above right] at (6.45,.08) {$h(\lambda)$};
+ \fill (4.95,.35) circle (1.5pt);
+ \node[above] at (4.95,.35) {$z$};
+\end{tikzpicture}
+\caption{The map $h$ sends the exterior of the circle to the exterior of the domain; the supporting line is shown schematically. Increasing the circle radius gives the outward normal $q(\lambda)$; convexity places every $z\in\overline\Omega$ on the inner side of the line.}
+\label{fig:normal}
+\end{figure}
diff --git a/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/build/main.tex b/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/build/main.tex
new file mode 100644
index 0000000000000000000000000000000000000000..5976502d77febf477933f36f6205c6e1aa6370ae
--- /dev/null
+++ b/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/build/main.tex
@@ -0,0 +1,623 @@
+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage{amsmath,amssymb,amsthm,mathtools}
+\usepackage{lmodern}
+\usepackage[margin=1.05in]{geometry}
+\usepackage{microtype,booktabs,tikz}
+\usetikzlibrary{arrows.meta}
+\usepackage[colorlinks=true,allcolors=blue]{hyperref}
+\newcommand{\C}{\mathbb C}
+\newcommand{\T}{\mathbb T}
+\newcommand{\D}{\mathbb D}
+\newcommand{\HS}[1]{\lvert #1\rvert_2}
+\DeclareMathOperator{\tr}{tr}
+\DeclareMathOperator{\Rea}{Re}
+\DeclareMathOperator{\diag}{diag}
+\DeclareMathOperator{\dist}{dist}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\theoremstyle{remark}\newtheorem{remark}[theorem]{Remark}
+\numberwithin{equation}{section}
+\setlength{\emergencystretch}{2em}
+\hypersetup{pdftitle={A direct proof of the complete Crouzeix inequality},pdfauthor={OpenAI}}
+\pdftrailerid{}
+\title{A direct proof of the complete Crouzeix inequality}
+\author{OpenAI}
+\date{September 26, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We give a direct proof of the sharp constant-two numerical-range inequality
+for matrix-valued polynomials in all finite base and coefficient dimensions.
+This resolves the complete Crouzeix conjecture in its matrix formulation,
+including matrices whose numerical ranges are points or line segments.
+\end{abstract}
+\section{Introduction}\label{sec:intro}
+The numerical range of a matrix $A\in M_n(\C)$ is
+\[
+ W(A)=\{u^*Au:u\in\C^n,\ u^*u=1\}.
+\]
+It is a compact convex set containing the spectrum. For a normal matrix,
+the spectral theorem controls polynomial evaluation by values on the
+spectrum. For a nonnormal matrix, the spectrum alone cannot provide the
+same control: even a nonzero nilpotent matrix has spectrum $\{0\}$.
+The numerical range retains enough information to give a universal bound.
+
+We allow matrix coefficients. If
+$P(z)=\sum_{k=0}^d B_kz^k$ with $B_k\in M_m(\C)$, set
+\[
+ P[A]=\sum_{k=0}^d A^k\otimes B_k.
+\]
+The base space $\C^n$ is the first tensor factor. All operator norms
+come from the Euclidean inner products and their Hilbert tensor products.
+The complete Crouzeix conjecture asks whether the constant two holds
+uniformly in the coefficient dimension $m$, the base dimension, and the degree.
+
+\begin{theorem}\label{thm:main}
+For every pair of positive integers $n,m$, every $A\in M_n(\C)$,
+every nonnegative integer $d$, and every polynomial
+$P(z)=\sum_{k=0}^d B_kz^k$ with $B_k\in M_m(\C)$,
+\begin{equation}\label{eq:main}
+ \|P[A]\|\le2\max_{z\in W(A)}\|P(z)\|.
+\end{equation}
+The constant two cannot be decreased uniformly over $n,m$, and $d$.
+\end{theorem}
+
+The estimate includes point and segment numerical ranges. Its complete
+character is the uniformity in $m$: a scalar proof need not survive
+matrix amplification, because coefficient matrices need not commute.
+The argument below keeps their multiplication order, using the two
+products $FG$ and $GF$ separately.
+
+\subsection*{Earlier bounds and related arguments}
+Crouzeix formulated the scalar constant-two question in 2004 and proved a
+universal complete bound of $11.08$ in 2007
+\cite{Crouzeix2004,Crouzeix2007}.
+Positive boundary representations developed by Delyon and Delyon
+\cite{DelyonDelyon1999} supply an important part of the numerical-range
+method. Crouzeix and Palencia combined a positive double-layer potential
+with the Cauchy transform of conjugate boundary data to obtain the
+complete bound $1+\sqrt2$ \cite{CrouzeixPalencia2017}; Ransford and
+Schwenninger simplified its final norm argument
+\cite{RansfordSchwenninger2018}.
+For all base matrices of order two, Badea, Crouzeix, and Delyon obtained
+the complete constant two. In the nonnormal case, they make the conformal
+image of the matrix a contraction by a similarity with condition number at
+most two, using the numerical-range ellipse \cite{BadeaCrouzeixDelyon2006}.
+
+Recent scalar proofs include work of Jin \cite{Jin2026},
+Lorist and Schwenninger \cite{LoristSchwenninger2026}, and Luo
+\cite{Luo2026}.
+Lorist and Schwenninger's Theorem~3 applies to scalar rational functions
+of bounded Hilbert-space operators; their Section~3(iv) explains why the
+commutation used in that proof does not automatically persist under
+amplification. \AA hag, Czy{\.{z}}, and Virtanen prove the complete
+inequality for base matrices of order at most three, with arbitrary
+coefficient dimension \cite[Theorem~1.1]{AhagCzyzVirtanen2026}.
+The scalar results fix $m=1$, while the latter complete result restricts
+the base dimension; Theorem~\ref{thm:main} leaves these dimensions independent.
+
+The use of Faber polynomials in numerical-range functional calculus has
+precedents in \cite{Beckermann2005,BeckermannCrouzeix2014}.
+Lorist and Schwenninger use a norm-attaining vector in their scalar
+argument \cite[Lemma~1]{LoristSchwenninger2026}, while Luo chooses an
+explicit top singular pair \cite[Section~3, equations~(21)--(22)]{Luo2026}.
+In their complete constant-two proof for weighted shift matrices, Crouzeix
+and Greenbaum use a scalar extremal function and a norm-attaining vector
+after a similarity reduction
+\cite[Section~3, proof of Theorem~2]{CrouzeixGreenbaum2025}.
+Here a Faber coefficient estimate controls the two diagonal functionals obtained by placing a test
+polynomial on each side of the polynomial being estimated.
+
+A separate structural proof of the complete inequality is given in
+\cite{OpenAI2026Complete}. For a bounded convex domain $\Omega$ with
+regular real-analytic boundary and $W(A)\subset\Omega$, it constructs a
+positive definite $S$ with $\|S\|\|S^{-1}\|\le2$ such that
+$S f(A)S^{-1}$ is a contraction, where $f:\Omega\to\D$ is an interior
+conformal map and $\D=\{z\in\C:|z|<1\}$. The same $S$, obtained from a
+minimizing metric, gives one continuous normalized positive boundary density representing
+the calculus of $SAS^{-1}$ for every coefficient dimension and every
+matrix function holomorphic near $\overline\Omega$
+\cite[Theorems~1.2--1.3 and Proposition~2.1]{OpenAI2026Complete}.
+The direct proof below estimates one polynomial at a time. Its exterior
+kernel, marginal identities, and positive block comparison have counterparts
+in \cite[Propositions~3.1 and~3.3 and Lemma~5.1]{OpenAI2026Complete};
+Sections~\ref{sec:coefficients}--\ref{sec:matrix} prove the specific
+coefficient, resolvent, and block estimates used here.
+
+\subsection*{How the coefficients give the sharp bound}
+First place $W(A)$ strictly inside an analytic convex domain $\Omega$.
+An exterior conformal map $h$ parametrizes its boundary by the unit circle.
+Expanding the Cauchy kernel at infinity defines scalar polynomials $b_k$ of degree
+$k$. For $k\ge1$, the boundary value $b_k\circ h$ has the positive Fourier
+term $\lambda^k$ and an otherwise strictly negative expansion. In
+Section~\ref{sec:coefficients}, a nonnegative kernel with integral one in
+each variable makes the map from the positive part to the negative part an
+$L^2$ contraction. Thus, for $G=\sum_k b_kC_k$, its squared boundary
+Hilbert--Schmidt norm lies between the square sum of its coefficient norms
+and that sum with weight two on every nonconstant coefficient.
+
+For a polynomial $F$ normalized by its boundary operator norm, choose a
+unit top singular pair $x,y$ for $F[A]$. The resolvent
+$R(\lambda)=\lambda h'(\lambda)(h(\lambda)I-A)^{-1}$ supplies the
+coefficient arrays representing the three functionals
+$y^*G[A]x$, $x^*G[A]x$, and $y^*G[A]y$. The first is bounded by a
+coefficient Hilbert norm. Testing it on $FG$ and $GF$ separately gives
+bounds for the other two. Their squared dual coefficient norms have reciprocal
+weights: one on the constant coefficient and one half on the others.
+These are exactly the weights in the squared Fourier norms of the corresponding
+Hermitian diagonal compressions, after a common factor of four.
+
+Section~\ref{sec:matrix} completes the comparison. The numerical-range
+condition makes $R+R^*$ positive, so its compression to a two-by-two
+block bounds the off-diagonal boundary norm by the two diagonal norms.
+The cross term retains all coefficients of the first functional, including
+its constant coefficient. The resulting inequality gives the sharp bound
+two. Section~\ref{sec:limit} then shrinks analytic convex neighborhoods
+to an arbitrary numerical range and proves sharpness with a two-by-two
+nilpotent matrix. Appendix~\ref{sec:collar} supplies the exterior collar
+using classical conformal mapping and reflection theorems.
+\section{Exterior coordinates and coefficient estimates}\label{sec:coefficients}
+In this section $\Omega$ is a bounded open convex set with regular real-analytic Jordan boundary. We work with a supplied exterior conformal map $h$ that is one-to-one on $|\lambda|>r$, including infinity on the Riemann sphere, for some $0<r<1$. It maps $|\lambda|>1$ onto $\C\setminus\overline\Omega$ and has a simple pole at infinity, with expansion
+\begin{equation}\label{eq:h}
+ h(\lambda)=c\lambda+c_0+O(\lambda^{-1}),\qquad c\ne0.
+\end{equation}
+Appendix~\ref{sec:collar} supplies such a coordinate for every domain under consideration. Here we use it to compare polynomial coefficients with boundary norms.
+
+The map $h$ sends $\T=\{\lambda:|\lambda|=1\}$ onto $\partial\Omega$. Put $q(\lambda)=\lambda h'(\lambda)$. On $\T$, the vector $q(\lambda)$ is an outward normal and $iq(\lambda)$ is a positively oriented tangent. Hence
+\begin{equation}\label{eq:support}
+ \Re\bigl(\overline{q(\lambda)}(h(\lambda)-z)\bigr)\ge0
+ \quad(\lambda\in\T,\ z\in\overline\Omega).
+\end{equation}
+Figure~\ref{fig:normal} illustrates this supporting-line relation.
+
+\input{figures/normal.tex}
+
+Every circle integral below uses $d\tau(e^{it})=dt/(2\pi)$.
+
+We use the unnormalized Hilbert--Schmidt norm $\HS C=(\tr(C^*C))^{1/2}$, which makes each finite-dimensional matrix space a Hilbert space.
+
+\begin{theorem}[Exterior Faber coefficients and the positive kernel]\label{lem:faber}
+Let $\Omega$ be a bounded convex domain with regular real-analytic Jordan boundary. Let $h$ be meromorphic and one-to-one on $|\lambda|>r$ including infinity, where $0<r<1$, with a simple pole at infinity as in \eqref{eq:h}, and mapping $|\lambda|>1$ onto $\C\setminus\overline\Omega$. Put $q(\lambda)=\lambda h'(\lambda)$ and define $b_k(z)$ by expansion at infinity:
+\begin{equation}\label{eq:faber}
+ \frac{q(\lambda)}{h(\lambda)-z}=\sum_{k\ge0}b_k(z)\lambda^{-k}.
+\end{equation}
+Then $b_0=1$, and each $b_k$ is a polynomial of degree $k$ with leading coefficient $c^{-k}$. There are scalars $s_{kj}$, $k,j\ge1$, such that
+\begin{equation}\label{eq:boundary-basis}
+ b_k(h(\mu))=\mu^k+\sum_{j\ge1}s_{kj}\mu^{-j}\quad(k\ge1).
+\end{equation}
+The function
+\begin{equation}\label{eq:s}
+ s(\lambda,\mu)=\frac{q(\lambda)}{h(\lambda)-h(\mu)}-\frac{\lambda}{\lambda-\mu}
+ =\sum_{k,j\ge1}s_{kj}\lambda^{-k}\mu^{-j}
+\end{equation}
+extends holomorphically across the diagonal and both infinities in $|\lambda|,|\mu|>r$. For every $r<\rho<1$ there is a constant $C_\rho$ with
+$|s_{kj}|\le C_\rho\rho^{k+j}$. The series converges normally on
+the exterior product collar: it converges absolutely and uniformly on
+every compact subset, including subsets meeting either infinity.
+In particular it converges absolutely and uniformly on $\T^2$. The continuous kernel
+\[
+ K(\lambda,\mu)=1+s(\lambda,\mu)+\overline{s(\lambda,\mu)}
+\]
+is nonnegative and has integral one in each variable with the other fixed.
+\end{theorem}
+
+\begin{proof}
+Expanding the denominator in \eqref{eq:faber} after factoring $c\lambda$ shows that its $k$th coefficient is a polynomial of degree $k$ with leading term $c^{-k}z^k$. The constant coefficient is one. In particular, the $b_k$ form a basis of the scalar polynomials.
+
+For the joint extension, consider
+\[
+ H(\lambda,\mu)=\frac{h(\lambda)-h(\mu)}{\lambda-\mu}.
+\]
+Write $u=\lambda^{-1}$, $v=\mu^{-1}$ and $h(\lambda)=c\lambda+g(u)$, where $g$ is holomorphic on $|u|<r^{-1}$. Then
+\begin{equation}\label{eq:H}
+ H=c-uv\frac{g(u)-g(v)}{u-v}.
+\end{equation}
+The divided difference of $g$ is jointly holomorphic, with diagonal value $g'(u)$, so this formula also covers either or both infinities. Off the diagonal and away from the axes, $H$ is nonzero by injectivity of $h$. On the diagonal it equals $h'(\lambda)\ne0$; on either axis it equals $c$. Thus it has no zeros on the reciprocal bidisk.
+
+Direct differentiation gives $s=\lambda\partial_\lambda H/H=-u\partial_uH/H$. This function is holomorphic and vanishes on both reciprocal axes. Its Taylor series has precisely the strictly mixed powers in \eqref{eq:s}. For any $r<\rho<1$, Cauchy estimates on the closed reciprocal bidisk
+of radius $\rho^{-1}$ give a constant $C_\rho$ with
+$|s_{kj}|\le C_\rho\rho^{k+j}$. On a compact subset of the exterior
+product collar, choose $\sigma>r$ below all finite coordinate moduli,
+and then $r<\rho<\min(1,\sigma)$. The resulting geometric double
+series dominates the Laurent series uniformly there; terms vanish on
+the reciprocal axes. This proves normal convergence throughout the
+product collar and absolute uniform convergence on the boundary. Comparing coefficients at $\lambda=\infty$ in \eqref{eq:s}, first for fixed finite $\mu$ in the exterior collar, yields \eqref{eq:boundary-basis}.
+
+For distinct $\lambda,\mu\in\T$, one has $2\Re(\lambda/(\lambda-\mu))=1$, and therefore
+\[
+ K(\lambda,\mu)=2\Re\frac{q(\lambda)}{h(\lambda)-h(\mu)}\ge0
+\]
+by \eqref{eq:support}. Continuity gives the same sign on the diagonal. Every term of $s$ and of its conjugate has nonzero frequency in each variable. Their integrals in either variable vanish, proving both normalizations.
+\end{proof}
+
+To identify the basis, let $\Phi=h^{-1}$ in the exterior. Equation~\eqref{eq:boundary-basis}, which holds throughout the exterior collar, gives $b_k(z)-\Phi(z)^k=O(z^{-1})$ at infinity. Thus $b_k$ is the polynomial part of $\Phi^k$, the Faber polynomial associated with the chosen exterior coordinate. The usual normalization chooses the phase of $h$ so that $c>0$ \cite{Beckermann2005,BeckermannCrouzeix2014}.
+
+These mixed coefficients are differentiated Grunsky coefficients.
+The classical normalization gives a weighted coefficient inequality
+\cite[equations~(1.1)--(1.3)]{BeckermannStylianopoulos2018};
+the unweighted estimate used here follows from convexity and the two
+angular marginal identities.
+
+The two normalizations in Theorem~\ref{lem:faber} use the same exterior circle measure. Together they control the negative Fourier part of a boundary polynomial by its positive part, as follows.
+
+\begin{proposition}[The coefficient norm estimate]\label{prop:coefficients}
+For every positive integer $m$ and every $M_m(\C)$-valued polynomial $G(z)=\sum_{k=0}^N b_k(z)C_k$,
+\begin{equation}\label{eq:coefficient-estimate}
+ \sum_{k=0}^N\HS{C_k}^2
+ \le\int_\T\HS{G(h(\mu))}^2\,d\tau(\mu)
+ \le\HS{C_0}^2+2\sum_{k=1}^N\HS{C_k}^2.
+\end{equation}
+\end{proposition}
+\begin{proof}
+For $u\in L^2(\T;\mathcal H)$ with values in a Hilbert space $\mathcal H$, define
+\[
+ (Tu)(\mu)=\int_\T K(\lambda,\mu)u(\lambda)\,d\tau(\lambda).
+\]
+Positivity and the normalization in $\lambda$ imply
+$\|(Tu)(\mu)\|^2\le\int K(\lambda,\mu)\|u(\lambda)\|^2d\tau(\lambda)$.
+Integrating in $\mu$ and using the normalization in that variable shows that $T$ is an $L^2$ contraction.
+
+For $u(\lambda)=\sum_{k=1}^N C_k\lambda^k$, only the $s$ term contributes to the integral, so
+\begin{equation}\label{eq:kernel-fourier-action}
+ (Tu)(\mu)=\sum_{j\ge1}\left(\sum_{k=1}^Ns_{kj}C_k\right)\mu^{-j}.
+\end{equation}
+All interchanges are justified by the absolute uniform convergence in Theorem~\ref{lem:faber}. In particular,
+\[
+ \sum_{j\ge1}\HS{\sum_{k=1}^Ns_{kj}C_k}^2\le\sum_{k=1}^N\HS{C_k}^2.
+\]
+By \eqref{eq:boundary-basis}, the boundary value of $G$ is the sum of its constant term $C_0$, its positive part $u$, and its negative part $Tu$. These have disjoint Fourier supports. Parseval's identity gives
+\[
+ \int_\T\HS{G\circ h}^2d\tau
+ =\HS{C_0}^2+\sum_{k=1}^N\HS{C_k}^2+
+   \sum_{j\ge1}\HS{\sum_{k=1}^Ns_{kj}C_k}^2,
+\]
+which proves both inequalities.
+\end{proof}
+For comparison with \cite[Proposition~3.1]{OpenAI2026Complete}, the
+companion's exterior map $G$ and variables $t,w$ correspond to
+$h,\lambda,\mu$, respectively.
+The mixed correction is $s(\lambda,\mu)=B(\mu,\lambda)$, so
+$s_{kj}=\beta_{jk}$ in that paper's notation. In particular,
+\eqref{eq:kernel-fourier-action} uses this coefficient orientation and
+does not require symmetry of the kernel.
+\section{Singular vectors and the amplified estimate}\label{sec:matrix}
+Fix $A\in M_n(\C)$ with $W(A)\subset\Omega$.
+To estimate the norm of one polynomial evaluation, we form coefficient
+functionals from its left and right singular vectors. The resolvent supplies
+their coefficient arrays, while its positive Hermitian part compares the
+resulting boundary norms. This Hermitian field is the numerical-range double-layer kernel of
+\cite{DelyonDelyon1999,CrouzeixPalencia2017}; we prove the required
+positivity for the original matrix $A$.
+
+\begin{theorem}[The resolvent expansion and double-layer positivity]\label{lem:resolvent}
+Let $\Omega,h,q$, and $(b_k)$ satisfy the hypotheses and conclusions of Theorem~\ref{lem:faber}. For any positive integer $n$ and $A\in M_n(\C)$ with $W(A)\subset\Omega$, on $\T$ let
+\begin{equation}\label{eq:R}
+ R(\lambda)=q(\lambda)(h(\lambda)I-A)^{-1}.
+\end{equation}
+Then
+\[
+ R(\lambda)=\sum_{k\ge0}b_k(A)\lambda^{-k},\qquad R(\lambda)+R(\lambda)^*\succeq0.
+\]
+For the fixed $A,h$, there are constants $C<\infty$ and $0<\theta<1$
+such that $\|b_k(A)\|\le C\theta^k$ for all $k\ge0$. Thus the series
+converges absolutely and uniformly on $\T$. In particular,
+$\int_\T R\,d\tau=I$ and $\int_\T(R+R^*)\,d\tau=2I$.
+\end{theorem}
+\begin{proof}
+Every eigenvalue of $A$ belongs to $W(A)$, by testing a unit eigenvector. Thus the exterior resolvent is defined on and outside the boundary. Compactness of the circle and strict spectral separation allow a smaller collar $|\lambda|>r_A$, with $r<r_A<1$, on which it remains holomorphic, including infinity where $R(\infty)=I$. Expansion at infinity gives the coefficients $b_k(A)$: this follows either by the geometric resolvent series for large $|h(\lambda)|$ or by substituting $A$ into the polynomial coefficient identities in \eqref{eq:faber}. Choose $r_A<\theta<1$. Cauchy estimates on the reciprocal circle of
+radius $\theta^{-1}$ give the stated geometric bound. They imply
+absolute uniform convergence on $\T$, and its constant coefficient
+$b_0(A)=I$ gives both integrals.
+
+For $B=h(\lambda)I-A$, direct congruence gives
+\[
+ B^*(R+R^*)B=\overline{q(\lambda)}B+q(\lambda)B^*.
+\]
+For each unit vector $v$, the quadratic form on the right is
+$2\Re(\overline{q(\lambda)}(h(\lambda)-v^*Av))$, which is nonnegative by \eqref{eq:support}. Invertibility of $B$ proves positivity.
+\end{proof}
+
+\begin{theorem}[The bound on an analytic convex domain]\label{thm:domain}
+For positive integers $n,m$, let $\Omega\subset\C$ be bounded, open, and convex with regular real-analytic Jordan boundary, and let $A\in M_n(\C)$ satisfy $W(A)\subset\Omega$. For every $M_m(\C)$-valued polynomial $F$ with $\|F(z)\|\le1$ on $\partial\Omega$, one has $\|F[A]\|\le2$.
+\end{theorem}
+\begin{proof}
+Choose $h$ from Lemma~\ref{lem:exterior-collar} and apply
+Theorem~\ref{lem:faber} to this coordinate.
+Put $\gamma=\|F[A]\|$. There is nothing to prove if $\gamma=0$. Otherwise choose unit singular vectors $x,y\in\C^n\otimes\C^m$ such that
+\begin{equation}\label{eq:singular}
+ F[A]x=\gamma y,\qquad F[A]^*y=\gamma x.
+\end{equation}
+Write $x=\sum_{i=1}^mXe_i\otimes e_i$ and $y=\sum_{i=1}^mYe_i\otimes e_i$, where $X,Y$ are $n$-by-$m$ matrices and $(e_i)$ is the standard basis of $\C^m$. For the resolvent in Theorem~\ref{lem:resolvent}, define matrix functions
+\[
+ P=X^*RX,\quad Q=Y^*RY,\quad M=Y^*RX,\quad L=X^*RY.
+\]
+Subscripts $k\ge0$ denote the coefficients of $\lambda^{-k}$, so $M_k=Y^*b_k(A)X$, for example. By Theorem~\ref{lem:resolvent}, all four series converge absolutely and uniformly on $\T$, and their coefficient sequences are square summable. Since $b_0=1$,
+\begin{equation}\label{eq:means}
+ P_0=X^*X=P_0^*,\qquad Q_0=Y^*Y=Q_0^*,\qquad L_0^*=M_0.
+\end{equation}
+Set $a^2=\sum_{k\ge0}\HS{M_k}^2$. We will bound the squared
+boundary $L^2$ Hilbert--Schmidt norms of $P+P^*$ and $Q+Q^*$
+above by $4a^2/\gamma^2$ each, and that of $M+L^*$ below by $a^2$.
+Positivity will then compare these quantities and force $\gamma\le2$.
+
+\emph{The off-diagonal coefficient functional.}
+For a polynomial $G=\sum_k b_kC_k$, direct expansion in the tensor basis gives
+\begin{equation}\label{eq:pairing}
+ y^*G[A]x=\sum_{k,i,j}(M_k)_{ij}(C_k)_{ij}.
+\end{equation}
+Indeed, $C_ke_j=\sum_i(C_k)_{ij}e_i$ in the coefficient factor, leaving $(Ye_i)^*b_k(A)Xe_j$ in the base factor. Equation~\eqref{eq:pairing} is the bilinear entry pairing of the two coefficient arrays. Cauchy--Schwarz gives
+\begin{equation}\label{eq:crossfunctional}
+ |y^*G[A]x|\le a\left(\sum_k\HS{C_k}^2\right)^{1/2}.
+\end{equation}
+Taking $G=F$ and using \eqref{eq:singular} shows that $a>0$.
+
+\emph{The two ordered tests.}
+Polynomial evaluation respects multiplication in its given order. Explicitly, if $F(z)=\sum_iD_iz^i$ and $G(z)=\sum_jE_jz^j$, then
+\[
+ (FG)[A]=\sum_{i,j}A^{i+j}\otimes D_iE_j=F[A]G[A],
+\]
+and the reversed product has coefficients $E_jD_i$. Hence
+\begin{equation}\label{eq:ordered-products}
+ y^*(FG)[A]x=\gamma x^*G[A]x,\qquad
+ y^*(GF)[A]x=\gamma y^*G[A]y.
+\end{equation}
+Write the full finite Faber expansions as
+\[
+ FG=\sum_k b_kU_k,\qquad GF=\sum_k b_kV_k,
+ \qquad \|H\|_\partial^2=\int_\T\HS{H(h(\mu))}^2\,d\tau(\mu).
+\]
+The functional in \eqref{eq:crossfunctional} uses all $M_k$, so it applies
+to these full expansions for every finite degree. Apply it to the two
+identities in \eqref{eq:ordered-products}. The lower coefficient estimate
+in Proposition~\ref{prop:coefficients} and the two boundary multiplier
+bounds give
+\begin{align*}
+ \gamma|x^*G[A]x|
+ &\le a\Bigl(\sum_k\HS{U_k}^2\Bigr)^{1/2}
+ \le a\|FG\|_\partial\le a\|G\|_\partial,\\
+ \gamma|y^*G[A]y|
+ &\le a\Bigl(\sum_k\HS{V_k}^2\Bigr)^{1/2}
+ \le a\|GF\|_\partial\le a\|G\|_\partial.
+\end{align*}
+Here $\HS{F(z)G(z)}\le\|F(z)\|\HS{G(z)}$ and
+$\HS{G(z)F(z)}\le\HS{G(z)}\|F(z)\|$ are used in their
+respective orders. The upper coefficient estimate for $G$ now gives
+\begin{equation}\label{eq:diagonal-functional}
+ \max\{|x^*G[A]x|,|y^*G[A]y|\}
+ \le\frac a\gamma
+ \left(\HS{C_0}^2+2\sum_{k\ge1}\HS{C_k}^2\right)^{1/2}.
+\end{equation}
+The entries paired with $C_k$ in these two functionals are respectively those of $P_k$ and $Q_k$, by the calculation in \eqref{eq:pairing}. The weighted coefficient norm on the right gives reciprocal weights on these two coefficient sequences:
+\begin{equation}\label{eq:weighted-dual}
+ \HS{P_0}^2+\tfrac12\sum_{k\ge1}\HS{P_k}^2\le a^2/\gamma^2,
+ \qquad
+ \HS{Q_0}^2+\tfrac12\sum_{k\ge1}\HS{Q_k}^2\le a^2/\gamma^2.
+\end{equation}
+For the first inequality, set
+$d_N=\HS{P_0}^2+\tfrac12\sum_{k=1}^N\HS{P_k}^2$
+and test \eqref{eq:diagonal-functional} with $C_0=\overline{P_0}$,
+$C_k=\overline{P_k}/2$ for $1\le k\le N$, and $C_k=0$ otherwise;
+bars mean entrywise conjugation. The paired value is $d_N$, and the squared
+weighted test norm is also $d_N$. Hence
+$d_N\le(a/\gamma)\sqrt{d_N}$, which gives $d_N\le a^2/\gamma^2$
+including when $d_N=0$. Letting $N\to\infty$ proves the first bound.
+The same finite tests with $Q_k$ prove the second.
+
+\emph{The boundary norms of the three blocks.}
+The compression of $R+R^*$ by $[X\ Y]$ has diagonal blocks $P+P^*$,
+$Q+Q^*$ and lower off-diagonal block $M+L^*$. We first compute their
+boundary norms. In the diagonal fields, \eqref{eq:means} makes the
+constant term twice $P_0$ or $Q_0$. For $k\ge1$, the coefficients $P_k$
+and $P_k^*$ occur at frequencies $-k$ and $k$ and have equal
+Hilbert--Schmidt norms; the same holds for $Q_k$ and $Q_k^*$.
+Parseval therefore gives four times the reciprocal weighted sum in
+\eqref{eq:weighted-dual}:
+\begin{equation}\label{eq:diagonal-L2}
+ \int_\T\HS{P+P^*}^2d\tau
+ =4\left(\HS{P_0}^2+\tfrac12\sum_{k\ge1}\HS{P_k}^2\right)
+ \le4a^2/\gamma^2,
+\end{equation}
+and the same identity and bound hold for $Q+Q^*$.
+
+For the cross field, $L_0^*=M_0$ makes the constant term $2M_0$.
+Its negative coefficients are $M_k$ for $k\ge1$ and its positive
+coefficients are $L_k^*$ for $k\ge1$. Consequently
+\begin{equation}\label{eq:cross-L2}
+ \int_\T\HS{M+L^*}^2d\tau
+ =4\HS{M_0}^2+\sum_{k\ge1}\bigl(\HS{M_k}^2+\HS{L_k}^2\bigr)
+ \ge a^2.
+\end{equation}
+
+\emph{The positive block comparison.}
+Theorem~\ref{lem:resolvent} gives $R+R^*\succeq0$. Its compression by the
+$n$-by-$2m$ matrix $[X\ Y]$ is therefore
+\[
+ \Delta=\begin{pmatrix}P+P^*&L+M^*\\M+L^*&Q+Q^*\end{pmatrix}\succeq0.
+\]
+For $J=\diag(I_m,-I_m)$, the matrix $J\Delta J$ is also positive, so
+\[
+ 0\le\tr\bigl(\Delta^{1/2}(J\Delta J)\Delta^{1/2}\bigr)
+ =\tr(\Delta J\Delta J)
+ =\HS{P+P^*}^2+\HS{Q+Q^*}^2-2\HS{M+L^*}^2.
+\]
+Integrating this inequality and using \eqref{eq:diagonal-L2} and \eqref{eq:cross-L2} yields
+\[
+ 2a^2\le2\int_\T\HS{M+L^*}^2d\tau
+ \le\int_\T\bigl(\HS{P+P^*}^2+\HS{Q+Q^*}^2\bigr)d\tau
+ \le8a^2/\gamma^2.
+\]
+Since $a>0$, this proves $\gamma\le2$.
+\end{proof}
+\section{Enclosing domains and arbitrary numerical ranges}\label{sec:limit}
+Theorem~\ref{thm:domain} applies on an analytic convex neighborhood. We now construct such neighborhoods even for compact convex sets with empty interior, and then shrink them to $W(A)$.
+
+\begin{lemma}[Analytic convex outer approximation]\label{lem:approx}
+Let $K\subset\C$ be nonempty, compact, and convex. There are bounded convex open domains $\Omega_j$ with regular real-analytic Jordan boundaries such that $K\subset\Omega_j$ and
+\[
+ \sup_{z\in\overline\Omega_j}\dist(z,K)\longrightarrow0.
+\]
+\end{lemma}
+\begin{proof}
+For $0<\delta<1$, choose a finite Euclidean $\delta$-net $D_\delta$ of unit directions that includes $1,-1,i,-i$. Let $s_K(\nu)=\max_{z\in K}\Re(\bar\nu z)$ be the support function. Choose $t_\delta>0$ with $|D_\delta|e^{-t_\delta\delta}<1$, and define
+\[
+ \Phi_\delta(z)=\sum_{\nu\in D_\delta}
+ \exp\bigl(t_\delta(\Re(\bar\nu z)-s_K(\nu)-\delta)\bigr),
+ \qquad \Omega_\delta=\{z:\Phi_\delta(z)<1\}.
+\]
+Each point of $K$ satisfies $\Phi_\delta<1$, so $K\subset\Omega_\delta$.
+If $\Phi_\delta(z)\le1$, each positive summand is at most one, and hence
+\[
+ \Re(\bar\nu z)\le s_K(\nu)+\delta\qquad(\nu\in D_\delta).
+\]
+In particular, the four axis directions give
+\[
+ \begin{aligned}
+ \min_{w\in K}\Re w-\delta&\le\Re z\le\max_{w\in K}\Re w+\delta,\\
+ \min_{w\in K}\Im w-\delta&\le\Im z\le\max_{w\in K}\Im w+\delta.
+ \end{aligned}
+\]
+Thus one fixed rectangle contains every closed sublevel for $0<\delta<1$.
+
+The Hessian is a sum of positive multiples of $\nu\nu^{\mathsf T}$ after identifying $\C$ with $\mathbb R^2$. The four axis directions ensure it is positive definite. Thus $\Phi_\delta$ is strictly convex and real analytic.
+
+The gradient cannot vanish at level one: a critical point of a differentiable convex function is a global minimum, whereas every point of $K$ has value less than one. The implicit function theorem therefore makes that level a regular real-analytic curve. The open sublevel set is bounded and convex with nonempty interior. Fix an interior point. Each ray from it crosses level one exactly once; convexity and boundedness give existence and uniqueness. Radial projection from the compact boundary to the circle is a continuous bijection, hence a homeomorphism. The boundary is a Jordan curve, and $\overline\Omega_\delta=\{\Phi_\delta\le1\}$.
+
+It remains to bound the distance of this sublevel from $K$. For
+$z\in\overline\Omega_\delta\setminus K$, let $w\in K$ be nearest to $z$,
+put $d=|z-w|$ and $\nu=(z-w)/d$, and choose $\mu\in D_\delta$ with
+$|\mu-\nu|\le\delta$. Choose $v\in K$ attaining $s_K(\mu)$.
+Since $w+t(v-w)\in K$ for $0\le t\le1$, minimality of $w$ gives
+$\Re(\bar\nu(v-w))\le0$.
+Using this inequality and the support inequality for $\mu$, we obtain
+\[
+ \begin{aligned}
+ (1-\delta)d
+ &\le\Re(\bar\mu(z-w))\\
+ &\le\Re(\bar\mu(v-w))+\delta\\
+ &\le\delta|v-w|+\delta
+ \le\delta\bigl(1+\operatorname{diam}K\bigr).
+ \end{aligned}
+\]
+The first and third inequalities use $|\mu-\nu|\le\delta$. Points of $K$
+have distance zero, so
+\[
+ \sup_{z\in\overline\Omega_\delta}\dist(z,K)
+ \le\frac{\delta(1+\operatorname{diam}K)}{1-\delta}\longrightarrow0.
+\]
+Taking, for example, $\delta_j=1/(j+1)$ completes the construction.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+Compactness of $W(A)$ follows from compactness of the unit sphere. We recall a short proof of the Toeplitz--Hausdorff convexity theorem \cite{Toeplitz1918,Hausdorff1919}. To join two numerical-range values, compress $A$ to the span of their unit testing vectors. If that span has dimension one there is nothing to prove. In dimension two, its quadratic form on unit vectors $(u,v)$ is a complex-valued real-affine function of
+\[
+ (2\Re(\bar uv),\ 2\Im(\bar uv),\ |u|^2-|v|^2),
+\]
+which ranges over the full unit sphere in $\mathbb R^3$. The linear part of a map from $\mathbb R^3$ to $\mathbb R^2$ has a nonzero kernel. Any point of the unit ball can be moved in a kernel direction to the sphere without changing its image. Thus the images of the sphere and ball coincide, and this image is convex. Its segment between the chosen values belongs to the numerical range of the compression and hence to $W(A)$.
+
+Apply Lemma~\ref{lem:approx} with $K=W(A)$. By scaling Theorem~\ref{thm:domain},
+\[
+ \|P[A]\|\le2\max_{z\in\overline\Omega_j}\|P(z)\|.
+\]
+If this maximum is zero, each entry of $P$ vanishes on an open set and the polynomial is identically zero, making the inequality immediate. Otherwise divide $P$ by the maximum to invoke that theorem. All $\overline\Omega_j$ lie in one compact set. Uniform continuity of the fixed function $z\mapsto\|P(z)\|$, the containment of $K$, and the uniform distance convergence imply convergence of these maxima to $\max_K\|P\|$. This proves \eqref{eq:main} without any interior assumption on $K$.
+
+For sharpness take $n=2$, $m=1$, $P(z)=z$, and
+\[
+ A=\begin{pmatrix}0&2\\0&0\end{pmatrix}.
+\]
+Its norm is two, while $|x^*Ax|=2|x_1x_2|\le1$ for every unit $x$, with equality at $x_1=x_2=1/\sqrt2$. Thus the right-hand maximum in \eqref{eq:main} is one. No smaller universal constant is possible.
+\end{proof}
+\appendix
+\section{The exterior conformal collar}\label{sec:collar}
+We record the classical conformal inputs and prove the continuation and global injectivity required in Section~\ref{sec:coefficients}.
+
+\begin{lemma}[Exterior coordinate and a univalent collar]
+\label{lem:exterior-collar}
+Let $\Omega\subset\mathbb C$ be a bounded convex domain whose boundary
+is a regular real-analytic Jordan curve. There are $0<r_0<1$ and a map
+$h$, meromorphic on $\{\lambda:|\lambda|>r_0\}\cup\{\infty\}$,
+which is one-to-one there as a map of the sphere, has a simple pole at
+infinity, and maps $|\lambda|>1$ conformally onto
+$\mathbb C\setminus\overline\Omega$. It maps the unit circle onto
+$\partial\Omega$ and has an expansion
+\[
+ h(\lambda)=c\lambda+c_0+\sum_{j\ge1}c_j\lambda^{-j},\qquad c\ne0.
+\]
+For $\lambda$ on the unit circle, $q(\lambda)=\lambda h'(\lambda)$
+is a nonzero outward normal vector at $h(\lambda)$. In particular,
+\[
+ \operatorname{Re}\bigl(\overline{q(\lambda)}(h(\lambda)-z)\bigr)
+ \ge0\qquad (z\in\overline\Omega).
+\]
+\end{lemma}
+
+\begin{proof}
+Fix $z_*\in\Omega$ and apply $z\mapsto1/(z-z_*)$, with infinity
+mapped to zero, to the spherical exterior
+$(\mathbb C\setminus\overline\Omega)\cup\{\infty\}$. Its image $U$ is
+a bounded simply connected domain containing zero, with a regular
+real-analytic Jordan boundary. Indeed, each ray from $z_*$ meets
+$\partial\Omega$ at a unique positive radius; inversion turns the
+exterior portions of these rays into a star-shaped domain about zero.
+The Riemann mapping theorem \cite[Chapter~6, Section~1.1, Theorem~1]{Ahlfors1979}
+supplies a conformal bijection
+$g:\mathbb D\to U$ with $g(0)=0$. Carath\'eodory's Jordan-domain theorem
+\cite[Chapter~6, Section~1.2]{Ahlfors1979} extends $g$ to a homeomorphism $\overline{\mathbb D}\to\overline U$.
+
+We justify the stronger boundary continuation needed below. A regular
+real-analytic boundary arc has a real-analytic parametrization with
+nonzero derivative. Its holomorphic continuation and the holomorphic
+inverse-function theorem provide a local coordinate in which the arc
+is a real interval and the domain is one side of that interval. Choose
+such coordinates at $\zeta\in\partial\mathbb D$ and
+$p=g(\zeta)\in\partial U$. Continuity of the boundary extension lets us
+shrink the source neighborhood so that $g$ maps its intersection with
+$\overline{\mathbb D}$ into the target chart.
+The composition representing $g$ in these
+coordinates is holomorphic on a half disk, continuous on its real
+interval, and real-valued there. Schwarz reflection extends it
+holomorphically across the interval
+\cite[Chapter~4, Section~6.5; Chapter~6, Sections~1.3--1.4]{Ahlfors1979}. Apply the same procedure to
+$g^{-1}$. Shrink the neighborhoods so that the reflected maps can be
+composed. Their compositions equal the identity on the original domain
+side, and hence throughout a smaller neighborhood by the identity theorem.
+Thus the continued $g$ has nonzero derivative at each boundary point.
+
+To patch the local continuations, choose around every
+$\zeta\in\partial\mathbb D$ a disk $B(\zeta,R_\zeta)$ carrying such a continuation, agreeing
+with $g$ on its intersection with $\mathbb D$. Select finitely many of
+the third-radius disks $V_i=B(\zeta_i,R_i/3)$ that cover the circle,
+and denote the larger disks by $U_i=B(\zeta_i,R_i)$. If
+$V_i\cap V_j\ne\varnothing$ and $R_i\ge R_j$, then
+$|\zeta_i-\zeta_j|<(R_i+R_j)/3\le2R_i/3<R_i$.
+The connected lens $U_i\cap U_j$ therefore contains $\zeta_j$ and
+meets $\mathbb D$. The two continuations coincide there by the identity
+theorem. They consequently patch with $g$ to a holomorphic map on
+$\mathbb D\cup\bigcup_iV_i$, a neighborhood of the closed disk.
+
+The extension is injective on a smaller neighborhood of the closed
+disk. Otherwise, distinct colliding pairs arbitrarily close to that
+compact set would have subsequential limits in the closed disk.
+Continuity and injectivity on the closed disk identify their limits.
+Both points of each pair would then eventually lie in one local
+inverse neighborhood of that common limit, a contradiction. By compactness,
+this injective neighborhood contains the disk $\{w:|w|<R_0\}$ for some $R_0>1$.
+Thus $g$ is univalent on that disk. Its sole zero there
+is $0$, and that zero is simple. Set
+\[
+ h(\lambda)=z_*+\frac1{g(1/\lambda)},\qquad r_0=R_0^{-1}.
+\]
+This gives all mapping and extension assertions, including the simple
+pole with coefficient $c=1/g'(0)\ne0$.
+
+Finally, at $\lambda=e^{i\theta}$ the radial and angular derivatives
+are $q(\lambda)$ and $iq(\lambda)$, respectively. They are perpendicular
+and nonzero. Increasing the radial parameter enters the exterior of
+$\overline\Omega$, so $q(\lambda)$ points outward. The supporting
+half-plane property of a smooth convex boundary proves the final
+inequality.
+\end{proof}
+\input{references.tex}
+\end{document}
diff --git a/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/build/references.tex b/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/build/references.tex
new file mode 100644
index 0000000000000000000000000000000000000000..5afd2ba6d247c20795e4aee5e5fd6bab84cc14ba
--- /dev/null
+++ b/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/build/references.tex
@@ -0,0 +1,105 @@
+\begin{thebibliography}{99}
+\raggedright
+
+\bibitem{AhagCzyzVirtanen2026}
+P.~\AA hag, R.~Czy{\.{z}}, and J.~Virtanen,
+\emph{The complete Crouzeix conjecture in dimension three and the
+Clou\^atre--Ostermann--Ransford conjecture},
+\href{https://arxiv.org/abs/2608.27346v4}{arXiv:2608.27346v4}, September 21, 2026.
+
+\bibitem{Ahlfors1979}
+L.~V. Ahlfors, \emph{Complex Analysis}, third ed., McGraw--Hill, New York, 1979.
+
+\bibitem{BadeaCrouzeixDelyon2006}
+C.~Badea, M.~Crouzeix, and B.~Delyon,
+Convex domains and $K$-spectral sets,
+\emph{Math. Z.} \textbf{252} (2006), 345--365.
+\href{https://doi.org/10.1007/s00209-005-0857-y}{doi:10.1007/s00209-005-0857-y}.
+
+\bibitem{Beckermann2005}
+B.~Beckermann, Image num{\'e}rique, GMRES et polyn{\^o}mes de Faber,
+\emph{C. R. Math.} \textbf{340} (2005), 855--860.
+\href{https://doi.org/10.1016/j.crma.2005.04.027}{doi:10.1016/j.crma.2005.04.027}.
+
+\bibitem{BeckermannCrouzeix2014}
+B.~Beckermann and M.~Crouzeix,
+Faber polynomials of matrices for non-convex sets,
+\emph{Jaen J. Approx.} \textbf{6} (2014), 219--231.
+\href{https://arxiv.org/abs/1310.1356v1}{arXiv:1310.1356v1}.
+
+\bibitem{BeckermannStylianopoulos2018}
+B. Beckermann and N. Stylianopoulos,
+\emph{Bergman Orthogonal Polynomials and the Grunsky Matrix},
+Constr. Approx. \textbf{47} (2018), 211--235.
+\href{https://doi.org/10.1007/s00365-017-9381-7}{doi:10.1007/s00365-017-9381-7}.
+
+\bibitem{Crouzeix2004}
+M.~Crouzeix, Bounds for analytical functions of matrices,
+\emph{Integral Equations Operator Theory} \textbf{48} (2004), 461--477.
+\href{https://doi.org/10.1007/s00020-002-1188-6}{doi:10.1007/s00020-002-1188-6}.
+
+\bibitem{Crouzeix2007}
+M.~Crouzeix, Numerical range and functional calculus in Hilbert space,
+\emph{J. Funct. Anal.} \textbf{244} (2007), 668--690.
+\href{https://doi.org/10.1016/j.jfa.2006.10.013}{doi:10.1016/j.jfa.2006.10.013}.
+
+\bibitem{CrouzeixGreenbaum2025}
+M.~Crouzeix and A.~Greenbaum,
+\emph{A new proof that the numerical range is a complete $2$-spectral
+set for weighted shift matrices},
+\href{https://arxiv.org/abs/2508.12768v1}{arXiv:2508.12768v1},
+August 18, 2025.
+
+\bibitem{CrouzeixPalencia2017}
+M.~Crouzeix and C.~Palencia,
+The numerical range is a $(1+\sqrt{2})$-spectral set,
+\emph{SIAM J. Matrix Anal. Appl.} \textbf{38} (2017), 649--655.
+\href{https://doi.org/10.1137/17M1116672}{doi:10.1137/17M1116672}.
+
+\bibitem{DelyonDelyon1999}
+B.~Delyon and F.~Delyon,
+Generalization of von Neumann's spectral sets and integral representation of operators,
+\emph{Bull. Soc. Math. France} \textbf{127} (1999), 25--41.
+\href{https://doi.org/10.24033/bsmf.2340}{doi:10.24033/bsmf.2340}.
+
+\bibitem{Hausdorff1919}
+F.~Hausdorff, Der Wertvorrat einer Bilinearform,
+\emph{Math. Z.} \textbf{3} (1919), 314--316.
+\href{https://doi.org/10.1007/BF01292610}{doi:10.1007/BF01292610}.
+
+\bibitem{Jin2026}
+S.~Jin, \emph{The numerical range is a $2$-spectral set},
+\href{https://www.preprints.org/manuscript/202607.1919/v4}{Preprints.org 202607.1919, version 4},
+August 7, 2026.
+
+\bibitem{LoristSchwenninger2026}
+E.~Lorist and F.~L. Schwenninger,
+\emph{A solution to Crouzeix's conjecture},
+\href{https://arxiv.org/abs/2608.03841v2}{arXiv:2608.03841v2}, August 17, 2026.
+
+\bibitem{Luo2026}
+Q.~Luo, \emph{Two boundary-certificate proofs of the Crouzeix theorem:
+Fiberwise Fourier recurrence and defect-Gram telescoping},
+\href{https://www.preprints.org/manuscript/202608.1661/v2}{Preprints.org 202608.1661, version 2},
+August 28, 2026.
+
+\bibitem{OpenAI2026Complete}
+OpenAI, \emph{The complete Crouzeix theorem: optimal similarity and a
+common positive boundary representation},
+OpenAI Math Release preprint
+\href{https://github.com/openai/math/blob/main/preprints/The-complete-Crouzeix-theorem-September-23-2026/paper.pdf}{OAI:The-complete-Crouzeix-theorem-September-23-2026},
+2026.
+
+\bibitem{RansfordSchwenninger2018}
+T.~Ransford and F.~L. Schwenninger,
+Remarks on the Crouzeix--Palencia proof that the numerical range is a
+$(1+\sqrt{2})$-spectral set,
+\emph{SIAM J. Matrix Anal. Appl.} \textbf{39} (2018), 342--345.
+\href{https://doi.org/10.1137/17M1143757}{doi:10.1137/17M1143757}.
+
+\bibitem{Toeplitz1918}
+O.~Toeplitz, Das algebraische Analogon zu einem Satze von Fej{\'e}r,
+\emph{Math. Z.} \textbf{2} (1918), 187--197.
+\href{https://doi.org/10.1007/BF01212904}{doi:10.1007/BF01212904}.
+
+\end{thebibliography}
diff --git a/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/paper.pdf b/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/paper.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..951d6770ed9d15865ed77ae9b828fed46e1c68d7
--- /dev/null
+++ b/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/paper.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:d9aa4ef2dccb1f044fdf78afd7a4c93f64e71bd60e53d0f0460329596f199bb6
+size 315777
diff --git a/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/README.md b/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/README.md
new file mode 100644
index 0000000000000000000000000000000000000000..be604e55f49a67d5dfdd9e02f58da084b01a7433
--- /dev/null
+++ b/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/README.md
@@ -0,0 +1,18 @@
+# [A directional zero-one law for finite-range-dependent random environments](directional-zero-one-finite-range.pdf)
+
+**Author:** OpenAI
+
+**Date:** October 5, 2026
+
+## Citation
+
+```bibtex
+@misc{OAI:A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026,
+  author = {{OpenAI}},
+  title = {{A directional zero--one law for finite-range-dependent random environments}},
+  howpublished = {OpenAI Math Release preprint
+                  \href{https://github.com/openai/math/blob/main/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/directional-zero-one-finite-range.pdf}{OAI:A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026}},
+  year = {2026}
+}
+```
+
diff --git a/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/build/manuscript/bridges.tex b/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/build/manuscript/bridges.tex
new file mode 100644
index 0000000000000000000000000000000000000000..d1d62ed536098b9e57ddfa7550ac2afcf879108e
--- /dev/null
+++ b/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/build/manuscript/bridges.tex
@@ -0,0 +1,302 @@
+\section{Coordinate bridges and opposed paths}
+\label{sec:bridges}
+
+By Proposition~\ref{prop:coordinate-reduction}, it remains to exclude
+coexistence in a coordinate direction. We therefore assume for the rest of
+the proof that both $A_{e_1}$ and $A_{-e_1}$ have positive probability.
+The two corresponding slab laws have finite mean width and finite mean
+radius, by Propositions~\ref{prop:mean-width} and
+\ref{prop:radius-moment}. We will arrange independent sequences of these
+slabs in opposite chronological directions. The contradiction will come
+from two estimates on how often the resulting paths approach each other.
+The bridge and opposed-path constructions follow the corresponding iid
+strategy in \cite[Sections~4.2--4.3]{OpenAI2026}. We give their marked
+versions here, including the separation arguments for the present
+environment law.
+
+For $\sigma\in\{+,-\}$, with $\sigma e_1$ denoting $e_1$ or $-e_1$, write
+\[
+ \widehat P_\sigma=\widehat P_{\sigma e_1},\qquad
+ D_\sigma=D_{\sigma e_1},\qquad p_\sigma=p_{\sigma e_1}>0,
+ \qquad \nu_\sigma=\nu_{\sigma e_1}.
+\]
+For a walk starting at the origin, let $T_j^\sigma$ be the first hit of
+signed coordinate height $\sigma x_1=j$. For integers $n\ge0$, set
+\begin{equation}
+ u_n^\sigma=P_0(T_n^\sigma<T_{-1}^\sigma),
+ \qquad F_\sigma(n)=\nu_\sigma(L>n).
+ \label{eq:hitting-width-tail}
+\end{equation}
+Thus $u_0^\sigma=1$, and $u_n^\sigma$ decreases to a limit at least
+$p_\sigma$: on $D_\sigma$, the signed height reaches every positive integer
+almost surely. Constants below may depend on the fixed environment law,
+$d,\kappa,R$, and the chosen script, but not on any of the scale parameters.
+
+\subsection{Bridges ending at prescribed cuts}
+
+Under $\widehat P_\sigma$, the widths of successive slabs form a renewal
+process on the nonnegative integers. Let $\bar u_n^\sigma$ be its renewal
+mass, including the initial renewal at zero. Equivalently, it is the
+probability of a cut at signed height $n$, with the initial boundary counted
+as a cut. To identify the corresponding finite path law, for $H\ge M$ let
+$\mathcal D_H^\sigma$ be the following event for the raw marked walk: it
+first hits $H-M$ before $-1$ in signed height, and from that hit it takes
+the prescribed $M$-step script. The script ends at its first hit of $H$.
+
+If $\bar u_H^\sigma>0$, let $\mathcal B_H^\sigma$ denote the law of the slab
+list through height $H$, conditional on a cut there. At $H=0$ this law is
+concentrated on the empty list. We call $\mathcal B_H^\sigma$ the
+\emph{exact-cut bridge law}.
+
+\begin{proposition}[Exact-cut bridges]
+\label{prop:exact-cut-bridges}
+The renewal masses satisfy
+\begin{equation}
+ \bar u_0^\sigma=1,\qquad
+ \bar u_j^\sigma=0\quad(1\le j<M),\qquad
+ \bar u_H^\sigma=\xi u_{H-M}^\sigma\quad(H\ge M).
+ \label{eq:bridge-renewal}
+\end{equation}
+For a list of slabs $(\gamma_1,\ldots,\gamma_m)$ with total width $H$, its
+mass under $\mathcal B_H^\sigma$ is
+\begin{equation}
+ \mathcal B_H^\sigma(\gamma_1,\ldots,\gamma_m)
+   =\frac{\prod_{i=1}^m W(\gamma_i)}{\bar u_H^\sigma}.
+ \label{eq:bridge-law}
+\end{equation}
+In particular, reversing the list order preserves this law; the steps
+within each word retain their original order.
+
+For $H\ge M$, concatenating the list gives the raw marked path through its
+first hit of $H$, conditioned on $\mathcal D_H^\sigma$. Each such path has
+a unique parsing into slabs. Concatenating any prescribed slab words, or
+successive bridges of prescribed widths, gives a raw path weight equal to
+the product of the constituent raw weights.
+\end{proposition}
+
+\begin{proof}
+The raw probability of $\mathcal D_H^\sigma$ is
+$\xi u_{H-M}^\sigma$. Its terminal script begins at a record and ends at
+a candidate of height $H$. By the script gap and
+Lemma~\ref{lem:separated-weights}, imposing a suffix that stays at signed
+height at least $H$ multiplies this probability by $p_\sigma$.
+Conversely, a cut at $H$ must be the end of the script beginning at the
+first hit of $H-M$: coordinate records are first hits, and each scripted
+step increases the signed height by one. Dividing by $p_\sigma$ to
+condition on $D_\sigma$ proves \eqref{eq:bridge-renewal}. No cut can have
+width less than $M$.
+
+The iid slab law from Proposition~\ref{prop:cut-law} now gives
+\eqref{eq:bridge-law}. Its numerator and the constraint on total width are
+unchanged when the list is reversed. Alternatively, for each fixed prefix
+through $H$ realizing $\mathcal D_H^\sigma$, the cut suffix supplies the
+same factor $p_\sigma$. Conditional on the cut at $H$ under
+$\widehat P_\sigma$, the prefix therefore has mass
+$W(\text{prefix})/\bar u_H^\sigma$. This is exactly its raw conditional
+mass given $\mathcal D_H^\sigma$.
+
+The parsing can be recovered from the finite prefix alone. Select the
+candidate ends whose heights are not undercut later in the prefix, and
+split at those ends. A suffix staying at or above $H$ cannot change this
+selection, because $H$ is a strict record height. Between successive
+selected candidates, every earlier candidate is undercut, which is
+precisely the defining condition for a slab. A script cannot straddle a
+selected boundary: it would overlap the script ending there. Thus these
+are exactly the slab boundaries, and the parsing is unique.
+
+Finally, active departures of a slab ending at signed height $b$ lie
+strictly below $b-M$. All departures of the following slabs lie at or
+above $b$. These row sets are more than $R$ apart in infinity distance.
+Applying Lemma~\ref{lem:separated-weights} successively, and using
+translation invariance, factors the weight of any prescribed
+concatenation. The same argument applies after grouping the slabs into
+bridges of prescribed widths. Each group boundary is reached on a first
+hit, so the grouping introduces no extra path multiplicity.
+\end{proof}
+
+The distinction between $u_n^\sigma$ and $\bar u_n^\sigma$ will matter.
+The former are ordinary hitting probabilities and decrease with $n$;
+the latter have a gap before the first possible script end. The next
+estimate retains the contribution of that gap.
+
+\begin{lemma}[Width tails control hitting differences]
+\label{lem:width-tail}
+For either sign and all integers $n'\ge n\ge0$,
+\begin{equation}
+ u_n^\sigma-u_{n'}^\sigma
+   \le C(n'-n)F_\sigma(n),
+ \qquad
+ \sum_{l\ge0}2^lF_\sigma(2^l)<\infty.
+ \label{eq:width-tail-comparison}
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+Fix a sign and suppress it. Partition according to the last renewal at
+or before $m$. The next width exceeds the distance from that renewal to
+$m$, so
+\[
+ \sum_{k=0}^m F(k)\bar u_{m-k}=1.
+\]
+Subtracting the identity at $m+1$, and using $\bar u_0=1$, gives
+\[
+ \sum_{k=0}^m F(k)
+       (\bar u_{m-k}-\bar u_{m+1-k})=F(m+1).
+\]
+By \eqref{eq:bridge-renewal}, every adjacent difference in this sum is
+nonnegative except the one with $m-k=M-1$, which equals $-\xi$.
+Take $m=n+M$. The exceptional index is $k=n+1$, while the term $k=0$
+equals $\xi(u_n-u_{n+1})$. Dropping the other nonnegative terms yields
+\[
+ \xi(u_n-u_{n+1})
+   \le F(n+M+1)+\xi F(n+1)
+   \le(1+\xi)F(n).
+\]
+Summation from $n$ to $n'-1$, using that $F$ decreases, proves the first
+bound. The second follows by grouping the ordinary tail sum
+$\sum_{k\ge0}F(k)=E_{\nu_\sigma}L<\infty$ into dyadic intervals.
+\end{proof}
+
+\subsection{Opposed tapes and common blocks}
+
+Take two independent sequences of iid relative slab words, with laws
+$\nu_+$ and $\nu_-$. We call these sequences the positive and negative
+\emph{tapes}, and denote their joint law and expectation by
+$\mathbf P$ and $\mathbf E$. Write
+$\pi x=(x_2,\ldots,x_d)$ for the lateral coordinate of $x$.
+Place the words in space as follows.
+\begin{itemize}[leftmargin=*,itemsep=3pt]
+\item The first positive word has its terminal site at $0$. Each
+subsequent positive word has its terminal site at the starting site of
+the preceding word. Thus tape order moves downward, whereas a finite
+initial portion is traversed chronologically by taking its words in
+reverse list order, always following the steps of each word forward.
+\item The negative words are concatenated in their original order,
+starting at $-e_1$. This produces a downward path with law
+$\widehat P_-$ translated to $-e_1$.
+\end{itemize}
+Backward placement of regeneration pieces also appears in Berger's
+backward-path construction \cite[Sections~2--3]{Berger2008}.
+We measure \emph{depth} by $z=-x_1$. A word of width $L$, preceded by
+total width $s$ on its tape, has all its departure depths in
+\begin{equation}
+                    (s,s+L]\cap\ZZ.
+ \label{eq:departure-depths}
+\end{equation}
+For the positive tape this excludes the terminal arrival at depth $s$.
+For the negative tape the same interval results from the initial
+one-level offset: its start has depth $s+1$, and its terminal arrival has
+depth $s+L+1$.
+
+A \emph{contact} is a positive departure site at infinity distance at
+most $R$ from some negative departure site. All departures count,
+regardless of their marks. The depth and, subsequently, the block index
+of a contact always refer to the positive departure. There is at least
+one contact: the last positive departure before its terminal site $0$
+is $-e_1$, the initial site of the negative path.
+
+To compare contacts at different depths, we group the two tapes using
+their common width renewals. Starting from width zero, end the first
+common block at the first positive integer that is a width sum on both
+tapes; continue with successive common width sums. The next proposition
+shows that this grouping gives iid finite pieces with the spatial
+integrability needed below. This is the intersection construction for
+independent renewal processes; see \cite{AlexanderBerger2016}.
+
+\begin{proposition}[Common blocks]
+\label{prop:common-blocks}
+The common blocks exist indefinitely, and their pairs of slab lists are
+iid. Their widths $K_i$ have a common law satisfying
+\[
+                  M\le K_i<\infty\quad\text{a.s.},
+             \qquad \mu:=\mathbf E K_i<\infty.
+\]
+The expected sum of the radii of all slabs on both sides of one common
+block is finite. In particular, if
+\begin{equation}
+ W'_0=0,\qquad W'_i=\sum_{j=1}^iK_j,
+ \label{eq:common-width-sums}
+\end{equation}
+then $W'_i/i\to\mu$ almost surely.
+\end{proposition}
+
+\begin{proof}
+Let $K$ be the first positive common width, allowing $K=\infty$ for now.
+For each fixed pair of finite prefixes that realize $K=k$, the fact
+that $k$ is their first common width is decided by those prefixes. Their
+unused tails are independent tapes with the initial laws. Enumerating
+these prefix pairs therefore gives the renewal recursion
+\[
+ c_0=1,\qquad
+ c_n=\sum_{k=1}^n\mathbf P(K=k)c_{n-k}\quad(n\ge1),
+ \qquad c_n=\bar u_n^+\bar u_n^-.
+\]
+Set $f(t)=\sum_{k\ge1}\mathbf P(K=k)t^k$ for $0<t<1$. Nonnegative
+power-series summation gives
+\[
+ \sum_{n\ge0}c_nt^n=\frac{1}{1-f(t)}.
+\]
+Equation~\eqref{eq:bridge-renewal} implies
+$c_n\ge\xi^2p_+p_->0$ for every $n\ge M$. Hence
+$\sum c_nt^n\ge\xi^2p_+p_-t^M/(1-t)$, and consequently
+$(1-f(t))/(1-t)$ stays bounded as $t\uparrow1$. First this forces
+$f(t)\to1$, proving $K<\infty$ almost surely. We can then write
+\[
+ \frac{1-f(t)}{1-t}
+  =\mathbf E\frac{1-t^K}{1-t}\ \uparrow\ \mathbf E K,
+\]
+which proves finite mean. The same prefix factorization at each common
+renewal proves that the successive pairs of lists are iid. The lower
+bound $K\ge M$ follows from the minimum slab width. The strong law gives
+the assertion about $W'_i$.
+
+Let $N_+$ be the number of positive words in the first common block.
+Since every width is at least one, $N_+\le K$. The event $\{N_+\ge i\}$
+is determined by the first $i-1$ positive widths and the entire negative
+tape: none of those positive width sums may be a negative width sum.
+It is therefore independent of the $i$th positive word and its radius
+$\mathcal R_i^+$. Tonelli's theorem gives
+\[
+ \mathbf E\sum_{i=1}^{N_+}\mathcal R_i^+
+  =\sum_{i\ge1}\mathbf P(N_+\ge i)E_{\nu_+}\mathcal R
+  =\mathbf E N_+\,E_{\nu_+}\mathcal R<\infty.
+\]
+The identical argument for the negative tape proves the claimed
+integrability, using Proposition~\ref{prop:radius-moment}.
+\end{proof}
+
+The notation $W'_i$ records cumulative width and is distinct from the
+word weight $W(\gamma)$. By \eqref{eq:departure-depths}, departures in
+common block $i$ on either tape have depths in
+$(W'_{i-1},W'_i]$. A contact in positive block $i$ can involve a negative
+departure only in block $i-1$, $i$, or $i+1$: skipping an entire common
+block would give a depth difference greater than $M>R$.
+
+For each common block let $S'_i\in\ZZ^{d-1}$ be the sum of the lateral
+displacements of all its positive and negative words, with both
+displacements measured in the words' original chronological directions.
+The $S'_i$ are iid and
+\begin{equation}
+                    \mathbf E\abs{S'_i}<\infty.
+ \label{eq:lateral-moment}
+\end{equation}
+Indeed, their norms are bounded by the sums of radii controlled in
+Proposition~\ref{prop:common-blocks}. As we pass downward through a
+block, the negative anchor moves by its chronological displacement,
+whereas the positive anchor moves by the negative of its chronological
+displacement. The lateral gap, negative anchor minus positive anchor,
+therefore increases by $S'_i$.
+
+We now have both a common block index for the two tapes and integrable
+iid increments for their lateral gap. The quantity on which the two
+remaining estimates will disagree is
+\begin{equation}
+ m(J)=\sum_{j=1}^J\mathbf P\bigl\{
+  \text{a contact occurs in a positive block of index in }
+                      (2^j,2^{j+1}]\bigr\},\qquad J\ge1.
+ \label{eq:contact-count}
+\end{equation}
+Thus $m(J)$ is the expected number of these dyadic block-index intervals
+that contain a contact. Section~\ref{sec:entropy} first gives an upper
+bound by comparing the dependent placement of the two tapes with
+independent bridges.
diff --git a/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/build/manuscript/conclusion.tex b/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/build/manuscript/conclusion.tex
new file mode 100644
index 0000000000000000000000000000000000000000..a2c50a14e758f7973e057282ef2fd02378117d58
--- /dev/null
+++ b/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/build/manuscript/conclusion.tex
@@ -0,0 +1,266 @@
+\section{Contacts at many depths and completion of the proof}
+\label{sec:contact-lower}
+
+We continue to assume coexistence in the coordinate direction and adapt the
+cut-averaging argument of \cite[Proposition~7.1]{OpenAI2026} to scripted cuts
+and positive-side proximity labels.  The first-contact estimate from
+Proposition~\ref{prop:first-contact} forces contacts in most groups
+of consecutive dyadic depth intervals.  We will first count these intervals and
+then use the finite mean width of the common blocks to convert depths into block
+indices.  The resulting lower bound contradicts
+Proposition~\ref{prop:contact-upper}.
+
+\begin{proposition}[Contact lower bound]\label{prop:contact-lower}
+Suppose that $P_0(A_{e_1})>0$ and $P_0(A_{-e_1})>0$, and let $m(J)$ be the
+contact statistic of the opposed tapes defined in \eqref{eq:contact-count}.
+There are constants $c>0$ and an integer $c_1\ge0$ such that, for all sufficiently
+large integers $J$,
+\begin{equation}\label{eq:contact-lower}
+ m(J+c_1)\ge \frac{cJ}{1+\log\log J}.
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+All constants in this proof may depend on the fixed environment law and the
+marked construction, but not on $J$ or the depth parameters.  Recall that the
+depth of a contact is the depth of its positive departure.  For an integer
+$l\ge0$, call $l$ a \emph{contact depth label} if there is a contact at an integer
+depth in $[2^l,2^{l+1})$.
+
+\paragraph{Conditioning on a distant cut.}
+Fix a large integer $J$, and set
+\begin{equation}\label{eq:lower-scales}
+ N=2^{2J},\qquad
+ G=\left\lceil 4\log_2\log(2+J)\right\rceil+3.
+\end{equation}
+Let $\mathcal C_N$ be the event that $N$ is a width renewal of the positive
+tape, and write
+\[
+ \mathbf Q_N=\mathbf P(\,\cdot\mid\mathcal C_N).
+\]
+By \eqref{eq:bridge-renewal},
+\begin{equation}\label{eq:lower-conditioning}
+ \mathbf P(\mathcal C_N)=\bar u_N^+\ge\xi p_+>0.
+\end{equation}
+Under $\mathbf Q_N$, follow the placed positive words from bottom to top by
+reversing the order of the slab list through this renewal, without reversing
+any word, and translate its bottom endpoint to the origin.  The product formula
+\eqref{eq:bridge-law} is invariant under this list
+reversal.  Thus the resulting chronological path $Y$ has bridge law
+$\mathcal B_N^+$.  If $E_{\rm lat}=\pi Y_{T_N}$, the same translation puts the
+negative tape at $(N-1,E_{\rm lat})$.  Write $\beta$ for this translated path.
+Its displacement path from its starting site has law $\widehat P_-$ and is
+independent of the positive list.  This is
+exactly the pair of paths in Proposition~\ref{prop:first-contact}.  In these
+coordinates the depth of a positive departure at $y$ is $N-y_1$.
+
+On $\mathcal C_N$, at most $N$ positive slabs are used before width $N$, since
+each width is at least $M\ge1$.  The norm of $E_{\rm lat}$ is bounded by the sum
+of their radii, and hence by the sum of the first $N$ radii of the unconditioned
+positive tape.  Proposition~\ref{prop:radius-moment},
+\eqref{eq:lower-conditioning}, and Markov's inequality give
+\begin{equation}\label{eq:lower-endpoint-tail}
+ \mathbf Q_N\{\abs{E_{\rm lat}}>N^2\}\le \frac{C}{N}.
+\end{equation}
+
+\paragraph{An interval without contacts forces a late first contact.}
+For $G<l\le J$, let $U_l'$ be the event that none of
+$l-G,l-G+1,\ldots,l$ is a contact depth label.  We will show that
+\begin{equation}\label{eq:uncovered-labels}
+ \sum_{l=G+1}^J\mathbf Q_N(U_l')=o(J).
+\end{equation}
+Consider the possible positive cut depths
+\[
+ \mathcal I_l=[2^l,\tfrac32\,2^l]\cap\ZZ.
+\]
+For $s\in\mathcal I_l$, put
+\begin{equation}\label{eq:lower-contact-parameters}
+ k=N-s,\qquad r=N-2^{l-G}.
+\end{equation}
+For all sufficiently large $J$, these integers satisfy
+$M\le k<r<N$ and $r-k\ge R$, uniformly over the indicated $l,s$.
+
+Suppose first that $s$ is an actual cut depth of the positive tape.  In the
+reversed list, its boundary is at height $k$.  Every preceding word remains
+strictly below its terminal height until its final arrival, and that arrival
+ends a record script.  Consequently the bridge first reaches $k$ along a
+prefix realizing $\mathcal D_k^+$: it first hits $k-M$ and then follows the
+prescribed $M$-step script.  All the subsequent words remain at or above $k$.
+These assertions hold for the concatenated bridge, because a word's initial
+boundary is a strict record of the preceding words and no script can straddle
+a boundary at which another script ends.
+
+The final positive departure before $T_N$ is $(N-1,E_{\rm lat})$, the initial
+site of $\beta$.  There is therefore a first contact along $Y$ between $T_k$
+and $T_N$.  Its depth is at most $s$.  On $U_l'$, there are no contacts at any
+depth in $[2^{l-G},2^{l+1})$; since $s<2^{l+1}$, this first contact must have
+depth less than $2^{l-G}$.  Its height is in particular at least $r$.  If also
+$\abs{E_{\rm lat}}\le N^2$, all the requirements of
+\eqref{eq:first-contact-event} are now satisfied.
+
+For each \emph{deterministic} $s\in\mathcal I_l$, let $\mathcal A_{l,s}$ denote
+the event \eqref{eq:first-contact-event} with the parameters
+\eqref{eq:lower-contact-parameters}.  This event includes the prefix condition
+$\mathcal D_k^+$, but we do not additionally require $s$ to be a cut of the
+tape.  Thus Proposition~\ref{prop:first-contact} applies directly under
+$\mathbf Q_N$, and gives
+\begin{equation}\label{eq:lower-deterministic-cut}
+ \mathbf Q_N(\mathcal A_{l,s})
+ \le C\log(2N)\,\phi(\Delta_{l,s}),\qquad
+ \Delta_{l,s}=u^+_{s-2^{l-G}}-u_s^+,
+\end{equation}
+where
+\[
+ \phi(x)=x\log^2(\mathrm e/x)\quad(0<x\le1),\qquad \phi(0)=0.
+\]
+Here and below $\mathrm e=\exp(1)$.  Lemma~\ref{lem:width-tail} implies
+\begin{equation}\label{eq:lower-delta-tail}
+ \Delta_{l,s}\le C2^{-G}a_l,\qquad
+ a_l=2^l F_+(2^{l-1}),\qquad
+ A:=\sum_{l\ge1}a_l<\infty,
+\end{equation}
+because $s-2^{l-G}\ge2^{l-1}$.
+The factor $2^{-G}$ is the ratio between the remaining contact depth
+$2^{l-G}$ and the cut depth scale $2^l$.
+
+\paragraph{Averaging over the cut locations.}
+Let $C_+(t)$ count the positive width renewals in $(0,t]$.  The positive widths
+are iid with finite mean by Propositions~\ref{prop:cut-law}
+and~\ref{prop:mean-width}; their strong law gives
+$C_+(t)=t/E_{\nu_+}L+o(t)$ almost surely.  It follows that the number of positive
+cuts in $\mathcal I_l$ is
+\[
+ \frac{2^{l-1}}{E_{\nu_+}L}+o(2^l)
+ \quad\text{almost surely as }l\longrightarrow\infty.
+\]
+Fix $0<c_0<1/(2E_{\nu_+}L)$, and let $\mathcal T_J$ be the event that every
+$\mathcal I_l$ with $G<l\le J$ contains at least $c_0 2^l$ positive cuts.  On
+almost every tape the displayed asymptotic supplies this bound for all
+sufficiently large $l$.  Since $G\to\infty$,
+\[
+ \mathbf P(\mathcal T_J^c)\longrightarrow0,
+ \qquad
+ \mathbf Q_N(\mathcal T_J^c)
+ \le\frac{\mathbf P(\mathcal T_J^c)}{\xi p_+}\longrightarrow0.
+\]
+In particular, no convergence rate in the strong law is needed for the
+changing conditional law $\mathbf Q_N$.
+
+Every cut counted in $\mathcal I_l$ lies before depth $N$.  On $U_l'$,
+$\mathcal T_J$, and $\{\abs{E_{\rm lat}}\le N^2\}$, each of these cuts supplies
+one of the events $\mathcal A_{l,s}$ by the preceding argument.  We obtain the
+pointwise inequality
+\begin{equation}\label{eq:cut-averaging}
+ \one_{U_l'}\one_{\mathcal T_J}\one_{\{\abs{E_{\rm lat}}\le N^2\}}
+ \le\frac{1}{c_0 2^l}\sum_{s\in\mathcal I_l}\one_{\mathcal A_{l,s}}.
+\end{equation}
+The right side involves only the deterministic-parameter probabilities already
+bounded in \eqref{eq:lower-deterministic-cut}.
+
+We next sum those bounds using only $\sum_l a_l<\infty$.  For large $J$,
+$C2^{-G}a_l\le\mathrm e^{-1}$ uniformly in $l$, and $\phi$ is increasing on
+$[0,\mathrm e^{-1}]$.  Hence \eqref{eq:lower-delta-tail} yields
+\begin{equation}\label{eq:lower-phi}
+ \phi(\Delta_{l,s})
+ \le C2^{-G}a_l\bigl((G+1)^2+\log_+^2(1/a_l)\bigr),
+\end{equation}
+where $\log_+ x=\max\{0,\log x\}$ and the right side is interpreted as zero
+when $a_l=0$.  Moreover,
+\begin{equation}\label{eq:lower-slow-tail}
+ \sum_{l=1}^J a_l\log_+^2(1/a_l)
+ \le C(A+J^{-1})\log^2 J.
+\end{equation}
+Indeed, terms with $a_l\ge J^{-2}$ contribute at most $4A\log^2 J$.
+For the remaining terms, the function $x\log^2(1/x)$ is increasing on
+$[0,J^{-2}]$ for large $J$, so each is at most $4J^{-2}\log^2 J$.
+
+Taking expectations in \eqref{eq:cut-averaging}, using
+$|\mathcal I_l|\le2^l$, and applying
+\eqref{eq:lower-endpoint-tail}, \eqref{eq:lower-deterministic-cut},
+\eqref{eq:lower-phi}, and \eqref{eq:lower-slow-tail}, we obtain
+\begin{align}
+ \sum_{l=G+1}^J\mathbf Q_N(U_l')
+ &\le J\mathbf Q_N(\mathcal T_J^c)+\frac{CJ}{N}\notag\\
+ &\quad+C\log(2N)\,2^{-G}
+       \bigl(A(G+1)^2+(A+J^{-1})\log^2 J\bigr)
+ =o(J).\label{eq:lower-uncovered-bound}
+\end{align}
+For the final equality, the first term is $o(J)$, while
+\eqref{eq:lower-scales} gives $\log(2N)=O(J)$,
+$2^{-G}=O((\log J)^{-4})$, and $G=O(\log\log J)$.
+The last term is therefore $O(J/(\log J)^2)$, and $CJ/N=o(J)$.  This proves
+\eqref{eq:uncovered-labels}, even when the summable sequence $(a_l)$ decays
+arbitrarily slowly.
+
+There are $J-G$ tested labels $l$.  By \eqref{eq:uncovered-labels} and Markov's
+inequality, with $\mathbf Q_N$-probability tending to one, at least $J/2$ of
+these tests have $U_l'$ false.  Each such test has a contact depth label in
+$\{l-G,\ldots,l\}$, and any one contact label can account for at most $G+1$
+tests.  Thus, if $\mathcal E_J$ is the event that at least $J/(2(G+1))$ of
+$1,\ldots,J$ are contact depth labels, then
+\begin{equation}\label{eq:lower-unconditioning}
+ \mathbf Q_N(\mathcal E_J)\longrightarrow1,
+ \qquad
+ \liminf_{J\to\infty}\mathbf P(\mathcal E_J)\ge\xi p_+>0.
+\end{equation}
+The second assertion follows by intersecting $\mathcal E_J$ with
+$\mathcal C_N$ and using \eqref{eq:lower-conditioning}.  We have therefore
+obtained the needed number of depth labels under the original tape law.
+
+\paragraph{From depths to common-block indices.}
+By Proposition~\ref{prop:common-blocks}, $\mu=\mathbf E K_1$ is finite and positive,
+and $W_i'/i\to\mu$ almost surely.  For an integer depth $z\ge1$, let $i(z)$ be
+its common-block index, so that
+\[
+ W_{i(z)-1}'<z\le W_{i(z)}'.
+\]
+The strong law implies, almost surely for all sufficiently large integer $z$,
+\begin{equation}\label{eq:depth-index-comparison}
+ \frac{z}{2\mu}\le i(z)<\frac{2z}{\mu}+1.
+\end{equation}
+Consequently there is a deterministic integer $c_1\ge0$ such that, almost
+surely for all sufficiently large $l$, every depth $2^l\le z<2^{l+1}$ has
+an index label $j$ satisfying
+\[
+ 2^j<i(z)\le2^{j+1},\qquad |j-l|\le c_1.
+\]
+The opposite choices of open endpoints for the depth and index intervals
+affect this comparison by at most one label and are included in $c_1$.
+Let $\mathcal V_J$ be the event that the comparison holds for every
+$l\ge\lceil\sqrt J\rceil$.  Its probability tends to one.  By
+\eqref{eq:lower-unconditioning}, the intersection $\mathcal E_J\cap\mathcal V_J$
+therefore has probability bounded below by a positive constant for all large
+$J$.
+
+On this intersection discard the depth labels below $\lceil\sqrt J\rceil$.
+This loses only $O(\sqrt J)=o(J/(G+1))$ labels.  Each remaining contact depth
+label gives a contact in a block-index interval labeled in
+$\{1,\ldots,J+c_1\}$, and each index label can receive at most $2c_1+1$
+distinct depth labels.  There are consequently at least $cJ/(G+1)$ such
+index labels on an event of probability bounded below.  Taking expectations
+in the definition \eqref{eq:contact-count} proves
+\[
+ m(J+c_1)\ge\frac{cJ}{G+1}.
+\]
+Since $G+1=O(1+\log\log J)$, this is \eqref{eq:contact-lower}.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+If both coordinate escape events $A_{e_1}$ and $A_{-e_1}$ had positive
+probability, Propositions~\ref{prop:contact-upper} and
+\ref{prop:contact-lower} would give, for all large $J$,
+\[
+ \frac{cJ}{1+\log\log J}
+ \le m(J+c_1)
+ \le \frac{C(J+c_1)}{\log(J+c_1)},
+\]
+which is impossible as $J\to\infty$.  The same argument applies after any
+permutation of the coordinate axes, so coexistence is impossible in every
+coordinate direction.
+
+Now fix any nonzero real direction $\ell$.  If $0<P_0(A_\ell)<1$,
+Proposition~\ref{prop:sign-union} gives positive probability to both
+$A_\ell$ and $A_{-\ell}$.  Proposition~\ref{prop:coordinate-reduction} then
+gives a coordinate direction with positive probability for both escape
+signs, contradicting what we have just proved.  Thus $P_0(A_\ell)\in\{0,1\}$.
+\end{proof}
diff --git a/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/build/manuscript/cuts.tex b/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/build/manuscript/cuts.tex
new file mode 100644
index 0000000000000000000000000000000000000000..7a94221098038677943d5003c268f690b2933ff9
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+++ b/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/build/manuscript/cuts.tex
@@ -0,0 +1,454 @@
+\section{Separated rows and regeneration words}\label{sec:cuts}
+
+We first construct independent successive pieces of a walk that stays
+above its initial level in a prescribed direction.  Finite-range dependence requires a
+gap between the rows used by consecutive pieces.  We create that gap by
+marking some transitions whose probabilities are constant, so that the
+walk can cross the gap without using its environmental rows.  The
+construction will also prove the directional sign-union statement and a
+finite mean for the spatial width of each piece.  Throughout this section
+the direction may be any fixed nonzero real vector.  We adapt the
+regeneration and record-count arguments of
+\cite[Section~2]{OpenAI2026}, proving here the separated-row identities
+needed for finite-range dependence.
+
+\subsection{Marks and the rows a path uses}
+
+We use the same splitting device as the independent-coin construction
+of Comets and Zeitouni \cite[Equation~(2.1)]{CometsZeitouni2004}:
+an environment-independent choice forces a step, and a residual choice
+restores the prescribed transition row.  Two distinguished marks will
+also prevent overlapping occurrences of the prescribed step sequence
+introduced below.
+
+Fix \(\lambda>0\) with \(2\lambda<\kappa\).  Enlarge the walk by giving
+each step a mark in \(\{0,1,*\}\).  Conditional on the environment and
+the marked past, a step from \(x\) has joint probabilities
+\begin{equation}\label{eq:marked-kernel}
+ \begin{aligned}
+ P_{x,\omega}(X_1=x+e,\text{ mark }0)&=\lambda,\\
+ P_{x,\omega}(X_1=x+e,\text{ mark }1)&=\lambda,\\
+ P_{x,\omega}(X_1=x+e,\text{ mark }*)&=\omega(x,e)-2\lambda,
+ \qquad e\in U.
+ \end{aligned}
+\end{equation}
+These probabilities are nonnegative and sum to one.  Summing over the
+marks recovers the original transition row, so every assertion about
+the unmarked path is unchanged.  We keep the notation
+\(P_{x,\omega}\) and \(P_x\) for the marked quenched and annealed laws.
+We call these laws \emph{raw} when no condition of staying on one side of
+a barrier is imposed.  The path filtration includes both positions and
+the marks of steps already taken.  All constructions take place on the
+full-measure set where the ellipticity inequalities hold at every site;
+the marked kernel may be defined arbitrarily elsewhere.
+
+A finite marked word \(\gamma\) consists of positions
+\(x_0,\ldots,x_t\), with \(x_{j+1}-x_j\in U\), and one mark for each
+of its \(t\) steps.  Its departure set and active departure set are
+\[
+ \Dep(\gamma)=\{x_0,\ldots,x_{t-1}\},\qquad
+ \operatorname{Act}(\gamma)
+ =\{x_j:0\le j<t,\ \text{step }j\text{ has mark }*\}.
+\]
+Let \(p_\omega(\gamma)\) be the product of the marked transition
+probabilities in \eqref{eq:marked-kernel}, and put
+\[
+ W(\gamma)=E_Q[p_\omega(\gamma)].
+\]
+Thus \(W(\gamma)\) is the annealed probability of the specified prefix.
+Stationarity makes it invariant under translation of all positions in
+the word.  The terminal arrival uses no row.  More generally, a visited
+row affects \(p_\omega(\gamma)\) only if it is an active departure.
+
+\begin{lemma}[Separated path weights]\label{lem:separated-weights}
+Let \(\gamma\) be a fixed finite marked word, and let
+\(B\subseteq\ZZ^d\) be a deterministic set.  With the convention that
+distance from an empty set is infinite, suppose that
+\[
+ \dist_\infty(\operatorname{Act}(\gamma),B)>R.
+\]
+If \(f\ge0\) is measurable with respect to the rows in \(B\), then
+\begin{equation}\label{eq:separated-weights}
+ E_Q[p_\omega(\gamma)f(\omega)]
+ =W(\gamma)E_Q[f(\omega)].
+\end{equation}
+The conclusion also holds when the active set is empty, without a
+separation requirement.
+
+For a deterministic set \(F\subseteq\ZZ^d\), the quenched law of the
+marked walk stopped on its first arrival in \(F\), including the arrival
+site, depends only on rows outside \(F\).  In particular, the quenched
+probability of any measurable infinite-path event that implies the walk
+never enters \(F\) depends only on those rows.  Such probabilities may
+therefore be used as \(f\) in \eqref{eq:separated-weights} whenever the
+exterior rows are separated from \(\operatorname{Act}(\gamma)\).
+\end{lemma}
+
+\begin{proof}
+Factors with marks \(0\) or \(1\) are constant.  All other factors in
+\(p_\omega(\gamma)\) are functions of rows in
+\(\operatorname{Act}(\gamma)\), with repeated departures from one row
+retaining that same row.  The finite-range assumption
+\eqref{eq:finite-range} gives independence from the rows in \(B\), also
+when \(B\) is infinite.  This proves \eqref{eq:separated-weights}; if
+the active set is empty, \(p_\omega(\gamma)\) itself is constant.
+
+Before the first arrival in \(F\), every transition departs from its
+complement.  Prefix probabilities for the stopped walk therefore use
+only exterior rows, including when the prefix ends with that arrival.
+These prefixes determine the stopped path law, so all its measurable
+event probabilities have the same row dependence.  Equivalently, one
+may kill the walk on arrival in \(F\).  An infinite-path event implying
+avoidance of \(F\) is determined by this killed trajectory: it is false
+on killed paths and retains its original meaning on surviving paths.
+This proves the last assertion.
+\end{proof}
+
+We will always fix the relevant finite words before applying this
+lemma.  The separated sets are then deterministic.  For example, two
+prescribed words with departure sets at distance greater than \(R\)
+have the same joint weight in independent environments as in a common
+environment with conditionally independent walks.  The lemma also
+applies to an avoidance event involving an entire infinite continuation.
+
+\subsection{Cuts with independent continuations}
+
+Fix a direction \(v\ne0\), rescale it so that
+\(\norm{v}_\infty=1\), and write \(h(x)=v\cdot x\).  Choose a
+coordinate step \(a_v\in U\) with \(h(a_v)=1\).  Fix an integer
+\(M>dR+1\), the same integer for every direction considered, and set
+\(\xi=\lambda^M\).  The prescribed \emph{script} in direction \(v\)
+consists of \(M\) steps \(a_v\), the first with mark \(0\) and all
+the others with mark \(1\).  Two occurrences of the script cannot
+overlap in a step: the initial \(0\) of a second occurrence would have
+to coincide with a \(1\) of the first.
+
+We call time zero a record.  A positive time is an \emph{ordinary
+record} if its height strictly exceeds all preceding heights.  A script
+starting at a record has a \emph{candidate} at its end.  A candidate
+time \(t\) is a \emph{cut} if
+\[
+ h(X_n)\ge h(X_t)\qquad(n\ge t).
+\]
+Each candidate is itself a strict record.  These definitions initially
+refer to the whole path starting at time zero.  For a walk started
+elsewhere we use heights relative to its starting site.  Write
+\[
+ D_v=\{h(X_n)\ge0\text{ for every }n\ge0\},\qquad
+ p_v=P_0(D_v),\qquad
+ \widehat P_v=P_0(\,\cdot\mid D_v)\quad(p_v>0).
+\]
+
+The script separates the rows of a record prefix from those of a
+continuation above its endpoint; Figure~\ref{fig:cut-gap} displays this
+distinction between visited and active sites.  Indeed, for all
+\(x,y\in\ZZ^d\),
+\begin{equation}\label{eq:height-distance}
+ |h(x)-h(y)|\le d\norm{x-y}_\infty.
+\end{equation}
+If a record prefix ends at height \(r\), every departure in that
+prefix has height strictly below \(r\).  The script has no active
+departures, whereas a continuation staying above its endpoint uses
+only rows of height at least \(r+M\).  The required separation follows
+from \(M>dR\).
+
+\begin{figure}[tb]
+\centering
+\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,font=\small]
+ \fill[blue!7] (-3.6,-1.65) rectangle (3.7,0);
+ \fill[green!8] (-3.6,2.15) rectangle (3.7,3.05);
+ \draw[dashed,gray] (-3.6,0)--(3.7,0);
+ \draw[dashed,gray] (-3.6,2.15)--(3.7,2.15);
+ \draw[->,gray] (-4.05,-1.65)--(-4.05,3.15) node[above] {$h$};
+ \node[left] at (-4.05,0) {$r$};
+ \node[left] at (-4.05,2.15) {$r+M$};
+ \draw[thick,blue!65!black]
+   (-2.6,-1.25)--(-2.1,-.8)--(-1.5,-1.1)--(-1.1,-.5)
+   --(-.6,-.75)--(0,0);
+ \foreach \x/\y in {-2.6/-1.25,-1.5/-1.1,-1.1/-.5}
+   \fill[blue!65!black] (\x,\y) circle (2pt);
+ \foreach \x/\y in {-2.1/-.8,-.6/-.75}
+   \draw[blue!65!black,fill=white] (\x,\y) circle (2pt);
+ \foreach \y/\yy/\m in {0/.43/0,.43/.86/1,.86/1.29/1,1.29/1.72/1,1.72/2.15/1}
+   \draw[->,thick,orange!85!black] (0,\y)--(0,\yy)
+      node[midway,right=3pt] {\scriptsize $\m$};
+ \foreach \y in {0,.43,.86,1.29,1.72,2.15}
+   \draw[orange!85!black,fill=white] (0,\y) circle (2pt);
+ \draw[thick,green!45!black,->]
+   (0,2.15)--(.65,2.5)--(1.2,2.3)--(1.85,2.8)--(2.5,2.55);
+ \node[align=left,anchor=west,blue!65!black] at (.9,-.75)
+   {prefix active rows\\[-1pt]have $h<r$};
+ \node[align=left,anchor=west,green!35!black] at (-3.4,2.6)
+   {continuation rows\\[-1pt]have $h\ge r+M$};
+ \node[align=left,anchor=west] at (.9,1.05)
+   {script departures\\[-1pt]use constant weights};
+ \draw[decorate,decoration={brace,amplitude=4pt}]
+   (-.48,0)--(-.48,2.15) node[midway,left=7pt] {$M$};
+\end{tikzpicture}
+\caption{The gap at a candidate, shown schematically in height
+coordinates.  Filled points indicate active prefix departures; orange
+arrows are the constant-weight script steps.  The height gap
+separates every active prefix row from every permitted continuation row
+by more than \(R\) in \(\ell^\infty\)-distance, although the script
+itself visits the intervening sites.}
+\label{fig:cut-gap}
+\end{figure}
+
+Consequently, conditional on any finite marked prefix ending at a
+record, the probability that the script occurs next and ends at a cut
+is exactly
+\begin{equation}\label{eq:script-probability}
+ \xi p_v.
+\end{equation}
+More precisely, if \(\gamma_0\) is that prefix and \(B\) is any
+event for a translated continuation from zero, then
+\begin{equation}\label{eq:record-suffix-factorization}
+ \begin{split}
+ &P_0\{\text{prefix }\gamma_0,\ \text{then the script,}\\
+ &\hspace{12mm}\text{then a translated continuation in }B\cap D_v\}\\
+ &\hspace{12mm}=W(\gamma_0)\xi P_0(B\cap D_v).
+ \end{split}
+\end{equation}
+Here and below a translated continuation records its positions
+relative to its own starting site.  Lemma~\ref{lem:separated-weights}
+proves this identity, using the forbidden halfspace below the script's
+endpoint and stationarity.
+
+We also record an elementary consequence of ellipticity that will be
+used repeatedly.  A walk started in a slab
+\(\{x:a\le h(x)\le b\}\), with \(a,b\) finite, exits that slab
+almost surely under every uniformly elliptic quenched law.  Choose an
+integer \(m>b-a\).  From any site of the slab, \(m\) consecutive
+steps \(a_v\) leave it, and their probability is at least
+\(\kappa^m\).  Conditional trials in successive blocks of \(m\)
+steps bound the probability of remaining for \(jm\) steps by
+\((1-\kappa^m)^j\).  In particular, on \(D_v\) the height supremum
+is almost surely infinite: remaining forever in \([0,b]\) is null for
+each positive integer \(b\).
+
+To describe the pieces between cuts, call a finite marked word
+\(\gamma\) from zero a \emph{regeneration word} if it has the
+following properties:
+\begin{enumerate}[label=(\roman*),leftmargin=*]
+ \item all its heights are nonnegative;
+ \item it ends at a candidate;
+ \item every earlier candidate in the word is undercut later within
+       the word, meaning that a subsequent height is strictly smaller
+       than that candidate's height.
+\end{enumerate}
+Records and candidates in this definition are computed within the
+word.  Its \emph{width} \(L(\gamma)\) is its terminal height.  The
+last script starts at a record of height \(L(\gamma)-M\), so
+\begin{equation}\label{eq:word-height-support}
+ \begin{gathered}
+ L(\gamma)\ge M,\qquad
+ 0\le h(x)<L(\gamma)\quad(x\in\Dep(\gamma)),\\
+ h(x)<L(\gamma)-M\quad(x\in\operatorname{Act}(\gamma)).
+ \end{gathered}
+\end{equation}
+We sometimes call a regeneration word a \emph{slab}, referring to this
+height support.
+
+\begin{proposition}[Regeneration law]\label{prop:cut-law}
+Fix a normalized real direction \(v\) and suppose \(p_v>0\).
+\begin{enumerate}[label=(\roman*),leftmargin=*]
+ \item Under the raw law \(P_0\), on
+       \(\{\sup_n h(X_n)=\infty\}\) there is a cut almost surely.
+ \item Under \(\widehat P_v\), there are infinitely many cuts.  With
+       time zero counted as an initial boundary, the relative marked
+       words between successive boundaries are independent and
+       identically distributed.  Their common law \(\nu_v\) assigns
+       mass
+       \begin{equation}\label{eq:regeneration-word-law}
+        \nu_v(\{\gamma\})=W(\gamma)
+       \end{equation}
+       to each regeneration word \(\gamma\).  These finite words
+       concatenate to the entire conditioned walk.
+ \item Under the raw law, conditional on any realized prefix through
+       the first cut, the translated suffix has law
+       \(\widehat P_v\).  This assertion is restricted to the event
+       that a cut exists.
+\end{enumerate}
+\end{proposition}
+
+\begin{proof}
+First test for a scripted cut from time zero.  Failure is detected in
+finite time: either the next \(M\) steps are not the script, or they
+are the script and the subsequent path first falls below its endpoint.
+After a detected failure, wait for a strict record above the whole
+maximum attained by the detection time, and test again.  The successive
+test starts, when finite, are stopping times in the marked path
+filtration.  Equation~\eqref{eq:script-probability}, applied to every
+finite record prefix, gives conditional success probability
+\(\xi p_v\) at each test.  Hence
+\[
+ P_0\{\text{the first }j\text{ tests occur and all fail}\}
+ \le (1-\xi p_v)^j.
+\]
+On infinite height supremum, every detected failure is followed by
+another finite test start.  Letting \(j\) tend to infinity proves (i).
+The slab-exit observation implies that the first cut is finite almost
+surely under \(\widehat P_v\).  The tests need not locate the earliest
+cut; their only role is to establish that some cut exists.
+
+We now identify the law at the actual first cut by finite prefixes.
+Suppose \(\gamma\) is a
+regeneration word ending at \(z\).  A path beginning with \(\gamma\)
+whose suffix stays at or above \(h(z)\) has its first cut at that
+endpoint: all earlier candidates have already been undercut.  It also
+satisfies the original \(D_v\).  Conversely, a path in \(D_v\)
+through its first cut gives such a word.  Indeed, every earlier
+candidate has smaller height than the terminal strict record.  If it
+were not undercut before the terminal record, it could not be undercut
+afterwards either, since the suffix stays above that record; it would
+already be a cut.
+
+For any measurable suffix event \(B\), the row separation in
+\eqref{eq:word-height-support} and
+Lemma~\ref{lem:separated-weights} therefore give
+\[
+ \begin{split}
+ &P_0\{\text{first-cut prefix }\gamma,\
+          \text{ translated suffix in }B\}\\
+ &\hspace{25mm}=W(\gamma)P_0(B\cap D_v).
+ \end{split}
+\]
+Dividing by \(p_v\) proves that, under \(\widehat P_v\), the
+first word has mass \(W(\gamma)\) and its translated suffix has law
+\(\widehat P_v\), independently of that word.  There are only
+countably many finite marked words, and the first cut is finite almost
+surely, so these masses sum to one.
+
+The cut is a strict record, hence records of its translated suffix are
+also records of the full path.  A script cannot straddle this boundary:
+such a script would overlap the script that has just ended.  Thus the
+cuts of the suffix are precisely the subsequent cuts of the full path.
+Iterating the preceding factorization proves the iid assertion in
+(ii).  Every word is finite and contains at least \(M\) steps, so the
+successive cut times tend to infinity and the words cover the whole
+path.
+
+For (iii), the prefix through the raw first cut may have negative
+heights, but it still ends with the script from a record, and
+every preceding candidate is undercut within the prefix.  Its active
+departures lie strictly below the start of the terminal script.  The
+same separated-weight calculation gives raw prefix probability
+\(W(\gamma)p_v\) and, with suffix restriction \(B\), probability
+\(W(\gamma)P_0(B\cap D_v)\).  Their ratio is
+\(\widehat P_v(B)\).  All of these arguments enumerate finite
+prefixes; none uses a Markov property at a cut selected using the
+future.
+\end{proof}
+
+\subsection{Escape signs and finite mean width}
+
+The regeneration law gives two consequences needed in the remainder of
+the proof.  First, a positive escape probability in one direction rules
+out oscillation between arbitrarily high and low levels.  Second, the
+widths of the regeneration words have finite mean.  Counting records,
+rather than integer height levels, will give the latter conclusion for
+real directions as well.
+
+\begin{proposition}[Directional sign union]\label{prop:sign-union}
+For every fixed \(v\ne0\),
+\begin{equation}\label{eq:sign-union}
+ P_0(A_v)>0\quad\Longrightarrow\quad
+ p_v>0\ \text{ and }\ P_0(A_v\cup A_{-v})=1.
+\end{equation}
+More precisely, when \(p_v>0\), infinite height supremum implies
+\(A_v\) almost surely, and finite height supremum implies \(A_{-v}\)
+almost surely.
+\end{proposition}
+
+\begin{proof}
+Assume first that \(p_v>0\).  On infinite supremum there is a first
+cut, and its suffix has the iid word law of
+Proposition~\ref{prop:cut-law}.  The successive slab bases increase in
+height by at least \(M\); within each slab the path stays above its
+base.  Since each slab is finite, \(h(X_n)\to+\infty\).
+
+For finite supremum, fix integers \(m,b\ge0\).  Consider the event
+that all heights are at most \(m\), but heights at least \(-b\) are
+visited infinitely often.  From each such visit, a script of
+\(m+b+1\) consecutive steps \(a_v\), with no mark restriction,
+crosses \(m\) with conditional probability at least
+\(\kappa^{m+b+1}\).  Take successive visit trials at least
+\(m+b+1\) times apart.  The probability that the first \(j\) trials
+occur and all fail to cross \(m\) is at most
+\((1-\kappa^{m+b+1})^j\).  The event in question is therefore null.
+Taking the countable union over \(m,b\) shows that finite supremum
+forces \(h(X_n)\to-\infty\).
+
+It remains to obtain \(p_v>0\) from a positive probability of
+\(A_v\).  On \(A_v\), the height sequence tends to infinity, so its
+global minimum is attained at some finite time, even if its possible
+values are not discrete.  Thus for some deterministic \(n\ge0\)
+and \(x\in\ZZ^d\), the event
+\[
+ \{X_n=x,\ h(X_{n+j})\ge h(x)\text{ for all }j\ge0\}
+\]
+has positive annealed probability.  If \(q_x(\omega)\) denotes the
+quenched probability from \(x\) of staying above \(h(x)\), the
+quenched Markov property at time \(n\) expresses this probability as
+\(E_Q[P_{0,\omega}(X_n=x)q_x(\omega)]\).  It is bounded above by
+\(E_Qq_x=p_v\), by stationarity.  Therefore \(p_v>0\), and the
+two alternatives just proved give \eqref{eq:sign-union}.
+\end{proof}
+
+\begin{proposition}[Finite mean spatial width]\label{prop:mean-width}
+For every normalized real direction \(v\) with \(p_v>0\), the
+regeneration law of Proposition~\ref{prop:cut-law} satisfies
+\begin{equation}\label{eq:finite-mean-width}
+ E_{\nu_v}L<\infty.
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+Let \(S(\gamma)\) be the number of ordinary records in a regeneration
+word \(\gamma\), excluding its initial boundary and including its
+terminal record.  Each increase of the running maximum is at most one,
+because every step has height increment at most one.  Therefore
+\(L(\gamma)\le S(\gamma)\).  The record indices of successive cuts,
+with index zero at the initial boundary, are sums of iid positive
+integer increments with law \(S\): the terminal height of each word
+is the maximum so far, so record counts concatenate across boundaries.
+
+For \(i\ge0\), let \(\rho_i\) be the time of ordinary record
+index \(i\) in a raw walk, with \(\rho_0=0\) and
+\(\rho_i=\infty\) if that record is never reached.  Put
+\[
+ D'_{v,i}=\{\rho_i<\infty,\ h(X_j)\ge0
+                       \text{ for }0\le j\le\rho_i\}.
+\]
+A script from record \(i\) creates exactly \(M\) consecutive
+strict records.  Hence a cut of record index \(i+M\), on a path in
+\(D_v\), occurs exactly when the prefix realizes \(D'_{v,i}\),
+the script follows, and its suffix stays above its endpoint.  Summing
+\eqref{eq:record-suffix-factorization} over the possible record
+prefixes and dividing by \(p_v\) gives
+\begin{equation}\label{eq:record-renewal-lower}
+ \widehat P_v\{i+M\text{ is a cut index}\}
+   =\xi P_0(D'_{v,i})\ge\xi p_v.
+\end{equation}
+The inequality holds because on \(D_v\) the supremum is infinite
+almost surely, so every ordinary record is reached.
+
+Let \(N_{\mathrm{rec}}(n)\) count the cut indices in \(\{1,\ldots,n\}\).
+Equation~\eqref{eq:record-renewal-lower} implies
+\[
+ \liminf_{n\to\infty}
+ \frac{E_{\widehat P_v}N_{\mathrm{rec}}(n)}{n}\ge\xi p_v>0.
+\]
+If \(E_{\nu_v}S=\infty\), however, the partial sums of iid copies
+of \(S\), divided by their number, tend to infinity almost surely.
+To see this, apply the strong law to \(\min(S,K)\) for each positive
+integer \(K\), then let \(K\to\infty\).  Inverting these partial
+sums shows \(N_{\mathrm{rec}}(n)/n\to0\) almost surely.  Since this
+ratio lies in \([0,1]\), bounded convergence gives the same limit in
+expectation, a contradiction.  Thus \(E_{\nu_v}S<\infty\), and
+\(L\le S\) proves \eqref{eq:finite-mean-width}.
+\end{proof}
+
+The estimate concerns spatial width; it uses no moment assumption on
+the time spent in a word.  We next show that coexistence of the two
+escape signs also forces a finite mean for the full spatial radius.
diff --git a/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/build/manuscript/entropy.tex b/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/build/manuscript/entropy.tex
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+\section{An entropy bound on contacts}\label{sec:entropy}
+
+Continue to assume coexistence in the coordinate direction, and use the two
+tapes and their common blocks from Section~\ref{sec:bridges}.  We prove that
+the contact count in \eqref{eq:contact-count} satisfies
+\(m(J)=O(J/\log J)\).  The argument compares a portion of the actual tapes
+with independent exact-cut bridges.  Contacts are unlikely for the independent
+bridges, whereas the information needed to pass from those bridges to the
+actual tapes can be bounded by the growth of a lateral-displacement entropy.
+We adapt the comparison and contact argument of
+\cite[Proposition~5.1 and Lemmas~5.2--5.3]{OpenAI2026}, using first proximity
+arrivals to separate departure rows; all estimates needed here are proved
+below.
+All constants in this section may depend on the fixed environment law and
+the fixed marking parameters, but not on the scales \(n\) or \(J\).
+
+\subsection{Entropy of the lateral displacement}
+
+For a countable random variable \(V\) with masses \(p(v)\), its Shannon
+entropy \cite{Shannon1948} is
+\[
+ \Ent(V)=\sum_v p(v)\log\frac1{p(v)}.
+\]
+All logarithms in this section are natural.  For probability laws \(P,Q'\)
+on the same countable set, their relative entropy
+\cite{KullbackLeibler1951} is
+\[
+ D(P\Vert Q')=\sum_v P(v)\log\frac{P(v)}{Q'(v)}.
+\]
+A zero mass contributes zero, and a positive \(P\)-mass at a zero
+\(Q'\)-mass gives infinite relative entropy.  Conditional entropy averages
+the entropy of the conditional law.  Conditional mutual information is
+defined by
+\[
+ I(U;V\mid C)
+ =\sum_c P(C=c)
+ D\bigl(P_{U,V\mid C=c}\,\Vert\,
+          P_{U\mid C=c}\otimes P_{V\mid C=c}\bigr).
+\]
+We use these definitions even when \(U\) and \(V\) are entire finite path
+lists, whose entropies need not be finite.
+
+The log-sum inequality gives nonnegativity of relative entropy and its
+decrease under taking a function of the variables.  Factoring joint masses
+gives the chain rules.  In particular, comparison with a conditional product
+law, while keeping the marginal of \(C\) fixed, gives
+\begin{align}
+ &D\bigl(P_{C,U,V}\,\Vert\,
+          P_C Q'_{U\mid C}Q'_{V\mid C}\bigr)\notag\\
+ &\qquad=I(U;V\mid C)
+   +\mathbb E\,D(P_{U\mid C}\Vert Q'_{U\mid C})
+   +\mathbb E\,D(P_{V\mid C}\Vert Q'_{V\mid C}).
+ \label{eq:conditional-product-entropy}
+\end{align}
+These relative-entropy identities hold on countable spaces: one may first
+use finite partitions and then take increasing limits, or factor the
+conditional masses directly, since the negative parts of relative-entropy
+sums are integrable.  When the variable whose entropy is being subtracted
+has finite entropy, the same rules give the usual identity between an
+entropy difference and mutual information.  Thus conditioning reduces
+entropy in all the finite-entropy expressions below.
+
+Put \(q=d-1\), and recall the iid lateral increments \(S_i'\in\ZZ^q\) of
+the common blocks.  Define
+\[
+ h_j'=\Ent(S_1'+\cdots+S_j'),\qquad j\ge1.
+\]
+Equation~\eqref{eq:lateral-moment} gives \(\mathbf E|S_1'|<\infty\).
+Consequently
+\begin{equation}
+ h_j'\le C+q\log(j+1).
+ \label{eq:lateral-entropy-bound}
+\end{equation}
+Indeed, on \(\ZZ^q\) compare the displacement law with the probability
+weights
+\[
+ r_j(x)=c_j^{-1}\exp\bigl(-|x|/(j+1)\bigr),
+ \qquad
+ c_j=\sum_{x\in\ZZ^q}\exp\bigl(-|x|/(j+1)\bigr)
+       \le C(j+1)^q.
+\]
+The expected negative logarithm of \(r_j\) is bounded by
+\(\log c_j+j\mathbf E|S_1'|/(j+1)\).  Nonnegativity of relative entropy,
+or its finite-partition version followed by a limit, bounds the Shannon
+entropy by this finite quantity.  This proves both finiteness and
+\eqref{eq:lateral-entropy-bound}.  The sequence \(h_j'\) is nondecreasing:
+conditioning \(S_1'+\cdots+S_{j+1}'\) on the independent last summand
+gives entropy \(h_j'\), while removing that conditioning can only increase
+entropy.
+
+\subsection{Random chunks and their reference law}
+
+Fix an integer \(n\ge1\).  Independently of the tapes, choose independent
+integer counts with laws
+\begin{equation}
+ \begin{split}
+ I_0,I_1,I_3&\text{ uniform on }\{n,\ldots,2n-1\},\\
+ I_2&\text{ uniform on }\{7n,\ldots,8n-1\}.
+ \end{split}
+ \label{eq:chunk-counts}
+\end{equation}
+Discard a buffer of \(I_0\) common blocks, and record its lateral gap
+\[
+ d_0=\sum_{j=1}^{I_0}S_j'.
+\]
+Divide the next blocks into three consecutive chunks containing
+\(I_1,I_2,I_3\) blocks.  For chunk \(i\), let \(A_i,B_i\) be its positive
+and negative slab lists, both in downward tape order, and let \(H_i\) be
+their common width.  Set
+\[
+ \mathcal A=(A_1,A_2,A_3),\qquad
+ \mathcal B=(B_1,B_2,B_3),\qquad H=(H_1,H_2,H_3).
+\]
+The lists contain relative marked words; they do not include their spatial
+placements.  The three chunks are independent of one another and of
+\((I_0,d_0)\).  To see this, condition on the four counts: the chunks and
+the buffer then use disjoint portions of an iid common-block sequence;
+averaging over the independent counts preserves this product structure.
+
+Let \(X_A,X_B\in\ZZ^q\) be the sums of the lateral displacements of all
+words in \(\mathcal A,\mathcal B\), respectively, measured in the words'
+original chronological directions.  Define
+\[
+ Z=d_0+X_A,\qquad T=I_0+I_1+I_2+I_3.
+\]
+Write \(P^{(n)}\) for the resulting law of
+\((H,Z,\mathcal A,\mathcal B)\).  The widths, the buffer displacement,
+and the chunk displacements have finite first moments by
+Proposition~\ref{prop:common-blocks}.  The lattice comparison just used
+therefore gives finite entropy for
+\(H,d_0,X_A,X_B,Z\), with \(n\) fixed.
+
+Each tuple \((H,Z,\mathcal A,\mathcal B)\) determines a pair of placed
+paths as follows.  Put \(N=H_1+H_2+H_3\).  Start the positive path at
+height zero and lateral position zero, and concatenate chunks \(3,2,1\),
+reversing the order of each positive list.  Every word is still traversed
+in its original chronological direction.  Start the negative path at
+height \(N-1\) and lateral position \(Z\), and concatenate chunks
+\(1,2,3\) in downward order.  Let \(E_{\rm mid}\) be the event that a
+positive departure in chunk \(2\) is within \(\ell^\infty\)-distance
+\(R\) of a negative departure in chunk \(2\).
+
+Under \(P^{(n)}\), these are exactly the original three chunks after a
+common translation.  At the top of the chunks their lateral gap was
+\(d_0\), and the positive path gains displacement \(X_A\) from bottom to
+top.  The negative starting coordinate relative to the positive bottom is
+therefore \(d_0+X_A=Z\).  The one-level offset in their heights is the
+offset in the tape construction.  Figure~\ref{fig:opposed-chunks} shows
+this placement and the distinction between tape order and chronological
+order.
+
+\input{opposed-chunks-figure}
+
+Define a second law \(Q^{(n)}\) on these same variables.  Keep the
+\(P^{(n)}\)-marginal of \((H,Z)\); conditional on \((H,Z)\), sample the
+six lists independently, with
+\[
+ A_i\sim\mathcal B_{H_i}^+,
+ \qquad B_i\sim\mathcal B_{H_i}^-,
+ \qquad i=1,2,3.
+\]
+These exact-cut bridge laws are defined by \eqref{eq:bridge-law}.
+Since \(H_i\ge Mn\), their normalizing constants have the positive lower
+bounds supplied by \eqref{eq:bridge-renewal}.
+Thus the reference law keeps the three widths and the negative starting
+position, and independently resamples the six relative lists.  The same
+placement rule defines \(E_{\rm mid}\) under both laws.
+
+\begin{proposition}[Entropy comparison]\label{prop:entropy-comparison}
+For every integer \(n\ge1\), the actual chunk law and the reference law
+above satisfy
+\begin{equation}
+ D(P^{(n)}\Vert Q^{(n)})\le C+h_{14n}'-h_n'.
+ \label{eq:entropy-comparison}
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+All entropies and expectations in this proof refer to the original
+tape-and-count experiment.
+First omit \(Z\), and let \(D_0\) denote the relative entropy between
+the resulting marginals of \(P^{(n)}\) and \(Q^{(n)}\).  A pair of lists
+of common width \(h\) determines the number of its common renewals,
+counting the terminal width \(h\) and excluding zero, and hence determines
+its parsing into common blocks.  If that number is in the count window
+for chunk \(i\), the actual unconditioned probability of the list pair is
+\[
+ \frac1n
+ \prod_{\gamma\in A_i}W(\gamma)
+ \prod_{\gamma\in B_i}W(\gamma).
+\]
+If the number is outside the window, the probability is zero.  This follows
+from the common-block construction and the slab weights: the specified
+pair of lists determines exactly one allowed count.  In contrast, the
+product of the two bridge laws at width \(h\) has the same product of
+weights divided by \(\bar u_h^+\bar u_h^-\).  Thus, conditional on
+\(H_i=h\), the density relative to the independent bridges is, on the
+actual support, exactly
+\begin{equation}
+ \frac{\bar u_h^+\bar u_h^-}
+      {nP^{(n)}(H_i=h)}.
+ \label{eq:chunk-pair-density}
+\end{equation}
+Independence of the three actual chunks now gives
+\begin{align}
+ I(\mathcal A;\mathcal B\mid H)
+ &\le D_0\notag\\
+ &=\sum_{i=1}^3
+   \left(\Ent(H_i)-\log n
+          +\mathbb E\log(\bar u_{H_i}^+\bar u_{H_i}^-)\right)
+ \le\sum_{i=1}^3(\Ent(H_i)-\log n)
+ \le C.
+ \label{eq:chunk-entropy-cost}
+\end{align}
+The information inequality follows from
+\eqref{eq:conditional-product-entropy}.  For the last bound, use
+\(\mathbb E H_i=\mathbf E K_1\,\mathbb E I_i=O(n)\) and compare the
+law of \(H_i\) with weights proportional to \(e^{-h/n}\) on the
+nonnegative integers.  The normalizer is \(O(n)\), so
+\(\Ent(H_i)\le\log n+C\).
+
+Reintroducing \(Z\) costs its conditional information with the lists:
+\begin{equation}
+ \begin{split}
+ D(P^{(n)}\Vert Q^{(n)})
+ &=D_0+I(Z;\mathcal A,\mathcal B\mid H)\\
+ &=D_0+\Ent(Z\mid H)-\Ent(d_0).
+ \end{split}
+ \label{eq:buffer-information}
+\end{equation}
+Here the lists determine \(X_A\), and the buffer is independent of the
+lists and widths; conditional on those data, \(Z\) is a translate of
+\(d_0\).  Likewise, given \((\mathcal A,H)\), the variable \(Z\) is
+conditionally independent of \(\mathcal B\).  Conditional data processing
+and \eqref{eq:chunk-entropy-cost} imply
+\[
+ I(Z;\mathcal B\mid H)
+ \le I(\mathcal A;\mathcal B\mid H)\le D_0.
+\]
+Every variable whose entropy is evaluated in the next calculation has
+finite entropy; the conditioning lists need not:
+\begin{align}
+ \Ent(T,Z+X_B)
+ &\ge\Ent(T,Z+X_B\mid\mathcal B,H)
+   =\Ent(T,Z\mid\mathcal B,H)\notag\\
+ &\ge\Ent(Z\mid H)-D_0
+       +\Ent(T\mid Z,\mathcal A,\mathcal B,H)\notag\\
+ &=\Ent(Z\mid H)-D_0+\Ent(I_0\mid d_0).
+ \label{eq:buffer-entropy-lower}
+\end{align}
+For the final identity, each pair of chunk lists determines its common
+block count, hence \(I_1,I_2,I_3\).  The lists and \(Z\) also determine
+\(d_0=Z-X_A\).  Independence from the buffer then identifies the
+remaining conditional entropy of \(T\) with \(\Ent(I_0\mid d_0)\).
+
+Subtract \(\Ent(I_0,d_0)=\Ent(d_0)+\Ent(I_0\mid d_0)\) from
+\eqref{eq:buffer-entropy-lower} and use
+\eqref{eq:buffer-information}.  The result is
+\begin{equation}
+ D(P^{(n)}\Vert Q^{(n)})
+ \le 2D_0+\Ent(T,Z+X_B)-\Ent(I_0,d_0).
+ \label{eq:two-entropy-costs}
+\end{equation}
+Now \(Z+X_B\) is the sum of the lateral increments through the first
+\(T\) common blocks.  Both \(T\) and \(I_0\) are independent of the
+underlying block sequence.  It follows that
+\begin{align*}
+ \Ent(T,Z+X_B)-\Ent(I_0,d_0)
+ &=\Ent(T)+\mathbb E h_T'-\log n-\mathbb E h_{I_0}'\\
+ &\le C+h_{14n}'-h_n'.
+\end{align*}
+Indeed, \(T<14n\), \(I_0\ge n\), and \(T\) has at most \(4n\)
+possible values.  Monotonicity of \(h_j'\) proves the displayed bound.
+Finally, the term \(2D_0\) is bounded uniformly in \(n\) by
+\eqref{eq:chunk-entropy-cost}, proving
+\eqref{eq:entropy-comparison}.
+\end{proof}
+
+\subsection{Contacts under the reference law}
+
+The entropy comparison will bound the contact count once we show that
+\(E_{\rm mid}\) contains the relevant actual contacts and is unlikely
+under the reference law.
+For every realization of the counts, a contact in a positive common block
+with index in \((4n,8n]\) implies \(E_{\rm mid}\).  In fact, chunk \(2\)
+starts after at most \(4n-2\) common blocks and ends at or beyond block
+\(9n\).  The negative departure realizing a contact is in the same or a
+neighboring block, by the common-block geometry in
+Section~\ref{sec:bridges}; all those blocks remain inside chunk \(2\).
+Consequently
+\begin{equation}
+ \mathbf P\{\text{a contact in a positive block indexed in }(4n,8n]\}
+ \le P^{(n)}(E_{\rm mid}).
+ \label{eq:middle-contact-inclusion}
+\end{equation}
+
+\begin{proposition}[Reference contact bound]\label{prop:reference-contact}
+For every integer \(n\ge1\),
+\begin{equation}
+ Q^{(n)}(E_{\rm mid})\le\delta_n,
+ \qquad
+ \delta_n=\min\{1,Cn(F_+(n)+F_-(n))\}.
+ \label{eq:reference-contact}
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+Fix \((H,Z)\) in the support of its retained marginal.  Conditional on
+these data, all six reference lists are independent.  Reversing the
+positive list order preserves its bridge law by
+\eqref{eq:bridge-law}.  Retain the positive path formed by chunks \(3\)
+and \(2\), and call it \(\alpha\).  It runs from the origin through its
+first hit of height \(N-H_1\).  For the negative path formed by chunks
+\(1\) and \(2\), retain only the prefix through its first arrival within
+distance \(R\) of \(\Dep(\alpha)\); call the retained prefix \(\beta\)
+and its terminal site \(y'\).  On \(E_{\rm mid}\) such an arrival exists
+no later than a negative departure in chunk \(2\), and hence before the
+two negative chunks terminate.
+This first proximity need not itself join the two middle chunks; the
+outer chunk widths will nevertheless put both visits at progress at
+least \(n\).
+
+The geometry puts this encounter away from both starting sides.  The two
+negative chunks stay at or below \(N-1\) and end at their first hit of
+\(H_3-1\).  Therefore
+\[
+ H_3\le y_1'\le N-1.
+\]
+Choose a departure \(y\) of \(\alpha\) within distance \(R\) of \(y'\).
+Since \(\alpha\) ends on its first hit of \(N-H_1\),
+\begin{equation}
+ H_3-R\le y_1\le N-H_1-1,
+ \qquad N-1-y_1'\ge H_1-R.
+ \label{eq:reference-contact-heights}
+\end{equation}
+The inequalities \(H_1,H_3\ge Mn\) and \(M>R+1\) show that both visits
+occur at or after the respective path has reached signed progress \(n\)
+from its start.  Up to these visits neither path has hit either absolute
+height \(-1\) or \(N\).
+
+We next express the probability of these prefixes using two raw walks in
+one environment.  The unused positive chunk \(1\) and negative chunk
+\(3\) integrate to one.  The four remaining bridge normalizations,
+\[
+ \bar u_{H_3}^+\bar u_{H_2}^+
+ \bar u_{H_1}^-\bar u_{H_2}^-,
+\]
+have a fixed positive lower bound by \eqref{eq:bridge-renewal}.  Within
+each sign, concatenating the two bridges gives the product of their raw
+weights, by the separation of active departures established in
+Section~\ref{sec:bridges}.
+
+For fixed \(\alpha\) and a retained negative prefix \(\beta\), sum over
+all allowed negative completions to the terminal height \(H_3-1\).
+Their total raw weight is at most \(W(\beta)\): full words to that height
+end on its first hit, so the corresponding continuation events are
+disjoint.  Nor is there multiplicity from the lists.  The intermediate
+chunk boundary is a first hit of its prescribed height, and the bridge
+parsing into slabs is unique.  The same uniqueness holds for the positive
+concatenation.  Thus, after dropping any further completion restrictions,
+the conditional contact probability is at most a fixed constant times a
+sum of products
+\begin{equation}
+ W(\alpha)W(\beta)
+ =\mathbb E_Q\bigl[p_\omega(\alpha)p_\omega(\beta)\bigr].
+ \label{eq:reference-prefix-transfer}
+\end{equation}
+The equality holds for each retained pair: every departure of \(\beta\)
+precedes its first proximity arrival and is therefore at distance strictly
+greater than \(R\) from every departure of \(\alpha\).  The terminal
+arrival \(y'\) uses no departure row.  Lemma~\ref{lem:separated-weights}
+then applies, with repeated departures within either path retained in
+their respective weights.
+
+The right side of \eqref{eq:reference-prefix-transfer} is the probability
+of the specified prefixes for two raw marked walks, started at \(0\) and
+\((N-1,Z)\), independently conditional on a shared environment of law
+\(Q\).  Summing these probabilities introduces no overcounting.  A full
+positive trajectory has at most one prefix through its first hit of the
+fixed height \(N-H_1\); that prefix determines \(\Dep(\alpha)\), and a
+full negative trajectory has at most one prefix through its first arrival
+within distance \(R\) of that set.
+
+Only in this shared-environment experiment do we classify an encounter by
+quenched exit probabilities.  For \(0\le x_1\le N-1\), define
+\[
+ q_x^\omega=P_{x,\omega}(T_{-1}<T_N),
+\]
+where \(T_a\) here is the first hit of absolute first-coordinate height
+\(a\).  Slab exit is almost surely finite, so
+\(1-q_x^\omega=P_{x,\omega}(T_N<T_{-1})\).  At the sites in
+\eqref{eq:reference-contact-heights}, if \(q_y^\omega\ge1/2\), the
+positive walk has visited, after reaching progress \(n\), a site with
+probability at least \(1/2\) of exiting through its starting side's
+barrier \(-1\).  If \(q_y^\omega<1/2\), join \(y'\) to \(y\) by a
+coordinate path of at most \(dR\) steps inside the slab.  Uniform
+ellipticity gives
+\[
+ 1-q_{y'}^\omega
+ \ge\kappa^{dR}(1-q_y^\omega)
+ \ge\kappa^{dR}/2.
+\]
+In that case the negative walk has visited, after reaching its signed
+progress \(n\), a site with probability at least \(\kappa^{dR}/2\) of
+exiting through its own starting side's barrier \(N\).
+
+For either sign \(\sigma\), consider the raw marginal event that, after
+its first hit of signed progress \(n\) and before slab exit, the walk
+visits a site whose quenched probability of exit through the starting
+side's barrier is at least \(c=\kappa^{dR}/2\).  With the environment
+fixed, stop at the first such visit.  The probability of the indicated
+exit after this visit is at least \(c\).  Integrating the resulting
+inequality over environments bounds the visit probability by
+\begin{equation}
+ c^{-1}(u_n^\sigma-u_N^\sigma).
+ \label{eq:reference-visit-bound}
+\end{equation}
+Indeed, from each of the two starting sites, the barriers have signed
+relative heights \(-1,N\).  The difference
+\(u_n^\sigma-u_N^\sigma\) is exactly the raw probability of hitting
+signed height \(n\) and then exiting at \(-1\) before \(N\): a path
+reaching \(N\) must first reach \(n\), and exit from the finite slab is
+almost surely finite.  Stationarity identifies the negative walk's
+translated raw probabilities with \(u_n^-,u_N^-\).
+
+Every retained pair belongs to at least one of these two raw visit events.
+Combining \eqref{eq:reference-prefix-transfer}--\eqref{eq:reference-visit-bound}
+therefore gives
+\[
+ Q^{(n)}(E_{\rm mid}\mid H,Z)
+ \le C\sum_{\sigma\in\{+,-\}}(u_n^\sigma-u_N^\sigma)
+ \le CN(F_+(n)+F_-(n)),
+\]
+where the last inequality is Lemma~\ref{lem:width-tail}.  This estimate
+uses the raw marginal visit probabilities after the prefix transfer; it
+does not condition an exit probability on the contact event.  Finally,
+the retained marginal has
+\[
+ \mathbb E N=\mathbf E K_1\,\mathbb E(I_1+I_2+I_3)=O(n).
+\]
+Averaging over \((H,Z)\) and also using the trivial probability bound
+one proves \eqref{eq:reference-contact}.
+\end{proof}
+
+\subsection{Summing over scales}
+
+\begin{proposition}[Contact upper bound]\label{prop:contact-upper}
+For the two independent opposed tapes under coordinate coexistence, the
+contact count defined in \eqref{eq:contact-count} satisfies
+\begin{equation}
+ m(J)=O\!\left(\frac{J}{\log J}\right)
+ \qquad(J\to\infty).
+ \label{eq:contact-upper}
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+Apply data processing in Proposition~\ref{prop:entropy-comparison} to
+the indicator of \(E_{\rm mid}\).  If \(p=P^{(n)}(E_{\rm mid})\) and
+\(q_n=Q^{(n)}(E_{\rm mid})\), binary relative entropy satisfies
+\[
+ D(\operatorname{Bern}(p)\Vert\operatorname{Bern}(q_n))
+ \ge p\log(1/q_n)-\log2.
+\]
+Together with Proposition~\ref{prop:reference-contact}, this yields, when
+\(\delta_n>0\),
+\begin{equation}
+ P^{(n)}(E_{\rm mid})\log(1/\delta_n)
+ \le C+h_{14n}'-h_n'.
+ \label{eq:contact-entropy-price}
+\end{equation}
+If \(\delta_n=0\), then \(q_n=0\), and the finite relative entropy in
+\eqref{eq:entropy-comparison} forces
+\(P^{(n)}(E_{\rm mid})=0\).  The same observation handles \(q_n=0\)
+when \(\delta_n>0\).
+
+For \(l\ge2\), take \(n=2^{l-2}\).  The width-tail summability in
+Lemma~\ref{lem:width-tail} implies
+\[
+ \sum_{l\ge2}\delta_{2^{l-2}}<\infty.
+\]
+Among \(2\le l\le J\), at most \(O(\sqrt J)\) indices can have
+\(\delta_{2^{l-2}}>J^{-1/2}\).  At every other index with positive
+\(\delta_{2^{l-2}}\), the logarithm in
+\eqref{eq:contact-entropy-price} is at least \(\tfrac12\log J\).
+The corresponding entropy differences have a telescoping bound:
+\begin{align*}
+ \sum_{l=2}^J
+   \bigl(h_{14\cdot2^{l-2}}'-h_{2^{l-2}}'\bigr)
+ &\le\sum_{l=2}^J
+   \bigl(h_{2^{l+2}}'-h_{2^{l-2}}'\bigr)\\
+ &\le4h_{2^{J+2}}'=O(J),
+\end{align*}
+by monotonicity and \eqref{eq:lateral-entropy-bound}.  Since
+\((4n,8n]=(2^l,2^{l+1}]\), use
+\eqref{eq:middle-contact-inclusion}, sum
+\eqref{eq:contact-entropy-price} over the remaining indices, and bound
+each exceptional probability by one.  Adding the initial scale gives
+\[
+ m(J)\le 1+O(\sqrt J)+O(J/\log J)=O(J/\log J),
+\]
+as claimed.
+\end{proof}
diff --git a/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/build/manuscript/history.tex b/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/build/manuscript/history.tex
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+++ b/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/build/manuscript/history.tex
@@ -0,0 +1,60 @@
+\subsection{Background and relation to earlier work}
+
+Directional zero--one laws are a basic part of the transience theory of
+random walks in random environments. Kalikow's work
+\cite{Kalikow1981}, together with the regeneration formulation of
+Sznitman and Zerner \cite[Lemma~1.1]{SznitmanZerner1999}, established
+the sign-union law for uniformly elliptic iid environments,
+\[
+ P_0(A_\ell\cup A_{-\ell})\in\{0,1\}.
+\]
+This leaves open which sign is chosen when the union has probability
+one. In two dimensions, Zerner and Merkl \cite{ZernerMerkl2001}
+proved the full directional zero--one law for iid elliptic environments;
+Zerner \cite{Zerner2007} subsequently gave a revised proof. The planar
+arguments exploit intersections of paths. The question in higher
+dimensions requires a different way to exclude two possible escape signs.
+
+Regeneration is a central tool for this problem and for laws of large
+numbers. Sznitman and Zerner
+\cite[Theorem~1.4 and Corollary~1.5]{SznitmanZerner1999} constructed
+independent increments after suitable record times in an iid environment.
+Results on asymptotic directions further distinguish a deterministic
+axis from the choice of a sign on that axis: see Simenhaus
+\cite{Simenhaus2007} and Drewitz and Ram\'irez
+\cite{DrewitzRamirez2010}. A limiting direction or a conditional law of
+large numbers does not by itself establish a directional zero--one law.
+
+For dependent environments, Comets and Zeitouni
+\cite{CometsZeitouni2004} used environment-independent forced steps to
+construct approximate regeneration under mixing assumptions, with
+additional hypotheses yielding a law of large numbers. That forcing
+device is an antecedent of the marked scripts used here. Rassoul-Agha
+\cite{RassoulAgha2005} derived a law of large numbers under Gibbs mixing
+assumptions and a directional zero--one hypothesis. Guo
+\cite{Guo2014} proved conditional velocity results under a mixing
+condition that includes finite-range-dependent environments. These
+conclusions concern velocity and do not determine the probabilities of
+the two directional escape events. Exact finite-range independence is
+also stronger than decay of local correlations: Bramson, Zeitouni, and
+Zerner \cite[Theorem~3]{BramsonZeitouniZerner2006} constructed stationary,
+uniformly elliptic, polynomially mixing environments in dimension at
+least three with positive probability of escape in each of two opposite
+directions.
+
+The companion article \cite[Theorem~1.1]{OpenAI2026} proves the iid
+directional zero--one law under strict ellipticity: every transition
+probability is positive, but no uniform lower bound is required. Here we
+extend the uniformly elliptic case to finite-range dependence. The proof
+adapts the radius-and-contact strategy developed in
+\cite[Sections~2--7]{OpenAI2026}. In that strategy, coexistence first
+forces integrability of spatial regeneration radii. Two independent
+sequences of regeneration pieces are then placed in an opposed arrangement,
+and entropy and endpoint-posterior estimates give incompatible bounds on
+their contacts. We retain this architecture and prove all required
+estimates for the present environment law. The finite-range extension
+depends on keeping track of the rows actually used by a marked path,
+replacing intersections by proximity within the dependence range, and
+transferring only the stopped path laws justified by separated rows.
+No conditional density assumption, independent-label representation,
+positive speed, or moment bound on regeneration times is imposed.
diff --git a/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/build/manuscript/main.tex b/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/build/manuscript/main.tex
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+++ b/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/build/manuscript/main.tex
@@ -0,0 +1,197 @@
+\documentclass[11pt]{article}
+\usepackage[margin=1in]{geometry}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs}
+\usepackage{microtype,enumitem,xcolor,tikz}
+\usetikzlibrary{arrows.meta,positioning,calc,decorations.pathreplacing,patterns}
+\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref}
+\hypersetup{pdftitle={A directional zero-one law for finite-range-dependent random environments},pdfauthor={OpenAI}}
+\setlength{\emergencystretch}{2em}
+\setlist{itemsep=3pt,topsep=5pt}
+\numberwithin{equation}{section}
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{proposition}[theorem]{Proposition}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Corollary}
+\theoremstyle{definition}
+\newtheorem{definition}[theorem]{Definition}
+\theoremstyle{remark}
+\newtheorem{remark}[theorem]{Remark}
+\newcommand{\RR}{\mathbb R}
+\newcommand{\ZZ}{\mathbb Z}
+\newcommand{\EE}{\mathbb E}
+\newcommand{\PP}{\mathbb P}
+\newcommand{\one}{\mathbf 1}
+\newcommand{\abs}[1]{\lvert #1\rvert}
+\newcommand{\norm}[1]{\lVert #1\rVert}
+\newcommand{\Ent}{\mathscr H}
+\DeclareMathOperator{\Dep}{Dep}
+\DeclareMathOperator{\dist}{dist}
+\title{A directional zero--one law for\newline finite-range-dependent random environments}
+\author{OpenAI}
+\date{October 5, 2026}
+\begin{document}
+\maketitle
+\begin{abstract}
+We prove a directional zero--one law for uniformly elliptic nearest-neighbor
+random walks in stationary, ergodic, finite-range-dependent environments on
+$\ZZ^d$, $d\ge3$. For every fixed nonzero real direction, the probability
+of escape in that direction, averaged over the environment, is either zero
+or one. Finite-range dependence is imposed on the full transition rows:
+collections of rows at distance greater than a fixed range are independent,
+including collections indexed by infinite deterministic sets.
+\end{abstract}
+
+\section{Introduction}\label{sec:introduction}
+
+A random walk in a random environment encounters the same transition
+probabilities whenever it returns to a site. This persistence distinguishes
+the model from a walk whose transition probabilities are resampled at each
+step. Directional transience asks whether the walk eventually moves beyond
+every hyperplane perpendicular to a fixed direction. The directional
+zero--one question is whether this escape event can have probability
+strictly between zero and one after averaging over the environment.
+
+\subsection{Model and main theorem}
+
+Fix $d\ge3$ and let $U=\{\pm e_1,\ldots,\pm e_d\}$ be the coordinate
+unit steps. Set
+\[
+ \Delta_U=\left\{p\in[0,1]^U:\sum_{e\in U}p(e)=1\right\},
+ \qquad \Omega=\Delta_U^{\ZZ^d},
+\]
+with its product Borel sigma-field. An environment $\omega\in\Omega$
+assigns the row $\omega(x,\cdot)$ to each site $x$. Let $Q$ be a Borel
+probability law on $\Omega$ satisfying the following assumptions.
+
+\begin{enumerate}[label=\textup{(\roman*)}]
+\item \emph{Stationarity and ergodicity.}
+For $z\in\ZZ^d$, define
+$(\theta_z\omega)(x,e)=\omega(x+z,e)$. The law $Q$ is invariant under
+every $\theta_z$, and every event invariant modulo null sets under all
+these translations has probability zero or one.
+\item \emph{Uniform ellipticity.}
+For some deterministic $\kappa>0$,
+\[
+ Q\bigl(\omega(x,e)\ge\kappa
+          \text{ for every }x\in\ZZ^d,\ e\in U\bigr)=1.
+\]
+\item \emph{Finite-range dependence.}
+For some deterministic integer $R\ge0$, the sigma-fields
+\[
+ \mathcal F_A=\sigma\bigl(\omega(x,e):x\in A,\ e\in U\bigr),
+ \qquad \mathcal F_B=\sigma\bigl(\omega(x,e):x\in B,\ e\in U\bigr)
+\]
+are independent whenever the nonempty deterministic sets
+$A,B\subseteq\ZZ^d$ satisfy
+\begin{equation}\label{eq:finite-range}
+ \dist_\infty(A,B):=
+ \inf_{a\in A,\,b\in B}\norm{a-b}_\infty>R.
+\end{equation}
+This condition includes infinite sets.
+\end{enumerate}
+
+For fixed $\omega$, let $P_{x,\omega}$ be the law of the Markov chain
+started at $x$ with transitions
+\[
+ P_{x,\omega}(X_{n+1}=y+e\mid X_n=y)=\omega(y,e).
+\]
+Its annealed law is $P_x=\int P_{x,\omega}\,Q(d\omega)$.
+For each fixed $\ell\in\RR^d\setminus\{0\}$, write
+\[
+ A_\ell=\{X_n\cdot\ell\longrightarrow+\infty\}.
+\]
+
+\begin{theorem}[Directional zero--one law]\label{thm:main}
+Under assumptions \textup{(i)--(iii)}, for every fixed
+$\ell\in\RR^d\setminus\{0\}$,
+\[
+ P_0(A_\ell)\in\{0,1\}.
+\]
+\end{theorem}
+
+The dependence range and ellipticity constant may vary with the law $Q$.
+The direction may be irrational; it is fixed before taking probability.
+The proof uses stationarity and finite-range independence directly, without
+an additional appeal to ergodicity. Its independence arguments concern
+separated sigma-fields of rows, rather than any representation of the
+environment in terms of independent labels.
+
+\input{history}
+
+\subsection{Proof strategy and additional ingredients}
+
+The argument rules out coexistence of escape in opposite directions.
+For a fixed direction, we construct independent regeneration pieces:
+finite path segments ending at new record heights that the subsequent
+path never undercuts. This construction proves that positive probability
+of escape in that direction makes the two escape signs exhaustive.
+
+Finite-range dependence creates two obstacles to the usual iid reasoning.
+Disjoint paths can use correlated rows, and an observed path can change the
+law of nearby unobserved rows. We split each transition into two marked
+choices of constant weight and a remaining choice. A prescribed sequence
+of the constant-weight choices moves beyond the dependence range without
+using any further row information. This leaves a region of genuinely
+independent rows for a continuation that stays beyond its endpoint.
+The prescribed sequences cannot overlap, so the regeneration pieces have
+an unambiguous concatenation law. Conditioning at their endpoints is
+proved by enumeration of finite prefixes, since those endpoints are
+selected using the future.
+
+Under coexistence, the regeneration pieces have finite mean spatial
+radius. The proof uses a large transverse excursion to splice in an
+oppositely directed continuation, separated from the observed path by a
+tilted hyperplane and a short marked connector. Repeating the construction
+would place uniformly positive probability in disjoint windows for the
+walk's final directional supremum. This contradiction gives the radius
+bound. It yields opposite deterministic limiting rays and reduces an
+arbitrary real direction to a coordinate direction.
+
+For that coordinate, place independent positive and negative sequences
+of regeneration pieces in an opposed arrangement. A contact means that
+their departure sites come within distance $R$. Common regeneration
+heights divide the arrangement into iid pairs of lists. Finite spatial
+moments bound the entropy of their cumulative lateral displacement.
+Comparing long chunks with independent bridges---concatenated regeneration
+pieces conditioned on prescribed total widths---then shows that among
+the first $J$ dyadic intervals of block indices, the expected number
+containing a contact is $O(J/\log J)$.
+
+The opposite estimate uses endpoint alignment. Start a negative path,
+conditioned to stay below its starting height, just below a positive
+bridge's endpoint, and search for their first contact along the positive
+path. Posterior probabilities for a finite set of possible lateral
+endpoints form a vector of martingales. The sum of their running
+maxima has an exponential tail on the scale of the logarithm of the number
+of endpoints. The marked segment ending an observed prefix at a lower
+height permits comparison with a fresh path law up to its first exit
+from a specified slab. The posterior
+functions retain their original meaning; only the stopped path marginal
+is transferred. Combining this estimate with nearby-point comparisons of
+quenched exit probabilities forces $\Omega(J/\log\log J)$ contact scales,
+contradicting the entropy bound.
+
+The finite-range adaptations are the marked separation, killing on entry
+to neighborhoods of radius $R$, and comparison of stopped posterior
+processes across the resulting gaps between active rows.
+We prove in full the vector-martingale estimate of
+\cite[Lemma~6.1]{OpenAI2026}; it controls the cost of selecting a path
+through an endpoint constraint.
+
+Section~\ref{sec:cuts} establishes regeneration and mean widths.
+Section~\ref{sec:radius} proves spatial integrability and reduces to
+coordinate coexistence. Sections~\ref{sec:bridges} and~\ref{sec:entropy}
+construct the opposed arrangement and its contact upper bound.
+Section~\ref{sec:posteriors} proves the endpoint estimate, and
+Section~\ref{sec:contact-lower} obtains the contradictory lower bound.
+
+\input{cuts}
+\input{radius}
+\input{bridges}
+\input{entropy}
+\input{posteriors}
+\input{conclusion}
+\input{../references/references}
+\end{document}
diff --git a/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/build/manuscript/opposed-chunks-figure.tex b/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/build/manuscript/opposed-chunks-figure.tex
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@@ -0,0 +1,61 @@
+\begin{figure}[t]
+\centering
+\begin{tikzpicture}[x=1cm,y=.86cm,>=Stealth,
+ every node/.style={font=\small},
+ posword/.style={draw=blue!60!black,fill=blue!6,line width=.6pt},
+ negword/.style={draw=orange!65!black,fill=orange!7,line width=.6pt},
+ guide/.style={densely dashed,gray!65,line width=.4pt}]
+ \node[align=center] at (3.1,8.05)
+   {Positive tape\\list order: $A_1,A_2,A_3$};
+ \node[align=center] at (7.0,8.05)
+   {Negative tape\\list order: $B_1,B_2,B_3$};
+ \draw[posword] (2.2,.7) rectangle (4,2.7);
+ \draw[posword,fill=blue!15] (2.2,2.7) rectangle (4,4.9);
+ \draw[posword] (2.2,4.9) rectangle (4,6.7);
+ \draw[negword] (6.1,.42) rectangle (7.9,2.42);
+ \draw[negword,fill=orange!17] (6.1,2.42) rectangle (7.9,4.62);
+ \draw[negword] (6.1,4.62) rectangle (7.9,6.42);
+ \node at (3.1,1.7) {$A_3$};
+ \node at (3.1,3.8) {$A_2$};
+ \node at (3.1,5.8) {$A_1$};
+ \node at (7,1.42) {$B_3$};
+ \node at (7,3.52) {$B_2$};
+ \node at (7,5.52) {$B_1$};
+ \draw[->,very thick,blue!60!black] (4.35,1.05)--(4.35,6.32);
+ \draw[->,very thick,orange!65!black] (5.75,6.07)--(5.75,.77);
+ \node[rotate=90,font=\footnotesize,fill=white,inner sep=2pt]
+   at (4.35,3.6) {chronological order};
+ \node[rotate=90,font=\footnotesize,fill=white,inner sep=2pt]
+   at (5.75,3.4) {chronological order};
+ \foreach \yy/\ll in {.7/0,2.7/H_3,4.9/{N-H_1},6.7/N} {
+   \draw[guide] (1.72,\yy)--(2.2,\yy);
+   \node[anchor=east] at (1.64,\yy) {$\ll$};
+ }
+ \foreach \yy/\ll in {.42/{-1},2.42/{H_3-1},4.62/{N-H_1-1},6.42/{N-1}} {
+   \draw[guide] (7.9,\yy)--(8.38,\yy);
+   \node[anchor=west] at (8.46,\yy) {$\ll$};
+ }
+ \fill[blue!60!black] (3.1,.7) circle (1.7pt);
+ \fill[blue!60!black] (3.1,6.7) circle (1.7pt);
+ \fill[orange!65!black] (7,6.42) circle (1.7pt);
+ \node[anchor=north,align=center,font=\footnotesize] at (3.1,.22)
+   {positive start\\$(0,0)$};
+ \node[anchor=north,align=center,font=\footnotesize] at (7,.02)
+   {negative terminal};
+ \node[anchor=south,font=\footnotesize,fill=white,inner sep=2pt]
+   at (7,6.47) {start $(N-1,Z)$};
+ \node[anchor=south,font=\footnotesize,fill=white,inner sep=2pt]
+   at (3.1,6.75) {end $(N,X_A)$};
+\end{tikzpicture}
+\caption{The three chunks in the opposed arrangement, after translation
+to put the positive starting site at $(0,0)$, with $N=H_1+H_2+H_3$.
+Both tape lists are indexed from top to bottom. The positive path follows
+chunks $3,2,1$, reversing each slab list while retaining the forward steps
+of every word; the negative path follows chunks $1,2,3$ in their original
+order. The one-level offset makes the departure heights in corresponding
+chunks agree. Under the actual law $P^{(n)}$, the lateral gap at the top
+is $Z-X_A=d_0$. The shaded middle chunks define $E_{\rm mid}$. Rectangles
+show height ranges, not path traces; widths, heights, and lateral
+placements are schematic.}
+\label{fig:opposed-chunks}
+\end{figure}
diff --git a/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/build/manuscript/posteriors.tex b/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/build/manuscript/posteriors.tex
new file mode 100644
index 0000000000000000000000000000000000000000..fdd3c0da5af79703680b60e7ac57618a1875701d
--- /dev/null
+++ b/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/build/manuscript/posteriors.tex
@@ -0,0 +1,394 @@
+\section{First contacts near an aligned endpoint}\label{sec:posteriors}
+
+Proposition~\ref{prop:contact-upper} bounds the number of scales at which
+the opposed tapes meet.  To obtain a competing lower bound, we consider a
+positive bridge and a relative negative path sampled independently of
+it.  We place the negative path one level below the bridge's endpoint.
+The bridge's final scripted departure is exactly the
+negative starting site, so there is always a contact one level below
+the endpoint.  We shall show that,
+after a prescribed script at a lower height, their first contact is
+unlikely to be delayed until much higher up the bridge.
+
+The lateral starting point of the negative path is determined by the
+positive endpoint.  We keep this dependence in the conditional
+probabilities of the possible endpoints, and estimate their running
+maxima together.  The martingale and endpoint-posterior arguments below
+adapt \cite[Lemma~6.1 and Proposition~6.2]{OpenAI2026}.  The following
+lemma gives the needed bound.
+
+\begin{lemma}[Maxima of a finite martingale family]\label{lem:posterior-max}
+Let \(m\ge1\), and let \((w_i(e))_{i\ge0}\), \(1\le e\le m\), be
+nonnegative martingales for the same filtration \((\mathcal F_i)_{i\ge0}\), with
+\(\sum_{e=1}^m w_i(e)\le1\) almost surely for every \(i\).  Set
+\[
+ A_i(e)=\max_{0\le j\le i}w_j(e),\qquad
+ A_i=\sum_{e=1}^m A_i(e),\qquad
+ A_\infty=\lim_{i\to\infty}A_i,
+ \qquad B=\log(m+1).
+\]
+Write also \(A_\infty(e)=\lim_i A_i(e)\).
+There are absolute constants \(c,C>0\) such that
+\begin{equation}\label{eq:posterior-exponential}
+ \EE\exp(cA_\infty/B)\le C.
+\end{equation}
+Moreover, on any joint probability space with this martingale marginal,
+every event \(V\) of probability \(p\) satisfies
+\begin{equation}\label{eq:event-weighted-max}
+ \EE[A_\infty\one_V]
+ \le CBp\log(\mathrm e/p),
+\end{equation}
+where the right-hand side is defined to be zero at \(p=0\).
+The event \(V\) need not belong to the martingale filtration.
+\end{lemma}
+
+The logarithmic first-moment bound underlying this lemma follows by
+coordinatewise integration of the scalar maximal inequality; the same
+posterior calculation appears in
+\cite[Section~1.1]{KesselheimMolinaroPattonSingla2026}.
+The passage from bounded conditional future increases to exponential
+integrability is related to the energy inequalities for increasing
+processes in \cite[Theorem~4 and its corollary]{Kikuchi1992}.
+We give a direct threshold proof.
+
+\begin{proof}
+Fix a time \(i\).  Conditional on the filtration at that time, the
+nonnegative-martingale maximal inequality
+\cite[Chapter~V, first section, \S2, Theorem~1]{Ville1939}
+bounds the probability that
+coordinate \(e\) subsequently exceeds \(u>0\) by \(w_i(e)/u\).
+For completeness, on a finite horizon this follows by stopping at the
+first crossing, using the martingale identity and nonnegativity on the
+complement of the crossing event; increasing the horizon gives the
+infinite-horizon bound.  Since each coordinate is at most one,
+integration over \(u\in[A_i(e),1]\) gives
+\[
+ \EE[A_\infty(e)-A_i(e)\mid\mathcal F_i]
+ \le w_i(e)\log\frac1{A_i(e)}
+ \le w_i(e)\log\frac1{w_i(e)}.
+\]
+If \(w_i(e)=0\), its future values vanish almost surely conditionally,
+and the expected increase is zero.  Add the missing mass
+\(1-\sum_e w_i(e)\) to form a probability vector.  Its entropy is at
+most \(\log(m+1)\), so
+\begin{equation}\label{eq:maximum-remaining-increase}
+ \EE[A_\infty-A_i\mid\mathcal F_i]\le B.
+\end{equation}
+
+We also have \(A_0\le1\) and
+\(A_i-A_{i-1}\le\sum_e w_i(e)\le1\).
+Consider the thresholds \(b_j=1+j(2B+1)\), \(j\ge1\).
+The probability of strictly crossing the first threshold is at most
+\(1/2\), by \eqref{eq:maximum-remaining-increase} and Markov's
+inequality.  At the first crossing of any threshold, the overshoot is
+at most one.  Crossing the next threshold therefore requires a further
+increase greater than \(2B\), whose conditional probability is at most
+\(1/2\).  This argument at a crossing time follows by splitting over
+its possible finite values.  It gives a geometric tail for
+\(A_\infty\) at the thresholds \(b_j\), and hence
+\eqref{eq:posterior-exponential}.
+
+For \(p>0\), conditional Jensen gives
+\[
+ \exp\!\left(\frac cB\EE[A_\infty\mid V]\right)
+ \le \EE[\exp(cA_\infty/B)\mid V]\le C/p.
+\]
+Taking logarithms and multiplying by \(p\) proves
+\eqref{eq:event-weighted-max}, after changing the absolute constant.
+\end{proof}
+
+We now formulate the first-contact estimate.  For a path in absolute
+coordinates, write \(T_j\) for its first hit of the hyperplane
+\(\{x:x_1=j\}\), specifying the path when needed.  Recall that
+\(\mathcal B_N^+\) is the positive exact-cut bridge law: its
+concatenated path has the raw marked law through \(T_N\), conditioned
+on \(\mathcal D_N^+\).  The event \(\mathcal D_k^+\) prescribes the
+\(M\)-step script immediately after the first hit of \(k-M\), reached
+before height \(-1\).
+
+\begin{proposition}[First contact above a scripted prefix]
+\label{prop:first-contact}
+Assume \(p_+p_->0\), and let \(M\le k<r<N\) be integers with
+\(r-k\ge R\).  Start a positive bridge \(Y\) with law
+\(\mathcal B_N^+\) at the origin, and put
+\(E_{\mathrm{lat}}=\pi Y_{T_N}\).
+Independently of \(Y\), sample a relative marked path with law
+\(\widehat P_-\), and translate it to start at
+\((N-1,E_{\mathrm{lat}})\); denote the resulting path by \(\beta\).
+Define
+\[
+ \tau=\inf\{i:T_k(Y)\le i<T_N(Y),\quad
+       \|Y_i-\beta_j\|_\infty\le R\text{ for some }j\ge0\},
+\]
+with \(\inf\varnothing=\infty\).  Let
+\begin{equation}\label{eq:first-contact-event}
+ \mathcal C_{k,r,N}=
+ \left\{
+ \begin{gathered}
+  \mathcal D_k^+(Y),\quad \tau<T_N(Y),\quad (Y_\tau)_1\ge r,\\
+  \min_{T_k(Y)\le i\le\tau}(Y_i)_1\ge k,
+  \quad |E_{\mathrm{lat}}|\le N^2
+ \end{gathered}
+ \right\}.
+\end{equation}
+Here \(\mathcal D_k^+(Y)\) is a condition on the prefix through the
+first hit of \(k\).  There is a constant \(C\), independent of
+\(k,r,N\), such that, with
+\(\Delta=u_{r-k}^+-u_{N-k}^+\),
+\begin{equation}\label{eq:first-contact-bound}
+ P(\mathcal C_{k,r,N})
+ \le C\log(2N)\,\Delta\log^2(\mathrm e/\Delta).
+\end{equation}
+The right-hand side is defined to be zero when \(\Delta=0\).
+\end{proposition}
+
+The small parameter in this estimate is controlled by the width tail:
+Lemma~\ref{lem:width-tail} gives
+\[
+ \Delta\le C(N-r)F_+(r-k).
+\]
+Thus the remaining height \(N-r\) is compared with the width tail at
+the progress \(r-k\) already made above the scripted prefix.
+
+\begin{proof}
+We first express the event using finite positive prefixes and the full
+negative path.  This permits a comparison in a shared environment
+without treating \(\tau\) as a stopping time.  The endpoint
+conditional probabilities remain attached to the positive prefixes;
+Lemma~\ref{lem:posterior-max} will control their contribution.
+
+\emph{Prefix probabilities.}
+Let
+\[
+ \mathcal E_N=\{e\in\ZZ^{d-1}:|e|\le N^2\}.
+\]
+Enumerate the marked prefixes \(a_0\) that realize
+\(\mathcal D_k^+\), ending at their first hit \(z\) of height \(k\).
+For each such \(a_0\), enumerate all finite marked words \(a\) from
+\(z\) whose vertices have heights in \([k,N-1]\) and whose terminal
+site \(y=y(a)\) has height at least \(r\).
+For a marked prefix \(c\) and \(e\in\mathcal E_N\), define
+\begin{equation}\label{eq:endpoint-prefix-posterior}
+ w(c;e)=P_0\bigl(\mathcal D_N^+,\ \pi X_{T_N}=e
+                  \mid X\text{ with marks begins with }c\bigr).
+\end{equation}
+This is a deterministic function of \(c\) and \(e\).
+All active departures in \(a_0\) lie strictly below height \(k-M\),
+whereas all departures in \(a\) have height at least \(k\).
+Lemma~\ref{lem:separated-weights} therefore gives
+\[
+ W(a_0a)=W(a_0)W(a).
+\]
+Consequently the raw probability of the prefix \(a_0a\) and the
+endpoint event in \eqref{eq:endpoint-prefix-posterior} is
+\(W(a_0)W(a)w(a_0a;e)\).
+
+For a raw negative path starting at \((N-1,e)\), let \(D^-\) denote
+the event that all its heights are at most \(N-1\).
+Let \(\mathcal V(a)\) be the event that this path satisfies \(D^-\),
+avoids the closed \(R\)-neighborhood of \(\Dep(a)\) for all time,
+and visits the closed \(R\)-neighborhood of \(y(a)\).
+Thus \(\mathcal V(a)\) asserts that the first contact along the word
+\(a\) occurs at its terminal site.  If that site was an earlier
+departure of \(a\), the event is empty.
+The raw bridge interpretation and the conditioning by \(D^-\) give
+the exact identity
+\begin{equation}\label{eq:first-contact-prefix-sum}
+ \begin{split}
+ P(\mathcal C_{k,r,N})
+ =\frac1{\bar u_N^+p_-}
+ \sum_{a_0}W(a_0)
+ \sum_{e\in\mathcal E_N}\sum_a
+ W(a)w(a_0a;e)P_{(N-1,e)}(\mathcal V(a)).
+ \end{split}
+\end{equation}
+Indeed, a pair of paths in \(\mathcal C_{k,r,N}\) determines a unique
+\(a_0\) and a unique prefix \(a\) ending at its first contact after
+\(T_k\).  Conversely every term represents exactly these conditions.
+All word families are countable, and every summand is nonnegative.
+
+\emph{A shared environment for the unconditioned prefixes.}
+Fix \(a_0\) for the moment.  For a word \(a\), put
+\[
+ B(a)=\{x\in\ZZ^d:\dist_\infty(x,\Dep(a))\le R\}.
+\]
+The quenched probability of \(\mathcal V(a)\) can be computed from
+the negative walk killed on arrival in \(B(a)\).  It requires eternal
+survival, the upper-height restriction, and a visit near \(y(a)\).
+It therefore uses only rows in \(B(a)^c\), including when these
+requirements concern the entire infinite path.  Those rows are more
+than distance \(R\) from every active departure of \(a\).
+Finite-range independence gives
+\begin{equation}\label{eq:first-contact-transfer}
+ W(a)P_{(N-1,e)}(\mathcal V(a))
+ =E_Q\left[p_\omega(a)
+           P_{(N-1,e),\omega}(\mathcal V(a))\right].
+\end{equation}
+
+The right-hand side has the following interpretation.  Sample a fresh
+environment with law \(Q\).  In it, run a raw comparison path
+\(\widetilde X\) from \(z\) and a raw negative path
+\(\widetilde\beta^{\,e}\) from \((N-1,e)\), independently
+conditional on the environment.  Denote its law and expectation by
+\(P_e^{\mathrm c},E_e^{\mathrm c}\), and write
+\(P^{\mathrm c},E^{\mathrm c}\) for the common
+\((\omega,\widetilde X)\) marginal, which does not depend on \(e\).
+For fixed \(a_0,e\), the inner sum in
+\eqref{eq:first-contact-prefix-sum} becomes
+\begin{equation}\label{eq:transferred-prefix-sum}
+ E_e^{\mathrm c}\left[
+  \sum_a w(a_0a;e)
+  \one_{\{\widetilde X\text{ begins with }a\}}
+  \one_{\{\widetilde\beta^{\,e}\in\mathcal V(a)\}}
+ \right].
+\end{equation}
+For every realized pair, at most one summand is nonzero: its terminal
+site must be the first site of \(\widetilde X\) within distance
+\(R\) of the full range of \(\widetilde\beta^{\,e}\).
+This uniqueness is a pathwise fact and uses no stopping-time property
+of the first contact.
+
+\emph{The stopped posterior law.}
+The factors \(w(a_0a;e)\) in
+\eqref{eq:transferred-prefix-sum} still come from the original raw
+positive law conditioned on \(a_0\).  In that law, let
+\(\mathcal G_i\) record the marked continuation for \(i\) steps,
+and form the martingales
+\[
+ w_i(e)=P_0(\mathcal D_N^+,\ \pi X_{T_N}=e
+                  \mid a_0,\mathcal G_i),
+ \qquad e\in\mathcal E_N.
+\]
+They are nonnegative and have sum at most one.  Stop the continuation
+on its first hit of height \(k-1\) or \(N\).  Every departure before
+that stop has height at least \(k\), so it is separated from the
+active departures of \(a_0\).  The stopping arrival uses no row.
+By factoring each finite stopped-prefix probability, the conditional
+stopped continuation therefore has exactly the same path law as
+\(\widetilde X\) stopped at
+\[
+ \zeta=T_{k-1}(\widetilde X)\wedge T_N(\widetilde X).
+\]
+Both stops are almost surely finite by uniform ellipticity and exit
+from a slab of bounded height.
+
+Evaluate the same deterministic functions
+\eqref{eq:endpoint-prefix-posterior} along the comparison path until
+\(\zeta\), and freeze them afterwards.  More explicitly, if
+\(\widetilde X_{[0,j]}\) denotes its marked prefix of length \(j\),
+set
+\[
+ \widetilde w_i(e)
+ =w\bigl(a_0\widetilde X_{[0,i\wedge\zeta]};e\bigr).
+\]
+Equality of the stopped path laws shows that this vector has the
+stopped martingale law just described.  Its values remain the original
+conditional probabilities given \(a_0\); they are not conditional
+probabilities given the fresh environment.  In particular, the value
+at a hit of \(k-1\) retains its original meaning, and no identification
+of continuation laws beyond that hit is needed.
+
+Define
+\[
+ M_*(e)=\max_{0\le i\le\zeta}\widetilde w_i(e),
+ \qquad S_*=\sum_{e\in\mathcal E_N}M_*(e).
+\]
+All endpoint coordinates are functions of the same stopped comparison
+path.  We can therefore estimate their sum \(S_*\) on an event while
+retaining only a logarithmic dependence on the number of endpoints.
+Applying Lemma~\ref{lem:posterior-max} and using
+\(\log(|\mathcal E_N|+1)\le C\log(2N)\) gives, for any event \(V\)
+depending on the comparison path and its environment,
+\begin{equation}\label{eq:stopped-posterior-event}
+ E^{\mathrm c}[S_*\one_V]
+ \le C\log(2N)\,P^{\mathrm c}(V)
+                  \log\frac{\mathrm e}{P^{\mathrm c}(V)}.
+\end{equation}
+The constant is uniform over \(a_0\).  We have thus controlled the
+endpoint factors while preserving a fresh shared environment in which
+to estimate contacts.
+
+\emph{Quenched exit probabilities.}
+For an interior site \(y\), meaning \(k\le y_1<N\), write
+\[
+ q_y^\omega=P_{y,\omega}(T_{k-1}<T_N).
+\]
+Ellipticity gives \(0<q_y^\omega<1\).  For
+\(t=2^{-j}\), \(j\ge1\), let \(V_t\) be the event that
+\(\widetilde X\), at or after its first hit of height \(r\) and
+before \(\zeta\), visits a site with \(q_y^\omega\ge t\).
+With the environment fixed, stop at the first such visit.  The
+conditional chance of exiting at \(k-1\) is then at least \(t\).
+After averaging, stationarity and almost sure slab exit give
+\begin{equation}\label{eq:first-contact-large-q}
+ \begin{split}
+ tP^{\mathrm c}(V_t)
+ &\le P_z(T_r<T_{k-1}<T_N)\\
+ &=u_{r-k}^+-u_{N-k}^+=\Delta.
+ \end{split}
+\end{equation}
+
+Suppose a retained prefix ends at \(y\) with
+\(q_y^\omega\in[t,2t)\).  Its endpoint lies at height at least
+\(r\), so \(V_t\) occurs.  The negative path visits a site \(y'\)
+within distance \(R\) of \(y\).  Because \(r-k\ge R\) and the
+negative path satisfies \(D^-\), this site is also interior:
+\(k\le y'_1\le N-1\).  A coordinate path from \(y\) to \(y'\)
+of length at most \(dR\) stays inside the slab.  Following this path
+and then exiting through the lower boundary shows that
+\[
+ q_y^\omega\ge\kappa^{dR}q_{y'}^\omega,
+ \qquad q_{y'}^\omega\le c_Rt,
+ \qquad c_R=2\kappa^{-dR}.
+\]
+
+For each fixed environment, the probability that a raw path from
+\((N-1,e)\) satisfies \(D^-\) and visits any interior site with
+\(q_{y'}^\omega\le c_Rt\) is at most \(c_Rt\), uniformly in \(e\).
+Indeed, stop on the first visit to that set.  To satisfy \(D^-\)
+after this visit, the path must next leave the slab through \(k-1\),
+except on the null event of never exiting it.  The strong Markov property
+bounds the conditional probability of this next exit by
+\(q_{y'}^\omega\le c_Rt\) at the first-visit site, regardless of
+earlier visits below \(k-1\).
+
+\emph{Summing the exit-probability bins.}
+We classify the transferred sum
+\eqref{eq:transferred-prefix-sum} according to
+\(q_{y(a)}^\omega\in[t,2t)\).  This classification is made in the
+shared environment, after \eqref{eq:first-contact-transfer} has been
+applied.  A retained prefix ends before \(\zeta\), so its posterior
+factor is at most \(M_*(e)\).  By uniqueness of the retained prefix,
+its contribution in this bin is pointwise bounded by \(M_*(e)\)
+times the indicators of \(V_t\) and of the negative-path event in
+the preceding paragraph.  Conditional on the environment, that
+negative path is independent of \(\widetilde X\), and its event has
+probability at most \(c_Rt\).  Summing over \(e\), whose comparison
+path and environment have the same marginal, bounds this bin's total
+contribution for fixed \(a_0\) by
+\begin{equation}\label{eq:first-contact-bin}
+ Ct\,E^{\mathrm c}[S_*\one_{V_t}].
+\end{equation}
+
+If \(\Delta=0\), \eqref{eq:first-contact-large-q} makes every term
+zero.  Suppose \(\Delta>0\).  For \(t<\Delta\), use
+\(E^{\mathrm c}S_*\le C\log(2N)\) and sum the geometric series in
+\(t\).  The contribution is at most \(C\log(2N)\Delta\).
+For \(t\ge\Delta\), combine
+\eqref{eq:stopped-posterior-event} and
+\eqref{eq:first-contact-large-q}.  Since
+\(p\mapsto p\log(\mathrm e/p)\) is increasing on \([0,1]\), each
+such bin contributes at most
+\[
+ C\log(2N)\,\Delta\log(\mathrm e t/\Delta).
+\]
+There are \(O(\log(\mathrm e/\Delta))\) bins in this second range,
+so all bins together contribute at most
+\(C\log(2N)\Delta\log^2(\mathrm e/\Delta)\).
+Finally, the first-hit prefixes \(a_0\) are disjoint, hence
+\(\sum_{a_0}W(a_0)\le1\), and
+\eqref{eq:bridge-renewal} gives
+\(\bar u_N^+p_-\ge\xi p_+p_->0\).
+Substitution in \eqref{eq:first-contact-prefix-sum} proves
+\eqref{eq:first-contact-bound}.
+\end{proof}
diff --git a/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/build/manuscript/radius.tex b/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/build/manuscript/radius.tex
new file mode 100644
index 0000000000000000000000000000000000000000..849aaa1832ef2402a2cf418e2a75c6f2af7772a6
--- /dev/null
+++ b/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/build/manuscript/radius.tex
@@ -0,0 +1,446 @@
+\section{Spatial moments and reduction to a coordinate direction}
+\label{sec:radius}
+
+The finite mean width from Proposition~\ref{prop:mean-width} controls a
+regeneration word only in its direction of progress.  We now show that
+coexistence of the two escape signs also controls its spatial extent.
+This will give two limiting rays and reduce the problem to a coordinate
+direction.  The radius argument adapts
+\cite[Proposition~3.1]{OpenAI2026}, with a marked connector and a
+separated infinite continuation supplying the finite-range version.
+
+For a regeneration word $\gamma$ with position sequence
+$(x_0,\ldots,x_m)$, written relative to $x_0=0$, define its radius by
+\[
+  \mathcal R(\gamma)=\max_{0\le j\le m}|x_j|,
+\]
+where $|\cdot|$ denotes Euclidean norm.  In particular, the terminal
+arrival contributes to the radius.  Distances used for finite-range
+independence remain $\ell^\infty$ distances.
+
+\subsection{An integrable spatial radius}
+
+We first record an elementary estimate that controls all initial sums at
+once.  Stationarity, rather than independence, is enough for this estimate.
+It is the stationary-sequence form of Hopf's maximal ergodic inequality;
+see Garsia~\cite{Garsia1965}.  We include a disjoint-interval proof.
+
+\begin{lemma}[A maximal estimate for initial averages]
+\label{lem:stationary-maximal}
+Let $(Z_i)_{i\ge1}$ be a stationary sequence of nonnegative integrable
+random variables.  For every integer $n\ge1$ and every $a>0$,
+\[
+  \PP\left\{\max_{1\le k\le n}\frac1k\sum_{i=1}^k Z_i>a\right\}
+  \le \frac{\EE Z_1}{a}.
+\]
+\end{lemma}
+
+\begin{proof}
+Call an integer $j\ge1$ bad if some interval starting at $j$, of length
+at most $n$, has average greater than $a$.  Among the bad starts in
+$\{1,\ldots,m\}$, select the leftmost one and a witnessing interval.
+Continue by selecting the leftmost bad start strictly to the right of
+the selected interval, and repeat.  These disjoint intervals cover all
+bad starts in $\{1,\ldots,m\}$ and lie in $\{1,\ldots,m+n\}$.
+Their total length is at least the number of those bad starts, while
+the sum of the $Z_i$ over them is at least $a$ times their total length.
+Consequently
+\[
+  a\,\#\{j\le m:j\text{ is bad}\}\le\sum_{i=1}^{m+n}Z_i.
+\]
+Taking expectations, using stationarity, dividing by $m$, and letting
+$m\to\infty$ proves the assertion.
+\end{proof}
+
+\begin{proposition}[Radius moment under coexistence]
+\label{prop:radius-moment}
+Let $v\ne0$ satisfy $\|v\|_\infty=1$.  If
+$P_0(A_v)>0$ and $P_0(A_{-v})>0$, then the regeneration-word laws of
+Proposition~\ref{prop:cut-law} satisfy
+\begin{equation}
+  E_{\nu_v}\mathcal R+E_{\nu_{-v}}\mathcal R<\infty.
+  \label{eq:radius-moment}
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+Both nonbacktracking probabilities $p_v,p_{-v}$ are positive by
+Proposition~\ref{prop:sign-union}.  For $\sigma\in\{+,-\}$ write
+\[
+  P_\sigma^*=\widehat P_{\sigma v},\qquad
+  E_\sigma^*=E_{P_\sigma^*}.
+\]
+Here $\sigma v$ means $v$ or $-v$, respectively.  Under either law, let
+$L_i,\mathcal R_i$ be the width and radius of its $i$th regeneration
+word, and put
+\[
+  H_0=0,\qquad H_k=\sum_{i=1}^k L_i.
+\]
+An unsubscripted $L$ or $\mathcal R$ denotes the corresponding variable
+for one word.  All constants chosen below are independent of the scale
+parameters $i_0$ and $a$.
+
+Suppose that \eqref{eq:radius-moment} fails.  The truncated mean
+\[
+  I(t)=\sum_{\sigma\in\{+,-\}}E_\sigma^*\min(\mathcal R,t),
+  \qquad t\ge0,
+\]
+is then nondecreasing and unbounded.  It is concave, $I(0)=0$, and
+$I(t)=o(t)$: the last assertion follows by dominated convergence from
+the almost-sure finiteness of each regeneration word.
+
+We will obtain a fixed positive probability that the final supremum of
+one of the two signed heights lies in a bounded window, and then place
+these windows arbitrarily far apart.  The unbounded truncated mean
+will produce a large spatial excursion after a controlled initial
+segment, and hence a record for a tilted linear functional.  A marked
+connector will let us append a separated continuation in the opposite
+direction while preserving the height maximum already attained.
+
+\paragraph{The scale and a controlled initial segment.}
+Proposition~\ref{prop:mean-width} and the strong law allow us to choose
+$c,C>0$ and $C_0\ge0$ so that, under each sign law, with probability
+greater than $0.95$,
+\begin{equation}
+  ck-C_0\le H_k\le Ck+C_0\qquad(k\ge0).
+  \label{eq:radius-height-bounds}
+\end{equation}
+Indeed, choose $c$ below both mean widths and $C$ above both, and then
+choose a deterministic $C_0$ controlling the finitely many initial
+deviations with the stated probability.  Next choose an integer
+$C_1\ge1$, a number $b>1/c$, a number $B>1$, and $\eta>0$, in that
+order, with
+\begin{equation}
+  C_1c>C+4,\qquad
+  \frac{B-1}{\sqrt d}-b(C+2)>2,\qquad
+  4C_1\eta<0.15.
+  \label{eq:radius-constants}
+\end{equation}
+
+For $a>0$ sufficiently large, let $N=N(a)$ be the first positive integer
+such that
+\[
+  I(aN)\ge\eta a.
+\]
+It exists because $I$ is unbounded, and $N(a)\to\infty$ because
+$I(t)=o(t)$.  For large $a$, minimality and concavity give
+\begin{equation}
+  \eta a\le I(aN)
+  \le\frac{N}{N-1}I(a(N-1))<2\eta a.
+  \label{eq:radius-scale}
+\end{equation}
+
+Let $\mathcal J_m$ be the event that
+\eqref{eq:radius-height-bounds} holds through $m$ and
+\[
+  \sum_{i=1}^k\mathcal R_i\le ak\qquad(1\le k\le m).
+\]
+For both signs and all sufficiently large $a$,
+\begin{equation}
+  P_\sigma^*(\mathcal J_{C_1N})>0.8.
+  \label{eq:radius-controlled}
+\end{equation}
+To see this, set $n=C_1N$ and truncate each radius at $an$.
+Its mean is at most
+\[
+  I(an)\le C_1 I(aN)<2C_1\eta a.
+\]
+Lemma~\ref{lem:stationary-maximal} bounds by $2C_1\eta$ the probability
+that any initial average of the truncated radii, through $n$, exceeds
+$a$.  Also
+\[
+  P_\sigma^*\{\mathcal R_i>an\text{ for some }i\le n\}
+  \le nP_\sigma^*(\mathcal R>an)
+  \le\frac{E_\sigma^*\min(\mathcal R,an)}a
+  <2C_1\eta.
+\]
+Combining these estimates with the probability of
+\eqref{eq:radius-height-bounds} proves
+\eqref{eq:radius-controlled}.
+
+\paragraph{A large excursion following that segment.}
+Choose an integer $i_0$ large enough that
+\begin{equation}
+\begin{gathered}
+  \sum_\sigma\sum_{i\ge i_0}P_\sigma^*(L>i)<\frac{\eta}{4B},\\
+  c(i-1)-C_0\ge\frac{ci}{2},\qquad
+  C(i-1)+C_0+i\le(C+2)i\qquad(i\ge i_0).
+\end{gathered}
+\label{eq:radius-initial-index}
+\end{equation}
+The first condition follows from integrability of the widths.
+With $i_0$ fixed, take $a$ sufficiently large that $N\ge i_0$ and
+$I(Bai_0)<\eta a/4$, as well as all preceding large-$a$ requirements.
+For $i_0\le i\le N$ define
+\[
+  \mathcal L_i=\{\mathcal R_i>Bai,\ L_i\le i\},\qquad
+  \Lambda_\sigma=\sum_{i=i_0}^N P_\sigma^*(\mathcal L_i).
+\]
+Integration of the decreasing radius tails gives
+\begin{align*}
+  \sum_\sigma\sum_{i=i_0}^N P_\sigma^*(\mathcal R>Bai)
+  &\ge\frac{I(Ba(N+1))-I(Bai_0)}{Ba},\\
+  \sum_\sigma\sum_{i=1}^N P_\sigma^*(\mathcal R>Bai)
+  &\le\frac{I(BaN)}{Ba}.
+\end{align*}
+The first right-hand side is at least $3\eta/(4B)$, by
+\eqref{eq:radius-scale} and monotonicity of $I$.  The second is at most
+$2\eta$, by concavity.  Subtracting the width-tail bound in
+\eqref{eq:radius-initial-index} therefore yields
+\begin{equation}
+  \frac{\eta}{2B}\le\Lambda_++\Lambda_-\le2\eta.
+  \label{eq:radius-large-excursion-mass}
+\end{equation}
+
+Choose a sign $\sigma$ for which $\Lambda_\sigma\ge\eta/(4B)$, and count
+the indices for which both $\mathcal J_{i-1}$ and $\mathcal L_i$ occur.
+Independence of the $i$th word from its predecessors and
+\eqref{eq:radius-controlled} give a mean of at least
+$0.8\Lambda_\sigma$.  The second moment is at most
+$\Lambda_\sigma+\Lambda_\sigma^2$, by domination by the sum of the
+independent indicators $\one_{\mathcal L_i}$.
+Cauchy--Schwarz and \eqref{eq:radius-large-excursion-mass} imply that,
+with probability at least a constant $\delta>0$ independent of $i_0,a$,
+there is an index $i\in[i_0,N]$ with
+\begin{equation}
+  \mathcal J_{i-1}\cap\mathcal L_i.
+  \label{eq:radius-selected-excursion}
+\end{equation}
+
+\paragraph{A record in a tilted direction.}
+For this sign write $h(x)=\sigma v\cdot x$.  On
+\eqref{eq:radius-selected-excursion}, all preceding positions have norm
+at most $a(i-1)$, whereas some position $X$ in word $i$ satisfies
+$|X|>(B-1)ai$.  Hence, for some coordinate step $u\in U$,
+\[
+  u\cdot X>\frac{(B-1)ai}{\sqrt d}.
+\]
+Throughout this word, the running maximum of $h$ lies between
+$H_{i-1}$ and $H_i$, and therefore in
+$[ci/2,(C+2)i]$.  Define the linear functional
+\[
+  Y(x)=u\cdot x-ba h(x).
+\]
+Before word $i$, nonnegativity of $h$ gives $Y\le a(i-1)$.
+At the indicated position, \eqref{eq:radius-constants} gives $Y>2ai$.
+There is consequently a strict $Y$-record whose running $h$-maximum
+belongs to the deterministic window
+\begin{equation}
+  \mathcal W=[ci_0/2,(C+2)N].
+  \label{eq:radius-window}
+\end{equation}
+
+For each fixed $u$, let $T$ be the first positive time at which the
+position is a strict $Y$-record, the running $h$-maximum belongs to
+$\mathcal W$, and the entire prefix has nonnegative $h$-heights.
+These conditions make $T$ a stopping time.  Since there are $2d$
+choices of $u$, one of them satisfies
+$P_\sigma^*(T<\infty)\ge\delta/(2d)$.  Fix that choice and set
+$p_* = \min(p_v,p_{-v})$.  Under the raw law,
+\begin{equation}
+  P_0(T<\infty)\ge\frac{p_*\delta}{2d}.
+  \label{eq:radius-record-probability}
+\end{equation}
+
+We have found a record with a uniformly positive probability, while its
+running height maximum lies in a window that can be moved arbitrarily
+far out.  We next continue from this record so that the final height
+maximum stays in the same window.  The continuation must remain
+separated from the observed prefix for its annealed probability to
+factor.
+
+\paragraph{A separated continuation in the opposite direction.}
+Choose a coordinate step $e$ with $h(e)=-1$.  From $X_T$ force $m_0$
+successive steps $e$, all with mark $0$, where $m_0$ is a fixed positive
+integer satisfying
+\begin{equation}
+  m_0b>1+bC_0+Rdb+1.
+  \label{eq:radius-connector-length}
+\end{equation}
+The choice of $m_0$ depends only on the fixed constants $b,C_0,R,d$,
+and is independent of $i_0,a$.
+These steps have weight $\lambda^{m_0}$.  At their endpoint
+$z=X_T+m_0e$, consider the translated raw event consisting of
+$D_{-\sigma v}$ and $\mathcal J_{C_1N}$ for the opposite regeneration
+sequence, with the required cuts finite.  Its raw probability is at
+least $0.8p_*$ by \eqref{eq:radius-controlled} and
+Proposition~\ref{prop:cut-law}.
+
+Every position of the continuation from $z$ is separated by more than
+$R$ from the prefix through $T$.  There are two parts to this geometric
+assertion.
+During opposite word $k+1$, for $0\le k<C_1N$, displacement from $z$
+in the $u$ coordinate is at least $-a(k+1)$, by the initial-sum radius
+bound in $\mathcal J_{C_1N}$; displacement in $h$ is at most $-ck+C_0$.
+Its total $Y$-increment from $X_T$, including
+the forced steps, is therefore at least
+\begin{align}
+  &m_0(ba-1)-a(k+1)+ba(ck-C_0)\notag\\
+  &\hspace{12mm}
+  =a\bigl(m_0b-1-bC_0+(bc-1)k\bigr)-m_0.
+  \label{eq:radius-tilted-gap}
+\end{align}
+Since $bc>1$, \eqref{eq:radius-connector-length} makes this quantity
+greater than $R(1+dba)$ for all sufficiently large $a$, uniformly in
+$k$.  Every prefix position has $Y$-value at most $Y(X_T)$, while
+\[
+  |Y(x)-Y(y)|\le(1+dba)\|x-y\|_\infty.
+\]
+Thus the first $C_1N$ opposite words are at distance greater than $R$
+from every prefix position.
+
+At the end of these words the absolute $h$-coordinate is at most
+\begin{equation}
+  (C+2)N-m_0-cC_1N+C_0<-dR
+  \label{eq:radius-tail-gap}
+\end{equation}
+for large $a$, by \eqref{eq:radius-constants}.  All subsequent positions
+stay at or below this height, since this endpoint is a cut in direction
+$-\sigma v$.  The prefix has nonnegative $h$-heights and
+$|h(x)-h(y)|\le d\|x-y\|_\infty$, so the entire remaining tail is
+also separated from the prefix.  The tilted inequality protects the
+initial opposite words; the height inequality protects everything
+after their last cut.  No bound on $Y$ is required for this remaining tail.
+
+We now justify the probability calculation for this infinite
+continuation.  Fix a finite marked prefix $\gamma$ realizing $T$, and
+let $A_\gamma$ be its set of positions, including its terminal arrival.
+The preceding deterministic bounds show that every path in the
+specified continuation event avoids the closed $R$-neighborhood of
+$A_\gamma$.  Consequently its quenched probability can be computed
+using the walk killed on first arrival in that neighborhood.  By
+Lemma~\ref{lem:separated-weights}, this probability depends only on
+rows outside the neighborhood, even though the event specifies future
+cuts and an infinite nonbacktracking tail.  Those rows are separated
+from the active departures of $\gamma$.
+
+The forced connector has constant quenched weight and reveals no rows.
+The weight of $\gamma$, the connector, and the continuation therefore
+factors as
+\[
+  W(\gamma)\lambda^{m_0}
+  P_0\{D_{-\sigma v},\ \mathcal J_{C_1N}
+                 \text{ for its regeneration sequence}\}
+  \ge 0.8p_*\lambda^{m_0}W(\gamma).
+\]
+Here stationarity identifies the last factor with the raw probability
+from $z$, and the almost-sure existence of the indicated cuts under
+$D_{-\sigma v}$ is understood.  This use of separated rows is made
+after fixing $\gamma$; no independence for a randomly chosen region is
+assumed.
+
+The connector decreases $h$, and the opposite continuation stays at
+or below its own initial height.  Thus the final supremum of $h$ is
+exactly the running supremum at $T$ and belongs to $\mathcal W$.
+Summing over the disjoint prefixes realizing $T$ and using
+\eqref{eq:radius-record-probability} yields
+\begin{equation}
+  P_0\left\{\sup_{n\ge0}\sigma v\cdot X_n\in\mathcal W\right\}
+  \ge\rho,
+  \qquad
+  \rho:=\frac{0.8\lambda^{m_0}p_*^2\delta}{2d}>0.
+  \label{eq:radius-final-supremum}
+\end{equation}
+
+Finally, choose $i_0$ and then $a$ successively so that the resulting
+windows \eqref{eq:radius-window} are pairwise disjoint.  This is
+possible by taking each new lower endpoint beyond the preceding upper
+endpoint before choosing its $a$.  The sign $\sigma$ may change from
+one window to the next, but for either fixed sign the corresponding
+final-supremum events are disjoint.  Their probabilities, summed over
+all windows and both signs, are at most $2$, contradicting the uniform
+lower bound \eqref{eq:radius-final-supremum}.  This proves
+\eqref{eq:radius-moment}.
+\end{proof}
+
+\subsection{Opposite limiting rays}
+
+The radius moment controls the path between cuts, so the ordinary
+strong law for regeneration displacements also describes the full
+trajectory.  The limiting-ray argument and the coordinate reduction
+follow \cite[Section~4.1]{OpenAI2026}.
+
+\begin{lemma}[Limiting rays under coexistence]
+\label{lem:opposite-rays}
+Under the coexistence hypotheses of Proposition~\ref{prop:radius-moment},
+there are deterministic unit vectors $r_+,r_-$ such that, for
+$\sigma\in\{+,-\}$,
+\[
+  |X_n|\longrightarrow\infty,\qquad
+  \frac{X_n}{|X_n|}\longrightarrow r_\sigma
+  \quad P_0\text{-almost surely on }A_{\sigma v}.
+\]
+They satisfy $r_\sigma\cdot(\sigma v)>0$ and $r_-=-r_+$.
+\end{lemma}
+
+\begin{proof}
+The terminal displacement of a word of law $\nu_{\sigma v}$ is
+integrable by Proposition~\ref{prop:radius-moment}.  Write its mean as
+$b_\sigma'\in\RR^d$.  Since
+\[
+  b_\sigma'\cdot(\sigma v)=E_{\nu_{\sigma v}}L>0,
+\]
+this vector is nonzero.  Under $\widehat P_{\sigma v}$, the $k$th cut
+location divided by $k$ tends almost surely to $b_\sigma'$.
+Integrability also gives $\mathcal R_k/k\to0$ almost surely: for each
+$\varepsilon>0$, the sum of
+$\widehat P_{\sigma v}(\mathcal R_k>\varepsilon k)$ is finite, and
+Borel--Cantelli applies.  Between consecutive cuts the displacement
+from the earlier cut is bounded by the next radius.  Since the number
+of completed words tends to infinity along the trajectory, it follows
+that $|X_n|\to\infty$ and
+\[
+  \frac{X_n}{|X_n|}\longrightarrow
+  r_\sigma:=\frac{b_\sigma'}{|b_\sigma'|}.
+\]
+Under the raw law on $A_{\sigma v}$, the first cut is finite and its
+translated suffix has law $\widehat P_{\sigma v}$ by
+Proposition~\ref{prop:cut-law}; hence the same conclusion holds there.
+
+The signs of the projections on $v$ make $r_+$ and $r_-$ distinct.
+Suppose they are not opposite.  Then $w=r_++r_-$ is nonzero and
+\[
+  w\cdot r_+=w\cdot r_-=1+r_+\cdot r_->0.
+\]
+Proposition~\ref{prop:sign-union} gives
+$P_0(A_v\cup A_{-v})=1$, so the limiting rays imply $P_0(A_w)=1$.
+Rescale $w$ to have $\ell^\infty$ norm one and apply the cut
+construction in that direction.
+
+Under $\widehat P_w$, the limiting ray exists almost surely because
+this law is a conditioning of the raw law just considered.  Its value
+is unchanged on deleting any finite number of regeneration words and
+translating the remaining path: such operations do not affect a
+limiting direction when positions tend to infinity in norm.
+Consequently it is a tail random variable of the iid words under
+$\widehat P_w$: for every $k$, its value can be computed from the words
+after $k$.  The independent tail zero--one law makes it
+deterministic.  Since $P_0(A_w)=1$, the raw walk has a first cut in
+direction $w$ almost surely, with translated suffix law
+$\widehat P_w$.  Its limiting ray must therefore have that same
+deterministic value almost surely.  This contradicts the two distinct
+rays $r_+,r_-$, each occurring with positive raw probability.  Hence
+$r_-=-r_+$.
+\end{proof}
+
+\begin{proposition}[Reduction to a coordinate direction]
+\label{prop:coordinate-reduction}
+If $0<P_0(A_v)<1$ for some fixed nonzero real direction $v$, then there
+is a coordinate $i\in\{1,\ldots,d\}$ such that
+\[
+  P_0(A_{e_i})>0\qquad\text{and}\qquad P_0(A_{-e_i})>0.
+\]
+\end{proposition}
+
+\begin{proof}
+After normalizing $v$, Proposition~\ref{prop:sign-union} shows that
+both signs of escape in direction $v$ have positive probability.
+Choose a coordinate with nonzero projection on the opposite rays
+from Lemma~\ref{lem:opposite-rays}.  Along one ray this coordinate
+tends to $+\infty$, and along the other it tends to $-\infty$.
+Each ray occurs with positive probability, giving the assertion.
+\end{proof}
+
+To prove Theorem~\ref{thm:main}, it therefore remains to exclude
+coexistence in a coordinate direction.  After permuting coordinates,
+we may take that direction to be $e_1$.
diff --git a/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/build/references/references.tex b/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/build/references/references.tex
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+++ b/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/build/references/references.tex
@@ -0,0 +1,117 @@
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+\emph{The zero-one law for planar random walks in i.i.d. random environments revisited},
+Electronic Communications in Probability \textbf{12} (2007), 326--335.
+\href{https://doi.org/10.1214/ECP.v12-1314}{doi:10.1214/ECP.v12-1314}.
+
+\bibitem{ZernerMerkl2001}
+Martin P. W. Zerner and Franz Merkl,
+\emph{A zero-one law for planar random walks in random environment},
+Annals of Probability \textbf{29} (2001), no.~4, 1716--1732.
+\href{https://doi.org/10.1214/aop/1015345769}{doi:10.1214/aop/1015345769}.
+
+\end{thebibliography}
diff --git a/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/directional-zero-one-finite-range.pdf b/preprints/A-directional-zero-one-law-for-finite-range-dependent-random-environments-October-5-2026/directional-zero-one-finite-range.pdf
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diff --git a/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/build/01-introduction.tex b/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/build/01-introduction.tex
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+\section{Introduction}\label{sec:introduction}
+
+A walk in a fixed environment is a Markov chain. Averaging over an
+independent environment at every lattice site creates a different
+process: revisiting a site also revisits information about its transition
+probabilities. The directional zero--one problem asks whether the
+annealed probability of escape in one prescribed direction can lie
+strictly between zero and one. We resolve this problem positively in
+every dimension $d\ge3$ for iid nearest-neighbor environments with
+strictly positive transition probabilities. No common positive lower
+bound on those probabilities is assumed.
+
+\subsection{The model and the theorem}
+
+Let $d\ge3$ be an integer, let $e_1,\ldots,e_d$ be the coordinate basis,
+and set
+\[
+ U=\{\pm e_1,\ldots,\pm e_d\},\qquad
+ \mathcal P_U=\left\{p\in[0,1]^U:\sum_{e\in U}p(e)=1\right\},
+ \qquad \Omega=\mathcal P_U^{\ZZ^d}.
+\]
+Equip $\Omega$ with its product Borel $\sigma$-field.
+An environment $\omega\in\Omega$ assigns a transition vector
+$\omega(x,\cdot)$ to each site $x$. We call this vector the \emph{row}
+at $x$. Let $\mu$ be a probability measure on $\mathcal P_U$, and let
+$\PP=\mu^{\otimes\ZZ^d}$. Thus the rows are independent and identically
+distributed; entries within one row need not be independent. Assume only
+\begin{equation}\label{eq:ellipticity}
+ \mu\bigl(\{p\in\mathcal P_U:p(e)>0\text{ for every }e\in U\}\bigr)=1.
+\end{equation}
+We call this \emph{strict ellipticity}. By countability, it gives
+$\omega(x,e)>0$ simultaneously for all sites and steps, almost surely.
+Unlike uniform ellipticity, it permits the smallest row entry to be
+arbitrarily close to zero.
+
+The path space is $\mathcal X=(\ZZ^d)^{\NN_0}$ with coordinate
+maps $X_n$ and the $\sigma$-field $\sigma(X_n:n\ge0)$. For $x\in\ZZ^d$ and a fixed environment, the
+\emph{quenched law} $P_{x,\omega}$ is the Markov-chain law specified by
+\[
+ P_{x,\omega}(X_0=x)=1,\qquad
+ P_{x,\omega}(X_{n+1}=X_n+e\mid X_0,\ldots,X_n)=\omega(X_n,e).
+\]
+The \emph{annealed law} and its expectation are
+\[
+ P_x(B)=\int P_{x,\omega}(B)\,\PP(d\omega),\qquad E_x.
+\]
+When both environment and path are needed, we use the joint law
+$\mathsf P_x(d\omega,dX)=\PP(d\omega)P_{x,\omega}(dX)$.
+The path filtration is $\mathcal F_n=\sigma(X_0,\ldots,X_n)$;
+on the joint space the larger filtration
+$\mathcal G_n=\sigma(\omega,X_0,\ldots,X_n)$ also reveals the entire
+environment. Quenched Markov arguments use $\mathcal G_n$.
+The endpoint-posterior martingales later in the proof use
+$\mathcal F_n$, with the environment integrated out.
+An unconditioned annealed path law will also be called the \emph{raw law}.
+
+For a nonzero vector $\ell\in\RR^d$, define
+\[
+ A_\ell=\{X_n\cdot\ell\longrightarrow+\infty\}.
+\]
+\begin{theorem}\label{thm:main}
+Let $d\ge3$, let $\PP$ be an independent and identically distributed
+law of nearest-neighbor transition rows on $\ZZ^d$ satisfying
+\eqref{eq:ellipticity}, and let $\ell\in\RR^d\setminus\{0\}$.
+Then
+\[
+ P_0(A_\ell)\in\{0,1\}.
+\]
+\end{theorem}
+
+The direction is any fixed real vector; no rationality assumption is
+made, and no assertion of one exceptional null set uniform over all
+directions is needed. No inverse-transition or logarithmic moment,
+drift, balance, reversibility, or speed condition supplements
+\eqref{eq:ellipticity}. In the proof, a finite first moment for a
+\emph{spatial radius} is derived under the hypothesis that both signs
+have positive probability. No first moment for a regeneration duration,
+positive speed or ballisticity assertion is used or obtained from that
+estimate.
+
+\subsection{Prior work}
+
+The directional question goes back to Kalikow~\cite{Kalikow1981};
+see also the historical discussion in~\cite{Zerner2007}.
+The classical sign-union zero--one law concerns
+$P_0(A_\ell\cup A_{-\ell})$, not $P_0(A_\ell)$.
+It originates in the uniformly elliptic theory of
+Kalikow~\cite{Kalikow1981}; its strict-ellipticity extension is
+\cite[Proposition~3]{ZernerMerkl2001}, as recalled in
+\cite[Equation~(1)]{Zerner2007}. Zerner and Merkl proved the
+one-sign directional law in dimension two for iid strictly elliptic
+environments~\cite[Theorem~1]{ZernerMerkl2001}, and Zerner gave a shorter
+proof~\cite{Zerner2007}. Slonim extended the planar iid law to bounded jumps
+when almost surely there is a unique infinite communicating class reachable
+from every site~\cite[Theorem~2.1]{Slonim2024}. Theorem~\ref{thm:main} establishes
+the directional zero--one conjecture for nearest-neighbor walks in every
+higher dimension under strict ellipticity.
+
+The difficulty in higher dimensions is that the planar arguments force
+oppositely directed paths to meet by geometry; their quantitative
+counterpart must be supplied differently here. Berger proved that in
+dimensions $d\ge5$ an iid uniformly elliptic walk has at most one nonzero
+limiting velocity~\cite[Theorem~1.1]{Berger2008}. This allows a second
+velocity equal to zero and leaves open the possibility of opposite
+directional escape at zero speed. Independence is also substantive:
+Bramson, Zeitouni, and Zerner constructed stationary, polynomially mixing,
+uniformly elliptic environments in $d\ge3$ with positive probabilities of
+linear escape in opposite directions~\cite[Theorem~3]{BramsonZeitouniZerner2006}.
+Their Open Problem~3 formulates the higher-dimensional question for iid
+uniformly elliptic environments.
+
+A non-uniformly elliptic special case is provided by Dirichlet-distributed
+rows. Sabot and Tournier obtained positive-probability transience in coordinate
+directions with positive corresponding mean drift, using reversal identities
+specific to that distribution~\cite[Theorem~1]{SabotTournier2011}.
+Bouchet proved the fixed-direction zero--one law in this class for $d\ge3$
+by studying an accelerated walk and an invariant law for the environment
+viewed from the particle~\cite[Corollary~4]{Bouchet2013}. Tournier identified
+the asymptotic direction as that of the initial drift when this drift is
+nonzero~\cite[Corollary~1]{Tournier2015}. The argument below applies to arbitrary iid
+strictly positive rows and uses only translations of its path pieces.
+
+The regeneration structure of Sznitman and Zerner~\cite{SznitmanZerner1999}
+is a central tool. We give the factorization and width arguments needed
+below, including the distinction between real projected heights and
+integer record indices. Simenhaus proved that distinct deterministic
+asymptotic directions occurring with positive probability must be
+opposite~\cite[Proposition~1]{Simenhaus2007}. His existence theorem for
+an asymptotic direction assumes transience on a nonempty open set of
+directions. Drewitz and Ram\'{i}rez obtained a corresponding two-ray
+description under sign-union transience on an open set, equivalently in
+$d$ linearly independent directions~\cite[Theorem~1.8]{DrewitzRamirez2010}.
+Those hypotheses do not follow here from the initial assumption in one
+direction. In our contradiction argument, the spatial radius estimate
+supplies the two rays first; a self-contained argument then proves
+their antipodality. The remaining estimates concern contacts between
+two arrangements of regeneration pieces.
+
+\subsection{Proof strategy}
+
+Two fresh-row arguments make the proof work without a common positive
+lower bound on the transition probabilities. First, finite-width slab escape
+(Lemma~\ref{lem:finite-supremum}) is
+proved using recurrent-site trials and first visits to new columns,
+in the spirit of \cite[Lemma~4]{ZernerMerkl2001}. Second, scripts launched
+from a tilted record use pairwise distinct, previously unused rows.
+Their annealed cost is a product of positive row averages, rather than
+a product of uniform quenched lower bounds.
+
+Suppose, toward a contradiction, that $0<P_0(A_\ell)<1$.
+A slab-exit argument and fresh-half-space regeneration first show that
+the path escapes in one of the two signs almost surely.
+At a strict record from which the path
+never falls below its new height, the future factors from the past.
+The path between successive such records is a finite \emph{regeneration
+piece}. Under conditioning never to backtrack, these pieces are
+independent and identically distributed. Their mean height width is
+finite, even for an irrational direction.
+
+The first main step, Proposition~\ref{prop:radius}, proves that
+coexistence of the two signs also forces finite mean spatial radius of
+a piece. A very large lateral
+excursion would create a record in a tilted direction. From that record,
+a fresh continuation escaping in the opposite sign has a probability
+bounded below independently of the excursion scale. This would give a
+fixed positive probability to arbitrarily large finite height maxima,
+contradicting the tail of their distribution.
+The radius bound gives deterministic limiting rays on the two escape
+events; the rays must be opposite. We may therefore reduce to
+coexistence in one coordinate direction.
+
+We then sample two independent sequences of coordinate regeneration
+pieces. Place positive pieces downward by successively anchoring their
+\emph{terminal} points, and place negative pieces downward in their
+ordinary chronological order. Individual path words retain their
+original orientation. Heights at which both sequences have a piece
+boundary divide the arrangement into independent common blocks.
+A \emph{contact} is a site from which both arranged paths depart.
+For each dyadic interval of common-block indices, consider whether a
+contact occurs. Let $m(J)$ be the sum of these probabilities over the
+first $J$ such intervals, as defined precisely in
+\eqref{eq:contact-count}.
+
+The two final steps give
+\[
+ m(J)=O\!\left(\frac{J}{\log J}\right),\qquad
+ m(J)\ge c\,\frac{J}{\log\log J}\quad\text{for large }J,
+\]
+with $c>0$, which are incompatible.
+For the upper bound, Proposition~\ref{prop:entropy-upper} compares the
+arrangement with independent bridges of prescribed heights. Randomizing
+the number of blocks keeps the cost of replacing each chunk by bridges
+of its realized height bounded.
+An independent stretch of blocks before the comparison interval turns the
+lateral alignment cost into an increment of displacement entropy. These
+increments telescope over dyadic scales, while finite mean widths make
+the bridge contact probabilities summable on those scales. A
+relative-entropy bound converts these two estimates into the stated bound
+for the expected number of contact intervals.
+
+For the lower bound, view a positive bridge in chronological order and
+start the negative path just below its random endpoint. A union bound
+over all possible endpoints would be too expensive. Instead, the
+conditional probabilities of the possible endpoints form a vector of
+annealed martingales. Lemma~\ref{lem:posterior-maxima} bounds the sum of
+their running maxima by an exponential tail on a logarithmic scale.
+In Proposition~\ref{prop:contact-bound}, first-contact prefixes are transferred to
+one shared environment before they are classified by quenched exit
+probabilities; the posterior functions remain in the unrevealed
+annealed path filtration. This separation controls the endpoint-dependent
+alignment without imposing a martingale property after the environment
+is revealed. A range of dyadic height intervals without a contact forces
+a late first contact near the top. Applying the estimate at the many
+regeneration boundaries in that range yields many contact height scales.
+The finite mean common-block width then converts the height
+count into the block-index count $m(J)$.
+
+Section~\ref{sec:regeneration} proves the path-weight and regeneration
+facts. Section~\ref{sec:radius} establishes the spatial moment bound.
+Section~\ref{sec:arrangement} reduces to coordinate heights and defines
+the common arrangement. Sections~\ref{sec:entropy} and
+\ref{sec:posterior} prove the entropy and endpoint-posterior estimates.
+Section~\ref{sec:contradiction} gives the lower contact count and
+completes the theorem.
diff --git a/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/build/02-regeneration.tex b/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/build/02-regeneration.tex
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+\section{Regeneration in a real direction}\label{sec:regeneration}
+
+Following the regeneration approach of Sznitman and
+Zerner~\cite{SznitmanZerner1999}, we construct finite path pieces whose
+successive translates are independent.  Their terminal times are selected using the entire future, so the usual
+stopping-time Markov property does not establish this independence.  An exact
+factorization of finite path weights will do so.  Counting ordinary records
+then gives a finite mean projected width, even when the projected heights do
+not lie in a discrete subgroup of $\RR$.
+
+\subsection{Departure rows and path weights}
+
+A finite nearest-neighbor word is a sequence
+$\gamma=(x_0,\ldots,x_m)$ with $x_{j+1}-x_j\in U$.  Its departure set and
+annealed weight are
+\[
+ \operatorname{Dep}(\gamma)=\{x_0,\ldots,x_{m-1}\},\qquad
+ W(\gamma)=\EE\!\left[
+       \prod_{j=0}^{m-1}\omega(x_j,x_{j+1}-x_j)\right],
+\]
+where the expectation in this display is over the environment.  Thus
+$W(\gamma)=P_{x_0}(X_j=x_j,\ 0\le j\le m)$.  A row visited repeatedly
+contributes repeatedly to the same product inside the expectation.  The
+terminal site contributes no row unless the word departed from it earlier.
+Translation invariance gives $W(\gamma+z)=W(\gamma)$.
+
+\begin{lemma}[Transfer through unused departure rows]
+\label{lem:departure-transfer}
+Let $\gamma=(x_0,\ldots,x_m)$ be a finite nearest-neighbor word and let
+$f(\omega)$ be a nonnegative measurable function of the rows outside
+$\operatorname{Dep}(\gamma)$.  Then
+\begin{equation}\label{eq:departure-transfer}
+ \EE\!\left[
+   \prod_{j=0}^{m-1}\omega(x_j,x_{j+1}-x_j)f(\omega)\right]
+       =W(\gamma)\EE[f].
+\end{equation}
+In particular, two prescribed words with disjoint departure sets have the
+same joint weight when sampled under independent annealed laws as when
+sampled independently conditional on one shared environment.
+
+The identity also applies when $f(\omega)$ is the quenched probability of an
+event for another walk stopped on its first arrival in
+$\operatorname{Dep}(\gamma)$.  Events that require this other walk to avoid
+that set forever are allowed.
+\end{lemma}
+
+\begin{proof}
+The product in \eqref{eq:departure-transfer} depends only on rows in
+$\operatorname{Dep}(\gamma)$, which are independent of all the rows defining
+$f$.  This proves the identity.  For two words, conditional independence in
+the shared environment gives the product of their quenched weights, and
+\eqref{eq:departure-transfer} factors its environment expectation.
+
+A trajectory stopped on arrival in a set does not use the row at that
+arrival site.  Its finite stopped-prefix probabilities therefore depend
+only on rows outside the set.  The same is true of every measurable event
+of the stopped trajectory, by the monotone class theorem.  This includes
+events on which the stopping time is infinite.
+\end{proof}
+
+The transfer is made for unclassified path weights.  A condition involving
+a quenched exit probability can depend on rows on both sides of the
+separation and cannot simply be inserted into the independent side of
+\eqref{eq:departure-transfer}.  Later we will first apply this identity and
+then classify sites in the shared environment.
+
+\subsection{Fresh records and true cuts}
+
+Fix $v\in\RR^d\setminus\{0\}$, normalized by $\|v\|_\infty=1$, and write
+$h(x)=x\cdot v$.  Define
+\[
+ D_v=\{h(X_n)\ge0\text{ for all }n\ge0\},\qquad
+ p_v=P_0(D_v),\qquad
+ \widehat P_v=P_0(\,\cdot\mid D_v)
+\]
+when $p_v>0$.  All statements in Section~\ref{sec:regeneration} concern this fixed real
+direction.  We write $\widehat E_v$ for expectation under
+$\widehat P_v$.  No arithmetic assumption on the coordinates of $v$ is
+imposed.
+
+The next lemma follows the repeated-trial approach of Zerner and
+Merkl~\cite[Lemma~4]{ZernerMerkl2001}. We give the argument to specify
+which transition rows remain unused at each trial.
+
+\begin{lemma}[Finite height supremum]\label{lem:finite-supremum}
+For this fixed real direction, almost surely under $P_0$,
+\[
+ \sup_{n\ge0}h(X_n)<\infty
+ \quad\Longrightarrow\quad h(X_n)\longrightarrow-\infty.
+\]
+In particular, a finite height interval cannot contain the path forever.
+The corresponding statements hold from every translated starting site,
+and hold quenched for almost every environment, simultaneously for all
+such starting sites.
+\end{lemma}
+
+\begin{proof}
+Fix nonnegative integers $b,k$, a signed coordinate step $a\in U$ with
+$h(a)=1$, and an integer $m>b+k$.  It suffices to rule out
+\[
+ E_{b,k}=\{h(X_n)\le b\text{ for every }n,
+                  \ h(X_n)\ge-k\text{ infinitely often}\}.
+\]
+Partition the lattice into columns, the cosets of $\ZZ a$.  Every column
+has only finitely many sites with height in $[-k,b]$: its successive
+heights differ by exactly one, regardless of the other coordinates of
+$v$.
+
+First suppose that only finitely many columns are visited at heights at
+least $-k$.  On $E_{b,k}$, some fixed site $x$ of height in $[-k,b]$ is
+then visited infinitely often.  Fix an environment whose transition
+entries are all positive.  At every visit to this particular site the
+quenched probability of the script consisting of $m$ steps $a$ is
+\[
+ q_x(\omega)=\prod_{j=0}^{m-1}\omega(x+ja,a)>0.
+\]
+Choose the first visit as a trial start, and subsequently choose the
+first visit at least $m$ steps after the preceding start.  These are
+stopping times.  By the quenched strong Markov property, the probability
+that $r$ such trials are finite and all fail is at most
+$(1-q_x(\omega))^r$.  Infinitely many visits provide infinitely many
+trials, while a successful script would cross above $b$.  Thus the
+specified event has quenched probability zero.  A countable union over
+$x$ and integration over the environment handle this first case.  There
+is no need to bound $q_x(\omega)$ uniformly in $x$ or $\omega$.
+
+For the other case, consider the first visit, for each column, to its
+portion of height at least $-k$.  These eligibility decisions are
+determined by the observed path prefix.  Select the first eligible time
+as a trial start, and thereafter the first eligible time at least $m$
+steps after the preceding trial start.  If infinitely many columns have
+such visits, infinitely many trial starts remain after these finite
+exclusions.  At an eligible time, every earlier visit in the same
+column had height below $-k$.  Consequently the $m$ scripted departure
+sites, all at height at least $-k$, are distinct and have not appeared
+earlier.  Conditional on any fixed eligible prefix, their rows retain
+their original independent laws.  The annealed conditional probability
+of the script is therefore exactly
+\[
+ q_a^m,\qquad q_a:=\EE[\omega(0,a)]>0.
+\]
+The probability that $r$ selected trials are finite and all fail is at
+most $(1-q_a^m)^r$.  On $E_{b,k}$ every one fails, so the second case is
+also null.  This argument uses no conditioning on the future event
+$E_{b,k}$ when evaluating a trial probability.
+
+Taking the countable union over $b,k$ proves the implication: a path
+from zero with finite height supremum that does not tend to minus
+infinity belongs to some $E_{b,k}$.  Translation gives the same
+annealed conclusion from every lattice site.  Integrating the quenched
+probabilities of these null events, and taking a countable intersection
+over starting sites and integer bounds, gives the last assertion.
+\end{proof}
+
+\begin{lemma}\label{lem:positive-nonbacktracking}
+If $P_0(A_v)>0$, then $p_v>0$.
+\end{lemma}
+
+\begin{proof}
+On $A_v$, the sequence $h(X_n)$ tends to $+\infty$ and therefore attains a
+global minimum.  Thus $A_v$ is contained in the countable union, over
+$n\ge0$ and $x\in\ZZ^d$, of the events
+\[
+ \{X_n=x,\ h(X_{n+k})\ge h(x)\text{ for every }k\ge0\}.
+\]
+At least one such event has positive probability.  Put
+$q_v(x,\omega)=P_{x,\omega}(h(X_k)\ge h(x)\text{ for all }k\ge0)$.
+The quenched Markov property at the deterministic time $n$ gives
+\[
+ \EE\big[P_{0,\omega}(X_n=x)q_v(x,\omega)\big]>0.
+\]
+Consequently $\EE[q_v(x,\omega)]>0$, and translation invariance identifies
+this expectation with $p_v$.  This argument does not use a Markov property
+at the time of the global minimum.
+\end{proof}
+
+The ordinary strict-record times are the stopping times
+\[
+ \rho_0=0,\qquad
+ \rho_{i+1}=\inf\{n>\rho_i:h(X_n)>h(X_{\rho_i})\},
+\]
+with all subsequent times set to infinity if one is infinite.  A finite
+$\rho_i$, $i\ge1$, is a \emph{true cut} if
+\[
+ h(X_n)\ge h(X_{\rho_i})\quad\text{for every }n\ge\rho_i.
+\]
+Truth of a cut is not measurable at its arrival time.
+Whenever true cuts exist, $\tau_1,\tau_2,\ldots$ enumerate them in time
+order.
+
+At an ordinary strict record, every earlier departure has height strictly
+below the current height.  The closed upper half-space is therefore
+unused.  Applying Lemma~\ref{lem:departure-transfer} to each finite record
+prefix and translating its endpoint gives
+\begin{equation}\label{eq:fresh-record}
+ P_0\big(h(X_{\rho_i+k})\ge h(X_{\rho_i})\text{ for all }k\ge0
+          \mid\cF_{\rho_i}\big)=p_v
+       \quad\text{on }\{\rho_i<\infty\}.
+\end{equation}
+This is an identity in the annealed path filtration.  Conditional on the
+whole environment, the corresponding probability would instead be
+$q_v(X_{\rho_i},\omega)$.
+
+For a word $\gamma=(0=x_0,\ldots,x_m=z)$, say that it is a
+\emph{regeneration word in direction $v$} if $m\ge1$ and the following two
+conditions hold:
+\begin{enumerate}
+ \item $0\le h(x_j)<h(z)$ for every $0\le j<m$;
+ \item whenever $1\le j<m$ is a strict record within the word, there is a
+       $k$ with $j<k<m$ and $h(x_k)<h(x_j)$.
+\end{enumerate}
+The second condition says that every intermediate record has already
+failed before the final arrival.  Define the width and radius of such a
+word by
+\[
+ L(\gamma)=h(z)>0,\qquad
+ R(\gamma)=\max_{0\le j\le m}\|x_j\|_2.
+\]
+Both are finite, but the width need not be an integer.
+
+\begin{lemma}[Regeneration words]\label{lem:slabs}
+Suppose $p_v>0$.  Under $\widehat P_v$, there are almost surely infinitely
+many true cuts
+$0=\tau_0<\tau_1<\tau_2<\cdots$, with time zero included as an initial
+boundary.  The relative words between consecutive cuts are independent
+and identically distributed, with probability law $\nu_v$ given by
+\[
+ \nu_v(\gamma)=W(\gamma)
+\]
+on the regeneration words in direction $v$.  The departure heights within
+each word belong to $[0,L)$ relative to its starting height.
+
+Under the unconditioned law, on $\{\sup_n h(X_n)=\infty\}$ a first true
+strict record exists almost surely.  Conditional on its finite prefix,
+the translated future has law $\widehat P_v$.
+\end{lemma}
+
+\begin{proof}
+We first establish existence using stopping-time trials.  Test the first
+ordinary strict record.  If the path later falls below its height, wait
+until the first strict record above the entire path seen at that failure
+time, and test again.  The candidates and the times at which failures are
+detected are stopping times.  By \eqref{eq:fresh-record}, the probability
+that the first $k$ tested candidates are finite and fail is at most
+$(1-p_v)^k$.  On $\{\sup_n h(X_n)=\infty\}$, a new candidate is available
+after every detected failure.  Hence a true cut exists there almost surely.
+
+Under $\widehat P_v$, the height supremum is infinite almost surely by
+Lemma~\ref{lem:finite-supremum}: its alternative conclusion of
+escape to minus infinity is incompatible with $D_v$.  In particular the
+first true cut is finite under $\widehat P_v$.  This application of
+Lemma~\ref{lem:finite-supremum} is
+valid before any independence or integrability of slabs has been proved.
+
+We now compute its law.  Fix a regeneration word
+$\gamma=(0,\ldots,z)$ of length $m$ and a measurable event $B$ for the
+translated infinite suffix.  The first true-cut word is $\gamma$ exactly
+when this prefix occurs and the suffix from $z$ belongs to $D_v$.
+For necessity, an earlier record not invalidated by time $m$ could never
+be invalidated by a future remaining above $h(z)$, so it would be an
+earlier true cut.  For sufficiency, the word's intermediate records have
+all failed, while its terminal record is true.
+
+The finite prefix uses only rows below $h(z)$, whereas its suffix
+constrained by $D_v$ uses only rows at or above $h(z)$.  Thus
+\[
+ P_0\big((X_0,\ldots,X_m)=\gamma,\ \tau_1=m,
+         \ (X_{m+k}-z)_{k\ge0}\in B\big)
+       =W(\gamma)P_0(B\cap D_v).
+\]
+The event on the left implies the original $D_v$.  Division by $p_v$
+therefore gives
+\begin{equation}\label{eq:slab-factorization}
+ \widehat P_v\big(\Gamma_1=\gamma,
+               \ (X_{\tau_1+k}-X_{\tau_1})_{k\ge0}\in B\big)
+          =W(\gamma)\widehat P_v(B).
+\end{equation}
+Here $\Gamma_1$ denotes the first relative word.  Since $\tau_1$ is almost
+surely finite, summing over all such words proves that their weights sum
+to one.  Equation~\eqref{eq:slab-factorization} gives an independent suffix
+with the original conditioned law, and iteration proves the iid
+assertion.  The departure-height claim is part of the word definition.
+
+For the unconditioned first true cut, the same word enumeration allows a
+prefix that goes below zero.  Its endpoint is still a strict global
+record, and each earlier strict record must have failed before that
+endpoint.  The fresh-suffix factor is again $P_0(B\cap D_v)$, so its
+conditional translated law is $\widehat P_v$.  No Markov property at a
+future-dependent cut has been used.
+\end{proof}
+
+We call these relative words \emph{slabs}.  The preceding proof gives
+their complete law, not just independence of their displacements.  This
+will allow us to arrange whole slabs in different spatial positions.
+
+\subsection{Mean width and the two signs}
+
+The record heights can have arbitrarily small positive increments in a
+real direction.  The next argument instead renews in the integer index
+of an ordinary record.
+
+\begin{lemma}[Finite mean width]\label{lem:mean-width}
+Suppose $p_v>0$ and $\|v\|_\infty=1$.  If $S$ is the number of ordinary
+strict records after the initial boundary of a slab with law $\nu_v$,
+including its terminal record, then
+\[
+ 0<\mathbb E_{\nu_v}L\le\mathbb E_{\nu_v}S\le p_v^{-1}<\infty.
+\]
+\end{lemma}
+
+\begin{proof}
+For $i\ge1$, let
+\[
+ B_i=\{\rho_i<\infty,\ h(X_n)\ge0\text{ for }0\le n\le\rho_i\}.
+\]
+Freshness at $\rho_i$ and cancellation of the conditioning probability
+give
+\begin{equation}\label{eq:record-renewal-mass}
+ \widehat P_v(i\text{ is a true-record index})
+   =\frac{P_0(B_i,\ \rho_i\text{ is true})}{p_v}
+   =P_0(B_i)\ge p_v.
+\end{equation}
+The last inequality follows because $D_v$ forces an infinite supremum
+and hence reaches every ordinary record index before dropping below zero.
+
+At a cut, its height is the maximum of the entire past.  The subsequent
+ordinary records are therefore exactly the records of the translated
+suffix.  By Lemma~\ref{lem:slabs}, the successive true-record indices are
+sums of iid positive integers $S_1,S_2,\ldots$ with the law of $S$.  Let
+\[
+ N(n)=\#\{k\ge1:S_1+\cdots+S_k\le n\}.
+\]
+Summing \eqref{eq:record-renewal-mass} yields
+$\widehat E_v[N(n)/n]\ge p_v$.
+If $\mathbb E_{\nu_v}S=\infty$, the strong law applied to every bounded
+truncation $S_j\wedge a$ gives
+$(S_1+\cdots+S_k)/k\to\infty$.  Inverting these sums gives
+$N(n)/n\to0$ almost surely.  Since $0\le N(n)/n\le1$, dominated
+convergence contradicts the preceding lower bound.  Thus the mean of
+$S$ is finite.  The ordinary strong law and the same inversion now give
+$N(n)/n\to1/\mathbb E_{\nu_v}S$; dominated convergence proves
+$\mathbb E_{\nu_v}S\le1/p_v$.
+
+Each ordinary record height gain is at most one: its final step has
+height increment at most $\|v\|_\infty=1$, and starts no higher than the
+preceding maximum.  Adding the gains within one slab gives $L\le S$.
+Finally, $L>0$ almost surely, so its mean is strictly positive.
+\end{proof}
+
+Lemma~\ref{lem:mean-width} controls record count and projected width, not the number of
+time steps in a slab.
+
+\begin{lemma}[The two directional signs]\label{lem:sign-union}
+For every fixed nonzero real direction $v$, if $P_0(A_v)>0$, then
+\[
+ P_0(A_v\cup A_{-v})=1.
+\]
+More precisely, up to null sets, infinite height supremum implies $A_v$
+and finite height supremum implies $A_{-v}$.
+\end{lemma}
+
+\begin{proof}
+Normalize $v$ as above.  Lemma~\ref{lem:positive-nonbacktracking} gives
+$p_v>0$.  On infinite height supremum, Lemma~\ref{lem:slabs} provides a
+first true cut followed by iid slabs.  Their positive widths have finite,
+strictly positive mean by Lemma~\ref{lem:mean-width}, so the successive
+base heights tend to infinity.  Every later slab stays at or above its
+base, and consequently $h(X_n)\to+\infty$.
+
+The finite-supremum implication is exactly
+Lemma~\ref{lem:finite-supremum}.  These two alternatives exhaust
+all paths and prove the sign union.
+\end{proof}
+
+Thus a probability strictly between zero and one for $A_v$ supplies
+positive probabilities for both signs.  In Section~\ref{sec:radius}, that
+coexistence assumption will give the stronger spatial estimate
+$\mathbb E_{\nu_v}R+\mathbb E_{\nu_{-v}}R<\infty$.
diff --git a/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/build/03-radius.tex b/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/build/03-radius.tex
new file mode 100644
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+++ b/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/build/03-radius.tex
@@ -0,0 +1,365 @@
+\section{Finite spatial radii under coexistence}\label{sec:radius}
+
+The regeneration construction controls the advance of a slab in its chosen
+direction.  We now control its full spatial extent, assuming that escape in
+both directions has positive probability.  The idea is to turn a very large
+slab into a new record for a slightly tilted linear functional.  From that
+record a fresh path escaping in the opposite direction can preserve the
+current height maximum forever.  If mean spatial radii were infinite, this
+would give a fixed positive probability that the final maximum is finite
+but arbitrarily large, which is impossible.
+
+\begin{proposition}[Spatial radius under coexistence]\label{prop:radius}
+Let the environment consist of iid strictly positive transition rows,
+and let
+\(v\in\RR^d\setminus\{0\}\).  Suppose
+\(P_0(A_v)>0\) and \(P_0(A_{-v})>0\).  For each sign
+\(\sigma\in\{+,-\}\), let \(\nu_{\sigma v}\) be the slab law from
+Lemma~\ref{lem:slabs}.  If \(R\) is the maximum Euclidean distance from
+a slab's initial site, including its terminal site, then
+\[
+ \int R\,d\nu_v+\int R\,d\nu_{-v}<\infty.
+\]
+\end{proposition}
+
+This is a spatial moment statement.  It asserts no moment bound on the
+number of steps in a slab.
+
+We first record a maximal inequality that will control all initial portions
+of a sequence of slabs at once. It is a consequence of Hopf's maximal
+ergodic theorem; see Garsia's proof~\cite{Garsia1965}. The interval argument
+below gives the precise stationary-sequence form that we use.
+
+\begin{lemma}[Initial averages]\label{lem:initial-averages}
+Let \((Z_j)_{j\ge1}\) be a stationary sequence of nonnegative integrable
+random variables.  For every \(a>0\) and positive integer \(n\),
+\[
+ P\left(\max_{1\le k\le n}\frac1k\sum_{j=1}^kZ_j>a\right)
+ \le \frac{EZ_1}{a}.
+\]
+\end{lemma}
+
+\begin{proof}
+Call an integer \(s\) a bad start if an interval beginning at \(s\), of
+length at most \(n\), has sum greater than \(a\) times its length.
+Among starts \(1,\ldots,M\), take the leftmost bad start, choose a
+witness interval, skip all its indices, and repeat.  The chosen intervals
+are disjoint, cover every bad start, and lie in \(\{1,\ldots,M+n\}\).
+Their total length is at least the number of bad starts.  Therefore
+\[
+ a\,\#\{s\le M:s\text{ is bad}\}
+ \le \sum_{j=1}^{M+n}Z_j.
+\]
+Taking expectations, using stationarity, and letting \(M\to\infty\)
+proves the assertion.
+\end{proof}
+
+\begin{proof}[Proof of Proposition~\ref{prop:radius}]
+The construction has two parts.  A displacement allowance \(a\) and a
+number \(N\) of slabs will be chosen so that most initial portions have
+total radius at most \(a\) times their length, but some slab has a much
+larger excursion with probability bounded below.  That excursion supplies
+a record for a small tilt of the opposite height direction.  The
+continuation then avoids the old prefix in two successive regions: it
+stays beyond the tilted record for a fixed multiple of \(N\) slabs, and
+after their final true cut it stays at negative height.  The estimates
+below arrange these two separations with a continuation probability
+independent of \(a\) and \(N\).
+
+Rescale \(v\) so that \(\|v\|_\infty=1\).  For either sign, under
+\(\widehat P_{\sigma v}\) let \((L_i,R_i)_{i\ge1}\) denote the widths
+and radii of its iid slabs, and put \(H_k=\sum_{i=1}^kL_i\), with
+\(H_0=0\).  Write \(E_\sigma\) for expectation in this slab sequence.
+Lemma~\ref{lem:mean-width} gives
+\(0<E_\sigma L<\infty\).  A point in slab \(k+1\) has signed height
+at least \(H_k\), and the displacement of any point through slab
+\(k\) has norm at most \(\sum_{j=1}^kR_j\).
+
+\paragraph{Scales with controlled initial portions.}
+Suppose, to obtain a contradiction, that
+\[
+ I(t):=\sum_{\sigma\in\{+,-\}}E_\sigma\min(R,t)
+ \longrightarrow\infty.
+\]
+The function \(I\) is increasing and concave, \(I(0)=0\), and
+\(I(t)=o(t)\).  The last assertion follows by bounded convergence from
+\(R<\infty\) almost surely.
+
+By the two width strong laws we may fix \(c>0\) and finite
+\(C,C_0\ge0\) such that, in each sign,
+\begin{equation}\label{eq:radius-height-bounds}
+ \widehat P_{\sigma v}
+ \bigl(ck-C_0\le H_k\le Ck+C_0\text{ for every }k\ge0\bigr)>.95.
+\end{equation}
+Indeed choose \(c\) below both means and \(C\) above both means, then
+choose \(C_0\) to bound the almost surely finite maximal deviations
+with the displayed probability.  Fix, in this order, an integer \(C_1\),
+numbers \(b,B\), and \(\eta>0\), so that
+\begin{equation}\label{eq:radius-constants}
+ C_1c>C+4,\qquad b>1/c,\qquad
+ \frac{B-1}{\sqrt d}-b(C+2)>2,\qquad 4C_1\eta<.15.
+\end{equation}
+The multiplier \(C_1\) gives the opposite continuation enough slabs to
+descend below height zero.  The coefficient \(b\) makes its height
+descent dominate its lateral displacement in the tilted functional;
+\(B\) ensures that a large excursion creates a new tilted record.
+Finally, \(\eta\) controls the probability that an initial portion exceeds
+its displacement allowance.  These constants remain fixed throughout
+the proof.
+
+For large \(a\), let \(N=N(a)\) be the smallest positive integer
+such that \(I(aN)\ge\eta a\).  Such an integer exists.  Since
+\(I(am)/a\to0\) for every fixed \(m\), we have \(N(a)\to\infty\).
+For sufficiently large \(a\), minimality and concavity give
+\begin{equation}\label{eq:radius-scale}
+ \eta a\le I(aN)
+ \le \frac{N}{N-1}I(a(N-1))<2\eta a.
+\end{equation}
+
+In either sign and for \(m\ge0\), define \(\mathcal R_m\) to be the
+event that, for every \(k\le m\),
+\[
+ ck-C_0\le H_k\le Ck+C_0,
+ \qquad \sum_{j=1}^kR_j\le ak.
+\]
+We claim that
+\begin{equation}\label{eq:radius-good-prefix}
+ \widehat P_{\sigma v}(\mathcal R_{C_1N})>.8.
+\end{equation}
+To see this, put \(n=C_1N\), \(A=an\), and truncate each radius to
+\(\widehat R_j=\min(R_j,A)\).  Concavity and
+\eqref{eq:radius-scale} give
+\[
+ E_\sigma\widehat R\le I(A)\le C_1 I(aN)<2C_1\eta a.
+\]
+Lemma~\ref{lem:initial-averages} bounds by \(2C_1\eta\) the
+probability that any of the first \(n\) averages of the truncated
+radii exceeds \(a\).  Also
+\[
+ \widehat P_{\sigma v}(\exists j\le n:R_j>A)
+ \le n\widehat P_{\sigma v}(R>A)
+ \le \frac{nE_\sigma\min(R,A)}{A}
+ \le2C_1\eta.
+\]
+Intersecting these estimates with \eqref{eq:radius-height-bounds}
+proves \eqref{eq:radius-good-prefix}.  In particular,
+\(\widehat P_{\sigma v}(\mathcal R_{i-1})>.8\) for \(i\le N\),
+and \(\mathcal R_{i-1}\) depends only on the first \(i-1\) slabs.
+
+\paragraph{A large slab after a controlled initial portion.}
+Choose a lower index \(i_0\) so large that
+\begin{equation}\label{eq:radius-width-tail}
+ \sum_\sigma\sum_{i\ge i_0}\widehat P_{\sigma v}(L>i)
+ <\frac{\eta}{4B}.
+\end{equation}
+This is possible by integrability of the widths.  Increase \(i_0\)
+if necessary so that, for every \(i\ge i_0\),
+\begin{equation}\label{eq:radius-index-margins}
+ c(i-1)-C_0\ge ci/2,
+ \qquad C(i-1)+C_0+i\le(C+2)i.
+\end{equation}
+After fixing \(i_0\), take \(a\) sufficiently large that \(N\ge i_0\)
+and \(I(Bai_0)<\eta a/4\).
+
+For \(i_0\le i\le N\), call
+\[
+ G_i=\{R_i>Bai,\ L_i\le i\}
+\]
+the large-slab event, and put
+\(\lambda_\sigma=\sum_{i=i_0}^N\widehat P_{\sigma v}(G_i)\).
+Integrating the decreasing radius tails gives
+\begin{align*}
+ \sum_\sigma\sum_{i=i_0}^N\widehat P_{\sigma v}(R>Bai)
+ &\ge \frac{I(Ba(N+1))-I(Bai_0)}{Ba},\\
+ \sum_\sigma\sum_{i=1}^N\widehat P_{\sigma v}(R>Bai)
+ &\le\frac{I(BaN)}{Ba}\le2\eta.
+\end{align*}
+For the last bound, use \(I(BaN)\le B I(aN)\).  For the lower
+bound, use \(I(Ba(N+1))\ge I(aN)\ge\eta a\), then subtract
+\eqref{eq:radius-width-tail}.  The result is
+\[
+ \frac{\eta}{2B}\le\lambda_++\lambda_-\le2\eta.
+\]
+Thus one sign, which may depend on the scale, satisfies
+\(\lambda_\sigma\ge\eta/(4B)\).
+
+In this sign consider
+\[
+ Z=\sum_{i=i_0}^N\1_{\mathcal R_{i-1}}\1_{G_i}.
+\]
+Independence of the \(i\)-th slab from its preceding slabs gives
+\(E_\sigma Z\ge.8\lambda_\sigma\).  The summands of \(Z\) need
+not be independent, but \(Z\le\sum_i\1_{G_i}\), whose summands
+are independent.  Hence
+\[
+ E_\sigma Z^2\le\lambda_\sigma+\lambda_\sigma^2,
+ \qquad
+ \widehat P_{\sigma v}(Z>0)
+ \ge\frac{.64\lambda_\sigma}{1+\lambda_\sigma}
+ \ge\frac{.64\eta}{4B(1+2\eta)}=:\delta>0.
+\]
+The constant \(\delta\) is independent of \(i_0\) and \(a\).
+
+We have therefore found, with a probability independent of the scale,
+a slab whose spatial excursion is much larger than the accumulated
+displacement before it, while its height width is at most its index.
+The next step converts that excursion into a stopping time from which
+an opposite-going path can use fresh environment rows.
+
+\paragraph{A tilted record.}
+Write \(h(x)=\sigma v\cdot x\) for the sign just selected.  On
+\(\mathcal R_{i-1}\cap G_i\), the initial site of slab \(i\) has
+norm at most \(a(i-1)\).  Some point of the slab consequently has
+norm greater than \((B-1)ai\).  At that point some signed coordinate
+\(u\in\{\pm e_1,\ldots,\pm e_d\}\) satisfies
+\[
+ u\cdot X>\frac{B-1}{\sqrt d}ai.
+\]
+Throughout this slab its running \(h\)-maximum is at least
+\(H_{i-1}\ge ci/2\), and at most
+\(H_i\le(C+2)i\), by \eqref{eq:radius-index-margins}.
+
+Define the linear functional
+\[
+ Y(x)=u\cdot x-ba h(x).
+\]
+Before slab \(i\), nonnegative height and \(\mathcal R_{i-1}\)
+give \(Y(X)\le a(i-1)\).  At the selected point,
+\[
+ Y(X)>\left(\frac{B-1}{\sqrt d}-b(C+2)\right)ai>2ai.
+\]
+There is thus a strict \(Y\)-record while the running \(h\)-maximum
+lies in the deterministic window
+\begin{equation}\label{eq:radius-window}
+ W=[ci_0/2,(C+2)N].
+\end{equation}
+
+For each fixed choice of \(\sigma,u,i_0,a\), let \(T\) be the first
+positive integer time such that the following three conditions hold:
+the entire prefix has nonnegative \(h\)-height; \(Y(X_T)\) is a
+strict record; and the running \(h\)-maximum belongs to \(W\).
+These are conditions on the observed prefix, so \(T\) is a stopping
+time for \((\cF_n)\).  It need not be a true regeneration time.
+The preceding construction and a union over the \(2d\) signed
+coordinates show that some \(u\) satisfies
+\[
+ \widehat P_{\sigma v}(T<\infty)\ge\frac{\delta}{2d}.
+\]
+Set \(p_*:=\min(p_v,p_{-v})>0\).  Removing the conditioning yields
+\begin{equation}\label{eq:radius-launch-probability}
+ P_0(T<\infty)\ge\frac{p_*\delta}{2d}.
+\end{equation}
+The site \(X_T\) has not appeared in its earlier prefix, since it is
+a strict \(Y\)-record.
+
+\paragraph{Preserving the maximum with a fresh continuation.}
+Since \(\|v\|_\infty=1\), there is a nearest-neighbor step \(e\)
+with \(h(e)=-1\).  From \(X_T\) force \(M\) repetitions of this
+step, where the fixed integer \(M\) will be chosen shortly.  Each
+step increases \(Y\) by at least \(ba-1\).  For all sufficiently
+large \(a\), its successive sites have strictly increasing
+\(Y\)-coordinates, starting at the strict record \(X_T\).
+Consequently every scripted departure site is distinct and unused by
+the prefix.  Conditional on any fixed prefix realizing \(T\), the
+script has the exact raw probability
+\[
+ \bigl(\EE[\omega(0,e)]\bigr)^M\ge\alpha^M,
+ \qquad \alpha:=\min_{e'\in U}\EE[\omega(0,e')]>0.
+\]
+Here independence is across departure rows, not across the entries in
+a single row.  The constant \(\alpha\) is positive because \(U\) is
+finite and every transition entry is positive almost surely.  It is
+not a lower bound on the transition entries in individual environments.
+
+At the script endpoint consider a translated opposite-going template
+with law \(\widehat P_{-\sigma v}\), and require
+\(\mathcal R_{C_1N}\) for its first \(C_1N\) true-cut slabs.
+This requirement includes existence of those cuts and concerns the
+entire template; they are not declared to be stopping times.  It has
+conditional-template probability greater than \(.8\).  At any point
+of its slab \(k+1\), where \(0\le k<C_1N\), the coordinate
+displacement in direction \(u\) is at least \(-a(k+1)\), while
+the displacement in the original height \(h\) is at most
+\(-ck+C_0\).  Its total \(Y\)-increment from the launching record
+is consequently at least
+\begin{equation}\label{eq:radius-template-separation}
+ \begin{split}
+ M(ba-1)-a(k+1)+ba(ck-C_0)
+ &=a\bigl(Mb-1-bC_0+(bc-1)k\bigr)-M.
+ \end{split}
+\end{equation}
+Choose \(M\) once and for all so that \(Mb>1+bC_0+1\).
+Since \(bc>1\), the expression in
+\eqref{eq:radius-template-separation} is strictly positive for all
+\(k\) in question and all sufficiently large \(a\).  Thus these
+template sites avoid the entire original prefix, whose \(Y\)-values
+are at most the launching record.
+
+There are two additional separations to check.  First, the whole
+opposite template stays at or below its own initial \(h\)-height,
+whereas every script departure lies strictly above that height.
+It therefore avoids all script departures, including in its infinite
+tail.  Second, after \(C_1N\) template slabs its absolute
+\(h\)-height is at most
+\[
+ (C+2)N-M-cC_1N+C_0<0
+\]
+for large \(N\).  This endpoint is a true opposite cut, so its
+remaining tail stays below zero.  That tail avoids the original
+prefix, all of whose heights are nonnegative.  No assumption that
+the script itself stays above zero is needed.
+
+These inequalities make the use of fresh rows exact.  Condition on a
+fixed finite prefix realizing \(T\), and then on the forced script.
+The conditional environment differs from the original iid law only
+at departure sites of these two finite words.  Under an independent
+raw law, the probability of
+\(D_{-\sigma v}\cap\mathcal R_{C_1N}\) is at least
+\(.8p_{-\sigma v}\).  Every path in that event avoids all the old
+departure sites after translation to the script endpoint, by the
+preceding separations.  Lemma~\ref{lem:departure-transfer} therefore
+identifies its probability with the corresponding fresh-row
+probability after the prefix and script.  More explicitly, kill both
+the continuation and an independent raw chain on arrival at the same
+set of old departure sites.  The finite-word identity gives equality
+of their cylinder probabilities, hence of their killed path laws.
+Now restrict to \(D_{-\sigma v}\cap\mathcal R_{C_1N}\), whose paths
+never hit that set.  Repeated sites within the template are retained
+in its raw path weights.
+
+It follows that, from every qualifying prefix, the script followed by
+the required continuation has conditional probability at least
+\(.8p_*\alpha^M\).  This argument uses finite-prefix conditioning
+and departure-row factorization, not a Markov property at a
+future-dependent true cut.  The script decreases \(h\), and the
+template never exceeds its own starting height.  Hence the final
+supremum of \(h\) on this continuation is its running supremum at
+\(T\), which belongs to \(W\).
+
+\paragraph{The finite final maximum.}
+Together with \eqref{eq:radius-launch-probability}, this proves that
+for every sufficiently large \(i_0\), after choosing \(a\) large
+enough, some sign \(\sigma\) satisfies
+\begin{equation}\label{eq:radius-final-maximum}
+ P_0\left(\sup_{n\ge0}\sigma v\cdot X_n
+       \in[ci_0/2,(C+2)N(a)]\right)
+ \ge \rho,
+ \qquad
+ \rho:=\frac{.8p_*^2\alpha^M\delta}{2d}>0.
+\end{equation}
+The constant \(\rho\) does not depend on the scale.  For each sign let
+\(M_\sigma=\sup_{n\ge0}\sigma v\cdot X_n\), allowing the value
+\(+\infty\).  Equation~\eqref{eq:radius-final-maximum} implies
+\[
+ \sum_{\sigma\in\{+,-\}}
+ P_0\bigl(ci_0/2\le M_\sigma<\infty\bigr)\ge\rho
+\]
+for every sufficiently large \(i_0\).  Both terms on the left tend to
+zero as \(i_0\to\infty\), a contradiction.  Thus \(I(t)\) is bounded, and
+monotone convergence proves Proposition~\ref{prop:radius}.
+\end{proof}
+
+Proposition~\ref{prop:radius} supplies integrable slab displacements as well as
+integrable errors between slab endpoints.  Section~\ref{sec:arrangement} uses these
+spatial facts to compare the two limiting directions and then to arrange
+the paths along a common coordinate height.
diff --git a/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/build/04-arrangement.tex b/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/build/04-arrangement.tex
new file mode 100644
index 0000000000000000000000000000000000000000..61a85d9cfd15dbe7b8edb6c902331e14b22bade2
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+++ b/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/build/04-arrangement.tex
@@ -0,0 +1,330 @@
+\section{Rays and a common renewal arrangement}\label{sec:arrangement}
+
+The radius estimate first reduces a mixed real direction to a mixed
+coordinate direction.  Integer heights then let us place two independent
+slab sequences in a common system of layers.  We will compare two bounds
+for contacts between these sequences.
+
+\subsection{Reduction to a coordinate direction}
+
+\begin{proposition}\label{prop:coordinate-reduction}
+Suppose that the annealed walk satisfies
+$0<P_0(A_v)<1$ for some $v\in\RR^d\setminus\{0\}$.
+There is a coordinate $i$ for which
+\[
+ P_0(A_{e_i})>0\qquad\text{and}\qquad P_0(A_{-e_i})>0.
+\]
+\end{proposition}
+
+\begin{proof}
+By Lemma~\ref{lem:sign-union}, both signs of $v$ have positive
+probability and $P_0(A_v\cup A_{-v})=1$.
+Normalize $\|v\|_\infty=1$.
+For $\sigma\in\{+,-\}$, let $D_j^\sigma$ be the displacement of
+the $j$th slab of the $\sigma v$ sequence, and let $R_j^\sigma$ be
+its radius.  Proposition~\ref{prop:radius} and the bound
+$|D_j^\sigma|\le R_j^\sigma$ give the mean vector
+\[
+ b_\sigma=E_\sigma D_1^\sigma,
+ \qquad b_\sigma\cdot(\sigma v)=E_\sigma L>0.
+\]
+In particular $b_\sigma\ne0$.
+The strong law gives convergence of the $k$th slab endpoint divided
+by $k$ to $b_\sigma$.  Moreover, for each $\varepsilon>0$,
+\[
+ \sum_{k\ge1}\widehat P_{\sigma v}(R_k^\sigma>\varepsilon k)<\infty,
+\]
+so $R_k^\sigma/k\to0$ almost surely.  Between the $k$th and
+$(k+1)$st endpoints, the displacement from the former is at most
+$R_{k+1}^\sigma$.  The number of completed slabs tends to infinity.
+Thus the entire trajectory, after its first true cut, has limiting
+unit direction
+\[
+ r_\sigma=\frac{b_\sigma}{|b_\sigma|}.
+\]
+Lemma~\ref{lem:slabs} transfers this conclusion to the raw law on
+$A_{\sigma v}$.  This argument uses slab count rather than elapsed
+time and requires no moment of a slab's duration.
+
+The two rays are distinct, since their projections on $v$ have
+opposite signs.  The following argument proves the special case of
+Simenhaus's antipodality principle needed here~\cite[Proposition~1]{Simenhaus2007}.
+We claim that $r_-=-r_+$.  Otherwise
+$w=r_++r_-$ satisfies
+\[
+ w\cdot r_+=w\cdot r_-=1+r_+\cdot r_->0.
+\]
+The preceding endpoint and radius estimates then imply $A_w$ on
+both original directional events, so $P_0(A_w)=1$.
+Under $\widehat P_w$, the limiting ray therefore exists almost
+surely.  By Lemma~\ref{lem:slabs}, this law is represented by an
+iid sequence of finite relative slab words.  Deleting finitely
+many words removes a finite time prefix and translates all remaining
+positions by a fixed vector.  Since $A_w$ holds, positions tend to
+infinity in norm, and these operations preserve the limiting ray.
+The ray is consequently a tail variable of the iid word sequence.
+The tail zero--one law, applied to a countable determining family
+of Borel subsets of the unit sphere, makes it deterministic.
+No displacement moment in direction $w$ is needed here.
+
+The raw walk has a first true $w$ cut almost surely, and its
+conditional future there has law $\widehat P_w$ by
+Lemma~\ref{lem:slabs}.  Its raw limiting ray must therefore be that
+same deterministic ray.  This contradicts the positive
+probabilities of the two distinct rays $r_+$ and $r_-$.
+The claim follows.  Choose a coordinate with nonzero projection
+on $r_+$; on $A_v$ and $A_{-v}$ the corresponding coordinate
+tends to opposite infinities, proving
+Proposition~\ref{prop:coordinate-reduction}.
+\end{proof}
+
+It now suffices to rule out a mixed coordinate direction.  Relabel
+coordinates so that it is $e_1$, and write $p_\pm=p_{\pm e_1}>0$.
+Write $D_\sigma=D_{\sigma e_1}$ and
+$\widehat P_\sigma=\widehat P_{\sigma e_1}$ for either sign.
+The slab laws $\nu_\pm=\nu_{\pm e_1}$ have positive integer widths;
+$E_\pm$ will now denote expectation for these coordinate slab laws.
+Lemma~\ref{lem:mean-width} and Proposition~\ref{prop:radius},
+applied to this coordinate, give
+\begin{equation}\label{eq:coordinate-moments}
+ E_\pm L<\infty,\qquad E_\pm R<\infty.
+\end{equation}
+All heights in the remaining proof are integer coordinate heights.
+In particular, a walk started inside a finite coordinate-height
+interval exits it almost surely: a path confined to that interval
+would have finite height supremum without tending to negative
+infinity, contrary to Lemma~\ref{lem:finite-supremum}.
+Translation invariance and integration give this exit property
+quenched for almost every environment, simultaneously for all
+integer barriers and starting sites.  Only a countable intersection
+is needed.  This property does not require uniform ellipticity.
+
+\subsection{Exact-cut bridges and renewal tails}
+
+For a sign $\sigma\in\{+,-\}$ define, with starting point zero,
+\[
+ T_j^\sigma=\inf\{n\ge0:\sigma X_n\cdot e_1=j\},\qquad
+ u_j^\sigma=P_0(T_j^\sigma<T_{-1}^\sigma),\qquad
+ F_\sigma(j)=\nu_\sigma(L>j),\quad j\ge0.
+\]
+When only one sign is being discussed we omit $\sigma$.
+A relative slab word starts at zero; its raw weight is the function
+$W$ defined in Section~\ref{sec:regeneration}.
+
+\begin{lemma}[Exact-cut bridges]\label{lem:exact-bridge}
+For either sign, the probability under $\widehat P_\sigma$ of a
+slab cut at height $H\ge0$ is $u_H$.  The numbers $u_H$ satisfy
+\[
+ u_0=1,\qquad 1\ge u_H\ge u_{H+1}\ge p_\sigma.
+\]
+Conditional on a cut at $H$, the list of slabs through that cut has
+law $\mathcal B_H^\sigma$ given by
+\begin{equation}\label{eq:exact-bridge-list}
+ \mathcal B_H^\sigma(\gamma_1,\ldots,\gamma_m)
+ =\frac{1}{u_H^\sigma}\prod_{i=1}^m W(\gamma_i),
+ \qquad \sum_{i=1}^m L(\gamma_i)=H.
+\end{equation}
+Concatenation gives the raw path stopped on its first hit of $H$,
+conditional on reaching $H$ before $-1$, in the signed height.
+Reversing the order of the list preserves $\mathcal B_H^\sigma$;
+every word is still traversed in its original chronological order.
+\end{lemma}
+
+\begin{proof}
+On $\{T_H<T_{-1}\}$, all departures before $T_H$ have height less
+than $H$.  The future non-backtracking event from the fresh endpoint
+has probability $p_\sigma$.  Its factor cancels the denominator
+in conditioning on $D_\sigma$, giving the cut probability $u_H$.
+The hitting events are nested in $H$, and a non-backtracking walk
+reaches every height, proving the inequalities.
+
+A finite path first hitting $H$ before $-1$ has a unique slab
+decomposition.  Its cuts are the strict record times whose heights
+are never undercut before the terminal time, together with the
+initial boundary.  Between two successive cuts all intermediate
+strict records are invalidated before the second cut.  These are
+exactly the qualifying words of Lemma~\ref{lem:slabs}.
+Conversely, concatenating any such list of total width $H$ gives
+one of these bridges.  Distinct words have disjoint departure-height
+intervals, so their raw weights multiply.  Their total mass is
+$u_H$, which proves~\eqref{eq:exact-bridge-list} and the bridge
+identification.  Reversing the list preserves its total width and
+the product of its word weights.  It is a bijection on admissible
+lists, proving the last assertion.  At $H=0$ the list is empty.
+\end{proof}
+
+\begin{lemma}[Renewal-tail comparison]\label{lem:renewal-tail}
+For either sign and all integers $n'\ge n\ge0$,
+\begin{equation}\label{eq:renewal-tail}
+ 0\le u_n-u_{n'}\le(n'-n)F(n).
+\end{equation}
+Moreover,
+\begin{equation}\label{eq:dyadic-width-tail}
+ \sum_{l\ge0}2^lF(2^l)<\infty.
+\end{equation}
+\end{lemma}
+
+\begin{proof}
+The slab cut heights form a renewal process with increments $L$.
+Partitioning according to its last cut at or before $n$ gives
+\[
+ \sum_{k=0}^n F(k)u_{n-k}=1.
+\]
+Subtracting this identity at $n+1$ from the identity at $n$ yields
+\[
+ \sum_{k=0}^n F(k)(u_{n-k}-u_{n+1-k})=F(n+1).
+\]
+Every term on the left is nonnegative, and $F(0)=1$.
+Thus $u_n-u_{n+1}\le F(n+1)\le F(n)$; telescoping proves
+\eqref{eq:renewal-tail}.  Finally
+$\sum_{k\ge0}F(k)=EL<\infty$, and monotonicity of $F$ gives
+\eqref{eq:dyadic-width-tail} by grouping this sum into dyadic
+intervals.
+\end{proof}
+
+\subsection{Two tapes placed from a common top}
+
+Berger's backward-path construction also assembles regeneration slabs
+from a terminal reference point~\cite[Sections~2--3]{Berger2008}. In the
+arrangement below, placement proceeds downwards on both tapes, while
+each word retains its chronological orientation.
+
+Take independent iid tapes $(A_i)_{i\ge1}$ and $(B_i)_{i\ge1}$
+with respective word laws $\nu_+$ and $\nu_-$.
+Their joint law and expectation will be denoted by $\mathbf P$
+and $\mathbf E$.  For a relative word $\gamma$ let $d(\gamma)$
+be its terminal displacement, and let $a(\gamma)\in\ZZ^{d-1}$
+be the last $d-1$ coordinates of that displacement.
+
+Set the positive top endpoint at zero.  Place $A_1$ with its
+terminal there, then place $A_2$ with its terminal at the start of
+$A_1$, and continue downwards.  Thus the placed copy of $A_i$ is
+translated by $-\sum_{r\le i}d(A_r)$.  Place the negative tape
+chronologically starting from $-e_1$, so the placed copy of $B_i$
+is translated by $-e_1+\sum_{r<i}d(B_r)$.
+Only translations are used; the steps inside each word retain
+their chronological orientation.
+
+For a site $x$, call $-x_1$ its \emph{progress below the top}, or \emph{depth}.
+A word of width $L$ preceded by total tape width $s$ has all its
+departures in the progress interval
+\begin{equation}\label{eq:departure-progress}
+ (s,s+L]\cap\ZZ
+\end{equation}
+on either tape.  The one-level offset of the negative starting
+point is what makes these intervals agree.  A \emph{contact} is
+a site that is a departure of both placed tapes.  The last positive
+step into the top is axial, so its departure is $-e_1$, the first
+negative departure.  There is always a contact at progress one.
+
+Write $H_i^+=\sum_{r\le i}L(A_r)$ and
+$H_j^-=\sum_{r\le j}L(B_r)$, with $H_0^+=H_0^-=0$.
+Successive positive common values of these two increasing
+sequences divide both tapes into \emph{common blocks}.
+This is the intersection of two independent renewal processes, a standard
+construction discussed, for example, in~\cite{AlexanderBerger2016}.
+Lemma~\ref{lem:common-blocks} proves that all such blocks exist and provides the moments
+needed for the entropy comparison.
+
+\begin{lemma}[Common blocks]\label{lem:common-blocks}
+The paired lists in successive common blocks are iid.  If $K$ is
+the total width of one block, then
+\begin{equation}\label{eq:common-width-moment}
+ \mathbf E K\le\frac{1}{p_+p_-}<\infty.
+\end{equation}
+Let $V_+$ and $V_-$ be the sums of the actual chronological
+lateral displacements of the positive and negative words in that
+block.  Then
+\begin{equation}\label{eq:common-lateral-moment}
+ \mathbf E(|V_+|+|V_-|)<\infty.
+\end{equation}
+When passing down through this block, the negative-minus-positive
+lateral gap increases by
+\[
+ S=V_++V_-\in\ZZ^{d-1},\qquad \mathbf E|S|<\infty.
+\]
+\end{lemma}
+
+\begin{proof}
+Initially allow the first positive common width $K$ to be infinite.
+If it is reached after $i$ positive and $j$ negative words, that
+event is determined by those two finite prefixes: their widths
+agree, and no earlier positive partial sums agree.  The unused
+tails are therefore independent tapes with their original laws,
+independent of the completed block.  To see this without any
+stopping-time convention, condition separately on each pair
+$(i,j)$, factor the law of the unused tails, and sum over $(i,j)$.
+Repeating the argument gives the renewal property at every common
+cut that exists.
+
+Let $c_n=u_n^+u_n^-$ be the probability that $n$ is a common cut,
+including $c_0=1$.  Partitioning according to the last common cut
+at or before $n$ gives the elementary identity
+\[
+ \sum_{i=0}^n c_{n-i}\mathbf P(K>i)=1.
+\]
+Indeed, after a common cut at $n-i$, the unused pair of tapes has
+its original law, and $K>i$ says precisely that there is no further
+common cut through $n$.  The identity remains valid if the next
+gap is infinite.  Since $c_j\ge p_+p_->0$, it follows that
+\[
+ (p_+p_-)\sum_{i=0}^n\mathbf P(K>i)\le1.
+\]
+Letting $n$ tend to infinity proves
+\eqref{eq:common-width-moment}.  In particular $K$ is finite
+almost surely.  Restarting at each successive common cut now
+justifies the infinite iid common-block sequence.
+
+Let $N_+$ be the number of positive words in the first common
+block.  The event $\{N_+\ge i\}$ says that none of
+$H_1^+,\ldots,H_{i-1}^+$ belongs to the negative renewal set.
+It depends only on the preceding positive words and the negative
+tape, and is independent of the whole next word $A_i$.
+Tonelli's Theorem therefore gives
+\[
+ \mathbf E\sum_{i=1}^{N_+}R(A_i)
+ =\sum_{i\ge1}\mathbf P(N_+\ge i)E_+R
+ =\mathbf E N_+\,E_+R
+ \le\mathbf E K\,E_+R.
+\]
+The last inequality uses the integer bound $L\ge1$.
+The same argument applies to the negative tape.  Since
+$|a(\gamma)|\le R(\gamma)$, this proves
+\eqref{eq:common-lateral-moment} and the moment bound for $S$.
+
+Finally, across the block the positive lower anchor is its old
+anchor minus $V_+$ in lateral coordinates, whereas the negative
+anchor moves by $V_-$.  Their negative-minus-positive difference
+increases by $V_++V_-$, as asserted.
+\end{proof}
+
+Let $K_j$ be the successive common widths and put
+$W_0=0$, $W_j=\sum_{i=1}^jK_i$.  By
+\eqref{eq:departure-progress}, both tapes' departures in block
+$j$ have progress in $(W_{j-1},W_j]$.  Consequently a contact
+always belongs to the same unique common block on both tapes.
+Figure~\ref{fig:arrangement} summarizes the two orientations.
+\begin{figure}[tbp]
+ \centering
+ \input{figures/opposed-arrangement}
+ \caption{The opposed arrangement, schematically and not to scale.
+ (a) Common departure bands; arrows indicate chronological orientation,
+ while placement proceeds downwards on both tapes. (b) Endpoint progress
+ for a word of width $L$ preceded by total width $s$, with the one-level
+ offset restored. Filled points are starts and open points are terminal
+ arrivals. Word interiors and lateral positions are not depicted;
+ the arrows do not assert monotonicity of the words.}
+ \label{fig:arrangement}
+\end{figure}
+For integers $0\le a<b$, denote by $\mathcal C(a,b]$ the event
+of a contact in at least one block with index $a<j\le b$.
+The quantity to be estimated is
+\begin{equation}\label{eq:contact-count}
+ m(J)=\sum_{j=1}^J\mathbf P\bigl(\mathcal C(2^j,2^{j+1}]\bigr),
+ \qquad J\ge1.
+\end{equation}
+It is the expected number of dyadic block-index intervals containing
+a contact.  Section~\ref{sec:entropy} bounds it above using the
+finite first moment of the lateral increment $S$; the remaining
+sections obtain an incompatible lower bound from the forced contact
+near the top and the renewal-tail estimate.
diff --git a/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/build/05-entropy.tex b/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/build/05-entropy.tex
new file mode 100644
index 0000000000000000000000000000000000000000..68e4e705a22261293f16c0563a2be962d87514de
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@@ -0,0 +1,397 @@
+\section{An entropy upper bound for contacts}\label{sec:entropy}
+
+We continue under the assumption that both coordinate signs have positive
+probability.  The opposed arrangement and its common blocks are those of
+Section~\ref{sec:arrangement}.  We will prove
+\begin{proposition}\label{prop:entropy-upper}
+Suppose that both coordinate signs have positive directional-transience
+probability. In the opposed arrangement of Section~\ref{sec:arrangement},
+let $m(J)$ be the expected number of dyadic common-block index intervals
+$(2^j,2^{j+1}]$, $1\le j\le J$, containing a departure-site contact,
+as in~\eqref{eq:contact-count}. Then
+\[
+ m(J)=O\!\left(\frac{J}{\log J}\right)\qquad(J\longrightarrow\infty).
+\]
+\end{proposition}
+The argument compares a piece of the opposed arrangement with independent
+bridges of prescribed heights.  Contacts are rare under the bridge law.
+The cost of this comparison is controlled by the entropy gained when more
+independent common-block displacements are added.
+
+\subsection{A randomized comparison with independent bridges}
+
+Fix an integer $n\ge1$.  We want to cover the common blocks with
+indices in $(4n,8n]$ by a middle portion of the arrangement.  We will
+surround it by two outer portions, each of height at least $n$, so
+that any middle contact occurs well inside a finite slab.  A buffer
+above all three portions supplies an independent lateral gap.
+
+We randomize the number of blocks in each portion as well as in the
+buffer.  Choosing a chunk count uniformly from $n$ possibilities
+will offset the cost of specifying its random height.  Randomizing
+the buffer count will let us compare its entropy with that of the whole
+experiment without paying an additional $\log n$.
+Independently of the common-block tape, choose four independent
+integer counts
+\begin{equation}\label{eq:entropy-count-windows}
+ \begin{split}
+ U,I_1,I_3&\ \hbox{uniform on }\{n,\ldots,2n-1\},\\
+ I_2&\ \hbox{uniform on }\{7n,\ldots,8n-1\}.
+ \end{split}
+\end{equation}
+Read a buffer of $U$ common blocks, followed by three chunks of
+$I_1,I_2,I_3$ common blocks.  Let $d_0$ be the lateral gap after the
+buffer, namely the sum of its gap increments.  For chunk $i$, let
+$A_i$ and $B_i$ be the positive and negative slab lists in their
+downward tape order, and let $H_i$ be their common total width.  Write
+\[
+ H=(H_1,H_2,H_3),\qquad
+ \mathcal A=(A_1,A_2,A_3),\qquad
+ \mathcal B=(B_1,B_2,B_3).
+\]
+The three chunks are independent, and $(U,d_0)$ is independent of all
+their data.  To see this despite their random boundaries, condition
+on the four external counts and use the product law of the common
+blocks; the distribution of a chunk depends only on its own count.
+The first and third count windows ensure the required outer heights.
+The longer middle window ensures that the middle chunk starts before
+block $4n$ and ends after block $8n$.
+
+Set $q=d-1$, and let $X_{A,i},X_{B,i}\in\ZZ^q$ be the sums of the
+lateral displacements of the words in $A_i,B_i$, measured in the words' actual chronological
+directions.  The total positive displacement determines where its
+bottom lies relative to its top.  Accordingly, define
+\begin{equation}\label{eq:entropy-alignment-variables}
+ X_A=\sum_{i=1}^3X_{A,i},\quad X_B=\sum_{i=1}^3X_{B,i},\quad
+ Z=d_0+X_A,\quad T=U+I_1+I_2+I_3.
+\end{equation}
+Here is the placement that explains the alignment variable $Z$ and
+will define the same contact event under the actual and reference
+laws.  Set $N_*=H_1+H_2+H_3$.  Place the positive chunks
+from height $0$ and lateral position $0$, in chronological order
+$3,2,1$, reversing the list order within each chunk.  Each word is
+still traversed in its original direction.  Place the negative chunks
+in chronological order $1,2,3$ from height $N_*-1$ and lateral position
+$Z$.  Let $E_{\mathrm{mid}}$ be a departure contact between the two
+middle chunks.
+
+For these actual chunk data, this placement reproduces the infinite
+opposed arrangement after its buffer, up to translation.  For example,
+at the top of the middle chunk the lateral gap, negative minus positive, is
+\[
+ Z+X_{B,1}-(X_{A,3}+X_{A,2})
+   =d_0+X_{A,1}+X_{B,1},
+\]
+as in the original arrangement; the subsequent list placements also
+agree.  From \eqref{eq:entropy-count-windows}, the middle chunk starts
+after at most $4n-2$ common blocks and ends after at least $9n$ blocks.
+It therefore contains the deterministic common-block interval
+$(4n,8n]$.
+
+Denote the actual law of $(H,Z,\mathcal A,\mathcal B)$ by $P_n$.
+Define a reference law $Q_n$ as follows.  First sample $(H,Z)$ with its
+actual joint marginal.  Given these values, sample the six lists
+independently, with respective exact-cut bridge laws
+$\mathcal B_{H_i}^+$ and $\mathcal B_{H_i}^-$ from
+Lemma~\ref{lem:exact-bridge}.  The reference retains the dependence
+between $H$ and $Z$; it changes only the conditional law of the lists.
+Apply the same placement rule to its newly sampled lists.  Thus $E_{\mathrm{mid}}$ is one fixed
+measurable event in both experiments, and its actual probability
+bounds the original contact probability in $(4n,8n]$.
+
+\subsection{The entropy cost of the comparison}
+
+We use entropy and relative entropy in their classical information-theoretic
+sense~\cite{Shannon1948,KullbackLeibler1951}, recalling the identities and
+bounds needed for the countable path spaces below.
+
+For a countable random variable $V$, write
+\[
+ \mathcal H(V)=-\sum_v P(V=v)\log P(V=v),
+\]
+with $0\log0=0$.  Conditional entropy is the average of the corresponding
+conditional entropies.  For probability measures $P,Q$ on a countable set,
+their relative entropy is
+\[
+ D(P\Vert Q)=\sum_x P(x)\log\frac{P(x)}{Q(x)},
+\]
+with value $+\infty$ if $P$ is not absolutely continuous with respect to
+$Q$.  Conditional mutual information $I(V;W\mid H)$ is the average,
+over $H$, of the relative entropy between the joint conditional law and
+the product of its two marginals.
+
+We use the chain rule for relative entropy, obtained by factoring joint
+probabilities into marginal and conditional probabilities.  In particular,
+if a reference law for $(V,W)$ conditional on $H$ factors between $V$ and
+$W$, the averaged conditional relative entropy is at least
+$I(V;W\mid H)$.  Nonnegativity of relative entropy follows from Jensen's
+inequality applied to $-\log$.  The same factorization, followed by
+nonnegativity, shows that applying a measurable map cannot increase
+relative entropy.  These facts apply to countable path lists without
+assuming that the lists themselves have finite entropy; equivalently one
+may first use finite partitions and pass to their increasing limit.
+
+Let $S_1,S_2,\ldots$ be the iid lateral gap increments of the common
+blocks, and set
+\[
+ h_k=\mathcal H(S_1+\cdots+S_k),\qquad k\ge1.
+\]
+Lemma~\ref{lem:common-blocks} gives $\mathbf E|S_1|<\infty$.  For a
+$\ZZ^q$-valued variable with finite first moment, comparison with the
+probability weights proportional to $\exp(-|x|/(k+1))$ gives
+\begin{equation}\label{eq:gap-entropy-growth}
+ h_k\le C+q\log(k+1).
+\end{equation}
+Indeed the normalizing sum is at most $C_q(k+1)^q$, and the expected
+comparison energy is at most $k\mathbf E|S_1|/(k+1)$.  Also $h_k$ is
+nondecreasing: condition on the last independent summand, use translation
+invariance of entropy, and then remove the conditioning.
+
+By Lemma~\ref{lem:common-blocks}, the sum of the slab radii in a common
+block has finite expectation.  Thus each tape's block displacement,
+and not only their sum $S$, has a finite first absolute moment.  In
+addition the common width has finite mean.  All count, height, and
+displacement variables used below therefore have first moments of
+order $n$, and their joint and conditional entropies are finite.
+No entropy bound for the full path lists is needed.
+
+\begin{lemma}\label{lem:entropy-comparison}
+There is a constant $C$, independent of $n$, such that
+\[
+ D(P_n\Vert Q_n)\le C+h_{14n}-h_n.
+\]
+\end{lemma}
+\begin{proof}
+There are two costs to control: imposing the random chunk heights,
+and then retaining the aligned starting displacement $Z$.  We first
+condition on $H$ alone.  For a specified
+pair of slab lists of common total width $h$, let $r$ be the number of
+their common positive cumulative widths, including $h$.  The pair is
+uniquely parsed into its first $r$ common blocks.  If $r$ belongs to the
+count window for chunk $i$, its actual unconditioned mass is
+\[
+ \frac1n\prod_{a\in A_i}W(a)\prod_{b\in B_i}W(b).
+\]
+It has zero mass otherwise.  The independent exact-cut bridge pair at
+height $h$ has mass given by the same product divided by $u_h^+u_h^-$.
+Thus the conditional density of the actual pair, on its support, is
+\begin{equation}\label{eq:count-window-density}
+ \frac{u_h^+u_h^-}{nP_n(H_i=h)}.
+\end{equation}
+Common cuts here are counted in the original downward list order,
+before any list is reversed for chronological placement.
+
+Let $D_0$ be the averaged conditional relative entropy of
+$(\mathcal A,\mathcal B)$ given $H$ with respect to these independent
+bridges.  The chunks are independent, so \eqref{eq:count-window-density}
+and $u_h^\pm\le1$ give
+\[
+ D_0\le\sum_{i=1}^3\bigl(\mathcal H(H_i)-\log n\bigr)\le C.
+\]
+For the last bound, $\mathbf E H_i\le8n\mathbf E K$, and a positive
+integer variable with mean $a$ has entropy at most $1+\log(a+1)$,
+by comparison with a geometric distribution.  Since the reference
+conditional on $H$ factors between the tapes,
+\begin{equation}\label{eq:conditional-tape-information}
+ I(\mathcal A;\mathcal B\mid H)\le D_0.
+\end{equation}
+
+Introducing $Z$ while retaining its actual marginal costs precisely
+its conditional mutual information with the lists.  The chain rule
+and the independence of the buffer give
+\begin{equation}\label{eq:endpoint-information-cost}
+ \begin{split}
+ D(P_n\Vert Q_n)
+ &=D_0+I(Z;\mathcal A,\mathcal B\mid H)\\
+ &=D_0+\mathcal H(Z\mid H)-\mathcal H(d_0).
+ \end{split}
+\end{equation}
+Conditional on $\mathcal A,H$, the variable $Z$ is obtained by adding
+the independent $d_0$.  Data processing in
+\eqref{eq:conditional-tape-information} therefore yields
+$I(Z;\mathcal B\mid H)\le D_0$.
+
+The direct bound $\mathcal H(Z\mid H)=O(\log n)$ would be too
+large after summing over scales.  Instead, add the negative tape's
+displacement to $Z$.  Their sum is the total common-block gap, whose
+entropy can be compared with that of the buffer.  Conditioning on
+the negative tape costs at most $D_0$, as just proved:
+
+\begin{align}
+ \mathcal H(T,Z+X_B)
+ &\ge \mathcal H(T,Z+X_B\mid\mathcal B,H)
+  =\mathcal H(T,Z\mid\mathcal B,H)\notag\\
+ &\ge \mathcal H(Z\mid H)-D_0
+       +\mathcal H(T\mid Z,\mathcal A,\mathcal B,H)\notag\\
+ &=\mathcal H(Z\mid H)-D_0+\mathcal H(U\mid d_0).
+ \label{eq:buffer-entropy-chain}
+\end{align}
+For the last equality, the two lists in each chunk determine its common
+count.  Knowing these lists and $Z$ therefore fixes both $T-U$ and
+$d_0=Z-X_A$.  The buffer pair is independent of the chunks.
+
+The vector $Z+X_B$ is the sum of the first $T$ common gap increments.
+The count $T$, like $U$, is independent of the underlying iid tape;
+we are not conditioning on the observed chunks in the following
+identities.  Consequently
+\[
+ \mathcal H(T,Z+X_B)=\mathcal H(T)+\mathbf E h_T,
+ \qquad
+ \mathcal H(U,d_0)=\log n+\mathbf E h_U.
+\]
+Here $n\le U<2n$ and $10n\le T\le14n-4$, so $T$ takes at most $4n$
+values.  Monotonicity of $h_k$ gives
+\begin{equation}\label{eq:random-count-entropy-difference}
+ \mathcal H(T,Z+X_B)-\mathcal H(U,d_0)
+ \le\log4+h_{14n}-h_n.
+\end{equation}
+Subtract $\mathcal H(d_0)$ in \eqref{eq:buffer-entropy-chain} and use
+$\mathcal H(U,d_0)=\mathcal H(d_0)+\mathcal H(U\mid d_0)$.
+Together with \eqref{eq:endpoint-information-cost} and
+\eqref{eq:random-count-entropy-difference}, this gives explicitly
+\[
+ D(P_n\Vert Q_n)\le 2D_0+\log4+h_{14n}-h_n.
+\]
+The bound for $D_0$ is uniform in $n$, so its doubled contribution
+is absorbed into $C$.  This proves
+Lemma~\ref{lem:entropy-comparison}.
+\end{proof}
+
+\subsection{Rare contacts under the reference law}
+
+Comparing opposed paths in one environment and classifying intersection
+sites by quenched escape probabilities already appears in the planar
+argument of Zerner~\cite[Section~2]{Zerner2007}. We prove the finite-barrier
+estimate needed here.
+
+\begin{lemma}\label{lem:reference-contact}
+For the reference experiment just defined,
+\[
+ Q_n(E_{\mathrm{mid}})
+ \le\delta_n:=\min\{1,Cn(F_+(n)+F_-(n))\}.
+\]
+\end{lemma}
+\begin{proof}
+Fix $(H,Z)$ in its retained support; in particular $H_1,H_3\ge n$.
+Write $N=N_*$.  By list-reversal invariance in
+Lemma~\ref{lem:exact-bridge}, the placed chronological chunks have
+their corresponding raw bridge laws.
+
+Retain the complete positive lower and middle chunks, concatenated
+into a path $\alpha$ from height $0$ to its first hit of $N-H_1$.
+Retain the negative upper and middle chunks only up to their first
+arrival at a departure site of $\alpha$.  A middle contact ensures
+that this arrival occurs before the negative middle endpoint at
+height $H_3-1$.  The contact site $y$ must therefore satisfy
+\begin{equation}\label{eq:reference-contact-height}
+ H_3\le y_1\le N-H_1-1.
+\end{equation}
+Thus each path has made at least $n$ progress in its own signed height
+before reaching the contact, and neither has crossed the global
+barriers $-1,N$.
+
+We explain the comparison with two paths in one environment before
+using any quenched exit probabilities.  The unused positive upper
+chunk and negative lower chunk integrate to one.  The remaining four
+bridge normalizations are bounded by $p_+^{-2}p_-^{-2}$.  Consecutive
+bridges within a sign have disjoint departure layers, so their raw
+weights multiply to the raw weight of their concatenation.  Summing
+over the negative completion after its retained prefix, and dropping
+its remaining bridge requirements, costs at most that prefix's raw
+weight: in each fixed environment the total conditional completion
+probability is at most one, and this inequality can then be averaged.
+This argument allows the completion to revisit its prefix's departure
+rows and does not factor their weights independently.
+The first-hit chunk boundaries determine the splits, so no
+extra multiplicity is introduced.
+
+The retained negative prefix has no departure site in
+$\operatorname{Dep}(\alpha)$: its terminal point is the first arrival
+there.  Repeated sites within either path cause no difficulty.
+Lemma~\ref{lem:departure-transfer} therefore identifies the product
+of their raw weights with the weight of the two paths sampled
+independently conditional on a common iid environment.  For each
+$\alpha$ the first contact fixes the negative prefix uniquely, and
+$\alpha$ itself is stopped at a fixed first-hit height.  We may thus
+sum these joint prefix weights without overcounting, and extend the
+paths as raw trajectories in the common environment.
+
+Only now, in that shared environment, define for interior sites
+\[
+ Q_y^\omega=P_{y,\omega}(T_{-1}<T_N),
+\]
+where the barriers are absolute first-coordinate heights.  If
+$Q_y^\omega\ge1/2$ at the contact, the positive raw path has visited
+such a site after hitting $n$ and before exiting the global slab.
+Conditional on the environment, stop it at its first such visit.
+The probability of a subsequent exit at $-1$ is at least $1/2$.
+Consequently the annealed probability of this visit event is at most
+\[
+ 2P_0(T_n<T_{-1}<T_N)=2(u_n^+-u_N^+).
+\]
+If $Q_y^\omega<1/2$, the negative marginal gives the analogous bound
+$2(u_n^--u_N^-)$.  Indeed the almost-sure finite-height-interval
+exit property established in Section~\ref{sec:arrangement} holds
+quenched for all these starting sites and barriers outside one
+null set.  The probability of the other exit is therefore greater
+than $1/2$, with no common lower bound on the transition rows.
+Measured from the negative start $N-1$, the
+global barriers $N$ and $-1$ have signed relative heights $-1$ and
+$N$, respectively.  Lateral translation by $Z$ does not change its
+annealed law.
+
+Combining these alternatives with the bounded normalizations gives,
+uniformly in $(H,Z)$,
+\[
+ Q_n(E_{\mathrm{mid}}\mid H,Z)
+ \le C\sum_{\sigma\in\{+,-\}}(u_n^\sigma-u_N^\sigma)
+ \le CN\bigl(F_+(n)+F_-(n)\bigr),
+\]
+where the last step is Lemma~\ref{lem:renewal-tail}.  Finally
+$\mathbf E N\le12n\mathbf E K$.  Average over the retained joint
+marginal of $(H,Z)$ and take the minimum with $1$.
+\end{proof}
+
+\subsection{Summing over scales}
+
+\begin{proof}[Proof of Proposition~\ref{prop:entropy-upper}]
+For $l\ge2$, take $n=2^{l-2}$ and abbreviate $\delta_l=\delta_n$.
+The event $E_{\mathrm{mid}}$ contains the original contact event in
+common blocks $(2^l,2^{l+1}]$.  If $\delta_l=0$, absolute continuity
+from Lemma~\ref{lem:entropy-comparison} makes its actual probability
+zero.  Otherwise the binary relative-entropy inequality gives
+\begin{equation}\label{eq:binary-contact-entropy}
+ P_n(E_{\mathrm{mid}})\log(1/\delta_l)
+ \le C+h_{14n}-h_n.
+\end{equation}
+For completeness, applying data processing to an event with actual
+and reference probabilities $a,b$ gives binary divergence
+$a\log(a/b)+(1-a)\log((1-a)/(1-b))$.  It is at least
+$a\log(1/b)-\log2$.  Use $b\le\delta_l$ and absorb $\log2$ into $C$.
+
+Width integrability gives
+\[
+ \sum_{l\ge2}\delta_l<\infty.
+\]
+Among $2\le l\le J$, at most $C\sqrt J$ indices have
+$\delta_l>J^{-1/2}$.  They contribute at most $C\sqrt J$ to $m(J)$.
+For every other index with positive $\delta_l$, the logarithm in
+\eqref{eq:binary-contact-entropy} is at least $\tfrac12\log J$.
+Since $14n\le16n$, monotonicity and bounded overlap give
+\[
+ \sum_{l=2}^J\bigl(h_{14\cdot2^{l-2}}-h_{2^{l-2}}\bigr)
+ \le\sum_{l=2}^J\bigl(h_{2^{l+2}}-h_{2^{l-2}}\bigr)
+ \le4h_{2^{J+2}}=O(J),
+\]
+using \eqref{eq:gap-entropy-growth}.  Summing
+\eqref{eq:binary-contact-entropy}, and adding the initial scale, proves
+\[
+ m(J)\le1+C\sqrt J+\frac{CJ}{\log J}
+       =O(J/\log J).
+\]
+\end{proof}
+
+The estimate uses the same departure contacts as the original
+arrangement.  The next section bounds the chance of a first contact
+far above a true cut; that bound will force more contact scales than
+Proposition~\ref{prop:entropy-upper} permits.
diff --git a/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/build/06-posterior-contact.tex b/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/build/06-posterior-contact.tex
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+\section{Contacts with a path anchored at a random endpoint}
+\label{sec:posterior}
+
+The lower contact bound requires a different comparison from the one used in
+Section~\ref{sec:entropy}.  A positive bridge now determines the starting
+point of the negative path.  We control this dependence by retaining the
+conditional probabilities of the bridge's possible endpoints.  The needed
+estimate for these probabilities is a finite-vector martingale inequality.
+
+\subsection{The sum of posterior maxima}
+
+The first-moment calculation integrates the nonnegative-martingale maximal
+inequality, whose classical form appears in
+\cite[Chapter~V, Premi\`ere section, \S2, Theorem~1]{Ville1939}.
+The same coordinatewise calculation for posterior vectors appears in
+Kesselheim, Molinaro, Patton, and Singla~\cite[Section~1.1]{KesselheimMolinaroPattonSingla2026}.
+To obtain an exponential moment, we also control conditional future
+increases and the size of each jump. This is the increasing-process
+energy mechanism of~\cite[Theorem~4 and its corollary]{Kikuchi1992};
+we give an elementary threshold proof with the required filtration explicit.
+
+\begin{lemma}[Posterior maxima]\label{lem:posterior-maxima}
+Let $m\ge1$, and let $(w_i(e))_{i\ge0}$, $1\le e\le m$, be nonnegative
+martingales for a filtration $(\mathcal F_i)$, with
+$\sum_{e=1}^m w_i(e)\le1$ almost surely.  Define
+\[
+ A_i(e)=\max_{0\le j\le i}w_j(e),\qquad
+ A_i=\sum_{e=1}^m A_i(e).
+\]
+Write $A_\infty(e)=\lim_{i\to\infty}A_i(e)$ and
+$A_\infty=\sum_e A_\infty(e)=\lim_{i\to\infty}A_i$.
+There are absolute constants $c,C>0$ such that
+\begin{equation}\label{eq:posterior-exponential}
+ \EE\exp\left(\frac{cA_\infty}{\log(m+1)}\right)\le C.
+\end{equation}
+Moreover, if $V$ is any event on a joint probability space with this
+martingale-path marginal and $p=P(V)$, then
+\begin{equation}\label{eq:posterior-event-weight}
+ \EE[A_\infty\1_V]
+ \le C\log(m+1)\,p\log(e/p),
+\end{equation}
+where the right side is zero when $p=0$.  The event $V$ need not belong to
+any $\mathcal F_i$.
+\end{lemma}
+
+\begin{proof}
+Write $B=\log(m+1)$.  We first show that, at every almost surely finite
+stopping time $\rho$,
+\begin{equation}\label{eq:posterior-future-increase}
+ \EE[A_\infty-A_\rho\mid\mathcal F_\rho]\le B.
+\end{equation}
+For a fixed coordinate put $a=A_\rho(e)$ and $p_e=w_\rho(e)$, so that
+$0\le p_e\le a\le1$.  Conditional maximal inequality and integration over
+levels give, when $a>0$,
+\[
+ \EE[A_\infty(e)-a\mid\mathcal F_\rho]
+ \le\int_a^1\frac{p_e}{u}\,du
+ =p_e\log(1/a)\le p_e\log(1/p_e).
+\]
+If $a=0$, the martingale has conditional expectation zero at every later
+time and contributes zero.  The conditional maximal inequality follows
+first for bounded stopping times and finite horizons; localization and
+monotone convergence give the displayed version.  Summing over $e$ and
+adding the missing mass $1-\sum_e p_e$ bounds the resulting sum by the
+entropy of a probability vector on $m+1$ elements, hence by $B$.  This
+proves~\eqref{eq:posterior-future-increase}.
+
+At each time the increase of a coordinate maximum is at most its current
+value.  Consequently
+\[
+ A_0\le1,\qquad 0\le A_i-A_{i-1}\le\sum_e w_i(e)\le1.
+\]
+Set $b=2B+1$ and consider the successive thresholds $1+kb$, $k\ge1$.
+At their first strict crossings, the overshoot is at most one.  After a
+crossing, reaching the next threshold requires a further increase greater
+than $b-1=2B$.  Conditional Markov inequality
+and~\eqref{eq:posterior-future-increase} therefore bound the conditional
+probability of this next crossing by $1/2$.  The first crossing has the
+same bound since $A_0\le1$.  Applying this argument at finite horizons
+and then letting the horizon increase yields
+\[
+ P(A_\infty>1+kb)\le2^{-k}.
+\]
+Because $B\ge\log2$, this is an exponential tail on scale $B$, and its
+integral proves~\eqref{eq:posterior-exponential}.
+
+Finally, for $p>0$, conditional Jensen inequality gives
+\[
+ \exp\left(\frac{c\,\EE[A_\infty\mid V]}{B}\right)
+ \le \EE[\exp(cA_\infty/B)\mid V]\le C/p.
+\]
+Multiplication by $p$ and an adjustment of the absolute constant give
+\eqref{eq:posterior-event-weight}.  This last argument uses only a
+marginal exponential moment, so it applies to events in a larger joint
+space without any assertion that the martingales remain martingales in
+an enlarged filtration.
+\end{proof}
+
+\subsection{A first-contact estimate}
+
+The comparison of opposing paths through unvisited rows and the use of
+quenched exit probabilities are related to Zerner's planar
+argument~\cite[Section~2]{Zerner2007}. Here the negative path starts at
+a random endpoint of the positive bridge; the posterior estimate controls
+the dependence created by that alignment.
+
+We continue to use integer height $x_1$, and write $\pi x\in\ZZ^{d-1}$
+for the lateral coordinates of $x\in\ZZ^d$.  Recall that
+$u_n^+=P_0(T_n<T_{-1})$ and $p_\pm>0$ are the non-backtracking
+probabilities.  The constants below may depend on the environment law and
+the dimension.  We assume only strict positivity of the transition rows;
+no common ellipticity bound is imposed.
+
+Fix an integer $N\ge3$.  Let $Y$ have the annealed bridge law from $0$ to its first
+arrival at height $N$, conditioned on $T_N<T_{-1}$, and put
+$E=\pi Y_{T_N}$.  Independently sample a negative path $B$ with law
+$\widehat P_{-e_1}$.  Translate it to the random starting site $(N-1,E)$:
+\[
+ \beta_j=(N-1,E)+B_j,\qquad j\ge0.
+\]
+Thus the relative negative path is independent of the positive bridge;
+its absolute position depends on the bridge endpoint.  A departure of
+$Y$ means an index $i<T_N$.  Every site of the infinite path $\beta$ is
+a departure site.  For a negative path started at height $N-1$, write
+$D^-$ for the event that it never rises above $N-1$.
+
+\begin{proposition}[First contact after a prescribed height]
+\label{prop:contact-bound}
+In the preceding experiment, let $0<k<r<N$ be integers and define
+\[
+ \tau=\inf\{i\ge T_k:i<T_N,\ Y_i\in\{\beta_j:j\ge0\}\},
+ \qquad
+ \Delta=u^+_{r-k}-u^+_{N-k}.
+\]
+An empty infimum is $+\infty$.  Then
+\begin{multline}\label{eq:first-contact-bound}
+ P\left(T_r\le\tau<T_N,\
+       \min_{T_k\le i\le\tau}(Y_i)_1\ge k,\
+       \|E\|\le N^2\right)\\
+ \le C\log(2N)\,\Delta\log^2(e/\Delta),
+\end{multline}
+with right side zero when $\Delta=0$.
+\end{proposition}
+
+The estimate is useful when the remaining gap $N-r$ is small relative
+to the height $r-k$ already crossed. Lemma~\ref{lem:renewal-tail} gives
+$\Delta\le (N-r)F_+(r-k)$; Section~\ref{sec:contradiction} arranges
+precisely this separation of scales.
+
+The time $\tau$ refers to the entire negative trajectory and is not
+asserted to be a stopping time for the positive path.  We will enumerate
+finite prefixes to use its first-contact property.  Martingale estimates
+will instead be stopped at deterministic height barriers.
+
+\begin{proof}
+Let $\mathcal E_N=\{e\in\ZZ^{d-1}:\|e\|\le N^2\}$.  Its cardinality
+satisfies
+\begin{equation}\label{eq:endpoint-cardinality}
+ \log(|\mathcal E_N|+1)\le C_d\log(2N).
+\end{equation}
+We first express the event in~\eqref{eq:first-contact-bound} by raw path
+weights, then transfer those weights to a shared environment, and only
+afterward introduce quenched exit probabilities.
+
+\paragraph{The lower prefix and the endpoint likelihood.}
+Enumerate a word $a_0$ from $0$ to its first arrival at height $k$, before
+height $-1$.  Its departure sites all have height less than $k$; write
+$z$ for its terminal site.  For each $e\in\mathcal E_N$, enumerate a word
+$a$ from $z$ to the first contact arrival.  The required words stay in
+heights $k,\ldots,N-1$ and reach height $r$ before or at their terminal
+time.  Let $\operatorname{Dep}(a)$ denote the departure sites of $a$,
+excluding its terminal site unless that site occurred earlier as a
+departure.
+
+For any such positive prefix define
+\[
+ w(a_0a;e)=P_0\bigl(T_N<T_{-1},\ \pi X_{T_N}=e
+                   \mid X\text{ begins with }a_0a\bigr).
+\]
+This is a deterministic function of the two words and $e$.  The two
+departure sets of $a_0$ and $a$ are disjoint, so the positive prefix and
+completion have raw probability
+\begin{equation}\label{eq:prefix-completion-weight}
+ W(a_0)W(a)w(a_0a;e).
+\end{equation}
+For a raw negative walk started at $(N-1,e)$, let $\mathcal B(a)$ be the
+event that it never rises above $N-1$, avoids $\operatorname{Dep}(a)$
+at every time, and visits the terminal site of $a$.  The first-contact
+condition is exactly this avoidance and visit condition.  In particular,
+if the terminal site of $a$ occurred earlier as a departure, the event
+$\mathcal B(a)$ is empty.  Summing
+\eqref{eq:prefix-completion-weight} times the raw probability of
+$\mathcal B(a)$, over these words and endpoints, and dividing by
+$u_N^+p_-$ gives the probability on the left
+of~\eqref{eq:first-contact-bound}.
+
+\paragraph{Transfer and unique prefix counting.}
+For a fixed word $a$, the quenched probability of $\mathcal B(a)$ uses
+no rows at $\operatorname{Dep}(a)$.  Indeed the walk can be killed on
+first arrival at this set, before any row there is used.  This remains
+valid for the infinite avoidance event by passage from finite paths.
+The departure-row identity of Lemma~\ref{lem:departure-transfer} therefore
+gives
+\begin{equation}\label{eq:unclassified-lower-transfer}
+ W(a)P_{(N-1,e)}(\mathcal B(a))
+ =\EE\left[p_\omega(a)
+       P_{(N-1,e),\omega}(\mathcal B(a))\right],
+\end{equation}
+where $p_\omega(a)$ is the quenched product of transition probabilities
+along the word.  This identity has not been restricted by any
+environment-dependent exit-probability condition.
+
+Keep $W(a_0)$ as a scalar outside the new experiment.  In a fresh
+i.i.d. environment sample a raw walk $Z$ from $z$ and a raw negative
+walk from $(N-1,e)$, independently conditional on the environment.  We
+call $Z$ the comparison path.  Formula~\eqref{eq:unclassified-lower-transfer}
+expresses the sum over $a$ as an integral over these two full paths.
+For each realized pair, at most one prefix $a$ of $Z$ satisfies the
+avoidance and visit conditions: its terminal is the first point of $Z$
+in the range of the entire negative path.  This proves uniqueness
+pathwise, without using an optional-stopping assertion at contact.
+The restrictions that $a$ stays above $k-1$ and reaches $r$ are retained
+at this stage.
+
+\paragraph{Stopped posteriors on the comparison path.}
+Fix $a_0$ and consider the original raw positive law conditional on this
+lower prefix.  In its natural continuation filtration, with no
+environment revealed, the coordinates
+\[
+ w_i(e)=P_0\bigl(T_N<T_{-1},\ \pi X_{T_N}=e
+             \mid a_0,\text{ continuation through time }i\bigr),
+ \qquad e\in\mathcal E_N,
+\]
+are nonnegative martingales whose sum is at most one.  Stop them on
+first arrival at height $k-1$ or $N$.  All departures up to this stop
+have height at least $k$.  These rows are fresh under conditioning on
+$a_0$, so the stopped continuation has precisely the same path law as
+$Z$ stopped at
+\[
+ \sigma=T_{k-1}(Z)\wedge T_N(Z).
+\]
+Arrival at the lower boundary uses a row at height $k$, not a departure
+row at $k-1$.  Thus the equality includes the terminal arrival, as
+illustrated in Figure~\ref{fig:fresh-strip}.  The stopping time is
+almost surely finite by
+Lemma~\ref{lem:finite-supremum}: a path that never leaves this finite
+height interval would have finite height supremum without tending to
+negative height infinity.  The corresponding quenched assertion holds
+for almost every environment, simultaneously for all lattice starting
+sites and integer height intervals.
+
+Use these same posterior functions on the observed prefixes of $Z$, and
+freeze them at $\sigma$.  At arrival at $k-1$ their values need not be
+zero: they still describe success before the original lower barrier
+$-1$, and that completion may use rows affected by $a_0$.  We neither
+recompute them in the fresh environment nor continue them through a
+departure below $k$.  Put
+\[
+ M(e)=\sup_{0\le i\le\sigma}w_i(e),\qquad
+ S_*=\sum_{e\in\mathcal E_N}M(e).
+\]
+Lemma~\ref{lem:posterior-maxima}
+and~\eqref{eq:endpoint-cardinality}, applied in the path filtration of
+the stopped comparison walk, imply uniformly in $a_0$ that
+\begin{equation}\label{eq:comparison-event-weight}
+ \EE[S_*\1_V]
+ \le C\log(2N)\,P(V)\log(e/P(V))
+\end{equation}
+for every event $V$ on the joint space of the fresh environment and $Z$.
+In particular, $\EE S_*\le C\log(2N)$.  The event-weight inequality,
+rather than a martingale assertion after revealing the environment,
+permits the next step.
+
+\begin{figure}[tbp]
+ \centering
+ \input{figures/posterior-fresh-strip}
+ \caption{Fresh rows for the stopped comparison path. The lower prefix
+ first reaches height $k$ at $z$, and all its departure rows lie below
+ $k$. The continuation, shown with two schematic exit alternatives,
+ stops on arrival at height $k-1$ or $N$. It has used only rows with
+ heights $k,\ldots,N-1$, so its stopped path law agrees with the fresh
+ comparison law. The original endpoint posteriors are frozen at this
+ stop, not recomputed under a new completion law. Lateral positions
+ are schematic.}
+ \label{fig:fresh-strip}
+\end{figure}
+
+\paragraph{Quenched exit-probability bins.}
+For $k\le y_1<N$ define
+\[
+ Q_y^\omega=P_{y,\omega}(T_{k-1}<T_N).
+\]
+Almost surely every transition at every site is strictly positive.
+For each interior site $y$, a finite sequence of downward axial steps
+reaches $k-1$ before $N$ with positive quenched probability; a finite
+sequence of upward axial steps reaches $N$ before $k-1$ with positive
+quenched probability.  Thus $0<Q_y^\omega<1$, without a common bound
+away from either endpoint.  Partition these values into dyadic bins
+$[a,2a)$, where $a=2^{-j}$ and $j\ge1$, and write
+$\mathcal D_a^\omega=\{y:k\le y_1<N,\ a\le Q_y^\omega<2a\}$
+for the sites in one bin.
+Let $V_a$ be the event that, at or after its first arrival at height
+$r$ and before $\sigma$, the comparison path visits an interior site with
+$Q_y^\omega\ge a$.  With the environment fixed, stop at the first such
+visit.  The quenched Markov property gives
+\begin{align}
+ aP(V_a)
+ &\le P_z(T_r<T_{k-1}<T_N)\notag\\
+ &=u^+_{r-k}-u^+_{N-k}=\Delta.
+ \label{eq:comparison-bin-probability}
+\end{align}
+The equality follows from nested first-hit events and almost sure exit
+from the height interval, again by Lemma~\ref{lem:finite-supremum}.
+These events use the fresh comparison law, not the law conditioned on
+the lower prefix.
+
+For almost every environment, the raw negative walk has probability at
+most $2a$ of both satisfying $D^-$ and hitting $\mathcal D_a^\omega$.
+To see this, take its first visit to that set.  On $D^-$ the negative walk
+has finite first-coordinate supremum, so Lemma~\ref{lem:finite-supremum} implies that its
+first coordinate tends to $-\infty$.  The quenched form of that lemma
+holds outside a single environment-null set for all lattice starting
+sites.  Consequently, after that first bin visit it must hit $k-1$
+before $N$, even if it had previously visited $k-1$.  Its conditional
+probability is at most $Q_y^\omega<2a$ by the quenched Markov property.
+
+We now use the unique prefix established above.  In the transferred
+integral, classify its contact site by its $Q^\omega$ bin and bound its
+endpoint factor $w(a_0a;e)$ by $M(e)$.  Dropping the exact-contact
+condition leaves $V_a$ for the comparison path and
+$D^-\cap\{\hbox{the negative path hits }\mathcal D_a^\omega\}$
+for the negative path.  There is no sum over competing prefixes.
+After integrating the negative path, every endpoint $e$ leaves the same
+fresh joint marginal $\PP(d\omega)P_{z,\omega}(dZ)$.  Thus the bin's
+total contribution, summed over endpoints, is at most
+\begin{equation}\label{eq:bin-contribution}
+ \begin{split}
+ &\sum_{e\in\mathcal E_N}\EE\left[
+     M(e)\1_{V_a}
+     P_{(N-1,e),\omega}
+     (D^-\cap\{\hbox{the path hits }\mathcal D_a^\omega\})\right]\\
+ &\hspace{35mm}\le 2a\,\EE\left[\1_{V_a}\sum_{e\in\mathcal E_N}M(e)\right]
+  =2a\,\EE[S_*\1_{V_a}].
+ \end{split}
+\end{equation}
+
+If $\Delta=0$, every $V_a$ is null
+by~\eqref{eq:comparison-bin-probability}, proving the required zero
+bound.  Suppose $\Delta>0$.  For $a<\Delta$, sum
+\eqref{eq:bin-contribution} using $\EE S_*\le C\log(2N)$ and
+$\sum_{a<\Delta}a\le2\Delta$.  For $a\ge\Delta$, use
+\eqref{eq:comparison-event-weight} and
+$P(V_a)\le\Delta/a$ to bound the contribution by
+\[
+ C\log(2N)\,\Delta\log(ea/\Delta).
+\]
+There are $O(\log(e/\Delta))$ such dyadic bins, and their logarithms
+sum to $O(\log^2(e/\Delta))$.  Thus, for each fixed $a_0$, all endpoint
+and upper-prefix contributions together are at most
+\[
+ C\log(2N)\,\Delta\log^2(e/\Delta).
+\]
+Finally $\sum_{a_0}W(a_0)=u_k^+\le1$, and
+$1/(u_N^+p_-)\le1/(p_+p_-)$.  Restoring these factors proves
+\eqref{eq:first-contact-bound}.  The constants lose only these fixed
+non-backtracking probabilities and the endpoint-cardinality factor.
+In particular, the argument has not required a lower bound for
+$Q_y^\omega$ or an inverse-transition moment.
+\end{proof}
+
+The estimate controls a contact anywhere after the prescribed height
+$r$.  It therefore applies to a long interval without subdividing the
+positive path by its successive record heights.  In the next section a
+missing interval of contact depths forces exactly such a late first
+contact.
diff --git a/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/build/07-contradiction.tex b/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026/build/07-contradiction.tex
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+\section{Many contact scales and the contradiction}
+\label{sec:contradiction}
+
+We now apply Proposition~\ref{prop:contact-bound} to the opposed
+arrangement.  An interval of depths containing no contact forces the first
+contact above each positive cut in that interval to occur close to the
+top.  The first-contact estimate makes this unlikely on average over the
+cuts.  We first count scales of \emph{depth}; the finite mean common-block
+width will then convert this count into the block-index count used in
+Proposition~\ref{prop:entropy-upper}.
+
+\begin{proposition}\label{prop:contact-lower}
+Suppose that both coordinate signs have positive directional-transience
+probability.  In the opposed arrangement of
+Section~\ref{sec:arrangement}, let $m(J)$ be the expected number of
+dyadic common-block index intervals $(2^j,2^{j+1}]$, $1\le j\le J$,
+containing a departure-site contact, as in~\eqref{eq:contact-count}.
+There are constants $c>0$ and $J_0<\infty$ such that
+\[
+ m(J)\ge \frac{cJ}{\log\log(3+J)}\qquad(J\ge J_0).
+\]
+\end{proposition}
+
+\begin{proof}
+All constants below may depend on the two slab laws.  Choose a large
+integer $J$ and put
+\begin{equation}\label{eq:lower-parameters}
+ N=2^{2J},\qquad
+ G=\left\lceil4\log_2\log(2+J)\right\rceil+3.
+\end{equation}
+For sufficiently large $J$, we have $3\le G<J$.  Depth means distance
+below the initial top of the arrangement.
+The choice $2^{-G}=O((\log J)^{-4})$ compensates for the logarithmic
+losses below, while $G=O(\log\log J)$ leaves enough contact scales.
+
+\paragraph{A bridge ending at a prescribed depth.}
+Let $\mathcal C_N$ be the event that the positive tape has a cut at
+depth $N$, and write
+\[
+ \mathbf Q_N=\mathbf P(\,\cdot\mid\mathcal C_N).
+\]
+The renewal identity gives
+$\mathbf P(\mathcal C_N)=u_N^+\ge p_+>0$.
+Under $\mathbf Q_N$, reverse the list of positive slabs through this cut
+and translate its bottom to the origin.  Call the resulting chronological
+path $Y$, and let $T_a$ denote its first hit of height $a$.
+Lemma~\ref{lem:exact-bridge} says that $Y$ has law
+$P_0(\,\cdot\mid T_N<T_{-1})$, stopped at $T_N$.
+The steps within each slab retain their original order.  Denote the
+lateral endpoint by $E$, so that the endpoint is $(N,E)$.  The negative
+trajectory starts at $(N-1,E)$ and, relative to that starting point,
+has the independent law $\widehat P_{-e_1}$.
+
+The endpoint has a useful elementary tail bound.  On $\mathcal C_N$
+there are at most $N$ positive slabs before depth $N$, since their
+widths are positive integers.  The norm of their total lateral
+displacement is at most the sum of their radii, hence at most the sum
+of the first $N$ positive tape radii.  Proposition~\ref{prop:radius}
+and Markov's inequality give
+\begin{equation}\label{eq:lower-endpoint-tail}
+ \EE_{\mathbf Q_N}|E|
+ \le \frac{N E_+R}{p_+},
+ \qquad
+ \mathbf Q_N\{|E|>N^2\}\le \frac{C}{N}.
+\end{equation}
+This uses a spatial radius moment only.
+
+\paragraph{An uncovered scale forces a late first contact.}
+Call $l\ge0$ a contact depth label if a contact occurs at some integer
+depth in $[2^l,2^{l+1})$.  For $G<l\le J$, let $U_l$ be the event
+that none of the labels $l-G,l-G+1,\ldots,l$ is a contact depth label.
+Equivalently, there is no contact at depths in
+$[2^{l-G},2^{l+1})$.
+
+Fix a positive cut depth
+\[
+ s\in I_l:=\bigl[2^l,\tfrac32\,2^l\bigr]\cap\ZZ,
+ \qquad k=N-s.
+\]
+After the list reversal, this cut occurs at the bridge's first hit
+$T_k$, and the remaining bridge stays at or above height $k$.
+Indeed, every earlier slab has departure heights below $k$, while
+every later slab has departure heights at least $k$.  This is a
+property of the words, without a stopping-time assertion about the
+cut.
+
+There is a contact after $T_k$ and before $T_N$: the last positive
+departure is $(N-1,E)$, the starting departure site of the negative
+trajectory.  Let $\tau$ be the first contact since $T_k$, defined
+pathwise against the entire negative trajectory.  Its depth is at
+most $s<2^{l+1}$.  On $U_l$, it must therefore be strictly less than
+$2^{l-G}$.  With
+\[
+ r=N-2^{l-G},
+\]
+the nearest-neighbor path must have reached height $r$ before this
+contact.  Thus
+\begin{equation}\label{eq:uncovered-first-contact}
+ T_r\le\tau<T_N,
+ \qquad Y_n\cdot e_1\ge k\quad(T_k\le n\le\tau).
+\end{equation}
+No restriction on drawdowns from the running maximum is needed.
+
+For each \emph{deterministic} $s\in I_l$, let $\mathcal A_{l,s}$ be
+the event in~\eqref{eq:uncovered-first-contact}, together with
+$|E|\le N^2$, whether or not $s$ is a cut.  Proposition~
+\ref{prop:contact-bound}, with upper stopping height $N$, applies
+to this event.  Its width-probability difference satisfies
+\begin{align}
+ \Delta_{l,s}
+ &=u^+_{s-2^{l-G}}-u^+_s\notag\\
+ &\le 2^{l-G}F_+(s-2^{l-G})
+ \le 2^{-G}a_l,
+ \qquad a_l:=2^l F_+(2^{l-1}),
+ \label{eq:lower-delta}
+\end{align}
+because $s-2^{l-G}\ge2^{l-1}$ and
+Lemma~\ref{lem:renewal-tail} applies.  The same Lemma gives
+\begin{equation}\label{eq:lower-summable-tail}
+ A:=\sum_{l\ge1}a_l<\infty.
+\end{equation}
+Writing $\phi(x)=x\log^2(e/x)$ for $0<x\le1$ and $\phi(0)=0$,
+the first-contact estimate is
+\begin{equation}\label{eq:lower-fixed-cut-bound}
+ \mathbf Q_N(\mathcal A_{l,s})
+ \le C\log(2N)\,\phi(\Delta_{l,s}).
+\end{equation}
+
+\paragraph{Averaging over the cuts.}
+Let $\mu_+=E_+L$.  The renewal-count strong law yields, almost surely
+under the positive tape law,
+\[
+ \#\{\hbox{positive cuts in }I_l\}
+ =\frac{2^{l-1}}{\mu_+}+o(2^l)
+ \qquad(l\longrightarrow\infty).
+\]
+Fix $c_0=1/(4\mu_+)$, and let $\mathcal B_J$ be the event that
+each interval $I_l$, $G<l\le J$, contains at least $c_0 2^l$
+positive cuts.  Since $G=G(J)\to\infty$, the eventual almost-sure
+estimate implies $\mathbf P(\mathcal B_J^c)=o(1)$.  Moreover,
+\begin{equation}\label{eq:cut-density-conditioning}
+ \mathbf Q_N(\mathcal B_J^c)
+ \le \frac{\mathbf P(\mathcal B_J^c)}{p_+}=o(1).
+\end{equation}
+In particular, no independence between the cut at $N$ and the density
+event is required.
+
+On $\mathcal B_J\cap\{|E|\le N^2\}$, every uncovered label $l$
+produces $\mathcal A_{l,s}$ for each actual cut $s\in I_l$.
+Consequently,
+\begin{equation}\label{eq:cut-averaging}
+ \1_{U_l}\1_{\mathcal B_J}\1_{\{|E|\le N^2\}}
+ \le \frac1{c_0 2^l}\sum_{s\in I_l}\1_{\mathcal A_{l,s}}.
+\end{equation}
+The right side has dropped the cut requirement on $s$.  This permits
+the deterministic-barrier estimate~\eqref{eq:lower-fixed-cut-bound};
+the number of summands is $O(2^l)$ and cancels the averaging factor.
+
+We next check that the logarithms in this estimate do not require a
+quantitative tail rate.  For $J\ge3$, put
+\[
+ B_J=\sum_{l=1}^J a_l\log_+^2(1/a_l),
+ \qquad \log_+x:=\max\{\log x,0\},
+\]
+where a zero summand is interpreted as zero.  Terms with
+$a_l\ge J^{-2}$ contribute at most $4A\log^2 J$.  On
+$(0,J^{-2})$, the function $x\log^2(1/x)$ is increasing, so the
+remaining at most $J$ terms contribute at most $4J^{-1}\log^2 J$.
+Therefore
+\begin{equation}\label{eq:weighted-tail-log}
+ B_J\le 4(A+J^{-1})\log^2 J.
+\end{equation}
+
+For sufficiently large $J$, $2^{-G}a_l\le e^{-1}$ uniformly in
+$l$, since $a_l\le A$.  The function $\phi$ is increasing on
+$[0,e^{-1}]$.  Thus~\eqref{eq:lower-delta} gives, uniformly over
+$s\in I_l$,
+\[
+ \phi(\Delta_{l,s})\le C2^{-G}a_l
+ \bigl(G+1+\log_+(1/a_l)\bigr)^2.
+\]
+Taking expectations in~\eqref{eq:cut-averaging}, then using
+\eqref{eq:lower-endpoint-tail},
+\eqref{eq:lower-fixed-cut-bound}, and
+\eqref{eq:cut-density-conditioning}, gives
+\begin{align}
+ \sum_{l=G+1}^J\mathbf Q_N(U_l)
+ &\le J\mathbf Q_N(\mathcal B_J^c)+\frac{CJ}{N}
+   +C\log(2N)\,2^{-G}\bigl[A(G+1)^2+B_J\bigr]\notag\\
+ &=o(J).
+ \label{eq:few-uncovered}
+\end{align}
+Indeed, $\log(2N)=O(J)$, while the choice~
+\eqref{eq:lower-parameters} gives
+$2^{-G}=O(\log^{-4}(2+J))$.  By~\eqref{eq:weighted-tail-log},
+the last term is $O(J/\log^2(2+J))$.
+
+\paragraph{Removing the conditioning.}
+Let $H_J$ count contact depth labels in $\{1,\ldots,J\}$.
+Equation~\eqref{eq:few-uncovered} and Markov's inequality imply
+that with $\mathbf Q_N$-probability tending to one, at most $J/4$
+of the tested labels are uncovered.  Since $G=o(J)$, at least
+$J/2$ are then covered.  A single contact depth label $h$ can cover
+only the labels $h,h+1,\ldots,h+G$.  Hence
+\begin{equation}\label{eq:height-count-conditioned}
+ \mathbf Q_N\left\{H_J\ge\frac{J}{2(G+1)}\right\}
+ \longrightarrow1.
+\end{equation}
+For large $J$, all depths counted here are less than $2^{J+1}<N$.
+On $\mathcal C_N$ they are therefore fully represented in the
+positive list ending at depth $N$.  Undoing the translation and list
+reversal leaves exactly the contacts in the original opposed
+arrangement.  Thus~\eqref{eq:height-count-conditioned} implies
+\begin{equation}\label{eq:height-count-unconditioned}
+ \mathbf P\left\{H_J\ge\frac{J}{2(G+1)}\right\}
+ \ge p_+-o(1).
+\end{equation}
+Only a probability bounded away from zero is asserted after removing
+the conditioning.  This is enough for an expectation lower bound.
+
+\paragraph{From depths to common-block indices.}
+Write $K_1,K_2,\ldots$ for the common-block widths and
+$W_i=K_1+\cdots+K_i$, with $W_0=0$.  By
+Lemma~\ref{lem:common-blocks}, $\mu:=\mathbf E K_1<\infty$ and
+$W_i/i\to\mu$ almost surely.  In particular, eventually
+\[
+ \frac\mu2 i\le W_i\le2\mu i.
+\]
+An integer depth $z\ge1$ belongs to the unique block
+\[
+ i(z)=\min\{i\ge1:W_i\ge z\},
+ \qquad W_{i(z)-1}<z\le W_{i(z)}.
+\]
+The departure-height convention makes this true even at a common
+cut depth.  For every sufficiently large $z$, the strong-law bounds
+give
+\begin{equation}\label{eq:depth-index-comparison}
+ \frac{z}{2\mu}\le i(z)<\frac{2z}{\mu}+1.
+\end{equation}
+The finite initial blocks, however large their widths, disappear once
+$z$ exceeds their cumulative width.
+
+Define the dyadic index label of a block $i>1$ by
+$j(i)=\lceil\log_2 i\rceil-1$, so that
+$2^{j(i)}<i\le2^{j(i)+1}$.  It follows from
+\eqref{eq:depth-index-comparison} that there is a deterministic
+integer $c_1$, depending only on $\mu$, such that eventually
+\begin{equation}\label{eq:bounded-label-shift}
+ |j(i(z))-l|\le c_1
+ \quad\hbox{whenever }2^l\le z<2^{l+1}.
+\end{equation}
+Almost surely this holds above a finite random depth.  Therefore
+the probability that it holds for every contact with
+$\lceil\sqrt J\rceil\le l\le J$ tends to one.
+
+Intersect this event with the event in
+\eqref{eq:height-count-unconditioned}.  The intersection still has
+probability at least $p_+-o(1)$.  Discarding the first
+$\lceil\sqrt J\rceil$ depth labels loses
+$o(J/(G+1))$ labels.  Each remaining contact depth label supplies a
+contact block whose index label lies within $c_1$ of it.  Conversely,
+one index label can account for at most $2c_1+1$ depth labels.
+Thus, on the intersection, at least $c_2J/(G+1)$ of the index labels
+$1,\ldots,J+c_1$ contain a contact, for a fixed $c_2>0$ and large
+$J$.  Taking expectations yields
+\[
+ m(J+c_1)\ge\frac{c_3J}{G+1}.
+\]
+Since $G+1=O(\log\log(3+J))$, replacing $J+c_1$ by the argument
+of $m$ proves the proposition.
+\end{proof}
+
+\begin{proof}[Proof of Theorem~\ref{thm:main}]
+Fix any nonzero real direction $\ell$.  Suppose that
+$0<P_0(A_\ell)<1$.  Lemma~\ref{lem:sign-union} implies that
+$P_0(A_{-\ell})>0$.  The spatial-radius estimate and
+Proposition~\ref{prop:coordinate-reduction} then supply a coordinate
+direction in which both signs have positive probability.  Reflecting
+and relabeling that coordinate gives the setting of
+Propositions~\ref{prop:entropy-upper} and~\ref{prop:contact-lower}.
+They imply simultaneously
+\[
+ m(J)=O\!\left(\frac{J}{\log J}\right)
+ \quad\hbox{and}\quad
+ m(J)\ge\frac{cJ}{\log\log(3+J)}
+\]
+for all sufficiently large $J$, which is impossible.  Thus
+$P_0(A_\ell)\in\{0,1\}$.
+
+The reduction applies to every fixed real $\ell$ without rational
+approximation.  Strict ellipticity supplied the fresh-row probabilities;
+the spatial-radius moment was derived under coexistence.  Neither a
+moment of a slab's duration nor a speed assumption entered the argument.
+\end{proof}